Melting and solidification

�

BEN-GURION UNIVERSITY OF THE NEGEV

FACULTY OF ENGINEERING SCIENCES DEPARTMENT OF MECHANICAL ENGINEERING

��

�

�

�

�

�

MELTING AND SOLIDIFICATION OF A PHASE-CHANGE MATERIAL

WITH INTERNAL FINS �

�

�

�

�

�

Thesis Submitted in Partial Fulfillment of the Requirements for the M.Sc. Degree

�

�

�

�

�

�

BY: Vartan SHATIKIAN �

� �

�

JUNE 2004

�

BEN-GURION UNIVERSITY OF THE NEGEV

FACULTY OF ENGINEERING SCIENCES DEPARTMENT OF MECHANICAL ENGINEERING

MELTING AND SOLIDIFICATION OF A PHASE-CHANGE MATERIAL

WITH INTERNAL FINS

Thesis Submitted in Partial Fulfillment of the Requirements for the M.Sc. Degree

BY: Vartan SHATIKIAN

SUPERVISORS: Prof. Ruth LETAN Dr. Gennady ZISKIND

Author:

Supervisors:

Chairman of Graduate Studies Committee:

JUNE 2004

II�

Abstract

The present study explores numerically the processes of melting and solidification of

the phase-change material (PCM) in a latent heat storage system with internal partitions

or plate-type fins. The heat is transferred to the system through its horizontal base, to

which conducting vertical fins are attached. Phase change material is stored between the

fins. The material used in the simulations is a commercially available paraffin wax, and

the fins are made of aluminum. The latent heat storage system is open at the top to

ambient air.

The study includes three main steps. First, the processes of melting and solidification

are studied in a relatively large partitioned heat storage unit, yielding detailed information

on the phase and temperature distributions inside the PCM and the partitions during the

transient phase-change processes. Then, a detailed parametric investigation is performed

for melting in a relatively small heat sink, 5mm to 10mm high, where the fin thickness

varies from 0.15mm to 1.2mm, and the thickness of the PCM layers between the fins

varies from 0.5mm to 4mm, while the base temperature varies from 6°C to 24°C above

the mean melting temperature of the PCM. Finally, melting and solidification in the sink

are explored for the case where the temperature of the sink base changes periodically.

Transient two-dimensional simulations are performed using the Fluent 6.0 software.

The melting temperature of the wax, 23–25°C, is incorporated in the simulations along

with its other properties, including the latent and sensible specific heat, thermal

conductivity and density in solid and liquid states.

Numerical simulations yield temperature evolution in the fins and the PCM. Time-

dependent heat fluxes are calculated separately for the processes of heat transfer from the

base to the PCM both directly and indirectly, through the fins. The computational results

show how the transient phase-change process, expressed in terms of the molten volume

fraction of the PCM, depends on the thermal and geometrical parameters of the system,

including the temperature difference between the base and the solid PCM, thickness of

the fins and the PCM layer between them, and the height of the fins.

Dimensional analysis is performed based on such dimensionless parameters as the

Fourier number, which takes into account the transient heat conduction, and the Stefan

number, which reflects the phase change. Comparison with the literature is presented and

discussed.

III�

Acknowledgments

My sincerely gratitude and appreciation is due to Prof. Ruth Letan for her kindness,

generous guidance and assistance.

I am most grateful to Dr. Gennady Ziskind who instructed me during this study. His

excellent guidance, support and understanding throughout my study are extremely

valued.

Special thanks to Mr. Vadim Dubovsky for his friendly encouragement and

invaluable help in my study.

I would like to thank the Ehud Ben-Amitai foundation and Ben-Amitai family for

their generous gift.

I would like to express my gratitude to Salim and Rachel Banin fund for their

financial help.

�

Contents

Page

�. Introduction 1

2. Literature survey 3

2.1 Phase-change materials 3

2.2 Theoretical background 5

2.3 Transient melting in an enclosure 11

2.4 Periodic melting and solidification in an enclosure 29

2.5 Phase change with internal fins 37

3. Numerical study 47

3.1 Physical model 47

3.2 Computational procedure 52

4. Numerical results 56

4.1 Preliminary results 56

4.2 Detailed study 65

5. Analysis of results 80

5.1 Heat-transfer rates 80

5.2 Melt fractions 86

5.3 Temperatures 87

5.4 Dimensional analysis 95

6. Summary 112

7. References 114

Appendices

A1. Periodic melting 116

A2. Velocity fields in the system 120

A3. Different time step calculations 121

A4. ASME article 122

IV

�

Nomenclature

a amplitude, ºC

A cross-section area, m2

cp specific heat at constant pressure, J/kg ºC

Fo Fourier number, =2l

t�

�s specific sensible enthalpy, J/kg

� � specific enthalpy change due to the phase change, J/kg

h heat transfer coefficient, W/m2 ºC � specific total enthalpy, J/kg

k thermal conductivity, W/m ºC

l characteristic length, m

l f fin length, m

l t fin thickness, m

lb spacing between the fins, m

L specific latent heat, J/kg

m mass, kg

Nu Nusselt number, =k

hl

P perimeter of the cross-section area, m

q heat transfer rate, W

q´´ heat flux, W/m2

q´´´ rate of heat generation, W/m3

Ra Rayleigh number, =� �

�� 3lTTg mw �

s* dimensionless melt or solid layer thickness

Ste Stefan number, =� �

L

TTc fPl 0�

t time, s

T temperature, ºC

V

�

u velocity components, m/s

V volume, m3

x, y Cartesian coordinates, m

Greek letters � thermal diffusivity, m2/s

� thermal expansion coefficient, K-1

� liquid fraction

�� phase shift � efficiency of the fin

� dimensionless temperature, =

mw

m

TT

TT

�

�

� dynamic viscosity, kg/m s � density, kg/m3

� period, s-1

Subscripts

amb ambient

cr critical

f fin

i component

l liquid

m melting

ref reference value

s solid

w wall

0 initial

VI

�

List of figures

Page

Figure 2.1. Classification of energy storage materials (Zalba et al., 2003). 3

Figure 2.2. Melting front positions for melting from a vertical isothermal

plate: a) SteFo = 0.0005, b) SteFo = 0.002, c) SteFo = 0.006,

d) SteFo = 0.010 (from Bertrand et al., 1999). 10

Figure 2.3. Schematic diagram of the test arrangement,

(Hale and Viskanta, 1979). 11

Figure 2.4. Comparison of measured and predicted (Neumann model)

solid-liquid interface position during solidification from below,



solid-liquid interface position during melting from above,



solid-liquid interface position during solidification from above,



solid-liquid interface position during melting from below,


Figure 2.8. Comparison of the predicted melting front profiles

with experimental results, (Ho and Viskanta, 1984). 15

Figure 2.9. Correlation of melt fraction from vertical wall of rectangular

cavity, (Ho and Viskanta, 1984). 15

Figure 2.10. System geometry, (Gadgil and Gobin, 1984). 16

Figure 2.11. Time evolution of the melting front position,

(Gadgil and Gobin, 1984). 17

Figure 2.12. Melting rates for enclosures of different aspect ratios

having same T, (Gadgil and Gobin, 1984). 18

Figure 2.13. Schematic diagram of the test cell, top view (a)

and front view (b), (Gau and Viskanta, 1986). 19

VII

�

Figure 2.14. Interface shape traced at preselected times during melting

from a side using pour-out method (Twh=38 °C, A=0.714),

(Gau and Viskanta, 1986). 19

Figure 2.15. Correlation of the average Nusselt number in terms of

the characteristic Rayleigh number Rac, (Gau and Viskanta, 1986). 20

Figure 2.16. Scale drawing showing the experimental apparatus

and instrumentation, (Zhang and Bejan, 1989). 21

Figure 2.17. View of the inside surface of the heated plate during

a later convection stage, (Zhang and Bejan, 1989). 22

Figure 2.18. The history of the temperature distribution along the heated plate,

(Zhang and Bejan, 1989). 23

Figure 2.19. Correlation of Nusselt number, (Zhang and Bejan, 1989) 23

Figure 2.20. Experiment cell (all dimensions are in mm),

(Pal and Joshi, 2000). 24

Figure 2.21. Comparison between computed and experimental solid-liquid

interfaces, (Pal and Joshi, 2000). 26

Figure 2.22. Time-wise variation of local Nusselt number on the heated wall

(computed) for a power level of 30 W, (Pal and Joshi, 2000). 27

Figure 2.23. Time-wise variation of average Nusselt number,

(Pal and Joshi, 2000). 28

Figure 2.24. Time-wise variation of melting rates, (Pal and Joshi, 2000). 28

Figure 2.25. Schematic diagram of the physical configuration and

coordinate system, (Ho and Chu, 1993). 29

Figure 2.26. Temporal variations of the imposed enthalpy oscillations

at the hot wall (a), the heat transfer rates (b) and the melting

rate (c), (Ho and Chu, 1993). 30

Figure 2.27. Influence of the time period on the heat transfer rates and

the melting rate, (Ho and Chu, 1993). 30

Figure 2.28. Effect of the Rayleigh number on the heat transfer rates and

the melting rate, (Ho and Chu, 1993). 31

Figure 2.29. Dependence of temporal variations of the heat transfer rates

VIII

�

and the melting rate on the subcooling factor, (Ho and Chu, 1993). 31

Figure 2.30. Schematic diagram of the physical configuration and

coordinate system, (Ho and Chu, 1994). 32

Figure 2.31. Histories of melting rate and heat transfer rates under

a fixed large-amplitude wall-temperature oscillation with

different time period, (Ho and Chu, 1994). 33

Figure 2.32. Schematic diagram of the experimental equipment,

(Casano and Piva, 1999). 34

Figure 2.33. Comparison between computed (lines) and measured

(symbols) temperature distributions for tests of different periods

(a: 4 h; b: 8 h; c: 16 h), (Casano and Piva, 1999). 36

Figure 2.34. Schematic of the test cell, (Eftekhar et al., 1984). 37

Figure 2.35. Photographs of the liquid-solid interface, (Eftekhar et al., 1984). 38

Figure 2.36. Digitized location of the interface, (Eftekhar et al., 1984). 38

Figure 2.37. Variation of fsNu as a function of Ra/Ste, (Eftekhar et al., 1984). 39

Figure 2.38. Latent heat storage vessel schematic, (Inaba et al., 2003). 40

Figure 2.39. Time history of Q/Qt: a) effect of fin thickness,

b) effect of fin pitch, and c) effect of cooling wall temperature,

(Inaba et al., 2003). 41

Figure 2.40. Semi-infinite phase change material storage with a fin,

(Lamberg and Siren, 2003). 42

Figure 2.41. The results of case studies 1–3 and the analytical solution

when t=3600 s in a semi-infinite n-octadecane storage,

Tw–Tm=20 _C, (Lamberg and Siren, 2003). 43

Figure 3.1. Physical model. 47

Figure 3.2. Computational model. 52

Figure 4.1. Phase distribution of PCM during melting as a function of time. 57

Figure 4.2. Heat transfer rate vs. time for PCM and partition during melting. 58

Figure 4.3. Temperature evolution in the partition during melting. 59

IX

�

Figure 4.4. Temperature evolution in the PCM during melting. 60

Figure 4.5. Phase distribution of PCM during solidification as

a function of time. 61

Figure 4.6. Heat transfer rate vs. time for PCM and partition

during solidification. 62

Figure 4.7. Temperature evolution in the partition during solidification. 63

Figure 4.8. Temperature evolution in the PCM during solidification. 63

Figure 4.9. Total heat flux from the base, case 2. 65





Figure 4.14. Heat fluxes in different geometries, �

T = 6 °C. 68


T = 12 °C. 68


T = 18 °C. 69


T = 24 °C. 69

Figure 4.18. Melt fraction, case 2. 70





Figure 4.23. Melt fraction, �

T = 6 °C. 73


T = 12 °C. 73


T = 18 °C. 74


T = 24 °C. 74

Figure 4.27. Temperature distribution on the symmetry line of PCM, case 2. 75





Figure 4.32. Temperature evolution of the fin, �

T = 6 °C. 78

X�

�


T = 12 °C. 78


T = 18 °C. 79


T = 24 °C. 79

Figure 5.1. Heat transfer rates from base, case 2: a. T = 6 ºC;

b. T = 12 ºC; c. T = 18 ºC; d. T = 24 ºC. 80


b. T = 12 ºC; c. T = 18 ºC; d. T = 24 ºC. 81


b. T = 12 ºC; c. T = 18 ºC; d. T = 24 ºC. 82


b. T = 12 ºC; c. T = 18 ºC; d. T = 24 ºC. 83


b. T = 12 ºC; c. T = 18 ºC; d. T = 24 ºC. 84

Figure 5.6. Evolution of the melting process of the PCM and

temperature distribution in the fin for case 2:a. T = 6 ºC;

b. T = 12 ºC; c. T = 18 ºC; d. T = 24 ºC. 87



b. T = 12 ºC; c. T = 18 ºC; d. T = 24 ºC. 88



b. T = 12 ºC; c. T = 18 ºC; d. T = 24 ºC. 89



b. T = 12 ºC; c. T = 18 ºC; d. T = 24 ºC. 90



b. T = 12 ºC; c. T = 18 ºC; d. T = 24 ºC. 91

Figure 5.11. Temperature distribution along the fin at base temperature

XI�

�

T = 24 ºC, Melt fraction =0.5. 93

Figure 5.12. Temperature distribution along the fin at base temperature

T = 24 ºC, Melt fraction =0.9. 93

Figure 5.13. Melt fraction of the PCM at different

base temperatures, case 2. 94









Figure 5.18. Correlation for melt fraction of case 2. 97





Figure 5.23. Dimensionless heat flux from the base, case 2. 100





Figure 5.28. Dimensionless temperature of the PCM, case 2. 103





Figure 5.33. Melt fraction evolution in all cases considered in this work. 106

Figure 5.34. Correlation for melt fraction taking into account

XII�

�

the fin thickness. 109

Figure 5.35. Correlation for melt fraction taking into account the

fin thickness, for maximum T. 110

Figure A1.1 a) evolution of melt front, and b) temperature distribution

in the unit, a =12 ºC, = 150 sec. 115


in the unit, a = 12 ºC, =180 sec. 115


in the unit, a = 18 ºC, = 120 sec. 116


in the unit, a =18 ºC, =150 sec. 116

Figure A1.5. Temperature distribution of the PCM, (a=12 ºC, �=150 sec). 117




Figure A2. Velocity fields for case 2 (a) V/V0=0.3, b) V/V0=0.6)

and case 4 (c) V/V0=0.6). 120

Figure A3. Heat .transfer rate at different time step calculations for case 3

with T = 18 ºC. 121

List of tables

Page

Table 2.1. Comparison of organic and inorganic materials

for heat storage, (Zalba et al., 2003). 5

Table 2.2. Summary of reviewed literature. 44

Table 3.1. Geometry parameters. 48

Table 3.2. Properties of aluminum and air used for computation. 48

Table 3.3. Properties of the PCM used for computation. 49

Table 3.4. Properties of PCM used in the detailed study, cases 2–6. 51

XIII�

�

1. Introduction

Thermal energy storage systems that utilize latent heat of phase-change materials

(PCMs) have received significant attention, because of their large heat storage capacity.

Phase-change heat transfer problems are relevant both to the storage of thermal energy

from intermittent source such as the sun and to various processes in geophysics and

technology.

Phase-change materials including organic paraffins, metallic alloys and inorganic

salts undergo reversible phase transformation. Due to their isothermal behavior during the

melting and solidification processes, such materials can be used in such diversified

applications as latent heat storage in building or thermal control in electronic modules.

A latent heat storage system is preferable to sensible heat storage in applications with

a small temperature swing because of its nearly isothermal storing mechanism and high

storage density, based on the enthalpy of phase change (latent heat).

In a latent heat thermal storage (LHTS) system, during phase change the solid–liquid

interface moves away from the heat transfer surface. In the case of solidification,

conduction is the sole transport mechanism, and in the case of melting, natural

convection occurs in the melt layer and this generally increases the heat transfer rate

compared to the solidification process. During the phase-change process, the surface heat

flux decreases due to the increasing thermal resistance of the growing layer of the molten

or solidified medium. This thermal resistance is significant in most applications and

especially when the organic phase-change materials are used, because the latter have

rather low thermal conductivity. The decrease of the heat transfer rate calls for the usage

of proper heat transfer enhancement techniques in the LHTS systems.

It is possible to enhance the internal heat transfer of PCM with various means,

including fins, metal honeycombs, metal matrices (wiremesh), high conductivity particles

or graphite.

The use of finned surfaces with different configurations has been proposed by

various researchers, (Velraj et al. (1999), Zalba et al., (2003)), as an efficient means to

improve the charge/discharge capacity of a LHTS system. Several other heat transfer

enhancement techniques for LHTS systems also have been studied, including inserting a

�

metal matrix into a PCM, using PCM with dispersed high conductivity particles, and

micro-encapsulation of PCM.


the phase-change material (PCM) in a latent heat storage system with internal partitions

or plate-type fins attached to a horizontal base which is heated or cooled. Phase change

material is stored between the fins, and it melts or solidifies as a result of heat exchange

with the base, the conducting partitions, and the ambient air.

The study includes three main steps. First, the processes of melting and solidification

are studied in a relatively large partitioned heat storage unit, yielding detailed information

on the phase and temperature distributions inside the PCM and the partitions during the

transient phase-change processes. Then, a detailed parametric investigation is performed

for melting in a relatively small heat sink, 5mm to 10mm high, where the fin thickness

varies from 0.15mm to 1.2mm, and the thickness of the PCM layers between the fins

varies from 0.5mm to 4mm, while the base temperature varies from 6°C to 24°C above

the mean melting temperature of the PCM. Finally, melting and solidification in the sink

are explored for the case where the temperature of the sink base changes periodically.

The material used in the simulations is a commercially available paraffin wax, and

the fins are made of aluminum. The melting temperature of the wax, 23–25°C, is

incorporated in the simulations along with its other properties, including the latent and

sensible specific heat, thermal conductivity and density in solid and liquid states.

Transient two-dimensional numerical simulations are performed using the Fluent 6.0

software. The computational results show how the transient phase-change process,

expressed in terms of the molten volume fraction of the PCM, depends on the thermal

and geometrical parameters of the system, including the temperature difference between

the base and the solid PCM, thickness of the fins and the PCM layer between them, and

the height of the fins. Dimensional analysis is performed based on such dimensionless

parameters as the Fourier number, which takes into account the transient heat conduction,

and the Stefan number, which reflects the phase change.

�

2. Literature survey

As mentioned in the Introduction, phase change processes are encountered in the

nature and in various fields of science and engineering. Accordingly, a very large body of

literature exists on the variety of phase-change problems. This survey is restricted to the

works relevant to the present investigation� It includes a review of the phase-change

materials which are or could be used in heat storage applications, theoretical background

related to mathematical modeling of phase change processes, and numerical methods

used for the problems of solid-liquid phase change. Then, the existing literature on phase-

change in enclosed spaces is discussed. The presented works concern both transient and

periodic melting and solidification. Finally, the recent studies on phase change problems

with internal cooling fins are surveyed.

2.1 Phase-change materials

Classification of phase-change materials used in latent-heat energy storage systems is

presented in figure 2.1.

Figure 2.1. Classification of energy storage materials (Zalba et al., 2003).

�

One can see from figure 2.1 that, in general, heat can be stored in gas-liquid, solid-

liquid, solid-gas, and solid-solid phase transitions. The liquid-gas and solid-gas

transitions are characterized by large volume change, thus requiring heavy and

complicated pressure vessels or special design features. Solid-solid transformation, such

as transition of the rhombic form of sulfur to the monoclinic form, is of interest because

the energy exchange at that transformation can be significant and a number of materials

display this phenomenon in a temperature range near their melting point. Solid-liquid

transition is of great importance because most classes of materials undergo this type of

transition without exhibiting large volume change, while releasing or absorbing

significant quantities of energy.

