Can hot water freeze faster than cold water? (2024)

[Physics FAQ] - [Copyright]

Original by Monwhea Jeng (Momo),
Department of Physics, University of California, 1998.

Yes—a general explanation

History of the Mpemba Effect

More-detailed explanations

References

Hot water can in fact freeze faster than cold water for a wide range of experimentalconditions. This phenomenon is extremely counterintuitive, and surprising even tomost scientists, but it is in fact real. It has been seen and studied in numerous experiments. Although this phenomenon has been knownfor centuries, and was described by Aristotle, Bacon, and Descartes [1–3], it was not introduced to the modern scientificcommunity until 1969, by a Tanzanian high school pupil named Mpemba. Both the earlyscientific history of this effect, and the story of Mpemba's rediscovery of it, areinteresting in their own right — Mpemba's story in particular providing a dramaticparable against making snap judgements about what is impossible. This is described separately below.

The phenomenon that hot water may freeze faster than cold is often called the Mpembaeffect. Because, no doubt, most readers are extremely skeptical at this point, weshould begin by stating precisely what we mean by the Mpemba effect. We start withtwo containers of water, which are identical in shape, and which hold identical amounts ofwater. The only difference between the two is that the water in one is at a higher(uniform) temperature than the water in the other. Now we cool both containers,using the exact same cooling process for each container. Under some conditions theinitially warmer water will freeze first. If this occurs, we have seen the Mpembaeffect. Of course, the initially warmer water will not freeze before the initiallycooler water for all initial conditions. If the hot water starts at 99.9°C, andthe cold water at 0.01°C, then clearly under those circ*mstances, the initially coolerwater will freeze first. But under some conditions the initially warmer waterwill freeze first: if that happens, you have seen the Mpemba effect. But you willnot see the Mpemba effect for just any initial temperatures, container shapes, or coolingconditions.

This seems impossible, right? Many sharp readers may have already come up with acommon proof that the Mpemba effect is impossible. The proof usually goes somethinglike this. Say that the initially cooler water starts at 30°C and takes 10minutes to freeze, while the initially warmer water starts out at 70°C. Now theinitially warmer water has to spend some time cooling to get to get down to 30°C, andafter that, it's going to take 10 more minutes to freeze. So since the initiallywarmer water has to do everything that the initially cooler water has to do, plus a littlemore, it will take at least a little longer, right? What can be wrong with thisproof?

What's wrong with this proof is that it implicitly assumes that the water ischaracterized solely by a single number — its average temperature. But ifother factors besides the average temperature are important, then when the initiallywarmer water has cooled to an average temperature of 30°C, it may look very differentthan the initially cooler water (at a uniform 30°C) did at the start. Why?Because the water may have changed when it cooled down from a uniform 70°C to anaverage 30°C. It could have less mass, less dissolved gas, or convectioncurrents producing a non-uniform temperature distribution. Or it could have changedthe environment around the container in the refrigerator. All four of these changesare conceivably important, and each will be considered separately below. So theimpossibility proof given above doesn't work. And in fact the Mpemba effect has beenobserved in a number of controlled experiments [5,7–14]

It is still not known exactly why this happens. A number of possible explanationsfor the effect have been proposed, but so far the experiments do not show clearly which,if any, of the proposed mechanisms is the most important one. While you will oftenhear confident claims that X is the cause of the Mpemba effect, such claims are usuallybased on guesswork, or on looking at the evidence in only a few papers and ignoring therest. Of course, there is nothing wrong with informed theoretical guesswork or beingselective in which experimental results you trust; the problem is that different peoplemake different claims as to what X is.

Why hasn't modern science answered this seemingly simple question about cooling water?The main problem is that the time it takes water to freeze is highly sensitive to a numberof details in the experimental setup, such as the shape and size of the container, theshape and size of the refrigeration unit, the gas and impurity content of the water, howthe time of freezing is defined, and so on. Because of this sensitivity, whileexperiments have generally agreed that the Mpemba effect occurs, they disagree over theconditions under which it occurs, and thus about why it occurs. As Firth [7] wrote "There is a wealth of experimental variation in theproblem so that any laboratory undertaking such investigations is guaranteed differentresults from all others."

