MODELING CHAR OXIDATION AS A FUNCTION OF PRESSURE ...
MODELING CHAR OXIDATION AS A FUNCTION OF PRESSURE ...
MODELING CHAR OXIDATION AS A FUNCTION OF PRESSURE ...
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= tanh(M T )<br />
M T<br />
= 1<br />
M T<br />
MT = L O ′ ′ r<br />
2<br />
⎛ 1<br />
⎝ tanh(3MT ) ⎜<br />
( )<br />
in C s<br />
in Cartesian coordinates (2.26)<br />
1 ⎞<br />
− ⎟ in spherical coordinates (2.27)<br />
3MT ⎠<br />
C s<br />
[ ∫ De ( C)<br />
O r in′ ′ ( C)dC<br />
0<br />
]<br />
17<br />
− 1<br />
2<br />
(2.28)<br />
where L is the characteristic length of the particle (Aris, 1957), which is equal to V p/S g, C<br />
is the local concentration of oxidizer in the particle, r in ′ is the intrinsic molar reaction rate<br />
in any form, o is the stoichiometric coefficient of oxygen for each mole of carbon<br />
consumed, which converts the carbon consumption rate into oxygen consumption rate,<br />
and D e is the effective diffusivity, which can be a function of oxygen concentration, but is<br />
assumed to be spatially uniform (but still allowed to vary temporally) in this study for<br />
simplicity. In particular, the general modulus for the m-th order rate equation (Eq. 2.23)<br />
becomes (Bischoff, 1965):<br />
M T = L<br />
(m + 1)<br />
2<br />
okmC m −1<br />
s<br />
De (2.29)<br />
This general modulus has been widely used in the chemical engineering literature<br />
(Laurendeau, 1978; Hill, 1977; Bischoff, 1965; Aris, 1975; Carberry, 1976; Fogler, 1992;<br />
Froment and Bischoff, 1979; Levenspiel, 1993; Levenspiel, 1999; Mehta and Aris, 1971).<br />
For the Langmuir rate equation, a general modulus was obtained in this study by<br />
substituting Eq. 2.17 into Eq. 2.28:<br />
M T = L<br />
o k 1<br />
KC s<br />
2De 1 + KCs 1<br />
−<br />
2 [KCs − ln(1 + KCs )] . (2.30)