Characterization of Cocoa butter saturated with ... · Characterization of Cocoa butter saturated...

Characterization of Cocoa butter saturated with

supercritical CO 2:

Experimental set-up and Modelling

Pedro Miguel Almeida dos Santos

Dissertação para obtenção do Grau de Mestre em Engenharia Química

Júri

Presidente: Prof. João Carlos Moura Bordado (IST) Orientadores: Prof. Henrique Aníbal Santos de Matos (IST) Profª Elisabeth Rodier (EMAC) Vogais: Profª Maria de Fátima da Costa Farelo (IST)

Novembro de 2008

1

Acknowledgements

This master thesis is the result of six months traineeship at Ecole Des Mines D’Albi-Carmaux

and 2 months at Instituto Superior Técnico. In the important last step of my master degree, I want to

dedicate some words to the people that helped me to arrive here and to accomplish my main

objective.

I want to express my sincerely and specially gratitude to Professor Henrique Matos, from

Instituto Superior Técnico, for all the support and help that gave during theses last months. Professor

Henrique Matos helped me a lot to find solutions in difficult moments.

I want to express my gratitude to the Professor Fátima Farelo, from Instituto Superior Técnico,

that makes possible this traineeship with her contacts and for trust in my capabilities, I really hope

have been equal to the task.

I also want to express my gratitude to my traineeship supervisor Doctor Elisabeth Rodier and

Brice Calvignac, from Ecole Des Mines D’Albi-Carmaux, not only to the help and suggestions that they

provided, but also for them reception.

The six months in Albi were a big adventure, where I met people that became my friends. In

this way, I want to express my gratitude to: Francisco, Lorenzo, Haruna, Mokrane, Sory, Hassen and

especially to Hugo Gonçalves, Márcio Martins and Fabienne Martins, which made me feel near home.

As I referred, in a long trip to arrive here, I had the help of many friends that I want to express

my gratitude. They are: Miguel Fíuza, Guilherme Gonçalves, Rita Pereira and especially to André

Neves, David Portugal and Cristiano Monteiro, my partner in the Albi adventure.

I also want to express my sincerely gratitude to Alexandre Chambel and especially PhD

student Luis Padrela. Without the help of Luis Padrela it would be impossible to finish my traineeship.

At last, but not least, I want to express my gratitude to those that I consider the most important

persons in my life, my family and girlfriend: António Santos (father), Teodora Santos (mother), Rui

Pedro Santos (brother) and Raquel Reis (girlfriend). Without them, I wouldn’t have accomplished my

objectives.

For you all, thank you very much.

2

Abstract

The main objective of this work was to study and to characterize cocoa butter and the binary

system cocoa butter/supercritical carbon dioxide (SC-CO2). In this way, it was measured the density

and the solubility of the heavy phase in equilibrium at high pressures at 40 and 50ºC, as the density

and compressibility of cocoa butter.

The density measurements were done trough an autoclave with a sapphire window that can

be under high pressures and an apparatus to measure the density, which bases in oscillating U-tube

principle1. The solubility was measured using a synthetic method, through a cell with variable volume.

At last, the compressibility measurements were done using a porosimeter of mercury.

The obtained results were modelled in function of pressure. For the density and

compressibility of the binary system (cocoa butter/SC-CO2) and of cocoa butter it was used the Tait

Equation and the Modified Tait Equation. The solubility was modelled by the Peng-Robinson Equation

of State with two mixing rules: Van der Waals (vdW) and Panagiotopoulos-Reid (P&R). For the

solubility calculations of the cocoa butter model-compound was considered the triglyceride 1-

palmitoyl-2-oleoyl-3stearoyglycerol (POS) since it is the major triglyceride of cocoa butter.

The thermo physical properties of the cocoa butter were predicted by four estimation methods:

Ambrose, Joback, Constatinou-Gani and Fedors.

Key words: Cocoa Butter; Supercritical Carbon Dioxide (SC-CO2), Density, Solubility, Compressibility,

Tait Equations, Modified Tait Equations, Peng-Robinson Equation of State, Properties estimation

methods.

1 method of density measurement based on the law of harmonic oscillation

3

Resumo

O principal objectivo deste trabalho foi estudar e caracterizar a manteiga de cacau e o

sistema binário manteiga de cacau/dióxido de carbono supercrítico. Desta forma, foi medida a

densidade e a solubilidade da fase líquida em equilíbrio a pressões elevadas, às temperaturas de 40

e 50ºC, assim como a densidade e compressibilidade da manteiga de cacau.

As medições da densidade foram efectuadas num autoclave de alta pressão, provido de uma

sapphire window, e de um densímetro, que se baseia no oscillating U-tube principle2. Por sua vez, a

solubilidade foi medida através de um método sintético, recorrendo a uma célula de volume variável.

Por fim, a compressibilidade foi medida por um porosimetro de mercúrio.

Os resultados obtidos foram modelados em função da pressão. A densidade e

compressibilidade do sistema binário (manteida de cacau/SC-CO2) e da manteiga de cacau foram

modeladas através das Tait Equations e Modified Tait equations. Por sua vez, a solubilidade foi

modelada com a Equação de Estado de Peng-Robinson, recorrendo a duas regras de mistura: Van

der Waals (vdW) and Panagiotopoulos-Reid (P&R). Para os cálculos de solubilidade considerou-se

que as propriedades da manteiga de cacau se deviam ao triglicérido em maior concentração: – o

triglicérido 1-palmitoyl-2-oléoyl-3stéaroyglycérol (POS).

As propriedades da manteiga de cacau foram estimadas através de métodos dos estimação:

Ambrose, Joback, Constatinou-Gani and Fedors.

Palavras-Chave: Manteiga de Cacau; Dióxido de carbono Supercrítico (SC-CO2), Densidade,

Solubilidade, Compressibilidade, Tait Equations, Modified Tait Equations, Equação de Estado de

Peng-Robinson, Methodos de Estimação de propriedades.


4

Table of Contents

Acknowledgements ................................................................................................................................... 1

Abstract...................................................................................................................................................... 2

Resumo ..................................................................................................................................................... 3

Table of Contents ...................................................................................................................................... 4

List of Tables ............................................................................................................................................. 5

List of Figures ............................................................................................................................................ 6

Abbreviations and acronyms ..................................................................................................................... 9

1. Introduction .......................................................................................................................................... 11

2. Literature Review ................................................................................................................................. 13

2.1 Lipids ............................................................................................................................................. 13

2.1.1. Cocoa Butter ......................................................................................................................... 15

2.2. Supercritical Fluids ....................................................................................................................... 16

2.2.1 Critical Point ........................................................................................................................... 16

2.2.2 Supercritical fluids and its properties ..................................................................................... 18

2.2.3. Carbon Dioxide: the most used Supercritical fluid ................................................................ 18

2.3. Study of the binary mixture: Cocoa butter/Supercritical CO2 ....................................................... 20

2.3.1. Phase Equilibria .................................................................................................................... 20

2.3.1.1. The Phase rule of Gibbs [7] ................................................................................................ 20

2.3.1.2. Classification of phase equilibria and fluid equilibria [7] ..................................................... 21

2.3.2. Van Konynenburg and Scott Classificaion [6], [7] .................................................................... 23

2.3.3. Representation and modelling of the mixture ....................................................................... 26

2.3.3.1. Equations of State ............................................................................................................. 26

2.3.3.1.1. Cubic Equations of State ................................................................................................ 26

2.3.3.1.2. Peng-Robinson Equation ................................................................................................ 28

2.3.3.1.3. Application to mixtures .................................................................................................... 28

2.3.3.1.4. Estimation techniques [11] ................................................................................................ 30

2.3.3.1.3. Acentric Factor [11] ........................................................................................................... 32

2.3.3.2. Tait Equations [13] ............................................................................................................... 33

2.3.3.3. Study and Determination of properties concerning SC-CO2 and Cocoa butter/Lipids ...... 36

3. Measures and modelling of the properties of the binary cocoa butter/SC-CO2 .................................. 37

3.1. Density measurements ................................................................................................................ 37

3.1.1. Material ................................................................................................................................. 37

3.1.2. Experimental Setup .............................................................................................................. 37

3.1.3. Experimental Procedure ....................................................................................................... 38

3.1.3.1. Calibration of the Densimeter DMA HPM .......................................................................... 39

3.1.4. Results and Analysis ................................................................................................................. 39

3.1.4.1. Density of cocoa butter ...................................................................................................... 39

5

3.1.4.2. Density of CO2-saturated cocoa butter .............................................................................. 40

3.1.4.3. Comparison of cocoa butter density and CO2-saturated cocoa butter density.................. 41

3.1.4.4. Comparison of experimental data with bibliographic data ................................................. 42

3.1.4.5. Compressibility of cocoa butter ......................................................................................... 43

3.1.5. Modelling of density and compressibility with Tait and Modified Tait Equation ........................ 44

3.1.5.1 Tait Equation ....................................................................................................................... 44

3.1.5.2. The Modified Tait Equation ................................................................................................ 48

3.2. Solubility measurements .............................................................................................................. 53

3.2.1. Material ................................................................................................................................. 53

3.2.2. Experimental Setup .............................................................................................................. 54

3.2.3. Experimental Procedures ..................................................................................................... 56

3.2.3.1. Calibration .......................................................................................................................... 57

3.2.4. Results and Discussion ............................................................................................................. 57

3.2.4.1. Solubility of SC-CO2 in cocoa butter .................................................................................. 57

3.2.4.2. Comparison of experimental data with bibliography data .................................................. 58

3.2.5. Modelling of solubility with Peng-Robinson Equation of State .................................................. 59

3.2.5.1. Properties Estimation ......................................................................................................... 59

3.2.5.2. Interaction Parameters ...................................................................................................... 60

3.2.5.3. Solubility Modelling ............................................................................................................ 64

3.2.5.4. Density prediction with Peng-Robinson Equation of State ................................................ 66

4. Conclusions and Perspectives ............................................................................................................ 68

5. References .......................................................................................................................................... 69

6. Appendix .............................................................................................................................................. 72

6.1. Appendix A ................................................................................................................................... 72

6.2. Appendix B ................................................................................................................................... 73

6.3. Appendix C ................................................................................................................................... 78

6.4. Appendix D ................................................................................................................................... 79

6.5. Appendix E ................................................................................................................................... 80

6.6. Appendix F ................................................................................................................................... 81

List of Tables Table 1: The polymorphic forms of cocoa butter and melting points [3] ................................................... 16

Table 2: Advantages and Disadvantages of Van der Waals Cubic Equations of State [14] ..................... 27

Table 3: Mixing Rules and Combining Rules Used in Two-Constant Cubic Equations of State ............. 29

Table 4: Temperature-Independent C Values for Liquids ....................................................................... 35

Table 5:Temperature-Depent C Values for Liquid .................................................................................. 36

Table 6: Parameters of Tait Equation for cocoa butter at 40 and 50ºC. ................................................. 45

Table 7: Parameters of Tait Equation for CO2-cocoa butter at 40 and 50ºC. ......................................... 47

6

Table 8: Parameters of the Modified Tait Equation for cocoa butter at 40 and 50ºC and the respective

errors. ...................................................................................................................................................... 49

Table 9: Parameters of the Modified Tait Equation for CO2-saturated cocoa butter at 40 and 50ºC and

the respective errors. ............................................................................................................................... 51

Table 10: Estimated TB, TC and PC .......................................................................................................... 59

Table 11: Acentric Factor estimated trough the definition and Lee Kesler vapour pressure relations.... 60

Table 12: Interactions parameters od vdW and P&R mixing rules, obtained by the minimization of

equation (36) ........................................................................................................................................... 61

Table 13: Interactions parameters od vdW and P&R mixing rules, obtained by the minimization of

equation (36) with the solubility data from Kokot et all [3] ....................................................................... 62

Table 14: The interaction parameters obtained by the minimization of the equation (36) and by the

correlation at 323,15K. ............................................................................................................................ 63

Table 15: The interaction parameters obtained by the minimization of the equation (36) and by the

correlation at 323,15 and 313,15 K ......................................................................................................... 64

Table 16: Interaction Parameters and respective AAD using solubility data from Venter et all [4] ......... 66

Table A. 1: The CO2 physical-chemical properties [21] ............................................................................. 72

Table B. 1: Ambrose Group Contributions for Critical Constants [11] ....................................................... 73

Table B. 2: Joback Group Contributions for Critical Properties, the Normal Boiling Poin, and the

Freezing Point [11]..................................................................................................................................... 74

Table B. 3: Fedors Group Contributions for Critical Temperature [11] ...................................................... 75

Table B. 4: First-Order Groups and their Contributions for the Physical Properties [16] .......................... 76

Table B. 5: First-Order Groups and their Contributions for the Physical Properties [16] .......................... 77

Table C. 1: Calibration Table ................................................................................................................... 78

Table C. 2: Density of Cocoa Butter at 40 and 50ºC ............................................................................... 78

Table C. 3: Density of CO2/saturated Cocoa Butter at 40 and 50ºC ....................................................... 78

Table D. 1: Compressibility calculations for Tait Equations for Cocoa Butter ......................................... 79

Table D. 2: Compressibility calculations for Tait Equations for CO2/Saturated Cocoa Butter ................ 79

Table E. 1: Solubility of CO2 in Cocoa butter at 40 and 50ºC ................................................................. 80

Table F 1: Density predicted by the Peng-Robinson Equation of State at 40 and 50ºC ......................... 81

List of Figures Figure 1: (a) – General chemical structure of a TAG; (b) – Chemical structure of a saturated fatty acid [2]. ........................................................................................................................................................... 13

7

Figure 2: Classification of triacylglycerides [2]. ....................................................................................... 14

Figure 3: (a) A Gibbs energy-temperature relationship and (b) transformation pathways of three typical

polymorphs in TAG [1] ............................................................................................................................ 15

Figure 4: Phase diagrams of a pure compound: (a) three-dimensional cut;(b) P-T cut;(c) isotherm line [10]. .......................................................................................................................................................... 16

Figure 5: Possible location of one – and two phase equilibria around a three-phase equilibrium in a

P,x-section [7]. ........................................................................................................................................ 22

Figure 6: Two phase equilibria – Pure component boiling point ........................................................... 22

Figure 7: Two phase equilibria: a – Critical Point; b – Azeotropic point [7]. ........................................... 23

Figure 8: The six basic types of fluid phase behaviour according to the classification of Van

Koynenburg and Scott [7]. ...................................................................................................................... 23

Figure 9: (a) - P-T projection of a binary mixture of type I according to the classification of Van

Koynenburg and Scott; (b) – phase diagram P,T of a binary mixture at lower pressures .................... 24

Figure 10: Classification of various type of equations of state, with a selection of equations for each

group. In this classification, Van der walls Equations of State are those cubic and noncubic equations

that consider the compressibility factor as Z=Zrep+Zatt. ......................................................................... 26

Figure 11: (a) – Pressure dependence of isothermal secant bulk modulus for different solids; (b) –

Pressure dependence of the isothermal secant bulk modulus for hydrocarbons (□) n-Hexane at 337K;

(○) n-heptane at 303K; (●) mesitylene at 298 K. ................................................................................... 34

Figure 12: Schematic diagram of device ............................................................................................... 37

Figure 13: (a) – PAAR reactor; (b) – Densimeter and PID controller; (c) – ISCO Pump ...................... 38

Figure 14: The density of pure cocoa butter at different pressures at 40 and 50ºC ............................. 40

Figure 15: The density of CO2-saturated cocoa butter at different pressures at 40and 50ºC. .............. 40

Figure 16: Density of CO2-saturated cocoa butter at different pressures at 40. ................................... 41

Figure 17: The density of CO2-saturated cocoa butter and pure cocoa butter at 50ºC......................... 41

Figure 18: The density of CO2-saturated cocoa butter of the present work and from bibliography [4]. . 42

Figure 19: Volume accumulated of cocoa butter solid in function of pressure. ..................................... 43

Figure 20: (a) photo of cocoa butter before the compressibility experience; (b) photo of cocoa butter

after the compressibility experience. ..................................................................................................... 43

Figure 21: The fit of the experimental of cocoa butter results at 40 and 50ºC with Tait Equation ........ 44

Figure 22: The fit of the last three experimental points of cocoa butter results at 40 and 50ºC with Tait

Equation................................................................................................................................................. 45

Figure 23: The fit of the experimental of CO2-saturated cocoa butter results at 40 and 50ºC with Tait

Equation................................................................................................................................................. 46

Figure 24: The fit of the experimental of CO2-saturated cocoa butter results at 40 and 50ºC with Tait

Equation for pressures above 100 bar .................................................................................................. 47

Figure 25: The fit of the experimental of solid cocoa butter results with Tait Equation: (a) in all range

pressure; (b) at high pressures. ............................................................................................................ 48

Figure 26: Representation of calculated

−ρ

ρρ 0and

calculated

−ρ

ρρ 0of cocoa butter in function of ( )pB +log at

40ºC. ...................................................................................................................................................... 50

8


−ρ

ρρ 0and

calculated

−ρ

ρρ 0 of cocoa butter in function of ( )pB +log at

50ºC. ...................................................................................................................................................... 50


−ρ

ρρ 0and

calculated

−ρ

ρρ 0of CO2-cocoa butter in function of ( )pB +log

at 40ºC ................................................................................................................................................... 51


−ρ

ρρ 0and

calculated

−ρ

ρρ 0of CO2-cocoa butter in function of ( )pB +log

at 50ºC ................................................................................................................................................... 52

