Environmental Engineering Reference
In-Depth Information
comb
ustion reaction proceeds in a turbulent flow where eddies of fuel and air meet
and mix with each other down to the molecular level. Since reaction occurs only at
the interface of eddies of fuel and air, the gas inside the eddies is not involved in
the reaction. Therefore the combustion rate is proportional to the dissipation rate of
turbulent energy because interfaces where fuel and air come into contact appear one
after another during the process of dissipation of turbulence, namely, the process of
large eddies collapsing into small ones. When the
model is adopted as a turbu-
lence model, the dissipation rate of turbulence is proportional to
k-
ε
. When the
similarity of the concept is applied to species concentration, which is a scalar
quantity, the dissipation of turbulent eddies is considered to correspond to the decay
of concentration fluctuation. Regarding a scalar quantity
ε
/
k
φ
, its fluctuation intensity
(secondary moment around a mean value
˜
g
φ
) is defined as follows:
(
)
2
φ
˜
g
=−
φ
(3.3)
The governing equation for
g
can be obtained by the analogy of turbulence
energy,
, from the Navier-Stokes equation. In the first
eddy-breakup model originally proposed by Spalding, the time-averaged reaction
rate was modeled to be proportional to the product of turbulence dissipation rate
and intensity of concentration fluctuations as given below:
k
, and its dissipation rate,
ε
1
(
)
12
R
=−
C
ρ ε
k g
(3.4)
fu,EBU
EBU
where
ρ
is density and
C
is an empirical constant. In the eddy dissipation model
EB
U
of Magnussen and Hjertager,
ve
model, concentration itself is used instead of its fluctuation intensity. They consid-
ered that combustion occurs only when burned product is involved as a heat source
in addition to fuel and oxygen, and thus the combustion reaction rate is regulated
by the smallest amount among the three. Then the reaction rate is expressed as
follows:
which is also based on the same thinking as the abo
2
[
]
(
)
(
)
˜
˜
˜
R
=−
ρ ε
k
min
Am
,
Am
i A m
,
1+
i
(3.5)
fu
fu
ox
pr
= 2, both being empirical
constants. The symbol min[ ] means that the smallest values between the brackets
is taken.
It is a common practice to use the smaller value of the combustion reaction rates
given by Equations 3.1 and 3.5, considering that the value given by Equation 3.5 is
based on a mixing controlled rate and, as such, cannot exceed the value based on a
homogeneous mixture.
A problem with this model is that the reaction rate is largely determined by the
state of turbulence, and hence the influence of chemical reactions does not appear
explicitly. If concentrations and state of turbulence are the same at different points
where
i
is stoichiometric oxygen/fuel ratio and
A
= 4 and
A
Search WWH ::
Custom Search