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 $(15-\xi_1)$ 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 $(7.6+\xi_1)$ 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.
Due on April 9 2014
This assignment is the hardest along with Assignment #8. It's natural that you cannot find the solution to the problems easily. When you encounter a hard problem that you cannot solve at first, it's very important not to quit and to make an extra effort in order to somehow come up with some solution that is reasonable. When in doubt, ask a question on my website or come see me in my office for some hints. Such will help you develop good engineering skills.
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