Written Nov, 1998 by Monwhea Jeng (Momo),
Department of Physics, University of California

# Can hot water freeze faster than cold water?

Yes — a general explanation
History of the Mpemba Effect
More-detailed explanations
References

## Yes — a general explanation

Hot water can in fact freeze faster than cold water for a wide range of experimental conditions.  This phenomenon is extremely counterintuitive, and surprising even to most scientists, but it is in fact real.  It has been seen and studied in numerous experiments.  While this phenomenon has been known for centuries, and was described by Aristotle, Bacon, and Descartes [1–3], it was not introduced to the modern scientific community until 1969, by a Tanzanian high school pupil named Mpemba.  Both the early scientific history of this effect, and the story of Mpemba's rediscovery of it, are interesting in their own right — Mpemba's story in particular providing a dramatic parable 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 Mpemba effect.  Because, no doubt, most readers are extremely skeptical at this point, we should begin by stating precisely what we mean by the Mpemba effect.  We start with two containers of water, which are identical in shape, and which hold identical amounts of water.  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 the initially warmer water will freeze first.  If this occurs, we have seen the Mpemba effect.  Of course, the initially warmer water will not freeze before the initially cooler water for all initial conditions.  If the hot water starts at 99.9°C, and the cold water at 0.01°C, then clearly under those circumstances, the initially cooler water will freeze first.  But under some conditions the initially warmer water will freeze first: if that happens, you have seen the Mpemba effect.  But you will not see the Mpemba effect for just any initial temperatures, container shapes, or cooling conditions.

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

What's wrong with this proof is that it implicitly assumes that the water is characterized solely by a single number — its average temperature.  But if other factors besides the average temperature are important, then when the initially warmer water has cooled to an average temperature of 30°C, it may look very different than 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 an average 30°C.  It could have less mass, less dissolved gas, or convection currents producing a non-uniform temperature distribution.  Or it could have changed the environment around the container in the refrigerator.  All four of these changes are conceivably important, and each will be considered separately below.  So the impossibility proof given above doesn't work.  And in fact the Mpemba effect has been observed in a number of controlled experiments [5,7–14]

It is still not known exactly why this happens.  A number of possible explanations for 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 often hear confident claims that X is the cause of the Mpemba effect, such claims are usually based on guesswork, or on looking at the evidence in only a few papers and ignoring the rest.  Of course, there is nothing wrong with informed theoretical guesswork or being selective in which experimental results you trust; the problem is that different people make 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 number of details in the experimental setup, such as the shape and size of the container, the shape and size of the refrigeration unit, the gas and impurity content of the water, how the time of freezing is defined, and so on.  Because of this sensitivity, while experiments have generally agreed that the Mpemba effect occurs, they disagree over the conditions under which it occurs, and thus about why it occurs.  As Firth [7] wrote "There is a wealth of experimental variation in the problem so that any laboratory undertaking such investigations is guaranteed different results 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 upon cooling to the initial temperature of the initially cooler water.  What follows below is a short description of the four related mechanisms that have been suggested to explain the Mpemba effect.  More ambitious readers can follow the links to more complete explanations of the mechanisms, as well as counter-arguments and experiments that the mechanisms cannot explain.  It seems likely that there is no one mechanism that explains the Mpemba effect for all circumstances, but that different mechanisms are important 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 lower temperature.  One experiment [12] found that its initially hot water supercooled less than its initially cold water.  This would mean that the initially warmer water might freeze first because it would freeze at a higher temperature than the initially cooler water.  If true, this would not fully explain the Mpemba effect, because we would still need to explain why initially warmer water supercools less than initially cooler water.

In short, hot water does freeze sooner than cold water under a wide range of circumstances.  It is not impossible, and has been seen to occur in a number of experiments.  But despite claims often made by one source or another, there is no well-agreed explanation for how this phenomenon occurs.  Different mechanisms have been proposed, but the experimental evidence is inconclusive.  For those wishing to read more on the subject, Jearl Walker's article in Scientific American [13] is very readable and has suggestions on how to do home experiments on the Mpemba effect, while the articles by Auerbach [12] and Wojciechowski [14] are two of the more modern papers on the effect.

