Environmental Engineering Reference
In-Depth Information
Formula (5.34) corresponds to the dependence (5.26) for the rate of heat re-
lease near the maximum. In a typical case at
T
D
T
m
all the “fuel” is used, and
0. Then all the above arguments are valid, because the primary portion
of the heat release takes place at temperatures
T
,where
f
(
T
m
)
D
α
(
T
m
T
)
1, that is,
where
1. Then, using the new form of the function
f
(
T
)nearthe
maximum of
Z
(
T
), we transform (5.34) for the velocity of the thermal wave into
the form
α
(
T
T
)
t
Z
T
m
1
c
p
N
2
u
D
f
(
T
)
dT
.
(5.35)
T
m
T
0
T
This formula is called the Zeldovich formula.
We can analyze the problem from another standpoint. We take an expression for
Z
(
T
) such that, in the appropriate limits, it would agree with (5.29) and (5.30). The
simplest expression of this type has the form
u
Z
D
(
T
T
0
)
f
1
exp[
α
(
T
m
T
)]
g
.
If we insert this into (5.28),
f
(
T
)isgivenby
u
2
T
0
)
p
1
f
(
T
)
c
p
N
D
2
d
dT
Z
2
uZ
D
(
T
exp
α
(
T
m
T
)
h
1
T
))
i
p
1
u
2
2
C
α
T
0
)
2
exp[
exp(
α
(
T
m
(
T
α
(
T
m
T
)] .
In the region
1, the first term is small compared with the second one
and one can ignore it. Then the comparison of this expression with that in (5.26) in
the temperature region
α
(
T
T
0
)
(
T
m
T
)
1and
T
m
T
T
m
T
0
gives the velocity
α
of the thermal wave as
s
2
T
m
T
m
E
a
f
(
T
m
)
c
p
N
u
D
.
T
0
E
a
/
T
m
. This equation agrees exactly with (5.34) because of the identical
assumptions used for construction of the solution in both cases.
Onecanusethismethodforthealternativecasewhenthefunction
f
(
T
) has an
exponential dependence far from
T
m
,andgoestozeroat
T
where
α
D
D
T
m
.Forexample,
we take the approximate dependence
f
(
T
)
D
A
(
T
m
T
)exp[
α
(
T
m
T
)] .
An approximate solution of (5.28) constructed on the basis of (5.29) and (5.30) has
the form
T
0
)
q
1
u
Z
D
(
T
e
α
(
T
m
T
)
[
α
(
T
m
T
)
C
1] .
