Short Notes: One Dimensional Heat Conduction | Short Notes for Mechanical Engineering PDF Download

 Page 1


One Dimensional Heat Conduction 
Conduction is the transfer of energy from the more energetic particles of a 
substance to the adjacent less energetic ones as result of interactions between 
the particles
Steady and Unsteady State Conduction
• Steady state conduction is the form of conduction that happens when the 
temperature difference driving the conduction is constant, so that, the spatial 
distribution of temperatures in the conducting object does not change any 
further.
• In general, during any period in which temperatures are changing in time at 
any place within an object, the mode of thermal energy flow is termed 
transient conduction or nonsteady state conduction.
One-Dimensional Heat Flow
The term 'one-dimensional' is applied to heat conduction problem when:
1. Only one space coordinate is required to describe the temperature distribution 
within a heat conducting body;
2. Edge effects are neglected;
3. The flow of heat energy takes place along the coordinate measured normal to 
the surface.
Fourier’s Law of Heat Conduction
Consider steady conduction through a large plane wall of thickness Ax = L 
and surface area A. The temperature difference across the wall is AT = T2 - 
T1.
Page 2


One Dimensional Heat Conduction 
Conduction is the transfer of energy from the more energetic particles of a 
substance to the adjacent less energetic ones as result of interactions between 
the particles
Steady and Unsteady State Conduction
• Steady state conduction is the form of conduction that happens when the 
temperature difference driving the conduction is constant, so that, the spatial 
distribution of temperatures in the conducting object does not change any 
further.
• In general, during any period in which temperatures are changing in time at 
any place within an object, the mode of thermal energy flow is termed 
transient conduction or nonsteady state conduction.
One-Dimensional Heat Flow
The term 'one-dimensional' is applied to heat conduction problem when:
1. Only one space coordinate is required to describe the temperature distribution 
within a heat conducting body;
2. Edge effects are neglected;
3. The flow of heat energy takes place along the coordinate measured normal to 
the surface.
Fourier’s Law of Heat Conduction
Consider steady conduction through a large plane wall of thickness Ax = L 
and surface area A. The temperature difference across the wall is AT = T2 - 
T1.
• It has been experimentally observed that the rate of heat conduction through 
a layer is
proportional to the temperature difference between the layer and the heat tran 
sfer area, but it is inversely proportional to the thickness of the layer.
rate of heat transfer o c
(surface area)(temperature difference) 
thickness
Q * Co»d
\W)
where constant proportionality k is the thermal conductivity of the material.
• In the limiting case where Ax— > 0, the equation above reduces to the 
differential form
, , dT
Q Cond = - k A — — 
dx
m
which is called Fourier's law of heat conduction.
• The term A = Cross-sectional area perpendicular to heat flow direction 
and dT/dx is called the temperature gradient, which is the slope of the 
temperature curve (the rate of change of temperature T with length x).
Thermal Conductivity(k [W/mK])
• The thermal conductivity is defined as the rate of heat transfer through a unit 
thickness of material per unit area per unit temperature difference.
• Thermal conductivity changes with temperature and is determined through 
experiments and is the measure of a material's ability to conduct heat.
• The thermal conductivity of certain materials show a dramatic change at 
temperatures near absolute zero, when these solids become 
superconductors.
• An isotropic material is a material that has uniform properties in all 
directions. Insulators are materials used primarily to provide resistance to
Page 3


One Dimensional Heat Conduction 
Conduction is the transfer of energy from the more energetic particles of a 
substance to the adjacent less energetic ones as result of interactions between 
the particles
Steady and Unsteady State Conduction
• Steady state conduction is the form of conduction that happens when the 
temperature difference driving the conduction is constant, so that, the spatial 
distribution of temperatures in the conducting object does not change any 
further.
• In general, during any period in which temperatures are changing in time at 
any place within an object, the mode of thermal energy flow is termed 
transient conduction or nonsteady state conduction.
One-Dimensional Heat Flow
The term 'one-dimensional' is applied to heat conduction problem when:
1. Only one space coordinate is required to describe the temperature distribution 
within a heat conducting body;
2. Edge effects are neglected;
3. The flow of heat energy takes place along the coordinate measured normal to 
the surface.
Fourier’s Law of Heat Conduction
Consider steady conduction through a large plane wall of thickness Ax = L 
and surface area A. The temperature difference across the wall is AT = T2 - 
T1.
