2014 Intermediate Thermodynamics Midterm Exam  
What are your favourite days to have the thermodynamics midterm quiz?
Friday April 18th  15% of the voters. 9% of the votes.    9
Sunday April 20th  7% of the voters. 4% of the votes.    4
Monday April 21st  12% of the voters. 7% of the votes.    7
Tuesday April 22nd  5% of the voters. 3% of the votes.    3
Wednesday April 23rd  10% of the voters. 6% of the votes.    6
Thursday April 24th  22% of the voters. 13% of the votes.    13
Friday April 25th  39% of the voters. 22% of the votes.    23
Sunday April 27th  64% of the voters. 37% of the votes.    38
Total votes: 103. Total voters: 59.
Please select your favourite days to have the thermodynamics midterm quiz. You can select up to 3 days. Based on your votes we will choose an optimal time for the midterm next Monday. Please vote before next Monday :)
Midterm Quiz
April 27th 2014
19:00 — 21:00
Question #1
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 1D energy conservation in differential form corresponds to: $$ \frac{\partial}{\partial t} \rho \left( e + \frac{1}{2} v^2 + gy \right) + \frac{\partial}{\partial y} \rho v\left( h + \frac{1}{2} v^2 + gy \right) = \frac{\rho \delta Q}{m \Delta t}-\frac{\rho \delta W}{m \Delta t}$$
Question #2
Consider nitrogen gas (N$_2$) at room temperature and atmospheric pressure. Do the following:
(a)  Find the average number of N$_2$ molecules striking the container walls per second per square meter.
(b)  A 1 m$^3$ glass bulb contains N$_2$ gas at a temperature of $300$ K and at a pressure of 1 atmosphere. The glass bulb, which is to be used in conjunction with some other experiment, is itself enclosed in a large evacuated chamber. Unfortunately the glass bulb has, unknown to the experimenter, a small pinhole about $10^{-4}$ cm radius. To assess the importance of this hole, estimate the time required for 1% of the N$_2$ molecules to escape from the bulb into the surrounding vacuum.
Question #3
Consider air being heated as it flows through a constant-area duct. At the duct entrance, the air has a pressure of 2 bars, a temperature of 300 K and a speed of 90 m/s. At the duct exit, the air has a pressure of 1.5 bars and a speed of 350 m/s. Do the following:
(a)  Find the temperature of the air at the duct exit
(b)  Determine the heat transfer per unit mass of air flowing through the duct in J/kg.
Question #4
In a gas turbine engine, the combustion products are expanded to ambient pressure through a nozzle as follows:
The gas constant and the specific heat at constant pressure of the combustion products is of $415$ J/kgK and of $1800$ J/kgK respectively and can be taken as constant throughout the nozzle. The properties at the nozzle entrance are as follows: $P_1=10$ bar, $T_1=2000$ K, $v_1=40$ m/s. Knowing that friction induces a change in specific entropy between the nozzle exit and entrance of $s_2-s_1=264.6$ J/kgK and that the pressure at the nozzle exit corresponds to $P_2=1$ bar, do the following:
(a)  Find the polytropic coefficient $n$ for this process (recall the polytropic relationship $P/\rho^n={\rm constant}$).
(b)  Which common thermodynamic process (isentropic, adiabatic, reversible, isochoric, isobaric, etc) is closest to the process taking place in the nozzle?
(c)  For cross-sectional areas at the nozzle entrance and exit equal to $A_1=1$ m$^2$ and $A_2=0.9$ m$^2$ respectively, determine the flow speed at the nozzle exit.
2.  $3.17 \times 10^{27} ~{\rm particules/m^2 s}$; 284 days.
3.  875 K, 635 kJ/kg.
4.  1.2, isentropic, 303 m/s.
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