1. What is the Monte Carlo method?

The Monte Carlo method is one of a class of methods for solving complex mathematical problems, which is based on the use of random numbers. In general, the Monte Carlo method is a generalized mathematical technique for calculating integrals. For radiation problems, it involves simulating the propagation of individual beams of radiation through the medium [1,2].

The origin and direction of each beam is selected randomly. The distribution of directions however, has to conform to the physical emission from a surface. Each beam is individually tracked until it has been absorbed, which may take place after several reflections off the walls. If reflection is from a [GLOSS]diffuse surface[/GLOSS], a new random number determines the direction in which the beam continues. Finally, the heat absorbed in each surface element or gas volume is counted and summed to yield a heat balance. The Monte Carlo technique can also simulate reflection off [GLOSS]specular surface[/GLOSS]s, as well as directional wall properties, and spectral gas effects.  In modelling radiation transfer through scattering media, the [GLOSS]phase function[/GLOSS] can be as complex as necessary to represent the non-[GLOSS]isotropic[/GLOSS] scattering

The Monte Carlo method has been used successfully to solve multi-dimensional radiation problems. It is easy to apply and is adaptable to complex configurations, including geometries where shadowing can occur (i.e. where a surface partially obscures the view between two other surfaces).  A disadvantage is that it can require large computing times although this is less of a problem with advances in computer processing speeds. A further issue is that errors are statistical rather than numerical, so there is no guaranteed convergence to the exact solution, although in general, increasing the number of beams does improve the solution.  The errors are inversely proportional to the square root of the number of tries. For example, increasing the number of beams by a factor of 4 can halve the relative error.

The method can be applied to determine radiation view factors or [GLOSS]direct exchange areas[/GLOSS] and [GLOSS]total exchange areas[/GLOSS] for enclosures. In this case random beams are tracked through the enclosure to determine simply the proportion of beams originating from one surface or gas element that is intercepted by a target surface or gas element. Tucker and Ward [3] applied this approach to determine exchange factors for use in long furnace Zone models and Souza [4] and Lawson [5] extended the method to 3-D Zone models.

Unlike Flux methods, the Monte Carlo method is not numerically compatible with CFD flow equations. If used with CFD models for solving radiation in combusting flows, then the Monte Carlo method is applied after the flow, turbulence, mixing and kinetic equations have been solved. The CFD model creates an initial temperature and composition field. The Monte Carlo model then solves the radiation field to derive new source terms for the CFD differential equations, which are then re-solved, and the cycle repeated. Thus the Monte Carlo solution is applied as a ‘post processor’ of the CFD solution.

Sources

1.        Howell J.R., ‘Application of Monte Carlo to Heat Transfer Problems’, in Advances in Heat Transfer (T.F.Irvine and J.P.Hartnett, eds.), vol.5, Ac.Press, 1968.

2.       Steward F.R. and Cannon P., The Calculation of Radiative Heat flux in a Cylindrical Furnace Using the Monte Carlo Method, Intl.J.Heat and Mass Transfer, vol. 14, No.2, pp.245-262, 1971.

3.       Tucker R.J. and Ward J., Use of a Monte Carlo Technique for the Determination of Radiation Exchange Areas in Long Furnace Models, Proc.8^th Intl. Heat Transfer Conference, San Francisco, 1986.

4.       Correia S.A.C., Ward J. & Sousa J.L.V.A., Comparison of the Predictions from a Range of Multi-Zone Mathematical Models of a Continuously Operated Metal Reheating Furnace, Proceedings 12^th Intl. Heat Transfer Conference, Grenoble, France, vol. 4, pp. 807-812, 2002.

5.       Khan Y.U., Lawson D.A. and Tucker R.J., Radiative Heat Transfer Calculations for Non-grey Surfaces, Numerical Methods for Thermal Problems (ed. Lewis), IX, Pineridge Press, 1995, pp.351-361.