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.

Heat is generated in a 2.5-cm-square copper rod at the rate of $35.3 \mathrm{MW} / \mathrm{m}^3$. The rod is exposed to a convection environment at $20^{\circ} \mathrm{C}$, and the heat-transfer coefficient is $4000 \mathrm{~W} / \mathrm{m}^2 \cdot{ }^{\circ} \mathrm{C}$. Calculate the surface temperature of the rod.

Rework Derive an expression for the temperature distribution in a plane wall in which distributed heat sources vary according to the linear relation

$$\dot{q}=\dot{q}_w\left[1+\beta\left(T-T_w\right)\right]$$

where $\dot{q}_w$ is a constant and equal to the heat generated per unit volume at the wall temperature $T_w$. Both sides of the plate are maintained at $T_w$, and the plate thickness is $2 L$. supposing that the plate is subjected to a convection environment on both sides of temperature $T_{\infty}$ with a heat-transfer coefficient $h. T_w$ is now some reference temperature not necessarily the same as the surface temperature.

Repeat the above problem, supposing that the outer surface is adiabatic while the inner surface temperature is maintained at $T_i$.

Consider a shielding wall for a nuclear reactor. The wall receives a gamma-ray flux such that heat is generated within the wall according to the relation

$$\dot{q}=\dot{q}_0 e^{-a x}$$

where $\dot{q}_0$ is the heat generation at the inner face of the wall exposed to the gamma-ray flux and $a$ is a constant. Using this relation for heat generation, derive an expression for the temperature distribution in a wall of thickness $L$, where the inside and outside temperatures are maintained at $T_i$ and $T_0$, respectively. Also, obtain an expression for the maximum temperature in the wall.

A plane wall $6.0 \mathrm{~cm}$ thick generates heat internally at the rate of $0.3 \mathrm{MW} / \mathrm{m}^3$. One side of the wall is insulated, and the other side is exposed to an environment at $93^{\circ} \mathrm{C}$. The convection heat transfer coefficient between the wall and the environment is $570 \mathrm{~W} / \mathrm{m}^2 \cdot{ }^{\circ} \mathrm{C}$. The thermal conductivity of the wall is $21 \mathrm{~W} / \mathrm{m} \cdot{ }^{\circ} \mathrm{C}$. Calculate the maximum temperature in the wall.

A circumferential fin of the rectangular profile is constructed of stainless steel with $k=43 \mathrm{~W} / \mathrm{m} \cdot{ }^{\circ} \mathrm{C}$ and a thickness of $1.0 \mathrm{~mm}$. The fin is installed on a tube having a diameter of $3.0 \mathrm{~cm}$ and the outer radius of the fin is $4.0 \mathrm{~cm}$. The inner tube is maintained at $250^{\circ} \mathrm{C}$ and the assembly is exposed to a convection environment having $T_{\infty}=35^{\circ} \mathrm{C}$ and $h=45 \mathrm{~W} / \mathrm{m}^2 \cdot{ }^{\circ} \mathrm{C}$. Compute the heat lost by the fin.

Derive an expression for the temperature distribution in a plane wall in which distributed heat sources vary according to the linear relation

$$\dot{q}=\dot{q}_w\left[1+\beta\left(T-T_w\right)\right]$$

where $\dot{q}_w$ is a constant and equal to the heat generated per unit volume at the wall temperature $T_w$. Both sides of the plate are maintained at $T_w$, and the plate thickness is $2 L$.

A 5-cm-diameter steel pipe is covered with a 1-cm layer of insulating material having $k=0.22 \mathrm{~W} / \mathrm{m} \cdot{ }^{\circ} \mathrm{C}$ followed by a 3 -cm-thick layer of another insulating material having $k=0.06 \mathrm{~W} / \mathrm{m} \cdot{ }^{\circ} \mathrm{C}$. The entire assembly is exposed to a convection surrounding condition of $h=60 \mathrm{~W} / \mathrm{m}^2 \cdot{ }^{\circ} \mathrm{C}$ and $\mathrm{T}_{\infty}=15^{\circ} \mathrm{C}$. The outside surface temperature of the steel pipe is $400^{\circ} \mathrm{C}$. Estimate the heat lost by the pipe-insulation assembly for a pipe length of $20 \mathrm{~m}$. Express in Watts.

A $6.0$-cm-diameter pipe whose surface temperature is maintained at $210^{\circ} \mathrm{C}$ passes through the center of a concrete slab $45 \mathrm{~cm}$ thick. The outer surface temperatures of the slab are maintained at $15^{\circ} \mathrm{C}$. Using the flux plot, calculate the heat loss from the pipe per unit length.

A large slab of concrete is suddenly exposed to a constant radiant heat flux of $900 \mathrm{~W} / \mathrm{m}^2$. The slab is initially uniform in temperature at $20^{\circ} \mathrm{C}$. Determine the temperature at a depth of $10 \mathrm{~cm}$ in the slab after a time of $9 \mathrm{~h}$.

A thick concrete wall having a uniform temperature of $54^{\circ} \mathrm{C}$ is suddenly subjected to an airstream at $10^{\circ} \mathrm{C}$. The heat-transfer coefficient is $10 \mathrm{~W} / \mathrm{m}^2 \cdot{ }^{\circ} \mathrm{C}$. Determine the temperature in the concrete slab at a depth of $7 \mathrm{~cm}$ after $30 \mathrm{~min}$.

The inner wall of a rocket motor combustion chamber receives $50,000 \mathrm{Btu} / \mathrm{h}_{\mathrm{ft}^2}$ by radiation from a gas at $5000^{\circ} \mathrm{F}$. The convection heat transfer coefficient between the gas and the wall is $20 \mathrm{Btu} / \mathrm{h} \mathrm{ft}^2 \mathrm{~F}$. If the inner wall of the combustion chamber is at a temperature of $1000^{\circ} \mathrm{F}$, determine (a) the total thermal resistance of a unit area of the wall in $\mathrm{htt}^2{ }^{\circ} \mathrm{F} / \mathrm{Btu}$ and (b) the heat flux. Also draw the thermal circuit.

An electronic device that internally generates $600 \mathrm{~mW}$ of heat has a maximum permissible operating temperature of $70^{\circ} \mathrm{C}$. It is to be cooled in $25^{\circ} \mathrm{C}$ air by attaching aluminum fins with a total surface area of $12 \mathrm{~cm}^2$. The convection heat transfer coefficient between the fins and the air is $20 \mathrm{~W} / \mathrm{m}^2 \mathrm{~K}$. Estimate the operating temperature when the fins are attached in such a way that (a) there is a contact resistance of approximately $50 \mathrm{~K} / \mathrm{W}$ between the surface of the device and the fin array and (b) there is no contact resistance (in this case, the construction of the device is more expensive). Comment on the design options.