Heating It Hotter Made It Cooler — The Boiling Curve and Critical Heat Flux

In 1934, in Tokyo, Nukiyama raised the current through a nichrome wire and watched the bubbles. Between the most violent boiling and the moment the wire melted, he saw nothing. In that gap the heat flux does not rise with the excess temperature.

Under Temperature control, dragging the ΔTe slider walks the operating point along the curve. Switch to Heat-flux control and one heat flux splits into two wall temperatures.

Push the flux past the peak and the point jumps from the nucleate branch to the film branch. That jump is where Nukiyama's wire melted.

The Boiling Curve — Four Regimes in One Graph#

Boiling is evaporation at a solid-liquid boundary. The wall heat flux is written

$q'' = h\,(T_s - T_{sat}) = h\,\Delta T_e$

$T_s$ is the wall temperature, $T_{sat}$ the saturation temperature (water, 1 atm, 100°C), and $\Delta T_e$ the excess temperature. $h$ is the convection coefficient.

A larger excess temperature looks like it should raise $q''$ monotonically. On log-log axes four regimes show up in order.

• Free convection ($\Delta T_e \lesssim 5$°C): almost no bubbles. Buoyancy-driven natural convection carries the heat.
• Nucleate boiling ($5 \lesssim \Delta T_e \lesssim 30$°C): bubbles rise from nucleation sites. $q''$ is steepest here.
• Transition boiling ($30 \lesssim \Delta T_e \lesssim 120$°C): a vapor film starts to cover the wall. $q''$ falls.
• Film boiling ($\Delta T_e \gtrsim 120$°C): a continuous vapor film blankets the wall. Radiation joins in and $q''$ rises again.

The point in the figure walks those four regimes. Hold the flux fixed and the transition branch is not a stable operating point. The point jumps to nucleate boiling or to film boiling.

Nucleate Boiling — Bubbles Pump the Heat#

Nucleate boiling cools well less because of the bubble itself than because cold liquid rushes into the spot the bubble left. Liquid near the wall keeps getting replaced.

Rohsenow wrote the flux in this regime as

$\mu_l$ is the liquid viscosity, $h_{fg}$ the latent heat, $\rho_l,\rho_v$ the liquid and vapor densities, $\sigma$ the surface tension, $c_{p,l}$ the liquid specific heat, and $Pr_l$ the liquid Prandtl number. $C_{s,f}$ is an experimental constant for the surface-liquid pair. Water on platinum is 0.013.

$q''_s$ rises with the cube of $\Delta T_e$. A small increase in excess temperature moves a lot of flux. Reactor cores, CPU cold plates, and boilers aim at this narrow band. Latent heat leaving by diffusion is a different path. The same latent heat drying a surface by diffusion still runs when the wall is below saturation.

Critical Heat Flux — The Cliff Zuber Predicted#

The cube does not last. Pack the bubbles tightly enough and they merge into a blanket. Liquid can no longer reach the wall. Just before that, $q''$ peaks. That peak is the critical heat flux (CHF). Water at 1 atm exceeds 1 MW/m².

Zuber wrote CHF from the Kelvin–Helmholtz breakup of the vapor columns.

The symbols match the previous relation, and $\rho_v$ is the vapor density. Wall temperature and surface material are absent. CHF is fixed by fluid properties alone.

On a heat-flux-controlled surface, crossing CHF throws the operating point from nucleate boiling to film boiling. Holding the same flux in film boiling takes a wall hundreds of degrees hotter. Metal melts first. That is the jump Nukiyama saw.

Film Boiling and Leidenfrost#

At the end of transition boiling, $q''$ reaches a minimum again. That point is the Leidenfrost point. Beyond it a continuous vapor film covers the wall and liquid never touches the solid. Heat crosses the vapor by conduction and radiation only. Vapor conducts far worse than liquid.

On a moderately hot pan a droplet boils and vanishes. On a very hot pan it sits on its own vapor film and lasts longer.

Film boiling on a horizontal cylinder has this shape.

$D$ is the diameter, $\nu_v, k_v$ the vapor kinematic viscosity and conductivity, and $C$ a shape constant (0.62 for a horizontal cylinder, 0.67 for a sphere). $h'_{fg} = h_{fg} + 0.8\,c_{p,v}(T_s - T_{sat})$ is latent heat corrected for vapor superheat. Excess temperature sits to the one-fourth power in the denominator, so a hotter wall makes $h$ duller, not sharper.

