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# Formula Sheet: Heat Transfer Mechanical Engineering Notes | EduRev

## Mechanical Engineering : Formula Sheet: Heat Transfer Mechanical Engineering Notes | EduRev

``` 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
? ) (

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