Thermodynamics of Phase Equilibria in Food Engineering 9780128115572, 0128115572, 9780128115565

Thermodynamics of Phase Equilibria in Food Engineering is the definitive book on thermodynamics of equilibrium applied t

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Thermodynamics of Phase Equilibria in Food Engineering
 9780128115572, 0128115572, 9780128115565

Table of contents :
Content: Part A: Introduction: 1. Phase equilibria in the food industry / Camila Gambini Pereira --
2. Fundamentals of phase equilibria / Camila Gambini Pereira --
3. Classical models, part 1 : Cubic equations of state and applications / Vladimir F. Cabral, Marcelo Castier, Lúcio Cardozo-Filho --
4. Classical models, part 2 : Activity coefficient models and applications / Leandro Danielski, Luiz Stragevitch --
5. Advanced models : Association theories and models / Fèlix Llovell. Part B: Phase equilibria and their applications: 6. Vapor-liquid equilibrium in food processes / Camila Gambini Pereira --
7. Liquid-liquid and vapor-liquid-liquid equilibrium in food processes / Guilherme J. Maximo, Marcela C. Ferreira, Simone Shiozawa, Larissa C.B.A. Bessa, Antonio J.A. Meirelles, Eduardo A.C. Batista --
8. Solid-liquid equilibrium in food processes / Guilherme J.C. Maximo, Natália D.D. Cararerto, Mariana C. Costa --
9. Equilibrium in pressurized systems (sub and supercritical) / Mariana Fortunatti Montoya, Francisco A. Sánchez, Selva Pereda. Part C: Advanced topics: 10. Phase transition in foods / Camila Gambini Pereira --
11. Molecular thermodynamics of protein systems / Robin A. Curtis --
12. Equilibrium in colloidal systems / Maria G. Semenova --
13. Equilibrium in electrolyte systems / Oscar Rodríguez, Elena Gómez, Noelia Calvar, Eugénia A. Macedo --
14. Phase equilibrium of organogels / Juliana N.R. Ract, Richtier G. da Cruz, Camila Gambini Pereira --
15. Thermodynamics of reactions in food systems / Saartje Hernalsteens, Camila Gambini Pereira.

Citation preview

Thermodynamics of Phase Equilibria in Food Engineering

Thermodynamics of Phase Equilibria in Food Engineering

Edited by

CAMILA GAMBINI PEREIRA Department of Chemical Engineering, Federal University of Rio Grande do Norte, Natal, Brazil

Academic Press is an imprint of Elsevier 125 London Wall, London EC2Y 5AS, United Kingdom 525 B Street, Suite 1650, San Diego, CA 92101, United States 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom Copyright r 2019 Elsevier Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress ISBN: 978-0-12-811556-5 For Information on all Academic Press publications visit our website at https://www.elsevier.com/books-and-journals

Publisher: Andre Gerhard Wolff Acquisition Editor: Nina Rosa de Araujo Bandeira Editorial Project Manager: Mariana L. Kuhl Production Project Manager: Denny Mansingh Cover Designer: Greg Harris Typeset by MPS Limited, Chennai, India

PREFACE Food engineering is a well-established field of engineering that also brings with it knowledge from science (biological and food sciences) and applied chemistry. The multidisciplinary character of this field implies that there is an interconnection of each segment so that the desired result in the processing of a food is obtained. Foods are rich source of compounds with wide applications in the food, pharmaceutical and chemical industries. Colorings, flavorings, and several functional compounds are the basis for many products. These various types of molecules in different states found in materials or systems demand the use of specific conditions when working with food. In the industry, the separation and isolation of these compounds is comprised of a number of separation steps. Single or multistage, these processes require important information for their implementation: the knowledge of phase equilibria of the involved compounds. Moreover, structural and sensorial changes resulting from food processing, including thermal treatments, can be explained by the effects of phase transition and/or biochemical reactions. The complexity of foods is still related to the inter and intramolecular interactions observed between different types of compounds. Separation of macromolecules, such as proteins and carbohydrates, is a special subject of discussion between thermodynamic researchers. Systems containing electrolytes, such as ionic liquid solvents, have been investigated in solutions containing food ingredients. Furthermore, new materials have been developed with the aim of improving the sensorial and stability characteristics in food, as is the case of organogels. As seen, the implications of the diversities inherent in food processing from the point of view of product engineering and process engineering reflect the high specificity that one must have when processing such products. And in this sense, working with food engineering to obtain quality products within the required specifications has become a challenge for both the industry and scientists in this area. Over the last years, the advances of thermodynamic models and their utilization in food systems have expanded the studies of researchers and food engineers. Parallel to this, within the industry, one notices the involvement of a wide range of information so that there is the correct domain in the processing of a given food. In this scenery, phase xi

Preface

equilibrium plays a fundamental role in food engineering, both in the separation processes as well as in the development of new formulations and quality control of food products. The purpose of this book is to present and discuss both the theory of phase equilibrium and its applications in the food industry. The core idea is to provide support to researchers from academia and industry to better understand the behavior of various food systems, and to develop ways to optimize/model or predict the obtaining of food compounds or desired products. The book contains 15 chapters which essentially focus on phase equilibria and their several uses aimed at the food industry and the products coming from there. The book is divided into three large sections. In Part A: Introduction (Chapters 1 5), the theoretical basis of thermodynamics of phase equilibria is presented, with an in-depth discussion of equations of state, activity coefficient models, associative thermodynamic models and applications. In Part B of the book: Phase Equilibria and Their Applications (Chapters 6 9), the chapters develop a specific discussion of the different possibilities of phase equilibrium applied to food systems: vapor liquid equilibrium (VLE), liquid liquid equilibrium (LLE), liquid liquid vapor equilibrium (LLVE), solid liquid equilibrium (SLE), and sub/supercritical equilibrium. In Part C of the book: Advanced Topics (Chapters 10 15), the emphasis is on the advanced applications of phase equilibria in specific food systems, such as studies on phase transition, molecular analysis of proteins, evaluation of equilibrium in colloidal and electrolyte systems, phase equilibria containing organogels, and finalizing with a discussion on the thermodynamic aspects of reactions found in food systems. Moreover, throughout the book, case studies addressing key topics of equilibrium thermodynamics are presented, providing a more in-depth discussion of phase equilibrium applications within the food industry. This book is intended for engineers and food scientists who face simple or complex situations involving phase equilibria found in food products and processing. This book aims to contribute to broadening the understanding of these systems and to assist in the development of products and process designs in the various areas and sectors related to foods. Camila Gambini Pereira Department of Chemical Engineering, Federal University of Rio Grande do Norte, Natal, Brazil

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ABOUT THE EDITOR Camila Gambini Pereira is professor of food and chemical engineering, at the Department of Chemical Engineering, Federal University of Rio Grande do Norte (UFRN), Brazil, where she has been teaching Thermodynamics, Unit Operations, and Transport Phenomena since 2006. She received a PhD in food engineering from University of Campinas in 2005 and an MSc from the same university in 2000. In 2013, she carried out research activities in the Department of Thermodynamics and Molecular Simulation, IFP Energies Nouvelles, France, whose cooperation has been extended in the following years. Dr. Pereira is the leader of the Research Group “Supercritical Technology applied to Natural Products and Biodiesel Production,” and responsible for the Laboratory of Separation Processes in Food at UFRN. She is the author of several publications, 11 chapters on thermodynamics, separation processes and fundamental engineering calculations, and co-editor of a book on Fundamentals of Food Engineering. Her current research areas are food, pharmaceuticals, chemicals, biofuel, environment, and biotechnology.

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ACKNOWLEDGMENTS Firstly, I would like to thank all contributors for their involvement and dedication in participating in this assignment. I am deeply grateful to all of them. I also thank the Department of Chemical Engineering of the Federal University of Rio Grande do Norte for allowing me to use part of the working hours at university to organize this book. Furthermore, during the editing, I have been assisted by many colleagues who aided directly and indirectly in the conception of this book. I particularly thank M. Angela de Almeida Meireles, John M. Prausnitz, Joa˜o A.P. Coutinho, Selva Pereda, and Jean-Charles de Hemptinne for their helpful advices and comments. Finally, I would especially like to thank Xavier Courtial for the valuable and productive discussions held throughout the preparation of this book. To all, my sincere gratitude and appreciation. Camila Gambini Pereira Natal, Brazil

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LIST OF CONTRIBUTORS Eduardo A.C. Batista

School of Food Engineering, University of Campinas, Campinas, Brazil Larissa C.B.A. Bessa

School of Food Engineering, University of Campinas, Campinas, Brazil Vladimir F. Cabral

Department of Food Engineering, Universidade Estadual de Maringa´, Maringa´, Brazil Noelia Calvar

Department of Chemical Engineering, University of Porto, Porto, Portugal Nata´lia D.D. Carareto

Institut de Sciences et Technologie de Valenciennes, Universite´ de Valenciennes et du Hainaut-Cambre´sis, Cambrai, France Lu´cio Cardozo-Filho

Nu´cleo de Pesquisa, Centro Universita´rio Fundac¸a˜o de Ensino Octa´vio Bastos, Sa˜o Joa˜o da Boa Vista, Brazil; Department of Chemical Engineering, Universidade Estadual de Maringa´, Maringa´, Brazil Marcelo Castier

Chemical Engineering Program, Texas A&M University at Qatar, Doha, Qatar Mariana C. Costa

Department of Process and Product Design, University of Campinas, Campinas, Brazil Robin A. Curtis

University of Manchester, MIB, Manchester, United Kingdom Richtier G. da Cruz

Department of Agri-Food Industry, University of Sa˜o Paulo, Piracicaba, Brazil Leandro Danielski

Department of Chemical Engineering, Federal University of Pernambuco, Recife, Brazil Marcela C. Ferreira

School of Food Engineering, University of Campinas, Campinas, Brazil Elena Go´mez

Department of Chemical Engineering, University of Porto, Porto, Portugal Saartje Hernalsteens

Department of Chemical Engineering, Federal University of Sa˜o Paulo, Sa˜o Paulo, Brazil

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List of Contributors

Fe`lix Llovell

Department of Chemical Engineering and Materials Science, Universitat Ramon Llull, Barcelona, Spain Euge´nia A. Macedo

Department of Chemical Engineering, University of Porto, Porto, Portugal Guilherme J. Maximo

School of Food Engineering, University of Campinas, Campinas, Brazil Antonio J.A. Meirelles

School of Food Engineering, University of Campinas, Campinas, Brazil Mariana F. Montoya

Planta Piloto de Ingenierı´a Quı´mica, CONICET, Universidad Nacional del Sur, Bahı´a Blanca, Argentina Selva Pereda

Planta Piloto de Ingenierı´a Quı´mica, CONICET, Universidad Nacional del Sur, Bahı´a Blanca, Argentina; Thermodynamics Research Unit, University of KwaZulu-Natal, Durban, South Africa Juliana N.R. Ract

Department of Biochemical and Pharmaceutical Technology, University of Sa˜o Paulo, Sa˜o Paulo, Brazil Oscar Rodrı´guez

Department of Chemical Engineering, Universidade de Santiago de Compostela, Santiago de Compostela, Spain Francisco A. Sa´nchez

Planta Piloto de Ingenierı´a Quı´mica, CONICET, Universidad Nacional del Sur, Bahı´a Blanca, Argentina Maria G. Semenova

N. M. Emanuel Institute of Biochemical Physics of Russian Academy of Sciences, Moscow, Russian Federation Simone Shiozawa

School of Food Engineering, University of Campinas, Campinas, Brazil Luiz Stragevitch

Department of Chemical Engineering, Federal University of Pernambuco, Recife, Brazil

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NOMENCLATURE a

a ai aij aijk amn ap aw aassoc adisp ahc ares A

Ai Aij Anm Aφ A Ai A2 A22 ; A44 A24 ASAi ASAbb i ASAsc i b b b0 bi bij bp

strength parameters for the intermolecular forces in the principle of corresponding states, attractive parameter between the molecules, constant in the excess Gibbs energy polynomial, radius of emulsion droplets (which are treated as the hard spheres), anion, stoichiometric coefficient of reaction molar Helmholtz energy density activity of component, attractive parameter for the pure component i attractive cross parameter, binary interaction parameter for species i, and chemical potential derivative for component i with respect to j ternary interaction parameter for species i, j; and k group interaction parameter for groups m and n parameter of the LJ equation (soft-SAFT) water activity association interactions’ contribution to the Helmholtz residual energy dispersive interactions’ contribution to the Helmholtz residual energy hard-chain (reference) contribution to the Helmholtz residual energy Helmholtz residual energy Helmholtz energy, binary interaction parameter in Margules, RedlichKister, and van Laar equations; constant in Antoine equation, constant in the expression for the ideal gas heat capacity, crystalline form, solvent, dimensionless attractive parameter, crystalline form, solvent, reagent of reaction associating sites in PC-SAFT, acceptor sites binary interaction parameter for species i and j fitting parameters of the double-series attractive pressure DebyeHu¨ckel parameter molar Helmholtz energy partial molar Helmholtz energy of component i osmotic second virial coefficient in molal scale units second virial coefficients in the molal scale characterizing the like pair interactions: biopolymer1biopolymer1 and biopolymer2biopolymer2, respectively cross second virial coefficient in the molal scale for the unlike biopolymer1biopolymer2 pair interaction g accessible surface area of atom type i accessible surface area of atom type i in backbone accessible surface area of atom type i in side chain covolume parameter, constant in the excess Gibbs energy polynomial, empirical proportionality factor, stoichiometric coefficient of reaction volume of one lattice cell reference van der Waals volume parameter covolume parameter for the pure component i covolume cross-parameter parameter of the LJ equation (soft-SAFT)

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Nomenclature

b1;i bc2 B

Bbound Bchain Bfree Bi Bii Bij Bjj Bmetastable Bex 2 exp B2 Bτ2 c

ci C

Ci C iii C jjj Ciij ; Cijj ; Cijk CP CV CM CC CPIG CPL CPS ChM ChC fus ΔCp;i d dc;i dp dw

xx

number of bound water molecules about surface atom type i reduced second virial coefficient at critical temperature second virial coefficient, binary interaction parameter in Margules, RedlichKister, and van Laar equations; constant in Antoine equation, constant in the expression for the ideal gas heat capacity, dimensionless covolume, reagent of reaction parameter of the HOC model parameter of the HOC model molecular volume in the HOC model associating sites in PC-SAFT, donor sites second virial coefficients of pure components i cross coefficient between molecules ij in the Virial EoS, number of molecules i in domain of molecule j second virial coefficients of pure components j parameter of the HOC model excluded volume contribution to second virial coefficient second virial coefficient in experimental units Baxter model contribution to second virial coefficient constant in the excess Gibbs energy polynomial, volume-translation parameter of a pure component, influence parameter, cation, stoichiometric coefficient of reaction 1/3 of the number of external degrees of freedom per molecule of component i, moles of component i per unit volume third virial coefficient, third parameter in the Patel and Teja EoS, binary interaction parameter in Margules and RedlichKister equations; constant in Antoine equation, constant in the expression for the ideal gas heat capacity, biopolymer concentration, product of reaction solid phase which contains the compound formed by peritectic reaction, mass of i per unit volume, concentration of component i third virial coefficients of pure components i third virial coefficients of pure components j cross coefficient between molecules iij; iji; and ijk in the Virial EoS, respectively constant pressure heat capacity constant volume heat capacity solid solution rich in myristic acid solid solution rich in capric acid ideal gas heat capacity at constant pressure heat capacity of the compound i in the liquid phase at constant pressure heat capacity of the compound i in the solid phase at constant pressure solid phase formed after metatectic reaction rich in myristic acid solid phase formed after metatectic reaction rich in capric acid variation of heat capacity of compound i between liquid and solid phases hard sphere diameter, stoichiometric coefficient of reaction GCA-EoS critical diameter of compound i mass density of protein solution mass density of water

Nomenclature

dcs D

D0 Dm E Ea f f0 f ð0Þ ; f ð1Þ f ð2Þ fi

mass density of co-solvent solution fourth virial coefficient, binary interaction parameter in RedlichKister equation, constant in the expression for the ideal gas heat capacity, constant in the expression for the ideal gas heat capacity, diffusion coefficient, product of reaction diffusion coefficient at solute infinite dilution solute mutual diffusion coefficient constant in the expression for the ideal gas heat capacity, enhancement factor energy of activation fugacity, McMillanMayer free energy per unit volume fugacity in the standard state parameters of the Tsonopoulos model, parameters of the Van Ness and Abbott model parameter of the Tsonopoulos model

fi 0

set of universal functions defined at reduced temperature in the principle of corresponding states, fugacity of pure component i fugacity of pure component i in the standard state

fi L

fugacity of pure component i in the subcooled liquid or liquid state

fi

S

f sat f^i

fugacity of pure component i in the solid state fugacity at the saturation condition fugacity of pure component i in a mixture

α f^i

fugacity of pure component i in a α phase

β f^i

fugacity of pure component i in a β phase

MI f^i

fugacity of pure component i in an ideal mixture

L f^i

fugacity of pure component i in a liquid phase

S f^i

fugacity of pure component i in a solid phase

V f^i AB

fugacity of pure component i in a vapor phase

f

Mayer f-function

F Fobj

number of degrees of freedom, feed stream, McMillanMayer free energy objective function, in terms of vapor pressure, molar volume of saturated liquid or saturated liquid density data points LLE objective function SLE objective function VLE objective function activity coefficient objective function nondimensional excess Gibbs energy function; radial distribution function / attractive parameter (GCA-EoS) GCA-EoS critical diameter of compound i characteristic energy of the ij interaction in the NRTL model, grand canonical pair distribution function between i and j solvation free energy of peptide backbone

FLLE FSLE FVLE ; FVLE;σ Fγ g gi gij gbb

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Nomenclature

gisc gmix G G0 Gv Gdep

ΔG 0f ΔG  h H Hij Hijcp

solvation free energy for side-group of amino acid i mixture characteristic attractive energy per total segments (GCA-EoS) Gibbs energy, complex shear modulus storage modulus loss modulus Gibbs free energy of the depletion interactions between a pair of colloidal particles KirkwoodBuff integral between particles i and j function of the reference density (soft-SAFT) solute solvation free energy Gibbs energy in the standard state solute standard state chemical potential molar Gibbs energy molar excess Gibbs energy molar Gibbs energy of an ideal mixture molar Gibbs energy of an ideal-gas mixture partial molar Gibbs energy of component i variation of molar Gibbs energy standard-state molar Gibbs energy change on reaction standard-state molar Gibbs energy change on reaction at reference temperature (T0) standard-state molar Gibbs energy change of formation molar Gibbs energy of activation Planck’s constant enthalpy, hydrodynamic function Henry’s constant Henry’s constant in terms of concentration

Hijm

Henry’s constant in terms of molality

H HE Hi

molar enthalpy molar excess enthalpy partial molar enthalpy of component i

Gij Gp Gs G0 G20 G GE GIM G IGM Gi ΔG ΔG 0 ΔG 00

E

Hi

excess partial molar enthalpy of component i

N Hi

H1 ; H2 ΔH ΔH 0 ΔH 00 ΔH 0f ΔH fus i ΔH sub i vap

ΔH i

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partial molar enthalpy of component i at infinite dilution liquid crystal structured in hexagonal forms variation of molar enthalpy standard-state molar enthalpy change on reaction standard-state molar enthalpy change on reaction at reference temperature (T0) standard-state molar enthalpy of formation, molar heat of formation molar enthalpy change on fusion of pure component i, heat of fusion of pure component i molar enthalpy change on sublimation of pure component i, heat of sublimation of pure component i molar enthalpy change on vaporization of pure component i, heat of vaporization of pure component i

Nomenclature

ΔH  Ix I2 k kB kD kij kref ks kT K Keq K Ki Kp Kp;i Kij; Kji Ki;w KiN K AB uf Ki K0 I1 ; I2 J Ji Lα Lβ Lβ1 L2 lij m m^ mi mkl mw m0 seg mi M

Mi n

molar enthalpy of activation ionic stretch in mole fraction reverse cubic phase adjustment constant in the Gordon and Taylor equation, reaction rate Boltzmann constant diffusion coefficient interaction parameter binary interaction parameter between components (or groups) i and j, dispersive energy binary parameter rate constant at reference pressure Pref sedimentation coefficient interaction parameter isothermal compressibility unfolding equilibrium constant, salting out constant chemical equilibrium constant chemical equilibrium constant for the first stage a reaction containing a transition state partition ratio of compound i, equilibrium ratio, K-factor, K-value local partition coefficient (note: this is different from partition ratio) local partition coefficient about atom type i fitting parameters partition ratio of compound i, in mass basis partition ratio of compound i at infinite dilution volume of association unfolding equilibrium constant salting out constant integrals of the perturbation theory (PC-SAFT) pseudo-ionization energy pseudo-ionization energy of component i liquid crystal structured in a lamellar form (sub-α phase) liquid crystal structured in a lamellar form (mesophase gel, α-gel) liquid crystal structured in a lamellar form (new α-gel) isotropic phase, reverse micellar solution phase binary interaction parameters, size binary parameter (equivalent to η 1 1) ratio of molar volumes of species 2 and 1 in the FloryHuggins equation, series number of terms of the double-series attractive pressure, chain length molality moles of component i per mole of water, concentration in the molal scale of the i-component temperature independent characteristic parameter of ASOG groups k and l number of kilograms of water standard-state molality for the biopolymer number of segments in the chain molar mass (molecular weight) of component i; number of lattice sites, number of associating sites, number-averaged molar mass of the biopolymer molar mass of component i number of components, series number of terms of the double-series attractive pressure, refractive index, biopolymer number density

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Nomenclature

nk nkl nsites N NAv Nc Ni Nρ NG Np

Osm P P atr P az Pc Pcij Pi Pref P0 P sat Pisat Pisub Pivap Pt;i P0 POY POY i ΔP q q ~ q

0 q qi 0

qi Q Qk r hr 0 i ri

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number of chemical groups, number of occurrences of the chemical group k in the molecule for the LorentzBerthelot combining rules temperature independent characteristic parameter of ASOG groups k and l number of association sites number of moles, number of molecules, number undistinguishable molecules, number of repeating units Avogadro’s number number of components number of moles of component i number of data points in terms of saturated liquid density number of different functional groups in a molecule. number of data points in terms of activity coefficient, pressure, mole fraction of component i in the liquid and vapor phases, temperature, vapor pressure, molar volume of saturated liquid. osmolality pressure double-series attractive pressure pressure at the azeotropic point critical pressure critical pressure of the mixture containing components i and j partial pressure of component i reference pressure reference pressure saturation pressure saturation pressure of component i sublimation pressure of component i vapor pressure of component i triple point pressure of component i reference pressure Poynting correction factor Poynting correction factor of compound i average deviation on pressure magnitude of scattering vector, fitting constant in the Kwei equation total number of surface segments UNFAC quantity UNIFAC-FV quantity, mass fractionbased UNIQUAC equation quantity van der Waals surface area parameter of component i; ASOG quantity; measure of the volume of molecule i in the van Laar equation, relative surface of group j, internal partition function of component i van der Waals surface area parameter of component i per unit mass quadrupole moment van der Waals surface area parameter of group k number of additional constraints (reactions, critical points, and others), radius of the associating site, distance of the emulsion droplet separation, center-to-center separation UNIFAC-FV quantity; mass fractionbased UNIQUAC equation quantity van der Waals volume parameter of component i

