What is the Stefan problem?

A heat-conduction problem, like melting or freezing, in which the solid–liquid boundary (front) moves with time. Latent heat is absorbed or released at the front, whose position is itself unknown. In the simplest case, with temperature varying in only one phase, Neumann's exact solution exists and the front advances with the square root of time.

How far can the quasi-steady approximation be used?

Assuming a linear liquid temperature profile works well at small Stefan number (ratio of sensible to latent heat). In the example it ran 1.6% fast at 0.1 and 3.2% at 0.2, but 14% at 1 and 25% at 2. Use the exact solution when sensible heat is large.

What to watch when solving phase change numerically?

The enthalpy method avoids tracking the front and is convenient, but each time a cell finishes melting, temperature and heat flux change stepwise — a 'staircase' oscillation. In the example, front error was just 0.02 mm on 25 cells, yet wall heat flux oscillated ±2%, dropping to ±0.1% on 100 cells. Choose the grid for the quantity you need.

Stefan problem

Category: Phase Change | Revised 2026-10-01

Stefan Problem

Theory: conduction with a moving front

Overview

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I want to calculate how fast a phase-change material melts. How does it differ from ordinary conduction?

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The difference is that the boundary (front) between melted liquid and remaining solid moves with time. The front absorbs latent heat to melt, so heat is spent advancing the front and barely reaches beyond. Problems where the front position is part of the answer are called Stefan problems. The simplest case has an exact solution, often used to verify numerical codes. Start with the exact solution to see how fast the front moves.

Neumann's exact solution

$$ s(t) = 2\lambda\sqrt{\alpha t}, \qquad \lambda\,e^{\lambda^2}\,\mathrm{erf}(\lambda) = \frac{Ste}{\sqrt{\pi}}, \qquad Ste = \frac{c\,(T_w - T_m)}{L} $$

Solid initially at melting point $T_m$, melted by a wall held at $T_w$. The front $s$ advances with the square root of time, with coefficient $\lambda$ set solely by the Stefan number $Ste$ (sensible-to-latent heat ratio). $\alpha$ is the liquid thermal diffusivity.

The enthalpy method

$$ \frac{\partial H}{\partial t} = \nabla\cdot(k\nabla T), \qquad T = \begin{cases} T_m & 0 \le H \le \rho L \\ T_m + (H - \rho L)/(\rho c) & H > \rho L \end{cases} $$

Solving for enthalpy $H$ (heat content including latent heat) instead of temperature handles phase change without tracking the front. A melting cell stays at the melting point while its enthalpy rises by the latent heat.

Coffee Break Trivia

A problem born of polar ice thickness

The problem is named after the Slovenian-born physicist Stefan, who around 1889 studied how fast polar sea ice thickens. Comparing with Arctic expedition data, he showed ice thickness grows with the square root of time. An exact solution of the same form had reportedly been presented earlier by Neumann in lectures, and it is now often called the Neumann solution. It is the starting point for moving-front heat problems: freezing, casting solidification, thermal storage and welding.

Worked examples

Example 1: exact versus quasi-steady

Assuming a linear liquid profile (quasi-steady) gives $\lambda = \sqrt{Ste/2}$:

Stefan number	0.01	0.1	0.2	0.5	1	2
Exact λ	0.0706	0.2200	0.3064	0.4648	0.6201	0.8006
Quasi-steady error	+0.2%	+1.6%	+3.2%	+7.6%	+14.0%	+24.9%

Example 2: melting paraffin

Paraffin (conductivity 0.2 W/mK, density 800 kg/m³, specific heat 2,000 J/kgK, latent heat 200 kJ/kg, melting point 50 °C) melted by a 70 °C wall (Stefan 0.2, convection ignored). The front reaches 13.0 mm in 1 h, 26.0 mm in 4 h and 63.7 mm in 24 h; wall heat flux at 4 h is 158.7 W/m².

Example 3: enthalpy-method accuracy

Cells (50 mm)	Front at 4 h	Error	Wall-flux oscillation
25	26.022 mm	0.021 mm	−1.8% to +2.6%
50	26.009 mm	0.008 mm	—
100	26.004 mm	0.003 mm	±0.1%
200	26.002 mm	0.001 mm	—

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The front is almost exact even on 25 cells, yet the heat flux oscillates.

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In the enthalpy method a cell stays pinned at the melting point until it finishes melting, then starts to warm. So the temperature gradient seen from the wall — the heat flux — changes stepwise each time a cell melts. The front position, from summing melted fractions, follows the overall energy balance and comes out accurately. But for wall heat flux or a point's temperature history, refine the grid or smooth by spreading latent heat over a small temperature range around the melting point.

Analysis workflow

Compute the Stefan number to see whether sensible or latent heat dominates.
Get a first estimate of front motion from the 1D exact solution.
Use the enthalpy or apparent-heat-capacity method and verify against the exact solution under the same conditions.
Where liquid natural convection matters (e.g. PCM melting), solve the flow too.
For alloys with a melting range, include the mushy zone.

Coffee Break Trivia

“It melted twice as fast as calculated”

A thermal-storage unit's paraffin, designed with conduction-only calculations, melted about twice as fast as predicted in the prototype. Warmed liquid rose and melted the upper solid first: natural convection in the liquid carried far more heat than conduction alone. Melting in a vertical container differs completely from a horizontal one. The exact Stefan solution is an important verification starting point, but real devices need the flow solved as well.

Common mistakes

Mistakes and fixes

Mistake	Effect	Fix
Quasi-steady at large Stefan number	Front too fast	Exact or numerical solution
Heat flux on a coarse grid	Staircase oscillation	Finer grid or smoothing
Ignoring liquid convection	Melting time overestimated	Solve the flow
Ignoring volume change	Voids and pressure missed	Account for density change
Ignoring supercooling	Solidification onset wrong	Measure and include supercooling

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I'd like to try related calculations.

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Try the PCM melting simulator, Chvorinov solidification-time tool and sensible/latent heat ratio tool. Related topics include the enthalpy method, melting and solidification, the mushy zone and phase-change materials.

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