Mpemba Effect Research Papers
Analysis of our ‘Mpemba style’ data and the data from other studies
Figure 1 plots the variation in the time t0, to cool samples to 0 °C, with the initial temperature from a variety of studies including our ‘Mpemba-type’ experiments. We have attempted to represent a broad selection of published experimental data regarding the Mpemba effect. We note that the data from the careful experiments of 29 reporting the time to cool to 0 °C (their Fig. 5), which exhibited no evidence of the Mpemba effect, could not be included due to difficulties in accurately obtaining data from their printed figure. Their results for the time to for the ice layer to grow to a depth of 25 mm cannot be fairly included in our analysis, since we exclude the freezing process; however, we discuss these results when drawing our conclusions. 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. 1 it is difficult to draw any conclusions from the data, except that broadly speaking the cooling time increases with initial temperature. The only exception, which reports data (across a broad range of temperatures) that exhibit a decreasing trend in cooling time with increasing initial temperature, is that of Mpemba & Osborne8.
Figure 2 shows the variation in the cooling time t0, scaled by the convective time scale, with the temperature averaged Rayleigh number from the various studies detailed in Fig. 1 (for details of the convective time scale and the temperature averaged Rayleigh number see the Methods section). Some of the studies included in Fig. 2 did not explicitly provide all the details required to scale the data, and in such cases we made reasonable estimates based on the information provided (details of which are also provided in our Methods section). The experimental conditions vary widely between the eight independent studies from which data are included within the figure. There is no obvious systematic bias for the cooling times based on the geometry of the cooling vessel, despite the aspect ratio of width to height, D/H, varying by a factor of fifteen and the depth of water being cooled varying by a factor of eight within the data — indicating that the geometry may be appropriately reflected by the length scales within the temperature averaged Rayleigh number RaT. 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. The first set we describe as ‘convectively dominated’ data (marked by the solid symbols in Fig. 2) which broadly consists of samples where the base was insulated or cooling from below was inhibited in some manner (see the legend in Fig. 2 for details). 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 second dataset we describe as ‘stably cooled’ (marked by the blue hollow symbols in Fig. 2) which consists of data for which the heat flux through the base of the sample is expected to have been significant (e.g. where the sample was placed directly on a cooling plate), and the cooling is expect to have promoted stably stratified sample of water (at least above 4 °C).
The data within each individual dataset exhibit a broadly consistent trend, with the cooling time increasing with RaT 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. 1 using a number of alternative definitions for the Rayleigh number, for example taking all parameters at the initial conditions or combining individually temperature-averaged parameters to form the Rayleigh number, cf. 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
(for example, see ref. 31) where Nu = Q/(κΔT/H) is the Nusselt number, with κ the thermal diffusivity of the fluid, Q being proportional to the flux of heat and ΔT being a characteristic temperature difference between the fluid and the cooled surface. The time rate of change of temperature for a given sample is then proportional to the heat flux, i.e. Q, and given that Ra∼βΔTgH3/(κv), from equation (2) we can write
where β and v are the coefficient of thermal expansion and the kinematic viscosity of the fluid, and A is the cooled surface area of the fluid. Hence
where and are the initial and final characteristic temperature differences (between the fluid and the cooled surface). Thus
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. 2 (obtained at initial Rayleigh numbers up to O(1010)), fits well with the trend predicted by (5) suggesting that the experimental data can be regarded as high Rayleigh number. As such, if the data plotted in Fig. 2 are shown not to exhibit the Mpemba effect, as indeed we go on to argue, then one must expect that data obtained at higher Rayleigh numbers would also not exhibit the Mpemba effect.
Analysis of the occurrence of the Mpemba effect
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. To illustrate when the Mpemba effect may be reported to have been observed we consider the average rate at which heat is transferred Q from the initially hot QH and initially cold QC samples, where for a given sample Q = ΔE/t0 = (Ei − E0)/t0∝ ΔT/t0 = (Ti − T0)/t0 with Ei and E0 denoting the initial and final enthalpy of the samples, respectively.
