Updated 23-JAN-2003 to include additional discussion and references.

Updated 7-JUN-2000 to correct errors in curve fits for CO_{2} and H_{2}O.

Updated 5-FEB-1998 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 turbulence-chemistry 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 4.3-micron band of CO_{2} [3-5].

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=5.669e-08 W/m
^{2}K^{4}is the Steffan-Boltzmann 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 emission-only 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 (m-1atm-1). 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 | -1.86840E-05 | -5.8169 |

A fourth-order polynomial in temperature is used for CH_{4}:

a_{p,ch4} = 6.6334 – 0.0035686*T + 1.6682e-08*T^{2} + 2.5611e-10*T^{3} – 2.6558e-14*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 = 2.95775e-4

c3 = -4.25732e-7

c4 = 2.02894e-10

else

c0 = 10.09

c1 = -0.01183

c2 = 4.7753e-6

c3 = -5.87209e-10

c4 = -2.5334e-14

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 Narrow-Band 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 117:4-31 (1999).

Frank, J. H., Barlow, R. S., and Lundquist, C., Proc. Combust. Inst. 28:447-454 (2000).

Zhu, X. L., Gore, J. P., Karpetis, A. N., and Barlow, R. S., Combust. Flame 129:342-345 (2002).

Coelho, P. J., Teerling, O. J., Roekaerts, D., “Spectral Radiative Effects and Turbulence-Radiation-Interaction in Sandia Flame D,” in TNF6_Proceedings.pdf (2002).

Barlow, R. S., Karpetis, A. N., Frank, J. H., and Chen, J.-Y., Combust. Flame 127:2102-2118 (2001).