Consider the velocity and temperature profiles for airflow in a tube with diameter of 8cm can be expressed as

$$\begin{array}{l}{u(r)=0.2\left[\left(1-(r / R)^{2}\right]\right.} \\ {T(r)=250+200(r / R)^{3}}\end{array}$$

with units in m/s and K, respectively. If the convection heat transfer coefficient is $100 \mathrm{W} / \mathrm{m}^{2} \cdot \mathrm{K},$ determine the mass flow rate and surface heat flux using the given velocity and temperature profiles. Evaluate the air properties at $20^\circ C$ and 1 atm.

An ASTM B335 nickel alloy rod, with a diameter of $5 \mathrm{~mm}$ and a length of $10 \mathrm{~cm}$, is used to boil water at $1 \mathrm{~atm}$. The rod is immersed in the water horizontally, and it has an emissivity of $0.3$. The maximum use temperature for ASTM B335 nickel alloy rod is $427^{\circ} \mathrm{C}$ (ASME Code for Process Piping, ASME B31.3-2014). Find the highest rate of heat transfer that can be supplied from the ASTM B335 rod to boil the water without heating the rod surface above the maximum use temperature.

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.

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.

Consider a very long rectangular fin attached to a flat surface such that the temperature at the end of the fin is essentially that of the surrounding air, i.e. $20^{\circ} \mathrm{C}$. Its width is $5.0 \mathrm{~cm}$; thickness is $1.0 \mathrm{~mm}$; thermal conductivity is $200 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$; and base temperature is $40^{\circ} \mathrm{C}$. The heat transfer coefficient is $20 \mathrm{~W} / \mathrm{m}^{2}$. $K$. Estimate the fin temperature at a distance of $5.0 \mathrm{~cm}$ from the base and the rate of heat loss from the entire fin.

A finned-tube heat exchanger operates with water in the tubes and airflow across the fins. The water inlet temperature is $130^{\circ} \mathrm{F}$ while the incoming air temperature is $75^{\circ} \mathrm{F}$. An overall heat-transfer coefficient of $57 \mathrm{~W} / \mathrm{m}^2 \cdot{ }^{\circ} \mathrm{C}$ is experienced when the water-flow rate is $0.5 \mathrm{~kg} / \mathrm{s}$. The area of the exchanger is $52 \mathrm{~m}^2$ and the airflow is $2.0 \mathrm{~kg} / \mathrm{s}$ for the given conditions. Determine the exit water temperature, expressed in ${ }^{\circ} \mathrm{C}$.

A steam turbine operating at steady state develops 9750 hp. The turbine receives 100,000 pounds of steam per hour at $400 \text { lbf/in. }^{2}$ and $600^{\circ} \mathrm{F}.$ At a point in the turbine where the pressure is $60 \mathrm{lbf} / \mathrm{in} .^{2}$ and the temperature is $300^{\circ} \mathrm{F},$ steam is bled off at the rate of 25,000 lb/h. The remaining steam continues to expand through the turbine, exiting at $2 \mathrm{lbf} / \mathrm{in}^{2}$ and 90% quality. a. Determine the rate of heat transfer between the turbine and its surroundings, in Btu/h. b. Devise and evaluate an exergetic efficiency for the turbine. Kinetic and potential energy effects can be ignored. Let $T_{0}=77^{\circ} \mathrm{F}, p_{0}=1 \mathrm{atm}.$

The response x(t) of a system to a forcing function u(t) is determined by the differential equation model $\frac{\mathrm{d}^{2} x}{\mathrm{d} t^{2}}+2 \frac{\mathrm{d} x}{\mathrm{d} t}+5 x=3 \frac{\mathrm{d} u}{\mathrm{d} t}+2 u$ (a) Determine the transfer function characterizing the system. (b) Write down the characteristic equation of the system. What is the order of the system? (c) Determine the transfer function poles and zeros, and illustrate them diagrammatically in the s plane.

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.

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) $31.7^\circ C$

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.

A -395 kg/s of water is to be cooled from $90^{\circ} \mathrm{C}$ to $70^{\circ} \mathrm{C}$ in a shell and tube heat exchanger having four shell passes and eight tube passes. The cooling fluid is also water with a flow rate of $5 \mathrm{~kg} / \mathrm{s}$ that enters at a temperature of $50^{\circ} \mathrm{C}$. The overall heat-transfer coefficient is $800 \mathrm{~W} / \mathrm{m}^2,{ }^{\circ} \mathrm{C}$. Determine the total area of the heat exchanger assuming all four shells are the same size.

One lb of water contained in a piston-cylinder assembly, initially saturated vapor at 1 atm, is condensed at constant pressure to saturated liquid. Evaluate the heat transfer, in Btu, and the entropy production, in $\mathrm{B} \mathrm{tu} /^{\circ} \mathrm{R},$ for a. the water as the system. b. an enlarged system consisting of the water and enough of the nearby surroundings that heat transfer occurs only at the ambient temperature, $80^\circ F.$ Assume the state of the nearby surroundings does not change during the process of the water, and ignore kinetic and potential energy.

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.