Question

A system consisting of 1 kg of an ideal gas at 5 bar pressure and 0.02 m³ volume executes a cyclic process comprising the following three distinct operations : (i) Reversible expansion to 0.08 m³ volume, 1.5 bar pressure, presuming pressure to be a linear function of volume (p = a + bV), (ii) Reversible cooling at constant pressure and (iii) Reversible hyperbolic compression according to law pV = constant. This brings the gas back to initial conditions. 

(i) Sketch the cycle on p-V diagram. 

(ii) Calculate the work done in each process starting whether it is done on or by the system and evaluate the net cyclic work and heat transfer.

Answer 1

Given : m = 1 kg ; p₁ = 5 bar ; 

V₁ = 0.02 m³ ; 

V₂ = 0.08 m³ ; 

p₂ = 1.5 bar. 

(i) p-V diagram : p-V diagram of the cycle is shown in 

(ii) Work done and heat transfer :

Process 1-2 (Linear law) : 

p = a + bV

The values of constants a and b can be determined from the values of pressure and volume at the state points 1 and 2.

5 = a + 0.02b ...(i) 

1.5 = a + 0.08b ...(ii)

From (i) and (ii) we get, b = – 58.33 and a = 6.167

This is work done by the system.

Process 2 – 3 (constant pressure) : 

p₃ = p₂ = 1.5 bar

The volume V₃ can be worked out from the hyperbolic compression 3–1, as follows :

Process 3 – 1 (hyperbolic process) :

This is the work done on the system

Net work done, W_net = W_1–2 + W_2–3 + W_3–1 

= 19.5 + (– 1.995) + (– 12.05) = 5.445 kJ.