View
1
Download
0
Category
Preview:
Citation preview
Introduction Prof. Maeno’s experiment Mathematical model Model results (Tcold =253 K) Conclusions
Natural convection-driven evaporationand 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’s experiment Mathematical model Model results (Tcold =253 K) 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’s experiment Mathematical model Model results (Tcold =253 K) 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’s experiment Mathematical model Model results (Tcold =253 K) 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’s experiment Mathematical model Model results (Tcold =253 K) 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’s experiment Mathematical model Model results (Tcold =253 K) 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’s experiment Mathematical model Model results (Tcold =253 K) 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’s experiment Mathematical model Model results (Tcold =253 K) Conclusions
Background
Hot water can freeze faster than cold waterObserved by Aristotle, Bacon and DescartesRediscovered in the 20th century by Erasto Mpemba(whilst making ice-cream at school in Tanzania)Proposed mechanisms: evaporative cooling, effect ofdissolved gases, supercooling, natural convectionReview by Jeng1 highlights difficulties with: definition ofproblem; interpretation of experimental resultsFurther confusion because: the effect may not occur at allunder certain conditions; when it occurs, it might be due toa combination of mechanismsSurprisingly, next to no models exist
1Jeng, M., The Mpemba effect: when can hot water freeze faster than cold?, American Journal of Physics, 74
(2006) 514-522
Introduction Prof. Maeno’s experiment Mathematical model Model results (Tcold =253 K) Conclusions
Background
Hot water can freeze faster than cold waterObserved by Aristotle, Bacon and DescartesRediscovered in the 20th century by Erasto Mpemba(whilst making ice-cream at school in Tanzania)Proposed mechanisms: evaporative cooling, effect ofdissolved gases, supercooling, natural convectionReview by Jeng1 highlights difficulties with: definition ofproblem; interpretation of experimental resultsFurther confusion because: the effect may not occur at allunder certain conditions; when it occurs, it might be due toa combination of mechanismsSurprisingly, next to no models exist
1Jeng, M., The Mpemba effect: when can hot water freeze faster than cold?, American Journal of Physics, 74
(2006) 514-522
Introduction Prof. Maeno’s experiment Mathematical model Model results (Tcold =253 K) Conclusions
Background
Hot water can freeze faster than cold waterObserved by Aristotle, Bacon and DescartesRediscovered in the 20th century by Erasto Mpemba(whilst making ice-cream at school in Tanzania)Proposed mechanisms: evaporative cooling, effect ofdissolved gases, supercooling, natural convectionReview by Jeng1 highlights difficulties with: definition ofproblem; interpretation of experimental resultsFurther confusion because: the effect may not occur at allunder certain conditions; when it occurs, it might be due toa combination of mechanismsSurprisingly, next to no models exist
1Jeng, M., The Mpemba effect: when can hot water freeze faster than cold?, American Journal of Physics, 74
(2006) 514-522
Introduction Prof. Maeno’s experiment Mathematical model Model results (Tcold =253 K) Conclusions
Background
Hot water can freeze faster than cold waterObserved by Aristotle, Bacon and DescartesRediscovered in the 20th century by Erasto Mpemba(whilst making ice-cream at school in Tanzania)Proposed mechanisms: evaporative cooling, effect ofdissolved gases, supercooling, natural convectionReview by Jeng1 highlights difficulties with: definition ofproblem; interpretation of experimental resultsFurther confusion because: the effect may not occur at allunder certain conditions; when it occurs, it might be due toa combination of mechanismsSurprisingly, next to no models exist
1Jeng, M., The Mpemba effect: when can hot water freeze faster than cold?, American Journal of Physics, 74
(2006) 514-522
Introduction Prof. Maeno’s experiment Mathematical model Model results (Tcold =253 K) Conclusions
Background
Hot water can freeze faster than cold waterObserved by Aristotle, Bacon and DescartesRediscovered in the 20th century by Erasto Mpemba(whilst making ice-cream at school in Tanzania)Proposed mechanisms: evaporative cooling, effect ofdissolved gases, supercooling, natural convectionReview by Jeng1 highlights difficulties with: definition ofproblem; interpretation of experimental resultsFurther confusion because: the effect may not occur at allunder certain conditions; when it occurs, it might be due toa combination of mechanismsSurprisingly, next to no models exist
