Generalization of the k-moment method using the maximum entropy principle. Application to the NBKM and full spectrum SLMB gas radiation models
Résumé
The k-moment method is generalized by applying the maximum entropy principle to get several estimates of the k-distribution function on any kind of spectral interval as a function of the first two moments of the absorption coefficient. Corresponding formulations of the blackbody weighted band averaged transmission function of a gaseous uniform path are obtained. Different constraints involving the first and second order positive, first order negative and logarithmic moments are introduced together with a physical meaning whenever it is possible. Different sets of these constraints are considered to get maximum entropy estimates of the distributions functions: the Dirac, exponential, Gamma, inverse Gaussian and reciprocal inverse Gaussiank-distribution functions. Analytical formulas are provided for each of these distributions and for their associated transmission function, as a function of the mean and variance of the absorption coefficient. The methodology can be applied considering any spectral interval: narrow, wide, the full spectrum, continuous or not. Thus the resulting associated transmission and cumulative k-distribution functions can be utilized in the frame of a large variety of gas radiation models. Hence the k-moment method using the maximum entropy principle is assessed in the frame of the NBKM and full spectrum SLMB gas radiation models. A series of test cases implying comparisons with reference Line-by-Line results exhibits which maximum entropy k-distributions are likely to give the best estimations of narrow band or total emitted intensities, curves-of-growth of the total emission function and full spectrum cumulative k-distribution functions. In particular, the inverse Gaussian and Gamma k-distributions seem most of the time to perform very well.