1. 13. Reactor Engineering: Reactor Design
Chemical engineering 170

2. Review: Reaction Rates
Ways to increase reaction rates:
Rate laws (𝐴+𝐵→𝐶+𝐷 ):
𝑟 𝑟𝑥𝑛,𝐴 = 𝑘 𝑟 𝑐 𝐴 𝑛 𝑐 𝐵 𝑚 (Under what assumptions?)
𝑟 𝑟𝑥𝑛,𝐴 units: 𝑚𝑜𝑙𝑒𝑠 𝑡𝑖𝑚𝑒 𝑉
If you have 𝑟 𝑐𝑜𝑛𝑠,𝐴 from a mole balance, how is it related to 𝑟 𝑟𝑥𝑛,𝐴 ?

3. Reaction Rate: Temperature
The Arrhenius Equation:
𝑘 𝑟 =𝐴 𝑒 − 𝐸 𝐴 𝑅𝑇

4. 𝑘 𝑟∗ = 𝑘 𝑟 𝑐 𝐵 𝑟 𝑟𝑥𝑛,𝐴 = 𝑘 𝑟∗ 𝑐 𝐴 Reaction Rate: 𝑐 𝐵 ≫ 𝑐 𝐴
Consider reaction 𝐴+𝐵→𝐶 with rate law 𝑟 𝑟𝑥𝑛,𝐴 = 𝑘 𝑟 𝑐 𝐴 𝑐 𝐵
If 𝑐 𝐵 ≫ 𝑐 𝐴 …
𝑐 𝐵 ≈𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
𝑘 𝑟∗ = 𝑘 𝑟 𝑐 𝐵
𝑟 𝑟𝑥𝑛,𝐴 = 𝑘 𝑟∗ 𝑐 𝐴

5. Batch Reactors Vs. Continuous Reactors

6. Two Types of continuous reactor
Plug Flow Reactor (PFR)
Continuously-Stirred Tank Reactor (CSTR)
reactants
products
reactants
products
Concentrations change across reactor
Reactant concentrations drop
Product concentrations rise
All concentrations assumed:
Constant
Same as outlet concentration

7. Reactor Design Equations
To design a reactor, we derive governing equations
Today we will find governing equations for:
A batch reactor
A CSTR
For reactions obeying 𝑟 𝑟𝑥𝑛,𝐴 = 𝑘 𝑟∗ 𝑐 𝐴

8. Batch Reactor 𝑟 𝑐𝑜𝑛𝑠,𝐴 =− 𝑑 𝑐 𝐴 𝑑𝑡 𝑉 𝑟 𝑐𝑜𝑛𝑠,𝐴 = 𝑟 𝑟𝑥𝑛,𝐴 𝑉
Is this steady-state?
𝑟 𝑐𝑜𝑛𝑠,𝐴 =− 𝑑 𝑐 𝐴 𝑑𝑡 𝑉
𝑟 𝑐𝑜𝑛𝑠,𝐴 = 𝑟 𝑟𝑥𝑛,𝐴 𝑉
𝑑 𝑐 𝐴 𝑐 𝐴 =− 𝑘 𝑟∗ 𝑑𝑡
𝑐 𝐴0 𝑐 𝐴 𝑑 𝑐 𝐴 𝑐 𝐴 =− 𝑘 𝑟∗ 0 𝑡 𝑑𝑡
𝐴+𝐵→𝐶
𝑟 𝑟𝑥𝑛,𝐴 = 𝑘 𝑟∗ 𝑐 𝐴

9. Batch Reactor
𝑙𝑛 𝑐 𝐴 𝑐 𝐴0 =− 𝑘 𝑟∗ 𝑡
𝑐 𝐴 = 𝑐 𝐴0 𝑒 −𝑘 𝑟∗ 𝑡

10. Example: Finding rate constant
With the reaction we’ve been using—if initial concentration of A in a batch reactor is 𝜇𝑀 and after 10 minutes of reaction the concentration of A is 5 𝜇𝑀 what is 𝑘 𝑟∗ ?
What would we need to know to find 𝑘 𝑟 ?
If 𝑐 𝐵 is 9160 𝜇𝑀, what’s 𝑘 𝑟 ?

