Thermodynamic Modelling of Liquid–liquid Equilibria Using the Nonrandom Two-Liquid Model and Its Applications
Zheng Li
Submitted in total fulfilment of the requirements of the degree of Doctor of Philosophy
August 2015
Department of Chemical and Biomolecular Engineering
The University of Melbourne
The way ahead is long and adventurous,
I shall keep on exploring.
路漫漫其修远兮，吾将上下而求索。
— Yuan Qu (屈原)
Abstract
Solvent extraction is a separation technique widely used in a variety of industrial applications. The basis of separation by this technique is the distribution of a solute between two immiscible solvents, which fundamentally is a phenomenon of thermodynamic phase equilibrium. As a result, the thermodynamic modelling of liquid–liquid equilibria (LLE) is a significant problem of solvent extraction.
In general, there are two approaches to calculate phase equilibrium: minimizing the Gibbs free energy combined with the Tangent Plane Distance (TPD) criterion for stability test and solving the isoactivity equations. Compared with the first approach, the second is easier, however, it strongly depends on initial estimation and may lead to erroneous results which correspond to maxima, local minima and saddle points of the Gibbs free energy. Therefore, the primary aim of this thesis is to understand the solution structure of the isoactivity equations of LLE and develop a procedure to determine the correct, physically realistic solution. This thesis has three parts: firstly, understanding the isoactivity equations of LLE using the nonrandom two-liquid (NRTL) model, the most popular thermodynamic model; secondly, regression of NRTL parameters using particle swarm optimization (PSO) and insights into the model’s capabilities in correlating LLE data; thirdly, application of the symmetric eNRTL model and the developed PSO method to the modelling of phenol extraction.
The solution structure of the isoactivity equations for ternary and quaternary LLE systems using the NRTL model under two types of mass balance constraints were investigated. The first constraint specifies the concentration of components (one component in a ternary system and two components in a quaternary system) in one phase. In this case, the three isoactivity equations of a ternary LLE system were presented in a three dimensional space as three surfaces with their intersection lines extracted. Three types of solutions were revealed, namely exact solutions, symmetric solutions and approximate solutions. These analyses were called Solution Structure Categorization (SSC). Results yielded by SSC further led to development of a procedure to identify the correct solution of LLE for ternary and quaternary systems.
The second constraint specifies the total amount of each component in a system. In this case, the SSC method was again applied and it was found that all solutions of
I isoactivity equations can be categorized into two types when converted into mole fractions: one correct solution and a number of symmetric solutions representing a homogeneous phase. A procedure based on solving isoactivity equations to determine the correct solution was also proposed, which was shown to be simple and effective for a number of ternary and quaternary LLE systems from a wide range of literature sources. The new procedure is recommended to be used as a parallel procedure to minimization of Gibbs free energy combined with stability test by the TPD criterion.
The NRTL model has binary interaction parameters and non-randomness parameters that need to be regressed before the model can be used. The particle swarm optimization (PSO) method was successfully used to regress the NRTL parameters from liquid–liquid equilibria (LLE) data and the resulting parameters showed smaller root-mean square deviations (RMSD) compared with literature values. Analysis of the results revealed that multiple groups of parameters with sufficiently small RMSDs can be found for the same set of LLE data. The activities calculated using these parameters and their corresponding predicted mole fractions can be far beyond the reasonable range of activity, demonstrating that the NRTL model does not always represent the intrinsic activities of components with these parameters.
Finally, extraction of phenol by toluene in the presence of sodium hydroxide was investigated with varying pH and varying concentration of sodium hydroxide to mimic extraction of alkaloids as acidity constant of phenol is close to that of many alkaloids, for example morphine. The phase equilibrium was modelled by the symmetric eNRTL model using the developed PSO method and the correlation agreed well with the experimental results.
II Declaration
This is to certify that:
(i) the thesis comprises only my original work towards the PhD except where indicated in the Preface,
(ii) due acknowledgement has been made in the text to all other material used,
(iii) the thesis is fewer than 100 000 words in length, exclusive of tables, maps, bibliographies and appendices.
Zheng Li August 2015
III Preface
Results in Sections 2.2 and 2.3 have been published, or are waiting for publication, in the following forms:
Li, Z.; Mumford, K. A.; Shang, Y.; Smith, K. H.; Chen, J.; Wang, Y.; Stevens, G. W., Analysis of the nonrandom two-liquid model for prediction of liquid– liquid equilibria. J. Chem. Eng. Data 2014, 59, (8), 2485-2489.
Li, Z.; Mumford, K. A.; Smith, K. H.; Chen, J.; Wang, Y.; Stevens, G. W., Reply to “Comments on ‘Analysis of the nonrandom two-liquid model for prediction of liquid–liquid equilibria’”. J. Chem. Eng. Data 2015, 60, (5), 1530-1531.
Li, Z.; Mumford, K. A.; Smith, K. H.; Chen, J.; Wang, Y.; Stevens, G. W., Solution structure of isoactivity equations using the nonrandom two-liquid model for liquid–liquid equilibrium calculations. In preparation.
Results in Sections 3.2, 3.3 and 3.4 have been included in the following journal paper:
Li, Z.; Smith, K.H.; Mumford, K.A.; Wang, Y.; Stevens, G.W., Regression of NRTL parameters from ternary liquid–liquid equilibria using particle swarm optimization and discussions. Fluid Phase Equilibria 2015, 398, 36-45.
Results in Sections 4.2, 4.3 and 4.4 have been presented in the following journal paper and refereed conference proceedings:
Li, Z.; Mumford, K. A.; Shang, Y.; Smith, K. H.; Chen, J.; Wang, Y.; Stevens, G. W., Extraction of phenol by toluene in the presence of sodium hydroxide. Sep. Sci. Technol 2014, 49, (18), 2913-2920.
Li, Z.; Mumford, K. A.; Smith, K. H.; Stevens, G. W., Experimental and model study on the extraction of phenol by toluene. The 20th international solvent extraction conference. Wurzburg, Germany, September 2014.
IV Acknowledgement
Gratitude fills my heart when I am able to summarize my work and write this thesis. I was lucky to have the opportunity to meet many awesome people from all walks of life during my PhD study, without their support and help I would not have gone through the PhD journey. I would like to sincerely thank them for their support, encouragement and friendship.
First and foremost, I would like to express my sincere acknowledgement to Prof. Geoffrey Stevens for recruiting and supervising me for my PhD study. Since receiving the very first email of mine for enquiring a potential PhD position, he has been patiently and thoughtfully supporting my study and associated affairs. His continuous support and encouragement is the source of power pushing me forward.
Dr. Kathryn Smith and Dr. Kathryn Mumford started to help me as my co-supervisors when my project was changed to be thermodynamic modelling, which was a hard decision to make. They devoted lots of time to discuss with me and provided me with constructive suggestions. Their expertise and generous help made my new project went much smoother. For this, I owe them big.
Equal thanks go to Dr. Jilska Perera who co-supervised me for study on kinetics of solvent extraction in the first year, during which I was trained for a series of experimental techniques and critical thinking. These skills were very useful in my following studies.
I would like to thank Mr. Justin Fox in the engineering workshop and Mr. Leslie Gamel from workshop of physics for constructing experimental equipments. Mr. David Danaci is thanked for his generous help in the trouble shooting of leakage. Thanks to Mr. Edward Nagul and Ms. Kezia Kezia for kind help in developing HPLC methods. Mr. Indrawan Indrawan and Ms. Alita Aguiar are thanked for their patient training for instrument utilization and assistance in apparatus setup. I am also grateful for Mr. Di Che and Dr. Guang Wen for their help in debugging MATLAB programs.
Co-authors of my papers, including Prof. Jian Chen at Tsinghua University, Mr. Yidan Shang at the RMIT University and Dr. Yong Wang who shares the same office
V with me, are acknowledged for their valuable contributions. Prof. Sandra Kentish is sincerely thanked for her precious and detailed comments on my work regarding a membrane model, despite this work is not included in the thesis. Acknowledgement also goes to Prof. Ray Dagastine for supervising me working as a demonstrator, which was a pleasant experience.
I would also like to thank Prof. Hans-Jörg Bart for allowing me to visit his lab at Technische Universität Kaiserslautern. Ms. Hanin Jildeh, Mr. Felix Gebauer and their colleagues in the lab warmly welcomed me. I feel grateful for their hospitality.
I had the opportunity to visit Profs. Wei Qin and Jichu Yang at Tsinghua University. Their work in recovery of salt brine resources inspired me deeply. I also had a chance to discuss some potential future research projects with Profs. Weiyang Fei and Yujun Wang from the same university. I would like to sincerely thank these professors for their precious time.
Friendships with many awesome people, including Dr. Davide Ciceri, Dr. Judy Lee, Dr. William Lum, Dr. Apple Koh, Dr. Jiwei Cui, Dr. Qiang Fu, Dr. Alex Duan, Dr. Zhou Zhang, Mr. Lachlan Mason, Dr. Faxin Xiao, Mr. Guopong Hu, Ms. Cathy Chen and so on, have been spiritual support for me. People come and go, but long live friendship.
Ms. Annmaree Sharkey, Dr Michelle de Silva, Ms. Louise Baker and especially Ms. Tabitha Cesnak, are sincerely acknowledged for their administrative support in managing my candidature.
Importantly, I would like to thank the University of Melbourne and China Scholarship Council for providing me with scholarships that made my study possible.
Last but not least, my deep gratitude goes to my family for their understanding and support. I give my particular thank you to Qian, my beloved wife, for her companionship, being patient to me and so much more she has done for me.
VI Table of Contents
Abstract ...... I
Declaration ...... III
Preface ...... IV
Acknowledgement ...... V
List of Figures ...... XI
List of Tables ...... XIII
Nomenclature ...... XIV
Chapter 1 Introduction ...... 1
1.1 Solvent Extraction ...... 2
1.1.1 Reactive Extraction ...... 2
1.1.2 Nonreactive Extraction ...... 7
1.2 Thermodynamics of Liquid–liquid Equilibria ...... 12
1.2.1 Criteria for Phase Equilibria ...... 12
1.2.2 Distribution Ratio ...... 15
1.2.3 Chemical Equilibrium and Equilibrium Constant ...... 16
1.2.4 Equilibrium Constant of Mixed Reference States ...... 17
1.3 Two Approaches to Phase Equilibria Calculations ...... 18
1.3.1 Minimizing the Gibbs Free Energy ...... 19
1.3.2 Solving the Isoactivity Equations ...... 20
1.4 Thermodynamic Models ...... 21
1.4.1 The SXFIT ...... 22
1.4.2 Wilson Model ...... 25
1.4.3 The NRTL Model and Its Expansions ...... 27
VII 1.4.4 The UNIQUAC and UNIFAC Model ...... 34
1.4.5 Conductor-Like Screening Model (COSMO) ...... 38
1.4.6 Summary of Thermodynamic Models ...... 41
1.5 Summary and Thesis Outline ...... 43
Chapter 2 Analysis of the Nonrandom Two-Liquid Model for
Prediction of Liquid–liquid Equilibria ...... 45
2.1 Computational Method ...... 46
2.1.1 Equations for Computation ...... 47
2.2 Specifying Concentrations of Components ...... 47
2.2.1 Isoactivity Equations in a 3D Space ...... 48
2.2.2 Evaluation of Solutions ...... 51
2.2.3 Convexity of the Gibbs Free Energy ...... 52
2.2.4 Development of a New Procedure ...... 53
2.3 Specifying Total Amount of Each Component ...... 57
2.3.1 Categorization of Solutions ...... 58
2.3.2 A New Calculation Procedure ...... 64
2.3.3 Comparison with Minimizing the Gibbs Free Energy ...... 65
2.4 Conclusions ...... 65
Chapter 3 Regression of Nonrandom Two-Liquid Parameters
Using Particle Swarm Optimization and Discussions ...... 67
3.1. Introduction ...... 68
3.2 Parameter Regression Using the PSO Algorithm ...... 69
3.2.1 Particle Swarm Optimization ...... 69
3.2.2 Optimization Method ...... 70
3.2.3 Regression Results ...... 72
3.3 Representation of LLE by the NRTL Model ...... 75
VIII 3.3.1 Local and Global Minimum ...... 75
3.3.2 Representation of Activities of LLE ...... 76
3.4 Correlating Normalized Random Data ...... 78
3.4.1 Generation of Normalized Random Data ...... 79
3.4.2 Correlation and Discussions ...... 81
3.5 Conclusions ...... 86
Chapter 4 Phenol Extraction by Toluene in the Presence of
Sodium Hydroxide ...... 87
4.1 Introduction ...... 88
4.2 Experimental ...... 89
4.2.1 UV-Vis Spectrophotometry ...... 89
4.2.2 Distribution Ratio of Phenol ...... 90
4.3. Results and Discussions ...... 92
4.3.1 Dissociation Equilibrium of Phenol with Sodium Hydroxide ...... 92
4.3.2 Spectrophotometric Properties of Phenol ...... 93
4.3.3 Oxidation of Phenol by Oxygen ...... 94
4.3.4 Distribution Ratio of Phenol with Varying pH ...... 96
4.3.5 Distribution Ratio of Phenol with Varying Sodium Hydroxide Concentration ...... 97
4.4. The Modelling of Phase Equilibrium ...... 99
4.5 Conclusions ...... 101
Chapter 5 Conclusions and Outlook ...... 103
5.1 Main Conclusions ...... 104
5.2 Outlook ...... 105
References ...... 107
Appendices ...... 120
IX A. Calculation Examples ...... 120
B. MATLAB Code for Solving Isoactivity Equations ...... 130
C. MATLAB Code for Regression of NRTL Parameters ...... 135
D. Mathematica Code for Plotting Isoactivity Equations ...... 144
E. List of Publications During Ph.D. Study ...... 147
X List of Figures
Figure 1.1 Structure of CYANEX® 272 (a) and the complex formed with cobalt (b). 4
Figure 1.2 A reaction scheme of cobalt extraction by CYANEX® 272 ...... 6
Figure 1.3 Cobalt and Nickel extraction isotherms...... 6
Figure 1.4 Calculated activity coefficient of hexane and heptane at 343.15 K ...... 8
Figure 1.5 Distribution ratio of phenol between water and toluene...... 9
Figure 1.6 LLE of n-Hexane (1) + Benzene (2) + Sulfolane (3) at 298.15 K ...... 11
Figure 1.7 Two types of cells in a binary mixture according to two-liquid theory ... 28
Figure 1.8 Three types of cells in an electrolyte system ...... 30
Figure 2.1 Surfaces of solutions of isoactivity equations ...... 49
Figure 2.2 Solutions of the isoactivity equations ...... 50
II Figure 2.3 Residuals of Eqs. (2.1a)~(2.2b) with respect to the solutions along x1 .. 52
Figure 2.4 Leading principle minors of Hessian matrix of the Gibbs free energy .... 53
Figure 2.5 1000 solutions of Eqs (2.1a)~(2.2b) with random initial estimations ...... 56
Figure 2.6 1000 solutions of the isoactivity equations under constraints of Eq. (2.6) ...... 59
Figure 2.7 Gibbs free energy of the 419 solutions ...... 60
Figure 2.8 Surfaces of solutions of isoactivity equations ...... 62
Figure 2.9 Solutions of the isoactivity equations under constraints of Eq. (2.6) ...... 63
Figure 3.1 Flowchart of the PSO algorithm ...... 70
Figure 3.2 Correlation of LLE data using Fa as objective function ...... 72
Figure 3.3 Initial RMSDs ...... 73
Figure 3.4 Decrease of RMSD with the number of iterations ...... 74
Figure 3.5 Representation of activities by different groups of parameters ...... 77
Figure 3.6 Phase diagrams of normalized random data ...... 82
Figure 3.7 Correlations of the regular LLE data by the NRTL model...... 84
XI Figure 3.8 Correlations of the normalized random data by the NRTL model ...... 85
Figure 4.1 Structure of morphine and phenol ...... 88
Figure 4.2 HPLC spectra of phenol and toluene ...... 91
Figure 4.3 Equilibrium of phenol reaction with sodium hydroxide ...... 92
Figure 4.4 The UV-Vis spectra of toluene, phenol and phenolate anion ...... 94
Figure 4.5 pH variation of aqueous phenol solution before and after equilibrium .... 95
Figure 4.6 Distribution ratio of phenol with varying pH ...... 96
Figure 4.7 Distribution ratio with varying concentration of sodium hydroxide ...... 98
XII List of Tables
Table 1.1 Dimerization constant of CYANEX® 272 ...... 4
Table 1.2 Equilibrium constant of cobalt extraction by CYANEX® 272 ...... 5
Table 1.3 Hansen Solubility Parameters at 25 °C ...... 11
Table 1.4 Standard states for solvents and solutes ...... 13
Table 2.1 Experimental data and three exact solutions ...... 55
Table 2.2 Solutions for isoactivity equations in moles ...... 61
Table 2.3 Solutions for isoactivity equations in mole fractions ...... 61
Table 2.4 Selected calculation examples for the proposed procedure ...... 64
Table 2.5 Comparison of the two procedures for LLE calculations ...... 65
Table 3.1 Comparison of regression results with literature values ...... 75
Table 3.2 Regressed parameters of the NRTL model for ternary LLE systems ...... 75
Table 3.3 Three groups of parameters for the same LLE data ...... 76
Table 3.4 Normalized random data No. 1 ...... 79
Table 3.5 Normalized random data No. 2 ...... 79
Table 3.6 Normalized random data No. 3 ...... 80
Table 3.7 Normalized random data No. 4 ...... 80
Table 3.8 Normalized random data No. 5 ...... 81
Table 3.9 Regressed NRTL parameters for data correlation ...... 83
Table 4.1 Molar absorptivity of phenol ...... 93
Table 4.2 Properties of species used in this study ...... 95
Table 4.3 Liquid–liquid equilibrium data (mole fraction) ...... 97
Table 4.4 Regressed parameters for the extraction system: phenol (1) + toluene (2) + sodium phenolate (3) + water (4) ...... 101
XIII Nomenclature
General Notations a Activity aeff Effective contact area between two surface segments amn Group interaction parameter
A Solute; Helmholtz Energy
Ai Stoichiometric formula of species i; Surface area of component i
Aϕ Debye–Hückel coefficient
B Solute c1 , c2 Acceleration factors ci Concentration of component i
Φ Cca Single salt parameter
Co Cobalt
D Distribution ratio
D0 Distribution constant
EHB Hydrogen bonding energy
Emisfit Electrostatic energy due to surface charge mismatch
EvdW van der Waals (vdW) interactions between surface segments
Fa Objective function with respect to activity
Fx Objective function with respect to mole fraction fi Fugacity of component i fγ Debye–Hückel term
G Gibbs free energy g Molar Gibbs free energy gji Interaction energy between molecules j and i
XIV H Hydrogen; Enthalpy
H2A2 Dimer of an extractant
HA Monomer of an extractant
I Ionic strength k Constriction factor kB Boltzmann constant
K Equilibrium constant
K2 Dimerization constant
M Molecular weight ni Mole amount of component i
N The number of components
NA Avogadro’s number,
Pi Pressure of component i
* Pi Saturation pressure of component i p(σ) The σ-profile of a mixture or a component q External surface area parameter of pure-component
Qe The electron charge
Qk Group area parameters r Molecular size parameter of pure-component
R Gas constant
R0 Radius of the Hansen Sphere
R1 , R2 Random numbers distributed within [0, 1]
Ra The distance between a solvent and a solute in HSP
RED Relative energy difference
Rk Group volume parameter
S Entropy; Solution of isoactivity equation
XV T Temperature uji Energy of interaction
U Internal energy
Umn Interaction energy between groups m and n
V Volume
Vi Molecular volume (or molar volume) of Component i
V0 Partial volume of an ethylene unit
νi Molar volume of component i; Moving velocity of particle i
ω Inertia weight xi Mole fraction of component i; Location of particle i xji Local mole fraction of component j around component i
j xik Experimental mole fraction of components
j xik Calculated mole fraction of components
Xm Mole fraction of group m yi Mole fraction of component i in gas phase zi Mole fraction of component i in the system; z The number of charges of an ionic species; Coordination number z Composition of a system
Greek Letters
αij Nonrandomness parameter
β Single salt parameter
γ Activity coefficient
Γk Group residual activity coefficient
δ Hansen Solubility Parameter
ε Dielectric constant
XVI ζi Local volume fraction of component i
θ Area fractions
Θm Area fraction of group m
λij(I) A function of ionic strength, indicating short-range forces
μi Chemical potential of component i
μijk Triple ion interaction parameter
μS(σ) σ-potential
ξi Mole number of component i
ρ The closest approach parameter
σ Screening charge density; Residual of an equation
τji Energy parameters
υi Stoichiometric number of species i
(i) υk The number of groups of type k in molecules i
ϕ Osmotic coefficient of water
φ Phase φ
φi Volume Fraction of component i
Φij Mixing parameters
* Φi Segment fraction
Ψijk Mixing parameters
Ψmn Group interaction parameter
Subscripts a Anion aq Aqueous phase c Molarity basis; Cation
Co Cobalt d Dispersion
XVII h Hydrogen bonding i Component i j Component j k The number of tie line l Component l m Molality basis; Molecule
M Cationic species
N Neutral species org Organic phase p Polar
S Average value of a parameter of the solvent
T Total w Water x Mole fraction basis
X Anionic species
Superscripts
0 Reference state
I Phase I
II Phase II c Combinatorial contribution
E Excess (molar) Gibbs free energy
ID Ideal solution j Phase j lc Local interaction
M Mixing r Residual contribution
XVIII α Phase α
β Phase β
π The number of phase o Standard state
+ Positive charge
Abbreviations
COSMO-RS COnductor like Screening MOdel for Real Solvents eNRTL Electrolyte Nonrandom Two-Liquid
GA Genetic Algorithm
HPLC High Performance Liquid Chromatography
HSP Hansen Solubility Parameter
ICP Inductively Coupled Plasma
LLE Liquid–Liquid Equilibrium
NMR Nuclear Magnetic Resonance
NRTL Nonrandom Two-Liquid
PDH Pitzer–Debye–Hückel
PS Pattern Search
PSO Particle Swarm Optimization
QSPR Quantitative Structure–Property Relationship
RMSD Root-Mean Square Deviations
SA Simulated Annealing
SSC Solution Structure Categorization
TBP Tributyl Phosphate
TPD Tangent Plane Distance
UNIFAC UNIQUAC Functional-group Activity Coefficient
UNIQUAC Universal Quasi-Chemical Equation
XIX UV Ultraviolet (light) vdW van der Waals
VLE Vapour–Liquid Equilibrium
XX Chapter 1
Introduction
Solvent extraction is a separation technique widely used in a variety of industrial applications. The modelling of liquid–liquid equilibria is a fundamental problem of solvent extraction. This chapter introduces principles of solvent extraction, briefly reviews the chemical thermodynamics of phase equilibria and some popular thermodynamic models, with special emphasis given to the nonrandom two-liquid (NRTL) model that is used in this thesis. Finally, an outline of the thesis is presented.
1 1.1 Solvent Extraction
Solvent extraction is an effective separation technique widely used in a variety of applications, ranging from separations in analytical chemistry[1, 2] to industrial processes in hydrometallurgy[1, 3-6], pharmaceutical[7-9], food engineering[9-11] and waste treatment[8, 12, 13]. Solvent extraction takes advantage of the solubility difference of a solute in two immiscible liquid phases (usually one organic and one aqueous phase) in contact with each other to attain separation. Solute A, which initially is dissolved in only one of the two phases, gradually distributes between the two phases with the process of reaction or diffusion at the interface and eventually reaches equilibrium. Concentrations of solute A in organic and aqueous phases are [A]org and
[A]aq respectively and the distribution ratio, DA (also called the distribution coefficient), of the solute is defined as the ratio of “the total concentration of the substance in the organic phase to its total concentration in the aqueous phase, usually measured at equilibrium”[1].
[]A org DA (1.1) []A aq
If a second solute B is present, the corresponding distribution ratio is indicated by DB and so forth. The solutes A and B can be separated if DA and DB are different.
The solubility difference of a solute in two phases due to the difference in the strength of interactions between molecules of the solute and those of the two solvents is the basis of separation. The distribution of a solute in two phases is a fundamental problem of phase equilibrium. The thermodynamics of solvent extraction deals with the final state of phase equilibrium.
1.1.1 Reactive Extraction
Solvent extraction can be classified into reactive extraction and non-reactive extraction according to whether chemical reactions are involved in the process. Inorganic compounds, particularly metals, are hydrophilic, their solubilities in organic solvents usually are very low and even non-detectable. Extraction of these elements can be achieved by formation of lipophilic complexes through coordination with organic compounds[14], which are named extractants. An extractant can either be
2 directly used to recover elements from an aqueous solution or be diluted in a diluent to obtain desirable operational properties such as appropriate viscosity. In some cases, additives, including modifiers, surfactants and a second extractant can be used for improved selectivity, ease of operation, better kinetics and enhanced interfacial properties[4]. The organic phase generally contains (1) a diluent, (2) the extractant, (3) the formed complexes and in some cases (4) other additives. By contrast, the aqueous phase has the elements to be recovered, impurities to be separated and dissolved extractant, which usually has very small solubility in water. The extraction reactions are believed to happen at the interface[3] and the interfacial region and the organic phase composition control the transfer of metal in solvent extraction process. As an example, consider extraction of cobalt by CYANEX® 272 (bis(2,4,4-trimethylpentyl) phosphinic acid, Figure 1.1a) in the presence of nickel.
CYANEX® 272 forms dimers and may exhibit different dimerization constants in different diluents (Table 1.1). The dimerization equilibrium can be expressed as follows:
HAHA22 2 (1.2) with dimerization constant defined as:
HA 22 (1.3) K2 2 HA where HA represents bis(2,4,4-trimethylpentyl) phosphinic acid (Figure 1.1a), the active ingredient of CYANEX® 272.
The mechanism of cobalt extraction by CYANEX® 272 has been proposed by various researchers[15-18].
Co222 H A Co HA H (1.4) 2 2 2 2
The chemical equilibrium of this extraction reaction can be expressed as
2 Co HA H 2 2 KCo 2 (1.5) Co2 H A 22
3 where KCo is the apparent equilibrium constant and the square brackets denote concentration of species within the brackets and the unit is usually mol∙L-1. A collection of equilibrium constant for this extraction reaction is presented in Table 1.2.
Figure 1.1 Structure of CYANEX® 272 (a) and the complex formed with cobalt (b) The structure of the cobalt complex formed with CYANEX® 272 has been proposed to be tetrahedral[15, 18], as shown in Figure 1.1b. The reactions occur in the interfacial region and the complexes formed diffuse into the organic bulk phase. These processes are shown schematically in Figure 1.2.
Table 1.1 Dimerization constant of CYANEX® 272
Organic Diluent K2 Analytical Method Ref. Comments Vapour Pressure Toluene Mainly Dimers [16] - Osmometry Chloroform 184.3 ICP [19] - (CHCL3) Chloroform 206.4 31P NMR [19] - (CDCL3) Colorimetric Kerosene 190 [20] - Method Two-phase Kerosene 15849 [21] Inconsistent Titration Technique
The apparent equilibrium constant calculated from Eq. (1.5) at the same temperature (such as No. 2 and 6 at 298 K in Table 1.2) may differ dramatically, as in this equation the nonideality of species due to the intermolecular interactions are neglected. A general definition for the intrinsic equilibrium constant can be expressed as follows:
4 22 2 aa γ γ Co HA HH Co HA H Co HA 2222 2 2 KCo 2= 2 2 (1.6) Co2 H A aa22 22 γ γ Co HAHA2 2 Co 2 2 where a is the activity of the corresponding species and γ is the activity coefficient. The activity coefficients describe the nonideality of species due to interactions. These coefficients cannot be measured directly and require thermodynamic models for their correlation. Theoretically the intrinsic equilibrium constant should be a constant and independent of concentration and composition of the system and depends only on the reference states chosen. Therefore the most important part of thermodynamic modelling of a solvent extraction process is to correlate the activity coefficients. The thermodynamic principles and the available models from the literature will be reviewed in Sections 1.3 and 1.4 respectively.
