Page 1 Short notes for Heat transfer Fo u r ie r ’s Law of Heat Conduction • Q = Heat transfer in given direction. • A = Cross-sectional area perpendicular to heat flow direction. • dT = Temperature difference between two ends of a block of thickness dx • dx = Thickness of solid body • = Temperature gradient in direction of heat flow. General Heat Conduction Equation • Carterisan Coordinates (side parallel to x, y and z-directions) q g = Internal heat generation per unit volume per unit time t = Temperature at left face of differential control volume k x, k y, k z = Thermal conductivities of the material in x, y and z-directions respectively c = Specific heat of the material ? = Density of the material a = Thermal diffusivity dt = Instantaneous time. o For homogeneous and isotropic material o For steady state condition (P oi ss on ’s equation) o For steady state and absence of internal heat generation (Laplace equation) o For unsteady heat flow with no internal heat generation Page 2 Short notes for Heat transfer Fo u r ie r ’s Law of Heat Conduction • Q = Heat transfer in given direction. • A = Cross-sectional area perpendicular to heat flow direction. • dT = Temperature difference between two ends of a block of thickness dx • dx = Thickness of solid body • = Temperature gradient in direction of heat flow. General Heat Conduction Equation • Carterisan Coordinates (side parallel to x, y and z-directions) q g = Internal heat generation per unit volume per unit time t = Temperature at left face of differential control volume k x, k y, k z = Thermal conductivities of the material in x, y and z-directions respectively c = Specific heat of the material ? = Density of the material a = Thermal diffusivity dt = Instantaneous time. o For homogeneous and isotropic material o For steady state condition (P oi ss on ’s equation) o For steady state and absence of internal heat generation (Laplace equation) o For unsteady heat flow with no internal heat generation • Cylindrical Coordinates o For homogeneous and isotropic material, o For steady state unidirectional heat flow in radial direction with no internal heat generation, • Spherical Coordinates o For homogeneous and isotropic material o For steady state uni-direction heat flow in radial direction with no internal heat generation, • Thermal resistance of hollow cylinders • Thermal Resistance of a Hollow Sphere • Heat Transfer through a Composite Cylinder Page 3 Short notes for Heat transfer Fo u r ie r ’s Law of Heat Conduction • Q = Heat transfer in given direction. • A = Cross-sectional area perpendicular to heat flow direction. • dT = Temperature difference between two ends of a block of thickness dx • dx = Thickness of solid body • = Temperature gradient in direction of heat flow. General Heat Conduction Equation • Carterisan Coordinates (side parallel to x, y and z-directions) q g = Internal heat generation per unit volume per unit time t = Temperature at left face of differential control volume k x, k y, k z = Thermal conductivities of the material in x, y and z-directions respectively c = Specific heat of the material ? = Density of the material a = Thermal diffusivity dt = Instantaneous time. o For homogeneous and isotropic material o For steady state condition (P oi ss on ’s equation) o For steady state and absence of internal heat generation (Laplace equation) o For unsteady heat flow with no internal heat generation • Cylindrical Coordinates o For homogeneous and isotropic material, o For steady state unidirectional heat flow in radial direction with no internal heat generation, • Spherical Coordinates o For homogeneous and isotropic material o For steady state uni-direction heat flow in radial direction with no internal heat generation, • Thermal resistance of hollow cylinders • Thermal Resistance of a Hollow Sphere • Heat Transfer through a Composite Cylinder • Heat Transfer through a Composite Sphere • Critical Thickness of Insulation: o In case of cylinder, where, k 0 = Thermal conductivity, and h = Heat transfer coefficient o The drop in temperature across the wall and the air film will be proportional to their resistances, = hL/k. • Steady Flow of Heat along a Rod Circular fin ?