Short Notes: Unsteady Heat Conduction | Short Notes for Mechanical Engineering PDF Download

 Page 1


Unsteady Heat Conduction 
If the temperature of a body does not vary with time, it is said to be in steady state. 
But if there is an abrupt change in its surface temperature, it attains an equilibrium 
temperature or a steady state after some period. During this period, the 
temperature varies with time and body is said to be in the unsteady or transient 
state. This phenomenon is known as Unsteady or transient heat conduction.
Lumped System Analysis
• Interior temperatures of some bodies remain essentially uniform at all times 
during a heat transfer process.
• The temperature of such bodies is only a function of time, T = T(t).
• The heat transfer analysis based on this idealization is called lumped system 
analysis.
Consider a body of the arbitrary shape of mass m, volume V , surface area A, density 
p and specific heat Cp initially at a uniform temperature Ti. •
• At time t = 0, the body is placed into a medium at temperature T°° (T°° >Ti) wit 
h a heat transfer coefficient h. An energy balance of the solid for a time
Page 2


Unsteady Heat Conduction 
If the temperature of a body does not vary with time, it is said to be in steady state. 
But if there is an abrupt change in its surface temperature, it attains an equilibrium 
temperature or a steady state after some period. During this period, the 
temperature varies with time and body is said to be in the unsteady or transient 
state. This phenomenon is known as Unsteady or transient heat conduction.
Lumped System Analysis
• Interior temperatures of some bodies remain essentially uniform at all times 
during a heat transfer process.
• The temperature of such bodies is only a function of time, T = T(t).
• The heat transfer analysis based on this idealization is called lumped system 
analysis.
Consider a body of the arbitrary shape of mass m, volume V , surface area A, density 
p and specific heat Cp initially at a uniform temperature Ti. •
• At time t = 0, the body is placed into a medium at temperature T°° (T°° >Ti) wit 
h a heat transfer coefficient h. An energy balance of the solid for a time
interval dt can be expressed as
heat transfer into the body during dt = the increase in the energy of
the body during dt 
h A (T « > - T) dt = m Cp dT
• With m = pV and change of variable dT = d(T - T°°), we find:
t - t, ere,
• Integrating from t = 0 to T = Ti
• Using above equation, we can determine the temperature T(t) of a body at 
time t, or alternatively, the time t required for the temperature to reach a 
specified value T(t).
Note that the temperature of a body approaches the ambient temperature 
T» exponentially.
• A large value of b indicates that the body will approach the environment 
temperature in a short time.
• b is proportional to the surface area but inversely proportional to the mass 
and the specific heat of the body.
• The total amount of heat transfer between a body and its surroundings over a 
time interval is: Q = m Cp [T(t) - Ti]
The behaviour of lumped systems can be interpreted as a thermal time constant 
as shown in fig. below:
Page 3


Unsteady Heat Conduction 
If the temperature of a body does not vary with time, it is said to be in steady state. 
But if there is an abrupt change in its surface temperature, it attains an equilibrium 
temperature or a steady state after some period. During this period, the 
temperature varies with time and body is said to be in the unsteady or transient 
state. This phenomenon is known as Unsteady or transient heat conduction.
Lumped System Analysis
• Interior temperatures of some bodies remain essentially uniform at all times 
during a heat transfer process.
• The temperature of such bodies is only a function of time, T = T(t).
• The heat transfer analysis based on this idealization is called lumped system 
analysis.
Consider a body of the arbitrary shape of mass m, volume V , surface area A, density 
p and specific heat Cp initially at a uniform temperature Ti. •
• At time t = 0, the body is placed into a medium at temperature T°° (T°° >Ti) wit 
h a heat transfer coefficient h. An energy balance of the solid for a time
interval dt can be expressed as
heat transfer into the body during dt = the increase in the energy of
the body during dt 
h A (T « > - T) dt = m Cp dT
• With m = pV and change of variable dT = d(T - T°°), we find:
t - t, ere,
• Integrating from t = 0 to T = Ti
• Using above equation, we can determine the temperature T(t) of a body at 
time t, or alternatively, the time t required for the temperature to reach a 
specified value T(t).
