aspen polymers vol2 v7 tutorial
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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.
All other brand and product names are trademarks or registered trademarks of their respective companies.
This document is intended as a guide to using AspenTech's software. This documentation contains AspenTech proprietary and confidential information and may not be disclosed, used, or copied without the prior consent of AspenTech or as set forth in the applicable license agreement. Users are solely responsible for the proper use of the software and the application of the results obtained.
Although AspenTech has tested the software and reviewed the documentation, the sole warranty for the software may be found in the applicable license agreement between AspenTech and the user. ASPENTECH MAKES NO WARRANTY OR REPRESENTATION, EITHER EXPRESSED OR IMPLIED, WITH RESPECT TO THIS DOCUMENTATION, ITS QUALITY, PERFORMANCE, MERCHANTABILITY, OR FITNESS FOR A PARTICULAR PURPOSE.
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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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iv Contents
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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vi Contents
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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Contents vii
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
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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.
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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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1 Thermodynamic Properties of Polymer Systems 7
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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8 1 Thermodynamic Properties of Polymer Systems
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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1 Thermodynamic Properties of Polymer Systems 9
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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10 1 Thermodynamic Properties of Polymer Systems
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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1 Thermodynamic Properties of Polymer Systems 11
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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12 1 Thermodynamic Properties of Polymer Systems
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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1 Thermodynamic Properties of Polymer Systems 13
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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14 1 Thermodynamic Properties of Polymer Systems
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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1 Thermodynamic Properties of Polymer Systems 15
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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16 1 Thermodynamic Properties of Polymer Systems
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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1 Thermodynamic Properties of Polymer Systems 17
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)
All vapor properties: Fugacity coefficient, enthalpy, entropy, Gibbs free energy, density
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18 1 Thermodynamic Properties of Polymer Systems
Properties Calculated
Model (Property method)
Used For
PC-SAFT (POLYPCSF)
All vapor properties: fugacity coefficient, enthalpy, entropy, Gibbs free energy, density
Copolymer PC-SAFT (PC-SAFT)
All vapor properties: Fugacity coefficient, enthalpy, entropy, Gibbs free energy, density
Liquid
Vapor pressure PLXANT
Antoine
(All activity coefficient property methods)
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
(All activity coefficient property methods)
Enthalpy, entropy
Molar Volume Rackett for monomers, Van Krevelen for polymers and oligomers from segments
Tait molar model for polymers and oligomers
(All activity coefficient property methods)
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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1 Thermodynamic Properties of Polymer Systems 19
Properties Calculated
Model (Property method)
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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20 1 Thermodynamic Properties of Polymer Systems
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:
From the Data Browser, click Properties.
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1 Thermodynamic Properties of Polymer Systems 21
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.
An object manager appears.
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:
From the Data Browser, click Properties.
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22 1 Thermodynamic Properties of Polymer Systems
Several subfolders appear.
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.
Click Next to proceed.
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:
From the Data Browser, click Properties.
Several subfolders appear.
Click Molecular Structure.
An object manager appears.
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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1 Thermodynamic Properties of Polymer Systems 23
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.
Click Next to proceed.
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.
An object manager appears.
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.
Click Next to proceed.
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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24 1 Thermodynamic Properties of Polymer Systems
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.
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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.
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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.
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1 Thermodynamic Properties of Polymer Systems 25
Huang, S. H., & Radosz, M. (1990). Equation of State for Small, Large, Polydisperse, and Associating Molecules. Ind. Eng. Chem. Res., 29, 2284.
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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.
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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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26 1 Thermodynamic Properties of Polymer Systems
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.
Topics covered include:
• 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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2 Equation-of-State Models 29
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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30 2 Equation-of-State Models
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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2 Equation-of-State Models 31
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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32 2 Equation-of-State Models
( )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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2 Equation-of-State Models 33
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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34 2 Equation-of-State Models
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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2 Equation-of-State Models 35
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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36 2 Equation-of-State Models
***~~~
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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2 Equation-of-State Models 37
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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38 2 Equation-of-State Models
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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2 Equation-of-State Models 39
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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40 2 Equation-of-State Models
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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2 Equation-of-State Models 41
Parameter Name / Element
Symbol Default Lower Limit
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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42 2 Equation-of-State Models
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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2 Equation-of-State Models 43
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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44 2 Equation-of-State Models
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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2 Equation-of-State Models 45
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.
