ame 60634 int. heat trans. d. b. go 1 1-d steady conduction: plane wall governing equation:...
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AME 60634 Int. Heat Trans.
D. B. Go 1
1-D Steady Conduction: Plane WallGoverning Equation:
Dirichlet Boundary Conditions:
qx kAdT
dx
kA
LTs,1 Ts,2
Solution:
Heat Flux:
Heat Flow:
temperature is not a function of k
T(x) Ts,1 Ts,2 Ts,1 x
L
q x kdT
dx
k
LTs,1 Ts,2
Notes:• A is the cross-sectional area of the wall perpendicular to the heat flow• both heat flux and heat flow are uniform independent of position (x)• temperature distribution is governed by boundary conditions and length
of domain independent of thermal conductivity (k)
heat flux/flow are a function of k
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1-D Steady Conduction: Cylinder WallGoverning Equation:
Dirichlet Boundary Conditions:
T(r1) Ts,1 ; T(r2) Ts,2
Notes:• heat flux is not uniform function of position (r)• both heat flow and heat flow per unit length are uniform independent of
position (r)
Solution:
Heat Flux:
Heat Flow:
T(r) Ts,1 Ts,2
ln r1 r2 ln
r
r2
Ts,2
q r kdT
dr
k Ts,1 Ts,2 r ln r2 r1
qr kAdT
dr2rL q r
2Lk Ts,1 Ts,2 ln r2 r1
q r qr
L
2k Ts,1 Ts,2 ln r2 r1 heat flow per unit length
heat flux is non-uniform
heat flow is uniform
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1-D Steady Conduction: Spherical ShellGoverning Equation:
Dirichlet Boundary Conditions:
T(r1) Ts,1 ; T(r2) Ts,2
Solution:
Heat Flux:
Heat Flow:
T(r) Ts,1 Ts,1 Ts,2 1 r1 r 1 r1 r2
q r kdT
dr
k Ts,1 Ts,2 r2 1 r1 1 r2
qr kAdT
dr4r2 q r
4k Ts,1 Ts,2 1 r1 1 r2
Notes:• heat flux is not uniform function of position (r)• heat flow is uniform independent of position (r)
heat flux is non-uniform
heat flow is uniform
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Thermal Resistance
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Thermal Circuits: Composite Plane Wall
Circuits based on assumption of (a) isothermal surfaces normal to x direction
or (b) adiabatic surfaces parallel to x direction
Rtot LE
kE A
kF A
2LF
kG A
2LG
1
LH
kH A
Rtot
2LE
kE A
2LF
kF A
2LH
kH A
1
2LE
kE A
2LG
kG A
2LH
kH A
1
1
Actual solution for the heat rate q is bracketed by these two approximations
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Thermal Circuits: Contact Resistance
In the real world, two surfaces in contact do not transfer heat perfectly
R t,c TA TB
q x Rt ,c
R t,cAc
Contact Resistance: values depend on materials (A and B), surface roughness, interstitial conditions, and contact pressure typically calculated or looked up
Equivalent total thermal resistance:
Rtot LA
kA Ac
R t,c
Ac
LB
kB Ac
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fig_02_04
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fig_02_05
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Fins: The Fin Equation• Solutions
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Fins: Fin Performance Parameters• Fin Efficiency
– the ratio of actual amount of heat removed by a fin to the ideal amount of heat removed if the fin was an isothermal body at the base temperature
• that is, the ratio the actual heat transfer from the fin to ideal heat transfer from the fin if the fin had no conduction resistance
• Fin Effectiveness– ratio of the fin heat transfer rate to the heat transfer rate that would exist
without the fin
• Fin Resistance– defined using the temperature difference between the base and fluid as
the driving potential
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Fins: Efficiency
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Fins: Efficiency
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Fins: Arrays
• Arrays– total surface area
– total heat rate
– overall surface efficiency
– overall surface resistance
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Fins: Thermal Circuit• Equivalent Thermal Circuit
• Effect of Surface Contact Resistance
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Fins: Overview• Fins
– extended surfaces that enhance fluid heat transfer to/from a surface in large part by increasing the effective surface area of the body
– combine conduction through the fin and convection to/from the fin
• the conduction is assumed to be one-dimensional
• Applications– fins are often used to enhance convection when h is
small (a gas as the working fluid)– fins can also be used to increase the surface area
for radiation– radiators (cars), heat sinks (PCs), heat exchangers
(power plants), nature (stegosaurus)
Straight fins of (a) uniform and (b) non-uniform cross sections; (c) annularfin, and (d) pin fin of non-uniform cross section.
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Fins: The Fin Equation• Solutions