So with the limited number of experiments done, often under very different conditions,none of the proposed mechanisms can be confidently proclaimed as "the" mechanism.Above we described four ways in which the initially warmer water could have changed uponcooling to the initial temperature of the initially cooler water. What follows belowis a short description of the four related mechanisms that have been suggested to explainthe Mpemba effect. More ambitious readers can follow the links to more completeexplanations of the mechanisms, as well as counter-arguments and experiments that themechanisms cannot explain. It seems likely that there is no one mechanism thatexplains the Mpemba effect for all circ*mstances, but that different mechanisms areimportant under different conditions.

1. Evaporation — As the initially warmer water cools to the initial temperature of the initially cooler water, it may lose significant amounts of water to evaporation. The reduced mass will make it easier for the water to cool and freeze. Then the initially warmer water can freeze before the initially cooler water, but will make less ice. Theoretical calculations have shown that evaporation can explain the Mpemba effect if you assume that the water loses heat solely through evaporation [11]. This explanation is solid, intuitive, and evaporation is undoubtedly important in most situations. But it is not the only mechanism. Evaporation cannot explain experiments that were done in closed containers, where no mass was lost to evaporation [12]. And many scientists have claimed that evaporation alone is insufficient to explain their results [5,9,12].
2. Dissolved Gasses — Hot water can hold less dissolved gas than cold water, and large amounts of gas escape upon boiling. So the initially warmer water may have less dissolved gas than the initially cooler water. It has been speculated that this changes the properties of the water in some way, perhaps making it easier to develop convection currents (and thus making it easier to cool), or decreasing the amount of heat required to freeze a unit mass of water, or changing the boiling point. There are some experiments that favor this explanation [10,14], but no supporting theoretical calculations.
3. Convection — As the water cools it will eventually develop convection currents and a non-uniform temperature distribution. At most temperatures, density decreases with increasing temperature, and so the surface of the water will be warmer than the bottom: this has been called a "hot top." Now if the water loses heat primarily through the surface, then water with a "hot top" will lose heat faster than we would expect based on its average temperature. When the initially warmer water has cooled to an average temperature the same as the initial temperature of the initially cooler water, it will have a "hot top", and thus its rate of cooling will be faster than the rate of cooling of the initially cooler water at the same average temperature. Got all that? You might want to read this paragraph again, paying careful distinction to the difference between initial temperature, average temperature, and temperature. While experiments have seen the "hot top", and related convection currents, it is unknown whether convection can by itself explain the Mpemba effect.
4. Surroundings — A final difference between the cooling of the two containers relates not to the water itself, but to the surrounding environment. The initially warmer water may change the environment around it in some complex fashion, and thus affect the cooling process. For example, if the container is sitting on a layer of frost which conducts heat poorly, the hot water may melt that layer of frost, and thus establish a better cooling system in the long run. Obviously explanations like this are not very general, since most experiments are not done with containers sitting on layers of frost.

Finally, supercooling may be important to the effect.Supercooling occurs when the water freezes not at 0°C, but at some lowertemperature. One experiment [12] found that its initiallyhot water supercooled less than its initially cold water. This would mean that theinitially warmer water might freeze first because it would freeze at a higher temperaturethan the initially cooler water. If true, this would not fully explain the Mpembaeffect, because we would still need to explain why initially warmer water supercools lessthan initially cooler water.

In short, hot water does freeze sooner than cold water under a wide range ofcirc*mstances. It is not impossible, and has been seen to occur in a number ofexperiments. But despite claims often made by one source or another, there isno well-agreed explanation for how this phenomenon occurs. Different mechanisms havebeen proposed, but the experimental evidence is inconclusive. For those wishing toread more on the subject, Jearl Walker's article in Scientific American [13] is very readable and has suggestions on how to do homeexperiments on the Mpemba effect, while the articles by Auerbach [12] and Wojciechowski [14] are two ofthe more modern papers on the effect.

The fact that hot water freezes faster than cold has been known for manycenturies. The earliest reference to this phenomenon dates back to Aristotle in 300B.C. The phenomenon was later discussed in the medieval era, as European physicistsstruggled to come up with a theory of heat. But by the 20th century the phenomenonwas only known as common folklore, until it was reintroduced to the scientific communityin 1969 by Mpemba, a Tanzanian high school pupil. Since then, numerous experimentshave confirmed the existence of the "Mpemba effect", but have not settled on any singleexplanation.

