Engineering task and calculation objective
The L2.1 module calculates the phase fractions of gas-liquid flows according to the VDI Heat Atlas (VDI-Wärmeatlas), 12th edition 2019 — the standard German reference for heat transfer and two-phase flow. The central quantity is the volumetric gas fraction (void fraction) ε — the share of the flow cross-section occupied by the gas phase. It differs from the flow quality ẋ, the mass-based gas fraction, because gas and liquid flow at different velocities (slip). The module calculates the void fraction in the cross-section as well as the pressure-drop components derived from it: the static-head (geodetic) pressure drop, the acceleration pressure drop and the friction pressure drop.
Being able to calculate the void fraction of a two-phase flow is a prerequisite for many design tasks: density and weight of the inventory in evaporator and riser tubes, natural-circulation calculations for steam generators, the static-head share in riser lines, level and inventory determination, and the assessment of flow patterns. The module treats pipes and channels as well as longitudinally flowed tube bundles and accounts for the inclination of the flow path relative to the horizontal.
The computational basis is the drift-flux model: the void fraction follows from the homogeneous gas fraction, corrected by a distribution parameter that captures the interaction of void and velocity profiles, and by the drift velocity with which bubbles rise relative to the mixture due to buoyancy. Fluid properties such as the densities of both phases and the surface tension enter directly.
Standard and calculation basis: VDI-Wärmeatlas, 12. Auflage 2019
Calculation workflow
- Define the flow path and fluid properties: The geometry (pipe/channel or longitudinally flowed tube bundle), cross-sectional area, hydraulic diameter, length of the flow path and inclination angle to the horizontal are defined, together with the fluid properties: densities of gas and liquid and the surface tension.
- Form the flow quality and mass flux: From the mass flow rates of gas and liquid, the total mass flow rate, the mass flux and the mass-based gas fraction ẋ follow. Optionally, the gas fraction can be assumed constant or linearly varying along the flow path, for example for heated evaporator tubes.
- Calculate the homogeneous gas fraction: As a reference quantity, the volumetric gas fraction of the homogeneous model is determined — the value that would result if both phases flowed at the same velocity. It follows solely from the quality and the density ratio of the phases.
- Evaluate the distribution parameter and drift velocity: According to the drift-flux model, the distribution parameter C0 (profile shape of void fraction and velocity across the cross-section) and the weighted drift velocity of the bubbles are calculated from fluid properties, mass flux and inclination. This yields the real void fraction ε, which lies below the homogeneous value in upward flow because of slip.
- Determine the pressure-drop components: With the void fraction, the mean mixture density follows and from it the static-head pressure drop over the height of the flow path; with varying void fraction, the acceleration pressure drop from the momentum change is added. Together with the friction pressure drop, the total pressure drop of the two-phase flow results.
Input quantities
| Quantity | Symbol | Unit |
|---|---|---|
| Mass flow | M | kg/s |
| Mass flux | m | kg/m²s |
| Gas fraction | x | – |
| Density gas | ρg | kg/m³ |
| Density liquid | ρl | kg/m³ |
| Cross section area | A | m² |
| Gas part | ε | – |
| Homogenous gas fraction | εhom | – |
| Distribution parameter | C0 | – |
| Drift velocity | ugj | m/s |
| Hydraulic diameter | dh | m |
| Surface tension | σ | N/m |
| Geodetic pressure loss | Δpg | Pa |
| Angle | φ | ° |
| Length | L | m |
| Mass flow liquid | Ml | kg/s |
| Mass flow gas | Mg | kg/s |
| Tube = 1 Tube bundle = 2 | Rohrbündel | – |
| Gas fraction constant = 1 linear in flow direction = 2 | Strömrichtun | – |
| Gas fraction high | xh | – |
| Gas fraction low | xn | – |
| Acceleration pressure drop | Δpa | Pa |
| Pressure drop | Δp | Pa |
| Friction pressure drop | Δpr | Pa |
Worked example
In an evaporator tube, a mixture flows with a flow quality of ẋ = 0.1 (10 % vapor by mass). The density of the gas phase is 5 kg/m³, that of the liquid 900 kg/m³. In this worked example, find the volumetric void fraction according to the homogeneous model (equal phase velocities) as the starting quantity of the drift-flux calculation.
Given values
| Flow quality ẋ | 0.1 |
| Gas density ρG | 5 kg/m³ |
| Liquid density ρL | 900 kg/m³ |
Solution
Homogeneous gas fraction
In the homogeneous model, both phases flow at the same velocity; the void fraction follows from the volume flows:
εhom = 1 / [1 + ((1 − ẋ)/ẋ) · (ρG/ρL)]
εhom = 1 / [1 + (0.9/0.1) · (5/900)] = 1 / (1 + 0.05) = 0.952
Interpretation
Although only 10 % of the mass is vapor, the gas fills about 95 % of the cross-section — a consequence of the 1:180 density ratio. The mean homogeneous mixture density is only ρhom = ε·ρG + (1−ε)·ρL = 0.952·5 + 0.048·900 ≈ 47.6 kg/m³.
The drift-flux model of the module subsequently corrects this value downward using the distribution parameter and the drift velocity, since the gas flows faster than the liquid in upward flow.
Result
| Homogeneous void fraction εhom | 0.952 |
| Homogeneous mixture density | approx. 47.6 kg/m³ |
All values are illustrative. The applicable standard and project-specific boundary conditions remain authoritative.
Frequently asked questions
What is the difference between flow quality ẋ and void fraction ε?
The flow quality ẋ is the mass share of the gas in the total mass flow rate; the void fraction ε is the volume or cross-sectional share of the gas phase. Because of the low gas density, even a small ẋ corresponds to a large ε: at ẋ = 0.1, ε can exceed 0.9. Confusing the two quantities is one of the most common errors in two-phase calculations — for the mixture density and the static-head pressure drop, ε alone is decisive.
Why does the homogeneous model deliver excessive void fractions in vertical upward flow?
The homogeneous model assumes equal velocities of both phases. In reality, the gas rises faster due to buoyancy (drift) and additionally concentrates in the fast core flow (distribution parameter C0 > 1). Both effects shorten the residence time of the gas in the cross-section and lower the actual void fraction below the homogeneous value. The drift-flux model corrects precisely these two effects.
What is the acceleration pressure drop needed for?
If the gas fraction increases along the flow path — for example in a heated evaporator tube — the mixture density drops and the mixture must be accelerated. The momentum change required for this appears as an additional pressure drop. At high mass fluxes and strong evaporation, this component can reach the order of magnitude of the friction component; in adiabatic flows with constant gas fraction it vanishes.
What role does the inclination angle of the flow path play?
The inclination angle acts twice: it determines the static-head share via the elevation difference and, via the buoyancy component, it influences the drift velocity and the flow pattern. In horizontal lines, the mixture stratifies at low velocities; in vertical upward flow, drift takes full effect; in downward flow, its sign reverses. The correlations of the module account for the inclination explicitly.