Evaluation of Coefficients for the Weighted Sum of Gray Gases Model

Two gray gas components were found adequate in [4, 6] whereas three components were employed in [3, 5-7]. A second order polynomial and six gray gas components were utilized for carbon dioxide in [8]. Four gray gases were employed in [9] for water vapor. Incorporation of four or more gray gases in the zone method may result in significant computational effort. Since the temperature and partial pressure-path length product ranges varied over which the coefficients are applicable, it is not meaningful to compare the magnitudes of the absorption and polynomial coefficients, the order of polynomials, as well as the number of gray gas components and emissivities. There exists, therefore, a lack of well-defined and representative coefficients for employment in the weighted sum of gray gases model. This deficiency is partially attributed to the absence of representative total emissivity and absorptivity values. The objective of this investigation is to develop a set of absorption and polynomial coefficients for the weighted sum of gray gases model. Emphasis is placed on acquiring coefficients for carbon dioxide and water vapor, as well as mixtures of these gases for a wide range of partial pressurepath length products and temperatures. This investigation also examines the number of gray gas components and polynomial orders required to represent the total emissivity and absorptivity values, taking into account the intended application of evaluation of radiant exchange. In order to develop these coefficients, total emissivities and absorptivity values must be available. Radiative gas property models capable of yielding total emissivity and absorptivity values are examined first. From this discussion, a model is selected to provide the total properties. Following this, the weighted sum of gray gases model and procedure for computing the coefficients are briefly described. This discussion is then succeeded by that for the results of this investigation. Radiative Gas Property Models Radiative gas property models which yield total emissivities and absorptivities may be conveniently classified into four general categories associated with narrow-band models [10], exponential wide-band models [11, 12], weighted sum of gray gases models [3-9], and charts and correlations [1, 2, 10, 11, 13-17]. It should be noted that some methods may rely on results of other models. Comparisons of total emissivities and absorptivities evaluated from the various models are available in [8, 9, 14, 15, 17, 18]. A total emissivity chart prepared by
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Evaluation of Coefficients for the Weighted Sum of Gray Gases Model .
The weighted sum of gray gases model postulates that total emissivity and absorptivity may be represented by the sum of a gray gas emissivity weighted with a temperature dependent factor. The gray gas emissivity is expressed in terms of a temperature-independent absorption coefficient, absorbing gas partial pressure, and path length. The weighting factors are given by polynomials in gas temperature with associated polynomial coefficients. For absorptivity, a second polynomial for the irradiation temperature is introduced. A regression scheme is employed to fit the model to total emissivity and absorptivity values obtained from the exponential wide-band model. Absorption and polynomial coefficients are reported for carbon dioxide, water vapor, and mixtures of these gases. The model with these coefficients more accurately represents the total properties over a wider range of temperatures and partial pressure-path length products than previously available coefficients.
Contributed by the Heat Transfer Division and presented at the 20th ASME/AIChE National Heat Transfer Conference, Milwaukee, Wisconsin, August 2-5, 1981. Manuscript received by the Heat Transfer Division, October 1,1981. Paper No. 81-HT-55.
Z. F. Shen
Research Assistant.
J. N. Friedman
Research Assistant. University of Iowa, Iowa City, Iowa 52242
Introduction Evaluation of radiant exchange within a gas as well as between a gas and surrounding surfaces may be accomplished by the zone method [1]. The zone method is attractive since the computational effort associated with the spectral and spatial integrations common to radiant exchange analyses is minimized by introduction of the weighted sum of gray gases model for the radiative properties of total emissivity and absorptivity. This model represents these properties by a summation of a number of terms, each given by the multiplication of a weighting factor and a gray emissivity. The weighting factors are expressed in terms of polynomials in temperature with associated coefficients. Gray gas emissivities are expressed in terms of a temperature independent gray gas absorption coefficient and the product of the partial pressure of the absorbing gas and path length. Total emissivity and absorptivity may then be evaluated once absorption and polynomial coefficients are available. In several energy systems, radiant exchange in gases resulting from combustion of a hydrocarbon fuel is to be evaluated. The systems generally exist at a total pressure of one atmosphere and exhibit temperatures ranging from about 600 to 2400 K. The gases contain carbon dioxide and water vapor where the radiant participation by these gases depends on several parameters of which gas and irradiation temperatures, total and absorbing gas partial pressures, and path lengths are the most important. Partial pressures of carbon dioxide and water vapor are near 0.1 atm for stoichiometric combustion of oil and 0.1 and 0.2 atm, respectively, for stoichiometric combustion of methane [2]. The path length range is approximately 0.1 to 10 m as characteristic of combustion chambers and boilers. The partial pressure-path length products, therefore, acquires values from 0.01 to 10 atm-m. Values for absorption and polynomial coefficients must be available for these cited parameteric ranges as well as for carbon dioxide, water vapor, and mixtures of these gases. Absorption and polynomial coefficients are available from several sources [3-9]. Coefficients reported in [3-7] are applicable for Pw/Pc = 1. In addition, coefficients were also presented in [4-6] for Pw/Pc = 2. For these coefficients, Pc = 0.1 atm and P r = 1 atm. Linear and third-order polynomials were utilized in [4-6] and [3, 7], respectively.