2008 - computational mass transfer method for chemical process simulation

8/13/2019 2008 - Computational Mass Transfer Method for Chemical Process Simulation

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Chinese Journal of Chemical Engineering , 16(4) 497 502 (2008)

PERSPECTIVES

Computational Mass Transfer Method for Chemical Process Simulation *

YUAN Xigang ( )** and YU Guocong ( ) State Key Laboratory of Chemical Engineering, Tianjin University, Tianjin 300072, China

Abstract The recent works on the development of computational mass transfer (CMT) method and its applica-tions in chemical process simulation are reviewed. Some development strategies and challenges in future researchare also discussed.Keywords computational mass transfer, turbulent mass transfer diffusivity, chemical process simulation

1 INTRODUCTION

As a result of the development of computer tech-nology beginning in the sixties of last century, thecomputational fluid dynamics (CFD) and the subse-quent computational heat transfer (CHT) were initi-ated on the basis of closure of the differential equa-tions of momentum and heat transfer respectively bythe fluid dynamic researchers and mechanical engi-neers. The computational method provides a soundfoundation for predicting the velocity and temperaturefields in the engineering areas with wide application.

Nevertheless, in the chemical engineering work, the prediction of concentration field is equally important,and the task of such development, which may be re-garded as computational mass transfer (CMT), isnaturally relied on the investigation by the chemicalengineers. Recently, works on CMT have been reportedcovering its basic ground and various applications.

Historically, the early research on the numericalsimulation of concentration distribution was reportedin 1960s, almost at the same time when CFD was de-veloped [1, 2] , but the simulation of concentration andtemperature distributions depended largely on the pat-tern of velocity distribution by CFD.

The computational mass transfer under investiga-tion now aims to the prediction of concentration dis-tribution of complex fluid systems with simultaneousmass, heat and momentum transports and/or chemical(or biochemical) reactions in chemical processes.More specifically, the CMT method should be used to

predict the distributions of concentration, velocity andtemperature as well as the transport parameters andoperating efficiency simultaneously for the chemicalequipment.

The key problem of establishing the CMT is tofind a method of closure for the mass transfer differ-ential equation, just like those for closing the momen-tum and the heat transfer differential equations. Theearly approach of this problem is by the use of em-

pirical turbulent mass transfer diffusivity [3] to calcu-late the concentration distribution in distillation col-umn. Subsequently, the two equations method for theclosure of mass transfer differential equation was de-veloped a few years ago [4, 5] . Since then, the framework of CMT was established including the funda-

mental equations and its applications to chemical en-gineering equipment. Undoubtedly, further investiga-tion needs to be done in order that CMT, CFD andCHT are generally to be recognized and used as thethree essentials of computational transport.

This article addresses the recent work on the CMT

method development and its applications in chemicalengineering. Some development strategies and thechallenges of future research are also discussed.

2 BASIC EQUATIONS OF MASS TRANSFERAND ITS CLOSURE

For incompressible fluids, the mass transferequation of a component species with instantaneousconcentration c can be expressed as [6] :

2

i ci i i

c c cu D S

t x x x

+ = +

(1)

where u is instantaneous velocity, D is the moleculardiffusivity and cS is the source term.

For the turbulent flow, if substituting c C c= + and u U u= + into Eq. (1), in which C and U are thetime-average values, we obtain the time-average trans-

port equation for the concentration scalar as follows:

j j c j j j

C C C U D u c S

t x x x

+ = +

(2)

In the foregoing equation, a new unknown term ju c ,the second order covariance of the velocity and con-centration appears and may be regarded as the Reynolds

mass flux, analogous to Reynolds stress in the CFD.Similar to the Boussinesq postulate made in turbulent modeling for isotropic fluid dynamics, the newvariable can be expressed as proportional to the gra-dient of average concentration [7]:

t,i

j i i j

C u c D

x

=

(3)

where D t,i is termed as turbulent diffusivity of component i, which is not a constant but related with thevelocity, temperature, component concentration andstructure of the flow field. Hence, the evaluation ofthe turbulent diffusivity plays a vital role in solving

Received 2008-05-26, accepted 2008-06-03.* Supported by the National Natural Science Foundation of China (20736005).