Table 1.2 Equilibrium constant of cobalt extraction by CYANEX® 272
-3 No. Keq Purified Solvent T(K) I(mol∙m ) Notes Ref. Assume 1 6.76×10-8 N Hexane Room 650 monomeric [22] extractant 2 2.63×10-8 298 -8 Assume 3 5.50×10 Esso 308 N 640 dimeric [17] -8 DX 3641 4 9.77×10 318 extractant 5 2.34×10-7 333 Assume 6 (6±1) ×10-8 Y Toluene 298 300 dimeric [18] extractant 7 1.78×10-8 293 8 3.72×10-8 303 9 5.25×10-8 308 Assume N Kerosene 640 dimeric [23] -8 10 6.92×10 313 extractant 11 1.00×10-7 318 12 2.51×10-7 333 Assume 13 6.0 ×10-8 Y Toluene 298 300 dimeric [24] extractant -8 14 2.54×10 298 Assume 15 6.23×10-8 dimeric -8 extractant; 16 6.14×10 Shellsol 308 N 634 [25] -8 2046 17 7.60×10 Repeated in 18 1.67×10-7 318 triplicate at 19 3.23×10-7 328 308K
5
Figure 1.2 A reaction scheme of cobalt extraction by CYANEX® 272
Figure 1.3 Cobalt and Nickel extraction isotherms Data source is Cytec technical report[26]. Experimental conditions: 15% CYANEX® 272 extractant, 10% p-nonylphenol in kermac 470B diluents; 1.6 g/L cobalt, 77 g/L nickel and
0.31 g/L calcium as sulphates; at 50 °C with pH controlled by NH4OH.
Nickel can form an octahedral complex with CYANEX® 272 and the general formula may be represented as Ni(HA2)2(H2A2)x(H2O)2-x, where x=0, 1 and 2, and the value of x depends on the concentration of CYANEX® 272[15, 18]. Since the complexes formed between CYANEX® 272 and different metals have different solubility properties, they have the potential to be separated. Extraction isotherms of cobalt and nickel are
6 presented in Figure 1.3. As is shown, the extraction percentage of cobalt and nickel differ significantly at the same pH, thus the two elements can be separated at appropriate pH.
1.1.2 Nonreactive Extraction
Unlike inorganic compounds, organic compounds are usually hydrophobic, which means they are more soluble in organic solvents than in water. The solubility difference of solutes in two phases, rather than the ability to form complexes, is the basis for separation of these compounds.
The solubility of a solute in a solvent depends on how similar the properties of the two compounds are, i.e. the strength of molecular interactions between them. The most simple case is an ideal solution. Raoult’s law (Eq. (1.7)) says that the partial vapour pressure of each component of an ideal mixture of liquids is equal to the vapour pressure of the pure component multiplied by its mole fraction in the mixture[27].
* Pi x i P i (1.7) where Pi and xi are the partial vapour pressure and mole fraction of component i * respectively, Pi is the saturated vapour pressure of component i. It can be seen that the compositions in an ideal solution have no interactions with each other, in other words the activity coefficients of components are unity. This is why they are called “ideal”. Ideal solution is a theoretical concept and no real system perfectly satisfies this definition. However, there are systems that behave close to ideal solution, such as binary mixtures of hexane and heptane, propan-1-ol and propan-2-ol, etc. The activity coefficients of heptane and hexane mixture determined by the nonrandom two-liquid (NRTL) model using vapour–liquid equilibrium (VLE) data of this system[28] is shown in Figures 1.4. It can be seen that the calculated activity coefficients are close to unity, indicating very weak interactions between the two compounds. As a result, compounds in an ideal solution cannot be separated by solvent extraction because they behave similarly in terms of solubility.
The equilibrium of nonreactive extraction can be expressed as follows since there is no reaction involved.
7 AAaq org (1.8) where A is the solute, the subscripts aq and org indicate aqueous and organic phase respectively. The solute A distributes between the two phases and the distribution ratio has been given by Eq. (1.1). Under some conditions, the distribution ratio equals the ratio of solubilities of the solute in the organic solvent and water. Accordingly, the distribution ratio is a constant that is independent of the concentrations of compounds, namely the distribution constant. This is the Nernst distribution law[1] and can be expressed in Eq. (1.9).
Figure 1.4 Calculated activity coefficient of hexane and heptane at 343.15 K Data and parameters are from Lee et al.[28], the NRTL model is used.
The distribution ratio of phenol between water and toluene in the presence of sodium phenate is presented in Figure 1.5[29]. It can be seen that the distribution ratio of phenol changes only slightly with changing phenol concentration when the sodium phenate concentration is 1.903 mole/kg, while the distribution ratio changes dramatically with initial phenol concentration at other sodium phenate concentrations. This variation in distribution ratio can be explained by the varying strength of interactions among the species in the equilibrium system. More specifically, the activity coefficient, which describes the discrepancy from ideal solution, changes with composition. The distribution ratio DA in Eq. (1.1) is a constant for pure solvents (mutual solubilities of solvents are small) with small concentration of solute since the
8 activity coefficient of solute is close to unity under these conditions. A more general equation for distribution ratio that incorporates activity coefficients can be expressed as follows.
o γAorg,, []A orgγ Aorg DDAA = (1.9) γA,, aq[]A aqγ A aq
0 where γ is activity coefficient. Theoretically, DA should always be a constant and DA also a constant when the ratio of activity coefficients is constant. The constant DA has been found to be valid for a range of extraction systems, the third line in Figure 1.5 (1.903 mole/kg sodium phenate) is an example.
Figure 1.5 Distribution ratio of phenol between water and toluene
The different solubilities of a solute in different solvents rest on the affinity between the solute and solvents. The basic idea for solubilities has been “like dissolves like”. On the basis of this principle, the theory of Hansen solubility parameters (HSP) has been developed. The principle of the Hansen three dimensional solubility parameters can be expressed in the following equation[30]:
2 2 2 2 δTδ d δ p δ h (1.10) where δT is the total Hansen solubility parameter and δd, δp and δh are the solubility parameters of the compound due to dispersion, polar and hydrogen bonding,
9 respectively. The closer the solubility parameters of the solute is to that of the solvent, the greater the solubility of the solute in the solvent.
The relative energy difference (RED) can be used to determine whether a solute and a solvent are miscible. RED numbers smaller than 1 indicate the solute and the solvent are miscible, RED numbers equal to or close to 1 is a boundary condition and higher [30] RED numbers mean low affinity of the two components .
RRRED a / 0 (1.11) where R0 is the radius of the Hansen sphere, which can be determined by optimizing the RED values of the solvent with a group of solutes whose Hansen solubility parameters are known to let the Hansen sphere contain the miscible solutes and [30, 31] excludes the immiscible solutes . Ra is the distance between the solvent (subscript “1”) and the solute (subscript “2”) in Hansen solubility parameters and can be defined as[30]:
2 2 22 Ra4δ d1 δ d 2 δ p 1 δ p 2 δ h 1 δ h 2 (1.12)
Either Ra or the RED numbers can be used to rank solutes with respect to a specific solvent. The Hansen solubility parameters of the solvents and solutes involved in this section are presented in Table 1.3. It can be seen that hexane and heptane have very close dispersion energy (δd) and both of them are non-polar and they do not form hydrogen bonding. The high similarity between the two solvents as reflected in the Hansen solubility parameters agree well with the fact that their mixture behaves close to an ideal solution (Figure 1.4). As for the distribution of phenol between toluene and water, the Ra values between phenol and toluene and that between phenol and water have been calculated, the former (13.7) is found to be much smaller than the later (29.7), indicating higher affinity between phenol and toluene than that between phenol and water, thus the distribution ratio is higher than unity, as is shown by the top line in Figure 1.5 (0 mole/kg sodium phenate).
In addition, from the view of HSP, reactive extraction modifies the solubility properties of the solutes (metal ions) by forming complexes with extractants to obtain higher affinity with the organic solvent thus achieving separations. In fact, the theory of HSP has been a useful tool for solvent selection for a range of applications, despite quantitative calculation of phase equilibrium being beyond its capability. However,
10 the theory has been applied in quantitative thermodynamic models as it is discussed in section 1.4.1.
Table 1.3 Hansen Solubility Parameters at 25 °C
Hansen Solubility Parameter Solute/Solvent Reference δd δp δh δT n-Hexane 14.9 0 0 14.9 n-Heptane 15.3 0 0 15.3 Phenol 18 5.9 14.9 24.1 [32] Toluene 18 1.4 2 18.2 Water 15.5 16 42.4 47.9
Figure 1.6 LLE of n-Hexane (1) + Benzene (2) + Sulfolane (3) at 298.15 K
Distribution of a solute between two solvents actually is an equilibrium problem of a ternary system. Not only does the solute have a distribution ratio, the two solvents also distribute themselves between the two phases, each of which is a mixture of three components. However, the distribution ratios of these components between the two phases are usually not constants due to the interactions among these species being nonlinear. As a result, comprehensive modelling of the extraction system is required to understand the solvent extraction equilibrium. The modelling of the ternary liquid– liquid equilibrium (LLE) system “n-Hexane (1) + Benzene (2) + Sulfolane (3)” using the NRTL model from Chen et al.[33] is presented in Figure 1.6 as an example. The
11 results show that the experimental equilibrium tie-lines agree well with the calculations. More discussions on extraction equilibrium following strict thermodynamic principles are given in sections 1.2 and 1.3, popular thermodynamic models are reviewed in section 1.4.
1.2 Thermodynamics of Liquid–liquid Equilibria
1.2.1 Criteria for Phase Equilibria
The chemical potential μi governs mass transfer in a closed system and at equilibrium [34] μi must be uniform throughout the entire systems . The chemical potential uniformity at equilibrium for a heterogeneous closed system consisting π phases and N components at constant pressure and temperature is given as follows[34, 35]
()()()12 π μiμ i μ i (1.13) where i is the component under consideration. While it is difficult to compute an absolute value of the chemical potential, computing the changes of the chemical potential that accompany changes in temperature, pressure and compositions is feasible[34]. Chemical potential is a function of fugacity[34].
o fi μiiμ RT ln o (1.14) fi
o o where μi and fi represent chemical potential and fugacity of the reference state.
o o Once fi is chosen, μi is fixed. Computation of the relative chemical potential requires a reference state whose chemical potential is used as a reference point. As a result, selection of the reference state influences the computation process and the final results. In practice a few standard reference states that are hypothetical but convenient for calculation are used and they are listed in Table 1.4.
Take a system with phases α and β as an example. The chemical potentials for the two phases are
α α α o fi μiiμ RT ln α o (1.15) fi and
12 β β β o fi μiiμ RT ln β o (1.16) fi
Substituting Eq.(1.15) and (1.16) into Eq. (1.14) at equilibrium, yields
α β α offiiβ o μii++RT lnα o μ RT ln β o (1.17) ffii
Table 1.4 Standard states for solvents and solutes
Activity Activity Name Meaning of the standard state Ref. coefficient Rational activity The pure substance at standard γ x γ (x →1) =1 x,i i (type I) pressure x,i i The hypothetical state of solute at mi γ () standard molality o1 , m,i o Molality basis mi 1 mol kg m γm,i (mi→0) =1 i (type II) standard pressure and exhibiting [27] infinite dilute solution behaviour [36] The hypothetical state of solute at c standard concentration γ ()i c,i o Concentration o3 c m1 mol dm , standard pressure γc,i (ci→0) =1 i basis (type III) i and exhibiting infinite dilute solution behaviour
We consider the reference states in two cases: the two phases use the same reference state and the reference states are at the same temperature but different pressures and compositions. We can get the same result in Eq. (1.18) under both conditions after mathematical rearrangements of Eqs. (1.15)-(1.17).
α β ffii= (1.18)
It means that the equilibrium condition in terms of the abstract chemical potential can be replaced by equality of fugacities in two phases without losing generality. The fugacity further defines the activity of component i at certain temperature, pressure and composition as follows:
f T,, P x a T,, P x i (1.19) i o 0 0 fi T,, P x
Here P0 and x0 are the pressure and composition of the reference state respectively. It is worth noting that at equilibrium the fugacities of a component in different phases are equal regardless of the reference states chosen for each phase, however, the
13 activities of a component in different phases are equal only when the same reference state is used for all the phases. The activity and concentration can be related by activity coefficient.
aci iγ i (1.20) where ci is concentration of component i, it can be of molarity (ci), molality (mj) or mole fraction (xj) scale; γi is activity coefficient of the component i. Activity coefficient describes the discrepancy of the component’s behaviour from that in the reference state, where the activity coefficient is 1. The principle for selecting a reference state is no more than convenience. Three reference states that have well- defined thermodynamic properties are conventionally used and deemed as standard states for solvents and solutes (Table. 1.4). In the standard states, not only is the activity coefficient of the substance unity, but the activity of the substance is also unity. However, if other reference states are used, when the activity coefficient of a substance is unity, the activity of the substance is not necessarily unity and vice versa. Usually the same reference state is used for all the phases in a system for convenience of calculations and thus the isoactivity equations can be used to calculate phase equilibria[37, 38]. In the cases where different reference states are used in different phases within an equilibrium system, the definitions and limitations of each reference state should be understood. Some examples of utilization of mixed reference states are given in section 1.2.4.
Another criterion for phase equilibrium can be derived when a system is described in terms of energy. The energy of a system as a function of composition should be convex where the minimum point corresponds to a stable equilibrium state. The energy criterion can be given in different forms of energy-like potential functions including minimum of internal energy (U), enthalpy (H), Helmholtz energy (A), the Gibbs free energy (G) and maximum of entropy (S)[35]. Among these criteria the most useful is the Gibbs free energy as it relates directly to the activity coefficient. The chemical potential is the partial molar Gibbs energy and the Gibbs energy of a mixture can be expressed as follows:
NN
G ni G i n iμ i (1.21) ii11
14 The molar Gibbs free energy can be obtained from Eq. (1.21) by dividing the total amount of components on both sides of the equation.
N
gx iiμ (1.22) i1
The molar Gibbs free energy can be considered in two parts representing the molar Gibbs free energy of an ideal solution gID and the excess molar Gibbs free energy gE.
NNN ID E 0 g g g xiμ i RT x iln x i+ RT x i ln γ i (1.23) i1 i 1 i 1
N 0 In the calculations, the item xiiμ is excluded from Eq.(1.23) and the resulting i1 energy function is called Gibbs energy of mixing. The key of the thermodynamic modeling of phase equilibrium is calculation of the activity coefficient γi, which has been correlated by a number of thermodynamic models that are reviewed in section 1.4.
1.2.2 Distribution Ratio
The distribution ratio of a solute in two phases at equilibrium has been defined by Eq. (1.1) and we rewrite it in Eq. (1.24).
ci, org Di (1.24) ci, aq where ci,org and ci,aq are total analytical concentration of the solute i in the organic and aqueous phase respectively. Molarity units are usually used. Substituting Eqs. (1.19) and (1.20) into Eq. (1.24), we have
o ffi,,, org i aqγ i aq Di o (1.25) fi,, aqfi, org γ i org
The reference state for the organic phase is the hypothetical ideal solution of species i in the organic solvent at concentration of 1 mol/m3 and the solution behaves identical to that of infinite dilution. The reference state for the aqueous phase states similarly but the solvent is water. As has been discussed above, at equilibrium ffi, org i, aq , thus Eq. (1.25) is simplified to be
15 o fi,, aqγ i aq Di o (1.26) fi, org γi, org
At low concentrations, the solutes behave similar to behaviours in infinite dilute solutions, i.e. γi,, aqγ i org 1 , hence the distribution ratio is a constant. When the concentrations are high, interactions between molecules are intense and thus activity coefficient can vary significantly, and the distribution ratio changes accordingly. However, the distribution ratio could still be a constant if the ratio of the two activity coefficients is close to a constant, as it is shown in the example of phenol extraction by toluene in the presence of sodium phenate in section 1.1.2. When the concentrations of the solute in Eq. (1.24) are replaced by its activities, a more general equation which is similar to Eq.(1.9) is produced.
o 0 afi,, org i aq Di o (1.27) ai, aq fi, org
0 Di is always a constant regardless of solute concentration, the constant only depends on the reference states chosen for the solute in both phases. If the same reference state is used for both the organic and the aqueous solution, Eq. (1.27) becomes isoactivity equation that is commonly used in liquid–liquid equilibrium (LLE) correlations[38, 39].
1.2.3 Chemical Equilibrium and Equilibrium Constant
The condition of chemical equilibrium at constant pressure and temperature is also the minimum of the Gibbs potential function G(P, T, n). At its minimum, the differential of the function is zero.
N
dG SdT VdP μiidn 0 (1.28) i1 where S and V are the entropy and volume of the system, N is the number of species and ni is the amount of species i. Consider a general chemical reaction
N
υiiA 0 (1.29) i1
Here Ai is the formula of the species i, υi is the stoichiometric number of the species and N is the number of species involved in the reaction. As the temperature and
16 pressure are constant, the condition of equilibrium can be simplified to Eq. (1.30) (details refer to Keszei[27]).
N o υi(μ iRTln a i ) 0 (1.30) i1
The equilibrium constant is defined as
N υi
Kaai (1.31) i1
This equation relates the equilibrium condition to the stoichiometric equation. As an example, the chemical equilibrium of cobalt extraction by CYANEX® 272 has been given in Eq. (1.6). Combination of Eqs. (1.30) and (1.31) yields the general equation for equilibrium constant
N υ μ o i i i1 Ka exp (1.32) RT
It is indicated that the equilibrium constant depends only on the reference states chosen. Once the reference states are fixed, the equilibrium constant is fixed at a certain temperature.
1.2.4 Equilibrium Constant of Mixed Reference States
Many commonly used thermodynamic models use a mixture of reference states, a selection of examples are discussed briefly here. Some of these models are further introduced in section 1.4.
(1). Pitzer model. Pitzer developed a model for excess Gibbs free energy of aqueous solutions containing electrolytes[40-42]. Differentiation of the excess Gibbs free energy with respect to mass of water and mass of ions gives the osmotic coefficient of water and logarithm of ions’ activity coefficient respectively. To regress the parameters in the equations, the vapour pressure of water or osmotic coefficient of the solution is used[43]. However in this model the reference state for ions is of Type II, while the reference state for water is pure water, which belongs to Type I reference state (see table 1.4).
17 (2). SXFIT. To model the thermodynamic equilibrium of solvent extraction, Baes[44] developed the computer program of SXFIT, which uses the Pitzer model to characterise the aqueous phase and the Hildebrand-Scott model for species in the organic phase. In addition to the variation in the reference states used in the Pitzer model, the Hildebrand-Scot model refers to the pure component of species as reference state. Moreover, unit conversion is involved in this model due to the fact that molarity scale is generally used in solvent extraction while molality unit is used in the Pitzer model.
(3). eNRTL model. The nonrandom two-liquid (NRTL) model[45] is popular for calculation of phase equilibrium of non-electrolytes. The reference state for this model is the same as that used in most models for non-electrolytes, that is, the pure component. Expansion of the NRTL model to electrolyte systems has resulted in the eNRTL model[46] which continues to use pure component for solvents as reference states and uses the “pure completely dissociated liquid electrolyte” as reference state for the electrolytes. However, such a hypothetical state does not exist as cation and anion must coexist to maintain charge neutrality. Following the eNRTL model, the symmetric eNRTL model[47] was developed. This model defines “the pure fused salt state of each electrolyte component” as reference state for ionic species. While still being hypothetical, the reference state is more reasonable than the one used in eNRTL model. This hypothetical reference state makes the whole model consistent in the reference states and simplifies equations. The theory and the equations look symmetric and the model has been applied to a range of systems.
(4). Li’s model. Li et al.[48] developed a model for electrolyte systems which treats a mixed solution as non-electrolyte solution plus charge interactions. In this model, “working reference state” is used in addition to traditional reference states. The working reference state can either be a traditional reference state or be a state of a multi-component system. The flexibility in choosing reference states largely simplifies the model’s calculations.
1.3 Two Approaches to Phase Equilibria Calculations
The focus of this thesis is the modelling of liquid–liquid equilibria (LLE) without chemical reactions, i.e. nonreactive extraction. The minimum of the Gibbs free energy
18 and the equality of chemical potential of a component in all phases are two criteria for equilibria of nonreactive extraction, as discussed in section 1.2.1. Accordingly, we have two general approaches to calculate LLE[37, 38, 49, 50]: (1) minimizing the Gibbs free energy and (2) solving the isoactivity equations.
1.3.1 Minimizing the Gibbs Free Energy
The first approach is searching for the global minimum of the Gibbs free energy of a closed system with π phases at specified temperature and pressure, which can be reduced to minimizing the following function[50, 51]
π N φ φ φ (1.33) Min g niln x iγ i n φ i φ11i with the inequality constraints given by
π-1 φ (i=1, 2, …, N) (1.34) nzii φ1 and
φ ni 0 (i=1, 2, …, N; φ=1, 2, 3, …,π) (1.35)
φ where ni is the mole number of component i in phase φ with the total mole number of materials being unity in the entire system, zi is the mole fraction of component i in the π π π π φ system. The variables ni , xi and γi (T, x ) are considered as functions of ni (i=1, …, N; φ=1, …, π-1).
The global minimum of the Gibbs free energy is the necessary and sufficient condition of phase equilibrium. However, the global minimum of Eq. (1.33), which is a complex nonlinear function with a number of variables, is difficult to obtain and any minimization algorithm may be stuck in local minima due to the mathematical complexity, leading to incorrect phase equilibrium[37, 38, 49, 52]. Therefore, it is essential to have a stability test accompanied with this approach. The tangent plane distance (TPD) criterion initially proposed as a theorem by Baker et al.[53] and then applied numerically by Michelsen[54] has been a popular stability analysis. It has been demonstrated that the tangent plane to the Gibbs energy surface at composition zφ should at no other points intersect the Gibbs energy surface. This requirement can be written as
19 N (1.36) F ξ 1 ξiln ξ i ln γ iξ h i 1 0 ξ i1
N where ξi is the mole number with the corresponding mole fraction as x ξ ξ , i i j1 j and
hziln i ln γ i z (i=1, 2, …, N) (1.37)
Eq. (1.36) requires that the Gibbs energy surface at all points lie above the tangent plane and this is achieved when F(ξ) is non-negative in all compositions. While testing all the trial compositions is impossible, it is sufficient to test all the stationary points (all the maxima, minima and saddle points). As a result, the major challenge for the stability test is locating all the stationary points of F(ξ), which requires a robust numerical method. This stability analysis verifies whether a system is stable or not. In the latter case, an estimation of composition for an additional phase is provided, then the number of phase is increased by one and the phase equilibrium is calculated again by minimizing the Gibbs free energy. This approach continues in a stepwise manner until a stable phase distribution is found[50, 51, 55]. This is a popular yet computationally demanding approach because both minimization of the Gibbs free energy and the stability test by TPD deals with minimizing complex non-linear functions.
1.3.2 Solving the Isoactivity Equations
The second approach utilises uniformity of chemical potential of a component in all the phases as discussed in section 1.2.1. The equality of chemical potential can be replaced by equality of fugacities, which could be further simplified to be the isoactivity equation as given in Eq. (1.38) for a two-phase system when the same reference state is used for both phases. In fact, the reference state used in phase equilibrium calculations of non-electrolytes is usually the pure substance at certain temperature and pressure, i.e. type I reference state in Table 1.4.
I II aaii (i=1, 2, …, N) (1.38) where ai is the activity of component i as defined by Eq. (1.19), the superscripts represent two phases.
20 These isoactivity equations must be constrained by the material balance, which can be given by specifying the total amount of each component as expressed in the following equation:
I II nnii ni (i=1, 2, …, N) (1.39)
I II where ni and ni are mole number of component i in phase I and II respectively and ni is the total amount of component i in the system. We thus have 2N equations with 2N variables. Alternatively, we may specify the concentrations of (N-2) components in one of the phases and use the composition constraints in both phases, which can be expressed as follows:
N φ xi 1 (i=1, 2, …, N; φ=I, II) (1.40) i1
φ where xi is the mole fraction of component i in phase φ. In this case, we have (N+2) equations with (N+2) variables.
Solving the isoactivity equations is easier than minimizing the Gibbs free energy in terms of the computational effort. However, the equality of activities is the necessary but not sufficient condition for stable phase equilibrium. As a result, erroneous solutions corresponding to local minima, maxima or saddle points of Gibbs free energy may be obtained[37, 49, 56]. Checking the convexity of the Gibbs free energy as proposed by Sørensen et al.[37] is able to rule out maxima and saddle points of the Gibbs free energy but fails to identify whether a solution in a convex surface is the global minimum of the Gibbs free energy. Additionally, the equation-solving method finds only one solution for a given initial guess and the next initial guess cannot be improved from the previous failed one. These difficulties must be overcome if this method is to be used as a universal means of calculating LLE and the key is a deep understanding of the solution structure of the isoactivity equations. This has been the motivation of the study in this thesis. A comprehensive analysis of the solution structure of isoactivity equations using the NRTL model is presented in Chapter 2.
1.4 Thermodynamic Models
Since the 1950s, intensive attention has been given to the development of thermodynamic models to quantify deviation of liquid mixtures from ideal solution. The deviations are usually characterised by activity coefficients. These models are
21 mainly based on developing excess-Gibbs-energy equations[41, 45, 57-59] and cover both aqueous electrolyte and organic non-electrolyte systems. A review of these kinds of models is given by Prausnitz et al.[59]. Initially these models are correlative and require a group of binary interaction parameters to be regressed from available experimental data, and then the correlations can be extended to conditions beyond where the data for parameter regression are obtained. Gradually these binary parameters are generalized to a wider range of data or are attributed to basic functional groups of molecules thus entitling these models with predictive capabilities. In addition, thermodynamic models based on quantum mechanics are emerging with the development of powerful computers. In this section, we briefly review some of the popular thermodynamic models that are used in the modelling of solvent extraction, including correlative, predictive and quantum-mechanics-based models.
1.4.1 The SXFIT
The SXFIT is the latest version of a series of computer programs developed by Baes[44] to model complex solvent extraction systems. The SXFIT uses the Pitzer[40-42, 60, 61] and the Hildebrand-Scott models[44] to calculate the activity coefficients of species in the aqueous and the organic phase respectively.
The Pitzer model. Among all the models for electrolyte solutions, the Pitzer model[40- 42, 60-62] is perhaps the most commonly used one. In this model, the total excess Gibbs E energy G for a solution containing nw kg of solvent (water) and ni, nj, …, moles of solute species i, j, …, is given by[40]
G E 11 nw f I λ ijI n ij n 2 μ ijkijkn n n ... (1.41) RT nwwij n ijk and the ionic strength I is
1 2 I mii z (1.42) 2 i where zi is the number of charges on the ith solute and mi is the molality of the ith solute, which is given as mi=ni/nw. f(I) is a function of ionic strength representing the effect of long-range electrostatic forces; λij(I) is also a function of ionic strength and it indicates the effect of short-range forces between species i and j. A term for triple ion interactions, μijk, which may be significant at high concentrations, is included but any
22 dependence of μijk on ionic strength is ignored. The λ and μ matrices are assumed to be symmetric, that is, λij=λji.
