Phase change problems in industry

Phase change problems, where a substance changes its state of matter (liquid, gas, solid), are ubiquitous in industrial processes. Applications range from casting in metallurgy to boiling and freezing in food manufacture. One of the challenges in modelling phase change is tracking the interface that separates the phases, as it moves over time. Here is a brief summary of two examples of such problems (also known as Stefan problems), one on the casting of silicon, the other on frying crisps.

Solidification of silicon

Silicon, a brittle shiny metalloid, has an enormous range of applications, most notably in electronics and solar cells. In the casting process of silicon, liquid silicon is poured into a metal mould and allowed to cool and solidify. It is well known that the cooling rates and mould size affect the microstructure, and this determines the quality of the final product. In general, the faster the cooling, the smaller the grain size.

During a 3 month internship at Elkem, Norway, I researched how the shape of the cast affected the cooling rates, and consequently the grain quality [1,2]. By formulating a moving-boundary Stefan problem for the solidification process, I studied both thin slab-shaped and wedge-shaped geometries.

The image to the right is a close up picture of silcon that has been solidified in a wedge-shaped geometry. The cooling near the edges was much faster than in the interior, resulting in larger grain sizes within.

Predicting lift-off time during crisp frying

When frying potato crisps, it is typically observed that the dough, which is submerged in hot oil, after some critical time increases its buoyancy and floats to the surface. The lift-off time is a useful metric in ensuring that the crisps are properly cooked.

During a workshop with a crisp company, together with a group of applied mathematicians, we created a multiphase mathematical model for the frying of potato crisps, where water inside the dough is evaporated from both the top and bottom surfaces of the crisp at two receding evaporation fronts [3]. The vapour created at the top of the crisp bubbles away to the surface, whereas the vapour released from the bottom surface forms a buoyant, yet insulating, blanket layer.

We used our mathematical model to predict the change in the crisp density as a function of time, and investigated how lift-off time depends on the different parameters of the problem.

[1] Benham, G.P., Hildal, K., Please, C.P. and Van Gorder, R.A. Solidification of silicon in a one-dimensional slab and a two-dimensional wedge. International Journal of Heat and Mass Transfer, 98, 530-540, (2016). [pdf]

[2] Benham, G.P., Hildal, K., Please, C.P. and Van Gorder, R.A. Penetration of molten silicon into a bed of fines. International Communications in Heat and Mass Transfer, 75, 323-327, (2016). [pdf]

[3] Babb, T., Benham, G.P., Gonzalez-Farina, R., Kiradjiev, K.B., Lee, W.T., Tibos, S. Predicting lift-off time when deep-frying potato dough snacks, arXiv preprint arXiv:1910.04458. (2019). [pdf]