1. CHEE 321: Chemical Reaction Engineering Module 4: Finding Rate Laws (Chapter 5, Fogler)

2. Topics 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

3. Common Reactor Types for Obtaining Rate Laws
Batch Reactor
used primarily for homogeneous reactions
Differential Reactor
used primarily for heterogeneous (solid-fluid catalytic) reactions

4. Rate Law from Batch Reactor

5. Typical Experimental Data Available from Batch Reactor Experiments
Time (min) t0 t1 t2 t3 t4 t5
Concentration (mol/L) CA0 CA1 CA2 CA3 CA4 CA5 OR
Time (min) t0 t1 t2 t3 t4 t5
Pressure (kPa) PT0 PT1 PT2 PT3 PT4 PT5

6. Rate 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

7. Differential Method of Determining Rate Law from Batch Reactor

8. Differential Method for Obtaining Rate Law from Batch Reactor
1. General Mole Balance
2. Rate Law
3. Stoichiometry
V=Vo
For constant volume or constant density system
4. Combine

9. Rate Law from Batch Reactor - Differential Method
Taking the logarithm of combined equation
Reaction order () can be found from slope of log-log plot of - dCA/dt and CA
dCA dt p CA p

10. Typical Experimental Data Available from Batch Reactor Experiments
Time (min) t0 t1 t2 t3 t4 t5
Concentration (mol/L) CA0 CA1 CA2 CA3 CA4 CA5 OR
Time (min) t0 t1 t2 t3 t4 t5
Pressure (kPa) PT0 PT1 PT2 PT3 PT4 PT5

11. Methods for Calculating dCA/dt

12. Methods for Calculating dCA/dt from [CA 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 measured
Step 2: Plot - CA / t vs t

13. Methods of Calculating dCA/dt from [CA 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

14. Methods of Calculating dCA/dt from [CA 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
First Point
Interior Points
Last Point

15. Integral Method of Determining Rate Law from Batch Reactor

16. Integral Method for Obtaining Rate Law from Batch Reactor
1. General Mole Balance
2. Rate Law
3. Stoichiometry
V=Vo
For constant density system
4. Combine

17. Integral Method for Obtaining Rate Law from Batch Reactor
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.

18. Reaction Order and Rate Constant
Zero-order
First-order
Second-order

19. Method of Excess and Method of Initial Rates

20. Method of Excess - Rate Law for Reactions Involving Two Reactants
Reaction: A + B  Products
Method of Excess: Experiment is carried out under conditions such that one species is in excess.
e.g. If B is in excess
The rate law can be written as follows:
We follow the same method for determining k' and  as we have discussed previously

21. Method of Initial Rates
For reversible reactions, both forward and backward reaction may become significant. In such case, the methods discussed earlier may not be suitable
Method of Initial Rates may provide the solution
Initially or at time t=0, CA=CA0 and the rate is given by(-rA0) = kCA0
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’
Slope = 

22. Rate Law from Differential Reactor

23. Differential Reactor FA0 FAe Catalyst Bed Inert Filling
Catalyst weight =W
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.

24. Method for Obtaining Rate Law from Differential Reactor
Reaction: a A  b B
1. General Mole Balance
- in terms of molar flow rates
- in terms of concentration
- in terms of conversion (X) and molar flow rate of product (FB)

25. Method for Obtaining Rate Law from Differential Reactor
- For constant volumetric flow rates
Concentration of product
2. Rate Law
Where,

26. Method for Obtaining Rate Law from Differential Reactor
3. Stoichiometery
4. Combine
Kinetic parameters, k and a, can be obtained by fitting the experimental data
Known from experiment

27. Other Types of Reactors used in Determining Rate Laws

28. Integral Fixed Bed Reactor
Ease of construction
less prone to rate data being affected by channeling/bypassing of some areas of catalyst bed

29. Stirred Batch Reactor Catalyst dispersed as slurry
Good fluid-solid contact
Sampling can be problematic

30. Stirred Contained Solids Reactor
Also called “spinning basket reactor”
Good fluid-solid contact