$($a$)$ A toroidal shape of diameter $D=50mm$ and cross$-$sectional area $A_c=4m{m}^2$ is of thermal conductivity $k=2\mathrm{.}3\frac{W}{m}*K.$ The

surface of the torus is exposed to a coolant corresponding to a convection coefficient of $h=50\frac{W}{m^2}*K.$

$($b$)$ A long, hot AISI $304$ stainless steel bar of rectangular cross section has dimensions $w=3mm,W=7mm,$ and $L=100mm.$

The bar is subjected to a coolant that provides a heat transfer coefficient of $h=15\frac{W}{m^2}*K$ at all exposed surfaces.

$($c$)$ A long extruded aluminum $($Alloy $2024)$ tube of inner and outer dimensions $w=20mm$ and $W=25\mathrm{.}5mm,$ respectively, is

suddenly submerged in water, resulting in a convection coefficient of $h=37\frac{W}{m^2}*K$ at the four exterior tube surfaces. The tube is

plugged at both ends, trapping stagnantair inside the tube.

$($d$)$ An $L=198\mathrm{.}1-mm-$long solid stainless steel rod of diameter $D=16mm$ and mass $m=0\mathrm{.}328kg$ is exposed to a convection

coefficient of $h=15\frac{W}{m^2}*K.$

$($e$)$ A solid sphere of diameter $D=11mm$ and thermal conductivity $k=120\frac{W}{m}*K$ is suspended in a large vacuum oven with

internal wall temperatures of $T_{\text{s}\text{u}\text{r}}=20C.$ The initial sphere temperature is $T_i=100C,$ and its emissivity is $=0\mathrm{.}73.$

$($f$)$ A long cylindrical rod of diameter $D=21mm,$ density $=2300k\frac{g}{m^3},$ specific heat $c_p=1750\frac{J}{k}g*K,$ and thermal conductivity

$k=16\frac{W}{m}*K$ is suddenly exposed to convective conditions with $T_{}=20C.$ The rod is initially at a uniform temperature of $T_i=$

$200C$ and reaches a spatially averaged temperature of $T=100C$ at $t=225s.$

Note: Characteristic lengths can be calculated as $L_{c1}=\frac{V}{A_s},$ or they can be taken conservatively as the dimension corresponding to

the maximum spatial temperature difference, $L_{c2}.$

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