Natural Convection-Driven Evaporation and the Mpemba Effect

Introduction Prof. Maeno’sexperiment Mathematicalmodel Model results (Tcold =253K) Conclusions

Natural convection-driven evaporation and the Mpemba effect

M. Vynnycky

Mathematics Applications Consortium for Science and Industry (MACSI), Department of Mathematics and Statistics, University of Limerick, Limerick, Ireland

University of Brighton, 8 May 2012 Introduction Prof. Maeno’sexperiment Mathematicalmodel Modelresults (Tcold =253K) Conclusions

Outline of my talk

Short history and background Prof. Maeno’s experiment (Hokkaido University) Mathematical model for the experiment Results Conclusions Introduction Prof. Maeno’sexperiment Mathematicalmodel Modelresults (Tcold =253K) Conclusions

Outline of my talk

Background

Hot water can freeze faster than cold water Observed by Aristotle, Bacon and Descartes Rediscovered in the 20th century by Erasto Mpemba (whilst making ice-cream at school in Tanzania) Proposed mechanisms: evaporative cooling, effect of dissolved gases, supercooling, natural convection Review by Jeng1 highlights difﬁculties with: deﬁnition of problem; interpretation of experimental results Further confusion because: the effect may not occur at all under certain conditions; when it occurs, it might be due to a combination of mechanisms Surprisingly, next to no models exist 1 Jeng, M., The Mpemba effect: when can hot water freeze faster than cold?, American Journal of Physics, 74 (2006) 514-522 Introduction Prof. Maeno’sexperiment Mathematicalmodel Modelresults (Tcold =253K) Conclusions

Background

Evaporative cooling

Lumped-mass model (0-D) by Kell2 Recently re-analyzed3 We want to move towards 1D,2D,3D models Consider Prof. Maeno’s experiment4

2 Kell, G. S., The freezing of hot and cold water, American Journal of Physics, 37 (1969) 564-565 3 Vynnycky M. and Mitchell S. L., Evaporative cooling and the Mpemba effect, Heat and Mass Transfer, 46 (2010) 881-890 4 N. Maeno, S. Takahashi, A. Sato, Y. Kominami, H. Konishi, S. Omiya, Veriﬁcation experiments of Mpemba effect, Seppyo 74 (2012) 1-13. Introduction Prof. Maeno’sexperiment Mathematicalmodel Modelresults (Tcold =253K) Conclusions

Evaporative cooling

Lumped-mass model (0-D) by Kell2 Recently re-analyzed3 We want to move towards 1D,2D,3D models Consider Prof. Maeno’s experiment4

Evaporative cooling

Lumped-mass model (0-D) by Kell2 Recently re-analyzed3 We want to move towards 1D,2D,3D models Consider Prof. Maeno’s experiment4

Evaporative cooling

Lumped-mass model (0-D) by Kell2 Recently re-analyzed3 We want to move towards 1D,2D,3D models Consider Prof. Maeno’s experiment4

Evaporative cooling

Lumped-mass model (0-D) by Kell2 Recently re-analyzed3 We want to move towards 1D,2D,3D models Consider Prof. Maeno’s experiment4

2 Kell, G. S., The freezing of hot and cold water, American Journal of Physics, 37 (1969) 564-565 3 Vynnycky M. and Mitchell S. L., Evaporative cooling and the Mpemba effect, Heat and Mass Transfer, 46 (2010) 881-890 4 N. Maeno, S. Takahashi, A. Sato, Y. Kominami, H. Konishi, S. Omiya, Veriﬁcation experiments of Mpemba effect, Seppyo 74 (2012) 1-13. Introduction Prof. Maeno’s experiment Mathematicalmodel Modelresults(Tcold =253K) Conclusions

Conﬁguration

Styrofoam, three suspended thermocouples, holes containing hot and cool water, TWINBIRD SCDF25 deep freezer Introduction Prof. Maeno’s experiment Mathematicalmodel Modelresults(Tcold =253K) Conclusions

Schematic Introduction Prof. Maeno’s experiment Mathematicalmodel Modelresults(Tcold =253K) Conclusions

Some experimental results (I)

350

340

330

320

310 hot water

300 Temperature [K] 290 cool water

280

270 0 500 1000 1500 2000 2500 Time [s]