Figure 30: Scheme of cell with variable volume of Di Andreth et al [29] ................................................. 53

Figure 31: Scheme of the phase equilibria device ................................................................................ 54

Figure 32: Visible cell with variable volume (VCVV) ............................................................................. 55

Figure 33: The solubility of CO2 in cocoa butter at different pressures at 40 and 50ºC ........................ 57

Figure 34: The solubility of CO2 in cocoa butter at 40ºC from different references (Venter et al. [4],

Kokot et al. [3] and Calvignac et al. [37]). .............................................................................................. 58

Figure 35: Representation of the interaction parameters kij and lij in function of the temperature and

correlation in a second-order polynomial. ............................................................................................. 63

Figure 36: Heavy phase composition in a weight- diagram of the system CO2/cocoa butter at 313.15 K

(a) and 323.15 K (b): ● experimental points; ── calculated line with interactions parameters of this

work; ── calculated line with interactions parameters obtained from the correlation. .......................... 64

Figure 37: Heavy phase composition in a weight- diagram of the system CO2/cocoa butter at 313.15 K

and 323.15 K : ● experimental points at 313.15K; ● experimental points at 323.15K; ── calculated line

with interactions parameters of this work at 313.15 K; ── calculated line with interactions parameters

of this work at 323.15 K. ........................................................................................................................ 65

Figure 38: Heavy phase composition in a weight- diagram of the system CO2/cocoa butter at 313.15

with solubility data from Venter et al [4]: ● experimental points at 313.15K ── calculated line with

interactions parameters of this work at 313.15 K. ................................................................................. 66

Figure 39: Density measurements at 40 and 50ºC and Density predicted by the Peng-Robinson

Equation of State at 40 and 50ºC .......................................................................................................... 67

9

Abbreviations and acronyms

BWR – Benefict-Webb-Rubin

CCB – Cocoa Butter

CO2 – Carbon Dioxide

EoS – Equation of State

F – Number of freedom degrees

GAME – Gas Assisted Mechanical Expression

GNQ – General Nonquadratic

KM – Kwak-Mansoori

KM1 – Kwak-Mansoori modification 1

KM2 – Kwak-Mansoori modification 2

KTK – Kurihara et al

N – Number of Components

PC – Critical Pressure

PGSS – Production and fractionation of fine Particles from Gas Saturated

POP – 1-palmitoyl-2-oleoyl-3-stearoyglycerol

POS – 1,3-stearoyl-2-oleoyglycerol

P&R – Panagiopoulos-Reid

PR – Peng Robinson

PT – Triple point Pressure

PTV – Patel-Teja-Valderrama

RESS – Rapid Expansion of Supercritical Solutions

RK – Redlich-Kwong

SC-CO2 – Supercritical Carbon Dioxide

SCF – Supercritical Fluids

SCFE – Supercritical Fluids Extraction

SOS –1,3-palmitoy-2oleoylglycerol

SRK – Soave-Redlich-Kwong

TAG – Trialcylglycerol

TB – Boiling Temperature

TC – Critical Temperature

TF – Melting temperature

TT – Triple point Temperature

TR – Reduced Temperature

vdW – Van der Waals

WS – Wong-Sandlers

µ - Chemical potential

ω - Acentric Factor

10

ρ - Density

Π - Number of phases

niα - number of components i in phase α

N – Number of components

∆HF – Fusion Enthalpy

11

1. Introduction Cocoa butter is a vegetable fat, characterized by a complex structure that contains different

types of triglycerides. This compound is very used in food industry, more specifically in the

manufacturing of chocolate, which is the responsible of a good taste of chocolate.

In the development of chemical processes that involves supercritical fluids, it is very important

to know solubility data or phase equilibria data (thermodynamic behaviour). The main application of

supercritical fluids is extraction (SCFE), but recently this kind of fluids started to be applied as solvents

in micronisation processes, as the process for the production and fractionation of fine particles from

gas saturated (PGSS), Crystallisation from supercritical fluids; Rapid expansion of supercritical

solutions (RESS); and fast antisolvent recrystallisation. In this line, the possible application of

micronisation processes in cocoa butter with supercritical carbon dioxide in food, cosmetic and

pharmaceutical industries took to many studies and investigations in order to know well the binary

system.

Despite, the novelty of micronisation processes, SCFE still to be the main application of the

supercritical fluids. This kind of extraction can be found, especially, in food industry, where it is used to

extract resin, aromas, fats, oils, etc.

Beyond of this, there is a new process of extraction of cocoa butter from the cocoa beans,

which is gas assisted mechanical expression (GAME). This process is a mechanical process

(pressing) helped by supercritical carbon dioxide (SC-CO2). In this process the high solubility of SC-

CO2 allows to increase the yield of cocoa butter [4].

The supercritical fluids properties, such as good solvent power due to a density near of liquids,

low viscosity near the one of gas, low surface tension and diffusivities and mass transfer near to the

gases, makes this kind of fluids subjects of many investigations. Carbon dioxide is the most used

supercritical fluid because of many advantages, like low price. It is to expect that in several years the

knowledge of supercritical fluids will increase a lot.

Objectives

In the present work the objective is to study and to characterize the binary system cocoa

butter/SC-CO2. Therefore, it was measured the density and the solubility of the heavy phase in

equilibrium at high pressures at 40 and 50ºC, as the density and compressibility of cocoa butter. For

the measurements of density it was used an autoclave with a saphira window that can be under high

pressures and an apparatus to measure the density, which bases in oscillating U-tube principle3. The

solubility was measured with a synthetic method, trough a cell with variable volume. At last, for

measure the compressibility it was used a porosimeter of mercury.

The last step of the work was to model the experimental data in function of pressure. For the

density and compressibility of the binary system (cocoa butter/SC-CO2) and of cocoa butter it was

used the Tait Equation and the Modified Tait Equation. For the solubility, it was used the Peng-

Robinson Equation of State with two mixing rules: Van der Waals (vdW) and Panagiopoulos-Reid


12

(P&R). For the solubility calculations of the cocoa butter model-compound was considered the

triglyceride 1-palmitoyl-2-oleoyl-3stearoyglycerol (POS) since it is the major triglyceride of cocoa

butter.

The thermo physical properties of the cocoa butter were predicted by the following estimation

methods: Ambrose, Joback, Constatinou-Gani and Fedors methods.

13

2. Literature Review

2.1 Lipids Lipids are one of the main nutrients, such as proteins and carbohydrates. They can be found

in biological tissues, where they play a dominant role in biological functionality with proteins,

carbohydrates, etc. Lipids usually are treated as fats and this word is used to mean both fats and oils.

Despite this fact, solid fats and vegetable oils aren’t alike because fats usually are solid at room

temperature and they come from both animal and plants sources and normally oils come from plants

and are liquid at room temperature. On another hand, considering their composition they are alike

because both are made up of fatty acid molecules and a molecule of glycerol [2] .

Fats (or lipids) have a great applicability in a lot of industries, such as: pharmaceuticals,

cosmetics, foods, etc, due to their physical and chemical characteristics. As an example, Cocoa butter

is used in manufacturing of chocolate, which is the main responsible of the taste of chocolate.

Fats can be essentially represented by triacylglycerols (TAGs), but also by diacylglycerol and

monoacylglycerols, with both saturated and unsaturated fatty acid chains [1] . TAGs are formed if all of

the OH groups of a glycerol molecule are esterified by fatty acid moieties, as it is shown in figure 1.

The fatty acids, as the glycerol molecule, are responsible for the physical characteristics of TAGs

molecules.

(a) (b)

Figure 1: (a) – General chemical structure of a TAG; (b) – Chemical structure of a saturated fatty acid [2].

The two essential features of the fatty acids structure are: the hydrocarbon chain and a

carboxylic acid group (figure 1-b). The hydrocarbon chains are typically linear and usually contain an

even number of carbons that can go up to 30 carbons but more commonly 12-24. However, the

hydrocarbon chains can have double bounds that change their shape and turn them into kinked ones.

Remind that when hydrocarbon chain hasn’t double bounds it is called saturated (straight chain);

otherwise, if it has double bounds it is called unsaturated. The fats usually present a wide range of

physical properties which are influenced by the degree of unsaturation, the length of the carbon chain,

the isomeric form of the fatty acids, the molecular configurations of TAG molecules, and the

polymorphic state of the fat [2].

There exists a lot of ways to classify the fats. They can be classified according to: fatty acid

chain lengths, degree of unsaturation, dominant polymorphic form, source, consumption and those

fatty acids species that dominate this particular fat.

According to the degree of unsaturation of fatty acid, TAG molecules can be classified as:

Saturated, Mono unsaturated, Polyunsaturated or Super-unsaturated (figure 2).

14

Figure 2: Classification of triacylglycerides [2].

The saturated fatty acids don’t have double bonds present (or “kinks”) and they are shaped

like a straight line. TAG molecules that are composed by this kind of fatty acids can easily align

themselves in a close packing to form a compact mass. For this reason, saturated TAG is usually solid

at room temperature and doesn’t easily become oxidised. Monounsaturated fatty acids present one

double bond (“kink”) in their chains. If the unsaturated fatty acids are on the central carbons or on

terminal carbons (sn-2 or sn-3 respectively) of the glycerol molecule, they will interfere with the

disposition of the TAG molecule, or in other words, they will interfere with the close packing of TAG

molecules. In addition, the melting point decreases, crystallization becomes more difficult and their

susceptibility to oxidise increase. Polyunsaturated fatty acids, which have two kinks, make the TAG

molecule be attached to have a higher reactivity. The super-unsaturated fatty acids have more than

three “kinks”, which is translated into an increase in reactivity of TAG molecules, in relation to

polyunsaturated. According to this, the fats saturated with a long chair (highly saturated) usually have

higher melting points than those that have a lot of unsaturations (number of carbon double bonds) or a

little chain of fatty acids.

The unsaturated fatty acids, which influences TAGs behaviour, can have different isomeric

forms (cis-trans form), which have different melting points. Normally, they are in cis-form, but can be

found in trans-form.

TAG molecules, as other lipids and long chain compounds, present polymorphism. Depending

of temperature, pressure, solvent, etc., they can present different crystalline forms. There are three

basic polymorphs of TAG crystals, which are: α, β’and β that are obtained from mono-acid TAG

molecules. The α is the least stable, β’ is metastable and β is the most stable. However, there is some

specific TAG’s whose β form isn’t observed and the β’ is the most stable. Other metastable phases

can be found, depending of the acyl chain composition (as γ, δ and multiple β’). In figure 3 is

presented the thermodynamic stability relationships of the three typical TAG polymorphs [1].

15

(a) (b)

Figure 3: (a) – A Gibbs energy-temperature relationship and (b) – transformation pathways of three typical

polymorphs in TAG [1]

In figure 3, it is possible to watch the variation of Gibbs energy with the temperature (a) and

the possible transformation pathways (b).

2.1.1. Cocoa Butter Cocoa butter is a vegetable fat (from the seeds of Theobroma cacao), which is used in the

manufacturing of chocolate, present at levels up to 40% [3]. It is composed by different kind of

triglycerides and its composition varies depending of the origin. Cocoa butter is responsible for the

taste of chocolate, due to its properties.

The main application of cocoa butter is the production of chocolate and other confectionery

products. However, this fat is also used in the cosmetics industries and pharmaceutical industries as a

basis for suppositories [26].

The properties of cocoa butter, as all the products composed by triglycerides, depend on the

arrangement of the fatty acids in the triglycerides. Cocoa butter has a high content of symmetrical

monounsaturated triglycerides that have the unsaturated fatty acid in position sn-2 and saturated fatty

acid in the other positions (sn-1 and 3). The most abundant triglycerides in cocoa butter, which have

those characteristics cited before, are: 1-palmitoyl-2-oleoyl-3-stearoyglycerol (POS); 1,3-stearoyl-2-

oleoyglycerol (POP) and 1,3-palmitoy-2oleoylglycerol (SOS), which account respectively for 34-45, 21-

29.5 and 12.2-21.5% of the total triglycerides. These triglycerides are the main responsible of all the

physical characteristics [3], [4] .

The main triglycerides, which compose cocoa butter, are polymorphics and they can solidify in

different crystallographic forms. Hereby, the process of solidification of cocoa butter is more complex

than in the case of other fats. It can crystallize in five different polymorphic forms, with different

physical properties (as the melting point). The different polymorphic forms are presented in table 1, in

order of increasing stability.

16

Table 1: The polymorphic forms of cocoa butter and melting points [3]

Polymorphic form Melting point (ºC)

γ 16-18

α 21-24

β1 27-29

β 34-35

β2 36-37

Usually, β is the most stable phase in triglycerides, but in cocoa butter the most stable is the

phase β2. However, phase β is always stable below its melting point, but its kinetics of nucleation and

growth is very slow. Hereby, under direct cooling a less stable phase is formed.

2.2. Supercritical Fluids

2.2.1 Critical Point

Critical point is the point (temperature and pressure) where two phases in equilibrium can

became identical. As the critical conditions are reached, all the properties of the fluid change, until

achieve the critical point. At the critical point, the interface between the coexisting liquid and vapour

phases disappear [5]. When the fluid is above critical point, it is called a supercritical fluid and it

presents features between liquids and gases. Supercritical fluids can have a solvent power near

liquids and have higher compressibility than the liquids one (near gases). Besides, they have transport

properties between liquids and gases. Due to these features, there are on course many investigations

in order to explore the advantages of the supercritical fluids and to apply these advantages in industry [6].

The phase diagrams are one of the best ways to define and to introduce the critical point and

the supercritical fluids. In this kind of diagrams it is possible to see the equilibrium condition between

the thermodynamically-distinct phases, or in other words, it is showed the relation of phases with

temperature, pressure and volume (and composition in case of mixtures) – figure 4.

Figure 4: Phase diagrams of a pure compound: (a) – three-dimensional cut; (b) – P-T cut; (c) – isotherm line [10].

17

According to the figure 4, depending of the pressure and temperature, a pure component can

be liquid, solid or gas. At high pressures and low temperatures it will be solid; at low pressures and

high temperatures it will be gas and it will be liquid at intermediate pressure and temperature. In the

diagrams of the figure 4, it is possible to find a point where the equilibrium liquid-gas ends. This point

is the critical point and it has critical coordinates, critical pressure (Pc) and critical temperature (Tc) [6].

Above this point, the fluid is called supercritical fluid, and it presents all the features that were already

referred.

The determination of the critical points is done by solving the two conditions for a critical point

as derived by Gibbs. For a system with N components these conditions are [7]:

(1)

(2)

Where, a is the molar Helmholtz energy and it can be obtained from a state equation using the

relation ( ) Pa xT −=∂∂ ,/ ν , as all the derivatives in these two determinants. For pure compounds, only

the positions (1, 1) are left. In this way, we have the conditions for a critical point of a pure component

– equation (3) [7].

( ) ( ) 0// 22 =∂∂=∂∂ TT PP νν (3)

This thermodynamic relation describes the fact that the isothermal line presents a tangent at the

critical point (figure 4), or in other words, equation (3) describes the point of inflexion of the critical

isotherm at the critical point. According to the equation (3), the isothermal compressibility (KT) goes

then to infinity as [7].

TT P

K

∂∂−≡ ν

ν1

(4)

18

As it was already referred, all the physical properties change with the approach of the critical

point. A lot of properties become larger when they approach of critical point, such as heat capacity at

constant pressure (Cp), heat capacity at constant volume (Cv), thermal conductivity and isothermal

compressibility. On the other hand, the heat of vaporization vanishes.

2.2.2 Supercritical fluids and its properties

Usually, a gas can condensate with the increase of pressure and a liquid can vaporize with the

increase of temperature. However, at temperatures and pressures above the critical point (Pc and Tc)

these phenomena don’t happen because it corresponds to the supercritical domain [6]. The state of

the molecules in the vicinity of critical point is characterized by one competition between systems

ordered by intermolecular forces and scattered systems due to kinetic energies of agitation [6]. Due to

these facts, supercritical fluids have physical properties between liquids and gases, such as:

• Good solvent power due to a density near of liquids;

• Low viscosity near the one of gas, which facilitates the transfer of the quantity of movement;

• Low surface tension that permit them to penetrate in the porous structures with great facility;

• Diffusivities and mass transfer near to the gases;

• Density near to liquids.

Another peculiarity of supercritical fluids is that they are fluids with versatile solvent power.

They can change their proprieties with the change of pressure and temperatures, especially with the

change of pressure [6].

In the last few decades many researches have been made in order to discover and to know

the potential applications of supercritical fluids, mainly in the food processing, like the

extraction/fractionation of oils, fatty acids [4]. So, one of the main applications of the supercritical fluids

is the extraction (SCFE). This kind of extraction can solve some problems of the conventional

extraction methods, like distillation, avoiding the high temperatures and, therefore the product

degradation. Moreover, this SCFE is a technique rather expensive yet.

