Chapter 7

Solar radiation is incident on the glass cover of a solar collector at a rate of \(700 \mathrm{~W} / \mathrm{m}^{2}\). The glass transmits 88 percent of the incident radiation and has an emissivity of \(0.90\). The entire hot water needs of a family in summer can be met by two collectors \(1.2-\mathrm{m}\) high and \(1-\mathrm{m}\) wide. The two collectors are attached to each other on one side so that they appear like a single collector $1.2-\mathrm{m} \times 2-\mathrm{m}$ in size. The temperature of the glass cover is measured to be \(35^{\circ} \mathrm{C}\) on a day when the surrounding air temperature is \(25^{\circ} \mathrm{C}\) and the wind is blowing at $30 \mathrm{~km} / \mathrm{h}$. The effective sky temperature for radiation exchange between the glass cover and the open sky is \(-40^{\circ} \mathrm{C}\). Water enters the tubes attached to the absorber plate at a rate of $1 \mathrm{~kg} / \mathrm{min}$. Assuming the back surface of the absorber plate to be heavily insulated and the only heat loss to occur through the glass cover, determine \((a)\) the total rate of heat loss from the collector, \((b)\) the collector efficiency, which is the ratio of the amount of heat transferred to the water to the solar energy incident on the collector, and \((c)\) the temperature rise of water as it flows through the collector.

The outer surface of an engine is situated in a place where oil leakage can occur. When leaked oil comes in contact with a hot surface that has a temperature above its autoignition temperature, the oil can ignite spontaneously. Consider an engine cover that is made of a stainless steel plate with a thickness of \(1 \mathrm{~cm}\) and a thermal conductivity of $14 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$. The inner surface of the engine cover is exposed to hot air with a convection heat transfer coefficient of $7 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$ at a temperature of \(333^{\circ} \mathrm{C}\). The engine outer surface is cooled by air blowing in parallel over the \(2-\mathrm{m}\)-long surface at $7.1 \mathrm{~m} / \mathrm{s}\(, in an environment where the ambient air is at \)60^{\circ} \mathrm{C}$. To prevent fire hazard in the event of an oil leak on the engine cover, a layer of thermal barrier coating \((\mathrm{TBC})\) with a thermal conductivity of \(1.1 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) is applied on the engine cover outer surface. Would a TBC layer with a thickness of $4 \mathrm{~mm}\( in conjunction with \)7.1 \mathrm{~m} / \mathrm{s}$ air cooling be sufficient to keep the engine cover surface from going above $180^{\circ} \mathrm{C}$ to prevent fire hazard? Evaluate the air properties at \(120^{\circ} \mathrm{C}\).

Metal plates $\left(k=180 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \rho=2800 \mathrm{~kg} / \mathrm{m}^{3}\right.\(, \)1 \mathrm{~m}\( and a thickness of \)2 \mathrm{~cm}\( exiting an oven are then conveyed through a \)10-\mathrm{m}$-long cooling chamber at a speed of \(5 \mathrm{~mm} / \mathrm{s}\). The plates enter the cooling chamber at an initial temperature of \(155^{\circ} \mathrm{C}\). In the cooling chamber, the plates are cooled with \(10^{\circ} \mathrm{C}\) air blowing in parallel over them. To prevent thermal burns, it is necessary to design the cooling process so that the plates exit the cooling chamber at a relatively safe temperature. Determine the air velocity such that the temperature of the plates exiting the cooling chamber is $45^{\circ} \mathrm{C}$ or less. Assume combined laminar and turbulent flow (verify this assumption). Hint: Use lumped system analysis to determine the required cooling time (verify the application of this method to this problem).

Mercury at \(25^{\circ} \mathrm{C}\) flows over a 3 -m-long and 2 -m-wide flat plate maintained at \(75^{\circ} \mathrm{C}\) with a velocity of $0.01 \mathrm{~m} / \mathrm{s}$. Determine the rate of heat transfer from the entire plate.

Liquid mercury at \(250^{\circ} \mathrm{C}\) is flowing in parallel over a flat plate at a velocity of \(0.3 \mathrm{~m} / \mathrm{s}\). Surface temperature of the \(0.1\)-m-long flat plate is constant at \(50^{\circ} \mathrm{C}\). Determine \((a)\) the local convection heat transfer coefficient at \(5 \mathrm{~cm}\) from the leading edge and \((b)\) the average convection heat transfer coefficient over the entire plate.

