Question 2.6: A cylinder contains 2.4 kg of a certain fluid at an initial ......
A cylinder contains 2.4 kg of a certain fluid at an initial pressure of 16 bar. The fluid can expand reversibly behind a piston according to a law pV^{2.6} = constant until the volume is 0.12 m³. The fluid is then cooled reversibly at constant pressure until the piston regains its original position; heat is then supplied reversibly with the piston firmly locked in position until the pressure rises to the original value of 16 bar. Calculate the net work done by the fluid, for an initial volume of 0.04 m³.
Given m = 2.4 kg
p_1 = 18 bar = 18 × 101.325 kPa
V_1 = 0.04 m^3; V_2 = 0.12 m^3
law: pV^{2.6} = c
During the process 1–2:
p_1V^{2.6}_1 = p_2V^{2.6}_2 → p_2 = 1.035 × 101.325 kPa
Work done, W_{12} = \int\limits_{1}^{2}{pdV} = \int\limits_{0.04}^{0.12}{\frac{c}{V^2}dV } = 6.755 kJ/kg
During the process 2–3:
Work done, W_{23} = \int\limits_{2}^{3}{pdV} = 0.083 \times 101.325 = 8.41 kJ/kg
During the process 3–1:
Work done, W_{31} = 0
Total work equals \sum{W} = m(W_{12} + W_{23} + W_{31}) = 2.4(6.755 – 8.41) = –3.972 kJ.
Note: W_{12} is negative because of compression work, work is being done on the system.