Aspen Polymers Vol2 V7 Tutorial

Aspen Polymers

User Guide Volume 2: Physical Property Methods & Models

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Version Number: V7.0 July 2008

Copyright (c) 2008 by Aspen Technology, Inc. All rights reserved.

Aspen Polymers™, Aspen Custom Modeler®, Aspen Dynamics®, Aspen Plus®, Aspen Properties®, aspenONE, the aspen leaf logo and Plantelligence and Enterprise Optimization are trademarks or registered trademarks of Aspen Technology, Inc., Burlington, MA.

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Contents iii

Contents

Introducing Aspen Polymers ...................................................................................1 About This Documentation Set ......................................................................... 1 Related Documentation................................................................................... 2 Technical Support .......................................................................................... 3

1 Thermodynamic Properties of Polymer Systems..................................................5 Properties of Interest in Process Simulation ....................................................... 5

Properties for Equilibria, Mass and Energy Balances................................... 6 Properties for Detailed Equipment Design ................................................ 6 Important Properties for Modeling........................................................... 6

Differences Between Polymers and Non-polymers ............................................... 7 Modeling Phase Equilibria in Polymer-Containing Mixtures .................................... 9

Vapor-Liquid Equilibria in Polymer Solutions ............................................. 9 Liquid-Liquid Equilibria in Polymer Solutions............................................11 Polymer Fractionation ..........................................................................12

Modeling Other Thermophysical Properties of Polymers.......................................12 Available Property Models...............................................................................13

Equation-of-State Models .....................................................................14 Liquid Activity Coefficient Models ...........................................................15 Other Thermophysical Models ...............................................................15

Available Property Methods.............................................................................16 Thermodynamic Data for Polymer Systems .......................................................19 Specifying Physical Properties .........................................................................19

Selecting Physical Property Methods.......................................................19 Creating Customized Physical Property Methods.......................................20 Entering Parameters for a Physical Property Model ...................................20 Entering a Physical Property Parameter Estimation Method........................21 Entering Molecular Structure for a Physical Property Estimation .................22 Entering Data for Physical Properties Parameter Optimization ....................23

References ...................................................................................................23

2 Equation-of-State Models ..................................................................................27 About Equation-of-State Models ......................................................................27 Phase Equilibria Calculated from EOS Models.....................................................29

Vapor-Liquid Equilibria in Polymer Systems.............................................30 Liquid-Liquid Equilibria in Polymer Systems.............................................30

Other Thermodynamic Properties Calculated from EOS Models.............................30 Physical Properties Related to EOS Models in Aspen Polymers..............................32 Sanchez-Lacombe EOS Model .........................................................................34

Pure Fluids .........................................................................................34 Fluid Mixtures Containing Homopolymers................................................36 Extension to Copolymer Systems...........................................................37


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Sanchez-Lacombe EOS Model Parameters ...............................................40 Specifying the Sanchez-Lacombe EOS Model ...........................................42

Polymer SRK EOS Model.................................................................................42 Soave-Redlich-Kwong EOS ...................................................................43 Polymer SRK EOS Model Parameters ......................................................45 Specifying the Polymer SRK EOS Model ..................................................47

SAFT EOS Model ...........................................................................................47 Pure Fluids .........................................................................................47 Extension to Fluid Mixtures ...................................................................52 Application of SAFT..............................................................................53 Extension to Copolymer Systems...........................................................55 SAFT EOS Model Parameters.................................................................57 Specifying the SAFT EOS Model .............................................................59

PC-SAFT EOS Model.......................................................................................59 Sample Calculation Results ...................................................................60 Application of PC-SAFT.........................................................................62 Extension to Copolymer Systems...........................................................63 PC-SAFT EOS Model Parameters ............................................................65 Specifying the PC-SAFT EOS Model ........................................................66

Copolymer PC-SAFT EOS Model.......................................................................67 Description of Copolymer PC-SAFT.........................................................67 Copolymer PC-SAFT EOS Model Parameters ............................................76 Option Codes for PC-SAFT ....................................................................78 Sample Calculation Results ...................................................................79 Specifying the Copolymer PC-SAFT EOS Model ........................................82

References ...................................................................................................83

3 Activity Coefficient Models ................................................................................87 About Activity Coefficient Models .....................................................................87 Phase Equilibria Calculated from Activity Coefficient Models.................................88

Vapor-Liquid Equilibria in Polymer Systems.............................................88 Liquid-Liquid Equilibria in Polymer Systems.............................................90

Other Thermodynamic Properties Calculated from Activity Coefficient Models.........90 Mixture Liquid Molar Volume Calculations .........................................................92 Related Physical Properties in Aspen Polymers...................................................93 Flory-Huggins Activity Coefficient Model ...........................................................94

Flory-Huggins Model Parameters ...........................................................97 Specifying the Flory-Huggins Model........................................................98

Polymer NRTL Activity Coefficient Model ...........................................................98 Polymer NRTL Model ............................................................................99 NRTL Model Parameters .....................................................................102 Specifying the Polymer NRTL Model .....................................................103

Electrolyte-Polymer NRTL Activity Coefficient Model .........................................103 Long-Range Interaction Contribution....................................................105 Local Interaction Contribution .............................................................107 Electrolyte-Polymer NRTL Model Parameters..........................................111 Specifying the Electrolyte-Polymer NRTL Model......................................114

Polymer UNIFAC Activity Coefficient Model......................................................114 Polymer UNIFAC Model Parameters ......................................................117 Specifying the Polymer UNIFAC Model ..................................................117

Polymer UNIFAC Free Volume Activity Coefficient Model....................................117 Polymer UNIFAC-FV Model Parameters .................................................119


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Contents v

Specifying the Polymer UNIFAC- FV Model ............................................119 References .................................................................................................119

4 Thermophysical Properties of Polymers ..........................................................121 About Thermophysical Properties...................................................................121 Aspen Ideal Gas Property Model ....................................................................123

Ideal Gas Enthalpy of Polymers ...........................................................124 Ideal Gas Gibbs Free Energy of Polymers ..............................................124 Aspen Ideal Gas Model Parameters ......................................................125

Van Krevelen Liquid Property Models..............................................................127 Liquid Enthalpy of Polymers ................................................................128 Liquid Gibbs Free Energy of Polymers...................................................130 Heat Capacity of Polymers ..................................................................131 Liquid Enthalpy and Gibbs Free Energy Model Parameters .......................131

Van Krevelen Liquid Molar Volume Model ........................................................136 Van Krevelen Liquid Molar Volume Model Parameters .............................137

Tait Liquid Molar Volume Model .....................................................................140 Tait Model Parameters .......................................................................141

Van Krevelen Glass Transition Temperature Correlation ....................................141 Glass Transition Correlation Parameters................................................142

Van Krevelen Melt Transition Temperature Correlation......................................142 Melt Transition Correlation Parameters .................................................143

Van Krevelen Solid Property Models ...............................................................143 Solid Enthalpy of Polymers .................................................................143 Solid Gibbs Free Energy of Polymers ....................................................144 Solid Enthalpy and Gibbs Free Energy Model Parameters........................144 Solid Molar Volume of Polymers...........................................................144 Solid Molar Volume Model Parameters ..................................................145

Van Krevelen Group Contribution Methods ......................................................145 Polymer Property Model Parameter Regression ................................................146 Polymer Enthalpy Calculation Routes with Activity Coefficient Models..................147 References .................................................................................................150

5 Polymer Viscosity Models ................................................................................151 About Polymer Viscosity Models.....................................................................151 Modified Mark-Houwink/van Krevelen Model....................................................152

Modified Mark-Houwink Model Parameters ............................................154 Specifying the MMH Model ..................................................................158

Aspen Polymer Mixture Viscosity Model ..........................................................158 Multicomponent System .....................................................................158 Aspen Polymer Mixture Viscosity Model Parameters ................................159 Specifying the Aspen Polymer Mixture Viscosity Model ............................161

Van Krevelen Polymer Solution Viscosity Model................................................161 Quasi-Binary System .........................................................................161 Properties of Pseudo-Components........................................................162 Van Krevelen Polymer Solution Viscosity Model Parameters .....................163 Polymer Solution Viscosity Estimation ..................................................164 Polymer Solution Glass Transition Temperature .....................................165 Polymer Viscosity at Mixture Glass Transition Temperature......................166 True Solvent Dilution Effect ................................................................167 Specifying the van Krevelen Polymer Solution Viscosity Model .................167

Eyring-NRTL Mixture Viscosity Model..............................................................167


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Multicomponent System .....................................................................168 Eyring-NRTL Mixture Viscosity Model Parameters ...................................169 Specifying the Eyring-NRTL Mixture Viscosity Model ...............................169

Polymer Viscosity Routes in Aspen Polymers ...................................................170 References .................................................................................................170

6 Polymer Thermal Conductivity Models.............................................................171 About Thermal Conductivity Models ...............................................................171 Modified van Krevelen Thermal Conductivity Model ..........................................173

Modified van Krevelen Thermal Conductivity Model Parameters ................174 Van Krevelen Group Contribution for Segments .....................................176 Specifying the Modified van Krevelen Thermal Conductivity Model ............179

Aspen Polymer Mixture Thermal Conductivity Model .........................................180 Specifying the Aspen Polymer Mixture Thermal Conductivity Model...........180

Polymer Thermal Conductivity Routes in Aspen Polymers ..................................181 References .................................................................................................181

A Physical Property Methods..............................................................................183 POLYFH: Flory-Huggins Property Method ........................................................183 POLYNRTL: Polymer Non-Random Two-Liquid Property Method ..........................185 POLYUF: Polymer UNIFAC Property Method .....................................................187 POLYUFV: Polymer UNIFAC Free Volume Property Method.................................189 PNRTL-IG: Polymer NRTL with Ideal Gas Law Property Method ..........................191 POLYSL: Sanchez-Lacombe Equation-of-State Property Method .........................193 POLYSRK: Polymer Soave-Redlich-Kwong Equation-of-State Property Method .....195 POLYSAFT: Statistical Associating Fluid Theory (SAFT) Equation-of-State Property Method......................................................................................................196 POLYPCSF: Perturbed-Chain Statistical Associating Fluid Theory (PC-SAFT) Equation-of-State Property Method .............................................................................198 PC-SAFT: Copolymer PC-SAFT Equation-of-State Property Method.....................200

B Van Krevelen Functional Groups .....................................................................202 Calculating Segment Properties From Functional Groups ...................................202

Heat Capacity (Liquid or Crystalline) ....................................................202 Molar Volume (Liquid, Crystalline, or Glassy).........................................203 Enthalpy, Entropy and Gibbs Energy of Formation ..................................203 Glass Transition Temperature..............................................................204 Melt Transition Temperature ...............................................................204 Viscosity Temperature Gradient...........................................................204 Rao Wave Function............................................................................204

Van Krevelen Functional Group Parameters.....................................................205 Bifunctional Hydrocarbon Groups.........................................................205 Bifunctional Oxygen-containing Groups.................................................208 Bifunctional Nitrogen-containing Groups ...............................................210 Bifunctional Nitrogen- and Oxygen-containing Groups.............................211 Bifunctional Sulfur-containing Groups...................................................212 Bifunctional Halogen-containing Groups................................................212


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C Tait Model Coefficients ....................................................................................215

D Mass Based Property Parameters....................................................................217

E Equation-of-State Parameters .........................................................................218 Sanchez-Lacombe Unary Parameters .............................................................218

POLYSL Polymer Parameters ...............................................................218 POLYSL Monomer and Solvent Polymers ...............................................219

SAFT Unary Parameters ...............................................................................220 POLYSAFT Parameters........................................................................220

F Input Language Reference ..............................................................................223 Specifying Physical Property Inputs................................................................223

Specifying Property Methods ...............................................................223 Specifying Property Data ....................................................................225 Estimating Property Parameters ..........................................................227

Index ..................................................................................................................228


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Introducing Aspen Polymers 1

Introducing Aspen Polymers

Aspen Polymers (formerly known as Aspen Polymers Plus) is a general-purpose process modeling system for the simulation of polymer manufacturing processes. The modeling system includes modules for the estimation of thermophysical properties, and for performing polymerization kinetic calculations and associated mass and energy balances.

Also included in Aspen Polymers are modules for:

• Characterizing polymer molecular structure

• Calculating rheological and mechanical properties

• Tracking these properties throughout a flowsheet

There are also many additional features that permit the simulation of the entire manufacturing processes.

About This Documentation Set The Aspen Polymers User Guide is divided into two volumes. Each volume documents features unique to Aspen Polymers. This User Guide assumes prior knowledge of basic Aspen Plus capabilities or user access to the Aspen Plus documentation set. If you are using Aspen Polymers with Aspen Dynamics, please refer to the Aspen Dynamics documentation set.

Volume 1 provides an introduction to the use of modeling for polymer processes and discusses specific Aspen Polymers capabilities. Topics include:

• Polymer manufacturing process overview - describes the basics of polymer process modeling and the steps involved in defining a model in Aspen Polymers.

• Polymer structural characterization - describes the methods used for characterizing components. Included are the methodologies for calculating distributions and features for tracking end-use properties.

• Polymerization reactions - describes the polymerization kinetic models, including: step-growth, free-radical, emulsion, Ziegler-Natta, ionic, and segment based. An overview of the various categories of polymerization kinetic schemes is given.

• Steady-state flowsheeting - provides an overview of capabilities used in constructing a polymer process flowsheet model. For example, the unit


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2 Introducing Aspen Polymers

operation models, data fitting tools, and analysis tools, such as sensitivity studies.

• Run-time environment - covers issues concerning the run-time environment including configuration and troubleshooting tips.

Volume 2 describes methodologies for tracking chemical component properties, physical properties, and phase equilibria. It covers the physical property methods and models available in Aspen Polymers. Topics include:

• Thermodynamic properties of polymer systems – describes polymer thermodynamic properties, their importance to process modeling, and available property methods and models.

• Equation-of-state (EOS) models – provides an overview of the properties calculated from EOS models and describes available models, including: Sanchez-Lacombe, polymer SRK, SAFT, and PC-SAFT.

• Activity coefficient models – provides an overview of the properties calculated from activity coefficient models and describes available models, including: Flory-Huggins, polymer NRTL, electrolyte-polymer NRTL, polymer UNIFAC.

• Thermophysical properties of polymers – provides and overview of the thermophysical properties exhibited by polymers and describes available models, including: Aspen ideal gas, Tait liquid molar volume, pure component liquid enthalpy, and Van Krevelen liquid and solid, melt and glass transition temperature correlations, and group contribution methods.

• Polymer viscosity models – describes polymer viscosity model implementation and available models, including: modified Mark-Houwink/van Krevelen, Aspen polymer mixture, and van Krevelen polymer solution.

• Polymer thermal conductivity models - describes thermal conductivity model implementation and available models, including: modified van Krevelen and Aspen polymer mixture.

Related Documentation A volume devoted to simulation and application examples for Aspen Polymers is provided as a complement to this User Guide. These examples are designed to give you an overall understanding of the steps involved in using Aspen Polymers to model specific systems. In addition to this document, a number of other documents are provided to help you learn and use Aspen Polymers, Aspen Plus, and Aspen Dynamics applications. The documentation set consists of the following:

Installation Guides

Aspen Engineering Suite Installation Guide

Aspen Polymers Guides

Aspen Polymers User Guide, Volume 1


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Introducing Aspen Polymers 3

Aspen Polymers User Guide, Volume 2 (Physical Property Methods & Models)

Aspen Polymers Examples & Applications Case Book

Aspen Plus Guides

Aspen Plus User Guide

Aspen Plus Getting Started Guides

Aspen Physical Property System Guides

Aspen Physical Property System Physical Property Methods and Models

Aspen Physical Property System Physical Property Data

Aspen Dynamics Guides

Aspen Dynamics Examples

Aspen Dynamics User Guide

Aspen Dynamics Reference Guide

Help

Aspen Polymers has a complete system of online help and context-sensitive prompts. The help system contains both context-sensitive help and reference information. For more information about using Aspen Polymers help, see the Aspen Plus User Guide.

Technical Support AspenTech customers with a valid license and software maintenance agreement can register to access the online AspenTech Support Center at:

http://support.aspentech.com

This Web support site allows you to:

• Access current product documentation

• Search for tech tips, solutions and frequently asked questions (FAQs)

• Search for and download application examples

• Search for and download service packs and product updates

• Submit and track technical issues

• Send suggestions

• Report product defects

• Review lists of known deficiencies and defects

Registered users can also subscribe to our Technical Support e-Bulletins. These e-Bulletins are used to alert users to important technical support information such as:

• Technical advisories


http://support.aspentech.com/

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4 Introducing Aspen Polymers

• Product updates and releases

Customer support is also available by phone, fax, and email. The most up-to-date contact information is available at the AspenTech Support Center at http://support.aspentech.com.


http://support.aspentech.com/

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1 Thermodynamic Properties of Polymer Systems 5

1 Thermodynamic Properties of Polymer Systems

This chapter discusses thermodynamic properties of polymer systems. It summarizes the importance of these properties in process modeling and outlines the differences between thermodynamic properties of polymers and those of small molecules.

Topics covered include:

• Properties of Interest in Process Simulation, 5

• Differences Between Polymers and Non-polymers, 7

• Modeling Phase Equilibria in Polymer-Containing Mixtures, 9

• Modeling Other Thermophysical Properties of Polymers, 12

• Available Property Models, 13

• Available Property Methods, 16

• Thermodynamic Data for Polymer Systems, 19

• Specifying Physical Properties, 19

Properties of Interest in Process Simulation Steady-state or dynamic process simulation is, in most instances, a form of performing simultaneous mass and energy balances. Rigorous modeling of mass and energy balances requires the calculation of phase and chemical equilibria and other thermophysical properties. In addition to the steps governed by equilibrium, there are rate-limited chemical reactions, and mass and heat transfer limited unit operations in a given process. Therefore, a fundamental understanding of the reaction kinetics and transport phenomena involved is a prerequisite for its modeling.

In process modeling, in addition to the properties needed for performing mass and energy balances and evaluating time dependent characteristics, detailed equipment design requires the calculation of additional thermophysical properties for equipment sizing. For detailed discussion of all these issues, please refer to references available in the literature (Bicerano, 1993; Bokis et


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6 1 Thermodynamic Properties of Polymer Systems

al., 1999; Chen & Mathias, 2002; Poling et al., 2001; Prausnitz et al., 1986; Reid et al., 1987; Sandler, 1988, 1994; Van Krevelen, 1990; Van Ness, 1964; Walas, 1985).

Properties for Equilibria, Mass and Energy Balances Often chemical and phase equilibria play the most fundamental role in mass and energy balance calculations. There are two ways of calculating chemical and phase equilibria. The classical route is to evaluate fugacities or activities of the components in the different phases, and find, at given conditions, the compositions that obey the equilibrium requirement of equality of fugacities for all components in all phases.

Fugacities or activities are quantities related to Gibbs free energy, and often it is more convenient to evaluate a fugacity coefficient or an activity coefficient rather than the fugacity and activity directly. Chapter 2 and Chapter 3 provide details on the calculation of these quantities.

Another method of calculating chemical and phase equilibria consists of searching for the minimum total of the mixture Gibbs free energies for the different phases involved. This is the Gibbs free energy minimization. This technique can be used to calculate simultaneous phase and chemical equilibria. Gibbs free energy minimization is discussed in Aspen Physical Property System Physical Property Methods and Models.

In performing energy balances, the interest is in changes in the energy content of a system, a section of a plant or a single unit, in a process. Depending upon the nature of the system, either an enthalpy H (usually for flow systems such as heat exchangers, flash towers in which pressure changes are modest) or an internal energy U (for systems such as closed batch reactors) balance is performed. These balances are often expressed as heat duty of a unit, yet the data on substances are usually measured as constant pressure heat capacity ( )pTH ∂∂ / , or as constant volume heat

capacity ( )VTU ∂∂ where T is the temperature, p is the pressure, and V is

the volume. Consequently, it is necessary to calculate temperature derivatives of enthalpy and internal energy.

Properties for Detailed Equipment Design Mixture density is required for equipment sizing. To calculate the efficiency of pumps and turbines, entropy is needed. Entropy is usually derived from enthalpy and Gibbs free energy. For detailed heat-exchanger design, viscosity and thermal conductivity of the mixture are needed. In detailed rating or design of column trays or packing, surface tension may be needed in addition to viscosity. Finally, diffusion coefficients are used to calculate mass transfer rates.

Important Properties for Modeling The most important properties for process simulation are:


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Thermodynamic properties Transport properties

Fugacity (or thermodynamic potential) Viscosity

Gibbs free energy Thermal conductivity

Internal energy, or CV Surface tension

Enthalpy, or CP Diffusivity

Entropy

Density

Differences Between Polymers and Non-polymers The word polymer derives from the Greek words poly ≡ many and meros ≡ part. A polymer consists of a large number of segments (repeating units of identical structure). Because of their structure, polymers exhibit thermodynamic properties significantly different than those of standard molecules (solvents, monomers, other additive solutes), consequently different property models are required to describe their behavior. For example, polymers being orders of magnitude larger molecules, have substantially more spatial conformations than the small molecules. This affects equilibrium properties such as the entropy of mixing, as well as non-equilibrium properties like viscosity. Unlike conventional molecules, polar interactions (between dipoles, quadrapoles etc., also called London-van-der-Waals or dispersion forces) among the segments of a single molecule play a role in thermodynamic behavior of polymers and their mixtures. Moreover, when polymer molecules interact with conventional small molecules, due to their large size, only a fraction of segments of the polymer molecule may be involved rather than the whole molecule. All these segment-segment and segment-conventional molecule interactions are influenced by the spatial conformations mentioned above.

Besides the different spatial conformations a single polymer molecule can have, they also exhibit chain length distributions, isomerism for each chain length due to distributions of branching and co-monomer composition, and stereo chemical configuration of segments in a chain.

Detailed discussion of these issues is beyond the scope of this document. However, excellent sources are available in the literature (Bicerano 1993; Brandup & Immergut, 1989; Cotterman & Prausnitz, 1991; Folie & Radosz, 1995; Fried, 1995; Ko et al., 1991; Kroschwitz, 1990; Sanchez, 1992; Van Krevelen, 1990). A simplified overview is presented here from a modeling point of view.

Polymer Polydispersity

When modeling polymer phase equilibrium, one must take into account the basic polymer characteristics briefly mentioned above. First, no polymer is ‘pure’. Rather, a polymer is a mixture of components with differing chain


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length, chain composition, and degree of branching. In other words, polymers are polydisperse. For the purposes of property calculations, this makes a polymer a mixture of an almost infinite number of components. In the calculation of phase equilibria of polymer solutions, some physical properties of the solution, such as vapor pressure depression, can be related to average polymer structure properties. On the other hand, physical properties of the polymer itself, for example distribution of the polymer over different phases or fractionation, cannot be related to the average polymer structure properties. It is also impossible to take each individual component into account; therefore, compromise approximations are made to incorporate information about polydispersity in polymer process modeling (Behme et al., 2003).

Long-chain polymers have very low vapor pressures and are considered nonvolatile. Short-chain polymers may be volatile, and these species can be treated as oligomers as discussed later in this section. The nonvolatile nature of polymers must be taken into account in developing models to describe polymer phase behavior, or when a model developed for conventional molecules is extended for use with polymers. Polymers cannot exhibit a critical point either, since they decompose before they reach their critical temperatures.

In the pure condensed phase, polymers can be a liquid-like melt, amorphous solid, or a semi-crystalline solid. Due to their possible semi-crystalline nature in the solid state, polymeric materials may exhibit two major types of transition temperatures from solid to liquid. A completely amorphous solid is characterized by glass transition temperature, Tg , at which it turns into melt

from amorphous solid.

A semi-crystalline polymer is not completely crystalline, but still contains unordered amorphous regions in its structure. Such a polymer, upon heating, exhibits both a Tg and a melting temperature, Tm , at which phase transition

of crystalline portion of the polymer to melt occurs. Thus, a semi-crystalline polymer may be treated as a glassy solid at temperatures below Tg , a

rubbery solid between Tg and Tm , and a melt above Tm .

The knowledge of state of aggregation of polymer in the condensed phase is important because all thermophysical characteristics change from one condensed state to another. For example, monomers and solvents are soluble in melt and in amorphous solid polymer, but crystalline areas are inert and do not participate in phase equilibrium. Other thermodynamic properties such as heat capacity, density, etc. are also significantly different in each phase.

Another very important characteristic of the polymers is their viscoelastic nature, which affects their transport properties enormously. The models to characterize viscosity of polymers or diffusion of other molecules in polymers must, therefore, be unique.

Oligomers

In process modeling, we also deal with oligomers. An oligomer is a substance that contains only a few monomeric segments in its structure, and its thermophysical properties are somewhere between a conventional molecule and a polymer. They can be considered like a heavy hydrocarbon molecule,


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and they act like one. In most cases they can be simulated as a heavy conventional molecule. Aspen Polymers (formerly known as Aspen Polymers Plus) permits a substance to be defined as oligomer, apart from standard molecules and polymers.

Modeling Phase Equilibria in Polymer-Containing Mixtures In modeling phase equilibrium of polymer mixtures, there are two broad categories of problems that are particularly important. The first is the solubility of monomers, other conventional molecules used as additives, and solvents in a condensed phase containing polymers. The second is the phase equilibrium when two polymer-containing condensed phases are in coexistence.

Vapor-Liquid Equilibria in Polymer Solutions A good example of the first case is the devolatilization of monomers, solvents and other conventional additives from a polymer. The issue here is to determine the extent of solubility of conventional molecules in the polymer at a given temperature and pressure. The polymer may be a melt, an amorphous solid, or a semi-crystalline solid.

An amorphous polymer is treated as a pseudo-liquid. If the polymer is semi-crystalline, then one would compute overall solubility based on the solubility in the amorphous polymer and the fraction of amorphous polymer in the total polymer phase.

This problem is somewhat similar to a vapor-liquid equilibrium (VLE) of conventional systems. The thermodynamic model selected can be tested by investigating pressure-composition phase diagrams of polymer-solvent pairs at constant temperature. For example:


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PIB-N-Pentane Binary System (Data from compilation of Wohlfarth, 1994)

Usually a flash algorithm is used to model the devolatilization process. Proven vapor-liquid equilibrium flash algorithms have been widely used for polymer systems. In these flash algorithms calculations can be done with a number of options such as specified temperature and pressure, temperature and vapor fraction (dew point or bubble point), pressure and vapor fraction, pressure and heat duty, and vapor fraction and heat duty. It is important to stress that in such calculations polymers are considered nonvolatile while solvents, monomers and oligomers are distributed between vapor and liquid phases.

Another example in this category is modeling of a polymerization reaction carried out in a liquid solvent with monomer coming from the gas phase. It is important to know the solubility of the monomer gas in the reaction solution, as this quantity directly controls the polymerization reaction kinetics in the liquid phase. In such a case, the mixture may contain molecules of a conventional solvent, dissolved monomer, other additive molecules, and the polymer either as dissolved in solution or as a separate particle phase swollen with solvent, monomer and additive molecules. Interactions of various conventional molecules in the solution with the co-existing polymer molecules have direct effect on the solubility of the monomer gas in the solution. Again, the phase equilibrium problem can be considered as a VLE (polymer dissolved in solution) or as a vapor-liquid-liquid equilibrium (VLLE; polymer in a separate phase swollen with conventional molecules).


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Liquid-Liquid Equilibria in Polymer Solutions Liquid-liquid phase equilibrium (LLE)between two polymer containing phases is also important in modeling polymer processes. The overall thermodynamic behavior of two co-existing liquid phases is shown here:

LCST-UCST Behavior of Polymer Mixtures (Folie & Radosz, 1995)

In the figure, the space under the saddle is the region where liquid-liquid phase split occurs. Above that region, only a single homogeneous fluid phase exists. Various two-dimensional temperature-composition projections are also shown in the figure. In these projections, several phase behavior types common in polymer-solvent systems are indicated. For example, at certain pressures, polymer-solvent mixtures exhibit two distinctly different regions of immiscibility.

These regions are characterized by the upper critical solution temperature (UCST) and the lower critical solution temperature (LCST). UCST characterizes the temperature below which a homogeneous liquid mixture splits into two distinct phases of different composition. This phase behavior is rather common, and it is observed in many kinds of mixtures of conventional molecules and polymers. LCST represents the temperature above which a formerly homogeneous liquid mixture splits into two separate liquid phases. This thermally induced phase separation phenomenon is observed in mixtures of conventional molecules only when strong polar interactions exist (such as aqueous solutions). However, for polymer-solvent mixtures the existence of a LCST is the rule, not the exception (Sanchez, 1992).

In polymerization processes, especially those carried out at high pressures in the gas phase, such as LDPE production, it is important to estimate the boundaries of these regions of immiscibility. It is directly pertinent to modeling of reaction kinetics whether the reactive mixture remains a homogeneous fluid phase or splits into two liquid phases.


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Polymer Fractionation Another process where LLE behavior plays a role is polymer fractionation. A classical method of fractionating a polydisperse polymer is to dissolve the polymer completely in a 'good' solvent and then progressively add small amounts of a poor solvent (or antisolvent). Upon addition of the antisolvent, a second phase, primarily consisting of lowest-molecular weight polymers, will form. The system can be modeled as an LLE system.

Existing liquid-liquid equilibrium and vapor-liquid-liquid equilibrium flash algorithms cannot be applied to solve these LLE systems with nonvolatile polymers, unless the polymers are treated as oligomers with 'some' volatility.

These flash algorithms are based on solving a set of nonlinear algebraic equations derived from the isofugacity relationship for each individual component. Such an isofugacity relationship cannot be mathematically established for nonvolatile polymer components. In such cases, using the Gibbs free energy minimization technique usually offers a more robust way of estimating the number of existing phases and their compositions.

Modeling Other Thermophysical Properties of Polymers Correlations for other important thermophysical properties of pure polymers such as heat capacity, density, and viscosity are essentially empirical in nature. Van Krevelen developed an excellent group contribution methodology to predict a wide variety of thermophysical properties for polymers, using polymer molecular structure, in terms of functional groups, and polymer compositions (Van Krevelen, 1990). These relations are basically applicable to random linear copolymers.

Group contribution techniques cannot be applied to polymers containing exotic structural units, if no experimental data is available for estimating contributions for functional groups not studied previously. To overcome these limitations, Bicerano developed a new generation of empirical quantitative structure-property relationships in terms of topological variables (Bicerano, 1993).

Correlations for predicting thermophysical properties of polymer mixtures are not well established. Typically, pure component properties are first estimated for polymers, monomers, and solvents by various techniques. Properties of polymer solutions are then calculated with mass fraction or segment-based molar fraction mixing rules. This methodology seems to work well for calorimetric properties and volumetric properties.

On the other hand, different empirical mixing rules are needed for transport properties. This is because polymers are viscoelastic, while conventional components exhibit Newtonian behavior, which poses a challenge in developing mixing rules for viscosity of polymer-solvent mixtures.


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Available Property Models Aspen Polymers contains several key property models specifically developed for polymer systems. These models consist of two classes:

• Solution thermodynamic models for polymer phase equilibrium calculations (activity coefficient models and equations of state)

• Models for other thermophysical properties (molar volume, enthalpy and heat capacity, entropy, Gibbs free energy, and transport properties)

These models, which are described individually in later chapters, have been incorporated into several physical property methods. A summary of the available thermodynamic and transport property models is provided here:

Model Description

Enthalpy, Gibbs free energy, heat capacity, and density models

Van Krevelen Models Calculates thermophysical properties of polymers using group contribution

Tait Model Calculates molar volume of polymers

Aspen Ideal Gas Property Model

Extends the ideal gas model to calculate the ideal gas properties of polymers. It is used together with equations of state to calculate thermodynamic properties of polymer systems

Transport property models

Modified Mark-Houwink/Van Krevelen Model

Calculates viscosity of polymers

Aspen Polymer Mixture Viscosity Model

Correlates liquid viscosity of polymer solutions and mixtures from pure component liquid viscosity and mass fraction mixing rules

Van Krevelen Polymer Solution Viscosity Model

Calculates liquid viscosity of polymer solutions

Eyring-NRTL Mixture Viscosity Model

Correlates liquid viscosity of polymer solutions and mixtures from pure component liquid viscosity and NRTL term to capture non-ideal mixing behavior

Modified van Krevelen Thermal Conductivity Model

Calculates thermal conductivity of polymers

Aspen Polymer Mixture Thermal Conductivity Model

Uses the modified van Krevelen thermal conductivity model with existing Aspen Plus thermal conductivity models to calculate thermal conductivity of mixture containing polymers

Activity coefficient models

Polymer NRTL Model Extends the non-random two liquid theory to polymer systems. It accounts for interactions with polymer segments and is well suited for copolymers

Electrolyte-Polymer NRTL Model

Integrates the electrolyte NRTL model and the polymer NRTL model. It computes activity coefficients for polymers, solvents, and ionic species

Flory-Huggins Model Represents non-ideality of polymer systems. Based on the well-known model developed by Flory and Huggins

Polymer UNIFAC and Polymer UNIFAC-FV Models

Extends the UNIFAC group contribution method to polymer systems taking into account polymer segments. They are predictive models

Equations of State


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Model Description

Sanchez-Lacombe Tailors the well-known equation of state model, based on the lattice theory, to polymer mixtures

Polymer SRK Extends the SRK equation of state to cover polymer mixtures

SAFT Provides a rigorous thermodynamic model for polymer systems based on the perturbation theory of fluids

PC-SAFT Provides an improved SAFT model based on perturbation theory

Copolymer PC-SAFT A complete PC-SAFT model applicable to complex fluids, including normal fluids, water and alcohols, polymers and copolymers, and their mixtures.

Phase equilibrium calculations are the most important aspect of thermodynamics. The basic relationship for every component in the vapor and liquid phases of a mixture at equilibrium is:

li

vi ff = (1.1)

Where:

vif = Fugacity of component i in the vapor phase

lif = Fugacity of component i in the liquid phase

Similarly, the liquid-liquid equilibrium condition is:

21 li

li ff = (1.2)

Where:

1lif = Fugacity of component i in the liquid phase 1

2lif = Fugacity of component i in the liquid phase 2

Applied thermodynamics provides two methods for representing the fugacities from the phase equilibrium relationship: equation-of-state models and liquid activity coefficient models.

Equation-of-State Models In modeling polymer systems at high pressures, the activity coefficient models suffer from certain shortcomings. For example, most of them are applicable only to incompressible liquid solutions, and they fail to predict the LCST type phase behavior that necessitates pressure dependence in a model (Sanchez, 1992). To overcome these difficulties an equation of state (EOS) is needed. Another advantage of using an equation of state is the simultaneous calculation of enthalpies and phase densities along with phase equilibrium from the same model.

The literature describes many polymer-specific equations-of-state. Currently, the most widely used EOS for polyolefin systems are the Sanchez-Lacombe EOS (Sanchez & Lacombe, 1976, 1978), Statistical Associating Fluid Theory EOS (SAFT) (Chapman et al., 1989; Folie & Radosz, 1995; Huang & Radosz,


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1990, 1991; Xiong & Kiran, 1995), and Perturbed-Chain Statistical Associating Fluid Theory EOS (PC-SAFT) (Gross & Sadowski, 2001, 2002). In addition, well-known cubic equations-of-state for systems with small molecules are being extended for polymer solutions (Kontogeorgis et al., 1994; Orbey et al., 1998a, 1998b; Saraiva et al., 1996). Presently, Aspen Polymers offers Sanchez-Lacombe EOS, an extension of the Soave-Redlich-Kwong (SRK) cubic equation of state to polymer-solvent mixtures (Polymer SRK EOS), the SAFT EOS, and the PC-SAFT EOS.

The Sanchez-Lacombe, SAFT, and PC-SAFT equations of state are polymer specific, whereas the polymer SRK model is an extension of a conventional cubic EOS to polymers. Polymer specific equations of state have the advantage of describing polymer components of the mixture more accurately. The Sanchez-Lacombe, SAFT, and PC-SAFT equations of state are implemented in Aspen Polymers for modeling systems containing homopolymers as well as copolymers. The details of the individual EOS models are given in Chapter 2.

Liquid Activity Coefficient Models In general, the activity coefficient models are versatile and accommodate a high degree of solution nonideality into the model. On the other hand, when applied to VLE calculations, they can only be used for the liquid phase and another model (usually an equation of state) is needed for the vapor phase. They are used for the calculation of fugacity, enthalpy, entropy and Gibbs free energy but are rather cumbersome for evaluation of calorimetric and volumetric properties. Usually other empirical correlations are used in parallel for the calculations of densities when an activity coefficient model is used in phase equilibrium modeling.

Many activity coefficient models can be used in polymer process modeling. Aspen Polymers offers the Flory-Huggins model (Flory, 1953), the Non-Random Two-Liquid Activity Coefficient model adopted to polymers (Chen, 1993), the Polymer UNIFAC model, and the UNIFAC free volume model (Oishi & Prausnitz, 1978). The two UNIFAC models are predictive while the Flory-Huggins and Polymer-NRTL model are correlative. Between the correlative models, the Flory-Huggins model is only applicable to homopolymers because its parameter is polymer-specific. The Polymer-NRTL model is a segment-based model that allows accurate representation of the effects of copolymer composition and polymer chain length. The details of the individual activity coefficient models are given in Chapter 3.

Other Thermophysical Models Aspen Polymers offers models for the calculations of enthalpy, Gibbs free energy, entropy, molar volume (density), viscosity, and thermal conductivity of pure polymers. It also extends the existing Aspen Ideal Gas Property Model to cover polymers, oligomers, and segments.

Van Krevelen (1990) physical property models are used to evaluate enthalpy, Gibbs free energy, and molar volume in both liquid and solid states, glass transition and melting point temperatures. For molar volume, another alternative is the Tait model (Danner & High, 1992).


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Aspen Polymers offers methods for estimation of zero-shear viscosity of polymer melts, for concentrated polymer solutions, and also for polymer solutions and mixtures over the entire range of composition. Melt viscosity is calculated using the modified Mark-Houwink/Van Krevelen model (Van Krevelen, 1990). Concentrated polymer solution viscosity is calculated using the van Krevelen polymer solution viscosity model. Liquid viscosity of polymer solutions and mixtures is correlated using the Aspen polymer viscosity mixture model (Song et al., 2003).

Aspen Polymers offers a modified van Krevelen model to calculate thermal conductivity of polymers. Liquid thermal conductivity of polymer solutions and mixtures is calculated using the modified van Krevelen model for polymers with existing Aspen Plus models for non-polymer components.

When an equation of state is used for calculation of enthalpy, entropy and Gibbs free energy, it provides only departure values from ideal gas behavior (departure functions). Therefore, in estimating these properties from an equation of state, the ideal gas contribution must be added to the departure functions obtained from the equation of state model. For this purpose, the ideal gas model already available in Aspen Plus for monomers and solvents was extended to polymers and oligomers and made available in Aspen Polymers.

Available Property Methods Following the Aspen Physical Property System, the methods and models used to calculate thermodynamic and transport properties in Aspen Polymers are packaged in property methods. Each property method contains all the methods and models needed for a calculation. A unique combination of methods and models for calculating a property is called a route. For details on the Aspen Physical Property System, see the Aspen Physical Property System Physical Property Methods and Models documentation.

You can select a property method from existing property methods in Aspen Polymers or create a custom-made property method by modifying an existing property method. The property methods already available in Aspen Polymers are listed here (Appendix A lists the entire physical property route structure for all polymer specific property methods):

Property method

Description

POLYFH Uses the Flory-Huggins model for solution thermodynamic property calculations and van Krevelen models for polymer thermophysical property calculations. The Soave-Redlich-Kwong equation-of-state is used to calculate vapor-phase properties of mixtures.

POLYNRTL Uses the polymer NRTL model for solution thermodynamic property calculations and van Krevelen models for polymer thermophysical property calculations. The Soave-Redlich-Kwong equation-of-state is used to calculate vapor-phase properties of mixtures.


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Property method

Description

POLYUF Uses the polymer UNIFAC model for solution thermodynamic property calculations and van Krevelen models for polymer thermophysical property calculations. The Soave-Redlich-Kwong equation-of-state is used to calculate vapor-phase properties of mixtures.

POLYUFV Uses the polymer UNIFAC model with a free volume correction for solution thermodynamic property calculations and van Krevelen models for polymer thermophysical property calculations. The Soave-Redlich-Kwong equation-of-state is used to calculate vapor-phase properties of mixtures.

PNRTL-IG Uses the ideal-gas equation-of-state to calculate vapor-phase properties of mixtures. This is a modified version of the standard POLYNRTL property method.

POLYSL Uses the Sanchez-Lacombe equation of state model for thermodynamic property calculations.

POLYSRK Uses an extension of the Soave-Redlich-Kwong equation of state to polymer systems, with the MHV1 mixing rules and the polymer NRTL excess Gibbs free energy model, for thermodynamic property calculations.

POLYSAFT Uses the statistical associating fluid theory (SAFT) equation of state for thermodynamic property calculations.

POLYPCSF Uses the perturbed-chain statistical associating fluid theory (PC-SAFT) equation of state for thermodynamic property calculations.

PC-SAFT Uses the perturbed-chain statistical associating fluid theory (PC-SAFT) equation of state for thermodynamic property calculations. The association term is included and no mixing rules are used for copolymers.

The following table describes the overall structure of the property methods in terms of the properties calculated for the vapor and liquid phases. Additionally, the models used for the property calculations are given.

Properties Calculated

Model (Property method)

Used For

Vapor

Departure functions, fugacity coefficient, molar volume

Soave-Redlich-Kwong

(All activity coefficient property methods)

All vapor properties: Fugacity coefficient, enthalpy, entropy, Gibbs free energy, density

Sanchez-Lacombe (POLYSL) All vapor properties: Fugacity coefficient, enthalpy, entropy, Gibbs free energy, density

Polymer SRK (POLYSRK) All vapor properties: Fugacity coefficient, enthalpy, entropy, Gibbs free energy, density

SAFT (POLYSAFT)



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Used For

PC-SAFT (POLYPCSF)

All vapor properties: fugacity coefficient, enthalpy, entropy, Gibbs free energy, density

Copolymer PC-SAFT (PC-SAFT)


Liquid

Vapor pressure PLXANT

Antoine


Activity Coefficient

Flory-Huggins (POLYFH) Fugacity, Gibbs free energy, enthalpy, entropy

Polymer NRTL (POLYNRTL) Fugacity, Gibbs free energy, enthalpy, entropy

Polymer UNIFAC (POLYUF) Fugacity, Gibbs free energy, enthalpy, entropy

UNIFAC free volume (POLYUFV)

Fugacity, Gibbs free energy, enthalpy, entropy

Vaporization enthalpy

Watson for monomers, Van Krevelen for polymers and oligomers from segments


Enthalpy, entropy

Molar Volume Rackett for monomers, Van Krevelen for polymers and oligomers from segments

Tait molar model for polymers and oligomers


Density

Departure functions, fugacity coefficient, molar volume

Sanchez-Lacombe (POLYSL) All liquid properties: Fugacity coefficient, enthalpy, entropy, Gibbs free energy, density

Polymer SRK (POLYSRK) All liquid properties: Fugacity coefficient, enthalpy, entropy, Gibbs free energy, density

SAFT (POLYSAFT) All liquid properties: Fugacity coefficient, enthalpy, entropy, Gibbs free energy, density

PC-SAFT (POLYPCSF) All liquid properties: fugacity coefficient, enthalpy, entropy, Gibbs free energy, density

Copolymer PC-SAFT (PC-SAFT)

All liquid properties: fugacity coefficient, enthalpy, entropy, Gibbs free energy, density

Viscosity Aspen Polymer Mixture Viscosity Model

Liquid viscosity of polymer solutions and mixture


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Used For

Thermal Conductivity

Aspen Polymer Mixture Thermal Conductivity Model

Liquid thermal conductivity of polymer solutions and mixtures

Thermodynamic Data for Polymer Systems The data published in the literature for pure polymers and for polymer solutions is very limited in comparison to the enormous amount of vapor-liquid equilibrium data available for mixtures of small molecules (Wohlfarth, 1994). The AIChE-DIPPR handbooks of polymer solution thermodynamics (Danner & High, 1992) and diffusion and Thermal Properties of Polymers and Polymer Solutions (Caruthers et al., 1998) provide computer databases for pure polymer pressure-volume-temperature data, finite concentration VLE data, infinite dilution VLE data, binary liquid-liquid equilibria data, and ternary liquid-liquid equilibria data. The DECHEMA polymer solution data collection contains data for VLE, solvent activity coefficients at infinite dilution, and liquid-liquid equilibrium (Hao et al., 1992).

Another data source for polymer properties is the compilation of Wohlfarth (1994). Wohlfarth compiled VLE data for polymer systems in three groups: vapor pressures of binary polymer solutions (or solvent activities), segment-based excess Gibbs free energies of binary polymer solutions, and weight fraction Henry-constants for gases and vapors in molten polymers.

In another useful source, Barton (1990) presented a comprehensive compilation of cohesion parameters for polymers as well as polymer-liquid Flory-Huggins interaction parameter χ.

Finally, Polymer Handbook (Brandup & Immergut, 1989; Brandup et al., 1999) brought together data and correlations for many properties of polymers and polymer solutions.

Specifying Physical Properties Following is an explanation of common procedures for working with physical properties in Aspen Polymers.

Selecting Physical Property Methods For an Aspen Polymers simulation, you must specify the physical property method(s) to be used. Aspen Polymers provides many built-in property methods. You can either select one of these built-in property methods, or customize your own property method. Additionally, you can choose a property method for the entire flowsheet, part of a flowsheet, or a unit.


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To select a built-in property method for the entire flowsheet:

From the Data Browser, double-click Properties.

From the Properties folder, click Specifications.

On the Specifications sheet, specify Process type and Base method.

You can also specify property methods for flowsheet sections.

Once you have chosen a built-in property method, the property routes and models used are resolved for you. You can use any number of property methods in a simulation.

Creating Customized Physical Property Methods Occasionally, you may prefer to construct new property methods customized for your own modeling needs.

To create customized property methods:

From the Data Browser, click Properties.

From the Properties folder, click Property Methods.

An object manager appears.

Click New.

In the Create new ID dialog box, enter property method ID and click OK.

Now you are ready to customize Routes and/or Models used in the property method you created. In general, to create a custom-made property method you select a base method and modify it.

To customize routes:

On the, Routes sheet, select a base method to be modified for customization.

A Property versus Route ID table is automatically filled in depending on your choice.

Click the Route ID that you want to change. From the list, select the new route ID.

The new route ID is highlighted.

To customize the models:

Click the Models tab.

In the Models form, from the Property versus Model name table, click the model name to be replaced and select the new model name from the list.

The new model name is highlighted.

Entering Parameters for a Physical Property Model Frequently you need to enter pure model parameters for a pure-component or mixture physical property model.

To enter pure model parameters:



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Several subfolders appear.

Click Parameters.

The following folders appear:

o Pure Component

o Binary Interaction

o Electrolyte Pair

o Electrolyte Ternary

o UNIFAC Group

o UNIFAC Group Binary

o Results

Following is a description of pure component parameter entry. Other parameter entries are completed in a similar manner.

To enter component parameters:

Click Pure Component.


Click New.

A New Pure Component Parameters form appears.

Use the New Pure Component Parameters form to select the type of the pure component parameter. The selections are:

• Scalar (default)

• T-dependent correlation

• Nonconventional

To prepare a New Pure Component Parameters form:

Select the type of the parameter (for example, click Scalar). On the same component parameter form, click the name box and either enter

a name, or accept the default, and click OK.

The parameter form is ready for parameter entry.

To enter a parameter:

Click the Parameters box, and click the name of the parameter.

Click the Units box.

Enter the proper unit for the parameter.

Click the Component column.

Enter the parameter value.

Click Next to proceed.

Entering a Physical Property Parameter Estimation Method If a parameter value for a physical property model is missing, you can request property parameter estimation.

To use parameter estimation:



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Click Estimation.

A Setup sheet appears.

There are three estimation options available in the Setup sheet:

• Do not estimate any parameters (default)

• Estimate all missing parameters

• Estimate only the selected parameters

o Pure component scalar parameters

o Pure component temperature-dependent property correlation parameters

o Binary interaction parameters

o UNIFAC group parameters

In the default option, no parameters are estimated during the simulation. If you select the second option, all missing parameters are estimated according to a preset hierarchy of the Aspen Plus simulator. If you select either of these first two options, the task is complete and you can continue by clicking Next

.

If you select the option to estimate only selected parameters, you must complete additional steps:

In the object manager, click Estimate only the selected parameters option.

All parameter types are selected automatically.

Clear all parameter types that you do not want estimated.

Click the parameter tab in the object manager for the parameters you want to estimate.

Fill in the parameter form by selecting the names of components, parameters, and estimation methods etc. from the lists.


Entering Molecular Structure for a Physical Property Estimation If a particular component is not in the component databank, or its structure is to be defined for a particular physical property estimation method, then you need to supply the molecular structure information. There are several ways to provide this information:



Click Molecular Structure.


All of the components selected for the current simulation are listed in the object manager. Click the name of the component structure you want to enter. Click Edit.


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A Molecular Structure Data Browser appears. Three options are available in the data-browser as forms for structure definition: o General (default form)

o Functional group

o Formula

Select the method you want to use and define the molecule according to the method selected.


Entering Data for Physical Properties Parameter Optimization If data is available for a particular physical property, this data can be used to fit a property model available in Aspen Polymers.

In order to accomplish this data fit, first the data must be supplied to the system:

From the Data Browser, double-click Properties.

Click Data.


Click New.

A Create a new ID form appears.