The temperature recorded at the uppermost thermocouple vs. time. The initial water temperatures are 348 K and 310 K, and the surrounding air temperature is 263 K (No Mpemba effect). Introduction Prof. Maeno’s experiment Mathematicalmodel Modelresults(Tcold =253K) Conclusions

Some experimental results (II)

360

350

340

330

320

310 hot water

Temperature [K] 300 cool water 290

280

270 0 500 1000 1500 2000 2500 Time [s]

The temperature recorded at the uppermost thermocouple vs. time. The initial water temperatures are 351 K and 318 K, and the surrounding air temperature is 260 K (Mpemba effect!). Introduction Prof. Maeno’s experiment Mathematicalmodel Modelresults(Tcold =253K) Conclusions

Some experimental results (III)

360

350

340

330

320

310 hot water

Temperature [K] 300

290 cool water

280

270 0 500 1000 1500 2000 2500 Time [s]

The temperature recorded at the centre thermocouple vs. time. The initial water temperatures are 348 K and 310 K, and the surrounding air temperature is 263 K (No Mpemba effect). Introduction Prof. Maeno’s experiment Mathematicalmodel Modelresults(Tcold =253K) Conclusions

Some experimental results (IV)

350

340

330

320

310 hot water

300 Temperature [K] cool water 290

280

270 0 500 1000 1500 2000 2500 Time [s]

Some experimental results (V)

360

350

340

330

320

310 hot water

Temperature [K] 300

290 cool water

280

270 0 500 1000 1500 2000 2500 Time [s]

The temperature recorded at the lowest thermocouple vs. time. The initial water temperatures are 348 K and 310 K, and the surrounding air temperature is 263 K (No Mpemba effect). Introduction Prof. Maeno’s experiment Mathematicalmodel Modelresults(Tcold =253K) Conclusions

Some experimental results (VI)

360

350

340

330

320

310 hot water

Temperature [K] 300 cool water 290

280

270 0 500 1000 1500 2000 2500 Time [s]

The temperature recorded at the lowest thermocouple vs. time. The initial water temperatures are 351 K and 318 K, and the surrounding air temperature is 260 K (Mpemba effect!). Introduction Prof. Maeno’s experiment Mathematicalmodel Modelresults(Tcold =253K) Conclusions

Yu-to-Mizu-Kurabe diagram

102 C]

◦ 348K/310K/263K

101

100 351K/318K/260K Temperature of hot water [

10−1 10−1 100 101 102 Temperature of cool water [◦C]

Each curve in the diagram denotes a hot and cool water pair, indicating hot water/cool water/air cooling temperatures). The Mpemba effect occurs if a curve goes below the diagonal line. Introduction Prof. Maeno’s experiment Mathematicalmodel Modelresults(Tcold =253K) Conclusions

General comments on experiments

Water at a given initial temperature and exposed to a given ambient temperature did not necessarily take the same amount of time to reach freezing point Consequently, it is difﬁcult to reproduce the Mpemba effect with this experiment, although it seems to be there We try to interpret the results with a mathematical model Introduction Prof. Maeno’s experiment Mathematicalmodel Modelresults(Tcold =253K) Conclusions

General comments on experiments

Water at a given initial temperature and exposed to a given ambient temperature did not necessarily take the same amount of time to reach freezing point Consequently, it is difﬁcult to reproduce the Mpemba effect with this experiment, although it seems to be there We try to interpret the results with a mathematical model Introduction Prof. Maeno’sexperiment Mathematical model Model results (Tcold =253K) Conclusions

What to model?

(A) Natural convection of air/water vapour mixture above the liquid water (B) Natural convection of the liquid water itself (C) The downward motion of the gas/water interface as water evaporates from the liquid pool (D) Thermal radiation from the pool surface Introduction Prof. Maeno’sexperiment Mathematical model Model results (Tcold =253K) Conclusions

What to model?