Various materials have been investigated for the energy storage systems based on the

solid-liquid phase change. In order to be suitable for heat storage, these materials should

satisfy a number of conditions related to their properties:

1) Thermal properties:

Phase change temperature fitted to application;

High change of enthalpy near the temperature of use;

Sufficient thermal conductivity in both liquid and solid phases.

2) Physical properties:

Low density variation;

High density;

Small or no undercooling.

3) Chemical properties:

Stability;

Absence of phase separation;

Compatibility with container materials;

Non-toxicity, non-flammability, friendliness to environment.

4) Economic properties:

Low price and abundance.

A comparison of the advantages and disadvantages of organic and inorganic materials

is shown in Table 2.1.

�

Table 2.1. Comparison of organic and inorganic materials for heat storage

(Zalba et al., 2003).

Organic Inorganic

Advantages Advantages

No corrosiveness Greater phase change enthalpy

Low or no undercooling

Chemical and thermal stability

Disadvantages Disadvantages

Lower phase change enthalpy Undercooling

Low thermal conductivity Corrosion

Inflammability Phase separation

Phase segregation, lack of thermal stability

2.2 Theoretical background and mathematical modeling

Solid-liquid phase-change (melting or solidification) heat transfer phenomena are

accompanied by a phase transformation of the medium and by either absorption or

release of thermal energy in the active zone. The energy absorbed or released from the

surrounding system is commonly transferred by conduction or convection. The essential

and common features of a system undergoing solid-liquid phase-change are that an

interface exists separating two regions of differing thermophysical properties and that a

moving surface, at which energy is absorbed or released, separates the two phases. To

solve the problem, it is necessary to determine the manner and rate at which the solid-

liquid interface moves with time. For this reason, problems of this type are referred to as

moving boundary problems. Because of the motion of the solid-liquid interface, the

problems posed are nonlinear, so that few exact analytical solutions are available, and

those that do exist are primarily for one-dimensional planar geometry. Following a

review by Viskanta (1983), no exact analytical solutions exist for a finite dimension

system when both phases are present, including planar geometry.

2.2.1 One-dimensional phase-change heat transfer problems

The analytical methods available can be classified as follows: rigorous (exact),

moment (integral), calculus of variations, perturbation and others. Exact analytical

solutions of the phase-change problems are known only for a very limited number of

physical situations with simple geometries and boundary conditions: e.g., a semi-infinite

slab with boundary conditions of the first kind (constant temperature), commonly referred

to as the Neumann problem (Carslaw and Jaeger, 1959), or semi-infinite solid with a

boundary condition of the second kind (constant heat flux).

When an exact solution does not exist, various numerical approaches can be applied.

In particular, the finite difference methods for the numerical solution of one-dimensional

phase-change heat transfer problems were reviewed by Viskanta (1983), who concludes

that one-dimensional problems are amenable to efficient and accurate numerical solution

with only moderate effort.

2.2.2 Methods of solution for multidimensional phase-change heat transfer problems

Multidimensional problems are of greater practical importance but are much harder

to solve than one-dimensional problems. A variety of numerical methods have been

proposed in the literature for solving multidimensional phase-change heat transfer

problems, including highly versatile finite difference and finite element methods.

The available finite difference methods for phase-change problems have been

broadly divided by Viskanta (1983) into two groups, based on the choice of the

dependent variables used: (A) the temperature-based methods, and (B) the enthalpy-

based methods.

A. Temperature-Based Methods:

In this group the temperature is the sole dependent variable, and energy conservation

equations are written separately in the liquid and solid regions. The temperatures in the

two phases are coupled through an energy balance at the solid-liquid interface. The

existing solutions can be divided into four main categories:

1. Explicit finite–difference scheme

2. Implicit finite–difference scheme

3. Moving boundary immobilization method

4. Isotherm migration method

These categories are briefly described below.

1. Explicit finite–difference scheme

A large number of one-dimensional and two-dimensional problems have been solved

using the explicit finite–difference method. The main drawback of this method is the

necessity to place restriction on the maximum step size to avoid numerical instability.

This is particularly critical when the Stefan number (heatLatent

solidofheatSensibleSte � ) is

small. Even though two problems, one with Ste=0.1 and the other with Ste=0.01, may

have nearly identical solutions, the stability criterion for an interior node ( � Ste /N,

were and are the step sizes, and N=2, 4, 8 for one-, two-, and three-dimensional

problems, respectively) forces one to use for Ste=0.1 that is one-tenth that for

Ste=0.01. A severe penalty is also paid when is reduced to improve the accuracy of the

results.

One way to handle the latent heat effect numerically is to include it as an energy

source or sink in the energy equation. An apparent specific heat (called also weak

solution technique) is defined to account for the enthalpy change, including latent heat.

The apparent specific heat concept assumes or approximates the latent heat effect as

taking place over a small but finite temperature range. This technique has the advantage

of also being applicable to impure substances (mixtures, solutions, etc.) for which the

latent heat is released over a finite temperature range.

2. Implicit finite–difference method

For certain problems it is possible that the restriction imposed on the time step will

lead to a prohibitively large number of computations. For such problems, it would be

preferable to attempt a solution by an implicit finite-difference scheme, which is not

subject to a stability restriction. Unfortunately, the use of implicit schemes usually yields

a system of nonlinear finite-difference equations for the unknown temperatures, which

must be solved by iterative methods. Following the Viskanta’s review (1983), this is the

major reason why the implicit finite-difference schemes have been less popular than the

�

explicit finite-difference schemes in the numerical solution of phase-change heat transfer

problems in the past.

3. Moving boundary immobilization method

This finite-difference scheme, also based on retaining the temperature as the

dependent variable, employs a transformation of the spatial variables to cause the moving

boundary to be stationary in the transformed coordinate system. In the transformed space,

the solid and the liquid each occupy a fixed region, but the governing differential

equations contain parameters of the interface location. However, in general, the

transformed coordinate system will not be orthogonal. As a consequence, the derivation

of a finite-difference representation of the conservation equations involves features that

are not found in conventional diffusion problems where orthogonal coordinate systems

are the rule. Also, the equations are usually more complicated and must be solved

iteratively.

4. Isotherm migration method

In the isotherm migration method the position of an isotherm is examined as a

function of time. The main advantage of the isotherm migration method is that the phase-

change interface is followed exactly and directly calculated, and that the region

represented is the only region in which a temperature change has occurred and no

interpolation of isotherms is needed. Finally, property reevaluation at each time step for

temperature dependent properties is eliminated, as the same properties are carried along

with each isotherm. However, there are several disadvantages of the method. First, the

governing equation becomes nonlinear. Second, a starting solution is often required,

necessitating a small time step during the numerical solution. Third, the magnitude of the

isotherm displacement may be multi-valued for certain boundary and initial conditions.

Finally, if the temperature at the boundary rises and then falls, the same temperature may

occur at two different positions at the same time, which presents a difficulty in tracking

the interface by temperature.

�

B. Enthalpy-Based Methods:

In the enthalpy models the enthalpy is used as a dependent variable along with

temperature. The basic equations for phase-change heat transfer written explicitly in

terms of enthalpy are also referred to as a weak solution formulation. The basic equation

of the enthalpy model is:

dVqAdTkAddVdt

d

VSSV�� '''��

(2.1)

where � is the density, � is the specific enthalpy, v is the velocity field, k is the thermal

conductivity, � is the gradient operator, T is the temperature, q´´´ is the rate of heat

generation, A is the area, S is the external surface, and V is the volume.

The velocity field is set equal to zero to avoid difficulties, and the solution methods

are based on it. The phase boundary position is not explicitly determined, and the

problem is made equivalent to one of heat conduction without phase change. Although

the interface is not tracked explicitly, its position can be approximately determined from

knowledge of enthalpy distribution. The enthalpy method avoids the energy balance and

the discontinuity at the solid-liquid interface. The advantages of the method are that one

does not need to track the interface, and the analyses can be performed without assuming

anything about the nature of the substance. Of course, for obtaining the solution a

specification of the properties of the substance will be necessary.

Bertrand et al. (1999) present the first results of comparison exercise of numerical

solution of phase-change problem that include the simulation of coupled natural

convection and melting from an isothermal vertical wall. The exercise is restricted to the

simulation of phase change of pure substances, driven by laminar natural convection in

2D enclosures. Among the 10 sets of results received, most contributions have used fixed

grid or “enthalpy” methods (FG), including that incorporated in a software package

Fluent, and two have used a front-tracking of the transformed grid procedure (TG). The

comparison covers two ranges of Prandtl numbers, corresponding to the melting of

metals or organic materials. The results of the test cases are presented in detail and show

that, while qualitative agreement is obtained in most situations, it is still relevant to

�

proceed to thorough numerical comparisons before assessing the accuracy of the different

algorithms. A necessary general remark concerning the present exercise is that the choice

of the problem itself (fusion of a pure substance) is such that the front-tracking methods

are better adapted to the problem than the fixed grid procedures. The front-tracking

methods would however fail to simulate situations where the transition from the liquid to

the solid phase is not a macroscopic surface, and enthalpy methods are to be used in most

solidification problems where a solid-liquid interfacial region is present between both

phases.

Figure 2.2. Melting front positions for melting from a vertical isothermal plate:

a) SteFo = 0.0005, b) SteFo = 0.002, c) SteFo = 0.006, d) SteFo = 0.010

(from Bertrand et al., 1999).

��

Review of the recent literature that deals with the phase-change processes in rectangular

enclosures is presented for three subjects:

1). Transient melting in an enclosure.

2). Periodic melting and solidification in a rectangular enclosure.

3). Phase change with internal fins.

2.3 Transient melting in an enclosure

Hale and Viskanta (1979) performed an experimental and analytical study of melting

and solidification of several PCMs such as stearic acid, sodium phosphate dodecahydrate,

sodium sulfate decahydrate, and n-octadecane, cooled or heated from above or below. A one-

dimensional analysis was performed using an empirical correlation to account for the effect

of natural convection. Comparison of experimental data for n-octadecane with predictions

based on Neumann and other analyses which account for natural convection heat transfer at

the solid-liquid interface was shown for stable and unstable conditions.

Figure 2.3. Schematic diagram of the test arrangement

(Hale and Viskanta, 1979).

The test cell where melting and freezing took place was placed inside a temperature

control chamber which was maintained as close to the fusion temperature of the test material

as possible without initiating the melting/freezing process. This was accomplished with two

heat exchangers, which were installed in the chamber through which water from a constant

temperature bath was circulated. Two copper heat exchangers formed the top and bottom

��

walls of the test cell and served as nearly isothermal heat sources. The vertical walls were

made of plexiglass. The depth of the test cell was purposely made much smaller than the

height or the width in order to minimize the non-uniformities in the melting front position

perpendicular to the faces of the cell.

When a liquid is cooled from bellow or a solid is heated from above, the fluid layer is

stable, and the Neumann model for one-dimensional phase-change heat transfer should be

appropriated. Figure 2.4 shows a comparison of the liquid-solid interface position predicted

from the Neumann analysis with experimental data during solidification from below. Here s*

is dimensionless melt or solid layer thickness, and � is dimensionless time, � =SteFo, where

� ��

�� ifPL TTc

Ste ;

2l

tFo

�� .

Figure 2.4. Comparison of measured and

predicted (Neumann model) solid-liquid

interface position during solidification from

below (Hale and Viskanta, 1979).



interface position during melting from

above (Hale and Viskanta, 1979).

At early times the analysis predicts a faster rate of solidification than experimentally

observed, but at later times there is good agreement between predictions and data.

��

The predicted and measured solid-liquid interface position for melting from above is

compared in Fig. 2.5. The agreement between data and analysis is reasonably good for early

times, but at later times the observed melt layer thickness was greater than that predicted.

In the unstable situation of freezing from above (Fig 2.6) natural convection at the solid-

liquid interface may decrease the rate of solidification at the later times if the Rayleigh

number is sufficiently large.



interface position during solidification from

above (Hale and Viskanta, 1979).



interface position during melting from

below (Hale and Viskanta, 1979).

In melting from below (Fig. 2.7) natural convection develops rapidly and greatly

influences heat transfer and the motion of the phase-change boundary during the process.

The findings of this study demonstrate the importance of natural convection in solid-

liquid phase-change heat transfer under unstable conditions.

Ho and Viskanta (1984) reported basic heat transfer data during melting of n-octadecane

from an isothermal vertical wall of a rectangular cavity which measures 130 mm high, 50

mm wide, and 50 mm deep. In order to accommodate the volume expansion associated with

the phase change process from the solid to the liquid, a small air gap was maintained between

the top of the material and the insulated top surface of the cavity. The right hand side vertical

wall of the cavity was heated by circulating a working fluid from a constant temperature bath

through the channels milled inside the heated vertical plate. The other walls were insulated.

��

The local heat transfer coefficients at the heated wall were measured using a

shadowgraph method. Experimentally, the shadowgraphic technique for heat transfer

measurements involves identification of the heat source surface as a reference position as

well as the recording of the deflection of the light beam on the screen after its passage

through the cell. To make the deflection of the light beam (which passes through the this

region adjacent to the heat source surface) more visible against the dark background, a plate

with a narrow aperture was placed between the collimating lens and the test cell. This

aperture blocked off the light entering the test cell, except that which passed in the immediate

vicinity of the heat source surface.

A numerical simulation of the corresponding two-dimensional melting in the presence of

natural convection was performed, and the numerical prediction was compared with

experimental data. A comparison between the experimentally determined and predicted

melting front profiles at different times is illustrated in Fig. 2.8. The solid and dashed lines

depict the measured and predicted melting fronts, respectively. At early time, the agreement

is excellent except at the very bottom and top regions of the cavity. The slightly greater

measured melting rate at the bottom edge could be partly attributed to heat conduction along

the bottom wall of the cavity due to the removal of initial singularity in the solution by a false

start. As for the discrepancy at the top free surface, it is primarily attributed to the neglect in

the mathematical model of the volume expansion due to the phase change from the solid to

the liquid. As melting continues, the effect of this density-induced overflow motion appears

to become more significant as indicated by the greater discrepancy between the measured

and predicted melting front position, particularly in the vicinity of the free surface.

Ho and Viskanta studied the evolution of the melt front and measured heat transfer

coefficients on the heated wall. Figure 2.9 indicates the time-wise variation of the molten

volume fraction, V/V0, evaluated by integrating the melting front contours. It appears that all

the data may collapse quite well into a single curve as a function of the Stefan number Ste,

the Fourier number Fo, and the Rayleigh number Ra:

� ��

� 3lTTgRa mw �

� .

��

Figure 2.8. Comparison of the predicted melting front profiles with experimental results

(Ho and Viskanta, 1984).

Figure 2.9. Correlation of melt fraction from vertical wall of rectangular cavity

(Ho and Viskanta, 1984).

Gadgil and Gobin (1984) simulated numerically two-dimensional melting of a solid

phase change material in a rectangular enclosure heated from one side. The simulations were

carried out by dividing the process in a large number of quasi-static steps. In each quasi-static

step, steady-state natural convection in the liquid phase is calculated by directly solving the

��

governing equations of motion with a finite difference technique. This is used to predict the

shape and motion of the solid-liquid boundary at the beginning of the next step.

Figure 2.10. System geometry

(Gadgil and Gobin, 1984).

A two-dimensional rectangular enclosure that shown in figure 2.10, contains the PCM

initially at its fusion temperature, Tf. The top and bottom surface of the enclosure are

adiabatic, the bottom surface is no-slip, while the top is filled by air gap. One vertical wall is

raised to a hot temperature, and the other wall is isothermal at fusion temperature of the

PCM. The solid-liquid interface (henceforth called the melting front) has a shape similar to

that shown in figure 2.10. The shape of the melting front is an extremely sensitive indicator

of the variation of the local heat transfer rate to the solid and is used as a test for the

numerical simulation procedure.

An unevenly spaced 20 � 12 grid was selected for use after solving a test problem of

high Ra convection in a rectangular enclosure (Ra=109) and comparing the solution with that

��

obtained with a specially constructed high resolution 31 � 25 grid. The predictions show

good agreement, particularly in the boundary layer regions, which are of main interest from

the viewpoint of heat transfer at the vertical walls.

Predictions from the numerical simulations for two different enclosure aspect ratios

(A=2.44 and 5.25) are shown for two values of Ra (Ra=108 and 109) in figure 2.11. The

position of the melting front at different dimensionless times was indicated.

Figure 2.11. Time evolution of the melting front position


It is practical interest to investigate the influence of the aspect ratio, A, on the melting

curve for such a case, since this has bearing on optimizing the dimensions of latent heat

storage elements. Results from one such investigation of Gadgil and Gobin, (1984) are

presented in figure 2.12, where the fraction of PCM in the liquid phase is plotted against

dimensionless time for three different values of A and Ra (with Ra/A3 being the same in all

three cases). The results show that A has a strong influence on the melting curve, an increase

in A from 1.13 to 5.25 causing more than sixteen-fold decrease in the time required to melt

80% of the PCM.

��

Figure 2.12. Melting rates for enclosures of different aspect ratios having same T,


The effect of free convection on the shape and motion of the melting front of gallium

from a vertical wall was investigated by Gau and Viskanta (1986). Melting and solidification

experiments were performed in a rectangular test cell (Fig. 2.13), which had inside

dimensions of .8.89 cm in height, 6.35 cm in width, and 81 cm in depth. The two end walls,

which served as the heat source/sink, were made of multipass heat exchange machined from

a copper plate.

Their results reported clearly that for larger aspect ratio cavities (A), melting of the solid

near the top may be greatly promoted, and melting near the bottom of the test cell may be

completely terminated, due to free convection in the melt. The interface shape traced at

preselected times is shown in figure 2.14 for different aspect ratios. At very early times,

before the buoyancy-driven flow was initiated or when fluid motion was still very weak, the

interface shape was flat and parallel to the heated wall of the test cell. Heat transfer was

dominated by conduction. As the heating progressed the buoyancy-driven convection in the

melt started to develop and continued to intensify. This is evidenced by the appearance of a

nonuniform melt layer receding from the top to the bottom of the test cell.

��

Figure 2.13. Schematic diagram of the test cell, top view (a) and front view (b): (1) heat

source and/or sink, (2) Plexiglas wall, (3) air gap, (4) phase-change material, (5)

thermocouple rack, (6) thermocouples along the walls, (7) small-diameter thermocouples, (8)

hole for filling material, and (9) constant temperature baths (Gau and Viskanta, 1986).

Figure 2.14. Interface shape traced at preselected times during melting from a side

using pour-out method (Twh=38 °C, A=0.714) (Gau and Viskanta, 1986).

��

The experimental data of heated-area-averaged heat transfer coefficients for melting of

gallium from a vertical wall were favorably correlated and compared with numerical data.

Since convection-dominated quasi-steady melting does not appear to have been fully

reached in this experiments no attempt was made to correlate the data in a form of a standard

natural convection. Even though the entire melting process is transient in nature, and one of

the boundaries of the heat sink (solid-liquid interface) is not planar, an average melt layer

thickness has been used as a characteristic length lC in correlating of heat transfer data. The

list-squares fit of the data was found to yield the following empirical equation

274.0

0631.0 ��

��

��

t

C

l

C

Ste

Ra

k

hlNu (2.2)

A comparison of the data with the equation is presented in figure 2.15. All the data points

collapse nicely onto a single line. The correlation is valid in the conduction, transition, and

natural convection regimes.

Figure 2.15. Correlation of the average Nusselt number in terms of the

characteristic Rayleigh number RaC (Gau and Viskanta, 1986).

In spite of the large thermal conductivity, the experimental results have clearly

established the important role played by natural convection during melting of a pure metal

from a vertical wall. The phase-change boundary, the melting rate, and heat transfer were

��

greatly affected by buoyancy driven natural convection. Laminar flow persisted throughout

the melting process, and nearly quasi-steady melting was attainted at later stages.

