Despite the non-specific nature of the effect, the Mpemba effect has been the subject of numerous articles in international broadsheet newspapers e. As such the Mpemba effect cannot simply be disregarded. Moreover, both The Telegraph 2 and The Daily Mail 4 have reported that the scientific work of a group of chemists working in Singapore 5 provides a molecular mechanism to explain the effect.
These findings, and those of ref. While the findings of these studies are of great interest in their own right, we note that small-scale molecular effects are parameterised within the thermal and fluid properties of water — properties which are known to a reasonable degree of accuracy and do indeed vary with temperature.
Such findings therefore offer a route to explaining the Mpemba effect only if they result in a meaningful hysteresis in the thermal or fluid properties of water. A model exhibiting a hysteresis in the cooling of water is presented by ref. It is our aim to present an investigation of the history behind the Mpemba effect, examine the scientific evidence for it, consider the underlying physical mechanisms for the effect and determine whether the effect actually exists in any meaningful manner.
This lack of clarity is reflected by the level of discrepancies in the literature, which offers a number of different explanations. Broadly speaking, when two samples of water are cooled to the same temperature, in the same manner with the two samples being identical except for their initial temperature, and the initially hotter sample cools in less time, one can consider the Mpemba effect to have been observed.
Observations of hot water freezing in less time than cold water date back to classical science. Aristotle 10 noted that the ancient Greeks of Pontus exploited the effects when they encamped on the ice to fish, and similar observations have been repeated by Bacon 11 and Descartes More modern awareness of this apparent anomaly range from the accidental experiments of the Tanzanian school boy, Mpemba after whom the phenomenon is popularly known 8 , to the competition calling for explanations of the phenomenon by the RSC.
The Mpemba effect is an oft cited scientific anomaly and has been widely used in high-school and undergraduate physics projects 13 , The effect may appear anomalous since on first consideration one might regard the first law of thermodynamics to be breached.
An interpretation of the first law is that the change in the internal energy of a closed system is equal to the amount of heat supplied accounting for any work done on or by the system. Thus, in the absence of work, for a constant heat flux one naturally expects hot water to take longer to cool to freezing than cooler water. However, typically the cooling does not occur in environments which can be regarded as inducing a constant heat flux, instead most cooling occurs in near constant temperature environments.
One example of this being the widespread domestic formation of ice-cubes within ice-trays, for which the ice-trays typically sit on a cold plate within a freezer and are cooled by the thermostatically controlled freezer which acts to maintain an approximately constant temperature.
Hence, an ice-tray filled with warm water experiences a larger temperature difference and, therefore, a larger initial heat flux compared with an ice-tray filled with cooler water. Moreover, in the presence of an initially hot sample the freezer may remain on, and doing work, to drive the cooling for longer. This, however, by no means explains the Mpemba effect — the hot water must take some time to cool to the initial temperature of the cooler sample of water, after which all else being equal one would expect the further cooling of the warm sample to take the same time as the cooling of the colder sample.
Hence the warm water, in total, would take longer to cool. Thus for the Mpemba effect to be observed there must be some difference in the chemistry of the samples or the physics of their cooling either initially or when at equivalent temperatures — understanding and examining the various mechanisms that might give rise to such differences remains the focus of scientific debate.
The winning entry to the RSC competition, for example, cites four factors as possibly contributing to the Mpemba effect, namely: a evaporation, b dissolved gases, c mixing by convective currents, and d supercooling 7.
No doubt all four processes affect the cooling rate of water, albeit to differing extents, and crucially their effects may be strongly coupled.
For example, in two volumes of water, only differing in initial temperature and then cooled in identical conditions, one would expect that different convective currents might develop. Therefore, for significantly different initial temperatures the characteristic times that a given water particle remains in contact with an imperfection in the container or impurity within the water e.
Thus it can be reasoned that the observed variations in the extent to which supercooling occurs must arise, at least in part, due to differences in convective currents and the relative levels of dissolved gases further affected if evaporation occurs. Hence all the factors which have been proposed to individually cause the Mpemba effect may alter the extent of supercooling required to cause water to freeze. In so doing any effects associated with the supercooling of water are entirely contained within the second stage.
We restrict our definition of the Mpemba effect to the first stage of the process, i. The cooling and freezing of water has intrigued some great scientific minds. Hence many people, when they want to cool hot water quickly, begin by putting it in the sun.
