thermal energy storage in copper foams filled with ......phase change materials (pcm) such as...

Thermal Energy Storage in Copper Foams filled with Paraffin Wax

by

Pathik Himanshu Vadwala

A thesis submitted in conformity with the requirements for the degree of Master of Applied Science

Mechanical & Industrial Engineering University of Toronto

© Copyright by Pathik Vadwala 2011

ii

Thermal Energy Storage in Copper Foams filled with Paraffin Wax

Pathik Vadwala

Master of Applied Science

Mechanical & Industrial Engineering

University of Toronto

2011

Abstract

Phase change materials (PCM) such as paraffin wax are known to exhibit slow thermal

response due to their relatively low thermal conductivity. In this study, experiments were

carried out to investigate a method of enhancing thermal conductivity of paraffin wax by

making use of high porosity open cell metal foams. By adding metal foam, thermal

conductivity of PCM‟s was shown to increase by 16-18 times that of pure paraffin wax. The

use of open cell metal foam material for thermal energy storage application was also

investigated by designing and testing different thermal energy storage systems (TESS) - with

and without metal foam. The effect of copper metal foam on heat transfer during melting and

solidification was analysed by determining the convective heat transfer coefficient. Lastly, a

numerical code was developed to predict the temperature field within PCM while melting.

iii

Acknowledgments

This research project conducted at the Heat Transfer and Combustion Laboratory in

collaboration with the Center for Advanced Coating Technologies (CACT) at the University

of Toronto was made possible by the help and support of many individuals and through the

financial support by Natural Sciences and Engineering Research Council (NSERC) of

Canada.

First of all I would like to express my sincere thanks to my advisor, Professor Sanjeev

Chandra for his guidance during the course of this project and for giving me the opportunity

to prove my potential in the last two years. This project would not have been a reality

without his enormous support and co-operation. I am also grateful to Professor Javad

Mostaghimi, for providing valuable guidance in every aspect of this project. I would like to

express my thanks to Ryan Mendell and the entire MIE machine shop, for fabricating

various experimental parts needed for this study.

Lastly, I would like to extend a special acknowledgment to my parents and brother

Himanshu, Paragi and Jay Vadwala. Their sacrifices and motivation have been the primary

reason for my position today. A heartfelt thank you goes out to all my friends in the

department, laboratory and at home.

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Table of Contents

Abstract .................................................................................................................................................. ii

Acknowledgments ................................................................................................................................ iii

Table of Contents .................................................................................................................................. iv

List of Tables ........................................................................................................................................ vi

List of Figures ..................................................................................................................................... viii

Nomenclature ........................................................................................................................................ xi

Chapter 1................................................................................................................................................ 1

Introduction ........................................................................................................................................... 1

1.1 Introduction ................................................................................................................................. 1

1.2 Literature Review ........................................................................................................................ 3

1.3 Research Objectives..................................................................................................................... 7

1.4 Organization of Thesis ................................................................................................................. 8

Chapter 2................................................................................................................................................ 9

Thermal Conductivity Enhancement of Phase Change Materials Using Metal Foams ......................... 9

2.1 Introduction ................................................................................................................................. 9

2.2 Experimental Test Apparatus .................................................................................................... 10

2.3 Results and Discussion .............................................................................................................. 16

2.4 Comparison with Theoretical Model ......................................................................................... 22

2.5 Photographic observation of melting front ................................................................................ 25

Chapter 3.............................................................................................................................................. 28

Thermal Energy Storage System with Metal Foam ............................................................................. 28

3.1 Introduction ............................................................................................................................... 28

3.2 Experimental Test Apparatus and Procedure............................................................................. 29

3.3 Results and Discussion .............................................................................................................. 35

3.4 Thermal Resistance .................................................................................................................... 48

3.5 Three Stage Thermal Energy Storage System ........................................................................... 53

Chapter 4............................................................................................................................................. 61

Numerical Simulation of Temperature Profile using Enthalpy Method .............................................. 61

4.1 Introduction ............................................................................................................................... 61

4.2 Mathematical Model .................................................................................................................. 62

4.3 Numerical Solution .................................................................................................................... 64

v

4.4 Verification of the Computational Model .................................................................................. 72

Chapter 5.............................................................................................................................................. 76

Conclusions ......................................................................................................................................... 76

References ........................................................................................................................................... 78

Appendix A.......................................................................................................................................... 84

Schematics ....................................................................................................................................... 84

Appendix B .......................................................................................................................................... 87

Thermal Conductivity Measurement Data ....................................................................................... 87

Appendix C .......................................................................................................................................... 93

Thermal Energy Storage System Data ............................................................................................. 93

Appendix D........................................................................................................................................ 120

Results from Enthalpy Code .......................................................................................................... 120

vi

List of Tables

Table 2.1: Thermal diffusivity values at different temperatures.......................................................... 21

Table 2.2: Effective Density and Specific Heat values for the foam-wax system ............................... 21

Table 4.1 Various physical properties used in the numerical code.................................................... 72

Table B1: Temperature results for 65 ..................................................................................... 88





Table C1: Temperature variation while melting (without metal foam) at constant temperature of

105 at the top copper plate ............................................................................................................... 94

Table C2: Temperature results for solidification at 20.0 L/min (no metal foam) ................................ 95





Table C7: Temperature variation while melting (with metal foam on wax side) at constant heat flux

of 50 V (2800 W/m2) ......................................................................................................................... 100

Table C8: Temperature results for solidification at 20.0 L/min (foam on wax side) ........................ 101

Table C9: Temperature results for solidification at 30.0 L/min (foam on wax side) ........................ 102

Table C10: Temperature results for solidification at 40.0 L/min (foam on wax side) ...................... 103



Table C13: Temperature variation while melting (with metal foam on both- air and wax side) at

constant heat flux of 50 V (2800 W/m2) ............................................................................................ 106

Table C14: Temperature results for solidification at 20.0 L/min (foam on both sides) .................... 107





Table C19: Thermal resistance analysis data (without metal foam) .................................................. 112

Table C20: Thermal resistance analysis data (metal foam on wax side) ........................................... 113

Table C21: Thermal resistance analysis data (metal foam on both sides) ......................................... 114

vii

Table C22: Temperature results for solidification at 20.0 L/min (three stage TESS) ....................... 115





Table D1: Experimental temperature results for melting (with metal foam) at a depth of 1 cm at

constant heat flux of 50 V (2800W/m2) ............................................................................................. 121

Table D2: Temperature results from numerical code (phase change at a single temperature) for

melting (with metal foam) at a depth of 1 cm at constant heat flux of 50 V (2800W/m2) ................ 122

Table D3: Temperature results from numerical code (phase change over a temperature range) for

melting (with metal foam) at a depth of 1 cm at constant heat flux of 50 V (2800W/m2) ................ 123

viii

List of Figures

Figure 2.1 : Schematic of the experimental apparatus showing the test sample and instrumentation . 11

Figure 2.2 : Photograph of the experimental apparatus showing test sample and instrumentation ..... 13

Figure 2.3 : Detailed view of Test sample showing the placement of thermocouples ........................ 13

Figure 2.4 : Experimentally obtained temperature results for 85 plotted as a function of time

at different axial distance along the length of the test sample. The experimental uncertainty is 2.6

in the temperature measurement. ......................................................................................................... 17

Figure 2.5 : Experimentally computed thermal diffusivity values from Equation (2.1) for different

surface temperatures and different distance along the length of the test sample. The thermal

diffusivity values falling between two bounds are only taken into consideration. .............................. 19

Figure 2.6 : A unit cell structure of tetrakaidecahedron with six squares and eight hexagons [5] ...... 22

Figure 2.7 : Experimental pictures showing propagation of melting front of wax without metal foam

............................................................................................................................................................. 25

Figure 2.8 : Experimental pictures showing propagation of melting front of wax with metal foam .. 26

Figure 3.1: Schematic of the TESS apparatus showing the instrumentation and air supply

configuration ........................................................................................................................................ 30

Figure 3.2: TESS apparatus with instrumentation ............................................................................... 33

Figure 3.3: Detailed view of TESS showing the placement of thermocouples used for measuring inlet

and outlet temperature of air ................................................................................................................ 33

Figure 3.4: TESS with no Metal Foam ................................................................................................ 34

Figure 3.5: TESS with Metal Foam on wax side ................................................................................. 34

Figure 3.6: Experimentally obtained temperature results plotted as a function of time while melting at

a distance of 1 & 2 cm along the length of wax compartment for both – with foam and without foam

case. The experimental uncertainty is 2.6 in the temperature measurement. ................................ 38

Figure 3.7: Experimentally obtained temperature results plotted as a function of time during melting

at a distance of 1 & 2 cm along the length of wax compartment with metal foam. The experimental

uncertainty is 2.6 in the temperature measurement. ...................................................................... 39

Figure 3.8: Experimentally obtained temperature results plotted as a function of time while

solidifying at a distance of 1 & 2 cm along the length of wax compartment for both – with foam and

without foam case when the mass flow rate of air is 40.0 L/min. The experimental uncertainty is

2.6 in the temperature measurement. ............................................................................................ 40

Figure 3.9: Experimentally obtained results for inlet and outlet temperature of air plotted as a

function of time for both-with foam and without foam case when the mass flow rate of air is 40.0

L/min. The experimental uncertainty is 2.6 in the temperature measurement. ............................. 41

Figure 3.10: Photograph showing metal foam on both sides along with copper plates and

thermocouples ...................................................................................................................................... 42

ix

Figure 3.11: TESS with metal foam on both sides .............................................................................. 42


solidifying at a distance of 1 cm along the length of wax compartment for all the three cases when the

mass flow rate of air is 40.0 L/min. The experimental uncertainty in temperature measurement is

2.6 . ................................................................................................................................................. 43


function of time for all the three cases when the mass flow rate of air is 40.0 L/min. The experimental

uncertainty is 2.6 in the temperature measurement. ...................................................................... 44

Figure 3.14: Experimentally computed Nusselt number with respect to Reynolds number for all the

three cases. ........................................................................................................................................... 45

Figure 3.15: Experimentally computed power extracted with respect to flow rate of air for all the

three cases of TESS ............................................................................................................................. 47

Figure 3.16: Experimentally computed convective heat transfer coefficient with respect to flow rate

of air for all the three cases of TESS ................................................................................................... 47

Figure 3.17: Computed value of solid wax resistance with respect to non-dimensional

solidification time for all three cases of TESS ................................................................................. 51

Figure 3.18: Computed value of Overall Thermal Resistance R with respect to non-dimensional

solidification time for all three cases of TESS ................................................................................. 52

Figure 3.19: A cross section view of 3 stage TESS showing the 3 layers of foam separated by copper

plates .................................................................................................................................................... 54

Figure 3.20: Top view of the TESS showing the hole for thermocouple fitting.................................. 54

Figure 3.21: Photograph showing the flexible silicon rubber heaters that were used for melting wax56

Figure 3.22: Three stage TESS apparatus with instrumentation.......................................................... 56

Figure 3.23: Experimentally obtained results for outlet temperature of air plotted as a function of

time. The mass flow rate of air in L/min is indicated in the parenthesis. The experimental uncertainty

is 2.6 in the temperature measurement .......................................................................................... 57

Figure 3.24 : Experimentally computed Nusselt number with respect to Reynolds number for all the

three stage TESS .................................................................................................................................. 59

Figure 3.25: Experimentally computed average power extracted – Q (W) with respect to flow rate of

air ......................................................................................................................................................... 60

Figure 3.26: Experimentally computed average convective heat transfer coefficient – (W/m2K)

with respect to flow rate of air ............................................................................................................. 60

Figure 4.1: Discretization domain for one-dimensional phase change problem with boundary

conditions............................................................................................................................................. 65

Figure 4.2 Flowchart ........................................................................................................................... 71

Figure 4.3 : Experimentally obtained temperature results plotted as a function of time, during melting

at a distance of 1 cm along the length of wax compartment with metal foam, along with the computed

x

temperature values from C++ code, for phase change at a single fixed temperature. The experimental

uncertainty is 2.6 in the temperature measurement ....................................................................... 73

Figure 4.4 : Experimentally obtained temperature results plotted as a function of time, during melting

at a distance of 1 cm along the length of wax compartment with metal foam, along with the computed

temperature values from C++ code, for phase change at a single fixed temperature and phase change

over a temperature range. The experimental uncertainty is 2.6 in the temperature measurement. 75

Figure A1: Two stage TESS drawing .................................................................................................. 85

Figure A2: Three stage TESS drawing ................................................................................................ 86

xi

Nomenclature

E = Energy (J)

m = Mass flow rate of fluid (L/min)

= Specific Heat (J/kg.K)

L = Latent heat of fusion (KJ/kg)

e = Error associated with given instrument

= Uncertainty in a measurement

erfc = Complimentary error function

k = Thermal conductivity (W/mK)

= Thermal diffusivity (m2/sec)

= Porosity

= Density (kg/m3)

= Convective heat transfer coefficient of air (W/m2K)

= Cross-sectional area of TESS (m2)

Q = Power (W)

Re = Reynolds number

Nu = Nusselt number

= Hydraulic diameter (m)

= Characteristics length (m)

u = Mean velocity of air (m/sec)

= Dynamic viscosity (Nsec/m2)

xii

= Temperature of the fluid ( )

= Melting temperature of the PCM ( )

= Temperature of fluid at the center of TESS ( )

= Convective resistance on air-side (K/W)

= Solid wax resistance (K/W)

R = Overall thermal resistance (K/W)

= Width of the copper plate (m)

= Thermal conductivity of copper plate (W/mK)

= Thermal conductivity of wax (W/mK)

= Thickness of solidified layer of wax that varies with time (m)

= Time required for solidification front to reach the distance (sec)

= Time required for total solidification (sec)

H = Total volumetric enthalpy (J/kg)

h = Sensible enthalpy (J/kg)

= Liquid fraction

Fo = Finite difference Fourier number

= Space increment (m)

= Time increment (sec)

= Non-dimensional solidification time -

CHAPTER 1. INTRODUCTION

1

Chapter 1

Introduction

1.1 Introduction

Thermal energy storage is of critical importance in many engineering applications. As solar

energy is available only during daytime, its application requires an efficient storage system

so that the energy gathered during daytime can be utilised later at night. Similar problems

arise in waste heat recovery systems where heat availability and utilisation periods are often

different, requiring thermal energy storage [1,4]. The most commonly used method of

thermal energy storage is the sensible heat method. The sensible heat storage refers to energy

systems that store thermal energy without changing phase. Sensible heat devices store

energy by raising the temperature of the storage medium. The amount of energy stored

depends on the temperature change and specific heat of the material and can be expressed as:

∫

where m is the mass and is the specific heat. As specific heat of a material is generally

two orders of magnitude smaller than its latent heat, sensible heat storage requires a much

larger volume of material to store the same amount of energy as compared to latent heat

storage [6]. Hence, sensible heat storage devices are very heavy and bulky in size. Heating of

a material that undergoes phase change (usually melting) is called latent heat storage. The


2

amount of energy stored depends on the mass and latent heat of fusion of the material and

can be expressed as:

where L is the latent heat of fusion. Materials used for latent heat storage are referred to as

phase change materials (PCMs). Latent heat storage is more attractive as it provides a high

energy storage density and can absorb or release energy at a constant temperature [3]. Hence

the use of PCMs for thermal energy storage has been of great interest in recent years in fields

such as waste heat recovery, solar energy utilization and passive cooling of electronic

devices [1-4]. PCM‟s are generally divided into three main categories: organic, inorganic

and eutectic compounds. Low temperature PCM‟s (< 200 ) (organic, inorganic or eutectic)

are mainly used in waste heat recovery systems and buildings, while high temperature

PCM‟s (> 200 ) (inorganic or eutectic) are used in solar power plants and other high

temperature applications [35].

Materials to be used for phase change thermal energy storage must have high latent heat of

fusion and high thermal conductivity. They should also have a melting temperature lying in a

practical range of operation, melt uniformly, be chemically inert, low cost, non toxic and non

corrosive [1,4]. As paraffin wax possesses most of these properties it attracts considerable

attention as a PCM. However, paraffin waxes have inherently low thermal conductivity and

so it takes considerable time to melt and solidify, which reduces the overall power of the

thermal storage device and thereby restrict their application [2]. Several methods have been

proposed to increase the thermal conductivity of paraffin wax including addition of


3

enhancers such as metallic fillers, finned tubes and aluminum shavings [4]. However, these

enhancers add significant weight and cost to the storage system and some of them are

incompatible with PCM‟s [2]. High porosity open cell metal foams can also serve as

thermal conductivity enhancers as they are available in copper, graphite, aluminum or nickel

foam whose thermal conductivity is very high (>80 W/mK) and have low bulk density and

are chemically inert [3]. The purpose of the present study was to determine the feasibility of

using metal foams to enhance the heat transfer capability of phase change materials. In the

present work, 10 pores per inch (PPI) copper foam (95% porous) was impregnated with

paraffin wax that melts in the temperature range of 42 - 50 .

1.2 Literature Review

Extensive research has been carried out to improve the thermal response of PCM by adding

different high thermal conductivity enhancers. In this section a summary of the relevant

research regarding PCM‟s is presented.

