2019 Intermediate Thermodynamics Final Exam  
Saturday June 15th 2019
16:00 — 19:00


Question #1
Starting from $\vec{F}=m\vec{a}$ applied on a gas particule, show that $$ P=\rho R T=N k_{\rm B} T= \frac{n k_{\rm B} T}{V} = \frac{\overline{R}T}{\overline{v}}$$ with $$ T\equiv \frac{m \overline{q^2}}{3 k_{\rm B}},~~~~R\equiv \frac{k_{\rm B}}{m},~~~~\overline{R}\equiv N_{\rm A} k_{\rm B}$$ Outline the definition of the pressure $P$, the density $\rho$, the number density $N$, and the molar specific volume $\overline{v}$.
Question #2
Consider a 1 m$^3$ tank in which air is contained in three different zones separated by membranes, as follows:
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Initially, the air within the three zones has the following properties: $$ \begin{array}{llll} \hline ~ & \rm Zone~A & \rm Zone~B & \rm Zone~C \\ \hline P & \rm 1~bar & \rm 2~bar & \rm 3~bar \\ T & \rm 300~K & \rm 300~K & \rm 300~K\\ V & \rm 0.2~m^3 & \rm 0.5~m^3 & \rm 0.3~m^3 \\ \hline \end{array} $$ The membranes are suddenly ruptured, mixing occurs between the zones, and after a large amount of time the properties of the air become uniform throughout the tank. Assuming no heat transfer from the air to the tank walls, calculate:
(a)  The final temperature and pressure of the mixed air
(b)  The change in entropy of the air within the tank in J/K (that is, find the difference between the entropy of the mixed air and the sum of the entropies of the air within the 3 zones)
Question #3
It is desired to cool 1300 kg/min of water from 40$^\circ$C to 26$^\circ$C using a cooling tower. The cooling tower receives 800 m$^3$/min of air at 1 bar with dry-bulb and wet-bulb temperatures of 29$^\circ$C and 21$^\circ$C, respectively. If the evaporation rate from the water stream is 1600 kg/hr, determine the temperature of the exit air stream.
Question #4
Consider a rigid insulated container containing a uniform mixture of H$_2$, O$_2$, and N$_2$ gases. It is known that the oxygen and hydrogen are in stoichiometric proportions and that the initial pressure and temperature of the mixture is of 1 atm and 500 K respectively. A spark ignites the mixture. After the combustion process is completed, the temperature is measured to be of 2000 K. Do the following:
(a)  Determine the ratio between the hydrogen mass and the oxygen mass before combustion occurs.
(b)  Determine the ratio between the nitrogen mass and the water vapor mass after the combustion is completed.
(c)  Determine the final pressure of the mixture.

Question #5
A gas flows in a vertical duct at steady state. The gas enters the duct with a velocity of 1 m/s, a temperature of 100 K, and a pressure of 1 bar. The duct cross-sectional area (perpendicular to the flow) is of 1 m$^2$ at the entrance and of 3 m$^2$ at the exit. The exit of the duct is located at an elevation 1000 m higher than the entrance. The pressure of the gas at the duct exit is of 0.5 bar. Heat is added/removed such that the temperature of the gas within the duct remains constant. The gas obeys the following equation of state: $$ P=\rho R T - \frac{a}{v} $$ with $a=8000$ J/kg and $R=300$ J/kgK. Do the following:
(a)  Find the velocity of the flow at the duct exit.
(b)  Find the amount of heat added to the flow in Watts assuming a calorically perfect gas model.
(c)  Find the amount of heat added to the flow in Watts using a real gas model.
Hints: you have to use the specified equation of state to find the enthalpies in (c). The gravitational acceleration constant is $g=9.81~$m/s$^2$.
Question #6
Three hundred kilograms (300 kg) of H$_2$O is put into a 1 m$^3$ tank. No substance other than H$_2$O is present within the tank. The tank rests for a very long time in a room where the temperature is of 25$^\circ$C. A valve is suddenly opened and the vapor within the tank gets slowly transferred into a large reservoir where the pressure is zero. While the valve is opened, 50 kg of water vapor is transferred from the tank to the reservoir. Knowing the latter, do the following:
(a)  Find the pressure inside the tank just prior to the valve being opened.
(b)  Find the mass of water vapor and the mass of liquid water inside the tank just prior to the valve being opened.
(c)  Find the mass of the water vapor and the mass of liquid water inside the tank after the mass transfer to the reservoir has been completed.
(d)  Find the temperature of the liquid and vapor inside the tank after the mass transfer is completed.
(e)  Find the pressure inside the tank after the mass transfer is completed.
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