1

Theory, experimentation and numerical modelling of thermal-fluid processes in PCMs

Part I

Gennady Ziskind

Heat Transfer LaboratoryDepartment of Mechanical Engineering

Ben-Gurion University of the NegevBeer-Sheva, Israel

1

Motivation

2

Large heat-storage capacity of phase-change

materials (PCMs) makes them attractive for use

in various thermal energy storage systems

where their latent heat is utilized.

There exists a wide range of applications for

such systems, from energy storage in buildings

to electronics cooling and thermal management

of air- and spacecraft.

2

Application: heat storage facilities

3

Application: cooling of electronics

4

3

Broad variety of issues

5

One should start with material properties

and arrive at storage configurations

• Importance of details

• Adequacy of the model

• Validity of the model

PCM research

6

Theoretical and experimental studies in the field of phase-

change fundamentals and applications have yielded

extensive literature on

• basic studies of phase-change phenomena (Viskanta 1983,

Bejan 2004)

• phase change materials (Zalba et al. 2003)

• mathematical modeling (Crank 1984, Alexiades and Solomon 1993, Hu

and Argyropoulos 1996, Gupta 2003)

• numerical techniques (Voller 1996, Lacroix 2002)

• experimentation (Fukusako and Yamada 1999, Kowalewski 2003)

• solar energy applications (Kenisarin and Mahkamov 2007)

• heat transfer enhancement (Ismail 2002)

4

Books since 2000

7

A. Faghri, Y. Zhang, Transport

Phenomena in Multiphase Systems,

2006.

Outline of this talk

8

• Heat transfer in phase change processes - recent

developments

• Geometries suggested for:

- latent-heat thermal energy

storage (LHTES) systems

- electronics cooling

• Numerical and experimental approaches that could

lead to reliable results and their generalization

Kenisarin and Mahkamov (2007)

5

Beyond the scope

• Phase-change slurries and micro-encapsulated PCM

• PCM-soaked graphite matrices

• Enhancement techniques other then fins(metal honeycombs, metal matrices, conductive particles…)

• Water-ice systems(i.e. cases in which solid is denser than liquid)

9

Classical formulationOne-dimensional phase change (solidification)

10

SLx x

0 SLx x 2

2

1S S

S

T T

tx

SLx x2

2

1L L

L

T T

tx

S SL L SL mT ( x ,t ) T ( x ,t ) T

S SL L SL SLS L

T ( x ,t ) T ( x ,t ) dxk k L

x x dt

2c

Ste tl

p m wc T T

SteL

Dimensionless time Stefan number

• Even one-dimensional conduction-governed phase

change is a non-linear problem

6

Two-dimensional phase change

11

TC

OL

D

TH

OT

Solid

Insulated

Tmelt

Liquid

Insulated

melt flow

Bertrand et al. (1999)Among others:

E.M. Sparrow

R. Viskanta

A. Bejan

Physical picture - special features

• Non-linear motion of the solid-liquid interface• Buoyancy effects in the melt

• Volume expansion/contraction• “Close-contact” melting

• Phase-change over an extended temperature range • Subcooling/superheating (enthalpy hysteresis)

These features are often modeled in a simplified manner or even ignored – justified or not?

12

7

Methods of our studies

13

Numerical:• Enthalpy formulation

• Enthalpy-porosity approach for the phase-change region

inside the PCM

• Volume-of-fluid (VOF) model for the PCM-air system

with a moving internal interface

Experimental:• Commercially available paraffin-type materials

• Temperature-controlled environment

Verification and validation

• “Verification” ~ solving the equations right

• “Validation” ~ solving the right equations

Patrick J. Roache

Verification and validation in computational science

and engineering (1998, p. 23)

Accordingly, careful examination is performed in every case

• Grid size/structure and time-step dependence

• Comparison with experiments

Not only overall results but also patterns are compared

14

8

Melting

15

Geometries:

• Cylindrical shells (tubes)

• Spherical shells

• Finned systems

Features:

• Time to complete the process

• Melting patterns

• Instant melt fractions and heat fluxes

Objective:

• Generalization

Vertical tube

16

L. Fraiman, E. Benisti ,G. Ziskind, R. Letan

ASME Biennial ESDA Conference, Haifa 2008 Sparrow and Broadbent (1982)

Pal and Joshi (2001)

(tall enclosure)

9

Summary of experimental results

17

• Melting time from 2.5 to about 70 minutes

Generalized results and correlation

18

9 4

1 1 1 3/

MF X / .

10

Narrow vertical tube

19

Observed:

• Solid collapse

• Solid leaning

• Solid rotation

L. Katsman, V. Dubovsky, G. Ziskind, R. Letan

ASME - JSME Thermal Engineering Conference, Vancouver 2007

Simulation and analysis

20

H. Shmueli, G. Ziskind, R. Letan

Int. J. Heat Mass Transfer 53 (2010) 4082-4091

11

Melt fraction prediction

21

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75

Time, min

Melt fra

ction

Experimental

C=10 5̂

C=10 7̂

C=10 8̂

C=10 9̂

C=10 1̂0

C=10 8̂, kl=0.15W/m K

VC

VpDt

VD

3

22 )1(

H. Shmueli, G. Ziskind, R. Letan

Int. J. Heat Mass Transfer 53 (2010) 4082-4091

• One minute takes approx. 24 hours of simulation

Effect of numerical schemes

22

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 2 4 6 8 10 12 14 16 18 20 22 24 26

Time, min

Melt fra

ctio

n

PISO+Body-Force-Weighted

SIMPLE+PRESTO!

PISO+PRESTO!

Experimental

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 5 10 15 20 25 30 35 40 45 50 55 60 65

Time, min

Melt fra

ctio

n

Experimental

Body-Force-Weighted

PRESTO!

300

305

310

315

320

325

330

0 50 100 150 200 250

Time, sec

Mean tem

pera

ture

, K

PRESTO!

body force weighted

Transient single phase

natural convection only

The differences are due

to the mushy zone

12

Spherical shell: Experimental

23

When the solid is fixed by some

mechanical obstacle, both melting

patterns and qualitative characteristics

of the process become entirely different

(Khodadadi and Zhang 2001, compare

also Ettouney et al. 2005, Felix Regin et

al. 2006)

• Laboratory studies could lead to results rather different

from those in real systems

E. Assis, L. Katzman, G. Ziskind, R. Letan

Int. J. Heat Mass Transfer 50 (2007) 1790-1804

24

Spherical shell: Modeling

Fomin and Saitoh (1999)

(based on Bareiss and Beer 1984(

13

Melt fraction: numerical vs. experimental

0.0

0.2

0.4

0.6

0.8

1.0

0 5 10 15 20 25 30 35 40

Melt

fra

cti

on

Time, min

Experimental

Numerical

Shell diameter: 80mm

Glass thickness: 2mm

Glass conductivity: 0.81W/moC

Temperature difference: 10oC

25

Complete numerical solution

26

Evolution of the solid fraction

Flow field in the upper

part of the melt

Flow field in the

gap between the

solid and the shell

Rise of the interface

E. Assis, L. Katzman, G. Ziskind, R. Letan

Int. J. Heat Mass Transfer 50 (2007) 1790-1804

14

Melt fraction correlation

27

• Stefan numbers smaller than 0.1 are characteristic

to latent-heat thermal storage systems

28

15

Larger Stefan numbers

Conto

urs

of

Liq

uid

Fra

ctio

n (

mix

ture

) (T

ime=1.9

996e+02)

FLU

EN

T 6

.1 (

axi,

dp, segre

gate

d, vof,

lam

, unste

ady)

Jan 0

2, 2006

1.0

0e+00

9.5

0e-0

1

9.0

0e-0

1

8.5

0e-0

1

8.0

0e-0

1

7.5

0e-0

1

7.0

0e-0

1

6.5

0e-0

1

6.0

0e-0

1

5.5

0e-0

1

5.0

0e-0

1

4.5

0e-0

1

4.0

0e-0

1

3.5

0e-0

1

3.0

0e-0

1

2.5

0e-0

1

2.0

0e-0

1

1.5

0e-0

1

1.0

0e-0

1

5.0

0e-0

2

0.0

0e+00

Velo

city

Vecto

rs C

olo

red B

y V

elo

city

Magnitu

de (

mix

ture

) (

m/s

) (

Tim

e=1.9

996e+02)

FLU

EN

T 6

.1 (

axi,

dp, segre

gate

d, vof,

lam

, unste

ady)

