A long Pyroceram rod (ρ = 2600 kg/m³, c_p= 808 J/kg·K, k = 3.98 W/m·K, and α = 1.89 × 10^€“6m²/s) with diameter of 10 mm has an initial uniform temperature of 1000°C. The Pyroceram rod is allowed to cool in ambient temperature of 25°C and convection heat transfer coefficient of 80 W/m2·K. If the Pyroceram rod is allowed to cool for 3 min, determine the temperature at the center of the rod using

(a) Table 18€“2

(b) The Heisler chart (Fig. 18€“15).

Transcribed Image Text:

TABLE 18-2 Coefficients used in the one-term approximate solution of transient one- dimensional heat conduction in plane walls, cylinders, and spheres (Bi = hLIk for a plane wall of thickness 2L, and Bi = radius r) hrjk for a cylinder or sphere of Plane Wall Cylinder A1 Sphere A1 Bi A1 λι λι 0.01 0.0998 1.0017 0.1412 1.0025 0.1730 1.0030 0.1995 0.2814 1.0033 1.0050 1.0099 0.02 0.1410 0.1987 0.2445 1.0060 0.04 1.0066 0.3450 1.0120 0.06 0.2425 1.0098 0.3438 1.0148 0.4217 1.0179 0.08 1.0130 1.0197 0.4860 0.2791 0.3960 1.0239 0.1 0.3111 1.0161 0.4417 1.0246 0.5423 1.0298 0.2 0.4328 1.0311 0.6170 1.0483 0.7593 1.0592 0.9208 0.3 0.5218 1.0450 0.7465 1.0712 1.0880 1.0931 0.4 0.5932 1.0580 0.8516 1.0528 1.1164 0.5 0.6533 1.0701 0.9408 1.1143 1.1656 1.1441 0.6 0.7051 1.0814 1.0184 1.1345 1.2644 1.1713 1.3525 1.4320 0.7 0.7506 1.0918 1.0873 1.1539 1.1978 1.1016 1.1107 1.2236 0.8 0.7910 1.1490 1.1724 1.5044 0.9 0.8274 1.2048 1.1902 1.2071 1.2488 1.0 0.8603 1.1191 1.2558 1.5708 1.2732 2.0 1.0769 1.1785 1.5995 1.3384 2.0288 1.4793 3.0 1.1925 1.2102 1.7887 1.4191 2.2889 1.6227 1.7202 4.0 1.2646 1.2287 1.9081 1.4698 2.4556 5.0 1.3138 1.2403 1.9898 1.5029 2.5704 1.7870 1.8338 6.0 1.3496 1.2479 2.0490 1.5253 2.6537 1.2532 7.0 1.3766 2.0937 1.5411 2.7165 1.8673 8.0 1.3978 1.2570 2.1286 1.5526 2.7654 1.8920 9.0 1.4149 1.2598 2.1566 1.5611 2.8044 1.9106 2.8363 10.0 1.4289 1.2620 2.1795 1.5677 1.9249 1.2699 20.0 1.4961 2.2880 1.5919 2.9857 1.9781 30.0 1.5202 1.2717 2.3261 1.5973 3.0372 1.9898 40.0 1.5325 1.2723 2.3455 1.5993 3.0632 1.9942 50.0 1.5400 1.5552 1.2727 2.3572 1.6002 1.6015 3.0788 1.9962 2.3809 3.1102 100.0 1.2731 1.9990 1.2732 1.5708 2.4048 1.6021 3.1416 2.0000 To - To Oo = 1.0 0.7 Sphere 0.5 0.4 0.3 hro Bị 100 0.2 0.1 0.07 0.05 0.04 0.03 0.02 0.01 0.007 0.005 0.004 0.003 0.002 0.001 3 4 5 6 7 8 9 10 T= atlr? 0.5 1.0 1.5 2 2.5 20 30 40 50 100 150 200 250 Too Initially T= T, (a) Midpoint temperature. From M. P. Helsler, "Temperature Charts for Induction and Constant Temperature Heating," Trans. ASME 69, 1947, Fig. 9, p. 231. Reprinted by permission of ASME International. Го T-T Ho To- Omax 1.0 rir, = 0.2. 1.0 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 0.6 0.5 0.5 0.4 0.4 0.3H'ö 0.2 H0.g 0.3 0.2 0.1 0.1 1.0LHI Sphere Sphere 0. 10 5 10-4 10-3 10-2 10 102 103 104 0.01 0.1 1.0 10 100 10 Bi?t = h?at/k? Bi hr. (b) Temperature distribution. (c) Heat transfer. From M. P. Helsler, "Temperature Charts for Induction and Constant Temperature Heating," Trans. ASME 69, 1947, Fig. 12, p. 232. Reprinted by permission of ASME International. From H. Gröber et al.