The use of finned tubes with different configurations has been proposed by various

researchers. Lacroix and Benmadda [53] studied the behaviour of a vertical rectangular

cavity filled with PCM. They found that both solidification and melting rates were improved

by long fins. Velraj et al [54] studied the impact of internal longitudinal fins on a cylindrical

vertical tube filled with paraffin wax. They concluded that adding fins reduces the

solidification time by a factor of 1/n, where n is the number of fins. They also pointed out


4

that for lower Biot numbers, addition of fins makes the surface heat flux more uniform,

whereas for higher Biot numbers the addition of fins improves the magnitude of surface heat

flux and appreciably reduces the solidification time. Stritih et al [11] studied the heat transfer

characteristics of a latent-heat storage unit with and without a finned surface. They

developed a correlation giving the dimensionless Nusselt number as a function of Rayleigh

number. A comparison of the equations for melting and freezing shows that natural

convection is present during melting and increases heat transfer, whereas during

solidification conduction is the dominant form of heat transfer. They concluded that heat

transfer during solidification is greater if fins are included and a 40% reduction in

solidification time is observed with fins.

Mettawee and Assassa [15] placed aluminum powder in the PCM for a compact PCM solar

collector. Solar energy was stored in the PCM and was discharged to cold water flowing in

pipes located inside the PCM. The propagation of melting and freezing fronts was studied

during the charging and discharging process. It was found that the addition of aluminum

powder in wax reduced the charging time by 60%. In the discharging process, it was found

that the useful heat gained was increased by adding aluminum powder in the wax. Bugaje et

al. [14] found that the thermal response of paraffin wax was enhanced by the use of metal

matrices embedded within the body of wax. Significant reductions in melting and freezing

times were obtained by the use of aluminum sheet metal. Melting times were reduced by

factors of up to 2.2 and freezing times reduced by factors of up to 4.2. It was also found that

thermal response enhancement is greater during freezing than melting as conduction plays a

greater role in freezing while natural convection becomes significant during melting.


5

Py et al [8] did some research on a new supported PCM made of paraffin impregnated in a

compressed expanded natural graphite (CENG) matrix and found thermal conductivities in

the range of 4 to 70 W/mK while that of paraffin wax is 0.24 W/mK. It was also found that

CENG induced a decrease in overall melting and solidification time. Zhong et al [13] used

CENG matrices with different densities to see the increase in thermal response of paraffin

wax. To predict the performance of paraffin wax/CENG composites as a thermal energy

storage system, their structure, thermal conductivity and latent heat were characterized.

Results indicated that the thermal conductivity of the composites can be 28-180 times that of

pure paraffin wax. Mesalhy et al [49] studied numerically and experimentally the effect of

porosity and thermal properties of a porous medium filled with PCM. In their model, the

governing partial differential equations describing the melting of phase-change material

inside porous matrix were obtained from volume averaging of the main conservation

equation of mass, momentum and energy. From their model it was found that the best

technique to enhance the response of PCM is to use a solid matrix with high porosity and

high thermal conductivity. Model results indicate that estimated value of the average output

power using carbon foam of porosity 97% is about five times greater than that for using pure

PCM‟s.

One intrinsic problem of a graphite matrix is its anisotropy in which the thermal conductivity

depends on direction [35]. To solve this problem, some metal materials with high thermal

conductivities were used by several researchers to enhance the heat transfer performance of

the PCM‟s. Zhao et al. [35] did experimental investigation on the solid/liquid phase change

in which paraffin wax was embedded in high porosity (> 85%) open cell copper metal

foams. The test samples were electrically heated on the bottom surface with a constant heat


6

flux. They found that the addition of metal foam increases the overall heat transfer by 3-10

times during the melting process. They also found that the temperature gradient in metal

foam sample is significantly reduced compared to pure PCM.


7

1.3 Research Objectives

The objectives of this research are:

To measure the enhancement in thermal conductivity of paraffin wax when used with

10 pores per inch (PPI) copper metal foam with porosity ( ) = 0.95.

To evaluate the use of high porosity copper metal foam material in thermal energy

storage application

To design and test different thermal energy storage systems (TESS), with and

without copper metal foam, and to analyze the effect of copper metal foam on

heat transfer, during charging and discharging process, by determining the

convective heat transfer coefficient

Develop a Nusselt number correlation based on geometric parameters to describe

the experimental results

To build an actual TESS device and test its performance

Develop a numerical code to determine the temperature field within PCM during

charging i.e. melting process


8

1.4 Organization of Thesis

The present chapter gives a general background, literature review and objectives of this

research thesis.

Chapter 2 explains the experimental apparatus and methodology used to determine the

enhancement in thermal conductivity of paraffin wax when used with copper metal foam.

The thermal conductivity value determined from the experiments is then compared with a

theoretical model developed by Boomsma and Poulikakos [5]. A photographic study of

propagation of the melting front of wax – with and without metal foam - is carried out to

determine the enhancement in rate of heat transfer.

Chapter 3 describes the design and analysis of a TESS and compares the performance of

TESS (with and without metal foam). The effect of addition of metal foam on thermal

resistance is analysed. The final part of the chapter describes the design and performance

analysis of a TESS device.

Chapter 4 describes the numerical model developed to determine the temperature field in

PCM while melting. It is concluded that the assumption of single temperature phase change

is not valid and so the code is modified such that it considers phase change over a

temperature range.

Chapter 5 lists the conclusions drawn from the present research.

At the end, references, and the appendices containing schematics and raw data obtained

during experimental runs are included.

CHAPTER 2. THERMAL CONDUCTIVITY ENHANCEMENT OF PHASE CHANGE

MATERIALS USING METAL FOAMS

9

Chapter 2

Thermal Conductivity Enhancement of Phase Change Materials Using Metal Foams

2.1 Introduction

Phase change materials such as paraffin wax are known to exhibit slow thermal response due

to their relatively low thermal conductivity [1,4]. In this study, experiments were carried out

to investigate a method of enhancing thermal conductivity of paraffin wax by making use of

high porosity open cell metal foams. In the present work, 10 pores per inch (PPI) copper

foam (95% porous) was impregnated with paraffin wax that melts in the temperature range

of 42 – 50 . The copper foam was heated at one end with a constant temperature boundary

condition and time varying temperatures were measured along its length. Thermal

conductivity was measured by modeling the test sample as a semi-infinite medium. The

experimental results were then compared with a theoretical model proposed by Boomsma

and Poulikakos [5]. In addition, photographic observation of propagation of melting front of

wax was done, for both pure wax and foam-wax system, to determine the enhancement in

melting rate of wax for the same time and surface temperature conditions when used with

copper metal foam.



10

2.2 Experimental Test Apparatus

The experimental apparatus used to test the foam-wax composite system is shown

schematically in Figure 2.1. A rectangular copper foam with a thickness of 20 mm and a

pore density of 10 PPI (Dalian Thrive Mining Co. Ltd, Dalian, China) was cut using an

electric-discharge machine (AD325L CNC Wire EDM, Sodnick, Japan) to the dimensions of

260 mm x 20 mm x 20 mm (L x W x H). In order to provide a heating surface to the foam,

and to ensure maximum attainable heat transfer to the foam, a T-shaped copper plug was

soldered to the flat face of the foam by using a Sn-alloy soldering paste (Loctite RP15,

Henkel AG and Company, Dusseldorf, Germany). The foam and plug were inserted into a

300 x 25 x 25 mm square polycarbonate tube (McMaster-Carr, OH, US). Before inserting

the foam, 2 mm diameter holes were drilled on one face of the tube at distances of 1, 2, 3, 5,

7, 9, 11 and 28 mm from one end of the polycarbonate tube. Eight Type-K thermocouples

with a 0.25 mm diameter junction were inserted through the holes drilled in the

polycarbonate tube to measure the temperature variation along the length of the foam. The

holes were then covered with high temperature cement (CC High Temperature, Omega,

Stamford, CT) to prevent leakage of wax. The thermocouple voltages were recorded by a

National Instruments Data Acquisition (DAQ) unit and transmitted directly to a personal

computer. The output voltages of the thermocouples were recorded in real-time using a

personal computer equipped with LabVIEW Signal Express v3.0.



11

Figure 2.1 : Schematic of the experimental apparatus showing the test sample and

instrumentation



12

To ensure a leak-proof joint between copper plug and the tube, ultra blue RTV silicon gasket

(Permatex) was applied on the mating surfaces of plug and the tube. The silicon gasket was

allowed to dry for 24 hours and after that paraffin wax (melting temperature 50C, thermal

conductivity 0.21 W/mK) was poured into the polycarbonate tube in liquid state and allowed

to solidify. A 7 mm diameter and 40 mm long blind hole was drilled in the center of the

copper plug to insert the cartridge heater (Model CSH-101120/120V, Omega, Stamford,

CT). The heater was 6.35 mm (1/4 in.) in diameter and is 38.1 mm (1.5 in.) long. It had an

electrical resistance of 130 and produced a maximum power output of 110.76 W at 120 V.

Power to the heater was provided by a variable transformer which in turn was connected to a

temperature controller. To prevent any heat loss to the surroundings the entire test sample

was covered using 38.1 mm (1.5 in.) thick fibreglass insulation with a aluminized outer

surface, having an average thermal conductivity of 0.038 W/mK (Micro-Flex, Johns

Manville Corporation, Denver, CO). A 3 mm diameter and 78 mm long blind hole was

drilled at one edge of the plug to insert the thermocouple probe to maintain a constant

surface temperature of the plug. A Type-K (chromel-alumel) thermocouple probe with a

304SS sheath of 0.813 mm diameter (Model TJ36-CASS-032-G-6, Omega, Stamford, CT)

was inserted in the plug. The other end of the thermocouple was connected to a temperature

controller (Model CN2110, Omega, Stamford, CT) which helped to maintain a constant

surface temperature by controlling the power output to the heater.



13

Figure 2.2 : Photograph of the experimental apparatus showing test sample and

instrumentation

Figure 2.3 : Detailed view of Test sample showing the placement of thermocouples

Temperature Controller

Controller

Data Acquisition

Controller

Test Sample S

S

Thermocouple wires



14

The uncertainty in the temperature measurement is measured using the root sum squares

(RSS) method. In RSS, the bias and precision elemental errors within the instrumentation are

combined to determine the realistic value for uncertainty. The uncertainty associated with

data acquisition system is:

√

√

The type K thermocouples used in these experiments have a standard limit of error of

. The resolution of LabView was set at , thus the zero order uncertainty

associated with it was . Hence, the uncertainty associated with thermocouples is:

√

√



15

Hence the overall uncertainty associated with temperature measurement is the combined

error of thermocouple and data acquisition system:

√

√



16

2.3 Results and Discussion

The analysis of metal foam impregnated with paraffin wax uses the standard solutions used

in analysis of heat conduction in semi-infinite medium described by Cengel [37]. Several

assumptions regarding the heat transfer that were made in the analysis are as follows:

The porosity is constant throughout the length of the foam

Natural convection and radiation heat transfer effects inside the porous medium are

neglected and the heat transfer is one dimensional: heat transfer occurs only in the

direction normal to the surface

The physical properties of the solid and fluid phases are assumed to be the same and

constant over the entire temperature range; i.e. volume change is neglected

The solid and fluid phases are in local thermal equilibrium

The length of the test sample is long enough to assume it is a semi-infinite medium

A medium is said to be semi-infinite if a step change in temperature at one end (x = 0) does

not change the temperature at the other end (x = l ) during the time of observation. To

demonstrate that this was a reasonable assumption in these experiments, time varying

temperatures at different points along the length of the column are plotted in Figure 2.4 for

the case Ts = 85 .



17

Figure 2.4 : Experimentally obtained temperature results for 85 plotted as a function

of time at different axial distance along the length of the test sample. The experimental

uncertainty is 2.6 in the temperature measurement.

It can be seen from Figure 2.4 that even after 1500 seconds the temperature at a distance of

x = 26 cm does not change at all. Thus, the assumption that the test sample is a semi –

infinite is valid during this time interval.

Based on the assumptions listed above the temperature variation along the length of foam for

a specified surface temperature, Ts = constant is given by Cengel [37] as

(

√ )

10

20

30

40

50

60

70

80

0 200 400 600 800 1000 1200 1400

Tem

pe

ratu

re (

c)

Time (sec)

x = 1cm

x = 2 cm

x = 3 cm

x = 5 cm

x = 7 cm

x = 9 cm

x = 11 cm

x = 26 cm



18

where is the transient temperature measured by a thermocouple at distance x and

time t from the face of the test sample using the data acquisition unit. is the initial

temperature of the test sample i.e. ambient temperature and is the surface temperature

which is kept constant with the help of a temperature controller. So the only unknown in

Equation (2.1) is the thermal diffusivity which can therefore be directly calculated.

Equation (2.1) is valid only when there is a step change in surface temperature. In the present

case the copper plug takes at least 4 to 5 minutes to reach the constant-surface temperature

condition and because of that, the temperature recorded by the thermocouples is lower than

the true temperature at which the foam-wax system would have been if its surface

temperature was increased instantaneously. As the measured value of temperature is always

lower than the true value, the numerator on the left hand side of Equation (2.1) is always

smaller than in reality. As thermal diffusivity is inversely proportional to the temperature

difference, the value we calculate from Equation (2.1) is expected to be smaller than the true

value.

The values of thermal diffusivity are measured at = 65, 70, 75, 80 and 85C. The values of

thermal diffusivity at different temperatures and location are shown in Figure 2.5. Thermal

diffusivity is plotted on a log scale whereas time is plotted on a normal scale. It can be seen

from the graph that the thermal diffusivity value is initially higher and after about 200

seconds it reaches a constant value. Thermal diffusivity value starts dropping once steady

state is reached because at steady state is a constant value and so the temperature

terms on left hand side of Equation (2.1) are all constant. Thus, at steady state, thermal



19

diffusivity is inversely proportional to time and as time is always increasing thermal

diffusivity value starts dropping once steady state is reached. Hence, to ignore the initial and

end effects and for simplicity, only the values that are within the two bounds shown in

Figure 2.5 are taken into consideration.

Figure 2.5 : Experimentally computed thermal diffusivity values from Equation (2.1) for

different surface temperatures and different distance along the length of the test sample. The

thermal diffusivity values falling between two bounds are only taken into consideration.

The thermal diffusivity values that are within the bounds were averaged for each

temperature. After that, they were averaged over different temperatures to find the overall

average thermal diffusivity value for the foam-wax composite system. This overall average

value was used to find the effective thermal conductivity of the foam-wax system.



20

Effective thermal conductivity is calculated by using the relation:

where = thermal diffusivity, = density and = specific heat. The effective value of

density and specific heat that will be used in Equation (2.2) was computed by accounting for

the volume fraction of each substance, giving the resulting relation for density and specific

heat based on porosity as:

where = porosity of metal foam. The subscripts and s are used for fluid and solid phases

respectively. The subscript denotes the effective value of a property. The value of for

10 PPI copper foam is 0.95 [52].



21

Table 2.1 shows the computed values of thermal diffusivity whereas table 2.2 shows the

computed values of effective density and specific heat using Equation (2.3) and (2.4)

Temperature (C) Average value of thermal

diffusivity - (m2/sec)

65 1.05 x 10-6

70 1.02 x 10-6

75 9.78 x 10-7

80 1.05 x 10-6

85 1.01 x 10-6

Overall average value of

thermal diffusivity 1.02 x 10

-6

Table 2.1: Thermal diffusivity values at different temperatures

Material Density

(kg/m3)

Effective

Density

(kg/m3)

Material Specific Heat

(J/kg.K)

Effective Specific

Heat (J/kg.K)

Copper 8933 [37] 446.65 Copper 385 [37] 19.25

Wax 930 [1] 883.5 Wax 2900 [1] 2755

Total Effective Density 1330.15 Total Effective Specific

Heat 2774.25

Table 2.2: Effective Density and Specific Heat values for the foam-wax system

The effective thermal conductivity is then calculated by substituting the values, computed in

Table 2.1 and 2.2 into Equation (2.2). The effective thermal conductivity computed by using

Equation (2.2) is 3.8 W/mK



22

2.4 Comparison with Theoretical Model

The calculated thermal conductivity can be compared with a theoretical model developed by

Boomsma and Poulikakos [5] who developed a model to estimate effective thermal

conductivity of a porous metal foam, based on the idealizing its structure as being a series of

cells with the shape of a tetrakaidecahedron. The complete cell of tetrakaidecahedron

consists of six squares and eight hexagons. It is the idealized shape that most of the foam

cells would attain because of the nature of foam manufacturing process. [5,56]

Figure 2.6 : A unit cell structure of tetrakaidecahedron with six squares and eight hexagons

[5]



23

They developed the following equations to calculate the effective thermal conductivity :

√√ ( √ )

( √ )

( ) ( )

(√ )

( √ ) (√ ( √ ))

√

where = porosity of the metal foam (i.e. 0.95), e = dimensionless cubic node length, =

thermal conductivity of solid phase (i.e. Copper foam, 400W/mK), thermal

conductivity of fluid phase (i.e. paraffin wax, 0.21 W/mK), = effective thermal

conductivity of the composite system. A detailed derivation of these equations is given in

reference [5]. The value of e is determined by the authors to be a constant equal to 0.339.



24

The theoretical value of effective thermal conductivity is then computed by making use of

Equations (2.4) to (2.9) and the value comes out to be 5.02 W/mK, which is within 25% of

the value calculated in the previous section. Besides the reason stated in the previous section,

about the time required for heater to reach the constant surface temperature condition, the

difference between experimental and theoretical value of thermal diffusivity can be due to

the following factors :

The porous medium is considered to be uniform but in reality the porosity varies

along the length. Also the model is highly dependent on porosity. A small increase in

porosity causes a large decrease in thermal conductivity.