Jan 0

2, 2006

2.6

5e-0

2

2.5

2e-0

2

2.3

9e-0

2

2.2

5e-0

2

2.1

2e-0

2

1.9

9e-0

2

1.8

6e-0

2

1.7

2e-0

2

1.5

9e-0

2

1.4

6e-0

2

1.3

3e-0

2

1.1

9e-0

2

1.0

6e-0

2

9.3

0e-0

3

7.9

7e-0

3

6.6

5e-0

3

5.3

3e-0

3

4.0

0e-0

3

2.6

8e-0

3

1.3

6e-0

3

3.1

7e-0

5

29

Nusselt number: generalized heat flux

30

16

Latent vs. total

31

0

2

4

6

8

10

12

0 5 10 15 20 25 30

Time, min

Hea

t tr

an

sfer

ra

te,

W

qMelt

qTotal

Diameter = 40mm

T = 2oC

a .T=2°C

0

2

4

6

8

10

12

0 2 4 6 8 10 12 14

Time, min

Hea

t tr

an

sfer

rate

, W

qMelting

qTotal

Diameter = 40mm

T = 6oC

b .T=6°C

0

2

4

6

8

10

12

0 2 4 6 8 10 12

Time, min

Hea

t tr

an

sfer

ra

te,

W

qMelt

qTotal

Diameter = 40mm

T = 10oC

c .T=10°CE. Assis, G. Ziskind, R. Letan

18th ISTP, Daejeon 2007

32

Finned systems

Ismail (2002)

Nagano (2004) Velraj et al. (1997)

• Suggested to overcome the low thermal conductivity of PCM

17

33

Detailed parametric study

34

Computational domain

V. Shatikian, G. Ziskind, R. Letan

Int. J. Heat Mass Transfer 48 (2005) 3689-3706

18

Examples

35

Dimensional analysis

36

Fin efficiency accounted for

Melt fraction vs. FoSte

19

Effect of convection

37

Ra included

Heat flux at the base

Figure 1. Definition of the problem.

Table 1. Geometry parameters.

Case lf, mm lt, mm lb, mm

1 10 1.2 4

2 10, 15, 20 0.6 2

3 10 0.3 1

l f

lb

lt

Air

q''w

PC

M

fin computational domain

base

38

V. Shatikian, G. Ziskind, R. Letan

Int. J. Heat Mass Transfer 51 (2008) 1488-1493

20

Heat accumulation: latent share

10mm fin 15mm fin

39

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 10 20 30 40 50 60 70

Time, s

Qla

t / Q

acc

25kW/m^2

50kW/m^2

75kW/m^2

0

10

20

30

40

50

60

0 10 20 30 40 50 60 70

Time, s

Tb

as

e - T

m ,

oC

25kW/m^2

50kW/m^2

75kW/m^2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 10 20 30 40 50 60 70 80 90 100

Time, s

Qla

t /

Qa

cc

25kW/m^2

50kW/m^2

75kW/m^2

0

10

20

30

40

50

60

0 10 20 30 40 50 60 70 80 90 100

Time, s

Tb

as

e -

Tm

, o

C

25kW/m^2

50kW/m^2

75kW/m^2

V. Shatikian, G. Ziskind, R. Letan

European Thermal Sciences Conference, Eindhoven 2008

Solidification

40

Geometries:

• Cylindrical shells (tubes)

• Spherical shells

Features:

• Time to complete the process

• Instant melt fractions and heat fluxes

• Shrinkage and void formation

Objective:

• Generalization

• Void prediction

• Subcooling modeling

21

Cylindrical shell: cooling direction

41

Adiabatic bottom

Cooled bottom

Experimental

Liquid

Air

Shell

H

H*

D

Model

V. Dubovsky, E. Assis, E. Kochavi, G. Ziskind, R. Letan

European Thermal Sciences Conference, Eindhoven 2008

Shamsundar and

Sparrow (1976)

Cylindrical shell: analysis

42

Same T

Same size

Melt fraction vs. time

Melt fraction vs. time

Melt fraction vs. Fo

Melt fraction vs. FoSte

22

Cylindrical shell: water freezing

43

Spherical shell: Modeling

44

Shell

Opening Air

PCM

E. Assis, G. Ziskind, R. Letan

ASME Journal of Heat Transfer 131 (2009) 024502

• In the literature, the shell is

initially filled with the liquid

completely

• This leads to “concentric”

solidification

• In reality, a completely filled

shell is hardly possible

23

Spherical shell: analysis

45

Glaich and Finkelshtein (2003)

Conto

urs

of

Liq

uid

Fra

ctio

n (

mix

ture

) (T

ime=4.8

000e+03)

FLU

EN

T 6

.1 (

axi,

dp, segre

gate

d, vof,

lam

, unste

ady)

Mar

16, 2006

1.0

0e+00

9.5

0e-0

1

9.0

0e-0

1

8.5

0e-0

1

8.0

0e-0

1

7.5

0e-0

1

7.0

0e-0

1

6.5

0e-0

1

6.0

0e-0

1

5.5

0e-0

1

5.0

0e-0

1

4.5

0e-0

1

4.0

0e-0

1

3.5

0e-0

1

3.0

0e-0

1

2.5

0e-0

1

2.0

0e-0

1

1.5

0e-0

1

1.0

0e-0

1

5.0

0e-0

2

0.0

0e+00

Conto

urs

of

Liq

uid

Fra

ctio

n (

mix

ture

) (T

ime=6.4

220e+03)

