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## Q. 19.13

A dynamic model of a continuous-flow, biological chemostat has the form

\begin{aligned}\dot{X} &=0.063 C-D x \\\dot{C} &=0.9 S[X-C]-0.7 C-D C \\\dot{S} &=-0.9 S[X-C]+D[10-S]\end{aligned}

where $X$ is the biomass concentration, $S$ is the substrate concentration, and $C$ is a metabolic intermediate concentration. The dilution rate, $D$, is an independent variable, which is defined to be the flow rate divided by the chemostat volume.

Determine the value of $D$, which maximizes the steady-state production rate of biomass, $f$, given by

$f=D X$

## Verified Solution

Assuming steady state behavior, the optimization problem is,

$\max f=D e$

Subject to

\begin{aligned}&0.063 c-D e=0 (1)\\&0.9 s e-0.9 s c-0.7 c-D c=0 (2)\end{aligned}

$\begin{gathered}-0.9 s e+0.9 s c+10 D-D s=0 (3) \\D, e, s, c \geq 0\end{gathered}$

where $f=f(D, e, c, s)$

Excel-Solver is used to solve this problem,

 c D e s Initial values 1 1 1 1 Final values 0.479031 0.045063 0.669707 2.079784 max f = 0.030179 Constraints 0.063 c –D e 2.08E-09 0.9 s e – 0.9 s c – 0.7 c – Dc -3.10E-07 -0.9 s e + 0.9 s c + 10D – Ds 2.88E-07

Table S19.13. Excel solution

Thus the optimum value of $D$ is equal to $0.045 h ^{-1}$