with X corresponding to the biomass concentration (in kg-biomass kgsubstrate –1 ).
The deceleration of growth as X approaches X max could be due to nutrient
limitations, accumulation of inhibitory products, or maximum packing densities based on steric limitations [91] although in practice the value of X max is
usually simply taken directly from experimental biomass profiles. Note that
m max is also typically determined empirically. However, with the assumption
that each fungal hypha branches to give identical daughter hyphae which extend at the same rate as the mother hypha, these macroscopic parameters can
be related to microscopic parameters such as hyphal branching frequencies and
hyphal extension rates [92, 93].
Linear and exponential kinetics can also apply for a significant part of the
growth phase. During an exponential growth phase m remains at m max , whereas
during a linear growth phase the growth rate itself is a constant. In this case
there is a need to impose an external condition to prevent predictions of infinite
growth; for example, that the growth rate equals zero when the biomass concentration reaches a particular value. In any case, linear or exponential growth
profiles rarely persist for the whole of the fermentation time, and it may be
necessary to divide the fermentation time into different phases. For example, to
describe a fungal growth profile with early exponential growth followed by an
extended period during which the growth rates slowly decelerates, the following
empirical equations can be used [94]. First, during the exponential phase:
dX
5 = m max X
(6)
dt
At time t a there is a switch to deceleration phase kinetics, described by
dX
5 = m max L · e –k (t – t a )
(7)
dt
where the factor L describes an instantaneous decrease in the number of
actively extending hyphal tips as the fungus enters the deceleration phase.
During the deceleration phase there is a further first-order decay in the number
of actively extending hyphal tips, with first order rate constant k.
The appropriate form of an empirical equation can only be decided after the
collection of experimental data. This should be done with small substrate
masses so that interparticle heat and mass transfer processes are not limiting.
The integrated form of the growth equation can be fitted against the biomass
fermentation profile by linear or non-linear regression to extract the parameter
values. In doing this, it must be clear whether the biomass is expressed in absolute concentration terms (i.e., kg-biomass m –3 ) or as in relative concentration
terms (i.e., kg-biomass kg-dry-matter –1 ). Since the amount of dry matter
decreases during the fermentation due to the loss of carbon in the form of CO 2 ,
these two expressions cannot simply be converted from one to the other by
multiplying by a constant [95].
Biochemical Engineering Aspects of Solid State Bioprocessing
85
The deceleration of growth as X approaches X max could be due to nutrient
limitations, accumulation of inhibitory products, or maximum packing densities based on steric limitations [91] although in practice the value of X max is
usually simply taken directly from experimental biomass profiles. Note that
m max is also typically determined empirically. However, with the assumption
that each fungal hypha branches to give identical daughter hyphae which extend at the same rate as the mother hypha, these macroscopic parameters can
be related to microscopic parameters such as hyphal branching frequencies and
hyphal extension rates [92, 93].
Linear and exponential kinetics can also apply for a significant part of the
growth phase. During an exponential growth phase m remains at m max , whereas
during a linear growth phase the growth rate itself is a constant. In this case
there is a need to impose an external condition to prevent predictions of infinite
growth; for example, that the growth rate equals zero when the biomass concentration reaches a particular value. In any case, linear or exponential growth
profiles rarely persist for the whole of the fermentation time, and it may be
necessary to divide the fermentation time into different phases. For example, to
describe a fungal growth profile with early exponential growth followed by an
extended period during which the growth rates slowly decelerates, the following
empirical equations can be used [94]. First, during the exponential phase:
dX
5 = m max X
(6)
dt
At time t a there is a switch to deceleration phase kinetics, described by
dX
5 = m max L · e –k (t – t a )
(7)
dt
where the factor L describes an instantaneous decrease in the number of
actively extending hyphal tips as the fungus enters the deceleration phase.
During the deceleration phase there is a further first-order decay in the number
of actively extending hyphal tips, with first order rate constant k.
The appropriate form of an empirical equation can only be decided after the
collection of experimental data. This should be done with small substrate
masses so that interparticle heat and mass transfer processes are not limiting.
The integrated form of the growth equation can be fitted against the biomass
fermentation profile by linear or non-linear regression to extract the parameter
values. In doing this, it must be clear whether the biomass is expressed in absolute concentration terms (i.e., kg-biomass m –3 ) or as in relative concentration
terms (i.e., kg-biomass kg-dry-matter –1 ). Since the amount of dry matter
decreases during the fermentation due to the loss of carbon in the form of CO 2 ,
these two expressions cannot simply be converted from one to the other by
multiplying by a constant [95].
Biochemical Engineering Aspects of Solid State Bioprocessing
85