A dynamic model of a continuous-flow, biological chemostat has the form Ẋ = 0.063C − Dx Ċ = 0.9S [X − C] − 0.7C − DC Ṡ = −0.9S [X − C] + D [10 − S] where X is the biomass concentration, S is the substrate concentration, and C is a metabolic intermediate concentration. The dilution rate, D, is an

Question

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

[latex]\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}[/latex]

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

Determine the value of [latex]D[/latex], which maximizes the steady-state production rate of biomass, [latex]f[/latex], given by

[latex]f=D X[/latex]

Accepted Answer

Assuming steady state behavior, the optimization problem is,

[latex]\max f=D e[/latex]

Subject to

[latex]\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}[/latex]

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

where [latex]f=f(D, e, c, s)[/latex]

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 [latex]D[/latex] is equal to [latex]0.045 h ^{-1}[/latex]

Question 19.13: A dynamic model of a continuous-flow, biological chemostat h...

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