The M-1 Radiation Model: Evaluation for Predicting Thermal Radiation From Fires

The M-1 radiation model is a thermal radiation transport model that is derived from a maximum entropy approximation to the radiative transport equation. It involves the solution of four hyperbolic equations for conservation of radiative energy. The M-1 model has similarities to the classical diffusion approximations (like P-1), but is able to better predict directed flux. Consequently, shadowing and long-range transport can be well resolved for a fraction of the cost of methods with exponentially increasing accuracy costs like the method of discrete ordinates and Monte Carlo ray-tracing. The M-1 method is mostly used historically in astronomical radiation transport, but has recently been shown to work for combustion applications of smaller scale. Past work has shown it to give good comparisons to fire problems with length scales of interest. Because of the potential for the model to predict radiation transport more cost-effectively, it is being examined for implementation as an option in our fire codes. We present the theory behind the model. The Eddington factor is used to partition directed and diffuse radiation. It is normally modeled since it is derived from a transcendental functional relationship. We analyze Eddington factor models presented in previous work, and present a new model that we show to be superior in most ways to all the previously presented models. Some 1-dimensional calculations are also shown that illustrate the potential accuracy and challenges with implementing the M-1 model. Such challenges include the specification of boundary conditions and the development of robust solver methods.