Compressible Gas Flow Hydraulics: Weymouth, Panhandle A & B Equations

Executive Summary & Engineering Relevance

Long-distance natural gas transmission pipelines operate across hundreds to thousands of kilometers at high pressures (typically 50 to 120 bar / 700 to 1,750 psia). Unlike liquid pipeline hydraulics where fluid density remains essentially constant, gas density decreases steadily along the line as friction lowers the static pressure.

From mass conservation, as the gas expands ($P_1 \rightarrow P_2$), its volumetric flow rate and actual linear velocity accelerate toward the pipeline discharge:

$$v(x) = \frac{\dot{m}}{\rho(x) \cdot A} \propto \frac{1}{P(x)}$$

This expansion effect invalidates incompressible Darcy-Weisbach formulations. Instead, pipeline design engineers rely on specialized steady-state isothermal compressible flow equations derived by integrating the fundamental momentum balance with real-gas equations of state. The three historical industry-standard correlations are: 1. Weymouth Equation (1912): Sized for gathering networks, short lateral lines, and high-pressure transmission where friction factor varies inversely with diameter ($f \propto D^{-1/3}$). 2. Panhandle A Equation (1940s): Formulated for medium-to-large diameter pipelines operating at moderate Reynolds numbers ($5 \times 10^6 \le Re \le 1.1 \times 10^7$). 3. Panhandle B Equation (1956): The standard benchmark for modern large-diameter, high-pressure, fully turbulent cross-country transmission corridors ($Re > 1.4 \times 10^7$).

In addition to line sizing, operators must calculate the Line Pack—the total volume of standard gas stored dynamically within the pipeline envelope—which acts as a vital commercial buffer against diurnal demand swings and upstream compressor trips.

1. Governing Differential Momentum Balance for Compressible Gas Flow

Consider a horizontal pipe element of diameter $D$, cross-sectional area $A$, and length $dx$:

$$-A \, dP - \tau_w (\pi D \, dx) = \dot{m} \, dv$$

For high-pressure, long-distance lines ($L > 1\text{ km}$), acceleration pressure drop is negligible compared to wall friction ($v \ll c$, where $c$ is the acoustic sonic velocity). Substituting the real-gas equation of state $\rho = \frac{P M}{Z R T}$:

$$-dP = \frac{f_D \rho v^2}{2 D} dx = \frac{f_D \dot{m}^2 Z R T}{2 D A^2 M P} dx$$

Separating variables and integrating from $x = 0$ ($P = P_1$) to $x = L$ ($P = P_2$) under isothermal conditions ($T = \text{const}$):

$$\int_{P_2}^{P_1} P \, dP = \frac{f_D \dot{m}^2 R T \bar{Z}}{2 D A^2 M} \int_0^L dx$$

$$\frac{P_1^2 - P_2^2}{2} = \frac{f_D \dot{m}^2 R T \bar{Z} L}{2 D A^2 M}$$

Expressed in terms of standard volumetric flow rate ($Q_{std}$ at base conditions $T_b, P_b$):

$$Q_{std} = C_{\text{units}} \cdot \left(\frac{T_b}{P_b} \right) \cdot \left[ \frac{P_1^2 - P_2^2}{G \cdot T \cdot L \cdot \bar{Z} \cdot f_D} \right]^{0.5} \cdot D^{2.5}$$

Where: * $P_1, P_2$: Absolute inlet and outlet pressures. * $G$: Gas specific gravity relative to air ($G = M_{\text{gas}} / 28.964$). * $T$: Absolute flowing temperature ($K$ or $^\circ R$). * $\bar{Z}$: Average gas compressibility factor. * $f_D$: Darcy friction factor. * $D$: Pipe inside diameter. * $L$: Pipeline length.

2. Weymouth, Panhandle A, and Panhandle B Formulations

Because the Darcy friction factor $f_D$ is itself a non-linear function of Reynolds number and wall roughness (Colebrook-White), early pipeline engineers introduced empirical power-law friction approximations that allowed direct analytical solution.

