The discrete transfer method In the discrete transfer method (DTM) of Lockwood and Shah (1981), the

13.4.2 The discrete transfer method In the discrete transfer method (DTM) of Lockwood and Shah (1981), the

solution proceeds by first discretising the radiation space into homogeneous surface and volume elements. Rays are emitted from the centre of each boundary surface element, with position vector r, in directions determined

by discretising the 2 π hemispherical solid angle above the surface into finite solid angles, δΩ. Shah (1979) chose to divide the hemisphere into N θ equal

polar angles and N φ equal azimuthal angles such that N Ω =N θ N φ and π

δθ = δφ = (13.25) 2N θ

These features are illustrated in Figures 13.6 and 13.7. In vector terms a ray is traced through the centre of each solid angle element, in the direction −s k , until it strikes a boundary at r L =r−s k L k , such that the ray path length is L k = |r − r L |. To calculate the contribution due to the solid angle element to the incident intensity at the origin, the ray is followed back, starting at r L , to origin r. The intensity distribution along its path is solved with the recurrence relation

n+1 =I n e + S(1 − e ) (13.26) where n and n + 1 designate successive boundary locations, separated by a dis-

I − βδs

− βδs

tance δs, as the ray passes through each medium control volume. The source function S includes the scattering integral in its angular discretised form:

S = (1 − ω)I b +

I − (s i ) Φ(s, s i )d Ω i

= (1 − ω)I b +

∑ I −,ave (s i ) Φ(s, s i ) δΩ i

4 π i=1

where the averaged intensity I −,ave (s i ) is taken as the arithmetic mean of the entering and leaving radiant intensities for each ray passing through the cell volume within the finite solid angle δΩ. The finite solid angle is evaluated for

each angular sub-division as δΩ =

sin θ dφ dθ = 2 sin θ sin(δθ/2)δφ

Source function S is assumed to be constant over the interval. A cell-averaged (directionally independent) value is taken for S. This considerably simplifies

the analysis, but as a consequence of this approximation the DTM does not have the ability to describe scattering anisotropy.

430 CHAPTER 13 CALCULATION OF RADIATIVE HEAT TRANSFER

Figure 13.6 Angular discretisation and representative ray selection in the discrete transfer method: (a) angular discretisation in the azimuthal direction; (b) angular discretisation in the polar direction; (c) selection of ray directions for a single d φ angular

sector

13.4 FOUR POPULAR RADIATION CALCULATION TECHNIQUES 431

Figure 13.7 An illustration of the discrete transfer method

The initial intensity value of each ray is evaluated at the originating surface element and is given by

(13.30) Equation (13.26) is applied in the direction towards the origin of each ray,

q + = εE s + (1 − ε)q −

and the incident radiative heat flux q − is evaluated by summing contributions over all solid angles, assuming that the intensity is constant over each finite solid angle. This gives

q − = ∑ I − (s) s . n δΩ = ∑ I − ( θ, φ) cos(θ) sin(θ) sin(δθ)δφ (13.31)

where N R is the number of incident rays arriving at the surface element. Since q + in (13.30) depends on the value of q − , an iterative solution is required, unless the surfaces are black ( ε = 1). In an iterative calculation the initial intensity leaving a surface is evaluated on the basis of its own temper- ature only. Estimates of the incident radiative heat flux will be available after the first iteration to calculate corrected intensities leaving boundary surfaces using (13.29) and (13.30). The process is repeated until the difference between successive values of the negative flux is within a specified limit.

When the calculation has converged the net radiative heat flow out of each surface element with area A i is computed from

(13.32) DTM calculates the radiative source for each medium element via an energy

Q si =A i (q + −q − )

balance. Lockwood and Shah (1981) evaluated the radiative source associated with the passage of each ray through a volume n by means of

432 CHAPTER 13 CALCULATION OF RADIATIVE HEAT TRANSFER

δQ gk = (I n+1 −I n )A s ( −s k . n)d Ω k

= (I n+1 −I n )A s cos θ k sin θ k sin( δθ k ) δφ k (13.33) where A s is the area of the surface element from which the ray was emitted.

Summing the individual source contributions from all the N rays passing through a volume element, and then dividing this value by its volume, ∆V, gives the divergence of radiative heat flux as

S h,rad =∇.q r =

∆V ∑ k =1

δQ g,k

Each DTM ray originates from a certain surface element and selects its initial intensity on the basis of the properties of this element. However, the incremental solid angle, represented by the ray, may partly cover an adjacent surface element. Since the influence of the initial intensity decays exponentially along its transmission path, the resulting inaccuracy is likely to be small.

The accuracy of the DTM depends on two factors, surface discretisation (number of surface elements used to fire rays) and angular discretisation (number of rays used per point). Mathematical expressions for the errors resulting from these discretisation practices have been derived for simple situations (Versteeg et al, 1999a, b). The studies using transparent media

show that the decay rate of the angular discretisation error ε H with increas- ing ray number N R depends on the smoothness of the irradiation. For smooth irradiating intensity ε H ∝ 1/N R , for piecewise sources ε H ∝ 1/ N R and for intensity fields field with derivative discontinuities ε H ∝ 1/N R . The surface discretisation error is generally small compared with ε H and can

be reduced by refinement of the surface mesh. Our predictive experience is that when an adequately refined mesh and a sufficiently large number of rays are used the standard method produces results which are comparable in accuracy with Monte Carlo solutions. Our tests have shown that, for non-scattering, absorption/emission-only problems, the DTM is around 500 times faster than MC in terms of CPU times. For isotropic scattering, absorbing and emitting problems DTM is about 10 times faster than MC calculations (see Henson and Malalasekera, 1997a).

Further details of the basic method can be found in Shah (1979), Lockwood and Shah (1981) and Henson (1998). The discrete transfer method is mathematically simple and also applicable to any geometrical configuration. When a fast and efficient ray tracing algorithm is used the method is computationally efficient. It is well established in application to combustion-related problems, and extensions of the method to isotropic scattering problems and non-grey calculations have been demonstrated in Carvalho et al. (1991) and Henson and Malalasekera (1997b). The method is, however, not suitable for anisotropic scattering applications. Several enhancements and modifications to the original method have been proposed,

e.g. Cumber (1995) and Coelho and Carvalho (1997).

13.4 FOUR POPULAR RADIATION CALCULATION TECHNIQUES 433