56 for each of the following cases determine an appropriate characteri

Question

5.6 For each of the following cases, determine an appropriate characteristic length Le and the corresponding Biot number Bithat is associated with the transient thermal response of the solid object. State whether the lumped capacitance approximation is valid. If temperature information is not provided, evaluate properties at T= 300 K.
(a) A toroidal shape of diameter D = 50 mm and cross-sectional area Ac = 5 mm² is of thermal conductivity k = 2.3 W/m K. The surface of the torus is exposed to a coolant corresponding to a convection coefficient of h = 50 W/m2 K. (b) A long, hot AISI 304 stainless steel bar of rectangular cross section has dimensions w = 3 mm, W = 5 mm, and L = 100 mm.The bar is subjected to a coolant that provides a heat transfer coefficient of h = 15 W/m2 . K at all exposed surfaces. (c) A long extruded aluminum (Alloy 2024) tube of inner and outer dimensions w = 20 mm and W = 24 mm, respectively, is suddenly submerged in water, resulting in a convection coefficient of h = 37 W/m2 . K at the four exterior tube surfaces.The tube is plugged at both ends, trapping stagnant air inside the tube. (d) An L = 300-mm-long solid stainless steel rod of diameter D= 13 mm and mass m = 0.328 kg is exposed to a convection coefficient of h = 30 W/m2 . K. (e) A solid sphere of diameter D = 13 mm and thermal conductivity k = 130 W/m · Kis suspended in a large vacuum oven with internal wall temperatures of Tsur = 18°C. The initial sphere temperature is T; = 100°C, and its emissivity is e = 0.75. (f) A long cylindrical rod of diameter D = 20 mm, density p =2300 kg/m3, specific heat c, = 1750 J/kg · K, and thermal Cp conductivity k = 16 W/m · K is suddenly exposed to convective conditions with T = 20°C. The rod is initially at a uniform temperature of T; = 200°C and reaches a spatially averaged temperature of T = 100°C at t = 225 s. (g) Repeat part (f) but now consider a rod diameter of D = 200mm.