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Geometrical Variations of Heat Conduction Equation
P M V Subbarao
Professor
Mechanical Engineering Department
IIT Delhi
Same Physics… But Shape Matters…..
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The Heat Conduction Equation
pC
trgT
t
T
),(
..
For Isotropic Material:
For a homogeneous Isotropic material:
pC
txgT
t
T
),(2
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Mass Diffusion Equation
Using the analogy
),(.. trCDt
C
For a homogeneous & isotropic material:
),(2 trCDt
C
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This is a general form of heat conduction equation.
Valid for all geometries.
Selection of geometry depends on nature of application.
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General conduction equation in Cartesian Coordinate System
xq xxq
yyq
yqzzq
zq
Rate of energy generation
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),(. 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
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General conduction equation based on Polar
Cylindrical Coordinates
):,,(1
2tzrg
z
Tk
z
Tk
rr
Tkr
rt
TC p
),( txgTkt
TC p
zrkk ,,
),,(1
2zrg
z
Tk
z
Tk
rr
Tkr
rt
TC p
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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
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Thermally Heterogeneous Materials
zyxkk ,,
),(. txgTkt
TC p
),,,( tzyxgzzT
k
y
yT
k
xxT
k
t
TC p
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),,,(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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Primitive Pottery Kilns
Time
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Thermal Conductivity of Brick Masonry Walls
1,thR
1,2, 2 thth RR
3,thR
3,4, 2 thth RR
1pC
3pC
12 pp CC
34 pp CC
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Geographical Applications of Thermal Mapping
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One Dimensional Heat Conduction problems
P M V Subbarao
Associate Professor
Mechanical Engineering Department
IIT Delhi
Simple ideas for complex Problems…
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Desert Housing & Composite Walls
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),(. txgTkt
TC p
General conduction equation in Cartesian Coordinate System
• 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.
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General Steady-State One-Dimensional Conduction
Rate of energy generation
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Steady Heat transfer through a plane slab
02
2
dx
TdA
0),,,(2
2
tzyxgx
Tk
No heat generation
211 CxCTCdx
dT
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Wall Surfaces with Convection
2112
2
0 CxCTCdx
dT
dx
TdA
Boundary conditions:
110
)0(
TThdx
dTk
x
22 )(
TLThdx
dTk
Lx
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Wall with isothermal Surface and Convection Wall
2112
2
0 CxCTCdx
dT
dx
TdA
Boundary conditions:
1)0( TxT
22 )(
TLThdx
dTk
Lx
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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
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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:
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Composite Walls
• The overall thermal resistance is given by
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Desert Housing & Composite Walls