Parameter Name / Element
Symbol Default Lower Limit
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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46 2 Equation-of-State Models
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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2 Equation-of-State Models 47
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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48 2 Equation-of-State Models
• 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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2 Equation-of-State Models 49
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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50 2 Equation-of-State Models
• 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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2 Equation-of-State Models 51
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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52 2 Equation-of-State Models
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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2 Equation-of-State Models 53
( )( )∑∑
∑∑=
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 /
• Segment number, m
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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54 2 Equation-of-State Models
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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2 Equation-of-State Models 55
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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56 2 Equation-of-State Models
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.
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(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
The cross energy parameter for a copolymer-copolymer pair in the mixture can be calculated as:
000212121
)1( pppppp uuku −=
Where:
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2 Equation-of-State Models 57
021 ppu = Cross energy parameter for a copolymer-copolymer pair
21 ppk = Binary parameter for a copolymer-copolymer pair
01pu = Energy parameter of pure copolymer 1p
02pu = 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
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.
SAFT EOS Model Parameters The following tables list the SAFT EOS model name and model parameters implemented in Aspen Polymers:
Parameter Name / Element
Symbol Default Lower Limit
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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58 2 Equation-of-State Models
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:
• 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 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:
• Specified for each polymer-solvent pair
• Specified for each segment-solvent pair
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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2 Equation-of-State Models 59
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 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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60 2 Equation-of-State Models
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 number, m
• 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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2 Equation-of-State Models 61
• 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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62 2 Equation-of-State Models
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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2 Equation-of-State Models 63
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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64 2 Equation-of-State Models
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
nM = Number average molecular weight of the copolymer
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.
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
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
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
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2 Equation-of-State Models 65
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
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.
PC-SAFT EOS Model Parameters The following table lists the PC-SAFT EOS model parameters implemented in Aspen Polymers:
Parameter Name / Element
Symbol Default Lower Limit
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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66 2 Equation-of-State Models
Parameter Input and Regression Three unary parameters, PCSFTR, PCSFTU, and PCSFTV 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 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:
• Specified for each polymer-solvent pair
• Specified for each segment-solvent pair
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.
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.
Specifying the PC-SAFT EOS Model See Specifying Physical Properties in Chapter 1.
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2 Equation-of-State Models 67
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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68 2 Equation-of-State Models
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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2 Equation-of-State Models 69
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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70 2 Equation-of-State Models
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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2 Equation-of-State Models 71
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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72 2 Equation-of-State Models
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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2 Equation-of-State Models 73
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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74 2 Equation-of-State Models
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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2 Equation-of-State Models 75
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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76 2 Equation-of-State Models
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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2 Equation-of-State Models 77
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:
Parameter Name / Element
Symbol Default Lower Limit
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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78 2 Equation-of-State Models
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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2 Equation-of-State Models 79
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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80 2 Equation-of-State Models
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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2 Equation-of-State Models 81
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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82 2 Equation-of-State Models
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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2 Equation-of-State Models 83
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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84 2 Equation-of-State Models
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.
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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.
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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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2 Equation-of-State Models 85
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86 2 Equation-of-State Models
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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.
Topics covered include:
• 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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3 Activity Coefficient Models 89
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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90 3 Activity Coefficient Models
∞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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3 Activity Coefficient Models 91
∑−=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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92 3 Activity Coefficient Models
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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3 Activity Coefficient Models 93
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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94 3 Activity Coefficient Models
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
An X indicates applicable to Pure or Mixture.
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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3 Activity Coefficient Models 95
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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96 3 Activity Coefficient Models
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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3 Activity Coefficient Models 97
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.
Parameter Name / Element
Symbol Default Lower Limit
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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98 3 Activity Coefficient Models
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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3 Activity Coefficient Models 99
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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100 3 Activity Coefficient Models
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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3 Activity Coefficient Models 101
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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102 3 Activity Coefficient Models
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.
