[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-seo-671666-105":3,"detail-sidebar-cat-0-en-105":79,"doc-detail-671666-en":129},{"code":4,"msg":5,"data":6},0,"ok",{"site_id":7,"language":8,"slug":9,"title":10,"keywords":11,"description":12,"schema_data":13,"social_meta":72,"head_meta":74,"extra_data":76,"updated_unix":78},105,"en","heat-transfer-equation-sheet-thermal-conduction-convection-radiation-and-heat-equations","Heat Transfer Equation Sheet - Thermal conduction, convection, radiation, and heat equations","","Heat Transfer Equation Sheet consolidates core heat-transfer relationships used for engineering analysis across conduction, convection, and radiation. It provides Fourier’s law for conduction, Newton’s law of cooling for convection, and blackbody radiation plus net radiation exchange formulations using emissivity and the Stefan–Boltzmann constant. It further includes energy-balance (control volume/surface), heat equation forms in Cartesian, cylindrical, and spherical coordinates, thermal circuit relations, lumped-capacitance guidance, and convection correlations using Reynolds and Nusselt numbers for internal and free convection cases, including heat-exchanger gain/loss equations.",{"@graph":14,"@context":71},[15,34,54],{"@type":16,"itemListElement":17},"BreadcrumbList",[18,23,27,31],{"item":19,"name":20,"@type":21,"position":22},"https://docshare.wps.com","Home","ListItem",1,{"item":24,"name":25,"@type":21,"position":26},"https://docshare.wps.com/document/","Document",2,{"item":28,"name":29,"@type":21,"position":30},"https://docshare.wps.com/document/technology/","Technology",3,{"item":32,"name":10,"@type":21,"position":33},"https://docshare.wps.com/document/heat-transfer-equation-sheet-thermal-conduction-convection-radiation-and-heat-equations/671666/",4,{"url":32,"name":10,"@type":35,"image":36,"author":41,"headline":10,"publisher":44,"fileFormat":47,"inLanguage":8,"description":12,"dateModified":48,"datePublished":48,"encodingFormat":47,"isAccessibleForFree":49,"interactionStatistic":50},"DigitalDocument",{"url":37,"@type":38,"width":39,"height":40},"https://docshare.wps.com/thumbnails/heat-transfer-equation-sheet-thermal-conduction-convection-radiation-and-heat-equations/671666.png","ImageObject",300,407,{"name":42,"@type":43},"pop