An $18\mathrm{.}5L$ chemostat is used to produce a mixed$-$growth product from a limiting substrate, $S,$ and operates at steady

state. Determine the rate of product, P$,$ formation in g$/$L$-$h $($i$.$e$.,$ DP$)$ and in total g$/$h$,$ using the following

assumptions and parameters.

Assumptions: $1.$ Monod kinetics apply. Assume that substrate is converted primarily to cell $($biomass$)$

so you can relate $$ to $S$ while assuming $S$ used to form product is negligible.

The Luedeking $-$ Piret equation can be applied to determine product formation.

Sterile feed.

Negligible cell death $(k_d=0).$

Dilution rate, $D=0\mathrm{.}77{}_{\text{m}\text{a}\text{x}}$

$Y_{\frac{x}{s}}=0\mathrm{.}52g\frac{x}{g}S$

$K_S=87\mathrm{.}6m\frac{g}{L}$

${}_{\text{m}\text{a}\text{x}}=1\mathrm{.}25h^{-1}$

$S_o=690m\frac{g}{L}$

$=0\mathrm{.}47mg\frac{P}{g}x$

$=0\mathrm{.}63mg\frac{P}{g}x-h$

$($assumption $1$ means that all substrate goes into making new cells$-$ OK to do for an approximation, but not

exactly correct, since a product is being formed$).$