Parameter Name / Element
Symbol Default Lower Limit
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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3 Activity Coefficient Models 103
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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104 3 Activity Coefficient Models
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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3 Activity Coefficient Models 105
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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106 3 Activity Coefficient Models
∑=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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3 Activity Coefficient Models 107
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 =
)exp( jijijiG τα−=
RTgg iiji
ji
)( −=τ
Where:
jC = jz for ions and unity for molecules
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
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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108 3 Activity Coefficient Models
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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3 Activity Coefficient Models 109
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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110 3 Activity Coefficient Models
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛−+
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛−+
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛−+
=
∑∑
∑∑ ∑
∑∑
∑∑∑
∑∑
∑ ∑
∑∑
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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3 Activity Coefficient Models 111
∑∑∑ ++=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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112 3 Activity Coefficient Models
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:
Parameter Name / Element
Symbol Default Lower Limit
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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3 Activity Coefficient Models 113
Parameter Name / Element
Symbol Default Lower Limit
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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114 3 Activity Coefficient Models
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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3 Activity Coefficient Models 115
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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116 3 Activity Coefficient Models
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
−−=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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3 Activity Coefficient Models 117
Polymer UNIFAC Model Parameters The input parameters for the Polymer UNIFAC model are given here:
Parameter Name / Element
Symbol Default Lower Limit
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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118 3 Activity Coefficient Models
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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3 Activity Coefficient Models 119
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.
References Aspen Physical Property System Physical Property Methods and Models. Cambridge, MA: Aspen Technology, Inc.
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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120 3 Activity Coefficient Models
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.
Topics covered include:
• 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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4 Thermophysical Properties of Polymers 123
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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124 4 Thermophysical Properties of Polymers
( )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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4 Thermophysical Properties of Polymers 125
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:
Parameter Name / Element
Symbol Default Lower Limit
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
• Specified for segments that compose a polymer or oligomer component
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126 4 Thermophysical Properties of Polymers
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
refT = Reference temperature (298.15 K)
)(*, 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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4 Thermophysical Properties of Polymers 127
)(*, 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.
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.
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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128 4 Thermophysical Properties of Polymers
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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4 Thermophysical Properties of Polymers 129
( )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)
refT = Reference temperature (298.15 K)
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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130 4 Thermophysical Properties of Polymers
mT = Melt transition temperature of the polymer
gT = Glass transition temperature of the polymer
xc = Mass-fraction crystallinity
refT = Reference temperature (298.15 K)
( )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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4 Thermophysical Properties of Polymers 131
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
refT = Reference temperature (298.15 K)
( )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:
Parameter Name / Element
Symbol Default Lower Limit
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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132 4 Thermophysical Properties of Polymers
Parameter Name / Element
Symbol Default Lower Limit
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:
• Specified for each polymer or oligomer component
• Specified for segments that compose a polymer or oligomer component
• Calculated automatically using van Krevelen group contribution techniques.
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4 Thermophysical Properties of Polymers 133
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:
Nseg = Number of segment types in the copolymer
AX = Mole segment fraction of segment type A in the copolymer
refT = Reference temperature (298.15 K)
)(*, 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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134 4 Thermophysical Properties of Polymers
)(* 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:
kn = Number of occurrences of functional group k in the segment from the segment database or user-specified segment structure
refT = Reference temperature (298.15 K)
)(*, 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
refT , from van Krevelen database
)(* refksub THΔ = Heat of sublimation 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
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4 Thermophysical Properties of Polymers 135
)(* 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:
refT = Reference temperature (298.15 K)
( )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:
• Specified for each oligomer component (polymer)
• Specified for each segment that composes an oligomer component (polymer)
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.
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136 4 Thermophysical Properties of Polymers
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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4 Thermophysical Properties of Polymers 137
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 :
Parameter Name / Element
Symbol Default Lower Limit
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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138 4 Thermophysical Properties of Polymers
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:
Nseg = Number of segment types in the copolymer
nM = Number average molecular weight of the copolymer
segM = Average molecular weight of segments in the copolymer
AX = Mole segment fraction of segment type A 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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4 Thermophysical Properties of Polymers 139
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
kn = Number of occurrences of functional group k in the segment from the segment database or user-specified segment structure
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 (polymer)
• Specified for each segment that composes an oligomer component (polymer)
• Specified for each oligomer component on a molar basis or mass basis
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140 4 Thermophysical Properties of Polymers
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.
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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4 Thermophysical Properties of Polymers 141
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.
Parameter Name / Element
Symbol Default Lower Limit
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.
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.
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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142 4 Thermophysical Properties of Polymers
Where:
AgT , = Glass transition temperature for segment type A
kn = Number of occurrences of functional group k in the segment from the segment database or user-specified segment structure
kgY , = Glass transition temperature of functional group k, from van Krevelen database
kM = Molecular weight of functional group k
gT = Glass transition temperature of the polymer
Nseg = Number of segment types in the copolymer
AX = Mole segment fraction of segment type A in the copolymer
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:
Parameter Name / Element
Symbol Default Lower Limit
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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4 Thermophysical Properties of Polymers 143
kn = Number of occurrences of functional group k in the segment from the segment database or user-specified segment structure
kmY , = Melt transition temperature of functional group k, from van Krevelen database
kM = Molecular weight of functional group k
mT = Melt transition temperature of the polymer
Nseg = Number of segment types in the copolymer
AX = Mole segment fraction of segment type A in the copolymer
AM = Molecular weight of segment type A
kmY , values for functional groups are given in Appendix B.
