For each of the following cases, determine an appropriate characteristic length L c and the corresponding Biot

Question:

For each of the following cases, determine an appropriate characteristic length L_c and the corresponding Biot number Bi that 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 A_c=5mm² 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 = 12 mm and thermal conductivity k = 120 W/m · K is suspended in a large vacuum oven with internal wall temperatures of T_sur=20°C. The initial sphere temperature is T_i= 100°C, and its emissivity is ε = 0.73.
(f) A long cylindrical rod of diameter D = 20 mm, density 2300 kg/m3, specific heat c_p= 1750 J/kg · K, and thermal 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_i = 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 = 200 mm.