Engineering task and calculation objective
This module calculates the pressure drop in flow boiling through evaporator tubes according to Section H3.3 of the VDI-Wärmeatlas (VDI Heat Atlas, 12th edition, 2019). In evaporator tubes of steam generators, falling film and circulation evaporators, refrigerant evaporators and thermosiphon reboilers, a two-phase mixture flows whose vapor content increases along the tube. Anyone who wants to calculate the two-phase pressure drop must capture three contributions: the frictional pressure drop, the static (geodetic) pressure drop and the acceleration pressure drop caused by the expansion of the evaporating mixture.
Since vapor content, densities and phase velocities change along the tube length, the evaporator tube is numerically divided into segments and the pressure drop is integrated stepwise – the module controls this via the number of iteration steps. For the description of the two-phase flow (homogeneous model or heterogeneous model with slip and volumetric vapor fraction) and for the frictional pressure drop, several model approaches documented in the VDI Heat Atlas are available for selection.
The result is particularly critical for the design of natural circulation systems: there, the circulation mass flow settles exactly where driving head and pressure drop are in equilibrium – an accurate two-phase pressure drop calculation is therefore a prerequisite for a stable evaporator design.
Standard and calculation basis: VDI-Wärmeatlas, 12. Auflage 2019
Calculation workflow
- Define boundary conditions and fluid properties: Tube geometry (diameter, length, inclination), mass flux and the saturation properties of both phases (densities, viscosities, surface tension) at the operating pressure are entered. Vapor mass fractions at inlet and outlet define the evaporation range.
- Select models: The model selection defines how the two-phase flow is described (e.g. homogeneous model or heterogeneous model with slip and volumetric vapor fraction) and which approach is used for the frictional pressure drop (e.g. two-phase multiplier method).
- Divide the tube into segments: The tube is divided into segments according to the chosen number of iteration steps; in each segment the calculation uses the local vapor mass fraction, which progresses between the inlet and outlet values.
- Calculate local pressure gradients: For each segment, the local frictional pressure drop from the two-phase model, the local static pressure drop from the mean mixture density and tube inclination, and the change in momentum flux as the basis of the acceleration contribution are determined.
- Sum up the contributions: Integration over all segments yields the frictional, static and acceleration pressure drops; their sum is the total pressure drop of the evaporator tube. The breakdown shows which mechanism dominates at the actual operating point.
Input quantities
| Quantity | Symbol | Unit |
|---|---|---|
| Density liquid phase | ρl | kg/m³ |
| Density vapour phase | ρg | kg/m³ |
| Diameter | di | m |
| Vapor mass fraction | ẋ | - |
| Mass velocity | ṁ | kg/(m²·s) |
| Froude number | Fr | - |
| Dyn. viscosity liquid phase | μl | mPa·s |
| Dyn. viscosity vapour phase | μg | mPa·s |
| Friction factor | ζL0 | - |
| Local frictional pressure drop | (dp/dl)Reibung | Pa/m |
| Two phase multiplicator | Φ | - |
| Friction factor | ζ | - |
| Angle of inclination of the tubes | θ | ° |
| Void fraction in homogeneous flow | ξ | - |
| Local static pressure drop | (dp/dl)statisch | Pa/m |
| Impulse flow | İ | N |
| Length of boiler tube | L | m |
| Vapor mass fraction inlet | ẋein | - |
| Vapor mass fraction outlet | ẋaus | - |
| Number of iteration steps | n | – |
| Frictional pressure drop | ΔpReibung | Pa |
| Static pressure drop | Δpstatisch | Pa |
| Kinetic pressure drop | ΔpBeschleunigung | Pa |
| Total pressure drop | Δpgesamt | Pa |
Calculation options
Model to describe two-phase flow
homogeneous · heterogeneous
Model for calculating the frictional pressure loss
from model · heck
Frequently asked questions
Why can't the pressure drop simply be calculated single-phase with averaged fluid properties?
During boiling, vapor content and mixture density change by orders of magnitude along the tube; moreover, the frictional pressure drop of a two-phase mixture – expressed by the two-phase multiplier – lies well above that of the pure liquid flow. A single-phase calculation with mean values systematically underestimates the pressure drop and does not capture the acceleration contribution at all.
What distinguishes the homogeneous from the heterogeneous two-phase model?
The homogeneous model treats vapor and liquid as a mixture with a common velocity and averaged properties – simple, but only good for finely dispersed flows with little slip. The heterogeneous model allows different phase velocities (slip) and requires a relation for the volumetric vapor fraction; in particular it describes the static pressure contribution at low mass fluxes considerably better.
When is the acceleration pressure drop relevant?
Whenever the vapor content increases strongly over the tube length, i.e. at high heat fluxes, low pressures (large density differences between vapor and liquid) and high mass fluxes. It results from the change in momentum flux between inlet and outlet and can make up a noticeable share of the total pressure drop in evaporators; it must always be accounted for as pressure-reducing in the flow direction.
What is the role of the number of iteration steps?
It sets the resolution of the tube discretization. Since the pressure gradients depend non-linearly on the local vapor content, too coarse a division leads to integration errors, especially in the range of small vapor contents where the mixture density changes fastest. In practice, the number of steps is increased until the total pressure drop no longer changes appreciably.