1Jeng, M., The Mpemba effect: when can hot water freeze faster than cold?, American Journal of Physics, 74
(2006) 514-522
Introduction Prof. Maeno’s experiment Mathematical model Model results (Tcold =253 K) Conclusions
Background
Hot water can freeze faster than cold waterObserved by Aristotle, Bacon and DescartesRediscovered in the 20th century by Erasto Mpemba(whilst making ice-cream at school in Tanzania)Proposed mechanisms: evaporative cooling, effect ofdissolved gases, supercooling, natural convectionReview by Jeng1 highlights difficulties with: definition ofproblem; interpretation of experimental resultsFurther confusion because: the effect may not occur at allunder certain conditions; when it occurs, it might be due toa combination of mechanismsSurprisingly, next to no models exist
1Jeng, M., The Mpemba effect: when can hot water freeze faster than cold?, American Journal of Physics, 74
(2006) 514-522
Introduction Prof. Maeno’s experiment Mathematical model Model results (Tcold =253 K) Conclusions
Background
Hot water can freeze faster than cold waterObserved by Aristotle, Bacon and DescartesRediscovered in the 20th century by Erasto Mpemba(whilst making ice-cream at school in Tanzania)Proposed mechanisms: evaporative cooling, effect ofdissolved gases, supercooling, natural convectionReview by Jeng1 highlights difficulties with: definition ofproblem; interpretation of experimental resultsFurther confusion because: the effect may not occur at allunder certain conditions; when it occurs, it might be due toa combination of mechanismsSurprisingly, next to no models exist
1Jeng, M., The Mpemba effect: when can hot water freeze faster than cold?, American Journal of Physics, 74
(2006) 514-522
Introduction Prof. Maeno’s experiment Mathematical model Model results (Tcold =253 K) Conclusions
Background
Hot water can freeze faster than cold waterObserved by Aristotle, Bacon and DescartesRediscovered in the 20th century by Erasto Mpemba(whilst making ice-cream at school in Tanzania)Proposed mechanisms: evaporative cooling, effect ofdissolved gases, supercooling, natural convectionReview by Jeng1 highlights difficulties with: definition ofproblem; interpretation of experimental resultsFurther confusion because: the effect may not occur at allunder certain conditions; when it occurs, it might be due toa combination of mechanismsSurprisingly, next to no models exist
1Jeng, M., The Mpemba effect: when can hot water freeze faster than cold?, American Journal of Physics, 74
(2006) 514-522
Introduction Prof. Maeno’s experiment Mathematical model Model results (Tcold =253 K) 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
2Kell, G. S., The freezing of hot and cold water, American Journal of Physics, 37 (1969) 564-565
3Vynnycky M. and Mitchell S. L., Evaporative cooling and the Mpemba effect, Heat and Mass Transfer, 46
(2010) 881-8904
N. Maeno, S. Takahashi, A. Sato, Y. Kominami, H. Konishi, S. Omiya, Verification experiments of Mpembaeffect, Seppyo 74 (2012) 1-13.
Introduction Prof. Maeno’s experiment Mathematical model Model results (Tcold =253 K) 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
2Kell, G. S., The freezing of hot and cold water, American Journal of Physics, 37 (1969) 564-565
3Vynnycky M. and Mitchell S. L., Evaporative cooling and the Mpemba effect, Heat and Mass Transfer, 46
(2010) 881-8904
N. Maeno, S. Takahashi, A. Sato, Y. Kominami, H. Konishi, S. Omiya, Verification experiments of Mpembaeffect, Seppyo 74 (2012) 1-13.
Introduction Prof. Maeno’s experiment Mathematical model Model results (Tcold =253 K) 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
2Kell, G. S., The freezing of hot and cold water, American Journal of Physics, 37 (1969) 564-565
3Vynnycky M. and Mitchell S. L., Evaporative cooling and the Mpemba effect, Heat and Mass Transfer, 46
(2010) 881-8904
N. Maeno, S. Takahashi, A. Sato, Y. Kominami, H. Konishi, S. Omiya, Verification experiments of Mpembaeffect, Seppyo 74 (2012) 1-13.
Introduction Prof. Maeno’s experiment Mathematical model Model results (Tcold =253 K) 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
2Kell, G. S., The freezing of hot and cold water, American Journal of Physics, 37 (1969) 564-565
3Vynnycky M. and Mitchell S. L., Evaporative cooling and the Mpemba effect, Heat and Mass Transfer, 46
(2010) 881-8904
N. Maeno, S. Takahashi, A. Sato, Y. Kominami, H. Konishi, S. Omiya, Verification experiments of Mpembaeffect, Seppyo 74 (2012) 1-13.
