{"group":{"id":1,"name":"Community","lockable":false,"created_at":"2012-01-18T18:02:15.000Z","updated_at":"2025-12-14T01:33:56.000Z","description":"Problems submitted by members of the MATLAB Central community.","is_default":true,"created_by":161519,"badge_id":null,"featured":false,"trending":false,"solution_count_in_trending_period":0,"trending_last_calculated":"2025-12-14T00:00:00.000Z","image_id":null,"published":true,"community_created":false,"status_id":2,"is_default_group_for_player":false,"deleted_by":null,"deleted_at":null,"restored_by":null,"restored_at":null,"description_opc":null,"description_html":null,"published_at":null},"problems":[{"id":2805,"title":"Radiation Heat Transfer — View Factors (1)","description":"View factors (aka configuration factors) are utilized in some radiation heat transfer models to estimate heat transfer rates between surfaces. In particular, the thermal energy leaving a given surface is applied to other surfaces, as appropriate, based on how much the hot surface \"sees\" the other surfaces. As such, view factors are purely geometrical in nature. A range of view factor formulae are available \u003chttp://www.thermalradiation.net/tablecon.html here\u003e.\r\n\r\nFor this problem, calculate the view factor from surface 1 to surface 2 (F_1-2) for two directly opposed, infinitely long plates of the same finite width given the height, which may be a vector of values:\r\n\r\n\u003c\u003chttp://www.thermalradiation.net/images/C-1fig.gif\u003e\u003e\r\n\r\n\u003c\u003chttp://www.thermalradiation.net/images/C-1eq.gif\u003e\u003e","description_html":"\u003cp\u003eView factors (aka configuration factors) are utilized in some radiation heat transfer models to estimate heat transfer rates between surfaces. In particular, the thermal energy leaving a given surface is applied to other surfaces, as appropriate, based on how much the hot surface \"sees\" the other surfaces. As such, view factors are purely geometrical in nature. A range of view factor formulae are available \u003ca href = \"http://www.thermalradiation.net/tablecon.html\"\u003ehere\u003c/a\u003e.\u003c/p\u003e\u003cp\u003eFor this problem, calculate the view factor from surface 1 to surface 2 (F_1-2) for two directly opposed, infinitely long plates of the same finite width given the height, which may be a vector of values:\u003c/p\u003e\u003cimg src = \"http://www.thermalradiation.net/images/C-1fig.gif\"\u003e\u003cimg src = \"http://www.thermalradiation.net/images/C-1eq.gif\"\u003e","function_template":"function F = view_factor(H)\r\n  F = 1;\r\nend","test_suite":"%%\r\nH = 1;\r\ny_correct = sqrt(2) - 1;\r\nassert(isequal(view_factor(H),y_correct))\r\n\r\n%%\r\nH = 3;\r\ny_correct = sqrt(10) - 3;\r\nassert(isequal(view_factor(H),y_correct))\r\n\r\n%%\r\nH = [0.5 2 4 10];\r\ny_correct = [sqrt(1.25)-0.5, sqrt(5)-2, sqrt(17)-4, sqrt(101)-10];\r\nassert(isequal(view_factor(H),y_correct))","published":true,"deleted":false,"likes_count":6,"comments_count":3,"created_by":26769,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":193,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":26,"created_at":"2014-12-31T02:15:48.000Z","updated_at":"2026-03-22T14:06:29.000Z","published_at":"2014-12-31T02:15:48.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/image\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/media/image1.gif\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/image\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/media/image2.gif\"}],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eView factors (aka configuration factors) are utilized in some radiation heat transfer models to estimate heat transfer rates between surfaces. In particular, the thermal energy leaving a given surface is applied to other surfaces, as appropriate, based on how much the hot surface \\\"sees\\\" the other surfaces. As such, view factors are purely geometrical in nature. 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Heat Transfer — View Factors (4)","description":"View factors (aka configuration factors) are utilized in some radiation heat transfer models to estimate heat transfer rates between surfaces. In particular, the thermal energy leaving a given surface is applied to other surfaces, as appropriate, based on how much the hot surface \"sees\" the other surfaces. As such, view factors are purely geometrical in nature. A range of view factor formulae are available \u003chttp://www.thermalradiation.net/tablecon.html here\u003e.\r\n\r\nFor this problem, calculate the view factor from surface 1 (an infinitely long plate) to surfaces 2 (n rows of in-line pipes):\r\n\r\n\u003c\u003chttp://www.thermalradiation.net/images/C-7fig.gif\u003e\u003e\r\n\r\nThe view factor for one row of pipes is F_1-2 (the first equation):\r\n\r\n\u003c\u003chttp://www.thermalradiation.net/images/C-7eq.gif\u003e\u003e\r\n\r\nThe second equation is utilized for more than one row of pipes. Any of the variables can be a vector. Also, note that D = d/b, where d is the pipe diameter and b is the center-to-center spacing between pipes.","description_html":"\u003cp\u003eView factors (aka configuration factors) are utilized in some radiation heat transfer models to estimate heat transfer rates between surfaces. In particular, the thermal energy leaving a given surface is applied to other surfaces, as appropriate, based on how much the hot surface \"sees\" the other surfaces. As such, view factors are purely geometrical in nature. A range of view factor formulae are available \u003ca href = \"http://www.thermalradiation.net/tablecon.html\"\u003ehere\u003c/a\u003e.\u003c/p\u003e\u003cp\u003eFor this problem, calculate the view factor from surface 1 (an infinitely long plate) to surfaces 2 (n rows of in-line pipes):\u003c/p\u003e\u003cimg src = \"http://www.thermalradiation.net/images/C-7fig.gif\"\u003e\u003cp\u003eThe view factor for one row of pipes is F_1-2 (the first equation):\u003c/p\u003e\u003cimg src = \"http://www.thermalradiation.net/images/C-7eq.gif\"\u003e\u003cp\u003eThe second equation is utilized for more than one row of pipes. Any of the variables can be a vector. Also, note that D = d/b, where d is the pipe diameter and b is the center-to-center spacing between pipes.