AME$60634$$ Int.$HeatTrans.$ Thermal Circuits: Contact...
Transcript of AME$60634$$ Int.$HeatTrans.$ Thermal Circuits: Contact...
![Page 1: AME$60634$$ Int.$HeatTrans.$ Thermal Circuits: Contact ...sst/teaching/AME60634/lectures/AME60634_… · AME$60634$$ Int.$HeatTrans.$ D.B.Go 1 Thermal Circuits: Contact Resistance](https://reader034.fdocuments.in/reader034/viewer/2022042205/5ea78b2fa4304424c961067c/html5/thumbnails/1.jpg)
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
![Page 2: AME$60634$$ Int.$HeatTrans.$ Thermal Circuits: Contact ...sst/teaching/AME60634/lectures/AME60634_… · AME$60634$$ Int.$HeatTrans.$ D.B.Go 1 Thermal Circuits: Contact Resistance](https://reader034.fdocuments.in/reader034/viewer/2022042205/5ea78b2fa4304424c961067c/html5/thumbnails/2.jpg)
AME 60634 Int. Heat Trans.
D. B. Go 2
![Page 3: AME$60634$$ Int.$HeatTrans.$ Thermal Circuits: Contact ...sst/teaching/AME60634/lectures/AME60634_… · AME$60634$$ Int.$HeatTrans.$ D.B.Go 1 Thermal Circuits: Contact Resistance](https://reader034.fdocuments.in/reader034/viewer/2022042205/5ea78b2fa4304424c961067c/html5/thumbnails/3.jpg)
AME 60634 Int. Heat Trans.
D. B. Go 3
![Page 4: AME$60634$$ Int.$HeatTrans.$ Thermal Circuits: Contact ...sst/teaching/AME60634/lectures/AME60634_… · AME$60634$$ Int.$HeatTrans.$ D.B.Go 1 Thermal Circuits: Contact Resistance](https://reader034.fdocuments.in/reader034/viewer/2022042205/5ea78b2fa4304424c961067c/html5/thumbnails/4.jpg)
AME 60634 Int. Heat Trans.
D. B. Go 4
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.
![Page 5: AME$60634$$ Int.$HeatTrans.$ Thermal Circuits: Contact ...sst/teaching/AME60634/lectures/AME60634_… · AME$60634$$ Int.$HeatTrans.$ D.B.Go 1 Thermal Circuits: Contact Resistance](https://reader034.fdocuments.in/reader034/viewer/2022042205/5ea78b2fa4304424c961067c/html5/thumbnails/5.jpg)
AME 60634 Int. Heat Trans.
D. B. Go 5
Fins: The Fin Equation • Solutions
![Page 6: AME$60634$$ Int.$HeatTrans.$ Thermal Circuits: Contact ...sst/teaching/AME60634/lectures/AME60634_… · AME$60634$$ Int.$HeatTrans.$ D.B.Go 1 Thermal Circuits: Contact Resistance](https://reader034.fdocuments.in/reader034/viewer/2022042205/5ea78b2fa4304424c961067c/html5/thumbnails/6.jpg)
AME 60634 Int. Heat Trans.
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
![Page 7: AME$60634$$ Int.$HeatTrans.$ Thermal Circuits: Contact ...sst/teaching/AME60634/lectures/AME60634_… · AME$60634$$ Int.$HeatTrans.$ D.B.Go 1 Thermal Circuits: Contact Resistance](https://reader034.fdocuments.in/reader034/viewer/2022042205/5ea78b2fa4304424c961067c/html5/thumbnails/7.jpg)
AME 60634 Int. Heat Trans.
D. B. Go 7
![Page 8: AME$60634$$ Int.$HeatTrans.$ Thermal Circuits: Contact ...sst/teaching/AME60634/lectures/AME60634_… · AME$60634$$ Int.$HeatTrans.$ D.B.Go 1 Thermal Circuits: Contact Resistance](https://reader034.fdocuments.in/reader034/viewer/2022042205/5ea78b2fa4304424c961067c/html5/thumbnails/8.jpg)
AME 60634 Int. Heat Trans.
D. B. Go 8
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
&
'(
)(
![Page 9: AME$60634$$ Int.$HeatTrans.$ Thermal Circuits: Contact ...sst/teaching/AME60634/lectures/AME60634_… · AME$60634$$ Int.$HeatTrans.$ D.B.Go 1 Thermal Circuits: Contact Resistance](https://reader034.fdocuments.in/reader034/viewer/2022042205/5ea78b2fa4304424c961067c/html5/thumbnails/9.jpg)
AME 60634 Int. Heat Trans.
D. B. Go 9
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
![Page 10: AME$60634$$ Int.$HeatTrans.$ Thermal Circuits: Contact ...sst/teaching/AME60634/lectures/AME60634_… · AME$60634$$ Int.$HeatTrans.$ D.B.Go 1 Thermal Circuits: Contact Resistance](https://reader034.fdocuments.in/reader034/viewer/2022042205/5ea78b2fa4304424c961067c/html5/thumbnails/10.jpg)
AME 60634 Int. Heat Trans.
D. B. Go 10
Fins: Efficiency
![Page 11: AME$60634$$ Int.$HeatTrans.$ Thermal Circuits: Contact ...sst/teaching/AME60634/lectures/AME60634_… · AME$60634$$ Int.$HeatTrans.$ D.B.Go 1 Thermal Circuits: Contact Resistance](https://reader034.fdocuments.in/reader034/viewer/2022042205/5ea78b2fa4304424c961067c/html5/thumbnails/11.jpg)
AME 60634 Int. Heat Trans.
D. B. Go 11
Fins: Efficiency
![Page 12: AME$60634$$ Int.$HeatTrans.$ Thermal Circuits: Contact ...sst/teaching/AME60634/lectures/AME60634_… · AME$60634$$ Int.$HeatTrans.$ D.B.Go 1 Thermal Circuits: Contact Resistance](https://reader034.fdocuments.in/reader034/viewer/2022042205/5ea78b2fa4304424c961067c/html5/thumbnails/12.jpg)
AME 60634 Int. Heat Trans.
D. B. Go 12
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( )
![Page 13: AME$60634$$ Int.$HeatTrans.$ Thermal Circuits: Contact ...sst/teaching/AME60634/lectures/AME60634_… · AME$60634$$ Int.$HeatTrans.$ D.B.Go 1 Thermal Circuits: Contact Resistance](https://reader034.fdocuments.in/reader034/viewer/2022042205/5ea78b2fa4304424c961067c/html5/thumbnails/13.jpg)
AME 60634 Int. Heat Trans.
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 )