There are social evolutionary theories that propose that a critical factor in the progress and advancement of human societies was, is, and will be the production of, and access to, adequate food supplies. Despite the accelerating increase in the internationalization of food trade, the supply chain is far from perfect. Improvement in the production, ­processing, storage, and distribution of food is still an unfinished ­business.

Yet consumers worldwide are becoming more sophisticated and knowledgeable. Foods that used to be local or regional are now coming to be known worldwide. Consumers demand more, and expect more. Safe food is the primary expectation, but still people all over the world experience food poisoning with unacceptable human and economic consequences. Consumers also demand and expect “better” foods. Quality expectations in terms of nutritional value and sensory attributes are getting more stringent. At the same time, there is fierce competition in the marketplace with thousands of new products being introduced every year, and thousands failing and being replaced. This creates serious economic pressures to reduce cost, to increase production, processing, and distribution efficiency, and to optimize their integration. On top of all this, there is a growing concern about the effect of human activities, including agriculture and food production, on the environment, with its consequences on the sustainability of these endeavors.

This is a complex scene where the players (the food industry, regulatory agencies, scientists, and consumers) have sometimes cooperating and sometimes adversarial roles. Yet, our overall objectives are the same: to provide a safe, excellent-quality (in terms of nutritional value and sensory properties), affordable, convenient, optimally distributed, readily available, and sustainable food supply.

In this complex, sometimes conflicting, and ever-changing environ­ment, the introduction of new technologies, new methods, and new products is becoming even more challenging. The food industry, with its dependence on volume sales and its small profit margin, is conservative and reluctant to change. Yet the changing needs of the marketplace, and increasing and conflicting pressures from consumers, are felt best by the industry. The increasing adoption of nonthermal technologies is therefore an indication that they fulfill a need, a niche in the marketplace.

Dense phase carbon dioxide (DPCD) is a nonthermal method of food processing. Its application is quite different from, for example, supercritical extraction with CO2. In extraction, the typical solubility of materials in CO2is in the order of 1%. This requires large volumes of CO2to be used. In contrast, processing with DPCD requires much less CO2. The typical solubility of CO2in many types of liquid foods is in the order of 5%. Even if more CO2was added, the excess would not dissolve and therefore would not be effective in, for example, microbial reduction or enzyme inactivation. The pressures used are at least one order of magnitude less than those typically used in ultra-high pressure (UHP) processing. There is no noticeable temperature increase with DPCD due to pressurization. The small weight percentage of CO2used assures this. The typical process temperatures are less than 40°C.

DPCD temporarily reduces the pH of the liquid foods with effects on microorganisms and enzymes. Because oxygen is removed from the environment, and because temperature is not high during the short process time (typically about 5 min in continuous systems), nutrients, antioxidant activity, and vitamins are preserved much better than in the case of thermal treatments. The economics of the operation have been studied, and look promising. Although the capital and operating costs are still higher than those of thermal treatments, they are much lower than those of other nonthermal technologies (e.g., UHP operations).

With all its potential for safe foods without nutrient or quality loss, DPCD still has not achieved commercial operation status. A significant number of scientific publications demonstrate its effectiveness against microorganisms and enzymes, and the rate of increase of publications is accelerating. Some of those recent publications are excellent reviews of the technology and its applications. Yet there has been no book that brought together this increasing body of knowledge. This book benefits from the experience and knowledge of individual and groups of international scientists and members of the industry. The editors are indebted to the contributors: the book could not have become a reality without their expertise, experience, and willingness to contribute to this effort.

On behalf of all authors, the editors hope that this book would benefit researchers working in the area of DPCD, and in other nonthermal or traditional technologies with the potential to be used in conjunction with DPCD. We also hope that the industry, the regulatory agencies, and, most importantly, consumers would find interesting and useful information in the book. We believe that DPCD will find its niche in the safe and minimal processing of foods, and will take its place among other commercial nonthermal food- and pharmaceutical-processing operations.

Murat O. Balaban
Giovanna Ferrentino


Murat O. Balaban

Department of Chemical and Materials Engineering

University of Auckland

Auckland, New Zealand

Patricia Ballestra

Department of Bioengineering

IUT Périgueux – Bordeaux IV

Périgueux, France

Thelma Calix

Department of Food Science

Zamorano University

Francisco Morazán, Honduras

Maria G. Corradini

Instituto de Tecnología

Facultad de Ingeniería y Ciencias Exactas

Universidad Argentina de la Empresa

Ciudad de Buenos Aires, Argentina

Luu Thai Danh

School of Chemical Engineering

University of New South Wales

Sydney, Australia

Frank Devlieghere

Department of Food Safety and Food Quality

Laboratory of Food Microbiology and Food Preservation


Ghent University

Ghent, Belgium

Kathy Elst

Business Unit Separation and Conversion Technology

Flemish Institute for Technological Research (VITO)

