A pin fin of uniform, cross-sectional area is fabricated of an aluminum alloy $(k=160 \mathrm{W} / \mathrm{m} \cdot \mathrm{K})$. The fin diameter is D=4 mm, and the fin is exposed to convective conditions characterized by $h=220 \mathrm{W} / \mathrm{m}^{2} \cdot \mathrm{K}$. It is reported that the fin efficiency is $\eta_{f}=0.65$. Determine the fin length L and the fin effectiveness $\varepsilon_{f}$. Account for tip convection.

A long, circular aluminum rod is attached at one end to a heated wall and transfers heat by convection to a cold fluid. (a) If the diameter of the rod is tripled, by how much would the rate of heat removal change? (b) If a copper rod of the same diameter is used in place of the aluminum, by how much would the rate of heat removal change?

Aluminum fins of triangular profile are attached to a plane wall whose surface temperature is $250^{\circ} \mathrm{C}$. The fin base thickness is 2 mm, and its length is 6 mm. The system is in ambient air at a temperature of $20^{\circ} \mathrm{C}$, and the surface convection coefficient is $40 \mathrm{W} / \mathrm{m}^{2} \cdot \mathrm{K}$. (a) What are the fin efficiency and effectiveness? (b) What is the heat dissipated per unit width by a single fin?

An uninsulated, thin-walled pipe of 100-mm diameter is used to transport water to equipment that operates outdoors and uses the water as a coolant. During particularly harsh winter conditions, the pipe wall achieves a temperature of $-15^{\circ} \mathrm{C}$ and a cylindrical layer of ice forms on the inner surface of the wall. If the mean water temperature is $3^{\circ} \mathrm{C}$ and a convection coefficient of $2000 \mathrm{W} / \mathrm{m}^{2} \cdot \mathrm{K}$ is maintained at the inner surface of the ice, which is at $0^{\circ} \mathrm{C}$, what is the thickness of the ice layer?

The temperature of a flowing gas is to be measured with a thermocouple junction and wire stretched between two legs of a sting, a wind tunnel test fixture. The junction is formed by butt-welding two wires of different material, as shown in the schematic. For wires of diameter $D=125 \quad \mu \mathrm{m}$ and a convection coefficient of $h=700 \mathrm{W} / \mathrm{m}^{2} \cdot \mathrm{K}$, determine the minimum separation distance between the two legs of the sting, $L=L_{1}+L_{2}$, to ensure that the sting temperature does not influence the junction temperature and, in turn, invalidate the gas temperature measurement. Consider two different types of thermocouple junctions consisting of (i) copper and constantan wires and (ii) chromel and alumel wires. Evaluate the thermal conductivity of copper and constantan at at T=300 K. Use $k_{\mathrm{Ch}}=19 \mathrm{W} / \mathrm{m} \cdot \mathrm{K}$ and $k_{\mathrm{Al}}=29 \mathrm{W} / \mathrm{m} \cdot \mathrm{K}$ for the thermal conductivities of the chromel and alumel wires, respectively.

A one-dimensional plane wall of thickness 2=100 mm experiences uniform thermal energy generation of $q=1000 \mathrm{W} / \mathrm{m}^{3}$ and is convectively cooled at $x=\pm 50 \mathrm{mm}$ by an ambient fluid characterized by $T_{\infty}=20^{\circ} \mathrm{C}$. If the steady-state temperature distribution within the wall is $T(x)=a\left(L^{2}-x^{2}\right)+b$ where $a=10^{\circ} \mathrm{C} / \mathrm{m}^{2}$ and $b=30^{\circ} \mathrm{C}$, what is the thermal conductivity of the wall? What is the value of the convection heat transfer coefficient, h?

Determine the percentage increase in heat transfer associated with attaching aluminum fins of rectangular profile to a plane wall. The fins are 50 mm long, 0.5 mm thick, and are equally spaced at a distance of 4 mm (250 fins/m). The convection coefficient associated with the bare wall is $40 \mathrm{W} / \mathrm{m}^{2} \cdot \mathrm{K}$, while that resulting from attachment of the fins is $30 \mathrm{W} / \mathrm{m}^{2} \cdot \mathrm{K}$.

Two long copper rods of diameter D=10 mm are soldered together end to end, with solder having a melting point of $650^{\circ} \mathrm{C}$. The rods are in air a $25^{\circ} \mathrm{C}$ with a convection coefficient of $10 \mathrm{W} / \mathrm{m}^{2} \cdot \mathrm{K}$. What is the minimum power input needed to effect the soldering?

Annular aluminum fins of rectangular profile are attached to a circular tube having an outside diameter of 50 mm and an outer surface temperature of $200^{\circ} \mathrm{C}$. The fins are 4 mm thick and 15 mm long. The system is in ambient air at a temperature of $20^{\circ} \mathrm{C}$, and the surface convection coefficient is $40 \mathrm{W} / \mathrm{m}^{2} \cdot \mathrm{K}$. (a) What are the fin efficiency and effectiveness? (b) If there are 125 such fins per meter of tube length, what is the rate of heat transfer per unit length of tube?