** To whom correspondence should be addressed. E-mail: [email protected]


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Chin. J. Chem. Eng., Vol. 16, No. 4, August 2008498

the equations.Traditionally, the turbulent diffusivity is usually

estimated empirically by simple analogy to the turbu-lent viscosity by assuming that the turbulent masstransfer is simply related to the turbulent momentumtransfer. This empirical analogy leads to a simple ex-

pression for the turbulent diffusivity as t, t t/ j D Sc = ,where t is a turbulent viscosity which can be ac-quired by using a CFD model such as k - model, andSct is the Schmidt number, usually assuming to be aconstant taken between 0.7 and 1.0. Wang et al . [3] used this approach combined with the volume aver-aged pseudo-single-liquid model to obtain thethree-dimensional concentration profile on the trays ofan industrial scale distillation column and evaluate thecorresponding tray efficiencies.

Another empirical approach to estimate D t,i isfrom the Peclet Number ( Pe ), which is equal to the

product of a characteristic velocity and characteristic

length divided by the turbulent diffusivity. The Pecletnumber, traditionally applied to represent the

back-mixing property of the fluid, is usually deter-mined experimentally by the inert tracer technique in anon-reactive flow and correlated as function of Rey-nolds number, physical properties and the characteris-tic dimension of equipment. Such method, however, isstill empirical.

The two empirical approaches mentioned abovefor estimating D t,i need no additional differentialequation, and are termed as zero-equation models forthe closure of mass transfer equation Eq. (1).

The main shortcoming of zero-equation modelsis that D

t,i is considered only related with the fluid

velocity fluctuation. Theoretically, D t,i is also de- pendent on the concentration fluctuation. Based onthis idea and in reference to the CFD and CHT treat-ments, Liu [4, 5] suggested that D t,i is proportional tothe product of characteristic velocity and characteristiclength, or mathematically:

12

t t m D C k L= (4)

where k is the turbulent kinetic energy, equal to/ 2i iu u , and its square root represents the characteris-

tic velocity, Lm is the characteristic length represented

by k 1/2

/ m where the mixed time scale m is taken asgeometric average c in which and c are the

dissipation time scale of the velocity and concentra-tion fluctuations. In CFD, we have / k = where is the dissipation rate of concentration fluctuation.

Similarly, we may let 2 /c cc = where2c is the

variance of concentration fluctuation and c is the dis-sipation rate of concentration fluctuation. Then Eq. (4)

becomes:1

2 2

t tc

kc D C k

= (5)

It can be seen that in Eq. (5), there are four vari-ables besides the proportional constant C t. The vari-

ables k and can be obtained by the k - model in CFD.

The other two variables 2c and c are obtained bysolving corresponding equations. Liu et al . [4, 5] de-

rived the exact 2c and c equations and the modeling

forms, which were further simplified by Sun et al . [8] as follows:

2c equation:

2 2 2t

t2 2

ii i c i

ci i

Dc c cU D

t x x x

C C D

x x

+ = + (6)

c equation:

t

1 22

c

c c ci

i i i

cc i c c

i

DU D

t x x xC

C cu C x k c

+ = +

(7)

where D and D t are the molecular and turbulent diffu-sivities respectively, with c,

c , C 1 and C 2 as con-

stants. Eqs. (6), (7) and (2) constitute the 2c - c modelfor the closure of Eq. (3), and are the fundamental partof the computational mass transfer.

The foregoing 2c - c model involves the variablesk and which can be solved by the CFD method forturbulent flow consisting of five equations, namely thecontinuity equation, the moment equation, the Bous-sinesqs D t equation, the k equation and the equation.

