Chemistry Reference
In-Depth Information
1.2 Thermoneutral Reaction
In this case, the thermal conductivity equation is
d
2
T
dx
2
+
M
δ
dT
dx
−
Bi
δ
2
(
T
−
T
∞
)=0
(1.7)
with the boundary conditions (1.3) and (1.4).
The solution for this equation is
T
∞
) exp
1 +
1 +
4Bi
M
2
x
δ
.
M
2
T
=
T
∞
+(
T
S
−
−
(1.8)
The temperature distribution in the sample described by Eq. (1.8) is similar to the
Michelson temperature distribution [11]
T
∞
) exp
.
xU
a
T
=
T
∞
+(
T
S
−
−
The difference in the exponents is associated with the effect of heat emission from
the sample's lateral surface. The temperature determined by Eq. (1.8) is denoted as
T
M
below.
In the case of a zero-order reaction, the rate of linear pyrolysis can be estimated
by using the following equation [9]:
exp
dx
.
∞
E
R
T
(
U
,
x
)
U
=
k
0
−
(1.9)
0
To calculate the integral, one can use the expansion of the exponential term ac-
cording to the Frank-Kamenetsky method [8] and the power series expansion of
T
=
T
(
x
). This simplification, which is valid for small
x
, will be applied to solve
this problem using other methods. Far from the limit
x
= 0, the kinetic function
k
0
exp(
E
/
R
T
) is close to zero. Thus, these simplifications do not cause significant
errors in the final expression for the integral Eq. (1.9).
Then, from Eq. (1.9),
−
R
T
S
exp(
2
k
0
δ
−
E
/
R
T
S
)
T
∞
)M
1 +
1 + 4Bi
/
M
2
U
=
E
(
T
S
−
and
2
R
T
S
exp(
E
/
R
T
S
)
2
k
0
δ
T
∞
)
1 +
1 + 4Bi
/
M
2
.
M =
(1.10)
aE
(
T
S
−
