Water at $0^{\circ} \mathrm{C}$ releases $333.7 \mathrm{~kJ} / \mathrm{kg}$ of heat as it freezes to ice $\left(\rho=920 \mathrm{~kg} / \mathrm{m}^3\right)$ at $0^{\circ} \mathrm{C}$. An aircraft flying under icing conditions carries a heat transfer coefficient of $150 \mathrm{~W} / \mathrm{m}^2 \cdot \mathrm{K}$ between the air and wing surfaces. What temperature must the wings be kept at to prevent ice from forming on them during icing conditions at a rate of $1 \mathrm{~mm} / \mathrm{min}$ or less?

Combustion gases passing through a $5$-cm-internaldiameter circular tube are used to vaporize wastewater at atmospheric pressure. Hot gases enter the tube at $115\ \mathrm{kPa}$ and $250^{\circ} \mathrm{C}$ at a mean velocity of $5 \mathrm{~m} / \mathrm{s}$ and leave at $150^{\circ} \mathrm{C}$. If the average heat transfer coefficient is $120 \mathrm{~W} / \mathrm{m}^2 \cdot \mathrm{K}$ and the inner surface temperature of the tube is $110^{\circ} \mathrm{C}$, find (a) the tube length and (b) the rate of evaporation of water. Use air properties for the combustion gases.

An air-cooled condenser is used to condense isobutane in a binary geothermal power plant. The isobutane is condensed at $85^\circ C$ by air $(c_p = 1.0 kJ/kg \cdot K)$ that enters at $22^\circ C$ at a rate of 18 kg/s. The overall heat transfer coefficient and the surface area for this heat exchanger are $2.4 kW/m^2 \cdot K$ and $1.25 m^2,$ respectively. The outlet temperature of air is (a) $45.4^\circ C$(b) $40.9^\circ C$ (c) $37.5^\circ C$ (d) $34.2^\circ C$ (e) $31.7^\circ C$

Hot water flows through a PVC $(k=0.092 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})$ pipe whose inner diameter is $2 \mathrm{~cm}$ and whose outer diameter is $2.5 \mathrm{~cm}$. The temperature of the interior surface of this pipe is $35^{\circ} \mathrm{C}$, and the temperature of the exterior surface is $20^{\circ} \mathrm{C}$. The rate of heat transfer per unit of pipe length is (a) $22.8 \mathrm{~W} / \mathrm{m}$ (b) $38.9 \mathrm{~W} / \mathrm{m}$ $(c)$ $48.7 \mathrm{~W} / \mathrm{m}$ (d) $63.6 \mathrm{~W} / \mathrm{m}$ (e) $72.6 \mathrm{~W} / \mathrm{m}$

Water is boiled at $150^\circ C$ in a boiler by hot exhaust gases $(c_p = 1.05 kJ/kg \cdot ^\circ C)$ that enter the boiler at $400^\circ C$ at a rate of 0.4 kg/s and leaves at $200^\circ C.$ The surface area of the heat exchanger is $0.64 m^2.$ The overall heat transfer coefficient of this heat exchanger is (a) $940 W/m^2 \cdot K$ (b) $1056 W/m^2 \cdot K$ (c) $1145 W/m^2 \cdot K$ (d) $1230 W/m^2 \cdot K$ (e) $1393 W/m^2 \cdot K$

Air with a mass flow rate of $5\ \mathrm{lb} / \mathrm{s}$ enters a horizontal nozzle operating at steady state at $800^{\circ} \mathrm{R}, 50\ \mathrm{lbf} / \mathrm{in}^2$ and a velocity of $10\ \mathrm{ft} / \mathrm{s}$. At the exit, the temperature is $570^{\circ} \mathrm{R}$ and the velocity is $1510\ \mathrm{ft} / \mathrm{s}$. Using the ideal gas model for air, determine

(a) the area at the inlet, in $\mathrm{ft}^2$, and

(b) the heat transfer between the nozzle and its surroundings, in Btu per lb of air flowing.

Water is the working fluid in an ideal Rankine cycle. Superheated vapor enters the turbine at 10 MPa, $480^\circ C,$ and the condenser pressure is 6 kPa. Isentropic efficiencies of the turbine and pump are 84% and 73%, respectively. Determine for the cycle a. the heat transfer to the working fluid passing through the steam generator, in kJ per kg of steam flowing. b. the thermal efficiency. c. the heat transfer from the working fluid passing through the condenser to the cooling water, in kJ per kg of steam flowing.

