﻿ BENCHMARK SOLUTIONS FOR VERIFICATION OF RADIATION SOLUTIONS

C.  BENCHMARK SOLUTIONS FOR VERIFICATION OF RADIATION SOLUTIONS

In this section, benchmark solutions are presented for comparison verification of numerical codes. Besides the tables presented here, additional benchmark solutions are in Table 11-2;  Burns et al. (1995c);  Hsu and Tan (1996); Hsu and Farmer (1997);  Tan et al. (2000), and Coelho et al. (2003).

TABLE C-1  Incident radiation  and z-component of radiative flux qz in cube of side length 2c exposed to uniform diffuse incident radiation qi on bottom surface z = −c with nonhomogeneous gray extinction coefficient b = (0.05/c) + (0.45/c) {[1−(x2/c2)] [1−(y2/c2)][1−(z2/c2)]}2, isotropic scattering, and scattering albedo w. Results tabulated along line z, x = y = 0. Results are by numerical quadrature using 17 quadrature points (QM17) [Wu et al. (1996)] (origin of coordinates is at cube center).

 z/(2c) w = 1.0 w = 0.5 w= 1.0 w = 0.5 −0.49529 2.0674 2.0205 0.9466 0.9746 −0.47534 1.9955 1.9427 0.9410 0.9691 −0.44012 1.8730 1.8107 0.9246 0.9525 −0.39076 1.7066 1.6302 0.8883 0.9150 −0.32884 1.5055 1.4120 0.8251 0.8478 −0.25635 1.2837 1.1745 0.7359 0.7508 −0.17562 1.0588 0.9416 0.6311 0.6355 −0.08924 0.8487 0.7345 0.5259 0,5197 0.00000 0.6667 0.5657 0.4325 0.4180 0.08924 0.5196 0.4374 0.3566 0.3374 0.17562 0.4081 0.3451 0.2987 0.2778 0.25635 0.3280 0.2813 0.2560 0.2357 0.32884 0.2729 0.2384 0.2252 0.2066 0.39076 0.2365 0.2100 0.2033 0.1867 0.44012 0.2131 0.1917 0.1882 0.1733 0.47534 0.1989 0.1803 0.1785 0.1647 0.49529 0.1914 0.1744 0.1734 0.1602

TABLE C-2 Integrated intensity  and surface heat flux distributions qz, in cylinder with diameter 2ro = height zo exposed to uniform collimated incident flux qi = 1 on top surface z = 0 (positive z extends vertically downward) with nonabsorbing gray homogeneous isotropic scattering, scattering coefficient σs, and optical thickness tzo = sszo = 0.25. Results tabulated along radius 0 < r < ro at z1 and z2, and along axial position 0 < z < zo at r1 and r2. Results by numerical quadrature using 17 quadrature points (QM17) [Hsu et al. (1999)]a

 tr/tro 0.015625 1.08356 0.85828 0.04490 0.81969 0.015625 1.08356 1.04819 0.02810 0.078125 1.08344 0.85819 0.04484 0.81964 0.078125 1.09101 1.04359 0.03485 0.140625 1.08313 0.85790 0.04464 0.81944 0.140625 1.08377 1.03439 0.03913 0.203125 1.08260 0.85742 0.04434 0.81915 0.203125 1.07618 1.02194 0.04052 0.265625 1.08188 0.85678 0.04390 0.81873 0.265625 1.06548 1.00914 0.04168 0.328125 1.08096 0.85595 0.04337 0.81824 0.328125 1.05352 0.99588 0.04252 0.390625 1.07980 0.85490 0.04271 0.81761 0.390625 1.04008 0.98229 0.04302 0.453125 1.07844 0.85370 0.04194 0.81689 0.453125 1.02603 0.96805 0.04293 0.515625 1.07688 0.85233 0.04106 0.81609 0.515625 1.01082 0.95352 0.04252 0.578125 1.07509 0.85076 0.04006 0.81519 0.578125 0.99534 0.93892 0.04194 0.640625 1.07303 0.84899 0.03892 0.81417 0.640625 0.97851 0.92381 0.04088 0.703125 1.07059 0.84686 0.03749 0.81286 0.703125 0.96127 0.90835 0.03942 0.765625 1.06747 0.84405 0.03557 0.81106 0.765625 0.94304 0.89258 0.03771 0.828125 1.06363 0.84070 0.03334 0.80902 0.828125 0.92268 0.87656 0.03583 0.890625 1.05879 0.83654 0.03064 0.80660 0.890625 0.90140 0.85956 0.03307 0.984375 1.04819 0.82775 0.02527 0.80204 0.984375 0.85828 0.82775 0.02462

atr1=tro/64, tr2=63tro/64, tz1=tzo/64, tz2=63tzo/64.