Page 1 Chapter 12 Radiation Heat Transfer 12-1 Chapter 12 RADIATION HEAT TRANSFER View Factors 12-1C The view factor F i j ? represents the fraction of the radiation leaving surface i that strikes surface j directly. The view factor from a surface to itself is non-zero for concave surfaces. 12-2C The pair of view factors F i j ? and F j i ? are related to each other by the reciprocity rule A F A F i ij j ji ? where A i is the area of the surface i and A j is the area of the surface j. Therefore, A F A F F A A F 1 12 2 21 12 2 1 21 ? ? ? ? ? 12-3C The summation rule for an enclosure and is expressed as F i j j N ? ? ? ? 1 1 where N is the number of surfaces of the enclosure. It states that the sum of the view factors from surface i of an enclosure to all surfaces of the enclosure, including to itself must be equal to unity. The superposition rule is stated as the view factor from a surface i to a surface j is equal to the sum of the view factors from surface i to the parts of surface j, F F F 1 2 3 1 2 1 3 ? ? ? ? ? ( , ) . 12-4C The cross-string method is applicable to geometries which are very long in one direction relative to the other directions. By attaching strings between corners the Crossed-Strings Method is expressed as F i i j ? ? ? ? ? ? Crossed strings Uncrossed strings string on surface 2 Page 2 Chapter 12 Radiation Heat Transfer 12-1 Chapter 12 RADIATION HEAT TRANSFER View Factors 12-1C The view factor F i j ? represents the fraction of the radiation leaving surface i that strikes surface j directly. The view factor from a surface to itself is non-zero for concave surfaces. 12-2C The pair of view factors F i j ? and F j i ? are related to each other by the reciprocity rule A F A F i ij j ji ? where A i is the area of the surface i and A j is the area of the surface j. Therefore, A F A F F A A F 1 12 2 21 12 2 1 21 ? ? ? ? ? 12-3C The summation rule for an enclosure and is expressed as F i j j N ? ? ? ? 1 1 where N is the number of surfaces of the enclosure. It states that the sum of the view factors from surface i of an enclosure to all surfaces of the enclosure, including to itself must be equal to unity. The superposition rule is stated as the view factor from a surface i to a surface j is equal to the sum of the view factors from surface i to the parts of surface j, F F F 1 2 3 1 2 1 3 ? ? ? ? ? ( , ) . 12-4C The cross-string method is applicable to geometries which are very long in one direction relative to the other directions. By attaching strings between corners the Crossed-Strings Method is expressed as F i i j ? ? ? ? ? ? Crossed strings Uncrossed strings string on surface 2 Chapter 12 Radiation Heat Transfer 12-2 12-5 An enclosure consisting of six surfaces is considered. The number of view factors this geometry involves and the number of these view factors that can be determined by the application of the reciprocity and summation rules are to be determined. Analysis A seven surface enclosure (N=6) involves N 2 2 6 ? ? 36 view factors and we need to determine 15 2 ) 1 6 ( 6 2 ) 1 ( ? ? ? ? N N view factors directly. The remaining 36-15 = 21 of the view factors can be determined by the application of the reciprocity and summation rules. 12-6 An enclosure consisting of five surfaces is considered. The number of view factors this geometry involves and the number of these view factors that can be determined by the application of the reciprocity and summation rules are to be determined. Analysis A five surface enclosure (N=5) involves N 2 2 5 ? ? 25 view factors and we need to determine N N ( ) (5 ) ? ? ? ? 1 2 5 1 2 10 view factors directly. The remaining 25-10 = 15 of the view factors can be determined by the application of the reciprocity and summation rules. 12-7 An enclosure consisting of twelve surfaces is considered. The number of view factors this geometry involves and the number of these view factors that can be determined by the application of the reciprocity and summation rules are to be determined. Analysis A twelve surface enclosure (N=12) involves 144 ? ? 