Figure P22.25 shows the kinetic interactions governing the concentrations of a bacteria culture
and their nutrition source (substrate) in a continuously stirred flow-through bioreactor.
The mass balances for the bacteria biomass, X (gC/m³), and the substrate concentration, S
(gC/m³), can be written as
\frac{dX}{dt} = \left(k_{g,max}\frac{S}{K_S+S} - k_d - k_r - \frac{1}{\tau_w}\right)X
\frac{dS}{dt} = -\frac{1}{Y}k_{g,max}\frac{S}{K_S+S}X + k_dX - \frac{1}{\tau_w}(S - S_{in})
where t = time (h), $k_{g,max}$ = maximum bacterial growth rate (/d), $K_S$ = half-saturation constant
(gC/m³), $k_d$ = death rate (/d), $k_r$ = respiration rate (h), Q = flow rate (m³/h), V = reactor volume
(m³), Y = yield coefficient (gC-cell/gCsubstrate), and $S_{in}$ = inflow substrate concentration
(mgC/m³). Simulate how the substrate, bacteria, and total organic carbon (X + S) change over
time in this reactor for three residence times: (a) $\tau_w$ = 20 h, (b) $\tau_w$ = 10 h, and (c) $\tau_w$ = 5 h. Employ
the following parameters for the simulation: X(0) = 100 gC/m³, S(0) = 0, $k_{g,max}$ = 0.2/hr, $K_S$ = 150
gC/m³, $k_d$ = $k_r$ = 0.01/hr, Y = 0.5 gC-cell/gC-substrate, V = 0.01 m³, and $S_{in}$ = 1000 gC/m³, and
display your results graphically.
Figure P22.25 Continuously stirred flow-through bioreactor to grow a bacterial culture.
