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Chemical Reaction Engineering Module

Application Library Manual

C o n t a c t I n f o r m a t i o n

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Part number: CM021605

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Created in COMSOL Multiphysics 5.3a

F i n d i n g K i n e t i c A r r h en i u s Pa r ame t e r s U s i n gPa r ame t e r E s t ima t i o n

This model is licensed under the COMSOL Software License Agreement 5.3a.All trademarks are the property of their respective owners. See www.comsol.com/trademarks.

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Introduction

This example shows how to use the Parameter Estimation feature in the Reaction Engineering interface to find the Arrhenius parameters of a first order reaction. Inspiration for this example is taken from Ref. 1.

Note: This example requires the Optimization Module.

Model Definition

Benzene diazonium chloride in the gas phase decomposes to benzene chloride and nitrogen according to:

The reaction is first order with the rate:

where the temperature dependent rate constant given by:

Above, A is the frequency factor (SI unit: 1/s) and E is the activation energy (SI unit: J/mol).

In order to evaluate the Arrhenius parameters, A and E, a set of experiments was conducted using a perfectly mixed isothermal batch system with constant volume. The concentration of benzene diazonium chloride was monitored as function of time for the temperatures; T = 313 K, 319 K, 323 K, 328 K, and 333 K.

The model optimizes A and E at these temperatures with the Parameter Estimation feature for simulations utilizing the isothermal constant volume Batch reactor type. Five experimental data sets are available in the model file as comma separated value files (csv-files).

N

N2+N

ClCl

k

r kcPhN2Cl=

k A ERgT-----------–

exp=

2 | F I N D I N G K I N E T I C A R R H E N I U S P A R A M E T E R S U S I N G P A R A M E T E R E S T I M A T I O N

Results and Discussion

Parameter estimation calculations give the values A = 1.27·1016 (SI unit: 1/s) and E = 116 (SI unit: kJ/mol) for the frequency factor and activation energy, respectively. Figure 1 plots the model results and the associated experimental data points.

Figure 1: Model results and experimental data for PhN2Cl concentration as a function of time.


Notes About the COMSOL Implementation

The parameter estimation solver is more efficient in finding an optimal parameter set if the model experiences similar sensitivity with respect to changes in parameter values. In this problem a parameter Aex is therefore defined, that is to be estimated together with the activation energy, E, such that the rate constant is written as:

The frequency factor A is then evaluated as:

The data indicates that the rate constant is of the order ~1·10-3 (1/s) at T = 323 K. Taking this into account and using an initial guess for the activation energy of 150 kJ/mol, an initial guess is set for Aex = 49.

Reference

1. H.S. Fogler, Elements of Chemical Reaction Engineering, 4th ed., p. 95, Prentice Hall, 2005.

Application Library path: Chemical_Reaction_Engineering_Module/Tutorials/activation_energy

Modeling Instructions

From the File menu, choose New.

N E W

In the New window, click Model Wizard.

M O D E L W I Z A R D

1 In the Model Wizard window, click 0D.

2 In the Select Physics tree, select Chemical Species Transport>Reaction Engineering (re).

3 Click Add.

4 Click Study.

k Aex( ) ERgT-----------–

exp⋅exp=

A Aex( )exp=


5 In the Select Study tree, select Preset Studies>Time Dependent.

6 Click Done.

G L O B A L D E F I N I T I O N S

Add a set of model parameters by importing their definitions from a data text file.

Parameters1 In the Model Builder window, under Global Definitions click Parameters.

2 In the Settings window for Parameters, locate the Parameters section.

3 Click Load from File.

4 Browse to the model’s Application Libraries folder and double-click the file activation_energy_parameters.txt.

R E A C T I O N E N G I N E E R I N G ( R E )

Reaction 11 On the Reaction Engineering toolbar, click Reaction.

2 In the Settings window for Reaction, locate the Reaction Formula section.

3 In the Formula text field, type PhN2Cl=>PhCl+N2.

Initial Values 11 In the Model Builder window, under Component 1 (comp1)>Reaction Engineering (re)

click Initial Values 1.

2 In the Settings window for Initial Values, locate the Volumetric Species Initial Value section.

3 In the table, enter the following settings:

Now, add a Parameter Estimation feature, define parameters and set initial values.

Parameter Estimation 11 On the Reaction Engineering toolbar, click Parameter Estimation.

2 In the Settings window for Parameter Estimation, locate the Estimation Parameters section.

Species Concentration (mol/m^3)

PhN2Cl c_init_PhN2Cl


3 In the Parameter table, enter the following settings:

4 Click Add.


Create separate Experiment features for the data collected at different temperatures (T_iso).

Experiment 11 On the Reaction Engineering toolbar, click Attributes and choose Experiment.

2 In the Settings window for Experiment, locate the Experimental Data section.

3 Click Browse.

4 Browse to the model’s Application Libraries folder and double-click the file activation_energy_experiment313K.csv.

5 Click Import.


7 Locate the Experimental Parameters section. Click Add.


Parameter Estimation 1In the Model Builder window, under Component 1 (comp1)>Reaction Engineering (re) click Parameter Estimation 1.



Parameter Initial value Scale Lower bound Upper bound

Aex 49 1


E 150e3[J/mol] 1

Data column Use Model variables Unit Weight

conc_PhN2Cl_313K √ c_PhN2Cl 1 1

Parameter names Parameter expressions

T_iso 313


3 Click Browse.


5 Click Import.







3 Click Browse.


5 Click Import.







T_iso 319




T_iso 323





3 Click Browse.


5 Click Import.







3 Click Browse.


5 Click Import.





T_iso 328






Next, return to the Reaction Engineering feature and introduce the parameters to the reaction model.

9 In the Model Builder window, click Reaction Engineering (re).

10 In the Settings window for Reaction Engineering, locate the Energy Balance section.

11 In the T text field, type T_iso.

1: PhN2Cl=>PhCl+N21 In the Model Builder window, under Component 1 (comp1)>Reaction Engineering (re)

click 1: PhN2Cl=>PhCl+N2.

2 In the Settings window for Reaction, locate the Rate Constants section.

3 Select the Use Arrhenius expressions check box.

4 In the Af text field, type exp(Aex).

5 In the Ef text field, type E.

S T U D Y 1

Step 1: Time Dependent1 In the Model Builder window, under Study 1 click Step 1: Time Dependent.

2 In the Settings window for Time Dependent, locate the Study Settings section.

3 In the Times text field, type range(0,50,5000).

Optimization1 On the Study toolbar, click Optimization.

2 In the Settings window for Optimization, locate the Optimization Solver section.

3 From the Method list, choose Levenberg-Marquardt.

Solution 1 (sol1)1 On the Study toolbar, click Show Default Solver.

2 In the Model Builder window, expand the Solution 1 (sol1) node.

3 In the Model Builder window, expand the Study 1>Solver Configurations>

Solution 1 (sol1)>Optimization Solver 1 node, then click Time-Dependent Solver 1.


T_iso 333


4 In the Settings window for Time-Dependent Solver, click to expand the Output section.

5 From the Times to store list, choose Specified values.

6 On the Study toolbar, click Compute.

Follow the following steps to plot the solutions.

R E S U L T S

Experiment 1 Group1 In the Model Builder window, under Results click Experiment 1 Group.

2 In the Settings window for 1D Plot Group, type Experiment Group 313 K in the Label text field.

3 Click to expand the Title section. From the Title type list, choose None.

4 Locate the Plot Settings section. Select the x-axis label check box.

5 In the associated text field, type Time (s).

6 Select the y-axis label check box.

7 In the associated text field, type Concentration PhN2Cl (mol/m3).

Experiment 1 Data1 In the Model Builder window, expand the Results>Experiment Group 313 K node, then

click Experiment 1 Data.

2 In the Settings window for Table Graph, click to expand the Legends section.

3 From the Legends list, choose Manual.


Global 11 In the Model Builder window, under Results>Experiment Group 313 K click Global 1.

2 In the Settings window for Global, locate the Data section.

3 From the Parameter selection (T_iso) list, choose From list.

4 In the Parameter values (T_iso) list, select 313.

5 Click Replace Expression in the upper-right corner of the y-axis data section. From the menu, choose Model>Component 1>Reaction Engineering>comp1.re.c_PhN2Cl -

Concentration.

Legends

Experiment at 313 K

6 Click Replace Expression in the upper-right corner of the x-axis data section. From the menu, choose Model>Solver>t - Time.

7 Click to expand the Legends section. Select the Show legends check box.



10 On the Experiment Group 313 K toolbar, click Plot.

11 Click the Zoom Extents button on the Graphics toolbar.










2 In the Settings window for Table Graph, locate the Legends section.






Legends

Simulation at 313 K

Legends

Experiment at 319 K




Concentration.


















Legends

Simulation at 319 K

Legends

Experiment at 323 K







Concentration.

















Legends

Simulation at 323 K








Concentration.













Legends

Experiment at 328 K

Legends

Simulation at 328 K












Concentration.






In the last step, output the estimated parameter to a table.

Derived ValuesOn the Results toolbar, click Evaluate All.

TA B L E

1 Go to the Table window.

E is found to be 1.16e5 J/mol and Aex is evaluated to 36.9.

Legends

Experiment at 333 K

Legends

Simulation at 333 K


R E S U L T S

Concentration (re)In the Model Builder window, under Results right-click Concentration (re) and choose Delete.



Homogen eou s Cha r g e Comp r e s s i o n I g n i t i o n o fMe t h an e




Introduction

Homogeneous Charge Compression Ignition (HCCI) engines are being considered as an alternative to traditional spark- and compression-ignition engines. As the name implies, a homogeneous fuel/oxidant mixture is auto-ignited by compression with simultaneous combustion occurring throughout the cylinder volume. Combustion temperatures under lean burn operation are relatively low, resulting in low levels of NOx emission. Furthermore, the fuel’s homogeneous nature, as well as the combustion process itself, lead to low levels of particulate matter being produced.

Although HCCI combustion shows promise, the method has several recurring problems: an important one to be addressed is ignition timing. This example examines the HCCI of methane, investigating ignition trends as a function of initial temperature, initial pressure, and fuel additives.

This example solves the mass and energy balances describing the detailed combustion of methane in a variable-volume system. The large amount of kinetic and thermodynamic data required to set up the problem is easily available by importing the relevant files into the Reaction Engineering interface.

Model Definition

It is difficult to form the uniform mixtures required for HCCI with conventional diesel fuel. Natural gas fuels, on the other hand, readily produce homogeneous mixtures and have the potential to serve as HCCI fuels. This example considers the combustion of methane, as described by the GRI-3.0 mechanism, incorporating a detailed reaction mechanism of 53 species taking part in 325 reactions. The files describing the reaction kinetics and thermodynamics of the GRI-3.0 mechanism are available on the Internet (Ref. 1), and you can import the files into the Reaction Engineering interface.

VA R I A B L E VO L U M E R E A C T O R

This model represents the combustion cylinder with a perfectly mixed batch system of variable volume, a predefined reactor type available with the Reaction Engineering interface. Figure 1 shows an engine cylinder and it includes the relevant parameters to

2 | H O M O G E N E O U S C H A R G E C O M P R E S S I O N I G N I T I O N O F M E T H A N E

calculate the instantaneous cylinder volume.

Figure 1: The volume of a combustion cylinder can be expressed as a function of time with the slider-crank relationship. The diagram shows the key geometric parameters. La is the length of the crank arm, Lc is the length of the connecting rod, D equals the cylinder diameter, and α is the crank angle.

The volume change as a function of time is described by the slider-crank equation:

where V is the cylinder volume (SI unit: m3), Vc is the clearance volume (SI unit: m3), CR equals the compression ratio, and R denotes the ratio of the connecting rod to the crank arm (Lc/La). Further, α is the crank angle (SI unit: rad), which is also a function of time

where N is the engine speed in rpm, and t is the time (SI unit: s).

The engine specifications are:

Equation 1 includes the clearance volume Vc which is calculated from

ENGINE SPECIFICATION VARIABLE NAME VALUE

Bore D 13 cm

Stroke S 16 cm

Connecting rod Lc 26.93 cm

Crank arm La 8 cm

Engine speed N 1500 rpm

Compression ratio CR 15

α

La

Lc

D

VVc------ 1 CR 1–( )

2----------------------- R 1 α R2 αsin( )2

––cos–+[ ]+=

α 2πN60------------t=


(1)

Vs is the volume swept by the piston during a cycle from the equation

Figure 2 shows the calculated cylinder volume as a function of the crank angle. The piston is initially at bottom dead center (BDC), corresponding to a crank angle of −180 degrees.

Figure 2: Cylinder volume as a function of crank angle. The crank angle is defined as being zero at top dead center (TDC).

M E T H A N E C O M B U S T I O N R E A C T I O N

The kinetic and thermodynamic data for methane combustion is available in the form of CHEMKIN data input files. The files are imported into the Reaction Engineering interface within the Reversible Reaction Group and Species Thermodynamics (belongs to Species Group) features. This automatically sets up the mass and energy balances for a batch reactor of constant volume.

VcVs

CR 1–( )-----------------------=

VsπD2

4-----------S=


In this example methane is combusted under lean conditions, that is, supplying more than the stoichiometric amount of oxidizer. The stoichiometric requirement of the oxidizer (air) to combust methane is found from the overall reaction:

Assuming that the composition of air is 21% oxygen and 79% nitrogen, the stoichiometric air-fuel ratio is

(2)

The equivalence ratio relates the actual air-fuel ratio to the stoichiometric requirements

(3)

This model sets the equivalence ratio to Φ = 0.5.

From Equation 2 and Equation 3 it is possible to calculate the molar fraction of fuel in the reacting mixture as

and subsequently the initial concentration is


Figure 3 shows the cylinder pressure as a function of time when a methane-air mixture is compressed and ignites. The piston starts at bottom dead center (BDC) and reaches top

CH4 + O2 N2 CO2 H2O N2+( )2 2 7.523.76 + +

A F⁄( )stoicmairmfuel--------------

stoic

4.76 2 Mair⋅⋅1 Mfuel⋅

------------------------------------= =

ΦA F⁄( )stoic

A F⁄( )----------------------------=

xfuel1

4.76 2⋅ Φ 1+⁄-------------------------------------=

cfuelxfuelpinitRgTinit-------------------------=


dead center (TDC) after 0.02 s. At BDC the pressure is set to 1.5 × 105 Pa, Φ is 0.5, and the compression ratio is CR = 15. The initial temperature is varied from 400 K to 800 K.

Figure 3: Pressure traces illustrating the compression and ignition of fuel in an engine cylinder. The initial temperature varies between 400 K and 800 K.

Consistent with literature results, methane does not ignite at an initial temperature of 400 K (Ref. 2). Furthermore, the induction delay decreases with increasing initial temperature. The induction delay time can be evaluated from the pressure gradient. For instance, the induction delay is 0.0193 s when Tinit = 500 K.

Figure 4 illustrates the pressure traces as the initial pressure varies from 1 × 105 Pa to 3 × 105 Pa. The initial temperature is 500 K. An increase in pressure means an increase in


the species concentrations in the fuel-air mixture, resulting in the expected advance in ignition times.

Figure 4: Increased initial gas pressure advances ignition times.

As mentioned, a significant challenge to the realization of HCCI engines is ignition control. In this regard, combustion at TDC is suggested as the optimum timing (Ref. 3). These results show that the inlet temperature of the fuel-air mixture is a potential tuning parameter for ignition. However, relatively high inlet temperatures are often required for proper timing. This adversely affects engine performance because the trapped mass as well as the volumetric efficiency decreases. An alternative that facilitates ignition is to mix small amounts of additives into the fuel-air mixture (Ref. 4). These additives chemically activate the reaction mixture even at relatively low temperatures. This approach alleviates the requirements of high intake temperatures. Figure 5 shows how small amounts of


formaldehyde (CH2O) cause ignition at an initial temperature of 400 K, which is a temperature insufficient to induce combustion with a pure methane fuel.

Figure 5: Small amounts of formaldehyde stimulate ignition of the fuel-air mixture.

The increased reactivity observed in the presence of CH2O is explained by the opening of a new chemical pathway leading to the formation of hydroxyl radicals. Specifically, CH2O reacts with O2 to produce H2O2:

H2O2, in turn, decomposes to reactive OH radicals, which subsequently react violently with the fuel molecules to cause ignition:

The results in the following figures show the species molar fractions of CH2O, HO2, H2O2, and OH during the combustion of methane. Figure 6 shows molar fraction plots for the case when 0.13% CH2O is added to the fuel; Figure 7 is the equivalent species plots for the case when pure methane is combusted. In each case conditions are tuned to

O2+CH2O HO2 CHO+

+ CH2O H2O2 CHO+HO2

+ M OH M+H2O2 2


produce ignition near TDC, thus providing a reference point for comparing the species concentrations.

Figure 6: Selected species molar fractions as a function of crank angle. 0.13 molar percent CH2O is added to the reacting mixture, which is initially at 400 K and 1.5 bar.


Figure 7: Selected species molar fraction as a function of crank angle. Only methane is combusted. The initial temperature is 469 K and the initial pressure is1.5 bar.

The implications of the CH2O reaction path are apparent by comparing Figure 6 and Figure 7: CH2O stimulates the production of HO2 and H2O2, which in turn produce OH radicals in amounts critical to fuel ignition.

References

1. G.P. Smith, D.M. Golden, M. Frenklach, N.W. Moriarty, B. Eiteneer, M. Goldenberg, C.T. Bowman, R.K. Hanson, S. Song, W.C. Gardiner, Jr., V. V. Lissianski, and Z. Qin, GRI-Mech 3.0 home page, http://www.me.berkeley.edu/gri_mech/.

2. S.B. Fiveland and D.N. Assanis, “A four-stroke homogeneous charge compression ignition engine simulation for combustion and performance studies, “SAE Paper 2000-01-0332, 2000.

3. D.L. Flowers, S.M. Aceves, C.K. Westbrook, J.R. Smith, and R.W. Dibble, “Detailed Chemical Kinetic Simulation of Natural Gas HCCI Combustion: Gas Composition Effects


http://www.me.berkeley.edu/gri_mech/

and Investigation of Control Strategies,” J. Eng. Gas Turbine Power, vol. 123, no. 2, pp. 433–439, 2001.

4. M.H. Morsy, “Ignition control of methane fueled homogeneous charge compression ignition engines using additives,” Fuel, vol. 86, no. 4, pp. 533–540, 2007.

Application Library path: Chemical_Reaction_Engineering_Module/Ideal_Tank_Reactors/compression_ignition

Notes about the COMSOL Implementation

The kinetic and thermodynamic data required for this model are available on the Internet. Find the GRI-Mech 3.0 input files at (Ref. 1):

http://www.me.berkeley.edu/gri_mech/version30/text30.html.

Download the reaction mechanism and rate coefficient file (grimech30.dat), as well as thermodynamic data file (thermo30.dat) and store them on your computer so you can import these into the Reaction Engineering interface.



N E W





3 Click Add.

4 Click Study.


6 Click Done.


http://www.me.berkeley.edu/gri_mech/version30/text30.html


In the Model Builder window’s toolbar, click the Show button and select Discretization in the menu.


Import the model parameters from a text file.




4 Browse to the model’s Application Libraries folder and double-click the file compression_ignition_parameters.txt.

Import also some necessary variables, amongst these the cylinder volume function, from a text file.

D E F I N I T I O N S

Variables 11 In the Model Builder window, under Component 1 (comp1) right-click Definitions and

choose Variables.

2 In the Settings window for Variables, locate the Variables section.


4 Browse to the model’s Application Libraries folder and double-click the file compression_ignition_variables.txt.


1 In the Model Builder window, under Component 1 (comp1) click Reaction Engineering (re).

2 In the Settings window for Reaction Engineering, locate the Reactor section.

3 From the Reactor type list, choose Batch.

4 Locate the Energy Balance section. From the list, choose Include.

>Use uniform scaling of the concentration variables to improve the computational performance.

5 Click to expand the Discretization section. Select the Uniform scaling of concentration variables check box.

6 Locate the Mass Balance section. In the Vr text field, type Vol.


Reversible Reaction Group 1Import the reaction kinetics data, available as a CHEMKIN file (grimech30.dat).

1 On the Reaction Engineering toolbar, click Reversible Reaction Group.

2 In the Settings window for Reversible Reaction Group, click to expand the CHEMKIN import for kinetics section.

3 Locate the CHEMKIN Import for Kinetics section. Select the Import CHEMKIN data check box.

4 Click Browse.

5 Browse to the model’s Application Libraries folder and double-click the file grimech30.dat.

6 Click Import.

Species Thermodynamics 1Import also the thermodynamic data, available as a CHEMKIN file (thermo30.dat).

1 In the Model Builder window, expand the Component 1 (comp1)>

Reaction Engineering (re)>Species Group 1 node, then click Species Thermodynamics 1.

2 In the Settings window for Species Thermodynamics, click to expand the CHEMKIN import for thermodynamic data section.

3 Locate the CHEMKIN Import for Thermodynamic Data section. Click Browse.

4 Browse to the model’s Application Libraries folder and double-click the file thermo30.dat.

5 Click Import.



2 In the Settings window for Initial Values, locate the General Parameters section.

3 In the T0 text field, type T_init.

4 Locate the Volumetric Species Initial Value section. In the table, enter the following settings:


CH2O c_CH2O_0

CH4 c_CH4_0


S T U D Y 1

Step 1: Time Dependent>Set up the time dependent study, modify the default tolerance settings to improve the accuracy of the solution.

1 In the Model Builder window, under Study 1 click Step 1: Time Dependent.


3 In the Times text field, type 0 0.026.

4 From the Tolerance list, choose User controlled.

5 In the Relative tolerance text field, type 1e-6.


2 In the Model Builder window, expand the Solution 1 (sol1) node, then click Time-

Dependent Solver 1.

3 In the Settings window for Time-Dependent Solver, click to expand the Absolute tolerance section.

4 Locate the Absolute Tolerance section. From the Tolerance method list, choose Manual.

5 In the Absolute tolerance text field, type 1.0E-7.

6 Clear the Update scaled absolute tolerance check box.


R E S U L T S

Global 11 In the Model Builder window, expand the Concentration (re) node, then click Global 1.

2 In the Settings window for Global, click Replace Expression in the upper-right corner of the y-axis data section. From the menu, choose Component 1>Reaction Engineering>

comp1.re.c_N2 - Concentration.

3 Click Add Expression in the upper-right corner of the y-axis data section. From the menu, choose Component 1>Reaction Engineering>comp1.re.c_CH4 - Concentration.

N2 c_N2_0

O2 c_O2_0



4 Click Add Expression in the upper-right corner of the y-axis data section. From the menu, choose Component 1>Reaction Engineering>comp1.re.c_O2 - Concentration.

5 Click Add Expression in the upper-right corner of the y-axis data section. From the menu, choose Component 1>Reaction Engineering>comp1.re.c_CH2O - Concentration.

6 Click to expand the Coloring and style section. Locate the Coloring and Style section. In the Width text field, type 2.

7 Click to expand the Legends section. From the Legends list, choose Manual.


9 On the Concentration (re) toolbar, click Plot.


The following steps create a plot of the pressure versus time.

1D Plot Group 31 On the Home toolbar, click Add Plot Group and choose 1D Plot Group.

2 In the Settings window for 1D Plot Group, click to expand the Title section.

3 From the Title type list, choose None.

Global 11 Right-click 1D Plot Group 3 and choose Global.


comp1.re.p - Pressure.

3 On the 1D Plot Group 3 toolbar, click Plot.

The following steps reproduce Figure 6.


2 In the Settings window for 1D Plot Group, type Mole fraction in the Label text field.

3 Locate the Title section. From the Title type list, choose None.

Legends

N2

CH4

O2

CH2O



5 In the associated text field, type Crank angle (deg).


7 In the associated text field, type Molar fraction.

Global 11 Right-click Mole fraction and choose Global.


comp1.re.c_OH - Concentration.

3 Locate the y-Axis Data section. In the table, enter the following settings:

4 Locate the x-Axis Data section. From the Parameter list, choose Expression.

5 In the Expression text field, type comp1.crank_angle.




Mole fraction1 In the Model Builder window, under Results click Mole fraction.

2 In the Settings window for 1D Plot Group, locate the Axis section.

3 Select the y-axis log scale check box.

4 On the Mole fraction toolbar, click Plot.

Global 21 In the Model Builder window, under Results>Mole fraction right-click Global 1 and choose

Duplicate.

2 In the Settings window for Global, locate the y-Axis Data section.

Expression Unit Description

comp1.re.c_OH/(comp1.re.p/(R_const*comp1.re.T)) 1 xOH

Legends

xOH




Global 31 Right-click Results>Mole fraction>Global 2 and choose Duplicate.




Global 41 Right-click Results>Mole fraction>Global 3 and choose Duplicate.








comp1.re.c_H2O2/(comp1.re.p/(R_const*comp1.re.T)) 1 xH2O2

Legends

xH2O2


comp1.re.c_HO2/(comp1.re.p/(R_const*comp1.re.T)) 1 xHO2

Legends

xHO2


comp1.re.c_CH2O/(comp1.re.p/(R_const*comp1.re.T)) 1 xCH2O

Legends

xCH2O


3 Select the Manual axis limits check box.

4 In the x minimum text field, type -30.

5 In the x maximum text field, type 30.

6 In the y minimum text field, type 1e-8.

7 In the y maximum text field, type 1e-2.

8 Click to expand the Legend section. From the Position list, choose Lower right.


The following steps reproduce Figure 7. First change the temperature and the initial CH2O concentration, then resolve.


Parameters1 In the Model Builder window, expand the Global Definitions node, then click Parameters.



S T U D Y 1

On the Home toolbar, click Compute.

R E S U L T S



To reproduce Figure 3, create a parametric sweep over the intial temperature parameter.

S T U D Y 1

Parametric Sweep1 On the Study toolbar, click Parametric Sweep.

2 In the Settings window for Parametric Sweep, locate the Study Settings section.

3 Click Add.

Name Expression Value Description

T_init 469[K] 469 K Initial temperature at BDC

x_CH2O 0 0 Initial CH2O mole fraction



5 In the Model Builder window, click Study 1.

6 In the Settings window for Study, locate the Study Settings section.

7 Clear the Generate default plots check box.


R E S U L T S

1D Plot Group 31 In the Model Builder window, under Results click 1D Plot Group 3.

2 In the Settings window for 1D Plot Group, locate the Data section.

3 From the Data set list, choose Study 1/Parametric Solutions 1 (sol2).


5 Click to expand the Legend section. Locate the Axis section. Select the Manual axis limits check box.

6 In the x minimum text field, type 0.012.

7 In the x maximum text field, type 0.026.

8 In the y minimum text field, type 0.

9 In the y maximum text field, type 1.2e7.

10 Locate the Legend section. From the Position list, choose Upper left.

Global 11 In the Model Builder window, under Results>1D Plot Group 3 click Global 1.

2 In the Settings window for Global, click to expand the Coloring and style section.

3 Locate the Coloring and Style section. In the Width text field, type 2.



Parameter name Parameter value list Parameter unit

T_init (Initial temperature at BDC)

400 450 500 600 800

Legends

T_init=400 K

T_init=450 K




ParametersTo reproduce Figure 4, sweep instead over the initial pressure parameter. Set the initial temperature to 500 K first.

1 In the Model Builder window, under Global Definitions click Parameters.



S T U D Y 1

Parametric Sweep1 In the Model Builder window, under Study 1 click Parametric Sweep.



4 On the Home toolbar, click Compute.

R E S U L T S



3 In the y maximum text field, type 2.5e7.

T_init=500 K

T_init=600 K

T_init=800 K




p_init (Initial pressure at BDC) {1 1.5 2 3}*1e5

Legends



2 In the Settings window for Global, locate the Legends section.




ParametersTo reproduce Figure 5, sweep instead over the initial CH2O mole fraction. Set the initial temperature to 400 K first.

1 In the Model Builder window, under Global Definitions click Parameters.



S T U D Y 1

Parametric Sweep1 In the Model Builder window, under Study 1 click Parametric Sweep.




Legends

p_init=1E5 Pa

p_init=1.5E5 Pa

p_init=2E5 Pa

p_init=3E5 Pa




x_CH2O (Initial CH2O mole fraction)

0 0.001 0.01 0.05


R E S U L T S


2 In the Settings window for 1D Plot Group, type Reactor pressure in the Label text field.

3 Locate the Axis section. In the y maximum text field, type 1.8e7.

Global 11 In the Model Builder window, under Results>Reactor pressure click Global 1.

2 In the Settings window for Global, locate the Legends section.


4 On the Reactor pressure toolbar, click Plot.

Legends

x_CH20=0

x_CH20=0.001

x_CH20=0.01

x_CH20=0.05



S t a r t u p o f a Con t i n uou s S t i r r e d T ank R e a c t o r




Introduction

The hydrolysis of propylene oxide into propylene glycol is an important chemical process with 400,000 metric tons produced worldwide each year. Propylene glycol finds wide application as a moisturizer in foods, pharmaceuticals, and cosmetics.

In this example the startup phase of a continuous stirred tank reactor (CSTR) used to produce propylene glycol is modeled. The non-isothermal process is described by a set of coupled mass and energy balances that are easily set up and solved in the Chemical Reaction Engineering Module. The model highlights the use of the predefined CSTR reactor type in the Reaction Engineering interface and also shows how to enter the thermodynamic data needed for energy balances.

This example reproduces results found in Ref. 1.

Model Description

Propylene glycol (PrOH) is produced from the reaction of propylene oxide (PrO) with water (W) in the presence of an acid catalyst:

The reaction rate (SI unit: mol/(m3·s)) is first order with respect to the activity of propylene oxide:

where the rate constant is temperature dependent according to the Arrhenius expression:

(1)

The Arrhenius parameters in Equation 1 are A1 = 4.71·109 (SI unit: 1/s) and E1 = 75,358 (SI unit: J/mol).

H2O+O H2SO4 OHHO

r1 k1cPrO–=

k1 A1E1

RgT------------–

exp=

2 | S T A R T U P O F A C O N T I N U O U S S T I R R E D TA N K R E A C T O R

The liquid phase reaction takes place in a continuous stirred tank reactor (CSTR) equipped with a heat-exchanger. Methanol (MeOH) is also added to the mixture but does not react. It is further assumed that the reactor volume is constant over time.

Figure 1: A perfectly mixed CSTR for the production of propylene glycol. The CSTR is a predefined reactor type in the Chemical Reaction Engineering Module.

The time evolution of the non-isothermal reacting system is given by several coupled balance equations. The species mass balances are:

(2)

In Equation 2, ci is the species molar concentration (SI unit: mol/m3), Vr denotes the reactor volume (SI unit: m3), Ri is the species rate expression (SI unit: mol/(m3·s)), and vf is the volumetric flow rate of the feed inlet (SI unit: m3/s). v is the volumetric flow of the species exiting the reactor and is defined as:

vp is the volumetric production rate, arising due to differences in molar mass, Mi, and densities, ρi, of the species.

For an incompressible and ideally mixed reacting liquid, the energy balance is:

where Cp,i is the species molar heat capacity (SI unit: J/(mol·K)), and T is the temperature (SI unit: K). On the right-hand side, Q represents the heat due to chemical reaction (SI unit: J/s), and Qext denotes heat added to the system (SI unit: J/s), for instance by a heat exchanger. The last term signifies heat added as species flow through the reactor. In this term, hi is the species molar enthalpy (SI unit: J/mol).

Vrdci

dt-------- vf cf i, vci RiVr+–=

v vf i, vp+ vf i, V+ r

RiMiρi--------------

i= =

Vr ciCp i,dTdt--------

i Q Q+ ext vf i, cf i, hf i, hi–( )

i+=


This example assumes that the species heat capacities, Cp,i, represent an average over the temperature interval. The associated species’ enthalpies are then given by:

where hi(Tref) is the standard heat of formation at the reference temperature Tref.

The heat of reaction is given by:

where Hj is the enthalpy of reaction (SI unit: J/mol), and rj denotes the reaction rate(SI unit: mol/(m3·s)).

The heat added by the heat exchanger is given by:

where F is the molar flow rate (SI unit: mol/s), U is the overall heat transfer coefficient(SI unit: J/(K·m2·s)), and A represents the heat exchange area (SI unit: m2). The subscript x refers to the heat exchanger medium, which in this case is water. Tx is the inlet temperature of the heat exchanger medium.

The following table summarizes additional parameters describing the reactor setup and process conditions:

PARAMETER VALUE DESCRIPTION

Vr 1.89 m3 Reactor volume

vf 3.47·10-3 m3/s Volumetric flow rate

cf,PrO 2903 mol/m3 Concentration of PrO in feed stream

cf,W 36291 mol/m3 Concentration of W in feed stream

cf,MeOH 3629 mol/m3 Concentration of MeOH in feed stream

c0,W 55273 mol/m3 Initial concentration of W in the reactor

ρPrO 830 kg/m3 Density of PrO

ρW 1000 kg/m3 Density of W

ρPrOH 1036 kg/m3 Density of PrOH

ρMeOH 792 kg/m3 Density of MeOH

hi Cp i, T Tref–( ) hi Tref( )+=

Q Vr Hjrj

j–=

Qext FxCp x, Tx T–( ) 1 UA–FxCp x,------------------ exp–⋅=


The model described here is readily set up and solved using the predefined CSTR reactor with constant volume in the Reaction Engineering interface available in the Chemical Reaction Engineering Module.

Cp,PrO 146.5 J/(mol·K) Heat capacity of PrO

Cp,W 75.4 J/(mol·K) Heat capacity of W

Cp,PrOH 192.6 J/(mol·K) Heat capacity of PrOH

Cp,MeOH 81.6 J/(mol·K) Heat capacity of MeOH

Cpx 75.4 J/(mol·K) Heat capacity of heat exchanger medium

href,PrO -153.5·103 J/mol Enthalpy of formation of PrO at Tref

href,W -286.1·103 J/mol Enthalpy of formation of W at Tref

href,PrOH -525.6·103 J/mol Enthalpy of formation of PrOH at Tref

href,MeOH -238.6 J/mol Enthalpy of formation of MeOH at Tref

Tf 297 K Feed stream temperature

T0 297 K Initial reactor temperature

Tref 293 K Reference temperature

Tx 289 K Temperature of heat exchanger medium at inlet

Fx 126 mol/s Heat exchanger medium molar flow

UA 8441 J/(s·K) Heat exchange parameter

PARAMETER VALUE DESCRIPTION



Figure 2 shows the concentration of PrO (SI unit: mol/m3) as a function of reaction time.

Figure 2: Concentrations of reactant PrO (mol/m3) after 4 hours of operation.

The corresponding development of the reactor temperature is shown in Figure 3.


Figure 3: Reactor temperature (K) after 4 hours of operation.

Initially both the reactant concentration and the temperature oscillate around their respective steady-state values (472 mol/m3 and 337 K, respectively). The model predicts that the reactor temperature passes a maximum value higher than the steady-state temperature. From a safety perspective it is therefore relevant to look closer at possible sets of initial conditions to see if process operation limits are violated. In the process modeled here, it is undesirable to exceed a reactor temperature of 355 K to avoid undesirable side reactions and not damage reactor equipment. Figure 4 shows the concentration-


temperature phase plane for three initial condition scenarios: (cPrO = 0, T0 = 297 K), (cPrO = 0, T0 = 340 K), and (cPrO = 1400, T0 = 340 K).

Figure 4: Trajectories in the concentration-temperature phase plane for three sets of initial conditions.

The plot shows that all investigated initial conditions converge to the same steady state. However, starting with cPrO = 1400 mol/m3 and T0 = 340 K leads to violation of the temperature safety limits.

Reference

1. H.S. Fogler, Elements of Chemical Reaction Engineering, 3rd ed., Prentice Hall PTR, Example 9-4, pp. 553–559, 1999.

Application Library path: Chemical_Reaction_Engineering_Module/Tutorials/cstr_startup


Safety limit


N E W





3 Click Add.

4 Click Study.


6 Click Done.


Add a set of model parameters by importing their definitions from a data text file provided with the Applications Library.




4 Browse to the model’s Application Libraries folder and double-click the file cstr_startup_parameters.txt.


Similarly, variables for the concentration-dependent and temperature-dependent enthalpies are available in a text file.


choose Variables.



4 Browse to the model’s Application Libraries folder and double-click the file cstr_startup_variables.txt.

Select the Reactor Type -CSTR, constant volume for a liquid mixture and include the Energy Balance, i.e. nonisothermal conditions apply.





3 From the Reactor type list, choose CSTR, constant volume.


5 In the Qext text field, type Q_xch.

6 Click to expand the Mixture properties section. Locate the Mixture Properties section. From the Phase list, choose Liquid.

7 Locate the Mass Balance section. In the Vr text field, type Vr_tank.



Add the reaction. Note that the reaction in this example is of first order in regard to PrO, not the default stoichiometric reaction order.


3 In the Formula text field, type PrO+W=>PrOH.

4 Locate the Reaction Rate section. From the Reaction rate list, choose User defined.

5 In the r text field, type re.kf_1*re.c_PrO[mol/m^3].

6 Locate the Rate Constants section. Select the Use Arrhenius expressions check box.

7 In the Af text field, type Af_reaction.

8 In the Ef text field, type Ea_reaction.

Species 11 On the Reaction Engineering toolbar, click Species.

2 In the Settings window for Species, locate the Species Name section.

3 In the Species name text field, type MeOH.

Species: PrO1 In the Model Builder window, under Component 1 (comp1)>Reaction Engineering (re)

click Species: PrO.

2 In the Settings window for Species, locate the General Parameters section.

3 In the M text field, type Mn_pro.

4 In the ρ text field, type rho_pro.


Species: W1 In the Model Builder window, under Component 1 (comp1)>Reaction Engineering (re)

click Species: W.


3 In the M text field, type Mn_w.

4 In the ρ text field, type rho_w.

Species: PrOH1 In the Model Builder window, under Component 1 (comp1)>Reaction Engineering (re)

click Species: PrOH.


3 In the M text field, type Mn_proh.

4 In the ρ text field, type rho_proh.

Species: MeOH1 In the Model Builder window, under Component 1 (comp1)>Reaction Engineering (re)

click Species: MeOH.


3 In the M text field, type Mn_meoh.

4 In the ρ text field, type rho_meoh.

Feed Inlet 11 On the Reaction Engineering toolbar, click Feed Inlet.

Define the inlet feed stream of the CSTR.

2 In the Settings window for Feed Inlet, locate the Feed Inlet Properties section.

3 In the vf text field, type v_feed.

4 In the Tf text field, type Tfeed.

5 Locate the Feed Inlet Concentration and Enthalpy section. In the Feed inlet concentration

and enthalpy table, enter the following settings:

Species Concentration (mol/m^3) Enthalpy (J/mol) Enthalpy (true--species feature, false--User defined )

MeOH cfeed_meoh hf_meoh

PrO cfeed_pro hf_pro





3 In the T0 text field, type Tinit.


Species: PrO1 In the Model Builder window, under Component 1 (comp1)>Reaction Engineering (re)

click Species: PrO.

2 In the Settings window for Species, click to expand the Species thermodynamic expressions section.

3 Locate the Species Thermodynamic Expressions section. From the Species enthalpy list, choose User defined.

4 In the Cp text field, type cp_pro.

5 In the h text field, type h_pro.

Species: W1 In the Model Builder window, under Component 1 (comp1)>Reaction Engineering (re)

click Species: W.

2 In the Settings window for Species, locate the Species Thermodynamic Expressions section.

3 From the Species enthalpy list, choose User defined.

4 In the Cp text field, type cp_w.

5 In the h text field, type h_w.

PrOH 0 0

W cfeed_w hf_w


PrO cinit_pro

W cinit_w

Species Concentration (mol/m^3) Enthalpy (J/mol) Enthalpy (true--species feature, false--User defined )


Species: PrOH1 In the Model Builder window, under Component 1 (comp1)>Reaction Engineering (re)

click Species: PrOH.



4 In the Cp text field, type cp_proh.

5 In the h text field, type h_proh.

Species: MeOH1 In the Model Builder window, under Component 1 (comp1)>Reaction Engineering (re)

click Species: MeOH.



4 In the Cp text field, type cp_meoh.

5 In the h text field, type h_meoh.

S T U D Y 1



3 From the Time unit list, choose h.

4 In the Times text field, type 4.

First, compute the temperature and concentrations.


The following instructions generate Figure 2 and Figure 3.

R E S U L T S

Concentration (re)1 In the Model Builder window, under Results click Concentration (re).

2 In the Settings window for 1D Plot Group, locate the Plot Settings section.


4 In the associated text field, type Concentration PrO (mol/m3).



comp1.re.c_PrO - Concentration.


4 Click to expand the Legends section. Clear the Show legends check box.


Global 11 In the Model Builder window, expand the Temperature (re) node, then click Global 1.

2 In the Settings window for Global, locate the Coloring and Style section.

3 In the Width text field, type 2.

4 Locate the Legends section. Clear the Show legends check box.

S T U D Y 1

Next, compute the corresponding solutions for a set of initial temperatures and propylene-oxide concentrations.



3 Click Add.


5 Click Add.



The following instructions generate Figure 4.


Tinit (Initial reactor temperature) 297[K] 340[K] 340[K]


cinit_pro (Initial concentration, propylene oxide)

0 0 1400[mol/m^3]


R E S U L T S

Concentration (re) 11 In the Model Builder window, under Results click Concentration (re) 1.

2 In the Settings window for 1D Plot Group, type Concentration vs Temperature (re) in the Label text field.

Global 11 In the Model Builder window, expand the Results>Concentration vs Temperature (re)

node, then click Global 1.


comp1.re.c_PrO - Concentration.

3 Click Replace Expression in the upper-right corner of the x-axis data section. From the menu, choose Component 1>Reaction Engineering>comp1.re.T - Temperature.


5 Locate the Legends section. From the Legends list, choose Manual.



8 On the Concentration vs Temperature (re) toolbar, click Plot.

Temperature (re) 1In the Settings window for 1D Plot Group, type Temperature vs Time (re) in the Label text field.

Global 11 In the Model Builder window, expand the Temperature (re) 1 node, then click Results>

Temperature vs Time (re)>Global 1.




Legends

Tinit=297 K, cinit_PrO=0 mol/m3


6 On the Temperature vs Time (re) toolbar, click Plot.

Legends






S e p a r a t i o n Th r ough D i a l y s i s




Introduction

Dialysis is a widely used separation method. An example is hemodialysis, where membranes are used as artificial kidneys for people with renal failure. Other applications include the recovery of caustic colloidal hemicellulose during viscose manufacturing as well as the removal of alcohol from beer (Ref. 1).

In the dialysis process specific components are preferentially transported through a membrane. The process is diffusion-driven, that is, components diffuse through a membrane due to concentration differences between the dialysate and the permeate sides of the membrane. Separation between solutes is achieved as a result of the different diffusion rates across the membrane, which arise from differences in molecular size and solubility.

This example examines a process aimed at lowering the concentration of a contaminant component in an aqueous product stream. The dialysis equipment is made of a hollow fiber module, where a large number of hollow fibers act as the membrane. It focuses on the transport of the contaminant in the hollow fiber and through its wall.

Figure 1 shows a diagram of the hollow fiber assembly within a dialysis module where the dialysate flows inside while the permeate flows on the outside in a co-current manner. The contaminant is transported through the fiber walls to the permeate side. Species with a higher molecular weight are retained in the dialysate side, due to their low solubility and diffusivity through the membrane.

Figure 1: The hollow fiber assembly in a dialysis module.

Dialysate feed

2 | S E P A R A T I O N T H R O U G H D I A L Y S I S

Model Definition

This example models a piece of hollow fiber through which the dialysate flows with a fully developed laminar parabolic velocity profile. The fiber is surrounded by a permeate that flows laminarly in the same direction as the dialysate.The dialysate, the permeate, and the membrane are all examined in the results. The model domain is shown in Figure 2.Here, the angular gradients are considered negligible, so an axisymmetrical approximation can be used.

Figure 2: Illustrations of the hollow fiber setup with the dialysate and permeate, and of the model domain.

Model domain

Model domain

Symmetry axis

Hol

low

fibe

r

Perm

eate

Dia

lysa

te

Dialysate Permeate

Mem

bran

e


You can draw a hexagonal-shaped unit cell of the fiber assembly as in Figure 3:

Figure 3: Hexagonal-shaped unit cell of the fiber assembly.

As a simplification, the hexagon is approximated as a circle in the model.

The contaminant is transported by diffusion and convection within the two liquids, whereas diffusion is the only transport mechanism through the membrane. The mass transport is modeled with the Transport of Diluted Species interface. To analyze the convective flux, the Laminar Flow interface is utilized, assuming that the flow is laminar.

The contaminant must dissolve into the membrane in order to be transported through it. The interface conditions between the liquids and the membrane are described by the dimensionless partition coefficient K:

(1)

where ci denotes the concentration of the contaminant (SI unit: mol/m3). The subscripts and superscripts describe the location in the dialysis fiber as displayed in Figure 4. This figure also shows a schematic concentration profile.

Hollow fiber

Unit cell

Circular approximation

Kc2

d

c1d-----

c2p

c3p-----= =


Figure 4: The concentration profile across the membrane (see Equation 1). Note that there are discontinuities in the concentration profile at the membrane boundaries.

To obtain a well-posed problem, an appropriate set of boundary conditions must be defined. Figure 5 displays the boundaries that need to be accounted for. Note that the boundaries are discontinuous and boundary conditions need to be set on both sides of the membrane at each liquid interface. Equation 1, set as Pointwise Constraints at the two boundaries, implements these boundary conditions in the model.

Figure 5: The boundaries accounted for in the model.

PermeateDialysate Membrane

c1d

c2d

c3p

c2p

Dialysate Permeate

Mem

bran

e

Permeate,outlet

Permeate,inlet

permeate|membrane

No flux

No flux

No fluxDialysate,inlet

Symmetrymembrane|dialysate

dialysate|membrane

membrane|permeate

Dialysate,outlet


Danckwerts’ inflow conditions are set at the inlet to the dialysate and permeate. At the outlet, the convective contribution to the mass transport is assumed to be much larger than the diffusive contribution and is modeled by setting outflow conditions. Symmetry applies at the leftmost boundary for this axisymmetrical model geometry and no flux is set at the membrane edges and the rightmost boundary, since no species pass these.

S U M M A R Y O F I N P U T D A T A

The input data are listed in the table:

Results

The surface plot in Figure 6 visualizes the concentration distribution throughout the three model domains in 3D: the dialysate liquid inside the hollow fiber (nearest the center), the membrane, and the permeate liquid outside the hollow fiber. As the plot shows, the concentration inside the hollow fiber decreases markedly over the first 10 mm from the inlet. The maximum concentration in the permeate occurs close to the outlet. If there is a risk of deposition on the fiber’s outer surface due to a high concentration of filtrated

PROPERTY VALUE DESCRIPTION

D 10-9 m2/s Diffusion coefficient, liquids

Dm 10-9 m2/s Diffusion coefficient, membrane

Rhf 0.2 mm Inner radius, hollow fiber

Lm 0.28 mm Thickness, membrane

Lpc 0.7 mm Width, concentric permeate channel

H 21 mm Length, fiber

Uave_dia 0.5 mm/s Average velocity, dialysate

Uave_per 0.8 mm/s Average velocity, permeate

K 0.7 Partition coefficient

c0 1 M Inlet concentration, dialysate


species, it is largest at the location of this maximum. The figure also shows the developing diffusion layers on both sides of the fiber wall.

Figure 6: Concentration in the three domains as seen from the inlet of the fiber.


Concentration jumps arise at the boundaries between the domains. This is shown in Figure 7 where the concentration profile at the middle of the fiber length is plotted along the radius of the model geometry.

Figure 7: Concentration across the three domains at the middle of the fiber length and at the outlet.

Modeling in COMSOL Multiphysics

Manually defined point-wise constraints are used to model the discontinuities in the concentration profile at the boundaries between the liquid and membrane phases. The constraint expressions are defined according to Equation 1.

When defining these constraints the constraint forces need to be set to ensure a continuous flux over the phase boundaries. This is done by using test() operators, operating on the dependent variables that are solved for in each domain, for example test(c1). What the test() terms in the force expressions represent depends on the partial differential equation solved for in each domain. In this model, the test() terms in the force expression on the liquid-membrane boundaries represent the diffusive flux.


References

1. M. Mulder, Basic Principles of Membrane Technology, 2nd ed., Kluwer Academic Publishers, 1998.

2. R.B. Bird, W.E. Stewart, and E.N. Lightfoot, Transport Phenomena, John Wiley & Sons, 1960.

Application Library path: Chemical_Reaction_Engineering_Module/Mixing_and_Separation/dialysis



N E W



1 In the Model Wizard window, click 2D Axisymmetric.

2 In the Select Physics tree, select Chemical Species Transport>

Transport of Diluted Species (tds).

3 Click Add.

4 In the Concentrations table, enter the following settings:



6 Click Add.


8 In the Select Physics tree, select Fluid Flow>Single-Phase Flow>Laminar Flow (spf).

9 Click Add.

10 Click Study.

c1

c2


11 In the Select Study tree, select Preset Studies for Selected Physics Interfaces>Stationary.

12 Click Done.


Load parameters from a text-file.




4 Browse to the model’s Application Libraries folder and double-click the file dialysis_parameters.txt.

Draw the geometry and make selections.

G E O M E T R Y 1

1 In the Model Builder window, under Component 1 (comp1) click Geometry 1.

2 In the Settings window for Geometry, locate the Units section.

3 From the Length unit list, choose mm.

Rectangle 1 (r1)1 On the Geometry toolbar, click Primitives and choose Rectangle.

2 In the Settings window for Rectangle, locate the Size and Shape section.

3 In the Width text field, type Rhf.

4 In the Height text field, type H.

5 Right-click Rectangle 1 (r1) and choose Build Selected.




3 In the Width text field, type Lm.


5 Locate the Position section. In the r text field, type Rhf.






3 In the Width text field, type Lpc.


5 Locate the Position section. In the r text field, type Rhf+Lm.



Form Union (fin)In the Model Builder window, under Component 1 (comp1)>Geometry 1 right-click Form Union (fin) and choose Build Selected.

Explicit Selection 1 (sel1)1 On the Geometry toolbar, click Selections and choose Explicit Selection.

2 In the Settings window for Explicit Selection, type Dialysate and Permeate in the Label text field.

3 On the object fin, select Domains 1 and 3 only.


2 In the Settings window for Explicit Selection, type Membrane in the Label text field.

3 On the object fin, select Domain 2 only.


Variables 11 On the Home toolbar, click Variables and choose Local Variables.

Add a common variable c_all for all concentrations.

2 In the Settings window for Variables, locate the Geometric Entity Selection section.

3 From the Geometric entity level list, choose Domain.

4 From the Selection list, choose Dialysate and Permeate.

5 Locate the Variables section. In the table, enter the following settings:

Name Expression Unit Description

c_all c1 mol/m³ Concentration





4 From the Selection list, choose Membrane.


Model transport by convection and diffusion in the dialysate and permeate, and diffusion in the membrane.

TR A N S P O R T O F D I L U T E D S P E C I E S ( T D S )

1 In the Model Builder window, under Component 1 (comp1) click Transport of Diluted Species (tds).

2 In the Settings window for Transport of Diluted Species, type Transport of Diluted Species - Dialysate and Permeate in the Label text field.

3 Locate the Domain Selection section. From the Selection list, choose Dialysate and Permeate.

TR A N S P O R T O F D I L U T E D S P E C I E S - D I A L Y S A T E A N D P E R M E A T E ( T D S )

On the Physics toolbar, click Transport of Diluted Species (tds) and choose Transport of Diluted Species - Dialysate and Permeate (tds).

Transport Properties 11 In the Model Builder window, expand the Component 1 (comp1)>

Transport of Diluted Species - Dialysate and Permeate (tds) node, then click Transport Properties 1.

2 In the Settings window for Transport Properties, locate the Diffusion section.

3 In the Dc1 text field, type D.

Initial Values 11 In the Model Builder window, under Component 1 (comp1)>Transport of Diluted Species -

Dialysate and Permeate (tds) click Initial Values 1.

2 In the Settings window for Initial Values, locate the Initial Values section.

3 In the c1 text field, type c0_dia.


c_all c2 mol/m³ Concentration


Initial Values 21 Right-click Component 1 (comp1)>Transport of Diluted Species -

Dialysate and Permeate (tds)>Initial Values 1 and choose Duplicate.

2 Select Domain 3 only.


4 In the c1 text field, type c0_per.

Inflow 11 On the Physics toolbar, click Boundaries and choose Inflow.

2 Select Boundary 2 only.

3 In the Settings window for Inflow, locate the Concentration section.

4 In the c0,c1 text field, type c0_dia.

5 Locate the Boundary Condition Type section. From the list, choose Flux (Danckwerts).




4 In the c0,c1 text field, type c0_per.


Outflow 11 On the Physics toolbar, click Boundaries and choose Outflow.

2 Select Boundaries 3 and 9 only.

3 In the Model Builder window’s toolbar, click the Show button and select Advanced Physics Options in the menu.

Pointwise Constraint 11 On the Physics toolbar, click Boundaries and choose Pointwise Constraint.

Select a pointwise boundary constraint to account for the transport between the domains.


3 In the Settings window for Pointwise Constraint, locate the Pointwise Constraint section.

4 From the Apply reaction terms on list, choose User defined.

5 In the Constraint expression text field, type c2-K*c1.

6 In the Constraint force expression text field, type test(c2)-test(c1).


TR A N S P O R T O F D I L U T E D S P E C I E S 2 ( T D S 2 )

1 In the Model Builder window, under Component 1 (comp1) click Transport of Diluted Species 2 (tds2).

2 In the Settings window for Transport of Diluted Species, type Transport of Diluted Species - Membrane in the Label text field.

3 Locate the Domain Selection section. From the Selection list, choose Membrane.

4 Locate the Transport Mechanisms section. Clear the Convection check box.

TR A N S P O R T O F D I L U T E D S P E C I E S - M E M B R A N E ( T D S 2 )

On the Physics toolbar, click Transport of Diluted Species 2 (tds2) and choose Transport of Diluted Species - Membrane (tds2).

Transport Properties 11 In the Model Builder window, under Component 1 (comp1)>Transport of Diluted Species -

Membrane (tds2) click Transport Properties 1.


3 In the Dc2 text field, type Dm.


Membrane (tds2) click Initial Values 1.


3 In the c2 text field, type c0_mem.

A D D M A T E R I A L

1 On the Home toolbar, click Add Material to open the Add Material window.

2 Go to the Add Material window.

3 In the tree, select Liquids and Gases>Liquids>Water.

4 Click Add to Component in the window toolbar.

5 On the Home toolbar, click Add Material to close the Add Material window.

L A M I N A R F L O W ( S P F )

1 In the Settings window for Laminar Flow, locate the Domain Selection section.

2 From the Selection list, choose Dialysate and Permeate.

3 In the Model Builder window, click Laminar Flow (spf).


Inlet 11 On the Physics toolbar, click Boundaries and choose Inlet.


3 In the Settings window for Inlet, locate the Boundary Condition section.

4 From the list, choose Laminar inflow.

5 Locate the Laminar Inflow section. In the Uav text field, type Uave_dia.





5 Locate the Laminar Inflow section. In the Uav text field, type Uave_per.

Outlet 11 On the Physics toolbar, click Boundaries and choose Outlet.


M U L T I P H Y S I C S

Flow Coupling 1 (fc1)On the Physics toolbar, click Multiphysics Couplings and choose Global>Flow Coupling.

M E S H 1

Distribution 11 In the Model Builder window, under Component 1 (comp1) right-click Mesh 1 and choose

Mapped.

2 Right-click Mapped 1 and choose Distribution.

3 Select Boundaries 1, 4, 7, and 10 only.

4 In the Settings window for Distribution, locate the Distribution section.

5 From the Distribution properties list, choose Predefined distribution type.

6 In the Number of elements text field, type 250.

7 In the Element ratio text field, type 25.

Distribution 21 Right-click Mapped 1 and choose Distribution.







7 Select the Symmetric distribution check box.










4 Click the Zoom Box button on the Graphics toolbar.






10 Select the Reverse direction check box.

Solve the model in two steps. First, the Laminar Flow interface and thereafter the Transport

of Diluted Species interfaces.

S T U D Y 1

Step 1: Stationary1 In the Model Builder window, under Study 1 click Step 1: Stationary.

2 In the Settings window for Stationary, locate the Physics and Variables Selection section.


3 In the table, clear the Solve for check box for Transport of Diluted Species - Dialysate and

Permeate (tds) and Transport of Diluted Species - Membrane (tds2).

Step 2: Stationary 21 On the Study toolbar, click Study Steps and choose Stationary>Stationary.


3 In the table, clear the Solve for check box for Laminar Flow (spf).


Plot the concentration distribution in Figure 6. For this axisymmetric geometry, the Revolution 2D data set is used.

R E S U L T S

Concentration (tds2) 11 In the Model Builder window, under Results click Concentration (tds2) 1.

2 In the Settings window for 3D Plot Group, type Concentration 2D Revolution - All in the Label text field.

Revolution 2D 21 On the Results toolbar, click More Data Sets and choose Revolution 2D.

2 In the Settings window for Revolution 2D, click to expand the Revolution layers section.

3 Locate the Revolution Layers section. In the Start angle text field, type -90.

4 In the Revolution angle text field, type 225.

Concentration 2D Revolution - All1 In the Model Builder window, expand the Results>Concentration 2D Revolution - All node,

then click Concentration 2D Revolution - All.


3 From the Data set list, choose Revolution 2D 2.

Surface1 In the Model Builder window, under Results>Concentration 2D Revolution - All click

Surface.

2 In the Settings window for Surface, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Component 1>Definitions>Variables>

c_all - Concentration.

If necessary, the view angle of the plot can be adjusted with the mouse.


Concentration 2D Revolution - All1 Click the Zoom Extents button on the Graphics toolbar.

2 On the Concentration 2D Revolution - All toolbar, click Plot.

Velocity (spf) 11 In the Model Builder window, expand the Concentration 2D Revolution - All node, then

click Results>Velocity (spf) 1.



4 On the Velocity (spf) 1 toolbar, click Plot.

Create cut lines at two locations along the fiber length to illustrate the concentration jump between the domains in Figure 7.

Cut Line 2D 11 On the Results toolbar, click Cut Line 2D.

2 In the Settings window for Cut Line 2D, locate the Line Data section.

3 In row Point 1, set z to H/2.

4 In row Point 2, set r to Rhf+Lm+Lpc and z to H/2.



3 In row Point 1, set z to H.

4 In row Point 2, set r to Rhf+Lm+Lpc and z to H.

1D Plot Group 81 On the Results toolbar, click 1D Plot Group.

2 In the Settings window for 1D Plot Group, type Concentration Jump in the Label text field.

Line Graph 11 Right-click Concentration Jump and choose Line Graph.

2 In the Settings window for Line Graph, type At H/2 in the Label text field.

3 Locate the Data section. From the Data set list, choose Cut Line 2D 1.

4 Locate the y-Axis Data section. In the Expression text field, type c_all.



6 In the Expression text field, type r.





Line Graph 21 Right-click Results>Concentration Jump>At H/2 and choose Line Graph.

2 In the Settings window for Line Graph, type At H in the Label text field.


4 Locate the y-Axis Data section. In the Expression text field, type c_all.


6 In the Expression text field, type r.


8 Locate the Legends section. Select the Show legends check box.



11 On the Concentration Jump toolbar, click Plot.

Fix the name of all plot groups.

Concentration (tds)1 In the Model Builder window, under Results click Concentration (tds).

2 In the Settings window for 2D Plot Group, type Concentration Surface - Dialysate and Permeate in the Label text field.

Concentration (tds) 11 In the Model Builder window, under Results click Concentration (tds) 1.

Legends

At half fiber length

Legends

At outlet


2 In the Settings window for 3D Plot Group, type Concentration 2D revolution - Dialysate and Permeate in the Label text field.

Concentration (tds2)1 In the Model Builder window, under Results click Concentration (tds2).

2 In the Settings window for 2D Plot Group, type Concentration Surface - Membrane in the Label text field.

Velocity (spf)1 In the Model Builder window, under Results click Velocity (spf).

2 In the Settings window for 2D Plot Group, type Velocity Surface in the Label text field.

Velocity (spf) 11 In the Model Builder window, under Results click Velocity (spf) 1.

2 In the Settings window for 3D Plot Group, type Velocity 2D Revolution in the Label text field.



Ch em i c a l R e a c t i o n s and S oo t Bu i l d -Up i n a D i e s e lF i l t e r




Introduction

This example deals with a model of a filter system for a diesel engine. A porous filter separates soot particles from exhaust gases passing through it, leading to the formation of a soot layer. Both catalytic and non-catalytic reactions suppress the layer’s build-up; carbon is oxidized to carbon monoxide and carbon dioxide, which both pass through the membrane.

Diesel filters are used to remove particulate matter in diesel-engine exhaust gases. A filter system’s efficiency and durability are closely related to the manner in which it removes soot deposits from the porous filter walls. For instance, one method is to remove the soot layer by means of non-catalytic reactions with oxygen. However, this scheme requires that the exhaust temperature increases above normal operating conditions. Another approach involves introducing cerium additives to the fuel. Cerium oxide species are subsequently present in the soot layer, acting as a catalyst in carbon-oxidation reactions. Under these conditions, it is possible to remove carbon deposits in the filter without increasing the exhaust temperature. The following model illustrates both of these working conditions.

Diesel filters are typically of a monolithic type with narrow channels running through a cylindrical structure. The silicon carbide filter under study is 15 cm long with a total of 2000 channels. Filter channels are open only at one end and are arranged in an alternating fashion in the monolith. The channels are separated by porous walls, as illustrated in Figure 1.

Figure 1: Front view of a channel section in a diesel filter. The A-channels are open while the B-channels are closed. The channels are separated by porous filter walls. A back view would show the opposite configuration. Gas therefore enters the filter through the A-channels and exits through the B-channels.

This application follows a two-part strategy to investigate the system in which these reactions take place. The first part is setting up an ideal model in the Reaction Engineering

Filter wall

Outlet

Inlet Open flow channel A

Open flow channel B

2 | C H E M I C A L R E A C T I O N S A N D S O O T B U I L D - U P I N A D I E S E L F I L T E R

interface with the plug flow reactor feature that assumes stationary conditions and accounts only for variations with reactor volume. This gives a rather qualitative understanding of the reactor system. The second part sets up a space- and time-dependent model in which variations in species composition, soot layer build up, and temperature are investigated in detail.

Model Definition

C H E M I C A L R E A C T I O N S

The soot layer in the diesel filter must be removed continuously or at intervals to keep the filter in working condition. Oxygen can react with carbon to form carbon monoxide and carbon dioxide. Further reaction pathways open when the diesel fuel is treated with additives containing cerium. The soot layer then contains cerium-oxide species that act as catalysts to oxidize the carbon, and oxygen regenerates the catalyst.

This model considers the following reactions taking place in the soot layer:

The reactions occur in the soot layer. The reaction rates (mol/(m3·s)) are given by:

+ O2

k1C CO2

+ 0.5O2

k2C CO

+ 4CeO2

k3C CO22Ce2O3 +

+ 2CeO2

k4C COCe2O3 +

0.5O2

k5Ce2O3 2CeO2+

r1 k1cO2=

r2 k2cO2=

r3 k3xCeO2=

r4 k4xCeO2=


where cO2 is the molar concentration of oxygen (SI unit: mol/m3), while xCeO2 and xCe2O3 are the molar fractions of the different cerium (catalytic) species.

P L U G F L O W M O D E L

The Reaction Engineering interface can automatically define the material and energy balances for a plug flow reactor at steady-state, as shown by the governing equation:

(1)

Fi is the molar flow and Ri includes all the reactions per unit volume of the channel in which species i is involved. Equation 1 thus gives a solution of an ideal model system only dependent on the accumulated volume, V, the species pass in the reactor. The plug flow model is solved for isothermal conditions assuming the reactions have negligible impact on temperature in the system since it contains an excess of nitrogen solvent.

The modeled system is schematically illustrated in Figure 2 and consists of one single cell and one single filter-unit. The exhaust gas enters Channel A and passes through a single filter wall to Channel B (note that a repetitive cell would have a filter wall on all sides). The plug flow model system treats just Channel A, the deposited particles, and the catalyst, and neglects any reaction in Channel B. Along the system boundaries, the filter is a distributed outlet. Because the pressure loss over the filter wall is comparably large, the velocity across

r5 k5xCe2O3cO2=

dFidV--------- Ri=


the wall along the length of the channel is considered to be constant. This implies that the velocity component along the channel decreases linearly from the inlet to the outlet.

Figure 2: Schematic illustration the plug flow system. The exhaust enters Channel A, goes through the filter wall, and exits through Channel B. Assume the velocity across the filter wall is constant along the x direction in the plug flow model, while the velocity component along the length of the channel decreases linearly with the distance from the inlet.

The general material balance for a small section of Channel A of length Δx gives the following equation for any species i:

(2)

Ni denotes the flux vector (SI unit: mol/(m2s)), Acs equals the channel’s cross-sectional area (SI unit: m2), L the reactor length (SI unit: m), and uf represents the velocity flow of species towards the filter (SI unit: m/s). At ideal conditions, Equation 2 can likewise be set up over the volume element (ΔV) through which the gas passes in Δx:

(3)

In Equation 3, vf represents a volumetric flow of species towards the filter (SI unit: m3/s), H the reactor height (SI unit: m). The reaction is limited to the soot layer on the filter surface. Because the kinetic data is given per unit volume of soot, the volume of soot per unit is estimated to be the same as the volume of the channel. Dividing Equation 3 by ΔV

x

Outlet

Inlet

Inlet

Modeled system

Channel A

Channel B

Velo

city

in

xdi

rect

ion

Ni x Δx+, Ni x,–( )Acs ciufΔVL-------- RiAcsΔx–+ 0=

Fi V ΔV+, Fi V,–( ) civfΔxH------- RiΔV–+ 0=


and if ΔV approaches zero gives:

(4)

From the assumption of a linearly decreasing flow along the channel, the volumetric flow in the reactor is given by:

where ku (SI unit: 1/s) denotes the proportionality constant and v0 (SI unit: m3/s) equals the inlet flow. V is the accumulated volume the species have passed in the tubular reactor. Note that the cross-sectional area is actually irrelevant; that is, the reactor volume, V, can just as well be interpreted as length and compositions, ci, as mol per reactor length. The second term on the left-hand side in Equation 4 is added to all species in the model using the Additional Source feature. The compositions of species along the plug flow reactor are expressed in terms of concentrations with inlet molar flows set as a simplification for all species. Another generalization is that all species are affected by the flow over the filter, even the solid components.

S P A C E - D E P E N D E N T M O D E L

This example removes a channel section from the monolith and places it into an experimental setup to study only a single filter unit with a single filter wall. Assuming that only a single filter wall exists and by thermally and mechanically insulating the upper and vertical channel walls, the system can be described with the 2D geometry shown in Figure 3.

dFidV--------- ciuf

1H-----+ Ri=

v kuV v0+=


Figure 3: Modeled domain including four sections: the inlet Channel A, soot layer, filter wall, and outlet Channel B. The units on the horizontal and vertical axes are meters and millimeters, respectively.

