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Chapter MECHANISMS OF HEAT TRANSFER CONDUCTION Definition • Conduction is the transfer of energy from the more energetic particles of a substance to the adjacent less energetic ones as a result of interaction between particles • Conduction can take place in solids, liquids or gases Fourier’s law 𝑇 + 𝑑𝑇 𝑇 𝑑𝑄 𝜕𝑇 𝑞= = −𝑘 𝑑𝐹𝑑𝜏 𝜕𝑛 Where: • 𝑞: heat flux 𝑊 𝑚 • 𝑘: thermal conductivity 𝑊 𝑚 𝐾 • 𝜕𝑇 : 𝜕𝑛 𝑛 𝑞 isothermal surface temperature gradient in direction of heat flux 𝐾 𝑚 • Thermal conductivity 𝑘 is a measure of the ability of a material to conduct heat It indicates how fast heat flows in a given material • Thermal conductivity strongly depends on temperature Energy balance 𝑒𝑛𝑒𝑟𝑔𝑦 𝑖𝑛𝑙𝑒𝑡 + 𝑒𝑛𝑒𝑟𝑔𝑦 𝑔𝑒𝑛𝑒𝑟𝑎𝑡𝑒𝑑 𝑤𝑖𝑡ℎ𝑖𝑛 𝑒𝑙𝑒𝑚𝑒𝑛𝑡 = 𝑖𝑛𝑡𝑒𝑟𝑛𝑎𝑙 𝑒𝑛𝑒𝑟𝑔𝑦 𝑐ℎ𝑎𝑛𝑔𝑒 + 𝑒𝑛𝑒𝑟𝑔𝑦 𝑜𝑢𝑙𝑒𝑡 • Energy generated: 𝑞𝑔𝑒𝑛 𝑑𝑥𝑑𝑦𝑑𝑧 • Internal energy change: Elemental volume 𝑑𝑣 = 𝑑𝑥 ∙ 𝑑𝑦 ∙ 𝑑𝑧 Cartesian coordinates 𝜕𝑇 𝑞𝑖 = −𝑘 𝜕𝑖 𝜕𝑇 𝜌𝑐𝑑𝑥𝑑𝑦𝑑𝑧 𝜕𝜏 Energy balance 𝑒𝑛𝑒𝑟𝑔𝑦 𝑖𝑛𝑙𝑒𝑡 + 𝑒𝑛𝑒𝑟𝑔𝑦 𝑔𝑒𝑛𝑒𝑟𝑎𝑡𝑒𝑑 𝑤𝑖𝑡ℎ𝑖𝑛 𝑒𝑙𝑒𝑚𝑒𝑛𝑡 = 𝑖𝑛𝑡𝑒𝑟𝑛𝑎𝑙 𝑒𝑛𝑒𝑟𝑔𝑦 𝑐ℎ𝑎𝑛𝑔𝑒 + 𝑒𝑛𝑒𝑟𝑔𝑦 𝑜𝑢𝑙𝑒𝑡 • Energy inlet 𝜕𝑇 𝑋 𝑎𝑥𝑖𝑠: −𝑘𝑑𝑦𝑑𝑧 𝜕𝑥 𝜕𝑇 𝑌 𝑎𝑥𝑖𝑠: −𝑘𝑑𝑥𝑑𝑧 𝜕𝑦 𝜕𝑇 𝑍 𝑎𝑥𝑖𝑠: −𝑘𝑑𝑥𝑑𝑦 𝜕𝑧 • Energy outlet 𝜕𝑇 𝜕 𝜕𝑇 𝑋 𝑎𝑥𝑖𝑠: − 𝑘 + 𝑘 𝑑𝑥 𝑑𝑦𝑑𝑧 𝜕𝑥 𝜕𝑥 𝜕𝑥 𝜕𝑇 𝜕 𝜕𝑇 𝑌 𝑎𝑥𝑖𝑠: − 𝑘 + 𝑘 𝑑𝑦 𝑑𝑥𝑑𝑧 𝜕𝑦 𝜕𝑦 𝜕𝑦 𝜕𝑇 𝜕 𝜕𝑇 𝑍 𝑎𝑥𝑖𝑠: − 𝑘 + 𝑘 𝑑𝑧 𝑑𝑥𝑑𝑦 𝜕𝑧 𝜕𝑧 𝜕𝑧 Fourier equation 𝜕 𝜕𝑇 𝜕 𝜕𝑇 𝜕 𝜕𝑇 𝜕𝑇 𝑘 + 𝑘 + 𝑘 + 𝑞𝑔𝑒𝑛 = 𝜌𝑐 𝜕𝑥 𝜕𝑥 𝜕𝑦 𝜕𝑦 𝜕𝑧 𝜕𝑧 𝜕𝜏 For constant thermal conductivity 