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PROBLEM 4.72
KNOWN
:
Thin metallic foil of thickness, t, whose edges are thermally coupled to a sink at temperature
T
sink
is exposed on the top surface to an ion beam heat flux,
¢¢
q
s
, and experiences radiation exchange with
he vacuum enclosure walls at T
t
sur
.
F
IND
:
Temperature distribution across the foil.
SCHEMATIC
:
ASSUMPTIONS
:
(1) Onedimensional, steadystate conduction in the foil, (2) Constant properties, (3)
Upper and lower surfaces of foil experience radiation exchange, (4) Foil is of unit length normal to the
age.
p
ANALYSIS
: The 10node network representing the foil is shown below.
From an energy balance on node n,
in
out
EE 0
−
=
±±
, for a unit depth,
abcde
qqqqq
′′′′′
++++=
0
()
(
)
(
)
( ) ( )( )
s
r,n
sur
n
n 1
n
sur
n
n
qxh xT
T k
tT
T
xh xT
T
x0
+−
′′
Δ+
Δ
−
+
−
Δ
−
+
−
Δ=
(1)
where the linearized radiation coefficient for node n is
( )
22
sur
n
sur
n
hT
T
T
εσ
=+
+
T
(2)
Solving Eq. (1) for T
n
find,
( )
( )
( )
2
nn
1
n
1
r
,
n
s
u
r
s
r
,
n
TTT
2
h
x
k
t
T
x
k
t
q
h
x
k
t
2
+
Δ
+
Δ
Δ
+
⎡⎤
⎢⎥
⎣⎦
⎡
⎤
⎢
⎥
⎣
⎦
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 Spring '10
 LEE,J.S.
 Heat Transfer

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