chee 321: chemical reaction engineering chee 321: chemical reaction engineering module 4: finding...
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CHEE 321: Chemical Reaction EngineeringCHEE 321: Chemical Reaction Engineering
Module 4: Finding Rate Laws(Chapter 5, Fogler)
Topics to be covered in this moduleTopics to be covered in this module
• Rate law from batch reactor– Differential Method
• Methods for calculating dCA/dt• Method Excess• Method of initial rates
– Integral Method
• Rate law from differential reactor
• Brief description of other reactor types employed for kinetics study
Common Reactor Types for Obtaining Common Reactor Types for Obtaining Rate LawsRate Laws
• Batch Reactor– used primarily for homogeneous
reactions
• Differential Reactor– used primarily for heterogeneous
(solid-fluid catalytic) reactions
Rate Law from Batch ReactorRate Law from Batch Reactor
Typical Experimental Data Available from Typical Experimental Data Available from Batch Reactor ExperimentsBatch Reactor Experiments
Time (min) t0 t1 t2 t3 t4 t5
Concentration (mol/L) CA0 CA1 CA2 CA3 CA4 CA5
Time (min) t0 t1 t2 t3 t4 t5
Pressure (kPa) PT0 PT1 PT2 PT3 PT4 PT5
OR
Rate Law from Batch ReactorRate Law from Batch Reactor
There are two general methods for obtaining rate law from batch reactor:
• Differential Method– Batch reactor data in differential form, i.e. dCA/dt or
dPA/dt, is analyzed.
• Integral Method– Batch reactor data in integral form, i.e. C(t), is analyzed
Differential Method of Determining Differential Method of Determining Rate Law from Batch ReactorRate Law from Batch Reactor
Differential Method for Obtaining Rate Differential Method for Obtaining Rate Law from Batch ReactorLaw from Batch Reactor
1. General Mole Balance
3. Stoichiometry
2. Rate Law
4. Combine
Vrdt
dNA
A )(
AA Ckr )(
)(1
AA r
dt
dN
V
AA Ck
dt
dC
V=Vo For constant volume or constant density system
1
10
100
1000
1 10 100 1000
CA
(- d
CA/d
t)Rate Law from Batch Reactor - Differential MethodRate Law from Batch Reactor - Differential Method
A
A Ckdt
dC
Reaction order () can be found from slope of log-log plot of - dCA/dt and CA
)ln()ln()ln( AA Ck
dt
dC
Taking the logarithm of combined equation
dCA
dt p
CA p pA
p
A
C
dtdC
k
Typical Experimental Data Available from Typical Experimental Data Available from Batch Reactor ExperimentsBatch Reactor Experiments
Time (min) t0 t1 t2 t3 t4 t5
Concentration (mol/L) CA0 CA1 CA2 CA3 CA4 CA5
Time (min) t0 t1 t2 t3 t4 t5
Pressure (kPa) PT0 PT1 PT2 PT3 PT4 PT5
OR
Methods for Calculating dCMethods for Calculating dCAA/dt/dt
Methods for Calculating dCMethods for Calculating dCAA/dt /dt
from [Cfrom [CAA vs t] data vs t] data
1. Graphical Method (Appendix A.2 of Fogler)
Step 1: Calculate CA and t Step 3: Read - dCA/dt at “t” for which corresponding CA has been measuredStep 2: Plot - CA / t vs t
Methods of Calculating dCMethods of Calculating dCAA/dt /dt
from [Cfrom [CAA vs t] data vs t] data
2. Polynomial Fit Method
Step 1: Fit CA vs t data using a polynomial of “n” th order
CA = ao +a1t +a2t2+… an tn
Step 2: Calculate dCA /dt
dCA /dt = a1 + 2 a2t+… (n-1) an tn-1
Methods of Calculating dCMethods of Calculating dCAA/dt /dt
from [Cfrom [CAA vs t] data vs t] data
3. Numerical Method (See Appendix A of Fogler)
Can be used when independent variable (in our case “t”) is equally spaced, i.e. t1-t0 = t2-t1=t3-t2 =tn-tn-1 = t
t
CCC
dt
dC AAA
t
A
2
43 210
0
First Point
t
CCC
dt
dC nAnAnA
t
A
n
2
34 )()1()2(
Last Point
t
CC
dt
dC iAiA
t
A
i
2)1()1(
Interior Points
Integral Method of Determining Integral Method of Determining Rate Law from Batch ReactorRate Law from Batch Reactor
Integral Method for Obtaining Rate Law Integral Method for Obtaining Rate Law from Batch Reactorfrom Batch Reactor
1. General Mole Balance
3. Stoichiometry
2. Rate Law
4. Combine
Vrdt
dNA
A )(
AA Ckr )(
)(1
AA r
dt
dN
V
tt
t
C
C A
A dtkC
dCA
A 00
V=Vo For constant density system
Integral Method for Obtaining Rate Law Integral Method for Obtaining Rate Law from Batch Reactorfrom Batch Reactor
tt
t
C
C A
A dtkC
dCA
A 00
In the integral method, the reaction order is hypothesized (or guessed) and the preceding equation is then integrated. The hypothesis is verified against experimental data.