Natural convection-driven evaporation

AIR z

r (a) the AIR + AIR + WATER VAPOUR WATER VAPOUR evaporation of a

WATER cold water into

(a) hotter air

AIR + z WATER VAPOUR (b) the evaporation of r hot water into AIR AIR a colder air

WATER

(b) Introduction Prof. Maeno’sexperiment Mathematical model Model results (Tcold =253K) Conclusions

Natural convection-driven evaporation

AIR z

r (a) the AIR + AIR + WATER VAPOUR WATER VAPOUR evaporation of a

WATER cold water into

(a) hotter air

AIR + z WATER VAPOUR (b) the evaporation of r hot water into AIR AIR a colder air

WATER

(b) Introduction Prof. Maeno’sexperiment Mathematical model Model results (Tcold =253K) Conclusions

Natural convection-driven evaporation

AIR z

r (a) the AIR + AIR + WATER VAPOUR WATER VAPOUR evaporation of a

WATER cold water into

(a) hotter air

AIR + z WATER VAPOUR (b) the evaporation of r hot water into AIR AIR a colder air

WATER

(b) Introduction Prof. Maeno’sexperiment Mathematical model Model results (Tcold =253K) Conclusions

Hot water, cold air

vertical plume

boundary layer

water Introduction Prof. Maeno’sexperiment Mathematical model Model results (Tcold =253K) Conclusions

Modelling considerations (I)

Axisymmetry Computational ﬂuid dynamics (CFD) for a full model Moving boundary problem Laminar ﬂow: Rayleigh numbers for the air/water vapour 6 mixture and the water (Rag and Ral ) are around 10 The water vapour is not dilute in the air; hence, we need the full Stefan-Maxwell equations for binary diffusion and convection Introduction Prof. Maeno’sexperiment Mathematical model Model results (Tcold =253K) Conclusions

Modelling considerations (I)

Modelling considerations (II)

Earlier computations 5 suggest that thermally insulating boundary conditions give (more or less) conduction only in the water, even though Ral is nominally high So we aim for 1D conduction in the water; however, the water is cooled and evaporates due to the boundary layer We need to ﬁnd the Nusselt and Sherwood numbers for ﬂow past a horizontal hot circular disk The boundary layer is known qualitatively to be of thickness −1/5 Rag for horizontal surfaces; however, the computations have never been done before, i.e. we have to do them

5 Vynnycky, M. and Kimura, S., Towards a natural-convection model for the Mpemba effect, Proceedings of the 19th International Symposium on Transport Phenomena, Reykjavik, Iceland, 17-20 August 2008, Paper 216 (CD-ROM). Introduction Prof. Maeno’sexperiment Mathematical model Model results (Tcold =253K) Conclusions

Modelling considerations (II)

Modelling considerations (III)

The velocity in the boundary layer is mass-averaged Dufour effect (diffusion thermo): species diffusion affects heat transfer ‘Blowing’ effect, i.e. the normal velocity at the gas-water interface is non-zero Cooling time scale is much longer than the boundary-layer convection time scale; hence, the boundary layer appears as if quasi-steady Introduction Prof. Maeno’sexperiment Mathematical model Model results (Tcold =253K) Conclusions

Modelling considerations (III)

Flow structure

vertical plume

boundary layer

water

Velocity vectors, and boundary-layer and vertical plume structure Introduction Prof. Maeno’sexperiment Mathematical model Model results (Tcold =253K) Conclusions

Governing equations (water)

∂ρ(l) + ∇ · ρ(l)u(l) = 0, (1) ∂t ∂u(l) ρ(l) + u(l) ·∇ u(l) = −∇p(l) + µ(l)∇2u(l) − ρ(l)gk, ∂t ! (2) ∂T ρ(l)c(l) + u(l) ·∇T = ∇ · k (l)∇T . (3) p ∂t Introduction Prof. Maeno’sexperiment Mathematical model Model results (Tcold =253K) Conclusions

Governing equations (gas phase above the water): I

Conservation of species:

∂ (g) ρ ωa + ∇ · na = 0, (4) ∂t ∂ (g) ρ ωw + ∇ · nw = 0, (5) ∂t where (g) (g) ni = ρ ωi u +ji , i = a, w; (6) (g) na and nw are the air and water vapour mass ﬂuxes, ρ is the (g) mixture density, ωi is the mass fraction, u is the mass-averaged velocity and the vector ji describes the diffusion-driven transport. Also, ωa + ωw = 1. Introduction Prof. Maeno’sexperiment Mathematical model Model results (Tcold =253K) Conclusions

