Power to dissipate : 50 W Typical dimension : L= 3cm Physical properties of air : density 1 kg/m3
specific heat 1000 J/kg.K
thermal conductivity 0.025 W/m.K
If we rely only on diffusion in air, how much power can we dissipate from the flat chip without fins?
Geometrical effects in diffusion problems, the case of radiator fins
How much power can we dissipate by adding a radiator with fins ?
Is there an optimum length for fins given their thickness e and the heat transfer coefficient h between the fins and the surrounding air ?
Physical properties of aluminum (material of the fins) : thermal conductivity 240 W/m.K
e H
T
1T
0x T(x)
h
Write the heat balance in an element of fin and derive an equation for T(x) What are the assumptions on the temperature field within
the fin that can be done using the hypothesis e << H ?
What is the temperature distribution within the fin and the resulting heat flux to the air ?
What are the boundary conditions for T(x) ?
J
D=
MdT dx
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
M
d
2T (x)
dx
2e = 2h [T (x) T
0]
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
0 = dJ
D(x)
dx e + 2h [T (x) T
0]
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
J
D(x) e = J
D(x + dx) e + 2h [T (x) T
0] dx
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
d
2T (x)
dx
2= 2h
e
M[T (x) T
0] = [T (x) T
0]
`
2<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Heat flux within the fin
Heat flux balances in an element of fin between x and x+dx :
` =
r e
M2h
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
T (x) T
0= A exp(x/`) + B exp( x/`)
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
T
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>(0) = T
1M
dT
dx
x=H= h[T (H ) T
0]
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
H
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>` T
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>(x)
x!1! T
0T
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>(x) T
0= (T
1T
0) exp( x/`)
Boundary conditions : x=0
x=H, assuming H < l
x=H, assuming
J
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit> L= 2h(T
1T
0) exp( x/`)
J
total=
Z
H 02h(T
1T
0) exp( x/`) dx
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
J
total⇡
Z
10
2h(T
1T
0) exp( x/`) dx = 2h`(T
1T
0)
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
J
total⇡ p
2h e
M(T
1T
0)
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Local heat flux to the air :
Total heat flux to the air :