The model includes four model sections: the inlet Channel A, the soot layer on the surface of the filter wall, the filter wall, and the outlet Channel B.

The filter is 15 cm long, and the height of a single channel is 1.27 mm. The filter wall is 0.40 mm thick. Diesel exhaust gas enters the inlet channel, and the gas containing soot particles filters off through the porous SiC filter wall, thereby leaving a soot layer on top of this wall. The exhaust exits the filter through the outlet channel. You must treat the top surface of the soot layer as a moving sub-boundary that grows and shrinks according to the amount of soot.

To solve this model, use the following physics interfaces in the space-dependent model:

• Free and Porous Media Flow

• Transport of Diluted Species (material balance)

• Heat Transfer in Fluids (energy balance)

• Moving Mesh (ale) (handles the moving top boundary of the soot layer)

In the Free and Porous Media Flow interface, free flow is defined in both the inlet and outlet channels, whereas the Porous Matrix Properties are defined in the soot layer and the filter wall. The reaction kinetics are taken from the plug flow model with the Generate Space-Dependent Model feature and are stored in a Chemistry interface in the space-dependent model.

Inlet channel A

Outlet channel B

Filter wall

Soot layer


There are two issues worth mentioning regarding the domain equations and boundary conditions used in the model implementation:

• The Moving Mesh interface needs an expression for the mesh velocity at the top surface of the soot layer. This expression comes from:

Here, vy is the mesh velocity, v is the y-component of the flow velocity at the top surface of the soot layer, Mc equals the molar weight of carbon, Rc denotes the sum of all carbon-consuming reaction rates (averaged across the layer thickness), and δsl equals the soot layer’s thickness.

• Note that the properties of the porous filter wall and soot layer are a combination of those of solids and fluids. For example, in the energy balance, the conduction term should include the conductivities of the solid and the fluid, while the convection term should contain only the properties of the fluid because the solid does not move. The accumulation (time-dependent) term should include a mixture of both properties in the following way:

where ρmix is the mixture’s density (SI unit: kg/m3), Cp,mix is its heat capacity (SI unit: J/(kg·K)), kmix equals its thermal conductivity (SI unit: W/(m·K)), and εsolid the void fraction (porosity).


C H E M I C A L R E A C T I O N S A N D P L U G F L O W M O D E L

Start by looking at the plug flow model for Channel A. Figure 4 and Figure 5 depict the composition of the reacting species along the channel length at two different

vyρpartρsoot-------------–

vMcδslρsoot---------------RC–=

ρmix ρsolid 1 εsolid–( ) ρgasεsolid+=

Cpmix Cpsolid 1 εsolid–( ) Cpgasεsolid+=

kmix ksolid 1 εsolid–( ) kgasεsolid+=


temperatures, one at 653 K, below “catalyzer ignition” temperature, and one at 705 K where the catalyst is ignited.

Figure 4: Composition of reacting species along the length of the reactor at 653 K.


Figure 5: Composition along the length of the reactor at 703 K.

At 703 K oxygen becomes depleted at approximately 0.12 m into the reactor. As the temperature increases, the rate of oxidation increases until the depletion of oxygen in the reactor. This decreases the rate of CeO2 regeneration substantially and Ce2O3 becomes the dominating cerium-containing species.

The impact of temperature can also be investigated with the ratio of the non-catalyzed to the catalyzed reaction rates for the carbon monoxide and carbon dioxide producing reactions. In the following figures the reaction rate ratios r1/r3, carbon dioxide


production, and r2/r4, carbon monoxide production, rates are depicted for the two different temperatures.

Figure 6: Relationship between the non-catalytic and catalytic carbon-oxidation reactions.

Figure 6 shows that the catalytic reactions by far dominate the oxidation of carbon. This is expected at these relatively moderate temperatures. The oxidation of carbon to carbon dioxide has a higher fraction of non-catalyzed oxidation than that of carbon to carbon monoxide. The dominance of the catalytic reactions decreases with temperature.



Figure 7 shows the initial velocity distribution in the channel unit cell. The flow regime shows laminar behavior.

Figure 7: Velocity magnitude in a filter unit cell.


Figure 8 shows the pressure distribution across the channel pair. The main pressure drop is observed across the soot layer deposited on top of the porous membrane.

Figure 8: Pressure distribution in a filter unit cell.

The oxidation of carbon is overall an exothermic reaction, although the catalytic steps are endothermic. The regeneration of the catalyst is heavily exothermic and makes up for the endothermic properties of the catalytic oxidation reaction. Figure 9 shows the temperature distribution in the system after 300 s for an inlet temperature of 550 K. The exhaust gases leave the filter at somewhat elevated temperatures due to the net exothermicity of the reactions.


Figure 9: Temperature distribution in the filter for an inlet temperature of 550 K.

The following illustration (Figure 10) shows the soot layer along the length of the reactor at an inlet temperature of 550 K. The base-line corresponds to the initial soot layer thickness of 50 μm. The artifact at x = 0 appears because you do not allow the soot layer


to grow or decrease at this position. Under the present conditions, carbon oxidation is not sufficient to keep the soot layer from growing.

Figure 10: The soot layer’s thickness at 0 to 300 s for 50 s intervals for an inlet temperature of 550 K.

Reference

1. G. Konstantas and A.M. Stamatelos, “Computer Aided Engineering of Diesel Filter Systems,” Joint Meeting of the Greek and Italian Sections of the Combustion Institute, http://www.mie.uth.gr/labs/ltte/pubs/Combust_Inst_Corfou_013.pdf.

Application Library path: Chemical_Reaction_Engineering_Module/Reactors_with_Porous_Catalysts/diesel_filter




http://www.mie.uth.gr/labs/ltte/pubs/Combust_Inst_Corfou_013.pdf

N E W





3 Click Add.

4 Click Study.

5 In the Select Study tree, select Preset Studies>Stationary Plug Flow.

6 Click Done.


Load model parameters from a text file.




4 Browse to the model’s Application Libraries folder and double-click the file diesel_filter_parameters.txt.



choose Variables.

Add the variable for the volumetric flow in the tubular reactor.







v1 v_inlet+ku*re.Vr m³/s Volumetric flow field in tubular reactor


3 From the Reactor type list, choose Plug flow.

4 Locate the Energy Balance section. In the T text field, type T0.

5 Click to expand the Calculate transport properties section. Locate the Calculate Transport Properties section. Select the Calculate mixture properties check box.

6 Locate the Mass Balance section. From the Volumetric rate list, choose User defined.

7 In the v text field, type v1.



3 In the Formula text field, type c+o2=>co2.

4 Click Apply.

Modify the reaction rates in order to take the difference in domain thickness between the catalyzing layer (0.05 mm thick) and the channel (1.27 mm). This is achieved by multiplying all reaction rates with the constant Sa. Alternatively, edit the Arrhenius parameter for the Frequency factor by multiplying the value with the constant Sa.


6 In the Af text field, type 1e13*Sa*1[m^4/(s*mol)].

7 In the Ef text field, type 165e3[J/mol].


9 In the r text field, type re.kf_1*re.c_o2*1[mol/m^3].

In this model, user-defined reaction rate expressions are used.



3 In the Formula text field, type c+0.5o2=>co.

4 Click Apply.


6 In the Af text field, type 5.5e10*Sa*1[m/s].



9 In the r text field, type re.kf_2*re.c_o2.




3 In the Formula text field, type c+4ceo2=>2ce2o3+co2.

4 Click Apply.


6 In the Af text field, type 4.5e11*Sa*1[m^13/(s*mol^4)].



9 In the r text field, type re.kf_3*re.c_ceo2*1[mol^4/m^12].



3 In the Formula text field, type c+2ceo2=>ce2o3+co.

4 Click Apply.


6 In the Af text field, type 4e8*Sa*1[m^7/(s*mol^2)].



9 In the r text field, type re.kf_4*re.c_ceo2*1[mol^2/m^6].



3 In the Formula text field, type ce2o3+0.5o2=>2ceo2.

4 Click Apply.


6 In the Af text field, type 1e12*Sa*1[m/s].



9 In the r text field, type re.kf_5*re.c_o2*re.c_ce2o3*1[m^3/mol].


Species: cThe carbon composition is locked in the model.

1 In the Model Builder window, under Component 1 (comp1)>Reaction Engineering (re) click Species: c.


3 In the M text field, type 12e-3[kg/mol].

4 Click to expand the Species concentration/activity section. Locate the Species Concentration/Activity section. Select the Locked concentration/activity check box.

Species: co21 In the Model Builder window, under Component 1 (comp1)>Reaction Engineering (re)

click Species: co2.



Species: co1 In the Model Builder window, under Component 1 (comp1)>Reaction Engineering (re)

click Species: co.



Species: ceo21 In the Model Builder window, under Component 1 (comp1)>Reaction Engineering (re)

click Species: ceo2.



Species: ce2o31 In the Model Builder window, under Component 1 (comp1)>Reaction Engineering (re)

click Species: ce2o3.



Nitrogen is also present in the gas as an inert.



Constant pressure in channel A is assumed. N2 is simply inert and does not act as a solvent in the first part of the model.


3 In the Species name text field, type n2.

4 Locate the General Parameters section. In the M text field, type 28e-3[kg/mol].





Additional Source 11 On the Reaction Engineering toolbar, click Additional Source.

All species are driven towards or through the filter as a simplification.

2 In the Settings window for Additional Source, locate the Additional Rate Expression section.

3 In the Volumetric species table, enter the following settings:

Species Molar flow rate (mol/s)

c c_c_inlet*v_inlet

ce2o3 c_ce2o3_inlet*v_inlet

ceo2 c_ceo2_inlet*v_inlet

co F_co_inlet

co2 F_co2_inlet

n2 F_n2_inlet

o2 F_o2_inlet

Species Additional rate expression (mol/(m^3*s))

ce2o3 -uf/H*re.c_ce2o3

ceo2 -uf/H*re.c_ceo2

co -uf/H*re.c_co

co2 -uf/H*re.c_co2


S T U D Y 1

Step 1: Stationary Plug Flow1 In the Model Builder window, under Study 1 click Step 1: Stationary Plug Flow.

2 In the Settings window for Stationary Plug Flow, locate the Study Settings section.

3 In the Volumes text field, type 0.149.

Using 0.149 avoids describing the stagnant zone close to the dead-end of the tubular reactor.

To study the model at two different temperatures, use a parametric sweep to vary T0.



3 Click Add.



To produce the plots shown in Figures 4, 5, and 6 follow these steps.

R E S U L T S

Molar Flow Rate (re)1 In the Model Builder window, under Results click Molar Flow Rate (re).

2 In the Settings window for 1D Plot Group, type Concentrations plug flow model in the Label text field.



5 In the associated text field, type L (m).

n2 -uf/H*re.c_n2

o2 -uf/H*re.c_o2


T0 (Initial temperature) 653 703



Global 11 In the Model Builder window, expand the Results>Concentrations plug flow model node,

then click Global 1.




5 Click Delete.

6 Click Delete.




10 Locate the Data section. From the Parameter selection (T0) list, choose First.

11 On the Concentrations plug flow model toolbar, click Plot.

12 From the Parameter selection (T0) list, choose Last.

13 On the Concentrations plug flow model toolbar, click Plot.

Concentrations plug flow model 11 In the Model Builder window, under Results right-click Concentrations plug flow model

and choose Duplicate.


comp1.re.c_o2 mol/m^3 Concentration

comp1.re.c_co2 mol/m^3 Concentration

comp1.re.c_co mol/m^3 Concentration

comp1.re.c_ceo2 mol/m^3 Concentration

comp1.re.c_ce2o3 mol/m^3 Concentration

Legends

O2

CO2

CO

CeO2

Ce2O3

2 In the Settings window for 1D Plot Group, type Rate comparison plug flow model in the Label text field.

3 Locate the Plot Settings section. Select the y-axis label check box.

4 In the associated text field, type Rate ratio comparison (-).

Global 11 In the Model Builder window, expand the Results>Rate comparison plug flow model node,

then click Global 1.


3 From the Parameter selection (T0) list, choose First.

4 Click Replace Expression in the upper-right corner of the y-axis data section. From the menu, choose Component 1>Reaction Engineering>comp1.re.r_1 - Reaction rate.



7 Locate the Legends section. In the table, enter the following settings:

Global 21 Right-click Results>Rate comparison plug flow model>Global 1 and choose Duplicate.


3 From the Parameter selection (T0) list, choose Last.


5 On the Rate comparison plug flow model toolbar, click Plot.


comp1.re.r_1/comp1.re.r_3 1

comp1.re.r_2/comp1.re.r_4 1

Legends

r_1/r_3 at 653 K

r_2/r_4 at 653 K

Legends

r_1/r_3 at 703 K

r_2/r_4 at 703 K



Generate Space-Dependent Model 1Use the Generate Space-Dependent Model feature to generate a 2D model that solves mass, energy, and momentum balances within the system.

1 On the Reaction Engineering toolbar, click Generate Space-Dependent Model.

2 In the Settings window for Generate Space-Dependent Model, locate the Component Settings section.

3 From the Component to use list, choose 2D: New.

4 Locate the Physics Interfaces section. Find the Heat transfer subsection. From the list, choose Heat Transfer in Fluids: New.

5 Find the Fluid flow subsection. From the list, choose Free and Porous Media Flow: New.

6 Locate the Space-Dependent Model Generation section. Click Create/Refresh.

C O M P O N E N T 2 ( C O M P 2 )

In the Model Builder window, click Component 2 (comp2).

A D D P H Y S I C S

1 On the Home toolbar, click Add Physics to open the Add Physics window.

2 Go to the Add Physics window.

Also, add a Moving Mesh interface in order to later be able to model the soot layer in the system.

3 In the tree, select Mathematics>Deformed Mesh>Moving Mesh (ale).


M OV I N G M E S H ( A L E )

On the Home toolbar, click Add Physics to close the Add Physics window.


In the Model Builder window, expand the Component 2 (comp2)>Definitions node.

Axis1 In the Model Builder window, expand the Component 2 (comp2)>Definitions>View 1

node, then click Axis.

2 In the Settings window for Axis, locate the Axis section.

3 From the View scale list, choose Automatic.


G E O M E T R Y 1 ( 2 D )

Start by drawing the Outlet Channel, B as shown in Figure 3.



3 In the Width text field, type 0.16.

4 In the Height text field, type 1.27e-3.

Next, draw the soot layer and filter wall as follows.




4 In the Height text field, type 4.5e-4.

5 Locate the Position section. In the y text field, type 1.27e-3.

6 Click to expand the Layers section. In the table, enter the following settings:

Finally move the Rectangle 1 as shown below to create the inlet channel.

Move 1 (mov1)1 On the Geometry toolbar, click Transforms and choose Move.


3 Select the object r1 only.

4 In the Settings window for Move, locate the Input section.

5 Select the Keep input objects check box.

6 Locate the Displacement section. In the x text field, type -1e-2.

7 In the y text field, type 1.72e-3.

Form Union (fin)1 In the Model Builder window, under Component 2 (comp2)>Geometry 1(2D) right-click

Form Union (fin) and choose Build Selected.


Layer name Thickness (m)

Layer 1 4e-4


Make geometry selections.


2 In the Settings window for Explicit Selection, type Inlet in the Label text field.

3 Locate the Entities to Select section. From the Geometric entity level list, choose Boundary.

4 On the object fin, select Boundary 1 only.


2 In the Settings window for Explicit Selection, type Outlet in the Label text field.


4 On the object fin, select Boundary 15 only.


2 In the Settings window for Explicit Selection, type Channel A in the Label text field.



2 In the Settings window for Explicit Selection, type Soot layer in the Label text field.



2 In the Settings window for Explicit Selection, type Filter wall in the Label text field.



2 In the Settings window for Explicit Selection, type Channel B in the Label text field.


Most reaction kinetics definitions were set up in the Reaction Engineering interface, but some changes need to be done directly in the Chemistry interface.


C H E M I S T R Y 1 ( C H E M )

The Chemistry interface should also calculate several material properties. This is activated in the following manner.

1 In the Model Builder window, under Component 2 (comp2) click Chemistry 1 (chem).

2 In the Settings window for Chemistry, click to expand the Calculate thermodynamic properties section.

3 Locate the Calculate Thermodynamic Properties section. Select the Calculate thermodynamic properties check box.

Adjust the reaction kinetics.

1: c+o2=>co21 In the Model Builder window, expand the Chemistry 1 (chem) node, then click 1: c+o2=>

co2.


3 In the Af text field, type 1e13.

4 Locate the Reaction Thermodynamic Properties section. From the Enthalpy of reaction list, choose User defined.

5 In the H text field, type -3.96e5[J/mol].

Set the parameters necessary to compute mixture transport properties. Also, the additional source should be inactivated here, since the filter is now included in the model geometry.

Species: o21 In the Model Builder window, under Component 2 (comp2)>Chemistry 1 (chem) click

Species: o2.

2 In the Settings window for Species, click to expand the Additional source section.

3 Locate the Additional Source section. Clear the Additional source check box.

4 Click to expand the Species transport expressions section. Locate the Species Transport Expressions section. In the σ text field, type 3.467[angstrom].

5 In the ε/kb text field, type 106.7[K].

Species: co21 In the Model Builder window, under Component 2 (comp2)>Chemistry 1 (chem) click

Species: co2.

2 In the Settings window for Species, locate the Additional Source section.

3 Clear the Additional source check box.


4 Locate the Species Transport Expressions section. In the σ text field, type 3.941[angstrom].


2: c+0.5o2=>co1 In the Model Builder window, under Component 2 (comp2)>Chemistry 1 (chem) click

2: c+0.5o2=>co.


3 In the Af text field, type 5.5e10.



Species: co1 In the Model Builder window, under Component 2 (comp2)>Chemistry 1 (chem) click

Species: co.

2 In the Settings window for Species, locate the Additional Source section.

3 Clear the Additional source check box.



3: c+4ceo2=>2ce2o3+co21 In the Model Builder window, under Component 2 (comp2)>Chemistry 1 (chem) click

3: c+4ceo2=>2ce2o3+co2.




5 In the H text field, type 6.72e4[J/mol].

All solid phase species are constant in the space-dependent model. Set these as locked.

Species: ceo21 In the Model Builder window, under Component 2 (comp2)>Chemistry 1 (chem) click

Species: ceo2.


2 In the Settings window for Species, click to expand the Species concentration/activity section.

3 Locate the Species Concentration/Activity section. Select the Locked concentration/activity check box.

Species: ce2o31 In the Model Builder window, under Component 2 (comp2)>Chemistry 1 (chem) click

Species: ce2o3.

2 In the Settings window for Species, locate the Species Concentration/Activity section.

3 Select the Locked concentration/activity check box.

4: c+2ceo2=>ce2o3+co1 In the Model Builder window, under Component 2 (comp2)>Chemistry 1 (chem) click

4: c+2ceo2=>ce2o3+co.




5 In the H text field, type 1.21e5[J/mol].

5: ce2o3+0.5o2=>2ceo21 In the Model Builder window, under Component 2 (comp2)>Chemistry 1 (chem) click

5: ce2o3+0.5o2=>2ceo2.





Select nitrogen as a solvent. When a species has been selected as solvent, the physical properties of the gas phase mixture will be taken to be those of the solvent species. This fits the Transport in Diluted Species interface particularly well.

Additionally, add the thermodynamic properties for the solvent.

Species: n21 In the Model Builder window, under Component 2 (comp2)>Chemistry 1 (chem) click

Species: n2.

2 In the Settings window for Species, locate the Species Type section.


3 From the Species type list, choose Solvent.



6 Click to expand the Species thermodynamic expressions section. Locate the Species Thermodynamic Expressions section. In the Tmid text field, type 3000[K].

7 In the Thi text field, type 3000[K].

8 In the alow, 1 text field, type 3.298677.

9 In the alow, 2 text field, type 0.14082404e-2.

10 In the alow, 3 text field, type -0.03963222e-4.

11 In the alow, 4 text field, type 0.05641515e-7.

12 In the alow, 5 text field, type -0.02444854e-10.

13 In the alow, 6 text field, type -0.10208999e4.

14 In the alow, 7 text field, type 3.950372.

Last, set the concentrations of the constant/locked species.

15 In the Model Builder window, click Chemistry 1 (chem).

16 In the Settings window for Chemistry, locate the Species Matching section.


Move on to the rest of the interfaces.


On the Physics toolbar, click Chemistry 1 (chem) and choose Transport of Diluted Species (tds).


2 In the Settings window for Transport of Diluted Species, locate the Transport Mechanisms section.

Species Species type Molar concentration Reaction rate

c Constant, locked c_c_inlet cLock

ce2o3 Constant, locked c_ce2o3_inlet cLock

ceo2 Constant, locked c_ceo2_inlet cLock

n2 Constant, solvent F_n2_inlet/v_inlet solvent


3 Select the Mass transfer in porous media check box.

Remove the species that are not involved in the mass transport; solid phase species and the solvent.

4 Click to expand the Dependent variables section. Locate the Dependent Variables section. In the Number of species text field, type 3.


Transport Properties 11 In the Model Builder window, expand the Transport of Diluted Species (tds) node, then

click Transport Properties 1.


3 In the Dco2 text field, type comp2.chem.D_o2.

4 In the Dcco2 text field, type comp2.chem.D_co2.

5 In the Dcco text field, type comp2.chem.D_co.

Initial Values 11 In the Model Builder window, under Component 2 (comp2)>

Transport of Diluted Species (tds) click Initial Values 1.


3 In the co2 text field, type 0.

4 In the cco2 text field, type 0.

5 In the cco text field, type 0.

The reactions should only take place in the soot layer.

Reactions 11 In the Model Builder window, under Component 2 (comp2)>

Transport of Diluted Species (tds) click Reactions 1.

2 In the Settings window for Reactions, locate the Domain Selection section.

3 From the Selection list, choose Soot layer.

4 Locate the Reaction Rates section. From the Rco2 list, choose Rate expression for species o2 (chem).

5 From the Rcco2 list, choose Rate expression for species co2 (chem).

co2

cco2


6 Click to expand the Reacting volume section. Locate the Reacting Volume section. From the list, choose Pore volume.

Inflow 1Set the concentrations at the inflow boundary to be same as the initial concentrations.

1 In the Model Builder window, under Component 2 (comp2)>

Transport of Diluted Species (tds) click Inflow 1.

2 In the Settings window for Inflow, locate the Boundary Selection section.

3 From the Selection list, choose Inlet.

4 Locate the Concentration section. In the c0,co2 text field, type F_o2_inlet/v_inlet.

5 In the c0,cco2 text field, type F_co2_inlet/v_inlet.

6 In the c0,cco text field, type F_co_inlet/v_inlet.

Outflow 11 In the Model Builder window, under Component 2 (comp2)>

Transport of Diluted Species (tds) click Outflow 1.

2 In the Settings window for Outflow, locate the Boundary Selection section.

3 From the Selection list, choose Outlet.

4 In the Model Builder window, click Transport of Diluted Species (tds).

Porous Media Transport Properties 11 On the Physics toolbar, click Domains and choose Porous Media Transport Properties.

2 Select Domains 3 and 4 only.

3 In the Settings window for Porous Media Transport Properties, locate the Model Input section.

4 From the T list, choose Temperature (ht).

5 Locate the Matrix Properties section. From the εp list, choose User defined. In the associated text field, type poro.

6 Locate the Convection section. From the u list, choose Velocity field (fp).

7 Locate the Diffusion section. In the DF, co2 text field, type comp2.chem.D_o2.

8 In the DF, cco2 text field, type comp2.chem.D_co2.

9 In the DF, cco text field, type comp2.chem.D_co.

10 From the Effective diffusivity model list, choose No correction.

Continue with the Heat Transfer in Fluids interface, several properties are available from the Chemistry interface.


H E A T TR A N S F E R I N F L U I D S 1 ( H T )

1 In the Model Builder window, under Component 2 (comp2) click Heat Transfer in Fluids 1 (ht).

2 In the Settings window for Heat Transfer in Fluids, locate the Physical Model section.

3 Select the Heat transfer in porous media check box.

Fluid 11 In the Model Builder window, expand the Heat Transfer in Fluids 1 (ht) node, then click

Fluid 1.

2 In the Settings window for Fluid, locate the Heat Conduction, Fluid section.

3 From the k list, choose Thermal conductivity (chem).

4 Locate the Thermodynamics, Fluid section. From the ρ list, choose Density (chem).

5 From the Cp list, choose Mass-averaged mixture specific heat (chem).

Initial Values 11 In the Model Builder window, under Component 2 (comp2)>Heat Transfer in Fluids 1 (ht)


2 In the Settings window for Initial Values, type Tin in the T text field.

Heat Source 11 In the Model Builder window, under Component 2 (comp2)>Heat Transfer in Fluids 1 (ht)

click Heat Source 1.

2 In the Settings window for Heat Source, locate the Domain Selection section.


4 Locate the Heat Source section. From the Q0 list, choose Heat source of reactions (chem).

Temperature 11 In the Model Builder window, under Component 2 (comp2)>Heat Transfer in Fluids 1 (ht)

click Temperature 1.

2 In the Settings window for Temperature, locate the Boundary Selection section.


4 Locate the Temperature section. In the T0 text field, type Tin.

Outflow 11 In the Model Builder window, under Component 2 (comp2)>Heat Transfer in Fluids 1 (ht)

click Outflow 1.




4 In the Model Builder window, click Heat Transfer in Fluids 1 (ht).

Porous Medium 11 On the Physics toolbar, click Domains and choose Porous Medium.


3 In the Settings window for Porous Medium, locate the Model Input section.

4 From the pA list, choose Absolute pressure (fp).

5 From the u list, choose Velocity field (fp).

6 Locate the Heat Conduction, Fluid section. From the k list, choose Thermal conductivity (chem).



9 From the γ list, choose User defined. Locate the Immobile Solids section. In the θp text field, type 1-poro.

10 Locate the Heat Conduction, Porous Matrix section. From the kp list, choose User defined. In the associated text field, type k_s.

11 Locate the Thermodynamics, Porous Matrix section. From the ρp list, choose User defined. In the associated text field, type rho_s.

12 From the Cp, p list, choose User defined. In the associated text field, type Cp_s.

Porous Medium 21 Right-click Porous Medium 1 and choose Duplicate.


3 In the Settings window for Porous Medium, locate the Heat Conduction, Porous Matrix section.

4 In the kp text field, type k_m.

5 Locate the Thermodynamics, Porous Matrix section. In the ρp text field, type rho_m.

6 In the Cp, p text field, type Cp_m.


Temperature Coupling 1 (tc1)On the Physics toolbar, click Multiphysics Couplings and choose Global>

Temperature Coupling.


F R E E A N D PO R O U S M E D I A F L OW 1 ( F P )

1 In the Model Builder window’s toolbar, click the Show button and select Discretization in the menu.

2 In the Model Builder window, under Component 2 (comp2) click Free and Porous Media Flow 1 (fp).

3 In the Settings window for Free and Porous Media Flow, click to expand the Discretization section.

4 From the Discretization of fluids list, choose P2+P1.

This uses second-order elements for the velocity components and linear elements for the pressure field.

Fluid Properties 11 In the Model Builder window, expand the Free and Porous Media Flow 1 (fp) node, then

click Fluid Properties 1.

2 In the Settings window for Fluid Properties, locate the Fluid Properties section.

3 From the ρ list, choose Density (chem).

4 From the μ list, choose Dynamic viscosity (chem).

Inlet 11 In the Model Builder window, under Component 2 (comp2)>

Free and Porous Media Flow 1 (fp) click Inlet 1.

2 In the Settings window for Inlet, locate the Boundary Selection section.


4 Locate the Boundary Condition section. From the list, choose Laminar inflow.

5 Locate the Laminar Inflow section. In the Uav text field, type 2.5[m/s].

Outlet 11 In the Model Builder window, under Component 2 (comp2)>

Free and Porous Media Flow 1 (fp) click Outlet 1.

2 In the Settings window for Outlet, locate the Boundary Selection section.


Fluid and Matrix Properties 11 On the Physics toolbar, click Domains and choose Fluid and Matrix Properties.

2 In the Settings window for Fluid and Matrix Properties, locate the Domain Selection section.



4 Locate the Fluid Properties section. From the ρ list, choose Density (chem).


6 Locate the Porous Matrix Properties section. From the εp list, choose User defined. In the associated text field, type poro.

7 From the κ list, choose User defined. In the associated text field, type kappa_s.

Fluid and Matrix Properties 21 Right-click Fluid and Matrix Properties 1 and choose Duplicate.

2 In the Settings window for Fluid and Matrix Properties, locate the Domain Selection section.

3 From the Selection list, choose Filter wall.

4 Locate the Porous Matrix Properties section. In the κ text field, type kappa_m.

The soot layer changes in size and the model takes this into account with the Moving Mesh interface where boundary 10 (top of soot layer) is prescribed to shift and Free Deformation is allowed in Channel A and the soot layer.

M OV I N G M E S H ( A L E )

In the Model Builder window, under Component 2 (comp2) click Moving Mesh (ale).

Free Deformation 11 On the Physics toolbar, click Domains and choose Free Deformation.


Prescribed Mesh Displacement 21 On the Physics toolbar, click Boundaries and choose Prescribed Mesh Displacement.

2 Select Boundaries 8, 13, and 14 only.

3 In the Settings window for Prescribed Mesh Displacement, locate the Prescribed Mesh Displacement section.

4 Clear the Prescribed y displacement check box.

Prescribed Mesh Velocity 11 On the Physics toolbar, click Boundaries and choose Prescribed Mesh Velocity.

An expression including the reactions that consume the soot particles is added to the top of the soot layer.


3 In the Settings window for Prescribed Mesh Velocity, locate the Prescribed Mesh Velocity section.


4 Clear the Prescribed x velocity check box.

5 In the vy text field, type -paco/pade*v+Mc/pade*((y-Y)+des)*(comp2.chem.r_1+comp2.chem.r_2+comp2.chem.r_3+ comp2.chem.r_4).

M E S H 1

Mapped 11 In the Model Builder window, under Component 2 (comp2) right-click Mesh 1 and choose

Mapped.

2 In the Settings window for Mapped, locate the Domain Selection section.




Distribution 11 Right-click Component 2 (comp2)>Mesh 1>Mapped 1 and choose Distribution.












Distribution 11 In the Model Builder window, right-click Mesh 1 and choose Mapped.










5 In the Model Builder window, click Mesh 1.

6 In the Settings window for Mesh, click Build All.

In a model described by a strongly coupled system of equations, it is often a good practice to solve the problem sequentially to avoid convergence issues. Start by solving for the stationary flow. After that, solve the full time-dependent system, with the dynamic nature of the soot layer handled by the Moving Mesh interface.

S T U D Y 2

Step 1: Stationary1 In the Model Builder window, expand the Study 2 node, then click Step 1: Stationary.




Step 2: Time Dependent1 On the Study toolbar, click Study Steps and choose Time Dependent>Time Dependent.

Physics interface Solve for Discretization

Reaction Engineering (re) physics

Chemistry 1 (chem) physics

Transport of Diluted Species (tds) physics

Heat Transfer in Fluids 1 (ht) physics

Free and Porous Media Flow 1 (fp) √ physics

Moving Mesh (ale) physics

Multiphysics couplings Solve for

Temperature Coupling 1 (tc1)


2 In the Settings window for Time Dependent, locate the Physics and Variables Selection section.

3 In the table, clear the Solve for check box for Reaction Engineering (re) and Free and

Porous Media Flow 1 (fp).

4 Locate the Study Settings section. In the Times text field, type range(0,50,300).

Setup proportional Scale values for Dependent Variables 2




Solution 5 (sol5)>Dependent Variables 2 node, then click Concentration (comp2.cco).

4 In the Settings window for Field, locate the Scaling section.

5 From the Method list, choose Manual.

6 In the Model Builder window, under Study 2>Solver Configurations>Solution 5 (sol5)>

Dependent Variables 2 click Concentration (comp2.cco2).




Dependent Variables 2 click Concentration (comp2.co2).




Dependent Variables 2 click Temperature (comp2.T).


14 From the Method list, choose Initial value based.

To avoid discontinuous inlet for Time Dependent study add a step function to gradually increase the Inflow concentration


Step 1 (step1)1 On the Home toolbar, click Functions and choose Global>Step.

2 In the Settings window for Step, locate the Parameters section.

3 In the Location text field, type 0.5.


4 Click to expand the Smoothing section. In the Size of transition zone text field, type 1.


Inflow 11 In the Model Builder window, expand the Transport of Diluted Species (tds) node, then

click Inflow 1.


3 In the c0,co2 text field, type F_o2_inlet/v_inlet*step1(t[1/s]).

4 In the c0,cco2 text field, type F_co2_inlet/v_inlet*step1(t[1/s]).

5 In the c0,cco text field, type F_co_inlet/v_inlet*step1(t[1/s]).

S T U D Y 2


To produce the plots shown in Figures 7, 8, 9, and 10 follow these steps.

R E S U L T S

Surface1 In the Model Builder window, expand the Temperature (ht1) node, then click Surface.

2 In the Settings window for Surface, locate the Expression section.

3 In the Expression text field, type T.

4 On the Temperature (ht1) toolbar, click Plot.

Streamline 11 In the Model Builder window, under Results right-click Velocity (fp1) and choose

Streamline.

2 In the Settings window for Streamline, locate the Streamline Positioning section.

3 From the Positioning list, choose Magnitude controlled.

4 Locate the Coloring and Style section. From the Color list, choose White.

5 On the Velocity (fp1) toolbar, click Plot.


Contour1 In the Model Builder window, expand the Pressure (fp1) node.

2 Right-click Contour and choose Disable.


Surface 11 In the Model Builder window, under Results right-click Pressure (fp1) and choose Surface.


3 In the Expression text field, type p.

4 On the Pressure (fp1) toolbar, click Plot.

Temperature (ht1)1 In the Model Builder window, under Results click Temperature (ht1).

2 On the Temperature (ht1) toolbar, click Plot.



2 In the Settings window for 1D Plot Group, type Soot layer thickness in the Label text field.

3 Locate the Data section. From the Data set list, choose Study 2/Solution 5 (sol5).

4 From the Time selection list, choose Interpolated.

5 Click Range.

6 In the Range dialog box, type 0 in the Start text field.

7 In the Step text field, type 50.

8 In the Stop text field, type 300.

9 Click Replace.




13 In the associated text field, type L (m).


15 In the associated text field, type y-displacement (m).

Line Graph 11 Right-click Soot layer thickness and choose Line Graph.



4 In the Settings window for Line Graph, locate the y-Axis Data section.


5 In the Expression text field, type y-Y.



8 On the Soot layer thickness toolbar, click Plot.


Delete 2D Plot Group 3 since it is a duplicate of Concentration (tds).

2D Plot Group 3In the Model Builder window, under Results right-click 2D Plot Group 3 and choose Delete.

For future use of Study 1 disable the interfaces that are used in the space-dependent model.

S T U D Y 1

Step 1: Stationary Plug Flow1 In the Settings window for Stationary Plug Flow, locate the Physics and Variables Selection

section.

2 In the table, clear the Solve for check box for the following interfaces:

Physics interface

Chemistry 1 (chem)

Transport of Diluted Species (tds)

Heat Transfer in Fluids 1 (ht)

Free and Porous Media Flow 1 (fp)

Moving Mesh (ale)



D i s s o c i a t i o n i n a Tubu l a r R e a c t o r




Introduction

Tubular reactors are often used in continuous large-scale production, for example in the petroleum industry. One key design and optimization parameter is the conversion, or the amount of reactant that reacts to form the desired product. In order to achieve high conversion, process engineers optimize the reactor design: its length, width and heating system. An accurate reactor model is a very useful tool, both at the design stage and in tuning an existing reactor.

Figure 1: Dissociation reaction in a tubular reactor.

This example deals with a gas-phase dissociation process, species A reacts to form B (see Figure 1). The following physics interfaces are used:

• Laminar Flow with compressible formulation.

• Transport of Concentrated Species.

• Heat Transfer in Fluids.

Model Definition

K E Y I N S T R U C T I V E E L E M E N T S

This model illustrates several attractive features in the Chemical Reaction Engineering Module:

• The use of the Transport of Concentrated Species to account for multicomponent diffusion.

• How to couple the variable density to a Laminar Flow interface.

• Implementation of temperature- and composition−dependent reaction kinetics.

A

A+B

A 2B

2 | D I S S O C I A T I O N I N A TU B U L A R R E A C T O R

• The use of a mapped mesh, which is structured, to discretize a long and thin geometry, typical for tubular reactors

• The setup of heat balances and how to couple these to both the mass balances and the velocity field

H A N D L I N G E X P A N D I N G F L O W — C O M P R E S S I B L E F L O W F O R M U L A T I O N

Each mole of the reactant, A, reacts to form two moles of the product, B:

This leads to a volumetric expansion of the gas mixture as the reaction proceeds. The fluid’s change in density influences the gas velocity in the reactor, causing an acceleration as the reaction proceeds.

In order to model the flow, use a compressible formulation of the Navier−Stokes equations, defined according to the following equations:

Here ρ denotes the solution’s density (SI unit: kg/m3), u is the velocity vector (SI unit: m/s), p gives the pressure (SI unit: Pa), μ represents the solution’s viscosity (SI unit: kg/(m·s), or Pa·s), and I denotes the identity matrix.

The density is varying and depends on the pressure, temperature and composition according to the ideal gas law. This is automatically handled through the “ideal gas” option for the density in the Transport of Concentrated Species interface.

The model applies the Laminar Flow interface, which solves the above equations, describing the momentum balances and the continuity (mass conservation) for fluids with variations in density.

C O N V E C T I O N A N D D I F F U S I O N I N M U L T I C O M P O N E N T S Y S T E M S —

TR A N S P O R T O F C O N C E N T R A T E D S P E C I E S

As the dissociation reaction proceeds, the composition of the mixture changes from pure A at the inlet to a mixture of A and B.

The total mass flux is strongly influenced by the flux of each species. In addition, several molecular interactions occur; A interacts with B and other A molecules, B interacts with A and other B molecules. This implies that the simple Fick’s law formulation, with one

A 2B→

ρ u ∇⋅( )u ∇ pI– μ u u∇( )T+∇( ) 2

3---μ ∇ u⋅( )I–+⋅=

∇ ρu( )⋅ 0=


constant diffusivity for each species is not applicable here. In a concentrated multicomponent mixture you must account for all possible interactions, and the flux is dependent on the fluid’s local composition. Simple Fick diffusivity accounts only for the interaction between solvent and solute. In the Transport of Concentrated Species with the Maxwell-Stefan or Mixture-Averaged diffusion equations, multicomponent diffusivities describe the interactions between all components in the system.

Since a change in a gas mixture composition affects the density, the species transport equation needs to be coupled with the flow equations (Laminar Flow, Navier-Stokes in this case).

Now consider a mathematical formulation of this discussion. The mass-balance equation for each species is

with the source terms given by the reaction kinetics follow the equations

where kf denotes the forward reaction rate constant (SI unit: s-1), cA represents the concentration of A (SI unit: mol/m3), and ni is the mass flux vector for species i (SI unit: kg/(m2·s)). Because the reaction is a pure dimerization, it is inherent that MB equals half of MA.

As mentioned earlier, it is possible to rewrite the mass-balances equations for each species by replacing one of the species’ mass balance with a total mass balance. A solution with two species follows following equation:

Because the system consists only of two species, the sum of wA and wB is always unity, and the sum of the reaction terms is zero. The above equation now becomes

t∂∂ ρwA( ) ∇ nA⋅+ RA=

t∂∂ ρwB( ) ∇ nB⋅+ RB=

RA kfcAMA–=

RB 2kfcAMB=

t∂∂ ρ wA wB+( )( ) ∇ nA nB+( )⋅+ RA RB+=

t∂∂ ρ ∇ nA nB+( )⋅+ 0=


which is the total mass-balance equation.

The reaction rate is described by an Arrhenius law according to

where A0, the pre-exponential factor, is set to 41.3 s-1, Ea, the activation energy, is set to 30 kJ/mol, R is the gas constant, 8.314 J/(mol·K), and T the temperature (SI unit: K).

The rate of production of species B is thus dependent on both composition and temperature. However, in the first model the gas is assumed to be isothermal, making the rate vary only with composition.

G E O M E T R Y

The geometry of the tubular reactor is rotationally symmetric, and it is possible to reduce the model from 3D to a 2D axisymmetric problem. This means that you only have to model half of the tube cross section, as illustrated in Figure 2.

Figure 2: Model geometry.

kf A0Ea–

RT---------- exp=

Inlet

Outlet

z

r = 0

r


B O U N D A R Y C O N D I T I O N S

Laminar Flow interfaceThe flow in the reactor is driven by a pressure drop. The pressure at the inlet, pin, is slightly higher than that at the outlet, where the pressure is set to atmospheric pressure

The walls are represented by no slip boundary conditions u = 0.

Transport of Concentrated Species interfaceAt the inlet, the mass fraction of A is set close to unity(0.99). The outlet boundary condition is a convective flux condition.

The convective flux condition implies that diffusive flux for the species is zero perpendicular to the boundary. This is a common assumption when modeling the outlet in tubular reactors.

No-flux conditions—referred to as insulation/symmetry in COMSOL Multiphysics—apply at all other boundaries: n · nA = 0.

M E S H

In this example, a mapped (structured) mesh is a good choice due to the reactor’s regular shape. The use of a structured mesh is especially suitable when the requirements for the mesh density is uneven. In this example a denser mesh is required in the inlet region and the reactor wall. This is achieved by specifying the edge element distribution, as you see in Modeling Instructions.

Study 1—Results and Discussion for Isothermal Conditions

Under isothermal conditions, the Laminar Flow and Transport of Concentrated Species interfaces in COMSOL Multiphysics are applied to solve the coupled model of the compressible Navier-Stokes equations and the Maxwell-Stefan convection and conduction equations.

Figure 3 shows the velocity magnitude for the isothermal case at different cross-sections of reactor. The velocity increases along the axis direction (z) because of the volume expansion of gas mixture during the proceeding of reaction. The maximum of velocity is found at the center of the tube due to the no slip on the side surface.

p patm=


Figure 3: Velocity magnitude for the isothermal case.

Figure 4 shows the mass fraction of species B for the isothermal case at different cross-sections of reactor. The closer to the side surface, the lower is the convective flow velocity, which gives rise to the higher mass fraction of species B towards the tube surface. The average mass fraction of species B at the outlet is 64.1%.


Figure 4: Mass fraction of species B for the isothermal case.

The average conversion rate depends on flow rate profile, density distribution, and velocity field. It is defined as

The average conversion rate at the outlet under isothermal conditions is 64.4%.

Model Definition—Non-Isothermal Model

Now it is time to expand the model by including an energy-balance equation modeling the temperature. In the previous model, the temperature was constant and set to 473 K. Now assume that the gas enters the reactor at room temperature, 293 K, and that the reactor walls are heated to 473 K to accomplish heating of the gas and reaction. This means that the gas is heated by the walls as the gas flows along the reactor. In addition, the heat of reaction is also included, acting as a source term. For the current dimerization reaction the

γB

XBρu n⋅ sdρu n⋅ sd

----------------------------------=


heat of reaction is -80 kJ/kg. This means that the fluid mixture is heated as the reaction proceeds.

The influence of the temperature on the reaction rate is significant. The rate follows an Arrhenius law, it increases exponentially with temperature. Thus, the reaction rate increases as the fluid flows through the reactor and is heated by the walls and by the heat of reaction.

The energy-balance equation is

where k is the thermal conductivity (SI unit: W/(m·K)), Cp is the specific heat capacity (SI unit: J/(kg·K)), and Q is the heat source term (SI unit: W/m).

The heat source term, Q, is given by the heat of reaction, ΔH (−80 000 J/kg), and reaction rate, R:

The material properties are specified to be those of the mixture. In this example assume that the heat capacity, Cp, and conductivity, k, are similar to those of propane, which is present in the Material Library.

The boundary conditions for the energy balance are similar to those of the mass balances. At the inlet, the gas temperature is specified, in this case to 293 K.

The default Axial symmetry condition gives a zero temperature gradient at the symmetry boundary: n · ∇T = 0. At the outlet, apply a convective flux condition: .

Model the reactors heated walls by applying a heat flux condition on the wall, using a heat transfer coefficient, h, (50 W/(m2·K)) for the heating fluid, and the heating temperature, Tf, which is 473 K. The condition is:

Study 2—Results and Discussion for Non−Isothermal Conditions

For the non-isothermal case, add a Heat Transfer in Fluids interface to the isothermal model.

Figure 5 shows the velocity magnitude for the non-isothermal case at different cross-sections of reactor. The velocity magnitude for the non-isothermal case is slightly smaller

ρCp t∂∂T ∇ k∇T–( )⋅+ Q ρCp u ∇T⋅( )–=

Q ΔHRA–=

n T∇⋅ 0=

n k∇T( )⋅ h Tf T–( )=


than that for the isothermal case (see Figure 3). This is due to the lower reaction rate caused by the lower temperature.

Figure 5: Velocity magnitude for the non-isothermal case.

Figure 6 shows the mass fraction of species B for the non−isothermal case at different cross-sections of reactor. At the region close to the side wall, the mole fraction is much higher than that in the central region due to the higher temperature close to the wall. The overall mole fraction is lower than that for the isothermal conditions (see Figure 4) because of the low temperature in the reactor. The average mass fraction of species B at the outlet is 26.4% and the average conversion rate at the outlet is 24.2% under non−isothermal conditions.


Figure 6: Mass fraction of species B for the non-isothermal case.

Figure 7 shows the temperature distribution under non-isothermal conditions. The temperature is much higher close to the reactor wall. This temperature profile has a significant impact on the reaction rate in the reactor, see Figure 6


Figure 7: Temperature distribution for non-isothermal case.

Application Library path: Chemical_Reaction_Engineering_Module/Reactors_with_Mass_and_Heat_Transfer/dissociation



N E W

1 In the New window, click Model Wizard.

In this problem, it is easier to add all physics from the start and disable the ones not used in all parts of the investigation.





3 Click Add.


Transport of Concentrated Species (tcs).

5 Click Add.

6 In the Select Physics tree, select Heat Transfer>Heat Transfer in Fluids (ht).

7 Click Add.

8 Click Study.


10 Click Done.


Load the model parameters from a text file.




4 Browse to the model’s Application Libraries folder and double-click the file dissociation_parameters.txt.


Load the variables from a text file.




4 Browse to the model’s Application Libraries folder and double-click the file dissociation_variables.txt.

G E O M E T R Y 1

The model geometry is simply a rectangle.



3 In the Width text field, type W0.


4 In the Height text field, type L0.

5 Click Build All Objects.



Average 1 (aveop1)1 On the Definitions toolbar, click Component Couplings and choose Average.

You will use this operator later in the results analysis. With the check box Compute

integral in revolved geometry enabled, it automatically performs a surface integration by multiplication with 2*pi*r.

2 In the Settings window for Average, locate the Source Selection section.

3 From the Geometric entity level list, choose Boundary.



Since the density variation is large, the flow cannot be regarded as incompressible. Therefore select Compressible flow.

1 In the Model Builder window, under Component 1 (comp1) click Laminar Flow (spf).

2 In the Settings window for Laminar Flow, locate the Physical Model section.

3 From the Compressibility list, choose Compressible flow (Ma<0.3).

Define the pressure reference level in the interface properties.

4 Find the Reference values subsection. In the pref text field, type p_atm.

Fluid Properties 11 In the Model Builder window, under Component 1 (comp1)>Laminar Flow (spf) click

Fluid Properties 1.


3 From the ρ list, choose Density (tcs/cdm1).

4 From the μ list, choose User defined. In the associated text field, type eta.




4 From the list, choose Pressure.

5 Locate the Pressure Conditions section. In the p0 text field, type p_in.



3 In the Settings window for Outlet, locate the Pressure Conditions section.

4 Select the Normal flow check box.

TR A N S P O R T O F C O N C E N T R A T E D S P E C I E S ( T C S )

1 In the Model Builder window, under Component 1 (comp1) click Transport of Concentrated Species (tcs).

2 In the Settings window for Transport of Concentrated Species, locate the Transport Mechanisms section.

3 From the Diffusion model list, choose Maxwell-Stefan.

4 Click to expand the Dependent variables section. Locate the Dependent Variables section. In the Mass fractions table, enter the following settings:

5 Locate the Species section. From the From mass constraint list, choose wB.

Transport Properties 11 In the Model Builder window, under Component 1 (comp1)>

Transport of Concentrated Species (tcs) click Transport Properties 1.

2 In the Settings window for Transport Properties, locate the Model Input section.

3 In the T text field, type Tf.

4 From the pA list, choose Absolute pressure (spf).

5 Locate the Convection section. From the u list, choose Velocity field (spf).

6 Locate the Density section. In the MwA text field, type MA.

7 In the MwB text field, type MB.

8 Locate the Diffusion section. In the Dik table, enter the following settings:

9 In the Model Builder window, click Transport of Concentrated Species (tcs).

wA

wB

1 DA

DA

1


Reaction Sources 11 On the Physics toolbar, click Domains and choose Reaction Sources.

2 Click in the Graphics window and then press Ctrl+A to select all domains.

3 In the Settings window for Reaction Sources, locate the Reactions section.

4 From the RwA list, choose User defined. In the associated text field, type -Ra.



3 In the Settings window for Inflow, locate the Inflow section.

4 In the ω0,wA text field, type wA_in.



The initial condition is set to a constant value T_atm in the heat transfer problem, even it is not solved in the first study. This is done because the reaction rate and density must have a valid temperature as input.

H E A T TR A N S F E R I N F L U I D S ( H T )

Initial Values 11 In the Model Builder window, under Component 1 (comp1)>Heat Transfer in Fluids (ht)


2 In the Settings window for Initial Values, type Tf in the T text field.

M E S H 1

A mapped mesh is suitable for fluid flow.


Mapped.














S T U D Y 1


2 In the Settings window for Study, type Study - Isothermal Model in the Label text field.

3 Locate the Study Settings section. Clear the Generate default plots check box.

S T U D Y - I S O T H E R M A L M O D E L

Turn off the Heat Transfer interface in the study to first examine the dissociation at isothermal conditions.

Step 1: Stationary1 In the Model Builder window, under Study - Isothermal Model click Step 1: Stationary.


3 In the table, clear the Solve for check box for the Heat Transfer in Fluids interface.


R E S U L T S

Line Average 11 On the Results toolbar, click More Derived Values and choose Average>Line Average.


3 In the Settings window for Line Average, click Replace Expression in the upper-right corner of the Expressions section. From the menu, choose Component 1>

Transport of Concentrated Species>wB - Mass fraction.

4 Click New Table.


Global Evaluation 11 On the Results toolbar, click Global Evaluation.

2 In the Settings window for Global Evaluation, locate the Expressions section.


4 Click Table 1 - Line Average 1 (wB).

3D Plot Group 11 On the Results toolbar, click More Data Sets and choose Revolution 2D.

2 On the Results toolbar, click 3D Plot Group.

3 In the Settings window for 3D Plot Group, type Velocity, isothermal in the Label text field.

Slice 11 In the Model Builder window, under Results right-click Velocity, isothermal and choose

Slice.

2 In the Settings window for Slice, locate the Plane Data section.

3 From the Plane list, choose xy-planes.

4 In the Planes text field, type 10.

5 Locate the Coloring and Style section. Clear the Color legend check box.

Slice 21 Right-click Results>Velocity, isothermal>Slice 1 and choose Duplicate.


3 From the Plane list, choose yz-planes.


5 Locate the Coloring and Style section. Select the Color legend check box.


7 On the Velocity, isothermal toolbar, click Plot.


9 In the Model Builder window, expand the Results>Views node.

Velocity, isothermalIn the Model Builder window, expand the Results>Views>View 3D 2 node.


aveop1(w*spf.rho*wB)/aveop1(w*spf.rho) 1


Velocity, isothermal 11 Right-click Results>Velocity, isothermal and choose Duplicate.

2 In the Settings window for 3D Plot Group, type Mass fraction, B, isothermal in the Label text field.

Slice 11 In the Model Builder window, expand the Mass fraction, B, isothermal node, then click

Slice 1.

2 In the Settings window for Slice, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Component 1>


Slice 21 In the Model Builder window, under Results>Mass fraction, B, isothermal click Slice 2.



3 On the Mass fraction, B, isothermal toolbar, click Plot.


Continue setting up the nonisothermal model. Start by selecting a new study

A D D S T U D Y

1 On the Home toolbar, click Add Study to open the Add Study window.

2 Go to the Add Study window.

3 Find the Studies subsection. In the Select Study tree, select Preset Studies>Stationary.

4 Click Add Study in the window toolbar.

S T U D Y 2

1 In the Settings window for Study, type Study - Nonisothermal Model in the Label text field.

2 On the Home toolbar, click Add Study to close the Add Study window.


Fluid 11 In the Settings window for Fluid, locate the Heat Conduction, Fluid section.

2 From the k list, choose User defined. In the associated text field, type k_mix.


3 Locate the Thermodynamics, Fluid section. From the ρ list, choose Density (tcs/cdm1).

4 From the Cp list, choose User defined. In the associated text field, type Cp_mix.

5 From the γ list, choose User defined.

Heat Source 11 On the Physics toolbar, click Domains and choose Heat Source.


3 In the Settings window for Heat Source, locate the Heat Source section.

4 In the Q0 text field, type Qr*Ra.



Temperature 11 On the Physics toolbar, click Boundaries and choose Temperature.


3 In the Settings window for Temperature, locate the Temperature section.

4 In the T0 text field, type T_atm.

Heat Flux 11 On the Physics toolbar, click Boundaries and choose Heat Flux.


3 In the Settings window for Heat Flux, locate the Heat Flux section.

4 In the q0 text field, type Ua*(Tf-T).






In the nonisothermal case, add temperature and flow couplings with the Multiphysics node.



Flow Coupling 2 (fc2)1 On the Physics toolbar, click Multiphysics Couplings and choose Global>Flow Coupling.

2 On the Physics toolbar, click Multiphysics Couplings and choose Global>Flow Coupling.

3 In the Settings window for Flow Coupling, locate the Coupled Interfaces section.

4 From the Destination list, choose Heat Transfer in Fluids (ht).

S T U D Y - N O N I S O T H E R M A L M O D E L

1 In the Model Builder window, click Study - Nonisothermal Model.




R E S U L T S

Line Average 1In the Model Builder window, expand the Results>Derived Values node.

Line Average 21 Right-click Line Average 1 and choose Duplicate.

2 In the Settings window for Line Average, locate the Data section.

3 From the Data set list, choose Study - Nonisothermal Model/Solution 2 (sol2).


Global Evaluation 21 In the Model Builder window, under Results>Derived Values right-click

Global Evaluation 1 and choose Duplicate.

2 In the Settings window for Global Evaluation, locate the Data section.




2 In the Settings window for Revolution 2D, locate the Data section.



Velocity, isothermal 11 In the Model Builder window, under Results right-click Velocity, isothermal and choose

Duplicate.

2 In the Settings window for 3D Plot Group, type Velocity, nonisothermal in the Label text field.

Velocity, nonisothermal1 In the Model Builder window, under Results click Velocity, nonisothermal.



Slice 11 In the Model Builder window, expand the Velocity, nonisothermal node, then click Slice 1.

2 On the Velocity, nonisothermal toolbar, click Plot.

3 Click the Go to View 3D 2 button on the Graphics toolbar.


Mass fraction, B, isothermal 11 In the Model Builder window, under Results right-click Mass fraction, B, isothermal and

choose Duplicate.

2 In the Settings window for 3D Plot Group, type Mass fraction B, nonisothermal in the Label text field.

Mass fraction B, nonisothermal1 In the Model Builder window, under Results click Mass fraction B, nonisothermal.



4 On the Mass fraction B, nonisothermal toolbar, click Plot.



Mass fraction B, nonisothermal 11 Right-click Results>Mass fraction B, nonisothermal and choose Duplicate.

2 In the Settings window for 3D Plot Group, type Temperature in the Label text field.

3 On the Temperature toolbar, click Plot.

Slice 11 In the Model Builder window, expand the Temperature node, then click Slice 1.


2 In the Settings window for Slice, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Component 1>Heat Transfer in Fluids>

Temperature>T - Temperature.

Slice 21 In the Model Builder window, under Results>Temperature click Slice 2.








Deg r ada t i o n o f DNA i n P l a sma




Introduction

Gene therapy is one biotechnology example of a clinical application where it is possible to produce proteins in vivo, using the body’s own mechanisms for protein production. Major issues in gene delivery involve the transport of plasmid DNA (pDNA) to target sites and the conversion between different forms of pDNA.

This example uses the Parameter Estimation feature with the Reaction Engineering interface to find the rate constants of three consecutive reactions involved in a DNA degradation process.

Note: This application requires the Optimization Module.

Model Description

pDNA can be used to express proteins in the human body, proteins that can have therapeutic effects. pDNA exists in three forms—a supercoiled form (SC), an open-circular form (OC), and a linear form (L)—each with varying protein-expression rates. These pDNA-forms interconvert and degrade with time, which means a patient’s therapy benefits from knowledge about the distribution of pDNA-forms over time.

The protein expression rate for the SC form is greater than the one for the OC form, which in turn is significantly greater than that for the L form. The kinetic model in this study assumes that the pDNA-forms interconvert and decompose according to the mechanism in Figure 1.

Figure 1: Kinetic model of plasmid DNA interconversion and decomposition. Supercoiled pDNA (SC) converts to an open-circular form (OC), which in turn converts to a linear form (L). The linear pDNA decomposes to form linear fragments (F).

This example proposes a set of irreversible reactions in which an SC-form pDNA converts to the OC form and then to the L form. Then the L-form decomposes into a number of linear fragments, collectively denoted as F.

The three irreversible reactions in Figure 1 translate into these reaction rate expressions:

LSC FOCk1 k3k2

r1 k1cSC=

2 | D E G R A D A T I O N O F D N A I N P L A S M A

The rate constants k1 through k3 are found by parameter estimation, making use of the experimental data summarized in the table:


The following rate constants are calculated from the experimental data and proposed reaction mechanism: k1 = 9.6 × 10-3 (1/s), k2 = 4.8 × 10-4 (1/s), and k3 = 9.6 × 10-

4 (1/s). In addition, the initial concentration of the SC species is estimated to 9.7 ng/µl.

TABLE 1: EXPERIMENTAL CONCENTRATION DATA

TIME (s) cSC (ng/μl) cOC (ng/μl) cL (ng/μl)

5 9.3 0.5 0

60 5.0 4.1 0.1

120 3.5 6.5 0.3

180 1.1 7.0 0.5

300 0.5 8.1 0.8

420 0.1 8.0 1.2

600 0 7.8 1.7

900 0 7.1 2.4

1200 0 6.3 2.5

1800 0 4.5 2.6

2400 0 3.0 2.0

3000 0 2.1 1.8

3600 0 1.5 1.2

r2 k2cOC=

r3 k3cL=


Figure 2 shows the experimental values in the same plot as the simulation results. Clearly, the assumptions of the kinetic model are in agreement with the experimental findings.

Figure 2: A plot resulting from reading in experimental data and comparing it to simulation results.

The estimated rate constants show that the supercoiled pDNA rapidly transforms into the open-circular form with a half-life of approximately 1.2 minutes:

The open-circular and linear pDNA decay with half-lives of 24.1 and 12.0 minutes, respectively. As mentioned, the supercoiled pDNA has the highest protein-expression rate of the three forms. However, because the SC form has a half-life of only 1.2 minutes, it is likely that it decomposes during transport to the therapeutic target sites. These findings imply that you have to find ways to hinder the relatively fast decay of SC.

Reference

1. B.E. Houk, G. Hochhaus, and J.A. Hughes, “Kinetic modeling of plasmid DNA degradation in rat plasma,” AAPS Pharmsci, vol. 1, no. 3, pp. 15–20, 1999.

t1 2⁄2ln

k---------=


Application Library path: Chemical_Reaction_Engineering_Module/Ideal_Tank_Reactors/dna_degradation



N E W





3 Click Add.

4 Click Study.


6 Click Done.


Read model parameters from a text file.




4 Browse to the model’s Application Libraries folder and double-click the file dna_degradation_parameters.txt.

Start by entering the reaction properties in the Reaction Engineering interface.



2 In the Settings window for Reaction Engineering, click to expand the Mixture properties section.

3 Locate the Mixture Properties section. From the Phase list, choose Liquid.




3 In the Formula text field, type SC=>OC.



3 In the Formula text field, type OC=>L.



3 In the Formula text field, type L=>F.

Species 1The species are dissolved in water. Add water as a solvent.

1 On the Reaction Engineering toolbar, click Species.


3 In the Species name text field, type H2O.

4 Locate the Species Type section. From the Species type list, choose Solvent.

Parameter Estimation 1Choose a Parameter Estimation feature to optimize the three reaction constants in the reactions entered in the interface.

1 On the Reaction Engineering toolbar, click Parameter Estimation.

Select the parameters to be estimated and provide an initial guess. The parameter c_SC_init will be used to estimate the initial concentration of the species SC. Note that the experimental data have compositions in ng/μl unit and only first order reaction constants are optimized. Therefore, the same unit will apply also for the model compositions.

2 In the Settings window for Parameter Estimation, locate the Estimation Parameters section.



4 Click Add.


6 Click Add.


8 Click Add.


Experiment 1Select an Experiment feature to import experimental data to which the simulation will be optimized.

1 On the Reaction Engineering toolbar, click Attributes and choose Experiment.


3 Click Browse.

4 Browse to the model’s Application Libraries folder and double-click the file dna_degradation_experiment1.csv.

5 Click Import.



k1 1e-3 1


k2 1e-3 1


k3 1e-3 1


c_SC_init 10 1


Conc. SC √ c_SC 1 1

Conc. OC √ c_OC 1 1

Conc. L √ c_L 1 1


Initial Values 1Note that the unit ng/l should be used due to the unit of the imported data in the Parameter Estimation feature. For water, enter the value 1000 ng/μl.

1 In the Model Builder window, under Component 1 (comp1)>Reaction Engineering (re) click Initial Values 1.



Set all reaction rate constants to the same names chosen in the Parameter Estimation feature.

1: SC=>OC1 In the Model Builder window, under Component 1 (comp1)>Reaction Engineering (re)

click 1: SC=>OC.


3 In the kf text field, type k1.

2: OC=>L1 In the Model Builder window, under Component 1 (comp1)>Reaction Engineering (re)

click 2: OC=>L.



3: L=>F1 In the Model Builder window, under Component 1 (comp1)>Reaction Engineering (re)

click 3: L=>F.



In the Study node, add an optimization step to finalize the optimization settings.


H2O 1000

SC c_SC_init


S T U D Y 1



3 In the Times text field, type 0 4000.



3 From the Method list, choose Levenberg-Marquardt.

4 In the Optimality tolerance text field, type 1.0E-4.

5 Locate the Results While Solving section. Select the Plot check box.


Follow these steps to create Figure 2. The experimental and simulation data should match.

R E S U L T S


2 In the Settings window for 1D Plot Group, type Concentrations in the Label text field.





7 In the associated text field, type Concentration (ng/\mu l).

Experiment 1 Data1 In the Model Builder window, expand the Results>Concentrations node, then click

Experiment 1 Data.

2 In the Settings window for Table Graph, type Experimental Data in the Label text field.





Global 11 In the Model Builder window, under Results>Concentrations click Global 1.

2 In the Settings window for Global, type Simulation Data in the Label text field.

3 Click Replace Expression in the upper-right corner of the y-axis data section. From the menu, choose Component 1>Reaction Engineering>comp1.re.c_SC - Concentration.

4 Click Add Expression in the upper-right corner of the y-axis data section. From the menu, choose Component 1>Reaction Engineering>comp1.re.c_OC - Concentration.

5 Click Add Expression in the upper-right corner of the y-axis data section. From the menu, choose Component 1>Reaction Engineering>comp1.re.c_L - Concentration.

6 Click to expand the Coloring and style section. Locate the Coloring and Style section. Find the Line style subsection. From the Line list, choose Cycle.





11 On the Concentrations toolbar, click Plot.


Output the values of the estimated parameters to a table.

Derived ValuesOn the Results toolbar, click Evaluate All.

Last plot group is not used and is therefore removed.

Legends

Experiment SC

Experiment OC

Experiment L

Legends

Simulation SC

Simulation OC

Simulation L


ConcentrationsIn the Model Builder window, under Results right-click Concentration (re) and choose Delete.



Drug R e l e a s e f r om a B i oma t e r i a l Ma t r i x




Introduction

Biomaterial matrices for drug release are useful for in vivo tissue regeneration. The following example describes the release of a drug from a biomaterial matrix to damaged cell tissue. Specifically, a nerve guide delivers a regenerating drug to damaged nerve ends.

This application examines detailed drug-release kinetics with rate expressions handling drug dissociation/association reactions as well as matrix degradation by enzyme catalysis. The enzyme reaction is described by Michaelis-Menten kinetics. The model enables investigation of design parameters governing the rate of drug release such as drug-to-biomaterial affinity, biomaterial degradation, drug loading, and the influence of geometry and composition of the biomaterial matrix.

Model Definition

The model consists of two parts. One part uses the batch reactor type in the Reaction Engineering interface to specify the reacting system in a perfectly mixed environment, that is, no space dependency is assumed. The purpose of this part is to study the reaction kinetics. The other part is space dependent and is generated from the Reaction Engineering interface. It utilizes the Transport of Diluted Species interface and serves to investigate the drug transport into a region containing damaged nerve ends.

Figure 1 shows the full 3D geometry as well as the 2D modeling domain, reduced by axial symmetry and a mirror plane, for the space-dependent model. The biomaterial holding the drug is assumed to have a strictly cylindrical shape. The three distinct areas (domains) are:

• The nerve-cell tissue

• The biomaterial matrix

• The surrounding medium

2 | D R U G R E L E A S E F R O M A B I O M A T E R I A L M A T R I X

Figure 1: The full 3D geometry (left) and the equivalent modeling domains reduced to 2D by axial symmetry (right). The regions are: the nerve-cell tissue, the biomaterial matrix, and the surrounding medium.

In the biomaterial matrix, a drug molecule, d, binds to a peptide, p, which in turn is anchored to the matrix, m. Matrix-bound species are labeled mpd and mp, respectively, the latter referring to a species where no drug is bound to the peptide. Species mpd and mp are active only in the nerve cell tissue domain.

Two mechanisms release the drug from the matrix. First, the drug can simply dissociate from the matrix site mp. Second, matrix degradation by an enzyme, e, originating from the cell-tissue domain, leads to release of the drug-peptide species, pd, from which the drug subsequently dissociates. The unbound species p, d, pd, and e are active in the entire model domain. Figure 2 illustrates the complete reaction scheme.

Nerve cell tissue

Biomaterial tissue

Surroundings


Figure 2: Reaction scheme describing drug dissociation/association reactions (horizontal) and matrix-degradation reactions (vertical).