So the inhabitants of Pontus when they encamp on the ice to fish they cut a hole in the ice and then fish pour warm water round their reeds that it may freeze the quicker, for they use the ice like lead to fix the reeds. It would, therefore, seem that Aristotle and the peoples of ancient Greece believed that warming water did make it freeze faster.
The second book section L in ref. No further discussion nor details are provided. While making ice-cream one pupil placed his mixture of milk and sugar in the freezer without first boiling it; another pupil, Mpemba, worried that he would not find space in the freezer and put his boiling mixture straight into the freezer without first allowing it to cool.
Mpemba did not brush this curious observation aside, instead he asked friends some of whom made a living selling ice-cream and apparently exploited the time saving effects of this anomalous behaviour and teachers to explain his observations but to no avail. Mpemba eventually asked a visiting lecturer from the University of Dar es Salaam to explain his observations. Certain subsequent studies report being unable to observe the effect, for example, Ahtee 19 who examined the fraction of ice formed and Hsu 9 who considered the time taken for the samples to form solid ice.
However, other studies report being able to reproduce the effect, typically, using domestic style ice formation. Numerous differences exist between the experimental conditions of these various studies.
These variations include: altering the nature of the cooling supplied, e. Despite this wealth of experimental data, detailed analysis is typically lacking; for example, almost all studies present the absolute sample temperature rather than the sample temperature relative to the cooling environment and typically no consideration is given to the volume mass of water being cooled nor the geometry of the cooling vessel. A notable exception is the study of Maciejewski 17 who analysed his data in terms of nondimensional parameters, the Grashof Gr , Prandtl Pr and Rayleigh Ra numbers, concluding that the key parameter is GrPr 3 and that the Mpemba effect may be driven by convection.
A number of studies have proposed physical models for the freezing of water in connection with the Mpemba effect. Vynnycky and co-workers include an experimental observation of the Mpemba effect based on temperature measurements near the water surface. Vynnycky and Kimura 29 present results from a detailed experimental examination, and a theoretical model, for the cooling of water in the context of the Mpemba effect.
Their experimental results reporting the time at which solidification begins, show no evidence to support the Mpemba effect. They attribute such effects to supercooling and they go on to suggest that their experimental data indicates that supercooling is more likely to occur with lower initial temperatures — a suggestion that would promote Mpemba-like effects in water.
Recent advances in the understanding of the bonding of water molecules have been suggested as a potential route to explaining the Mpemba effect which requires a hysteresis within the molecular interactions dependent on the initial temperature. The results of the model are qualitatively compared to an experimental observation of the Mpemba effect documented as part of the competition organised by the RSC 7. The advances in the understanding of the molecular interactions within water, and clathrate hydrates, may be of some significance in understanding the Mpemba effect.
Should it be shown to be necessary it would, indeed, be a result of real significance; for example, standard reference tables for the properties of water would need to be updated to account for not only the current temperature but also the route to the said temperature. We have attempted to represent a broad selection of published experimental data regarding the Mpemba effect. The mass of water, the geometry of its container and indeed the nature of the cooling varied widely between the different datasets and this variation is reflected in the spread of the data.
From Fig. Figure 2 shows the variation in the cooling time t 0 , scaled by the convective time scale, with the temperature averaged Rayleigh number from the various studies detailed in Fig. Some of the studies included in Fig. The experimental conditions vary widely between the eight independent studies from which data are included within the figure.
There is, however, an obvious bias in the cooling times based on the nature of the cooling and we broadly split the data into two datasets. In such cases there is no direct heat transfer between the freezer base or cooling plate and the sample of water is predominately cooled through the sides or top of the sample and unstable density stratifications are promoted.
In such cases, the heat transfer is inhibited by the addition of insulation and hence the cooling times are typically increased, despite the increased role of convection. The data from Fig. The black solid line marks the scaling for high-Rayleigh number convective cooling, 5.
The data within each individual dataset exhibit a broadly consistent trend, with the cooling time increasing with Ra T and the datasets are best-fit in a least squares sense by a power law of approximately. This suggests that the cooling times follow. We note that we scaled the data in Fig. Equation 7. The different definitions of the Rayleigh number that we tested all resulted in the various datasets exhibiting trends well approximated by 1.
Considerations of high Rayleigh number convection, in which the assumption that the heat flux is independent of the depth of the fluid, imply that. The time rate of change of temperature for a given sample is then proportional to the heat flux, i. We note that crucially, in deriving 5 we assumed that the convection exhibited behaviour associated with that of asymptotically high Rayleigh number convection.
