$$ Q_\text{base} = 4.5965 \times 10^{-3} \cdot \left(\frac{T_b}{P_b}\right)^{1.0788} \cdot \left( \frac{P_1^2 - e^s \cdot P_2^2}{G^{0.8539} \cdot T_f \cdot L_e \cdot Z} \right)^{0.5394} \cdot D^{2.6182} \cdot E $$
$$ L_e = \frac{L \cdot \left(e^s - 1\right)}{s} \qquad \qquad s = 0.0684 \cdot G \cdot \left(\frac{H_2 - H_1}{T_f \cdot Z}\right) $$
| \( Q_\text{base} \) | = Volume flow rate at base conditions, m³/d |
| \( T_b \) | = Base temperature, K |
| \( P_b \) | = Base pressure, kPa |
| \( P_1 \) | = Upstream pressure, kPa |
| \( P_2 \) | = Downstream pressure, kPa |
| \( E \) | = Pipeline efficiency (dimensionless) |
| \( G \) | = Gas gravity (air = 1) |
| \( Z \) | = Gas compressibility factor (dimensionless) |
| \( T_f \) | = Flowing gas temperature, K |
| \( L_e \) | = Equivalent pipe length, km |
| \( L \) | = Pipe segment length, km |
| \( D \) | = Pipe inside diameter, mm |
| \( s \) | = Elevation adjustment parameter |
| \( H_1 \) | = Upstream elevation, m |
| \( H_2 \) | = Downstream elevation, m |
Note: Valid for Reynolds numbers 5-11 million, 300-1500 mm pipes, 5500-10300 kPa
$$ Q_\text{base} = 1.002 \times 10^{-2} \cdot \left(\frac{T_b}{P_b}\right)^{1.02} \cdot \left( \frac{P_1^2 - e^s \cdot P_2^2}{G^{0.961} \cdot T_f \cdot L_e \cdot Z} \right)^{0.51} \cdot D^{2.53} \cdot E $$
$$ L_e = \frac{L \cdot \left(e^s - 1\right)}{s} \qquad \qquad s = 0.0684 \cdot G \cdot \left(\frac{H_2 - H_1}{T_f \cdot Z}\right) $$
| \( Q_\text{base} \) | = Volume flow rate at base conditions, m³/d |
| \( T_b \) | = Base temperature, K |
| \( P_b \) | = Base pressure, kPa |
| \( P_1 \) | = Upstream pressure, kPa |
| \( P_2 \) | = Downstream pressure, kPa |
| \( E \) | = Pipeline efficiency (dimensionless) |
| \( G \) | = Gas gravity (air = 1) |
| \( Z \) | = Gas compressibility factor (dimensionless) |
| \( T_f \) | = Flowing gas temperature, K |
| \( L_e \) | = Equivalent pipe length, km |
| \( L \) | = Pipe segment length, km |
| \( D \) | = Pipe inside diameter, mm |
| \( s \) | = Elevation adjustment parameter |
| \( H_1 \) | = Upstream elevation, m |
| \( H_2 \) | = Downstream elevation, m |
Note: Valid for Reynolds numbers 4-40 million, >900 mm pipes, >6900 kPa
$$ v = \sqrt{\frac{2 \cdot \Delta P_f \cdot D}{f \cdot L \cdot \rho}} \qquad Q_\text{actual} = v \cdot A \cdot E \qquad Q_\text{base} = Q_\text{actual} \cdot \frac{P_1 \cdot T_b \cdot Z}{P_b \cdot T_f} $$
$$ \Delta P_f = P_1 - P_2 - \rho \cdot g \cdot (H_2 - H_1) \qquad A = \frac{\pi D^2}{4} $$
| \( v \) | = Gas velocity, m/s |
| \( \Delta P_f \) | = Frictional pressure drop, Pa |
| \( f \) | = Darcy friction factor (dimensionless) |
| \( \rho \) | = Gas density at flowing conditions, kg/m³ |
| \( Q_\text{actual} \) | = Actual volumetric flow rate, m³/s |
| \( Q_\text{base} \) | = Volume flow rate at base conditions, m³/s |
| \( A \) | = Pipe cross-sectional area, m² |
| \( E \) | = Pipeline efficiency (dimensionless) |
| \( g \) | = Gravitational acceleration, 9.81 m/s² |
| \( P_1, P_2 \) | = Upstream and downstream pressures, Pa |
| \( P_b \) | = Base pressure, Pa |
| \( T_b \) | = Base temperature, K |
| \( T_f \) | = Flowing gas temperature, K |
| \( Z \) | = Gas compressibility factor (dimensionless) |
| \( D \) | = Pipe inside diameter, m |
| \( L \) | = Pipe segment length, m |
| \( H_1, H_2 \) | = Upstream and downstream elevations, m |
Gas velocity full expanded formula:
$$ v = \frac{Q}{A} = \frac{Q_\text{base} \times \frac{\rho_\text{base}}{\rho}}{\frac{\pi D^2}{4}} = \frac{4 \times Q_\text{base} \times \rho_\text{base}}{\pi D^2 \times \rho} = \frac{4 \times Q_\text{base} \times P_\text{base} \times Z \times T}{\pi D^2 \times P \times T_\text{base}} $$
Where:
| \( v \) | = Gas velocity, m/s |
| \( Q \) | = Actual volumetric flow rate at flowing conditions, m³/s |
| \( Q_\text{base} \) | = Volumetric flow rate at base conditions, m³/s |
| \( \rho \) | = Gas density at flowing conditions, kg/m³ |
| \( \rho_\text{base} \) | = Gas density at base conditions, kg/m³ |
| \( A \) | = Pipe cross-sectional area, m² |
| \( P_\text{base} \) | = Base pressure, kPa |
| \( P \) | = Flowing pressure, kPa |
| \( T_\text{base} \) | = Base temperature, K |
| \( T \) | = Flowing temperature, K |
| \( Z \) | = Gas compressibility factor (dimensionless) |
| \( D \) | = Pipe inside diameter, m |
Average pressure and gas volume inside the pipeline (base conditions):
$$ P_\text{avg} = \frac{2}{3} \times \frac{P_1^3 - P_2^3}{P_1^2 - P_2^2} \qquad V_\text{base} = \left( \frac{\pi D^2}{4} \right) \times L \times \frac{P_\text{avg} \times T_\text{base}}{P_\text{base} \times T \times Z} $$
Where:
| \( P_\text{avg} \) | = Average gas pressure in the pipeline, kPa |
| \( V_\text{base} \) | = Gas volume at base conditions, m³ |
| \( P_1 \) | = Upstream pressure, kPa |
| \( P_2 \) | = Downstream pressure, kPa |
| \( L \) | = Pipe length, m |
| \( D \) | = Pipe inside diameter, m |
| \( P_\text{base} \) | = Base pressure, kPa |
| \( T_\text{base} \) | = Base temperature, K |
| \( T \) | = Flowing gas temperature, K |
| \( Z \) | = Gas compressibility factor (dimensionless) |
| \( \pi \) | = Pi constant (≈ 3.1416) |