Answer:
Step-by-step explanation:
A well-mixed fermenter 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 substrate in the feed is si. The equation for the rate of cell growth is: rx = k1 x and the equation for the rate of substrate consumption is: rs = k2 x where k1 and k2 are rate constants with dimensions T-1 , rx and rs have dimensions M L -3T -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, k1, 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 x0 = 0.5 g l-1 k1 = 0.33 h-1
e) Set up a differential equation for the mass balance of 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?
Solutions
Expert Solution
Given: Rate of cell growth is rx=k1x and Rate of substrate consumption is rs=k2x
Assumptions:
It is a well mixed fermenter. For a well mixed fermenter, concentration of cells and substrate at the outlet and inside the fermenter remain same.
Density of the broth remains constant at the inlet and outlet.
Cell lysis is negligible.
where F is the Volumetric flow rate at the feed and product stream
xi is the Concentration of cells in the feed
si is the Concentration of substrate in the feed
V is the Volume of broth in the fermenter
x is the Concentration of cells
s is the Concentration of substrate
(a) The general equation for unsteady state mass balance is
where dM/dt is the Rate of change of mass with time
i is the Mass flowrate of species in inlet stream
0 is the Mass flowrate of species in outlet stream
RG is the Rate of species generated
RC is the Rate of species consumed
In this case, the inlet stream has no cells, i=0. Mass flowrate of cells in product stream, 0=Fx. Rate of cell generation, RG=rxV. The rate of cells consumed, RC=0 as cell lysis is negligible.
The unsteady state mass balance for cell is
where rx is the Rate of cell growth
As flowrate and density of liquid is constant, the volume of liquid in fermenter remains constant.
Divide by V throughout the equation
This equation is the differential form of unsteady state mass balance for cells.