The theoretical model does not take into account the thermal resistance between the

foam and copper plug.



25

2.5 Photographic observation of melting front

The objective of this chapter is to investigate the propagation of melting front of wax with

and without the use of metal foam. The photographs are taken at various values of for

both cases to compare the amount of wax melted. The photographs are taken 45 minutes

after the heater was turned on, by which time the melting front was stationary.

65 70 75

80 85

Figure 2.7 : Experimental pictures showing propagation of melting front of wax without

metal foam



26

The photographs shown in Figure 2.7 are taken at various temperatures, for melting front of

wax, without the use of metal foam. It can be seen clearly from the photographs that the

maximum amount of wax that can be melted without the use of metal foam at 85C is around

1.2 cm. Temperature is not increased beyond 85C as it results in softening of the

polycarbonate tube.

65 70 75

80 85

Figure 2.8 : Experimental pictures showing propagation of melting front of wax with metal

foam



27

The photographs shown in Figure 2.8 are taken at various temperatures with the use of metal

foam. It can be seen from the pictures that the maximum amount of wax that can be melted

at 85C is around 3 cm which is 2.5 times more than that can be melted without the use of

metal foam. The thermal conductivity of paraffin wax is very low (0.21 W/mK) and because

of that the amount of wax that can be melted is also very low (for a given time and

temperature boundary condition). However, when wax is used with metal foam it increases

the overall thermal conductivity of the foam-wax composite system and so a greater amount

of wax can be melted with the same temperature boundary condition with the use of metal

foam. Also, as metal foam is 95% porous, only 5% reduction in storage volume of wax is

observed in foam-wax composite system.

CHAPTER 3. THERMAL ENERGY STORAGE WITH METAL FOAM

28

Chapter 3

Thermal Energy Storage System with Metal Foam

3.1 Introduction

The purpose of a thermal energy storage device is to overcome the time difference between

the availability of and demand for thermal energy. These systems are used in order to store

thermal energy during a period when the supply is sufficient or cheaper, to be discharged

when the supply becomes insufficient or expensive [6]. A latent heat storage system is a

practical device that promises high thermal storage density because the phase change

material (PCM) can absorb or release a large amount of heat during melting or solidification

process [17]. One of the major drawbacks of current PCM‟s is their low thermal conductivity

and so it takes considerable time to melt and solidify, which in turn reduces the rate at which

thermal energy can be stored and extracted and so it restricts their application. One way of

improving the thermal conductivity is to make use of high porosity (> 85%) open cell metal

foams which enhances heat transfer due to their high thermal conductivity and high surface

area density. The aim of this section is to evaluate the use of high porosity metal foam in

thermal energy storage application.


29

3.2 Experimental Test Apparatus and Procedure

One of the objectives of this research is to demonstrate the use of metal foam material for

thermal energy storage. The objective is to fabricate a small scale thermal energy storage

device, both with and without metal foam, and to carry out necessary experimentation to

determine the effect of metal foam. The experimental apparatus consists of the air facility,

thermal energy storage system (TESS) and instrumentation. A schematic representation of

the experimental setup is shown in Figure 3.1. Compressed air provided by the laboratory

was connected to a pressure regulator attached to a globe valve, which when adjusted, can

control the amount of air released to the test section. Measurement of mass flow rate was

done by an electronic gas mass flow-meter (Model FMA1842, Omega Company, Stamford,

CT) for a flow rate range of 20.0 L/min to 60.0 L/min. The air enters the test section via

ductwork mated to a 9.5 mm (0.375 in.) tee-junction. The direction of air flow was from left

to right in Figure 3.1 and progresses downstream where it exits to the surrounding. Both inlet

and outlet tee-junctions were fitted with 13 mm (0.5 in.) Type K thermocouple pipe plug

probes to measure the inlet and outlet temperature of air respectively. The TESS was made

out of 6.35mm thick polycarbonate sheet (McMaster Carr, OH, USA). The dimensions of the

TESS were 212 mm x 62.5 mm x 62.5 mm (L x W x H). Two rectangular slots were milled

on the inside of the TESS box to accommodate the copper plates. The purpose of the

copper plate was to form two separate compartments, the top compartment to store wax and

the bottom one to pass air through it, as well as to provide a heating surface to melt wax. The

height of wax compartment was 20 mm whereas that of air compartment was 25 mm.


30

F

igure

3.1

: S

chem

atic

of

the

TE

SS

appar

atus

show

ing t

he

inst

rum

enta

tion a

nd a

ir s

upply

confi

gura

tion


31

A flexible silicon rubber heater (Model SRFG – 208/10-P, Omega Company, Stamford, CT)

with dimensions of 203.2 mm x 50.8 mm (2 in. X 8in.) approximately the same size as the

test sample was used for melting the wax during the charging process. It had an electrical

resistance of 82.0 Ω and produces a maximum power output of 160 W at 115 V. Power to

the heater was provided by a variable transformer that was used to supply 50 V to the heater

to ensure that the TESS was not heated rapidly. Three Type-K (chromel-alumel)

thermocouples, two at the copper plates and one in the middle of the wax section were used

to measure the transient variation in temperature during charging as well as discharging

process. In order to ensure good contact between thermocouple and the copper plate,

thermocouple wires were welded to the copper plate using a Hotspot TC welder. The

thermocouple welded to the top copper plate, which had heater attached to it, was used as an

input to the temperature controller (Model CN2110, Omega, Stamford, CT) to maintain a

constant temperature boundary condition on the top surface. A small hole was drilled in the

middle of the wax compartment to attach a 0.813 mm (0.032 in.) compression fitting. A

Type-K thermocouple probe with a 304 SS sheath of 0.813 mm diameter (Model TJ36-

CASS-032-G-6, Omega Company, Stamford, CT) was inserted until the probe tip was

approximately at the mid-point of the sample width.

In order to ensure minimum heat loss to the surroundings the entire test sample was covered

using 38.1 mm (1.5 in.) thick fibreglass insulation with an aluminized outer surface, having

an average thermal conductivity of k = 0.038 W/mK (Micro-Flex, Johns Manville

Corporation, Denver, CO). The pipe plug thermocouples along with the thermocouple at the

middle and bottom of wax compartment were connected to a National Instruments Data

Acquisition (DAQ) unit that transmits the temperature readings of the thermocouple in real-


32

time to a personal computer equipped with a graphical programming environment to record

the temperature readings (LabVIEW Signal Express V 3.0, National Instruments

Corporation, Austin, TX). In order to ensure maximum attainable heat transfer from the

copper plate to the foam struts, a Sn-alloy soldering paste (Loctite RP 15, Henkel AG and

Company, Dusseldorf, GER) was applied to bond the two components (copper foam and

copper plate) together. The solder paste was applied on both copper plates (with

thermocouples welded) and then metal foam was sandwiched between them. To ensure good

contact between the foam and copper plates, the entire sample was clamped using two C-

clamps. After that, the entire sample along with C-clamps was placed in the oven and heated

till the oven reached a temperature of 225 . The sample was left to cool overnight in the

oven.

Molten wax was poured into the TESS device and allowed to solidify. Due to the difference

in solid and liquid densities of wax, the volume of wax decreased upon solidification leaving

an air gap at the top. When the wax was melted again, it expanded in volume and filled the

air gap. A typical experimental run started off by turning on the heater to melt the wax.

Temperatures at the center and bottom of the wax were recorded with the data acquisition

device. Once the wax had completely melted the heater was switched off and the compressed

air supply turned on. Mass flow rate of air was controlled by adjusting the globe valve. The

inlet and outlet temperature of air were measured by the pipe plug thermocouple probes and

were recorded by the data acquisition device. The compressed air supply was kept on for an

hour. This experimental procedure was repeated for different flow rates of air ranging from

20.0 L/min to 60.0 L/min.


33

Figure 3.2: TESS apparatus with instrumentation

Figure 3.3: Detailed view of TESS showing the placement of thermocouples used for

measuring inlet and outlet temperature of air

Temperature Controller Data

Acquisition Air Mass Flow meter

TC for measuring air temperature

Heater


34

Figure 3.4: TESS with no Metal Foam

Figure 3.5: TESS with Metal Foam on wax side


35

3.3 Results and Discussion

The heat transfer characteristics of the TESS can be analyzed using standard heat exchanger

correlations described by Cengel [37]. Several assumptions were made in the following

analysis:

Natural convection and radiation heat transfer effects inside the porous medium are

neglected i.e. heat transfer is considered to be only through conduction

Physical properties of the fluid and solid phases remain constant throughout the

temperature range

Conduction resistance from the soldering paste and copper plate are neglected

Physical properties of the fluid are evaluated at bulk mean temperature and 1 atm

pressure

The heat transfer coefficient is considered to be constant along the length of foam

Convection is defined as the heat transfer from a solid surface to a fluid in the presence of

bulk fluid motion. Despite the complexity of convection, the rate of convection heat transfer

is observed to be proportional to the temperature difference and is conveniently expressed by

Newton‟s law of cooling as [37]

( )


36

From an energy balance the heat transfer to the fluid (air) flowing through the channel is

equal to the increase in energy of the fluid

( ) ( )

The value of convective heat transfer coefficient - h is computed by using equation (3.3) in

which is the temperature of copper plate that is measured experimentally with the

thermocouple. The concept of similitude allows one to define dimensionless numbers that

provides a means to compare systems with varying dimensions and flow parameters. In

laminar flow, the hydrodynamic entry length as described by Kays and Crawford [55] is

approximately:

The hydrodynamic entry length for TESS from Equation (3.4) is found to be 2.85 m at the

maximum flow rate of 60.0 L/min. In the present study as the length of TESS is only 200

mm the flow is still developing and so the fluid does not feel the effect of presence of

polycarbonate wall on the other side. Hence, the fluid motion can be considered as flow over

a flat plate and the characteristic length is taken to be length of the copper plate i.e. Lc = 200

mm. The factors controlling forced convection are defined as:

Nusselt number


37

Reynolds number

where the mean velocity is determined from

When metal foam was used, TESS was able to store approximately 180 ml of wax. As

density of wax is 930 kg/m3, the total mass of wax used was approximately 0.168 kg. The

latent heat of wax is 190 KJ/kg and so the TESS was able to store 31.8 KJ of energy. The

volume occupied by wax was around 1.9 x 10-4

m3 and the remaining volume i.e. 1 x 10

-5 m

3

was occupied by metal foam. As density of copper is 8933 kg/m3 the mass of metal foam

used was approximately 0.08933 kg. Using the same ideology, if it is required to store 1 MJ

of energy then the TESS should be designed such that it can hold 5.3 kg wax and that would

require a total storage volume of 6.3 x 10-3

m3. The mass of metal foam used will be

approximately 2.8 kg.


38


melting at a distance of 1 & 2 cm along the length of wax compartment for both – with foam

and without foam case. The experimental uncertainty is 2.6 in the temperature

measurement.

As paraffin wax melts in the temperature range of 42 - 50 , it is considered to have melted

completely when the temperature at depth of 2 cm reaches 55 . As seen from the figure,

without the use of metal foam, paraffin wax takes around 5000 seconds to melt completely

whereas with the use of metal foam it takes around 1800 seconds. The time required to melt

approximately the same amount of wax when using metal foams is reduced to 36% of that

necessary without metal foam. Also, the temperature gradient in wax without metal foam is

significantly higher than with metal foam. This reduction of temperature gradient can be

attributed to the significant increase in effective thermal conductivity due to addition of

20

24

28

32

36

40

44

48

52

56

60

64

68

72

76

0 1000 2000 3000 4000 5000

Tem

pe

ratu

re (

c)

Time (sec)

T @ 1cm (no foam)

T @ 2cm (no foam)

T @ 1cm (foam)

T @ 2cm (foam)


39

metal foam so that, the temperature distribution in the foam-wax system is more uniform

than it is in pure wax.

Figure 3.7: Experimentally obtained temperature results plotted as a function of time during

melting at a distance of 1 & 2 cm along the length of wax compartment with metal foam.

The experimental uncertainty is 2.6 in the temperature measurement.

As discussed earlier, the temperature gradient while melting is significantly lower than that

in a pure wax system. Also, from the graph it can be seen that the phase change starts when

the slope of the temperature profile starts decreasing and phase change is complete when the

slope starts increasing again. From the graph, with the help of tangents drawn to the curve, it

can be seen that phase change starts around 42 and it ends around 50 .

20

22

24

26

28

30

32

34

36

38

40

42

44

46

48

50

52

54

56

58

60

0 200 400 600 800 1000 1200 1400 1600

Tem

pe

ratu

re (

c)

Time (sec)

T @ 1cm

T @ 2cm

Beginning of Phase change

End of Phase change


40


solidifying at a distance of 1 & 2 cm along the length of wax compartment for both – with

foam and without foam case when the mass flow rate of air is 40.0 L/min. The experimental

uncertainty is 2.6 in the temperature measurement.

It can be seen that the foam-wax system solidifies more uniformly than the pure wax system.

As discussed earlier, the effective thermal conductivity of foam-wax system is significantly

higher than that of pure wax, which in turn reduces the thermal resistance to heat transfer

and so the temperature gradient is very small (<5 ). The temperature of wax at a depth of 1

cm reaches 60 when used with metal foam, whereas without it the temperature reaches

70 . As wax, with and without metal foam, is at different initial temperature at the

beginning of solidification, it is difficult to say whether metal foam helps to increase heat

transfer during solidification or not.

24

28

32

36

40

44

48

52

56

60

64

68

72

76

0 1000 2000 3000

Tem

pe

ratu

re (

c)

Time (sec)

T @ 1cm (no foam)

T @ 2cm (no foam)

T @ 1cm (foam onwax side)

T @ 2cm (foam onwax side)


41


function of time for both-with foam and without foam case when the mass flow rate of air is

40.0 L/min. The experimental uncertainty is 2.6 in the temperature measurement.

It can be seen that the outlet temperature of air with metal foam is slightly higher than that of

a pure wax system. This small increase in outlet temperature of air shows that addition of

metal foam on wax side does not help much in terms of heat transfer to air. Addition of

metal foam on wax side only helped to increase heat transfer during melting, but not during

solidification. Hence, in order to increase the heat transfer to air, it becomes necessary to add

metal foam on the air side as well. The addition of high porosity metal foam provides a large

solid-to-fluid surface area, combined with a high thermal conducting metallic phase, such as

copper, would allow for enhanced heat transfer by conducting heat from the metallic struts to

the air flowing through them.

20

21

22

23

24

25

26

27

28

29

30

31

32

0 1000 2000 3000

Tem

pe

ratu

re (

c)

Time (sec)

T outlet (no foam)

T inlet

T outlet (foam-waxside)


42

Figure 3.10: Photograph showing metal foam on both sides along with copper plates and

thermocouples

Figure 3.11: TESS with metal foam on both sides

Thermocouples

Copper plates


43


solidifying at a distance of 1 cm along the length of wax compartment for all the three cases

when the mass flow rate of air is 40.0 L/min. The experimental uncertainty in temperature

measurement is 2.6 .

As discussed earlier, addition of foam on wax side does not significantly affect heat transfer

to air, so foam was added on the air side, to determine its impact on heat transfer. As wax, at

the beginning of solidification, is at uniform initial temperature, at least for foam on wax and

for foam on wax as well as air side, it would be easier to conclude if addition of foam on air

side helps to increase heat transfer to air or not. It can be seen from Figure 3.12 that, after an

hour, wax is at a temperature of 24 when metal foam is used on both – wax and air side,

whereas at the same time wax is at a temperature of 34 when metal foam is used only on

the wax side. Thus, wax solidifies much faster with foam on both wax and air side. Addition

of metal foam on air side reduces the thermal resistance of heat transfer to air and so wax

20

24

28

32

36

40

44

48

52

56

60

64

68

72

76

0 1000 2000 3000

Tem

pe

ratu

re (

c)

Time (sec)

T @ 1cm (no foam)

T @ 1cm (foam-waxside)

T @ 1cm (foam-wax &air side)


44

was able to lose heat and hence cool much faster than in case of pure wax or with metal

foam on wax side.

Figure 3.13: Experimentally obtained results for inlet and outlet temperature of air plotted as

a function of time for all the three cases when the mass flow rate of air is 40.0 L/min. The

experimental uncertainty is 2.6 in the temperature measurement.

A significant increase in outlet temperature of air is seen for the case of metal foam on both

sides as compared to the previous two cases – pure wax and metal foam on wax side. This

significant increase in outlet temperature of air is mainly due to:

1. Increase in the convective heat transfer coefficient – h due to the tortuous path

provided by metal foam to incoming air that causes the flow to be more turbulent

20

22

24

26

28

30

32

34

36

38

40

42

44

0 1000 2000 3000

Tem

pe

ratu

re (

c)

Time (sec)

T outlet (no foam)

T inlet

T outlet (foam-waxside)

T outlet (foam-wax& air side)


45

2. Metal foams provide a large solid-to-fluid surface area and the presence of high

thermal conducting metallic phase enhances heat transfer by conducting heat from

metallic struts to the passing air

Figure 3.14: Experimentally computed Nusselt number with respect to Reynolds number for

all the three cases.

