Chapter 28, Problem 16P

Bacteria growing in a batch reactor utilize a soluble food source (substrate) as depictedin Fig.P28.16. The up take of the substrate is represented by a logistic model with Michaelis-Menten limitation. Death of the bacteria produces detritus which is subsequently converted to the substrate by hydrolysis. In addition, the bacteria also excrete some substrate directly. Death, hydrolysis, and excretion are all simulated as first-order reactions. Mass balances can be written as

$\begin{array}{l}\frac{dX}{dt}={\mu }_{\text{max}}\left(1-\frac{X}{K}\right)\left(\frac{S}{K_s+S}\right)X-k_{d}X-k_{e}X\\ \frac{dC}{dt}=k_{d}X-k_{h}C\\ \frac{dS}{dt}=k_{e}X+k_{h}C-{\mu }_{\text{max}}\left(1-\frac{X}{K}\right)\left(\frac{S}{K_s+S}\right)X\end{array}$

where X, C and $S=$ the concentrations $\left(\text{mg}/\text{L}\right)$ of bacteria, detritus, and substrate, respectively; ${\mu }_{\text{max}}=$ maximum growth rate $\left(/\text{d}\right),K=$ the logistic carrying capacity $\left(\text{mg/L}\right);K_s=$ the Michaelis-Menten half-saturation constant $\left(\text{mg}/\text{L}\right),$ $k_d=\text{death rate }\left(/\text{d}\right)\text{;}{k}_e\text{= excretion rate }\left(\text{/d}\right)\text{; }$ $\text{and }{k}_h\text{= hydrolysis rate }\left(\text{/d}\right)$. Simulate the concentrations from $t=0$ to 100 given the initial conditions $X\left(0\right)=1\text{mg}/\text{L, }S\left(0\right)=100\text{mg/L,}\text{ and }C\left(0\right)=0\text{ mg/L}\text{. }$ Employ the following parameters in your calculation: ${\mu }_{\text{max}}=10\text{/d,}K=10\text{ mg/L, }$ $k_d=0\text{ 1/d, }{k}_e=0,$ $\text{ 1/d, and }{k}_h=0.\text{ 1/d}$.