Engineering & Science
Lockhart-Martinelli Parameter (X_tt) & Chisholm Two-Phase Multiplier (φ_L²)
Calculate gas-liquid two-phase pipe frictional pressure drop multiplier φ_L^2 = 1 + C / X_tt + 1 / X_tt^2 from vapor quality x, phase densities, and viscosities.
Why flashing just 15% steam by mass multiplies liquid frictional pressure drop over 160-fold!
In geothermal wells, thermosiphon reboilers, and evaporator headers, low-density vapor accelerates mixtures dramatically—making single-phase Darcy-Weisbach friction hopelessly wrong! Robert W. Lockhart and Raymond C. Martinelli (Chem. Eng. Prog., 1949) correlated two-phase frictional pressure gradient (dP/dz)_TP = φ_L^2 * (dP/dz)_L against the turbulent-turbulent Martinelli parameter X_tt = ((1 - x) / x)^0.9 * (ρ_g / ρ_l)^0.5 * (μ_l / μ_g)^0.1, which Duncan Chisholm (1967) expressed analytically as φ_L^2 = 1 + C / X_tt + 1 / X_tt^2 (with Chisholm constant C = 20 for turbulent liquid + turbulent gas, 12 viscous-turbulent, 10 turbulent-viscous, 5 viscous-viscous)!
Watch atmospheric steam-water at vapor mass quality x = 0.15 (ρ_l = 958.4 kg/m^3, ρ_g = 0.598 kg/m^3, μ_l = 0.000282 Pa·s, μ_g = 0.0000123 Pa·s, C = 20): X_tt = 0.1628 and Chisholm multiplier φ_L^2 reaches 161.60!
| Steam-Water Vapor Quality x (1 atm, C = 20) | Martinelli Parameter X_tt | Two-Phase Multiplier φ_L² | Hydraulic Consequence |
|---|---|---|---|
| x = 0.15 (15% Vapor Mass Quality) | 0.1628 | 161.60× liquid-only ΔP | High Annular-Mist Acceleration |
| x = 0.05 (5% Early Flash) | 0.4832 | 46.63× liquid-only ΔP | Bubbly-to-Slug Flow Transition |
| Invalid x <= 0 or x >= 1 | Single-Phase | "Quality x Must Be Between 0 and 1" | Domain Guard |
Chisholm Two-Phase Frictional Pressure-Drop Multiplier φ_L²
Computes 1 + C / X_tt + 1 / X_tt^2 from vapor quality x (B2), liquid/gas densities B3:B4, liquid/gas viscosities B5:B6, and Chisholm regime constant C (B7).
=IF(OR(B2 <= 0, B2 >= 1, B3 <= 0, B4 <= 0, B5 <= 0, B6 <= 0, B7 <= 0), "Quality x Must Be Between 0 and 1", ROUND(1 + (B7 / ((((1 - B2) / B2) ^ 0.9) * ((B4 / B3) ^ 0.5) * ((B5 / B6) ^ 0.1))) + (1 / (((((1 - B2) / B2) ^ 0.9) * ((B4 / B3) ^ 0.5) * ((B5 / B6) ^ 0.1)) ^ 2)), 2))Turbulent-Turbulent Lockhart-Martinelli Parameter X_tt
Evaluates ((1 - x) / x)^0.9 * (ρ_g / ρ_l)^0.5 * (μ_l / μ_g)^0.1.
=IF(OR(B2 <= 0, B2 >= 1, B3 <= 0, B4 <= 0, B5 <= 0, B6 <= 0), "Quality x Must Be Between 0 and 1", ROUND((((1 - B2) / B2) ^ 0.9) * ((B4 / B3) ^ 0.5) * ((B5 / B6) ^ 0.1), 4))Try the example
Use the original values or change them to test your understanding.