The time-dependent mass balance per species is described by

(1)

where Dik (SI unit: m2/s) is the diffusion coefficient for species i in the respective medium. Further, Rik (SI unit: mol/(m3·s)) is the rate expression for species i in the medium. In the matrix, all the reactions described in Figure 2 are possible, leading to the

m

p

m

pd +

m

p

m

m

pE

m

pEE

k1r

k1f

k2r k2

f

k3f

E

d

d

p

d

d +

k2r k2

f

k1r

k1f

k3f

Dissociation

Degradation

ci∂t∂------- ∇+ D– ik∇ci( )⋅ Rik=


following rate expressions:

The rate terms RMMmp and RMMmpd refer to the Michaelis-Menten kinetics describing the enzyme catalyzed degradation of the matrix:

with

RMMmp describes the disappearance of mp sites and the production of p species. RMMmpd describes the disappearance of mpd sites and the production of pd species. Vmax is the maximum rate and KM the Michaelis-Menten constant. In the cell region and in the surrounding medium only dissociation/association reactions occur, leading to the rate expressions

Rd2 k– 1f cd cmp cp+( ) k1

r cmpd cpd+( )+=

Rp2 k– 1f cd cp k1

r cpd RMMmp+ +=

Rpd2 k1f cd cp k1

r cpd– RMMmpd+=

Rmp2 k– 1f cd cmp k1

r cmpd RMMmp–+=

Rmpd2 k1f cd cmp k1

r cmpd– RMMmpd–=

RMMmpVmax cmpKM cmp+------------------------=

RMMmpdVmax cmpdKM cmpd+---------------------------=

Vmax k3f ce=

KMk3

f k+ 2r

k2f

------------------=

Rd1 Rd3= Rp1 Rp3 k– 1f cd cp k1

r cpd+= = =

Rpd1 Rpd3= k1f cd cp k1

r cpd–=


The boundary condition is axial symmetry along the rotational axis and insulation/symmetry elsewhere. Values for diffusion coefficients and rate constants come from the literature (Ref. 1).


Figure 3 shows the concentration transients of the reacting species in a perfectly mixed (space-independent) system.

Figure 3: Concentrations of all reacting species (mol/ m3) as functions of time (s).

The effect of enzyme degradation is clearly visible, with matrix-bound peptide species (mp and mpd) decreasing and free peptide species (p and pd) increasing with time. The matrix is completely degraded after approximately 5000 seconds. As the drug and peptide species have the same association/dissociation kinetics, no matter the peptide is free or matrix-bound, the steady-state concentration of drug is constant during the degradation process.


Solving the space-dependent mass balances of Equation 1 results in concentration distributions of all participating species as functions of time. Figure 4 shows the concentration of the drug across the modeling domain.

Figure 4: Drug concentration after 6000 s.


As mentioned earlier, the enzyme originates from the nerve-cell tissue. From Figure 5, where the total drug release is shown, it is clear that matrix degradation has a directing effect on the drug release toward the damaged cell region..

Figure 5: Concentration profiles describing the total drug concentration (cd + cpd) across the modeling domain. Profiles were collected at times up to 6000 s.


Figure 6 visualizes the biomaterial matrix degradation. The plotted total matrix site concentrations (cmp + cmpd) shows how the degradation front passes through the biomaterial geometry.

Figure 6: Concentration profiles describing the total matrix site concentration (cmp + cmpd) with profiles collected at times up to 6000 s.

The detailed reaction/transport description in this model allows for the investigation of many design parameters relevant to bioengineering. This case presents the effect of matrix degradation on drug release as a function of time and geometry. Furthermore, it is straightforward to study the influence of the drug/peptide affinity by varying the rate constants kf

1 and kr1, or the influence of drug loading by varying the cmp:cmpd ratio. Of

course, the ability to examine alternative geometries and mixed biomaterial domains gives even more design flexibility.

Reference

1. D.J. Maxwell, B.C. Hicks, S. Parsons, and S.E. Sakiyama-Elbert, “Development of rationally designed affinity-based drug delivery systems”, Acta Biomat., vol. 1, no. 1, pp. 101–113, 2005.


Application Library path: Chemical_Reaction_Engineering_Module/Reactors_with_Mass_Transfer/drug_release



N E W





3 Click Add.

4 Click Study.


6 Click Done.


Read global parameters from a text file.




4 Browse to the model’s Application Libraries folder and double-click the file drug_release_parameters.txt.


First, model the reaction behavior of drug release within the biomaterial matrix, regarding it as a perfectly mixed batch reactor.




3 In the Formula text field, type d+mp<=>mpd.

4 Click Apply.

5 Locate the Rate Constants section. In the kf text field, type kf_d.

6 In the kr text field, type kr_d.



3 In the Formula text field, type d+p<=>pd.

4 Click Apply.





3 In the Species name text field, type e.

4 On the Reaction Engineering toolbar, click Species.


6 In the Species name text field, type h2o.


Additional Source 1Add additional Michaelis-Menten reaction rates for species p, pd, mp and mpd.

1 On the Reaction Engineering toolbar, click Additional Source.



S T U D Y 1





R E S U L T S

Concentration (re)Follow these steps to create Figure 3.

1 In the Model Builder window, under Results click Concentration (re).


e ce_init

h2o c_solv

mp cmp_init

mpd cmpd_init


mp -kf_mm*re.c_e*re.c_mp/(Km+re.c_mp)

mpd -kf_mm*re.c_e*re.c_mpd/(Km+re.c_mpd)

p kf_mm*re.c_e*re.c_mp/(Km+re.c_mp)

pd kf_mm*re.c_e*re.c_mpd/(Km+re.c_mpd)


2 In the Settings window for 1D Plot Group, type Concentration within biomatrix material, OD model in the Label text field.

3 Locate the Axis section. Select the x-axis log scale check box.

4 Click to expand the Legend section. From the Position list, choose Middle left.

Global 11 In the Model Builder window, expand the Results>

Concentration within biomatrix material, OD model node, then click Global 1.

2 In the Settings window for Global, click to expand the Title section.





7 On the Concentration within biomatrix material, OD model toolbar, click Plot.


Start setting up the space-dependent model by exporting the settings of the Reaction

Engineering interface with the Generate Space-Dependent Model feature.


Generate Space-Dependent Model 11 On the Reaction Engineering toolbar, click Generate Space-Dependent Model.

2 In the Settings window for Generate Space-Dependent Model, locate the Component Settings section.

3 From the Component to use list, choose 2Daxi: New.

4 Locate the Study Type section. From the Study type list, choose Time dependent.

Legends

Drug

Matrix-bound peptide

Matrix-bound drug-peptide

Peptide

Drug-peptide

Enzyme



G E O M E T R Y 1 ( 2 D A X I )

1 In the Model Builder window, expand the Component 2 (comp2) node, then click Geometry 1(2Daxi).






4 In the Height text field, type 9.








5 Locate the Position section. In the r text field, type 1.



Explicit 11 On the Definitions toolbar, click Explicit.

2 In the Settings window for Explicit, type Biomaterial matrix in the Label text field.



2 In the Settings window for Explicit, type Nerve cell tissue in the Label text field.




2 In the Settings window for Explicit, type Surroundings in the Label text field.



2 In the Settings window for Explicit, type Nerve and surroundings in the Label text field.


The automatically generated Chemistry interface is based on the conditions for the biomaterial matrix.


1 In the Model Builder window, under Component 2 (comp2) click Chemistry 1 (chem).

2 In the Settings window for Chemistry, type Chemistry - Biomaterial matrix in the Label text field.

3 Locate the Model Inputs section. In the T text field, type T_in.

C O M P O N E N T 2 ( C O M P 2 )

Add a new Chemistry interface for the drug dissociation taking place in the nerve cell tissue and surroundings.

A D D P H Y S I C S



3 In the tree, select Chemical Species Transport>Chemistry (chem).


5 On the Home toolbar, click Add Physics to close the Add Physics window.

C H E M I S T R Y 2 ( C H E M 2 )

1 In the Settings window for Chemistry, type Chemistry - Nerve and Surroundings in the Label text field.

2 Locate the Model Inputs section. In the T text field, type T_in.


C H E M I S T R Y - N E R V E A N D S U R R O U N D I N G S ( C H E M 2 )

The reaction is the same as one of the reactions in the biomatrix domain. Use e.g. capital letter notations to distinguish this reaction.

Reaction 11 On the Physics toolbar, click Domains and choose Reaction.


3 In the Formula text field, type D+P<=>PD.

4 Click Apply.



Species 11 On the Physics toolbar, click Domains and choose Species.


3 In the Species name text field, type E.

4 On the Physics toolbar, click Domains and choose Species.




8 In the Model Builder window, click Chemistry - Nerve and Surroundings (chem2).



With the molar masses added it is possible to compute several transport properties outside the scope of this example.

C H E M I S T R Y - B I O M A T E R I A L M A T R I X ( C H E M )

On the Physics toolbar, click Chemistry - Nerve and Surroundings (chem2) and choose Chemistry - Biomaterial matrix (chem).


E Constant, free species cE Constant

H2O Constant, solvent c_solv solvent

Species: d1 In the Model Builder window, expand the Chemistry - Biomaterial matrix (chem) node,

then click Species: d.


3 In the M text field, type Mnd.

Species: mp1 In the Model Builder window, under Component 2 (comp2)>Chemistry -

Biomaterial matrix (chem) click Species: mp.


3 In the M text field, type Mnp.

Species: mpd1 In the Model Builder window, under Component 2 (comp2)>Chemistry -

Biomaterial matrix (chem) click Species: mpd.


3 In the M text field, type Mnpd.

Species: p1 In the Model Builder window, under Component 2 (comp2)>Chemistry -

Biomaterial matrix (chem) click Species: p.



Species: pd1 In the Model Builder window, under Component 2 (comp2)>Chemistry -

Biomaterial matrix (chem) click Species: pd.



Species: e1 In the Model Builder window, under Component 2 (comp2)>Chemistry -

Biomaterial matrix (chem) click Species: e.


3 In the M text field, type Mne.


Species: h2o1 In the Model Builder window, under Component 2 (comp2)>Chemistry -

Biomaterial matrix (chem) click Species: h2o.


3 In the M text field, type Mnh2o.

C H E M I S T R Y - N E R V E A N D S U R R O U N D I N G S ( C H E M 2 )

On the Physics toolbar, click Chemistry - Biomaterial matrix (chem) and choose Chemistry -

Nerve and Surroundings (chem2).

Species: D1 In the Model Builder window, under Component 2 (comp2)>Chemistry -

Nerve and Surroundings (chem2) click Species: D.


3 In the M text field, type Mnd.

Species: P1 In the Model Builder window, under Component 2 (comp2)>Chemistry -

Nerve and Surroundings (chem2) click Species: P.



Species: PD1 In the Model Builder window, under Component 2 (comp2)>Chemistry -

Nerve and Surroundings (chem2) click Species: PD.



Species: E1 In the Model Builder window, under Component 2 (comp2)>Chemistry -

Nerve and Surroundings (chem2) click Species: E.


3 In the M text field, type Mne.

Species: H2O1 In the Model Builder window, under Component 2 (comp2)>Chemistry -

Nerve and Surroundings (chem2) click Species: H2O.



3 In the M text field, type Mnh2o.

Continue setting the mass transport properties in the biomaterial matrix in the Transport

of Diluted Species interface.


On the Physics toolbar, click Chemistry - Nerve and Surroundings (chem2) and choose Transport of Diluted Species (tds).


2 In the Settings window for Transport of Diluted Species, type Transport of Diluted Species - Biomaterial matrix in the Label text field.

3 Locate the Domain Selection section. From the Selection list, choose Biomaterial matrix.


TR A N S P O R T O F D I L U T E D S P E C I E S - B I O M A T E R I A L M A T R I X ( T D S )

On the Physics toolbar, click Transport of Diluted Species (tds) and choose Transport of Diluted Species - Biomaterial matrix (tds).


Transport of Diluted Species - Biomaterial matrix (tds) node, then click Transport Properties 1.


3 In the Dcd text field, type Dd.

4 In the Dcmp text field, type Dmp.

5 In the Dcmpd text field, type Dmpd.

6 In the Dcp text field, type Dp.

7 In the Dcpd text field, type Dpd.

8 In the Dce text field, type De.


Biomaterial matrix (tds) click Initial Values 1.


3 In the ce text field, type 0.

4 In the Model Builder window, click Transport of Diluted Species - Biomaterial matrix (tds).


Concentration 11 On the Physics toolbar, click Boundaries and choose Concentration.


3 In the Settings window for Concentration, locate the Concentration section.

4 Select the Species cd check box.

5 In the c0,cd text field, type cD.

6 Select the Species ce check box.

7 In the c0,ce text field, type cE.

Here the concentrations are dependent variables that will be taken from a second Transport of Diluted Species interface that will be created later.

8 Select the Species cp check box.

9 In the c0,cp text field, type cP.

10 Select the Species cpd check box.

11 In the c0,cpd text field, type cPD.

A D D P H Y S I C S

1 On the Physics toolbar, click Add Physics to open the Add Physics window.


3 In the tree, select Chemical Species Transport>Transport of Diluted Species (tds).


5 On the Physics toolbar, click Add Physics to close the Add Physics window.

In the second Transport of Diluted Species interface the mass transport within the cell nerve tissue and surroundings is set up.

TR A N S P O R T O F D I L U T E D S P E C I E S 2 ( T D S 2 )

1 In the Settings window for Transport of Diluted Species, type Transport of Diluted Species - Nerve and Surroundings in the Label text field.

2 Locate the Domain Selection section. From the Selection list, choose Nerve and surroundings.





TR A N S P O R T O F D I L U T E D S P E C I E S - N E R V E A N D S U R R O U N D I N G S ( T D S 2 )

Transport Properties 11 In the Model Builder window, under Component 2 (comp2)>Transport of Diluted Species -

Nerve and Surroundings (tds2) click Transport Properties 1.


3 In the DcD text field, type DsD.

4 In the DcE text field, type DsE.

5 In the DcP text field, type DsP.

6 In the DcPD text field, type DsPD.

Reactions 11 On the Physics toolbar, click Domains and choose Reactions.


3 From the Selection list, choose Nerve and surroundings.

4 Locate the Reaction Rates section. From the RcD list, choose Rate expression for species D (chem2).

5 From the RcP list, choose Rate expression for species P (chem2).

6 From the RcPD list, choose Rate expression for species PD (chem2).

7 From the RcE list, choose Rate expression for species E (chem2/E).

Transport Properties 21 On the Physics toolbar, click Domains and choose Transport Properties.

2 In the Settings window for Transport Properties, type Transport Properties - Nerve in the Label text field.

3 Locate the Domain Selection section. From the Selection list, choose Nerve cell tissue.

4 Locate the Diffusion section. In the DcD text field, type DnD.

5 In the DcE text field, type DnE.

6 In the DcP text field, type DnP.

cD

cP

cPD

cE


7 In the DcPD text field, type DnPD.

Initial Values 21 On the Physics toolbar, click Domains and choose Initial Values.

2 In the Settings window for Initial Values, type Initial Values 2 - Nerve in the Label text field.

3 Locate the Domain Selection section. From the Selection list, choose Nerve cell tissue.

4 Locate the Initial Values section. In the cE text field, type cnE_init.

M E S H 1

Size1 In the Model Builder window, under Component 2 (comp2) right-click Mesh 1 and choose

Size.

2 In the Settings window for Size, locate the Element Size section.

3 From the Predefined list, choose Extra fine.

Size 11 In the Model Builder window, under Component 2 (comp2)>Mesh 1 click Size 1.

2 In the Settings window for Size, locate the Geometric Entity Selection section.


4 Click Paste Selection.

5 In the Paste Selection dialog box, type 4,7,9 in the Selection text field.

6 Click OK.


8 Click the Custom button.

9 Locate the Element Size Parameters section. Select the Maximum element size check box.

10 In the associated text field, type 0.1.

Boundary Layer Properties1 In the Model Builder window, right-click Mesh 1 and choose Free Triangular.

2 Right-click Mesh 1 and choose Boundary Layers.

3 In the Settings window for Boundary Layer Properties, locate the Boundary Selection section.


5 In the Paste Selection dialog box, type 4,7,9 in the Selection text field.


6 Click OK.


Load the common concentration variables in domain 2 from a text file.


choose Variables.

2 In the Settings window for Variables, type Variables - Biomaterial matrix in the Label text field.

3 Locate the Geometric Entity Selection section. From the Geometric entity level list, choose Domain.


5 Locate the Variables section. Click Load from File.

6 Browse to the model’s Application Libraries folder and double-click the file drug_release_variables1.txt.

Variables 21 In the Model Builder window, right-click Definitions and choose Variables.

Load the common concentration variables in domain 1 and 3 from a text file.

2 In the Settings window for Variables, type Variables - Nerve and surroundings in the Label text field.

3 Locate the Geometric Entity Selection section. From the Geometric entity level list, choose Domain.


5 Locate the Variables section. Click Load from File.

6 Browse to the model’s Application Libraries folder and double-click the file drug_release_variables2.txt.

S T U D Y 2

Step 1: Time Dependent1 In the Model Builder window, expand the Study 2 node, then click

Step 1: Time Dependent.





The Figure 4 is generated in the following manner.

R E S U L T S

Concentration (tds) 11 In the Model Builder window, expand the Results>Data Sets node, then click Results>

Concentration (tds) 1.

2 In the Settings window for 3D Plot Group, type Concentration distribution drug in the Label text field.

3 Locate the Data section. From the Data set list, choose Revolution 2D 1.

4 From the Time (s) list, choose 6000.

Surface1 In the Model Builder window, expand the Results>Concentration distribution drug node,

then click Surface.


3 In the Expression text field, type c_drug.


5 On the Concentration distribution drug toolbar, click Plot.

The concentration profiles across parts of the modeling domains, as in Figures 5 and 6, require cut line data sets.



3 In row Point 1, set Z to 3.

4 In row Point 2, set R to 6 and z to 3.





Concentration (tds)Create Figure 5 following these steps.


2 In the Settings window for 1D Plot Group, type Total drug concentration profiles across modeling domain in the Label text field.


4 From the Time selection list, choose From list.

5 In the Times (s) list, choose 0, 600, 1200, 1800, 2400, 3000, 3600, 4200, 4800, 5400, and 6000.



8 In the associated text field, type cdrug + cpeptide-drug (mol/m3).

Line Graph 11 Right-click Total drug concentration profiles across modeling domain and choose

Line Graph.


3 In the Expression text field, type c_peptidedrug+c_drug.



6 On the Total drug concentration profiles across modeling domain toolbar, click Plot.


Create Figure 6 following these steps.


2 In the Settings window for 1D Plot Group, type Matrix site concentration profiles in the Label text field.


4 From the Time selection list, choose From list.

5 In the Times (s) list, choose 0, 600, 1200, 1800, 2400, 3000, 3600, 4200, 4800, 5400, and 6000.

8 In the associated text field, type cmatrix-bound species (mol/m3).


Line Graph 11 Right-click Matrix site concentration profiles and choose Line Graph.


3 In the Expression text field, type cmp+cmpd.



6 On the Matrix site concentration profiles toolbar, click Plot.


Concentration distribution drug 11 In the Model Builder window, under Results right-click Concentration distribution drug


2 In the Settings window for 3D Plot Group, type Matrix site concentration in the Label text field.

Surface1 In the Model Builder window, expand the Results>Matrix site concentration node, then

click Surface.


3 In the Expression text field, type cmp+cmpd.


5 On the Matrix site concentration toolbar, click Plot.

The following steps show how you can set up animations of your model results.

Animation 11 On the Results toolbar, click Animation and choose File.

2 In the Settings window for Animation, type Animation - Concentration distribution drug in the Label text field.

3 Locate the Scene section. From the Subject list, choose Concentration distribution drug.

4 Locate the Target section. From the Target list, choose Player.

5 Right-click Animation - Concentration distribution drug and choose Play.

Animation 21 On the Results toolbar, click Animation and choose File.

2 In the Settings window for Animation, type Animation - Matrix site concentration in the Label text field.

3 Locate the Scene section. From the Subject list, choose Matrix site concentration.

4 Locate the Target section. From the Target list, choose Player.

5 Right-click Animation - Matrix site concentration and choose Play.

Remove the unused plot groups.


Concentration (tds)In the Model Builder window, under Results right-click 3D Plot Group 3 and choose Delete.

2D Plot Group 6In the Model Builder window, under Results right-click Concentration (tds) and choose Delete.


Concentration (tds2) 11 In the Model Builder window, under Results right-click 3D Plot Group 7 and choose

Delete.

2 Right-click Concentration (tds2) and choose Delete.

3 Right-click Concentration (tds2) 1 and choose Delete.

For future use of the 0D model, turn off the space-dependent interfaces in the Study 1 node.

S T U D Y 1

Step 1: Time Dependent1 In the Settings window for Time Dependent, locate the Physics and Variables Selection

section.



Physics interface

Chemistry - Biomaterial matrix (chem)

Transport of Diluted Species - Biomaterial matrix (tds)

Chemistry - Nerve and Surroundings (chem2)

Transport of Diluted Species - Nerve and Surroundings (tds2)



T r a n s po r t i n an E l e c t r o k i n e t i c V a l v e




Introduction

This tutorial presents an example of pressure-driven flow and electrophoresis in a microchannel system.

Researchers often use a device similar to the one in this example as an electrokinetic sample injector in biochips to obtain well-defined sample volumes of dissociated acids and salts and to transport these volumes. The model presents a study of a pinched injection cross valve during the focusing, injection, and separation stages. Inspiration for the model comes from a study by Ermakov and others (Ref. 1). Focusing is obtained through pressure-driven flow of the sample and buffer solution, which confines the sample in the focusing channel. When the system reaches steady state, the pressure-driven flow is turned off and an electric field is applied along the channels. This field drives the dissociated sample ions in the focusing zone at right angles to the focusing channel and through the injection channel. A clean separation of the sample ions is important, so the model examines the effect on ion separation of different configurations of the electric field.

This specific case does not account for electro-osmosis because the channel surfaces are subjected to a treatment that minimizes the extension of the electric double layer.

Note: This model requires either the CFD Module, the Microfluidics Module, or the Subsurface Flow Module

Model Definition

Figure 1 shows a 2D cross section of the geometry in the xz-plane and points out the different channels and boundaries. The horizontal channel serves as the focusing channel, while the vertical channel is the injection channel. The actual model is in 3D with rectangular pipes whose corners are rounded. For geometry dimensions refer to Table 1 below.

TABLE 1: MODEL DIMENSIONS

HORIZONTAL CHANNEL VERTICAL CHANNEL CROSSING AREA

Dimensions (μm)

- x 340 20 28

- y 20 20 20

- z 20 340 28

Position (μm)

2 | TR A N S P O R T I N A N E L E C T R O K I N E T I C V A L V E

Figure 1: The focusing stage involves pressure-driven flow of both the sample and the buffering solution. The device applies an electric field over the focusing channel.

The device operation and hence the modeling procedure takes place in two stages: focusing and injection.

In the focusing stage, the device injects a buffering solution through pressure-driven convection into the vertical channels from the top and bottom. At the same time, it forces the sample solution through the horizontal focusing channel (see Figure 1). The buffering solution neutralizes the acids contained in the sample except for a very thin region

- x -100 0 -4

- y 0 0 0

- z 0 -200 -4

Rounding (μm)

- radius 4 4 4

- direction in in out

TABLE 1: MODEL DIMENSIONS

HORIZONTAL CHANNEL VERTICAL CHANNEL CROSSING AREA

Focusing channel

Injection channel

Focusing zone

Sample inlet

Buffer inlet

Outlet

Buffer inlet


confined to the crossing between the horizontal and vertical channels. This means that the dissociated ions are only in a needle-shaped region in the focusing zone.

Next, in the injection stage the device turns off the convective flow and then applies a vertical field to migrate the sample from the focusing channel to the injection point at the lower end of the vertical channel. The sample ions are negatively charged and migrate in opposite direction to the electric field. This model studies two different configurations (See Table 2) for the applied electric field. In the first configuration (Injection stage, Mode A) electric field is only applied in the vertical direction. In the second configuration (Injection stage, Mode B) the electric field is applied in both the horizontal and vertical directions (Figure 2).The horizontal field focuses the sample during the initial part of the injection stage in order to obtain a well-separated sample.

Figure 2: During the injection stage, the device turns off convective flow and applies an electric field. The horizontal field avoids the broadening of the sample, while the vertical field injects the sample into the vertical channel in the direction opposite to the electric field.

Electric field vectors

Migration of the sample opposite to the field


The model assumes that the charged sample concentration is very low compared to other ions dissolved in the solution. This implies that the sample concentration does not influence the solution’s conductivity and that you can neglect the concentration gradients of the charge-carrying species, which are present in a much higher concentration than the sample ions. Such an electrolyte is known as a supporting electrolyte.

Several equations describe the model: the Stokes flow equations, the equation for current balance, and a mass balance using the Nernst-Planck equation. This model uses the steady-state solution for the focusing stage as the initial condition for the injection stages.

Now consider the formulation of the model equations.

T H E F O C U S I N G S T A G E

The Stokes flow equations give the global mass a momentum balance in the focusing stage:

In these equations, h denotes the dynamic viscosity (SI unit: kg/(m·s)), u is the velocity (SI unit: m/s), p is the pressure (SI unit: Pa).

The total balance of charges for a supporting electrolyte comes from the divergence of the current-density vector, which in a supporting electrolyte is given by Ohm’s law:

Here κ is the electrolyte’s conductivity (SI unit: S/m) and V is the potential (SI unit: V). The balance of current at steady state then becomes

TABLE 2: APPLIED ELECTRIC FIELD CONFIGURATION

INLET FOCUSING STAGE INJECTION STAGE, MODE A

INJECTION STAGE, MODE B

Sample inlet Electric potential, V = -1V

Electric insulation Electric potential, V = -1V

Outlet Ground Electric insulation Ground

Upper buffer inlet

Electric insulation Electric potential, V = -3.2V

Electric potential, V = -3.2V

Lower buffer inlet

Electric insulation Ground Ground

0 ∇ pI– η ∇u ∇u( )T+( )+[ ]⋅=

∇ u⋅ 0.=

i κ V∇–=

∇ i⋅ 0=


which gives

The flux vector for the sample ions comes from the Nernst-Planck equation

which leads to the following mass balance equation at steady state for species i:

Here ci is the concentration (SI unit: mol/m3), Di represents the diffusivity (SI unit: m2/s), zi equals the charge number (which equals 1 for this model), umi is the mobility (SI unit: s·mol/kg), and F is Faraday’s constant (SI unit: C/mol).

For the pressure-driven flow, assume that the flow has fully developed laminar form in all inlets, that all sides have no slip conditions, and that the fluid flows freely out from the end of the focusing channel.

The boundary conditions for the charge balance determine the potential at the respective inlet and outlet boundary

where i denotes the index for each boundary. This model also assumes that all wall boundaries are insulating:

The boundary conditions for the mass balance of the sample during the focusing stage appear below. The equation

gives the concentration at the inlet of the sample, while the equation

gives the concentration of the buffer at both boundaries of the vertical channel. At the outlet boundary, convection and migration are the dominating transport mechanisms (that is, diffusion is negligible), so that

∇ κ V∇–( )⋅ 0=

Ni Di∇ci– ziumiFci V∇– ciu+=

∇ Di∇ci– ziumiFci V∇– ciu+( )⋅ 0=

V V0 i,=

V∇ n⋅ 0=

c cin=

c cbuffer=

Ni n⋅ ziumiFci V∇– ciu+( ) n⋅=


T H E I N J E C T I O N S T A G E

In the injection and separation stages, the device turns the flow off and changes the configuration of the electric field. You again solve the charge-balance equations but with new boundary conditions:

The mass balance for the dilute species comes from a time-dependent mass balance:

The model assumes that the convective contribution is zero.

The boundary conditions for the current-balance equation imply that the potential is locked at all boundaries except for the walls,

Further assume the walls are electrically insulated, which yields

As opposed to the focusing state, the boundary conditions for the mass balance are changed. In the injection stage, set the concentration at the inlet boundary:

For all other boundaries, assume that migration is the dominating transport mechanism, so that:

The time-dependent solution requires an initial condition for the mass balance, which you obtain from the steady-state solution of the focusing stage:


This example analyzes the focusing stage and two configurations for the injection stages. Recall that the first injection-stage configuration (Mode A) applies the electric field only over the injection channel while the inlet and outlet boundaries of the focusing channel

∇ κ V∇–( )⋅ 0=

t∂∂c ∇+ Di∇ci– ziumiFci V∇–( )⋅ 0=

V V0 i,=

V∇ n⋅ 0=

c cin=

Ni n⋅ ziumiFci V∇–( ) n⋅=

c t 0=( ) cfocus=


are insulated; the second injection-stage configuration (Mode B) applies the electric field over both channels.

Figure 3 shows the steady-state concentration distribution during the focusing stage along with the distribution at the beginning of the injection stage. Note that the vertical flows from the upper and lower injection channels focus the concentration on a very narrow region near the crossing area of the channels. Further away from the crossing area, however, the concentration spreads again more equally over the channel.

Figure 3: The steady-state concentration distribution during the focusing stage and prior to the injection stage.

Figure 4 and Figure 5 compare the concentration distribution for the two configurations at two times, specifically 0.06 s and 0.12 s after the beginning of the injection stage. The figures on the left show that for Mode A the concentration boundary is practically stationary in the horizontal direction. Consequently, the vertical electric field can continuously draw ions from the focusing channel, which results in poor separation and a poorly defined sample volume of the substance. For Mode B the situation is very different. The horizontal electric field draws the concentration boundary to the left, and the


channels separate rapidly. Consequently, this scheme draws a well-defined sample volume of the substance into the injection channel.

Figure 4: The concentration distribution at a time 0.06 s after starting the injection stage for the Mode A configuration (left) and Mode B configuration (right).

Figure 5: The concentration distribution at a time 0.12 s after starting the injection stage for the Mode A configuration (left) and Mode B configuration (right).

It is also possible to observe the difference between the two configurations if you look at the concentration along a line through the middle of the injection channel, examining it at several times after the start of the injection stage (Figure 6). The maximum concentration moves down the injection channel with time. The peaks are higher in the upper axis corresponding to Mode A, but they are much wider than for Mode B. A considerable amount of concentration appears at the left of the peak, and the sample remains attached to the focusing area—resulting in an unwanted distortion of the sample package. The narrow peaks of Mode B, on the other hand, form nice bell curves


throughout the downward transport in the injection channel, resulting in a well-defined sample package.

Figure 6: Concentration profile for Mode A (top) and Mode B (bottom) along the injection channel at various time steps: 0 s, 0.06 s, 0.12 s, 0.18 s, 0.24 s, 0.30 s, 0.36 s, 0.42 s, 0.48 s,


0.54 s, and 0.6 s after initialization of the injection stage. The origin of the x-axis marks the centerline of the focusing channel.

This study illustrates that modeling is extremely valuable in the investigation of electrophoretic transport. You can vary the configuration of the potential to obtain even better focusing and injection stages for the valve under study.

Notes About the COMSOL Implementation

I N T E R F A C E S

In COMSOL Multiphysics you define the model with the following physics interfaces:

• The Creeping Flow interface solves the fluid flow in the channels governed by Stokes equations.

• The Electric Currents interface solves the equation for current balance.

• The Transport of Diluted Species interface solves the Nernst-Planck equation.

C O M P U T I N G T H E S O L U T I O N

The operation of the actual device proceeds in two stages, the focusing stage and the injection stage. This model simulates two settings of the injection stage so in total it works in three phases.

The first phase defines the domain settings and boundary conditions for the focusing phase. Then the model solves the interfaces sequentially with a nonlinear solver in the following sequence:

1 Creeping Flow interface

2 Electric Currents interface

3 Transport of Diluted Species interface

Each step uses the solution from the previous one. The model stores the last solution for use as the initial value for the consequent modeling.

In the second phase you change the domain settings and boundary conditions to handle the injection stage Mode A. In a real device you would turn off the convective flow; in the model you simulate this by setting the velocity parameters of the Electrokinetic Flow interface to zero. Thus it uses no information from the Laminar Flow interface.

Solving the second phase starts from the stored solution of the first phase, and the model solves the Electric Currents interface with a nonlinear solver. Then you select a time-


dependent solver and solve the Transport of Diluted Species interface. This solution is the result for the injection stage Mode A.

In the third phase you again change domain settings and boundary conditions but this time for the injection stage Mode B; you then solve for the final solution the same way as in the second phase.

Reference

1. S.V. Ermakov, S.C. Jacobson, and J.M. Ramsey, Technical Proc 1999 Int. Conf. on Modeling and Simulation of Microsystems, Computational Publications, 1999.

Application Library path: Chemical_Reaction_Engineering_Module/Electrokinetic_Effects/electrokinetic_valve



N E W




2 In the Select Physics tree, select AC/DC>Electric Currents (ec).

3 Click Add.



5 Click Add.

6 In the Select Physics tree, select Fluid Flow>Single-Phase Flow>Creeping Flow (spf).

7 Click Add.

8 Click Study.


10 Click Done.



1 On the Home toolbar, click Parameters.

Add the model parameters from a text file.




4 Browse to the model’s Application Libraries folder and double-click the file electrokinetic_valve_parameters.txt.

G E O M E T R Y 1

Import the geometry.

1 On the Geometry toolbar, click Insert Sequence.

2 Browse to the model’s Application Libraries folder and double-click the file electrokinetic_valve_geom_sequence.mph.

Work Plane 1 (wp1)1 In the Model Builder window, under Component 1 (comp1)>Geometry 1 click

Work Plane 1 (wp1).

2 In the Settings window for Work Plane, click Build All Objects.

Use the material node to make your own electrolyte fluid material.

M A T E R I A L S

Material 1 (mat1)1 In the Model Builder window, under Component 1 (comp1) right-click Materials and

choose Blank Material.

2 Right-click Material 1 (mat1) and choose Rename.

3 In the Rename Material dialog box, type Electrolyte fluid in the New label text field.

4 Click OK.

5 In the Settings window for Material, locate the Material Contents section.



Choose all the necessary features in the interfaces to model the focusing stage and injection stages for both mode A and B. Later in the study node, you can select which of the features that are solved for.

E L E C T R I C C U R R E N T S ( E C )

Electric Potential 11 On the Physics toolbar, click Boundaries and choose Electric Potential.

2 In the Settings window for Electric Potential, type Electric Potential - Focusing stage and Injection stage mode B in the Label text field.

3 Locate the Boundary Selection section. From the Selection list, choose Sample inlet.

4 Locate the Electric Potential section. In the V0 text field, type V_appS.

Ground 11 On the Physics toolbar, click Boundaries and choose Ground.

2 In the Settings window for Ground, type Ground - Focusing stage and Injection stage mode B in the Label text field.

3 Locate the Boundary Selection section. From the Selection list, choose Outlet.

Electric Potential 21 On the Physics toolbar, click Boundaries and choose Electric Potential.

2 In the Settings window for Electric Potential, type Electric Potential - Injection stage in the Label text field.

3 Locate the Boundary Selection section. From the Selection list, choose Upper buffer inlet.

4 Locate the Electric Potential section. In the V0 text field, type V_appUB.

Ground 21 On the Physics toolbar, click Boundaries and choose Ground.

Property Variable Value Unit Property group

Electrical conductivity sigma 1[S/m] S/m Basic

Relative permittivity epsilonr 1 1 Basic

Density rho 1e3[kg/m^3] kg/m³ Basic

Dynamic viscosity mu 1e-3[Pa*s] Pa·s Basic


2 In the Settings window for Ground, type Ground - Injection stage in the Label text field.

3 Locate the Boundary Selection section. From the Selection list, choose Lower buffer inlet.

The use of higher order elements, set in the Discretization section of Transport of Diluted Species, improves the accuracy of the results significantly for low Reynolds number flows such as those in this model.





3 Select the Migration in electric field check box.

4 Click to expand the Discretization section. From the Concentration list, choose Quadratic.



2 In the Settings window for Transport Properties, type Transport Properties - Focusing stage in the Label text field.

3 Locate the Model Input section. In the T text field, type T.

4 Locate the Convection section. From the u list, choose Velocity field (spf).

5 Locate the Migration in Electric Field section. From the V list, choose Electric potential (ec).

6 Locate the Diffusion section. In the Dc text field, type D.

7 Locate the Migration in Electric Field section. In the zc text field, type z_c.

Transport Properties - Focusing stage 11 In the Model Builder window, under Component 1 (comp1)>

Transport of Diluted Species (tds) right-click Transport Properties - Focusing stage and choose Duplicate.

2 In the Settings window for Transport Properties, type Transport Properties - Injection stage in the Label text field.

3 Locate the Domain Selection section. From the Selection list, choose All domains.


4 Locate the Convection section. From the u list, choose User defined.


2 In the Settings window for Concentration, type Concentration at sample inlet in the Label text field.


4 Locate the Concentration section. Select the Species c check box.

5 In the c0,c text field, type c_in.


2 In the Settings window for Concentration, type Concentration at buffer inlets in the Label text field.

3 Locate the Boundary Selection section. From the Selection list, choose Buffer inlets.

4 Locate the Concentration section. Select the Species c check box.




Flux 11 On the Physics toolbar, click Boundaries and choose Flux.

2 In the Settings window for Flux, type Migration at inlets and outlets - Injection stage in the Label text field.

3 Locate the Boundary Selection section. From the Selection list, choose Migration at inlets and outlets - Injection stage.

4 Locate the Inward Flux section. Select the Species c check box.

5 In the N0,c text field, type -tds.nmflux_c.

The predefined boundary variable tds.nmflux_c gives the outward normal electrophoretic flux, N_i.n.

C R E E P I N G F L O W ( S P F )

In the Model Builder window, under Component 1 (comp1) click Creeping Flow (spf).



2 In the Settings window for Inlet, type Inlet, sample in the Label text field.



5 Locate the Laminar Inflow section. In the Uav text field, type u_a.

6 In the Lentr text field, type 1e-4.


2 In the Settings window for Inlet, type Inlets, buffer in the Label text field.

3 Locate the Boundary Selection section. From the Selection list, choose Buffer inlets.


5 Locate the Laminar Inflow section. In the Uav text field, type w_a.

6 In the Lentr text field, type 1e-4.




4 Locate the Pressure Conditions section. Select the Normal flow check box.

With the settings in the Discritization section, a user-controlled mesh with less elements than the physics-controlled mesh can be used.

M E S H 1


Free Tetrahedral.



4 Locate the Element Size Parameters section. In the Maximum element size text field, type 29.

5 In the Minimum element size text field, type 5.


6 Click Build All.

Solve the mode A injection in five steps. Choose the features solved for by modifying the physics tree and variables in each study step as shown in the following steps:

S T U D Y 1

In the Settings window for Study, type Study - for mode A in the Label text field.

S T U D Y - F O R M O D E A

Step 1: Stationary1 In the Model Builder window, expand the Study - for mode A node, then click

Step 1: Stationary.

2 In the Settings window for Stationary, type Stationary - Focusing stage in the Label text field.

3 Locate the Physics and Variables Selection section. In the table, clear the Solve for check box for Electric Currents (ec) and Transport of Diluted Species (tds).


2 In the Settings window for Stationary, type Stationary 2 - Focusing stage in the Label text field.

3 Locate the Physics and Variables Selection section. Select the Modify physics tree and variables for study step check box.

4 In the Physics and variables selection tree, select Component 1 (comp1)>


5 Click Disable in Solvers.


Creeping Flow (spf).



Electric Currents (ec)>Electric Potential - Injection stage.

9 Click Disable.


Electric Currents (ec)>Ground - Injection stage.

11 Click Disable.






Electric Currents (ec).






Transport of Diluted Species (tds)>Transport Properties - Injection stage.

9 Click Disable.


Transport of Diluted Species (tds)>Migration at inlets and outlets - Injection stage.

11 Click Disable.


2 In the Settings window for Stationary, type Stationary - Injection stage in the Label text field.



Electric Currents (ec)>Electric Potential - Focusing stage and Injection stage mode B.

5 Click Disable.


Electric Currents (ec)>Ground - Focusing stage and Injection stage mode B.

7 Click Disable.









2 In the Settings window for Time Dependent, type Time Dependent - Injection stage in the Label text field.

3 Locate the Study Settings section. In the Times text field, type range(0,0.06,0.6).

4 Locate the Physics and Variables Selection section. In the table, clear the Solve for check box for Electric Currents (ec) and Creeping Flow (spf).


R E S U L T S

Electric Potential (ec)Plot the concentration at the surface of the channels at selected times. Begin with t = 0, which corresponds to the end of the focusing stage and Figure 3.


2 In the Settings window for 3D Plot Group, type Concentration - Mode A in the Label text field.

3 Locate the Data section. From the Data set list, choose Study - for mode A/

Solution 1 (sol1).


Surface1 In the Model Builder window, expand the Results>Concentration - Mode A node, then

click Surface.

2 In the Settings window for Surface, locate the Coloring and Style section.

3 Clear the Color legend check box.

4 On the Concentration - Mode A toolbar, click Plot.

Concentration - Mode A1 In the Model Builder window, under Results click Concentration - Mode A.


3 From the Time (s) list, choose 0.06.



This plot should look like that in the left panel of Figure 4.



The plot in the Graphics window should now look like that in the left panel of Figure 5.

Next, set up a plot for the concentration along a line through the middle of the injection channel.


2 In the Settings window for Cut Line 3D, locate the Data section.

3 From the Data set list, choose Study - for mode A/Solution 1 (sol1).

4 Locate the Line Data section. In row Point 1, set X to 10, y to 10 [um], and z to 200 [um].

5 In row Point 2, set X to 10, y to 10 [um], and z to -200 [um].

Plot the cut line to verify that you have entered the correct points; it should run along the center of the lower injection channel and extend past the crossing.

6 Click Plot.



2 In the Settings window for 1D Plot Group, type Concentration line plot mode A in the Label text field.


Line Graph 11 Right-click Concentration line plot mode A and choose Line Graph.

2 In the Settings window for Line Graph, click Replace Expression in the upper-right corner of the y-axis data section. From the menu, choose Component 1>

Transport of Diluted Species>c - Concentration.


4 In the Expression text field, type 10[um]-z.

5 On the Concentration line plot mode A toolbar, click Plot.

Compare the results with the upper plot in Figure 6.

Now change the boundary conditions to correspond to injection mode B, then set up a solver and compute the solution before generating plots to compare with those for injection mode A.

A D D S T U D Y





Solve the mode B injection in five steps. Choose the features solved for by modifying the physics tree and variables in each study step as shown in the following steps:

S T U D Y 2

In the Settings window for Study, type Study - for mode B in the Label text field.

S T U D Y - F O R M O D E B

Step 1: Stationary1 On the Home toolbar, click Add Study to close the Add Study window.

2 In the Model Builder window, under Study - for mode B click Step 1: Stationary.


4 In the table, clear the Solve for check box for Electric Currents (ec) and Transport of

Diluted Species (tds).


5 In the Label text field, type Stationary - Focusing stage.








6 Click Disable.


Electric Currents (ec)>Ground - Injection stage.

8 Click Disable.











Electric Currents (ec).



Transport of Diluted Species (tds)>Transport Properties - Injection stage.

7 Click Disable.



Transport of Diluted Species (tds)>Migration at inlets and outlets - Injection stage.

9 Click Disable.





2 In the Settings window for Stationary, type Stationary - Injection stage in the Label text field.

3 Locate the Physics and Variables Selection section. In the table, clear the Solve for check box for Transport of Diluted Species (tds) and Creeping Flow (spf).


2 In the Settings window for Time Dependent, type Time Dependent - Injection stage in the Label text field.

3 Locate the Study Settings section. In the Times text field, type range(0,0.06,0.6).

4 Locate the Physics and Variables Selection section. In the table, clear the Solve for check box for Electric Currents (ec) and Creeping Flow (spf).


R E S U L T S

Electric Potential (ec) 1Plot the concentration at the surface of the channels at selected times for mode B.


2 In the Settings window for 3D Plot Group, type Concentration - Mode B in the Label text field.

3 Locate the Data section. From the Time (s) list, choose 0.06.

4 On the Concentration - Mode B toolbar, click Plot.

Compare the results with those on the right side of Figure 4.



6 On the Concentration - Mode B toolbar, click Plot.

Compare the results with those on the right side of Figure 5.

Data SetsAdd a second Cut Line 3D node for Injection stage mode B.



3 From the Data set list, choose Study - for mode B/Solution 6 (sol6).

4 Locate the Line Data section. In row Point 1, set X to 10, y to 10 [um], and z to 20 [um].

5 In row Point 2, set X to 10, y to 10 [um], and z to -200 [um].


2 In the Settings window for 1D Plot Group, type Concentration line plot mode B in the Label text field.


Line Graph 11 Right-click Concentration line plot mode B and choose Line Graph.




4 In the Expression text field, type 10[um]-z.

5 On the Concentration line plot mode B toolbar, click Plot.

Compare the results with the lower plot in Figure 6.

Several plot groups are not used and can be removed.

Concentration (tds)In the Model Builder window, under Results right-click Electric Potential (ec) and choose Delete.

Velocity (spf)In the Model Builder window, under Results right-click Concentration (tds) and choose Delete.


Pressure (spf)In the Model Builder window, under Results right-click Velocity (spf) and choose Delete.

Electric Potential (ec) 1In the Model Builder window, under Results right-click Pressure (spf) and choose Delete.

Concentration (tds) 1In the Model Builder window, under Results right-click Electric Potential (ec) 1 and choose Delete.

Pressure (spf) 11 In the Model Builder window, under Results right-click Concentration (tds) 1 and choose

Delete.

2 Right-click Velocity (spf) 1 and choose Delete.

3 Right-click Pressure (spf) 1 and choose Delete.



Eng i n e Coo l a n t P r op e r t i e s




Introduction

The engine block of a car includes a cooling jacket to remove excess heat from combustion. The cooling jacket consists of open spaces in the cylinder block and the cylinder head. When the engine is running, a coolant fluid is pumped through the jacket to keep the engine from overheating. Optimizing the heat removal is important to minimize coolant boiling, prevent engine failure, and, more recently, improve overall efficiency through waste heat recovery. This example demonstrates how the Thermodynamics feature can be used to evaluate the performance of different engine coolants.

Although pure water works well as a coolant, to prevent freezing at low temperatures, a mixture of ethylene glycol and water is normally used to lower the freezing point. The Thermodynamics feature, available when licensed to the Chemical Reaction Engineering Module, is used here to show how the boiling point, density, viscosity, thermal conductivity, and heat capacity also depend on the composition of the coolant mixture and how changes in these properties affect the cooling process.

Model Definition

Figure 1 shows the flow pattern inside the cooling jacket of a representative four cylinder engine. Solving a fully coupled nonisothermal turbulent flow problem with temperature, pressure, and composition dependent coolant properties in this complex geometry typically requires a significant number of computer hours. One approach to obtain a reliable approximate solution in a shorter time is to use the functionality available in the Thermodynamics feature to investigate the coolant property behavior and determine where simplifying assumptions can be made. The consequences of these assumptions can

2 | E N G I N E C O O L A N T P R O P E R T I E S

be investigated efficiently in a simplified geometry in order to provide confidence in their use in more complex geometries.

Figure 1: The coolant flow inside the cooling jacket of a four cylinder engine.

Here a simplified 2D axially symmetric geometry, shown in Figure 2, is considered as an engine coolant test apparatus. Coolant is introduced at a specified flow rate in the bottom of the device, the coolant hits a solid steel part and is then deflected into a larger flow domain. A heat flux is applied on the outer boundary of the larger section. The resulting temperature is measured at steady state in the solid structure near the coolant outflow at the top.

To solve for the fluid flow and heat in the test apparatus, the current model uses the Single-Phase Flow and the Heat Transfer in Fluids interfaces. The interfaces are coupled using a Nonisothermal Flow multiphysics feature, and the k-ε model is used to model the fluid flow turbulence.

The properties of the coolant fluid are defined using the Thermodynamics feature. This is done by first defining and adding a Property Package node to the Thermodynamics feature. Included in the Property Package are the relevant chemical species, in this case ethylene glycol and water. The package node in turn can be used to compute property functions for transport properties and thermodynamic properties, both for pure species as and for a resulting mixture. In this example a Chemistry interface is also added to the


model. When coupling the Chemistry interface to the Property Package, the mixture functions relevant for fluid flow, mass transfer and heat transfer processes are automatically defined. In this case, functions for the density, the viscosity, the thermal conductivity and the heat capacity of the coolant mixture are created. In this model the pressure dependence of the mixture properties are assumed to be small. The properties are easily set to be evaluated at a constant pressure using the Chemistry interface.

The analysis of the coolant properties are performed in three steps. First the mixture properties are evaluated by plotting the Property Package functions. Then the phase envelope of the coolant vapor-liquid system is visualized by plotting the equilibrium temperatures (for boiling and condensation) as a function of the composition. The required equilibrium functions are defined by adding an Equilibrium Calculation feature to the Property Package. Using the equilibrium functions the phase envelope for two different pressures are compared.

The fluid flow and heat transfer of the coolant mixture inside the test apparatus is then solved for. Results for pure water, and a 50 volume percent mixture of ethylene glycol in water are compared. For these chemicals, a 50 volume percent mixture corresponds to 52.7 mass percent and 24.4 mole percent. Finally the results are used to compute average mixture properties.


Figure 2: Axially symmetric engine coolant test apparatus.

Results

Figure 3 shows the temperature and composition dependence of the heat capacity. Similar graphs are generated for density, viscosity, and thermal conductivity. Studying these graphs reveals that the addition of ethylene glycol increases the density and viscosity, but decreases the thermal conductivity and heat capacity when compared with pure water. It should be

Outlet

Inlet

Insulation

Heat flux

Insulation

Temperature measurement point

Solid


expected that a 50 volume percent mixture will yield more pressure drop and require a higher flow rate to achieve the same cooling effect as that of pure water.

Figure 3: Heat capacity as a function of temperature and composition for ethylene glycol water mixtures.

Figure 4 shows the phase envelope for ethylene glycol-water mixtures produced using the Equilibrium Calculation feature of the Property Package. A car coolant system typically


operates at about 2 atm pressure. Here we can see that a 50 volume percent (24.4 mole percent) mixture should boil at slightly higher than 400 K at this pressure.

Figure 4: Phase envelope for the equilibrium temperature of ethylene glycol-water mixtures at two pressures.


Figure 5 shows the flow pattern inside the test apparatus with water entering at 1 m/s. The coolant flow of 42 l/min and a heat input of 50 kW used here in the test apparatus are on the same order of magnitude as a conventional car cooling system.

Figure 5: Flow patterns inside the test apparatus with water at 1 m/s.

Figure 6 shows that, indeed, an ethylene glycol-water mixture will provide less cooling than pure water at a fixed flow rate. And also that about 15 percent more coolant flow is required to produce the same cooling as pure water. It can also be seen that some boiling


of the coolant (at T > 400 K) is expected in the recirculation zones in the outer corners of the apparatus.

Figure 6: Temperature within the test apparatus for three cases: (a) water at 1 m/s, (b) 50 volume percent ethylene glycol at 1 m/s, and (c) 50 volume percent ethylene glycol at 1.15 m/s.

a

b c


Table 1 provides a comparison of results for pressure drop, outlet temperature, and outlet density.

1 Using constant mixture properties.

Considering the graphical results for the various coolant properties, it might be reasonable to use approximate averages for the relatively small temperature range considered, between 353 and 400 K. In Figure 7 the resulting heat capacity for the pure water and the two ethylene glycol-water mixture cases is plotted. As seen before, the heat capacity differs significantly when comparing pure water and the mixture. But, the individual variation for each coolant however is seen to be small, about 2% for this mixture property and location.

TABLE 1: SIMULATION RESULTS

Weight fraction, ethylene glycol

Velocity (m/s)

Pressure drop (Pa)

Outlet temperature (K)

Outlet density (kg/m)

0 1 554 368 961

0.527 1 624 372 1007

0.527 1.15 821 369 1008

0.5271 1 607 372 1010


Analyzing the density in the same manner, the variation can be seen to be in the same order of magnitude.

Figure 7: Coolant heat capacity plotted along a vertical cut line at half the radius of the test apparatus chamber.

Using the solution for a mixture with 50 volume percent ethylene glycol in water, the following average values are computed; density = 1010 kg/m3, viscosity = 9.11·10-4 Pa·s, thermal conductivity = 0.574 W/(m·K), and heat capacity = 3480 J/(kg·K). Figure 8 shows a comparison of the temperature results obtained using these approximations with those using the fully coupled temperature dependent properties in our test device. The similarity between these results is probably sufficient to justify the use of the approximate average values in a cooling jacket model with a realistic geometry. Solving the flow and


heat transfer equations requires considerably less computational effort for the constant average property value case.

Figure 8: Comparison of temperature within our test apparatus for 50 volume percent ethylene glycol in water at 1 m/s using: (a) temperature dependent properties, (b) approximate average properties.

Reference

1. http://www.engineeringtoolbox.com/ethylene-glycol-d_146.html

a

b


http://www.engineeringtoolbox.com/ethylene-glycol-d_146.html

Application Library path: Chemical_Reaction_Engineering_Module/Thermodynamics/engine_coolant_properties



N E W




2 In the Select Physics tree, select Chemical Species Transport>Chemistry (chem).

3 Click Add.

4 Click Study.

5 In the Select Study tree, select Preset Studies>Stationary.

6 Click Done.





4 Browse to the model’s Application Libraries folder and double-click the file engine_coolant_properties_parameters.txt.

G E O M E T R Y 1



3 In the Width text field, type r_p.

4 In the Height text field, type l_p.


5 Click Build Selected.



3 In the Width text field, type r_c.

4 In the Height text field, type l_c.

5 Locate the Position section. In the z text field, type zpos_c.




3 In the Width text field, type r_s1.

4 In the Height text field, type l_s1.

5 Locate the Position section. In the z text field, type zpos_s.


G E O M E T R Y 1



3 In the Width text field, type r_s2.

4 In the Height text field, type l_s2.

5 Locate the Position section. In the z text field, type zpos_s.


Union 1 (uni1)1 On the Geometry toolbar, click Booleans and Partitions and choose Union.

2 Select the objects r3 and r4 only.

3 In the Settings window for Union, locate the Union section.

4 Clear the Keep interior boundaries check box.




2 Select the objects r2 and r1 only.




Fillet 1 (fil1)1 On the Geometry toolbar, click Fillet.

2 On the object uni2, select Points 6, 7, 9, and 10 only.

3 On the object uni1, select Point 5 only.

4 In the Settings window for Fillet, locate the Radius section.

5 In the Radius text field, type 0.3[cm].



1 In the Model Builder window, right-click Global Definitions and choose Thermodynamics.

2 Right-click Global Definitions>Thermodynamics and choose Property Package.

3 In the Species filter field, type ethylene glycol.

4 Select ethylene glycol (107-21-1, C2H6O2) in the list of available species.

5 Click Add Selected to add the species to the Selected species list.

6 In the Species filter field, type water.

7 Select water (7732-18-5, H2O) in the list of available species.

8 Click Add Selected to add the species to the Selected species list.

9 Click Next in the window toolbar.

10 In the Settings window for Select Phase, choose Vapor-liquid from the list of phases to include in the system.


12 In the Settings window for Select Thermodynamic Model, choose UNIFAC VLE from the list.

13 Click Finish in the window toolbar.

C H E M I S T R Y ( C H E M )

1 In the Model Builder window, expand the Property Package 1 (pp1) node, then click Component 1 (comp1)>Chemistry (chem).


2 In the Settings window for Chemistry, locate the Model Inputs section.

3 In the T text field, type Tc.

4 In the p text field, type pRef.



3 In the Species name text field, type EG.

4 On the Physics toolbar, click Domains and choose Species.


6 In the Species name text field, type W.

7 In the Model Builder window, click Chemistry (chem).

8 In the Settings window for Chemistry, click to expand the Calculate thermodynamic properties section.

9 Click to expand the Calculate transport properties section. Locate the Calculate Transport Properties section. Select the Calculate mixture properties check box.

10 Click to expand the Mixture properties section. Locate the Mixture Properties section. Select the Thermodynamics check box.

11 From the Phase list, choose Liquid.

12 Locate the Species Matching section. From the Mixture type list, choose Concentrated species.


Note that when coupled to a Property Package under Thermodynamics, the Chemistry interface automatically creates functions for transport and thermodynamic properties and adds them to the corresponding Property Package. The functions are created when the interface is fully coupled, that is all species in the Chemistry interface has been matched to a corresponding species in the Property Package.

Species Species type Mass fraction

From Thermodynamics Reaction rate

EG Constant, free species

w_EG C2H6O2 Constant

W Constant, free species

w_W H2O Constant


S T U D Y 1

Now, compute and plot the properties of the glycol-water coolant as defined by the Thermodynamics functions.


2 In the Settings window for Study, type Study 1: Mixture properties parameterization in the Label text field.


S T U D Y 1 : M I X T U R E P R O P E R T I E S P A R A M E T E R I Z A T I O N

Step 1: Stationary1 In the Model Builder window, expand the Study 1: Mixture properties parameterization

node, then click Step 1: Stationary.

2 In the Settings window for Stationary, click to expand the Study extensions section.

3 Locate the Study Extensions section. Select the Auxiliary sweep check box.

4 Click Add.

5 Click to select row number 1 in the table.


7 Click Add.


9 From the Sweep type list, choose All combinations.



In the Model Builder window, expand the Global Definitions>Thermodynamics node.

1D Plot Group 1On the Home toolbar, click Add Plot Group and choose 1D Plot Group.


x_EG 0 0.244 1


Tc (Coolant temperature) range(273,10,473)


R E S U L T S

1D Plot Group 1On the 1D Plot Group 1 toolbar, click Global.


Mixture: Density 5 (chempp1rho_mixture)Go to the mixture density function, labeled Mixture: Density 5, that has been added under Property Package 1. Copy the function name chempp1rho_mixture to use it plotting.

R E S U L T S









5 In the associated text field, type Temperature (K).


7 In the associated text field, type Density (kg/m3).

8 Right-click Results>1D Plot Group 1 and choose Rename.

9 In the Rename 1D Plot Group dialog box, type Density in the New label text field.

10 Click OK.

Density 1Right-click Results>Density and choose Duplicate.


chempp1rho_mixture(Tc,pRef,x_EG,x_W)


Mixture: Viscosity 7 (chempp1eta_mixture)Go to the thermal conductivity function, labeled Mixture: Viscosity 7, that has been added under Property Package 1. Copy the function name chempp1eta_mixture to use for plotting.

R E S U L T S

Global 11 In the Model Builder window, expand the Results>Density 1 node, then click Global 1.



4 Click the y-Axis Log Scale button on the Graphics toolbar.

5 On the Density 1 toolbar, click Plot.

Density 11 In the Model Builder window, under Results click Density 1.

2 In the Settings window for 1D Plot Group, locate the Title section.





7 In the associated text field, type Viscosity (Pa*s).

8 Right-click Results>Density 1 and choose Rename.

9 In the Rename 1D Plot Group dialog box, type Viscosity in the New label text field.

10 Click OK.

Viscosity 1Right-click Results>Viscosity and choose Duplicate.


chempp1eta_mixture(Tc,pRef,x_EG,x_W)



Mixture: Density 6 (chempp1k_mixture)Go to the thermal conductivity function, labeled Mixture: Thermal conductivity 6, that has been added under Property Package 1. Copy the function name chempp1k_mixture to use for plotting.

R E S U L T S

Global 11 In the Model Builder window, expand the Viscosity 1 node, then click Global 1.



4 On the Viscosity 1 toolbar, click Plot.

Viscosity 11 In the Model Builder window, under Results click Viscosity 1.

2 In the Settings window for 1D Plot Group, locate the Legend section.

3 From the Position list, choose Middle right.





8 In the associated text field, type Thermal conductivity (W/(m*K)).

9 Right-click Results>Viscosity 1 and choose Rename.

10 In the Rename 1D Plot Group dialog box, type Thermal Conductivity in the New label text field.

11 Click OK.

Thermal Conductivity 1Right-click Results>Thermal Conductivity and choose Duplicate.


chempp1k_mixture(Tc,pRef,x_EG,x_W)



Mixture: Heat capacity 3 (chempp1Cp_mixture)Go to the thermal conductivity function, labeled Mixture: Heat capacity 3, that has been added under Property Package 1. Copy the function name chempp1Cp_mixture to use for plotting.

R E S U L T S

Global 11 In the Model Builder window, expand the Thermal Conductivity 1 node, then click

Global 1.



4 On the Thermal Conductivity 1 toolbar, click Plot.

Thermal Conductivity 11 In the Model Builder window, under Results click Thermal Conductivity 1.






7 In the associated text field, type Heat capacity (J/(kg*K)).

8 Locate the Legend section. From the Position list, choose Upper left.

9 Right-click Results>Thermal Conductivity 1 and choose Rename.

10 In the Rename 1D Plot Group dialog box, type Heat Capacity in the New label text field.

11 Click OK.

Now, use the Thermodynamics node to define an equlibrium function. This will be used to visualize the phase envelope of the coolant mixture.


chempp1Cp_mixture(Tc,pRef,x_EG,x_W)



Property Package 1 (pp1)1 In the Model Builder window, under Global Definitions>Thermodynamics right-click

Property Package 1 (pp1) and choose Equilibrium Calculation.

2 In the Settings window for Select Species, Click Add All.


4 In the Settings window for Equilibrium Specifications, choose mol from the [[basis]] list.

5 Find the Equilibrium conditions subsection. From the First condition list, choose Pressure.

6 From the Second condition list, choose Phase fraction.



Analytic 1 (an1)1 On the Home toolbar, click Functions and choose Global>Analytic.

2 In the Settings window for Analytic, type Phase envelope in the Label text field.

3 In the Function name text field, type T_x_y.

4 Locate the Definition section. In the Expression text field, type Flash1_1_Temperature(p,n,x_EG,x_W).

5 In the Arguments text field, type p, n, x_EG, x_W.

6 Locate the Units section. In the Arguments text field, type Pa,1,mol/mol,mol/mol.

7 In the Function text field, type K.

A D D S T U D Y




4 Click Add Study.

S T U D Y 2


2 In the Model Builder window, under Study 2 click Step 1: Stationary.

3 In the Settings window for Stationary, locate the Study Extensions section.

4 Select the Auxiliary sweep check box.


5 Click Add.



8 Click Add.


10 Click Add.


12 From the Sweep type list, choose All combinations.

13 In the Settings window for Study, type Study 2: Phase envelope parameterization in the Label text field.



Plot the phase envelope, for the two pressures used, as a function of the mole fraction of ethylene glycol.

1D Plot Group 5On the Home toolbar, click Add Plot Group and choose 1D Plot Group.

R E S U L T S

1D Plot Group 51 In the Settings window for 1D Plot Group, locate the Data section.

2 From the Data set list, choose Study 2: Phase envelope parameterization/Solution 2 (sol2).



5 In the associated text field, type Mole fraction ethylene glycol.


x_EG range(0,0.01,1)


n (Phase fraction) 0 1


pRef (Coolant pressure) 1[atm] 2[atm] Pa




8 On the 1D Plot Group 5 toolbar, click Global.




4 Locate the x-Axis Data section. From the Axis source data list, choose x_EG.




3 From the Position list, choose Upper left.


5 In the Rename 1D Plot Group dialog box, type Phase Envelope in the New label text field.

6 Click OK.

Now add the physics interfaces and apply boundary conditions.

A D D P H Y S I C S



3 In the tree, select Fluid Flow>Nonisothermal Flow>Turbulent Flow>Turbulent Flow, k-ε.

4 Find the Physics interfaces in study subsection. In the table, enter the following settings:



T_x_y(pRef, n, x_EG, x_W)

Studies Solve

Study 1: Mixture properties parameterization

Study 2: Phase envelope parameterization


TU R B U L E N T F L O W, K - ε ( S P F )





3 In the Settings window for Inlet, locate the Velocity section.

4 In the U0 text field, type Vel.




In the Model Builder window, under Component 1 (comp1) click Heat Transfer in Fluids (ht).



3 In the Settings window for Inflow, locate the Upstream Properties section.

4 In the Tustr text field, type Tc.

5 In the pustr text field, type pRef.





3 In the Settings window for Heat Flux, locate the Heat Flux section.

4 Click the Heat rate button.

5 In the P0 text field, type P0.

Solid 11 On the Physics toolbar, click Domains and choose Solid.



Add material properties for the solid steel part from Materials.




3 In the tree, select Built-In>Structural steel.


M A T E R I A L S

Structural steel (mat1)1 In the Settings window for Material, locate the Geometric Entity Selection section.

2 In the list, select 1.

3 Click Remove from Selection.



The default behavior in the current physics interfaces is to use properties from the domain material. Here instead apply properties defined by the Chemistry interface. This is in turn coupled to a Property Package node, implying that the automatically created thermodynamics functions will be used.


Fluid Properties 11 In the Settings window for Fluid Properties, locate the Fluid Properties section.



Fluid 11 In the Model Builder window, under Component 1 (comp1)>Heat Transfer in Fluids (ht)

click Fluid 1.







Go to the Chemistry interface to specify that the computed temperature should be used when evaluating thermodynamics functions. The pressure differences are assumed to have an insignificant influence on the results. The pressure input is therefore kept at the reference pressure.


1 In the Model Builder window, under Component 1 (comp1) click Chemistry (chem).


3 From the T list, choose Temperature (nitf1).

Now add a Stationaryl> study to solve for the flow and heat transfer in the test apparatus using pure water as the coolant. Use two stationary study steps. The first step solves for the flow only. This serves as initial conditions for the second step, which in turn solves for both flow and heat transfer.

A D D S T U D Y






S T U D Y 3

Step 1: Stationary1 In the Settings window for Stationary, locate the Study Extensions section.


3 Click Add.



Physics Solve

Heat Transfer in Fluids (ht)


w_EG (Mass fraction, ethylene glycol)

0


6 In the Settings window for Study, type Study 3: Water in the Label text field.

Stationary 2On the Study toolbar, click Study Steps and choose Stationary>Stationary.

S T U D Y 3 : WA T E R

Step 2: Stationary 21 In the Settings window for Stationary, click to expand the Study extensions section.

2 Locate the Study Extensions section. Select the Auxiliary sweep check box.

3 Click Add.



Study step 2 uses the converged solution for the flow as initial conditions. In this case Anderson acceleration can be enabled to reduce the simulation time.


3 In the Model Builder window, expand the Study 3: Water>Solver Configurations>

Solution 3 (sol3)>Stationary Solver 2 node, then click Segregated 1.

4 In the Settings window for Segregated, locate the General section.

5 From the Stabilization and acceleration list, choose Anderson acceleration.


R E S U L T S

Velocity (spf) 1Delete some superfluous plot groups.

1 In the Model Builder window, under Results select the following plot groups 2D Plot

Group 6, 3D Plot Group 7, Isothermal Contours (ht), and Temperature, 3D (ht).

2 Right-click the selection and choose Delete.

Create a 2D plot group for the temperature.



0




3 From the Data set list, choose Study 3: Water/Solution 3 (sol3).

4 On the 2D Plot Group 12 toolbar, click Surface.

Surface 11 In the Model Builder window, under Results>2D Plot Group 12 click Surface 1.




5 Click to expand the Range section. Select the Manual color range check box.

6 In the Maximum text field, type 400.







5 In the associated text field, type r (m).


7 In the associated text field, type Temperature (K) and click Plot.

A D D S T U D Y

Add a new Study to solve for the flow and heat transfer in the test apparatus when using a coolant mixture composed of equal volumes of ethylene glycol and water. Use two study steps; One using the same inlet velocity as for the pure water case, and one where the flow rate of the ethylene/water mixture is increased by 15%.