Python — Drawing the Boiling Curve#

Rohsenow nucleate boiling and the Zuber CHF go on one table, using water properties at 1 atm.

import numpy as np
 
# Water, saturated properties at 1 atm
g    = 9.81       # m/s^2
rho_l = 957.9     # kg/m^3
rho_v = 0.5956
h_fg  = 2257e3    # J/kg
sigma = 0.0589    # N/m
mu_l  = 279e-6    # N.s/m^2
cp_l  = 4217.0    # J/kg.K
Pr_l  = 1.76
Csf, n = 0.013, 1.0   # water-platinum
 
def rohsenow_nucleate_flux(dTe):
    """Rohsenow nucleate boiling heat flux q'' (W/m^2)."""
    bracket = np.sqrt(g * (rho_l - rho_v) / sigma)
    ja = cp_l * dTe / (Csf * h_fg * Pr_l**n)
    return mu_l * h_fg * bracket * ja**3
 
def zuber_critical_heat_flux():
    """Zuber critical heat flux (W/m^2)."""
    return 0.149 * h_fg * rho_v * (
        sigma * g * (rho_l - rho_v) / rho_v**2) ** 0.25
 
chf = zuber_critical_heat_flux()
print(f"CHF (Zuber)      = {chf/1e6:.3f} MW/m^2")
 
# Scan the nucleate branch and find where it reaches CHF
for dTe in [5, 10, 20, 30, 40]:
    q = rohsenow_nucleate_flux(dTe)
    flag = "  <-- exceeds CHF (Rohsenow invalid)" if q > chf else ""
    print(f"dTe={dTe:3d} C : q''={q/1e6:6.3f} MW/m^2{flag}")

The run prints this.

CHF (Zuber)      = 1.259 MW/m^2
dTe=  5 C : q''= 0.017 MW/m^2
dTe= 10 C : q''= 0.137 MW/m^2
dTe= 20 C : q''= 1.094 MW/m^2
dTe= 30 C : q''= 3.692 MW/m^2  <-- exceeds CHF (Rohsenow invalid)

Drag the surface-temperature slider from 90°C to 400°C and the droplet shifts from boiling on the wall to riding on a vapor film.

Near 130°C it throws off bubbles and disappears quickly. Past 200°C it is hotter and yet lasts longer. The 30°C row in the table is the nucleate end already past CHF. The film on this slider is the other side of the same curve.

Eight times the 10°C row, 0.137, is 1.096. The 20°C row is 1.094. Twenty-seven times is 3.699, and the 30°C row is 3.692. Doubling and tripling the excess temperature lands on the cube, inside the printed table.

Divide flux by excess temperature and $h$ comes out: 3400 W/m²K at 5°C, 13700 at 10°C, 54700 at 20°C. Double the excess temperature and $h$ quadruples. If $q''$ goes as the cube, $h$ goes as the square.

The ratio of the CHF 1.259 to the 20°C flux 1.094 is 1.15. Its cube root is about 1.048, and $20\times 1.048$ sits near 21°C. Rohsenow meets the Zuber ceiling near 21°C, not on the 30°C row. The 30°C value 3.692 is 2.93 times 1.259. That row is a number printed after the correlation has left its range.

Eight times the 5°C row is 0.136, against the printed 0.137. Dividing 3.692 by 30 gives 123067 W/m²K. Multiplying 54700 by 2.25 gives 123075. The square still holds at the printed digits, on a row that has already left the ceiling.

Rohsenow crossing CHF at 30°C does not mean the algebra is wrong. It has left its range. The real curve bends down at CHF. Used together, the two relations have to enforce 1.259 MW/m² as a ceiling.

The Language of Dimensionless Numbers — Jakob and Bond#

The Jakob number is the sensible heat the liquid takes up, divided by the latent heat.

$Ja = \frac{c_{p,l}\,\Delta T_e}{h_{fg}}$

A small $Ja$ means latent heat dominates sensible heat. Water's latent heat is large, so $Ja$ stays small in boiling. The term inside the Rohsenow brackets is this $Ja$.

The Bond number is buoyancy divided by surface tension.

$Bo = \frac{g(\rho_l - \rho_v)\,L^2}{\sigma}$

A bubble's length $L$ comes from that balance. The diameter scales as $D_b \propto \sqrt{\sigma / [g(\rho_l-\rho_v)]}$. That is why surface tension and the density difference sit inside the fourth root of the CHF relation. Bubble size and departure frequency set the ceiling on the flux.

Designing on the 30°C row that already left the correlation#

At 20°C the flux is already 87% of the ceiling, and the two relations meet near 21°C. Write 3.692 from the 30°C row down as a nucleate design flux and the wall cannot accept that flux in nucleate boiling. The operating point jumps to film boiling, and the wall temperature that holds the same $q''$ climbs by hundreds of degrees.

Whether a vapor film still lets friction predict the heat flux is a separate question. The analogy that reads heat flux off wall friction alone belongs to a wall the liquid still touches. Once a film is in the way, this table does not say whether that analogy survives.