Introduction Prof. Maeno’s experiment Mathematical model Model results (Tcold =253 K) 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
2Kell, G. S., The freezing of hot and cold water, American Journal of Physics, 37 (1969) 564-565
3Vynnycky M. and Mitchell S. L., Evaporative cooling and the Mpemba effect, Heat and Mass Transfer, 46
(2010) 881-8904
N. Maeno, S. Takahashi, A. Sato, Y. Kominami, H. Konishi, S. Omiya, Verification experiments of Mpembaeffect, Seppyo 74 (2012) 1-13.
Introduction Prof. Maeno’s experiment Mathematical model Model results (Tcold =253 K) Conclusions
Configuration
Styrofoam, three suspended thermocouples, holes containinghot and cool water, TWINBIRD SCDF25 deep freezer
Introduction Prof. Maeno’s experiment Mathematical model Model results (Tcold =253 K) Conclusions
Schematic
Introduction Prof. Maeno’s experiment Mathematical model Model results (Tcold =253 K) Conclusions
Some experimental results (I)
Time [s]
Tem
pera
ture
[K]
cool water
hot water
0 500 1000 1500 2000 2500270
280
290
300
310
320
330
340
350
The temperature recorded at the uppermost thermocouple vs.time. The initial water temperatures are 348 K and 310 K, andthe surrounding air temperature is 263 K (No Mpemba effect).
Introduction Prof. Maeno’s experiment Mathematical model Model results (Tcold =253 K) Conclusions
Some experimental results (II)
Time [s]
Tem
pera
ture
[K]
cool water
hot water
0 500 1000 1500 2000 2500270
280
290
300
310
320
330
340
350
360
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 Mathematical model Model results (Tcold =253 K) Conclusions
Some experimental results (III)
Time [s]
Tem
pera
ture
[K]
cool water
hot water
0 500 1000 1500 2000 2500270
280
290
300
310
320
330
340
350
360
The temperature recorded at the centre thermocouple vs. time.The initial water temperatures are 348 K and 310 K, and thesurrounding air temperature is 263 K (No Mpemba effect).
Introduction Prof. Maeno’s experiment Mathematical model Model results (Tcold =253 K) Conclusions
Some experimental results (IV)
Time [s]
Tem
pera
ture
[K]
cool water
hot water
0 500 1000 1500 2000 2500270
280
290
300
310
320
330
340
350
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 Mathematical model Model results (Tcold =253 K) Conclusions
Some experimental results (V)
Time [s]
Tem
pera
ture
[K]
cool water
hot water
0 500 1000 1500 2000 2500270
280
290
300
310
320
330
340
350
360
The temperature recorded at the lowest thermocouple vs. time.The initial water temperatures are 348 K and 310 K, and thesurrounding air temperature is 263 K (No Mpemba effect).
Introduction Prof. Maeno’s experiment Mathematical model Model results (Tcold =253 K) Conclusions
Some experimental results (VI)
Time [s]
Tem
pera
ture
[K]
cool water
hot water
0 500 1000 1500 2000 2500270
280
290
300
310
320
330
340
350
360
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 Mathematical model Model results (Tcold =253 K) Conclusions
Yu-to-Mizu-Kurabe diagram
Temperature of cool water [◦C]
Tem
pera
ture
ofho
twat
er[◦
C]
351K/318K/260K
348K/310K/263K
10−1 100 101 10210−1
100
101
102
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 Mathematical model Model results (Tcold =253 K) Conclusions
General comments on experiments
Water at a given initial temperature and exposed to a givenambient temperature did not necessarily take the sameamount of time to reach freezing point
Consequently, it is difficult to reproduce the Mpemba effectwith this experiment, although it seems to be there
We try to interpret the results with a mathematical model
Introduction Prof. Maeno’s experiment Mathematical model Model results (Tcold =253 K) Conclusions
General comments on experiments
Water at a given initial temperature and exposed to a givenambient temperature did not necessarily take the sameamount of time to reach freezing point
Consequently, it is difficult to reproduce the Mpemba effectwith this experiment, although it seems to be there
We try to interpret the results with a mathematical model
Introduction Prof. Maeno’s experiment Mathematical model Model results (Tcold =253 K) Conclusions
General comments on experiments
Water at a given initial temperature and exposed to a givenambient temperature did not necessarily take the sameamount of time to reach freezing point
Consequently, it is difficult to reproduce the Mpemba effectwith this experiment, although it seems to be there
We try to interpret the results with a mathematical model
Introduction Prof. Maeno’s experiment Mathematical model Model results (Tcold =253 K) Conclusions
General comments on experiments
Water at a given initial temperature and exposed to a givenambient temperature did not necessarily take the sameamount of time to reach freezing point
Consequently, it is difficult to reproduce the Mpemba effectwith this experiment, although it seems to be there
We try to interpret the results with a mathematical model
Introduction Prof. Maeno’s experiment Mathematical model Model results (Tcold =253 K) Conclusions
What to model?