2.2.3. Carbon Dioxide: the most used Supercritical fluid

The carbon dioxide (CO2) is the most used supercritical fluid because of many advantages,

such as: low cost; low toxicity; nonflammable; if recycled doesn’t contribute to pollution and is also

rather safe to handle. Despite these facts, CO2 isn’t the best solvent. The solvent power of SCCO2 can

be summarized by a few rules:

1. it dissolves non-polar or slightly polar compounds;

2. the solvent power for low molecular weight compounds is high and decreases with

increasing molecular weight;

3. SC-CO2 has high affinity with oxygenated organic compounds of medium molecular

weight;

4. free fatty acids and their glycerides exhibit low solubilities;

5. pigments are even less soluble;

19

6. water has a low solubility at temperatures below 100ºC;

7. proteins, polysaccharides, sugars and mineral salts are insoluble;

8. SC-CO2 is capable of separating compounds that are less volatile, have a higher

molecular weight and/or more polar, as pressure increases. [9]

9. The carbons aliphatic chains are more soluble than the aromatic compounds;

10. The double bonds increase the solubility in SC-CO2;

11. Hydrocarbons with more kinks are more soluble than the straight lines;

12. The polar functional groups decrease the solubility [6];

Carbon dioxide critical coordinates are easily obtained at laboratory scale, and they are:

• TC=31.1ºC

• PC=73.8 bars

Despite this, SC-CO2 has other advantages at ecological level. It can be used as an alternative for the

organic solvents to generate divided solids [36]. Additionally, CO2 is a gas at atmospheric pressure and

temperature, which allows not leaving residues in the product and avoids other expensive process

stages, as drying [6] .

Carbon dioxide has a very low viscosity, ten times lower than the water. Therefore, carbon

dioxide has a high Reynolds number (Re), which justifies the high capacity to transfer the convective

heat (transfer of convective heat is proportional to Re) of SC-CO2 [6]. Additionally, despite of a lower

surface tension of carbon dioxide than the one of organic solvents, it has high diffusivity due to low

viscosity, which allows SC-CO2 to penetrate in the microspores of complex structures [6].

2.2.3.1 Applications

Supercritical fluids (SCF) have a lot of applications due to its specific properties, but the main

application is in the area of supercritical fluid extraction (SCFE), which uses the variation of solubility

with temperature and pressure of SCF. This application is the most known and it is used in a lot of

industries, such as, food industries in the extraction of coffee, spices, aromas, fats and oils and in

pharmaceutical industries in the isolation of particular components or active ingredients. Currently,

due to all the investigation over this field the application of SCF extends to a lot of areas beyond

SCFE. It is possible to find SCF in: fractionation, chromatography, chemical and biochemical

reactions, generation of divided solids, etc. All this domains, as other, can be found in textiles, paper,

petrochemical and fine chemistry industries.

However, SCF has a big disadvantage that is the need to work at high pressures. This fact

increases a lot the cost of processes involving SCF, which prevents the spread of SCF in more

industries. In this way, beyond SCFE, the applications more advantageous are: the fractionation of

liquid mixtures of fats, polymers (etc) due to the high selectivity of SCF for one molecule; SCF, as

water and SC-CO2, can be used as reactium medium; chromatography with SCF eluent (complement

almost indispensable for chromatography in gaseous phase and in liquid phase under high pressure);

and, at last, SCF is very used in solid treatments, such as, transformation and purification/

fractionation of polymers, productions of fine powders, fibres, liposomes, microencapsulation (RESS,

20

PGSS, SAS), painting covering, porous material (mousses, aero gels, ceramic’s) and impregnation of

a lot of matrices, as polymers, paper,etc.

2.3. Study of the binary mixture: Cocoa butter/Supe rcritical CO 2

2.3.1. Phase Equilibria

2.3.1.1. The Phase rule of Gibbs [7]

The equilibrium conditions of phase equilibria can be derived with the second law of

thermodynamics and Gibbs energy (G). According to this law, the total Gibbs energy of a closed

system, at constant temperature and pressure, is minimum at equilibrium. Combining with the

condition that the total number of moles of a component i is constant in a closed system

( ntconstani∑ =α

α , where αin is the number of moles of component i in phase α), it can be easily

derived the equilibrium conditions of a system of Π phases and N components (5).

Niforiii ,...,2,1... ==== πβα µµµ (5)

Where, µ is the chemical potential. At the equilibrium, the chemical potentials of the different

phases of each component must to be equal. The chemical potential is defined by the following

equation (6).

ijnTP

i

ii

i n

gn

≠

∂

∂=

∑

,,

α

αα

αµ (6)

Where, g is the molar Gibbs energy, αin is the number of moles of component i in phase α. If it is

assumed the condition∑ =i

ix 1α , the mole fraction at component j becomes to a dependent variable.

In this way, since αµ i is a function of P, T and (N-1) mole fractions, the equation (5) represents N(Π-1)

equations in 2+Π(N-1) variables. Then, the number of degrees of freedom (F) is obtained through the

subtraction of the number of equations from the number of variables – equation (7) – Phase rule of

Gibbs.

NF +Π−= 2 (7)

21

Through this rule, if a number of freedom degrees of the system is chosen, provided that g of

all phases (a function of pressure, temperature and composition) is known, all the thermodynamic

properties of a system with N compounds and Π phases can be calculated.

This phase rule is considered the most complete rule. This equation is only applicable to

systems in equilibrium and there can be only one gas phase, due to the mutual solubility of gases in

each other, but there can be multiple liquid (immiscible liquids) and solid phases [12].

2.3.1.2. Classification of fluid equilibria [7]

The cocoa butter is a mixture of triglycerides and for this reason it is very difficult to classify

and characterize the equilibrium of cocoa butter with SC-CO2 considering every triglyceride. Hereby, it

will be assumed that the behaviour of cocoa butter is very near to the one of a major triglyceride of

cocoa butter (POS-1-palmitoyl-2-oléoyl-3stéaroyglycérol).

Considering that cocoa butter is made of by POS and is in equilibrium with CO2, the phase

rule of Gibbs (equation (7)) simplifies to equation (8), which is the phase rule for a binary system

(N=2).

Π−= 4F (8)

According to equation (8), a binary system at equilibrium can have 4 phases (Π=4) at

maximum, and the maximum number of degrees of freedom needed to describe the system is 3 (F=3).

It can then be concluded that all phases of the system cocoa butter/CO2 can be represented in a

three-dimentional space (P, T and x).

At equilibrium, the phases, which participate to the equilibrium, present the same T and P but

usually different x (mass composition). In other words, this means that the four-phase equilibrium

(F=0) is described by four points in P,T,x space, a three-phase equilibrium (F=1) by three curves, a

two-phase equilibrium (F=2) by two planes and one phase (F=3) state by a region. In turn, the critical

state and azeotropic state are represented by one curve.

Usually, the essentials of phase diagrams are represented in a P, T projection but it can be

represented in isothermal P, x sections, isobaric T, x sections or isoplethic P, T sections. In the first

type mentioned (P, T projection), just the non-variant and monovariant (F= 0 and 1) can be

represented. In this kind of projection, the four-phase equilibrium is represented by one point at the

intersection of four three-phases and the three phase equilibrium is represented by one curve, as

critical and azeotropic curve. There is also a point, which is the interception of a three-phase curve

and a critical curve and is called critical point, which is characterized as the end of the three phase

curve and critical curve.

In a P/T, x projection, the three-phase equilibrium is represented by three points. These points

give the composition of each phase, which are at equilibrium at one pressure or temperature. In figure

5 it is possible to find an example of three phase equilibrium αβγ in a P, x section.

22

Figure 5: Possible location of one – and two phase equilibria around a three-phase equilibrium in a P,x-section [7].

As it is seen in figure 5, there are three two-phase regions (α+β, β+γ and α+γ) and three one-

phase region (α, β and γ) in the vicinity of the three-phase equilibrium (three black points). These facts

are in agreement with the theory of transformations, which says that the one and two-phase regions

have to be arranged in the vicinity of the three-phase equilibrium. Figure 5 is an example of P, x

section, but it would be observed the same thing if it was a T, x section, because they basically look

the same.

In figure 6 it is presented a case of a vapour-liquid equilibrium (P, x section).

Figure 6: Two phase equilibria – Pure component boiling point

The equilibrium between the vapour and liquid phases, as it was already observed in figure 5,

are represented by curves. In the case of vapour-liquid equilibrium this curves are called binodal

curves. At a given pressure (or temperature, in case of T, x section) a mixture with a composition

between the two binodal curves will split in two phases, liquid and gaseous phase. On other hand, the

mixture outside the binodal curves will be liquid or vapour (one phase region). When there’s no

azeotrope or if it isn’t above the critical temperature, the binodal curves intercept at the pure

component vapour pressures. Otherwise, they can intercept at the azeotropic point (figure 7-a) or st

the critical point (figure 7-b). These two cases represent extremes in pressure or temperature in P, x

or T, x sections, respectively.

23

(a) (b)

Figure 7: Two phase equilibria: (a) – Critical Point; (b) – Azeotropic point [7].

2.3.2. Van Konynenburg and Scott Classificaion [6], [7]

The classification proposed by Van Konynenburg and Scott was established in 1980, being

today a reference and a very used classification. According to this classification, there are six basic

types of fluid phase behaviour. All these types have been found experimentally, except the type VI,

and they can be predicted with a van der Waals equation (state equation). With this classification, it is

possible to predict, in a qualitatively way, the behaviour of the phases.

In the figure 8 it is presented the P-T projections of the six basic types of fluid phase

behaviour.

Figure 8: The six basic types of fluid phase behaviour according to the classification of Van Koynenburg and

Scott [7].

In figure 8, it is possible to observe two kinds of curves. The curves lg are the vapour pressure

curves of the pure compounds and end at the critical point (l=g). The other curves (bold lines) are

divided in: the curves l=g, l1=g, and l2=g that are vapour-liquid critical curves and the curves l1=l2 that

are curves where two liquid phases become critical.

The point of interception of a critical curve with a three-phase curve (l2l1g) is a critical endpoint.

There are two kinds of critical endpoints: upper critical points (UCEP - ▲) and lower critical endpoints

24

(LCEP - ▼). The UCEP is the one who presents the highest temperature of a three-phase curve and

LCEP is the critical endpoint with lower temperature of a three-phase curve.

• Type I

As it is possible to see in figure 8, the type I phase behaviour has only one critical curve. The

vapour-liquid critical curve (l=g) runs continuously from the critical point of a component to the critical

point of the other component. This kind of behaviour is only observed for compounds chemically

similar.

In the figure 9, it is put side by side the phase diagrams at lower pressure of a simple binary

system and the system at high pressures of type I, in order to compare, clarify and situate the ideas.

Figure 9: (a) – P-T projection of a binary mixture of type I according to the classification of Van Koynenburg and

Scott; (b) – phase diagram P,T of a binary mixture at lower pressures

The points C1 and C2 are the critical points, as it was already shown in figure 8, they are

connected by a vapour-critical line. Each point of this line corresponds to a critical point of a binary

mixture with one composition.

In the figure 9-b, the curve that corresponds to mass fraction x=0 is the equivalent to the curve

1 of the high pressure diagram (figure 9-a) – the bubble point curve. The other curve (x=1) correspond

to the curve 2 of the high pressure diagram (figure 9-a) – dew point curve. The two curves intercept

each other at critical point (point of figure 9-b) and between both curves it is an equilibrium biphasic

liquid-vapour.

• Type II

The phase behaviour of type II is similar to type I, with the exception that, at lower

temperatures, the liquid mixtures of the compounds aren’t completely miscible in every proportion.

These facts are due to the weak intermolecular interactions, which lead to the separation of phases.

In figure 8 it is possible to see that in type II, next to a continuous l=g critical curve, at low

temperatures there is also l2=l1 critical curve and a three-phase curve l2l1g, which intercept in a UCEP.

The critical curve l2=l1 runs steeply to high pressure and represents upper critical solution

temperatures.

• Type III

The phase behaviour type III is an example of mixtures with a large zone of immiscibility liquid-

liquid. The critical curve moves to elevated temperatures and interferes with the vapour-liquid critical

25

curve. It is then possible to conclude that the critical curve doesn’t have to connect the critical points of

the two pure compounds, but it can present two branches. In figure 8 it is possible to see one branch,

which starts in critical point of compound more volatile (the upper curve) and ends at UCEP, ore

where the liquid and gas phase have the same composition. The UCEP results of the interception of

the three-phase line l1l2g with the l1=g. The other branch starts at the critical point of the compound

less volatile and increases to high pressures. In this critical line it is found l2=g and l1=l2.

The branch of l2=g/l1=l2 can have the shape present in figure 8, but it is also possible that this

curve goes from the critical point of the component less volatile at high pressure via a temperature

minimum (dP/dT is always positive) [7].

• Type IV

The type IV phase behaviour is a combination between type II and V behaviour. This system

presents two branches of the l2l1g, three branches of the critical curve and three critical endpoints. As

result of the interception of the critical curves with the three-phase curves result three critical end

points: two UCEP and one LCEP (critical endpoint with lower temperature of a three-phase curve).

The two three phase curves mean that there are two zones where exist two liquid phases not

miscible. The first one is at low temperatures and ends at the first UCEP. With the Pressure and

temperature increase, there is another zone of immiscibility, which starts at LCEP and finishes at the

second UCEP.

• Type V

The type V phase behaviour is characterized by present near to Tc of the more volatile

compound a three phase curve (l1l2g). This curve starts at LCEP (interception of l2=g and l2l1g) and

ends at UCEP (interception of l1=g and l2l1g).

In turn, the critical curve shows two branches. The first one (l1=g) starts at critical point of the

more volatile compound and finishes at UCEP. The second (l2=g/l2=l1) starts at critical point of the less

volatile compound and finishes at LCEP.

• Type VI

The binary mixtures, which present phase behaviour of type VI, have two critical curves. The

critical curve l=g starts at critical point of one compound and ends at the critical point of the other

compound. The other one (l2=l1) starts and finishes at the critical endpoints: LCEP and UCEP.

The three-phase curve intercept two times the critical curve l2=l1, resulting the critical

endpoints.

Beyond these six types of phase behaviour, there are many more types of fluid phase

behaviour, a lot of them have been found computationally but not in the real systems [7]. There is also

fluid phase equilibrium with solids that isn’t presented here.

The type of behaviour of a mixture depends essentially of the compounds, as an example: a

mixture that has behaviour of type I probably has two compounds with critical points not very far.

26

In binary systems composed by light gas and lower members of the n-alkane series are

normally described trough the type II phase behaviour. With the increase of the number of carbons,

the binary are usually described by the type IV phase behaviour, followed by the type III phase

behaviour, for high carbon numbers. In case of the systems CO2/n-alkanes, lower n-alkane molecules

(n≤12 carbons) usually show a phase behaviour of type II; n-alkane show phase behaviour of type IV

with n=13 and phase behaviour of type III for molecules with n≥14 [7].

2.3.3. Representation and modelling of the system

To model the compressibility of cocoa butter it was used the original Tait Equation, which is

known to represent very satisfactorily the compressibility of solids under high pressures. For the

density of cocoa butter and of the mixture it was used the Modified Tait Equation as Tait Equation.

For the solubility, the equations of state were chosen to predict the solubility of the mixture. In

this way, it will be presented a resume about the equations of state and which one it should be

chosen.

To apply the equations of state the critical parameters, which are unknown, are needed.

Hereby, it is also included some estimation methods for these parameters.

2.3.3.1. Equations of State

The objectives of the State Equations are to correlate the data that already exist and to predict

the data in regions where experimental results are not available. An ideal state equation would predict

with high accuracy the phase equilibria under all conditions and it would be theoretically based.

However, there’s no such state equation, or other methods that can treat all situations [7].

In figure 10 a classification of the various types of equations of state is presented.

Figure 10: Classification of various type of equations of state, with a selection of equations for each group. In this

classification, Van der walls Equations of State are those cubic and noncubic equations that consider the compressibility factor as Z=Zrep+Zatt.

2.3.3.1.1. Cubic Equations of State

The van der Waals equation of state was proposed more than one century ago and it was a

very important step for the development of science. Since first proposed by van der Waals, many

modifications have been proposed in order to improve the predictions of volumetric, thermodynamic

27

and phase equilibrium properties. Despite van der Waals equation isn’t the best equation to describe

in an accurate way most of the cases, it was a great contribution to this field. The great innovation of

van der Waals and his equations was to consider the volume occupied by the molecules, thus

substituting the volume V by V-b, where b represents the volume occupied by the molecules [14].

After van der Waals equation and the many modifications suggested, it was proposed a

generic cubic equation (equation (9) e (10)).

( )VTPbV

RTP att ,−

−= (9)

( ) ( ) ( )dVcdVV

aVTPatt −++

=, (10)

Where, P is the pressure, V the molar volume, a, b, c and d can be constants or functions of

temperature and some fluid properties, such as acentric factor, critical compressibility, normal boiling

point, critical temperature, etc.

Presently, all cubic equations of state (as van der Waals equation) are considered special

cases of the equation (9). In this way, despite the existence of many different types of equations as

several new applications, cubic equations are similar to equation (9) and are still used in

semiquantitative predictions of the equilibrium phenomenon in process design and in simulations.

The popularity of the cubic equations of state is due to several reasons. In table 2 is presented

their advantages and disadvantages.

