Heisler
Páginas: 20 (4884 palabras)
Publicado: 28 de marzo de 2012
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5S.1 Graphical 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). Theresults apply for Fo 0.2 and can conveniently be represented in graphical forms that illustrate the functional dependence 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 through 5S.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 transientprocess. If To is known for particular values of Fo and Bi, Figure 5S.2 may be used to determine the corresponding temperature at any location off the midplane. Hence Figure 5S.2 must be used in conjunction with Figure 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.2 would then be used todetermine the surface temperature from knowledge of To. The
100 50 1.0 0.7 0.5 0.4 0.3 0.2 0 0.1 0 0.1 0.3 0.5 1 2
θo To – T∞ θ* = __ = _______ o θ i Ti – T∞
0.1 0.07 0.05 0.04 0.03 0.02
30 20 10 9 7 6 2.5 3
2.0 1.4 4
0.8 1.0 3
1.0 0.7 0.5 0.4 0.3 0.2 9 1.0 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 3 2.5 2.0 1.8 1.4 1.2 1.6 4 6 8 10 121416 18 2022 24262830405060708090 6 5 4 8 7 14 10 12 25 20 3090 100 50 60 70 80 35 40 45
Bi –1 = k/hL
18 16
0.01 0.007 0.005 0.004 0.003 0.002 0.001 0
0.05 0 1 2 3
110 130 150
300 400 500 600 700
t* = (α t/L2) = Fo
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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5S.1
Representations of One-Dimensional,Transient Conduction
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1.0 0.2 0.9 0.8 0.7 0.6 0.6 0.5 0.4 0.8 0.3 0.2 0.9 0.1 1.0 0 0.01 0.02 0.05 0.1 0.2 0.4
x/L
T – T∞ θ __ = _______ θ o To – T∞
0.5 1.0
2 3 5
10 20
50 100
(k/hL) = Bi –1
FIGURE 5S.2 Temperature distribution in a plane wall of thickness 2L [1]. Used with permission.
procedure would be inverted if the problem were one of determining the timerequired for the surface to reach a prescribed temperature. Graphical results for the energy transferred from a plane wall over the time interval t are presented in Figure 5S.3. These results were generated from Equation 5.46. The dimensionless energy transfer Q/Qo is expressed exclusively in terms of Fo and Bi. Results for the infinite cylinder are presented in Figures 5S.4 through 5S.6, and those forthe sphere are presented in Figures 5S.7 through 5S.9, where the Biot number is defined in terms of the radius ro.
1.0 0.9
0.7 0.6
= 0. 0.00 001 2 0.0 05 0.01 0.02 0.05 0.1 0.2
0.8
20
h L /k
0.5
0.4 0.3 0.2 0.1 0 –5 10 10–4
Bi =
Qo 0.5
Q ___
1
10–3
10–2
10–1
2
10
5
1
10
102
50
103
104
( )
h2α t ____ = Bi2 Fo k2
FIGURE 5S.3Internal energy change as a function of time for a plane wall of thickness 2L [2]. Adapted with permission.
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5S.1
Representations of One-Dimensional, Transient Conduction
1.0 0.7 0.5 0.4 0.3 0.2 3.5 0.6 0.8 0 0.1 0 1 2
0.1
18 12 8 6 5 4 3.0 4
100 50 30
2.0 1.2 1.6
1.0 0.7 0.5 0.4 0.3 0.2
25 20 18 16 14 12 10 9 8 750 6 5 4 3.5 2.0 2.53.0 30 35 40 45 60
2.5 3
Bi –1 = k/hro
100 90 80 70
θ* = θo = _______ __ To – T∞ o θ i Ti – T∞
0.07 0.05 0.04 0.03 0.02 0.01 0.007 0.005 0.004 0.003 0.002 0.001 0 1 2
0.6 0.5 0.4 0.3 0.2 0.1 0
2.0 1.6 1.4 1.2 1.0 0.8
2.5
1.8
3
4 6 8 10 12 14 16 18 20 22 24 26 28 30 40 50 60 70 80 90100 115 130 150200
2 (α t /r o) = Fo
300
FIGURE 5S.4...
