The discrete ordinates method In the discrete ordinates method the equation of transfer is solved for a set of

13.4.4 The discrete ordinates method In the discrete ordinates method the equation of transfer is solved for a set of

n different directions in the total of 4 π solid angle, and the integrals over directions are replaced by numerical quadrature. Thus the equation of trans- fer is approximated by

dI(r, s n

= κI b (r) − βI(r, s i ) + s ∑ w j I − (s j ) Φ(s i ,s j )

ds

4 π j =1

i = 1, 2, . . . , n (13.35) where w j are quadrature weights associated with the directions s j (see

Modest, 2003). The equation is subject to the boundary condition

∑ w j I − (r w ,s j ) |n . s j | π (13.36)

j =1

The angular ordinates s j = ξi + ηj + µk and angular weights w j are available in Lathrop and Carlson (1965), Fiveland (1991), and the basis of obtaining the weights is discussed further in Modest (2003). The order of the S N

approximation is denoted by S 2 ,S 4 ,S 6 ...S N . The total number of direc- tions used (n) is related to N through the relation n = N(N + 2). Direction cosines ξ, η or µ and weights w j for basic discrete ordinates approximations are shown in Table 13.1. Only the positive direction cosines

Table 13.1 Ordinate directions and weights for S 2 and S 4 approximations

S N approximation

Ordinates

Weights

µ µ S 2 (symmetric)

0.5773 1.57079 S 2 (non-symmetric)

S 4 0.2959

434 CHAPTER 13 CALCULATION OF RADIATIVE HEAT TRANSFER

and weights are shown. The most basic discrete ordinate approximation is S 2 . In S 2 , two different direction cosines are used to define a principal direc- tion. Therefore one direction is used in an eighth of a sphere. In total 2(2 +2) or 8 directions are used per sphere. There are two S 2 representations: sym- metric and non-symmetric. The symmetric representation uses equal values for all direction cosines whereas the non-symmetric representation uses dif-

ferent values. Figure 13.8a illustrates the non-symmetric S 2 representation

in one-eighth of a sphere.

Figure 13.8 Discrete ordinates in one-eighth of a sphere: (a) illustration of the S 2 (non-symmetric) representation; (b) illustration of the S 4 representation

The next improved approximation is S 4 . Here four different direction cosine values are used. They are ±0.2959 and ±0.9082 for ξ, µ or η. Using two different positive values we can generate three principal directions in one-eighth of a sphere as illustrated in Figure 13.8b. Only three directions are shown; all other directions may be obtained by using appropriate negative

13.4 FOUR POPULAR RADIATION CALCULATION TECHNIQUES 435

and positive values, giving a total of 4(4 +2) = 24 permutations. The angular discretisation represents the solid angle subtended by a sphere. Therefore, the sum of the weights for all directions should be equal to the solid angle of

a sphere, i.e. 4 π. This can be easily verified in Table 13.1 for the S 2 and S 4

approximations.

More detailed tabulations of ordinates including S N approximations with N > 4 can be found in Siegel and Howell (2002) and Modest (2003). The angular approximation transforms the original integro-differential equation into a set of coupled differential equations. For Cartesian co-ordinates equation (13.34) may be discretised as follows:

+ βI i = βS i i = 1, 2, . . . , n (13.37)

where ξ i , η i and µ i are the direction cosines of direction i and

S i = (1 − ω)I b +

j I j Φ ij i = 1, 2, . . . , n (13.38)

j =1

The set of coupled differential equations is solved by discretisation using the finite volume method: see Modest (2003). For example, consider the two-dimensional control volume shown in Figure 13.9. The components

dI i /dx, dI i /dy, dI i /dz of the intensity gradient and other terms of equation (13.37) are integrated over the control volume, applying the usual finite volume approximations. Using the areas shown in Figure 13.9 we obtain

ξ i (I E i A E −I W i A W ) + η i (I N i A N −I S i A S )

=− βI P i ( ∆V ) + βS P i ( ∆V ) i = 1, 2, . . . , n (13.39) where I P i and S P i are volume averages of the intensity and source function.

The intensity I P i at the centre of the cell is approximated as

Figure 13.9 A general two-dimensional geometry to illustrate the discrete ordinates method

The parameter γ is a weighting factor used to relate cell edge intensities to the volume average intensity. The weighting factor γ is a constant 0 ≤ γ ≤ 1. The most widely used diamond and step difference schemes are obtained by setting γ to 0.5 and 1.0, respectively.

436 CHAPTER 13 CALCULATION OF RADIATIVE HEAT TRANSFER

The solution marches through the computational domain as follows. At a boundary some I i values are known from boundary conditions. Consider for example the two-dimensional corner cell shown in Figure 13.9. I W i and I S i are known for this cell. Using equations (13.40a) and (13.40b) we may write

γI E i =I P i − (1 − γ)I W i

(13. 41a)

(13.41b) Substitution of equations (13.41a and b) into equation (13.39) and rearrang-

γI N i =I P i − (1 − γ)I S i

ing gives

β(∆V )γS P i + ξ i A WE I W i + η i A SN I

(13.42a)

β(∆V)γ + ξ i A E + η i A N

where A WE = γA W + (1 − γ)A E (13.42b)

(13.42c) Equation (13.42a) gives a means of calculating I P i from boundary intensities

and A SN = γA S + (1 − γ)A N

I W i and I S i . Once I P i is calculated, I E i and I N i can be obtained from equations (13.41a and 13.41b), and the process can move to the next cell where newly calculated values are used as boundary values for the next cell and so on. Starting from a boundary where the intensity is known the domain can be swept to find unknown intensities along each ordinate direction. The above procedure has to be repeated for all ordinate directions. For negative ordinate directions the process starts from the north and east boundaries. Iteration is required as initial boundary intensities are based only on approximate values (usually calculated using surface temperatures) and can only be updated once all incoming intensities are known.

Equation (13.42a) may be generalised to three dimensions as follows: β(∆V)γS P i +| ξ i |A x I x i ,i +| η i |A y I y i ,i +| µ i |A z I

(13.43a) β(∆V)γ + |ξ i |A x +| η i |A y +| µ i |A z

z i ,i

where ∆V is the volume of the cell. Absolute values of direction cosines are used in the above equation to indicate that the equation is valid for both positive and negative direction cosines, and

(13.43c) where subscripts i and e denote entering and exit faces of a control volume,

A z = (1 − γ)A z e + γA z i

respectively, and A x ,A y and A z refer to x, y and z control volume areas in

a three-dimensional Cartesian control volume. As before, we start from boundary cells to calculate I Pi and progress into inner cells. Once all the directional intensities are known, the radiative flux at a surface may be calculated from

q − (r) = I − (r, s)n . sd Ω= ∑ w i I

i (r, s i )n . s i (13.44)

i =1

13.5 ILLUSTRATIVE EXAMPLES

The radiative source term for the enthalpy equation may be calculated from

S h,rad =∇.q r (r) =4 πκI b − κ ∑ w i I i (r, s)

i =1

Further details of the method and its derivation can be found in Modest (2003), Fiveland (1982, 1988, 1991), Fiveland and Jessee (1994), Jamaluddin and Smith (1988) and Hyde and Truelove (1977). The standard discrete ordinates method is suitable for Cartesian and axisymmetric geometries, but it is not directly applicable to non-orthogonal and unstructured grids. However, a modified version of the discrete ordinates method that is suitable for complex geometries has been demonstrated by Charette et al. (1997) and Sakami et al. (1996, 1998).