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ME 3345 Heat Transfer
Week_3_1
Conduction with heat generation
Extended Surfaces
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1 2 [K/W]tT T
Rq
Review of Thermal Resistance
2 1ln( / )
2t
r r
R Lk
1 21/ 1/
4t
r rR
k
One-dimensional, steady state, and constant k Without heat generation
t
LR
kA
1 2( ) lnT r C r C
1 2( )T x C x C
12( )
CT r C
r
0d dT
kdx dx
0
d dT
krdr dr
2 0d dT
krdr dr
Plane Wall
Cylindrical
Spherical
Definition:
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21 2
10 ( ) ln
4
d dT q qr T r r C r C
r dr dr k k
Cylindrical:
Spherical:
2 2 122
10 ( )
6
Cd dT q qr T r r C
dr dr k k r r
22
1 220 ( )
2
d T q qT x x C x C
k kdx
Plane Wall:
With Heat Generation
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21 2( )
2qT x x C x C k
Temperature Distribution
Plane Wall with Heat Generation
Symmetric
vs.
Adiabatic
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Example 3.6.A plane wall is a composite of two materials, A and B. The
wall of material A has uniform heat generation 1.5 x 106W/m3, kA= 75W/m K, and thicknessLA= 50 mm. The wall material B has no generation
with kB= 150 W/m K and thicknessLB= 20 mm. The inner surface of
material A is well insulated, while the outer surface of material B is cooled
by a water stream with T = 30C and h= 1000 W/m2 K.
Determine the temperature distribution and T0, T1, T2.
Example 3.6
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(a) parabolic in A(b) zero slope at wall
(c) linear in B
(d) slope change
(e) large gradient near surface/ 2B Ak k
How will qxchange?
It linearly increases in A
and remains the same in B.
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Can we use thermal resistance in A?
"g out AE E q qL
/ 1/B BL k h
2 2" ( )AqLq h T T T T
h
1
1BA
B
L
T T qL k h 2 22
( ) 12
sqL xT x T
k L
2
0 1(0)2
A
A
qLT T T
k
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The influence of h
Maximum temperature is often important for design consideration.
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21 2
1 0 ( ) ln4
d dT q qr T r r C r C r dr dr k k
Cylindrical:
For a solid cylinder, there is only one boundary.
T(r)
Apply symmetric B.C.
10
20
20
0
0 0
( ) 4
4
r
s
dTC
dr
qT r T r k
qrT T
k
r
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Example: An insulated wire of diameter D = 2mm
and uniform temperature T has an electrical resistance of
0.01 /m and a current of 20A. The insulation has an outerdiameter of 3mm and thermal conductivity of K=0.01 W/mK.
A) If heat is loss through convection, what is the surface
temperature of the rod and the insulation?
B) What is the rate of heat transfer per unit length at r =0, 0.5mm,
and 1.5 mm if the power density in the rod is 3x105W/m3?
D = 2mm
D = 3mm
h = 5 W/m2KT = 20 C
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( )conv surface sq A h T T
How to enhance heat transfer
(without increasing the temperature difference) ??
Fins - Extended Surfaces
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( )conv surface sq A h T T
How to enhance heat transfer
(without increasing the temperature difference) ??
(1) Increase hby strong forced convection (use fan, usewater instead of air, spray or inject water, etc.
(2) Increase the surface areaA. The second is often
achieved by using fins.
Fins - Extended Surfaces
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Mobile PentiumProcessors
Extruded Heat Sink
Automobile Radiator
Examples of Extended Surfaces
Radiator (household heating)
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Simple Structures:
We will perform the analysis for simple cases and discuss
engineering methods to deal with complicated geometry.
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How much performance increase
Space
Weight/ Material
Manufacturing processCost
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(a) Rectangular fin. (b) Pin fin.
Fins of Uniform Cross Section
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Analysis of Heat Transfer Enhancement
The application of extended surfaces for heat transferenhancement must be carefully considered. This processes
induces additional manufacturing costs and complexity.
Thus, we must find a way to quantify the added benefits
of using extended surfaces to justify their application.
A) Determine the rate of heat transfer from an extended
surface. Involves finding the temperature distribution
in the fin structure.
B) Define some measure of efficiencyfor extended
surfaces. Use this as a basis for determining when to use them.