2 2 12 N view factors and we need to determine N N ( ) ( ) ? ? ? ? 1 2 12 12 1 2 66 view factors directly. The remaining 144-66 = 78 of the view factors can be determined by the application of the reciprocity and summation rules. 2 1 4 5 3 6 5 4 3 2 1 2 1 3 9 11 12 10 4 5 8 6 7 Page 3 Chapter 12 Radiation Heat Transfer 12-1 Chapter 12 RADIATION HEAT TRANSFER View Factors 12-1C The view factor F i j ? represents the fraction of the radiation leaving surface i that strikes surface j directly. The view factor from a surface to itself is non-zero for concave surfaces. 12-2C The pair of view factors F i j ? and F j i ? are related to each other by the reciprocity rule A F A F i ij j ji ? where A i is the area of the surface i and A j is the area of the surface j. Therefore, A F A F F A A F 1 12 2 21 12 2 1 21 ? ? ? ? ? 12-3C The summation rule for an enclosure and is expressed as F i j j N ? ? ? ? 1 1 where N is the number of surfaces of the enclosure. It states that the sum of the view factors from surface i of an enclosure to all surfaces of the enclosure, including to itself must be equal to unity. The superposition rule is stated as the view factor from a surface i to a surface j is equal to the sum of the view factors from surface i to the parts of surface j, F F F 1 2 3 1 2 1 3 ? ? ? ? ? ( , ) . 12-4C The cross-string method is applicable to geometries which are very long in one direction relative to the other directions. By attaching strings between corners the Crossed-Strings Method is expressed as F i i j ? ? ? ? ? ? Crossed strings Uncrossed strings string on surface 2 Chapter 12 Radiation Heat Transfer 12-2 12-5 An enclosure consisting of six surfaces is considered. The number of view factors this geometry involves and the number of these view factors that can be determined by the application of the reciprocity and summation rules are to be determined. Analysis A seven surface enclosure (N=6) involves N 2 2 6 ? ? 36 view factors and we need to determine 15 2 ) 1 6 ( 6 2 ) 1 ( ? ? ? ? N N view factors directly. The remaining 36-15 = 21 of the view factors can be determined by the application of the reciprocity and summation rules. 12-6 An enclosure consisting of five surfaces is considered. The number of view factors this geometry involves and the number of these view factors that can be determined by the application of the reciprocity and summation rules are to be determined. Analysis A five surface enclosure (N=5) involves N 2 2 5 ? ? 25 view factors and we need to determine N N ( ) (5 ) ? ? ? ? 1 2 5 1 2 10 view factors directly. The remaining 25-10 = 15 of the view factors can be determined by the application of the reciprocity and summation rules. 12-7 An enclosure consisting of twelve surfaces is considered. The number of view factors this geometry involves and the number of these view factors that can be determined by the application of the reciprocity and summation rules are to be determined. Analysis A twelve surface enclosure (N=12) involves 144 ? ? 2 2 12 N view factors and we need to determine N N ( ) ( ) ? ? ? ? 1 2 12 12 1 2 66 view factors directly. The remaining 144-66 = 78 of the view factors can be determined by the application of the reciprocity and summation rules. 2 1 4 5 3 6 5 4 3 2 1 2 1 3 9 11 12 10 4 5 8 6 7 Chapter 12 Radiation Heat Transfer 12-3 12-8 The view factors between the rectangular surfaces shown in the figure are to be determined. Assumptions The surfaces are diffuse emitters and reflectors. Analysis From Fig. 12-6, 24 . 0 5 . 0 2 1 1 5 . 0 2 1 31 3 ? ? ? ? ? ? ? ? ? ? ? ? F W L W L and 29 . 0 1 2 2 5 . 0 2 1 ) 2 1 ( 3 2 1 3 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? F W L L W L We note that A 1 = A 3. Then the reciprocity and superposition rules gives 0.24 ? ? ? ? ? ? 