Mol, Belgium

Osman Erkmen

Department of Food Engineering

University of Gaziantep

Gaziantep, Turkey

Giovanna Ferrari

Chemical and Food Engineering Department

ProdAl – Centro Regionale di Competenza sulle Produzioni Agroalimentari

University of Salerno

Fisciano, Salerno, Italy

Giovanna Ferrentino

Department of Materials Engineering and Industrial Technologies

University of Trento


Neil R. Foster

School of Chemical Engineering

University of New South Wales

Sydney, Australia

Linsey Garcia-Gonzalez

Business Unit Separation and Conversion Technology

Flemish Institute for Technological Research (VITO)

Mol, Belgium

Ireneo Kikic

Department of Materials and Natural Resources

University of Trieste

Trieste, Italy

Xiaojun Liao

National Engineering Research Center for Fruit and Vegetable Processing

College of Food Science and Nutritional Engineering

China Agricultural University

Beijing, China

Raffaella Mammucari

School of Chemical Engineering

University of New South Wales

Sydney, Australia

John S. Novak

Food Lab City of New York Public Health Laboratory

New York, NY, USA

Micha Peleg

Department of Food Science

University of Massachusetts

Amherst, MA USA

Massimo Poletto

Department of Chemical and Food Engineering

University of Salerno

Fisciano (SA), Italy

Sara Spilimbergo

Department of Materials Engineering and Industrial Technologies

University of Trento

Trento, Italy

Wen Hui Teoh

Department of Chemical Engineering

Faculty of Engineering

University of Malaya, Malaysia


School of Chemical Engineering

University of New South Wales

Sydney, Australia

Luc Van Ginneken

Business Unit Separation and Conversion Technology

Flemish Institute for Technological Research (VITO)

Mol, Belgium

James T.C. Yuan

Global Beverage R&D


Valhalla, NY, USA

1  Introduction to Dense Phase Carbon Dioxide Technology

Giovanna Ferrentino and Murat O. Balaban

Abstract: The introduction aims to bring together accumulated knowledge in the areas of supercritical and dense phase CO2 technology. A summary is given of the areas covered by the book’s chapters. Recognized experts in their fields discuss the following topics: solubility of CO2 in liquids; the effects of supercritical and dense phase CO2 on microorganisms, including vegetative cells and spores; the application of supercritical and dense phase CO2 on juices, beverages, and dairy products; progress in the use of pressurized CO2 in pharmaceuticals. Finally an outlook regarding the future of the technology is presented.

Keywords: overview; dense phase; carbon dioxide; nonthermal processing.

Nonthermal technologies have gained increasing importance in recent years as potentially valuable processes to replace or complement the traditional technologies, currently used for preserving foods and other biological materials. Traditionally, many foods are thermally processed by subjecting them to a temperature range from 60°C to more than 100°C for few seconds to several minutes (Jay 1992). During thermal treatments, heat transferred to the food kills vegetative cells of microorganisms, yeast, and molds, and also inactivates spores depending on the severity of the applied conditions. This process also inactivates many undesirable enzymes in foods that cause quality loss. However, thermal treatment may cause unwanted reactions leading to undesirable changes or formation of ­by-products in the food.

Thermally processed foods can undergo organoleptic changes and a cooked flavor accompanied by a loss of vitamins, essential nutrients, and flavors.

Increased consumer demand for safe, nutritious, fresh-like food ­products with a high organoleptic quality and an extended shelf life resulted in the concept of preserving foods using nonthermal methods.

During nonthermal processing, the temperature of the food is held below temperatures normally used in thermal processing. Therefore, the quality degradation expected from high temperatures is reduced and some vitamins, essential nutrients, and flavors are expected to undergo minimal or no changes (Barbosa-Cánovas 1998). High hydrostatic ­pressure (HHP), dense phase carbon dioxide (DPCD), oscillating magnetic fields, high-intensity pulsed electric fields, intense light pulses, ­irradiation, cold plasma, chemicals, biochemicals, and hurdle technology are some of the possibilities in the area of nonthermal technologies. Compared to traditional techniques, these avoid drawbacks such as loss of flavors and ­nutrients, production of undesirable side reactions, as well as changes in physical, mechanical, and optical properties of the food treated.