𝑞𝑔𝑒𝑛 𝜕𝑇 𝑘 𝜕2𝑇 𝜕2𝑇 𝜕2𝑇 + 2+ + = 𝜌𝑐 𝜕𝑥 𝜕𝑦 𝜕𝑧 𝜌𝑐 𝜕𝜏 𝑞𝑔𝑒𝑛 𝜕𝑇 𝑎𝛻 𝑇 + = 𝜌𝑐 𝜕𝜏 𝑎 thermal diffusivity The larger the value of 𝑎, the faster heat will diffuse through the material Fourier equation Cylindrical coordinates 𝑞𝑔𝑒𝑛 𝜕𝑇 𝑘 𝜕 𝑇 𝜕𝑇 𝜕 𝑇 𝜕 𝑇 + + 2+ + = 𝜌𝑐 𝜕𝑟 𝑟 𝜕𝑟 𝑟 𝜕𝜙 𝜕𝑧 𝜌𝑐 𝜕𝜏 Fourier equation Spherical coordinates 𝑞𝑔𝑒𝑛 𝜕𝑇 𝑘 𝜕2 𝜕 𝜕𝑇 𝜕2𝑇 𝑟𝑇 + sin 𝜃 + 2 + = 2 𝜌𝑐 𝑟 𝜕𝑟 𝑟 sin 𝜃 𝜕𝜃 𝜕𝜃 𝑟 𝑠𝑖𝑛 𝜃 𝜕𝜙 𝜌𝑐 𝜕𝜏 Fourier equation • For steady state one dimensional heat flux without heat sources 𝜕2𝑇 =0 𝜕𝑥 • For steady state one dimensional heat flux with heat sources 𝜕2𝑇 𝑘 + 𝑞𝑔𝑒𝑛 = 𝜕𝑥 • For unsteady state one dimensional heat flux with heat sources 𝑘 𝜕 𝑇 𝑞𝑔𝑒𝑛 𝜕𝑇 + = 𝜌𝑐 𝜕𝑥 𝜌𝑐 𝜕𝜏 • For steady state three dimensional heat flux without heat sources 𝜕2𝑇 𝜕2𝑇 𝜕2𝑇 + 2+ =0 𝜕𝑥 𝜕𝑦 𝜕𝑧 Laplace equation 𝛻2𝑇 = Plane wall 1 1 = + + 𝑅𝐵𝐶𝐷 𝑅𝐵 𝑅𝐶 𝑅𝐷 1 = + 𝑅𝐹𝐺 𝑅𝐹 𝑅𝐺 𝑅 = 𝑅𝐴 + 𝑅𝐵𝐶𝐷 + 𝑅𝐸 + 𝑅𝐹𝐺 𝑇1 − 𝑇5 𝑞= 𝑅 Cylinder Boundary condition 𝑇 𝑟1 = 𝑇1 𝑇 𝑟2 = 𝑇2 𝑇2 𝑟2 𝑟1 𝑇1 𝑇1 − 𝑇2 𝑞= 𝑟2 ln 2𝜋𝑘 𝑟1 𝑑𝑇 Heat flux 𝑞 = −𝑘 𝑑𝑟 𝑞: heat flux per cylinder length 𝑊 𝑚 Cylinder Temperature profile 𝑞𝑔𝑒𝑛 𝜕𝑇 𝑘 𝜕 𝑇 𝜕𝑇 𝜕 𝑇 𝜕 𝑇 + + 2+ + = 𝜌𝑐 𝜕𝑟 𝑟 𝜕𝑟 𝑟 𝜕𝜙 𝜕𝑧 𝜌𝑐 𝜕𝜏 0 𝜕 𝑇 𝜕𝑇 + =0 𝜕𝑟 𝑟 𝜕𝑟 ln 𝑟 𝑟1 𝑇 𝑟 = 𝑇1 − 𝑇1 − 𝑇2 ln 𝑟2 𝑟1 ln 𝑟 𝑟1 𝑇1 − 𝑇 𝑟 = ln 𝑟2 𝑟1 𝑇1 − 𝑇2 Cylinder 𝑞 𝑅𝐴 𝑇1 𝑞= 𝑞= 𝑟2 ln 𝑟1 2𝜋𝑘𝐴 𝑅𝐵 𝑇2 𝑅𝐶 𝑇3 𝑇4 𝑇1 − 𝑇2 𝑇2 − 𝑇3 𝑇3 − 𝑇4 = = 𝑟 𝑟 𝑟 ln ln ln 2𝜋𝑘𝐴 𝑟1 2𝜋𝑘𝐵 𝑟2 2𝜋𝑘𝐶 𝑟3 𝑇1 − 𝑇4 𝑇1 − 𝑇4 = 𝑟3 𝑟4 𝑅𝐴 + 𝑅𝐵 + 𝑅𝐶 + ln + ln 𝑟2 𝑟3 2𝜋𝑘𝐵 2𝜋𝑘𝐶 