Compressible Flow Basics

Incompressible vs Compressible Flow

For liquids, density is essentially constant regardless of pressure — flow is incompressible and Darcy-Weisbach with constant density works. For gases and vapors, density changes significantly with pressure, so the analysis changes.

Rule of thumb: If pressure drop < 10% of absolute inlet pressure, incompressible methods (using average density) give acceptable accuracy. Above 10%, use compressible flow equations. Above ~40% of absolute pressure, the flow can choke.

Key Gas Flow Concepts

Mach Number

Where:

• v = gas velocity (m/s)

• c = speed of sound (m/s) in the gas

• γ = ratio of specific heats (1.4 for air, 1.3 for natural gas, 1.33 for steam)

• R = specific gas constant (J/kg·K)

• T = absolute temperature (K)

• Subsonic: Ma < 1

• Sonic: Ma = 1 (choked flow)

• Supersonic: Ma > 1 (converging-diverging nozzles)

Speed of Sound in Common Gases

Choked Flow

When gas velocity reaches the speed of sound (Ma = 1) at some point in the system (typically at a restriction or pipe exit), flow becomes "choked" — reducing downstream pressure further does NOT increase flow rate. The mass flow is limited by upstream conditions only.

For an orifice/nozzle at choked conditions:

Choked flow is common in:

• Safety/relief valves discharging to atmosphere
• Control valves at high pressure drop
• Leaks from high-pressure gas systems
• Blowdown/vent systems

Isothermal Gas Flow in Long Pipes

For long pipelines at roughly constant temperature (isothermal), compressible pressure drop uses the Weymouth or Panhandle equations, or the general flow equation:

In terms of volumetric flow at standard conditions (e.g., Nm³/h), this is the basis for natural gas pipeline hydraulic calculations.

Adiabatic Flow (Fanno Flow)

For insulated pipes with no heat transfer, flow follows Fanno-line relationships. Key features:

• For subsonic flow (Ma < 1): friction accelerates the flow toward Ma = 1, pressure decreases
• For supersonic flow: friction decelerates toward Ma = 1
• Maximum pipe length exists before choking occurs (L_max for given inlet Ma)

Simplified Approach — General Gas Pressure Drop

For approximate gas line sizing with less than ~30% pressure drop, use the Darcy equation with average density:

1. Calculate density at average pressure: ρ = P_avg/(R × T)
2. Calculate velocity at average conditions: v = m/(ρ × A)
3. Apply Darcy-Weisbach: ΔP = f × (L/D) × (ρ_avg × v²/2)

For more accurate results when ΔP > 10%, iterate since density changes along the pipe.

Worked Example: Compressed Air Line

200 m of 100mm steel pipe, 1000 Nm³/h of air at 7 barg inlet, 20°C.

1. Inlet P = 8 bar abs; estimate outlet P ≈ 7.5 bar abs; P_avg = 7.75 bar
2. ρ_avg = 775,000 / (287 × 293) = 9.22 kg/m³
3. Q_actual = (1000/3600) × (1.013/7.75) × (293/273) = 0.0386 m³/s
4. v = Q/A = 0.0386 / (π × 0.102²/4) = 4.7 m/s
5. Re = ρvD/μ = 9.22 × 4.7 × 0.102 / (1.8×10⁻⁵) = 246,000 → turbulent
6. f ≈ 0.019 (for ε/D = 0.00045)
7. ΔP = 0.019 × (200/0.102) × (9.22 × 4.7²/2) = 0.019 × 1961 × 102 = ~38 kPa ≈ 0.38 bar

Final: outlet pressure ≈ 7.0 - 0.38 = 6.6 barg. Since ΔP is 0.4 bar / 8.0 bar = 5%, incompressible approximation is reasonable.

Practical Design Guidelines for Gas Piping

Density and Flow at Standard Conditions

Gas flow rates are usually stated at standard/reference conditions:

• Nm³/h: Normal cubic meters per hour at 0°C, 1.013 bar abs
• SCFM: Standard cubic feet per minute at 60°F, 14.7 psia

Convert between actual and standard flow:

Unit Conversion Reference

Example: 1000 Nm³/h = 588.6 SCFM. At 7 barg and 20°C this becomes Q_actual = 1000 × (1.013/8.013) × (293/273) = 135.5 m³/h. See the Engineering Unit Conversion Guide for the full unit reference.