S T U D Y 4




4 Click Add.


6 Click Add.


Step 2: Stationary 11 Right-click Study 4>Step 1: Stationary and choose Duplicate.





2 In the Settings window for Stationary, click to expand the Values of dependent variables section.

3 Locate the Values of Dependent Variables section. Find the Initial values of variables solved for subsection. From the Settings list, choose User controlled.

4 From the Method list, choose Solution.

5 From the Study list, choose Study 3: Water, Stationary 2.



0.527


Vel (Pipe inlet velocity) 1


Vel 1.15



Apply Anderson acceleration also for the cases using ethylene glycol. Note that the previous solution, using water as coolant, was used for the initial conditions.






6 In the Model Builder window, collapse the Study 4>Solver Configurations>

Solution 5 (sol5)>Stationary Solver 1 node.






11 In the Settings window for Study, type Study 4: Glycol and Water in the Label text field.



S T U D Y 4 : G L Y C O L A N D WA T E R

Solution 5 (sol5)1 In the Model Builder window, under Study 4: Glycol and Water>Solver Configurations>

Solution 5 (sol5) click Solution Store 2 (sol6).

2 In the Settings window for Solution Store, type Vel = 1.0 m/s in the Label text field.

3 In the Model Builder window, click Solution 5 (sol5).

4 In the Settings window for Solution, type Vel = 1.15 m/s in the Label text field.

Plot the temperature for the glycol/water mixture to reproduce the plots in Figure 6.

R E S U L T S

Temperature1 In the Model Builder window, under Results click Temperature.



3 From the Data set list, choose Study 4: Glycol and Water /Vel = 1.0 m/s (sol6).





Create a cut line plot line to evaluate the heat capacity throughout the chamber section of the apparatus.



3 In row Point 1, set R to r_c*0.5.

4 In row Point 2, set R to r_c*0.5.

5 In row Point 2, set Z to 1.

6 Locate the Data section. From the Data set list, choose Study 3: Water/Solution 3 (sol3).

Cut Line 2D 21 Right-click Cut Line 2D 1 and choose Duplicate.



Cut Line 2D 31 Right-click Results>Data Sets>Cut Line 2D 2 and choose Duplicate.




2 On the 1D Plot Group 13 toolbar, click Line Graph.

Line Graph 11 In the Model Builder window, under Results>1D Plot Group 13 click Line Graph 1.

2 In the Settings window for Line Graph, locate the Data section.

3 From the Data set list, choose Cut Line 2D 1.

4 Locate the y-Axis Data section. In the Expression text field, type ht.Cp.







Line Graph 21 Right-click Results>1D Plot Group 13>Line Graph 1 and choose Duplicate.






2 In the Settings window for Line Graph, locate the Legends section.









Legends

Water

Legends

Glycol/Water

Legends

Glycol/Water, Vel = 1.15



6 In the Rename 1D Plot Group dialog box, type Heat Capacity, Chamber Cut Line in the New label text field.

7 Click OK.

Compute the average mixture property values.

Surface Average 11 On the Results toolbar, click More Derived Values and choose Average>Surface Average.

2 In the Settings window for Surface Average, locate the Data section.



5 Locate the Expressions section. In the table, enter the following settings:

6 Click Evaluate.

Store the average property values as parameters.






ht.rho kg/m^3 Density

ht.Cp J/(kg*K) Heat capacity at constant pressure

ht.krr W/(m*K) Thermal conductivity, rr component

spf.mu Pa*s Dynamic viscosity


rhoC 1010[kg/m^3] 1010 kg/m³ Average constant density

CpC 3480[J/kg/K] 3480 J/(kg·K) Average constant heat capacity

kC 0.574[W/m/K] 0.574 W/(m·K) Average constant conductivity

muC 9.11e-4[Pa*s] 9.11E-4 Pa·s Average constant viscosity


C O M P O N E N T 1 ( C O M P 1 )

In the Model Builder window, expand the Component 1 (comp1) node.


Finally compute the flow and heat in the test apparatus using the average values for the mixture properties.

Fluid 11 In the Model Builder window, expand the Component 1 (comp1)>

Heat Transfer in Fluids (ht) node, then click Fluid 1.


3 From the k list, choose User defined. In the associated text field, type kC.

4 Locate the Thermodynamics, Fluid section. From the ρ list, choose User defined. In the associated text field, type rhoC.

5 From the Cp list, choose User defined. In the associated text field, type CpC.


On the Physics toolbar, click Heat Transfer in Fluids (ht) and choose Turbulent Flow, k-

ε (spf).

Fluid Properties 11 In the Model Builder window, expand the Component 1 (comp1)>Turbulent Flow, k-ε (spf)

node, then click Fluid Properties 1.


3 From the μ list, choose User defined. In the associated text field, type muC.

A D D S T U D Y





S T U D Y 5


2 In the Settings window for Stationary, click to expand the Values of dependent variables section.


3 Locate the Values of Dependent Variables section. Find the Initial values of variables solved for subsection. From the Settings list, choose User controlled.


5 From the Study list, choose Study 3: Water, Stationary 2.

6 In the Settings window for Study, type Study 5: Glycol and Water, Constant Properties in the Label text field.



R E S U L T S

Study 5: Glycol and Water, Constant Properties /Solution 7 (sol7)Plot the temperature for the case with average values for the mixture properties.

Temperature1 In the Model Builder window, expand the Study 5: Glycol and Water, Constant Properties>

Solver Configurations node, then click Results>Temperature.


3 From the Data set list, choose Study 5: Glycol and Water, Constant Properties /

Solution 7 (sol7).


Compute the outlet temperature, the average pressure drop, and the average outlet density.

Point Evaluation 11 On the Results toolbar, click Point Evaluation.

2 Select Point 6 only.

3 In the Settings window for Point Evaluation, locate the Expressions section.



6 Click Evaluate.

7 In the Settings window for Point Evaluation, locate the Data section.


T



9 Click New Table.



12 Click New Table.



Solution 7 (sol7).

15 Click New Table.

TA B L E

Go to the Table window.

Line Average 2On the Results toolbar, click More Derived Values and choose Average>Line Average.

R E S U L T S

Line Average 21 Select Boundary 2 only.

2 In the Settings window for Line Average, locate the Expressions section.



5 Click Table 2 - Point Evaluation 1 (T).









p



Solution 7 (sol7).


TA B L E

Go to the Table window.

Line Average 3On the Results toolbar, click More Derived Values and choose Average>Line Average.

R E S U L T S

Line Average 31 Select Boundary 11 only.

2 In the Settings window for Line Average, locate the Expressions section.












Solution 7 (sol7).



ht.rho



Ch em i c a l V apo r Depo s i t i o n o f GaA s




Introduction

This example illustrates the modeling of a reactor for chemical vapor deposition (CVD). CVD is an important process for the electronics industry in which a thin film is grown on a substrate by allowing molecules and molecular fragments to adsorb and react on a surface. Combining detailed chemical reaction kinetics with transport models of a CVD reactor allows for realistic modeling of the deposition process. Such simulations in turn minimize the large number of expensive and time-consuming trial runs typically required for a reactor design.

In the CVD process described here, triethyl-gallium (Ga(C2H5)3) first decomposes into a gas phase. The reaction products, along with arsine (AsH3), then adsorb and react on a substrate to form GaAs layers. The CVD system is modeled using momentum, energy, and mass balances including a detailed description of the gas phase and adsorption kinetics (Ref. 1).

The model highlights the usability of the Reaction Engineering and Chemistry interfaces together with the Reversible Reaction Group feature for simulation of reaction/transport systems in well-mixed (0D) and space-dependent reactors.

In the Reaction Engineering interface you can easily study the transient behavior of different sets of reactions in a perfectly mixed system. The Chemistry interface collects reaction kinetics and calculates transport and thermal parameters, which can seamlessly be coupled with other interfaces. In this application, you also utilize the Reversible Reaction Group feature for CHEMKIN import and organization of the complex system of bulk and surface reactions that are involved in the CVD process. The space-dependent reactor model accounts for mass transport, heat transfer and fluid flow in the CVD reactor using the Transport of Diluted Species, Heat Transfer in Fluids, and Laminar Flow interfaces.

Note: This application requires the Chemical Reaction Engineering Module. and either the Heat Transfer Module or the CFD Module.

Model Definition

C H E M I S T R Y

In this model, the reaction kinetics together with most species transport and thermal properties are imported from CHEMKIN files using the Reversible Reactions Group feature available in either the Reaction Engineering or Chemistry interface. The

2 | C H E M I C A L V A P O R D E P O S I T I O N O F G A A S

CHEMKIN reaction kinetics file includes the following reactions making up the CVD process.

1 The gas phase decomposition of Ga(C2H5)3:

(1)

(2)

(3)

2 Gas phase radical reactions:

(4)

(5)

(6)

(7)

(8)

(9)

3 Growth of GaAs at the surface by the adsorption of gas phase species and the subsequent reaction of the surface-bonded molecular fragments. These surface reactions involve the Ga and As species. SA and SG represent surface sites, corresponding to dangling bonds of As or Ga atoms, respectively.

(10)

(11)

Ga(C2H5)3 C2H5+k1 Ga(C2H5)2

Ga(C2H5)2 Ga(C2H5)H C2H4+k2

Ga(C2H5)H GaH2 C2H4+k3

+ H2 C2H6 H+f

rC2H5

k4

k4

+ 2CH3C2H5 Hk5

+ H2 CH4 +CH3 Hk6

C2H62CH3k7

C2H4 + C2H5k8H

2H2+ H22Hk9

Ga(C2H5)3k10

+ 2C2H5+ SA GaC2H5*

Ga(C2H5)H + SA Ga(C2H5)H*f

r

k11

k11


(12)

(13)

(14)

4 Surface reactions of carbon and hydrogen fragments:

(15)

(16)

(17)

(18)

(19)

(20)

5 Surface reactions leading to GaAs growth:

(21)

(22)

The reaction rates (SI unit: mol/(m3·s)) corresponding to the chemistry just described involve the mass action law

Here, and denote the forward and reverse rate constants, respectively. The concentration of species i is denoted ci (SI unit: mol/m3). The stoichiometric coefficients

GaH2k12

Ga* + 2H+ SA

Ga*Ga(C2H5)H*k13

+ HC2H5+

AsH3k14

As* + 3H+ SG

+ SAC2H5 C2H5A*

f

r

k15

k15

+ SGC2H5 C2H5G*

f

r

k16

k16

k17+C2H5A* C2H4HA*

k18+C2H5G* C2H4HG*

k19+SAHA* H

k20+SGHG* H

k21GaAs + C2H5GaC2H5* As*+ + SA + SG

k22GaAsGa* As*+ + SA + SG

rj kjf ci

ν– j kjr ci

νij

i prod∈∏–

i react∈∏=

kjf kj

r


are denoted νij, and are defined as negative for reactants and positive for products. The temperature dependence of the reaction rates is included through Arrhenius expressions for the rate constants:

In this equation, A denotes the frequency factor, T the temperature (K), n the temperature exponent, E the activation energy (SI unit: J/mol), and Rg the ideal gas constant, 8.314 J/(mol·K). The frequency factor is expressed in the units , where α is the order of the reaction.

With the CHEMKIN import, the chemical species automatically adapts the following labels, where _1(ads) indicates adsorbed surface species and _Ga_ indicates adsorption at gallium (Ga) sites instead of the more common arsenic (As) sites Figure 1.

Figure 1: Species labels used in the model.

k ATn ERgT-----------–

exp=

(m3/mol)α 1–

/s

Ga(C2H5)3

GaC2H5*

Ga(C2H5)H

Ga(C2H5)H*

GaH2

Ga*

AsH3

As*

H2

C2H6

H

C2H5Ga(C2H5)2

C2H4

CH3

CH4

H*

C2H5*

GaAs

GaC6H15

GaC4H10

GaC2H6

GaH2

GaC2H6

GaC2H6

Ga_Ga_1(ads)

AsH3_1(ads)

As_1(ads)

GaAs

C2H6

C2H5

C2H4

CH4

CH3

H2

H

C2H5_1(ads), C2H5_Ga_1(ads)

H_1(ads), H_Ga_1(ads)


M O D E L A N A L Y S I S

The analysis follows these steps: First, study of the reaction kinetics in an ideal batch reactor using the Reaction Engineering interface. Afterwards, setup of a space-dependent model with the following interfaces to investigate the effects of momentum, heat, and mass transport within the system:

• Chemistry

• Transport of Diluted Species

• Heat Transfer in Fluids

• Laminar Flow

Figure 2 shows the CVD reactor model geometry. The reactor is 40 cm long and 10 cm high. Located in the center is the substrate, 5 cm across and tilted 10° with respect to the vertical position. Gas enters the reactor at the inlet with a velocity of 0.4 m/s and at a pressure of 4000 Pa.

Figure 2: The modeling domain consists of the CVD reactor and the substrate surface.

Wall

Wall

OutletInlet Substrate



As noted, the first step in the modeling process is to enter the complete set of gas phase reactions, Equation 1 to Equation 9, into the Reaction Engineering interface for analysis. Figure 3 shows the species concentrations as functions of time in a perfectly mixed batch reactor kept at 900 K.

Figure 3: The complete set of gas phase reactions including decomposition reactions of gallium species as well as radical reactions. The chemistry occurs in a perfectly mixed batch reactor held at 900K. Radical species are not shown in the graph.

As a test, omit the radical reactions given by Equation 4 to Equation 9 from the set of gas phase reactions. Once again analyze the kinetics of the reactions describing gallium species decomposition (Equation 1 to Equation 3) at 900 K. The results appear in Figure 4.


Figure 4: A reduced set of gas phase reactions including only the decomposition reactions of gallium species. Reactions occur in a perfectly mixed system held at 900 K.

Reducing the gas phase reaction set does not affect the reactions of the gallium species. However, excluding the radical reactions has a considerable influence on the carbon-species distribution. For the reduced reaction set, ethene and ethyl radicals are the main carbon products; for the full reaction set the main products are ethene and methane. The various species have different characteristics with respect to surface adsorption and reaction. Furthermore, the net concentration of carbon species is higher for the full reaction set. Both these factors can significantly influence the growth of surface layers. For a first study of geometrical effects on the reacting system, you can bring the reduced reaction model into the actual geometry of the CVD reactor and then solve the space-dependent problem.

The first results from the space-dependent model are displayed below. Figure 5 shows the fluid velocity and Figure 6 the temperature distribution in the reactor domain. The gas mixture enters the reactor with a velocity of 0.4 m/s and a temperature of 300 K with the substrate held at a constant temperature of 900 K. Notice the large effect that the heating plate has on the temperature and the expansion this causes in the fluid. This effect is seen in the average velocity, which increases downstream after the position of the substrate.


Figure 5: The gas phase velocity in the reactor domain.

Figure 6: The temperature distribution in the reactor domain.

In Figure 7 shows the concentration distribution of the triethyl-gallium species in the reactor domain, while Figure 8 displays the concentration profile along the reactor


centerline for triethyl-gallium together with that of the final product gallium hydride. Triethyl-gallium is stable at the inlet temperature (300 K) and then rapidly decomposes near the hot substrate.

Figure 7: Concentration distribution of triethyl-gallium in the reactor domain.


Figure 8: Concentration profiles of triethyl-gallium (blue line) and gallium hydride (green line) along the reactor centerline.

Figure 9 shows the arsine concentration change along the reactor centerline. This species does not decompose in the gas phase. The decrease in concentration at the substrate surface (at the 0 length coordinate) is due to the adsorption of arsine at the surface.


Figure 9: Composition change of arsine along the reactor centerline. Arsine is adsorbed at the substrate surface, which is located at the center of the length scale.

Figure 10 and Figure 11 depict a few of the transport properties calculated in the Chemistry node which are coupled to the physics interfaces of the space-dependent model. Figure 10 shows the diffusivity of triethyl-gallium (bottom) and arsine (top). Figure 11 shows the thermal conductivity of the hydrogen carrier gas. All variables are plotted as functions of temperature.


Figure 10: The diffusivities of triethyl-gallium (bottom) and arsine (top) as functions of temperature.

Figure 11: The thermal conductivity of the hydrogen carrier gas.


Reference

1. N.K. Ingle, C. Theodoropoulos, T.J. Mountziaris, R.M. Wexler, and F.T.J. Smith, “Reaction kinetics and transport phenomena underlying the low-pressure metalorganic chemical vapor deposition of GaAs,” J. Crystal Growth, vol. 167, pp. 543–556, 1996.

Application Library path: Chemical_Reaction_Engineering_Module/Reactors_with_Mass_and_Heat_Transfer/gaas_cvd



N E W





3 Click Add.

4 Click Study.


6 Click Done.






4 Browse to the model’s Application Libraries folder and double-click the file gaas_cvd_parameters.txt.

The 25 reactions describing the GaAs vapor deposition are available in a kinetics CHEMKIN file. Both bulk and surface reactions are present in this process.



Use the Reversible Reaction Group to import the kinetics CHEMKIN file.

Reversible Reaction Group 11 On the Reaction Engineering toolbar, click Reversible Reaction Group.



4 Click Browse.

5 Browse to the model’s Application Libraries folder and double-click the file gaas_cvd_reaction_kinetics.txt.

6 Click Import.

Species Group 1First, investigate the bulk reactions at 900 K.


Reaction Engineering (re)>Species Group 1 node, then click Reaction Engineering (re).


3 In the T text field, type 900[K].

Remove the imported reactions associated with surface reactions and move reaction 9 to the model builder tree.

Reversible Reaction Group 11 In the Model Builder window, under Component 1 (comp1)>Reaction Engineering (re)

click Reversible Reaction Group 1.

2 In the Settings window for Reversible Reaction Group, locate the CHEMKIN Import for Kinetics section.

3 Clear the Import CHEMKIN data check box.

4 Click to expand the Move reaction and species section. Locate the Move Reaction and Species section. In the Move reaction (with the number) from table text field, type 9.

5 Click Create Reaction.

Remove reactions 10 to 25 from the Reaction table by clicking Delete button.

Reaction 9 contains H2 which is the solvent in this process. With the placement of the reaction in the model builder tree, H2 can now be accessed and set as solvent.


Species: H21 In the Model Builder window, under Component 1 (comp1)>Reaction Engineering (re)

click Species: H2.



Initial Values 1Initially, only GaC6H15 and H2 exist in the reactor.




S T U D Y 1


R E S U L T S


2 In the Settings window for 1D Plot Group, type Concentrations full reaction set (re) in the Label text field.

Global 1Select the species concentrations that are plotted in Figure 3.

1 In the Model Builder window, expand the Results>Concentrations full reaction set (re) node, then click Global 1.


comp1.re.c_GaC6H15 - Concentration.

3 Click Add Expression in the upper-right corner of the y-axis data section. From the menu, choose Component 1>Reaction Engineering>comp1.re.c_C2H4 - Concentration.


GaC6H15 c_GaC6H15_init

H2 c_H2_init


4 Click Add Expression in the upper-right corner of the y-axis data section. From the menu, choose Component 1>Reaction Engineering>comp1.re.c_GaH2 - Concentration.


6 Click Add Expression in the upper-right corner of the y-axis data section. From the menu, choose Component 1>Reaction Engineering>comp1.re.c_CH4 - Concentration.


8 Find the Line markers subsection. From the Marker list, choose Cycle.



11 Click the x-Axis Log Scale button on the Graphics toolbar.


Concentrations full reaction set (re)1 In the Model Builder window, under Results click Concentrations full reaction set (re).



4 Click to expand the Axis section. Select the Manual axis limits check box.




Legends

GaC6H15

C2H4

GaH2

C2H6

CH4


8 On the Concentrations full reaction set (re) toolbar, click Plot.

To reduce the model before simulating the process in a 2-dimensional model, study whether it is possible to remove the non-gallium species and reactions and yet obtain approximately the same results.

To do so, modify the existing reaction model by first removing reactions of non-gallium species from the Reversible Reaction Group. Then re-solve the mass balances and compare the results with the full reaction model.

In order not to lose the previous solution, which is to be used for comparison, copy the solution.

S T U D Y 1

Solution 1 (sol1)1 In the Model Builder window, expand the Study 1>Solver Configurations node.

2 Right-click Solution 1 (sol1) and choose Solution>Copy.

Solution 1 - Copy 1 (sol2)1 In the Model Builder window, under Study 1>Solver Configurations right-click Solution 1 -

Copy 1 (sol2) and choose Rename.

2 In the Rename Solution dialog box, type complete_set in the New label text field.

3 Click OK.


Reversible Reaction Group 11 In the Settings window for Reversible Reaction Group, locate the Reaction Table section.



1: 2H+H2=>2H2In the Model Builder window, under Component 1 (comp1)>Reaction Engineering (re)>

1: 2H+H2= right-click 2H2 and choose Disable.

R E S U L T S

Concentrations full reaction set (re)1 In the Model Builder window, expand the Component 1 (comp1)>

Reaction Engineering (re)>Reversible Reaction Group 1 node, then click Results>

Concentrations full reaction set (re).



3 From the Data set list, choose Study 1/complete_set (sol2).

S T U D Y 1


Select the species concentrations that are plotted in Figure 4.

R E S U L T S


2 In the Settings window for 1D Plot Group, type Concentrations reduced reaction set (re) in the Label text field.

Global 11 In the Model Builder window, expand the Results>

Concentrations reduced reaction set (re) node, then click Global 1.


comp1.re.c_GaC6H15 - Concentration.

3 Click Add Expression in the upper-right corner of the y-axis data section. From the menu, choose Component 1>Reaction Engineering>comp1.re.c_GaC4H10 - Concentration.


5 Click Add Expression in the upper-right corner of the y-axis data section. From the menu, choose Component 1>Reaction Engineering>comp1.re.c_GaC2H6 - Concentration.


7 Click Add Expression in the upper-right corner of the y-axis data section. From the menu, choose Component 1>Reaction Engineering>comp1.re.c_GaH2 - Concentration.


9 Find the Line markers subsection. From the Marker list, choose Cycle.




12 Click the x-Axis Log Scale button on the Graphics toolbar.


Concentrations reduced reaction set (re)1 In the Model Builder window, under Results click Concentrations reduced reaction set (re).



4 Locate the Axis section. Select the Manual axis limits check box.




8 On the Concentrations reduced reaction set (re) toolbar, click Plot.

A comparison of Figure 3 and Figure 4 reveals that the gallium-related reactions remain approximately the same. This means that you can go on to set up a space-dependent CVD model based on the reduced model instead of the one comprising all bulk species.

Move on to the space-dependent model. This CVD model is in 2D and you set up the necessary reactions using the Chemistry interface and the Reversible Reaction Group feature. Mass transport, heat transfer, and fluid flow are accounted for with Transport of

Diluted Species, Heat Transfer in Fluids, and Laminar Flow interfaces, respectively.

9 On the Home toolbar, click Component and choose Add Component>2D.

First, draw the 2D geometry.

G E O M E T R Y 1

In the Model Builder window, expand the Concentrations reduced reaction set (re) node, then click Component 2 (comp2)>Geometry 1.

Legends

GaC6H15

GaC4H10

C2H5

GaC2H6

C2H4

GaH2





4 In the Height text field, type 0.1.

5 Locate the Position section. From the Base list, choose Center.



2 In the Settings window for Rectangle, locate the Position section.

3 In the y text field, type -0.025.

4 Locate the Size and Shape section. In the Width text field, type 1e-3.


6 Locate the Rotation Angle section. In the Rotation text field, type -10.


Difference 1 (dif1)1 On the Geometry toolbar, click Booleans and Partitions and choose Difference.


3 In the Settings window for Difference, locate the Difference section.

4 Find the Objects to add subsection. Clear the Active toggle button.

5 Find the Objects to subtract subsection. Select the Active toggle button.


Form Union (fin)On the Geometry toolbar, click Build All.

Select the Chemistry interface and the Reversible Reaction Group feature to set up all necessary reaction kinetics and define some species parameters.

A D D P H Y S I C S



3 In the tree, select Chemical Species Transport>Chemistry (chem).





Reversible Reaction Group 11 On the Physics toolbar, click Domains and choose Reversible Reaction Group.

Aside from CHEMKIN import of reaction kinetics, also use CHEMKIN import of transport and thermal properties. In this manner, several thermal and transport properties available in the Chemistry interface can be utilized in the other interfaces.



4 Click Browse.

5 Browse to the model’s Application Libraries folder and double-click the file gaas_cvd_reaction_kinetics.txt.

6 Click Import.


8 In the Settings window for Chemistry, click to expand the Calculate transport properties section.

9 Locate the Calculate Transport Properties section. Select the Calculate mixture properties check box.

Species Group 11 In the Model Builder window, under Component 2 (comp2)>Chemistry (chem) click

Species Group 1.

2 In the Settings window for Species Group, click to expand the CHEMKIN section.

3 Click Browse.

4 Browse to the model’s Application Libraries folder and double-click the file gaas_cvd_transp.txt.

5 Click Import.

Species Thermodynamics 11 In the Model Builder window, expand the Species Group 1 node, then click

Species Thermodynamics 1.

2 In the Settings window for Species Thermodynamics, click to expand the CHEMKIN import for thermodynamic data section.


3 Locate the CHEMKIN Import for Thermodynamic Data section. Click Browse.

4 Browse to the model’s Application Libraries folder and double-click the file gaas_cvd_thermo.txt.

5 Click Import.

Remove the bulk reactions that were considered redundant in the 0D model investigation of the system.

Reversible Reaction Group 11 In the Model Builder window, under Component 2 (comp2)>Chemistry (chem) click

Reversible Reaction Group 1.

2 In the Settings window for Reversible Reaction Group, locate the CHEMKIN Import for Kinetics section.

3 Clear the Import CHEMKIN data check box.

4 Locate the Reaction Table section. Click to select row number 4 in the table.


Continue with moving the reactions containing surface species to the model builder tree. This enables access to all surface species. Additionally, do the same with the hydrogen reaction (reaction 17).

5 Click to expand the Move reaction and species section. Locate the Move Reaction and Species section. In the Move reaction (with the number) from table text field, type 9.

6 Click Create Reaction.

Repeat this with reactions 10 to 25.

Disable reaction 17 ( Ga_1(ads)+As_Ga_1(ads)=>GaAs ) and select H2 as solvent.

17: Surface: Ga_1(ads)+As_Ga_1(ads)=>GaAsIn the Model Builder window, under Component 2 (comp2)>Chemistry (chem)>

17: Surface: Ga_1(ads)+As_Ga_1(ads)= right-click GaAs and choose Disable.

Species: H21 In the Model Builder window, under Component 2 (comp2)>Chemistry (chem) right-click

Species: H2 and choose Enable.



The surface species concentrations are considered constant. To account for this, lock the concentrations for these.


Species: GaAs1 In the Model Builder window, under Component 2 (comp2)>Chemistry (chem) click

Species: GaAs.



Repeat the same lock operation for GaC2H5_1_surf, GaC2H6_1_surf, Ga_1_surf, As_Ga_1_surf, C2H5_1_surf, C2H5_Ga_1_surf, H_1_surf, and H_Ga_1_surf.

Set the constant (locked) concentrations in the Chemistry interface main node.




7 In the Model Builder window, collapse the Chemistry (chem) node.

C O M P O N E N T 2 ( C O M P 2 )

Add a Transport of Diluted Species interface to model the mass transport of the bulk species diluted in solvent. When available, use the transport parameters from the Chemistry interface.

A D D P H Y S I C S




GaAs Constant, locked 0 cLock

H2 Constant, solvent c_H2_init solvent

As_Ga_1(ads) Constant, locked c_Assurf_Ga cLock

C2H5_1(ads) Constant, locked c_C2H5surf cLock

C2H5_Ga_1(ads) Constant, locked c_C2H5surf_Ga cLock

GaC2H5_1(ads) Constant, locked c_GaC2H5surf cLock

GaC2H6_1(ads) Constant, locked 0 cLock

Ga_1(ads) Constant, locked c_Gasurf cLock

H_1(ads) Constant, locked c_Hsurf cLock

H_Ga_1(ads) Constant, locked c_Hsurf_Ga cLock


3 In the tree, select Chemical Species Transport>Transport of Diluted Species (tds).




In the Settings window for Transport of Diluted Species, click to expand the Dependent variables section.


1 In the Model Builder window, expand the Component 2 (comp2)>Chemistry (chem) node, then click Component 2 (comp2)>Transport of Diluted Species (tds).

2 In the Settings window for Transport of Diluted Species, locate the Dependent Variables section.


4 Click Add Concentration.







cGaC4H10

cGaC4H10

cC2H5

cGaC4H10

cC2H5

cH

cGaC4H10

cC2H5

cH

cC2H4









cGaC4H10

cC2H5

cH

cC2H4

cAsH3

cGaC4H10

cC2H5

cH

cC2H4

cAsH3

cGaH2

cGaC4H10

cC2H5

cH

cC2H4

cAsH3

cGaH2

cGaC2H6

cGaC4H10

cC2H5

cH

cC2H4

cAsH3

cGaH2



Transport of Diluted Species (tds) click Transport Properties 1.


3 In the DcGaC4H10 text field, type chem.D_GaC4H10.

4 In the DcC2H5 text field, type chem.D_C2H5.

5 In the DcH text field, type chem.D_H.

6 In the DcC2H4 text field, type chem.D_C2H4.

7 In the DcAsH3 text field, type chem.D_AsH3.

8 In the DcGaH2 text field, type chem.D_GaH2.





3 From the Selection list, choose All domains.

4 Locate the Reaction Rates section. From the RcGaC4H10 list, choose Rate expression for species GaC4H10 (chem).

5 From the RcC2H5 list, choose Rate expression for species C2H5 (chem).

6 From the RcH list, choose Rate expression for species H (chem).

7 From the RcC2H4 list, choose Rate expression for species C2H4 (chem).

8 From the RcAsH3 list, choose Rate expression for species AsH3 (chem).

9 From the RcGaH2 list, choose Rate expression for species GaH2 (chem).

10 From the RcGaC2H6 list, choose Rate expression for species GaC2H6 (chem).

11 From the RcGaC6H15 list, choose Rate expression for species GaC6H15 (chem).

Flux 1Choose the Flux feature at the substrate surface and set the surface reaction rates to model the deposition on the substrate.

1 On the Physics toolbar, click Boundaries and choose Flux.

cGaC2H6

cGaC6H15


2 Select Boundaries 4–7 only.

3 In the Settings window for Flux, locate the Inward Flux section.

4 Select the Species cGaC4H10 check box.

5 In the N0,cGaC4H10 text field, type chem.Rsurf_GaC4H10.

6 Select the Species cC2H5 check box.

7 In the N0,cC2H5 text field, type chem.Rsurf_C2H5.

8 Select the Species cH check box.

9 In the N0,cH text field, type chem.Rsurf_H.

10 Select the Species cC2H4 check box.

11 In the N0,cC2H4 text field, type chem.Rsurf_C2H4.

12 Select the Species cAsH3 check box.

13 In the N0,cAsH3 text field, type chem.Rsurf_AsH3.

14 Select the Species cGaH2 check box.

15 In the N0,cGaH2 text field, type chem.Rsurf_GaH2.








4 In the c0,cAsH3 text field, type c_AsH3_in.

5 In the c0,cGaC6H15 text field, type c_GaC6H15_in.







3 In the cAsH3 text field, type c_AsH3_in.

4 In the cGaC6H15 text field, type c_GaC6H15_in.

C O M P O N E N T 2 ( C O M P 2 )

Add a Heat Transfer in Fluids interface to model the heat transfer and heat generation in the reactor. When available, use the thermal parameters from the Chemistry interface.

A D D P H Y S I C S



3 In the tree, select Heat Transfer>Heat Transfer in Fluids (ht).





click Fluid 1.









4 From the Q0 list, choose Heat source of reactions (chem).




4 In the T0 text field, type T_in.


4 In the T0 text field, type T_surf.





2 In the Settings window for Initial Values, type T_in in the T text field.

C O M P O N E N T 2 ( C O M P 2 )

Add a Laminar Flow interface to model the fluid flow. When available, use the fluid parameters from the Chemistry interface.

A D D P H Y S I C S



3 In the tree, select Fluid Flow>Single-Phase Flow>Laminar Flow (spf).





2 From the Compressibility list, choose Compressible flow (Ma<0.3).

3 Find the Reference values subsection. In the pref text field, type 0[atm].


Fluid Properties 1.


3 From the ρ list, choose Density (chem).




4 In the U0 text field, type u_in.




4 In the p0 text field, type p_0.


Initial Values 11 In the Model Builder window, under Component 2 (comp2)>Laminar Flow (spf) click

Initial Values 1.


3 In the p text field, type p_0.


Finish the space-dependent model setup by coupling the interfaces.

On the Physics toolbar, click Laminar Flow (spf) and choose Chemistry (chem).

1 In the Model Builder window, under Component 2 (comp2) click Chemistry (chem).



4 From the p list, choose Absolute pressure (spf).

Set the density dependent on both pressure and temperature.

5 Clear the Reference pressure check box.

6 Click to expand the Mixture properties section. Locate the Mixture Properties section. From the Mixture density list, choose User defined.

7 In the ρ text field, type chem.p/R_const/chem.T*chem.M_H2.



Flow Coupling 2 (fc2)1 On the Physics toolbar, click Multiphysics Couplings and choose Global>Flow Coupling.

2 On the Physics toolbar, click Multiphysics Couplings and choose Global>Flow Coupling.

3 In the Settings window for Flow Coupling, locate the Coupled Interfaces section.

4 From the Destination list, choose Heat Transfer in Fluids (ht).

Temperature Coupling 2 (tc2)1 On the Physics toolbar, click Multiphysics Couplings and choose Global>


2 On the Physics toolbar, click Multiphysics Couplings and choose Global>


3 In the Settings window for Temperature Coupling, locate the Coupled Interfaces section.

4 From the Destination list, choose Laminar Flow (spf).

M E S H 1

1 In the Model Builder window, under Component 2 (comp2) click Mesh 1.

2 In the Settings window for Mesh, locate the Mesh Settings section.

3 From the Element size list, choose Finer.

4 Click Build All.

S T U D Y 1


section.


R O O T

Solve the model for stationary conditions by selecting the Stationary study type.

Physics interface

Chemistry: Chemistry (chem)

Transport of Diluted Species: Transport of Diluted Species (tds)

Heat Transfer in Fluids: Heat Transfer in Fluids (ht)

Laminar Flow: Laminar Flow (spf)


A D D S T U D Y



3 Find the Studies subsection. In the Select Study tree, select Preset Studies.

4 Find the Physics interfaces in study subsection. In the table, clear the Solve check box for the Reaction Engineering (re) interface.




S T U D Y 2

Step 1: StationaryOn the Home toolbar, click Compute.

R E S U L T S

Velocity (spf)To create Figure 5, follow these steps:

1 In the Model Builder window, under Results click Velocity (spf).

2 In the Settings window for 2D Plot Group, click to expand the Color legend section.

3 Locate the Color Legend section. From the Position list, choose Bottom.

4 On the Velocity (spf) toolbar, click Plot.


Temperature (ht)To reproduce Figure 6, do the following

1 In the Model Builder window, under Results click Temperature (ht).

2 In the Settings window for 2D Plot Group, locate the Color Legend section.

3 From the Position list, choose Bottom.

4 On the Temperature (ht) toolbar, click Plot.


Concentration (tds)You can reproduce Figure 7 as follows:

1 In the Model Builder window, under Results click Concentration (tds).


2 In the Settings window for 2D Plot Group, type Concentration distribution GaC6H15 in the Label text field.

3 Locate the Color Legend section. From the Position list, choose Bottom.

Surface1 In the Model Builder window, expand the Results>Concentration distribution GaC6H15

node, then click Surface.

2 In the Settings window for Surface, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Component 2>

Transport of Diluted Species>cGaC6H15 - Concentration.


4 On the Concentration distribution GaC6H15 toolbar, click Plot.

In order to produce the remaining figures, illustrating various results along the reactor centerline, use the CutLine2D data set.



3 In row Point 1, set x to -0.2.

4 In row Point 2, set x to 0.2.

5 Click Plot.


2 In the Settings window for 1D Plot Group, type Concentration profiles GaC6H15 and GaH2 in the Label text field.


Line Graph 11 Right-click Concentration profiles GaC6H15 and GaH2 and choose Line Graph.

2 In the Settings window for Line Graph, type GaC6H15 in the Label text field.

3 Click Replace Expression in the upper-right corner of the y-axis data section. From the menu, choose Component 2>Transport of Diluted Species>cGaC6H15 - Concentration.

4 Click Replace Expression in the upper-right corner of the x-axis data section. From the menu, choose Component 2>Geometry>Coordinate>x - x-coordinate.






Line Graph 21 In the Model Builder window, under Results right-click

Concentration profiles GaC6H15 and GaH2 and choose Line Graph.

2 In the Settings window for Line Graph, type GaH2 in the Label text field.

3 Click Replace Expression in the upper-right corner of the y-axis data section. From the menu, choose Component 2>Transport of Diluted Species>cGaH2 - Concentration.





Concentration profiles GaC6H15 and GaH21 In the Model Builder window, under Results click

Concentration profiles GaC6H15 and GaH2.



4 On the Concentration profiles GaC6H15 and GaH2 toolbar, click Plot.



2 In the Settings window for 1D Plot Group, type Concentration profile AsH3 change in the Label text field.

Legends

GaC6H15

Legends

GaH2


4 Click to expand the Title section. From the Title type list, choose Automatic.

Line Graph 11 Right-click Concentration profile AsH3 change and choose Line Graph.


Transport of Diluted Species>cAsH3 - Concentration.

3 Locate the y-Axis Data section. In the Expression text field, type cAsH3-c_AsH3_init.

4 Click to expand the Title section. From the Title type list, choose Manual.

5 In the Title text area, type Change in arsine, AsH3, concentration compared to initial conditions in the reactor.



8 On the Concentration profile AsH3 change toolbar, click Plot.

The Chemistry node calculates the diffusivities, the thermal conductivity, and other fluid properties, including their temperature dependence. Next, plot the diffusivities along the reactor centerline for two of the species as functions of the temperature.


2 In the Settings window for 1D Plot Group, type Diffusivities vs temperature in the Label text field.


Line Graph 11 Right-click Diffusivities vs temperature and choose Line Graph.

2 In the Settings window for Line Graph, type GaC6H15 in the Label text field.

3 Click Replace Expression in the upper-right corner of the y-axis data section. From the menu, choose Component 2>Transport of Diluted Species>tds.Dav_cGaC6H15 -

Average diffusion coefficient.


5 Click Replace Expression in the upper-right corner of the x-axis data section. From the menu, choose Component 2>Heat Transfer in Fluids>Temperature>T - Temperature.




10 On the Diffusivities vs temperature toolbar, click Plot.

Line Graph 21 In the Model Builder window, under Results right-click Diffusivities vs temperature and

choose Line Graph.

2 In the Settings window for Line Graph, type AsH3 in the Label text field.

3 Click Replace Expression in the upper-right corner of the y-axis data section. From the menu, choose Component 2>Transport of Diluted Species>tds.Dav_cAsH3 -

Average diffusion coefficient.







Diffusivities vs temperature1 In the Model Builder window, under Results click Diffusivities vs temperature.



4 Locate the Grid section. Select the Manual spacing check box.

5 In the x spacing text field, type 100.

6 In the y spacing text field, type 1e-3.

7 Click to expand the Legend section. From the Position list, choose Upper left.

8 On the Diffusivities vs temperature toolbar, click Plot.

Legends

GaC6H15

Legends

AsH3



2 In the Settings window for 1D Plot Group, type Thermal conductivity H2 in the Label text field.


Line Graph 11 Right-click Thermal conductivity H2 and choose Line Graph.

2 In the Settings window for Line Graph, click Replace Expression in the upper-right corner of the y-axis data section. From the menu, choose Component 2>Heat Transfer in Fluids>

Material properties>ht.kmean - Mean effective thermal conductivity.




Thermal conductivity H21 In the Model Builder window, under Results click Thermal conductivity H2.



4 On the Thermal conductivity H2 toolbar, click Plot.


2 In the Settings window for 2D Plot Group, type Concentration distribution H in the Label text field.

Surface 11 In the Model Builder window, expand the Results>Concentration distribution H node,

then click Surface 1.


Transport of Diluted Species>cH - Concentration.



I b up r o f e n S y n t h e s i s




Introduction

Kinetic analysis of catalytic reactions is essential for understanding rate behavior as well as the reaction mechanism. Developing knowledge of intrinsic reaction kinetics and of rate equations is central to reaction engineering studies aimed at improving reactor design.

This example illustrates the reaction kinetics of a complex chemistry occurring in a perfectly stirred tank reactor. The homogeneous catalysis of 1-(4-isobutylphenyl) ethanol into the anti-inflammatory drug ibuprofen serves as the example chemistry. The model determines concentrations of reactants, intermediates, and products as functions of time for the network of chemical reactions.

The chemistry in this example involves homogeneous catalysis. As this terminology suggests, the catalyst and the reacting species are in the same phase. Most commonly, a liquid reaction mixture contains a soluble metalorganic complex that affects the catalysis. Organometallic catalysts can often be fine-tuned with respect to reaction activity and selectivity. Because these relatively expensive catalysts produce highly-refined reaction products, they commonly find application in fine chemicals or pharmaceutics.

The model focuses on the use of the Chemical Reaction Engineering Module for a kinetics investigation. You easily enter chemical reaction formulas from the keyboard, then the Reaction Engineering interface automatically generates rate expressions and material balances. It solves the equations, and you postprocess results directly in the COMSOL Desktop.

Model Description

Analyzing chemical kinetics involves solving the set of ordinary differential equations corresponding to individual steps in a network of reactions. This example illustrates the kinetics of ibuprofen synthesis. Figure 1 shows the reaction steps displayed in a catalytic

2 | I B U P R O F E N S Y N T H E S I S

cycle (Ref. 1).

Figure 1: Catalytic cycle of ibuprofen synthesis.

Prior to entering the cycle, the starting material, 1-(4-isobutylphenyl)ethanol, is first dehydrated to form 4-isobutylstyrene. This species subsequently undergoes the addition of HCl to produce the active substrate 1-(4-isobutylphenyl)ethyl chloride. The palladium catalyst must also go through an initial transformation, from L2PdCl2 (L=triphenylphosphine) to anionic L2PdCl, before becoming active. The activated catalyst then assists in the carbonylation and hydrolysis of 1-(4-isobutylphenyl)ethyl chloride, producing ibuprofen.

The following reactions represent the catalytic cycle:

L2PdCl

CO

H2O

OH

L2PdCl2

Cl

L2PdCl2COClPdL2

COOH

H+

H+

HCl

-

-

Cl-


(1)

(2)

(3)

(4)

(5)

(6)

(7)

Reaction 1 involves the dehydration of the reactant alcohol to form the corresponding alkene. Reaction 2 describes the hydrohalogenation of alkene, resulting in the active substrate 1-(4-isobutylphenyl)ethyl chloride. Reaction 3 shows the dehydrohalogenation of the active substrate, assisted by a base, B. Reaction 4 describes the transformation of the precatalytic species L2PdCl2 into the active anionic catalyst L2PdCl. In Reaction 5 the active substrate undergoes oxidative addition to the L2PdCl catalyst. Reaction 6 summarizes the carbonylation, and Reaction 7 describes the hydrolysis of the metalorganic species, leading to the formation of ibuprofen and regeneration of the catalyst.

In order to make species notation more manageable, this example uses the following labels:

OH

+ H+k1

+ H2O+ H+

++ H+ Cl-Clk2

++ H+ Cl- + B+ BCl k3

+ CO + H2OL2PdCl2k4

L2PdCl + +Cl- + CO2

-

2H+

L2PdCl +k5

L2PdCl2

Cl--

COk6

+ Cl-L2PdCl2 + L2PdClCO

-

H2Ok7

+ H+ + L2PdCl+L2PdClCOCOOH

-


Making use of these notations, the reaction rates corresponding to Reaction 1 through Reaction 7 are:

(8)

(9)

(10)

(11)

(12)

(13)

(14)

species abbreviation species abbreviationOH

L2PdCl2

L2PdCl

L2PdCl2

Cl

L2PdClCO

-

-

roh

ren

rhcl

pd1

pd2

pd3

pd4COOH

ibu

r1 k1crohcH=

r2 k2crencHcCl=

r3 k3crhclcB=

r4 k4cpd1cCOcH2O=

r5 k5cpd2crhcl=

r6 k6cpd3cCO=

r7 k7cpd4cH2O=


The Reaction Engineering interface automatically generates these expressions and displays them immediately when you enter the chemical reaction formulas. By default, the software assumes that the chemistry takes place isothermally in a perfectly stirred batch reactor.

Figure 2: A perfectly stirred batch reactor where the reactant alcohol is carbonylated to form ibuprofen by means of palladium catalysis.

With no inflow or outflow from the reactor, the change of species concentrations with time is a function only of the reaction rates:

(15)

The Reaction Engineering automatically generates the mass balance in Equation 15 for each of the species, i, in the reactions, j, accounting for the stoichiometry in the reaction formulas, ν, and solves these equations.

The model investigates two reaction conditions. The first simulation (Case 1) solves for the seven reaction displayed previously. In Case 2 you modify the reaction network with an additional reaction, altering the simulation results. Assume that the reactant alcohol and product ibuprofen (a carboxylic acid) react reversibly, forming an ester:

The results of the two simulations are compared to gain insight in the process implications.

td

dci νijrj

j 1=

nr

=

OH

+ H+k8

+ H2OH+COOH

+k8 O

O+

r

f


Results

C A S E 1

Figure 3 shows the concentration profiles for reactants and products over time.

Figure 3: Species concentrations (mol/m3) as a function of time (s).

Clearly, after approximately two hours the process has run to completion.


C A S E 2

This expansion of the original case adds a reversible reaction between the reactant alcohol and the product ibuprofen to form an ester. Figure 4 shows the concentration transients.

Figure 4: Species concentrations (mol/m3) as a function of time (s).

In the course of the reaction, ester forms as an intermediary product. In order to achieve the same final concentration of ibuprofen for Case 2 as in Case 1, the process must run for at least 12 hours.

In conclusion, this example illustrates the use of the Chemical Reaction Engineering Module for analyzing the kinetics of a complex reaction network. When you enter the chemical-reaction formulas into the physics interface, the Reaction Engineering interface automatically sets up the corresponding rate expressions and material balances. You can modify simulation conditions effortlessly, for instance by activating/deactivating individual reactions or by changing initial conditions.

Reference

1. R.V. Chaudhari, A. Seayad, and S. Jayasree, “Kinetic modeling of homogeneous catalytic processes,” Catalysis Today, vol. 66, pp. 371–380, 2001.


Application Library path: Chemical_Reaction_Engineering_Module/Ideal_Tank_Reactors/ibuprofen_synthesis



N E W





3 Click Add.

4 Click Study.


6 Click Done.


Start by reading in a set of global parameters.




4 Browse to the model’s Application Libraries folder and double-click the file ibuprofen_synthesis_parameters.txt.

First, set up the model for the first case.








3 In the Formula text field, type roh+H+=>ren+H2O+H+.

4 Locate the Rate Constants section. In the kf text field, type kreac_1.



3 In the Formula text field, type ren+H++Cl-=>rhcl.




3 In the Formula text field, type rhcl+B=>ren+H++Cl-+B.




3 In the Formula text field, type pd1+CO+H2O=>pd2+Cl-+2H++CO2.




3 In the Formula text field, type pd2+rhcl=>pd3.




3 In the Formula text field, type pd3+CO=>pd4+Cl-.





3 In the Formula text field, type pd4+H2O=>pd2+H++ibu.






S T U D Y 1

Run case 1 for three hours.



3 In the Times text field, type 0 3600*3.


2 Click Compute.



Save a copy of the solution for case 1.


B cB_0

CO cCO_0

Cl- cClion_0

H+ cHion_0

H2O cH2O_0

pd1 cpd1_0

roh croh_0


Solution 1 - Copy 1 (sol2)1 In the Model Builder window, under Study 1>Solver Configurations click Solution 1 -

Copy 1 (sol2).

2 In the Settings window for Solution, type Solution 1 - Case 1 in the Label text field.

R E S U L T S


2 In the Settings window for 1D Plot Group, type Concentration - Case 1 in the Label text field.

3 Locate the Data section. From the Data set list, choose Study 1/Solution 1 - Case 1 (sol2).


Global 11 In the Model Builder window, expand the Results>Concentration - Case 1 node, then click

Global 1.

2 In the Settings window for Global, click to expand the Coloring and style section.




Legends

roh

H+

ren

H2O

Cl-

rhcl

B

pd1

CO

pd2

CO2

pd3

pd4

ibuprofen

6 On the Concentration - Case 1 toolbar, click Plot.

The plot in the Graphics window should look like that in Figure 3.

Continue with the model for case 2.




3 In the Formula text field, type ibu+roh+H+<=>ester+H2O+H+.

4 Locate the Rate Constants section. In the kf text field, type kfreac_8.

5 In the kr text field, type krreac_8.

S T U D Y 1

Run case 2 for 12 hours.



3 In the Times text field, type 0 3600*12.


Solution 1 (sol1)Save a copy of the solution for case 2.

In the Model Builder window, under Study 1>Solver Configurations right-click Solution 1 (sol1) and choose Solution>Copy.


Copy 1 (sol3).

2 In the Settings window for Solution, type Solution 1 - Case 2 in the Label text field.

R E S U L T S


2 In the Settings window for 1D Plot Group, type Concentration - Case 2 in the Label text field.

3 Locate the Data section. From the Data set list, choose Study 1/Solution 1 - Case 2 (sol3).


Global 11 In the Model Builder window, expand the Results>Concentration - Case 2 node, then click

Global 1.





6 On the Concentration - Case 2 toolbar, click Plot.

Compare the plot in the Graphics window with that in Figure 4.

Legends

roh

H+

ren

H2O

Cl-

rhcl

B

pd1

CO

pd2

CO2

pd3

pd4

ibuprofen

ester


I s o e l e c t r i c S e p a r a t i o n




Introduction

This modeling example applies the Electrophoretic Transport and the Laminar Flow interfaces to model isoelectric separation in a free flow electrophoresis device. A stream containing four different proteins is divided into separated component streams by means of migrative transport in an electric field.

Free flow electrophoresis can be used to separate macromolecules, such as proteins perpendicular to the flow of the carrier fluid. If a pH and potential gradient is applied across the carrier flow, then molecules can be focused along their isoelectric points. The isoelectric point is the pH at which a molecule has zero net charge. The concept of isoelectric focusing is illustrated in Figure 1.

Figure 1: Ampholytic molecules can be focused around their isoelectric point by means of migrative transport in an electric field.

Molecules with a positive net charge travel in the direction of the electric field, along the pH gradient, until they reach the isoelectric point. At this instance the migrative transport

Isoelectric point

Migrative transport

Migrative transport

Electric field

pH gradient

2 | I S O E L E C T R I C S E P A R A T I O N

is switched off as the molecules net charge is zero. Similarly, anionic species travel in the direction opposite of the electric field.

Model Definition

The model geometry is shown in Figure 2. It represents the separation region in an isoelectric focusing chip. A laminar carrier stream transports a mixture of four proteins, one weak acid and one weak base, injected at the bottom of the cell.


The Electrophoretic Transport interface supports ionic species transport by diffusion, convection and migration in electric fields. The mass balance that is solved in the interface is as follows:

In the equation, ci denotes the concentration of species i (SI unit: mol/m3), Di is the diffusion coefficient of species i (SI unit: m2/s), u is the fluid velocity (SI unit: m/s), F refers to Faraday’s constant (SI unit: s·A/mol), φ denotes the electric potential (SI unit: V), zi is the charge number of the ionic species (unitless), and um,i its ionic mobility (SI unit: s·mol/kg). The Electrphoretic Transport interface also adds an equation for the

Outlet boundary

Inlet boundary

Laminar flow profile

Electrode 0 V 0.15 Vheight

∇ Di ci zium,iFci φ∇–∇–( ) u ∇ci⋅+⋅ 0=


charge transport in the electrolyte, based on an assumption of electroneutrality, and a set of chemical equilibria describing the water self- ionization reaction and the dissociation reactions of weak acids and bases.

The fluid flow is set up with a Laminar Flow interface, solving for the Navier-Stokes equations.


Figure 3 shows the pH in the cell. The pH gradients in the x-direction increase towards the outlet.

Figure 3: pH plot.


Figure 4 shows the electrolyte conductivity. The conductivity gets lower close to the electrode surfaces and towards the outlet due to depletion of charge carriers.[

Figure 4: Conductivity


Figure 5 shows the electrolyte potential outside the area between the electrodes the potential gradients are very low, which means that the migrative flux close to the inlet and outlet is small.[

Figure 5: Electrolyte Potential


Figure 6 shows the concentration of Protein 1, which has an isoelectric point of 4.7. The protein is concetrated towards the right electrode.[

Figure 6: Molar concentration of Protein 1.


Figure 7 shows the concentration of Protein 2, with an isoelectric point of 6.1. This protein reaches its maximum outlet concentration somewhere between the center and the right electrode.[

Figure 7: Molar concentration of Protein 2


Figure 8 and Figure 9 show the weak acid and base concentrations, respectively. The weak acid, with a negative average charge, is transported to the right and the base, with a positive average charge, is concentrated transported to the left in the electric field.[

Figure 8: Weak acid concentration.


[

Figure 9: Weak base concentration.


Finally, Figure 10 and Figure 11 show the pH and protein concentration profiles at the outlet. [

Figure 10: pH profile at outlet.


[

Figure 11: Protein concentrations at outlet.

Application Library path: Chemical_Reaction_Engineering_Module/Electrokinetic_Effects/isoelectric_separation


In this tutorial we couple Electrophoretic Transport to Laminar Flow in 2D. Use the Model Wizard to select the space dimension and the physics interfaces. (Studies will be added to the model at a later stage.)


N E W






Electrophoretic Transport (el).

3 Click Add.


5 Click Add.

6 Click Done.






4 Browse to the model’s Application Libraries folder and double-click the file isoelectric_separation_parameters.txt.

G E O M E T R Y 1

Now draw the geometry as a union of three rectangles.



3 In the Width text field, type W.




3 In the Width text field, type W.

4 In the Height text field, type W.

5 Locate the Position section. In the y text field, type -W.


Rectangle 3 (r3)1 Right-click Rectangle 2 (r2) and choose Duplicate.

2 In the Settings window for Rectangle, locate the Position section.


3 In the y text field, type H.



The finished geometry should now look like this:

Explicit Selection 1 (sel1)Add selections for the inlet and outlet boundaries. These will be used later when setting up the physics.

1 On the Geometry toolbar, click Selections and choose Explicit Selection.

2 In the Settings window for Explicit Selection, type Inlet in the Label text field.


4 On the object r2, select Boundary 1 only.


2 In the Settings window for Explicit Selection, type Outlet in the Label text field.



4 On the object r3, select Boundary 3 only.

M A T E R I A L S

To set up flow equations for the electrolyte we will use some material parameters for water from the Material Library.

1 On the Home toolbar, click Windows and choose Add Material from Library.



2 In the tree, select Built-In>Water, liquid.

3 Click Add to Component 1.


E L E C T R O P H O R E T I C TR A N S P O R T ( E L )

Now start setting up the Electrophoretic Transport.

1 In the Settings window for Electrophoretic Transport, locate the Transport Mechanisms section.

2 Select the Convection check box.

Potential 1This cell is operated potentiostatically (at a constant potential).

1 On the Physics toolbar, click Boundaries and choose Potential.


Potential 21 On the Physics toolbar, click Boundaries and choose Potential.


3 In the Settings window for Potential, locate the Electrolyte Potential section.

4 In the φl, bnd text field, type V0.

Protein 1Now add the various species in the electrolyte: 4 proteins, one weak acid and one weak base.

1 On the Physics toolbar, click Domains and choose Protein.

2 In the Settings window for Protein, locate the Protein section.

3 In the Species name text field, type p1.


4 In the Average charge text field, type iep_1-el.pH.

5 Locate the Diffusion and Migration section. In the D text field, type D_p.

Inflow 11 On the Physics toolbar, click Attributes and choose Inflow.



4 Locate the Concentration section. In the c0 text field, type cp_in.


Protein 1In the Model Builder window, under Component 1 (comp1)>Electrophoretic Transport (el) click Protein 1.

Outflow 11 On the Physics toolbar, click Attributes and choose Outflow.



Protein 1In this tutorial we assume that only the isoelectric point varies between the proteins. Therefore you can just duplicate the first protein to create the second protein, and then and change the average charge.

Protein 21 Right-click Protein 1 and choose Duplicate.




Protein 31 Right-click Component 1 (comp1)>Electrophoretic Transport (el)>Protein 2 and choose

Duplicate.





Protein 41 Right-click Component 1 (comp1)>Electrophoretic Transport (el)>Protein 3 and choose

Duplicate.




Weak Acid 1Finish the electrophoretic transport settings by adding one weak acid and one weak base.

1 On the Physics toolbar, click Domains and choose Weak Acid.

2 In the Settings window for Weak Acid, locate the Weak Acid section.

3 In the Species name text field, type wa.

4 In the pKa text field, type pKa_wa.

5 Locate the Diffusion and Migration section. In the um text field, type mob_wa.

Initial Concentration 11 In the Model Builder window, expand the Weak Acid 1 node, then click

Initial Concentration 1.

2 In the Settings window for Initial Concentration, locate the Initial Concentration section.

3 In the c text field, type cwa_in.

Weak Acid 1In the Model Builder window, under Component 1 (comp1)>Electrophoretic Transport (el) click Weak Acid 1.




4 Locate the Concentration section. In the c0 text field, type cwa_in.

Inflow 11 In the Model Builder window, under Component 1 (comp1)>

Electrophoretic Transport (el)>Weak Acid 1 click Inflow 1.

2 In the Settings window for Inflow, locate the Boundary Condition Type section.

3 From the list, choose Flux (Danckwerts).


Weak Acid 1In the Model Builder window, under Component 1 (comp1)>Electrophoretic Transport (el) click Weak Acid 1.




Weak Base 11 On the Physics toolbar, click Domains and choose Weak Base.

2 In the Settings window for Weak Base, locate the Weak Base section.

3 In the Species name text field, type wb.

4 In the pKa text field, type pKa_wb.

5 Locate the Diffusion and Migration section. In the um text field, type mob_wb.

Initial Concentration 11 In the Model Builder window, expand the Weak Base 1 node, then click

Initial Concentration 1.

2 In the Settings window for Initial Concentration, locate the Initial Concentration section.

3 In the c text field, type cwb_in.

Weak Base 1In the Model Builder window, under Component 1 (comp1)>Electrophoretic Transport (el) click Weak Base 1.




4 Locate the Concentration section. In the c0 text field, type cwb_in.

Weak Base 1In the Model Builder window, under Component 1 (comp1)>Electrophoretic Transport (el) click Weak Base 1.






Now set up the Laminar Flow interface. Only the boundary conditions need to be specified here since most settings are taken from the Materials node (Water), o.

1 In the Model Builder window, under Component 1 (comp1) click Laminar Flow (spf).




4 Locate the Velocity section. In the U0 text field, type Uave.





The velocity used in the Electrophoretic Transport interface is specified using Flow Coupling multiphysics node.


M E S H 1

Now set up the mesh. A mapped mesh is suitable since we are using a rectangular geometry.


Mapped.






Size1 In the Model Builder window, under Component 1 (comp1)>Mesh 1 click Size.



4 Locate the Element Size Parameters section. In the Maximum element size text field, type 5e-4.

5 Click Build All.

Your finished mesh should look like this:

R O O T

The model is now ready for solving. In this model the flow is not affected the physics of the Electrophoretic Transport interface. Therefore solve the Laminar Flow first in a separate study.

1 On the Home toolbar, click Windows and choose Add Study.

A D D S T U D Y







S T U D Y 1

1 In the Settings window for Study, type Study 1 - Flow Calculation in the Label text field.


R E S U L T S

Velocity (spf)Default plots for the velocity and pressure are generated automatically as follows:

Physics Solve

Electrophoretic Transport (el)


Pressure (spf)

R O O T

Add a second study to solve for the Electrophoretic Transport, using the solution of the first Study as input for the velocity (and pressure).

1 Click Windows and choose Add Study.

A D D S T U D Y


2 Find the Studies subsection. In the Select Study tree, select Preset Studies.


4 Find the Studies subsection. In the Select Study tree, select Preset Studies>

Stationary with Initialization.



Physics Solve

Laminar Flow (spf)


S T U D Y 2

In the Settings window for Study, type Study 2 - Separation Calculation in the Label text field.

S T U D Y 2 - S E P A R A T I O N C A L C U L A T I O N

Step 1: Current Distribution Initialization1 In the Model Builder window, under Study 2 - Separation Calculation click

Step 1: Current Distribution Initialization.

2 In the Settings window for Current Distribution Initialization, click to expand the Values of dependent variables section.

3 Locate the Values of Dependent Variables section. Find the Values of variables not solved for subsection. From the Settings list, choose User controlled.


5 From the Study list, choose Study 1 - Flow Calculation, Stationary.


R E S U L T S

pH (el)A default pH plot should now have been generated, looking like this:


Electrolyte Conductivity (el)The default electrolyte conductivity plot should look like this:

Electrolyte Potential (el)The default electrolyte potential plot should look like this:


Molar Concentration - p1 (el)There should also be six default Molar Concentration plots present; one for each species in the Electrophoretic Transport interface:


Molar Concentration - p2 (el)




Molar Concentration - wa (el)


Molar Concentration - wb (el)

1D Plot Group 12Now proceed to plot the pH at the outlet.

1 On the Home toolbar, click Add Plot Group and choose 1D Plot Group.

2 In the Settings window for 1D Plot Group, type pH at Outlet in the Label text field.

3 Locate the Data section. From the Data set list, choose Study 2 - Separation Calculation/

Solution 2 (sol2).

Line Graph 11 Right-click pH at Outlet and choose Line Graph.

2 In the Settings window for Line Graph, locate the Selection section.



5 In the Expression text field, type x.


6 On the pH at Outlet toolbar, click Plot.

The pH plot should look as follows:

1D Plot Group 13Finally, create plot of the protein concentrations at the outlet as follows:


2 In the Settings window for 1D Plot Group, type Protein Concentrations at Outlet in the Label text field.

3 Locate the Data section. From the Data set list, choose Study 2 - Separation Calculation/

Solution 2 (sol2).

Line Graph 11 Right-click Protein Concentrations at Outlet and choose Line Graph.



4 Click Replace Expression in the upper-right corner of the y-axis data section. From the menu, choose Component 1>Electrophoretic Transport>Protein 1>el.c_p1 -

Concentration.







10 On the Protein Concentrations at Outlet toolbar, click Plot.

Line Graph 21 Right-click Results>Protein Concentrations at Outlet>Line Graph 1 and choose Duplicate.


3 In the Expression text field, type el.c_p2.










Protein Concentrations at Outlet1 In the Model Builder window, under Results click Protein Concentrations at Outlet.


Legends

Protein 1

Legends

Protein 2

Legends

Protein 3

Legends

Protein 4



4 Locate the Legend section. From the Position list, choose Lower middle.

5 On the Protein Concentrations at Outlet toolbar, click Plot.

The plot should look like this:



L am i n a r S t a t i c M i x e r




Introduction

In static mixers, also called motionless or in-line mixers, a fluid is pumped through a pipe containing stationary blades. This mixing technique is particularly well suited for laminar flow mixing because it generates only small pressure losses in this flow regime. This example studies the flow in a twisted-blade static mixer.

Model Definition

This model studies the mixing of one species dissolved in water at room temperature. The geometry consists of a tube with three twisted blades of alternating rotations (Figure 1).

Figure 1: Depiction of a laminar static mixer containing three blades with alternating rotations.

The tube’s radius, R, is 3 mm; the length is 14R, and the length of each blade is 3R. The inlet flow is laminar with an average velocity of 10 mm/s. At the outlet, the model specifies a constant reference pressure of 0 Pa. The Laminar Flow interface is used in 3D and solves the Navier-Stokes equations:

(1)

Here μ denotes the dynamic viscosity (SI unit: kg/(m·s)), u is the velocity (SI unit: m/s), ρ represents the fluid density (SI unit: kg/m3), and p denotes the pressure (SI unit: Pa).

Outlet

Inlet

ρ u ∇⋅( )u ∇ pI– μ ∇u ∇u( )T+( )+[ ]⋅=

∇ u⋅ 0=

2 | L A M I N A R S T A T I C M I X E R

These fluid flow properties are not affected by any change in concentration of the dissolved species and are imported from the Material Library.

The species transport, which is defined in the Transport of Diluted Species interface, consists of convection and diffusion. At the outlet an Outflow boundary condition is used to prescribe vanishing diffusion in the normal direction. At the inlet, a discontinuous concentration profile is assumed. The inlet concentration is defined as:

(2)

where the line x = 0 separates the two inlet sides. Because the convective term leads to instabilities in the solution, a fine mesh is required to obtain a stable solution for the concentration field.

The Reynolds numbers in the mixer based on the average velocity and the diameter of the pipe is about 60. This indicates that the flow is laminar and fluid flow does not (Equation 1) do not require a particularly dense mesh near the walls. The Peclet number for the mass transport on the other hand is significantly higher.

This means that the concentration gradient will be thinner than the shear layers in the flow. Consequently a higher resolution is needed for the mass transport compared to that for fluid flow. Since the concentration does not influence the fluid flow, you can therefore first solve the Navier-Stokes equations on a coarse mesh, and then map the solution onto a finer mesh and solve for the mass transport. In this model the resolution is further increased by using second order elements for the species concentration.

Results

Figure 2 shows a slice plot of the concentration in the mixer. The slice at the bottom shows the blue and red halves of the fluid with and without the dissolved species, respectively. As the fluid flows upward through the system, the two solutions are mixed producing a more homogeneous concentration profile at the outlet.

cinletc0 x 0<

0 x 0≥

=

Peudp

D---------- 1200= =

Figure 2: Slice plot of the concentration at different distances from the inlet.


Figure 3 shows the flow field responsible for the mixing. The streamlines clearly reveal the twisting motion in the fluid that is induced by the mixer blades.

Figure 3: Slice plots of the velocity magnitude field inside the mixer. The streamlines show the flow direction.

You can also visualize the mixing through a series of cross-section plots. Figure 4 contains such a series of plots showing the concentration in the mixer’s cross section along the


direction of the flow. The results show that most of the mixing takes place where the blades change rotational direction (the three middle figures).

Figure 4: Cross-sectional plots of the concentration at different distances from the inlet. The nine plots shows the concentration at z =- 2 mm to z = 30 mm in steps of 4 mm.

References

1. R. Perry and D. Green, Perry’s Chemical Engineering Handbook, 7th ed., McGraw-Hill, 1997.

2. J.M. Coulson and J.F. Richardson, Chemical Engineering, vol. 1, 4th ed., Pergamon Press, 1990.

Application Library path: Chemical_Reaction_Engineering_Module/Mixing_and_Separation/laminar_static_mixer




N E W





3 Click Add.



5 Click Add.

6 Click Study.


8 Click Done.

G E O M E T R Y 1






4 Browse to the model’s Application Libraries folder and double-click the file laminar_static_mixer_parameters.txt.

A Step function is required to create a discontinuous boundary condition at the inlet of the mixer.

