where rmax is the maximum reaction rate, βs is the dimensionless substrate surface
concentration ( ) and α 0 , α 1 and α 2 are kinetic constants, the expressions of
which are given in Table 4.2.
Thus substituting equations (5) and (9) in equation (6) yields an expression for the
overall rate of reaction, robs:
(10)where μ is the dimensionless substrate modulus ( ), from which the
dimensionless substrate concentration at the support surface, βS, can be calculated.
The dependence of robs/rmax on βB for different values of μ is shown in Figure 4.1 for
the case of Michaelis-Menten kinetics. Similar plots have been obtained by other authors
(Horvath and Engasser, 1974).
When the kinetic constant α 2 is not equal to zero, equation (10) leads to a third-order
polynomial. In the acceptable domain of 0<βS<βB three different values satisfy equation
(10) at certain values of βB and μ. The possibility of multiple steady states is illustrated
by Figure 4.2 where both the rate of surface reaction and the transport rate of the
substrate are plotted against βS. As the two rates are equal at steady state, the intersections
of the two curves yield the possible substrate surface concentrations that comply with
equation (10). For a Michaelis-Menten kinetics (α 2 =0) there is only one solution in the
acceptable domain, but for certain values of α 2 , μ and βB, there can be up to three
different mathematical solutions for βS. Figure 4.2 shows a situation with three possible
solutions βS1, βS2 and βS3. The stability of the rate at βS1 can be established by considering
a small increase in the surface concentration (Moo-Young and Kobayashi,
Table 4.2 Expressions for reaction rates and
coefficients α 0 , α 1 and α 2 accounting for external or
internal mass transfer effects. When accounting for
external mass transfer effects the transport
parameter is given by and SL=SB and PL=PB.
When accounting for internal mass transfer effects
the transport parameter is given by and
SL=SS and PL=PS. KS and KP are the substrate and
product inhibition constants, respectively.
Michaelis-Menten kinetics:
Substrate inhibition:
Multiphase bioreactor design 96