The use of dense phase carbon dioxide (DPCD) has been proposed as an alternative nonthermal pasteurization technique for foods (Spilimbergo et al. 2002), for the first time in the 1950s by Fraser (1951) and Foster et al. (1962) who reported the disruption of bacterial cells by the rapid decompression of carbon dioxide (CO2) from a pressure of 500 lbf/in2 (about 3.45 MPa) to ambient pressure. DPCD involves mostly the supercritical state of CO2, but may also involve subcritical gases and sometimes liquids under pressure. In the DPCD technique, food is contacted with (pressurized) sub- or supercritical CO2 for a certain amount of time in batch, semibatch, or continuous equipment. The DPCD technique presents some advantages over HHP due to its milder process conditions. The pressures applied are much lower (generally < 30 MPa) compared to those used in HHP (300–1200 MPa). In addition, capital expenditure is considerably lower. In 1969 Swift & Co. (Chicago, IL) obtained the first US patent for food product pasteurization with CO2 at “super atmospheric” pressure. Since 1980, others demonstrated the bacteriostatic action and inhibitory effect of CO2 on the growth and metabolism of some microorganisms. Pseudomonas was found to be very sensitive while other types, such as Lactobacillus and Clostridium, were less sensitive. However, it was with the work published by Kamihira et al. (1987) that the inhibitory effect of CO2 under pressure started to be addressed systematically. These authors tested the sterilizing effect of CO2 in supercritical, liquid and gaseous phase on wet and dry Escherichia coli, Staphilococcus aureus and conidia of Aspergillus niger by using a supercritical fluid extraction apparatus. Since then, many studies investigated the effects of CO2 on pathogenic and spoilage organisms, vegetative cells and spores, yeasts and molds, and enzymes. It has been proven that this technique can be considered a cold pasteurization method that affects microorganisms and enzymes, using CO2 under pressures below 50 MPa without exposing foods to the adverse effects of heat. Thus foods retain their physical, nutritional and sensory qualities.

DPCD pasteurization of liquid foods is operational and almost ready to be employed on a commercial scale. Most of the commercialization efforts have been performed so far by Praxair Inc. (Burr Ridge, IL, US). Based on the technology, licensed from the University of Florida (Balaban et al. 1995; Balaban 2004a, b), Praxair developed a continuous process ­system which utilizes the DPCD as a nonthermal process alternative to thermal pasteurization (Connery et al. 2005). This system has been registered under the trademark “Better Than Fresh (BTF).” Praxair ­constructed four mobile BTF units for processing about 1.5 L/min of liquid foods for demonstration purposes. In addition, a commercial-scale unit of 150 L/min has also been constructed (Connery et al. 2005) and tested in an orange juice–processing plant in Florida. For the continuous treatment of liquid foods, pilot-scale equipment was also manufactured by Mitsubishi Kakoki Co. (Tokyo, Japan) on behalf of and according to the patents owned by Shimadzu Co. (Kyoto, Japan) (Osajima et al. 1997, 1999a, b). This equipment consisted of a vessel of 5.8 L through which CO2 and ­liquid foodstuff were simultaneously pumped at maximum flow rates of 3.0 kg/h and 20 kg/h, respectively. In 2003, the apparatus was made available only to research laboratories in Japan (private ­communication, Shimadzu Belgium). At the moment, we have no information available on further commercialization efforts of Shimadzu in the field of DPCD processing for liquid foods. On the basis of their own ­patent (Sims 2000), PoroCrit LLC (Berkeley, CA, US) also developed a membrane contactor consisting of several hollow-fiber membrane ­modules for the ­continuous DPCD pasteurization of liquid foods, mainly beverages, juices, milk, and wine.

As for all non-thermal technologies, the most important issue involved in the commercialization of DPCD process is the regulatory approval. Foods processed thermally or nonthermally must comply with the safety regulations set forth by the US Food and Drug Administration prior to being marketed or consumed. For example, the regulations for thermally processed low-acid canned foods are contained in Title 21, Part 113 of the US Code of Federal Regulations, entitled “Thermally Processed Low-Acid Foods Packaged in Hermetically Sealed Containers.” The ­regulations in Title 21 were established to evaluate (1) the adequacy of the equipment and procedures to perform safe processing operations, (2) the adequacy of record keeping proving safe operation, (3) justification of the adequacy of process time and temperature used, and (d) the qualifications of ­supervisory staff responsible for thermal-processing and container closure operations (Teixeira 1992). However, the validation of DPCD as a nonthermal method and the determination of compliance regulations necessary for commercialization are complex and challenging. The progress in the validation needs to be encouraged to address the regulatory needs in the near future.

This volume attempts to bring together the accumulated knowledge in the area of DPCD. Experts in many areas have contributed to this book regarding the following topics:

It is sincerely hoped that the reader will find the book valuable in bringing information, research results, and most importantly an extensive ­bibliography in the nonthermal field of DPCD.

