Pipeline Hydraulics

Average Pressure Calculations

Representative pressure for gas pipeline flow equations. The two-thirds average (GPSA method) accounts for non-linear density variation in compressible flow.

Launch Calculator Compare Methods

Arithmetic Average

Pavg = (P₁+P₂)/2

Valid for ΔP < 10% or liquid flow

Two-Thirds Average (GPSA)

Pavg = ⅔[(P₁³−P₂³)/(P₁²−P₂²)]

Standard for compressible gas flow

Primary Use

Flow Equations

Weymouth, Panhandle, General Flow Eq.

Contents

Quick Reference

  • ΔP < 10%: Arithmetic OK
  • ΔP 10-40%: Two-Thirds recommended
  • ΔP > 40%: Two-Thirds required

1. Why Average Pressure?

Gas flow equations require fluid properties (density, viscosity, compressibility factor Z) evaluated at a representative pressure. Since pressure varies along a pipeline due to friction losses, we need a single "average" value that, when used in flow equations, gives the correct result.

Pressure profile along pipeline showing P₁ at inlet, P₂ at outlet, and average pressure positions.
Pressure profile along pipeline with arithmetic vs two-thirds average positions.

For compressible flow, pressure does not decrease linearly. Gas density decreases as pressure drops, causing velocity to increase. This non-linear behavior means the simple arithmetic average underestimates the effective average pressure.

Key Insight: The two-thirds average gives a value slightly higher than the arithmetic mean, better representing the pressure-weighted conditions along the pipeline.

2. Calculation Methods

Arithmetic Average

The simplest approach—just the midpoint between inlet and outlet pressures.

Arithmetic Average: P_avg = (P₁ + P₂) / 2 Where: P₁ = Inlet absolute pressure (psia) P₂ = Outlet absolute pressure (psia) When to use: • Liquid flow (incompressible) • Gas flow with ΔP < 10% of inlet • Quick preliminary estimates

Two-Thirds Average (GPSA Method)

The industry-standard method for compressible gas flow, derived from integrating the pressure-flow relationship.

Two-Thirds Average (GPSA): P_avg = (2/3) × [(P₁³ - P₂³) / (P₁² - P₂²)] Algebraically equivalent to: P_avg = (2/3) × [P₁ + P₂ - (P₁ × P₂)/(P₁ + P₂)] When to use: • All compressible gas flow calculations • Required when ΔP > 40% of inlet pressure • Recommended when ΔP > 10% Example: P₁ = 1000 psia, P₂ = 800 psia P_avg = (2/3) × [(1000³ - 800³) / (1000² - 800²)] P_avg = (2/3) × [488,000,000 / 360,000] P_avg = (2/3) × 1355.56 P_avg = 903.7 psia Compare: Arithmetic = (1000 + 800)/2 = 900 psia Difference: 0.4%

3. Mathematical Derivation

The two-thirds formula comes from the general flow equation for compressible flow, where flow rate Q is proportional to √(P₁² - P₂²).

Derivation of Two-Thirds Formula: For isothermal gas flow, Q ∝ √[(P₁² - P₂²)] The average of P² along the pipeline: (P²)_avg = (P₁² + P₂²) / 2 The correct P_avg that represents flow conditions is: P_avg = (2/3) × (P₁² + P₁×P₂ + P₂²) / (P₁ + P₂) Using factorization: P₁³ - P₂³ = (P₁ - P₂)(P₁² + P₁×P₂ + P₂²) P₁² - P₂² = (P₁ - P₂)(P₁ + P₂) Therefore: P_avg = (2/3) × [(P₁³ - P₂³) / (P₁² - P₂²)]
P² integration concept showing area under curve for two-thirds average derivation.
P² vs pipeline length showing integration concept for two-thirds average.

4. Method Comparison

The table below shows how the two methods compare across different pressure drop scenarios.

P₁ (psia) P₂ (psia) ΔP (%) Arithmetic Two-Thirds Difference
10009505%975.00975.210.02%
100090010%950.00950.880.09%
100080020%900.00903.700.41%
100070030%850.00858.821.04%
100060040%800.00816.672.08%
100050050%750.00777.783.70%
100040060%700.00742.866.12%
Practical Guidance (per GPSA):
• ΔP < 10%: Either method acceptable (difference < 0.1%)
• ΔP 10-40%: Two-thirds recommended for accuracy
• ΔP > 40%: Two-thirds required; arithmetic gives significant error
Comparison chart showing arithmetic vs two-thirds average methods across pressure drop ranges.
Arithmetic vs two-thirds average comparison across pressure drop percentages.

5. Atmospheric Pressure

Gauge pressure must be converted to absolute pressure before applying average pressure formulas. At high elevations, atmospheric pressure decreases significantly.

US Standard Atmosphere 1976: P_atm = 14.696 × (1 - 6.8753×10⁻⁶ × h)^5.2561 Where: P_atm = Atmospheric pressure (psia) h = Elevation above sea level (ft) 14.696 = Sea level standard pressure (psia) 6.8753×10⁻⁶ = Lapse rate constant (per ft) 5.2561 = Barometric exponent

Atmospheric Pressure vs Elevation

Elevation (ft) P_atm (psia) % of Sea Level Typical Location
014.696100.0%Sea level
1,00014.17396.4%Houston, TX
3,00013.17189.6%Calgary, AB
5,00012.22883.2%Denver, CO
7,00011.34077.2%—
10,00010.10768.8%Mountain passes
Atmospheric pressure decrease with elevation per US Standard Atmosphere 1976.
Atmospheric pressure vs elevation per US Standard Atmosphere 1976.
Practical Impact: At 5,000 ft elevation, atmospheric pressure is ~2.5 psi lower than sea level. For a gauge pressure of 100 psig, this represents a 2% difference in absolute pressure—significant for custody transfer calculations.

6. Practical Applications

Use in Flow Equations

Average pressure is used to calculate gas properties (Z, ρ, μ) in flow equations like Weymouth, Panhandle A/B, and the General Flow Equation.

General Flow Equation: Q = C × √[(P₁² - P₂²) / (G × T_avg × L × Z_avg × f)] Where Z_avg and gas density are evaluated at P_avg (two-thirds) Weymouth Equation: Q = 433.5 × E × (T_b/P_b) × √[(P₁² - P₂²)/(G × T_f × L × Z)] × D^2.667 The compressibility Z should be calculated at P_avg.

Common Errors to Avoid

Error Impact Correction
Using arithmetic mean for large ΔP Underestimates P_avg by 2-6% Use two-thirds when ΔP > 10%
Using gauge instead of absolute ~2% error at low pressures P_abs = P_gauge + P_atm
Ignoring elevation for P_atm ~2.5 psi error at 5000 ft Calculate P_atm from elevation

Industry Standards Reference

Standard Application
GPSATwo-thirds formula for gas flow
AGA Report No. 3Orifice metering calculations
Crane TP-410Fluid flow reference
US Std Atmosphere 1976Atmospheric pressure model

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