Friction Factor Substitution in Classical Equations:
+-----------------------------------------------------------------------------+
| Equation      | Friction Approximation  | Best Operating Window             |
+-----------------------------------------------------------------------------+
| Weymouth      | f = 0.032 / D^(1/3)     | D ≤ 16", High dP, Gathering lines |
| Panhandle A   | f = 0.0854 / Re^0.1461  | Re = 5×10⁶ to 1.1×10⁷             |
| Panhandle B   | f = 0.015 / Re^0.0392   | Re = 1.4×10⁷ to 4×10⁷, D ≥ 20"    |
+-----------------------------------------------------------------------------+

2.1 The Weymouth Equation (Imperial Units)

Thomas R. Weymouth (1912) assumed that relative roughness increases in smaller pipes, formulating $f_D = 0.032 / D_{\text{in}}^{1/3}$. Substituting into the integrated equation:

$$Q = 433.5 \cdot E \cdot \left(\frac{T_b}{P_b} \right) \cdot \left[ \frac{P_1^2 - P_2^2}{G \cdot T_R \cdot L_{\text{mi}} \cdot \bar{Z}} \right]^{0.50} \cdot D_{\text{in}}^{2.667}$$

Where: * $Q$: Standard capacity (SCFD, standard cubic feet per day). * $E$: Pipeline efficiency factor (typically $0.88 - 0.95$, dimensionless). * $T_b = 520^\circ R$ ($60^\circ F$), $P_b = 14.73\text{ psia}$. * $D_{\text{in}}$: Inside diameter in inches ($D^{8/3} = D^{2.667}$). * $L_{\text{mi}}$: Length in miles.

2.2 Panhandle A Equation (1940s)

Developed by the Panhandle Eastern Pipe Line Company for smooth, long-distance transmission lines operating under moderate Reynolds numbers:

$$Q = 435.87 \cdot E \cdot \left(\frac{T_b}{P_b} \right)^{1.0788} \cdot \left[ \frac{P_1^2 - P_2^2}{G^{0.8539} \cdot T_R \cdot L_{\text{mi}} \cdot \bar{Z}} \right]^{0.5394} \cdot D_{\text{in}}^{2.6182}$$

2.3 Panhandle B Equation (Revised Panhandle, 1956)

Panhandle B is the most widely adopted pipeline standard for large-diameter ($D \ge 20\text{ inches}$), high-flow, fully turbulent cross-country lines:

$$Q = 737.0 \cdot E \cdot \left(\frac{T_b}{P_b} \right)^{1.02} \cdot \left[ \frac{P_1^2 - P_2^2}{G^{0.961} \cdot T_R \cdot L_{\text{mi}} \cdot \bar{Z}} \right]^{0.510} \cdot D_{\text{in}}^{2.530}$$

2.4 Pipeline Efficiency Factor ($E$)

Real pipelines deviate from theoretical formulas due to internal pipe corrosion, welding bead intrusion, bends, liquid condensate accumulation, and scale. Engineers apply an efficiency factor $E$: * New, clean, internally epoxy-lined pipe: $E = 0.95 - 0.98$ * Standard clean carbon steel transmission line: $E = 0.90 - 0.92$ * Old pipe with mild scale or sand deposits: $E = 0.80 - 0.85$ * Wet gas gathering line with periodic liquid holdup: $E = 0.70 - 0.75$

3. Average Line Pressure ($P_{avg}$) & Compressibility ($Z$)

Because pressure drop along a compressible gas line is non-linear—steepest near the outlet where velocity is highest—the simple arithmetic average $(P_1 + P_2)/2$ significantly underpredicts the effective pressure.

Pressure Gradient Along Pipeline:
Pressure P(x)
  ^
P1+---\
  |    \\
Pavg - - -\\ - - - - - - - - (2/3 Rule Effective Mean)
  |        \\
  |          \\
P2+-----------+--------> Distance x
  0           L

The 2/3 Rule for Average Pipeline Pressure:

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

Average Compressibility ($\bar{Z}$) via the CNGA Correlation:

For pipeline hydraulics where rapid compositional flashes are impractical, the California Natural Gas Association (CNGA) correlation calculates $\bar{Z}$ as a function of specific gravity $G$ and $P_{avg}$:

$$\bar{Z} = \frac{1}{1 + \frac{P_{avg} \cdot 3.444 \times 10^5 \cdot 10^{1.785 G}}{T_R^{3.825}}}$$

For lean natural gas ($G \approx 0.60 - 0.68$) at 60–80 bar, $\bar{Z}$ typically ranges between 0.82 and 0.89. Neglecting $Z$ (assuming ideal gas $Z = 1.0$) introduces a 12% to 18% error in pipeline capacity!

