Mass Transfer
461
It should be stressed that one cannot use the molecular diffusivity directly in Eq. (10.41) for two
reasons; a) Diffusion only takes place in the void volume of the pellet, and b) The length needed for
transport is increased due to the twisted path of the pores relative to the radial coordinate. This is
called
tortuosity.
Both these factors tend to give an effective diffusivity significantly lower than the
molecular diffusivity.
The void volume of the pellet can be expressed as:
(10.42)
The void volume is a function of the biomass concentration in the pellet, and of the matrix material
used to immobilize the cell system. Thus if the biomass grows within the pellet, it takes up an
increasing fraction of the void volume, resulting in a decreasing mass transfer. The effect of a
longer diffusion pathlength can schematically be described by an overall tortousity factor,
zl,
defined as the ratio between the length of the straight path and the twisted pores. The tortuosity
factor is usually in the range 1.5-5. The effective diffusivity is thus obtained as:
(10.43)
The effective diffusivity in a pellet is therefore lower than the free diffusivity by at least a factor of
2
.
In the case of large diffusing molecules, such as proteins, the pores may have diameters of the same
order of magnitude as that of the diffusing species. In such a case, so-called hindered diffusion
takes place and further corrections are needed in calculating the effective diffusivity (see e.g.
Blanch and Clark, 1997, for further discussion).
At steady-state conditions, we have
D A<,rV2s A + qA =
0
(10.44)
Eq. 10.44 is in general difficult to solve analytically, but can rather easily be solved numerically
provided of course, that an adequate expression for
qA
is known. In such a case the concentration
profile of the substrate in the pellet can be found. The analytical solutions for the simple cases of
zeroth and first order kinetics are presented in Note 10.7. These cases are of interest, since the
Monod kinetic expression gives an effective reaction order between zero and one.
Note 10.7. Finding the concentration profile for a zeroth and a first order reaction.
For very simple kinetic expressions and highly symmetric geometries, it is possible to find an analytical
solution to Eq. (10.44). We will here consider the simple case of a spherical particle, with radius
Rp
for
which Eq. (10.44) can be written:
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