Bacteria growing in a batch reactor utilize a soluble food source (substrate) as depicted in Fig. P28.16. The uptake 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

Where X, C, and S = the concentrations (mg/L) of bacteria, detritus, and substrate, respectively; mmax = maximum growth rate (/d), K = the logistic carrying capacity (mg/L); Ks = the Michaelis-Menten half-saturation constant (mg/L), kd = death rate (/d); ke = excretion rate (/d); and kh = hydrolysis rate (/d). Simulate the concentrations from t = 0 to 100 d, given the initial conditions X(0) = 1 mg/L, S(0) = 100 mg/L, and C(0) = 0 mg/L. Employ the following parameters in your calculation: mmax = 10/d, K = 10 mg/L, Ks = 10 mg/L, kd = 0.1/d, ke = 0.1/d, and kh = 0.1/d.

dX dt dC dt dS dt