4. Stored Line Pack Inventory Calculation

Line pack is the physical inventory of natural gas stored inside the pressurized pipeline. Operators "pack" the line during off-peak night hours by raising discharge pressure, and "draft" the line during peak industrial day hours.

The geometric pipe volume is: $$V_{\text{geom}} = \frac{\pi}{4} D^2 \cdot L$$

Applying the real gas law at average line conditions ($P_{avg}, T_{avg}, \bar{Z}$) relative to standard base conditions ($P_{std} = 1.01325\text{ bar a}, T_{std} = 288.15\text{ K}$):

$$V_{\text{pack}} = V_{\text{geom}} \cdot \left(\frac{P_{avg}}{P_{std}} \right) \cdot \left(\frac{T_{std}}{T} \right) \cdot \left(\frac{1}{\bar{Z}} \right) \quad [\text{Sm}^3]$$

In imperial units (MMSCF): $$V_{\text{pack}} = \frac{\pi}{4} \left(\frac{D_{\text{in}}}{12} \right)^2 \cdot (L_{\text{mi}} \times 5280) \cdot \left(\frac{P_{avg,\text{psi}}}{14.73} \right) \cdot \left(\frac{520}{T_R} \right) \cdot \left(\frac{1}{\bar{Z}} \right)$$

5. Comprehensive Step-by-Step Industrial Worked Example

Design Problem Statement:

A 24-inch natural gas transmission pipeline connects an onshore processing gas terminal to an electrical power generation hub: * Nominal Size: DN600 (24-inch Sch 40), Inside Diameter $D = 574.0\text{ mm}$ ($22.60\text{ inches}$) * Pipeline Length: $L = 85.0\text{ km}$ ($52.82\text{ miles}$) * Inlet Compressor Station Pressure: $P_1 = 75.0\text{ bar absolute}$ ($1,087.8\text{ psia}$) * Minimum Delivery Pressure at Power Plant: $P_2 = 48.0\text{ bar absolute}$ ($696.2\text{ psia}$) * Gas Properties: Specific Gravity $G = 0.620$ (Lean methane rich), Operating Temp $T = 20.0^\circ\text{C}$ ($68.0^\circ\text{F} = 527.67^\circ\text{R}$) * Pipeline Efficiency Factor: $E = 0.92$

Determine: 1. Average line pressure ($P_{avg}$) and average compressibility factor ($\bar{Z}$). 2. Standard flow capacity ($Q$) in both MMSCFD and standard cubic meters per day ($\text{Sm}^3/\text{day}$) using the Panhandle B equation. 3. Actual gas velocity at the pipeline outlet ($v_{out}$). 4. Total stored line pack gas inventory ($V_{\text{pack}}$).

Step 1: Average Operating Pressure ($P_{avg}$)

$$P_1 = 1,087.8\text{ psia}, \quad P_2 = 696.2\text{ psia}$$

$$P_1 + P_2 = 1,784.0\text{ psia}$$ $$P_1 \cdot P_2 = 1,087.8 \times 696.2 = 757,326.4$$ $$\frac{P_1 P_2}{P_1 + P_2} = \frac{757,326.4}{1,784.0} = 424.51\text{ psia}$$

$$P_{avg} = \frac{2}{3} \left[ 1,784.0 - 424.51 \right] = \frac{2}{3} (1,359.49) = \mathbf{906.33\text{ psia}} \quad (62.49\text{ bar a})$$

Step 2: Average Compressibility Factor ($\bar{Z}$)

Using the CNGA correlation: $$T_R = 527.67^\circ\text{R}$$ $$T_R^{3.825} = (527.67)^{3.825} \approx 2.458 \times 10^{10}$$ $$10^{1.785 \times 0.620} = 10^{1.1067} = 12.785$$

$$\text{Term} = \frac{906.33 \times 3.444 \times 10^5 \times 12.785}{2.458 \times 10^{10}} = \frac{3.991 \times 10^9}{2.458 \times 10^{10}} = 0.16237$$

$$\bar{Z} = \frac{1}{1 + 0.16237} = \mathbf{0.8603}$$

Step 3: Flow Capacity Calculation (Panhandle B)