Differentiation of equation (1.41) over ni gives activity coefficient of species i.
ex 1 G ln γi (1.43) RT ni
Expansion of Eq. (1.43) with respect to cation M, anion X and neutral species N are respectively as follows:
2 ln γMz M F m a22 B Ma ZC Ma m c Mc m aMcaψ a c a (1.44)
mmaaMaaψ z M m caca m C 2 m nnMλ ... a a c a n
2 ln γXz X F m c22 B cX ZC cX m a Xa m cψ cXa c a c (1.45)
mmc cψ cc X z X m c m a C ca 2 m nλ n X ... c c c a n
lnγN2m cλ Nc m aλ Na m nλ Nn ... (1.46) c a n where the subscripts M, X and N refer to the activity coefficient of the cation, anion and neutral species to be calculated respectively, and subscripts c, a and n refer to the cation, anion and neutral species presented in the mixed electrolyte solution respectively, Φij and Ψijk are the mixing parameters. The quantity F includes the Debye–Hückel term fγ and they are defined as follows:
γ F f mc m a B ca m c m c cc m a m a aa (1.47) c a c c a a
γ I 2 f=1 A ln b I (1.48) 1 bI b
Here Aϕ is the Debye–Hückel coefficient, Aϕ=0.3915 and b=1.2 at 25 °C.
(0) (1) (2) Φ , ca and Cca contain the single salt parameters βca , βca , βca , Cca׳The terms Bca, B and they are expressed in the following equations:
0 1 2 Bca=β caβ cag α12I β ca g α I (1.49)
23 2 g x = 1 1 xexp x (1.50) x2
1 B= β12 g α I β g α I (1.51) caI ca 12 ca
2 x2 g x =2 1 1 x exp x (1.52) x 2
Cca Cca = (1.53) 2 zzca where α1 and α2 are coefficients. The function Z is defined as:
Z= mzii (1.54) i
The above activity coefficients are related to the osmotic coefficient of water, which E is the derivative of G over nw, as expressed in the following way:
A I3// 2/ 1 bI 1 2 m m B ZC c a ca ca c a 2 1 m m m ψ m m m ψ cc cc acca aa aa ccaa (1.55) mi c c a a a c i 1 2 mmn cλ nc mm n aλ na mm n nλ nn m nλ nn ... n c n ann 2 n
The osmotic coefficient of water and water activity has the following relationship[44]:
lnaMw 0 . 001 w mi (1.56) i where aw and Mw are activity and molecular weight of water respectively.
With the water activity or osmotic coefficient of an electrolyte solution, the interaction parameters in the Pitzer model can be regressed, which could be further used to characterise other solutions that contain these electrolytes. Parameters for a collection of single salt and mixed electrolyte solutions have been regressed by Kim et al.[43, 63] and these parameters are good references.
The Hildebrand-Scott model. This model is used to calculate activity coefficient of species in the organic phase in the SXFIT. The formula is given in Eq. (1.57). The
24 first term on the right is adopted from Hildebrand and Scott[64], while the last three terms represent the effect of differing molecular volumes to activity coefficient[44].
2 VVVi i i lnγiiδ δ ln 1 (1.57) RTVV
V xVii (1.57a) i
δ φiiδ (1.57b) i
xVii φi (1.57c) xVii i
1/2 3/2 1/2 3/2 where Vi is the molar volume, δi is Hansen solubility parameter (J cm , 1 J cm = 1/2 2.0455 MPa ) as discussed in section 1.1.2, xi is the mole fraction of species i
(solute or solvent) and φi is volume fraction.
Based on molar volume and solubility parameters, which are available in literature for most solvents and many solutes, the Hildebrand-Scott model is easy to use. However, when solubility parameters are not available, for instance when new complexes are formed in solvent extraction, fitting solubility parameters of these complexes makes the model more arbitrary since the fitted Hansen solubility parameters do not necessarily represent the true values. On other hand, the Pitzer model for the aqueous solution is not accurate for high concentrations although it is good for moderate concentrations. In addition, two types of reference states are used respectively for the Pitzer model (type III, see table 1.4) and the Hildebrand-Scott model (type I, see table 1.4), which may bring confusions to the equilibrium constant of extraction reactions. Overall, SXFIT is a useful program for modelling solvent extraction and it has been applied to the modelling of a range of solvent extraction systems including the extraction of water and hydrochloric acid by tri-n-butyl phosphate (TBP)[65], [66] [67] extraction of ZnCl2 by TBP , separation of cobalt and nickel amongst others.
1.4.2 Wilson Model
The Wilson model[57] uses the concept of “local composition” and derives the equations on the basis of two assumptions. First, it is assumed that the free energy of mixing is similar to the Flory-Huggins equation:
25 g M xiiln ζ (1.58) RT i where ζi is the “local” volume fraction of component i about a central molecule of the same type.
Second, the distribution of molecules about a central molecule is assumed to comply with the following rule:
x xjexp g ji RT ji (1.59) xki x kexp g ki RT where xji is the “local” mole fraction of component j around i and gji is the interaction energy between molecules j and i.
The free energy of mixing can be derived as a function of mole fraction xi and interaction energy gji. Then the excess energy of mixing is expressed as:
g E xiln1 x j A ji (1.60) RT ij and Aji is defined as
(1.61) Aji1 V j V iexp g ji g ii / RT where Vj is the molar volume of component j, Aji ≠ Aij and gji = gij .
The activity coefficient of component i can be obtained as follows by differentiation of Eq. (1.60).
lnγi ln1 x j A ji 1 x j 1 A ij / 1 x k A kj (1.62) j j k
The Wilson model has two binary interaction parameters for each pair of components and is able to predict multi-component properties from binary data. However, the representation of systems with negative deviations from ideality is not as good as that of systems with positive deviations. More importantly, the Wilson model is unable to predict limited miscibility thus should only be used for completely miscible systems where a single liquid phase is present[34]. This drawback significantly limits its usage in solvent extraction.
26 1.4.3 The NRTL Model and Its Expansions
The NRTL model. The Wilson model has been shown to be useful for a variety of single phase liquid mixtures. However, the model has two arbitrary steps in the derivation: (1) the relation between the local mole fractions and (2) the introduction to the Flory-Huggins equation[45]. The NRTL model[45] adapts the concept of “local composition” and derives the equation of excess Gibbs energy in a somewhat more rigorous way.
The non-randomness of mixing is taken into consideration in the NRTL model and the relation between the local mole fractions is given by modifying Eq. (1.59) for a binary mixture:
x x exp α g RT 21 2 12 21 (1.63) xx11 1exp α 12g 11 RT where α12 is a constant characteristic of the non-randomness of the mixture. Interchanging subscripts 1 and 2, we also have
x x exp α g RT 12 1 12 12 (1.64) xx22 2exp α 12g 22 RT
The local mole fractions are related by
xx21 11 1 (1.65)
xx12 22 1 (1.66)
Rearrangement of Eqs. (1.63) to (1.66) yields the local mole fractions
x2expα 12 g 21 g 11 RT x21 (1.67) xx1 2exp α 12 g 21 g 11 RT
x1expα 12 g 12 g 22 RT x12 (1.68) xx2 1exp α 12 g 12 g 22 RT
The two-liquid theory of Scott which assumes that there are two types of cells in a binary mixture—one for molecules 1 and one for molecules 2 (as shown in Figure 1.7)—is used. The residual Gibbs free energy for cells containing molecules 1 and molecule 2 at their centres are respectively given by
27 1 g x11 g 11 x 21 g 21 (1.69)
2 g x12 g 12 x 22 g 22 (1.70)
Figure 1.7 Two types of cells in a binary mixture according to two-liquid theory
The residual Gibbs free energy of transferring x1 molecules from a cell of pure liquid
1 into a cell 1 of the solution and that of transferring x2 molecules from a cell of pure liquid 2 into a cell 2 of the solution are the two contributions to the molar excess Gibbs energy. We therefore have
E 12 g x1 g g 11 x 2 g g 22 (1.71)
Substitution of Eqs. (1.65)-(1.70) into Eq. (1.71) produces the molar excess Gibbs free energy of a binary mixture, which is readily generalized to that of solutions containing multiple components, as follows:
N τ Gx E N ji ji j g j1 xi N (1.72) RT i1 Gxki k k 1
τ ji g ji g ii RT (1.73)
G jiexp α jiτ ji RT (1.74) where N is the number of components in the system, τji ≠ τji and αji = αji.
The activity coefficient is obtained by appropriate differentiation of Eq. (1.72):
28 N N τ Gx x τ G ji ji j N xG l lj lj j1 j ij l1 ln γiNNN τ ij (1.75) j1 Gki x k G kj x k G kj x k k1 k 1 k 1
In this model, each pair of molecules have two adjustable energy parameters (τji and τji) and one non-randomness parameter (αji) that can be either adjustable or fixed to characterise the non-ideality of solutions. The model is able to correlate VLE and LLE systems with satisfactory accuracy and has the capability to predict equilibria of ternary systems from binary data, including strongly non-ideal mixtures, especially partially immiscible systems. Besides, LLE of quaternary and even quinary systems have been predicted successfully with parameters fitted from ternary LLE data[33, 68]. Furthermore, a recent study generalised the NRTL model via a theory-framed quantitative structure-property relationship (QSPR) modelling approach and thus provided the model with more predictive capabilities. The newly developed QSPR model used 30 significant descriptors as inputs and was claimed to have smaller prediction errors than the UNIFAC-1988-LLE model[69] for binary systems.
The eNRTL model. The electrolyte NRTL (eNRTL) model[46, 70, 71] expands the NRTL model to cover electrolytes by taking into account the long range forces induced by the charged species. The presence of both molecular species and ionic species introduces two extra types of interactions, that is, ion-ion and ion-molecular interactions, in addition to molecular-molecular interactions. To characterise the interaction forces in the electrolyte systems, two basic assumptions are made: (1) the local composition of cations around cations is zero, and similarly for anions, which is equivalent to assuming that repulsive forces between ions of like charge are extremely large; (2) the distribution of cations and anions around a central solvent molecule is such that the net local ionic charge is zero. These two assumptions are referred to as like-ion repulsion and local electroneutrality respectively. We thus have three types of cells in an electrolyte system with cation, anion and molecule at the centre, respectively, as shown in Figure 1.8.
The eNRTL model considers the excess Gibbs free energy as the sum of two contributions, one is the short range forces between each pair of all the species and the other is the long range forces between each pair of ions. The short range force is accounted by the same local composition treatment as the NRTL model using the pure
29 solvent and “pure completely dissociated liquid electrolyte” (the reference state may be hypothetical or it may actually exist) as reference states for molecules and electrolytes. While the long range force is dealt with by the Pitzer-Debye-Hückel (PDH) expression which uses the infinite dilution as reference state. Hence, the short range force is required to be normalized by infinite dilution to obtain an unsymmetric expression which would be consistent with the long range force. In terms of the interaction parameters, each pair of species (cation c, anion a and molecule m) are characterised by two energy parameters and one non-randomness parameter. This treatment results in significant complexity making it difficult to use. However, the model has been a bridge for further development of a more robust model, that is, the symmetric eNRTL model.
Figure 1.8 Three types of cells in an electrolyte system
The Symmetric eNRTL model. This model retains the like-ion repulsion and local electroneutrality hypotheses. It is a symmetric reformulation of the eNRTL model with the reference states chosen to be pure liquids for solvents and pure fused salts for electrolytes and these reference states are consistently used for both local interaction term and long range interaction term.
The local interactions are accounted for by the local composition model similar to the NRTL model and the long range ion-ion interactions are accounted for by the symmetric PDH formula which is extended to multi-component electrolyte systems. Consequently the excess Gibbs free energy is written as follows:
GGGE E,, lc E PDH (1.76) where GE,lc and GE,PDH represent the contributions from the local and long range interactions respectively. Accordingly, the activity coefficient is written as:
30 lc PDH lnγi lnγ i ln γ i , (i=m, c, a) (1.77)
Each of the two terms will be discussed in the following sections.
The excess Gibbs free energy from local interactions of a single salt electrolyte system can be expressed as:
XG τ XG τ XG τ G E, lc iimim iicic iiaia XXXi i c i a (1.78) m c a nRTm Xi G im c X i G ic a X i G ia i i c i a with
ni Xi C i x i C i , (i=m, c, a) (1.79) n
n ni n m n c n a (1.80) i m c a where the first, second and third term in Eq. (1.78) is the contribution when a molecule, a cation and an anion is at the centre of a cell, respectively, as shown in
Figure 1.8. In Eq. (1.79), Ci=zi (charge number) for ionic species and Ci=1 for molecular components. G and τ are local binary quantities related to each other by the NRTL non-randomness parameter α:
G exp ατ (1.81)
The activity coefficient of component i from local interactions can be derived from differentiation of Eq. (1.78) and the results are:
XG τ XG τ i im imXG i im im ln γ lc i m mmτ i m mm XGXGXGi imm i im i im i i i (1.82) XG τ XG τ XGXGi ic ic i ia ia c mcτ i c a ma τ i a mc ma caXGXGXGXGiic iic iia iia i c i c i a i a
XG τ 1 XG i im im ln γ lc m cmτ i c cm zcm X i G im X i G im ii (1.83) XG τ XG τ i ic icXG i ia ia + i c a caτ i a ca XGXGXGi ica i ia i ia i c i a i a
31 XG τ 1 XG i im im ln γ lc m amτ i a am zam X i G im X i G im ii (1.84) XG τ XG τ i ia iaXG i ic ic + i a c acτ i c ac XGXGXGi iac i ic i ic i a i c i c
There are three types of binary interactions in the model: molecule-molecule, molecule-electrolyte and electrolyte-electrolyte. Accordingly, the non-randomness parameters and energy parameters include:
(a,ca (1.85׳a=αc׳ca, αca,c,׳αca=׳m, αm,ca=αca,m, αca,ca׳αm=׳αmm
(a,ca (1.86׳a≠τc׳ca, τca,c,׳τca≠׳m, τm,ca≠τca,m, τca,ca׳τm≠׳τmm
However, the Eqs. (1.82)-(1.84) require the following binary parameters:
αmc, αma, αcm, αam, αac, αca (1.87)
τmc, τma, τcm, τam, τac, τca (1.88)
These parameters are calculated from the parameters in Eqs. (1.85) and (1.86) through a group of mixing rules, details are given by Song et al.[47].
For electrolyte systems containing multiple electrolytes, the symmetric reference state can be generalised from pure fused salt as follows:
γca x m 01 (1.89)
This is a hypothetical symmetric reference state where the multi-component electrolytes form an “ideal solution”. Thus the activity coefficient of ionic species in a multi-component electrolyte system can be normalized as:
0 lc lc,, Ixx lc I lnγc lnγ c ln γ c (1.90)
0 lc lc,, Ixx lc I lnγa lnγ a ln γ a (1.91) with
0 lc,, Ixx lc I lnγc ln γ c x m 0 (1.92)
0 lc,, Ixx lc I lnγa ln γ a x m 0 (1.93)
32 lc, I x lc, I x lc lc where ln γc and ln γa have the same formula as ln γc and ln γa shown in the
0 Eqs. (1.83) and (1.84), Ix is the ionic strength and Ix represents Ix at the reference states.
To account for the long range ion-ion interactions, the symmetric PDH formula[72] has been extended from single electrolyte systems to multi-component electrolyte systems. The excess Gibbs free energy due to the long range ion-ion interactions for single electrolyte system is given as:
ex, PDH 4AI 1 ρI 12/ G x x ln 12/ (1.94) nRT ρ 1 ρ I 0 x with
32/ 12/ 2 1 2πN A Qe A = (1.95) 3 v εkTB
12 1 2 1 2 Ix x i z i x c z c x a z a (1.96) 2i 2 c 2 a where Aϕ is the Debye–Hückel coefficient, Ix is the ionic strength, ρ is the closest approach parameter, NA is Avogadro’s number, Qe is the electron charge, kB is the Boltzmann constant, v is the molar volume and ε is the dielectric constant of the solvent.
0 The term Ix in Eq. (1.94) is based on a single electrolyte with its pure fused salt as the reference state and it is generalised to multi-component electrolyte system with the state expressed in Eq. (1.89) as reference state as follows:
011 0 2 0 2 Ix x c z c x a z a (1.97) 22ca with
0 xc xc (1.98) xxca ca
0 xa xa (1.99) xxca ca
33 The activity coefficient can be derived by differentiation of GE,PDH, the results for molecular and ionic components are, respectively, as follows:
32/ PDH 2A I x ln γi 12/ , (i=m) (1.100) 1 ρI x
2z 2 1ρI 1/// 2 z 2 I 1 2 2 I 3 2 i x i x x ln 12/ 12/ 0 ρ 1 ρ I 1 ρI x PDH x ln γ A , (i=c, a, ) (1.101) i 12/ 0 0 2IIxx I x 12/ n 0 1 ρ I ni x
I 0 The calculation of the term n x is given by Song et al.[47]. ni
Substitution of Eqs.(1.82)-(1.84), (1.90)-(1.93), (1.100) and (1.101) into Eq. (1.77) gives the general expression for activity coefficient of both ionic and molecular components in multi-component electrolyte systems. The equation reduces to the NRTL model when the concentration of electrolyte is zero. Therefore this model along with the NRTL model provides a thermodynamically consistent frame for modelling solvent extraction since both molecular and ionic components use symmetric reference states in these models. The thermodynamic consistency is an advantage compared with the SXFIT which uses inconsistent reference states for aqueous and organic phases. Besides, the NRTL model is supposed to be superior to the Hildebrand-Scott model as it is able to deal with highly non-ideal systems, this is perhaps the reason why the NRTL model is more popular. Moreover, the symmetric eNRTL model is more versatile in terms of characterising mixed systems in the presence of both molecular and ionic components. Despite the advantages of the symmetric eNRTL model, further development is required to simplify the equations. .
1.4.4 The UNIQUAC and UNIFAC Model
UNIQUAC model. A few years after proposal of the NRTL model, another model was derived by extending and generalizing the quasi-chemical theory of Guggenheim. The resulting universal quasi-chemical equation (UNIQUAC)[58] turned out to be applicable to a wide range of non-electrolyte systems, including polymer solutions and the model reduces to several well-known equations including the NRTL and
34 Wilson model when well-defined simplifications are made. The excess molar Gibbs free energy of the UNIQUAC model has two parts, namely a combinatorial part and a residual part. The combinatorial part describes the dominant entropic contribution and depends on the composition and the sizes and shapes of the molecules. Thus, it requires only pure-component data which is given as constant. The residual part accounts for the enthalpy of mixing and depends also on the intermolecular forces on top of the composition and pure-component data. Therefore, the residual part contains the two binary adjustable parameters for each molecular pair. The equation for the UNIQUAC model is
E EE g ggcombinatorial residual (1.102) RTRTRT with
E g NN* combinatorial iiz θ =xiln q i x i ln * (1.103) RTii11 xii2
E g NN residual = qxi ilnθ jτ ji (1.104) RT ij11 and
uuji ii τ exp (1.105) ji RT
Here τji (τji≠τij) is binary parameter for a j-i pair of molecules, uji is the energy of interaction, z is the coordination number and is usually set to be 10, parameters r, q are pure-component molecular-structure constants depending on molecular size ׳and q Anderson empirically .׳and external surface areas. In the original equation, q=q to obtain a better fit to systems containing water and lower ׳adjusted the values of q alcohols that involve hydrogen bonding[73]. For components other than water or lower .applies ׳alcohols, q=q
* are given by ׳The segment fraction Φi and area fractions θ and θ
* rxii i = N (1.106a)
rxjj j1
35 qxii θi = N (1.106b)
qxjj j1
qx ii θi = N (1.106c) qxjj j1
According to the above equations, the activity coefficient can be written as:
cr lnγi lnγ i ln γ i (1.107)
**N c iz θ i i lnγi ln q i ln * l i x j l j (1.108) xxi2 i i j1
NNθ τ r j ij lnγiqq i i ln θ jτ ji q i N (1.109) jj11 θkτ kj k 1 where the superscripts c and r in Eq.(1.107) refer to combinatorial and residual respectively, and
z lj r j q j r j 1 (1.110) 2
The UNIQUAC model uses only two adjustable binary parameters for each pair of molecules, albeit it requires pure-component constants. The model is applicable to a wide range of mixtures for both VLE and LLE with accuracy comparable to that of the NRTL model. However, similar to the NRTL model, the UNIQUAC model is also a correlative model with little predictive capability.
UNIFAC model. Following the UNIQUAC model, a group-contribution method was developed. The method, namely UNIFAC model (UNIQUAC Functional-group Activity Coefficient)[74], combines the solution-of-functional-groups concept with the extension of the UNIQUAC method and contains two adjustable parameters for each pair of functional groups. Essentially the UNIFAC model is similar to the UNIQUAC model except for that the former splits the interactions between molecules to that between functional groups. This treatment significantly reduces the number of interaction parameters since the number of functional groups is much fewer than that of chemical compounds. More importantly, with parameters characterising
36 interactions between pairs of functional groups obtained from existing equilibrium systems, the model allows calculation of other systems that are not involved in the parameter regression. Therefore, the UNIFAC model is a predictive model for non- electrolytes.
The activity coefficient in the UNIFAC model also has two contributions from combinatorial part and residual part, thus Eq. (1.107) is still valid. The combinatorial part uses Eq. (1.108) directly with parameters ri and qi calculated as the sum of volume and area parameters, Rk and Qk, of functional groups, given below:
i ri υ kR k (1.111a) k
i qi υ kQ k (1.111b) k
(i) where υk is the number of groups of type k in molecules i. Group volume and area parameters Rk and Qk are obtained from the van der Waals group volume and surface area[58].
The residual part of the activity coefficient is given as:
r ii lnγiυ k ln k ln k (1.112) k
(i) where Γk is the group residual activity coefficient and Γk is the residual activity coefficient of group k in a reference solution containing only molecules of type i.
The group activity coefficient Γk is expressed as follows in a way similar to Eq. (1.109) (i) and it also holds for Γk :
(1.113) lnk Q k1 ln m mk m km n nm m m n where Θm is the area fraction of group m, and the sums are over all different groups.
Θm is calculated in a manner similar to that for θi:
QXmm m (1.114) QXnn n where Xm is the mole fraction of group m in the mixture.
The group interaction parameter Ψmn is written as:
37 UUmn nn mn exp exp aT mn (1.115) RT where Umn is the energy of interaction between groups m and n, amn (amn≠ anm) is the group interaction parameter that must be evaluated from experimental phase equilibrium data.
Group parameters obtained from a wide range of experimental phase equilibrium systems for a collection of functional groups were given by Fredenslund et al.[74]. With these parameters, activity coefficients in infinite dilution, binary and ternary VLE systems can be predicted with satisfactory accuracy. However, for LLE systems the UNIFAC model only provides approximate results[74]. The UNIFAC model has gone through a number of modifications by various researchers[75-79], both its coverage in functional groups and accuracy have been improved.
1.4.5 Conductor-Like Screening Model (COSMO)
COSMO-RS (COnductor like Screening MOdel for Real Solvents)[80, 81] is also a semi-theoretical model, whereas, it is essentially different from the models reviewed above since it is based on information obtained from quantum chemistry and does not require binary interaction parameters. As an initial step, COSMO-RS performs a quantum chemistry calculation using COSMO[82] to obtain the screening charge density σ on the surface of molecules, which is the most time consuming step and fortunately the information can be stored in a data base for subsequent use. The screening charge density σ is then combined into a statistical thermodynamic methodology to calculate the chemical potentials of molecules, which are the basis of calculations of thermodynamic properties including activity coefficients and vapour pressure.
Throughout the COSMO-RS, the interactions arise from the screening charge density of each molecule, namely the σ-profile. The σ-profile of a mixture, pS(σ), is assumed to be the sum of the σ-profiles of all the components weighted by mole fractions:
pSσ xp i i σ (1.116) i
38 Electrostatic interactions between two contacting molecules arise when they have This specific interaction energy is the .׳differing screening charge densities, σ and σ result of surface charge density mismatching, and is given as:
α 2 E σ,σ a σ σ (1.117) misfit eff 2 where aeff is the effective contact area between two surface segments and is is an ׳considered as adjustable since there is no simple way to determine its value, α adjustable parameter.
Hydrogen bonding can also be described by two adjacent σ-profiles. Hydrogen bonding donors have strong negative screening charge densities, on the contrary, hydrogen bonding acceptors have strong positive screening charge densities. The hydrogen bonding energy is given as:
E min0 ; min 0; σ σ max 0 ; σ σ (1.118) HB donor HB acceptor HB where cHB and σHB are adjustable parameters.
In addition, the van der Waals (vdW) interactions between surface segments are taken into account in the following way:
EavdW effτ vdWτ vdW (1.119)
vdW are element-specific parameters that depend only on the element׳where τvdW and τ type. These parameters have been optimized for nine elements including H, C, N, O,
F, S, Cl, Br and I. EvdW is independent of neighbourhood interactions and therefore is not interaction energy. It is added to the energy of reference state. As a result, EvdW does not contribute to activity of molecules and only influences vaporization.
The chemical potential of a surface segment with screening charge density σ in an ensemble described by the normalized σ-profile p(σ) is given by:
RT a μ σ lnp σ expeff μ σ E σ ,σ E σ ,σ dσ S S S misfit HB (1.120) aeff RT
μS(σ) is called σ-potential, it is a characteristic function of each system. μS(σ) occurs on both sides of Eq. (1.120), it can only be solved by iteration.
39 The chemical potential of a component i is the integration of the σ-potential over the surface of i:
c r c μiμ i μ i μ i p iσ μ S σ dσ (1.121)
c r where μi and μi are the combinatorial and residual contribution to the chemical c potential, respectively. μi can be expressed as:
c z Ai μiS RT λ ln ALL 12 (1.122) 2 A0 with
VVii L1 1 ln (1.123) VVSS
VAVAi S i S L2 1 ln (1.124) VAVAS i S i where Vi and Ai are the molecular volume and surface area of component i, VS and AS are the average molecular volume and average surface area of the solvent (mixture),
V0 is the partial volume of an ethylene unit, z is the coordination number. The
cHB , σHB and z) together with element-specific ,׳adjustable parameters (aeff, α [81, 83] . vdW) have been optimized using reliable experimental data׳parameters (τvdW and τ
The calculated chemical potential is a “pseudo-chemical potential” which is the standard chemical potential minus RTln(xi). In other words, it is the excess chemical potential from that of ideal solution.
The activity coefficient of a component can be calculated from chemical potential easily, as shown below:
lnγii μ / RT (1.125)
The COSMO-RS resembles the UNIFAC in the statistical thermodynamics and both of them have combinatorial and residual contributions. However, the COSMO-RS does not require binary interaction parameters determined from experimental data as the UNIFAC does, instead, it uses surface charge densities obtained from quantum chemistry calculations. In the core region of parameterization of the UNIFAC model where a wealth of reliable data is available, COSMO-RS is inferior to the UNIFAC in
40 accuracy. However, COSMO-RS is capable to predict behaviour of systems where rare or even no data is available since it is free of interaction parameters. Parallel to the COSMO-RS, a similar model, namely COSMO-SAC (SAC denotes segment activity coefficient)[84-87], is also developed based on the same principle, that is, the surface charge density of molecules. Both of them are good tools for fast solvent screening.