The data investigating the Mpemba effect, plotted in Fig. As such, if the data plotted in Fig. The above analysis, although informative as to the physics of cooling water, does not explicitly address when the Mpemba effect has been observed. In order to establish a single observation of the Mpemba effect, one must compare two experiments which are identical in every manner except for a difference in the initial temperatures of the water samples.
One can then state that the Mpemba effect may be regarded to have been observed if the sample of water initially at the higher temperature reaches the desired cooling temperature first.
Hence, any data lying above this line may be reasonably reported as an observation of the Mpemba effect. The variation in the ratio of mean heat transfer rates with initial temperature or equivalently enthalpy for pairs of otherwise identical samples of hot and cold water.
Examining Fig. Data from a number of studies do lie on or just above Mpemba effect line. Notably, these data tend to be towards the left hand end of the horizontal axis, i. This suggests that any inaccuracies in the measurement of temperature may be significant. None of the data of Thomas 14 lie far above the Mpemba effect line. Indeed, Fig. In addition to our data deduced by comparing temperatures recorded at equal heights within the hotter and cooler samples, Fig. These data show observations which lie above the Mpemba effect line and as such could, quite incorrectly, be described as being observations of the Mpemba effect if sufficient care had not been taken in our experiments.
The vertical and horizontal location of this data within the figure encompasses the region that includes all the data reporting to be observations of the Mpemba effect in other studies.
We note that in studies reporting observations of the Mpemba effect the authors are either unable to produce the effect in a repeatable manner or details pertaining to the precise height of the temperature measurements were not reported.
The only study which includes observations beyond the region covered by our data shown in Fig. We have made efforts to contact both of the authors, Mr Erasto B. Mpemba and Dr Denis Osborne. In our attempts to contact Dr Osborne we were saddened to be informed of his death in September It seems that throughout his life, Dr Osborne continued to make extremely positive contributions to both science and politics.
We have so far failed in our attempt to contact Mr Mpemba although we understand he was the principal game officer in the Tanzanian Ministry of Natural Resources and Tourism, Wildlife Division he is now retired. We conclude that despite our best efforts, we were not able to make observations of any physical effects which could reasonably be described as the Mpemba effect. Moreover, we have shown that all data with the only exceptions coming from a single study reporting to be observations of the Mpemba effect within existing studies fall just above the Mpemba effect line, i.
We have shown Fig. Indeed, all the data which lie just above the Mpemba effect line in Fig. In this case, of course, the freezer will have worked harder during the given amount of time, extracting more heat from hot water. One way [described in Jearl Walker's book The Flying Circus of Physics Wiley, ] depends on the fact that hot water evaporates faster, so that if you started with equal masses of hot and cold water, there would soon be less of the hot water to freeze, and hence it would overtake the cold water and freeze first, because the lesser the mass, the shorter the freezing time.
The other way it could happen in the case of a flat-bottomed dish of water placed in a freezer is if the hot water melts the ice under the bottom of the dish, leading to a better thermal contact when it refreezes. Still feeling skeptical? Fred W. Decker, a meteorologist at Oregon State University in Corvallis, encourages readers to settle the question for themselves:. Use a given setting on an electric hot plate and clock the time between start and boiling for a given pot containing, say, one quart of water; first start with the water as cold as the tap will provide and then repeat it with the hottest water available from that tap.
I'd wager the quart of water initially hot will come to a boil in much less time than the quart of water initially cold. Take into the chamber two quart-volume milk bottles filled with water, one from a hot tap and the other from a cold tap outside the chamber. Time them to freezing, and I would wager again that the initially colder water will freeze sooner than the initially hot water. Decker concludes that "much folklore results from trying to answer such a question under conditions that do not make 'all other things equal,' which the foregoing experiments do.
Already a subscriber? Sign in. Thanks for reading Scientific American. This apparent quirk of nature is the "Mpemba effect," named after the Tanzanian high school student, Erasto Mpemba, who first observed it in The Mpemba effect occurs when two bodies of water with different temperatures are exposed to the same subzero surroundings and the hotter water freezes first.
Evaporation is the strongest candidate to explain the Mpemba effect. As hot water placed in an open container begins to cool, the overall mass decreases as some of the water evaporates. With less water to freeze, the process can take less time. And evaporation may not be the only reason the water can freeze more quickly. There may be less dissolved gas in the warmer water, which can reduce its ability to conduct heat, allowing it to cool faster.
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