Chemistry Reference
In-Depth Information
1.3 Reaction with Low Thermal Effect
In this case, considering Eq. (1.2) in combination with inequality Eq. (1.5), one can
assume that for the entire region
exp
.
M
δ
dT
dx
>>
Qk
0
ac
E
R
T
−
Now, to solve Eq. (1.2), one can use the perturbation method on the assumption that
the temperature profile and the pyrolysis rate for a reaction with low thermal effect
are quite similar to those of a thermoneutral reaction (Fig. 1.2). In other words,
T
(
x
)=
T
M
(
x
)+
Δ
T
(
x
)
(1.11)
and
M = M
M
+
Δ
M
,
(1.12)
where M
M
is determined by Eq. (1.10).
When Eqs. (1.11) and (1.12) are substituted into Eqs. (1.2) and (1.7) is subtracted,
the result is
exp
d
2
T
dx
2
Δ
+
M
δ
d
Δ
T
dx
+
Δ
M
δ
dT
M
dx
±
Qk
0
ac
E
R(
T
M
+
Bi
δ
−
−
2
Δ
T
= 0
.
Δ
T
)
This can be rearranged using the approach applied in the case of a thermoneutral
reaction. Taking into account that
T
<<
R
T
S
E
Δ
,
(1.13)
one can obtain the following equation with respect to
Δ
T
:
d
2
T
dx
2
Δ
+
M
δ
d
Δ
T
dx
+
Δ
M
δ
dT
M
dx
1 +
1 + 4Bi
/
M
2
M
2
⎡
⎤
exp
exp
Qk
0
ac
E
R
T
S
E
(
T
S
−
T
∞
)
x
δ
⎣
−
⎦
±
−
M
M
R
T
S
exp
E
T
R
T
S
−
Δ
Bi
δ
×
2
Δ
T
= 0
(1.14)
R
Fig. 1.2
Temperature profiles
inthesamplefora
thermoneutral reaction (
1
), an
exothermic reaction (
2
), an
endothermic reaction (
3
)
x
