Can natural convection alone explain the Mpemba effect?

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Abstract

The Mpemba effect is popularly summarized by the statement that “hot water can freeze faster than cold water”, and has been observed experimentally since the time of Aristotle; however, there remains no definitive explanation for the effect. Here, we consider experimentally and theoretically the freezing of water in a rectangular vessel, with a view to investigating natural convection as a possible mechanism. The experimental and theoretical results are, in general, found to agree well; however, in combination, the results suggest that, whereas natural convection gives the correct general timescale for freezing, supercooling adjusts the actual time required. Moreover, the effect of supercooling leads to a spread in the experimental freezing times, giving results that constitute evidence of the occurrence of the Mpemba effect, even though the model results by themselves do not.

Introduction

The observation that “hot water can freeze faster than cold water” has been made on numerous occasions in history ever since the time of Aristotle [1]. It was rediscovered in the 20th century by a Tanzanian student, Erasto Mpemba; subsequently, it was reported by Mpemba and Osborne [2], and thereafter has been termed the Mpemba effect. At first sight, the phenomenon appears to be highly counterintuitive, and various hypotheses have been proposed in order to explain it. Amongst these are:

  • evaporative cooling, whereby more of a water sample at an initially higher temperature will evaporate, resulting in the sample’s mass decreasing more quickly and therefore reaching freezing temperature first [3], [4], [5], [6], [7];

  • dissolved gases within water samples, which alter their physical properties and, thence, cooling characteristics [8], [9];

  • supercooling, whereby the fact that impurities contained in water can affect the temperature at which ice will nucleate, either homogeneously or heterogeneously [10], [11], [12];

  • natural convection, whereby the motion of water at an initially higher temperature could lead to enhanced heat transfer, and in particular more rapid cooling, than would water that is initially at a temperature that is closer to ambient [13], [14].

In 2012, the UK’s Royal Society of Chemistry even sponsored a competition to solicit explanations for the phenomenon.1

Recent reviews of the topic and its history are given by Jeng [15] and Maeno [16], who point out that many of the experiments that have been carried out to demonstrate the effect have been unsatisfactory in their conception, with insufficient care being taken regarding what is actually meant by the statement that “hot water can freeze faster than cold water”, i.e. whether it means:

  • (A)

    the time taken for ice to first appear;

  • (B)

    or, rather, the time taken for the entire cooling sample to freeze.

In addition, which point in space in a water sample that is chosen to make the decision as regards the start and end of freezing should be specified, since this is different depending on the freezing method. For example, when water in a container is exposed to cold air, the decision should be made based on the temperature at the water surface in the case of criterion (A), but it should be made at the center or bottom in the case of (B). On the other hand, when the container is kept on a cold metal block, the bottom should be chosen in the case of (A), and the center or surface in the case of (B). Of course, a further complicating feature is that the effect may have more than one cause, and that it may not occur at all under certain conditions.

The development of theoretical models to assist with the interpretation of experimental results is a potentially attractive, yet largely unexplored, activity in the context of the Mpemba effect. Of all the papers cited above, only the ones by Kell [3] and Vynnycky and Maeno [7], both in the context of evaporative cooling, provide any form of simultaneous experimental and theoretical evidence for the effect. Kell’s model [7] has recently been analyzed in detail by Vynnycky and Mitchell [5], although see also the erratum by Vynnycky et al. [6]. Even more recently, Vynnycky and Maeno [7] considered the cooling of a shallow pool of water whose upper surface was exposed to air. Both experimental and model results hinted strongly at the occurrence of the Mpemba effect, which occurs because a considerably greater fraction of the initial water mass evaporates if the initial water temperature is higher; there is, therefore, less water to cool, and hence it can cool more quickly. The model results are quantified in Fig. 15 in [7], and indicate that the initial water temperature must be greater that around 318 K for the effect to be observable when the ambient temperature is 253 K. Even so, both experimental and modeling approaches encountered difficulties: the experimental results suffered from a low degree of reproducibility; the mathematical model, although retaining all of the essential physics, had to be significantly reduced in order to increase its computational tractability.

The purpose of this work therefore is to initiate an investigation, using both experiments and modeling, of a different situation where the Mpemba effect is thought to occur: the cooling of water in a closed container. In this situation, provided that the water that is used has been distilled and degassed, and the cooling is controlled in a known way, a leading candidate for the effect would be natural convection; in this sense, the situation is much simpler than in [7], where evaporation occurs in tandem with natural convection and thermal radiation. In fact, Maciejewski [13] carried out a large series of experiments in which water was frozen in a cylinder; a variety of initial water and cooling conditions were tried, and in some cases water that was initially at a higher temperature did appear to freeze completely more quickly than water at a lower temperature. The mechanism proposed was an instability associated with the natural convection, possibly turbulent, that will occur within the vessel as a result of temperature gradients. In particular, the results showed that whilst all experimental runs could be universally correlated as regards the time taken for the bulk liquid to reach 277 K, the temperature at which the density of water reaches a maximum, two clusters emerged as regards the time taken for complete freezing. A necessary, though not sufficient, condition for the phenomenon to occur was that the Rayleigh number had to be high enough, although even then, it was still possible that colder water would freeze faster. Although this instability was said to be associated with turbulent motion of the liquid, the critical Rayleigh number given was around 2 × 106, which ought to be still well within the laminar flow regime.

