UNDERSTANDING THE k-DISTRIBUTION AND Ck

UNDERSTANDING THE k-DISTRIBUTION AND C-k METHODS IN LESS THAN 10 MINUTES (MAYBE A LITTLE MORE) This document is a note on the full course on k-distribution methods. It can be useful if you need to remind quickly what the main ideas behind this technique are. If you want to cite it, please use the following reference in which more details can be found: V.P. Solovjov, B.W. Webb, F. André, “Radiative Properties of Gases”, in Handbook of Thermal Science and Engineering, Springer, pp. 1-74, 2017.

An extended version of the present note (in French) is also freely available: https://www.researchgate.net/publication/316087006_Rayonnement_des_gaz_des_spectres_de_rai es_aux_modeles_en_k-distributions_-_Cours_ETR2017

Ambartzumian introduced the principle of the so-called k-distribution modeling in 1932. During many years, the same technique was applied, mostly in atmospheric sciences, under the name of “Ambarztumian’s method”. The statistical view, which is now widely admitted, only appeared in 1968.

Narrow band k-distributions in uniform media The objective of all approximate models of gas radiation is to provide methods to reduce the computational cost compared to a LBL (Line-By-Line) calculation by averaging the high resolution radiative properties of the gas over more or less wide spectral bands. There are several kinds of approximate models, classified in terms of the width of the spectral interval used for the averaging: 1/ narrow bands over which the Planck function is treated as constant (a few to a few dozen of cm-1) ; 2/ wide bands (from a hundred to thousands of cm-1) ; 3/ global models, that aim at defining radiative properties averaged over the full spectrum over which the Planck function takes significant values. In the case of narrow bands, as considered here but the same ideas apply to the full spectrum with minor modifications, the objective is thus to evaluate, for instance, the band averaged transmissivity of a gas path of length L (definition is given at the top right of FIGURE 1).

FIGURE 1. From a single gray gas (GG) to a weighted sum-of-gray gases

If the gas is gray (1 GG in FIGURE 1) viz. the absorption coefficient does not depend on the wavenumber, the exponential term can be put outside the spectral integral. In this case, the transmissivity follows the Beer’s law. If the gas is non-gray but can be represented as two gray

gases (2 GG in FIGURE 1), the transmissivity cannot be represented as a single exponential but as a weighted sum of such exponentials. Each of them is associated to a gray gas and follows the Beer’s law. The weight corresponding to a given gray gas with absorption coefficient k (or k’) is the fraction of the spectral interval over which the absorption coefficient takes the value k (or k’). These fractions of the spectral interval can be interpreted as the probability P for the absorption coefficient to take the particular value k (or k’). This probability only depends on the size of the spectral interval over which the absorption coefficient takes a particular value and does not depend on the way these spectral locations are distributed over the wavenumber axis (bottom of FIGURE 1). The same idea can be applied with any integer number n of gray gases (see n GG, FIGURE 2). However, as n approaches infinity, it is not possible to define anymore the probability for the absorption coefficient to take a single prescribed value. In this case, the transmissivity needs to be written as an integral that involves the derivative of the probability for the spectral absorption coefficient to take a value lower than a given k (bottom in FIGURE 2). This derivative is called the k-distribution function. The transmissivity is the Laplace transform of the k-distribution function.

FIGURE 2. From a weighted sum-of-gray gases to a k-distribution

The method to evaluate the probability law P in the case of a discrete set of LBL data (what we always have in practice) is illustrated in FIGURE 3 (top). At the bottom of the same FIGURE 3, examples of cumulative k-distributions g  k   P   k  are given.

FIGURE 3. Cumulated/cumulative k-distribution function. Top: how to evaluate the cumulative distribution in the case of a LBL dataset evaluated over a finite set of wavenumbers; bottom: examples of cumulative k-distributions, g(k)

Cumulative k-distribution functions: 1/ are strictly increasing (and thus invertible), 2/ take values between 0 and 1. Accordingly, for any X between these limits, one can find a real k (X) such that the value taken by the cumulative k-distribution function for this particular k is X. This is depicted in FIGURE 4.This property allows application of a change of variable to evaluate the band averaged transmissivity, as illustrated in the same FIGURE 4.

FIGURE 4. Inversion of the cumulative k-distribution function

The main interest of the change of variable from k to k (X) is to allow the use of simple quadrature formulas (Gauss-Legendre for instance) to perform the spectral integration over the [0,1] interval (see the black arrows in FIGURE 4). This approximation of the integral by a quadrature formula is the last equation at the bottom of FIGURE 4. In this equation, wi and Xi are the weights and abscissas of the numerical quadrature. Quadratures orders I lower than 10 are frequently encountered for narrow band calculations in combustion applications. In the atmosphere or for global calculations, the use of higher values of I are common. Once formulated in term of quadratures, the k-distribution method mostly simplifies into a weighted sum-of-gray gases model (compare FIGURE 2 and FIGURE 4).

Narrow band k-distributions over non-uniform gas paths Up to now, we have only considered uniform gaseous paths. In many cases, the gases that one needs to treat are the seats of gradients of temperature, species concentrations and sometimes pressure. In these situations, adaptations of the previous principles are required. They are quite recent (1979) and referred to as the Correlated-k (C-k) method. The C-k modeling consists in assuming that gas spectra in two distinct states (with respective absorption coefficients  and  ' in FIGURE 5) can be related to each other by a strictly increasing function f. These spectra are thus correlated, or more precisely co-monotonic since f is strictly increasing. The common monotonicity is due to the fact that as the derivative of f is strictly positive whatever the value of its argument, the derivatives of the two absorption coefficients with respect to the wavenumber evaluated at the same spectral location share the same sign. Doing so, a gray gas in one state remains gray in another state. Accordingly, the strictly increasing transformation f preserves the gray gases probabilities. This is depicted in FIGURE 5 (top). The main strength of the C-k method is that function f does not need to be specified explicitly. Indeed, the simple fact that the two probabilities g  k   P   k  and g'  k'   P  '  k'  f  k  are assumed the same allows to relate implicitly a value k of the absorption coefficient of the gas in thermophysical state 1 to its corresponding value in the second state, which is in this case f (k). This is shown in the case of n gray gases in FIGURE 5 (top). The bottom of FIGURE 5 illustrates the use of the C-k model for non-uniform calculations.

Extension of this idea to an infinite number of gray gases follows the same steps as used in the uniform case. It can be noticed that writing this general case has no real practical interest since in practice only finite numbers of gray gases are used (see the bottom right of FIGURE 4 and FIGURE 5). The same idea can also be applied to any number of uniform sub-paths along a non-uniform path, viz. more than two gas path lengths as used in the bottom of FIGURE 5.

FIGURE 5. the C-k method