Step 1 (step1)1 On the Home toolbar, click Functions and choose Global>Step.


3 In the To text field, type 5.


4 Click to expand the Smoothing section. In the Size of transition zone text field, type 3e-4.

Create the geometry. To simplify this step, insert a prepared geometry sequence. On the Geometry toolbar, click Insert Sequence. Browse to Application Library folder and double-click the file laminar_static_mixer.mph. Then click Build all on the Geometry toolbar.







M A T E R I A L S

Water, liquid (mat1)The first material you add applies to all domains by default, so you do not need to change any settings.


In the Laminar Flow interface, set Inlet and outlet boundary conditions together with the initial values.





5 Locate the Laminar Inflow section. In the Uav text field, type u_av.

This gives a parabolic inlet velocity profile appropriate for fully developed laminar flow with mean velocity u_av.



Apply quadratic basis functions for the concentration to further increase the resolution.





3 In the Settings window for Transport of Diluted Species, click to expand the Discretization section.

4 From the Concentration list, choose Quadratic.

In the Transport of Diluted interface, apply inlet and outlet boundary conditions together with initial values. Use the velocity from the Laminar Flow interface as the for the convective transport.




3 In the Dc text field, type D.



Select Inflow at the inlet, and use the Step function to put a discontinuous concentration there.


3 In the c0,c text field, type step1(x[1/mm]).




The Outflow feature assumes that the flow through the outlet is governed by the convection.




Two meshes are required: A course one for the Laminar Flow interface and a finer one for the Transport of Diluted Species interface.

M E S H 1


2 From the Element size list, choose Extra coarse.

3 Click Build All.

C O M P O N E N T 1 ( C O M P 1 )

Mesh 2On the Mesh toolbar, click Add Mesh.

M E S H 2

Size1 In the Model Builder window, under Component 1 (comp1)>Meshes right-click Mesh 2 and

choose Free Tetrahedral.



4 Locate the Element Size Parameters section. In the Maximum element size text field, type 0.65.

5 In the Minimum element size text field, type 0.35.

6 Click Build All.

The study needs to be solved in two Stationary steps. First, the Laminar Flow is computed with Mesh 1. Second, the Transport of Diluted Species is calculated for Mesh 2.

S T U D Y 1

Step 1: Stationary1 On the Study toolbar, click Study Steps and choose Stationary>Stationary.


3 In the table, clear the Solve for check box for the Transport of Diluted Species interface.

Step 2: Stationary 21 In the Model Builder window, under Study 1 click Step 2: Stationary 2.


3 In the table, clear the Solve for check box for the Laminar Flow interface.



R E S U L T S

Velocity (spf)To reproduce Figure 2 that visualizes a slice plot of the concentration within the mixer, follow these steps.

Slice1 In the Model Builder window, expand the Results>Concentration (tds) node, then click

Slice.


3 From the Plane list, choose XY-planes.



Since quadratic elements was applied for the concentration, the resolution of the plot can be increased.

6 Click to expand the Quality section. From the Resolution list, choose Finer.

7 On the Concentration (tds) toolbar, click Plot.

Velocity (spf)To create Figure 3 that displays the velocity field, follow these steps.

1 In the Model Builder window, expand the Results>Velocity (spf) node.

Slice1 Right-click Velocity (spf) and choose Streamline.




Streamline 11 In the Model Builder window, under Results>Velocity (spf) click Streamline 1.



4 In the Min distance text field, type 0.025.

5 In the Max distance text field, type 0.1.

6 Locate the Coloring and Style section. From the Line type list, choose Tube.


7 In the Tube radius expression text field, type 0.05.

8 From the Color list, choose Yellow.

9 Select the Radius scale factor check box.

10 In the associated text field, type 2.


Finally, reproduce the series of cross-sectional concentration plots for different z-coordinates shown in Figure 4 with the following steps.

Cut Plane 11 On the Results toolbar, click Cut Plane.

2 In the Settings window for Cut Plane, locate the Plane Data section.


4 In the Z-coordinate text field, type -2.


2 In the Settings window for 2D Plot Group, type Cross-sectional concentration plot in the Label text field.

3 Locate the Data section. From the Data set list, choose Cut Plane 1.

Surface 11 Right-click Cross-sectional concentration plot and choose Surface.




Increase the resolution also for this plot.

4 Click to expand the Quality section. From the Resolution list, choose Finer.

5 On the Cross-sectional concentration plot toolbar, click Plot.

Cut Plane 11 In the Model Builder window, under Results>Data Sets click Cut Plane 1.


3 In the Z-coordinate text field, type 2.


Cross-sectional concentration plotRepeat these steps for z-coordinate 6, 10, 14, 18, 22, 26, and 30 to reproduce the remaining plots in Figure 4.























Concentration (tds) 1In the Model Builder window, under Results right-click Concentration (tds) 1 and choose Delete.



Hyd r od e a l k y l a t i o n i n a Memb r an e R e a c t o r




IntroductionAt high temperatures and pressures, and in the presence of hydrogen, toluene can be demethylated to produce benzene. Furthermore, benzene can react reversibly to produce biphenyl. The following example illustrates the simulation of the hydrodealkylation process, carried out in a membrane reactor. This reactor arrangement allows for continuous addition of hydrogen to the process, increasing the selectivity for the desired benzene product.

The example shows how you can easily modify the predefined Plug Flow reactor type in the Reaction Engineering interface to set up a membrane reactor model. You also learn how to create a Property Package from Thermodynamics to get different thermodynamic and physical property functions for each compound and their mixture. After all species in Reaction Engineering have been coupled to a corresponding species in a Property Package, the required species and mixture properties will automatically be created and added to the package.

Model Definition

Two important reactions occur in the thermal hydrodealkylation (HDA) of toluene. The main reaction involves toluene reacting with hydrogen to produce benzene and methane:

(1)

The dealkylation reaction rate is first order in the toluene concentration and half order in the hydrogen concentration:

At the same time, biphenyl is reversibly formed from benzene:

(2)

The rate of the coupling reaction follows the mass action law:

k1f

CH4+

CH3

+ H2

r1 k1cC7H8 cH2=

k2r

k2f

2 + H2

r2 k2f cC6H6

2 k2r cH2cC12H10–=

2 | H Y D R O D E A L K Y L A T I O N I N A M E M B R A N E R E A C T O R

In the above rate expressions, the rate constants follow Arrhenius type behavior:

The values of the frequency factors and activation energies (J/mol) are taken from the literature (Ref. 1 and Ref. 2) and are presented in Table 1.

The chemical reactions given in Equation 1 and Equation 2 suggest that maintaining high concentrations of hydrogen would be beneficial to ensure a high benzene yield. Such process conditions can be achieved using a membrane reactor. As illustrated schematically in Figure 1, hydrogen can be supplied continuously across the porous membrane.

Figure 1: Hydrogen is continuously supplied to the reactor through a porous membrane.

The species mass balance for hydrogen in the membrane reactor is given by:

where F is the molar flow rate (SI unit: mol/s) in the reactor, V is the reactor volume (SI unit: m3), R is the species rate expression (SI unit: mol/(m3·s)), and f is the molar flow rate per unit volume (SI unit: mol/(m3·s)) across the membrane. The velocity of the hydrogen gas across the porous membrane can be described by Darcy’s law:

where K (SI unit: m3/(N·s)) is a proportionality constant, pshell (SI unit: Pa) is the gas pressure on the shell side of the membrane, and preactor (SI unit: Pa) the pressure on the reactor side. The molar flow rate per unit volume across the membrane then becomes:

TABLE 1: ARRHENIUS PARAMETERS

FREQUENCY FACTOR ACTIVATION ENERGY

Forward reaction 1 5.67e9 228.2e3

Forward reaction 2 1e8 167.5e3

Reverse reaction 2 1e8 149.8e3

k Ae

ERgT----------–

=

C6H5CH3 + H2

H2 C6H6 + CH4

C12H10

dFH2dV-------------- RH2 fH2+=

u K pshell preactor–( )=


Above, a is the membrane surface area per unit volume (SI unit: m2/m3), and cshell is the concentration of hydrogen (SI unit: mol/m3) on the shell side.

Except for hydrogen, the other chemical species in the reactor do not pass through the membrane and their material balances thus follow the standard plug flow equations:

(3)

The adiabatic energy balance for the reactor is given by:

(4)

In Equation 4, Cmix represents the mixture (reacting system) molar heat capacity (SI unit: J/(mol·K)), and Q denotes the heat due to chemical reaction (SI unit: J/(m3·s)):

where Hj is the heat produced by reaction j, calculated from:

(5)

In Equation 5 hi represents the species partial molar enthalpy (SI unit: J/mol) and νij the stoichiometric coefficients.

The last term in the energy balance accounts for the energy transfer associated with the flow of hydrogen across the membrane:

The Reaction Engineering interface automatically sets up and solves Equation 3 and Equation 4 when you select the predefined Plug flow reactor type. To adjust the default model to account for hydrogen entering the reactor through the membrane, the flow term fH2 has to be specified and included into the H2 material balance.

Solving the energy balance, Equation 4, requires the input of mixture molar heat capacities Cmix (SI unit: J/(mol·K)), and the partial molar enthalpies, hi (SI unit: J/mol), of the

fH2 uacshell=

dFidV--------- Ri=

FiCmixdTdV--------

i Q Qmem+=

Q Hjrj

j–=

Hj νijhj

i=

Qmem fH2hH2=


reacting species. In this example, these thermodynamic properties are calculated from the property package, accessed using the Thermodynamics feature.


In a first simulation, the reactor is assumed to be a standard tubular reactor, that is, without hydrogen entering through the reactor circumference. The reactor is fed with equal molar flows (10 mol/s) of hydrogen and toluene. At the inlet the reactant gas is held at 1200 K and 2 atmospheres. The result is shown in Figure 2.

Figure 2: Molar flow rates (mol/s) as function of reactor volume (m3) for a tubular reactor design.


A second model simulates the membrane reactor, with a continuous supply of hydrogen through the membrane. Figure 3 shows the corresponding concentration distributions.

Figure 3: Molar flow rates (mol/s) as function of reactor volume (m3) for a membrane reactor design with a continuous supply of hydrogen.

Clearly, the membrane reactor produces benzene with greater selectivity.

This example is summarized in how to model a membrane reactor by modifying the predefined reactor equations for the Plug flow reactor type. Furthermore, calculations of thermodynamic properties in the model are performed by a property pack under Thermodynamics.

References

1. K.C. Hou and H.B. Palmer, “The Kinetics of Thermal Decomposition of Benzene in a Flow System,” J. Phys. Chem., vol. 69, no. 3, pp. 863–868, 1965.

2. S.E. Shull and A.N. Hixson, I&EC Process Design and Development, vol. 5, p. 147, 1966.


Application Library path: Chemical_Reaction_Engineering_Module/Thermodynamics/membrane_hda



N E W





3 Click Add.

4 Click Study.


6 Click Done.


In this model, a Property Package will be created from Thermodynamics. The Property

Package includes all thermodynamic properties (such as enthalpy, entropy, etc) which are needed in the reacting system. The properties from Thermodynamics are coupled to Reaction Engineering automatically.

1 In the Model Builder window, under Component 1 (comp1) right-click Reaction Engineering (re) and choose Thermodynamics.


1 On the Reaction Engineering toolbar, click Thermodynamics and choose Property Package.

Select species hydrogen, methane, benzene, toluene and biphenyl.

2 In the Model Builder window, under Global Definitions click Thermodynamics.

3 In the Settings window for Select Species, select hydrogen (1333-74-0, H2) in the Species list.


4 Click Add Selected.

5 In the Species list, select methane (74-82-8, CH4).


7 In the Species list, select benzene (71-43-2, C6H6).


9 In the Species list, select toluene (108-88-3, C7H8).


11 In the Species list, select biphenyl (92-52-4, C12H10).



Select phases Vapor-liquid.

14 In the Settings window for Select Phase, choose Vapor-liquid from the list.



Property Package 1 (pp1)Create a mixture function for enthalpy of formation to introduce a heat source used later in Reaction Engineering.

1 In the Model Builder window, under Global Definitions>Thermodynamics right-click Property Package 1 (pp1) and choose Mixture Property.

2 In the Settings window for Select Properties, select Enthalpy of formation (J/mol) in the list.



5 In the Settings window for Select Species, Click Add All.




Mixture: Enthalpy of formation 1 (EnthalpyF_benzene_biphenyl_hydrogen_methane_toluene_Vapor11)Change the function name to hF_mixture.


1 In the Model Builder window, under Global Definitions>Thermodynamics>

Property Package 1 (pp1) click Mixture: Enthalpy of formation 1 (EnthalpyF_benzene_biphenyl_hydrogen_methane_toluen

e_Vapor11).

2 In the Settings window for Mixture Property, type hF_mixture in the Function name text field.

Load model parameters by importing their definitions from a text file.




4 Browse to the model’s Application Libraries folder and double-click the file membrane_hda_parameters.txt.


Add variable definitions from a text file.


choose Variables.



4 Browse to the model’s Application Libraries folder and double-click the file membrane_hda_variables.txt.

Note that some expressions use functions in Thermodynamics and variables defined in the Reaction Engineering interface. These variables need to be specified with the scope of the Reaction Engineering node. For example, re.T specifies the temperature variable defined by the Reaction Engineering node with the identifier re.







5 In the Q text field, type Q_mem.

This accounts for the heat transferred into the reactor due to the flow across the membrane.

6 Click to expand the Mixture properties section. Locate the Mixture Properties section. In the p text field, type p_reactor.



3 In the Formula text field, type C6H5CH3+H2=>C6H6+CH4.



6 In the Ef text field, type 228.2e3.


8 In the r text field, type re.kf_1*re.c_C6H5CH3*(re.c_H2/1[mol/m^3])^0.5*1[mol/m^3].

Change the default kinetic expression by modifying the reaction order for hydrogen.



3 In the Formula text field, type 2C6H6<=>C12H10+H2.




7 In the Ar text field, type 1e8.

8 In the Er text field, type 149.8e3.

Additional Source 1You will now add an Additional Source feature to Reaction Engineering to the membrane reactor model.

1 On the Reaction Engineering toolbar, click Additional Source.


The f_H2 term corresponds to the flow of hydrogen across the membrane.

Initial Values 1The plug flow reactor requires you to input the inlet molar flow.



3 In the T0,in text field, type T_inlet.


Couple all species in Reaction Engineering to corresponding species in the created Property Package. For a coupled species, its thermodynamic properties will be set automatically to the coupled property package. When all species in Reaction Engineering are coupled (fully coupled), the properties of the reacting system (heat capacity, molar volume, etc) will be calculated from the property package.



7 Locate the Mixture Properties section. Select the Thermodynamics check box.

8 Locate the Species Matching section. In the table, enter the following settings:


H2 f_H2


C6H5CH3 10

H2 10

Species From Thermodynamics

C12H10 C12H10

C6H5CH3 C7H8

C6H6 C6H6

CH4 CH4

H2 H2


In the present example the total reactor volume is 1m3, so default solver settings can be used.

S T U D Y 1

You have now set up a model for a nonisothermal tubular reactor. For the tubular reactor, the Additional Source feature is disabled. First solve this model and review the results, then move on to study the related membrane reactor model with the Additional Source feature.

1 In the Model Builder window, right-click Study 1 and choose Rename.

2 In the Rename Study dialog box, type Tubular reactor in the New label text field.

3 Click OK.

TU B U L A R R E A C T O R

Step 1: Stationary Plug Flow1 In the Model Builder window, expand the Tubular reactor node, then click

Step 1: Stationary Plug Flow.

2 In the Settings window for Stationary Plug Flow, locate the Physics and Variables Selection section.

3 Select the Modify model configuration for study step check box.


Reaction Engineering (re)>Additional Source 1.

5 Click Disable.


R E S U L T S

Molar Flow Rate (re)1 In the Model Builder window, under Results right-click Molar Flow Rate (re) and choose

Rename.

2 In the Rename 1D Plot Group dialog box, type Molar Flow Rate, tubular reactor in the New label text field.

3 Click OK.

4 In the Settings window for 1D Plot Group, click to expand the Legend section.

5 From the Position list, choose Upper middle.

6 On the Molar Flow Rate, tubular reactor toolbar, click Plot.



The default plot shows the molar flow rates of all species as a function of the reactor volume.

Temperature (re)1 In the Model Builder window, under Results right-click Temperature (re) and choose

Rename.

2 In the Rename 1D Plot Group dialog box, type Temperature (re), tubular reactor in the New label text field.

3 Click OK.



6 On the Temperature (re), tubular reactor toolbar, click Plot.


R O O T

On the Home toolbar, click Windows and choose Add Study.

A D D S T U D Y


2 Find the Studies subsection. In the Select Study tree, select Preset Studies>

Stationary Plug Flow.


S T U D Y 2

1 In the Model Builder window, right-click Study 2 and choose Rename.

2 In the Rename Study dialog box, type Membrane reactor in the New label text field.

3 Click OK.

Note: the Additional Source feature is included in membrane reactor model.


R E S U L T S

Molar Flow Rate (re)1 In the Model Builder window, under Results right-click Molar Flow Rate (re) and choose

Rename.


2 In the Rename 1D Plot Group dialog box, type Molar Flow Rate, membrane reactor in the New label text field.

3 Click OK.

4 In the Settings window for 1D Plot Group, click to expand the Legend section.


6 On the Molar Flow Rate, membrane reactor toolbar, click Plot.


Temperature (re)1 In the Model Builder window, under Results right-click Temperature (re) and choose

Rename.

2 In the Rename 1D Plot Group dialog box, type Temperature (re), membrane reactor in the New label text field.

3 Click OK.



6 On the Temperature (re), membrane reactor toolbar, click Plot.




M i c r o c h ann e l H - C e l l




Introduction

This example models an H-microcell for separation through diffusion. In Figure 1, it is shown how the cell puts two different laminar streams in contact for a controlled period of time. The contact surface is well defined, and by controlling the flow rate it is possible to control the amount of species transported from one stream to the other through diffusion. This example was originally formulated by Albert Witarsa under Professor Bruce Finlayson’s supervision at the University of Washington in Seattle. It was part of a graduate course in which the assignment consisted of using mathematical modeling to evaluate the potential of patents in the field of microfluidics.

The model utilizes the Laminar Flow and Transport of Diluted Species interfaces to fully capture the separation within the cell.

Figure 1: Diagram of the H-microcell (dimensions in meters).

Model Definition

The geometry of the microcell is shown in Figure 2. Due to symmetry, the model geometry can be simplified to half of it.

2 | M I C R O C H A N N E L H - C E L L

The design is aimed to eliminate upsets in the flow field when the two streams, A and B, are united. Thus, the cell enables mixing of A and B solely through diffusion. A system allowing convection would mix all species equally and lead to loss of control over the separation abilities.

Figure 2: Model geometry as given by Albert Witarsa’s and Professor Finlayson. To avoid any type of convective mixing, the design must smoothly let both streams come in contact with each other. Due to symmetry, it is sufficient to model half the geometry.

The simulations involve solving the fluid flow in the H-cell. According to the specifications, the flow rate at the inlet is roughly 0.1 mm/s. This implies a low Reynolds number, well inside the region of laminar flow:

(1)

Equation 1 gives a Reynolds number of 0.001 for a water solution and the channel dimensions given in Figure 1. This value is typical for microchannels. Additionally, this indicates that it is easy to get a numerical solution of the full momentum balance and continuity equations with a reasonable number of elements. The Laminar Flow interface can set up and solve the Navier-Stokes equations at steady state:

Here ρ denotes density (SI unit: kg/m3), u is the velocity (SI unit: m/s), μ denotes viscosity (SI unit: Pa·s), and p equals pressure (SI unit: Pa).

Separation in the H-cell involves species in relatively low concentrations compared to the solvent, in this case water. This means that the solute molecules interact only with water molecules, and it is safe to use Fick’s law to describe the diffusive transport in the cell. Use the Transport of Diluted Species interface to set up and solve the appropriate stationary mass-balance equation:

Full geometry Symmetry Half of the cell

Re dρuμ-----------=

1·10 5– 1·103 1·10 4–⋅ ⋅1·10 3–

-------------------------------------------------------=

ρ u ∇⋅( )u ∇ pI– μ ∇u ∇u( )T+( ) 2

3---μ ∇ u⋅( )I–+⋅=

∇ ρu( )⋅ 0=


(2)

In this equation, D denotes the diffusion coefficient (SI unit: m2/s) and c represents the concentration (SI unit: mol/m3). In this model, you use the parametric solver to solve Equation 2 for three different values of D—1 × 10−11 m2/s, 5 × 10−11 m2/s, and 1 × 10−10 m2/s—to simulate the mixing of different species.

You solve two versions of the model:

• In the first version, you assume that a change in concentration does not influence the fluid’s density and viscosity. This implies that it is possible to first solve for the fluid flow and then for the mass transport.

• In the second version, you include a correction term in the viscosity that depends quadratically on the concentration in Equation 1:

(3)

Here α is a constant of dimension (SI unit: m6/mol2). An influence of concentration on viscosity of this kind is usually observed in solutions of larger molecules. In this case the flow and mass transport equations have to be solved simultaneously.

Last, consider the boundary conditions. All boundaries are visualized in Figure 3.

Figure 3: Model domain boundaries.

For the Laminar Flow interface:

• At the inlets and outlets, Pressure conditions apply along with vanishing viscous stress. Setting the pressure at the outlets to zero, the pressure at the inlets represents the

∇ D∇– c cu+( )⋅– 0=

μ μ0 1 α c2+( )=

Outlet B

Outlet A

Inlet A

Inlet B

Symmetry plane

Wall


pressure drop over the cell. These inlet and outlet conditions comply with the H-cell being a part of a channel system of constant width, which justifies the assumption of developed flow.

• At the walls, No slip conditions state that the velocity is zero.

• At the symmetry plane, using the Symmetry boundary condition sets the velocity component in the normal direction of the surface to zero.

For the Transport of Diluted Species:

• At the inlets, use the Concentration boundary condition to set concentration. At inlets A and B the concentration are 1 mol/m3 and 0 mol/m3, respectively.

• At the outlets, apply the convective flux condition through the Outflow boundary condition, stating that the diffusive transport perpendicular to the boundary normal is negligible. This condition will thus eliminate concentration gradients in the flow direction.

• Model the symmetry plane and cell walls with the No Flux condition. This equation states that the flux of species perpendicular to the boundary equals zero.

Results

Figure 4 shows the velocity field for the whole microcell. The flow is symmetric and is not influenced by the concentration field.

Figure 4: Flow velocity field.


Figure 5 shows the concentration distribution for the species with the highest simulated diffusivity.

Figure 5: Concentration distribution for a species with diffusivity 10−10 m2/s.

Because of the relatively large diffusion coefficient, the degree of mixing is almost perfect. The species with a diffusion coefficient ten times smaller shows a different result. The


concentration distribution in Figure 6 shows that the diffusion coefficient for the species too low for achieving significant mixing of the streams.

Figure 6: Concentration distribution for a species with diffusivity 1·10−11m2/s.

The simulation clearly shows that the H-cell can separate lighter molecules from heavier ones. A cascade of H-cells can achieve a very high degree of separation.

In some cases, especially those involving solutions of macromolecules, the macromolecule concentration has a large influence on the liquid’s viscosity. In such situations, the model needs to be fully coupled and solve both interfaces simultaneously. Figure 7 shows the results of such a simulation. Here, the changes in viscosity have caused an asymmetry in the velocity. As a consequence of the modified flow field, the transport of molecules to outlet B is also different from the constant flow field case (Figure 8 vs Figure 5).


Figure 7: Velocity field. The viscosity varies with the concentration according to Equation 3 with α = 0.5 (m3/mol)2. The figure shows that the velocity field is affected by variations in concentration..

Figure 8: Concentration distribution for the species with diffusivity 1·10−10m2/s for the case where the fluid viscosity varies with concentration. Comparison with the plot in Figure 5 shows that fewer molecules of the species are transported to outlet B.


Application Library path: Chemical_Reaction_Engineering_Module/Mixing_and_Separation/microchannel_h_cell



N E W





3 Click Add.



5 Click Add.

6 Click Study.


8 Click Done.

Create the geometry. To simplify this step, insert a prepared geometry sequence.


10 Browse to the application’s Application Library folder and double-click the file microchannel_h_cell.mph.



This completes the geometry modeling stage. The geometry should now look like that in the figure below.






4 Browse to the model’s Application Libraries folder and double-click the file microchannel_h_cell_parameters.txt.

The material properties of water are available within the Material Library.








For microfluidic flows, second order elements are more accurate and computationally efficient than the linear elements. Increase the element order for both interfaces in the model.



2 In the Settings window for Laminar Flow, click to expand the Discretization section.

3 From the Discretization of fluids list, choose P2+P1.



2 In the Settings window for Transport of Diluted Species, click to expand the Discretization section.

3 From the Concentration list, choose Quadratic.


On the Physics toolbar, click Transport of Diluted Species (tds) and choose Laminar Flow (spf).





5 Locate the Pressure Conditions section. In the p0 text field, type p0.






Symmetry 11 On the Physics toolbar, click Boundaries and choose Symmetry.




Transport of Diluted Species (tds) node, then click Transport Properties 1.







4 Select the Species c check box.

5 In the c0,c text field, type c0.









M E S H 1


2 From the Element size list, choose Coarse.


3 Right-click Component 1 (comp1)>Mesh 1 and choose Edit Physics-Induced Sequence.



3 From the Predefined list, choose Coarser.


Size 21 In the Model Builder window, right-click Mesh 1 and choose Size.


3 From the Geometric entity level list, choose Point.

4 Select Points 19, 20, 25, and 26 only.

5 Locate the Element Size section. Click the Custom button.


7 In the associated text field, type 4.5E-6.

Free Tetrahedral 11 In the Model Builder window, under Component 1 (comp1)>Mesh 1 click

Free Tetrahedral 1.

2 In the Settings window for Free Tetrahedral, click Build Selected.

3 Click Build All.

Use two Stationary solver steps. In the first only the Laminar Flow interface is solved.

S T U D Y 1

Step 1: Stationary1 In the Settings window for Stationary, locate the Physics and Variables Selection section.

2 In the table, clear the Solve for check box for the Transport of Diluted Species interface.

Add a second stationary step that solves the Transport of Diluted Species interface using the previously computed flow field.




3 In the table, clear the Solve for check box for the Laminar Flow interface.

Within this study step, enable the parametric continuation solver to solve the mass transport problem for several diffusion coefficients.

4 Click to expand the Study extensions section. Locate the Study Extensions section. Select the Auxiliary sweep check box.

5 Click Add.



By default, the solution from the first step will be applied to the (fluid) variables not solved for in the second step.

2 In the Settings window for Solution, click Compute.

Create a Mirror 3D Data Set that enables plotting of the whole geometry, i.e. both sides of the symmetry plane.

R E S U L T S

Mirror 3D 11 On the Results toolbar, click More Data Sets and choose Mirror 3D.

2 In the Settings window for Mirror 3D, locate the Plane Data section.


4 In the z-coordinate text field, type 1e-5.


Velocity (spf)To generate Figure 4 follow these steps:

Slice1 In the Model Builder window, expand the Velocity (spf) node, then click Slice.

2 In the Settings window for Slice, locate the Expression section.

3 From the Unit list, choose mm/s.



D (Diffusion constant) 1e-10 5e-11 1e-11


Slice 21 Right-click Results>Velocity (spf)>Slice and choose Duplicate.

2 In the Settings window for Slice, click to expand the Title section.


4 Locate the Plane Data section. From the Plane list, choose zx-planes.


6 Click to expand the Inherit style section. Locate the Inherit Style section. From the Plot list, choose Slice.


Slice 31 Right-click Results>Velocity (spf)>Slice 2 and choose Duplicate.




5 From the Entry method list, choose Coordinates.

6 In the z-coordinates text field, type 1e-5.


Arrow Volume 11 In the Model Builder window, under Results right-click Velocity (spf) and choose

Arrow Volume.

2 In the Settings window for Arrow Volume, locate the Arrow Positioning section.

3 Find the x grid points subsection. In the Points text field, type 14.

4 Find the y grid points subsection. In the Points text field, type 21.

5 Find the z grid points subsection. In the Points text field, type 3.

6 Locate the Coloring and Style section. From the Color list, choose Black.




3 From the Data set list, choose Mirror 3D 1.

4 Locate the Plot Settings section. From the Color list, choose Gray.

5 Click the Show Grid button on the Graphics toolbar.


Slice1 In the Model Builder window, expand the Velocity (spf) node, then click Slice.


3 From the Unit list, choose m/s.

4 Click to expand the Quality section. From the Resolution list, choose Fine.

5 Click to expand the Inherit style section. Locate the Coloring and Style section. From the Color table list, choose RainbowLight.

Slice 21 In the Model Builder window, under Results>Velocity (spf) click Slice 2.



Slice 31 In the Model Builder window, under Results>Velocity (spf) click Slice 3.



Color Expression 11 In the Model Builder window, under Results>Velocity (spf) right-click Arrow Volume 1 and

choose Color Expression.

2 In the Settings window for Color Expression, locate the Coloring and Style section.

3 From the Color table list, choose RainbowLight.

Arrow Volume 11 In the Model Builder window, under Results>Velocity (spf) click Arrow Volume 1.

2 In the Settings window for Arrow Volume, locate the Coloring and Style section.

3 Select the Scale factor check box.


5 Click to expand the Inherit style section. Locate the Inherit Style section. From the Plot list, choose Slice.



8 Click Plot.

Proceed to plot the concentration distribution Figure 5.





4 From the Parameter value (D) list, choose 1E-10.

SliceIn the Model Builder window, expand the Concentration (tds) node.

Slice 21 Right-click Slice and choose Duplicate.

2 In the Settings window for Slice, locate the Title section.


4 Locate the Plane Data section. From the Plane list, choose zx-planes.


6 Locate the Inherit Style section. From the Plot list, choose Slice.


Slice 31 Right-click Results>Concentration (tds)>Slice 2 and choose Duplicate.




5 From the Entry method list, choose Coordinates.

6 In the z-coordinates text field, type 1e-5.



Concentration (tds)Generate Figure 6 using the diffusion coefficient value 1e−11 [m2/s].

1 In the Model Builder window, under Results click Concentration (tds).


3 From the Parameter value (D) list, choose 1E-11.




Add a correction term in the viscosity.


On the Physics toolbar, click Transport of Diluted Species (tds) and choose Laminar Flow (spf).


Fluid Properties 1.


3 From the μ list, choose User defined. In the associated text field, type mu*(1+alpha*c^2).

When viscosity is concentration dependent you must solve all equations simultaneously. Add a second study for the fully coupled problem.

A D D S T U D Y






S T U D Y 2




Create Figure 7 and Figure 8 by following these steps.

R E S U L T S


2 In the Settings window for Mirror 3D, locate the Plane Data section.


4 In the z-coordinate text field, type 1e-5.



Velocity (spf) 11 In the Model Builder window, under Results right-click Velocity (spf) and choose

Duplicate.




5 On the Velocity (spf) 1 toolbar, click Plot.


Concentration (tds) 21 In the Model Builder window, under Results right-click Concentration (tds) and choose

Duplicate.




5 On the Concentration (tds) 2 toolbar, click Plot.




Op t im i z a t i o n o f a C a t a l y t i c M i c r o r e a c t o r




Introduction

In this application, a solution is pumped through a catalytic bed, where a reactant undergoes chemical reaction as it gets in contact with the catalyst. The purpose of the example is to maximize the total reaction rate for a given total pressure difference across the bed by finding an optimal catalyst distribution. The distribution of the porous catalyst determines the total reaction rate in the bed. A large amount of catalyst results in a low flow rate through the bed while less catalyst gives a high flow rate but low conversion of the reactant.

This modeling example is based on Ref. 1.


Model Definition

The model geometry is shown in Figure 1. The reactor consists of an inlet channel, a fixed catalytic bed, and an outlet channel.


Reacting domain

Symmetry boundary

OutletInlet

2 | O P T I M I Z A T I O N O F A C A T A L Y T I C M I C R O R E A C T O R

The optimal catalyst distribution should maximize the average reaction rate, which is expressed as the integral of the local reaction rate, r (SI unit: mol/(m3·s)), over the domain, Ω. This is equivalent to minimizing the negative of this average reaction rate:

Assuming a first-order catalytic reaction with respect to the reactant species, the local reaction rate is determined by

(1)

where ε denotes the volume fraction of solid catalyst, c refers to the concentration (SI unit: mol/m3), and ka is the rate constant (SI unit: 1/s).

The mass transport is described by the convection and diffusion equation

where u denotes the velocity vector (SI unit: m/s) and D is the diffusion coefficient (SI unit: m2/s). The Navier-Stokes equations describe the fluid flow:

(2)

The coefficient α(ε) depends on the distribution of the porous catalyst as

(3)

where Da is the Darcy number; L is the length scale (SI unit: m); and q is a dimensionless parameter, the interpretation of which is discussed in the next section.

From Equation 3, the direct conclusion is that when ε equals 1, α equals zero and Equation 2 reduces to the ordinary Navier-Stokes equations. In this case the reaction rate is zero; see Equation 1.

To summarize, the optimization problem is

(4)

where

min 1vol(Ω )-------------------– r Ωd

Ω

r ka 1 ε–( )c=

∇ D∇c–( )⋅ r u ∇c⋅–=

ρ u ∇⋅( )u ∇– p ∇+ μ ∇u ∇u( )T+( ) α ε( )u–⋅=

∇ u⋅ 0=

α ε( ) μDa L2⋅------------------- q 1 ε–( )

q ε+--------------------⋅=

minε

1vol(Ω )------------------– ka 1 ε–( )c( ) Ωd

Ω


and physical boundary conditions apply.

C O N V E X O P T I M I Z A T I O N P R O B L E M S

One of the most important characteristics of an optimization problem is whether or not the problem is convex. This section therefore briefly describes this property. For a more general discussion of the subject, see for example Ref. 2.

A set C is said to be convex if for any two members x, y of C, the following relation holds:

that is, the straight line between x and y is fully contained in C. A convex function is a mapping f from a convex set C such that for every two members x, y of C

(5)

An optimization problem is said to be convex if the following conditions are met:

• the design domain is convex

• the objective and constraints are convex functions

The importance of convexity follows simply from the result that if x* is a local minimum to a convex optimization problem, then x* is also a global minimum. This is easily proven by simply assuming that there is a y such that f(y) < f(x*), and then using Equation 5.

This particular optimization problem is nonlinear, because a change in ε implies a change in the concentration, c. Because of this implicit dependence, it is very difficult to determine whether or not the objective is convex. There is therefore no guarantee that the optimal solution you obtain is globally optimal or unique. In the best of cases, running the optimization gives a good local optimum.

The parameter q can be used to smoothen the interfaces between the catalyst and the open channel. To see the effect of this parameter, rewrite Equation 3 as

ρ u ∇⋅( )u ∇– p ∇+ μ ∇u ∇u( )T+( ) α ε( )u–⋅=

∇ u⋅ 0=

∇ D∇c–( )⋅ r u ∇c⋅–=

0 ε 1≤ ≤

tx 1 t–( )y C for every t 0 1,[ ]∈∈+

f tx 1 t–( )y+( ) t f x( ) 1 t–( )f y( ) for every t 0 1,[ ]∈+≤

α ε( ) μDa L2⋅------------------- 1 ε–

1 εq---+

-------------⋅=

It follows that when q approaches infinity, α is the (inverse) porosity. On the other hand, lowering the value of q decreases the magnitude of α.

Figure 2: q(1 − ε)/(q − ε) plotted as a function of ε for different values of q.

Figure 2 shows q(1 − ε)/(q − ε) plotted as a function of ε for different values of q. This plot shows that lowering the value of q, increases the convexity of the force coefficient. For a low q value, an increase in ε around 0.5, imposes a small increase of the force coefficient, while for a higher value of q, a change in ε imposes an almost equal change for the whole range. Therefore, for a lower q value, the solution is not sharp at the interfaces. On the other hand, for small values of ε, the force term decreases rapidly when q is small, and thus affects the flow field to a much wider extent. In the limit when q approaches infinity, α as a function of ε is a straight line.


Figure 3 shows the velocity field in the empty channel, this is the starting point for the optimization.


Figure 3: Velocity field in the open channel.

Figure 4: Distribution of the porous catalyst seen in black and open channel in white.


Figure 4 shows the distribution of the porous catalyst in black and the open channels in white. This result shows that, optimally, the supply of the reactant should be distributed over a large area of the reactor. Note also that the amount of open channel volume is significant.

Figure 5 shows the concentration distribution in the reactor. This plot shows how the porous catalyst is fed with the reactant through the open channels. The plot naturally resembles that of Figure 4.

Figure 5: Concentration distribution in the reactor after optimization.

Let

where nflow refers to the normal to the boundary ∂Ωi in the flow direction (that is, pointing in to the domain at the inlet and out from the domain at the outlet). Then Fi is a measurement of the flow of the species with concentration c through the boundary ∂Ωi per unit length in the transverse dimension. The conversion, X, of the reactant is defined as

Fi nflow D∇c– cu+( )⋅ sd∂Ωi

=

XFin Fout–

Fin--------------------------=


In this case, the conversion of the reactant is around 50%.

Figure 6 shows the velocity field in the reactor. The porous catalyst slows down the flow significantly compared to Figure 3.

Figure 6: Velocity field in the reactor after optimization.

References

1. F. Okkels and H. Bruus, “Scaling Behavior of Optimally Structured Catalytic Microfluidic Reactors,” Phys. Rev. E, vol. 75, pp. 016301 1–4, 2007.

2. S.G. Nash and A. Sofer, Linear and Nonlinear Programming, McGraw-Hill, 1995.

Application Library path: Chemical_Reaction_Engineering_Module/Reactors_with_Mass_Transfer/microreactor_optimization




N E W





3 Click Add.



5 Click Add.

6 In the Select Physics tree, select Mathematics>Optimization and Sensitivity>

Optimization (opt).

7 Click Add.

8 Click Study.


10 Click Done.


Load parameters from a text file.




4 Browse to the model’s Application Libraries folder and double-click the file microreactor_optimization_parameters.txt.

G E O M E T R Y 1

Next, create the geometry. The reactor consists of three domains: the inlet channel, the reacting domain, and the outlet channel (see Figure 1).







3 In the Width text field, type 2*L.

4 In the Height text field, type L.




4 In the Height text field, type 3*L.

5 Locate the Position section. In the x text field, type 2*L.




4 In the Height text field, type L.

5 Locate the Position section. In the x text field, type 8*L.



The geometry should now look like that in Figure 1.


Define integration couplings to use for calculating the conversion of the reactant.

Integration 1 (intop1)1 On the Definitions toolbar, click Component Couplings and choose Integration.

2 In the Settings window for Integration, locate the Source Selection section.








Variables 11 On the Definitions toolbar, click Local Variables.



Here, tds.tfluxx_c is the COMSOL Multiphysics variable for the x-component of the total flux.













Volume Force 11 On the Physics toolbar, click Domains and choose Volume Force.



F_in intop1(tds.tflux_cx) mol/(m·s) Molar flow at inlet

F_out intop2(tds.tflux_cx) mol/(m·s) Molar flow at outlet

X (F_in-F_out)/F_in Conversion of reactant


phi k_a*(1-epsilon)*c Local reaction rate

alpha (eta/(Da*L^2))*q*(1-epsilon)/(q+epsilon)

Drag-force coefficient


3 In the Settings window for Volume Force, locate the Volume Force section.

4 Specify the F vector as





5 Locate the Pressure Conditions section. In the p0 text field, type delta_p.













3 In the Settings window for Reactions, locate the Reaction Rates section.

4 In the Rc text field, type -phi.


-alpha*u x

-alpha*v y




4 In the c0,c text field, type c_in.




This completes the setup of the physics. Now set up the optimization problem.

O P T I M I Z A T I O N ( O P T )

Define the control variable epsilon, select its shape, and constrain its values to the interval [0, 1].

1 In the Model Builder window, under Component 1 (comp1) click Optimization (opt).

Control Variable Field 11 On the Physics toolbar, click Domains and choose Control Variable Field.


Because the porous catalyst is only used in the reacting domain you can deactivate the inlet and outlet channels.

3 In the Settings window for Control Variable Field, locate the Control Variable section.

4 In the Control variable name text field, type epsilon.

5 In the Initial value text field, type 1.

6 Locate the Discretization section. From the Element order list, choose Linear.

Control Variable Bounds 11 In the Model Builder window, right-click Control Variable Field 1 and choose

Control Variable Bounds.

2 In the Settings window for Control Variable Bounds, locate the Bounds section.

3 In the Upper bound text field, type 1.

Next, define the objective function.

Integral Objective 11 On the Physics toolbar, click Domains and choose Integral Objective.


3 In the Settings window for Integral Objective, locate the Objective section.


4 In the Objective expression text field, type -phi/vol.

This example requires a fine mesh, both to solve the physics problem and to resolve the topology optimization problem.


Couple the interfaces with the Multiphysics node.


M E S H 1


2 From the Element size list, choose Finer.

3 Click Build All.

S T U D Y 1

Although you can choose to solve the optimization problem directly, it can be useful to check that the solution for the PDE problem looks sound before starting the optimization.


R E S U L T S

Velocity (spf)The first default plot (see Figure 3) shows the velocity field in the reactor.

Now solve the optimization problem.

S T U D Y 1


2 In the Settings window for Stationary, click to expand the Results while solving section.

3 Locate the Results While Solving section. Select the Plot check box.

This setting gives a plot of the evolving velocity distribution in the Graphics window.



Choose the SNOPT solver rather than MMA since the objective function contains an implicit trade-off between a high flow rate and a high active catalyst area. This makes the objective severely nonlinear.


3 From the Method list, choose SNOPT.

4 In the Optimality tolerance text field, type 1.0E-6.




Solution 1 (sol1)>Optimization Solver 1 node, then click Stationary 1.

4 In the Settings window for Stationary, locate the General section.



R E S U L T S

Velocity (spf)The velocity field in the reactor after optimization should resemble that in Figure 6.

Concentration (tds)The third default plot shows the concentration distribution in the reactor after optimization (Figure 5).

To reproduce the plot in Figure 4, modify the default plot with the following steps.


2 In the Settings window for 2D Plot Group, type Distribution of porous catalyst in the Label text field.


4 In the Title text area, type Distribution of porous catalyst.


Surface 11 In the Model Builder window, expand the Results>Distribution of porous catalyst node,

then click Surface 1.

2 In the Settings window for Surface, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Component 1>Optimization>epsilon -

Control variable epsilon.

3 Locate the Coloring and Style section. From the Color table list, choose GrayScale.

4 Clear the Color legend check box.

5 On the Distribution of porous catalyst toolbar, click Plot.


7 Right-click Results>Distribution of porous catalyst>Surface 1 and choose Rename.

8 In the Rename Surface dialog box, type Porous Catalyst in the New label text field.

9 Click OK.


Derived ValuesTo display the result for the conversion rate, continue as follows.


2 In the Settings window for Global Evaluation, click Replace Expression in the upper-right corner of the Expressions section. From the menu, choose Component 1>Definitions>

Variables>X - Conversion of reactant.

3 Click Evaluate.

TA B L E


The value appears in the Table window below the Graphics window.



Non i s o t h e rma l P l u g F l ow Rea c t o r




Introduction

This example considers the thermal cracking of acetone, which is a key step in the production of acetic anhydride. The gas phase reaction takes place under nonisothermal conditions in a plug flow reactor. As the cracking chemistry is endothermic, control over the temperature in the reactor is essential in order to achieve reasonable conversion. Furthermore, it is illustrated how to affect the conversion of acetone by means of a heat exchanger supplying energy to the system and how the conversion of acetone is affected by mixing inert into the feed inlet stream.

The example details the use of the predefined Plug flow reactor type in the Reaction Engineering interface of the Chemical Reaction Engineering Module. This model reproduces the results in Ref. 1.

Model Definition

The model simulates the step in the gas-phase production of acetic anhydride where cracking of acetone (A) into ketene (K) and methane (M) occurs:

The rate of reaction is:

where the rate constant is given by the Arrhenius expression:

For the decomposition of acetone described above, the frequency factor is A1 = 8.2·1014, and the activation energy is E1 = 284.5 (kJ/mol). Together with the reacting species nitrogen can also be present as an inert in the system.

CH3COCH3 CH2CO CH4+

(A) (K) (M)

r1 k1cA=

k1 A1E1

RgT-----------–

exp=

2 | N O N I S O T H E R M A L P L U G F L O W R E A C T O R

The chemical reaction takes place under nonisothermal conditions in a plug flow reactor, illustrated schematically in Figure 1.

Figure 1: The Plug flow reactor.

Species mass balances are set up as:

(1)

where Fi is the species molar flow rate (mol/s), V is the reactor volume (m3), and Ri is the species rate expression (mol/(m3·s)). Concentrations needed to evaluate rate expressions are described by:

where v is the volumetric flow rate (m3/s). By default, the Reaction Engineering interface treats gas phase reacting mixtures as being ideal. Under this assumption the volumetric flow rate is given by:

where p is the pressure (Pa), Rg denotes the ideal gas constant (8.314 J/(mol·K)), and T is the temperature (K).

The reactor energy balance is:

(2)

In Equation 2, Cp,i represents the species molar heat capacity (J/(mol·K)), and ws is the shaft work per unit volume (J/(m3·s)). Q denotes the heat due to chemical reaction (J/(m3·s)) and is described by:

dFidV--------- Ri=

ciFiv-----=

v Fi

i

RgTp-----------=

Fi

i Cp i,

dTdV-------- ws Q Qext+ +=


where Hj is the heat produced by reaction j. The term Qext represents external heat added or removed from the reactor. The present model treats both adiabatic reactor conditions, that is:

and the situation where the reactor is equipped with a heat exchanger jacket. In the latter case Qext is given by:

where U is the overall heat transfer coefficient (J/(m2·s·K)), a is the effective heat transfer area per unit of reactor volume (1/m), and Tamb is the temperature of the heat exchanger medium (K). The Reaction Engineering interface automatically sets up and solves Equation 1 and Equation 2 as the predefined Plug flow reactor type is selected. As input to the balance equations you need to supply the chemical reaction formula, the Arrhenius parameters, the species thermodynamic properties, as well as the inlet molar feed of reactants.

Working with Thermodynamic Polynomials

The Reaction Engineering interface uses the following set of polynomials as default expressions describing species thermodynamic properties:

(3)

(4)

(5)

Here, Cp,i denotes the species’ heat capacity (J/(mol·K)), T the temperature (K), and Rg the ideal gas constant, 8.314 (J/(mol·K)). Further, hi is the species’ molar enthalpy (J/mol), and si represents its molar entropy (J/(mol·K)). A set of seven coefficients per species are taken as input for the polynomials above. The coefficients a1 through a5 relate to the species heat capacity, the coefficient a6 is associated with the species enthalpy of

Q Hjrj

j–=

Qext 0=

Qext Ua Tamb T–( )=

Cp i, Rg a1 a2T a3T2 a4T3 a5T4+ + + +( )=

hi Rg a1Ta22------T2 a3

3------T3 a4

4------T4 a5

5------T5 a6+ + + + +

=

si Rg a1 Tln a2Ta32------T2 a4

3------T3 a5

4------T4 a7+ + + + +

=


formation (at 0 K), and the coefficient a7 comes from the species entropy of formation (at 0 K).

The format outlined by Equation 3, Equation 4, and Equation 5 is referred to as CHEMKIN or NASA format (Ref. 2). This is a well established format, and database resources list the needed coefficients for different temperature intervals. Many web-based data resources exist for CHEMKIN thermodynamic data and otherwise. For a collection of databases, see Ref. 3. It is therefore often a straightforward task to assemble the thermodynamic data required and then import it into your reaction model.

C R E A T I N G A T H E R M O D Y N A M I C D A T A F I L E

CHEMKIN thermodynamic files typically contain blocks of data, one block for each species. Such a data block is illustrated in Figure 2 for gas phase acetone.

Figure 2: Thermodynamic data block on the CHEMKIN format.

The first line starts with the species label, given with a maximum of 18 characters, followed by a 6 character comment space. Then follows a listing of the type and number of atoms that constitute the species. The G comments that this is gas phase data. The line ends with listing of three temperatures, defining two temperature intervals. Lines 2 through 4 contain two sets of the polynomial coefficients, a1 through a7. The first set of coefficients is valid for the upper temperature interval, and the second for the lower interval.


A complete CHEMKIN thermodynamics file has the structure illustrated in Figure 3.

Figure 3: Thermodynamic data file on the CHEMKIN format.

The keyword thermo always starts the text file. The subsequent line lists three temperatures, defining two temperature intervals. These intervals are used if no intervals are provided in the species specific data. Then follows the species data blocks, in any order. The keyword end is the last entry in the file.

The data file reproduced in Figure 3 is used in the present model. It was constructed by creating a text-file with species data blocks from the Sandia National Labs Thermodynamics Resource.

U S I N G S U B S E T S O F T H E R M O D Y N A M I C C O E F F I C I E N T S

In some instances, not all coefficients a1 through a7 are available. Coefficients may also be on a format differing from the NASA or CHEMKIN style. Input parameters can still be put on the form of Equation 3, Equation 4, and Equation 5, as illustrated below.

For species A, K, M, and N2, Ref. 1 lists polynomials for the heat capacity on the form:

Furthermore, the species enthalpies of formation at Tref = 298 K are provided:

SPECIES a1' a2' a3' h(298)

A 26.63 0.183 -45.86e-6 -216.67e3

K 20.04 0.0945 -30.95e-6 -61.09e3

Cp i, a1' a2'T a3'T2

+ +=


The data in the table above can be correlated with the polynomials given by Equation 3 and Equation 4 by noting the relations:

(6)

(7)

Using Equation 6 and Equation 7, you find the coefficients to enter in the Reaction Engineering interface.

Read more about how to work with thermodynamic data in the section Transport Properties in the Chemical Reaction Engineering Module User’s Guide.


A 5 m3 plug flow reactor is simulated both for adiabatic conditions and with heat exchange. The reactor is fed with pure acetone at 1035 K. 18.8 mol/m3 acetone is set to enter with a volumetric flow of 2 m3/s. Figure 4 shows the reactor temperature as a function of the reactor volume for both cases. In the adiabatic case, with no additional heating, the reactor temperature decreases along the reactor length since the cracking reaction is endothermic. On the other hand, the temperature profile as a heat exchanger supplies energy increases after first passing through a minimum. This is explained by Figure 5 where the reaction rates are shown. The minimum is due to the energy demand of the cracking reaction being dominant initially. However, as the reaction rate drops off,

M 13.39 0.077 -18.71e-6 -71.84e3

N2 6.25 8.78e-3 -2.1e-8 0

SPECIES a1 a2 a3 a6

A 3.20 2.20e-2 -5.52e-6 -2.79e4

K 2.41 1.14e-2 -3.72e-6 -8.54e3

M 1.61 9.26e-3 -2.25e-6 -9.87e3

N2 0.752 1.06e-3 -2.53e-9 -2.71e2

SPECIES a1' a2' a3' h(298)

anan'Rg-------= n 1 … 5, ,=

a6h Tref( )

Rg------------------- a1Tref

a22------Tref

2 a33------Tref

3+ +

–=


the heat exchanger starts to heat up the system. The heating rate decreases with the temperature difference between the heat exchanger medium and reacting fluid.

Figure 4: The reactor temperature (K) as a function of reactor volume (m3) for the two investigated cases.


Figure 5: The reaction rates (mol/(m3·s)) as functions of reactor volume (m3)


Figure 6 shows the conversion of acetone. The reactor conversion is considerably lower at the reactor outlet in the adiabatic case than with heat exchange.

Figure 6: The conversion of acetone (%) as a function of reactor volume (m3) for the two investigated cases.

If the feed inlet stream contains an additional amount of nitrogen inert the conversion of A becomes more efficient at adiabatic conditions. This is shown for three different molar fractions of acetone in the inlet in Figure 7. The explanation is that the inert acts as a heat supply to the endothermic reaction.


Figure 7: The conversion of acetone (%) as a function of reactor volume (m3) for 100 mol%. 50 mol%, and 10 mol% of acetone in the feed inlet stream.

References

1. H.S. Fogler, Elements of Chemical Reaction Engineering, 3rd ed., Prentice Hall PTR, example 8-7, pp. 462–468, 1999.

2. S. Gordon and B.J. McBride, Computer Program for Calculation of Complex Chemical Equilibrium Compositions, Rocket Performance, Incident and Reflected Shocks, and Chapman-Jouquet Detonations, NASA-SP-273, 1971.

3. See, for example http://www.comsol.com/chemical-reaction-engineering-module#specs

Application Library path: Chemical_Reaction_Engineering_Module/Tutorials/nonisothermal_plug_flow


http://www.comsol.com/chemical-reaction-engineering-module#specs

http://www.comsol.com/chemical-reaction-engineering-module#specs



N E W





3 Click Add.

4 Click Study.


6 Click Done.


Add a set of model parameters by importing their definitions from a data text file provided with the Application Library.




4 Browse to the model’s Application Libraries folder and double-click the file nonisothermal_plug_flow_parameters.txt.






5 Click to expand the Mixture properties section. Locate the Mixture Properties section. In the p text field, type P_reactor.




3 In the Formula text field, type A=>K+M.


5 In the Af text field, type Af_reaction.

6 In the Ef text field, type Ef_reaction.



3 In the Species name text field, type N2.

Set inlet properties derived from the volumetric flow and concentration of A at the inlet.




3 In the T0,in text field, type T_inlet.


Import material and thermal properties for the reaction with the CHEMKIN Import for

Species property.


6 In the Settings window for Reaction Engineering, click to expand the CHEMKIN import for species property section.

7 Locate the CHEMKIN Import for Species Property section. Click Browse.

8 Browse to the model’s Application Libraries folder and double-click the file nonisothermal_plug_flow_thermo.txt.

9 Click Import.

Also add a variable expression for monitoring the conversion of A. This will be used later on when post-processing the solution.


A Finlet_A

N2 Finlet_N2




choose Variables.



Solve first for adiabatic conditions.

S T U D Y 1

Step 1: Stationary Plug Flow1 In the Model Builder window, under Study 1 click Step 1: Stationary Plug Flow.

2 In the Settings window for Stationary Plug Flow, locate the Study Settings section.

3 In the Volumes text field, type range(0,0.05,5).


Save a copy of the adiabatic solution.




Copy 1 (sol2).

2 In the Settings window for Solution, type Adiabatic in the Label text field.

Add a heat source from a heat exchanger.




3 In the Q text field, type Ua*(T_x-re.T).

Solve for the heat exchanged plug flow reactor


X_A (re.F0_A-re.F_A)/re.F0_A*100 Conversion, A


S T U D Y 1


Solution 1 (sol1)In the Model Builder window, under Study 1>Solver Configurations right-click Solution 1 (sol1) and choose Solution>Copy.


Copy 1 (sol3).

2 In the Settings window for Solution, type With heat exchange in the Label text field.

Remove the heat exchange and solve for a Parametric Sweep of three molar fractions of A, 0 mol%, 50 mol%, and 90 mol%, at the inlet at adiabatic conditions.




3 In the Q text field, type 0.

S T U D Y 1



3 Click Add.



Parametric Solutions 1 (sol4)In the Model Builder window, under Study 1>Solver Configurations right-click Parametric Solutions 1 (sol4) and choose Solution>Copy.

Parametric Solutions 1 - Copy 1 (sol8)1 In the Model Builder window, expand the Study 1>Solver Configurations node, then click

Parametric Solutions 1 - Copy 1 (sol8).


A_frac (Inlet mole fraction A) 1 0.5 0.1


2 In the Settings window for Solution, type With inert in the Label text field.

Start with Figure 4, where the temperature change is displayed for the adiabatic and heat exchanged cases.

R E S U L T S

The following instructions generate Figure 4 through Figure 7.

Temperature (re)1 In the Model Builder window, under Results click Temperature (re).




5 In the associated text field, type Reactor volume (m3).

6 Click to expand the Legend section. From the Position list, choose Middle right.

Global 11 In the Model Builder window, expand the Temperature (re) node, then click Global 1.


3 From the Data set list, choose Study 1/Adiabatic (sol2).




Global 21 Right-click Results>Temperature (re)>Global 1 and choose Duplicate.


3 From the Data set list, choose Study 1/With heat exchange (sol3).



Legends

Adiabatic

Legends

With heat exchange

6 On the Temperature (re) toolbar, click Plot.

Continue with Figure 5, displaying the reaction rate for the adiabatic and heat exchanged cases.

Molar Flow Rate (re) 11 On the Home toolbar, click Add Plot Group and choose 1D Plot Group.

2 In the Settings window for 1D Plot Group, type Reaction rate (re) in the Label text field.




Global 11 Right-click Results>Reaction rate (re) and choose Global.







Global 21 In the Model Builder window, under Results>Reaction rate (re) click Global 2.






Legends

Adiabatic



8 On the Reaction rate (re) toolbar, click Plot.

Next is Figure 6 illustrating the conversion of acetone for the adiabatic and heat exchanged cases.

Molar Flow Rate (re)1 In the Model Builder window, under Results click Molar Flow Rate (re).

2 In the Settings window for 1D Plot Group, type Conversion A (re) in the Label text field.





7 In the associated text field, type Conversion A (%).

8 Locate the Legend section. From the Position list, choose Middle right.

Global 11 In the Model Builder window, expand the Results>Conversion A (re) node, then click

Global 1.



4 Click Replace Expression in the upper-right corner of the y-axis data section. From the menu, choose Component 1>Definitions>Variables>comp1.X_A - Conversion, A.




Global 21 Right-click Results>Conversion A (re)>Global 1 and choose Duplicate.


Legends

With heat exchange

Legends

Adiabatic





6 On the Conversion A (re) toolbar, click Plot.

Last set up Figure 7 to plot the conversion of acetone with varied inlet molar fractions of inert for the adiabatic case.


2 In the Settings window for 1D Plot Group, type Conversion A with inert (re) in the Label text field.






8 In the associated text field, type Conversion A (%).

9 Locate the Legend section. From the Position list, choose Lower right.

Global 11 Right-click Results>Conversion A with inert (re) and choose Global.


3 From the Data set list, choose Study 1/With inert (sol8).

4 Click Replace Expression in the upper-right corner of the y-axis data section. From the menu, choose Component 1>Definitions>Variables>comp1.X_A - Conversion, A.




Legends

With heat exchange

Legends

100 mol% A at inlet


8 On the Conversion A with inert (re) toolbar, click Plot.

Temperature (re) 1In the Model Builder window, under Results right-click Temperature (re) 1 and choose Delete.

50 mol% A at inlet

10 mol% A at inlet

Legends



Op t ima l C oo l i n g o f a Tubu l a r R e a c t o r




Introduction

Maximizing product yield is a main task in chemical reaction engineering. This can be especially challenging if the desired product, once formed, can be consumed by further reactions. The following example investigates such a series reaction as it occurs in a tubular reactor. This starts by setting up the tightly coupled mass and energy balance equations describing the reactor applying predefined physics interfaces within the Chemical Reaction Engineering Module. This is followed by the addition of an Optimization Study node to compute the temperature conditions in the reactor that maximize the production of the intermediary product.


Model Definition

Two consecutive reactions take place in a tubular reactor. A heat exchanger jacket, run in cocurrent mode, is used to control the reaction rates and hence the product distribution in the reactor.

Figure 1: Reactions occur in a tubular reactor equipped with a heat exchanger jacket, run in cocurrent mode.

Temperature control in the reactor involves a delicate balance, where on the one hand, energy has to be supplied to the system to achieve acceptable reaction rates. On the other hand, the energy transfer to the reacting stream must be limited so that the desired intermediate product is not consumed by further reaction. The situation is further complicated by the fact that the temperature of the reacting stream is not only affected by

Cooling stream

Reacting stream

2 | O P T I M A L C O O L I N G O F A TU B U L A R R E A C T O R

the heat transfer from the heat exchanger jacket, but also by the endothermic nature of the reactions. The idea for this challenge in reactor optimization is taken from a literature example (Ref. 1), although the present reactor model is considerably more detailed.

The model is set up in 1D, coupling mass and energy balances in the reactor tube with an energy balance for the heat exchanger jacket. Streams in both the tube and jacket are treated as plug flows.

C H E M I S T R Y

Two consecutive reaction occur in water (hydrolysis), where the desired product is species B:

The following rate equations apply:

where the rate constants are temperature dependent according to the Arrhenius relation:

The kinetic parameters are summarized in the table below:

M A S S T R A N S P O R T

The mass transport is modeled by the convection-diffusion equation at steady-state using the Transport of Diluted Species interface:

J AJ [1/S] EJ [J/MOL]

1 1.6e8 75e3

2 1e15 125e3

A Bk1

B Ck2

r1 k1cA=

r2 k2cB=

kj AjEj

RgT-----------–

exp=

∇ Di∇– ci( )⋅ u ∇ci⋅+ Ri=


In this equation, ci denotes the concentration (SI unit: mol/m3) and Di is the diffusivity (SI unit: m2 /s). Ri is the rate expression for species i (SI unit: mol/(m3·s)). The velocity u (SI unit: m/s) of the fluid in the reactor is represented by a constant profile:

At the inlet, the concentration of the reactant A is 700 mol/m3. At the outlet, it is assumed that convective mass transport is dominant:

E N E R G Y T R A N S P O R T - R E A C T O R

The energy transport in the reactor is modeled with the Heat Transfer in Fluids interface in which the following equation is solved:

Above, k is the thermal conductivity (SI unit: W/(m·K)) and T the temperature of the reacting stream (SI unit: K). ρ is the density (SI unit: kg/m3) and Cp the heat capacity (SI unit: J/(kg·K)). The reacting species are diluted in water, and hence, the physical properties of the reacting mixture are assumed to be those of water.

The heat source due to reaction, Qrxn (SI unit: W/m3), is calculated from the reaction rates and the enthalpies of reaction:

Both reactions are endothermic, with ΔH1 = 200 kJ/mol and ΔH2 = 100 kJ/mol. Furthermore, the heat transferred from the reactor to the cooling jacket is given by:

Here, U is the overall heat transfer coefficient (SI unit: J/(K·m2·s)), and A represents the heat exchange area per unit volume (SI unit: m2/m3).

The temperature of the reacting fluid at the inlet is 400 K. At the outlet, it is assumed that convective heat transport is dominant:

u 0.0042 m/s=

∇ Di∇– ci( )⋅ 0=

∇ k– ∇T( )⋅ ρCpu ∇T⋅+ Qrxn Qj+=

Qrxn ΔHjrj–

j 1 2–==

Qj UA T Tj–( )–=

∇ k– ∇T( )⋅ 0=


E N E R G Y T R A N S P O R T - C O O L I N G J A C K E T

Water serves as the cooling medium in the jacket, and the energy transport is given by the following equation, which is set up and solved with the Heat Transfer in Fluids interface:

The cooling stream is assumed to have plug flow character, and hence a constant velocity profile:

The optimal temperature of the cooling fluid at the inlet is to be found such that the maximum concentration of species B is achieved at the outlet.


In a first simulation, the inlet temperatures of the jacket stream and the reacting stream are set to be equal, at 400 K. In a second simulation, an optimization calculation is performed to find the inlet temperature of the jacket stream that maximizes the concentration of the desired intermediary product (B) at the reactor outlet. Comparisons between the two cases follow below.

Figure 2 shows the concentration of reacting species as a function of the reactor length when the inlet temperature of the jacket stream is 400 K. Figure 3 shows concentration curves for the optimal inlet temperature of the jacket stream, found to be 334 K. Clearly, when the inlet temperature is 400 K the conversion of reactant A is high, but at the same time, the selectivity for the desired product B is unfavorable. Under the optimized conditions, the concentration of B at the reactor outlet is 352 mol/m3, to be compared to a concentration of 153 mol/m3 when the inlet temperature is 400 K.

∇ k– ∇Tj( )⋅ ρCpuj ∇T⋅+ Qj–=

uj 0.001 m/s=


Figure 2: Species concentrations (blue c_A, green c_B, red c_C) as function of reactor position when the inlet temperature of the cooling fluid is 400 K.


Figure 3: Species concentrations (blue c_A, green c_B, red c_C) as function of reactor position when the inlet temperature of the cooling fluid is 334 K.


Plots of reacting stream and jacket stream temperatures are shown in Figure 4 and Figure 5. The jacket stream heats up the reacting stream when its inlet temperature is kept at 400 K.

Figure 4: Temperature distribution for the reacting stream (blue) and jacket stream (green) when the inlet temperature of the jacket stream is 400 K.

In contrast, the jacket stream cools the reacting stream when its inlet temperature is 334 K.


.

Figure 5: Temperature distribution for the reacting stream (blue) and jacket stream (green) when the inlet temperature of the jacket stream is 334 K.

The reaction rates are illustrated in Figure 6 and Figure 7. When the inlet temperature of the jacket stream is 400 K, the rate at which B is consumed (r2) dominates over the production rate (r1) from a point approximately 0.65 m down the reactor. This effect is


due to heat being transferred from the jacket stream, counteracting the cooling effect of the endothermic reactions.

Figure 6: Rate of the production r1 (blue) and rate consumption r2 (green) of species B when the inlet temperature of the cooling fluid is 400 K.


At an inlet temperature of 334 K, the combined effect of cooling by the jacket stream and energy consumption due to reaction work together to quench the system, resulting in increased concentrations levels of B at the outlet.

Figure 7: Rate of the production r1 (blue) and rate consumption r2 (green) of species B when the inlet temperature of the cooling fluid is 334 K.

Reference

1. T.F. Edgar and D.M. Himmelblau, Optimization of Chemical Processes, McGraw-Hill, 1988.

Application Library path: Chemical_Reaction_Engineering_Module/Reactors_with_Mass_and_Heat_Transfer/optimal_cooling




N E W





3 Click Add.



5 Click Add.

6 In the Number of species text field, type 3.



9 Click Add.


11 Click Add.

12 In the Temperature text field, type Tj.

13 Click Study.


15 Click Done.





4 Browse to the model’s Application Libraries folder and double-click the file optimal_cooling_parameters.txt.

cA

cB

cC


G E O M E T R Y 1

Interval 1 (i1)1 On the Geometry toolbar, click Interval.

2 In the Settings window for Interval, locate the Interval section.

3 In the Right endpoint text field, type L_r.

4 Right-click Interval 1 (i1) and choose Build Selected.

















Transport of Diluted Species (tds) node, then click Component 1 (comp1)>

Chemistry (chem).




cB_out intop1(cB) mol/m³ Outlet concentration





3 In the Formula text field, type A=>B.

4 Click Apply.


6 In the Af text field, type A1.

7 In the Ef text field, type E1.


9 In the H text field, type H1.



3 In the Formula text field, type B=>C.

4 Click Apply.


6 In the Af text field, type A2.



9 In the H text field, type H2.

Species: A1 In the Model Builder window, under Component 1 (comp1)>Chemistry (chem) click

Species: A.


3 In the M text field, type Mn_A.

Species: B1 In the Model Builder window, under Component 1 (comp1)>Chemistry (chem) click

Species: B.



3 In the M text field, type Mn_B.

Species: C1 In the Model Builder window, under Component 1 (comp1)>Chemistry (chem) click

Species: C.