Note that the temperature of a body approaches the ambient temperature 
T» exponentially.
• A large value of b indicates that the body will approach the environment 
temperature in a short time.
• b is proportional to the surface area but inversely proportional to the mass 
and the specific heat of the body.
• The total amount of heat transfer between a body and its surroundings over a 
time interval is: Q = m Cp [T(t) - Ti]
The behaviour of lumped systems can be interpreted as a thermal time constant 
as shown in fig. below:
1
• where Rt is the resistance to convection heat transfer and Ct is the lumped 
thermal capacitance of the solid. Any increase in Rt or Ct will cause a solid to 
respond more
slowly to changes in its thermal environment and will increase the time respo 
nd required to reach thermal equilibrium.
Criterion for Lumped System Analysis
• Lumped system approximation provides a great convenience in heat transfer 
analysis. We
want to establish a criterion for the applicability of the lumped system analysi 
s. A characteristic length scale is defined as:
Lc= V/A
• A nondimensional parameter, the Biot number, is defined:
„ hLc 
k
. h&T convection at the surface of the body
^ j ' ' |
— AT conduction within the body
4 '
L. ! k conduction resistance within the bodv
B i = — — = ---------- :---------: ----------------------------------- --------
1 / h convection resistance at the surface of the body
• The Biot number is the ratio of the internal resistance (conduction) to the 
external resistance to heat convection.
• Lumped system analysis assumes a uniform temperature distribution 
throughout the body, which implies that the conduction heat resistance is 
zero. Thus, the lumped system analysis is exact when Bi = 0.
• It is generally accepted that the lumped system analysis is applicable if: Bis
0.1
• Therefore, small bodies with high thermal conductivity are good candidates 
for lumped system analysis. Note that assuming h to be constant and 
uniform is an approximation
Fourier number:
Page 4


Unsteady Heat Conduction 
If the temperature of a body does not vary with time, it is said to be in steady state. 
But if there is an abrupt change in its surface temperature, it attains an equilibrium 
temperature or a steady state after some period. During this period, the 
temperature varies with time and body is said to be in the unsteady or transient 
state. This phenomenon is known as Unsteady or transient heat conduction.
Lumped System Analysis
• Interior temperatures of some bodies remain essentially uniform at all times 
during a heat transfer process.
• The temperature of such bodies is only a function of time, T = T(t).
• The heat transfer analysis based on this idealization is called lumped system 
analysis.
Consider a body of the arbitrary shape of mass m, volume V , surface area A, density 
p and specific heat Cp initially at a uniform temperature Ti. •
• At time t = 0, the body is placed into a medium at temperature T°° (T°° >Ti) wit 
h a heat transfer coefficient h. An energy balance of the solid for a time
interval dt can be expressed as
heat transfer into the body during dt = the increase in the energy of
the body during dt 
h A (T « > - T) dt = m Cp dT
• With m = pV and change of variable dT = d(T - T°°), we find:
t - t, ere,
• Integrating from t = 0 to T = Ti
• Using above equation, we can determine the temperature T(t) of a body at 
time t, or alternatively, the time t required for the temperature to reach a 
specified value T(t).
Note that the temperature of a body approaches the ambient temperature 
T» exponentially.
• A large value of b indicates that the body will approach the environment 
temperature in a short time.
• b is proportional to the surface area but inversely proportional to the mass 
and the specific heat of the body.
• The total amount of heat transfer between a body and its surroundings over a 
time interval is: Q = m Cp [T(t) - Ti]
The behaviour of lumped systems can be interpreted as a thermal time constant 
as shown in fig. below:
1
• where Rt is the resistance to convection heat transfer and Ct is the lumped 
thermal capacitance of the solid. Any increase in Rt or Ct will cause a solid to 
respond more
slowly to changes in its thermal environment and will increase the time respo 
nd required to reach thermal equilibrium.