31 13 31 3 13 1 A F F F A F 05 . 0 24 . 0 29 . 0 32 32 32 31 ) 2 1 ( 3 ? ? ? ? ? ? ? ? ? ? ? ? ? F F F F F Finally, 0.05 ? ? ? ? ? ? 32 23 3 2 F F A A W = 2 m (2) L2 = 1 m L1 = 1 m L3 = 1 m A 3 (3) A 2 A 1 (1) Page 4 Chapter 12 Radiation Heat Transfer 12-1 Chapter 12 RADIATION HEAT TRANSFER View Factors 12-1C The view factor F i j ? represents the fraction of the radiation leaving surface i that strikes surface j directly. The view factor from a surface to itself is non-zero for concave surfaces. 12-2C The pair of view factors F i j ? and F j i ? are related to each other by the reciprocity rule A F A F i ij j ji ? where A i is the area of the surface i and A j is the area of the surface j. Therefore, A F A F F A A F 1 12 2 21 12 2 1 21 ? ? ? ? ? 12-3C The summation rule for an enclosure and is expressed as F i j j N ? ? ? ? 1 1 where N is the number of surfaces of the enclosure. It states that the sum of the view factors from surface i of an enclosure to all surfaces of the enclosure, including to itself must be equal to unity. The superposition rule is stated as the view factor from a surface i to a surface j is equal to the sum of the view factors from surface i to the parts of surface j, F F F 1 2 3 1 2 1 3 ? ? ? ? ? ( , ) . 12-4C The cross-string method is applicable to geometries which are very long in one direction relative to the other directions. By attaching strings between corners the Crossed-Strings Method is expressed as F i i j ? ? ? ? ? ? Crossed strings Uncrossed strings string on surface 2 Chapter 12 Radiation Heat Transfer 12-2 12-5 An enclosure consisting of six surfaces is considered. The number of view factors this geometry involves and the number of these view factors that can be determined by the application of the reciprocity and summation rules are to be determined. Analysis A seven surface enclosure (N=6) involves N 2 2 6 ? ? 36 view factors and we need to determine 15 2 ) 1 6 ( 6 2 ) 1 ( ? ? ? ? N N view factors directly. The remaining 36-15 = 21 of the view factors can be determined by the application of the reciprocity and summation rules. 12-6 An enclosure consisting of five surfaces is considered. The number of view factors this geometry involves and the number of these view factors that can be determined by the application of the reciprocity and summation rules are to be determined. Analysis A five surface enclosure (N=5) involves N 2 2 5 ? ? 25 view factors and we need to determine N N ( ) (5 ) ? ? ? ? 1 2 5 1 2 10 view factors directly. The remaining 25-10 = 15 of the view factors can be determined by the application of the reciprocity and summation rules. 12-7 An enclosure consisting of twelve surfaces is considered. The number of view factors this geometry involves and the number of these view factors that can be determined by the application of the reciprocity and summation rules are to be determined. Analysis A twelve surface enclosure (N=12) involves 144 ? ? 2 2 12 N view factors and we need to determine N N ( ) ( ) ? ? ? ? 1 2 12 12 1 2 66 view factors directly. The remaining 144-66 = 78 of the view factors can be determined by the application of the reciprocity and summation rules. 2 1 4 5 3 6 5 4 3 2 1 2 1 3 9 11 12 10 4 5 8 6 7 Chapter 12 Radiation Heat Transfer 12-3 12-8 The view factors between the rectangular surfaces shown in the figure are to be determined. Assumptions The surfaces are diffuse emitters and reflectors. Analysis From Fig. 12-6, 24 . 0 5 . 0 2 1 1 5 . 0 2 1 31 3 ? ? ? ? ? ? ? ? ? ? ? ? F W L W L and 29 . 0 1 2 2 5 . 0 2 1 ) 2 1 ( 3 2 1 3 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? F W L L W L We note that A 1 = A 3. Then the reciprocity and superposition rules gives 0.24 ? ? ? ? ? ? 31 13 31 3 13 1 A F F F A F 05 . 0 24 . 0 29 . 0 32 32 32 31 ) 2 1 ( 3 ? ? ? ? ? ? ? ? ? ? ? ? ? F F F F F Finally, 0.05 ? ? ? ? ? ? 