Kinetics of the Steady-State Bacterial
Culture.    I.   Mathematical Model
E. J. MARTIN, Public Health Service Trainee
D. R. WASHINGTON, Associate Professor
Department of Environmental Engineering
Rensselaer Polytechnic Institute
Troy, New York
SUMMARY
A mathematical model for the continuous flow steady-state bacterial culture
is presented which permits the experimental determination of carbon transfer rates
within the system by use of radioactive tracer techniques. The transfer rates are
specific for hydraulic loading rates, feed concentrations, type of organism and the
substrate. The model is established with a view toward elucidating the effect of
changes in dilution rate on the carbon transfer rates.
PRESENT THEORY
The continuous culture of microorganisms, particularly the behavior in the
steady-state, is difficult to treat experimentally since the parameters normally
monitored to observe the functioning of biological systems (e.g., flow rate, substrate concentration, etc.) remain constant with time as long as there are no environmental disturbances. Research has indicated the need for the application
of radioisotope techniques to the continuous culture to study the changes occurring
within the system.
The theory of such systems has been presented in the past and is composed of
equations describing both the organism balance and substrate balance within the
culture vessel (1,2,3).
Mass Balance
The rate of change of mass of organisms in the reactor can be described by
the balance equation (individual terms are rates)
mass change = growth - loss in effluent - death
or
rtM.'
MMa'-DMa,-f(S1, Ma') (1)
dMa'
dt
where Ma' is the total active organism weight in the culture vessel at time, t,
u is the specific growth rate, D is the dilution rate equal to flow rate divided by
volume of the reactor (V) and S-^ is the concentration of organic material in solution in the reactor at time, t.
The function describing the "death" contribution has been discussed by
Moser (4) and Powell (5) and is generally taken to be merely some function of the
mass of organisms present, even though it is probably a function of the concentration of substrate (Sj) as well.   The death rate of the organisms could be ef-
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