11. Continuously-stirred tank Reactor (CSTR)
A, B, C B
Is this steady-state?
Overall mass balance: 1 2 3
𝑚 𝑚 2 = 𝑚 𝑜𝑢𝑡
𝜌 𝑉 1 + 𝜌 𝑉 2 = 𝜌 𝑉 𝑜𝑢𝑡
Assume densities about the same
𝑉 𝑉 2 = 𝑉 𝑜𝑢𝑡
𝐴+𝐵→𝐶
𝑟 𝑟𝑥𝑛,𝐴 = 𝑘 𝑟∗ 𝑐 𝐴

12. Continuously-stirred tank Reactor (CSTR)
Mole balance on A: A
A, B, C B 1 2
𝑛 𝐴,1 = 𝑛 𝐴,𝑜𝑢𝑡 + 𝑟 𝑐𝑜𝑛𝑠,𝐴
𝑐 𝐴,1 𝑉 1 = 𝑐 𝐴,3 𝑉 3 + 𝑘 𝑟∗ 𝑐 𝐴,3 𝑉 𝑅
Mole balance on B: 3
𝑛 𝐵,2 = 𝑛 𝐵,3 + 𝑟 𝑐𝑜𝑛𝑠,𝐵
𝐴+𝐵→𝐶
𝑐 𝐵,2 𝑉 2 = 𝑐 𝐵,3 𝑉 3 +
𝑣𝑒𝑟𝑦 𝑠𝑚𝑎𝑙𝑙
(≈0)
𝑟 𝑟𝑥𝑛,𝐴 = 𝑘 𝑟∗ 𝑐 𝐴

13. 𝑐 𝐴,1 𝑉 1 = 𝑐 𝐴,3 𝑉 3 + 𝑘 𝑟∗ 𝑐 𝐴,3 𝑉 𝑅 𝑐 𝐵,2 𝑉 2 = 𝑐 𝐵,3 𝑉 3
CSTR
𝑐 𝐴,1 𝑉 1 = 𝑐 𝐴,3 𝑉 3 + 𝑘 𝑟∗ 𝑐 𝐴,3 𝑉 𝑅
𝑐 𝐵,2 𝑉 2 = 𝑐 𝐵,3 𝑉 3
𝑘 𝑟∗ = 𝑐 𝐵,3 𝑘 𝑟

14. Residence time Residence time 𝝉 : 1 2 3 𝜏= 𝑉 𝑅 𝑉 3 𝐴+𝐵→𝐶A
A, B, C B
Residence time 𝝉 :
The average time a molecule spends inside a reactor
Sort of like 𝑡 in the batch reactor
For our set-up: 1 2 3
𝜏= 𝑉 𝑅 𝑉 3
𝐴+𝐵→𝐶
𝑟 𝑟𝑥𝑛,𝐴 = 𝑘 𝑟∗ 𝑐 𝐴

15. 𝑐 𝐴 = 𝑐 𝐴,1 𝑉 1 𝑉 3 1+ 𝑘 𝑟∗ 𝜏 𝑘 𝑟∗ = 𝑘 𝑟 𝐶 𝐵,2 𝑉 2 𝑉 3 𝜏= 𝑉 𝑅 𝑉 3
CSTR Discussion
Can rearrange equations into this expression for concentration of A in the reactor:
𝑐 𝐴 = 𝑐 𝐴,1 𝑉 𝑉 𝑘 𝑟∗ 𝜏
𝑘 𝑟∗ = 𝑘 𝑟 𝐶 𝐵,2 𝑉 𝑉 3
𝜏= 𝑉 𝑅 𝑉 3
𝑤ℎ𝑒𝑟𝑒:
Discuss with your neighbor what will happen to the concentration of A exiting the reactor if:
Reactor volume is increased
Input concentration of B is increased
Temperature is increased
Input flowrate of either A is increased (this will also increase total output flowrate!)