2  Thermodynamics of Solutions of CO2 with Effects of Pressure and Temperature

Sara Spilimbergo and Ireneo Kikic

Abstract: Knowledge of the thermodynamics of CO2 solutions under pressure is fundamental to the investigation of both the inactivation mechanism and the efficiency of the process as a function of the process parameters of temperature and pressure. In order to evaluate the properties of the solutions during DPCD treatment, it is essential to have reliable high-pressure experimental data and accurate thermodynamic models over a broad range of ­conditions. A large number of experimental and theoretical studies have been conducted on CO2 solubility in pure water; however, the literature lacks studies concerning the phase behavior of CO2–water solutions or CO2–solid components in food applications. In the present chapter. a general ­survey of the published knowledge concerning the thermodynamics of CO2–water phase systems under pressure is given focusing on both the theoretical aspects and the applications, for electrolyte and non-electrolyte ­models.

Keywords: thermodynamic model; liquid–vapor phase equilibria; equation of state.

2.1 Introduction

The phase behaviour of the system carbon dioxide (CO2) + water is of great industrial and scientific interest. For instance, in the petroleum industry many natural gases contain acidic gases which have to be removed, such as CO2 and hydrogen sulphide. Another industrial application is the decontamination of wastewater streams containing dissolved acidic gases. The capture of CO2, a greenhouse gas, has become a great concern: different storage techniques have been considered such as the storage of CO2 in deep saline reservoirs. Also, in geochemical applications, especially in the analysis of CO2-bearing fluid inclusions in minerals, the accurate description of solubility of CO2 in pure-water solutions is required. Recently, the increasing attention to CO2 processing in food applications, in particular in microbial inactivation of food products at low temperature compared to the traditional thermal treatments, makes the knowledge of a CO2–water phase system under pressure fundamental to investigate both the inactivation mechanism and the efficiency of the process as a function of the operating parameters of temperature and pressure.

Therefore, it is essential to have reliable high-pressure experimental data and an accurate thermodynamic model over a broad range of conditions to evaluate the solubility in different conditions. For most of these applications, temperatures (T) up to 100°C and pressures (P) up to 100 MPa are particularly relevant.

A large number of experimental and theoretical studies have been conducted on CO2 solubility in pure water. However, to the best of our knowledge, literature lacks studies concerning phase behaviour of CO2–liquid solutions or CO2–solid components in food applications. The objective of the present chapter is to give a general survey of the published knowledge concerning the thermodynamics of CO2–water phase systems under ­pressure, focusing on the theoretical aspects and the applications, for both electrolyte and non-electrolyte models.

2.2 Thermodynamics of liquid–vapour phase equilibria

According to the second law of thermodynamics, the total Gibbs energy of a closed system at constant temperature and pressure is minimum at equilibrium. If this condition is combined with the condition that the total number of moles of component i is constant in a closed system,

(2.1)  image

Where image is the number of moles of component i in phase α, it can be derived that for a system of Π phases and N components, the equilibrium conditions expressed in terms of chemical potential (μi) are as follows (Smith et al. 2005):


for i = 1 to N.

The chemical potential of component i in phase α is defined by

(2.3)  image

where g is the molar Gibbs energy. Since image is a function of P, T and (N-1) mole fractions (the additional condition image makes one of the mole ­fractions a dependent variable), represents N(Π - 1) equations in N + Π(N - 1) variables. Therefore the number of degrees of freedom F is


is the phase rule of Gibbs. According to this rule a state with Π phases in a system with N components is determined (all intensive thermodynamic properties can be calculated) if the values of F variables are fixed, provided that g values of all phases as functions of pressure, temperature and composition are known.

For practical applications (calculation of the composition of phases) the explicit dependence of chemical potentials on composition, temperature and pressure is needed. For this reason auxiliary functions such as the fugacity coefficient and the activity coefficient are used. These functions are closely related to the Gibbs energy and assume the ideal gas and the ideal solution behaviour in the reference state.

The fugacity of component i in a mixture image is defined by


at constant T (where the symbol ˆ indicates the value of the quantity in a mixture) with

(2.5b)  image

According to this definition, image is equal to the partial pressure Pi in the case of an ideal gas. The fugacity coefficient image is defined by

(2.6)  image

and is a measure for the deviation from ideal gas behaviour.

The fugacity coefficient (image can be calculated from an equation of state by one of the following expressions (Prausnitz et al. 1999):



According to , the equilibrium relation can be replaced by

(2.9)  image

for i = 1 to N.

This approach for the calculation of the fugacities in mixtures can be used for gaseous and condensed phases. This approach followed for all the phases at equilibrium is named the image method since the fugacities of both phases are calculated with or using an equation of state.

However, the calculation of the integral in needs an ­equation of state for the description of volumetric properties of the phase considered from low density (ideal gas) to the actual density of the phase. For this reason an alternative method for the calculation of the fugacities in condensed-phase mixtures is proposed. In this case, the reference is not the behaviour of an ideal gas mixture (that normally is far away from that of a solid or liquid mixture) but that of an ‘ideal mixture’. Excess functions are defined to describe the departure of the properties of a real mixture from the ideal behaviour.

