Engineering task and calculation objective
This module calculates heat transfer in flow boiling beyond the boiling crisis according to Section H3.7 of the VDI Heat Atlas (VDI-Wärmeatlas, 12th edition 2019), the standard German reference work for heat transfer. In the so-called post-dryout regime the liquid no longer wets the tube wall: depending on the vapor quality, either inverted annular film boiling or mist flow (droplets in superheated vapor) prevails. The heat transfer coefficient here is markedly lower than in nucleate boiling, and the wall temperature correspondingly high.
This calculation is needed wherever evaporator tubes are operated beyond dryout, whether by design or under upset conditions: in once-through steam generators, waste-heat boilers, in the design of emergency-cooling scenarios, or when re-rating overheated tube sections. Anyone who wants to calculate heat transfer after the boiling crisis must know, in addition to the system pressure, above all the Leidenfrost temperature — it defines the wall temperature above which rewetting of the wall is impossible.
The VDI Heat Atlas explicitly accounts for the thermodynamic non-equilibrium between superheated vapor and droplets not yet evaporated, which largely governs the actual heat transfer in mist flow.
Standard and calculation basis: VDI-Wärmeatlas, 12. Auflage 2019
Calculation workflow
- Determine the flow pattern after the boiling crisis: Based on system pressure, mass flux, and vapor quality, the module establishes whether inverted film boiling (vapor film at the wall, liquid core) or mist flow (dispersed droplets in the vapor) is present — the validity ranges of the individual correlations are stored in the module.
- Determine the Leidenfrost temperature: The Leidenfrost temperature is calculated from the pressure-dependent relations of the VDI Heat Atlas. If the wall temperature lies above it, the wall cannot be rewetted and post-dryout heat transfer governs; below it, a return to wetted boiling is possible.
- Calculate heat transfer in film boiling or mist flow: The heat transfer correlations are evaluated for the respective regime: in film boiling via conduction through the vapor film with a radiation contribution, in mist flow via convective transfer to the superheated vapor including the heat transported to the entrained droplets.
- Account for thermodynamic non-equilibrium: In mist flow the droplets evaporate more slowly than equilibrium would demand; the vapor superheats while the actual vapor quality lags behind the equilibrium quality. This coupling is taken into account iteratively when determining the wall temperature and heat flux.
- Assess the wall temperature: The wall temperature follows from the heat flux and the calculated heat transfer coefficient; it is checked against the allowable material temperature of the tube.
Input quantities
| Quantity | Symbol | Unit |
|---|---|---|
| System pressure | p | Pa |
| Mass velocity | ṁ | kg/(m²·s) |
| Tube diameter | d | m |
| Acceleration due to gravity | g | m/s² |
| Critical pressure | pc | Pa |
| Critical temperature | ϑc | °C |
| Boiling temperature | ϑs | °C |
| Enthalpy | h' | J/kg |
| Enthalpy | h'' | J/kg |
| Heat of evaporation | Δhv | J/kg |
| Density | ρ' | kg/m³ |
| Density | ρ'' | kg/m³ |
| Specific heat capacity | cp'' | J/(kg·K) |
| Thermal conductivity | λ'' | W/(m·K) |
| Dynamic viscosity | η'' | mPa·s |
| Surface tension | σ | mN/m |
| Liquid temperature | ϑl | °C |
| Density | ρl | kg/m³ |
| Specific heat capacity | cpl | J/(kg·K) |
| Thermal conductivity | λl | W/(m·K) |
| Dynamic viscosity | ηl | mPa·s |
| Mean gas temperature | ϑg | °C |
| Density | ρg | kg/m³ |
| Specific heat capacity | cpg | J/(kg·K) |
Calculation options
System pressure
Droplet flow in thermodynamic nonequilibrium · Droplet flow in thermodynamic equilibrium · Film boiling during flow through vertical tubes · Leidenfrost temperature · Characteristic boiling curve for heat transfer after the boiling crisis · Film boiling during flow over horizontal tube
Frequently asked questions
What does the Leidenfrost temperature mean in practice?
The Leidenfrost temperature is the wall temperature above which a stable vapor film persists between liquid and wall, so that the wall is no longer wetted. For design this means: above this temperature, the poor post-dryout heat transfer persists even if the heat flux drops again — only after falling below the Leidenfrost temperature does the heat transfer jump back to wetted boiling (hysteresis).
Why is an equilibrium calculation not sufficient in mist flow?
After dryout, the vapor absorbs the heat directly from the wall and superheats, while the droplets evaporate only with a delay. The real vapor quality is therefore lower and the vapor temperature higher than at equilibrium. A pure equilibrium calculation can underestimate the wall temperature considerably — which is why the models of the VDI Heat Atlas work with thermodynamic non-equilibrium.
Is the radiation contribution relevant in film boiling?
In film boiling, wall temperatures of several hundred degrees Celsius occur; from roughly 300 to 400 °C of wall superheat, radiation through the vapor film contributes noticeably to the overall heat transfer and is superimposed on the convective components. It should therefore not be neglected when re-rating hot tubes.
Do the correlations apply equally to horizontal and vertical tubes?
The basic correlations are formulated for vertical tubes with upward flow. In horizontal tubes, phase separation leads to circumferentially non-uniform heat transfer — the top of the tube dries out first. The VDI Heat Atlas states the respective validity ranges; the module displays these conditions ('Valid for') explicitly.