=pd Page 4 Short notes for Heat transfer Fo u r ie r ’s Law of Heat Conduction • Q = Heat transfer in given direction. • A = Cross-sectional area perpendicular to heat flow direction. • dT = Temperature difference between two ends of a block of thickness dx • dx = Thickness of solid body • = Temperature gradient in direction of heat flow. General Heat Conduction Equation • Carterisan Coordinates (side parallel to x, y and z-directions) q g = Internal heat generation per unit volume per unit time t = Temperature at left face of differential control volume k x, k y, k z = Thermal conductivities of the material in x, y and z-directions respectively c = Specific heat of the material ? = Density of the material a = Thermal diffusivity dt = Instantaneous time. o For homogeneous and isotropic material o For steady state condition (P oi ss on ’s equation) o For steady state and absence of internal heat generation (Laplace equation) o For unsteady heat flow with no internal heat generation • Cylindrical Coordinates o For homogeneous and isotropic material, o For steady state unidirectional heat flow in radial direction with no internal heat generation, • Spherical Coordinates o For homogeneous and isotropic material o For steady state uni-direction heat flow in radial direction with no internal heat generation, • Thermal resistance of hollow cylinders • Thermal Resistance of a Hollow Sphere • Heat Transfer through a Composite Cylinder • Heat Transfer through a Composite Sphere • Critical Thickness of Insulation: o In case of cylinder, where, k 0 = Thermal conductivity, and h = Heat transfer coefficient o The drop in temperature across the wall and the air film will be proportional to their resistances, = hL/k. • Steady Flow of Heat along a Rod Circular fin ?=pd • Generalized Equation for Fin Rectangular fin • Heat balance equation if A c constant and A s 8 P(x) linear • General equation of 2 nd order ? = c1e mx + c2e -mx o Heat Dissipation from an Infinitely Long Fin (l ? 8) ? Heat transfer by conduction at base o Heat Dissipation from a Fin Insulated at the End Tip o Heat Dissipation from a Fin loosing Heat at the End Tip Page 5 Short notes for Heat transfer Fo u r ie r ’s Law of Heat Conduction • Q = Heat transfer in given direction. • A = Cross-sectional area perpendicular to heat flow direction. • dT = Temperature difference between two ends of a block of thickness dx • dx = Thickness of solid body • = Temperature gradient in direction of heat flow. General Heat Conduction Equation • Carterisan Coordinates (side parallel to x, y and z-directions) q g = Internal heat generation per unit volume per unit time t = Temperature at left face of differential control volume k x, k y, k z = Thermal conductivities of the material in x, y and z-directions respectively c = Specific heat of the material ? = Density of the material a = Thermal diffusivity dt = Instantaneous time. o For homogeneous and isotropic material o For steady state condition (P oi ss on ’s equation) o For steady state and absence of internal heat generation (Laplace equation) o For unsteady heat flow with no internal heat generation • Cylindrical Coordinates o For homogeneous and isotropic material, o For steady state unidirectional heat flow in radial direction with no internal heat generation, • Spherical Coordinates o For homogeneous and isotropic material o For steady state uni-direction heat flow in radial direction with no internal heat generation, • Thermal resistance of hollow cylinders • Thermal Resistance of a Hollow Sphere • Heat Transfer through a Composite Cylinder • Heat Transfer through a Composite Sphere • Critical Thickness of Insulation: o In case of cylinder, where, k 0 = Thermal conductivity, and h = Heat transfer coefficient o The drop in temperature across the wall and the air film will be proportional to their resistances, = hL/k. • Steady Flow of Heat along a Rod Circular fin ?=pd • Generalized Equation for Fin Rectangular fin • Heat balance equation if A c constant and A s 8 P(x) linear • General equation of 2 nd order ? = c1e mx + c2e -mx o Heat Dissipation from an Infinitely Long Fin (l ? 8) ? Heat transfer by conduction at base o Heat Dissipation from a Fin Insulated at the End Tip o Heat Dissipation from a Fin loosing Heat at the End Tip • Fin Efficiency • Fin efficiency is given by • If l ? 8 (infinite length of fin), • If finite length of fin, • Fin Effectiveness • Lumped Parameter System Q = - ? ? Ta T hA dt dT VCp ? ? ? ? ? ? ? ? dt VCp hA Ta T dT ? ) (Read More

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