• It has been experimentally observed that the rate of heat conduction through 
a layer is
proportional to the temperature difference between the layer and the heat tran 
sfer area, but it is inversely proportional to the thickness of the layer.
rate of heat transfer o c
(surface area)(temperature difference) 
thickness
Q * Co»d
\W)
where constant proportionality k is the thermal conductivity of the material.
• In the limiting case where Ax— > 0, the equation above reduces to the 
differential form
, , dT
Q Cond = - k A — — 
dx
m
which is called Fourier's law of heat conduction.
• The term A = Cross-sectional area perpendicular to heat flow direction 
and dT/dx is called the temperature gradient, which is the slope of the 
temperature curve (the rate of change of temperature T with length x).
Thermal Conductivity(k [W/mK])
• The thermal conductivity is defined as the rate of heat transfer through a unit 
thickness of material per unit area per unit temperature difference.
• Thermal conductivity changes with temperature and is determined through 
experiments and is the measure of a material's ability to conduct heat.
• The thermal conductivity of certain materials show a dramatic change at 
temperatures near absolute zero, when these solids become 
superconductors.
• An isotropic material is a material that has uniform properties in all 
directions. Insulators are materials used primarily to provide resistance to
heat flow. They have low thermal conductivity 
General Heat Conduction Equation
Carterisan Coordinates (side parallel to x, y and z-directions)
• qg = Internal heat generation per unit volume per unit time
• f = Temperature at left face of the differential control volume
• kx, ky , kz = Thermal conductivities of the material in x, y and z-directions 
respectively
• c = Specific heat of the material
• p = Density of the material
• a = Thermal diffusivity
k_
pc
• dT = Instantaneous time.
• For homogeneous and isotropic material.
kx = k ^ = k. = k, a = — 
pc
&t + £‘t + _ l & t
cbt* £y‘ 8z2 k a St
• For steady state condition (Poisson’s equation),
6~t + d't + d~t ^
£x' £ y ~ £z'
• For steady state and absence of internal heat generation (Laplace equation),
• For unsteady heat flow with no internal heat generation,
d^t + £:t + £*t _ 1 £t 
£yr 5 ), : a £t
Cylindrical Coordinates
• For homogeneous and isotropic material,
£‘ t 1 £t 1 c^r £‘ t + _ 1 £t
_£r‘ r £r r ‘ £<p‘ 5r: _ k t t £ l
For steady state unidirectional heat flow in radial direction with no internal 
heat generation,
Page 4


One Dimensional Heat Conduction 
Conduction is the transfer of energy from the more energetic particles of a 
substance to the adjacent less energetic ones as result of interactions between 
the particles
Steady and Unsteady State Conduction
• Steady state conduction is the form of conduction that happens when the 
temperature difference driving the conduction is constant, so that, the spatial 
distribution of temperatures in the conducting object does not change any 
further.
• In general, during any period in which temperatures are changing in time at 
any place within an object, the mode of thermal energy flow is termed 
transient conduction or nonsteady state conduction.
One-Dimensional Heat Flow
The term 'one-dimensional' is applied to heat conduction problem when:
1. Only one space coordinate is required to describe the temperature distribution 
within a heat conducting body;
2. Edge effects are neglected;
3. The flow of heat energy takes place along the coordinate measured normal to 
the surface.
Fourier’s Law of Heat Conduction
Consider steady conduction through a large plane wall of thickness Ax = L 
and surface area A. The temperature difference across the wall is AT = T2 - 
T1.
• It has been experimentally observed that the rate of heat conduction through 
a layer is
proportional to the temperature difference between the layer and the heat tran 
sfer area, but it is inversely proportional to the thickness of the layer.
rate of heat transfer o c
(surface area)(temperature difference) 
thickness
Q * Co»d
\W)
where constant proportionality k is the thermal conductivity of the material.
• In the limiting case where Ax— > 0, the equation above reduces to the 
differential form
, , dT
Q Cond = - k A — — 
dx
m
which is called Fourier's law of heat conduction.
• The term A = Cross-sectional area perpendicular to heat flow direction 
and dT/dx is called the temperature gradient, which is the slope of the 
temperature curve (the rate of change of temperature T with length x).
