61. In a fed-batch culture, 200 g·L−1 glucose solution is added at a flow
rate of 50 L·h−1. The initial culture volume (at quasi steady state)
and the initial cell concentration are 600 L and 20 g·L−1,
respectively. The yield coefficient (Yx/s) is
0.5 g cell mass·g substrate−1.
The cell concentration (g·L−1) at quasi steady state at
t = 8 h is
Fed-Batch Culture Cell Concentration
The correct cell concentration at t=8 h is 52 g·L⁻¹
Fed-batch cultures maintain quasi-steady state cell concentrations by balancing substrate addition with biomass growth based on yield coefficients.
🔬 Problem Breakdown
In fed-batch at quasi-steady state, cell concentration x remains constant despite volume increase from feed, as biomass grows proportionally to substrate input.
X₀ = 20 × 600 = 12,000 gFeed Rate:
F × S₀ = 50 × 200 = 10,000 g/h glucoseCell Production:
10,000 × 0.5 = 5,000 g/h (Yₓ/ₛ = 0.5)Total biomass at 8 h: Xₜ = 12,000 + 5,000 × 8 = 52,000 g
Total volume at 8 h: Vₜ = 600 + 50 × 8 = 1,000 L
Final concentration: x = 52,000 / 1,000 = 52 g·L⁻¹
📋 Options Analysis
| Option | Value | Why Incorrect/Correct |
|---|---|---|
| A | 40 | Underestimates growth; ignores full 40,000 g cells from 8 h feed (ignores yield × substrate). |
| B | 52 | Matches exact calculation: total biomass 52,000 g in 1,000 L. |
| C | 60 | Overestimates; perhaps assumes constant volume or higher yield. |
| D | 68 | Furthest error; might double-count volume or yield effects. |
⚗️ Key Equation
Quasi-steady state formula:
x = x₀ + (F · Yₓ/ₛ · S₀ · t) / (V₀ + F · t)
Simplifies to 52 g·L⁻¹ for given parameters
This approach is standard in bioprocess engineering for predicting biomass in nutrient-limited fed-batch processes.