Nomenclature

0

ri R Rh Rk Rθ R124 s s0 sv S Si Sij Siξ Ski Sq S0 S Si ΔS ΔS 0 ΔS 00 ΔS  T Tb TB Tc T ci T cj Tcij Tco TDSC Te Tm Tm;i Tmicro Tg Tg 0 Tgel TP Tr Ttr Tt Tj ΔT u

van der Waals volume parameter of component i per unit mass universal gas constant hydrodynamic radius contribution of group k to the chain length of the molecule, van der Waals volume parameter of group k excess Rayleigh ratio chain fraction in GC-PPC-SAFT sedimentation coefficient of solute sedimentation coefficient of solute at infinite dilution entropy per unit volume entropy solubility of component i solvent selectivity for species i relative to species j, separation factor solubility of component i in concentration scale ξ ASOG quantity; UNIFAC-FV quantity structure factor solubility in the pure solvent molar entropy partial molar entropy of component i variation of molar entropy standard-state molar entropy change on reaction standard-state molar entropy change on reaction at reference temperature (T0) molar entropy of activation temperature boiling temperature temperature in the binodal line by TLI or OM critical temperature critical temperature of component i critical temperature of component j critical temperature of the mixture containing components i and j crystallization temperature sol-to-gel transition temperature by DSC eutectic point temperature melting temperature melting temperature of pure component i sol-to-gel transition temperature by optical microscopy glass transition temperature glass transition temperature in the maximally freeze-concentrated solution sol-to-gel transition temperature peritectic temperature reduced temperature temperature of solidsolid transition triple point temperature component i reference temperature of the group j average deviation on temperature speed of sound

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Nomenclature

uðnÞ uαβ uij

vi hυ~ i υi υ~ i V V Vc V ci

n-body interaction potential groupgroup parameter for the volume of association (GC-PPC-SAFT) average interaction energy for species i and j in the UNIQUAC model, groupgroup correction for the volume of association (GCA-EoS)/pair potential between tangentially bonded segments energy interaction parameter for groups m and n in UNIFAC van der Waals dispersion parameter in PC-SAFT internal energy molar internal energy partial molar internal energy of component i internal energy change on vaporization at the normal boiling point for component i partial specific volume of component i reduced volume of the mixture liquid specific volume of pure component i reduced volume of component i volume molar volume critical volume critical volume of component i

V cj Vcij

critical volume of component j critical volume of the mixture containing components i and j

Vm

volume per mole water

Vi

molar volume of the component i



molar volume of the solvent

umn ui =kB U U Ui vap ΔU i

L

liquid molar volume

VV

vapor molar volume

V Si

solid molar volume of component i

Vi

partial molar volume of component i

E Vi N Vi 

excess partial molar volume of component i

V V1

partial molar volume of the mixture in a reaction in the transition state liquid crystal structured in a cubic form

ΔV 0

standard-state molar volume change on reaction, molar volume change

ΔV 

activation molar volume

ΔV mix

molar volume of mixing

wi wαβ wij w ðnÞ w soft w el

mass fraction of component i groupgroup parameter for the volume of association (GC-PPC-SAFT) groupgroup correction for the energy of association (GCA-EoS) n-body potential of mean force soft contribution to two-body potential of mean force electrostatic contribution to two-body potential of mean force

V

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partial molar volume of component i at infinite dilution

Nomenclature

wsr wex 0 w1 Δw xi xp xμp m X Xm y yi Y Δy z zi z1 z2 Z Z ci Z cj Zcij

short-range contribution to two-body potential of mean force excluded volume contribution to two-body potential of mean force mass fraction of the solute in the condition of maximally freezeconcentrated solution average deviation on mass fractions liquid phase mole fraction of component i in the mixture peritectic composition, fraction of segments in the chain that contains an effective quadrupole number of dipoles on the molecule mole fraction of component i not bonded at site A, effective mole fraction group mole fraction of group m cavity distribution function vapor phase mole fraction of component i in the mixture packing fraction average deviation on vapor phase compositions number of possible neighbors, coordination number volume fraction of component i in the van Laar equation, absolute value for the ionic charge of the ion i absolute value for the ionic charge for the cation absolute value for the ionic charge for the anion compressibility factor, protein net charge number critical compressibility factor of component i critical compressibility factor of component j critical compressibility factor of the mixture containing components i and j

GREEK SYMBOLS α

β0 βi β1 β2

isobaric thermal expansion coefficient, parameters of generalized cubic equation of state, crystal form alpha, binary factors (GCA), energy parameter (CPA), nonrandomness factor gel phase alpha, liquid crystal structured in a lamellar form (mesophase gel) NRTL binary nonrandomness parameter for species i and, GCA-EoS nonrandomness interaction parameter between groups i and j, relative volatility parameters of generalized cubic equation of state, crystal form beta, inverse reduced temperature ( 5 1=kB T ) crystal form beta prime ion free energy parameter in Record model crystal form beta one crystal form beta two

βAB β-gel Δ ΔAB ρ ρi

association volume parameter (CPA) coagel depletion layer thickness specific sitesite function density, closet-approach parameter molecules of component i per unit volume

α-gel αij β

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Nomenclature

ε εj

energy of each intermolecular interaction, dispersive energy, extent of reaction, reaction coordinate extent of reaction j, reaction coordinate of reaction j

εAB εki;lk ε=k εAB =k εAiBi =kB κT κAB κAiBi κAαBβ κki;lk μ μi μr μ0i

energy of association association energy between site k in group i and site l in group j association-energy parameter in GC-PPC-SAFT association-energy parameter in GC-PPC-SAFT association-energy parameter in PC-SAFT isothermal compressibility coefficient association volume parameter (SAFT) association volume parameter (PC-SAFT) association volume parameter (GC-PPC-SAFT) association volume between site k in group i and site l in group j dynamic viscosity, dipole moment chemical potential of component i reduced dipole moment standard state chemical potential of component i

μ0i;ξ

standard state chemical potential of component i on concentration scale ξ

μmix i γ γi γLi γSi γN i γi;ξ

mixing chemical potential that is determined by the system composition parameters of generalized cubic equation of state, crystal form gamma, surface (interfacial) tension activity coefficient of the component i activity coefficient of the component i in liquid phase activity coefficient of the component i in solid phase activity coefficient of the component i at infinite dilution asymmetric activity coefficient for i on concentration scale ξ

γi;ξ

activity coefficient for component i on concentration scale ξ

γN i;ξ

activity coefficient for i at infinite dilution on concentration scale ξ

Γk

group activity coefficient for group k

Γ ðkiÞ

group activity coefficient for group k in pure component i

δi η ηa ηk θ θi θ θE θi

solubility parameter of the component i fitting parameter, kinematic viscosity, packing fraction, size binary parameter apparent viscosity ASOG quantity; UNIFAC-FV quantity general thermodynamic property, angle of detection in light scattering experiment, surface fraction of a group area fraction of component i in UNIQUAC and UNIFAC molar property molar excess property molar property of component i

θi Δθmix Θm

partial molar property of component i in a mixture mixing property van der Waals area fraction for UNIFAC group m; ASOG quantity

xxviii

Nomenclature

λ

ν1 ν2 ν FH i σ σi σy σP τ τ ij ϕi φ φi

fitting parameter, thermal conductivity, parameter of the PitzerDebyeHu¨ckel extended equation free-volume term coefficients energy of interaction for species i and j in Wilson equation binary interaction parameter for species i and j in Wilson equation number of phases stoichiometric coefficient of molecule or ion i number of groups j in a molecule i number of UNIFAC groups of type k in molecule i; number of atoms, other than hydrogen atoms, in ASOG group k of molecule i stoichiometric coefficient for the cation stoichiometric coefficient for the anion number of atoms, other than hydrogen atoms, in the molecule i segment diameter, molecular diameter, surface tension, protein diameter segment diameter in PC-SAFT experimental uncertainties associated with vapor phase mole fraction experimental uncertainties associated with pressure adhesion parameter in Baxter model, Boltzmann factors NRTL parameter, UNIQUAC parameter volume fraction of component i fugacity coefficient fugacity coefficient of component i

φsat i

fugacity coefficient at saturation condition

φsat i

pure component i fugacity coefficient at saturation condition

φ0i app φi

specific volume of component i at infinite dilution

λi λij Λij π νi ν ij ν ki

φ^ i L φ^

apparent specific volume of component i fugacity coefficient of component i in a mixture

i

liquid phase mixture fugacity coefficient of component i

V φ^ i

vapor phase mixture fugacity coefficient of component i

Φ Φi χ Ψ Ψ mn Ω ξ ξi

parameter of the PitzerDebyeHu¨ckel extended equation volume fraction of component i in Regular Solution Theory and UNIQUAC and UNIFAC models interaction parameter in the FloryHuggins equation parameter of the PitzerDebyeHu¨ckel extended equation group interaction parameter for groups m and n in UNIFAC number of accessible microscopic states, the set of angular orientation variables reduced density, dispersive energy binary parameter (equivalent to 1kij) concentration of component i on scale ξ

ξ Vi κ λ λi λB ω

concentration of component i on scale ξ DebyeHu¨ckel screening parameter wavelength of light used in scattering experiment thermal wavelength of component i Bjerrum length acentric factor, weighting factor

xxix

Nomenclature

ωi ωj ωij Π E Γ1 Γ μ1 ;μ3 Γ P;μ3

acentric factor of component i acentric factor of component j acentric factor of the mixture containing components i and j osmotic pressure, osmotic coefficient activity coefficient derivative surface concentration of the i-component of the system dialysis equilibrium preferential interaction parameter preferential interaction parameter from osmolality measurements

SUBSCRIPTS A C c cal e exp i; j, 1,2,. . .,n g gel G j k l m n obj P p ρ s w t; T

diluent component, solute critical calculated electrolyte experimental the specific compound i, j, 1, 2, or n, in a mixture glass gel group component; data point group group melting, group group objective pressure polar density segments, solvent mass base, water triple point

SUPERSCRIPTS A, B assoc cal comb disp E exp fus fv hs

xxx

specific sit in a compound association calculated combinatorial dispersive excess experimental fusion free-volume hard-sphere

Nomenclature

ice id liquid mix polar phd ref res sat seg total vap I II L LR S SR α; β; π; δ . . . ψ N 

ice ideal liquid mixture polar PitzerDebyeHu¨ckel reference residual saturation condition segment total vapor liquid or solid phase liquid or solid phase liquid phase long-range interactions solid phase short-range interactions phases α; β; π,δ . . . ψ, respectively infinite dilution Henry’s conversion

ABBREVIATIONS A AA ASA ASOG AOCS BHPB10 BMA BzA C CAPD CBE CBS CBR CN COSMO CPA CW DAG DB DEX DFT DGT DHW

arachidic acid acrylic acid accessible surface area analytical solution of groups American Oil Chemists Society 3,5-bis-(5-hexylcarbamoyl-pentyloxy)-benzoic acid decyl ester butyl methacrylate benzyl alcohol capric acid computer-aided product design cocoa butter equivalents cocoa butter substitutes cocoa butter replacers carbon number conductor-like screening model cubic plus association candelilla wax diacylglycerol double bounds number dextran density functional theory density gradient theory domestic hot water

xxxi

Nomenclature

DMAEMA DSC DVE DOP EB EFV EO EoS FAME FA FDA FFA FAL FHSO G GC GC-EoS GCA GCM GCVOL GK-FV GMO GMS GRAS HMF HOC HPP HR IG IGM IM IUPAC L Li Ln LB LC LC1 LC2 LDL LJ LLE LLVE LMF LMW M

xxxii

dimethylamino-ethyl methacrylate differential scanning calorimetry dynamic viscoelastic experiments dioctyl phthalate 2-ethyl butyryl chloride UNIFAC entropic-free-volume ethyoleate equation of state fatty acid methyl ester fatty acid Food and Drug Administration free fatty acid free fatty alcohol fully hydrogenated soybean oil gadoleic acid group contribution group contribution equation of state group contribution 1 association group contribution method group contribution method for the prediction liquid density UNIFAC with entropic-free-volume and StavermanGuggenheim combinatorial glycerol monooleate glycerol monostearate generally recognized as safe high melting fraction Hayden-O’Connell high pressure processing technology HuangRadosz ideal gas ideal-gas mixture ideal mixture International Union of Pure and Applied Chemistry liquid phase, lauric acid linoleic acid linolenic acid LorentzBerthelot local composition liquid crystal phase one liquid crystal phase two low-density lipoprotein LennardJones liquidliquid equilibrium liquidliquidvapor equilibrium low melting fraction low molecular weight myristic acid

Nomenclature

MAG MMF NMR MPP NRTL O O/W OM OPVR P Po PADB PADBA PC PCB PCM PEG PEO PHCT POO POP POS POY PUFA PPC PPG PPP PR PS PVA PVME PVP PSty-DVB QM REA RG S SAFT SC SCF SCFSE SFC SFE SLE SVE SOA

monoacylglycerol medium-melting fraction wide-line or pulsed nuclear magnetic resonance spectroscopy modified poly phenol UNIFAC nonrandom two-liquid oleic acid oil/water optical microscopy para-phenylene vinylene palmitic acid palmitoleic acid polymer obtained by polymerization of AA, DMAEMA, and BMA polymer obtained by polymerization of AA, DMAEMA, BMA, and allyl alcohol perturbed chain Polychlorinated biphenyls phase change material polyethylene glycol polyethylene oxide perturbed hard chain theory 1-palmitoyl-2,3-dioleoylglycerol 1,3-dipalmitoyl-2-dioleoylglycerol 1-palmitoyl-3-stearoyl-2-oleoylglycerol Poynting correction factor polyunsaturated fatty acid polar perturbed chain polypropylene glycol tripalmitin PengRobinson palm stearin polyvinyl alcohol polyvinyl methyl ether polyvinyl pyrrolidone poly-(styrene-divinylbenzene) resin quantum mechanics ricinelaidic acid renormalization group solid phase, stearic acid statistical associating fluid theory supercritical supercritical fluids supercritical fluids-solute equilibrium solid fat content supercritical fluid extraction solid-liquid equilibrium solid-liquid equilibrium 1-stearoyl-2-oleoyl-3-arachidonylglycerol

xxxiii

Nomenclature

SO SOS SPC SRK SSS SSL SW trD TAG TLI TG TPT1 TREN UNIFAC UNIFAC-FV UNIQUAC V vdW vdW-FV VLE VLLE W/O WE WS

xxxiv

refined soybean oil 1,3-distearoyl-2-oleoylglycerol simplified perturbed chain SoaveRedlichKwong tristearin sodium stearoyl lactylate square well trans-decalin triacylglycerol transmitted light intensity transglutaminases first-order thermodynamic perturbation theory tris(2-aminoethyl)amine UNIQUAC functional-group activity coefficient UNIFAC free-volume universal quasi-chemical vapor phase van der Waals UNIFAC van der Waals free-volume vaporliquid equilibrium vaporliquidliquid equilibrium water/oil wax ester WolbachSandler

CHAPTER 1

Phase Equilibria in the Food Industry Camila Gambini Pereira

1.1 INTRODUCTION The main intention of the food industry is to ensure the supply of safe and healthy food to consumers. Within the universe of food and processes of transformation and generation of new products, a wide range of possibilities is observed in the processing of a food. By virtue of the knowledge of the high added value of food constituents that are of interest both for the food industry, and for the pharmaceutical, chemical, biotechnological, biofuel and other related industries, processes have been applied in order to separate such compounds or to mantain them in the final food product. Among the different unit operations found in the food industries, the separation processes (distillation, extraction, absorption, evaporation, etc.) are those where this proposal can be found in the various industrial sectors. Foods, in essence, are multicomponent and often multiphase systems. In this sense, food engineers need to understand not only the process but also the variations and modifications that such a system may undergo during processing. For this reason, the structural characteristics of the raw material, its composition, and its potential variations need to be taken into account in the preparation of the material and in the definition of the type of process and operating conditions. The phase equilibria in foods is presented in two aspects: in the food product and in the food processing. In the first, the target object is in food. The legal and consumer requirements and the quality of a food product are governed by the sensorial and microbiological aspects. This is related to the presence of water and other constituents, such as carbohydrates, proteins, lipids, and other minor compounds responsible for the nutritional and sensorial characteristics of food, such as essential oils. The molecular interaction of these compounds, the chemical and biochemical reactions, and microbial Thermodynamics of Phase Equilibria in Food Engineering

© 2019 Elsevier Inc. All rights reserved.

3

Thermodynamics of Phase Equilibria in Food Engineering

growth that may exist in foods guide the physical and quality properties of the final product. Moreover, this entire framework is related to the equilibrium or non-equilibrium condition in the material. In the second, the phase equilibrium can be directly linked to the industrial process. In this, the question focuses on the process itself and the effects that the phase equilibrium have on the obtaining or removal of certain components. The separation processes have become great allies in the recovery of bioactive compounds, and in the manufacture of highquality products in the food industry. Even in industrial processes where the removal of a specific compound is not involved, as in freezing for instance, thermal effects can cause physical changes in food, which may be related to the melting point or glass transition. In all these cases, the thermodynamics of equilibrium is present. In this scenario, this book aims to provide a theoretical basis on thermodynamics of equilibrium linked to systems and processes found in the food industry. In order to support and guide the reader, this chapter gives an overview of the main sectors of the food industry and food products in which the phase equilibrium condition is required.

1.2 EQUILIBRIUM AND THE REAL WORLD Phase equilibrium is present in our life every day in many situations: in the preparation of coffee, in the dissolution of sugar in a juice, or in the tasting of a creamy ice cream. Furthermore, for a previously prepared food, the phase equilibrium was already present in certain stages of its processing (as in the edible oil processing). The knowledge of phase equilibrium is of great importance, but many times this is only perceived when unusual products are found (powdered chocolate that does not dissolve in hot milk, precipitation of particles in juices and soft drinks, etc.). In addition, the sensorial attributes of a given product—color, taste, aroma, texture—are directly related to the equilibrium condition between the food constituents and the storage state of the product. What is more, many food products are characterized by the existence of multiphases, as exemplified in Fig. 1.1 for milk, and their conditions are dependent on the physical properties of the substances and molecular interactions observed in each case. Some of them are colloids, as is the case of ice cream. Although apparently ice cream is a homogeneous creamy fluid, it consists of four phases: ice (crystals), air (bubbles), a continuous viscous aqueous phase (containing dissolved particles of sugar and 4

Phase Equilibria in the Food Industry

Fat globules Casein micelles

Fat globule

Figure 1.1. Representative image of milk structure.

proteins), and a dispersed phase (fatty compounds). Table 1.1 presents some examples of similar food products. In terms of food processing, the specificity in a process increases with the particularities found in the different varieties of a given product. The coffee industry is one example: how many kinds of coffee products can be found? Caffeinated, decaffeinated, different blends, aromatized, etc. From each one, different kinds of processing are created, either to insert a new compound, or to remove another one. In this sense, the understanding of equilibrium has been essential in the analysis and optimization of food processes. Table 1.1. Foods containing multiphases in their structures Product

Continuous phase

Liquid (water 1 solutes) Mayonnaise Liquid (water 1 solutes) Butter Solid (fat) Milk

Ice cream

Chocolate

Liquid (water 1 proteins and sugar) Solid (fat)

Dispersed phase

Other phases

Liquid (fat droplets 1 protein aggregates) Liquid (fat)



Liquid (water 1 solutes) Liquid (oil)

  Air (bubbles); ice (crystals)

Liquid (milk 1 solutes) Variations: • solid (nuts) • air (bubbles) in aerated products • liquid (liquor) 5

Thermodynamics of Phase Equilibria in Food Engineering

1.3 THERMODYNAMICS OF EQUILIBRIUM FOR PROCESSES AND DESIGN IN THE FOOD INDUSTRY One of the main obstacles in the design of efficient food processing is the difficulty to accurately predict the phase behavior and related properties. Food by itself is considered a complex system, formed by different classes of molecules (volatile compounds, proteins, carbohydrates, lipids, organic acids, and minerals), often thermolabile. Active and bioactive, aromatic, reactive, complexed, and grouped compounds are part of this broad range of substances frequently found in food. These substances can also be in free form or linked to other molecules or structures. The diversity of components present in each group of chemical species increases the complexity in working with food systems. The separation of α-carotene and β-carotene, for instance, is not an easy task. This characteristic is not only observed for isomers, compounds with a similar molecular weight and boiling temperature cannot be separated by simple distillation. Nevertheless, the knowledge of phase equilibria is fundamental in the separation of these compounds. Furthermore, food, which can be presented in different states of matter, may undergo transformations as a single phase or multiphase system. The presence of multicompounds chemically and physically dissimilar in food and their function on the food products make these compound materials useful when employed in different formulations for different applications. For this reason, foods are important sources for obtaining valuable substances for the food industry and also for other industrial sectors, including pharmaceutical, cosmetic, agrochemical, and fine chemical industries. The complexity of food is more than only considering the different compounds present in a specific raw material. Obtaining a product with desirable characteristics for consumers is a very important factor in the food industry. Sensory properties of a food product are affected by how the microstructures (crystals of ice or fat, droplets of aqueous solution or oil) were formed and how they will be broken down during the mastication. Ice creams, chocolates, margarines are products whose formation of microstructures is essential to obtain a product with desired specifications and quality. The control of solidification and melting of these structures reflects the quality of the product and it is strongly related to the phase equilibrium. The thermodynamics of phase equilibria applied to food processes and products aims to provide essential information for the manufacturing of the different types of raw materials, in different states of aggregation,

6

Phase Equilibria in the Food Industry

which can lead to the production of a variety of products. Because of the requirement of forming a product with a macroperception of a single phase (as in the homogeneity of ice cream, without ice crystals being perceptible during swallowing), or the need for the formation of multiple stages (as bi or tri-phases in liquid make-up or body oils, marketed in cosmetic products), the knowledge of phase behavior is essential to obtain a product as specified. Within the industry, the greatest the challenges of food engineers are to design a process of transformation while keeping the desired characteristic of the raw material and eliminating or reprocessing the by-products with lower energy consumption and higher global process efficiency. The environmental regulations and public health requirements are also factors that are currently considered. Nowadays, the engineer must have a global view and at the same time a detailed understanding of the process and the final product. Certainly, phase equilibria have arrived in food engineering to give support in various operational steps: either in the recovery, solubilization, or encapsulation of bioactive compounds, in the removal of undesirable compounds, in the processing or development of food products, or in the analysis and quality control of the final product. The unit operations of mass transfer with or without heat transfer are based on the contact of two phases initially at non-equilibrium. In the food industry, several transforming operations make use of separation processes (extraction, distillation, evaporation, absorption, stripping, and others). Some examples are shown in Table 1.2. In all of them, the knowledge of phase equilibria is required. From the molecular point of view, the thermodynamics of phase equilibria also may be used to understand the molecular interactions of biomolecules like proteins, providing an important tool in biocatalytic processes. Phase equilibrium is also crucial for the development of various other manufacturing procedures such as emulsification, granulation, and crystallization. Moreover, the industry has made frequent use of virtual plants (simulators) to evaluate new processes or optimize what already exists. For this, information about the phase equilibria is used as input data for the initialization of the calculations. Independent of where the focus of the process is or how it will be performed, the role of phase equilibria is to provide tools to indicate the best condition for obtaining a desired product and/or to assist in the design and simulation of a process of interest within the industry. Briefly, the diverse applications of phase equilibria are discussed in the followed section.

7

Table 1.2. Main unit operations of mass and heat transfer, their use within the food industrya, and the phase equlibrium correlated. Unit operation

Distillation

Employ

• Obtaining distilled spirits • Obtaining volatile oils from aromatic condimentary and medicinal plants • Deacidification of fixed oils • Deodorization of edible fats and oils • Removal of carotenoids from vegetable oils Crystallization • Production of sugar • Manufacturing of chocolate • Obtaining citric acid Freezing • Obtaining ice creams • Manufacturing of frozen foods Evaporation • Concentration of juices • Obtaining syrup • Production of concentrated milk • Production of milk cream/caramel Drying • Obtaining dried or semi-dried materials • Encapsulation of bioactive compounds Lyophilization • Obtaining dried milk • Production of powder coffee Liquidliquid • Deacidification of vegetable oils extraction • Recovery of proteins • Purification of organic acids Absorption/ • Cleaning of fermentation gases in alcohol stripping and sugar mills • Deodorization and deacidification of vegetable oils (physical refining) • Recovery of hexaneb from miscella after extraction of vegetable oils • Recovery of aromas Adsorption • Clarification of juices • Removal of tannins • Removal of undesirable compounds (odor/color) • Water treatment Supercritical fluid • Separation of biocompounds (essential oils, Extraction carotenoids, flavonoids, vitamins, and others) • Isolation of colorants • Extraction of aromas • Removal of toxic substances (such as PCBs)

Phase equilibrium

VLE

SLE

SLE/glass transition VLE

VLE SVE LLE

VLE

SLE SLE SVE/SLE SLE SCFSE

where VLE is vaporliquid equilibrium; SLE is solidliquid equilibrium; SVE is solidvapor equilibrium; LLE is liquidliquid equilibrium; SCFSE is supercritical fluid—solute equilibrium; PCB is polychlorinated biphenyl. a Solid liquid extraction is not considered here because, due to the existence of physical barriers of the solid matrix, the equilibrium observed is not true. For this operation, the calculations are performed considering operational equilibrium relations. b Mixture of saturated 6C hydrocarbons.