The experiments were performed at different flow rates of air for all the three cases and so it

was necessary to combine the data from these different experimental runs and transfer them

onto one graph to show the net effect of addition of metal foam. Hence, a graph of Nusselt as

a function of Reynolds number is plotted for all the three cases in which Nusselt number and

R² = 0.9949

R² = 0.9911

0

200

400

600

800

1000

1200

1400

1600

1800

2000 4000 6000 8000 10000 12000

Nu

sse

lt N

um

be

r

Reynolds Number

Nu(no foam)

Nu(foam-wax side)

Nu(foam-wax & airside)


46

Reynolds number are calculated using Equation (3.5) and (3.6) respectively. It can be seen

that Nusselt numbers for pure wax and foam on wax side are almost identical for all

Reynolds number. The reason being that addition of metal foam on wax side only helped to

reduce the melting time of wax and not much gain was obtained in terms of heat transfer to

air. But, by adding foam on the air side a significant increase in outlet temperature of air was

seen and that resulted in much higher Nusselt number for the same Reynolds number. The

Nusselt number varies linearly with Reynolds number for all the three cases with an R2 value

of at least 99%.

Figures 3.15 & 3.16 show the average power extracted – Q (W) and average convective heat

transfer coefficient – (W/m2K) for different flow rates of air for all three cases of TESS.

These values are averaged over the entire time range for which the air supply is on i.e. 3600

seconds.


47

Figure 3.15: Experimentally computed power extracted with respect to flow rate of air for all

the three cases of TESS

Figure 3.16: Experimentally computed convective heat transfer coefficient with respect to

flow rate of air for all the three cases of TESS

2

4

6

8

10

12

14

20 30 40 50 60

Po

we

r e

xtra

cte

d (

W)

Flow-rate of air (L/min)

No foam

Foam-wax side

Foam-wax & airside

0

20

40

60

80

100

120

140

160

180

200

220

20 30 40 50 60

Co

nve

ctiv

e h

eat

tra

nsf

er

coe

ffic

ien

t -

h (

W/m

2 K

)

Flow rate of air (L/min)

No Foam

Foam-wax side

Foam-wax &air side


48

3.4 Thermal Resistance

During the solidification process, the solidification front of wax is moving towards the top

copper plate i.e. solidification front is moving towards the outer surface whereas heat

transfer is in the opposite direction i.e. inwards through the already solidified PCM.

Therefore, during solidification, the thermal resistance of solid wax increases [8]. Hence, it

is essential to determine this thermal resistance and see how it varies with time. The power

extracted from the TESS P varies inversely as the sum of the thermal resistance due to

copper plate ( ), convective resistance on air side ( ) and resistance due to solidified layer

of wax ( ) so that:

The resistance due to the copper plate, per unit area, is given by:

The convective resistance on the air side, per unit area, is given by:

and the solid wax resistance, per unit area, that varies with time as:


49

where is the thickness of the solidification front in m. = 0 corresponds to the

beginning of solidification and corresponds to end of solidification. Though

varies with time, for comparison purposes, is considered to be 10 i.e. constant

for all the three cases. The value of h used in Equation (3.10) is determined experimentally

by making use of Equation (3.3). Also, as the copper plate is very thin (3 mm) would be

very small and hence for further analysis it is neglected. These resistances being in series,

the overall thermal resistance is expressed by

The equation used for calculation of the front position with respect to time is the heat energy

balance assuming that all the heat produced by the solidification at is withdrawn

towards the external fluid i.e. air [10,12]. Thus:

Substituting the expression for P from Equation (5.7) one gets,

Integrating both sides we get,


50

∫ (

)

Upon integration and substituting the limits we get an expression for solidification time

required to reach length as:

*

+

The time required for complete solidification of wax is calculated by substituting the value

of as l and is labelled as . The value of l for all the three cases is the length of wax i.e.

0.02 m. Thus,

*

+

As the time required for solidification would be different for all three cases of TESS, it

becomes essential to non-dimensionalise it so that they can be compared against each other.

The dimensionless solidification time =


51

The effect of increase in wax resistance during the solidification process is illustrated by

plotting the solid wax resistance against the non-dimensional solidification time . is

calculated by making use of Equation (3.11) and substituting different values of from 0 to

0.02 m.

Figure 3.17: Computed value of solid wax resistance with respect to non-dimensional

solidification time for all three cases of TESS

Without foam, the solid wax contribution to the overall thermal resistance increases sharply

during the first 20% of the solidification time. Without foam, after total solidification, solid

wax resistance was approximately 0.095 (K/W). Adding metal foam on the wax side, after

total solidification, solid wax resistance was approximately 0.005 (K/W). As shown earlier,

adding metal foam on both wax and air side did not help to reduce the melting time of wax

0.0000.0050.0100.0150.0200.0250.0300.0350.0400.0450.0500.0550.0600.0650.0700.0750.0800.0850.0900.0950.1000.105

0 0.2 0.4 0.6 0.8 1

Solid

wax

re

sist

ance

- R

w (

K/W

)

Non-dimensional

solidification time ()

No foam

Foam-waxsideFoam-wax &air side


52

and so the solid wax resistance at total solidification is same as that for the case of metal

foam on wax side. However, adding metal foam on air side helped to decrease the overall

thermal resistance R and this decrease can be seen when overall thermal resistance R is

plotted against non-dimensional solidification time as shown in Figure 3.18. The overall

thermal resistance is calculated by making use of Equation (3.12).

Figure 3.18: Computed value of Overall Thermal Resistance R with respect to non-

dimensional solidification time for all three cases of TESS

0.0000.0050.0100.0150.0200.0250.0300.0350.0400.0450.0500.0550.0600.0650.0700.0750.0800.0850.0900.0950.1000.1050.1100.1150.120

0 0.2 0.4 0.6 0.8 1

Ove

rall

Th

erm

al

Re

sist

ance

- R

(K

/W)

Non-dimensional

solidification time ()

No Foam

Foam-wax side

Foam-wax & airside


53

3.5 Three Stage Thermal Energy Storage System

The aim of the study described in this section was to build an actual TESS device and test its

performance. The TESS fabricated before were made out of polycarbonate and had only two

stages i.e. it contained wax on one side and air on the other. The main drawback of two stage

TESS was that the metal foam section through which air is passed had only its top surface in

contact with wax whereas the bottom surface was in contact with the polycarbonate wall and

so not all the available surface area of metal foam was used for heat transfer. The TESS

described here had three channels of metal foam stacked on each other separated by copper

plates. The top and the bottom channel of foam were filled with paraffin wax whereas air

was passed through the middle channel of foam. Also, in order to prevent the formation of

hot spots along the corners of TESS, compressed air was admitted through three separate

inlets instead of just one.

The three stage TESS was made out of 4.5 mm thick, hollow Aluminum box (McMaster

Carr, OH,USA). The dimensions of the TESS were 200 mm x 76.2 mm (3 in.) x 76.2 mm (3

in.) (L x W x H). Two rectangular slots were milled on the inside of the Aluminum box to

accommodate the copper plates. The copper plates helped to form three separate

compartments as shown in Figure 3.19. Two small holes were drilled, one on top and the

other on bottom surface, in the middle of the Aluminum box, to attach a 0.813 mm (0.032

in.) compression fitting as shown in Figure 3.20. A Type-K thermocouple probe with a 304

SS sheath of 0.813 mm diameter (Model TJ36-CASS-032-G-6, Omega Company, Stamford,

CT) was inserted until the probe touched the copper plate.


54

Figure 3.19: A cross section view of 3 stage TESS showing the 3 layers of foam separated

by copper plates

Figure 3.20: Top view of the TESS showing the hole for thermocouple fitting


55

These two thermocouples were used to measure the transient temperature variation as well as

to ensure the complete melting of wax. Two square end plates (76.2 mm x 76.2 mm) were

made out of 5 mm thick aluminum to seal the two faces of TESS. To introduce air supply,

three holes were drilled along the center line of the aluminum end plates to attach three 13.0

mm compression fittings. In order to attach the end plates to the aluminum box, sixteen 3.0

mm diameter holes were drilled on both, the end plates as well as the mating surfaces of the

aluminum box, so that the end plates can be bolted firmly to the aluminum box and thereby

forming a rigid structure. A flexible silicon rubber heater (Model SRFG – 304/10-P, Omega

Company, Stamford, CT) with dimensions of 7.62 mm x 10.16 mm (3 in. x 4 in.) is used to

melt wax during the heating process. Four such heaters are used, two on each side as shown

in Figure 3.21, with a gap between them to provide an opening for the thermocouple. All

four heaters were connected in parallel. The heater has an electrical resistance of 110.0 Ω

and produces a maximum power output of 120 W at 115 V. Power to the heater is provided

by a variable transformer that is used to supply 50 V to the heater to ensure that the TESS is

not heated rapidly

The remaining instrumentation (i.e. thermocouple probes for measuring inlet and outlet

temperature of air, air Mass Flow meter and the air circulation loop) was the same as used

for testing the two-stage TESS described in the previous section. The three stage TESS was

able to contain approximately 250 ml of wax in each compartment. As there were two such

wax compartments, the total volume of wax used was approximately 500 ml.


56

Figure 3.21: Photograph showing the flexible silicon rubber heaters that were used for

melting wax

Figure 3.22: Three stage TESS apparatus with instrumentation

End caps

Heater


57

Figure 3.23: Experimentally obtained results for outlet temperature of air plotted as a

function of time. The mass flow rate of air in L/min is indicated in parentheses. The

experimental uncertainty is 2.6 in the temperature measurement.

It can be seen from Figure 3.23 that even after 5400 seconds, the outlet air is at a higher

temperature as compared to two stage TESS for the same mass flow rate condition. The

difference in temperature is approximately 7-9 as seen from Figure 3.13. The three stage

TESS is broader and has wax compartments on both sides as compared to the two stage

TESS. Hence, the three stage TESS will be able to store more than twice the amount of wax

and thereby latent energy, as compared to the previous case. Thus, as the available energy

for extraction is higher, the outlet air is at a higher temperature for the three stage TESS.

However, it should be noted that the air still reaches a maximum temperature of around 46

for both two stage and three stage TESS.

18

20

22

24

26

28

30

32

34

36

38

40

42

44

46

48

50

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

Tem

pe

ratu

re (

c)

Time (sec)

T inlet

T outlet(20)

T outlet(30)

T outlet(40)

T outlet(50)

T outlet(60)


58

This shows that addition of one more wax compartment does not help to increase the

maximum attainable temperature of the TESS. However, it does help to supply air at a high

temperature for a longer duration of time. Hence, it can be concluded that the number of wax

or air compartments depends on the application. If an application requires fast energy

extraction or high temperature supply of air for a short period of time (< 1 hour) then it is

economical to have only one wax compartment and the number of air compartments depend

on how fast the energy needs to be extracted. On the other hand, if an application requires

high temperature supply of air for a long period of time (> 1 hour) then it is economical to

have more than one wax compartment and the total number of wax compartments depend on

the duration for which the high temperature air needs to be supplied.

As the experiments were performed for different flow rates of air ranging from 20.0 L/min to

60.0 L/min, a graph of Nusselt vs. Reynolds number is plotted similar to the one for two

stage TESS. Nusselt number and Reynolds number are calculated using Equation (3.5) and

(3.6) respectively. It can be seen that Nusselt number varies linearly with Reynolds number

with an R2 value of at least 99%


59

Figure 3.24 : Experimentally computed Nusselt number with respect to Reynolds number for

all the three stage TESS

Figures 3.25 & 3.26 show the average power extracted – Q (W) and average convective heat

transfer coefficient – (W/m2K) for different flow rates of air. These values are averaged

over the entire time range for which the air supply is on i.e. 5400 seconds.

R² = 0.9987

0

200

400

600

800

1000

1200

3000 4000 5000 6000 7000 8000 9000 10000

Nu

sse

lt N

um

be

r

Reynolds Number


60

Figure 3.25: Experimentally computed average power extracted – Q (W) with respect to flow

rate of air

Figure 3.26: Experimentally computed average convective heat transfer coefficient –

(W/m2K) with respect to flow rate of air

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

20 25 30 35 40 45 50 55 60 65

Ave

rage

Po

we

r e

xtra

cte

d (

W)

Flow-rate of air (L/min)

20

30

40

50

60

70

80

90

100

110

120

130

140

150

160

20 25 30 35 40 45 50 55 60 65

Ave

rage

co

nve

ctiv

e h

eat

tr

ansf

er

coe

ffic

ien

t (W

/m2 K

)

Flow rate of air (L/min)

CHAPTER 4. NUMERICAL SIMULATION OF TEMPERATURE PROFILE USING

ENTHALPY METHOD

61

Chapter 4

Numerical Simulation of Temperature Profile using Enthalpy Method

4.1 Introduction

Phase change of materials is an example of a boundary value problem, named after the

physicist Jozef Stefan, who introduced the general class of such problems in 1889. Very few

analytical solutions are available in closed form and the ones that are available are for one-

dimensional heat transfer in an infinite or a semi-infinite region [17]. When the PCM

changes state, both liquid and solid phases are present and they are separated by a moving

interface between them. There have been several numerical methods developed to deal with

the problem of phase change but the most attractive and common ones are the enthalpy

methods. The enthalpy method simplifies the phase-change problem since the governing

equations are same for the two phases and the method does not require explicit treatment of

conditions on the phase change boundary. The fact that the temperature and liquid fraction

fields are decoupled and thus can be calculated separately, makes it relatively easy to

implement [9].


ENTHALPY METHOD

62

4.2 Mathematical Model

The two-stage TESS (with metal foam) problem as described in the previous section is

solved numerically during melting. In order to check the validity of the code, temperature

profile that is obtained from the numerical code would be compared with the experimental

temperature results obtained at a depth of 1 cm shown in Figure 3.7. In the present analysis

the problem for phase change is solved numerically under the following assumptions:

The effects of natural convection within the melt are neglected and heat transfer is

considered to be one dimensional i.e. heat transfer occurs only in the direction

normal to the surface

The PCM (paraffin wax) is assumed to have a definite melting point ( = 50 ) i.e.

phase change is isothermal

The physical properties of the solid and fluid phases of PCM are assumed to be same

and constant over the entire temperature range i.e. volume change is neglected

Thermal resistance across the copper plate is neglected

Lateral sides of the TESS are well insulated

The problem is solved without considering the presence of metal foam and the

physical properties such as density and specific heat are calculated by accounting for

the volume fraction of each substance as described in Equations (2.3) and (2.4).

Thermal conductivity value derived from Boomsma‟s model [5] is used in the code


ENTHALPY METHOD

63

As a result of the above assumptions, the enthalpy formulation for the conduction-controlled

phase change can be written as [9]

(

)

where H is the total volumetric enthalpy, which is the sum of sensible and latent heat:

and where

∫

In the case of isothermal phase change, the liquid fraction is given by [9]:

{

Substituting Equation (4.2) into Equation (4.1) we get:

(

)


ENTHALPY METHOD

64

Equation (4.5) together with equations (4.3) and (4.4) and the appropriate initial and

boundary conditions represents the mathematical model of conduction controlled isothermal

phase change.

4.3 Numerical Solution

For the problem of one-dimensional phase change, Equation (6.5) reduces to:

(

)

Thus,

(

)

The fully implicit discretization equation for an internal node ‘i’ can be written as


ENTHALPY METHOD

65

Figure 4.1: Discretization domain for one-dimensional phase change problem with boundary

conditions

Consider first the case when control volume ‘i’ is fully solid or fully liquid. In that case,

from the definition of sensible enthalpy Equation (4.3) and the liquid fraction Equation (4.4),

we have [9]:

and

After introducing Equations (4.9) and (4.10), Equation (4.8) reduces to a heat diffusion

equation:


ENTHALPY METHOD

66

Backward Euler discretization in time gives:

After rearranging Equation (4.12) we get,

where coefficients e = g = -Fo, f = 1+(2 x Fo) and are introduced for the sake of

computational simplicity. Subscript ‘t’ refers to the previous time step whereas ‘t+1’ refers

to the current time step. Fo is a finite difference Fourier number:

As the phase change is considered to be isothermal, the coefficients of Equation (4.13) are

modified to e = g = 0, f =1 and for the control volumes which are undergoing phase

change

Now, consider the case in which melting is taking place at node ‘i’. When melting is taking

place, the liquid fraction i.e. lies strictly in the interval [0,1]. For an isothermal phase

change [9]:


ENTHALPY METHOD

67

and from Equation (4.3):

Equation (4.8) reduces to:

Backward differencing of the liquid fraction term gives:

Equation (4.18) updates the liquid fraction field within the control volume that is undergoing

a phase change. Equation (4.18) shows that the liquid fractions are updated from the

temperature field and not from the sensible enthalpy field. Hence, the temperature and liquid

fraction field are decoupled.

It is necessary to keep track of when the phase change initiates and ends so that the

coefficients of Equation (4.13) are modified accordingly. At the end of each time step, the

check for start and end of phase change is performed throughout the entire domain in the

following manner [9]:


ENTHALPY METHOD

68

START of melting:

For a given time step, if and

, it indicates that within this time step,

melting has started at node i [9]. In that case, the coefficients of Equation (4.13) are modified

to e = g = 0, f =1 and . However, when melting has just begun Equation (4.18) will

have the following form [9]:

After that i.e. in the next time step, Equation (4.18) would be used to update the liquid

fraction instead of (4.19).

