Melting of polar ice caps is a topic of current interest due to global warming and its impact. But not long back in human history, in times of lesser pollution and implication, the inverse problem of solidification or growth of polar ice was of interest. During mid nineteenth century Arctic expeditions to study polar ice caps intended to obtain proper first hand information about those regions for scientific investigation. British and German expeditions to the Arctic regions recorded the time of growth of ice and the air temperature on several occasions, sitting in ships, either frozen up in winter quarters, or drifting with the ice. Data from such expeditions led to the formulation and partial solution of, what is known today as the Stefan problem or the moving boundary problem.
The working of Stefan’s diathermometer to measure the thermal conductivity of gases and how that knowledge helped him in predicting the T power fouth radiation law was explained earlier in two separate notes. This note is the third part that recounts Stefan’s analytical contribution to the understanding of solidification.
We shall use content from the recent Crepeau’s article  that recounts Stefan’s achievements and also two other sources, a historical perspective of the Stefan problem by Vuik, available as pdf on the web  and the scale analysis from Bejan’s text book .
In a nut shell Stefan’s problem of melting has to do with finding for known initial and boundary conditions (in terms of temperature), the ice growth rate or how thick ice grows in time. Say, the water in the Arctic ocean is at the freezing point (see figure below). For certain days the air above the water would be at a temperature much below freezing and would remain that way. So the water begins to freeze. Can one predict the growth rate of ice? The answer Stefan gave in his paper was, the thickness of ice growth depends on the square root of the time involved. Or the square of the thickness is a linear function of time. Of course, Stefan went on to write four papers on this topic detailing his analysis (relaxing many of the earlier assumptions) and compared his calculations with actual polar ice growth rate data taken during the British and German expeditions around 1865.
Before briefing on these history, here is a brief math on the simplified one dimensional melting or moving boundary problem that Stefan considered. The analysis can be found in standard modern heat transfer text books. What I show in the picture is from my class notes but adapted from Bejan’s text book , as it discusses the problem through scale analysis.
(click on image for a larger version)
In the figure above, the movement of the solidification front is governed by the conservation of energy at the x = d(t) plane. The Control Volume (CV) – marked by the dashed rectangle – is assumed to move downwards at a speed equal to the solidification front progress dd/dt so that the liquid flow rate into the CV can be imagined to equal the solid ‘flow rate’ leaving the CV. Therefore the energy balance (Eq. (1) in figure) follows as a straightforward first approximation. In Eq. (1) A is the frontal area (see schematic at the left bottom), hf is the specific enthalpy of the liquid, hs is the specific enthalpy of solid, k is the thermal conductivity of the liquid. In Stefan’s initial analysis, he assumed the air-ice interface at the top to remain at a constant temperature. Since the liquid that is entering the CV is solidifying, the energy released in this process can be expressed as the LHS in Eq. (1), basically mass times the latent heat of fusion times the solidification front velocity. This is equated to the conduction heat transfer through the ice (the driving thermal gradient for the soidification). Stefan assumed this conduction to result in an approximate linear temperature profile (Eq. (2)) resulting in the simplification of Eq.(1) to Eq. (3), which upon an integration and rearrangement gives Eq. (4), the conclusion that the solidification thickness or ice pack growth depends on the square root of time. Observe the rest of the parameters on the RHS of Eq. (4) is already known. (although I write that the rest of the parameters are known, the thermal conductivity of ice that Stefan used was, according to , 20 percent smaller than what is the currently accepted value.)
Stefan did much beyond this. He relaxed the assumption of fixed T for air-ice interface and made it a function of time. The revised model gave better comparison with the recorded data of both the British and German expeditions. This comparison with actual data is given in Vuik’s article . Stefan then proceeded to formulate the transient heat diffusion model with a moving boundary at one end (ice-water interface) and solved it with reasonable assumptions via a transcendental equation. The first approximation of the equation of course, would result in the simplified result we discussed above. This analysis is detailed nicely in Crepeau’s article  and also in Vuik’s article  and I skip it for brevity of this note.
Now for some history. Firstly, the Stefan problem was analysed even before Stefan investigated it. To quote from :
Unbeknownst to Stefan, some work on the moving boundary problem had already been done. In 1762, Joseph Black, a professor of medicine at the University of Glasgow in Scotland, studied the icewater phase change problem and identified the phenomenon of latent heat, while Franz Neumann presented solutions to the moving boundary problem in a series of lectures given around 1860. However, his work was not published until 1901 by Weber H. Weber, Die partiellen Differential-Gleichung der Mathematischen Physik, nach Riemanns Vorlesungen II, Braunshweig, 1901, pp. 118122.
Another interesting fact I learn from  is
Because Stefan’s journal of choice, the Sitzungberichte der Kaiserlichen Akademie der Wissenschaften of Vienna was not widely distributed and his results were considered important, his entire paper was reprinted in the Annalen der Physik und Chemie in 1891 J. Stefan, Ueber die Theorie der Eisbildung, insbesondere ber die Eisbildung im Polarmeere, Annalen der Physik und Chemie 42 (1891), pp. 269286., which had a higher circulation. For this reason dual references to this same work exist.
According to Crepeau , Soviet scientists Dacev and Rubenstein were those who gave the name The problem of Stefan in late 1940s, while attempting to solve the generalized version of it. The ratio of sensible heat to latent heat of a (phase change) material written as Ste = cP?T/hsf (first by Lock in 1969 – from ) is nowadays identified as the Stefan number. Interest in the moving boundary problem is picking up over the last several decades – average number of publications per year increase from 0.1 in 1931-40 to 55 in 1981-82 [see 2]. Searching Scirus, the portal collecting several hundred journals in the sciences and engineering, with a key phrase as Stefan Problem for the last decade (1999 – 2009) picks up 634 peer reviewed articles.
Melting obviously is the reverse of the solidification problem discussed. In figure above, flip vertically the bottom left schematic for the Stefan melting problem; accordingly, the left top solidification curves need to be flipped horizontally (T0 > Tm). Melting problems find engineering interest for instance in the investigation of phase change material applications such as composite heat sinks to cool electronics and several manufacturing processes. Nowadays however the phase change process is solved using numerical methods in complex geometries. The benchmarking is nevertheless done with the Stefan analytical result, as the example figure below shows.
In summary, after going through the works of Josef Stefan – the thermal conductivity measurements for gases, radiation law from experiments, and the analysis of Arctic ice growth, one is awed by what natural talent coupled with enthusiasm and perseverance can achieve as first rate science. I think Stefan is one of those who remained faithful to who a scientist is in one important quality – not wondering about the external classification of their work by others, but worked on anything and everything that pricks their curiosity and interest with whatever resources they could mobilize. And as it happens often, have nevertheless managed to contribute both to fundamental understanding and particular applications of a subject.
Narasimhan, A., (2013), "The Scientific Legacy of Josef Stefan," Chapter 11, pp. 200-220, in Jožef Stefan: His Scientific Legacy on the 175th Anniversary of His Birth, ed. John Crepeau, Bentham Press. [DOI: 10.2174/97816080547701130101 | Product Link]
- Crepeau, J. (2007). Josef Stefan: His life and legacy in the thermal sciences Experimental Thermal and Fluid Science, 31 (7), 795-803 DOI: 10.1016/j.expthermflusci.2006.08.005
- C. Vuik, Some historical notes on the Stefan problem, Nieuw Archief voor Wiskunde, 4e series, 11 (1993) 157167. Available as pdf.
- A. Bejan, Heat Transfer (1995), John Wiley, NY.