FLU

EN

T 6

.1 (

axi,

dp, segre

gate

d, vof,

lam

, unste

ady)

Mar

16, 2006

1.0

0e+00

9.5

0e-0

1

9.0

0e-0

1

8.5

0e-0

1

8.0

0e-0

1

7.5

0e-0

1

7.0

0e-0

1

6.5

0e-0

1

6.0

0e-0

1

5.5

0e-0

1

5.0

0e-0

1

4.5

0e-0

1

4.0

0e-0

1

3.5

0e-0

1

3.0

0e-0

1

2.5

0e-0

1

2.0

0e-0

1

1.5

0e-0

1

1.0

0e-0

1

5.0

0e-0

2

0.0

0e+00

Conto

urs

of

Liq

uid

Fra

ctio

n (

mix

ture

) (T

ime=9.5

903e+03)

FLU

EN

T 6

.1 (

axi,

dp, segre

gate

d, vof,

lam

, unste

ady)

Apr

09, 2006

1.0

0e+00

9.5

7e-0

1

9.1

3e-0

1

8.7

0e-0

1

8.2

6e-0

1

7.8

3e-0

1

7.3

9e-0

1

6.9

6e-0

1

6.5

2e-0

1

6.0

9e-0

1

5.6

5e-0

1

5.2

2e-0

1

4.7

8e-0

1

4.3

5e-0

1

3.9

1e-0

1

3.4

8e-0

1

3.0

4e-0

1

2.6

1e-0

1

2.1

7e-0

1

1.7

4e-0

1

1.3

0e-0

1

8.7

0e-0

2

4.3

5e-0

2

0.0

0e+00

C=105

Spherical shell: shrinkage and voids

46

Yotvat and Zelikover (2008)

C=108

Revankar and Croy (2007)

FoSte ~ T/D2

• There exists a critical value

of FoSte and the central void

is formed when this value is

exceeded

24

47

Generalization

48

Melt fraction vs. time

Melt fraction vs. FoSte

25

Spherical shell: attempt of analysis

0.0

0.2

0.4

0.6

0.8

1.0

0 30 60 90 120 150 180

Time, min

melt

fra

cti

on

Casting

Shell Diameter = 80mm

T = 20 oC

0.0

0.2

0.4

0.6

0.8

1.0

0 10 20 30 40 50 60 70

time, min

mel

t fr

act

ion

Casting Shell Diameter: 40mm

ΔT = -10 oC, -20

oC

49

FoSte ~ T/D2

• There exists a critical value

of FoSte and the central void

is formed when this value is

exceeded

Melting revisited

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20 25 30 35 40

Time, min

melt f

raction

Melting after

solidification

Diameter: 80mm

T = 6oC

Mushy zone: 2oC

50

Assis (2007)

26

Subcooling

51

-15

-12

-9

-6

-3

0

3

6

9

12

15

18

21

24

27

30

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000 6500 7000

Tem

pera

ture

, ˚

C

Time , s

66% Ga, 20.5% In, 13.5% SnMelting Point = 11 oCTamb = -15 oCΔT = -26 oC

"Solidification"

B A

Subcooling

C D

Simplified modeling

52

• Systems where some features may be

discarded, e.g. matrices, impregnated boards,

narrow layers

• Systems for which quick estimates are

important to see the preferable case, which may

be fine-tuned at the next step

27

Example (1)

53

Y. Kozak, G. Ziskind

Applied Thermal Engineering (2013) under review

Models vs. experiments

54

28

(Example (2

55

V. Dubovsky, G. Ziskind, R. Letan

Applied Thermal Engineering (2011)

Parameter variation

56

29

Example (3)

57

Y. Kozak, T. Rozenfeld, G. Ziskind

ASME Energy Sustainability Conference (2013)

Close-contact melting

58

30

Closing remarks

59

• “New level of realism” in the simulations – resulting in

a detailed picture of the processes – is possible

• In many cases, a full simulation is essential because

otherwise the results do not reflect the reality and their

use can be misleading

• Still, it is very important to be flexible and apply a variety

of approaches

• Validation vs. experiments is indispensable

• Modeling challenges:

properties of real materials

in melting - mushy zone

in solidification - void formation and subcooling

• "If you cannot solve a problem, then there is an easier problem you cannot solve: find it"

60

George Pólya (1887-1985)

31

• Tafasta meruba lo tafasta (Hebrew: תפשת מרובה לא תפשת)

If you have seized a lot, you have not seized

• The general meaning is that an over-ambitious claim defeatsitself: the intended analogy is to one who grabs more than he canhold

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