Melt Transition Correlation Parameters The glass transition model parameters are given here:
Parameter Name / Element
Symbol Default Lower Limit
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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144 4 Thermophysical Properties of Polymers
lH *, = Enthalpy of the polymer in the liquid phase
cH *, = Enthalpy of the polymer in the crystalline phase
mT = Melt transition temperature of the polymer
gT = Glass transition temperature of the polymer
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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4 Thermophysical Properties of Polymers 145
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
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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146 4 Thermophysical Properties of Polymers
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 (polymer)
• Specified for each segment that composes an oligomer component (polymer)
• 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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4 Thermophysical Properties of Polymers 147
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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148 4 Thermophysical Properties of Polymers
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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4 Thermophysical Properties of Polymers 149
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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150 4 Thermophysical Properties of Polymers
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.
References Aspen Physical Property System Physical Property Methods and Models. Cambridge, MA: Aspen Technology, Inc.
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.
Topics covered include:
• 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
An X indicates applicable to Pure or Mixture.
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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5 Polymer Viscosity Models 153
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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154 5 Polymer Viscosity Models
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†:
Parameter Name / Element
Symbol Default Lower Limit
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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5 Polymer Viscosity Models 155
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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156 5 Polymer Viscosity Models
(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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5 Polymer Viscosity Models 157
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
AM = Molecular weight of segment type A
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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158 5 Polymer Viscosity Models
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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5 Polymer Viscosity Models 159
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:
2ln/ rijrijrijrijijij TeTdTcTbak ++++=
2''''' ln/ rijrijrijrijijij TeTdTcTbal ++++=
with
refr T
TT =
Where:
refT = Reference temperature and the default value = 298.15 K
Aspen Polymer Mixture Viscosity Model Parameters The binary parameters for the Aspen polymer mixture viscosity model are listed here:
Parameter Name / Element
Symbol Default Lower Limit
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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160 5 Polymer Viscosity Models
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 .
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.
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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5 Polymer Viscosity Models 161
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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162 5 Polymer Viscosity Models
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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5 Polymer Viscosity Models 163
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:
Parameter Name / Element
Symbol Default Lower Limit
Upper Limit
MDS Units Keyword
Comments
MULVK/1 Ki 2.5 0 10 --- --- Unary
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164 5 Polymer Viscosity Models
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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5 Polymer Viscosity Models 165
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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166 5 Polymer Viscosity Models
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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5 Polymer Viscosity Models 167
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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168 5 Polymer Viscosity Models
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
,
,
)exp( jijijiG τα−=
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
jiα = NRTL non-random factor
jiτ = Interaction parameter
Binary Parameters The binary parameter, ijτ , allows complex temperature dependence:
2ln/ rijrijrijrijijij TeTdTcTba ++++=τ
with
refr T
TT =
Where:
refT = Reference temperature and the default value = 298.15 K
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5 Polymer Viscosity Models 169
Eyring-NRTL Mixture Viscosity Model Parameters The parameters for the Eyring-NRTL mixture viscosity model are listed here:
Parameter Name / Element
Symbol Default Lower Limit
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.
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.
Specifying the Eyring-NRTL Mixture Viscosity Model See Specifying Physical Properties in Chapter 1.
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170 5 Polymer Viscosity Models
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.
Topics covered include:
• 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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6 Polymer Thermal Conductivity Models 173
An X indicates applicable to Pure or Mixture.