## History of the Mpemba Effect

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

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

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

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

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

Later, in the 1600s, it was apparently common knowledge that hot water would freeze faster than cold.  In 1620 Bacon wrote "Water slightly warm is more easily frozen than 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 sooner than other water" [3].

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

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

In 1963, Mpemba was making ice cream at school, which he did by mixing boiling milk with sugar.  He was supposed to wait for the milk to cool before placing it the refrigerator, but in a rush to get scarce refrigerator space, put his milk in without cooling it.  To his surprise, he found that his hot milk froze into ice cream before that of other pupils.  He asked his physics teacher for an explanation, but was told that 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 of his who made and sold ice cream in Tanga town.  His friend told Mpemba that when making ice cream, he put the hot liquids in the refrigerator to make them freeze faster.  Mpemba found that other ice cream sellers in Tanga had the same practice.

Later, when in high school, Mpemba learned Newton's law of cooling, that describes how hot bodies are supposed to cool (under certain simplifying assumptions).  Mpemba asked his teacher why hot milk froze before cold milk when he put them in the freezer.  The teacher answered that Mpemba must have been confused.  When Mpemba kept arguing, the teacher said "All I can say is that is Mpemba's physics and not the universal physics" and from then on, the teacher and the class would criticize Mpemba's mistakes in mathematics and physics by saying "That is Mpemba's mathematics" or "That is Mpemba's physics." But when Mpemba later tried the experiment with hot and cold water in the 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.  Dr Osborne said that he could not think of any explanation, but would try the experiment later.  When back in his laboratory, he asked a young technician to test Mpemba's claim.  The technician later reported that the hot water froze first, and said "But we'll keep on repeating the experiment until we get the right result." But repeated tests 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 Kell independently wrote a paper on hot water freezing sooner than cold water.  Kell showed that if one assumed that the water cooled primarily by evaporation, and maintained a uniform temperature, the hot water would lose enough mass to freeze first [11].  Kell thus argued that the phenomenon (then a common urban legend in Canada) was real and could be explained by evaporation.  But he was unaware of Osborne's experiments, which had measured the mass lost to evaporation and found it insufficient to explain the effect.  Subsequent experiments were done with water in a closed container, eliminating the effects of evaporation, and still found that the hot water froze first [14].

Subsequent discussion of the effect has been inconclusive.  While quite a few experiments have replicated the effect [4,6–13], there has been no consensus on what causes the effect.  The different possible explanations are discussed above.  The effect has repeatedly a topic of heated discussion in the "New Scientist", a popular science magazine.  The letters have revealed that the effect was known by laypeople around the world long before 1969.  Today, there is still no well-agreed explanation of the Mpemba effect.

## More-detailed explanations

### Evaporation

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

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

### Dissolved Gasses

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

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

One problem with this explanation is that many experiments pre-boiled both the initially hot and initially cold water, precisely to eliminate the effect of dissolved gasses, and yet they still saw the effect [5,13].  Two somewhat unsystematic experiments found that varying the gas content of the water made no substantial difference to the Mpemba effect [9,12].

### Convection

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

How does this explain the Mpemba effect?  Well, the initially hot water will cool rapidly, and quickly develop convection currents and so the temperature of the water will vary greatly from the top of the water to the bottom.  On the other hand, the initially cool water will have a slower rate of cooling, and will thus be slower to develop significant convection currents.  Thus, if we compare the initially hot water and initially cold water at the same average temperature, it seems reasonable to believe that the initially hot water will have greater convection currents, and thus have a faster rate of cooling.  To consider a concrete example, suppose that the initially hot water starts at 70°C, and the initially cold water starts at 30°C.  When the initially 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 the water is probably much warmer than 30°C, and it will thus lose heat faster than the initially cold water for the same average temperature.  Got that?  This explanation is pretty confusing, so you might want to go back and read the last two paragraphs 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 temperature versus time for both the initially hot and initially cold water, then for some average temperatures the initially hot water will be cooling faster than the initially cold water.  So the cooling curve of the initially hot water will not simply reproduce the cooling curve of the initially cold water, but will drop faster when in the same temperature range.