Sphere Boundary condition Heat flux 𝑇 𝑟1 = 𝑇1 𝑇 𝑟2 = 𝑇2 𝑇1 − 𝑇2 𝑞= 1 − 4𝜋𝑘 𝑟1 𝑟2 Sphere Temperature profile 𝑞𝑔𝑒𝑛 𝜕𝑇 𝑘 𝜕2 𝜕 𝜕𝑇 𝜕2𝑇 𝑟𝑇 + sin 𝜃 + 2 + = 2 𝜌𝑐 𝑟 𝜕𝑟 𝑟 sin 𝜃 𝜕𝜃 𝜕𝜃 𝑟 𝑠𝑖𝑛 𝜃 𝜕𝜙 𝜌𝑐 𝜕𝜏 𝜕2 𝑟𝑇 = 𝑟 𝜕𝑟 0 𝑇1 − 𝑇 𝑟 𝑟2 𝑟1 − 𝑟 = 𝑇1 − 𝑇2 𝑟 𝑟1 − 𝑟2 STEADY STATE WITH HEAT SOURCES Plane wall 𝑞𝑔𝑒𝑛 𝑇1 𝑇2 𝛿 Boundary condition 𝑇 = 𝑇1 𝑇 𝛿 = 𝑇2 Plane wall Temperature profile 𝜕 𝑇 𝑞𝑔𝑒𝑛 + =0 𝜕𝑥 𝑘 𝑞𝑔𝑒𝑛 𝛿 𝑥 𝑥 𝑇 𝑥 = 𝑇1 + − 2𝑘 𝛿 𝛿 𝑥 − 𝑇1 − 𝑇2 𝛿 𝑞𝑔𝑒𝑛 𝛿 𝑥 𝑇1 − 𝑇 𝑥 𝑥 = = 1− 𝛿 𝑇1 − 𝑇2 𝛿 2𝑘 𝑇1 − 𝑇2 𝑥 1− 𝛿 𝑞𝑔𝑒𝑛 𝛿 𝑘 𝑇1 − 𝑇2 𝑇 𝑥 𝑚𝑎𝑥 = 𝑇 𝑥𝑜 = 𝑇1 + + − 𝑇1 − 𝑇2 8𝑘 2𝑞𝑔𝑒𝑛 𝛿 𝑥𝑜 𝑘 𝑇1 − 𝑇2 = − 𝛿 𝑞𝑔𝑒𝑛 𝛿 Plane wall Heat flux 𝑑𝑇 𝑞 = −𝑘 𝑑𝑥 𝑘 𝑞 𝑥 = 𝑇1 − 𝑇2 𝛿 𝑞𝑔𝑒𝑛 𝛿 1+ 𝑘 𝑇1 − 𝑇2 𝑥 − 𝛿 Cylinder Boundary condition 𝑇 𝑟1 = 𝑇1 𝑇 𝑟2 = 𝑇2 Cylinder 𝜕 𝑇 𝜕𝑇 𝑞𝑔𝑒𝑛 + + =0 𝜕𝑟 𝑟 𝜕𝑟 𝑘 Temperature profile 𝑞𝑔𝑒𝑛 2 𝑇 − 𝑇 − 𝑟 − 𝑟 𝑞𝑔𝑒𝑛 2 4𝑘 𝑇 𝑟 = 𝑇1 − 𝑟 − 𝑟1 − ln 𝑟 4𝑘 ln 𝑟2 𝑟1 𝑞𝑔𝑒𝑛 𝑇1 − 𝑇2 − 𝑟2 − 𝑟12 4𝑘 + ln 𝑟1 ln 𝑟2 𝑟1 𝑞𝑔𝑒𝑛 𝑟12 𝑇1 − 𝑇 𝑟 = 𝑇1 − 𝑇2 4𝑘 𝑇1 − 𝑇2 𝑟 𝑟1 𝑞𝑔𝑒𝑛 𝑟22 −1 + 1− 4𝑘 𝑇1 − 𝑇2 𝑟1 1− 𝑟2 ln 𝑟 𝑟1 ln 𝑟2 𝑟1 Cylinder Heat flux 𝑑𝑇 𝑞 = −𝑘 𝑑𝑟 𝑞𝑔𝑒𝑛 𝑟22 − 𝑟12 𝑞𝑔𝑒𝑛 𝑟 𝑘 𝑇1 − 𝑇2 − 𝑞= + 𝑟2 𝑟 ln 𝑟1 Sphere Boundary condition 𝑇 𝑟1 = 𝑇1 𝑇 𝑟2 = 𝑇2 Sphere Temperature profile 𝑇1 − 𝑇 𝑟 = 𝑇1 − 𝑇2 𝑞𝑔𝑒𝑛 𝜕2 𝑟𝑇 + =0 𝑟 𝜕𝑟 𝑘 𝑞𝑔𝑒𝑛 𝑟 𝑟23 − 𝑟13 𝑞𝑔𝑒𝑛 𝑟22 − 𝑟12 𝑟𝑟2 − − 𝑟1 𝑟2 − 6𝑘 𝑇1 − 𝑇2 6𝑘 𝑇1 − 𝑇2 𝑟 𝑟1 − 𝑟2 𝑞𝑔𝑒𝑛 𝑟 + 6𝑘 𝑇1 − 𝑇2 𝑞𝑔𝑒𝑛 𝑟 𝑟1 𝑟2 𝑘 𝑇1 − 𝑇2 𝑞𝑔𝑒𝑛 Heat flux 𝑞 = + − 𝑟1 + 𝑟2 𝑟 𝑟2 − 𝑟1

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