The earliest known reference to this phenomenon is by Aristotle, who wrote:

"The fact that water has previously been warmed contributes to its freezingquickly; for so it cools sooner. Hence many people, when they want to coolhot water quickly, begin by putting it in the sun. . ." [1,4]

He wrote these words in support of a mistaken idea which he calledantiperistasis. Antiperistasis is defined as "the supposed increase in the intensityof a quality as a result of being surrounded by its contrary quality, for instance, thesudden heating of a warm body when surrounded by cold" [4].

Medieval scientists believed in Aristotle's theory of antiperistasis, and also soughtto explain it. Not surprisingly, scientists in the 1400s had trouble explaining howit worked, and could not even decide whether (as Aristotle claimed in support ofantiperistasis), human bodies and bodies of water were hotter in the winter than in thesummer [4]. Around 1461, the physicist Giovanni Marliani,in a debate over how objects cooled, said that he had confirmed that hot water frozefaster than cold. He said that he had taken four ounces of boiling water, and fourounces of non-heated water, placed them outside in similar containers on a cold winterday, and observed that the boiled water froze first. Marliani was, however, unableto explain this occurrence [4].

Later, in the 1600s, it was apparently common knowledge that hot water would freezefaster than cold. In 1620 Bacon wrote "Water slightly warm is more easily frozenthan quite cold" [2], while a little later Descartes claimed"Experience shows that water that has been kept for a long time on the fire freezes soonerthan other water" [3].

In time, a modern theory of heat was developed, and the earlier observations ofAristotle, Marliani, and others were forgotten, perhaps because they seemed socontradictory to modern concepts of heat. But it was still known as folkloreamong many non-scientists in Canada [11], England [15–21], the food processing industry [23], andelsewhere.

It was not reintroduced to the scientific community until 1969, 500 years afterMarliani's experiment, and more than two millennia after Aristotle's "Meteorologica I" [1]. The story of its rediscovery by a Tanzanian high schoolpupil named Mpemba is written up in the New Scientist [4].The story provides a dramatic parable cautioning scientists and teachers againstdismissing the observations of non-scientists and against making quick judgements aboutwhat is impossible.

In 1963, Mpemba was making ice cream at school, which he did by mixing boiling milkwith sugar. He was supposed to wait for the milk to cool before placing it therefrigerator, but in a rush to get scarce refrigerator space, put his milk in withoutcooling it. To his surprise, he found that his hot milk froze into ice cream beforethat of other pupils. He asked his physics teacher for an explanation, but was toldthat he must have been confused, since his observation was impossible.

Mpemba believed his teacher at the time. But later that year he met a friend ofhis who made and sold ice cream in Tanga town. His friend told Mpemba that whenmaking ice cream, he put the hot liquids in the refrigerator to make them freezefaster. Mpemba found that other ice cream sellers in Tanga had the samepractice.

Later, when in high school, Mpemba learned Newton's law of cooling, that describes howhot bodies are supposed to cool (under certain simplifying assumptions). Mpembaasked his teacher why hot milk froze before cold milk when he put them in thefreezer. The teacher answered that Mpemba must have been confused. When Mpembakept arguing, the teacher said "All I can say is that is Mpemba's physics and not theuniversal physics" and from then on, the teacher and the class would criticize Mpemba'smistakes in mathematics and physics by saying "That is Mpemba's mathematics" or "That isMpemba's physics." But when Mpemba later tried the experiment with hot and cold water inthe biology laboratory of his school, he again found that the hot water froze sooner.

Earlier, Dr Osborne, a professor of physics, had visited Mpemba's high school.Mpemba had asked him to explain why hot water would freeze before cold water. DrOsborne said that he could not think of any explanation, but would try the experimentlater. When back in his laboratory, he asked a young technician to test Mpemba'sclaim. The technician later reported that the hot water froze first, and said "Butwe'll keep on repeating the experiment until we get the right result." But repeatedtests gave the same result, and in 1969 Mpemba and Osborne wrote up their results [5].

In the same year, in one of the coincidences so common in science, Dr Kellindependently wrote a paper on hot water freezing sooner than cold water. Kellshowed that if one assumed that the water cooled primarily by evaporation, and maintaineda uniform temperature, the hot water would lose enough mass to freeze first [11]. Kell thus argued that the phenomenon (then a commonurban legend in Canada) was real and could be explained by evaporation. But hewas unaware of Osborne's experiments, which had measured the mass lost to evaporation andfound it insufficient to explain the effect. Subsequent experiments were done withwater in a closed container, eliminating the effects of evaporation, and still found thatthe hot water froze first [14].