Thermal Conductivity(k [W/mK])
• The thermal conductivity is defined as the rate of heat transfer through a unit 
thickness of material per unit area per unit temperature difference.
• Thermal conductivity changes with temperature and is determined through 
experiments and is the measure of a material's ability to conduct heat.
• The thermal conductivity of certain materials show a dramatic change at 
temperatures near absolute zero, when these solids become 
superconductors.
• An isotropic material is a material that has uniform properties in all 
directions. Insulators are materials used primarily to provide resistance to
heat flow. They have low thermal conductivity 
General Heat Conduction Equation
Carterisan Coordinates (side parallel to x, y and z-directions)
• qg = Internal heat generation per unit volume per unit time
• f = Temperature at left face of the differential control volume
• kx, ky , kz = Thermal conductivities of the material in x, y and z-directions 
respectively
• c = Specific heat of the material
• p = Density of the material
• a = Thermal diffusivity
k_
pc
• dT = Instantaneous time.
• For homogeneous and isotropic material.
kx = k ^ = k. = k, a = — 
pc
&t + £‘t + _ l & t
cbt* £y‘ 8z2 k a St
• For steady state condition (Poisson’s equation),
6~t + d't + d~t ^
£x' £ y ~ £z'
• For steady state and absence of internal heat generation (Laplace equation),
• For unsteady heat flow with no internal heat generation,
d^t + £:t + £*t _ 1 £t 
£yr 5 ), : a £t
Cylindrical Coordinates
• For homogeneous and isotropic material,
£‘ t 1 £t 1 c^r £‘ t + _ 1 £t
_£r‘ r £r r ‘ £<p‘ 5r: _ k t t £ l
For steady state unidirectional heat flow in radial direction with no internal 
heat generation,
Constant
Spherical Coordinates
• For homogeneous and isotropic material,
A 1 g U n f* l 1 9' - 1 9 1
r1 an- 6 do' rsin£d0( 06) r-Or) Or) k a 0-
• For steady state unidirectional heat flow in radial direction with no internal 
heat generation,
M 4 I - 0
r dry dr)
Heat Generation in Solids
• Conversion of some form of energy into heat energy in a medium is called 
heat generation. Heat generation leads to a temperature rise throughout the 
medium.
• Heat generation is usually expressed per unit volume (W/m3)
• The maximum temperature Tm a x in a solid that involves uniform heat 
generation will occur at a location furthest away from the outer surface when 
the outer surface is maintained at a constant temperature, Ts
• Consider a solid medium of surface area A, volume V , and constant thermal co 
nductivity k,
where heat is generated at a constant rate of g' per unit volume. Heat is transf 
erred from the solid to the surrounding medium at T„.
Under steady conditions, the energy balance for the solid can be expressed as:
• rate of heat transfer from the solid = rate of energy generation 
within the solid
• Q' = g' V
Newton’s law of cooling, Q'= hA (Ts - T°°), combining these equations, a 
relationship for the surface temperature can be found:
Page 5


One Dimensional Heat Conduction 
Conduction is the transfer of energy from the more energetic particles of a 
substance to the adjacent less energetic ones as result of interactions between 
the particles
Steady and Unsteady State Conduction
• Steady state conduction is the form of conduction that happens when the 
temperature difference driving the conduction is constant, so that, the spatial 
distribution of temperatures in the conducting object does not change any 
further.
• In general, during any period in which temperatures are changing in time at 
any place within an object, the mode of thermal energy flow is termed 
transient conduction or nonsteady state conduction.
One-Dimensional Heat Flow
The term 'one-dimensional' is applied to heat conduction problem when:
1. Only one space coordinate is required to describe the temperature distribution 
within a heat conducting body;
2. Edge effects are neglected;
3. The flow of heat energy takes place along the coordinate measured normal to 
the surface.
Fourier’s Law of Heat Conduction
Consider steady conduction through a large plane wall of thickness Ax = L 
and surface area A. The temperature difference across the wall is AT = T2 - 
T1.
• It has been experimentally observed that the rate of heat conduction through 
a layer is
proportional to the temperature difference between the layer and the heat tran 
sfer area, but it is inversely proportional to the thickness of the layer.
rate of heat transfer o c
(surface area)(temperature difference) 
thickness
Q * Co»d
\W)
where constant proportionality k is the thermal conductivity of the material.