1.4.6 Summary of Thermodynamic Models
A few popular thermodynamic models covering non-electrolyte and electrolyte species have been reviewed in this section. The Wilson model first used the local composition concept and satisfactorily correlated a variety of VLE systems with two binary parameters for each pair of molecules, however, it failed to correlate LLE systems where the behaviour of components are highly non-ideal. The inheritor of the local composition concept, the NRTL model, incorporated the two-liquid theory and introduced the non-randomness parameters, it successfully correlated both VLE and LLE systems including highly non-ideal systems. The expansions of the NRTL model, particularly the symmetric eNRTL model, extended the model to electrolyte components with thermodynamically consistent reference states. Therefore, the combination of the NRTL model with the symmetric eNRTL model is a good framework for modelling both non-reactive and reactive LLE systems containing both electrolytes and non-electrolytes.
The SXFIT uses the Hildebrand-Scott model and the Pitzer model for the organic and aqueous phase respectively. The Hildebrand-Scott model might be inferior to the NRTL and the UNIQUAC model in terms of correlating data. However, its simplicity of only requiring the Hansen solubility parameters makes it attractive for modelling reactive solvent extraction where the binary interaction parameters involving complexes formed by extractant and metal ions are not available. On the other hand, treating Hansen solubility parameters of the complexes as adjustable parameters in SXFIT introduces more uncertainty to the modelling considering that the fitted Hansen solubility parameters may not represent their real values. The Pitzer model is popular for modelling the activity coefficients of electrolytes, but is not suitable for non-electrolytes. Therefore, it cannot be used for mixed solvents. Additionally, the model requires ternary parameters at high ionic strength. Furthermore, when
41 combined with the Hildebrand-Scott model in modelling solvent extraction, the reference states for the two phases are inconsistent.
The UNIQUAC model is established on the basis of quasi-chemical analysis with two adjustable parameters for each pair of molecules. It is comparable to the NRTL model in terms of correlating equilibrium data. It is also the basis for development of the UNIFAC model, which is a predictive model characterising the non-electrolyte systems by distributing the interactions to functional groups. With well regressed parameters from a wide range of experimental data, the UNIFAC model usually produces satisfactory predictions.
COSMO-RS has similarity with the UNIFAC model in terms of statistical thermodynamics, however, it is supposed to be more rigorous than other models because it starts from the surface charge density of molecules obtained from quantum chemistry calculations. Besides, it does not require binary interaction parameters and theoretically can predict systems involving any component (non-electrolytes) regardless of whether experimental data is available or not.
In summary, these models differ in the following ways:
(1). The Pitzer model and the symmetric eNRTL models are suitable for electrolyte systems, whilst the other models presented are suitable for non- electrolyte systems;
(2). The Pitzer model uses unsymmetric reference state (infinite dilution), whilst the other models presented use symmetric reference state (pure component);
(3). The UNIFAC model and COSMO-RS are predictive models, whilst the other models presented are correlative;
(4). The COSMO-RS combines information from quantum chemistry calculations with statistical thermodynamics, the others are based only on statistical thermodynamics;
(5). The Hildebrand-Scott model and the COSMO-RS model do not require interaction parameters whilst the others do.
42 When modelling systems containing only non-electrolytes, any of these models can be used except for Pitzer model, albeit their capabilities differ in terms of handling strongly non-ideal systems. However, either the Pitzer model or the symmetric eNRTL model should be used for characterising electrolyte systems. For the modelling of solvent extraction systems containing both electrolytes and non- electrolytes, two separate models for electrolyte and non-electrolyte systems must be used. Considering that the NRTL model is perhaps the most popular thermodynamic model with good accuracy for a wide range of non-electrolyte equilibrium systems and the symmetric eNRTL model is its consistent expansion to electrolyte systems, combination of these two models can be a good framework for the modelling of solvent extraction systems, no matter if it is reactive or nonreactive. Preference of the symmetric eNRTL model over the Pitzer model is because the latter uses unsymmetric reference state which is inconsistent with other models, and also it requires ternary interaction parameters for high ionic strength solutions. Throughout this study, the NRTL model is used for non-electrolyte systems and the symmetric eNRTL model is applied to aqueous electrolyte systems.
1.5 Summary and Thesis Outline
Solvent extraction is the distribution of a solute between two immiscible solvents and it is a thermodynamic equilibrium. Reactive and nonreactive solvent extractions are two types of solvent extraction and the difference is whether there are chemical reactions at the interface of the two phases. The principles of the modelling of solvent extraction following chemical thermodynamics and a few popular thermodynamic models have been reviewed. The focus of this thesis is calculating phase equilibrium of nonreactive solvent extraction. The NRTL model and the symmetric eNRTL model have been chosen for this study and the rationality for this selection has been given.
Minimization of the Gibbs free energy and solving the isoactivity equations are two approaches for calculation of phase equilibria. The first means has been well established and widely used, however, it requires significant computational efforts. The second approach, despite its simplicity, suffers from obtaining erroneous solutions and thus is less preferred. The aim of this study is to make the equation- solving method feasible by investigating the characteristics of solutions of these isoactivity equations, while maintaining its simplicity.
43 This thesis has three main chapters. In chapter 2, the solution structure of isoactivity equations using the NRTL model is investigated through a serious of novel analyses, which lead to a new procedure for identifying correct solution of LLE. Chapter 3 optimizes the NRTL parameters using the particle swarm optimization (PSO) and discusses the model’s capabilities and limitations. Following these studies, chapter 4 applies the NRTL model and the symmetric eNRTL model to the modelling of phenol extraction in the presence of sodium hydroxide, which is important in both pharmaceutical and environmental engineering. Finally, a brief summary of the thesis is given in chapter 5.
44 Chapter 2
Analysis of the Nonrandom Two-Liquid Model for Prediction of Liquid–liquid Equilibria
This chapter investigates the solution structure of isoactivity equations using the NRTL model for liquid–liquid equilibria (LLE) calculations. Two types of mass balance constraints to the isoactivity equations, specifying concentrations of (N-2) components in one phase and specifying the total amount of each component, are considered. The isoactivity equations are analysed by plotting them in a three dimensional space, followed by a series of mathematical calculations. Three categories of solutions are revealed under the first constraint, and two categories of solutions are identified under the second constraint. The solution structure categorization (SSC) analysis leads to development of two procedures for determining correct phase equilibrium solutions. These procedures are shown to be effective for a wide range of LLE systems.
45 2.1 Computational Method
Section 1.3 introduced two approaches for calculation of LLE: (1) minimization of the Gibbs free energy and (2) solving isoactivity equations under mass balance constraints (or the K-value method). Minimization of the Gibbs free energy is the necessary and sufficient condition for phase equilibrium, but the global minimum is difficult to obtain and calculations can be time consuming and tedious[37, 38, 52], additionally the incorrect phase distributions may arise if local or constrained minima are obtained[52]. The tangent plane distance (TPD) criteria summarized by Michelsen[54] proved to be a powerful method to determine phase stability. Thus the minimization of Gibbs free energy combined with TPD method to determine the correct phase equilibrium became a popular procedure[51]. Stability test and minimizing the Gibbs free energy is a stepwise process to find a stable solution[51, 55]. Minimizing the Gibbs free energy can be computationally expensive[49] due to complexity of the nonlinear thermodynamic models and the number of variables. The TPD stability test is also a challenging minimization problem as it requires locating all the stationary points of the function of the distance between the Gibbs free energy surface and the tangent plane (F(ξ), see section 1.3.1) of a composition. Overall, this approach is computationally demanding.
Solving isoactivity equations under mass balance constraints is an alternative and much easier method in terms of computational effort, however, its results depend on the initial guess and thus the method suffers from possible erroneous solutions, which correspond to local minima, maxima or saddle points of the Gibbs free energy[37, 49]. Checking the convexity of the Gibbs free energy surface suggested by Sørensen et al.[37] excludes maxima and saddle points, but fails to rule out local minima, as demonstrated in the following discussions. As a result, the equation-solving method has been largely abandoned in favour of minimizing the Gibbs free energy combined with the TPD method. The TPD method can also be used to test the stability of results yielded by the equation-solving method, however, the next trial initial guess would not be improved from the previous one if it had failed. Therefore the equation-solving loop with trial initial guess and stability test by TPD method would be tedious and depends on how good the initial guess is. This difficulty is caused by the complexity of the equations and lack of knowledge of the solutions of the equations. To make the
46 equation-solving method feasible, a thorough understanding of the structure of all the solutions of isoactivity equations is indispensible. This chapter aims to investigate the structure of solutions of the isoactivity equations and make the equation-solving method work while maintaining its simplicity.
2.1.1 Equations for Computation
The isoactivity equations for LLE have been given in Eq. (1.38), here we rewrite it in Eq. (2.1):
I I II II xiγ i x iγ i (2.1)
For a ternary LLE system, Eq. (2.1) expands to be:
I I II II x1γ 1 x 1γ 1 (2.1a)
I I II II x2γ 2 x 2γ 2 (2.1b)
I I II II x3γ 3 x 3γ 3 (2.1c)
As it has been discussed in section 1.3.2, there are two types of mass balance constraints for the isoactivity equations: (1) specifying concentrations of (N-2) components in one phase and (2) specifying total amount of each component. The solution structures of these isoactivity equations under the two types of constraints are discussed in the following sections.
2.2 Specifying Concentrations of Components
Consider a two-phase LLE system, the mass balance constraints can be written as follows by specifying concentrations of (N-2) components in one phase:
N I xi 1 (2.2a) i1
N II xi 1 (2.2b) i1
For a system with N components and two phases, we have 2N variables, which are mole fractions of each component in any of the two phases. We specify concentrations of (N-2) components in one phase and then we have (N+2) equations (N equations of isoactivity and two equations for constraints) with (N+2) variables. Thus, the system
47 is fixed. These isoactivity equations and mass balance constraints are the basic equations for both correlation and prediction of LLE. In terms of a ternary system, we have five equations and five variables (concentration of one component in one phase is specified); as for a quaternary system, we have six equations and six variables (concentrations of two components in one phase are specified). The thermodynamic model used in this study is the NRTL model and its equations have been given in section 1.4.3.
Calculation of LLE involves solving isoactivity equations under mass balance constraints. In particular for a ternary system, Eqs. (2.1a) to (2.2b) are to be solved with a specified concentration of one component in one phase. With an initial estimation, a solution can be found by numerically solving the equations, which is then checked by convexity of Gibbs free energy surface to avoid maxima and inflection points. The detailed procedure is given by Sørensen et al.[37]. However, this convexity check is not able to distinguish the global minimum and local minima of the Gibbs free energy, the latter correspond to erroneous solutions. Therefore, a discussion on the structure of solutions of the equations above is warranted.
2.2.1 Isoactivity Equations in a 3D Space
Expansion of Eqs. (2.1a) to (2.1c) by inserting the NRTL model for activity coefficients leads to lengthy equations that are difficult to obtain explicit analytical solutions, hence only numerical solutions can be expected. These equations may be I II simplified by eliminating x2 and x3 by rearranging mass balance equations, yielding I II II three isoactivity equations with three variables of x3 , x1 and x2 . For convenience, [38] we use the system of “CCl4 (1) + (2-propanol) (2) + H2O (3)” from Sørensen et al. I as an example of experimental data, and specify x1 =0.56812, thus the system is fixed. Each of Eqs. (2.1a) to (2.1c) has multiple solutions that make a surface in the three I II II dimensional space with x3 , x1 and x2 as coordinates. The surfaces are plotted in I II II I II Figure 2.1. In the plot, x3 , x1 and x2 are allowed to range within 0~1 and x2 and x3 are not confined to be positive. As a result, some data points on the surfaces may I II correspond to negative values of x2 or x3 due to the mass balance constraint (Eq. (2.2a) and (2.2b)), however this does not influence the overall analysis. Intersection lines of any two surfaces and three surfaces together are shown in Figure 2.2.
48
Figure 2.1 Surfaces of solutions of isoactivity equations (a), (b) and (c) are the solutions of Eqs. (2.1a), (2.1b) and (2.1c) respectively.
49
Figure 2.2 Solutions of the isoactivity equations (a) Red, green and blue surfaces are solutions of Eqs. (2.1a), (2.1b) and (2.1c) respectively; intersection lines of red and green surfaces, red and blue surfaces, green and blue surfaces are shown in (b), (c), and (d); intersection lines are presented together in (e). Black and magenta points in (e) are solutions E and A in Table 2.1.
50 Intersection lines of each pair of surfaces represent the mutual solutions of the two corresponding equations, and similarly, data points on the overlap region of all the three intersection lines satisfy all the three equations and hence are solutions of the three isoactivity equations. As can be seen, the straight lines in Figures 2.2b, 2.2c and 2.2d absolutely overlap in Figure 2.2e. Data points on the overlapping straight line I II represent symmetric solutions, i.e. xi =xi . The black point E in Figure 2.2e is the I experimental data with x1 =0.56812, the magenta point A is the solution of the equations that is closest to the experimental point. In addition to symmetric solutions and the magenta point, the three lines in Figure 2.2e largely overlap in the “arc shape” II I II area. With a given value of x1 , the values of x3 and x2 that best satisfy the isoactivity equations can be found, combinations of the three values is a data point on the overlapping arc. Starting from the magenta point A, a number of such points can II II be found by repeating the procedure with a small increase of x1 (∆x1 =0.00001). We use the residuals of Eqs. (2.1a) to (2.2b) to evaluate the quality of each data point.
2.2.2 Evaluation of Solutions
The residual of each equation with respect to a solution is defined as:
value of the left side of an equation σ= -1 (2.3) value of the right side of an equation
Residual of the five equations with respect to the calculated solutions along the “arc shape” line in Figure 2.2e are shown in Figure 2.3, which jointly with Figure 2.2e reveals three categories of solutions. Firstly, three “exact solutions” are found, which are labelled as A, B and C, residuals of them are close to zero. Point A in Figure 2.3 corresponds to the magenta point in Figure 2.2e. Secondly, a number of “symmetric solutions” are found, which are represented by the straight line in Figure 2.2e. Symmetric solutions depicted by the dotted rectangular in Figure 2.3 are data points on the arc and close to point P, which is the intersection point of the straight line and the arc in Figure 2.2e. Thirdly, there are a number of “approximate solutions” found with varying residuals. Any of the solutions can be found by solving the isoactivity equations, depending on the numerical method and initial estimation used. In short, many solutions of the isoactivity equations are found and categorized into the three types described above. Hereafter, this analysis is referred to Solution Structure Categorization (SSC).
51
II Figure 2.3 Residuals of Eqs. (2.1a)~(2.2b) with respect to the solutions along x1
[38] I The system is “CCl4(1)+(2-propanol)(2)+H2O(3) ” from Sørensen et al. , x1 =0.56812.
2.2.3 Convexity of the Gibbs Free Energy
The equilibrium state is stable and unique for a given system, thus only one of the solutions described above is correct. Conventionally, checking the convexity of the Gibbs free energy is used to exclude erroneous solutions[37, 38]. The molar Gibbs free energy of mixing in one phase is equal to the sum of an ideal and an excess [37] contribution :
g gID g E xi ln x i x i ln γ i (2.4) RTRTRT ii
Convexity of a function of many variables can be determined by the Hessian matrix, which is expressed as follows for the Gibbs free energy of a ternary system in one phase.
2g 2 g 2 g 2 x x x x x 1 1 2 1 3 2g 2 g 2 g H 2 (2.5) x2 x 1 x 2 x 2 x 3 2g 2 g 2 g 2 x3 x 1 x 3 x 2 x 3
52 The Hessian matrix of a convex function is positive semidefinite[88], i.e. the leading principle minors of the Hessian matrix (H1, H2 and H3) are non-negative. The leading principle minors of the Gibbs free energy for the system of “CCl4 (1) + (2-propanol)
(2) + H2O (3)” are presented in Figure 2.4. Only when all the six lines are above zero, are the Gibbs free energy functions of both phases convex. As can be seen, a number of solutions fall in the convex region of the Gibbs free energy. As a result, convexity check of the Gibbs free energy could exclude solutions that correspond to maxima or inflection points, but this method cannot recognize the correct prediction from erroneous ones when many solutions have convex surface of the Gibbs free energy, even if the solutions differ much in values. Therefore, the prediction of LLE largely depends on the initial estimation. If the initial estimation is poor, an erroneous prediction is highly likely to be obtained.
Figure 2.4 Leading principle minors of Hessian matrix of the Gibbs free energy
[38] I The system is “CCl4 (1) + (2-propanol) (2) + H2O (3)” from Sørensen et al. , x1 =0.56812. When all the six lines are above zero, the Gibbs free energy functions of both phases are convex. Convex regions are pointed out by arrows.
2.2.4 Development of a New Procedure
To determine the correct prediction, a new procedure is required. From the residuals presented in Figure 2.3, we can see that despite a large number of approximate solutions, the number of exact solutions is very limited (usually no more than three). The correct prediction should be an exact solution, and symmetric solutions can be
53 recognized easily. Based on this knowledge, we solve the equations 1000 times with randomly produced initial estimations with values between 0~1, followed by a filtering of solutions by residuals to exclude approximate solutions. 10-7 is found to be an effective residual criterion to exclude approximate solutions and adequately filterable for exact solutions. Figure 2.5 shows the solutions found by this method for the system of “CCl4 (1) + (2-propanol) (2) + H2O (3)”, excluded solutions are given the value of -0.1. The exact solutions are successfully found, together with symmetric solutions, which can be easily recognized as overlap of red circle and blue square markers. When multiple exact solutions coexist, all of them can be found using this method. In this particular case, three exact solutions are found, which are presented in Table 2.1 together with the experimental data. E is the experimental data, A, B and C are the three solutions in Figure 2.5 and correspond to points A, B and C in Figure 2.3 II respectively. In the water rich phase, the mole fraction of carbon tetrachloride (x1 ) II should be low, while mole fraction of water (x3 ) should be high, thus solution A is easily picked out as the correct prediction, which actually is very close to the experimental value. In fact, convexity check of the Gibbs free energy excludes solution B, but doesn’t exclude solution C, as can be seen from Figure 2.4. Since the correct solution is unique, convexity check is not essential in this method.
A two-phase solvent extraction system contains two immiscible or partly miscible solvents, their mutual solubility can be used to distinguish the correct solution when multiple exact solutions coexist. Consider a two-phase system with phase I being
I component 1 rich and phase II being component 3 rich. In general x3 will be much
I II II smaller than x1 and x1 will be much smaller than x3 , otherwise the two phases would merge. This is the basis of this criterion. In the cases when water is involved in a LLE system, the mole fraction of water is dominant in the water rich phase. These solubility characteristics are simple tools to exclude erroneous solutions when more than two “exact solutions” are found, as illustrated by the example of excluding solutions B and C. Moreover, checking convexity of the Gibbs free energy (the convex region is usually narrow) is a supplement to the solubility criterion. Together these criteria form strict constraints to enable the correct solution to be identified. It was found that there are usually no more than three exact solutions (in many cases only one) for a LLE system and these solutions are always significantly different, it is
54 the large difference among these solutions that makes the mutual solubility criterion useful.
This new procedure for determining the correct prediction is thus summarized into five steps:
(1). solve the isoactivity equations under mass balance constraints 1000 times with random initial estimations;
(2). eliminate approximate solutions using the residual criterion;
(3). identify the exact solutions from symmetric solutions;
(4). pick the correct prediction out according to mutual solubility of solvents;
(5). check convexity of the Gibbs free energy (optional).
The proposed new procedure performs two tasks: predicting LLE and determining the correct prediction. Although this procedure comprises five steps, the primary step is to solve the equations multiple times, the remaining steps are straightforward. Of course, the TPD method can be used to test the few exact solutions to find the correct one. The most significant contribution of this new procedure is finding the very small number of potential solutions, which can be further determined by either the mutual solubility criterion or the TPD method. This knowledge of the solution structure provides the equation-solving method with more certainty, rather than relying on trial initial guesses.
Table 2.1 Experimental data and three exact solutions
I I I II II II Solutions x1 x2 x3 x1 x2 x3 Source E 0.56812 0.31166 0.12022 0.00160 0.09869 0.89970 Exp A 0.56812 0.31325 0.11863 0.01878 0.09490 0.88632 Cal B 0.56812 0.30783 0.12405 0.22885 0.37519 0.39596 Cal C 0.56812 0.30833 0.12355 0.42487 0.36664 0.20849 Cal
55
Figure 2.5 1000 solutions of Eqs (2.1a)~(2.2b) with random initial estimations
[38] I The system is “CCl4 (1) + (2-propanol) (2) + H2O (3) ” from Sørensen et al. , x1 =0.56812.
I II I II I II (a): x1 , x1 ; (b): x2 , x2 ; (c): x3 , x3 . Overlap of blue square marker and red circle marker means symmetric solution. Excluded solutions are assigned to be -0.1.
56 The graphical analysis method for analyzing the NRTL model is readily transferable to analyzing ternary LLE systems based on other thermodynamic models, such as the UNIQUAC model[58]. However, generalization of this method to studying multiple components (more than three) is difficult. Based on information of structure of solutions revealed from ternary LLE system, the proposed procedure is capable of investigating LLE systems with multiple components. In fact, the procedure of looking for correct predictions is a process of analyzing the structure of solutions. A number of ternary and quaternary LLE systems from different sources[33, 38, 51, 89] were investigated for the structure of solutions using the proposed procedure and the NRTL model. In many cases, multiple solutions were found and failure of excluding erroneous solutions by convexity check of the Gibbs free energy was observed for both ternary and quaternary LLE. On the contrary, the procedure presented here correctly predicted LLE for all the tested systems.
Statistically, all types of solutions of isoactivity equations using any thermodynamic model can be found by solving the equations multiple times, and exact solutions are highly likely to be included because exact solutions are easier to be found than approximate ones. Therefore, the procedure is applicable to analyzing LLE systems using various thermodynamic models.
2.3 Specifying Total Amount of Each Component
In addition to specifying concentrations of components in a LLE calculation, the total quantity of each component can also be specified, which may be presented as follows.
III nii+ n ni (i=1, 2, 3) (2.6)
I II where ni and ni are moles of component i in phase I and II respectively and ni is the total moles of component i. To illustrate the use of this constraint, the same ternary LLE example as used in the above discussions has been applied by specifying the initial amounts to be: n1=0.56972 (0.56812+0.00160); n2=0.41035 (0.31166+0.09869); n3=1.01992 (0.12022+0.89970). The same thermodynamic model, i.e. the NRTL model, is used.
57 2.3.1 Categorization of Solutions
Ternary isoactivity equations has been solved under the constraints of Eq. (2.6), with initial estimates randomly produced in the range 0~ni (i=1, 2 and 3), 1000 times with results presented in Figure 2.6. Solutions that are out of range 0~ni (i=1, 2 and 3) or with large residues (residue as defined in Eq.(2.3)) are excluded and assigned as -0.1. 419 solutions are found and a subset of the solutions is also provided in Table 2.2 together with the experimental composition E. These solutions distribute quite evenly in the range of moles of components, no significant character is observed. Therefore, further analysis from the Gibbs free energy is necessary.
The Gibbs free energy values of these solutions are calculated using the following equations and results are presented in Figure 2.7.
Ggj n j ( j=I, II) (2.7) RTRT
3 jj nn i (i=1, 2 and 3; j=I, II) (2.8) i1
GGGI II (2.9) RT RT RT where R is gas constant and T is absolute temperature, the molar Gibbs free energy is given in Eq.(2.4). Interestingly the distributions of GI/RT and GII/RT are similar as if one is the reflection of another and the G/RT converges into two values: -0.40987 and -0.56729 (refer to Figure 2.7). Conversion of these solutions from moles into mole fractions (Table 2.3) reveals that all the solutions except S.1 and S.2 (Table 2.2) are in fact the same solution, i.e. solution S.B in Table 2.3. The solution S.B is a symmetric solution and it represents a homogenous single phase, this is why the Gibbs free energy values for all of these solutions have the same value. The solutions S.1 and S.2 are the same solution, with the compositions in two phases reversed. They are represented as S.A in Table 2.3. Solutions represented by S.B (S.3 onwards in Table 2.2) have larger Gibbs free energy values than the solutions S.1 and S.2 and thus can be excluded either by their symmetry or by the Gibbs free energy. Thus S.A (S.1 or S.2) is the correct solution.
58
Figure 2.6 1000 solutions of the isoactivity equations under constraints of Eq. (2.6)
I II I II I II (a): n1 , n1 ; (b): n2 , n2 ; (c): n3 , n3 . The solutions S.1 and S.2 occurred 82 and 93 times respectively and they are pointed out in the figures. Excluded solutions are assigned to be -0.1.
59
Figure 2.7 Gibbs free energy of the 419 solutions
(a): GI/RT; (b): GII/RT; (c): G/RT. The solutions S.1 and S.2 occurred 82 and 93 times in 1000 solutions respectively and they are pointed out in the figures.
60
Table 2.2 Solutions for isoactivity equations in moles
I I I II II II Solutions n1 n2 n3 n1 n2 n3 G/RT E 0.56812 0.31166 0.12022 0.00160 0.09869 0.89970 - S.1 0.55064 0.31389 0.12149 0.019078 0.096462 0.89844 -0.56729 S.2 0.019078 0.096462 0.89844 0.55064 0.31389 0.12149 -0.56729 S.3 0.19321 0.13916 0.34589 0.37651 0.27119 0.67403 -0.40987 S.4 0.024096 0.017356 0.043137 0.54562 0.39299 0.97678 -0.40987 S.5 0.067562 0.048663 0.12095 0.50216 0.36169 0.89897 -0.40987 … … … … … … … …
Table 2.3 Solutions for isoactivity equations in mole fractions
I I I II II II Solutions x1 x2 x3 x1 x2 x3 E 0.56812 0.31166 0.12022 0.00160 0.09869 0.89970 S.A 0.55845 0.31834 0.12321 0.018815 0.095133 0.88605 S.B 0.28486 0.20518 0.50996 0.28486 0.20518 0.50996
The solutions of the isoactivity equations can also be analysed by a plot in a three dimensional space using SSC analysis, as presented in section 2.2 (The Mathematica code for plotting equations has been given in Appendix D). Similar plots developed from this work are presented in Figures 2.8 and 2.9 for single solution surfaces and surfaces intersected with each other. These figures reveal the categories of solutions: (1) the straight line in the centre of Figure 2.9e represents symmetric solutions (S.B); (2) S.1 and S.2 are the same solution (solution S.A) with their compositions in two phases reversed. E is the experimental value, which is closely adjacent to S.1. In this case, there is only one “exact solution”. Examination of LLE systems from a range of data sources[33, 38, 51, 89] reveals that the plots of isoactivity equations for ternary systems are similar to Figure 2.9e and the “exact solution” is unique for both ternary and quaternary systems. In other words, the seemingly multiple solutions are in fact the symmetric solutions which can be identified easily when expressed in mole fractions. The unique “exact solution” is exactly the correct solution we are searching for a LLE system. Even if multiple “exact solutions” were found, the erroneous ones can be excluded easily by comparing the Gibbs free energy.
61
Figure 2.8 Surfaces of solutions of isoactivity equations (a), (b) and (c) are the solutions of Eqs. (2.1a), (2.1b) and (2.1c) under constraint of Eq. (2.6) respectively.
62
Figure 2.9 Solutions of the isoactivity equations under constraints of Eq. (2.6)
Red, green and blue surfaces in (a) are solutions of Eqs. (2.1a), (2.1b) and (2.1c) under constraint of Eq. (2.6) respectively; intersection lines of red and green surfaces, red and blue surfaces, green and blue surfaces are shown in (b), (c), and (d); intersection lines are presented together in (e), black and magenta points in (e) are solutions E, S.1 and S.2 in Table 2.2, the straight line in the centre represents symmetric solutions.
