Q.60 An enzyme converts substrate A to product B. At a given liquid feed stream of flow rate 25 L·min−1 and
feed substrate concentration of 2 mol·L−1, the volume of continuous stirred tank reactor needed for 95%
conversion will be __________ L.
Given the rate equation:
−rA = 0.1CA /(1 + 0.5CA)
where −rA is the rate of reaction in mol·L−1·min−1 and
CA is the substrate concentration in mol·L−1.
Assumptions: Enzyme concentration is constant and does not undergo any deactivation during the reaction.
Result: Required CSTR volume = 4987 L
Problem Statement
The volume of a continuous stirred tank reactor (CSTR) is to be calculated
for achieving 95% conversion of substrate A.
The reaction follows enzyme kinetics similar to the
Michaelis–Menten model, and the system operates at steady state.
Given Data
- Feed volumetric flow rate, F = 25 L·min-1
- Inlet substrate concentration, CA0 = 2 mol·L-1
- Desired conversion, X = 0.95
- Rate equation:
-rA = 0.1 CA / (1 + 0.5 CA)
Design Equation for CSTR
The CSTR design equation is:
V = F (CA0 X) / (−rA)exit
For a CSTR, the reaction rate is evaluated at the exit concentration.
Step-by-Step Solution
Step 1: Exit Concentration of Substrate
CAs = CA0(1 − X)
= 2 × (1 − 0.95)
= 0.1 mol·L-1
Step 2: Reaction Rate at Exit
−rA = (0.1 × 0.1) / (1 + 0.5 × 0.1)
= 0.01 / 1.05
= 0.00952 mol·L-1·min-1
Step 3: Molar Flow of Substrate
Feed molar flow = F × CA0
= 25 × 2
= 50 mol·min-1
Moles converted per minute = 50 × 0.95
= 47.5 mol·min-1
Step 4: Reactor Volume Calculation
V = 47.5 / 0.00952
= 4987 L
Final Answer
Required CSTR volume = 4987 L
Enzyme Kinetics Insight
The rate expression follows a Monod/Michaelis–Menten type form:
−rA = Vmax CA / (Km + CA)
where Vmax = 0.1 mol·L-1·min-1 and
Km = 2 mol·L-1.
Common Mistakes Avoided
- Using average reaction rate instead of exit rate (invalid for CSTR)
- Forgetting molar flow term F·CA0
- Confusing CSTR design with PFR integral design
Bioreactor Applications
This calculation is essential in enzyme reactor design for
pharmaceutical and fermentation industries.
CSTRs typically require larger volumes than PFRs for the same conversion,
especially for Michaelis–Menten kinetics.
GATE Exam Tip
CSTR design shortcut:
V = F CA0 X / (−rA)exit