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:
xc = Mass fraction crystallinity
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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174 6 Polymer Thermal Conductivity Models
⎟⎟⎠
⎞⎜⎜⎝
⎛ −⎟⎟⎠
⎞⎜⎜⎝
⎛ −+⎟
⎟⎠
⎞⎜⎜⎝
⎛ −+
⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟
⎟⎠
⎞⎜⎜⎝
⎛−+⎟
⎟⎠
⎞⎜⎜⎝
⎛ −+=
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:
Parameter Name / Element
Symbol Default Lower Limit
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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6 Polymer Thermal Conductivity Models 175
Parameter Name / Element
Symbol Default Lower Limit
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:
• 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 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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176 6 Polymer Thermal Conductivity Models
A
Nseg
AAAA
Nseg
AAi MXMX ∑∑= /*,*, σσ λλ
Where:
Nseg = Number of segment types in the copolymer
AX = Mole segment fraction of segment type A in the copolymer
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:
kn = Number of occurrences of functional group k in the segment from the segment database or user-specified segment structure
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
kM = Molecular weight of functional group k
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6 Polymer Thermal Conductivity Models 177
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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178 6 Polymer Thermal Conductivity Models
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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6 Polymer Thermal Conductivity Models 179
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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180 6 Polymer Thermal Conductivity Models
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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6 Polymer Thermal Conductivity Models 181
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
References Aspen Physical Property System Physical Property Methods and Models. Cambridge, MA: Aspen Technology, Inc.
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182 6 Polymer Thermal Conductivity Models
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
Route ID Model Name Description
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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A Physical Property Methods 185
Liquid
Property Name
Route ID Model Name Description
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
Route ID Model Name Description
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
van Krevelen, Polynomial
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
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
MUVMX MUVMX02 MUV2DNST Dean-Stiel
KVMX KVMX02 KV2STLTH Stiel-Thodos
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186 A Physical Property Methods
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
Route ID Model Name Description
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
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
SIGLMX SIGLMX01 SIG2HSS Hakim-Steinberg-Stiel, Power Law Mixing
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 HL0DVK*, GL0DVK van Krevelen, DIPPR
VL VLDVK VL0DVK, VL0RKT van Krevelen, Rackett
DL DL01 DL0WCA Wilke-Chang-Andrade
MUL MULMH MUL0MH Modified Mark-Houwink/van Krevelen, Andrade
KL KL0VKDP KL0VKDP Modified van Krevelen, Sato-Riedel/DIPPR
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A Physical Property Methods 187
* Optional van Krevelen/DIPPR model, HL0DVKD, available (see Pure Component
Liquid Enthalpy Models in Chapter 4).
Solid
Property Name
Route ID Model Name Description
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
van Krevelen, Polynomial
POLYUF: Polymer UNIFAC Property Method The following table lists the physical property route structure for the POLYUF 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
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
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188 A Physical Property Methods
Vapor
MUV MUV01 MUV0BROK Chapman-Enskog-Brokaw
KV KV01 KV0STLTH Stiel-Thodos
Liquid
Property Name
Route ID Model Name Description
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
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
SIGLMX SIGLMX01 SIG2HSS Hakim-Steinberg-Stiel, Power Law Mixing
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 HL0DVK*, GL0DVK van Krevelen, DIPPR
VL VLDVK VL0DVK, VL0RKT van Krevelen, Rackett
DL DL01 DL0WCA Wilke-Chang-Andrade