END of melting:

For a given time step, if > 1 and

[9], it indicates that within this time step, PCM

has melted completely at node i. In that case coefficients of Equation (4.13) are again set to

e = g = -Fo, f = 1+(2 x Fo). However, in the time step when phase change boundary moves

from node i to i+1, will have the following form [9]:

(

)

After that i.e. in the next time step, would be just


ENTHALPY METHOD

69

Boundary conditions

The boundary nodes are discretized using the transient finite difference discretization

scheme as described in Cengel [37]. The node at x = 0 is subjected to a constant heat flux

boundary condition with Q being 28 W, whereas the node at x = n-1 is subjected to a

convective boundary condition with a convective heat transfer coefficient of h = 20 W/m2K

and as 22 . It is to be noted that the code is written in C++ and so the first node is

numbered 0 whereas the last node is numbered n-1. The final discretized equation of

boundary nodes is as follows:

Node at x = 0:

(

)

(

)

(

)

Node at x = n-1:

(

)

(

) (

)


ENTHALPY METHOD

70

The implementation of computational model would be as follows:

Coefficients e, f, g of Equation (4.13) are set to e = g = -Fo, f = 1+(2 x Fo) and

For nodal points, if and

, melting has started in this time step and

the coefficients of (4.13) are changed to e = g = 0, f =1, and liquid fraction

is updated using Equation (4.19)

For nodal points if

, coefficients of Equation (4.13) are

changed to e = g = 0, f =1, and liquid fraction is updated using Equation

(4.18)

For nodal points if

, coefficients of Equation (4.13) are again

changed to e = g = -Fo, f = 1+(2 x Fo) and will be used as described in Equation

(4.20)

The set of linear algebraic equations is solved using the Tri-Diagonal Matrix

Algorithm

The flow chart for the computational procedure is given in Fig. 4.2


ENTHALPY METHOD

71

Figure 4.2 Flowchart


ENTHALPY METHOD

72

4.4 Verification of the Computational Model

In order to validate the computational model, the numerical results would be compared with

the experimental ones. The experimental results used for this purpose are the one described

in Figure 3.7 for melting of wax with metal foam. The physical properties used in the

numerical code are listed in Table 3.1. The computational procedure described in 4.3 is for

an isothermal phase change at a single fixed temperature. So, the numerical results for phase

change at a single fixed temperature i.e. 50 will be discussed first.

Number of nodes 81

Length 0.02m

Thermal conductivity 5.02 W/mK [5]

Density 1330.15 kg/m3 (2.3)

Specific Heat 2774.25 J/kg.K (2.4)

Latent Heat 190000 J/kg [1]

Convective heat transfer coefficient 20 W/m2K

del t 1 sec

Melting temperature 50

Table 4.1 Various physical properties used in the numerical code


ENTHALPY METHOD

73

The experimental values of temperature at a depth of 1 cm is shown in Figure 4.3 along with

the error bars. The error bars are determined from the uncertainty in temperature

measurement that is found to be 2.6 as showed in the previous section. Figure 4.3 also

show the results from the numerical code for an isothermal phase change i.e. = 50 .

Figure 4.3 : Experimentally obtained temperature results plotted as a function of time, during

melting at a distance of 1 cm along the length of wax compartment with metal foam, along

with the computed temperature values from C++ code, for phase change at a single fixed

temperature. The experimental uncertainty is 2.6 in the temperature measurement

It can be seen from Figure 4.3 that the results from the numerical code initially agrees well

with the experimental results i.e. before start of phase change. But once phase change starts,

the code shows a large discrepancy in the temperature values. The code shows that the phase

2022242628303234363840424446485052545658606264

0 500 1000 1500 2000

Tem

pe

ratu

re (

c)

Time (sec)

Expt. result

Num. Result-Single step


ENTHALPY METHOD

74

change will continue till 1800 seconds and only after that, the temperature will increase. The

reason for the discrepancy is the fact that the phase change is assumed to begin at 50 but in

reality phase change is complete at 50 . The PCM used is paraffin wax which is a mixture

of various alkanes that melt at different temperatures. In other words, as the wax is not pure

it melts over a temperature range and so the phase change starts and ends earlier than

predicted by the code. So, it becomes essential to modify the code to incorporate the

condition that phase change takes place over a temperature range rather than at a single fixed

temperature. It can be inferred from the experimental results that the phase change starts

once the slope of the temperature profile starts decreasing and it ends once the slope starts

increasing again as shown in Figure 3.7. It can be seen that phase change starts at 42 and it

ends at 50 .

It is to be noted that even though the wax melts over a temperature range it is difficult to

measure exactly what fraction of wax would melt at a certain temperature. Hence, for

simplification purposes it was assumed that the wax would melt linearly over the

temperature range of 42-50 . It was assumed that the wax would melt in 4 steps of 25%

each i.e. once the temperature reaches 42 , 25% of wax would melt and the temperature of

wax would remain fixed at 42 till 25% of wax has melted. Once 25% of wax has melted

temperature will continue to rise till it reaches 45.5 at which another 25% of wax would

melt. Again temperature would remain constant till second 25% of wax has melted and then

increase to 48 at which another 25% of wax would melt. This will continue till the

temperature reaches 50.5 at which 100% of wax would have melted. Thus, the phase

change is still considered to be isothermal just that the wax will melt in equal fractions over

a temperature range instead of the entire melting taking place at a single fixed temperature.


ENTHALPY METHOD

75

The results of this modification can be seen in Figure 4.4 which shows the results of both -

phase change taking place at a single fixed temperature and phase change taking place over a

temperature range.

Figure 4.4 : Experimentally obtained temperature results plotted as a function of time, during

melting at a distance of 1 cm along the length of wax compartment with metal foam, along

with the computed temperature values from C++ code, for phase change at a single fixed

temperature and phase change over a temperature range. The experimental uncertainty is

2.6 in the temperature measurement.

It can be seen from Figure 4.4 that when phase change is assumed to take place over a

temperature range the numerical results are in much better agreement with the experimental

results. Even though the numerical values of temperature are marginally higher, the code still

follows the same pattern as the experimental curve.

2022242628303234363840424446485052545658606264

0 500 1000 1500 2000

Tem

pe

ratu

re (

c)

Time (sec)

Expt. values

Num. Result-Single step

Num. Result-Four steps

CHAPTER 5. CONCLUSIONS

76

Chapter 5

Conclusions

The use of metal foam in thermal energy storage application was evaluated by designing and

testing different thermal energy storage systems, with and without copper metal foam. The

following conclusions were drawn from the observations made during the study:

The equivalent thermal conductivity of a foam-wax composite was found to be 3.8

W/mK which was 18 times higher than that of pure paraffin wax (0.21 W/mK)

The experimental value of thermal conductivity was compared with a theoretical

model developed by Boomsma and Poulikakos [5] and the experimental value was

found to be within 25% of the theoretical one

Copper foam reduced the time required to melt approximately the same amount of

wax to 36% of that without the use of metal foam. The temperature gradients in

TESS (with metal foam) while melting and solidification were significantly lower

than that in a pure wax system.

The addition of metal foam on the wax side of the TESS helped to significantly

increase heat transfer during melting but did not increase heat transfer to air during

cooling. Hence, metal foam should be added to both wax and air sides to increase

heat recovery by air.

The outlet temperature of air passing through the TESS increases significantly when

metal foam is placed on both wax and air sides.

CHAPTER 5. CONCLUSIONS

77

The transient temperature variation in PCM can be modelled using the enthalpy

method. It is important to accurately to model the temperature range over which wax

melts rather than assume a single fixed temperature.

REFERENCES

78

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APPENDIX A: SCHEMATICS

84

Appendix A

Schematics

The following schematics are detailed drawings of different types of Thermal Energy

Storage System (TESS) fabricated for use in this study. The first drawing corresponds

to two stage TESS as described in Chapter 3. The second drawing corresponds to three

stage TESS also described in Chapter 3. All dimensions are in mm except wherever

specified.


85

Figure A1: Two stage TESS drawing


86

Figure A2: Three stage TESS drawing

APPENDIX B: THERMAL CONDUCTIVITY MEASUREMENT DATA

87

Appendix B

Thermal Conductivity Measurement Data

The raw temperature data obtained from the Thermal conductivity measurement

experiments described in Chapter 2 is given in this appendix in tabular form. The data

is grouped as follows: The table is named by the constant temperature condition at

which time varying temperatures are measured. In each table the first column lists the

time at which temperature is measured. The next columns lists the axial temperature

values.


88

Temperature at axial locations (C)

Time Ti 1 cm 2 cm 3 cm 5 cm 7 cm 9 cm 11 cm 26 cm

1 19.40 26.2 25.2 24.60 24.50 24.20 24.00 23.80 22.90

64 19.40 34.40 28.20 25.30 25.80 24.20 24.10 23.70 22.70

128 19.40 43.50 32.60 26.90 27.80 24.50 24.50 23.80 22.70

192 19.40 47.50 36.00 29.10 29.50 24.80 25.00 23.80 22.60

256 19.40 49.70 38.50 31.00 30.80 25.30 25.40 23.90 22.40

320 19.40 51.00 40.60 32.60 31.80 25.90 25.70 24.00 22.30

384 19.40 51.90 42.00 34.20 32.80 26.40 26.30 24.20 22.20

448 19.40 52.50 43.30 35.40 33.60 26.90 26.60 24.40 22.10

512 19.40 53.00 44.30 36.40 34.40 27.40 27.00 24.60 22.00

576 19.40 53.50 45.00 37.30 35.00 27.80 27.30 24.80 22.10

640 19.40 53.80 45.70 38.00 35.50 28.20 27.60 24.90 21.90

704 19.40 54.00 46.20 38.60 36.10 28.70 27.90 25.20 21.90

768 19.40 54.10 46.60 39.20 36.50 29.10 28.20 25.20 21.90

832 19.40 54.40 47.00 39.70 36.90 29.40 28.50 25.50 21.90

896 19.40 55.00 47.50 40.20 37.30 29.70 28.80 25.70 21.80

960 19.40 55.30 47.90 40.50 37.70 30.00 29.00 25.90 21.90

1024 19.40 55.40 48.20 41.00 38.10 30.30 29.30 26.00 21.90

1088 19.40 55.30 48.30 41.20 38.30 30.60 29.50 26.20 21.70

1152 19.40 55.10 48.40 41.50 38.50 30.90 29.60 26.40 21.70

1216 19.40 54.80 48.40 41.70 38.60 31.10 29.90 26.50 21.70

1280 19.40 54.90 48.50 41.70 38.80 31.20 30.00 26.70 21.70

1344 19.40 54.60 48.40 41.80 38.90 31.50 30.10 26.70 21.60

1408 19.40 54.70 48.50 41.90 39.10 31.50 30.30 26.90 21.60

1472 19.40 54.70 48.50 42.00 39.20 31.70 30.50 27.00 21.60

Table B1: Temperature results for 65


89



1 19.40 23.79 22.96 22.41 22.26 21.74 21.66 21.45 20.53

64 19.40 31.93 25.87 23.02 23.39 21.82 21.77 21.33 20.50

128 19.40 40.93 30.31 24.72 25.47 22.04 22.19 21.52 20.53

192 19.40 46.17 34.14 26.91 27.32 22.61 22.72 21.64 20.53

256 19.40 49.06 37.02 29.09 28.87 23.22 23.25 21.71 20.50

320 19.40 51.05 39.48 31.01 30.11 23.75 23.74 21.90 20.46

384 19.40 52.49 41.45 32.74 31.35 24.39 24.23 22.21 20.46

448 19.40 53.38 42.90 34.27 32.25 25.07 24.76 22.39 20.50

512 19.40 54.12 44.20 35.54 33.18 25.67 25.29 22.66 20.46

576 19.40 54.52 45.16 36.59 34.05 26.24 25.63 22.96 20.46

640 19.40 55.07 46.05 37.45 34.76 26.84 26.12 23.27 20.46

704 19.40 55.55 46.76 38.35 35.39 27.41 26.54 23.49 20.42

768 19.40 55.88 47.31 39.05 36.03 27.82 26.99 23.79 20.42

832 19.40 56.29 47.87 39.69 36.63 28.35 27.33 24.06 20.46

896 19.40 56.55 48.35 40.25 37.07 28.80 27.71 24.32 20.50

960 19.40 56.77 48.68 40.69 37.52 29.25 28.04 24.55 20.42

1024 19.40 56.92 49.05 41.18 38.05 29.59 28.35 24.82 20.46

1088 19.40 56.95 49.35 41.51 38.31 29.93 28.65 25.00 20.46

1152 19.40 57.03 49.57 41.88 38.68 30.31 29.02 25.23 20.42

1216 19.40 57.06 49.72 42.11 39.02 30.61 29.25 25.49 20.53

1280 19.40 57.03 49.79 42.33 39.20 30.87 29.47 25.72 20.53

1344 19.40 57.10 49.98 42.55 39.43 31.13 29.70 25.91 20.53

1408 19.40 57.54 50.12 42.78 39.69 31.36 29.89 26.10 20.65

1472 19.40 57.73 50.38 42.96 39.95 31.62 30.08 26.29 20.61



90



1 19.40 23.11 22.32 22.11 22.18 22.08 22.08 22.09 21.82

64 19.40 31.40 25.49 22.83 23.47 22.12 22.15 22.05 21.75

128 19.40 41.23 30.09 24.53 25.43 22.46 22.61 22.09 21.71

192 19.40 48.32 34.77 26.98 27.70 23.03 23.14 22.17 21.67

256 19.40 52.49 38.47 29.58 29.58 23.60 23.74 22.32 21.59

320 19.40 55.00 41.56 31.91 31.16 24.28 24.39 22.55 21.56

384 19.40 56.81 43.90 34.05 32.58 25.07 24.99 22.77 21.56

448 19.40 58.06 45.87 35.81 33.86 25.79 25.56 23.11 21.56

512 19.40 59.01 47.39 37.34 34.87 26.47 26.08 23.30 21.44

576 19.40 59.67 48.72 38.72 35.84 27.18 26.61 23.68 21.41

640 19.40 60.12 49.64 39.84 36.74 27.82 27.14 23.91 21.37

704 19.40 60.85 50.49 40.92 37.56 28.54 27.59 24.17 21.33

768 19.40 61.62 51.34 41.66 38.38 29.10 28.12 24.48 21.33

832 19.40 62.06 52.08 42.48 39.09 29.63 28.42 24.78 21.33

896 19.40 62.43 52.67 43.26 39.65 30.12 28.95 25.12 21.29

960 19.40 62.72 53.26 43.85 40.28 30.64 29.32 25.38 21.29

1024 19.40 62.87 53.59 44.45 40.69 31.09 29.70 25.68 21.29

1088 19.40 63.05 53.92 44.78 41.14 31.47 30.00 25.95 21.18

1152 19.40 63.09 54.22 45.26 41.55 31.99 30.45 26.17 21.22

1216 19.40 63.16 54.44 45.63 41.96 32.30 30.64 26.44 21.22

1280 19.40 63.35 54.66 45.82 42.29 32.63 31.05 26.66 21.22

1344 19.40 63.27 54.92 46.23 42.55 32.93 31.20 26.93 21.18

1408 19.40 63.35 54.96 46.41 42.81 33.19 31.54 27.12 21.25

1472 19.40 63.42 55.03 46.60 43.07 33.42 31.80 27.42 21.18



91



1 19.40 23.70 22.80 22.30 22.20 21.70 21.60 21.40 20.80

64 19.40 31.70 25.70 22.90 23.40 21.80 21.80 21.50 20.80

128 19.40 41.40 30.20 24.60 25.40 22.10 22.30 21.70 20.80

192 19.40 49.40 34.90 26.90 27.80 22.70 22.90 21.70 20.70

256 19.40 53.90 38.80 29.40 29.80 23.50 23.60 22.00 20.80

320 19.40 56.80 42.10 31.70 31.50 24.30 24.20 22.20 20.70

384 19.40 58.60 44.60 33.80 33.00 25.10 24.90 22.50 20.70

448 19.40 59.90 46.60 35.70 34.30 25.90 25.50 22.70 20.70

512 19.40 61.10 48.40 37.30 35.40 26.60 26.00 23.00 20.70

576 19.40 61.90 49.80 38.80 36.60 27.30 26.70 23.40 20.70

640 19.40 62.60 51.00 40.00 37.40 28.10 27.30 23.80 20.80

704 19.40 63.20 51.90 41.10 38.30 28.80 27.80 24.00 20.80

768 19.40 63.60 52.80 42.00 39.10 29.40 28.30 24.40 20.70

832 19.40 63.90 53.40 42.90 39.70 29.90 28.70 24.70 20.80

896 19.40 64.00 53.90 43.50 40.40 30.50 29.20 25.00 20.80

960 19.40 64.30 54.40 44.10 40.90 31.00 29.60 25.30 20.80

1024 19.40 64.40 54.80 44.60 41.40 31.50 30.00 25.70 20.80

1088 19.40 64.50 55.10 45.10 41.80 31.80 30.50 25.90 20.80

1152 19.40 64.60 55.40 45.50 42.20 32.30 30.70 26.20 20.70

1216 19.40 65.30 55.80 45.90 42.70 32.70 31.10 26.40 20.80

1280 19.40 65.70 56.20 46.30 43.10 33.00 31.40 26.70 20.80

1344 19.40 66.10 56.60 46.70 43.50 33.40 31.60 26.90 20.90

1408 19.40 66.20 56.80 47.00 43.80 33.70 31.90 27.20 20.80

1472 19.40 66.30 57.10 47.40 44.10 34.00 32.30 27.40 20.90



92



1 19.40 22.85 21.90 21.43 21.31 21.02 20.94 20.84 20.57

64 19.40 31.03 24.74 21.96 22.64 21.06 21.09 20.96 20.53

128 19.40 40.75 29.18 23.44 24.68 21.44 21.62 21.07 20.46

192 19.40 49.76 34.21 25.82 27.21 22.04 22.34 21.22 20.50

256 19.40 54.96 38.40 28.53 29.39 22.80 23.06 21.41 20.53

320 19.40 58.20 41.97 31.05 31.31 23.63 23.78 21.75 20.53

384 19.40 60.63 44.94 33.41 32.88 24.50 24.46 22.02 20.50

448 19.40 62.61 47.42 35.58 34.38 25.33 25.18 22.39 20.46

512 19.40 64.45 49.68 37.64 35.80 26.20 25.86 22.70 20.50

576 19.40 65.62 51.53 39.39 37.11 27.03 26.46 23.08 20.46

640 19.40 66.54 53.00 40.88 38.19 27.86 27.14 23.45 20.46

704 19.40 67.20 54.22 42.29 39.28 28.69 27.82 23.91 20.46

768 19.40 67.79 55.21 43.41 40.13 29.37 28.31 24.29 20.50

832 19.40 68.23 56.21 44.45 40.99 30.12 28.95 24.63 20.50

896 19.40 68.52 56.87 45.37 41.70 30.72 29.40 25.08 20.53

960 19.40 68.85 57.46 46.19 42.33 31.28 29.89 25.38 20.57

1024 19.40 69.14 57.94 46.82 42.89 31.84 30.30 25.80 20.50

1088 19.40 69.22 58.45 47.41 43.44 32.37 30.79 26.10 20.57

1152 19.40 69.33 58.82 47.93 43.89 32.93 31.05 26.48 20.57

1216 19.40 69.55 59.11 48.41 44.33 33.31 31.50 26.74 20.61

1280 19.40 69.66 59.33 48.78 44.74 33.76 31.80 27.04 20.65

1344 19.40 69.80 59.66 49.23 45.15 34.09 32.18 27.38 20.68

1408 19.40 69.84 59.89 49.52 45.45 34.43 32.52 27.68 20.65

1472 19.40 69.84 60.03 49.71 45.71 34.77 32.85 27.95 20.68


APPENDIX C: THERMAL ENERGY STORAGE SYSTEM DATA

93

Appendix C

Thermal Energy Storage System Data

The raw temperature data obtained from the Thermal energy storage system

experiments described in Chapter 3 is given in this appendix in tabular form. The data

