Question

A piston-cylinder apparatus contains 4 kg of saturated liquid water at 35˚C. The piston’s cross sectional area is 0.06 m2 , and initially it is resting on stops, enclosing a volume of 0.03 m3 . To raise the piston, a pressure of 300kPa has to be applied. Heat is transferred until the volume is 0.075 m3 , at which point the piston encounters a linear spring. To deflect the spring 1 m, a force of 360 kN is required. More heat is transferred until the final pressure is 7 MPa. How much work is done in the process? How much heat is required?

Answer 1

W = W1 + W2

P1 = 300000 Pa

V1 = 0.03 m³

V2 = 0.075 m³

W1 = P1* (V2-V1) = 300000 * (0.075 - 0.03) = 13500 J

For spring:

P2 = 7 x 10⁶ Pa

A = 0.06 m²

k_s = 360000 N/m

Δx = (P2 - P1) * A / k_s = 7 x 10⁶ * 0.06 / 360000 = 1.12 m

W2 = 1/2 * ks * Δx² = 224450 J

Total work done = W1 + W2 = 13500 + 224450 = 237950 J

From steam table for saturated liquid water:

Initial intenal energy, U1 = 146.67 * 1000 * 4 = 586680 J

Final specific volume = (V2 + A * Δx)/M = (0.075 + 0.06 * 1.12)/4 = 0.0356 m³/kg

Final temperature = 270 ^oC (from steam table for the calculated saturated vapor specific volume)

Final intenal energy, U2 = 2593.7 * 1000 * 4 = 10374800 J

ΔU = U2 - U1 = 10374800 - 586680 = 9788120 J

Q = ΔU + W

Total heat required, Q = 9788120 J + 237950 J = 10026070 J