where a is the recycle ratio equal to the recycle volume divided by the feed volume, C is the concentration factor (cell concentration in the recycle divided by the effluent concentration) related to the efficiency of the separation step, and X0, X1, and X2 are the cell concentrations in the feed, recycle, and separator effluent streams, respectively. Note that the low substrate concentrations in waste treatment create suboptimal growth environments in which cell death cannot be neglected.
Assuming the system is at steady state (dX1/dt = 0) and that the feed is sterile (X0 = 0), equation 44 yields l = (1 +
The dilution rate is no longer equal to the specific growth rate; in fact, because C > 1 and a < 1, the dilution rate is greater than the specific growth rate.
A material balance on the limiting substrate, again neglecting maintenance and product formation, may be written
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