\u003c/p\u003e","function_template":"function F = view_factor4(d,b,n)\r\n  F = 1;\r\nend","test_suite":"%%\r\nd = 1;   b = 2;   n = 1;\r\ny_correct = 0.6576;\r\nF = view_factor4(d,b,n);\r\nassert(F \u003c (y_correct + 1e-4))\r\nassert(F \u003e (y_correct - 1e-4))\r\n\r\n%%\r\nd = 2;   b = 10;   n = [1 2 4 8 16 32];\r\ny_correct = [0.2941    0.5017    0.7517    0.9383    0.9962    1.0000];\r\nF = view_factor4(d,b,n);\r\nfor i = 1:numel(y_correct)\r\n assert(F(i) \u003c (y_correct(i) + 1e-4))\r\n assert(F(i) \u003e (y_correct(i) - 1e-4))\r\nend\r\n\r\n%%\r\nd = [0.25 0.5 1 1.25 2.5 4 5];\r\nb = [4 10 2.5 2 3 10 11];\r\nn = [1 2 2 1 1 5 5];\r\ny_correct = [0.0962    0.1486    0.7950    0.7792    0.9353    0.9810    0.9908];\r\nF = view_factor4(d,b,n);\r\nfor i = 1:numel(y_correct)\r\n assert(F(i) \u003c (y_correct(i) + 1e-4))\r\n assert(F(i) \u003e (y_correct(i) - 1e-4))\r\nend","published":true,"deleted":false,"likes_count":1,"comments_count":3,"created_by":26769,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":39,"test_suite_updated_at":"2015-01-16T02:04:37.000Z","rescore_all_solutions":false,"group_id":37,"created_at":"2015-01-15T01:47:46.000Z","updated_at":"2026-02-08T12:38:27.000Z","published_at":"2015-01-15T01:47:45.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/image\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/media/image1.gif\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/image\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/media/image2.gif\"}],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eView factors (aka configuration factors) are utilized in some radiation heat transfer models to estimate heat transfer rates between surfaces. In particular, the thermal energy leaving a given surface is applied to other surfaces, as appropriate, based on how much the hot surface \\\"sees\\\" the other surfaces. As such, view factors are purely geometrical in nature. A range of view factor formulae are available\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://www.thermalradiation.net/tablecon.html\\\"\u003e\u003cw:r\u003e\u003cw:t\u003ehere\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor this problem, calculate the view factor from surface 1 (an infinitely long plate) to surfaces 2 (n rows of in-line pipes):\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe view factor for one row of pipes is F_1-2 (the first equation):\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId2\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe second equation is utilized for more than one row of pipes. Any of the variables can be a vector. Also, note that D = d/b, where d is the pipe diameter and b is the center-to-center spacing between pipes.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray 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Heat Transfer — View Factors (3)","description":"View factors (aka configuration factors) are utilized in some radiation heat transfer models to estimate heat transfer rates between surfaces. In particular, the thermal energy leaving a given surface is applied to other surfaces, as appropriate, based on how much the hot surface \"sees\" the other surfaces. As such, view factors are purely geometrical in nature. A range of view factor formulae are available \u003chttp://www.thermalradiation.net/tablecon.html here\u003e.\r\n\r\nFor this problem, calculate the view factor from surface 1 to surface 2 (F_1-2) for two infinitely long plates of different finite widths and having a common edge given the angle between the plates:\r\n\r\n\u003c\u003chttp://www.thermalradiation.net/images/C-5fig.gif\u003e\u003e\r\n\r\n-\r\n\r\n\u003c\u003chttp://www.thermalradiation.net/images/C-5a%20eq.gif\u003e\u003e\r\n\r\nAny of the variables can be a vector. Also, note that A = a/b, and is different than A_1 or A_2. Finally, the angle is provided in degrees.","description_html":"\u003cp\u003eView factors (aka configuration factors) are utilized in some radiation heat transfer models to estimate heat transfer rates between surfaces. In particular, the thermal energy leaving a given surface is applied to other surfaces, as appropriate, based on how much the hot surface \"sees\" the other surfaces. As such, view factors are purely geometrical in nature. A range of view factor formulae are available \u003ca href = \"http://www.thermalradiation.net/tablecon.html\"\u003ehere\u003c/a\u003e.\u003c/p\u003e\u003cp\u003eFor this problem, calculate the view factor from surface 1 to surface 2 (F_1-2) for two infinitely long plates of different finite widths and having a common edge given the angle between the plates:\u003c/p\u003e\u003cimg src = \"http://www.thermalradiation.net/images/C-5fig.gif\"\u003e\u003cp\u003e-\u003c/p\u003e\u003cimg src = \"http://www.thermalradiation.net/images/C-5a%20eq.gif\"\u003e\u003cp\u003eAny of the variables can be a vector. Also, note that A = a/b, and is different than A_1 or A_2. Finally, the angle is provided in degrees.\u003c/p\u003e","function_template":"function F = view_factor3(a,b,alpha)\r\n  F = 1;\r\nend","test_suite":"%%\r\na = 1;   b = 2;   alpha = 135;\r\ny_correct = 0.0505;\r\nF = view_factor3(a,b,alpha);\r\nassert(F \u003c (y_correct + 1e-4))\r\nassert(F \u003e (y_correct - 1e-4))\r\n\r\n%%\r\na = 4;   b = 1;   alpha = [45 90 135 180];\r\ny_correct = [0.8160    0.4384    0.1200         0];\r\nF = view_factor3(a,b,alpha);\r\nfor i = 1:numel(y_correct)\r\n assert(F(i) \u003c (y_correct(i) + 1e-4))\r\n assert(F(i) \u003e (y_correct(i) - 1e-4))\r\nend\r\n\r\n%%\r\na = [1 2 5 10 1 2 5 10];\r\nb = [2 5 2 5 2 5 2 5];\r\nalpha = [60 60 60 60 120 120 120 120];\r\ny_correct = [0.3170    0.2641    0.6603    0.6340    0.0886    0.0755    0.1888    0.1771];\r\nF = view_factor3(a,b,alpha);\r\nfor i = 1:numel(y_correct)\r\n assert(F(i) \u003c (y_correct(i) + 1e-4))\r\n assert(F(i) \u003e (y_correct(i) - 1e-4))\r\nend","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":26769,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":52,"test_suite_updated_at":"2015-01-01T19:53:58.000Z","rescore_all_solutions":false,"group_id":26,"created_at":"2014-12-31T02:39:43.000Z","updated_at":"2026-02-19T10:53:46.000Z","published_at":"2014-12-31T02:50:28.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/image\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/media/image1.gif\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/image\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/media/image2.gif\"}],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eView factors (aka configuration factors) are utilized in some radiation heat transfer models to estimate heat transfer rates between surfaces. In particular, the thermal energy leaving a given surface is applied to other surfaces, as appropriate, based on how much the hot surface \\\"sees\\\" the other surfaces. As such, view factors are purely geometrical in nature. A range of view factor formulae are available\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://www.thermalradiation.net/tablecon.html\\\"\u003e\u003cw:r\u003e\u003cw:t\u003ehere\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor this problem, calculate the view factor from surface 1 to surface 2 (F_1-2) for two infinitely long plates of different finite widths and having a common edge given the angle between the plates:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e-\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId2\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eAny of the variables can be a vector. Also, note that A = a/b, and is different than A_1 or A_2. Finally, the angle is provided in degrees.