Transferência de Calor 6 Período da Graduação de Engenharia Quím

a b c d e f g h Unidade em metros K50Wm K ValorTemp100 a ValorTemp50 b ValorTemp100 c Valor Temp100 d Valor Temp100 e Valor Temp100 f fluxo de calor0 g fluxo de calor0 h Caso 1 APSGylles Temp max 100 102 990 960 930 900 870 840 810 780 750 720 690 660 630 600 570 540 510 480 588 min APSGylles Temp X Temp X Temp Y Temp Distance APSGylles Temp 100 102 990 960 930 900 870 840 810 780 750 720 690 660 630 600 570 540 510 480 588 min APSGylles ValorTemp100 a ValorTemp50 b ValorTemp100 c Valor Temp80 d Valor Temp100 e Valor Temp100 f fluxo de calor0 g fluxo de calor0 h Caso 2 APSGylles Temp max 100 102 990 960 930 900 870 840 810 780 750 720 690 660 630 600 570 540 510 480 588 min APSGylles Temp X Temp X Temp Y Temp Distance APSGylles Temp 100 102 990 960 930 900 870 840 810 780 750 720 690 660 630 600 570 540 510 480 588 min APSGylles fluxo de calor0 a ValorTemp50 b ValorTemp100 c Valor Temp80 d Valor Temp100 e Valor Temp100 f fluxo de calor0 g fluxo de calor0 h Caso 3 APSGylles Temp max 100 102 990 960 930 900 870 840 810 780 750 720 690 660 630 600 570 540 510 480 588 min APSGylles Temp X Temp X Temp Y Temp Distance APSGylles Temp 100 102 990 960 930 900 870 840 810 780 750 720 690 660 630 600 570 540 510 480 588 min APSGylles fluxo de calor0 a ValorTemp50 b ValorTemp100 c Valor Temp80 d Valor Temp100 e fluxo de calor0 f fluxo de calor0 g fluxo de calor0 h Caso 4 APSGylles Temp max 100 102 990 980 980 930 870 870 810 780 750 720 700 690 630 600 570 540 510 488 488 min Temp APSGylles Temp 87 84 81 78 75 0 03 06 09 12 15 18 21 X Temp APSGylles Temp 90 85 80 75 60 65 70 75 80 85 2 3 Y Temp APSGylles Temp Distance Temp 95 90 85 80 75 70 65 60 85 05 1 15 2 25 3 35 4 Distance 2 fluxo de calor0 a ValorTemp50 b ValorTemp100 c Valor Temp80 d Valor Temp100 e fluxo de calor0 f Valor Temp0 g fluxo de calor0 h Caso 5 APSGylles Temp max 100 100 950 900 850 800 750 700 650 600 450 300 00 00 00 00 min Temp APSGylles Temp 80 75 70 65 60 55 50 45 32 03 06 09 12 15 18 210 X Temp APSGylles Temp 70 60 50 40 30 20 10 0 2 3 Y Temp APSGylles Temp Distance Temp 90 80 70 60 50 40 30 20 0 05 1 15 2 25 3 35 4 45 Distance 2 Caso 6 ValorTemp80 a ValorTemp50 b ValorTemp100 c Valor Temp20 d Valor Temp100 e fluxo de calor0 f Valor Temp0 g Valor Temp30 h APSGylles Temp max 100 100 950 900 850 800 750 700 650 600 550 500 450 400 350 300 250 200 150 100 5 00 min Temp APSGylles Temp 55 50 45 40 35 30 25 20 03 06 09 12 15 18 21 X Temp APSGylles Temp 80 70 60 50 40 30 20 10 0 2 3 Y Temp APSGylles Temp Distance Temp 90 80 70 60 50 40 30 20 0 05 1 15 2 25 3 35 4 45 Distance 2 Caso 7 ValorTemp80 a ValorTemp50 b ValorTemp100 c Valor Temp20 d Valor Temp100 e fluxo de calor0 f Valor Temp0 g Valor Temp100 h APSGylles Temp max 100 100 95 95 90 90 85 85 80 80 75 75 70 70 65 65 60 60 55 55 50 50 45 45 40 40 35 35 30 30 25 25 20 20 15 15 10 10 5 5 min Caso 8 ValorTemp80 a ValorTemp50 b fluxo de calor100 c Valor Temp20 d fluxo de calor100 e fluxo de calor0 f fluxo de calor0 g Valor Temp100 h APSGylles Temp max 101 105 100 95 95 90 85 85 80 75 75 70 65 65 60 55 55 50 45 40 35 30 30 25 25 20 20 15 15 10 10 5 5 min Caso 9 ValorTemp180 a ValorTemp50 b fluxo de calor000 c Valor Temp20 d fluxo de calor000 e fluxo de calor0 f fluxo de calor0 g Valor Temp100 h APSGylles Temp max 180 180 170 160 150 140 130 120 110 100 90 80 70 60 50 40 30 20 20 15 10 10 5 5 min Caso 10 fluxo de calor0 a ValorTemp50 b fluxo de calor000 c Valor Temp200 d fluxo de calor0 e fluxo de calor0 f fluxo de calor0 g Valor Temp80 h APSGylles Temp max 200 min 500 Caso 11 fluxo de calor0 a ValorTemp50 b Valor Temp80 c Valor Temp200 d fluxo de calor0 e fluxo de calor0 f fluxo de calor0 g fluxo de calor0 h APSGylles Temp max 200 min 580 Caso 12 ValorTemp0 a ValorTemp50 b Valor Temp80 c Valor Temp200 d Valor Temp0 e ValorTemp0 f Valor Temp0 g ValorTemp0 h APSGylles Temp max 200 min 001 Caso 13 fluxo de calor0 a ValorTemp50 b Valor Temp80 c Valor Temp200 d fluxo de calor0 e fluxo de calor0 f Valor Temp0 g ValorTemp0 h APSGylles Temp max 200 