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:

Assume that the reactor is well mixed that all fluid elements are assumed to be identical. The analysis starts with the mass balance equation:

Rate of Change = Input - Output ± Reaction:

  where x = organism concentration in g/L at time t
  V =the constant volume of the vessel
  F = feed rate
  µ = specific growth rate coefficient
  x_i=organism concentration in the feed

Always there is no organisms in the feed stream, then the equation becomes:

Where D is the dilution rate, difined as F/V.

A mass balance for the the growth-limiting nutrient  gives:

  M = maintenance coefficient for endogenous metabolism
  Y_x/s = yield coefficient, mass of cells per mass of nutrient consumed

At steady state, there is no change in the concentration of the nutrient, and u=D, so:

Solving for x after substituting D for µ gives:

If there is product, the mass balance of the product is:

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.

Estimating Monod Model parameters

Estimating Yield Coefficients
 

Last modify: 05/02/2000, by Xuezhen Kang