Formula Sheet: Heat Transfer | Heat Transfer - Mechanical Engineering PDF Download

 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
? ) (