Chapter 2
Heat Conduction Equation
2126
A spherical liquid nitrogen container is subjected to specified temperature on the inner surface and
convection on the outer surface.
The mathematical formulation, the variation of temperature, and the rate
of evaporation of nitrogen are to be determined for steady onedimensional heat transfer.
Assumptions
1
Heat conduction is steady and onedimensional since there is no change with time and
there is thermal symmetry about the midpoint.
2
Thermal conductivity is constant.
3
There is no heat
generation.
Properties
The thermal conductivity of the tank is given to be
k
= 18 W/m
°C. Also,
h
fg
= 198 kJ/kg for
nitrogen.
Analysis
(
a
) Noting that heat transfer is onedimensional in the radial
r
direction, the mathematical
formulation of this problem can be expressed as
0
2
dr
dT
r
dr
d
and
T r
T
( )
1
1
196
C
k
dT r
dr
h T r
T
(
)
[
(
)
]
2
2
(
b
)
Integrating the differential equation once with respect to
r
gives
r
dT
dr
C
2
1
Dividing both sides of the equation above by
r
to bring it to a readily integrable form and then integrating,
dT
dr
C
r
1
2
T r
C
r
C
( )
1
2
where
C
1
and
C
2
are arbitrary constants.
Applying the boundary conditions give
r
=
r
1
:
T r
C
r
C
T
(
)
1
1
1
2
1
r
=
r
2
:
T
C
r
C
h
r
C
k
2
2
1
2
2
1
Solving for
C
C
1
2
and
simultaneously gives
C
r
T
T
r
r
k
hr
C
T
C
r
T
T
T
r
r
k
hr
r
r
1
2
1
2
1
2
2
1
1
1
1
1
2
1
2
2
1
1
1
(
)
and
Substituting
C
C
1
2
and
into the general solution, the variation of temperature is determined to be
)
/
1
.
2
05
.
1
(
8
.
549
C
)
196
(
1
.
2
2
1
.
2
)
m
1
.
2
)(
C
W/m
25
(
C
W/m
18
2
1
.
2
1
C
)
20
196
(
1
1
1
)
(
2
1
2
1
2
2
1
2
1
1
1
1
1
1
1
1
r
r
T
r
r
r
r
hr
k
r
r
T
T
T
r
r
C
r
C
T
r
C
r
T
(
c
) The rate of heat transfer through the wall and the rate of evaporation of nitrogen are determined from
(
)
(
)
(
)
( .
(
)
.
(
)( .
)
Q
kA
dT
dx
k
r
C
r
kC
k
r
T
T
r
r
k
hr
4
4
4
1
4
18
21
196
20
1
21
2
18
25
21
2
1
2
1
2
1
2
1
2
W / m
C
m)
C
W / m
C
W / m
C
m
261,200 W (to the tank since negative)
2
,
,
m
Q
h
fg
261 200
198 000
J / s
J / kg
1.32 kg / s
270
r
1
r
2
h
T
r
196°C
N
2
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Chapter 2
Heat Conduction Equation
2127
A spherical liquid oxygen container is subjected to specified temperature on the inner surface and
convection on the outer surface.
The mathematical formulation, the variation of temperature, and the rate
of evaporation of oxygen are to be determined for steady onedimensional heat transfer.
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 Spring '08
 BENARD
 Heat Transfer

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