8

Phase Equilibria in the Food Industry

1.4 OVERVIEW OF PHASE EQUILIBRIA IN THE FOOD PROCESSING The food industry has many processes that involve knowledge of equilibrium. Some of them are already well-known, others are being improved or under development. Due to its importance, the thermodynamics of phase equilibria have been indispensable in the development of food processes and products. In this section, a discussion regarding the various segments of the food industry and their connections with the phase equilibrium is outlined.

1.4.1 Beverage Industries The beverage industry covers a wide range of beverages ranging from water to distilled spirits like whisky. They can be carbonated, fermented, clarified, and distilled, and can be produced from fruits, cereal, seeds, herbs, or vegetables. The wide diversity of this sector is due to the wide variety of raw materials and manufacturing processes. In all these cases, the phase equilibrium can be present in certain processing steps, as shown in Table 1.3. Drinks in a broad manner can be consumed hot, cold, or at room temperature, depending on the consumer’s preference. One of the main issues concerning non-alcoholic beverages is the presence of biocompounds and aromas from fruits, herbs, teas, and coffee with high added value, for which the use of an efficient process to avoid losses during processing is fundamental. For this reason, the recovery of these compounds during the industrial process (essential oils, for instance) or from the waste of the beverage industry (from fruit peel and bagasse, for example) has been carried out to reinsert them in the final product or even to be used as raw material in the formulation of new products in the pharmaceutical and cosmetic or other food industries. Similarly, in the production of concentrated fruit juices by evaporation, many substances responsible for the aroma are removed together with the water vapor, decreasing the sensorial quality of the concentrated beverage. In this case, the flavor substances are recovered by a particular technique (see Table 1.3) and added again in the concentrated product, resulting in a final product with sensory characteristics closer to those of natural juice. An alcoholic drink is a beverage that contains ethanol in its formulation. These beverages are divided into three general classes: fermented (beer, wine, and others), distilled (cachac¸a, vodka, gin, rum, whisky, tequila, and others), and liqueur (anisette, sambuca, curacao, and others). 9

Thermodynamics of Phase Equilibria in Food Engineering

Table 1.3. Processes in the beverage industry and correlated phase equilibria Beverage

Non-Alcoholic

Juices

Soft drinks Sparkling water Coffee

Tea Alcoholic

Fermented beverages

Distilled Spirits

Processa

Type of equilibrium

Extraction of aromas Adsorption of essential oils Distillation or stripping of aromas Adsorption (clarification) Concentration Absorption of gas Absorption of gas

LLE, SCFSE SVE VLE SLE VLE VLE VLE

Extraction (decaffeination) Drying Lyophilization Decaffeination

SCFSE VLE SVE VLE, SCFSE

Carbonatation Cleaning of fermentation gases in alcohol production Clarification of the must Distillation

VLE VLE VLE VLE, VLLE

where VLLE is Vapor-liquid-liquid equilibrium. a Solid liquid extraction is not being considered here because the equilibrium observed in this operation is not true. See the footnote “a” presented in Table 1.2.

In fermented drinks, ethanol is produced by fermentation, while in distilled drinks, ethanol is obtained by distillation of fermented must from cereal grain, fruit, vegetable molasse, or another source of carbohydrate. Liqueur is a class of alcoholic beverages that is made from distilled spirit and is flavored with other products, such as fruits, cream, herbs, and spices, which, in most cases, contains between 35% and 45% volume of ethanol [1]. Table 1.4 presents the main alcoholic beverages consumed in the world and their characteristics. The two general schemes of alcoholic distillation for the production of cachac¸a and neutral spirits, which are the basis for the obtention of beverages such as vodka, gin, some types of whiskeys, and some liqueurs, are presented in Figs. 1.2 and 1.3, these are considering industrial and pilot plant cases, according to the literature [1215]. In the alcoholic beverages industry, during fermentation some compounds responsible for sensorial profile (taste, aroma, color) of alcoholic beverages are produced. These compounds, called congeners, are comprised 10

Table 1.4. Types of fermented and distilled beverages [111] Alcoholic beverages

Processing or characteristics/raw material

Fermented Beer Fermentation of malted grain Wine Fermentation of ripe grapes Special wines: Champagne Type of sparkling light wine made in Champagne Vermouth Type of aperitif wine containing grape wine or another fruit wine, fortified with additional alcohol and with addition of extracts of herbs and spices Sake Wine produced from rice Cider Wine produced from the juice of the apple Perry

Country

Alcohol content (% ethanol, v/v), at 298.15 K

Congener teor (g/hL)

Methyl alcohol (g/hL)

Several Several

0.59.0c 814c

 

 

Francea

1013





Italy

.14





Japana Several

1820 1.28.5c 4.08.0e 2.08.0c 4.08.0e

 

 





Wine produced from the juice of the pears Several

(Continued)

Table 1.4. (Continued) Alcoholic beverages

Processing or characteristics/raw material

Distilled spirits

Neutral spirit obtained from fermentation Several, but of rye, potato, cassava, molasse, or grains main ones are Russian and Poland Alcoholic distillation of fermented Brazila sugarcane or sugarcane must From fermented juice of sugarcane or Caribbean sugar beet or molasse (sugar juice cooked); aged in oak barrels Neutral grain spirit or rectified neutral Several, but the grain spirit or their mixture or made by main ones are distilling the fermented extract of malted Scotland and cereal grains (corn, rye, maize, wheat, Ireland barley) or molasse; aged in wooden casks Neutral spirit flavored with volatile Several compounds from juniper

Vodka

Cachac¸a Rum

Whisky

Gin

Country

Brandy

Distillation of wine of fruits, such as grapes Several

Cognac

Variety of brandy, alcoholic distillation of fermented juice of grapes Variety of brandy, alcoholic distillation of Muscat wines Distillation from fermented blue agave juice

Pisco Tequila

Francea

Alcohol content (% ethanol, v/v), at 298.15 K

Congener teor (g/hL)

Methyl alcohol (g/hL)

3540c .37.5d 3554e

, 50e

, 10d

3848e

200650e

,200e

3550c .37.5d 3554e .40d 3854e .37f

$ 225d 40500e

3540c .37.5d 3554e 4060c .50d, .37f 3840

, 50e

$ 300e (for pure malt whisky)

$ 125d

, 200d

200650e

30300e

Chile and Perua 3046 Mexicoa

3042c 3654e

Liqueur

Francea,b Italy Curacao, various Frangelico Wild hazelnuts blended with berries, herbs, Italy and flowers Ja¨germeister Fifty-six roots, herbs, and fruits Germany Pastis Star anise Francea

Absinthe Amaretto Curac¸ao

Sambuca

Mixture of herbs including wormwood Almond Orange

Anise

“” not informed. a In compliance with the law and regulations of original country. b It is currently illegal in EU, but it is available from other countries [1]. c Generally used. d EU regulation [7]. e Brazilian regulation [3,4]. f FDA [8].

Italy

6085 B28 2040 24 35 4550c .40d .38

Thermodynamics of Phase Equilibria in Food Engineering

Degassing

Cachaça Fermented must

Stillage

Pot still

(A)

(B)

Figure 1.2. Flow diagram of the distillation process to produce cachaça: (A) bath, (B) column. Based on [12,13]. Lights

LCS

Bottom products

Heads

Neutral spirit Fermented must Fusels

CC

RC

Bottom product

Fusel oil

Figure 1.3. Flow diagram of the separation process to produce neutral spirits, where CC is the concentration column, RC is the rectification column, and LCS is the light component separator. Based on [12,14,15].

mainly of alcohols, acids, esters, aldehydes, and ketones. After distillation, these compounds are still present, even in small quantities (10261024 mg/ L) [16,17], providing the characteristics of each distilled spirit. However, the legislation establishes limits for the presence of some of these compounds in the final beverage, and for this reason, during the distillation process, the removal of the excess amount of these compounds must be carried out. 14

Phase Equilibria in the Food Industry

Table 1.5. Physical properties of some congeners [18] Compound

M (kg/kmol)

Tb (K)

Tc (K)

Pc (MPa)

ω

Acetaldehyde 1-Propanol 2-Propanol 2-Methyl-1-propanol 3-Methyl-1-butanol Acetic acid Ethyl acetate 2-Methyl-1 butanol

294.0 60.096 60.096 72.123 88.15 60.053 88.106 88.15

294.0 370.35 355.41 380.81 404.35 391.05 350.21 401.85

466.0 536.8 508.3 574.8 577.2 591.9 523.3 575.4

5.55 5.17 4.764 4.295 3.930 5.786 3.88 3.94

0.2907 0.6204 0.6669 0.5857 0.5890 0.4665 0.3664 0.5736

The control of the presence and amount of such compounds is a very important factor in the quality and specifications of each alcoholic beverage. The evaluation of an industrial plant requires physical and thermodynamic properties, including phase equilibria data. Some of these properties of the main congeners are presented in Table 1.5.

1.4.2 Volatile or Essential Oil Industries Volatile or essential oils have an important application in different industrial sectors. Traditionally, these compounds have been employed as aroma in the perfumery industry and, also, as compounds with medicinal properties in folk medicine since ancient times. In fact, a wide variety of substances with potential beneficial health effects has been identified in this group, which makes them important for food and other related products. Essential oils are mixture of volatile substances produced by the secondary metabolism of plants and are mainly obtained from aromatic, condimentary, and medicinal plants. They are chemically classified as phenylpropanoids and terpenoids. The phenylpropanoids are formed from shikimic acid and are often the principal volatile component characteristic of certain species. The main phenylpropanoids found in aromatic species are eugenol, chavicol, estragole, eugenol methyl ether, and others such as anethole, ferulic acid, and myristicin [19,20]. The terpenoids are predominant and encompass a wide variety of substances. They are constituted of isoprene units (C5H8): hemiterpenes (C5H8), monoterpenes (C10H16), sesquiterpenes (C15H24), diterpenes (C20H32), triterpenes (C30H48), tetraterpenes (C40H64) [21,22]. Most terpenoids are formed through the mevalonic acid metabolic pathway, but they can also originate 15

Thermodynamics of Phase Equilibria in Food Engineering

Figure 1.4. Chemical structure of some essential oils.

from methylerythritol 4-phosphate and are found in several parts of the plant. The chemical structure of some essential oils is illustrated in Fig. 1.4. The importance of these compounds is related to their intense flavor and the bioactivities that are attractive for the cosmetic, pharmaceutical, and food industries. The essential oils can be obtained from the flowers, bark, stem, leaves, roots, fruits, and other parts of the plant (see Table 1.6) by various methods. In industry, it is mainly performed by distillation and steam distillation, but other processes are also used: such as supercritical extraction, solvent extraction, expression (or pressing, used mainly in citrus), enfleurage (used by the perfumery industries to extract volatile oils from flowers), and hydrodistillation (used on few plants on an industrial scale, mainly used on laboratory scale). The high volatility of these compounds is the main reason why the industry prefers to use distillation as a means of obtaining these compounds. However, some compounds have very close volatilities; for this reason, data of phase equilibrium behavior under different operating conditions is so important. In the other cases (except for pressing), the solubility of essential oils in the solvent is also a decisive parameter to be taken into account in the process. 16

Phase Equilibria in the Food Industry

Table 1.6. Natural plants used for the obtention of essential oils in the food, cosmetic, and pharmaceutical industries [23,32] Type of material

Raw material

Botanical name

Dried bud

Cloves

Eugenia caryophyllata

Juniperus drupacea L.

Seeds

Juniper berry Cumin

Cuminum cyminum L.

Grass

Citronella

Cymbopogon winterianus

Flowers Leaf Evergreen leaves Bark Wood Roots Rhizome Bulb Fresh fruit Fresh fruit peel Dried fruit

Main compounds

Eugenol, eugenol acetate, caryophyllene Jasmine Jasminum grandiflorum L. Benzyl acetate, cis-phytol Sage Salvia officinalis L. Camphor, thyjone Cedar leaf Thuja occidentalis L. α-Thujone, β-thujone, fenchone Cinnamon Cinnaomum zehlanicum Linalool, α-pinene, α-phellandrene Sandalwood Santalum album Santalol Angelica β-Phellandrene, p-cymene Ginger Zingiber officinale Nerolidol Roscoe Onion Allium cepa Dipropyl disulphide, propionaldehyde Lime Citrus limonia Osbeck Limonene Orange Citrus sinensis Osbeck Limonene α-Pinene, thymol methyl ether, camphor γ-Terpinene, p-cymene, β-pinene Citronellal, citronellol, geraniol

1.4.3 Vegetable Oil Industry The industrialization of oil seeds constitutes an important agroindustrial sector, due to the use of their products in the food, cosmetic, pharmaceutical, and biofuel industries. Crude vegetable oils consist of triacylglycerols (TAGs) (96%98%), diacylglycerols (DAGs), monoacylglycerols (MAGs), free fatty acids (FFAs), phospholipids, and other minor components, such as vitamins (tocopherols and tocotrienols), sterols (such as phytosterols) and sterol esters, pigments (carotenes and chlorophylls), metals, and traces of other substances [33]. TAGs are composed of three fatty acids, which are substituted in the hydroxyl (alcoholic) sites of a glycerin (glycerol) backbone, as shown in Fig. 1.5, where Ri represents the hydrocarbon chain of each fatty acid. Most fatty acids have carbon chains varying from 12 to 18 carbon atoms, and they may be either saturated or unsaturated, as indicated in Table 1.7. 17

Thermodynamics of Phase Equilibria in Food Engineering

Figure 1.5. Formation reaction of a triacylglycerol.

Table 1.7. Main fatty acids present in vegetable oils [18,3436] Symbol C z:y

a

C4:0 C6:0 C8:0 C10:0 C12:0 C14:0 C16:0 C16:1(9) C18:0 C18:1(9) C18:2(9,12) C18:3 (9,12,15) C20:0 C20:1 C20:4 (5,8,11,14) a

Usual name (Abbreviation)

Official name (IUPAC)

Butyric acid Caproic acid Caprylic acid Capric acid (C) Lauric acid (L) Myristic acid (M) Palmitic acid (P) Palmitoleic acid (Po) Stearic acid (S) Oleic acid (O) Linoleic acid (Li)

Butanoic acid Hexanoic acid Octanoic acid Decanoic acid Dodecanoic acid Tetradecanoic acid Hexadecanoic acid cis-Hexadec-9-enoic acid Octadecanoic acid cis-Octadec-9-enoico acid cis,cis-Octadec-9,12-dienoico acid Linolenic acid (Ln) cis,cis,cis-Octadec-9,12,15trienoic acid Arachidic acid (A) Eicosanoic acid Gadoleic acid (Ga) (Z)-icos-9-enoic acid Arachiconic acid (An) 5,8,11,14-Eicosatetraenoic acid

Tm (K)

267.95 270.15 289.65 304.75 316.89 327.37 335.66 273.65 342.75 286.53 268.15 262.05 348.23 296.65 223.65

C z:y: where z, number of carbons and y, number of double bonds.

Therefore, there are different possibilities of distribution of the different fatty acid chains in the mixed TAG molecules. As a result, vegetable oils are normally a complex mixture of many different TAGs, as observed in the composition of some oils indicated in Table 1.8. 18

Table 1.8. Composition (%) of vegetable oils in terms of TAGs [3739] TAG

Soybean oila

Sunflower oila

Rapeseed oila

Cottonseed oila

TAG

Palm oilb,c

Coconut oilb

Palm kern oilb

LOP LLiP PPoP MLiP LOLi LLiLi POP PLiP PPoLi MLiLi POS POO PLiS POLi PLnO PLiLi PoLiLi PLeLi POA PLiA SOO SOLi OOO SLiLi

      1.4 3.14   0.89 5.35  11.74  14.59  1.91   1.77 6.99 2.83 

       1.27  

      0.52    0.58 8.07  5.79 2.75      2.68  35.21 

0.09 0.26 0.62 1.16 0.19 0.23 3.67 13.74 1.93 1.11 0.54  3.91 14.3  26.58 1.3  0.07 0.39  1.31  4.75

LMO MPLi LMP LOO LPO LPP 1 MMO OOLi MMP MOO PMO 1 POLi POLi PPLi MPP OOO POO PPO PPP SOO PSO PPS SSO

 2.4     0.7 1.8   10.1 9.8 0.6 4.1 24.2 31.1 5.9 2.3 5.1 0.9 0.5

2.4  5.5 1.1 1.6 1.9  0.2 0.8 2.1    0.6 0.3 0.7 0.6    

4.6  4.6 3.8 4.3   0.7 2   0.6  1.4 1.9 1.1 0.1 0.4 0.4  

2.02  7.63  11.52     0.67 3.64 2.72 

(Continued)

Table 1.8. (Continued) TAG

Soybean oila

Sunflower oila

Rapeseed oila

Cottonseed oila

OOLi OLiLi OOLn LiLiLi OLiLn LiLiLn OOA OLiA OOGa OLiGa LiLiA

13.58 16.82  17.12  1.88     

16.89 29.8  23.82       

23.54  14.42  4.15  0.63  1.04 0.6 

 10.43  12.88  0.28  0.09   0.17

a

Estimated (% in mass). % Area. Refined, bleached, and deodorized palm oil.

b c

TAG

Palm oilb,c

Coconut oilb

Palm kern oilb

Phase Equilibria in the Food Industry

In addition to direct consumption by the consumer, vegetable oils have been used as a source of nutraceutical compounds of importance for the pharmaceutical industry. Compounds like phytosterols, isoflavonoids, omega 3 (such as linolenic acid, eicosatrienoic acid, docosapentaenoic acid), omega 6 (such as linoleic acid, arachidonic acid), and omega 9 (such as oleic acid, elaidic acid) have regulatory action in the human organism with inhibitory effects on various diseases. Thus, studies on the extraction methods of these compounds have gained visibility among researchers in the area. In the oil industry, several unit operations are involved to produce an edible oil. The processing is comprised of two major steps: (1) the extraction to obtain the crude oil and the bran and (2) the refining (chemical or physical) whose purpose is the removal of a series of impurities from the crude oil to obtain an edible oil. During the refining step, which includes the processes of degumming, neutralization, clarification, and deodorization, undesirable substances in the crude oil are removed, such as phospholipids, pigments, oxidized materials, flavors, trace metals, and sulfur compounds. However, these processes also remove valuable substances, such as antioxidants and vitamins, being important to establish conditions that minimize the loss of such compounds. In addition, other industrial procedures can be used to obtain related products, such as hydrogenation in the production of margarine, or recovering the phospholipids to manufacture lecithin. The main steps where the phase equilibrium is present are listed in Table 1.9. Table 1.9. Processes in the vegetable oil industry and correlated phase equilibria Product

Processa

Type of equilibrium

Vegetable oil

Distillation of miscella (solvent removal) Deacidification (distillation, liquidliquid extraction) Clarification (adsorption) Deodorization (distillation, stripping, adsorption)

VLE VLE, LLE SLE VLE, SLE (Continued)

21

Thermodynamics of Phase Equilibria in Food Engineering

Table 1.9. (Continued) Product

Processa

Type of equilibrium

Related products (a) Bioactive compounds esters, FA vitamin E, esters, FAs FAs Sterols (phytosterols)

Fractioning (crystallization) Extraction Distillation Extraction

SLE LLE, SCFSE VLE LLE, SCFSE

(b) By-product Margarine Hydrogenated oil Isolated proteins

Hydrogenation (emulsion) Hydrogenation (emulsion) Extraction

Chem. Eq. Chem. Eq. LLE

where Chem. Eq. is chemical equilibrium. a Solid liquid extraction is not being considered here because the equilibrium observed in this operation is not true. See the footnote “a” presented in Table 1.2.

1.4.4 Dairy Industry and Derivatives The milk industry is quite diverse not only for producing milk and its derivatives (yoghurt, cheeses, butter, etc.) but also because these products are the base for other foods (ice cream, cream, concentrated milk, etc.), and functional compounds (milk proteins, lactose, etc.). Milk is an emulsion oil-in-water type, i.e., a multidispersed phase of fatty material in fluid that contains water and other substances (presented in a colloidal or suspension form), including carbohydrates, proteins, vitamins, salts, minerals. Although whole milk is composed of a large amount of water (87.4%), many of the applications of this food in related industries come from the presence of the other constituents that are part of the milk solids, the proportion of which is shown in Fig. 1.6 [40]. One of the main constituents is the milk proteins, which are a rich source of precursors of biologically active peptides. The main protein is casein (actually, a mixture of four proteins: αS1-, αS2-, β-, and κ-casein). The remainder consists mainly of milk whey proteins, mainly α-lactalbumin and β-lactoglobulin (70%80% of the total whey proteins) [40,41], which have a high value for the pharmaceutical industry, and several smaller proteins, such as enzymes which also have high added value.

22

Phase Equilibria in the Food Industry

Figure 1.6. Composition of milk solids (on dry basis). Based on [40].

Milk fat is a complex mixture of lipids composed mainly of TAGs (97%98%), and a minority of phospholipids, free sterols, DAGs, MAGs, and FFAs, such as vitamins A, D, E, and K [40]. About 62% of the fatty acids are saturated, 30% monounsaturated, 4% polyunsaturated, and 4% are other minor types of fatty acids [42]. Lactose is the main carbohydrate and the solid component that is present in greater quantity in milk (38.1% of solid milk, dry basis). It is a disaccharide consisting of galactose and glucose. Although this sugar has low relative sweetness (two to four times the amount of lactose is required to produce the sweetness of sucrose), among of the main characteristics of this sugar is its ability to facilitate browning, induce crystallization, stabilize proteins, enhance flavor, and improve mouth-feel texture in some products and/or viscosity [43]. As seen, milk is a very nutritious food product, which can be consumed in its original form or processed. As well as this, the functional characteristics of milk constituents, emulsifying ability and foam formation of milk proteins, for instance, are important for the formulation of other products. For such constituents to be utilized by other industries, different processes of separation, fractionation, and isolation are used for their recovery. Some examples are listed in Table 1.10, being evidenced when the phase equilibrium is present. It is worth remembering that the use of separation processes is not only important for obtaining a specific component. Drying and evaporation are examples of separation processes whose end product in the milk industry is not a single compound but a new product.

23

Thermodynamics of Phase Equilibria in Food Engineering

Table 1.10. Processes and type of phase equilibrium present in the processing of milk products and by-products Product

Process

Type of equilibrium

Milk powder Milk cream Butter (emulsion) Concentrated milk Sweetened condensed milk Protein

Drying, evaporation Homogenization Homogenization Evaporation Evaporation Solubilization Extraction Adsorption Purification (liquidliquid extraction, ion-exchange resin) Crystallization Fractionation (crystallization, dry, solvent fractionation) Crystallization

VLE LLE LLE VLE VLE SLE LLE SLE LLE, SLE

Isolated protein Milk fat Lactose

SLE SLE, VLE, LLE SLE

1.5 CONCLUDING REMARKS The food industry is comprised of a wide range of food segments, often containing a series of industrial steps for each segment. In this chapter, it was possible to visualize how the phase equilibrium is inserted into this context, and how important it is to have data that can aid in the processing and optimization of a certain stage of the process, and in obtaining a particular product. In the next chapters of this book, a theoretical basis of the equations that govern the phase equilibrium will be presented. These equations are used to determine phase behavior, as well as the exposition of the various types of equilibrium, and specific cases are given in the section of advanced phase equilibrium studies.

REFERENCES [1] Lea AGH, Piggott JR. Fermented beverage production. 2nd ed. New York, NY: Springer; 2003. [2] Vamam AH, Sutherland JP. Beverages: technology, chemistry and microbiology. Salisbury, UK: Chapman & Hall; 1994. [3] MAPA. Ministe´rio da Agricultura, Pecua´ria e Abastecimento. Regulamento da Lei nu´mero 8.918, de 14 de julho de 1994.

