Environmental Engineering Reference
In-Depth Information
The boundary condition for this equation is that it should coincide with (5.29) far
from the maximum
Z
.Inthisregiononecanignorethethirdtermin(5.28),which
is the approximation that leads to (5.29). This is valid under the condition
α
(
T
T
0
)
1 .
(5.32)
Now we seek the solution of (5.28) at
T
>
T
. Near the maximum of
Z
(
T
),
Z
is
given by (5.31), so (5.28) has the form
dZ
dT
D
Z
max
T
f
1
exp[
α
(
T
T
)]
g
.
T
0
This equation is valid in the region where the second term in (5.31) is significantly
smaller than the first one. According to condition (5.32), this condition is fulfilled
at temperatures where exp[
1. Therefore, because of the exponential
dependence of the second term in (5.28), there is a temperature region where the
second term in (5.28) gives a scant contribution to
Z
, but nevertheless determines
its derivative. This property will be used below. On the basis of the above infor-
mation and the dependence given in (5.26), we find that the solutions of (5.28) at
T
α
(
T
T
)]
>
T
are
s
2
f
(
T
m
)
c
p
N
p
1
Z
D
exp[
α
(
T
T
m
)] ,
α
s
2
f
(
T
m
)
c
p
N
dZ
dT
D
exp[
α
(
T
T
m
)]
p
1
.
α
exp[
α
(
T
T
m
)]
Comparing the expressions for
dZ
/
dT
near the maximum, one can see that
they can be connected if condition (5.32) is satisfied. Then solution (5.30) is valid
in the temperature region
T
T
m
up to temperatures near the maximum
of
Z
. Connecting values of
dZ
/
dT
in regions where
<
T
α
(
T
T
)
1andwhere
α
(
T
m
T
)
1, we obtain
s
2
f
(
T
m
)
c
p
N
Z
max
D
(
T
T
m
)exp[
α
(
T
T
m
)] .
α
Next, from (5.31) it follows that
Z
max
f
(
T
)/(
uc
p
N
). Comparing these
expressions, one can find the temperature
T
corresponding to the maximum of
Z
(
T
) and hence to the velocity of the thermal wave. We obtain
D
Z
(
T
)
D
α
(
T
T
0
)
D
exp[
α
(
T
m
T
)] ,
(5.33)
2
s
2
T
m
T
m
f
(
T
m
)
c
p
NE
a
u
D
.
(5.34)
T
0
. Relation (5.33) together with condition (5.32) gives
T
m
T
We used (5.26) for
α
T
T
0
. This was taken into account in (5.32).
