2013 Thermodynamics Final Exam  
June 9th, 2013
14:00 — 17:00

Question #1
If the $PvT$ behavior of a gas is given by the Berthelot equation of state, show that the change in enthalpy of the gas during an isothermal process from state 1 to state 2 can be written as: $$ \bar{h}_2-\bar{h}_1=\frac{3a}{T}\left(\frac{1}{\bar{v}_1}-\frac{1}{\bar{v}_2} \right)+ \bar{R} T \left(\frac{\bar{v}_2}{\bar{v}_2-b}-\frac{\bar{v}_1}{\bar{v}_1-b} \right)$$
Question #2
An air stream at 34$^\circ$C with a wet-bulb temperature of 15$^\circ$C and a mass flow rate of 1 kg/s passes through an evaporative cooler until the temperature is 21$^\circ$C. It is then heated to 30$^\circ$C. Determine the amount of liquid water added in the first section if the liquid water temperature is of $30^\circ$C. Also determine the heat added in the second section of the equipment.
Question #3
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: $$ \frac{{\rm d}}{{\rm d}t} \int_V \rho\left( e +\frac{1}{2} \vec{v}\cdot\vec{v}+g y \right){\rm d} V + \int_S \rho (\vec{v}\cdot\vec{n}) \left(h +\frac{1}{2} \vec{v}\cdot\vec{v}+gy \right){\rm d}S=\dot{Q}-\dot{W}$$
Question #4
Octane (C$_{8}$H$_{18}$) is expanded isentropically from a pressure of 124.5 bars and a temperature of 683 K to a pressure of 49.8 bars. Determine the temperature after the expansion process using (i) an ideal model, and (ii) the generalized correction charts.
Question #5
A stream of air from indoors ($T_{\rm dry}=20^\circ$C, $T_{\rm wet}=10^\circ$C) is mixed with a stream of air from outdoors ($T_{\rm dry}=30^\circ$C, $T_{\rm wet}=10.5^\circ$C). Knowing that the mass flow rate of the stream from indoors is equal to the one of the stream from outdoors, find the dry bulk and wet bulb temperatures of the mixed air. The mixed air is then diverted into a humidifier where liquid water at 20$^\circ$C is injected in the air and vaporized. Determine the maximum mass flow rate of liquid water (per mass flow rate of air) that can be vaporized in the mixed air and determine the resulting dry bulk temperature and wet bulb temperature.
Question #6
Consider a mixture of O$_2$ and H$_2$ in gaseous form at 1 atm and 300 K in a closed container, separated by a piston from dry air at 300 K and 1 atm:
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Before combustion takes place, the volume of the O$_2$-H$_2$ mixture is of 0.1 m$^3$ while the volume of the air is of 0.2 m$^3$. Knowing that the oxygen and hydrogen are in stoichiometric conditions, determine the pressure and temperature of the combustion products after a spark ignites the mixture (you can assume that the piston does not move during the combustion process). After the combustion has taken place, the piston moves and compresses the air on its right. Determine the final temperature and pressure of the air on the right of the piston assuming that throughout the combustion and compression processes, no heat transfer takes place through the piston or with the surroundings.
2.  $0.0054~{\rm kg/s}$, $9.11~{\rm kW}$.
4.  $526~{\rm K}$, $609~{\rm K}$.
5.  $25^\circ{\rm C}$, $10.2^\circ{\rm C}$, $10.3^\circ{\rm C}$, $10.3^\circ{\rm C}$, $0.006$.
6.  $9841~{\rm K}$, $22~{\rm atm}$, $6.24~{\rm atm}$, $506~{\rm K}$.
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