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"},{\"partUri\":\"/media/image1.gif\",\"contentType\":\"image/gif\",\"content\":\"data:image/gif;base64,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\"},{\"partUri\":\"/media/image2.gif\",\"contentType\":\"image/gif\",\"content\":\"data:image/gif;base64,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\"}]}"},{"id":2806,"title":"Radiation Heat Transfer — View Factors (2)","description":"View factors (aka configuration factors) are utilized in some radiation heat transfer models to estimate heat transfer rates between surfaces. In particular, the thermal energy leaving a given surface is applied to other surfaces, as appropriate, based on how much the hot surface \"sees\" the other surfaces. As such, view factors are purely geometrical in nature. A range of view factor formulae are available \u003chttp://www.thermalradiation.net/tablecon.html here\u003e.\r\n\r\nFor this problem, calculate the view factor from surface 1 to surface 2 (F_1-2) for two directly opposed, infinitely long plates of different finite widths given the distance between the plates:\r\n\r\n\u003c\u003chttp://www.thermalradiation.net/images/C-2fig.gif\u003e\u003e\r\n\r\n\u003c\u003chttp://www.thermalradiation.net/images/C-2eq.gif\u003e\u003e\r\n\r\nEach variable may be a vector of values. Also, note that B = b/a and C = c/a.","description_html":"\u003cp\u003eView factors (aka configuration factors) are utilized in some radiation heat transfer models to estimate heat transfer rates between surfaces. In particular, the thermal energy leaving a given surface is applied to other surfaces, as appropriate, based on how much the hot surface \"sees\" the other surfaces. As such, view factors are purely geometrical in nature. A range of view factor formulae are available \u003ca href = \"http://www.thermalradiation.net/tablecon.html\"\u003ehere\u003c/a\u003e.\u003c/p\u003e\u003cp\u003eFor this problem, calculate the view factor from surface 1 to surface 2 (F_1-2) for two directly opposed, infinitely long plates of different finite widths given the distance between the plates:\u003c/p\u003e\u003cimg src = \"http://www.thermalradiation.net/images/C-2fig.gif\"\u003e\u003cimg src = \"http://www.thermalradiation.net/images/C-2eq.gif\"\u003e\u003cp\u003eEach variable may be a vector of values. Also, note that B = b/a and C = c/a.\u003c/p\u003e","function_template":"function F = view_factor2(a,b,c)\r\n  F = 1;\r\nend","test_suite":"%%\r\na = 1;   b = 1;   c = 2;\r\ny_correct = 0.6847;\r\nF = view_factor2(a,b,c);\r\nassert(F \u003c (y_correct + 1e-4))\r\n\r\n%%\r\na = 4;   b = 1;   c = [2 4 8];\r\ny_correct = [0.2409    0.4450    0.7057];\r\nF = view_factor2(a,b,c);\r\nfor i = 1:numel(y_correct)\r\n assert(F(i) \u003c (y_correct(i) + 1e-4))\r\n assert(F(i) \u003e (y_correct(i) - 1e-4))\r\nend\r\n\r\n%%\r\na = [1 2 5 10];\r\nb = [2 5 2 5];\r\nc = [6 6 10 10];\r\ny_correct = [0.9435    0.7582    0.7036    0.4384];\r\nF = view_factor2(a,b,c);\r\nfor i = 1:numel(y_correct)\r\n assert(F(i) \u003c (y_correct(i) + 1e-4))\r\n assert(F(i) \u003e (y_correct(i) - 1e-4))\r\nend","published":true,"deleted":false,"likes_count":2,"comments_count":0,"created_by":26769,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":54,"test_suite_updated_at":"2015-01-01T19:59:48.000Z","rescore_all_solutions":false,"group_id":26,"created_at":"2014-12-31T02:27:10.000Z","updated_at":"2026-02-19T10:52:03.000Z","published_at":"2014-12-31T02:27:10.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/image\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/media/image1.gif\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/image\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/media/image2.gif\"}],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eView factors (aka configuration factors) are utilized in some radiation heat transfer models to estimate heat transfer rates between surfaces. In particular, the thermal energy leaving a given surface is applied to other surfaces, as appropriate, based on how much the hot surface \\\"sees\\\" the other surfaces. As such, view factors are purely geometrical in nature. A range of view factor formulae are available\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://www.thermalradiation.net/tablecon.html\\\"\u003e\u003cw:r\u003e\u003cw:t\u003ehere\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor this problem, calculate the view factor from surface 1 to surface 2 (F_1-2) for two directly opposed, infinitely long plates of different finite widths given the distance between the plates:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId2\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eEach variable may be a vector of values. Also, note that B = b/a and C = c/a.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"},{\"partUri\":\"/media/image1.gif\",\"contentType\":\"image/gif\",\"content\":\"data:image/gif;base64,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\"},{\"partUri\":\"/media/image2.gif\",\"contentType\":\"image/gif\",\"content\":\"data:image/gif;base64,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\"}]}"},{"id":2833,"title":"Radiation Heat Transfer — View Factors (5)","description":"View factors (aka configuration factors) are utilized in some radiation heat transfer models to estimate heat transfer rates between surfaces. In particular, the thermal energy leaving a given surface is applied to other surfaces, as appropriate, based on how much the hot surface \"sees\" the other surfaces. As such, view factors are purely geometrical in nature. A range of view factor formulae are available \u003chttp://www.thermalradiation.net/tablecon.html here\u003e.\r\n\r\nFor this problem, calculate the view factor from surface 1 (an infinitely long plate) to surfaces 2 (n rows of in-line pipes):\r\n\r\n\u003c\u003chttp://www.thermalradiation.net/images/C-7fig.gif\u003e\u003e\r\n\r\nThe view factor for one row of pipes is F_1-2 (the first equation):\r\n\r\n\u003c\u003chttp://www.thermalradiation.net/images/C-7eq.gif\u003e\u003e\r\n\r\nThe second equation is utilized for more than one row of pipes. Any of the variables can be a vector. Also, note that D = d/b, where d is the pipe diameter and b is the center-to-center spacing between pipes.