3 In the M text field, type Mn_C.





5 Locate the General Parameters section. In the M text field, type Mn_H2O.





On the Physics toolbar, click Chemistry (chem) and choose Transport of Diluted Species (tds).



2 In the Settings window for Transport Properties, locate the Convection section.

3 Specify the u vector as

4 Locate the Diffusion section. In the DcA text field, type D.

5 In the DcB text field, type D.

6 In the DcC text field, type D.


H2O Constant, solvent c_solv solvent

u x






4 In the c0,cA text field, type cA_in.






4 From the RcA list, choose Rate expression for species A (chem).

5 From the RcB list, choose Rate expression for species B (chem).

6 From the RcC list, choose Rate expression for species C (chem).



Heat Transfer in Fluids (ht) node, then click Heat Transfer in Fluids (ht).

2 In the Settings window for Heat Transfer in Fluids, type Heat Transfer in Fluids - Reactor in the Label text field.

H E A T TR A N S F E R I N F L U I D S - R E A C T O R ( H T )

On the Physics toolbar, click Heat Transfer in Fluids (ht) and choose Heat Transfer in Fluids -

Reactor (ht).

Fluid 11 In the Model Builder window, under Component 1 (comp1)>Heat Transfer in Fluids -

Reactor (ht) click Fluid 1.

2 In the Settings window for Fluid, locate the Model Input section.


u x


4 In the Model Builder window, click Heat Transfer in Fluids - Reactor (ht).




4 In the T0 text field, type T_in.






4 In the Q0 text field, type -UA*(T-Tj)+chem.Qtot.

H E A T TR A N S F E R I N F L U I D S 2 ( H T 2 )


Heat Transfer in Fluids 2 (ht2) node, then click Heat Transfer in Fluids 2 (ht2).

2 In the Settings window for Heat Transfer in Fluids, type Heat Transfer in Fluids - Cooling jacket in the Label text field.

H E A T TR A N S F E R I N F L U I D S - C O O L I N G J A C K E T ( H T 2 )

On the Physics toolbar, click Heat Transfer in Fluids 2 (ht2) and choose Heat Transfer in Fluids - Cooling jacket (ht2).


Cooling jacket (ht2) click Fluid 1.



4 In the Model Builder window, click Heat Transfer in Fluids - Cooling jacket (ht2).

uj x





4 In the T0 text field, type Tj_in.






4 In the Q0 text field, type UA*(T-Tj).

M E S H 1


Edge.



4 Click Build All.

S T U D Y 1


R E S U L T S

1D Plot Group 1Go through the steps below to save a copy of the solution where the coolant temperature is 400 K at the inlet.

S T U D Y 1





Copy 1 (sol2).

2 In the Settings window for Solution, type Tj_in 400K in the Label text field.

R E S U L T S


2 In the Settings window for 1D Plot Group, type Concentration Tj_in 400K in the Label text field.

3 Locate the Data section. From the Data set list, choose Study 1/Tj_in 400K (sol2).


5 In the associated text field, type x-coordinate (m).

Line Graph1 In the Model Builder window, expand the Results>Concentration Tj_in 400K node, then

click Line Graph.

2 In the Settings window for Line Graph, type A in the Label text field.






A 11 Right-click Results>Concentration Tj_in 400K>A and choose Duplicate.

2 In the Settings window for Line Graph, type B in the Label text field.

3 Locate the y-Axis Data section. In the Expression text field, type cB.


Legends

A

Legends

B


B 11 Right-click Results>Concentration Tj_in 400K>B and choose Duplicate.

2 In the Settings window for Line Graph, type C in the Label text field.

3 Locate the y-Axis Data section. In the Expression text field, type cC.


5 On the Concentration Tj_in 400K toolbar, click Plot.

Temperature (ht)1 In the Model Builder window, under Results click Temperature (ht).

2 In the Settings window for 1D Plot Group, type Temperature Tj_in 400K in the Label text field.


Line Graph1 In the Model Builder window, expand the Results>Temperature Tj_in 400K node, then

click Line Graph.

2 In the Settings window for Line Graph, type Reactor in the Label text field.






Reactor 11 Right-click Results>Temperature Tj_in 400K>Reactor and choose Duplicate.

2 In the Settings window for Line Graph, type Cooling jacket in the Label text field.

3 Click Replace Expression in the upper-right corner of the y-axis data section. From the menu, choose Component 1>Heat Transfer in Fluids - Cooling jacket>Temperature>Tj -

Temperature.

Legends

C

Legends

Reactor



5 On the Temperature Tj_in 400K toolbar, click Plot.

Temperature (ht2)1 In the Model Builder window, under Results click Temperature (ht2).

2 In the Settings window for 1D Plot Group, type Rate of Production Tj_in 400K in the Label text field.


Line Graph1 In the Model Builder window, expand the Results>Rate of Production Tj_in 400K node,

then click Line Graph.

2 In the Settings window for Line Graph, type Reaction 1 in the Label text field.


4 Click Replace Expression in the upper-right corner of the y-axis data section. From the menu, choose Component 1>Chemistry>chem.r_1 - Reaction rate.





Reaction 1.11 Right-click Results>Rate of Production Tj_in 400K>Reaction 1 and choose Duplicate.

2 In the Settings window for Line Graph, type Reaction 2 in the Label text field.

3 Click Replace Expression in the upper-right corner of the y-axis data section. From the menu, choose Component 1>Chemistry>chem.r_2 - Reaction rate.



Legends

Cooling jacket

Legends

Reaction 1



7 On the Rate of Production Tj_in 400K toolbar, click Plot.

Now, solve the optimization problem.

S T U D Y 1



3 From the Method list, choose BOBYQA.

4 Locate the Objective Function section. In the table, enter the following settings:

5 From the Type list, choose Maximization.

6 Locate the Control Variables and Parameters section. Click Add.



R E S U L T S

Concentration Tj_in 400K 11 In the Model Builder window, under Results right-click Concentration Tj_in 400K and

choose Duplicate.

2 In the Settings window for 1D Plot Group, type Concentration Tj_in Opti in the Label text field.


4 On the Concentration Tj_in Opti toolbar, click Plot.

Legends

Reaction 2

Expression Description Evaluate for

comp1.cB_out Outlet concentration stat

Parameter name Initial value Scale Lower bound

Upper bound

Tj_in (Inlet temperature, jacket) 400[K] 1


Temperature Tj_in 400K 11 In the Model Builder window, under Results right-click Temperature Tj_in 400K and

choose Duplicate.

2 In the Settings window for 1D Plot Group, type Temperature Tj_in Opti in the Label text field.


4 On the Temperature Tj_in Opti toolbar, click Plot.

Rate of Production Tj_in 400K 11 In the Model Builder window, under Results right-click Rate of Production Tj_in 400K and

choose Duplicate.

2 In the Settings window for 1D Plot Group, type Rate of Production Tj_in Opti in the Label text field.


4 On the Rate of Production Tj_in Opti toolbar, click Plot.


R E S U L T S

Objective Table 2Scroll down the table to find the resulting values of the inlet temperature.



A Mu l t i s c a l e 3D Pa c k ed Bed Rea c t o r




Introduction

The packed bed reactor is used in heterogeneous catalytic processes and is one of the most common reactors in the chemical industry. Its basic design is a column filled with porous catalyst particles, and in some cases the reactor also has a specially designed bottom plate through which the reaction mixture enters. The catalyst particles can be contained within a supporting structure, such as tubes or channels, or they can be packed in one single compartment in the reactor.

Figure 1: An example of the macroscale (bed volume with entry holes) and the microscale (pellet) of a packed bed reactor.

The bed with the packed catalyst particles makes the modeling of mass transport and reactions in the reactor a challenge. The challenge is that species transport and reaction occur in dimensions of different orders of magnitude:

• In the macropores between the dumped pellets, and

• inside the catalyst pellets in micropores.

Inflow

Microscale: Concentration in porous pellet

Pellet radius rpe

cpe

Macroscale: Concentration in fluid passing through bed

Outflow

2 | A M U L T I S C A L E 3 D P A C K E D B E D R E A C T O R

As such, the problem is regarded as a multiscale problem. The COMSOL feature Reactive Pellet Bed, available with the Transport of Diluted Species interface, is dedicated to these multiscale problems.

The structure between particles in the bed is described as a macroporous material of meter dimensions. The particle radii are often in the order of 1 mm. The pores inside the catalyst particles form the microscale structure of the bed. The pore radii in the particles are often between 1 and 10 microns. There are two porosities that are important: bed porosity (macroscale) and pellet porosity (microscale). Sometimes such models are called double-porosity models.

When a pressure drop is applied across the bed, flow and convection of the fluid is initiated in the bed. The transport of chemicals inside the pellets are dominated by diffusion.

This model is an extension to the 1D example, Packed Bed Reactor, which contains more complex reactions.

Model Definition

A model geometry made up of one eighth of the reactor in Figure 1 can be used due to symmetry. The geometry is shown in Figure 2.

Figure 2: The packed bed reactor simulation geometry. Symmetry observations enables modeling of 1/8 of he true geometry. The results will be expanded to the true geometry with aid of a sector dataset.

The reversible catalytic chemical reaction occurs inside a pellet. The reactant species A and B forms a product C:


A + B ↔ 2C

The reaction kinetics are assumed to be equimolecular and are set up with the Chemistry interface. The automatic reaction rate can thus be used and has the following form:

where k is the rate factor (SI unit: m3/(mol·s)) with the superscripts f and r denoting the forward and reverse reaction, respectively. ci is the concentration (SI unit: mol/m3) of species i. The forward reaction constant is defined with the inbuilt Arrhenius expression and the reverse is computed with the equilibrium constant of the reaction.

The mass transport of the reacting species in the reactor is modeled with the Transport of Diluted Species interface, which accounts for diffusion, convection, and reaction in diluted solutions. The species are assumed to be diluted in water.

The reaction inside the pellets is added to the mass balances in the Transport of Diluted Species interface with the Reactive Pellet Bed feature. This feature has a predefined extra dimension (1D) on the normalized radius (r = rdim/rpe) of the pellet particle. The mesh on the extra dimension has a default of 10 elements with a cubic root sequence distribution. If spherical pellets are selected, the following spherical diffusion/reaction equation is set up and solved along the pellet radius for each species i:

(1)

Here, r is a dimensionless radial coordinate that goes from 0 (center) to 1 (pellet surface), rpe is the pellet radius, and N the number of pellets per unit volume of bed. The advantage of formulating Equation 1 on a dimensionless 1D geometry is that the pellet radius can be changed without changing the geometry limits.

Dpe is an effective diffusion coefficient (SI unit: m2/s) and Rpe, i is the reaction source term (SI unit: mol/(m3·s)). Note that the latter term is taken per unit volume of porous pellet material.

At the pellet-fluid interface, a film condition assumption is made. The flux of mass across the pellet-fluid interface into the pellet is possibly rate determined by the resistance to mass transfer on the bulk fluid side. The resistance is expressed in terms of a film mass transfer coefficient, hDi, such that:

, (2)

r kfcAcB krcC2

–=

4πN r2rpe2εpe

cpe,i∂t∂

------------- ∂∂r----- r2Dpe,i

∂cpe,i∂r--------------–

+ r2rpe2Rpe,i=

Ni,inward hD i, ci cpe i,–( )=


where Ni, inward is the molar flux from the free fluid into a pellet and has the unit moles/(m2· s). The mass transfer coefficient to calculated automatically as described in the section Theory for the Reactive Pellet Bed in the Chemical Reaction Engineering Module User’s Guide.

The pressure drop in the reactor is also accounted for and is modeled with the Darcy’s Law interface.

In Table 1 the model parameters are tabulated.

TABLE 1: SUMMARY OF INPUT DATA

PROPERTY VALUE DESCRIPTION

Hr 1 [m] Height of the packed bed reactor

Rr 0.2 [m] Radius of packed bed reactor

ρb 0.51 [g/cm3] Density of packed bed

ρpe 0.68 [g/cm3] Density individual pellet

εb 1-ρb/ρpe Macroscale porosity (of bed)

εpe 0.70 (-) Microscale porosity (of pellet)

rpe 0.5 [mm] Pellet radius (spherical shape)

Dpe,A 1.5e-9 [m2/s] Diffusion coefficient of A in pellet

Dpe,B 2e-9 [m2/s] Diffusion coefficient of B in pellet

Dpe,C 0.5e-9 [m2/s] Diffusion coefficient of C in pellet

A 2e12 [m3/(mol s)] Frequency factor reaction

E 75000[J/mol] Activation energy reaction

Keq0 1000 Equilibrium reaction constant

kappa 1.88e-10[m2] Permeability of Bed

CA_in 1[mol/m3] Inlet concentration A

CB_in 1[mol/m3] inlet concentration B

CC_in 0[mol/m3] inlet concentration C

DA 1e-8 [m2/s] Diffusion coefficient of A in bed

DB 1.5e-8 [m2/s] Diffusion coefficient of B in bed

DC 0.5e-8 [m2/s] Diffusion coefficient of C in bed

pDarcy 0.4 [atm] Inlet pressure offset



The following figures display the results at 180 s. Figure 3 shows the velocity distribution in the fluid between the pellets.

Figure 3: Velocity distribution on the macroscale.

Figure 4 shows the macroscale concentration of the reactant A in the bed column fluid. The species is consumed due to the catalytic chemical reaction in the pellets.


Figure 4: Concentration of reactant A.

Streamline plots can be useful to get an understanding of the flow pattern. It can be seen from Figure 5 that no recirculation occurs at the entry holes. The fluid is evenly spread out in the bed chamber as it enters the holes in the bottom place.


Figure 5: The streamlines show how the fluid enters the holes and then spread out in the bed volume as it. The colors of the lines represent the reactant concentration in moles/m3.

A line plot of the concentration in a pellet at a certain position in the bed is interesting in order to understand the local reaction. Figure 6 shows the position at which the pellet line plot is sampled: (x = 0.5, z = 0, y = 0), and Figure 7 is the line plot of both the reactant and the product inside a pellet in the same position.


Figure 6: Coordinate at which the pellet plot is sampled: Center line of reactor and at 0.5 m height.

Figure 7: Concentration of species A, B and C within the pellet at 0.5 m bed height.


In Figure 8, the concentrations of the species are shown along the reactor height in the center of the geometry. Both the concentrations in the bed and averaged in the pellets are shown and illustrates the local reaction in detail. The species C concentration profiles portray a reaction intense zone within the reactor. A closer look at this zone shows it expanding towards the outlet with time.

Figure 8: Concentration of the species in reactor bed and averaged within the pellets along the reactor height.

Figure 9 shows a 3D concentration plot of the product C within the pellet at the sampled coordinate. It can be seen that the concentration is higher closer to the center of the pellet, where products build up and from where these diffuse into the bulk gas.


Figure 9: Concentration of species C at 0.5 m bed height.

Application Library path: Chemical_Reaction_Engineering_Module/Reactors_with_Porous_Catalysts/packed_bed_reactor_3d


Start by adding the necessary physics interfaces for a 3D model.


N E W






3 Click Add.



5 Click Add.



8 In the Select Physics tree, select Fluid Flow>Porous Media and Subsurface Flow>

Darcy’s Law (dl).

9 Click Add.

10 Click Study.

11 In the Select Study tree, select Preset Studies for Selected Physics Interfaces>

Time Dependent.

12 Click Done.

G E O M E T R Y 1

Add the model parameters from a text file.





4 Browse to the model’s Application Libraries folder and double-click the file packed_bed_reactor_3d_parameters.txt.

G E O M E T R Y 1

Now create the geometry. You can simplify this by inserting a prepared geometry sequence from a file with prepared geometry selections. You can read the instructions for building the geometry in the appendix.


cA

cB

cC


2 Browse to the model’s Application Libraries folder and double-click the file packed_bed_reactor_3d_geom_sequence.mph.

3 On the Geometry toolbar, click Build All.

Specify the material properties. Some properties can be found in the COMSOL built-in materials, other are manually entered.

Assume the reaction mixture has mainly aqueous properties.








Start with the Chemistry interface and create the needed reaction kinetics expressions by typing in the reaction formulas.

1 In the Settings window for Chemistry, click to expand the Mixture properties section.




3 In the Formula text field, type A+B<=>2C.

4 Click Apply.


6 Locate the Rate Constants section. Select the Specify equilibrium constant check box.

7 Select the Use Arrhenius expressions check box.

8 In the Af text field, type A.

9 In the Ef text field, type E.

10 Locate the Equilibrium Settings section. In the Keq0 text field, type Keq0.

The molar masses for the reacting species can be entered for possible future use. For example, if the mass-based Concentrations feature is used in the Transport of Diluted

Species interface, it can pick up the molar mass values from the Chemistry node automatically.

Species: A1 In the Model Builder window, under Component 1 (comp1)>Chemistry (chem) click

Species: A.



Species: B1 In the Model Builder window, under Component 1 (comp1)>Chemistry (chem) click

Species: B.



Species: C1 In the Model Builder window, under Component 1 (comp1)>Chemistry (chem) click

Species: C.


3 In the M text field, type Mn_C.

The reactive species are diluted in water. For completeness, add the solvent H2O, which does not partake in the reactions. It can be used later if the model is extended.





5 Locate the General Parameters section. In the M text field, type Mn_solvent.

Now tell the Chemistry interface which concentrations to use as input for the rate expressions. Select the pellet concentrations. The entries will at this stage appear yellow since the Reactive Pellet Bed feature is not yet created.





Select the Define variables in extra dimension check box because the Chemistry is coupled to the Reactive Pellet Bed feature which is defined in extra dimension.

9 Click to expand the Extra dimension section. Locate the Extra Dimension section. Select the Define variables in extra dimension check box.

Continue with the Transport of Diluted Species interface to set up the mass transport model.


On the Physics toolbar, click Chemistry (chem) and choose Transport of Diluted Species (tds).



3 Select the Mass transfer in porous media check box.




3 From the u list, choose Darcy’s velocity field (dl).

4 Locate the Diffusion section. In the DcA text field, type DA.

5 In the DcB text field, type DB.

6 In the DcC text field, type DC.

Add the Reactive Pellet Bed feature. A predefined extra dimension is attached to this feature. The extra dimension is 1D on the radial coordinate of the pellet particle of which the radius is normalized to 1. The mesh for the extra dimension has a default of 10 elements with a cubic root sequence distribution.



A Variable, from Reaction tds.rpb1.cpe_cA chem.R_A

B Variable, from Reaction tds.rpb1.cpe_cB chem.R_B

C Variable, from Reaction tds.rpb1.cpe_cC chem.R_C

H2O Constant, solvent C_solvent solvent


Reactive Pellet Bed 11 On the Physics toolbar, click Domains and choose Reactive Pellet Bed.

2 In the Settings window for Reactive Pellet Bed, locate the Domain Selection section.


4 Locate the Bed Parameters section. In the ρb text field, type rho_b.

5 In the ρpe text field, type rho_pe.

The densities for pellet and bed are used to calculate the bed porosity.

6 Locate the Pellet Shape and Size section. In the rpe text field, type r_pe.

7 Locate the Pellet Parameters section. In the εpe text field, type epsilon_pe.

8 From the Diffusion model list, choose User defined.

9 In the Dpeff, cA text field, type DAp.

10 In the Dpeff, cB text field, type DBp.

11 In the Dpeff, cC text field, type DCp.

Use a film theory condition to account for any film resistance to mass transfer between the bulk fluid and the pellet. Use spherical pellets.

12 Locate the Pellet-Fluid Surface section. From the Coupling type list, choose Film resistance (mass flux).

13 Locate the Pellet Discretization section. In the Nelem text field, type 6.

Use the reaction rates calculated in the Chemistry interface.

Reactions 11 In the Model Builder window, expand the Reactive Pellet Bed 1 node, then click

Reactions 1.


3 From the RcA list, choose Rate expression for species A (chem).

4 From the RcB list, choose Rate expression for species B (chem).

5 From the RcC list, choose Rate expression for species C (chem).




4 Locate the Concentration section. In the c0,cA text field, type CA_in.

5 In the c0,cB text field, type CB_in.


6 In the c0,cC text field, type CC_in.




D A R C Y ’ S L A W ( D L )

Last, enter the model specifications for the Darcy’s Law interface to compute the convective flow in the reactor.

Fluid and Matrix Properties 1Take the porosity directly from the Reactive Pellet Bed feature.

1 In the Model Builder window, under Component 1 (comp1)>Darcy’s Law (dl) click Fluid and Matrix Properties 1.

2 In the Settings window for Fluid and Matrix Properties, locate the Matrix Properties section.

3 From the εp list, choose User defined. In the associated text field, type tds.rpb1.epsilon_b.

4 From the κ list, choose User defined. In the associated text field, type kappa.

5 In the Model Builder window, click Darcy’s Law (dl).

Pressure 11 On the Physics toolbar, click Boundaries and choose Pressure.

2 In the Settings window for Pressure, locate the Boundary Selection section.





4 Locate the Pressure section. In the p0 text field, type p_Darcy.

This completes the set up of the model equations describing the reacting flow and heat transfer in the packed bed reactor. Before solving the problem numerically, the geometry needs to be meshed.

First create a free triangular mesh at the reactor inlet and sweep that mesh along the x-direction (the height) of the reactor.


M E S H 1

Free Triangular 11 In the Model Builder window, under Component 1 (comp1) right-click Mesh 1 and choose

More Operations>Free Triangular.

2 In the Settings window for Free Triangular, locate the Boundary Selection section.

3 From the Selection list, choose Bottom plate.

Size 11 Right-click Component 1 (comp1)>Mesh 1>Free Triangular 1 and choose Size.


3 From the Predefined list, choose Fine.

Distribution 11 In the Model Builder window, right-click Mesh 1 and choose Swept.

2 Right-click Swept 1 and choose Distribution.





7 Click Build All.

Since this is a one-way coupled problem, it can be solved in two steps in order to consume less memory: First solve the Darcy’s law interface for the velocity, which is a stationary problem. Then solve the Transport of Diluted Species interface with a time dependent study step.

S T U D Y 1

Step 1: Time Dependent1 In the Settings window for Time Dependent, locate the Study Settings section.


3 Locate the Physics and Variables Selection section. In the table, clear the Solve for check box for Darcy’s Law (dl).




3 In the table, clear the Solve for check box for Chemistry (chem) and Transport of Diluted

Species (tds).

4 In the Model Builder window, right-click Step 2: Stationary and choose Move Up.


R O O T

In the Model Builder window’s toolbar, click the Show button and select Advanced Results Options in the menu.

R E S U L T S

Create views for plotting different angles of the geometry.

1 In the Model Builder window, expand the Results node.

View 3D 51 Right-click Views and choose View 3D.

2 In the Settings window for View 3D, type Column view in the Label text field.

View 3D 61 Right-click Views and choose View 3D.

2 In the Settings window for View 3D, type Pellet view in the Label text field.

Data SetsCreate a data set that can be used to plot the column with a sector cut-out for better view.

Sector 3D 11 On the Results toolbar, click More Data Sets and choose Sector 3D.

2 In the Settings window for Sector 3D, locate the Axis Data section.

3 In row Point 2, set X to 1 and z to 0.

4 Locate the Symmetry section. In the Number of sectors text field, type 8.

5 From the Sectors to include list, choose Manual.

6 In the Number of sectors to include text field, type 5.

Adjust the view angle of the plot with the mouse. Optionally, the go to the Views -> Column view under Results and select the Lock camera check-box to save the view.

First create Figure 3 showing the velocity distribution in the reactor.

Pressure (dl)1 In the Model Builder window, under Results click Pressure (dl).


2 In the Settings window for 3D Plot Group, type Velocity in the Label text field.

3 Locate the Data section. From the Data set list, choose Sector 3D 1.

4 Locate the Plot Settings section. From the View list, choose Column view.

Slice1 In the Model Builder window, expand the Results>Velocity node, then click Slice.

2 In the Settings window for Slice, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Component 1>Darcy’s Law>

Velocity and pressure>dl.U - Darcy’s velocity magnitude.

3 Locate the Expression section. Clear the Description check box.

4 Locate the Plane Data section. In the Planes text field, type 8.

5 On the Velocity toolbar, click Plot.


Continue with Figure 4 illustrating the concentration of species A in the reactor.


2 In the Settings window for 3D Plot Group, type Concentration, surface in the Label text field.



5 On the Concentration, surface toolbar, click Plot.


Data SetsTo create the first x-y plot, a new data set is required for that purpose. Also, the special syntax comp1.atxd3 is needed to plot variables within the pellet.

Study 1/Solution 1 (sol1)In the Model Builder window, expand the Results>Data Sets node.

Study 1/Solution 1 (3) (sol1)1 Right-click Study 1/Solution 1 (sol1) and choose Duplicate.

2 In the Settings window for Solution, locate the Solution section.

3 From the Component list, choose Extra Dimension from Reactive Pellet Bed 1 (tds_rpb1_xdim).


2 In the Model Builder window, right-click 1D Plot Group 6 and choose Line Graph.

3 In the Settings window for 1D Plot Group, type Pellet x-y plot in the Label text field.


5 In the Title text area, type Pellet concentrations @ x=0.5[m], y=0, z=0.

6 Locate the Data section. From the Data set list, choose Study 1/Solution 1 (3) (sol1).

The syntax comp1.atxd3(0.5,0,0,comp1.tds.rpb1.cpe_cA) means that you visualize the internal pellet concentration 0.5 m from the inlet, and in the center of the column.

Line Graph 11 In the Model Builder window, under Results>Pellet x-y plot click Line Graph 1.

2 In the Settings window for Line Graph, type A in the Label text field.

3 Locate the Selection section. From the Selection list, choose All domains.

4 Locate the y-Axis Data section. In the Expression text field, type comp1.atxd3(0.5,0,0,comp1.tds.rpb1.cpe_cA).

5 Select the Description check box.

6 In the associated text field, type pellet (comp1.tds.rpb1.cpe_cA) @ x=0.5[m], y=0, z=0.





A 11 Right-click Results>Pellet x-y plot>A and choose Duplicate.

2 In the Settings window for Line Graph, type B in the Label text field.

3 Locate the y-Axis Data section. In the Expression text field, type comp1.atxd3(0.5,0,0,comp1.tds.rpb1.cpe_cB).

Legends

Pellet, cA

4 In the Description text field, type pellet (comp1.tds.rpb1.cpe_cB) @ x=0.5[m], y=0, z=0.


B 11 Right-click Results>Pellet x-y plot>B and choose Duplicate.

2 In the Settings window for Line Graph, type C in the Label text field.

3 Locate the y-Axis Data section. In the Expression text field, type comp1.atxd3(0.5,0,0,comp1.tds.rpb1.cpe_cC).

4 In the Description text field, type pellet (comp1.tds.rpb1.cpe_cC) @ x=0.5[m], y=0, z=0.


6 On the Pellet x-y plot toolbar, click Plot.

Pellet x-y plot1 In the Model Builder window, under Results click Pellet x-y plot.


3 From the Time selection list, choose Last.


5 In the associated text field, type Normalized Pellet Radius (r/rpe).


7 In the associated text field, type Concentration (mol/m3).

8 On the Pellet x-y plot toolbar, click Plot.


Next plot is created to visualize the difference in species’ average concentrations in the pellets and the reactor bed in the same plot. The figure requires a new data set.

1D Plot Group 71 On the Results toolbar, click Cut Line 3D.

2 On the Results toolbar, click 1D Plot Group.

Legends

Pellet cB

Legends

Pellet cC

3 In the Settings window for 1D Plot Group, type Concentration comparison in the Label text field.

4 Locate the Data section. From the Time selection list, choose Last.



7 In the Title text area, type Comparison between concentration in bed and average concentration in pellets.


9 In the associated text field, type Height coordinate from bottom at center of reactor (m).


11 In the associated text field, type Concentration (mol/m3).


Line Graph 11 Right-click Concentration comparison and choose Line Graph.

2 In the Settings window for Line Graph, type A, bed in the Label text field.

3 Locate the y-Axis Data section. In the Expression text field, type cA.

4 Click to expand the Coloring and style section. Locate the Coloring and Style section. Find the Line style subsection. From the Line list, choose Dashed.

5 From the Color list, choose Blue.





A, bed 11 Right-click Results>Concentration comparison>A, bed and choose Duplicate.

2 In the Settings window for Line Graph, type B, bed in the Label text field.

3 Locate the y-Axis Data section. In the Expression text field, type cB.

4 Locate the Coloring and Style section. From the Color list, choose Green.

Legends

cA

B, bed 11 Right-click Results>Concentration comparison>B, bed and choose Duplicate.

2 In the Settings window for Line Graph, type C, bed in the Label text field.

3 Locate the y-Axis Data section. In the Expression text field, type cC.

4 Locate the Coloring and Style section. From the Color list, choose Red.


C, bed 11 Right-click Results>Concentration comparison>C, bed and choose Duplicate.

2 In the Settings window for Line Graph, type A, pellet in the Label text field.

3 Locate the y-Axis Data section. In the Expression text field, type tds.rpb1.avecpe_cA.

4 Locate the Coloring and Style section. Find the Line style subsection. From the Line list, choose Solid.



A, pellet 11 Right-click Results>Concentration comparison>A, pellet and choose Duplicate.

2 In the Settings window for Line Graph, type B, pellet in the Label text field.

3 Locate the y-Axis Data section. In the Expression text field, type tds.rpb1.avecpe_cB.


5 From the Color list, choose Green.

Legends

cB

Legends

cC

Legends

Pellet average, cA

B, pellet 11 Right-click Results>Concentration comparison>B, pellet and choose Duplicate.

2 In the Settings window for Line Graph, type C, pellet in the Label text field.

3 Locate the y-Axis Data section. In the Expression text field, type tds.rpb1.avecpe_cC.


5 From the Color list, choose Red.


7 On the Concentration comparison toolbar, click Plot.


Last, create the figure showing the concentration of species C in a pellet. The pellet is located at (0.5,0,0) and is visualized with a new data set.

Cut Point 3D 11 On the Results toolbar, click Cut Point 3D.

Next step is to create the image showing the coordinates of the visualized pellet.

2 In the Settings window for Cut Point 3D, locate the Point Data section.

3 In the X text field, type 0.5.

4 In the Y text field, type 0.

5 In the Z text field, type 0.

6 Click Plot.




3 Locate the Revolution Layers section. In the Start angle text field, type -90.

Legends

Pellet average, cB

Legends

Pellet average, cC

4 In the Revolution angle text field, type 180.



3 Locate the Revolution Layers section. In the Revolution angle text field, type 270.


2 In the Settings window for 3D Plot Group, type Pellet 3D plot in the Label text field.

3 Locate the Data section. From the Data set list, choose Revolution 2D 2.

4 Locate the Plot Settings section. From the View list, choose Pellet view.

5 Clear the Plot data set edges check box.

Surface 11 Right-click Pellet 3D plot and choose Surface.


3 In the Expression text field, type comp1.atxd3(0.5,0,0,comp1.tds.rpb1.cpe_cC).


5 In the Title text area, type Pellet concentration of species C (mol/m3) at x=0.5[m], y=0, z=0.

Adjust the view angle of the plot with the mouse. Optionally, the go to the Views - Pellet under Results and select the Lock camera check-box to save the view.

6 On the Pellet 3D plot toolbar, click Plot.



2 In the Settings window for 3D Plot Group, type Concentration, isosurface in the Label text field.



Slice 11 In the Model Builder window, expand the Results>Concentration, isosurface node.

2 Right-click Slice 1 and choose Disable.

Isosurface 11 In the Model Builder window, under Results right-click Concentration, isosurface and

choose Isosurface.

2 In the Settings window for Isosurface, locate the Levels section.

3 In the Total levels text field, type 10.


5 On the Concentration, isosurface toolbar, click Plot.


2 In the Settings window for 3D Plot Group, type Concentration, slice in the Label text field.



5 On the Concentration, slice toolbar, click Plot.

Pressure (dl) 11 In the Model Builder window, under Results click Pressure (dl) 1.


3 From the View list, choose Column view.


5 On the Pressure (dl) 1 toolbar, click Plot.

Concentration, surfaceCreate a streamline plot. For high plot performance it is good to make them start on a cut plane above the bottom.

Concentration, surface 11 In the Model Builder window, under Results right-click Concentration, surface and choose

Duplicate.

2 In the Settings window for 3D Plot Group, type Streamlines in the Label text field.

Surface1 In the Model Builder window, expand the Results>Streamlines node.

2 Right-click Surface and choose Disable.



2 In the Settings window for Cut Plane, locate the Data section.

3 From the Data set list, choose Sector 3D 1.

4 Locate the Plane Data section. In the X-coordinate text field, type 0.005.

Streamline 11 In the Model Builder window, under Results right-click Streamlines and choose

Streamline.


3 In the Points text field, type 1000.

4 From the Along curve or surface list, choose Cut Plane 1.


6 In the Tube radius expression text field, type cA[m^4/mol].

7 Select the Radius scale factor check box.

8 In the associated text field, type .004.

Color Expression 11 Right-click Results>Streamlines>Streamline 1 and choose Color Expression.

You can zoom in by pressing down the middle mouse button and moving the mouse forward. Hold down the Ctrl-button to dolly in the camera position.

2 On the Streamlines toolbar, click Plot.




Po l yme r i z a t i o n i n Mu l t i j e t T ubu l a r R e a c t o r




Introduction

Production processes for polymers often involve turbulent flows and rapid reaction kinetics. The sophisticated interplay between fluid dynamics and fast chemical reactions can significantly impact the reactor performance, and thereby affect conversion and yield. Furthermore, the turbulent fluid mixing and its effects on the reaction can influence the average length of polymer chains, the molecular weight distribution, cross-linking, and chain-branching. All these properties are important for the integrity of the final material. This example demonstrates a polyester reactor, with multiple inlets, and includes heat transfer and temperature dependent kinetics. It employs the eddy dissipation concept (EDC), a model for the mean reaction rate in turbulent flows.

Note: This application requires both the Chemical Reaction Engineering Module and the CFD Module.

Model Definition

G E O M E T R Y

The geometry of the inlet section of a multijet tubular reactor is illustrated in Figure 1.

Figure 1: Inlet section of a multijet tubular reactor. Monomer A (diol) enters through the axial inlets while monomer B (diacid) enters through the radial ports.

Two reacting monomers enter through separate inlet ports. Monomer A enters through the axial inlets while monomer B enters through the radial ports.

B

A

2 | P O L Y M E R I Z A T I O N I N M U L T I J E T TU B U L A R R E A C T O R

C H E M I S T R Y

Condensation reactions are fundamental to the production of many important polymers, such as polyamides, polyesters, polyurethanes, and silicones.

This model simulates a polyester reactor. Condensation polymerization of monomers A (a diol) and B (a diacid), forms the polyester linkage, L (Ref. 1, Ref. 2). The reactions take place in the presence of a solvent catalyst, S.

The catalytic species, S, is temporarily trapped in an intermediary H2O complex, S · C, where C represents the complex-forming water in the irreversible reaction

(1)

The regeneration of solvent is governed by the reversible reaction

(2)

The reaction rates for each chemical reaction is determined by the law of mass action and the eddy dissipation concept (EDC) model. The law of mass action gives the rates (mol/(m3·s))

(3)

and

(4)

TABLE 1: SPECIES USED ON THE MODEL

NAME DESCRIPTION

A Diol monomer

B Diacid monomer

L Polyester linkage (product)

S Solvent catalyst (TiCl3)

C Complexating water

k1f

2A B S+ + → L 2SC+

k2f

A SC+ ⇔ S AC+

k2r

r1 k1f cA

2 cBcS=

r2 k2f cAcSC k2

r cScAC–=


for reactions Equation 1and Equation 2, respectively, where the rate constants are given by the Arrhenius expression

(5)

In Equation 5, Aj is the frequency factor and Ej the activation energy (J/mol) for the jth reaction. The table below lists the values of the Arrhenius parameters for the reactions. The rates are adjusted for turbulent conditions according to the EDC model: If the time scale of the turbulent mixing is larger than the reaction kinetics derived by the law of mass action above, the turbulent mixing will be rate determining. For detailed information, see the section Eddy Dissipation Model in the CFD Module User’s Guide.

TR A N S P O R T

The 3D model geometry is illustrated in Figure 1.

Velocities and PressureThe velocities at the radial and axial inlets are set to 5 m/s. Furthermore, a constant pressure is set at the outlet and logarithmic wall functions are specified at the solid walls.

Mass TransportConcentration boundary conditions apply at the inlets:

(6)

The catalytic solvent S is set as solvent in the mass transport model.

Energy TransportThe reactor is assumed to be insulated at the walls and all inlet streams are specified to 440 K temperature.

kj Aj

EjRgT-----------–

exp=

cA 1200= mol/m3 at axial inlets

cB 1000= mol/m3 at radial inlets


Summary of Input Data

For the rate expressions in Equation 3 and Equation 4 the following data is used (Ref. 1):

The material properties and boundary conditions used are (Ref. 1 and Ref. 2).

Modeling in COMSOL

• For the 3D model, the Reacting Flow, Turbulent interface is used for the mass transport, reactions, and fluid flow simulation. The Heat Transfer in Fluids interface is used to do the heat transfer simulation including the heat of reactions, coupled with the reacting flow.

TABLE 2: KINETIC DATA

QUANTITY FREQUENCY FACTOR ACTIVATION ENERGY TURBULENTPARAMETERS α AND β

Forward Reaction 1 25.6 61.3[kJ/mol] 4, 0.5

Forward reaction 2 3.9e3 56.8[kJ/mol] 4, 0.5

Reverse reaction 2 4.7e3 102[kJ/mol] 4, 0.5

TABLE 6-1: INPUT DATA

PROPERTY VALUE

Diffusivity 1e-8[m^2/s]

Density of catalyst solvent 2640[kg/m^3]

Heat capacity of catalyst solvent 2550[J/kg/K]

Inlet velocity 5[m/s]

Inlet temperature 440[K]

Molar mass, monomer A 48[g/mol]

Molar mass, monomer B 104[g/mol]

Molar mass, complexating H2O 18[g/mol]

Molar mass, polymer L 164[g/mol]

Molar mass, catalyst S 154[g/mol]

Molar mass, catalytic species complex SC 172[g/mol]

Molar mass, species complex AC 66[g/mol]

Heat of reaction, Reaction 1 100[kJ/mol]

Heat of reaction, Reaction 2 40[kJ/mol]


S T A G E D S O L U T I O N

Since the chemical reactions are strongly depending on the fluid movement, the fully coupled system may be difficult to converge in the first iterations due insufficient start guesses on the velocity field. Therefore the following staged solutions is used. Each study step uses the converged solution from the previous step as a start guess:

1 Velocity and pressure only.

2 Velocity, pressure, concentrations distribution including reactions. Isothermal.

3 Temperature only, including heat of reaction.

4 All variables.

G E O M E T R Y

Thanks to symmetry observations, a sector of one 1/20 of the geometry shown in Figure 1 is modeled. The modeling results are rotated to the full geometry by sector datasets.

M E S H

The mesh is calibrated to resolve the shear layers that appear near the inlets of the reactor. Further downstream where the flow profile is expected to be more uniform, a simpler extruded mesh is used to save time and memory.



Results of the flow field calculations are presented first. Figure 2 shows the velocity field in the multijet tubular reactor, plotted in two perpendicular planes through the reactor.

Figure 2: Velocity field (m/s) in the multijet tubular reactor.

The plot illustrates the impinging axial and radial jets.


Plotting the streamlines of the velocity field provides additional information, indicating flow paths. Figure 3 shows such a plot. Closer inspection at the entrance of the reactor reveals several recirculation zones.

Figure 3: Streamlines of the velocity field shows some recirculation behavior near the inlet orifices. The concentration of reactants decrease rapidly at after the inlet stretch.


Next, mass is transported with the calculated flow field. Once monomer A comes into contact with the radial streams of monomer B, polymerization starts. Figure 4 shows the concentration field of monomer A.

Figure 4: Concentration distribution of monomer A (mol/m3).

Figure 5 shows isosurfaces for the polymer linkage L concentration. Isolevels at the entrance of the reactor clearly mark the positions of where the inlet streams mix. However, the azimuthal concentration gradients increase quickly with axial position, indicating that


inlet streams are well mixed for reaction to take place approximately 5 cm down the reactor.

Figure 5: Isosurfaces for the concentration of L (mol/m3).

As mentioned above, recirculation is evident in the entrance of the reactor. Recirculation will increase the effective residence time of the reactor. Figure 6 shows the concentration of polymer linkage, cL, with a surface slice plot. The turbulence time scale that is accounted for in the reaction kinetic model according to the EDC concept affects the prediction of reactor turnover.


Figure 6: Concentration distribution of polymer linkage, cL (mol/m3).

Clearly, the concentration of L is relatively low in the recirculation region. In polymerization processes, increasing linkage concentration can lead to dramatic changes in the properties of the reacting fluid, particularly viscosity. This in turn may cause fouling or even reactor failure.

Figure 7 shows the concentration of product L in a cross section plot along the axis of the reactor. The recirculation effects in the beginning of the multijet tubular model are evident. Results also point to the influence of mixing on the reaction rate. The mixing in the space-dependent reactor is influenced by the detailed flow field.


Figure 7: Concentration of polymer linkage, cL, as a function of axial position in the reactor. The space-dependent model accounts for recirculation effects near the reactor inlet.


The total condensation chemistry is endothermic. Figure 8 displays the resulting temperature field in the reactor.

Figure 8: Temperature distribution in the multijet tubular reactor. The inlet temperatures of radial and axial streams are 440 K.

The endothermic reactions efficiently cool down the reacting flow.

References

1. N.H. Kolhapure, J.N. Tilton, and C.J. Pereira, “Integration of CFD and condensation polymerization chemistry for a commercial multi-jet tubular reactor,” Chem. Eng. Sci., vol. 59, p. 5177, 2004.

2. http://en.wikipedia.org/wiki/Polyester.

Application Library path: Chemical_Reaction_Engineering_Module/Reactors_with_Mass_and_Heat_Transfer/polymerization_multijet




N E W




2 In the Select Physics tree, select Chemical Species Transport>Reacting Flow>

Turbulent Flow>Turbulent Flow, k-ε.

3 Click Add.

4 In the Added physics interfaces tree, select Transport of Concentrated Species (tcs).


6 In the Mass fractions table, enter the following settings:


8 Click Add.

9 Click Study.


11 Click Done.





4 Browse to the model’s Application Libraries folder and double-click the file polymerization_multijet_parameters.txt.

wA

wB

wS

wL

wSC

wAC


G E O M E T R Y 1

Start by creating the geometry. You can simplify this by inserting a prepared geometry sequence from file. You can read the instruction for building the geometry in the appendix.


2 Browse to the model’s Application Libraries folder and double-click the file polymerization_multijet_geom_sequence.mph.

Mesh Control Domains 1 (mcd1)1 On the Geometry toolbar, click Virtual Operations and choose Mesh Control Domains.


Mesh Control Faces 1 (mcf1)1 On the Geometry toolbar, click Virtual Operations and choose Mesh Control Faces.

2 On the object mcd1, select Boundary 11 only.

Mesh Control Domains 1 (mcd1)1 In the Model Builder window, under Component 1 (comp1)>Geometry 1 click

Mesh Control Domains 1 (mcd1).

2 In the Settings window for Mesh Control Domains, locate the Input section.

3 Find the Domains to include subsection. Select the Active toggle button.

4 On the Geometry toolbar, click Build All.


Fluid Properties 11 In the Model Builder window, under Component 1 (comp1)>Turbulent Flow, k-ε (spf) click

Fluid Properties 1.


3 From the μ list, choose User defined. In the associated text field, type 0.001*(1.17817558982837+(-298[K]+T)/223[K])^(-3.758)[Pa*s].

Initial Values 11 In the Model Builder window, under Component 1 (comp1)>Turbulent Flow, k-ε (spf) click

Initial Values 1.


3 In the k text field, type 7e-8.

4 In the ep text field, type 1e-11.







4 In the U0 text field, type 5.

5 Locate the Turbulence Conditions section. In the IT text field, type 0.05.

6 In the LT text field, type 0.01[m].






3 From the Diffusion model list, choose Fick’s law.

4 Locate the Species section. From the From mass constraint list, choose wS.





4 Locate the Density section. From the ρ list, choose User defined. In the associated text field, type rho_S.

5 In the MwA text field, type MwA.

6 In the MwB text field, type MwB.

7 In the MwS text field, type MwS.

8 In the MwL text field, type MwL.

9 In the MwSC text field, type MwSC.


10 In the MwAC text field, type MwAC.

11 Locate the Diffusion section. In the DfwA text field, type D.

12 In the DfwB text field, type D.

13 In the DfwS text field, type D.

14 In the DfwL text field, type D.

15 In the DfwSC text field, type D.

16 In the DfwAC text field, type D.


Transport of Concentrated Species (tcs) click Initial Values 1.


3 In the ω0,wA text field, type 1e-6.

4 In the ω0,wB text field, type 1e-6.

5 In the ω0,wL text field, type 1e-6.

6 In the ω0,wSC text field, type 1e-6.

7 In the ω0,wAC text field, type 1e-6.




4 From the Mixture specification list, choose Molar concentrations.

5 In the c0,wA text field, type 1200[mol/m^3].

6 In the c0,wB text field, type 1e-3[mol/m^3].

7 In the c0,wL text field, type 1e-3[mol/m^3].

8 In the c0,wSC text field, type 1e-3[mol/m^3].

9 In the c0,wAC text field, type 1e-3[mol/m^3].




4 From the Mixture specification list, choose Molar concentrations.


5 In the c0,wA text field, type 1e-3[mol/m^3].

6 In the c0,wB text field, type 1000[mol/m^3].

7 In the c0,wL text field, type 1e-3[mol/m^3].

8 In the c0,wSC text field, type 1e-3[mol/m^3].

9 In the c0,wAC text field, type 1e-3[mol/m^3].







3 In the Settings window for Reaction, locate the Reaction Rate section.

4 In the νwA text field, type -2.

5 In the νwB text field, type -1.

6 In the νwS text field, type -1.

7 In the νwL text field, type 1.

8 In the νwSC text field, type 2.


10 In the Af text field, type 25.6.


12 In the Ar text field, type 0.

13 Locate the Turbulent Flow section. From the Turbulent-reaction model list, choose Eddy-

dissipation.




4 In the νwA text field, type -1.


5 In the νwS text field, type 1.

6 In the νwSC text field, type -1.

7 In the νwAC text field, type 1.




11 In the Ar text field, type 4.7e3.

12 In the Er text field, type 102e3.


dissipation.



click Fluid 1.



4 From the u list, choose Velocity field (spf).

5 Locate the Heat Conduction, Fluid section. From the k list, choose User defined. In the associated text field, type 0.21+Cp_S*spf.muT/0.72.

6 Locate the Thermodynamics, Fluid section. From the ρ list, choose Density (tcs/cdm1).

7 From the Cp list, choose User defined. In the associated text field, type Cp_S.




2 In the Settings window for Initial Values, type 440[K] in the T text field.




4 In the T0 text field, type 440[K].









4 In the Q0 text field, type -100[kJ/mol]*tcs.treac1.r-40[kJ/mol]*tcs.treac2.r.

M E S H 1


Free Tetrahedral.


3 From the Calibrate for list, choose Fluid dynamics.

Free Tetrahedral 11 In the Model Builder window, under Component 1 (comp1)>Mesh 1 click

Free Tetrahedral 1.

2 In the Settings window for Free Tetrahedral, locate the Domain Selection section.


4 Select Domains 1 and 3–5 only.

Size 11 Right-click Component 1 (comp1)>Mesh 1>Free Tetrahedral 1 and choose Size.


3 From the Geometric entity level list, choose Edge.

4 Select Edges 13, 14, 22, 23, 31, 33–35, 38, 40, 42, and 43 only.

5 Locate the Element Size section. From the Calibrate for list, choose Fluid dynamics.



Swept 11 In the Model Builder window, right-click Mesh 1 and choose Swept.

2 In the Settings window for Swept, locate the Domain Selection section.



Distribution 11 Right-click Component 1 (comp1)>Mesh 1>Swept 1 and choose Distribution.





Boundary Layer Properties1 In the Model Builder window, right-click Mesh 1 and choose Boundary Layers.

2 Select Boundaries 2, 3, 6, 7, 9, 10, 14, 19, and 22–24 only.

3 In the Settings window for Boundary Layer Properties, locate the Boundary Layer Properties section.

4 In the Number of boundary layers text field, type 6.

5 In the Thickness adjustment factor text field, type 2.5.

6 Click Build All.

S T U D Y 1



3 In the Physics interface table, clear the Solve for check box for Transport of Concentrated

Species (tcs) and Heat Transfer in Fluids (ht).

4 In the Multiphysics table, clear the Solve for check box for Reacting Flow 1 (rf1).



3 In the Physics interface table, clear the Solve for check box for Turbulent Flow, k-ε (spf) and Heat Transfer in Fluids (ht).




3 In the Physics interface table, clear the Solve for check box for Turbulent Flow, k-ε (spf) and Transport of Concentrated Species (tcs).

4 In the Multiphysics table, clear the Solve for check box for Reacting Flow 1 (rf1).

Solution 1 (sol1)1 On the Study toolbar, click Study Steps and choose Stationary>Stationary.

2 On the Study toolbar, click Show Default Solver.

3 In the Model Builder window, expand the Solution 1 (sol1) node, then click Stationary Solver 1.

4 In the Settings window for Stationary Solver, locate the General section.

5 In the Relative tolerance text field, type 0.0001.

6 In the Model Builder window, under Study 1>Solver Configurations>Solution 1 (sol1) click Stationary Solver 2.













R E S U L T S

In the Model Builder window, expand the Results node.



2 In the Settings window for Sector 3D, locate the Data section.

3 From the Data set list, choose Study 1/Solution Store 3 (sol4).

4 Locate the Axis Data section. In row Point 2, set X to 1 and z to 0.

5 Locate the Symmetry section. In the Number of sectors text field, type 20.

6 From the Transformation list, choose Rotation and reflection.

7 Find the Radial direction of reflection plane subsection. In the X text field, type 0.

8 In the Z text field, type 1.

9 Click Plot.



2 In the Settings window for Cut Plane, locate the Data section.


4 Locate the Plane Data section. From the Plane list, choose XY-planes.

5 Click Plot.




4 Locate the Line Data section. In row Point 2, set X to 0.4.

5 Click Plot.


Figure 2 is created with the following steps.


2 In the Settings window for 3D Plot Group, type Velocity, xy plane in the Label text field.



Slice 11 Right-click Velocity, xy plane and choose Slice.




5 On the Velocity, xy plane toolbar, click Plot.


Figure 4 showing the monomer A concentration is reproduced in the following way.

Velocity, xy plane 11 In the Model Builder window, under Results right-click Velocity, xy plane and choose

Duplicate.

2 In the Settings window for 3D Plot Group, type Concentration, A in the Label text field.

Slice 11 In the Model Builder window, expand the Results>Concentration, A node, then click

Slice 1.


Transport of Concentrated Species>tcs.c_wA - Molar concentration.


Slice 21 Right-click Results>Concentration, A>Slice 1 and choose Duplicate.


3 From the Plane list, choose ZX-planes.



6 On the Concentration, A toolbar, click Plot.


Figure 6 showing the polymer linkage L concentration is reproduced in the following way.

Concentration, A 11 In the Model Builder window, under Results right-click Concentration, A and choose

Duplicate.


2 In the Settings window for 3D Plot Group, type Concentration, L in the Label text field.

Slice 11 In the Model Builder window, expand the Results>Concentration, L node, then click

Slice 1.


Transport of Concentrated Species>tcs.c_wL - Molar concentration.


Slice 2In the Model Builder window, under Results>Concentration, L right-click Slice 2 and choose Disable.

Concentration, L1 Click the Zoom Extents button on the Graphics toolbar.

2 In the Model Builder window, under Results click Concentration, L.

3 On the Concentration, L toolbar, click Plot.

Concentration, AFigure 8 shows the temperature within the reactor and is created with these steps.

Concentration, A 11 In the Model Builder window, under Results right-click Concentration, A and choose

Duplicate.


Slice 11 In the Model Builder window, expand the Results>Temperature node, then click Slice 1.



Slice 21 In the Model Builder window, under Results>Temperature click Slice 2.






Use a 1D Plot Group to create Figure 7, showing the axial concentration distribution of L.


2 In the Settings window for 1D Plot Group, type Concentration, L (axial) in the Label text field.


Line Graph 11 Right-click Concentration, L (axial) and choose Line Graph.




4 On the Concentration, L (axial) toolbar, click Plot.


Figure 3 shows the velocity profiles with the aid of a streamline plot and can be reproduced following these steps.


2 In the Settings window for 3D Plot Group, type Velocity streamlines in the Label text field.


Streamline 11 Right-click Velocity streamlines and choose Streamline.


3 In the Points text field, type 150.


5 In the Tube radius expression text field, type tcs.c_wA+tcs.c_wB.

Color Expression 11 Right-click Results>Velocity streamlines>Streamline 1 and choose Color Expression.


2 In the Settings window for Color Expression, locate the Expression section.

3 In the Expression text field, type tcs.c_wA+tcs.c_wB.

4 On the Velocity streamlines toolbar, click Plot.


Adjust the view angle of the plot with the mouse.

Velocity streamlines1 In the Model Builder window, under Results click Velocity streamlines.

2 Click Plot.

Figure 7 shows the isosurface concentration of L. Follow these step to create this figure.


2 In the Settings window for 3D Plot Group, type Concentration, L (isosurface) in the Label text field.


Isosurface 11 Right-click Concentration, L (isosurface) and choose Isosurface.

2 In the Settings window for Isosurface, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Component 1>


3 Locate the Levels section. In the Total levels text field, type 8.


5 On the Concentration, L (isosurface) toolbar, click Plot.

Appending — Geometry Modeling Instructions


N E W




2 Click Done.


G E O M E T R Y 1

Cylinder 1 (cyl1)1 On the Geometry toolbar, click Cylinder.

2 In the Settings window for Cylinder, locate the Size and Shape section.

3 In the Radius text field, type 0.005.


5 Locate the Position section. In the x text field, type 0.01318.

6 In the z text field, type 0.0205.


Rotate 1 (rot1)1 On the Geometry toolbar, click Transforms and choose Rotate.

Click the object to select it.

2 Select the object cyl1 only.

3 In the Settings window for Rotate, locate the Rotation Angle section.

4 In the Rotation text field, type -19.2.

5 Locate the Point on Axis of Rotation section. In the x text field, type 0.01318.


7 Locate the Axis of Rotation section. From the Axis type list, choose y-axis.


Cylinder 2 (cyl2)1 On the Geometry toolbar, click Cylinder.

2 In the Settings window for Cylinder, locate the Size and Shape section.



5 Locate the Position section. In the x text field, type -0.03.


7 Locate the Axis section. From the Axis type list, choose x-axis.


Extrude 1 (ext1)1 On the Geometry toolbar, click Extrude.

Select the far edge of the lying cylinder to add face 4 to the list.


2 On the object cyl2, select Boundary 4 only.

3 In the Settings window for Extrude, locate the Distances section.


5 Click to expand the Scales section. In the table, enter the following settings:

The scales creates a slightly tapered cylinder section.

6 In the Settings window for Extrude, click Build Selected.


Now click both geometry parts to add them to the selection list.

2 Click in the Graphics window and then press Ctrl+A to select both objects.



Distances (m)

0.016

Scales xw Scales yw

.9 .9



Work Plane 1 (wp1)1 On the Geometry toolbar, click Work Plane.

2 In the Settings window for Work Plane, locate the Plane Definition section.

3 From the Plane list, choose zx-plane.

Partition Objects 1 (par1)1 On the Geometry toolbar, click Booleans and Partitions and choose Partition Objects.

2 Select the object uni1 only.

3 In the Settings window for Partition Objects, locate the Partition Objects section.

4 From the Partition with list, choose Work plane.


6 Click the Select Domains button on the Graphics toolbar.

7 On the object par1, select Domain 2 only.

8 On the Geometry toolbar, click Delete.

Work Plane 2 (wp2)1 On the Geometry toolbar, click Work Plane.

2 In the Settings window for Work Plane, locate the Plane Definition section.


3 From the Plane list, choose yz-plane.

4 Click Show Work Plane.

Circle 1 (c1)1 On the Work Plane toolbar, click Primitives and choose Circle.

2 In the Settings window for Circle, locate the Size and Shape section.


4 In the Sector angle text field, type 18.

5 Locate the Rotation Angle section. In the Rotation text field, type 90.


Work Plane 2 (wp2)In the Model Builder window, under Component 1 (comp1)>Geometry 1 click Work Plane 2 (wp2).

Extrude 2 (ext2)1 On the Geometry toolbar, click Extrude.

2 In the Settings window for Extrude, locate the Distances section.



Form Union (fin)1 In the Model Builder window, under Component 1 (comp1)>Geometry 1 click

Form Union (fin).

2 In the Settings window for Form Union/Assembly, click Build Selected.

Distances (m)

.1

.3



P r o t e i n Ad s o r p t i o n




This application simulates an ion-exchange column for protein adsorption. The fluid phase contains four components: two proteins, solvent, and one salt. The adsorption/desorption kinetics is described by two equilibrium reactions where proteins displace ions adsorbed at the surface and vice versa.

The example highlights how reactions at chemical equilibrium can be studied in a 0D reactor system in Reaction Engineering. In addition, it also shows how the kinetics from the 0D setup is exported to a 3D model where the reacting surface in the column can be studied in detail. The 3D model incorporates mass transport through diffusion and convection, and the reactions at the surface of the ion-exchange mass with Transport of Diluted Species, Free and Porous Media Flow, and Surface Reactions interfaces.

Introduction Protein Ion-Exchange

The binding of proteins to ion exchangers can be described within the steric mass action approximation (SMA). This approach assumes that the adsorption of a protein can be considered as an exchange reaction of the protein with a given number of adsorbed ions.

Figure 1: Proteins A and B displacing salt ions S at an ion-exchanger surface.

The equilibrium describing the adsorption/desorption reactions is

(1)

Here, S denotes the salt ion, P stands for either protein A and B. Once P is adsorbed, P(ads), salt ions are displaced, reducing the concentration of adsorbed salt ions, S(ads).

S(ads)+P P(ads)+S⇔

2 | P R O T E I N A D S O R P T I O N

Model Definition

The system is modeled both in 0D and in 3D. The former model setup is adequate to investigate the kinetics of the equilibrium reactions within the column. The latter makes it possible to study the surface of the ion-exchange beads that make up the porous structure of the ion-exchange mass. In Figure 2, the two model approaches are presented.

Figure 2: Ion-exchange column and model geometries. The 0D model approximates the entire column, while the 3D geometry is a detailed representation of a section at the top of the column.

The 0D model utilizes the reactor type CSTR, constant volume in the Reaction Engineering interface. Ideal conditions are assumed within the reactor, meaning that well-mixed conditions with no concentration gradients apply. The reaction kinetics are described by Equation 1 for proteins A and B are entered directly into the interface as equilibrium surface reactions. The equilibrium constants for reaction 1, adsorption of A, is K1

eq = 2 and for reaction 2, adsorption of B, is K2eq = 5. These are entered as well,

relating the concentrations as followed

(2)

Outlet

Inlet

3D

0D

CSTR constant volume

Outlet, salt

Inlet, proteins

ion exchangeresin (beads)

K1eq cScA ads( )

cAcS ads( )-----------------------=


(3)

In order to compute the concentrations, both protein surface concentrations need to be set as dependent in the Equilibrium Species Vector section.

The proteins enter the reactor with a Feed Inlet with concentrations that vary in accordance to a 10 s Gaussian pulse with a maximum of 0.05 mol/m3. The outlet flow rate regulates so that the volume of the reactor is constant. Initially, no protein is available in the column and the ion-exchange mass is set to be completely adsorbed with salt, i.e. the initial site density, γ0, is equal to the initial surface concentration of S, S(ads).

For the 3D model, as shown in Figure 3, only one quarter of the top section of the column is simulated due to symmetry. The proteins enter the top with a constant concentration of 0.05 mol/m3. In the column initially, the pores (the bulk) is filled with solvent and the ion-exchange beads are saturated with adsorbed salt.

Figure 3: 3D model geometry.

The reaction kinetics are taken directly from the 0D model with the Generate Space-Dependent Model feature and are collected in a Chemistry interface that also supplies computed diffusion coefficients and fluid density for the 3D setup.

K2eq cScB ads( )

cBcS ads( )-----------------------=


The mass balances describing the species mass transport in the pores of the column are set up with the Transport of Diluted Species interface with diffusion and convection accounted for

(4)

On the right hand side of Equation 4 no reaction source is present for the bulk. However, the surface of the ion-exchange beads produces a reaction source that needs to be coupled to this equation. This is simply done with a Surface Reaction interface. Such an interface is always automatically generated with the use of the Generate Space-Dependent Model feature if surface reactions are present.

The 3D model also computes the convective velocity, u, with a Free and Porous Media Flow interface. The velocity is based on the assumption that the velocity, U, within the reactor is 0.1 mm/s and that the inlet is open to the surroundings, i.e. exposed to the atmospheric pressure.

S T U D Y S E T T I N G S

The 3D problem is solved in a two-step study. First, the Free and Porous Media Flow interface is first solved with a Stationary study step. Second, the rest of the interfaces (Chemistry, Transport of Diluted Species, and Surface Reactions) are solved with a Time Dependent study step. A Fully Coupled Direct solver is required in the second step to obtain stable computations.

Study 1—Space−independent

Figure 4 shows how the concentrations of the reacting species change with the time. Initially, only adsorbed salt species are present in the column. The concentrations of proteins A and B are seen to change with the Gaussian concentration pulse feed inlet. A stronger adsorption affinity of protein B compared to protein A is readily observed. Note also how the concentration of bulk salt species S increases as the proteins adsorb at the

ci∂t∂------- ∇+ D– i∇ci( ) u ∇ci⋅+⋅ 0=


surface. Towards the end of the pulse, most proteins have been adsorbed in the column and the bulk salt species have exited the system.

Figure 4: Concentrations of the reacting species as functions of time (s).

Study 2—Space−dependent

The 3D model is solved for 30 s and a selection of the results are displayed in this section. Figure 5, Figure 6, Figure 7, and Figure 8 show the behavior of protein B after 5 s and 30 s.

A comparison between Figure 5 and Figure 6, in which the bulk concentration is shown, indicate that the beads at the center of the column are less accessible for adsorption or that


the protein is more rapidly adsorbed at the center, both phenomena lowering the bulk concentration there.

Figure 5: Bulk concentration of B at 5 s.


Figure 6: Bulk concentration of B at 30 s.

To get a proper understanding of the ion-exchange beads, a comparison of the adsorbed B surface concentration at the two times is also made with the results in Figure 7 and Figure 8. The lower surface concentration of B in the center suggests that less B is adsorbed there and that the porous structure obstructs the species transport. Note that the


bead surface cannot adsorb more than the (initial) site density of the ion-exchange material, γ0 (SI unit: mol/m2).

Figure 7: Surface concentration of B at 5 s.


Figure 8: Surface concentration of B at 30 s.


In Figure 9, the velocity field is displayed. It shows that the porous structure causes a quite distorted velocity field. The exception is at the walls where the flow is less obstructed due to the relatively large gap between beads and wall.

Figure 9: Velocity field in the pores of the column section.

Application Library path: Chemical_Reaction_Engineering_Module/Mixing_and_Separation/protein_adsorption



N E W






3 Click Add.

4 Click Study.


6 Click Done.


Import the model parameters from a file.




4 Browse to the model’s Application Libraries folder and double-click the file protein_adsorption_parameters.txt.

Gaussian Pulse 1 (gp1)Reactants are injected to the system in a pulse. Choose a Gaussian Pulse function to describe the injection.

1 On the Home toolbar, click Functions and choose Global>Gaussian Pulse.

2 In the Settings window for Gaussian Pulse, locate the Parameters section.

3 In the Location text field, type 5.

4 In the Standard deviation text field, type 1.5.

Use the Gaussian Pulse function to set up a time-dependent pulse variable with an amplitude of 1.



choose Variables.




pulse11 3.76*gp1(t/1[s]) Pulse with amplitude 1



Select a constant volume CSTR to model the ion-exchange column in 0D.



3 From the Reactor type list, choose CSTR, constant volume.


Reaction 1Set up the equilibrium reaction for adsorption/desorption of protein A.

1 On the Reaction Engineering toolbar, click Reaction.


3 In the Formula text field, type S(ads)+A=S+A(ads).

4 Click Apply.



In similar fashion, set up the equilibrium reaction for adsorption/desorption of protein B.


3 In the Formula text field, type S(ads)+B=S+B(ads).

4 Click Apply.


Replace the default dependent variables and replace these with A(ads) and B(ads). This omits any cyclic dependence of the reaction rates.


7 In the Settings window for Reaction Engineering, click to expand the Equilibrium species vector section.

8 Locate the Equilibrium Species Vector section. In the Predefined dependent species (separated by ’,’) text field, type A(ads),B(ads).

9 Locate the Mass Balance section. From the Volumetric rate list, choose User defined.

Set vp=0, to neglect the volumetric production rate.

10 In the vp text field, type 0.


11 Click to expand the Surface reaction area section. Select the reactive surface area available in the reactor. This regulates, when multiplied with the site density, how much the ion-exchange column can adsorb.

12 Locate the Surface Reaction Area section. In the Ar text field, type Arsurf.

Initial Values 1In the Initial Values feature, set all species except adsorbed S (S(ads)) to zero. Also, set the initial density of surface sites for the ion-exchange mass. Note that the latter property sets the upper limit of how many moles can be adsorbed. Assume that each species takes up one reactive site.


2 In the Settings window for Initial Values, click to expand the Surface species initial value section.

3 Locate the Surface Species Initial Value section. In the Γs text field, type G0.

4 In the Surface species initial concentration table, enter the following settings:

Feed Inlet 1Last, in the OD model, add the Feed Inlet feature containing bulk species A and B. Add the concentration with the Gaussian pulse in the feed stream.

1 On the Reaction Engineering toolbar, click Feed Inlet.


3 In the vf text field, type vfp.

4 Locate the Feed Inlet Concentration section. In the Feed inlet concentration table, enter the following settings:

Solve the OD model for 10s.

Species Surface concentration (mol/m^2) Site occupancy number (1)

S(ads) CS0surf 1


A CAmax_inlet*pulse11

B CBmax_inlet*pulse11


S T U D Y 1



3 In the Times text field, type range(0,0.1,10).


Create Figure 2 in which all species concentrations are displayed.

R E S U L T S


2 In the Settings window for 1D Plot Group, type Concentrations 0D model in the Label text field.




6 In the associated text field, type Concentration (mol/m-3) | Surface concentration (mol/dm-2).

Global 11 In the Model Builder window, expand the Results>Concentrations 0D model node, then

click Global 1.


comp1.re.c_A - Concentration.

3 Click Add Expression in the upper-right corner of the y-axis data section. From the menu, choose Component 1>Reaction Engineering>comp1.re.c_B - Concentration.

4 Click Add Expression in the upper-right corner of the y-axis data section. From the menu, choose Component 1>Reaction Engineering>comp1.re.c_S - Concentration.

5 Click Add Expression in the upper-right corner of the y-axis data section. From the menu, choose Component 1>Reaction Engineering>comp1.re.csurf_A_surf -

Surface concentration.

