dgre

1. Solutions of the Conduction Equation
P M V Subbarao
Associate Professor
Mechanical Engineering Department
IIT Delhi
An Idea Generates More Mathematics….
Mathematics Generate Mode Ideas…..

2. The Conduction Equation
)
,
(
'
'
. t
r
g
q
t
H





  )
,
(
.
. t
r
g
T
k
t
T
Cp 







Incorporation of the constitutive equation into the energy
equation above yields:
Dividing both sides by Cp and introducing the thermal
diffusivity of the material given by
s
m
m
s
m
C
k
p



2



3. Thermal Diffusivity
• Thermal diffusivity includes the effects of properties like
mass density, thermal conductivity and specific heat
capacity.
• Thermal diffusivity, which is involved in all unsteady heat-
conduction problems, is a property of the solid object.
• The time rate of change of temperature depends on its
numerical value.
• The physical significance of thermal diffusivity is
associated with the diffusion of heat into the medium
during changes of temperature with time.
• The higher thermal diffusivity coefficient signifies the
faster penetration of the heat into the medium and the less
time required to remove the heat from the solid.

4. p
p C
t
r
g
T
C
k
t
T


)
,
(
.
. 
















This is often called the heat equation.
 
p
C
t
r
g
T
t
T


)
,
(
.
. 





For a homogeneous material:
p
C
t
x
g
T
t
T


)
,
(
2






5. This is a general form of heat conduction equation.
Valid for all geometries.
Selection of geometry depends on nature of application.

6. General conduction equation based on
Cartesian Coordinates
x
q x
x
q 

y
y
q 

y
q
z
z
q 

z
q

7. )
,
(
. t
x
g
T
k
t
T
Cp 






For an isotropic and homogeneous material:
)
,
(
2
t
x
g
T
k
t
T
Cp 





)
:
,
,
(
2
2
2
2
2
2
t
z
y
x
g
z
T
y
T
x
T
k
t
T
Cp 



















8. General conduction equation based on Polar
Cylindrical Coordinates
)
:
,
,
(
1
2
2
2
2
2
t
z
r
g
z
T
T
r
r
T
r
r
k
t
T
Cp 

 


























9. General conduction equation based on Polar
Spherical Coordinates
)
:
,
,
(
sin
1
sin
sin
1
1
2
2
2
2
2
2
2
t
r
g
T
r
T
r
r
T
r
r
r
k
t
T
Cp 







 

































X
Y

10. Thermal Conductivity of Brick Masonry Walls

11. Thermally Heterogeneous Materials
 
z
y
x
k
k ,
,

)
,
(
. t
x
g
T
k
t
T
Cp 






)
,
,
,
( t
z
y
x
g
z
z
T
k
y
y
T
k
x
x
T
k
t
T
Cp 







































12. )
,
,
,
(
2
2
2
2
2
2
t
z
y
x
g
z
T
k
z
T
z
k
y
T
k
y
T
y
k
x
T
k
x
T
x
k
t
T
Cp 



























More service to humankind than heat transfer rate calculations

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17. One Dimensional Heat Conduction problems
P M V Subbarao
Associate Professor
Mechanical Engineering Department
IIT Delhi
Simple ideas for complex Problems…

18. Desert Housing & Composite Walls

19. Steady-State One-Dimensional Conduction
• Assume a homogeneous medium with invariant thermal conductivity ( k
= constant) :
• For conduction through a large wall
the heat equation reduces to:
)
,
,
,
(
2
2
t
z
y
x
g
x
T
k
x
T
x
k
t
T
Cp 











)
,
,
,
(
2
2
t
z
y
x
g
x
T
k
t
T
Cp 






One dimensional Transient conduction with heat generation.

20. Steady Heat transfer through a plane slab
0
2
2

dx
T
d
A
0
)
,
,
,
(
2
2




t
z
y
x
g
x
T
k
No heat generation
2
1
1 C
x
C
T
C
dx
dT






21. Isothermal Wall Surfaces
Apply boundary conditions to solve for
constants: T(0)=Ts1 ; T(L)=Ts2
2
1
1 C
x
C
T
C
dx
dT





The resulting temperature distribution
and varies linearly with x.

22. Applying Fourier’s law:
heat transfer rate:
heat flux:
Therefore, both the heat transfer rate and heat
flux are independent of x.

23. Wall Surfaces with Convection
2
1
1
2
2
0 C
x
C
T
C
dx
dT
dx
T
d
A 





Boundary conditions:
 
1
1
0
)
0
( 



 T
T
h
dx
dT
k
x
 
2
2 )
( 



 T
L
T
h
dx
dT
k
L
x

24. Wall with isothermal Surface and Convection Wall
2
1
1
2
2
0 C
x
C
T
C
dx
dT
dx
T
d
A 





Boundary conditions:
1
)
0
( T
x
T 

 
2
2 )
( 



 T
L
T
h
dx
dT
k
L
x

25. Electrical Circuit Theory of Heat Transfer
• Thermal Resistance
• A resistance can be defined as the ratio of a driving
potential to a corresponding transfer rate.
i
V
R


Analogy:
Electrical resistance is to conduction of electricity as thermal
resistance is to conduction of heat.
The analog of Q is current, and the analog of the
temperature difference, T1 - T2, is voltage difference.
From this perspective the slab is a pure resistance to heat
transfer and we can define

26. q
T
R
R
T
q th
th





W
K
m
W
K
m
m
kA
L
L
T
T
kA
T
T
q
T
R
s
s
s
s
cond
th /
1
.
2
1
2
2
1







 





 
W
K
m
W
K
m
hA
T
T
hA
T
T
q
T
R
s
s
conv
th /
1
.
1
2
2










27.  
W
K
m
W
K
m
A
h
T
T
A
h
T
T
q
T
R
r
surr
s
r
surr
s
rad
th /
1
.
1
2
2








28. The composite Wall
• The concept of a thermal
resistance circuit allows
ready analysis of problems
such as a composite slab
(composite planar heat
transfer surface).
• In the composite slab, the
heat flux is constant with x.
• The resistances are in series
and sum to Rth = Rth1 + Rth2.
• If TL is the temperature at the
left, and TR is the
temperature at the right, the
heat transfer rate is given by
2
1 th
th
R
L
th R
R
T
T
R
T
q






29. Wall Surfaces with Convection
2
1
1
2
2
0 C
x
C
T
C
dx
dT
dx
T
d
A 





Boundary conditions:
 
1
1
0
)
0
( 



 T
T
h
dx
dT
k
x
 
2
2 )
( 



 T
L
T
h
dx
dT
k
L
x
Rconv,1 Rcond Rconv,2
T1 T2

30. Heat transfer for a wall with dissimilar
materials
• For this situation, the total heat flux Q is made up of the heat flux
in the two parallel paths:
• Q = Q1+ Q2
 with the total resistance given by:

31. Composite Walls
• The overall thermal resistance is given by

32. Desert Housing & Composite Walls