(A) Natural convection of air/water vapour mixtureabove the liquid water
(B) Natural convection of the liquid water itself
(C) The downward motion of the gas/waterinterface as water evaporates from the liquid pool
(D) Thermal radiation from the pool surface
Introduction Prof. Maeno’s experiment Mathematical model Model results (Tcold =253 K) Conclusions
What to model?
(A) Natural convection of air/water vapour mixtureabove the liquid water
(B) Natural convection of the liquid water itself
(C) The downward motion of the gas/waterinterface as water evaporates from the liquid pool
(D) Thermal radiation from the pool surface
Introduction Prof. Maeno’s experiment Mathematical model Model results (Tcold =253 K) Conclusions
What to model?
(A) Natural convection of air/water vapour mixtureabove the liquid water
(B) Natural convection of the liquid water itself
(C) The downward motion of the gas/waterinterface as water evaporates from the liquid pool
(D) Thermal radiation from the pool surface
Introduction Prof. Maeno’s experiment Mathematical model Model results (Tcold =253 K) Conclusions
What to model?
(A) Natural convection of air/water vapour mixtureabove the liquid water
(B) Natural convection of the liquid water itself
(C) The downward motion of the gas/waterinterface as water evaporates from the liquid pool
(D) Thermal radiation from the pool surface
Introduction Prof. Maeno’s experiment Mathematical model Model results (Tcold =253 K) Conclusions
What to model?
(A) Natural convection of air/water vapour mixtureabove the liquid water
(B) Natural convection of the liquid water itself
(C) The downward motion of the gas/waterinterface as water evaporates from the liquid pool
(D) Thermal radiation from the pool surface
Introduction Prof. Maeno’s experiment Mathematical model Model results (Tcold =253 K) Conclusions
Natural convection-driven evaporation
WATER
a
AIR + WATER VAPOUR
AIR
r
z
AIR + WATER VAPOUR
WATER
a
AIR + WATER VAPOUR
AIR
r
z
AIR
(a)
(b)
(a) theevaporation ofcold water intohotter air
(b) theevaporation ofhot water intocolder air
Introduction Prof. Maeno’s experiment Mathematical model Model results (Tcold =253 K) Conclusions
Natural convection-driven evaporation
WATER
a
AIR + WATER VAPOUR
AIR
r
z
AIR + WATER VAPOUR
WATER
a
AIR + WATER VAPOUR
AIR
r
z
AIR
(a)
(b)
(a) theevaporation ofcold water intohotter air
(b) theevaporation ofhot water intocolder air
Introduction Prof. Maeno’s experiment Mathematical model Model results (Tcold =253 K) Conclusions
Natural convection-driven evaporation
WATER
a
AIR + WATER VAPOUR
AIR
r
z
AIR + WATER VAPOUR
WATER
a
AIR + WATER VAPOUR
AIR
r
z
AIR
(a)
(b)
(a) theevaporation ofcold water intohotter air
(b) theevaporation ofhot water intocolder air
Introduction Prof. Maeno’s experiment Mathematical model Model results (Tcold =253 K) Conclusions
Hot water, cold air
boundary layer
vertical plume
water
Introduction Prof. Maeno’s experiment Mathematical model Model results (Tcold =253 K) Conclusions
Modelling considerations (I)
Axisymmetry
Computational fluid dynamics (CFD) for a full model
Moving boundary problem
Laminar flow: Rayleigh numbers for the air/water vapourmixture and the water (Rag and Ral ) are around 106
The water vapour is not dilute in the air; hence, we needthe full Stefan-Maxwell equations for binary diffusion andconvection
Introduction Prof. Maeno’s experiment Mathematical model Model results (Tcold =253 K) Conclusions
Modelling considerations (I)
Axisymmetry
Computational fluid dynamics (CFD) for a full model
Moving boundary problem
Laminar flow: Rayleigh numbers for the air/water vapourmixture and the water (Rag and Ral ) are around 106
The water vapour is not dilute in the air; hence, we needthe full Stefan-Maxwell equations for binary diffusion andconvection
Introduction Prof. Maeno’s experiment Mathematical model Model results (Tcold =253 K) Conclusions
Modelling considerations (I)
Axisymmetry
Computational fluid dynamics (CFD) for a full model
Moving boundary problem
Laminar flow: Rayleigh numbers for the air/water vapourmixture and the water (Rag and Ral ) are around 106
The water vapour is not dilute in the air; hence, we needthe full Stefan-Maxwell equations for binary diffusion andconvection
Introduction Prof. Maeno’s experiment Mathematical model Model results (Tcold =253 K) Conclusions
Modelling considerations (I)
Axisymmetry
Computational fluid dynamics (CFD) for a full model
Moving boundary problem
Laminar flow: Rayleigh numbers for the air/water vapourmixture and the water (Rag and Ral ) are around 106
The water vapour is not dilute in the air; hence, we needthe full Stefan-Maxwell equations for binary diffusion andconvection
Introduction Prof. Maeno’s experiment Mathematical model Model results (Tcold =253 K) Conclusions
Modelling considerations (I)
Axisymmetry
Computational fluid dynamics (CFD) for a full model
Moving boundary problem
Laminar flow: Rayleigh numbers for the air/water vapourmixture and the water (Rag and Ral ) are around 106
The water vapour is not dilute in the air; hence, we needthe full Stefan-Maxwell equations for binary diffusion andconvection
Introduction Prof. Maeno’s experiment Mathematical model Model results (Tcold =253 K) Conclusions
Modelling considerations (I)
Axisymmetry
Computational fluid dynamics (CFD) for a full model
Moving boundary problem
Laminar flow: Rayleigh numbers for the air/water vapourmixture and the water (Rag and Ral ) are around 106
The water vapour is not dilute in the air; hence, we needthe full Stefan-Maxwell equations for binary diffusion andconvection
Introduction Prof. Maeno’s experiment Mathematical model Model results (Tcold =253 K) Conclusions
Modelling considerations (II)
Earlier computations 5 suggest that thermally insulatingboundary conditions give (more or less) conduction only inthe water, even though Ral is nominally high
So we aim for 1D conduction in the water; however, thewater is cooled and evaporates due to the boundary layer
We need to find the Nusselt and Sherwood numbers forflow past a horizontal hot circular disk
The boundary layer is known qualitatively to be of thicknessRa−1/5
g for horizontal surfaces; however, the computationshave never been done before, i.e. we have to do them
5Vynnycky, 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’s experiment Mathematical model Model results (Tcold =253 K) Conclusions
Modelling considerations (II)
Earlier computations 5 suggest that thermally insulatingboundary conditions give (more or less) conduction only inthe water, even though Ral is nominally high
So we aim for 1D conduction in the water; however, thewater is cooled and evaporates due to the boundary layer
We need to find the Nusselt and Sherwood numbers forflow past a horizontal hot circular disk
The boundary layer is known qualitatively to be of thicknessRa−1/5
g for horizontal surfaces; however, the computationshave never been done before, i.e. we have to do them
5Vynnycky, 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’s experiment Mathematical model Model results (Tcold =253 K) Conclusions
Modelling considerations (II)
Earlier computations 5 suggest that thermally insulatingboundary conditions give (more or less) conduction only inthe water, even though Ral is nominally high
So we aim for 1D conduction in the water; however, thewater is cooled and evaporates due to the boundary layer
We need to find the Nusselt and Sherwood numbers forflow past a horizontal hot circular disk
The boundary layer is known qualitatively to be of thicknessRa−1/5
g for horizontal surfaces; however, the computationshave never been done before, i.e. we have to do them
5Vynnycky, 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’s experiment Mathematical model Model results (Tcold =253 K) Conclusions
Modelling considerations (II)
Earlier computations 5 suggest that thermally insulatingboundary conditions give (more or less) conduction only inthe water, even though Ral is nominally high
So we aim for 1D conduction in the water; however, thewater is cooled and evaporates due to the boundary layer
We need to find the Nusselt and Sherwood numbers forflow past a horizontal hot circular disk
The boundary layer is known qualitatively to be of thicknessRa−1/5
