Intermediate Thermodynamics Assignment 4 — Conservation of Energy  
$\xi$ is a parameter related to your student ID, with $\xi_1$ corresponding to the last digit, $\xi_2$ to the last two digits, $\xi_3$ to the last three digits, etc. For instance, if your ID is 199225962, then $\xi_1=2$, $\xi_2=62$, $\xi_3=962$, $\xi_4=5962$, etc. Keep a copy of the assignment — the assignment will not be handed back to you. You must be capable of remembering the solutions you hand in.
Question #1
Starting from Newton's law $\vec{F}=m\vec{a}$, the first law of thermo ${\rm d}(mh) - V {\rm d}P=\delta Q -\delta W$, and the mass conservation equation in differential form, show that the energy conservation in control volume form corresponds to: $$ \frac{{\rm d}}{{\rm d}t} \int_V \rho\left( e +\frac{1}{2} \vec{v}\cdot\vec{v}+g y \right){\rm d} V + \int_S \rho (\vec{v}\cdot\vec{n}) \left(h +\frac{1}{2} \vec{v}\cdot\vec{v}+gy \right){\rm d}S=\dot{Q}-\dot{W}$$ Note: this questions is worth double the points awarded to the other questions.
Question #2
Two adiabatic tanks are interconnected through a valve. Tank A contains $\rm 0.2~m^3$ of air at 40 bar and $\rm 90^\circ C$. Tank B contains $\rm 2~m^3$ of air at 1 bar and $\rm 30^\circ C$. The valve is opened until the pressure in A drops to $12.5$ bar. At this instant, determine (a) the temperatures and pressures in both tanks and (b) the amount of mass that has left tank A.
Question #3
A 4 m$^3$ storage tank (see schematic below) containing 2 m$^3$ of liquid is to be pressurized with air from a large, high-pressure reservoir through a valve at the top of the tank to permit rapid ejection of the liquid:
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The air in the reservoir is maintained at 100 bar and 300 K. The gas space above the liquid contains initially air at 1 bar and 280 K. When the pressure in the tank reaches 5 bar, the liquid transfer valve is opened and the liquid is ejected at the rate of 0.2 m$^3$/min while the tank pressure is maintained at 5 bar. What is the air temperature when the pressure reaches 5 bar and when the liquid has been drained completely? Hints: neglect heat interaction at the gas-liquid and gas-tank boundaries. It may be assumed that the gas above the liquid is well mixed and that air is a perfect gas.
Question #4
A tank containing 45 kg of liquid water initially at 45$^\circ$C has one inlet and one exit with equal mass flow rates. Liquid water enters at 45$^\circ$C and a mass flow rate of 270 kg/h. A cooling coil immersed in the water removes energy at a rate of $7.6$ kW. The water is well mixed by a paddle wheel so that the water temperature is uniform throughout. The power input to the water from the paddle wheel is 0.6 kW. The pressures at the inlet and exit are equal and all kinetic and potential energy effects can be ignored. Determine the variation of water temperature with time.
Question #5
A spaceship cabin may be considered to be a rigid pressurized vessel, which contains the atmosphere required to support the life of its occupants. A puncture in the cabin when it is in space must be detected quickly, so that the cabin occupants can seal the puncture. The time interval that elapses before a dangerously low pressure is reached is a critical quantity for designing the warning devices and countermeasure systems. Derive an equation giving an estimate for the pressure $P$ after a puncture as a function of the time $t$, the initial pressure $P_{\rm i}$, the initial temperature $T_{\rm i}$, the area $A$ of the puncture, and the volume $V$ of the cabin. Consider the atmosphere of the cabin to be a perfect gas. If the cabin atmosphere is air at an initial temperature of 300 K, determine the ratio $P/P_{\rm i}$ as a function of time with the ratio $(A/V)$ as a parameter. Assume that for air, $\gamma=1.4$ and $R=287.4$ J/kg K.
Question #6
Consider air being heated as it flows through a constant-area duct. At the duct entrance, the air has a pressure of 2 bars, a temperature of 300 K and a speed of 90 m/s. At the duct exit, the air has a pressure of 1.5 bars and a speed of 350 m/s. Do the following:
(a)  Find the temperature of the air at the duct exit
(b)  Determine the heat transfer per unit mass of air flowing through the duct in J/kg.
Question #7
Consider air entering a duct at station 1 and exiting at station 2, as follows:
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Knowing that the gravitational acceleration $g$ is 9.8 m/s$^2$, that the flow speed at the entrance $q_1$ is 150 m/s, that the cross-sectional areas $A_1$ and $A_2$ are equal to 1.0 m$^2$ and 1.2 m$^2$ respectively, that the height difference $\Delta y$ is equal to 200 m, that the temperature at the entrance is of 300 K, and that the pressure at the entrance and exit of the duct is equal to 1 atm and 1.05 atm respectively, do the following:
(a)  calculate the temperature at the exit, $T_2$
(b)  calculate the flow speed at the exit, $q_2$
2.  4.35 kg, 3.75 bar, 393 K.
3.  382 K, 400 K.
5.  $\displaystyle \frac{P}{P_{\rm i}}=\left( 1+\frac{35 A}{256 V} (\gamma-1)t\sqrt{3RT_{\rm i}}\right)^\frac{6-70\gamma}{35\gamma-35}$
6.  875 K, 635 kJ/kg
7.  302 K, 121 m/s.
Due on Wednesday April 10th at 16:30. Do Questions #3, #6, and #7 only.
There was a typo in the formulation of Question 2, and this is why the answers for Q2 seemed wrong. Check the problem formulation again.
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