Updated 23JAN2003 to include additional discussion and references.
Updated 7JUN2000 to correct errors in curve fits for CO_{2} and
H_{2}O.
Updated 5FEB1998 to include information on CH_{4} and CO, as well
as CO_{2} and H_{2}O.
Most of the current TNF target flames could not be described as
strongly radiating flames. However, thermal radiation from all
but the most highly diluted hydrogen jet flames reduces the local
temperatures sufficiently to affect the production rate of NO.
Detailed treatment of radiative transfer within a turbulent flame
is computationally very expensive. In order to include the effect
of radiation in turbulent combustion models, without significantly
increasing computational expense, a highly simplified treatment
of radiative heat loss is needed. Because the primary focus of
the TNF Workshop has been on the modeling of turbulencechemistry
interaction, we have agreed adopted an optically thin radiation
model, with radiative properties based on the RADCAL model by Grosshandler
of NIST [1].
The characteristics of some flames selected for the workshop
are such that a model based upon the assumption of optically thin
radiative heat loss should yield reasonable accuracy. This
has
been demonstrated for the simple hydrogen jet flames [2]. However,
there is some evidence that the optically thin model significantly
over predicts radiative losses from the CH4 flames in the TNF
library, due to strong absorption by the 4.3micron band of
CO_{2} [35].
Radiation mainly affects the NO predictions in the TNF target
flames. In general, one can expect an adiabatic flame calculation
to over predict NO levels (if all other submodels are correct),
while an optically thin model is expected to under predict NO levels.
The answer corresponding to a detailed radiation model should be
somewhere between these limits. At present, the majority of modelers
in the TNF Workshop are satisfied with this limitation of the present
radiation model. Further discussion of radiation in the TNF hydrocarbon
flames is included in the TNF5 and TNF6 Proceedings.
The model described here is also documented in [6], which may
be used as reference.
Under the assumptions that these flames are optically thin, such
that each radiating point source has an unimpeded isotropic view
of the cold surroundings, the radiative loss rate per unit volume
may be calculated as:
Q(T,species) = 4 SUM{p_{i}*a_{p,i}}
*(T^{4}Tb^{4})
where
 sigma=5.669e08 W/m^{2}K^{4} is the SteffanBoltzmann constant,
 SUM{ } represents a summation over the species included in
the radiation calculation,
 p_{i} is the partial pressure of species i in atmospheres (mole
fraction times local pressure),
 a_{p,i} is the Planck mean
absorption coefficient of species i,
 T is the local flame temperature (K), and
 Tb is the background temperature (300K unless otherwise
specified in the experimental data).
Note that the T_{b} term is not consistent with an emissiononly
model. It is included here to eliminate the unphysical possibility
of calculated temperatures in the coflowing air dropping below
the ambient temperature. In practice, this term has a negligible
effect on results.
CO_{2} and H_{2}O are the most important radiating species for hydrocarbon
flames. As an example, Jay Gore reports that the peak temperature
in a strained laminar flame calculation decreased by 50K when radiation
by CO_{2} and H_{2}O was included. Inclusion of CH_{4} and CO dropped the
peak temperature by another 5K. On the rich side of the same flames
the maximum effect of adding the CH_{4} and CO radiation was an 8K
reduction in temperature (from 1280 K to 1272 K for a particular
location). These were OPPDIF calculations (modified to include
radiation) of methane/air flames with 1.5 cm separation between
nozzles and 8 cm/s fuel and air velocities.
In the context of the expected accuracy of the optically thin
model, it would appear that the inclusion of CH_{4} and CO radiation
is not essential for calculations of methane flames. Radiation
from CO may be important for CO/H_{2} flames and methanol flames.
We do not see a need to repeat expensive calculations to include
this detail. However, we suggest that all four species be included
in future calculations.
Curve fits for the Planck mean absorption coefficients for H_{2}O,
CO_{2}, CH_{4}, and CO are given below as functions of temperature. These
are fits to results from the RADCAL program [1].
Curve Fits: The following expression should be used to calculate
a_{p} for H_{2}O, and CO_{2} in units of (m1atm1). These curve fits were
generated for temperatures between 300K and 2500K and may be very
inaccurate outside this range.
a_{p} = c0 + c1*(1000/T) + c2*(1000/T)^{2} + c3*(1000/T)^{3} + c4*(1000/T)^{4}
+ c5*(1000/T)^{5}
The coefficients are:

H_{2}O 
CO_{2} 
c0 
0.23093 
18.741 
c1 
1.12390 
121.310 
c2 
9.41530 
273.500 
c3 
2.99880 
194.050 
c4 
0.51382 
56.310 
c5 
1.86840E05 
5.8169 
A fourthorder polynomial in temperature is used for CH_{4}:
a_{p,ch4} = 6.6334  0.0035686*T + 1.6682e08*T^{2} + 2.5611e10*T^{3}
 2.6558e14*T^{4}
A fit for CO is given in two temperature ranges:
a_{p,co} = c0+T*(c1 + T*(c2 + T*(c3 + T*c4)))
if( T .le. 750 ) then
c0 = 4.7869
c1 = 0.06953
c2 = 2.95775e4
c3 = 4.25732e7
c4 = 2.02894e10
else
c0 = 10.09
c1 = 0.01183
c2 = 4.7753e6
c3 = 5.87209e10
c4 = 2.5334e14
endif
Acknowledgments: Nigel Smith (AMRL), Jay Gore (Purdue University),
JongMook Kim (Purdue University), and Qing Tang (Cornell) provided
information for this radiation model.
References:
Grosshandler, W. L., RADCAL: A NarrowBand Model
for Radiation Calculations in a Combustion Environment, NIST
technical note
1402, 1993.
Barlow, R. S., Smith, N. S. A., Chen, J.Y., and Bilger,
R. W., Combust. Flame 117:431 (1999).
Frank, J. H., Barlow,
R. S., and Lundquist, C., Proc. Combust. Inst. 28:447454 (2000).
Zhu,
X. L., Gore, J. P., Karpetis, A. N., and Barlow, R. S., Combust.
Flame 129:342345 (2002).
Coelho, P. J., Teerling, O. J., Roekaerts,
D., “Spectral
Radiative Effects and TurbulenceRadiationInteraction in Sandia
Flame D,” in TNF6_Proceedings.pdf (2002).
Barlow, R.
S., Karpetis, A. N., Frank, J. H., and Chen, J.Y., Combust.
Flame 127:21022118 (2001).
