Known Air executes a thermodynamic cycle consisting of three processes: Process 1–2, v = constant; Process 2–3, T = constant; Process 3–1, p = constant. Values are given for T_{1}, p_{1}, \text { and } p_{2}.
Find Sketch the cycle on p–v coordinates and determine T_{2} \text { and } v_{3}.
Schematic and Given Data:
Engineering Model
1. The air is a closed system.
1 2. The air behaves as an ideal gas.
3. The piston is the only work mode.
Analysis a. The cycle is shown on p–v coordinates in the accompanying figure. Note that since p = RT/v and temperature is constant, the variation of p with v for the process from 2 to 3 is nonlinear.
b. Using pv = RT, the temperature at state 2 is
T_{2}=p_{2} v_{2} / R
To obtain the specific volume v_{2} required by this relationship, note that v_{2}=v_{1}, so
2 v_{2}=R T_{1} / p_{1}
Combining these two results gives
T_{2}=\frac{p_{2}}{p_{1}} T_{1}=\left(\frac{2 atm }{1 atm }\right)\left(540^{\circ} R \right)=1080^{\circ} R
c. Since pv = RT, the specific volume at state 3 is
v_{3}=R T_{3} / p_{3}
Noting that T_{3}=T_{2}, p_{3}=p_{1}, \text { and } R=\bar{R} / M
v_{3}=\frac{\bar{R} T_{2}}{M p_{1}}
=\left(\frac{1545 \frac{ ft \cdot lbf }{ lbmol \cdot{ }^{\circ} R }}{28.97 \frac{ lb }{ lbmol }}\right) \frac{\left(1080^{\circ} R \right)}{\left(14.7 lbf / in ^{2}\right)}\left|\frac{1 ft ^{2}}{144 in ^{2}}\right|
=27.2 ft ^{3} / lb
where the molecular weight of air is from Table A-1E.
1 Table A-1E gives p_{ c }=37.2 atm , T_{ c }=239^{\circ} R for air. Therefore, p_{ R 2}=0.054, T_{ R 2}=4.52. Referring to Fig. A-1, the value of the compressibility factor at this state is Z \approx 1. The same conclusion results when states 1 and 3 are checked. Accordingly, pv = RT adequately describes the p–v–T relation for the air at these states.
2 Carefully note that the equation of state pv = RT requires the use of absolute temperature T and absolute pressure p.
Skills Developed
Ability to…
• evaluate p–v–T data using the ideal gas equation of state.
• sketch processes on a p–v diagram.
Quick Quiz
Is the cycle sketched in Fig. E3.8 a power cycle or a refrigeration cycle? Explain. Ans. A power cycle. As represented by enclosed area 1-2-3-1, the net work is positive.
