Engineering task and calculation objective
This module calculates the heat transfer during boiling in free convection without bubble formation according to the VDI Heat Atlas (VDI-Wärmeatlas, 12th edition 2019, section H2). This regime — often called silent boiling or convective boiling — lies at the beginning of the boiling curve: the wall temperature is above the saturation temperature of the liquid, but the wall superheat is not yet sufficient to activate vapor bubbles at the heating surface. The heat is then removed from the heating surface solely by single-phase natural convection; evaporation takes place only at the free liquid surface.
In practice, this regime matters for lightly loaded evaporator surfaces, heated vessels, and immersion heaters at small temperature differences, as well as for determining the onset of nucleate boiling: only when the convective heat transfer calculated with this module can no longer remove the heat flux does the mechanism switch to nucleate boiling. The calculation uses the classical natural convection correlations based on the Grashof and Prandtl numbers, evaluated with the characteristic length of the heating surface (tube diameter or plate height).
If you want to calculate pool boiling heat transfer, this module determines the convective branch of the boiling curve — Nusselt number, heat transfer coefficient, and heat flux as a function of the temperature difference between wall and saturation temperature.
Standard and calculation basis: VDI-Wärmeatlas, 12. Auflage 2019
Calculation workflow
- Define the boiling state and fluid properties: Starting from the boiling pressure and the corresponding saturation temperature, the properties of the boiling liquid are evaluated: density, specific heat capacity, thermal conductivity, dynamic and kinematic viscosity, and the thermal expansion coefficient.
- Determine geometry and temperature difference: The characteristic length is the diameter for a horizontal tube and the height for a vertical plate. The driving temperature difference ϑ_w − ϑ_s is formed from the wall temperature and the saturation temperature.
- Calculate the Grashof and Prandtl numbers: The Grashof number is formed from the expansion coefficient, gravitational acceleration, temperature difference, characteristic length, and kinematic viscosity; together with the Prandtl number, this yields the product Gr·Pr (Rayleigh number) as the governing parameter of natural convection.
- Evaluate the Nusselt number of natural convection: From Gr·Pr, the Nusselt number is calculated with the natural convection correlation for the respective geometry — proportional to (Gr·Pr)^(1/4) for a laminar boundary layer and to (Gr·Pr)^(1/3) for a turbulent one.
- Heat transfer coefficient and heat flux: The Nusselt number, thermal conductivity, and characteristic length give the heat transfer coefficient; multiplied by the temperature difference, this yields the heat flux of convective boiling. Comparison with the nucleate boiling heat flux shows which mechanism governs.
Input quantities
| Quantity | Symbol | Unit |
|---|---|---|
| Boiling pressure | ps | Pa |
| Boiling temperature | ϑs | °C |
| Density | ρ' | kg/m³ |
| Specific heat capacity | cp' | J/(kg·K) |
| Coefficient of expansion | β' | 1/K |
| Thermal conductivity | λ' | W/(m·K) |
| Dynamic viscosity | η' | mPa·s |
| Kinematic viscosity | ν' | m²/s |
| Length of flow path (Tube: Diameter, Plate: Height) | L | m |
| Wall temperature | ϑw | °C |
| Acceleration due to gravity | g | m/s² |
| Temperature difference (ϑw - ϑs) | Δϑ | K (diff) |
| Prandtl number | Pr | - |
| Grashof number | Gr | - |
| Grashof number ∙ Prandtl number | Gr∙Pr | - |
| Nusselt number | Nu (3);(4) | - |
Calculated results
| Quantity | Symbol | Unit |
|---|---|---|
| Heat transfer coefficient | αK | W/(m²·K) |
| Heat flux | q̇K | W/m² |
Frequently asked questions
How can you tell whether the regime is still silent boiling or already nucleate boiling?
The wall superheat is decisive: for water at atmospheric pressure, nucleate boiling typically sets in only at a wall superheat of a few kelvin. Computationally, both branches are determined — natural convection with this module and nucleate boiling with the pool boiling sections — and compared: the mechanism with the higher heat transfer determines the actual state. The onset of nucleate boiling lies at the intersection of the two curves.
Why are the single-phase natural convection correlations used here, even though boiling occurs?
As long as no bubbles form at the heating surface, the heat transport there is purely single-phase: the superheated liquid rises by buoyancy, and it evaporates only at the free surface (surface evaporation). The heat transfer at the heating surface therefore follows exactly the laws of natural convection with Gr·Pr; the enthalpy of vaporization does not enter the calculation of the wall heat transfer.
What influence does the boiling pressure have on the convective boiling regime?
The pressure acts through the fluid properties and through the position of the transition point: with increasing pressure, the wall superheat needed to activate bubbles decreases, because the nucleation sites become active earlier. The convective regime therefore becomes narrower — at high pressures, boiling switches to nucleate boiling even at small temperature differences, whereas in vacuum evaporators the convective regime persists over larger temperature differences.
Is the wall temperature or the heat flux the correct input quantity?
For temperature-controlled heating surfaces (steam or thermal oil heating), the wall temperature is the natural specification, and the heat flux follows from the calculation. For electrically heated surfaces, the heat flux is imposed; the wall temperature must then be determined iteratively so that the convectively removed heat flux matches the imposed one. In both cases, the fluid properties should be evaluated at the mean boundary layer temperature.