(8)
Solutions of this equation are generally obtained by numerical methods. The analytical
integration of the equation is however possible in particular situations, such as in the case
of intrinsic first order (low substrate concentrations, Sf substantially lower than Ks) and
zero order reactions (high substrate concentrations, Sf substantially higher than Ks). In
those cases, as indicated in Figure 10.4, Equation (8) will be reduced to:
(9)
or to;
(10)
As shown in Figure 10.4, the reaction rate constants for the two limiting cases are:
(11)
(12)
So far, we have only described the reaction and diffusion phenomena inside the microbial
layer. In the more general case, where the external mass transfer (in the liquid medium)
may also control the consumption rate, the mass transfer resistance in the liquid film next
to the biofilm surface should be incorporated in the model. In steady state conditions, the
rate of external mass transfer will be equal to the overall rate of diffusion and reaction in
the biofilm. Therefore, if Je is the external mass transfer rate of substrate in the liquid:
(13)
Multiphase bioreactor design 304