\r\n\r\n*Note: This problem is identical to the previous problem (4) except that the variables will be provided in vectors of varying sizes.* The variables will need to be combined to produce all the applicable combinations. For example, if the sizes of d, b, and n are 2, 5, and 1, respectively, there will be ten total variable combinations (and answers). As another example, if the sizes of d, b, and n are 4, 3, and 2, respectively, there will be 24 total variable combinations. This simulates a parametric design study. Also, because all possible combinations can be generated in various orders, the answers will be sorted; the order that you output answers does not matter.","description_html":"\u003cp\u003eView factors (aka configuration factors) are utilized in some radiation heat transfer models to estimate heat transfer rates between surfaces. In particular, the thermal energy leaving a given surface is applied to other surfaces, as appropriate, based on how much the hot surface \"sees\" the other surfaces. As such, view factors are purely geometrical in nature. A range of view factor formulae are available \u003ca href = \"http://www.thermalradiation.net/tablecon.html\"\u003ehere\u003c/a\u003e.\u003c/p\u003e\u003cp\u003eFor this problem, calculate the view factor from surface 1 (an infinitely long plate) to surfaces 2 (n rows of in-line pipes):\u003c/p\u003e\u003cimg src = \"http://www.thermalradiation.net/images/C-7fig.gif\"\u003e\u003cp\u003eThe view factor for one row of pipes is F_1-2 (the first equation):\u003c/p\u003e\u003cimg src = \"http://www.thermalradiation.net/images/C-7eq.gif\"\u003e\u003cp\u003eThe second equation is utilized for more than one row of pipes. Any of the variables can be a vector. Also, note that D = d/b, where d is the pipe diameter and b is the center-to-center spacing between pipes.\u003c/p\u003e\u003cp\u003e\u003cb\u003eNote: This problem is identical to the previous problem (4) except that the variables will be provided in vectors of varying sizes.\u003c/b\u003e The variables will need to be combined to produce all the applicable combinations. For example, if the sizes of d, b, and n are 2, 5, and 1, respectively, there will be ten total variable combinations (and answers). As another example, if the sizes of d, b, and n are 4, 3, and 2, respectively, there will be 24 total variable combinations. This simulates a parametric design study. Also, because all possible combinations can be generated in various orders, the answers will be sorted; the order that you output answers does not matter.\u003c/p\u003e","function_template":"function F = view_factor5(d,b,n)\r\n  F = 1;\r\nend","test_suite":"%%\r\nd = [1 2];\r\nb = [3 5 6 7 8];\r\nn = 1;\r\ny_correct = [0.1885    0.2142    0.2479    0.2941    0.3613    0.4077    0.4675    0.4675    0.5472    0.8154];\r\nF = sort(view_factor5(d,b,n));\r\nfor i = 1:numel(y_correct)\r\n assert(F(i) \u003c (y_correct(i) + 1e-4))\r\n assert(F(i) \u003e (y_correct(i) - 1e-4))\r\nend\r\n\r\n%%\r\nd = [2 2.2 2.4 2.5];\r\nb = [3.2 3.4 3.6];\r\nn = 2;\r\ny_correct = [0.9182    0.9352    0.9455    0.9512    0.9594    0.9659    0.9720    0.9738    0.9767    0.9831    0.9857    0.9905];\r\nF = sort(view_factor5(d,b,n));\r\nfor i = 1:numel(y_correct)\r\n assert(F(i) \u003c (y_correct(i) + 1e-4))\r\n assert(F(i) \u003e (y_correct(i) - 1e-4))\r\nend\r\n\r\n%%\r\nd = [1 1.1 1.2];\r\nb = [3 3.2];\r\nn = 1:5;\r\ny_correct = [0.4416    0.4675    0.4803    0.5080    0.5179    0.5472    0.6882    0.7165    0.7299    0.7579    0.7676    0.7950    0.8259    0.8490    0.8596    0.8809    0.8879    0.9028    0.9072    0.9196    0.9270    0.9414    0.9457    0.9460    0.9572    0.9580    0.9621    0.9712    0.9740    0.9810];\r\nF = sort(view_factor5(d,b,n));\r\nfor i = 1:numel(y_correct)\r\n assert(F(i) \u003c (y_correct(i) + 1e-4))\r\n assert(F(i) \u003e (y_correct(i) - 1e-4))\r\nend","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":26769,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":37,"test_suite_updated_at":"2015-01-16T02:08:47.000Z","rescore_all_solutions":false,"group_id":37,"created_at":"2015-01-15T02:16:48.000Z","updated_at":"2026-02-08T12:40:09.000Z","published_at":"2015-01-15T02:16:48.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/image\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/media/image1.gif\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/image\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/media/image2.gif\"}],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eView factors (aka configuration factors) are utilized in some radiation heat transfer models to estimate heat transfer rates between surfaces. In particular, the thermal energy leaving a given surface is applied to other surfaces, as appropriate, based on how much the hot surface \\\"sees\\\" the other surfaces. As such, view factors are purely geometrical in nature. A range of view factor formulae are available\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://www.thermalradiation.net/tablecon.html\\\"\u003e\u003cw:r\u003e\u003cw:t\u003ehere\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor this problem, calculate the view factor from surface 1 (an infinitely long plate) to surfaces 2 (n rows of in-line pipes):\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe view factor for one row of pipes is F_1-2 (the first equation):\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId2\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe second equation is utilized for more than one row of pipes. Any of the variables can be a vector. Also, note that D = d/b, where d is the pipe diameter and b is the center-to-center spacing between pipes.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eNote: This problem is identical to the previous problem (4) except that the variables will be provided in vectors of varying sizes.\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e The variables will need to be combined to produce all the applicable combinations. For example, if the sizes of d, b, and n are 2, 5, and 1, respectively, there will be ten total variable combinations (and answers). As another example, if the sizes of d, b, and n are 4, 3, and 2, respectively, there will be 24 total variable combinations. This simulates a parametric design study. 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:[{"id":2805,"title":"Radiation Heat Transfer — View Factors (1)","description":"View factors (aka configuration factors) are utilized in some radiation heat transfer models to estimate heat transfer rates between surfaces. In particular, the thermal energy leaving a given surface is applied to other surfaces, as appropriate, based on how much the hot surface \"sees\" the other surfaces. As such, view factors are purely geometrical in nature. A range of view factor formulae are available \u003chttp://www.thermalradiation.net/tablecon.html here\u003e.