24

Phase Equilibria in the Food Industry

[4] MAPA. Ministe´rio da Agricultura, Pecua´ria e Abastecimento. Regulamento te´cnico para fixac¸a˜o dos padro˜es de padro˜es de identidade e qualidade para bebidas alcoo´licas destiladas, comercializadas em todo territo´rio nacional. Instruc¸a˜o Normativa no15, % de 31 de marc¸o de 2011. hhttp://www.agricultura.gov.br/i (accessed in July 2017). [5] Encyclopedia Britannica. hhttps://www.britannica.comi (accessed in July 2017). [6] Pino JA. Characterization of rum using solid-phase microextraction with gas chromatographymass spectrometry. Anal Nutr Clin Methods Food Chem 2007;104:4218. [7] Regulation (EC) no. 110/2008 of the European Parliament and of the Council of 15 January 2008. On the definition, description, presentation, labelling and the protection of geographical indications of spirit drinks and repealing Council Regulation (EEC) No 1576/89. [8] FDA. National Agency for Food and Drug Administration and Control Act 1993. Spirit Drinks Regulations 2005. [9] Comite´ interprofessionnel du vin de champagne. The Champagne harvest—late, but potentially outstanding. hwww.champagne.fri (accessed in July 2017). [10] Culbert J, Verdonk N, Ristic R, Mantilla SO, Lane M, Pearce K, et al. Understanding consumer preferences for Australian sparkling wine vs. French champagne. Beverages 2016;2(3):112. [11] Piggot JR. Alcoholic beverages. Sensory evaluation and consumer research. Cambridge, UK: Woodhead Publishing Limited; 2012. [12] Valderrama JO, Toselli LA, Fau´ndezd CA. Advances on modeling and simulation of alcoholic distillation. Part 2: Process simulation. Food Biop Proces 2012;90:83240. [13] Batista FRM, Meirelles AJA. Computer simulation applied to studying continuous spirit distillation and product quality control. Food Control 2011;22:1592603. [14] Gaiser M, Bell GM, Lim AW, Roberts NA, Faraday BF, Schulz RA, et al. Computer simulation of a continuous whisky still. J Food Eng 2002;51:2731. [15] Piggott JR, Conner JM. Whiskies. In: Lea AGH, Piggott JR, editors. Fermented beverage production. 2nd ed. New York, NY: Springer; 2003. [16] Valderrama JO, Huerta R, Alarco´n R. Base de Datos de Propiedades de Sustancias para Procesos de Destilacio´n de Vinos. Inf Tecnol. 2002;13(4):15566. [17] Valderrama JO, Fau´ndez CA. Modelado del Equilibrio Lı´quido-Vapor en Mezclas Binarias y Ternarias de Intere´s en Destilacio´n Vı´nica. Inf Tecnol 2003;14(1):8392. [18] DIPPR. Design Institute for Physical Properties Data Bank In AIChE: [2005, 2008, 2009, 2010]. [19] Polya G. Biochemical targets of plant bioactive compounds. A pharmacological reference guide to sites of action and biological effects. Florida: CRC Press; 2003. [20] Anand A, Jayaramaiah RH, Beedkar SD, Singh PA, Joshi RS, Mulani FA, et al. Comparative functional characterization of eugenol synthase from four different Ocimum species: implications on eugenol accumulation. Biochim Biophys Acta 2016;1864:153947. [21] Guenther E. The essential oils—Vol 1: history—origin in plants—production—analysis. Jepson Press; 2007. [22] Simo˜es C.M.O., Schenkel E.P., Gosmann G., Auler Mentz JCPM, Petrovick PR. Farmacognosia: daPlanta ao Medicamento. Ed. Universidade/UFRGS/Ed. da UFSC: Rio Grande do Sul, Brazil, 1999. [23] Silva TK. A manual on essential oil industry. Vienna, Austria: United Nations Industrial Development Organization; 1995. [24] Simon DZ, Beliveau J, Aube G. Cedarleaf oil (Thuja occidentalis L.) extracted by hydrodiffusion and steam distillation. A comparison of oils produced by both processes. Int J Crude Drug Res 1987;25:46.

25

Thermodynamics of Phase Equilibria in Food Engineering

[25] Lima MP, Zoghbi MGB, Andrade EHA, Silva TMD, Fernandes CS. Volatile constituents from leaves and branches of Cinnamomum zeylanicum Blume (Lauraceae). Acta Amaz. 2005;35(3):3636. [26] Romeilah RM, Fayed SA, Mahmoud GI. Chemical compositions, antiviral and antioxidant activities of seven essential oils. J Appl Sci Res 2010;6(1):5062. [27] Nykanen I, Nykanen L. Composition of angelica root oils obtained by supercritical CO, extraction and steam distillation. J Ess Oil Res 1991;3:22936. [28] Tu NTM, Thanh LX, Une A, Ukeda H, Sawamura M. Volatile constituents of Vietnamese pummelo, orange, tangerine and lime peel oils. Flavour Fragrance J 2002;17:16974. [29] Borges P, Pino J. The isolation of volatile oil from cumin seeds by steam distillation. Nahrung 1993;37(2):1236. [30] Gulfraz M, Mehmood S, Minhas N, Jabeen N, Kausar R, Jabeen K, et al. Composition and antimicrobial properties of essential oil of Foeniculum vulgare. Afr J Biotechnol 2008;7(24):43648. [31] El-Ghorab A, Shaaban HA, El-Massry KF, Shibamoto T. Chemical composition of volatile extract and biological activities of volatile and less-volatile extracts of juniper berry (Juniperus drupacea L.) Fruit. J Agric Food Chem 2008;56:50215. [32] Zhang H, Qiu M, Chen Y, Chen J, Sun Y, Wang C, et al. Plant terpenes. Phytochemistry and pharmacognosy. Encyclopedia of life support systems (EOLSS) 2011. [33] Gupta MK. Practical guide to vegetable oil processing. Urbana, IL: AOCS Press; 2008. [34] Belitz HD, Grosch W, Schieberle P. Food chemistry. 4th ed. Springer-Verlag Berlin Heidelberg; 2009. [35] Shahidi F. Bailey’s industrial oil and fat products. 6th ed. Hoboken, New Jersey: John Wiley & Sons, Inc; 2005. [36] Vesely V, Chudozilov LK. Sur les acides gadole´ique et se´lachole´ique synthe´tiques. Collect. Czech. Chem. Commun. 1930;2:95107. [37] Belting PC, Gmehling J, Bo¨lts R, Rarey J, Ceriani R, Chiavone-Filho O, et al. Measurement, correlation and prediction of isothermal vaporliquid equilibria of different systems containing vegetable oils. Fluid Phase Equilibr 2015;395:1525. [38] Ceriani R, Meirelles AJA. Predicting vaporliquid equilibria of fatty systems. Fluid Phase Equilibr 2004;215:22736. [39] Tan CP, Man YBC. Differential scanning calorimetric analysis of edible oils: comparison of thermal properties and chemical composition. J Am Oil Chem Soc 2000;77(2):14356. [40] Chandan R. Dairy-based ingredients. Eagan Press. Minnesota, 1997. [41] Walstra P, Wouters JTM, Geurts TJ. Dairy science and technology. 2nd ed. New York, NY: Taylor & Francis; 2006. [42] Miller GM, Jarvis JK, McBean LD. National Dairy Council Handbook of dairy foods and nutrition. 2nd ed. Florida: CRC Press; 2000. [43] Pomeranz Y. Functional properties of food components. 2nd ed. Academic Press, Florida; 1991.

26

CHAPTER 2

Fundamentals of Phase Equilibria Camila Gambini Pereira

2.1 INTRODUCTION The importance of phase equilibria in food engineering has been evidenced in Chapter 1. The variety of applications demonstrates the considerable interest in knowing the equilibrium conditions necessary in obtaining a successful process and an expected product. As previously shown, many operations in the manufacture of food products involve the contact of different phases. This panorama reveals how essential the equilibrium data are for the different process stages. Indeed, the design, optimization, and simulation of these processes require reliable phase equilibrium data and models to ensure a realistic analysis. The principles of phase equilibrium are based on the stability of the phases. But before starting a discussion on this theme, there are some important concepts to be elucidated. In fact, what is equilibrium? And what is phase? The equilibrium state means no tendency to spontaneously change in the system properties, i.e., the properties of the system are independent of time, since no work, and no heat and mass transfer cross the boundary of the system. It is important here to point out that equilibrium state is different from steady state. For the equilibrium state, the homogeneity of intensive properties is required throughout the singlephase system, i.e., the intensive properties must be uniform (they have the same value point to point). For the steady state, these requisites are not imposed, that is, there may be a system in the steady state that is not in equilibrium. For physics and thermodynamics, phase is defined as state of matter: solid, liquid, gas, or supercritical. A simple system with two phases, α and β, is represented in Fig. 2.1. In the case of the focus of this book, food systems can be arranged into one, two, or more phases, depending on the type of raw material,

Thermodynamics of Phase Equilibria in Food Engineering

© 2019 Elsevier Inc. All rights reserved.

27

Thermodynamics of Phase Equilibria in Food Engineering

T P

Phase α c

x1α , x2α , x3α , x4α ,....xnα

c Phase β x1β , x2β , x3β , x4β ,....xnβ

Figure 2.1. Schematic representation of a system with two phases at temperature T and pressure P.

the industrial processing step, and characteristics of the final product. As well as this, the processing can modify the original state of the material according to operational conditions and the accomplished process. For this reason, the understanding of the changes observed in the involved phases during the processing becomes necessary. There is another concept that is very useful for someone who works day-to-day with various sets of thermodynamic data. During the industrial processing, many times the engineer needs to act quickly to solve problems found in the fabrication of a determined product online. Of course, we are not talking about mechanical problems, or even about instability or maladjustment of one or more devices. However, physical problems, such as the failure to obtain the desired separation, or the accumulation (precipitation) of material on the walls of the pipe, or the failure to obtain a homogeneous fluid during processing, can be avoided. Sometimes the engineer has several pieces of process information in front of them, in this case, what information should he/she choose? Or better, what process data can he/she trust? There are many questions; however, a simple piece of information can expedite the resolution of the problem and/or facilitate finding the 28

Fundamentals of Phase Equilibria

solution. This information can be obtained by the Gibbs phase rule, Eq. (2.1), which establishes the number of intensive properties that are needed to identify the thermodynamic state in a multicomponent system. F 522π1n2r

(2.1)

where F is the number of degrees of freedom, π is the number of phases, n is the number of compounds present in the system, and r is the number of additional constraints (reactions, and critical points, for instance). Knowing the number of intensive properties, and having this information (only considering reliable data), the engineer can obtain information to solve the problem or to direct him/her to find the solution. The thermodynamic equations and correlations can lead to obtaining new data that can assist in the setting process. Consider the system represented in Fig. 2.1, with two phases (α and β). Each phase is constituted of n compounds, whose mole fraction of each component is given by xα1 ; xα2 ; xα3 ; xα4 ; . . .; xαn for phase α and xβ1 ; xβ2 ; xβ3 ; xβ4 ; . . .; xβn for phase β. Here, the number of independent intensive properties necessary to define the system is directly related to the number of compounds involved. Observe that Eq. (2.1) is very useful and can aid in different circumstances. For instance, for a biphasic (two phases) and binary system (two compounds), non-reactive, a combination of two intensive properties (T and P, or T and xαn , or xαn and xβn , etc.) is enough to define the properties of the system.

2.2 FROM CLASSICAL THERMODYNAMICS TO PHASE EQUILIBRIUM In the real industrial world, the task of many engineers is to transcribe the information obtained during the processing into properties and data to be inserted into mathematical models, or vice versa. Although this is a challenge for many, thermodynamics is an important ally to assist in this function. The Classical Thermodynamics is a rich science. Many of the fundamental equations provide the theoretical basis to formulate new equations and correlations which are responsible to link theory to real data. These equations are essential for expressing the thermodynamic abstraction in a piece of information that translates the problem of phase equilibria. The representation of this connection between the reality and a value (property) numerically exact is schematically illustrated in Fig. 2.2. 29

Thermodynamics of Phase Equilibria in Food Engineering

Figure 2.2. Representation of the use of thermodynamics in solving problems within the industry.

2.2.1 Fundamental Relationships Several thermodynamic books show an extensive description of the fundamental properties [14]. The discussion about the development of these properties is not the focus of this section. The purpose here is to present briefly the fundamental equations and correlations of thermodynamic properties. Throughout this book, these equations will be used to designate the theoretical basis to transcribe the real information into a property, sometimes apparently abstract, or vice versa, and moreover, these equations serve as the basis to formulate new equations. Table 2.1 summarizes the main equations derived from Classical Thermodynamics with a wide application in process engineering and in phase equilibrium studies. Table 2.1. Important equations from Classical Thermodynamics Definitions

H 5 U 1 PV A 5 U 2 TS G 5 U 1 PV 2 TS 5 H 2 TS 5 A 1 PV Fundamental thermodynamic functions

dU 5 TdS 2 PdV 1 GdN

or

dU 5 TdS 2 PdV 1

dH 5 TdS 1 VdP 1 GdN

or

dH 5 TdS 1 VdP 1

dA 5 2 SdT 2 PdV 1 GdN

or

dG 5 2 SdT 1 VdP 1 GdN

or

X X

(2.2) (2.3) (2.4)

μi dNi

(2.5)

μ dNi (2.6) Xi dA 5 2 SdT 1 PdV 1 μ dNi (2.7) X i dG 5 2 SdT 1 VdP 1 μi dNi (2.8) (Continued)

30

Fundamentals of Phase Equilibria

Table 2.1. (Continued)

Derivatives relationsa     @A @U 2P 5 5 (2.9) @V T @V S     @H @G V5 5 (2.10) @P S @P T     @U @H T5 5 (2.11) @S V @S P     @A @G 2S 5 5 (2.12) @T V @T S Other properties

Maxwell relations     @S @P 5 @V T @T V     @T @V 5 @P S @S P     @T @P 52 @V S @S V     @S @V 52 @P T @T P

   @U @S 5T @T V @T V     @H @S CP 5 5T @T P @T P   1 @V α5 V @T P   1 @V κT 5 2 V @P T

(2.13) (2.14) (2.15) (2.16)



CV 5

(2.17) (2.18) (2.19) (2.20)

where H is the enthalpy; U is the internal energy; P is the pressure; V is the volume; A is the Helmholtz energy; T is the temperature; G is the Gibbs energy; S is the entropy; N is the number of moles; μi is the chemical potential; CV is the constant volume heat capacity; CP is the constant pressure heat capacity); α is the isobaric thermal expansion coefficient; kT is the isothermal compressibility coefficient.       a Remember that @X=@Y Z;N 5 @X =@Y Z , where the thermodynamic property X 5 X=N .

2.2.2 Thermodynamic Properties of Mixtures An important topic of thermodynamics and something that is fundamental in the studies of phase equilibrium is related to the properties of mixtures. For that, first we will introduce the definition of Partial Molar Thermodynamic Property, referred to θ i. This property corresponds to the representative portion of the thermodynamic property of a component i within the property of the mixture as a whole (θ):   @N θ θi 5 (2.21) @Ni T ;P;Nj6¼i

31

Thermodynamics of Phase Equilibria in Food Engineering

with Nθ 5

X

Ni θ i

(2.22)

or rewriting Eq. (2.21), the molar property of the mixture (θ) can be determined by X θ5 xi θ i (2.23) where Ni is the number of moles of the component i, N is the total number of moles, and xi is the mole fraction of the component i in the mixture. In other words, at a given temperature and pressure, each component   i has a specific set of properties U i ; V i ;H i ; . . . that are responsible for part of the total property of the mixture U ; V ; H ; . . . . Thus, the property of the mixture is the result of the weighted sum of the partial molar properties of each component in the mixture. This is true for any thermodynamic property (see Table 2.2), being important in the characterization of the behavior of different mixtures. Note that θ i 6¼ θ i . By the definition: θ i is the molar property of pure component i (fixed value, at constant T and P), while θ i is the partial molar property of the component i in the mixture, dependent on the kind of mixture and composition (at constant T and P). However, it is possible to calculate the property of the mixture θ using the molar property of pure component i, since an additional term related to molecular interaction between molecules, defined as mixing property (Δθ mix ), is considered in this calculation, as indicated by the following equation: X θ5 xi θ i 1 Δθ mix (2.30)

Table 2.2. Partial molar properties of the component i in the mixture (θ i )

  @N U Ui 5 (2.24) @Ni T;P;Nj6¼i   @N V Vi5 (2.25) @Ni T;P;Nj6¼i   @N H Hi 5 (2.26) @Ni T;P;Nj6¼i

32

  @N A Ai 5 (2.27) @Ni T ;P;Nj6¼i   @N S Si 5 (2.28) @Ni T ;P;Nj6¼i   @N G Gi 5 (2.29) @Ni T ;P;Nj6¼i

Fundamentals of Phase Equilibria

Thereby, both Eqs. (2.23) and (2.30) can be used to calculate the property of the mixture. For instance, for a binary system, containing the species 1 and 2, the volume of the mixture can be calculated by V 5 x1 V 1 1 x2 V 2

(2.31)

V 5 x1 V 1 1 x2 V 2 1 ΔV mix

(2.32)

or by and from Eqs. (2.31) and (2.32), the volume of mixing (ΔV mix ) can be expressed by     ΔV mix 5 x1 V 1 2 V 1 1 x2 V 2 2 V 2 (2.33) and then, expanding to a mixture containing i compounds for any thermodynamic property of mixing, one has: X   Δθ mix 5 xi θ i 2 θ i (2.34) i

The consequence of Eq. (2.21) is its application to determine a specific partial molar property through the following equation: X @θ   θi 5 θ 2 xk (2.35)  @x k T ;P;xj6¼i;k k6¼i For binary systems, Eq. (2.35) reduces to θ 1 5 θ 1 x2

dθ dx1

(2.36)

θ 2 5 θ 2 x1

dθ dx1

(2.37)

The thermodynamic properties of mixtures are quite relevant in food systems, due to the different behaviors resulting from molecular interactions often found in these systems. Herein, the mixing properties of several mixtures have been investigated recently [59]. The variety of functional groups found in the compounds involved, which consequently results in the occurrence of different interactions, is the reason for the diverse behavior observed in these systems. Fig. 2.3 illustrates an example of mixing effects on system properties. Herein, the enthalpy and volume of mixing of olive oil in different solvents change with the composition and type of solvent present in the binary mixture.

33

Thermodynamics of Phase Equilibria in Food Engineering

(A) 4000 3500

ΔHmix (J/mol)

3000 2500 2000 1500 1000 500 0

0

0.2

0.4

0.6

0.8

1

x1 (B) 4 3

ΔVmix (cm3/mol)

2 1 0 –1 –2 –3 –4

0

0.2

0.4

0.6

0.8

1

x1

Figure 2.3. Mixing properties in function of concentration of the system solvent (1) olive oil (2) at 298.15 K: (A) enthalpy of mixing [8], and (B) volume of mixing [9]: methanol (V), ethanol (x), 1-propanol (&), 2-propanol (▲), 1-butanol (K), 2-butanol (’).

Moreover, among the equations presented in Table 2.2, the partial molar Gibbs free energy is the most important partial molar property in the studies of phase equilibrium, since the chemical potential of component i (μi ) is defined through Eq. (2.38). Additional discussion is presented in Section 2.3.     @G @N G μi 5 5 5 Gi (2.38) @Ni T ;P;Nj6¼i @Ni T ;P;Nj6¼i 34

Fundamentals of Phase Equilibria

In the case of mixtures, different scenarios can be found. Amongst them, some special characteristics can be observed in ideal systems. Determining the best reference assists in quality and accuracy of system data. In Ideal-Gas Mixtures (IGMs), the density of the mixture is so low that the interaction between the species can be neglected. Thereby, the behavior observed in ideal gas can be extended for this type of mixture. Three fundamental features of ideal gas can be listed: internal energy and enthalpy are a function only of temperature, i.e., U 5 f ðTÞ and H 5 f ðTÞ; and the equation of state (EoS) specific for an ideal gas is defined by PV IG 5 NRT or PV IG 5 RT. For ideal-gas mixtures, the equation of state of ideal gas can also be used: PV 5 NRT , where N 5 N1 1 N2 1 N3 1 ? 1 Nn. Therefore, the volume of the ideal-gas mixture can be calculated by V IGM 5

NRT ðN1 1 N2 1 N3 1 ? 1 Nn ÞRT 5 P P

(2.39)

From the definition of the partial molar property expressed in Eq. (2.21), the partial molar volume of the ideal gas presented in this mixture can be defined by   @N V IGM IGM Vi 5 (2.40) @Ni T ;P;Nj6¼i Using Eq. (2.39), the partial molar volume of a gas presented in the ideal-gas mixture results in the following equation:    @ðN1 1N2 1?1Ni 1?1Nn Þ RT =P RT IGM Vi 5 5 (2.41) @Ni P T ;P;Nj6¼i so IGM

Vi

5 V IG i 5

RT P

(2.42)

This indicates that the volume occupied by the pure ideal gas is the same as the volume occupied by a gas presented in an ideal-gas mixture.1 1

Although ideal-gas mixture (IGM) and mixture of ideal gas (MIG) are conceptually different [1,10], in terms of properties of mixtures this difference is considered null, since in both cases, the density of the mixture is low enough to treat the interaction between moleIGM MIG cules as zero or almost nonexistent, resulting in V i 5 V i 5 V IG i . Then, in terms of calculation, “IGM” and “MIG” may be considered similar.