Enter a name for the data form or accept the default.

In the same form, select the data type: o MIXTURE

o PURE-COMP

Following is a description for pure component data entry. Similar steps are required for mixture data entry.

Select a property from the Property list.

Select a component from the Component list. Click the Data tab.

Enter the data in proper units.

Note that the numbers in the first row in the data form indicate estimated standard deviation in each piece of data. They are automatically filled in, but you can edit those figures if necessary.


References Aspen Physical Property System Physical Property Methods and Models. Cambridge, MA: Aspen Technology, Inc.

Barton, A. F. M. (1990). CRC Handbook of Polymer-Liquid Interaction Parameters and Solubility Parameters. Boca Raton, FL: CRC Press, Inc.


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Behme, S., Sadowski, G., Song, Y., & Chen, C.-C. (2003). Multicomponent Flash Algorithm for Mixtures Containing Polydisperse Polymers. AIChE J., 49, 258-268.

Bicerano J. (1993). Prediction of Polymer Properties. New York: Marcel Dekker, Inc.

Bokis, C. P., Orbey, H., & Chen, C.-C. (1999). Properly Model Polymer Processes. Chem. Eng. Prog., 39, 39-52.

Brandup, J., & Immergut, E. H. (Eds.) (1989). Polymer Handbook, 3rd Ed. New York: John Wiley & Sons.

Brandup, J., Immergut, E. H., & Grulke, E. A. (Eds.) (1999). Polymer Handbook, 4th Ed. New York: John Wiley & Sons.

Caruthers, J. M., Chao, K.-C., Venkatasubramanian, V., Sy-Siong-Kiao, R., Novenario, C. R., & Sundaram, A. (1998) . Handbook of Diffusion and Thermal Properties of Polymers and Polymer Solutions. New York: American Institute of Chemical Engineers.

Chapman, W. G., Gubbins, K. E., Jackson, G., & Radosz, M. (1989). Fluid Phase Equilibria, 52, 31.

Chen, C.-C. (1993). A Segment-Based Local Composition Model for the Gibbs Energy of Polymer Solutions. Fluid Phase Equilibria, 83, 301-312.

Chen, C.-C. (1996). Molecular Thermodynamic Model for Gibbs Energy of Mixing of Nonionic Surfactant Solutions. AIChE Journal, 42, 3231-3240.

Chen, C.-C., & Mathias, P. M. (2002). Applied Thermodynamics for Process Modeling. AIChE Journal, 48, 194-200.

Cotterman, R. L., & Prausnitz, J. M. (1991). Continuous Thermodynamics for Phase-Equilibrium Calculations in Chemical Process Design. In Kinetics and Thermodynamic Lumping of Multicomponent Mixtures. New York: Elsevier.

Danner R. P., & High, M. S. (1992). Handbook of Polymer Solution Thermodynamics. New York: American Institute of Chemical Engineers.

Flory, P. J. (1953). Principles of Polymer Chemistry. London: Cornell University Press.

Folie, B., & Radosz, M. (1995). Phase Equilibria in High-Pressure Polyethylene Technology. Ind. Eng. Chem. Res., 34, 1501-1516.

Fried, J. R. (1995). Polymer Science and Technology. Englewood Cliffs, NJ: Prentice-Hall International.

Gross, J., & Sadowski, G. (2001). Perturbed-Chain SAFT: An Equation of State Based on a Perturbation Theory for Chain Molecules. Ind. Eng. Chem. Res., 40, 1244-1260.

Gross, J., & Sadowski, G. (2002). Modeling Polymer Systems Using the Perturbed-Chain Statistical Associating Fluid Theory Equation of State. Ind. Eng. Chem. Res., 41, 1084-1093.

Hao W., Elbro, H. S., & Alessi, P. (1992). Part 1: Vapor-Liquid Equilibrium; Part 2: Solvent Activity Coefficients at Infinite Dilution; Part 3: Liquid-Liquid Equilibrium, Chemistry Data Series, Vol. XIV. In Polymer Solution Data Collection. Frankfurt: DECHEMA.


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Huang, S. H., & Radosz, M. (1990). Equation of State for Small, Large, Polydisperse, and Associating Molecules. Ind. Eng. Chem. Res., 29, 2284.

Huang, S. H., & Radosz, M. (1991). Equation of State for Small, Large, Polydisperse, and Associating Molecules: Extension to Fluid Mixtures. Ind. Eng. Chem. Res., 30, 1994.

Ko, G. H., Osias, M., Tremblay, D. A., Barrera, M. D., & Chen, C.-C. (1991). Process Simulation in Polymer Manufacturing. Computers & Chemical Engineering, 16, S481-S490.

Koningsveld, R., & Kleintjens, L. A. (1971). Liquid-Liquid Phase Separation in Multicomponent Polymer Systems. X. Concentration Dependence of the Pair-Interaction Parameter in the System Cyclohexane-Polystyrene. Macromolecules, 4, 637-641.

Kontogeorgis, G. M., Harismiadis, V. I., Frendenslund, Aa., & Tassios, D. P. (1994). Application of the van der Waals Equation of State to Polymers. I. Correlation. Fluid Phase Equilibria, 96, 65-92.

Kroschwitz, J. I. (Ed.). (1990). Concise Encyclopedia of Polymer Science and Engineering. New York: Wiley.

Oishi, T., & Prausnitz, J. M. (1978). Estimation of Solvent Activity in Polymer Solutions Using a Group Contribution Method. Ind. Eng. Chem. Process Des. Dev., 17, 333-335.

Orbey, H., Bokis, C. P., & Chen, C.-C. (1998a). Polymer-Solvent Vapor-Liquid Equilibrium: Equations of State versus Activity Coefficient Models. Ind. Eng. Chem. Res., 37, 1567-1573.

Orbey, H., Bokis, C. P., & Chen, C.-C. (1998b). Equation of State Modeling of Phase Equilibrium in the Low-Density Polyethylene Process: The Sanchez-Lacombe, Statistical Associating Fluid Theory, and Polymer-Soave-Redlich-Kwong Equation of State. Ind. Eng. Chem. Res., 37, 4481-4491.

Poling, B. E., Prausnitz, J. M., & O’Connell, J. P. (2001). The Properties of Gases and Liquids, 5th Ed. New York: Mc Graw-Hill.

Prausnitz, J. M., Lichtenthaler, R. N., & de Azevedo, E. G. (1986). Molecular Thermodynamics of Fluid Phase Equilibria, 2nd Ed, Englewood Cliffs, NJ: Prentice-Hall.

Qian, C., Mumby, S. J., & Eichinger, B. E. (1991). Phase Diagram of Binary Polymer Solutions and Blends. Macromolecules, 24, 1655-1661.

Reid, R. C., Prausnitz, J. M., & Poling, B. E. (1987). The Properties of Gases and Liquids, 4th Ed. New York: McGraw-Hill.

Sanchez, I. C., & Lacombe, R. H. (1976). An Elementary Molecular Theory of Classical Fluids. Pure Fluids. J. Phys. Chem., 80, 2352-2362.

Sanchez, I. C., & Lacombe, R. H. (1978). Statistical Thermodynamics of Polymer Solutions. Macromolecules, 11, 1145-1156.

Sanchez, I. C. (1992). Polymer Phase Separation. In Encyclopedia of Physical Science and Technology, 13. New York: Academic Press.

Sandler, S. I. (1994). Models for Thermodynamic and Phase Equilibria Calculations. New York: Marcel-Dekker.


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Sandler, S. I. (1988). Chemical and Engineering Thermodynamics, 2nd Ed. New York: J. Wiley & Sons.

Saraiva A., Kontogeorgis, G. M., Harismiadis, V. I., Fredenslund, Aa., & Tassios, D. P. (1996). Application of the van der Waals Equation of State to Polymers IV. Correlation and Prediction of Lower Critical Solution Temperatures for Polymer Solutions. Fluid Phase Equilibria, 115, 73-93.

Song, Y., Mathias, P. M., Tremblay, D., & Chen, C.-C. (2003). Viscosity Model for Polymer Solutions and Mixtures. Ind. Eng. Chem. Res., 42, 2415.

Van Ness, H. C. (1964). Classical Thermodynamics of Non-Electrolyte Solutions. Oxford: Pergamon Press.

Van Krevelen, D. W. (1990). Properties of Polymers, 3rd Ed. Amsterdam: Elsevier.

Walas, S. M. (1985). Phase Equilibria in Chemical Engineering. Boston: Butterworth-Heinemann.

Wohlfarth, C. (1994). Vapor-Liquid Equilibrium Data of Binary Polymer Solutions: Vapor Pressures, Henry-Constants and Segment-Molar Excess Gibbs Free Energies. Amsterdam: Elsevier.

Xiong, Y., & Kiran, E. (1995). Comparison of Sanchez-Lacombe and SAFT Model in Predicting Solubility of Polyethylene in High-Pressure Fluids. J. of Applied Polymer Science, 55, 1805-181.


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2 Equation-of-State Models 27

2 Equation-of-State Models

This chapter discusses thermodynamic properties of polymer systems from equation-of-state models (EOS) used in Aspen Polymers (formerly known as Aspen Polymers Plus). EOS models are used to calculate molar volumes, fugacity coefficients, enthalpy, entropy, and Gibbs free energy departures, for both pure components and mixtures.


• About Equation-of-State Models, 27

• Phase Equilibria Calculated from EOS Models, 29

• Other Thermodynamic Properties Calculated from EOS Models, 30

• Physical Properties Related to EOS Models in Aspen Polymers, 32

• Sanchez-Lacombe EOS Model, 34

• Polymer SRK EOS Model, 42

• SAFT EOS Model, 47

• PC-SAFT EOS Model, 59

• Copolymer PC-SAFT EOS Model, 67

About Equation-of-State Models In modeling polymer systems at high pressures, activity coefficient models suffer from certain shortcomings. For example, most of them are applicable only to incompressible liquid solutions, and they fail to predict the lower critical solution temperature (LCST) type phase behavior that necessitates pressure dependence in a model (Sanchez, 1992). In contrast to activity coefficient models, equations-of-state models do not suffer from these shortcomings. EOS models are able to predict both upper critical solution temperature (UCST) and LCST types of phase behavior in polymer solutions. EOS models are valid over the entire fluid region, from the dilute-gas to the dense-liquid region, and, therefore, are not limited to incompressible liquids. Thus, unlike activity coefficient models, EOS are able to evaluate the physical properties of any fluid phase, liquid and/or vapor, such as fugacity coefficient, molar volume, enthalpy, entropy, and Gibbs free energy departures. In addition, EOS are developed as pure-component models and subsequently extended to mixtures, thus providing information for both pure components and mixtures.


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28 2 Equation-of-State Models

There are a large number of equations of state for polymers and polymer solutions in the literature, which can be classified in the following categories:

• Cell models

• Lattice models

• Hole models

• Tangent sphere models

Detailed discussions of these models are beyond the scope of this chapter. Refer to available literature for this purpose (Lambert et al., 2000; Rodgers, 1993; Wei & Sadus, 2000). Currently, the most widely used EOS for polymer systems are the:

• Sanchez-Lacombe EOS (Sanchez & Lacombe, 1976, 1978)

• Statistical Associating Fluid Theory EOS (SAFT) (Chapman et al., 1989; Huang & Radosz, 1990, 1991)

• Perturbed-Chain Statistical Associating Fluid Theory EOS (PC-SAFT) (Gross & Sadowski, 2001, 2002a)

• Copolymer PC-SAFT (Gross and Sadowski, 2001, 2002a, 2002b; Gross et al., 2003; Becker et al., 2004; Kleiner et al., 2006)

Although many details are different, these segment-based polymer equations of state that were derived from statistical thermodynamics share a common formulation. That is, each pure component in the polymer mixture is characterized by three segment-based parameters: segment number, segment size or volume, and segment energy. In addition, well-known cubic equations-of-state for systems with small molecules are being extended for polymer solutions (Kontogeorgis et al., 1994; Orbey et al., 1998a, 1998b; Saraiva et al., 1996).

Presently, Aspen Polymers offers:

• Sanchez-Lacombe EOS

• An extension of the Soave-Redlich-Kwong (SRK) cubic equation of state to polymer-solvent mixtures (Polymer SRK EOS)

• SAFT EOS

• PC-SAFT EOS

• Copolymer PC-SAFT EOS

The Sanchez-Lacombe, SAFT, and PC-SAFT equations of state are polymer specific, whereas the polymer SRK model is an extension of a conventional cubic EOS to polymer systems. Copolymer PC-SAFT is a complete PC-SAFT model applicable to complex fluids, including normal fluids, water and alcohols, polymers and copolymers, and their mixtures. Polymer specific equations of state have the advantage of describing polymer components of the mixture more accurately; these EOS models are implemented in Aspen Polymers for modeling systems containing homopolymers as well as copolymers. The details of the individual EOS models are given in the following sections.


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Phase Equilibria Calculated from EOS Models Phase equilibrium calculations, as given by Equations 1.1 and 1.2, are critical for accurate simulations (For more information, see Chapter 1.):

li

vi ff = for vapor-liquid equilibria (1.1)

21 li

li ff = for liquid-liquid equilibria (1.2)

The equation of state can be related to the fugacity through fundamental thermodynamic equations:

pyf ivi

vi ϕ= (2.1)

pxf ili

li ϕ= (2.2)

With

ααα

∂∂ϕ m

V

nVTii ZVd

VRT

np

RTij

ln1ln,,

−⎥⎥⎦

⎤

⎢⎢⎣

⎡−⎟⎟

⎠

⎞⎜⎜⎝

⎛−= ∫∞

≠

(2.3)

Where:

viϕ = Fugacity coefficient of component i in the vapor phase

liϕ = Fugacity coefficient of component i in the liquid phase

iy = Mole fraction of component i in the vapor phase

ix = Mole fraction of component i in the liquid phase

p = P , system pressure, calculated using an EOS model

α = Vapor phase ( v ) or liquid phase ( l )

R = Universal gas constant

T = System temperature

V = Total volume of the mixture

ni = Mole number of component i

mZ =

nRTpVZ = , compressibility factor of the mixture

n = ∑i

in , total mole number of the mixture

Equations 2.1 and 2.2 are identical except for the phase to which the

variables apply. The fugacity coefficient αϕ i is obtained from the equation of

state, represented by p in Equation 2.3.


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Vapor-Liquid Equilibria in Polymer Systems The relationship for vapor-liquid equilibrium (VLE) is obtained by substituting Equations 2.1 and 2.2 in Equation 2.1 and dividing by p :

ilii

vi xy ϕϕ = (2.4)

In principle, Equation 2.4 applies to each component in the mixture. In practice, however, the polymer components in VLE are considered nonvolatile. Therefore, fugacity coefficients are needed from the equation of state only for solvents, monomers and oligomers. The mole fraction of the polymers in the liquid phase at VLE can be determined by the mass balance condition.

Liquid-Liquid Equilibria in Polymer Systems The liquid-liquid phase equilibrium (LLE) in polymer systems is also important in modeling polymer processes, and the calculation is more complicated than that in VLE as the polymer components are present in two-coexisting liquid phases. From Equation 2.2, the equation-of-state model can be applied to liquid-liquid equilibria:

2211 li

li

li

li xx ϕϕ = (2.5)

and also to vapor-liquid-liquid equilibria:

2211 li

li

li

lii

vi xxy ϕϕϕ == (2.6)

Where:

1liϕ = Fugacity coefficient of component i in the liquid phase 1l

2liϕ = Fugacity coefficient of component i in the liquid phase 2l

1lix = Mole fraction of component i in the liquid phase 1l

2lix = Mole fraction of component i in the liquid phase 2l

It is important to address the fact that fugacity coefficients in all phases are calculated from the same equation of state model. They are all functions of composition, temperature, and pressure.

Other Thermodynamic Properties Calculated from EOS Models The equation of state can be related to other properties through fundamental thermodynamic equations. These properties (called departure functions) are


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relative to the ideal gas properties of the same mixture at the same condition:

• Enthalpy departure:

( )

( ) ( )1−+−+

⎟⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛ −−=− ∫∞

migmm

V

igigmm

ZRTSST

VVlnRTdV

VRTpHH

(2.7)

• Entropy departure:

( ) ∫∞⎟⎠⎞

⎜⎝⎛+⎥

⎦

⎤⎢⎣

⎡−⎟

⎠⎞

⎜⎝⎛−=−

V

igv

igmm V

VRdVVR

TpSS ln

∂∂

(2.8)

• Gibbs free energy departure:

( ) ( )1ln −+⎟⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛ −−=− ∫∞ m

V

igigmm ZRT

VVRTdV

VRTpGG

(2.9)

• Molar volume:

• Solve ( )mVTp , for mV

Where:

mH = Molar enthalpy of the mixture

mS = Molar entropy of the mixture

mG = Molar Gibbs free energy of the mixture

mV = Molar volume of the mixture

igmH = Molar ideal gas enthalpy of the mixture

igmS = Molar ideal gas entropy of the mixture

igmG = Molar ideal gas Gibbs free energy of the mixture

igV = refp

RT, molar ideal gas volume

refp = Reference pressure (1 atm)

The departure functions given by the previous equations are calculated from the same equation of state and apply to both vapor and liquid phases. They also apply to both pure components and mixtures. Once the departure functions are known from the equation of state, the thermodynamic properties of a system (pure or mixture) in both vapor and liquid phases can be computed as follows:

( )igm

vm

igm

vm HHHH −+= (2.10)

( )igm

lm

igm

lm HHHH −+= (2.11)


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( )igm

vm

igm

vm SSSS −+= (2.12)

( )igm

lm

igm

lm SSSS −+= (2.13)

( )igm

vm

igm

vm GGGG −+= (2.14)

( )igm

lm

igm

lm GGGG −+= (2.15)

Vapor and liquid volume are computed by solving )V,T(p m for mV or by using

an empirical correlation.

The molar ideal gas properties of the mixture are computed by the summation over the components in the mixture. For instance, the molar ideal gas enthalpy of the mixture in both vapor and liquid phases is calculated as follows:

∑=i

igii

igm HyH *, in vapor phase (2.16)

∑=i

igii

igm HxH *, in liquid phase (2.17)

Where:

igiH *, = Ideal gas molar enthalpy of component i

The ideal gas properties for non-polymer components are well established in the Aspen Plus databanks and related results are retrieved automatically when an equation-of-state model is chosen in a calculation (for details, see Aspen Physical Property System Physical Property Methods and Models). Aspen Polymers extends the Aspen ideal gas property model to handle polymer components in the mixture. For a detailed description of the Aspen Ideal Gas Property Model, see Chapter 4.

Physical Properties Related to EOS Models in Aspen Polymers The following properties are related to equation-of-state models in Aspen Polymers:

Property Name

Symbol Description

PHIVMX viϕ Vapor fugacity coefficient of a component in a mixture

PHILMX liϕ Liquid fugacity coefficient of a component in a mixture

HVMX vmH Vapor mixture molar enthalpy

HLMX lmH Liquid mixture molar enthalpy

SVMX vmS Vapor mixture molar entropy


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Property Name

Symbol Description

SLMX lmS Liquid mixture molar entropy

GVMX vmG Vapor mixture molar Gibbs free energy

GLMX lmG Liquid mixture molar Gibbs free energy

VVMX vmV Vapor mixture molar volume

VLMX lmV Liquid mixture molar volume

PHIV vi*,ϕ Vapor pure component fugacity coefficient

PHIL li*,ϕ Liquid pure component fugacity coefficient

HV viH *, Vapor pure component enthalpy

HL liH *, Liquid pure component enthalpy

SV viS *, Vapor pure component entropy

SL liS *, Liquid pure component entropy

GV vi*,μ Vapor pure component Gibbs free energy

GL li*,μ Liquid pure component Gibbs free energy

VV viV *, Vapor pure component molar volume

VL liV *, Liquid pure component molar volume

DHVMX igm

vm HH − Vapor mixture molar enthalpy departure

DHLMX igm

lm HH − Liquid mixture molar enthalpy departure

DSVMX igm

vm SS − Vapor mixture molar entropy departure

DSLMX igm

lm SS − Liquid mixture molar entropy departure

DGVMX igm

vm GG − Vapor mixture molar Gibbs free energy departure

DGLMX igm

lm GG − Liquid mixture molar Gibbs free energy departure

DHV igi

vi HH *,*, − Vapor pure component molar enthalpy departure

DHL igi

li HH *,*, − Liquid pure component molar enthalpy departure

DSV igi

vi SS *,*, − Vapor pure component molar entropy departure

DSL igi

li SS *,*, − Liquid pure component molar entropy departure

DGV igi

vi

*,*, μμ − Vapor pure component molar Gibbs free energy departure

DGL igi

li

*,*, μμ − Liquid pure component molar Gibbs free energy departure


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The following table provides an overview of the equation-of-state models available in Aspen Polymers. This table lists the Aspen Physical Property System model names, and their possible use in different phase types, for pure components and mixtures. Details of individual models are presented in the next sections of this chapter.

EOS Models Model Name

Phase(s) Pure Mixture Properties Calculated

POLYSL ESPLSL0 v and l X — PHIV, PHIL, DHV, DHL, DSV, DSL, DGV, DGL, VV, VL

ESPLSL v and l — X PHIVMX, PHILMX, DHVMX, DHLMX, DSVMX, DSLMX, DGVMX, DGLMX, VVMX, VLMX

POLYSRK ESPLRKS0 v and l X — PHIV, PHIL, DHV, DHL, DSV, DSL, DGV, DGL, VV

ESPLRKS v and l — X PHIVMX, PHILMX, DHVMX, DHLMX, DSVMX, DSLMX, DGVMX, DGLMX, VVMX

POLYSAFT ESPLSFT0 v and l X — PHIV, PHIL, DHV, DHL, DSV, DSL, DGV, DGL, VV, VL

ESPLSFT v and l — X PHIVMX, PHILMX, DHVMX, DHLMX, DSVMX, DSLMX, DGVMX, DGLMX, VVMX, VLMX

POLYPCSF ESPCSFT0 v and l X — PHIV, PHIL, DHV, DHL, DSV, DSL, DGV, DGL, VV, VL

ESPCSFT v and l — X PHIVMX, PHILMX, DHVMX, DHLMX, DSVMX, DSLMX, DGVMX, DGLMX, VVMX, VLMX

PC-SAFT ESPSAFT0 v and l X — PHIV, PHIL, DHV, DHL, DSV, DSL, DGV, DGL, VV, VL

ESPSAFT v and l — X PHIVMX, PHILMX, DHVMX, DHLMX, DSVMX, DSLMX, DGVMX, DGLMX, VVMX, VLMX

An X indicates applicable to Pure or Mixture.

Sanchez-Lacombe EOS Model This section describes the Sanchez-Lacombe equation-of-state (EOS) model for polymers and polymer solutions. This EOS is used through the POLYSL property method.

Pure Fluids According to the lattice theory of Sanchez and Lacombe (1976), a pure fluid is viewed as a mixture of molecules and holes, confined on the sites of a lattice. Each segment of the chain, as well as each hole, occupies one lattice site. The


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total number of lattice sites for a binary mixture of N m-mers and N0 empty sites is:

mNNNr += 0

The total volume of the system is:

( ) *0 vmNNV +=

Where:

v* = Volume of a lattice site

m = Number of segments per chain

Sanchez and Lacombe defined a reduced density as the fraction of occupied lattice sites:

mNNmN+

==0

*~

ρρρ

With

**

mvM

=ρ

VNM

=ρ

Where:

*ρ = Scale factor for density

ρ = Mass density

M = Molecular weight (for polymer components this is the number average molecular weight)

Sanchez and Lacombe used the Flory-Huggins expression for the combinatorial entropy of a binary mixture on an incompressible lattice, replacing one component with holes. For the energy, they only considered segment-segment interactions (in other words, segment-hole and hole-hole pair interactions were set equal to zero), and assumed that the segments and the holes are randomly distributed in the lattice. They developed an expression for the Gibbs free energy of a chain fluid on a lattice. By minimizing the Gibbs free energy expression, Sanchez and Lacombe derived the SL EOS:

Sanchez-Lacombe EOS Equation

( ) 0~11~1ln~~~ 2 =⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −+−++ ρρρ

mTP (2.18)

Where the reduced quantities are defined by:


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***~~~

ρρρ ===

PpP

TTT (2.19)

The scale factors, T P* *, and *ρ are related to lattice variables by:

Sanchez-Lacombe Parameters

**

*

**

**

mvM

vP

kT === ρεε

(2.20)

In the above expressions:

m = Number of segments per chain

*ε = Characteristic interaction energy per segment

v* = Closed-packed volume of a segment

k = Boltzmann's constant

A pure fluid is characterized completely by three molecular parameters: *ε , *v , and m, or equivalently, the scale factors T * , *P , and *ρ . These

parameters are obtained by fitting pure component experimental data, usually data along the saturation curve. Some additional characteristics of the SL EOS are:

• The SL EOS has an explicit size or shape dependency through the molecular parameter m. Thus, it takes into account the chain-like structure of long-chain molecules, such as heavy paraffins and polymers.

• SL is more accurate than most cubic equations of state of the van der Waals type (Redlich-Kwong, Peng-Robinson, Redlich-Kwong-Soave, etc.) in calculating liquid volumes.

• SL is not accurate at the critical point of pure fluids; the vapor-liquid equilibrium coexistence curve predicted by the SL EOS is too sharp near critical conditions. Therefore, when experimental vapor pressure data are being regressed, temperatures closer than 15-20°C of the critical point should be omitted.

• Unlike most cubic EOS, the SL EOS does not satisfy a corresponding states principle, except for large molecules ( )∞→m . This is related directly to the fact that the repulsive part of the EOS scales with molecular size through the parameter m.

• For polymer molecules, m is very large. This means that polymeric liquids of high molecular weight satisfy a corresponding states principle.

• Since vapor pressure data are unavailable for polymer liquids, the molecular parameters are determined by fitting experimental liquid volume data.

Fluid Mixtures Containing Homopolymers The SL EOS for multicomponent fluid mixtures containing homopolymers is identical to the pure-component equation, Equation 2.18 (Sanchez & Lacombe, 1978). The difference is that the parameters become composition


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dependent through mixing rules. These mixing rules are written in terms of volume fractions, rather than mole fractions:

Sanchez-Lacombe Mixing Rules

∑∑=i j

ijijjimix

mix vv

***

* 1 εφφε (2.21)

∑∑=i j

ijjimix vv ** φφ (2.22)

∑=i i

i

mix mmφ1

(2.23)

∑ ⎟⎟⎠

⎞⎜⎜⎝

⎛=

j jj

j

ii

i

i

vwv

w

**

**

ρ

ρφ (2.24)

Where:

iφ = Volume fraction of component I

iw = Weight fraction of component i

The cross parameters are calculated by:

[ ]( )ijjjiiij vvv η−+= 121 *** (2.25)

( )ijjjiiij k−= 1*** εεε (2.26)

In two expressions above, kij and ijη are binary interaction parameters that

are fitted to experimental VLE and LLE data. Both parameters are symmetric. If no data are available, they are set equal to zero.

The SL EOS is able to predict the thermodynamic properties of multicomponent mixtures through pure-component and binary interaction parameters only.

Extension to Copolymer Systems The same equation, Equation 2.18, is used for mixtures containing copolymers. The pure-component parameters for copolymers can be input directly or can be calculated in terms of the relative composition of the segments or repeat units that form the copolymer. The mixing rule for the closed-packed volume parameter of the copolymer is:

Pure Parameters

∑ ∑=Nseg

A

Nseg

BABBAp vv ** φφ


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Where:

Aφ and Bφ = Volume fractions of the segments that form the copolymer (calculated using an equation similar to the third Sanchez-Lacombe mixing rule given by Equation 2.24)

Nseg = Number of distinct segment types present in the copolymer chain, and

[ ] )1(21 ***

ABBBAAAB vvv η−+=

Where:

vAA* and vBB

* = Characteristic volume parameters of the segments A and B

ABη = Factor that accounts for differences in molecular size

Similarly, for the energy parameter of the copolymer:

∑ ∑=Nseg

A

Nseg

BABABBA

pp v

v**

** 1 εφφε

With:

)1(***ABBBAAAB k−= εεε

Where:

*AAε and *

BBε = Characteristic energy parameters for the segments A and B

ABk = Correction to the geometric-mean rule

Finally, for the molecular size of the copolymer:

∑=Nseg

A A

A

p mmφ1

Where:

Am = Characteristic size parameter of segment A in the copolymer

The characteristic parameters ε* , v* , and m for the segments A and B are obtained from data on the homopolymers A and B, respectively.

McHugh and coworkers (Hasch et al., 1992) have shown that the correction terms ABη and ABk have little effect on calculated copolymer phase behavior. For this reason, these two binary parameters are not used in the model and have not been made available for user input. The SL EOS is able to predict UCST and LCST types of phase immiscibility.

If parameters T * , P* , and *ρ are provided for the polymer or oligomer, then these have highest priority and are used for calculations. If they are not known, usually in the case of copolymers, the user must provide these parameters for the segments that compose the copolymer.


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Binary Parameters There are three types of binary parameters: solvent-solvent, solvent-segment and segment-segment. Assuming the binary parameter between different segments is zero, the cross energy parameter for a solvent-copolymer pair can be calculated as:

)1(***iAAAii

Nseg

AAip kX −= ∑ εεε

Where:

*ipε = Cross energy parameter for a solvent-copolymer pair

AX = Segment mole fraction or weight fraction of segment type A in the copolymer. The default is the segment mole fraction and the segment weight fraction can be selected via “Option Code” in Aspen Plus

*AAε = Energy parameter of segment type A in the copolymer, determined

from data on the homopolymer A

iAk = Binary parameter for a solvent-segment pair, determined from VLE or LLE data of the solvent (i)- homopolymer (A) solution

The cross energy parameter for a copolymer-copolymer pair in the mixture can be calculated as:

***212121

)1( pppppp k εεε −=

Where:

*21 ppε = Cross energy parameter for a copolymer-copolymer pair

21 ppk = Binary parameter for a copolymer-copolymer pair

*1pε = Energy parameter of pure copolymer 1p

*2pε = Energy parameter of pure copolymer 2p

The binary interaction parameter, ijk , allows complex temperature

dependence:

2ln/ rijrijrijrijijij TeTdTcTbak ++++=

with

refr T

TT =

Where:

refT = Reference temperature and the default value = 298.15 K


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Similarly, the cross volume parameter for solvent-copolymer pairs and copolymer-copolymer pairs can be calculated as:

2/)1)(( ***iAAAii

Nseg

AAip vvXv η−+= ∑

2/))(1( ***212121 pppppp vvv +−= η

Where:

*ipv = Cross volume parameter for a solvent-copolymer pair

*AAv = Volume parameter of segment type A in the copolymer, determined

from data on the homopolymer A

iAη = Binary parameter for a solvent-segment pair, determined from VLE or LLE data of the solvent (i)- homopolymer (A) solution

*21 ppv = Cross volume parameter for a copolymer-copolymer pair

21 ppη = Binary parameter for a copolymer-copolymer pair

*1pv = Volume parameter of pure copolymer 1p

*2pv = Volume parameter of pure copolymer 2p

The binary interaction parameter, ijη , allows complex temperature

dependence:

2''''' ln/ rijrijrijrijijij TeTdTcTba ++++=η

Similar to pure component parameters, if the user provides the binary parameters for solvent-copolymer pairs, then these have highest priority and are used directly for calculations. However, if the user provides the binary parameters for solvent-segment pairs, the values for solvent-copolymer pairs will be calculated.

Sanchez-Lacombe EOS Model Parameters The following table lists the Sanchez-Lacombe model parameters implemented in Aspen Polymers:

Parameter Name / Element

Symbol Default Lower Limit

Upper Limit

MDS Units Keyword

Comments

SLTSTR T * --- --- --- X TEMP Unary

SLPSTR P* --- --- --- X PRESSURE Unary

SLRSTR *ρ --- --- --- X DENSITY Unary

SLKIJ/1 ija 0.0 --- --- X --- Binary,

Symmetric

SLKIJ/2 ijb 0.0 --- --- X --- Binary,


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Upper Limit

MDS Units Keyword

Comments

Symmetric

SLKIJ/3 ijc 0.0 --- --- X --- Binary,

Symmetric

SLKIJ/4 ijd 0.0 --- --- X --- Binary,

Symmetric

SLKIJ/5 ije 0.0 --- --- X --- Binary,

Symmetric

SLKIJ/6 refT 298.15 --- --- X TEMP Binary,

Symmetric

SLETIJ/1 'ija 0.0 --- --- X --- Binary,

Symmetric

SLETIJ/2 'ijb 0.0 --- --- X --- Binary,

Symmetric

SLETIJ/3 'ijc 0.0 --- --- X --- Binary,

Symmetric

SLETIJ/4 'ijd 0.0 --- --- X --- Binary,

Symmetric

SLETIJ/5 'ije 0.0 --- --- X --- Binary,

Symmetric

SLETIJ/6 refT 0.0 --- --- X TEMP Binary,

Symmetric

Parameter Input and Regression All three unary parameters, SLTSTR, SLPSTR, and SLRSTR can be:

• Specified for each polymer or oligomer component

• Specified for segments that compose a polymer or oligomer component

These options are shown in priority order. For example, if parameters are provided for a polymer component and for the segments, the polymer parameters are used and the segment parameters are ignored. If the user provides the parameters for segments only, the values for polymers will be calculated from the segment parameters.

Both binary parameters, SLKIJ and SLETIJ, can be:

• Specified for each polymer-solvent pair

• Specified for each segment-solvent pair

These options are also shown in priority order. For example, if the binary parameters are provided for both polymer-solvent pairs and segment-solvent pairs, the polymer-solvent parameters are used and the segment-solvent pairs are ignored. If the user provides the binary parameters for segment-solvent pairs only, the values for polymer-solvent pairs will be calculated.

Appendix E lists pure component parameters for some solvents (monomers) and homopolymers. If the pure parameters are not available for components in a calculation, the user can perform an Aspen Plus Regression Run (DRS) to obtain these pure component parameters. For non-polymer components, the


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pure component parameters are usually obtained by fitting experimental vapor pressure and liquid molar volume data. To obtain pure homopolymer (or segment) parameters, experimental data on liquid density should be regressed. Once the pure component parameters are available for a homopolymer, ideal-gas heat capacity parameters, CPIG, should be regressed for the same homopolymer using experimental liquid heat capacity data (for details on ideal-gas heat capacity for polymers, see Aspen Ideal Gas Property Model section in Chapter 4).

In addition to pure component parameters (SLTSTR, SLPSTR, and SLRSTR), the binary parameters (SLKIJ and SLETIJ) for each solvent-solvent pair or each solvent-polymer (segment) pair can be regressed using vapor-liquid equilibrium (VLE) data in the form of TPXY data in Aspen Plus.

Note: In a Data Regression Run, a polymer component must be defined as an OLIGOMER type, and the number of each type of segment that forms the oligomer (or polymer) must be specified.

Missing Parameters If the user does not provide all three unary parameters for a defined component or segment, the following nominal values are assumed:

• SLTSTR = 415 (K)

• SLPSTR = 3000 (bar)

• SLRSTR = 736 (kmol/cum)

Specifying the Sanchez-Lacombe EOS Model See Specifying Physical Properties in Chapter 1.

Polymer SRK EOS Model This section describes the Polymer SRK equation-of-state model available in the POLYSRK physical property method. The polymer SRK EOS model is an extension of the popular cubic SRK EOS to mixtures containing polymers. From a modeling point of view, this model is considered similar to the PSRK EOS model available in Aspen Plus for conventional mixtures. Like the PSRK model, for mixture applications this model uses a Huron-Vidal-type mixing rule that incorporates an excess energy (Gibbs or Helmholtz) term. The detailed discussion of these types of mixing rules can be found elsewhere (see Aspen Physical Property System Physical Property Methods and Models, see also Orbey, et al., 1998a and 1998b; Fischer & Gmehling, 1996). Here, the basic characteristics of the model are summarized from a modeling perspective.

The excess Gibbs free energy can be written from an EOS using rigorous thermodynamics, and it can be equated to the same property from an activity coefficient model:


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RTG

xxRT

G E

iii

iii

EEOS γγϕϕ =≡−= ∑∑ lnlnln

Where:

EEOSG = Excess Gibbs free energy from an EOS model

ϕ = Mixture fugacity coefficient

iϕ = Fugacity coefficient of component i in a mixture

iγ = Activity coefficient of component i in a mixture

EGγ = Excess Gibbs free energy from an activity coefficient model

The above equality can only be written at a selected reference pressure. A reference for pressure is needed since the Gibbs free energy from an EOS is pressure dependent but the same term from an activity coefficient is not. Thus, an algebraically explicit equality can only be established at a single reference pressure.

The usual alternatives for the reference pressure are either 0=p or ∞=p . There is much debate as to which selection is better (Fischer & Gmehling, 1996; Orbey & Sandler, 1995, 1997), and it is beyond the scope of this documentation.

In general, the combination of an EOS with an activity coefficient model by equating the Gibbs free energy terms leads to a general functional relation between a and b parameters of a cubic EOS in the form:

( )EEiii AGxba

bRTa

γγ or ,,,Γ=

Where:

a = Cubic EOS parameter of a mixture

b = Cubic EOS parameter of a mixture

ia = Cubic EOS parameter of component i

ib = Cubic EOS parameter of component i

EAγ = Excess Helmholtz free energy

Soave-Redlich-Kwong EOS The functional form Γ depends on the selection of reference pressure. Holderbaum and Gmehling (1991) used this approach for the SRK EOS to develop the following relation at the limit of low (atmospheric) pressure:

)()(bvv

Tabv

RTp+

−−

=

Holderbaum and Gmehling Approach


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abRT

xa

RTbGRT

xbbi

i

i

E

iiii

= − +⎛

⎝⎜

⎞

⎠⎟∑∑ 1546. lnγ

For the co-volume parameter, b, the linear mixing rule ∑=i

iibxb was used.

With the Holderbaum and Gmehling approach (see previous equation), this completely defines a and b parameters of the SRK EOS for any mixture, provided that an activity coefficient model is selected to represent the molar

excess Gibbs free energy term GE

γ . In the original PSRK EOS, the UNIFAC predictive model was used for this purpose. For the polymer SRK model here, the POLYNRTL model proposed for polymer mixtures is used (for details, see the Polymer-NRTL Activity Coefficient Model section in Chapter 3). Consequently, the same mixture interaction parameters used in the POLYNRTL model are used in the polymer SRK model, only this time in the EOS format.

In modeling polymer containing mixtures with the polymer SRK EOS, one needs values of the critical temperature, the critical pressure, and component-specific constants of Mathias and Copeman (1983) for each

constituent of the mixture to evaluate pure component ai and bi 's. (For more details on the Mathias-Copeman constants for the SRK EOS, See Aspen Physical Property System Physical Property Methods and Models). Only the final results are presented here:

Mathias-Copeman Constants

ic

ici p

RTb

,

,08664.0=

iic

ici p

TRa α

,

2,

2

42748.0=

235.0,3

25.0,2

5.0,1 ])1()1()1(1[ iririri TcTcTc −+−+−+=α

Where:

icT , = Critical temperature of component i

icp , = Critical pressure of component i

irT , = icTT ,/

321 ,, ccc = Mathias-Copeman constants of a component

For conventional components, values of the pure component constants are readily available and stored in the Aspen Plus databanks. For oligomers and polymers, these parameters are not available. To overcome this drawback, some estimation techniques have been suggested by several researchers

based on the available experimental values for Tc and cp for alkanes up to

about C20 (See works of Tsonopoulos & Tan, 1993; Teja et al., 1990). The


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user needs to supply these constants for the polymers and oligomers using the guidelines given in the Polymer SRK EOS Model Parameters section on page 45.

Most two-parameter cubic equations of state (SRK, Peng-Robinson, etc.) cannot predict the molar volumes in the liquid phase accurately. To overcome this difficulty, the Rackett model is used to overwrite the liquid molar volume predictions of the EOS in PSRK property method in Aspen Plus. In the case of the polymer SRK EOS, the van Krevelen liquid molar volume model (See Chapter 4) is used for the polymer and oligomer components; the Rackett equation is still used for conventional components. Mixture liquid molar volumes are calculated using the ideal-mixing assumption. For details, see Mixture Liquid Molar Volume Calculations in Chapter 3.

Polymer SRK EOS Model Parameters To use the polymer SRK EOS, several pure component parameters are

required, including the critical constants Tc , cp and the Mathias-Copeman constants. The following tables show the polymer SRK EOS model unary parameters implemented in Aspen Polymers. The conventional components are available from the Aspen Plus data bank. For oligomers and polymers, the user needs to provide them using unary parameter forms.



Upper Limit

MDS Units Keyword

Comments

TCRKS cT

TC 2 7000 --- TEMP Unary

PCRKS cp

PC 105 108 --- PRESSURE Unary

RKSMCP/1 1c

0 --- --- X --- Unary

RKSMCP/2 2c

--- --- --- X --- Unary

RKSMCP/3 3c

--- --- --- X --- Unary

Critical Constants for Polymers Polymers are not supposed to vaporize, and, therefore, for the critical temperature of the polymers a high value is recommended (typically T Kc > 1000 ). For the same reason, a relatively low critical pressure is needed

(26 /10 mNpc < ). For all of the Mathias-Copeman parameters for oligomers

and polymers, zero is recommended due to unavailability of information on polymer vapor pressure, though the user may overwrite them. For oligomers, critical temperatures lower than those used for polymers and critical pressures higher than that of polymers could be used.

Depending upon the magnitude of these choices, some oligomer may appear in the vapor phase. For the selection of these constants for oligomers, the works of Tsonopoulos and Tan (1993) and Teja et al. (1990) can be used as a


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guideline. The Tc and cp profiles obtained by Tsonopoulos and by Teja for alkane hydrocarbons are shown here:

In some cases, the choices for the critical constants for polymers and oligomers may affect the VLE calculations significantly. This largely depends on the nature of the solvents present and the temperature and pressure at which the phase calculations are made. None of the parameters listed previously are automatically supplied by Aspen Polymers for oligomers and polymers. The user needs to enter them using unary parameter forms.

The default option for the excess energy model used in the polymer SRK model is the polymer NRTL activity coefficient model. Therefore, the same binary interaction parameters needed for the polymer NRTL model are required in this application. The polymer NRTL model is described in Chapter 3. The user may overwrite this choice by creating a custom property method

selecting another activity coefficient model for the evaluation of GE

γ term in the polymer SRK model. In this case, the mixture parameters of the selected GE

γ model need to be supplied.


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Specifying the Polymer SRK EOS Model See Specifying Physical Properties in Chapter 1.

SAFT EOS Model This section describes the Statistical Associating Fluid Theory (SAFT). This equation-of-state model is used through the POLYSAFT property method. The SAFT EOS is a rigorous thermodynamic model for polymer systems based on the perturbation theory of fluids. The equation of state accounts explicitly for the molecular repulsions, the chain connectivity, dispersion (attractive) forces, and specific interactions via hydrogen bonding.

TheSAFT EOS was developed by Gubbins and co-workers (Chapman et al., 1990), and was first used for engineering calculations by Huang and Radosz (1990, 1991). This EOS currently represents a state-of-the-art engineering tool for the thermodynamic properties and phase equilibria correlation and prediction of polymer-containing systems.

Recent research efforts by various research groups worldwide have demonstrated the applicability of SAFT to a variety of polymer systems . Among others, these include:

• Low-density polyethylene (Folie & Radosz, 1995; Xiong & Kiran, 1995)

• Polystyrene (Pradham et al., 1994)

• Poly(ethylene-propylene) copolymer (Chen et al., 1992)

• Polyisobutylene (Gregg et al., 1994)

• Poly(ethylene-methyl acrylate) copolymers (Lee et al., 1996)

• Poly(ethylene-acrylic acid) copolymers (Hasch & McHugh, 1995; Lee et al., 1994)

The above researchers, together with others in the field of polymer thermodynamics, have found that the SAFT equation of state is able to correlate accurately the thermodynamic properties and phase behavior of both pure-components and their mixtures. In addition, SAFT has shown remarkable predictive capability, which is a very important feature for modeling industrial applications.

Although SAFT of Huang and Radosz (1990, 1991) is a homopolymer model, the version implemented in Aspen Polymers has some features that make the model convenient to use for copolymer property modeling.

Pure Fluids The SAFT model is a molecularly-based equation of state, which means that it evaluates the properties of fluids based on interactions at the molecular level. This way the model is able to separate and quantify the effects of molecular structure and interactions on bulk properties and phase behavior. Examples of such effects are:

• Molecular size and shape (e.g., chain length)

• Association energy (e.g., hydrogen bonding)


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• Attractive (e.g., dispersion) energy

In developing any equation of state based on theoretical considerations, a model fluid has to be selected. In the case of SAFT, Chapman et al. (1990) chose a model fluid that is a mixture of equal-sized spherical segments interacting with square-well potential. To make the model fluid more realistic, two kinds of bonds where also considered between the segments: covalent-like bonds that form chain molecules, and hydrogen bonds. As a result, the model fluid can represent a wide variety of real fluids such as:

• Small nearly-spherical species (methane, ethane, etc.)

• Chain molecules (alkanes, polymers)

• Associating species (alkanols)

Reduced density term

The reduced density η of the fluid (segment packing fraction) is defined as:

3

6md

N AV ρπ

η = (2.27)

Where:

ρ = Molar density

m = Number of segments in each molecule

d = Effective segment diameter (temperature dependent)

AVN = Avogadro constant

This equation can be rewritten as:

omvτρη = (2.28)

With

30

6d

Nv AV

τπ

= (2.29)

Where:

τ = Constant equal to 0.74048

v 0 = Segmental molar volume at closed-packing (the volume occupied by a mole of closely packed segments), in units of cc per mole of segments

From the previous two equations, it follows that v 0 is temperature dependent, since it depends on the temperature dependent diameter d. Thus, it is convenient to define a temperature-independent segmental molar volume at T=0, denoted voo . This parameter will be referred to as the segment volume. Chen and Kreglewski (1977) solved the Barker-Henderson integral equation of the diameter d (which depends on the square-well potential), and proposed the following expression between vo and voo :

33exp1 ⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ −−=

kTuCvv

oooo (2.30)


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In the above equation, u ko / is the square-well depth, a temperature-independent energy parameter, referred to as the segment energy, in Kelvins. Chen and Kreglewski (1977) set the constant C=0.12, and used the following temperature dependence of the dispersion energy of interaction between segments:

Chen and Kreglewski temperature dependence of dispersion energy

u ue

kTo= +⎡⎣⎢

⎤⎦⎥

1 (2.31)

Where:

e/k = Constant (values will be provided later)

SAFT was proposed by Gubbins, Radosz, and co-workers (Chapman et al., 1990). The main idea in SAFT is perturbation theory. In perturbation theory, the fluid is simulated using a reference fluid. The reference fluid is usually a well-understood and well-described fluid (such as the hard-sphere fluid). Any deviations between the properties of the real and the reference fluid are referred to as perturbations. These authors used a reference fluid that incorporates both the chain length (molecular size and shape) and the molecular association (whenever applicable). (In most pre-existing engineering equations of state, the much simpler hard-sphere fluid had been used as the reference fluid).

To derive the equation of state for the reference fluid, Chapman et al. (1990) needed expressions for the Helmholtz free energy for the chain and association effects. These researchers used Wertheim’s expressions for chain and hydrogen bonding, which are based on cluster expansion theory (Wertheim, 1984; 1986a,b). (As a reminder, equation of state developers often derive expressions for the Helmholtz free energy for convenience reasons. Most properties of interest, such as the system pressure, can be easily obtained via simple algebraic differentiation of the Helmholtz free energy.)

As mentioned above, the reference equation of state in SAFT accounts for the hard-sphere, chain, and association effects. The effects of other kinds of intermolecular forces, such as dispersion forces, are usually weaker, and are treated through a perturbation term. Chapman et al. (1990) used an expression similar to that of Alder et al. for the square-well potential (Alder et al., 1972).

The statistical associating fluid theory results in an expression of the residual

Helmholtz free energy, ares per mole, defined as:

),,(),,(),,( NVTaNVTaNVTa idealres −= (2.32)

Where:

),,( NVTa = Total Helmholtz energy per mole at the same temperature and volume as:

),,( NVTaideal = Ideal-gas Helmholtz energy per mole

In SAFT, the residual Helmholtz free energy ares is a sum of three contributions:


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• aseg represents segment-segment interactions (hard-sphere repulsions and attractive or dispersion forces)

• achain is due to the presence of covalent chain-forming bonds among the segments

• aassoc is present when the fluid exhibits hydrogen bonding interactions among the segments

The general expression for the Helmholtz free energy in SAFT is given by:

assocchainsegres aaaa ++= (2.33)

Segment contribution per mole of molecules

The segment contribution aseg per mole of molecules is given by:

( )disphsseg aama += (2.34)

The two contributions represent the segmental hard-sphere and dispersion interactions. These two quantities are given by:

Hard-Sphere Term

( )2

2

134η

ηη−−

=RTahs

(2.35)

Dispersion Term

∑∑ ⎥⎦⎤

⎢⎣⎡

⎥⎦⎤

⎢⎣⎡=

i j

ji

ij

disp

kTuD

RTa

τη

(2.36)

The hard-sphere term is the well-known Carnahan-Starling expression for the hard-sphere fluid (Carnahan & Starling, 1972). The dispersion term is a fourth-order perturbation expansion of the Helmholtz free energy, initially fitted by Alder et al. (1972) to molecular dynamics simulation data for the square-well fluid. In the dispersion term, Dij are universal constants. In

SAFT, Huang and Radosz (1990) used the Dij constants that were proposed

by Chen and Kreglewski (1977), who re-fitted Alder’s expression to very accurate experimental data for argon.