6 Click Add Expression in the upper-right corner of the y-axis data section. From the menu, choose Component 1>Reaction Engineering>comp1.re.csurf_B_surf -

7 Click Add Expression in the upper-right corner of the y-axis data section. From the menu, choose Component 1>Reaction Engineering>comp1.re.csurf_S_surf -






12 On the Concentrations 0D model toolbar, click Plot.

Continue setting up a space-dependent model to investigate the top section of the ion-exchange column. Start by adding a solvent to the system. The solvent in these types of systems is often water.







comp1.re.csurf_A_surf*1e4 mol/m^2 Surface concentration per square dm

comp1.re.csurf_B_surf*1e4 mol/m^2 Surface concentration per square dm

comp1.re.csurf_S_surf*1e4 mol/m^2 Surface concentration per square dm

Legends

A

B

S

Surface A

Surface B

Surface S


5 Locate the General Parameters section. In the M text field, type MH2O.

6 In the ρ text field, type rho_H2O.





For a liquid system with solvent, several transport parameters can be computed within the Chemistry interface. Activate this functionality by following these steps.


5 In the Settings window for Reaction Engineering, click to expand the Calculate transport properties section.

6 Locate the Calculate Transport Properties section. Select the Calculate mixture properties check box.

7 From the Dynamic viscosity list, choose User defined.

8 In the μ text field, type myH2O.

To compute the diffusion coefficients correctly, enter densities and molar masses in the bulk species nodes.

Species: A1 In the Model Builder window, under Component 1 (comp1)>Reaction Engineering (re)

click Species: A.


3 In the M text field, type MA.

4 In the ρ text field, type rho_p.

Species: S1 In the Model Builder window, under Component 1 (comp1)>Reaction Engineering (re)

click Species: S.


3 In the M text field, type MS.


H2O CH2O


4 In the ρ text field, type rho_S.

Species: B1 In the Model Builder window, under Component 1 (comp1)>Reaction Engineering (re)

click Species: B.


3 In the M text field, type MB.

4 In the ρ text field, type rho_p.

Enter also molar mass in the surface species nodes.

Surface species: S(ads)1 In the Model Builder window, under Component 1 (comp1)>Reaction Engineering (re)

click Surface species: S(ads).


3 In the M text field, type MS.

Surface species: A(ads)1 In the Model Builder window, under Component 1 (comp1)>Reaction Engineering (re)

click Surface species: A(ads).


3 In the M text field, type MA.

Surface species: B(ads)1 In the Model Builder window, under Component 1 (comp1)>Reaction Engineering (re)

click Surface species: B(ads).


3 In the M text field, type MB.

Use the Generate Space-dependent Model feature to set up the space-dependent model. Select a 3D geometry and Transport of Diluted Species and Free and Porous Media Flow interfaces together with a Time Dependent study type.


2 In the Settings window for Generate Space-Dependent Model, locate the Study Type section.

3 From the Study type list, choose Time dependent.


4 Locate the Physics Interfaces section. Find the Fluid flow subsection. From the list, choose Laminar Flow: New.


S T U D Y 1

Turn off the new interfaces in Study 1 so it can be used for the 0D model later if necessary.


2 In the Settings window for Time Dependent, locate the Physics and Variables Selection section.


G E O M E T R Y 1 ( 3 D )

Insert the 3D geometry sequence from a file.

1 In the Model Builder window, expand the Component 2 (comp2) node, then click Geometry 1(3D).


3 Browse to the model’s Application Libraries folder and double-click the file protein_adsorption_geom_sequence.mph.

Cylinder 1 (cyl1)1 In the Model Builder window, under Component 2 (comp2)>Geometry 1(3D) click

Cylinder 1 (cyl1).

2 In the Settings window for Cylinder, click Build All Objects.



On the Physics toolbar, click Reaction Engineering (re) and choose Transport of Diluted Species (tds).

Start with the Transport of Diluted Species interface.

Physics interface

Chemistry 1 (chem)


Surface Reactions 1 (sr)

Free and Porous Media Flow 1 (fp)


Transport Properties 1Couple the velocity field to the Free and Porous Media Flow interface.





A and B enter the column at the top. Select Danckwerts (Flux) for a more stable computation.

Inflow 11 In the Model Builder window, under Component 2 (comp2)>

Transport of Diluted Species (tds) click Inflow 1.



4 Locate the Concentration section. In the c0,cA text field, type CAmax_inlet.

5 In the c0,cB text field, type CBmax_inlet.


Outflow 1Set an Outflow condition at the bottom of the geometry. This means that the transport along the z-direction (the height) of the reactor is dominated by convection.


Transport of Diluted Species (tds) click Outflow 1.


3 From the Selection list, choose Outlet column section.

Surface Equilibrium Reaction 1The surface of the beads is the boundary where reactions take place.


Transport of Diluted Species (tds) click Surface Equilibrium Reaction 1.

2 In the Settings window for Surface Equilibrium Reaction, locate the Boundary Selection section.

3 From the Selection list, choose Array 1.


Surface Equilibrium Reaction 21 In the Model Builder window, under Component 2 (comp2)>

Transport of Diluted Species (tds) click Surface Equilibrium Reaction 2.

2 In the Settings window for Surface Equilibrium Reaction, locate the Boundary Selection section.


Despite the fact that only solvent makes up the fluid within the column initially, adding initial trace concentrations of all dependent species gives a more stable solution to the equilibrium reactions.




3 In the cA text field, type CAmax_inlet*0.01.

4 In the cB text field, type CBmax_inlet*0.01.

5 In the cS text field, type 1e-7.

S U R F A C E R E A C T I O N S 1 ( S R )

In the surface reactions interface make sure that the reactions take place on the surface of the beads.

On the Physics toolbar, click Transport of Diluted Species (tds) and choose Surface Reactions 1 (sr).

1 In the Model Builder window, under Component 2 (comp2) click Surface Reactions 1 (sr).

2 In the Settings window for Surface Reactions, locate the Boundary Selection section.


Reactions 11 In the Model Builder window, expand the Surface Reactions 1 (sr) node, then click

Reactions 1.

2 In the Settings window for Reactions, locate the Boundary Selection section.


4 In the Model Builder window, click Reactions 1.

5 In the Settings window for Reactions, locate the Boundary Selection section.



For the same reason as in the Transport of Diluted Species interface, add initial trace concentrations of species A and B.

Initial Values 11 In the Model Builder window, under Component 2 (comp2)>Surface Reactions 1 (sr) click

Initial Values 1.


3 In the csA text field, type CS0surf*0.01.

4 In the csB text field, type CS0surf*0.01.

L A M I N A R F L O W 1 ( S P F )

On the Physics toolbar, click Surface Reactions 1 (sr) and choose Laminar Flow 1 (spf).

1 In the Model Builder window, under Component 2 (comp2) click Laminar Flow 1 (spf).


3 From the Compressibility list, choose Incompressible flow.

Wall 11 In the Model Builder window’s toolbar, click the Show button and select

Advanced Physics Options in the menu.

2 In the Model Builder window, expand the Laminar Flow 1 (spf) node, then click Wall 1.

3 In the Settings window for Wall, click to expand the Constraint settings section.

4 Locate the Constraint Settings section. From the Apply reaction terms on list, choose All physics (symmetric).

At the inlet, or the top of the column, the atmopheric pressure is applied since the column is open to the surroundings.

Inlet 11 In the Model Builder window, under Component 2 (comp2)>Laminar Flow 1 (spf) click

Inlet 1.



4 Locate the Boundary Condition section. From the list, choose Pressure.

5 Click to expand the Constraint settings section.

The average velocity through the xy-plane of the geometry is assumed to be constant everywhere along the column height. Set a constant velocity within the column, in this case, at the bottom of the 3D model geometry.


Outlet 11 In the Model Builder window, under Component 2 (comp2)>Laminar Flow 1 (spf) click

Outlet 1.


3 From the Selection list, choose Outlet column section.

4 Locate the Boundary Condition section. From the list, choose Velocity.

5 Locate the Velocity section. In the U0 text field, type U_column.

6 In the Model Builder window, click Laminar Flow 1 (spf).


2 In the Settings window for Symmetry, locate the Boundary Selection section.

3 From the Selection list, choose Symmetry.



M E S H 1

Select a mesh that resolves the surface of the beads well.









5 In the associated text field, type 9E-2.

Free Tetrahedral 11 In the Model Builder window, right-click Mesh 1 and choose Free Tetrahedral.









5 In the associated text field, type 5E-1.

This model requires that a two-step study node is used. The first step solves the stationary solution of the Free and Porous Media Flow interface and the second the time-dependent solution for the rest of the interfaces.

S T U D Y 2







4 From the Tolerance list, choose User controlled.


6 Locate the Physics and Variables Selection section. In the table, clear the Solve for check box for the Free and Porous Media Flow: Free and Porous Media Flow 1 (fp1) interface.

Physics interface

Reaction Engineering: Reaction Engineering {re}

Chemistry: Chemistry 1 {chem1}

Transport of Diluted Species: Transport of Diluted Species {tds}

Surface Reactions: Surface Reactions 1 {sr1}


Step 2: StationaryIn the Model Builder window, under Study 2 right-click Step 2: Stationary and choose Move Up.


Replace the Segregated solver with a Fully Coupled one since this fits the quick shifts in concentration originating from the equilibrium reactions in the system. Also apply a Direct solver to further improve the stability of the computations.

2 In the Model Builder window, expand the Solution 2 (sol2) node, then click Time-

Dependent Solver 1.

3 In the Settings window for Time-Dependent Solver, click to expand the Absolute tolerance section.

4 Locate the Absolute Tolerance section. From the Tolerance method list, choose Manual.

5 In the Absolute tolerance text field, type 1e-4.

6 Right-click Study 2>Solver Configurations>Solution 2 (sol2)>Time-Dependent Solver 1 and choose Fully Coupled.

7 In the Settings window for Fully Coupled, locate the General section.

8 From the Linear solver list, choose Direct.




12 Clear the Generate convergence plots check box.


R E S U L T S

Setup two Sector 3D data sets to rotate the 3D geometry according to its symmetry.


2 In the Settings window for Sector 3D, locate the Symmetry section.

3 In the Number of sectors text field, type 4.

4 From the Sectors to include list, choose All.

5 Click Plot.


Sector 3D 21 In the Model Builder window, right-click Sector 3D 1 and choose Duplicate.

2 In the Settings window for Sector 3D, locate the Symmetry section.

3 From the Sectors to include list, choose Manual.

4 In the Number of sectors to include text field, type 3.

5 In the Start sector text field, type 3.

6 Click Plot.

The following step creates Figures 3-6.


2 In the Settings window for 3D Plot Group, type Bulk concentration B in the Label text field.

Surface 11 Right-click Bulk concentration B and choose Surface.

2 In the Settings window for Surface, locate the Data section.



5 Click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Component 2>Transport of Diluted Species>cB - Concentration.


7 On the Bulk concentration B toolbar, click Plot.





12 On the Bulk concentration B toolbar, click Plot.

Bulk concentration B 11 In the Model Builder window, under Results right-click Bulk concentration B and choose

Duplicate.

2 In the Settings window for 3D Plot Group, type Surface concentration B in the Label text field.


Surface 11 In the Model Builder window, expand the Results>Surface concentration B node, then

click Surface 1.

2 In the Settings window for Surface, locate the Data section.



5 Click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Component 2>Surface Reactions 1>cs_B - Surface concentration.

6 Click the Transparency button on the Graphics toolbar.


8 On the Surface concentration B toolbar, click Plot.



11 Click the Transparency button on the Graphics toolbar.


13 On the Surface concentration B toolbar, click Plot.

3D Plot Group 4Last, create Figure 7 to visualize the velocity field within the 3D geometry.


2 In the Settings window for 3D Plot Group, type Velocity field in the Label text field.

3 Locate the Plot Settings section. Clear the Plot data set edges check box.

Streamline 11 Right-click Velocity field and choose Streamline.



4 In the Min distance text field, type 0.05.

5 In the Max distance text field, type 0.15.


Color Expression 11 Right-click Results>Velocity field>Streamline 1 and choose Color Expression.

2 In the Settings window for Color Expression, locate the Coloring and Style section.


3 From the Color table list, choose Twilight.

Surface 11 In the Model Builder window, under Results right-click Velocity field and choose Surface.

2 In the Settings window for Surface, click to expand the Inherit style section.

3 Locate the Inherit Style section. From the Plot list, choose Streamline 1.

Selection 11 Right-click Results>Velocity field>Surface 1 and choose Selection.

2 In the Settings window for Selection, locate the Selection section.

3 From the Selection list, choose All boundaries.









12 Select Boundaries 3 and 5–213 only.

13 On the Velocity field toolbar, click Plot.




S y n g a s Combu s t i o n i n a Round - J e t Bu r n e r




Introduction

This model simulates turbulent combustion of syngas (synthesis gas) in a simple round jet burner. Syngas is a gas mixture, primarily composed of hydrogen, carbon monoxide and carbon dioxide. The name syngas relates to its use in creating synthetic natural gas.

The model set up corresponds to the one studied by Couci et al. in Ref. 1. The temperature and composition resulting from the nonpremixed combustion in the burner setup have also been experimentally investigated by Barlow and coworkers (Ref. 2 and Ref. 3) as a part of the International Workshop on Measurement and Computation of Turbulent Nonpremixed Flames (Ref. 4). The model is solved in COMSOL Multiphysics by combining a Reacting Flow and a Heat Transfer in Fluids interface.

Model Definition

The burner studied in this model consists of a straight pipe placed in a slight co-flow. The gas phase fuel is fed through the pipe using an inlet velocity of 76 m/s, while the co-flow velocity outside of pipe is 0.7 m/s. At the pipe exit, the fuel gas mixes with the co-flow, creating an unconfined circular jet. The gas fed through the tube consists of three compounds typical of syngas; carbon monoxide (CO), hydrogen (H2) and nitrogen (N2). The co-flow gas consists of air. At the pipe exit, the fuel is ignited. Since the fuel and oxidizer enter the reaction zone separately, the resulting combustion is of the non-premixed type. A continuous reaction requires that the reactants and the oxidizer are mixed to stoichiometric conditions. In this set-up the turbulent flow of the jet effectively mixes the fuel from the pipe with the co-flowing oxygen. Furthermore the mixture needs to be continuously ignited. In this burner the small recirculation zones generated by the pipe wall thickness provide the means to decelerate hot product gas. The recirculation zones hereby promote continuous ignition of the oncoming mixture and stabilizes the flame at the pipe orifice. In experiments (Ref. 4) no lift-off or localized extinction of the flame has been observed.

In the current model, the syngas combustion is modeled using two irreversible reactions:

(1)

This assumption of a complete oxidation of the fuel corresponds to one of the approaches used in Ref. 1. The mass transport in the reacting jet is modeled by solving for the mass fractions of six species; the five species participating in the reactions and nitrogen N2 originating in the co-flowing air.

CO 0.5O2+ CO2→

H2 0.5O2+ H2O→

2 | S Y N G A S C O M B U S T I O N I N A R O U N D - J E T B U R N E R

The Reynolds number for the jet, based on the inlet velocity and the inner diameter of the pipe, is approximately 16700, indicating that the jet is fully turbulent. Under these circumstances, both the mixing and the reactions processes in the jet are significantly influenced by the turbulent nature of the flow. To account for the turbulence when solving for the flow field, the k-ω turbulence model is applied.

Taking advantage of the symmetry, a two-dimensional model using a cylindrical coordinate system is solved.

TU R B U L E N T R E A C T I O N R A T E

When using a turbulence model in a Reacting Flow interface, the production rate (SI unit: kg/(m3·s)) of species i resulting from reaction j is modeled as the minimum of the mean-value-closure reaction rate and the eddy-dissipation-model rate:

The mean-value-closure rate is the kinetic reaction rate expressed using the mean mass fractions. This corresponds to the characteristic reaction rate for reactions which are slow compared to the turbulent mixing, or the reaction rate in regions with negligible turbulence levels. This can be quantified through the Damköhler number, which compares the turbulent time scale (τT) to the chemical time scale (τc). The mean-value-closure is appropriate for low Damköhler numbers:

The reaction rate defined by the eddy-dissipation model (Ref. 5) is:

(2)

where τT (SI unit: s) is the mixing time scale of the turbulence, ρ is the mixture density (SI unit: kg/m3), ω is the species mass fraction, ν denotes the stoichiometric coefficients, and M is the molar mass (SI unit: kg/mol). Properties of reactants of the reaction are indicated using a subscript r, while product properties are denoted by a subscript p.

The eddy-dissipation model assumes that both the Reynolds and Damköhler numbers are sufficiently high for the reaction rate to be limited by the turbulent mixing time scale τT. A global reaction can then at most progress at the rate at which fresh reactants are mixed, at the molecular level, by the turbulence present. The reaction rate is also assumed to be

Rij νijMi min rMVC j, rED j,,⋅=

Da =τTτc------ 1«

rED,jαj

τT------ρ min min

ωrνrjMr--------------- β

ωpνpjMp----------------

p,⋅=


limited by the deficient reactant; the reactant with the lowest local concentration. The model parameter β specifies that product species is required for reaction, modeling the activation energy. For gaseous non-premixed combustion the model parameters have been found to be (Ref. 5):

,

In the current model the molecular reaction rate of the reactions is assumed to be infinitely fast. This is achieved in the model by prescribing unrealistically high rate constants for the reactions. This implies that the production rate is given solely by the turbulent mixing in Equation 2.

It should be noted that the eddy-dissipation model is a robust but simple model for turbulent reactions. The reaction rate is governed by a single time scale, the turbulent mixing time-scale. For this reason, the reactions studied should be limited to global one step (as in Equation 1), or two step reactions.

H E A T O F R E A C T I O N

The heat of reaction, or change in enthalpy, following each reaction is defined from the heat of formation of the products and reactants:

The heat of formations for each species is given in Table 1 (based on Ref. 6). Since the heat of formation of the products is lower than that of the reactants, both reactions are exothermic and release heat. The heat release is included in the model by adding a Heat Source feature to the Heat Transfer in Fluids interface. The heat source (SI unit: W/m3) applied is defined as:

TABLE 1: SPECIES ENTHALPY OF FORMATION AND HEAT CAPACITY

SPECIES ΔHf (cal/mol)

T = 298 K

Cp (cal/(mol·K)

T = 300 K

Cp (cal/(mol·K)

T = 1000 K

Cp (cal/(mol·K)

T = 2000 K

N2 0 6.949 7.830 8.601

H2 0 6.902 7.209 8.183

O2 0 7.010 8.350 9.032

H2O -57.80 7.999 9.875 12.224

α 4= β 0.5=

ΔHr ΔHfproducts ΔHf

reactants–=

q rED,1ΔHr1 rED,2ΔHr2+=


H E A T C A P A C I T Y

The temperature in the jet increases significantly due to the heat release following the reactions, this is one of the defining features of combustion. For an accurate prediction of the temperature it is important to account for the temperature dependence of the species heat capacities. In the model, interpolation functions for the heat capacity at constant pressure, Cp,i (SI unit: cal/(mol·K)), for each species are defined using the values at three different temperatures given in Table 1. The heat capacity of the mixture, cp,mix (SI unit: J/(kg·K)), is computed as a mass fraction weighted mean of the individual heat capacities:

S O L U T I O N P R O C E D U R E

The syngas combustion model is solved in three steps.

1 Use an initial submodel to solve for isothermal turbulent flow in a straight pipe with the same diameter as the burner. The fully developed flow at the pipe outlet is then used as inlet condition for the burner.

2 Solve for the turbulent and reacting, but isothermal, flow in the round jet burner configuration.

3 Include the heat transfer and solve for the fully coupled reacting flow, using the previous solution as initial condition.

Using several solution steps is vital for a robust solution procedure when solving models with a high degree of coupling. This is the case for turbulent reacting flow including heat transfer.


The resulting velocity field in the non-isothermal reacting jet is visualized in Figure 1. The expansion and development of the hot free jet is clearly seen. The turbulent mixing in the outer parts of the jet acts to accelerate fluid originating in the co-flow, and incorporate it

CO -26.420 47.259 6.950 7.948

CO2 -94.061 51.140 8.910 12.993

TABLE 1: SPECIES ENTHALPY OF FORMATION AND HEAT CAPACITY

SPECIES ΔHf (cal/mol)

T = 298 K

Cp (cal/(mol·K)

T = 300 K

Cp (cal/(mol·K)

T = 1000 K

Cp (cal/(mol·K)

T = 2000 K

cp, mixωiCp i,

Mi-----------------

i=


in the jet. This is commonly referred to as entrainment and can be observed in the co-flow streamlines which bend towards the jet downstream of the orifice.

Figure 1: The velocity magnitude and flow paths (streamlines) of the reacting jet.

The temperature in the jet is shown in Figure 2 where a revolved data set has been used to emphasize the structure of the round jet. The maximum temperature in the jet is seen to be approximately 1960 K. The carbon dioxide mass fraction in the reacting jet is plotted in Figure 3. The formation of CO2 takes place in the outer shear layer of the jet. This is where the fuel from the pipe encounters oxygen in the co-flow and reacts. The reactions are promoted by the turbulent mixing in the jet shear layer. It is also seen that the CO2 formation starts just outside of the pipe. This is also the case for the temperature increase in Figure 2. This implies that there is no lift-off and the flame is attached to the pipe.

In Figure 4, Figure 5, and Figure 6 the results reached in the model are compared with the experimental results of Barlow and coworkers (Ref. 2, Ref. 3, and Ref. 4). In Figure 4 the jet temperature is further examined and compared with the experiments. In the left panel the temperature along the centerline is plotted. It is seen that the maximum temperature predicted in the model is close to that in the experiment. However in the model the temperature profile is shifted in the downstream direction. This is most likely due to the fact that radiation has not been included in the model.


In the right panel of Figure 4 temperature profiles at 20 and 50 pipe diameters downstream of the pipe exit are compared with the experiments. The axial velocity of the jet is compared with the experimental results in Figure 5, using the same down stream positions. The axial velocity is found to compare well with the experimental values at both positions.

In Figure 6 the species concentration along the jet centerline is analyzed and compared with the experimental results. For some species, N2, and CO2, the axial mass fraction development agrees well with the experimental results. For the fuel species CO and H2 a fair agreement is observed. For the remaining species, O2 and H2O, the trend appears correct but the profiles are shifted downstream, as was the case with the temperature. The reason for the discrepancy in the mass fractions can in part be attributed to the fact that radiation is not included, but the accuracy is probably also significantly influenced by the simplified reaction scheme and the eddy-dissipation model.

Figure 2: Jet temperature shown using a revolved data set.


Figure 3: CO2 mass fraction in the reacting jet.

Figure 4: Jet temperature along the centerline (left), and radially at two different positions downstream of the pipe exit (right) scaled by the inlet temperature. The centerline and radial distance is scaled by the inner diameter of the pipe. Model results are plotted using lines, while experimental results are indicated using symbols. The downstream positions are defined in terms of the inner diameter of the pipe (d).

z/d = 50

z/d = 20


Figure 5: Axial velocity at two different positions downstream of the pipe exit, scaled by the inlet velocity. The radial distance is scaled by the inner diameter of the pipe. Model results are plotted using lines, while experimental results are indicated using symbols.

Figure 6: Species mass fractions along the jet centerline. The centerline distance is scaled by the

z/d = 50

z/d = 20

CO

N2

H2

H2O

O2

CO2


inner diameter of the pipe. Model results are plotted using lines, while experimental results are indicated using symbols.

References

1. A. Cuoci, A. Frassoldati, G. Buzzi Ferraris, T. Faravelli, E. Ranzi, “The ignition, combustion and flame structure of carbon monoxide/hydrogen mixtures. Note 2: Fluid dynamics and kinetic aspects of syngas combustion,” Int. J. Hydrogen Energy, vol. 32, pp. 3486–3500, 2007

2. R. S. Barlow, G. J. Fiechtner, C. D. Carter, and J.-Y. Chen, “Experiments on the Scalar Structure of Turbulent CO/H2/N2 Jet Flames,” Comb. and Flame, vol. 120, pp. 549–569, 2000.

3. M. Flury, Experimentelle Analyse der Mischungstruktur in turbulenten nicht vorgemischten Flammen, Ph.D. Thesis, ETH Zurich, 1998.

4. R. S. Barlow et al., “Sandia/ETH-Zurich CO/H2/N2 Flame Data - Release 1.1,” http://www.sandia.gov/TNF/DataArch/SANDchnWeb/SANDchnDoc11.pdf, 2002.

5. B.F. Magnussen and B.H. Hjertager, “On Mathematical Modeling of Turbulent Combustion with Special Emphasis on Soot Formation and Combustion,” 16th Symp. (Int.) on Combustion. Comb. Inst., Pittsburg, Pennsylvania, pp.719–729, 1976.

6. A. Frassoldati, T. Faravelli, and E. Ranzi, “The Ignition, Combustion and Flame Structure of Carbon Monoxide/Hydrogen Mixtures. Note 1: Detailed Kinetic Modeling of Syngas Combustion Also in Presence of Nitrogen Compounds,” Int. J. Hydrogen Energy, vol. 32, pp. 3471–3485, 2007.

Application Library path: Chemical_Reaction_Engineering_Module/Reactors_with_Mass_and_Heat_Transfer/round_jet_burner



N E W



http://www.sandia.gov/TNF/DataArch/SANDchnWeb/SANDchnDoc11.pdf



2 In the Select Physics tree, select Fluid Flow>Single-Phase Flow>Turbulent Flow>

Turbulent Flow, k-ω (spf).

3 Click Add.

4 Click Study.


6 Click Done.





4 Browse to the model’s Application Libraries folder and double-click the file round_jet_burner_params.txt.

G E O M E T R Y 1



3 In the Width text field, type Di/2.

4 In the Height text field, type Di*200.




For easier visualization of the slender geometry, disable preserve aspect ratio for the view.

View 1In the Model Builder window, expand the Component 1 (comp1)>Definitions node.

Axis1 In the Model Builder window, expand the View 1 node, then click Axis.

2 In the Settings window for Axis, locate the Axis section.


3 From the View scale list, choose Automatic.


Apply fluid properties for the pipe simulation. An approximate density can be used.

TU R B U L E N T F L O W, K - ω ( S P F )

Fluid Properties 11 In the Model Builder window, under Component 1 (comp1)>Turbulent Flow, k-ω (spf)



3 From the ρ list, choose User defined. In the associated text field, type 1.

4 From the μ list, choose User defined. In the associated text field, type mu_mix.



3 In the Settings window for Inlet, locate the Turbulence Conditions section.

4 In the LT text field, type 0.07*Di.

5 Locate the Velocity section. In the U0 text field, type Ujet.





M E S H 1


Mapped.
















Now add a second model for the round reacting jet simulation.

8 On the Home toolbar, click Component and choose Add Component>2D Axisymmetric.

G E O M E T R Y 2

In the Model Builder window, under Component 2 (comp2) click Geometry 2.

A D D P H Y S I C S



3 In the tree, select Chemical Species Transport>Reacting Flow>Turbulent Flow>

Turbulent Flow, k-ω.

4 Find the Physics interfaces in study subsection. In the table, clear the Solve check box for Study 1.




wCO

wO2

wCO2

wH2

wH2O

wN2



9 In the tree, select Heat Transfer>Heat Transfer in Fluids (ht).

10 Find the Physics interfaces in study subsection. In the table, clear the Solve check box for Study 1.



C O M P O N E N T 2 ( C O M P 2 )

On the Home toolbar, click Windows and choose Add Multiphysics.

A D D M U L T I P H Y S I C S

1 Go to the Add Multiphysics window.

2 In the tree, select No Coupling Features Available for the Selected Physics Interfaces.

3 Find the Select the physics interfaces you want to couple subsection. In the table, enter the following settings:

4 In the tree, select Fluid Flow>Nonisothermal Flow>Turbulent Flow>Turbulent Flow, k-ω.

5 Find the Multiphysics couplings in study subsection. In the table, enter the following settings:


7 On the Home toolbar, click Add Multiphysics.


1 In the Model Builder window, under Component 2 (comp2)>Multiphysics click Nonisothermal Flow 1 (nitf1).

2 In the Settings window for Nonisothermal Flow, locate the Material Properties section.

3 From the Specify density list, choose Custom.

4 From the ρ list, choose Density (tcs/cdm1).

5 Locate the Flow Heating section. Select the Include work done by pressure changes check box.

Physics Couple

Transport of Concentrated Species (tcs)

Studies Solve

Study 1


A D D S T U D Y










4 Browse to the model’s Application Libraries folder and double-click the file round_jet_burner_vars.txt.

G E O M E T R Y 2



3 In the Width text field, type GeomW.

4 In the Height text field, type GeomH.




3 In the Width text field, type Pth.

4 In the Height text field, type Pl.

5 Locate the Position section. In the r text field, type Di/2.


Chamfer 1 (cha1)1 On the Geometry toolbar, click Chamfer.

2 On the object r2, select Points 3 and 4 only.


3 In the Settings window for Chamfer, locate the Distance section.

4 In the Distance from vertex text field, type Pth*0.15.

5 Right-click Chamfer 1 (cha1) and choose Build Selected.


Bézier Polygon 1 (b1)1 On the Geometry toolbar, click Primitives and choose Bézier Polygon.


3 In the Settings window for Bézier Polygon, locate the Polygon Segments section.

4 Find the Added segments subsection. Click Add Linear.

5 Find the Control points subsection. In row 1, set r to GeomW.

6 In row 2, set r to GeomW*1.5 and z to GeomH.


8 Find the Control points subsection. In row 2, set r to GeomW.

9 Right-click Bézier Polygon 1 (b1) and choose Build Selected.


2 Select the objects b1 and r1 only.



5 Right-click Union 1 (uni1) and choose Build Selected.

Difference 1 (dif1)1 On the Geometry toolbar, click Booleans and Partitions and choose Difference.

2 Select the object uni1 only.

3 In the Settings window for Difference, locate the Difference section.

4 Find the Objects to subtract subsection. Select the Active toggle button.

5 Select the object cha1 only.

6 Right-click Difference 1 (dif1) and choose Build Selected.





4 Find the Control points subsection. In row 1, set r to Di/2 and z to Pl+0.3e-3.

5 In row 2, set r to Di and z to GeomH.





4 Find the Control points subsection. In row 1, set r to Di/2+Pth and z to Pl+0.3e-3.

5 In row 2, set r to 0.04 and z to GeomH.


Form Union (fin)In the Model Builder window, under Component 2 (comp2)>Geometry 2 right-click Form Union (fin) and choose Build Selected.

Mesh Control Edges 1 (mce1)1 On the Geometry toolbar, click Virtual Operations and choose Mesh Control Edges.

2 On the object fin, select Boundaries 6 and 11 only.

3 Right-click Mesh Control Edges 1 (mce1) and choose Build Selected.


Form Composite Edges 1 (cme1)1 On the Geometry toolbar, click Virtual Operations and choose Form Composite Edges.

2 On the object mce1, select Boundaries 3 and 11 only.

3 Right-click Form Composite Edges 1 (cme1) and choose Build Selected.

That concludes the geometry for the reacting jet. Now define a coupling variable that can be used to apply the outlet conditions from the previous model to the inlet of the current.


In the Model Builder window, under Component 1 (comp1) click Definitions.

Linear Extrusion 1 (linext1)1 On the Definitions toolbar, click Component Couplings and choose Linear Extrusion.


3 In the Settings window for Linear Extrusion, locate the Source Vertices section.

4 Select the Active toggle button.





8 Click to expand the Destination section. From the Destination geometry list, choose Geometry 2.

9 Locate the Destination Vertices section. Select the Active toggle button.




TU R B U L E N T F L O W, K - ω 2 ( S P F 2 )

Fluid Properties 11 In the Model Builder window, under Component 2 (comp2)>Turbulent Flow, k-ω 2 (spf2)



3 From the μ list, choose User defined. In the associated text field, type mu_mix.




4 In the U0 text field, type comp1.linext1(w).

5 Locate the Turbulence Conditions section. Click the Specify turbulence variables button.

6 In the k0 text field, type comp1.linext1(k).

7 In the ω0 text field, type comp1.linext1(om).




4 Click the Velocity field button.


5 Specify the u0 vector as

6 Locate the Turbulence Conditions section. In the IT text field, type 0.01.

7 In the LT text field, type 0.1*Di.








3 From the Diffusion model list, choose Fick’s law.

4 Locate the Species section. From the From mass constraint list, choose wN2.

Transport Properties 1Apply the temperature from the heat transfer interface.





4 Locate the Density section. In the MwCO text field, type M_CO.

5 In the MwO2 text field, type M_O2.

6 In the MwCO2 text field, type M_CO2.

7 In the MwH2 text field, type M_H2.

8 In the MwH2O text field, type M_H2O.

9 In the MwN2 text field, type M_N2.

0 r

Ucf z





3 In the ω0,wCO text field, type 0.

4 In the ω0,wO2 text field, type wcf_O2.

5 In the ω0,wCO2 text field, type 0.

6 In the ω0,wH2 text field, type 0.

7 In the ω0,wH2O text field, type 0.




4 From the Mixture specification list, choose Mole fractions.

5 In the x0,wCO text field, type x0_CO.

6 In the x0,wO2 text field, type x0_O2.

7 In the x0,wCO2 text field, type x0_CO2.

8 In the x0,wH2 text field, type x0_H2.

9 In the x0,wH2O text field, type x0_H2O.




4 In the ω0,wCO text field, type 1e-5.

5 In the ω0,wO2 text field, type wcf_O2.

6 In the ω0,wCO2 text field, type 1e-5.

7 In the ω0,wH2 text field, type 1e-5.

8 In the ω0,wH2O text field, type 1e-5.







4 In the νwCO text field, type -1.

5 In the νwO2 text field, type -0.5.

6 In the νwCO2 text field, type 1.

Apply an unrealistically high reaction rate to model the reactions as infinitely fast. In this case the reaction rate will be given by the turbulent mixing.

7 Locate the Rate Constants section. In the kf text field, type 1e100.


dissipation.

Regularization makes the reaction system much easier to converge.

9 Click to expand the Regularization section. Select the Rate expressions check box.

Reaction 21 Right-click Reaction 1 and choose Duplicate.


3 In the νwCO text field, type 0.

4 In the νwCO2 text field, type 0.

5 In the νwH2 text field, type -1.

6 In the νwH2O text field, type 1.

Use the tabulated heat capacities to create interpolation functions, one for each species.


Interpolation 1 (int1)1 On the Home toolbar, click Functions and choose Local>Interpolation.

2 In the Settings window for Interpolation, locate the Definition section.

3 In the Function name text field, type Cp_CO.


t f(t)

300 47.259


5 Locate the Interpolation and Extrapolation section. From the Interpolation list, choose Piecewise cubic.

6 Locate the Units section. In the Arguments text field, type K.

7 In the Function text field, type cal/mol/K.

Plot the resulting interpolation function.

8 Click Plot.

Interpolation 2 (int2)1 Right-click Interpolation 1 (int1) and choose Duplicate.


3 In the Function name text field, type Cp_CO2.


Interpolation 3 (int3)1 Right-click Component 2 (comp2)>Definitions>Interpolation 2 (int2) and choose

Duplicate.


3 In the Function name text field, type Cp_H2.



Duplicate.

1000 6.950

2000 7.948

t f(t)

300 51.140

1000 8.910

2000 12.993

t f(t)

300 6.902

1000 7.209

2000 8.183

t f(t)



3 In the Function name text field, type Cp_H2O.



Duplicate.


3 In the Function name text field, type Cp_N2.



Duplicate.


3 In the Function name text field, type Cp_O2.


Define the mixture heat capacity. It is computed as the mass average of the species capacities. Also define the enthalpy change for each of the reactions included.

Variables 11 In the Model Builder window, under Component 2 (comp2)>Definitions click Variables 1.

t f(t)

300 7.999

1000 9.875

2000 12.224

t f(t)

300 6.949

1000 7.830

2000 8.601

t f(t)

300 7.010

1000 8.350

2000 9.032




Now setup the heat transfer interface.


On the Physics toolbar, click Transport of Concentrated Species (tcs) and choose Heat Transfer in Fluids (ht).


click Fluid 1.


3 From the k list, choose User defined. In the associated text field, type k_mix.

4 Locate the Thermodynamics, Fluid section. From the Fluid type list, choose Ideal gas.

5 From the Gas constant type list, choose Mean molar mass.

6 From the Cp list, choose User defined. In the associated text field, type Cp_mix.



2 In the Settings window for Initial Values, type T0 in the T text field.

3 In the Model Builder window, click Heat Transfer in Fluids (ht).



3 In the Settings window for Inflow, locate the Upstream Properties section.


Cp_mix tcs.wr_wCO*Cp_CO(T)/M_CO+tcs.wr_wCO2*Cp_CO2(T)/M_CO2+tcs.wr_wH2*Cp_H2(T)/M_H2+tcs.wr_wH2O*Cp_H2O(T)/M_H2O+tcs.wr_wN2*Cp_N2(T)/M_N2+tcs.wr_wO2*Cp_O2(T)/M_O2

J/(kg·K) Heat capacity, mixture

dH_R1 dH_CO2-(dH_CO+0.5*dH_O2) J/mol Enthalpy change reaction 1

dH_R2 dH_H2O-(dH_H2+0.5*dH_O2) J/mol Enthalpy change reaction 2


4 In the Tustr text field, type T0.






4 In the Q0 text field, type -(dH_R1*tcs.treac1.r+dH_R2*tcs.treac2.r).

M E S H 2


2 In the Model Builder window, under Component 2 (comp2) right-click Mesh 2 and choose Edit Physics-Induced Sequence.




4 Locate the Element Size Parameters section. In the Maximum element size text field, type 0.05.

5 In the Maximum element growth rate text field, type 1.12.

6 In the Resolution of narrow regions text field, type 5.




4 Locate the Element Size Parameters section. Select the Resolution of narrow regions check box.



Size 21 In the Model Builder window, right-click Mesh 2 and choose Size.




4 Locate the Geometric Entity Selection section. From the Geometric entity level list, choose Boundary.




8 Select the Maximum element growth rate check box.


Free Triangular 11 In the Model Builder window, under Component 2 (comp2)>Mesh 2 click

Free Triangular 1.

2 In the Settings window for Free Triangular, click to expand the Scale geometry section.

3 Locate the Scale Geometry section. In the z-direction scale text field, type 0.5.

Boundary Layer Properties 11 In the Model Builder window, expand the Boundary Layers 1 node, then click

Boundary Layer Properties 1.



4 Click Build All.

Solve the fully developed turbulent pipe flow set up in Component 1.

S T U D Y 1



Use the second study to solve the axially symmetric jet flow in Component 2 using the fully developed turbulent outlet profiles as inlet conditions for the pipe.


S T U D Y 2

Step 1: Stationary1 In the Model Builder window, expand the Solution 1 (sol1) node, then click Study 2>

Step 1: Stationary.


3 In the table, clear the Solve for check box for Turbulent Flow, k-ω (spf) and Heat Transfer

in Fluids (ht).

4 Click to expand the Values of dependent variables section. Locate the Values of Dependent Variables section. Find the Values of variables not solved for subsection. From the Settings list, choose User controlled.


6 From the Study list, choose Study 1, Stationary.

Adjust the CFL-number controller parameters to speed up convergence. Also, solve the temperature coupled with the velocity and pressure for increased robustness and convergence rate.




Solution 2 (sol2)>Stationary Solver 1 node.


Stationary Solver 1 click Segregated 1.


6 In the PID regulator-Proportional text field, type 0.65.

7 In the PID regulator-Derivative text field, type 0.025.

8 In the Target error estimate text field, type 0.1.


Stationary Solver 1>Segregated 1 click Velocity u2, Pressure p2.

10 In the Settings window for Segregated Step, locate the General section.

11 Under Variables, click Add.

12 In the Add dialog box, in the Variables list, choose Wall temperature,

downside (comp2.nitf1.TWall_d), Wall temperature, upside (comp2.nitf1.TWall_u), and Temperature (comp2.T).


13 Click OK.


Stationary Solver 1>Segregated 1 click Segregated Step 2.

15 In the Settings window for Segregated Step, click to expand the Method and termination section.

16 Locate the Method and Termination section. In the Number of iterations text field, type 2.

17 In the Damping factor text field, type 0.4.


Stationary Solver 1>Segregated 1 right-click Segregated Step 3 and choose Disable.

Solution 2 (sol2)1 In the Model Builder window, collapse the Study 2>Solver Configurations>

Solution 2 (sol2) node.

2 In the Model Builder window, click Solution 2 (sol2).


Now move on to post process the result from the nonisothermal jet. Start by creating a mirrored 2D data set as well as a revolved 3D data set.

R E S U L T S


2 In the Settings window for Mirror 2D, locate the Data section.

3 From the Data set list, choose Study 2/Solution 2 (4) (sol2).

Revolution 2D 11 In the Model Builder window, under Results>Data Sets click Revolution 2D 1.


3 Locate the Revolution Layers section. In the Revolution angle text field, type 180.

Revolution 2D 31 Right-click Results>Data Sets>Revolution 2D 1 and choose Duplicate.

2 In the Settings window for Revolution 2D, locate the Data section.


Create two cut lines at fixed heights from the pipe exit.





4 Locate the Line Data section. From the Line entry method list, choose Point and direction.

5 Find the Point subsection. In the y text field, type Pl+20*Di.

6 Click to expand the Advanced section. Find the Space variable subsection. In the x text field, type r_mirr20.

Cut Line 2D 21 Right-click Cut Line 2D 1 and choose Duplicate.


3 Find the Point subsection. In the y text field, type Pl+50*Di.

4 Locate the Advanced section. Find the Space variable subsection. In the x text field, type r_mirr50.

Now apply the mirror data set to the existing plot groups.

Velocity (spf2)1 In the Model Builder window, under Results click Velocity (spf2).



4 On the Velocity (spf2) toolbar, click Plot.

5 In the Model Builder window, expand the Velocity (spf2) node.

Streamline 11 Right-click Results>Velocity (spf2) and choose Streamline.


3 From the Positioning list, choose Uniform density.

4 In the Separating distance text field, type 0.035.

5 Locate the Coloring and Style section. From the Color list, choose Gray.



Pressure (spf2)1 In the Model Builder window, under Results click Pressure (spf2).




4 On the Pressure (spf2) toolbar, click Plot.

Wall Resolution (spf2)1 In the Model Builder window, under Results click Wall Resolution (spf2).



4 On the Wall Resolution (spf2) toolbar, click Plot.

Mass Fraction (tcs)1 In the Model Builder window, under Results click Mass Fraction (tcs).



4 Locate the Plot Settings section. From the Color list, choose White.

5 On the Mass Fraction (tcs) toolbar, click Plot.

Surface1 In the Model Builder window, expand the Mass Fraction (tcs) node, then click Surface.


3 In the Expression text field, type wCO2.

4 On the Mass Fraction (tcs) toolbar, click Plot.


Velocity (spf2) 11 In the Model Builder window, under Results click Velocity (spf2) 1.



Temperature, 3D (ht)1 In the Model Builder window, under Results click Temperature, 3D (ht).



4 On the Temperature, 3D (ht) toolbar, click Plot.


Import the experimental data files. The files corresponds to the ones published online (Ref. 2) by R. Barlow and co-workers. The name of the model, round_jet_burner, has been prepended to the file names.

Table 11 On the Results toolbar, click Table.

2 In the Settings window for Table, locate the Data section.

3 Click Import.

4 Browse to the model’s Application Libraries folder and double-click the file round_jet_burner_chnAclY.fav.

TA B L E


2 Right-click Table 1 and choose Rename.

3 In the Rename Table dialog box, type Centerline data in the New label text field.

4 Click OK.

R E S U L T S



3 Click Import.

4 Browse to the model’s Application Libraries folder and double-click the file round_jet_burner_chnAd20Y.fav.

TA B L E



3 In the Rename Table dialog box, type z/Di = 20, radial data in the New label text field.

4 Click OK.

R E S U L T S




3 Click Import.

4 Browse to the model’s Application Libraries folder and double-click the file round_jet_burner_chnAd50Y.fav.

TA B L E



3 In the Rename Table dialog box, type z/Di = 50, radial data in the New label text field.

4 Click OK.

R E S U L T S



3 Click Import.

4 Browse to the model’s Application Libraries folder and double-click the file round_jet_burner_seq1420.dat.

TA B L E



3 In the Rename Table dialog box, type z/Di = 20, radial velocity data in the New label text field.

4 Click OK.

R E S U L T S



3 Click Import.

4 Browse to the model’s Application Libraries folder and double-click the file round_jet_burner_seq1450.dat.


TA B L E



3 In the Rename Table dialog box, type z/Di = 50, radial velocity data in the New label text field.

4 Click OK.

R E S U L T S




Line Graph 11 Right-click 1D Plot Group 13 and choose Line Graph.



4 In the Expression text field, type T/T0.


6 In the Expression text field, type (z-Pl)/Di.

7 Click to expand the Coloring and style section. Locate the Coloring and Style section. From the Color list, choose Black.




Table Graph 11 In the Model Builder window, under Results right-click 1D Plot Group 13 and choose

Table Graph.

2 In the Settings window for Table Graph, locate the Data section.

3 From the x-axis data list, choose r(mm).

4 From the Plot columns list, choose Manual.

Legends

Model


5 In the Columns list, select T(K).

6 Click to expand the Preprocessing section. Find the x-axis column subsection. From the Preprocessing list, choose Linear.

7 In the Scaling text field, type 1/(Di*1000).

8 Find the y-axis columns subsection. From the Preprocessing list, choose Linear.

9 In the Scaling text field, type 1/T0.

10 Locate the Coloring and Style section. Find the Line style subsection. From the Line list, choose None.

11 From the Color list, choose Black.

12 Find the Line markers subsection. From the Marker list, choose Square.

13 From the Positioning list, choose In data points.




1D Plot Group 131 Right-click 1D Plot Group 13 and choose Rename.

2 In the Rename 1D Plot Group dialog box, type T @ centerline in the New label text field.

3 Click OK.


5 Select the x-axis label check box.

6 In the associated text field, type (z-Pl)/Di.


8 In the associated text field, type T/T0.




12 In the y minimum text field, type 0.5.

13 In the y maximum text field, type 8.

Legends

Exp.



15 In the Title text area, type Temperature along the centerline.

16 On the T @ centerline toolbar, click Plot.



3 From the Data set list, choose None.

Line Graph 11 Right-click 1D Plot Group 14 and choose Line Graph.



4 Locate the y-Axis Data section. In the Expression text field, type T/T0.


6 In the Expression text field, type r_mirr20/Di.

7 Click to expand the Coloring and style section. Locate the Coloring and Style section. From the Color list, choose Black.







4 Locate the x-Axis Data section. In the Expression text field, type r_mirr50/Di.

5 Locate the Coloring and Style section. Find the Line style subsection. From the Line list, choose Dashed.


Legends

z/Di = 20, Model

Legends

z/Di = 50, Model



Table Graph 11 In the Model Builder window, under Results right-click 1D Plot Group 14 and choose

Table Graph.


3 From the Table list, choose z/Di = 20, radial data.

4 From the x-axis data list, choose r(mm).

5 From the Plot columns list, choose Manual.

6 In the Columns list, select T(K).

7 Click to expand the Preprocessing section. Find the x-axis column subsection. From the Preprocessing list, choose Linear.

8 In the Scaling text field, type 1/(Di*1000).

9 Find the y-axis columns subsection. From the Preprocessing list, choose Linear.

10 In the Scaling text field, type 1/T0.

11 Click to expand the Coloring and style section. Locate the Coloring and Style section. Find the Line style subsection. From the Line list, choose None.

12 From the Color list, choose Black.

13 Find the Line markers subsection. From the Marker list, choose Square.

14 From the Positioning list, choose In data points.




Table Graph 21 Right-click Results>1D Plot Group 14>Table Graph 1 and choose Duplicate.


3 From the Table list, choose z/Di = 50, radial data.

4 Locate the Coloring and Style section. Find the Line markers subsection. From the Marker list, choose Triangle.

Legends

z/Di = 20, Exp



1D Plot Group 141 In the Model Builder window, under Results right-click 1D Plot Group 14 and choose

Rename.

2 In the Rename 1D Plot Group dialog box, type T @ z/Di = 20, 50 in the New label text field.

3 Click OK.


5 From the Title type list, choose Manual.

6 In the Title text area, type Temperature downstream of the pipe exit.


8 In the associated text field, type r/Di.


10 In the associated text field, type T/T0.




14 In the y minimum text field, type 0.5.


16 On the T @ z/Di = 20, 50 toolbar, click Plot.

T @ z/Di = 20, 50.11 In the Model Builder window, under Results right-click T @ z/Di = 20, 50 and choose

Duplicate.

2 Right-click T @ z/Di = 20, 50.1 and choose Rename.

3 In the Rename 1D Plot Group dialog box, type uz @ z/Di = 20, 50 in the New label text field.

4 Click OK.

Line Graph 11 In the Model Builder window, expand the Results>uz @ z/Di = 20, 50 node, then click

Line Graph 1.

Legends

z/Di = 50, Exp



3 In the Expression text field, type w2/Ujet.

Line Graph 21 In the Model Builder window, under Results>uz @ z/Di = 20, 50 click Line Graph 2.


3 In the Expression text field, type w2/Ujet.

Table Graph 11 In the Model Builder window, under Results>uz @ z/Di = 20, 50 click Table Graph 1.


3 From the x-axis data list, choose Fblgr.

4 From the Table list, choose z/Di = 20, radial velocity data.

5 In the Columns list, select uz.

6 Locate the Preprocessing section. Find the y-axis columns subsection. From the Preprocessing list, choose Linear.

7 In the Scaling text field, type 1/Ujet.

Table Graph 21 In the Model Builder window, under Results>uz @ z/Di = 20, 50 click Table Graph 2.


3 From the x-axis data list, choose Fblgr.

4 From the Table list, choose z/Di = 50, radial velocity data.

5 In the Columns list, select uz.

6 Locate the Preprocessing section. Find the y-axis columns subsection. From the Preprocessing list, choose Linear.

7 In the Scaling text field, type 1/Ujet.

uz @ z/Di = 20, 501 In the Model Builder window, under Results click uz @ z/Di = 20, 50.


3 In the Title text area, type Axial velocity downstream of the pipe exit.

4 Locate the Plot Settings section. In the y-axis label text field, type uz/Ujet.

5 Locate the Axis section. In the x minimum text field, type -10.



7 In the y minimum text field, type -0.25.

8 In the y maximum text field, type 1.25.

9 On the uz @ z/Di = 20, 50 toolbar, click Plot.


T @ centerline 11 In the Model Builder window, under Results right-click T @ centerline and choose

Duplicate.

2 Right-click T @ centerline 1 and choose Rename.

3 In the Rename 1D Plot Group dialog box, type CO, N2 @ centerline in the New label text field.

4 Click OK.


6 In the Title text area, type Mass fraction along the centerline.

7 Locate the Plot Settings section. In the y-axis label text field, type wCO, wN2.

8 Locate the Axis section. In the y minimum text field, type -0.05.


10 On the CO, N2 @ centerline toolbar, click Plot.


Line Graph 11 In the Model Builder window, expand the Results>CO, N2 @ centerline node, then click

Line Graph 1.


3 In the Expression text field, type wCO.


Table Graph 11 In the Model Builder window, under Results>CO, N2 @ centerline click Table Graph 1.


3 In the Columns list, select YCO.

Legends

CO, Model


4 Locate the Preprocessing section. Find the y-axis columns subsection. In the Scaling text field, type 1.



Line Graph 21 In the Model Builder window, under Results>CO, N2 @ centerline right-click Line Graph 1



3 In the Expression text field, type wN2.

4 Click to expand the Coloring and style section. Locate the Coloring and Style section. Find the Line style subsection. From the Line list, choose Dashed.


Table Graph 21 In the Model Builder window, under Results>CO, N2 @ centerline right-click

Table Graph 1 and choose Duplicate.


3 In the Columns list, select YN2.

4 Locate the Coloring and Style section. Find the Line markers subsection. From the Marker list, choose Triangle.



CO, N2 @ centerline 11 In the Model Builder window, under Results right-click CO, N2 @ centerline and choose

Duplicate.

2 Right-click CO, N2 @ centerline 1 and choose Rename.

Legends

CO, Exp.

Legends

N2, Model

Legends

N2, Exp


3 In the Rename 1D Plot Group dialog box, type H2, H2O @ centerline in the New label text field.

4 Click OK.

Line Graph 11 In the Model Builder window, expand the Results>H2, H2O @ centerline node, then click

Line Graph 1.


3 In the Expression text field, type wH2.


Table Graph 11 In the Model Builder window, under Results>H2, H2O @ centerline click Table Graph 1.


3 In the Columns list, select YH2.


Line Graph 21 In the Model Builder window, under Results>H2, H2O @ centerline click Line Graph 2.


3 In the Expression text field, type wH2O.


Table Graph 21 In the Model Builder window, under Results>H2, H2O @ centerline click Table Graph 2.


3 In the Columns list, select YH2O.

Legends

H2, Model

Legends

H2, Exp.

Legends

H2O, Model



H2, H2O @ centerline1 In the Model Builder window, under Results click H2, H2O @ centerline.


3 In the y-axis label text field, type wH2, wH2O.

4 Locate the Axis section. In the y maximum text field, type 0.15.

5 In the y minimum text field, type -0.02.

6 Locate the Legend section. From the Position list, choose Upper right.

7 On the H2, H2O @ centerline toolbar, click Plot.

H2, H2O @ centerline 11 In the Model Builder window, right-click H2, H2O @ centerline and choose Duplicate.

2 Right-click H2, H2O @ centerline 1 and choose Rename.

3 In the Rename 1D Plot Group dialog box, type O2, CO2 @ centerline in the New label text field.

4 Click OK.

Line Graph 11 In the Model Builder window, expand the Results>O2, CO2 @ centerline node, then click

Line Graph 1.


3 In the Expression text field, type wO2.


Table Graph 11 In the Model Builder window, under Results>O2, CO2 @ centerline click Table Graph 1.


3 In the Columns list, select YO2.

Legends

H2O, Exp

Legends

O2, Model



Line Graph 21 In the Model Builder window, under Results>O2, CO2 @ centerline click Line Graph 2.


3 In the Expression text field, type wCO2.


Table Graph 21 In the Model Builder window, under Results>O2, CO2 @ centerline click Table Graph 2.


3 In the Columns list, select YCO2.


O2, CO2 @ centerline1 In the Model Builder window, under Results click O2, CO2 @ centerline.


3 In the y-axis label text field, type wO2, wCO2.

4 Locate the Axis section. In the y minimum text field, type -0.05.

5 In the y maximum text field, type 0.4.

6 On the O2, CO2 @ centerline toolbar, click Plot.

7 Click Plot.

Legends

O2, Exp.

Legends

CO2, Model

Legends

CO2, Exp



S em i b a t c h Po l yme r i z a t i o n




Introduction

As reactant monomer converts into polymer chains, the density of the reacting mixture often changes notably. This example looks at how this effect impacts the total production of polymer in a process. The liquid phase polymerization takes place in a semibatch reactor, where two operating conditions are compared. In the first scenario, the feed of monomer to the reactor is turned off once the maximum volume capacity is reached. In a second scenario, the feed of monomer is allowed to continuously compensate for the volume change due to chemical reaction.

The model illustrates the use of the Semibatch reactor type, that is predefined in the Reaction Engineering interface in the Chemical Reaction Engineering Module. It also shows how to set timed events, in this case, controlling the reactant feed to the reactor. This example reproduces results found in Ref. 1.

Model Definition

A liquid phase polymerization can be modeled as a first order irreversible reaction:

In the above equations, M denotes the monomer, P the polymer, r is the reaction rate (SI unit: mol/(m3·s)), k is the rate constant (SI unit: 1/s), and cM is the concentration of the monomer. This process takes place in the presence of water.

The polymerization takes place in a semibatch reactor with a volume capacity of 20 m3. Initially the reactor is charged with 10 m3 of water. Pure monomer enters the reactor with a volumetric flow rate of vf = 1 m3/min.

k1M P

r1 k1cM=

2 | S E M I B A T C H P O L Y M E R I Z A T I O N

Figure 1 shows a schematic representation of the semibatch reactor.

Figure 1: The Semibatch reactor is a predefined reactor type in the Reaction Engineering interface.

The following mass balance describes the semibatch reactor:

(1)

In Equation 1, ci is species molar concentration (SI unit: mol/m3), cf,i is the species concentration (SI unit: mol/m3) of the associated feed stream vf,i (SI unit: m3/s), and Ri denotes the species rate expression (SI unit: mol/(m3·s)). Vr labels the reactor volume (SI unit: m3) and is a function of time. For ideal mixtures:

where vp is the volumetric production rate due to chemical reaction:

(2)

In Equation 2, νij is the stoichiometric coefficient of species i in reaction j, Mi denotes the species molecular weight (SI unit: kg/mol), ρi is the species density (SI unit: kg/m3), and rj is the reaction rate (SI unit: mol/(m3·s)) of reaction j.

In the present example, the density of the monomer is 800 kg/m3, 1100 kg/m3 for the polymer, and 1000 kg/m3 for water. Hence, as polymer is being formed, the volume of

d ciVr( )dt

-------------------- vf i, cf i, R+ iVr=

dVr

dt---------- vf i, vp+=

vp νijMiρi-------rjVr

i

j=


the reacting mixture decreases (vp is negative). The model investigates two operating conditions:

• Operating condition 1 — The monomer feed (1 m3/min) is shut off once the reactor volume reaches 20 m3, which occurs after 11.2 minutes. The reaction is then allowed to go to completion.

• Operating condition 2 — The monomer feed is adjusted to keep the reactor filled while the reaction goes to completion. This is accomplished by setting the volumetric feed equal to −vp, for t > 11.2 minutes.

Results

Figure 2 illustrates the volumetric flow rate of the feed stream, vf.

Figure 2: The volumetric flow rate of the feed stream (m3/s) as function of time (minutes) for operating condition 1 (solid line) and 2 (dash-dotted line).


Figure 3 shows the reactor volume as function of the run time, illustrating the two operating conditions listed above.

Figure 3: The reactor volume (m3) as function of time (minutes) for operating condition 1 (solid line) and 2 (dash-dotted line).

Figure 4 shows the total mass of monomer in the reactor, mM (kg), as evaluated by the expression:

mM cMVrMM=


Figure 4: The total monomer mass in the reactor volume (kg) as function of time (minutes) for operating condition 1 (solid line) and 2 (dash-dotted line).

It is straightforward to compare the amount of produced polymer as a result of the different operating conditions. For both cases the reaction has run to completion after approximately 100 minutes. At this time, the total volume of the reacting mixture is 18.2 m3 for operating condition 1 and 20 m3 for operating condition 2.

The relative increase in polymer production using operating condition 2 compared to condition 1 is then:

Reference

1. J.B. Rawlings and J.G. Ekerdt, Chemical Reactor Analysis and Design Fundamentals, Nob Hill Publishing, example 4.3, pp. 139–144, 2004.

mP 2, mP 1,–

mP 1,---------------------------------

cP 2, Vr 2, MP cP 1, Vr 1, MP–

cP 1, Vr 1, MP---------------------------------------------------------------------- 10.5 8.8–

8.8-------------------------- 19.3%= = =


Application Library path: Chemical_Reaction_Engineering_Module/Ideal_Tank_Reactors/semibatch_polymerization



N E W





3 Click Add.

4 Click Study.


6 Click Done.


Add a set of model parameters by importing their definitions from a text file provided with the Model Library.




4 Browse to the model’s Application Libraries folder and double-click the file semibatch_polymerization_parameters.txt.

Step 1 (step1)Add a step function that regulates the volumetric feed rate during the operations.

1 On the Home toolbar, click Functions and choose Global>Step.


3 In the Location text field, type t_cond.





3 From the Reactor type list, choose Semibatch.




3 In the Formula text field, type M=>P.

4 Locate the Rate Constants section. In the kf text field, type kf_reaction.

Species: M1 In the Model Builder window, under Component 1 (comp1)>Reaction Engineering (re)

click Species: M.


3 In the M text field, type Mm_M.

4 In the ρ text field, type density_M.

Species: P1 In the Model Builder window, under Component 1 (comp1)>Reaction Engineering (re)

click Species: P.


3 In the M text field, type Mm_P.

4 In the ρ text field, type density_P.

The reaction takes place in water.




4 Locate the General Parameters section. In the M text field, type Mm_H2O.

5 In the ρ text field, type density_H2O.



Add the two filling conditions as variables dependent on the step function and the monomer mass expression. This is retrieved from a data text file provided with the Model

Library.


choose Variables.



4 Browse to the model’s Application Libraries folder and double-click the file semibatch_polymerization_variables.txt.




3 In the vf text field, type vfs.





3 In the Vr0 text field, type Vr_init.



H2O cinlet_H2O

M cinlet_M


H2O cinit_H2O


S T U D Y 1





R E S U L T S


2 In the Settings window for Global, locate the x-Axis Data section.

3 From the Parameter list, choose Expression.

4 In the Expression text field, type t.

5 From the Unit list, choose min.


S T U D Y 1

Use the Parametric Sweep feature to investigate the difference when the compensating fill is turned on.



3 Click Add.



R E S U L T S

Concentration (re) 11 In the Model Builder window, under Results click Concentration (re) 1.


fill (Control parameter for filling)

0 1


2 In the Settings window for 1D Plot Group, type Volumetric feed rate in the Label text field.

Global 11 In the Model Builder window, expand the Results>Volumetric feed rate node, then click

Global 1.


comp1.re.sumvf - Sum of volume flows.


4 In the Title text area, type Global: Volumetric feed rate (m3/s).




8 Click to expand the Coloring and style section. Locate the Coloring and Style section. Find the Line style subsection. From the Line list, choose Cycle.





13 On the Volumetric feed rate toolbar, click Plot.


2 In the Settings window for 1D Plot Group, type Reactor volume in the Label text field.

3 Locate the Data section. From the Data set list, choose Study 1/

Parametric Solutions 1 (sol2).


Global 11 Right-click Reactor volume and choose Global.

Legends

Operating condition 1


comp1.re.Vr - Reactor volume.




6 Locate the Coloring and Style section. Find the Line style subsection. From the Line list, choose Cycle.





11 On the Reactor volume toolbar, click Plot.


2 In the Settings window for 1D Plot Group, type Monomer mass in the Label text field.

3 Locate the Data section. From the Data set list, choose Study 1/

Parametric Solutions 1 (sol2).

Global 11 Right-click Monomer mass and choose Global.

2 In the Settings window for Global, click Replace Expression in the upper-right corner of the y-axis data section. From the menu, choose Component 1>Definitions>Variables>

comp1.m_mon - Monomer mass.




6 Locate the Coloring and Style section. Find the Line style subsection. From the Line list, choose Cycle.


Legends







11 On the Monomer mass toolbar, click Plot.

Legends





S t e am Re f o rme r




Introduction

This example illustrates the modeling of a steam reformer, serving a stationary fuel cell unit with hydrogen. The tightly coupled system of mass, energy, and momentum equations used to describe the system is readily set up using the predefined physics interfaces of the Chemical Reaction Engineering Module.

Model Definition

In fuel cell power generators, a steam reformer unit typically produces the hydrogen needed for the cell stack. Figure 1 shows the geometry of such a system. The reformation chemistry occurs in a porous catalytic bed where energy is supplied through heating tubes to drive the endothermic reaction system. The reactor is enclosed in an insulating jacket.

Figure 1: Geometry of the steam reformer unit.

Insulating jacket

Heating tubes

Catalytic bed

2 | S T E A M R E F O R M E R

Propane and steam are mixed in stoichiometric amounts and enter through the inlet of the reactor. For heating purposes, hot gases from a burner are passed in the opposite direction, through a number of tubes perforating the reactor bed.

Figure 2: Making use of symmetry, the modeling domain is reduced to a quarter of the full geometry.

In the reformer, water and propane react to form hydrogen and carbon dioxide:

An overall kinetic model has been established from experiments (Ref. 1), where the reaction rate (SI unit: mol/(m3·s)) has been found to be first order in the propane concentration:

(1)

The rate constant is temperature dependent according to:

where A is 7·105 (SI unit: 1/s) and E is 83.14 (SI unit: kJ/mol).

F L U I D F L OW — R E F O R M E R B E D

The flow of gaseous species through the reformer bed is described by Darcy’s law:

Insulating solid

Free flow domain

Porous domainCombustion product gases

H2O+C3H8

+ 6H2O 10H2 3CO2+C3H8k

r kcC3H8=

k A ERgT-----------–

exp=


Here, ρ denotes the gas density (SI unit: kg/m3), η the viscosity (SI unit: Pa·s), κ the permeability of the porous medium (SI unit: m2), and psr is the pressure in the reformer bed (SI unit: Pa). The Darcy’s law equation is in this example solved with the Darcy’s law interface.

The inlet and outlet boundary conditions describe a 75 Pa pressure drop across the bed. All other boundaries are impervious, corresponding to the condition:

E N E R G Y TR A N S P O R T — R E F O R M E R B E D

A one-equation approach is used to describe the average temperature distribution in the porous bed:

The volumetric heat capacity of the bed is given by:

In the above equations, the indices “f” and “s” denote fluid and solid phases, respectively, and ε is the volume fraction of the fluid phase. Furthermore, Tsr is the temperature (SI unit: K) and ksr the thermal dispersion (SI unit: W/(m·K)) of the reformer bed. Q represents a heat source (SI unit: W/m3), and u the fluid velocity (SI unit: m/s). The equation is modeled using the Heat Transfer in Fluids interface.

Assuming that the porous medium is homogeneous and isotropic, the steady-state equation becomes

(2)

The heat source due to reaction is

where, r is given by Equation 1. The steam reformation of propane is endothermic, with an enthalpy of reaction of ΔHr = 4.1·105 J/mol.

∇ ρ κη--- ∇psr–

⋅ 0=

κη---∇psr– n⋅ 0=

ρCp( )t

∂Tsr∂t------------ ∇+ ksr∇Tsr–( )⋅ ρCp( )fu ∇Tsr⋅+ Q=

ρCp( )t ε ρCp( )f 1 ε–( ) ρCp( )s+=

∇ ksr∇Tsr–( )⋅ ρCp( )fu ∇Tsr⋅+ Q=

Q ΔHr r⋅=


Equation 2 also accounts for the conductive heat transfer in the insulating jacket. As no reactions occur in this domain, the description reduces to:

where ki is the thermal conductivity (W/(m·K)) of the insulating material.

The temperature of the gas is 700 K at the inlet. At the outlet, it is assumed that convective heat transport is dominant:

The heat exchange between the bed and the tubes is described by:

(3)

where ht is the heat transfer coefficient (SI unit: W/(m2·K)) and T is the temperature (K) of the heating tubes. A similar expression describes the heat flux from the insulating jacket to the surroundings:

where hj is the heat transfer coefficient of the jacket (SI unit: W/(m2·K)) and Tamb is the ambient temperature (K).

M A S S TR A N S P O R T — R E F O R M E R B E D

The mass-balance equations for the model are the Maxwell-Stefan diffusion and convection equations at steady state:

In the equations above, ρ denotes the density (SI unit: kg/m3), ωi is the mass fraction of species i, xj is the molar fraction of species j, is the ij component of the multicomponent Fick diffusivity (SI unit: m2/s). denotes the generalized thermal diffusion coefficient (SI unit: kg/(m·s)), T is the temperature (SI unit: K), and Ri the reaction rate (SI unit: kg/(m3·s)). The mass-balances are set up and solved with the Transport of Concentrated Species interface.