If the process involves heat effect, such as thechemical absorption or exothermic catalytic chemicalreaction, the effect of temperature distribution cannot

be ignored. In this case, the CHT model, which in-volves the heat transport equation, the Boussinesqs

thermal diffusivity Dh equation, the2t equation and

the t equation, should be accompanied.

3 COMPUTATIONAL MASS TRANSFEREQUATION SYSTEM

Computational mass transfer equation systemshould be solved by coupling the momentum, heat andmass transport; therefore it consists of three parts:

(i) Equation set of mass transfer and its closure,including Eqs. (2), (3), (6) and (7), for predicting theconcentration distribution.

(ii) Equation set of CFD including five equationsas mentioned above for the purpose to obtain the k , and velocity distribution.

(iii) Equation set for heat transfer: including four

equations as mentioned above for the purpose to ob-tain the temperature distribution.These three parts should be solved simultane-

ously as the concentration field and temperature fieldare both affected by the velocity field. It is noted that


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under some nearly isothermal conditions, such as in asingle distillation tray, the heat transfer equations can be neglected for simplification. It can be seen thatCMT methodology involves solving a large number ofdifferential equations and thus the computer load isheavy, but it helps to use more efficient commercialsoftware.

The interrelationship of equation system can berepresented by the diagram shown in Fig. 1.

4 THE MODELING OF CHEMICAL PROCESSBY COMPUTATIONAL MASS TRANSFER

Most chemical processes are under the conditionof two flowing phases, such as gas/vapor-liquid flowin the distillation column or the catalytic reactor.

In such cases of two-phase flow, the modelingequations of each phase should be established. Thenumber of equations is then double and the computa-tion becomes extremely complicated. Methods ofsimplification are being sought. Roughly speaking, theycan be classified into three categories: pseudo-single-fluidmodel, mixed-fluid model and two-fluid model.

The pseudo-single-fluid model, proposed origi-nally for distillation trays, assumes that the character-istic of two-phase flow can be represented by an

equivalent single phase flow with the account for theinteracting force by the other. The interaction between phases, represented by a proper constitutive equation,is inserted to the CFD equations as a source term. Thismodel has been used successfully by many investiga-tors [3, 5, 9, 10] for the simulation of two-phase flow intray/packed distillation column. The advantage of thismodel is that it is simple and able to simulatecounter-current or cross-current two-phase flow, pro-vided the proper expression of source terms.

The mixed-fluid model assumes the two-phasemixture be an equivalent mixture of coexisting gasand liquid with weighted average velocity and other

parameters. Besides, the interfacial action still need to be accounted for. The advantage of the mixed fluidmodel is particularly suitable to co-current flows.

The two-fluid model formulates the respectiveequations for each interacting fluid and consequently

the number of equations needed to be solved is dou- bled, although it is the best method for the numericalsimulation. Its advantage is able to obtain the concen-tration distribution for each phase simultaneously.

For the chemical process with solid phase, suchas in the case of packed column or catalytic reactor,the simulation can be done with the help of using theRepresentative Elementary Volume (REV) method [11] .By this approach, the gas-liquid-solid three-phasespace is divided into a number of virtual REV, the sizeof which is large enough to represent the character ofthree phases configuration, but yet small enough in theflow field as an elementary volume of discretization,as shown in Fig. 2. This approach has been success-fully used for simulating the transport processes in

packed absorption/distillation column [9, 10] and cata-lytic reactor [12] .

Figure 2 A representative elementary volume

5 APPLICATIONS

5.1 Distillation

For the distillation process using tray columns,the temperature of fluid on each tray can be consid-ered nearly constant and the equations in the heattransfer part of CMT methodology may be neglectedin order to simplify the computation. However, thetemperature of each individual tray is changing from

the bottom to the top of the column. With such idea,Sun et al . [5, 8] simulated an industrial scale sieve traycolumn for cyclohexane and n-heptane separation [13] in FRI (Fractionation Research Institute) by using

Figure 1 Equation system for CMT


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CMT method with 2 - cc model to obtain the veloc-ity and concentration distributions of each tray and itsMurphree tray efficiency. Fig. 3 gives some computa-tional results. It is seen from the figure that the simu-

lated result in outlet concentration of each tray isclosely checked by the experiment except on tray No.4 where experimental error is obvious.

