Homework #3
Name___
solution
____
Problem 1.
A 1.5 kW resistance heater wire whose thermal conductivity is k =
10.4 BTU/h·ft
0
F has a radius of r
0
= 0.06 in. and a length of L = 15 in., and is
used for space heating.
Assuming constant thermal conductivity and one
dimensional heat transfer, express the mathematical formulation (the differential
equation and boundary conditions) of this heat conduction problem during
steady operation.
The surface temperature of the rod is 220
0
F
Solution:
K = 10.4
BTU/h ft
0
F
r
0
= 0.06 in
L = 15 in
kW
h
BTU
3412.14
1.5kW x
=
Q
= 5,118.21
BTU/h
Differential equation:
0
1
=
+
⎟
⎠
⎞
⎜
⎝
⎛
k
g
dr
dT
r
dr
d
r
Boundary conditions
@r=0 (Thermal symmetry)
0
)
0
(
=
dr
dT
@r=r
0
(specified heat flux)
L
r
A
Q
q
h
BTU
r
0
2
21
.
118
,
5
0
π
=
=
=1.303 x 10
5
BTU/(hr·ft
2
)
Also @r=r
0
(specified temperature) T(r
0
) = 220
0
F
Solve the differential equation and obtain a temperature distribution in the
heater wire.
What is the temperature at the center of the rod?
The heat flux boundary condition at r = r
0
is not helpful, as it is redundant with
the balance of
.
If we apply a surface temperature of T
V
g
Q
=
s
, then we can solve
the differential equation.
1
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 Spring '08
 Staff
 Thermodynamics, Heat, Heat Transfer, 1 cm

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