[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-seo-671662-105":3,"detail-sidebar-cat-0-en-105":80,"doc-detail-671662-en":130},{"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":73,"head_meta":75,"extra_data":77,"updated_unix":79},105,"en","heat-transfer-basic-heat-transfer-rate-equations","Heat Transfer - Basic Heat Transfer Rate Equations","","Heat transfer covers three fundamental mechanisms—conduction, convection, and radiation—and provides the core rate equations used to quantify energy transport. It details Fourier’s law for conduction, Newton’s law of cooling for convection, and the radiation relation with emissivity and the Stefan–Boltzmann constant. The document further derives conduction through plane and cylindrical walls, introduces thermal resistance and series resistances, and expands to steady conduction with internal heat generation and transient conduction using the lumped capacitance 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expresses heat transfer rate using thermal resistance and states that resistances in series are added to form a total resistance.","https://schema.org",{"og:url":32,"og:type":74,"og:title":10,"og:site_name":45,"og:description":12},"article",{"robots":76,"canonical":32},"index,follow",{"doc_id":78,"site_id":7},671662,1793577982,{"code":4,"msg":81,"data":82},"success",[83,87,91,95,100,104,109,114,119,122,126],{"id":22,"doc_module":4,"doc_module_name":25,"category_name":84,"show_sort_weight":85,"slug":86},"Story & 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EQUATIONS Conduction  \nFourier’s Law of Conduction  \nQo = -kA ~~ d~~T~~dx~~, where Qo = rate of heat transfer (W)  \nk = the thermal conductivity [W/(m·K)]  \nA = the surface area perpendicular to direction of heat transfer (m2)  \nConvection  \nNewton’s Law of Cooling  \nQo = hA_Tw-T3i, where  \nh = the convection heat transfer coefficient of the fluid [W/(m2·K)]  \nA = the convection surface area (m2) Tw = the wall surface temperature (K)  \nT∞ = the bulk fluid temperature (K)  \nRadiation  \nThe radiation emitted by a body is given by Qo = fvAT4, where  \nε = the emissivity of the body  \nσ = the Stefan-Boltzmann constant = 5.67 × 10-8 W/(m2·K4)  \nA = the body surface area (m2) T = the absolute temperature (K)  \nCONDUCTION  \nConduction Through a Plane Wall  \nT1  \nQo = -kA^T2L-T1h, where  \nA = wall surface area normal to heat flow (m2) L = wall thickness (m)  \nT1 = temperature of one surface of the wall (K) T2 = temperature of the other surface of the wall (K)  \nConduction Through a Cylindrical Wall  \nT1 Q  \nT2  \nCylinder (Length = L)  \nQo = 2rkL^T1 -T2 h ln d ~~ ~~r~~ 2r~~1~~ ~~ n  \nCritical Insulation Radius rcr = k~~insuhlation~~3  \nThermal Resistance (R)  \nQo = DTRtotal  \nh∞  \nr insulation  \nkinsulation  \nResistances in series are added: Rtotal = RR, where  \nPlane Wall Conduction Resistance (K/W): R = L~~k~~A~~ ~~ , where L = wall thickness  \nCylindrical Wall Conduction Resistance (K/W): R = ln2r~~r~~k~~2~~1Ln  , where  \nL = cylinder length  \nConvection Resistance (K/W) : R = ~~h1~~A Composite Plane Wall  \nFluid 1 T∞ 1  \nh 1  \n2  \n1  \nh 1AQ  \nLA  \nkA A  \nLB  \nkBA  \n1  \nh2 A  \nT∞ 1 T1 T2 T3 T∞2  \n8􀀴 HEAT TRANSFER  \nTo evaluate Surface or Intermediate Temperatures: Qo = T~~1 ~~R-AT~~2~~ = T~~2 ~~R~~ ~~T~~3~~  \nSteady Conduction with Internal Energy Generation  \nThe equation for one-dimensional steady conduction is ~~d~~d2Tx~~2~~ + Qoge~~k~~n = 0, where  \nQo gen = the heat generation rate per unit volume (W/m3)  \nFor a Plane Wall  \n−L 0 L  \nT^ xh = Qogen~~2k~~L2 d1-~~x~~L2~~2 ~~ n + c T~~s2~~-~~2 ~~T~~s 1~~ m b~~x~~Ll + c T~~s 1 ~~-~~2 ~~T~~s2 ~~ m Qo\"1 + Qo\"2 = 2Qo gen L, where  \nQo\" = the rate of heat transfer per area (heat flux) (W/m2) Qo\"1 = kb ~~ d~~T~~dx ~~ l-L and Qo\"2 = kb ~~ d~~T~~dx ~~ lL  \nFor a Long Circular Cylinder  \n~~1r~~ ~~ddr~~ br ~~ dd~~T~~r ~~ l + Qoge~~k~~n = 0  \nT]rg = Qog~~4~~en~~k~~2r0 f1-~~r ~~2~~2~~r0~~ ~~ p + Ts  \nQo l = rr20Qo gen, where  \nQo l = the heat transfer rate from the cylinder per unit length of the cylinder (W/m)  \nTransient Conduction Using the Lumped Capacitance  Method  \nThe lumped capacitance method is valid if Biot number, Bi = ~~hk~~VAs % 1, where  \nh = the convection heat transfer coefficient of the fluid [W/(m2·K)]  \nV = the volume of the body (m3)  \nk = thermal conductivity of the body [W/(m·K)] As = the surface area of the body (m2)  \nFluid  \nConstant Fluid Temperature  \nIf the temperature may be considered uniform within the body at any time, the heat transfer rate at the body surface is given by  \nQo = hAs ^T-T3 h =-tV^cP hb ~~ dd~~T~~t ~~ l, where T = the body temperature (K)  \nT∞ = the fluid temperature (K)  \nρ = the density of the body (kg/m3)  \ncP = the heat capacity of the body [J/(kg·K)] t = time (s)  \nThe temperature variation of the body with time is  \nT-T3 = _Tib = hA~~t~~V~~sc~~P  \n-T3ie-bt, where where b = ~~1x~~ and x = time constant^ sh  \nThe total heat transferred (Qtotal) up to time t is Qtotal = tVcP _Ti -Ti, where  \nTi = initial body temperature (K)  \nHEAT TRANSFER 85  \nVariable Fluid Temperature  \nIf the ambient fluid temperature varies periodically according to the equation  \nT3 = T3, mean + ~~12~~ _T3, max-T3, mini cos ^~th  \nThe temperature of the body, after initial transients have died away, is  \nT = b~~ ;  12 ~~_T3,a2x+-T32,~~ ~~miniE cos =~t -tan-1 c ~~ ~~~bmG + T3, mean  \nFins  \nFor a straight fin with uniform cross section (assuming negli","cbCainicSvBbWUZx","https://ap.wps.com/l/cbCainicSvBbWUZx","pdf",947594,"English","# Heat Transfer\n## Conduction\n## Convection\n## Radiation\n## Thermal Resistance and Conduction Through Walls\n## Steady/Transient Conduction and Lumped Capacitance\n## Fins\n## Convection Correlations and Flow Types","[{\"question\":\"What are the three modes of heat transfer covered in the document?\",\"answer\":\"The document lists conduction, convection, and radiation as the three modes of heat transfer.\"},{\"question\":\"Which equations are used to calculate conduction, convection, and radiation heat transfer rates?\",\"answer\":\"Conduction uses Fourier’s law, convection uses Newton’s law of cooling, and radiation uses an equation based on emissivity and the Stefan–Boltzmann constant.\"},{\"question\":\"How does the document relate thermal resistance to overall heat transfer?\",\"answer\":\"It expresses heat transfer rate using thermal resistance and states that resistances in series are added to form a total resistance.\"}]","Heat Transfer - Basic Heat Transfer Rate Equations | PDF",15]