During solidification of gallium from a sidewall, the interface shape was controlled not

only by the natural convection circulation in the melt but also by the combined

crystallographic effects and anisotropy of the gallium crystal.

Zhang and Bejan (1989) are report their study of the time-dependent melting within a

relatively tall enclosure (height/width = 5) filled with paraffin, due to a constant heat flux

boundary condition. The first part describes experimental measurements with n-octadecane

conducted in a 74 cm tall enclosure. The second part of the study describes the liquid

superheated effect analytically, by means of “matched boundary layers” solution for the

convection regime of the heat transfer and the melting process.

The opposite enclosure wall was maintained at a uniform temperature. Experiments were

performed to determine the heat transfer coefficients and melting rates for various power

dissipations. Based on an analytical approach involving boundary layer theory, a steady state

heat transfer correlation was developed.

The vertical enclosure in which the experiments were performed is shown in figure 2.16.

Figure 2.16. Scale drawing showing the experimental apparatus and instrumentation

(Zhang and Bejan, 1989).

��

The enclosure is defined by two parallel vertical plates positioned 14.6 cm apart. The

top, bottom and sidewalls are made out of acrylic plexiglas 2.5 cm thick. The height of the

enclosure space is 73.7 cm. The length is 56 cm: this dimension is large enough to minimize

the three-dimensional (end) effects on the flow and two-phase interface.

The heat flux Rayleigh numbers of these experiments are of the order of 1013, and the

liquid flow pattern is weakly turbulent. Zhang and Bejan (1989) presented in figure 2.17 that

the boundary layer flow near the heated plate is turbulent.

Figure 2.17. View of the inside surface of the heated plate during a later convection stage


There are three important features to note in figure 2.18. First, there is a relatively short

time interval, during which the temperature of the heated plate rises from the ambient level to

the melting point of the phase change material. Immediately following the start-up time

interval is a period when the temperature of the plate is uniform and rises linearly in time.

Finally, figure 2.18 shows how the wall temperature reaches a plateau in the convection

melting regime.

��

Figure 2.18. The history of the temperature distribution along the heated plate


The analysis of these experiments suggests that the Nu data can be correlated by a

unique relation between Nu/Ra1/5 and �/Ra1/5 (Fig. 2.19).

491.05/1�

Ra

Nu (2.3)

Figure 2.19. Correlation of Nusselt number


��

Finally, the Nusselt number correlation for the convection regime is:

� � 4/13

411.0��

�

��

� ��

��

� mavg TTlgNu (2.4)

Pal and Joshi (2000) studied computationally and experimentally melting of an organic

phase change material (PCM) n-triacontane (C30H62) in a side heated tall enclosure of aspect

ratio 10, by a uniformly dissipating heat source. An implicit enthalpy-porosity approach was

utilized for computational modeling of the melting process. Experimental visualization of

melt front locations was also performed. Correlations of heat transfer rate and melt fraction

were obtained.

The experimental cell of this study is shown in figure 2.19. One inner wall used as

uniformly dissipating heat source. Other walls were insulated. A total of 17 copper-

constantan thermocouples of 0.13 mm diameter were used in the experiments (Fig. 2.20).

Figure 2.20. Experiment cell (all dimensions are in mm)

(Pal and Joshi, 2000).

Before the experiments, the enclosure was slowly filled with molten PCM at a

temperature of 95 C. The PCM was poured in 1 cm layers at a time, allowing the earlier

��

layer to solidify, to ensure absence of any entrapped air bubble. Experiments were

subsequently conducted with all the thermocouples initially uniform within ! 0.15 C. Power

to the heater was set and controlled by a voltage controller. Experiments were continued until

the maximum heater temperature rose to 105 C, the highest allowable operating temperature

for the adhesives used in the setup. Tests were performed from 15 to 60 W at a step of 5 W.

The fluid flow was assumed laminar due to the small velocities (~5 mm/s). Two-dimensional

transport was considered, following experimental observations of the solid-liquid interface,

which was uniform over a large region in the depth-wise (z) direction. For all solids, the

thermophysical properties were assumed to be independent of temperature. The density of

molten PCM was assumed to be constant, except for the buoyancy force term in the y-

momentum equations, according to the Boussinesq approximation for natural convection

flows.

The computational results of evolution of melt front, and the velocity vectors are

presented in figure 2.21a-d, for a power level of 15 W (heat flux =0.093 W/cm2). The

corresponding experimental solid-liquid interface locations are visualized in figure2.21e-h.

As melting continues, buoyancy driven convection in the molten PCM starts to develop (Fig.

2.21a, e). During this stage, the hot and light molten PCM moves upward along the heated

wall, turns around and moves towards the solid-liquid interface. This flow impinges on the

solid-liquid interface, and transfers heat to the solid, which enhances melting.

Figure 2.22 presents the time-wise variation of local instantaneous Nusselt number

following melting at various locations on the heated wall for a power level of 30 W (Ste =

0.81). Starting with an infinite value at the initiation of melting Nusselt numbers drop very

rapidly during the initial conduction dominated stage (I). This was followed by a sudden

change in slope, which marked the beginning of a transition regime (II). During this phase,

heat transfer was due to combined conduction and convection effects. Almost a constant

value of Nu at a given location followed the transition stage. During the convection

dominated stage of melting (III), Nu increased from the bottom to the top. As melting

approached completion (IV), Nu started to reduce with time.

��

Figure 2.21. Comparison between computed and experimental solid-liquid interfaces.

The molten PCM appears to be dark and the solid PCM is seen as the white region


��

Figure 2.22. Time-wise variation of local Nusselt number on the heated wall (computed) for

a power level of 30 W (Pal and Joshi, 2000).

This reduction was caused by sensible heating of the PCM, and lack of heat transfer

from the system since all the boundaries were insulated. The difference in Nu along the

heater began to reduce, as melting approached completion, indicating uniformity in

temperature and a decreased strength of natural convection at this stage.

The heat transfer data were scaled with Ra1/5. This scaling reduced the normalized heat

transfer and molten volume fraction data very well, except at the post-melt stage (Figs. 2.23

and 2.24).

��

Figure 2.23. Time-wise variation of average Nusselt number


Figure2.24. Time-wise variation of melting rates


4.05

1

01559.0��

�

��

�

�

FoSte

RaNu�

��

�

�

��

�

��

�

51

4865

957.043.1 Ra

FoSte

e"

��

2.4 Periodic melting and solidification in an enclosure

A numerical study of periodic melting in a rectangular enclosure with an oscillatory

heated-wall temperature was reported by Ho and Chu (1993, 1994), who adopted the

enthalpy method to model the latent heat absorption at the melting front.

The mathematical model of two-dimensional melting of solid PCM (Ho and Chu, 1993)

is presented in figure 2.25. The solid PCM is assumed to be initially subcooled at a uniform

temperature Tc. At time t=0, the left-hand vertical wall is isothermally heated with a

sinusoidal perturbation of enthalpy � about a mean hot wall temperature Th, with amplitude,

A, and frequency, f, were � �ftA #2sin1�� . The time-dependent hot wall temperature

remains above the fusion temperature of the PCM at all times. The right-hand vertical wall is

kept isothermal at Tc, and the horizontal walls of the enclosure are assumed adiabatic.

Figure 2.25. Schematic diagram of the physical configuration and coordinate system

(Ho and Chu, 1993).

The set of governing differential equations was solved numerically by means of a finite

difference method. The differential equations were discretized by using second-order central

differencing for the spatial derivatives except the convective terms, for which the second

��

upwind scheme was adopted. The time derivatives were approximated by a forward

difference. The authors used a 41�41 uniform mesh system for the simulations as a result of

a series of tests calculations for the grid dependency.

Ho and Chu (1993) studied influence of the temporal variations of the imposed enthalpy

oscillations (Fig. 2.26), influence of the time period (Fig. 2.27), effect of the Rayleigh

number (Fig. 2.28) on the heat transfer rates and the melting rate, and dependence of

temporal variations of the heat transfer rates and the melting rate on the subcooling factor

(Fig. 2.29).

Figure 2.26. Temporal variations of the

imposed enthalpy oscillations at the hot wall

(a), the heat transfer rates (b) and the melting

rate (c) (Ho and Chu, 1993).

Figure 2.27. Influence of the time period

on the heat transfer rates and the melting

rate (Ho and Chu, 1993).

��

Figure 2.28. Effect of the Rayleigh number

on the heat transfer rates and the melting rate

(Ho and Chu, 1993).

Figure 2.29. Dependence of temporal

variations of the heat transfer rates and the

melting rate on the subcooling factor

(Ho and Chu, 1993).

The periodic mean values of the heat transfer rates as well as the melting rate are

strongly affected by the Rayleigh number and the subcooling factor, but are rather insensitive

to the oscillation amplitude or the time period of the imposed wall temperature perturbation.

Moreover, the oscillation components of the heat transfer rates as well as the melting rate

respond differently to the variation of amplitude and period of imposed wall temperature

perturbation, the Rayleigh number, the subcooling factor, and the Stefan number.

Ho and Chu (1993) found that a steady periodic melting regime arises following a period

of transient oscillatory melting, and that the heat transfer rates at the vertical heated and

cooled walls as well as the melting rate exhibit a regular temporal oscillation at a frequency

equal to the imposed wall temperature perturbation but with phase difference.

��

Ho and Chu (1994) also reported that there might coexist three solid-liquid interfaces

during the sustained periodic solid-liquid phase change process inside the enclosure

depending on the time-periodic amplitude oscillatory wall temperature.

In this work they presented a numerical simulations of multiple moving boundaries arising in

the natural convection-dominated melting process of a pure phase-change material from a

vertical wall of a square enclosure, as illustrated in figure 2.30. The hot wall was subjected to

oscillatory large-amplitude temperature perturbation

� �ftATT hh #2sin�� (2.4)

Other walls were in a low temperature. Top and bottom were adiabatic.

Figure 2.30. Schematic diagram of the physical configuration and coordinate system

(Ho and Chu, 1994).

In figure 2.31 the influence of varying time period of the imposed large-amplitude

(A=1.5) wall-temperature oscillation on the temporal variation of the melting rate, V* , and

the average heat transfer rates, Qh and Qc, at the vertical walls is conveyed. A marked

amplification of the induced steady oscillation amplitudes of both the melting rate and heat

transfer rate at the cold wall arises, as can be seen in figure 2.31; while an adverse effect

occurs for the heat transfer rate at the hot wall. This implies that the effect of varying time

��

period on the melting rate and the heat transfer rate has little or no bearing on whether the

imposed wall-temperature oscillation amplitude is greater or not.

Figure 2.31. Histories of melting rate and heat transfer rates under a fixed large-

amplitude wall-temperature oscillation with different time period

(Ho and Chu, 1994).

Casano and Piva (2002) presented a numerical and experimental investigation of a

periodic phase-change process dominated by heat conduction. In the experimental

arrangement a plane slab of PCM (n-octadecane) is periodically heated from above. A

schematic diagram of the experimental equipment is shown in figure 2.32. The test cell

consists of a polycarbonate round duct (outer diameter 150 mm, inner diameter 140 mm and

height 210 mm) filled with the phase-change medium. The bottom is kept at a temperature

lower than the melting point of the test material by means of a refrigeration system.

��

Figure 2.32. Schematic diagram of the experimental equipment: ET, external trigger;

PS, power supply; IPR, ice point reference; SU, switch control unit;DV, digital voltmeter

(Casano and Piva, 2002).

To remove easily the heat flow during the experiment, the refrigeration system uses

thermoelectric coolers. A uniform distribution of temperature over the whole surface of the

bottom wall is obtained by means of nine thermoelectric modules (dimension 40 $ 40 $ 4

mm3) connected in series. In order to reduce the heat transfer to the environment, the wall of

the cylinder is insulated with a 20 mm thick layer of foam rubber surrounded by a 100 mm

thick layer of expanded polystyrene. The test volume (height 51.1 mm) is closed on the top

by the heater. The heat flow is obtained by dissipating via the Joule effect an assigned power

in a resistor (450 W of nominal power at 220 V). Such a resistor is placed inside a copper

disk soldered at the end of a cylindrical copper container.

The experimental tests were carried out so as to obtain cyclic processes of melting and

freezing in a sample of n-octadecane. The experiments always included a preliminary settling

period when the sample, previously frozen in a refrigerator, was kept solid. When the

measured temperature of the sample became practically uniform, enabling the reaching of

steady-state conditions to be assured, the heating system was activated and the periodic

heating process started.

Having demonstrated the numerical validity of the code by comparison with results for

an analytical solution, the physical validity of the mathematical model may be studied by

Joule effect heater

Peltier cells refrigerator

��

comparison of predictions with experimental data. The results of the comparison are reported

for three experimental tests. The runs differ according to the period of the sinusoidal power

dissipated in the heater (4, 8 and 16 h for Test a, b and c respectively). The temperature was

sampled every 4 min, the experiments lasting for 24 h in the case of Test a, and 48 h for Tests

b and c. For these trials numerical simulation was carried out by using the measured top and

bottom temperature as the boundary conditions. A linear interpolation of the sampled data

was used to calculate the boundary values needed in the computation but not available

experimentally. A uniform grid of 257 control volumes (x= 0.2 mm) was used, together

with a time step equal to t= 0.2 s, necessary to satisfy the stability condition.

In figure 2.33, for the selected runs, the measured and computed temperature

distributions are compared. Results for the three tests show that only Test b reaches steady

periodic conditions; Tests a and c are too short for this steadiness to be achieved. For all the

tests reported in figure 2.33, the presence of an interface separating an upper zone, where the

material is liquid, from a lower zone at the solid state, is clearly evidenced. The liquid zone is

characterized by wide oscillations of temperature, reaching the maximum amplitude at the

heated surface (equal to 5, 8 and 15 °C for Tests a, b and c respectively).

The solid region is characterized by moderate oscillations of temperature, practically

negligible in a test and increasing with the period. This is because the greater part of the heat

flux introduced in the sample via the heater is used for the advancement of the interface and

only a limited part of this flux is still available for the heating of the remaining solid. From a

qualitative point of view the numerical model gives trends of temperature totally similar to

those experimental, both in the solid and in the liquid phase. The start of the melting process,

and of the following freezing and melting cycles, appears clearly and the wide oscillations of

temperature in the liquid are also well evident.

The analysis of the energy behavior of the system gives interesting results. The energy

stored in the system oscillates in time with the energy introduced. A regular behavior is

shown by dimensionless mean value of the energy stored, decreasing for increasing Fourier

numbers and also for increasing Stefan numbers. For small values of the Fourier number the

dimensionless amplitude of the oscillations of the energy stored in the system is constant,

independent of the values of the Stefan number. In this case the system is able to act as a

damper of entering energy oscillations and the heat flux emerging from the output surface is

��

almost constant. Conversely, for increasing Fourier numbers the amplitude strongly depends

on the Stefan number. Furthermore, the amplitude shows a maximum, which is more marked

for large values of the Stefan number. It means that for large period of oscillation there are

situations where the system does not exert any damping effect.

Figure 2.33. Comparison between computed (lines) and measured (symbols)

temperature distributions for tests of different periods (a: 4 h; b: 8 h; c: 16 h)

(Casano and Piva, 2002).

��

2.5 Phase change with internal fins

Eftekhar et al. (1984) reported about the heat transfer enhancement in a thermal storage

system consisting of vertically arranged fins between a heated and cooled horizontal finned-

tube arrangement. The high thermal expansion coefficient and low viscosity of paraffin wax,

at temperature about 50°C, are utilized to induce natural convection in the liquid phase even

at small thickness.

In this study, the heat source and heat sink are two parallel plates separated by vertical

fins. The heated plate was located below the cooled plate so that the liquid layer eventually

becomes unstable. This instability is expected to cause natural convection along the fins and

at the solid-liquid interface, which, in turn, will influence the rate of production of liquid.

The thermal storage device, figure 2.34, is 53.5 mm high, 61.5 mm long and 56 mm

wide. The device is divided by partitions into three equal sections along its length. The

middle cell is used for the experimental study. The side compartments serve as insulation

along both sides of the middle cell.

Figure 2.34. Schematic of the test cell

(Eftekhar et al., 1984).

��

Diffused back lighting and a camera with a long focal lens are used to photographically

record the liquid-solid interface. The enlarged photographs of the liquid-solid-interface were

digitized and scaled down to their proper coordinates. The photographs and digitized location

of the interface for Tu=35.7°C and Tb=53°C are presented in figures. 2.35 and 2.36.

Figure 2.35. Photographs of the liquid-solid interface


Figure 2.36. Digitized location of the interface


��

Eftekhar et al. calculate the heat transfer coefficient at the interface from experimentally

determined instantaneous position of the moving boundary. They obtained a correlation of

experimental data by including the contribution of the surface area of the interface, fs, and the

influence of conduction on the solid phase, fc. The value of fc2/3 fs Nue is plotted as a function

of Ra/Ste in figure 2.37 and can be expressed as

� � 311 /027.0 SteRafNu S

�� . (2.5)

Figure 2.37. Variation of fsNu as a function of Ra/Ste


Inaba et al. (2003) presents a numerical study that dealt with the enhancement of latent

heat release by using plate type fins mounted on the vertical cooling surface in the

rectangular vessel packed with molten salt as a latent heat storage material. The nitric type

molten salt having high melting point of over 140 ºC was selected as a latent heat storage

material for drying as well as hot water supply. The nitric molten salt has many advantages

such as greater latent heat, higher thermal conductivity and lower viscosity than the paraffin

��

wax. Inaba et al. (2003) found that the fin thickness and pitch exerted an influence on

solidification heat transfer in a liquid layer of a nitric molten salt. The numerical results

elucidated the flow pattern, velocity profile and heat transfer rate in the melted liquid layer.

Figure 2.38 shows the schematic of the latent storage vessel used by Inaba et al. (2003).

Rectangular tubes were arranged vertically in the heat storage vessel and the plate fins were

installed horizontally between those tubes. The rectangular cavity, that was surrounded by

the heating surface of the tubes and the plate fins, was filled with nitric molten salt as a latent

heat storage material.

Figure 2.38. Latent heat storage vessel schematic

(Inaba et al., 2003).

The effect of the fin thickness FT on the heat transfer characteristics under the heat

release process was examined for some fin thickness under the condition of fin pitch FP=0.25

and Ste=0.462. The relationship between Q/Qt and dimensionless time period is shown in

figure 2.39a. The value of Q/Qt means the ratio of accumulated heat Q at a given time to total

amount of theoretical heat Qt of the heat storage vessel. The value of Q/Qt decreases with an

increase in FT. Inaba et al. (2003) explains this behavior by the fact that the heat

transportation ability of the attached fins increases by increasing FT. Effect of the fin pitch

and effect of cooling wall temperature on heat transfer characteristics were also presented

(Fig. 2.39b, c).

��

The results obtained by Inaba et al. revealed that the amount of transported heat through

the fins increased with an increase in fin-thickness, and the heat release completion time

period is shortened. Inaba et al. also found that the effect of increasing the cooling area per

unit latent heat storage material on the heat flux at the cooling wall with a decrease in fin

pitch excelled that of the convection heat transfer. As a result, the completion time of heat

release process decreased with a decrease in the fin-pitch.

a b

c

Figure 2.39. Time history of Q/Qt: a) effect of fin thickness,

b) effect of fin pitch, and c) effect of cooling wall temperature

(Inaba et al., 2003).

��

Lamberg and Siren (2003) studied the melting process in a semi-infinite PCM storage

with a thin fin (see Fig. 2.40). The storage is 2-dimensional and it is semi-infinite both in the

x-direction (0 � x %&) and y-direction (0 � y %&) and the length of the fin approaches infinity.

The end-wall with a constant temperature and the fin act as heat sources in the melting

process.