Table 2: Advantages and Disadvantages of Van der Waals Cubic Equations of State [14]

Advantages Disadvantages

a

Third degree in volume, which makes calculations

relatively simple to perform a

Actual PVT data tend to follow a fourth-degree equation instead of a cubic equation

b Present correct limiting behaviour: as V→b;P→∞ in

all van der Walls type equations b Both the repulsive and attractive terms are

inaccurate, as shown by molecular simulations

c

Kown inaccuracies of both the repulsive and

attractive terms are cancelled when Equations of

State are used to calculate fluid properties, in

particular VLE

c Cubic equations cannot represent all properties

of a fluid in all different ranges of P and T

d

For most applications, cubic Equations of State can

be tuned to give accurate values for any volumetric

or thermodynamic property

d

Temperature dependency of the force constant a is not well

established; co-volume b seems to be density-dependent,

but the dependence is unknown

e extension to mixtures is relatively easy using

mixing and combining rules of any complexity

e

Because interactions between unlike molecules are unknown,

most mixing and combining rules are empirical, and

interaction parameters are usually required

f cubic equations are suitable for the application of

modern mixing rules that include Gibbs free energy models or concentration-dependent parameters

f

In applications to complex mixtures, several interaction

parameters might be required, even with the use of modern

mixing rules

28

2.3.3.1.2. Peng-Robinson Equation

Many modifications were proposed, one of them was proposed by Peng and Robinson [15] who

improved a previous modification (Soave’s equation) by recalculating the term α(TR,ω) (introduced

before by Redlich-Kwong [14]) and modifying the volume dependency of the attractive term. They

suggested the following equation [15].

( ) ( )bVbbVV

Ta

bV

RTP

−++−

−= )( (11)

Where, a is “specific attraction” and b is a multiple of the molecular volume. The variables are

obtained through the expressions below.

( )

( ) ( )[ ]2

25.05.2

26992.054226.137464.007780.0

11,45724.0

,)(

ωω

ωα

ωα

−−==

−+==

=

mP

RTb

TmTP

RTa

TaTa

C

C

RRC

Cc

RC

(12)

Where, R is the specific gas constant, Tc and Pc are, respectively, critical temperature and

pressure, TR the reduced temperature and ω is the acentric factor.

With the equation (11) it was obtained better results for liquid volumes were obtained, as well

as better representations of vapour-liquid equilibrium for many mixtures [14]. The Peng-Robinson

equation (PR) and the Soave-Redlich-Kwong (SRK) equation are the most popular cubic equations of

state. Presently, it is very used in research, simulations and optimizations in which thermodynamic

and vapour-liquid equilibria are required.

2.3.3.1.3. Application to mixtures

The cubic state equation can be used to calculate vapour-liquid equilibrium involving mixtures.

In other words, the same equation that is used for the pure fluids can be used for mixtures. For that, it

is needed to get mixing parameters to calculate the values of a and b for mixtures.

van der Waals suggested the classical mixture rules, which were used in many applications,

and are still very common:

∑∑=i j

ijji axxa (13)

∑∑=i j

ijji bxxb (14)

For the parameters aij and bij it is needed combination rules, which are usual given by the following

equations:

)1( ijjiij kaaa −= (15)

)1(2 ij

jiij l

bbb −

+= (16)

29

Where, kij and lij are the binary interaction parameters. These parameters are obtained

through fitting the experimental data, minimizing the error between the experimental values and

calculated ones. Usually, the value of lij is considered equal to zero; therefore the value kij is more

important in these equations.

The combining rule for the parameter a is based in the intermolecular potential theory, which

ruled that the attractive parameter in the intermolecular interaction for the interaction between an

unlike pair of molecules is given by a relationship similar to equation (15). The parameter b (repulsive

parameter) is obtained with the equation (16). However, these combining rules considered that the

molecules were hard spheres. This isn’t a good approximation because most of molecules are non-

spherical, don’t have hard-body and there’s not a one-to-one (direct) relationship between the

attractive part of the intermolecular potential and a parameter in equation of state [7], justifying the fact

of lower accuracy obtained, especially with highly polar mixtures, associated mixtures and other

complex systems.

In this way, more complex combining rules have been developed in order to obtain more

accurate results. In the following table a summary of some combining mixture rules is presented.

Table 3: Mixing Rules and Combining Rules Used in Two-Constant Cubic Equations of State [14] Mixing/Combining rule Formulas

Van der Waals (vdW)

one parameter: kij two parameters: kij, lij

Panagiotopoulos-Reid (P&R) two parameters: kij, kji

three parameters: kij, kji, lj

general nonquadratic (GNQ)

two parameters: δi, δj parameters: δi, δj, βi

Kwak-Mansoori (KM)

three parameters: kij, βij , lij

Kwak-Mansoori modification 1 (KM-1)

three parameters: kij, lij, δi (one solute)

Kwak-Mansoori modification 2 (KM-2)

three parameters: δi, δj, βi (one solute)

Kurihara et al. (KTK)

three parameters: η1, η2, η3

Wong-Sandler

one parameter: kij

two parameters: kij, li (one solute)

In the present work, the mixture has a supercritical fluid (SC-CO2) and Cocoa butter.

According to the bibliography, the equations of state and mixing rules that usually produce better

30

results are the Peng-Robinson Equation and Patel-Teja-Valderrama with the Panagiotopoulos-Reid

(P&R) and Wong-Sandler (WS) mixing rules [14]. This doesn’t mean that these equations will give

better results with the mixture cocoa butter/CO2, but only that for systems with supercritical fluids one

usually gets a good fit. In this way, will be used the Peng-Robinson equation with the

Panagiotopoulos-Reid (PR) and van der Waals mixing rules in order to analyse and compare the

results.

2.3.3.1.4. Estimation techniques [11]

The critical point, or critical properties (Critical temperature – TC, Critical pressure – PC, Critical

Volume –VC) are very used data. However, a lot of them are still unkown and as such it is necessary

to use some estimation techniques in order to predict them.

• Ambrose method

In the Ambrose method, the critical coordinates are determined through a group contribution

technique, using the following relations:

( )[ ]1242.11

−∑∆++= TBC TT (17)

( ) 2339.0

−∑∆+= PC MP (18)

∑∆+= VCV 40 (19)

Where, TB is the normal boiling point (at 1 atm) and M is the molecular weight. The ∆ quantities are

evaluated by summing contributions of various atoms or groups of atoms (Appendix B). The units

used are Kelvin, bar, and cubic centimeter per mole, respectively.

For perfluorinated compounds or for compounds containing only halogens the constant 1.242

of equation (17) would be replaced by 1.570 and the constant 0.339 of equations (18) would be

replaced by 1.000.

• Joback modification of Lydersen’s method

Lydersen (in 1955) developed the first successful group contribution method to estimate the

critical coordinates of pure compounds. Then, Joback evaluated the Lydersen’s method and added

several functional groups and determined the values of the group contributions, in order to improve the

previous method. The relations proposed are:

( )[ ] 12965.0584.0

−

∑ ∑∆−∆+= TTBC TT (20)

( ) 20032.0113.0

−∑∆−+= PAC nP (21)

∑∆+= VCV 5.17 (22)

∑∆+= TBT 2.198 (23)

31

Where, nA is the number of atoms in the molecule. The others variables have the same meaning as for

Ambrose method (∆ contributions are in Appendix B), and the same units (Kelvin, bar, and cubic

centimeter per mole).

Fedors method

The Fedors method is a group contribution method too. However, it isn’t as accurate as the

methods presented before and it is just valid for critical temperatures. Despite this, it has the

advantage of not requiring the normal boiling point to determine the critical temperature (TC). The

relation proposed is:

∑∆= TCT log535 (24)

Where, Kelvin is the unit of TC and the contributions values (∆) are presented in Appendix B.

• Constatinou and Gani Method [16]

The Constatinou-Gani Method is also a group contribution method, but it presents some

differences from the above. In this method, the structure of a compound is characterized by two type

of groups:

• First-Order groups – simple functional groups;

• Second-Order groups – functional groups as buildings blocks.

The other methods used already the first-order groups to estimate the properties of the compounds.

The innovation of this method was the addition of the second-order groups. These groups have the

purpose to provide more structural information about the portions of the molecular structure of a

compound, where the description of the first-order group is insufficient. Moreover, it was intended to

improve the accuracy, reliability, the range of the applicability and overcome some of the

disadvantages of the first-order group. With this method it is possible to distinguish isomers. In table

B4 and B5 (Appendix B) the contributions of the first- and second-order groups are presented.

The properties are estimation according to the equation below.

( ) ∑∑ +=j

jjii

i DMWCNXf (25)

Where, Ci is the contribution of the first order group type-i which occurs Ni times and Dj is the

contribution of second-order group type-j which occurs Mj times. The f(X) is a simple function of the

property X (boiling temperature, critical temperature, etc) and W is assigned to unity in second-level

estimation, where both first and second-order group contributions are involved and to zero in the basic

lever (first-order approximation), where only the contributions of first-order groups are employed.

There is lack of lipids data, and because of this it is needed to estimate all the properties of

lipids. With this, it appeared a big problem: which method we should choose. As there are only a few

results of estimations, it is difficult to compare estimated results and it is impossible to compare these

32

results (which can be obtained) with experimental data as there isn’t any. In this way, it is needed a

good critical sense.

According to the literature [17], for fatty acids esters the best estimations obtained (without

experimental TB) were:

• TB – Joback method;

• Tc – Joback method;

• Pc – Constatinou and Gani .

However, for the fatty acids, the Constatinou and Gani method showed better results for TB

and for TC without experimental data of TB, comparing with other methods that used experimental TB.

Despite of this, in the literature (Weber,1999) [18] it was chosen the Ambrose method because this

method presented lower errors for molecules with high molecular-weight (like triglycerides)

In other literature [11], it was suggested to use Ambrose and Joback method, when a reliable

value of Tb is known, and the Fedors relation, when Tb isn’t known. But the accuracy of Fedors

method depends of the compounds. However, this reference was written before the existence of the

Constatinou and Gani method.

In conclusion, the choice of the method will depend on the particular case. A good analysis

should be done by comparing some experimental data with the properties estimation obtain by the

methods. This comparison/validation could be direct (in a measured property) or by a indirect via (in

case of a calculated property).

2.3.3.1.3. Acentric Factor [11]

The acentric factor is defined through equation (26):

( ) 000.17.0log −=−= TratPrvpω (26)

Where, ω is the acentric factor,

rvpP is the reduced vapour pressure (PR=P/PC) at reduced temperature

of 0,7 (TR=T/TC).

The acentric factor (ω) is one of the most common pure component constants and it

represents the acentricity or nonspshericity of a molecule. In this way, it is possible to conclude that for

the monoatomic gases ω is essentially zero and is still small for methane. The acentric factor

increases with the increase of the molecular weight of hydrocarbons, as well as with the polarity.

Usually, the ω is very used as a parameter to measure the complexity of a molecule with

respect to both the geometry and polarity. However, the large values of ω for some polar compounds

(ω > 0.4) aren’t meaningful in the context of the original meaning of this property.

Usually, when the acentric factor of a compound is needed and isn’t known, it has to be

estimated after having estimated the critical constants (Tc and Pc) and then determine the reduced

vapour pressure at Tr= 0.7. However, there are other estimation techniques, which allow to determine

ω. The acentric factor can be estimated trough the definition (equation 26) or with Lee-Kesler

vapour pressure relations [11].

33

2.3.3.2. Tait Equations [13]

The isothermal density data for liquids up to elevated pressures are widely represented by the

equation (27) – Modified Tait Equation.

( )( )

++=

−

0

0 logpB

pBC

ρρρ (27)

Where, ρ is the density of the liquid, p the pressure, A and C are two parameters obtained trough the

fit of data. The subscript 0 refers to low pressure, usually 0.1 MPa or saturation pressure.

This equation represented in terms of volume gives (28).

( )( )

++=−

00

0 logv

vv

pB

pBC (28)

Where, v is the volume and the other parameters has the same mean of the equation (27).

The Equation (27) reproduces very satisfactorily liquid density measurements over a wide

pressure range, and it is known as Tait equation. However, this equation isn’t the original Tait

equation. This equation results of a modification of the true Tait equation (equation (29)).

p

A

+=

−π0

0

pv

vv (29)

Where, v is the volume, p the additional pressure, A and π are parameters to determine. This equation

was developed in the XIX century (1871) to determine the compressibility of water in order “To

investigate the Physical Conditions of the Deep Sea, in the great Ocean-basins, (…)in regard to

Depth, Temperature, Circulation, Specific Gravity, and Penetration of Light (…)”.

In 1967, Hayward made a comparative study of compressibility equations and suggested the

use of equation (30) (inverse of equation (29)), in order to fit the data with a straight line – Isothermal

secant bulk modulus.

A

p

Avv

pv+=

−π

0

0 (30)

34

The Tait equation in this form represents very well the compressibility data of solids at high

pressures (up to a few gigapascals of pressure) – figure 11-a. However, for liquids the equation (29)

isn’t so good. As it is possible to observe in figure 11-b, equation (29) fits the experimental

measurements within the estimated uncertainty over a more limited pressure range. In the case

presented in the figure 10-b, the linear dependence of the equation (29) extends only to just above

150 MPa.

(a) (b)

Figure 11: (a) – Pressure dependence of isothermal secant bulk modulus for different solids; (b) – Pressure dependence of the isothermal secant bulk modulus for hydrocarbons (□) n-Hexane at 337K; (○) n-heptane at

303K; (●) mesitylene at 298 K.

Tammann was the responsible for the creation of the modified Tait equation (equation (27)).

First, he replaced the Tait’s average compressibility with the corresponding differential coefficient –

equation (31).

pB

A

p

v

+=

∆∆

(31)

Then, he integrated the equation and obtained the equation (32).

( )

+−=B

pBAvv

ln10

(32)

Finally, replaced A by C, included p0 and, through a rearrangement, it was obtained the

equation (28) – Modified Tait Equation. Through this equation it is possible the equation (27).

Haydard, who proposed the equation (30), considered that the modified Tait equation hadn’t

advantages over the equation (30) because it didn’t fit the experimental data for water so satisfactorily

as the equation (30). However, the new modified Tait equation was well accepted and it is very used

to represent high-pressure density for liquids and liquid mixtures. This equation presents good results

especially until 150 MPa, where the equation (30) has some limitations.

• Parameters B and C

According to the results obtained for the hydrocarbons, it was suggested that the parameter C

was independent of temperature. In turn, the parameter B decreases with the increase of temperature.

35

The first C values suggested were: C=0.2173 for n-alkanes and C=0.2058 for aliphatic hydrocarbons.

These C values were suggested in order to be constant for a given series. Despite the fact that these

values show a reasonable agreement, the number of figures given implies a significant difference in C

for different groups of compounds. However, as it is possible to observe in table 4, there is evidence

that for liquids in general the parameter C isn’t constant, where C ranges from 0.172 to 0.25.

Table 4: Temperature-Independent C Values for Liquids Compound T (K) P max C

Cyclohexane 298-348

303-393

100

250

0.1988

0.2095

Chlorobenzene 298-358 100 0.2159

Bromobenzene 298-358 100 0.2159

Nitrobenzene 298-358 100 0.2159

Aniline 298-358 100 0.2159

1-Chlorobutane 303 500 0.173

1,2-Dichloroethane 298-398 101 0.232

Tetrachloroethane 298 100 0.2126

Acetone 298 100 0.2356

Diethyl ether 273-353 100 0.207

Ethyl Acetate 253-313 150 0.202

Glycerol 223-353 280 0.2568

Pentan-1, 5-diol 253-308 280 0.3146

Polyisobutylene 326-383 100 0.2006

Poly (vinyl acetate) 337-393 100 0.2409

In turn, in table 5 it is possible to observe that for water the parameter C varies strongly with

temperature. Through this, it can be concluded that when the density data is extended to low

temperatures it is needed to consider the variation of the parameter C with temperature –

Temperature-Dependent C Values for Liquids.

36

Table 5 :Temperature-Depent C Values for Liquid

Compound T (K) P max C

Water 283

348

200

200

0.2691

0.3467

Bromobutane 203

373

550

550

0.2176

0.2648

Acetonitrile 253

313

300

300

0.194

0.24

Bromobenzene 278

323

200

275

0.2209

0.2251

An expression to calculate the parameter B was developed – equation (33). This equation is to

be used for temperatures below 0.66 times the reduced temperature (Tr) and is adjusted for the

hydrocarbon number (Cn).

( )62210 −−++= nrr CTaTaaB (33)

This equation presented good results to n-alkane. The parameters a0, a1 and a2 depend of the kind of

compounds.

2.3.3.3. Study and Determination of properties conc erning SC-CO 2 and Cocoa butter/Lipids

In the last years a lot of investigations have been taken in order to study the behaviour of lipids

(fatty acids, triglycerides) with SCF. All these investigations, specially the characterization of systems,

are very important steps for future applications.

Venter et al. [4] , and Kokot et al. [3] studied the system cocoa butter- SCCO2 in order to

characterize this binary system. These two investigations had different purposes. While Venter wanted

to characterize this system to apply it in a mechanical process of extraction (GAME), Kokot wanted to

apply in in PGSS. In this two works, it is possible to have access to density, solubility, melting

temperatures, etc.

Beyond these investigations, there are much more works done in this area, with others fats

and oils. A lot of them don’t have just as objective the characterization of one system, but also

modelling. For that, the equations of state are a powerful tool to estimate the system behaviour out of

the measured conditions.

37

3. Measures and modelling of the properties of the binary cocoa butter/SC-CO 2

3.1. Density measurements

3.1.1. Material

The cocoa butter used was acquired from Gerkens Cacao (Wormer, The Netherlands).

The Liquid CO2 was acquired from Air Liquid SA, France with the purity 99,995%.

3.1.2. Experimental Setup

In the figure 12 it is presented the scheme of the diagram of the experimental device.

