2014 Intermediate Thermodynamics Final Exam  
June 8th, 2014
18:00 — 21:00

Question #1
Starting from $\psi\equiv h-Ts$, $\phi\equiv e-Ts$ and the $T{\rm d}s$ equations, prove the following two Maxwell relations: $$ \left(\frac{\partial s}{\partial v}\right)_T=\left( \frac{\partial P}{\partial T}\right)_v ~~~{\rm and}~~~ \left(\frac{\partial s}{\partial P}\right)_T=-\left( \frac{\partial v}{\partial T}\right)_P$$
Question #2
A fixed amount of air initially at $25^\circ$C, 1 atm, and 50% relative humidity is cooled at constant pressure to 10$^\circ$C. Determine whether condensation occurs. Evaluate the amount of water condensed (kg of water condensed per kg of dry air), or if there is no condensation determine the relative humidity at the final state. Validate your answers by using the psychrometric chart.
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 rigid nonconducting tank with a volume of $120$ cubic meters is divided into two equal parts by a thin membrane. Hydrogen gas is contained on one side of the membrane at $3.5$ bar and 80$^\circ$C. The other side is a perfect vacuum. The membrane is suddenly ruptured, and the H$_2$ gas fills the tank following the polytropic process $P V^{1.2}={\rm constant}$. What is the entropy change of the hydrogen? Consider hydrogen to be a perfect gas ($R=4.124$ kJ/kgK, $c_p=14.307$ kJ/kgK, $c_v=10.183$ kJ/kgK).
Question #5
Hydrogen is mixed with oxygen and nitrogen and stored in a rigid container at a pressure of 1 atm and $25^\circ$C. A spark ignites the mixture and combustion occurs. After the chemical process has reached equilibrium, an analysis of the products reveals that they are composed of 1 kg of $\rm H_2 O$, 4 kg of $\rm N_2$, and no oxygen or hydrogen. Knowing that the volume of the container is not altered by the combustion process, determine the following:
(a)  the mass of hydrogen, oxygen, and nitrogen prior to combustion
(b)  the temperature of the products
(c)  the pressure of the products
(d)  the specific entropy change between the mixture of the products and the mixture of the reactants (in kJ/kgK)
Question #6
In a typical rocket engine, oxygen (O$_2$) is used to cool the walls of the engine before being mixed with the fuel in the combustion chamber. For this purpose, consider a duct where oxygen flows with a mass flow rate of 10 kg/s. At the entrance of the duct, the oxygen has a pressure of 152 bar and a temperature of 193 K. Knowing that the heat transfer rate to the oxygen flowing in the duct is of 1.23 MW, and that the duct is designed such that the heat addition process takes place isobarically, determine the temperature and pressure of the oxygen at the exit of the duct using (i) an ideal model, and (ii) the generalized correction charts.
2.  0.0022.
3.  382 K, 400 K.
4.  20.83 kJ/K.
5.  4 kg, 0.111 kg, 0.889 kg, 3360 K, 9.91 atm, 1.66 kJ/kgK.
6.  327 K, 152 bar, 262 K, 152 bar.
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