Engineering task and calculation objective
This module calculates the maximum (critical) heat flux q̇crit of nucleate boiling per Chapter H2.5.1 of the VDI Heat Atlas (VDI-Wärmeatlas, 12th German edition, 2019). The critical heat flux marks the boiling crisis: above this load, the vapor generated can no longer be removed from the heated surface fast enough, a closed vapor film forms, the heat transfer coefficient collapses, and the wall temperature jumps abruptly upward (burnout).
For the design of evaporators, reboilers, and electrically heated internals, q̇crit is the hard safety limit — particularly with an imposed heat flux, such as electrical heating or condensation of high-pressure steam, exceeding it can thermally destroy the heating surface. Engineers who want to calculate the critical heat flux use here the hydrodynamically based relations of Kutateladze and Zuber with the factor A₄ = 0.13 … 0.16.
The module provides three evaluation variants (Eqs. 27, 28, and 29 with correction K₂) and accounts for the geometry effect of small heaters via a characteristic length (e.g. radius or fin height); the property data of both phases, the surface tension, and the enthalpy of vaporization at the boiling point are required.
Standard and calculation basis: VDI-Wärmeatlas, 12. Auflage 2019
Calculation workflow
- Determine the operating point and property data: At the boiling pressure and the corresponding boiling temperature, the property data of both phases are determined: density of liquid and vapor, enthalpy of vaporization, surface tension, and thermal conductivity, viscosity, and Prandtl number of the liquid.
- Evaluate the basic hydrodynamic relation: The critical heat flux follows from the stability limit of the vapor outflow: q̇_crit = A₄ · Δh_v · ρ''^0.5 · [σ·g·(ρ' − ρ'')]^0.25. The factor A₄ lies between 0.13 and 0.16 and covers the scatter of the measured data.
- Check the geometry effect: For small heaters (thin wires, fins, small tube radii), the characteristic length is compared with the Laplace length; the dimensionless ratio and the correction factor K₂ (Eq. 29) capture the deviation from the large flat heater.
- Select the variant and adopt the result: The module calculates q̇_crit per equations 27, 28, and 29 in parallel; via the selection parameter, the variant governing the geometry is adopted as the result.
- Set the design margin: The operating heat flux is chosen with an adequate safety margin below q̇_crit, since near the crisis even small disturbances (pressure fluctuations, local accumulations) can trigger the transition to film boiling.
Input quantities
| Quantity | Symbol | Unit |
|---|---|---|
| Boiling pressure | ps | Pa |
| Boiling temperature | ϑs | °C |
| Density | ρ' | kg/m³ |
| Specific heat capacity | cp' | J/(kg·K) |
| Thermal conductivity | λ' | W/(m·K) |
| Dynamic viscosity | η' | mPa·s |
| Kinematic viscosity | ν' | m²/s |
| Prandtl number | Pr | – |
| Surface tension | σ | mN/m |
| Density | ρ'' | kg/m³ |
| Heat of evaporation | Δhv | J/kg |
| Characteristic length (e.g. radius; fin height) | L | m |
| Acceleration due to gravity | g | m/s² |
| Factor A4 with A4 = 0.13 ... 0.16 | A4 (27) | – |
| Ratio | L' (29a) | - |
| Correction factor | K2 (29) | - |
Calculated results
| Quantity | Symbol | Unit |
|---|---|---|
| Option (1), (2) or (3) | qKrit | – |
| Critical heat flux acc. to equa. 27 (1) | q̇crit,27 (27) | W/m² |
| Critical heat flux acc. to equa. 29 with correction K2 (3) | q̇crit,L (29) | W/m² |
| Critical heat flux | q̇crit | W/m² |
| Critical heat flux acc. to equa. 28 (2) | q̇crit,28 (28) | W/m² |
Worked example
For water boiling at atmospheric pressure (p = 1.013 bar, ϑs = 100 °C) on a large horizontal heated surface, determine the maximum heat flux of nucleate boiling in this worked example.
Given values
| Fluid | Water at 1.013 bar / 100 °C |
| Enthalpy of vaporization Δhv | 2,257 kJ/kg |
| Liquid density ρ' | 958.4 kg/m³ |
| Vapor density ρ'' | 0.598 kg/m³ |
| Surface tension σ | 0.0589 N/m |
| Gravitational acceleration g | 9.81 m/s² |
| Factor A4 | 0.145 (mean of the range 0.13 … 0.16) |
Solution
Basic relation per Kutateladze/Zuber
q̇crit = A4 · Δhv · (ρ'')0.5 · [σ · g · (ρ' − ρ'')]0.25
Evaluate the buoyancy-capillary term
σ · g · (ρ' − ρ'') = 0.0589 · 9.81 · (958.4 − 0.598) = 553.4 (SI units) → [553.4]0.25 = 4.85
Vapor density term
(ρ'')0.5 = 0.5980.5 = 0.773
Critical heat flux
q̇crit = 0.145 · 2,257,000 · 0.773 · 4.85 ≈ 1.23 · 106 W/m² = 1.23 MW/m²
With the band A4 = 0.13 … 0.16, q̇crit ≈ 1.10 … 1.35 MW/m² is obtained — consistent with the well-known literature value for water at 1 bar.
Result
| Critical heat flux q̇crit | ≈ 1.23 MW/m² |
| Band (A4 = 0.13 … 0.16) | ≈ 1.10 … 1.35 MW/m² |
All values are illustrative. The applicable standard and project-specific boundary conditions remain authoritative.
Frequently asked questions
What happens physically when the critical heat flux is exceeded?
The vapor production at the wall becomes so large that the counter-current flow of departing vapor and incoming liquid becomes hydrodynamically unstable (Taylor/Helmholtz instability). The liquid can no longer reach the wall, a closed vapor film insulates the heated surface, and with an imposed heat flux the wall temperature rises abruptly by several hundred kelvin — up to the point of burning through the wall.
Why is the limit more critical with electrical heating than with steam heating?
With an imposed heat flux (electrical heating, radiation), the system forces the duty independently of the wall temperature — the jump into film boiling leads directly to extreme overtemperature. With an imposed wall temperature (steam or liquid heating with limited temperature), only the transferred duty drops at the transition; the plant loses capacity but is not destroyed.
How do I choose the factor A4 within the range 0.13 to 0.16?
The classical Zuber value for the large flat heated surface is about 0.13; Kutateladze gives roughly 0.16; measured data scatter in between. For a conservative design (smallest q̇_crit), use 0.13; if you are after the most probable value, use the middle of the range and document the band.
Does the calculated limit also apply to tube bundles?
No, it applies to the individual heater in a large pool of liquid. In bundles, the vapor streams of neighboring tubes obstruct one another, so the crisis occurs considerably earlier; for kettle reboilers, bundle-specific load limits must additionally be checked.