MONTE CARLO METHOD FOR EXCHANGE AMONG DIFFUSEGRAY SURFACES
Leading to: Net radiation method in radiative transfer
1. INTRODUCTION
Radiation problems are ideally suited for using the Monte Carlo method. Here, we examine radiant exchange between surfaces in the absence of a participating medium.
The general expression for the radiant emissive power of a surface element is
(1) 
In this expression, ε(λ, T, θ, φ, r) is the monochromatic directional emissivity, I_{λb} is the blackbody spectral intensity, and θ and φ are the angular directions (Fig. 1).
Figure 1. Spherical coordinate system.
Equation (1) places no restriction on the wavelength, direction, or temperature dependence of the emissive power. However, from this point, the discussion is limited to application of the method to diffusegray surfaces. Extension to directionalspectral surfaces can be explored in the references.
2. DIRECT SIMULATION MONTE CARLO
Direct simulation Monte Carlo simulates natural radiation phenomena by assuming that the emitted radiant energy is an ensemble of energy bundles, each satisfying the required physical laws. Equation (1) can then be rearranged to yield appropriate probability functions for diffusegray surfaces, i.e., ε(λ, T, θ, φ, r) = ε(T, r), as
(2) 
in which
(3) 
(4) 
(5) 
(6) 
The notation dP indicates a probability density function (PDF) and R denotes a cumulative distribution function (CDF). When I_{bλ} in the definition of R_{λ} is replaced by E_{bλ} = πI_{bλ}, Eq. (3) can be represented for any given temperature as
(7) 
where E_{b(0  λT)} denotes the blackbody emissive power of the spectral range between λ = 0 and any λ. The f(λT) = E_{b(0  λT)}/σT^{4} is a universal function of λ (see Thermopedia section, Blackbody). Then the relation R_{λ} = f(λT) with 0 < f(λT) < 1 represents a universal function that can be used to define the probable wavelength of a photon bundle. The radiation emitted from a surface element is envisioned as the sum of the energy of N photon bundles. The probability functions now can be used to determine the probable wavelength, direction, and energy assigned to each bundle. The wavelength of an energy bundle is determined by assigning to the CDF R_{λ} the value of a random number drawn from a set of uniformly distributed random numbers between 0 and 1. The value of R_{λ} that has been set equal to the drawn random number corresponds to a specific λT for a given temperature T. This random number is equal to the fraction of photon bundles that have wavelengths smaller than the corresponding λ. This procedure of relating the random number to the CDF of a function assures that after enough such samples are chosen from the CDF, the original PDF will be reproduced.
Random numbers drawn from the same set and assigned to and R_{φ} give the angular direction from Eqs. (4) and (5). For the nth photon bundle, let λ_{n}, θ_{n}, and φ_{n} denote the wavelength and directions of emission. The energy portion of the nth photon bundle is the weighting function, Eq. (6), divided by N,
(8) 
If the surface is gray, then the weighting function w ≠ F(λ), and Eq. (8) can be used directly for photon bundles of equal weight leaving a gray (or black) surface. For diffuse surfaces, Eqs. (4) and (5) are used without individually weighting the photon bundle energies with respect to the angle of emission. Thus, for a diffusegray surface, all bundles can be assigned the same weight. Analytical functions relating the desired variables to random numbers are given in Table 1 for diffusegray surfaces. Treatment of nondiffuse surfaces is covered in the references.
Table 1. Inverse probability functions for diffusegray surfaces
Angle θ (diffuse surface): θ = sin^{1} [√R_{θ} ] 
Angle φ (diffuse surface): φ = 2π(R_{φ}) 
Wavelength λ: 
For a black or gray surface, for an error of less than ±1% over a broad range for (λT) from 750 μK (R_{λ} = 5.96×10^{6}) to 65,000 μK (R_{λ} = 0.99957), use the relation

3. MONTE CARLO PROCEDURE
The random variables that have the same characteristics as the available random numbers are related to R_{λ}, R_{θ}, and R_{φ}. Consider two random numbers, R_{1} and R_{2}, chosen from a uniform distribution in the range 0 ≤ R ≤ 1 and each to be used only once. The following relations yield the probable directions (θ,φ) of a photon bundle emitted from a surface element:
(9) 
(10) 
However, computation of the probable wavelength requires additional effort. An empirical equation for the inverse probability function λT = f^{1}(R_{λ}) is given in Table 1. A random number R, when introduced as the argument of the inverse function, f^{1}(R) yields the value of λT.