This shows that the initially hot water goes faster, but of course it also has farther to go.  So whether it actually finishes first (that is, reaches 0°C first), is not clear from the above discussion.  To know which one finishes first would require theoretical modelling of the convection currents (hopefully for a range of container shapes and sizes), which has not been done.  So convection alone may be able to explain 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 done that 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 decreasing temperature, and we will get a "cold top." This makes the situation even more complicated.

### Surroundings

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

For example, one explanation for the Mpemba effect is that if the container is resting on a thin layer of frost, than the container holding the cold water will simply sit on the surface of the frost, while the container with the hot water will melt the frost, and then be sitting on the bottom of the freezer.  The hot water will then have better thermal contact with the cooling systems.  If the melted frost refreezes into an ice bridge between 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 most scientific experiments are careful enough not to rest the container on a layer of frost in a freezer, but instead place the container on a thermal insulator, or in a cooling bath.  So while this proposed mechanism may or may not have some relevance to some home experiments, it's irrelevant for most published results.

### Supercooling

Finally, supercooling may be important to the effect.  Supercooling occurs when water freezes not at 0°C, but at some lower temperature.  This happens because the statement that "water freezes at 0°C" is a statement about the lowest energy state of the water: at less than 0°C, the water molecules "want" to be arranged as an ice crystal.  This means that they will stop zooming around randomly as a liquid, and instead form a solid ice lattice.  But they don't know how to form themselves into an ice lattice, but need some small irregularity or nucleation site to tell them how to arrange themselves.  Sometimes, when water is cooled below 0°C, the molecules will not see a nucleation site for some time, and then water will cool below 0°C without freezing.  This happens quite often.  One experiment found that initially hot water would supercool only a little (say to about −2°C), while initially cold water would supercool more (to around −8°C) [12].  If true, this could explain the Mpemba effect because the initially cold water would need to "do more work"; — that is, get colder — in order to freeze.

But this also cannot be considered "the" sole explanation of the Mpemba effect.  First of all, as far as I know, this result has not been independently confirmed.  The experiment described above [12] only had a limited 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, but replace one mystery with another.  Why should initially hot water supercool less than initially 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 to be.  One explanation is that the initially hot water has less dissolved gas than the initially cold water, and that this affects its supercooling properties (see Dissolved Gasses for more on this).  The problem with this explanation is that one would expect that since the hot water has less dissolved gas, and thus fewer nucleation sites, it would supercool more, not less.  Another explanation is that when the initially hot water has cooled down to 0°C (or less), its temperature distribution throughout the container varies more than the initially cold water (see Convection for more on this).  Since temperature shear induces freezing [26], the initially hot water supercools less, and thus freezes sooner.

Third, this explanation cannot work in all of the experiments, because many of the experimenters chose to look not at the time to form a complete block of ice, but the time for some part of the water to reach 0°C [7,10,13] (or perhaps 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 defined the Mpemba effect differently.  Since the precise time of supercooling is inherently unpredictable (see e.g. [26]), many experiments have chosen to measure not the time for the sample to actually become ice, but the time for which the sample's equilibrium ground state is ice; that is, the time when the top of the sample reached 0°C [7,10,13].  The supercooling argument does not 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, pgs 246–7 (Sept. 1971)
• 14.  B. Wojciechowski, "Freezing of Aqueous Solutions Containing Gases", Cryst. Res. Technol., 23, #7, pgs 843–8 (1988)

### General discussion on the Mpemba Effect

• 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

### Related Articles

• 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)