Criterion for Lumped System Analysis
• Lumped system approximation provides a great convenience in heat transfer 
analysis. We
want to establish a criterion for the applicability of the lumped system analysi 
s. A characteristic length scale is defined as:
Lc= V/A
• A nondimensional parameter, the Biot number, is defined:
„ hLc 
k
. h&T convection at the surface of the body
^ j ' ' |
— AT conduction within the body
4 '
L. ! k conduction resistance within the bodv
B i = — — = ---------- :---------: ----------------------------------- --------
1 / h convection resistance at the surface of the body
• The Biot number is the ratio of the internal resistance (conduction) to the 
external resistance to heat convection.
• Lumped system analysis assumes a uniform temperature distribution 
throughout the body, which implies that the conduction heat resistance is 
zero. Thus, the lumped system analysis is exact when Bi = 0.
• It is generally accepted that the lumped system analysis is applicable if: Bis
0.1
• Therefore, small bodies with high thermal conductivity are good candidates 
for lumped system analysis. Note that assuming h to be constant and 
uniform is an approximation
Fourier number:
t - r,
.(-•Sr F °)
Fo =
A _
/ * . :
= e
t — t.
O = - h A (t— ta)e(- B:Fo) o r Q = PVc(tt - t . ) [«“* * - 1 ]
• The temperature of a body in the unsteady state can be calculated at any time 
only when Biot number < 0.1.
Characteristic Length: Characteristic length is denoted by lc.
lc _
Volume (V)
Surface area exposed to surrounding (A)
Characteristic Length for different Section
C h aracteristic len g th 
fo r sphere
4
- ttR3 -
l - 3 - R
‘ 3
C h aracteristic len g th 
fo r so lid cylinder
t yR2 L R 
‘c~ 2t v R(L+R) ^ t > > ‘ ~ 2
C h aracteristic len g th 
fo r cube
i = ± = L 
c 611 6
C h aracteristic len g th 
fo r rectan g u lar p late
^ Ibt t
C h aracteristic len g th 
fo r h o llo w cy lin d er
2Ttrf + 2 ttW + 2-TrOo “ ! f )
Transient Conduction in Large Plane Walls, Long Cylinders, and Spheres
• The lumped system approximation can be used for small bodies of highly 
conductive materials.
• But, in general, the temperature is a function of position as well as time.
• Consider a plane wall of thickness 2L, a long cylinder of radius ro, and a 
sphere of radius ro initially at a uniform temperature Ti
Plane wall
Long cylinder Sphere
Page 5


Unsteady Heat Conduction 
If the temperature of a body does not vary with time, it is said to be in steady state. 
But if there is an abrupt change in its surface temperature, it attains an equilibrium 
temperature or a steady state after some period. During this period, the 
temperature varies with time and body is said to be in the unsteady or transient 
state. This phenomenon is known as Unsteady or transient heat conduction.
Lumped System Analysis
• Interior temperatures of some bodies remain essentially uniform at all times 
during a heat transfer process.
• The temperature of such bodies is only a function of time, T = T(t).
• The heat transfer analysis based on this idealization is called lumped system 
analysis.
Consider a body of the arbitrary shape of mass m, volume V , surface area A, density 
p and specific heat Cp initially at a uniform temperature Ti. •
• At time t = 0, the body is placed into a medium at temperature T°° (T°° >Ti) wit 
h a heat transfer coefficient h. An energy balance of the solid for a time
interval dt can be expressed as
heat transfer into the body during dt = the increase in the energy of
the body during dt 
h A (T « > - T) dt = m Cp dT
• With m = pV and change of variable dT = d(T - T°°), we find:
t - t, ere,
• Integrating from t = 0 to T = Ti
• Using above equation, we can determine the temperature T(t) of a body at 
time t, or alternatively, the time t required for the temperature to reach a 
specified value T(t).
Note that the temperature of a body approaches the ambient temperature 
T» exponentially.
• A large value of b indicates that the body will approach the environment 
temperature in a short time.
