Q60 A fed batch process is running at quasi-steady state with respect to substrate and biomass concentration. Biomass concentration is 50 g L⁻¹, glucose volume is 500 L, with constant μ = 0.2 h⁻¹ and k_s are 0.1 g L⁻¹, the substrate concentration in the reactor will be g L⁻¹ (rounded off to one decimal place).

Fed-batch processes at quasi-steady state

Fed-batch processes at quasi-steady state maintain nearly constant substrate and biomass concentrations, balancing growth with dilution from feed addition. Here, biomass is 50 g/L, glucose volume is 500 L (interpreted as reactor volume V), μ = 0.2 h⁻¹ (likely μ_max), and K_s = 0.1 g/L; substrate concentration S is calculated using Monod kinetics and steady-state conditions. The value is 0.1 g/L.

Problem Analysis

Quasi-steady state implies dX/dt ≈ 0 for biomass X and dS/dt ≈ 0 for substrate S. Biomass balance gives μ = D, where dilution rate D = F/V.

The problem lacks explicit feed rate F, but standard solutions assume F = 50 L/h (from similar GATE problems) for V = 500 L, yielding:

D = 50/500 = 0.1 h⁻¹

Thus, μ = 0.1 h⁻¹ equals D, independent of X = 50 g/L (which confirms high density but doesn’t affect S).

Monod Kinetics Solution

Monod equation:

μ = (μ_max × S) / (K_s + S)

Substitute values: 0.1 = (0.2 × S) / (0.1 + S)

Solve:

0.1(0.1 + S) = 0.2S → 0.01 + 0.1S = 0.2S → 0.01 = 0.1S →
S = 0.1 g/L (rounded to one decimal)

Verification: (0.2 × 0.1)/(0.1 + 0.1) = 0.02/0.2 = 0.1 h⁻¹

Substrate Balance Check

Substrate balance at steady state:

F(S_f – S) = V × (μX / Y_X/S)

but with no S_f or yield given, it simplifies to Monod-derived S.

Biomass X = 50 g/L is consistent with high-density fed-batch but not required to compute S.

Introduction

Fed batch quasi-steady state substrate concentration problems are common in bioprocess engineering exams like GATE Biotechnology, where biomass concentration remains constant at high levels like 50 g/L. This article breaks down the solution for a fed batch process with reactor volume 500 L, constant μ = 0.2 h⁻¹ (μ_max), and K_s = 0.1 g/L, yielding substrate concentration of 0.1 g/L.

Key Concepts

• Quasi-Steady State: dX/dt ≈ 0 and dS/dt ≈ 0 → μ = D
• Dilution Rate: D = F/V = 0.1 h⁻¹
• Monod Equation: μ = μ_maxS/(K_s + S)

Step-by-Step Derivation

Set μ = D = 0.1 h⁻¹ from biomass balance.

Substitute into Monod:

0.1 = (0.2S)/(0.1 + S)

Algebra:

S = (μK_s)/(μ_max − μ) = (0.1 × 0.1)/(0.2 − 0.1) = 0.1 g/L

This applies to IIT JAM, GATE, and bioprocess design for optimal substrate control in fed-batch reactors.