468
Chapter 10
The volumetric substrate consumption rate (-qs) is shown in Fig. 10.12. As could be expected from the
concentration profile, the rate is not constant throughout the pellet, and the effectiveness factor is clearly
smaller than one. The effectiveness factor is calculated numerically from Eq. (4) and Eq. (5).
Qs ,obs
4«R j/3
(4)
7hff =
Qs.abs
Qs(S s)
(5)
The numerically obtained value is
= 0.63.
Alternatively, an approximate solution may be derived using the generalized Thiele modulus (Eq. 10.55). In
this case, we get
3(1 +
a)^2De^ssk(i - a
ln (l + 1
l a))
or with inserted values
4 > ^ .= l-2 9
(7)
The approximate effectiveness factor calculated by Eq. (10.53) is 0.66, which is a satisfactory
approximation to the value obtained by numerically solving the complete equations.____________________
Until now we have considered external and internal mass transport separately. However, in general
the substrate must first traverse the external film or boundary layer and subsequently diffuse into
the pellet where the reaction occurs. The mass balance of Eq. (10.44) of course still holds, but the
outer boundary condition is now written
- D
= * , ( *
= R
p ) - S
a ,i )
(10.56)
This boundary condition (sometimes called a Neumann boundary condition) basically states that
there can be no accumulation of substrate at the pellet interface.
The
overall effectiveness factor, r}Iot,
is defined as the observed reaction rate divided by the reaction
rate that would occur at bulk liquid condition e.g.
n,ol
Q A ,o b s
Q Asaj)
(10.57)
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