-
W-8
5S.1Graphical Representation of One-Dimensional, Transient
Conduction in the Plane Wall, Long Cylinder, and Sphere
In Sections 5.5 and 5.6, one-term approximations have been
developed for transient,one-dimensional conduction in a plane wall
(with symmetrical convection conditions)and radial systems (long
cylinder and sphere). The results apply for Fo! 0.2 and
canconveniently be represented in graphical forms that illustrate
the functional depen-dence of the transient temperature
distribution on the Biot and Fourier numbers.
Results for the plane wall (Figure 5.6a) are presented in
Figures 5S.1 through5S.3. Figure 5S.1 may be used to obtain the
midplane temperature of the wall, T(0, t) !To(t), at any time
during the transient process. If To is known for particular values
of Fo and Bi, Figure 5S.2 may be used to determine the
corresponding temperature atany location off the midplane. Hence
Figure 5S.2 must be used in conjunction withFigure 5S.1. For
example, if one wishes to determine the surface temperature (x*
"#1) at some time t, Figure 5S.1 would first be used to determine
To at t. Figure 5S.2would then be used to determine the surface
temperature from knowledge of To. The
302010976
50100
32.5
2.0
1.41.00.80.50.30.1
0
0 1 2 3 40.1
0.2
0.3
0.40.5
0.7
1.0
Bi 1 = k/hL
0 1 2 3 4 6 8 10121416 202224262830405060708090 110 130 150 300
400 500 600 7000.001
0.002
0.0030.0040.0050.007
0.01
0.02
0.030.040.050.07
0.1
0.20.30.40.50.71.0
00.05
0.10.2
0.3
0.40.50.60.70.8
1.0
1.21.4
1.6
1.8
2.02.5
3
456
78
9 10 1214
16
18
2030
35 40 45
50 6070 80
90 100
* o =
o =
To
T
__
____
___
i
Ti
T
t* = ( t/L2) = Fo18
25
FIGURE 5S.1 Midplane temperature as a function of time for a
plane wall of thickness 2L [1]. Used with permission.
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-
procedure would be inverted if the problem were one of
determining the time requiredfor the surface to reach a prescribed
temperature.
Graphical results for the energy transferred from a plane wall
over thetime interval t are presented in Figure 5S.3. These results
were generated fromEquation 5.46. The dimensionless energy transfer
Q/Qo is expressed exclusively interms of Fo and Bi.
Results for the infinite cylinder are presented in Figures 5S.4
through 5S.6, andthose for the sphere are presented in Figures 5S.7
through 5S.9, where the Biotnumber is defined in terms of the
radius ro.
5S.1 ! Representations of One-Dimensional, Transient Conduction
W-9
0.21.0
0.4
0.6
0.8
0.9
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
00.01 0.02 0.05 0.1 0.2 0.5 1.0 2 3 5 10 20 50 100
(k/hL) = Bi 1
=
T
T
__
___
____
o
To
T
x/L
FIGURE 5S.2 Temperature distribution in a plane wall of
thickness 2L [1]. Used with permission.
0.00
20.
005
0.01
0.02
0.05
0.1
0.2
0.5 1 2 5 10 20 50
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Q___Qo
h2 t ____k2(
Bi=
hL/k
=0.
001
)105 104 103 102 101 1 10 102 103 104
= Bi2 Fo
FIGURE 5S.3 Internal energy change as a function of time for a
plane wall of thickness 2L [2]. Adapted with permission.