32 23 3 2 F F A A W = 2 m (2) L2 = 1 m L1 = 1 m L3 = 1 m A 3 (3) A 2 A 1 (1) Chapter 12 Radiation Heat Transfer 12-4 12-9 A cylindrical enclosure is considered. The view factor from the side surface of this cylindrical enclosure to its base surface is to be determined. Assumptions The surfaces are diffuse emitters and reflectors. Analysis We designate the surfaces as follows: Base surface by (1), top surface by (2), and side surface by (3). Then from Fig. 12-7 (or Table 12-1 for better accuracy) 38 . 0 1 1 21 12 2 2 2 1 1 1 ? ? ? ? ? ? ? ? ? ? ? ? ? F F r r L r r r r L 1 : rule summation 13 12 11 ? ? ? F F F 62 . 0 1 38 . 0 0 13 13 ? ? ? ? ? ? ? F F ? ? 0.31 ? ? ? ? ? ? ? ? ? ? ? ? ? ) 62 . 0 ( 2 1 2 2 : rule y reciprocit 13 1 1 2 1 13 1 2 1 13 3 1 31 31 3 13 1 F r r r F L r r F A A F F A F A Discussion This problem can be solved more accurately by using the view factor relation from Table 12-1 to be 1 1 2 2 2 2 1 1 1 1 ? ? ? ? ? ? r r L r R r r L r R 382 . 0 1 1 4 3 3 4 3 1 1 1 1 1 1 5 . 0 2 2 2 1 5 . 0 2 1 2 2 2 1 12 2 2 2 1 2 2 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? R R S S F R R S 618 . 0 382 . 0 1 1 12 13 ? ? ? ? ? F F ? ? 0.309 ? ? ? ? ? ? ? ? ? ? ? ? ? ) 618 . 0 ( 2 1 2 2 : rule y reciprocit 13 1 1 2 1 13 1 2 1 13 3 1 31 31 3 13 1 F r r r F L r r F A A F F A F A (2) (3) (1) L D Page 5 Chapter 12 Radiation Heat Transfer 12-1 Chapter 12 RADIATION HEAT TRANSFER View Factors 12-1C The view factor F i j ? represents the fraction of the radiation leaving surface i that strikes surface j directly. The view factor from a surface to itself is non-zero for concave surfaces. 12-2C The pair of view factors F i j ? and F j i ? are related to each other by the reciprocity rule A F A F i ij j ji ? where A i is the area of the surface i and A j is the area of the surface j. Therefore, A F A F F A A F 1 12 2 21 12 2 1 21 ? ? ? ? ? 12-3C The summation rule for an enclosure and is expressed as F i j j N ? ? ? ? 1 1 where N is the number of surfaces of the enclosure. It states that the sum of the view factors from surface i of an enclosure to all surfaces of the enclosure, including to itself must be equal to unity. The superposition rule is stated as the view factor from a surface i to a surface j is equal to the sum of the view factors from surface i to the parts of surface j, F F F 1 2 3 1 2 1 3 ? ? ? ? ? ( , ) . 12-4C The cross-string method is applicable to geometries which are very long in one direction relative to the other directions. By attaching strings between corners the Crossed-Strings Method is expressed as F i i j ? ? ? ? ? ? Crossed strings Uncrossed strings string on surface 2 Chapter 12 Radiation Heat Transfer 12-2 12-5 An enclosure consisting of six surfaces is considered. The number of view factors this geometry involves and the number of these view factors that can be determined by the application of the reciprocity and summation rules are to be determined. Analysis A seven surface enclosure (N=6) involves N 2 2 6 ? ? 36 view factors and we need to determine 15 2 ) 1 6 ( 6 2 ) 1 ( ? ? ? ? N N view factors directly. The remaining 36-15 = 21 of the view factors can be determined by the application of the reciprocity and summation rules. 12-6 An enclosure consisting of five surfaces is considered. The number of view factors this geometry involves and the number of these view factors that can be determined by the application of the reciprocity and summation rules are to be determined. Analysis A five surface enclosure (N=5) involves N 2 2 5 ? ? 25 view factors and we need to determine N N ( ) (5 ) ? ? ? ? 1 2 5 1 2 10 view factors directly. The remaining 25-10 = 15 of the view factors can be determined by the application of the reciprocity and summation rules. 12-7 An enclosure consisting of twelve surfaces is considered. The number of view factors this geometry involves and the number of these view factors that can be determined by the application of the reciprocity and summation rules are to be determined. Analysis A twelve surface enclosure (N=12) involves 144 ? ? 