Engineering task and calculation objective
This module calculates film condensation of metal vapors according to Section J1.6 of the VDI Heat Atlas (VDI-Wärmeatlas, 12th edition 2019), the standard German reference work for heat transfer. Liquid metals such as sodium, potassium, or mercury have extremely small Prandtl numbers and very high thermal conductivities: the conductive resistance of the condensate film, which governs the heat transfer for ordinary fluids, is almost meaningless here. Instead, the molecular-kinetic resistance at the phase interface becomes governing — classical Nusselt theory therefore overestimates the heat transfer, in some cases by orders of magnitude.
This calculation is needed for liquid-metal loops, for example in sodium heat transfer systems, alkali-metal heat pipes, or metallurgical condensation processes. The module first determines the heat transfer coefficient according to Nusselt and then corrects it via the phase-change number and the accommodation coefficient, which specifies what fraction of the vapor molecules striking the phase interface actually condenses.
The results are the interface temperature, the corrected heat transfer coefficient, and the mass flux and heat flux at the phase interface — the basis for designing metal-vapor condensers per the VDI Heat Atlas.
Standard and calculation basis: VDI-Wärmeatlas, 12. Auflage 2019
Calculation workflow
- Define the vapor and wall states: Inputs are the pressure and temperature of the (possibly superheated) metal vapor, the saturation temperature corresponding to the pressure, the wall temperature, as well as the molar mass and enthalpy of vaporization of the substance.
- Evaluate the properties of the liquid metal: Thermal conductivity, dynamic viscosity, and specific heat capacity of the liquid metal are determined at the mean film temperature; from these follows the Prandtl number, which for liquid metals is typically far below 0.1.
- Calculate the heat transfer according to Nusselt: First, the heat transfer coefficient of the condensate film is determined per classical Nusselt film theory — it describes only the conductive resistance of the film and would be unrealistically high for metal vapors.
- Apply the molecular-kinetic correction: Using the accommodation coefficient, the molar mass, and the vapor pressure at the interface temperature, the mass transfer resistance at the phase interface is calculated from kinetic gas theory; the phase-change number yields the correction factor for the heat transfer coefficient.
- Iterate the interface temperature and flux densities: Since the vapor pressure at the phase interface depends on the initially unknown interface temperature, this temperature is determined iteratively; at the converged state, the corrected heat transfer coefficient, the mass flux at the phase interface, and the heat flux follow.
Input quantities
| Quantity | Symbol | Unit |
|---|---|---|
| Pressure of the superheated vapor | p∞ | Pa |
| Temperature of the superheated vapor | ϑ∞ | °C |
| Heat of evaporation | Δhv | J/kg |
| Wall temperature | ϑW | °C |
| Molar mass | M | kg/kmol |
| Dynamic viscosity of the liquid metal | ηF | mPa·s |
| Thermal conductivity of the liquid metal | λF | W/(m·K) |
| Specific heat capacity of the liquid metal | cpF | J/(kg·K) |
| Heat transfer coefficient acc. to Nusselt | α | W/(m²·K) |
| Prandtl number of the liquid metal | PrF | - |
| Accommodation coefficient | σAkk | - |
| Critical pressure pcrit | Pc | Pa |
| Critical temperature ϑcrit | Tc | °C |
| Acentric factor | ω | - |
| Constant | A | - |
| Constant | B | - |
| Constant | C | - |
| Constant | W1 | - |
| Constant | W2 | - |
| Constant | W3 | - |
| Constant | W4 | - |
| Condensation temperature at the pressure p∞ | ϑS,∞ | °C |
| Dampfdruckgleichung | Dampfdruckgleichung | – |
Calculated results
| Quantity | Symbol | Unit |
|---|---|---|
| Vapor pressure at phase layer temperature | pPh | Pa |
| Mean film temperature | ϑF | °C |
| Phase change number | Ph | - |
| Correction factor for the heat transfer coefficient | φ (55) | - |
| Corrected heat transfer coefficient | α+ (55) | W/(m²·K) |
| Mass flux at phase layer | ṁ (52) | kg/(m²·s) |
| Heat flux | q̇ | W/m² |
| Phase layer temperature | ϑPh (53) | °C |
Calculation options
Dampfdruckgleichung
Ambrose-Walton equation · Antoine equation · Vapor pressure equation 1
Frequently asked questions
Why does classical Nusselt theory fail for metal vapors?
Nusselt theory assumes saturation equilibrium at the phase interface and that the entire resistance lies in the condensate film. For liquid metals, however, the film is nearly resistance-free because of the high thermal conductivity; the limiting factor is the molecular-kinetic process of condensation itself. A temperature jump develops between vapor and interface that only the kinetic correction captures — without it, the heat transfer is overestimated by orders of magnitude.
What is the accommodation or condensation coefficient?
It specifies the fraction of vapor molecules striking the liquid surface that actually condenses instead of being reflected. For very clean metal surfaces it is close to 1, but it drops considerably with contaminants and oxide layers. Since it enters the interfacial resistance quadratically, or at least strongly nonlinearly, it is the largest source of uncertainty in the calculation — published values scatter substantially.
Does the absolute pressure play a special role?
Yes. The molecular-kinetic resistance grows strongly with decreasing pressure, because fewer molecules strike the phase interface. Metal vapor condensation often takes place at low pressures (e.g. in heat pipes), so the interfacial resistance frequently dominates there; at higher pressures the result approaches the Nusselt solution again.