Smoothing Monte Carlo Exchange Factors through Constrained Maximum Likelihood Estimation

Complex radiant enclosure problems are often best treated by calculating exchange factors through Monte Carlo simulation. Because of its inherent statistical nature, however, these exchange factor estimations contain random errors that cause violations of the reciprocity rule of exchange factors, and consequently the second law of thermodynamics. Heuristically adjusting a set of exchange factors to satisfy reciprocity usually results in a violation of the summation rule of exchange factors and the first law of thermodynamics. This paper presents a method for smoothing exchange factors based on constrained maximum likelihood estimation. This method works by finding the set of exchange factors that maximizes the probability that the observed bundle emissions and absorptions would occur subject to the reciprocity and summation rules of exchange factors as well as a nonnegativity constraint. The technique is validated by using it to smooth the sets of exchange factors corresponding to two three-dimensional radiant enclosure problems.