It is very useful from a practical point of view to define as ‘ideal’ the behaviour of a mixture when it is possible to predict it from the knowledge of the properties of the pure components involved at the same temperature and pressure.

The activity ai is defined as the ratio of image and the fugacity of component i in the standard state at the same P and T:


In the ideal solution:

(2.11)  image

The activity coefficient of component i, γi, measures the deviation from ideal solution behaviour:

(2.12)  image

so the fugacity of a solid or liquid solution can be written as

(2.13)  image

The activity coefficient γi can be calculated from the molar excess Gibbs energy gE:

(2.14)  image

The standard-state fugacity of the liquid or solid component image is ­usually the fugacity of the pure solid or liquid component at the same temperature and pressure and is related to the sublimation pressure image or vapour pressure image, respectively.

On the sublimation curve of a pure component, we have


where superscripts s and V indicate a solid or vapour phase, respectively.


where image is the molar volume of pure solid i.

Combining and , we get


A similar derivation is possible for a liquid:

(2.18)  image

At low pressure the fugacity coefficients and the exponential terms are close to 1, so

(2.19)  image

From this assumption for the standard state fugacity, it follows that the activity coefficient of component i is equal to unity when the composition xi = 1 (i.e. for the pure component i).

This assumption is convenient if the component, at the temperature and pressure conditions of the mixture, exists in the same physical state. This is neither realistic nor practical when considering liquid mixtures if one of the components is supercritical. In this case the more convenient reference state is not that of pure component but that of the component infinitely diluted. The fugacity of the component i in an ideal mixture when xi → 0 is expressed by


where Hi is the Henry constant and is a function of temperature, pressure and the solvent–solute pair in the system. At high concentration there is no linear proportionality between image and the molar fraction . can also be used for high concentrations considering a new activity coefficient image:

(2.21)  image

This new coefficient is different from γi: when xi→ 0, image → 1 while when xi→ 1,γi → 1. It is possible to write


If the fugacities of the phases at equilibrium are calculated using the equation of state approach through for one phase and the activity coefficient approach ( for the other phase, the method is named the γ φ or activity–fugacity coefficient approach.

2.2.1 Calculation of γ

In typical mixtures, the ideal (or Raoult law) approximation provides no more than a rough approximation and it is valid only when the components are similar. The activity coefficient, therefore, plays a key role in the calculation of vapour–liquid equilibria.

Classical thermodynamics gives information on the effect on the activity coefficients of pressure (related to the partial molar volume) and of temperature (related to the partial molar enthalpy).

The Gibbs–Duhem equation, however, is a useful tool for correlating and extending limited experimental data due to the fact that, in a mixture, the activity coefficients of the individual components are not independent of one another but are related by a differential equation. For a binary mixture, the equation is


This equation means that in a binary mixture, activity coefficient data for one component can be used to predict the activity coefficient of the other component. Alternatively, with extensive experimental data for both activity coefficients as a function of composition, it is possible to test the data for thermodynamic consistency by determining whether or not the data obey . In the case of limited data, the integral form of the Gibbs–Duhem equation provides an equation to extend the information. To do so, it is necessary to assume a mathematical expression of the excess energy as a function of composition. The numerical values of the constants are calculated from the fitting of the limited data. Normally these constants, independent from composition, depend on temperature. Once the values of the constants are known, the activity coefficients can be calculated by differentiation.

Different expressions are proposed for the evaluation of the excess energy (Prausnitz et al. 1999): usually for moderately non-ideal systems, all equations containing two or more parameters give good results.

The older expressions are those proposed by Van Laar and by Margules (see the list following in this section). These expressions are ­mathematically easier to handle than the modern and new models (the Wilson, Nonrandom Two Liquids (NRTL) and Universal Quasi-Chemical (UNIQUAC) equations), but their use is often limited to the correlation of data for binary mixtures alone (for this reason, in the list for these models, the equations are reported for binary systems).

The Wilson equation, with two parameters, gives good results for strongly non-ideal binary mixtures and it is often used for vapour–liquid calculations: it is simpler than the UNIQUAC equation and contains only two parameters per binary mixtures, whereas the NRTL equation requires the knowledge of three parameters. Also the dilute region is represented with reasonable accuracy. The main deficiency of the Wilson equation is the impossibility to represent mixtures which exhibit a miscibility gap. For the systems that have incomplete miscibility and as a consequence are very non-ideal, the NRTL and UNIQUAC equations are very useful. The NRTL equation contains three parameters but very often the numerical value of the nonrandomness parameter αji is fixed (at the value 0.2 or 0.3). The UNIQUAC equation has the advantage that it uses only two parameters (with often a lesser dependence on temperature) and since the primary concentration variable is a surface fraction (rather than mole fraction), it is applicable to solutions containing small or large molecules.