Subsequent discussion of the effect has been inconclusive. While quite a fewexperiments have replicated the effect [4,6–13], therehas been no consensus on what causes the effect. The different possible explanationsare discussed above. The effect has repeatedly a topicof heated discussion in the "New Scientist", a popular science magazine. The lettershave revealed that the effect was known by laypeople around the world long before1969. Today, there is still no well-agreed explanation of the Mpemba effect.

One explanation of the effect is that as the hot water cools, it loses mass toevaporation. With less mass, the liquid has to lose less heat to cool, and so itcools faster. With this explanation, the hot water freezes first, but only becausethere's less of it to freeze. Calculations done by Kell in 1969 [11] showed that if the water cooled solely by evaporation, andmaintained a uniform temperature, the warmer water would freeze before the coolerwater.

This explanation is solid, intuitive, and undoubtedly contributes to the Mpemba effectin most physical situations. But many people have incorrectly assumed that itis therefore "the" explanation for the Mpemba effect. That is, they assume that theonly reason hot water can freeze faster than cold is because of evaporation, and that allexperimental results can be explained by the calculations in Kell's article.But the experiments currently do not bear this belief out. While experimentsshow evaporation to be important [13], they do not show that itis the only mechanism behind the Mpemba effect. A number of experimenters haveargued that evaporation alone is insufficient to explain their results [5,9,12]; in particular, the original experiment by Mpemba andOsborne measured the mass lost to evaporation, and found it substantially less that theamount predicted by Kell's calculations [5,9]. And mostconvincingly, an experiment by Wojciechowski observed the Mpemba effect in a closedcontainer, where no mass was lost to evaporation.

Another explanation argues that the dissolved gas usually present in water is expelledfrom the initially hot water, and that this changes the properties of the water in someway that explains the effect. It has been argued that the lack of dissolved gas maychange the ability of the water to conduct heat, or change the amount of heat needed tofreeze a unit mass of water, or change the freezing point of the water by some significantamount. It is certainly true that hot water holds less dissolved gas than coldwater, and that boiled water expels most dissolved gas. The question is whether thiscan significantly affect the properties of water in a way that explains the Mpembaeffect. As far as I know, there is no theoretical work supporting this explanationfor the Mpemba effect.

Indirect support can be found in two experiments that saw the Mpemba effect in normalwater which held dissolved gasses, but failed to see it when using degassed water [10,14]. But an attempt to measure the dependence ofthe enthalpy of freezing on the initial temperature and gas content of the water wasinconclusive [14].

One problem with this explanation is that many experiments pre-boiled both theinitially hot and initially cold water, precisely to eliminate the effect of dissolvedgasses, and yet they still saw the effect [5,13]. Twosomewhat unsystematic experiments found that varying the gas content of the water made nosubstantial difference to the Mpemba effect [9,12].

It has also been proposed that the Mpemba effect can be explained by the fact that thetemperature of the water becomes non-uniform. As the water cools, temperaturegradients and convection currents will develop. For most temperatures, the densityof water decreases as the temperature increases. So over time, as water cools wewill develop a "hot top" — the surface of the water will be warmer than the averagetemperature of the water, or the water at the bottom of the container. If the waterloses heat primarily through the surface, then this means that the water should lose heatfaster than one would expect based just on looking at the average temperature of thewater. And for a given average temperature, the heat loss should be greater the moreinhom*ogenous the temperature distribution is (that is, the greater the range of thetemperatures seen as we go from the top to the bottom).

How does this explain the Mpemba effect? Well, the initially hot water will coolrapidly, and quickly develop convection currents and so the temperature of the water willvary greatly from the top of the water to the bottom. On the other hand, theinitially cool water will have a slower rate of cooling, and will thus be slower todevelop significant convection currents. Thus, if we compare the initially hot waterand initially cold water at the same average temperature, it seems reasonable to believethat the initially hot water will have greater convection currents, and thus have a fasterrate of cooling. To consider a concrete example, suppose that the initially hotwater starts at 70°C, and the initially cold water starts at 30°C. When theinitially cold water is at an average 30°C, it is also a uniform 30°C.But when the initially hot water reaches an average 30°C, the surface of thewater is probably much warmer than 30°C, and it will thus lose heat faster than theinitially cold water for the same average temperature. Got that? Thisexplanation is pretty confusing, so you might want to go back and read the last twoparagraphs again, paying careful attention to the difference between initial temperature,average temperature, and surface temperature.

