High-Pressure Gas Pipeline Hydraulics: Weymouth vs. Panhandle

In long-distance natural gas transmission pipelines spanning 50 to 500 kilometers, standard incompressible pipe flow equations (like Darcy-Weisbach) fail because the density and volumetric flow rate change drastically as pressure drops along the line.

To accurately size pipelines and compressor station spacing, midstream engineers rely on specialized compressible flow equations: Weymouth, Panhandle A, and Panhandle B.

1. The Fundamental Gas Flow Formulations

All pipeline flow equations share the fundamental integrated form derived from the mechanical energy balance for compressible flow:

$$Q = C \cdot E \cdot \left(\frac{T_b}{P_b}\right)^{a} \cdot \left[\frac{P_1^2 - P_2^2}{G^b \cdot T_{avg} \cdot L \cdot Z_{avg}}\right]^{c} \cdot D^d$$

Where: * $Q$ is standard flow rate (SCFD or MMSCFD). * $E$ is pipeline flow efficiency (typically $0.85$ to $0.95$). * $P_1, P_2$ are inlet and delivery pressures (psia or bar). * $D$ is internal pipe diameter (inches or mm). * $L$ is pipeline length (miles or km). * $G$ is gas specific gravity relative to air ($MW / 28.964$). * $T_{avg}$ is average gas temperature in Rankine or Kelvin. * $Z_{avg}$ is average gas compressibility factor.

A. Weymouth Equation

Developed for gathering systems and high-pressure transmission lines: $$Q_{SCFD} = 433.5 \cdot E \cdot \left(\frac{T_b}{P_b}\right) \cdot \left[\frac{P_1^2 - P_2^2}{G \cdot T_{avg} \cdot L \cdot Z_{avg}}\right]^{0.5} \cdot D^{2.667}$$

• Characteristics: Assumes fully turbulent flow where the friction factor depends solely on pipe diameter ($f = 0.032 / D^{1/3}$). It yields the most conservative (lowest) flow prediction.

B. Panhandle A Equation

Developed for moderate-pressure, medium-diameter lines ($Re = 5 \times 10^6$ to $1.1 \times 10^7$): $$Q_{SCFD} = 435.87 \cdot E \cdot \left(\frac{T_b}{P_b}\right)^{1.0788} \cdot \left[\frac{P_1^2 - P_2^2}{G^{0.8539} \cdot T_{avg} \cdot L \cdot Z_{avg}}\right]^{0.5394} \cdot D^{2.6182}$$

C. Panhandle B Equation

The industry benchmark for modern high-pressure, large-diameter, smooth transmission lines ($Re = 4 \times 10^6$ to $4 \times 10^7$): $$Q_{SCFD} = 737.0 \cdot E \cdot \left(\frac{T_b}{P_b}\right)^{1.02} \cdot \left[\frac{P_1^2 - P_2^2}{G^{0.961} \cdot T_{avg} \cdot L \cdot Z_{avg}}\right]^{0.510} \cdot D^{2.530}$$

2. Average Pipeline Pressure & Line Pack

Because the pressure gradient along a gas pipeline is non-linear (steeper near the outlet as gas expands), the average operating pressure ($P_{avg}$) is calculated as:

$$P_{avg} = \frac{2}{3} \left( P_1 + P_2 - \frac{P_1 \cdot P_2}{P_1 + P_2} \right)$$

Line Pack Storage

Line Pack is the total volume of standard gas stored inside the pipeline at any given moment:

$$V_{pack} = V_{geom} \cdot \left(\frac{P_{avg}}{P_b}\right) \cdot \left(\frac{T_b}{T_{avg} \cdot Z_{avg}}\right)$$

Where $V_{geom} = \frac{\pi}{4} D^2 L$ is the geometric interior pipe volume. Line pack acts as a natural gas accumulator, allowing pipeline operators to meet sudden surge demands without immediately adjusting wellhead production.

3. Erosional Velocity Limits (API 14E)

Because gas accelerates as pressure drops, the maximum velocity occurs at the pipeline exit. To prevent pipe wall erosion from micro-particulates, API RP 14E establishes the erosional velocity threshold:

$$V_e = \frac{C}{\sqrt{\rho_{gas}}}$$

Where $C = 100$ (for continuous service) to $125$ (for intermittent service), and $\rho_{gas}$ is real gas density at discharge conditions.