Governing equations (gas phase above the water): II

Conservation of momentum:

∂u(g) ρ(g) + u(g) ·∇ u(g) = ∇ · σ(g) − ρ(g)gk, (7) ∂t ! where σ is the total stress tensor, given by

tr 2 σ(g) = −p(g)I+µ(g) ∇u(g) + ∇u(g) − µ(g) ∇ · u(g) I. (8) 3 Introduction Prof. Maeno’sexperiment Mathematical model Model results (Tcold =253K) Conclusions

Governing equations (gas phase above the water): III

Conservation of heat: ∂T ρ(g)c(g) + c(g)n + c(g) n ·∇T = ∇ · k (g)∇T , (9) p ∂t p,a a p,w w (g) where cp,i for i = a, w are the speciﬁc heat capacities for each (g) species, cp is given by

(g) (g) (g) cp = cp,aωa + cp,w ωw ,

and k (g) is the mixture thermal conductivity, which will also, in general, depend on the local mass fraction and temperature. Introduction Prof. Maeno’sexperiment Mathematical model Model results (Tcold =253K) Conclusions

Strategy

Solve (A) to ﬁnd Nu = Nu(Rag), Sh = Sh(Rag); steady-state boundary layer equations6 Neglect (B); just assume conduction in the water Solve for (C) and (D) with a 1D Stefan problem7

6 M. Vynnycky and N. Maeno, Axisymmetric natural convection-driven evaporation of water: analysis and numerical solution, submitted to Int. J. Heat Mass Tran. (2012) 7 M. Vynnycky and N. Maeno, Axisymmetric natural convection-driven evaporation of hot water and the Mpemba effect, submitted to Int. J. Heat Mass Tran. (2012) Introduction Prof. Maeno’sexperiment Mathematical model Model results (Tcold =253K) Conclusions

Strategy

Solve (A) to ﬁnd Nu = Nu(Rag), Sh = Sh(Rag); steady-state boundary layer equations6 Neglect (B); just assume conduction in the water Solve for (C) and (D) with a 1D Stefan problem7

Strategy

Solve (A) to ﬁnd Nu = Nu(Rag), Sh = Sh(Rag); steady-state boundary layer equations6 Neglect (B); just assume conduction in the water Solve for (C) and (D) with a 1D Stefan problem7

Strategy

Solve (A) to ﬁnd Nu = Nu(Rag), Sh = Sh(Rag); steady-state boundary layer equations6 Neglect (B); just assume conduction in the water Solve for (C) and (D) with a 1D Stefan problem7

Time to reach freezing (t0) vs initial water temperature (Tw,i )

8000

7000

6000

5000

[s] 4000 0 t

3000

2000 Tcold = 253 K Tcold = 258 K 1000 Tcold = 263 K Tcold = 268 K 0 260 280 300 320 340 360 380 Tw,i [K] Introduction Prof. Maeno’sexperiment Mathematicalmodel Model results (Tcold =253 K) Conclusions

Water height at t0 (h0) vs Tw,i

Tcold = 253 K

[mm] 7

0 Tcold = 258 K h Tcold = 263 K 6 Tcold = 268 K

4 260 280 300 320 340 360 380 Tw,i [K] Introduction Prof. Maeno’sexperiment Mathematicalmodel Model results (Tcold =253 K) Conclusions

Water height (h) vs time (t) for different Tw,i

9.5

8.5 [mm] h

8 Tw,i = 303 K Tw,i = 320 K Tw,i = 337 K 7.5 Tw,i = 354 K

7 0 1000 2000 3000 4000 5000 t [s] Introduction Prof. Maeno’sexperiment Mathematicalmodel Model results (Tcold =253 K) Conclusions

Surface temperature (Thot ) vs t for different Tw,i

360

350 Tw,i = 303 K 340 Tw,i = 320 K T = 337 K 330 w,i Tw,i = 354 K 320 [K] hot

T 310

300

290

280

270 0 1000 2000 3000 4000 5000 t [s] Introduction Prof. Maeno’sexperiment Mathematicalmodel Model results (Tcold =253 K) Conclusions

Water temperature (T ) vs distance from base (z)