At any rate, if the above argument is right, then when we plot the average temperatureversus time for both the initially hot and initially cold water, then for some averagetemperatures the initially hot water will be cooling faster than the initially coldwater. So the cooling curve of the initially hot water will not simply reproduce thecooling curve of the initially cold water, but will drop faster when in the sametemperature range.

This shows that the initially hot water goes faster, but of course it also has fartherto go. So whether it actually finishes first (that is, reaches 0°C first), isnot clear from the above discussion. To know which one finishes first would requiretheoretical modelling of the convection currents (hopefully for a range of containershapes and sizes), which has not been done. So convection alone may be able toexplain the Mpemba effect, but whether it actually does is not currently known.Experiments on the Mpemba effect have often reported a "hot top" [5,8,10], as we would expect. Experiments have been donethat looked at the convection currents of freezing water [27,28],but their implications for the Mpemba effect are not entirely clear.

It should also be noted that the density of water reaches a maximum at four°C. So below four°C, the density of water actually decreases with decreasingtemperature, and we will get a "cold top." This makes the situation even morecomplicated.

The initially hot water may change the environment around it in some way that makes itcool faster later on. One experiment reported significant changes in the data simplyupon changing the size of the freezer that the container sat in [7]. So conceivably it is important not just to know aboutthe water and the container, but about the environment around it.

For example, one explanation for the Mpemba effect is that if the container is restingon a thin layer of frost, than the container holding the cold water will simply sit on thesurface of the frost, while the container with the hot water will melt the frost, and thenbe sitting on the bottom of the freezer. The hot water will then have better thermalcontact with the cooling systems. If the melted frost refreezes into an ice bridgebetween the freezer and the container, the thermal contact may be even better.

Obviously, even if this argument is true, it has fairly limited utility, since mostscientific experiments are careful enough not to rest the container on a layer of frost ina freezer, but instead place the container on a thermal insulator, or in a coolingbath. So while this proposed mechanism may or may not have some relevance to somehome experiments, it's irrelevant for most published results.

Finally, supercooling may be important to the effect. Supercooling occurs whenwater freezes not at 0°C, but at some lower temperature. This happens becausethe statement that "water freezes at 0°C" is a statement about the lowest energy stateof the water: at less than 0°C, the water molecules "want" to be arranged as an icecrystal. This means that they will stop zooming around randomly as a liquid, andinstead form a solid ice lattice. But they don't know how to form themselvesinto an ice lattice, but need some small irregularity or nucleation site to tell them howto arrange themselves. Sometimes, when water is cooled below 0°C, the moleculeswill not see a nucleation site for some time, and then water will cool below 0°Cwithout freezing. This happens quite often. One experiment found thatinitially hot water would supercool only a little (say to about −2°C), whileinitially cold water would supercool more (to around −8°C) [12]. If true, this could explain the Mpemba effect becausethe initially cold water would need to "do more work"; — that is, get colder —to freeze.

But this also cannot be considered "the" sole explanation of the Mpembaeffect. First of all, as far as I know, this result has not been independentlyconfirmed. The experiment described above [12] only had alimited number of trials, so the results found could have been a statistical fluke.

Second, even if the results are true, they do not fully explain the Mpemba effect, butreplace one mystery with another. Why should initially hot water supercool less thaninitially cold water? After all, once the water has cooled to the lower temperature,one would generally expect that the water would not "remember" what temperature it used tobe. One explanation is that the initially hot water has less dissolved gas than theinitially cold water, and that this affects its supercooling properties (see Dissolved Gasses for more on this). The problem with thisexplanation is that one would expect that since the hot water has less dissolved gas, andthus fewer nucleation sites, it would supercool more, not less. Another explanationis that when the initially hot water has cooled down to 0°C (or less), its temperaturedistribution throughout the container varies more than the initially cold water (see Convection for more on this). Since temperature shear inducesfreezing [26], the initially hot water supercools less, and thusfreezes sooner.