\r\n\r\nFor this problem, calculate the view factor from surface 1 to surface 2 (F_1-2) for two directly opposed, infinitely long plates of the same finite width given the height, which may be a vector of values:\r\n\r\n\u003c\u003chttp://www.thermalradiation.net/images/C-1fig.gif\u003e\u003e\r\n\r\n\u003c\u003chttp://www.thermalradiation.net/images/C-1eq.gif\u003e\u003e","description_html":"\u003cp\u003eView factors (aka configuration factors) are utilized in some radiation heat transfer models to estimate heat transfer rates between surfaces. In particular, the thermal energy leaving a given surface is applied to other surfaces, as appropriate, based on how much the hot surface \"sees\" the other surfaces. As such, view factors are purely geometrical in nature. A range of view factor formulae are available \u003ca href = \"http://www.thermalradiation.net/tablecon.html\"\u003ehere\u003c/a\u003e.\u003c/p\u003e\u003cp\u003eFor this problem, calculate the view factor from surface 1 to surface 2 (F_1-2) for two directly opposed, infinitely long plates of the same finite width given the height, which may be a vector of values:\u003c/p\u003e\u003cimg src = \"http://www.thermalradiation.net/images/C-1fig.gif\"\u003e\u003cimg src = \"http://www.thermalradiation.net/images/C-1eq.gif\"\u003e","function_template":"function F = view_factor(H)\r\n  F = 1;\r\nend","test_suite":"%%\r\nH = 1;\r\ny_correct = sqrt(2) - 1;\r\nassert(isequal(view_factor(H),y_correct))\r\n\r\n%%\r\nH = 3;\r\ny_correct = sqrt(10) - 3;\r\nassert(isequal(view_factor(H),y_correct))\r\n\r\n%%\r\nH = [0.5 2 4 10];\r\ny_correct = [sqrt(1.25)-0.5, sqrt(5)-2, sqrt(17)-4, sqrt(101)-10];\r\nassert(isequal(view_factor(H),y_correct))","published":true,"deleted":false,"likes_count":6,"comments_count":3,"created_by":26769,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":193,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":26,"created_at":"2014-12-31T02:15:48.000Z","updated_at":"2026-03-22T14:06:29.000Z","published_at":"2014-12-31T02:15:48.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/image\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/media/image1.gif\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/image\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/media/image2.gif\"}],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eView factors (aka configuration factors) are utilized in some radiation heat transfer models to estimate heat transfer rates between surfaces. In particular, the thermal energy leaving a given surface is applied to other surfaces, as appropriate, based on how much the hot surface \\\"sees\\\" the other surfaces. As such, view factors are purely geometrical in nature. A range of view factor formulae are available\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://www.thermalradiation.net/tablecon.html\\\"\u003e\u003cw:r\u003e\u003cw:t\u003ehere\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor this problem, calculate the view factor from surface 1 to surface 2 (F_1-2) for two directly opposed, infinitely long plates of the same finite width given the height, which may be a vector of values:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" 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Heat Transfer — View Factors (4)","description":"View factors (aka configuration factors) are utilized in some radiation heat transfer models to estimate heat transfer rates between surfaces. In particular, the thermal energy leaving a given surface is applied to other surfaces, as appropriate, based on how much the hot surface \"sees\" the other surfaces. As such, view factors are purely geometrical in nature. A range of view factor formulae are available \u003chttp://www.thermalradiation.net/tablecon.html here\u003e.\r\n\r\nFor this problem, calculate the view factor from surface 1 (an infinitely long plate) to surfaces 2 (n rows of in-line pipes):\r\n\r\n\u003c\u003chttp://www.thermalradiation.net/images/C-7fig.gif\u003e\u003e\r\n\r\nThe view factor for one row of pipes is F_1-2 (the first equation):\r\n\r\n\u003c\u003chttp://www.thermalradiation.net/images/C-7eq.gif\u003e\u003e\r\n\r\nThe second equation is utilized for more than one row of pipes. Any of the variables can be a vector. Also, note that D = d/b, where d is the pipe diameter and b is the center-to-center spacing between pipes.","description_html":"\u003cp\u003eView factors (aka configuration factors) are utilized in some radiation heat transfer models to estimate heat transfer rates between surfaces. In particular, the thermal energy leaving a given surface is applied to other surfaces, as appropriate, based on how much the hot surface \"sees\" the other surfaces. As such, view factors are purely geometrical in nature. A range of view factor formulae are available \u003ca href = \"http://www.thermalradiation.net/tablecon.html\"\u003ehere\u003c/a\u003e.\u003c/p\u003e\u003cp\u003eFor this problem, calculate the view factor from surface 1 (an infinitely long plate) to surfaces 2 (n rows of in-line pipes):\u003c/p\u003e\u003cimg src = \"http://www.thermalradiation.net/images/C-7fig.gif\"\u003e\u003cp\u003eThe view factor for one row of pipes is F_1-2 (the first equation):\u003c/p\u003e\u003cimg src = \"http://www.thermalradiation.net/images/C-7eq.gif\"\u003e\u003cp\u003eThe second equation is utilized for more than one row of pipes. Any of the variables can be a vector. Also, note that D = d/b, where d is the pipe diameter and b is the center-to-center spacing between pipes.\u003c/p\u003e","function_template":"function F = view_factor4(d,b,n)\r\n  F = 1;\r\nend","test_suite":"%%\r\nd = 1;   b = 2;   n = 1;\r\ny_correct = 0.6576;\r\nF = view_factor4(d,b,n);\r\nassert(F \u003c (y_correct + 1e-4))\r\nassert(F \u003e (y_correct - 1e-4))\r\n\r\n%%\r\nd = 2;   b = 10;   n = [1 2 4 8 16 32];\r\ny_correct = [0.2941    0.5017    0.7517    0.9383    0.9962    1.0000];\r\nF = view_factor4(d,b,n);\r\nfor i = 1:numel(y_correct)\r\n assert(F(i) \u003c (y_correct(i) + 1e-4))\r\n assert(F(i) \u003e (y_correct(i) - 1e-4))\r\nend\r\n\r\n%%\r\nd = [0.25 0.5 1 1.25 2.5 4 5];\r\nb = [4 10 2.5 2 3 10 11];\r\nn = [1 2 2 1 1 5 5];\r\ny_correct = [0.0962    0.1486    0.7950    0.7792    0.9353    0.9810    0.9908];\r\nF = view_factor4(d,b,n);\r\nfor i = 1:numel(y_correct)\r\n assert(F(i) \u003c (y_correct(i) + 1e-4))\r\n assert(F(i) \u003e (y_correct(i) - 1e-4))\r\nend","published":true,"deleted":false,"likes_count":1,"comments_count":3,"created_by":26769,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":39,"test_suite_updated_at":"2015-01-16T02:04:37.000Z","rescore_all_solutions":false,"group_id":37,"created_at":"2015-01-15T01:47:46.000Z","updated_at":"2026-02-08T12:38:27.000Z","published_at":"2015-01-15T01:47:45.