(a) Velocity distribution for liquid phase

(b) Concentration distribution for cyclohexane

(c) Comparison with experiments for trays outlet concentrations

Figure 3 Computation results by CMT with 2 - c c modelfor distillation tray [5]

The simulation by Liu et al . [14] using the CMTmethod to a commercial scale distillation column

packed with 50.8 mm pall ring in FRI [15] for C 6-nC7 separation is shown in Fig. 4, in which the radial con-centration distribution and HETP are clearly seen.

5.2 Chemical absorption

The chemical absorption process usually involvesheat effect, and all three parts of CMT equation sys-tem should be applied. Liu et al . [9, 10] simulated theabsorption of CO 2 by using NaOH and MEA (mono-ethanolamine) separately as absorbent in packed col-

umns. With the 2 - cc , k - and2

t-t models to beused in CMT formulation, their simulated results in-clude not only radial and axial concentration, tem-

perature and velocity distributions but also the masstransfer diffusivity, thermal diffusivity and the en-hancement factor. All those are in agreement with theexperimental data.

5.3 Catalytic reaction

Liu et al . [9] applied the CMT methodology to amore complicated case, an exothermic fixed bed cata-lytic reactor with cooling jacket for the synthesis ofvinyl acetate from acetic acid and acetylene [16] . Likein the case of chemical absorption, the concentration,temperature and velocity distributions as well as theirrespective diffusivities are all obtained at once asshown in Fig. 5. It can be seen that there are notabledifferences among the values of the three kinds ofdiffusivities in both radial and axial directions.Thereby no simple relationship is seen to be existing

between these diffusivities. In other words, theSchmidt number Sc and the Peclet number Pe are notconstant throughout. Hence the traditional empiricalassumption of assuming constant Sc and Pe is unjusti-fied and may lead to serious error.

(a) Concentration distribution for C 6

(b) Comparison with experiments for HETP

Figure 4 Computation results by CMT with 2 - c c model

for packed distillation column [14]


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Chin. J. Chem. Eng., Vol. 16, No. 4, August 2008 501

(a) Turbulent viscosity

(b) Turbulent diffusivity of acetylene

(c) Turbulent thermal diffusivity

Figure 5 Transport properties of the fixed bed reactor [14]

Thus, an obvious advantage of CMT methodol-ogy is able to predict all the relevant parameterswithout using empirical assumption or correlation.

6 CHALLENGES AND RESEARCH PER-SPECTIVES

The success of CMT development reveals thefact that the computational technique could be usedmore extensively to the chemical engineering area inorder to achieve the goal of simulating chemical proc-ess and equipment on the basis of more theoretical ap-

proach rather than experimental and/or empirical ones.In that direction, we are facing a number of chal-

lenges, shown as follows.

6.1 Interface mass-transfer

At present, the evaluation of mass transfer rate between phases is based on the film concept and em-

pirical correlation. The theoretical prediction frommicroscopic viewpoint is a challenge to the chemicalengineers and scientists.

Interfacial turbulence The controlling factor ofmass transfer rate is usually considered to be the re-sistance between the surface and the main fluid, andthe facial effect is always being neglected. It has beenreported that in many cases, interfacial mass-transferis accompanied by interfacial convection (or interfa-cial turbulence). When the liquid surface tension gra-dient on the interface formed by the concentrationdifference increases to a certain extent, the interfaceturns out to be unstable, and consequently creates in-terface convection which is regarded as Marangonieffect [17, 18] . Similarly, the convection induced bydensity gradient is called as Rayleigh effect [19] . It has

been proven that the existence of the interface convec-tion has an obvious effect on interface mass-transfer[20] , for instance, it may increase the rate by several

folds. Thus, further research on interface convection phenomenon and its impact to the mass-transfer rate isan important fundamental research work.