Sometimes there is a total absence of experimental data to fit for the evaluation of the parameters used in the models for the activity coefficients. Then, a completely predictive model must be used. These models are based on the groups’ contributions concept. Each molecule in the mixture is considered as formed by functional groups. The behaviour of the mixture can be predicted by knowing the interaction between the functional groups in the mixture. The interaction between functional groups, called amn, is evaluated by studying the experimental data of known ­mixtures and it is assumed to be the same in every mixture in which the functional groups are considered. This allows predicting the equilibrium for systems with no experimental data. The UNIFAC (Fredenslund et al. 1977) and ASOG (Kojima and Tochigi 1979) are the most important models based on the functional groups method for mixtures. The most used is the UNIFAC model because it can be applied to a great number of compounds. The UNIFAC is the version based on the contribution of the functional groups of the UNIQUAC model, while the ASOG uses the Wilson model. To use the UNIFAC model it is necessary to identify the functional groups of the molecules in the system and evaluate the values of the parameters from its tables. The equations describing this model are very similar to that one of the UNIQUAC model:

Van Laar equation

(2.24a)  image

(2.24b)  image

Margules equation


(2.25b)  image

Wilson equation


(2.26b)  image


(2.26c)  image

NRTL equation


(2.27b)  image



(2.27d)  image

UNIQUAC equation

(2.28)  image

(2.29a)  image



(2.29c)  image

UNIFAC equation

(2.30a)  image

where φi (volume fraction) and υi (surface fraction) are expressed as in the UNIQUAC equation, but the values of ri and qi of the components are calculated by the addition of the corresponding contributions (Rk and Qk) of the constituent groups:

(2.30b)  image

and νk(i) is the number of the k functional group in the i species.

(2.30c)  image

(2.30d)  image

(2.30e)  image

(2.30f)  image

where anm is the energy interaction between the m and n functional groups. Γk(i) is the residual activity coefficient of the k functional group if the ­solution contains only i species. This means that for the i pure species, the residual term is equal to zero.

2.2.2 Calculation of ϕ

The equation of state (EOS) is an analytical relationship between pressure P, temperature T and molar volume V:

(2.31)  image

Starting from this equation, the calculation of volumetric and thermodynamic properties of a pure component or of a mixture is possible. Substituting this equation in or , the fugacity ­coefficients and as a consequence equilibrium between different phases are also ­calculated.

The equation of state models for the calculation of fugacities can be divided into classes on the basis of different criteria. One is based on the degree of the polynomial used in developing the equation of state in terms of volume. In this case it is possible to divide the EOS in cubic and ­non-cubic models.

An interesting observation, that can be useful for the classification, is that a common feature of the equations of state is that it is possible to ­recognize separate contributions resulting from repulsive and attractive interactions.

Following these criteria, the EOS can be separated into three families:

(1) Family of virial equation of state.
(2) Family of Van der Waals–type EOS where the contribution of repulsive and attractive forces is present.
(3) Molecular-based equation of state.

In the virial equation of state, the compressibility factor z is given as the power expansion of the density ρ:

(2.32)  image

where B and C are the second and third virial coefficients that are a ­function of temperature for a pure fluid or a function of temperature and composition for a mixture.

This theoretical equation was empirically modified by different authors often introducing a large number of constants. These modifications can be useful for the evaluation of pure-component properties, but their ­extension to mixtures is generally questionable.

For a mixture, the composition dependence is expressed as

(2.33)  image

(2.34)  image

where Bii is the second virial coefficient for the pure component i and the terms with different indices (ij) are calculated from the average of the second virial coefficients of component i and j. Sometimes, an empirical correction to this average, based on the introduction of a binary interaction parameter kij, is used. This empirical modification improves the performance of the equation of state but its value must be determined by fitting experimental data.

It is important to emphasize that virial equation of states cannot be applied for the calculation of vapour–liquid equilibrium using the φ-φ approach.

The van der Waals family EOS are derived by modification of the classical van der Waals equation that represents the first attempt to describe the coexistence of a liquid and vapour phase:

(2.35)  image

or, in term of the compressibility factor z,

(2.36)  image

In these equations the pressure (or the compressibility factor z) is given as a sum of two different contributions: the first, containing the co-volume b, represents the effect of repulsive forces, and the second, containing the ‘a’ parameter, takes into account the influence of attractive forces. The numerical values of the parameters can be calculated from the critical coordinates, from fixing the critical constraints or from vapour pressures and liquid or vapour densities. The modification of the Van der Waals equation by Redlich and Kwong (1949), who introduced a different temperature dependence and a slightly different volume dependency in the attractive term, is very important since it opened the way to a better description of the temperature-dependent properties like virial coefficients:

(2.37)  image

where a(T) = a/T½.

