Inputs for the evaluation of mixture emissivity with different absorptive mole fraction of CO2 as a function of gas temperature. Fundamentally, the total absorptivity from non-gray gases can have a completely different behavior as compared to the total emissivity if the source temperature differs from the gas temperature. However, if the mixture is assumed to be gray, the total absorptivity is constrained to be equal to the total emissivity, an assumption generally invoked without any mathematical verification in many fire applications for example, this practice is found in one of the most commonly used fire simulation codes 5 [25] certified by the U.
Department of Energy. Calculated total emissivity of soot-gas mixture using OpenSC. Inputs for the evaluation of mixture emissivity and absorptivity as a function of gas temperature with 3 different source temperatures for 2 soot conditions. In contrast to the evaluation of total emissivity, the source temperature is now important.
For a gas mixture, the total emissivity and the total absorptivity as a function of gas temperature are plotted in Fig. Three different source temperatures K, K, and K for the total absorptivity are considered in the example.
Since the total emissivity is independent of source temperature, there is only one curve for the total emissivity. As shown in the figure, the total absorptivity is significantly different than the total emissivity both in terms of its numerical value and its dependence on the gas temperature. In general, the total absorptivity is an increasing function of gas temperature, which is completely opposite to the trend for the total emissivity.
Physically, this trend is expected because the peak of the Planck function for lower source temperature is shifted to the more active absorption band for H2O in the long wavelength region 4. The total absorptivity for a soot-gas mixture is presented in Fig. In the limit of a pure soot mixture, the total absorptivity is only a function of the source temperature. Absorptivity of gas mixture no soot at different source temperature Emissivity of the same mixture is also plotted for comparison.
Absorptivity of soot-gas mixture at different source temperature Emissivity of the same mixture is also plotted for comparison. Quantitatively, comparison between the total emissivity and the total absorptivity as shown in Fig. Therefore, their accuracy can be highly uncertain. In this section, results generated from OpenSC will be used to assess the predictions obtained from these approximations or simplified methods provided in handbooks [14,15].
Specifically, the accuracy of the emissivity chart model will be examined. In general, the total emissivity charts refer to those appearing in Edwards [15] which are the modified emissivity charts first introduced by Hottel and his co-workers [6] based on their experimental measurements. Following the calculation guidelines provided in [14], it is assumed that the pressure correction factors are 1.
Based on Eq. The benchmark emissivity obtained from the LBL method [26,27], as shown in dashed lines, is provided for reference. For the emissivity chart model, the overlapping effect for H2O and CO2 is simulated using the approximation given in Eq. For OpenSC, the narrow- band absorption coefficient of the species generated by RADCAL which accounts for the overlapping effect of the species with the narrow-band model is used to evaluate the spectral transmissivity and the total emissivity is obtained by direct integration over all wavelengths.
The evaluation of mixture emissivity by OpenSC is thus expected to have the same level of accuracy as that for a pure gas and can serve as an approximate benchmark to compare to the emissivity chart model. Results in Fig. Therefore, the suitability of the ad-hoc approach for the emissivity chart model to properly account for the effect of mixture, as represented in Eq.
Total mixture emissivity with a range of optical thickness a as a function of CO2 fraction for gas temperature at K left and b as a function for gas temperature for CO2 fraction of 0. Total emissivity of soot particulates obtained based on Eq. As expected, soot emissivity from both sets of data is increasing function of increasing temperature. Since the soot absorption coefficient used in Eq. Compared to the pure mixture results presented in Fig. Physically, the inverse temperature behavior of the soot absorption coefficient is not captured by Eq.
In general, the discrepancy between OpenSC and the emissivity chart model is significant. Total emissivity for soot particulates in logarithmic scale. Total soot-gas mixture emissivity a with a range of optical thickness as a function of gas temperature for soot concentration of 1e-7 m and CO2 fraction of 0. Since the numerical integration requires no additional assumptions, the OpenSC prediction for the absorptivity is expected to have the same degree of accuracy as that for the total emissivity.
OpenSC can thus serve as an approximate benchmark solution to assess the accuracy of Eq. Comparison between results generated from OpenSC and the approximate absorptivity generated by Eq.
It is apparent that while Eq. It can be seen that similar discrepancy appears for both radiative properties as a function of gas temperature. In summary, OpenSC and the general emissivity chart have the same order of accuracy in predicting the total emissivity of a mixture consisting only one absorptive gas species H2O or CO2.
Using the OpenSC results as benchmark, the emissivity chart model and its ad-hoc modifications for mixture and absorptivity are shown to be in significant error.
To highlight the error of the emissivity chart model associated with the example calculations presented in Figs. For the total emissivity shown in Figs. For total absorptivity shown in Figs.
The mathematical formulation and the neural network correlations used in OpenSC are described. Quantitative comparisons show that the total emissivity is generally different from the total absorptivity and they have completely different dependence on mixture conditions.
Therefore, the gray assumption of equal total emissivity and total absorptivity is generally not valid. The emissivity chart approach suggested by the SFPE Fire Protection Handbooks for the evaluation of mixture emissivity and mixture absorptivity are assessed.
In comparison to the results generated from OpenSC, the emissivity chart approach generally under-predicts the mixture emissivity. For mixture absorptivity, the discrepancy in between OpenSC and the emissivity approaches is substantial. We would also like to thank Dr. Jiann C. Yang from NIST for his constructive comments and valuable suggestions to this manuscript. Modest, M. The treatment of nongray properties in radiative heat transfer: from past to present.
Journal of Heat Transfer, 6 , Wang, L. Theory Model. Rothman, L. Transfer, 15 , pp. Grosshandler, W. Hottel, H. McAdams, ed. McGrattan, K. Fire dynamics simulator version 5 , technical reference guide. NIST special publication, 5.
Edwards, D. Tam, W. Yuen, W. Analysis of radiative heat transfer in inhomogeneous nonisothermal media using neural networks. Journal of Thermophysics and Heat Transfer, 30 4 , Assessment of radiative heat transfer characteristics of a combustion mixture in a three-dimensional enclosure using RAD-NETT with application to a fire resistance test furnace. International Journal of Heat and Mass Transfer, 68, Highlights in 0.
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