OCR Text |
Show problem is to assume that the surface intensities can be related by lp = o:le + (1 - o)lw = 0:1n + (1 - 0:)1, = o:lt + (1 - 0:)1b, (25) where 0: is a weighting factor. When 0:=1, the scheme is a step differencing scheme [15]. However, most of the works were done with 0: = ~ which is called the diamond differencing scheme [14]. The intensity lp at grid node can be calculated through the explicit relation with known I's, for example with lw, 18 and lb: (26) I e -- Ip _ 1 - 0:1w , 0: 0: (27) (28) 1 1 - 0: It = 2. - --lb. 0: 0: (29) The basic procedure for calculating the unknown intensities is to start with the known boundary intensities and progress to the next control volume in the same row. When the first row has been visited, the calculation is continued to the next row where the calculated intensities in the first row can be used as the intensities of the faces in the second row. The calculation is progressed until all control volumes are visited. A set of known values of intensities in Eq. (26) can be determined through the direction cosines. All of the equations for the intensity calculation were derived for positive signs of J.l, 17 and e· Before the development of the discrete ordinates equation, the selection of the quadra-ture set which contains the direction cosines (P,m, 17m, em) and the associated weight factor Wm becomes a critical step because the integral equations are replaced by sums over the appropriate directions in the quadrature set. The set contains a total of M directions where m = 1, ... , M. Therefore the boundary condition and the equations for area and volume averaged heat fluxes become Iwall,leaving(O) = twaulb ( Twall ) + 1 - t wall ~L.J I n· 1n1'' m I I incident (0' )W ,m , 7r D.O'm<O (30) 8 |