1/24/06
5:35 PM
Page W-8
W-8
5S.1 Graphical 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). Theresults apply for Fo 0.2 and can conveniently be represented in graphical forms that illustrate the functional dependence 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 through 5S.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 transientprocess. If To is known for particular values of Fo and Bi, Figure 5S.2 may be used to determine the corresponding temperature at any location off the midplane. Hence Figure 5S.2 must be used in conjunction with Figure 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.2 would then be used todetermine the surface temperature from knowledge of To. The
100 50 1.0 0.7 0.5 0.4 0.3 0.2 0 0.1 0 0.1 0.3 0.5 1 2
θo To – T∞ θ* = __ = _______ o θ i Ti – T∞
0.1 0.07 0.05 0.04 0.03 0.02
30 20 10 9 7 6 2.5 3
2.0 1.4 4
0.8 1.0 3
1.0 0.7 0.5 0.4 0.3 0.2 9 1.0 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 3 2.5 2.0 1.8 1.4 1.2 1.6 4 6 8 10 121416 18 2022 24262830405060708090 6 5 4 8 7 14 10 12 25 20 3090 100 50 60 70 80 35 40 45
Bi –1 = k/hL
18 16
0.01 0.007 0.005 0.004 0.003 0.002 0.001 0
0.05 0 1 2 3
110 130 150
300 400 500 600 700
t* = (α t/L2) = Fo
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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5S.1
Representations of One-Dimensional,Transient Conduction
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1.0 0.2 0.9 0.8 0.7 0.6 0.6 0.5 0.4 0.8 0.3 0.2 0.9 0.1 1.0 0 0.01 0.02 0.05 0.1 0.2 0.4
x/L
T – T∞ θ __ = _______ θ o To – T∞
0.5 1.0
2 3 5
10 20
50 100
(k/hL) = Bi –1
FIGURE 5S.2 Temperature distribution in a plane wall of thickness 2L [1]. Used with permission.
procedure would be inverted if the problem were one of determining the timerequired for the surface to reach a prescribed temperature. Graphical results for the energy transferred from a plane wall over the time interval t are presented in Figure 5S.3. These results were generated from Equation 5.46. The dimensionless energy transfer Q/Qo is expressed exclusively in terms of Fo and Bi. Results for the infinite cylinder are presented in Figures 5S.4 through 5S.6, and those forthe sphere are presented in Figures 5S.7 through 5S.9, where the Biot number is defined in terms of the radius ro.
1.0 0.9
0.7 0.6
= 0. 0.00 001 2 0.0 05 0.01 0.02 0.05 0.1 0.2
0.8
20
h L /k
0.5
0.4 0.3 0.2 0.1 0 –5 10 10–4
Bi =
Qo 0.5
Q ___
1
10–3
10–2
10–1
2
10
5
1
10
102
50
103
104
( )
h2α t ____ = Bi2 Fo k2
FIGURE 5S.3Internal energy change as a function of time for a plane wall of thickness 2L [2]. Adapted with permission.
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5S.1
Representations of One-Dimensional, Transient Conduction
1.0 0.7 0.5 0.4 0.3 0.2 3.5 0.6 0.8 0 0.1 0 1 2
0.1
18 12 8 6 5 4 3.0 4
100 50 30
2.0 1.2 1.6
1.0 0.7 0.5 0.4 0.3 0.2
25 20 18 16 14 12 10 9 8 750 6 5 4 3.5 2.0 2.53.0 30 35 40 45 60
2.5 3
Bi –1 = k/hro
100 90 80 70
θ* = θo = _______ __ To – T∞ o θ i Ti – T∞
0.07 0.05 0.04 0.03 0.02 0.01 0.007 0.005 0.004 0.003 0.002 0.001 0 1 2
0.6 0.5 0.4 0.3 0.2 0.1 0
2.0 1.6 1.4 1.2 1.0 0.8
2.5
1.8
3
4 6 8 10 12 14 16 18 20 22 24 26 28 30 40 50 60 70 80 90100 115 130 150200
2 (α t /r o) = Fo
300
FIGURE 5S.4...
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