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/image\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/media/image1.gif\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/image\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/media/image2.gif\"}],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eView factors (aka configuration factors) are utilized in some radiation heat transfer models to estimate heat transfer rates between surfaces. In particular, the thermal energy leaving a given surface is applied to other surfaces, as appropriate, based on how much the hot surface \\\"sees\\\" the other surfaces. As such, view factors are purely geometrical in nature. A range of view factor formulae are available\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://www.thermalradiation.net/tablecon.html\\\"\u003e\u003cw:r\u003e\u003cw:t\u003ehere\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor this problem, calculate the view factor from surface 1 (an infinitely long plate) to surfaces 2 (n rows of in-line pipes):\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe view factor for one row of pipes is F_1-2 (the first equation):\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId2\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe second equation is utilized for more than one row of pipes. Any of the variables can be a vector. 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Heat Transfer — View Factors (3)","description":"View factors (aka configuration factors) are utilized in some radiation heat transfer models to estimate heat transfer rates between surfaces. In particular, the thermal energy leaving a given surface is applied to other surfaces, as appropriate, based on how much the hot surface \"sees\" the other surfaces. As such, view factors are purely geometrical in nature. A range of view factor formulae are available \u003chttp://www.thermalradiation.net/tablecon.html here\u003e.\r\n\r\nFor this problem, calculate the view factor from surface 1 to surface 2 (F_1-2) for two infinitely long plates of different finite widths and having a common edge given the angle between the plates:\r\n\r\n\u003c\u003chttp://www.thermalradiation.net/images/C-5fig.gif\u003e\u003e\r\n\r\n-\r\n\r\n\u003c\u003chttp://www.thermalradiation.net/images/C-5a%20eq.gif\u003e\u003e\r\n\r\nAny of the variables can be a vector. Also, note that A = a/b, and is different than A_1 or A_2. Finally, the angle is provided in degrees.","description_html":"\u003cp\u003eView factors (aka configuration factors) are utilized in some radiation heat transfer models to estimate heat transfer rates between surfaces. In particular, the thermal energy leaving a given surface is applied to other surfaces, as appropriate, based on how much the hot surface \"sees\" the other surfaces. As such, view factors are purely geometrical in nature. A range of view factor formulae are available \u003ca href = \"http://www.thermalradiation.net/tablecon.html\"\u003ehere\u003c/a\u003e.\u003c/p\u003e\u003cp\u003eFor this problem, calculate the view factor from surface 1 to surface 2 (F_1-2) for two infinitely long plates of different finite widths and having a common edge given the angle between the plates:\u003c/p\u003e\u003cimg src = \"http://www.thermalradiation.net/images/C-5fig.gif\"\u003e\u003cp\u003e-\u003c/p\u003e\u003cimg src = \"http://www.thermalradiation.net/images/C-5a%20eq.gif\"\u003e\u003cp\u003eAny of the variables can be a vector. Also, note that A = a/b, and is different than A_1 or A_2. Finally, the angle is provided in degrees.\u003c/p\u003e","function_template":"function F = view_factor3(a,b,alpha)\r\n  F = 1;\r\nend","test_suite":"%%\r\na = 1;   b = 2;   alpha = 135;\r\ny_correct = 0.0505;\r\nF = view_factor3(a,b,alpha);\r\nassert(F \u003c (y_correct + 1e-4))\r\nassert(F \u003e (y_correct - 1e-4))\r\n\r\n%%\r\na = 4;   b = 1;   alpha = [45 90 135 180];\r\ny_correct = [0.8160    0.4384    0.1200         0];\r\nF = view_factor3(a,b,alpha);\r\nfor i = 1:numel(y_correct)\r\n assert(F(i) \u003c (y_correct(i) + 1e-4))\r\n assert(F(i) \u003e (y_correct(i) - 1e-4))\r\nend\r\n\r\n%%\r\na = [1 2 5 10 1 2 5 10];\r\nb = [2 5 2 5 2 5 2 5];\r\nalpha = [60 60 60 60 120 120 120 120];\r\ny_correct = [0.3170    0.2641    0.6603    0.6340    0.0886    0.0755    0.1888    0.1771];\r\nF = view_factor3(a,b,alpha);\r\nfor i = 1:numel(y_correct)\r\n assert(F(i) \u003c (y_correct(i) + 1e-4))\r\n assert(F(i) \u003e (y_correct(i) - 1e-4))\r\nend","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":26769,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":52,"test_suite_updated_at":"2015-01-01T19:53:58.000Z","rescore_all_solutions":false,"group_id":26,"created_at":"2014-12-31T02:39:43.000Z","updated_at":"2026-02-19T10:53:46.000Z","published_at":"2014-12-31T02:50:28.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/image\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/media/image1.gif\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/image\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/media/image2.gif\"}],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eView factors (aka configuration factors) are utilized in some radiation heat transfer models to estimate heat transfer rates between surfaces. In particular, the thermal energy leaving a given surface is applied to other surfaces, as appropriate, based on how much the hot surface \\\"sees\\\" the other surfaces. As such, view factors are purely geometrical in nature. A range of view factor formulae are available\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://www.thermalradiation.net/tablecon.html\\\"\u003e\u003cw:r\u003e\u003cw:t\u003ehere\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor this problem, calculate the view factor from surface 1 to surface 2 (F_1-2) for two infinitely long plates of different finite widths and having a common edge given the angle between the plates:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e-\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId2\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eAny of the variables can be a vector. Also, note that A = a/b, and is different than A_1 or A_2. Finally, the angle is provided in degrees.