Legend:

1 – Diaphragm pump 6 – PAAR Reactor (Autoclave) 11 – Densimeter 2 – Heat Exchanger 7 – Heating Jacket 12 – Module of acquisition 3 – Pump 8 – Purge 13 – Thermostatic bath of Water 4 – Thermostatic bath of Water 9 – Isolator Jacket 14 – Fed of CO2

5 – Stirrer 10 – PID regulator

Figure 12: Schematic diagram of device The autoclave (6 – PAAR Reactor – figure 13 a) has a capacity of 1.2 L and can be operated

under high pressures. This cell has a stirrer (5), three sapphire windows (in order to observe the

behaviour of the mixture) and a heating jacket (Radiospare - 7) covered by an isolator (9), which

controls the temperature and allows to operate at constant temperature (±1ºC), after receiving the

information of the controller. The temperature was measured with a thermocouple (Watlow). The

constant pressure of the autoclave was maintained by a Pump (ISCO - figure 13 c) that works as a

piston. When the pressure of the cell is lower than that of the set up it increases the pressure feeding

more CO2 and it does the inverse when the pressure is above of the set point. The temperature in this

pump is maintained by a thermostatic bath of water (JULABO – 4).

The densimeter (11) is an oscillating densimeter (DMA HPM, Anton Paar, Graz, Austria –

figure 13 b), which measures the density according to the oscillating U-tube principle that is based on

the law of harmonic oscillation [20]. Therefore, the device measures the period of the oscillation, which

is related with the density (according to the calibration) of the material that is inside of the U-tube. The

results are obtained in the module of acquisition (12). The temperature of the densimeter is controlled

38

by a thermostatic bath of water (13). The pressure and temperature at the entrance of the densimeter

are controlled by a PID regulator (10).

(a) (b) (c)

Figure 13: (a) – PAAR reactor; (b) – Densimeter and PID controller; (c) – ISCO Pump

3.1.3. Experimental Procedure

• Density measurements of the mixture Cocoa Butter/CO 2

The experimental work was carried out at 40 and 50ºC, at 50, 80, 100, 150, 200 and 250 bar.

The experience started at 50ºC.

The first step of the experimental work was to measure the quantity of cocoa butter (weight)

required for the experience. The quantity of cocoa butter used in all the experiment work was

approximately 380 g. This quantity didn’t have to be a rigorously quantity, just had to ensure that it had

enough cocoa butter for all the experience. For the measurement of cocoa butter, it was used a glass

cell. After the measurement of the quantity cocoa butter, the glass cell was put in a hot water bath at

45ºC (temperature above the melting temperature), in order to melt all the material.

Once melted, the cell (with cocoa butter) was placed inside of autoclave (6 – PAAR reactor).

After this, the agitation was turned on at velocity number 2 (2 of 5 velocities). The next step was to

heat the autoclave to 50ºC. Then, the heating device was turned on and the set point defined to 50ºC.

After achieved the preset temperature, CO2 was fed into the autoclave through the diaphragm pump.

During the feeding of the CO2 the valve V-5 and V-6 (valves of entrance and exit of densimeter) were

closed and the ISCO pump was empty (volume 0 mL). The feeding of CO2 ended when the pressure

of the autoclave reached 60 bar. At this moment, the ISCO pump was turned on in constant pressure

mode, defined to 50 bar. The ISCO pump started to work, removing CO2 from the autoclave. The

ISCO pump ensured the constant pressure during all the experience. When the pressure was constant

at 50 bar and the flow indicated in ISCO pump was null, the agitation was stopped. The next step was

to achieve the equilibrium, which was obtained when the pressure and temperature stop to vary.

At last, when the equilibrium was achieved, the valve V-5 was opened, and after the pressure

stops to vary, the valve V-6 was opened 4 or 5 times, in order to purge the densimeter and ensure that

entire system was at the same conditions.

For 80 bar, the cell was fed with CO2 by the diaphragm pump (1) until 80 bar and the ISCO

pump was completely refilled with CO2 (260 mL). When the pressure of the cell was approximately 80

39

bar, the diaphragm pump was turned off and it was turned on the ISCO pump selecting the constant

pressure option of ISCO pump, and pressure was set to 80 bars. After this, the procedure was the

same as the previous procedure.

After 80 bar, to increase the pressure it was just needed to increase the set point of ISCO

pump in the constant pressure option. All the next steps were similar to other pressures.

For 40ºC the procedure was exactly the same, with the change of the set point of the heating

device (T=40ºC).

All the measurements were taken at least 3 times.

• Density measurements of Cocoa Butter

The experimental procedure to measure the cocoa butter density was simple. The first step

was to fill the ISCO pump with cocoa butter and to connect this pump to the densimeter. Then, it was

just needed to define the pressure and select the constant pressure option.

The procedure to take the measurements of cocoa butter was similar to the previous. After

defining the pressure and achieve the equilibrium, the system had to be purged (open the valve V-6) 4

or 5 times, in order to guarantee that all the system was in the same condition. Then, it was just

needed to wait that the pressure and the flow of ISCO pump stopped to vary.

All the measurements were taken at least three times.

3.1.3.1. Calibration of the Densimeter DMA HPM

The calibration of the densimeter DMA HPM was carried out with water and CO2. In this way,

the device was filled with CO2 or water in the calibration. In the calibration of the apparatus, programs

were created for each temperature and pressure, for example, program 40C50B, which corresponds

to 40ºC and 50 bar. Then, for each temperature and pressure the period of the oscillation was taken.

The values of the period were correlated to a value of density (CO2 or water, according to the fluid),

according to the data presented on NIST web site [21] (http://webbook.nist.gov/chemistry/). In Appendix

C the table of calibration for 40ºC and 50ºC is presented (Appendix C – Table C.1).

The NIST web site is a web site where it is possible to find fluid data. The density data of

water were from the reference (Wagner and all) [22] and the CO2 data were from the reference (Span

et al.) [23].

3.1.4. Results and Analysis

3.1.4.1. Density of cocoa butter

Figure 14 presents the density of pure cocoa butter, at different pressures, at 40ºC and 50ºC.

In Table C. 2 are presented the results obtained (Appendix C).

Comparing the density at the two temperatures, it is visible that the density decreases with the

temperature. In all range of pressures, the density at 40ºC is higher than 50ºC

40

Figure 14: The density of pure cocoa butter at different pressures at 40 and 50ºC

In figure 14, it is possible to observe that the cocoa butter density increases linearly with the

increase of pressure, as it was expected. This behaviour was expected because with the increase of

pressure the quantity of material in the same volume increases, therefore the density obviously

increases. The slope of the correlations obtained have a similar value (same order of magnitude),

which means that the increase of pressure makes the same effect for the two temperatures.

The regression lines for both temperatures have a good correlation coefficient (R2), which

shows the linear behaviour of the density with the pressure. However, the third point (100 bar) at both

temperatures shows a high error, compared with the other points. This fact can be due to an error in

the calibration at this pressure due to the proximity of the critical point of carbon dioxide.

3.1.4.2. Density of CO 2-saturated cocoa butter

The figure below presents the results of the density of CO2/saturated cocoa butter as function

of the pressure. In Table C.3 the results obtained are presented (Appendix C).

Figure 15: The density of CO2-saturated cocoa butter at different pressures at 40and 50ºC.

In figure 15, it is shown the behaviour of the mixture CO2-cocoa butter. As it is

possible to observe, at both temperatures the density seems to increase linearly with the pressure, as

it was seen for cocoa butter (figure 14). Beyond this, the behaviour of the density in function of

pressure at both temperatures follow the same behaviour, as it proved by the same slope of the linear

regression. Through this, it is possible to conclude that the increase of density just depends on the

increase of CO2 pressure. However, the density at 40ºC is higher in all range of pressures, like in the

case of pure cocoa butter.

Analyzing the influence of CO2 pressure in the density increase, it is evidenced that the

magnitude of the density increase is small, proved by the small slope of the linear regression (0,0001).

However, this behaviour was already been observed for triglycerides in previous studies with cocoa

butter [4] and corn oil [24]. Venter et al. [4] also observed this behaviour in soybean oil, coconut oil, palm

kernel oil, castor oil, linseed oil, olive oil and palm oil.

The correlation coefficients present high values, especially in the case of 50ºC (R2=0,9946).

The mixture at 40ºC shows some points which are a little far from the regression line (higher errors),

more specifically the points at 0, 125 and 250 bars. To add to this, it seems that the last four points are

41

very well aligning in one different direction of the first three points. In this way, in figure 16 is presented

the density correlated to a second degree polynomial.

Figure 16: Density of CO2-saturated cocoa butter at different pressures at 40.

Analyzing figure 16, it is possible to observe the fact evidenced above. It is clear that the

density measurements are better correlated with a second degree polynomial. This same behaviour

was also observed by Venter et al [4] . According to these authors [4], the density increases linearly for

pressures above 50 bar, but it is also observed that the density points at pressures between 50 and

100 bar seems to have a different alignment. This behaviour can be attributed to difference of the

properties of CO2 near to the critical point and at higher pressures (CO2 properties changes more

gradually).

3.1.4.3. Comparison of cocoa butter density and CO 2-saturated cocoa butter density

In order to analyse the influence of the CO2 in the density of the mixture CO2-saturated cocoa

butter it is present in figure 17 the density of the pure cocoa butter and the mixture at 50ºC.

Figure 17: The density of CO2-saturated cocoa butter and pure cocoa butter at 50ºC.

In figure 17 is possible to observe the influence of CO2 in the density of the studied mixture.

The density of the mixture is higher in all range of pressures; however the difference between the

densities isn’t constant. This difference increases with the increase of pressure. This fact is justified by

the increase of solubility of CO2 in cocoa butter with the increase of pressure. This fact was observed

during the experimental procedure, when the liquid volume of the heavy phase increased with the

increase of pressure. Through this, it is possible to conclude that the increase of CO2 quantity in the

mixture increases the density.

This behaviour was also observed in corn oil [24] and it was considered by the author as

surprising because for all the pressures and temperatures the density of CO2 is lower the density of

cocoa butter. However, the density doesn’t depend only of the density of the compounds present in a

mixture. Therefore, it is also related with the volume expansion and with the arrangement of the

molecules of CO2 in the middle of the triglycerides of cocoa butter.

42

As it was already referred, during the experimental procedure there was a volume expansion

of the heavy phase but this expansion wasn’t much high. Venter et al [4] was observed the same

behaviour, which means that probably exist high volumes between the triglycerides (high molecules

with long chains of carbons – cocoa butter review). Consequently, the CO2 are placed in this free

volume leading to a low volume expansion. Adding to this, the arrangement of the molecules in the

mixture also influences the volume expansion, as the density. In way of conclusion, in the same

volume there would be more molecules due to replacing free volumes by the molecules of CO2 that

obviously increase the density.

3.1.4.4. Comparison of experimental data with bibli ographic data

The next figure will present the experimental data obtained in this work and data from

bibliography [4] in order to compare the different results.

Figure 18: The density of CO2-saturated cocoa butter of the present work and from bibliography [4].

The density values of the bibliography reference [4] at 40ºC increase linearly with the pressure

for pressures above 50 bar. This behaviour was also verified for corn oil [24]. These behaviours are

probably due to the proximity of the critical region of carbon dioxide (TC=31.1ºC and PC=73.8 bar),

which is characterized by a high instability.

In the case of experimental data obtained in this work, this behaviour isn’t so evident.

However, as it has already been discussed above, it seems that the last four points aren’t well align

with the first three points (figure 18), which is possibly justified by the same reason, proximity of critical

point of carbon dioxide. At 50ºC this behaviour wasn’t verified, probably due to the fact of being farther

away of the critical point. Beyond this, in Venter and all [4] , it was verified the same behaviour for high

temperatures (80 and 100ºC) – linearly increase of density with the increase of pressure, in all range

of pressures.

Analyzing the experimental values and comparing with the data of Venter and all [4] , it is clear

that the density values of the bibliography reference are higher than the values obtained during the

experimental procedure. With the exception of the first point (at 0 bar), where the difference is

minimal, all the other points has an average difference of 1.7%, which is a small error. Despite the fact

of the supplier of cocoa butter for both experience had been the same (Gerkens Cacao), the

difference between the values could be justified by a possible difference in a composition of the cocoa

butter because the composition of cocoa butter depends on the origin of the cocoa beans. A possible

difference in the composition would influence the solubility of CO2 in cocoa butter, influencing by this

way the results. Other fact that should be mentioned is that it isn’t known the precision of the results of

bibliography data [4], or how the measurements were done. In the experimental work, all the

43

measurements were done after achieving the equilibrium. The criteria used to guarantee that

equilibrium had already been achieved were to keep sure that the pressure, temperature and density

values don’t vary. Beyond this, all the measurements were made at least three times, in order to

minimize possible errors.

In order to analyse in deep way the results obtained, they will be related with the solubility data

to verify its validity.

3.1.4.5. Compressibility of cocoa butter

The compressibility of cocoa butter was also studied. To determine the compressibility of solid

cocoa butter it was used mercury porosimeter. In this device, the mercury is introduced into the

sample of solid material and it start to fill the pores of the material with the increase of pressure

(compression of the material). While the pressure is increased the mercury enters in small pores, until

filling them all. Trough the volume of the porous it is possible to study the compressibility of a material

(defined by (1/V).dV/dP).

In next figure it is presented the accumulated volume (Vcu-mL/g) in function of pressure. The

accumulated volume is the volume filled by mercury due to the compression of the solid. The angle of

contact between mercury and the sample of cocoa butter was 130ºC.

Figure 19: Volume accumulated of cocoa butter solid in function of pressure.

Analyzing the figure 19 it is possible to identify two different zones, the first one before 10 bar

and the second one after. This change of behaviour is probably due the fusion of the material with the

increase of pressure. This fact was evidenced with the analysis and comparison of the sample before

and after the experience – figure 20.

(a) (b)

Figure 20: (a) – photo of cocoa butter before the compressibility experience; (b) – photo of cocoa butter after the compressibility experience.

Figure 20 shows that the sample of cocoa butter started to melt during the experience. The

figure (b) presents a dark colour due to the mercury that stayed in the porous of the material.

The compressibility of cocoa butter can be related with the mixture volume expansion. As it is

possible to observe, the compressibility of cocoa butter increases especially after 10 bar, when it

44

starts to melt. As it was already referred, cocoa butter is solid at room temperature and it is composed

by three triglycerides (large molecules with large chains). The solid compounds are characterized by

an organized structure with small volumes between their molecules (depending of the compound).

With melting, the organized structure is destroyed and the free volume increases, along with

compressibility (phenomenon observed in figure 19). This increase of compressibility depends on the

attraction forces, molecules size, etc. Usually the compounds with large molecules, such as cocoa

butter have, high compressibility due to the free space between the molecules.

3.1.5. Modelling of density and compressibility wit h Tait and Modified Tait Equation

3.1.5.1 Tait Equation

The Tait Equation, as mentioned in the bibliographic review, is known by representing very

well the compressibility data of solids at high pressures. For liquids, this equation fits the experimental

measurements over a more limited pressure range. However, the Tait Equation (under the form of

equation (30)) will be tested for the solid and liquid cocoa butter and for the mixture of CO2-saturated

cocoa butter.

The Tait equation under the form of equation (30) is a linear pressure equation, which

represents the inverse of compressibility in function of pressure. For that it is required the specific

volume, which is the inverse of the density. In this way, with density data is possible to determine the

compressibility.

• Cocoa Butter at 40 and 50ºC

Figure 21 shows the fit of the experimental results of liquid cocoa butter at 40 and 50ºC with

the Tait Equation. The calculated values are in appendix D (table D.1).

y = -14.163x + 19191

R2 = 0.1148

y = -13.864x + 21320

R2 = 0.1265

0

5000

10000

15000

20000

25000

30000

0 50 100 150 200 250 300

P (bar)

P.v

0/(v

0-v

)

Tait Equation at 40ºC Tait Equation at 50ºC

Figure 21: The fit of the experimental of cocoa butter results at 40 and 50ºC with Tait Equation

Analyzing the figure 21, it is clear that Tait equation doesn’t fit the results obtained, revealing a

small regression coefficient at both temperatures. The highest error was verified at 100 bar, as

expected (highest deviations in figure 14).

45

According to Dymond and Malhotra [13], the Tait Equation represents well the compressibility

at high pressures. Adding to this, the last three points are aligning in the same direction. In this way,

the next figure presents the modelling of the last three points.

y = 10.085x + 13480

R2 = 0.8868

y = -23.215x + 22808

R2 = 0.9492

100

5100

10100

15100

20100

25100

100 120 140 160 180 200 220 240 260

P (bar)

P.v

0/(v

0-v)

Tait Equation at 40ºC Tait Equation at 50ºC

Figure 22: The fit of the last three experimental points of cocoa butter results at 40 and 50ºC with Tait Equation

Figure 22 shows that, considering only the last three points, the Tait equation fits in a

reasonable way the experimental results. However, the trends verified at 40 and 50ºC are different.

According to the figure 22, the compressibility at 40ºC increases with the increase of pressure, and at

50ºC decrease with the increase of pressure. However, it was expected to verify that the

compressibility decreases with the increase of pressure at both temperatures, as it was verified at

50ºC.

The compressibility is a material capacity to compress. Therefore, with the increase of

pressure (compression), this capacity decreases because the free volumes between the molecules

decrease. In this way, it was expected to obtain in a figure 22 (where the inverse of compressibility is

represented) two correlations with positive slopes.

The highest deviations were obtained at 100 bar. These deviations were already observed in

figure 14, and they can be justified by an error in the calibration at this pressure due to the proximity of

the critical point of carbon dioxide.