Water vapor at \(250^{\circ} \mathrm{C}\) is flowing with a velocity of $5 \mathrm{~m} / \mathrm{s}\( in parallel over a \)2-\mathrm{m}$-long flat plate where there is an unheated starting length of \(0.5 \mathrm{~m}\). The heated section of the flat plate is maintained at a constant temperature of \(50^{\circ} \mathrm{C}\). Determine \((a)\) the local convection heat transfer coefficient at the trailing edge, \((b)\) the average convection heat transfer coefficient for the heated section, and \((c)\) the rate of heat transfer per unit width for the heated section.

A \(15-\mathrm{mm} \times 15-\mathrm{mm}\) silicon chip is mounted such that the edges are flush in a substrate. The chip dissipates \(1.4 \mathrm{~W}\) of power uniformly, while air at \(20^{\circ} \mathrm{C}\) ( $\left.1 \mathrm{~atm}\right)\( with a velocity of \)25 \mathrm{~m} / \mathrm{s}$ is used to cool the upper surface of the chip. If the substrate provides an unheated starting length of \(15 \mathrm{~mm}\), determine the surface temperature at the trailing edge of the chip. Evaluate the air properties at $50^{\circ} \mathrm{C}$.

In cryogenic equipment, cold gas flows in parallel over the surface of a \(1-\mathrm{m}\)-long plate. The gas velocity is \(4 \mathrm{~m} / \mathrm{s}\) at a temperature of \(-70^{\circ} \mathrm{C}\). The gas has a thermal conductivity of \(0.01979 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\), a kinematic viscosity of \(9.319 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}\), and a Prandtl number of \(0.7440\). A stainless steel (ASTM A437 B4B) bolt is embedded in the plate at midlength. The minimum temperature suitable for the ASTM A437 B4B stainless steel bolt is \(-30^{\circ} \mathrm{C}\) (ASME Code for Process Piping, ASME B31.3-2014, Table A-2M). To keep the bolt from getting below its minimum suitable temperature, the plate is subjected to a uniform heat flux of $250 \mathrm{~W} / \mathrm{m}^{2}$. Determine whether the heat flux to the plate is sufficient to keep the bolt above the minimum suitable temperature of \(-30^{\circ} \mathrm{C}\).

Air is flowing in parallel over the upper surface of a flat plate with a length of \(4 \mathrm{~m}\). The first half of the plate length, from the leading edge, has a constant surface temperature of \(50^{\circ} \mathrm{C}\). The second half of the plate length is subjected to a uniform heat flux of $86 \mathrm{~W} / \mathrm{m}^{2}$. The air has a free-stream velocity and temperature of \(2 \mathrm{~m} / \mathrm{s}\) and \(10^{\circ} \mathrm{C}\), respectively. Determine the local convection heat transfer coefficients at $1 \mathrm{~m}\( and \)3 \mathrm{~m}$ from the leading edge. As a first approximation, assume the boundary layer over the second portion of the plate with uniform heat flux has not been affected by the first half of the plate with constant surface temperature. Evaluate the air properties at a film temperature of \(30^{\circ} \mathrm{C}\). Is the film temperature \(T_{f}=30^{\circ} \mathrm{C}\) applicable at \(x=3 \mathrm{~m}\) ?

Hot gas flows in parallel over the upper surface of a \(2-\mathrm{m}\)-long plate. The velocity of the gas is \(17 \mathrm{~m} / \mathrm{s}\) at a temperature of \(250^{\circ} \mathrm{C}\). The gas has a thermal conductivity of \(0.03779 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\), a kinematic viscosity of \(3.455 \times 10^{-5} \mathrm{~m}^{2} / \mathrm{s}\), and a Prandtl number of \(0.6974\). Two copper-silicon (ASTM B98) bolts are embedded in the plate: the first bolt at \(0.5 \mathrm{~m}\) from the leading edge, and the second bolt at \(1.5 \mathrm{~m}\) from the leading edge. The maximum use temperature for the ASTM B98 copper-silicon bolt is \(149^{\circ} \mathrm{C}\) (ASME Code for Process Piping, ASME B31.3-2014, Table A-2M). A cooling device removes the heat from the plate uniformly at \(2000 \mathrm{~W} / \mathrm{m}^{2}\). Determine whether the heat being removed from the plate is sufficient to keep the bolts below the maximum use temperature of \(149^{\circ} \mathrm{C}\).