AME 60634 Int. Heat Trans. Thermal Circuits: Contact Resistance
In the real world, two surfaces in contact do not transfer heat perfectly
ʹ′ʹ′ =
Rt,c
Rʹ′ʹ′
TA − TB
⇒ Rt,c = t,c
qʹ′xʹ′
Ac
€
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 =
D. B. Go Rʹ′ʹ′
LA
L
+ t,c + B
k A Ac Ac k B Ac
1 €
AME 60634 Int. Heat Trans. D. B. Go 2 AME 60634 Int. Heat Trans. D. B. Go 3 AME 60634 Int. Heat Trans. •
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 nonuniform cross section.
D. B. Go 4 AME 60634 Int. Heat Trans. •
D. B. Go Fins: The Fin Equation
Solutions
5 AME 60634 Int. Heat Trans. Bessel Equations
Form of Bessel equation of order ν
d2y
dy
2 2
2
x
+
x
+
m
x
−
ν
y=0
(
)
2
dx
dx
2
with solution
d2y
dy
2 2
2
x
+
x
+
m
x
−
ν
y=0
(
)
2
dx
dx
2
Jν = Bessel function of first kind of order ν
Yν = Bessel function of second kind of order ν
Form of modified Bessel equation of order ν
d2y
dy
x
+
x
− (m2 x 2 − ν 2 ) y = 0
2
dx
dx
2
with solution
y ( x ) = C1Iν ( mx ) + C2 Kν ( mx )
Iν = modified Bessel function of first kind of order ν
Kν = modified Bessel function of second kind of order ν
D. B. Go 6 AME 60634 Int. Heat Trans. D. B. Go 7 AME 60634 Int. Heat Trans. Bessel Functions – Recurrence Relations
&
ν
( mWν −1 ( mx ) − Wν ( mx ) W = J,Y, I
(
d!
x
#
W
mx
=
" ν ( )$ '
dx
( −mW ( mx ) − ν W ( mx ) W = K
ν −1
ν
()
x
OR
&
ν
( −mWν +1 ( mx ) + Wν ( mx ) W = J,Y, K
(
d!
x
#
W
mx
=
'
(
)
" ν
$
dx
ν
(
mWν +1 ( mx ) + Wν ( mx ) W = I
()
x
AND
&
ν
mx
Wν −1 ( mx ) W = J,Y, I
(
d! ν
#
x
W
mx
=
)$ '
ν(
"
ν
dx
() −mx Wν −1 ( mx ) W = K
D. B. Go 8 AME 60634 Int. Heat Trans. •
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
ηf ≡
•
qf
q f ,max
=
qf
hA f θ b
Fin Effectiveness
– ratio of the fin heat transfer rate to the heat transfer rate that would exist
without the fin
€
εf ≡
•
€
qf
q f ,max
Fin Resistance
– defined using the temperature difference between the base and fluid as
the driving potential
Rt, f ≡
D. B. Go qf
R
=
= t,b
hAc,bθ b Rt, f
θb
1
=
q f hA f η f
9 AME 60634 Int. Heat Trans. D. B. Go Fins: Efficiency
10 AME 60634 Int. Heat Trans. D. B. Go Fins: Efficiency
11 AME 60634 Int. Heat Trans. •
Fins: Arrays
Arrays
– total surface area
At = NA f + Ab
N ≡ number of fins
Ab ≡ exposed base surface (prime surface)
– total heat rate
€
θ
€
qt = Nη f hA f θ b + hAbθ b = ηo hAtθ b = b
Rt,o
– overall surface efficiency
€
NA f
ηo = 1−
1− η f )
(
At
– overall surface resistance
€
€
D. B. Go Rt,o =
θb
1
=
qt hAtηo
12 AME 60634 Int. Heat Trans. •
Fins: Thermal Circuit
Equivalent Thermal Circuit
•
Effect of Surface Contact Resistance
θ
qt = ηo(c )hA f θ b = b
Rt,o(c )
$ ηf '
&1− )
% C1 (
% R$$ (
C1 = 1− η f hA f ' t,c *
& Ac,b )
NA f
ηo(c ) = 1−
At
Rt,o =
D. B. Go 1
hAtηo(c )
13