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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