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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
Tr
T
()
11
196
==
− °
C
−=
−
∞
k
dT r
dr
hTr
T
[( )
]
2
2
(
b
)
Integrating the differential equation once with respect to
r
gives
r
dT
dr
C
2
1
=
r
1
r
2
h
T
∞
r
196°C
N
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 Fall '10
 Dr.DanielArenas
 Thermodynamics, Convection, Mass, Heat

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