35

Thermodynamics of Phase Equilibria in Food Engineering

For ideal-gas mixture, except in the moment of the collision, there is no effective interaction between the molecules; for this reason, the internal energy partial molar of compound present in this ideal-gas mixture is also a function of temperature only, then   IGM  n X @U ðT ;N Þ @  IGM IG U i ðT ;xÞ5 5 Nk U IG i ðT Þ5U i ðT Þ  @Ni @N i T ;P;Nj6¼i k51 T ;P;Nj6¼i (2.43) IGM Vi

From the equalities given by Eqs. (2.42) and (2.43), 5 V IG i and IG 5 U i , the partial molar enthalpy of the component i present in an ideal-gas mixture leads to

IGM Ui

IGM

Hi

IGM

5Ui

IGM

1 PV i

IG IG 5 U IG i 1 PV i 5 H i

(2.44)

As a consequence of Eqs. (2.42)(2.44) and (2.34), the values of volume, enthalpy, and internal energy of mixing of an ideal-gas mixture are equal to zero. For the other properties (entropy, Helmholtz free energy and Gibbs free energy), as the occurrence of mixing is an irreversible event, the entropy of the system varies, compared to the reference which is the ideal gas. Consequently, Helmholtz free energy and Gibbs free energy are also changed. Using the thermodynamic relations (Table 2.1), these last partial molar properties are defined, as well as the mixing properties of ideal-gas mixture (Δθ IGM ), as presented in Table 2.3. Parallel to the concept of ideal-gas mixture, the ideality of a system can be adopted to real systems. These systems, named as Ideal Mixtures (IMs) or Ideal Solutions, are constituted of different species whose forces exerted between themselves are the same. Although these substances

Table 2.3. Thermodynamic properties of an ideal-gas mixture, at constant T and P IGM

Vi

5 V IG i

IGM U i 5 U IG i IGM H i 5 H IG i IGM S i 5 S IG i 2 Rlnxi IGM A i 5 A IG i 1 RT lnxi IGM G i 5 G IG i 1 RT lnxi

36

(2.42)

ΔV IGM 5 0

(2.48)

(2.43)

(2.49)

ΔU

IGM

50

(2.44)

ΔH

IGM

50

(2.45)

ΔS

(2.46)

ΔA

(2.47)

ΔG

IGM

52R

IGM IGM

5 RT 5 RT

X X X

(2.50) xi lnxi

(2.51)

xi lnxi

(2.52)

xi lnxi

(2.53)

Fundamentals of Phase Equilibria

(molecules i and j) are different, they have very similar chemical structures, thereby the molecular interactions of the type ij or ji are very close to the molecular interactions of the type ii and jj. As a result, the properties of the species present in an ideal mixture are similar to the properties of the pure species. It is noted that in ideal-gas mixtures, the compounds present have the same behavior to that of an ideal gas, whereas for ideal mixtures, the species behave as if they were pure compounds. Therefore, in analogy to ideal-gas mixture, the properties of ideal mixture are defined, as presented in Table 2.4. For real systems, a mixture can be ideal or not. The consequence of this ends up falls upon the analysis of the mixture behavior and also on the equilibrium. Excess properties or excess functions (θ E ) are thermodynamic properties of mixtures that indicate how much the properties of a mixture are in excess to those of ideal solution at the same conditions of temperature, pressure, and composition, that is, θ E 5 θ 2 θ IM

(2.66)

and also E

IM

θi 5 θi 2 θi

(2.67)

Using the thermodynamic properties of ideal mixtures (Table 2.4) in Eq. (2.67), the functions in terms of the excess thermodynamic properties are defined according to the expressions indicated in Table 2.5. For the studies of phase equilibria, the most useful excess property is the Gibbs energy since this property is directly related to activity coefficient. This correlation will be discussed in Section 2.3. Note that θ E and Δθ mix are not necessarily equal. Even though this fact is evidenced in different books [1013], there is still some Table 2.4. Thermodynamic properties of an ideal mixture, at constant T and P IM

Vi 5Vi IM Ui 5Ui IM Hi 5Hi IM S i 5 S i 2 Rlnxi IM A i 5 A i 1 RT lnxi IM G i 5 G i 1 RT lnxi

(2.54)

ΔV IM 5 0

(2.60)

(2.55)

(2.61)

ΔU

IM

50

(2.56)

ΔH

IM

50

(2.57)

ΔS

(2.58)

ΔA

(2.59)

ΔG

IM

52R

IM IM

5 RT 5 RT

X X X

(2.62) xi lnxi

(2.63)

xi lnxi

(2.64)

xi lnxi

(2.65)

37

Thermodynamics of Phase Equilibria in Food Engineering

Table 2.5. Excess properties of real mixtures, at constant T and P

X   xi V i 2 V i X   E U i 5 U i 2 U i and U E 5 xi U i 2 U i X   E H i 5 H i 2 H i and H E 5 xi H i 2 H i X X   E xi lnxi S i 5 S i 2 S i 1 Rlnxi and S E 5 xi S i 2 S i 1 R X  X  E E A i 5 A i 2 A i 2 RT lnxi and A 5 xi A i 2 A i 2 RT xi lnxi X X   E xi lnxi G i 5 G i 2 G i 2 RT lnxi and G E 5 xi G i 2 G i 2 RT E

Vi 5Vi 2Vi

and

VE5

(2.68) (2.69) (2.70) (2.71) (2.72) (2.73)

misconception of considering experimental methods used for the calculation of Δθ mix as being those described to calculate θ E . However, this association is not always applicable. Although for volume, enthalpy, and internal energy, these quantities are numerically equal, the same cannot be said about the other intensive properties. In fact, by definition, excess properties are different from mixing properties, i.e., Δθ mix indicates the property variation observed when different substances are mixed to form a particular solution; and θ E indicates how much the property of real solution differs from the property of the ideal solution, in the same conditions of temperature, pressure, and composition. Comparing Eqs. (2.34) and (2.71)(2.73), it is clear that Δθ mix is different from θ E . For this reason, it is important to understand the definition and purpose of the use of these properties. In terms of process, what matters is the effect that these properties have on the system. Thus, the numerical equality of the mixing properties and the excess properties is accepted for the volume, enthalpy, and internal energy properties. Another important property for real fluids is the Residual Property (θ res ), defined as the difference between the property of real fluid and the property of the fluid with ideal-gas behavior, that is, θ res 5 θ 2 θ IG

(2.74)

Observe that residual properties are different from excess properties, not only in terms of equations but also mainly in terms of concept. The difference is clear and easy to identify: the reference of the residual properties is the ideal gas, while the reference for excess properties is the ideal mixture. The residual properties have been quite helpful for mixtures of nonideal gases. The relevance of these properties is more evident when it is applied to describe the behavior of fluids taking as a reference the 38

Fundamentals of Phase Equilibria

behavior of ideal gases. An example is its application in the structuring of the equations of state which use residual properties and thermodynamic correlations to formulate the basis of the equation. Further details of the use of residual properties are found in Chapter 3.

2.3 PRINCIPLES OF PHASE EQUILIBRIA 2.3.1 The Essence of Phase Equilibrium As in many books concerning chemical thermodynamics, the section of fundamentals of phase equilibria starts with definitions of Gibbs, as a simple correlation, or a common definition. Nevertheless, the concepts established by Gibbs allowed us to understand the thermodynamics from a new standpoint. In fact, his concepts led the classical thermodynamics to new directions, bringing about the beginning of Modern Thermodynamics. The work “On the Equilibrium of Heterogeneous Substances” [14], considered one of the greatest achievements in this area of the 19th century, is the basis of chemical and molecular thermodynamics. The purpose here is to present briefly the concepts introduced by Gibbs and to formulate the equilibrium conditions, providing a framework for the studies of phase equilibrium in food systems. For the formulation of phase equilibrium, consider the system represented by Fig. 2.1, containing two phases (α and β). Each phase is constituted of n compounds and is characterized by specific properties (T α ; P α ; U α ; V α ; Sα ; . . . for phase α and T β ; P β ; U β ; V β ; Sβ ; . . . for phase β). The transfer of matter and energy between phases can occur. Since the system is closed, with fixed limits, and thermally isolated, i.e., it is an isolated system, from the First Law of Thermodynamics, the total number of moles, total volume, and total internal energy of the system are constant, i.e.,: dN 5 0 5 dN α 1 dN β ; dV 5 0 5 dV α 1 dV β , and dU 5 0 5 dU α 1 dU β . From the Second Law of Thermodynamics, as dS $ 0, there are two possibilities for this system: being away from equilibrium (dS . 0) or in the equilibrium (dS 5 0). Moreover, as entropy is a crescent function (dS $ 0), in equilibrium the entropy of the system (S or S) reaches a maximum value. Subsequently, since the total entropy is the sum of the entropies of each phase, one obtains dS 5 dSα 1 dSβ

(2.75)

39

Thermodynamics of Phase Equilibria in Food Engineering

It is possible to evaluate the variations of total entropy assessing the individual entropies of each phase (dSα and dSβ ). From Eq. (2.5), the entropy of the system can be rewritten, resulting in 1 P 1X μi dNi ; with i 5 1; 2; 3 . . . n (2.76) dS 5 dU 1 dV 2 T T T Using Eq. (2.76) for both phases, dSα 5

1 Pα 1 X α α α dU 1 dV α 2 α μi dNi α α T T T

(2.77)

dSβ 5

1 Pβ 1 X β β β dU 1 dV β 2 β μi dNi β β T T T

(2.78)

Again, if the system is isolated (dN 5 0; dV 5 0 and dU 5 0), then dN α 5 dN β ; dV α 5 2dV β and dU α 5 2dU β . Thereby, applying these equalities and by replacing Eqs. (2.77) and (2.78) in Eq. (2.76), the total entropy of the system results in !    α  β β X μα 1 1 P P μ i dS 5 2 β dU α 1 2 β dV α 2 2 iβ dNiα Tα T Tα T Tα T (2.79) α

α

α

However, N 6¼ 0; dV 6¼ 0 and dU 6¼ 0. Therefore, in order to be in equilibrium, the entropy of the system needs to be constant (dS 5 0), and this is established when, in Eq. (2.79), the followed equations are satisfied simultaneously: 1 1 2 β 50 α T T

or

Tα 5 Tβ

(2.80)

Pα Pβ 2 50 Tα Tβ

or

Pα 5 Pβ

(2.81)

μαi μβi 2 50 Tα Tβ

or

μαi 5 μβi

(2.82)

The equalities expressed by Eqs. (2.80)(2.82) represent the necessary conditions to define the equilibrium between phases α and β. These conditions are also extended to systems constituted of many phases (α, β, δ,. . .,ψ). Moreover, according to Eq. (2.38), μi 5 G i , and so, Eq. (2.82) 40

Fundamentals of Phase Equilibria α

β

can also be written as G i 5 G i . Thus, the equilibrium conditions for the system containing many phases are Tα 5 Tβ 5 Tδ 5 ? 5 Tψ

(2.83)

Pα 5 Pβ 5 Pδ 5 ? 5 Pψ

(2.84)

μαi 5 μβi 5 μδi 5 ? 5 μψi

or

α

β

δ

ψ

Gi 5 Gi 5 Gi 5 ? 5 Gi

(2.85)

2.3.2 The Property Called Fugacity The property μi is, by definition, the partial molar Gibbs free energy, Eq. (2.38), and, therefore, is a state function. However, the determination of this property is not direct. For this reason, this property is generally replaced by another which easily leads to physical perception. Lewis [15], in 1901, introduced the concept of fugacity in order to appoint a property of real fluid through an evaluation performed from the ideal gas. The formulation of this new property is based on the analysis of molar Gibbs free energy. Considering 1 mol of an ideal gas at constant temperature, the integration of Eq. (2.8) between an arbitrary condition P and another P0 (as reference) results in P (2.86) P0 However, Eq. (2.86) was defined only for its application to ideal gas. For real fluids, in analogy to Eq. (2.86), Lewis defined a new property, fugacity (f ), being written as G IG 2 G IG;0 5 RT ln

G 2 G 0 5 RT ln

f f0

(2.87)

where f 0 is the fugacity of the pure component in the reference state. Observe that the reference state applied in Eq. (2.87) is arbitrary, thus, using the same reference state of Eq. (2.86), it becomes G 2 G IG;0 5 RT ln

f P0

(2.88)

and subsequently, the subtraction of Eq. (2.88) from Eq. (2.86) results in the equation used for the fugacity calculation: G 2 G IG 5 RT ln

f P

(2.89) 41

Thermodynamics of Phase Equilibria in Food Engineering

Eq. (2.89) provides the definition of fugacity which has the aim to indicate the limit of the ideality of the system. This property, with the pressure unit, represents how much the behavior of a real substance deviates from the ideal gas behavior. At low pressures (P-0), the fugacity becomes equal to the system pressure (f -P), this is because the pressure is low enough so that the fluid has the behavior of an ideal gas. The ratio f =P is known as fugacity coefficient (φ), and consequently, at low pressure φ-1. The definition of fugacity can also be extended to mixtures. To this end, the thermodynamic properties of mixtures are considered by inserting the concepts of partial molar properties. In this case, the fugacity of component i in a mixture (f^i ) can be determined by IGM

Gi 2 Gi

5 RT ln

f^i yi P

(2.90)

where yi is the mole fraction of component i in the mixture. And similarly, the fugacity coefficient of component i in a mixture (φ^ i ) is defined as f^ φ^ i 5 i yi P

(2.91)

The importance of the fugacity for phase equilibrium is that various models are developed from its definition, providing easy understanding through physical analysis of this property. Considering Eq. (2.90), and since μi 5 G i, the conditions for phase equilibrium of a multicomponent biphasic system2. Eqs. (2.80)(2.82), can also be rewritten as

μαi 5 μβi

2

or

α

β

Gi 5 Gi

or

Tα 5 Tβ

(2.80)

Pα 5 Pβ

(2.81)

α β f^i 5 f^i ;

with i 5 1; 2; 3; . . .; n (2.92)

If the system is constituted of only one component, the chemical equilibrium is established simply by the properties of the pure compound in both phases: G α 5 G β or f α 5 f β.

42

Fundamentals of Phase Equilibria

2.3.3 Equilibrium and Stability In the beginning of this chapter, it was shown that a system remains in the equilibrium state as long as the external vicinity does not disturb it. In other words, the system can go into non-equilibrium states when it receives an external perturbation. But how “fragile” is the equilibrium state? In fact, the equilibrium state of a system can be of four types: stable, unstable, metastable, neutral. An easy way to explain the differences between them is to correlate them with the behavior of a sphere in different circumstances, as represented in Fig. 2.4. In a stable equilibrium state, when an external perturbation is applied, the system can suffer slight changes, but it returns to the initial state naturally. In an unstable equilibrium state, the opposite occurs; if any small disturbance is applied, the system will change its thermodynamic state. In a metastable equilibrium state, the system is “stable” if small perturbations are applied; however, large disturbances will change its thermodynamic state. In the neutral equilibrium state, any disturbances will change its thermodynamic state; however, its potential energy will remain unchanged. Actually, most real systems are considered metastable. Nevertheless, mostly, the perturbations are so small compared to the magnitude of the barriers to changes that the system can be considered stable [2]. Although this classification does not indicate the final thermodynamic conditions, it provides information about how sensitive the system is in relation to small perturbations. Thus, in practice, metastable systems can be treated as stable, when the perturbations are small enough to not lead to changes in the properties of the system.

Figure 2.4. Representation of equilibrium states. 43

Thermodynamics of Phase Equilibria in Food Engineering

For the thermodynamic equilibrium state discussed in Section 2.3.1, it was implicitly assumed that the equilibrium state was stable. However, the stability of an equilibrium is not a fixed situation; the stable equilibrium is only one of the four possible types of states of equilibrium that are available for real systems. For this reason, in order to define if the system is in a stable equilibrium state, the criteria of equilibrium and stability must be analyzed. In the previous section, the conditions for the thermodynamic equilibrium were derived when considering a closed system, at fixed volume, and internal energy. By the analysis performed for this system, the equilibrium occurs when the total entropy of the system is constant (dS 5 0), i.e., this is the criterion used to reach the equilibrium state. However, dS 5 0 is not sufficient to assure that the entropy reaches a maximum value (requisite for the stable equilibrium state). Actually, the condition dS 5 0 can be obtained at a point of maximum, point of minimum, or inflection point. Thus, to ensure a maximum entropy value, it is necessary that d2 S , 0. In other words, the criterion of stability provides the requirement for the equilibrium to be stable, and that is defined by the second derivative of the function analyzed. Then, as entropy is a crescent function, the equilibrium is established when dS 5 0, being a stable equilibrium (maximum value of S) when d 2 S , 0. (Note: if dS 5 0 is at a minimum point, the equilibrium is unstable; or if it is at an inflection point, the equilibrium is metastable). It is worth mentioning that the same definition of the equilibrium conditions, Eqs. (2.80)(2.82), can be obtained using different systems, in order words, the way to achieve the equilibrium can vary; however, the final conditions to obtain the equilibrium will always be the same. Thus, for systems with distinct constraints, the criteria of equilibrium and stability can vary, as indicated in Table 2.6. Table 2.6. Criteria for equilibrium and stability [1,2]

44

Constraints

Equilibrium criterion

Fixed N, V, U

dS 5 0

Fixed N, Fixed N, Fixed N, Fixed N,

dU 5 0 dH 5 0 dA 5 0 dG 5 0

V, S P, S V, T P, T

Stability criterion

d2 S , 0 d2 U . 0 d2 H . 0 d2 A . 0 d2 G . 0

(2.93) (2.94) (2.95) (2.96) (2.97)

Fundamentals of Phase Equilibria

2.3.4 Fugacity of Species: Pure and in Mixture 2.3.4.1 Fugacity of a Pure Compound The fugacity of a pure component is a fundamental property in the phase equilibrium evaluation, and it may be calculated in different ways. Using Eq. (2.8) in molar base, dG 5 2 SdT 1 V dP

(2.98)

The integration of Eq. (2.98) at constant temperature results in ð P2 GðT ; P2 Þ 2 GðT; P1 Þ 5 V dP

(2.99)

P1

Similarly, applying for an ideal gas: G ðT; P2 Þ 2 G IG ðT; P1 Þ 5

ð P2

V IG dP

(2.100)

P1

If P1 5 0, GðT ; P1 5 0Þ 5 G IG ðT ; P1 5 0Þ, then by subtracting Eq. (2.100) from Eq. (2.99), it becomes ð P2 IG G ðT ; P2 Þ 2 G ðT ; P2 Þ 5 V 2 V IG dP (2.101) 0

or simply GðT ; PÞ 2 G IG ðT ; PÞ 5

ðP 0

V2

RT dP P

(2.102)

Considering Eqs. (2.102) and (2.89), the fugacity and fugacity coefficient of a pure component can be calculated by ð f 1 P RT lnφ 5 ln 5 V2 dP (2.103) P RT 0 P Eq. (2.103) is the general equation used to calculate the fugacity or fugacity coefficient of a pure component at a given temperature and pressure. Indeed, this equation is more commonly used than Eq. (2.89), due to the ease of obtaining data of the molar volume of the compound in function of temperature and pressure. In addition to experimental data, the molar volume of the pure compound can be calculated by using an equation of state, which has an advantage over Eq. (2.89), mainly when it comes to performing calculations for modeling and simulation of processes. One easy example of volumetric data implemented in Eq. (2.103) is the use of compressibility factor (Z). As equations of state can be 45

Thermodynamics of Phase Equilibria in Food Engineering

expressed in terms of compressibility factor, the resolution of this equation is simple. If Z 5 PV =RT , the Eq. (2.103) becomes ðP f ðZ 2 1Þ dP (2.104) lnφ 5 ln 5 P P 0 The equations of state are basically arranged into four classes: ideal gas equation, Virial equation and extensions, cubic and non-cubic equations of state from van der Waals theory, and equations of state based on molecular principles. Due to the importance of the Virial equation, a special supplement discussing its formulation is available at the end of this chapter. The main equations of state applied to food systems are shown throughout this book, and detailed in Chapter 3 and Chapter 5. For the application of Eq. (2.103), the equation of state must be set to V 5 f ðP; T Þ. However, most equations of state are defined as P 5 f ðV ; T Þ, being Ð necessary Ð to perform the change of the integration variable from V dP to PdV . Thus, as dPV 5 V dP 1 PdV and Z 5 PV =RT , Eq. (2.103) can be rewritten as   ð f 1 V RT lnφ 5 ln 5 (2.105) 2 P dV 2 lnZ 1 ðZ 2 1Þ P RT V N V or f lnφ 5 ln 5 P

ðV N V

ðZ 2 1Þ dV 2 lnZ 1 ðZ 2 1Þ V

(2.106)

Eqs. (2.103)(2.106) can be applied to calculate the fugacity of a pure substance in gaseous, liquid, or solid state. Nevertheless, their use is only possible if the reference state is appropriate for the requested application. For liquid substances, for instance, when an equation of state is not applicable, experimental data of molar volume as a function of pressure can be used, V 5 f ðP; TÞ. In this case, from Eq. (2.103), the following equation is obtained: ðP fL V 1 lnφL 5 ln 5 2 dP (2.107) P P 0 RT The integration of Eq. (2.107) from a reference state where the pressure is zero, P 5 0 (gas state), up to system pressure, P 5 P (liquid state), considering the trasnsition of phase change (“PhCh”) from gas to liquid state, P 5 Pvap, results in

46

Fundamentals of Phase Equilibria

fL ln 5 P

    ðP  L ð P vap  V V 1 f V 1 1 dP 1 ln dP 2 2 P P Ph2Ch P RT P vap RT 0 (2.108)

Analyzing each independent term of Eq. (2.108): the first integration is relative to the value of lnðf =PÞ at the saturation point (lnφsat ); the second term is relative to the phase change and as GL ðT ; PÞ 5 G V ðT at the phase the term  ; PÞ   change, L V ln f =P Ph2Ch 5 G ðT; PÞ 2 G ðT ; PÞ =RT is null; and for the third term, the integration can be replaced by the sum of integrations of the internal terms. Thus, Eq. (2.108) is reduced to ðP VL f L 5 P vap φsat exp dP (2.109) P vap RT or f 5P L

vap



f =P



exp sat;T

ðP

VL dP P vap RT

(2.110)

From Eqs. (2.109) and (2.110), two pieces of information are relevant. The first is related to the saturation coefficient, φsat , whose value is very close to unity for typical non-associating species in conditions where the temperature is well below the critical point. For species that associate such as carboxylic acids, the deviation of the ideality even at low pressures may be considered. The calculation of the saturation coefficient φsat can be performed from vapor-phase volumetric data, using the principle of corresponding states, or an equation of state applied to Eq. (2.105) to compute this value. The second aspect is the exponential term of Eqs. (2.109) and (2.110), called Poynting correction factor (POY). This factor corrects the effect of pressure when the pressure of the system differs from the vapor pressure of the component and has an important use in the calculation of fugacities at very high pressures. Thus, returning to Eq. (2.110), since φsat 5 f sat =P sat, the calculation of the fugacity of a pure liquid species can be performed using the equation in the general form: ð P  VL L sat f 5 f exp dP (2.111) P vap RT

47

Thermodynamics of Phase Equilibria in Food Engineering

As the component is a liquid substance, it can be assumed that this is an incompressible fluid, and then Eq. (2.111) becomes  L  V ðP 2 P vap Þ L sat f 5 f exp (2.112) RT The same path can be used to calculate the fugacity of a pure substance in the solid state, resulting in the following equation: " #   N ð P J11 1 X S sat f J f 5P exp V dP (2.113) P sat;T RT J51 P J The summation is necessary since for a large range of different pressures it is possible to find more than one crystalline solid state, up to N. At low pressures, the saturation pressure at the sublimation condition of the solid is small; then in this condition, Eq. (2.113) results in f S 5 P sat . For moderate or high pressures, the fugacity of the solid pure is calculated by  S  V ðP 2 P sat Þ S sat f 5 P exp (2.114) RT 2.3.4.2 Fugacity of Compound i in Multicomponent Mixtures In multicomponent mixtures, the fugacity of the species is generally calculated using Eq. (2.90), or similarly as developed for pure substances, it can be defined in volumetric terms: ð IGM Gi 2 Gi 1 P f^i IGM ^ lnφi 5 ln 5 5 V i 2 V i dP (2.115) RT 0 yi P RT The notation generally used is identified by a superscript: “V” for vapor, “L” for liquid, “S” for solid, and “SC” for supercritical states. For the mole fraction, the notation used is yi and xi for the mole fraction of the component i in the light and heavy phases, respectively. In the case of vaporliquid equilibrium, for example, one has

48

^V ^φV 5 f i i yi P

ðfor vapor phaseÞ

(2.116)

^L ^φL 5 f i i xi P

ðfor liquid phaseÞ

(2.117)

Fundamentals of Phase Equilibria

For condensed materials, when an equation of state is not applied or when the ideal-gas mixture cannot be used as a reference for the mixture, Eq. (2.117) is not appropriate, and then, other reference is usually considered: ideal mixtures. Therefore, in the case of liquid mixtures, similarly to Eq. (2.115), the following expression is defined: ð L IM Gi 2 Gi 1 P f^i IM lnγi 5 ln 0 5 V i 2 V i dP (2.118) 5 RT 0 xi fi RT L where γi is the activity coefficient of component i, f^i is the fugacity of component i in the liquid mixture, fi0 is the fugacity of the pure component i in a standard state of reference (which is generally the pure component i at the same conditions of temperature and pressure of the mixture, i.e., fi 0 5 fi ). The consequence of Eq. (2.118) is the relaL tion f^i 5 xi γi fi0 , which reveals the behavior of component i in solutions. Similar considerations can be made to solid mixtures, and in this S case, the expression f^i 5 xSi γSi fi0;S is of great importance in studies of phase equilibrium involving solid phases. In parallel, another thermodynamic property can be defined when different chemical compounds are in mixtures. The ratio of the fugacity of component i in the mixture (f^i ) and the fugacity of the pure component i in its standard state (fi0 ) at the same temperature and pressure of the mixture is called activity of the component i (ai ):

ai 5

f^i fi0

(2.119)

The application of Eqs. (2.118) and (2.119) in liquid mixtures allows to evaluate the system at a given condition of T, P, and xi, and indicates their proximity to ideal behavior of solutions. This is evidenced by the activity (from Eq. (2.119), this becomes ai 5 xiγi) and activity coefficient values. For ideal mixtures, the values of activity and activity coefficient are ai 5 xi and γi 5 1, respectively. This result refers to the LewisRandall rule, whose relationship between the fugacity of a component i present in an ideal mixture and the fugacity of the pure component i is defined by expression IM f^i 5 xi fi0

