Problem 58

Abstract

KNOWN: Initial temperature and properties of a solid sphere. Surface temperature after immersion in a fluid of prescribed temperature and convection coefficient. FIND: (a) Time to reach surface temperature, (b) Effect of thermal diffusivity and conductivity on thermal response. SCHEMATIC: ASSUMPTIONS: (1) One-dimensional, radial conduction, (2) Constant properties. ANALYSIS: (a) For k = 15 W/m⋅K, the Biot number is () () 2 o h r 3 300 W m K 0.05 m 3 Bi 0.333 k 1 5 WmK ⋅ = = = ⋅. Hence, the lumped capacitance method cannot be used. From Equation 5.53a, () () * 1 2 1 1 * i 1 sin r T T C exp Fo T T r ζ ζ ζ ∞ ∞ − = − −. At the surface, r * = 1. From Table 5.1, for Bi = 1.0, ζ 1 = 1.5708 rad and C 1 = 1.2732. Hence, () 2 60 75 sin 90 0.30 1.2732 exp 1.5708 Fo 25 75 1.5708 − = = − − o exp(-2.467Fo) = 0.370 2 o t Fo 0.403 r α = = () 2 2 o 5 2 0.05 m r t 0.403 0.403 100s 10 m s α − = = =