The rate of digestion in the complex stomach can be modeled as either an enzyme catalyzed digestion or an autocatalytic microbial fermentation. In autocatalytic digestion, microbes break down the feed, and in the process produces more microbes along with a digestion product P.
or more precisely
The rate law for the rate of disappearance of feed in the stomach is of the following form. [See Chapter 7 and Summary Notes]1
Here CA is the concentration of hippo feed in (kg/m3) , CB is the concentration of microbes (kg microbes/ m3 of stomach), and KM is the Monod constant (kg/m3). We note that -rAM1 is in a basis of mass per unit time per unit volume rather than moles per unit time per unit volume, which is the basis for -rA. Letting X be the conversion of feed in the stomach, in kg converted per kg feed we can arrive at X.
（The derivation of -rAM1 as a function of conversion is shown below：）
Where, v0 is the entering volumetric flow rate (m3/kg), mAO is the entering mass flow rate (kg/day), CAO is the entering concentration of hippo feed in (kg/m3), ΘB is the ratio of microbes to hippo feed, YM is the yield coefficient (dimensionless) and KM is the Monod constant (kg/m3). Substituting for the concentrations CA and CB:
This function can then be written as:
Where a, b, c and e are all positive constants.
mAO ：Normally flow rates are given in a molar basis. However, in this case it is easier to use a mass flow rate because referring to a hippo feed rate in terms of moles is impractical. The justification for the substitution of molar for mass flow rate follows:
dFA=rA*dV
where FA is the molar flow rate of A in Moles of A per unit time(e.g. mol/s). rA is the rate of formation in Moles of A formed per unit time per unit volume (e.g. mol/dm3/s).
Multiplying by the molecular weight of A: MWA*dFA=d(MWA*FA)=(rA*MWA)*dV
Then dmA=rMA*dV
where，mA is the mass flow rate of A in mass flow rate of A, mass per unit time.(e.g., kg/s); rMA is the mass rate of formation of A, mass of A per unit time per unit volume (e.g., kg/dm3/s).
thus, X= moles of A reacted per mole of a feed = mass of A reacted per mass or A fed.
Proof X=1-FA/FAO
Multiply numerator and denominator by the molecular weight of A: X=1-FA/FAO=1-MWA*FA/(FAO*MWA)=1-mA/mAO
The derivation of -rAM1 as a function of conversion is shown below:
Where, v0 is the entering volumetric flow rate (m3/kg), mAO is the entering mass flow rate (kg/day), CAO is the entering concentration of hippo feed in (kg/m3), ΘB is the ratio of microbes to hippo feed, YM is the yield coefficient (dimensionless) and KM is the Monod constant (kg/m3). Substituting for the concentrations CA and CB:
This function can then be written as: