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Evaporative cooling and the Mpemba effect

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An Erratum to this article was published on 22 February 2011

Abstract

The Mpemba effect is popularly summarized by the statement that “hot water can freeze faster than cold”, and has been observed experimentally since the time of Aristotle; however, there exist almost no theoretical models that predict the effect. With a view to initiating rigorous modelling activity on this topic, this paper analyzes in some depth the only available model in literature, which considers the potential role of evaporative cooling and treats the cooling water as a lumped mass. Certain omissions in the original work are highlighted and corrected, and results are obtained for a wide range of operating conditions—in particular, initial liquid temperature and cooling temperature. The implications and importance of the results of the model for experimental design are discussed, as are extensions of the model to handle more realistic 1-, 2- and 3-dimensional configurations.

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Abbreviations

A 0, A 1, A 2 :

Dimensionless constants

c p :

Specific heat capacity of water

C p :

Dimensionless specific heat capacity of water

[c p ]:

Characteristic scale for c p

k :

Mass transfer proportionality constant

\( \mathcal{L} \) :

Dimensionless constant (L e (T 0)/L f )

L e :

Latent heat of evaporation of water

[L e ]:

Characteristic scale for L e

L f :

Latent of freezing

m :

Mass of water vapour

m liq :

Mass of water

m liq,i :

Initial mass of water

M :

Mass of ice

p a :

Water vapour pressure of the surroundings

P a :

Dimensionless water vapour pressure of the surroundings

p v :

Water vapour pressure

P v :

Dimensionless water vapour pressure

[p v ]:

Characteristic scale for p v

Q1, Q2:

Polynomials in exp(Stθ)

St :

Stefan number, [c p ](T m  − T 0)/[L e ]

t :

Time

[t]:

Characteristic time scale

t f :

Time taken for cooling sample to freeze

t f,total :

Total time taken for two cooling samples to freeze

t 0 :

Time taken for cooling sample to reach 0°C

t 0,total :

Total time taken for two cooling samples to reach 0°C

T :

Temperature

T i :

Initial water temperature

T crit i :

Dimensional solutions to Eqs. (51) and ( 52)

T m :

Maximum water temperature (100°C)

T 0 :

Minimum water temperature (0°C)

α ± :

Dimensionless constants

θ :

Dimensionless temperature

θ i :

Dimensionless initial water temperature

θ criti,f :

Solution to Eq. (52)

θ criti,0 :

Solution to Eq. (51)

Λ:

Dimensionless latent heat of evaporation of water

μ :

Dimensionless mass of water vapour

μ ice :

Dimensionless mass of ice

μ liq :

Dimensionless mass of water

μ liq,i :

Dimensionless initial mass of water

ν :

Dimensionless constant

τ :

Dimensionless time (t/[t])

τ f :

Dimensionless time taken for cooling sample to freeze

τ f,total :

Total dimensionless time taken for two cooling samples to freeze

τ 0 :

Dimensionless time taken for cooling sample to reach 0°C

τ 0,total :

Total dimensionless time taken for two cooling samples to reach 0°C

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Acknowledgments

The authors acknowledge the support of the Mathematics Applications Consortium for Science and Industry (www.macsi.ul.ie) funded by the Science Foundation Ireland Mathematics Initiative Grant 06/MI/005.

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Authors and Affiliations

  1. Mathematics Application Consortium for Science and Industry (MACSI), Department of Mathematics and Statistics, University of Limerick, Limerick, Republic of Ireland

    M. Vynnycky & S. L. Mitchell

Authors
  1. M. Vynnycky
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  2. S. L. Mitchell
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Correspondence to M. Vynnycky.

Additional information

An erratum to this article can be found at http://dx.doi.org/10.1007/s00231-010-0759-3

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Vynnycky, M., Mitchell, S.L. Evaporative cooling and the Mpemba effect. Heat Mass Transfer 46, 881–890 (2010). https://doi.org/10.1007/s00231-010-0637-z

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