MUL MULMH MUL0MH Modified Mark-Houwink/van Krevelen, Andrade
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).
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A Physical Property Methods 189
Solid
Property Name
Route ID Model Name Description
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
van Krevelen, Polynomial
POLYUFV: Polymer UNIFAC Free Volume Property Method The following table lists the physical property route structure for the POLYUFV 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
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
MUV MUV01 MUV0BROK Chapman-Enskog-Brokaw
KV KV01 KV0STLTH Stiel-Thodos
Liquid
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190 A Physical Property Methods
Property Name
Route ID Model Name Description
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
KLMX KLMXVKDP KL0VKDP, KL2VR Vredeveld mixing, Modified van Krevelen, Sato-Riedel/DIPPR
DLMX DLMX02 DL1WCA Wilke-Chang-Andrade
SIGLMX SIGLMX01 SIG2HSS Hakim-Steinberg-Stiel, Power Law Mixing
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 HL0DVK*, GL0DVK van Krevelen, DIPPR
VL VLDVK VL0DVK, VL0RKT van Krevelen, Rackett
DL DL01 DL0WCA Wilke-Chang-Andrade
MUL MULMH MUL0MH Modified Mark-Houwink/van Krevelen, Andrade
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
Route ID Model Name Description
HSMX HSMXDVK HS0DVK Ideal mixing, van Krevelen
GSMX GSMXDVK GS0DVK Ideal mixing, van Krevelen
SSMX SSMXDVK HS0DVK, GS0DVK
Ideal mixing, van Krevelen
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A Physical Property Methods 191
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
van Krevelen, Polynomial
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
Route ID Model Name Description
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
MUVMX MUVMX02 MUV2DNST Dean-Stiel
KVMX KVMX02 KV2STLTH Stiel-Thodos
DVMX DVMX02 DV1DKK Dawson-Khoury-Kobayashi
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
DV DV01 DV0CEWL Chapman-Enskog-Wilke-Lee
MUV MUV01 MUV0BROK Chapman-Enskog-Brokaw
KV KV01 KV0STLTH Stiel-Thodos
Liquid
Property Name
Route ID Model Name Description
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192 A Physical Property Methods
Liquid
Property Name
Route ID Model Name Description
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
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
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
GL GLDVK GL0DVK van Krevelen, DIPPR
SL SLDVK HL0DVK*, GL0DVK van Krevelen, DIPPR
VL VLDVK VL0DVK, VL0RKT van Krevelen, Rackett
DL DL01 DL0WCA Wilke-Chang-Andrade
MUL MULMH MUL0MH Modified Mark-Houwink/van Krevelen, Andrade
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
Route ID Model Name Description
HSMX HSMXDVK HS0DVK Ideal mixing, van Krevelen,
GSMX GSMXDVK GS0DVK Ideal mixing, van Krevelen
SSMX SSMXDVK HS0DVK, GS0DVK Ideal mixing, van Krevelen
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A Physical Property Methods 193
Solid
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 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
Route ID Model Name Description
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
MUVMX MUVMX02 MUV2DNST Dean-Stiel
KVMX KVMX02 KV2STLTH Stiel-Thodos
DVMX DVMX02 DV1DKK Dawson-Khoury-Kobayashi
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
DV DV01 DV0CEWL Chapman-Enskog-Wilke-Lee
MUV MUV01 MUV0BROK Chapman-Enskog-Brokaw
KV KV01 KV0STLTH Stiel-Thodos
Liquid
Property Name
Route ID Model Name Description
PHILMX PHILMXSL ESPLSL Sanchez-Lacombe
HLMX HLMXSL ESPLSL Sanchez-Lacombe
GLMX GLMXSL ESPLSL Sanchez-Lacombe
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194 A Physical Property Methods
Liquid
Property Name
Route ID Model Name Description
SLMX SLMXSL ESPLSL Sanchez-Lacombe
VLMX VLMXSL ESPLSL Sanchez-Lacombe
MULMX MULMX13 MUPOLY, MULMH
Aspen, Modified Mark-Houwink/van Krevelen, Andrade
KLMX KLMXVKTR KLMXVKTR Vredeveld mixing , Modified van Krevelen, TRAPP
DLMX DLMX02 DL1WCA Wilke-Chang-Andrade
SIGLMX SIGLMX01 SIG2HSS Hakim-Steinberg-Stiel, Power Law Mixing
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
DL DL01 DL0WCA Wilke-Chang-Andrade
MUL MULMH MUL0MH Modified Mark-Houwink/van Krevelen, Andrade
KL KL0VKTR KL0VKTR Modified van Krevelen, TRAPP
Solid
Property Name
Route ID Model Name Description
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
van Krevelen, Polynomial
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A Physical Property Methods 195
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
Route ID Model Name Description
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
MUVMX MUVMX02 MUV2DNST Dean-Stiel
KVMX KVMX02 KV2STLTH Stiel-Thodos
DVMX DVMX02 DV1DKK Dawson-Khoury-Kobayashi
