Engineering task and calculation objective
The LOMA module determines the flow pattern of a two-phase liquid–gas or liquid–vapor flow in a pipe and plots the operating point on the Lockhart-Martinelli flow pattern map. The calculation is based on chapter H3.4 of the VDI Wärmeatlas (VDI Heat Atlas) covering flow boiling of saturated liquids. Whether bubble, plug, stratified, wavy or annular flow develops largely decides which correlations for heat transfer and pressure drop are applicable at all.
In practice, flow pattern determination is needed when designing evaporators, circulation reboilers, boiler tubes and two-phase piping in plant engineering. Critical conditions such as stratified flow with a dry upper pipe surface, or plug flow with pressure pulsations, can be identified early and avoided by design. To calculate a two-phase flow, LOMA first determines the Martinelli parameter as the central characteristic quantity and reads the position on the map from it.
The module calculates all required dimensionless parameters from the physical properties of both phases, the vapor quality and the mass flux, and assigns the operating point to a flow regime on the map.
Standard and calculation basis: VDI Wärmeatlas
Calculation workflow
- Collect physical properties and operating data: For the operating state, the pressure, mass flow rate or mass flux, vapor quality and the physical properties of both phases are required: density and dynamic viscosity of the liquid and of the gas/vapor at saturation conditions.
- Determine the apparent single-phase pressure gradients: Following the Lockhart-Martinelli approach, the frictional pressure gradient is calculated that would occur if each phase flowed through the pipe alone. For each phase, it is checked whether the flow is laminar or turbulent.
- Form the Martinelli parameter X: The Martinelli parameter is the square root of the ratio of the two single-phase pressure gradients. X > 1 indicates liquid-dominated flow, X < 1 gas-dominated flow. For the common turbulent/turbulent case, a closed-form expression exists in terms of vapor quality, density ratio and viscosity ratio.
- Calculate the additional ordinate parameters of the map: Depending on the region of the map, further dimensionless numbers are formed that weigh inertial, gravitational and surface forces against each other, such as Froude-based parameters for the transition from stratified to wavy and intermittent flow.
- Plot the operating point on the flow pattern map: The point formed by the Martinelli parameter and the associated ordinate value is entered into the diagram. The regime it falls into gives the flow pattern, e.g. bubble, plug, slug, stratified, wavy or annular flow.
- Assess the result: The identified flow pattern is assessed with regard to operational safety: annular flow is usually desirable for evaporator tubes, whereas stratified flow with a dry upper pipe surface or pulsating plug flow is critical in heated tubes.
Input quantities
| Quantity | Symbol | Unit |
|---|---|---|
| Density | ρL ρG | kg/m³ |
| Density | ρL ρG | kg/m³ |
| Dynamic viscosity | ηL ηG | mPa·s |
| Dynamic viscosity | ηL ηG | mPa·s |
| Angle of inclination | Θ | ° |
| Vapour mass fraction | ẋ | – |
| Mass flux | ṁ | kg/(m²·s) |
| Inner diameter of tube | d | m |
| Surface tension | σ | N/m |
Calculated results
| Quantity | Symbol | Unit |
|---|---|---|
| Pressure drop coefficient | ζL ζG | – |
| Pressure drop coefficient | ζL ζG | – |
| Press. drop coeff. at phase interface | ζPh | – |
| Void fraction | ε | – |
| Reynolds number | ReL ReG | – |
| Reynolds number | ReL ReG | – |
| Parameter | (ReL∙Fr'G)0.5 | – |
| Parameter | (FrGm)0.5 | – |
| Parameter | (Fr∙Eu)L0.5 | – |
| Parameter | (Fr∙Eu)G | – |
| Parameter | (We/Fr)L | – |
| Martinelli parameter | X | – |
| Liquid height | h | m |
| Reference liquid height | hL | – |
| Periphery | UL UG | – |
| Periphery | UL UG | – |
| Width of phase interface | Ui | – |
| Cross-sectional area | fL fG | – |
| Cross-sectional area | fL fG | – |
| Unwetted arc | φ | ° |
| or | Bogen | rad |
| Factor | ψ | ° |
| or | Faktor | rad |
| Parameter for stratified flow | ReLFr'G | – |
Worked example
A saturated water–steam mixture flows in a horizontal evaporator tube at 10 bar (saturation temperature approx. 180 °C) with a vapor quality of x = 0.2. Required is the Martinelli parameter Xtt (both phases turbulent) as input for the flow pattern map — a worked example of how to calculate two-phase flow according to the Lockhart-Martinelli method.
Given values
| Pressure (saturation) | 10 bar |
| Vapor quality x | 0.2 |
| Density of water ρ' | 887 kg/m³ |
| Density of saturated steam ρ'' | 5.15 kg/m³ |
| Dynamic viscosity of water η' | 0.150 mPa·s |
| Dynamic viscosity of steam η'' | 0.015 mPa·s |
Solution
Formula for the Martinelli parameter (turbulent/turbulent)
For the case where both phases flow turbulently, the closed form applies:
Xtt = ((1 − x)/x)0.9 · (ρ''/ρ')0.5 · (η'/η'')0.1
Calculate the individual factors
((1 − 0.2)/0.2)0.9 = 40.9 = 3.482
(5.15/887)0.5 = 0.0762
(0.150/0.015)0.1 = 100.1 = 1.259
Martinelli parameter
Xtt = 3.482 · 0.0762 · 1.259 ≈ 0.33
Since Xtt < 1, the vapor phase dominates the frictional pressure drop. Together with the gas-side ordinate parameter, the point is plotted on the flow pattern map; at sufficiently high mass flux, the operating point typically lies in the annular flow regime, which is desirable for heated evaporator tubes.
Result
| Martinelli parameter Xtt | ≈ 0.33 |
| Assessment | gas-dominated flow (Xtt < 1) |
All values are illustrative. The applicable standard and project-specific boundary conditions remain authoritative.
Frequently asked questions
For which flow situations does the Lockhart-Martinelli diagram apply?
The classic flow pattern map applies to adiabatic or quasi-adiabatic gas–liquid flows in horizontal pipes with only moderately varying vapor quality. In flow boiling, the vapor quality changes along the pipe, so the operating point migrates across the map; the map must then be applied section by section to local conditions. For vertical pipes, different flow pattern maps with different transition boundaries apply.
What does the Martinelli parameter X mean physically?
X compares the frictional pressure gradients that the liquid and the gas would each generate flowing alone in the pipe. For X well below 1, the gas phase dominates, indicating annular or spray flow; for X well above 1, the liquid dominates and bubble or plug flow is more likely. X is also an input to two-phase pressure drop correlations via the two-phase multiplier.
Why is the flow pattern so important for evaporator design?
Heat transfer correlations for flow boiling assume a wetted heating surface. With stratified or wavy flow in a horizontal tube, the upper pipe surface can dry out, causing the local heat transfer to collapse and the wall temperature to rise. Plug and slug flow additionally cause pressure and mass flow pulsations that can impair control and mechanical integrity.
What are typical sources of error in applying the method?
Common errors are using physical properties at the wrong reference temperature instead of at saturation, confusing vapor quality with volumetric gas fraction, and applying the turbulent/turbulent formula even though one phase is laminar. Transferring the map to strongly foaming or viscous systems is also inadmissible, since the transition boundaries were determined for air–water-like systems.