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.