2013 Thermodynamics Midterm Exam  
Thermodynamics
Midterm Quiz
May 3rd 2013
18:30 — 20:30

NO NOTES OR BOOKS; USE THERMODYNAMICS TABLES THAT WERE DISTRIBUTED; ANSWER ALL 4 QUESTIONS; ALL QUESTIONS HAVE EQUAL VALUE.
05.04.14
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 totally enclosed cylinder is perfectly insulated thermally around its side and one end and contains a frictionless thermally non-conducting piston:
question2.jpg  ./download/file.php?id=1425&sid=ff9991c3d817964887ecf1bd3a7b34cc  ./download/file.php?id=1425&t=1&sid=ff9991c3d817964887ecf1bd3a7b34cc
The other end of the cylinder is perfectly conducting thermally. Initially, each side of the piston contains 0.0283 m$^3$ of air and 0.0283 m$^3$ of argon, respectively at 21$^\circ$C and $103$ kN/m$^2$. The argon side is completely insulated. If by adding heat to the air side the argon is compressed until its pressure reaches 206 kN/m$^2$:
(a)  What is the internal energy change of the argon?
(b)  How much heat was transferred to the air?
Question #3
Consider three water jets entering a mixing chamber as follows:
question03.png  ./download/file.php?id=1426&sid=ff9991c3d817964887ecf1bd3a7b34cc  ./download/file.php?id=1426&t=1&sid=ff9991c3d817964887ecf1bd3a7b34cc
The properties of the water jets entering the chamber correspond to: $$ \begin{array}{llll} \hline ~ & \rm Jet~1 & \rm Jet~2 & \rm Jet~3 \\ \hline \dot{m} & \rm 1~kg/s & \rm 2~kg/s & \rm 3~kg/s \\ T & \rm 300~K & \rm 310~K & \rm 330~K\\ P & \rm 1~atm & \rm 1~atm & \rm 1~atm \\ \hline \end{array} $$ The chamber is sufficiently long that the 3 water jets mix completely with each other. This results in the water exiting the chamber having uniform properties. Knowing that the mixing chamber loses heat to the environment at a rate of 200 kW, determine the following:
(a)  The final temperature of the mixed water
(b)  The rate of change in entropy of the water within the chamber in W/K (that is, find the difference between the entropy of the mixed water and the sum of the entropies of the incoming 3 water jets).
Question #4
Consider air entering a duct at station 1 and exiting at station 2, as follows:
question04.png  ./download/file.php?id=1427&sid=ff9991c3d817964887ecf1bd3a7b34cc  ./download/file.php?id=1427&t=1&sid=ff9991c3d817964887ecf1bd3a7b34cc
Knowing that the gravitational acceleration $g$ is 9.8 m/s$^2$, that the flow speed at the entrance $q_1$ is 150 m/s, that the cross-sectional areas $A_1$ and $A_2$ are equal to 1.0 m$^2$ and 1.2 m$^2$ respectively, that the height difference $\Delta y$ is equal to 200 m, that the temperature at the entrance is of 300 K, and that the pressure at the entrance and exit of the duct is equal to 1 atm and 1.05 atm respectively, do the following:
(a)  calculate the temperature at the exit, $T_2$
(b)  calculate the flow speed at the exit, $q_2$
Answers
2.  $1401~{\rm J}$, $13670~{\rm J}$.
3.  $310.4~{\rm K}$, $-614.5~{\rm W/K}$.
4.  $302~{\rm K}$, $121~{\rm m/s}$.
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