Continuous Stirred-Tank Fermenter — Unsteady-State Cell Balance: 04-BS-13, December 2019, Q2

Question 2: Continuous Stirred-Tank Fermenter — Unsteady-State Cell Balance (20 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.

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The source figure for this question shows both an inlet stream (feed $F$, $x_i=0$, $s_i$) and an outlet stream (effluent $F$, $x$, $s$) leaving the vessel, and part (b) asks for a steady-state relation among $F$, $k_1$, and $V$ — that only makes sense if liquid also leaves the vessel at rate $F$ (constant volume). A true "fed-batch" reactor has no outflow and V grows without bound, so no F–$k_1$–V steady state could exist. This is solved as the continuous (chemostat-style) stirred-tank case implied by the figure and by part (b), with the "fed-batch" label in the question stem treated as a wording slip rather than followed literally.

Given. Constant-volume, well-mixed vessel; inflow $F$ at $x_i=0$; outflow $F$ at concentration $x$; $r_x=k_1x$ (growth), $r_s=k_2x$ (substrate consumption, not needed for parts (a)–(c)); initial condition $x(0)=x_0$.

Find. (a) the unsteady-state cell balance; (b) the F–$k_1$–V relation at steady state; (c) $x(t)$.

Approach. Write a cell-number balance over the vessel (accumulation = in $-$ out $+$ generation), simplify using constant V and sterile feed ($x_i=0$), set the accumulation term to zero for steady state, then integrate the first-order linear ODE with the given initial condition.

Fig. 2: control volume for the fermenter — sterile feed in at $F,\,x_i=0,\,s_i$; effluent out at $F,\,x,\,s$; constant liquid volume $V$.

(a) Unsteady-state cell balance. Accumulation of cells in the vessel equals the net inflow of cells plus the rate of cell generation by growth: $$\frac{d(Vx)}{dt}=Fx_i-Fx+Vr_x.$$ Since the feed is sterile ($x_i=0$), $V$ is constant, and $r_x=k_1x$: $$V\frac{dx}{dt}=-Fx+Vk_1x\ \Longrightarrow\ \boxed{\dfrac{dx}{dt}=\left(k_1-\dfrac{F}{V}\right)x}.$$
(b) Steady-state relationship. At steady state $dx/dt=0$ with $x\neq0$ (washout, $x=0$, is the trivial steady state and is excluded as it is of no operational interest), so the bracketed term must vanish: $$k_1-\frac{F}{V}=0\ \Longrightarrow\ \boxed{F=k_1V}.$$ Physically: the volumetric feed/dilution rate $F/V$ (the "dilution rate," often written $D$) must exactly equal the specific growth-rate constant $k_1$ for the cell concentration to stop changing.
(c) Solving the ODE. $dx/dt=(k_1-F/V)x$ is separable/linear first-order with constant coefficient $(k_1-F/V)$: $$\frac{dx}{x}=\left(k_1-\frac{F}{V}\right)dt\ \Longrightarrow\ \ln x = \left(k_1-\frac{F}{V}\right)t+\ln x_0\ \Longrightarrow\ \boxed{x(t)=x_0\exp\!\left[\left(k_1-\frac{F}{V}\right)t\right]}.$$ If $F/V>k_1$ (dilution faster than growth) the culture washes out ($x\to0$); if $F/V

Quantity	Result
Unsteady-state cell balance	$dx/dt=(k_1-F/V)x$
Steady-state relation	$F=k_1V$ (dilution rate $F/V$ equals $k_1$)
Cell concentration vs. time	$x(t)=x_0\exp[(k_1-F/V)t]$

Topic: Unsteady-state cell mass balance on a continuous (chemostat-style) stirred-tank bioreactor.

Key relations: general balance "accumulation = in $-$ out + generation"; dilution rate $D=F/V$; at steady state the specific growth rate must equal the dilution rate, $\mu=D$ (here $\mu=k_1$, a constant since the kinetics are first-order in $x$ alone, not Monod-limited).

Why this works: with a sterile feed the only source of cells is growth inside the vessel and the only sink is washout in the effluent; because $r_x$ is directly proportional to $x$ (not saturating), the balance reduces to a single linear ODE with constant coefficient $(k_1-F/V)$, giving pure exponential growth or decay depending on the sign of that coefficient.

Common pitfall: taking the question's "fed-batch" label literally and omitting the outflow term — that would leave no F-dependence at all and no steady state could exist for part (b), contradicting the question's own request for an $F$–$k_1$–$V$ relationship.

Source: Shuler & Kargi, Bioprocess Engineering: Basic Concepts, Ch. 6 (mass balances on continuous stirred-tank bioreactors, dilution rate, washout).

Question 2 of 9: Continuous Stirred-Tank Fermenter — Unsteady-State Cell Balance

Question 2: Continuous Stirred-Tank Fermenter — Unsteady-State Cell Balance (20 marks)