The chain and association terms in SAFT are the result of Wertheim’s thermodynamic theory of polymerization. This section does not deal with associating species, and, therefore, the association term will be neglected. The chain term, which represents the Helmholtz free energy increment due to the formation of covalent bonds, is given by the following expression (Chapman et al., 1990):

Chain Term

( ) segchain

dgmRT

a )(ln1−= (2.37)

Where g d seg( ) is the value of the segmental radial distribution function at a

distance equal to the effective segment diameter d. In other words, g d seg( ) is


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the radial distribution function at the surface of the segment, or the contact value. As explained by Chapman et al. (1990) and Huang and Radosz (1990), this equation is derived from the association theory by replacing the hydrogen bonds with covalent, chain-forming bonds. As mentioned above, in SAFT, the

segments are approximated by hard spheres, and thus, g d seg( ) can be approximated by the hard-sphere radial distribution function (Carnahan & Starling, 1972):

( )31211

)()(η

η

−

−=≈ hsseg dgdg (2.38)

Therefore, the chain contribution to the free energy in SAFT can be rewritten as:

( )( )31

211

ln1η

η

−

−−= m

RTachain

(2.39)

Compressibility Factor

The compressibility factor Z can be easily obtained by taking the molar volume derivative of the residual Helmholtz free energy; the resulting SAFT equation of state has the form:

assocchainseg ZZZRTPvZ +++== 1 (2.40)

Where:

( ) ⎥⎥⎦

⎤

⎢⎢⎣

⎡⎥⎦⎤

⎢⎣⎡

⎥⎦⎤

⎢⎣⎡+

−−

= ∑∑i j

ji

ijseg

kTujDmZ

τη

ηηη

3

2

124

(2.41)

( )( ) ⎟

⎠⎞

⎜⎝⎛ −−

−−=

ηη

ηη

2111

25

12

mZ chain (2.42)

The contribution from association, Zassoc , is not considered for the time being, and thus this term will be zero.

The SAFT equation of state presented above has been used to correlate vapor pressures and liquid densities of over 100 real fluids by Huang and Radosz (1990). For each fluid, three parameters were fitted to the experimental data:

• Segment volume, voo

• Segment energy, u ko /

• Segment number, m

Estimated parameters for these fluids are given in Appendix E.


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Extension to Fluid Mixtures Huang and Radosz (1991) extended the SAFT equation of state to treat multicomponent fluid mixtures. In doing so, they took advantage of the fact that SAFT was based on theoretical arguments and, therefore, the extension of the equation of state from pure components to mixtures is straightforward, based on statistical mechanical considerations.

For the extension of the hard-sphere term to mixtures, Huang and Radosz (1991) used the theoretical result of Mansoori et al. for the Helmholtz free energy of a mixture of hard spheres, which is given by the following expression (Mansoori et al., 1971):

Helmholtz free energy of a mixture of hard spheres

( ) ( )( )

( )( )

( )⎥⎥⎦

⎤

⎢⎢⎣

⎡−

⎥⎥⎦

⎤

⎢⎢⎣

⎡−−

−

−+= 32

3

32

0233

2321321

32 1ln

1336 ζ

ζζ

ζζζ

ζζζζζζζπρRT

a hs

With

( )∑=i

kiiii

Avk dmx

Nρ

πζ

6

Note that the Helmholtz free energy equation reduces to the same result for pure components, as given by the segment contribution equation and the hard-sphere equation, given by Equation 2.32, in the limit of xi of unity.

In a similar fashion, the chain contribution for fluid mixtures is a direct extension of the pure-component result:

( ) ( )( )∑ −=i

hsiiiiii

chain

dgmxRT

a ln1

Where gii is the radial distribution function of two species i in a mixture of spheres, evaluated at the hard-sphere contact. This value was derived from statistical mechanics by Mansoori et al. (1971), and has the form:

( ) ( )( ) ( )3

3

22

2

23

2

3 122

123

11

ζζ

ζζ

ζ −⎥⎦

⎤⎢⎣

⎡+−

+−

=≈ iiiihsiiii

segiiii

dddgdg

For the dispersion (attractive) term in SAFT, Huang and Radosz (1991) used several approaches for its extension to fluid mixtures. One of these approaches, the conformal solution approach (which has been considered by most researchers who have applied SAFT to engineering calculations) is discussed here. According to the conformal solution, or van der Waals one-fluid (vdW1) theory, a fluid mixture is approximated by a hypothetical pure fluid having the same molecular energy and size (volume). The vdW1 theory leads to the vdW1 mixing rules. For the energy parameter in SAFT, the vdW1 mixing rule is:

Dispersion Energy Mixing Rule


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( )( )∑∑

∑∑=

i jij

ojiji

i jij

oijjiji

vmmxx

vkTu

mmxx

kTu

With

( ) ( ) ( ) 33/13/1

2 ⎥⎥⎦

⎤

⎢⎢⎣

⎡ += j

oi

o

ijo vv

v

( )ijjjiiij kuuu −= 1

Where kij is an empirical binary parameter, fitted to experimental VLE or LLE

data. In the absence of mixture data, kij is equal to zero.

Finally, the molecular size is taken into account via the segment number m. For mixtures, it is calculated as:

ii

imxm ∑=

Application of SAFT Huang and Radosz (1991) have proposed a comprehensive parameterization of the SAFT equation of state based on the work by Topliss (1985), which facilitates the coding of the SAFT individual terms and their derivatives with respect to density and composition. This approach has been followed in Aspen Polymers. All individual terms and their derivatives are provided in the Huang and Radosz (1991) paper, and will not be reproduced here.

To apply SAFT to real fluid systems, three pure-component (unary) parameters need to be provided for each species:

• Segment volume, voo

• Segment energy, u ko /


These parameters are estimated by fitting vapor-pressure and liquid-density experimental data for the pure components. Huang and Radosz (1990) have evaluated pure-component parameters for about 100 species; these parameters are also tabulated in Appendix E for convenience. In case the component of interest is not included in the list of components with already available parameters, the user needs to set up a regression run (DRS), and use vapor-pressure and liquid density experimental data to estimate the

necessary parameters voo, u ko / , and m.

For the components that Huang and Radosz (1991) regressed experimental data and obtained parameters, they reported percent average absolute deviations in vapor pressures and liquid densities. The quality of their fit is very good, as can be usually expected for a reasonable, three parameter equation of state. However, the advantage of SAFT is the behavior of its


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parameters. This means that the SAFT unary parameters follow expected trends, which makes their estimation possible in the absence of experimental data. This is very important because engineers are often dealing with polydisperse, poorly defined pseudocomponents of real fluid mixtures, whose parameters cannot be fitted due to the absence of experimental information. The fact that the parameter values are well-behaved and suggest predictable trends upon increasing the molar mass of components in the same homologous series gives SAFT a predictive capability in the absence of experimental data.

SAFT Parameter Generalization

To understand this important concept better, it helps to remember what the

three SAFT parameters represent. The segment energy (u ko / ) and the

segment volume ( voo) are segmental parameters, which suggests that they

should remain fairly constant between components in the same homologous series. The third parameter (m) represents the number of segments on the chain; this implies that m should be proportional to the molecular mass. In the case of normal alkanes, Huang and Radosz proposed the following generalized correlations for the pure-component parameters:

nMmr = (2.43)

nMm 046647.070402.0 += (2.44)

noo Mmv 55187.0888.11 += (2.45)

[ ]n

oM

ku 013341.0exp886.260.210 −−= (2.46)

In the above expression, r is the ratio of the size parameter m divided by the polymer number-average molecular weight, nM . This is a more convenient

parameterization for SAFT, since the size of the polymer (and thus the size parameter m) changes during polymerization. Entering the r parameter instead of the size parameter m gives more flexibility to the user. Note, the r parameter cannot be used for conventional components. It can only be used for polymers, oligomers, and segments.

Equations 2.44–2.46 are implemented for calculating missing parameters of

components in a simulation. The units of voo are cm mole3 / , and the units of

u ko / are in Kelvin. The last two equations given above suggest that as nM

becomes a very large number (polymer components), voo and u ko / will

assume some limiting values. Huang and Radosz (1991) also have proposed generalized correlations for other kinds of organic compounds, such as polynuclear aromatics, n-alkylbenzenes, and others. These can be found in the original reference, and will not be reproduced here.

As mentioned earlier, the temperature dependence of the energy u in SAFT is given by the Chen and Kreglewski equation, Equation 2.31. In that equation, the parameter e/k is a constant that was related to the acentric factor and the critical temperature by Chen and Krewlewski (1977). Since, in SAFT, the


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energy parameter is between segments rather than components, Huang and Radosz set e/k=10 for all components. They only proposed a few exceptions for some small molecules: e/k=0 for argon; 1 for methane, ammonia, and

water; 3 for nitrogen; 4.2 for carbon monoxide; 18 for chlorine; 38 for CS2 ;

40 for CO2 ; and 88 for SO2 .

The three unary parameters voo, u ko / , and m for each component represent

the necessary user input to apply SAFT to real fluid systems (together with the value of e/k). For fine-tuning of mixture phase behavior, the binary

parameter kij can be regressed to available phase equilibrium data from the

literature and/or the lab.

Huang and Radosz’s (1990, 1991) version of SAFT is a homopolymer EOS model. We have implemented an empirical way of treating copolymers in Aspen Polymers using SAFT. The user can enter or regress both pure-component parameters and the binary parameter on the segment basis, rather than the polymer molecule itself. The parameters to be regressed for

the segments are the segment ratio (r), the segment volume ( 00v ), and

segment energy ( 0u ). The binary parameter, ijk , can be regressed for

segment-solvent pairs, instead of polymer-solvent pairs. Aspen Polymers then uses a segment mole fraction or weight fraction average mixing rule to calculate the copolymer SAFT pure-component parameters and the binary parameter for copolymer-solvent pairs based on the segment parameters and the copolymer composition. The details are described next.

Extension to Copolymer Systems The same expressions are used for mixtures containing copolymers. The pure-component parameters for copolymers can be input directly or calculated in terms of the relative composition of the segments or repeat units that form the copolymer. The mixing rules for the characteristic parameters of the copolymer are:

Pure Parameters

∑=Nseg

AAAp vXv 0000

∑=Nseg

AAAp uXu 00

∑=Nseg

AAAnp rXMm

Where:

00pv = Average segment volume for the copolymer

0pu = Average segment energy for the copolymer


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pm = Average segment number for the copolymer

Nseg = Number of distinct segment types present in the copolymer

AX = Segment mole fraction or weight fraction of segment type A in the copolymer; the default is the segment mole fraction and the segment weight fraction can be selected via “Option Code” in Aspen Plus

00Av = Segment volume for segment A, determined from data on the

homopolymer A

0Au = Segment energy for segment A, determined from data on the

homopolymer A

Ar = Segment ratio parameter for segment A, determined from data on the homopolymer A

nM = Number average molecular weight of the copolymer

If parameters 00pv , 0

pu and pm are provided for the polymer or oligomer, then

these have highest priority and are used for calculations. If they are not

known, usually in the case of copolymers, the user must provide 00Av , 0

Au and

Ar for the segments that compose the copolymer.


)1(000iAAAii

Nseg

AAip kuuXu −= ∑

Where:

0ipu = Cross energy parameter for a solvent-copolymer pair

0iiu = 0

iu , energy parameter for pure solvent i

0AAu = 0

Au

iAk = Binary parameter for a solvent-segment pair; it is determined from VLE or LLE data of the solvent (i)- homopolymer (A) solution


000212121

)1( pppppp uuku −=

Where:


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021 ppu = Cross energy parameter for a copolymer-copolymer pair


01pu = Energy parameter of pure copolymer 1p

02pu = Energy parameter of pure copolymer 2p


dependence:


with

refr T

TT =

Where:



SAFT EOS Model Parameters The following tables list the SAFT EOS model name and model parameters implemented in Aspen Polymers:



Upper Limit

MDS Units Keyword

Comments

SAFTM m --- --- --- X --- Unary

SAFTV voo

--- --- --- X MOLE-VOLUME

Unary

SAFTU u ko / --- --- --- X TEMP Unary

SAFTR r --- --- --- X --- Unary

SFTEPS e/k 10.0 --- --- --- --- Unary

SFTKIJ/1 ija 0.0 --- --- X --- Binary,

Symmetric

SFTKIJ/2 ijb 0.0 --- --- X --- Binary,

Symmetric

SFTKIJ/3 ijc 0.0 --- --- X --- Binary,

Symmetric

SFTKIJ/4 ijd 0.0 --- --- X --- Binary,

Symmetric


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SFTKIJ/5 ije 0.0 --- --- X --- Binary,

Symmetric

SFTKIJ/6 refT 298.15 --- --- X TEMP Binary,

Symmetric

Parameter Input and Regression Three unary parameters, SAFTR, SAFTU, and SAFTV can be:



These options are shown in priority order. For example, if property parameters are provided for a polymer component and for the segments, the polymer parameters are used and the segment parameters are ignored. If the user provides the parameters for segments only, the values for polymers will be calculated.

For each non-polymer component, these three parameters, SAFTM, SAFTV, and SAFTU must be specified. Note that SAFTR cannot be used for non-polymer components and can only be used for polymers, oligomers, and segments. The parameter SFTEPS has a default value of 10, which applies to most species, including polymers, oligomers, and segments (see text for some exceptions)

The binary parameter, SFTKIJ, can be:



These options are shown in priority order. For example, if the binary parameters are provided for both polymer-solvent pairs and segment-solvent pairs, the polymer-solvent parameters are used and the segment-solvent pairs are ignored. If the user provides the binary parameters for segment-solvent pairs only, the values for polymer-solvent pairs will be calculated.

Appendix E lists pure component parameters for some solvents (monomers) and homopolymers. If the pure parameters are not available for components in a calculation, the user can perform an Aspen Plus Regression Run (DRS) to obtain these pure component parameters. For non-polymer components, the pure component parameters are usually obtained by fitting experimental vapor pressure and liquid molar volume data. To obtain pure homopolymer (or segment) parameters, experimental data on liquid density should be regressed. Once the pure component parameters are available for a homopolymer, ideal-gas heat capacity parameters, CPIG, should be regressed for the same homopolymer using experimental liquid heat capacity data (for details on ideal-gas heat capacity for polymers, see Aspen Ideal Gas Property Model in Chapter 4). In addition to pure component parameters, SAFTU, SAFTV, and SAFTM or SAFTR, the binary parameter, SFTKIJ, for each solvent-solvent pair or each solvent-polymer (segment) pair, can be regressed using vapor-liquid equilibrium (VLE) data in the form of TPXY data in Aspen Plus.


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Missing Parameters If the user does not provide all three unary parameters for a defined conventional component or segment, the following approximated values are assumed:

• SAFTV and SAFTU will be calculated from Equations 2.45 and 2.46, respectively.

• For a conventional component, SAFTM will be calculated from Equation 2.44.

• For a segment, SAFTR will be set to a nominal value of 0.046647.

Specifying the SAFT EOS Model See Specifying Physical Properties in Chapter 1.

PC-SAFT EOS Model This section describes the Perturbed-Chain Statistical Associating Fluid Theory (PC-SAFT). This equation-of-state model is used through the POLYPCSF property method. The PC-SAFT EOS model was developed by Gross and Sadowski (2001, 2002). It was based on the well-established SAFT EOS, with some modifications on the expressions for the dispersion forces.

PC-SAFT represents an improved version of the very successful SAFT EOS. Therefore, its applicability includes fluid systems of small and/or large molecules over a wide range of temperature and pressure conditions. The big advantage of this EOS method is that it can represent the thermodynamic properties of polymer systems very well. In addition, it is better than other chain equations of state (Sanchez-Lacombe, SAFT) in describing the properties of conventional chemicals. In fact, its accuracy is comparable to, and often better than, the Peng-Robinson EOS or other similar cubic equations of state for small molecules.

The perturbation term in SAFT takes into account the attractive (dispersion) interactions between molecules. In PC-SAFT, Gross and Sadowski used the Barker-Henderson second-order perturbation theory of spherical molecules and extended it to chain molecules. The idea is that the perturbation theory concept applies to segments that are connected to chains rather than between disconnected segments, which is the case in SAFT. This is equivalent to considering attractive (dispersion) interactions between the connected segments instead of disconnected ones. For example:


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PC-SAFTSAFT

This concept offers a more realistic picture of how chain molecules, such as hydrocarbons, oligomers, and polymers, behave in a solution.

In SAFT, the perturbation (attractive) contribution is a series expansion in terms of reciprocal temperature, and each coefficient depends on density and composition. PC-SAFT expresses the attractive term of the equation as a sum of two terms (first- and second-order perturbation terms):

RTA

RTA

RTApert

21 +=

Where A denotes the Helmholtz free energy. The Helmholtz free energy is used frequently in statistical thermodynamics to express equations of state because most properties of interest, such as the system pressure, can be obtained by proper differentiation of A. The coefficients 1A and 2A have a dependence on density and composition, as well as molecular size. Gross and Sadowski (2000) obtained all the necessary constants that appear in the coefficients of the previous equation by regression of thermophysical properties of pure n-alkanes. They are reported in their original publication and thus they will not be reproduced here.

Similarly to SAFT, there are three pure-component parameters for each chemical substance:


• Segment diameter, σ • Segment energy, ε

These parameters are obtained by fitting experimental vapor pressure and liquid molar volume data for pure components. Also, a ijk binary interaction

parameter is used to fit phase equilibrium binary data; this parameter defaults to zero if not supplied.

Sample Calculation Results From the work of Gross and Sadowski, we can draw the following conclusions:

• PC-SAFT has better predictive capability for the VLE of hydrocarbon systems than SAFT.

• PC-SAFT has better predictive capability for the VLE of polymer/solvent solutions at low pressures than SAFT.


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• It also can predict the LLE of polymer solutions at high pressures better than SAFT.

• Although PC-SAFT somewhat overpredicts the critical point of pure substances, the predicted critical point is much closer to the measured value in PC-SAFT than in SAFT.

• The correlative capability of PC-SAFT is superior, especially for the phase equilibria of polymer solutions at high pressures.

The following figures demonstrate some of these remarks:

Methane-Butane VLE at 21.1 C. Predictions using ijk =0.


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0

20

40

60

80

100

120

0.0 0.2 0.4 0.6 0.8 1.0xEthane

P (bar)

PC-SAFT, kij=0

SAFT, kij=0

Raemer,Sage,1962

Ethane-Decane VLE at 238 C. Predictions using ijk =0.

Application of PC-SAFT Each species must have a set of three pure-component parameters (segment number, m, segment diameter, σ, and segment energy, ε) so the PC-SAFT EOS can calculate all its thermodynamic properties. A databank called POLYPCSF contains both pure and binary parameters available from literature; it is must be used with the property method POLYPCSF. The pure parameters available for segments are stored in the SEGMENT databank.

For components not found in the databanks, a pure-component multi-property parameter fit must be performed. In this case, you must create a Data Regression run type, create data sets for the vapor pressure, the liquid density, and the liquid heat capacity of the species of interest, and then create a regression case that regresses the PC-SAFT pure component parameters.

Note: Always supply starting values for the PC-SAFT parameters in the data regression.

Pure component parameters have been provided by Gross and Sadowski (2002) for selected polymers. They have also shown that PC-SAFT parameters follow well-behaved trends (similar to SAFT). Therefore, the parameters for a linear polyethylene can be estimated by extrapolating those of n-alkanes. The following generalized expressions are proposed by Gross and Sadowski (2001):

072.4=σ 02434.0/ == nMmr K67.269/ =kε

In the above expression, r is the ratio of the size parameter m divided by the polymer number-average molecular weight, nM . This is a more convenient


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parameterization for PC-SAFT, since the size of the polymer (and thus the size parameter m) is often unknown until after polymerization. Entering the r parameter instead of the size parameter m gives more flexibility to the user. Note, the r parameter cannot be used for conventional components. It can only be used for polymers, oligomers, and segments. The previous equation is implemented for calculating the missing parameters of components in a simulation.

The current version of PC-SAFT by Gross and Sadowski (2001, 2002) is a homopolymer EOS model. We have implemented an empirical way of treating copolymers in Aspen Polymers using PC-SAFT. The user can enter or regress both pure-component parameters and the binary parameter on the segment basis, rather than the polymer molecule itself. The parameters to be regressed for the segments are the segment ratio, r, the segment diameter, σ, and the segment energy, ε/k. The binary parameter, ijk , can be regressed

for segment-solvent pairs, instead of polymer-solvent pairs. A segment mole fraction /or weight fraction average mixing rule is then used by Aspen Polymers to calculate the copolymer PC-SAFT pure-component parameters and the binary parameter for copolymer-solvent pairs based on the segment parameters and the copolymer composition. The details are described next.

Extension to Copolymer Systems The same expressions are used for mixtures containing copolymers. The pure-component parameters for copolymers can be input directly or calculated in terms of the relative composition of the segments or repeat units that form the copolymer. The mixing rules for the characteristic parameters of the copolymer are:

Pure Parameters

∑=Nseg

AAAp X σσ

∑=Nseg

AAAp X εε

∑=Nseg

AAAnp rXMm

Where:

pσ = Average segment diameter for the copolymer

pε = Average segment energy for the copolymer

pm = Average segment number for the copolymer

Nseg = Number of distinct segment types present in the copolymer


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AX = Segment mole fraction or weight fraction of segment type A in the copolymer; the default is the segment mole fraction and the segment weight fraction can be selected via “Option Code” in Aspen Plus

Aσ = Segment diameter for segment A, determined from data on the homopolymer A

Aε = Segment energy for segment A, determined from data on the homopolymer A

Ar = Segment ratio parameter for segment A, determined from data on the homopolymer A


If parameters pσ , pε and pm are provided for the polymer or oligomer, then

these have highest priority and are used for calculations. If they are not known, usually in the case of copolymers, the user must provide Aσ , Aε and

Ar for the segments that compose the copolymer.


)1( iAAAii

Nseg

AAip kX −= ∑ εεε

Where:

ipε = Cross energy parameter for a solvent-copolymer pair

iiε = iε , energy parameter for pure solvent i

AAε = Aε

iAk = Binary parameter for a solvent-segment pair; it is determined from VLE or LLE data of the solvent (i)- homopolymer (A) solution


212121)1( pppppp k εεε −=

Where:

21 ppε = Cross energy parameter for a copolymer-copolymer pair


1pε = Energy parameter of pure copolymer 1p


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2pε = Energy parameter of pure copolymer 2p


dependence:


with

refr T

TT =

Where:



PC-SAFT EOS Model Parameters The following table lists the PC-SAFT EOS model parameters implemented in Aspen Polymers:



Upper Limit

MDS Units Keyword

Comments

PCSFTM m --- --- --- X --- Unary

PCSFTV σ --- --- --- X --- Unary

PCSFTU ε/k --- --- --- X TEMP Unary

PCSFTR r --- --- --- X --- Unary

PCSKIJ/1 ija 0.0 --- --- X --- Binary,

Symmetric

PCSKIJ/2 ijb 0.0 --- --- X --- Binary,

Symmetric

PCSKIJ/3 ijc 0.0 --- --- X --- Binary,

Symmetric

PCSKIJ/4 ijd 0.0 --- --- X --- Binary,

Symmetric

PCSKIJ/5 ije 0.0 --- --- X --- Binary,

Symmetric

PCSKIJ/6 refT 298.15 --- --- X TEMP Binary,

Symmetric


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Parameter Input and Regression Three unary parameters, PCSFTR, PCSFTU, and PCSFTV can be:



These options are shown in priority order. For example, if property parameters are provided for a polymer component and for the segments, the polymer parameters are used and the segment parameters are ignored. If the user provides the parameters for segments only, the values for polymers will be calculated.

For each non-polymer component, these three parameters (PCSFTM, PCSFTU, and PCSFTV) must be specified. Note that PCSFTR cannot be used for non-polymer components and can only be used for polymers, oligomers, and segments.

The binary parameter, PCSKIJ, can be:



These options are shown in priority order. For example, if the binary parameters are provided for both polymer-solvent pairs and segment-solvent pairs, the polymer-solvent parameters are used and the segment-solvent pairs are ignored. If the user provides the binary parameters for segment-solvent pairs only, the values for polymer-solvent pairs will be calculated.

The databank POLYPCSF contains both unary and binary PC-SAFT parameters available from literature; it must be used with the POLYPCSF property method. If the pure parameters are not available for components in a calculation, the user can perform an Aspen Plus Regression Run (DRS) to obtain these pure component parameters. For non-polymer components, the pure component parameters are usually obtained by fitting experimental vapor pressure and liquid molar volume data. To obtain pure homopolymer (or segment) parameters, experimental data on liquid density should be regressed. Once the pure component parameters are available for a homopolymer, ideal-gas heat capacity parameters, CPIG, should be regressed for the same homopolymer using experimental liquid heat capacity data (for details on ideal-gas heat capacity for polymers, see Aspen Ideal Gas Property Model in Chapter 4). In addition to pure component parameters, PCSFTU, PCSFTV, and PCSFTM or PCSFTR, the binary parameter, PCSKIJ, for each solvent-solvent pair or each solvent-polymer (segment) pair, can be regressed using vapor-liquid equilibrium (VLE) data in the form of TPXY data in Aspen Plus.


Specifying the PC-SAFT EOS Model See Specifying Physical Properties in Chapter 1.


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Copolymer PC-SAFT EOS Model This section describes the Copolymer Perturbed-Chain Statistical Associating Fluid Theory (PC-SAFT). This equation-of-state model is used through the PC-SAFT property method.

The copolymer PC-SAFT represents the completed PC-SAFT EOS model developed by Sadowski and co-workers (Gross and Sadowski, 2001, 2002a, 2002b; Gross et al., 2003; Becker et al., 2004; Kleiner et al., 2006). Unlike the PC-SAFT EOS model (POLYPCSF) in Aspen Plus, the copolymer PC-SAFT includes the association and polar terms and does not apply mixing rules to calculate the copolymer parameters from its segments. Its applicability covers fluid systems from small to large molecules, including normal fluids, water, alcohols and ketones, polymers and copolymers and their mixtures.

Description of Copolymer PC-SAFT

Fundamental equations

The copolymer PC-SAFT model is based on the perturbation theory. The underlying idea is to divide the total intermolecular forces into repulsive and attractive contributions. The model uses a hard-chain reference system to account for the repulsive interactions. The attractive forces are further divided into different contributions, including dispersion, polar and association. Using a generated function, ψ , the copolymer PC-SAFT model in general can be written as follows:

polarassocdisphc ψψψψψ +++= (2.47)

where hcψ , dispψ , assocψ , and polarψ are contributions due to hard-chain fluids, dispersion, association, and polarity, respectively.

The generated functionψ is defined as follows:

∫ −==ρ

ρρψ

0

)1( dZRTa

m

res

(2.48)

where resa is the molar residual Helmholtz energy of mixtures, R is the gas constant, T is the temperature, ρ is the molar density, and mZ is the

compressibility factor; resa is defined as:

,...),,(,...),,( iig

ires xTaxTaa ρρ −= (2.49)

where a is the Helmholtz energy of a mixture and iga is the Helmholtz energy of a mixture of ideal gases at the same temperature, density and composition ix . Once ψ is known, any other thermodynamic function of

interest can be easily derived. For instance, the fugacity coefficient iϕ is

calculated as follows:


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mm

xTjjj

xTii ZZ

xx

xjkij

ln1ln,,,,

−−+⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

−⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+=≠≠

∑ρρ

ψψψϕ (2.50)

with

ixTmZ

,

1 ⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+=ρψρ (2.51)

where ix∂

∂ψis a partial derivative that is always done to the mole fraction

stated in the denominator, while all other mole fractions are considered constant.

Applying ψ to Equations 2.7, 2.8, and 2.9, departure functions of enthalpy, entropy, and Gibbs free energy can be obtained as follows:

Enthalpy departure:

( ) ⎥⎦⎤

⎢⎣⎡ −+

∂∂

−=− )1( migmm Z

TTRTHH ψ

(2.52)

Entropy departure:

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛−⎥⎦

⎤⎢⎣⎡ +

∂∂

−−=− refmigmm p

pRZT

TRSS lnlnψψ (2.53)

Gibbs free energy departure:

( ) ( )[ ] ⎟⎟⎠

⎞⎜⎜⎝

⎛+−−+=− refmm

igmm p

pRTZZRTGG lnln1ψ (2.54)

The following thermodynamic conditions must be satisfied:

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+=− ∑ refi

ii

igmm p

pRTxRTGG lnlnϕ (2.55)

( ) ( ) ( )igmm

igmm

igmm SSTHHGG −−−=− (2.56)

Hard-chain fluids and chain connectivity

In PC-SAFT model, a molecule is modeled as a chain molecule by a series of freely-jointed tangent spheres. The contribution from hard-chain fluids as a reference system consists of two parts, a nonbonding contribution (i.e., hard-sphere mixtures prior to bonding to form chains) and a bonding contribution due to chain formation:

chainhshc m ψψψ += (2.57)

where m is the mean segment in the mixture, hsψ is the contribution from

hard-sphere mixtures on a per-segment basis, and chainψ is the contribution


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due to chain formation. Both m and hsψ are well-defined for mixtures containing polymers, including copolymers; they are given by the following equations:

ii

imxm ∑= (2.58)

⎥⎥⎦

⎤

⎢⎢⎣

⎡−⎟⎟

⎠

⎞⎜⎜⎝

⎛−+

−+

−= )1ln(

)1()1(31

3023

32

233

32

3

21

0

ξξξξ

ξξξ

ξξξ

ξψ hs (2.59)

∑=α

αii mm (2.60)

3,2,1,0,6

== ∑∑ ndzmx niii

iin α

ααρπξ (2.61)

i

ii m

mz α

α = (2.62)

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛−−=

kTd i

iiα

ααε

σ 3exp12.01 (2.63)

where αim , ασ i , and αε i , are the segment number, the segment diameter,

and the segment energy parameter of the segment typeα in the copolymer component i , respectively. The segment number αim is calculated from the

segment ratio parameter αir :

ααα iii Mrm = (2.64)

where αiM is the total molecular weight of the segment type α in the

copolymer component i and can be calculated from the segment weight fraction within the copolymer:

iii MwM αα = (2.65)

where αiw is the weight fraction of the segment type α in the copolymer

component i , and iM is the molecular weight of the copolymer component

i .

Following Sadowski and co-worker’s work (Gross et al., 2003; Becker et al., 2004), the contribution from the chain connectivity can be written as follows:

∑ ∑∑= =

−−=i

iihs

iiiiiichain dgBmx )(ln)1( ,,

1 1, βαβα

γ

α

γ

ββαψ (2.66)

with

11 1

, =∑∑= =

γ

α

γ

ββα iiB (2.67)


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33

22

2

23

2

3,, )1(

2)1(

3)1(

1)(ξξ

ξξ

ξ βα

βα

βα

βαβαβα −⎟

⎟⎠

⎞⎜⎜⎝

⎛

++

−⎟⎟⎠

⎞⎜⎜⎝

⎛

++

−=

ji

ji

ji

jiji

hsji dd

dddd

dddg (2.68)

where βα iiB , is defined as the bonding fraction between the segment type α

and the segment type β within the copolymer component i , γ is the number of the segment types within the copolymer component i , and

)( ,, βαβα jihs

ji dg is the radial distribution function of hard-sphere mixtures at

contact.

However, the calculation for βα iiB , depends on the type of copolymers. We

start with a pure copolymer system which consists of only two different types of segments α and β ; Equation 2.66 becomes:

[ ])(ln)(ln)()(ln)1( ββββββαβαββααβααααααψ dgBdgBBdgBm hshshschain +++−−= (2.69)

with

1=+++ βββααβαα BBBB (2.70)

βα mmm += (2.71)

We now apply Equations 2.69-2.71 to three common types of copolymers; a) alternating, b) block, and c) random.

For an alternating copolymer, βα mm = ; there are no αα or ββ adjacent

sequences. Therefore:

1,0 ==== αββαββαα BBBB (2.72)

)(ln)1( αβαβψ dgm hschain −−= (2.73)

For a block copolymer, there is only one αβ pair and the number of αα and

ββ pairs depend on the length of each block; therefore:

0,1

1,11

,11

=−

=−

−=

−−

= βααββ

ββα

αα Bm

Bm

mB

mm

B (2.74)

For a random copolymer, the sequence is only known in a statistical sense. If the sequence is completely random, then the number of αβ adjacent pairs is proportional to the product of the probabilities of finding a segment of type α and a segment of type β in the copolymer. The probability of finding a

segment of type α is the fraction of α segments αz in the copolymer:

mm

z a=α (2.75)

The bonding fraction of each pair of types can be written as follows:

βαβααββββααα zCzBBCzBCzB ==== ,, 22 (2.76)


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where C is a constant and can be determined by the normalization condition set by Equation 2.70; the value for C is unity. Therefore:

βαβααββββααα zzBBzBzB ==== ,, 22 (2.77)

A special case is the Sadowski’s model for random copolymer with two types of segments only (Gross et al., 2003; Becker et al., 2004). In this model, the bonding fractions are calculated as follows:

When αβ zz <

0,1,1

=−−=−

== βββααβααβ

βααβ BBBBmm

BB (2.78)

When βα zz <

βααβββααα

βααβ BBBBmm

BB −−==−

== 10,1

(2.79)

The generalization of three common types of copolymers from two types of different segments to multi types of different segments γ within a copolymer is straightforward.

For a generalized alternative copolymer, γβαmmmm r ==== ... ; there are no

adjacent sequences for the same type of segments. Therefore,

1,)1(

+=−

= αβγαβ m

mB (2.80)

γβαγ

γαβ ==

−−

= ,1,)1(m

mB (2.81)

1,0 +>= αβαβB (2.82)

αβαβ ≤= ,0B (2.83)

1)1()1(

)1(,1

1

11,

1 1=

−−

+−−

=+= ∑∑∑−

=+

= = mm

mmBBB

γγ

γγ

γ

γ

ααα

γ

α

γ

βαβ (2.84)

For a generalized block copolymer, there is only one pair for each adjacent type of segment pairs ( βα ≠ ) and the number of pairs for a same type depends on the length of the block; therefore:

γαααα ,...2,1,

11

=−−

=mm

B (2.85)

1,1

1+=

−= αβαβ m

B (2.86)

1,0 +>= αβαβB (2.87)


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αβαβ <= ,0B (2.88)

11)1(1

11

1

11,

11 1=⎥

⎦

⎤⎢⎣

⎡−+−

−=+= ∑∑∑∑∑

=

−

=+

== =

γ

αα

γ

ααα

γ

ααα

γ

α

γ

βαβ γm

mBBB (2.89)

For a generalized random copolymer, the sequence is only known in a statistical sense. If the sequence is completely random, then the number of αβ adjacent pairs is proportional to the product of the probabilities of finding

a segment of type α and a segment of type β in the copolymer. The

probability of finding a segment of type α is the fraction of α segments αz

in the copolymer:

γααα ,...2,1, ==

mm

z (2.90)

The bonding fraction of each pair of types can be written as follows:

γβαβααβ ,...2,1, == zCzB (2.91)

where C is a constant and can be determined by the normalization condition set by Equation 2.67. Therefore,

11 11 1

== ∑∑∑∑= == =

γ

α

γ

ββα

γ

α

γ

βαβ zzCB (2.92)

That is,

∑∑= =

= γ

α

γ

ββα

1 1

1

zzC (2.93)

Put C into Equation 2.91, we obtain:

γβαγ

α

γ

ββα

βααβ ,...2,1,,

1 1

==

∑∑= =

zz

zzB (2.94)

Dispersion term

The equations for the dispersion term are given as follows:

YICmXIdisp2112 πρπρψ −−= (2.95)

∑ ∑ ⎟⎟⎠

⎞⎜⎜⎝

⎛=

ijji

jijijiji kT

zzmmxxX 3,

,βα

βα

αββα σ

ε (2.96)

3,

2,

βαβα

αββα σ

εji

jijijij

iji kT

zzmmxxY ⎟⎟⎠

⎞⎜⎜⎝

⎛= ∑∑ (2.97)


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17

11 )(),( −

=∑= l

ll mamI ηη (2.98)

17

12 )(),( −

=∑= l

ll mbmI ηη (2.99)

[ ]

1

2

32

4

1

1

)2)(1()2122720()1(

)1()4(21

1

−

−

⎭⎬⎫

⎩⎨⎧

−−−+−

−+−

−+=

⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂++=

ηηηηηη

ηηη

ρρ

mm

ZZChc

hc

(2.100)

llll am

mm

mam

maa 321211

⎟⎠⎞

⎜⎝⎛ −

⎟⎠⎞

⎜⎝⎛ −

+−

+= (2.101)

llll bm

mm

mbm

mbb 321211

⎟⎠⎞

⎜⎝⎛ −

⎟⎠⎞

⎜⎝⎛ −

+−

+= (2.102)

3ξη = (2.103)

where βασ ji , and βαε ji , are the cross segment diameter and energy

parameters, respectively; only one adjustable binary interaction parameter,

βακ ji , is introduced to calculate them:

)(21

, βαβα σσσ jiji += (2.104)

2/1,, ))(1( βαβαβα εεκε jijiji −= (2.105)

In above equations, the model constants la1 , la2 , la3 , lb1 , lb2 , and lb3 are

fitted to pure-component vapor pressure and liquid density data of n-alkanes (Gross and Sadowski, 2001).

Association term for copolymer mixtures – 2B model

The association term in PC-SAFT model in general needs an iterative procedure to calculate the fraction of a species (solvent or segment) that are bounded to each association-site type. Only in pure or binary systems, the fraction can be derived explicitly for some specific models. We start with general expressions for the association contribution for copolymer systems as follows:

⎟⎟⎠

⎞⎜⎜⎝

⎛+−= ∑ ∑∑ 2

12

lnα

αα

α

ψi

ii

AA

i A

Ai

assoc XXNx (2.106)

where A is the association-site type index, αiAN is the association-site number of the association-site type A on the segment type α in the

copolymer component i , and αiAX is the mole fraction of the segment type


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α in the copolymer component i that are not bonded with the association-site type A ; it can be estimated as follows:

∑ ∑∑ Δ+=

j B

BABBj

A

jiji

i

XNxX

β

βαβα

α

ρ11

(2.107)

with

⎥⎥⎦

⎤

⎢⎢⎣

⎡−⎟

⎟⎠

⎞⎜⎜⎝

⎛=Δ 1exp)( 3

,,, kTdg

jijiji

BA

jiBA

jihs

jiBA

βαβαβα εσκ βαβαβα (2.108)

where βακ ji BA is the cross effective association volume and βαε ji BA

is the cross association energy; they are estimated via simple combination rules:

3

)()(

2/)( ⎥⎥⎦

⎤

⎢⎢⎣

⎡

+=

βα

βα

σσ

σσκκκ βαβα

ji

jiABABBA jiji (2.109)

2

)()( βαβα εεε

jiji

ABABBA +

= (2.110)

where ακ iAB)( and αε iAB)( are the effective association volume and the association energy between the association-site types A and B , of the segment type α in the copolymer component i , respectively.

The association-site number of the site type A on the segment type α in the copolymer component i is equal to the number of the segment type α in the copolymer component i ,

α

α

α

αα

α

MMw

MM

NN iiii

Ai === (2.111)

where αiN is the number of the segment type α in the copolymer

component i and αM is the molecular weight of the segment type α . In

other words, the association-site number for each site type within a segment is the same; therefore, we can rewrite Equations 2.107 and 2.108 as follows:

∑ ∑ ∑ ⎟⎟⎠

⎞⎜⎜⎝

⎛+−=

i A

AA

iiassoc

ii

XXNxα

α

ααψ

21

2ln (2.112)

∑ ∑ ∑ Δ+=

j B

BABjj

A

jij

i

XNxX

ββ

βαβ

α

ρ11

(2.113)

To calculate αiAX , Equation 2.113 has to be solved iteratively for each association-site type associated with a species in a component. In practice, further assumption is needed for efficiency. The commonly used model is the so-called 2B model (Huang and Radosz, 1990). It assumes that an associating species (solvent or segment) has two association sites, one is designed as the


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site type A and another as the site type B . Similarly to the hydrogen bonding, type A treats as a donor site with positive charge and type B as an acceptor site with negative charge; only the donor-acceptor association bonding is permitted and this concept applies to both pure systems (self-association such as water) and mixtures (both self-association and cross-association such as water-methanol). Therefore, we can rewrite Equations 2.112 and 2.113 as follows:

∑ ∑ ⎥⎦⎤

⎢⎣⎡ ++−=

i

BABAii

assoc iiii XXXXNxα

αααααψ 1)(

21)ln( (2.114)

∑ ∑ Δ+=

j

BABjj

A

jij

i

XNxX

ββ

βαβ

α

ρ11

(2.115)

∑ ∑ Δ+=

j

ABAjj

B

jij

i

XNxX

ββ

βαβ

α

ρ11

(2.116)

It is easy to show that

βαβα jiji BAAB Δ=Δ (2.117)

Therefore

αα ii BA XX = (2.118)

∑ ∑ ⎟⎟⎠

⎞⎜⎜⎝

⎛+−=

i

AA

iiassoc

ii

XXNxα

α

ααψ

21

2ln2 (2.119)

∑ ∑ Δ+=

j

BAAjj

A

jij

i

XNxX

ββ

βαβ

α

ρ11

(2.120)

Polar term

The equations for the polar term are given by Jog et al (2001) as follows:

23

2

/1 ψψψ

ψ−

=polar (2.121)

∑ ∑−=ij ji

jijpipjijiji d

xxzzmmxxkTI

3,

22

22

2 )()()(

)(9

2

βα

βα

αββαβα

μμηρπψ (2.122)

∑ ∑=ijk kikjji

kjikpjpipkjikjikji ddd

xxxzzzmmmxxxkTI

γαγββα

γβαγ

αβγβαγβα

μμμηρπψ,,,

222

33

22

3 )()()()(

)(1625

(2.123)

2/)(, βαβα jiji ddd += (2.124)


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In the above equations, )(2 ηI and )(3 ηI are the pure fluid integrals and αμ i

and αipx )( are the dipole moment and dipolar fraction of the segment type α

within the copolymer component i , respectively. Both ( )kTi /2αρμ and

( )3,

2 / βααμ jii kTd are dimensionless. In terms of them, we can have:

∑ ∑ ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−=

ij ji

jijpipjijiji kTdkT

xxzzmmxxI 3,

22

22 )()(9

2

βα

β

αβ

αβαβα

μρμπψ (2.125)

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛×

= ∑ ∑

γαγββα

γβα

γαβγ

βαγβα

μρμρμ

πψ

kikjji

kji

ijkkpjpipkjikjikji

ddkTdkTkT

xxxzzzmmmxxxI

,,,

222

3

2

3 )()()(1625

(2.126)

Rushbrooke et al. (1973) have shown that

2*

3*2***

2 )5236.01(1078.03205.03618.01)(

ρρρρρ

−+−−

=I (2.127)

2**

2***

320059.059056.0111658.062378.01)(

ρρ

ρρρ+−

−+=I (2.128)

πηρ 6* = (2.129)

In terms of η , )(2 ηI and )(3 ηI are computed by the expressions:

2

32

2 )1(75097.016904.169099.01)(

ηηηηη

−+−−

=I (2.130)

2

2

3 73166.012789.1142523.019133.11)(

ηηηηη

+−−+

=I (2.131)

Copolymer PC-SAFT EOS Model Parameters Pure parameters. Each non-association species (solvent or segment) must have a set of three pure-component parameter; two of them are the segment diameter σ and the segment energy parameter ε . The third parameter for a solvent is the segment number m and for a segment is the segment ratio parameter r . For an association species, two additional parameters are the

effective association volume )( ABκ and the association energy )( ABε . For a polar species, two additional parameters are the dipole moment μ and the

segment dipolar fraction px .


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Binary parameters

There are three types of binary interactions in copolymer systems: solvent-solvent, solvent-segment, and segment-segment. The binary interaction parameter βακ ji , allows complex temperature dependence:

2,,,,,, ln/ rjirjirjirjijiji TeTdTcTba βαβαβαβαβαβακ ++++= (2.132)

with

refr T

TT = (2.133)

where refT is a reference temperature and the default value is 298.15 K.

The following table lists the copolymer PC-SAFT EOS model parameters implemented in Aspen Plus:



Upper Limit

MDS Units Keyword

Comments

PCSFTM m — — — X — Unary

PCSFTV σ — — — X — Unary

PCSFTU k/ε — — — X TEMP Unary

PCSFTR r — — — X — Unary

PCSFAU kAB /ε — — — X TEMP Unary

PCSFAV ABκ — — — X — Unary

PCSFMU μ --- --- --- X DIPOLE MOMENT

Unary

PCSFXP px --- --- --- X --- Unary

PCSKIJ/1 βα jia , 0.0 — — X — Binary,

Symmetric

PCSKIJ/2 βα jib , 0.0 — — X — Binary,

Symmetric

PCSKIJ/3 βα jic , 0.0 — — X — Binary,

Symmetric

PCSKIJ/4 βα jid , 0.0 — — X — Binary,

Symmetric

PCSKIJ/5 βα jie , 0.0 — — X — Binary,

Symmetric

PCSKIJ/6 refT 298.15 — — X TEMP Binary,

Symmetric

Parameter input and regression

Since the copolymer PC-SAFT is built based on the segment concept, the unary (pure) parameters must be specified for a solvent or a segment. Specifying a unary parameter for a polymer component (homopolymer or copolymer) will be ignored by the simulation. For a non-association and non-


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polar solvent, three unary parameters PCSFTM, PCSFTU, and PCSFTV must be specified. For a non-association and non-polar segment, these three unary parameters PCSFTR, PCSFTU, and PCSFTV must be specified. For an association species (solvent or segment), two additional unary parameters PCSFAU and PCSFAV must be specified. For a polar species (solvent or segment), two additional unary parameters PCSFMU and PCSFXP must be specified.

The binary parameter PCSKIJ can be specified for each solvent-solvent pair, or each solvent-segment pair, or each segment-segment pair. By default, the binary parameter is set to be zero.

A databank called PC-SAFT contains both unary and binary PC-SAFT parameters available from literature; it must be used with the PC-SAFT property method. The unary parameters available for segments are stored in the SEGMENT databank. If unary parameters are not available for a species (solvent or segment) in a calculation, the user can perform an Aspen Plus Data Regression Run (DRS) to obtain unary parameters. For non-polymer components (mainly solvents), the unary parameters are usually obtained by fitting experimental vapor pressure and liquid molar volume data. To obtain unary parameters for a segment, experimental data on liquid density of the homopolymer that is built by the segment should be regressed. Once the unary parameters are available for a segment, the ideal-gas heat capacity parameter CPIG may be regressed for the same segment using experimental liquid heat capacity data for the same homopolymer. In addition to unary parameters, the binary parameter PCSKIJ for each solvent-solvent pair, or each solvent-segment pair, or each segment-segment pair, can be regressed using vapor-liquid equilibrium (VLE) data in the form of TPXY data in Aspen Plus.

Note: In Data Regression Run, a homopolymer must be defined as an OLIGOMER type, and the number of the segment that builds the oligomer must be specified.

Option Codes for PC-SAFT The copolymer PC-SAFT has three option codes.

Option code 1. The user can use this option code to specify the copolymer type. The default type is the random copolymer (0). Other types are the alternative copolymer (1) and the block copolymer (2). All other values are assigned to the random copolymer.

Option code 2. This option code is restricted to the Sadowski’s copolymer model in which a copolymer must be built only by two different types of segments (Gross and Sadowski, 2003; Becker et al., 2004). In order to use the Sadowski’s copolymer model, this option code must be set to one.

Option code 3. The user can use this option code to turn off the association term from the copolymer PC-SAFT model by setting a non-zero value.


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Sample Calculation Results In Figure 1, Aspen Plus applies the PC-SAFT EOS model to calculate both vapor-liquid and liquid-liquid equilibria for methanol-cyclohexane mixtures at p = 1.013 bar. This mixture exhibits an azeotropic vapor-liquid equilibrium at higher temperatures and shows a liquid-liquid equilibrium at lower temperatures. Both pure and binary parameters used are taken directly from the paper by Gross and Sadowski (2002b). The results show that the PC-SAFT model with the association term included can correlate phase equilibrium data well for associating mixtures.

Figure 1. Isobaric vapor-liquid and liquid-liquid equilibria of methanol-cyclohexane at p = 1.013 bar. Experimental data are taken from Jones and Amstell (1930) and Marinichev and Susarev (1965).

Figure 2 shows a model calculation for HDPE-Hexane mixtures. This system exhibits both lower critical solution temperature (LCST) and upper critical solution temperature (UCST) at p = 50 bar. The pure parameters are taken directly from papers Gross and Sadowski (2001; 2002a). The binary parameter between hexane and ethylene segment is set to 0.012. The phase equilibrium calculations are carried by Flash3 block with Gibbs flash algorithm in Aspen Plus.


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Liquid-liquid equilibria of HDPE-Hexane

0

50

100

150

200

250

0 0.1 0.2 0.3 0.4 0.5 0.6

HDPE weight fraction

Tem

pera

ture

(C)

UCST

LCST

Figure 2. Liquid-liquid equilibria of HDPE-Hexane mixtures in a weight fraction-pressure plot by PC-SAFT EOS model. It shows both lower critical solution temperature (LCST) and upper critical solution temperature (UCST).


