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Heat Transfer Handbook part 26

Heat Transfer Handbook part 26. The Heat Transfer Handbook provides succinct hard data, formulas, and specifications for the critical aspects of heat transfer, offering a reliable, hands-on resource for solving day-to-day issues across a variety of applications. | PERIODIC CONDUCTION 241 and the steady periodic temperature distribution is given by Bi j e 5 e- X cos 2nr - Vn X - p 3.333 Bi2 2Bi 2 1 2 where e T - T-.- 3.334a a . X x 3.334b 2naz t 3.334c 2n. . 3 Bi h 2 m 3.334d a P arctan i r 3.334e Note that as h to Bi to and P 0 eqs. 3.333 reduce to eq. 3.328 with 7 m Ti .The presence of the factor Bi Bi 2Bi 2 1 2 in eq. 3.333 shows that convection enhances the damping effect and that it also increases the phase angle by an amount P arctan 1 1 Bi . 3.9.5 Finite Plane Wall with Periodic Surface Temperature Consider a plane wall of thickness L with the face at x 0 insulated and the face at x L subjected to a periodic temperature change of hie form T L t Ti a cos mt 3.335 where Ti is the initial temperature of the wall and the instated boundary condttfon at x 0 gives dT 0 t 0 3.336 dx The steady periodic solution is T x t Ti a L -1 L cos Lt Jx I . l 3.337 L 2 J I 2 l1 2a JJ where the numerical values of 1 as a function of x L and w 2a 1 2L are supplied in Table 3.13 and 2 which is also a function of x L and w 2a 1 2L is given by 9 9 - 69 92 arctan-------------- 3.338 9 a 6 - c d 242 CONDUCTION HEAT TRANSFER TABLE 3.13 Values of the Amplitude Decay Function as a Function of x L and a w 2a 1 2L Ơ x L 0 0.125 0.250 0.375 0.500 0.625 0.750 0.875 1.000 0.0 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.5 0.98 0.98 0.98 0.98 0.98 0.98 0.98 0.99 1.00 1.0 0.77 0.77 0.77 0.78 0.79 0.81 0.85 0.91 1.00 1.5 0.47 0.47 0.47 0.48 0.52 0.58 0.68 0.83 1.00 2.0 0.27 0.27 0.28 0.30 0.36 0.45 0.58 0.77 1.00 4.0 0.04 0.04 0.05 0.08 0.13 0.22 0.37 0.64 1.00 8.0 0.00 0.00 0.01 0.01 0.02 0.05 0.14 0.36 1.00 œ 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 where ộa cos 1 2 L cosh 1 2 L ộ6 cos ộc sin ộd sin rn X1 2 x 2az 1 2 J 2a rn X1 2 x 2a cosh sinh sinh rn X1 2 x 2a 1 2 2a rn X1 2 x 2a 3.339a 3.339b 3.339c 3.339d 3.9.6 Infinitely Long Semi-infinite Hollow Cylinder with Periodic Surface Temperature Consider an infinitely long cylinder of inside radius ri .