Consider a culture of bacteria that secrete a product in a chemostat operated at steady

state. The specific growth rate of biomass is adequately described by the Monod equation,

and the rate of product formation is given by the Leudeking and Piret equation:

$r_P=(+)x$

This system is well characterized, such that the following constants are known:

$Y_{\frac{x}{S}}=0\mathrm{.}4\frac{g}{g},{}_{\text{m}\text{a}\text{x}}=0\mathrm{.}7h^{-1},K_S=0\mathrm{.}2\frac{g}{L},=0\mathrm{.}2\frac{g}{g}$

$Y_{\frac{P}{S}}=0\mathrm{.}5\frac{g}{g},=0\mathrm{.}3\frac{g}{g}-h,F=14\frac{L}{h}r,V=100L$

The liquid feed to the chemostat is sterile and contains $10\frac{g}{L}$ of the limiting growth

substrate. Assume negligible maintenance substrate consumption. Find the dilution rate and

concentrations of substrate, biomass, and product in the exit stream.A continuous culture follows the inhibited cell growth kinetics given below:

$r_x=\frac{{}_{\text{m}\text{a}\text{x}}sx}{K_s+s+i\frac{K_s}{K_i}}$

Experiments are done with dilution rates varying from $0\mathrm{.}05h^{-1}$ to $0\mathrm{.}50h^{-1}$ under uninhibited

and inhibited conditions $(i=0\mathrm{.}05\frac{g}{l}).$

$($i$)$ Plot $x$ and $s$ as a function of $D$ for uninhibited and inhibited conditions

$($ii$)$ Estimate $D_{\text{m}\text{a}\text{x}}$ for uninhibited and inhibited conditions

$($iii$)$ Develop expressions for the ratios $\frac{s}{s_i}$ and $x\frac{D}{(xD{)}_i}$

$($iv$)$ Graphically estimate cell productivity and optimum dilution rate for uninhibited and

inhibited conditions.

Data: $S_0=10\frac{g}{1}$;$x_0=0$;$K_S=1\frac{g}{1}$;$K_i=0\mathrm{.}01\frac{g}{1}$;${}_{\text{m}\text{a}\text{x}}=0\mathrm{.}5h^{-1}$;$Y_{\frac{x}{S}}=0\mathrm{.}1g$ cells $\frac{?}{g}$ substrate.