g for horizontal surfaces; however, the computationshave never been done before, i.e. we have to do them
5Vynnycky, 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’s experiment Mathematical model Model results (Tcold =253 K) Conclusions
Modelling considerations (II)
Earlier computations 5 suggest that thermally insulatingboundary conditions give (more or less) conduction only inthe water, even though Ral is nominally high
So we aim for 1D conduction in the water; however, thewater is cooled and evaporates due to the boundary layer
We need to find the Nusselt and Sherwood numbers forflow past a horizontal hot circular disk
The boundary layer is known qualitatively to be of thicknessRa−1/5
g for horizontal surfaces; however, the computationshave never been done before, i.e. we have to do them
5Vynnycky, 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’s experiment Mathematical model Model results (Tcold =253 K) Conclusions
Modelling considerations (III)
The velocity in the boundary layer is mass-averaged
Dufour effect (diffusion thermo): species diffusion affectsheat transfer
‘Blowing’ effect, i.e. the normal velocity at the gas-waterinterface is non-zero
Cooling time scale is much longer than the boundary-layerconvection time scale; hence, the boundary layer appearsas if quasi-steady
Introduction Prof. Maeno’s experiment Mathematical model Model results (Tcold =253 K) Conclusions
Modelling considerations (III)
The velocity in the boundary layer is mass-averaged
Dufour effect (diffusion thermo): species diffusion affectsheat transfer
‘Blowing’ effect, i.e. the normal velocity at the gas-waterinterface is non-zero
Cooling time scale is much longer than the boundary-layerconvection time scale; hence, the boundary layer appearsas if quasi-steady
Introduction Prof. Maeno’s experiment Mathematical model Model results (Tcold =253 K) Conclusions
Modelling considerations (III)
The velocity in the boundary layer is mass-averaged
Dufour effect (diffusion thermo): species diffusion affectsheat transfer
‘Blowing’ effect, i.e. the normal velocity at the gas-waterinterface is non-zero
Cooling time scale is much longer than the boundary-layerconvection time scale; hence, the boundary layer appearsas if quasi-steady
Introduction Prof. Maeno’s experiment Mathematical model Model results (Tcold =253 K) Conclusions
Modelling considerations (III)
The velocity in the boundary layer is mass-averaged
Dufour effect (diffusion thermo): species diffusion affectsheat transfer
‘Blowing’ effect, i.e. the normal velocity at the gas-waterinterface is non-zero
Cooling time scale is much longer than the boundary-layerconvection time scale; hence, the boundary layer appearsas if quasi-steady
Introduction Prof. Maeno’s experiment Mathematical model Model results (Tcold =253 K) Conclusions
Modelling considerations (III)
The velocity in the boundary layer is mass-averaged
Dufour effect (diffusion thermo): species diffusion affectsheat transfer
‘Blowing’ effect, i.e. the normal velocity at the gas-waterinterface is non-zero
Cooling time scale is much longer than the boundary-layerconvection time scale; hence, the boundary layer appearsas if quasi-steady
Introduction Prof. Maeno’s experiment Mathematical model Model results (Tcold =253 K) Conclusions
Flow structure
boundary layer
water
vertical plume
Velocity vectors, and boundary-layer and vertical plumestructure
Introduction Prof. Maeno’s experiment Mathematical model Model results (Tcold =253 K) Conclusions
Governing equations (water)
∂ρ(l)
∂t+∇ ·
(
ρ(l)u(l))
= 0, (1)
ρ(l)
(
∂u(l)
∂t+(
u(l)· ∇
)
u(l)
)
= −∇p(l) + µ(l)∇
2u(l)− ρ(l)gk,
(2)
ρ(l)c(l)p
(
∂T∂t
+ u(l)· ∇T
)
= ∇ ·
(
k (l)∇T
)
. (3)
Introduction Prof. Maeno’s experiment Mathematical model Model results (Tcold =253 K) Conclusions
Governing equations (gas phase above the water): I
Conservation of species:
∂
∂t
(
ρ(g)ωa
)
+∇ · na = 0, (4)
∂
∂t
(
ρ(g)ωw
)
+∇ · nw = 0, (5)
whereni = ρ(g)ωiu(g)+ji , i = a,w ; (6)
na and nw are the air and water vapour mass fluxes, ρ(g) is themixture density, ωi is the mass fraction, u(g) is themass-averaged velocity and the vector ji describes thediffusion-driven transport. Also, ωa + ωw = 1.
