ch05_supp

# ch05_supp - c05_supl.qxd 5:35 PM Page W-8 W-8 5S.1...

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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). The results apply for Fo 0.2 and can conveniently 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 5.6 a ) 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 ) T o ( t ), at any time during the transient process. If T o 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 T o at t . Figure 5S.2 would then be used to determine the surface temperature from knowledge of T o . The 30 20 10 9 7 6 50 100 3 2.5 2.0 1.4 1.0 0.8 0.5 0.3 0.1 0 0 1 2 3 4 0.1 0.2 0.3 0.4 0.5 0.7 1.0 Bi –1 = k / hL 0 1 2 3 4 6 8 10121416 202224262830405060708090 110 130 150 300 400 500 600 700 0.001 0.002 0.003 0.004 0.005 0.007 0.01 0.02 0.03 0.04 0.05 0.07 0.1 0.2 0.3 0.4 0.5 0.7 1.0 0 0.05 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.5 3 4 5 6 7 8 9 10 12 14 16 18 20 30 35 40 45 50 60 70 80 90 100 * o = o = T o T __ _______ i T i T θ θ θ t * = ( t / L 2 ) = Fo 18 25 α F IGURE 5S.1 Midplane temperature as a function of time for a plane wall of thickness 2 L [1]. Used with permission. c05_supl.qxd 1/24/06 5:35 PM Page W-8

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procedure would be inverted if the problem were one of determining the time required 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/Q o 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 for the sphere are presented in Figures 5S.7 through 5S.9, where the Biot number is defined in terms of the radius r o . 5S.1 Representations of One-Dimensional, Transient Conduction W-9 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 0 0.01 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 T o T θ θ x/L F IGURE 5S.2 Temperature distribution in a plane wall of thickness 2 L [1]. Used with permission. 0.002 0.005 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 ___ Q o h 2 t ____ k 2 ( B i = h L / k = 0 . 0 0 1 ) 10 –5 10 –4 10 –3 10 –2 10 –1 1 10 10 2 10 3 10 4 α = Bi 2 Fo F IGURE 5S.3 Internal energy change as a function of time for a plane wall of thickness 2 L [2]. Adapted with permission.
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