where Ds is the substrate diffusivity; ε is the void fraction in the porous matrix; and τ is a
tortuosity factor that takes into account the pore geometry and by definition is larger than
unity.
Considering, again, the reaction S → vP, at steady state behaviour, no accumulation of
substrate or product within the support occurs. Thus,
(18)Rearranging this equation and defining the transport parameter as results in an
expression for the local product concentration:
(19)
which can be substituted in the expression for r(S, P) in equation (13) to obtain an
equation in terms of S only.
As for the case of external mass transfer limitation, the rate of Michaelis-Menten type
reactions can be defined as:
(20)
where β is the dimensionless substrate concentration (β=S/Km) and α 0 , α 1 and α 2 are
kinetic constants, the expressions of which are given in Table 4.2.
The solution of equation (16) gives the concentration profile of the substrate, which
allows the calculation of the overall reaction rate within the immobilisation support. The
analytical solutions of these equations are easily obtained for first-order or zero order
reactions, but numerical solutions are required for Michaelis-Menten type reactions. The
above equations, in these cases, are usually rewritten in terms of dimensionless variables:
(21)In this equation z is the dimensionless position in the porous support, given by z=x/L
where L is the characteristic length of the support particle (the radius of a spherical or a
cylindrical pellet or the thickness or half-thickness of a rectangular membrane for
asymmetric and symmetric boundary conditions, respectively). is the Thiele modulus for
the substrate defined by:
(22)Most authors use different definitions for the Thiele modulus depending on the geometry
of the support leading to some confusion when interpreting and using the graphs thereby
derived (Aris, 1957; Moo-Young and Kobayashi, 1972; Smith, 1981). For generalisation
and clarification purposes we chose to use the definition above regardless the geometry
of the support.
For reversible Michaelis-Menten kinetics, equation (21) becomes:
Multiphase bioreactor design 104