• In the limiting case where Ax— > 0, the equation above reduces to the 
differential form
, , dT
Q Cond = - k A — — 
dx
m
which is called Fourier's law of heat conduction.
• The term A = Cross-sectional area perpendicular to heat flow direction 
and dT/dx is called the temperature gradient, which is the slope of the 
temperature curve (the rate of change of temperature T with length x).
Thermal Conductivity(k [W/mK])
• The thermal conductivity is defined as the rate of heat transfer through a unit 
thickness of material per unit area per unit temperature difference.
• Thermal conductivity changes with temperature and is determined through 
experiments and is the measure of a material's ability to conduct heat.
• The thermal conductivity of certain materials show a dramatic change at 
temperatures near absolute zero, when these solids become 
superconductors.
• An isotropic material is a material that has uniform properties in all 
directions. Insulators are materials used primarily to provide resistance to
heat flow. They have low thermal conductivity 
General Heat Conduction Equation
Carterisan Coordinates (side parallel to x, y and z-directions)
• qg = Internal heat generation per unit volume per unit time
• f = Temperature at left face of the differential control volume
• kx, ky , kz = Thermal conductivities of the material in x, y and z-directions 
respectively
• c = Specific heat of the material
• p = Density of the material
• a = Thermal diffusivity
k_
pc
• dT = Instantaneous time.
• For homogeneous and isotropic material.
kx = k ^ = k. = k, a = — 
pc
&t + £‘t + _ l & t
cbt* £y‘ 8z2 k a St
• For steady state condition (Poisson’s equation),
6~t + d't + d~t ^
£x' £ y ~ £z'
• For steady state and absence of internal heat generation (Laplace equation),
• For unsteady heat flow with no internal heat generation,
d^t + £:t + £*t _ 1 £t 
£yr 5 ), : a £t
Cylindrical Coordinates
• For homogeneous and isotropic material,
£‘ t 1 £t 1 c^r £‘ t + _ 1 £t
_£r‘ r £r r ‘ £<p‘ 5r: _ k t t £ l
For steady state unidirectional heat flow in radial direction with no internal 
heat generation,
Constant
Spherical Coordinates
• For homogeneous and isotropic material,
A 1 g U n f* l 1 9' - 1 9 1
r1 an- 6 do' rsin£d0( 06) r-Or) Or) k a 0-
• For steady state unidirectional heat flow in radial direction with no internal 
heat generation,
M 4 I - 0
r dry dr)
Heat Generation in Solids
• Conversion of some form of energy into heat energy in a medium is called 
heat generation. Heat generation leads to a temperature rise throughout the 
medium.
• Heat generation is usually expressed per unit volume (W/m3)
• The maximum temperature Tm a x in a solid that involves uniform heat 
generation will occur at a location furthest away from the outer surface when 
the outer surface is maintained at a constant temperature, Ts
• Consider a solid medium of surface area A, volume V , and constant thermal co 
nductivity k,
where heat is generated at a constant rate of g' per unit volume. Heat is transf 
erred from the solid to the surrounding medium at T„.
Under steady conditions, the energy balance for the solid can be expressed as:
• rate of heat transfer from the solid = rate of energy generation 
within the solid
• Q' = g' V
Newton’s law of cooling, Q'= hA (Ts - T°°), combining these equations, a 
relationship for the surface temperature can be found:
T = T +
g 'V
hA
• Using the above relationship, the surface temperature can be calculated for a 
plane wall of thickness 2L, a long cylinder of radius r0 , and a sphere of radius 
r0 , as follows:
s'L
T = T - s
‘. p l a n e w a ll ® ^
- . c y l i n d e r
= T +
S J ';
T
“ . s p h e r e
= T -
2h 
3 h
• Note that the rise in temperature is due to heat generation. Using the 
Fourier’s law, we can derive a relationship for the centre (maximum) 
temperature of long cylinder of radius r0.
r, dT
- H — = g K
After integrating,
ATTO = r 0- r , =
A. = 2 m -L Vr - t it L
4 J fc
where T0 is the centerline temperature of the cylinder (Tmax). Using the approach, 
the maximum temperature can be found for plane walls and spheres
A T,
m a s . c y l i n d e r
AT,
m a s . p L a n e w a l l
3 rS 
4k
_ g L 
2k
AT,
m a s , i p h e r e
g - r< ,
6k