(2.120)

In the case of solutions, Eq. (2.118) is quite useful in the calculations of phase equilibria involving liquid phases, since this equation makes use  E IM  of excess properties θ i 5 θ i 2 θ i . The consequence of this is the 49

Thermodynamics of Phase Equilibria in Food Engineering

development of mathematical expressions, known as GE models or activity coefficient models, for the determination of the activity coefficient. In this way, from Eq. (2.118) one arrives at E

G lnγi 5 i RT

(2.121)

E

and as G i is a partial molar property, Eq. (2.23) can be applied, then, one also has X G E 5 RT xi lnγi (2.122) By including excess properties, Eqs. (2.121) and (2.122) incorporate the concept of mixing properties in the calculation of fugacity. GE models have this function to represent the behavior of mixtures by analyzing the activity coefficient and to establish a link between this behavior and its application in the phase equilibrium domain. The details of the different GE models are presented in Chapter 4. An analysis of Eq. (2.122) in a binary mixture, for instance, assists in describing the kind of mixing observed in a given system. For the mixture containing species 1 and 2, Eq. (2.122) becomes GE 5 x1 lnγ1 1 x2 lnγ2 RT

(2.123)

In the concentration limits of the system, when x1 -1 and x2 -0, the mixture is almost pure for compound 1. Considering this condition in Eq. (2.123), it can be verified that the excess property tends to zero. This value is coherent with the definition of this property. Then, in a solution almost pure of species 1: G E -0 and γ1 -1. Note also that Gibbs energy is a function of temperature and pressure. Thus, the activity coefficient can be calculated from derivation of thermodynamic properties. One of the advantages of this is that the activity coefficient can be obtained experimentally from the evaluation of the excess properties of the mixture, through Eqs. (2.124) and (2.125):   E @lnγi Hi 52 (2.124) @T P;x RT 2 

50

@lnγi @P

 T ;x

E

5

Vi RT

(2.125)

Fundamentals of Phase Equilibria

2.3.5 Phase Equilibrium Calculations In the previous section, the formulation of phase equilibrium was defined by considering a two-phase system containing n components. According to the analysis performed, the phase equilibrium is only established when thermal, mechanical, and chemical equilibria are constant. Nonetheless, one of the main issues in phase equilibrium calculations is to determine the necessary conditions so that the equality between fugacities, Eq. (2.92), is established. For this, it is necessary to quantify the fugacities of the species involved in the system, being for the pure species or in mixture. This task can be easy or not, and depends on the species involved and the conditions fixed. In order to solve Eq. (2.92), one has to define the type of equilibrium that is working (VLE, LLE, etc.) and also to consider the different possible equations to be applied. Indeed, there are several possibilities to compute the value of the fugacity. Here, it is important to know the system well to indicate the best state of reference to be used. Consider, for example, a system comprised of the liquid and vapor phases, with n compounds. The constraints for the vaporliquid equilibrium of this system are TV 5 TL

(2.126)

PV 5 PL

(2.127)

V L (2.128) f^i 5 f^i The resolution of Eqs. (2.126)(2.128) involves different possibilities. As a matter of fact, the key point for all phase equilibria problems is the equality of the fugacities of components i. Thus, for Eq. (2.128), the fugacity of component i of vapor phase can be computed from Eq. (2.116), while for the liquid phase two equations are possible from Eqs. (2.117) and (2.118), as represented by the schematic design in Fig. 2.5.

fˆiV = fˆi L ˆf V = y φ ˆV i i i P

fˆi L = x i φˆ iL P

⇒ φ − φ approach

fˆi L = xi γi f i

⇒ γ − φ approach

Figure 2.5. Representation of the calculation approaches of vaporliquid equilibrium.

51

Thermodynamics of Phase Equilibria in Food Engineering

Due to the existence of two ways to calculate the vaporliquid equilibrium, it is usual to indicate the calculation procedure according to the type of approach applied, relating to the coefficient used in the calculations: φ 2 φ (“phiphi”) or γ 2 φ (“gammaphi”). The advantage in making use of the phiphi approach is the ease in working with equations of state. Additionally, this method can be used for a wide range of temperatures and pressures, including supercritical conditions. Using the same equation of state in both phases provides greater accuracy for systems near the critical point. Nevertheless, when an equation of state cannot be used for liquid phase, the calculation of the fugacity of component i in the liquid phase is performed using Eq. (2.118), which leads to the method of calculation known as gammaphi approach. This way of calculation is a traditional method used with success in systems with non-ideality in the liquid phase. It may be applied to a wide variety of mixtures, although its use is restricted. This limitation is due to GE models which were formulated at non-elevated pressure conditions. The selection of the appropriate approach (phiphi, gammaphi, or gammagamma) will depend on the kind of system, type of data (TP, Tx, Ty, etc.), type of equilibrium: vaporliquid, liquidliquid, vaporliquidliquid, solidliquid, and condition: at low or high pressures. Another example is the equilibrium among condensed phases, such as liquidliquid equilibrium, which makes use of the gammagamma method. The application of these approaches will be discussed throughout Chapters 69.

2.4 PHASE DIAGRAMS The representation of phase equilibria can be performed through phase diagrams. These can exhibit the behavior of phases both for pure substances and for multicomponent mixtures. The typical phase diagrams for a pure species are shown in Fig. 2.6, in terms of TP (A) and PV (B) data. In Fig. 2.6A, each curve represents the coexistence of phases: (1) vaporization curve: between liquid and vapor phases; (2) fusion curve: between solid and liquid phases; and (3) sublimation curve: between solid and vapor phases. The coexistence of all three phases is represented by a unique point, the triple point, characterized by a unique temperature and pressure. The point indicated as the critical point is the last condition of phase change between the liquid and vapor phases. At temperatures and 52

Fundamentals of Phase Equilibria

Pressure

(A)

Liquid

Supercritical fluid

Solid Pc -

cp

Gas

Vapor

tp

Tc Temperature

Pressure

(B)

Tc

Liquid

Pc Solid

Supercritical fluid cp L+V

Triple line

Gas Vapor

S+V Molar volume

Figure 2.6. Schematic phase diagrams for a pure substance. (A) PT and (B) PV, where tp is the triple point and cp is the critical point.

pressures higher than this point there is no more distinction between a gas and a liquid, and the fluid is in the supercritical region. These regions can also be observed in Fig. 2.6B. In this last diagram, it is also possible to observe the two-phase regions (S 1 L, L 1V; S 1 V) relative to phase transition, and the triple point indicated by the “triple line”. From Fig. 2.6, it can be observed that for each temperature, there is an only one specific value of saturation pressure. An analysis of two conditions in any one of these three curves will provide an important correlation in thermodynamics. If one considers any phase coexistence curve, and two conditions of temperature (T1 and T2), this leads to two values 53

Pressure

Thermodynamics of Phase Equilibria in Food Engineering

Phase α

sat

P2

Phase β

sat

P1

T1

T2

Temperature

Figure 2.7. Representation of ClausiusClapeyron equation formulation.

of saturation pressure (P1sat and P2sat ), as in Fig. 2.7. Using Eq. (2.98) and applying the equilibrium condition, dG α 5 dG β , one obtains 2S α dT α 1 V α dP α 5 2 S β dT β 1 V β dP β

(2.129)

Since dT α 5 dT β and dP α 5 dP β , Eq. (2.129) is reduced to ðS 2 S α ÞdT 5 ðV β 2 V α ÞdP, or simply   @P ΔS (2.130) 5 @T G ΔV β

By applying Eq. (2.4), G 5 H 2 T S, to both phases results in ΔH 5 T ΔS. Then, replacing this and Eq. (2.130) in Eq. (2.129), this becomes   @P ΔH (2.131) 5 @T G TΔV Eq. (2.131), called late the temperature liquidvapor phase ΔV  V V 5 RT =P, becomes

54

the ClausiusClapeyron equation, is used to correand saturation pressure of phase changes. In the change, for instance, since at low pressures the integration of Eq. (2.131) with defined limits 

ð T2 vap  P2 ΔH vap ln vap 5 dT 2 P1 T1 RT

(2.132)

Fundamentals of Phase Equilibria

Eq. (2.132) is interesting to define the vapor pressure at two specific points, as indicated in Fig. 2.7. Generally, it is assumed that ΔH vap is constant or independent of temperature, in the analyzed range of temperatures. This consideration provides the simplest expression:  vap    P ΔH vap 1 1 ln 2vap 5 2 2 (2.133) T2 T1 R P1 On the other hand, a generalized form of Eq. (2.131), obtained from an indefinite integration instead of definite integral can also be used. For liquidvapor change, this integration results in lnP vap 5 2

ΔH vap 1C RT

(2.134)

It can be noted that, when plotting lnPvap vs 1/T, the term ΔH vap , which is the latent heat of vaporization, is obtained from the slope of Eq. (2.134), ΔH vap =R. This same reasoning can be used for the other latent heats of phase change. As well as Eq. (2.134), other similar equations provided by Antoine, Riedel, HarlecherBraum, Wagner (Table 2.7), correlate the temperature and saturation pressure of several pure compounds. Another important piece of information from Fig. 2.6 is relative to the fusion curve inclination in the PT diagram. For most chemical species, the slope of this curve is positive. An exception is water, whose fusion curve has a negative slope. This occurs because although the latent

Table 2.7. Correlationsa of saturation pressure and temperature [2]

Antoine equation Riedel equation HarlecherBraum equation Wagner equation

B T 1C B lnP vap 5 A 1 1 ClnT 1 DT 6 T B P vap lnP vap 5 A 1 1 ClnT 1 D 2 T T Aτ 1 Bτ 1:5 1 Cτ 3 1 Cτ 6 vap lnP 5 ; 12τ with τ 5 1 2 Tr lnP vap 5 A 2

(2.135) (2.136) (2.137) (2.138)

a

The constants  A, B, C, and D are specific for each substance and equation applied, where Tr 5 T=Tc .

55

Thermodynamics of Phase Equilibria in Food Engineering

heat of fusion is positive (ΔH fus w . 0), the variation of volume in phase fus liquid change is negative (ΔV w , 0), due to ρice and then w , ρw ice liquid V w . V w . The consequence of this is the negative signal in Eq. (2.131), resulting in a negative slope in the fusion curve. For binary mixtures, there are different possibilities of diagrams, depending on the type of equilibrium (VL, LL, SL, etc.). Due to the degrees of freedom, generally the system is represented by a diagram type TP, Txy or Pxy. In all of them, the phase diagram is characterized by monophasic region (homogeneous) and multiphasic region (heterogeneous). A typical vaporliquid equilibrium diagram of a binary system is represented in Fig. 2.8, which illustrates the phase behavior of a by-product of sugar manufacturing (diacetyl) used as a flavoring component in beer, wine, and dairy products, in equilibrium with a solvent used in its recovery. The curves represent the condition of the liquidvapor saturation. The upper curve is the saturated liquid curve (curve A), also called the bubble point curve, and the lower curve is the saturated vapor curve (curve B), called the dew point curve. In the region up to curve A and down of curve B, the system is homogeneous (only one phase), and in the region between curves A and B, the system is heterogeneous (two phases). The tie-line describes the exact composition of the vapor and liquid phases in equilibrium for a given temperature and pressure condition. The choice of the type of diagram to be used (TP, Txy, etc.) for a particular type of equilibrium will depend on the experimental viability of 60

P1vap

50 Pressure (kPa)

(A) 40

(B) Tie-line

30

20 P2vap 10 0

0

0.2

0.4 0.6 x1, y1

0.8

1

Figure 2.8. Phase diagram of the vaporliquid equilibrium (Pxy) of acetone (1)diacetyl (2) at 313.15 K. Experimental data from [16]. Continuous solid lines are to guide the eyes.

56

Fundamentals of Phase Equilibria

data, and application. For instance, for the evaluation of the temperature effect on the volatility of compounds contained in essential oils, the phase diagram Txy is more appropriate. On the other hand, for the analysis of distillation processes, the most applicable phase diagram is x vs y (type-xy). The representation of liquidliquid equilibrium and solidliquid equilibrium is also possible in binary diagrams (Figs. 2.9 and 2.10, respectively). 345 340

One phase

Temperature (K)

335 330 325

Two phases

320 315 310 305 300

0.0

0.2

0.4

0.6

0.8

1.0

x1

Figure 2.9. Phase diagram of the liquidliquid equilibrium (Tx1) of ethanol (1)soybean oil (2) at 101.3 kPa. Experimental data from [17]. Continuous solid line is to guide the eyes. 455

Temperature (K)

445 435

L

425 415 Te 405

S1 + S2

395 385 0.0

S1 + L

S2 + L

0.2

0.4

0.6

0.8

1.0

x1

Figure 2.10. Phase diagram of the solidliquid equilibrium (Txy) of tartaric acid (1)citric acid (2) at 101.3 kPa, where Te is the eutectic point temperature. Experimental data from [18]. Continuous solid lines are to guide the eyes. 57

Thermodynamics of Phase Equilibria in Food Engineering

In the case of liquidliquid equilibrium, the curve that delimitates the two-phase region is called the binodal curve. Besides the representation of the different types of equilibrium, there are other relevant points that can also be observed in phase diagrams: (1) the effect of temperature and pressure on phase behavior of a system as a whole; (2) the influence of composition on dissolution of a specific compound in the mixture; (3) the existence of azeotropes, eutectics, and peritectics; (4) multibiphasic regions; and (5) multiple homogeneous regions (crystals). These aspects are important not only in terms of representation but mainly also for the application. Processes involving temperature changes (dissolution, distillation), high pressures (supercritical extraction, high pressure processing), multiphasic regions (colloids), and azeotropic systems (as in waterethanol), for instance, are often found in the food industry and the representation of equilibrium by phase diagrams assists in understanding changes that occur in the processing of various food systems. These issues will be discussed throughout this book. For ternary systems, the diagram is generally represented by an equilateral triangle (Fig. 2.11), whose vertices represent the pure substances. Similar to the binary phase diagrams, there is a homogeneous region [represented by (a)] and a heterogeneous region [represented by (b)].

Figure 2.11. Ternary diagram of the system water1-propanolβ-citronellol at 298.15 K. Adapted with permission from [19].

58

Fundamentals of Phase Equilibria

The representation of systems containing more than three compounds is also possible. Generally, it is performed considering two or more compounds as a “single compound” or “pseudocompound.” One example is the system R(1) limoneneethanolwaterPGTween 60, illustrated in Fig. 2.12, where two single-compounds were considered: R(1) limoneneethanol and waterPG. Besides the diagrams represented by Fig. 2.12, 3D diagrams are also found to represent systems containing more than three compounds, but due to the difficulties of obtaining data, 2D diagrams are most commonly used. The existence of more than two heterogeneous regions is also possible (Fig. 2.13). In fact, the influence of temperature and/or the composition of specific components can be so strong that they can change completely the behavior of the mixture. For instance, the formation of microemulsions water/oil (W/O). Emulsions are defined as a mixture with two immiscible liquids, wherein one is usually dispersed in the other, like an immersed droplet. The inversion of (W/O) to (O/W) due to increased oil content in the solution which generates the formation of nanoemulsions is much sought after in food, cosmetic, and pharmaceutical products. The newly formed systems (nanoemulsions) are much more stable than the original form (macroemulsions), providing good characteristics for the desired products, such as the formation of nanocapsules containing biocompounds that can be degraded during the ingestion orally or by gastrointestinal track [14].

Figure 2.12. Ternary diagram of the system water/PGR(1)-limonene/EtOHTween 60. Reprinted with permission from [20].

59

Thermodynamics of Phase Equilibria in Food Engineering

Oil

Water-in-oil μe

O O μe

μe

W

L2

μe μe

Bicontinuous μe

W

D L1 Water Micelle

Oil-in-water μe

Surfactant Water

Oil

Figure 2.13. Ternary diagram of the system oilwatersurfactant with more than one heterogeneous region. Reprinted with permission from [21].

The purpose of this section was to summarize some main features of phase diagrams to familiarize you with the terms and phase behavior. More phase diagrams and details on the calculations of phase equilibrium will be discussed in the next chapters for each type of equilibrium, such as vaporliquid, liquidliquid, vaporliquidliquid, solidliquid, supercriticalliquid, and special systems, as in colloid and electrolyte systems.

2.5 EQUILIBRIUM AND NON-EQUILIBRIUM IN FOODS The definition of equilibrium is based on the condition of minimum of Gibbs energy, as a consequence of the system properties do not change over time, while the vicinity does not affect it. When any change happens, the system goes into non-equilibrium3 condition and continues in this state while the time-dependent external force is acting on the system. 3

Attention: a non-equilibrium state is different from an “unstable equilibrium” state (defined in Section 2.3.2).

60

Fundamentals of Phase Equilibria

However, the term “equilibrium” can be applied in different contexts. The definition based on the minimum of energy is well accepted for microscopic systems, whose analysis is done considering a fixed number of compounds with well defined properties. Nevertheless, in macroscopic systems, the properties can be constant over time or not. If the macroscopic system receives input and output currents, the properties of the system will change and the global system will not be in equilibrium. Nevertheless, the tendency imposed by the Second Law of Thermodynamics indicates that every system goes naturally to the equilibrium state, even if it needs a long period of time. In other words, any open system can reach the equilibrium state if the surroundings, in conditions of steady state, do not affect it, i.e., since all intensive variables of the system remain uniform. As explained in Section 2.1, the steady state is the condition in which the properties are constant over time, also known as the time-invariant state. However, a system in steady state is not necessarily in equilibrium. For instance, in the thermal treatment of milk flowing in a heat exchanger, material flows at a steady state; however, it is not in equilibrium. The system in steady state can be in equilibrium when the system is considered uniform, i.e., when there is no internal gradients of concentration, velocity, temperature, and pressure. Consider the flow of fluid inside of a pipeline where the effects of the wall and its vicinity do not affect the properties of the fluid, and yet there is no work and heat flow involved in the process. If an element of fluid is flowing with the same velocity of the fluid around it and since there is no mass flow into or out of this small system, the element of fluid can be considered as a small closed system, with the same considerations previously made for the equilibrium (Section 2.3). Therefore, the system can be considered in equilibrium. The application of these concepts in food systems aids in understanding the different events observed in these materials. Foods are a mixture of several dissimilar compounds present in different aggregation states (dissolved molecules, emulsions, colloidal dispersions, crystals, fiber network, etc.) and in many works this complex structure, as well as biological organisms, is considered a system in non-equilibrium state. However, although they globally are far from the equilibrium, these systems are locally in equilibrium. The definition of equilibrium may also appear from another point of view. Many times, the events that occur in the macroscopic systems are much more effective than in the molecular scale, mantaining the global system analyzed in a condition of equilibrium. 61

Thermodynamics of Phase Equilibria in Food Engineering

And for these cases, the analysis is performed considering the global equilibrium. Another important discussion on this theme involves when and why to apply the equilibrium conditions. In the conception of thermodynamic equilibrium, the time for reaching the equilibrium is disregarded. This is one of differences between equilibrium and non-equilibrium state. This previous analysis of the problem may define when and why certain equations and models shown in this book will be applied. For instance, if the idea is to evaluate the procedure of preparation of ice cream, the presence of a multicomponent with different freezing temperatures brings to us an analysis performed for a range of operating conditions that change during processing. This requires information about the properties of the fluid that vary over time, indicating an analysis of a system typically in nonequilibrium. On the other hand, if the goal is to provide the best condition to solubilize a vitamin in a yogurt, for instance, the analysis here involves the study of equilibrium. Indeed, even if the process to solubilize takes a period of time, in this case it is more important to evaluate the conditions of equilibrium, rather than to examine the period to reach it. The analysis of systems from the microscopic point of view is object of study in Statistical Thermodynamics, in which the observation is performed taking into account the behavior of microscopic entities (e.g. molecules, particles, atoms) presented in the system. The molecular models developed by this field incorporate molecular parameters which allow a more accurate assessment of the interactions between the molecules that form part of the system. In food systems, this kind of evaluation is becoming more widely used, mainly in phase equilibrium studies involving complex molecular organizations, containing multifunctional compounds. In this book, a discussion about molecular models is presented in Chapters 4 and 5, with applications which are exemplified in the other chapters of the book. In contrast, looking at another aspect, Chapter 10 points out the evaluation of the state of non-equilibrium through the discussion of phase transition in foods. Here, the study of glass transition, which considers the changes during the formation of the glassy form, will be approached. For the food industries that have operational stages of separation the assessment of each case is made considering the thermodynamic equilibrium as a condition necessary to carry out the process. In fact, the application of equilibrium is a mandatory condition of the separation process, and its calculation helps in the analysis and optimization of such operations. For these reasons, the knowledge of the equilibrium conditions has become indispensable in the work of food engineer. 62

Fundamentals of Phase Equilibria

2.6 SPECIAL SUPPLEMENT Virial Equation of State SpS.1 The Virial Equation In Section 2.3, it was indicated that among the different types of equations of states developed to describe the volumetric properties of chemical species, Virial EoS holds a special position. Although with a limited range of application (not recommended for systems at high densities4, the importance of Virial EoS is related to the theoretical basis that is established from the concepts of statistical thermodynamics [22]. This equation of state is a polynomial serial expansion defined in terms of pressure or inverse volume (density), as indicated in Eqs. (2.139) and (2.140).         P 2   P 3 PV P 2 2 z5 1 D23BC 12B 1? 511B 1 C 2B RT RT RT RT (2.139) or z5

PV B C D 511 1 2 1 3 1? RT V V V

(2.140)

where the coefficients B, C, D are called the second, third, and fourth virial coefficients, respectively. The main characteristic of Virial EoS is that it assigns a physical meaning to the constants of the equation. The virial coefficients are directly related to interactions between the molecules: the second virial coefficient (B) represents interaction between molecule pairs, the third virial coefficient (C) represents interaction of three molecules, the fourth virial coefficient (D) for four molecules, and so on. The Virial EoS truncated after the second term, or in some cases after the third term, has shown reasonable results in systems at low and moderate pressures; however, it can be used only for the vapor phase. One of the advantages is its application in multicomponent systems. In mixtures, the second and third virial coefficients are determined by the following equations: XX B5 yi yj Bij (2.141) i

4

j

See the discussion at the end of this section.