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Figure 3 shows the vapor-liquid equilibrium of the mixture water-acetone at p = 1.703 bar. The dashed line represents PC-SAFT calculations where water is treated as an associating component and acetone as a polar component; the cross association in the mixture is not considered ( 15.0−=ijκ ). The solid line

represents PC-SAFT calculations where the cross association between water and acetone is accounted for ( 055.0−=ijκ ) using a simple approach by

Sadowski & Chapman et al. (2006). In this approach, the association energy and effective volume parameters of the non-associating component (acetone) are set to zero and to the value of the associating component (water), respectively. Further, the polar component is represented by the three pure-component parameters without using the dipolar model.

Figure 3. Vapor-liquid equilibrium of the mixture water-acetone at p = 1.703 bar. Experimental data are taken from Othmer and Morley (1946).


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Figure 4 shows the liquid-liquid equilibria of polypropylene (PP)-n-pentane at three temperatures in a pressure-weight fraction plot. The weight average molecular weight of PP is 2.2/,/4.50 == nww MMmolkgM . Both pure and

binary parameters used are taken directly from the paper by Gross and Sadowski (2002a).

Liquid-liquid equilibria of PP-n-Pentane

0

10

20

30

40

50

60

70

80

0 0.05 0.1 0.15 0.2 0.25 0.3

PP weight fraction

Pres

sure

bar

PC-SAFTData (T=187 C)

Data (T=177 C)Data (T=197 C)

Figure 4. Liquid-liquid equilibria of PP-n-Pentane at three different temperatures. Comparison of experimental cloud points (Martin et al., 1999)

to PC-SAFT calculations ( 0137.0=ijκ ). The polymer was assumed to be

monodisperse at molkgM w /4.50= .

Specifying the Copolymer PC-SAFT EOS Model See Specifying Physical Properties in Chapter 1.


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References Alder, B. J., Young, D. A., & Mark, M. A. (1972). Studies in Molecular Dynamics. X. Corrections to the Augmented van der Waals Theory for the Square-Well Fluid. J. Chem. Phys., 56, 3013.

Aspen Physical Property System Physical Property Methods and Models. Cambridge, MA: Aspen Technology, Inc.

Behme, S., Sadowski, G., Song, Y., & Chen, C.-C. (2003). Multicomponent Flash Algorithm for Mixtures Containing Polydisperse Polymers. AIChE J., 49, 258.

Becker, F., Buback, M., Latz, H., Sadowski, G., & Tumakaka, F. (2004). Cloud-Point Curves of Ethylene-(Meth)acrylate Copolymers in Fluid Ethene up to High Pressures and Temperatures – Experimental Study and PC-SAFT Modeling. Fluid Phase Equilibria, 215, 263-282.

Behme, S., Sadowski, G., Song, Y., & Chen, C.-C. (2003). Multicomponent Flash Algorithm for Mixtures Containing Polydisperse Polymers. AIChE J., 49, 258.

Carnahan, N. F., & Starling, K. E. (1972). Intermolecular Repulsions and the Equation of State for Fluids. AIChE J., 18, 1184.

Chapman, W. G., Gubbins, K. E., Jackson, G., & Radosz, M. (1989). Fluid Phase Equilibria, 52, 31.

Chapman, W. G., Gubbins, K. E., Jackson, D., & Radosz, M. (1990). A New Reference Equation of State for Associating Liquids. Ind. Eng. Chem. Res., 29, 1709.

Chen S.-J., Economou, I. G., & Radosz, M. (1992). Density-Tuned Polyolefin Phase Equilibria. 2. Multicomponent Solutions of Alternating Poly(Ethylene-Propylene) in Subcritical and Supercritical Solvents. Experiment and SAFT Model. Macromolecules, 25, 4987.

Chen, S. S., & Kreglewski, A. (1977). Applications of the Augmented van der Waals Theory of Fluids I. Pure Fluids. Ber. Bunsenges. Phys. Chem., 81, 1048.

Fischer, K., & Gmehling, J. (1996). Further development, status and results of the PSRK method for the prediction of vapor-liquid equilibria and gas solubilities. Fluid Phase Eq., 121, 185.

Folie, B., & Radosz, M. (1995). Phase Equilibria in High-Pressure Polyethylene Technology. Ind. Eng. Chem. Res., 34, 1501.

Gregg, C. J., Stein, F. P., & Radosz, M. (1994). Phase Behavior of Telechelic Polyisobutylene (PIB) in Subcritical and Supercritical Fluids. 1. Inter- and Intra-Association Effects for Blank, Monohydroxy, and Dihydroxy PIB(1K) in Ethane, Propane, Dimethyl Ether, Carbon Dioxide, and Chlorodifluoromethane. Macromolecules, 27, 4972.

Gross, J., & Sadowski, G. (2001). Perturbed-Chain SAFT: An Equation of State Based on a Perturbation Theory for Chain Molecules. Ind. Eng. Chem. Res., 40, 1244-1260.


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Gross, J., & Sadowski, G. (2002a). Modeling Polymer Systems Using the Perturbed-Chain Statistical Associating Fluid Theory Equation of State. Ind. Eng. Chem. Res., 41, 1084-1093.

Gross, J., & Sadowski, G. (2002b). Application of the Perturbed-Chain SAFT Equation of State to Associating Systems. Ind. Eng. Chem. Res., 41, 5510-5515.

Gross, J., Spuhl, O., Tumakaka, F., & Sadowski, G. (2003). Modeling Copolymer Systems Using the Perturbed-Chain SAFT Equation of State. Ind. Eng. Chem. Res., 42, 1266-1274.

Hasch, B. M., & McHugh, M. A. (1995). Calculating Poly(ethylene-co-acrylic acid)-Solvent Phase Behavior with the SAFT Equation of State. J. Pol. Sci.:B: Pol. Phys., 33, 715.

Hasch, B. M, Meilchen, M. A., Lee, S.-H., & McHugh, M. A. (1992). High-Pressure Phase Behavior of Mixtures of Poly(Ethylene-co-Methyl Acrylate) with Low-Molecular Weight Hydrocarbons. J. Pol. Sci., 30, 1365-1373.

Holderbaum, T., & Gmehling, J. (1991). PSRK: A group contribution equation of state based on UNIFAC. Fluid Phase Eq., 70, 251.

Huang, S. H., & Radosz, M. (1990). Equation of State for Small, Large, Polydisperse, and Associating Molecules. Ind. Eng. Chem. Res., 29, 2284.

Huang, S. H., & Radosz, M. (1991). Equation of State for Small, Large, Polydisperse, and Associating Molecules: Extension to Fluid Mixtures. Ind. Eng. Chem. Res., 30, 1994.

Jog, P. K., Sauer, S. G., Blaesing, J., & Chapman, W. G. (2001), Application of Dipolar Chain Theory to the Phase Behavior of Polar Fluids and Mixtures. Ind. Eng. Chem. Res., 40, 4641.

Jones, D. C., & Amstell, S. (1930). J. Chem. Soc., 1316.

Kleiner, M., Tumakaka, F., Sadowski, G., Dominik, A., Jain, S., Bymaster, A., & Chapman, W. G. (2006). Thermodynamic Modeling of Complex Fluids using PC-SAFT. Final Report for Consortium of Complex Fluids. Universität Dortmund & Rice University.

Kleiner, M., Tumakaka, F., Sadowski, G., Latz, H., & Buback, M. (2006).Phase Equilibria in Polydisperse and Associating Copolymer Solutions: Poly(ethane-co-(meth)acrylic acid) – Monomer Mixtures. Fluid Phase Equilibria, 241, 113-123.

Kontogeorgis, G. M., Harismiadis, V. I., Frendenslund, Aa., & Tassios, D. P. (1994). Application of the van der Waals Equation of State to Polymers. I. Correlation. Fluid Phase Equilibria, 96, 65-92.

Lambert, S. M., Song, Y., & Prausnitz, J. M. (2000). Equations of State for Polymer Systems. In J. V. Sengers, R. F. Kayer, C. J. Peters, and H. J. White, (Eds.), Equations of State for Fluids and Fluid Mixtures. (pp. 523-588) New York: Elsevier Science.

Lee, S.-H., Hasch, B. M., & McHugh, M. A. (1996). Calculating Copolymer Solution Behavior with Statistical Associating Fluid Theory. Fluid Phase Equil., 117, 61.


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Lee, S.-H., LoStracco, M. A., & McHugh, M. A. (1994). High-Pressure, Molecular-Weight Dependent Behavior of (Co)polymer-Solvent Mixtures: Experiments and Modeling. Macromolecules, 27, 4652.

Mansoori, G. A., Carnahan, N. F., Starling, K. E., & Leland, T. W. J. (1971). J. Chem. Phys., 54, 1523.

Marinichev, A.N., & Susarev, M.P. (1965). Zh. Prikl. Khim., 38, 1619.

Martin, T. M., Lateef, A. A., & Roberts, C. B. (1999). Measurements and modeling of cloud point behavior for polypropylene/n-pentane and polypropylene/n-pentane/carbon dioxide mixtures at high pressures. Fluid Phase Equilibria, 154, 241.

Mathias, P. M., & Copeman, T. W. (1983). Extension of the Peng-Robinson equation of state to complex mixtures: evaluation of the various forms of the local composition concept. Fluid Phase Eq., 13, 91.

Orbey, H., Bokis, C. P., & Chen, C.-C. (1998a). Polymer-Solvent Vapor-Liquid Equilibrium: Equations of State versus Activity Coefficient Models. Ind. Eng. Chem. Res., 37, 1567-1573.

Orbey, H., Bokis, C. P., & Chen, C.-C. (1998b). Equation of State Modeling of Phase Equilibrium in the Low-Density Polyethylene Process: The Sanchez-Lacombe, Statistical Associating Fluid Theory, and Polymer-Soave-Redlich-Kwong Equation of State. Ind. Eng. Chem. Res., 37, 4481-4491.

Orbey, H., & Sandler, S. I. (1995). On the combination of equation of state and excess free energy models. Fluid Phase Eq., 111, 53.

Orbey, H., & Sandler, S. I. (1997). A comparison of Huron-Vidal type mixing rules of compounds with large size differences, and a new mixing rule. Fluid Phase Eq., 132, 1.

Othmer, D. F., & Morley, F. R. (1946). Composition of Vapors from Boiling Binary Solutions – Apparatus for Determinations under Pressure. Ind. Eng. Chem., 38, 751-757.

Pradham, D., Chen, C.-K., & Radosz, M. (1994). Fractionation of Polystyrene with Supercritical Propane and Ethane: Characterization, Semibatch Solubility Experiments, and SAFT Simulations. Ind. Eng. Chem. Res., 33, 1984.

Rodgers, P. A. (1993). Pressure-Volume-Temperature Relationships for Polymeric Liquids: A Review of Equations of State and Their Characteristic Parameters for 56 Polymers. J. of Applied Polymer Science, 48, 1061-1080.

Rushbrooke, G. S., & Stell, G., Hoye, J. S. (1973), Molec. Phys., 26, 1199.

Sanchez, I. C., & Lacombe, R. H. (1976). An Elementary Molecular Theory of Classical Fluids. Pure Fluids. J. Phys. Chem., 80, 2352-2362.

Sanchez, I. C., & Lacombe, R. H. (1978). Statistical Thermodynamics of Polymer Solutions. Macromolecules, 11, 1145-1156.

Sanchez, I. C. (1992). Polymer Phase Separation. In Encyclopedia of Physical Science and Technology, 13. New York: Academic Press.

Saraiva A., Kontogeorgis, G. M., Harismiadis, V. I., Fredenslund, Aa., & Tassios, D. P. (1996). Application of the van der Waals Equation of State to Polymers IV. Correlation and Prediction of Lower Critical Solution Temperatures for Polymer Solutions. Fluid Phase Equilibria, 115, 73-93.


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Teja, A. S., Lee, R. J., Rosenthal, D.J., & Anselme, M. (1990). Correlation of the critical properties of alkanes and alkanols. Fluid Phase Eq., 56, 153.

Topliss, R. J. (1985). Techniques to Facilitate the Use of Equations of State for Complex Fluid-Phase Equilibria. Ph.D. Dissertation, University of California, Berkeley.

Tsonopoulos, C., & Tan, Z. (1993). The critical constants of normal alkanes from methane to polyethylene. II. Application of the Flory theory. Fluid Phase Eq., 83, 127.

Wei, Y. S., & Sadus, R. J. (2000). Equations of State for Calculation of Fluid-Phase Equilibria. AIChE Journal, 46, 169.

Wertheim, M. S. (1984). Fluids with Highly Directional Attractive Forces. II. Thermodynamic Perturbation Theory and Integral Equations. J. Stat. Phys., 35, 35.

Wertheim, M. S. (1986a). Fluids with Dimerizing Hard Spheres, and Fluid Mixtures of Hard Spheres and Dispheres. J. Stat. Phys., 85, 2929.

Wertheim, M. S. (1986b). Fluids with Highly Directional Attractive Forces. IV. Equilibrium Polymerization. J. Stat. Phys., 42, 477.

Xiong, Y., & Kiran, E. (1995). Comparison of Sanchez-Lacombe and SAFT Model in Predicting Solubility of Polyethylene in High-Pressure Fluids. J. Appl. Pol. Sci., 55, 1805.


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3 Activity Coefficient Models 87

3 Activity Coefficient Models

This chapter discusses thermodynamic properties of polymer systems from activity coefficient models. Activity coefficient models are used in Aspen Polymers (formerly known as Aspen Polymers Plus) to calculate liquid activity coefficients, liquid excess Gibbs free energy, liquid excess enthalpy, and liquid excess entropy of mixtures.


• About Activity Coefficient Models, 87

• Phase Equilibria Calculated from Activity Coefficient Models, 88

• Other Thermodynamic Properties Calculated from Activity Coefficient Models, 90

• Mixture Liquid Molar Volume Calculations, 92

• Related Physical Properties in Aspen Polymers, 93

• Flory-Huggins Activity Coefficient Model, 94

• Polymer NRTL Activity Coefficient Model, 98

• Electrolyte-Polymer NRTL Activity Coefficient Model, 103

• Polymer UNIFAC Activity Coefficient Model, 114

• Polymer UNIFAC Free Volume Activity Coefficient Model, 117

About Activity Coefficient Models In general, the activity coefficient models are versatile, accommodating a high degree of solution nonideality into the model. On the other hand, when applied to VLE calculations, they can only be used for the liquid phase and another model (usually an equation of state) is needed for the vapor phase.

Activity coefficient models usually perform well for systems of polar compounds at low pressures and away from the critical region. They are the best way to represent highly non-ideal liquid mixtures at low pressures. They are used for the calculation of fugacity, enthalpy, entropy and Gibbs free energy. Usually an empirical correlation is used in parallel for the calculations of density when an activity coefficient model is used in phase equilibrium modeling.


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88 3 Activity Coefficient Models

There are a large number of activity coefficient models for use in polymer process modeling. Aspen Polymers offers:

• Flory-Huggins model (Flory, 1953)

• Non-Random Two-Liquid (NRTL) Activity Coefficient model adopted to polymers (Chen, 1993)

• Polymer UNIFAC model

• UNIFAC free volume model (Oishi & Prausnitz, 1978)

• The two UNIFAC models are predictive while the Flory-Huggins and Polymer-NRTL models are correlative. Between the correlative models, the Flory-Huggins model is only applicable to homopolymers because its parameter is polymer-specific. The Polymer-NRTL model is a segment-based model that allows accurate representation of the effects of copolymer composition and polymer chain length.

Phase Equilibria Calculated from Activity Coefficient Models The activity coefficient model can be related to the fugacity of liquid phase through fundamental thermodynamic equation:

liii

li fxf *,γ=

Where:

lif

= Fugacity of component i in the liquid phase

ix = Mole fraction of component i in the liquid phase

iγ = Activity coefficient of component i in the liquid phase

lif*,

= Liquid phase reference fugacity of component i

In the equation above, the activity coefficient, iγ , represents the deviation of

the mixture from ideality, and the liquid phase reference fugacity, lif*, , is

defined as that of the pure liquid i at the temperature and pressure of the mixture. The activity coefficient, iγ , is obtained from an activity coefficient

model, as shown in the following sections.

Vapor-Liquid Equilibria in Polymer Systems In the activity coefficient approach, the basic vapor-liquid equilibrium relationship is represented by:

liiii

vi fxpy *,γϕ = (3.1)


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The vapor phase fugacity coefficient, viϕ , is computed from an equation of

state (see Chapter 2).The liquid activity coefficient, γ i , is computed from an activity coefficient model.

Liquid Phase Reference Fugacity

The liquid phase reference fugacity, lif*, , is generally expressed as:

li

li

vi

li pf *,*,*,*, θϕ= (3.2)

With

⎟⎠⎞

⎜⎝⎛= ∫

p

p

li

li l

i

dpVRT *,

*,*, 1expθ (3.3)

Where:

vi*,ϕ

= Fugacity coefficient of pure component i at the system temperature and the vapor pressure of component i, as calculated from the vapor phase equation of state

pil*,

= Liquid vapor pressures of component i at the system temperature

li*,θ

= Poynting correction of component i for pressure

liV *,

= Liquid molar volume of component i at T and p

However, Equations 3.2 and 3.3 are applicable only to solvents, light polymers and oligomers (volatile) in the mixture. For other components such as heavy polymers and oligomers (nonvolatile) and dissolved gases in the mixture, the liquid phase reference fugacity, l

if*, , has to be computed in

different ways:

• For nonvolatile polymers or oligomers (used in Data Regression) : These components exist only in the liquid phase. Therefore, the vapor-liquid equilibrium condition given by Equation 3.1 does not apply to them. Their mole fractions in the liquid phase at VLE can be determined by the mass balance condition.

• For dissolved gases: Light gases (such as O2 and N 2 ) are usually supercritical at the temperature and pressure of the solution. In this case pure component vapor pressure is meaningless and, therefore, cannot serve as the reference fugacity. The reference state for a dissolved gas is redefined to be at infinite dilution and at the temperature and pressure of

the mixtures. The liquid phase reference fugacity, f il*, , becomes Hi (the

Henry's constant for component i in the mixture).

The activity coefficient, γ i , is converted to the infinite dilution reference state through the relationship:

( )∞=ii i γγγ *

(3.4)

Where:


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∞i

γ

= Infinite dilution activity coefficient of component i in the mixture

By this definition γi

* approaches unity as xi approaches zero. The phase

equilibrium relationship for dissolved gases becomes:

iiiivi Hxpy *γϕ = (3.5)

To compute Hi , you must supply the Henry's constant for the dissolved-gas component i in each subcritical solvent component.

Liquid-Liquid Equilibria in Polymer Systems The basic liquid-liquid-vapor equilibrium relationship is:

pyfxfx ivi

li

li

li

lli

li i

ϕγγ == *,*, 2211

(3.6)

For liquid-liquid equilibria, the vapor phase term can be omitted, and the pure component liquid fugacity cancels out:

2211 li

li

li

li xx γγ = (3.7)

Where:

1liγ

= Activity coefficient of component i in the liquid phase 1l

2liγ

= Activity coefficient of component i in the liquid phase 2l

1lix

= Mole fraction of component i in the liquid phase 1l

2lix

= Mole fraction of component i in the liquid phase 2l

Unlike Equation 3.1 for vapor-liquid equilibria, Equation 3.7 applies to each component of mixtures in two-coexisting liquid phases.

Other Thermodynamic Properties Calculated from Activity Coefficient Models The activity coefficient model can be related to other properties through fundamental thermodynamic equations. These properties (called excess liquid functions) are relative to the ideal liquid mixture at the same condition:

• Excess molar liquid Gibbs free energy:

∑=i

iilE

m xRTG γln, (3.8)

• Excess molar liquid enthalpy:


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∑−=i

ii

lEm T

xRTH∂

γ∂ ln2,

(3.9)

• Excess molar liquid entropy:

⎥⎦⎤

⎢⎣⎡

∂∂

+−= ∑ TTxRS i

ii

ilE

mγ

γln

ln,

(3.10)

Where:

lEmG ,

= Excess molar liquid Gibbs free energy of the mixture

lEmH ,

= Excess molar liquid enthalpy of the mixture

lEmS ,

= Excess molar liquid entropy of the mixture

The excess liquid functions given by Equations 3.8–3.10 are calculated from the same activity coefficient model. In practice, however, the activity coefficient iγ is often derived first from the excess liquid Gibbs free energy of

a mixture from an activity coefficient model:

ijnpTi

lEm

i nnG

RT≠

⎥⎦

⎤⎢⎣

⎡∂

∂=

,,

, )(1lnγ

(3.11)

With

idmixingmixing

lEm GGnG Δ−Δ=,

(3.12)

ii

iidmixing xnG ln∑=Δ

(3.13)

Where:

n = Total mole number of the mixture

in = Mole number of component i in the mixture

mixingGΔ

= Liquid Gibbs free energy of mixing; it is defined as the difference between the Gibbs free energy of the mixture and that of the pure components

idmixingGΔ

= Ideal Gibbs free energy of mixing

Once the excess liquid functions are known, the thermodynamic properties of liquid mixtures can be computed as follows:

∑ +=i

lEm

lii

lm HHxH ,*,

(3.14)

ii

ilE

ml

ii

ilm xxRTGxG ln,*, ∑∑ ++= μ

(3.15)

( )lm

lm

lm GH

TS −=

1 (3.16)


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Where:

lmH = Liquid mixture molar enthalpy

lmG = Liquid mixture molar Gibbs free energy

lmS = Liquid mixture molar entropy

liH *,

= Liquid pure component enthalpy

li*,μ

= Liquid pure component Gibbs free energy

In Equations 3.14 and 3.15, the first terms are the ideal mixing terms and the second terms come from the excess functions. The last term in Equation 3.15 represents the Gibbs free energy of mixing for ideal gases. For non-polymer components, Aspen Plus provides the standard correlation model such as the DIPPR method to calculate l

iH *, and li*,μ . For more information, see Aspen

Physical Property System Physical Property Methods and Models. Aspen Polymers provides the van Krevelen liquid property models to calculate l

iH *,

and li*,μ for polymers, oligomers, and segments. For more information, see

Chapter 4.

Mixture Liquid Molar Volume Calculations In Aspen Plus, when an activity coefficient model or a cubic equation-of-state model is used, an empirical correlation method is used in parallel for calculating liquid density of both pure components and mixtures. This concept is extended to cover polymer and oligomer components and polymer mixtures in Aspen Polymers. The liquid molar default route uses the van Krevelen model or the Tait model to calculate the liquid molar volume of pure polymers, oligomers, and segments. The Rackett model is used to calculate the liquid molar volume of non-polymer components. The mixture liquid molar volume is calculated using the ideal mixing rule:

lp

pp

sm

lm VxVV *,∑+=

With

),,( ' pTxRackettV ss

m =

∑=s

sss xxx /'

1=+ ∑∑p

ps

s xx

Where:


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lmV = Liquid mixture molar volume

smV = Liquid polymer-free mixture molar volume

lpV *, = Liquid molar volume of a polymer or oligomer component in the

mixture

px = Liquid mole fraction of a polymer or oligomer component in the mixture

sx = Liquid mole fraction of a solvent component in the mixture

'sx = Liquid mole fraction of a solvent component in the polymer-free

mixture

The liquid polymer-free mixture molar volume, smV , is calculated using the

Rackett model. For more information, see Aspen Physical Property System Physical Property Methods and Models. The liquid molar volume of a polymer or oligomer component, l

pV *, , is calculated using either the van Krevelen

model or the Tait model. For more information, see Chapter 4.

Related Physical Properties in Aspen Polymers The following properties are related to activity coefficient models in Aspen Polymers:

Property Name

Symbol Description

GAMMA iγ

Liquid activity coefficient of a component in a mixture

HLMX lmH

Liquid mixture molar enthalpy

SLMX lmS

Liquid mixture molar entropy

GLMX lmG

Liquid mixture molar Gibbs free energy

HLXS lEmH ,

Liquid mixture molar excess enthalpy

GLXS lEmG ,

Liquid mixture molar excess Gibbs free energy

SLXS lEmS ,

Liquid mixture molar excess entropy

The following table provides an overview of the activity coefficient models available in Aspen Polymers. This table lists the Aspen Physical Property System model names, and their possible use in liquid phase for mixtures. Details of individual models are presented next.

Models Model Name

Phase(s) Pure Mixture Properties Calculated


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Flory-Huggins GMFH l — X GAMMA, HLXS, GLXS, SLXS

Polymer NRTL GMNRTLP l — X GAMMA, HLXS, GLXS, SLXS

Electrolyte-Polymer NRTL

GMEPNRTL l — X GAMMA, HLXS, GLXS, SLXS

Polymer UNIFAC GMPOLUF l — X GAMMA, HLXS, GLXS, SLXS

Polymer UNIFAC Free Volume

GMUFFV l — X GAMMA, HLXS, GLXS, SLXS


Flory-Huggins Activity Coefficient Model This section describes the Flory-Huggins activity coefficient model available in the POLYFH physical property method. The Flory-Huggins model gives good results if the interaction parameter χ is known accurately at the particular physical states of the system, i.e., temperature, composition, and polymer molecular weight. According to the Flory-Huggins theory, the χ parameter should be independent of polymer concentration and of polymer molecular weight. In reality, it is shown to vary significantly with both.

The model works well if the interaction parameter at a low solvent concentration is used to estimate the activity coefficient at a higher solvent concentration. However, extrapolations to low solvent concentrations using χ based on a higher solvent concentration can lead to significant errors.

Finally, the Flory-Huggins model is not very accurate for polar systems, and unless it is used with a cubic-equation-of-state, it should not be used for phase equilibrium calculations at high pressures.

Flory (1941) and Huggins (1941) independently derived an expression for the combinatorial entropy of mixing of polymer molecules with monomer molecules based on the lattice theory of fluids. This statistical approach, widely used for liquid mixtures, takes into account the unequal size of the molecules and the linkage between flexible segments on the polymer chains. The enthalpy of mixing and the energetic interactions between the molecules are quantified through an interaction parameter χ for each molecule-molecule pair. (See Polymer NRTL Activity Coefficient Model on page 98 for a relationship of χ to NRTL interaction parameters.)

Consider a binary mixture with components differing significantly in molecular size: a polymer and a spherical solvent. To obtain the mixing properties of this system, Flory and Huggins applied a lattice model to this system. The combinatorial and non-combinatorial properties of the mixture are derived by arranging both polymer and solvent on the lattice. Each solvent molecule occupies one lattice site. Each polymer molecule is divided into m flexible segments and each segment occupies one lattice site.


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Gibbs free energy of mixing

Based on statistical arguments and several assumptions, the Gibbs free energy of mixing is derived as follows for a binary system:

( )mnnmRT

Gmixing2121122

211 lnln +⎟⎟

⎠

⎞⎜⎜⎝

⎛++=

Δφφχφ

φφφ

(3.17)

With:

21

11 mnn

n+

=φ (3.18)

21

22 mnn

mn+

=φ (3.19)

Where:

12χ = Molecular interaction parameter

m = Number of segments in the polymer molecule

n1 = Number of moles of solvent in the mixture

n2 = Number of polymer molecules in the mixture

1φ , 2φ = Mole fractions on a segment basis

If m is set equal to the ratio of molar volumes of polymer and solvent, then

1φ and 2φ are the volume fractions.

If m is set equal to the ratio of molecular weight of polymer and solvent, then

1φ and 2φ are the weight fractions.

Therefore, the Gibbs free energy of mixing equation, Equation 3.17, is a generalized form that can be expanded to three different equations with φ being the segment-based mole fraction, volume fraction or weight fraction, depending on how m is defined. These three equations can be accessed in the Flory-Huggins model using option codes.

Option codes 1, 2, and 3, correspond to the weight basis, segment mole basis and volume basis, respectively. Option code 2 (segment basis) is the default.

A large portion of experimental polymer solution phase equilibria data in the open literature are reported using a volume fraction basis. The volume fraction basis allows users to directly apply interaction parameters from literature to their simulation. There are, however, situations where neither the segment-based mole fraction basis nor the volume fraction basis are appropriate. This is the case for many industrial processes of polymer mixtures. In such situations composition is usually known on a weight basis. Unlike segment mole fraction, component weight fraction remains consistent regardless of how the polymer segments are defined.

Multicomponent Mixtures


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The derivation of Flory and Huggins has been extended to cover multiple components (Tompa, 1956):

ii

iij

jiijii i

imixing mnmRT

G∑∑∑ ⎟⎟

⎠

⎞⎜⎜⎝

⎛Δ

<

+ln = φφχφφ

(3.20)

From this equation, one can derive the activity coefficient of a component (for example, Equation 3.11):

⎥⎥⎦

⎤

⎢⎢⎣

⎡+⎟

⎟⎠

⎞⎜⎜⎝

⎛−−+= ∑∑

>k

jkjjkij

jjji

i

ii m

mx

φφχχφφ

γ 11lnln (3.21)

Where:

xi = Mole fraction of component I

ijχ = Interaction binary parameter

In the above equations, note that iφ can be calculated on three different

basis: segment-based mole fraction, volume fraction, and weight fraction, as given in the next table for three option codes. However, im is treated

independently as a pure component characteristic size parameter regardless of what option basis is used for calculating iφ ; it is related to the degree of

polymerization by:

iiii Psm ε*= (3.22)

Where:

Pi = Degree of polymerization

is and iε = Empirical parameters

si and ε i account for deviation of the component characteristic size from its degree of polymerization. Users may use these parameters singly or in combination to adjust the component characteristic size. By default Pi is 1.0 for small molecules.

The binary interaction parameter, ijχ , accounts for the enthalpic effects on

mixing. It is strongly temperature dependent:

2ln/ rijrijrijrijijij TeTdTcTba ++++=χ (3.23)

with

refr T

TT =

Where:

refT = Reference temperature and the default value = 1 K for compatibility with previous releases.


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A summary of equations for the three options for concentration basis of the Flory-Huggins model is given here :

Option Description Concentration Characteristic Size

1 Mass Basis:

iw = Mass fraction

iM = Number average molecular

weight for polymer/oligomer; molecular weight for conventional component

i

jjj

iii w

MnMn

==∑

φ

iiii Psm ε*=

2 Segment mole fraction basis:

in = Number of moles

iP = Number average chain length ∑

=

jjj

iii Pn

Pnφ

iiii Psm ε*=

3 Volume basis:

iV = Molar volume ( kmolm /3 )

iv = Specific volume ( kgm /3 )

iw = Mass fraction

∑∑==

jjj

ii

jjj

iii vw

vwVn

Vnφ

iiii Psm ε*=

Note that for monomers and solvents, 0.1== POLDPPi unless changed by the

user. si and ε i are defaulted to be unity for all components. For option code 2

(segment-based mole fraction), Equation 3.21 reduces to the original Flory-Huggins equation for the solvent activity coefficient.

Flory-Huggins Model Parameters The following table lists the input parameters for the Flory-Huggins model. These parameters would normally be regressed from experimental data.



Upper Limit

MDS Units Keyword

Comments

FHCHI/1 ija

0.0 -100 100 X --- Binary,

Symmetric

FHCHI/2 ijb

0.0 -1E6 1E6 X --- Binary,

Symmetric

FHCHI/3 ijc

0.0 -1E6 1E6 X --- Binary,

Symmetric

FHCHI/4 ijd

0.0 -1E6 1E6 X --- Binary, Symmetric

FHCHI/5 ije

0.0 -1E6 1E6 X --- Binary, Symmetric

FHCHI/6 refT

1.0 -1E6 1E6 X TEMP Binary,

Symmetric


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FHSIZE/1 si 1.0 1E-15 1E15 X --- Unary

FHSIZE/2 iε

1.0 -1E10 1E10 X --- Unary

POLDP* Pi 1.0 1.0 1E10 --- --- Unary

* The actual degree of polymerization is used for polymer components.

Specifying the Flory-Huggins Model See Specifying Physical Properties in Chapter 1.

Polymer NRTL Activity Coefficient Model This section describes the Polymer NRTL activity coefficient model available in the POLYNRTL physical property method. The polymer NRTL activity coefficient model is an extension of the NRTL model for low molecular weight compounds (Chen, 1993; Renon & Prausnitz, 1968). The main difference between this model and the Flory-Huggins model is that in the polymer NRTL activity coefficient model the binary interaction parameters are relatively independent of polymer concentration and polymer molecular weight. Furthermore, in the case of copolymers, the polymer NRTL binary parameters are independent of the relative composition of the repeat units on the polymer chain. This model can be used in a correlative mode at low and moderate pressures for a wide variety of fluids, including polar systems.

The current model does not address the free volume term or the so-called equation-of-state term, and strong orientational interactions, such as hydrogen bonding, as part of the entropy of mixing. As a result, the models cannot be used to represent lower critical solution temperature.

The polymer NRTL model is a segment-based local composition model for the Gibbs free energy of mixing of polymer solutions. It represents a synergistic combination of the Flory-Huggins description for the entropy of mixing molecules of different sizes and the Non-Random Two Liquid theory for the enthalpy of mixing solvents and polymer segments. It reduces to the well-known NRTL equation if no polymers are present in the system.

The NRTL model is known to be one of the most widely used activity coefficient models. It has been used to represent phase behavior of systems with nonelectrolytes and electrolytes. The polymer NRTL model is an extension of the NRTL model from systems of small molecules to systems with both small molecules and macromolecules. It requires the solvent-solvent, solvent-segment, and segment-segment binary parameters. The solvent-solvent binary parameters can be readily obtained from systems of monomeric molecules. Many such solvent-solvent binary parameters are available in the literature. Furthermore, the solvent-segment binary


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parameters have the desirable characteristic that they are relatively independent of temperature, chain length, and polymer concentration.

The polymer NRTL model provides a flexible thermodynamic framework to correlate the phase behavior of polymer solutions. The model can be used to represent vapor-liquid equilibrium and liquid-liquid equilibrium of polymer systems.

Polymer NRTL Model In the Polymer NRTL model (GMNRTLP), the Gibbs free energy of mixing of a polymer solution is expressed as the sum of the entropy of mixing, based on the Flory-Huggins equation, and the enthalpy of mixing, based on the Non-Random Two Liquid theory.

The reference states for the polymer NRTL equation are pure liquids for solvents and a hypothetical segment aggregate state for polymers. In this hypothetical aggregate state, all segments are surrounded by segments of the same type. The following is the equation for the Gibbs free energy of mixing:

RS

RTH

RTG FH

mixingNRTLmixingmixing Δ

−Δ

=Δ

Gibbs free energy of mixing

II

I

jjij

jjijij

ip,i

pp

jjsj

jjsjsj

ss

mixing

lnn

Gx

Gxrn

Gx

Gxn

RTG

φ

ττ

∑

∑∑

∑∑∑∑

∑

+

+=Δ

With:

∑∑∑

=

J jJjJ

IIiI

i rX

rXx

,

,

∑=

JJ

II n

nX

)exp( jijijiG τα−=

RTgg iiji

ji

)( −=τ

∑=

JJJ

III mn

mnφ

Where:


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I and J = Component based indices

i and j = Segment based indices

s = Solvent component

p = Polymer component

sn = Number of mole of solvent component s

pn

= Number of mole of polymer component p

ix = Segment based mole fraction for segment based species i

XI = Mole fraction of component I in component basis

ri I, = Number of segment type i in component I

jiα = NRTL non-random factor

jiτ = Interaction parameter

g ji = Energies of interaction between j-i pairs of segment based species

gii = Energies of interaction between i-i pairs of segment based species

nI = Number of moles of component I

φ I = Volume fraction (approximated as segment mole fraction) of component I

Im = Ratio of polymer molar volume to segment molar volume of component I

The species i and j can be solvent molecules or segments.

The excess Gibbs free energy expression is obtained by subtracting the ideal Gibbs free energy of mixing from the Gibbs free energy of mixing equation:

⎟⎟⎠

⎞⎜⎜⎝

⎛+

+=

∑

∑∑

∑∑∑∑

∑

I

I

II

jjij

jjijij

ipi

pp

jjsj

jjsjsj

ss

lEm

Xn

Gx

Gxrn

Gx

Gxn

RTnG

φ

ττ

ln

,

,

The activity coefficient of each component in the polymer solution can also be considered as the sum of two contributions:

FHI

NRTLII γγγ lnlnln +=

With:

∑ ⎟⎟⎠

⎞⎜⎜⎝

⎛−+⎟⎟

⎠

⎞⎜⎜⎝

⎛=

J J

JI

I

IFHI m

mX

φφγ 1lnln


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Where:

Jm = Characteristic size of component J

Jm is related to the degree of polymerization by:

JJJJ Psm ε*=

Where:

JP = Degree of polymerization

Js and Jε = Empirical parameters

Js and Jε account for deviation of the component characteristic size from its

degree of polymerization. These parameters can be used singularly or in combination to adjust the component characteristic size. By default JP is 1.0

for small molecules.

Solvent Activity Coefficient

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−+=

∑∑

∑ ∑∑∑

=

kkjk

kkjkjk

sjj

kkjk

sjj

kksk

jjsjsj

NRTLsI Gx

Gx

GxGx

Gx

Gx ττ

τγln

Polymer Activity Coefficient

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−+=

∑∑

∑ ∑∑∑

∑=

kkjk

kkjkjk

ijj

kkjk

ijj

kkik

jjijij

ipi

NRTLpI Gx

Gx

GxGx

Gx

Gxr

ττ

τγ ,ln

The activity coefficient of a polymer component given by this last equation needs to be further normalized so that NRTL

pI =γ becomes unity as 1→pX (i.e.,

pure polymer); it can be easily done as follows:

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−+−

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−+=

∑∑

∑ ∑∑∑

∑

∑∑

∑ ∑∑∑

∑=

kkjpk

kkjkjpk

ijj

kkjpk

ijpj

kkipk

jjijipj

ipi

kkjk

kkjkjk

ijj

kkjk

ijj

kkik

jjijij

ipi

NRTLpI

Gx

Gx

GxGx

Gx

Gxr

Gx

Gx

GxGx

Gx

Gxr

,

,

,

,

,

,

,

,ln

ττ

τ

ττ

τγ

With

∑=

jpj

pipi r

rx

,

,,

Where:


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pix , = Segment mole fraction of type i in polymer component p

It is often useful for the case of homopolymers to establish a relationship between the NRTL interaction parameters and the Flory-Huggins χ parameter:

( ) ( )IJIJ

IJIJ

JIJI

JIJIIJ G

GG

Gφφ

τφφ

τχ

++

+=

Where:

IJχ = Solvent-polymer Flory-Huggins binary interaction parameter

NRTL Model Parameters The polymer NRTL model requires two binary interaction parameters, τij and

τ ji , for the solvent-solvent interactions, the solvent-segment interactions,

and the segment-segment interactions. These binary interaction parameters become the correlation variables in representing the thermodynamic properties of polymer solutions. The binary interaction parameters have the following features:

• The model automatically retrieves the NRTL binary interaction parameters from the Aspen Plus databank for standard components when they are available.

• The binary parameters allow complex temperature dependence:

τij ijij

ij ijabT

e T f T= + + +ln

• The non-randomness factor ijα is allowed to be temperature dependent:

α ij ij ijc d T= + −( . )27315

Typically, the temperature dependency is weak and ijα is mainly

influenced by ijc . The default value for ijc is 0.3, and ijα increases as the

association between molecules increases.

The input parameters for the polymer NRTL model are summarized in the following table. These parameters are normally regressed from experimental data.



Upper Limit

MDS Units Keyword

Comments

NRTL/1 aij 0 --- --- X --- Binary, Asymmetric

NRTL/2 bij 0 --- --- X TEMP Binary, Asymmetric

NRTL/3 cij 0.3 --- --- X --- Binary, Symmetric


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NRTL/4 dij 0 --- --- X 1/TEMP Binary, Symmetric

NRTL/5 eij 0 --- --- X --- Binary, Asymmetric

NRTL/6 fij 0 --- --- X 1/TEMP Binary, Asymmetric

NRTL/7 minT 0 --- --- X TEMP Unary

NRTL/8 maxT 1000 --- --- X TEMP Unary

FHSIZE/1 si 1.0 1E-15 1E15 X --- Unary

FHSIZE/2 iε

1.0 -1E10 1E10 X --- Unary

POLDP† iP ‡ 1.0 1.0 1E10 --- --- Unary

† The number-average degree of polymerization is used for polymer and oligomer components.

‡ For monomers, unless changed by the user, 0.1== POLDPPi .

Specifying the Polymer NRTL Model See Specifying Physical Properties in Chapter 1.

Electrolyte-Polymer NRTL Activity Coefficient Model The Electrolyte-Polymer Non-Random Two-Liquid (EP-NRTL) activity coefficient model is an integration of the electrolyte NRTL model for electrolytes (Chen et al., 1982, 1999; Chen & Evans, 1986) and the polymer NRTL model (Chen, 1993) for oligomers and polymers. The model is used to compute activity coefficients for polymers, solvents, and ionic species (Chen & Song 2004).

This integrated electrolyte-polymer NRTL model is designed to represent the excess Gibbs free energy of aqueous organic electrolytes and complex chemical systems with the presence of oligomers, polymers and electrolytes. The model incorporates the segment-based local composition concept of the polymer NRTL model into the electrolyte NRTL model. From the Gibbs free energy expression, one can compute activity coefficients for various species as functions of compositions and molecular structure of oligomers, polymers, solvents, and electrolytes.

As an integrated model, the electrolyte-polymer NRTL model reduces to the electrolyte NRTL model in the absence of polymers or oligomers. The model reduces to the polymer NRTL model in the absence of electrolytes. Furthermore, the model reduces to the original NRTL model (Renon & Prausnitz, 1968) if neither electrolytes nor polymers or oligomers are present.


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As such, this model is a very versatile activity coefficient model. Note that this model does not address the solution nonideality of polyelectrolytes, which are further characterized by counterion condensation (Manning, 1979), an intramolecular phenomenon that closely resembles micelle formation.

The excess Gibbs free energy expression for the electrolyte-polymer NRTL model contains three contributions:

• Long-range ion-ion interactions that exist beyond the immediate neighborhood of an ionic species

• Local interactions that exist at the immediate neighborhood of any species

• Entropy of mixing polymeric species as described by the Flory-Huggins equation.

The model uses pure liquid at the system temperature and pressure as the reference state for solvents. For ions, the reference state is at infinite dilution in water at the system temperature and pressure. In the case of mixed-solvent electrolytes, the Born equation is added to account for the Gibbs free energy of transfer of ionic species from the infinite dilution state in the mixed solvent to the infinite dilution state in aqueous phase (Mock et al., 1986).

To account for the long-range ion-ion interactions, the model uses the unsymmetric Pitzer-Debye-Hückel (PDH) expression (Pitzer, 1973). To account for the local interactions, the model uses the segment-based local composition (lc) concept as given by the polymer NRTL expression. This local composition term is first developed as a symmetric expression that envisions a hypothetical reference state of pure, completely dissociated, segment-based liquid species. It is then normalized using “infinite-dilution activity coefficient in water” terms for each solute species, including ions, in order to obtain an expression based on the unsymmetric convention.

The model retains the two fundamental assumptions regarding the local composition of electrolyte solutions:

• The like-ion repulsion assumption: this states that the local composition of cations around cations is zero (and likewise for anions around anions). Here cations refer to either monomeric cations or cationic segments. The same is true for anions.

• The local electroneutrality assumption: this states that the distribution of cations and anions around a central molecular species is such that the net local ionic charge is zero. As before, here cations and anions refer to either monomeric ones or ionic segments.

In summary, the integrated model has four terms, which are discussed later in this chapter:

• Pitzer-Debye-Hückel term

• Born term

• Local composition term

• Flory-Huggins term

RTg

RTg

RTg

RTg

RTg

RTG FHexlcexBornexPDHexexlE

m,*,*,*,**,*

+++==


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Note: Using * to denote an unsymmetric reference state is well accepted in electrolyte thermodynamics and will be maintained here. In this case, * does not refer to a pure component property, as it does in other sections of this document.

Following this equation, the ionic activity coefficient is the sum of four terms, which are discussed later in this chapter:

• Pitzer-Debye-Hückel term activity coefficient

• Born term activity coefficient

• Local composition term activity coefficient

• Flory-Huggins term activity coefficient

FHI

lcI

Borni

PDHII

***** lnlnlnlnln γγγγγ +++=

Mean ionic activity coefficients and molality scale mean ionic activity coefficients can then be computed by the following expressions:

( )*** lnln1ln aaccac

γυγυυυ

γ ++

=±

( )( )1000/1lnlnln ** mM acBm υυγγ ++−= ±±

Where:

*±γ = Mean ionic activity coefficient

*m±γ = Molality scale mean ionic coefficient

cυ = Cationic stoichiometric coefficient

aυ = Anionic stoichiometric coefficient

= Molecular weight of the solvent B

m = Molality

Long-Range Interaction Contribution The Pitzer-Debye-Hückel formula, normalized to mole fractions of unity for solvent and zero for electrolytes, is used to represent the long-range interaction contribution:

Pitzer-Debye-Hückel Term

( )21

21

1ln41000,*

xx

B

PDHex

IIA

MRTg ρ

ρϕ +⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−=

With

23

21 2

31

10002

⎟⎟⎠

⎞⎜⎜⎝

⎛ε

⎟⎠⎞

⎜⎝⎛ π

=ϕ kTQdNAw

eA


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∑=i

iix zxI 22

1

Where:

= Debye-Hückel parameter

= Ionic strength (mole fraction scale)

ρ = "Closest approach" parameter

= Avogadro's number

d = Density of solvent

= Electron charge

= Dielectric constant of water

T = Temperature

k = Boltzmann constant

= Segment-based mole fraction of component i (i can be a monomeric species or a segment)

= Charge number of component i

Pitzer-Debye-Hückel Term Activity Coefficient

Taking the appropriate derivative of the Pitzer-Debye-Hückel term, an expression for the activity coefficient can then be derived:

( )⎥⎥⎦

⎤

⎢⎢⎣

⎡

+−

++⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−=

21

23

21

21

21

12

1ln21000ln

22*

x

xxix

i

B

PDHi I

IIzI

zA

M ρρ

ργ ϕ

For oligomeric ions, we sum up the contributions from various ionic segments of species I:

∑ ∑+=c a

PDHIaIa

PDHcIc

PDHI rr *

,,*

,* lnlnln γγγ

Where:

Icr , = Number of cationic segments in species I

Iar , = Number of anionic segments in species I

Born Term

The Born equation is used to account for the Gibbs free energy of transfer of ionic species from the infinite dilution state in a mixed-solvent to the infinite dilution state in aqueous phase:

2

22,*

10112

−

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

∑i

iii

w

eBornex

r

zx

kTQ

RTg

εε


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Where:

ε = Mixed-solvent dielectric constant

= Born radius

Born Term Activity Coefficient

The expression for the activity coefficient can be derived from the Born term:

222

* 10112

ln −⎟⎟⎠

⎞⎜⎜⎝

⎛−=

i

i

w

eBorni r

zkT

Qεε

γ

Local Interaction Contribution The local interaction contribution is accounted for by the Non-Random Two Liquid theory. The basic assumption of the NRTL model is that the nonideal entropy of mixing is negligible compared to the heat of mixing, and, indeed, this is the case for electrolyte systems. This model was adopted because of its algebraic simplicity and its applicability to mixtures that exhibit liquid phase splitting. The model does not require specific volume or area data.

The effective local mole fractions and of species j and i, respectively,

in the neighborhood of i are related by:

jii

j

ii

ji GXX

XX

⎟⎟⎠

⎞⎜⎜⎝

⎛=

With

jjj CxX =


RTgg iiji

ji

)( −=τ

Where:

jC = jz for ions and unity for molecules



g ji = Energies of interaction between j-i pairs of segment based species

gii = Energies of interaction between i-i pairs of segment based species

and are energies of interaction between species j and i, and i and i,

respectively. Both and ijα are inherently symmetric ( and

).


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Similarly,

kijik

j

ki

ji GXX

XX

,⎟⎟⎠

⎞⎜⎜⎝

⎛=

With

kijikijieG kiji,,

,τα−=

RTgg kiji

kiji

−=,τ

Where:

kiji ,α = Nonrandomness factor

Local Composition Term

The local composition term for multicomponent systems is:

∑ ∑ ∑∑

∑

∑∑

∑∑∑

∑∑

∑∑

+

+

=

a ck

cakak

jcajacajaj

caI

Ia

kackck

jacjcacjcj

aa

cc

IIc

kkmk

jjmjmj

mm

IIm

lcexmix

GX

GXYXr

GX

GXYXr

GX

GXXr

RTg

,

,,

,

,

,,

,

,

,

τ

τ

τ

With

∑=

''

aa

aa X

XY

∑=

''

cc

cc X

XY

cama

acm Y ,αα ∑=

camc

cam Y ,αα ∑=

∑=a

mcaacm GYG ,

∑=c

mcacam GYG ,


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cm

cmcm

Gα

τ)ln(

−=

am

amam

Gα

τ)ln(

−=

cmacmc αα =,

amcama αα =,

)( ,,,

,, cammca

acmc

mcacmacmc ττ

αα

ττ −−=

)( ,,,

,, cammca

cama

mcaamcama ττ

αα

ττ −−=

)exp()exp( ,,,, acmccmacmcacmcacmcG τατα −=−=

)exp()exp( ,,,, camaamcamacamacamaG τατα −=−=

Where:

j & k = Any species

Imr , = Number of molecular segments in species I

Icr , = Number of cationic segments in species I

Iar , = Number of anionic segments in species I

To compute the local composition term for the activity coefficients of polymeric species, we first compute local composition contributions for each of the segments. The segment contributions to the activity coefficients from molecular segments, cationic segments, and anionic segments are given in the next three equations.