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
DV DV01 DV0CEWL Chapman-Enskog-Wilke-Lee
MUV MUV01 MUV0BROK Chapman-Enskog-Brokaw
KV KV01 KV0STLTH Stiel-Thodos
Liquid
Property Name
Route ID Model Name Description
PHILMX PHILMXPS ESPLRKS Polymer SRK
HLMX HLMXPS ESPLRKS Polymer SRK
GLMX GLMXPS ESPLRKS Polymer SRK
SLMX SLMXPS ESPLRKS Polymer SRK
VLMX VLMXVKRK VL2VKRK Ideal mixing, van Krevelen, Rackett
MULMX MULMX13 MUPOLY, MULMH
Aspen, Modified Mark-Houwink/van Krevelen, Andrade
KLMX KLMXVKTR KLMXVKTR Vredeveld mixing, Modified van Krevelen, TRAPP
DLMX DLMX02 DL1WCA Wilke-Chang-Andrade
SIGLMX SIGLMX01 SIG2HSS Hakim-Steinberg-Stiel, Power Law Mixing
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196 A Physical Property Methods
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
DL DL01 DL0WCA Wilke-Chang-Andrade
MUL MULMH MUL0MH Modified Mark-Houwink/van Krevelen, Andrade
KL KL0VKTR KL0VKTR Modified van Krevelen, TRAPP
Solid
Property Name
Route ID Model Name Description
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 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
Route ID Model Name Description
PHIVMX PHIVMXSF ESPLSFT SAFT
HVMX HVMXSF ESPLSFT SAFT
GVMX GVMXSF ESPLSFT SAFT
SVMX SVMXSF ESPLSFT SAFT
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A Physical Property Methods 197
VVMX VVMXSF ESPLSFT SAFT
MUVMX MUVMX02 MUV2DNST Dean-Stiel
KVMX KVMX02 KV2STLTH Stiel-Thodos
DVMX DVMX02 DV1DKK Dawson-Khoury-Kobayashi
PHIV PHIVSF ESPLSFT0 SAFT
HV HVSF ESPLSFT0 SAFT
GV GVSF ESPLSFT0 SAFT
SV SVSF ESPLSFT0 SAFT
VV VVSF ESPLSFT0 SAFT
DV DV01 DV0CEWL Chapman-Enskog-Wilke-Lee
MUV MUV01 MUV0BROK Chapman-Enskog-Brokaw
KV KV01 KV0STLTH Stiel-Thodos
Liquid
Property Name
Route ID Model Name Description
PHILMX PHILMXSF ESPLSFT SAFT
HLMX HLMXSF ESPLSFT SAFT
GLMX GLMXSF ESPLSFT SAFT
SLMX SLMXSF ESPLSFT SAFT
VLMX VLMXSF ESPLSFT SAFT
MULMX MULMX13 MUPOLY, MULMH
Aspen, Modified Mark-Houwink/ van Krevelen, Andrade
KLMX KLMXVKTR KLMXVKTR Vredeveld mixing, Modified van Krevelen, TRAPP
DLMX DLMX02 DL1WCA Wilke-Chang-Andrade
SIGLMX SIGLMX01 SIG2HSS Hakim-Steinberg-Stiel, Power Law Mixing
PHIL PHILSF ESPLSFT0 SAFT
HL HLSF ESPLSFT0 SAFT
GL GLSF ESPLSFT0 SAFT
SL SLSF ESPLSFT0 SAFT
VL VLSF ESPLSFT0 SAFT
DL DL01 DL0WCA Wilke-Chang-Andrade
MUL MULMH MUL0MH Modified Mark-Houwink/van Krevelen, Andrade
KL KL0VKTR KL0VKTR Modified van Krevelen, TRAPP
Solid
Property Name
Route ID Model Name Description
PHILMX PHILMXSF ESPLSFT SAFT
HLMX HLMXSF ESPLSFT SAFT
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198 A Physical Property Methods
GLMX GLMXSF ESPLSFT SAFT
SLMX SLMXSF ESPLSFT SAFT
VLMX VLMXSF ESPLSFT SAFT
MULMX MULMX13 MUPOLY, MULMH
Aspen, Modified Mark-Houwink/ van Krevelen, Andrade
KLMX KLMXVKTR KLMXVKTR Vredeveld mixing, Modified van Krevelen, TRAPP
DLMX DLMX02 DL1WCA Wilke-Chang-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
Route ID Model Name Description
PHIVMX PHIVMXPC ESPCSFT PCSAFT
HVMX HVMXPC ESPCSFT PCSAFT
GVMX GVMXPC ESPCSFT PCSAFT
SVMX SVMXPC ESPCSFT PCSAFT
VVMX VVMXPC ESPCSFT PCSAFT
MUVMX MUVMX02 MUV2DNST Dean-Stiel
KVMX KVMX02 KV2STLTH Stiel-Thodos
DVMX DVMX02 DV1DKK Dawson-Khoury-Kobayashi
PHIV PHIVPC ESPCSFT0 PCSAFT
HV HVPC ESPCSFT0 PCSAFT
GV GVPC ESPCSFT0 PCSAFT
SV SVPC ESPCSFT0 PCSAFT
VV VVPC ESPCSFT0 PCSAFT
DV DV01 DV0CEWL Chapman-Enskog-Wilke-Lee
MUV MUV01 MUV0BROK Chapman-Enskog-Brokaw
KV KV01 KV0STLTH Stiel-Thodos
Liquid
Property Name
Route ID Model Name Description
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A Physical Property Methods 199
Liquid
Property Name
Route ID Model Name Description
PHILMX PHILMXPC ESPCSFT PCSAFT
HLMX HLMXPC ESPCSFT PCSAFT
GLMX GLMXPC ESPCSFT PCSAFT
SLMX SLMXPC ESPCSFT PCSAFT
VLMX VLMXPC ESPCSFT PCSAFT
MULMX MULMX13 MUPOLY, MULMH
Aspen, Modified Mark-Houwink/van Krevelen, Andrade
KLMX KLMXVKTR KLMXVKTR Vredeveld mixing, Modified van Krevelen, TRAPP
DLMX DLMX02 DL1WCA Wilke-Chang-Andrade
SIGLMX SIGLMX01 SIG2HSS Hakim-Steinberg-Stiel, Power Law Mixing
PHIL PHILPC ESPCSFT0 PCSAFT
HL HLPC ESPCSFT0 PCSAFT
GL GLPC ESPCSFT0 PCSAFT
SL SLPC ESPCSFT0 PCSAFT
VL VLPC ESPCSFT0 PCSAFT
DL DL01 DL0WCA Wilke-Chang-Andrade
MUL MULMH MUL0MH Modified Mark-Houwink/van Krevelen, Andrade
KL KL0VKTR KL0VKTR Modified van Krevelen, TRAPP
Solid
Property Name
Route ID Model Name Description
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
van Krevelen, Polynomial
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200 A Physical Property Methods
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
Route ID Model Name Description
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
MUVMX MUVMX02 MUV2DNST Dean-Stiel
KVMX KVMX02 KV2STLTH Stiel-Thodos