Introduction Prof. Maeno’s experiment Mathematical model Model results (Tcold =253 K) Conclusions
Governing equations (gas phase above the water): II
Conservation of momentum:
ρ(g)
(
∂u(g)
∂t+(
u(g)· ∇
)
u(g)
)
= ∇ · σ(g)− ρ(g)gk, (7)
where σ is the total stress tensor, given by
σ(g) = −p(g)I+µ(g)[
∇u(g) +(
∇u(g))tr]
−23µ(g)
(
∇ · u(g))
I. (8)
Introduction Prof. Maeno’s experiment Mathematical model Model results (Tcold =253 K) Conclusions
Governing equations (gas phase above the water): III
Conservation of heat:
ρ(g)c(g)p
∂T∂t
+(
c(g)p,ana + c(g)
p,wnw
)
· ∇T = ∇ ·
(
k (g)∇T
)
, (9)
where c(g)p,i for i = a,w are the specific heat capacities for each
species, c(g)p is given by
c(g)p = c(g)
p,aωa + c(g)p,wωw ,
and k (g) is the mixture thermal conductivity, which will also, ingeneral, depend on the local mass fraction and temperature.
Introduction Prof. Maeno’s experiment Mathematical model Model results (Tcold =253 K) 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
6M. 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 Mpembaeffect, submitted to Int. J. Heat Mass Tran. (2012)
Introduction Prof. Maeno’s experiment Mathematical model Model results (Tcold =253 K) 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
6M. 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 Mpembaeffect, submitted to Int. J. Heat Mass Tran. (2012)
Introduction Prof. Maeno’s experiment Mathematical model Model results (Tcold =253 K) 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
6M. 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 Mpembaeffect, submitted to Int. J. Heat Mass Tran. (2012)
Introduction Prof. Maeno’s experiment Mathematical model Model results (Tcold =253 K) 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
6M. 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 Mpembaeffect, submitted to Int. J. Heat Mass Tran. (2012)
Introduction Prof. Maeno’s experiment Mathematical model Model results (Tcold =253 K) Conclusions
Time to reach freezing (t0) vs initial water temperature(Tw ,i)
Tw ,i [K]
t 0[s
]
Tcold = 253 KTcold = 258 KTcold = 263 KTcold = 268 K
260 280 300 320 340 360 3800
1000
2000
3000
4000
5000
6000
7000
8000
Introduction Prof. Maeno’s experiment Mathematical model Model results (Tcold =253 K) Conclusions
Water height at t0 (h0) vs Tw ,i
Tw ,i [K]
h 0[m
m]
Tcold = 253 KTcold = 258 KTcold = 263 KTcold = 268 K
260 280 300 320 340 360 3804
5
6
7
8
9
10
Introduction Prof. Maeno’s experiment Mathematical model Model results (Tcold =253 K) Conclusions
Water height (h) vs time (t) for different Tw ,i
t [s]
h[m
m]
Tw ,i = 303 KTw ,i = 320 KTw ,i = 337 KTw ,i = 354 K
0 1000 2000 3000 4000 50007
7.5
8
8.5
9
9.5
10
Introduction Prof. Maeno’s experiment Mathematical model Model results (Tcold =253 K) Conclusions
Surface temperature (Thot) vs t for different Tw ,i
t [s]
Tho
t[K
]
Tw ,i = 303 KTw ,i = 320 KTw ,i = 337 KTw ,i = 354 K
0 1000 2000 3000 4000 5000270
280
290
300
310
320
330
340
350
360
Introduction Prof. Maeno’s experiment Mathematical model Model results (Tcold =253 K) Conclusions
Water temperature (T ) vs distance from base (z)
z [mm]
T[K
]
t/[t] = 10−2
t/[t] = 10−1
t/[t] = 1
0 2 4 6 8 10275
280
285
290
295
300
305
310
315
320
Introduction Prof. Maeno’s experiment Mathematical model Model results (Tcold =253 K) Conclusions
Temperature difference (bottom-top, ∆T ) vs t fordifferent Tw ,i
t [s]
∆T
[K]
Tw ,i = 303 KTw ,i = 320 KTw ,i = 337 KTw ,i = 354 K
0 1000 2000 3000 4000 50000
1
2
3
4
5
6
7
8
9
Introduction Prof. Maeno’s experiment Mathematical model Model results (Tcold =253 K) Conclusions
Comparison with experiment (I)
20 40 60 80 1000
10
20
30
40
50
60
70
80
90
100
Temperature of cool water [C]
Tem
pera
ture
of h
ot w
ater
[C]
Mpemba effect
310/348/263 (no)318/351/260 (yes)T_{cold}=253 KT_{cold}=263 K
The number of possible hot/cool water pairs increases withTcold . However, the model predicts that the Mpemba effect
should be seen for both experiments, whereas it was actuallyseen for only one of them.