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⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−+

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−+

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−+

=

∑∑

∑∑ ∑

∑∑

∑∑∑

∑∑

∑ ∑

∑∑

kcakak

kcakacakak

camca c

kcakak

camaac

kackck

kackcackck

acmc

kackck

acmcc

c aa

kkmk

kkmkmk

mmm

kkmk

mmm

kkmk

jjmjmj

lcm

GX

GX

GXGX

Y

GX

GX

GXGX

Y

GX

GX

GXGX

GX

GX

,

,,

,,

,

,

,,

,,

,

'

''

'' '

''

ln

ττ

ττ

ττ

τγ

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−+

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−+

=

∑∑

∑∑ ∑

∑∑

∑ ∑

∑∑

∑

′

′′

′′ ′

′

kackak

kackaackak

accaa c

kackak

accaac

kkmk

kkBkmk

cmm

kkmk

cmm

kackck

kackcackck

aa

lcc

c

GX

GX

GXGX

Y

GX

GX

GXGX

GX

GXY

z

,

,,

,,

,'

,

,,

ln1

ττ

ττ

τγ

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−+

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−+

=

∑∑

∑∑ ∑

∑∑

∑ ∑

∑∑

∑

′

′′

′′ ′

′

kcakck

kcakccakck

caacc a

kcakck

caacca

kkmk

kkmkmk

amm

kkmk

amm

kcakak

kcakacakak

cc

lca

a

GX

GX

GXGX

Y

GX

GX

GXGX

GX

GXY

z

,

,,

,,

,'

,

,,

ln1

ττ

ττ

τγ

Local Composition Term Activity Coefficient

The local composition term for the activity coefficient of a species I is then computed as the sum of the individual segment contributions:


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∑∑∑ ++=m

lcmIm

a

lcaIa

c

lccIc

lcI rrr γγγγ lnlnlnln ,,,

For electrolytes, we are interested in unsymmetric convention activity coefficients. Therefore, we need to compute “infinite dilution activity coefficients” for ionic segments and molecular segments. They are then used to compute the unsymmetric activity coefficients of oligomeric ions:

lcI

lcI

lcI

∞γ−γ=γ lnlnln *

Flory-Huggins Term

To account for the entropy of mixing from polymeric species, we also compute the Flory-Huggins term:

∑ ∑ ⎟⎠

⎞⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=

I III

I

II

FHex

mnx

xRT

g φln,

∑ ⎟⎟⎠

⎞⎜⎜⎝

⎛−+⎟⎟

⎠

⎞⎜⎜⎝

⎛=

J J

JI

I

IFHI m

mx

φφγ 1lnln

with

∑∑∑ ++=a

Iac

Icm

ImI rrrm ,,,

∑=

JJJ

III mx

mxφ

Flory-Huggins Term Activity Coefficient

The unsymmetric activity coefficients from the Flory-Huggins term are:

IIFH

I mm −+=γ∞ 1lnln

FHI

FHI

FHI

∞γ−γ=γ lnlnln *

Electrolyte-Polymer NRTL Model Parameters The adjustable parameters for the EP-NRTL model include the:

• Pure component dielectric constant coefficient of nonaqueous solvents and molecular segments

• Born radius of ionic monomeric species or ionic segments

• Segment-based NRTL parameters for molecule-molecule, molecule-electrolyte, and electrolyte-electrolyte pairs

The pure component dielectric constant coefficients of nonaqueous solvents and Born radius of ionic species are required only for mixed-solvent electrolyte systems. The temperature dependency of the dielectric constant of solvent B is:


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ε B B BB

T A BT C

( ) = + −⎛⎝⎜

⎞⎠⎟

1 1

Each type of NRTL parameter consists of both the nonrandomness factor, α , and the energy parameter, τ . The temperature dependency relations of the NRTL parameters are:

• Molecule-Molecule Binary Parameters:

τBB BBBB

BB BBA BT

F T G T' ''

' 'ln( )= + + +

• Electrolyte-Molecule Pair Parameters:

τca B ca Bca B

ca B

ref

refCD

TE T T

TT

T, ,,

,( ) ln= + +

−+ ⎛

⎝⎜⎞⎠⎟

⎡

⎣⎢

⎤

⎦⎥

τB ca B caB ca

B ca

ref

refCD

TE T T

TT

T, ,,

,( ) ln= + +

−+ ⎛

⎝⎜⎞⎠⎟

⎡

⎣⎢

⎤

⎦⎥

• Electrolyte-Electrolyte Pair Parameters:

For the electrolyte-electrolyte pair parameters, the two electrolytes must share either one common cation or one common anion:

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛+

−++= ref

ref

accaacca

accaacca TT

TTTE

TD

C ln)(',

',',',τ

τca ca ca caca ca

ca ca

ref

refCD

TE T T

TT

T' , ' ' ' , ' '' , ' '

' , ' '( ) ln= + +

−+ ⎛

⎝⎜⎞⎠⎟

⎡

⎣⎢

⎤

⎦⎥

Where:

T ref = Reference temperature (298.15K)

Note that all of these interacting species (c, a, B, etc.) should be only monomeric species or segments.

The following table lists the EP-NRTL activity coefficient model parameters:



Upper Limit

MDS Units Keyword

Comments

Dielectric Constant Parameters *

CPDIEC/1 AB --- --- --- X --- Unary

CPDIEC/2 BB 0.0 --- --- X --- Unary

CPDIEC/3 CB 298.15 --- --- X TEMP Unary

Ionic Born Radius Parameters

RADIUS ir --- 1E-11 1E-9 --- LENGTH Unary

Molecule-Molecule Binary Parameters

NRTL/1 'BBA 0 --- --- X --- Binary,

Asymmetric


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Upper Limit

MDS Units Keyword

Comments

NRTL/2 BBB' 0 --- --- X TEMP Binary, Asymmetric

NRTL/3 'BBα 0.3 --- --- X --- Binary,

Symmetric

NRTL/4 --- --- --- --- --- --- ---

NRTL/5 'BBF 0 --- --- X --- Binary,

Asymmetric

NRTL/6 'BBG 0 --- --- X 1/TEMP Binary,

Asymmetric

NRTL/7 minT 0 --- --- X TEMP Unary

NRTL/8 maxT 1000 --- --- X TEMP Unary

Electrolyte-Molecule Pair Parameters **

GMELCC Cca B, 0.0 -100 100 X --- Binary, Asymmetric

GMELCD Dca B, 0.0 -3E4 3E4 X TEMP Binary, Asymmetric

GMELCE BcaE , 0.0 -100 100 X --- Binary,

Asymmetric

GMELCN Bca,α 0.2 0.01 5 X --- Binary,

Symmetric

Electrolyte-Electrolyte Pair Parameters

GMELCC ',cacaC 0.0 -100 100 X --- Binary,

Asymmetric

accaC ', 0.0 -100 100 X --- Binary,

Asymmetric

GMELCD Dca ca' , ' ' 0.0 -3E4 3E4 X TEMP Binary, Asymmetric

Dc a c a' , ' ' 0.0 -3E4 3E4 X TEMP Binary, Asymmetric

GMELCE Eca ca' , ' ' 0.0 -100 100 X --- Binary, Asymmetric

Ec a c a' , ' ' 0.0 -100 100 X --- Binary, Asymmetric

GMELCN '',' cacaα 0.2 0.01 5 X --- Binary,

Symmetric

acac '','α 0.2 0.01 5 X --- Binary,

Symmetric

*

If dielectric constant parameters are missing for a solvent, the dielectric constant of water is automatically assigned.

** If an electrolyte-molecule parameter is missing, the following defaults are used:

Electrolyte-water

-4

Water- 8


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electrolyte

Electrolyte-solvent

-2

Solvent-electrolyte

10

Electrolyte-solute

-2

Solute-electrolyte

10

Option Codes

The primary version of EPNRTL implemented is for aqueous solutions; that is, for ions, the reference state is at infinite dilution in water. The version for handling mixed-solvent electrolyte systems is also available by using Option Codes in the Aspen Plus Interface.

Option Codes in EPNRTL model

0 Aqueous solutions

1 Mixed-solvent solutions

Specifying the Electrolyte-Polymer NRTL Model See Specifying Physical Properties in Chapter 1.

Polymer UNIFAC Activity Coefficient Model This section describes the polymer UNIFAC activity coefficient model available in the POLYUF physical property method. The polymer UNIFAC model is an extension of the UNIFAC group contribution method for standard components to polymer systems (Fredenslund et al., 1975, 1977; Hansen et al., 1991). It is a predictive method of calculating phase equilibria, and, therefore, it should be used only in the absence of experimental information. The UNIFAC method yields fairly accurate predictions. It becomes less reliable, however, in the dilute regions, especially for highly non-ideal systems (systems that exhibit strong association or solvation).

Although the UNIFAC approach is a good predictive method, it should not be used as a substitute to reducing good experimental data to calculate phase equilibria. In general, higher accuracy can be obtained from empirical models when these models are used with binary interaction parameters obtained from experimental data.

Finally, the method is only applicable in the temperature range of 300-425 K (Danner & High, 1992). Extrapolation outside this range is not recommended. The group parameters are not temperature-dependent; consequently, predicted phase equilibria extrapolate poorly with respect to temperature.


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The polymer UNIFAC model calculates liquid activity coefficients for the POLYUF property method. This UNIFAC model is the same as the UNIFAC model in Aspen Plus for monomer systems except that this model obtains functional group information from segments and polymer component attributes.

The equation for the original UNIFAC liquid activity coefficient model is made up of a combinatorial and residual term:

Ri

Ci γγγ lnlnln +=

⎟⎟⎠

⎞⎜⎜⎝

⎛−+−−+=

i

i

i

i

i

i

i

iCi

zxx θ

φθφφφ

γ 1ln2

1lnln

Where the molecular volume and surface fractions are:

∑∑== nc

jjj

ii

nc

jjj

iii

qzx

qzx

rx

rx

2

2 and iθφ

With:

nc = Number of components in the mixture

The coordination number z is set to 10.

The parameters ri and qi are calculated from the group volume and area parameters:

∑∑ ==ng

kkki

ng

kkkii QqRr νν i and

Where:

νki = Number of groups of type k in molecule i

ng = Number of groups in the mixture

The residual term is:

[ ]∑ Γ−Γ=ng

k

ikkki

Ri lnlnln νγ

Where:

ln Γk = Activity coefficient of a group at mixture composition

Γki = Activity coefficient of group k in a mixture of groups corresponding

to pure i

The parameters Γk and Γki are defined by:


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⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

−−=Γ ∑ ∑∑

ng

m

ng

mng

nnmn

kmmmkmkk Q

τθ

τθτθln1ln

With:

∑= ng

mmm

kk

k

QzX

QzX

2

2θ

And:

Tbmn

mne /−=τ

The parameter Xk is the group mole fraction of group k in the liquid:

∑∑

∑=

nc

j

ng

mjmj

nc

jjkj

k

x

xX

ν

ν


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Polymer UNIFAC Model Parameters The input parameters for the Polymer UNIFAC model are given here:



Upper Limit

MDS Units Keyword

Comments

UFGRP ...,, miki vv --- --- --- X --- Unary

GMUFQ Qk --- --- --- X --- Unary

GMUFR Rk --- --- --- X --- Unary

GMUFB bkn --- --- --- X --- Unary

The parameter UFGRP stores the UNIFAC functional group number and number of occurrences of each group. UFGRP is stored in the Aspen Polymers segment databank for polymer segments, and in the Aspen Plus pure component databank for standard components. For non-databank components, enter UFGRP on the Properties Molec-Struct.Func-Group form. See Aspen Physical Property System Physical Property Data, for a list of the UNIFAC functional groups.

Specifying the Polymer UNIFAC Model See Specifying Physical Properties in Chapter 1.

Polymer UNIFAC Free Volume Activity Coefficient Model This section describes the polymer UNIFAC free volume activity coefficient model available in the POLYUFV physical property method. The polymer UNIFAC free volume activity coefficient model (UNIFAC-FV) is the same as the polymer UNIFAC model, with the exception that it contains a term to account for free-volume (compressibility) effects. Thus, the two methods have similar applicability (see Polymer UNIFAC Activity Coefficient Model on page 114). The UNIFAC-FV model can be used with more confidence for predictions at higher pressures than the polymer UNIFAC model. Nonetheless, both methods are predictive, and should not be used to substitute correlative models (such as Flory-Huggins or POLYNRTL) with fitted binary parameters.

Oishi and Prausnitz (1978) modified the UNIFAC model (Fredenslund et al., 1975, 1977) to include "a contribution for free volume difference between the polymer and solvent molecules." Oishi and Prausnitz suggested that the UNIFAC combinatorial contribution does not account for the free volume differences between the polymer and solvent molecules. While this difference is usually not significant for small molecules, it could be important for


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polymer-solvent systems. They added the free volume contribution derived from the Flory equation of state to the original UNIFAC model to arrive at the following expression for the weight fraction activity coefficient of a solvent in a polymer:

FVi

Ri

Ci γγγγ lnlnlnln ++=

⎟⎟⎠

⎞⎜⎜⎝

⎛−+−−+=

i

i

i

i

i

i

i

iCi

zxx θ

φθφφφ

γ 1ln2

1lnln

[ ]∑ Γ−Γ=ng

k

ikkki

Ri lnlnln νγ

Free-Volume Contribution

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −⎟⎟

⎠

⎞⎜⎜⎝

⎛−−

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−

−=

−−

131

31

31

~11~~

1~1~

ln3ln im

ii

m

ii

FVi V

VV

CV

VCγ

i

ii br

VV

01517.0~ =

∑∑=

ii

iim xrb

xVV

01517.0~

Where:

Ci = 1.1

b = 1.28

ir = Volume parameter for component i

Vi = Specific volume of component i, cubic meters per kilogram mole, calculated from Rackett equation for solvents and from Tait equation for polymers.

See Chapter 4 for a description of the Tait equation.

The combinatorial and residual contributions, γ C and γ R , are identical to those in the polymer UNIFAC model (see Polymer UNIFAC Activity Coefficient Model on page 114).

The Oishi-Prausnitz modification of UNIFAC is currently the most used method available to predict solvent activities in polymers. Required for the Oishi-Prausnitz method are the densities of the pure solvent and pure polymer at the temperature of the mixture and the structure of the solvent and polymer. The Tait equation is used to calculate molar volume for polymers (see Chapter 4 for a description of the Tait equation).

Molecules that can be constructed from the groups available in the UNIFAC method can be treated. At present, groups are available to construct alkanes, alkenes, alkynes, aromatics, water, alcohols, ketones, aldehydes, esters, ethers, amines, carboxylic acids, chlorinated compounds, brominated


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compounds, and a few other groups for specific molecules. The Oishi-Prausnitz method has been tested only for the simplest of these structures, and these groups should be used with care.

Polymer UNIFAC-FV Model Parameters The UNIFAC free volume parameters are the same as those required for the polymer UNIFAC model (see Polymer UNIFAC Model Parameters on page 117). In addition, parameters for the Tait liquid molar volume model are required for free volume calculations (see Chapter 4 for a description of the Tait liquid molar volume model).

Specifying the Polymer UNIFAC- FV Model See Specifying Physical Properties in Chapter 1.


Flory, P. J. (1953). Principles of Polymer Chemistry. London: Cornell University Press.

Chen, C.-C. (1993). A Segment-Based Local Composition Model for the Gibbs Energy of Polymer Solutions. Fluid Phase Equilibria, 83, 301-312.

Chen, C.-C. (1996). Molecular Thermodynamic Model for Gibbs Energy of Mixing of Nonionic Surfactant Solutions. AIChE Journal, 42, 3231-3240.

Chen, C-C., Britt, H. I., Boston, J. F., & Evans, L. B. (1982). Local Composition Model for Excess Gibbs Energy of Electrolyte Systems. AIChE J., 28, 588.

Chen, C-C., & Evans, L. B. (1986). A Local Composition Model for the Excess Gibbs Energy of Aqueous Electrolyte Systems. AIChE J., 32, 444.

Chen, C-C., Mathias, P. M., & Orbey, H. (1999). Use of Hydration and Dissociation Chemistries with the Electrolyte-NRTL Model. AIChE Journal, 45, 1576.

Chen, C-C., Song Y. (2004). Generalized Electrolyte-NRTL Model for Mixed-Solvent Electrolyte Systems. AIChE Journal, 50, 1928.

Danner, R. P., & High, M. S. (1992). Handbook of Polymer Solution Thermodynamics. New York: American Institute of Chemical Engineers.

Flory, P. J. (1941). Thermodynamics of High Polymer Solutions. J. Chem. Phys., 9, 660.

Fredenslund, Aa., Jones, R. L., & Prausnitz, J. M. (1975). AIChE J., 21, 1086.

Fredenslund, Aa., Gmehling, J., & Rasmussen, P. (1977). Vapor-Liquid Equilibria using UNIFAC. Amsterdam: Elsevier.


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Hansen, H. K., Rasmussen, P., Fredenslund, Aa., Schiller, M., & Gmehling, J. (1991). Vapor-Liquid Equilibria by UNIFAC Group Contribution. 5 Revision and Extension. Ind. Eng. Chem. Res., 30, 2352-2355.

Huggins, M. L. (1941). Solutions of Long Chain Compounds. J. Phys. Chem., 9, 440.

Manning, G.S. (1979). Counterion Binding in Polyelectrolyte Theory. Acc. Chem. Res., 12, 443.

Mock, B., Evans, L. B., & Chen, C.-C. (1986). Thermodynamic Representation of Phase Equilibria of Mixed-Solvent Electrolyte Systems. AIChE Journal, 32, 1655.

Oishi, T., & Prausnitz, J. M. (1978). Estimation of Solvent Activity in Polymer Solutions Using a Group Contribution Method. Ind. Eng. Chem. Process Des. Dev., 17, 333-335.

Pitzer, K.S. (1973). Thermodynamics of Electrolytes. I. Theoretical Basis and General Equations. J. Phys. Chem., 77, 268.

Renon, H., & Prausnitz, J. M. (1968). Local Compositions in Thermodynamic Excess Functions for Liquid Mixtures. AIChE J., 14, 135-144.

Tompa, H. (1956). Principles of Polymer Chemistry. London: Butterworths.


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4 Thermophysical Properties of Polymers 121

4 Thermophysical Properties of Polymers

This chapter discusses thermophysical properties of polymers. These properties are needed when an equation-of-state model (see Chapter 2) or an activity coefficient model (see Chapter 3) is used to calculate thermodynamic properties of mixtures containing polymers. In general, Aspen Polymers (formerly known as Aspen Polymers Plus) provides various property models to estimate thermophysical properties of polymers; these models are implemented as polynomial expressions so that they can be used in a predictive mode (such as Van Krevelen Group Contribution Methods, explained on page 145), or in a correlative mode (in case experimental data are available for parameter estimation). Note that these models only apply to polymers, oligomers, and segments. Models for conventional components are already available in Aspen Plus.


• About Thermophysical Properties, 121

• Aspen Ideal Gas Property Model, 123

• Van Krevelen Liquid Property Models, 127

• Van Krevelen Liquid Molar Volume Model, 136

• Tait Liquid Molar Volume Model, 140

• Van Krevelen Glass Transition Temperature Correlation, 141

• Van Krevelen Melt Transition Temperature Correlation, 142

• Van Krevelen Solid Property Models, 143

• Van Krevelen Group Contribution Methods, 145

• Polymer Property Model Parameter Regression, 146

• Polymer Enthalpy Calculation Routes with Activity Coefficient Models, 147

About Thermophysical Properties As discussed in Chapter 1, due to their structure, polymers exhibit thermophysical properties significantly different than those of conventional


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122 4 Thermophysical Properties of Polymers

components. Consequently, different property models are required to describe their behavior. Aspen Polymers provides models to estimate polymer enthalpy, Gibbs free energy and molar volume. These properties are essential for heat and mass balance calculations of mixtures containing polymers. Aspen Polymers also provides models to estimate some of the unique properties of polymer components (such as the glass transition temperature and melt transition temperature).

The following tables list the properties available for polymers and the models available for calculating these properties in Aspen Polymers:

Property Name

Symbol Description

HL liH *, Liquid pure component enthalpy

GL li*,μ Liquid pure component Gibbs free energy

SL liS *, Liquid pure component entropy

VL liV *, Liquid pure component molar volume

TGVG gT Glass Transition temperature

TMVG mT Melt Transition temperature

HS siH *, Solid pure component enthalpy

GS si*,μ Solid pure component Gibbs free energy

VS siV *, Solid pure component molar volume

SS siS *, Solid pure component entropy

Property Models Model Name Properties Calculated

Aspen Ideal Gas Property Model HIG, GIG, CPIG

Van Krevelen/DIPPR Model HL0DVK, HL0DVKD

HL

Van Krevelen/DIPPR Model GL0DVK GL

Van Krevelen/Rackett Model VL0DVK VL

Tait/Rackett Model VL0TAIT VL

Van Krevelen Model TG0DVK TGVK

Van Krevelen Model TM0DVK TMVK

Van Krevelen/Standard Model HS0DVK HS

Van Krevelen/Standard Model GS0DVK GS

Van Krevelen/Rackett Model VS0DVK VS


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For conventional components, standard models are already available in Aspen Plus, and, therefore, no details are presented here. (See Aspen Physical Property System Physical Property Methods and Models for more information). Instead, we focus on describing the calculation of thermophysical properties of polymers, oligomers, and segments.

Polymer properties except gT and mT are calculated using different routes,

depending on whether an equation-of-state model or an activity coefficient model is used. For instance, when an equation-of-state model is used, only the Aspen Ideal Gas Property Model is needed to calculate the polymer ideal gas properties to the departure functions. When an activity coefficient model is used, the van Krevelen property models (van Krevelen, 1990) are used to calculate polymer enthalpy, Gibbs free energy and molar volume. In most cases, the van Krevelen models provide separate correlations for the crystalline phase and the liquid phase.

Depending on the temperature region being considered, above the melt transition temperature, between the melt and glass transition temperature, or below the glass transition temperature, one or both correlations may apply. When the temperature region is between the melt transition temperature and the glass transition temperature, the contribution of each correlation is determined by the degree of crystallinity, which is one of the models input parameters. Correlations for estimating the melt and glass transition temperature are also provided. The entropy of polymers in both liquid and solid phases is calculated using the rigorous thermodynamic equations:

)(1 *,*,*, li

li

li H

TS μ−=

)(1 *,*,*, si

si

si H

TS μ−=

As the models presented in the remainder of this chapter relate only to polymers, oligomers, and segments, the index i is dropped for simplicity.

Aspen Ideal Gas Property Model As shown in Chapter 2, equations of state provide information concerning ideal gas departure functions. Therefore, in estimating enthalpy, entropy, and Gibbs free energy with an equation of state, the ideal gas contribution must be added to the departure functions obtained from the equation of state. The ideal gas model already available in Aspen Plus for conventional components is extended to handle polymers and oligomers.

First, we apply Equations 3.11 and 3.15 to pure polymer components to calculate the liquid enthalpy and Gibbs free energy of polymers:

( )igligl HHTHH *,*,*,*, )( −+=


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( )igligl T *,*,*,*, )( μμμμ −+=

Where:

igH *, = Ideal gas molar enthalpy of polymers

ig*,μ = Ideal gas molar Gibbs free energy of polymers

igl HH *,*, − = Polymer molar enthalpy departure, calculated from an equation-of-state model

igl *,*, μμ − = Polymer molar Gibbs free energy departure, calculated from an equation-of-state model

Both departure functions, igl HH *,*, − and igl *,*, μμ − , are calculated from the same equation of state.

Ideal Gas Enthalpy of Polymers The ideal gas enthalpy of a polymer at temperature T is given by the following equation:

dTCpTHTHT

T

igrefigig

ref∫+= *,*,*, )()(

Where:

refT = Reference temperature (298.15 K)

( )refig TH *, = Heat of formation of the polymer at the ideal-gas state and refT

igCp*, = Ideal-gas heat capacity of the polymer

Ideal Gas Gibbs Free Energy of Polymers Similarly, the ideal gas Gibbs free energy of a polymer at temperature T is given by the following equation:

)()(

)()(

*,

*,*,*,*,

refigref

T

T

igT

T

igrefigig

TSTT

dTT

CpTdTCpTTrefref

−−

−+= ∫∫μμ

With

ref

refigrefigrefig

TTTHTS )()()(

*,*,*, μ−

=

Where:

)(*, refig Tμ = Gibbs free energy of formation of the polymer at the ideal-gas

state and refT

)(*, refig TS = Entropy of formation of the polymer at the ideal-gas state


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and refT

In the ideal gas model, the quantities )(*, refig TH and )(*, refig Tμ are constants for polymers and oligomers. They can be estimated using Van Krevelen Group Contribution Methods (see page 145). They can also be adjusted to fit the data of the polymer. However, the ideal-gas heat capacity

of polymers, igCp*, , is temperature-dependent and is implemented as polynomial expressions:

87

56

45

34

2321

*, )(CTC

TCTCTCTCTCCTCp ig

≤≤+++++=

or

7109*, 11)( CTTCCTCp Cig <+=

or

)(*, TCp ig = Linearly extrapolated using slope at 8C for 8CT >

Aspen Ideal Gas Model Parameters The following table lists the parameters used in the ideal gas model:



Upper Limit

MDS Units Keyword

CPIG/1 1C --- --- --- X MOLE-HEAT-CAPACITY

CPIG/2,…, 6 62 ,...,CC 0.0 --- --- X MOLE-HEAT-

CAPACITY, TEMP

CPIG/7 7C 0 --- --- X TEMP

CPIG/8 8C 1000 --- --- X TEMP

CPIG/9 9C --- --- --- X MOLE-HEAT-CAPACITY

CPIG/10, 11 1110 ,CC --- --- --- X MOLE-HEAT-

CAPACITY, TEMP

DHFVK ( )refig TH *, --- 10105×− 10105× --- MOLE-ENTHALPY

DGFVK ( )refig T*,μ --- 10105×− 10105× --- MOLE-ENTHALPY

--- refT 298.15 --- --- --- Kelvin

Parameter Input All three unary parameters, CPIG, DHFVK, and DGFVK can be:

• Specified for each polymer or oligomer component; or



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These options are shown in priority order. For example, if parameters are provided for a polymer component as well as for the segments, the polymer parameters are used and the segment parameters are ignored.

Parameters for Polymers If parameters are specified for the polymeric component, then these values are used directly, independent of the polymer composition. Otherwise, the parameters of a polymer are calculated using the polymer composition (segment fraction) and the parameters of segments:

)()( *,*, refigA

Nseg

AA

refig THXTH ∑=

)()( *,*, refigA

Nseg

AA

refig TXT μμ ∑=

)()( *,*, TCpXTCp igA

Nseg

AA

ig ∑=

Where:

Nseg = Number of segment types in the copolymer

AX = Mole segment fraction of segment type A in the copolymer


)(*, refigA TH = Ideal-gas enthalpy of formation of segment type A at refT

)(*, refigA Tμ = Ideal-gas Gibbs free energy of formation of segment type A

at refT

)(*, TCp igA = Ideal-gas heat capacity of segment type A

Van Krevelen Group Contribution for Segments If the parameters DHFVK and DGFVK are not entered for the segments, then these values are estimated using using Van Krevelen Group Contribution Methods (see page 145). That is, Aspen Polymers automatically retrieves functional group data of segments from the van Krevelen databank.

∑=k

refigkk

refigA THnTH )()( *,*,

∑=k

refigkk

refigA TnT )()( *,*, μμ

Where:

kn = Number of occurrences of functional group k in the segment from the segment database or user-specified segment structure


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)(*, refigk TH = Ideal-gas enthalpy of formation of functional group k at refT ,

from van Krevelen database

)(*, refigk Tμ = Ideal-gas Gibbs free energy of formation of functional group

k at refT , from van Krevelen database

In some cases, the parameters of functional groups may not be available in the databank. The contributions from these groups are ignored.

Ideal Gas Heat Capacity Parameters CPIG parameters are required for the model. If your model uses polymers and oligomers contained in the polymer segment databank, the CPIG parameters are calculated automatically. However, if the values are not in the databank you must either estimate or regress the CPIG parameters.

Parameter Regression You must estimate parameters if your polymer includes non-databank segments. You can perform an Aspen Plus Regression Run (DRS) to obtain these parameters. Additionally, you can use a DRS run to adjust the values for polymers that contain databank segments. This is useful for fitting available experimental data. Since the ideal-gas property model is used with an equation-of-state model, experimental data on liquid density of a polymer should be regressed first to obtain the EOS pure parameters for the polymer (or segments). In the data regression, these parameters can be:

• Specified for each oligomer component (polymer)

• Specified for each segment that composes an oligomer component (polymer)

Once the pure EOS parameters are available for a polymer, ideal-gas heat capacity parameters, CPIG, should be regressed for the same polymer using experimental liquid heat capacity data. Data on heat of formation and Gibbs free energy of formation, of the same polymer (segment), can then be used to obtain DHFVK and DGFVK by performing an Aspen Design Spec or Aspen Sensitivity.


Van Krevelen Liquid Property Models The activity coefficient and equation-of-state property methods calculate polymer liquid properties using a different structure. For example, equation-of-state property methods normally use an ideal gas reference state to estimate polymer properties. However, activity coefficient property methods use a liquid reference state. In Aspen Polymers, the activity coefficient


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property methods use simple polynomial equations to calculate polymer liquid properties of interest.

Liquid Enthalpy of Polymers By design, Aspen Polymers uses the liquid-phase property routes to calculate the properties of both liquid- and solid-phase polymers present in the mixed substream. Liquid enthalpy of polymer components is calculated first. The enthalpy and heat capacity of amorphous solid polymers are continuous with the liquid-phase polymer properties across the melting point, so the models do not distinguish between amorphous solid and liquid polymer.

Alternately, the crystalline polymer can be included in the CISOLID substream. The solid property model, described later in this chapter, is used to calculate the properties of polymer in the CISOLID substream.

Temperature-Enthalpy Relationship

The following figure summarizes the relationship between temperature and enthalpy for a polymer component:

Real GasIdeal Gas

Liquid

CrystallineSolid

AmorphousSolid

Semi-Crystalline

Solid

Tref Tmelt T

DHFORMDHFVK

DHCON

Temperature

Enth

alpy

DHSUB Hi*,l(T)

Hi*,v(T)

( )refigi TH *,

( )refisub TH *Δ ( )ref

icon TH *Δ

videp H *,Δ

( )refli TH *,

( )refci TH *,

( )meltifus TH *Δ

( )meltli TH *,

( )meltci TH *,

( )THivap*Δ

The key variables are:

( )refigi TH *, = Ideal gas heat of formation (DHFORM, DHFVK)

( )refli TH *, = Liquid phase reference enthalpy

( )refci TH *, = Crystal phase reference enthalpy

videpH *,Δ = Vapor phase enthalpy departure (DHV)


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( )reficon TH *Δ = Enthalpy of condensation (DHCON)

( )refisub TH *Δ = Enthalpy of sublimation (DHSUB)

( )meltifus TH *Δ = Heat of fusion at the melting point

( )THivap*Δ = Heat of vaporization (DHVL)

( )TH vi*, = Vapor-phase enthalpy (HV)

( )TH li*, = Liquid-phase enthalpy (HL)

( )meltli TH *, = Enthalpy of amorphous solid phase or liquid phase at the

melting point

( )meltci TH *, = Enthalpy of pure crystalline polymer at the melting point

meltT = Melt transition temperature (TMVK)


The crystalline polymer generally has a lower enthalpy and higher heat capacity than amorphous polymer. The van Krevelen enthalpy model accounts for this difference by using two sets of equations corresponding to the amorphous/liquid and crystalline phases. The net enthalpy is calculated using the mass fraction crystallinity and a mass-average mixing rule:

HL lH *,= for mTT >

)1(*,*,c

lc

c xHxH −+= for mg TTT ≤≤

cH *,= for gTT <

With:

( ) ∫+=T

T

lrefll

ref

dTCpTHH *,*,*,

( ) ( )refcon

refigrefl THTHTH **,*, )( Δ+=

( ) ∫+=T

T

crefcc

ref

dTCpTHH *,*,*,

( ) ( )refsub

refigrefc THTHTH **,*, )( Δ−=

Where:

HL = Net enthalpy of the polymer

lH *, = Enthalpy of the polymer in the liquid phase

cH *, = Enthalpy of the polymer in the crystalline phase


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mT = Melt transition temperature of the polymer

gT = Glass transition temperature of the polymer

xc = Mass-fraction crystallinity


( )refig TH *, = Heat of formation of the polymer at the ideal-gas state and refT

( )refcon TH *Δ = Heat of condensation of the polymer at refT

( )refsub TH *Δ = Heat of sublimation of the polymer at refT

lCp*, = Heat capacity of the polymer in the liquid phase

cCp*, = Heat capacity of the polymer in the crystalline phase

Note that superscript c refers to the crystalline state, superscript l refers to the liquid state, and the asterisk (*) refers to pure component properties.

Aspen Polymers uses the heat of condensation, ( )refcon TH *Δ , and heat of

sublimation, ( )refsub TH *Δ , as reference parameters to convert between the

ideal gas reference state and the condensed phase reference state.

Liquid Gibbs Free Energy of Polymers The liquid Gibbs free energy of polymers can be calculated using a similar approach:

GL l*,μ= for mTT >

)1(*,*,c

lc

c xx −+= μμ for mg TTT ≤≤

c*,μ= for gTT <

With:

)()(

)()(

*,

*,*,*,*,

reflref

T

T

lT

T

lrefll

TSTT

dTT

CpTdTCpTTrefref

−−

−+= ∫∫μμ

dTT

CpTdTCpTTT

T

cT

T

crefcc

refref∫∫ −+=

*,*,*,*, )()( μμ

( ) ( )refcon

refigrefl TTT **,*, )( μμμ Δ+=


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ref

reflreflrefl

TTTHTS )()()(

*,*,*, μ−

=

)()()( **,*, refsub

refigrefc TTT μμμ Δ−=

Where:

GL = Net Gibbs free energy of the polymer

l*,μ = Gibbs free energy of the polymer in the liquid phase

c*,μ = Gibbs free energy of the polymer in the crystalline phase


( )refig T*,μ = Gibbs free energy of formation of the polymer at the ideal-gas state and refT

( )refcon T*μΔ = Gibbs free energy of condensation of the polymer at refT

( )refsub T*μΔ = Gibbs free energy of sublimation of the polymer at refT

Heat Capacity of Polymers The liquid- and crystalline-phase heat capacities for polymeric components are calculated using the polynomial expressions:

32 TDTCTBACp lllll*, +++= for max,lmin,l TTT <<

32 TDTCTBACp ccccc*, +++= for max,cmin,c TTT <<

Liquid Enthalpy and Gibbs Free Energy Model Parameters The following table lists the liquid enthalpy and Gibbs free energy model parameters:



Upper Limit

MDS Units Keyword

CPLVK/1 lA Calculated† --- --- X MOLE-HEAT-CAPACITY

CPLVK/2 lB Calculated† --- --- X MOLE-HEAT-CAPACITY, TEMP

CPLVK/3 lC 0 --- --- X MOLE-HEAT-CAPACITY, TEMP

CPLVK/4 lD 0 --- --- X MOLE-HEAT-CAPACITY, TEMP

CPLVK/5 min,lT 0 --- --- X TEMP


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Upper Limit

MDS Units Keyword

CPLVK/6 max,lT 1000 --- --- X TEMP

CPCVK/1 cA Calculated† --- --- X MOLE-HEAT-CAPACITY

CPCVK/2 cB Calculated† --- --- X MOLE-HEAT-CAPACITY, TEMP

CPCVK/3 cC 0 --- --- X MOLE-HEAT-CAPACITY, TEMP

CPCVK/4 cD 0 --- --- X MOLE-HEAT-CAPACITY, TEMP

CPCVK/5 min,cT 0 --- --- X TEMP

CPCVK/6 max,cT 1000 --- --- X TEMP

DHFVK ( )refig TH *, --- 10105×− 10105× --- MOLE-ENTHALPY

DHCON ( )refcon TH *Δ

-7E6 10105× 10105×

--- MOLE-ENTHALPY

DHSUB ( )refsub TH *Δ

1.7E7 10105×− 10105×

--- MOLE-ENTHALPY

DGFVK ( )refig T*,μ --- 10105×− 10105× --- MOLE-ENTHALPY

DGCON ( )refcon T*μΔ -2.528E6 10105×− 10105× --- MOLE-ENTHALPY

DGSUB ( )refsub T*μΔ 5.074E6 10105×− 10105× --- MOLE-ENTHALPY

POLCRY xc 0.0 0 1 --- ---

TMVK mT --- 0 5000 X TEMP

TGVK gT --- 0 5000 X TEMP

--- refT 298.15 --- --- --- Kelvin

† The default values of these parameters are calculated using the van Krevelen group

contribution model as given by Equations 4.7–4.10 later in this chapter.

Parameter Input The parameters in the above table can be:



• Calculated automatically using van Krevelen group contribution techniques.


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These options are shown in priority order. For example, if parameters are provided for a polymer component and for the segments, the polymer parameters are used and the segment parameters are ignored.

Parameters for Polymers If parameters are specified for the polymeric component, then these values are used directly, independent of the copolymer composition. Otherwise, the parameters of a copolymer are calculated using the copolymer composition and the parameters of segments:

)()( *,*, refigA

Nseg

AA

refig THXTH ∑=

)()( ** refAcon

Nseg

AA

refcon THXTH Δ=Δ ∑

)()( ** refAsub

Nseg

AA

refsub THXTH Δ=Δ ∑

)()( *,*, refigA

Nseg

AA

refig TXT μμ ∑=

)()( ** refAcon

Nseg

AA

refcon TXT μμ Δ=Δ ∑

)()( ** refAsub

Nseg

AA

refsub TXT μμ Δ=Δ ∑

)()( *,*, TCpXTCp lA

Nseg

AA

l ∑=

)()( *,*, TCpXTCp cA

Nseg

AA

c ∑=

Where:




)(*, refigA TH = Ideal-gas enthalpy of formation of segment type A at refT

)(* refAcon THΔ = Heat of condensation of segment type A at refT

)(* refAsub THΔ = Heat of sublimation of segment type A at refT

)(*, refigA Tμ = Ideal-gas Gibbs free energy of formation of segment type A

at refT


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)(* refAcon TμΔ = Gibbs free energy of condensation of segment type A at

refT

)(* refAsub TμΔ = Gibbs free energy of sublimation of segment type A at refT

lACp*, = Heat capacity of segment type A in the liquid phase

cACp*, = Heat capacity of segment type A in the crystalline phase

Van Krevelen Group Contribution for Segments If you do not enter parameters for the segments, these values are estimated using using Van Krevelen Group Contribution Methods (see page 145). Aspen Polymers automatically retrieves functional group data for segments from the van Krevelen databank.

∑=k

refigkk

refigA THnTH )()( *,*, (4.1)

∑ Δ=Δk

refkconk

refAcon THnTH )()( ** (4.2)

∑ Δ=Δk

refksubk

refAsub THnTH )()( ** (4.3)

∑=k

refigkk

refigA TnT )()( *,*, μμ (4.4)

∑ Δ=Δk

refkconk

refAcon TnT )()( ** μμ (4.5)

∑ Δ=Δk

refksubk

refAsub TnT )()( ** μμ (4.6)

Where:



)(*, refigk TH = Ideal-gas enthalpy of formation of functional group k at

refT , from van Krevelen database

)(* refkcon THΔ = Heat of condensation of formation of functional group k at


)(* refksub THΔ = Heat of sublimation of formation of functional group k at


)(*, refigk Tμ = Ideal-gas Gibbs free energy of formation of functional

group k at refT , from van Krevelen database


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)(* refkcon TμΔ = Gibbs free energy of condensation of formation of

functional group k at refT , from van Krevelen database

)(* refksub TμΔ = Gibbs free energy of sublimation of formation of functional

group k at refT , from van Krevelen database

In some cases, the parameters of functional groups are not available in the databank. The contributions from these groups are ignored.

Missing parameters in heat capacity of segments are estimated using van Krevelen’s group contribution model:

( )reflk

kk

l TCpnA *,64.0 ××= ∑ (4.7)

( )reflk

kk

l TCpnB *,0012.0 ××= ∑ (4.8)

( )refck

kk

c TCpnA *,106.0 ××= ∑ (4.9)

( )refck

kk

c TCpnB *,003.0 ××= ∑ (4.10)

Where:


( )reflk TCp*, = Liquid molar heat capacity of functional group k at refT , from

van Krevelen database

( )refck TCp*, = Crystalline molar heat capacity of functional group k at refT ,

from van Krevelen database

Parameter Regression You must estimate parameters if your polymer includes non-databank segments. You can perform an Aspen Plus Regression Run (DRS) to obtain heat capacity parameters. Additionally, you can use a DRS run to adjust the values for polymers that contain databank segments. This is useful for fitting available experimental data. In the data regression, these parameters can be:





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Van Krevelen Liquid Molar Volume Model The molar volume of a polymeric component depends on the temperature and the physical state of the polymer, as shown here:

VgVc

Vl

Tglass Tmelt

Temperature

Mol

ar V

olum

e

Glassy

Crystalline

Semi-crystalline

Amorphous

Liquid

The polymer molar volume model uses the temperature and user-specified crystallinity to determine the phase regime of the polymer. The molar volume is calculated using the following equations:

VL lV *,= for mTT >

)1(*,*,c

lc

c xVxV −+= for mg TTT ≤≤

)1(*,*,c

gc

c xVxV −+= for gTT <

Where:

VL = Net molar volume of the polymer

lV *, = Molar volume of the polymer in the liquid phase

cV *, = Molar volume of the polymer in the crystalline phase

gV *, = Molar volume of the polymer in the glassy phase

xc = Mass fraction crystallinity

mT = Melt transition temperature

gT = Glass transition temperature


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Superscripts l, c, and g refer to the liquid, crystalline, and glassy states respectively.

lV *, , cV *, , and gV *, are calculated from the following expressions:

lll ATBV /)1(*, += for max,min, ll TTT << (4.11)

ccc ATBV /)1(*, += for max,min, cc TTT << (4.12)

gg

ggg ATCTBV /)1(*, ++= for max,min, gg TTT << (4.13)

Van Krevelen Liquid Molar Volume Model Parameters The following table lists the van Krevelen liquid molar volume model parameters :



Upper Limit

MDS Units Keyword

DNLVK/1 lA Calculated† --- --- X MOLE-DENSITY

DNLVK/2 lB Calculated† --- --- X 1/TEMP

DNLVK/3 min,lT 0 --- --- X TEMP

DNLVK/4 max,lT 1000 --- --- X TEMP

DNCVK/1 cA Calculated† --- --- X MOLE-DENSITY

DNCVK/2 cB Calculated† --- --- X 1/TEMP

DNCVK/3 min,cT 0 --- --- X TEMP

DNCVK/4 max,cT 1000 --- --- X TEMP

DNGVK/1 gA Calculated† --- --- X MOLE-DENSITY

DNGVK/2 gB Calculated† --- --- X 1/TEMP

DNGVK/3 gC Calculated† --- --- X 1/TEMP

DNGVK/4 min,gT 0 --- --- X TEMP

DNGVK/5 max,gT 1000 --- --- X TEMP

POLCRY xc 0.0 0 1 --- ---

TMVK mT --- 0 5000 X TEMP

TGVK gT --- 0 5000 X TEMP

† The default values of these parameters are calculated using the van Krevelen

group contribution model as given by Equations 4.14–4.16 later in this chapter.


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Parameter Input

The parameters lA , lB , cA , cB , gA , gB , and gC can be:

• Specified for each polymer or oligomer component on a mass or molar basis

• Specified for segments that compose a polymer or oligomer component on a molar basis

• Calculated automatically using van Krevelen group contribution techniques

These options are shown in priority order. For example, if parameters are provided for a polymer component and for the segments, the polymer parameters are used and the segment parameters are ignored. The mass based parameters take precedence over the molar based parameters.

Parameters for Polymers If parameters are specified for the polymeric component, then these values are used directly, independent of the copolymer composition. Otherwise, the parameters of a copolymer are calculated using the copolymer composition and the parameters of segments:

lA

Nseg

AA

seg

nl VXMM

V *,*, ∑=

cA

Nseg

AA

seg

nc VXMM

V *,*, ∑=

gA

Nseg

AA

seg

ng VXMM

V *,*, ∑=

With

A

Nseg

AAseg MXM ∑=

Where:



segM = Average molecular weight of segments in the copolymer


AM = Molecular weight of segment type A in the copolymer

lAV *, = Molar volume of segment type A in the copolymer in the liquid phase

cAV *, = Molar volume of segment type A in the copolymer in the crystalline

phase


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gAV *, = Molar volume of segment type A in the copolymer in the glassy

phase

Van Krevelen Group Contribution for Segments If the parameters are not entered for the segments, then these values are estimated using Van Krevelen Group Contribution Methods (see page 145). The van der Walls molar volume of a segment is calculated from contributions of functional groups in the segment:

kk

kVwnVw ∑=

Where:

Vw = Van der Waals molar volume of a segment


kVw = Van der Waals volume of functional group k, from van Krevelen database

The segment parameters lA , lB , cA , cB , gA , gB , and gC , are then calculated by the following equations:

VwAAA gcl

×===

3.11

(4.14)

3.1001.0

=== gcl BBB (4.15)

3.1105.5 4−×

=gC 1 (4.16)

In some cases, the parameters of functional groups are not available in the databank. The contributions from these groups are ignored.

Parameter Regression If the parameters in Equations 4.11–4.13 are not available for components, and cannot be estimated by van Krevelen group contribution, the user can perform an Aspen Plus Regression Run (DRS) to obtain these parameters. In some cases, a DRS run can also be used to adjust these parameters to fit available experimental data. In the data regression, these parameters can be:



• Specified for each oligomer component on a molar basis or mass basis


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Tait Liquid Molar Volume Model The Tait molar volume model is an empirical correlation of the molar volume of polymer and oligomer components with temperature and pressure. This model is especially useful when the model parameters are available in the literature, or can be estimated through experimental data regression. Due to the empirical nature of the model, it should be used only within the ranges of temperature and pressure that were used to obtain the model parameters for each polymer or oligomer.

The Tait model is applicable over a wide range of temperature and pressure, and it is particularly useful in cases where the effect of pressure is significant. In almost all cases, the average error with the Tait model was found to be within the reported experimental error (approximately 0.1%).

The Tait equation is a P-V-T relationship for pure polymers, which gives the best representation of P-V-T data for most polymers (Danner & High, 1992). This empirical equation uses a polynomial expression for the zero pressure isobar.

The Tait equation is used to calculate the molar volume of a polymer component as follows :

⎟⎟⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡+−×=

)(1ln1),0(*,

TBPCTVMV n

l

2210 )15.273()15.273(),0( −+−+= TATAATV

[ ])15.273(exp)( 10 −−= TBBTB

Where: visit

lV *, = Molar volume of the polymer in

kmolm /3

nM = Polymer molecular weight

),0( TV = Zero pressure isobar

C = 0.0894

P = Pressure in Pascals ( )P P Plower upper≤ ≤

T = Temperature in Kelvin ( )T T Tlower upper≤ ≤

A A A B B0 1 2 0 1, , , , = Specific constants

Values for several common polymers are given in Appendix C.


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Tait Model Parameters The following table lists the Tait model parameters. These parameters may be entered on the T-Dependent correlation Input form located in the Pure Component subfolder. Note that the Tait model parameters have to be specified for a polymer or oligomer component.



Upper Limit

MDS Units Keyword

VLTAIT/1 A0 --- --- --- --- MASS-VOLUME

VLTAIT/2 A1 --- --- --- --- MASS-VOLUME TEMP

VLTAIT/3 A2 --- --- --- --- MASS-VOLUME TEMP

VLTAIT/4 B0 --- --- --- --- PRESSURE

VLTAIT/5 B1 --- --- --- --- 1/TEMP

VLTAIT/6 Plower 0 --- --- --- PRESSURE

VLTAIT/7 Pupper 1000 --- --- --- PRESSURE

VLTAIT/8 Tlower 0 --- --- --- TEMP

VLTAIT/9 Tupper 1000 --- --- --- TEMP

Parameter Regression If the parameters are not available for a polymer component, the user can perform an Aspen Plus Regression Run (DRS) to obtain these parameters. In some cases, a DRS run can also be used to adjust these parameters to fit available experimental data.


Van Krevelen Glass Transition Temperature Correlation The van Krevelen correlations for the glass transition temperature are as follows:

∑∑=k

kkk

kgkAg MnYnT /,,

∑∑=Nseg

AAA

Nseg

AAgAAg MXTMXT /,


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Where:

AgT , = Glass transition temperature for segment type A


kgY , = Glass transition temperature of functional group k, from van Krevelen database

kM = Molecular weight of functional group k




AM = Molecular weight of segment type A

kgY , values for functional groups are given in Appendix B.

Glass Transition Correlation Parameters The glass transition model parameters are given here:



Upper Limit

MDS Units Keyword

TGVK gT , or --- 0 5000 X TEMP

AgT , --- 0 5000 X TEMP

Van Krevelen Melt Transition Temperature Correlation The van Krevelen correlations for the glass transition temperature are:

∑∑=k

kkk

kmkAm MnYnT /,,

∑∑=Nseg

AAA

Nseg

AAmAAm MXTMXT /,

Where:

AmT , = Melt transition temperature for segment type A


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kmY , = Melt transition temperature of functional group k, from van Krevelen database






kmY , values for functional groups are given in Appendix B.

Melt Transition Correlation Parameters The glass transition model parameters are given here:



Upper Limit

MDS Units Keyword

TMVK Tm , or --- 0 5000 X TEMP

AmT , --- 0 5000 X TEMP

Van Krevelen Solid Property Models The polymer properties at the solid state in Aspen Polymers can be calculated using the similar approach of that for the liquid state (see Van Krevelen Liquid Property Models on page 127).

