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

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**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

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**Common Reactor Types for Obtaining Rate Laws**

Batch Reactor used primarily for homogeneous reactions Differential Reactor used primarily for heterogeneous (solid-fluid catalytic) reactions

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**Rate Law from Batch Reactor**

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**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

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**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

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**Differential Method of Determining Rate Law from Batch Reactor**

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**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

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**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

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**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

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**Methods for Calculating dCA/dt**

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**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

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**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

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**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

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**Integral Method of Determining Rate Law from Batch Reactor**

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**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

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**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.

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**Reaction Order and Rate Constant**

Zero-order First-order Second-order

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**Method of Excess and Method of Initial Rates**

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**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

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**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 =

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**Rate Law from Differential Reactor**

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**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.

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**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)

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**Method for Obtaining Rate Law from Differential Reactor**

- For constant volumetric flow rates Concentration of product 2. Rate Law Where,

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**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

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**Other Types of Reactors used in Determining Rate Laws**

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**Integral Fixed Bed Reactor**

Ease of construction less prone to rate data being affected by channeling/bypassing of some areas of catalyst bed

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**Stirred Batch Reactor Catalyst dispersed as slurry**

Good fluid-solid contact Sampling can be problematic

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**Stirred Contained Solids Reactor**

Also called “spinning basket reactor” Good fluid-solid contact

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