Lamberg and Siren present a simplified analytical model based on a quasi-linear,

transient, thin-fin equation which predicts the solid–liquid interface location and temperature

distribution of the fin in the melting process with a constant imposed end-wall temperature.

The analytical results were compared to the numerical results and they show good agreement.

Figure 2.40. Semi-infinite phase change material storage with a fin

(Lamberg and Siren, 2003).

According to Lamberg and Siren (2003), melting occurs in two different regions, shown

in figure 2.40. In region 1, the only heat source is the constant temperature end-wall. Here the

fin does not influence the melting process. Heat transferred from the wall is first melting the

phase change material by conduction and later by natural convection.

In region 2, both the wall and the fin are transferring heat to the phase change material.

There are three stages in the melting process: pure conduction from the constant temperature

end-wall and the fin, conduction from the fin with some natural convection from the end-wall

and finally, only natural convection from the fin. Lamberg and Siren assumed that for Ra '

1708, natural convection dominates the heat transfer from the horizontal fin to the solid–

liquid interface. The fin tends to decrease natural convection from the end-wall due to the

decreasing temperature gradient in the liquid. After a short period the fin plays the most

important role in the heat transfer in region 2.

��

Lamberg and Siren compared the predictions of their analytical model with numerical

results in three different cases to predict the influence of the basic heat transfer modes on the

results. The first case was conduction plus natural convection from the fin, case 2 was pure

conduction from the fin, and case 3 was for solid material with a fin without phase change.

They also compared their model with exact Neumann solution. The results are presented in

figure 2.41.

Figure 2.41. The results of case studies 1–3 and the analytical solution when t=3600 s

in a semi-infinite n-octadecane storage, Tw–Tm=20°C

(Lamberg and Siren, 2003).

In case 1, where heat transfers from the fin to the solid–liquid interface by conduction

when Ra � 1708 and with natural convection when Ra>1708, the solid–liquid interface Sy is

slightly ahead of the interface of the analytical solution. At small Sy values the heat transfer

coefficient is relatively large. That is the reason for the solid–liquid interface location being

slightly ahead of the analytical solution.

In case 2, where heat transfers from the fin to the solid–liquid interface by pure

conduction, the interface location Sy is much behind the analytical solution. Natural

convection enhances heat transfer and accelerates melting. Therefore, it should be taken into

consideration. Otherwise the model considerably underestimates the solid–liquid interface

location.

In case 3, the heat from the fin and the constant temperature end-wall is conducted inside

the material without phase change. It is obvious that the initial temperature interface in which

the material’s temperature differs from its initial temperature is situated substantially ahead

of the solid–liquid interface of the analytical solution because there is no effect of latent heat

of fusion.

��

Table 2.2. Summary of reviewed literature. Author Title Model Study Boundary

conditions Material Solution

N. W. Hale, Jr., R. Viskanta

1980

Solid-liquid phase change heat transfer

and interface motion in materials

cooled or heated from above or

below

rectangular enclosure

analytical and experimental

Melting/solid fraction from a horizontal plate

(heat source/sink)

n-octadecane steoric aced

sodium phosphate

dodecahidrate sodium sulfat decahidrate

1-dimentional analytical solution of Neumann

C. J. Ho, R. Viskanta

1984

Heat transfer during melting from an

isothermal vertical wall

rectangular cavity

numerical and experimental

isothermal vertical wall

other sidewalls insulated

n-octadecane 2-dimentional Landau

coordinate transformation

A. Gadgil, D. Gobin

1984

Analysis of two-dimensional melting

in rectangular enclosures in presence of convection

rectangular enclosures

numerical top and bottom adiabatic, top with

the air gap, one wall in high

temperature, other wall in temperature

of fusion

n-octadecane quasi-static steps

C. Gau, R. Viskanta

1986

Melting and solidification of a pure metal on a

vertical wall

rectangular test cell

experimental two walls – heat source or sink,

other walls insulated

Gallium

Z. Zhang, A. Bejan

1989

Melting in an enclosure heated at

constant rate

rectangular enclosures

analytical and experimental

one wall in high temperature, other walls insulated

n-octadecane “matched boundary layer”

solution

��

Tab

le 2

.2. S

umm

ary

of r

evie

wed

lite

ratu

re (

cont

inue

).

Aut

hor

Tit

le

Mod

el

Stu

dy

Bou

ndar

y co

ndit

ions

M

ater

ial

Sol

utio

n

C.J

. Ho,

C

.H. C

hu

1993

Per

iodi

c m

elti

ng

wit

hin

a sq

uare

en

clos

ure

wit

h an

os

cill

ator

y te

mpe

ratu

re

squa

re

encl

osur

e nu

mer

ical

os

cill

ator

y ho

t w

all,

othe

r w

all i

n lo

w te

mpe

ratu

re,

top

and

bott

om

adia

bati

c

tin

enth

alpy

met

hod

C.J

. Ho,

C

.H. C

hu

1994

A s

imul

atio

n fo

r m

ulti

ple

mov

ing

boun

dari

es d

urin

g m

elti

ng in

side

an

encl

osur

e im

pose

d w

ith

cycl

ic w

all

tem

pera

ture

vert

ical

sq

uare

en

clos

ure

num

eric

al

Osc

illa

tory

larg

e am

plit

ude

tem

pera

ture

hot

w

all,

othe

r w

all i

n lo

w te

mpe

ratu

re,

top

and

bott

om

adia

bati

c

n-oc

tade

cane

en

thal

py m

etho

d

G. C

asan

o,

S. P

iva

2002

Exp

erim

enta

l and

nu

mer

ical

in

vest

igat

ion

of th

e st

eady

per

iodi

c so

lid-

liqu

id p

hase

-ch

ange

hea

t tra

nsfe

r

poly

carb

onat

e ro

und

duct

co

mpu

tati

onal

an

d ex

peri

men

tal

Bas

e is

kep

t in

low

er te

mpe

ratu

re,

Per

iodi

c he

ater

in

the

top,

W

alls

insu

late

d

n-oc

tade

cane

1-

dim

enti

onal

he

at c

ondu

ctio

n

J. E

ftek

har,

A

. Haj

i-S

heik

h,

D. Y

. S. L

ou

1984

Hea

t tra

nsfe

r en

hanc

emen

t in

a pa

raff

in w

ax

ther

mal

sto

rage

sy

stem

rect

angu

lar

encl

osur

e se

para

ted

by

vert

ical

fin

s

expe

rim

enta

l T

he h

eate

d pl

ate

in

the

bott

om, c

oole

d pl

ate

in th

e to

p.

Wal

ls a

re in

sula

ted.

para

ffin

wax

(S

UN

TE

CH

P

116)

��

Table 2.2. Summary of reviewed literature (continue). Author Title Model Study Boundary

conditions Material Solution

H. Inaba, K. Matsuo, A. Horibe

2003

Numerical simulation for fin

effect of a rectangular latent heat storage vessel packed with molten

salt under heat release process

Rectangular vertical tubes and the plate

fins were installed

horizontally between those

tubes.

numerical Cooling wall kept at the constant temperature. Other walls were

insulated

nitric molten salt

2-dimensional numerical solution

P. Lamberg, K. Siren

2003

Analytical model for melting in a

semi-infinite PCM storage with an

internal fin

semi-infinite PCM storage with a thin fin

analytical and numerical

The end-wall with a constant

temperature and the fin act as heat

sources

n-octadecane 2-dimensional analytical solution

��

3. Numerical study


phase-change material (PCM) in a heat storage unit with internal partitions or fins. The

physical model is presented in detail below.

3.1 Physical model

A schematic view of the physical model is shown in Fig. 3.1. The conducting

partitions or plate-type fins are attached to the horizontal bottom that acts as a heat source

or sink, which temperature, Tw, is uniform. The PCM fills the space above the bottom

between the partitions. From above, the PCM is exposed to the ambient air. Thus, it is

assumed that heat is transferred between the PCM and the bottom, the partitions, and the

ambient air. Heat transfer between the fin tips and the ambient is neglected.

Figure 3.1. Physical model.

l f

lb�

l t Air�

�

Tw

PC

M�

fin�

�

��

The height of the partitions is larger than the height of the solid PCM, which fills

85% of the space. This is done in order to allow the PCM to expand during the solid-

liquid phase transition, reflecting the large difference in solid and liquid density which

exists in reality.

The geometrical dimensions explored in the present study are summarized in

Table 3.1.

Table 3.1. Geometry parameters.

Case l f, mm l t, mm lb, mm

1 200 2 10

2 10 1.2 4

3 10 0.6 2

4 10 0.3 1

5 10 0.15 0.5

6 5 0.6 2

7 10 0.6 2

The partitions are made of aluminum, in order to ensure their high thermal

conductivity. The properties of aluminum and of the air are summarized in Table 3.2.

Note that the density-temperature relation is used for air.

Table 3.2. Properties of aluminum and air used for computation.

Materials Thermal

conductivity (W/m K)

Density (kg/m3)

Specific heat (J/kg K)

Aluminum 202.4 2719 871

Air 0.0242 498.301134.0102.1 25 �� TT 1006.43

In all the cases explored in this study, the model PCM was based on the properties of

commercially available paraffin wax (Rubitherm RT 25), as discussed below.

��

Preliminary simulations. The first simulations of phase-change processes were

performed for the unit with partitions 200 mm high and 2 mm thick (case 1, table 3.1).

The space between two adjacent fins was 10 mm.

PCM melting has been studied under the following conditions: The initial

temperature of the whole system was 20ºC, i.e. the PCM was slightly subcooled. At t=0,

the temperature of the bottom was changed to 42ºC, while the ambient air above the unit

was 27ºC.

PCM solidification has been studied under the following conditions: The initial

temperature of the whole system was 27ºC, i.e. the PCM was slightly overheated. At t=0,


was 27ºC.

The properties of the PCM were assumed temperature-independent. Thus, the only

change in its density was between the solid and liquid states. The properties of the PCM

for this case are provided in Table 3.3.

Table 3.3. Properties of the PCM used for computation.

Density (kg/m3)

Material Melting point (ºC)

Latent heat

(kJ/kg) Solid Liquid

Thermal conductivity

(W/m K)

Specific heat

(J/kg K)

Dynamic viscosity (kg m/s)

RT 25 23-25 206 800 750 0.15 2500 0.0072

PCM melting has been studied under the following conditions: The initial

temperature of the whole system was 20ºC, i.e. the PCM was slightly subcooled. At t=0,


was 27ºC.

PCM solidification has been studied under the following conditions: The initial

temperature of the whole system was 27ºC, i.e. the PCM was slightly overheated. At t=0,


was 27ºC.

The results of the preliminary simulations are presented in Section 4.

��

Detailed study. For a detailed study of phase-change with internal fins, we have

decided to choose a relatively small system, with the maximum fin length of 10mm. Such

system could serve, for example, as a PCM-based heat sink for electronic equipment.

The following parameter variations were chosen, as reflected in Table 3.1

(cases 2-6):

1). The fin thickness, which varied from 0.15 mm to 1.2 mm;

2). The PCM layer thickness between the fins, which varied from 0.5mm to 4mm.

The thickness ratio of the fin to the PCM layer was kept constant, so that the amount

of the PCM and aluminum in the sink were constant for a constant fin length;

3). In addition to the basic fin length of 10mm, shorter fins of 5mm have been

explored. The amount (height) of the PCM was reduced accordingly. However, it is

assumed that a heat sink can be built of two “stories,” making the amount of the PCM

and aluminum the same as for the basic fin length;

4). For each of cases 2-6 of Table 3.1, calculations were performed for four

differences between the temperature of the bottom and mean melting temperature of the

PCM, namely 6ºC, 12ºC, 18ºC and 24ºC. In the simulations, this difference was set at t=0

and kept constant through the entire process.

For the melting simulations, the initial temperature of the whole system was 20 ºC,

i.e. the PCM was slightly subcooled. The ambient air above the unit was kept at 27 ºC.

In the detailed study, density and dynamic viscosity of the PCM at liquid state

depend on temperature changes. The density is expressed as

� � 1��

f

f

TT�

�� (3.1),

were �f is the density of PCM at temperature of the melting, Tf is the melting

temperature, and � is the thermal expansion coefficient. The value of �=0.001 has been

chosen based on the analysis of the detailed data presented by Humphries and Griggs

(1977).

Following Reid et al. (1987), the dynamic viscosity of the liquid PCM has been

expressed as

��

��

��

��

T

BAexp( (3.2),

were A= - 4.25 and B = 1790 are coefficients. The properties of the PCM are summarized

in Table 3.4.

Table 3.4. Properties of PCM used in the detailed study, cases 2–6.

Periodic phase-change process. Finally, periodic phase-change processes were

simulated. Here, each fin was 0.6 mm wide and 10 mm long (case 7, Table 3.1) and the

space between the fins was 2 mm, i.e. the geometry was identical to that of case 3. The

material properties are presented in Tables 3.2 and 3.3.

The bottom of the unit was subjected to oscillatory temperature, Tw, according to the

following equation:

��

��

� �� #

taTT mw

2sin (3.3)

where a is the amplitude, i.e. the difference between the maximum wall temperature and

the mean melting temperature, Tm, of the PCM, is the period, t is the time elapsed from

the beginning of the process, and ��is the phase shift. The phase shift is used to account

for the fact that the initial temperature of the whole system was 20 ºC, i.e. below the

melting temperature, and thus the periodic process starts with the wall temperature

Tw=20ºC, as well.

The simulations were performed for two different temperature amplitudes, a=12ºC

and 18ºC. For the first amplitude, the periods used were =150s and 180s, while for

second amplitude, the periods were 120s and 150s. This change reflected the more rapid

processes encountered with the larger amplitude.

Material Melting point (ºC)

Latent heat

(kJ/kg)

Density (liquid state)

(kg/m3)

Thermal conductivity

(W/m K)

Specific heat

(J/kg K)

Dynamic viscosity (g m/s)

RT 25 23-25 206 � � 1296001.0

750

��T 0.15 2500 �

�

��

� ��T

179025.4exp

��

3.2 Computational procedure

Based on the symmetry of the system, the computational domain was defined by the

physical boundaries of the unit in the vertical y-direction, and by the symmetry planes of

the partition/fin and the PCM-filled channel in the horizontal x-direction, as shown in

figure 3.2. The origin of the coordinate system was taken at the intersection of the plane

of symmetry of the partition/fin and the lower boundary of the partitions and PCM. It was

assumed that the unit is both sufficiently long and its boundaries are well insulated in the

z-direction, so that a two-dimensional formulation has been used at this stage.

Figure 3.2. Computational model.

For the large system, case 1 of Table 3.1, a computational grid of 24�100=2400 cells

was used in the simulations. It included 4�100=400 cells in the 1mm thick and 200mm

long half-partition (0<x<1mm, 0<y<200mm), 20�95=1900 cells in the 5mm thick and

190mm long half channel filled by the PCM (1<x<6mm, 0<y<190mm), and 20�5=100

yAir�

PC

M�

Tw

� x

��

cells in the air above it (1<x<6mm, 190<y<200mm).

For the small system, cases 2-7 of Table 3.1, the computational grid was built of

13�100=1300 cells. This number was kept constant, while the size of the system itself

varied.

The numerical calculations were performed for the transient temperature and

velocity fields inside the unit, including both the PCM and the fins. The equations for the

air were solved only in the small domain bounded by the fin from the left, symmetry

plane from the right, PCM from below and the plane connecting the fin tips from above.

The basic conservation equations of continuity, momentum, and energy were solved

numerically, using the FLUENT 6.0 software. Laminar flow inside air and liquid PCM

was assumed. In order to describe the behavior of the PCM, a so-called “volume-of-

fluid” model has been activated. The model made it possible to calculate the processes

that occur inside the partitions (solid), PCM (solid/liquid), and air (fluid) simultaneously.

In addition, density changes in the PCM and the effect of gravity were taken into account.

The model makes it possible to account for the moving boundary due to the variation of

the PCM volume. As a result, the numerical model was rather close to reality.

For the mushy region, FLUENT applies the enthalpy-porosity approach, by which

the porosity in each cell is set equal to the liquid fraction in that cell. Accordingly, the

porosity is zero inside fully solid regions.

The governing equations for the solidification/melting problem are:

The continuity equation:

� � 0�))

ii

ux

� (3.4)

The momentum equations:

� � � � iijj

iij

ji S

x

p

xx

uuu

xu

t�

))

�))

)�

))

�)) 2

(�� (3.5)

The energy equation:

� � � � hii

ii

Sx

Tk

xu

xt��

�

��

�

))

))

�))

�))

�� (3.6)

��

where � is the density, k is the thermal conductivity, ( is the dynamic viscosity, Si and Sh

are the source terms, ui is the velocity component in the i - direction, xi is a Cartesian

coordinate, and � is the specific enthalpy.

The enthalpy is defined as a sum of the sensible enthalpy, �

s, and the enthalpy

change due to the phase change � �

. � =

�s +

� � (3.7)

The sensible enthalpy is defined as

dTcT

Tprefs

ref

�� (3.8)

where �

ref is the reference enthalpy at the reference temperature Tref, and cp is the specific

heat.

The enthalpy change due to the phase change is defined as

L*�� (3.9)

where L is the latent heat of the material, or specific enthalpy of fusion, and * is the liquid

fraction during the phase change which occurs over a range of temperatures, defined by

the following relations:

* = 0 if T < Ts

* = 1 if T > Tl

sl

s

TT

TT

�

��* if Ts < T < Tl (3.10)

Accordingly, the enthalpy change due to the phase change varies from zero for a solid to

L for a liquid.

The source term for energy equation is:

tLSh )

)�

*� (3.11)

The liquid fraction is also used to drive the velocity components to zero in the solid phase

of the PCM via the source terms Sx and Sy in the momentum equation (3.5) for the x- and

y-direction, respectively:

� � xx uBS *�� (3.12)

� � � �myy TTguBS �� *�* (3.13)

��

The function B becomes very large when * is zero and goes to zero as * tends to one. It is

defined based on the Carman-Kozeny relation for a porous medium as:

� � � �"**

*�

��

3

21CB (3.14)

with C = 1.6·106 and " = 10-3. This numerical artifact can be viewed as a way of modeling

the transition zone between the solid and liquid phases.

FLUENT solves for the temperature by iterations involving the energy equation

(3.6), and the liquid fraction relation, equation (3.10), using the approach of Voller and

Swaminathan (1991), who propose methods in which the phase change rate is linearized

as a truncated Taylor series, and old iteration values are then used to estimate the linear

term.

The common continuity, momentum, and energy equations were solved for the air in

the small domain above the PCM. Heat conduction equation was solved for the fin.

��

4. Results

The results presented here include the preliminary study of a partitioned storage unit,

and the detailed parametric study of a heat sink with internal fins. The results for periodic

phase change are presented in Appendix A1.

The preliminary study served as a basis for the detailed study performed later. For

this reason, its results are discussed here. The results of the detailed parametric study will

be analyzed in Section 5.

4.1 Preliminary results

As listed in Section 3.1, in the preliminary study the simulations of phase-change

processes were performed for the unit with conducting partitions 200 mm high and 2 mm

thick. The space between two adjacent partitions, filled with PCM, was 10 mm.

Melting. Initially, the whole system was at 20ºC. At t=0, the temperature of the

bottom was changed to 42ºC, while the ambient air above the unit was kept at 27ºC.

The results of simulations of the melting are shown in figures 4.1-4.4.

Figure 4.1 shows the results of melting simulations in the form of phase

distributions. The left picture represents the initial phase distribution, in which the PCM

was entirely solid (blue). The only fluid here is the air above the PCM (red). Since the

software can represent a solid-fluid distribution, it was impossible to show the liquid

PCM and the air by different colors, and they both are red in all other parts of figure 4.1.

Fortunately, air is above the PCM, while melting occurs mostly from below, therefore

such blue-red scale is quite understandable.