The inlet weight fraction of propane is 0.28. At the outlet, the convective flux condition is used:

∇ ki∇Tsr–( )⋅ 0=

n ksr∇Tsr–( )⋅ 0=

q ht Tsr T–( )=

q h– j Tsr Tamb–( )=

∇ ρωiu ρωi D̃ij ∇xj xj ωj–( )∇pp--------+

DiT∇T

T--------–

j 1=

n

–

⋅ Ri=

D̃ij

DiT


All other boundaries use the insulating or symmetry condition.

F L U I D F L OW — H E A T I N G TU B E S

The flow of heating gas in the tubes is described by the weakly compressible Navier-Stokes equations at steady-state:

where ρ denotes density (SI unit: kg/m3), u represents the velocity (SI unit: m/s), η denotes viscosity (SI unit: kg/(m· s)), and p equals the pressure in the tubes (SI unit: Pa). The Laminar Flow interface sets up and solves the Navier-Stokes equations and is here used to model the gas flow in the tubes.

The boundary conditions are

At the outlet, viscous stresses are ignored and the pressure is set to 1 atmosphere.

E N E R G Y TR A N S P O R T — H E A T I N G TU B E S

The energy transport in heating tubes is described by:

where kht is the thermal conductivity (SI unit: W/(m·K)) of the heating gas. The temperature of the gas is 900 K at the inlet. Also this energy transport is modeled with the Heat Transfer in Fluids interface.

At the outlet, it is assumed that convective heat transport is dominant:

The heat exchange between the bed and tubes is given by:

n ρωi D̃ij ∇xj xj ωj–( )∇pp--------+

j 1=

n

–

DT–

∇TT--------

⋅ 0=

ρu ∇⋅ u ∇ pI– η ∇u ∇u( )T+( ) 2η 3⁄( ) ∇ u⋅( )I–+[ ]⋅=

∇ ρu( )⋅ 0=

u n⋅ v0=

u 0 =

p pref =

inletwalls

outlet

∇ kht∇T–( )⋅ ρCpu ∇T⋅+ 0=

n kht∇T–( )⋅ 0=

q ht– Tsr T–( )=


This is the same heat flux as given by Equation 3, but with reversed sign.


Figure 3 shows the weight fraction of propane in the reformer bed. The inlet weight fraction is 0.28 while the fraction at the outlet is close to zero.

Figure 3: Weight fraction distribution of propane in the reformer bed.


A cross section plot through the center of the reformer reveals a concentration distribution in the bed. As the local reactivity is mainly controlled by the temperature, results indicate the heat supplied by the tubes is sufficient to efficiently make use of the entire catalytic bed.

Figure 4: Weight fraction distribution of propane in a cross section through the middle of the reformer bed.


Figure 5 shows the weight fractions of all reacting species in the bed, evaluated along the reactor centerline. The plot shows that the entire bed length is active in converting propane.

Figure 5: Weight fraction of reacting species as function of reactor position, plotted along the reactor centerline.

The energy exchange between the heating tubes and reformer bed is clearly illustrated in Figure 6. The gas of the heating tubes enters at 900 K and exits at approximately 674 K.


The gas temperature in the reformer bed is 700 K at the inlet, goes through a minimum, and finally exits with an average temperature of 795 K.

Figure 6: Temperature distributions in the reformer system, including the reformer bed, heating tubes and insulating wall.

A line plot through the center of the reactor shows how the temperature initially decreases due to the endothermic reformation reactions. When the reaction rate is reduced as a


function of lower temperature and propane content, the energy supplied by the heating tubes dictates the temperature evolution.

Figure 7: Bed temperature as a function of position, plotted along the reactor centerline.

Figure 8 shows the velocity fields of both the heating gas in the tubes and the reacting gas in the bed. The flow in the heating tubes is laminar and the parabolic velocity distribution


is clearly seen. The gas velocity in the porous bed increases significantly through the reactor, and, at the outlet, the gas velocity is approximately than twice that at the inlet.

Figure 8: Velocity fields of the heating tubes and the reformer bed.


The increased velocity is mainly due to gas expansion caused by chemical reaction and, to a lesser extent, by temperature increase. Figure 9 illustrates the associated density variations in the reformer bed, accounting for both composition and temperature effects.

Figure 9: Overall gas density in the reformer bed.

In summary, this example illustrates the simulation of a reactor described by fully coupled mass, energy and flow equations.

Reference

1. P.Gateau, Design of Reactors and Heat Exchange Systems to Optimize a Fuel Cell Reformer, Proceedings of the COMSOL User’s Conference Grenoble, 2007.

Application Library path: Chemical_Reaction_Engineering_Module/Reactors_with_Porous_Catalysts/steam_reformer




N E W





3 Click Add Physics.

4 In the Select Physics tree, select Fluid Flow>Porous Media and Subsurface Flow>

Darcy’s Law (dl).


6 In the Pressure text field, type p_sr.




10 In the Temperature text field, type T_sr.






15 Click Study.


17 Click Done.

G E O M E T R Y 1

Now create the geometry. To simplify this step, insert a prepared geometry sequence:

w_H2O

w_C3H8

w_H2

w_CO2



2 On the Geometry toolbar, click Import/Export and choose Insert Sequence.

3 Browse to the model’s Application Libraries folder and double-click the file steam_reformer.mph.

4 Click Build All on the Geometry toolbar.


A set of parameters useful when building the model are available in a text file.




4 Browse to the model’s Application Libraries folder and double-click the file steam_reformer_parameters.txt.





tcs.c_w_C3H8 is the concentration variable for C3H8 in the Transport of Concentrated

Species interface.




3 In the tree, select Liquids and Gases>Gases>Air.



rate tcs.c_w_C3H8*7e5[1/s]*exp(-83.14e3[J/mol]/R_const/T_sr)

mol/(m³·s) Reaction rate

dens_sr tcs.rho kg/m³ Density, reformer bed


M A T E R I A L S

Air (mat1)1 In the Settings window for Material, locate the Geometric Entity Selection section.

2 From the Selection list, choose heating tubes.



Since the density variation is substantial, the flow cannot be regarded as incompressible. Therefore set the flow to be weakly compressible.


2 From the Compressibility list, choose Weakly compressible flow.

3 Locate the Domain Selection section. From the Selection list, choose heating tubes.


4 Locate the Physical Model section. Find the Reference values subsection. In the pref text field, type p_ref.

Fluid Properties 11 In the Model Builder window, expand the Laminar Flow (spf) node, then click

Fluid Properties 1.

2 In the Settings window for Fluid Properties, locate the Model Input section.




3 From the Selection list, choose tubes inlet.


5 Locate the Laminar Inflow section. In the Uav text field, type u_in_ht.



3 From the Selection list, choose tubes outlet.

4 Locate the Pressure Conditions section. Select the Normal flow check box.




3 From the Selection list, choose tubes symmetry.

D A R C Y ’ S L A W ( D L )

1 In the Model Builder window, under Component 1 (comp1) click Darcy’s Law (dl).

2 In the Settings window for Darcy’s Law, locate the Domain Selection section.

3 From the Selection list, choose catalytic bed.


4 Locate the Physical Model section. In the pref text field, type p_ref.

Fluid and Matrix Properties 11 In the Model Builder window, expand the Darcy’s Law (dl) node, then click

Fluid and Matrix Properties 1.

2 In the Settings window for Fluid and Matrix Properties, locate the Fluid Properties section.

3 From the ρ list, choose User defined. In the associated text field, type dens_sr.

4 Locate the Matrix Properties section. From the κ list, choose User defined. In the associated text field, type 1e-9.

5 Locate the Fluid Properties section. From the μ list, choose User defined. In the associated text field, type visc_sr.

6 Locate the Matrix Properties section. From the εp list, choose User defined. In the associated text field, type 0.25.

7 In the Model Builder window, click Darcy’s Law (dl).



3 From the Selection list, choose bed inlet.

4 Locate the Pressure section. In the p0 text field, type p_in_sr.



3 From the Selection list, choose bed outlet.




3 From the Selection list, choose bed symmetry.


1 In the Model Builder window, under Component 1 (comp1) click Heat Transfer in Fluids (ht).

2 In the Settings window for Heat Transfer in Fluids, type Heat Transfer in Fluids - Steam tubes in the Label text field.

3 Locate the Domain Selection section. From the Selection list, choose heating tubes.

H E A T TR A N S F E R I N F L U I D S - S T E A M T U B E S ( H T )

On the Physics toolbar, click Heat Transfer in Fluids (ht) and choose Heat Transfer in Fluids -

Steam tubes (ht).

Fluid 11 In the Model Builder window, expand the Component 1 (comp1)>Heat Transfer in Fluids -

Steam tubes (ht) node, then click Fluid 1.




Initial Values 11 In the Model Builder window, under Component 1 (comp1)>Heat Transfer in Fluids -

Steam tubes (ht) click Initial Values 1.

2 In the Settings window for Initial Values, type T_in_ht in the T text field.

3 In the Model Builder window, click Heat Transfer in Fluids - Steam tubes (ht).



3 From the Selection list, choose tubes inlet.

4 Locate the Temperature section. In the T0 text field, type T_in_ht.




3 From the Selection list, choose tubes outlet.


2 In the Settings window for Heat Flux, locate the Boundary Selection section.

3 From the Selection list, choose tubes/bed.

4 Locate the Heat Flux section. Click the Convective heat flux button.

5 In the h text field, type ht.

6 In the Text text field, type T_sr.

H E A T TR A N S F E R I N F L U I D S 2 ( H T 2 )

1 In the Model Builder window, under Component 1 (comp1) click Heat Transfer in Fluids 2 (ht2).

2 In the Settings window for Heat Transfer in Fluids, type Heat Transfer in Fluids - Catalytic bed and jacket in the Label text field.

3 Locate the Domain Selection section. Click Clear Selection.


H E A T TR A N S F E R I N F L U I D S - C A T A L Y T I C B E D A N D J A C K E T ( H T 2 )

Solid 11 On the Physics toolbar, click Domains and choose Solid.


3 In the Settings window for Solid, locate the Heat Conduction, Solid section.

4 From the k list, choose User defined. In the associated text field, type k_foam.

5 Locate the Thermodynamics, Solid section. From the Cp list, choose User defined. In the associated text field, type Cp_foam.

6 From the ρ list, choose User defined. In the associated text field, type dens_foam.


Catalytic bed and jacket (ht2) click Fluid 1.


3 From the pA list, choose Absolute pressure (dl).

4 From the u list, choose Darcy’s velocity field (dl).


5 Locate the Heat Conduction, Fluid section. From the k list, choose User defined. In the associated text field, type k_sr.

6 Locate the Thermodynamics, Fluid section. From the ρ list, choose User defined. In the associated text field, type dens_sr.

7 From the Cp list, choose User defined. In the associated text field, type Cp_sr.

8 From the γ list, choose User defined. In the associated text field, type 1.4.

Initial Values 11 In the Model Builder window, under Component 1 (comp1)>Heat Transfer in Fluids -

Catalytic bed and jacket (ht2) click Initial Values 1.

2 In the Settings window for Initial Values, type T_in_sr in the Tsr text field.




4 Locate the Temperature section. In the T0 text field, type T_in_sr.






3 From the Selection list, choose tubes/bed.


5 In the h text field, type ht.

6 In the Text text field, type T.



3 From the Selection list, choose jacket/ambient.



5 In the h text field, type hj.

6 In the Text text field, type 300[K].


2 In the Settings window for Heat Source, locate the Domain Selection section.


4 Locate the Heat Source section. In the Q0 text field, type -rate*H_sr.



2 In the Settings window for Transport of Concentrated Species, locate the Domain Selection section.


Transport Properties 11 In the Model Builder window, expand the Transport of Concentrated Species (tcs) node,

then click Transport Properties 1.


3 From the pA list, choose Absolute pressure (dl).

4 Locate the Density section. In the MwH2O text field, type M_H2O.

5 In the MwC3H8 text field, type M_C3H8.

6 In the MwH2 text field, type M_H2.

7 In the MwCO2 text field, type M_CO2.


9 Locate the Convection section. From the u list, choose Darcy’s velocity field (dl).

10 Locate the Model Input section. From the T list, choose Temperature (ht2).

1 D_C3H8_H2O D_H2_H2O D_CO2_H2O

D_C3H8_H2O 1 D_H2_C3H8 D_CO2_C3H8

D_H2_H2O D_H2_C3H8 1 D_CO2_H2

D_CO2_H2O D_CO2_C3H8 D_CO2_H2 1





3 In the ω0,wC3H8 text field, type w_C3H8_in.

4 In the ω0,wH2 text field, type w_H2_in.

5 In the ω0,wCO2 text field, type w_CO2_in.

6 In the Model Builder window, click Transport of Concentrated Species (tcs).

Reaction Sources 11 On the Physics toolbar, click Domains and choose Reaction Sources.

2 In the Settings window for Reaction Sources, locate the Domain Selection section.


4 Locate the Reactions section. In the RwC3H8 text field, type -M_C3H8*rate.

5 In the RwH2 text field, type 10*M_H2*rate.

6 In the RwCO2 text field, type 3*M_CO2*rate.




4 Locate the Inflow section. In the ω0,wC3H8 text field, type w_C3H8_in.

5 In the ω0,wH2 text field, type w_H2_in.

6 In the ω0,wCO2 text field, type w_CO2_in.




M E S H 1





Free Quad 11 In the Model Builder window, right-click Mesh 1 and choose More Operations>Free Quad.

2 In the Settings window for Free Quad, locate the Boundary Selection section.


Distribution 11 Right-click Mesh 1 and choose Swept.

2 In the Model Builder window, under Component 1 (comp1)>Mesh 1 right-click Swept 1 and choose Distribution.






4 Locate the Element Size Parameters section. In the Maximum element size text field, type 2e-3.

5 In the Minimum element size text field, type 1e-3.

Boundary Layers 11 In the Model Builder window, right-click Mesh 1 and choose Boundary Layers.

2 In the Settings window for Boundary Layers, click to expand the Transition section.

3 Clear the Smooth transition to interior mesh check box.

Boundary Layer Properties1 In the Model Builder window, under Component 1 (comp1)>Mesh 1>Boundary Layers 1

click Boundary Layer Properties.

2 Select Boundaries 1, 4, 9, 13, and 17 only.



5 From the Thickness of first layer list, choose Manual.

6 In the Thickness text field, type 0.0003.


Boundary Layer Properties1 In the Model Builder window, under Component 1 (comp1)>Mesh 1 right-click

Boundary Layers 1 and choose Duplicate.

2 In the Model Builder window, expand the Component 1 (comp1)>Mesh 1>

Boundary Layers 2 node, then click Boundary Layer Properties.




Boundary Layer Properties1 In the Model Builder window, under Component 1 (comp1)>Mesh 1 right-click

Boundary Layers 2 and choose Duplicate.

2 In the Model Builder window, expand the Component 1 (comp1)>Mesh 1>

Boundary Layers 3 node, then click Boundary Layer Properties.

3 Select Boundaries 6, 8, 14–16, 18, 20, and 21 only.


5 In the Settings window for Boundary Layer Properties, click Build All.

S T U D Y 1


2 In the Model Builder window, expand the Study 1>Solver Configurations node.

3 In the Model Builder window, expand the Solution 1 (sol1) node, then click Stationary Solver 1.




Solution 1 (sol1)>Stationary Solver 1>Segregated 1 node, then click Segregated Step 1.


8 Locate the Method and Termination section. In the Damping factor text field, type 1.


Stationary Solver 1>Segregated 1 click Temperature T_sr.


10 In the Settings window for Segregated Step, locate the Method and Termination section.

11 In the Damping factor text field, type 1.


Stationary Solver 1>Segregated 1 click Velocity u, Pressure p.


14 Locate the Method and Termination section. In the Damping factor text field, type 1.


R E S U L T S

In the first part of the results processing, create the default plots giving Figure 3, Figure 4 and Figure 8.


2 In the Settings window for 3D Plot Group, type Velocity fields in the Label text field.

Slice1 In the Model Builder window, expand the Results>Velocity fields node, then click Slice.

2 In the Settings window for Slice, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Component 1>Darcy’s Law>

Velocity and pressure>dl.U - Darcy’s velocity magnitude.

3 Locate the Coloring and Style section. From the Color table list, choose Cyclic.

4 Click the Go to Default View button on the Graphics toolbar.

Surface 11 In the Model Builder window, under Results right-click Velocity fields and choose Surface.

2 In the Settings window for Surface, click to expand the Title section.


4 On the Velocity fields toolbar, click Plot.

This is Figure 8.


2 In the Settings window for 3D Plot Group, type Mass fraction propane mid-reactor in the Label text field.

3 Locate the Plot Settings section. Clear the Plot data set edges check box.


Slice1 In the Model Builder window, expand the Results>Mass fraction propane mid-reactor

node, then click Slice.


3 In the Expression text field, type w_C3H8.

4 Locate the Plane Data section. In the Planes text field, type 1.

5 Click the Go to YZ View button on the Graphics toolbar.

6 On the Mass fraction propane mid-reactor toolbar, click Plot.

This is Figure 4.

Mass Fraction (tcs) 11 In the Model Builder window, expand the Results>Mass Fraction (tcs) 1 node.

2 In the Settings window for 3D Plot Group, type Mass fraction propane in the Label text field.

3 In the Model Builder window, expand the Mass Fraction (tcs) 1 node.

Surface1 In the Model Builder window, expand the Results>Mass fraction propane node, then click

Surface.


3 In the Expression text field, type w_C3H8.

4 Click the Go to Default View button on the Graphics toolbar.


6 On the Mass fraction propane toolbar, click Plot.

This is Figure 3.

Next, create new plot groups to process results for the mass fractions and temperature distribution in the reactor (Figures 5, 6, 7, and 9).


2 In the Settings window for 1D Plot Group, type Mass fractions along centerline in the Label text field.


4 In the Title text area, type Mass fractions.



6 In the associated text field, type x (m).

Line Graph 11 Right-click Mass fractions along centerline and choose Line Graph.



4 In the Paste Selection dialog box, type 3 in the Selection text field.

5 Click OK.

6 In the Settings window for Line Graph, type C3H8 in the Label text field.

7 Locate the y-Axis Data section. In the Expression text field, type w_C3H8.







C3H8.11 Right-click Results>Mass fractions along centerline>C3H8 and choose Duplicate.

2 In the Settings window for Line Graph, type CO2 in the Label text field.

3 Locate the y-Axis Data section. In the Expression text field, type w_CO2.


CO2.11 Right-click Results>Mass fractions along centerline>CO2 and choose Duplicate.

2 In the Settings window for Line Graph, type H2 in the Label text field.

3 Locate the y-Axis Data section. In the Expression text field, type w_H2.

Legends

w_C3H8

Legends

w_CO2



H2.11 Right-click Results>Mass fractions along centerline>H2 and choose Duplicate.

2 In the Settings window for Line Graph, type H2O in the Label text field.

3 Locate the y-Axis Data section. In the Expression text field, type w_H2O.



6 On the Mass fractions along centerline toolbar, click Plot.

This is Figure 5.



Surface 11 Right-click Temperature and choose Surface.



Slice 11 In the Model Builder window, under Results right-click Temperature and choose Slice.


3 In the Expression text field, type T_sr.

4 Locate the Coloring and Style section. From the Color table list, choose Thermal.



This is Figure 6.


Legends

w_H2

Legends

w_H2O


2 In the Settings window for 1D Plot Group, type Temperature along centerline in the Label text field.


4 In the Title text area, type Temperature.


6 In the associated text field, type x (m).


8 In the associated text field, type T (K).

Line Graph 11 Right-click Temperature along centerline and choose Line Graph.




5 Click OK.


7 In the Expression text field, type T_sr.




11 On the Temperature along centerline toolbar, click Plot.


This is Figure 7.


2 In the Settings window for 3D Plot Group, type Gas density reformer bed in the Label text field.

Surface 11 Right-click Gas density reformer bed and choose Surface.


3 In the Expression text field, type dl.rho.

4 On the Gas density reformer bed toolbar, click Plot.



This is Figure 9.

Remove unused plot groups.

Pressure (dl)In the Model Builder window, under Results right-click Pressure (spf) and choose Delete.

Pressure (dl) 1In the Model Builder window, under Results right-click Pressure (dl) and choose Delete.

Temperature (ht)In the Model Builder window, under Results right-click Pressure (dl) 1 and choose Delete.

Isothermal Contours (ht)In the Model Builder window, under Results right-click Temperature (ht) and choose Delete.

Temperature (ht2)In the Model Builder window, under Results right-click Isothermal Contours (ht) and choose Delete.

Derived Values1 In the Model Builder window, under Results right-click Temperature (ht2) and choose

Delete.

2 Right-click Isothermal Contours (ht2) and choose Delete.

Finally, calculate the average outlet temperatures for the gas in the heating tubes and in the reformer bed.




4 Click Evaluate.




ht.ofl1.Tave K Weighted average temperature



4 Click Evaluate.


ht2.ofl1.Tave K Weighted average temperature



S t e f a n Tub e




Introduction

This example illustrates the use of the Maxwell-Stefan diffusion model available with the Transport of Concentrated Species interface. It models multicomponent gas-phase diffusion in a Stefan tube in 1D. In this case, it is a liquid mixture of acetone and methanol that evaporates into air.

The concentration profiles are modeled at steady-state and validated against experimental data by Taylor and Krishna (Ref. 6).

Model Definition

The Stefan tube, shown in Figure 1, is a simple device used for measuring diffusion coefficients in binary vapors.

Figure 1: Schematic diagram of a Stefan tube.

At the bottom of the tube is a pool of mixture. The vapor that evaporates from this pool diffuses to the top of the tube, where a stream of air, flowing across the top of the tube, keeps the mole fraction of diffusing vapor there to be zero. The mole fraction of vapor above the liquid interface is at equilibrium. Because there is no horizontal flux inside the tube, you can analyze the problem using a 1D model geometry representing the distance between the liquid mixture surface and the top of the tube. The system composition of acetone, methanol, and air has been extensively investigated; both diffusion coefficients and composition have been measured at various positions within Stefan tubes. This makes it an ideal system for this model.

Air

Gas/vapor phase

Liquid mixture

x

2 | S T E F A N TU B E

As a comparison, one experiment measured the mole fraction at the liquid interface to be xAc = 0.319 and xMe = 0.528 where the pressure, p, was 99.4 kPa and the temperature, T, was 328.5 K. The length of the diffusion path was 0.238 m. The respective Maxwell-Stefan diffusion coefficients, Dij, of the three binary pairs were calculated and are used in the model according to Table 1.

To model this problem, use the Transport of Concentrated Species interface with the Maxwell-Stefan diffusion model. It solves for the fluxes in terms of mass fractions for two of the three components. The mass fraction, ω, of the third is given by the two other ones. The three equations are:

where D is the diffusion coefficient (SI unit: m2/s), p is the pressure (SI unit: Pa), T is the temperature (SI unit: K), u is the velocity (SI unit: m/s), x and ω are mole and mass fractions, respectively, and the mixture density, ρmix (SI unit: kg/m3), is a function of the average mixture mole fraction, Mmix (SI unit: kg/mol), according to Equation 2:

(1)

(2)

TABLE 1: LABELS AND MAXWELL-STEFAN DIFFUSION COEFFICIENTS

COMPONENT LABEL Dij VALUE

Acetone 1 D12 8.48·10-6 m2/s

Methanol 2 D13 13.72·10-6 m2/s

Air 3 D23 19.91·10-6 m2/s

∇ ρω1 D1k xk∇ xk ωk–( ) p∇( ) p⁄( )+( )[ ]k–⋅ +

DT T∇( ) T⁄( ) ] R ρu ω1∇⋅( )–=

∇ ρω2 1( ) D2k xk∇ xk ωk–( ) p∇( ) p⁄( )+( )[ ]k–⋅ +

DT T∇( ) T⁄( ) ] R ρu ω2∇⋅( )–=

ω3 1 ω1– ω2–=

Mmix xiMi

i=

ρmixp

RT--------Mmix=


In this case, there is no imposed fluid velocity. However, there will appear a fluid velocity due to the diffusive fluxes. At the top of the tube the mass fractions are fixed, with the fraction of air being unity. At the bottom (at the liquid interface), the fractions are also fixed according to the previously mentioned experimental conditions. The fact that there is no air flux at the interface results in the following relation for the convective velocity, at steady state:

where ndiff,3 is the diffusive mass flux of air (SI unit: kg/(m2·s)).

Results

Both the modeled and experimental steady-state mole fractions as a function of position are shown in Figure 2.

Figure 2: Modeled and experimental (Ref. 6) steady-state mole fractions of: acetone, methanol, and air, in the Stefan tube.

We can see that the model reproduces the results from Ref. 6 well, which means the Maxwell-Stefan equations can describe the mass transport process in the system accurately.

undiff,3ω3ρ--------------=


The Maxwell-Stefan diffusion formulation includes the conservation of mass. In the absence of chemical reactions (source terms) and convective contributions, the Maxwell-Stefan formulation results in zero net mass flux. In this example, the convective term is included, which you can see in the velocity profile in Figure 3.

Figure 3: Velocity of the gas mixture in the Stefan tube.

References

1. C.F. Curtiss and R.B. Bird, “Multicomponent diffusion,” Ind. Eng. Chem. Res., vol. 38, p. 2515, 1999.

2. R.B. Bird, W. Stewart, and E. Lightfoot, Transport Phenomena, John Wiley & Sons, New York, 1960.

3. G.A.J. Jaumann, Wien. Akad. Sitzungsberichte (Math.-Naturw. Klasse), vol. 120, p. 385, 1911.

4. J.O. Hirschfelder, C.F. Curtiss, and R.B. Bird, Molecular Theory of Gases and Liquids, Wiley, USA, 1954.

5. E.N. Fuller, P.D. Schettler, and J.C. Giddings, Ind. Eng. Chem., vol. 58, p. 19, 1966.


6. R. Taylor and R. Krishna, Multicomponent Mass Transfer, John Wiley & Sons, NY, p. 21, 1993.

Application Library path: Chemical_Reaction_Engineering_Module/Mixing_and_Separation/stefan_tube



N E W






3 Click Add.

4 Click Add Mass Fraction.


6 Click Study.


8 Click Done.

G E O M E T R Y 1

Interval 1 (i1)1 On the Geometry toolbar, click Interval.

2 In the Settings window for Interval, locate the Interval section.

3 In the Right endpoint text field, type 0.238.


w1

w2

w3



Next, add a set of model parameters by importing their definitions from a data text file provided with the Application Library.




4 Browse to the model’s Application Libraries folder and double-click the file stefan_tube_parameters.txt.




3 From the Diffusion model list, choose Maxwell-Stefan.

4 Locate the Species section. From the From mass constraint list, choose w3.


6 Click to expand the Discretization section. From the Mass fraction list, choose Quadratic.

Transport Properties 11 In the Model Builder window, expand the Transport of Concentrated Species (tcs) node,

then click Transport Properties 1.


3 In the T text field, type T0.

4 In the pA text field, type p0.

5 Locate the Convection section. Specify the u vector as

6 Locate the Density section. In the Mw1 text field, type M_ace.

7 In the Mw2 text field, type M_met.

8 In the Mw3 text field, type M_air.

-tcs.dflux_w3x/(w3*tcs.rho) x






3 In the ω0,w1 text field, type w_ace0.

4 In the ω0,w2 text field, type w_met0.

Mass Fraction 11 On the Physics toolbar, click Boundaries and choose Mass Fraction.


3 In the Settings window for Mass Fraction, locate the Mass Fraction section.

4 Select the Species w1 check box.

5 In the ω0,w1 text field, type w_ace0.


7 In the ω0,w2 text field, type w_met0.

Mass Fraction 21 On the Physics toolbar, click Boundaries and choose Mass Fraction.


3 In the Settings window for Mass Fraction, locate the Mass Fraction section.



M E S H 1

1 In the Model Builder window, under Component 1 (comp1) click Mesh 1.


3 From the Element size list, choose Extra fine.

4 Click Build All.

1 D12 D13

D12 1 D23

D13 D23 1


S T U D Y 1


R E S U L T S


Import the experimental data for comparison.

2 In the Settings window for Table, type Experimental Mole Fractions in the Label text field.

3 Locate the Data section. Click Import.

4 Browse to the model’s Application Libraries folder and double-click the file stefan_tube_exp.csv.

In order to reproduce the plot in Figure 2, do the following:


2 In the Settings window for 1D Plot Group, type Mole Fraction in the Label text field.

Line Graph1 In the Model Builder window, expand the Results>Mole Fraction node, then click

Line Graph.


3 In the Settings window for Line Graph, click to expand the Title section.

4 Click Replace Expression in the upper-right corner of the y-axis data section. From the menu, choose Component 1>Transport of Concentrated Species>tcs.x_w1 - Mole fraction.

5 Locate the y-Axis Data section. Clear the Description check box.









Line Graph 21 Right-click Results>Mole Fraction>Line Graph and choose Duplicate.


Transport of Concentrated Species>tcs.x_w2 - Mole fraction.


Line Graph 31 Right-click Results>Mole Fraction>Line Graph 2 and choose Duplicate.


Transport of Concentrated Species>tcs.x_w3 - Mole fraction.


Table Graph 11 In the Model Builder window, under Results right-click Mole Fraction and choose

Table Graph.


3 From the x-axis data list, choose Column 1.

4 Locate the Coloring and Style section. Find the Line markers subsection. From the Marker list, choose Cycle.

5 Find the Line style subsection. From the Line list, choose None.



Legends

Acetone

Legends

Methanol

Legends

Air



Mole Fraction1 In the Model Builder window, under Results click Mole Fraction.


3 Select the x-axis label check box.

4 In the associated text field, type Position (m).

5 Click to expand the Legend section. From the Position list, choose Upper left.

To reproduce Figure 3, proceed as follows:


2 In the Settings window for 1D Plot Group, type Velocity Field in the Label text field.


4 In the associated text field, type Position (m).

Line Graph 11 Right-click Velocity Field and choose Line Graph.



Transport of Concentrated Species>Velocity field>tcs.u - Velocity field, x component.




7 On the Velocity Field toolbar, click Plot.

Legends

Acetone exp.

Methanol exp.

Air exp.



I d e a l S t i r r e d T ank R e a c t o r S y s t em




Introduction

In the chemical and biochemical industries, for instance in fermentation processes, reactors having well-mixed conditions and liquid level control are common.

This example shows how to use the Reaction Engineering interface to model a 0D ideal system of tank reactors in series, with controlled feed inlet and product outlet streams. The volume change in each reactor is monitored and controlled.

Model Definition

This model solves for a liquid-phase first order irreversible reaction A → B where

(1)

In Equation 1, r is the reaction rate (SI unit: mol/(m3·s)), k is the rate constant (SI unit: 1/min), and cA is the concentration of A.

The reaction takes place in a system of two ideal reactors in series. The first reactor tank has a volume capacity of vtank1 = 1 m3 and the second vtank2 = 1.5 m3 initially.

Initially the reactors are charged only with solvent. A is fed with solvent to the first tank with a volumetric flow rate of vf1 = 1 m3/min. The volumetric flow rate at the outlet of the first tank is set to vout1 = 0.9 m3/min. This whole stream is fed to the second tank in the system, vf2 = vout1. The second tank also has a fresh supply of A with solvent entering at a rate of vf2 = 0.5 m3/min. In a similar way as for the first reactor, the outlet flow is also set to vout2 = 1 m3/min. Figure 1 shows the reactor system in detail.

Figure 1: The reactor system.

The mass balance for species i in each reactor is shown in Equation 2:

r kcA=

vfresh2

Vtank1Vtank2

vout2

vf1

vf2

2 | I D E A L S T I R R E D TA N K R E A C T O R S Y S T E M

(2)

Vr denotes the reactor volume (SI unit: m3), Ri is the species rate expression (SI unit: mol/(m3·s)). Subscripts f and out indicate the feed inlet and the outlet, respectively. j and k are the number of feed inlet and outlet streams, respectively.

The volume depends on the volumetric production rate vp and the regulated inlet and outlet volumetric flow rates as shown in Equation 3:

(3)

The model incorporates two stop conditions: If any of the reactor volumes is 1 % or less of the initial volumes, the computations stop.


In Figure 2 and Figure 3, the concentrations of the species in the two tanks are shown.

Figure 2: Concentrations of A and B in tank 1 and tank 2.

d Vrci( )dt

-------------------- vf j, cf ij,j vout k, ci

k RiVr+–=

dVr

dt---------- vf j,

j vout k,

k vp+–=


The variation of the tank volumes is shown in Figure 3. With a known, diameter of the reactor, the liquid level can be calculated from this volume. Given the inlet and outlet flows, this system will fill up the tanks considerably.

Figure 3: Tank volumes.

The results show only some system aspects that can be investigated with the Reaction Engineering interface. This application can also be utilized as a starting point for studying, for instance, startup and steady-state process conditions.

Application Library path: Chemical_Reaction_Engineering_Module/Ideal_Tank_Reactors/tank_flow_system




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3 Click Add.

4 Click Add.

5 Click Study.


Time Dependent.

7 Click Done.


Add a set of global parameters by importing their definitions from a data text file.




4 Browse to the model’s Application Libraries folder and double-click the file tank_flow_system_parameters.txt.

Start with the first reactor. Select the CSTR constant mass/generic reactor type to model the first reactor.




3 From the Reactor type list, choose CSTR, constant mass/generic.






4 Click Apply.


Species: A1 In the Model Builder window, under Component 1 (comp1)>Reaction Engineering (re)

click Species: A.



Species: B1 In the Model Builder window, under Component 1 (comp1)>Reaction Engineering (re)

click Species: B.



Add a solvent (water) to the first tank.



3 In the Species name text field, type S.


5 Locate the General Parameters section. In the M text field, type Mn_S.

6 In the ρ text field, type density_S.

Generic in the Mass Balance section substitutes the constant mass condition. Choose this to set a constant value of the volumetric outlet rate, v.


8 In the Settings window for Reaction Engineering, locate the Mass Balance section.

9 From the Volumetric rate list, choose Generic.

10 Click Reset to Default.

11 In the v text field, type v_outlet1.




3 In the Vr0 text field, type Vr_init_tank1.


Feed Inlet 1Add one Feed Inlet stream to the first tank.



3 In the vf text field, type v_inlet.


Initial Values 1Set only solvent in the first tank initially.




Continue with the second reactor. Select the CSTR constant mass/generic reactor type to model the second reactor.

R E A C T I O N E N G I N E E R I N G 2 ( R E 2 )

On the Physics toolbar, click Reaction Engineering (re) and choose Reaction Engineering 2 (re2).

1 In the Model Builder window, under Component 1 (comp1) click Reaction Engineering 2 (re2).


3 From the Reactor type list, choose CSTR, constant mass/generic.



A cinlet_A1

S cinlet_S


S cinit_S





4 Click Apply.


Species: A1 In the Model Builder window, under Component 1 (comp1)>Reaction Engineering 2 (re2)

click Species: A.



Species: B1 In the Model Builder window, under Component 1 (comp1)>Reaction Engineering 2 (re2)

click Species: B.




Add a solvent (water) to the second tank.


3 In the Species name text field, type S.


5 Locate the General Parameters section. In the M text field, type Mn_S.

6 In the ρ text field, type density_S.

Selecting Generic in the Mass Balance section can substitute the constant mass condition and a constant value of the volumetric outlet rate, v, can be set instead.

7 In the Model Builder window, click Reaction Engineering 2 (re2).

8 In the Settings window for Reaction Engineering, locate the Mass Balance section.

9 From the Volumetric rate list, choose Generic.

10 Click Reset to Default.

11 In the v text field, type v_outlet2.


Initial Values 11 In the Model Builder window, under Component 1 (comp1)>Reaction Engineering 2 (re2)



3 In the Vr0 text field, type Vr_init_tank2.


Add a Feed Inlet stream to the second reactor equal to the whole outlet stream from the first reactor.

2 In the Settings window for Feed Inlet, type Feed Inlet 1 - from tank 1 in the Label text field.

3 Locate the Feed Inlet Properties section. In the vf text field, type v_outlet1.


Feed Inlet 2Add a second Feed Inlet stream to the second reactor to model the fresh feed.


2 In the Settings window for Feed Inlet, type Feed Inlet 2 - fresh in the Label text field.

3 Locate the Feed Inlet Properties section. In the vf text field, type v_fresh2.


Initial Values 1Only solvent exists in the second tank initially.


A re.c_A

B re.c_B

S re.c_S


A cfresh2_A

S cinlet_S


1 In the Model Builder window, under Component 1 (comp1)>Reaction Engineering 2 (re2) click Initial Values 1.



S T U D Y 1

Step 1: Time Dependent1 In the Model Builder window, expand the Study 1 node, then click

Step 1: Time Dependent.


3 From the Time unit list, choose min.



Stop the computations if any of the two reactor volumes is empty.


3 Right-click Time-Dependent Solver 1 and choose Stop Condition.

4 In the Settings window for Stop Condition, locate the Stop Expressions section.

5 Click Add.


7 Click Add.



S cinit_S

Stop expression Stop if Active Description

comp1.re.Vr<=Vr_init_tank1*0.01

true √ Stop expression 1

Stop expression Stop if Active Description

comp1.re2.Vr<=Vr_init_tank2*0.01

true √ Stop expression 2

9 Locate the Output at Stop section. From the Add solution list, choose Steps before and after stop.

10 Clear the Add warning check box.


Follow these steps to create Figure 2.

R E S U L T S


2 In the Settings window for 1D Plot Group, type Concentrations in tanks in the Label text field.


Global 11 In the Model Builder window, expand the Results>Concentrations in tanks node, then

click Global 1.

2 In the Settings window for Global, type Tank 1 in the Label text field.




Tank 1.11 Right-click Results>Concentrations in tanks>Tank 1 and choose Duplicate.


3 Click Replace Expression in the upper-right corner of the y-axis data section. From the menu, choose Component 1>Reaction Engineering 2>comp1.re2.c_A - Concentration.

4 Click Add Expression in the upper-right corner of the y-axis data section. From the menu, choose Component 1>Reaction Engineering 2>comp1.re2.c_B - Concentration.

Legends

A in tank 1

B in tank 1



6 On the Concentrations in tanks toolbar, click Plot.

Follow these steps to create Figure 3.

Concentration (re2)1 In the Model Builder window, under Results click Concentration (re2).

2 In the Settings window for 1D Plot Group, type Volume in tanks in the Label text field.


Global 11 In the Model Builder window, expand the Results>Volume in tanks node, then click

Global 1.


3 Click Replace Expression in the upper-right corner of the y-axis data section. From the menu, choose Component 1>Reaction Engineering>comp1.re.Vr - Reactor volume.




Tank 1.11 Right-click Results>Volume in tanks>Tank 1 and choose Duplicate.


3 Click Replace Expression in the upper-right corner of the y-axis data section. From the menu, choose Component 1>Reaction Engineering 2>comp1.re2.Vr - Reactor volume.


Legends

A in tank 2

B in tank 2

Legends

Tank 1

Legends

Tank 2


5 On the Volume in tanks toolbar, click Plot.



T ank S e r i e s w i t h F e edba c k Con t r o l




Introduction

This example illustrates a series of three consecutive CSTR reactors. A feedback loop continuously adjusts the inlet concentration of the first tank to keep the concentration at the outlet of the last reactor close to a set level. The model utilizes the Reaction Engineering interface in the Chemical Reaction Engineering Module.

Model Description

The following example reproduces the results in Ref. 1. Three CSTR reactors are connected in a series arrangement, as in Figure 1.

Figure 1: An example of three continuous stirred tank reactors (CSTRs) in a series.

The same unimolecular liquid reaction takes place in aqueous solution in each unit:

Under isothermal conditions and the assumption that the volume is constant, the balance equations for reactant A in each of the tanks become:

v0

cA0 v1

cA1 v2

cA2 v3

cA3

V1

V2

V3

A Bk

V1dcA1

dt------------- v0cA0 v1cA1– V1kcA1–=

V2dcA2

dt------------- v1cA1 v2cA2– V2kcA2–=

V3dcA3

dt------------- v2cA2 v3cA3– V3kcA3–=

2 | TA N K S E R I E S W I T H F E E D B A C K C O N T R O L

V denotes the reactor volume (SI unit: m3) and v is the volumetric flow rate for an inlet or outlet (SI unit: m3/min). The concentration of A is represented by cA (SI unit: mol/m3), while k is the rate constant (SI unit: 1/min).

These equations are modeled using the CSTR reactor with constant volume feature in the Reaction Engineering interface. The feed inlet streams connect the reactors to each other. It is assumed that the reactor holdups (volumes) are constant and that the reacting fluid has constant density. Thus, all volumetric flow rates are equal within the reactor system:

The volumetric flow rate is calculated from the known residence time, τ, from the following expression:

F E E D B A C K C O N T R O L

The model also considers adding a feedback control to the system where the product concentration, cA3, is monitored in the outlet stream leaving the third tank. Adjustments are made to the inlet concentration of the first tank cA0 to keep cA3 close to a set level,

. Figure 2 illustrates the control system.

.

Figure 2: An example of tanks in a series with feedback control.

The concentration of A in the inlet to the first reactor is now given by:

v0 v1 v2 v3 v= = = =

τ Vv----=

cA3set

cAD cA0

cA1

cA2

Feedbackcontroller

-1

cA3

cAM

Σ

ΣcA3

set

cA0 cAM cAD+=


The variable cAD is a disturbance concentration while cAM is the manipulated concentration changed by the controller. The value of cAM is based on the magnitude of the error and the integral of the error according to this expression:

(1)

Above, the error is defined by:

where Kc is the controller gain and τi the controller reset time. The term 800 mol/m3 in Equation 1 is the bias value of the controller, that is, the value of cAM at time zero.

According to Equation 1, the integral of the error needs to be evaluated for the feedback control. Noting that from:

it is clear that the integral can be evaluated by solving an ODE. The ODE is specified by adding a Global Equation, a Global ODEs and PDEs interface, to the model.

Results

Figure 3 shows the concentration of A (SI unit: mol/m3) in the three tanks as a function of time (SI unit: min). The initial concentration of A is 400 mol/m3 in tank 1, 200 mol/

cAM 800 mol

m3----------- Kc E 1

τi---- E td+

+=

E cA3set cA3–=

ddt------ E td

E=


m3 in tank 2, and 100 mol/m3 in tank 3. The system is “open loop,” that is, without feedback control. The reactors reach steady state after approximately 10 minutes.

Figure 3: Concentration transients for three tanks in series without feedback control.


Figure 4 illustrates the concentration transients in the “closed loop” system. The control system, regulating on the outlet concentration in the last unit, sets the inlet concentration of the first unit.

Figure 4: Concentration transients for three tanks in series with feedback control. cAM is the manipulated concentration.

The set concentration, , is 100 mol/m3. The feedback control appears to be reasonably tuned to keep the outlet concentration from tank 3 at the desired level.

Reference

1. W.L. Luyben, Process Modeling, Simulation and Control for Chemical Engineers 2nd ed., McGraw-Hill, pp. 119–124, 1990.

Application Library path: Chemical_Reaction_Engineering_Module/Ideal_Tank_Reactors/tankinseries_control

cA3set




N E W





Add three Reaction Engineering interfaces to model three CSTRs.

3 Click Add.

4 Click Add.

5 Click Add.

Add a Global ODEs and PDEs interface to model the closed loop system later.

6 In the Select Physics tree, select Mathematics>ODE and DAE Interfaces>

Global ODEs and DAEs (ge).

7 Click Add.

8 Click Study.


Time Dependent.

10 Click Done.


Add a set of model parameters by importing their definitions from a data text file.




4 Browse to the model’s Application Libraries folder and double-click the file tankinseries_control_parameters.txt.


On the Physics toolbar, click Global ODEs and DAEs (ge) and choose Reaction Engineering (re).



2 In the Settings window for Reaction Engineering, type tank1 in the Name text field.

The interface name will help you keep track of the variables that belong to the physics interface. In this case, the Reaction Engineering interface corresponds to a tank reactor, and to keep this in mind the interface name is changed to tank1.

3 Locate the Reactor section. From the Reactor type list, choose CSTR, constant volume.



R E A C T I O N E N G I N E E R I N G ( T A N K 1 )





Species: A1 In the Model Builder window, under Component 1 (comp1)>Reaction Engineering (tank1)

click Species: A.



4 In the ρ text field, type rho_spec.

Species: B1 In the Model Builder window, under Component 1 (comp1)>Reaction Engineering (tank1)

click Species: B.









5 Locate the General Parameters section. In the M text field, type Mn_solv.

6 In the ρ text field, type rho_solv.

Initial Values 11 In the Model Builder window, under Component 1 (comp1)>Reaction Engineering (tank1)





Add Feed Inlet streams at each tank containing the outlet compositions of the previous tank. The volumetric flow is constant in the tank system.


3 In the vf text field, type v_tanks.









A cinit_A_tank1

H2O c_solv


A cinlet_A

H2O c_solv


R E A C T I O N E N G I N E E R I N G 2 ( T A N K 2 )





Species: A1 In the Model Builder window, under Component 1 (comp1)>

Reaction Engineering 2 (tank2) click Species: A.




Species: B1 In the Model Builder window, under Component 1 (comp1)>

Reaction Engineering 2 (tank2) click Species: B.











Reaction Engineering 2 (tank2) click Initial Values 1.














R E A C T I O N E N G I N E E R I N G 3 ( T A N K 3 )






A cinit_A_tank2

H2O c_solv


A tank1.c_A

B tank1.c_B

H2O c_solv


Species: A1 In the Model Builder window, under Component 1 (comp1)>

Reaction Engineering 3 (tank3) click Species: A.




Species: B1 In the Model Builder window, under Component 1 (comp1)>

Reaction Engineering 3 (tank3) click Species: B.











Reaction Engineering 3 (tank3) click Initial Values 1.







A cinit_A_tank3

H2O c_solv



S T U D Y 1





R E S U L T S

Concentration (tank1)Store a copy of the solution for the open loop reactor system. This way you readily access the results for comparison with the closed loop system.

S T U D Y 1

On the Study toolbar, click Create Solution Copy.

S T U D Y 1

Solution 1 - Copy 1 (sol2)1 In the Model Builder window, expand the Study 1>Solver Configurations node, then click

Solution 1 - Copy 1 (sol2).

2 In the Settings window for Solution, type Open loop in the Label text field.

Follow the steps below to plot the concentration of species A in all three tanks for the open loop system.

R E S U L T S


2 In the Settings window for 1D Plot Group, type Open loop in the Label text field.


A tank2.c_A

B tank2.c_B

H2O c_solv


4 In the associated text field, type Concentration A (mol/m3).

5 Locate the Data section. From the Data set list, choose Study 1/Open loop (sol2).

Global 11 Right-click Open loop and choose Global.


comp1.tank1.c_A - Concentration.

3 Click Add Expression in the upper-right corner of the y-axis data section. From the menu, choose Component 1>Reaction Engineering 2>comp1.tank2.c_A - Concentration.

4 Click Add Expression in the upper-right corner of the y-axis data section. From the menu, choose Component 1>Reaction Engineering 3>comp1.tank3.c_A - Concentration.







11 On the Open loop toolbar, click Plot.


G L O B A L O D E S A N D D A E S ( G E )

Set up the feedback control to model the closed loop system using the Global ODEs and

PDEs interface and some variables.

On the Physics toolbar, click Reaction Engineering 3 (tank3) and choose Global ODEs and DAEs (ge).

Legends

Tank 1

Tank 2

Tank 3

Global Equations 11 In the Model Builder window, under Component 1 (comp1)>Global ODEs and DAEs (ge)

click Global Equations 1.

2 In the Settings window for Global Equations, locate the Global Equations section.


4 In the Settings window for Global Equations, locate the Units section.

5 Click Define Dependent Variable Unit.

6 In the Dependent variable quantity table, enter the following settings:

7 In the Settings window for Global Equations, locate the Units section.

8 Click Select Source Term Quantity.

9 In the Physical Quantity dialog box, In the associated text field, type id:concentration.

10 Click Filter.

11 In the tree, select General>Concentration (mol/m^3).

12 Click OK.



choose Variables.



Name f(u,ut,utt,t) (1) Initial value (u_0) (1)

Initial value (u_t0) (1/s)

Description

E_int E_intt-E 0 0

Dependent variable quantity Unit

Custom unit mol/m^3*s


E cset_A-tank3.c_A mol/m³ Measured error

cM_A 800+Kc*(E+E_int/tau1) mol/m³ Manipulated concentration

cinlet_A cM_A+cdisturb_A mol/m³ Inlet concentration


S T U D Y 1


Follow the steps below to plot the concentration of species A in all three tanks and at the inlet for the closed loop system.

R E S U L T S

Open loop 11 In the Model Builder window, under Results right-click Open loop and choose Duplicate.

2 In the Settings window for 1D Plot Group, type Closed loop in the Label text field.


Global 11 In the Model Builder window, expand the Results>Closed loop node, then click Global 1.

2 In the Settings window for Global, click Add Expression in the upper-right corner of the y-axis data section. From the menu, choose Component 1>Definitions>Variables>

comp1.cinlet_A - Inlet concentration.



5 On the Closed loop toolbar, click Plot.


Legends

Tank 1

Tank 2

Tank 3

At inlet to tank 1



Hyd r o c a r b on Deha l o g ena t i o n i n a T o r t u ou sM i c r o r e a c t o r




Introduction

Removing halogen groups from hydrocarbons is an important reaction step in several chemical processes. One application is water purification. Other examples involve organic synthesis, where the removal of halogen groups serves as a starting point for carbon-carbon coupling reactions. Typically, the carbon-halogen bond scission is activated by precious metal catalysts based on platinum or palladium.

The model presented here shows hydrocarbon dehalogenation as it occurs in a microreactor. The reactants are transported from the fluid bulk to the catalytic surfaces at the reactor walls, where they react. First you set up a space-independent model, analyzing two competing reactions, using the Reaction Engineering interface. Then, you export the reaction kinetics and set up and solve a space-dependent model of the microreactor.

Model Definition

The adsorption of halogenated hydrocarbons onto the surface of a platinum catalyst leads to cleavage of the carbon halogen bond. The hydrocarbon fragments then undergo either hydrogenation or coupling reactions. The scheme below illustrates the overall reactions for a brominated hydrocarbon species.

Figure 1: The dehalogenation of RBr can result either in hydrogenation or coupling of the hydrocarbon fragments.

The reaction rates are:

and

where the rate constants are given by the Arrhenius expression:

(1)

R-Hk1R-Br

[Pt]

R-R2R-Brk2

[Pt]

r1 k1cRBr=

r2 k2cRBr2

=

kj AjEj

RgT-----------–

exp=

2 | H Y D R O C A R B O N D E H A L O G E N A T I O N I N A TO R T U O U S M I C R O R E A C T O R

In Equation 1, A is the frequency factor, and E the activation energy (SI unit: J/mol). The bulk species are said to be dissolved in water. The table below lists the values of the Arrhenius parameters for the two reactions.

I D E A L R E A C T O R M O D E L

The mass balance equation for a flow-through reactor is given by

(2)

where F is the molar flow rate (SI unit: mol/s), V the reactor volume (SI unit: m3), and Ri the net reaction term (SI unit: mol/(m3·s)). If the reactor has constant cross-section and constant flow velocity, the left-hand side of Equation 2 can be rewritten as

The reactor mass balance thus becomes

(3)

where τ represents the residence time (SI unit: s). The assumption of constant flow velocity is valid for incompressible liquids or liquids where the effect of temperature on the density is small. Equation 3 is identical to the balance equation of the batch reactor, except that residence time replaces the reaction time. You can therefore make use of the Batch reactor type when solving the model in the Chemical Reaction Engineering Module.

The ideal reactor model assumes by default that reactions take place in the entire reactor volume. In the 3D microreactor model, reactions occur at catalytic surfaces located at the reactor walls. In order to make the ideal model represent a reactor with surface reactions, Equation 3 has to be scaled by the reactive area per reactor volume. Scaling the ideal reactor equations by the dimensions of the microreactor makes the 1D and 3D models comparable. The area to volume ratio is

TABLE 1: ARRHENIUS PARAMETERS

Frequency factor Activation energy

Reaction 1 2e-3[1/s] 10e3[J/mol]

Reaction 2 1e-3[m^2/(mol*s)] 30e3[J/mol]

dFi

dV--------- Ri=

dFi

dV--------- u

dcidx--------

dci

dτ--------= =

dci

dτ-------- Ri=

WLWLH--------------- 1

H-----=


where W is the width of the channel (SI unit: m), H the channel height (SI unit: m), and L the length of a reactive section (SI unit: m). The scaled ideal reactor equation is then

Note that the net reaction term (Ri) in this case represents surface reactions (SI unit: mol/(m2·s)).


The microreactor considered in this example consists of a tortuous channel, fitted with inlet and outlet adapter sections, as illustrated in Figure 2.

Figure 2: Microreactor geometry.

In the straight sections of the reactor, the channel walls are in part coated with platinum catalyst. As water with small amounts of a brominated hydrocarbon flows through the reactor, dehalogenation reactions occur at the catalytic surfaces.

M O M E N T U M B A L A N C E S

The flow in the channel is modeled with the Laminar Flow interface that solves the Navier-Stokes equations at steady state:

dci

dτ--------

Ri

H------=

Catalytic surface

Outlet

Inlet

ρ u ∇⋅( )u ∇ p– I η ∇u ∇u( )T+( ) 2η

3------- κdv– ∇ u⋅( )I–+⋅=

∇ ρu( )⋅ 0=


Here, η denotes the dynamic viscosity (SI unit: Ns/m2), u the velocity (SI unit: m/s), ρ the density of the fluid (SI unit: kg/m3), and p the pressure (SI unit: Pa).

A pressure difference drives the flow through the reactor, as indicated by the boundary conditions

Each pressure condition is specified along with a vanishing viscous stress condition at the boundary

At the wall the velocity is zero

M A S S B A L A N C E S

The mass balances are set up with the Transport of Diluted Species interface and solves the diffusion-convection equations at steady state:

Here Di denotes the diffusion coefficient (SI unit: m2/s), ci is the species concentration (SI unit: mol/m3), and u equals the velocity (SI unit: m/s).

The diffusivity of the reacting species is assumed to depend on the temperature according to

No reactions take place in the fluid bulk. Rather, the reactions take place on the catalytic surfaces. The boundary fluxes at the catalytic surfaces thus become

where Ri represents the reaction term.

Inlet conditions are equal to the inlet concentrations

p pinlet= inlet

p 0= outlet

η ∇u ∇u( )T+( )n 0=

u 0= walls

∇ Di∇– ci ciu+( )⋅ 0=

D 5 10 7– 2000 KT

-------------------– m2/sexp⋅ ⋅=

n Di∇– ci ciu+( )⋅ Ri=

c cin=


At the outlet, you can set the convective flux condition, assuming that the transport of mass across the boundary is dominated by convection

All other boundaries use the insulating condition


First review the results of the ideal reactor model, which you set up and solve using the Reaction Engineering interface.

Figure 3 through Figure 5 show concentration profiles of reactant and products as function of residence time, evaluated at 288 K, 343 K, and 363 K.

Figure 3: Concentration of the RBr, RH, and RR species as function of residence time. Reactions occur at 288 K.

n D∇– c( )⋅ 0=

n D∇– c cu+( )⋅ 0=


The hydrocarbon coupling reaction has the higher activation energy and is hence more temperature sensitive than the hydrogenation reaction (see Table 1). The concentration plots of the ideal reactors outline the effect quite clearly. At 288 K, the hydrogenation product RH is dominant, while at 363 K the coupling product is the more prominent. Notably, at 343 K, the concentration dependency on the reaction rates becomes accentuated, so that RBr dominates only at shorter τ and RH at longer τ.

Although the primary goal may be to remove the halogenated reactant, RBr, it may also be important to set reaction conditions in such a way that the most favorable by-product is formed. The present model shows how such design aspects can readily be investigated with the Chemical Reaction Engineering Module.

The next set of results refer to the space-dependent model of a tortuous microreactor.

Figure 6 shows the velocity of the laminar flow field in the reactor running at 363 K. The flow is driven by a pressure difference of 1500 Pa between inlet and outlet. The resulting maximum velocity is approximately 18 mm/s.

Figure 6: Velocity field in the reactor where the pressure difference between inlet and outlet is 1500 Pa.


Figure 7 shows the concentration distribution of the reactant RBr in the reactor. At relatively high temperature, 363 K, the outlet concentration is 7.4 mol/m3.

Figure 7: Concentration distribution of the halogenated reactant RBr. Transport properties and reaction rates are evaluated at 363 K.


Running the reactor at 288 K, the outlet concentration of RBr is 11.1 mol/m3.

Figure 8: Concentration distribution of the halogenated reactant RBr. Transport properties and reaction rates are evaluated at 288 K.

Judging from the results of the ideal reactor models, a more pronounced temperature effect would be expected, as both reaction rates and species diffusivities increase notably at higher temperatures. The reason for this apparently moderate influence on conversion is that the increased temperature also affects the flow.

Raising the temperature from 282 K to 363 K decreases the viscosity of water from 1.2·10-

3 to 3.2·10-4 Pa·s. This is automatically taken into account by the temperature-dependent fluid properties. As the flow through the reactor is driven by a constant pressure difference, the velocity increases as viscosity decreases. Results also show that the maximum fluid velocity at 363 K is greater than at 282 K, resulting in a shorter residence time.

Application Library path: Chemical_Reaction_Engineering_Module/Reactors_with_Mass_Transfer/tortuous_reactor




N E W





3 Click Add.

4 Click Study.


6 Click Done.





4 Browse to the model’s Application Libraries folder and double-click the file tortuous_reactor_parameters.txt.





You will solve for an isothermal system at three different temperatures.


As explained by Equation 3 , you can use a batch reactor type reactor to model a flow-through system if the velocity is constant. The Batch reactor type is the default selection of the Reaction Engineering interface.



In Reaction Engineering, a reaction is defined as surface reaction if there is any surface species participating in the reaction.

In order to set up the chemical reactions describing the dehalogenation process. A surface species AS(ads) is introduced into following two surface reactions.


3 In the Formula text field, type RBr+AS(ads)=>RH.


5 In the r text field, type re.kf_1*re.c_RBr/H*1[mol/m^2].

As explained in the Introduction, modifying the automatically generated reaction rates by dividing with the channel height parameter, H, allows you to compare this ideal 1D reactor with the 3D reactor that you model in the second part of this example.


7 In the Af text field, type A1*1[m^3/mol].




3 In the Formula text field, type 2RBr+AS(ads)=>RR.


5 In the r text field, type re.kf_2*re.c_RBr^2/H*1[mol/m^2].


7 In the Af text field, type A2*1[m^3/mol].






Surface species: AS(ads)The surface species AS(ads) acts as catalyst. Its concentration is constant.


1 In the Model Builder window, under Component 1 (comp1)>Reaction Engineering (re) click Surface species: AS(ads).







4 Click to expand the Surface species initial value section. Locate the Surface Species Initial Value section. In the Surface species initial concentration table, enter the following settings:

S T U D Y 1

With the selection of reactor type and the input of the chemical kinetics you are ready to solve the ideal reactor model. Following the steps below you will set up a parametric sweep to solve the model for three different temperatures.



3 Click Add.


H2O c_H2O

RBr c_RBr0

RH c_RH0

RR c_RR0

Species Surface concentration (mol/m^2) Site occupancy number (1)

AS(ads) c_As0_ads 1




R E S U L T S


2 In the Settings window for 1D Plot Group, type Concentrations 0D model in the Label text field.

Global 11 In the Model Builder window, expand the Results>Concentrations 0D model node, then

click Global 1.



This data set contains the stored results from the parametric sweep. You can review the results by choosing entries in the Parameters selection list and then clicking Plot button.

4 From the Parameter selection (T_iso) list, choose Manual.

5 In the Parameter indices (1-3) text field, type 1.

6 Locate the x-Axis Data section. From the Axis source data list, choose Inner solutions.


This selection will give you the (residence) time on the x-axis of the plot.



10 Locate the Data section. In the Parameter indices (1-3) text field, type 2.


12 In the Parameter indices (1-3) text field, type 3.



T_iso 288 343 363



Recall that you scaled the reaction rates to make the ideal reactor comparable to the 3D microreactor. Before you export the reaction kinetics to the 3D model, remove the effect of the scaling by setting the value of the parameter H to 1.

Parameters1 In the Model Builder window, expand the Global Definitions node, then click Parameters.




The Generate Space-Dependent Model feature automatically sets up physics interfaces for modeling space- and time-dependent systems. This process uses the model in the Reaction

Engineering interface as reference to set up transport interfaces, transferring variable names, reaction kinetics, as well as property expressions. By default the Generate Space-Dependent

Model feature sets up a Transport of Diluted Species interface to describe a space-dependent reacting system and a Chemistry node with all reaction and species properties. The default settings are used in this example.


2 In the Settings window for Generate Space-Dependent Model, locate the Physics Interfaces section.

3 Find the Fluid flow subsection. From the list, choose Laminar Flow: New.


Note how the model generation creates a new component node, Component 2. Expanding this node you will find the Transport of Diluted Species and Laminar Flow interfaces that have been set up automatically.

G E O M E T R Y 1 ( 3 D )

Import a geometry from the tortuous_reactor_geometry.mph file for the tortuous microreactor.

1 In the Model Builder window, expand the Component 2 (comp2) node, then click Geometry 1(3D).


H 1 1 Channel height scale parameter



3 Browse to the model’s Application Libraries folder and double-click the file tortuous_reactor_geom_sequence.mph.

4 Right-click Geometry 1(3D) and choose Build All Objects.








By default, the first material you add applies on all domains. Associating a material with the geometry makes the predefined property expressions of the material available to the physics interfaces. In this case the temperature dependent expressions for the density and viscosity of water will be automatically included in the definition of the Laminar Flow interface.


Species: RBr1 In the Model Builder window, expand the Chemistry 1 (chem) node, then click

Species: RBr.


3 In the M text field, type Mn_RBr.

Surface species: AS(ads)1 In the Model Builder window, under Component 2 (comp2)>Chemistry 1 (chem) click

Surface species: AS(ads).


3 In the M text field, type Mn_As.

Species: RH1 In the Model Builder window, under Component 2 (comp2)>Chemistry 1 (chem) click

Species: RH.



3 In the M text field, type Mn_RH.

Species: RR1 In the Model Builder window, under Component 2 (comp2)>Chemistry 1 (chem) click

Species: RR.


3 In the M text field, type Mn_RR.

Species: H2O1 In the Model Builder window, under Component 2 (comp2)>Chemistry 1 (chem) click

Species: H2O.


3 In the M text field, type Mn_H2O.


On the Physics toolbar, click Chemistry 1 (chem) and choose Transport of Diluted Species (tds).




3 In the DcRBr text field, type D.

4 In the DcRH text field, type D.

5 In the DcRR text field, type D.

Flux 1This applies the surface reactions as boundary fluxes. The Inward Flux text fields have been filled in automatically with variables from the Chemistry set up by the model generation process.


Transport of Diluted Species (tds) click Flux 1.

2 In the Settings window for Flux, locate the Boundary Selection section.

3 From the Selection list, choose Catalytic Surfaces.




2 In the Settings window for Concentration, locate the Boundary Selection section.


4 Locate the Concentration section. Select the Species cRBr check box.

5 In the c0,cRBr text field, type c_RBr0.

6 Select the Species cRH check box.

7 In the c0,cRH text field, type c_RH0.

8 Select the Species cRR check box.

9 In the c0,cRR text field, type c_RR0.





Recall that the reactions take place solely on the catalytic surfaces. Therefore the generated Reactions and Flux Discontinuity nodes can be removed.


Reactions 1In the Model Builder window, expand the Component 2 (comp2)>Laminar Flow 1 (spf) node.

Flux Discontinuity 11 Right-click Component 2 (comp2)>Transport of Diluted Species (tds)>Reactions 1 and

choose Delete.


Transport of Diluted Species (tds) right-click Flux Discontinuity 1 and choose Delete.


Fluid Properties 11 In the Settings window for Fluid Properties, locate the Model Input section.


Note that the fluid density and dynamic viscosity are taken from the Material node.

3 In the Model Builder window, click Laminar Flow 1 (spf).





4 Locate the Boundary Condition section. From the list, choose Pressure.

5 Locate the Pressure Conditions section. In the p0 text field, type delta_p.






This completes the set up of the physics interfaces. Following the steps below, discretize the reactor geometry with a mesh. With chemistry occurring on reactor walls, mass transport gradients are expected to be most pronounced perpendicular to the direction of the flow. Using a swept mesh will allow you to maintain high mesh resolution in the cross-section of the reactor channel while setting a lower resolution in the direction of the flow. Analyzing the chemistry and physics of your system and distributing the mesh accordingly will often allow you to reduce memory requirements and computational time.

M E S H 1






5 Click OK.







Swept 11 In the Model Builder window, right-click Mesh 1 and choose Swept.

2 In the Settings window for Swept, locate the Domain Selection section.


4 From the Selection list, choose Channel.

Distribution 11 Right-click Component 2 (comp2)>Mesh 1>Swept 1 and choose Distribution.



Free Tetrahedral 11 In the Model Builder window, right-click Mesh 1 and choose Free Tetrahedral.



4 From the Selection list, choose Adapter Sections.







7 In the Settings window for Mesh, click Build All.


The instructions below detail how to solve the model. As was the case for the perfectly mixed reactor model, you will set up a parametric sweep to solve for three different temperatures. The temperature affects not only reaction rates but also the diffusivity for the mass balance equations. The fluid flow is also affected as the fluid density and viscosity are temperature dependent.


S T U D Y 2



3 Click Add.


Set up two study steps to solve the model. The first step solves for the fluid flow. The second step solves for the mass transport, using the flow field calculated in the first step for the convective mass transport. There is no need to solve the fully coupled problem as the mass transport does not influence the fluid flow in this example.



3 In the table, clear the Solve for check box for Reaction Engineering and Transport of

Diluted Species.



3 In the table, clear the Solve for check box for Reaction Engineering and Laminar Flow 1.


R E S U L T S


2 In the Settings window for 3D Plot Group, type Concentration distribution RBr in the Label text field.

3 On the Concentration distribution RBr toolbar, click Plot.


The default surface plot shows the concentration of species RBr evaluated at 363 K. Select from the Parameter value list to look at other results.


T_iso 288 343 363


5 Locate the Data section. From the Parameter value (T_iso) list, choose 288.

6 On the Concentration distribution RBr toolbar, click Plot.


Slice1 In the Model Builder window, expand the Results>Velocity (spf1) node, then click Slice.






Three of the plot groups are not used and can be removed.

Pressure (spf1)1 In the Model Builder window, under Results right-click 3D Plot Group 2 and choose

Delete.

2 Right-click Concentration (tds) and choose Delete.

3 Right-click Pressure (spf1) and choose Delete.

Last, for future use of Study 1 and the 0D model, turn off all interfaces except Reaction

Engineering in the interface.

S T U D Y 1


section.


Physics interface

Chemistry 1 (chem)


Laminar Flow 1 (spf)


chemical reaction engineering module - comsol multiphysics...reaction engineering (re) reaction 1 1...

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