63 2.3.2 A New Calculation Procedure
It is worth mentioning that the method of “minimizing the Gibbs free energy” requires knowledge of the total amount of each component. In practice specifying the total amount of each component is more common than specifying concentration of components in one phase. With known amount of each component, we can summarize the procedure for determining the LLE solution using the equation-solving approach into the following steps based on the solution structure categorization (SSC) analysis:
(1). solve the isoactivity equations under mass balance multiple (e.g. 20) times with random estimations within the amount range of all components;
(2). exclude symmetric solutions;
(3). (i) if the solution is found, end;
(ii) if the solution is not found, repeat from step (1).
It has been found that the “exact solution” occurs at least 100 times in 1000 solutions with random estimation, i.e. the possibility of finding the correct solution is over 10% for any trial. Thus 20 times is recommended in step (1) of the procedure. Generally a solution can be found within five loops and the calculation time is 0.5~2 seconds on an ordinary personal computer. This technique has been successfully applied for a range of LLE systems, selected examples are shown in Table 2.4, details refer to Appendices A and B (MATLAB code).
Table 2.4 Selected calculation examples for the proposed procedure
LLE systems Number of No. “exact Ref. Type Components solutions”
1 Heptane (1)+Benzene (2)+Ethylbenzene (3)+Methanol (4) 1 [51] 2 n-Hexane (1)+n-Octane (2)+Benzene (3)+Sulfolane (4) 1 [33] Quaternary 3 1-octanol (1) + TAME (2) + Water (3) + Methanol (4) 1 [90] 4 Water (1) + Acrylic Acid (2) + Acetic Acid (3) + Cyclohexane (4) 1 [91] 5 Water (1) + Ethanol (2) + 1,1-Difluorothane (3) 1 [92] 6 Quinoline (1) + Furfural (2) + Water (3) 1 [93] 7 Water (1) + Acetic Acid (2) + Ethyl Acetate (3) 1 [94] Ternary 8 Acetic Acid Butyl Ester (1) + 2-Propanone (2) + 1,2-Ethanediol (3) 1 [38] 9 Heptane (1) + Methane Trichloro (2) + Aniline (3) 1 [38] 10 Water (1) + Phenol (2) + Benzene (3) 1 [89]
64 2.3.3 Comparison with Minimizing the Gibbs Free Energy
When the total amount of each component is known, we have two procedures for LLE calculations: combination of minimizing the Gibbs free energy with stability test by the TPD method (procedure I) and solving isoactivity equations under mass balance constraints followed by SSC analysis (procedure II). Both procedures have two parts: looking for potential solutions and determining the correct one. These procedures differ in a number of ways, as compared in Table 2.5.
Table 2.5 Comparison of the two procedures for LLE calculations
Minimizing the Gibbs free energy combined Equation-solving followed by SSC analysis with TPD Based on the global minimum of the Gibbs free Based on isoactivity equations and their energy solution structure Necessary and sufficient condition for phase Necessary but not sufficient condition for phase equilibrium equilibrium Looks for solution in a stepwise way and repeats Produces a number of candidate solutions at the procedure until the correct solution is found once and rules out the incorrect solutions Requires robust numerical method Low requirement in numerical method
Satisfaction of isoactivity equations is the necessary but not sufficient condition for stable phase equilibrium, however, it is almost the sufficient condition when it is followed by the SSC analysis. The main advantage of the approach of equation- solving followed by SSC analysis is the ease of calculation because it does not require robust numerical method.
2.4 Conclusions
The solutions of isoactivity equations using the NRTL model for LLE calculations under two types of mass balance constraints have been studied. When concentrations of (N-2) components in one phase are specified, three categories of solutions of the isoactivity equations have been found by a graphical analysis method, namely solution structure categorization (SSC). Failure of convexity check of the Gibbs free energy to exclude erroneous predictions was observed. A new procedure for determining the correct prediction of LLE as well as to investigate structure of solutions of isoactivity equations was proposed, which is applicable to LLE systems with multiple components.
65 With known total amount of each component as constraint, the SSC analysis reveals that there is only one correct solution and a number of symmetric solutions for both ternary and quaternary systems. This finding makes the equation-solving approach for LLE calculation feasible and a procedure for determining the correct solution is also proposed. This procedure is recommended to be used as a parallel procedure to minimizing the Gibbs free energy combined with the tangent plane distance (TPD) criterion to calculate LLE.
66 Chapter 3
Regression of Nonrandom Two-Liquid Parameters Using Particle Swarm Optimization and Discussions
The interaction parameters of thermodynamic models should be properly regressed before the models can be used. Regression of these parameters deals with non-linear functions that have multiple local minima and therefore it is difficult to obtain the global minimum. This chapter investigates the application of particle swarm optimization (PSO), a global optimization algorithm, to the search of global minimum in parameter regression using the NRTL model as an example. In addition, the capabilities and limitations of the NRTL model in correlating LLE data are discussed.
67 3.1. Introduction
The nonrandom two-liquid (NRTL) model[45] has been used widely to correlate and predict vapour–liquid equilibria (VLE)[59, 95] and liquid–liquid equilibria (LLE)[38, 59, 96, 97] of non-electrolytes, which have numerous applications including distillation and liquid–liquid extraction respectively. Extensions to this model for predicting phase equilibria of systems containing polymers[98, 99] and electrolytes[46, 47] have expanded the model’s versatility. Calculations of phase equilibria using the equation-solving approach using the NRTL model have been discussed in Chapter 2. This chapter deals with the regression of NRTL parameters.
To predict phase equilibria of multiple component systems, either solving the isoactivity equations or minimizing the Gibbs free energy can be used. Both the two approaches require known binary interaction parameters of the NRTL model, which can be regressed from experimental data of either VLE or LLE. As a result, regressing the NRTL parameters is the precondition for predicting phase equilibria. While regression of interaction parameters of the NRTL model from VLE is straightforward, regressing parameters from LLE, which minimizes the difference of experimental and calculated mole fractions, is more complicated due to of the need to solve nonlinear isoactivity equations. Although some researchers have obtained small root-mean square deviations (RMSD) of mole fractions using various algorithms[37, 51, 89, 100, 101], none of these methods can guarantee obtaining the global minimum. To obtain global minimum of RMSD, a robust algorithm is essential. The particle swarm optimization method (PSO)[102] is a powerful optimization method that has attracted extensive interest in recent years and has been used across a wide range of applications[103]. Having a number of entities (particles) interacting with each other and sharing information in the search space of a function, the swarm as a whole is more likely to move close to the global minimum of the function compared with the algorithms that have been used.
Genetic Algorithm (GA), another global optimization algorithm, has been successfully applied to parameter estimation of thermodynamic models[104-108], however, the use of PSO for this application was only reported by Khansary et al.[106] recently. It has been demonstrated that PSO is more computationally efficient while having the same effectiveness as GA in terms of optimization[109]. In addition, PSO
68 outperformed GA, Simulated Annealing (SA), Pattern Search (PS) and interior point (constrained nonlinear minimization) when both efficiency and cost are considered in the case study of efficiency optimization of a squirrel cage induction monitor[110]. Therefore, PSO has been selected as optimization algorithm in this study.
This chapter aims to investigate the utilization of the PSO method to regress the parameters of the NRTL model from ternary LLE data and discuss the capabilities and limitations of the model in representing the activities of components in LLE systems.
3.2 Parameter Regression Using the PSO Algorithm
3.2.1 Particle Swarm Optimization
The PSO was first proposed by Kennedy and Eberhart for optimization of nonlinear functions by simulating social behaviours such as bird flocks searching for corn[102]. It has since attracted extensive studies from a wide variety of applications due to its flexibility and ease of use and the algorithm has undergone many variations[103, 111]. The PSO optimizes a problem by iteratively improving the locations of candidate solutions (particles) in the search space. Each particle updates its location after evaluating the objective function (fitness) of its current location and the best known location (pbest) and the best location of the entire particles (gbest). The next iteration occurs after all the particles have been moved and “pbest” and “gbest” have been updated. Gradually the particle swarm is expected to move towards an optimum like a flock of birds collectively foraging for food. A flowchart illustrating the PSO algorithm is given in Figure 3.1. The MATLAB code for the program has been given in Appendix C.
The PSO algorithm applied in this study for update of moving velocity and location of each particle is as follows:
viω v i c1 R 1 p best x i c 2 R 2 g best x i (3.1)
xi x i k v i (3.2) where vi is the moving velocity of the particle, xi is the current location, ω is inertia weight, c1 and c2 are acceleration factors, R1 and R2 are two random numbers
69 distributed in the range of [0, 1], k is constriction factor. Values of the coefficients used in this study were selected following Eberhart et al.[112].
Figure 3.1 Flowchart of the PSO algorithm
3.2.2 Optimization Method
The isoactivity equations of components in two phases are applied for calculation of LLE of ternary systems:
I I II II xiγ i x iγ i (i=1, 2 and 3) (3.3) where xi and γi are the mole fraction and activity coefficient of component i respectively, I and II represent the two phases.
Total amount of each component is used as constraint of isoactivity equations and can be represented as follows:
III nii+ n ni (i=1, 2 and 3) (3.4) where ni is the total moles of component i. For the prediction of phase equilibria, a stability test of the solution[54] satisfying Eqs. (3.3) and (3.4) based on the tangent
70 plane criterion[54] or the procedure proposed in Chapter 2 is required. This test is not necessary for regression of parameters. The equations of the NRTL model used in the calculation have been given in Section 1.4.3.
Two types of objective functions[37] are usually used to regress the parameters:
3 M 2 I I II II Fxa ik γ ik x ik γ ik (3.5) ik11
32M 2 jj Fx x ik x ik (3.6) i1 j 1 k 1
j j where i is component, j is phase and k is the number of tie lines, xik and xik are the experimental and calculated mole fraction of components respectively.
While minimization of Fa (Eq. (3.5)) is straightforward, minimization of Fx (Eq. (3.6)) is complex as it involves solving isoactivity equations under mass balance constraints. Each iteration solves the equations for each tie line to calculate and then calculates the value of Fx, which in turn determines the direction of the next iteration through the algorithm of PSO. The iteration goes on until a criterion, which is the maximum number of iterations in this study, is met. The quality of regression is generally evaluated by the RMSD.
12/ 32M 2 jj xxik ik RMSD i11 j k 1 (3.7) 6M
For convenience of evaluating the quality of correlation, Eq. (3.5) is modified into Eq. (3.8).
3 10
Fa σ ik (3.8) ik11
where σik is defined as
I I II II I I II II xikikγ // x ikik γ 1, if x ikik γ x ikik γ 1 σik (3.9) xIIγ II//x Iγ I1, if x I γ Ix IIγ II 1 ikik ikik ikik ikik
71 3.2.3 Regression Results
To further illustrate the minimization, we use the ternary system of “Ethene Tetrachloro (1) + 2-propanol (2) + water (3)”[38] (P.148) as an example. In this example, all the non-randomness parameters (α12, α13, α23) are fixed as 0.2 and the energy parameters (τ12, τ21, τ13, τ31, τ23, τ32,) are searched in the range of [-15, 15]. All the data in a LLE system are used simultaneously in this example as well as examples in the following sections. Firstly, Fa was minimized using 200 particles that are generated randomly, each particle is a group of six energy parameters of the NRTL model. An example of calculation is given in Figure 3.2. It can be seen that Fa reaches a minimum (either local or global) after approximately 20 iterations. After a number of calculations, it was determined that 60 iterations are sufficient to reach a local minimum. The minimization procedure was then repeated 300 times and the resulting 300 groups of parameters (called “pre-regressed parameters”) were evaluated over RMSD and the results are presented in Figure 3.3a. It is shown that the parameters obtained from minimization of Fa do not necessarily result in small RMSD, which is the target of parameter regression. As a comparison, 300 groups of randomly generated parameters without any further manipulation were also evaluated using Eq. (3.7) with results presented in Figure 3.3b and as shown the RMSD can be very large.
Figure 3.2 Correlation of LLE data using Fa as objective function
Following optimization of Fa, Fx was optimized with either 30 pre-regressed particles or 30 randomly generated particles and each case was repeated 10 times. The results presented in Figure 3.4 clearly indicate that the pre-regressed particles are good initial estimates for the optimization of Fx although initially they may not have sufficiently
72 small RMSDs. Nine out of the 10 calculations using pre-regressed particles reached very similar RMSD within 1000 iterations, indicating that the RMSD calculated from this group of parameters is most likely the global minimum. On the contrary, the optimizations using 30 random particles reached very different and much larger RMSDs. The comparison demonstrates that the pre-regressed particles are superior over the random ones. Therefore, it is a good strategy to minimize Fx using pre- regressed particles obtained from minimization of Fa.
Figure 3.3 Initial RMSDs (a) pre-regressed particles; (b): random initial particles
Although selection of the number of particles and iterations appears arbitrary, there is some rationale behind their selection. For the optimization of Fa, 200 particles were chosen to ensure a rapid approach to a local minimum and 60 iterations aim to ensure the minimum reached is a local one rather than the global minimum thus to maintain diversity of the pre-regressed particles that are to be used for optimization of Fx. The
73 optimization of Fx used only 30 particles because using more particles is time consuming due to the involvement of solving nonlinear equations.
Figure 3.4 Decrease of RMSD with the number of iterations (a) iteration using pre-regressed particles; (b) iteration using random particles
The optimization method was further tested by regressing ternary LLE systems from literature with 30 pre-regressed particles and 1000 iterations. Comparison of the results with literature is given in Table 3.1 and the parameters are given in Table 3.2. Whether the non-randomness parameters are treated as fixed or adjustable is based on the treatment in the literature in comparison. Smaller RMSDs were obtained using the current method which demonstrates the strength of the PSO method used in this study. The PSO method demonstrated using the NRTL model as an example is readily transferable to regressing parameters of other thermodynamic models.
74 Table 3.1 Comparison of regression results with literature values
Method/software RMSD System Ref. Literature This study Literature This study Water(1) + Phenol(2) + Benzene(3), Not Given PSO 0.0036 0.0022 P.292[89] at 298.15 K Ethene Tetrachloro(1) + Levenberg- PSO 0.0078 0.0045 P.148[38] 2-propanol(2) + water(3) at 303.15 K Marquardt Heptane(1) + Ethylbenzene(2) + Simplex PSO 0.0029 0.0026 P.62[51] Methanol(3) at 293.15K Cyclohexane(1) + Enthylbenzene(2) + ASPEN Plus PSO 0.0230 0.0034 P.1718[100] Sulfolane (3) at 303.15 K
Table 3.2 Regressed parameters of the NRTL model for ternary LLE systems
Systems i-j αij 휏ij 휏ji
Water (1) 1-2 0.24962 5.5351 -0.96053 Phenol(2) 1-3 0.25426 5.4491 3.1784 Benzene (3) 2-3 0.30332 6.8088 -0.048392
Ethene Tetrachloro(1) 1-2 0.2 7.2316 2.9827 2-propanol(2) 1-3 0.2 5.0656 5.6934 Water(3) 2-3 0.2 0.26419 9.0067
Heptane(1) 1-2 0.4 -0.94695 -0.57120 Ethylbenzene(2) 1-3 0.4 2.0569 2.4497 Methanol(3) 2-3 0.4 1.2377 0.0067217
Cyclohexane(1) 1-2 0.3 1.0378 3.6815 Enthylbenzene(2) 1-3 0.2 4.6788 1.6054 Sulfolane(3) 2-3 0.3 1.2566 2.7125
3.3 Representation of LLE by the NRTL Model
3.3.1 Local and Global Minimum
Multiple groups of parameters were obtained for the ternary system of “Ethene Tetrachloro (1) + 2 propanol (2 )+ water (3)”[38] (P.148), two of them from this study and one from literature are presented in Table 3.3. Each group corresponds to a local minimum of RMSD and the smallest RMSD is most likely the global minimum. In terms of RMSD, all three groups of parameters are good. However, parameters in different groups show no relevance. If two pairs of binary interaction parameters in
75 different groups are exchanged, much larger RMSD will be yielded and the calculated mole fractions can even be negative which does not make sense. For example, if the energy parameters 휏12 and 휏21 in “Group 1” are replaced by the corresponding values in “Group 3”, some of the predicted mole fractions using the new group of parameters are negative and the RMSD obtained is 0.5557, which is much larger than both 0.0045 and 0.0078. This reveals the NRTL model’s theoretical limitation and more discussions are given in section 3.4.
Table 3.3 Three groups of parameters for the same LLE data
System Parameters Group 1 Group 2 Group 3
휏12 7.2316 5.8391 483.32/303.15
휏21 2.9827 2.9387 -116.92/303.15
휏13 5.0656 6.4053 2022.0/303.15 Ethene Tetrachloro (1) 휏31 5.6934 7.7491 2382.4/303.15 2-propanol (2) 휏 0.26419 0.025600 291.60/303.15 Water (3) 23
휏32 9.0067 7.9771 259.50/303.15 RMSD 0.0045 0.0076 0.0078 Reference This study This study P.148[38]
3.3.2 Representation of Activities of LLE
Equality of fugacity of a component in VLE can be simplified to be[95]
Py i xiiγ 0 (3.10) Pi where yi and xi are the mole fraction of component i in the vapour and liquid phase
0 respectively, P is the total pressure and Pi is the vapour pressure of the pure component i at the same temperature. It can be seen that the activity of a component in the liquid is equal to the ratio of its partial pressure and vapour pressure of the pure component. The mole fraction xi is within [0, 1], the activity coefficient can be larger or smaller than unity, thus mathematically we cannot derive the range of activity. In fact, it is unusual to find a component whose activity is larger than unity.
76
Figure 3.5 Representation of activities by different groups of parameters
77 Unlike correlation of VLE which directly correlates partial vapour pressures of components and compositions of liquid phase, correlation of LLE equates two sides of isoactivity equations without knowing the intrinsic activity of each component. As a result, representation of activities of LLE systems by the NRTL model needs to be examined. The three groups of parameters in Table 3.3 and their corresponding predicted mole fractions are used to calculate activities and the results are shown in Figure 3.5.
Surprisingly, activities calculated by the three groups of parameters differ significantly and many data points are clearly incorrect as their values are far beyond the range of [0, 1] (Figures 3.5a and 3.5b). Even the data points within [0, 1] most likely do not represent the intrinsic activities of component as the values of activities are not considered in the parameter regression. Consequently, correlation of LLE by the NRTL model is an empirical fitting that may not reflect the intrinsic activities of components, which are the properties we aim to correlate. This drawback of the NRTL model limits its application in predicting phase equilibria using parameters regressed from LLE. Expansions of the NRTL model[46, 47, 98, 99] may also suffer from the same weakness as they are generally obtained by adding additional items to the original equations of the model while maintaining the model’s theoretical concepts and core equations.
3.4 Correlating Normalized Random Data
The activity coefficient equation of the NRTL model uses a combination of exponential functions with three adjustable parameters, providing the model with excellent flexibility in correlating data. However, it has been demonstrated in Chapter 2 that the isoactivity equations of ternary LLE using the NRTL model represent many more data points besides the experimental equilibrium data due to sensitivity of the exponential functions. Indeed random data that are normalized to resemble experimental LLE data have been generated and shown to be able to be fitted equally well as correlations of experimental LLE data. This reveals the NRTL model’s empiricism and weak theoretical basis, which should be considered when using the model to calculate phase equilibria.
78 3.4.1 Generation of Normalized Random Data
Five arrays of random data in the range of 0 to 1 were produced by the mathematical software “MATLAB”. Each array comprises ten rows and six columns of data with every three columns of data representing a phase. The data in each phase were then normalized by Eq. (3.11) such that the sum of data in a row equals 1, thus resembling an experimental ternary LLE data set, where each data point represents the mole fraction of a component in one phase.
N j xik 1 (i=1, 2, 3; j=1, 2) (3.11) i1 where i is component, j is phase number and k is the number of rows. The five sets of normalized data are given in Tables 3.4-3.8 and the randomness of these data sets is clearly shown by the phase diagrams in Figure 3.6. To evaluate the correlations, five regular ternary LLE systems that also have ten rows of data have been taken from literatures[33, 38, 89] and are correlated as comparisons (refer to Table 3.9).
Table 3.4 Normalized random data No. 1
Phase I Phase II
x1 x2 x3 x1 x2 x3 0.3868 0.4331 0.1801 0.0359 0.2842 0.6799 0.3971 0.5335 0.0694 0.9403 0.0410 0.0187 0.1333 0.6224 0.2443 0.1416 0.7276 0.1308 0.2848 0.4569 0.2583 0.2618 0.3405 0.3977 0.5170 0.4643 0.0187 0.1253 0.2266 0.6481 0.5626 0.3476 0.0899 0.1399 0.7893 0.0708 0.1501 0.5665 0.2834 0.2327 0.4206 0.3466 0.0332 0.7131 0.2537 0.1050 0.3462 0.5488 0.1277 0.3931 0.4791 0.4940 0.1841 0.3219 0.3246 0.3900 0.2854 0.0997 0.0985 0.8017
Table 3.5 Normalized random data No. 2
Phase I Phase II
x1 x2 x3 x1 x2 x3 0.5004 0.0968 0.4028 0.4969 0.3088 0.1943 0.4737 0.5076 0.0187 0.0291 0.3491 0.6218 0.0657 0.4951 0.4392 0.1631 0.4510 0.3859 0.3915 0.2081 0.4004 0.0460 0.7920 0.1620
79 Table 3.5 Normalized random data No. 2 (continued)
Phase I Phase II
x1 x2 x3 x1 x2 x3 0.2995 0.3790 0.3215 0.2410 0.4637 0.2953 0.0978 0.1423 0.7599 0.4546 0.2704 0.2751 0.1929 0.2922 0.5149 0.3308 0.2122 0.4570 0.2948 0.4937 0.2115 0.2432 0.4957 0.2611 0.3981 0.3294 0.2725 0.4233 0.3160 0.2607 0.4604 0.4579 0.0817 0.0340 0.7450 0.2210
Table 3.6 Normalized random data No. 3
Phase I Phase II
x1 x2 x3 x1 x2 x3 0.8003 0.0408 0.1589 0.5117 0.3860 0.1023 0.1278 0.6101 0.2621 0.5316 0.3167 0.1516 0.3530 0.4001 0.2470 0.3143 0.3176 0.3682 0.2882 0.1622 0.5496 0.1676 0.2852 0.5471 0.2411 0.7363 0.0226 0.3701 0.2939 0.3360 0.3294 0.5819 0.0887 0.4285 0.3980 0.1735 0.3969 0.1515 0.4516 0.0653 0.3949 0.5398 0.0476 0.8646 0.0878 0.0693 0.1473 0.7834 0.5765 0.0672 0.3563 0.3271 0.3695 0.3034 0.8353 0.0075 0.1572 0.1579 0.4580 0.3840
Table 3.7 Normalized random data No. 4
Phase I Phase II
x1 x2 x3 x1 x2 x3 0.1130 0.4734 0.4136 0.1410 0.4457 0.4133 0.3687 0.3665 0.2648 0.3476 0.3987 0.2537 0.3295 0.3704 0.3001 0.5219 0.3870 0.0911 0.5622 0.2998 0.1379 0.3578 0.4432 0.1990 0.2080 0.6424 0.1496 0.2626 0.6813 0.0561 0.3816 0.3801 0.2383 0.1317 0.5413 0.3270 0.5809 0.1051 0.3141 0.1011 0.2481 0.6509 0.2738 0.0603 0.6658 0.4057 0.4611 0.1333 0.0003 0.7269 0.2727 0.4338 0.1958 0.3704 0.0451 0.7700 0.1849 0.4539 0.0965 0.4497
80 Table 3.8 Normalized random data No. 5
Phase I Phase II
x1 x2 x3 x1 x2 x3 0.3146 0.2183 0.4671 0.3597 0.3395 0.3008 0.3517 0.2915 0.3568 0.6338 0.0725 0.2937 0.3856 0.1152 0.4992 0.1881 0.2986 0.5133 0.5042 0.0630 0.4328 0.5320 0.2341 0.2339 0.5609 0.4093 0.0297 0.2021 0.3114 0.4865 0.5180 0.3861 0.0959 0.2930 0.0603 0.6467 0.5080 0.2071 0.2849 0.4065 0.0707 0.5228 0.1920 0.3427 0.4653 0.6164 0.2453 0.1383 0.0929 0.4440 0.4631 0.0412 0.3605 0.5983 0.2152 0.3011 0.4837 0.3870 0.5936 0.0194
3.4.2 Correlation and Discussions
Fa in Eq.(3.8) was used as the objective function to correlate all 10 sets of data using the NRTL model with all three parameters adjustable. Minimization of Fa was performed by the PSO method as illustrated in section 3.2. The regressed parameters and results are given in Table 3.9 and results are also presented in Figures 3.7 and 3.8.
Interestingly, the values of Fa for the normalized random data and the regular LLE data from literature are comparable, indicating that the two types of data are correlated equally well. This means that the NRTL model does not have any advantage in correlating regular experimental LLE data over correlating normalized random data. The normalized random data does not have any physical meaning in terms of molecular interactions, hence the regressed NRTL parameters are meaningless. Similarly, the parameters for the regular LLE systems are somewhat empirical, as can be seen from derivation of the model’s equations. The weak theoretical basis of the model explains the inconsistency of different groups of parameters. Therefore, the NRTL model’s weaknesses and limitations should be considered when using it to correlate and predict phase equilibria. Further work is required to strengthen the model from two aspects: first, a better parameter estimation strategy that considers not only RMSD but also thermodynamic consistency including the range of components’ activities should be explored; second, the model’s equations can be improved by applying more reliable assumptions.
81
Figure 3.6 Phase diagrams of normalized random data
82 Table 3.9 Regressed NRTL parameters for data correlation
System i-j 휏ij 휏ji αij Fa Ref.
n-Hexane(1) 1-2 9.1825 3.4202 0.44505 Benzene(2) 1-3 7.9969 5.6226 0.42441 0.594 P111[33] Sulfolane(3) 2-3 3.3058 0.71576 0.49549
Benzene Isopropyl(1) 1-2 -15.617 3.5251 0.41036 2-propanone(2) 1-3 15.537 6.2466 0.40480 1.578 P64[38] Formic Acid Amide(3) 2-3 5.7609 -14.755 0.47162
Ethene Tetrachloro(1) 1-2 3.4885 4.5286 0.26742 2-propanone(2) 1-3 13.538 10.178 0.33408 1.844 P148[38] Water(3) 2-3 4.1020 4.3232 0.32702
Ethene Trichloro(1) 1-2 4.4067 0.87823 0.25113 Hexanoic Acid 6-amino Lactam(2) 1-3 3.7488 7.9841 0.31627 0.907 P162[38] Water(3) 2-3 1.0684 4.8880 0.34126
Water (1) 1-2 4.2195 10.272 0.19513 Phenol (2) 1-3 11.022 5.0282 0.46432 4.048 P292[89] Benzene (3) 2-3 2.0239 2.2543 0.50000 1-2 3.3317 3.4104 0.42839 Normalized Random 1-3 4.4288 3.2477 0.43793 1.075 This study Data No. 1 2-3 2.8774 2.6955 0.43179 1-2 20.000 3.4461 0.34736 Normalized Random 1-3 10.675 3.8366 0.40511 0.683 This study Data No. 2 2-3 3.0290 2.6762 0.43372 1-2 18.214 3.1006 0.23501 Normalized Random 1-3 2.2269 2.1058 0.50000 1.748 This study Data No. 3 2-3 4.2231 1.7168 0.44643 1-2 2.8988 19.844 0.23173 Normalized Random 1-3 3.0419 4.3020 0.42781 1.331 This study Data No. 4 2-3 2.8322 2.4435 0.45078 1-2 3.0110 20.000 0.30478 Normalized Random 1-3 2.8082 2.8503 0.42566 0.736 This study Data No. 5 2-3 16.338 3.1631 0.31597
83
Figure 3.7 Correlations of the regular LLE data by the NRTL model
84
Figure 3.8 Correlations of the normalized random data by the NRTL model
85 3.5 Conclusions
The parameters of the NRTL model have been successfully regressed from LLE data using the particle swarm optimization (PSO) method and smaller RMSDs were obtained compared with literature results. Further analysis reveals that multiple groups of parameters with sufficiently small RMSDs can be found for the same set of LLE data and the activities calculated with these parameters and their corresponding predicted mole fractions can be far beyond the reasonable range of activity. This demonstrates that the NRTL model does not represent the intrinsic activities of components with parameters regressed from LLE. Moreover, five sets of normalized random data have been correlated by the NRTL model equally well as correlations of regular experimental LLE data. This questions the theoretical validity of the NRTL model and highlights the need to strengthen the model’s theory.
