$(25$ points$)$ A single cell protein grown on methanol $(M)$ in a sparged CSTR obeys the following kinetic parameters at $T=30C$ and $pH=6\mathrm{.}0$ :

${}_m=0\mathrm{.}53h^{-1},Y_{\frac{x}{M}}=0\mathrm{.}48g\frac{x}{g}C{H}_{3}OH$

carbon conversion efficiency $(C\frac{x}{C}M)=0\mathrm{.}57\frac{g}{g}$

$K_s=2\mathrm{.}0m\frac{g}{1}$

$m=$ maintenance coefficient $=0\mathrm{.}35g\frac{M}{g}x-hr$

The specified yield factors correspond to a dilution rate of $0\mathrm{.}52h^{-1}.$

a$.$ Develop the equations for CSTR growth that describe the rates of protein production and substrate utilization as a function of the dilution rate. Assume that the cell concentration and the oxygen concentration in the entering liquid stream are zero and that sufficient oxygen is provided by the sparger.

b$.$ Plot $x,Dx$ and $M$ versus $D$ and find $x_{\text{m}\text{a}\text{x}}$ and $(Dx{)}_{\text{m}\text{a}\text{x}}$ when $M_0=7\mathrm{.}96\frac{g}{?}.$