Solid Enthalpy of Polymers The solid enthalpy of a polymer component is calculated using the following equation:

HS )1(*,*,c

lc

c xHxH −+= for mg TTT ≤≤

cH *,= for T Tg<

Where:

HS = Net enthalpy of the polymer


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lH *, = Enthalpy of the polymer in the liquid phase

cH *, = Enthalpy of the polymer in the crystalline phase



xc = Mass-fraction crystallinity

For a detailed discussion of the above quantities, see Liquid Enthalpy of Polymers on page 128.

Solid Gibbs Free Energy of Polymers The solid Gibbs free energy of a polymer component is calculated using the following equation:

GS )1(*,*,c

lc

c xx −+= μμ for mg TTT ≤≤

c*,μ= for T Tg<

Where:

GS = Net Gibbs free energy of the polymer

l*,μ = Gibbs free energy of the polymer in the liquid phase

c*,μ = Gibbs free energy of the polymer in the crystalline phase

For a detailed discussion of the above quantities, see Liquid Gibbs Free Energy of Polymers on page 130.

Solid Enthalpy and Gibbs Free Energy Model Parameters The van Krevelen solid property model parameters are the same as those required for the van Krevelen liquid property models. For a detailed discussion, see Liquid Enthalpy and Gibbs Free Energy Model Parameters on page 131.

Solid Molar Volume of Polymers The solid molar volume of a polymer component is calculated using the following equation:

VS )1(*,*,c

lc

c xVxV −+= for mg TTT ≤≤

)1(*,*,c

gc

c xVxV −+= for gTT <

Where:

VS = Net molar volume of the polymer in the solid state


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lV *, = Molar volume of the polymer in the liquid phase

cV *, = Molar volume of the polymer in the crystalline phase

gV *, = Molar volume of the polymer in the glassy phase


For a detailed discussion of lV *, , cV *, , and gV *, , see Van Krevelen Liquid Molar Volume Model on page 136.

Solid Molar Volume Model Parameters The van Krevelen solid molar volume model parameters are the same as those required for the van Krevelen liquid molar volume model. For a detailed discussion, see Van Krevelen Liquid Molar Volume Model Parameters on page 137.

Van Krevelen Group Contribution Methods Based on the group contribution concept, the van Krevelen models use the

properties of functional groups to estimate heat capacity ( cl CpCp *,*, , ), and

molar volume ( gcl VVV *,*,*, , , ), for polymer segments, and, thereafter, of polymers and oligomers.

In Aspen Polymers, a polymer is defined in terms of its repeating units or segments. The van Krevelen models use the following approach to estimate properties for a system containing polymers:

• First, the segment properties are estimated using the properties of the functional groups that make up the segment(s). For example, for heat capacity, Cp, the segment property is calculated as the sum of the functional group values using:

∑=k

kkCpnCp **

Where subscript k refers to the functional group. Correlations for other properties are given in Appendix B.

If you are retrieving the segments from the SEGMENT databank, you do not need to supply functional groups. If you are not retrieving the segments from SEGMENT, or wish to override their databank functional group definition, you must supply their molecular structure in terms of van Krevelen functional groups.

• Next, the polymer properties are calculated using the properties of polymer segments, number average degree of polymerization, and segment composition.

• Finally, mixture properties for the whole component system (polymer, monomer, and solvents) are calculated.


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The following table illustrates this approach for acrylonitrile-butadiene-styrene (ABS). The van Krevelen functional groups available in Aspen Polymers are given in Appendix B.

Polymer Segments Functional Groups

ABS

Butadiene-R

CH2 CH CH CH<

CH2CH CHCH<

Styrene-R

CH2 CH

CH2CH<

Acrylonitrile-R

CH2 CHC N

CH2

CHC N

Polymer Property Model Parameter Regression As stated earlier in this chapter, the polymer property models, including Aspen Ideal Gas Property Model, van Krevelen Liquid and Solid Property Models, and Tait Liquid Molar Volume Model, are implemented as polynomial expressions in Aspen Polymers so that they can be used in a predictive mode (such as Van Krevelen Group Contribution Methods, explained on page 145), or in a correlative mode (in case experimental data are available for parameter estimation). Therefore, all polymer property model parameters can be adjusted to fit available experimental data. The user can perform an Aspen Plus Regression Run (DRS) to obtain these parameters. These parameters can be:



• Specified for each oligomer component on a molar basis or mass basis

Note: The Tait model parameters have to be specified for an oligomer component (polymer). In a Data Regression Run, a polymer component must be defined as an OLIGOMER type, and the number of each type of segment that forms the oligomer (or polymer) must be specified.


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Polymer Enthalpy Calculation Routes with Activity Coefficient Models When an activity coefficient model is used, Aspen Polymers 2006 will provide three new routes, DHL00P, DHL01P, and DHL09P, to calculate the liquid pure component enthalpy departure. In the Aspen Physical Property System, a route is defined as a unique combination of methods and models for calculating a property. The polymer mixture enthalpy is calculated from the ideal gas mixture enthalpy and the liquid mixture enthalpy departure as follows:

lEm

i

lii

igm

lm

igm

lm HHxHHHH ,*,)( +=−+= ∑

( ) lEm

igi

li

ii

igm

lm HHHxHH ,*,*, +−=− ∑

)( *,*,*,*, igi

li

igi

li HHHH −+=

or

HLXSHLxDHLMXHIGMXHLMXi

ii +=+= ∑

HLXSDHLxDHLMXi

ii += ∑

DHLHIGHL +=

Where:

Name Symbol Description

HLMX lmH Liquid mixture molar enthalpy

HIGMX igmH Ideal gas mixture molar enthalpy

DHLMX igm

lm HH − Liquid mixture molar enthalpy departure

HLXS lEmH , Liquid mixture molar excess enthalpy

HL liH *, Liquid pure component molar enthalpy

HIG igiH *, Ideal gas pure component molar enthalpy

DHL igi

li HH *,*, − Liquid pure component molar enthalpy departure

For an activity coefficient model, the calculation procedure for HLMX and HL is constituted in a unique route, respectively. The ideal gas pure or mixture molar enthalpy is automatically calculated using the same Aspen model


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regardless of the route used for HLMX or HL. HLXS is calculated directly from the activity coefficient model.

The new routes for the liquid pure component enthalpy departure are embedded in new routes for both HLMX and HL to ensure that both HLMX and HL are directly dependent on DHL. Changing the route for DHL will affect both HLMX and HL simultaneously. The route structure also insures that HLMX will reduce to HL when there is only a single component in the mixture (HLXS = 0).

For polymer/oligomer components, all three routes apply the same van Krevelen model to calculate the liquid pure component enthalpy departure. The difference lies in the way to calculate the liquid pure component enthalpy departure for conventional components. DHL00P uses Ideal gas law, Extended Antoine model, and Watson model to calculate the enthalpy departure. DHL01P uses Redlich-Kwong model, Extended Antoine model, and Watson model. And DHL09P uses the DIPPR liquid heat capacity correlation model. For PNRTL-IG, the default route is DHL00P and for all other activity coefficient models, the default route is DHL01P. Use DHL09P to calculate the liquid pure component enthalpy or heat capacity from the DIPPR correlation model for conventional components.

The following tables list the routes available in Aspen Polymers for liquid pure component enthalpy and polymer mixture enthalpy calculations with activity coefficient models:

Routes available for liquid pure component enthalpy (HL) Route ID Route ID for DHL Description

HLDVK0 DHL00P Using Ideal gas law, Extended Antoine model and Watson model for conventional components and van Krevelen model for polymer components.

HLDVK1 (default)

DHL01P Using Redlich-Kwong model (RK), Extended Antoine model and Watson model for conventional components and van Krevelen model for polymer components

DHL09P Using DIPPR model for conventional components and van Krevelen model for polymer components.

HLDVK Using Ideal gas law, Extended Antoine model and Watson model for conventional components and van Krevelen model for polymer components.

HL0DVKRK† Using Redlich-Kwong model (RK), Extended Antoine model, and Watson model for conventional components and van Krevelen model for polymer components

HL0DVKD Using DIPPR model for conventional components and van Krevelen model for polymer components.

† HL0DVKRK is the default route for HL in Aspen Polymers 2004.1 and earlier releases.

Routes available for polymer mixture enthalpy (HLMX) in POLYNRTL Route ID Route ID for DHL Description


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Route ID Route ID for DHL Description

HLMXP1 (default)

DHL01P Using Redlich-Kwong model (RK), Henry’s law, Extended Antoine model, Watson model, van Krevelen model, and polymer-NRTL model

HLMXP2 DHL01P Using Redlich-Kwong model (RK), Extended Antoine model, Watson model, van Krevelen model, and polymer-NRTL model

HLMXPRK† Using Redlich-Kwong model (RK), Henry’s law, Extended Antoine model, Watson model, van Krevelen model, and polymer-NRTL model

† HLMXPRK is the default route for HLMX in Aspen Polymers 2004.1 and earlier releases.

Routes available for polymer mixture enthalpy (HLMX) in POLYFH Route ID Route ID for DHL Description

HLMXFH1 (default)

DHL01P Using Redlich-Kwong model (RK), Henry’s law, Extended Antoine model, Watson model, van Krevelen model, and polymer-FH model

HLMXFH2 DHL01P Using Redlich-Kwong model (RK), Extended Antoine model, Watson model, van Krevelen model, and polymer-FH model

HLMXFHRK† Using Redlich-Kwong model (RK), Henry’s law, Extended Antoine model, Watson model, van Krevelen model, and polymer-FH model

† HLMXFHRK is the default route for HLMX in Aspen Polymers 2004.1 and earlier releases.

Routes available for polymer mixture enthalpy (HLMX) in POLYUF Route ID Route ID for DHL Description

HLMXUF1 (default)

DHL01P Using Redlich-Kwong model (RK), Henry’s law, Extended Antoine model , Watson model, van Krevelen model, and polymer-UNIFAC model

HLMXUF2 DHL01P Using Redlich-Kwong model (RK), Extended Antoine model, Watson model, van Krevelen model, and polymer-UNIFAC model

HLMXPURK† Using Redlich-Kwong model (RK), Henry’s law, Extended Antoine model, Watson model, van Krevelen model, and polymer-UNIFAC model

† HLMXPURK is the default route for HLMX in Aspen Polymers 2004.1 and earlier releases.

Routes available for polymer mixture enthalpy (HLMX) in POLYUFV Route ID Route ID for DHL Description


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Route ID Route ID for DHL Description

HLMXFV1 (default)

DHL01P Using Redlich-Kwong model (RK), Henry’s law, Extended Antoine model, Watson model, van Krevelen model, and polymer-UNIFAC-Free-Volume model

HLMXFV2 DHL01P Using Redlich-Kwong model (RK), Extended Antoine model, Watson model, van Krevelen model, and polymer-UNIFAC-Free-Volume model

HLMXFVRK † Using Redlich-Kwong model (RK), Henry’s law, Extended Antoine model, Watson model, van Krevelen model, and polymer-UNIFAC-Free-Volume model

† HLMXFVRK is the default route for HLMX in Aspen Polymers 2004.1 and earlier releases.

Routes available for polymer mixture enthalpy (HLMX) in PNRTL-IG Route ID Route ID for DHL Description

HLMXP00 (default)

DHL00P Using Ideal gas law, Henry’s law, Extended Antoine model, Watson model, van Krevelen model, and polymer-NRTL model

HLMXP02 DHL00P Using Ideal gas law, Extended Antoine model, Watson model, van Krevelen model, and polymer-NRTL model

HLMXP † Using Ideal gas law, Extended Antoine model, Watson model, van Krevelen model, and polymer-NRTL model

† HLMXP is the default route for HLMX in Aspen Polymers 2004.1 and earlier releases.


Bicerano, J. (1993). Prediction of Polymer Properties. New York: Marcel Dekker, Inc.

Danner, R. P., & High, M. S. (1992). Handbook of Polymer Solution Thermodynamics. New York: American Institute of Chemical Engineers.

Van Krevelen, D. W. (1990). Properties of Polymers, 3rd Ed. Amsterdam: Elsevier.


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5 Polymer Viscosity Models 151

5 Polymer Viscosity Models

This chapter describes the polymer viscosity models in Aspen Polymers (formerly known as Aspen Polymers Plus). Polymer melt viscosity is calculated using the Modified Mark-Houwink/van Krevelen model. Viscosity of polymer solutions and mixtures over the entire range of composition is calculated using the Aspen polymer mixture viscosity model. Polymer solution viscosity can also be calculated using the van Krevelen polymer solution viscosity model or the Eyring-NRTL mixture viscosity model.


• About Polymer Viscosity Models, 151

• Modified Mark-Houwink/van Krevelen Model, 152

• Aspen Polymer Mixture Viscosity Model, 158

• Van Krevelen Polymer Solution Viscosity Model, 161

• Eyring-NRTL Mixture Viscosity Model, 167

• Polymer Viscosity Routes in Aspen Polymers, 170

About Polymer Viscosity Models The modified Mark-Houwink/van Krevelen model is used to calculate the zero-shear viscosity of polymer melts. The effects of temperature and polymer molecular weight on viscosity are considered. The model can be used correlatively (in the presence of viscosity data for regression) or predictively, as proposed by van Krevelen. The Aspen polymer mixture viscosity model is used with good accuracy to correlate data over the entire concentration range, from pure polymer melt to polymer at infinite dilution. The Eyring-NRTL mixture viscosity model is also applicable to polymer mixture systems. For polymer solutions, the effect of polymer concentration can also be considered using the van Krevelen polymer solution viscosity model.

The following tables provide an overview of the available models for polymer systems in Aspen Polymers:

Property Name

Symbol Description

MUL li*,η Liquid viscosity of a component in a mixture


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152 5 Polymer Viscosity Models

MULMX lη Liquid viscosity of a mixture

Viscosity Models Model Name Pure Mixture Properties Calculated

Modified Mark-Houwink/ van Krevelen Model

MUL0MH X __ MUL

Aspen Polymer Mixture Viscosity Model

MUPOLY __ X MULMX

Van Krevelen Polymer Solution Viscosity Model

MUL2VK __ X MULMX

Eyring-NRTL Mixture Viscosity Model

EYRING __ X MULMX


Modified Mark-Houwink/van Krevelen Model The polymer melt viscosity varies with the polymer structural characteristics, state conditions, and shear history. Currently, the melt viscosity model available in Aspen Polymers considers the effects of polymer structure, polymer molecular weight and molecular weight averages, and temperature. This model combines two zero-shear viscosity correlations. The modified Mark-Houwink equation correlates polymer molecular weight and temperature effect; the van Krevelen method estimates viscosity-temperature function based on functional group properties. The Andrade/DIPPR model is used to calculate viscosity for conventional components (Andrade, 1930)

Polymer melt viscosity increases as polymer molecular weight increases. The classical Mark-Houwink equation correlates the viscosity-molecular weight dependency with a power-law expression. Polymer melt viscosity is also a strong function of temperature; it decreases as the temperature increases.

Modified Mark-Houwink Expression

The Modified Mark-Houwink (MMH) equation uses an Arrhenius expression to account for the viscosity-temperature relationship of polymers:

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=

ref

ref

ref

wref

li T

TRTE

TT

MM

1exp*, ηβα

ηη

Where:

li*,η = Zero-shear viscosity of a polymer component

refη = Zero-shear viscosity of the polymer at the specified reference temperature and molecular weight


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wM = Weight average molecular weight for the polymer

refM = Reference molecular weight of the polymer

refT = Reference temperature

α = Exponential factor accounting for the polymer molecular weight effect. This is a two parameter vector where

)1(α is used for crw MM >

)2(α is used for crw MM ≤

Mcr = Critical molecular weight of polymer, at which viscosity-molecular weight dependency changes. It corresponds to the polymer weight-average molecular weight at the turning point of a 0logη vs.

wMlog plot. For example:

ηE = Activation energy of viscous flow

R = Universal gas constant

T = Absolute temperature

β = Empirical temperature exponent

The weight average molecular weight, wM , of the polymer can be retrieved

from the polymer attribute MWW or calculated from its number average molecular weight and polydispersity index:

PDIMM nw *=

Where:

nM = Number average molecular weight of the polymer


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PDI = Polydispersity index of the polymer

The value for critical molecular weight is available for a limited number of POLYMER databank polymers (Van Krevelen, 1990). If the critical molecular weight for a polymer component is not available from the databank, you must supply it.

Modified Mark-Houwink Model Parameters The following table lists the MMH model parameters†:



Upper Limit

MDS Units Keyword

Comments

MULMH/1 refη --- 1010− 1010 --- VISCOSITY Unary

MULMH/2 ηE 0 0 1010 --- MOLE-

ENTHALPY Unary

MULMH/3 )1(α 3.4 0 20.0 --- --- Unary

MULMH/4 )2(α 1.0 0 20.0 --- --- Unary

MULMH/5 β 0 -5.0 5.0 --- --- Unary

MULMH/6 refT --- 200 5000 --- TEMP Unary

MULMH/7 refM --- 5000 1010 --- --- Unary

CRITMW Mcr --- 1.0 1010 --- -- Unary

HMUVK ηH --- 10-10 1010 --- (MOLE-

ENTHALPY)1

/3

Unary

TGVK gT --- 0 5000 --- TEMP Unary

POLPDI* PDI 1.0 1.0 1000 --- --- Unary

† MULMH must be created as a new parameter to enter data for it. Values for

MULMH must be entered in SI units.

* Only required for Data Regression (DRS) runs and oligomer components or when weight-average molecular weight is not included in the list of polymer component attributes.

Parameter Input and Regression All unary parameters have to be specified for each polymer or oligomer component.

The parameters ηE and β are related to the effect of temperature on

viscosity. The parameters α and crM are related to polymer molecular weight.

Except crM , values for ηE , β , and α can be regressed from experimental


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data and entered for any polymer or oligomer. Therefore, if viscosity data is available for a given polymer component, a Data Regression (DRS) simulation will return the MMH equation parameters.

Note: In DRS runs, the polymer must be treated as an oligomer and the number of each type of segment that forms the oligomer must also be specified.

In order to calculate the weight average molecular weight of the polymer in a DRS run, the polydispersity index of the polymer has to be specified using the pure component property POLPDI.

Van Krevelen Viscosity-Temperature Correlation If no MMH parameters are supplied to the MMH expression the Arrhenius term drops out:

α

ηη ⎟⎟⎠

⎞⎜⎜⎝

⎛=

cr

wcr

li M

MT )(*,

We set crref MM = . In this case, the )(Tcrη term is estimated using the van

Krevelen viscosity-temperature correlation.

The van Krevelen viscosity-temperature correlation estimates the crη based

on polymer structural information and glass transition temperature. The following figure shows the viscosity-temperature relationship for a number of common polymer components:

crη vs. T Graphical Correlation (Hoftyzer & Van Krevelen, 1976)

The zero-shear viscosity of various polymers exhibits similar T−η trends. If

gTT 2.1≤ , all the polymers follow a Williams-Landel-Ferry (WLF) relationship


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(Williams et al., 1955). At higher temperatures, different polymers follow different paths.

Van Krevelen modeled this behavior using a group contribution method. The principles of the van Krevelen method can be summarized as follows:

• The viscosity-temperature relationship of different polymer components can be represented by a number of master curves. These master curves are functions of three parameters: the polymer glass transition temperature, gT , the critical mass viscosity at [ ])2.1(2.1 gcrg TTT η= , and a

structural parameter, A.

• A new transport property called the viscosity-temperature gradient, ηH ,

is defined. Each functional group of a polymer molecule has a unique value for ηH that is mole-additive with respect to functional groups and

segments.

• ηH is used to compute )2.1( gcr Tη and A.

The van Krevelen master curves, which correlate the polymer viscosity-temperature relationship, are shown here:

crη vs. T Master Curves (Hoftyzer & Van Krevelen, 1976)

These master curves simulate the polymer viscosity-temperature behavior of the previous graphical correlation figure. The van Krevelen method calculates the critical mass viscosity at given temperature [ ])(Tcrη through the following

steps:

1 Compute the component viscosity-temperature gradient from van Krevelen functional group values. Aspen Polymers uses the following mixing rules to compute polymer component viscosity-temperature gradient from van Krevelen functional groups:

For segments:

Ak

Ngrp

kkA MHnH /,, ηη ∑=


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For polymers and oligomers:

A

Nseg

AAAA

Nseg

AA MXHMXH ∑∑= /,ηη

Where:

AH ,η = Viscosity-temperature gradient of segment type A

Ngrp = Number of types of groups in a segment

kn = Number of occurrences of group k in a segment

kH ,η = Viscosity-temperature gradient of group k, from van Krevelen database


ηH = Viscosity-temperature gradient of a polymer

Nseg = Number of types of segments in a polymer

AX = Mole segment fraction of segment type A in a polymer

2 )(∞ηE , the activation energy of viscous flow at high temperature, is

calculated from the polymer component viscosity-temperature gradient: 3)( ηη HE =∞

3 With )(∞ηE computed from group quantity, the following two parameters

that affect polymer melt viscosity are estimated using the following equations:

The critical mass viscosity at gTT 2.1= is calculated using the WLF

equation:

4.1)105.8052.0(

)()2.1(log5

−×−

∞=−

g

ggcr T

TET ηη

The structural parameter A is calculated using the following equation:

gRTE

A)(

3.21 ∞

= η

Tg may be provided for polymer components. If Tg is not supplied, the

van Krevelen estimate is used.

4 Given values for T Tg / and A, the value for the reduced viscosity is

obtained from the master curves shown in the previous figure and where:

)2.1()(

log1gcr

crg

TT

TT

Aη

η=⎟⎟

⎠

⎞⎜⎜⎝

⎛−

)2.1( gcr Tη is known from the previous step, therefore the final


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value for )(Tcrη can be calculated.

Specifying the MMH Model See Specifying Physical Properties in Chapter 1.

Aspen Polymer Mixture Viscosity Model Viscosity of polymer solutions,, and mixtures in general, depends on composition, molecular weight, temperature, and shear rate. Although a great deal of effort has been spent to describe the viscosity of dilute polymer solutions (intrinsic viscosity) as well as that of polymer melts, relatively little attention has been paid to the broad range of polymer composition between these extremes. It is this region, however, that is import when describing reacting mixtures. Aspen polymer mixture viscosity model (Song et al., 2003) can be used to correlate the entire concentration range from pure polymer melt to polymer at infinite dilution. This correlative model is essentially a new mixing rule for calculating the mixture viscosity from the pure component viscosities. It assumes that the viscosities of both pure polymer and non-polymeric components (solvents) are already available as input. This model uses two binary parameters to capture non-ideal mixing behavior. Our testing indicates that this model is very effective for polymeric and conventional chemical systems.

Multicomponent System The Aspen polymer mixture viscosity model is applicable to mixtures containing any number of components containing polymers. It expresses the zero-shear viscosity of the mixture as follows:

( )3

3/1*, lnlnlnln ⎥⎦

⎤⎢⎣

⎡++= ∑∑∑∑

≠> ijijijj

ii

ijijjiij

i

lii

l lwwwwkw ηηηη

Where:

lη = Zero shear viscosity of the mixture

li*,η = Zero shear viscosity of component i

iw = Weight fraction of component i

ijk = Symmetric binary parameter, ijji kk =

ijl = Antisymmetric binary parameter, ijji ll −=

ijηln = Cross binary term from viscosities of pure components


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The first term is the linear mixing, the second term is the binary symmetric-quadratic mixing, and the last term is the binary antisymmetric mixing. The cross binary term is chosen as follows:

|lnln|ln *,*, lj

liij ηηη −=

Therefore, 0ln →ijη when lj

li

*,*, ηη → . The antisymmetric mixing term

satisfies the invariant condition when a component is divided into two or more identical subcomponents (Mathias et al., 1991).

Binary Parameters There are two binary parameters, one symmetric, ijk , and one antisymmetric,

ijl . Both binary parameters allows complex temperature dependence:


2''''' ln/ rijrijrijrijijij TeTdTcTbal ++++=

with

refr T

TT =

Where:


Aspen Polymer Mixture Viscosity Model Parameters The binary parameters for the Aspen polymer mixture viscosity model are listed here:



Upper Limit

MDS Units Keyword

Comments

MUKIJ/1 ija 0 --- --- X --- Binary,

Symmetric

MUKIJ/2 ijb 0 --- --- X --- Binary,

Symmetric

MUKIJ/3 ijc 0 --- --- X --- Binary,

Symmetric

MUKIJ/4 ijd 0 --- --- X --- Binary,

Symmetric

MUKIJ/5 ije 0 --- --- X --- Binary,

Symmetric

MUKIJ/6 refT 298.15 --- --- X --- Binary,

Symmetric


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MULIJ/1 'ija 0 --- --- X --- Binary,

Antisymmetric

MULIJ/2 'ijb 0 --- --- X --- Binary,

Antisymmetric

MULIJ/3 'ijc 0 --- --- X --- Binary,

Antisymmetric

MULIJ/4 'ijd 0 --- --- X --- Binary,

Antisymmetric

MULIJ/5 'ije 0 --- --- X --- Binary,

Antisymmetric

MULIJ/6 refT 298.15 --- --- X --- Binary,

Antisymmetric

Parameter Input and Regression Both binary parameters, ijk and ijl , have to be specified for each component-

component pair. Their default values are zero.

If viscosity data is available for a polymer solution or a binary mixture, a Data Regression (DRS) simulation will return both binary parameters, ijk and ijl .


Polymer Solution Viscosity Correlation For polymer-solvent solutions, the Aspen polymer mixture viscosity model reduces to:

121212

*,*,

ln)1)](21([

lnln)1(ln

η

ηηη

ppp

lpp

lsp

l

wwwlk

ww

−−++

+−=

2/)ln(lnln *,*,12

ls

lp ηηη −=

Where:

lη = Zero shear viscosity of the polymer-solvent solution

ls*,η = Zero shear viscosity of the solvent

lp*,η = Zero shear viscosity of the polymer


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pw = Weight fraction of the polymer

12k = Symmetric binary parameter

12l = Antisymmetric binary parameter

Specifying the Aspen Polymer Mixture Viscosity Model See Specifying Physical Properties in Chapter 1.

Van Krevelen Polymer Solution Viscosity Model The viscosity of concentrated polymer solutions exhibits characteristics similar to those of polymer melts. The influence of parameters such as molecular mass, temperature and shear rate on viscosity are largely similar. The viscosity of a polymer solution is also a function of polymer concentration. A discontinuity is observed in polymer solution viscosity versus concentration profile at the so-called critical concentration. A solution is considered “concentrated” when the polymer weight concentration exceeds the critical concentration, typically at five percent by weight.

Historically, a clear distinction has been made in the literature between dilute polymer solutions and concentrated polymer solutions with regard to viscosity. In concentrated solutions, solvents reduce the solution viscosity by reducing the glass transition temperature, Tg , and through dilution effects.

This model extends the van Krevelen binary polymer solution viscosity correlations to multicomponent mixtures. The solution is treated as a quasi-binary mixture of polymer and solvent.

For mixtures without polymeric components, the Letsou-Stiel corresponding state correlation is used.

Quasi-Binary System The van Krevelen binary polymer solution viscosity model in Aspen Polymers treats a multicomponent polymer mixture as a quasi-binary system consisting of a pseudo-polymer component and a pseudo-solvent component. The pseudo-polymer component is a blend of all polymers and oligomers in the mixture that possesses properties averaged across the components present. The pseudo-solvent component is composed of all present non-polymeric species. The properties of the pseudo-solvent are averaged across the conventional species in the system.


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Properties of Pseudo-Components A weight-average mixing rule is used to compute pseudopolymer properties:

Weight-average mixing rule

Q w Q wBi ip i

i

Npol

i

Npol

===∑∑ /

11

Where:

QB = Property of the pseudo-polymer (the superscript B stands for the pseudo-polymer)

Npol = Total number of polymeric components in the system

QB represents any of the following quantities:

η0B = Zero-shear viscosity of the pseudo-polymer. The above mixing

rule for the pseudo-polymer viscosity is derived from the influence of polydispersity on zero-shear viscosity (Flory, 1943)

H Bη = Van Krevelen viscosity-temperature gradient of the pseudo-

polymer. Hη is additive for van Krevelen groups. The viscosity-

temperature gradient of the blend equals the weight-averaged viscosity-temperature gradient of all polymeric species

TgB = Glass transition temperature of the pseudo-polymer. The weight-

average mixing rule is derived for TgB by extending the Bueche

formula to polymer mixtures, with an assumption that the K constant is the same for all polymers (Bueche, 1962)

γ B = Power-law exponential factor that accounts for the real solvent dilution effects

Qpi = Property of polymer component i, and represents any of the following quantities:

η0i = Zero-shear viscosity of polymer i, computed from pure component viscosity models. It is a function of polymer molecular weight, temperature and polymer structure

H iη = Van Krevelen viscosity-temperature gradient of polymer i. It is estimated from the van Krevelen group contribution method (See Chapter 4)

Tgi = Glass transition temperature of polymer i. Tg values are user

specified or estimated from the van Krevelen group contribution method (see Chapter 4)

γ i = Power-law exponent for solvent dilution of polymer i. γ i is correlated to the molecular weight exponential factor α by γ α/ .≈ 15 usually varies from 4.0 to 5.6. For more information on α, see Van Krevelen Viscosity-Temperature Correlation on page 155.


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The same mixing rule applies to the solvent mixture for the properties of the pseudo-solvent:

Q w Q wSi si i

i

Nsol

i

Nsol

===∑∑ /

11

Where:

QS = Property of the pseudo-solvent (the superscript S stands for pseudo-component solvent)

Nsol = Total number of solvent components in the system

QS represents any of the following quantities:

TgS = Glass transition temperature of the pseudo-solvent component.

The mixing rule for TgS is an extension of the Bueche formula

(Bueche, 1962)

K S = Constant related to the component volume expansion coefficient

Qsi = Property of solvent component i and represents any of the following quantities:

Tgi = Glass transition temperature of solvent component i. In situations when the solvent Tg values are not available, user may use

component melting point for estimation: T Tg m≈ 2 3/ . Tg values

must be specified to for each solvent

Ki = Constant related to the component volume expansion coefficient:

Ki

s gs

p gp≈

−

−

α α

α α1

1, α1 is the volume expansion coefficient above Tg ,

and α g is the volume expansion coefficient below Tg . Ki is

defined as a solvent parameter. Typically, Ki has a value

between 1.0 and 3.0. If there is no data available to estimate Ki , a default value of 2.5 is suggested

With the above mixing rules, the two pseudo-component properties needed to compute solution viscosity are available. The van Krevelen binary solution model is applied to the quasi-binary solution to obtain the mixture viscosity.

Van Krevelen Polymer Solution Viscosity Model Parameters The parameters for the van Krevelen polymer solution viscosity model are listed here:



Upper Limit

MDS Units Keyword

Comments

MULVK/1 Ki 2.5 0 10 --- --- Unary


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MULVK/2 γ i 5.1 1 100 --- --- Unary

TGVK Tg --- 0 5000 --- TEMP Unary

Polymer Solution Viscosity Estimation In a binary solution of polymer and solvent, the solution viscosity decreases as the solvent concentration increases. This is caused by:

• Decrease of the viscosity of the pure polymer as a result of a decrease of the glass transition temperature

• Real dilution effect, which causes the viscosity of the solution to fall between that of the pure polymer and that of the pure solvent

For these reasons, the concentration dependency and temperature dependency of solution viscosity are strongly related. Polymer viscosity is much more significant than solvent viscosity. Therefore, in the van Krevelen solution viscosity model, the solvent viscosity is neglected.

To calculate the binary polymer solution viscosity, the van Krevelen model estimates Tg of the polymer mixture, calculates the mixture viscosity at given

temperature with the mixture glass point, then applies the true solvent dilution effect. The Tg effect and the real dilution effect are imposed on the

polymer viscosity only.

The polymer viscosity-temperature relationship is described in graphical form in the van Krevelen polymer melt viscosity correlation in the figure crη vs T

Graphical Correlation (see page 155). The steps used to calculate viscosity in the van Krevelen solution viscosity model are illustrated here:


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Polymer Solution Glass Transition Temperature Polymer viscosity varies with glass transition temperature. Addition of a solvent to the polymer lowers the glass transition temperature to the mixture glass point, Tg

m , and, therefore, lowers the polymer viscosity. This is the so-

called plasticizer effect. A theoretical treatment of the plasticizer effect has been developed by Bueche who gave the following equation for the glass transition temperature of a plasticized polymer (Bueche, 1962):


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TT w K T w

w K wgm g

BB

SgS

S

BS

S=

++

Where:

Tgm = Glass transition temperature of the mixture (superscript m stands

for the mixture)

wB = Total weight fraction of polymer in the mixture, w wB i

i

Npol

==∑

1

wS = Total weight fraction of solvent in the mixture, w wS i

i

Nsol

==∑

1

Polymer Viscosity at Mixture Glass Transition Temperature For a polymer-solvent binary mixture, the undiluted polymer viscosity at the mixture glass point is calculated from the van Krevelen viscosity-temperature relationship:

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛= A

TT

fT

mg

g

,2.1

log*

ηη

Where:

*η = Viscosity of the undiluted polymer with a new glass temperature

( )gT2.1η = Viscosity of the undiluted polymer at its own glass temperature

f = Van Krevelen graphical correlation for polymer melt viscosity

A = Structural factor related to the viscosity-temperature gradient of the polymer ηH by:

( )

AH

RTg= η

3

2 303.

For a quasi-binary system, the structural factor of pseudo-polymer, AB , is

used in the van Krevelen viscosity-temperature relationship. AB is calculated using the pseudo-polymer properties:

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛= B

mg

Bg

BB A

TT

fT

,2.1

log*

ηη

( )B

g

BB

RTH

A303.2

3η=


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where )2.1( Bg

B Tη is solved from the van Krevelen zero shear viscosity

graphical correlation of the pseudo-polymer B0η :

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛= B

Bg

Bg

B

B

AT

Tf

T,

2.1log 0

ηη

True Solvent Dilution Effect The influence of the solvent concentration can be described by a power-law equation:

pp

m wγηη *0 =

For a quasi-binary system, the mixture viscosity is:

BBB

m wγηη *0 =

Where:

m0η = Zero shear viscosity of the mixture

pγ = Exponential factor that accounts for polymer concentration

Bγ = Exponential factor that accounts for the pseudo-polymer concentration

Specifying the van Krevelen Polymer Solution Viscosity Model See Specifying Physical Properties in Chapter 1.

Eyring-NRTL Mixture Viscosity Model The Eyring-NRTL viscosity model (Novak et al., 2004) is a segment-based mixture model for correlating the viscosity of polymer mixtures, including copolymers. It represents a synergistic combination of the Eyring theory for fluid diffusion and flow and the NRTL model for local composition interaction. The segment-based approach provides a more physically realistic model for large molecules like polymers when diffusion and flow is viewed to occur by a sequence of individual segment jumps into vacancies rather than jumps of the entire large molecules. This model uses NRTL binary parameters to capture non-ideal mixing behavior.


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Multicomponent System The Eyring-NRTL mixture viscosity model is applicable to mixtures containing any number of components containing polymers. It expresses the zero-shear viscosity of the mixture as follows:

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛+=

∑∑

∑∑j

jij

jjijij

ii

i

lii

l

Gx

Gxxx

τηη *,lnln

With:

∑∑∑

=

J jJjJ

IIiI

i rX

rXx

,

,


Where:

lη = Zero shear viscosity of the mixture

li*,η = Zero shear viscosity of component i

I and J = Component based indices

i and j = Segment based indices

ix = Segment based mole fraction for segment based species i

XI = Mole fraction of component I in component basis

ri I, = Number of segment type i in component I



Binary Parameters The binary parameter, ijτ , allows complex temperature dependence:

2ln/ rijrijrijrijijij TeTdTcTba ++++=τ

with

refr T

TT =

Where:



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Eyring-NRTL Mixture Viscosity Model Parameters The parameters for the Eyring-NRTL mixture viscosity model are listed here:



Upper Limit

MDS Units Keyword

Comments

PVISC li*,η --- --- --- X VISCOSITY Unary

VNRTL/1 ija 0 --- --- X --- Binary,

Antisymmetric

VNRTL /2 ijb 0 --- --- X --- Binary,

Antisymmetric

VNRTL /3 ijc 0 --- --- X --- Binary,

Antisymmetric

VNRTL /4 ijd 0 --- --- X --- Binary,

Antisymmetric

VNRTL /5 ije 0 --- --- X --- Binary,

Antisymmetric

VNRTL /6 ijα 0.3 --- --- X --- Binary,

Symmetric

VNRTL /7 refT 298.15 --- --- X --- Binary,

Antisymmetric

Parameter Input and Regression

The input for pure component viscosity, li*,η , is optional. By default, the pure

component viscosity is automatically calculated by the modified Mark-Houwink/van Krevelen model. The binary parameters in ijτ and ijα have to

be specified for solvent-solvent pairs and solvent-segment pairs for data input or data regression. Their default values are zero.


Specifying the Eyring-NRTL Mixture Viscosity Model See Specifying Physical Properties in Chapter 1.


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Polymer Viscosity Routes in Aspen Polymers Aspen Polymers offers two routes, MULMX13 and MULMXVK, for calculating the polymer mixture viscosity. MULMX13 directly refers the Aspen polymer mixture viscosity model, MUPOLY, and is the default route employed by all polymer property methods. MULMXVK directly refers the van Krevelen polymer solution viscosity model, MUL2VK, and can be chosen as an option. A summary of routes appears in the following table:

Route Model Name Applicability Property Methods

MULMX13 MUPOLY default POLYFH, POLYNRTL, POLYUF, POLYUFV, POLYSL, POLYSRK, POLYSAFT, POLYPCSF

MULMXVK MUL2VK optional

References Andrade, E. N. da Costa (1930). Nature, 125, 309, 582.

Bueche. F. (1962). Physical Properties of Polymers. New York: Wiley.

Flory, P. J. (1943). J. Amer. Chem. Soc., 65, 372.

Hoftyzer, P. J., & Van Krevelen, D. W. (1976). Angew. Makromol. Chem., 54, 1.

Kim, D.-M., & Nauman, E. B. (1992). J. Chem. Eng. Data, 37, 427.

Mathias, P. M., Klotz, H. C., & Prausnitz, J. M. (1991). Fluid Phase Equilibria, 67, 31.

Novak, L. T., Chen, C.-C., & Song, Y. (2004). A Segment-Based Eyring-NRTL Viscosity Model for Mixtures Containing Polymers. Ind. Eng. Chem Res., 43, 6231.

Song, Y., Mathias, P. M., Tremblay, D., & Chen, C.-C. (2003). Viscosity Model for Polymer Solutions and Mixtures. Ind. Eng. Chem Res., 42, 2415.

Van Krevelen, D. W. (1990). Properties of Polymers, 3rd. Ed. Amsterdam: Elsevier.

Van Krevelen, D. W., & Hoftyzer, P.J. (1976). Angew. Makromol. Chem., 52, 101.

Williams, M. L., Landel, R. F., & Ferry, J. D. (1955). J. Am. Chem. Soc., 77, 3701.


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6 Polymer Thermal Conductivity Models 171

6 Polymer Thermal Conductivity Models

This chapter describes the polymer thermal conductivity models in Aspen Polymers (formerly known as Aspen Polymers Plus). Polymer thermal conductivity is calculated using the modified van Krevelen thermal conductivity model. The Aspen polymer mixture thermal conductivity model is used to calculate the thermal conductivity of mixtures containing polymers.


• About Thermal Conductivity Models, 171

• Modified van Krevelen Thermal Conductivity Model, 173

• Aspen Polymer Mixture Thermal Conductivity Model, 180

• Polymer Thermal Conductivity Routes in Aspen Polymers, 181

About Thermal Conductivity Models The thermal conductivity of a polymeric component generally depends on both temperature and pressure, as well as on the physical state of the polymer. The physical state and temperature dependence is depicted in the following figure:


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172 6 Polymer Thermal Conductivity Models

Ther

mal

Con

duct

ivity

Temperature

Tg Tm

glassy

liquidrubbery

semi-crystalline

crystalline

Ther

mal

Con

duct

ivity

Temperature

Tg Tm

glassy

liquidrubbery

semi-crystalline

crystalline

In the figure, gT and mT are the polymer glass and melt transition

temperatures respectively. The top boundary of the semi-crystalline region is the crystalline thermal conductivity curve. The bottom boundary of the semi-crystalline region is separated into two curves. To the left of the glass transition temperature is the glass thermal conductivity curve, and to the right is the liquid thermal conductivity curve.

Based on this description, the modified van Krevelen thermal conductivity model is used to calculate the thermal conductivity of polymers. Additionally, the effects of temperature and pressure are considered. The model can be used correlatively (in the presence of thermal conductivity data for regression) or predictively, as proposed by van Krevelen. The Aspen polymer mixture thermal conductivity model is used to calculate the polymer mixture thermal conductivity.

The following tables provide an overview of the available models for polymer systems in Aspen Polymers:

Property Name

Symbol Description

KL li*,λ Liquid thermal conductivity of a component in a

mixture

KLMX lλ Liquid thermal conductivity of a mixture

Thermal Conductivity Models

Model Name Pure Mixture Properties Calculated

Modified van Krevelen/DIPPR

KL0VKDP X __ KL

Modified van Krevelen/TRAPP

KL0VKTR X __ KL

Aspen Polymer Thermal Conductivity Mixture Model

KLMXVKTR __ X KLMX


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Modified van Krevelen Thermal Conductivity Model The polymer thermal conductivity is calculated using the following equation:

KL li*,λ= for

mTT >

)1(*,*, ϕλϕλ −+= li

ci for

mg TTT ≤<

)1(*,*, ϕλϕλ −+= gi

ci for

gTT ≤

Where:

KL = Net thermal conductivity of the polymer

li*,λ = Thermal conductivity of the polymer in the liquid phase

ci*,λ = Thermal conductivity of the polymer in the crystalline phase

gi*,λ = Thermal conductivity of the polymer in the glassy phase

ϕ = Crystalline weighting fraction

The superscripts l, c, and g refer to the liquid, crystalline, and glassy curves, respectively. The crystalline weighting fraction is given by Eirmann (1962):

⎟⎟⎠

⎞⎜⎜⎝

⎛−++

=

lgi

ci

clgi

ci

c

x

x

,*,

*,

,*,

*,

12

3

λλ

λλ

ϕ (6.1)

With

lgi

,*,λ li*,λ= for

mg TTT ≤<

gi*,λ= for

gTT ≤

Where:


Modified van Krevelen Equation

The thermal conductivities for the liquid, crystalline and glassy states are calculated using the modified van Krevelen equation:

σ

ψλλ σσ Frefii

*,,

*, = (6.2)

With


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⎟⎟⎠

⎞⎜⎜⎝

⎛ −⎟⎟⎠

⎞⎜⎜⎝

⎛ −+⎟

⎟⎠

⎞⎜⎜⎝

⎛ −+

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛−+⎟

⎟⎠

⎞⎜⎜⎝

⎛ −+=

σ

σ

σ

σσ

σ

σσ

σσ

σσσ

σ

σσψ

ref

ref

ref

ref

ref

ref

refrefref

ref

ref

PPP

TTT

EP

PPD

TTC

TTTB

TTT

A ln111

(6.3)

Where:

σ = l, c, or g

σλ*,i = Thermal conductivity of the polymer for state σ

σλ*,,refi = Thermal conductivity of the polymer for state σ at the reference

temperature and pressure

σrefT = Reference temperature for state σ

σrefP = Reference pressure for state σ

σσσσσ EDCBA ,,,, , and σF are dimensionless constants.

Modified van Krevelen Thermal Conductivity Model Parameters The following table lists the modified van Krevelen thermal conductivity model parameters:



Upper Limit

MDS Units Keyword

Comments

KLVKL/1 lrefi

*,,λ --- 10-6 1000 --- THERMAL-

CONDCTIVITY

Unary

KLVKL/2 lA 0 -1000 1000 --- --- Unary

KLVKL/3 lB 0 -1000 1000 --- --- Unary

KLVKL/4 lC 0 -1000 1000 --- --- Unary

KLVKL/5 lD 0 -1000 1000 --- --- Unary

KLVKL/6 lE 0 -1000 1000 --- --- Unary

KLVKL/7 lF 1 -100 100 --- --- Unary

KLVKL/8 lrefT 298.15 2 1000 --- TEMP Unary

KLVKL/9 lrefP 101325 1000 1010 --- PRESSURE Unary

KLVKC/1 crefi

*,,λ --- 10-6 1000 --- THERMAL-

CONDCTIVITY

Unary

KLVKC/2 cA 0 -1000 1000 --- --- Unary

KLVKC/3 cB 0 -1000 1000 --- --- Unary


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Upper Limit

MDS Units Keyword

Comments

KLVKC/4 cC 0 -1000 1000 --- --- Unary

KLVKC/5 cD 0 -1000 1000 --- --- Unary

KLVKC/6 cE 0 -1000 1000 --- --- Unary

KLVKC/7 cF 1 -100 100 --- --- Unary

KLVKC/8 crefT 298.15 2 1000 --- TEMP Unary

KLVKC/9 crefP 101325 1000 1010 --- PRESSURE Unary

KLVKG/1 grefi

*,,λ --- 10-6 1000 --- THERMAL-

CONDCTIVITY

Unary

KLVKG/2 gA 0 -1000 1000 --- --- Unary

KLVKG/3 gB 0 -1000 1000 --- --- Unary

KLVKG/4 gC 0 -1000 1000 --- --- Unary

KLVKG/5 gD 0 -1000 1000 --- --- Unary

KLVKG/6 gE 0 -1000 1000 --- --- Unary

KLVKG/7 gF 1 -100 100 --- --- Unary

KLVKG/8 grefT 298.15 2 1000 --- TEMP Unary

KLVKG/9 grefP 101325 1000 1010 --- PRESSURE Unary

POLCRY xc 0 0 1 --- --- Unary

TGVK gT

--- 0 5000 X TEMP Unary

TMVK mT

--- 0 5000 X TEMP Unary

Parameter Input The unary parameters can be:



These options are shown in priority order. For example, if the model parameters are provided for a polymer component and for the segments, the polymer parameters are used and the segment parameters are ignored.

Parameters for Polymers If parameters are specified for the polymeric component, then these values are used directly, independent of the copolymer composition. Otherwise, the polymer component thermal conductivity is calculated as the mass average of the segment contributions:


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A

Nseg

AAAA

Nseg

AAi MXMX ∑∑= /*,*, σσ λλ

Where:



AM = Molecular weight of segment type A in the copolymer

σλ*,A = Thermal conductivity of segment type A in state σ , estimated

using Equation 6.2

Van Krevelen Group Contribution for Segments If σλ*,

,refi is missing for a segment, all parameters for that segment in state σ

are estimated using van Krevelen group contributions (van Krevelen, 1990).

The first step is to estimate the segment reference temperature σrefT . For

liquid and glassy states, the segment reference temperatures are calculated from:

kk

kkgk

kVKg

gref

lref MnYnTTT ∑∑=== /,

Segment Reference Temperature Similarly, the segment reference temperature in crystalline state is calculated from:

kk

kkmk

kVKm

cref MnYnTT ∑∑== /,

Where:


VKgT = Van Krevelen estimate of segment glass transition temperature

VKmT = Van Krevelen estimate of segment melt transition temperature

kgY , = Glass transition contribution of functional group k from van Krevelen database

kmY , = Melt transition contribution of functional group k from van Krevelen database



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Note that van Krevelen group parameters are used to calculate σrefT even if

the user has provided component or segment TGVK and TMVK parameter values.

Segment Thermal Conductivity at 298K In order to estimate the segment reference thermal conductivity, the segment

thermal conductivity at 298.15 K, 298VKλ , is calculated first:

3

298

298

298

298298

⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡= VK

VK

VK

VKVK

VUR

VCpLλ

Where:

L = 11105 −x m (constant)

298VKCp = Van Krevelen estimate of segment molar heat capacity at 298.15 K (J/mol.K)

298VKV = Van Krevelen estimate of segment molar volume at 298.15 K /mol)(m3

298VKUR = Van Krevelen estimate of segment Rao wave function at 298.15

K .mol)/s(m 1/3310 /

The segment heat capacity, molar volume, and Rao function at 298.15 K are calculated from van Krevelen group contributions:

∑=k

lkk

VK CpnCp *,298

∑=k

kkVK VwnV 6.1298

∑=k

kkVK URnUR 298

Where:

lkCp*, = Liquid heat capacity contribution of functional group k from van

Krevelen database

kVw = Van der Walls volume contribution of functional group k from van Krevelen database

kUR = Rao wave function contribution of functional group k from van Krevelen database

Note that van Krevelen group parameters are used to calculate 298VKCp and 298VKV even if the user has provided component or segment CPLVK, CPLVKM,

DNLVK, or DNLVKM parameter values. Also, either lkCp*, , kVw , or kUR are

missing for any group comprising a segment, 298VKλ is set equal to 0.20 W/m-K.