86 Chapter 4
Phenol Extraction by Toluene in the Presence of Sodium Hydroxide
Phenol is a good model compound to study extraction of alkaloids as its acidity constant is close to that of many alkaloids, for example morphine. In addition, extraction of phenol is important environmentally as it is a water pollutant. In this study the distribution ratio of neutral phenol and total phenol between water and toluene with varying pH and varying concentration of sodium hydroxide is determined and discussed. The phase equilibrium is modelled by the symmetric eNRTL model using the PSO algorithm and the calculated values agree well with the experimental results.
87 4.1 Introduction
Extraction of natural products such as alkaloids from plants is a successful application of solvent extraction. In the process of extracting alkaloids, such as morphine (Figure [113] 4.1, pKa1≈8.05, pKa2≈9.50, at 25°C ), chemical bases are used to increase solubility[114] as well as to enhance selectivity. However, the dissociation/protonation of alkaloids and addition of bases increases the complexity of the extraction system. To quantitatively describe these extraction processes, thermodynamic modelling [115] based on experimental data is required. Phenol (pKa=9.99 at 25°C ), an easily accessible and cheap chemical, has a pKa value close to that of many alkaloids and as such is a good model compound to be used to mimic extraction of alkaloids.
Figure 4.1 Structure of morphine and phenol
Phenol is commonly used in industries to make precursors to plastics, phenolic resins, fertilizers, paints and drugs[116-118], and is produced globally at a rate of about 6 million tons per year and the trend is increasing[118]. Phenol is a water pollutant, which has negative impact on indigenous biota and human beings if introduced to aquatic ecosystems[116, 117]. Therefore removal of phenol from aquatic streams is environmentally important.
Extraction of phenol from aqueous solution has been the subject of many studies. Won et al.[119] studied the distribution of phenol between water and butyl acetate and methyl isobutyl ketone with varying mass fraction of phenol, results were explained by the theory of associated solutions. Gonzalez et al.[89] measured the distribution of phenol between water and a variety of solvents, the experimental data were correlated
88 by the NRTL model and the UNIQUAC model. Pinto et al.[120] evaluated the thermodynamic models of the NRTL, the UNIQUAC and the UNIFAC for prediction of phenol extraction using parameters from the “Aspen Databank” and literature and compared modelling results with experimental data of Gonzalez et al.[89]. It was found that the UNIFAC model provides the best predictions. Apart from traditional solvents, extraction of phenol by ionic liquids[121-123], CYANEX® 923[124, 125] and TBP[12, 126, 127] have also been explored, and new extractants are currently being synthesised[128]. Importantly, these investigations only concern extraction of neutral phenol, studies involving the dissociation of phenol is limited. As a weak electrolyte, phenol will partially react with hydroxide, which makes the system more complicated. Bins and co-workers[29] studied the distribution of phenol with sodium hydroxide in aqueous phase, the concentration of produced sodium phenolate was assumed to be the same as initial sodium hydroxide. It was found that the distribution ratio of phenol decreases with the increasing concentration of sodium phenolate. Greminger et al.[129] reported on the extraction of phenol by methyl isobutyl ketone and diisopropyl ether in the presence of sodium hydroxide in water, the distribution of phenol was correlated with a simple model considering the dissociation of phenol and the activity coefficients were roughly estimated using equilibrium distribution coefficient. Bízek et al.[130] developed a model that has five parameters to describe phenol extraction, however, the physical meanings of the parameters are ambiguous. Hence, a detailed modelling based on chemical thermodynamics to describe the process of extracting phenol from solutions containing hydroxides is still lacking.
This study investigates the extraction of phenol using toluene in the presence of sodium hydroxide in water. The distribution of phenol between the two phases with changing pH and changing concentration of sodium hydroxide are presented and the phase equilibrium is modelled by the symmetric eNRTL model using the PSO algorithm developed in chapter 3.
4.2 Experimental
4.2.1 UV-Vis Spectrophotometry
A CARY 100 UV-Vis spectrophotometer was used to measure the UV-Vis absorbance spectra of phenol, toluene and sodium phenolate in water. A pair of quartz
89 cuvettes from Starna with 10.0 mm path length were used as sample containers. Sodium phenolate was obtained by adding 1.0 M sodium hydroxide into dilute phenol solutions.
4.2.2 Distribution Ratio of Phenol
Distribution of phenol between water and toluene was measured by completing standard equilibrium shake-up experiments using 100 ml separation funnels. Two sets of experiments were completed: varying pH and varying concentration of sodium hydroxide. In this study phenol was always in excess when the influence of sodium hydroxide concentration was studied, therefore varying the concentration of sodium hydroxide does not necessarily change the pH. The aqueous solution was made of analytical reagent phenol (99.90%, Chem-Supply) and miliQ water. Toluene was analytical reagent (99.9%, Scharlau) and sodium hydroxide was semiconductor grade (99.99%, Sigma-Aldrich). Both water and toluene were degassed by vacuum to remove dissolved oxygen which has been shown to oxidize phenol, and the funnels were filled with nitrogen to exclude oxygen. The room temperature was controlled at 25±0.5 °C. Each experiment was conducted three times in parallel, both the average value and standard deviation of experimental data are presented in section 4.3.
For the first set of experiments, the initial concentration of phenol in water was 5.0×10-4 M and pH was adjusted by adding sodium hydroxide. For each experiment, 25 ml of phenol solution was contacted with 25 ml of toluene in a 100 ml separation funnel, and shaken for three hours to reach equilibrium. After equilibrium the samples were left to settle for one hour, during which the two phases separated. Phenol concentration in the aqueous solution was analysed using a UV-Vis spectrophotometer, while phenol in toluene was first stripped using 1 M sodium hydroxide (analytical grade, 97%, Chem-Supply) and then analysed using a UV-Vis spectrophotometer. Recovery of phenol from toluene using sodium hydroxide was found to be higher than 99.7%.
The second set of experiments was conducted with an initial phenol concentration of 0.78 M in the aqueous phase with varying concentrations of sodium hydroxide. The maximum concentration of sodium hydroxide was 0.505 M, thus phenol in aqueous phase was always in excess. At equilibrium, the mixture was separated and the
90 aqueous solution was neutralized and diluted using 0.01 M hydrochloric acid and the organic phase was diluted using methanol. Phenol concentrations were then measured by Agilent 1200 High Performance Liquid Chromatography (HPLC) using methanol and water (56:44) as mobile phase and the column used was Waters Symmetry® C18 5µm 3.9×150 mm. The HPLC spectra of phenol and toluene are shown in Figure 4.2.
Figure 4.2 HPLC spectra of phenol and toluene
(a): Organic phase; (b): Aqueous phase. The mobile phase was methanol and water (56:44) and the column used was Waters Symmetry® C18 5µm 3.9×150 mm.
91 4.3. Results and Discussions
4.3.1 Dissociation Equilibrium of Phenol with Sodium Hydroxide
As a weak electrolyte, phenol (PhOH) partly reacts with sodium hydroxide, the reaction is given in Eq. (4.1). Determination of species’ concentration in the aqueous phase is not possible using the UV-Vis spectrophotometer when solute concentration is high. Speciation in the aqueous phase can be obtained by solving Eqs. (4.2)-(4.5) for a number of initial phenol concentrations, results of calculations are presented in Figure 4.3. It is found that when the concentration of phenol is 0.1 M higher than that of sodium hydroxide, the ratio of [OH-]/[Na+] is so small that it is assumed sodium hydroxide reacts with phenol completely. Under these conditions, the initial concentration of sodium hydroxide approximately equals that of sodium phenolate.
Figure 4.3 Equilibrium of phenol reaction with sodium hydroxide
Phenol reaction with sodium hydroxide
PhOH OH PhO H2 O (4.1)
92 Dissociation of water
13. 995 H OH 10 (4.2)
Dissociation of phenol
PhO H 9. 99 10 (4.3) PhOH
Charge balance
Na H OH PhO (4.4)
Mass balance of phenol
PhO PhOH PhOH (4.5) initial
4.3.2 Spectrophotometric Properties of Phenol
In the aqueous phase, toluene, phenol and phenolate anions absorb ultra violet (UV) light in an overlapped wavelength range, refer to Figure 4.4. However, at 300 nm only the phenolate anion absorbs UV light. Therefore, absorbance at 300 nm can be used to determine the concentration of phenolate and then the total phenol concentration (neutral phenol and phenolate) can be measured by converting phenol into phenolate, which was achieved by diluting the aqueous solution with the same volume of 0.2 M sodium hydroxide solution. Molar absorptivities of phenol are presented in Table 4.1.
Table 4.1 Molar absorptivity of phenol
Molar absorptivity (M-1·cm-1) solvent Ref. 270 nm 287 nm 300 nm 1373 - - Water [131] 1450 - - Water [132, 133] 1430 - - Water [134] 1495 - - 0.01 M HCL [135] 1660 - - Water [136] - 2560 - 0.1 M NaOH [135] 1501 - - Water This study - 2609 1019 1 M NaOH This study
93
Figure 4.4 The UV-Vis spectra of toluene, phenol and phenolate anion
4.3.3 Oxidation of Phenol by Oxygen
Initially the distribution experiments were conducted under an atmospheric condition. Significant change of pH was observed, which was attributed to oxidation of phenol by oxygen and subsequent reactions[137, 138]. Phenol can be oxidized by dissolved oxygen in water and toluene, producing a variety of components including acids such as maleic acid, succinic acid, oxalic acid, acetic acid and so on, resulting in not only loss of phenol, but also a decrease in equilibrium pH. Solubility of oxygen in toluene is 9.22×10-4 by mole fraction (8.67×10-3 M) at 283.71K[139]. In water it is 5.95 ml/l -4 [140] (2.66×10 M) at standard temperature and pressure . These amounts of dissolved oxygen would be enough to oxidize all the phenol (5.0×10-4 M, for the first set of experiments) in the extraction system. Fortunately, oxidation proceeds slowly at moderate temperatures. Comparison of pH variation at equilibrium with and without oxygen is shown in Figure 4.5. Decrease of pH at equilibrium when oxygen is not isolated indicates significant oxidation of phenol by oxygen. Equilibrium distribution experiments were subsequently completed in an inert atmosphere using nitrogen as described in section 4.2.2.
94
Figure 4.5 pH variation of aqueous phenol solution before and after equilibrium (a): With oxygen in the extraction system; (b): without oxygen in the extraction system.
Table 4.2 Properties of species used in this study
Molecular Molar Density, Dissociation Solubility in a Species Weight, Mw Volume , ρ (25°C) constant water, xi (25°C) 3 3 (g/mol) Vi (cm /mol) (g/cm ) (25°C), pKa (Mole fraction) Toluene 92.139 106.85 0.8623 - 1.04×10-4 [141] Phenol 94.111 87.872 1.071 b 9.99 0.0178 [142]
c H2O 18.015 18.069 0.9970 15.738 - NaOH 39.997 - 2.13 - 0.3105
Notes: Values come from CRC Handbook of Chemistry & Physics[115] except for specified. a Calculated from molecular weight and density; b Extrapolated from measured value of liquid state at around
[143] c melting temperature from Badachhape et al. . Calculated from Kw=13.995 at 25°C.
95 4.3.4 Distribution Ratio of Phenol with Varying pH
Neutral phenol can be extracted into toluene, while dissociated phenol (sodium phenolate) remains in the aqueous phase. The distribution ratio of total phenol (DT) and neutral phenol (DN) are defined as follows:
concentration of phenol in organic phase D = (4.6) T concentration of neutral phenol and phenolate in aqueous phase
concentration of phenol in organic phase D = (4.7) N concentration of neutral phenol in aqueous phase
The distribution ratio of phenol between water and toluene determined experimentally and mathematically using apparent pKa of phenol over a range of pH is presented in Figure 4.6. With increasing pH, the distribution ratio of total phenol decreases mainly due to the conversion of phenol from neutral molecules to phenolate ions. Distribution of neutral phenol also decreases slightly with increasing pH. The reduction in measured distribution ratio of DN at high pH is due to the interaction of species in the aqueous phase caused by addition of sodium hydroxide. If DN is calculated using a constant pKa, the opposite trend to the experimental result is observed at high pH. This is clearly wrong and is a result of increased interactions of species in solution causing changes in species’ activities.
Figure 4.6 Distribution ratio of phenol with varying pH
96 4.3.5 Distribution Ratio of Phenol with Varying Sodium Hydroxide Concentration
To further investigate the influence of sodium hydroxide on the distribution ratio of phenol, we measured both DT and DN with varying concentration of sodium hydroxide.
The results are presented in Figure 4.7. The rapid decrease of DT is consistent with the results shown in Figure 4.6 and can again be explained by the conversion of neutral phenol into sodium phenolate. By contrast, DN decreases in a slower manner with some fluctuations. In theory, DN should be constant for dilute solutions. Disprepancy of species’ behavior from ideal solution can occur with an increase in solute concentration when the interactions among species are not neglegible. These discrepancies can be described by the NRTL model and the symmetric eNRTL model.
Table 4.3 Liquid–liquid equilibrium data (mole fraction)
Organic phase （I） Aqueous phase （II） Sodium Phenol(1) Toluene(2) Phenol(1) Water(4) Phenolate(3) 0.05897 0.94103 0.00374 0 0.99626 0.05239 0.94761 0.00397 0.00111 0.99492 0.04799 0.95201 0.00385 0.00209 0.99406 0.0436 0.95641 0.00377 0.00303 0.99319 0.04059 0.95941 0.00346 0.00393 0.99262 0.03529 0.96471 0.00347 0.00495 0.99158 0.03073 0.96927 0.00334 0.00593 0.99073 0.02836 0.97164 0.00272 0.00702 0.99026 0.02468 0.97532 0.00237 0.00805 0.99957 0.02060 0.97940 0.00214 0.00902 0.98884 0.01801 0.98199 0.00198 0.00966 0.98835
In order to quantitatively describe the distribution ratio of phenol in the presence of sodium hydroxide, liquid–liquid equilibrium data of all species in the two phases is required. Considering that the solubility of water in toluene and toluene in water is very small[141] especially when compared with the concentration of phenol and sodium hydroxide in this study, the water concentration in toluene is neglected as well as the toluene concentration in water. Moreover, volume additivity of phenol and
97 toluene is assumed. Addition of phenol into sodium hydroxide solution does not change the ionic strength of the solution, thus volume additivity of phenol and sodium hydroxide solution is also assumed. On the basis of these assumptions and the correlation of density for sodium hydroxide solution[144], the liquid–liquid equilibrium data is calculated from the equilibrium experiments (properties of species used in this study is provided in Table 4.2) with results given in Table 4.3.
Figure 4.7 Distribution ratio with varying concentration of sodium hydroxide (a): Neutral phenol; (b): Total phenol
98 4.4. The Modelling of Phase Equilibrium
At equilibrium the activities of phenol in the organic and aqueous phases are equal. The isoactivity equation of phenol is expressed as:
I I II II x1γ 1 x 1γ 1 (4.8) where x is the mole fraction, γ is activity coefficient and I and II represent the two phases. Constraints of Eq. (4.8) are mass balance in the two phases:
II xx121 (4.9)
x1 x 3 x 4 1 (4.10)
The NRTL model[45] and the symmetric eNRTL model are used to characterise the organic and aqueous phase respectively. These two thermodynamic models have been introduced in section 1.4.3, here we re-write some of the key equations. The equation for activity coefficient of component i in the NRTL model is given as follows:
N N τ Gx x τ G ji ji j N xG l lj lj ln γ j1 j ij τ l1 (4.11) iNNN ij j1 Gki x k G kj x k G kj x k k1 k 1 k 1 where N is the number of components in the system and
τ ji()/g ji g ii RT (4.12)
G jiexp α jiτ ji (4.13)
Here gji is energy of interaction between a j-i pair of molecules; 휏ji (휏ji≠휏ij) and αji (αji=
αij) are the energy parameter and non-randomness parameter respectively.
The aqueous phase contains salt, i.e. sodium phenolate, thus cannot be described by the NRTL model, but can be modelled by the symmetric eNRTL model[47], which is an expansion of the NRTL model. In this model, contribution to the excess Gibbs free energy has two parts: local and long-range interactions. Accordingly, the activity coefficient equation is derived as:
lc PDH ln γiln γ iln γ i , i=m, c, a (4.14)
99 where m, c, a represent molecule, cation and anion respectively.
In the symmetric eNRTL model, binary parameters are categorized into three types: molecule-molecule binary parameter, molecule-electrolyte binary parameter and electrolyte-electrolyte binary parameter, where electrolyte is meant to be a pair of cation c and anion a. In the equation of activity coefficient given by Song et al.[47], interaction parameters involving ions (cation or anion) are included, these parameters are expressed as functions of the three types of binary parameters by mixing rules. In the case of phenol extraction by toluene, only one salt (sodium phenolate) is involved, so the mixing rules are not necessary. As a result, the local contribution to activity coefficient can be expressed in the same way as the NRTL model.
The long-range contribution is accounted for by the extended Pitzer-Debye-Hückel (PDH) formula[72]. For a non-electrolyte component m, the activity coefficient can be expressed as:
32/ PDH 2AIφ x ln γi 12/ (4.15) 1 ρI x with
32/ 12/ 2 1 2πN A Qe Aφ (4.16) 3 ν εkTB
11 22 (4.17) Ix z c x c z a x a 22ca where Aφ is the Debye-Hückel parameter, Ix is the ionic strength, NA is Avogadro’s number, ρ is the closest approach parameter, Qe is the electron charge, kB is the Boltzmann constant, ν is the molar volume and ε is the dielectric constant of the solvent.
With the equations above, the experimental data in Table 4.3 can be correlated and the parameters regressed by the PSO method developed in Chapter 3. The regressed parameters are presented in Table 4.4. With these parameters and given the total amount of each compound, the equilibrium compositions of each phase can be calculated and thus the distribution ratio of phenol can be obtained. The calculated distribution ratio of neutral phenol and total phenol are shown alongside the
100 experimental data in Figure 4.7. It can be seen that the calculated values agree well with the experimental data.
Despite good consistency between the calculated values and the experimental results, the modelling results are limited because of the semi-empirical nature of the model. Firstly, if sodium hydroxide exists in the extraction system the accuracy of the calculated values would be lower because in the parameter regression sodium hydroxide is assumed to be absolutely consumed. Secondly, accuracy of the calculation can only be guaranteed for solute concentrations close to the values used in this study. These limitations are closely related to the model’s theoretical background as discussed in chapter 3.
Table 4.4 Regressed parameters for the extraction system: phenol (1) + toluene (2) + sodium phenolate (3) + water (4)
i-j αij 휏ij 휏ji 1-2 0.2 3.0859 -10.964 1-3 0.3 -12.462 -7.7032 1-4 0.2 6.3083 -8.2940 3-4 0.3 -14.584 -7.7580
4.5 Conclusions
The distribution ratio of total phenol and neutral phenol between water and toluene with varying pH and concentration of sodium hydroxide has been measured. Data was obtained using UV-Vis spectrophotometric properties of phenol and HPLC analysis. Liquid–liquid equilibrium data of the extraction system was then calculated and modelled by the NRTL model and the symmetric eNRTL model using PSO algorithm. The calculated values agree well with experimental results.
101
102 Chapter 5
Conclusions and Outlook
The three main parts of this thesis—analysis of the isoactivity equations, regression of NRTL parameters and the modelling of phenol extraction—have been presented in chapters 2, 3 and 4 respectively. This chapter summarizes the key conclusions and provides an outlook for further work.
103 5.1 Main Conclusions
The aim of this study is to analyse the isoactivity equations of LLE systems using the NRTL model to make the equation-solving approach capable of identifying the correct solution of phase equilibrium from multiple solutions. Toward this end, a series of analyses, entitled solution structure categorization (SSC), were carried out under two types of mass balance constraints and procedures for determining the correct solutions were proposed. Following this work, regression of the NRTL parameters, which is the precondition for using the model, was investigated using the particle swarm optimization (PSO) algorithm and limitations of the NRTL model were discussed. On the basis of this work, the NRTL model and its expansion (symmetric eNRTL model) were used to model the phase equilibrium of phenol extraction in the presence of sodium hydroxide and good correlation was obtained.
The main conclusions of this thesis are as follows:
(1). The structure of solutions of the isoactivity equations for ternary and quaternary LLE systems using the NRTL model under two types of mass balance constraints were investigated through SSC analysis:
(a). When the concentrations of components (one component in a ternary system and two components in a quaternary system) in one phase are specified, three types of solutions were revealed, namely exact solutions, symmetric solutions and approximate solutions. The number of “exact solutions” is usually no more than three and they differ significantly in values. The correct solution should be an “exact solution”.
(b). When the total amount of each component in a system is specified, all solutions of isoactivity equations can be categorized into two types when converted into mole fractions: one correct solution and a number of symmetric solutions representing a homogeneous phase.
(2). Procedures for determining the correct LLE solution under the two mass balance constraints were proposed on the basis of the SSC results. These procedures are simpler compared with minimizing the Gibbs free energy. The procedure, where total amount of each component is specified, is recommended to be used as an alternative of minimizing the Gibbs free energy.
104 (3). The NRTL parameters were successfully regressed by the particle swarm optimization (PSO) algorithm, which is a powerful global optimization algorithm, from liquid-liquid equilibria (LLE) data, and the resulting parameters showed smaller root-mean square deviations (RMSD) compared with results in literature.
(4). Multiple groups of parameters with sufficiently small RMSDs were found for the same set of LLE data. The activities calculated using these parameters and their corresponding predicted mole fractions can be far beyond the reasonable range of activity, clearly demonstrating limitations of the NRTL model. This is caused by the regression method that equates the two sides of isoactivity equations without considering the intrinsic activities of each component.
(5). The distribution of neutral phenol and total phenol was determined by UV-Vis spectrophotometry and HPLC with varying pH and concentration of sodium hydroxide. With the increasing pH or increasing sodium hydroxide concentration, distribution ratio of total phenol decreases due to conversion of neutral phenol into sodium phenolate, distribution of neutral phenol also decreases because of increasing interactions among species.
(6) Phase equilibrium of phenol in the extraction system of “Phenol (1) + Toluene (2) + Sodium Phenolate (3) +Water (4)” was correlated using the NRTL model and the symmetric eNRTL model for the organic and aqueous phase respectively. The calculations agreed well with the experimental results.
5.2 Outlook
While a systematic study of the LLE calculation using the NRTL model based on the equation-solving approach has been conducted, further studies are necessary for better understanding of the equation-solving approach for calculating phase equilibria. The recommended future studies are:
(1). Investigation of solution structures of isoactivity equations of LLE systems based on other thermodynamic models, such as the UNIQUAC model and the UNIFAC model. Despite that the categorization of solutions of the isoactivity equations seems to be universal, studies using other models is worthwhile.
105 (2). Extension of the SSC analysis to LLE systems with multiple phases. The systems discussed in the thesis have two phases, the suitability of SSC for multi-phase LLE systems requires examination.
(3). Extension of the SSC analysis to vapour-liquid equilibria (VLE) systems. Parallel to LLE, VLE is important in distillation. Therefore, it is worth investigating the solution structure of equations representing VLE.
(4). Quantification of differences between minimization of the Gibbs free energy combined with stability test and the SSC analysis for LLE calculations. This would provide us with deeper understanding of advantages and disadvantages of the two approaches, although a qualitative comparison of them is given in the thesis.
(5). Exploration of an algorithm for NRTL parameter regression from LLE data that takes the activities of components into consideration. This would solve the dilemma that the activities calculated with regressed parameters may beyond the reasonable range.
(6). The empiricism of the NRTL model revealed in this study may also apply to other thermodynamic models including the UNIQUAC model and the UNIFAC model etc. Investigations of these models’ capabilities and limitations are worthwhile.
(7). A comprehensive measurement of phase composition of the system “Phenol (1) + Toluene (2) + Sodium Phenolate (3) +Water (4)”, including water in the organic phase and toluene in water, thus obtain a more robust correlation covering a broader concentration range.
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119 Appendices
A. Calculation Examples
Example 1 Quaternary system: Heptane (1) + Benzene (2) + Ethylbenzene (3) + Methanol (4) at 293.15 K from García-Flores et al.[51].
n1=0.7033 (0.5444+0.1589); n2=0.1352 (0.0882+0.0470); n3=0.0897 (0.0592+0.0305); n4=1.0718 (0.3082+0.7636). Only one correct solution was found. S.A (correct solution) occurred 163 times and S.B (symmetric solution) occurred 100 times in 1000 solutions. E represents the experimental condition.
Table A1-1. Solutions for isoactivity equations in moles
I I I I II II II II Solutions n1 n2 n3 n4 n1 n2 n3 n4 G/RT E 0.5444 0.0882 0.0592 0.3082 0.1589 0.0470 0.0305 0.7636 - S.A 0.1615 0.0525 0.0262 0.7726 0.5418 0.0827 0.0635 0.2993 -0.9297 S.B 0.6032 0.1159 0.0769 0.9192 0.1001 0.0193 0.0128 0.1526 -0.9274 S.B 1.14E-05 2.19E-06 1.46E-06 1.74E-05 0.7033 0.1352 0.0897 1.0718 -0.9274 S.B 0.4549 0.0875 0.0580 0.6933 0.2484 0.0477 0.0317 0.3785 -0.9274 S.B 0.0960 0.0184 0.0122 0.1463 0.6073 0.1168 0.0775 0.9255 -0.9274 … … … … … … … … …
Table A1-2. Solutions for isoactivity equations in mole fractions
I I I I II II II II Solutions x1 x2 x3 x4 x1 x2 x3 x4 E 0.5444 0.0882 0.0592 0.3082 0.1589 0.0470 0.0305 0.7636 S.A 0.5488 0.0838 0.0644 0.3031 0.1594 0.0518 0.0258 0.7629 S.B 0.3517 0.0676 0.0449 0.5359 0.3517 0.0676 0.0449 0.5359
120 Example 2 Quaternary system: n-Hexane (1) + n-Octane (2) + Benzene (3) + Sulfolane (4) at 298.15 K from Chen et al.[33].
n1=0.313 (0.282+0.031); n2=0.183 (0.171+0.012); n3=0.950 (0.524+0.426); n4=0.554 (0.023+0.531). Only one correct solution was found. S.A (correct solution) occurred 103 times and S.B (symmetric solution) occurred 531 times in 1000 solutions. E represents the experimental condition.