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-
W-10 5S.1 ! Representations of One-Dimensional, Transient
Conduction
0.1
0.2
0.3
0.4
0.5
0.7
1.0
0 1 2 3 4 6 8 10 12 14 16 18 20 22 24 26 30 40 50 60 70 80 90
130 150200 3000.001
0.002
0.0030.0040.0050.007
0.01
0.02
0.030.040.050.07
0.1
* o =
o =
To
T
__
____
___
i
Ti
T
0 1 2 3 4
10028
0.2
0.30.40.5
0.71.0
0
0.1
0.20.3
0.40.5
0.6
0.8
1.0
1.2
1.41.6
1.8
2.0 2.5
2.02.53.0
3.54
5
6
7
8
9
10
12
14161820
25
30 35 40
45
50
60
70
80
90
100
Bi 1 = k/hro
( t/r2o) = Fo
3050100
18
12
8
6
5
4
3.0
3.5
2.52.0
1.61.20.8
0.6
0
115
FIGURE 5S.4 Centerline temperature as a function of time for an
infinite cylinder of radius ro [1]. Used with permission.
0.21.0
0.4
0.6
0.8
0.9
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
00.01 0.02 0.05 0.1 0.2 0.5 1.0 2 3 5 10 20 50 100
(k/hro) = Bi 1
=
T
T
__
___
____
o
To
T
r/ro
FIGURE 5S.5 Temperature distribution in an infinite cylinder of
radius ro [1]. Used withpermission.
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The foregoing charts may also be used to determine the transient
response of aplane wall, an infinite cylinder, or sphere subjected
to a sudden change in surfacetemperature. For such a condition it
is only necessary to replace T$ by the prescribedsurface
temperature Ts and to set Bi%1 equal to zero. In so doing, the
convection coef-ficient is tacitly assumed to be infinite, in which
case T$ " Ts.
5S.1 ! Representations of One-Dimensional, Transient Conduction
W-11
0.1
0.2
0.3
0.4
0.5
0.7
1.0
0 1 3 4 6 8 10 15 30 1300.001
0.002
0.0030.0040.0050.007
0.01
0.02
0.030.040.05
0.070.1
* o =
o =
To
T
__
____
___
i
Ti
T
0 1 2 30.2
0.30.40.5
0.7
1.0
t* = ( t/r 2o) = Fo
1825303550100
1412
98
76
5
4
3.53.02.62.01.40.750.20
20 40 50 70 90 250
00.05
0.10.2
0.350.5
0.751.2
1.4
1.6
1.8
2.02.22.4
2.62.8
3.5
45 6
78 9
1012
14
16
18
20
25
30
35
40
45
50
60 70 8090
100Bi 1 = k/hro
2 5 7 9 45 170 210
1.0
3.0
FIGURE 5S.7 Center temperature as a function of time in a sphere
of radius ro [1]. Used with permission.
0.00
20.
005
0.01
0.02
0.05
0.1
0.2
0.5 1 2 5 10 20 50
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Q___Qo
1 10 102 103 104
h2 t ____k2(
Bi=
hr o/k
=0.
001
105 104 103 102 101
) = Bi2 FoFIGURE 5S.6 Internal energy change as a function of
time for an infinite cylinder of radius ro [2]. Adapted with
permission.
c05_supl.qxd 1/24/06 5:35 PM Page W-11
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W-12 5S.1 ! Representations of One-Dimensional, Transient
Conduction
0.2
1.0
0.4
0.6
0.8
0.9
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
00.01 0.02 0.05 0.1 0.2 0.5 1.0 2 3 5 10 20 50 100
(k/hro) = Bi 1
=
T
T
__
___
____
o
To
T
r/ro
FIGURE 5S.8 Temperature distribution in a sphere of radius ro
[1]. Used with permission.
0.00
20.
005
0.01
0.02
0.05
0.1
0.2
0.5
1 2 5 10 20 50
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Q___Qo
101 1 10 102 103 104
h2 t ____k2(
Bi=
hr o/k
=0.
001
105 104 103 102
) = Bi2 FoFIGURE 5S.9 Internal energy change as a function of
time for a sphere of radius ro [2].Adapted with permission.
References
1. Heisler, M. P., Trans. ASME, 69, 227236, 1947.2. Grber, H.,
S. Erk, and U. Grigull, Fundamentals of Heat
Transfer, McGraw-Hill, New York, 1961.
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