X = cell mass
S = concentration of nutrient that is growth-limiting
r is the recycle ratio
F = flow rate of fresh feed
µ is the specific growth rate coefficient
The mass balance for cell mass for the bioreactor is:
Cells in - Cells out - Cells recycled + Cell growth = rate of accumulation
0 - ( 1 + r ) X F r F XR F µ X V V dX/dt
For industrial fermentations, the fresh feed has been sterilized and is devoid of living organisms. For biological waste treatment, the types of organisms in the sewage are different enough from those that thrive in the bioreactor that their contribution is negligible.
Note that the recycle flow rate is simply r times F. Adding this to the fresh feed rate of F give (1 + r ) times F for the flow rate out of the bioreactor. There are two different X's in the equation, but the XR in the recycle stream is the concentration factor, C, times the concentration X that enters the centrifuge. We will replace XR with C times X.
The next step is to divide each term by V while remembering that the dilution rate, D, is defined as F/V. Now we get:
0 - ( 1 + r ) X D + r F X C D + µ X = dX/dtThis is a useful equation, but we are also interested in the steady state where the dX/dt term is set equal to zero. Let's multiply and collect terms:
- X D - r X D + r F X C D + µ X = 0
Now we solve for µ to get:
µ = ( 1 + r - r C ) D
This is interesting because µ = D at steady state with no recycle.
With the use of the mass balance for limiting nutrient and the Monod equation, steady state solutions for cell mass and nutrient concentration as functions of dilution rate can be developed. These will be added to this page soon. In the meantime, go to the exercise that graphs these relationships.