Chemistry Reference
In-Depth Information
Fig. 1.1
To the problem
solution (see the text for the
explanations:
a
-fora
semi-infinite plate,
b
-fora
semi-infinite cylinder
b
situation where the temperature gradient in the transverse direction is much lower
than that in the longitudinal direction is considered (
T
y
<<
T
x
in the sample; in
other words, when the thermal resistance is concentrated outside the sample near
its lateral surface, and the Bio number, which is characteristic of the heat exchange,
Bi =
αδ
/
λ
<<
1, where
α
is the coefficient of outward heat exchange,
δ
is the
characteristic size of the sample, and
λ
is the thermal conductivity coefficient of the
sample).
At Bi
>
1, the inaccuracy caused by replacing
T
=
T
(
x
,
y
) with
T
=
T
(
x
) can
be eliminated by introducing an averaged temperature,
T
(according to the Frank-
Kamenetsky method [8]).
In this case,
T
should be replaced with
T
in the resulting solution presented
below, which does not change the qualitative character of the obtained relationships.
For a one-dimensional problem, the basic equation is
exp
d
2
T
dx
2
+
M
δ
dT
dx
±
Qk
0
ac
E
R
T
Bi
δ
−
−
2
(
T
−
T
∞
)=0
(1.2)
with the boundary conditions
T
|
x
=0
=
T
S
,
(1.3)
T
|
x
→
∞
=
T
∞
,
(1.4)
where
is the half-thickness of the plate (or half-radius of the cylinder), the
Michelson number M =
U
δ
/
a
, and Bi is the Bio number.
Let us consider the following cases:
δ
-
Thermoneutral reaction
Q
= 0
-
Reaction with low thermal effect
Q
c
(
T
S
−
<<
1
(1.5)
T
∞
)
-
Reaction with high thermal effect
Q
c
(
T
S
−
>>
1
(1.6)
T
∞
)