Worked Example: Natural Gas Pipeline Pressure Drop

A 5 km DN300 natural gas transmission line carries 50,000 Nm³/h at 60 bar gauge inlet, 20°C. Estimate the outlet pressure using the general flow equation:

1. Convert to standard mass flow: m = Q_std × ρ_std = (50,000/3600) × 0.80 = 11.1 kg/s
2. For DN300 (ID 0.303 m), f ≈ 0.012 (fully turbulent)
3. Apply the isothermal general flow equation for P₂: m² = (P₁² − P₂²) × (π² × D⁵) / (16 × R × T × L × f)
4. Rearranging: P₂ = √[P₁² − (16 × R × T × L × f × m²)/(π² × D⁵)] = √[(6.1013×10⁶)² − (16 × 518 × 293 × 5000 × 0.012 × 11.1²)/(π² × 0.303⁵)]
5. P₂ ≈ 5.79 MPa abs ≈ 57.9 bar abs ≈ 56.9 barg — about 3.5% drop over 5 km

For full design, iterate with the Weymouth/Panhandle correlations and Z factor corrections; use the Pressure Loss Calculator and Pressure Drop Calculator for automated sizing. Compare with Natural Gas Density & Properties for density values at line conditions.

Frequently Asked Questions

When should I use compressible flow equations instead of incompressible? When pressure drop exceeds ~10% of absolute inlet pressure. Below 10%, Darcy-Weisbach with average density is adequate. Above 40% pressure drop, choking is possible and full compressible analysis is required.

What is choked flow? When gas reaches sonic velocity (Ma = 1) at a restriction (valve, orifice, relief nozzle), downstream pressure changes no longer affect the flow rate — mass flow is set by upstream pressure, temperature, and throat area. Relief valves discharging to atmosphere are almost always choked.

How do I convert Nm³/h to SCFM? Multiply Nm³/h by 0.5886 (1 Nm³ = 35.31 SCF; 1 h = 60 min; 35.31/60 = 0.5886). Both are standard-condition volume flows; the density at those standard conditions differs (0°C vs 60°F), so the ratio is fixed.

What is the speed of sound in air and natural gas? 343 m/s for air at 20°C and about 446 m/s for natural gas (methane), because the lighter gas has a higher sound speed. Sonic velocity c = √(γRT) increases with temperature and gas lightness.

Why are gas velocities limited to 20 m/s in plant piping? To control noise (jet noise grows roughly with the 6th power of velocity), erosion of fittings, and pressure drop. Long pipelines use even lower velocities (5-10 m/s) for economic compressor sizing. Relief headers are the exception — they run near Mach.

What is the difference between Weymouth and Panhandle equations? Both are empirical isothermal pipeline flow correlations. Weymouth is conservative for short, high-pressure lines; Panhandle A/B better fit large-diameter, high-Reynolds transmission lines. Use the applicable equation within its tested Reynolds number range.

What is the isentropic relation between temperature and pressure in compressible flow? For an isentropic (reversible adiabatic) process, T₂/T₁ = (P₂/P₁)^((γ-1)/γ), where γ is the specific heat ratio (1.4 for air, ~1.3 for natural gas). This relation governs the temperature change when gas accelerates or decelerates without heat transfer or friction — the basis of isentropic nozzle and diffuser analysis. Real pipe flow adds friction, so actual temperature changes are larger than isentropic predictions.

How does Mach number affect pressure drop in gas piping? Below Mach 0.3, gas can be treated as incompressible and standard liquid pressure-drop methods apply. Above Mach 0.3, compressibility effects become significant: density changes along the pipe, the pressure gradient steepens, and the flow can choke at Mach 1.0. Practical plant piping keeps gas velocity below 20 m/s (typically Mach 0.05-0.1) to avoid excessive pressure drop and noise.

Pressure Ratings for Gas vs Liquid Piping

Gas piping has stricter requirements than liquid piping due to:

• Higher stored energy (compressible fluid releases energy on failure)
• ASME B31.3 requires more NDE for gas service above certain pressures
• Safety factors are higher for "Category D" vs "Category M" (toxic) services

Summary

For gases, density changes with pressure — use compressible flow equations when pressure drop exceeds 10% of absolute pressure. Mach number relates velocity to speed of sound. Choked flow (Ma = 1) limits maximum flow through restrictions regardless of downstream pressure. For long pipelines, isothermal equations (Weymouth, Panhandle) apply. For plant piping with moderate ΔP, Darcy-Weisbach with average density provides good results. Keep gas velocities below 20 m/s to limit noise and erosion.