The Mpemba effect can be reported as having been observed when the inequality QH/QC > ΔEH/ΔEC is satisfied, since QH/QC > ΔEH/ΔEC⇒tc > tH, where tc and tH denote the cooling time of the cold and hot samples, respectively. Figure 3(a) plots the variation in the ratio QH/QC with ΔEH/ΔEC (or equivalently ΔTH/ΔTC) for the various pairs of data shown in Fig. 1 and the results of our experiments of the ‘second-type’ (see the Methods section). Figure 3(b) highlights the results of our experiments of the ‘second-type’, with an allowance for spatial variation in the temperature measurements. The relationship QH/QC = ΔEH/ΔEC is marked by solid black lines within Fig. 3. Hence, any data lying above this line may be reasonably reported as an observation of the Mpemba effect.
Examining Fig. 3a shows that the majority of the data reported lie below the ‘Mpemba effect line’ (QH/QC = ΔEH/ΔEC) and hence the Mpemba effect was clearly not observed in these cases. 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.e. the temperature of the hotter sample is only marginally greater than that of the cooler sample. This suggests that any inaccuracies in the measurement of temperature may be significant. There are two datasets which are exceptions to this finding, namely, Mpemba & Osborne8 and Thomas14. None of the data of Thomas14 lie far above the Mpemba effect line. Indeed, Fig. 3b plots our data from our ‘second-type’ experiments, i.e. those designed to avoid any formation of ice, in which we recorded the temperatures at a range of different heights within each sample. In addition to our data deduced by comparing temperatures recorded at equal heights within the hotter and cooler samples, Fig. 3b includes the data (marked ) which we would have reported if the vertical positions at which we recorded the temperature were incorrectly measured by up to 1 cm. 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. Hence, if in any particular set of experiments the vertical position of the temperature measurements were incorrect, by just 1 cm, then from the data of those experiments one could (again, quite incorrectly) conclude that the Mpemba had been observed. 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. 3b is that of Mpemba & Osborne8, which includes observation that lie both far above the Mpemba effect line and also towards the right-hand end of the horizontal axis — we note that these data show significant scatter from any physically reasonable trend.
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 2014. 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 have been unable to deduce the source of any systematic error in the experimental procedure or experimental set-up of Mpemba & Osborne8 that could feasibly have led to such extreme data being recorded.
Discussion and Conclusions
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.e. the difference in the cooling times between the hot and cold samples is marginal. We have shown (Fig. 3) that much of the data reporting to be observations of the Mpemba effect were from studies not reporting the height at which temperatures were measured7,14,20,21,22,23 and that the conclusions drawn from these data could have been altered by simply recording temperatures without precisely monitoring the height. Indeed, all the data which lie just above the Mpemba effect line in Fig. 3 (including data for which the temperautre measurement height was carefully monitored and reported17,24,28) are, by the very nature of experiments, subject to some degree of uncertainty which may ultimately affect whether the observed results are recorded as an apparent observation of the Mpemba effect or not. To be precise regarding our meaning by this statement, let us now consider the reported observations of the Mpemba effect from, arguably, the two most careful sets of experiments within the literature28,29. The study28 does present data for one observation of the Mpemba effect but also reports obtaining “different cooling curves even if the initial temperatures were identical”, furthermore they state “[c]areful and precise experiments to probe the Mpemba effect can be tried by cooling hot and cool water in two similar containers simultaneously, but it is extremely difficult to obtain scientifically meaningful and reproducible results”. The study29 shows a potential observation of the Mpemba effect (in the times for the ice layer to grow to a thickness of 25 mm, their figure 19) for a single pair of initial temperatures (from a possible 21 initial temperature pairings), namely the pair of initial temperatures 10 °C and 15 °C. From data recorded at a fixed height (for example, 5 mm) the samples cooling from 15 °C exhibit a mean cooling time of approximately 95 minutes while those cooling from 10 °C the mean is approximately 105 minutes — hence in taking only the mean of the data for this particular temperature pairing one could describe the Mpemba effect as having been observed. However, the variation in notionally identical experiments is significant. At the same recording height, for samples cooling from 15 °C the recorded time spans the range 95–105 minutes while for samples cooling from 10 °C the recorded time spans the range 100–110 minutes. As such, the variation in notionally identical experiments is at least large enough to render any conclusion that the Mpemba effect has been observed in the mean data as highly questionable, and so this cannot be regarded as a meaningful observation of the effect.