63

Thermodynamics of Phase Equilibria in Food Engineering

C5

XXX i

j

yi yj yk Cijk

(2.142)

k

where Bij and Cijk represent the molecular interactions between ij and ijk molecules, respectively. In a binary mixture, these equations become B 5 y2i Bii 1 yi yj Bij 1 y2j Bjj

(2.143)

C 5 y3i Ciii 1 3y2i yj Ciij 1 3yi y2j Cijj 1 y3j Cjjj

(2.144)

The parameters Bii and Bjj are the second virial coefficients of the pure components i and j, and the parameter Bij, known as the cross coefficient, considers the interaction between molecules ij. The three coefficients are independent of the composition but are temperature dependent. The same idea is extended to the parameters Ciii , Cjjj , Ciij , and Cijj for the third virial coefficient. As most applications of Virial EoS have been done considering the equation truncated in the second term, the subsequent discussions will be performed allowing for this perspective. Different methods for the measurement of virial coefficients of pure compounds, many of them also being applied to mixtures, have been proposed by literature [2327]. Dymond and Smith [28] and Dymond et al. [29] provided an extensive compilation of data of virial coefficients and experimental methods. Other works also presented important data [30,31]. Although there are different ways to determine these parameters experimentally, most researchers have determined them by theoretical equations. The virial coefficients can be derived from molecular theory [28], nevertheless, it is not a simple task. Several groups have provided equations based on the concepts of potential functions or corresponding states [22]. Among the methods used to estimate the second virial coefficients, one of the most used is the HaydenO’Connell (HOC) [32] model. The calculation procedure of the HOC model consists of a series of complex equations, whose parameters include the properties of pure component such as the critical pressure, critical temperature, mean radius of gyration, dipole moment, solvation, and parameters of association. The main equation of the model considers that the second virial coefficient is a sum of different kinds of intermolecular forces, being represented by the expression B 5 Bfree 1 Bmetastable 1 Bbound 1 Bchem

64

(2.145)

Fundamentals of Phase Equilibria

where Bfree is related to the molecular structure (molecular volume), ðBmetastable 1 Bbond Þ provides the contribution from the potential energy from pairs of molecules more or less strongly bonded, and Bchem incorporates the factor from associating species. One of the characteristics of this model is the good representation of systems with strong association and solvation effects, as observed in carboxylic acids. On the other hand, in most cases, the methods are defined from the Principles of Corresponding States presenting essentially the general form X BPc 5 ai fi ðTr Þ (2.146) RTc i where ai represents the strength parameters for the intermolecular forces, and fi is a set of universal functions defined at reduced temperature. Among the models developed with this format, the model defined by Tsonopoulos [33] is one of the most widely employed for polar and nonpolar fluids. Derived from modifications in the Pitzer and Curl correlation [34], the Tsonopoulos model correlates the second virial coefficient with reduced properties from the expression BPc 5 f ð0Þ 1 ωf ð1Þ 1 f ð2Þ RTc

(2.147)

with f ð0Þ 5 0:1445 2

0:330 0:1385 0:0121 0:000607 2 2 2 Tr Tr2 Tr3 Tr8

f ð1Þ 5 0:0637 1

0:331 0:423 0:008 2 2 Tr2 Tr3 Tr8

f ð2Þ 5

a b 2 8 6 Tr Tr

(2.148)

(2.149)

(2.150)

where ω is the acentric factor, and a and b are parameters used for polar and hydrogen-bonded fluids (Table 2.8). Another simpler model applied to simple fluids, with values close enough to the Eqs. (2.147)(2.150), developed by Van Ness and Abbott [35] has also been applied, where: BPc 5 f ð0Þ 1 ωf ð1Þ RTc

(2.151)

65

Thermodynamics of Phase Equilibria in Food Engineering

Table 2.8. Tsonopoulos parameters (a and b) of Eq. (2.150), for polar and associating species Specie class

a

b

Ketones, esters, ethers, alkyl nitriles, aldehydes

22:112 3 1024 μr

0

Alcohols

0.0878

9:08 3 1023

0.0878 20.0109

1 6:957 3 104 μr 0.0525 0

Methanol Water

2 3:877 3 10221 μ8r

Where μr is the reduced dipole moment, calculated by μr 5 105 μPc =Tc2 and μ is the dipole moment at 293.15 K. Adapted from [22].

with f ð0Þ 5 0:083 2

0:442 Tr1:6

(2.152)

0:172 (2.153) Tr4:2 The estimation of the second virial coefficient using the equations described above, Eqs. (2.145)(2.146), can be used for pure species and mixtures. For mixtures, the second cross coefficients, Bij , is computed using the critical properties and acentric factor of the mixture determined by f ð1Þ 5 0:139 2

Pcij 5

Zcij RTcij V cij

(2.154)

Zcij 5

Zci 1 Zcj 2

(2.155)

 1=2   Tcij 5 Tci Tcj 1 2 kij !3 Vc1=3 1Vc1=3 i j Vcij 5 2 ωij 5

66

ω i 1 ωj 2

(2.156) (2.157)

(2.158)

Fundamentals of Phase Equilibria

In Eq. (2.156), kij is the interaction binary parameter that can be estimated from predictive equations [3640], or in most cases, it is estimated from the regression of experimental data of vaporliquid equilibrium. SpS.2 Fugacity Coefficient From the Virial Equation As detailed in Section 2.3.4, the fugacity coefficient of a component i in a mixture can be determined from the following equation: ð 1 P RT ^ dP (2.115) lnφi 5 Vi2 RT 0 P whose value of the partial molar volume of component i can be determined by an equation of state. If the Virial EoS truncated after the second term is considered, the combination of Eqs. (2.115), (2.139), and (2.141) results in ! X P 2 lnφ^ i 5 yj Bij 2 B (2.159) RT j or, if the Virial equation is truncated after the third virial coefficient, using Eq. (2.142), the fugacity coefficient equation of component i becomes 2X 3 XX lnφ^ i 5 yj Bij 1 y y Cijk 2 lnZ (2.160) 2V 2 j k j k V j Similarly to Virial EoS, other equations were developed in a series of density. Among them, the equations of BeattieBridgeman [41], BenedictWebbRubin (BWR) [42], modified BWR [43,44] are the most used among the equations of this type in the chemical industry. However, the Virial EoS still has a greater relevance. In food, this equation has been applied in the evaluation of systems in different ways. Since the virial coefficients are directly related to interactions between the molecules, one of its most important applications is in the evaluation of interactions of species, such as protein, lipids, and others [4547]. A discussion about its uses in the analysis of systems containing biopolymers is detailed in Chapter 12. Another important application is in studies involving phase equilibria. The use of Virial EoS in vaporliquid equilibrium has been observed in several systems containing functional compounds such as alcohols, esters, carboxylic acids, lipids, ketones, and others [4854]. One example of the application of this equation in the calculations of vaporliquid equilibrium can be observed in a Case Study in Chapter 6. 67

Thermodynamics of Phase Equilibria in Food Engineering 

Special Remark: The Limits of Application of Virial EoS Although the application of Virial EoS has been restricted to systems at low and moderate pressures, recent studies [51,5560] reveal that the Virial EoS using higher orders has provided a good description of properties of single and binary systems at high densities. The reason for the historical limited application of Virial EoS is more related to mathematical difficulties, which nowadays has been replaced by the aid of computational tools. A good example of this is the evaluation of the properties of CO2CH4 in supercritical conditions by the application of the Virial EoS in different orders [60]. The calculations using the virial EoS with the fourth or fifth virial coefficients showed a good representation of all properties of the binary mixture for densities up to the critical density. As well as this, the study revealed that the insertion of terms up to seventh order extended considerably the range of application of the Virial EoS [60]. In this sense, it is observed that the virial EoS can still provide good representations for different systems (associating and non-associating species) and a longer application range (low or high densities), since specific parameters and a greater number of coefficients of the original equation, Eq. (4.139) or (4.140), are considered.

REFERENCES [1] Tester JW, Modell M. Thermodynamics and its applications. New Jersey: PrenticeHall, Inc; 1986. [2] Sandler SI. Chemical and engineering thermodynamics. Danvers: John Wiley & Sons; 1989. [3] Smith JM, Van Ness HC, Abbott MM. Introduction to chemical engineering thermodynamics. 5th ed. New York, NY: McGraw-Hill; 1996. [4] Moran MJ, Shapiro HN. Fundamentals of engineering thermodynamics. 5th ed. Chichester: John Wiley & Sons; 2006. [5] Apelblat A, Wisniak J, Shapiro G. Physical properties of (jojoba oil 1 n-hexane) compared with other (vegetable oil 1 n-hexane) mixtures. J Chem Thermodyn 2008;40 (10):147784. [6] Gmehling J. Enthalpy of mixing of the mixture ethanolwater. “Dortmund Data Bank (DDB)—Thermophysical Properties Edition 2014” in Springer Materials. hhttp://materials.springer.com/thermophysical/docs/he_c11c174i (accessed in July 2017). [7] Mallikarjun MPN, Desai S, Shivabasappa KL. Enthalpy of mixing and vaporization of esters with alcohols (methyl acetatemethyl alcohol and ethyl acetate2-butyl alcohol) at 298 and 308 K. IUP J Chem 2012;5(1):6374. [8] Resa JM, Gonza´lez C, Fanega MA, Landaluce SO, Lanz J. Enthalpies of mixing, heat capacities, and viscosities of alcohol (C1C4) 1 olive oil mixtures at 298.15 K. J Food Eng 2002;51:11318.

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[9] EL-Hefnawy ME. Densities and refractive indices of binary mixtures of olive oil 1 alkanols at 298.15 K. J Solution Chem 2013;42:2399408. [10] Pereira CG, Meireles MAA. Chapter 6—phase equilibrium: mono and multicomponents. In: Meireles MAA, Pereira CG, editors. Fundamentals of food engineering (“Fundamentos de Engenharia de Alimentos”). Rio de Janeiro, Brazil: Ed Atheneu; 2013. [11] Gmehling J, Kolbe B, Kleiber M, Rarey J. Chemical thermodynamics for process simulation. Germany: Wiley-VCH Verlag & Co. KGaA, Boschstr; 2012. [12] Dahm K.D., Visco D.P. Fundamentals of chemical engineering thermodynamics, SI Edition., Stamford: Cengage Learning. [13] Walas SM. Phase equilibria in chemical engineering. Oxford: ButterworthHeinemann; 1985. [14] Gibbs JW. On the equilibrium of heterogeneous substances. In: The collected works of J. Willard Gibbs, vol. I. New Haven, CT: Yale University Press; 1906. [15] Lewis GN. The law of physico-chemical change. Proc Am Acad Arts Sci 1901;37 (3):4969. [16] Soni M, Ramjugernath D, Raal JD. Vaporliquid equilibrium for binary systems of diacetyl with methanol and acetone. J Chem Eng Data 2006;51:20837. [17] Andreatta EA, Arposio A, Ciparicci S, Longo MB, Francescato F, Gavotti L, Fontanessi M. Liquidliquid equilibrium in mixtures containing glycerol and mixtures containing vegetable oils. VII CAIQ2013, JASP 2013;113. [18] Meltzer V, Pincu E. Thermodynamic study of binary mixture of citric acid and tartaric acid. Cent Eur J Chem 2012;10:15849. [19] Li H, Han Y, Huang C, Yang C. (Liquid 1 liquid) equilibria for (water 1 1propanol or acetone 1 b-citronellol) at different temperatures. J Chem Thermodyn 2015;86:206. [20] Yaghmur A, Aserin A, Garti N. Phase behavior of microemulsions based on foodgrade nonionic surfactants: effect of polyols and short-chain alcohols. Colloids Surf, A: Physicochem Eng Aspects 2002;209:7181. [21] Garti N. Microemulsions as microreactor for food applications. Curr Opin Colloid Interface Sci 2003;8:197211. [22] Prausnitz JM, Lichtenthaler RN, Azevedo EG. Molecular thermodynamics of fluidphase equilibria. 2nd ed New Jersey: Prentice-Hall, Inc; 1986. [23] Sato T, Kiyoura H, Sato H, Watanabe K. Measurements of PVTx properties of refrigerant mixture HFC-32 1 HFC-125 in the gaseous phase. Int J Thermophys 1996;17:4354. [24] Blanke W, Weiss R. Virial coefficients of methaneethane mixtures in the temperature range from 0 to 60˚C determined with an automated expansion apparatus. Int J Thermophys 1995;16:64353. [25] Trusler JPM, Wakeham WA, Zarari MP. Second and third interaction virial coefficients of the (methane 1 propane) system determined from the speed of sound. Int J Thermophys 1996;17:3542. [26] Wormald CJ, Johnson PW. Quadrupole coupling in (carbon dioxide 1 dioxane)(g). The excess molar enthalpy and second virial cross-coefficient of (dioxane 1 carbon dioxide or propane)(g). J Chem Thermodyn 1999;31(8):108591. [27] Jaeschke M. Determination of the interaction second virial coefficients for the carbon dioxideethane system from refractive index measurements. Int J Thermophys 1987;8:8195. [28] Dymond JH, Smith EB. The virial coefficients of pure gases and mixtures; a critical compilation. Oxford: Clarendon Press; 1980. [29] Dymond JH, Wilhoit RC, Wong KC, Virial coefficients of pure gases. In: Frenkel M, Marsh KN, Marsh KN, LandoltBornstein: numerical data and functional

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[30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50]

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relationships in science and technology—new series physical chemistry. Heidelberg: Springer-Verlag Berlin Heidelberg; 2002. Mason EA, Spurling TH. The virial equation of state. New York, NY: Pergamon; 1968. Warowny W, Stecki J. The second cross virial coefficients of gaseous mixtures. Warszawa: PWN—Polish Scientific Publishers; 1979. Hayden JG, O’Connell JP. A generalized method for predicting second virial coefficients. Ind Eng Chem Process Des Dev 1975;14:20916. Tsnopoulos C. An empirical correlation of second virial coefficients. AICHE 1974;20:26372. Pitzer KS, Curl RF. The volumetric and thermodynamic properties of fluids. I. Theoretical basis and virial coefficients. J Am Chem Soc 1955;77:342733. Van Ness HC, Abbott MM. Classical thermodynamics of non electrolyte solutions. New York, NY: McGraw-Hill; 1982. Chueh PL, Prausnitz JM. Vaporliquid equilibria at high pressures: calculation of partial molar volumes in nonpolar liquid mixtures. Ind Eng Chem Fundam 1967;6:4928. Meng L, Duan YY, Li L. Correlations for second and third virial coefficients of pure fluids. Fluid Phase Equilib 2004;226:10920. Meng L, Duan YY. An extended correlation for second virial coefficients of associated and quantum fluids. Fluid Phase Equilib 2007;258:2933. Tsonopoulos C, Dymond JH. Second virial coefficients of normal alkanes, linear 1alkanols (and water), alkyl ethers, and their mixtures. Fluid Phase Equilib 1997;133:1134. Tsonopoulos C. Second virial cross-coefficients: correlation and prediction of kij. Adv Chem Ser 1979;182:14362. Beattie JA, Bridgeman OC. A new equation of state for fluids. I. Application to gaseous ethyl ether and carbon dioxide. J Am Chem Soc 1927;63:16657. Benedict M, Webb GB, Rubin LC. An empirical equation for thermodynamic properties of light hydrocarbons and their mixtures. Methane, ethane, propane and butane. J Chem Phys 1940;8:33445. Cox KW, Bono JL, Kwok YC, Arling KE. Multiproperty analysis: modified BWR equation from PVT and enthalpy data. Ind Eng Chem Fundam 1971;1:24550. Starling KE, Han MS. Thermo data refined for LPG. Part 14: Mixtures. Hydrocarbon Process 1972;51:12932. Chou T, Kim KS, Oster G. Statistical thermodynamics of membrane bendingmediated proteinprotein attractions. Biophys J 2001;80:107587. Sanchez-Gonzalez J, Cabrerizo-Vılchez MA, Galvez-Ruiz MJ. Evaluation of the interactions between lipids and b-globulin protein at the airliquid interface. Colloids Surf, B 1999;12:12338. Schorsch C, Clark AK, Jones MG, Norton IT. Behaviour of milk protein/polysaccharide systems in high sucrose. Colloids Surf, B 1999;12:31729. Cepeda EA. Isobaric vapor 2 liquid equilibrium for binary mixtures of 3-methyl-1butanol 1 3-methyl-1-butyl ethanoate and 1-pentanol 1 pentyl ethanoate at 101.3 kPa. J Chem Eng Data 2010;55(6):234954. Susial P, Sosa-Rosario A, Rios-Santana R. Vapourliquid equilibrium with a new ebulliometer: ester 1 alcohol system at 0.5 MPa. Chin J Chem Eng 2010;18 (6):10007. Mandagara´n BA, Campanella EA. Correlation of vaporliquid equilibrium data for acetic acidisopropanolwaterisopropyl acetate mixtures. Braz J Chem Eng 2006;23:93103.

Fundamentals of Phase Equilibria

[51] Susial R, Susial P. High pressure vaporliquid equilibria of methyl acetate or ethyl acetate with 2-propanol at 1.5 MPa. Experimental data and predictions. Chem Eng Res Des 2015;102:33744. [52] Fornari T, Bottini S, Brignole EA. Application of UNIFAC to vegetable oilalkane mixtures. JAOCS 1994;71(4):3915. [53] Soni M. Vapourliquid equilibria and infinite dilution activity coefficient measurements of systems involving diketones. PhD thesis. Durban, University of Natal; 2013. [54] Varasteh A, Tahmasebi HA, Gheshlaghi R. Modeling of vapour liquid equilibrium data for thyme essential oil based on UNIQUAC thermodynamic model. J Chem Eng Process Technol 2013;4:15. [55] Schultz AJ, Shaul KRS, Yang S, Kofke DA. Modeling solubility in supercritical fluids via the virial equation of state. J Supercrit Fluid 2010;55:47984. [56] Schultz AJ, Kofke DA, Harvey AH. Molecular-based virial coefficients of CO2H2O mixtures. AIChE J 2015;61:302937. [57] Aimoli CG, Maginn EJ, Abreu C. Force field comparison and thermodynamic property calculation of supercritical CO2 and CH4 using molecular dynamics simulations. Fluid Phase Equilib 2014;368:8090. [58] Joslin CG, Gray CG, Goldman S, Tomberli B, Li W. Solubilities in supercritical fluids from the virial equation of state. Mol Phys 1996;89:489503. [59] Benjamin KM, Singh JK, Schultz AJ, Kofke DA. Higher-Order virial coefficients of water models. J Phys Chem B 2007;111(39):1146373. [60] Yang S, Schultz AJ, Kofke DA. Thermodynamic properties of supercritical CO2/ CH4 mixtures from the virial equation of state. J Chem Eng Data 2016;61 (12):4296312.

71

CHAPTER 3

Classical Models Part 1: Cubic Equations of State and Applications Vladimir F. Cabral, Marcelo Castier and Lúcio Cardozo-Filho

3.1 INTRODUCTION The knowledge of phase equilibrium conditions plays an important role in food production since most of the processes in the food industry involve the coexistence of more than one phase. Furthermore, most foods are multiphase dispersions in which, for example, ice crystals, fat droplets, aqueous solutions of biopolymers, and air bubbles are combined to produce different phases. In general, the properties of these dispersions are highly dependent on the composition of each phase. For example, the perception of flavor is determined by how the molecules that impart flavor are distributed among the several phases present in the system. Therefore, in order to design and properly optimize food production processes, it is essential for the food engineer to describe or predict such phenomena. In Chapter 2, the key expressions for calculating fugacities using T and P or T and V as independent variables, Eqs. (2.103) and (2.105), were presented. Eq. (2.103) is used whenever the volumetric data is given in an explicit form in the volume, i.e., V 5 f(T, P, N), while Eq. (2.105) is used whenever the volumetric data is given in an explicit form in the pressure, i.e., P 5 f(T, V, N). The expressions that indicate this relation between volume, pressure, temperature, and mole number of the components are called Equations of State (EoS) and are useful in many process calculations. EoS are essential to describe fluid behavior and to evaluate physical properties of food products. Some EoSs are able to represent the gaseous and condensed phases at high- and low-pressure conditions. Some of

Thermodynamics of Phase Equilibria in Food Engineering

© 2019 Elsevier Inc. All rights reserved.

73

Thermodynamics of Phase Equilibria in Food Engineering

these equations are completely empirical, while other ones have some theoretical basis. The PVT behavior of a fluid reflects the microscopic interactions that occur between the atoms and molecules that comprise the pure chemical substances or their mixtures. Molecular simulation is the most rigorous tool to represent these microscopic interactions; however, due to the chemical complexity of food, this approach is still impractical for industrial process design. The force-fields of molecular interactions, necessary for the simulation, are not easily available for complex molecules and their mixtures. Therefore, EoSs and, in particular, cubic EoSs, remain very useful for practical purposes, despite their limitations. Cubic EoSs present simple expressions, which have easy mathematical resolution and are present in most of the commercial process simulation programs. Although there is a continuous evolution of these equations, many of them give relatively poor predictions of properties such as liquid density. Condensed phases are predominant in most foods, limiting their application to many food processes, even though this issue can be alleviated by using volume translation techniques. One of the main factors of success on the application of the cubic EoS, mainly in processes of the chemical industry, is the possibility of rapid evaluation of repulsive and attractive parameters from information about critical properties and acentric factor of pure substances. However, many functional substances present in foods are thermolabile and it is not possible to measure their critical properties. Thus, calculating the parameters via real values of critical properties and acentric factor is infeasible. One way to overcome this limitation is the direct calculation of the attractive and repulsive parameters by using experimental data of density and vapor pressure. Another is to use groupcontribution methods to predict pseudovalues for the critical properties and acentric factor. This chapter presents some of the most important cubic EoSs proposed in the literature for the description of vaporliquid, liquidliquid, and solidliquid equilibria of multicomponent systems. The use of these equations is considered for the cases of pure substances and mixtures, upon the presentation of mixture rules. Besides, discussions related to the diagrams using EoS, as well as their applications to food systems are presented.

74

Classical Models Part 1: Cubic Equations of State and Applications

3.2 CUBIC EQUATIONS OF STATE The van der Waals (vdW) equation, proposed in 1873 [1], was the first EoS able to represent the qualitative behavior of liquid and vapor phases simultaneously. There are several paths to using statistical mechanics to derive the vdW EoS for pure components and mixtures. The path used here is based on lattice statistics [2]. For simplicity, the case of pure components is considered. The starting point is the relationship between entropy (S) and the number of microscopic states accessible to an ergodic system of given number of molecules (N), volume (V), and internal energy (U). These specifications constitute what is called a microcanonical ensemble in statistical mechanics. The initial step is to split the space available to the pure fluid into M cells of equal volume, referred to as lattice sites. The volume of each lattice site should be enough to accommodate one molecule of the fluid and the number of lattice sites should be such that M $ N . After placing the molecules on the lattice, there will be (M 2 N) vacant sites. In such a model, changes to the fluid density are related to changes in the relative amounts of vacant and occupied lattice sites. Fig. 3.1 shows the schematic of a two-dimensional lattice partially occupied by molecules. The number of accessible microscopic states is represented by the symbol Ω. The relationship between S and Ω is SðN ; V ; U Þ 5 kB ln Ω

(3.1)

Figure 3.1. Two-dimensional lattice partially occupied by molecules.

75

Thermodynamics of Phase Equilibria in Food Engineering

where kB is the Boltzmann constant. In the absence of intermolecular interactions, all microscopic states that have the same volume and number of molecules will have the same internal energy. Thus, the number of microscopic states is the number of ways of placing N undistinguishable molecules and (M 2 N) undistinguishable vacant sites on the lattice, which is Ω5 It follows that

M! N!ðM 2 N Þ!

