Q.83 A continuous cell culture being carried out in a stirred tank reactor is described in terms of S. The cell mass concentration of the X substrate in the sterile feed stream is S_F = 10 g/L and yield coefficient Y_X/S = 0.5. (F – flow rate (ml/min) and feed stream and specific growth rate (h^-1) = 0.35 in the steady state. The concentration of S is g/L (up to 1 decimal point).

Answer: Sf = 10.0 g/L

(reported to one decimal place)

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Chemostat Basics

A chemostat maintains a steady-state continuous culture by balancing inflow and outflow
at a defined dilution rate:

D = F / V

Since the feed is sterile (X_i = 0), the steady-state biomass
balance simplifies to:

μX V = D X V

which reduces to:

(μ − D)X = 0

Given μ = 0.3 h⁻¹ and
D = 0.25 h⁻¹, μ ≠ D, so the only feasible steady-state solution is:

X = 0 g/L

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Mass Balances Explained

Biomass Balance (steady state)

At steady state (dX/dt = 0):

• Inflow of biomass = 0
• Outflow of biomass = D X V
• Generation of biomass = μ X V

This yields:

(μ − D)X = 0

Since μ > D, biomass cannot accumulate and washout occurs, giving
X = 0 g/L.

Substrate Balance (steady state)

At steady state (dS/dt = 0):

D(S₀ − S_f)V = (μ X V) / Y_X/S

Since X = 0, the consumption term becomes zero:

D(S₀ − S_f) = 0

Therefore:

S_f = S₀ = 10.0 g/L

The yield coefficient Y_X/S = 0.5 is unused because no growth occurs.

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Why No Growth Occurs

In standard chemostat theory with sterile feed, nonzero biomass at steady state requires
μ = D. When μ is fixed and greater than D (as in this problem),
cells are washed out of the reactor.

Since no Monod parameters (μ_max, K_S) are provided, μ cannot
adjust to match D. As a result, substrate flows through the system unchanged.

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Step-by-Step Solution

1. Check steady-state biomass condition: μ ≠ D → X = 0 g/L
2. With no biomass, no substrate is consumed
3. Exit substrate concentration equals feed concentration
4. S_f = 10.0 g/L

This problem illustrates key steady-state assumptions commonly tested in
CSIR NET Life Sciences and Biotechnology exams.

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Keywords: chemostat steady state, cell mass concentration, substrate yield Y_X/S,
dilution rate D, continuous culture, Sf calculation