is grouped as follows: The table is named by the melting or solidification period at

which time varying temperatures are measured. In each table the first column lists the

time at which temperature is measured. The next columns list the axial temperature

values.


94

Temperature (C)

Time 1 cm 2 cm

0 23.64 23.51

128 23.83 23.44

256 24.77 23.59

384 26.32 24.19

512 28.20 24.98

640 31.32 25.82

768 34.10 26.68

896 36.64 27.89

1024 38.92 28.87

1152 40.74 30.07

1280 42.27 31.39

1408 43.86 32.62

1536 46.05 33.86

1664 47.46 34.95

1792 49.12 36.03

1920 50.71 37.04

2048 52.30 37.94

2176 53.52 38.76

2304 54.62 39.50

2432 55.58 40.21

2560 56.43 40.92

2688 57.27 41.40

2816 58.01 42.03

2944 58.75 42.52

3072 59.41 43.04

3200 60.18 43.52

3328 60.91 44.00

3456 61.76 44.67

3584 62.68 45.26

3712 63.45 45.93

3840 64.33 46.56

3968 65.14 47.30

4096 65.94 48.08

4224 66.97 49.04

4352 67.85 50.04

4480 68.66 51.26

4608 69.53 52.51

4736 70.45 53.88

Table C1: Temperature variation while melting (without metal foam) at constant

temperature of 105 at the top copper plate


95

Temperature (C)

Time Inlet 1 cm 2 cm Outlet

0 21.67 71.81 55.57 22.83

128 21.56 72.25 53.80 30.33

256 21.41 71.44 52.55 30.71

384 21.18 69.86 51.74 30.93

512 21.14 68.00 50.74 30.78

640 20.99 65.94 49.74 30.67

768 20.88 63.89 48.74 30.56

896 20.88 61.87 47.89 30.29

1024 20.88 59.89 47.12 30.07

1152 20.80 58.16 46.34 29.88

1280 20.80 56.47 45.60 29.69

1408 20.80 54.96 44.78 29.47

1536 20.72 53.52 44.04 29.28

1664 20.65 52.41 43.33 29.02

1792 20.65 51.27 42.85 28.87

1920 20.61 50.42 42.14 28.68

2048 20.65 49.68 41.47 28.49

2176 20.57 48.98 40.92 28.30

2304 20.57 48.50 40.51 28.15

2432 20.57 48.27 39.99 27.92

2560 20.57 48.09 39.46 27.81

2688 20.61 47.94 38.98 27.74

2816 20.57 47.68 38.61 27.47

2944 20.57 47.46 38.16 27.25

3072 20.53 47.16 37.75 27.17

3200 20.61 46.90 37.41 27.06

3328 20.53 46.65 37.11 26.98

3456 20.65 46.42 36.74 26.83

Table C2: Temperature results for solidification at 20.0 L/min (no metal foam)


96

Temperature (C)


0 22.51 72.14 55.61 23.43

128 22.20 72.17 53.25 31.05

256 22.01 71.04 51.40 31.23

384 21.86 69.17 50.00 31.20

512 21.86 67.04 49.11 30.93

640 21.82 64.95 48.15 30.67

768 21.75 62.90 47.30 30.52

896 21.71 60.95 46.60 30.18

1024 21.67 59.11 45.75 29.88

1152 21.67 57.42 45.08 29.77

1280 21.63 55.95 44.41 29.47

1408 21.56 54.59 43.63 29.28

1536 21.56 53.44 42.81 29.05

1664 21.48 52.34 41.55 28.83

1792 21.48 51.49 41.59 28.53

1920 21.41 50.79 40.92 28.38

2048 21.41 50.20 40.32 28.15

2176 21.29 49.83 39.58 27.89

2304 21.26 49.57 38.91 27.74

2432 21.26 49.20 38.20 27.55

2560 21.18 48.87 37.79 27.32

2688 21.14 48.46 37.41 27.21

2816 21.10 48.09 36.93 27.02

2944 21.10 47.68 36.55 26.83

3072 21.10 47.39 36.22 26.64

3200 21.03 47.02 35.81 26.53

3328 21.03 46.65 35.54 26.42

3456 21.03 46.24 35.17 26.30

3584 21.03 45.90 34.87 26.11



97

Temperature (C)


0 22.01 72.32 56.16 22.98

128 21.75 72.58 52.66 30.67

256 21.52 71.48 50.48 30.41

384 21.33 69.39 49.00 30.07

512 21.26 67.08 47.75 29.69

640 21.22 64.77 46.71 29.43

768 21.18 62.60 45.75 29.05

896 21.14 60.44 44.89 28.83

1024 21.10 58.49 43.96 28.64

1152 21.10 56.72 43.18 28.30

1280 21.07 55.14 42.37 28.15

1408 21.14 53.67 41.44 27.96

1536 21.07 52.49 40.58 27.70

1664 21.10 51.42 39.32 27.51

1792 21.10 50.46 38.76 27.40

1920 21.18 49.86 37.97 27.21

2048 21.14 49.31 37.49 26.94

2176 21.07 48.98 36.96 26.91

2304 21.18 48.64 36.40 26.68

2432 21.22 48.31 35.96 26.68

2560 21.18 47.98 35.54 26.45

2688 21.14 47.76 35.13 26.38

2816 21.26 47.42 34.83 26.19

2944 21.14 47.13 34.42 26.04

3072 21.18 46.83 34.09 26.00

3200 21.22 46.39 33.86 25.89

3328 21.10 46.05 33.56 25.74

3456 21.14 45.68 33.26 25.55



98

Temperature (C)


0 21.90 72.17 56.16 22.56

128 21.26 72.03 51.26 30.14

256 20.99 70.38 48.86 29.58

384 20.76 68.07 47.41 29.02

512 20.53 65.61 46.04 28.60

640 20.50 63.26 45.00 28.19

768 20.38 60.99 44.00 27.81

896 20.27 58.86 42.70 27.51

1024 20.16 56.91 41.18 27.21

1152 20.27 55.14 39.20 26.91

1280 20.16 53.59 40.06 26.68

1408 20.12 52.26 38.31 26.45

1536 20.16 51.16 36.89 26.23

1664 20.12 50.20 36.48 26.00

1792 20.08 49.42 35.88 25.77

1920 20.08 48.79 35.36 25.55

2048 20.08 48.39 34.91 25.43

2176 20.00 48.05 34.42 25.25

2304 20.08 47.76 33.82 25.06

2432 20.08 47.31 33.41 24.91

2560 20.04 46.98 33.11 24.76

2688 20.04 46.61 32.74 24.64

2816 20.04 46.24 32.40 24.49

2944 20.08 45.87 32.14 24.34

3072 19.97 45.46 31.80 24.19



99

Temperature (C)


0 21.22 71.22 55.83 21.35

128 20.35 70.93 49.67 29.09

256 20.08 69.09 46.82 28.30

384 19.85 66.60 45.15 27.70

512 19.66 64.11 43.78 27.17

640 19.59 61.72 42.66 26.76

768 19.51 59.41 41.66 26.38

896 19.43 57.20 40.69 25.96

1024 19.40 55.43 39.73 25.66

1152 19.40 53.74 38.76 25.51

1280 19.32 52.23 37.00 25.13

1408 19.28 50.97 35.73 24.94

1536 19.32 49.94 35.28 24.72

1664 19.28 49.05 34.57 24.53

1792 19.32 48.42 33.86 24.30

1920 19.40 47.90 33.22 24.23

2048 19.40 47.57 32.59 24.00

2176 19.28 47.13 32.29 23.89

2304 19.40 46.72 32.02 23.66

2432 19.36 46.35 31.57 23.62

2560 19.32 45.98 31.20 23.43

2688 19.32 45.42 30.97 23.36

2816 19.32 45.05 30.71 23.21

2944 19.36 44.68 30.41 23.09

3072 19.36 44.27 30.26 23.02

3200 19.40 43.79 30.00 22.86

3328 19.32 43.38 29.54 22.79



100

Temperature (c)

Time 1 cm 2 cm

0 2.18E+01 2.17E+01

64 24.05521 22.37554

128 26.16963 23.81319

192 28.31753 25.32449

256 30.34838 26.94678

320 32.48782 28.7548

384 34.51069 30.59731

448 36.26823 32.43661

512 37.79909 34.08551

576 39.25332 35.50751

640 40.33345 36.8156

704 41.44977 37.86094

768 42.41639 38.86802

832 43.23369 39.7252

896 44.08755 40.65618

960 44.86665 41.47482

1024 45.71941 42.21855

1088 46.53456 42.88751

1152 47.34921 43.63036

1216 48.23737 44.37277

1280 49.27283 45.18891

1344 50.49225 46.04159

1408 51.93211 46.89371

1472 53.37067 47.96736

1536 55.02902 49.48385

1600 56.86986 51.55263

1664 58.92959 53.76614

Table C7: Temperature variation while melting (with metal foam on wax side) at

constant heat flux of 50 V (2800 W/m2)


101

Temperature (C)


0 21.63 61.24 59.62 22.71

128 21.79 58.93 56.57 30.75

256 21.71 56.72 54.54 31.53

384 21.56 54.62 52.70 31.83

512 21.52 52.78 51.00 31.80

640 21.41 51.01 49.23 31.57

768 21.41 49.35 47.71 31.27

896 21.26 48.09 46.86 31.01

1024 21.29 47.83 46.34 30.75

1152 21.26 47.68 45.93 30.56

1280 21.22 47.35 45.52 30.37

1408 21.22 46.90 45.19 30.26

1536 21.18 46.50 44.82 30.14

1664 21.18 46.02 44.34 29.96

1792 21.18 45.61 44.04 29.81

1920 21.18 45.20 43.63 29.62

2048 21.14 44.72 43.30 29.50

2176 21.22 44.27 42.89 29.39

2304 21.18 43.86 42.52 29.28

2432 21.18 43.38 42.11 29.13

2560 21.10 42.94 41.66 29.02

2688 21.14 42.49 41.25 28.87

2816 21.10 41.97 40.88 28.68

2944 21.10 41.56 40.47 28.56

3072 21.10 41.04 40.02 28.34

3200 21.14 40.59 39.61 28.23

3328 21.10 40.04 39.09 28.11

3456 21.03 39.55 38.68 27.92

Table C8: Temperature results for solidification at 20.0 L/min (foam on wax side)


102

Temperature (C)


0 21.37 61.39 58.59 21.80

128 21.44 59.19 56.01 30.56

256 21.26 56.61 53.88 31.27

384 21.18 54.22 51.81 31.27

512 21.03 52.04 49.82 31.01

640 20.95 50.01 47.93 30.67

768 20.95 49.16 47.15 30.33

896 20.84 48.94 46.63 30.14

1024 20.80 48.39 46.12 29.88

1152 20.76 47.83 45.60 29.73

1280 20.65 47.39 45.19 29.47

1408 20.72 46.87 44.71 29.32

1536 20.69 46.35 44.22 29.20

1664 20.69 45.90 43.74 29.05

1792 20.69 45.31 43.26 28.90

1920 20.57 44.79 42.85 28.71

2048 20.65 44.31 42.40 28.56

2176 20.65 43.75 41.88 28.41

2304 20.65 43.23 41.44 28.26

2432 20.61 42.68 40.88 28.11

2560 20.50 42.12 40.36 27.96

2688 20.57 41.41 39.84 27.77

2816 20.53 40.85 39.32 27.58

2944 20.50 40.22 38.72 27.36

3072 20.50 39.51 38.12 27.13

3200 20.46 38.88 37.53 26.98

3328 20.42 38.21 36.93 26.79

3456 20.38 37.46 36.29 26.64



103

Temperature (C)


0 21.48 60.29 57.01 22.07

128 21.60 58.27 54.61 30.56

256 21.44 55.58 52.44 31.01

384 21.26 52.93 50.22 30.82

512 21.10 50.71 48.37 30.52

640 21.07 49.98 47.52 30.18

768 21.03 49.64 46.82 29.96

896 20.91 49.09 46.38 29.73

1024 20.88 48.53 45.75 29.54

1152 20.91 47.94 45.23 29.43

1280 20.80 47.28 44.67 29.24

1408 20.76 46.61 44.11 29.05

1536 20.76 46.02 43.56 28.83

1664 20.76 45.35 43.00 28.64

1792 20.80 44.72 42.48 28.49

1920 20.72 44.09 41.77 28.34

2048 20.65 43.38 41.25 28.11

2176 20.65 42.75 40.66 27.89

2304 20.69 42.01 40.02 27.74

2432 20.72 41.23 39.39 27.47

2560 20.65 40.52 38.68 27.25

2688 20.69 39.74 38.05 27.06

2816 20.65 38.96 37.38 26.83

2944 20.69 38.13 36.67 26.60

3072 20.57 37.35 35.96 26.27

3200 20.53 36.53 35.21 26.08

3328 20.53 35.82 34.53 25.85

3456 20.53 35.07 33.86 25.51

3584 20.57 34.40 33.22 25.25



104

Temperature (C)


0 21.79 60.03 56.42 22.41

128 21.60 57.79 53.73 30.37

256 21.26 54.59 51.37 30.56

384 21.03 51.67 48.97 30.11

512 20.95 50.27 47.71 29.81

640 20.80 49.83 46.89 29.62

768 20.76 49.20 46.15 29.35

896 20.69 48.46 45.49 29.17

1024 20.61 47.61 44.78 28.79

1152 20.53 46.83 44.11 28.60

1280 20.42 46.13 43.44 28.41

1408 20.53 45.27 42.66 28.07

1536 20.38 44.50 42.07 27.89

1664 20.42 43.61 41.33 27.62

1792 20.46 42.79 40.58 27.47

1920 20.42 41.93 39.95 27.17

2048 20.42 41.08 39.02 26.91

2176 20.42 40.15 38.23 26.72

2304 20.46 39.25 37.45 26.45

2432 20.50 38.25 36.67 26.30

2560 20.46 37.31 35.84 26.04

2688 20.46 36.46 34.98 25.74

2816 20.53 35.56 34.24 25.51

2944 20.42 34.74 33.41 25.17

3072 20.53 33.91 32.77 25.02



105

Temperature (C)


0 21.26 61.24 57.34 21.50

128 20.95 58.41 53.91 29.92

256 20.65 54.96 51.15 29.84

384 20.46 51.78 48.41 29.32

512 20.38 49.53 46.56 28.83

640 20.23 48.90 45.56 28.49

768 20.16 48.24 44.82 28.15

896 20.16 47.50 44.04 27.85

1024 20.12 46.76 43.52 27.66

1152 20.04 46.05 42.85 27.51

1280 20.00 45.35 42.29 27.28

1408 20.04 44.64 41.66 27.06

1536 20.00 43.94 41.03 26.83

1664 19.93 43.16 40.43 26.68

1792 19.97 42.45 39.84 26.49

1920 19.93 41.64 39.17 26.23

2048 19.85 40.93 38.46 26.08

2176 19.85 40.04 37.79 25.77

2304 19.89 39.14 37.04 25.59

2432 19.89 38.40 36.29 25.28

2560 19.81 37.54 35.62 25.13

2688 19.89 36.57 34.80 24.87

2816 19.74 35.71 34.09 24.64

2944 19.74 34.92 33.34 24.42

3072 19.78 34.14 32.62 24.19

3200 19.74 33.35 31.95 23.96

3328 19.74 32.60 31.27 23.73

3456 19.70 31.85 30.60 23.55

3584 19.74 31.25 30.03 23.32



106

Temperature (c)

