Solutions of the Conduction Equation
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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…..
The Conduction Equation
),(''. trgqt
H
),(.. trgTkt
TC p
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
mm
s
m
C
k
p
2
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.
pp C
trgT
C
k
t
T
),(
..
This is often called the heat equation.
pC
trgT
t
T
),(
..
For a homogeneous material:
pC
txgT
t
T
),(2
This is a general form of heat conduction equation.
Valid for all geometries.
Selection of geometry depends on nature of application.
General conduction equation based on Cartesian Coordinates
xqxxq
yyq
yqzzq
zq
),(. txgTkt
TC p
For an isotropic and homogeneous material:
),(2 txgTkt
TC p
):,,(2
2
2
2
2
2
tzyxgz
T
y
T
x
Tk
t
TC p
General conduction equation based on Polar
Cylindrical Coordinates
):,,(1
2
2
2
2
2tzrg
z
TT
rr
Tr
rk
t
TC p
General conduction equation based on Polar Spherical Coordinates
):,,(sin
1sin
sin
112
2
2222
2trg
T
r
T
rr
Tr
rrk
t
TC p
X
Y
Thermal Conductivity of Brick Masonry Walls
Thermally Heterogeneous Materials
zyxkk ,,
),(. txgTkt
TC p
),,,( tzyxgz
zT
k
y
yT
k
xxT
k
t
TC p
),,,(2
2
2
2
2
2
tzyxgz
Tk
z
T
z
k
y
Tk
y
T
y
k
x
Tk
x
T
x
k
t
TC p
More service to humankind than heat transfer rate calculations
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Thermal Imaging of Brain
One Dimensional Heat Conduction problems
P M V Subbarao
Associate Professor
Mechanical Engineering Department
IIT Delhi
Simple ideas for complex Problems…
Desert Housing & Composite Walls
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
tzyxgx
Tk
x
T
x
k
t
TC p
),,,(2
2
tzyxgx
Tk
t
TC p
One dimensional Transient conduction with heat generation.
Steady Heat transfer through a plane slab
02
2
dx
TdA
0),,,(2
2
tzyxgx
Tk
No heat generation
211 CxCTCdx
dT
Isothermal Wall Surfaces
Apply boundary conditions to solve for constants: T(0)=Ts1 ; T(L)=Ts2
211 CxCTCdx
dT
The resulting temperature distribution is:
and varies linearly with x.
Applying Fourier’s law:
heat transfer rate:
heat flux:
Therefore, both the heat transfer rate and heat flux are independent of x.
Wall Surfaces with Convection
2112
2
0 CxCTCdx
dT
dx
TdA
Boundary conditions:
110
)0(
TThdx
dTk
x
22 )(
TLThdx
dTk
Lx
Wall with isothermal Surface and Convection Wall
2112
2
0 CxCTCdx
dT
dx
TdA
Boundary conditions:
1)0( TxT
22 )(
TLThdx
dTk
Lx
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
VR
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
q
TR
R
Tq th
th
WKmW
Kmm
kA
L
L
TTkA
TT
q
TR
ss
ss
condth /
1.2
12
21
WKmW
Km
hATThA
TT
q
TR
s
s
convth /
1.12
2
WKmW
Km
AhTTAh
TT
q
TR
rsurrsr
surrs
radth /
1.12
2
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
21 thth
RL
th RR
TT
R
Tq
Wall Surfaces with Convection
2112
2
0 CxCTCdx
dT
dx
TdA
Boundary conditions:
110
)0(
TThdx
dTk
x
22 )(
TLThdx
dTk
Lx
Rconv,1 Rcond Rconv,2
T1 T2
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:
Composite Walls
• The overall thermal resistance is given by
Desert Housing & Composite Walls
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