A semi$-$batch bio$-$reactor with the volumetric flowrates Fi $($in liters$/$hour$)$ for the inlet stream, has an initial liquid volume V $=2$ L for the contents inside, which is changing since there is only Fi$.$ Assume that the density of all material is constant at $=1$ g$/$L$.$ There are no cells or product in the inlet stream. The reactor is well mixed. Substrate is present in the inlet stream. The inlet concentration of substrate is Sf $=10$ g$/$L $($constant$).$ Take the concentrations of substrate, cells and products inside the reactor as S$,$ X and P$,$ all varying with time $($all in g$/$L$).$ The growth rate of the cells and products is defined as:r$\_$g$=$X where the specific growth rate $()$ of the cells is given by $(\_$m S$)/($K$\_$S$+$S$)$ with maximum specific growth rate $(\_$m$)$ as $0\mathrm{.}1$ h$-1$ and KS $=2$ g$/$L$.$ r$\_$P$=$Y$\_($P$/$X$)*$r$\_$g where Y$\_($P$/$X$)=0\mathrm{.}1.$ Also, take Y$\_($X$/$S$)$ and Y$\_($P$/$S$)$ both as $0\mathrm{.}2.$ Fi can be taken as $1$ L$/$h and the steady state values of X$,$ S and P can all be taken as $2$ g$/$L$.$ Derive a model for the reactor by doing the overall MB and component balances for the cell, substrate and products. Remember that the only variables in the system are V$,$ X$,$ P and S$.$