Table A2-1. Solutions for isoactivity equations in moles
I I I I II II II II Solutions n1 n2 n3 n4 n1 n2 n3 n4 G/RT E 0.282 0.171 0.524 0.023 0.031 0.012 0.426 0.531 - S.A 0.278 0.169 0.525 0.040 0.035 0.014 0.425 0.514 -1.510 S.B 0.134 0.078 0.406 0.237 0.179 0.105 0.544 0.317 -1.431 S.B 0.202 0.118 0.615 0.358 0.111 0.065 0.336 0.196 -1.431 S.B 0.246 0.144 0.747 0.436 0.067 0.039 0.203 0.118 -1.431 S.B 0.147 0.086 0.447 0.261 0.166 0.097 0.503 0.293 -1.431 … … … … … … … … … …
Table A2-2. Solutions for isoactivity equations in mole fractions
I I I I II II II II Solutions x1 x2 x3 x4 x1 x2 x3 x4 E 0.282 0.171 0.524 0.023 0.031 0.012 0.426 0.531 S.A 0.274 0.167 0.519 0.040 0.036 0.014 0.430 0.520 S.B 0.157 0.092 0.475 0.277 0.157 0.092 0.475 0.277
121 Example 3 Quaternary system: 1-octanol (1) + TAME (2) + Water (3) + Methanol (4) at 298.15 K from Arce et al.[90].
n1=0.1664 (0.1651+0.0013); n2=0.1930 (0.1849+0.0081); n3=1.0385 (0.3243+0.7142); n4=0.6022 (0.3258+0.2764). Parameters are from Table 4 of the paper[90]. Only one correct solution was found. S.A (correct solution) occurred 14 times and S.B (symmetric solution) occurred 121 times in 1000 solutions. E represents the experimental condition.
Table A3-1. Solutions for isoactivity equations in moles
I I I I II II II II Solutions n1 n2 n3 n4 n1 n2 n3 n4 G/RT E 0.1651 0.1849 0.3243 0.3258 0.0013 0.0081 0.7142 0.2764 - S.A 0.1657 0.1916 0.2498 0.4150 0.0007 0.0014 0.7887 0.1872 -2.9304 S.B 0.1624 0.1883 1.0133 0.5876 0.0040 0.0047 0.0252 0.0146 -2.8955 S.B 0.0136 0.0158 0.0848 0.0492 0.1528 0.1772 0.9537 0.5530 -2.8955 S.B 0.0098 0.0113 0.0609 0.0353 0.1566 0.1817 0.9776 0.5669 -2.8955 S.B 0.1577 0.1830 0.9845 0.5709 0.0087 0.0100 0.0540 0.0313 -2.8955 … … … … … … … … … …
Table A3-2. Solutions for isoactivity equations in mole fractions
I I I I II II II II Solutions x1 x2 x3 x4 x1 x2 x3 x4 E 0.1651 0.1849 0.3243 0.3258 0.0013 0.0081 0.7142 0.2764 S.A 0.1622 0.1875 0.2444 0.4060 0.0007 0.0014 0.8065 0.1914 S.B 0.0832 0.0965 0.5192 0.3011 0.0832 0.0965 0.5192 0.3011
122 Example 4 Quaternary system: Water (1) + Acrylic Acid (2) + Acetic Acid (3) + Cyclohexane (4) at 293.15 K from Zhang et al.[91].
n1=0.7428 (0.0175+0.7253); n2=0.3637 (0.1473+0.2164); n3=0.0619 (0.0155+0.0464); n4=0.8316 (0.8197+0.0119). Only one correct solution was found. S.A (correct solution) occurred 9 times and S.B (symmetric solution) occurred 611 times in 1000 solutions. E represents the experimental condition.
Table A4-1. Solutions for isoactivity equations in moles
I I I I II II II II Solutions n1 n2 n3 n4 n1 n2 n3 n4 G/RT E 0.0175 0.1473 0.0155 0.8197 0.7253 0.2164 0.0464 0.0119 - S.A 0.0037 0.1153 0.0132 0.8284 0.7391 0.2484 0.0487 0.0032 -2.782 S.B 0.0946 0.0463 0.0079 0.1059 0.6482 0.3174 0.0540 0.7257 -2.069 S.B 0.3731 0.1827 0.0311 0.4177 0.3697 0.1810 0.0308 0.4139 -2.069 S.B 0.4148 0.2031 0.0346 0.4644 0.3280 0.1606 0.0273 0.3672 -2.069 S.B 0.6692 0.3277 0.0558 0.7492 0.0736 0.0360 0.0061 0.0824 -2.069 … … … … … … … … … …
Table A4-2. Solutions for isoactivity equations in weight fractions
I I I I II II II II Solutions x1 x2 x3 x4 x1 x2 x3 x4 E 0.0175 0.1473 0.0155 0.8197 0.7253 0.2164 0.0464 0.0119 S.A 0.0038 0.1200 0.0138 0.8623 0.7111 0.2390 0.0468 0.0031 S.B 0.3714 0.1819 0.0310 0.4158 0.3714 0.1819 0.0310 0.4158
123 Example 5 Ternary system: Water (1) + Ethanol (2) + 1,1-Difluorothane (3) at 223.2 K and 6.08 Mpa from Nakayama et al.[92].
n1=0.8825 (0.8157+0.0668); n2=0.3096 (0.1649+0.1447); n3=0.8079 (0.0194+0.7885). Only one correct solution was found. S.A (correct solution) occurred 47 times and S.B (symmetric solution) occurred 135 times in 1000 solutions. E represents the experimental condition.
Figure A1. Solutions for isoactivity equations with the total amount of each component as constraint. S.1 and S.2 are two reversed solutions. The straight line in the middle represents symmetric solutions.
Table A5-1. Solutions for isoactivity equations in moles
I I I II II II Solutions n1 n2 n3 n1 n2 n3 G/RT E 0.8157 0.1649 0.0194 0.0668 0.1447 0.7885 - S.A 0.8278 0.1738 0.0281 0.0547 0.1358 0.7798 -1.1207 S.B 0.0375 0.0132 0.0344 0.8450 0.2964 0.7735 -0.9124 S.B 0.0016 0.0006 0.0015 0.8809 0.3090 0.8064 -0.9124 S.B 0.6352 0.2229 0.5815 0.2473 0.0867 0.2264 -0.9124 S.B 0.0008 0.0003 0.0007 0.8817 0.3093 0.8072 -0.9124 … … … … … … … …
Table A5-2. Solutions for isoactivity equations in mole fractions
I I I II II II Solutions x1 x2 x3 x1 x2 x3 E 0.8157 0.1649 0.0194 0.0668 0.1447 0.7885 S.A 0.8039 0.1688 0.0273 0.0563 0.1400 0.8037 S.B 0.4413 0.1548 0.4040 0.4413 0.1548 0.4040
124 Example 6 Ternary system: Quinoline (1) + Furfural (2) + Water (3) at 298.15 K from Sørensen et al.[93] (Page 208).
n1=0.15117 (0.15058+0.00059); n2=0.45748 (0.44457+0.01291); n3=1.39135 (0.40485+0.98650). Only one correct solution was found. S.A (correct solution) occurred 12 times and S.B (symmetric solution) occurred 89 times in 1000 solutions. E represents the experimental condition.
Figure A2. Solutions for isoactivity equations with the total amount of each component as constraint. S.1 and S.2 are two reversed solutions. The straight line in the middle represents symmetric solutions.
Table A6-1. Solutions for isoactivity equations in moles
I I I II II II Solutions n1 n2 n3 n1 n2 n3 G/RT E 0.15058 0.44457 0.40485 0.00059 0.01291 0.98650 - S.A 0.14832 0.45245 0.46869 0.00285 0.00503 0.92266 -0.5825 S.B 0.15063 0.45584 1.38637 0.00054 0.00164 0.00498 -0.4744 S.B 0.05835 0.17658 0.53704 0.09282 0.28090 0.85431 -0.4744 S.B 0.11921 0.36077 1.09723 0.03196 0.09671 0.29412 -0.4744 S.B 0.04155 0.12573 0.38238 0.10963 0.33175 1.00897 -0.4744 … … … … … … … …
Table A6-2. Solutions for isoactivity equations in mole fractions
I I I II II II Solutions x1 x2 x3 x1 x2 x3 E 0.15058 0.44457 0.40485 0.00059 0.01291 0.98650 S.A 0.13869 0.42307 0.43825 0.00306 0.00540 0.99154 S.B 0.07559 0.22874 0.69568 0.07559 0.22874 0.69568
125 Example 7 Ternary system: Water (1) + Acetic Acid (2) + Ethyl Acetate (3) at 298 K from Colombo et al.[94].
n1=1.43442 (0.87819+0.55623); n2=0.22615 (0.07766+0.14849); n3=0.33943 (0.04415+0.29528). Only one correct solution was found. S.A (correct solution) occurred 259 times and S.B (symmetric solution) occurred 53 times in 1000 solutions. E represents the experimental condition.
Figure A3. Solutions for isoactivity equations with the total amount of each component as constraint. S.1 and S.2 are two reversed solutions. The straight line in the middle represents symmetric solutions.
Table A7-1. Solutions for isoactivity equations in moles
I I I II II II Solutions n1 n2 n3 n1 n2 n3 G/RT E 0.87819 0.07766 0.04415 0.55623 0.14849 0.29528 - S.A 0.91852 0.08164 0.04522 0.51590 0.14451 0.29421 -1.1193 S.B 0.03277 0.00517 0.00776 1.40165 0.22098 0.33168 -1.1073 S.B 0.73670 0.11615 0.17433 0.69773 0.11000 0.16510 -1.1073 S.B 0.12934 0.02039 0.03061 1.30508 0.20576 0.30882 -1.1073 S.B 0.94156 0.14845 0.22280 0.49286 0.07770 0.11663 -1.1073 … … … … … … … …
Table A7-2. Solutions for isoactivity equations in mole fractions
I I I II II II Solutions x1 x2 x3 x1 x2 x3 E 0.87819 0.07766 0.04415 0.55623 0.14849 0.29528 S.A 0.87865 0.07810 0.04326 0.54043 0.15138 0.30820 S.B 0.71721 0.11308 0.16972 0.71721 0.11308 0.16972
126 Example 8 Ternary system: Acetic Acid Butyl Ester (1) + 2-Propanone (2) + 1,2-Ethanediol (3) at 304.15 K from Sørensen et al.[38] (Page 410).
n1=0.44754 (0.34433+0.10321); n2=0.71492 (0.45231+0.26261); n3=0.83754 (0.20336+0.63418). Only one correct solution was found. S.A (correct solution) occurred 206 times and S.B (symmetric solution) occurred 143 times in 1000 solutions. E represents the experimental condition.
Figure A4. Solutions for isoactivity equations with the total amount of each component as constraint. S.1 and S.2 are two reversed solutions. The straight line in the middle represents symmetric solutions.
Table A8-1. Solutions for isoactivity equations in moles
I I I II II II Solutions n1 n2 n3 n1 n2 n3 G/RT E 0.34433 0.45231 0.20336 0.10321 0.26261 0.63418 - S.A 0.37262 0.47237 0.19371 0.07492 0.24255 0.64383 -1.1551 S.B 0.43318 0.69198 0.81067 0.01436 0.02294 0.02688 -1.1389 S.B 0.20014 0.31971 0.37454 0.24740 0.39521 0.46300 -1.1389 S.B 0.01834 0.02929 0.03432 0.42920 0.68563 0.80323 -1.1389 S.B 0.33995 0.54306 0.63620 0.10759 0.17186 0.20134 -1.1389 … … … … … … … …
Table A8-2. Solutions for isoactivity equations in mole fractions
I I I II II II Solutions x1 x2 x3 x1 x2 x3 E 0.34433 0.45231 0.20336 0.10321 0.26261 0.63418 S.A 0.35874 0.45477 0.18649 0.07793 0.25232 0.66975 S.B 0.22377 0.35746 0.41877 0.22377 0.35746 0.41877
127 Example 9 Ternary system: Heptane (1) + Methane Trichloro (2) + Aniline (3) at 291.15 K from Sørensen et al.[38] (Page 32).
n1=0.69860 (0.43554+0.26306); n2=0.49064 (0.24689+0.24375); n3=0.81076 (0.31758+0.49318). Only one correct solution was found. S.A (correct solution) occurred 299 times and S.B (symmetric solution) occurred 68 times in 1000 solutions. E represents the experimental condition.
Figure A5. Solutions for isoactivity equations with the total amount of each component as constraint. S.1 and S.2 are two reversed solutions. The straight line in the middle represents symmetric solutions.
Table A9-1. Solutions for isoactivity equations in moles
I I I II II II Solutions n1 n2 n3 n1 n2 n3 G/RT E 0.43554 0.24689 0.31758 0.26306 0.24375 0.49318 - S.A 0.45031 0.24898 0.30629 0.24829 0.24166 0.50447 -1.7724 S.B 0.26662 0.18726 0.30943 0.43198 0.30339 0.50133 -1.7718 S.B 0.10303 0.07236 0.11958 0.59557 0.41828 0.69118 -1.7718 S.B 0.68203 0.47900 0.79153 0.01657 0.01164 0.01924 -1.7718 S.B 0.33562 0.23571 0.38951 0.36298 0.25493 0.42125 -1.7718 … … … … … … … …
Table A9-2. Solutions for isoactivity equations in mole fractions
I I I II II II Solutions x1 x2 x3 x1 x2 x3 E 0.43554 0.24689 0.31758 0.26306 0.24375 0.49318 S.A 0.44781 0.24760 0.30459 0.24968 0.24302 0.50730 S.B 0.34930 0.24532 0.40538 0.34930 0.24532 0.40538
128 Example 10 Ternary system: Water (1) + Phenol (2) + Benzene (3) at 298.15 K from Gonzalez et al.[89].
n1=1.4238 (0.4361+0.9877); n2=0.4319 (0.4205+0.0114); n3=0.1443 (0.1434+0.0009). Only one correct solution was found. S.A (correct solution) occurred 14 times and S.B (symmetric solution) occurred 126 times in 1000 solutions. E represents the experimental condition.
Figure A6. Solutions for isoactivity equations with the total amount of each component as constraint. S.1 and S.2 are two reversed solutions. The straight line in the middle represents symmetric solutions.
Table A10-1. Solutions for isoactivity equations in moles
I I I II II II Solutions n1 n2 n3 n1 n2 n3 G/RT E 0.4361 0.4205 0.1434 0.9877 0.0114 0.0009 - S.A 0.4340 0.4189 0.1426 0.9898 0.0130 0.0017 -0.5671 S.B 1.4006 0.4249 0.1419 0.0232 0.0070 0.0024 -0.4831 S.B 0.9866 0.2993 0.1000 0.4372 0.1326 0.0443 -0.4831 S.B 1.1378 0.3451 0.1153 0.2860 0.0868 0.0290 -0.4831 S.B 1.4238 0.4319 0.1443 1.39E-05 4.2E-06 1.4E-06 -0.4831 … … … … … … … …
Table A10-2. Solutions for isoactivity equations in mole fractions
I I I II II II Solutions x1 x2 x3 x1 x2 x3 E 0.4361 0.4205 0.1434 0.9877 0.0114 0.0009 S.A 0.4359 0.4208 0.1433 0.9854 0.0130 0.0017 S.B 0.7119 0.2159 0.0722 0.7119 0.2159 0.0722
129 B. MATLAB Code for Solving Isoactivity Equations
The MATLAB code has been provided for solving the isoactivity equations with the mass balance constraints of specifying total amount of each component. This code is also applicable for conditions of specifying concentrations of components in one phase with minor modifications. This program has a main code and a function handle.
The following is the main code: clear; tic; rand('state',sum(100*clock)*rand(1)); % Producing initial state for random number
n1=1.4238; n2=0.4319; n3=0.1443; % Specifying total amount of each component
D=[]; s=0; while (length(D)<1 && s<=9) % Criterion for stopping the program. s=s+1; % The number of loops.
N=20;
X0(:,1)=rand(1,N)*n1; X0(:,4)=rand(1,N)*n1; X0(:,2)=rand(1,N)*n2; X0(:,5)=rand(1,N)*n2; X0(:,3)=rand(1,N)*n3; X0(:,6)=rand(1,N)*n3; % Initialize random estimations.
for i =1:length(X0(:,1));
X00=X0(i,:); X=fsolve(@myfun_Random,X00,optimset('Display','off')); % Command for solving equations. S(i,1)=X(1); S(i,2)=X(2); S(i,3)=X(3); S(i,4)=X(4); S(i,5)=X(5); S(i,6)=X(6); % collection of solutions q=myfun_Random(X); % The residue of each solution. Q(i,:)=q; end
130 R=1e-7; % Threshold for filtering solutions according to residues. for i=1:N; for j=1:6 if (n1> S(i,1) && S(i,1)>0) && (n2> S(i,2) && S(i,2)>0) && (n3> S(i,3) && S(i,3)>0) && (n1> S(i,4) && S(i,4)>0) && (n2> S(i,5) && S(i,5)>0) && (n3> S(i,6) && S(i,6)>0) &&... % Judge solutions by mole fractions. (R >abs(Q(i,1))) && (R>abs(Q(i,2))) && (R>abs(Q(i,3))) && (R>abs(Q(i,4))) && (R>abs(Q(i,5))) && (R>abs(Q(i,6))); % Judge solutions by residual of equations.
S(i,j)=S(i,j); else S(i,j)=-0.1; end end end
n=1:N; figure(1); plot(n,S(:,1),'bs',n,S(:,4),'ro'); figure(2); plot(n,S(:,2),'bs',n,S(:,5),'ro'); figure(3); plot(n,S(:,3),'bs',n,S(:,6),'ro'); % Plot the results.
m=0; for i=1:N if S(i,1)>0 m=m+1; A(m,:)=S(i,:); B(m,:)=Q(i,:); end end % Collect the remaining solutions. for i=1:length(A(:,1)); C(i,1)=A(i,1)/(A(i,1)+A(i,2)+A(i,3)); C(i,2)=A(i,2)/(A(i,1)+A(i,2)+A(i,3)); C(i,3)=A(i,3)/(A(i,1)+A(i,2)+A(i,3)); C(i,4)=A(i,4)/(A(i,4)+A(i,5)+A(i,6)); C(i,5)=A(i,5)/(A(i,4)+A(i,5)+A(i,6)); C(i,6)=A(i,6)/(A(i,4)+A(i,5)+A(i,6)); end% Convert solutions from moles into mole fractions. m=0; for i=1:length(C(:,1)); if C(i,1)/C(i,4)>1.01 | C(i,1)/C(i,4)<0.99; m=m+1;
131 D(m,:)=C(i,:); E(m,:)=A(i,:); % Exclude symmetric solutions. end end end
F=unique (round(D.*100000)./100000,'rows'); % Filter out duplicate solutions in mole fractions. G=unique (round(E.*1000000)./1000000,'rows'); % Filter out duplicate solutions in moles. xlswrite('C:\Research-Solvent Extraction\Pharmaceutical extraction\Calculation\Papers\NRTL Analysis\More Discussions\Three Phases\two-phase\1\S.xlsx',S); xlswrite('C:\Research-Solvent Extraction\Pharmaceutical extraction\Calculation\Papers\NRTL Analysis\More Discussions\Three Phases\two-phase\1\A.xlsx',A); xlswrite('C:\Research-Solvent Extraction\Pharmaceutical extraction\Calculation\Papers\NRTL Analysis\More Discussions\Three Phases\two-phase\1\C.xlsx',C); xlswrite('C:\Research-Solvent Extraction\Pharmaceutical extraction\Calculation\Papers\NRTL Analysis\More Discussions\Three Phases\two-phase\1\D.xlsx',D); xlswrite('C:\Research-Solvent Extraction\Pharmaceutical extraction\Calculation\Papers\NRTL Analysis\More Discussions\Three Phases\two-phase\1\E.xlsx',E); xlswrite('C:\Research-Solvent Extraction\Pharmaceutical extraction\Calculation\Papers\NRTL Analysis\More Discussions\Three Phases\two-phase\1\F.xlsx',F); xlswrite('C:\Research-Solvent Extraction\Pharmaceutical extraction\Calculation\Papers\NRTL Analysis\More Discussions\Three Phases\two-phase\1\G.xlsx',G); % Save results. toc % Estimate the calculation time.
The following is the function handle: function q=myfun_Random(p) n_1=p(1); n_2=p(2); n_3=p(3); nn_1=p(4); nn_2=p(5); nn_3=p(6); % Six variables to be solved. x1=n_1/(n_1+n_2+n_3); x2=n_2/(n_1+n_2+n_3); x3=n_3/(n_1+n_2+n_3); y1=nn_1/(nn_1+nn_2+nn_3); y2=nn_2/(nn_1+nn_2+nn_3);
132 y3=nn_3/(nn_1+nn_2+nn_3); % Converting moles into mole fractions. t12=1276.2/298.15; t21=-141.25/298.15; t13=1410.6/298.15; t31=683.25/298.15; t23=-247.98/298.15; t32=940.37/298.15; % The energy parameters of the NRTL model. a12=0.3335; a13=0.2163; a23=0.3768; % The non-randomness parameters of the NRTL model. t11=0;t22=0;t33=0; G11=1;G22=1;G33=1;
G12=exp(-a12*t12); G21=exp(-a12*t21); G13=exp(-a13*t13); G31=exp(-a13*t31); G23=exp(-a23*t23); G32=exp(-a23*t32);% Intermediate variables. r_1=exp((t11*G11*x1+t21*G21*x2+t31*G31*x3)/(G11*x1+G21*x2+G31* x3)... +(x1*G11)/(G11*x1+G21*x2+G31*x3)*(t11- (x1*t11*G11+x2*t21*G21+x3*t31*G31)/(G11*x1+G21*x2+G31*x3))... +(x2*G12)/(G12*x1+G22*x2+G32*x3)*(t12- (x1*t12*G12+x2*t22*G22+x3*t32*G32)/(G12*x1+G22*x2+G32*x3))... +(x3*G13)/(G13*x1+G23*x2+G33*x3)*(t13- (x1*t13*G13+x2*t23*G23+x3*t33*G33)/(G13*x1+G23*x2+G33*x3))); r_2=exp((t12*G12*x1+t22*G22*x2+t32*G32*x3)/(G12*x1+G22*x2+G32* x3)... +(x1*G21)/(G11*x1+G21*x2+G31*x3)*(t21- (x1*t11*G11+x2*t21*G21+x3*t31*G31)/(G11*x1+G21*x2+G31*x3))... +(x2*G22)/(G12*x1+G22*x2+G32*x3)*(t22- (x1*t12*G12+x2*t22*G22+x3*t32*G32)/(G12*x1+G22*x2+G32*x3))... +(x3*G23)/(G13*x1+G23*x2+G33*x3)*(t23- (x1*t13*G13+x2*t23*G23+x3*t33*G33)/(G13*x1+G23*x2+G33*x3))); r_3=exp((t13*G13*x1+t23*G23*x2+t33*G33*x3)/(G13*x1+G23*x2+G33* x3)... +(x1*G31)/(G11*x1+G21*x2+G31*x3)*(t31- (x1*t11*G11+x2*t21*G21+x3*t31*G31)/(G11*x1+G21*x2+G31*x3))... +(x2*G32)/(G12*x1+G22*x2+G32*x3)*(t32- (x1*t12*G12+x2*t22*G22+x3*t32*G32)/(G12*x1+G22*x2+G32*x3))... +(x3*G33)/(G13*x1+G23*x2+G33*x3)*(t33- (x1*t13*G13+x2*t23*G23+x3*t33*G33)/(G13*x1+G23*x2+G33*x3)));
% Activity coefficient of components in phase one, mole fractions expressed as x1, x2, x3. rr_1=exp((t11*G11*y1+t21*G21*y2+t31*G31*y3)/(G11*y1+G21*y2+G31 *y3)... +(y1*G11)/(G11*y1+G21*y2+G31*y3)*(t11- (y1*t11*G11+y2*t21*G21+y3*t31*G31)/(G11*y1+G21*y2+G31*y3))...
133 +(y2*G12)/(G12*y1+G22*y2+G32*y3)*(t12- (y1*t12*G12+y2*t22*G22+y3*t32*G32)/(G12*y1+G22*y2+G32*y3))... +(y3*G13)/(G13*y1+G23*y2+G33*y3)*(t13- (y1*t13*G13+y2*t23*G23+y3*t33*G33)/(G13*y1+G23*y2+G33*y3))); rr_2=exp((t12*G12*y1+t22*G22*y2+t32*G32*y3)/(G12*y1+G22*y2+G32 *y3)... +(y1*G21)/(G11*y1+G21*y2+G31*y3)*(t21- (y1*t11*G11+y2*t21*G21+y3*t31*G31)/(G11*y1+G21*y2+G31*y3))... +(y2*G22)/(G12*y1+G22*y2+G32*y3)*(t22- (y1*t12*G12+y2*t22*G22+y3*t32*G32)/(G12*y1+G22*y2+G32*y3))... +(y3*G23)/(G13*y1+G23*y2+G33*y3)*(t23- (y1*t13*G13+y2*t23*G23+y3*t33*G33)/(G13*y1+G23*y2+G33*y3))); rr_3=exp((t13*G13*y1+t23*G23*y2+t33*G33*y3)/(G13*y1+G23*y2+G33 *y3)... +(y1*G31)/(G11*y1+G21*y2+G31*y3)*(t31- (y1*t11*G11+y2*t21*G21+y3*t31*G31)/(G11*y1+G21*y2+G31*y3))... +(y2*G32)/(G12*y1+G22*y2+G32*y3)*(t32- (y1*t12*G12+y2*t22*G22+y3*t32*G32)/(G12*y1+G22*y2+G32*y3))... +(y3*G33)/(G13*y1+G23*y2+G33*y3)*(t33- (y1*t13*G13+y2*t23*G23+y3*t33*G33)/(G13*y1+G23*y2+G33*y3))); % Activity coefficient of components in phase two, mole fractions expressed as y1, y2, y3. q(1)=(x1*r_1)/(y1*rr_1)-1; q(2)=(x2*r_2)/(y2*rr_2)-1; q(3)=(x3*r_3)/(y3*rr_3)-1; % Three isoactivity equations. q(4)=n_1+nn_1-n1; q(5)=n_2+nn_2-n2; q(6)=n_3+nn_3-n3; % Three mass balance constraints.
134 C. MATLAB Code for Regression of NRTL Parameters
The MATLAB code has been provided for regressing the NRTL parameters using the particle swarm optimization (PSO). The main program has two steps: optimizing the pre-regressed parameters and optimizing the final parameters. Three function handles are used in the main program.
The following program is the main code: tic; % This is the first step of the program, it produces the pre-regressed parameters. Array=[]; for sst=1:100 % The number of groups of parameters (particles) to be produced. sst c1 = 1.49445; c2 = 1.49445; maxgen=50; % The number of iterations. sizepop=200; % The size of population.
popmax=[15,15,15,15,15,15] ;% Upper limit of parameters. popmin=-popmax ;% Lower limit of parameters. Vmax=(popmax-popmin)/6;% Controlling moving velocity of each particle. Vmin=-Vmax; dim=6; for i=1:sizepop pop(i,:)=(popmax-popmin).*rand(1,dim)+popmin; V(i,:)=(Vmax-Vmin).*rand(1,dim)+Vmin; fitness(i)=myfun(pop(i,:)); end % Producing the initial locations and speeds of particles, and calculating their corresponding values with respect to the objective function.
