An electrical heater 100 mm long and 5 mm in diameter is inserted into a hole drilled normal to the surface of a large block of material having a thermal conductivity of $5 \mathrm{W} / \mathrm{m} \cdot \mathrm{K}$. Estimate the temperature reached by the heater when dissipating 50 W with the surface of the block at a temperature of 25 C.

Radioactive wastes $\left(k_{\mathrm{nw}}=20 \mathrm{W} / \mathrm{m} \cdot \mathrm{K}\right)$ are stored in a spherical, stainless steel $\left(k_{\mathrm{ss}}=15 \mathrm{W} / \mathrm{m} \cdot \mathrm{K}\right)$ container of inner and outer radii equal to $r_{i}=0.5 \mathrm{m}$ and $r_{o}=0.6 \mathrm{m}$. Heat is generated volumetrically within the wastes at a uniform rate of $\dot{q}=10^{5}$, and the outer surface of the container is exposed to a water flow for which h=$1000 \mathrm{W} / \mathrm{m}^{2} \cdot \mathrm{K}$ and $T_{\infty}=25^{\circ} \mathrm{C}$. (a) Evaluate the steady-state outer surface temperature, $T_{s, o}$. b) Evaluate the steady-state inner surface temperature, $T_{s, i}$. (c) Obtain an expression for the temperature distribution, T(r), in the radioactive wastes. Express your result in terms of $r_{i}, T_{s, i}, k_{\mathrm{rw}}$, and $\dot{q}$. Evaluate the temperature at r=0. (d) A proposed extension of the foregoing design involves storing waste materials having the same thermal conductivity but twice the heat generation $\left(\dot{q}=2 \times 10^{5} \mathrm{W} / \mathrm{m}^{3}\right)$ in a stainless steel container of equivalent inner radius $\left(r_{i}=0.5 \mathrm{m}\right)$. Safety considerations dictate that the maximum system temperature not exceed $475^{\circ} \mathrm{C}$ and that the container wall thickness be no less than t=0.04 m and preferably at or close to the original design (t=0.1 m). Assess the effect of varying the outside convection coefficient to a maximum achievable value of $h=5000 \mathrm{W} / \mathrm{m}^{2} \cdot \mathrm{K}$ (by increasing the water velocity) and the container wall thickness. Is the proposed extension feasible? If so, recommend suitable operating and design conditions for h and t, respectively.

An aluminum heat sink $(k=240 \mathrm{W} / \mathrm{m} \cdot \mathrm{K})$, used to cool an array of electronic chips, consists of a square channel of inner width w=25 mm, through which liquid flow may be assumed to maintain a uniform surface temperature of $T_{1}=20^{\circ} \mathrm{C}$. The outer width and length of the channel are W=40 mm and L=160 mm, respectively. If N=120 chips attached to the outer surfaces of the heat sink maintain an approximately uniform surface temperature of $T_{2}=50^{\circ} \mathrm{C}$ and all of the heat dissipated by the chips is assumed to be transferred to the coolant, what is the heat dissipation per chip? If the contact resistance between each chip and the heat sink is $R_{t, c}=0.2 \mathrm{K} / \mathrm{W}$, what is the chip temperature?