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Three-process gas cycle: net work, heat, type

Short answer

Net work = 10.5 (1-2) - 18.78 (2-3) = -8.28 kJ. Process 1-2 heat transfer is 36.9 kJ (added). Because net work is negative and cycle runs anticlockwise on P-v, it is a refrigerator cycle.

A Gas undergoes a thermodynamic cycle consisting of three process, Process 1-2 constant Pressure, P = 1.4 bars, v1 = 0.028 m³, w12 = 10.5 kJ Process 2-3, compression with Pv = constant, U3 = U2 Process 3-1, constant volume, U1-U3 = -26.4 kJ There are no significant change in kinetic and potential energy a) Sketch the system on a P-v diagram. b) Calculate Net work for a cycle, in kJ. c) Calculate the heat transfer of Process 1-2 in kJ d) Is this a Power cycle or a Refrigerator cycle?

Given: 1-2: constant P = 1.4 bar = 140 kPa, V₁ = 0.028 m³, W₁₂ = 10.5 kJ; 2-3: PV = constant (isothermal) with U₃ = U₂; 3-1: constant volume, U₁ − U₃ = −26.4 kJ.

Step 1: Volumes
W₁₂ = P (V₂ − V₁) ⇒ V₂ = V₁ + W₁₂/P = 0.028 + 10.5/140 = 0.103 m³
V₃ = V₁ = 0.028 m³ (3-1 is at constant volume)
P₃ = P₂ V₂/V₃ = 140 × 0.103/0.028 = 515 kPa

a) P-V diagram

 P (kPa)
 515 |3
     ||\
     || \    PV = C (2-3)
     ||  \
     ||   `-._
 140 |1--------2
     +---------------- V (m³)
     0.028     0.103

1 → 2 is a horizontal line to the right, 2 → 3 is a hyperbola going up and to the left, and 3 → 1 is a vertical line down. The cycle is traversed anticlockwise.

b) Net work

W₂₃ = P₂V₂ ln(V₃/V₂) = 140 × 0.103 × ln(0.028/0.103) = 14.42 × (−1.3025) = −18.78 kJ
W₃₁ = 0 (constant volume)

W_net = 10.5 − 18.78 + 0 = −8.28 kJ

c) Heat transfer in process 1-2

U₁ − U₃ = −26.4 ⇒ U₃ − U₁ = 26.4 kJ; with U₂ = U₃, U₂ − U₁ = 26.4 kJ
Q₁₂ = ΔU₁₂ + W₁₂ = 26.4 + 10.5 = 36.9 kJ (added)

(Check: Q₂₃ = W₂₃ = −18.78 kJ, Q₃₁ = −26.4 kJ; ΣQ = 36.9 − 18.78 − 26.4 = −8.28 kJ = W_net.)

d) Type of cycle

The net work is negative (work is supplied to the system) and the cycle runs anticlockwise on the P-V diagram, so it is a refrigerator (or heat pump) cycle, not a power cycle.

✓ Answer checked by Claude Opus 5.5 (high effort)

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