% Producing pbest and gbest, and their corresponding objective function values. [bestfitness bestindex]=min(fitness); zbest=pop(bestindex,:); gbest=pop; fitnessgbest=fitness; fitnesszbest=bestfitness;
% Iterations for optimization. for i=1:maxgen i for j=1:sizepop
% Update of velocity. V(j,:) = V(j,:) + c1*rand*(gbest(j,:) - pop(j,:)) + c2*rand*(zbest - pop(j,:)); V(j,find(V(j,:)>Vmax))=Vmax(find(V(j,:)>Vmax)); V(j,find(V(j,:) 135 % Update of location. pop(j,:)=pop(j,:)+0.2*V(j,:); pop(j,find(pop(j,:)>popmax))=popmax(find(pop(j,:)>popmax)); pop(j,find(pop(j,:) % Variation of particles for maintaining diversity of particles. pos=unidrnd(dim); if rand>0.95 pop(j,pos)=(popmax(pos)- popmin(pos))*rand(1,1)+popmin(pos); end % The objective function. fitness(j)=myfun(pop(j,:)); end for j=1:sizepop % Update of objective function value for each particle. if fitness(j) < fitnessgbest(j) gbest(j,:) = pop(j,:); fitnessgbest(j) = fitness(j); end % Update of the global best objective function value. if fitness(j) < fitnesszbest zbest = pop(j,:); fitnesszbest = fitness(j); end end yy(i)=fitnesszbest; end zbest fitnesszbest Array=[Array;zbest]; end dlmwrite('result.txt',Array); % Save the results, which are to be used in the second step. toc; clc clear tic; warning off load result.txt % Load the result obtain from the first step. c1 = 1.49445; c2 = 1.49445; % Two parameters. 136 maxgen=1000; % The number of iterations. sizepop=size(result,1); % The size of population. popmax=[15,15,15,15,15,15];% Limit of parameter range. popmin=-popmax ; Vmax=(popmax-popmin);% Limit of moving velocity of each particle. Vmin=-Vmax; dim=6; pop=result; xlswrite('C:\Research-Solvent Extraction\Pharmaceutical extraction\Calculation\Papers\NRTL parameter optimization\20140302NRTL Optimization-Chen 2000\weight factor\0.2\pop_1.xlsx',pop); % The initial particles are saved for comparison with final results. for i=1:sizepop V(i,:)=(Vmax-Vmin).*rand(1,dim)+Vmin; fitness(i)=myfunA(pop(i,:)); F(i)=fitness(i); % Producing the initial velocity of particles and their corresponding objective function values. end xlswrite('C:\Research-Solvent Extraction\Pharmaceutical extraction\Calculation\Papers\NRTL parameter optimization\20140302NRTL Optimization-Chen 2000\weight factor\0.2\Fitness_1.xlsx',F'); % Producing pbest and gbest, and their corresponding objective function values. [bestfitness bestindex]=min(fitness); zbest=pop(bestindex,:); gbest=pop; fitnessgbest=fitness; fitnesszbest=bestfitness; % Iterations for optimization. for i=1:maxgen i for j=1:sizepop % Update of velocity. V(j,:) = 0.2*V(j,:) + c1*rand*(gbest(j,:) - pop(j,:)) + c2*rand*(zbest - pop(j,:)); % w V(j,find(V(j,:)>Vmax))=Vmax(find(V(j,:)>Vmax)); V(j,find(V(j,:) % Update of locations. pop(j,:)=pop(j,:)+V(j,:); pop(j,find(pop(j,:)>popmax))=popmax(find(pop(j,:)>popmax)); pop(j,find(pop(j,:) % Variation of particles for maintaining diversity of particles. pos=unidrnd(dim); 137 if rand>0.95 pop(j,pos)=(popmax(pos)- popmin(pos))*rand(1,1)+popmin(pos); end % Oblective functions values of particles. fitness(j)=myfunA(pop(j,:)); end for j=1:sizepop % The best objective function value of a single particle. if fitness(j) < fitnessgbest(j) gbest(j,:) = pop(j,:); fitnessgbest(j) = fitness(j); end % The global best objective function value. if fitness(j) < fitnesszbest zbest = pop(j,:) % Display parameters. fitnesszbest = fitness(j); end end yy(i)=fitnesszbest % Display results. F(1,i)=yy(i); end zbest fitnesszbest % Save the results. xlswrite('C:\Research-Solvent Extraction\Pharmaceutical extraction\Calculation\Papers\NRTL parameter optimization\20140302NRTL Optimization-Chen 2000\weight factor\0.2\Fitness_2.xlsx',F'); % Save objective function values during iterations. plot(yy) title(['Optimization' 'Number of Iterations' num2str(maxgen)]); xlabel(' Number of Iterations ');ylabel('Objective Function'); % Plot the results. xlswrite('C:\Research-Solvent Extraction\Pharmaceutical extraction\Calculation\Papers\NRTL parameter optimization\20140302NRTL Optimization-Chen 2000\weight factor\0.2\pop_2.xlsx',pop); % Save locations of particles during iterations. toc; 138 The following program is the first function handle: function total=myfun(data) A=[ 0.8309 0.0137 0.1554 0.0825 0.0034 0.9141 0.7912 0.0403 0.1685 0.0850 0.0103 0.9047 0.7332 0.0778 0.1890 0.0872 0.0221 0.8907 0.6650 0.1123 0.2227 0.0945 0.0362 0.8693 0.5885 0.1413 0.2702 0.1107 0.0518 0.8375 0.5762 0.1450 0.2788 0.1149 0.0587 0.8264]; % Experimental data. tao=zeros(3,3); tao(1,2)=data(1); tao(2,1)=data(2); tao(1,3)=data(3); tao(3,1)=data(4); tao(2,3)=data(5); tao(3,2)=data(6); % Energy parameters to be regressed. a12=0.4; a13=0.4; a23=0.4; % Non-randomness parameters. They can be either fixed or regressed together with the energy parameters. G=ones(3,3); G(1,2)=exp(-a12*tao(1,2)); G(2,1)=exp(-a12*tao(2,1)); G(1,3)=exp(-a13*tao(1,3)); G(3,1)=exp(-a13*tao(3,1)); G(2,3)=exp(-a23*tao(2,3)); G(3,2)=exp(-a23*tao(3,2)); % Intermediate parameters. total=0; for ks=1:6 x=A(ks,1:3)'; y=A(ks,4:6)'; rI1=sum(tao(:,1).*G(:,1).*x)/sum(G(:,1).*x)+x(1)*G(1,1)/sum(G( :,1).*x)*(tao(1,1)-sum(x.*tao(:,1).*G(:,1))/sum(G(:,1).*x))... +x(2)*G(1,2)/sum(G(:,2).*x)*(tao(1,2)- sum(x.*tao(:,2).*G(:,2))/sum(G(:,2).*x))... +x(3)*G(1,3)/sum(G(:,3).*x)*(tao(1,3)- sum(x.*tao(:,3).*G(:,3))/sum(G(:,3).*x)); rI2=sum(tao(:,2).*G(:,2).*x)/sum(G(:,2).*x)+x(1)*G(2,1)/sum(G( :,1).*x)*(tao(2,1)-sum(x.*tao(:,1).*G(:,1))/sum(G(:,1).*x))... +x(2)*G(2,2)/sum(G(:,2).*x)*(tao(2,2)- sum(x.*tao(:,2).*G(:,2))/sum(G(:,2).*x))... +x(3)*G(2,3)/sum(G(:,3).*x)*(tao(2,3)- sum(x.*tao(:,3).*G(:,3))/sum(G(:,3).*x)); 139 rI3=sum(tao(:,3).*G(:,3).*x)/sum(G(:,3).*x)+x(1)*G(3,1)/sum(G( :,1).*x)*(tao(3,1)-sum(x.*tao(:,1).*G(:,1))/sum(G(:,1).*x))... +x(2)*G(3,2)/sum(G(:,2).*x)*(tao(3,2)- sum(x.*tao(:,2).*G(:,2))/sum(G(:,2).*x))... +x(3)*G(3,3)/sum(G(:,3).*x)*(tao(3,3)- sum(x.*tao(:,3).*G(:,3))/sum(G(:,3).*x)); rI=[rI1,rI2,rI3]; rII1=sum(tao(:,1).*G(:,1).*y)/sum(G(:,1).*y)+y(1)*G(1,1)/sum(G (:,1).*y)*(tao(1,1)-sum(y.*tao(:,1).*G(:,1))/sum(G(:,1).*y))... +y(2)*G(1,2)/sum(G(:,2).*y)*(tao(1,2)- sum(y.*tao(:,2).*G(:,2))/sum(G(:,2).*y))... +y(3)*G(1,3)/sum(G(:,3).*y)*(tao(1,3)- sum(y.*tao(:,3).*G(:,3))/sum(G(:,3).*y)); rII2=sum(tao(:,2).*G(:,2).*y)/sum(G(:,2).*y)+y(1)*G(2,1)/sum(G (:,1).*y)*(tao(2,1)-sum(y.*tao(:,1).*G(:,1))/sum(G(:,1).*y))... +y(2)*G(2,2)/sum(G(:,2).*y)*(tao(2,2)- sum(y.*tao(:,2).*G(:,2))/sum(G(:,2).*y))... +y(3)*G(2,3)/sum(G(:,3).*y)*(tao(2,3)- sum(y.*tao(:,3).*G(:,3))/sum(G(:,3).*y)); rII3=sum(tao(:,3).*G(:,3).*y)/sum(G(:,3).*y)+y(1)*G(3,1)/sum(G (:,1).*y)*(tao(3,1)-sum(y.*tao(:,1).*G(:,1))/sum(G(:,1).*y))... +y(2)*G(3,2)/sum(G(:,2).*y)*(tao(3,2)- sum(y.*tao(:,2).*G(:,2))/sum(G(:,2).*y))... +y(3)*G(3,3)/sum(G(:,3).*y)*(tao(3,3)- sum(y.*tao(:,3).*G(:,3))/sum(G(:,3).*y)); rII=[rII1,rII2,rII3]; % Activity coefficient in the NRTL model. for i=1:3 m(i)=x(i)*exp(rI(i))./(y(i)*exp(rII(i))); if m(i)<1 m(i)=1./m(i); else m(i)=m(i); end total=total+(abs(m(i)-1)); % Objective function of the first step. end end The following program is the second function handle: function RMSD=myfunA(dataIn) global canshu canshu=dataIn; A=[ 0.8309 0.0137 0.1554 0.0825 0.0034 0.9141 0.7912 0.0403 0.1685 0.0850 0.0103 0.9047 0.7332 0.0778 0.1890 0.0872 0.0221 0.8907 0.6650 0.1123 0.2227 0.0945 0.0362 0.8693 0.5885 0.1413 0.2702 0.1107 0.0518 0.8375 0.5762 0.1450 0.2788 0.1149 0.0587 0.8264]; 140 % Experimental data. for K=1:6 n1 = A(K,1)+A(K,4); save n1.mat n1 n2 = A(K,2)+A(K,5); save n2.mat n2 n3 = A(K,3)+A(K,6); save n3.mat n3 % Specify the total amount of each component. X=fsolve(@myfun_3Com,A(K,:),optimset('Display','off')); % Solving the isoactivity equations. C(K,1)=X(1,1)./(X(1,1)+X(1,2)+X(1,3)); C(K,2)=X(1,2)./(X(1,1)+X(1,2)+X(1,3)); C(K,3)=X(1,3)./(X(1,1)+X(1,2)+X(1,3)); C(K,4)=X(1,4)./(X(1,4)+X(1,5)+X(1,6)); C(K,5)=X(1,5)./(X(1,4)+X(1,5)+X(1,6)); C(K,6)=X(1,6)./(X(1,4)+X(1,5)+X(1,6)); % Convert the solutions from moles to mole fractions. end D=(A-C).^2; RMSD=(sum(D(:))/36)^0.5; % Calculate the RMSD. The following program is the third function handle: function q=myfun_3Com(p) global canshu load n1 load n2 load n3 % Load the total amount of each component. n11=p(1); n12=p(2); n13=p(3); n21=p(4); n22=p(5); n23=p(6); % The six variables to be solved. x1=n11./(n11+n12+n13); x2=n12./(n11+n12+n13); x3=n13./(n11+n12+n13); y1=n21./(n21+n22+n23); y2=n22./(n21+n22+n23); y3=n23./(n21+n22+n23); 141 % Moles are converted into mole fractions. a12=0.4; a13=0.4; a23=0.4; % Non-randomness parameters. B=canshu; t12=B(1); t21=B(2); t13=B(3); t31=B(4); t23=B(5); t32=B(6); % Energy parameters. t11=0;t22=0;t33=0; G11=1;G22=1;G33=1; G12=exp(-a12*t12); G21=exp(-a12*t21); G13=exp(-a13*t13); G31=exp(-a13*t31); G23=exp(-a23*t23); G32=exp(-a23*t32); % Intermediate variables. r_1=exp((t11.*G11.*x1+t21.*G21.*x2+t31.*G31.*x3)./(G11.*x1+G21 .*x2+G31.*x3)... +(x1.*G11)./(G11.*x1+G21.*x2+G31.*x3).*(t11- (x1.*t11.*G11+x2.*t21.*G21+x3.*t31.*G31)./(G11.*x1+G21.*x2+G31 .*x3))... +(x2.*G12)./(G12.*x1+G22.*x2+G32.*x3).*(t12- (x1.*t12.*G12+x2.*t22.*G22+x3.*t32.*G32)./(G12.*x1+G22.*x2+G32 .*x3))... +(x3.*G13)./(G13.*x1+G23.*x2+G33.*x3).*(t13- (x1.*t13.*G13+x2.*t23.*G23+x3.*t33.*G33)./(G13.*x1+G23.*x2+G33 .*x3))); r_2=exp((t12.*G12.*x1+t22.*G22.*x2+t32.*G32.*x3)./(G12.*x1+G22 .*x2+G32.*x3)... +(x1.*G21)./(G11.*x1+G21.*x2+G31.*x3).*(t21- (x1.*t11.*G11+x2.*t21.*G21+x3.*t31.*G31)./(G11.*x1+G21.*x2+G31 .*x3))... +(x2.*G22)./(G12.*x1+G22.*x2+G32.*x3).*(t22- (x1.*t12.*G12+x2.*t22.*G22+x3.*t32.*G32)./(G12.*x1+G22.*x2+G32 .*x3))... +(x3.*G23)./(G13.*x1+G23.*x2+G33.*x3).*(t23- (x1.*t13.*G13+x2.*t23.*G23+x3.*t33.*G33)./(G13.*x1+G23.*x2+G33 .*x3))); r_3=exp((t13.*G13.*x1+t23.*G23.*x2+t33.*G33.*x3)./(G13.*x1+G23 .*x2+G33.*x3)... +(x1.*G31)./(G11.*x1+G21.*x2+G31.*x3).*(t31- (x1.*t11.*G11+x2.*t21.*G21+x3.*t31.*G31)./(G11.*x1+G21.*x2+G31 .*x3))... +(x2.*G32)./(G12.*x1+G22.*x2+G32.*x3).*(t32- (x1.*t12.*G12+x2.*t22.*G22+x3.*t32.*G32)./(G12.*x1+G22.*x2+G32 .*x3))... 142 +(x3.*G33)./(G13.*x1+G23.*x2+G33.*x3).*(t33- (x1.*t13.*G13+x2.*t23.*G23+x3.*t33.*G33)./(G13.*x1+G23.*x2+G33 .*x3))); % Activity coefficients of components in phase one, mole fractions expressed as x1, x2, x3. rr_1=exp((t11.*G11.*y1+t21.*G21.*y2+t31.*G31.*y3)./(G11.*y1+G2 1.*y2+G31.*y3)... +(y1.*G11)./(G11.*y1+G21.*y2+G31.*y3).*(t11- (y1.*t11.*G11+y2.*t21.*G21+y3.*t31.*G31)./(G11.*y1+G21.*y2+G31 .*y3))... +(y2.*G12)./(G12.*y1+G22.*y2+G32.*y3).*(t12- (y1.*t12.*G12+y2.*t22.*G22+y3.*t32.*G32)./(G12.*y1+G22.*y2+G32 .*y3))... +(y3.*G13)./(G13.*y1+G23.*y2+G33.*y3).*(t13- (y1.*t13.*G13+y2.*t23.*G23+y3.*t33.*G33)./(G13.*y1+G23.*y2+G33 .*y3))); rr_2=exp((t12.*G12.*y1+t22.*G22.*y2+t32.*G32.*y3)./(G12.*y1+G2 2.*y2+G32.*y3)... +(y1.*G21)./(G11.*y1+G21.*y2+G31.*y3).*(t21- (y1.*t11.*G11+y2.*t21.*G21+y3.*t31.*G31)./(G11.*y1+G21.*y2+G31 .*y3))... +(y2.*G22)./(G12.*y1+G22.*y2+G32.*y3).*(t22- (y1.*t12.*G12+y2.*t22.*G22+y3.*t32.*G32)./(G12.*y1+G22.*y2+G32 .*y3))... +(y3.*G23)./(G13.*y1+G23.*y2+G33.*y3).*(t23- (y1.*t13.*G13+y2.*t23.*G23+y3.*t33.*G33)./(G13.*y1+G23.*y2+G33 .*y3))); rr_3=exp((t13.*G13.*y1+t23.*G23.*y2+t33.*G33.*y3)./(G13.*y1+G2 3.*y2+G33.*y3)... +(y1.*G31)./(G11.*y1+G21.*y2+G31.*y3).*(t31- (y1.*t11.*G11+y2.*t21.*G21+y3.*t31.*G31)./(G11.*y1+G21.*y2+G31 .*y3))... +(y2.*G32)./(G12.*y1+G22.*y2+G32.*y3).*(t32- (y1.*t12.*G12+y2.*t22.*G22+y3.*t32.*G32)./(G12.*y1+G22.*y2+G32 .*y3))... +(y3.*G33)./(G13.*y1+G23.*y2+G33.*y3).*(t33- (y1.*t13.*G13+y2.*t23.*G23+y3.*t33.*G33)./(G13.*y1+G23.*y2+G33 .*y3))); % Activity coefficients of components in phase two, mole fractions expressed as y1, y2, y3. q(1)=(y1.*rr_1)./(x1.*r_1)-1; q(2)=(y2.*rr_2)./(x2.*r_2)-1; q(3)=(y3.*rr_3)./(x3.*r_3)-1; % Three isoactivity equations. q(4)=n11+n21-n1; q(5)=n12+n22-n2; q(6)=n13+n23-n3; % Three mass balance constraits. 143 D. Mathematica Code for Plotting Isoactivity Equations The Mathematica code has been provided for plotting the isoactivity equations in a three dimensional space with the mass balance constraints of specifying total amount of each component. This code is also applicable for conditions of specifying concentrations of components in one phase with minor modifications. G11=1;G22=1;G33=1; t11=0;t22=0;t33=0; a12=0.3335;a13=0.2163;a23=0.3768; t12=1276.2/298.15;t21=-141.25/298.15; t13=1410.6/298.15;t31=683.25/298.15; t23=-247.98/298.15;t32=940.37/298.15; G12=Exp[-a12 t12];G21=Exp[-a12 t21]; G13=Exp[-a13 t13];G31=Exp[-a13 t31]; G23=Exp[-a23 t23];G32=Exp[-a23 t32]; Dx1=(x1 t11 G11+x2 t21 G21+x3 t31 G31)/(G11 x1+G21 x2+G31 x3); Dx2=(x1 t12 G12+x2 t22 G22+x3 t32 G32)/(G12 x1+G22 x2+G32 x3); Dx3=(x1 t13 G13+x2 t23 G23+x3 t33 G33)/(G13 x1+G23 x2+G33 x3); Dy1=(y1 t11 G11+y2 t21 G21+y3 t31 G31)/(G11 y1+G21 y2+G31 y3); Dy2=(y1 t12 G12+y2 t22 G22+y3 t32 G32)/(G12 y1+G22 y2+G32 y3); Dy3=(y1 t13 G13+y2 t23 G23+y3 t33 G33)/(G13 y1+G23 y2+G33 y3); nn1=0.4361+0.9877-n1; nn2=0.4205+0.0114-n2; nn3=0.1443-n3; x1=n1/(n1+n2+n3); x2=n2/(n1+n2+n3); x3=n3/(n1+n2+n3); y1=nn1/(nn1+nn2+nn3); y2=nn2/(nn1+nn2+nn3); y3=nn3/(nn1+nn2+nn3); r11=Dx1+x1 G11/(G11 x1+G21 x2+G31 x3)*(t11-Dx1)+x2 G12/(G12 x1+G22 x2+G32 x3)*(t12-Dx2)+ x3 G13/(G13 x1+G23 x2+G33 x3)*(t13-Dx3); r12=Dx2+x1 G21/(G11 x1+G21 x2+G31 x3)*(t21-Dx1)+ x2 G22/(G12 x1+G22 x2+G32 x3)*(t22-Dx2)+ x3 G23/(G13 x1+G23 x2+G33 x3)*(t23-Dx3); r13=Dx3+x1 G31/(G11 x1+G21 x2+G31 x3)*(t31-Dx1)+x2 G32/(G12 x1+G22 x2+G32 x3)*(t32-Dx2)+ x3 G33/(G13 x1+G23 x2+G33 x3)*(t33-Dx3); r21=Dy1+y1 G11/(G11 y1+G21 y2+G31 y3)*(t11-Dy1)+y2 G12/(G12 y1+G22 y2+G32 y3)*(t12-Dy2)+ y3 G13/(G13 y1+G23 y2+G33 y3)*(t13-Dy3); 144 r22=Dy2+y1 G21/(G11 y1+G21 y2+G31 y3)*(t21-Dy1)+y2 G22/(G12 y1+G22 y2+G32 y3)*(t22-Dy2)+ y3 G23/(G13 y1+G23 y2+G33 y3)*(t23-Dy3); r23=Dy3+y1 G31/(G11 y1+G21 y2+G31 y3)*(t31-Dy1)+y2 G32/(G12 y1+G22 y2+G32 y3)*(t32-Dy2)+ y3 G33/(G13 y1+G23 y2+G33 y3)*(t33-Dy3); z1[n1_,n2_,n3_]:=Log[x1/y1]+r11-r21; z2[n1_,n2_,n3_]:=Log[x2/y2]+r12-r22; z3[n1_,n2_,n3_]:=Log[x3]-Log[y3]+r13-r23; img1 = ContourPlot3D[z1[n1, n2, n3] == 0, {n1, 0.00001, 1.4238}, {n2, 0.00001, 0.4319}, {n3, 0.00001, 0.1443}, PlotRange -> {{0, 1.4238}, {0, 0.4319}, {0, 0.1443}}, MeshStyle -> None, ContourStyle -> {Opacity[0.9], Red}, BoundaryStyle ->None, AxesLabel -> {n1, n2, n3}] img2 = ContourPlot3D[z2[n1, n2, n3] == 0, {n1, 0.00001, 1.4238}, {n2, 0.00001, 0.4319}, {n3, 0.00001, 0.1443}, PlotRange -> {{0, 1.4238}, {0, 0.4319}, {0, 0.1443}}, MeshStyle -> None, ContourStyle -> {Opacity[0.9], Green}, BoundaryStyle -> None, AxesLabel -> {n1, n2, n3}] img3 = ContourPlot3D[z3[n1, n2, n3] == 0, {n1, 0.00001, 1.4238}, {n2, 0.00001, 0.4319}, {n3, 0.00001, 0.1443}, PlotRange -> {{0, 1.4238}, {0, 0.4319}, {0, 0.1443}}, MeshStyle -> None, ContourStyle -> {Opacity[0.9], Blue}, BoundaryStyle -> None, AxesLabel -> {n1, n2, n3}] line1 = ContourPlot3D[z2[n1, n2, n3] == 0, {n1, 0.00001, 1.4238}, {n2, 0.00001, 0.4319}, {n3, 0.00001, 0.1443}, PlotRange -> {{0, 1.4238}, {0, 0.4319}, {0, 0.1443}}, Mesh -> {{0}, {0}, {0}}, MeshFunctions -> Function[{n1, n2, n3}, z1[n1, n2, n3]], MeshStyle -> {Thick, Blue}, ContourStyle -> None, BoundaryStyle -> None, LabelStyle -> {Bold, Black, 18}] line2 = ContourPlot3D[z3[n1, n2, n3] == 0, {n1, 0.0001, 1.4238}, {n2, 0.0001, 0.4319}, {n3, 0.0001, 0.1443}, PlotRange -> {{0, 1.4238}, {0, 0.4319}, {0, 0.1443}}, Mesh -> {{0}, {0}, {0}}, MeshFunctions -> Function[{n1, n2, n3}, z1[n1, n2, n3]], MeshStyle -> {Thick, Green}, ContourStyle -> None, BoundaryStyle -> None, LabelStyle -> {Bold, Black, 18}] line3 = ContourPlot3D[z3[n1, n2, n3] == 0, {n1, 0.00001, 1.4238}, {n2, 0.00001, 0.4319}, {n3, 0.00001, 0.1443}, PlotRange -> {{0, 1.4238}, {0, 0.4319}, {0, 0.1443}}, Mesh -> {{0}, {0}, {0}}, MeshFunctions -> Function[{n1, n2, n3}, z2[n1, n2, n3]], MeshStyle -> {Thick, Red}, ContourStyle -> None, BoundaryStyle -> None, LabelStyle -> {Bold, Black, 18}] Show[img1, img2, img3, LabelStyle -> {Bold, Black, 18}] cal = ListPointPlot3D[{{0.4340, 0.4189, 0.1426}}, AxesLabel -> {n1, n2, n3}, PlotStyle -> Directive[Opacity[0.99], Magenta, PointSize[0.025]]] 145 sym = ListPointPlot3D[{{0.9898, 0.0130, 0.0017}}, AxesLabel -> {n1, n2, n3}, PlotStyle -> Directive[Opacity[0.99], Magenta, PointSize[0.025]]] Show[img1, img2, img3, cal, sym, LabelStyle -> {Bold, Black, 18}] 146 E. List of Publications During PhD. Study [1]. Li, Z.; Mumford, K. A.; Shang, Y.; Smith, K. H.; Chen, J.; Wang, Y.; Stevens, G. W., Analysis of the non-random two-liquid model for prediction of liquid–liquid equilibria. J. Chem. Eng. Data 2014, 59, (8), 2485-2489. (Selected as cover). [2]. Li, Z.; Mumford, K. A.; Shang, Y.; Smith, K. H.; Chen, J.; Wang, Y.; Stevens, G. W., Extraction of phenol by toluene in the presence of sodium hydroxide. Sep. Sci. Technol 2014, 49, (18), 2913-2920. [3] Li, Z.; Mumford, K.A.; Smith, K.H.; Chen, J.; Wang, Y.; Stevens, G.W., Reply to “Comments on ‘Analysis of the Nonrandom Two-Liquid Model for Prediction of Liquid– Liquid Equilibria’”. J. Chem. Eng. Data 2015, 60, (5), 1530-1531.. [4] Li, Z.; Smith, K.H.; Mumford, K.A.; Wang, Y.; Stevens, G.W., Regression of NRTL parameters from ternary liquid–liquid equilibria using particle swarm optimization and discussions. Fluid Phase Equilib. 2015, 398, 36-45. [5] Li, Z.; Smith, K.H.; Stevens, G.W., The use of environmentally sustainable bio-derived solvents in solvent extraction applications-a review. Chinese J Chem Eng. Accepted. doi:10.1016/j.cjche.2015.07.021. [6]. Li, Z.; Mumford, K. A.; Smith, K. H.; Chen, J.; Wang, Y.; Stevens, G. W., Solution structure of isoactivity equations using the non-random two-liquid model for liquid-liquid equilibrium calculations. In preparation. 147 Minerva Access is the Institutional Repository of The University of Melbourne Author/s: Li, Zheng Title: Thermodynamic modelling of liquid–liquid equilibria using the nonrandom two-liquid model and its applications Date: 2015 Persistent Link: http://hdl.handle.net/11343/57353 File Description: Thermodynamic Modelling of Liquid–liquid Equilibria Using the Nonrandom Two-Liquid Model and Its Applications