The equations described above have one limitation in common; none can account for microbial growth. As microorganisms consume a substrate, one portion is used for maintenance energy requirements and if enough substrate is available, the remainder will be used to support growth. As microorganisms grow and multiply they will exert an increasing demand on the remaining substrate, thereby changing the kinetics of decomposition. Three equations can be used to describe micro-bial growth: the exponential, logistic, and Monod equations.
When rapidly consuming substrate, microbes are known to grow exponentially, dN
dt where N is the number (biomass) of cells and p is the growth rate. Changes in biomass can be equated to changes in substrate by dividing the biomass by the yield (y), which is the mass of cells generated per mass of substrate consumed:
dt dt y
Substituting Eq. (11) for dN/dt into Eq. (12), we obtain the following equation in terms of substrate depletion, dS Nt
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