β-Galactosidase bound to DEAE-cellulose is used to hydrolyze lactose
to glucose and galactose in a plug flow bioreactor with a packed bed
of volume 100 liters and a voidage (ε) of 0.55. The K’m and
V’max for the immobilized enzyme are
0.72 g l−1 and
18 g l−1 h−1, respectively.
The lactose concentration in the feed stream is
20 g l−1, and a fractional conversion of 0.90 is desired.
Diffusional limitations may be ignored.
Q.54 The residence time required for the steady state
reactor operation will be
Introduction
Plug flow bioreactors (PFRs) employing immobilized enzymes are extensively used in biochemical engineering
for continuous substrate conversion. A common numerical problem involves estimating the
residence time required to achieve a desired conversion when the enzyme follows
Michaelis–Menten kinetics.
Problem Summary
- Packed bed volume, V = 100 L
- Voidage, ε = 0.55
- Michaelis constant, Km = 0.72 g L-1
- Maximum rate, Vmax = 18 g L-1 h-1
- Feed lactose concentration, S0 = 20 g L-1
- Desired conversion, X = 0.90
- Diffusional limitations are negligible
Step-by-Step Solution
Step 1: Exit Substrate Concentration
S = S0(1 − X) = 20 × (1 − 0.9) = 2 g L-1
Step 2: Plug Flow Reactor Design Equation
For a plug flow reactor with Michaelis–Menten kinetics:
τ = 1 / Vmax [ (S0 − S) + Km ln(S0/S) ]
Step 3: Substitution of Values
τ = 1 / 18 [ (20 − 2) + 0.72 ln(20/2) ]
τ = 1 / 18 [ 18 + 0.72 ln(10) ]
ln(10) ≈ 2.303
τ = 1 / 18 (18 + 1.658)
τ ≈ 1.09 h ≈ 1.1 h
Final Answer
Correct Option: (D) 1.1 h
Conclusion
By applying the plug flow reactor design equation with Michaelis–Menten kinetics,
the required residence time for 90% lactose conversion is approximately
1.1 hours. Therefore, option (D) is the correct answer.
This problem is highly relevant for exams such as GATE and CSIR-NET.