One can see from figure 4.1 that melting is initiated not only at the hot bottom of the

unit, but also at the partitions, due to their high thermal conductivity. As a result, the solid

PCM attains the shape of a downward-directed wedge.

It can also be seen from figure 4.1 that the upper boundary of the PCM is moving

upward with time, reflecting the increasing volume of liquid PCM in the course of

��

melting. While the PCM expands between the partitions, the air above it is pushed

outside.

0s 300s 600s 900s 1200s 1500s 1800s

Figure 4.1. Phase distribution of PCM during melting as a function of time.

Figure 4.2 shows the calculated heat transfer rates from the base to the PCM, from

the base to the partition, and from the partition to the PCM and air. The rates given are

x

y

0

��

for the computational domain introduced above. One can see from the figure that at each

time instant, the heat transfer rate from the base to the partition is much higher than that

going directly to the PCM.

0

50

100

150

200

250

0 500 1000 1500 2000

Time, s

Hea

t tra

nsfe

r ra

te, W

base to PCMpartition to PCM and airbase to partition

Figure 4.2. Heat transfer rate vs. time for PCM and partition during melting.

Results presented in figure 4.2 show that since the major part of the heat transferred

to the partition eventually reaches the PCM, the latter is heated mostly from the sides and

not from below. Moreover, the more time elapses, the thicker is the layer of liquid PCM

between the base and the wedge, as seen also in figure 4.1. Since this layer is a thermal

resistance for the melting process, the direct heat transfer from the base to the PCM

eventually becomes negligible.

Figure 4.2 also shows that the heat transfer rate from the base to the partition also

decreases with time, because of the decreasing temperature gradient. As the partition is

heated and the solid PCM “retreats,” the resistance to heat transfer increases.

The difference between the heat that enters the partition and the heat that leaves it is

due to the transient heating of the partition (increase in its internal energy). This

difference obviously decreases with time.

��

Further details of the melting are presented in figures 4.3 and 4.4. Figure 4.3 shows

the temperature inside the partition (at x=0) at various distances from the base

(0<y<200mm), for the same time instants, which have been represented by their heat

transfer rates in figure 4.2. One can clearly see that the temperature at any point inside the

partition increases with time. The distribution tends asymptotically to a steady state in

which heat input from the base would be balanced by heat output to the air, while the

PCM is completely liquid. One can see from the figure that under given conditions this

would take a significant amount of time.

Figure 4.4 shows the temperature at the plane of symmetry of the PCM (at x=6mm)

at various distances from the base (0<y<200mm), for the same time instants that have

been represented by their heat transfer rates in figure 4.2. Close to the upper boundary,

the temperature is that of air above the PCM.

One can clearly see that the temperature at any point inside the liquid PCM grows

with time, while the liquid region itself grows. These results correspond well to the phase

distributions in figure 4.1 and heat transfer rates in figure 4.2.

290

295

300

305

310

315

320

0 0.05 0.1 0.15 0.2

Distance from the base, m

Tem

pera

ture

, K

t=135

t=235

t=300

t=600

t=900

t=1200

t=1500

t=1800

t=2100

Figure 4.3. Temperature evolution in the partition during melting

(time is given in seconds).

�

290

295

300

305

310

315

320

0 0.05 0.1 0.15 0.2

Distance from the base , m

Tem

pera

ture

, K

t=135

t=235

t=300

t=600

t=900

t=1200

t=1500

t=1800

Figure 4.4. Temperature evolution in the PCM during melting

(time is given in seconds).

One can see from the figure that the temperature inside the PCM is growing with

time. The temperature patterns, however, are very different from those obtained for the

solid partition. From the beginning, PCM starts to melt close to the base, but only after it

has been heated from the subcooled state to the melting point temperature. This heating is

represented at various time instants by the curves found below the melting temperature of

23ºC (296K). While the heating of the solid PCM relatively far from the base continues,

melting takes place near the base, followed by heating of the liquid PCM. This process is

represented by a family of similar curves.

Above the PCM there is air heated from above by the ambient and cooled from

below by the PCM. The air is also heated by the partition, which protrudes above the

PCM. In figure 4.4, the region occupied by air can be clearly seen. This region shrinks

with time due to the PCM expansion as could be seen in figure 4.1.

��

Solidification. Initially, the whole system was at 27ºC. At t=0, the temperature of the

bottom was changed to 6ºC, while the ambient air above the unit was 27ºC.

The results of solidification simulations are shown in figures 4.5-4.8.

Figure 4.5 shows the results of solidification simulations in the form of phase

distributions. Once again, it was impossible to show the liquid PCM and the air by

different colors, and they both are red in the figure. Fortunately, solidification occurs

mostly from below, making the blue-red scale quite understandable.

It can be seen from Figure 4.5 that solidification is initiated not only at the cold

bottom of the unit, but also at the partitions, due to their high thermal conductivity.

100s - 600s 900s 1200s 1500s 1800s 2100s

Figure 4.5. Phase distribution of PCM during solidification as a function of time.

x

y

0

��

Figure 4.6 shows the calculated heat transfer rates from the PCM to the base, from

the partition to the base, and from the PCM and air to the partition. The rates given are

for the computational domain introduced above.

Figure 4.6 shows that the heat transfer rate from the partition to the base also

decreases with time, because of the decreasing temperature gradient.

It can be seen from figure 4.6 that at each time instant, the heat transfer rate from the

partition to the base is much higher than that coming directly from the PCM. Thus, the

latter is cooled mostly from the sides and not from below. Moreover, the more time

elapses, the thicker is the layer of solid PCM between the base and the wedge, see also

figure 4.5. Since this layer is a thermal resistance for the solidification process, the direct

heat transfer from the PCM to the base eventually becomes negligible.

-120

-100

-80

-60

-40

-20

0

0 500 1000 1500 2000 2500

Time, s

Hea

t tra

nsfe

r ra

te, W

partition to base

PCM to base

PCM and air to partition

Figure 4.6. Heat transfer rate vs. time for PCM and partition during solidification.

Further details of the solidification process are presented in figures 4.7 and 4.8.

Figure 4.7 shows the temperature inside the partition (at x=0) at various distances from

the base (0<y<200mm), for the same time instants, which have been represented by their

heat transfer rates in Figure 4.6. One can clearly see that the temperature at any point

inside the partition decreases with time. The distribution tends asymptotically to a steady

state in which heat output to the base would be balanced by heat input from the air, while

��

the PCM is completely solid. Like in melting, under given conditions it would take a

significant amount of time to reach that state.

275

280

285

290

295

300

305

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2Distance from the base, m

Tem

pera

ture

, K

t=100

t=200

t=300

t=600

t=900

t=1200

Figure 4.7. Temperature evolution in the partition during solidification (time is given in seconds).

275

280

285

290

295

300

305

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2


Tem

pera

ture

, K

t=100

t=200

t=300

t=600

t=900

t=1200

t=1500

t=1800

t=2100

Figure 4.8. Temperature evolution in the PCM during solidification (time is given in seconds).

��

Figure 4.8 shows the temperature at the plane of symmetry of the PCM (at x=6mm)

at various distances from the base (0<y<200mm), for the same time instants that have

been represented by their phase distributions in figure 4.5 and heat transfer rates in

figure 4.6.

From the beginning, PCM starts to solidify close to the base, after it has been cooled

to the temperature of solidification. This process is represented by a family of similar

curves, which correspond to different time instants. One can clearly see that the

temperature at any point inside the solid PCM decreases with time, while the solid region

itself grows. These results correspond well to the phase distributions in figure 4.5 and

heat transfer rates in figure 4.6.

The preliminary study presented above provides important information on the

evolution of phase-change processes in an energy storage system with internal partitions.

In particular, we obtained phase distributions and heat transfer rates for various time

instants, and the instantaneous temperature distributions inside the partition and the PCM.

It has been shown that the heat is transferred to the PCM mostly through the

conducting partitions, while the heat transfer between the heated bottom and the PCM

was relatively small and decreases with time. This influenced the appearance of the solid

phase in melting, giving it a wedge-like shape.

It turned out that under given conditions it would take a large amount of

computational time to obtain completely melted or solidified PCM even in one case. As

we planned a parametric study where various geometrical and thermal parameters of the

system had to be changed, it has been decided to switch to a similar but smaller system.

Its geometrical parameters were chosen based on a possible application of the system as a

heat sink for electronics cooling.

The results of a detailed study for that small system are presented in Section 4.2 and

analyzed in Section 5.

��

4.2 Detailed study

As discussed in Section 3.1, an extensive parametric investigation of the effect of

internal fins on PCM melting was carried out for five different geometries, listed as cases

2-6 in Table 3.1. Recall that the parameters varied in the study included the fin thickness,

l t, PCM thickness, lb, and fin length, l f :

- case 2: l t = 1.2mm, lb = 4mm, l f = 10mm;



- case 5: l t = 0.15mm, lb = 0.5mm, l f = 10mm;

- case 6: l t = 0.6mm, lb = 2mm, l f = 5mm.

For all five cases, four different base temperatures were explored, yielding

differences of �

T = 6ºC, 12ºC, 18ºC, and 24ºC between the base temperature and the

PCM melting temperature.

Heat fluxes. Figure 4.9 represents the calculated heat flux, q´´, as a function of time,

for case 2. Four curves in the figure correspond to four temperature differences explored.

Figures 4.10-4.13 do the same for cases 3-6, respectively.

Figure 4.14 represents the calculated heat flux, q´´, as a function of time, for the

temperature difference of 6ºC. Five curves in the figure correspond to the five

geometrical variants explored, i.e. cases 2-6. Figures 4.15-4.17 do the same for the �

T =12ºC, 18ºC, and 24ºC, respectively.

0

50

100

150

200

0 20 40 60 80 100 120 140

time, s

�T = 6°C

�T = 12 °C

�T = 18 °C

�T = 24 °C

Figure 4.9. Total heat flux from the base, case 2.

q", k

W/m

2

�

0

50

100

150

200

250

300

350

0 20 40 60 80 100 120 140

time, s

�T = 6°C

�T = 12 °C

�T = 18 °C

�T = 24 °C


0

50

100

150

200

250

300

350

0 20 40 60 80 100 120 140

time, s

�T = 6°C

�T = 12 °C

�T = 18 °C

�T = 24 °C


q", k

W/m

2 q"

, kW

/m2

�

0

50

100

150

200

250

300

350

0 20 40 60 80 100 120 140

time, s

�T = 6°C

�T = 12 °C

�T = 18 °C

�T = 24 °C


0

50

100

150

200

250

300

350

0 20 40 60 80 100 120 140

time, s

�T = 6 °C

�

T= 12 °C�

T = 18 °C�

T = 24 °C


q", k

W/m

2 q"

, kW

/m2

��

0

50

100

150

200

250

300

350

0 20 40 60 80 100 120 140

time, s

case 2

case 3

case 4

case 5

case 6


T = 6 °C.

0

50

100

150

200

250

300

350

0 20 40 60 80 100 120 140

time, s

case 2

case 3

case 4

case 5

case 6


T = 12 °C.

q", k

W/m

2 q"

, kW

/m2

��

0

50

100

150

200

250

300

350

0 20 40 60 80 100 120 140

time, s

case 2

case 3

case 4

case 5

case 6


T = 18 °C.

0

50

100

150

200

250

300

350

0 20 40 60 80 100 120 140

time, s

case 2

case 3

case 4

case 5

case 6


T = 24 °C.

q", k

W/m

2 q"

, kW

/m2

�

It is important to note that all the figures of this section have exactly the same time

scales, namely, from 0 to 140 seconds, making it possible to see the differences between

various cases even before a detailed analysis is done.

Melt fractions. Figure 4.18 represents the calculated melt fraction, V/V0, as a function

of time, for case 2. Four curves in the figure correspond to four temperature differences

explored. Figures 4.19-4.22 do the same for cases 3-6, respectively.

Figure 4.23 represents the calculated melt fraction, V/V0, as a function of time, for the

temperature difference of 6ºC. Five curves in the figure correspond to the five

geometrical variants explored, i.e. cases 2-6. Figures 4.24-4.26 do the same for the �


0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100 120 140

time, s

Mel

t fra

ctio

n

�T = 6 °C

�T = 12 °C

�T = 18 °C

�T = 24 °C

Figure 4.18. Melt fraction, case 2.

��

0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100 120 140

time, s

Mel

t fra

ctio

n

�T = 6 °C

�T = 12 °C

�T = 18 °C

�T = 24 °C


0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100 120 140

time, s

Mel

t fra

ctio

n

�T = 6 °C

�T = 12 °C

�T = 18 °C

�T = 24 °C


��

0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100 120 140

time, s

Mel

t fra

ctio

n

�T = 6 °C

�T = 12 °C

�T = 18 °C

�T = 24 °C


0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100 120 140

time, s

Mel

t fra

ctio

n

�T = 6 °C

�T = 12 °C

�T = 18 °C

�T = 24 °C


��

0

0.2

0.4

0.6

0.8

1

1.2

0 20 40 60 80 100 120 140

t [sec]

Mel

t fra

ctio

n

case 3

case 4

case 5

case 6

case 7


T = 6 °C.

0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100 120 140

time, s

Mel

t fra

ctio

n case 2

case 3

case 4

case 5

case 6


T = 12 °C.

��

0

0.2

0.4

0.6

0.8

1

1.2

0 20 40 60 80 100 120 140

t [sec]

Mel

t fra

ctio

n

case 3

case 4

case 5

case 6

case 7


T = 18 °C.

0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100 120 140

time, s

Mel

t fra

ctio

n case 2

case 3

case 4

case 5

case 6


T = 24 °C.

��

Temperatures. Figure 4.27 represents the calculated averaged mean temperature in

the plane of symmetry of the PCM, as a function of time, for case 2. Four curves in the

figure correspond to four temperature differences explored. Figures 4.28-4.31 do the

same for cases 3-6, respectively.

Figure 4.32 represents the calculated averaged mean temperature in the plane of

symmetry of the fin as a function of time, for the temperature difference of 6ºC. Five

curves in the figure correspond to the five geometrical variants explored, i.e. cases 2-6.

Figures 4.33-4.35 do the same for the �


290

295

300

305

310

315

320

325

0 20 40 60 80 100 120 140

time, s

T, K

�T = 6°C

�T = 12°C

�T = 18°C

�T = 24°C

Figure 4.27. Temperature distribution on the symmetry line of PCM, case 2.

�

290

295

300

305

310

315

320

325

0 20 40 60 80 100 120 140

time, s

T, K

�T = 6°C

�T = 12°C

�T = 18°C

�T = 24°C


290

295

300

305

310

315

320

325

0 20 40 60 80 100 120 140

time, s

T, K

�T = 6°C

�T = 12°C

�T = 18°C

�T = 24°C


�

290

295

300

305

310

315

320

325

0 20 40 60 80 100 120 140

time, s

T, K

�T = 6°C

�T = 12°C

�T = 18°C

�T = 24°C


290

295

300

305

310

315

320

325

0 20 40 60 80 100 120 140

time, s

T, K

�T = 6°C

12�

T = 18°C�

T = 24°C


��

290

295

300

305

310

315

320

325

0 20 40 60 80 100 120 140

time, s

T, K

case 2

case 3

case 4

case 5

case 6


T= 6 °C.

290

295

300

305

310

315

320

325

0 20 40 60 80 100 120 140

time, s

T, K

case 2

case 3

case 4

case 5

case 6


T = 12 °C.

��

290

295

300

305

310

315

320

325

0 20 40 60 80 100 120 140

time, s

T, K

case 2

case 3

case 4

case 5

case 6


T = 18 °C.

290

295

300

305

310

315

320

325

0 20 40 60 80 100 120 140

time, s

T, K

case 2

case 3

case 4

case 5

case 6


T = 24 °C.

��

5. Analysis of results

In this section, the results of the parametric study are discussed and analyzed. The

discussion concerns the calculated heat fluxes and heat transfer rates, melt fractions of the

PCM, and temperature evolution in the PCM and the fins. The analysis is performed in terms

of dimensionless groups.

5.1 Heat-transfer rates

Dependence of the heat fluxes on the geometrical and thermal parameters of the system

can be seen in figures 4.9-4.17. Recall that figure 4.9 represents the calculated heat flux, q´´, as

a function of time, for case 2 of Table 3.1. As expected, the heat flux transferred to the PCM is

maximal at the beginning and diminishes to zero when the melting is complete. One can see

that the higher the difference between the temperature of the bottom and the melting point of

the PCM, the shorter the melting time. As for the heat flux, it is initially higher for the larger

temperature differences T. However, for the larger T the heat flux decreases steeper with

time, because it corresponds to a more rapid melting process. Similar results appear also in

figures 4.10-4.13 for cases 3-6.

Figure 4.14 represents the calculated heat flux, q´´, as a function of time, for cases 2-6 of

Table 3.1 at T = 6°C. One can see from the figure that as the width of the system decreases

while its height is preserved, cases 2-5, the heat flux from the base increases. For the “ low”

system, case 6, the heat flux is lower. However, as mentioned in Section 3.1, the “ low” system

may be built in two levels, thus occupying the same volume as the “high” one. A comparison

of the results for cases 3 and 6, which have the same fin and PCM thickness, show that the

melting time will be much shorter in the two-level “ low” system than in the “high” one, for the

same mass and volume of the heat sink. Similar results appear also in figures 4.15-4.17 for T

= 12°C, 18°C, and 24°C.

As shown in the preliminary study, see Section 4.1, in the large system heat was

transferred to the PCM mostly through the conducting partitions, while the heat transfer

directly from the heated bottom was less significant. For the small system, this problem is

analyzed in figures 5.1-5.5 below. Since we’ve decided to show the heat-transfer rates rather

than the heat fluxes, each figure here stands for a specific case (2, 3, 4, 5, or 6) alone.

� �

��

050100

150

200

250

300

020

4060

8010

012

014

0

time,

s

q, W

q fin

q ba

se

050100

150

200

250

300

020

4060

8010

012

014

0

time,

s

q, W

q fin

q ba

se

a.

b.

050100

150

200

250

300

020

4060

8010

012

014

0

time,

s

q, W

q fin

q ba

se

050100

150

200

250

300

020

4060

8010

012

014

0

time,

sq, W

q fin

q ba

se

c.

d.

Fig

ure

5.1.

Hea

t tra

nsfe

r ra

tes

from

bas

e, c

ase

2:

a.

T =

6 º

C; b

. T

= 1

2 ºC

; c.

T =

18

ºC; d

. T

= 2

4 ºC

.

� �

��

050100

150

200

250

010

2030

4050

6070

time,

s

q, W

q fin

q ba

se

050100

150

200

250

010

2030

4050

6070

time,

s

q, W

q fin

q ba

se

a.

b.

050100

150

200

250

010

2030

4050

6070

time,

s

q, W

q fin

q ba

se

050100

150

200

250

010

2030

4050

6070

time,

sq, W

q fin

q ba

se

c.

d.

Fig

ure

5.2.

Hea

t tra

nsfe

r ra

tes

from

bas

e, c

ase

3:

a.

T =

6 º

C; b

. T

= 1

2 ºC

; c.

T =

18

ºC; d

. T

= 2

4 ºC

.

� �

��

050100

150

200

010

2030

4050

6070

time,

s

q, W

q fin

q ba

se

050100

150

200

010

2030

4050

6070

time,

s

q, W

q fin

q ba

se

a.

b.

050100

150

200

010

2030

4050

6070

time,

s

q, W

q fin

q ba

se

050100

150

200

010

2030

4050

6070

time,

sq, W

q fin

q ba

se

c.

d.

Fig

ure

5.3.

Hea

t tra

nsfe

r ra

tes

from

bas

e, c

ase

4:

a.

T =

6 º

C; b

. T

= 1

2 ºC

; c.

T =

18

ºC; d

. T

= 2

4 ºC

.

� �

��

020406080100

120

010

2030

4050

6070

time,

s

q, W

q fin

q ba

se

020406080100

120

010

2030

4050

6070

time,

s

q, W

q fin

q ba

se

a.

b.