Multi-component mass transfer Many chemi-cal processes deal with multi-component system. Thecharacteristic of such system is different from the bi-nary mixture as the molecular interaction is morecomplicated. For instance, the mass transfer efficiencyof a component in multi-component mixture may beeither much higher or much lower than one, which isentirely different from binary mixture. At present, thetheoretical ground to solve the multi-component masstransfer is by the application of Maxwell-Stefen equa-tion. Yet the solution of this equation is based onvarious simplified assumptions and suitable only forsome cases. Therefore, further investigation is needed.

6.2 Modeling of the multi-phase flow and rele-vant transfer parameters

As stated above, there are three categories ofmodels can be used to simulate multi-phase flow. Al-though the pseudo-single-phase model is simpler andsuitable to many cases as shown in previous sections,more accurate model is still need to investigate. Theexplicit two-phase flow approach, such as VOFmethod, can hardly be applied at present for simula-tion due to the increased heavy computation load.Certainly, the development of more efficient approachesto handle explicitly two-phase flow is of great interest.

Furthermore, the current CMT model depends onthe reliability of estimating such vital parameters asthe inter-phase contact area, the mass transfer rate, etc .The study on these aspects by theoretical orsemi-theoretical means is urged.

6.3 Anisotropic problem

In present CMT methodology, the closure of turbulent Navier-Stokes equation is by applying the Bous-sinesqs postulate and the isotropic assumption. But insome occasions, such as the boundaries have strongimpact on the fluid flow, the transport in porous mediumwith clear orientations, and the cyclic flow, etc ., the


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isotropic assumption is invalid, and may produce con-siderable error. Therefore it is necessary to developreliable approaches for the CMT under anisotropicflow, such as those employed in the CFD methodology.

6.4 Multi-scale modeling

As a trend, the depth modeling of chemical proc-ess should be considered in multi-scale. For instance,the interface convection occurs on a mesoscopic scale,which affects the macroscopic rate of mass transfer.The Lattice-Boltzmann method to be used in recentyears for simulating fluid flow and heat/mass transferin the complex systems displays significant advan-tages in describing the transport phenomena from par-ticle distribution to macroscopic scale. More under-standing in microscopic and mesoscopic levels ofmass transfer will make the modeling on the sound

scientific basis.6.5 Closure of mass transfer equation

The use of 2 - cc model for the closure of masstransfer differential equation has been proven to besuccessful as a base of CMT methodology. However,this is not the only way to close the mass transferequation or to solve the unknown parameter u c .More method of solution is under investigation.

Briefly, the perspectives and challenges discussedabove are mainly based on CMT investigation that has

been made. The exploration on novel strategies for higherlevel of modeling is still in need and of significance.

7 CONCLUSIONS

Computational mass transfer as a method forchemical process simulation integrates the advances inCFD and transport theories. The recent development

of 2 - cc model for closing the mass transfer differ-ential equation constitutes the ground work of CMTmethodology for chemical process simulation. The

feature of CMT method is able to simulate the con-centration, temperature and velocity fields as well asthe transport diffusivities and operating efficiency atonce without depending on the assumption of constantSc/ Pr number or empirical correlations. The presentCMT method has been applied successfully to distilla-tion, chemical absorption and exothermic catalyticreaction processes. However, the CMT method is nowstill in the developing stage, further investigation isnecessary in order to make it relying on the basis ofmore theoretical and less empirical in order to achievemore reliable and accurate simulation.

ACKNOWLEDGEMENTS

The authors acknowledge the assistance from the

staff in the State Key Laboratories of Chemical Engi-neering (Tianjin University) .

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2008 - computational mass transfer method for chemical process simulation

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