Conduction: Theory of Extended Surfaces -...
Transcript of Conduction: Theory of Extended Surfaces -...
Conduction: Theory of Extended Surfaces
Why extended surface?
)( TThAq s
q
h, T∞
Increasing h Increasing A 2
Fins as extended surfacesA fin is a thin component or appendage attached to a larger body or structure
In the context of heat transfer also, these are components protruding out of a heated (or cold) surface
Hot surface
Fin
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What happens in a fin?
An extended surface (combined conduction-convection system) is a solid within which heat transfer by conduction is assumed to be one dimensional,while heat is also transferred by convection (and/or radiation)from the surface in a direction transverse to that of conduction.
qconv (+ qrad)=hAs(T-T∞)
qcond
qconv= hAs(T-T∞)
qcond= -kAcdT/dx
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What happens in a fin?
An extended surface (combined conduction-convection system) is a solid within which heat transfer by conduction is assumed to be one dimensional,while heat is also transferred by convection (and/or radiation)from the surface in a direction transverse to that of conduction.
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Examples of Fins
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Question time
• Heat is transferred from hot water flowing inside a tube to cooling air flowing over the tube. To enhance heat transfer rate, which side should the fins be installed?
• Fins are most beneficial where his low
• Fin dimensions and k are criticaldesign parameters
Hot water @90C
Air @ 25C
Liquid side Air side7
Summary
• Extended surfaces help in enhancing heat dissipation
– Increases surface area for heat exchange
• Fins: most common embodiment of extended surface
– Can be of varied shape and forms
• Fin peformance = f (h, material, size)
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Fin Analysis
x
Tb
q kAdT
dxx C q q
dq
dxdxx dx x
x
dq h dA T Tconv S ( )( ), where dA is the surface area of the elementS
P: the fin perimeter
Ac: the fin cross-sectional area
Energy balance: )( TThPdx
dqqdqqq x
xconvdxxx
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Fin Analysis (cont.)
0)()( TThPdx
dTkA
dx
dc
TT 0)(
k
hP
dx
dA
dx
dc
)(xAA cc
ckA
hPmm
dx
d 22
2
2
,0
For a constant cross-section Ac
mxmx eCeCx 21)(
Need two boundary conditions0at xbBase:
Tip: 4 scenarios at x = L
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Temperature profilesCase Tip Condition Temp. Distribution Fin heat transfer
A Convection heat
transfer:
h(L)=-k(d/dx)x=L mLmk
hmL
xLmmk
hxLm
sinh)(cosh
)(sinh)()(cosh
MmL
mkhmL
mLmk
hmL
bsinh)(cosh
cosh)(sinh
B Adiabatic
(d/dx)x=L=0 mL
xLm
cosh
)(cosh
mLM b tanh
C Given temperature:
(L)=L
mL
xLmxLmb
L
sinh
)(sinh)(sinh)(
mL
mL
M b
L
bsinh
)(cosh
D Infinitely long fin
(L)=0
mxe M b
bCbb
C
hPkAMTT
kA
hPmTT
,)0(
, 2
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Example Problem
An Aluminum pot is used to boil water as shown below. The handle of the pot is 20-cm long, 3-cm wide, and 0.5-cm thick. The pot is exposed to room air at 25C, and the convection coefficient is 5 W/m2-K. Assume no heat transfer at the end of the handle. Question: can you touch the handle when the water is boiling? (k for Al = 237 W/m-K)
100 CT = 25 Ch = 5 W/ m2 Cx
Steps• Treat handle as fin• Identify tip condition
• Adiabatic• Calculate P, m, M• Get temp. profile• Calculate T at x=L= 20 cm
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Example (contd…)
Temperature distribution along the pot handle
0 0.05 0.1 0.15 0.285
90
95
100
T( )x
x
Temperature at the tip = 87.3 C Not safe to touch
Why?What can you do?
0 0.05 0.1 0.15 0.20
25
50
75
100
T x( )
x
Al handle Stainless Steel handle
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Question time?
Triangular fin
How can we get the temp profile?
1. What are the boundary conditions?
2. How will the temp. profile look like?
3. Any other way we can think of?
T = 75C T = 75C
xT= 25 Ch = 5 W/ m2-K
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Fin parameters: Effectiveness (ef)
Fin effectiveness (ef): how effective is the fin•Ratio of heat transferred in presence of fin to in its absence
,
f
f
c b b
q
hAe
>1 fin is effective<1 should not include fin
with , and /f ch k A Pe
Remember we wanted fins on the air side!!
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Fin parameters: Efficiency (hf)
Fin efficiency (hf): how close to ideal scenario is the fin•Ratio of heat transferred to that if entire fin were at base temp
,max
f f
f
f f b
q q
q hAh
Real situation
Tb x
Ideal situation
x
For a fin array with N fins,
AB: total base area Ab, At: base and tip area of finAf: surface area of a single fin (excluding tip)
qb
qf
btffbsf AANNAAhq h ))((
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Fin array in heat sinks
Parallel fins Radial fins Pin fins
Dovetail finsCircular heat sink
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Summary
• Solved ODE for 1-D heat conduction to get temp profile
– 4 different tip conditions
• Fin performance metrics – Effectiveness (ef) & Efficiency (hf)
• Fin arrays used to design heat sinks
– Commonly used in computing products
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Thank you!!
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