In addition to the transport of substrate S and bacteria X by the flow of the water (see Eqs. 6.60 and 6.61), a transport by dispersion will be considered:
dS d2S UmaxX S 0 = - w (1+nR) —— + Dx -^-o— ^^ (6.76)
The decay rate is neglected here. Two boundary conditions are needed for each balance. Normally, the following conditions are used for a "closed tube" (Danckwerths 1953). They will be written here only for Eq. (6.76) (see Fig. 6.7):
mixing point M
mixing point M
The same conditions for X can be used to solve Eq. (6.77). As Fig. 6.7 shows at x = 0, a discontinuity occurs in S. For very low Dx values, this discontinuity disappears (^ PFR), while for high Dx values, it reaches a maximum (^CSTR). Dispersion in the tube for x < 0 is neglected in this model. Under real conditions, a dS/dx < 0 would be correct at x = L. But for this "boundary condition" Eq. (6.76) cannot be solved. Danckwerths (1953) introduced Eq. (6.79) and expected a very small error at this point.
Equations (6.76) and (6.77) can be rewritten with dimensionless Peclet or Bodenstein numbers:
YX/S
PmaxX
For PeP 1 (high w(1 + nR), high L, low Dx), Eq. (6.81) and Eq. (6.82) come close to the balances for a PFR (Eqs. 6.60 and 6.61), for PeP 1 (low w(1 + nR), low L, high Dx) the ideal CSTR system is nearly reached.
For S P KS, the equations can be solved analytically (Lin 1979). For Monod kinetics, a numerical solution is needed (Fan 1970; Vasilin and Vasilyev 1978) even for the case of constant biomass concentration X.
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