Spherical glass beads coming out of a kiln are allowed to cool in a room temperature of 30°C. A glass bead with a diameter of 10 mm and an initial temperature of 400°C is allowed to cool for 3 min. If the convection heat transfer coefficient is 28 W/m²·K, determine the temperature at the center of the glass bead using

(a) Table 18€“2

(b) The Heisler chart (Fig. 18€“16). The glass bead has properties of ρ = 2800 kg/m³, c_p = 750 J/kg·K, and k = 0.7 W/m·K.

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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 = hrjk for a cylinder or sphere of radius r) Plane Wall Cylinder A1 Sphere A1 Bi A1 A1 A1 0.1730 0.01 0.0998 1.0017 0.1412 1.0025 1.0030 0.02 0.1410 1.0033 0.1995 1.0050 0.2445 1.0060 1.0099 0.3450 1.0120 1.0179 0.04 0.1987 1.0066 0.2814 0.06 0.2425 1.0098 0.3438 1.0148 0.4217 0.08 0.2791 1.0130 0.3960 1.0197 0.4860 1.0239 0.3111 0.1 1.0161 0.4417 1.0246 0.5423 1.0298 0.6170 0.7593 0.9208 0.2 0.4328 1.0311 1.0450 1.0483 1.0592 0.3 0.5218 0.7465 1.0712 1.0880 0.4 0.5932 1.0580 0.8516 1.0931 1.0528 1.1164 0.5 1.1441 0.6533 1.0701 1.0814 0.9408 1.1143 1.1656 1.1713 0.6 0.7051 1.0184 1.1345 1.2644 0.7 0.7506 1.0918 1.0873 1.1539 1.3525 1.1978 1.1016 1.1724 0.8 0.7910 1.1490 1.4320 1.2236 0.9 0.8274 1.1107 1.2048 1.1902 1.5044 1.2488 1.0 1.5708 0.8603 1.1191 1.2558 1.2071 1.2732 1.1785 2.0 1.0769 1.5995 1.3384 2.0288 1.4793 3.0 1.1925 1.2102 1.7887 1.4191 2.2889 1.6227 4.0 1.2646 1.2287 1.9081 1.4698 2.4556 1.7202 1.5029 5.0 1.3138 1.2403 1.9898 2.5704 1.7870 6.0 1.3496 1.2479 2.0490 1.5253 2.6537 1.8338 1.5411 7.0 1.3766 1.2532 2.0937 2.7165 1.8673 8.0 9.0 1.3978 1.2570 2.1286 1.5526 2.7654 1.8920 1.4149 1.2598 2.1566 1.5611 2.8044 1.9106 10.0 1.4289 1.2620 2.1795 1.5677 2.8363 1.9249 20.0 30.0 40.0 1.4961 1.2699 2.2880 1.5919 2.9857 1.9781 1.5202 1.2717 2.3261 1.5973 3.0372 1.9898 1.5325 1.2723 2.3455 1.5993 3.0632 1.9942 50.0 1.5400 1.2727 2.3572 1.6002 3.0788 1.9962 100.0 3.1102 1.5552 1.2731 2.3809 1.6015 1.9990 1.5708 1.2732 2.4048 1.6021 3.1416 2.0000