For each of the following cases, determine an appropriate characteristic length L
cand the corresponding Biot number B
ithat 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/m1
∙ 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 Tsur 20 C. The initial sphere temperature is Ti
= 100°C, and its emissivity is ε = 0.73.
(f) A long cylindrical rod of diameter D = 20 mm, density ρ = 2300 kg/m3, specific heat cp
= 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 Ti
= 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. Case