A well-mixed fermenter of volume V contains cells initially at concentration x0. A sterile feed enters the fermenter with volumetric flow rate F; fermentation broth leaves at the same rate. The concentration of the substrate in the feed is si. The equation for rate of cell growth is r_x = k_1 x and the equation for the rate of substrate consumption is: r_s = k_2 x where k1 and k2 are rate constants with dimensions T^(-1), r_x and r_s have dimensions M L^(-3) T^(-1) , and x is the concentration of cells in the fermenter.
(a) Derive a differential equation for the unsteady-state mass balance of cells.
(b) From this equation, what must be the relationship between F, k_1, and the volume of liquid in the fermenter V at steady state?
(c) Solve the differential equation to obtain an expression for cell concentration in the fermenter as a function of time.
(d) Use the following data to calculate how long it takes for the cell concentration in the fermenter to reach 4.0 g l^(-1) :
F = 2200 l h^(-1)
V = 10,000 l
x_0 = 0.5 g l^(-1)
k_1 = 0.33 h^(-1)
(e) Set up a differential equation for the mass balance of the substrate. Substitute the result for x from (c) to obtain a differential equation in which the only variables are substrate concentration and time. (Do you think you would be able to solve this equation algebraically?)
(f) At steady state, what must be the relationship between s and x?