020406080100

120

010

2030

4050

6070

time,

s

q, W

q fin

q ba

se

020406080100

120

010

2030

4050

6070

time,

sq, W

q fin

q ba

se

c.

d.

Fig

ure

5.4.

Hea

t tra

nsfe

r ra

tes

from

bas

e, c

ase

5:

a.

T =

6 º

C; b

. T

= 1

2 ºC

; c.

T =

18

ºC; d

. T

= 2

4 ºC

.

� �

��

020406080100

010

2030

4050

6070

time,

s

q, W

q fin

q ba

se

020406080100

010

2030

4050

6070

time,

s

q, W

q fin

q ba

se

a.

b.

020406080100

010

2030

4050

6070

time,

s

q, W

q fin

q ba

se

020406080100

010

2030

4050

6070

time,

sq, W

q fin

q ba

se

c.

d.

Fig

ure

5.5.

Hea

t tra

nsfe

r ra

tes

from

bas

e, c

ase

6:

a.

T =

6 º

C; b

. T

= 1

2 ºC

; c.

T =

18

ºC; d

. T

= 2

4 ºC

.

�

��

One can see from figures 5.1-5.5 that, like in the large system, the heat transfer rates from

the heated base reaches the PCM mostly through the fins. The results, however, are not

identical for the systems having different thickness. When the PCM layer between the fins is

thicker, the part of the total heat, which is transferred from the base directly to the PCM

increases, e.g. compare figure 5.1 for case 2 and figure 5.4 for case 5. This effect would have

been even more pronounced if we had considered two cases with different PCM thickness for

the same fin thickness. As in the preliminary study, as the layer of liquid PCM grows, the

direct heat transfer from the base to PCM becomes negligible.

For the same geometrical case, the difference between the base temperature and the

melting temperature of the PCM also affects the ratio of the heat transfer through the fins to the

heat transfer directly from the bottom. For example, figure 5.1a-d shows for case 2 that the

higher this temperature difference, the larger the part of heat transferred directly to the PCM.

One can see from figures 5.1, 5.2, and 5.5 that the heat transfer rate from the bottom

directly to the PCM does not always decrease with time, while the total heat transfer and the

heat transfer through the fins do decrease monotonically. If the layer of liquid PCM had grown

with time, the resistance to this transfer would have increased. We assume that this does not

happen due to the “sinking” of the denser solid PCM in the melt. This assumption is verified

below, where the melt fractions are analyzed.

5.2 Melt fractions

Dependence of the melt fractions on the geometrical and thermal parameters of the system

can be seen in figures 4.18-4.26.

Figure 4.18 represents the calculated melt fraction, V/V0, as a function of time, for case 2

of Table 3.1. One can see that the higher the difference between the temperature of the bottom

and the melting point of the PCM, the more rapid is the melt fraction growth. Similar results

appear also in figures 4.19-4.22 for cases 3-6.

Figure 4.23 represents the calculated melt fraction, V/V0, as a function of time, for cases 2-

6 of Table 3.1 at T = 6°C. One can see from the figure that as the width of the system

decreases while its height is preserved, cases 2-5, the melt fraction grows more rapidly. For the

“ low” system, case 6, the melt fraction grows more rapidly that for the “high” system with the

same width, case 3, but more slowly than the “high” system with the same mass, case 4.

Similar results appear also in figures 4.24-4.26 for T = 12°C, 18°C, and 24°C.

�

��

In order to illustrate the differences in the melting process for different geometries and base

temperatures, solid-liquid phase distributions are shown for cases 2-6 in figures 5.6–5.10,

respectively. Each figure shows the phase distributions at the instances when the melt fraction

equals 0.3, 0.6, and 0.9, for the temperature differences of (a) 6°C, (b) 12°C, (c) 18°C,

(d) 24°C. In addition, figures 5.6-5.10 include temperature distributions inside the fins for the

same time instances. Accordingly, each figure includes two color scales: one for the melt

fraction of PCM, 0 to 1, and one for the fin temperature, in degrees Kelvin.

One can see from figures 5.6, 5.7, and 5.10 that for the “wide” cases the melting front

moves generally parallel to the vertical fin. Simultaneously, the denser solid PCM “sinks” in

the melt and is partially melted by heat transferred from the bottom. This result corresponds to

the behavior of the heat transfer rate from the bottom in “wide” systems, as discussed above.

As the width decreases, see figures 5.8-5.9, the phase distribution becomes similar to that

observed in the preliminary study, where the wedge-like solid phase is at the top and the liquid

at the bottom.

Figures 5.6-5.10 also show that the temperature distribution in the fins depends on the

width. The fins are almost isothermal when the width is large, see figures 5.6, 5.7, and 5.10.

When the width is small, the temperature difference between the fin base and tip becomes

significant, as shown in figures 5.8, 5.9. This behavior affects the melting process. Further

results on fin temperature are discussed below.

5.3 Temperatures

Dependence of the temperature of the PCM on the geometrical and thermal parameters of

the system can be seen in figures 4.27-4.31.

Figure 4.27 represents the calculated averaged mean temperature in the plane of symmetry

of the PCM, as a function of time, for case 2 of Table 3.1. A typical temperature curve in the

figure reflects the process in the following manner: at the beginning, the temperature increases

rather quickly, corresponding to sensible heating of the subcooled solid PCM. Then, there is a

stage at which the curve is almost horizontal, corresponding to the phase change. Afterwards, a

steep increase in the temperature is observed once again, corresponding to sensible heating of

liquid PCM. Finally, a slowing increase is observed in the PCM temperature as it approached

the temperature of the heated bottom.

� �

��

Mel

t fra

ctio

n V

/V0=

0.3

V

/V0=

0.6

V

/V0=

0.9

Tem

pera

ture

Mel

t fra

ctio

n V

/V0=

0.3

V

/V0=

0.6

V

/V0=

0.9

Tem

pera

ture

a.

b.

Mel

t fra

ctio

n V

/V0=

0.3

V

/V0=

0.6

V

/V0=

0.9

Tem

pera

ture

Mel

t fra

ctio

n V

/V0=

0.3

V

/V0=

0.6

V

/V0=

0.9

Tem

pera

ture

c.

d.

F

igur

e 5.

6. E

volu

tion

of

the

mel

ting

pro

cess

of

the

PC

M a

nd te

mpe

ratu

re d

istr

ibut

ion

in th

e fi

n fo

r ca

se 2

:

a.

T =

6 º

C; b

. T

= 1

2 ºC

; c.

T =

18

ºC; d

. T

= 2

4 ºC

.

fin

PCM

� �

��

M

elt f

ract

ion

V

/V0=

0.3

V/V

0=0.

6 V

/V0=

0.9

Tem

pera

ture

Mel

t fra

ctio

n V

/V0=

0.3

V/V

0=0.

6 V

/V0=

0.9

Tem

pera

ture

a.

b.

M

elt f

ract

ion

V

/V0=

0.3

V/V

0=0.

6 V

/V0=

0.9

Tem

pera

ture

Mel

t fra

ctio

n V

/V0=

0.3

V/V

0=0.

6 V

/V0=

0.9

Tem

pera

ture

c.

d.

F

igur

e 5.

7. E

volu

tion

of

the

mel

ting

pro

cess

of

the

PC

M a

nd te

mpe

ratu

re d

istr

ibut

ion

in th

e fi

n fo

r ca

se 3

:

a.

T =

6 º

C; b

. T

= 1

2 ºC

; c.

T =

18

ºC; d

. T

= 2

4 ºC

.

� �

�

Mel

t fra

ctio

n V

/V0=

0.3

V

/V0=

0.6

V

/V0=

0.9

Tem

pera

ture

Mel

t fra

ctio

n V

/V0=

0.3

V/V

0=0.

6

V

/V0=

0.9

Tem

pera

ture

a.

b.

M

elt f

ract

ion

V/V

0=0.

3

V/V

0=0.

6

V/V

0=0.

9 T

empe

ratu

re M

elt f

ract

ion

V/V

0=0.

3

V

/V0=

0.6

V/V

0=0.

9 T

empe

ratu

re

c

.

d.

Fig

ure

5.8.

Evo

luti

on o

f th

e m

elti

ng p

roce

ss o

f th

e P

CM

and

tem

pera

ture

dis

trib

utio

n in

the

fin

for

case

4:

a.

T =

6 º

C; b

. T

= 1

2 ºC

; c.

T =

18

ºC; d

. T

= 2

4 ºC

.

� �

��

M

elt f

ract

ion

V/V

0=0.

3

V

/V0=

0.6

V/V

0=0.

9 T

empe

ratu

re

Mel

t fra

ctio

n

V/V

0=0.

3

V

/V0=

0.6

V/V

0=0.

9 T

empe

ratu

re

a

.

b.

M

elt f

ract

ion

V/V

0=0.

3

V

/V0=

0.6

V/V

0=0.

9 T

empe

ratu

re

Mel

t fra

ctio

n

V/V

0=0.

3

V

/V0=

0.6

V/V

0=0.

9 T

empe

ratu

re

c

.

d.

Fig

ure5

.9. E

volu

tion

of

the

mel

ting

pro

cess

of

the

PC

M a

nd te

mpe

ratu

re d

istr

ibut

ion

in th

e fi

n fo

r ca

se 5

:

a.

T =

6 º

C; b

. T

= 1

2 ºC

; c.

T =

18

ºC; d

. T

= 2

4 ºC

.

� �

��

M

elt f

ract

ion

V/V

0=0.

3

V/V

0=0.

6

V

/V0=

0.9

Tem

pera

ture

M

elt f

ract

ion

V/V

0=0.

3

V

/V0=

0.6

V/V

0=0.

9 T

empe

ratu

re

a

.

b.

M

elt f

ract

ion

V/V

0=0.

3

V/V

0=0.

6

V

/V0=

0.9

Tem

pera

ture

M

elt f

ract

ion

V/V

0=0.

3

V

/V0=

0.6

V/V

0=0.

9 T

empe

ratu

re

c

.

d.

Fig

ure

5.10

. Evo

luti

on o

f th

e m

elti

ng p

roce

ss o

f th

e P

CM

and

tem

pera

ture

dis

trib

utio

n in

the

fin

for

case

6:

a.

T =

6 º

C; b

. T

= 1

2 ºC

; c.

T =

18

ºC; d

. T

= 2

4 ºC

.

�

��

One can see that the higher the difference between the temperature of the bottom and

the melting point of the PCM, the shorter the time when the PCM is at an almost constant

temperature. It also can be seen that finally, the temperature of the PCM almost reaches

the base temperature in all cases.

Similar in general results appear also in figures 4.28-4.31 for cases 3-6. However,

one can see that when the system is “ thin,” the almost horizontal part of the curve can

disappear, see figure 4.30. This is because in such system there is no vertical melting

front characteristic for “ thick” systems, and melting does not occur simultaneously at

different distances from the bottom, leading to a very non-uniform temperature

distribution in the PCM.

Dependence of the mean temperature of the fin on the geometrical and thermal

parameters of the system can be seen in figures 4.32-4.35.

Figure 4.32 represents the calculated averaged mean temperature in the plane of

symmetry of the fin as a function of time, for cases 2-6 of Table 3.1 at T = 6°C. One can

see from the figure that as the width of the system increases while its height is preserved,

cases 2-5, the shorter is the time required for the mean temperature of the fin to reach the

base temperature. For the “ low” system, case 6, the fin almost at once reached the

temperature of the base. Similar results appear also in figures 4.33-4.35 for T = 12°C,

18°C, and 24°C.

As mentioned above, figures 5.6-5.10 include temperature distributions inside the

fins at certain time instances corresponding to the melt fractions of 0.3, 0.6, and 0.9,

showing the different behavior of thin and thick fins. This point is further illustrated in

figures 5.11 and 5.12 for the melt fraction of 0.5 and 0.9, respectively, and the difference

of T = 24 ºC between the base temperature and the melting temperature. In these

figures, the temperature inside the fin is presented as the difference between the actual

temperature and the temperature of the ambient air.

One can see that as the thickness of the system increases while its height is

preserved, cases 2-5, the difference between the temperature of the bottom and the tip of

the fin decreases. A comparison of the results for cases 3 and 6, which have the same fin

and PCM thickness, show that the “ low” system, case 6, has a smaller temperature

variation along its length.

�

��

-5

0

5

10

15

20

25

0 0.002 0.004 0.006 0.008 0.01

l f, m

case 2

case 3

case 4

case 5

case 6

Figure 5.11. Temperature distribution along the fin at base temperature T = 24 ºC,

for the melt fraction equal 0.5.

0

5

10

15

20

0 0.002 0.004 0.006 0.008 0.01

l f, m

case 2

case 3

case 4

case 5

case 6

Figure 5.12. Temperature distribution along the fin at base temperature T = 24 ºC,

for the melt fraction equal 0.9.

T-T

amb,

K

T-T

amb,

K

�

��

5.4 Dimensional analysis

The results discussed in Sections 5.1-5-3 show that melting process depends on the

studied geometrical and thermal parameters. Dimensional analysis is applied now in

attempt to obtain generalized results.

Following a common approach to the heat conduction problems, we define the

dimensionless Fourier number as Fo=�t/l2. Based on the analysis presented above, we

choose a half-thickness of the PCM layer, lb/2, as the characteristic length l.

Variation of the melt fraction as a function of the Fourier number is shown in figures

5.13–5.17 for cases 2–6, respectively. For the sake of comparison, all the figures are

drawn with the same dimensionless time scale, 0<Fo<50. One can see that for the thick

fin and PCM layer, complete melting is achieved at lower Fourier numbers than those

characteristic of the thin fin and PCM.

Within the same geometrical case, the higher the temperature difference, the smaller

the Fourier number that corresponds to complete melting.

0

0.2

0.4

0.6

0.8

1

0 10 20 30 40 50

Fo

Mel

t fra

ctio

n �T = 6 °C

�T = 12 °C

�T = 18 °C

�T = 24 °C

Figure 5.13. Melt fraction of the PCM at different base temperatures, case 2.

�

��

0

0.2

0.4

0.6

0.8

1

0 10 20 30 40 50

Fo

Mel

t fra

ctio

n �T = 6 °C

�T = 12 °C

�T = 18 °C

�T = 24 °C


0

0.2

0.4

0.6

0.8

1

0 10 20 30 40 50

Fo

Mel

t fra

ctio

n �T = 6 °C

�T = 12 °C

�T = 18 °C

�T = 24 °C


�

��

0

0.2

0.4

0.6

0.8

1

0 10 20 30 40 50

Fo

Mel

t fra

ctio

n �T = 6 °C

�T = 12 °C

�T = 18 °C

�T = 24 °C


0

0.2

0.4

0.6

0.8

1

0 10 20 30 40 50

Fo

Mel

t fra

ctio

n �T = 6 °C

�T = 12 °C

�T = 18 °C

�T = 24 °C


�

��

One can conclude from figures 5.13-5.17 that the Fourier number is not sufficient for

generalization even within the same geometrical case. This is because it cannot take into

account the phase change processes. For this reason, the Stefan number should be added

to the analysis. It is defined in our case as Ste = cpT/L, where cp is the specific sensible

heat of the PCM, T is the difference between the heated base temperature and the PCM

melting temperature, and L is the specific heat of fusion. In the present study Stefan

number varies from about 0.07 to 0.29.

Following Ho and Viskanta (1984), we combine the Fourier and Stefan numbers as

(SteFo)0.68.

Figures 5.18–5.22 presents melt faction of the PCM vs. (FoSte)0.68. For the sake of

comparison, all the figures are drawn once again with the same dimensionless time scale,

0<(FoSte)0.68<3. One can see from the figures that within each separate case, the curves

for different T almost coincide.

Note that the power of 0.68 has been proposed by Ho and Viskanta (1984) for

melting from a vertical plate. It works here because, as discussed above, we observed

melting mostly from the fins which leads to a vertical melting front. Comparison of

figures 5.18–5.22 shows that for the thinnest system, figure 5.21, the coincidence of

different curves is less good, reflecting the fact that in this case the melting front is

different.

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5 3

(FoSte)0.68

Mel

t fra

ctio

n �T = 6 °C

�T = 12 °C

�T = 18 °C

�T = 24 °C

Figure 5.18. Correlation for melt fraction of case 2.

�

��

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5 3

(FoSte)0.68

Mel

t fra

ctio

n �T = 6 °C

�T = 12 °C

�T = 18 °C

�T = 24 °C


0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5 3

(FoSte)0.68

Mel

t fra

ctio

n �T = 6 °C

�T = 12 °C

�T = 18 °C

�T = 24 °C


�

��

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5 3

(FoSte)0.68

Mel

t fra

ctio

n �T = 6 °C

�T = 12 °C

�T = 18 °C

�T = 24 °C


0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5 3

(FoSte)0.68

Mel

t fra

ctio

n �T = 6 °C

�T = 12 °C

�T = 18 °C

�T = 24 °C


�

��

The same way was used to correlate the heat flux and the temperature of PCM, as

shown in figures 5.23–5.27 and 5.28–5.32, respectively.

Figures 5.23–5.27 present a dimensionless heat flux as a function of (SteFo)0.68. The

dimensionless heat flux, ''q~ , was defined as the ratio of the instant heat flux transferred

from the base, q’ ’ , to the mean heat flux of the melting process:

� �At/Q

''q''q~

0� (5.2)

where � �ip TTmcQ w ��0 is the total heat transferred to the PCM during the melting

time t, and A is the total area of the base.

Figures 5.28–5.32 present dimensionless temperature of PCM, �, defined as

mw

m

TT

TT

�

��+ (5.3)

where T is the average temperature of the PCM, Tm is the melting temperature, and Tw is

the base temperature.

0

20

40

60

80

100

0 0.5 1 1.5 2 2.5 3

(FoSte)0.68

dim

ensi

onle

ss h

eat f

lux

�T = 6°C

�T = 12 °C

�T = 18 °C

�T = 24 °C

Figure 5.23. Dimensionless heat flux from the base, case 2.

�

��

0

20

40

60

80

100

0 0.5 1 1.5 2 2.5 3

(FoSte)0.68

dim

ensi

onle

ss h

eat f

lux

�T = 6°C

�T = 12 °C

�T = 18 °C

�T = 24 °C


0

20

40

60

80

100

0 0.5 1 1.5 2 2.5 3

(FoSte)0.68

dim

ensi

onle

ss h

eat f

lux

�T = 6°C

�T = 12 °C

�T = 18 °C

�T = 24 °C


�

��

0

20

40

60

80

100

0 0.5 1 1.5 2 2.5 3

(FoSte)0.68

dim

ensi

onle

ss h

eat f

lux

�T = 6°C

�T = 12 °C

�T = 18 °C

�T = 24 °C


0

20

40

60

80

100

0 0.5 1 1.5 2 2.5 3

(FoSte)0.68

dim

ensi

onle

ss h

eat f

lux

�T = 6 °C

�

T= 12 °C�

T = 18 °C�

T = 24 °C


�

��

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5 3

(FoSte)0.68

�

�T = 6 °C

�T = 12 °C

�T = 18 °C

�T = 24 °C

Figure 5.28. Dimensionless temperature of the PCM, case 2.

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5 3

(FoSte)0.68

�

�T = 6 °C

�T = 12 °C

�T = 18 °C

�T = 24 °C


�

��

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5 3

(FoSte)0.68

�

�T = 6 °C

�T = 12 °C

�T = 18 °C

�T = 24 °C


-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5 3

(FoSte)0.68

�

�T = 6 °C

�T = 12 °C

�T = 18 °C

�T = 24 °C


�

��

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5 3

(FoSte)0.68

�

�T = 6 °C

�T = 12 °C

�T = 18 °C

�T = 24 °C


One can see from figures 5.23-5.27 that the behavior of the dimensionless heat fluxes

for each separate case is similar to the behavior of the melt fractions.