out","Person",{"url":19,"name":45,"@type":46},"DocShare","Organization","application/pdf","2026-11-02",true,{"@type":51,"interactionType":52,"userInteractionCount":26},"InteractionCounter",{"@type":53},"ViewAction",{"@type":55,"mainEntity":56},"FAQPage",[57,63,67],{"name":58,"@type":59,"acceptedAnswer":60},"What equations summarize heat conduction, convection, and radiation on the sheet?","Question",{"text":61,"@type":62},"Conduction uses Fourier’s law, convection uses Newton’s law of cooling, and radiation uses blackbody emission with net radiation exchange relations based on emissivity and the Stefan–Boltzmann constant.","Answer",{"name":64,"@type":59,"acceptedAnswer":65},"How does the sheet handle energy balance for control volumes and surfaces?",{"text":66,"@type":62},"It lists conservation of energy as an energy balance for a control volume and a corresponding relation for a control surface, with additional specification for steady versus transient conditions.",{"name":68,"@type":59,"acceptedAnswer":69},"Which heat equation coordinate forms are included to find temperature distributions?",{"text":70,"@type":62},"The sheet provides heat equation forms in Cartesian, cylindrical, and spherical coordinates, plus a simplified version when thermal conductivity is constant.","https://schema.org",{"og:url":32,"og:type":73,"og:title":10,"og:site_name":45,"og:description":12},"article",{"robots":75,"canonical":32},"index,follow",{"doc_id":77,"site_id":7},671666,1793577992,{"code":4,"msg":80,"data":81},"success",[82,86,90,94,99,103,108,113,118,121,125],{"id":22,"doc_module":4,"doc_module_name":25,"category_name":83,"show_sort_weight":84,"slug":85},"Story & Novel",90,"story-novel",{"id":26,"doc_module":4,"doc_module_name":25,"category_name":87,"show_sort_weight":88,"slug":89},"Literature",80,"literature",{"id":33,"doc_module":4,"doc_module_name":25,"category_name":91,"show_sort_weight":92,"slug":93},"Exam",70,"exam",{"id":95,"doc_module":4,"doc_module_name":25,"category_name":96,"show_sort_weight":97,"slug":98},5,"Comic",60,"comic",{"id":100,"doc_module":4,"doc_module_name":25,"category_name":29,"show_sort_weight":101,"slug":102},6,50,"technology",{"id":104,"doc_module":4,"doc_module_name":25,"category_name":105,"show_sort_weight":106,"slug":107},7,"Healthcare",40,"healthcare",{"id":109,"doc_module":4,"doc_module_name":25,"category_name":110,"show_sort_weight":111,"slug":112},8,"Research & Report",30,"research-report",{"id":114,"doc_module":4,"doc_module_name":25,"category_name":115,"show_sort_weight":116,"slug":117},9,"Religion & Spirituality",20,"religion-spirituality",{"id":116,"doc_module":4,"doc_module_name":25,"category_name":119,"show_sort_weight":116,"slug":120},"World Cup","world-cup",{"id":122,"doc_module":4,"doc_module_name":25,"category_name":123,"show_sort_weight":122,"slug":124},10,"Lifestyle","lifestyle",{"id":126,"doc_module":4,"doc_module_name":25,"category_name":127,"show_sort_weight":95,"slug":128},19,"General","general",{"code":4,"msg":80,"data":130},{"doc_id":77,"user_id":131,"nickname":42,"user_avatar":132,"doc_module":4,"category_id":100,"category_name":29,"doc_title":10,"doc_description":12,"doc_content":133,"file_id":134,"file_url":135,"file_type":136,"file_size":137,"view_count":26,"is_deleted":4,"is_public":22,"is_downloadable":22,"audit_status":22,"page_count":122,"language":138,"language_code":8,"site_id":7,"html_lang":8,"table_of_contents":139,"faqs":140,"seo_title":141,"seo_description":12,"update_tm":78,"read_time":142},962090760408,"https://ap-avatar.wpscdn.com/davatar_085a072bc5b1113ac321206ff7593b45","HEAT TRANSFER EQUATION SHEET  \n\n| Heat Conduction Rate Equations (Fourier's Law)\u003Cbr>􀂾 Heat Flux : 􀝍′′􀯫 = −􀝇 􀯗􀯗􀯍􀯫~~ ~~ 􀯐􀯠2~~ ~~ k : Thermal Conductivity 􀯐􀯠∙~~ ~~􀯞\u003Cbr>􀂾 Heat Rate : 􀝍􀯫 = 􀝍′′􀯫􀜣 􀯖 􀜹 Ac : Cross-Sectional Area Heat Convection Rate Equations (Newton's Law of Cooling)\u003Cbr>􀂾 Heat Flux: 􀝍 ′′ = ℎ (􀜶􀯦 − 􀜶∞ ) 􀯐􀯠2~~ ~~ h : Convection Heat Transfer Coefficient 􀯠􀯐2∙~~ ~~􀯄\u003Cbr>􀂾 Heat Rate: 􀝍 = ℎ􀜣􀯦 (􀜶􀯦 − 􀜶∞ ) 􀜹 As : Surface Area 􀝉 2\u003Cbr>Heat Radiation emitted ideally by a blackbody surface has a surface emissive power: 􀜧􀯕 = 􀟪 􀜶􀯦4 􀯐􀯠2\u003Cbr>􀂾 Heat Flux emitted : 􀜧 = 􀟝􀟪􀜶􀯦4 􀯐􀯠2~~ ~~ where ε is the emissivity with range of 0 ≤ 􀟝 ≤ 1 and 􀟪 = 5.67 × 10 −8 􀯠􀯐2~~ ~~􀯄4 is the Stefan-Boltzmann constant\u003Cbr>􀂾 Irradiation: 􀜩􀯔􀯕􀯦 = 􀟙􀜩 but we assume small body in a large enclosure with 􀟝 = 􀟙 so that 􀜩 = 􀟝 􀟪 􀜶4􀯦􀯨􀯥\u003Cbr>􀂾 Net Radiation heat flux from surface: 􀝍′􀯥′􀯔􀯗 = 􀯤􀮺 = 􀟝􀜧􀯕 (􀜶􀯦 ) − 􀟙􀜩 = 􀟝􀟪(􀜶􀯦4 − 􀜶4􀯦􀯨􀯥 )\u003Cbr>􀂾 Net radiation heat exchange rate: 􀝍􀯥􀯔􀯗 = 􀟝􀟪􀜣􀯦 (􀜶􀯦4 − 􀜶4􀯦􀯨􀯥 ) where for a real surface 0 ≤ 􀟝 ≤ 1 This can ALSO be expressed as: 􀝍􀯥􀯔􀯗 = ℎ􀯥 􀜣 (􀜶􀯦 − 􀜶􀯦􀯨􀯥 ) depending on the application\u003Cbr>where ℎ􀯥 is the radiation heat transfer coefficient which is: ℎ􀯥 = 􀟝􀟪(􀜶􀯦 + 􀜶􀯦􀯨􀯥 ) (􀜶􀯦2 + 􀜶2􀯦􀯨􀯥 ) 􀯠􀯐2∙~~ ~~􀯄􀂾 TOTAL heat transfer from a surface: 􀝍 = 􀝍 􀯖􀯢􀯡􀯩 + 􀝍􀯥􀯔􀯗 = ℎ􀜣􀯦 (􀜶􀯦 − 􀜶∞ ) + 􀟝􀟪􀜣􀯦 (􀜶􀯦4 − 􀜶4􀯦􀯨􀯥 ) 􀜹 |\n| --- |\n| Conservation of Energy (Energy Balance)\u003Cbr>̇ ̇ ̇ ̇ ̇ ̇\u003Cbr>􀜧􀯜􀯡 + 􀜧􀯚 − 􀜧􀯢􀯨􀯧 = 􀜧􀯦􀯧 (Control Volume Balance) ; 􀜧􀯜􀯡 − 􀜧􀯢􀯨􀯧 = 0 (Control Surface Balance)̇\u003Cbr>where 􀜧􀯚 is the conversion of internal energy (chemical, nuclear, electrical) to thermal or mechanical energy, and ̇ ̇ 􀯗􀯍\u003Cbr>􀜧􀯦􀯧 = 0 for steady-state conditions. If not steady-state (i.e., transient) then 􀜧􀯦􀯧 = 􀟩􀜸􀜿􀯣 􀯗􀯧 |\n\nHeat Equation (used to find the temperature distribution)  \nHeat Equation (Cartesian): 􀰡􀰡􀯫 􁉀 􀝇 􀰡􀰡􀯍􀯫􁉁 + 􀰡􀰡􀯬 􁉀 􀝇 􀰡􀰡􀯍􀯬􁉁 + 􀰡􀰡􀯭 􁉀 􀝇 􀰡􀰡􀯍􀯭􁉁 +  = 􀟩􀜿􀯣 􀰡􀰡􀯍􀯧  \nIf 􀝇 is constant then the above simplifies to: 􀰡􀰡2􀯫􀯍2 + 􀰡􀰡2􀯬􀯍2 + 􀰡􀰡2􀯭􀯍2 +  = 1􀰈 􀰡􀰡􀯍􀯧 where 􀟙 = 􀰘􀯞􀯖􀳛 is the thermal diffusivity Heat Equation (Cylindrical): 1􀯥 􀰡􀰡􀯥 􁉀 􀝇􀝎 􀰡􀰡􀯍􀯥􁉁 + 􀯥12 􀰡􀰡􀰥 􁉀 􀝇 􀰡􀰡􀯍􀰥~~ ~~ 􁉁 + 􀰡􀰡􀯭 􁉀 􀝇 􀰡􀰡􀯍􀯭􁉁 +  = 􀟩􀜿􀯣 􀰡􀰡􀯍􀯧  \nHeat Eqn. (Spherical): 􀯥12 􀰡􀰡􀯥 􁉀 􀝇􀝎 2 􀰡􀰡􀯍􀯥􁉁 + 􀯥~~ ~~2~~ ~~s1in~~ ~~􀰏~~ ~~2 􀰡􀰡􀰥 􁉀 􀝇 􀰡􀰡􀯍􀰥~~ ~~ 􁉁 + 􀯥~~ ~~2~~ ~~1sin~~ ~~􀰏 􀰡􀰡􀰏 􁉀 􀝇 sin 􀟠 􀰡􀰡􀯍􀰏􁉁 +  = 􀟩􀜿􀯣 􀰡􀰡􀯍􀯧  \nThermal Circuits  \n􀯅 ln 􁉀􀳝􀳝21􁉁 (r11 −~~ ~~1r~~ ~~2)  \nPlane Wall: 􀜴 􀯧,􀯖􀯢􀯡􀯗 = 􀯞􀮺  Cylinder: 􀜴 􀯧,􀯖􀯢􀯡􀯗 =  2􀰗􀯞􀯅  Sphere: 􀜴 􀯧,􀯖􀯢􀯡􀯗 =  4􀰗􀯞   \n􀜴 =  1  􀜴 =  1 􀯧,􀯖􀯢􀯡􀯩 ℎ􀮺 􀯧,􀯥􀯔􀯗 ℎ􀳝􀮺  \nGeneral Lumped Capacitance Analysis  \n􀝍′′􀯦􀜣 􀯦,ℎ + 􀜧̇􀯚 − [ℎ(􀜶 − 􀜶∞ ) + 􀟝􀟪(􀜶4 − 􀜶4􀯦􀯨􀯥 )]􀜣 􀯦(􀯖 ,􀯥) = 􀟩􀜸􀜿 􀝀􀝀􀜶􀝐  \nRadiation Only Equation  \n􀝐 = 4~~ ~~􀰌~~ ~~􀮺􀰘􀳞,􀯏􀳝􀯖􀰙~~ ~~􀯍3􀳞􀳠􀳝 􁉄 ln 􁉚 􀯍􀯍􀳞􀳞􀳠􀳠􀳝􀳝~~ ~~􀯍􀯍 􁉚 − ln 􁉚 􀯍􀯍􀳞􀳞􀳠􀳠􀳝􀳝~~ ~~􀯍􀯍􀳔􀳔 􁉚 + 2 􁉂tan −1 􁉀 ~~ ~~􀯍􀯍􀳞􀳠􀳝~~ ~~ 􁉁 − tan−1 􁉀 ~~ ~~􀯍􀯍􀳞􀳠􀳔~~ ~~􀳝~~ ~~ 􁉁􁉃􁉅  \nHeat Flux, Energy Generation, Convection, and No Radiation Equation  \n􀯍􀯍􀳔~~ ~~−􁉀􀳌􀳍􁉀􁉁􀳌􀳍􁉁 = exp (−􀜽􀝐) ; where 􀜽 = 􁉀 ~~ ~~ℎ􀮺􀰘􀯏􀳞􀯖,~~ ~~􀳎~~ ~~ 􁉁 and 􀜾 = 􀯤′′􀳞􀮺~~ ~~􀳞􀰘􀯏,ℎ~~ ~~􀳒  \nConvection Only Equation  \n􀟠􀟠􀯜 = 􀜶􀜶􀯜~~ ~~−~~ ~~ = exp 􀵤− 􀵬ℎ􀟩􀜣􀜸􀯦􀜿􀵰 􀝐 􀵨  \n􀟬 􀯧 = 􁉀 ~~ ~~ℎ1􀮺􀳞~~ ~~ 􁉁 (􀟩􀜸􀜿) = 􀜴 􀯧 􀜥􀯧 ; 􀜳 = 􀟩􀜸􀜿 􀟠 􀯜 􁉂 1 − exp 􁉀 − 􀯧􀰛􀳟~~ ~~ 􁉁􁉃 ; 􀜳􀯠􀯔􀯫 = 􀟩􀜸􀜿 􀟠 􀯜  \n􀜤􀝅 = ℎ􀯅􀯞􀳎  \nIf there is an additional resistance either in series or in parallel, then replace ℎ with 􀜷 in all the above lumped capacitance equations, where  \n􀜷 = 􀯋~~ ~~1􀳟􀮺􀳞~~ ~~ 􁉂􀯠􀯐2∙~~ ~~􀯄~~ ~~ 􁉃 ; 􀜷 = overall heat transfer coefficient, 􀜴􀯧 = total resistance, 􀜣􀯦 = surface area.  \nConvection Heat Transfer  \n􀴥  \n􀰘􀯏􀯅 􀳎 􀯏􀯅 􀳎 􀴤 ℎ􀯅 􀳎  \n􀜴􀝁 =  􀰓  = ~~ ~~􀰔~~ ~~ [Reynolds Number] ; 􀜰􀝑 =  􀯞 􀳑  [Average Nusselt Number]  \nwhere 􀟩 is the density, 􀜸 is the velocity, 􀜮 􀯖 is the characteristic length, 􀟤 is the dynamic viscosity, 􀟥 is the kinematic viscosity,  is the mass flow  \n􀴤  \nrate, ℎ is the average convection coefficient, and 􀝇􀯙 is the fluid thermal conductivity.  \nInternal Flow  \n􀜴􀝁 = 4􀰗􀯠􀮽~~ ~~ [For Internal Flow in a Pipe of Diameter D]  \nFor Constant Heat Flux [􀝍 = 􀜿􀝋􀝊􀝏􀝐􀜽􀝊􀝐]: 􀝍 􀯖􀯢􀯡􀯩 = 􀝍 (􀜲 ∙ 􀜮) ; where P = Perimeter, L = Length  \nʺ  \n􀜶􀯠 (􀝔) = 􀜶􀯠, 􀯜 + 􀝍􀝉̇~~ ~~􀯦~~ ~~∙·~~ ~~􀜲􀜿~~ ~~􀯣~~ ~~ 􀝔  \nFor Constant Surface Temperature [􀜶􀯦 = 􀜿􀝋􀝊􀝏􀝐􀜽􀝊􀝐]:  \nIf there is only convection between ","cbCaibXYbfgNa0Y3","https://ap.wps.com/l/cbCaibXYbfgNa0Y3","pdf",1141333,"English","# Heat Conduction Rate Equations (Fourier's Law)\n# Heat Convection Rate Equations (Newton's Law of Cooling)\n# Heat Radiation and Net Radiation Exchange\n# Conservation of Energy (Energy Balance)\n# Heat Equations (Cartesian, Cylindrical, Spherical)\n# Thermal Circuits\n# Lumped Capacitance Analysis\n# Radiation Only and Convection-Only Equations\n# Convection Heat Transfer (Internal Flow and Correlations)\n# Free Convection Heat Transfer (Plates, Cylinders, Spheres)\n# Heat Exchangers (Heat Gain/Loss Equations)","[{\"question\":\"What equations summarize heat conduction, convection, and radiation on the sheet?\",\"answer\":\"Conduction uses Fourier’s law, convection uses Newton’s law of cooling, and radiation uses blackbody emission with net radiation exchange relations based on emissivity and the Stefan–Boltzmann constant.\"},{\"question\":\"How does the sheet handle energy balance for control volumes and surfaces?\",\"answer\":\"It lists conservation of energy as an energy balance for a control volume and a corresponding relation for a control surface, with additional specification for steady versus transient conditions.\"},{\"question\":\"Which heat equation coordinate forms are included to find temperature distributions?\",\"answer\":\"The sheet provides heat equation forms in Cartesian, cylindrical, and spherical coordinates, plus a simplified version when thermal conductivity is constant.\"}]","Heat Transfer Equation Sheet - Thermal conduction, convection, radiation, and heat equations | PDF",25]