cen58933_ch04

# The determination of the constants a1 and 1 usually

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Unformatted text preview: to determine the temperature anywhere in the medium. The determination of the constants A1 and 1 usually requires interpolation. For those who prefer reading charts to interpolating, the relations above are plotted and the one-term approximation solutions are presented in graphical form, known as the transient temperature charts. Note that the charts are sometimes difficult to read, and they are subject to reading errors. Therefore, the relations above should be preferred to the charts. The transient temperature charts in Figs. 4–13, 4–14, and 4–15 for a large plane wall, long cylinder, and sphere were presented by M. P. Heisler in 1947 and are called Heisler charts. They were supplemented in 1961 with transient heat transfer charts by H. Gröber. There are three charts associated with each geometry: the first chart is to determine the temperature To at the center of the geometry at a given time t. The second chart is to determine the temperature at other locations at the same time in terms of To. The third chart is to determine the total amount of heat transfer up to the time t. These plots are valid for 0.2. cen58933_ch04.qxd 9/10/2002 9:12 AM Page 219 219 CHAPTER 4 TABLE 4–1 TABLE 4–2 Coefficients used in the one-term approximate solution of transient onedimensional heat conduction in plane walls, cylinders, and spheres (Bi hL/k for a plane wall of thickness 2L, and Bi hro /k for a cylinder or sphere of radius ro ) The zeroth- and first-order Bessel functions of the first kind Plane Wall A1 1 Bi 0.01 0.02 0.04 0.06 0.08 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 20.0 30.0 40.0 50.0 100.0 0.0998 0.1410 0.1987 0.2425 0.2791 0.3111 0.4328 0.5218 0.5932 0.6533 0.7051 0.7506 0.7910 0.8274 0.8603 1.0769 1.1925 1.2646 1.3138 1.3496 1.3766 1.3978 1.4149 1.4289 1.4961 1.5202 1.5325 1.5400 1.5552 1.5708 1.0017 1.0033 1.0066 1.0098 1.0130 1.0161 1.0311 1.0450 1.0580 1.0701 1.0814 1.0918 1.1016 1.1107 1.1191 1.1785 1.2102 1.2287 1.2403 1.2479 1.2532 1.2570 1.2598 1.2620 1.2699 1.2717 1.2723 1.2727 1.2731 1.2732 Cylinder A1 1 0.1412 0.1995 0.2814 0.3438 0.3960 0.4417 0.6170 0.7465 0.8516 0.9408 1.0184 1.0873 1.1490 1.2048 1.2558 1.5995 1.7887 1.9081 1.9898 2.0490 2.0937 2.1286 2.1566 2.1795 2.2880 2.3261 2.3455 2.3572 2.3809 2.4048 Sphere 1.0025 1.0050 1.0099 1.0148 1.0197 1.0246 1.0483 1.0712 1.0931 1.1143 1.1345 1.1539 1.1724 1.1902 1.2071 1.3384 1.4191 1.4698 1.5029 1.5253 1.5411 1.5526 1.5611 1.5677 1.5919 1.5973 1.5993 1.6002 1.6015 1.6021 1 0.1730 0.2445 0.3450 0.4217 0.4860 0.5423 0.7593 0.9208 1.0528 1.1656 1.2644 1.3525 1.4320 1.5044 1.5708 2.0288 2.2889 2.4556 2.5704 2.6537 2.7165 2.7654 2.8044 2.8363 2.9857 3.0372 3.0632 3.0788 3.1102 3.1416 A1 1.0030 1.0060 1.0120 1.0179 1.0239 1.0298 1.0592 1.0880 1.1164 1.1441 1.1713 1.1978 1.2236 1.2488 1.2732 1.4793 1.6227 1.7202 1.7870 1.8338 1.8673 1.8920 1.9106 1.9249 1.9781 1.9898 1.9942 1.9962 1.9990 2.0000 Note that the case 1/Bi k/hL 0 corresponds to h → , which corresponds to the case of specified surface temperature T . That is, the case in which the surfaces of the body are suddenly brought to the temperature T at t 0 and kept at T at all times can be handled by setting h to infinity (Fig. 4–16). The temperature of the body changes from the initial temperature Ti to the temperature of the surroundings T at the end of the transient heat conduction process. Thus, the maximum amount of heat that a body can gain (or lose if Ti T ) is simply the change in the energy content of the body. That is, Qmax mCp(T Ti ) VCp(T Ti ) (kJ) (4-16) Jo( ) J1( ) 0.0 0.1 0.2 0.3 0.4 1.0000 0.9975 0.9900 0.9776 0.9604 0.0000 0.0499 0.0995 0.1483 0.1960 0.5 0.6 0.7 0.8 0.9 0.9385 0.9120 0.8812 0.8463 0.8075 0.2423 0.2867 0.3290 0.3688 0.4059 1.0 1.1 1.2 1.3 1.4 0.7652 0.7196 0.6711 0.6201 0.5669 0.4400 0.4709 0.4983 0.5220 0.5419 1.5 1.6 1.7 1.8 1.9 0.5118 0.4554 0.3980 0.3400 0.2818 0.5579 0.5699 0.5778 0.5815 0.5812 2.0 2.1 2.2 2.3 2.4 0.2239 0.1666 0.1104 0.0555 0.0025 0.5767 0.5683 0.5560 0.5399 0.5202 2.6 2.8 3.0 3.2 0.0968 0.1850 0.2601 0.3202 0.4708 0.4097 0.3391 0.2613 cen58933_ch04.qxd 9/10/2002 9:12 AM Page 220 220 HEAT TRANSFER To – T Ti – T 1.0 0.7 0.5 0.4 0.3 0.2 θo = k hL = 1 Bi = 0.6 0.4 0.7 0.5 0.3 35 7 6 25 30 16 3 2 1.8 1.6 1.4 1.2 0.05 2.5 0 2 50 40 20 18 5 4 0.2 0.1 1 45 9 8 8 0 12 10 0. 0.01 0.007 0.005 0.004 0.003 0.002 3 4 6 8 10 14 18 22 26 30 50 τ = α t/L2 70 100 120 150 T h (a) Midplane temperature (from M. P. Heisler) 300 Initially T = Ti 0 T– T To – T x/L = 0.2 1.0 0.9 0.9 0.4 Bi = h L /k 0.2 50 10 5 2 1 0.5 0.05 0.1 0.2 0.3 0.9 0.1 1.0 0 0.01 0.1 Bi = 0.4 0.8 0.00 5 0.01 0.02 0.5 0.00 1 0.00 2 0.6 0.5 0.3 x L 0.7 0.6 0.6 0.4 T h 0.8 0.8 0.7 600 700 2L Q Qmax 1.0 θ= 400 500 20 0.001 100 80 90 60 70 14 1.0 0.1 0.07 0.05 0.04 0.03 0.02 Plate 0.2 Plate 1.0 10 100 0.1 0 10–5 Plate 10– 4 10–3 10–2 1 k = Bi hL (b) Temperature distribution (from M. P. Heisler) 10–1 1 Bi 2 τ = h2 α t /k 2 10 102 103 104 (c) Heat transfer (from H. Gröber et al.) FIGURE 4–13 Transient t...
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