Mass balances for a fed-batch bioreactor are given by
$$\begin{align*} \frac{d(XV)}{dt} & = V r_g(X,S) \\ \frac{d(PV)}{dt} & = V r_P(X,S) \\ \frac{d(SV)}{dt} & = F S_f - \frac{1}{Y_{X/S}}V r_g(X,S) \end{align*}$$where $X$ is cell concentration, $P$ is product concentration, and $S$ is substrate concentration, all given in units of grams/liter. The reactor is fed with fresh substrate at concentration $S_f$ and flowrate $F(t)$ in liters per hour. The volume (in liters) is therefore changing
$$\frac{dV}{dt} = F(t)$$Rate $r_g(X,S)$ is the production of fresh cell biomass in units of grams/liter/hr. The cell specific growth is expressed as
$$r_g(X,S) = \mu(S)X$$where $\mu(S)$ is the cell specific growth rate. In the Monod model, the specific growth rate is a function of substrate concentration given by
$$\mu(S) = \mu_{max}\frac{S}{K_S + S}$$where $\mu_{max}$ is the maximum specific growth rate, and $K_S$ is the half saturation constant which is the value of $S$ for which $\mu = \frac{1}{2}\mu_{max}$.
For this model, the product is assumed to be a by-product of cell growth
$$r_P(X,S) = Y_{P/X}r_g(X,S)$$where $Y_{P/X}$ is the product yield coefficient defined as
$$Y_{P/X} = \frac{\mbox{mass of product formed}}{\mbox{mass of new cells formed}}$$The model further assumes that substrate is consumed is proportion to the mass of new cells formed where $Y_{X/S}$ is the yield coefficient for new cells
$$Y_{P/X} = \frac{\mbox{mass of new cells formed}}{\mbox{mass of substrate consumed}}$$One aspect of the fed-batch model is that volume is not constant, therefore the cell, product, and substrate concentrations are subject to a dilution effect. Mathematically, the chain rule of differential calculus provides a means to recast the state of model in terms of the intensive concentration variables $X$, $P$, and $S$, and extensive volume $V$.
$$\begin{align*} \frac{d(XV)}{dt} & = V\frac{dX}{dt} + X\frac{dV}{dt} = V\frac{dX}{dt} + F(t)X \\ \frac{d(PV)}{dt} & = V\frac{dP}{dt} + P\frac{dV}{dt} = V\frac{dP}{dt} + F(t)P \\ \frac{d(SV)}{dt} & = V\frac{dS}{dt} + S\frac{dV}{dt} = V\frac{dS}{dt} + F(t)S \end{align*}$$Rearranging and substituting into the mass balances gives
$$\begin{align*} \frac{dX}{dt} & = - \frac{F(t)}{V}X + r_g(X,S) \\ \frac{dP}{dt} & = - \frac{F(t)}{V}P + r_P(X,S) \\ \frac{dS}{dt} & = \frac{F(t)}{V}(S_f - S) - \frac{1}{Y_{X/S}}r_g(X,S) \\ \frac{dV}{dt} & = F(t) \end{align*}$$
%matplotlib inline
import numpy as np
import matplotlib.pyplot as plt
from scipy.integrate import odeint
# parameter values
mumax = 0.20 # 1/hour
Ks = 1.00 # g/liter
Yxs = 0.5 # g/g
Ypx = 0.2 # g/g
Sf = 10.0 # g/liter
# inlet flowrate
def F(t):
return 0.05
# reaction rates
def mu(S):
return mumax*S/(Ks + S)
def Rg(X,S):
return mu(S)*X
def Rp(X,S):
return Ypx*Rg(X,S)
# differential equations
def xdot(x,t):
X,P,S,V = x
dX = -F(t)*X/V + Rg(X,S)
dP = -F(t)*P/V + Rp(X,S)
dS = F(t)*(Sf-S)/V - Rg(X,S)/Yxs
dV = F(t)
return [dX,dP,dS,dV]
IC = [0.05, 0.0, 10.0, 1.0]
t = np.linspace(0,50)
sol = odeint(xdot,IC,t)
X,P,S,V = sol.transpose()
plt.plot(t,X)
plt.plot(t,P)
plt.plot(t,S)
plt.plot(t,V)
plt.xlabel('Time [hr]')
plt.ylabel('Concentration [g/liter]')
plt.legend(['Cell Conc.',
'Product Conc.',
'Substrate Conc.',
'Volume [liter]'])
<matplotlib.legend.Legend at 0x1152fb668>