In the next table it is presented the parameters of Tait Equation for cocoa butter at 40 and

50ºC, considering all the experimental points and the last three points.

Table 6: Parameters of Tait Equation for cocoa butter at 40 and 50ºC.

T (ºC) A ΠΠΠΠ

All experimental points 40 -0.072 -1537.796

50 -0.071 -1355.010

Last three Points 40 -0.0431 -982.468 50 0.099 1336.639

46

In the table 6 is shown the Tait equations parameters. Considering all the experimental points,

the parameters are similar for both temperatures. However, considering only the last three

experimental points, the parameters are different. This difference is explained by the different trend

evidenced in the graphic of figure 22. A deeper analyze isn’t possible because the Tait equation is an

empirical equation and its parameters don’t have physical meaning.

Through the obtained results, it is possible to conclude that the Tait Equation didn’t fit the

experimental results in all pressure range. However, it isn’t possible to conclude if Tait Equation just fit

well the experimental results at high pressures or if Tait equation didn’t fit the experimental in all

pressure range due the high deviations verified at 100 bar in figure 14 (possible error in calibration). In

this way, the Tait equation will be applied to CO2-saturated cocoa butter.

• CO2-saturated cocoa butter

In the next figure it is presented the inverse of compressibility as function of pressure of CO2-

saturated cocoa butter. The calculated values are in appendix D (table D.2).

y = 9.7467x + 3840.2

R2 = 0.9432

y = 5.1459x + 5767.2

R2 = 0.47

0

1000

2000

3000

4000

5000

6000

7000

8000

0 50 100 150 200 250 300

P (bar)

P.V

0/(

V0-V

)

Tait Equation 40ºC Tait Equation 50ºC Figure 23: The fit of the experimental of CO2-saturated cocoa butter results at 40 and 50ºC with Tait Equation

Through the figure 23 it is possible to observe that the Tait Equation fits well the experimental

data at 40ºC, with a high regression coefficient. However, analyzing in deep way, it is possible to verify

that the highest errors are at 80 and 100 bar at 50ºC and 80 bar at 40ºC. These high errors can be

justified by the same reason already referred (error in the calibration due to the proximity of CO2

critical point). At 40ºC, the density was measured at high pressure (120 bar), more distant from the

CO2 critical point. In this way, next figure shows the representation of the inverse of compressibility as

function of pressure of CO2-saturated cocoa butter for pressures above 100 bar.

47

y = 11.85x + 3403.9

R2 = 0.975

y = 6.9326x + 5312.4

R2 = 0.9515

0

1000

2000

3000

4000

5000

6000

7000

8000

0 50 100 150 200 250 300

P (bar)

P.V

0/(

V 0-V

)

Tait Equation 40ºC Tait Equation 50ºC Figure 24: The fit of the experimental of CO2-saturated cocoa butter results at 40 and 50ºC with Tait Equation for

pressures above 100 bar

For pressures above 100 bar, the Tait equation fits well the experimental results, with high

regression coefficients for both pressures. Through the last figures, it is possible to conclude that If

there wasn’t that deviation at 80 and 100 bar, the Tait equation would have fitted better all the

experimental results.

The figures 23 and 24 shows that the compressibility decreases with the increase of pressure

at both temperatures, as expected. However, according to figure 24, the decrease of compressibility is

sharper at lower temperatures (sharper slope).

The parameters of the Tait equation for the CO2-saturated cocoa butter are presented in the

table 7.

Table 7: Parameters of Tait Equation for CO2-cocoa butter at 40 and 50ºC. T (ºC) A ΠΠΠΠ

All experimental points 40 0.106 405.312 50 0.194 1120.698

Pressures Above 100 bar 40 0.084 287.249 50 0.144 766.293

In the table 7 is shown the Tait equations parameters for CO2-saturated cocoa butter. It is

possible to conclude that both parameters increase with the increase of pressure, especially Π

parameter. Once again, it isn’t possible to make a deeper analyze due to the empirical bases of the

Tait equation.

• Solid cocoa butter

At last, the Tait equation was applied for solid cocoa butter. Figure 23 presents the modelling

of cocoa butter.

48

0

500

1000

1500

2000

2500

3000

3500

0.01 0.1 1 10 100 1000 10000

P (bar)

P.V

0/(

V0-V

)

y = 0.4733x + 1220.1

R2 = 0.9931

1800

2000

2200

2400

2600

2800

3000

3200

3400

1600 2100 2600 3100 3600 4100 4600

P (bar)

P.V

0/(

V0-V

)

(a) (b) Figure 25: The fit of the experimental of solid cocoa butter results with Tait Equation: (a) – in all range pressure;

(b) – at high pressures. Figure 23 (a) presents the modelling of the compressibility of solid cocoa butter with the Tait

Equation in all pressure range. From the previous figure, it is possible to observe that the Tait

Equation doesn’t fit very well the experimental results in all pressure range. According to Dymond and

Malhotra [13], the Tait Equation (linear pressure equation – equation (30)) represents very well the

compressibility of solids at high pressures. Therefore, representing the compressibility of solid cocoa

butter at high pressures (pressures above 1700 bar – figure 23 (b)) it is obtained a good fitting of the

experimental results. However, the cocoa butter started to melt at 10 bar (figure 19 and 20), which

means that at high pressures there is not only solid but also liquid. Adding to this, the compressibility

behaviour changed exactly at 10 bar (figure 19 and figure 23 (a)).

The parameters of the Tait equation for the solid cocoa butter are presented in the table 7.

Table 8: Parameters of Tait Equation for solid cocoa butter A ΠΠΠΠ

0.000388 0.000820

As conclusion, the Tait equation fitted in an accurate way the compressibility of cocoa butter

and CO2-saturated cocoa butter for pressures above 100 bar. This fact was probably due to an error in

calibration at pressures near to the critical point of CO2, where its characteristics change in sharp way.

For the solid compressibility, the Tait equation didn’t fit well the experimental results because at 10 bar

the cocoa butter started to melt.

3.1.5.2. The Modified Tait Equation

The Modified Tait equation (equation (27)) is an equation used to modelling the density or

specific volume as a function of pressure for liquids or liquid mixtures at high pressures. Therefore,

this equation was used to modelling the density of cocoa butter and CO2-cocoa butter saturated at 40

and 50ºC, at high pressures.

The parameters B and C were determined trough the minimization of the absolute average

deviation equation (equation (34) – AAD) using the Solver function of Excel. The minimization of the

equation (34) was achieved by the variation of the parameters of modified Tait Equation.

49

∑

−

−−

−

=

measured

measuredcalculated

NAAD

ρρρ

ρρρ

ρρρ

0

00

1 (34)

Where, N is the number of points, ρ0 and ρ the density at atmospheric and high pressure,

calculated

−ρ

ρρ 0 is the calculation of the expression using the Modified Tait equation and measured

−ρ

ρρ 0 is

the calculation of the expression using the experimental data.

• Cocoa Butter

In next table it is shown the values of the parameters of the modified Tait equation of cocoa

butter at 40 and 50ºC and the respective mean errors.

Table 9: Parameters of the Modified Tait Equation for cocoa butter at 40 and 50ºC and the respective errors. T (ºC) C B AAD (%)

40 0.193 1384.867 4.0

50 0.193 1378.971 3.9

To obtain the parameters of the Modified Tait Equation it was needed to take out the second

point (100 bar) because it shows a high error (40%) at both temperatures and this same deviation

wasn’t observed in the mixture of CO2/saturated cocoa butter. This fact was already discussed above

(figure 14), ant it is possible due to an error in the calibration at this pressure due to the proximity of

critical point of carbon dioxide.

Analysing the values of the parameters, it is possible to see that C has the same value for

both temperatures and the value of B decreased with the increase of temperature, which shows that

the parameter B is more sensible to the temperature than C, as it was expected. The decrease of the

parameter B with the temperature was expected too.

The value of the parameter C is in the habitual range, but isn’t possible to conclude something

because these parameters don’t have physical meaning.

The following figures show the representation of

−ρ

ρρ 0 measured and calculated in function

of ( )pB +log at 40 and 50ºC.

50

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

3.15 3.16 3.17 3.18 3.19 3.2 3.21 3.22

log (B+P)

( ρρ ρρ- ρρ ρρ

o)/ ρρ ρρ

ModifiedTait Equation Experimenta datal

Figure 26: Representation of

calculated

−ρ

ρρ 0and

calculated

−ρ

ρρ 0of cocoa butter in function of ( )pB +log at 40ºC.

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

3.15 3.16 3.17 3.18 3.19 3.2 3.21 3.22

log (B+p)


0)/ ρρ ρρ

Modified Tait Equation Experimental data


calculated

−ρ

ρρ 0and

calculated

−ρ

ρρ 0 of cocoa butter in function of ( )pB +log at 50ºC.

As it is possible to see in the figures 26 and 27, the modified Tait equation fitted in a

reasonable way the experimental results. The second point (100 bar) presents the highest error.

• CO2-saturated cocoa butter

The modified Tait equation was also applied to the mixture CO2-saturated cocoa butter. In the

next table it is presented the values of the parameter and their errors.

51

Table 10 : Parameters of the Modified Tait Equation for CO2-saturated cocoa butter at 40 and 50ºC and the respective errors.

T (ºC) C B AAD (%)

40 0.09 142.317 2.7

50 0.45 1236.179 4.6

According to table 10, both parameters varied with temperature. However, the variations of the

parameters don’t just depend on the temperature variation, but also on the variation of the CO2

solubility in cocoa butter with temperature. The errors are very small for both temperatures, but are

lower at 40ºC. The values of the parameter C at 40ºC and 50ºC are out of the normal range (0,15-0,3)

but, as it was previously referred, it isn’t possible to conclude anything about these values because the

Tait equations and modified Tait Equation are empirical, with no physical meaning.

In the next figures are presented the representation of

−ρ

ρρ 0 measured and calculated as

function of ( )pB +log at 40 and 50ºC.

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

2.25 2.3 2.35 2.4 2.45 2.5 2.55 2.6 2.65

log (B+p)


0)/

ρρ ρρ



calculated

−ρ

ρρ 0and

calculated

−ρ

ρρ 0of CO2-cocoa butter in function of ( )pB +log at 40ºC

52

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

3.1 3.11 3.12 3.13 3.14 3.15 3.16 3.17 3.18

log (B+p)


0)/ ρρ ρρ



calculated

−ρ

ρρ 0and

calculated

−ρ

ρρ 0of CO2-cocoa butter in function of ( )pB +log at 50ºC

Through the figures 28 and 29, it is possible to conclude that the modified Tait equation is a

good tool to model the density in function of pressure (low errors).

Table 11 present a resume of the AAD (Equation (34)) of Modified Tait Equation and Tait

Equation, in order to compare both Equations.

Table 11: The AAD of Modified Tait Equation and AAD of Tait Equation

Compound/Mixture T (ºC) AAD of Modified Tait Equation (%)

AAD of Tait Equation (%)

AAD of Tait Equation (%) – pressures above 100 bar

Cocoa Butter 40 11.4 9.4 1.1

50 11.4 11.8 0.9

CO2-Saturated Cocoa Butter

40 2.7 2.7 1.5

50 4.6 4.5 0.9

Analyzing the table 11, it is evident that the AAD of Modified and Tait Equation (considering all

the experimental points) are very similar. These results are in agreement with Hayward [13], which

considered that the Modified Tait Equation didn’t have advantageous over the Tait Equation, referring

that it didn’t fit the experimental data so well.

The AAD obtained for cocoa butter are higher than the values obtained for CO2-saturated

cocoa butter. These high AAD values are due to the density measurements at 100 bar, where high

errors were obtained. The AAD of the Modified Tait equation without the experimental point at 100 bar

decrease to 4%.

The Tait equation presents small AAD values for pressures above 100 bar, especially for

cocoa butter. However, through this it isn’t possible to conclude if the Tait equation fits well the

experimental data for pressures above 100 bar or if the Tait equation didn’t fit well the experimental

results in all range of pressure due to a possible error in calibration.

53

The Tait Equation and the Modified Tait Equation revealed to be good tools for the modelling

of density and compressibility of CO2-saturated cocoa butter in all range of pressure. For cocoa butter,

high errors were obtained at 100 bar. However, both equations are empirical and don’t have physical

meaning, which complicate the evaluation of the parameters and of the equations

3.2. Solubility measurements

According to Fornari et al [27], the experimental techniques that are used to study multiphase

equilibria are divided in two classes: synthetic and analytical methods. In the present work, the phase

equilibria of cocoa butter/SC-CO2 were studied through a synthetic method.

• Synthetic Methods

The synthetic methods consist in the determination of all the conditions of pressure,

temperature and composition where a change of phase occurs without the need of extract samples

from the system.

The experimental procedure in this type of methods consists in putting known quantities of the

compounds in a cell with variable volume, and by varying the volume adjusts the pressure and

temperature of the system a homogenous phase (phase transition) is achieved. In this way, with visual

detection of bubble, dew and fusion points or transition for supercritical state it is possible to get

equilibria lines (P, T and x) [28].

Synthetic methods are more suitable to work at high pressures and near the critical point than

analytical methods. In these latter, complicated procedures are required to extract samples from the

cell, which can cause high perturbations in the system [27].

The visible cell with variable volume is the main feature of this method. This cell was

described for the first time by Di Andreth et al. [29] . These kinds of cells have one piston that is

actuated by a manual compressor and by a liquid that allows increasing or decreasing the volume of

the cell. In this way, it is possible to change the pressure of the cell without changing the temperature

and composition. In figure 30, the first cell with variable volume of Di Andreth et al [29] is shown.

Figure 30: Scheme of cell with variable volume of Di Andreth et al [29]

3.2.1. Material The cocoa butter used was acquired from Gerkens Cacao (Wormer, The Netherlands).

54

The Liquid CO2 was acquired from Air Liquid SA, Portugal with the purity 99,995%.

3.2.2. Experimental Setup In the figure 31 it is presented the scheme of the diagram of the experimental device. This

apparatus is a development of the apparatus used by Costa et all [30] .

Legend:

1 – CO2 Cylinder 7 – Manual compressor 2 – Pressure controller V – Valve 3 – Compressor T – Temperature sensor and indicator 4 – Storage Cylinder P – Pressure Sensor and indicator 5 – Vacuum Pump TC – Temperature controller 6 – Visible Cell with variable volume (VCVV)

Figure 31: Scheme of the phase equilibria device

The apparatus can be divided in three zones: Carbon dioxide (CO2) admission zone;

Calibrated volume zone and Phase Equilibria zone.

• Carbon dioxide admission zone

In this zone the CO2 from the storage cylinder (1) is compressed (where the vapour pressure

is below 6 MPa at room temperature) to the admission pressure, which is controlled by pressure

controller (2 – Tescom; 26-1722-24) of backpressure type. The pressure is measured by two Pressure

sensors (P1, P2 – VDO) and compressed by a compressor Newport/46-13421-2 (3).

55

• Calibrated volume zone

The calibrated volume zone is made of a storage cylinder (4 - Hoke; DOT 75cc), a pressure

sensor (P3 - Setra/206, Setra/300D) and temperature sensor (T1 - Omega/tipoT, Omega/DP462) and

it is under a thermostatic water bath (controlled by TC1 – HAAKE;E3), to ensure that the calibrated

volume zone is at experience temperature.

The volume of this zone was calibrated with distillate water, in order to know the exactly

volume of this zone (calibration). Knowing the volume it is possible to calculate through equation (35)

the CO2 quantities that are introduced in the cell (6 –VCVV).

WPTPTM

VV

Vm

fi

.,, −

=

(35)

Where, V is the calibrated volume, m the CO2 mass, MW the molar weight of carbon dioxide and VT,P

are the molar volume at initial pressure (Pi) and at final pressure (Pf). The molar volumes are obtained

through a correlation with tabulated values (IUPAC [31]).

This zone is also connected to a vacuum pump (5 – Edwards) trough V4, in order to guarantee

that there is no air in VCVV.

• Phase Equilibria zone

The phase equilibria zone is constituted by a visible cell with variable volume (VCVV – 6),

which is in an isothermic air box of acrylic. The air box temperature is controlled by a PID controller

(TC2 - BTC-9200, Tc Omega type T), one heating lamp of 200 W and two fans.

The VCVV (figure 32) is formed by a steel cylinder, one window of borosilicate glass, one

piston with o-rings of silicone and two thread closures isolated by Teflon. With this cell is possible to

study multiphase equilibria at pressures below 150 bar.

23.018.0

1/16'

1/8'

1/4'

22.0 25.0 146.0 14.0

215.0

60.0 32.024.020.0

15.0

40.0 30.0

8.0

3.5

4.5

2.8

14.0

4.5

3.0

3.520.00

6.014.0

3.03.0

Figure 32: Visible cell with variable volume (VCVV)

The pressure inside of the cell is regulated by a manual compressor (7 - HiP/86-6-5) that

compresses/uncompress the water and pushes/pulls the piston, which is placed inside of the cylinder.

56

In this way, the system pressure is controlled and through the window is possible to observe the

phase variations.

Under the VCVV is one magnetic stirrer in order to mix the contents of cell.

The solubility of CO2 in cocoa butter is determined through the achievement of the bubble

point. At pressures above the bubble point all the CO2 is dissolved in cocoa butter (one single liquid

phase) and at bubble point the first vapour of CO2 appears. Knowing the quantities of both compounds

it is possible to determine de solubility of CO2 in cocoa butter at each temperature and pressure.