Parameters: * $E = 0.92$ * $D_{\text{in}} = 22.60\text{ in}$ * $L_{\text{mi}} = 52.82\text{ mi}$ * $G = 0.620$

Base ratio term: $$\left(\frac{T_b}{P_b} \right)^{1.02} = \left(\frac{520}{14.73} \right)^{1.02} = (35.302)^{1.02} = 37.844$$

Pressure term: $$P_1^2 - P_2^2 = (1,087.8)^2 - (696.2)^2 = 1,183,309 - 484,694 = 698,615\text{ psia}^2$$

Denominator inside radical: $$G^{0.961} = (0.620)^{0.961} = 0.6311$$ $$\text{Denom} = 0.6311 \times 527.67 \times 52.82 \times 0.8603 = 15,133.6$$

Radical term: $$\left[ \frac{698,615}{15,133.6} \right]^{0.510} = (46.163)^{0.510} = 7.042$$

Diameter term: $$D_{\text{in}}^{2.530} = (22.60)^{2.530} = 2,618.5$$

Calculating total flow $Q$: $$Q = 737.0 \times 0.92 \times 37.844 \times 7.042 \times 2,618.5$$ $$Q = 678.04 \times 37.844 \times 7.042 \times 2,618.5 = \mathbf{472,900,000\text{ SCFD}} = \mathbf{472.9\text{ MMSCFD}}$$

Converting to metric standard cubic meters per day: $$Q = \frac{472,900,000}{35.31467} = \mathbf{13,391,000\text{ Sm}^3/\text{day}}$$

Step 4: Outlet Actual Gas Velocity

At the pipeline outlet ($P_2 = 48.0\text{ bar a}$, $T = 293.15\text{ K}$): $$Z_2 \approx 0.892$$ $$Q_{\text{act}} = \left(\frac{13,391,000}{86400} \right) \times \left(\frac{1.01325}{48.0} \right) \times \left(\frac{293.15}{288.15} \right) \times 0.892 = 154.99 \times 0.02111 \times 1.01735 \times 0.892 = 2.969\text{ m}^3/\text{s}$$

Cross-sectional area of DN600 pipe ($D = 0.574\text{ m}$): $$A = \frac{\pi}{4} (0.574)^2 = 0.2588\text{ m}^2$$

Outlet velocity: $$v_{\text{out}} = \frac{2.969\text{ m}^3/\text{s}}{0.2588\text{ m}^2} = \mathbf{11.47\text{ m/s}}$$

Engineering Verification: The outlet velocity of $11.5\text{ m/s}$ is well within the recommended transmission window of $8 - 15\text{ m/s}$, ensuring safe acoustic noise levels without excessive pipe wall erosion.

Step 5: Stored Line Pack Inventory

Geometric pipe volume: $$V_{\text{geom}} = A \cdot L = 0.2588\text{ m}^2 \times 85,000\text{ m} = 21,998\text{ m}^3$$

Total standard gas pack: $$V_{\text{pack}} = 21,998 \times \left(\frac{62.49}{1.01325} \right) \times \left(\frac{288.15}{293.15} \right) \times \left(\frac{1}{0.8603} \right)$$ $$V_{\text{pack}} = 21,998 \times 61.673 \times 0.9829 \times 1.1624 = \mathbf{1,550,000\text{ Sm}^3} \quad (54.7\text{ MMSCF})$$

Commercial Operational Value: The pipeline contains $1.55\text{ million Sm}^3$ of stored energy. If the upstream terminal compressors trip, the line pack can sustain the power plant at full load for nearly 3 hours without curtailment.

6. Comparison of Major Gas Pipeline Hydraulic Equations

References & Industry Standards

1. Weymouth, T. R. (1912). Problems in Natural Gas Engineering. Transactions of the American Society of Mechanical Engineers, 34, 185–234.
2. Panhandle Eastern Pipe Line Company. (1956). Panhandle B Flow Formula for High-Pressure Natural Gas Pipelines. Technical Report, Kansas City, MO.
3. American Gas Association. (1964). Steady Flow in Gas Pipelines (AGA Project NX-28). AGA Publishing, Arlington, VA.
4. Menon, E. S. (2005). Gas Pipeline Hydraulics. CRC Press / Taylor & Francis Group.
5. ASME. (2020). ASME B31.8: Gas Transmission and Distribution Piping Systems. American Society of Mechanical Engineers.