## Info

(16.4)

k, A

a The integrated form of these equations are given in Table 14.1. These equations are expressed in terms of absolute biomass concentration (e.g., g-dry-biomass g-IDS-1). b These parameters may later be expressed as functions of the environmental conditions.

a The integrated form of these equations are given in Table 14.1. These equations are expressed in terms of absolute biomass concentration (e.g., g-dry-biomass g-IDS-1). b These parameters may later be expressed as functions of the environmental conditions.

a relative concentration to an absolute basis, it would be necessary to know to what initial dry weight of substrate the removed sample corresponded. To do this it would be necessary to weigh the whole bioreactor contents and determine the moisture content of the bed just before each sampling time. It is not a simple matter to weigh the whole bioreactor, especially at large scale. It is easier to use the kinetic sub-model to predict the relative biomass concentration.

Such a conversion can be done in the following manner. If the total dry weight of solids in the bioreactor (D, kg) is given as:

where X is the total dry weight of biomass (kg) and S the total dry weight of residual substrate (kg), then for the absolute amount of biomass in the bioreactor (X, kg) we have:

dX = d(CxAD0) = D dCxA(166) dt dt o dt while for the "relative concentration" we have:

dt dt dt dt

Equation (16.6) can be substituted into Eq. (16.7) in order to eliminate the term dX/dt. The resulting equation can be rearranged to be explicit in dCXR/dt:

Equation (16.8) says that the change in relative concentration (kg-dry-biomass kg-dry-solids-1) during growth occurs due to growth itself in absolute terms, as described by the first term on the right-hand side, and due to the decrease in dry solids that occurs during growth, as described by the second term on the right-hand side. Growth leads to an overall loss of dry solids, and therefore dD/dt will be negative; given that this term is subtracted, its effect is to increase the relative concentration.

The rate of change in the total dry weight of solids is the sum of the rates of change in dry biomass and residual dry substrate:

dD dX dS

dt dt dt

The rate of consumption of the residual dry substrate is related to the rate of growth by the following equation:

dS 1 dX

where YXS is the true growth yield (kg-dry-biomass kg-dry-substrate-) and mS is the maintenance coefficient (kg-dry-substrate kg-dry-biomass-1 h-1).

Substituting Eq. (16.10) into Eq. (16.9) and using the distributive law to separate out dX/dt on the right hand side gives:

dD dt

0 0