3.2.3. Experimental Procedures

The experimental work was carried out at 40 and 50 ºC.

The first step was to wash the cell (6 – VCVV) and its accessories with ethanol, hot water and

compressed air. Special attention was required for the cleaning of the cell components.

The next step was to weigh the quantity of cocoa butter for the experience. As it was referred,

in the synthetic methods all the compound quantities added to the system have to be well known. In

this way, this measurement was taken with an analytical balance (METTLER H315) in order to know

the exact weight of cocoa butter. After weighing, the cocoa butter was placed in an oven at 50 ºC to

melt the cocoa butter.

The mounting of VCVV (6) started with the piston. The silicon o-rings were put in the piston

and tied through the screws, in order to guarantee that the o-rings were in contact with the wall of the

cell and to isolate the binary system of compressed water. After the piston, the back closure and

Teflon ring were placed and well tied, to avoid leaks.

Once melted, the cocoa butter was placed inside of the cell. The next step was to put the

Teflon o-rings, borosilicate glass window and the front closure in the same order of the figure 30. This

procedure was done very carefully, tightening enough all the screws to avoid possible leaks that can

affect the results. The last step of the mounting was to connect the cell to the other equipments (T2,

P4; compress system; CO2 feeding).

The next step was to switch on: the thermostatic bath to ensure that CO2 is at experience

temperature (40 or 50 ºC); the air box PID controller, fans and lamp to control the temperature of the

same (and VCVV); and the magnetic mixer.

After that, the storage cylinder (4) was filled with compressed CO2 when temperature was

stable. To fill the storage cylinder the compressor 3 was first switched on and then the valves V1 and

V2 were opened. When the storage cylinder was already filled with enough CO2, the valves V1 and V2

were closed and the compressor switched off.

Before the addition of CO2 to the cell, it was needed to make vacuum in the cell to guarantee

that the system didn’t have air. In this way, the valve V4 was opened and the vacuum pump was

switched on (5) during 5-10 minutes. During this procedure, it was possible to observe the formation of

bubbles in the liquid cocoa butter, which means that the air was being removed.

For the addition of CO2 the valves V3 and V6v were opened. The CO2 quantities were

determined by equation (35). After that, the pressure was regulated with the manual compressor (7).

The pressure was increased until the achievement of one single phase (homogenous system). The

mixture in the cell was mixed until achieving equilibrium, when pressure and temperature were

57

constant. After achieving the equilibrium, the pressure was decreased until the appearance of the first

vapour quantity (first bubble).

3.2.3.1. Calibration

The calibration of the storage cylinder volume (4 – calibrated volume zone) was carry out

using an analytical balance (with an error of 1mg) in order to minimize the error.

The storage cylinder was weighed empty and full of distillate water at 298K. This procedure

was repeated three times. Between each measurement, the cylinder was dried in an oven at 340 K, to

guarantee that the cylinder didn’t have water.

The same procedure was used for the calibration of tubes and other accessories of the

calibrated volume zone.

• Sensors Calibration

The sensors were calibrated by the determination of the critical point of CO2 (PC=73.8 bar;

TC=31.1ºC).

3.2.4. Results and Discussion

3.2.4.1. Solubility of SC-CO 2 in cocoa butter

Figure 33 shows the solubility of CO2 (weight percentage) in cocoa butter at different

pressures at 40ºC and 50 ºC. The results obtained are presented in table E.1(Appendix E).

20.00

60.00

100.00

140.00

180.00

0.00 5.00 10.00 15.00 20.00 25.00 30.00 35.00 40.00

w CO2 (%)

P (b

ar)

SC-CO2-CCB at 40 ºC SC-CO2-CCB at 50 ºC

Figure 33: The solubility of CO2 in cocoa butter at different pressures at 40 and 50ºC

Figure 33 shows the variation of the solubility of CO2 in cocoa butter with the pressure at 40

and 50ºC. As it is possible to observe, the solubility increases with the increase of pressure at both

temperatures and decreases with the increase of temperature. The increase of solubility is sharper at

40ºC than at 50ºC.

58

The increase of solubility with pressure is explained by the Henry’s law states. According to

this law, the solubility of a gas in a liquid is directly proportional to the pressure of that gas above the

surface of the solution. By increasing pressure, the CO2 molecules are induced to solubilise in cocoa

butter in order to relief the pressure that has been applied.

In another way, the increasing of temperature takes to a decrease of solubility. The increase of

temperature augments the kinetic energy, which increases the motion of the molecules that break

intermolecular bonds and escape from the solution.

The variation of the solubility with the pressure is in the agreement which what was already

referred. The increase of solubility justifies why the increase of density of the mixture CO2-cocoa butter

is sharper than for cocoa butter (figure 17). The increase of solubility was also verified through the low

volume expansion of the heavy phase during the density measurements.

3.2.4.2. Comparison of experimental data with bibli ography data

Figure 34 presents the experimental data obtained in this work at 40ºC and data from Venter

et al. [4] , Kokot et al. [3] and Calvignac et al. (data not published) [37] in order to compare the

different results.

Figure 34: The solubility of CO2 in cocoa butter at 40ºC from different references (Venter et al. [4] , Kokot et al. [3]

and Calvignac et al. [37]).

Analyzing the figure 34 it is possible to observe that the data points at low pressure are very

similar, with small differences. However, for high pressures the solubility differences increase, showing

a different trend.

The solubility data of Venter et al. [4] reaches a maximum of 36 wt % at 200 bar and remains

constant at higher pressures. According to Venter et al. [4] , the same behaviour was already observed

in seed oil, rapeseed oil and palm oil. In the other cases, the solubility doesn’t reach any maximum,

showing a growing trend. In Kokot et al. [3] the solubility at high pressures continues with a sharp

growing.

As it is possible to observe in figure 32, the solubility of CO2 in cocoa butter in Calvignac et al.

[37] is lower than the other data. The solubility shows a growing trend and it seems that is near to

reach a maximum. In this experimental work, the solubility shows a trend very similar to Kokot et al.

[3] .

A possible reason to explain the differences of the experimental data obtained of the

bibliography data is the cocoa butter used. Despite the fact that the supplier was the same of Venter et

al. [4] (Gerkens Cacao), the cocoa butter could be from different batches, which can influence de

composition. Besides that, the cocoa butter composition depends also of its origin and the age of the

plant. Another possible reason is the experimental method used. In the previous works, all the

59

solubility measurements were carried out by analytical methods, with an autoclave. In the present

work, the solubility was measured through a synthetic method, which can justify the differences

between the data obtained.

3.2.5. Modelling of solubility with Peng-Robinson E quation of State

The next step of this work is to model the solubility of CO2 in cocoa butter with the Peng-

Robinson Equations of State, which is recommended in Valderrama et al. [14] for mixtures with a

supercritical component. For that, the critical properties of cocoa butter (such as PC, TC, TB and ω) are

required.

The cocoa butter is a vegetable fat composed by three main triglycerides: POS, POP and

SOS, which account respectively for 34-45, 21-29.5 and 12.2-21.5% of the total triglycerides. For the

modelling it was assumed that the cocoa butter properties are the same of POS because it is the

triglyceride in majority. Besides, the three triglycerides are very similar, they are monounsaturated in

sn-2 carbon (oleic acid – which means that they have similar dispositions) and the carboxylic acids of

carbons sn-1 and 3 have similar properties (palmitic and stearic acid) [21].

There is a lack of information about triglycerides and the POS critical properties are unknown.

In this way, the properties of POS have to be estimated.

3.2.5.1. Properties Estimation

The POS properties were estimated through a program (Aspen Plus 2006.5) using four

methods, which are: Constatinou-Gani, Joback, Ambrose and Fedors methods. The table 12 shows

the results obtained.

Table 12: Estimated TB, TC and PC for POS and estimated TB, TC and PC for Tripalmitin, Triolein and Tristearin [18],

[32] Triglyceride Methods T B (K) TC (K) PC (bar)

POS

Constatinou-Gani 817.768 937.526 3.379

Joback 1636.180 3737.962 4

2.424 1868.246 5

Ambrose 1811.830 2

4.776 905.559 3

Fedors 762.400

Tripalmitin 889.140 [18] 5.093 [18] 864.210 [32] 947.100 3.968 [32]

Triolein 947.070 [18] 4.682 [18] 879.920 [32] 954.100 3.602 [32]

Tristearin 900.950 [18] 4.583 [18]

Comparing the values of the table 12 with the predicted values in Weber et al. [18] and Lim et

al. [32] of Tripalmitin, Triolein and Tristearin it is evident that the TB and TC values estimated by the

Constatinou-Gani method are more similar than those predicted by the other methods. Besides that,

according to Araújo et al. [17] the properties of fatty acids estimated with the Constatinou-Gani method 4 Using TB estimated by Joback method 5 Using TB estimated by Constatinou-Gani method

60

were more accurate than the values predicted by methods that used experimental boiling temperature.

Therefore, the chosen method to predict TB and TC was the Constatinou-Gani method.

The PC value predicted by the Constatinou-Gani method is also similar to those predicted in

Lim et al. [32] for Tripalmitin and Triolein. This same method was chosen to predict the PC of fatty acids

and fatty acid esters in Araújo et al. [17].

These choices had been made according to predicted and experimental values obtained for

other triglycerides. It is possible that the predicted value of other methods was better that the chosen

values.

For the estimation of the acentric factor (ω) the program Aspen Plus 2006.5 was used. The ω

estimation was done through the definition of ω (equation 26) and Lee Kesler vapour pressure

correlations. The estimation of acentric factor was done after the estimation of the other critical

properties due to its dependency on those critical values. The next table presents the estimated

values.

Table 13: Acentric Factor estimated through the definition and Lee Kesler vapour pressure relations for POS, and

Acentric Factor estimated for Tripalmitin, Triolein and Tristearin [18], [32] Triglyceride Method ωωωω

POS Lee Kesler 1.863678 Definition 1.220611

Tripalmitin 1.650000 1.819471

Triolein 1.800400 1.686230

Tristearin 1.737092

Analyzing table 11, it is possible to conclude that both values are higher values than expected.

According to the definition, the acentric factor measure the complexity of a molecule with respect to

both geometry and polarity. Therefore, large molecules (such as triglycerides) have usually high ω

values. Comparing both values with the bibliography values [18], [32] , the ω value obtained by the Lee

Kesler vapour pressure relations is more similar than the value obtained by the definition.

3.2.5.2. Interaction Parameters

In this work, the van der Waals (vdW) and Panagiotopoulos-Reid (P&R) were the mixing rules

used. The Peng-Robinson equation of state with P&R mixing rules is recommended by Valderrama et

al. [14] . This recommendation was based on literature information and on the author experience. In

this way, the P&R and vdW mixing rules were tested in order to compare the results obtained by both

mixing rules and to chose which mixture rules should be used.

The vdW mixing rules have two binary parameters and it is also known as a quadratic mixing

rule. The P&R mixing rule is nonquadratic with three binary parameters and is characterized by the

introduction of a second interaction parameter, by making the kij parameter concentration-dependent,

transforming by this way in a nonquadratic mixing rule – Table 3.

The binary parameters (interaction parameters) of the mixing rules are determined by the

minimization of the differences between predicted and experimental values. The values that minimize

61

these differences correspond to the optimum interaction parameters. These calculations (binary

parameters determination) were done with the program PE2000 (Phase Equilibria 2000), which was

developed in the Professor Brunner’s research group at the Tecnhical University of Hamburg-Harburg [33]. This program has been presented in many conferences and publications and often leads to a

better convergence than Aspen.

The Phase Equilibria 2000 is a program that offers more than 40 different equations of state

with up to 7 mixing rules, to correlate and predict phase equilibria. This program can be used for pure

components and for mixtures (binary and ternary systems).

The binary parameters were obtained through the minimization of the equation (36) – AAD2

absolute average deviation.

( )∑=

−=n

i

calcii xx

nAAD

1

2exp2 1 (36)

Where, xi is the mole fraction of the component I, exp and calc are the experimental and the calculated

values, respectively. Beyond this, it was only considered the liquid phase data for the minimization of

the equation (36) because the vapour phase wasn’t studied during the experimental work. Next table

presents the interactions parameters and the respective error for mixing rules, vdW and P&R, at 40

and 50ºC.

Table 14: Interactions parameters of vdW and P&R mixing rules, obtained by the minimization of equation (36) T (K) Mixing rule Interaction parameters AAD 2 (%)

313.15 vdW kij = 0.02841 lij = 0.04977 2.0 P&R kij = 0.00167 kji = 0.05639 lij = 0.04621 4.7

323.15 vdW kij = 0.05328 lij -0.00536 4.6 P&R kij = 0.02066 kji = 0.05489 lij = -0.00462 4.1

Analyzing and comparing the results obtained with the two mixing rules, it is evident that the

error obtained with vdW with two parameters at 313.15 K is lower than with P&R with three

parameters. At 323.15 K, the P&R mixing rule error is lower than the vdW error, but the values are

very similar. According to these results, despite P&R mixing rule having three parameters fitted tothe

experimental measurements it didn’t produce better results than the vdW mixing rules. In this way, the

vdW mixing rule was chosen for the modelling of the solubility of CO2 in cocoa butter with the Peng-

Robinson Equation of State due its simplicity and lower correlation deviations (AAD2).

Usually, it is considered that the interactions parameters are temperature independent.

However, as it is possible to observe in table 14, that the interaction parameters of both mixing rules

vary with temperature, which makes impossible to use the same interaction parameters at various

temperatures. Therefore, it is better to get a relation of interaction parameters with temperature.

In this experimental work, solubility measurements were done just at 40 and 50ºC. This

temperatures range is very small and it limits the possible application to other temperatures out of the

range. In Kokot et al. [3] , the solubility was measured at various temperatures (40, 60 and 80 ºC), with

results similar to those of this work at 40ºC. The high number of measurements and large temperature

62

range allow a better global view of the interactions parameters variation with temperature than the

solubility measurements of this work. Therefore, in order to get a parameters prediction in a large

temperature range, the interactions parameters obtained with data of Kokot et al. [3] were fitted in a

second degree polynomial (Equation (37) and figure 33). Besides that, through this it will be possible

to test and compare the solubility measurements at 50ºC of this work with the modelling using the

interaction parameters of table 12and predicted by the equation (37).

2210/ TaTaalk ijij ++= (37)

Next table presents the interaction parameters obtained using the vdW mixing rules and

solubility data of Kokot et al. [3] and the solubility data of this work at 313.15K. The parameters were

obtained by the same way of the parameters that are presented in the table 14.

Table 15: Interactions parameters od vdW and P&R mixing rules, obtained by the minimization of equation (36) with the solubility data from Kokot et all [3]

T (K) Work Interaction parameters AAD 2 (%)

313.15 Kokot et al. [3] kij = 0.02399 lij = 0.04402 12.1

This Work kij = 0.02841 lij = 0.04977 2.0

333.15 Kokot et al. [3] kij = 0.01247 lij = 0.01002 4.0

353.15 Kokot et al. [3] kij = 0.03652 lji = 0.05581 3.9

Comparing the interaction parameters obtained with solubility data of this work and Kokot et

al. [3] it is clear that the values are very similar, thus reinforcing what was concluded from figure 32.

Analyzing the values of the deviations obtained, it can be concluded that the Peng-Robinson equation

of state with vdW mixing rules gives a good correlation at all the temperatures of table 15. The highest

deviation was obtained at 313.15 K with experimental data of Kokot et al. [3] , probably due to the

large number of solubility measurements, which leads to a high deviation in the correlation. Besides, in

this particular case, other possible reasons has to be considered, namely the fact that the physical

properties of cocoa butter were taken as identical to of the main triglyceride (POS).

Figure 35 shows the variation of the interaction parameters (kij and lij) with the temperature,

using data from [3] . It is also possible to observe the fitting of the results with a second-order

polynomial (equation (37)).

63

y = 1E-04T2 - 0.0662T + 10.981

R2 = 1

y = 4E-05T2 - 0.0293T + 4.8439

R2 = 1

0

0.01

0.02

0.03

0.04

0.05

0.06

310 315 320 325 330 335 340 345 350 355 360

T (K)

Inte

ract

ion

para

met

er

kij lij

Figure 35: Representation of the interaction parameters kij and lij in function of the temperature and correlation in a second-order polynomial.

It is possible to observe in figure 35 that the kij and lij vary in similar way with the temperature,

but lij parameter varies in a sharper way. Through this correlation, it is possible to predict the values of

kij and lij in this temperature range. However, prediction is a last resource tool because it doesn’t

guarantee good results in all cases.

According to the similarly of the results obtained in this work and in Kokot et al. [3] at 315.15 K

and assuming that the correlation describes in a correct way the variation of the interactions

parameters with temperature, the kij and lij parameters were determined using the correlations

obtained (figure 33) at 325.15 K. The results obtained are presented in table 16.

Table 16: Interaction parameters obtained by the minimization of the equation (36) and by the correlation at 323.15K.

T (K) k ij l ij

323.15 Correlation 0.01378 0.01705 This work 0.05328 -0.00536

Relative Deviation (%) 74.1 418.0

As it is possible to observe in table 16, the interactions parameters obtained with the

correlation are very different from those obtained with the minimization of equation (36) (AAD2). These

large deviations can be due two different reasons:

• The correlation isn’t a good tool to describe the variation of the binary parameters with the

temperature;

• Incorrect experimental procedure at 323.15 K.

