There are several mathematical relationships of specific growth rate coefficient to concentration of growth-limiting nutrient, but the Monod equation is by far the most popular. The Monod equation is:
We will first study a process with such good mixing that all fluid elements are assumed to be identical. The analysis starts with the mass balance equation:
rate of change = input - output ± reaction
V dx/dt = 0 + µ x V - F x
If we neglect organisms in the feed stream, the F xo term is not needed. With aseptic operation, the feed is sterile.
Because all fluid elements are assumed to be identical, x in the vessel is the same as x in the overflow. A new term D, the dilution rate, is equal to F/V, thus dividing each term in the equation by V and dropping the zero term gives
µ x = D x
A mass balance for the nutrient in lowest proportions for growth the growth-limiting nutrient (same link as above) ) gives:
where:
Again the equation would benefit from dividing every term by V and substituting D for F/V. An equation is needed to relate µ to S, and we can use the Monod equation or one of the various inhibition equations.
At steady state, there is no change thus the derivatives in the differential equations disappear so that µ = D and
Substituting D for µ in the Monod Equation and solving for S gives:
Solving for x after substituting D for µ gives:
This is an old, familiar analysis for any continuous culture that meets the assumptions of perfect mixing, no organisms in the feed, and constant volume. The equations are fundamental except for the Monod equation that has no time dependency and should be applied with caution to transient states where there may be a time lag as µ responds to changing S.