320

315

310

305

300 − [K] t/[t]= 10 2 T 295 t/[t]= 10−1 t/[t]= 1 290

285

280

275 0 2 4 6 8 10 z [mm] Introduction Prof. Maeno’sexperiment Mathematicalmodel Model results (Tcold =253 K) Conclusions

Temperature difference (bottom-top, ∆T ) vs t for different Tw,i

7 Tw,i = 303 K 6 Tw,i = 320 K T = 337 K 5 w,i [K] Tw,i = 354 K T

∆ 4

0 0 1000 2000 3000 4000 5000 t [s] Introduction Prof. Maeno’sexperiment Mathematicalmodel Model results (Tcold =253 K) Conclusions

Comparison with experiment (I)

100

90 Mpemba effect

30 Temperature of hot water [C]

20 310/348/263 (no) 318/351/260 (yes) 10 T_{cold}=253 K T_{cold}=263 K 0 20 40 60 80 100 Temperature of cool water [C]

The number of possible hot/cool water pairs increases with Tcold . However, the model predicts that the Mpemba effect should be seen for both experiments, whereas it was actually seen for only one of them. Introduction Prof. Maeno’sexperiment Mathematicalmodel Model results (Tcold =253 K) Conclusions

Comparison with experiment (II)

Model overestimates time taken to reach freezing point, although order of magnitude is correct The inclusion of thermal radiation is necessary to ensure that freezing time is not highly overestimated Model doesn’t seem to capture the sensitivity of the experiments with respect to t0

At higher values of Tw,i, the model gives higher-than-observed levels of evaporation Introduction Prof. Maeno’sexperiment Mathematicalmodel Model results (Tcold =253 K) Conclusions

Comparison with experiment (II)

At higher values of Tw,i, the model gives higher-than-observed levels of evaporation Introduction Prof. Maeno’sexperiment Mathematicalmodel Model results (Tcold =253 K) Conclusions

Comparison with experiment (II)

At higher values of Tw,i, the model gives higher-than-observed levels of evaporation Introduction Prof. Maeno’sexperiment Mathematicalmodel Model results (Tcold =253 K) Conclusions

Comparison with experiment (II)

At higher values of Tw,i, the model gives higher-than-observed levels of evaporation Introduction Prof. Maeno’sexperiment Mathematicalmodel Model results (Tcold =253 K) Conclusions

Comparison with experiment (II)

At higher values of Tw,i, the model gives higher-than-observed levels of evaporation Introduction Prof. Maeno’sexperiment Mathematicalmodel Model results (Tcold =253 K) Conclusions

‘Over-evaporation’

INSULATION INSULATION WATER

Schematic of possible ﬂow streamlines as water evaporates Introduction Prof. Maeno’sexperiment Mathematicalmodel Model results (Tcold =253 K) Conclusions

Conclusions

Experiment and model, and the agreement between them, is far from perfect Experimental results seem to be highly irreproducible; sometimes, the Mpemba effect is seen, sometimes not The model shows the Mpemba effect, although it may be less valid when more of the water evaporates, due to simplifying assumptions A better (more difficult) model requires CFD to solve simultaneously for convection in the water flow, multicomponent flow in the boundary layer and the movement of the evaporation front Introduction Prof. Maeno’sexperiment Mathematicalmodel Model results (Tcold =253 K) Conclusions

Conclusions

Future work (natural convection only)

Experimental vessel Introduction Prof. Maeno’sexperiment Mathematicalmodel Model results (Tcold =253 K) Conclusions

Future work (natural convection only)

Time elapsed for the ice layer to grow 25 mm in thickness Introduction Prof. Maeno’sexperiment Mathematicalmodel Model results (Tcold =253 K) Conclusions

Acknowledgements

Science Foundation Ireland Mathematics Initiative Grant 06/MI/005 Prof. N. Maeno Introduction Prof. Maeno’sexperiment Mathematicalmodel Model results (Tcold =253 K) Conclusions

Acknowledgements

Science Foundation Ireland Mathematics Initiative Grant 06/MI/005 Prof. N. Maeno Introduction Prof. Maeno’sexperiment Mathematicalmodel Model results (Tcold =253 K) Conclusions

Acknowledgements

Science Foundation Ireland Mathematics Initiative Grant 06/MI/005 Prof. N. Maeno