Consider two long, slender rods of the same diameter but different materials. One end of each rod is attached to a base surface maintained at $100^{\circ} \mathrm{C}$, while the surfaces of the rods are exposed to ambient air at $20^{\circ} \mathrm{C}$. By traversing the length of each rod with a thermocouple, it was observed that the temperatures of the rods were equal at the positions $x_{\mathrm{A}}=0.15 \mathrm{m}$ and $x_{\mathrm{B}}=0.075 \mathrm{m}$, where x is measured from the base surface. If the thermal conductivity of rod A is known to be $k_{\mathrm{A}}=70 \mathrm{W} / \mathrm{m} \cdot \mathrm{K}$, determine the value of $k_{\mathrm{B}}$ for rod B.

A hair dryer may be idealized as a circular duct through which a small fan draws ambient air and within which the air is heated as it flows over a coiled electric resistance wire. (a) If a dryer is designed to operate with an electric power consumption of $P_{\mathrm{clec}}=500 \mathrm{W}$ and to heat air from an ambient temperature of $T_{i}=20^{\circ} \mathrm{C}$ to a discharge temperature of at what volumetric $T_{o}=45^{\circ} \mathrm{C}$, flow rate $\forall$ should the fan operate? Heat loss from the casing to the ambient air and the surroundings may be neglected. If the duct has a diameter of D=70 mm, what is the discharge velocity $V_{o}$ of the air? The density and specific heat of the air may be approximated as $\rho=1.10 \mathrm{kg} / \mathrm{m}^{3}$ and $c_{p}=1007\mathrm{J} / \mathrm{kg} \cdot \mathrm{K}$, respectively. (b) Consider a dryer duct length of L=150 mm and a surface emissivity of $\varepsilon=0.8$. If the coefficient associated with heat transfer by natural convection from the casing to the ambient air is h=4 $\mathrm{W} / \mathrm{m}^{2} \cdot \mathrm{K}$ and the temperature of the air and the surroundings is $T_{\infty}=T_{\mathrm{sur}}=20^{\circ} \mathrm{C}$, confirm that the heat loss from the casing is, in fact, negligible. The casing may be assumed to have an average surface temperature of $T_{\mathrm{s}}=40^{\circ} \mathrm{C}$.

Circular copper rods of diameter D=1 mm and length L=25 mm are used to enhance heat transfer from a surface that is maintained at $T_{s, 1}=100^{\circ} \mathrm{C}$.One end of the rod is attached to this surface (at x=0), while the other end (x=25 mm) is joined to a second surface, which is maintained at $T_{s, 2}=0^{\circ} \mathrm{C}$. Air flowing between the surfaces (and over the rods) is also at a temperature of $T_{\infty}=0^{\circ} \mathrm{C}$, and a convection coefficient of $h=100 \mathrm{W} / \mathrm{m}^{2} \cdot \mathrm{K}$ is maintained. (a) What is the rate of heat transfer by convection from a single copper rod to the air? (b) What is the total rate of heat transfer from a $1 \mathrm{m} \times 1 \mathrm{m}$ section of the surface at $100^{\circ} \mathrm{C}$, if a bundle of the rods is installed on 4-mm centers?

An isothermal silicon chip of width W=20 mm on a side is soldered to an aluminum heat sink $( k=180 \mathrm{W} / \mathrm{m} \cdot \mathrm{K})$ of equivalent width. The heat sink has a base thickness of $L_{h}=3 \mathrm{mm}$ and an array of rectangular fins, each of length $L_{f}=15 \mathrm{mm}$. Airflow at $T_{\infty}=20^{\circ} \mathrm{C}$ is maintained through channels formed by the fins and a cover plate, and for a convection coefficient of $h=100 \mathrm{W} / \mathrm{m}^{2} \cdot \mathrm{K}$, a minimum fin spacing of 1.8 mm is dictated by limitations on the flow pressure drop. The solder joint has a thermal resistance of $R_{t, c}^{\prime \prime}=2 \times 10^{-6} \mathrm{m}^{2} \cdot \mathrm{K} / \mathrm{W}$. (a) Consider limitations for which the array has N=11 fins, in which case values of the fin thickness t=0.182 mm and pitch S=1.982 mm are obtained from the requirements that W=(N-1)S+t and S-t=1.8 mm. If the maximum allowable chip temperature is $T_{c}=85^{\circ} \mathrm{C}$, what is the corresponding value of the chip power $q_{c} ?$ An adiabatic fin tip condition may be assumed, and airflow along the outer surfaces of the heat sink may be assumed to provide a convection coefficient equivalent to that associated with airflow through the channels. (b) With (S-t) and h fixed at 1.8 mm and 100 $\mathrm{W} / \mathrm{m}^{2} \cdot \mathrm{K}$, respectively, explore the effect of increasing the fin thickness by reducing the number of fins. With N=11 and S-t fixed at 1.8 mm, but relaxation of the constraint on the pressure drop, explore the effect of increasing the airflow, and hence the convection coefficient.

A thick oak wall, initially at $25^{\circ} \mathrm{C}$, is suddenly exposed to combustion products for which $T_{\infty}=800^{\circ} \mathrm{C}$ and $h=20 \mathrm{W} / \mathrm{m}^{2} \cdot \mathrm{K}$. (a) Determine the time of exposure required for the surface to reach the ignition temperature of $400^{\circ} \mathrm{C}$. (b) Plot the temperature distribution T(x) in the medium at t=325 s. The distribution should extend to a location for which $T \approx 25^{\circ} \mathrm{C}$.