OpenSpecEngineering

Finds the discharge temperature, the specific work and the power to compress a near-ideal gas in one stage, along an isentropic or a polytropic path. Process and rotating-equipment engineers use it for a first estimate of compressor size and driver rating, before a vendor’s selection.

How the power is calculated

The work per kilogram of gas along a path with exponent n is [1, 2]

w = [n/(n − 1)]·(Z·R·T₁/M)·[(P₂/P₁)^(n−1)/n − 1]
T₂ = T₁·(P₂/P₁)^(n−1)/n

where P₁ and P₂ are the absolute suction and discharge pressures, T₁ the absolute suction temperature, R = 8.314 J/(mol·K) the gas constant, M the molecular weight and Z the average compressibility factor. For the isentropic path n = k = c_p/c_v; for the polytropic path you enter n. The gas power is w times the mass flow, and the shaft power is the gas power divided by the efficiency: isentropic efficiency with the isentropic path, polytropic efficiency with the polytropic path [3].

Picking a gas fills its molecular weight and its ideal-gas k at 27 °C (300 K) [4]. Enter Z = 1 for an ideal gas.

Limits and common mistakes

• Enter absolute pressures, not gauge.
• On the isentropic path, T₂ is the ideal discharge temperature. The real gas leaves hotter, at about T₁ + (T₂ − T₁)/η. On the polytropic path, n already includes the losses, so T₂ is the real one.
• k falls as most gases get hotter. For hot suction or a large temperature rise, enter k at the mean temperature.
• Z is an average over the path. Near the critical point or at high pressure, the ideal-gas form loses accuracy.
• This is one stage. With a high ratio (above about 3 to 4) or a hot discharge, split the job into stages with intercooling and work each one.
• Shaft power excludes mechanical losses in bearings, seals and gearboxes. Add them before sizing the driver.

Worked example

A booster compressor takes 20,000 kg/h of methane at 2000 kPa(a) and 30 °C and delivers it at 5000 kPa(a). Take Z = 0.96 and an isentropic efficiency of 78 %. What shaft power is needed?

1. Choose Compression Path Isentropic.
2. Enter Suction Pressure 2000 kPa, Discharge Pressure 5000 kPa and Suction Temperature 30 °C.
3. Choose Gas Methane (CH₄), which fills k = 1.299 and M = 16.043.
4. Enter Average Compressibility Factor 0.96, Mass Flow 20000 and Efficiency 78.
5. Press Solve. The Solution shows: Pressure ratio P2/P1: 2.5
   Discharge temperature T2: 101.18 °C
   Specific work w: 153.856 kJ/kg
   Gas power (ideal path): 854.754 kW
   Shaft power (with efficiency): 1,095.84 kW

Check by hand. (k − 1)/k = 0.2302 and 2.5^0.2302 = 1.2348, so T₂ = 303.15 × 1.2348 = 374.3 K, or 101.2 °C. ZRT₁/M = 0.96 × 8.314 × 303.15/0.016043 = 150.8 kJ/kg, and 150.8 × 0.2348/0.2302 ≈ 153.8 kJ/kg. At 5.556 kg/s that is 855 kW of gas power, and 1,096 kW at the shaft. The real discharge is near 30 + 71.2/0.78 ≈ 121 °C, so one stage is fine here.