ame$60634$$ int.$heattrans.$ thermal circuits: contact...

AME 60634 Int. Heat Trans.

D. B. Go 1

Thermal Circuits: Contact Resistance

In the real world, two surfaces in contact do not transfer heat perfectly

€

ʹ′ ʹ′ R t,c =TA −TB

ʹ′ ʹ′ q x⇒ Rt ,c =

ʹ′ ʹ′ R t,cAc

Contact Resistance: values depend on materials (A and B), surface roughness, interstitial conditions, and contact pressure è typically calculated or looked up

Equivalent total thermal resistance:

€

Rtot =LA

kA Ac

+ʹ′ ʹ′ R t,c

Ac

+LB

kB Ac


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D. B. Go 3


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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) annular fin, and (d) pin fin of non-uniform cross section.


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Fins: The Fin Equation •  Solutions


D. B. Go 6

x2 d2ydx2

+ x dydx+ m2x2 −ν 2( ) y = 0

Bessel Equations

with solution x2 d2ydx2

+ x dydx+ m2x2 −ν 2( ) y = 0

Form of Bessel equation of order ν

Jν = Bessel function of first kind of order ν Yν = Bessel function of second kind of order ν

x2 d2ydx2

+ x dydx− m2x2 −ν 2( ) y = 0 with solution y x( ) =C1Iν mx( )+C2Kν mx( )

Form of modified Bessel equation of order ν

Iν = modified Bessel function of first kind of order ν Kν = modified Bessel function of second kind of order ν


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ddx

Wν mx( )!" #$=mWν−1 mx( )− ν

xWν mx( ) W = J,Y, I

−mWν−1 mx( )− νxWν mx( ) W = K

&

'((

)((

Bessel Functions – Recurrence Relations

OR

ddx

Wν mx( )!" #$=−mWν+1 mx( )+ ν

xWν mx( ) W = J,Y,K

mWν+1 mx( )+ νxWν mx( ) W = I

&

'((

)((

AND

ddx

xνWν mx( )!" #$=mxνWν−1 mx( ) W = J,Y, I

−mxνWν−1 mx( ) W = K

&

'(

)(


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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

€

η f ≡qfqf ,max

=qf

hAfθb

€

ε f ≡qfqf ,max

=qf

hAc,bθb=Rt,b

Rt, f

€

Rt, f ≡θbqf

=1

hAfη f


D. B. Go 10

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

€

At = NAf + Ab

€

qt = Nη f hAfθb + hAbθb =ηohAtθb =θbRt,o

€

Rt,o =θbqt

=1

hAtηo

€

N ≡ number of finsAb ≡ exposed base surface (prime surface)

€

ηo =1−NAf

At

1−η f( )


D. B. Go 13

Fins: Thermal Circuit •  Equivalent Thermal Circuit

•  Effect of Surface Contact Resistance

€

Rt,o =1

hAtηo(c )

€

ηo(c ) =1−NAf

At

1−η f

C1

$

% &

'

( )

€

C1 =1−η f hAf$ $ R t,c

Ac,b

%

& '

(

) *

€

qt =ηo(c )hAfθb =θbRt,o(c )

ame$60634$$ int.$heattrans.$ thermal circuits: contact...

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