Engineering task and calculation objective
This module calculates the reference value α₀ for nucleate boiling using the method of Stephan and Preußer according to the VDI Heat Atlas (VDI-Wärmeatlas, 12th edition 2019, section H2). In pool boiling in the nucleate boiling regime, the heat transfer coefficient is usually calculated in a normalized representation: starting from a reference value α₀ at defined reference conditions — reduced pressure p* = p/p_crit = 0.03 and a reference heat flux — the value at the actual operating point is scaled via a pressure function and a heat flux exponent. If no experimental reference value is available for a substance, α₀ must be calculated from the fluid property data — which is exactly what this module does.
The Stephan-Preußer correlation represents nucleate boiling heat transfer through dimensionless groups formed from the properties of the boiling liquid and its vapor at the reference pressure: the densities of both phases, thermal conductivity, thermal diffusivity, dynamic viscosity, surface tension, and enthalpy of vaporization. The characteristic length is the bubble departure diameter, which is calculated from the surface tension, the density difference, and the contact angle.
If you want to calculate nucleate boiling heat transfer for a substance without measured data — new working fluids, mixture components, or special substances in process engineering — this module delivers the code-conform reference value as the starting point for the boiling curve calculation according to the VDI Heat Atlas.
Standard and calculation basis: VDI-Wärmeatlas, 12. Auflage 2019
Calculation workflow
- Define the reference state: From the critical pressure of the substance, the reference pressure is determined as p* = p/p_crit = 0.03, together with the corresponding saturation temperature. All fluid properties are evaluated at this reference state.
- Capture the properties of both phases: Required are, for the liquid, the density, specific heat capacity, thermal conductivity, dynamic viscosity, and thermal diffusivity; for the vapor, the density; and, in addition, the surface tension and the enthalpy of vaporization at the reference pressure.
- Calculate the bubble departure diameter: The bubble departure diameter — the characteristic length of nucleate boiling — is determined from the surface tension, the density difference between liquid and vapor, gravitational acceleration, and the substance-dependent contact angle.
- Form the dimensionless groups: Using the reference heat flux, the dimensionless groups of the Stephan-Preußer correlation are formed — among them a boiling number built from the heat flux and the departure diameter, the vapor-to-liquid density ratio, a group containing the enthalpy of vaporization, and the Prandtl number of the liquid.
- Evaluate the Nusselt number and the reference value: The product of the dimensionless groups raised to their exponents gives the Nusselt number of nucleate boiling, referred to the departure diameter, and from it the heat transfer coefficient at the reference state — the sought reference value α₀. Via the pressure function F(p*), this value can then be converted to the actual operating pressure.
Input quantities
| Quantity | Symbol | Unit |
|---|---|---|
| Critical pressure | pc | Pa |
| Heat flux (Reference value) | q̇0 | W/m² |
| Acceleration due to gravity | g | m/s² |
| Contact angle for equation 12 | β | ° |
| Design pressure (p* =0.03) | p0.03 | Pa |
| Boiling temperature (p* =0.03) | ϑ0.03 | °C |
| Density | ρ' | kg/m³ |
| Specific heat capacity | cp' | J/(kg·K) |
| Thermal conductivity | λ' | W/(m·K) |
| Dynamic viscosity | η' | mPa·s |
| Thermal diffusivity | a' | m²/s |
| Surface tension | σ' | mN/m |
| Density | ρ'' | kg/m³ |
| Heat of evaporation | Δhv | J/kg |
| Bubble diameter at departure | d0 (12) | m |
| Characteristic | 1 π1 | - |
| Characteristic | 2 π2 | - |
| Characteristic | 3 π3 | - |
| Characteristic | 4 π4 | - |
| Characteristic | 5 π5 | - |
| Nusselt number | Nu (11) | - |
| Heat transfer coefficient | α0.03 | W/(m²·K) |
| Pressure function | F0.03 (8) | - |
Calculated results
| Quantity | Symbol | Unit |
|---|---|---|
| Heat transfer coefficient | α0 | W/(m²·K) |
Frequently asked questions
What is the reference value α₀ for, and why exactly at p* = 0.03?
The boiling curves of many substances can be represented in normalized form (α/α₀ as a function of q/q₀ and p*) in a nearly substance-independent way. The reference point at p* = 0.03 has become the established normalization state because measured values exist there for a very large number of substances and the pressure function of the VDI Heat Atlas is referred to this point. The reference value anchors the entire boiling curve; all operating points are scaled relative to it.
When should α₀ be calculated instead of using tabulated values?
The VDI Heat Atlas tabulates experimentally validated reference values for numerous substances — these are always preferable to a calculated value. The Stephan-Preußer correlation is the tool for substances without measured data: new refrigerants, organic intermediates, or special substances. Since it extrapolates from pure property data, larger uncertainties must be expected than with measured values; typical deviations of a few tens of percent are possible.
What role does the contact angle play in the calculation?
The contact angle between bubble and heating wall enters the calculation of the bubble departure diameter and characterizes the wetting behavior of the fluid/wall system. For water, a considerably larger contact angle is usually assumed than for well-wetting organic liquids and refrigerants. Since the departure diameter enters the Nusselt number as the characteristic length, an incorrect contact angle assumption affects α₀ directly.
Does the calculated value also apply to mixtures and to flow boiling?
No. The correlation applies to pool boiling of pure substances. For mixtures, the mass transfer resistance at the bubble interface reduces the heat transfer, in some cases considerably; dedicated mixture corrections exist for this. In flow boiling in a tube, the convective and nucleate boiling contributions are superimposed — there, α₀ serves only as a building block within the methods for boiling under forced convection.