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"},{\"partUri\":\"/media/image1.gif\",\"contentType\":\"image/gif\",\"content\":\"data:image/gif;base64,R0lGODlh4wB3AIAAAAAAAP///yH5BAAAAAAALAAAAADjAHcAAAL/jI+py+0Po5y02ouz3rz7D4biSJbmiZYAkLbuC4frGtf2jQesPuf+D+TQDr2g8YhU7BLFpPNpGyqbxyX0erIuZlpgFwv2SB1Ury6MFk+41XP6fRlH2GbDF47fYsoaFvcPKLd0l1coV8EXJ0RnR1j4uJaIuKG1YwmJuScp4cjJ9JmJ1/mwCTFK2sR4GopUSubasNrHKspIYeuZ5UaLFmiBS4oCzAsVKEt0SEa8vPgXB2vHLJ0BSA17PJ3dmIzIjYCtPd0DzlRKHs48dP4tuY6e7b79Ff8uTc9HX5/ebJWvv08p1b+Bufp4I4gQlMGDCRveg9Zw4EOGESV2gFix3kR//xlZbezYMd8wkBYDUiSJ7iNKhxcxrgRo8iVLkxxlwhHp0iYtnCd1ElPpU2PLnkE9Dq1ZNNzIpDKXMl3p9CnJqB+KIJVqiugUrdSi7QJ5dQ8Ra1x/ffsastXCUWW9NoKK9gbVV+3m6hkLN6xZt5oE+VL0FuUgH4Np8jDW9nDgtHzl2qVrydjhxJj8IL6MyxkvW3/Z6btaCVkmzpq36m0jQ2HcPJmv5ewlI5XVUKVFR9pc43SMx6hoR0HdlyPlILqVAX82fIqhKMlFuONdME1xUWudTwdxnTrZsNkvNt3OHUx37cFJjAds8/nr6MfTVx/RHH7SdbVTr3fOVH38LfVzS//VXxNiOZxnVHABdvYbVgDG1t8LBO70XlUNtvAgbtsxuN9eWKnWzX2xeEjJhkpEKOGDFcKEHGgninhXiiuymMIgx2BGY4023ohjjjrueGMSlnHFY5BCDklkkTY6MWEskxnJZJNOPgmiedVw8uJURgiYFV4welcFgi02tmVXPiZ5SJUZmcnAhHegGRGbaZZJiJsJyZmmZ8aFGaJ0c+ApRC+r0EkQoF+awmeexWAj6D+J8oBeoY0WKJ44yyxqJ4qQKkccJN5QKpZpV2qaCKePTmlGhjCQCuZoyKBK2CU3dSbqL1AyqZAjs95KQ6yTyAOdLqliIdmvmpbTqwmFvcGqrvwdfCrsoaYaImeUjjlKbbXWXottttpuy2233n7raAEAOw==\"},{\"partUri\":\"/media/image2.gif\",\"contentType\":\"image/gif\",\"content\":\"data:image/gif;base64,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\"}]}"},{"id":2806,"title":"Radiation Heat Transfer — View Factors (2)","description":"View factors (aka configuration factors) are utilized in some radiation heat transfer models to estimate heat transfer rates between surfaces. In particular, the thermal energy leaving a given surface is applied to other surfaces, as appropriate, based on how much the hot surface \"sees\" the other surfaces. As such, view factors are purely geometrical in nature. A range of view factor formulae are available \u003chttp://www.thermalradiation.net/tablecon.html here\u003e.\r\n\r\nFor this problem, calculate the view factor from surface 1 to surface 2 (F_1-2) for two directly opposed, infinitely long plates of different finite widths given the distance between the plates:\r\n\r\n\u003c\u003chttp://www.thermalradiation.net/images/C-2fig.gif\u003e\u003e\r\n\r\n\u003c\u003chttp://www.thermalradiation.net/images/C-2eq.gif\u003e\u003e\r\n\r\nEach variable may be a vector of values. Also, note that B = b/a and C = c/a.","description_html":"\u003cp\u003eView factors (aka configuration factors) are utilized in some radiation heat transfer models to estimate heat transfer rates between surfaces. In particular, the thermal energy leaving a given surface is applied to other surfaces, as appropriate, based on how much the hot surface \"sees\" the other surfaces. As such, view factors are purely geometrical in nature. A range of view factor formulae are available \u003ca href = \"http://www.thermalradiation.net/tablecon.html\"\u003ehere\u003c/a\u003e.\u003c/p\u003e\u003cp\u003eFor this problem, calculate the view factor from surface 1 to surface 2 (F_1-2) for two directly opposed, infinitely long plates of different finite widths given the distance between the plates:\u003c/p\u003e\u003cimg src = \"http://www.thermalradiation.net/images/C-2fig.gif\"\u003e\u003cimg src = \"http://www.thermalradiation.net/images/C-2eq.gif\"\u003e\u003cp\u003eEach variable may be a vector of values. Also, note that B = b/a and C = c/a.\u003c/p\u003e","function_template":"function F = view_factor2(a,b,c)\r\n  F = 1;\r\nend","test_suite":"%%\r\na = 1;   b = 1;   c = 2;\r\ny_correct = 0.6847;\r\nF = view_factor2(a,b,c);\r\nassert(F \u003c (y_correct + 1e-4))\r\n\r\n%%\r\na = 4;   b = 1;   c = [2 4 8];\r\ny_correct = [0.2409    0.4450    0.7057];\r\nF = view_factor2(a,b,c);\r\nfor i = 1:numel(y_correct)\r\n assert(F(i) \u003c (y_correct(i) + 1e-4))\r\n assert(F(i) \u003e (y_correct(i) - 1e-4))\r\nend\r\n\r\n%%\r\na = [1 2 5 10];\r\nb = [2 5 2 5];\r\nc = [6 6 10 10];\r\ny_correct = [0.9435    0.7582    0.7036    0.4384];\r\nF = view_factor2(a,b,c);\r\nfor i = 1:numel(y_correct)\r\n assert(F(i) \u003c (y_correct(i) + 1e-4))\r\n assert(F(i) \u003e (y_correct(i) - 1e-4))\r\nend","published":true,"deleted":false,"likes_count":2,"comments_count":0,"created_by":26769,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":54,"test_suite_updated_at":"2015-01-01T19:59:48.000Z","rescore_all_solutions":false,"group_id":26,"created_at":"2014-12-31T02:27:10.000Z","updated_at":"2026-02-19T10:52:03.000Z","published_at":"2014-12-31T02:27:10.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/image\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/media/image1.gif\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/image\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/media/image2.gif\"}],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eView factors (aka configuration factors) are utilized in some radiation heat transfer models to estimate heat transfer rates between surfaces. In particular, the thermal energy leaving a given surface is applied to other surfaces, as appropriate, based on how much the hot surface \\\"sees\\\" the other surfaces. As such, view factors are purely geometrical in nature. A range of view factor formulae are available\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://www.thermalradiation.net/tablecon.html\\\"\u003e\u003cw:r\u003e\u003cw:t\u003ehere\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor this problem, calculate the view factor from surface 1 to surface 2 (F_1-2) for two directly opposed, infinitely long plates of different finite widths given the distance between the plates:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId2\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eEach variable may be a vector of values. Also, note that B = b/a and C = c/a.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"},{\"partUri\":\"/media/image1.gif\",\"contentType\":\"image/gif\",\"content\":\"data:image/gif;base64,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\"},{\"partUri\":\"/media/image2.gif\",\"contentType\":\"image/gif\",\"content\":\"data:image/gif;base64,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\"}]}"},{"id":2833,"title":"Radiation Heat Transfer — View Factors (5)","description":"View factors (aka configuration factors) are utilized in some radiation heat transfer models to estimate heat transfer rates between surfaces. In particular, the thermal energy leaving a given surface is applied to other surfaces, as appropriate, based on how much the hot surface \"sees\" the other surfaces. As such, view factors are purely geometrical in nature. A range of view factor formulae are available \u003chttp://www.thermalradiation.net/tablecon.html here\u003e.