Time 1 cm 2 cm

0 2.18E+01 2.17E+01

64 24.05521 22.37554

128 26.16963 23.81319

192 28.31753 25.32449

256 30.34838 26.94678

320 32.48782 28.7548

384 34.51069 30.59731

448 36.26823 32.43661

512 37.79909 34.08551

576 39.25332 35.50751

640 40.33345 36.8156

704 41.44977 37.86094

768 42.41639 38.86802

832 43.23369 39.7252

896 44.08755 40.65618

960 44.86665 41.47482

1024 45.71941 42.21855

1088 46.53456 42.88751

1152 47.34921 43.63036

1216 48.23737 44.37277

1280 49.27283 45.18891

1344 50.49225 46.04159

1408 51.93211 46.89371

1472 53.37067 47.96736

1536 55.02902 49.48385

1600 56.86986 51.55263

1664 58.92959 53.76614

Table C13: Temperature variation while melting (with metal foam on both- air and

wax side) at constant heat flux of 50 V (2800 W/m2)


107

Temperature (C)


0 21.60 60.33 56.90 23.13

128 21.33 56.35 53.10 42.37

256 21.14 52.52 49.85 42.85

384 21.03 49.31 47.23 42.66

512 20.76 47.28 45.71 42.03

640 20.76 46.68 44.93 41.70

768 20.65 46.20 44.22 41.36

896 20.61 45.61 43.59 40.88

1024 20.46 44.94 42.89 40.47

1152 20.42 44.31 42.26 40.02

1280 20.38 43.68 41.59 39.69

1408 20.38 42.97 40.95 39.16

1536 20.31 42.23 40.32 38.68

1664 20.23 41.56 39.61 38.23

1792 20.27 40.74 38.91 37.63

1920 20.19 40.00 38.23 37.11

2048 20.19 39.22 37.56 36.55

2176 20.16 38.43 36.78 35.95

2304 20.12 37.50 36.07 35.24

2432 20.12 36.68 35.28 34.68

2560 20.04 35.86 34.61 33.97

2688 20.08 35.03 33.86 33.33

2816 20.08 34.21 33.11 32.73

2944 20.04 33.35 32.36 32.13

3072 20.00 32.53 31.61 31.42

3200 19.93 31.81 30.86 30.82

3328 19.97 31.06 30.22 30.18

3456 19.93 30.31 29.62 29.62

Table C14: Temperature results for solidification at 20.0 L/min (foam on both sides)


108

Temperature (C)


0 21 59.1134 56.08727 23.20519

128 21 54.77115 51.10956 43.51722

256 21 50.52919 47.63425 43.33152

384 21 47.83037 45.708 42.55128

512 20 47.23815 44.66961 41.99365

640 20 46.53456 43.77888 41.51016

768 20 45.71941 42.88751 40.95205

896 20 44.94083 42.06984 40.35643

1024 20 43.97621 41.25161 39.72324

1152 20 43.12227 40.47004 39.05243

1280 20 42.19339 39.61343 38.38122

1408 20 41.22659 38.71888 37.52297

1536 20 40.18453 37.74899 36.85083

1664 20 39.10425 36.85295 36.06615

1792 20 37.98564 35.95618 35.2809

1920 20 36.86589 34.909 34.45765

2048 20 35.63285 33.97316 33.55883

2176 20 34.51069 32.99902 32.73425

2304 20 33.46228 32.02398 31.8715

2432 20 32.45033 31.0856 31.00804

2560 20 31.43741 30.25913 30.25664

2688 20 30.64891 29.50724 29.54231

2816 20 29.82224 28.83007 28.86513

2944 20 29.03253 28.19013 28.26283

3072 20 28.43046 27.5498 27.58483

3200 20 27.67738 27.02218 27.09489

3328 20 27.11222 26.38114 26.64244

3456 20 26.58447 25.96615 26.07661

3584 20 26.09418 25.51325 25.66148



109

Temperature (C)


0 21.52 59.96 56.12 22.94

128 21.03 54.59 50.04 43.78

256 20.72 49.94 46.41 42.96

384 20.61 47.53 44.60 41.88

512 20.42 46.83 43.52 41.14

640 20.42 45.94 42.40 40.39

768 20.23 44.98 41.44 39.65

896 20.23 43.98 40.62 38.83

1024 20.12 42.83 39.58 38.05

1152 20.12 41.67 38.57 37.30

1280 20.04 40.52 37.53 36.37

1408 19.97 39.22 36.44 35.54

1536 19.97 37.87 35.40 34.57

1664 19.97 36.49 34.24 33.67

1792 19.93 35.15 33.07 32.73

1920 19.93 33.80 31.95 31.76

2048 19.93 32.56 30.90 30.75

2176 20.00 31.40 29.92 29.84

2304 19.93 30.39 29.13 28.98

2432 20.00 29.45 28.23 28.19

2560 20.00 28.58 27.51 27.51

2688 19.93 27.83 26.80 26.83

2816 19.97 27.11 26.23 26.23

2944 19.85 26.47 25.63 25.74

3072 19.93 25.83 25.14 25.17

3200 19.89 25.34 24.72 24.79

3328 19.85 24.89 24.19 24.30

3456 19.81 24.40 23.89 24.00



110

Temperature (C)


0 21.07 60.07 56.49 22.07

128 20.35 54.33 49.48 43.29

256 20.16 50.16 46.71 42.55

384 19.93 49.31 45.11 41.88

512 19.89 48.20 43.85 41.14

640 19.85 47.09 42.70 40.32

768 19.85 45.83 41.62 39.39

896 19.89 44.50 40.32 38.49

1024 19.78 42.90 39.05 37.52

1152 19.81 41.26 37.75 36.44

1280 19.74 39.48 36.33 35.24

1408 19.74 37.58 34.87 34.05

1536 19.70 35.82 33.41 32.85

1664 19.74 34.21 32.10 31.65

1792 19.78 32.75 30.86 30.52

1920 19.74 31.47 29.81 29.50

2048 19.78 30.31 28.75 28.56

2176 19.78 29.26 27.89 27.81

2304 19.81 28.32 27.10 27.02

2432 19.85 27.49 26.38 26.34

2560 19.81 26.74 25.78 25.74

2688 19.85 26.13 25.21 25.21

2816 19.85 25.49 24.72 24.72

2944 19.97 24.92 24.19 24.23

3072 19.97 24.43 23.85 23.92

3200 19.89 24.02 23.44 23.55

3328 19.89 23.64 23.13 23.28

3456 19.89 23.26 22.94 22.98



111

Temperature (C)


0 21.26 60.07 56.57 21.96

128 20.35 53.56 48.08 42.33

256 19.97 48.76 44.86 40.99

384 19.74 47.65 43.18 39.98

512 19.74 46.53 41.88 39.05

640 19.59 45.27 40.58 38.01

768 19.59 43.86 39.39 36.96

896 19.43 42.42 38.05 36.03

1024 19.40 40.78 36.74 34.91

1152 19.36 38.99 35.36 33.82

1280 19.36 36.98 33.82 32.55

1408 19.28 35.07 32.36 31.27

1536 19.24 33.31 30.94 30.14

1664 19.36 31.74 29.62 28.94

1792 19.32 30.24 28.53 28.00

1920 19.24 29.03 27.44 27.02

2048 19.28 27.94 26.46 26.19

2176 19.24 27.00 25.59 25.40

2304 19.24 26.09 25.02 24.72

2432 19.28 25.45 24.30 24.15

2560 19.21 24.66 23.81 23.58

2688 19.21 24.13 23.32 23.17

2816 19.21 23.60 22.87 22.71

2944 19.21 23.03 22.53 22.37

3072 19.21 22.66 22.15 22.03

3200 19.21 22.32 21.81 21.77

3328 19.24 22.05 21.58 21.58

3456 19.24 21.71 21.35 21.27

3584 19.24 21.41 21.16 21.12



112

The data from Thermal resistance analysis is presented in the following tables. The

value of is 0.95. For the no foam case, is 930 kg/m3 i.e. density of pure wax

whereas for the other two cases is 1330.15 kg/m3 as obtained from Equation (2.3).

The subscript „a‟ is for air side whereas „w‟ is for wax side.

0 0 0 0.02139 0 0 0.02139037

0.001 0.375957 0.042071 0.02139 0.004762 0.017741 0.02615228

0.002 0.751915 0.168286 0.02139 0.009524 0.039054 0.03091418

0.003 1.127872 0.378643 0.02139 0.014286 0.063938 0.03567609

0.004 1.50383 0.673143 0.02139 0.019048 0.092392 0.04043799

0.005 1.879787 1.051786 0.02139 0.02381 0.124418 0.0451999

0.006 2.255745 1.514571 0.02139 0.028571 0.160015 0.0499618

0.007 2.631702 2.0615 0.02139 0.033333 0.199183 0.05472371

0.008 3.00766 2.692571 0.02139 0.038095 0.241922 0.05948561

0.009 3.383617 3.407786 0.02139 0.042857 0.288232 0.06424752

0.01 3.759574 4.207143 0.02139 0.047619 0.338113 0.06900942

0.011 4.135532 5.090643 0.02139 0.052381 0.391565 0.07377133

0.012 4.511489 6.058286 0.02139 0.057143 0.448588 0.07853323

0.013 4.887447 7.110071 0.02139 0.061905 0.509183 0.08329514

0.014 5.263404 8.246 0.02139 0.066667 0.573348 0.08805704

0.015 5.639362 9.466071 0.02139 0.071429 0.641085 0.09281895

0.016 6.015319 10.77029 0.02139 0.07619 0.712392 0.09758085

0.017 6.391277 12.15864 0.02139 0.080952 0.787271 0.10234276

0.018 6.767234 13.63114 0.02139 0.085714 0.865721 0.10710466

0.019 7.143191 15.18779 0.02139 0.090476 0.947742 0.11186656

0.02 7.519149 16.82857 0.02139 0.095238 1.033334 0.11662847

Table C19: Thermal resistance analysis data (without metal foam)


113

0 0 0 0.02139 0 0 0.0213904

0.001 0.510638 0.003093 0.02139 0.000258 0.044868 0.0216481

0.002 1.021277 0.012371 0.02139 0.000515 0.090276 0.0219058

0.003 1.531915 0.027835 0.02139 0.000773 0.136224 0.0221636

0.004 2.042553 0.049485 0.02139 0.001031 0.182713 0.0224213

0.005 2.553191 0.07732 0.02139 0.001289 0.229741 0.022679

0.006 3.06383 0.11134 0.02139 0.001546 0.27731 0.0229368

0.007 3.574468 0.151546 0.02139 0.001804 0.325419 0.0231945

0.008 4.085106 0.197938 0.02139 0.002062 0.374069 0.0234522

0.009 4.595745 0.250515 0.02139 0.00232 0.423258 0.02371

0.01 5.106383 0.309278 0.02139 0.002577 0.472988 0.0239677

0.011 5.617021 0.374227 0.02139 0.002835 0.523258 0.0242254

0.012 6.12766 0.445361 0.02139 0.003093 0.574069 0.0244832

0.013 6.638298 0.52268 0.02139 0.003351 0.625419 0.0247409

0.014 7.148936 0.606186 0.02139 0.003608 0.67731 0.0249986

0.015 7.659574 0.695876 0.02139 0.003866 0.729741 0.0252564

0.016 8.170213 0.791753 0.02139 0.004124 0.782712 0.0255141

0.017 8.680851 0.893814 0.02139 0.004381 0.836224 0.0257718

0.018 9.191489 1.002062 0.02139 0.004639 0.890276 0.0260295

0.019 9.702128 1.116495 0.02139 0.004897 0.944868 0.0262873

0.02 10.21277 1.237113 0.02139 0.005155 1 0.026545

Table C20: Thermal resistance analysis data (metal foam on wax side)


114

0 0 0 0.004892 0 0 0.0048921

0.001 0.117411 0.003093 0.004892 0.000258 0.03361 0.0051499

0.002 0.234822 0.012371 0.004892 0.000515 0.068946 0.0054076

0.003 0.352233 0.027835 0.004892 0.000773 0.106006 0.0056653

0.004 0.469644 0.049485 0.004892 0.001031 0.144792 0.0059231

0.005 0.587055 0.07732 0.004892 0.001289 0.185303 0.0061808

0.006 0.704467 0.11134 0.004892 0.001546 0.22754 0.0064385

0.007 0.821878 0.151546 0.004892 0.001804 0.271502 0.0066963

0.008 0.939289 0.197938 0.004892 0.002062 0.317188 0.006954

0.009 1.0567 0.250515 0.004892 0.00232 0.364601 0.0072117

0.01 1.174111 0.309278 0.004892 0.002577 0.413738 0.0074694

0.011 1.291522 0.374227 0.004892 0.002835 0.464601 0.0077272

0.012 1.408933 0.445361 0.004892 0.003093 0.517188 0.0079849

0.013 1.526344 0.52268 0.004892 0.003351 0.571502 0.0082426

0.014 1.643755 0.606186 0.004892 0.003608 0.62754 0.0085004

0.015 1.761166 0.695876 0.004892 0.003866 0.685303 0.0087581

0.016 1.878577 0.791753 0.004892 0.004124 0.744792 0.0090158

0.017 1.995988 0.893814 0.004892 0.004381 0.806006 0.0092736

0.018 2.1134 1.002062 0.004892 0.004639 0.868946 0.0095313

0.019 2.230811 1.116495 0.004892 0.004897 0.93361 0.009789

0.02 2.348222 1.237113 0.004892 0.005155 1 0.0100468

Table C21: Thermal resistance analysis data (metal foam on both sides)


115

Temperature (C)


0 19.85 59.48 65.75 20.33

128 20.38 62.24 63.29 39.84

256 20.16 61.28 61.24 43.07

384 20.04 59.77 59.47 44.85

512 19.93 58.23 57.85 45.67

640 19.89 56.76 56.31 45.78

768 19.78 55.21 54.83 45.56

896 19.74 53.85 53.43 45.15

1024 19.78 52.60 52.14 44.45

1152 19.74 51.19 50.89 43.78

1280 19.70 50.01 49.67 43.15

1408 19.66 49.12 48.82 42.33

1536 19.59 48.61 48.34 41.58

1664 19.59 48.35 48.00 41.10

1792 19.51 48.05 47.71 40.69

1920 19.40 47.72 47.30 40.24

2048 19.43 47.50 47.00 39.98

2176 19.43 47.24 46.71 39.61

2304 19.47 46.94 46.41 39.39

2432 19.43 46.61 46.08 39.09

2560 19.43 46.42 45.82 38.90

2688 19.51 46.13 45.45 38.83

2816 19.51 45.76 45.23 38.61

2944 19.43 45.53 44.89 38.42

3072 19.32 45.13 44.60 38.19

3200 19.36 44.79 44.22 37.97

3328 19.36 44.46 43.89 37.75

3456 19.43 44.20 43.67 37.60

3584 19.43 43.94 43.41 37.45

3712 19.47 43.72 43.11 37.26

3840 19.43 43.38 42.81 37.07

3968 19.51 43.09 42.52 36.93

4096 19.55 42.79 42.18 36.78

4224 19.59 42.49 41.96 36.63

4352 19.59 42.16 41.59 36.37

4480 19.59 41.86 41.25 35.99

4608 19.59 41.52 40.95 35.92

4736 19.62 41.19 40.69 35.77

4864 19.70 40.82 40.32 35.51

4992 19.70 40.41 39.95 35.24

5120 19.70 40.07 39.65 35.09

5248 19.74 39.70 39.28 34.72

5376 19.81 39.33 38.87 34.53 Table C22: Temperature results for solidification at 20.0 L/min (three stage TESS)


116

Temperature (C)


0 20.35 58.16 60.02 20.86

128 20.50 57.61 57.82 41.47

256 20.12 56.10 55.76 44.33

384 20.00 54.37 53.99 45.60

512 19.93 52.67 52.29 45.78

640 19.78 51.12 50.67 45.45

768 19.78 49.64 49.26 44.74

896 19.62 48.35 48.08 44.07

1024 19.66 47.61 47.34 43.41

1152 19.62 47.13 46.78 42.70

1280 19.62 46.79 46.38 42.33

1408 19.66 46.42 46.04 41.99

1536 19.62 46.13 45.60 41.70

1664 19.70 45.79 45.26 41.44

1792 19.66 45.42 44.86 41.06

1920 19.62 45.02 44.41 40.77

2048 19.66 44.68 44.00 40.43

2176 19.62 44.24 43.67 40.13

2304 19.62 43.98 43.22 39.87

2432 19.70 43.53 42.89 39.54

2560 19.62 43.20 42.48 39.05

2688 19.62 42.75 42.03 38.83

2816 19.62 42.34 41.59 38.57

2944 19.59 41.93 41.25 38.19

3072 19.59 41.49 40.81 37.90

3200 19.55 41.08 40.36 37.67

3328 19.55 40.67 39.99 37.26

3456 19.55 40.18 39.61 36.89

3584 19.51 39.81 39.17 36.59

3712 19.59 39.33 38.76 36.25

3840 19.59 38.88 38.31 35.88

3968 19.55 38.47 37.82 35.51

4096 19.59 38.02 37.41 35.21

4224 19.59 37.58 36.93 34.72

4352 19.59 37.13 36.52 34.38

4480 19.51 36.64 36.07 34.05

4608 19.66 36.08 35.62 33.67

4736 19.62 35.60 35.13 33.30

4864 19.59 35.15 34.68 32.92

4992 19.62 34.74 34.31 32.47

5120 19.59 34.25 33.86 32.21

5248 19.59 33.72 33.34 31.68



117

Temperature (C)