One disadvantage of the method is that if the reaction order is not known a priori, several trial and errors may have to be done before an acceptable solution is achieved.
Reaction Order and Rate ConstantReaction Order and Rate Constant
Zero-order First-order Second-order
Method of Excess Method of Excess and and
Method of Initial RatesMethod of Initial Rates
Method of Excess - Rate Law for Reactions Involving Method of Excess - Rate Law for Reactions Involving Two ReactantsTwo Reactants
BAA CCkr )(
Method of Excess: Experiment is carried out under conditions such that one species is in excess.
BOB CC e.g. If B is in excess
AABOBAA CkCkCCCkr )(
BOCkk where,
We follow the same method for determining k' and as we have discussed previously
Reaction: A + B Products
The rate law can be written as follows:
Method of Initial RatesMethod of Initial RatesFor reversible reactions, both forward and backward reaction may become significant. In such case, the methods discussed earlier may not be suitable
1
10
100
1000
1 10 100 1000CA0
(rA
0)Slope =
Initially or at time t=0, CA=CA0 and the rate is given by(-rA0) = kCA0
Method of Initial Rates may provide the solution
Methodology• Conduct a number of experiments
at various initial concentrations (CA0) is carried out
• Next, plot (-rA0) vs CA0
• The slope = • Knowing the slope, one can
calculate the rate constant ‘k’
Rate Law from Differential ReactorRate Law from Differential Reactor
• Channeling must be avoided• Volumetric flow rate, inlet and outlet concentrations must be monitored• Heat release per unit volume is low, as such the reactor behaves isothermally.• The reactor is assumed to be gradientless, i.e., concentration is assumed to be
uniform in the catalyst bed.
Differential ReactorDifferential Reactor
FA0 FAe
Catalyst Bed
Inert Filling
Catalyst weight =W
Method for Obtaining Rate Law from Method for Obtaining Rate Law from Differential ReactorDifferential Reactor
1. General Mole Balance
0)(0 WrFF AAA W
FFr AA
A
0)(
- in terms of concentration
W
CvCvr AAo
A
0)(
- in terms of conversion (X) and molar flow rate of product (FB)
W
F
W
XFr BA
A 0)(
- in terms of molar flow rates
Reaction: a A b B
Method for Obtaining Rate Law from Method for Obtaining Rate Law from Differential ReactorDifferential Reactor
W
CvCvr AAo
A
0)(
- For constant volumetric flow rates
W
Cv
W
vCCvr BAAo
A
0)(
Concentration of product
2. Rate Law
avgAA Ckr )()(
Where, ]2
[)( 0 AAavgA
CCC
Method for Obtaining Rate Law from Method for Obtaining Rate Law from Differential ReactorDifferential Reactor
3. Stoichiometery
b
r
a
r
Xvv
BA
o
)()(
)1(
4. Combine
A
AA kCW
CvCv
00
Known from experiment
Kinetic parameters, k and , can be obtained by fitting the experimental data
Other Types of ReactorsOther Types of Reactorsused inused in
Determining Rate LawsDetermining Rate Laws
Integral Fixed Bed ReactorIntegral Fixed Bed Reactor
• Ease of construction
• less prone to rate data being affected by channeling/bypassing of some areas of catalyst bed
Stirred Batch ReactorStirred Batch Reactor
• Catalyst dispersed as slurry
• Good fluid-solid contact
• Sampling can be problematic
Stirred Contained Solids ReactorStirred Contained Solids Reactor
• Also called “spinning basket reactor”
• Good fluid-solid contact