Chemical, Process & Catalysis Engineering
Friedel Two-Phase Pressure-Drop Multiplier (Phi_LO^2) Formula: Froude & Weber Pipe Flow
Calculate two-phase gas-liquid frictional pressure drop dP_TP = Phi_LO^2 * dP_LO in horizontal and vertical upward pipelines using Lutz Friedel’s (1979) 25,000-point correlation incorporating Froude (Fr_H) and Weber (We_L) numbers.
Why Friedel (1979) is the industry gold standard when mu_L / mu_G < 1,000!
Classic Lockhart-Martinelli two-phase multipliers ignore pipe diameter, surface tension sigma (N/m), and mass flux G (kg/(m^2·s))! Regressing 25,000 steam-water, refrigerant, and hydrocarbon two-phase pipe measurements, Lutz Friedel (1979, European Two-Phase Flow Group Meeting, Ispra) defined Liquid-Only Multiplier Phi_LO^2 = (dP/dL)_TP / (dP/dL)_LO (where (dP/dL)_LO treats total mass flux G as flowing with liquid properties rho_L, mu_L and friction factor f_LO, versus All-Gas friction factor f_GO):
With vapor mass quality x (0 < x < 1), Homogeneous Density rho_H = 1 / (x / rho_G + (1 - x) / rho_L), Homogeneous Froude Number Fr_H = (G^2) / (9.80665 * D * (rho_H^2)), and Two-Phase Weber Number We_L = (G^2 * D) / (sigma * rho_H), Friedel’s correlation combines single-phase anchor E = (1 - x)^2 + (x^2) * (rho_L * f_GO) / (rho_G * f_LO), quality factor F = (x^0.78) * ((1 - x)^0.224), and property factor H = ((rho_L / rho_G)^0.91) * ((mu_G / mu_L)^0.19) * ((1 - mu_G / mu_L)^0.70) into Phi_LO^2 = E + (3.24 * F * H) / ((Fr_H^0.045) * (We_L^0.035))!
| Friedel Dimensionless Group (x = 0.15, rho_L/rho_G = 40, mu_G/mu_L = 0.05) | Formula | Evaluated Value |
|---|---|---|
| Single-Phase Limit Term E (f_GO/f_LO = 0.80) | (1 - x)^2 + x^2 * (rho_L/rho_G) * (f_GO/f_LO) | 1.4425 |
| Quality Slip Term F | x^0.78 * (1 - x)^0.224 | 0.2198 |
| Fluid Property Term H | (rho_L/rho_G)^0.91 * (mu_G/mu_L)^0.19 * (1 - mu_G/mu_L)^0.70 | 15.6948 |
| Friedel Multiplier Phi_LO^2 (at Fr_H = 18, We_L = 4,500) | E + 3.24 * F * H / (Fr_H^0.045 * We_L^0.035) | 8.73 |
Friedel Single-Phase Anchor Term E
Evaluates (1 - x)^2 + (x^2) * (rho_L / rho_G) * f_ratio across vapor quality x (B2 in 0..1), density ratio rho_L / rho_G > 1 (B3), and friction-factor ratio f_GO / f_LO (B5).
=IF(OR(B2 <= 0, B2 >= 1, B3 <= 1, B5 <= 0), "Requires 0 < x < 1, rho_L/rho_G > 1, and f_GO/f_LO > 0", ROUND(((1 - B2) ^ 2) + (B2 ^ 2) * B3 * B5, 4))Full Friedel (1979) Two-Phase Liquid-Only Multiplier Phi_LO^2
Evaluates E + (3.24 * F * H) / ((Fr_H^0.045) * (We_L^0.035)) across x (B2), rho_L/rho_G (B3), viscosity ratio mu_G/mu_L < 1 (B4), f_GO/f_LO (B5), Froude Fr_H (B6), and Weber We_L (B7).
=IF(OR(B2 <= 0, B2 >= 1, B3 <= 1, B4 <= 0, B4 >= 1, B5 <= 0, B6 <= 0, B7 <= 0), "Invalid Two-Phase Quality, Property Ratios, Fr_H, or We_L", ROUND(((1 - B2) ^ 2) + (B2 ^ 2) * B3 * B5 + (3.24 * ((B2 ^ 0.78) * ((1 - B2) ^ 0.224)) * ((B3 ^ 0.91) * (B4 ^ 0.19) * ((1 - B4) ^ 0.7))) / ((B6 ^ 0.045) * (B7 ^ 0.035)), 2))Try the example
Use the original values or change them to test your understanding.