The only exception to our above statements, the single study in which some data is reported that shows dramatically warmer samples cooling in substantially less time (i.e. data points that are far above the line QH/Qc = ΔTH/ΔTc in Fig. 3) is the data reported by Mpemba & Osborne8. If these data could be reproduced in a repeatable fashion and the underlying mechanism understood then it would be of real significance to a multitude of applications relying on the transfer of heat. For example ref. 8, report cooling a sample from 90 °C to freezing point in 30 minutes while a sample at 20 °C took 100 minutes to cool to freezing point, i.e. the average heat transfer rate during cooling was observed to increase by a factor of 15 by simply increasing the initial temperature of the sample. With the use of modern heat-exchangers such a result would have profound implications for the efficiency of any number of common industrial processes. However, over the subsequent 47 years, numerous studies have attempted to demonstrate the ‘effect’ on a scale comparable to that reported by Mpemba & Osborne. Despite these efforts, including our own, none have succeeded. We must therefore assert that this particular dataset may be fundamentally flawed and thus, unless it can be shown to be reproducible and repeatable, this dataset must be regarded as erroneous.
We must highlight that our primary focus has been to examine the cooling of water to the freezing point (observed under standard atmospheric conditions), i.e. an enthalpy equivalent of 0 °C. In so doing we have been able to show that much of the published experimental data exhibit a scaling behaviour associated with asymptotically high Rayleigh number convection. Thus one cannot expect to observe samples of hot water cooling to 0 °C faster than colder samples by carrying out experiments at higher Rayleigh numbers. Under our definition of the Mpemba effect, akin to the definition in the ‘original’ paper by Mpemba & Osborne8 (in which they documented “the time for water to start freezing”) we are forced to conclude that the ‘Mpemba effect’ is not a genuine physical effect and is a scientific fallacy.
If one extends the definition of the Mpemba effect to include the freezing process then one can examine the experimental evidence presented by a number of scientific studies which have sought to include the effect of freezing, e.g. refs 9,21,22,28 and 29. The freezing of water to ice is a thermodynamically intensive process. For example, the energy required to change the phase of a given mass of water at 0 °C, into ice at 0 °C is approximately equal to the energy required to cool the same mass of water from 80 °C to 0 °C in the liquid state. Intuition, therefore, guides one to expect the time to completely freeze a sample of water could depend only weakly on the initial water temperature. Moreover, freezing is initiated by a nucleation process and as such it is susceptible to variations at the smallest physical scales, e.g. imperfections in the surface of containers or impurities within the water samples — the physical scales of which are extremely difficult to control in even the most precise experiments. Such intuition is entirely born out in the experimental evidence, with no single study able to report repeatable observations of the Mpemba effect when the freezing process is included9,21,22,28,29. Experimental observations of a particular example of warm water cooling and freezing in less time than a particular example of initially cooler water have been made — what is yet to be reported is any experimental evidence that samples of water can be consistently cooled and frozen in less time (the time being less by a repeatable and statistically significant amount) by simply initiating the cooling from a higher temperature. As such we can conclude that even with the freezing process included within the definition of the Mpemba effect, the Mpemba effect is not observable in any meaningful way.
We are not gladdened by such a conclusion, indeed quite the opposite. The Mpemba effect has proved to be a wonderful puzzle with which to engage and interest people of all ages and backgrounds in the pursuit of scientific understanding. However, the role of scientists is to objectively examine facts and further knowledge by reporting the conclusions, and as such we feel compelled to disseminate our findings. Finally, we want to give hope to the educators who may have previously relied on the Mpemba effect as a useful tool with which to inspire their students. There are numerous genuine artefacts of science which can continue to provide such inspiration. For example, try filling two identical glasses, one with fresh water and one with salty water (both of equal temperature), place a few cubes of ice in each and observe which melts first — many students will be surprised by the result, finding it counter to their experience and intuition. Equally one could try placing a thin sheet of card on top of a glass of water, turn the glass upside down and then remove your hand from the card — watch as the atmospheric air pressure allows the water to be held in the glass — repeat this, replacing the card by just a rigid gauze with holes of up to a few millimetres and still the water will be held within the glass32. We hope that these examples serve to act as catalysts for those seeking other examples of genuine science and that these help to inspire scientific interest within future generations.