200 190 180 170 160 150 140 130 120 110 100 90 80 70 60 50 40 30 20 10 0 min APSGylles Temp X 0 03 08 09 12 15 18 21 Y Temp APSGylles Temp Y 0 05 1 15 2 25 3 30 25 Temp APSGylles Temp Distance 0 5 1 15 2 25 3 35 4 45 210 180 150 120 90 60 30 0 TABELA 42 Resumo de equações de diferenças finitas para pontos nodais Configuração Equação de Diferenças Finitas para Δx Δy Caso 1 Ponto nodal interior Caso 2 Ponto nodal em um vértice interno com convecção Caso 3 Ponto nodal em uma superfície plana com convecção Caso 4 Ponto nodal em um vértice externo com convecção Caso 5 Ponto nodal em uma superfície plana com fluxo térmico uniforme Para obter a equação de diferenças finitas para uma superfície adiabática ou superfície de simetria simplesmente coloque h ou q igual m m 1 m 1 m m m1 m m m m 1 PROBLEM 457 KNOWN Steadystate temperatures at selected nodal points of the symmetrical section of a flow channel with uniform internal volumetric generation of heat Inner and outer surfaces of channel experience convection FIND a Temperatures at nodes 1 4 7 and 9 b Heat rate per unit length Wm from the outer surface A to the adjacent fluid c Heat rate per unit length Wm from the inner fluid to surface B and d Verify that results are consistent with an overall energy balance SCHEMATIC ASSUMPTIONS 1 Steadystate twodimensional conduction 2 Constant properties ANALYSIS a The nodal finitedifference equations are obtained from energy balances on control volumes about the nodes shown in the schematics below Node 1 a b c d g q q q q E 0 3 1 2 1 T T T T 0 k y 2 k x 2 0 q x y 4 0 x y Δ Δ Δ Δ Δ Δ 2 1 2 3 T T T 2 q x 4k Δ 6 3 6 2 1 T 9547 1173 C 2 10 W m 25 25 10 m 4 10 W m K 1220 C Node 4 a b c d e f g q q q q q q E 0 2 4 i i 4 i 4 T T k x 2 h y 2 T T h x 2 T T y Δ Δ Δ Δ Continued PROBLEM 457 Cont 5 4 8 4 3 4 T T T T T T k y 2 k x k y x y x Δ Δ Δ Δ Δ Δ q 3 x y 4 0 Δ Δ 2 4 2 3 5 8 i i T T 2T T 2T 2 h x k T 3q x 2k Δ Δ i 6 2 h x k Δ 4 T 9450 C Node 7 a b c d g q q q q E 0 3 7 8 7 o o 7 T T T T k x 2 k y 2 h x 2 T T 0 q x y 4 0 y x Δ Δ Δ Δ Δ Δ Δ 2 7 3 8 o o o T T T h x k T q x 2k 2 h x k Δ Δ Δ T7 9580 C Node 9 a b c d g q q q q E 0 5 9 10 9 o o 9 T T T T k x k y 2 h x T T y y Δ Δ Δ Δ Δ 8 9 T T k y 2 q x y 2 0 x Δ Δ Δ Δ 2 9 5 8 10 o o o T T 05T 05T h x k T q x 2k 2 h x k Δ Δ Δ T9 7967 C b The heat rate per unit length from the outer surface A to the adjacent fluid qA is the sum of the convection heat rates from the outer surfaces of nodes 7 8 9 and 10 A o 7 o 8 o 9 o 10 o q h x 2 T T x T T x T T x 2 T T Δ Δ Δ Δ 2 qA 250 W m K 25 2 9580 25 25 8728 25 3 25 7967 25 25 2 7765 25 10 m K Continued PROBLEM 457 Cont qA 1117 W m c The heat rate per unit length from the inner fluid to the surface B B q is the sum of the convection heat rates from the inner surfaces of nodes 2 4 5 and 6 B i i 2 i 4 i 5 i 6 q h y 2 T T y 2 x 2 T T x T T x 2 T T Δ Δ Δ Δ Δ 2 qB 500 W m K 25 2 50 9547 25 2 25 2 50 9450 3 25 50 7979 25 2 50 7729 10 m K qB 1383 W m d From an overall energy balance on the section we see that our results are consistent since the conservation of energy requirement is satisfied in out gen A B gen E E E q q E 1117 1383 2500W m 0 where 6 3 6 2 Egen q 10 W m 25 50 25 50 10 m 2500 W m COMMENTS The nodal finitedifference equations for the four nodes can be obtained by using IHT Tool FiniteDifference Equations TwoDimensional Steadystate Options are provided to build the FDEs for interior corner and surface nodal arrangements including convection and internal generation The IHT code lines for the FDEs are shown below Node 1 interior node e w n s labeled 2 2 3 3 00 fd2dintT1T2T2T3T3kqdotdeltaxdeltay Node 4 internal corner node en orientation e w n s labeled 5 3 2 8 00 fd2dicenT4T5T3T2T8kqdotdeltaxdeltayTinfihiqa4 qa4 0 Applied heat flux Wm2 zero flux shown Node 7 plane surface node sorientation e w n labeled 8 8 3 00 fd2dpsursT7T8T8T3kqdotdeltaxdeltayTinfohoqa7 qa70 Applied heat flux Wm2 zero flux shown Node 9 plane surface node sorientation e w n labeled 10 8 5 00 fd2dpsursT9 T10 T8 T5kqdotdeltaxdeltayTinfohoqa9 qa9 0 Applied heat flux Wm2 zero flux shown