Chapter 5, Problem 5.5P

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 $T=300\text{K}\text{.}$

1. A toroidal shape of diameter $D=50\text{mm}$ and cross-sectional area $A_c=5{\text{mm}}^{\text{2}}$ is of thermal conductivity $k=2.3\text{W/m}\cdot \text{K}.$ The surface of the torus is exposed to a coolant corresponding to a convection coefficient of $h=50{\text{W/m}}^{\text{2}}\cdot \text{K}\text{.}$
2. A long, hot AISI 304 stainless steel bar of rectangular cross section has dimensions $w=3\text{mm,}W=5\text{mm,}$ and $L=100\text{mm}\text{.}$ The bar is subjected to a coolant that provides a heat transfer coefficient of $h=15{\text{W/m}}^{\text{2}}\cdot \text{K}$ at all exposed surfaces.
3. A long extruded aluminum (Alloy 2024) tube of inner and outer dimensions $w=20\text{mm}$ and $W=24\text{mm,}$ respectively, is suddenly submerged in water, resulting in a convection coefficient of $h=37{\text{W/m}}^{\text{2}}\cdot \text{K}$ at the four exterior tube surfaces. The tube is plugged at both ends. trapping stagnant air inside the tube.
4. An $L=300\text{-mm}$ -long solid stainless steel rod of diameter $D=13\text{mm}$ and mass $M=0.328\text{kg}$ is exposed to a convection coefficient of $h=30{\text{W/m}}^{\text{2}}\cdot \text{K}\text{.}$
5. A solid sphere of diameter $D=12\text{mm}$ and thermal conductivity $k=120\text{W/m}\cdot \text{K}$ is suspended in a large vacuum oven with internal wall temperatures of $T_{\text{sur}}=20°\text{C}\text{.}$ The initial sphere temperature is $T_i=100°\text{C,}$ and its emissivity is $\epsilon =\mathrm{0.73.}$
6. A long cylindrical rod of diameter $D=20\text{mm,}$ density $\rho =2300{\text{kg/m}}^{\text{3}},$ specific heat $c_p=1750\text{J/kg}\cdot \text{K,}$ and thermal conductivity $k=16\text{W/m}\cdot \text{K}$ is suddenly exposed to convective conditions with $T_{\infty }=20°\text{C}\text{.}$ The rod is initially at a uniform temperature of $T_i=200°\text{C}$ and reaches a spatially averaged temperature of $T=100°\text{C}$ at $t=225\text{s}\text{.}$
7. Repeat part (f) but now consider a rod diameter of $D=200\text{mm}\text{.}$