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Segment Reference Thermal Conductivity Liquid and Glassy States

The next step is to calculate the segment thermal conductivity at the reference temperature. First, we examine liquid and glassy states. Van Krevelen (1990) presents a generalized curve relating the thermal conductivity for liquid and glassy polymer, at an arbitrary temperature, to that at the glass transition temperature:

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛=

gg TTg

TT

λλ )(

(6.4)

Bicerano (1993) fit van Krevelen's curve to a pair of equations: one applicable for the glassy region (below gT ), and the other applicable in the liquid region

(above gT ):

( )22.0

)(⎟⎟⎠

⎞⎜⎜⎝

⎛=

gg TT

TT

λλ

for gTT ≤ (glass) (6.5)

( ) gg TT

TT 2.02.1)(

−=λλ

for gTT > (liquid) (6.6)

Since the segment thermal conductivity at 298.15 K is known, these expressions can be inverted to provide an estimate for ( )gTλ , and

equivalently, σλ*,ref for a segment:

⎟⎟⎠

⎞⎜⎜⎝

⎛−

=

VKg

VKgl

ref

T15.2982.02.1

298,*, λλ if VK

gT < 298.15 K (6.7)

22.0

298,*,

15.298⎟⎟⎠

⎞⎜⎜⎝

⎛=

VKg

VKgl

ref

T

λλ if VKgT ≥ 298.15 K (6.8)

Crystalline Polymer

For crystalline polymer, we use the following expression from van Krevelen (1990) relating liquid and crystalline thermal conductivity:

⎟⎟⎠

⎞⎜⎜⎝

⎛−+= 18.51 l

c

l

c

ρρ

λλ

(6.9)

Where:

cρ = Density of crystalline state

lρ = Density of liquid state


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Applying this expression at the segment melt temperature (the reference temperature for crystalline thermal conductivity), noting that the density ratio can be replaced by a molar volume ratio, and simplifying, we obtain:

( )( )

( )( ) 8.48.5 −⎟⎟

⎠

⎞⎜⎜⎝

⎛=

mc

ml

ml

mc

TVTV

TT

λλ

(6.10)

Where:

cV = Segment molar volume at mT and crystalline state

lV = Segment molar volume at mT and liquid state

Van Krevelen (1990) relates the molar volumes at an arbitrary temperature for liquid and crystalline polymer to the segment van der Waals volume, VW :

[ ])15.298(106.1)( 3 −+= − TVWTV l (6.11)

[ ])15.298(1045.0435.1)( 3 −×+= − TVWTV c (6.12)

The liquid thermal conductivity at the melting point can be related to the thermal conductivity at the glass point using Equation 6.6. The final expression for the crystalline reference thermal conductivity is therefore:

⎟⎟⎠

⎞⎜⎜⎝

⎛−×+−×+

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−= −

−

)15.298(1045.0435.1)15.298(1064.3392.2

2.02.1 3

3298*,

VKm

VKm

VKg

VKmVKc

ref TT

TT

λλ

(6.13)

Other Parameters No adequate method exists for estimating the pressure dependence of polymer thermal conductivity. Therefore, the estimated value of parameters

σD and σE is zero for all three polymer states. For liquid and glassy polymer, the estimated values of σA , σB , σC , and σF are set in order to be consistent with Equations 6.5 and 6.6. For crystalline polymer, we assume

no temperature dependence, and so 0=== ccc CBA , and 1=cF . The

reference pressure, σrefP , is set equal to 101325 Pa in all cases.

Specifying the Modified van Krevelen Thermal Conductivity Model See Specifying Physical Properties in Chapter 1.


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Aspen Polymer Mixture Thermal Conductivity Model The Aspen polymer mixture thermal conductivity model (KLMXVKTR) is used to calculate the thermal conductivity of mixtures containing polymers. This model uses the Vredeveld mixing rules for calculating the mixture thermal conductivity from the pure component thermal conductivities. It assumes that both pure polymer and non-polymeric components (solvents) are already available as input. For polymer components, it uses the modified van Krevelen model previously described for calculating thermal conductivity. For non-polymer components, it uses the TRAPP model to calculate the thermal conductivity.

Since the TRAPP model directly calculates the thermal conductivity of a polymer-free mixture, the Vredeveld mixing rule is written as:

[ ]

2/1

2'*,2*, )()(

−

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧+

=∑

∑∑

iii

ps

sss

lp

pp

l

Mxx

MxMx

λλλ

With

∑=

ss

ss x

xx '

Where:

lλ = Thermal conductivity of the mixture

lp*,λ = Thermal conductivity of polymer component p in the mixture

)( '*, xsλ = Thermal conductivity of the polymer-free mixture, calculated using the TRAPP model

'sx = Mole fraction of non-polymer component s in the polymer-free

mixture

ix = Mole fraction of component i

iM = Molecular weight of component I

Specifying the Aspen Polymer Mixture Thermal Conductivity Model See Specifying Physical Properties in Chapter 1.


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Polymer Thermal Conductivity Routes in Aspen Polymers Aspen Polymers offers two routes, KLMXVKDP and KMXVKTR, for calculating the polymer mixture thermal conductivity. The KMXVKTR route directly refers the Aspen polymer mixture thermal conductivity model described previously. The KLMXVKDP route uses the Vredeveld mixing rule to combine the modified van Krevelen thermal conductivity model for polymer components and the Sato-Reidel/DIPPR model for non-polymer components:

2/1

2*, )(

−

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

=∑∑

iii

il

i

ii

l

Mx

Mxλ

λ

Where:

lλ = Thermal conductivity of the mixture

li*,λ = Thermal conductivity of component i

ix = Mole fraction of component i

iM = Molecular weight of component i; it is the number average molecular weight for polymer components

The routes differ in the manner in which the thermal conductivity of non-polymer components is handled. The Sato-Reidel/DIPPR model includes only the temperature dependence, and should be used at low pressures. The TRAPP model is a corresponding states model that includes both temperature and pressure dependences, and is applicable to the high pressure region as well.

The following table provides a summary of the available routes:

Route Model Name Applicability Property Methods

KLMXVKDP KL0VKDP Low pressure (less than 20 bar)

POLYFH, POLYNRTL, POLYUF, POLYUFV

KLMXVKTR KLMXVKTR High pressure (greater than 20 bar)

POLYSL, POLYSRK, POLYSAFT, POLYPCSF



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Bicerano, J. (1993). Prediction of Polymer Properties. New York: Marcel Dekker.

Eirmann, V. K. (1962). Bestimmung der wärmeleitfähigkeit des amorphen und des kristallinen anteils von polyäthylen. Kolloid-Zeitschrift & Zeitschrift für Polymere, 180, 163-164.

Van Krevelen, D. W. (1990). Properties of Polymers, 3rd. Ed. Amsterdam: Elsevier.


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A Physical Property Methods 183

A Physical Property Methods

This appendix documents the physical property route structure for the polymer specific property methods:

• POLYFH: Flory-Huggins Property Method, 183

• POLYNRTL: Polymer Non-Random Two-Liquid Property Method, 185

• POLYUF: Polymer UNIFAC Property Method, 187

• POLYUFV: Polymer UNIFAC Free Volume Property Method, 189

• PNRTL-IG: Polymer NRTL with Ideal Gas Law Property Method, 191

• POLYSL: Sanchez-Lacombe Equation-of-State Property Method, 193

• POLYSRK: Polymer Soave-Redlich-Kwong Equation-of-State Property Method, 195

• POLYSAFT: Statistical Associating Fluid Theory (SAFT) Equation-of-State Property Method, 196

• POLYPCSF: Perturbed-Chain Statistical Associating Fluid Theory (PC-SAFT) Equation-of-State Property Method, 198

• PC-SAFT: Copolymer PC-SAFT Equation-of-State Property Method, 200

For each property method the property models used in the route calculations are described.

POLYFH: Flory-Huggins Property Method The following table lists the physical property route structure for the POLYFH property method:

Vapor

Property Name

Route ID Model Name Description

PHIVMX PHIVMX01 ESRK Redlich-Kwong

HVMX HVMX01 ESRK Redlich-Kwong

GVMX GVMX01 ESRK Redlich-Kwong

SVMX SVMX01 ESRK Redlich-Kwong

VVMX VVMX01 ESRK Redlich-Kwong


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184 A Physical Property Methods

MUVMX MUVMX02 MUV2DNST Dean-Stiel

KVMX KVMX02 KV2STLTH Stiel-Thodos

DVMX DVMX02 DV1DKK Dawson-Khoury-Kobayashi

PHIV PHIV01 ESRK0 Redlich-Kwong

HV HV02 ESRK0 Redlich-Kwong

GV GV01 ESRK0 Redlich-Kwong

SV SV01 ESRK0 Redlich-Kwong

VV VV01 ERSK0 Redlich-Kwong

DV DV01 DV0CEWL Chapman-Enskog-Wilke-Lee

MUV MUV01 MUV0BROK Chapman-Enskog-Brokaw

KV KV01 KV0STLTH Stiel-Thodos

Liquid

Property Name


PHILMX PHILMXFH GMFH, WHENRY, HENRY1, PL0XANT, ESRK0, VL0RKT, VL1BROC

Flory-Huggins, HENRY, Extended Antoine, Redlich-Kwong, Rackett, Brevi-O'Connell

HLMX HLMXFHRK GMFH, WHENRY, HENRY1, VL1BROC, DHL02PH

Flory-Huggins, van Krevelen, HENRY, Redlich-Kwong, Rackett, Brevi-O'Connell

GLMX GLMXFH GMFH, GL0DVK Flory-Huggins, van Krevelen, DIPPR

SLMX SLMXFHRK GMFH, WHENRY, HENRY1, VL1BROC, DHL02PH, GL0DVK

Flory-Huggins, van Krevelen, DIPPR

VLMX VLMXVKRK VL2VKRK Ideal mixing, van Krevelen, Rackett

MULMX MULMX13 MUPOLY,

MULMH

Aspen, Modified Mark-Houwink/van Krevelen, Andrade

KLMX KLMXVKDP KL0VKDP, KL2VR Vredeveld mixing, Modified van Krevelen, Sato-Riedel/DIPPR

DLMX DLMX02 DL1WCA Wilke-Chang-Andrade

PHIL PHIL04 PL0XANT, ESRK0, VL0RKT

Extended Antoine, Redlich-Kwong, Rackett

HL HLDVKRK HL0DVKRK* van Krevelen, Redlich-Kwong, Ideal gas

GL GLDVK GL0DVK van Krevelen, DIPPR

SL SLDVK HL0DVKRK*, GL0DVK

van Krevelen, Redlich-Kwong, Ideal gas

VL VLDVK VL0DVK, VL0RKT van Krevelen, Rackett

DL DL01 DL0WCA Wilke-Chang-Andrade

MUL MULMH MUL0MH Modified Mark-Houwink, Andrade


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Liquid

Property Name


KL KL0VKDP KL0VKDP Modified van Krevelen, Sato-Riedel/DIPPR

* Optional van Krevelen/DIPPR model, HL0DVKD, available (see Pure Component

Liquid Enthalpy Models in Chapter 4).

Solid

Property Name


HSMX HSMXDVK HS0DVK Ideal mixing, van Krevelen

GSMX GSMXDVK GS0DVK Ideal mixing, van Krevelen

SSMX SSMXDVK HS0DVK, GS0DVK

Ideal mixing, van Krevelen

VSMX VSMXDVK VS0DVK, VS0POLY

van Krevelen, Polynomial

HS HSDVK HS0DVK van Krevelen

GS GSDVK GS0DVK van Krevelen

SS SSDVK HS0DVK, GS0DVK

van Krevelen

VS VSDVK VS0DVK, VS0POLY


POLYNRTL: Polymer Non-Random Two-Liquid Property Method The following table lists the physical property route structure for the POLYNRTL property method:

Vapor

Property Name










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Liquid

Property Name


PHILMX PHILMXP GMNRTLP, WHENRY, HENRY, PL0XANT, ESRK0, VL0RKT, VL1BROC

Polymer NRTL, HENRY, Extended Antoine, Redlich-Kwong, Rackett, Brevi-O'Connell

HLMX HLMXPRK GMNRTLP, WHENRY, HENRY1, VL1BROC, DHL02PH

Polymer-NRTL, van Krevelen, HENRY, Redlich-Kwong, Rackett, Brevi-O'Connell

GLMX GLMXP GMNRTLP, GL0DVK Polymer NRTL, van Krevelen, DIPPR

SLMX SLMXPRK GMNRTLP, WHENRY, HENRY1, VL1BROC, DHL02PH, GL0DVK

Polymer NRTL, van Krevelen, HENRY, Redlich-Kwong, Brevi-O'Connell


MULMX MULMX13 MUPOLY, MULMH Aspen, Modified Mark-Houwink/van Krevelen, Andrade



SIGLMX SIGLMX01 SIG2HSS Hakim-Steinberg-Stiel, Power Law Mixing





SL SLDVK HL0DVK*, GL0DVK van Krevelen, DIPPR



MUL MULMH MUL0MH Modified Mark-Houwink/van Krevelen, Andrade



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Solid

Property Name











van Krevelen



POLYUF: Polymer UNIFAC Property Method The following table lists the physical property route structure for the POLYUF property method:

Vapor

Property Name
















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Vapor



Liquid

Property Name


PHILMX PHILMPUF GMPOLUF, WHENRY, HENRY, PL0XANT, ESRK0, VL0RKT, VL1BROC

Polymer UNIFAC, HENRY, Extended Antoine, Redlich-Kwong, Rackett, Brevi-O'Connell

HLMX HLMXPURK GMPOLUF, WHENRY, HENRY1, VL1BROC, DHL02PH

Polymer UNIFAC, van Krevelen, HENRY, Redlich-Kwong, Rackett, Brevi-O'Connell

GLMX GLMXPUF GMPOLUF, GL0DVK Polymer UNIFAC, van Krevelen, DIPPR

SLMX SLMXPURK GMPOLUF, WHENRY, HENRY1, VL1BROC, DHL02PH, GL0DVK

Polymer UNIFAC, van Krevelen, HENRY, Redlich-Kwong, Rackett, Brevi-O'Connell


MULMX MULMX13 MUPOLY, MULMH Aspen, Modified Mark-Houwink/ van Krevelen, Andrade
















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Solid

Property Name











van Krevelen



POLYUFV: Polymer UNIFAC Free Volume Property Method The following table lists the physical property route structure for the POLYUFV property method:

Vapor

Property Name

















Liquid


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Property Name


PHILMX PHILMUFV GMUFFV, WHENRY, HENRY, PL0XANT, ESRK0, VL0RKT, VL1BROC, VL0TAIT

UNIFAC-FV, HENRY, Extended Antoine, Redlich-Kwong, Rackett, Brevi-O'Connell

HLMX HLMXFVRK GMUFFV, WHENRY, HENRY1, VL1BROC, DHL02PH

UNIFAC-FV, van Krevelen, HENRY, Redlich-Kwong, Rackett, Brevi-O'Connell

GLMX GLMXUFV GMUFFV, GL0DVK UNIFAC-FV, van Krevelen, DIPPR

SLMX SLMXFVRK GMUFFV, WHENRY, HENRY1, VL1BROC, DHL02PH, GL0DVK

UNIFAC-FV, van Krevelen, HENRY, Redlich-Kwong, Rackett, Brevi-O'Connell

VLMX VLMXVKRK VL2VKRK Tait/van Krevelen, Rackett

MULMX MULMX13 MUPOLY, MULMH Aspen, Modified Mark-Houwink/ van Krevelen, Andrade













* Optional van Krevelen/DIPPR model, HL0DVKD, available (see Pure

Component Liquid Enthalpy Models in Chapter 4).

Solid

Property Name







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van Krevelen



PNRTL-IG: Polymer NRTL with Ideal Gas Law Property Method The following table lists the physical property route structure for the PNRTL-IG property method:

Vapor

Property Name


PHIVMX PHIVMX00 ESIG Ideal gas law

HVMX HVMX00 ESIG Ideal gas law

GVMX GVMX00 ESIG Ideal gas law

SVMX SVMX00 ESIG Ideal gas law

VVMX VVMX00 ESIG Ideal gas law




PHIV PHIV0 ESIG0 Ideal gas law

HV HV00 ESIG0 Ideal gas law

GV GV00 ESIG0 Ideal gas law

SV SV00 ESIG0 Ideal gas law

VV VV00 ESIG0 Ideal gas law




Liquid

Property Name



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Liquid

Property Name


PHILMX PHILMXPI GMNRTLP, WHENRY, HENRY, PL0XANT, ESIG0, VL0RKT, VL1BROC

Polymer NRTL, HENRY, Extended Antoine, Ideal gas law, Rackett, Brevi-O'Connell

HLMX HLMXP GMNRTLP, WHENRY, HENRY1, VL1BROC, DHL01PH

Polymer NRTL, van Krevelen, Ideal gas, Rackett, Brevi-O'Connell

GLMX GLMXP GMNRTLP, GL0DVK Polymer NRTL, van Krevelen, DIPPR

SLMX SLMXP GMNRTLP, WHENRY, HENRY1, VL1BROC, DHL01PH, GL0DVK

Polymer NRTL, van Krevelen, Ideal gas, Rackett, Brevi-O'Connell


MULMX MULMX13 MUPOLY, MULMH Aspen, Modified Mark-Houwink/van Krevelen, Andrade



SIGLMX SIGLMX01 SIG2HSS Hakim-Steinberg-Stiel, Power law mixing

PHIL PHIL00 PL0XANT, ESIG0, VL0RKT

Extended Antoine, Ideal gas law

HL HLDVKD HL0DVKD* van Krevelen, DIPPR









Solid

Property Name


HSMX HSMXDVK HS0DVK Ideal mixing, van Krevelen,


SSMX SSMXDVK HS0DVK, GS0DVK Ideal mixing, van Krevelen


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Solid

VSMX VSMXDVK VS0DVK, VS0POLY van Krevelen, Polynomial



SS SSDVK HS0DVK, GS0DVK van Krevelen

VS VSDVK VS0DVK, VS0POLY van Krevelen, Polynomial

POLYSL: Sanchez-Lacombe Equation-of-State Property Method The following table lists the physical property route structure for the POLYSL property method:

Vapor

Property Name


PHIVMX PHIVMXSL ESPLSL Sanchez-Lacombe

HVMX HVMXSL ESPLSL Sanchez-Lacombe

GVMX GVMXSL ESPLSL Sanchez-Lacombe

SVMX SVMXSL ESPLSL Sanchez-Lacombe

VVMX VVMXSL ESPLSL Sanchez-Lacombe




PHIV PHIVSL ESPLSL0 Sanchez-Lacombe

HV HVSL ESPLSL0 Sanchez-Lacombe

GV GVSL ESPLSL0 Sanchez-Lacombe

SV SVSL ESPLSL0 Sanchez-Lacombe

VV VVSL ESPLSL0 Sanchez-Lacombe




Liquid

Property Name


PHILMX PHILMXSL ESPLSL Sanchez-Lacombe

HLMX HLMXSL ESPLSL Sanchez-Lacombe

GLMX GLMXSL ESPLSL Sanchez-Lacombe


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Liquid

Property Name


SLMX SLMXSL ESPLSL Sanchez-Lacombe

VLMX VLMXSL ESPLSL Sanchez-Lacombe

MULMX MULMX13 MUPOLY, MULMH


KLMX KLMXVKTR KLMXVKTR Vredeveld mixing , Modified van Krevelen, TRAPP



PHIL PHILSL ESPLSL0 Sanchez-Lacombe

HL HLSL ESPLSL0 Sanchez-Lacombe

GL GLSL ESPLSL0 Sanchez-Lacombe

SL SLSL ESPLSL0 Sanchez-Lacombe

VL VLSL ESPLSL0 Sanchez-Lacombe



KL KL0VKTR KL0VKTR Modified van Krevelen, TRAPP

Solid

Property Name











van Krevelen




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POLYSRK: Polymer Soave-Redlich-Kwong Equation-of-State Property Method The following table lists the physical property route structure for the POLYSRK property method:

Vapor

Property Name


PHIVMX PHIVMXPS ESPLRKS Polymer SRK

HVMX HVMXPS ESPLRKS Polymer SRK

GVMX GVMXPS ESPLRKS Polymer SRK

SVMX SVMXPS ESPLRKS Polymer SRK

VVMX VVMXPS ESPLRKS Polymer SRK




PHIV PHIVPS ESPLRKS0 Polymer SRK

HV HVPS ESPLRKS0 Polymer SRK

GV GVPS ESPLRKS0 Polymer SRK

SV SVPS ESPLRKS0 Polymer SRK

VV VVPS ESPLRKS0 Polymer SRK




Liquid

Property Name


PHILMX PHILMXPS ESPLRKS Polymer SRK

HLMX HLMXPS ESPLRKS Polymer SRK

GLMX GLMXPS ESPLRKS Polymer SRK

SLMX SLMXPS ESPLRKS Polymer SRK




KLMX KLMXVKTR KLMXVKTR Vredeveld mixing, Modified van Krevelen, TRAPP




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Liquid

PHIL PHILPS ESPLRKS0 Polymer SRK

HL HLPS ESPLRKS0 Polymer SRK

GL GLPS ESPLRKS0 Polymer SRK

SL SLPS ESPLRKS0 Polymer SRK

VL VLDVK VL0DVK, VL0RKT

van Krevelen, Rackett




Solid

Property Name





VSMX VSMXDVK VS0DVK, VS0POLY van Krevelen, Polynomial




VS VSDVK VS0DVK, VS0POLY van Krevelen, Polynomial

POLYSAFT: Statistical Associating Fluid Theory (SAFT) Equation-of-State Property Method The following table lists the physical property route structure for the POLYSAFT property method:

Vapor

Property Name


PHIVMX PHIVMXSF ESPLSFT SAFT

HVMX HVMXSF ESPLSFT SAFT

GVMX GVMXSF ESPLSFT SAFT

SVMX SVMXSF ESPLSFT SAFT


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VVMX VVMXSF ESPLSFT SAFT




PHIV PHIVSF ESPLSFT0 SAFT

HV HVSF ESPLSFT0 SAFT

GV GVSF ESPLSFT0 SAFT

SV SVSF ESPLSFT0 SAFT

VV VVSF ESPLSFT0 SAFT




Liquid

Property Name


PHILMX PHILMXSF ESPLSFT SAFT

HLMX HLMXSF ESPLSFT SAFT

GLMX GLMXSF ESPLSFT SAFT

SLMX SLMXSF ESPLSFT SAFT

VLMX VLMXSF ESPLSFT SAFT


Aspen, Modified Mark-Houwink/ van Krevelen, Andrade




PHIL PHILSF ESPLSFT0 SAFT

HL HLSF ESPLSFT0 SAFT

GL GLSF ESPLSFT0 SAFT

SL SLSF ESPLSFT0 SAFT

VL VLSF ESPLSFT0 SAFT




Solid

Property Name


PHILMX PHILMXSF ESPLSFT SAFT

HLMX HLMXSF ESPLSFT SAFT


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GLMX GLMXSF ESPLSFT SAFT

SLMX SLMXSF ESPLSFT SAFT

VLMX VLMXSF ESPLSFT SAFT


Aspen, Modified Mark-Houwink/ van Krevelen, Andrade



POLYPCSF: Perturbed-Chain Statistical Associating Fluid Theory (PC-SAFT) Equation-of-State Property Method The following table lists the physical property route structure for the POLYPCSF property method:

Vapor

Property Name


PHIVMX PHIVMXPC ESPCSFT PCSAFT

HVMX HVMXPC ESPCSFT PCSAFT

GVMX GVMXPC ESPCSFT PCSAFT

SVMX SVMXPC ESPCSFT PCSAFT

VVMX VVMXPC ESPCSFT PCSAFT




PHIV PHIVPC ESPCSFT0 PCSAFT

HV HVPC ESPCSFT0 PCSAFT

GV GVPC ESPCSFT0 PCSAFT

SV SVPC ESPCSFT0 PCSAFT

VV VVPC ESPCSFT0 PCSAFT




Liquid

Property Name



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Liquid

Property Name


PHILMX PHILMXPC ESPCSFT PCSAFT

HLMX HLMXPC ESPCSFT PCSAFT

GLMX GLMXPC ESPCSFT PCSAFT

SLMX SLMXPC ESPCSFT PCSAFT

VLMX VLMXPC ESPCSFT PCSAFT






PHIL PHILPC ESPCSFT0 PCSAFT

HL HLPC ESPCSFT0 PCSAFT

GL GLPC ESPCSFT0 PCSAFT

SL SLPC ESPCSFT0 PCSAFT

VL VLPC ESPCSFT0 PCSAFT




Solid

Property Name













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PC-SAFT: Copolymer PC-SAFT Equation-of-State Property Method The following table lists the physical property route structure for the PC-SAFT property method:

Vapor

Property Name


PHIVMX PHIVMXPA ESPSAFT Copolymer PCSAFT

HVMX HVMXPA ESPSAFT Copolymer PCSAFT

GVMX GVMXPA ESPSAFT Copolymer PCSAFT

SVMX SVMXPA ESPSAFT Copolymer PCSAFT

VVMX VVMXPA ESPSAFT Copolymer PCSAFT




PHIV PHIVPA ESPSAFT0 Copolymer PCSAFT

HV HVPA ESPSAFT0 Copolymer PCSAFT

GV GVPA ESPSAFT0 Copolymer PCSAFT

SV SVPA ESPSAFT0 Copolymer PCSAFT

VV VVPA ESPSAFT0 Copolymer PCSAFT




Liquid

Property Name


PHILMX PHILMXPA ESPSAFT Copolymer PCSAFT

HLMX HLMXPA ESPSAFT Copolymer PCSAFT

GLMX GLMXPA ESPSAFT Copolymer PCSAFT

SLMX SLMXPA ESPSAFT Copolymer PCSAFT

VLMX VLMXPA ESPSAFT Copolymer PCSAFT






PHIL PHILPA ESPSAFT0 Copolymer PCSAFT


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Liquid

Property Name


HL HLPA ESPSAFT0 Copolymer PCSAFT

GL GLPA ESPSAFT0 Copolymer PCSAFT

SL SLPA ESPSAFT0 Copolymer PCSAFT

VL VLPA ESPSAFT0 Copolymer PCSAFT




Solid

Property Name













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202 B Van Krevelen Functional Groups

B Van Krevelen Functional Groups

This appendix lists the methods used to calculate segment and polymer property parameters and the van Krevelen functional group parameters used in these calculations. These functional groups are used by the van Krevelen property models.

Calculating Segment Properties From Functional Groups The van Krevelen property models use functional groups to calculate segment property parameters, which are in turn used to calculate polymer property parameters. The functional group parameters listed in Van Krevelen Functional Group Parameters on page 205 are used to calculate segment properties using the following correlations:

Heat Capacity (Liquid or Crystalline) ( ) ( )∑=

k

reflkk

refl TCpnTCp *,*,

( ) ( )∑=k

refckk

refc TCpnTCp *,*,

Where:

lCp*, = Liquid heat capacity of a segment

cCp*, = Crystalline heat capacity of a segment


kn = Number of occurrences of functional group k in a segment

lkCp*, = Liquid heat capacity for functional group k in Van Krevelen

Functional Group Parameters on page 205


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B Van Krevelen Functional Groups 203

ckCp*, = Crystalline heat capacity for functional group k in Van Krevelen


Molar Volume (Liquid, Crystalline, or Glassy)

∑=k

kkVnV **

)**(**gkk TCTBAVwV ++=

Where:

*V = Molar volume of a segment (liquid, crystalline, or glassy)

*kV = Molar volume of functional group k (liquid, crystalline, or glassy)

Vwk = Van der Waals volume of functional group k in Van Krevelen Functional Group Parameters on page 205

T = Temperature

gT

= Glass transition temperature

A, B, and C = Empirical constants and vary by phase

Enthalpy, Entropy and Gibbs Energy of Formation

( ) ( )∑=k

refigkk

refig THnTH *,*,

( ) ( )∑=k

refigkk

refig TSnTS *,*,

( ) ( ) ( )refigrefrefigrefig TSTTHT *,*,*, +=μ

Where:

( )refig TH *, = Ideal gas heat of formation of a segment

( )refig TS *, = Ideal gas entropy of formation of a segment

( )refig T*,μ = Ideal gas Gibbs energy of formation of a segment

( )refigk TH *, = Ideal gas heat of formation of functional group k in Van

Krevelen Functional Group Parameters on page 205

( )refigk TS *, = Ideal gas entropy of formation of functional group k in Van

Krevelen Functional Group Parameters on page 205


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Glass Transition Temperature

∑∑

=

kkk

kkgk

g Mn

YnT

,

Where:

gT = Glass transition temperature of a segment

kgY , = Glass transition parameter of functional group k in Van Krevelen Functional Group Parameters on page 205


Melt Transition Temperature

∑∑

=

kkk

kkmk

m Mn

YnT

,

Where:

mT = Melt transition temperature of a segment

kmY , = Melt transition parameter of functional group k in Van Krevelen Functional Group Parameters on page 205

Viscosity Temperature Gradient

∑=k

kk MHnH /,ηη

Where:

ηH = Viscosity-temperature gradient of a segment

kH ,η = Viscosity-temperature gradient of functional group k in Van Krevelen Functional Group Parameters on page 205

M = Molecular weight of a segment

Rao Wave Function )()( ref

kkk

ref TURnTUR ∑=

Where:

)( refTUR = Rao wave function of a segment


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)( refk TUR = Rao wave function of functional group k in Van Krevelen


Van Krevelen Functional Group Parameters This section shows the functional group parameters used to calculate segment properties. Function groups are listed by category:

• Hydrocarbon and hydrogen-containing groups

• Oxygen-containing groups

• Nitrogen-containing groups

• Sulfur-containing groups

• Halogen-containing groups

Source: Van Krevelen, D.W. (1990). Properties of Polymers, 3rd Ed. Amsterdam: Elsevier.

Bifunctional Hydrocarbon Groups The following table shows the bifunctional hydrocarbon group parameters. Estimated values appear in italic.

Functional Group Group No. M

k

g/

mo

l

Vw

cm3/

mo

l

Cp

k*

,c(T

ref )

J/

mo

l.K

Cp

k*

,l(T

ref )

J/

mo

l.K

Hk*

,ig(T

ref )

J/

mo

l

Sk*

,ig(T

ref )

J/

mo

l.K

Yg

,k

10

-

3(g

K/

mo

l)

Ym

,k

10

-

3(g

K/

mo

l)

Hη,

k

10

-

3(

/l)

UR

k

cm1

0/

3(s

CH2 100 14.03 10.23 25.35 30.4 -22,000 102 2,700 5,700 420 880

CH(CH3) (sym)

101 28.05 20.45 46.5 57.85 -48,700 215 8,000 13,000 1060 1,850

CH(CH3) (asym)

102 28.05 20.45 46.5 57.85 -48,700 215 8,000 13,000 1,060 1,850

CH(C5H9)

103 82.14 53.28 110.8 147.5 -73,400 548 30,700

45,763 2,180 4,600

CH(C6H11)

104 96.17 63.58 121.2 173. 9 -118,400

680 41,300

51,463 2,600 5,500

CH(C6H5)

105 90.12 52.62 101.2 144.15 84,300 287 36,100

48,000 3600 5,100

C(CH3)2

106 42.08 30.67 68.0 81.2 -72,000 330 8,500 12,100 1620 2,850

C(CH3)(C6H5)

107 104.14

62.84 122.7 167.5 61,000 402 51,000

54,000 4,160 6,100


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k

g/

mo

l

Vw

cm3/

mo

l

Cp

k*

,c(T

ref )

J/

mo

l.K

Cp

k*

,l(T

ref )

J/

mo

l.K

Hk*

,ig(T

ref )

J/

mo

l

Sk*

,ig(T

ref )

J/

mo

l.K

Yg

,k

10

-

3(g

K/

mo

l)

Ym

,k

10

-

3(g

K/

mo

l)

Hη,

k

10

-

3(

/l)

UR

k

cm1

0/

3(s

CH CH (cis) 108 26.04 16.94 37.3 42.8 76,000 76 3,800 8,000 760

1,400

CH CH (trans) 109 26.04 16.94 37.3 42.8 70,000 83 7,400 11,000 760

1,400

CH C(CH3) (cis)

110 40.06 27.16 60.05 74.22 42,000 183 8,100 10,000 1,190 2,150

CH C(CH3)

(trans) 111 40.06 27.16 60.05 74.22 36,000 190 9,100 13,000 1,190

2,150

C C

112 24.02 16.1 ---- ---- 230,000

-50 ---- ---- ---- 1,240

(cis) 113 82.14 53.34 103.2 147.5 -96,400 578

19,000 31,000 2,180

2,900

(trans) 114 82.14 53.34 103.2 147.5

-102,400

585 27,000

45,000 2,180 2,900

115 76.09 43.32 78.8 113.1

100,000 180

29,000 38,000 3200

4,100

116 76.09 43.32 78.8 113.1

100,000 180

25,000 31,000 ----

3,500

117 76.09 43.32 78.8 113.1

100,000 180 ---- ---- ----

3,450

CH3

CH3

118 104.14 65.62 126.8 166.8 33,000 394

54,000 67,000 4,820

6,150

CH3

119 90.12 54.47 102.75 140.1 66,500 287 35,000

45,000 4,010 5,500

CH2 (sym)

120 90.12 53.55 104.15 143.5 78,000 282 31,700

43,700 3,620 4,980

CH2 (asym)

121 90.12 53.55 104.15 143.5 78,000 282 31,700

43,700 3,620 4,980

CH2CH2

122 104.14 63.78 129.5 173.9 56,000 384

25,000 47,000 4,040

5,860

CH2

123 166.21

96.87 182.95 256.6 178,000

462 65,000

85,000 6,820 9,100

124

152.18

86.64 157.6 226.2 200,000

360 70,000

99,000 6,400 8,200


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k

g/

mo

l

Vw

cm3/

mo

l

Cp

k*

,c(T

ref )

J/

mo

l.K

Cp

k*

,l(T

ref )

J/

mo

l.K

Hk*

,ig(T

ref )

J/

mo

l

Sk*

,ig(T

ref )

J/

mo

l.K

Yg

,k

10

-

3(g

K/

mo

l)

Ym

,k

10

-

3(g

K/

mo

l)

Hη,

k

10

-

3(

/l)

UR

k

cm1

0/

3(s

125 228.22 130 236 339

299,000 538

118,000

173,000 9,900

12,650

Other Hydrogen-containing Groups The following table shows the other hydrogen-containing group parameters:

Functional Group

Group No. M

k

g/

mo

l

Vw

cm3/

mo

l

Cp

k*

,c(T

ref )

J/

mo

l.K

Cp

k*

,l(T

ref )

J/

mo

l.K

Hk*

,ig(T

ref )

J/

mo

l

Sk*

,ig(T

ref )

J/

mo

l.K

Yg

,k

10

-

3(

K/

l)

Ym

,k

10

-

3(

K/

l)

Hη,

k

10

-

3(

/l)

UR

k

cm1

0/

3(s

1/

3m

ol)

CH3 126 15.03 13.67 30.9 36.9 -46,000 95 2,900 1,519 810 1,400

C2H5 127 29.06 23.90 56.25 67.3 -68,000 197 5,600 3,952 1,230 2,280

nC3H7 128 43.09 34.13 81.6 97.7 -90,000 299 8,300 6,774 1,650 3,160

iC3H7 129 43.09 34.12 77.4 94.75 -94,700 310

10,900

14,519

1,870 3,250

tC4H9 130 57.11 44.34 99.0 118.1

-118,000

425 13,600

20,129

2,290 4,130

CH

131 13.02 6.78 15.9 20.95 -2,700 120 5,100 11,481 250 460

C

132 12.01 3.33 6.2 7.4 20,000 140 2,700 9,063 0 40

CH2 133 14.01 11.94 22.6 21.8 23,000 30 ---- ---- 0 ----

CH

134 13.02 8.47 18.65 21.8 38,000 38 1,900 4,000 380 745

C

135 12.01 5.01 10.5 15.9 50,000 50 3,300 4,481 0 255

C

136 12.01 6.96 ---- ---- 147,000 -20 ---- ---- ---- ----

CH C

137 25.03 13.48 29.15 37.3 88,000 88 ---- ---- 380 1,000

CH

138 13.02 8.05 ---- ---- 112,500 -32.5 ---- ---- ---- ----

C

139 12.01 8.05 ---- ---- 115,000 -25 ---- ---- ---- ----


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Functional Group

Group No. M

k

g/

mo

l

Vw

cm3/

mo

l

Cp

k*

,c(T

ref )

J/

mo

l.K

Cp

k*

,l(T

ref )

J/

mo

l.K

Hk*

,ig(T

ref )

J/

mo

l

Sk*

,ig(T

ref )

J/

mo

l.K

Yg

,k

10

-

3(

K/

l)

Ym

,k

10

-

3(

K/

l)

Hη,

k

10

-

3(

/l)

UR

k

cm1

0/

3(s

1/

3m

ol)

C C (cis)

140 24.02 10.02 21.0 31.8 100,000 100 ---- ---- 0 510

C C (trans)

141 24.02 10.02 21 31.8 94,000 107 ---- ---- ---- 510

CHar

142 13.02 8.06 15.4 22.2 12,500 26 ---- ---- ---- 830

Car

143 12.01 5.54 8.55 12.2 25,000 38 ---- ---- ---- 400

144 69.12 45.56 95.2

126.55

-70,700 428 28,000

34,281

1,930 4,140

145 83.15 56.79 105.6

152.95

-115,700

560 38,600

39,981

2,350 5,000

146 77.10 45.84 85.6 123.2 87,000 167

33,400

42,300 3,350 4,640

147 74.08 38.28 65.0 93.0 125,000 204 48,200

63,963 3,200 3,300

148 75.08 40.80 71.85 103.2 112,500 192

29,200

41,963 2,390 3,700

Bifunctional Oxygen-containing Groups The following table shows the bifunctional oxygen-containing group parameters:


k

g/

mo

l

Vw

cm3/

mo

l

Cp

k*

,c(T

ref )

J/

mo

l.K

Cp

k*

,l(T

ref )

J/

mo

l.K

Hk*

,ig(T

ref )

J/

mo

l

Sk*

,ig(T

ref )

J/

mo

l.K

Yg

,k

10

-

3(

K/

l)

Ym

,k

10

-

3(

K/

l)

Hη,

k

10

-

UR

k

cm1

0/

3(s

O

149 16.00 3.71/[5.1]1 16.8 35.6

-120,000

70 4,000 13,500

480 400

C

O

150 28.01 11.7 23.05 52.8 -132,000 40 9,000

12,000 970 900

O CO

151 44.01 15.2 46 65.0

-337,000 116

12,500

30,000

1,450 1,250


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k

g/

mo

l

Vw

cm3/

mo

l

Cp

k*

,c(T

ref )

J/

mo

l.K

Cp

k*

,l(T

ref )

J/

mo

l.K

Hk*

,ig(T

ref )

J/

mo

l

Sk*

,ig(T

ref )

J/

mo

l.K

Yg

,k

10

-

3(

K/

l)

Ym

,k

10

-

3(

K/

l)

Hη,

k

10

-

UR

k

cm1

0/

3(s

O CO

(acrylic)

152 44.01 17.0 46 65.0 -337,000

116 12,500

30,000

1,450

1,250

O C OO

153 60.01

18.9/[23.0]1 63 100.6

-457,000 186

20,000

30,000

3,150 1,600

C O C

O O

154 72.02 27 63 114 -584,070

156 22,000

35,000

2,420

2,150

CH(OH)

155 30.03 14.82 32.6 65.75 -178,700

170 13,000

37,500

539 1,050

CH(COOH)

156 58.04 26.52 65.6 119.85

-395,700

238 ---- 30,724

1,587

1,990

CH(HC=O)

157 42.14 21.92 ---- ---- -127,700

146 ---- 13,362

908 ----

COO

158 120.10 58.52 124.8 178.1

-237,000 296

38,000

50,000

4,170 5,350

O CH2 O

159 46.03 17.63 58.95 101.6 -262,000

242 10,700

32,700

1,380

1,680

Other Oxygen-containing Groups The following table shows the other oxygen-containing group parameters:


k

g/

mo

l

Vw

cm3/

mo

l

Cp

k*

,c(T

ref )

J/

mo

l.K

Cp

k*

,l(T

ref )

J/

mo

l.K

Hk*

,ig(T

ref )

J/

mo

l

Sk*

,ig(T

ref )

J/

mo

l.K

Yg

,k

10

-

3(

K/

l)

Ym

,k

10

-

3(

K/

l)

Hη,

k

10

-

UR

k

cm1

0/

3(s

1/

3m

ol)

OH

160 17.01 8.04 17.0 44.8 -176,000

50 ---- 1,477 289 630

OH

161 93.10 51.36 95.8 157.9 -76,000 230 ---- 39,477

3,489 4,730

C H

O

162 29.02 15.14 ---- ---- -125,000

26 ---- 1,881 658 ----

C OH

O

163 45.02 19.74 50 98.9 -393,000

118 ---- 19,243

1,337

1,530


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Bifunctional Nitrogen-containing Groups The following table shows the bifunctional nitrogen-containing group parameters:

Functional Group

Group No. M

k

g/

mo

l

Vw

cm3/

mo

l

Cp

k*

,c(T

ref )

J/

mo

l.K

Cp

k*

,l(T

ref )

J/

mo

l.K

Hk*

,ig(T

ref )

J/

mo

l

Sk*

,ig(T

ref )

J/

mo

l.K

Yg

,k

10

-

3(

K/

l)

Ym

,k

10

-

3(

K/

l)

Hη,

k

10

-

UR

k

cm1

0/

3(s

1/

3m

ol)

NH

164 15.02

8.08 14.25 31.8 58,000 120 7,000 18,000

680 875

CH(CN)

165 39.04

21.48 40.6 ---- 120,300 91.5 16,405

25,717

---- 1,750

CH(NH2)

166 29.04

17.32 36.55 ---- 8,800 222.5

---- 15,088

562 ----

NH

167 91.11 51.4 93.05 144.9 158,000 300

36,000

56,000

3,880 4,975

Other Nitrogen-containing Groups The following table shows the other nitrogen-containing group parameters:

Functional Group

Group No. M

k

g/

mo

l

Vw

cm3/

mo

l

Cp

k*

,c(T

ref )

J/

mo

l.K

Cp

k*

,l(T

ref )

J/

mo

l.K

Hk*

,ig(T

ref )

J/

mo

l

Sk*

,ig(T

ref )

J/

mo

l.K

Yg

,k

10

-

3(

K/

l)

Ym

,k

10

-

3(

K/

l)

Hη,

k

10

-3(g

/m

ol)

UR

k

cm1

0/

3(s

1/

3m

ol)

NH2 168

16.02

10.54 20.95 ---- 11,500 102.5

---- 3,607 312 ----

N

169 14.01

4.33 17.1 44.0 97,000 150 ---- 10,380

---- 65

Nar

170 14.01 ---- ---- ---- 69,000 50 ---- ---- ---- ----

C N

171 26.02

14.7 25 ---- 123,000 -28.5

---- 1,824 ---- 1,400

NH2

172 92.12 53.86 99.75 ---- 111,500

282.5 ----

41,607

3,512 ----

N

173 90.10 47.65 95.9 157.1 197,000 330 ----

48,380 ---- 4,165


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Bifunctional Nitrogen- and Oxygen-containing Groups The following table shows the bifunctional nitrogen- and oxygen-containing group parameters:

Functional Group

Group No. M

k

g/

mo

l

Vw

cm3/

mo

l

Cp

k*

,c(T

ref )

J/

mo

l.K

Cp

k*

,l(T

ref )

J/

mo

l.K

Hk*

,ig(T

ref )

J/

mo

l

Sk*

,ig(T

ref )

J/

mo

l.K

Yg

,k

10

-

3(g

.K/

mo

l)

Ym

,k

10

-

3(g

.K/

mo

l)

Hη,

k

10

-

UR

k

cm1

0/

3(s

C NH

O

174 43.03 19.56

1[18.1] 38 90.1 -74,000 160 15,000 45,000 1,650

1,700

O C NH

O

175 59.03 23 58 125.7 -279,000 -240 20,000 43,500

2,130

1,800

NH C NH

O

176 58.04 27.6 50 121.9 -16,000 280 20,000 60,000 2,330

2,000

CH(NO2)

177 59.03 23.58 57.5 ---- -44,200 263 ---- ---- ---- ----

C

NH

O

178 119.12

62.88 116.8

203.2 26,000 340 7,000 98,000 4,850

5,800

Other Nitrogen- and Oxygen-containing Groups The following table shows the other nitrogen- and oxygen-containing group parameters:

Functional Group

Group No. M

k

g/

mo

l

Vw

cm3/

mo

l

Cp

k*

,c(T

ref )

J/

mo

l.K

Cp

k*

,l(T

ref )

J/

mo

l.K

Hk*

,ig(T

ref )

J/

mo

l

Sk*

,ig(T

ref )

J/

mo

l.K

Yg

,k

10

-

3(g

.K/

mo

l)

Ym

,k

10

-

3(g

.K/

mo

l)

Hη,

k

10

-

UR

k

cm1

0/

3(s

CO

NH2

179 44.03 22.2 ---- ---- ---- ---- ---- 20,721 ---- ----

CO

N

180 42.02 16.0 ---- ---- ---- ---- ---- 48,380 ---- 965

NO2 181

46.01

16.8 41.9 ---- -41,500

143 ---- ---- ---- ----


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Bifunctional Sulfur-containing Groups The following table shows the bifunctional sulfur-containing group parameters:

Functional Group

Group No. M

k

g/

mo

l

Vw

cm3/

mo

l

Cp

k*

,c(T

ref )

J/

mo

l.K

Cp

k*

,l(T

ref )

J/

mo

l.K

Hk*

,ig(T

ref )

J/

mo

l

Sk*

,ig(T

ref )

J/

mo

l.K

Yg

,k

10

-

3(g

.K/

mo

l)

Ym

,k

10

-

3(g

.K/

mo

l)

Hη,

k

10

-

3(g

/m

ol)

×

(J/

mo

l)1

/3

UR

k

cm1

0/

3(s

1/

3m

ol)

S

182 32.06 10.8 24.05 44.8 40,000 -24 8,000 22,500 ---- 550

S S

183 64.12 22.7 48.1 89.6 46,000 -28 16,000 30,000 ---- 1,100

SO2 184 64.06 20.3 50 ----

-282,000

152 32,000 56,000 ---- 1,250

S CH2 S

185 78.15 31.8 73.45 120.0 58,000 54 ---- ---- ---- 1,980

Other Sulfur-containing Groups The following table shows the other sulfur-containing group parameters:

Functional Group

Group No. M

k

g/

mo

l

Vw

cm3/

mo

l

Cp

k*

,c(T

ref )

J/

mo

l.K

Cp

k*

,l(T

ref )

J/

mo

l.K

Hk*

,ig(T

ref )

J/

mo

l

Sk*

,ig(T

ref )

J/

mo

l.K

Yg

,k

10

-

3(g

.K/

mo

l)

Ym

,k

10

-

3(g

.K/

mo

l)

Hη,

k

10

-3(g

/m

ol)

× U

Rk

cm1

0/

3(s

SH

186 33.07 14.81 46.8 52.4 13,000

-33 ---- ---- ---- ----

Bifunctional Halogen-containing Groups The following table shows the bifunctional halogen-containing group parameters:

Functional Group

Group No. M

k

g/

mo

l

Vw

cm3/

mo

l

Cp

k*

,c(T

ref )

J/

mo

l.K

Cp

k*

,l(T

ref )

J/

mo

l.K

Hk*

,ig(T

ref )

J/

mo

l

Sk*

,ig(T

ref )

J/

mo

l.K

Yg

,k

10

-

3(g

.K/

mo

l)

Ym

,k

10

-

3(g

.K/

mo

l)

Hη,

k

10

-

3(g

/m

ol)

×

(J/

mo

l)1

/3

UR

k

cm1

0/

3(s

1/

3m

ol)

CHF

187 32.02 13.0 37.0 41.95 -197,700

114 12,400 17,400 ---- 950

CF2 188 50.01 15.3 49.0 49.4

-370,000

128 10,500 25,500 ---- 1,050

CHCl

189 48.48 19.0 42.7 60.75 -51,700 111 19,400 27,500 2,330 1,600

CCl2 190 82.92 27.8 60.4 87.0 -78,000 122 22,000 29,000 ---- 2,350

CH CCl

191 60.49 25.72 56.25 77.1 39,000 79 15,200 32,000 ---- 1,900


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Functional Group

Group No. M

k

g/

mo

l

Vw

cm3/

mo

l

Cp

k*

,c(T

ref )

J/

mo

l.K

Cp

k*

,l(T

ref )

J/

mo

l.K

Hk*

,ig(T

ref )

J/

mo

l

Sk*

,ig(T

ref )

J/

mo

l.K

Yg

,k

10

-

3(g

.K/

mo

l)

Ym

,k

10

-

3(g

.K/

mo

l)

Hη,

k

10

-

3(g

/m

ol)

×

(J/

mo

l)1

/3

UR

k

cm1

0/

3(s

1/

3m

ol)

CFCl

192 66.47 21.57 54.7 68.2 -224,000

125 28,000 32,000 ---- 1,700

CHBr

193 92.93 21.4 41.9 ---- -16,700 106 ---- ---- ---- 1,760

CBr2 194 171.84 32.5 58.8 ---- -8,000 112 ---- ---- ---- 2,640

CHI

195 139.93 27.1 38.0 ---- 37,300 79 ---- ---- ---- ----

CI2 196 265.83 44.0 51.0 ---- 100,000 58 ---- ---- ---- ----

Other Halogen-containing Groups The following table shows the other halogen-containing group parameters:

Functional Group

Group No. M

k

g/

mo

l

Vw

cm3/

mo

l

Cp

k*

,c(T

ref )

J/

mo

l.K

Cp

k*

,l(T

ref )

J/

mo

l.K

Hk*

,ig(T

ref )

J/

mo

l

Sk*

,ig(T

ref )

J/

mo

l.K

Yg

,k

10

-

3(g

.K/

mo

l)

Ym

,k

10

-

3(g

.K/

mo

l)

Hη,

k

10

-

3(g

/m

ol)

×

(J/

mo

l)1

/3

UR

k

cm1

0/

3(s

1/

3m

ol)

F

197 19.00 6.0 21.4 21.0 -195,000

-6 9,000 11,000 ---- 530

CF3 198 69.01 21.33 70.4 70.4

-565,000

122 19,500 36,500 ---- 1,580

CHF2 199 51.02 18.8 58.4 62.95

-392,700

108 21,400 28,400 ---- 1,480

CH2F

200 33.03 16.2 46.75 51.4 -217,000

96 11,700 16,700 ---- 1,410

Cl

201 35.46 12.2 27.1 39.8 -49,000 -9 17,500 22,000 2,080 1,265

CCl3 202 118.38 40 87.5 126.8

-127,000

113 39,500 51,000 ---- 3,615

CHCl2 203 83.93 31.3 69.8 100.55 100,700 102 36,900 49,500 4,410 2,865

CH2Cl

204 49.48 22.5 52.45 70.2 -71,700 93 20,200 27,700 2,500 2,145

Cl

205 111.55 55.3 105.9 152.9 51,000 171 46,500 60,000 5,280 5,365

Br

206 79.92 14.6 26.3 ---- -14,000 -14 35,000 11,500 ---- 1,300

CBr3 207 251.76 47.1 85.1 ---- -22,000 98 ---- ---- ---- 3,940

CHBr2 208 172.85 36.0 68.2 ---- -30,700 92 ---- ---- ---- 3,060

CH2Br

209 93.94 24.8 51.65 ---- -36,000 88 ---- ----- ---- 2,180

I

210 126.91 20.4 22.4 ---- 40,000 -41 ---- ---- ---- ----


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Functional Group

Group No. M

k

g/

mo

l

Vw

cm3/

mo

l

Cp

k*

,c(T

ref )

J/

mo

l.K

Cp

k*

,l(T

ref )

J/

mo

l.K

Hk*

,ig(T

ref )

J/

mo

l

Sk*

,ig(T

ref )

J/

mo

l.K

Yg

,k

10

-

3(g

.K/

mo

l)

Ym

,k

10

-

3(g

.K/

mo

l)

Hη,

k

10

-

3(g

/m

ol)

×

(J/

mo

l)1

/3

UR

k

cm1

0/

3(s

1/

3m

ol)

CI3 211 392.74 64.4 73.4 ---- 140,000 17 ---- ---- ---- ----

CHI2 212 266.84 47.5 60.4 ---- 77,300 38 ---- ---- ---- ----

CH2I

213 140.94 30.6 47.75 ---- 18,000 61 ---- ----- ---- ----


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C Tait Model Coefficients 215

C Tait Model Coefficients

This appendix lists parameters available for the Tait molar volume calculations for selected polymers.