DVMX DVMX02 DV1DKK Dawson-Khoury-Kobayashi
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
DV DV01 DV0CEWL Chapman-Enskog-Wilke-Lee
MUV MUV01 MUV0BROK Chapman-Enskog-Brokaw
KV KV01 KV0STLTH Stiel-Thodos
Liquid
Property Name
Route ID Model Name Description
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
MULMX MULMX13 MUPOLY, MULMH
Aspen, Modified Mark-Houwink/van Krevelen, Andrade
KLMX KLMXVKTR KLMXVKTR Vredeveld mixing, Modified van Krevelen, TRAPP
DLMX DLMX02 DL1WCA Wilke-Chang-Andrade
SIGLMX SIGLMX01 SIG2HSS Hakim-Steinberg-Stiel, Power Law Mixing
PHIL PHILPA ESPSAFT0 Copolymer PCSAFT
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A Physical Property Methods 201
Liquid
Property Name
Route ID Model Name Description
HL HLPA ESPSAFT0 Copolymer PCSAFT
GL GLPA ESPSAFT0 Copolymer PCSAFT
SL SLPA ESPSAFT0 Copolymer PCSAFT
VL VLPA ESPSAFT0 Copolymer PCSAFT
DL DL01 DL0WCA Wilke-Chang-Andrade
MUL MULMH MUL0MH Modified Mark-Houwink/van Krevelen, Andrade
KL KL0VKTR KL0VKTR Modified van Krevelen, TRAPP
Solid
Property Name
Route ID Model Name Description
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
van Krevelen, Polynomial
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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
refT = Reference temperature (298.15 K)
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
Functional Group Parameters on page 205
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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204 B Van Krevelen Functional Groups
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
kM = Molecular weight of functional group k
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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B Van Krevelen Functional Groups 205
)( refk TUR = Rao wave function of functional group k in Van Krevelen
Functional Group Parameters on page 205
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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206 B Van Krevelen Functional Groups
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
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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B Van Krevelen Functional Groups 207
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
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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208 B Van Krevelen Functional Groups
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:
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
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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B Van Krevelen Functional Groups 209
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
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:
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)
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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210 B Van Krevelen Functional Groups
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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B Van Krevelen Functional Groups 211
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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212 B Van Krevelen Functional Groups
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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B Van Krevelen Functional Groups 213
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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214 B Van Krevelen Functional Groups
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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220 E Equation-of-State Parameters
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
Formula Component T range, K
v , cm / moloo 3 m u / k, Ko
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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222 E Equation-of-State Parameters
Formula Component T range, K
v , cm / moloo 3 m u / k, Ko
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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F Input Language Reference 225
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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F Input Language Reference 227
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
parameters for property models 21
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
119 specifying model 117
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–
94 specifying model 42
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
parameters for property models 20
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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