ame$60634$$ int.$heattrans.$ 1-d steady conduction: plane...
TRANSCRIPT
AME 60634 Int. Heat Trans.
D. B. Go 1
1-D Steady Conduction: Plane Wall Governing Equation:
Dirichlet Boundary Conditions: T (0) = Ts,1 ; T (L) = Ts,2
d 2Tdx2
= 0
€
qx = −kA dTdx
=kAL
Ts,1 −Ts,2( )
Solution:
Heat Flux:
Heat Flow:
temperature is not a function of k
€
T(x) = Ts,1 + Ts,2 −Ts,1( ) xL
€
ʹ′ ʹ′ q x = −k dTdx
=kL
Ts,1 −Ts,2( )
Notes: • A is the cross-sectional area of the wall perpendicular to the heat flow • both heat flux and heat flow are uniform è independent of position (x) • temperature distribution is governed by boundary conditions and length of domain è independent of thermal conductivity (k)
heat flux/flow are a function of k
AME 60634 Int. Heat Trans.
D. B. Go 2
1-D Steady Conduction: Cylinder Wall Governing Equation:
Dirichlet Boundary Conditions:
€
T(r1) =Ts,1 ; T(r2) =Ts,2
Notes: • heat flux is not uniform è function of position (r) • both heat flow and heat flow per unit length are uniform è independent of position (r)
Solution:
Heat Flux:
Heat Flow: €
T(r) =Ts,1 −Ts,2ln r1 r2( )
ln rr2
⎛
⎝ ⎜
⎞
⎠ ⎟ + Ts,2
€
ʹ′ ʹ′ q r = −k dTdr
=k Ts,1 −Ts,2( )r ln r2 r1( )
€
qr = −kA dTdr
= 2πrL ʹ′ ʹ′ q r =2πLk Ts,1 −Ts,2( )ln r2 r1( )
€
ʹ′ q r =qr
L=2πk Ts,1 −Ts,2( )ln r2 r1( ) heat flow per unit length
heat flux is non-uniform
heat flow is uniform
1rddr
kr dTdr
!
"#
$
%&= 0⇒
ddr
r dTdr
!
"#
$
%&= 0
AME 60634 Int. Heat Trans.
D. B. Go 3
1-D Steady Conduction: Spherical Shell Governing Equation:
Dirichlet Boundary Conditions:
€
T(r1) =Ts,1 ; T(r2) =Ts,2
Solution:
Heat Flux:
Heat Flow:
€
T(r) = Ts,1 − Ts,1 −Ts,2( )1− r1 r( )1− r1 r2( )
€
ʹ′ ʹ′ q r = −k dTdr
=k Ts,1 −Ts,2( )
r2 1 r1( )− 1 r2( )[ ]
€
qr = −kA dTdr
= 4πr2 ʹ′ ʹ′ q r =4πk Ts,1 −Ts,2( )1 r1( )− 1 r2( )
Notes: • heat flux is not uniform è function of position (r) • heat flow is uniform è independent of position (r)
heat flux is non-uniform
heat flow is uniform
1r2
ddr
kr2 dTdr
!
"#
$
%&= 0⇒
ddr
r2 dTdr
!
"#
$
%&= 0
AME 60634 Int. Heat Trans.
D. B. Go 4
Thermal Resistance
AME 60634 Int. Heat Trans.
D. B. Go 5
Thermal Circuits: Composite Plane Wall
Circuits based on assumption of (a) isothermal surfaces normal to x direction
or (b) adiabatic surfaces parallel to x direction
€
Rtot =LEkEA
+kFA2LF
+kGA2LG
⎡
⎣ ⎢
⎤
⎦ ⎥
−1
+LHkHA
€
Rtot =
2LEkEA
+2LFkFA
+2LHkHA
⎛
⎝ ⎜
⎞
⎠ ⎟
−1
+2LEkEA
+2LGkGA
+2LHkHA
⎛
⎝ ⎜
⎞
⎠ ⎟
−1⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
−1
Actual solution for the heat rate q is bracketed by these two approximations
AME 60634 Int. Heat Trans.
D. B. Go 6
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
AME 60634 Int. Heat Trans.
D. B. Go 7
AME 60634 Int. Heat Trans.
D. B. Go 8
AME 60634 Int. Heat Trans.
D. B. Go 9
Fins: The Fin Equation • Solutions
AME 60634 Int. Heat Trans.
D. B. Go 10
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
AME 60634 Int. Heat Trans.
D. B. Go 11
Fins: Efficiency
AME 60634 Int. Heat Trans.
D. B. Go 12
Fins: Efficiency
AME 60634 Int. Heat Trans.
D. B. Go 13
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( )
AME 60634 Int. Heat Trans.
D. B. Go 14
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. Heat Trans.
D. B. Go 15
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.
AME 60634 Int. Heat Trans.
D. B. Go 16
Fins: The Fin Equation • Solutions