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Single-stage compressor power numerical

Short answer

Using pV^1.3 = C: indicated power = 16.4 kW; isothermal power = p1V1 ln(p2/p1) = 12.97 kW; isothermal efficiency = 79.2%. Delivery temperature = T1 (p2/p1)^((n‑1)/n) = 470 K (197 °C).

A single-stage reciprocating compressor takes in 4 m³/min of free air at 1 bar and 27 °C and delivers it at 7 bar. Compression follows pV1.3=CpV^{1.3} = C and the clearance is negligible. Calculate (a) the indicated power, (b) the isothermal power and the isothermal efficiency, and (c) the delivery temperature.

Given

p1=100p_1 = 100 kPa, V1=4V_1 = 4 m³/min =0.0667= 0.0667 m³/s, T1=300T_1 = 300 K, p2=700p_2 = 700 kPa, n=1.3n = 1.3.

p1V1=100×0.06667=6.667 kWp_1V_1 = 100 \times 0.06667 = 6.667\ \text{kW}

(a) Indicated (polytropic) power

Pressure-ratio factor: (p2/p1)n−1n=70.2308=1.5668(p_2/p_1)^{\frac{n-1}{n}} = 7^{0.2308} = 1.5668

P=nn−1 p1V1[(p2p1)n−1n−1]=1.30.3×6.667×(1.5668−1)=16.38 kW\begin{aligned} P &= \frac{n}{n-1}\,p_1V_1\left[\left(\frac{p_2}{p_1}\right)^{\frac{n-1}{n}} - 1\right]\\ &= \frac{1.3}{0.3} \times 6.667 \times (1.5668 - 1) = 16.38\ \text{kW} \end{aligned}

(b) Isothermal power and efficiency

Piso=p1V1ln⁡p2p1=6.667×ln⁡7=12.97 kWP_{iso} = p_1V_1 \ln\frac{p_2}{p_1} = 6.667 \times \ln 7 = 12.97\ \text{kW} ηiso=PisoP=12.9716.38=0.792\eta_{iso} = \frac{P_{iso}}{P} = \frac{12.97}{16.38} = 0.792

(c) Delivery temperature

T2=T1(p2p1)n−1n=300×1.5668=470 K=197 °CT_2 = T_1\left(\frac{p_2}{p_1}\right)^{\frac{n-1}{n}} = 300 \times 1.5668 = 470\ \text{K} = 197\ \text{°C}

Answer: (a) 16.4 kW; (b) 12.97 kW, ηiso\eta_{iso} = 79.2 %; (c) 470 K (about 197 °C).

✓ Answer checked by Claude Opus 5.5 (high effort)

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