Environmental Engineering Reference
In-Depth Information
Introducing normalized quantities for the coordinate and for the mass fraction of
oxygen,
Y
O
2
Y
O
2
,
∞
where r
p
is the particle radius and Y
O
2
,
∞
is the oxygen
mass fraction far outside the boundary layer. After substitution of the equation
for the source term Equation (9.3) and some manipulation,
ξ
= r/r
p
and
ψ
=
the conservation
Equation (9.1) becomes
Y
O
2
,
∞
r
p
1
ξ
1
r
p
∂
∂
ξ
∂
ψ
∂
ξ
2
D
eff
ξ
k
O
2
Y
O
2
,
∞
ψ
:
:
−
=0
ð
Eq
9
4
Þ
2
which can be rewritten as
−
1
ξ
2
∂
∂
ψ
∂
ξ
2
Th
2
ξ
ψ
=0
ð
Eq
:
9
:
5
Þ
∂
ξ
where
s
k
O
2
D
eff
Th
=r
p
ð
Eq
:
9
:
6
Þ
is a dimensionless number, called the
The Thiele modulus tells us
whether the reaction is controlled by internal diffusion or the (intrinsic) reaction rate.
If the Thiele modulus is large (
“
Thiele modulus.
”
1), the conversion of the char particle is controlled
by diffusion, while if the Thiele modulus is small (
1), the conversion is controlled
by chemical reaction. With the introduction of the Thiele modulus, the general solution
ψ
of the equation is
=
1
ξ
Ae
Th
ξ
+
Be
−
Th
ξ
ψ
ð
Eq
:
9
:
7
Þ
The boundary condition in the center of the particle is a symmetry condition, which
means that the first derivative of
ψ
is equal to zero. From this boundary condition, it
follows that
A
=
B
.
The boundary condition at the surface of the particle is that the molar flow of oxygen
through the boundary layer is equal to the diffusion of oxygen from the surroundings
into the particle:
−
r=r
p
D
eff
∂
Y
O
2
∂
−
=
k
g
Y
O
2
−
Y
O
2
,
∞
ð
Eq
:
9
:
8
Þ
r
where
k
g
is the mass transfer coefficient. Equation (9.8) can be written as
r=r
p
Y
O
2
,
∞
r
p
∂
ψ
∂
ξ
=
k
g
Y
O
2
,
∞
ψ
−
Y
O
2
,
∞
ð
Eq
:
9
:
9
Þ
−
D
eff
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