Ever since the days of Aristotle, people have made the counterintuitive observation that hot water sometimes freezes faster than cold water. In modern times, the observation has been named the Mpemba effect after Erasto Mpemba, an elementary school student living in what is now Tanzania in the early '60s. When making ice cream, Mpemba observed that using warmer milk causes the ice cream to freeze faster than when using colder milk.
In the last few decades, the Mpemba effect has been studied and observed in several physical systems besides water, including carbon nanotube resonators and ice-like water cages called clathrate hydrates. Despite these findings, the causes of the effect are not well-understood. Proposed explanations include the presence of impurities, hydrogen bonding, and supercooling. Even the mere existence of the Mpemba effect remains controversial, as one recent study found insufficient evidence to replicate a meaningful effect.
Now, their interest rekindled by a recent paper proposing a generic mechanism for similar effects, scientists Antonio Lasanta and coauthors from universities in Spain have returned to the question in a new study published in Physical Review Letters. In their work, the researchers theoretically demonstrate and investigate the Mpemba effect in granular fluids, such as those made of sand or other small particles.
Using simulations of granular systems and a simple kinetic theory approach, the researchers were able to determine that the initial conditions in which the system is prepared play a critical role in determining whether or not the system exhibits the Mpemba effect. Their analysis also enabled them to identify the initial conditions required in order for a granular system to exhibit the Mpemba effect.
"Our work shows that the existence of the Mpemba effect is very sensitive to the initial preparation of the fluid or, in other words, to its previous history," coauthor Andrés Santos at the University of Extremadura in Badajoz, Spain, told Phys.org. "In our opinion, this may explain the elusiveness and controversy of the Mpemba effect in water, as a consequence of the lack of control on the detailed initial preparation of the sample."
As the researchers showed, if a system is not prepared under certain initial conditions, then the colder system cools down more quickly than the warmer one, as expected, and there is no Mpemba effect.
"We theoretically showed, at least in the case of a gas, that a system's temperature evolution and thus its cooling and/or heating rate do not depend on initial temperature alone, but also on the previous history of the system that control the initial value of the additional variables," Santos said. "Therefore, it is perfectly possible that an initially heated system cools down quicker than a colder one with a different history."
As the researchers explained further, the simplicity of the Mpemba effect in granular fluids compared to water and other systems enabled them to reach this conclusion.
"Our results show that the Mpemba effect is a generic non-equilibrium phenomenon that appears if the evolution of temperature depends on other physical quantities that characterize the initial state of the system," Santos said. "In practice, such an initial state can be experimentally achieved if the system is taken by some physical procedure very far away from equilibrium (for instance, by a sudden heating impulse prior to cooling down). Our theoretical and computational work shows that the Mpemba effect is particularly simple in a granular gas, since, in practice, there is one single extra parameter controlling the Mpemba effect. This parameter is the kurtosis, which measures the deviation of the velocity distribution function from a Gaussian distribution."
With this new understanding, the researchers could estimate a range of initial temperatures for which the effect emerges and determine how different the initial values of this parameter must be in order for the Mpemba effect to appear.
The results also support predictions of the existence of an inverse Mpemba effect: when heated, a colder sample may reach a hot target temperature sooner than a warmer sample. The researchers plan to investigate this area and others in the future.
"On the theoretical side, we plan to carry out a similar study in the case of a molecular solute (where collisions are fully elastic) suspended in a solvent that produces a nonlinear drag force on the solute particles," Santos said. "Going back to granular fluids, we also want to analyze the impact of particle roughness and spin on the Mpemba effect. In the latter system, the simplest model would couple the temperature evolution to that of the parameter measuring the non-equipartition of energy between the translational and rotational degrees of freedom.
"On the experimental side, we think that reproducing in a laboratory the Mpemba effect in a granular gas would be a breakthrough. We are currently working on the design of an ad hoc experiment."
Explore further:Mpemba effect: Why hot water can freeze faster than cold
More information: Antonio Lasanta et al. "When the Hotter Cools More Quickly: Mpemba Effect in Granular Fluids." Physical Review Letters. DOI: 10.1103/PhysRevLett.119.148001. Also at arXiv:1611.04948 [cond-mat.soft]
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