Mass Transfer
465
71 -ff
o (ta n h (3 0 )
3 0 )
t10
Like many other dimensionless parameters, also the Thiele modulus has a physical significance.
We see that
O
2
=
Rlk
i
.
?
^ O
3
Rp^lSA,s
3
d a
(10.52)
i.e. ^ can (loosely) be interpreted as the ratio between the reaction rate at the surface and a
maximum “diffusion rate”. A high value of the Thiele modulus therefore suggests a fast reaction
compared to the diffusional transport, and the effectiveness factor will in such a case be low. For
other geometries, e.g. a slab-formed solid of half-thickness
L,
it can be shown that
tanhO
(10.53)
where the Thiele modulus is defined by
0
=
1
D
A,e
ff
(10.54)
As pointed out previously, it is for most kinetic expressions not possible to find an analytical
expression for the concentration profile or the effectiveness factor. Instead Eq. (10.44) must be
solved by numerical methods (e.g. by finite element, finite difference or collocation methods, see
Villadsen and Michelsen, 1978) to give the concentration gradient in the pellet, and from that
calculate the observed reaction rate by a quadrature. Aris (1975) demonstrated that Eq. (10.53)
gives a satisfactory approximation for the effectiveness factor for any reasonable kinetics and any
particle shape if the characteristic length A is defined as the ratio between pellet volume and pellet
exterior surface area, V/A, and provided that the following
generalized Thiele modulus
is used
m
~<1a(sa = sa,s)a
j
~
(10.55)
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