A stainless-steel sphere $\left[k=16 \mathrm{~W} / \mathrm{m} \cdot{ }^{\circ} \mathrm{C}\right]$ having a diameter of $4 \mathrm{~cm}$ is exposed to a convection environment at $20^{\circ} \mathrm{C}, h=15 \mathrm{~W} / \mathrm{m}^2 \cdot{ }^{\circ} \mathrm{C}$. Heat is generated uniformly in the sphere at the rate of $1.0 \mathrm{MW} / \mathrm{m}^3$. Calculate the steady-state temperature for the center of the sphere.

An electric current is used to heat a tube through which a suitable cooling fluid flows. The outside of the tube is covered with insulation to minimize heat loss to the surroundings, and thermocouples are attached to the outer surface of the tube to measure the temperature. Assuming uniform heat generation in the tube, derive an expression for the convection heat transfer coefficient on the inside of the tube in terms of the measured variables: voltage $E$, current $I$, outside tube wall temperature $T_0$, inside and outside radii $r_i$ and $r_o$, tube length $L$, and fluid temperature $T_f$.

The heater wire of Example 2-7 is submerged in a fluid maintained at $93^{\circ} \mathrm{C}$. The convection heat-transfer coefficient is $5.7 \mathrm{~kW} / \mathrm{m}^2 \cdot{ }^{\circ} \mathrm{C}$. Calculate the center temperature of the wire.

A 3.2-mm-diameter stainless-steel wire $30 \mathrm{~cm}$ long has a voltage of $10 \mathrm{~V}$ impressed on it. The outer surface temperature of the wire is maintained at $93^{\circ} \mathrm{C}$. Calculate the center temperature of the wire. Take the resistivity of the wire as $70 \mu \Omega \cdot \mathrm{cm}$ and the thermal conductivity as $22.5 \mathrm{~W} / \mathrm{m} \cdot{ }^{\circ} \mathrm{C}$.

A plate having a thickness of $4.0 \mathrm{~mm}$ has an internal heat generation of $200 \mathrm{MW} / \mathrm{m}^3$ and a thermal conductivity of $25 \mathrm{~W} / \mathrm{m} \cdot{ }^{\circ} \mathrm{C}$. One side of the plate is insulated and the other side is maintained at $100^{\circ} \mathrm{C}$. Calculate the maximum temperature in the plate.

Heat is generated uniformly in a stainless steel plate having $k=20 \mathrm{~W} / \mathrm{m} \cdot{ }^{\circ} \mathrm{C}$. The thickness of the plate is $1.0 \mathrm{~cm}$ and the heat-generation rate is $500 \mathrm{MW} / \mathrm{m}^3$. If the two sides of the plate are maintained at 100 and $200^{\circ} \mathrm{C}$, respectively, calculate the temperature at the center of the plate.

A 3.0-cm-thick plate has heat generated uniformly at the rate of $5 \times 10^5 \mathrm{~W} / \mathrm{m}^3$. One side of the plate is maintained at $200^{\circ} \mathrm{C}$ and the other side at $45^{\circ} \mathrm{C}$. Calculate the temperature at the center of the plate for $k=16 \mathrm{~W} / \mathrm{m} \cdot{ }^{\circ} \mathrm{C}$.

Two 5.0-cm-diameter aluminum bars, $2 \mathrm{~cm}$ long, have ground surfaces and are joined in compression with a $0.025-\mathrm{mm}$ brass shim at a pressure exceeding $20 \mathrm{~atm}$. The combination is subjected to an overall temperature difference of $200^{\circ} \mathrm{C}$. Calculate the temperature drop across the contact join.

Electric heater wires are installed in a solid wall having a thickness of $8 \mathrm{~cm}$ and $k=2.5 \mathrm{~W} / \mathrm{m} \cdot{ }^{\circ} \mathrm{C}$. The right face is exposed to an environment with $h=50 \mathrm{~W} / \mathrm{m}^2 \cdot{ }^{\circ} \mathrm{C}$ and $T_{\infty}=30^{\circ} \mathrm{C}$, while the left face is exposed to $h=75 \mathrm{~W} / \mathrm{m}^2 \cdot{ }^{\circ} \mathrm{C}$ and $T_{\infty}=50^{\circ} \mathrm{C}$. What is the maximum allowable heat-generation rate such that the maximum temperature in the solid does not exceed $300^{\circ} \mathrm{C}$?

The temperature distribution in a certain plane wall is

$$\frac{T-T_1}{T_2-T_1}=C_1+C_2 x^2+C_3 x^3$$

where $T_1$ and $T_2$ are the temperatures on each side of the wall. If the thermal conductivity of the wall is constant and the wall thickness is $L$, derive an expression for the heat generation per unit volume as a function of $x$, the distance from the plane where $T=T_1$. Let the heat-generation rate be $\dot{q}_0$ at $x=0$.

A certain semiconductor material has a conductivity of $0.0124 \mathrm{~W} / \mathrm{cm} \cdot{ }^{\circ} \mathrm{C}$. A rectangular bar of the material has a cross-sectional area of $1 \mathrm{~cm}^2$ and a length of $3 \mathrm{~cm}$. One end is maintained at $300^{\circ} \mathrm{C}$ and the other end at $100^{\circ} \mathrm{C}$, and the bar carries a current of $50 \mathrm{~A}$. Assuming the longitudinal surface is insulated, calculate the midpoint temperature in the bar. Take the resistivity as $1.5 \times 10^{-3} \Omega \cdot \mathrm{cm}$.

Discover the heat transfer per unit area through the composite wall. Assume one-dimensional heat flow.

$$\begin{aligned} k_A & =150 \mathrm{~W} / \mathrm{m} \cdot{ }^{\circ} \mathrm{C} \\ k_B & =30 \\ k_C & =50 \\ k_D & =70 \\ A_B & =A_D \end{aligned}$$

Specify fin efficiency.

A plane wall of thickness $2 L$ has an internal heat generation that varies according to $\dot{q}=\dot{q}_0 \cos a x$, where $\dot{q}_0$ is the heat generated per unit volume at the center of the wall $(x=0)$ and $a$ is a constant. If both sides of the wall are maintained at a constant temperature of $T_w$, derive an expression for the total heat loss from the wall per unit surface area.