Figures 5.28-5.32 show that the mean temperatures cannot be correlated this way. As

mentioned above, the mean temperature includes both molten and solid regions, and does

not reflect the details of the melting process. Note also that by equation (5.3) the initial

values of the dimensionless temperature are different for different levels of the base

temperature.

Figure 5.33 presents the melt fraction of the PCM for all cases explored in this study.

One can see from the figure that obtained curves are similar for different base

temperatures at each case. However, the curves for different geometry cases do not

coincide. One can note that the results for the “ thick” cases 2, 3 and 6 are rather close,

while a thinner case 4 detaches noticeably, and the thinnest case 5 is rather far from the

other cases. It thus appears that although the PCM layer thickness is accounted for in the

Fourier number, there are additional factors which cause the differences between the

cases.

� �

�

0

0.2

0.4

0.6

0.81

00.

51

1.5

22.

53

(FoS

te)0.

68

Melt fraction

� T =

6 °

C c

ase

2

� T =

12

°C� T

= 1

8 °C

� T =

24

°C

� T =

6 °

C c

ase

3

� T =

12

°C

� T =

18

°C

� T =

24

°C

� T =

6 °

C c

ase

4

� T =

12

°C

� T =

18

°C

� T =

24

°C

� T =

6 °

C c

ase

5

� T =

12

°C

� T =

18

°C

� T =

24

°C

� T =

6 °

C c

ase

6

� T =

12

°C

� T =

18

°C

� T =

24

°C

Fig

ure

5.33

. Mel

t fra

ctio

n ev

olut

ion

in a

ll c

ases

con

side

red

in th

is w

ork.

�

��

One can see from figures 5.6-5.10 and 5.11-5.12 that the temperature distribution

along the fin axis depends on the fin thickness. This result corresponds to the general

theory of the cooling fins, which establishes the relations between the temperature

distribution in the fin and the parameters like the fin cross-section area A, perimeter of

this area P, fin length l f, thermal conductivity of the fin material kf, and the heat transfer

coefficient h from the fin to the surroundings.

Recall that fin efficiency, ,, is defined as the ratio of the actual heat transfer to the

maximum possible heat transfer. The latter takes place when the whole fin is at the

temperature of the base. The lower the fin efficiency, the closer the local temperature

inside the fin to the temperature of the surroundings, which results in a decrease in the

heat transfer from the fin to the surroundings. We can conclude, therefore, that for the

“ thin” cases, fin efficiency is lower than for the “ thick” ones.

Fin efficiency depends on the parameter ff lAk/hP , which includes all

geometrical and thermal factors listed above. In particular, for a fin with an insulated tip,

the efficiency is given by

� �ff

ff

lAk/hP

lAk/hPtanh�, (5.4)

Some parameters included in equation (5.4) are easy to define. For example, in our

study, the fin material is the same in all cases, thus kf remains constant. For the adopted

fin shape, we have P/A=2/l t where l t is the fin thickness. We also know the fin length.

The heat transfer coefficient h, however, represents a major problem. For heating

from a side, which is typical for the “ thick” cases considered in this work, there is no

critical Rayleigh number, and flow exists for any finite Ra (analysis of Lamberg and

Siren (2003) for a horizontal fin is based on the critical Rayleigh number of 1708, as for

heating from below no convection exists below this value). As discussed by Raithby and

Hollands (1985), for the small Rayleigh numbers the velocities are small and essentially

parallel to the vertical boundaries, so that they contribute little to the heat transfer.

The maximum Rayleigh number in our study would correspond to case 2, where the

total thickness of the PCM layer is lb = 4mm, accompanied by the temperature difference

of �

T=24°C. For an idealization, let us find the (unattainable) upper limit of Ra assuming

that the fin is the hot wall and the PCM is the cold one at lb/2 = 2mm from the fin. Based

�

��

on the PCM properties listed in Section 3.1, this limit will be Ramax�2700. For cases 3-6,

the corresponding values of Ramax would be 338, 42, 5.3, and 338, respectively.

As discussed in Raithby and Hollands (1985), the Nusselt number can be estimated

based on the following expression:

� �

max

Ral

/lPr.

Ral

/lPr.

.

k

/lhNu

.

.

f

b.

.

.

f

b.

PCM

b

-----

.

-----

/

0

-----

1

-----

2

3

��

��

�

��

��

��

3010

0510

250

0360

0510

20840

2360

01

2 (5.5)

For Ramax�2700, the second and third terms in figure brackets yield 1.8 and 0.96.

Since even this value of Ra is unattainably high, we can conclude that in all the cases

considered, the Nusselt number will be about unity, corresponding to the conduction-

dominated regime at all times. The minimum value of the effective heat transfer

coefficient could be estimated for the final stages of melting based on lb/2, because as the

thickness of the molten layer increases in the process from zero to lb/2, h decreases. Since

h is time-dependent, it will cause the fin efficiency of equation (5.4) to increase with time

rather than have a constant value for a given case. For this reason, application of fin

efficiency in our analysis is problematic and remains out of the scope of the present work.

It is possible, however, to assess the influence of the fin thickness, l t, on the melt

fraction. From the analysis above, it could be expected that the thicker the fin, the more

effective the heat transfer through it to the PCM. By equation (5.4), the fin efficiency

depends on the square root of its thickness.

Figure 5.34 presents melt fraction as function of l t0.5(FoSte)0.68 . One can see from the

figure that the curves for different cases become even closer to each other and even

intersect. The only exclusion now is the result for the thinnest fin, case 5, for which the

melting process is rather different, as discussed above. The agreement is especially good

when the maximum temperature difference is applied, as shown in figure 5.35.

It is hard to expect better agreement between the different cases because of the

complexity of the problem, including its two-dimensionality, transient character, irregular

melting patterns, and heat transfer to the surroundings.

� �

��

0

0.2

0.4

0.6

0.81

00.

010.

020.

030.

040.

050.

060.

070.

080.

090.

1

l t0.

5 (F

oS

te)0.

68

Melt fraction

� T =

6 °

C c

ase

2

� T =

12

°C

� T =

18

°C

� T =

24

°C

� T =

6 °

C c

ase

3

� T =

12

°C

� T =

18

°C

� T =

24

°C

� T =

6 °

C c

ase

4

� T =

12

°C

� T =

18

°C

� T =

24

°C� T

= 6

°C

cas

e 5

� T =

12

°C

� T =

18

°C

� T =

24

°C

� T =

6 °

C c

ase

6

� T =

12

°C

� T =

18

°C

� T =

24

°C

Fig

ure

5.34

. Cor

rela

tion

for

mel

t fra

ctio

n ta

king

into

acc

ount

the

fin

thic

knes

s.

� �

��

0

0.2

0.4

0.6

0.81

00.

010.

020.

030.

040.

050.

060.

070.

080.

090.

1

l t0.

5 (F

oS

te)0.

68

Melt fraction

case

2

� T =

24

°C

case

3

� T =

24

°C

case

4

� T =

24

°C

case

5

� T =

24

°C

case

6

� T =

24

°C

Fig

ure

5.35

. Cor

rela

tion

for

mel

t fra

ctio

n ta

king

into

acc

ount

the

fin

thic

knes

s, f

or m

axim

um

T.

�

��

6. Summary

In the present work, the processes of melting and solidification of phase-change

material (PCM) with conducting partitions or fins have been studied numerically.

Transient two-dimensional simulations were performed using the Fluent 6.0 software. In

the simulations, a most complete formulation has been attempted, which takes into

account conduction inside the fins, conduction and convection in the PCM, volume

change of the PCM associated with phase transition, motion of solid phase in the liquid,

density variation in liquid PCM with temperature, heat transfer to the surrounding air and

convection in it.

The study included simulations of phase-change processes in a relatively large

partitioned storage unit, simulations of periodic phase-change process, and an extensive

parametric study of melting in a latent heat system with internal fins.

The detailed parametric investigation was performed for melting in a relatively small

heat sink, 5mm to 10mm high, where the fin thickness varies from 0.15mm to 1.2mm,

and the thickness of the PCM layers between the fins varies from 0.5mm to 4mm, while

the base temperature varies from 6°C to 24°C above the mean melting temperature of the

PCM.

The material used in the simulations was a commercially available paraffin wax, and

the fins were made of aluminum. The melting temperature of the wax, 23–25°C, was

incorporated in the simulations along with its other properties, including the latent and

sensible specific heat, thermal conductivity and density in solid and liquid states.

Detailed temperature and phase fields have been obtained as function of time,

showing evolution of the heat transfer in the system. The computational results have

clearly established how the melting rate, the melting front profiles, and the heat transfer

are affected by the geometry of the system and by the boundary conditions.

A dimensional analysis was performed on such parameters as the melting rate, heat

transfer rate and temperature of the PCM. The analysis was based on the use of Fourier

and Stefan numbers, which represent transient heat conduction and phase change,

respectively. The effect of fin thickness has been also explored.

�

��

In addition to their theoretical importance, the results of the present study will be

useful for development and design of PCM-based transient cooling systems for various

applications, including electronics cooling.

�

��

7. References

H. S. Carslaw, J. C. Jaeger, Conduction of Heat in Solids, 2nd ed., Oxford University

Press: London, (1959).

G. Casano, S. Piva, Experimental and numerical investigation of the steady periodic

solid-liquid phase-change heat transfer, Int. J. Heat Mass Transf. 45 (2002), pp.

4181–4190

J. Eftekhar, A. Haji-Sheikh, D. Y. S. Lou, Heat transfer enhancement in a paraffin wax

thermal storage system, Journal of Solar Energy Engineering 106 (1984), pp. 299-

306.

A. Gadgil, D. Gobin, Analysis of two-dimensional melting in rectangular enclosures in

presence of convection, ASME J. Heat Transf. 106 (1984), pp. 20–26.

C. Gau, R. Viskanta, Melting and solidification of a pure metal on vertical wall, ASME J.

Heat Transf. 108 (1986), pp. 174–181.

D. Gobin and P. Le Quéré, Melting driven by natural convection. A comparison exercise:

first results, Int. J. Therm. Sci. 38 (1999), pp. 5-26.

N.W. Hale Jr., R. Viskanta, Solid-liquid phase-change heat transfer and interface motion

in materials cooled or heated from above or below, Int. J. Heat and Mass Transfer 23

(1980), pp. 283–292.

C.J. Ho, C.H. Chu, Periodic melting within a square enclosure with an oscillatory

temperature, Int. J. Heat Mass Transf. 36 (1993), pp. 725–733.

C.J. Ho, C.H. Chu, A simulation for multiple moving boundaries during melting inside an

enclosure imposed with cyclic wall temperature, Int. J. Heat Mass Transf. 37 (1994),

pp. 2505–2516.

C.J. Ho, R. Viskanta, Heat transfer during melting from an isothermal vertical wall,

ASME J. Heat Transf. 106 (1984), pp. 12–19.

J.P. Holman, Heat Transfer, 5th ed., McGraw-Hill, (1981).

W. R. Humphries, E. I. Griggs, A design handbook for phase change thermal control and

energy storage devices, NASA Technical Paper, (1977), pp. 6-37.

�

��

H. Inaba, K. Matsuo, A. Horibe, Numerical simulation for fin effect of a rectangular

latent heat storage vessel packed with molten salt under heat release process, Heat

and Mass Transfer 39 (2003), pp. 231–237.

P. Lamberg, K. Siren, Analytical model for melting in a semi-infinite PCM storage with

an internal fin, Heat and Mass Transfer 39 (2003), pp. 167–176.

D. Pal, Y. K.Joshi, Melting in a side heated tall enclosure by a uniformly dissipating heat

source, Int. J. Heat Mass Transf. 44 (2001), pp. 375–387.

G. D. Raithby, K. G. Hollands, Natural Convection, in Handbook of Heat Transfer

Fundamentals, 2nd ed., Editors W. M. Rohsenow, J. P. Hartnett, E. N. Gani�, (1985),

chap. 6.

R. C. Reid, J. M. Prausnitz, B. E. Poling, The properties of gases and liquids, (1987), pp.

439-456.

V. Shatikian, V. Dubovsky, G. Ziskind, R. Letan, Simulations of PCM melting and

solidification in a partitioned storage unit, ASME Summer Heat Transfer Conference,

Las Vegas, Nevada, USA (2003).

R. Velraj, R.V. Seeniraj, B. Hafner, C. Faber, K.Schwarzer, Heat transfer

enhancement in a latent heat storage system, Solar Energy, 65 (1999), pp. 171-

180.

R. Viskanta, Phase-Change Heat Transfer, in Solar Heat Storage: Latent Heat Materials,

Volume I: Background and Scientific Principles, Editor George A. Lane, Ph.D.

(1983), chap. 5.

V. R. Voller, C. R. Swaminathan, General Source-Based Method for Solidification Phase Change, Numerical Heat Transfer B19, (1991), pp. 175.

B. Zalba, J. M. Mari�n, L. F. Cabeza, H. Mehling, Review on thermal energy storage with

phase change: materials, heat transfer analysis and applications, Applied Thermal

Engineering 23 (2003), pp. 251–283.

Z. Zhang, A. Bejan, Melting in an enclosure heated at constant rate, Int. J. Heat Mass

Transf. 32 (1989), pp. 1063–1076.

�

��

Appendices

A1. Periodic melting process

This Appendix presents the results which are preliminary for the study of periodic phase

change, which is planned for the future.

The computational results of evolution of melt front and temperature distribution in the

fin and in the PCM are presented in figures A1.1-A1.4 for different temperature variations.

One can see that for the same amplitude, the period of variation has a significant influence on

the physical picture of melting or solidification. For a small period of variation, not the whole

PCM succeeds to solidify, figures A1.1 and A1.3, and the capacity of the energy storage unit

is not used completely.

30 s 90 s 110 s 140 s 160 s 30 s 90 s 140 s 160 s a. b.

Figure A1.1. a) evolution of melt front, and b) temperature distribution in the unit. a =12 ºC, = 150 sec.

40 s 110 s 140 s 200s 40 s 110 s 140 s 200 s a. b.

Figure A1.2. a) evolution of melt front, and b) temperature distribution in the unit. a = 12 ºC, =180 sec.

�

��

30 s 70 s 90 s 120 s 130 s 150 s 30 s 90 s 120 s 150s

a. b.

Figure A1.3. a) evolution of melt front, and b) temperature distribution in the unit.

a = 18 ºC, = 120 sec.

30 s 90 s 110 s 140s 30 s 90 s 110 s 140 s

a. b.

Figure A1.4. a) evolution of melt front, and b) temperature distribution in the unit.

a =18 ºC, =150 sec.

Figures A1/5 – A1.8 shows the temperature evolution in the PCM for ten points inside

the PCM. The effect of period of oscillation can be observed in the figures. It can be seen,

that not all curves cross over to the region of melting temperature (296-298 K), denoted by to

horizontal lines.

�

��

280

285

290

295

300

305

310

315

0 50 100 150 200 250 300 350

t, sec

T, K

p1

p2

p3

p4

p5

p6

p7

p8

p9

p10

sinus

Figure A1.5. Temperature distribution of the PCM,

(a=12 ºC, �=150 sec).

280

285

290

295

300

305

310

315

0 50 100 150 200 250 300 350

t, sec

T, K

p1

p2

p3

p4

p5

p6

p7

p8

p9

p10

sinus


(a=12 ºC, �=180 sec).

�

��

275

280

285

290

295

300

305

310

315

320

0 50 100 150 200 250 300 350 400 450

t, sec

T, K

p1

p2

p3

p4

p5

p6

p7

p8

p9

p10

sinus


(a=18 ºC, �=120 sec).

275

280

285

290

295

300

305

310

315

320

0 50 100 150 200 250 300 350

t, sec

T, K

p1

p2

p3

p4

p5

p6

p7

p8

p9

p10

sinus


(a=18 ºC, �=150 sec).

�

��

A2. Velocity fields

Our numerical results indicate that convection in the liquid PCM depends not only on

the given geometrical parameters, but also on the melt fraction at certain time instant. This

point is illustrated in Figure A2, where the vector flow field inside the PCM is shown for the

following physical situations: case 2 at melt fractions of 0.3 (A2a) and 0.6 (A2b),

corresponding to Figure 5.6c (T =18 ºC), and case 4 at melt fraction of 0.6 (A2c),

corresponding to Figure 5.8c (T =18 ºC). For the same temperature difference, one can see

from Fig. A2a-b that in the wide case 2, the flow is weak at V/V0 =0.3 but rather strong at

V/V0 =0.6, while for the narrow case 4 (Fig. A2c) the flow is insignificant even at V/V0 =0.6.

That has been seen also in Figures 5.18 – 5.22, where the curves for various T coincided at

low melt fractions for both wide and narrow cases. At higher melt fractions the curves

diverged in the wide cases (2,3,6) but continued to stick together in the narrow ones (4,5).

Comparing the results of Figures 5.18 – 5.22 and A2, one can conclude that convection in the

liquid phase is an additional physical phenomenon that should be accounted for in the

analysis for relatively wide systems at high melt fractions.

a. b. c.

Figure A2. Velocity fields for case 2 (a) V/V0=0.3, (b) V/V0=0.6, and case 4 (c) V/V0=0.6).

m/s

�

��

A3. Different time step calculations

0

20

40

60

80

100

120

140

0 10 20 30 40 50 60 70

time, s

step=0.005 sec

step=0.01 sec

step=0.02 sec

Figure A3. Heat .transfer rate at different time step calculations for case 3 with T = 18 ºC.

Figure A3 represents the heat flux, q´´, as a function of time, for case 3 at T = 18°C.

For this case, the calculations were performed with three different time steps: 0.005, 0.01 and

0.02 seconds. One can see from the figure that there is practically no difference between the

results for 0.005s and 0.01s. On this basis, the calculations in the present work were

performed with the time step of 0.01s.

q", k

W/m

2

�

��

A4. V. Shatikian, V. Dubovsky, G. Ziskind, R. Letan, Simulation of PCM Melting and

Solidification in a Partitioned Storage Unit, Proceedings of 2003 ASME Summer Heat

Transfer Conference, Las Vegas, USA, July 2003.

�

Copyright © 2003 by ASME ��

Proceedings of 2003 ASME Summer Heat Transfer Conference

July 21–23, 2003, Las Vegas, Nevada, USA

HT2003-47167

SIMULATION OF PCM MELTING AND SOLIDIFICATION IN A PARTITIONED STORAGE UNIT

V. Shatikian, V. Dubovsky, G. Ziskind and R. Letan Heat Transfer Laboratory, Department of Mechanical Engineering,

Ben-Gurion University of the Negev, P.O.B. 653, Beer-Sheva 84105, Israel

ABSTRACT The present study explores numerically the processes of

melting and solidification of a phase change material (PCM). The material used was a commercially available paraffin wax, which is non-toxic, recyclable, chemically inert, non-corrosive and can withstand an unlimited number of cycles.

The phase-change material was stored in a rectangular box, open at the top. The bottom of the box could be heated or cooled. The inner space of the box was partitioned by vertical conducting plates attached to the bottom. Thus, heat was transferred to and from the PCM both through its melted/solidified layer and by conduction through the vertical plates.

Transient two-dimensional numerical simulations were performed using the Fluent 6.0 software. The melting temperature of the wax, 23-25ºC, was incorporated in the simulations along with its other properties, including the latent and sensible specific heat, thermal conductivity and density in solid and liquid states.

The simulations provided detailed temperature and phase fields inside the system as functions of time, showing evolution of the heat transfer in the system as the phase change material melts/solidifies. The dependence of the heat transfer rate on the properties of the system and on the PCM phase composition at various time instants is presented and discussed.

INTRODUCTION

Thermal storage units that utilize latent heat of phase change materials (PCM) have received significant attention in the recent years, because of their large heat storage capacity. Due to their isothermal behavior during the melting and solidification processes, those materials can be used in such diversified applications as latent heat storage in buildings or thermal control of electronic modules.

