Compressible Flow Basics | Industrial Engineering Studio

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.

Where Compressible Flow Matters

Natural gas pipelines, compressed air systems, steam lines, vent/flare systems, safety relief valve discharge, pneumatic conveying, and high-velocity gas vents.

Key Gas Flow Concepts

Mach Number

Ma = v / c = v / √(γ × R × T)

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

Gas	Speed of Sound at 20°C (m/s)	γ
Air	343	1.40
Natural gas (methane)	446	1.32
Hydrogen	1303	1.41
Steam (100°C)	472	1.33
Nitrogen	349	1.40
CO₂	267	1.30

Velocity Limits in Gas Pipes

Typical design practice limits gas velocity to 20 m/s in plant piping to control noise and erosion. Long-distance gas pipelines run at 5-10 m/s. High-pressure steam can go to 60 m/s. Vacuum systems often operate near sonic (Ma = 0.5-0.9).

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:

m_max = C_d × A × P₀ × √(γ / (R × T₀) × (2/(γ+1))^{(γ+1)/(γ−1)})

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

Relief Valves Must Be Sized for Choked Flow

When a safety valve discharges to atmosphere, choked flow typically occurs. API 520 provides formulas for relief valve sizing at choked conditions. Using incompressible formulas drastically underestimates required orifice area.

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:

m = √[(P₁² − P₂²) × (π²D⁵) / (16 × R × T × L × f)]

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

Weymouth vs Panhandle

The Weymouth formula is used for high-pressure gas transmission. The Panhandle A/B equations are preferred for large-diameter, high-flow lines. Both are empirical and apply within specific Reynolds number ranges.

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:

Calculate density at average pressure: ρ = P_avg/(R × T)
Calculate velocity at average conditions: v = m/(ρ × A)
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.

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Worked Example: Compressed Air Line

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

Inlet P = 8 bar abs; estimate outlet P ≈ 7.5 bar abs; P_avg = 7.75 bar
ρ_avg = 775,000 / (287 × 293) = 9.22 kg/m³
Q_actual = (1000/3600) × (1.013/7.75) × (293/273) = 0.0386 m³/s
v = Q/A = 0.0386 / (π × 0.102²/4) = 4.7 m/s
Re = ρvD/μ = 9.22 × 4.7 × 0.102 / (1.8×10⁻⁵) = 246,000 → turbulent
f ≈ 0.019 (for ε/D = 0.00045)
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

Gas Service	Typical Velocity	Max ΔP Guideline
Compressed air (plant)	6-10 m/s	0.1-0.3 bar per 100m
Natural gas (high-pressure transmission)	5-10 m/s	0.05-0.2 bar/km
Natural gas (distribution, low pressure)	10-20 m/s	<10% of line pressure
Saturated steam	15-40 m/s	0.1-0.5 bar per 100m
Superheated steam	30-60 m/s	Varies with pressure
Relief/flare headers	50-100% of Mach	Choked flow at outlet
Vacuum lines	up to 150 m/s	Near sonic at pump inlet

Steam Line Sizing

For steam lines, pressure drop reduces saturation temperature. Design steam lines so that end-of-line pressure meets process requirements. Typical target: 0.1-0.5 bar pressure drop per 100m. Steam velocity should be kept below 40 m/s (saturated) to avoid water hammer from entrained condensate.

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:

Q_actual = Q_std × (P_std/P_actual) × (T_actual/T_std)

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.

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