From the theoretical point of view, even the experiment in [13] is overly complicated as regards understanding fundamental heat and mass transfer, since the convection patterns that develop will be highly complex, in view of the initial formation of solidification fronts on all sides of the container. A further complication is that the cooling container that was used held both air and water, whereas it is only the freezing of the water that is of interest. A much simpler geometry for fundamental studies of solidification in the presence of convection is that of an enclosed rectangular container that is cooled on one side and insulated at all of the others; more specifically, one may expect the effect of natural convection to be maximal if the container is cooled from above. In this situation, there ought to be two cases, depending on the initial temperature of the water:

  • 1.

    if the initial water temperature is less than the temperature at which water reaches its density maximum, then we would not expect any convection at all, since the conduction solution is stable;

  • 2.

    if the initial temperature is greater, we can expect a stably stratified region just below the solidification front, below which there will be a region in which conduction is unstable and where there should initially be convection, as is known from Rayleigh–Bénard theory. Ultimately, this case becomes as case (1), with solidification being completed in conduction mode.

Although there have been numerous papers which have theoretically investigated coupled natural convection and ice formation in a rectangular geometry, the majority consider cooling applied on one of the vertical walls [4], [17], [18], [19], [20], [21]; a recent review of mathematical models and simulations for phase-change materials in general is given in [22]. Other relevant papers, which are more experimental in nature, are [23], [24], [25], [26], [27], [28]; however, Abegg et al. [29] have considered experimental work and modeling simultaneously for the cooling of water from above, although not systematically in the context of the Mpemba effect.

The layout of this paper is as follows. In Section 2, we give the details of an experiment, and the results obtained, to investigate the cooling and freezing of water from above. In Section 3, a mathematical model relevant to the experiment is formulated; in Section 4, the model is nondimensionalized, and in Section 5 scaling arguments are used to reduce it. Section 6 explains the numerical issues involved in applying a recent finite-element model for coupled transient natural convection and solidification [4] to the current problem; Section 7 gives the results of the model, as well as providing quantitative comparison with the results of Section 2. Conclusions are drawn in Section 8.

Section snippets

Apparatus

The experiment was carried out in a room held at a constant temperature of 288 K. The experimental apparatus, shown in Fig. 1, Fig. 2, consisted of two sets of acrylic plates; between each set, there was an insulating layer of air, whose temperature was controlled by means of a mini-cooler. The outer set of acrylic plates are 10 mm thick; for the inner set, plates of thickness 5 mm and 10 mm were prepared. The water sample to be frozen is bounded on five sides by the inner acrylic plates and

Mathematical model

We consider a two-dimensional (2D) time-dependent model; even though the experiment is three-dimensional (3D), we find that a 2D model is adequate enough to capture the details we require, as will be seen in the later results. Fig. 6 shows a schematic of a square vessel of water initially at temperature Tinit and occupying the region -W/2<x<W/2, 0<y<W, which is subsequently cooled by means of a plate at y=W that is kept at a temperature Tcold, where Tcold<Tmelt, the melting temperature of ice

Nondimensionalization

To nondimensionalize, we setX=xW,Y=yW,S=sW,τ=t[t],θair=Tair-TmeltTinit-Tmelt,θa,i=Ta,i-TmeltTinit-Tmelt,θa,o=Ta,o-TmeltTinit-Tmelt,θl=Tl-TmeltTinit-Tmelt,θs=Ts-TcoldTmelt-Tcold,U=uu,V=vu,P=pp.Suitable choices for the velocity scale [u] and the pressure scale [p] are[u]=klWρl,maxcpl,[p]=μklW2ρl,maxcpl.For the time scale [t], the main possibilities areρl,maxcplW2kl,ρl,maxΔHfW2ksTinit-Tmelt,ρacpaW2ka,ρscpsW2ks.For the data in Table 1, we have1.8×104s,800s,2.0×104s,2.6×103s,respectively. On the

Reduced model

Here, we use order of magnitude arguments to reduce the problem to one that needs only to be solved for X1/2,0Y1. Observe first that κla,κlsO1, so it appears that all terms in (27), (32) should be retained. However, the vertical acrylic plates are comparatively slender, suggesting that (32) would becomeκlaθa,jτ2θa,jX2.Now, since Xωa,j, where ωa,j1 for j=i,o, we obtain2θa,jX20,a similar argument applies to the horizontal acrylic plates.

Now, consider the air between the acrylic

Numerical issues

The remaining numerical task is then the solution of equations (27), (37), (38), (39), (40), subject to (41), (42), (46), (47), (48), (49), (59), (60), (61) and (71), (72), (73). There are, however, a number of issues to address before numerical implementation. First of all, since ρs<ρl, it clearly does not make physical sense to solve the equations as they stand on a fixed computational domain: the newly-formed ice occupies a greater volume than the initial water. Although one could try to

Results

First, in Section 7.1, we give detailed model results when Tinit=323 K; then, in Section 7.2, we compare model and experimental results.

Conclusions

This paper has investigated whether the Mpemba effect can be observed for a closed cubic container of water that is cooled from above, but effectively insulated on all other sides; in this situation, the only mechanism for the effect can be natural convection. Experiments were performed, and a two-dimensional time-dependent model was constructed that took into account heat conduction in the ice, heat conduction and convection in the water, and the release of latent heat at the ice–water

Conflict of interest

None declared.

Acknowledgment

The second author would like to acknowledge Mr. Tetsuya Miura, who conducted the experiments for his Master’s thesis.

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