cen58933_ch04

# 1 m 149 wm c hro k 0537 the coefficients 1 and a1 for

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Unformatted text preview: Table 4–1 to be 1 0.970, A1 1.122 Substituting these values into Eq. 4–14 gives 0 To Ti T T A1e 2 1 1.122e (0.970)2(1.07) 0.41 cen58933_ch04.qxd 9/10/2002 9:12 AM Page 228 228 HEAT TRANSFER and thus To T 0.41(Ti T) 200 0.41(600 200) 364°C The value of J1( 1) for 1 0.970 is determined from Table 4–2 to be 0.430. Then the fractional heat transfer is determined from Eq. 4–18 to be Q Qmax 1 2 J1( 1) 0 1 2 1 0.41 0.430 0.970 0.636 and thus Q 0.636Qmax 0.636 (47,354 kJ) 30,120 kJ Discussion The slight difference between the two results is due to the reading error of the charts. 4–3 Plane surface T h 0 x FIGURE 4–22 Schematic of a semi-infinite body. I TRANSIENT HEAT CONDUCTION IN SEMI-INFINITE SOLIDS A semi-infinite solid is an idealized body that has a single plane surface and extends to infinity in all directions, as shown in Fig. 4–22. This idealized body is used to indicate that the temperature change in the part of the body in which we are interested (the region close to the surface) is due to the thermal conditions on a single surface. The earth, for example, can be considered to be a semi-infinite medium in determining the variation of temperature near its surface. Also, a thick wall can be modeled as a semi-infinite medium if all we are interested in is the variation of temperature in the region near one of the surfaces, and the other surface is too far to have any impact on the region of interest during the time of observation. Consider a semi-infinite solid that is at a uniform temperature Ti. At time t 0, the surface of the solid at x 0 is exposed to convection by a fluid at a constant temperature T , with a heat transfer coefficient h. This problem can be formulated as a partial differential equation, which can be solved analytically for the transient temperature distribution T(x, t). The solution obtained is presented in Fig. 4–23 graphically for the nondimensionalized temperature defined as 1 (x, t) 1 T(x, t ) T Ti T T(x, t ) Ti T Ti (4-21) against the dimensionless variable x/(2 t) for various values of the parameter h t/k. Note that the values on the vertical axis correspond to x 0, and thus rept/k resent the surface temperature. The curve h corresponds to h → , which corresponds to the case of specified temperature T at the surface at x 0. That is, the case in which the surface of the semi-infinite body is suddenly brought to temperature T at t 0 and kept at T at all times can be handled by setting h to infinity. The specified surface temperature case is closely cen58933_ch04.qxd 9/10/2002 9:12 AM Page 229 229 CHAPTER 4 1.0 Ambient T(x, t) T ,h 0.5 0.4 T(x, t) – T 1 – ————— = 1 – θ(x, t) Ti – T 0.3 3 2 0.2 x 1 0.5 0.4 0.3 0.2 0.1 0.1 0.05 0.04 hα t k= 0.0 5 0.03 0.02 0.01 0 0.25 0.5 0.75 1.0 1.25 1.5 ξ = —x —– 2 αt FIGURE 4–23 Variation of temperature with position and time in a semi-infinite solid initially at Ti subjected to convection to an environment at T with a convection heat transfer coefficient of h (from P. J. Schneider, Ref. 10). approximated in practice when condensation or boiling takes place on the surface. For a finite heat transfer coefficient h, the surface temperature approaches the fluid temperature T as the time t approaches infinity. The exact solution of the transient one-dimensional heat conduction problem in a semi-infinite medium that is initially at a uniform temperature of Ti and is suddenly subjected to convection at time t 0 has been obtained, and is expressed as T(x, t) Ti T Ti erfc x 2 t hx k exp x h2 t erfc k2 2 t h t k (4-22) where the quantity erfc ( ) is the complementary error function, defined as erfc ( ) 1 2 e u2 du (4-23) 0 Despite its simple appearance, the integral that appears in the above relation cannot be performed analytically. Therefore, it is evaluated numerically for different values of , and the results are listed in Table 4–3. For the special case of h → , the surface temperature Ts becomes equal to the fluid temperature T , and Eq. 4–22 reduces to T(x, t) Ti Ts Ti erfc x 2 t (4-24) cen58933_ch04.qxd 9/10/2002 9:12 AM Page 230 230 HEAT TRANSFER TABLE 4–3 The complementary error function erfc ( ) 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30 0.32 0.34 0.36 erfc ( ) 1.00000 0.9774 0.9549 0.9324 0.9099 0.8875 0.8652 0.8431 0.8210 0.7991 0.7773 0.7557 0.7343 0.7131 0.6921 0.6714 0.6509 0.6306 0.6107 0.38 0.40 0.42 0.44 0.46 0.48 0.50 0.52 0.54 0.56 0.58 0.60 0.62 0.64 0.66 0.68 0.70 0.72 0.74 0.5910 0.5716 0.5525 0.5338 0.5153 0.4973 0.4795 0.4621 0.4451 0.4284 0.4121 0.3961 0.3806 0.3654 0.3506 0.3362 0.3222 0.3086 0.2953 erfc ( ) 0.76 0.78 0.80 0.82 0.84 0.86 0.88 0.90 0.92 0.94 0.96 0.98 1.00 1.02 1.04 1.06 1.08 1.10 1.12 0.2825 0.2700 0.2579 0.2462 0.2349 0.2239 0.2133 0.2031 0.1932 0.1837 0.1746 0.1658 0.1573 0.1492 0.1413 0.1339 0.1267 0.1198 0.1132 erfc ( ) 1.14 1.16 1.18 1.20 1.22 1.24 1.26 1.28 1.30 1.32 1.34 1.36 1.38 1.40 1.42 1.44 1.46 1.48 1.50 0.10...
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