Introduction Prof. Maeno’s experiment Mathematical model 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 ensurethat freezing time is not highly overestimated
Model doesn’t seem to capture the sensitivity of theexperiments with respect to t0At higher values of Tw ,i , the model giveshigher-than-observed levels of evaporation
Introduction Prof. Maeno’s experiment Mathematical model 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 ensurethat freezing time is not highly overestimated
Model doesn’t seem to capture the sensitivity of theexperiments with respect to t0At higher values of Tw ,i , the model giveshigher-than-observed levels of evaporation
Introduction Prof. Maeno’s experiment Mathematical model 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 ensurethat freezing time is not highly overestimated
Model doesn’t seem to capture the sensitivity of theexperiments with respect to t0At higher values of Tw ,i , the model giveshigher-than-observed levels of evaporation
Introduction Prof. Maeno’s experiment Mathematical model 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 ensurethat freezing time is not highly overestimated
Model doesn’t seem to capture the sensitivity of theexperiments with respect to t0At higher values of Tw ,i , the model giveshigher-than-observed levels of evaporation
Introduction Prof. Maeno’s experiment Mathematical model 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 ensurethat freezing time is not highly overestimated
Model doesn’t seem to capture the sensitivity of theexperiments with respect to t0At higher values of Tw ,i , the model giveshigher-than-observed levels of evaporation
Introduction Prof. Maeno’s experiment Mathematical model Model results (Tcold =253 K) Conclusions
‘Over-evaporation’
WATER
INSULATION INSULATION
Schematic of possible flow streamlines as water evaporates
Introduction Prof. Maeno’s experiment Mathematical model 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 beless valid when more of the water evaporates, due tosimplifying assumptions
A better (more difficult) model requires CFD to solvesimultaneously for convection in the water flow,multicomponent flow in the boundary layer and themovement of the evaporation front
Introduction Prof. Maeno’s experiment Mathematical model 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 beless valid when more of the water evaporates, due tosimplifying assumptions
A better (more difficult) model requires CFD to solvesimultaneously for convection in the water flow,multicomponent flow in the boundary layer and themovement of the evaporation front
Introduction Prof. Maeno’s experiment Mathematical model 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 beless valid when more of the water evaporates, due tosimplifying assumptions
A better (more difficult) model requires CFD to solvesimultaneously for convection in the water flow,multicomponent flow in the boundary layer and themovement of the evaporation front
Introduction Prof. Maeno’s experiment Mathematical model 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 beless valid when more of the water evaporates, due tosimplifying assumptions
A better (more difficult) model requires CFD to solvesimultaneously for convection in the water flow,multicomponent flow in the boundary layer and themovement of the evaporation front
Introduction Prof. Maeno’s experiment Mathematical model 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 beless valid when more of the water evaporates, due tosimplifying assumptions
A better (more difficult) model requires CFD to solvesimultaneously for convection in the water flow,multicomponent flow in the boundary layer and themovement of the evaporation front
Introduction Prof. Maeno’s experiment Mathematical model Model results (Tcold =253 K) Conclusions
Future work (natural convection only)
Experimental vessel
Introduction Prof. Maeno’s experiment Mathematical model 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’s experiment Mathematical model Model results (Tcold =253 K) Conclusions
Acknowledgements
Science Foundation Ireland Mathematics Initiative Grant06/MI/005
Prof. N. Maeno
Introduction Prof. Maeno’s experiment Mathematical model Model results (Tcold =253 K) Conclusions
Acknowledgements
Science Foundation Ireland Mathematics Initiative Grant06/MI/005
Prof. N. Maeno
Introduction Prof. Maeno’s experiment Mathematical model Model results (Tcold =253 K) Conclusions
Acknowledgements
Science Foundation Ireland Mathematics Initiative Grant06/MI/005
Prof. N. Maeno
Recommended