Thermodynamics

172 | Thermodynamicsthe gas, and (c) the fraction of this work done against the spring tocompress it.Solution A gas in a piston–cylinder device equipped with a linear springexpands as a result of heating. The final gas pressure, the total work done, andthe fraction of the work done to compress the spring are to be determined.Assumptions 1 The expansion process is quasi-equilibrium. 2 The spring islinear in the range of interest.Analysis A sketch of the system and the P-V diagram of the process areshown in Fig. 4–10.(a) The enclosed volume at the final state isThen the displacement of the piston (and of the spring) becomesThe force applied by the linear spring at the final state isThe additional pressure applied by the spring on the gas at this state isWithout the spring, the pressure of the gas would remain constant at200 kPa while the piston is rising. But under the effect of the spring, thepressure rises linearly from 200 kPa toat the final state.(b) An easy way of finding the work done is to plot the process on aP-V diagram and find the area under the process curve. From Fig. 4–10 thearea under the process curve (a trapezoid) is determined to beW area V 2 2V 1 12210.05 m 3 2 0.1 m 3x ¢VAF kx 1150 kN>m210.2 m2 30 kNP F A10.1 0.052 m30.25 m 2 0.2 m30 kN 120 kPa20.25 m200 120 320 kPa1200 3202 kPa310.1 0.052 m 3 1 kJ4a b 13 kJ21 kPa # m3k = 150 kN/mP, kPa320II200FIGURE 4–10Schematic and P-V diagram forExample 4–4.A = 0.25 m 2P 1 = 200 kPaV 1 = 0.05 m 3Heat0.05I0.1V, m 3