Consider an array of vertical rectangular fins which is to be used to cool an electronic device mounted in quiescent, atmospheric air at $T_{\mathrm{x}}=27^{\circ} \mathrm{C}$. Each fin has $L=20 \mathrm{~mm}$ and $H=150 \mathrm{~mm}$ and operates at an approximately uniform temperature of $T_s=77^{\circ} \mathrm{C}$. Viewing each fin surface as a vertical plate in an infinite, quiescent medium, calculate the rate of heat transfer from a fin by free convection. Comment on the effect of boundary layer formation on specifying the spacing between fins.

The general form of the Stefan-Maxwell equation for mass transfer of species i in an n-component ideal gas mixture along the z-direction for flux is given by

$$\frac{d y_{i}}{d z}=\sum_{j=1, j \neq i}^{n} \frac{y_{i} y_{j}}{D_{i j}}\left(\nu_{j}-\nu_{i}\right)$$

Show that for a gas-phase binary mixture of species A and B, this relationship is equivalent to Fick's rate equation:

$$N_{A, z}=-c D_{A B} \frac{d y_{A}}{d z}+y_{A}\left(N_{A, z}+N_{B, z}\right)$$

Oil in an engine is being cooled by air in a cross- flow heat exchanger, where both fluids are unmixed. Oil $\left(c_{p h}=2047 \mathrm{J} / \mathrm{kg} \cdot \mathrm{K}\right)$ flowing with a flow rate of 0.026 kg/s enters the tube side at $75^{\circ} \mathrm{C},$ while air $\left(c_{p c}=1007 \mathrm{J} / \mathrm{kg} \cdot \mathrm{K}\right)$ enters the shell side at $30^{\circ} \mathrm{C}$ with a flow rate of 0.21 kg/s. The overall heat transfer coefficient of the heat exchanger is $53 \mathrm{W} / \mathrm{m}^{2} \cdot \mathrm{K}$ and the total surface area is $1 m^2.$ If the correction factor is F = 0.96, determine the outlet temperatures of the oil and air.

A preheater involves the use of condensing steam at $100^{\circ} \mathrm{C}$ on the inside of a bank of tubes to heat air that enters at 1 atm and $25^{\circ} \mathrm{C}$. The air moves at 5 m/s in cross flow over the tubes. Each tube is 1 m long and has an outside diameter of 10 mm. The bank consists of 196 tubes in a square, aligned array for which $S_{T}=S_{L}=15 \mathrm{mm}$. What is the total rate of heat transfer to the air? What is the pressure drop associated with the airflow?

The convection heat transfer coefficient for a clothed person standing in moving air is expressed as $h=14.8 \mathrm{~V}^{0.69}$ for $0.15<V<1.5 \mathrm{~m} / \mathrm{s}$, where $V$ is the air velocity. For a person with a body surface area of $1.7 \mathrm{~m}^{2}$ and an average surface temperature of $29^{\circ} \mathrm{C}$, Find the rate of heat loss from the person in windy air at $10^{\circ} \mathrm{C}$ by convection for air velocities of $(a) 0.5 \mathrm{~m} / \mathrm{s}$, (b) $1.0 \mathrm{~m} / \mathrm{s}$, and $(c) 1.5 \mathrm{~m} / \mathrm{s}$.

Air is compressed in an axial-flow compressor operating at steady state from $27^{\circ} \mathrm{C},$ 1 bar to a pressure of 2.1 bar. The work required is 94.6 kJ per kg of air flowing. Heat transfer from the compressor occurs at an average surface temperature of $40^{\circ} \mathrm{C}$ at the rate of 14 kJ per kg of air flowing. The effects of motion and gravity can be ignored. Let $T_{0}=20^{\circ} \mathrm{C}, p_{0}=1$ bar. Assuming ideal gas behavior, (a) determine the temperature of the air at the exit, in $^\circ C,$ (b) determine the rate of exergy destruction within the compressor, in kJ per kg of air flowing, and (c) perform a full exergy accounting, in kJ per kg of air flowing, based on work input.

Nitrogen $(N_2)$ initially at $40^{\circ} \mathrm{F} \text { and } 14.7 \mathrm{lbf} / \mathrm{in}^{2}$ fills a closed, rigid, $6.5-ft^3$ tank fitted with a paddle wheel. During a process, the paddle wheel provides 5 Btu of energy transfer by work to the gas. The gas temperature reaches $80^\circ F$ at the end of the process. Assuming ideal gas behavior, determine the mass of nitrogen, in lb, and the heat transfer, in Btu. Kinetic and potential energy effects can be ignored.