(3.2)



 M! SðN ; M Þ 5kB ln Ω 5kB ln 5kB ½lnM!2ln N ! 2ln ðM 2N Þ! N!ðM 2 N Þ! (3.3) For large numbers, such as the numbers of molecules and vacant sites of a macroscopic system, the Stirling approximation gives that lnx!  xlnx 2 x. Applying this approximation to Eq. (3.3) the following expression is obtained: SðN ; M Þ 5 kB ½MlnM 2 N ln N 2 ðM 2 N Þ ln ðM 2 N Þ

(3.4)

To derive the pressure that corresponds to this entropy expression, it is useful to recall the relationship among the Helmholtz energy (A), internal energy (U), and entropy and the relationship between pressure (P) and the Helmholtz energy, Eqs. (2.3) and (2.9): A 5 U 2 TS

(2.3)

  @A P52 @V T ;N

(2.9)

It follows that       @A @U @S P52 52 1T 5 PU 1 PS (3.5) @V T ;N @V T ;N @V T;N     where P U 5 2 @U=@V T ;N and P S 5 T @S=@V T;N are the energetic and entropic contributions to the pressure, respectively. Omitting the algebraic details of the derivation, PS for the lattice model developed here is         @S @S @M kT Nb S P 5T 5T 5 2  ln 1 2 (3.6) @V T;N @M T ;N @V T ;N b V

76

Classical Models Part 1: Cubic Equations of State and Applications

where b 5 V/M is the volume of one lattice cell. It is now useful to recall that the Taylor series expansion of ln(1 2 x) around x 5 0 is 1 ln ð1 2 xÞ 5 2x 2 x2 2 ? (3.7) 2 Truncating the series in its quadratic term and combining this result with the PS expression, the following result is obtained:   2 !    kT Nb 1 Nb NkT Nb S 5 P 5  11 1 (3.8) b 2 V V V 2V       Consider the product 1 1 Nb =2V 3 1 2 Nb =2V . Its value tends to 1 when the ratio (N/V)  tends to zero,  that is, at low densities. It is then possible to replace 1 1 Nb =2V in the PS expression at low densities as follows: PS 5

NkT NkT RT   5  5   V 2b V 1 2 Nb =2V V 2 Nb =2 

(3.9)

The last term on the right-hand side of this equation expresses PS using the universal gas constant  (R), molar volume (V ) and the covolume (b), given by b 5 NAv b =2 , where NAv is Avogadro’s number. This PS expression was obtained assuming that the system’s energy was independent of the placement of molecules on the site. At the expense of theoretical rigor, assume that the molecules interact with up to z possible neighbors. Assuming that the molecules’ positions on the lattice are random, the average number of neighbors that interact with any given central molecule is (zN/M). If the energy of each intermolecular interaction is equal to ε, the total internal energy of the lattice is N zN zN 2 ε 1 zN 2 ε b ε5 5 (3.10) 2 M 2 M 2 V where the factor (N/2) prevents double counting the intermolecular interactions. It follows that   @U zN 2 ε b N2 a PU 5 2 5 5 2a 52 2 (3.11) @V T;N 2 V2 V2 V   where a 5 2 zεb =2 . The last term on the right-hand side expresses the energetic contribution to the pressure as a function of the molar volume. Adding the entropic and energetic contributions, the result is U5

RT a (3.12) 2 2 V 2b V which is the conventional way of writing the van der Waals EoS. P5

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Thermodynamics of Phase Equilibria in Food Engineering

In Eq. (3.12), b (covolume) is the volume correction parameter of the equation, which represents part of the molar volume that is inaccessible to a molecule of the fluid, due to the presence of the other ones. This contribution increases the prediction of the pressure given by Eq. (3.14) when compared to the ideal gas EoS at the same density and temperature. In addition, a(energy parameter) represents the attractive parameter between the molecules. The second term on the right-hand side, in the van der Waals EoS, is independent of the temperature and is related to the attractive forces between the molecules of the fluid. The pressure exerted by a gas inside a reservoir is a consequence of the collision of the molecules of the gas against the walls of the reservoir. Therefore, the pressure of the gas is directly related to the force and the frequency of these collisions. The assumption of the existence of the attractive forces between the molecules of gas induces a decrease of the force and of the frequency of the collisions resulting in a decrease of the pressure exerted by the gas. That is why this contribution decreases the pressure prediction given by Eq. (3.12). The van der Waals equation can be interpreted as a sum of two terms. The first one represents the repulsive interactions between the molecules and the second term represents the attractive interactions. The repulsive term considers repulsive forces that act when the molecules collide. In this situation, according to the van der Waals EoS, the molecules act like hard-spheres. However, molecular simulation data of hard-spheres show that the repulsive term of the van der Waals EoS does not represent quantitatively such kind of interaction at high densities. This is the root cause why cubic EoS have, in general, difficulty to predict accurate values of liquid phase properties. Much improvement can be obtained by substituting the repulsive term of the van der Waals EoS with the CarnahanStarling [3] equation for hard-spheres: z5

PV 1 1 ξ 1 ξ2 2 ξ3   5 RT 1 2 ξ3   V0 ξ 5 0:74 V  3 σ V 0 5 pffiffiffi NA 2

78

(3.13)

(3.14)

(3.15)

Classical Models Part 1: Cubic Equations of State and Applications

where ξ is the reduced density, σ is the molecular diameter, and NAv is Avogadro’s number. The attractive term can be also substituted by the double power series expansion of Alder et al. [4] and Sandler [5]   XX ε n  3 m atr P 52 Anm ρσ (3.16) kB T n m where ε is the well depth, ρ is the density, kB is Boltzmann constant, T is the temperature, m and n are the series number of terms, and Anm are fitting parameters. Such modifications significantly improve the representation of highpressure phase equilibrium data. However, the mathematical representation of these equations is not cubic, it requires a larger number of adjustable parameters and the extension to multicomponent systems becomes more complex. Such features make it difficult to use those sophisticated equations in applications to food processing. Parameters a and b for a pure substance may be obtained in the following ways: 1. The parameters may be adjusted directly from experimental data, usually vapor pressure and density of the liquid or the vapor. As in van der Waals’s EoS, these parameters do not have an explicit dependence on the temperature; this method may be obtained from data of only one temperature condition. For other equations (with parameters dependent on T), the calculation is made for a defined range of temperatures. 2. By calculating the parameters with critical point information of the pure substance. At this point, the critical isotherm line has zero slopes and changes concavity. This behavior is expressed mathematically as P 5 Pc 



@P @V

(3.17)



@2 P @V 2

50

(3.18)

50

(3.19)

Tc

 Tc

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Thermodynamics of Phase Equilibria in Food Engineering

For known values of Tc and Pc , this is a set with three equations and three unknowns: a, b, and Vc . For the van der Waals EoS, this system of equations has the following solution: a5

27R2 Tc2 64Pc

(3.20)

RTc 8Pc

(3.21)

3RTc 8Pc

(3.22)

b5 Vc 5

The van der Waals equation has historical interest because it was the first cubic EoS. Although it is not quantitatively accurate, this equation is able to reproduce a range of phenomena related to vaporliquid equilibrium qualitatively. From the vdW equation, a series of cubic state equations have been proposed, with variations in two directions: modifying the dependence with the temperature of the attractive term and modifying the dependence of the molar volume with the pressure. The modern development of the cubic EoS began in 1949 when Redlich and Kwong [6] proposed a modification of the van der Waals equation. Such modification consisted of introducing an explicit temperature and volume dependence in the attractive term of the vdW EoS, as shown here: P5

RT a 2 1=2 V 2 b T V ðV 1 bÞ

(3.23)

In fact, the main improvement of the RedlichKwong EoS was the incorporation of the term T1/2 into the denominator of the attractive term, providing good results for several gaseous systems; however, is unable to reproduce the vapor pressure of pure substances accurately and not even the volumetric properties of liquids. In this equation, the parameters a and b can be evaluated as previously described for the van der Waals EoS and they are expressed as a5

27R2 Tc2:5 64Pc

(3.24)

RTc 8Pc

(3.25)

b5

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Classical Models Part 1: Cubic Equations of State and Applications

The repulsive term in the RedlichKwong equation can be modified using the CarnahamStarling expression (3.13), resulting in the modified RedlichKwong EoS as follows: z5

PV 1 1 ξ 1 ξ2 2 ξ3 4aξ   2 5 3 3=2 RT bRT ð1 1 4ξÞ 12ξ

(3.26)

The molar volume in the attractive term was also substituted by ξ in order to make the calculations easier and show explicitly the use of molecular parameters. Fig. 3.2 shows the predicted compressibility factor of carbon dioxide at the temperature of 350 K and pressures up to 50.0 MPa, as predicted by the vdW, the RedlichKwong and the modified RedlichKwong EoSs. Table 3.1 presents the parameters used in the calculations of compressibility factor shown in Fig. 3.2. In conditions of low density (low pressure), the results of all equations are similar. However, at high density (high pressure), the repulsive terms in the RedlichKwong and in the modified RedlichKwong present different results. The introduction of molecular parameters into the modified RedlichKwong equation produces smaller values of the compressibility factor. This result agrees with the experimental data of the

Figure 3.2. Compressibility factor of carbon dioxide calculated using the van der Waals (vdW), the RedlichKwong (RK), and the modified RedlichKwong (mRK) equations at 350 K.

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Thermodynamics of Phase Equilibria in Food Engineering

Table 3.1. Critical properties and molecular diameter of carbon dioxidea and the values of attractive and repulsive parameters used in the modified RedlichKwong equation Pc (MPa)

Tc (K)

σ 3 10212 (m)

bPc =RTc

7.382

304.2

317.0

0.1050

a

aPc =ðRTc Þ2

pffiffiffiffiffi 0:4619 Tc

R (MPa cm3/K mol)

8.31451

Data from [7].

compressibility factor of CO2 as described by O’Connell and Haile [7]. However, it should be emphasized that Eq. (3.26) is not a cubic EoS. Most of the subsequent developments of cubic EoS are related to improvements of the attractive term, in an attempt to enhance the description of the PVT behavior. One of the standard strategy applied to the development of the EoS consisted of using of the conditions of the critical point, Eqs. (3.17)(3.19), to obtain an attractive parameter ac at the critical point, and introducing a correction dependent on the temperature in the attractive term, which has the following form: aðTÞ 5 ac αðT Þ

(3.27)

considering that αðTÞ is equal to 1 at the critical point and adjustable in other conditions so as to produce better predictions over a wide temperature. This strategy was first used by Wilson [8] and later by Soave [9] and PengRobinson [10]. Table 3.2 shows the EoSs proposed by these authors. The cubic EoSs proposed by Soave [9], also known as SoaveRedlichKwong (SRK) equation, and by PengRobinson (PR) [10] predict the vapor pressure of non-polar compounds very well, especially of light hydrocarbons. These equations have become important tools for the calculation of vaporliquid equilibrium at moderate and high pressures. Therefore, the SoaveRedlichKwong and Peng Robinson equations are widely used in industrial applications. These equations have the following advantages: (1) they require little information input (critical properties and acentric factor), (2) they are easily solved, (3) they are available in most of the commercial chemical process simulation programs, and (4) they provide good prediction of the liquidvapor equilibrium for non-polar compounds. In contrast, these equations have the disadvantages of not describing the following aspects accurately: (1) the volume of the liquid phase, (2) the properties of longchain molecules, and (3) the behavior of fluids near the critical region and at low pressures. These disadvantages limit the direct application of the cubic EoSs in various systems composed of long-chain molecules. 82

Table 3.2. Some cubic equations of state Author

Equation

Wilson [8]

P5 

RT aα 2  c  V 2b V V 1b

Parameters,a,b with aðTÞ 5 ac αðTÞ

0:42748R2 Tc2 ; Pc 2

0:08664RTc Pc 3   1 αðT Þ 5 Tr 41 1 ð1:57 1 1:62ωÞ 21 5 Tr

(3.28)

(3.29)

ac 5

b5

Soave RedlichKwong [9]

RT aα 2  c  P5  V 2b V V 1b

0:42748R2 Tc2 0:08664RTc ; b5 Pc Pc    2 αðT Þ 5 11 0:48011:574ω20:176ω2 12Tr1=2

PengRobinson [10]

RT aα 2  c   P5  V 2b V V 1b 1b V 2b

ac 5 0:45724

ac 5

 



 R2 Tc2 ; Pc

 b 5 0:07780

RTc Pc



αðT Þ 5 11 0:3746411:54226ω20:26992ω a

ω is the acentric factor.

b

  Tr is the reduced temperature (Tr 5 T =Tc ).

2



12Tr1=2

2

(3.30)

Thermodynamics of Phase Equilibria in Food Engineering

Although there are such limitations, the literature has presented several works illustrating goods results of the application of these equations (vdW, SRK, and PR) in the description of the behavior of food systems in different conditions. Several other modifications have been proposed in calculating the alpha function, see Table 3.3, improving the prediction of vapor pressure for pure species. These expressions require information about critical properties, the acentric factor, and adjustable parameters. In addition to the introduction of the alpha function, another strategy used to enhance cubic EoSs consists in the insertion of an additional parameter. For example, Patel and Teja [16] introduced a third parameter c in the attractive term, resulting in the following equation of state: P5

RT a 2 2 V 2 b V 1 ðb 1 cÞV 2 bc

(3.31)

Eq. (3.31) becomes the PengRobinson EoS if the parameter c is equal to b. This kind of equation is capable of predicting vapor pressure data and molar volume of the liquid more accurately than models such as the Soave and PengRobinson EoSs. Besides, with a proper selection of the parameters, these equations are also capable of representing the properties of polar fluids. However, this class of equation still fails to predict the properties of the fluids close to the critical condition. Another disadvantage of this kind of equation is that the third parameter, c, must be

Table 3.3. Alpha function for some cubic equations of state Authors

Graboski and Daubert Heyen Mathias and Copeman

Soave Stryjek and Vera

84

Function

h

pffiffiffiffii2 αðT Þ 5 11m T m 5 0:48508

1 1:55171ω 2 0:1561ω2   αðT Þ 5 exp Cð1 2 Trn Þ

pffiffiffiffiffi 2 pffiffiffiffiffi αðT Þ 5 1 1 C1 1 2 Tr 1 C2 12 Tr

pffiffiffiffiffi 3 1 C3 12 Tr   αðT Þ 5 1 1 mð1 2 Tr Þ 1 n Tr21 2 1   pffiffiffiffiffi2 αðT Þ 5 11k 12 Tr

Ref.

[11]

[12] [13]

[14] [15]

Classical Models Part 1: Cubic Equations of State and Applications

obtained through additional pure component data, besides requiring an additional equation that matches the molar composition of the system. Yet, this equation is present in most of the commercial process simulation programs. In order to show an application of cubic EoS to a fluid relevant to Food Engineering, isotherms in a pressureenthalpy diagram of refrigerant R134a were calculated over the temperature range from 423.15 to 673.15 K and a pressure range from 0.1 to 10 MPa using the Peng Robinson equation of state. The XSEOS [17] software (Thermodynamic Properties using Excess Gibbs Free Energy Models and EoS) can be used to calculate that diagram easily. We choose the reference state of R134a to be the ideal gas at T  5 298.15 K and P  5 0.1 MPa. We use the concept of residual properties to compute the enthalpy as follows: H ðT ; PÞ 5 ΔH IG 1 H R ðT ; PÞ 1 H R ðT  ; P  Þ where ΔH IG is the enthalpy change of ideal gas given by ð T IG CP IG ΔH 5 R dT T R

(3.32)

(3.33)

Using the following expression for the ideal gas heat capacity: CpIG R

5 A 1 BT 1 CT 2 1 DT 3 1 ET 4

Eq. (3.33) becomes  C 3  B 2 IG T 2 T 2 1 T 2 T 3 ΔH 5 R AðT 2 T  Þ 1 2 3     D 4 E 5 4 5 T 2T 1 T 2T 1 4 5

(3.34)

(3.35)

The residual enthalpy in the reference state, H R ðT  ; P  Þ, is equal to zero. Because the gas ideal state was used as the reference state, all residual properties are equal to zero in such condition. The residual enthalpy, H R ðT ; PÞ, is given by the following equation: # ð V 5V ðT ;PÞ "     @P H R ðT ; PÞ 5 PV 2 RT 1 T 2 P dV (3.36) @T V V 5N

85

Thermodynamics of Phase Equilibria in Food Engineering

The desired equation is obtained by substituting Eq. (3.30) in Eq. (3.36), resulting in " pffiffiffi #    T da=dT 2 a Z 1 1 1 2 B R pffiffiffi pffiffiffi  ln (3.37) H ðT ; PÞ 5 RT ðZ 2 1Þ 1 2 2b Z 1 12 2 B     where Z 5 PV =RT and B 5 Pb=RT . The scheme of calculation was the following: for each value of T  values for a and b were computed using Eq. (3.30), and B 5 Pb=RT and ΔH IG using Eq. (3.32). Then, at each pressure, PengRobinson EoS was solved  for the Rvolume V , from which is easy to calculate Z 5 PV =RT . Next, H ðT ; PÞ was computed using Eq. (3.37) and, finally, the enthalpy, H ðT; PÞ, was computed using Eq. (3.26). The critical properties, acentric factor, and constants of Eq. (3.34) for the refrigerant R134a are presented in Table 3.4. Fig. 3.3 shows the calculated isotherms in the pressureenthalpy diagram for refrigerant R134a. The values of enthalpy calculated in this case are in conditions at which the refrigerant is in the gaseous state. Such state can be confirmed by Fig. 3.4, which shows a diagram of the compressibility factor as a function of pressure for the same temperatures used in Fig. 3.3. Fig. 3.4 shows that for each temperature and pressure specification, there is only one real value of the compressibility factor, while the existence of multiple values would have meant the possibility of a phase transition between the gas and liquid states. In this case, note that the calculations were performed in a region of one phase. The other line of studies aimed at modification of cubic state equations is the change in dependence on the pressure of the molar volume, through the concept of Volume translation. The concept of volume translation consists of correcting the volume of the liquid provided by the equation of state (V EoS ) with the insertion of the translation term (V t ), considering empirical data (V exp ), through the correlation: V exp 5 V EoS 1 V t

(3.38)

Table 3.4. Critical properties, acentric factor, and constants in the equation for ideal gas heat capacity, Eq. (3.34), for the refrigerant R134a Physical properties [18]

Pc (MPa) 4.059

86

Tc (K) 374.26

ω 0.326

Constants of Eq. (3.34)

A 3.064

B 2.542 3 1022

C1 5.860 3 1026

D 23.339 3 1028

E 1.716 3 10211

Classical Models Part 1: Cubic Equations of State and Applications

Figure 3.3. Pressureenthalpy diagram for the gaseous refrigerant R134a over the temperature range from 423.15 to 673.15 K and pressure range from 0.1 to 10 MPa using the PengRobinson EoS.

Figure 3.4. Compressibility factor as a function of pressure for the gaseous refrigerant R134a over the temperature range from 423.15 to 673.15 K and pressure range from 0.1 to 10 MPa using the PengRobinson EoS.

87

Thermodynamics of Phase Equilibria in Food Engineering

Figure 3.5. Isotherms of n-hexane in the pressuremolar volume plane at 470 K. Values calculated with the original PengRobinson EoS (PR), and calculated values using the volume translation (PR VT). The horizontal dotted line is the EoS prediction for the vapor pressure of n-hexane at 470 K (Psat).

In this way, the limitations of cubic EoS on predicting liquid phase properties can be alleviated using the volume translation concept. Fig. 3.5 illustrates the concept with a pressuremolar volume diagram of n-hexane at 470 K, which is used by the oil industry in the extraction of edible oils. In this figure, the solid line is the isotherm calculated by direct application of the PengRobinson EoS. The dashed line represents a displaced isotherm, which is obtained by the horizontal translation of the original isotherm by a fixed value in the molar volume scale. The horizontal dotted line indicates the vapor pressure of n-hexane at 470 K. At the saturation conditions, the pressures of the liquid and vapor phases are equal to each other and equal to the vapor pressure of the pure substance at the given temperature, that is PV 5 PL 5 Psat. The graphical interpretation of the phase equilibrium criterion represented by these equations is known as the Maxwell equal area rule, according to which the closed areas between the isotherm and the saturation line, above and below of the saturation lines in Fig. 3.5, are equal. The elegance of the volume translation concept is that displacing the 88

Classical Models Part 1: Cubic Equations of State and Applications

isotherm horizontally does not affect the value of these areas. In other words, the predicted vapor pressure remains unchanged upon volume translation. In addition, a given volume translation may have a large percent effect on the molar liquid volume but little percent effect on the molar volume of the vapor phase, for which the original form of cubic EoS such as the PengRobinson and SoaveRedlichKwong models perform fairly well. In this way, it is possible to improve the results for the molar volume of the liquid phases, with small impact on the molar volume of the vapor, except close to critical points. In addition, volume translation has no impact on the phase equilibrium conditions, that is, the vapor pressure predicted by a cubic EoS with or without volume translation is identical. These results can be extended to mixtures in the sense that the results of phase equilibrium calculations for mixtures are unaffected by volume translation. Mathematically, the volume translation effect is obtained by adding a parameter c to the molar volume. For example, Eq. (3.42) shows the volume-translated PengRobinson EoS, which can be compared to the original PengRobinson EoS Eq. (3.30). Observe that a volumetranslated cubic EoS retains its cubic dependence on molar volume. P 5 

RT aα  2   c     (3.42) V 1c 2b V 1c V 1c 1b 1b V 1c 2b

The volume-translation parameter of a pure component, c, can be calculated from the value of the b parameter as follows: c 5 sb

(3.43)

where s is usually fitted from experimental liquid density data. For mixtures, the value of c is determined using a mixing rule: X c5 xi ci (3.44) i

where ci 5 si bi

(3.45)

Many other cubic EoS were proposed by using the strategies presented previously. Reviews of the subject by Sandler [5] and Valderrama [19] discuss the applications and the disadvantages of several cubic EoSs thoroughly.

89

Thermodynamics of Phase Equilibria in Food Engineering

3.3 MIXING RULES FOR THE CUBIC EQUATIONS OF STATE 3.3.1 Classical Mixing Rule The cubic EoSs previously presented for pure substances can be applied to mixtures if the parameters of the equation properly characterize the mixture. These parameters for mixtures are determined by using the socalled mixing rules. The most applied mixing rule for cubic EoS is the quadratic mixing rule proposed by vdW2, known as the main Classical Mixing Rule. This rule shows a quadratic composition dependence for the parameter a as well as for the parameter b of the cubic equations: XX a5 yi yj aij (3.46) i

b5

j

XX i

yi yj bij

(3.47)

j

where, it is necessary to define combining rules for the cross parameters aij and bij . The adopted equations are  pffiffiffiffiffiffiffi aij 5 ai aj 1 2 kij (3.48) bij 5

  1 bi 1 bj 1 2 lij 2

(3.49)

where ai and bi are the parameters for the pure component i and kij and lij are the binary interaction parameters, which can be fitted from experimental phase equilibrium data for mixtures. Again, such parameters are introduced to improve the results of the cubic equations for mixtures. Usually, it is admitted that lij 5 0 and, thus, Eq. (3.47) becomes X yi bi (3.50) b5 i

The combining rule, Eqs. (3.46) and (3.47), used in the mixing rules of van der Waals EoS is based on the direct relationship between the parameter a of the equation of state and the attractive parameter (ε) of the intermolecular interaction potential for a cross-interaction [20]. Besides, the parameter b is related to the volume of a hard sphere. Thus, the mixing rules for the cubic EoS are justified with the same arguments used to obtain the mixing rule of the second virial coefficient.

90

Classical Models Part 1: Cubic Equations of State and Applications

Despite these limitations, the van der Waals mixing rules can be used with relative accuracy for predicting the behavior of mixtures of components that are chemically similar, that is, mixtures whose molecules have similar size and chemical nature [21]. Other several mixing rules for cubic EoSs are available in the literature. The choice of the mixing rule can have a significant impact on the ability to describe the thermodynamic properties of the real systems.

3.3.2 EoS/GE mixing rules One of the most outstanding strategies is the combination of excess Gibbs energy models with cubic EoSs, also known as Equation of state/ excess Gibbs energy (EoS/GE mixing rules). The purpose of this approach is to obtain a mixing rule that reflects the variations in composition in the liquid phase that a G E model is capable to reproduce. The principle of this strategy is to match the correlated G E with the expression of this same property evaluated by a cubic EoS. Many strategies have been developed (Huron and Vidal [22]; Michelsen [23]; Dahl and Michelsen [24]; Boukouvalas et al. [25]; Tochigi et al. [26]; Orbey and Sandler [27]) to allow this connection. Table 3.5 presents the algebraic expressions for the PR a and b parameters used in some EoS/GE mixing rules. The results of these mixing rules depend on the G E model that is adopted in the equations presented in Table 3.5. Several G E models can be selected such as NRTL, UNIQUAC, and van Laar (described in Chapter 4). The Wong and Sandler [28] and HuronVidal [22] (HVO) mixing rules, chose the condition of P -N, at which the ratio V =b tends to 1 for the calculation of excess Gibbs energy. The WongSandler mixing rule [28] results in an equation of state capable of predicting a consistent behavior at high- and low-density conditions, without the need to include a density dependence in the mixing rule. In addition, this mixing rule is able to reproduce the quadratic dependence with the composition of the second virial coefficient. The modified HuronVidal mixing rule of first-order (MHV1) [23] and modified HuronVidal mixing rule of second-order (MHV2) [24] mixing rules adopt P-0 and V =b tends, respectively, to 1.235 and 1.632. The linear combination of the Vidal and Michelsen mixing rules (LCVM) mixing rule becomes the HVO or MHV1 depending on the value of arbitrary parameter λ.

91

Table 3.5. Algebraic expressions for the PR-EoS parameters a and b of some EoS/GE mixing rules Mixing rules

b

WongSandler



a and other parameters

pffiffiffiffiffiffiffi   ai aj  a 1 1 2 ki;j b2 5 bi 1 bj 2 RT i;j 2 RT

HuronVidal (original) (HVO)

b5

HuronVidal first-order (MHV1)

b5

" (3.51)

a 5 bRT

P

i yi bi

" a5b

#   X G E T; yi ai 1 yi 20:6223RT bi RT i

#   X ai G E T; yi 1 yi 20:6223RT bi i

P

i yi bi

a 5 ðbRT Þ

(3.52)

(3.53)

8