MODELING CHAR OXIDATION AS A FUNCTION OF PRESSURE ...

d 2 C 2 dC
2 +
dr r dr
O ′ ′ − r in (C)
= 0 (4.4)
De where r in ′ is the molar carbon consumption rate per unit particle volume as a function of
C, D e is the effective diffusivity, O is the stoichiometric coefficient of oxygen for each
mole of carbon consumed in the reaction, C is the local oxygen concentration as a function
of r, and r is the radial distance from the origin. The boundary conditions are
and
C = C s, at r = r s
dC
dr
30
(4.5)
= 0 , at r = 0. (4.6)
Substituting the Langmuir rate equation into Eq. (4.4) and normalizing the
resulting equation lead to:
d 2
2 d
2 +
d d − r 2
s
k1
De 1+ KCs = 0 (4.7)
where = C/C s , and = r/r s. If an intermediate modulus is defined as:
M 1 = r s
3
k 1
D e
Eq. (4.7) can be re-written as:
d 2
2 d
2 +
d d − 9M 2
1
1 + KCs . (4.8)
= 0 . (4.9)
By using similar techniques of Patankar (1980) and central differences for first and second
order derivatives, Eq. (4.9) is discretized to:
where
a P i = a E i + 1 + a W i −1 + b , i = 2, 3, 4, …, N (4.10)
a E =
a W =
1 1
+
(i +1) − (i) (i)
1 1
−
(i) − (i −1) (i)
, (4.11a)
, (4.11b)
a P = a W + a E , (4.11c)