2 2 12 N view factors and we need to determine N N ( ) ( ) ? ? ? ? 1 2 12 12 1 2 66 view factors directly. The remaining 144-66 = 78 of the view factors can be determined by the application of the reciprocity and summation rules. 2 1 4 5 3 6 5 4 3 2 1 2 1 3 9 11 12 10 4 5 8 6 7 Chapter 12 Radiation Heat Transfer 12-3 12-8 The view factors between the rectangular surfaces shown in the figure are to be determined. Assumptions The surfaces are diffuse emitters and reflectors. Analysis From Fig. 12-6, 24 . 0 5 . 0 2 1 1 5 . 0 2 1 31 3 ? ? ? ? ? ? ? ? ? ? ? ? F W L W L and 29 . 0 1 2 2 5 . 0 2 1 ) 2 1 ( 3 2 1 3 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? F W L L W L We note that A 1 = A 3. Then the reciprocity and superposition rules gives 0.24 ? ? ? ? ? ? 31 13 31 3 13 1 A F F F A F 05 . 0 24 . 0 29 . 0 32 32 32 31 ) 2 1 ( 3 ? ? ? ? ? ? ? ? ? ? ? ? ? F F F F F Finally, 0.05 ? ? ? ? ? ? 32 23 3 2 F F A A W = 2 m (2) L2 = 1 m L1 = 1 m L3 = 1 m A 3 (3) A 2 A 1 (1) Chapter 12 Radiation Heat Transfer 12-4 12-9 A cylindrical enclosure is considered. The view factor from the side surface of this cylindrical enclosure to its base surface is to be determined. Assumptions The surfaces are diffuse emitters and reflectors. Analysis We designate the surfaces as follows: Base surface by (1), top surface by (2), and side surface by (3). Then from Fig. 12-7 (or Table 12-1 for better accuracy) 38 . 0 1 1 21 12 2 2 2 1 1 1 ? ? ? ? ? ? ? ? ? ? ? ? ? F F r r L r r r r L 1 : rule summation 13 12 11 ? ? ? F F F 62 . 0 1 38 . 0 0 13 13 ? ? ? ? ? ? ? F F ? ? 0.31 ? ? ? ? ? ? ? ? ? ? ? ? ? ) 62 . 0 ( 2 1 2 2 : rule y reciprocit 13 1 1 2 1 13 1 2 1 13 3 1 31 31 3 13 1 F r r r F L r r F A A F F A F A Discussion This problem can be solved more accurately by using the view factor relation from Table 12-1 to be 1 1 2 2 2 2 1 1 1 1 ? ? ? ? ? ? r r L r R r r L r R 382 . 0 1 1 4 3 3 4 3 1 1 1 1 1 1 5 . 0 2 2 2 1 5 . 0 2 1 2 2 2 1 12 2 2 2 1 2 2 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? R R S S F R R S 618 . 0 382 . 0 1 1 12 13 ? ? ? ? ? F F ? ? 0.309 ? ? ? ? ? ? ? ? ? ? ? ? ? ) 618 . 0 ( 2 1 2 2 : rule y reciprocit 13 1 1 2 1 13 1 2 1 13 3 1 31 31 3 13 1 F r r r F L r r F A A F F A F A (2) (3) (1) L D Chapter 12 Radiation Heat Transfer 12-5 12-10 A semispherical furnace is considered. The view factor from the dome of this furnace to its flat base is to be determined. Assumptions The surfaces are diffuse emitters and reflectors. Analysis We number the surfaces as follows: (1): circular base surface (2): dome surface Surface (1) is flat, and thus F 11 0 ? . 1 1 : rule S ummation 12 12 11 ? ? ? ? F F F 0.5 ? ? ? ? ? ? ? ? ? 2 1 2 4 ) 1 ( A : rule y reciprocit 2 2 2 1 12 2 1 21 21 2 12 1 D D A A F A A F F A F ? ? 12-11 Two view factors associated with three very long ducts with different geometries are to be determined. Assumptions 1 The surfaces are diffuse emitters and reflectors. 2 End effects are neglected. Analysis (a) Surface (1) is flat, and thus F 11 0 ? . 1 ? ? ? ? 12 12 11 1 : rule summation F F F 0.64 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 2 ) 1 ( 2 A : rule y reciprocit 12 2 1 21 21 2 12 1 s D Ds F A A F F A F (b) Noting that surfaces 2 and 3 are symmetrical and thus F F 12 13 ? , the summation rule gives 0.5 ? ? ? ? ? ? ? ? ? ? ? ? ? 12 13 12 13 12 11 1 0 1 F F F F F F Also by using the equation obtained in Example 12-4, F L L L L a b b a a a 12 1 2 3 1 2 2 2 1 2 ? ? ? ? ? ? ? ? ? 0.5 2b a ? ? ? ? ? ? ? ? ? ? ? ? ? 2 1 A : rule y reciprocit 12 2 1 21 21 2 12 1 b a F A A F F A F (c) Applying the crossed-string method gives F F L L L L L a b b a 12 21 5 6 3 4 1 2 2 2 2 2 2 ? ? ? ? ? ? ? ? ? ? ? ( ) ( ) a b b a 2 2 (1) (2) D (1) (2) D (1) (3) (2) a L3 = b L4 = b L5 L6 L 2 = a L 1 = aRead More

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