Chemistry Reference
In-Depth Information
This gives
⎛
⎝
⎞
⎠
1
/
2
k
0
R
T
S
δ
2
exp(
−
E
/
R
T
S
)
aE
(
T
S
−
T
∞
)
1 +
1 + 4Bi
/
M
2
/
2
Q
/
2
c
M =
.
(1.31)
∓
This is the most general expression. Relationship Eq. (1.22), when obtained using
a perturbation method, gives an identical dependence for the linear decomposition
rate. The difference in the coefficients before term
Q
/
c
is explained by the double
series expansion of the exponential term.
The resulting Eq. (1.31) can be obtained in a shorter (though less rigorous) way
if the heat losses due to the warming-up of the sample and heat emission from the
lateral surface of the sample in the first zone are neglected. In this case, the thermal
conductivity equation (Eq. 1.23) for the first zone is
exp
= 0
,
d
2
T
dx
2
±
Qk
0
ac
E
R
T
−
while for the second zone it does not change. After mathematical manipulations sim-
ilar to those performed for Eqs. (1.24)-(1.30), and taking into account the equality
of the heat flows at the boundary between the two zones, one obtains
1 +
1 + 4Bi
/
M
2
2
1 +
1 + 4Bi
/
M
2
2
⎡
⎤
⎡
⎤
2
2
M
2
δ
M
δ
Q
c
⎣
⎦
⎣
(
T
S
−
⎦
(
T
S
−
T
∞
)
=
T
∞
)
∓
2
exp
.
2
Qk
0
R
T
S
acE
E
R
T
S
±
−
Equation (1.31) also represents the solution to this equation.
1.5 Analysis of the Basic Expression
for the Linear Pyrolysis Rate
Solution of Eq. (1.31) with respect to M results in a very bulky expression. It is more
convenient to analyze the physical meaning of relationship (1.31) for two limiting
cases:
1. High rates of linear decomposition (burning) and insignificant heat losses from
the lateral surface; that is
4Bi
M
2
<<
1
,
(1.32)