The Redlich-Kwong equation gives a somewhat better critical compressibility (Zc = 0.333 instead of 0.375 from the van der Waals equation), but is still not very accurate for the prediction of vapour pressures and liquid densities.

Soave’s modification (Soave 1972) of the temperature dependence of the a parameter, which resulted in accurate vapour pressure predictions (especially above 1 bar) for light hydrocarbons, led to cubic equations of state becoming important tools for the prediction of vapour–liquid equilibria at moderate and high pressures for nonpolar fluids.

(2.38)  image

where ω is the acentric factor defined as

(2.39)  image

Attractive terms used in cubic equations of state.

Equation Zatt
van der Waalsimage
Redlich-Kwong (1949)image
Soave (1972)image
Peng-Robinson (1976)image
Patel-Teja (1982)image
Trebble and Bishhnoi (1987)image

Peng and Robinson (1976) used a different volume dependency of the attractive term, which results in slightly improved liquid volumes and changed slightly the temperature dependence of a to give accurate vapour pressure predictions for hydrocarbons in the 6- to 10-carbon-number range.

(2.40)  image

The Peng–Robinson (PR) and the Soave–Redlich–Kwong (SRK) equations are widely used since they require little input (only critical properties and an acentric factor to calculate the generalized parameters) and require little computing power.

All these modifications and those proposed by different authors address the attractive part of the cubic equation of state. Some of these are reported in . Other expressions, based on some physical justifications, for the repulsive part, were also proposed, but no cubic equation of state can arise from the combination of some of these expressions.

The greatest use of cubic equation of state is for phase equilibrium calculations involving mixtures. This is most commonly done using the van der Waals one-fluid mixing rules,

(2.41)  image

(2.42)  image

In addition, combining rules are needed for the parameters aij and bij. The usual combining rules are



(2.44)  image

where kij and lij are the binary interaction parameters obtained by fitting experimental vapour–liquid equilibrium (VLE) data or VLE and density data. Generally, lij is set equal to zero, in which case we have the linear mixing rule for the b-parameter b = xi bii.

A shortcoming of the van der Waals classical mixing rules is that they are not applicable to the so-called asymmetric mixtures and to mixtures containing polar compounds. For that reason, different mixing rules have been proposed for the a-parameter, involving essentially a concentration dependence of the kij (Adachi and Sugie 1986; Panagiotopoulos and Reid 1986).

Since many mixtures of interest in the chemical industry exhibit much greater degrees of non-ideality, and have been traditionally described by activity coefficient (Gibbs energy) models, Huron and Vidal (1979) ­suggested a method to use excess Gibbs energy models to represent the mixing rule for the a-parameter of the equation of state.

The basic assumptions of the Huron–Vidal method are as follows:

The excess Gibbs energy GE calculated from a liquid-phase activity­   coefficient model, and the excess Gibbs energy GE calculated from the   equation of state, are equal at infinite pressure.

The co-volume b is equal to the molar volume V at infinite pressure.

The excess volume at infinite pressure is zero.

By using the linear mixing rule for the volume parameter b, the expression for the parameter a is


where Λ is a constant depending on the equation of state used (Λ is ln 2 for the Redlich–Kwong equation) and image is the value of the molar excess Gibbs energy at infinite pressure. For image it is possible to choose between the different models proposed in the literature (Prausnitz et al. 1999): the Wilson, NRTL or UNIQUAC equations. Huron and Vidal (1979) suggested using the NRTL model:

(2.46)  image

In this equation αij, τij, τij are adjustable parameters. When αij = 0, the van der Waals mixing rules are obtained.

This mixing rule, when combined with the Wilson or NRTL models, gives excellent results in describing the VLE of some highly non-ideal ­systems. However, the Huron–Vidal mixing rule has some theoretical and computational difficulties. The mixing rule may not be successful in describing nonpolar hydrocarbon mixtures, and this is a problem when a multicomponent mixture contains both polar and nonpolar components since all species must be represented by the same mixing rule. Furthermore, it is necessary to draw attention to the difficulties of this mixing rule in correlating low-pressure vapour–liquid equilibrium data.

Some efforts have been directed towards relaxing the infinite pressure limit in the Huron–Vidal model (Mollerup 1986; Michelsen 1990; Dahl and Michelsen 1990). The most successful of these is the so-called modified Huron–Vidal first order (MHV1) mixing rule, proposed by Michelsen (1990). In developing the new mixing rule, the Soave–Redlich–Kwong equation of state and the Huron–Vidal approach are used but with the equation of state and excess Gibbs energy models matched at liquid ­density and zero pressure at the temperature of interest:

(2.47)  image

for q1, a recommended value of −0.593 is suggested.

