Environmental Engineering Reference
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which can be written in dimensionless form as
ψ
ξ
= r p k g
D eff ψ
ð Þ
1
=Bi m ψ
ð Þ
1
ð
Eq
:
9
:
10
Þ
which defines Bi m , the Biot number for mass transfer. The Biot number is the ratio of
external mass transfer to internal mass transfer by diffusion. It can be considered as a
measure of the thickness of the boundary layer through which the molecules have to
diffuse to reach the particle. In contrast to the Thiele modulus, the Biot number relates
to the condition external to the particle. The Biot number has the same form as the
Sherwood number, which for isolated spheres in a gas flow can be estimated from
the Reynolds and Schmidt numbers as
Sh = 2r p k g
D
6Re 1 = 2 Sc 1 = 3
=2+0
:
ð
Eq
:
9
:
11
Þ
with Re = 2r p u
ν
and Sc = D :
After substitution of A with
, and substituting the
result into the boundary condition at the surface of the particle, the constant A can be
written as
B , deriving the equation for
ψ
Bi m
A =
B =
ð
Eq
:
9
:
12
Þ
ð
Bi m + Th
1
Þ
e Th + Th +1
ð
Bi m
Þ
e Th
The normalized oxygen mass fraction can now be plotted for different values of the
Thiele modulus and the Biot number in order to investigate the behavior of the char
combustion. To better visualize the influence of different parameters, with Equations
(9.2) and (9.11), the Biot number is rewritten as
Bi m = r p k g
D eff = D
Sh
2
= Sh
2
ð
Eq
:
9
:
13
Þ
ε
2
D eff
As theminimumof the Sherwood number Equation (9.11) is 2, the Biot number always
must be larger than 1. For a porous particle, e.g., char, the porosity is often around 0.5 or
higher, which gives a Biot number around 4 or less. The normalized mass fraction of
oxygen, plotted against the dimensionless particle radius, for different Biot number is
shown in Figure 9.3. As can be seen in the figure, the Biot number influences the mass
fraction of oxygen inside the particle. This is an expected result since a high Bi m means
that there is no external diffusion resistance; the rate of mass transfer of oxygen from the
surroundings to the surface of the particle ismuch higher than the rate ofmass transfer of
oxygen into the interior of the particle.
The normalized mass fraction of oxygen, plotted against the dimensionless particle
radius for different Thiele numbers, is shown in Figure 9.4. The result from the var-
iation of the Thiele modulus is more interesting; it visualizes under which conditions
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