0 19.59 58.75 60.61 19.87

128 19.78 57.90 57.93 42.81

262 19.59 55.88 55.42 46.23

390 19.47 53.81 53.21 46.78

518 19.32 51.78 51.26 46.37

646 19.32 49.86 49.30 45.48

774 19.21 48.24 47.82 44.59

902 19.21 47.35 46.93 43.74

1030 19.21 46.87 46.26 42.92

1158 19.28 46.42 45.75 42.48

1286 19.28 45.94 45.30 41.84

1414 19.36 45.61 44.82 41.51

1542 19.32 45.16 44.45 41.21

1670 19.36 44.72 43.93 40.84

1798 19.40 44.24 43.48 40.54

1926 19.40 43.86 43.07 40.13

2054 19.40 43.42 42.59 39.80

2182 19.43 42.97 42.07 39.43

2310 19.43 42.53 41.59 39.05

2438 19.43 42.04 41.14 38.68

2566 19.40 41.56 40.69 38.31

2694 19.43 41.04 40.25 37.93

2822 19.47 40.59 39.80 37.49

2950 19.47 40.11 39.24 37.15

3078 19.51 39.55 38.72 36.85

3206 19.51 39.07 38.27 36.33

3334 19.59 38.43 37.71 35.92

3462 19.51 37.91 37.23 35.51

3590 19.59 37.35 36.63 35.06

3718 19.59 36.79 36.14 34.64

3846 19.59 36.23 35.62 34.08

3974 19.59 35.60 35.06 33.71

4102 19.62 35.15 34.50 33.22

4230 19.74 34.55 34.05 32.81

4358 19.70 33.95 33.52 32.40

4486 19.70 33.42 32.96 31.83

4614 19.74 32.98 32.40 31.38

4742 19.70 32.38 32.02 31.01

4870 19.78 31.93 31.50 30.59

4998 19.78 31.40 31.01 30.18

5126 19.81 30.91 30.63 29.77

5254 19.89 30.57 30.26 29.43



118

Temperature (C)


0 20.04 57.83 60.65 20.36

128 20.12 57.35 57.60 44.67

256 19.85 55.29 54.87 47.00

384 19.78 53.08 52.51 47.23

512 19.59 50.82 50.26 46.63

640 19.55 48.90 48.30 45.60

768 19.55 47.28 46.97 44.33

896 19.51 46.53 46.04 43.33

1024 19.43 46.05 45.37 42.51

1152 19.43 45.57 44.86 41.92

1280 19.40 44.98 44.30 41.21

1408 19.40 44.50 43.67 40.77

1536 19.36 43.98 43.07 40.24

1664 19.40 43.53 42.66 39.84

1792 19.36 43.01 42.03 39.43

1920 19.32 42.45 41.51 39.02

2048 19.32 41.97 40.99 38.57

2176 19.32 41.41 40.47 37.90

2304 19.32 40.97 40.02 37.41

2432 19.28 40.41 39.46 36.93

2560 19.28 39.89 39.02 36.51

2688 19.28 39.33 38.38 36.07

2816 19.36 38.77 37.90 35.65

2944 19.32 38.21 37.38 35.36

3072 19.40 37.72 36.82 35.02

3200 19.43 37.02 36.26 34.57

3328 19.40 36.38 35.66 34.16

3456 19.51 35.75 35.02 33.63

3584 19.55 35.07 34.42 33.22

3712 19.55 34.40 33.79 32.70

3840 19.59 33.72 33.19 32.21

3968 19.62 33.09 32.59 31.72

4096 19.62 32.49 32.10 31.12

4224 19.66 31.96 31.54 30.67

4352 19.74 31.51 31.09 30.03

4480 19.78 31.10 30.63 29.73

4608 19.85 30.61 30.22 29.32

4736 19.89 30.20 29.85 28.98

4864 19.89 29.71 29.39 28.60

4992 19.81 29.30 28.94 28.19

5120 19.93 28.84 28.57 27.81

5248 19.89 28.39 28.00 27.47



119

Temperature (C)


0 20.23 58.60 60.87 20.78

128 20.23 57.13 57.08 46.00

256 19.97 54.66 53.99 47.63

384 19.78 51.90 51.18 47.34

512 19.74 49.46 48.86 46.23

640 19.70 47.61 47.19 44.85

768 19.62 46.57 46.15 43.63

896 19.66 46.02 45.30 42.66

1024 19.66 45.53 44.63 41.84

1152 19.62 45.02 44.15 41.32

1280 19.74 44.46 43.59 40.84

1408 19.70 44.01 43.07 40.36

1536 19.70 43.49 42.48 39.98

1664 19.66 43.05 41.96 39.50

1792 19.81 42.49 41.40 39.09

1920 19.78 41.90 40.84 38.61

2048 19.78 41.30 40.25 38.08

2176 19.74 40.56 39.69 37.60

2304 19.74 40.00 39.02 37.19

2432 19.74 39.22 38.27 36.70

2560 19.74 38.51 37.60 36.14

2688 19.74 37.72 36.82 35.47

2816 19.74 36.90 36.18 34.91

2944 19.78 36.12 35.36 34.31

3072 19.81 35.33 34.61 33.60

3200 19.93 34.62 34.05 33.00

3328 19.89 33.99 33.41 32.36

3456 19.89 33.24 32.74 31.87

3584 19.97 32.64 32.17 31.35

3712 19.93 32.00 31.50 30.75

3840 19.97 31.40 30.97 30.26

3968 20.00 30.80 30.41 29.73

4096 19.97 30.20 29.92 29.20

4224 20.00 29.75 29.32 28.71

4352 20.00 29.18 28.83 28.34

4480 20.12 28.73 28.45 27.85

4608 20.08 28.39 28.08 27.51

4736 20.16 28.09 27.74 27.21

4864 20.23 27.64 27.32 26.87

4992 20.19 27.30 27.02 26.53

5120 20.16 26.96 26.76 26.23

5248 20.23 26.58 26.38 25.96


APPENDIX D: RESULTS FROM ENTHALPY CODE

120

Appendix D

Results from Enthalpy Code

The numerical results obtained from the Enthalpy code is given in this appendix in

tabular form. The data is organized as follows: the table is named as either numerical

or experimental results. In each table, first column indicates the time at which

temperature is measured. The second column indicates the temperature values at a

depth of 1 cm. In the end, C++ enthalpy code for both single step and four-step phase

change is described.


121

Time Temperature @1 cm

0 21.82

64 24.06

128 26.17

192 28.32

256 30.35

320 32.49

384 34.51

448 36.27

512 37.80

576 39.25

640 40.33

704 41.45

768 42.42

832 43.23

896 44.09

960 44.87

1024 45.72

1088 46.53

1152 47.35

1216 48.24

1280 49.27

1344 50.49

1408 51.93

1472 53.37

1536 55.03

1600 56.87

1664 58.93

Table D1: Experimental temperature results for melting (with metal foam) at a depth of

1 cm at constant heat flux of 50 V (2800W/m2)


122


1 22.00

65 24.00

129 26.40

193 28.76

257 31.09

321 33.38

385 35.62

449 37.83

513 40.01

577 42.14

641 44.24

705 46.27

769 47.51

833 48.22

897 48.64

961 48.90

1025 49.05

1089 49.18

1153 49.26

1217 49.34

1281 49.39

1345 49.44

1409 49.55

1473 49.61

1537 49.67

1601 49.70

1665 49.78

1729 49.81

1793 49.92

Table D2: Temperature results from numerical code (phase change at a single

temperature) for melting (with metal foam) at a depth of 1 cm at constant heat flux of

50 V (2800W/m2)


123


1 22.00

65 24.00

129 26.40

193 28.76

257 31.09

321 33.38

385 35.62

449 37.83

513 39.77

577 41.25

641 42.28

705 43.27

769 44.25

833 44.98

897 45.86

961 46.87

1025 47.62

1089 48.49

1153 49.50

1217 50.42

1281 51.54

1345 52.91

1409 54.25

1473 56.13

1537 58.00

1601 59.83

1665 61.64

1729 63.41

1793 65.15

Table D3: Temperature results from numerical code (phase change over a temperature

range) for melting (with metal foam) at a depth of 1 cm at constant heat flux of 50 V

(2800W/m2)


124

The finite difference code for phase change at a single fixed temperature is as follows:

#include <cstdlib>

#include <iostream>

#include<math.h>

#include<iomanip>

#include<fstream>

using namespace std;

int n = 81;

double length = 0.02;

double lambda = 5.02;

double rho = 1330.15;

double cp = 2774.25;

double h = 20;

double Tinf = 22;

double L = (190000);

double delt = 1;

double Tm = 50;

void TDMA(double e[ ], double f[ ], double g[ ], double x[ ], double b[ ])

{

int i; // TDMA algorithm

for(i=1;i<n;i++)

{

e[i] = e[i]/f[i-1];

f[i] = f[i] - e[i]*g[i-1];

b[i] = b[i] - e[i]*b[i-1];

}

// Backward substitution

x[n-1] = b[n-1]/f[n-1];


125

for(i=n-2;i>=0;i--)

{

x[i] = b[i];

x[i] = ((x[i] - g[i]*x[i+1])/f[i]);

}

}

int main(int argc, char *argv[ ])

{

ofstream myfile;

myfile.open ("Enth_0steps18.txt");

int i,j,k,l,m,p,q;

double delx;

delx = length/(n-1);

double alpha = lambda/(rho*cp);

double Fo = alpha*delt/pow(delx,2);

double a = lambda/pow(delx,2);

double v = (rho*cp)/(2*delt);

double c = a+v;

double T[n], fl[n], fold[n];

double Told[n];

for(i=0;i<n;i++)

{

T[i] = 22;

fl[i] = 0;

fold[i] = 0;

}

double e[n];

double f[n];


126

double g[n];

double b[n];

int iter = 0;

for(j=0;j<(1800*2);j++)

{

e[0] = 0;

f[0] = -c; // West side boundary conditions

g[0] = a;

b[0] = (-2800/delx)-v*T[0];

for(i=1;i<n-1;i++)

{

e[i] = -Fo;

f[i] = 1+2*Fo;

g[i] = -Fo;

b[i] = T[i];

if(T[i]>=Tm && Told[i]<Tm)

{

e[i] = 0;

f[i] = 1;

g[i] = 0;

b[i] = Tm;

fl[i] = fold[i] + (Fo*cp/L)*(T[i-1] - 2*Tm + T[i+1]) - (cp/L)*(Tm-

Told[i]);

}

if(T[i]==Tm && fl[i]<1.0)

{

e[i] = 0;

f[i] = 1;


127

g[i] = 0;

b[i] = Tm;

fl[i] = fold[i] + (Fo*cp/L)*(T[i-1] - 2*Tm + T[i+1]);

}

if(fl[i]>=1 && fold[i]<1)

{

e[i] = -Fo;

f[i] = 1+2*Fo;

g[i] = -Fo;

b[i] = Tm - (L/cp)*(1-fold[i]);

}

}

e[n-1] = a; // East side boundary conditions

f[n-1] = -((h/delx)+c);

g[n-1] = 0;

b[n-1] = ((-h/delx)*Tinf)-(v*T[n-1]);

for(k=0;k<n;k++)

{

Told[k] = T[k];

fold[k] = fl[k];

}

TDMA(e,f,g,T,b);

myfile<<"\n"<<T[((n-1)/2)]<<"\t\t\t\t"<<fl[((n-1)/2)];

}

myfile.close();

system("PAUSE");

return EXIT_SUCCESS;

}


128

The finite difference code for phase change over a temperature range is as follows:

#include <cstdlib>

#include <iostream>

#include<math.h>

#include<iomanip>

#include<fstream>

using namespace std;

int n = 81;

double length = 0.02;

double lambda = 5.02;

double rho = 1330.15;

double cp = 2774.25;

double h = 20;

double Tinf = 22;

double L = (190000/4);

double delt = 1;

double Tm1 = 42.5;

double Tm2 = 45.5;

double Tm3 = 48.0;

double Tm4 = 50.5;

void TDMA(double e[ ], double f[ ], double g[ ], double x[ ], double b[ ])

{

int i; // TDMA algorithm

for(i=1;i<n;i++)

{

e[i] = e[i]/f[i-1];

f[i] = f[i] - e[i]*g[i-1];

b[i] = b[i] - e[i]*b[i-1];

}


129

// Backward substitution

x[n-1] = b[n-1]/f[n-1];

for(i=n-2;i>=0;i--)

{

x[i] = b[i];

x[i] = ((x[i] - g[i]*x[i+1])/f[i]);

}

}

int main(int argc, char *argv[ ])

{

ofstream myfile;

myfile.open ("Enth_4stepsSS.txt");

int i,j,k,l,m,p,q;

double delx;

delx = length/(n-1);

double alpha = lambda/(rho*cp);

double Fo = alpha*delt/pow(delx,2);

double a = lambda/pow(delx,2);

double v = (rho*cp)/(2*delt);

double c = a+v;

double T[n], fl[n], fold[n];

double Told[n];

for(i=0;i<n;i++)

{

T[i] = 22;

fl[i] = 0;

fold[i] = 0;

}

double e[n];


130

double f[n];

double g[n];

double b[n];

int iter = 0;

for(j=0;j<(1800*2);j++)

{

e[0] = 0;

f[0] = -c; // West side boundary conditions

g[0] = a;

b[0] = (-2800/delx)-v*T[0];

for(i=1;i<n-1;i++)

{

e[i] = -Fo;

f[i] = 1+2*Fo;

g[i] = -Fo;

b[i] = T[i];

if(T[i]>=Tm1 && Told[i]<Tm1)

{

e[i] = 0;

f[i] = 1;

g[i] = 0;

b[i] = Tm1;

fl[i] = fold[i] + (Fo*cp/L)*(T[i-1] - 2*Tm1 + T[i+1]) - (cp/L)*(Tm1-

Told[i]);

}

if(T[i]==Tm1 && fl[i]<0.25)

{

e[i] = 0;

f[i] = 1;


131

g[i] = 0;

b[i] = Tm1;

fl[i] = fold[i] + (Fo*cp/L)*(T[i-1] - 2*Tm1 + T[i+1]);

}

if(fl[i]>=0.25 && fold[i]<0.25)

{

e[i] = -Fo;

f[i] = 1+2*Fo;

g[i] = -Fo;

b[i] = Tm1 - (L/cp)*(0.25-fold[i]);

}


{

e[i] = 0;

f[i] = 1;

g[i] = 0;

b[i] = Tm2;


Told[i]);

}

if(T[i]==Tm2 && fl[i]<0.5)

{

e[i] = 0;

f[i] = 1;

g[i] = 0;

b[i] = Tm2;


}

if(fl[i]>=0.5 && fold[i]<0.5)


132

{

e[i] = -Fo;

f[i] = 1+2*Fo;

g[i] = -Fo;

b[i] = Tm2 - (L/cp)*(0.5-fold[i]);

}


{

e[i] = 0;

f[i] = 1;

g[i] = 0;

b[i] = Tm3;


Told[i]);

}

if(T[i]==Tm3 && fl[i]<0.75)

{

e[i] = 0;

f[i] = 1;

g[i] = 0;

b[i] = Tm3;


}

if(fl[i]>=0.75 && fold[i]<0.75)

{

e[i] = -Fo;

f[i] = 1+2*Fo;

g[i] = -Fo;

b[i] = Tm3 - (L/cp)*(0.75-fold[i]);

}


133


{

e[i] = 0;

f[i] = 1;

g[i] = 0;

b[i] = Tm4;


Told[i]);

}

if(T[i]==Tm4 && fl[i]<1)

{

e[i] = 0;

f[i] = 1;

g[i] = 0;

b[i] = Tm4;


}

if(fl[i]>=1 && fold[i]<1)

{

e[i] = -Fo;

f[i] = 1+2*Fo;

g[i] = -Fo;

b[i] = Tm4 - (L/cp)*(1-fold[i]);

}

}

e[n-1] = a; // East side boundary conditions

f[n-1] = -((h/delx)+c);

g[n-1] = 0;

b[n-1] = ((-h/delx)*Tinf)-(v*T[n-1]);


134

for(k=0;k<n;k++)

{

Told[k] = T[k];

fold[k] = fl[k];

}

TDMA(e,f,g,T,b);

myfile<<"\n"<<T[((n-1)/2)]<<"\t\t\t\t"<<fl[((n-1)/2)];

}

myfile.close();

system("PAUSE");

return EXIT_SUCCESS;

}

thermal energy storage in copper foams filled with ......phase change materials (pcm) such as...

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