Once all N photon bundles from a given surface i are used up, then a bookkeeping procedure determines the rate of radiant heat transfer. For instance, if the number of photon bundles emitted by the surface i is N_{i}, then the radiant emission from surface i absorbed by surface j is
(11) 
where e_{n}^{*} = e_{n} if absorption takes place by surface j and otherwise e_{n}^{*} = 0. The values of e_{n} are given by Eq. (8). If j = i, then Q_{ij} is the energy from surface i that is absorbed by surface i directly or after reflection from other surfaces. The subscript n denotes the first, second,..., nth energy bundles that left surface i.
The estimation of the random error in the radiation absorption requires computation of the variance s^{2} from
(12) 
The position from which an energy bundle departs may be determined at random, and the energy of a bundle departing that location is determined by Eq. (8). Hence, temperature may vary over the surface, and its value depends on the justdetermined position. As an alternative procedure, a surface may be subdivided into smaller elements, with each element treated as a different surface.
Whenever participating surfaces are black, i.e., ε = α = 1, the Monte Carlo energy exchange yields the configuration factors between these surfaces. According to the definition of the configuration factor,
(13) 
in which Q_{ij} is determined by the Monte Carlo procedure. Numerous computations of configuration factors using the Monte Carlo method are reported in the literature. Reviews or more indepth discussions of the Monte Carlo procedure in radiative heat transfer are in the references.
REFERENCES
Farmer, J. T. and Howell, J. R., Comparison of Monte Carlo strategies for radiative transfer in participating media, Advances in Heat Transfer, vol. 31, Hartnett, J. P. and Irvine, T. F. (eds.), pp. 333429, Academic Press, New York, 1998.
HajiSheikh, A. and Howell, J. R., Monte Carlo methods, Handbook of Numerical Heat Transfer, 2nd ed., Minkowycz, W. J., Sparrow, E. M., and Murthy, J. Y. (eds.), pp. 249296, Wiley, Hoboken, NJ, 2006.
Howell, J. R., The Monte Carlo Method in Radiative Heat Transfer, J. Heat Transfer, vol. 120, pp 547560, 1998.
Howell, J. R., Application of Monte Carlo to heat transfer problems, Adv. Heat Transfer, vol. 5, pp. 154, 1968.
Modest, M. F., Radiation Heat Transfer, 2^{nd} ed., Academic Press, San Diego, 2003.
Siegel, R. and Howell, J. R., Thermal Radiation Heat Transfer, 5th ed., Taylor and Francis, New York, 2010.
References
 Farmer, J. T. and Howell, J. R., Comparison of Monte Carlo strategies for radiative transfer in participating media, Advances in Heat Transfer, vol. 31, Hartnett, J. P. and Irvine, T. F. (eds.), pp. 333429, Academic Press, New York, 1998.
 HajiSheikh, A. and Howell, J. R., Monte Carlo methods, Handbook of Numerical Heat Transfer, 2nd ed., Minkowycz, W. J., Sparrow, E. M., and Murthy, J. Y. (eds.), pp. 249296, Wiley, Hoboken, NJ, 2006.
 Howell, J. R., The Monte Carlo Method in Radiative Heat Transfer, J. Heat Transfer, vol. 120, pp 547560, 1998.
 Howell, J. R., Application of Monte Carlo to heat transfer problems, Adv. Heat Transfer, vol. 5, pp. 154, 1968.
 Modest, M. F., Radiation Heat Transfer, 2^{nd} ed., Academic Press, San Diego, 2003.
 Siegel, R. and Howell, J. R., Thermal Radiation Heat Transfer, 5th ed., Taylor and Francis, New York, 2010.