The application of PCM to heat storage in buildings has been investigated by many researchers. An extensive review of available thermal energy storage technologies is presented by Hasnain [1]. Application of PCM to the solar energy storage is reviewed by Fath [2]. The work by Kurklu [3] discusses the three most-used phase change material groups based on salt hydrates, paraffins and polyethylene glycol. Kang et al. [4] describe and classify latent heat thermal energy storage systems according to their structural characteristics. With regard to the heat transfer mechanisms, a review of heat transfer enhancement methods is presented by Velraj et al. [5].

Recently, a number of studies have been published in the literature on PCM application in transient thermal control of electronic equipment. As noted by Pal and Joshi [6], most electronic systems are subjected to a complex combination of internal and external transient thermal loads [7], and the use of PCM would make it possible to maintain a nearly uniform temperature of the components. Among possible applications, one can mention portable systems, outdoor telecommunications enclosures, and processor chips [6].

The choice of the phase change material and the mechanisms of heat transfer are the most important factors in the development of a latent heat thermal energy storage system. In addition to such properties as the thermal conductivity of the material, its melting temperature and latent heat of fusion, stability of properties through the cycling process with a large number of cycles is essential.

Pal and Joshi [8] studied numerically application of phase change materials for the passive thermal control of avionics modules with transient thermal loads occurring during normal operation or due to the loss of primary cooling system. Different system configurations were suggested for two different PCM types, of which one was an organic paraffin and the second an eutectic alloy. It has been concluded that, while the organic

�


PCM has higher latent heat and hence a lesser mass of it is needed, its poor thermal conductivity (0.39W/m K in solid state) demands special arrangements when the incorporation in electronic systems is considered.

One of the most common methods used to improve thermal performance of PCM-based cooling systems is the use of extended surfaces, like partitions, cooling fins [9], or inclusions made of good heat conductors, usually metals.

In our previous study [10], extended surface heat transfer area was achieved experimentally, using an aluminum shell which consisted of 82 staggered separate channels filled with PCM. Each of the channels had a cross-section of 5.5mm�4mm and a length of 267mm. The PCM used was a paraffin wax, which is non-toxic, recyclable, chemically inert, non-corrosive and can withstand an unlimited number of cycles.

In the present study, the same PCM is involved, and the storage unit has partitions made of aluminum. Only the bottom of the unit is heated/cooled directly, while heat is transferred to the PCM mostly through the partitions. The problem of transient heat conduction through the partitions and PCM, accompanied by melting/solidification of the latter, is solved numerically.

We present first a physical model of the problem, including the details of the system and the properties of the PCM used. Then, the numerical procedure is described. The results for both melting and solidification are presented in terms of the phase distributions, heat transfer rates, and temperature distributions inside the partitions and the PCM for various time instants.

NOMENCLATURE cp specific heat at constant pressure (J/kg ºC) h sensible enthalpy (J/kg) H total enthalpy (J/kg) k thermal conductivity (W/m ºC) L latent heat (J/kg) S source term (W/m3) t time (s) T temperature (ºC) u velocity components (m/s) x Cartesian coordinate (m)

Greek letters �� liquid fraction �� density (kg/m3)

Subscripts i component ref reference value w wall

PHYSICAL MODEL The storage unit is shown in Fig. 1. It is a rectangular box

which bottom is the hot wall. Parallel partitions are attached to this wall. Each partition is 2mm thick and 200mm long. The space between two adjacent partitions is 10mm. The partitions are made of aluminum (k=202 W/m ºC).

Between the partitions, there is a phase-change material. The PCM used was paraffin wax (Rubitherm RT25) with the following properties: density in liquid state 750kg/m3, dynamic viscosity 0.0072kg/m·s, density in solid state 800kg/m3, enthalpy of melting 206kJ/kg, thermal conductivity 0.15W/m K, specific heat 2.5kJ/kg K, and melting temperature of 23-25°C.

Figure 1. Partitioned storage unit: a). overall schematic view; b). computational domain.

2 m

m� �

10 mm�Air�

�

Tw

190

mm

�PCM

�

Partition�

�

a.��

b.��

Air�

5 mm�1 mm�

200

mm

�

190

mm

�

PCM

�

Tw

� x

y

�


The PCM layers are 190mm high, which means that they are 10mm shorter than the partitions. This is done in order to compensate for the density changes during the phase-change processes. From above, the PCM is exposed to the ambient air.

At this stage, model problems of melting and solidification were considered: in both cases the initial temperature of the system, including the PCM, was uniform. It was slightly lower than the temperature of phase change for melting and slightly higher than the temperature of phase change for solidification.

PCM melting has been studied under the following conditions: The initial temperature of the whole system was 20ºC, i.e. the PCM was slightly subcooled. At t=0, the temperature of the bottom was changed to 42ºC, while the ambient air above the unit was 27ºC.

PCM solidification has been studied under the following conditions: The initial temperature of the whole system was 27ºC, i.e. the PCM was slightly overheated. At t=0, the temperature of the bottom was changed to 6ºC, while the ambient air above the unit was 27ºC.

NUMERICAL PROCEDURE

The numerical calculations were performed for the transient temperature and velocity fields inside the unit, including both the PCM and the partitions.

The basic conservation equations of continuity, momentum, and energy were solved numerically, using the FLUENT 6.0 software. Laminar flow inside air and liquid PCM was assumed. In order to describe the behavior of the PCM, a so-called “volume-of-fluid” model has been activated. The model made it possible to calculate the processes that occur inside the partitions (solid), PCM (solid/liquid), and air (fluid) simultaneously. In addition, density changes in the PCM and the effect of gravity were taken into account. The model makes it possible to account for the moving boundary due to the variation of the PCM volume. As a result, the numerical model was rather close to reality.

Based on the symmetry of the system, the computational domain was defined by the physical boundaries of the unit in the vertical y-direction, and by the symmetry planes of the partition and the PCM-filled channel in the horizontal x-direction, as shown in Fig. 1b. The origin of the coordinate system was taken at the intersection of the plane of symmetry of the partition and the lower boundary of the partitions and PCM. It was assumed that the unit is both sufficiently long and its boundaries are well insulated in the z-direction, so that a two-dimensional formulation has been used at this stage.

The computational grid of 24�100=2400 cells was used in the simulations, including 4�100=400 cells in the 1mm thick and 200mm long half-partition (0<x<1mm, 0<y<200mm), 20�95=1900 cells in the 5mm thick and 190mm long half channel filled by the PCM (1<x<6mm, 0<y<190mm), and 20�5=100 cells in the air above it (1<x<6mm, 190<y<200mm).

The partitions were taken as insulated from above, while the PCM was exposed to the air above it. The equations for the air were solved only in the small domain bounded by the

partition from the left, symmetry from the right, PCM from below and the plane connecting the partition tips from above. At that plane, a boundary condition corresponding to “pressure inlet/outlet” has been used, and the air above it was at atmospheric pressure and had a constant temperature of 27ºC. For air, the Boussinesq approximation was not used, and the density-temperature relation was provided as an input.

Since the temperature differences were relatively small, the single-phase density variations were neglected compared to the density change due to the phase change, and the constant but different PCM densities were assumed in the solid and liquid states, respectively. For the mushy region, FLUENT applies the enthalpy-porosity approach, by which the porosity in each cell is set equal to the liquid fraction in that cell. Accordingly, the porosity is zero inside fully solid regions.

The energy equation for the solidification/melting problem is:

� � � � Sx

Tk

xHu

xH

t iii

i

��

��

�

)

)

)

)�

)

)�

)

)�� (1)

where � is the density, S is the source term, , ui is the velocity component in the i - direction, xi is a Cartesian coordinate, and H is the enthalpy.

The enthalpy is defined as a sum of the sensible enthalpy, h, and the enthalpy change due to the phase change H.

The sensible enthalpy is defined as

dTchhT

Tpref

ref

�� (2)

where href is the reference enthalpy at the reference temperature Tref, and cp is the specific heat.

The enthalpy change due to the phase change is defined as

LH � � (3)

where L is the latent heat of the material, and � is the liquid fraction defined by the following relations:

� = 0 if T < Tsolidus � = 1 if T > Tliquidus

solidusliquidus

solidus

TT

TT

�

�� if Tsolidus < T < Tliquidus (4)

Accordingly, the enthalpy change due to the phase change

vary from zero for a solid to L for a liquid. FLUENT solves for the temperature by iterations involving

the energy equation, Eq. (1), and the liquid fraction relation, Eq. (4), using the approach of Voller and Swaminathan [11], who propose methods in which the phase change rate is linearized as a truncated Taylor series, and old iteration values are then used to estimate the linear term.

�


RESULTS AND DISCUSSION The results are shown in Figs. 2-5 for melting and in Figs.

6-9 for solidification. Melting. Figure 2 shows the results of melting simulations

in the form of phase distributions. The left picture represents the initial phase distribution, in which the PCM was entirely solid (blue). The only fluid here is the air above the PCM (red). Since the software can represent a solid-fluid distribution, it was impossible to show the liquid PCM and the air by different colors, and they both are red in all other parts of Fig. 2. Fortunately, air is above the PCM, while melting occurs mostly from below, therefore such blue-red scale is quite understandable.

0s 300s 600s 900s 1200s 1500s 1800s

Figure 2. Phase distribution of PCM during melting as a function of time.

One can see from Fig. 2 that melting is initiated not only at the hot bottom of the unit, but also at the partitions, due to their high thermal conductivity. As a result, the solid PCM attains the shape of a downward-directed wedge.

It can also be seen from Fig. 2 that the upper boundary of the PCM is moving upward with time, reflecting the increasing volume of liquid PCM in the course of melting.

Figure 3 shows the calculated heat transfer rates from the base to the PCM, from the base to the partition, and from the partition to the PCM and air. The rates given are for the computational domain introduced above. One can see from the figure that at each time instant, the heat transfer rate from the base to the partition is much higher than that going directly to the PCM. Since the major part of the heat transferred to the partition eventually reaches the PCM, the latter is heated mostly from the sides and not from below. Moreover, the more time elapses, the thicker is the layer of liquid PCM between the base and the wedge, as seen also in Fig. 2. Since this layer is a thermal resistance for the melting process, the direct heat transfer from the base to the PCM eventually becomes negligible.

Figure 3 also shows that the heat transfer rate from the base to the partition also decreases with time, because of the decreasing temperature gradient. As the partition is heated and the solid PCM “retreats,” the resistance to heat transfer increases.

The difference between the heat that enters the partition and the heat that leaves it is due to the transient heating of the partition (increase in its internal energy). This difference obviously decreases with time.

0

50

100

150

200

250

0 500 1000 1500 2000

Time, s

Hea

t tr

ansf

er r

ate,

W

base to PCMpartition to PCM and airbase to partition

Figure 3. Heat transfer rate vs. time for PCM and partition during melting.

Further details of the melting are presented in Figs. 4 and 5. Figure 4 shows the temperature inside the partition (at x=0) at various distances from the base (0<y<200mm), for the same time instants which have been represented by their heat transfer rates in Fig. 3. One can clearly see that the temperature at any point inside the partition grows with time. The distribution tends asymptotically to a steady state in which heat input from the

�


base would be balanced by heat output to the air, while the PCM is completely liquid. One can see from the figure that under given conditions this would take a significant amount of time.

290

295

300

305

310

315

320

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2


Tem

per

atu

re, K

t=135

t=235

t=300

t=600

t=900

t=1200

t=1500

t=1800

t=2100

Figure 4. Temperature evolution in the partition during melting.

290

295

300

305

310

315

320

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2


Tem

per

atu

re, K

t=135

t=235

t=300

t=600

t=900

t=1200

t=1500

t=1800

Figure 5. Temperature evolution in the PCM during melting.

Figure 5 shows the temperature at the plane of symmetry of the PCM (at x=6mm) at various distances from the base (0<y<200mm), for the same time instants that have been represented by their heat transfer rates in Fig. 3. Close to the upper boundary, the temperature is that of air above the PCM.

One can see from the figure that the temperature inside the PCM is growing with time. The temperature patterns, however, are very different from those obtained for the solid partition. From the beginning, PCM starts to melt close to the base, but only after it has been heated from the subcooled state to the melting point temperature. This heating is represented at various time instants by the curves found below the melting temperature of 23ºC (296K). While the heating of the solid PCM relatively far from the base continues, melting takes place near the base, followed by heating of the liquid PCM. This process is represented by a family of similar curves. One can clearly see that the temperature at any point inside the liquid PCM grows

with time, while the liquid region itself grows. These results correspond well to the phase distributions in Fig. 2 and heat transfer rates in Fig. 3.

Above the PCM there is air heated from above by the ambient and cooled from below by the PCM. The air is also heated by the partition which protrudes above the PCM. In Fig. 5, the region occupied by air can be clearly seen. This region shrinks with time due to the PCM expansion.

Solidification. Figure 6 shows the results of solidification simulations in the form of phase distributions. Once again, it was impossible to show the liquid PCM and the air by different colors, and they both are red in the figure. Fortunately, solidification occurs mostly from below, making the blue-red scale quite understandable.

100s - 600s 900s 1200s 1500s 1800s 2100s

Figure 6. Phase distribution of PCM during solidification as a function of time.

�


The results of Fig. 6 correspond to the simulation in which the solidus and liquidus temperatures were taken as 23°C and 25°C, respectively.

One can see from Fig. 6 that solidification is initiated not only at the hot bottom of the unit, but also at the partitions, due to their high thermal conductivity. As a result, the solid PCM attains a U-shape pattern.

Figure 7 shows the calculated heat transfer rates from the PCM to the base, from the partition to the base, and from the PCM and air to the partition. The rates given are for the computational domain introduced above. One can see from the figure that at each time instant, the heat transfer rate from the partition to the base is much higher than that coming directly from the PCM. Thus, the latter is cooled mostly from the sides and not from below. Moreover, the more time elapses, the thicker is the layer of solid PCM between the base and the wedge, see also Fig. 6. Since this layer is a thermal resistance for the solidification process, the direct heat transfer from the PCM to the base eventually becomes negligible.

Figure 7 also shows that the heat transfer rate from the partition to the base also decreases with time, because of the decreasing temperature gradient.

Further details of the solidification process are presented in Figs. 8 and 9. Figure 8 shows the temperature inside the partition (at x=0) at various distances from the base (0<y<200mm), for the same time instants which have been represented by their heat transfer rates in Fig. 7. One can clearly see that the temperature at any point inside the partition decreases with time. The distribution tends asymptotically to a steady state in which heat output to the base would be balanced by heat input from the air, while the PCM is completely solid. Like in melting, under given conditions it would take a significant amount of time to reach that state.

Figure 9 shows the temperature at the plane of symmetry of the PCM (at x=6mm) at various distances from the base (0<y<200mm), for the same time instants that have been represented by their phase distributions in Fig. 6 and heat transfer rates in Fig. 7.

From the beginning, PCM starts to solidify close to the base, after it has been cooled to the temperature of solidification. This process is represented by a family of similar curves which correspond to different time instants. One can clearly see that the temperature at any point inside the solid PCM decreases with time, while the solid region itself grows. These results correspond well to the phase distributions in Fig. 6 and heat transfer rates in Fig. 7.

CLOSURE In the present work, the processes of melting and

solidification of a phase change material (PCM) have been studied numerically. Transient two-dimensional simulations were performed using the Fluent 6.0 software. The material used as a model was a commercially available paraffin wax, stored in a rectangular box. The inner space of the box was partitioned by vertical conducting plates attached to the bottom, which could be

heated or cooled. As a result, heat was transferred to and from the PCM mostly by conduction through the vertical plates.

-120

-100

-80

-60

-40

-20

0

0 500 1000 1500 2000 2500

Time, s

Hea

t tr

ansf

er r

ate,

W

partition to basePCM to basePCM and air to partition

Figure 7. Heat transfer rate vs. time for PCM and partition during solidification.

275

280

285

290

295

300

305

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2Distance from the base, m

Tem

per

atu

re, K

t=100

t=200

t=300

t=600

t=900

t=1200

Figure 8. Temperature evolution in the partition during solidification.

275

280

285

290

295

300

305

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2


Tem

per

atu

re, K

t=100

t=200

t=300

t=600

t=900

t=1200

t=1500

t=1800

t=2100

Figure 9. Temperature evolution in the PCM during solidification.

�


Detailed temperature and phase fields have been obtained as functions of time, showing evolution of the heat transfer in the system as the phase change material melts/solidifies. The change in the heat transfer rate with time, and the instantaneous temperature distributions inside the partitions and the PCM have been presented and discussed.

REFERENCES [1] Hasnain, S.M., 1998, “Review on Sustainable Energy Storage Technologies, Part I: Heat Storage Materials and Techniques,“ Energy Conversion and Management 39, pp. 1127-1138. [2] Fath, H.E.S., 1998, “Assessment of Solar Thermal Energy Storage Technologies,” Renewable Energy 14, pp. 35-40. [3] Kurklu, A., 1998, “Energy Storage Applications in Greenhouses by Means of Phase Change Materials (PCMs): a Review,” Renewable Energy 13, pp. 89-103. [4] Kang, Y.B., Zhang, Y.P., Jiang, Y., and Zhu, Y.X., 1999, “A General Model for Analyzing the Thermal Characteristics of a Class of Latent Heat Thermal Energy Storage System,” Journal of Solar Energy Engineering–Transactions of the ASME 121, pp. 185-193. [5] Velraj, R., Seeniraj, R.V., Hafner, B., Faber, C., and Schwarzer, K., 1999, “Heat Transfer Enhancement in a Latent Heat Storage System,” Solar Energy 65, pp. 171-180.

[6] Pal, D., and Joshi, Y.K., 2001, “Melting in a Side Heated Tall Enclosure by a Uniformly Dissipating Heat Source,” International Journal of Heat and Mass Transfer 45, pp. 375-387. [7] Evans, A.G., He, M.Y., Hutchinson, J.W., and Shaw, M., 2001, “Temperature Distribution in Advanced Power Electronics Systems and the Effect of Phase Change Materials on Temperature Suppression During Power Pulses,” Journal of Electronic Packaging–Transactions of the ASME 123, pp. 211-217. [8] Pal, D., and Joshi, Y.K., 1997, “Application of Phase-Change Materials to Thermal Control of Electronic Modules,” Journal of Electronic Packaging–Transactions of the ASME 119, pp. 40-50. [9] Ismail, K.A.R., Alves C.L.F., and Modesto M.S., 2001, “Numerical and Experimental Study on the Solidification of PCM around a Vertical Axially Finned Isothermal Cylinder,” Applied Thermal Engineering 21, pp. 53-77. [10] Goldenberg, A., Abramzon, B., Dubovsky, V., Ziskind, G., and Letan, R., 2002, “Temperature Moderation in an Enclosed Space by a Portable PCM Heat Storage Unit,” Proc. 12th International Heat Transfer Conference, J. Taine et al., eds., Elsevier SAS, pp. 717-722. [11] Voller, V. R., and Swaminathan, C. R., 1991, “General Source-Based Method for Solidification Phase Change,” Numerical Heat Transfer B19, pp. 175.

�

�

��

��

��(PCM)��

��

��

��

� ��

��

��PCM�

��

�� !��

�� "��!�� PCM�� #�

��!��°C$��°C"#��

�� PCM� �%��

��

�� Fluent 6.0��

� ��°C"��"&�� PCM��

��

�� PCM��

��

�PCM��

�� PCM�� PCM��

�� PCM��

�� PCM��

��

��Fourier�� Stefan��

��

�

�

��

��

��

��

�

�

�� !� "#�$�

�$�&% �'% �(��$�)� �*� �

�

�

�

�

�

��

��

�

��

��

��

�

�

��

��

��

��

�

�

�� # �$�)� ��$�� "#��

�$� % �'% � ��

�

��

�

�

�

�

��

��

�

��

�

��

Documents

Melting and solidification