• b is proportional to the surface area but inversely proportional to the mass 
and the specific heat of the body.
• The total amount of heat transfer between a body and its surroundings over a 
time interval is: Q = m Cp [T(t) - Ti]
The behaviour of lumped systems can be interpreted as a thermal time constant 
as shown in fig. below:
1
• where Rt is the resistance to convection heat transfer and Ct is the lumped 
thermal capacitance of the solid. Any increase in Rt or Ct will cause a solid to 
respond more
slowly to changes in its thermal environment and will increase the time respo 
nd required to reach thermal equilibrium.
Criterion for Lumped System Analysis
• Lumped system approximation provides a great convenience in heat transfer 
analysis. We
want to establish a criterion for the applicability of the lumped system analysi 
s. A characteristic length scale is defined as:
Lc= V/A
• A nondimensional parameter, the Biot number, is defined:
„ hLc 
k
. h&T convection at the surface of the body
^ j ' ' |
— AT conduction within the body
4 '
L. ! k conduction resistance within the bodv
B i = — — = ---------- :---------: ----------------------------------- --------
1 / h convection resistance at the surface of the body
• The Biot number is the ratio of the internal resistance (conduction) to the 
external resistance to heat convection.
• Lumped system analysis assumes a uniform temperature distribution 
throughout the body, which implies that the conduction heat resistance is 
zero. Thus, the lumped system analysis is exact when Bi = 0.
• It is generally accepted that the lumped system analysis is applicable if: Bis
0.1
• Therefore, small bodies with high thermal conductivity are good candidates 
for lumped system analysis. Note that assuming h to be constant and 
uniform is an approximation
Fourier number:
t - r,
.(-•Sr F °)
Fo =
A _
/ * . :
= e
t — t.
O = - h A (t— ta)e(- B:Fo) o r Q = PVc(tt - t . ) [«“* * - 1 ]
• The temperature of a body in the unsteady state can be calculated at any time 
only when Biot number < 0.1.
Characteristic Length: Characteristic length is denoted by lc.
lc _
Volume (V)
Surface area exposed to surrounding (A)
Characteristic Length for different Section
C h aracteristic len g th 
fo r sphere
4
- ttR3 -
l - 3 - R
‘ 3
C h aracteristic len g th 
fo r so lid cylinder
t yR2 L R 
‘c~ 2t v R(L+R) ^ t > > ‘ ~ 2
C h aracteristic len g th 
fo r cube
i = ± = L 
c 611 6
C h aracteristic len g th 
fo r rectan g u lar p late
^ Ibt t
C h aracteristic len g th 
fo r h o llo w cy lin d er
2Ttrf + 2 ttW + 2-TrOo “ ! f )
Transient Conduction in Large Plane Walls, Long Cylinders, and Spheres
• The lumped system approximation can be used for small bodies of highly 
conductive materials.
• But, in general, the temperature is a function of position as well as time.
• Consider a plane wall of thickness 2L, a long cylinder of radius ro, and a 
sphere of radius ro initially at a uniform temperature Ti
Plane wall
Long cylinder Sphere
• We also assume a constant heat transfer coefficient h and neglect radiation. 
The formulation of the one-dimensional transient temperature distribution 
T(x,t) results in a partial differential equation (PDE), which can be solved 
using advanced mathematical methods. For the plane wall, the solution 
involves several parameters:
T = T(x, L, k, a, h, Tif T„) 
where a = k/pCp.
• By using dimensional groups,we can reduce the number of parameters.
0=0(x, Bi,r)
• To find the temperature solution for plane wall, i.e. Cartesian coordinate, we 
should solve the Laplace's equation with boundary and initial conditions:
g:r 1 cT
Sx1 a cl
¦ V x
Rec r =5 :10: =
So, we can write:
£-6 _ £6 
ex- ~ er
where,
${x.T) =
T[xA)-Ta
T ~ T m
dimensionless temperature
X — — dimensionless distance 
L
Bi - — Biot number
k
Fourier number