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    Process / two-phase flow

    Two-phase pressure drop

    A boiling or condensing stream is not one fluid. The liquid runs faster than the vapour because it is denser and the vapour is squeezed past it, so the mixture accelerates, the wall shear is set by the liquid film, and the pressure gradient can be ten times what either phase would give alone. The homogeneous and separated-flow models are computed side by side.

    X0.67RegimeSlugModel gap2.4×
    Pressure gradient
    3.2 kPa/m
    33× the liquid-only value
    Total drop
    32.0 kPa
    over 10.0 m
    Void fraction
    0.81
    by volume, Zivi correlation
    Mass flux
    255 kg/m²·s
    Slug or churn
    Two models of the same pipe
    The separated-flow model is closer to reality; the homogeneous one is what a first estimate uses.
    3.2 kPa/m
    Two models of the same pipe
    QuantityValueNote
    Mass flow0.500 kg/sas entered
    Quality0.200vapour mass fraction
    Diameter50 mmas entered
    Mass flux G254.6 kg/m²·sṁ / area
    Homogeneous density91.8 kg/m³1/(x/ρ_g + (1−x)/ρ_l)
    Homogeneous gradient1.34 kPa/mone fluid
    Liquid-only gradient0.10 kPa/mthe reference
    Lockhart-Martinelli X0.673the separated-flow parameter
    Two-phase multiplier32.9×1 + 20/X + 1/X²
    Separated gradient3.20 kPa/mφ² × liquid
    Total drop32.0 kPaover the pipe length
    Void fraction0.813Zivi slip correlation

    The separated-flow model gives 3.2 kPa/m against 1.3 kPa/m homogeneous — a factor of 2.4. At this quality the liquid is travelling at about 5.3 times the mixture velocity, which is why the homogeneous model underestimates the loss.

    Inspect the model
    Choose a quick result view, a scaled comparison, or the complete diagnostic set.
    Pressure gradient against quality
    Rises steeply with quality: the mixture gets bulkier and faster, so the gradient can climb by an order of magnitude.
    Pressure gradient against quality0.0001.352.714.065.416.772%2%5%5%10%10%20%20%40%40%70%70%90%90%kPa/m
    The two models
    The homogeneous model underestimates because it assumes the phases travel together.
    The two models0.0000.7551.512.273.023.780.097Liquid onlyLiquid only1.34HomogeneousHomogeneous3.20SeparatedSeparatedkPa/m
    Canonical equations
    The exact relationships implemented by this engine.

    X = ((1−x)/x)^0.9 (ρ_g/ρ_l)^0.5 (μ_l/μ_g)^0.1 · Separated flow.

    φ_l² = 1 + C/X + 1/X² · C = 20 turbulent.

    ΔP = φ_l² ΔP_liquid · Applied to the liquid-only value.

    α = 1/(1 + ((1−x)/x)(ρ_g/ρ_l) S) · S = (ρ_l/ρ_g)^0.25.

    engineering methodology

    A clear path from inputs to results.

    A boiling or condensing stream is not one fluid. The liquid runs faster than the vapour because it is denser and the vapour is squeezed past it, so the pressure gradient can be ten times what either phase would give alone.

    Step-by-step methodWorked exampleAssumptions explained

    How to use

    1. Enter the mass flow and vapour quality.
    2. Set the pipe diameter and length.
    3. Add the phase densities and viscosities.
    4. Compare the homogeneous and separated-flow models.

    The formula

    ΔP=ϕl2 ΔPl,X=(1−xx)0.9(ρgρl)0.5(μlμg)0.1\Delta P = \phi_l^2 \, \Delta P_l, \quad X = \left(\frac{1-x}{x}\right)^{0.9}\left(\frac{\rho_g}{\rho_l}\right)^{0.5}\left(\frac{\mu_l}{\mu_g}\right)^{0.1}

    Reference: Lockhart-Martinelli separated flow; Zivi void correlation.

    Worked example

    0.5 kg/s at 20% quality in a 50 mm pipe

    Inputs
    • 0.5 kg/s, 20% quality, 50 mm bore, 10 m long.
    • Liquid 900 kg/m³ and 0.0002 Pa·s; vapour 20 kg/m³ and 0.000015 Pa·s.
    Calculation path
    1. 1

      The mass flux is 255 kg/m²·s.

    2. 2

      The Lockhart-Martinelli parameter is 0.673.

    3. 3

      The two-phase multiplier is 1 + 20/0.673 + 1/0.673² = 32.9.

    4. 4

      The liquid-only gradient is 97 Pa/m, so the two-phase gradient is 3,195 Pa/m.

    5. 5

      The homogeneous model gives 1,337 Pa/m — less than half, because it assumes the phases travel together.

    Result

    0.673 X · 32.9 multiplier · 3.2 kPa/m.

    Interpreting the result

    The homogeneous model is only reasonable near the critical point, where the two phases have similar densities. Everywhere else it underestimates, often by a factor of two or more, and the gap widens with quality because the slip between the phases grows.

    Assumptions & limitations

    • Friction gradient only — no acceleration or elevation terms.
    • Lockhart-Martinelli with C = 20 (turbulent-turbulent).
    • Constant quality along the pipe.
    • Zivi void with a slip ratio of (ρ_l/ρ_g)^0.25.

    Frequently asked questions

    Why does quality matter so much?

    Because the vapour occupies most of the volume even at low mass fraction. At 20% quality the void fraction is already 0.81, so the mixture is far bulkier and much faster than the liquid alone.

    Why is the homogeneous model so far off?

    Because it assumes both phases move at the same velocity. In reality the vapour travels several times faster, which raises the wall shear and the acceleration.

    What about boiling along the pipe?

    Quality rises along the length, so the gradient must be integrated. A single-quality calculation is only valid for a short section or an evaporator with a small quality change.

    What is the Lockhart-Martinelli parameter?

    The square root of the ratio of liquid to vapour pressure gradients if each flowed alone. Small X means vapour-dominated, large X liquid-dominated.

    Does the pipe orientation matter?

    Enormously, for the total drop. The friction term here is orientation-independent, but the elevation term can dominate in a vertical riser.

    Methodology last reviewed 2026-10-22. Results are informational and not financial or engineering advice — see our disclaimer.