Third, this explanation cannot work in all of the experiments, because many of theexperimenters chose to look not at the time to form a complete block of ice, but the timefor some part of the water to reach 0°C [7,10,13] (orperhaps the time for a thin layer of frost to form on the top [17]). While [12] says that it is only a"true Mpemba effect" if the hot water freezes entirely first, other papers have definedthe Mpemba effect differently. Since the precise time of supercooling is inherentlyunpredictable (see e.g. [26]), many experiments have chosen tomeasure not the time for the sample to actually become ice, but the time for which thesample's equilibrium ground state is ice; that is, the time when the top of the samplereached 0°C [7,10,13]. The supercooling argument doesnot apply to these experiments.

References

Historical

• 1. Aristotle in E. W. Webster, "Meteorologica I", Oxford U. P., Oxford, 1923, pgs 348b–349a
• 2. Bacon F 1620 Novum Organum Vol VIII of "The Works of Francis Bacon" 1869 ed. J. Spedding, R. L. Ellis and D. D. Heath (New York) pp 235, 337, quoted in T.S. Kuhn 1970 "The Structure of Scientific Revolutions" 2nd edn (Chicago: University of Chicago Press), pg 16
• 3. Descartes R 1637, "Les Meteores" 164 published with "Discours de la Methode" (Leyden: Ian Marie) 1637, quoted in "Oeuvres de Descartes" Vol. VI 1902 ed. Adam and Tannery (Paris: Leopold Cerf) pg 238 (trans. F. C. Frank)
• 4. Clagett, Marshall, "Giovanni Marliani and Late Medieval Physics", AMS press, Inc., New York, 1967, pgs 72, 79, 94

Experiments on the Mpemba Effect

• 5. Mpemba and Osborne, "Cool", Physics Education 4, pgs 172–5 (1969)
• 6. Ahtee, "Investigation into the Freezing of Liquids", Phys. Educ. 4, pgs 379–80 (1969)
• 7. I. Firth, "Cooler?", Phys. Educ. 6, pgs 32–41 (1979)
• 8. E. Deeson, "Cooler—lower down", Phys. Educ. 6, pgs 42–44 (1971)
• 9. Osborne, "Mind on Ice", Phys. Educ. 14, pgs 414–17 (1979)
• 10. M. Freeman, "Cooler Still", Phys. Educ. 14, pgs 417–21 (1979)
• 11. G.S. Kell, "The Freezing of Hot and Cold Water", American Journal of Physics, 37, #5, pgs 564–5 (May 1969)
• 12. D. Auerbach, "Supercooling and the Mpemba effect: When hot water freezes quicker than cold", American Journal of Physics, 63, #10, pgs 882–5 (Oct 1995)
• 13. J. Walker, "The Amateur Scientist", Scientific American, 237, #3, pgs246–7 (Sept. 1971)
• 14. B. Wojciechowski, "Freezing of Aqueous Solutions Containing Gases", Cryst. Res. Technol., 23, #7, pgs 843–8 (1988)

• 15. New Scientist, 42, #652, 5 June 1969, pg 515
• 16. New Scientist, 2 Dec. 1995, pg 22
• 17. New Scientist, 42, #654, 19 June 1969, pgs 655–6
• 18. New Scientist, 43, #657, 10 July 1969, pgs 88–9
• 19. New Scientist, 43, #658, 17 July 1969, pgs 158–9
• 20. New Scientist, 43, #658, 25 Sept. 1969, pg 662
• 21. New Scientist, 44, #672, 23 Oct. 1969, pg 205
• 22. New Scientist, 45, #684, 15 Jan. 1970, pgs 125–6
• 23. New Scientist, 45, #686, 29 Jan. 1970, pgs 225–6
• 24. New Scientist, 2 Dec. 1995, pg 57
• 25. New Scientist, 16 Mar. 1996, pg 58

• 26. J. Elsker, "The Freezing of Supercooled Water", Journal of Molecular Structure, 250, pgs 245–51 (1991)
• 27. R.A. Brewster and B. Gebhart, "An experimental study of natural convection effects on downward freezing of pure water", Int. J. Heat Mass Trans. 31, #2, pgs 331–48 (1988)
• 28. R.S. Tankin and R. Farhadieh, "Effects of Thermal Convection currents on Formation of Ice", Int. J. Heat Mass Trans., 14, pgs 953–61 (1971)