2012 Intermediate Thermodynamics Final Exam (including solutions)  
Final Examination
June 21st, 2012
9:30 — 12:30

Question #1
For a system composed of $k=1..n$ bodies each with a temperature $T_k$, and after defining the entropy as $$ {\rm d} S_k\equiv \frac{\delta Q}{T_k} + \frac{\delta W}{T_k}$$ prove that, when $\delta W =0$, the entropy of the system always increases: $$ \sum_{k=1}^n {\rm d} S_k \ge 0$$
Question #2
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 #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 1786 kg/hr, determine the temperature of the exit air stream.
Question #4
Nitrogen ($T_{\rm c}=126.2$ K, $P_{\rm c}=3.4$ MPa) expands isothermally from 125$^\circ$C and 8.5 MPa to $3.5$ MPa. Determine the change in internal energy, enthalpy, and entropy of the gas, assuming that the nitrogen obeys the Van der Waals' equation of state. Compare your results with the values obtained assuming ideal-gas behavior.
Question #5
Starting from the Berthelot equation of state for a non-ideal gas: $$ P=\frac{\overline{R}T}{\overline{v}-b}-\frac{a}{T \overline{v}^2}$$ prove that: $$ a=\frac{27}{64} \frac{\overline{R}^2 T_{\rm c}^3}{P_{\rm c}}~~~{\rm and}~~~ b=\frac{1}{8} \frac{\overline{R} T_{\rm c}}{P_{\rm c}}$$
Question #6
Consider air and molecular hydrogen (H$_2$) in gaseous form entering the combustor of a ramjet engine.
(a)  Determine the air-fuel ratio on both a mass and a molar basis for complete combustion of hydrogen with the theoretical amount of air (stoichiometric air-fuel ratio).
(b)  At a given moment, the air enters the combustor at a velocity of 200 m/s and a temperature of 800 K, while the hydrogen is injected in the combustor at negligible velocity with a temperature of 298 K and in stoichiometric proportions:
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Complete combustion of the fuel then occurs in the combustor. Knowing that the temperature of the engine exhaust flow is of 600 K, find the velocity of the exhaust flow. Note: the engine exhaust flow has the same composition as the flow exiting the combustor — it consists of the combustion products of air and hydrogen.
2.  $\displaystyle \frac{P}{P_{\rm i}}=\left( 1+\frac{35 A}{256 V} (\gamma-1)t\sqrt{3RT_{\rm i}}\right)^\frac{6-70\gamma}{35\gamma-35}$
3.  305 K.
4.  $e_2-e_1= 7309~{\rm J/kg}$, $h_2-h_1=6613~{\rm J/kg}$, $s_2-s_1= 280.2~{\rm J/kgK}$, $(e_2-e_1)_{\rm perfect}= 0$, $(h_2-h_1)_{\rm perfect}= 0$, $(s_2-s_1)_{\rm perfect}= 263.5~{\rm J/kgK}$
5.  $a=27 \bar{R}^2 T_{\rm c}^3/(64 P_{\rm c})$, $b=\bar{R} T_{\rm c}/(8 P_{\rm c})$.
6b.  $2611~{\rm m/s}$.
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