These parameters are available in the POLYMER databank for the polymers listed:

Polymer /kg)(mA 3

0 /kg.K)(mA 3

1 )/kg.K(mA 23 2 0B (Pa) B1 (1/K)

P Range† (Mpa)

T Range† (K)

BR 1.0969E-03 7.6789E-07 -2.2216E-10 1.7596E+08 4.3355E-03 0.1-283 277-328

HDPE 1.1567E-03 6.2888E-07 1.1268E-09 1.7867E+08 4.7254E-03 0.1-200 415-472

I-PB 1.1561E-03 6.1015E-07 8.3234E-10 1.8382E+08 4.7833E-03 0.0-196 407-514

I-PMMA 7.9770E-04 5.5274E-07 -1.4503E-10 2.9210E+08 4.1960E-03 0.1-200 328-463

I-PP 1.2033E-03 4.8182E-07 7.7589E-10 1.4236E+08 4.0184E-03 0.0-196 447-571

LDPE 1.1004E-03 1.4557E-06 -1.5749E-09 1.7598E+08 4.6677E-03 0.1-200 398-471

LLDPE 1.1105E-03 1.2489E-06 -4.0642E-10 1.7255E+08 4.4256E-03 0.1-200 420-473

PAMIDE 7.8153E-04 3.6134E-07 2.7519E-10 3.4019E+08 3.8021E-03 0.0-177 455-588

PBMA 9.3282E-04 5.7856E-07 5.7343-10 2.2569E+08 5.3116E-03 0.1-200 295-473

PC 7.9165E-04 4.4201E-07 2.8583E-10 3.1268E+08 3.9728E-03 0.0-177 430-610

PCHMA 8.7410E-04 4.9035E-07 3.2707E-10 3.0545E+08 5.5030E-03 0.1-200 383-472

PDMS 1.0122E-03 7.7266E-07 1.9944E-09 8.7746E+07 6.2560E-03 0.0-100 298-343

PHENOXY 8.3796E-04 3.6449E-07 5.2933E-10 3.5434E+08 4.3649E-03 0.0-177 349-574

PIB 1.0890E-03 2.5554E-07 2.2682E-09 1.9410E+08 3.9995E-03 0.0-100 326-383


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216 C Tait Model Coefficients

PMMA 8.2396E-04 3.0490E-07 7.0201E-10 2.9803E+08 4.3789E-03 0.1-200 387-432

PMP 1.2078E-03 5.1461E-07 9.7366E-10 1.4978E+08 4.6302E-03 0.0-196 514-592

POM 8.3198E-04 2.7550E-07 2.2000E-09 3.1030E+08 4.4652E-03 0.0-196 462-492

POMS 9.3905E-04 5.1288E-07 5.9157E-11 2.4690E+08 3.6633E-03 0.1-180 413-471

PS-1 9.3805E-04 3.3086E-07 6.6910E-10 2.5001E+08 4.1815E-03 0.1-200 389-469

PTFE 4.6867E-04 1.1542E-07 1.1931E-09 4.0910E+08 9.2556E-03 0.0-392 604-646

PVAC 8.2832E-04 4.7205E-07 1.1364E-09 1.8825E+08 3.8774E-03 0.0-100 337-393

† Range of experimental data used in the determination of equation constants.

Source: Danner R. P., & High, M. S. (1992). Handbook of Polymer Solution Thermodynamics. New York: American Institute of Chemical Engineers. p. 3B-5.


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D Mass Based Property Parameters 217

D Mass Based Property Parameters

The Aspen Plus convention is to use mole based parameters for property model calculations. However, polymer property parameters are often more conveniently obtained on a mass basis. To satisfy the needs of users who may prefer the use of mass based parameters, in Aspen Polymers (formerly known as Aspen Polymers Plus) there is a corresponding mass based parameter for selected mole based parameters.

The following table shows a list of model parameters and their mass-based counterparts. Note that the mass based parameters should only be used for polymers and oligomers, and not for segments.

Mole Based Parameter

Mass Based Parameter

Description

CPCVK CPCVKM Crystalline heat capacity

CPLVK CPLVKM Liquid heat capacity

DGCON DGCONM Standard free energy of condensation

DGFORM DGFVKM Standard free energy on formation at 25°C

DBSUB DGSUBM Standard free energy of sublimation

DHCON DHCONM Standard enthalpy of condensation

DHFVK DHFVKM Standard enthalpy of formation at 25°C

DHSUB DHSUBM Standard enthalpy of sublimation

DNCVK DNCVKM Crystalline density

DNGVK DNGVKM Glass density

DNLVK DNLVKM Liquid density


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218 E Equation-of-State Parameters

E Equation-of-State Parameters

This appendix lists unary parameters used with the:

• Sanchez-Lacombe (POLYSL) equation of state model

• SAFT (POLYSAFT) equation of state model

The parameters are not automatically retrieved from databanks. These parameters are not unique in any way. Users may generate them through experimental data regression for the components of interest.

Sanchez-Lacombe Unary Parameters This section lists the POLYSL model parameters for polymers, monomers, and solvents.

POLYSL Polymer Parameters The following table shows the Sanchez-Lacombe (POLYSL) unary parameters for polymers:

Polymer T*, K P*, bar 3kg/m *,ρ T range, K P, up to bar

HDPE 649 4250 904 426-473 1000

LDPE 673 3590 887 408-471 1000

PDMS 476 3020 1104 298-343 1000

PBMA 627 4310 1125 307-473 2000

PHMA 697 4260 1178 398-472 2000

PIB 643 3540 974 326-383 1000

PMMA 696 5030 1269 397-432 2000

POMS 768 3780 1079 412-471 1600

PS 735 3570 1105 388-468 2000

PVAC 590 5090 1283 308-373 800


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E Equation-of-State Parameters 219

Source: Sanchez, I. C., & Lacombe, R. H. (1978). Statistical Thermodynamics of Polymer Solutions. Macromolecules, 11(6), pp. 1145-1156.

POLYSL Monomer and Solvent Polymers The following table shows the Sanchez-Lacombe (POLYSL) unary parameters for monomers and solvents:

Formula Component T*, K P*, bar ρ*, kg / m3

CCl4 Carbon Tetrachloride 535 8126 1788

CHCl3 Chloroform 512 4560 1688

CH3Cl Methyl chloride 487 5593 1538

CO2 Carbon dioxide † 277 7436 1629

CS2 Carbon disulfide 567 5157 1398

CH4 Methane 224 2482 500

CH4O Methanol 468 12017 922

C3H3N Acrilonitrile † 527 5930 868

C3H6O Acetone 484 5330 917

C3H6O2 Ethyl formate 466 4965 1076

C6H7N Aniline 614 6292 1115

C3H8O Propanol 420 8856 972

C3H8O Isopropyl alcohol 399 8532 975

CH3NO2 Nitromethane 620 9251 1490

C2HCl3 1,1,1-Trichloroethylene 516 3779 1518

C2H2Cl2 1,1-Dichloroethylene 488 5117 1722

C2H4 Ethylene † 291 3339 660

C2H4O2 Acetic acid 562 8613 1164

C2H6 Ethane 315 3273 640

C2H6O Ethanol 413 10690 963

C3H8 Propane 371 3131 690

C4H8O Methyl ethyl ketone 513 4468 913

C4H8O2 Ethyl acetate 468 4580 1052

C4H10 n-Butane 403 3222 736

C4H10 Isobutane 398 2878 720

C4H10O Tert-butyl alcohol 448 6931 952

C4H10O Diethyl ether 431 3627 870

C5H5N Pyridine 566 5492 1079

C5H10 Cyclopentane 491 3800 867

C5H12 n-Pentane 441 3101 755

C5H12 Isopentane 424 3080 765

C5H12 Neopentane 415 2655 744

C6H5Cl Chlorobenzene 585 4367 1206

C6H6 Benzene 523 4438 994

C6H6O Phenol 530 7934 1192


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Formula Component T*, K P*, bar ρ*, kg / m3

C6H14 n-Hexane 476 2979 775

C6H12 Cyclohexane 497 3830 902

C6H12O2 n-Butyl acetate 498 3942 1003

C7H8 Toluene 543 4023 966

C7H16 n-Heptane 487 3090 800

C8H8 Styrene † 563 3684 870

C8H10 p-Xylene 561 3810 949

C8H10 m-Xylene 560 3850 952

C8H10 o-Xylene 571 3942 965

C8H18 n-Octane 502 3080 815

C9H20 n-Nonane 517 3070 828

C10H18 trans-Decalin 621 3151 935

C10H18 cis-Decalin 631 3334 960

C10H22 n-Decane 530 3040 837

C11H24 n-Undecane 542 3030 846

C12H26 n-Dodecane 552 3009 854

C13H28 n-Tridecane 560 2989 858

C14H10 Phenanthrene 801 3769 1013

C14H30 n-Tetradecane 570 2959 864

C17H36 n-Heptadecane 596 2867 880

C20H42 n-Eicosane † 617 3067 961

H2O Water 623 26871 1105

H2S Hydrogen Sulfate 382 6129 1095

† Evaluated from vapor-pressure and liquid-density data

regression

Source: Sanchez, I. C., & Lacombe, R. H. (1978). Statistical Thermodynamics of Polymer Solutions. Macromolecules, 11(6), pp. 1145-1156.

SAFT Unary Parameters This section lists the POLYSAFT model parameters for solvents and polymers.

POLYSAFT Parameters The following table shows the SAFT (POLYSAFT) unary parameters for various non-associating fluids:

Formula Component T range, K

v , cm / moloo 3 m u / k, Ko


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E Equation-of-State Parameters 221



N2 Nitrogen --- 19.457 1.0 123.53

AR Argon --- 16.29 1.0 150.86

CO Carbon Monoxide 72-121 15.776 1.221 111.97

CO2 Carbon Dioxide 218-288 13.578 1.417 216.08

CL2 Chlorine 180-400 22.755 1.147 367.44

CS2 Carbon Disulfide 278-533 23.622 1.463 396.05

SO2 Sulfur Dioxide 283-413 22.611 1.133 335.84

CH4 Methane 92-180 21.576 1.0 190.29

C2H6 Ethane 160-300 14.460 1.941 191.44

C3H8 Propane 190-360 13.457 2.696 193.03

C4H10 n-Butane 220-420 12.599 3.458 195.11

C5H12 n-Pentane 233-450 12.533 4.091 200.02

C6H14 n-Hexane 243-493 12.475 4.724 202.72

C7H16 n-Heptane 273-523 12.282 5.391 204.61

C8H18 n-Octane 303-543 12.234 6.045 206.03

C9H20 n-Nonane 303-503 12.240 6.883 203.56

C10H22 n-Decane 313-573 11.723 7.527 205.46

C12H26 n-Dodecane 313-523 11.864 8.921 205.93

C14H30 n-Tetradecane 313-533 12.389 9.978 209.40

C16H34 n-Hexadecane 333-593 12.300 11.209 210.65

C20H42 n-Eicosane 393-573 12.0 13.940 211.25

C28H58 n-Octacosane 449-704 12.0 19.287 209.96

C36H74 n-Hexatriacontane 497-768 12.0 24.443 208.74

C44H90 n-Tetratetracontane 534-725 12.0 29.252 207.73

C5H10 Cyclopentane 252-483 12.469 3.670 226.70

C6H12 Methyl-cyclopentane 263-503 13.201 4.142 223.25

C7H14 Ethyl-cyclopentane 273-513 13.766 4.578 229.04

C8H16 Propyl-cyclopentane 293-423 14.251 5.037 232.18

C9H18 Butyl-cyclopentane 314-578 14.148 5.657 230.61

C10H20 Pentyl-cyclopentane 333-483 13.460 6.503 225.56

C6H12 Cyclohexane 283-513 13.502 3.970 236.41

C7H14 Methylcyclohexane 273-533 15.651 3.954 248.44

C8H16 Ethylcyclohexane 273-453 15.503 4.656 243.16

C9H18 Propylcyclohexane 313-453 15.037 5.326 238.51

C10H20 Butylcyclohexane 333-484 14.450 6.060 234.30

C11H22 Pentylcyclohexane 353-503 14.034 6.804 230.91

C6H6 Benzene 300-540 11.421 3.749 250.19

C7H8 Methyl-benzene 293-533 11.789 4.373 245.27

C8H10 Ethyl-benzene 293-573 12.681 4.719 248.79

C9H12 n-Propyl-benzene 323-573 12.421 5.521 238.66

C10H14 n-Butyl-benzene 293-523 12.894 6.058 238.19

C8H10 m-Xylene 309-573 12.184 4.886 245.88


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C12H10 Biphenyl 433-653 12.068 6.136 280.54

C10H8 Naphthalene 373-693 13.704 4.671 304.80

C11H10 1-Methyl-naphthalene 383-511 13.684 5.418 293.45

C12H12 1-Ethyl-naphthalene 393-563 12.835 6.292 276.18

C13H14 1-n-Propyl-naphthalene 403-546 13.304 6.882 266.82

C14H16 1-n-Butyl-naphthalene 413-566 13.140 7.766 252.11

C14H10 Phenanthrene 373-633 16.518 5.327 352.00

C14H10 Anthracene 493-673 16.297 5.344 352.65

C16H10 Pyrene 553-673 18.212 5.615 369.38

C2H6O Dimethyl-ether 179-265 11.536 2.799 207.83

C3H8O Methyl-ethyl-ether 266-299 10.065 3.540 203.54

C4H10O methyl-n-propyl-ether 267-335 10.224 4.069 208.13

C4H10O Diethyl-ether 273-453 10.220 4.430 191.92

C12H10O Phenyl-ether 523-633 12.100 6.358 276.13

C3H9N Trimethylamine 193-277 14.102 3.459 196.09

C12H10O Triethylamine 323-368 11.288 5.363 201.31

C3H6O Acetone 273-492 7.765 4.504 210.92

C4H8O Methy-ethyl ketone 257-376 11.871 4.193 229.99

C5H10O Methyl-n-propyl ketone 274.399 11.653 4.644 230.40

C5H10O Diethyl-ketone 275-399 10.510 4.569 235.24

C2H4 Ethylene 133-263 18.157 1.464 212.06

C3H6 Propylene 140-320 15.648 2.223 213.90

C4H8 1-Butene 203-383 13.154 3.162 202.49

C6H12 1-Hexene 213-403 12.999 4.508 204.71

CH3CL Chloromethane 213-333 10.765 2.377 238.37

CH2CL2 Dichloromethane 230-333 10.341 3.114 253.03

CHCL3 Trichloromethane 244-357 10.971 3.661 240.31

CCL4 Tetrachloromethane 273-523 13.730 3.458 257.46

C2H5CL Chloroethane 212-440 11.074 3.034 229.58

C3H7CL 1-Chloropropane 238-341 11.946 3.600 229.14

C4H7CL 1-Chlorobutane 262-375 12.236 4.207 227.88

C6H11CL 1-Chlorohexane 306-435 12.422 5.458 225.82

C6H5CL Chlorobenzene 273-543 13.093 3.962 276.72

PE Polyethylene (MW=25000) 413-473 12.0 1165.77 210.0

P(E&P) Polypropylene 263-303 12.0 822.68 210.0

Source: Huang, S. H., & Radosz, M. (1990). Equation of State for Small, Large, Polydisperse, and Associating Molecules. Ind. & Eng. Chem. Res., 29, pp. 2284-2294.


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F Input Language Reference 223

F Input Language Reference

This appendix describes the input language for specify polymer physical property inputs only. A complete input language reference for Aspen Polymers (formerly known as Aspen Polymers Plus) is provided in Appendix D of the Aspen Polymers User Guide, Volume 1.

Specifying Physical Property Inputs This section describes the input language for specifying physical property inputs.

Specifying Property Methods Following is the input language used to specify property methods.

Input Language for Property Methods

PROPERTIES opsetname keyword=value / opsetname [sectionid-list] keyword=value /...

Optional keywords: FREE-WATER SOLU-WATER HENRY-COMPS

HENRY-COMPS henryid cid-list

Input Language Description for Property Methods


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224 F Input Language Reference

The PROPERTIES paragraph is used to specify the property method(s) to be used in your simulation. In this paragraph properties may be specified for the entire flowsheet, for a flowsheet section, or for an individual unit operation block. Depending on the component system used, additional information may be required such as Henry's law information, water solubility correlation, free-water phase properties. The input language for specifying property methods is as follows.

opsetname Primary property method name (See Available Property Methods in Chapter 1).

sectionid-list List of flowsheet section IDs.

FREE-WATER Free water phase property method name (Default=STEAM-TA).

SOLU-WATER Method for calculating the K-value of water in the organic phase.

SOLU-WATER=0 Water solubility correlation is used, vapor phase fugacity for water calculated by free water phase property method

SOLU-WATER=1 Water solubility correlation is used, vapor phase fugacity for water calculated by primary property method

SOLU-WATER=2 Water solubility correlation is used with a correction for unsaturated systems, vapor phase fugacity for water calculated by primary property method

SOLU-WATER=3 Primary property method is used. This method is not recommended for water-hydrocarbon systems unless water-hydrocarbon interaction parameters are available. (Default)

HENRY-COMPS Henry's constant component list ID.

The HENRY-COMPS paragraph identifies lists of components for which Henry's law and infinite dilution normalization are used. There may be any number of HENRY-COMPS paragraphs since different lists may apply to different blocks or sections of the flowsheet.

henryid Henry's constant component list ID

cid-list List of component IDs

Input Language Example for Property Methods

HENRY-COMPS HC INI1

PROPERTIES POLYNRTL HENRY-COMPS=HC


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Specifying Property Data Following is the input language used to specify property data.

Input Language for Property Data

PROP-DATA PROP-LIST paramname [setno] / . . . PVAL cid value-list / value-list / . . . PROP-LIST paramname [setno] / . . . BPVAL cid1 cid2 value-list / value-list / . . . COMP-LIST cid-list CVAL paramname setno 1 value-list COMP-LIST cid2-list BCVAL paramname setno 1 cid1 value-list / 1 cid1 value-list / . . .

Physical property models require data in order to calculate property values. Once you have selected the property method(s) to be used in your simulation, you must determine the parameter requirements for the models contained in the property method(s), and ensure that they are available in the databanks. If the model parameters are not available from the databanks, you may estimate them using the Property Constant Estimation System, or enter them using the PROP-DATA or TAB-POLY paragraphs. The input language for the PROP-DATA paragraphs is as follows. Note that only the general structure is given, for information on the format for the input parameters required by polymer specific models see the relevant chapter of this User Guide.

Input Language Description for Property Data

PROP-LIST Used to enter parameter names and data set numbers.

PVAL Used to enter the PROP-LIST parameter values.

BPVAL Used to enter the PROP-LIST binary parameter values.

COMP-LIST Used to enter component IDs.

CVAL Used to enter the COMP-LIST parameter values.

BCVAL Used to enter the COMP-LIST binary parameter values.

paramname Parameter name

setno Data set number. For CVAL and BCVAL the data set number must be entered. For setno > 1, the data set number must also be specified in a new property method defined using the PROP-REPLACE paragraph. (For PROP-LIST, Default=1)

cid Component ID


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226 F Input Language Reference

cid1 Component ID of first component of binary pair

cid2 Component ID of second component of binary pair

value-list List of parameter values. For PROP-LIST, enter one value for each element of the property; for COMP-LIST, enter one value for each component in the cid-list.

cid-list List of component ID

Input Language Example for Property Data

PROP-DATA

IN-UNITS SI

PROP-LIST PLXANT / TB

PVAL HOPOLY -40.0 0 0 0 0 0 0 0 1D3 / 2000.0

PVAL COPOLY -40.0 0 0 0 0 0 0 0 1D3 / 2000.0

PROP-DATA

IN-UNITS SI

PROP-LIST MW

PVAL HOPOLY 1.0

PVAL COPOLY 1.0

PVAL ABSEG 192.17

PVAL ASEG 76.09

PVAL BSEG 116.08

PROP-DATA

IN-UNITS SI

PROP-LIST DHCONM / DHSUB / TMVK / TGVK

PVAL HOPOLY -3.64261D4 / 8.84633D4 / 1.0 / 0.0

PVAL COPOLY -3.64261D4 / 8.84633D4 / 1.0 / 0.0

PROP-DATA

IN-UNITS SI

PROP-LIST GMRENB / GMRENC

BPVAL MCH ASEG -92.0 / 0.2

BPVAL ASEG MCH 430.0 / 0.2


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Estimating Property Parameters Following is the input language used to estimate property parameters.

Input Language for Property Parameter Estimation

ESTIMATE [option]

STRUCTURES method SEG-id groupno nooccur / groupno nooccur /...

Input Language Description for Property Parameter Estimation

The main keywords for specifying property parameter estimation inputs are the ESTIMATE and the STRUCTURES paragraphs. A brief description of the input language for these paragraphs follows. For more detailed information please refer to the Aspen Physical Property System Physical Property Data documentation.

option Option=ALL Estimate all missing parameters (default)

method Polymer property estimation method name

SEG-id Segment ID defined in the component list

groupno Functional group number (group ID taken from Appendix B)

nooccur Number of occurrences of the group

Input Language Example for Property Parameter Estimation

ESTIMATE ALL

STRUCTURES

VANKREV ABSEG 115 1 ;-(C6H4)-

VANKREV BSEG 151 2 / 100 2 ; -COO-CH2-CH2-COO-

VANKREV ABSEG 115 1 / 151 2 / 100 2 ;-(C6H4)-COO-CH2-CH2-COO-


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228 Index

Index

A

ABS 146 Acrylonitrile-butadiene-styrene 146 Activation energy 157 Activity coefficients

Born 107 Electrolyte-Polymer NRTL model

103–14 Flory-Huggins model 94–98 Flory-Huggins term 111 for polymer activity 101 for polymer components 101 for solvents 101 ionic 105 liquid-liquid equilibria

calculations 90 local composition term 110 mixture liquid molar volume

calculations 92 model overview 87 models available 87–120 models list 93 phase equilibria calculations 88 Pitzer-Debye-Hückel 106 Polymer NRTL model 98–103 Polymer UNIFAC Free Volume

model 117–19 Polymer UNIFAC model 114–17 properties available 93 property models 13, 15 thermodynamic property

calculations 90 vapor-liquid equilibria

calculations 88 Adding

data for parameter optimization 23

molecular structure for property estimation 22

parameters for property models 20

property methods 20 Amorphous solid 8 Aspen polymer mixture viscosity

model See Polymer mixture viscosity model

Aspen Polymers activity coefficient models 87–

120, 93 activity coefficient properties 93 available polymer properties 122 available property methods 16–

19 available property models 13–16,

122 EOS models 34 EOS properties 32 equation of state models 27–86 input language for physical

properties 223–27 polymer thermal conductivity

models 171–82 polymer viscosity models 151–70 thermal conductivity routes 181 viscosity models 151 viscosity routes 170

AspenTech support 3 AspenTech Support Center 3

B

Binary antisymmetric mixing 159 Binary interaction parameters 96,

102 Binary parameters

for Eyring-NRTL 168 for PC-SAFT EOS 64 for polymer mixture viscosity

159 for SAFT EOS 56 for Sanchez-Lacombe EOS 39

Binary symmetric-quadratic mixing 159

Born


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Index 229

activity coefficient 107 equation 106

C

Calculating segment properties from

functional groups 202 solution viscosity 164

Carnahan-Starling expression 50 Components

liquid enthalpy model 147 Concentrated solution 161 Concentration basis 97 Copolymer PC-SAFT

description 67 model 67 option codes 78 parameters 76 property method (PC-SAFT) 200

Critical concentration 161 constants 45 mass viscosity 156 molecular weight 154

Crystalline weighting fraction 173 Custom

property methods 20 customer support 3

D

Data for optimizing parameters 23 parameter estimation 21 thermodynamic 19

Density of mixtures 6 property model 13, 15

Departure functions about 30 ideal gas 123

Devolatilization of monomers 9 Diffusion coefficients 6 Dilution effect 167 Dissolved gas 89

E

e-bulletins 3 Electrolyte-Polymer NRTL

adjustable parameters 111 applicability 103 assumptions 104

Born term 106 excess Gibbs free energy 104 Flory-Huggins term 111 for multicomponent systems 108 ionic activity coefficient 105 local composition term 108 local interaction contribution 107 long range interaction

contribution 105 model 103–14 model parameters 112 Pitzer-Debye-Hückel term 105 specifying model 114 terms 104

Energy balance 6 Enthalpy See also Solid enthalpy,

See also Liquid enthalpy calculation 203 departure 31 excess molar liquid 90 for amorphous polymer 129 for crystalline polymer 129 ideal gas 124 in systems 6 of mixing 94, 98 property model 13, 15 temperature relationship 128

Entropy calculation 203 departure 31 excess molar liquid 91 in equipment design 6 of mixing 94, 98, 111 of polymers 123

EP-NRTL See Electrolyte-Polymer NRTL

Equations of state Copolymer PC-SAFT model 67 liquid-liquid equilibria

calculations 30 model overview 27 models available 27–86, 34 parameters for 218–22 PC-SAFT model 59–66 phase equilibria calculations 29–

30 Polymer SRK model 42–47 properties available 32 property models 13, 14 SAFT model 47–59 Sanchez-Lacombe model 34–42 thermodynamic property

calculations 30–32


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230 Index

vapor-liquid equilibria calculations 30

Equilibria See also Phase equilibria calculating 6 liquid-liquid 11, 30, 90 polymer solutions 9–12 polymer systems 30, 88, 90 properties 6 vapor-liquid 9, 30, 88 vapor-liquid-liquid 10

Equipment design 6 Estimating


property parameters 227 solution viscosity 164 thermophysical properties 145

Excess liquid functions 90 molar liquid 90 molar liquid enthalpy 90 molar liquid entropy 91

Eyring-NRTL mixture viscosity model

about 167 applicability 167 binary parameters 168 for multicomponent systems 168 parameters 169 specifying 169

F

Flory-Huggins activity coefficient 111 applicability 94 binary interaction parameter 96 concentration basis 97 equation 111 for multiple components 96 Gibbs free energy of mixing 95 interaction parameter 94 model 94–98 model parameters 97 property method (POLYFH) 183–

85 specifying model 98

Fractionation 12 Fugacity 6 Functional groups

containing halogen 212 containing hydrocarbons 205 containing hydrogen 207 containing nitrogen 210 containing nitrogen and oxygen

211 containing oxygen 208 containing sulfur 212 parameters 205–14 van Krevelen 202–14

G

Gas dissolved 89

Gibbs free energy See also Solid Gibbs free energy, See also Liquid Gibbs free energy

calculation 203 departure 31 excess (EP-NRTL) 104 excess (NRTL) 100 excess (SRK) 42 ideal gas 124 minimization 6 of mixing (Flory-Huggins) 95 of mixing (Polymer NRTL) 99 of polymers 123 property model 13, 15

Glass transition model parameters 142

Glass transition temperature calculation 204 for amorphous solids 8 polymer mixture 166 polymer solution 165 Van Krevelen correlation 141

Group contribution Van Krevelen method 145

Group contribution method van Krevelen 145, 146

H

Heat capacity calculation 202 ideal gas 125 of polymers 131 parameters 127 property model 13, 15

Helmholtz free energy 49, 60


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Index 231

help desk 3

I

Ideal gas departure functions 123 enthalpy 124 Gibbs free energy 124 heat capacity 125 heat capacity parameters 127 model 123–27 model parameters 125 molar mixture properties 32 property model 15

Input language for physical properties 223–27 property data 225 property methods 223 property parameter estimation

227 Interaction contribution

local 107 long range 105

Interaction parameter 94 Internal energy 6

K

KLMXVKDP 181 KLMXVKTR 180 KMXVKTR 181

L

Lattice theory 34 LCST 11 Letsou-Stiel 161 Linear mixing 159 Liquid enthalpy

model parameters 131 of polymers 123, 128 pure component model 147

Liquid Gibbs free energy model parameters 131 of polymers 130

Liquid molar volume mixture calculations 92 model parameters (Tait) 141 model parameters (van

Krevelen) 137 Tait model 140–41 Van Krevelen model 136–40

Liquid Van Krevelen model 127–35 Liquid-liquid equilibrium 30, 90

Liquid-liquid phase equilibrium 11 LLE 11, 30, 90 Local composition

activity coefficient 110 equation 108

Local interaction contribution 107 Long range interaction contribution

105 Lower critical solution temperature

11

M

Mark-Houwink equation 152

Mark-Houwink/van Krevelen model 152–58 model applicability 151 model parameters 154 specifying model 158

Mass balance 6 Mass-based property parameters

217 Melt transition

model parameters 143 Melt transition temperature

calculation 204 Van Krevelen correlation 142

Melting temperature 8 Melts 8 Mixing

binary antisymmetric 159 binary symmetric-quadratic 159 linear 159

Mixture density 6 thermal conductivity 6 viscosity 6

Mixture liquid molar volume calculations 92

Mixture viscosity See also Eyring-NRTL mixture viscosity model, See also Polymer mixture viscosity model

Mixtures Eyring-NRTL viscosity model 167 glass transition temperature 166 thermal conductivity model 180 viscosity model 158–61

Modeling See Process modeling Models

activity coefficient 87–120 Copolymer PC-SAFT EOS 67


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232 Index

Electrolyte-Polymer NRTL activity coefficient 103–14

equation of state 27–86 Eyring-NRTL mixture viscosity

167 Flory-Huggins activity coefficient

94–98 ideal gas 123–27 Mark-Houwink/van Krevelen

152–58 mixture thermal conductivity 180 parameter regression 146 PC-SAFT EOS 59–66 polymer mixture viscosity 158–

61 Polymer NRTL activity coefficient

98–103 Polymer SRK EOS 42–47 polymer thermal conductivity

171–82 Polymer UNIFAC activity

coefficient 114–17 Polymer UNIFAC Free Volume

activity coefficient 117–19 polymer viscosity 151–70 pure component liquid enthalpy

147 SAFT EOS 47–59 Sanchez-Lacombe EOS 34–42 Tait liquid molar volume 140–41 Van Krevelen glass transition

temperature 141 Van Krevelen liquid 127–35 Van Krevelen liquid molar

volume 136–40 Van Krevelen melt transition

temperature 142 Van Krevelen polymer solution

viscosity 161–67 Van Krevelen solid 143 Van Krevelen thermal

conductivity 173–79 Modified Mark-Houwink equation

152 Molar liquid (excess) 90 Molar volume See also Liquid molar

volume calculation 203 calculations for liquid mixture 92 for polymer (Tait) 140

for polymers (van Krevelen) 136 from EOS models 31

Molecular structure entering for property estimation

22 Molecular weight

critical 154 weight average 153

Monomers devolatilization of 9

MULMX13 170 MULMXVK 170

N

Non-random two liquid See NRTL Nonvolatility 8, 89 NRTL See also Electrolyte-Polymer

NRTL, See also Polymer NRTL electrolye-polymer model 103–

14 polymer model 98–103

O

Oligomers 8 ideal gas model 123 nonvolatility 89

P

Parameters binary for Eyring-NRTL 168 binary for PC-SAFT 64 binary for polymer mixture

viscosity 159 binary for SAFT 56 binary for Sanchez-Lacombe 39 binary interaction 96, 102 calculating segment properties

202 electrolyte-electrolyte 112 electrolyte-molecule 112 entering for components 21 entering for property models 20 estimating for property models

21 estimating property 227 for Electrolyte-Polymer NRTL

model 112


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Index 233

for equation of state models 218–22

for Eyring-NRTL mixture viscosity model 169

for Flory-Huggins model 97 for glass transition 142 for ideal gas heat capacity 127 for ideal gas model 125 for liquid enthalpy 131 for liquid Gibbs free energy 131 for liquid molar volume (Tait)

141 for liquid molar volume (van

Krevelen) 137 for Mark Houwink/van Krevelen

model 154 for melt transition 143 for PC-SAFT model 65 for polymer mixture viscosity

model 159 for Polymer NRTL model 102 for polymer solution viscosity

model 163 for Polymer SRK model 45 for Polymer UNIFAC free volume

model 119 for Polymer UNIFAC model 117 for polymers (ideal gas) 126 for polymers (van Krevelen liquid

models) 133 for polymers (van Krevelen liquid

molar volume model) 138 for polymers (van Krevelen

thermal conductivity model) 175

for SAFT (POLYSAFT) 220 for SAFT model 57 for Sanchez-Lacombe (POLYSL)

218 for Sanchez-Lacombe model 36,

40–42 for segments (ideal gas) 126 for segments (thermal

conductivity) 176 for segments (van Krevelen

liquid molar volume) 139 for segments (van Krevelen

liquid) 134 for solid enthalpy 144 for solid Gibbs free energy 144 for solid molar volume (van

Krevelen) 145 for Tait model 215–16

for van Krevelen liquid model 131

for van Krevelen solid model 144 for van Krevelen thermal

conductivity model 174 input for Eyring-NRTL mixture

viscosity model 169 input for ideal gas model 125 input for Mark-Houwink model

154 input for PC-SAFT model 66 input for polymer mixture

viscosity model 160 input for SAFT model 58 input for Sanchez-Lacombe

model 41 input for van Krevelen liquid

models 132 input for van Krevelen liquid

molar volume model 138 input for van Krevelen thermal

conductivity model 175 interaction 94 mass-based 217 missing for SAFT model 59 missing for Sanchez-Lacombe

model 42 molecule-molecule 112 optimizing 23 regression 146 regression for Eyring-NRTL

mixture viscosity model 169 regression for ideal gas model

127 regression for Mark-Houwink

model 154 regression for PC-SAFT model 66 regression for polymer mixture

viscosity model 160 regression for SAFT model 58 regression for Sanchez-Lacombe

model 41 regression for Tait model 141 regression for van Krevelen

liquid model 135 regression for van Krevelen

liquid molar volume 139 Tait model 143 Van Krevelen estimation 145

PC-SAFT about 17 applicability 59 binary parameters 64


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234 Index

comparison to SAFT 60 copolymer model 67 for copolymer systems 63 implementation 62 model 59–66 model parameters 65 property method (POLYPCSF)

198–99 pure-component parameters 60 route structure 200 specifying model 66

Perturbation theory 49 Perturbed chain statistical

associating fluid theory See PC-SAFT

Phase equilibria calculated from activity

coefficients 88 calculated from EOS models 29–

30 modeling 9

Physical properties See also Properties

for activity coefficient models 93 for EOS models 32 input language 223 route structure 183 specifying 19–23

Pitzer-Debye-Hückel activity coefficient 106 equation 105

Plasticizer effect 165 PNRTL-IG 17, 191–93 Polydispersity 7 POLYFH 16, 183–85 Polymer mixture thermal

conductivity model 180 Polymer mixture viscosity model

about 158–61 applicability 151, 158 binary parameters 159 Eyring-NRTL 151 for multicomponent systems 158 parameters 159 specifying 161

Polymer NRTL activity coefficients 100 applicability 98 binary interaction parameters

102

excess Gibbs free energy 100 for homopolymer 102 Gibbs free energy of mixing 99 ideal gas property method

(PNRTL-IG) 191–93 model 98–103 model parameters 102 property method (POLYNRTL)

185–87 specifying model 103

Polymer solution glass transition temperature 165

Polymer solution viscosity model about 161–67 calculation steps 164 estimating values 164 for multicomponent mixtures 161 parameters 163 pseudo-component properties

162 quasi-binary system 161 specifying 167

Polymer SRK characteristics 42 cubic EOS parameters 43 equation 43 for polymer mixtures 44 model 42–47 model parameters 45 property method (POLYSRK)

195–96 specifying model 47

Polymer UNIFAC applicability 114 for solvent activity 118 free volume model 117–19 free volume model parameters

119 free volume property method

(POLYUFV) 189–91 model 114–17 model parameters 117 modification for free volume 117 property method (POLYUF) 187–

89 specifying free volume model


Polymer viscosity


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Index 235

at mixture glass transition temperature 166

Polymers amorphous 8 available properties 122 available property methods 16–

19 available property models 13–16,

122 critical constants 45 critical molecular weight 154 differences from non-polymers 7 entropy 123 fractionation 12 Gibbs free energy 123 glass transition for mixtures 166 glass transition for solution 165 heat capacity 131 ideal gas enthalpy 124 ideal gas Gibbs free energy 124 ideal gas heat capacity 125 ideal gas model 123 liquid enthalpy 123, 128 liquid Gibbs free energy 130 melt 8 melt viscosity 152 melts 151 modeling considerations 7 modeling mixture phase

equilibria 9 modeling thermophysical

properties 12 molar volume (Tait) 140 molar volume (van Krevelen)

136 nonvolatility 8, 89 parameter regression 146 parameters for van Krevelen

thermal conductivity model 175

polydispersity 7 semi-crystalline 8 solid enthalpy 143 solid Gibbs free energy 144 solid molar volume 144 solution viscosity 16, 158, 161,

164, 167 solution viscosity correlation 160 solutions 9–12 solvent activity 118 species 109 systems 30, 88, 90

temperature enthalpy relationship 128

thermodynamic data for systems 19

thermodynamic properties 5–26 thermophysical properties 121–

50 Van Krevelen group contribution

156 viscoelasticity 8 weight average molecular weight

153 POLYNRTL 16, 185–87 POLYPCSF

about 17 route structure 198–99

POLYSAFT about 17 model parameters 220 route structure 196–99

POLYSL about 17 model parameters 218 route structure 193–94

POLYSRK 17, 195–96 POLYUF 17, 187–89 POLYUFV 17, 189–91 Process modeling

liquid-liquid equilibria 11 phase equilibria for polymer

mixtures 9 polymer fractionation 12 properties of interest 5 property methods available 16–

19 property models available 13–16 thermophysical polymer

properties 12 vapor-liquid equilibria 9 vapor-liquid-liquid equilibria 10

Process simulation See Process Modeling

Properties See also Thermophysical properties, See also Thermodynamic properties

calculating segment from functional groups 202

estimating parameters 227 for activity coefficient models 93 for energy balance 6 for EOS models 32 for equilibria 6 for equipment design 6


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236 Index

for mass balance 6 for polymers 122 input language 223–27 mass-based 217 modeling thermophysical 12 models available 122 of interest in modeling 5 of polymer solution viscosity

components 162 of polymers 5–26, 121–50 specifying 19–23 specifying data 225 thermodynamic 5–26 thermophysical 121–50

Property methods available 16–19 customizing 20 input language 223 liquid phase calcualtions 18 PC-SAFT 17, 200 PNRTL-IG 17, 191–93 POLYFH 16, 183–85 POLYNRTL 16, 185–87 POLYPCSF 17, 198–99 POLYSAFT 17, 196–99 POLYSL 17, 193–94 POLYSRK 17, 195–96 POLYUF 17, 187–88, 187–89 POLYUFV 17, 189–91 properties calculated 17 selecting 19 vapor phase calcualtions 17

Property models available 13–16 Copolymer PC-SAFT 67 Electrolyte-Polymer NRTL 103–

14 entering molecular structure 22 entering parameters 20 estimating parameters 21 Eyring-NRTL mixture viscosity

167 Flory-Huggins 94–98 for activity coefficients 13, 15,

87–120 for density 13, 15 for enthalpy 13, 15 for equations of state 13, 14,

27–86 for Gibbs free energy 13, 15

for heat capacity 13, 15 for ideal gas 15, 123–27 for polymer thermal conductivity

171–82 for polymer viscosity 151–70 for solution thermodynamics 13 for thermophysical properties 13 for transport properties 13 Mark-Houwink/van Krevelen

152–58 mixture thermal conductivity 180 optimizing 23 PCSAFT 59–66 polymer mixture viscosity 158–

61 Polymer NRTL 98–103 Polymer UNIFAC 114–17 Polymer UNIFAC Free Volume

117–19 pure component liquid enthalpy

147 SAFT 47–59 Sanchez-Lacome 34–42 SRK 42–47 Tait liquid molar volume 140–41 Van Krevelen glass transition

temperature 141 Van Krevelen liquid 127–35 Van Krevelen liquid molar

volume 136–40 Van Krevelen melt transition

temperature 142 Van Krevelen polymer solution

viscosity 161–67 Van Krevelen solid 143 Van Krevelen thermal

conductivity 173–79 Pseudo-components 162

Q

Quasi-binary systems 161

R

Rao function calculation 204 from van Krevelen group

contribution 177 Reduced viscosity 157 Regressing


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Index 237

Eyring-NRTL mixture viscosity parameters 169

ideal gas parameters 127 liquid model parameters (van

Krevelen) 135 liquid molar volume parameters

139 Mark-Houwink parameters 154 PC-SAFT parameters 66 polymer mixture viscosity

parameters 160 polymer properties 146 SAFT parameters 58 Sanchez-Lacombe parameters 41 Tait model parameters 141

Routes calculations for property models

183–99 for thermal conductivity 181 for viscosity 170

S

SAFT applicability 47 binary parameters 56 comparison to PC-SAFT 60 for copolymer systems 55 for fluid mixtures 52 for pure fluids 47 implementation 53–55 model 47–59 model parameters 57, 220 property method (POLYSAFT)

196–99 required parameters 53 specifying model 59

Sanchez-Lacombe binary parameters 39 characteristics 36 equation 35 for copolymer systems 37 for fluid mixtures 36 for homopolymers 36 for pure fluids 34 model 34–42 model parameters 40–42, 218 molecular parameters 36 property method (POLYSL) 193–


Sato-Reidel/DIPPR model 181 Segments

calculating properties from functional groups 202

reference temperature (thermal conductivity) 176

reference thermal conductivity 178

thermal conductivity at 298K 177 Van Krevelen group contribution

(ideal gas) 126 Van Krevelen group contribution

(liquid molar volume) 139 Van Krevelen group contribution

(liquid) 134 Van Krevelen group contribution

(thermal conductivity) 176 Semi-crystalline solid 8 Simulation See Process Modeling Soave-Redlich-Kwong See Polymer

SRK Solid enthalpy

model parameters 144 of polymers 143

Solid Gibbs free energy model parameters 144 of polymers 144

Solid molar volume model parameters (van

Krevelen) 145 of polymers 144

Solids amorphous 8 semi-crystalline 8 Van Krevelen model 143

Solution viscosity See also Polymer solution viscosity model

Van Krevelen model 161–67 Solutions

critical concentration 161 glass transition temperature 165 viscosity estimation 164

Solvent dilution effect 167

Specifying data for parameter optimization

23 Electrolyte-Polymer NRTL model

114 Eyring-NRTL mixture viscosity

model 169 Flory-Huggins model 98 Mark-Houwink/van Krevelen

model 158


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238 Index

mixture thermal conductivity model 180

molecular structure for property estimation 22

parameter estimates for property models 21


PC-SAFT model 66 physical properties 19–23 physical properties (input

language) 223 polymer mixture viscosity model

161 Polymer NRTL model 103 polymer solution viscosity model

167 Polymer SRK model 47 Polymer UNIFAC free volume

model 119 Polymer UNIFAC model 117 property data 225 property methods 19 SAFT model 59 Sanchez-Lacombe model 42 Van Krevelen thermal

conductivity model 179 SRK See Polymer SRK Statistical associating fluid theory

See PC-SAFT , See SAFT support, technical 3 Surface tension 6

T

Tait equation 140 liquid molar volume model 140–

41 liquid molar volume model

parameters 141 model coefficients 215–16

technical support 3 Temperature

enthalpy relationship 128 glass transition 8, 141 glass transition calculation 204 lower critical solution 11 melt transition 142 melt transition calculation 204

melting 8 polymer mixture glass transition

166 polymer solution glass transition

165 segment reference 176 segment thermal conductivity

177 upper critical solution 11 Van Krevelen viscosity

correlation 155 viscosity gradient 204

Thermal conductivity for crystalline state 173 for equipment design 6 for glassy state 173 for liquid state 173 for segments at 298K 177 mixture model 180 model applicability 172 model overview 171 model parameters 174 models available 171–82 modified van Krevelen equation

173 pressure dependence 179 routes in Aspen Polymersl 181 segment reference (crystalline

state) 178 segment reference (glassy state)

178 segment reference (liquid state)

178 segment reference temperature

176 specifying mixture model 180 specifying van Krevelen model

179 temperature dependence 179 Van Krevelen model 173–79

Thermodynamic data for polymer systems 19

Thermodynamic properties See also Properties

activity 6 calculated from activity

coefficient models 90 calculated from EOS models 30–

32 density 6


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Index 239

enthalpy 6 enthalpy departure 31 entropy 6 entropy departure 31 excess molar liquid 90 excess molar liquid enthalpy 90 excess molar liquid entropy 91 fugacity 6 Gibbs free energy 6 Gibbs free energy departure 31 ideal gas 32 internal energy 6 molar volume 31 of polymers 5–26

Thermophysical properties See also Properties, See also Properties

estimating 145 modeling 12 of polymers 121–50 overview 121

Transport properties diffusivity 6 property models 13 surface tension 6 thermal conductivity 6 viscosity 6

TRAPP model 180 True solvent dilution effect 167

U

UCST 11 UNIFAC See also Polymer UNIFAC

polymer free volume model 117–19

polymer model 114–17 UNIFAC free volume

applicability 117 Upper critical solution temperature

11

V

Van der Waals for fluid mixture 52 volume 203

Van Krevelen equation for thermal conductivity

173 functional group parameters

205–14 functional groups 202–14 glass transition temperature 141

group contribution 145 group contribution for polymers

156 liquid model 127–35 liquid model parameters 131 liquid molar volume model 136–

40 liquid molar volume model

parameters 137 melt transition temperature 142 model for thermal conductivity

173–79 polymer solution viscosity model

161–67 solid model 143 solid model parameters 144 solid molar volume model

parameters 145 viscosity-temperature correlation

155 Van Krevelen group contribution

for segments (ideal gas) 126 for segments (liquid molar

volume) 139 for segments (liquid) 134 for segments (thermal

conductivity) 176 Vapor-liquid equilibrium 9, 30, 88 Vapor-liquid-liquid equilibrium 10 Viscoelasticity 8 Viscosity

at mixture glass transition temperature 166

critical mass 156 estimating 164 Eyring-NRTL mixture model 167 Mark-Houwink/van Krevelen

model 152–58 model overview 151 models available 151–70 models list 151 of mixtures 6 of polymer mixtures 167 of polymer solutions 158 of solutions 161 polymer melt 152 polymer mixture model 158–61 polymer solution 16 polymer solution correlation 160 reduced 157 routes in Aspen Polymers 170 temperature gradient calculation

204


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240 Index

Van Krevelen polymer solution model 161–67

Van Krevelen temperature correlation 155

zero-shear 16, 155 VLE 9, 30, 88 VLLE 10 Volatility 8 Volume fraction basis 95 Vredeveld mixing rule 180

W

web site, technical support 3 Weight average

mixing rule 162 molecular weight 153

Weight fraction crystalline 173

Williams-Landel-Ferry 155

Z

Zero-shear viscosity estimation methods 16 of mixtures 158, 168 of polymers 155


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