Fed-Batch Reactor — Volume and Pseudo-Steady-State Substrate Expressions: 20-Bio-A5 Systems Analysis & Control, May 2017, Q6

Question 6: Fed-Batch Reactor — Volume and Pseudo-Steady-State Substrate Expressions (a. 10 marks; b. 5 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. $V(0)=V_0$; feed rate $F(t)=F_0+\alpha t$ with feed substrate concentration $S_f$; growth kinetics $\mu_g=kS$; no cell death; true growth yield $Y_{X/S}^M$; no outflow (fed-batch).

Find. (a) $V(t)$; (b) $S(t)$ under the pseudo-steady-state (quasi-steady-state, QSS) approximation.

Fig. 4 — fed-batch reactor: an increasing feed rate $F(t)=F_0+\alpha t$ raises the liquid volume with no outflow; biomass and substrate concentrations evolve inside.

Approach. Part (a) is a direct integration of the overall liquid-volume balance. Part (b) applies the pseudo-steady-state approximation ($dS/dt\approx0$, i.e. $S$ adjusts on a much faster timescale than $V$ or the total biomass $XV$ change) to the coupled substrate and biomass balances, together with the standard fed-batch simplification that the feed is concentrated ($S_f\gg S$ at quasi-steady state — the whole point of running fed-batch).

(a) Overall liquid-volume balance. With no outflow, $dV/dt=F(t)=F_0+\alpha t$. Integrating from $V(0)=V_0$: $$\boxed{V(t)=V_0+F_0t+\frac{\alpha}{2}t^2.}$$
(b) Overall biomass and substrate balances. With no death and no outflow: $$\frac{d(XV)}{dt}=\mu_g\,XV, \qquad \frac{d(SV)}{dt}=F\,S_f-\frac{\mu_g\,XV}{Y_{X/S}^M}.$$
Apply the pseudo-steady-state approximation. Expanding $d(SV)/dt=V\,dS/dt+S\,dV/dt$ and setting $dS/dt\approx0$ gives $d(SV)/dt\approx S\,dV/dt=SF$. Substituting into the substrate balance: $$SF=FS_f-\frac{\mu_g XV}{Y_{X/S}^M} \ \Rightarrow\ \mu_g XV=Y_{X/S}^M F(S_f-S)\approx Y_{X/S}^M F S_f$$ (dropping $S$ next to $S_f$, consistent with $S$ staying small — that is what “quasi-steady” means here).
Integrate the total-biomass balance. The biomass balance becomes $d(XV)/dt=\mu_g XV=Y_{X/S}^M S_f F(t)$, a known function of time, so $$XV(t)=X_0V_0+Y_{X/S}^M S_f\int_0^t F\,dt'=X_0V_0+Y_{X/S}^M S_f\big(V(t)-V_0\big).$$
Solve for $S(t)$. Since $\mu_g=kS$ and $\mu_g XV=Y_{X/S}^M S_f F(t)$ from Step 3, $$kS\cdot XV(t)=Y_{X/S}^M S_f F(t) \ \Rightarrow\ \boxed{S(t)=\frac{Y_{X/S}^M S_f\big(F_0+\alpha t\big)}{k\Big[X_0V_0+Y_{X/S}^M S_f\big(V(t)-V_0\big)\Big]}.}$$

Result	Expression
(a) Reactor volume $V(t)$	$V_0+F_0t+\alpha t^2/2$
Total biomass $XV(t)$	$X_0V_0+Y_{X/S}^M S_f\big(V(t)-V_0\big)$
(b) Substrate conc. $S(t)$	$Y_{X/S}^M S_f F(t)\big/\big[k\,XV(t)\big]$

Concept & Theory

Topic. Fed-batch culture under the pseudo- (quasi-) steady-state approximation.

Key relations. $dV/dt=F(t)$; QSS substrate balance $\mu_g XV\approx Y_{X/S}^M S_f F(t)$; total biomass $XV(t)=X_0V_0+Y_{X/S}^M\int F\,dt$.

Why this works. At quasi-steady state the reactor's residual substrate concentration is small and self-adjusting: whatever $S$ is needed to make $\mu_g X$ exactly track the incoming substrate flux is what the system settles to almost instantly, so biomass growth becomes directly slaved to the feed-rate profile $F(t)$ rather than to any free kinetic parameter — this is precisely why fed-batch is used industrially, to hold $S$ low (avoiding substrate inhibition or overflow metabolism) while still controlling the growth rate through the feed schedule.

Common pitfall. Assuming $X$ (rather than $S$) is the pseudo-steady quantity — it is not: $X$ (or $XV$) keeps growing throughout the fed-batch, and only $S$ is pseudo-constant; also forgetting the $S\ll S_f$ simplification is an approximation (valid specifically because QSS keeps $S$ small), not an exact identity.

Source. Bailey & Ollis, Biochemical Engineering Fundamentals, Ch. 9 (fed-batch culture, quasi-steady-state analysis).

Question 6 of 6: Fed-Batch Reactor — Volume and Pseudo-Steady-State Substrate Expressions

Question 6: Fed-Batch Reactor — Volume and Pseudo-Steady-State Substrate Expressions (a. 10 marks; b. 5 marks)