In addition, an alternative mixing rule (referred as the second-order modified Huron–Vidal mixing rule, or MHV2) was also derived (Dahl and Michelsen 1990):

(2.48)  image

with q1 = −0.478, and q2 = −0.0047

These new mixing rules have the advantage that they allow the use of numerical parameters for the excess-Gibbs-energy models which were obtained by fitting low-pressure vapour–liquid equilibrium data. In particular, the MHV2 mixing rule was used in combination with the UNIFAC group contribution model with excellent results (Dahl et al. 1992).

However, these new mixing rules (based on both infinite- and zero-pressure limits) give, for the composition dependence on the second virial coefficient, results that are inconsistent with those obtained from statistical mechanics.

Wong and Sandler (1992; Orbey and Sandler 1995) used the Helmholtz excess energy to develop the following mixing rule that satisfies the second virial restriction:

(2.49)  image

where Λ is a constant dependent on the equation of state selected ( is equal to ln 2 for the Redlich–Kwong equation), and kij is a binary inter­action parameter.

(2.50)  image

These mixing rules were applied to perform critical-point calculations and the critical behaviour of some highly non-ideal systems (Castier and Sandler 1997a, b).

Statistical mechanics and computer simulations have contributed to the development of new generations of equations of state, the so-called molecular-based EOS which, in contrast with those discussed so far, have a sounder theoretical basis.

The original van der Waals idea was that the pressure in a fluid is the result of both repulsive forces or excluded volume effects, which increase as the molar volume decreases, and attractive forces which reduce the pressure. These assumptions can be justified since, having the molecules have a finite size, there would be a limiting molar volume, b, which could be achieved only at infinite pressure. At large separations, on the basis of the London dispersion theory, the attractive forces increase as r6, where r is the intermolecular distance. Since volume is proportional to r3, this provides some explanations for the attractive term in the van der Waals equation of state. Nevertheless, modern statistical mechanics has shown that neither the repulsive nor the attractive term in the equation is correct.

The noncubic equations of state are characterized by the use of a repulsive term that is based on the Carnahan–Starling expression (Carnahan and Starling 1969, 1972) derived assuming that molecules behave as hard spheres:


where η = b/4V.

The attractive part is generally based on that derived from the perturbed hard chain theory (PHCT) (Beret and Prausnitz 1975), or from the statistical associating fluid theory (SAFT) (Chapman et al. 1988, 1990). These ­theories consider that most molecules like polymers or high-molecular-weight compounds do not have a spherical but a more complex structure that is more similar to chains. The thermodynamic properties of the molecules are also more complex than those derived on the basis of hard-body considerations: the dependence on rotational and vibrational motions of the molecules must be taken in consideration.

These approaches were the precursors of many theoretical attractive terms and consequently of different equations of state. Different authors reported applications of PHCT models to polymer systems (Donohue and Prausnitz 1978; Liu and Prausnitz 1979, 1980) and to supercritical systems (Gregorowicz et al. 1991; Fermeglia and Kikic 1993). The SAFT equation of state was successively applied to asymmetric (Huang and Radosz 1991) and water–hydrocarbon systems (Economou and Tsonopoulos 1997).

These equations of state can be extended for the description of the properties of mixtures by introducing combination rules for the pure-component parameters. The main advantage is the reduced number of binary parameters required (normally one) that are unrelated to temperature. It is also interesting to emphasize that, in these equations, the binary parameter is a measure of interactions between the segments of the different molecules and it can be used for the evaluation of the properties of mixtures made with components belonging to the same families (Fermeglia and Kikic 1993).

2.2.3 Calculation of the liquid–vapour phase equilibria

The equilibrium equations describe the conditions of the thermodynamic stability of the system. The use of these equations is important to design a great number of unit operations based on the concept of the equilibrium stage and on the tendency of a given chemical species to be in a phase rather than in other one at equilibrium conditions. The measure of this tendency, for liquid–vapour equilibrium, is defined by the equilibrium ratio yi/xi:

(2.52)  image

where Ki is called equilibrium constant or vapour–liquid distribution coefficient. This parameter is a measure of the volatility of a species or its ­tendency to be in the vapour phase, if Ki > 1 the “i” species.

The Duhem theory (Sandler 1989) states that the equilibrium stage of a closed system can be described by pressure and temperature which are uniform in all the equilibrium phases. In this way it is possible to evaluate the equilibrium phase compositions by knowing the global compositions z1, z2, z3,….., zm of the m components of the system. This kind of calculation, applied to solve a liquid – vapour equilibrium problem, is defined as flash calculation. The algorithm to solve the problem is shown in .