In the small$-$scale pilot plant the yeast Saccharomyces cerevisiae is produced in

fed$-$batch cultivations, to test new feeding strategies. The process consists of two

phases: an initial batch phase and a subsequent fed$-$batch phase. In order to

maximise the biomass yield, the substrate concentration during the fed$-$batch phase

shall be kept close to $1\frac{g}{L}.$ Design an appropriate feeding strategy $($starting at $5h$

process time$),$ using the data from the batch phase below.

Initial volume of the medium in the ideally stirred tank reactor: $10L$

The process can be described with the following mathematical model:

$\frac{dS}{dt}=\frac{S_0*F_0}{V}-r_S*x-\frac{S*F_0}{V},r_S=\frac{r_{\text{S}\text{m}\text{a}\text{x}}*S}{K_S+S},r_x=Y_{xS}*r_S$

$\frac{dx}{dt}=r_x*x-\frac{x*F_0}{V},\frac{dV}{dt}=F_0-F_1$ fed$-$batch operation $F_1=0$

With $S_0=$ Substrate conc. in inlet feed solution, $F_0=$ Inlet flow rate, $V=$ Volume of

medium in reactor, $x=$ Biomass conc., $r_s=$ specific substrate uptake rate, $r_{\text{s}\text{m}\text{a}\text{x}}=$

maximum specific substrate uptake rate, $K_s=$ Substrate affinity constant, $S=$

Substrate conc. in reactor, $r_x=$ specific biomass formation rate, $Y_{xs}=$ Yield

coefficient of biomass on substrate, $F1=$ Outlet flow rate.

a$)$ Plot using the above data the time course of the batch phase of the cultivation in an

appropriate graph.

b$)$ Calculate the overall yield coefficient $(Y_{xs}),$ and the specific biomass formation rate

$(r_x)$ as a function of time.

c$)$ Calculate the specific substrate uptake rate $(r_s)$ at $t=2\mathrm{.}5h$ and estimate the

maximum specific substrate uptake rate $(r_{\text{s}\text{m}\text{a}\text{x}}),$ if $K_S=0\mathrm{.}2\frac{g}{L}.$