It was expected that the interaction parameters at 323.15 K of this work were similar to those

predicted by the correlation. However, there is the possibility of an incorrect experimental procedure, it

would be required to wait more time to achieve the total equilibrium. The solubility of a gas in a liquid

64

phase decreases with temperature (Henry’s law states), which leads to a large time to achieve

equilibrium. In this way, it would be admissible to repeat the experimental procedure at 325.15 K to

have a correct conclusion.

3.2.5.3. Solubility Modelling

The next step is to model the solubility of CO2 in cocoa butter with the Peng-Robinson

Equation of State, using the vdW mixing rules and the interaction parameters obtained by the

correlation (equation (37)) and by minimization of equation (36) (AAD2) – table 17.

Table 17: Interaction parameters obtained by the minimization of the equation (36) and by the correlation at 323.15 and 313.15 K

T (K) k ij l ij

313.15 Correlation 0.02399 0.04402 This work 0.02841 0.04977

323.15 Correlation 0.01378 0.01705 This work 0.05328 -0.00536

In the next figure is presented the modelling of the pressure as a function of pressure.

(a) (b)

Figure 36: Heavy phase composition in a weight- diagram of the system CO2/cocoa butter at 313.15 K (a) and 323.15 K (b): ● experimental points; ── calculated line with interactions parameters of this work; ── calculated

line with interactions parameters obtained from the correlation.

Analyzing the figure 34 – a, it is possible to observe that at 313.15K the modelling of solubility

with the interactions parameters obtained from correlation are similar, as it was already expected.

According to figure 36 – a, the solubility achieves a maximum at 110 bar. However, it seems that the

experimental points have a different trend. The large differences between the two models are verified

between 90 to 150 bar, in the maximum zone of solubility.

The models obtained at 323.15 K are very different (figure 34 – b). According to the relative

deviations of the interactions parameters (table 16), this behaviour was already expected. In both

models the solubility reaches a maximum, showing a different trend from the experimental

65

measurements. For a deeper analysis, it is presented the figure 35, where it is possible to observe the

solubility points at both temperatures, using the interaction parameters obtained by the minimization of

the equation (36) (AAD2).

Figure 37: Heavy phase composition in a weight- diagram of the system CO2/cocoa butter at 313.15 K and

323.15 K : ● experimental points at 313.15K; ● experimental points at 323.15K; ── calculated line with interactions parameters of this work at 313.15 K; ── calculated line with interactions parameters of this work at

323.15 K.

As it is possible to observe in figure 35, the models show a different trend from the

experimental points at both temperatures. According to the models, the solubility reaches a maximum

at 105 and 128 bar, at 313.15 and 323.15 K, respectively. After this maximum, the solubility starts to

decrease with the increase of pressure. At the same pressures, the Peng-Robinson equation of state

predicts that the vapour – liquid equilibria changes to liquid – liquid equilibria. However, during the

experimental procedure this phenomenon wasn’t observed, which justify the different trends of the

results and models.

Despite this phenomenon wasn’t observed during the experimental procedure, it was

observed and predicted by Weber et al. [18] , in the studies of vapour – liquid equilibria of tristearin,

tripalmitin and triolein in CO2. Besides, in this work it was assumed that cocoa butter properties were

the same as of POS (since this triglyceride is the major component and the other triglycerides are very

similar), which can’t correspond to reality. On other hand, Peng-Robinson Equation of State may not

be the best equation of state to describe this system. Adding to this, all the POS properties were

predicted by estimation methods, which mean that there is an inherent error.

The solubility data of Venter et al. [4] will be modelled with the Peng-Robinson Equation of

State in order to analyze if it would get better modelling with different results. The next table shows the

interactions parameters and its respective deviations (AAD2).

66

Table 18: Interaction Parameters and respective AAD using solubility data from Venter et all [4] T(K) k ij l ij AAD2 (%)

313,15 0,00770 -0,00028 6,7

The figure 36 shows the modelling of the solubility data [4] with the Peng-Robinson Equation of State.

Figure 38: Heavy phase composition in a weight- diagram of the system CO2/cocoa butter at 313.15 with solubility data from Venter et al [4] : ● experimental points at 313.15K ── calculated line with interactions

parameters of this work at 313.15 K.

Through the analysis of the figure 36 and table 18, it is clear that the modelling of the solubility

data from Venter et al. [4] presents higher deviations than the modelling of the results from this work.

Therefore, there is the possibility that it can achieve better models with other equations of state.

3.2.5.4. Density prediction with Peng-Robinson Equa tion of State

The final step of this work is to predict the density with the Peng-Robinson Equation of State

using the solubility data. In this way, the figure 37 shows the density experimental measurements and

the predicted density at 40 and 50ºC. In table F.1 the predicted densities are presented (Appendix F).

67

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

0 50 100 150 200 250 300

P (bar)

Den

sity

(g/

cm3)

Pred. Density at 40ºC Pred. Density at 50ºC Density at 40ºC Density at 50ºC

Figure 39: Density measurements at 40 and 50ºC and Density predicted by the Peng-Robinson Equation of State at 40 and 50ºC

As it is possible to observe in figure 37 the predicted densities at both temperatures have high

deviations from the densities measured. However, according to the modelling of the Peng-Robinson

Equation of State, it was already expected high errors.

The density predicted by the Peng-Robinson Equation of State increase with the pressure,

and decrease with the temperatures, as it was already expected. The increase of the predicted density

with the increasing pressure is sharper than the increase of density measurements.

As conclusion, triglycerides are large and complicated molecules with lack of information. Due

to this it is required to use estimation methods to determine the triglycerides properties, which leads to

some errors. On the other hand, to get a good modelling it is very important to know the molecules

properties, not only for the modelling itself, as to choose the better equations of state. The high

deviations verified can be due different reasons, such as:

• The Peng-Robinson Equation of State wasn’t the better equation of state to describe the

binary cocoa butter-CO2;

• The estimated methods weren’t the better ones to predict POS properties;

• Cocoa butter properties are different of POS properties.

• vdW mixing rules weren’t the better mixing rules to this binary system.

These justifications are just some possible answers to the deviations verified. Therefore, it would be

required a deep investigation to get the right answers.

68

4. Conclusions and Perspectives The main objectives of this work were to study and to characterize the binary cocoa butter/SC-

CO2, measuring the density and solubility of the heavy phase, as well as the density and

compressibility of the cocoa butter itself. At last, the results were modelled, the density and

compressibility by the Tait and Modified Tait Equations and the solubility by the Peng-Robinson

Equation of State.

According to the obtained results, the densities of the binary and cocoa butter increases with

the increase of pressure and decreases with the increase of temperature. However, the increase of

density in the binary is sharper than for cocoa butter, due to the increase of CO2 solubility. The cocoa

butter compressibility increases especially after starting to melt.

The Tait and Modified Tait Equation revealed to be a good tool to modelling the density and

compressibility in function of pressure. The better correlations (lower deviations) were obtained with

the Tait Equation. However, these equations are empirical, without physical mean, making difficult to

evaluate its parameters.

The CO2 solubility in cocoa butter increases with the increase of pressure and with the

decrease of temperature, as expected. These phenomena were also observed in the density

measurements, through the volume expansion of the heavy phase. The solubility behaviour was in the

agreement with the Henry’s law states.

The last step was to model the solubility in function of pressure with the Peng-Robinson EoS

using van der Waals (vdW) and Panagiotopoulos & Reid (P&R) mixing rules. Lower deviations were

obtained with the vdW mixing rules, which have only two interaction parameters. The Peng-Robinson

Equation modelling didn’t describe the obtained results, showing a different trend and predicting a

phase change that wasn’t observed. The density predicted using the correlated parameters for the

Peng-Robinson EoS had also high deviations from the density measurements.

Cocoa butter is a vegetable fat used in pharmaceutical, cosmetic and specially food industry.

Researches like this work are very important to improve industrial processes. Therefore, similar

studies should continue, in order to achieve better models.

69

5. References

[1]. Sato, K.; Ueno, S; Yano, J.; Molecular interactions and kinetic properties of fats, Progress in

Lipid Research 1999: 38, 91-116.

[2]. Ghotra, B. S.; Dyal, S. D.; Narine, S.S.; Lipid shortenings: a review, Food Research International

2002: 35, 1015-1048.

[3]. Kokot K.; Knez, Ž.; Bauman, D.; S-L-G (Solid- Liquid-Gas) phase transition of Cocoa butter in

Supercritical CO2, Acta Alimentaria 1999: Vol. 28 (2), 197-208.

[4]. Venter, M.J.; Willems, P.; Kareth, S.; Weidner, E.; Kuipers, N.J.; de Haan, A.B., Phase equilibria

and physical properties of CO2-satureted cocoa butter mixtures at elevated pressures, J.

Supercrit. Fluids 2006.

[5]. Walas, S.M.; Phase Equilibria in Chemical Engineering. University of Kansas, Butterworth

Publishers 1985.

[6]. Calderone, M.; Étude de la microencapsulation de particules fines par fluide supercritique: mise

au point, validation et comprehension phénoménologique d’un nouveau procédé, These, 2005.

[7]. Kikic, I.; Thermodynamic Properties Phase Equilibria, Socrates Intensive course 2007.

[8]. Güçlü-Üstündğ,Ö; Temelli, F.; Correlating the solubility behaviour of fatty acids, mono, di, and

triglycerides, and fatty acid esters in supercritical carbon dioxide. Ind. Eng. Chem. Res. 2000: 39

, 4756 – 4766.

[9]. Brunner, g.; Supercritical fluids: technology and application to food processing, Journal of Food

Engineering 2005: 67, 21-33.

[10]. Fages, J., Letournou, J.J., Sauceau, M., Rodier, E., Elaboration de solides divisés +ar fluids

supercritiques, Techniques de l’ingénieur, 2005: Vol. RE26, p.1-12.

[11]. Reid, R.C.; Prausnitz, J.M.; Poling, B. E.; The properties of Gases & Liquids, McGraw-Hill

Book Company 1987.

[12]. Ball, W. D.; Physical Chemistry, Brooks Cole 2003.

[13]. Dymond, J.H.; Malhotra, R.; The Tait Equation: 100 Years On International Journal of

Thermophysucs, 1988: Vol. 9, No. 6.

[14]. Valderrama, J. O.; The State of the Cubic Equations of State, Ind. Eng. Chem. Res., 2003: 42,

1603-1618.

[15]. Peng, D. Y.; Robinson, D. B.; A new Two-Constant Equation of State, Ind. Eng. Chem.

Fundam, 1976: Vol.15, No.1.

[16]. Constatinou, L.; Gani, R.; New Group Contribution Method for Estimating Properties of Pure

Compounds, AIChE, 1994: Vol.40, No. 10.

[17]. Araújo, M. E.; Meireles, M. A. A.; Improving phase equilibrium calcularion with Peng-Robinson

EOS for fats and oils related compounds/superctitical CO2 systems, Fluid Phase Equilibria,

2000: 169, 49-64.

[18]. Weber, W.; Petkov, S.; Brunner, G.; Vapour-Liquid-equilibria and calculations using the

Redlich-Kwong-Aspen-equation of state for tristearin, tripalmitin, and triolein in CO2 and

propane, Fluid Phase Equilibria, 1999: 158-160, 695-706.

70

[19]. Knox, D. E.; Solubilities in supercritical fluids, Pure Appl. Chem., 2005: Vol.77, No. 3, pp 513-

530.

[20]. http://www.anton-paar.com/ in August 2008

[21]. http://webbook.nist.gov/chemistry/ in May, June, July, August and October2008.

[22]. Wagner, W; Pruss, A., The LAPWS formulation 1995 for the thermodunamic properties of

ordinary water substance for general and scientific use, J. Phys. Chem. Ref. Data, 2002:

31,2,387-535.

[23]. Span, R.; Wagner, W., A New Equation of State for Carbon Dioxide Covering the Fluid Region

from the Triple-Point Temperature to 1100 K at Pressures up to 800 MPa, J.Phys. Chem. Ref.

Data, 1996: 25, 6, 1509-1596.

[24]. Tegetmereir, A.; Ditmar, D.; Fredenhagen A.; Eggers, R.; Density and volume of water and

triglyceride mixtures in contact with carbon dioxide, Chem. Eng. Process, 2000: 39, 399.

[25]. Venter, M. J.; Gas Assisted Mechanical Expression of Cocoa Nibs, PhD thesis, University of

Twente, The Netherlands, 2006;

[26]. Ullmann’s Encyclopedia of Industrial Chemistry; 5th edition; Vol. A7 and A10; VCH; 1989 to

1995;

[27]. Fornari, R., Alessi, P., Kikic, I. High pressure fluid phase equilibria: experimental methods and

systems investigated (1978-1987) Fluid Phase Equilibria, 1990: 57:1, 1-33.

[28]. Schneider G. M. Physico-chemical principles of supercritical fluid science and technology,

Supercritical Fluids, ed. Kiran E., Sengers, J. M. H., Kluver Academic Publishers, Dordrecht,

1994: 91-115.

[29]. Di Andreth, J. R., Paulaitis, M. E., Experimental study of three- and four-phase equilibria for

isopropanol-water-carbon dioxide mixtures at elevated pressures Fluid Phase Equilibria,

1987: 32, 1261-271.

[30]. Costa, M. A., Matos, H. A., Nunes da Ponte, M., Gomes de Azevedo, E. Binary and ternary

phase behaviour of α-pinene, β-pinene and supercritical ethene J. Chem. Eng. Data 41, 1996:

1104-1110.

[31]. Angus, S., Armstrong, B., de Reuck, K., (Eds.), International Thermodynamic Tables of the

Fluid State Vol. 3., Carbon Dioxide, IUPAC, Division of Physical Chemistry, Pergamon Press,

Oxford: 1976.

[32]. Lim, C.S., Manan, Z.A., Sarmidi, M.R., Simulation Modelling of the Phase Behaviour of Palm

Oil-Supercritical Carbon Dioxide, JAOCS, 2003:Vol. 80, no. 11.

[33]. http://www.tu-harburg.de/vt2/pe2000/HomePage.html in October and November 2008.

[34]. Martins e Costa, Miguel Ângelo; Equilíbrio de Fases para Sistemas Contendo Pinenos e

Etileno Supercrítico, Dissertação para obtenção do grau de mestre em Engenharia Química,

Instituto Superior Técnico, Lisboa, 1994.

[35]. Joaquim Rodrigues, Miguel Ângelo; Produção de Partículas para Libertação Controlada de

Fármacos Utilizando Fluidos Supercríticos, Tese de Doutoramento, Instituto Superior técnico,

2006.

71

[36]. Green Chemical Processing Using CO2, Ind. Eng. Chem. Res., 2003: 42, 1598-1602.

[37]. Calvignac, Brice; Internal Report – Ecole des Mines D’Albi-Carmaux, 2008.

72

6. Appendix

6.1. Appendix A

Table A. 1: The CO2 physical-chemical properties [21]

Chemical Formula CO2 Molar Mass 44.0095 g.mol-1

ωωωω 0.2250

TC 304.18 K PC 73.80 bar ρρρρC 10.6 mol.L-1 TT 216.58 K PT 5.185 bar TF 216.6 K TB 194.7 K

∆∆∆∆HF -393.52 kJ.mol-1

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6.2. Appendix B

Table B. 1: Ambrose Group Contributions for Critical Constants [11]

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Table B. 2: Joback Group Contributions for Critical Properties, the Normal Boiling Poin, and the Freezing Point [11]

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Table B. 3: Fedors Group Contributions for Critical Temperature [11]

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Table B. 4: First-Order Groups and their Contributions for the Physical Properties [16]

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Table B. 5: First-Order Groups and their Contributions for the Physical Properties [16]

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6.3. Appendix C

Table C. 1: Calibration Table

Table C. 2: Density of Cocoa Butter at 40 and 50ºC

Table C. 3: Density of CO2/saturated Cocoa Butter at 40 and 50ºC

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6.4. Appendix D

Table D. 1: Compressibility calculations for Tait Equations for Cocoa Butter

Table D. 2: Compressibility calculations for Tait Equations for CO2/Saturated Cocoa Butter

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6.5. Appendix E

Table E. 1: Solubility of CO2 in Cocoa butter at 40 and 50ºC

T (ºC) w CO2 w CO2 (%) P (bar)

40

0.090304 9.03036 32.83333

0.141293 14.12933 45

0.190778 19.07781 55.86667

0.2664 26.63996 83.73333

0.299244 29.92436 93.1

0.344423 34.44226 124.35

0.39438 39.43799 180.0333

50

0.078775 7.877536 52.2

0.142068 14.2068 92.6

0.052611 5.261063 50.66667

0.171162 17.11619 129.9333

0.230137 23.0137 198.6

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6.6. Appendix F

Table F 1: Density predicted by the Peng-Robinson Equation of State at 40 and 50ºC

T (ºC) P (bar) w CO 2 w CCB Density (g/cm 3)

40

32.83 0.090304 0.909696 0.489666 45.00 0.141293 0.858707 0.497073 55.87 0.190778 0.809222 0.50351 83.73 0.2664 0.7336 0.515654 93.10 0.299244 0.700756 0.520933

124.35 0.344423 0.655577 0.533293 180.03 0.39438 0.60562 0.553706

50

52.20 0.078775 0.921225 0.48741 50.67 0.052611 0.947389 0.482859 92.60 0.142068 0.857932 0.499312

129.93 0.171162 0.828838 0.506962 198.60 0.230137 0.769863 0.52332

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