Exercise $5\mathrm{.}9$: Penicillin Fermentation

A model for a batch reactor in which penicillin is produced by fermentation has been derived as follows $($Constantantinides et al$.1970)$ for cell production and penicillin synthesis, respectively:

$\frac{d{y}_1}{dt}=b_1{y}_1-\frac{b_1}{b_2}{y}_1^2,y_1(0)=0\mathrm{.}03$

$\frac{d{y}_2}{dt}=b_3{y}_1,y_2(0)=0\mathrm{.}0$

where

$y_1=$ dimensionless concentration $of$ cell mass

$y_2=$ dimensionless concentration $of$ penicillin

$t=$ dimensionless time, $0t1$

Experiments have determined that

$b_1=13\mathrm{.}1$

$b_2=0\mathrm{.}94$

$b_3=1\mathrm{.}71$

$b_1=13\mathrm{.}1$

$b_2=0\mathrm{.}94$

$b_3=1\mathrm{.}71$

a$.$ Solve this ODE$-$IVP using the Euler method and Excel $($not VBA$).$

b$.$ Solve this ODE$-$IVP using the second$-$order RK method and Excel $($not VBA$).$

c$.$ Solve this ODE$-$IVP using the Euler VBA program as given in Example $5\mathrm{.}7.$

d$.$ Solve this ODE$-$IVP using the second$-$order RK VBA program from Exercise $5\mathrm{.}8.$

In all cases, experiment with the time step to assure an accurate solution. Also, graph $y_1$ and $y_2$ versus time with appropriate annotations.

e$.$ Solve this ODE$-$IVP using fourth$-$order RK method and Excel $($not VBA$)$