\r\n\r\nFor this problem, calculate the view factor from surface 1 (an infinitely long plate) to surfaces 2 (n rows of in-line pipes):\r\n\r\n\u003c\u003chttp://www.thermalradiation.net/images/C-7fig.gif\u003e\u003e\r\n\r\nThe view factor for one row of pipes is F_1-2 (the first equation):\r\n\r\n\u003c\u003chttp://www.thermalradiation.net/images/C-7eq.gif\u003e\u003e\r\n\r\nThe second equation is utilized for more than one row of pipes. Any of the variables can be a vector. Also, note that D = d/b, where d is the pipe diameter and b is the center-to-center spacing between pipes.\r\n\r\n*Note: This problem is identical to the previous problem (4) except that the variables will be provided in vectors of varying sizes.* The variables will need to be combined to produce all the applicable combinations. For example, if the sizes of d, b, and n are 2, 5, and 1, respectively, there will be ten total variable combinations (and answers). As another example, if the sizes of d, b, and n are 4, 3, and 2, respectively, there will be 24 total variable combinations. This simulates a parametric design study. Also, because all possible combinations can be generated in various orders, the answers will be sorted; the order that you output answers does not matter.","description_html":"\u003cp\u003eView factors (aka configuration factors) are utilized in some radiation heat transfer models to estimate heat transfer rates between surfaces. In particular, the thermal energy leaving a given surface is applied to other surfaces, as appropriate, based on how much the hot surface \"sees\" the other surfaces. As such, view factors are purely geometrical in nature. A range of view factor formulae are available \u003ca href = \"http://www.thermalradiation.net/tablecon.html\"\u003ehere\u003c/a\u003e.\u003c/p\u003e\u003cp\u003eFor this problem, calculate the view factor from surface 1 (an infinitely long plate) to surfaces 2 (n rows of in-line pipes):\u003c/p\u003e\u003cimg src = \"http://www.thermalradiation.net/images/C-7fig.gif\"\u003e\u003cp\u003eThe view factor for one row of pipes is F_1-2 (the first equation):\u003c/p\u003e\u003cimg src = \"http://www.thermalradiation.net/images/C-7eq.gif\"\u003e\u003cp\u003eThe second equation is utilized for more than one row of pipes. Any of the variables can be a vector. Also, note that D = d/b, where d is the pipe diameter and b is the center-to-center spacing between pipes.\u003c/p\u003e\u003cp\u003e\u003cb\u003eNote: This problem is identical to the previous problem (4) except that the variables will be provided in vectors of varying sizes.\u003c/b\u003e The variables will need to be combined to produce all the applicable combinations. For example, if the sizes of d, b, and n are 2, 5, and 1, respectively, there will be ten total variable combinations (and answers). As another example, if the sizes of d, b, and n are 4, 3, and 2, respectively, there will be 24 total variable combinations. This simulates a parametric design study. Also, because all possible combinations can be generated in various orders, the answers will be sorted; the order that you output answers does not matter.\u003c/p\u003e","function_template":"function F = view_factor5(d,b,n)\r\n  F = 1;\r\nend","test_suite":"%%\r\nd = [1 2];\r\nb = [3 5 6 7 8];\r\nn = 1;\r\ny_correct = [0.1885    0.2142    0.2479    0.2941    0.3613    0.4077    0.4675    0.4675    0.5472    0.8154];\r\nF = sort(view_factor5(d,b,n));\r\nfor i = 1:numel(y_correct)\r\n assert(F(i) \u003c (y_correct(i) + 1e-4))\r\n assert(F(i) \u003e (y_correct(i) - 1e-4))\r\nend\r\n\r\n%%\r\nd = [2 2.2 2.4 2.5];\r\nb = [3.2 3.4 3.6];\r\nn = 2;\r\ny_correct = [0.9182    0.9352    0.9455    0.9512    0.9594    0.9659    0.9720    0.9738    0.9767    0.9831    0.9857    0.9905];\r\nF = sort(view_factor5(d,b,n));\r\nfor i = 1:numel(y_correct)\r\n assert(F(i) \u003c (y_correct(i) + 1e-4))\r\n assert(F(i) \u003e (y_correct(i) - 1e-4))\r\nend\r\n\r\n%%\r\nd = [1 1.1 1.2];\r\nb = [3 3.2];\r\nn = 1:5;\r\ny_correct = [0.4416    0.4675    0.4803    0.5080    0.5179    0.5472    0.6882    0.7165    0.7299    0.7579    0.7676    0.7950    0.8259    0.8490    0.8596    0.8809    0.8879    0.9028    0.9072    0.9196    0.9270    0.9414    0.9457    0.9460    0.9572    0.9580    0.9621    0.9712    0.9740    0.9810];\r\nF = sort(view_factor5(d,b,n));\r\nfor i = 1:numel(y_correct)\r\n assert(F(i) \u003c (y_correct(i) + 1e-4))\r\n assert(F(i) \u003e (y_correct(i) - 1e-4))\r\nend","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":26769,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":37,"test_suite_updated_at":"2015-01-16T02:08:47.000Z","rescore_all_solutions":false,"group_id":37,"created_at":"2015-01-15T02:16:48.000Z","updated_at":"2026-02-08T12:40:09.000Z","published_at":"2015-01-15T02:16:48.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/image\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/media/image1.gif\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/image\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/media/image2.gif\"}],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eView factors (aka configuration factors) are utilized in some radiation heat transfer models to estimate heat transfer rates between surfaces. In particular, the thermal energy leaving a given surface is applied to other surfaces, as appropriate, based on how much the hot surface \\\"sees\\\" the other surfaces. As such, view factors are purely geometrical in nature. A range of view factor formulae are available\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://www.thermalradiation.net/tablecon.html\\\"\u003e\u003cw:r\u003e\u003cw:t\u003ehere\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor this problem, calculate the view factor from surface 1 (an infinitely long plate) to surfaces 2 (n rows of in-line pipes):\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe view factor for one row of pipes is F_1-2 (the first equation):\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId2\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe second equation is utilized for more than one row of pipes. Any of the variables can be a vector. Also, note that D = d/b, where d is the pipe diameter and b is the center-to-center spacing between pipes.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eNote: This problem is identical to the previous problem (4) except that the variables will be provided in vectors of varying sizes.\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e The variables will need to be combined to produce all the applicable combinations. For example, if the sizes of d, b, and n are 2, 5, and 1, respectively, there will be ten total variable combinations (and answers). As another example, if the sizes of d, b, and n are 4, 3, and 2, respectively, there will be 24 total variable combinations. This simulates a parametric design study. 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