AspenCOMThermoV7 3 Ref

8/11/2019 AspenCOMThermoV7 3 Ref

1/358

Aspen HYSYS Thermodynamics COMInterface

Reference Guide


2/358

Version Number: V7.3March 2011

Copyright (c) 1981-2011 by Aspen Technology, Inc. All rights reserved.

Aspen HYSYS Thermodynamics COM Interface, Aspen HYSYS and the aspen leaf logo are registeredtrademarks of Aspen Technology, Inc., Burlington, MA.

All other brand and product names are trademarks or registered trademarks of their respectivecompanies.

This manual is intended as a guide to using AspenTechs software. This documentation containsAspenTech proprietary and confidential information and may not be disclosed, used, or copied withoutthe prior consent of AspenTech or as set forth in the applicable license agreement. Users are solelyresponsible 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 thesoftware 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.

Aspen Technology, Inc.200 Wheeler RoadBurlington, MA 01803-5501USAPhone: (781) 221-6400Website http://www.aspentech.com
http://www.aspentech.com/http://www.aspentech.com/


3/358

13

13

Technical Support

Online Technical Support Center....................................................... 14

Phone and E-mail .............................................................................. 15


4/358

14

14

Online Technical Support

CenterAspenTech customers with a valid license and softwaremaintenance agreement can register to access the OnlineTechnical Support Center at:

http://support.aspentech.com

You use the Online Technical Support Center to:

Access current product documentation. Search for technical 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. Search for and review known limitations. Send suggestions.

Registered users can also subscribe to our Technical Support e-Bulletins. These e-Bulletins proactively alert you to importanttechnical support information such as:

Technical advisories Product updates

Service Pack announcements Product release announcements
http://support.aspentech.com/http://support.aspentech.com/


5/358

15

15

Phone and E-mailCustomer support is also available by phone, fax, and e-mail forcustomers who have a current support contract for theirproduct(s). Toll-free charges are listed where available;otherwise local and international rates apply.

For the most up-to-date phone listings, please see the OnlineTechnical Support Center at:

http://support.aspentech.com

Support Centers Operating Hours

North America 8:00 - 20:00 Eastern time

South America 9:00 - 17:00 Local timeEurope 8:30 - 18:00 Central European time

Asia and Pacific Region 9:00 - 17:30 Local time
http://support.aspentech.com/http://support.aspentech.com/


6/358

16

16


7/358


8/358

iv

4.3 COSTALD Method............................................4-11

4.4 Viscosity ........................................................4-14

4.5 Thermal Conductivity.......................................4-18

4.6 Surface Tension ..............................................4-21

4.7 Insoluble Solids ..............................................4-22

5 References & Standard States ..............................5-1

5.1 Enthalpy Reference States................................. 5-2

5.2 Entropy Reference States .................................. 5-4

5.3 Ideal Gas Cp ................................................... 5-5

5.4 Standard State Fugacity.................................... 5-6

6 Flash Calculations.................................................6-1

6.1 Introduction .................................................... 6-2

6.2 T-P Flash Calculation ........................................ 6-3

6.3 Vapour Fraction Flash ....................................... 6-4

6.4 Flash Control Settings....................................... 6-7

7 Property Packages................................................7-1

7.1 Introduction .................................................... 7-2

7.2 Vapour Phase Models........................................ 7-2

7.3 Liquid Phase Models ........................................7-13

8 Utilities................................................................. 8-1

8.1 Introduction .................................................... 8-2

8.2 Envelope Utility................................................ 8-2

9 References ...........................................................9-1

Index.................................................................... I-1


9/358

Introducing Aspen HYSYS Thermodynamics COM Interface 1-1

1-1

1 Introducing Aspen HYSYS

Thermodynamics COM Interface

1.1 Introduction................................................................................... 2


10/358

1-2 Introduction

1-2

1.1 IntroductionThe use of process simulation has expanded from its origins inengineering design to applications in real time optimization,dynamic simulation and control studies, performancemonitoring, operator training systems and others. At everystage of the lifecycle there is a need for consistent results suchthat the modeling efforts can be leveraged in those manyapplications.

Accurate thermophysical properties of fluids are essential fordesign and operation in the chemical process industries. Theneed of having a good thermophysical model is widelyrecognized in this context. All process models rely on physical

properties to represent the behavior of unit operations, and thetransformations that process streams undergo in a process.Properties are calculated from models created and fine-tuned tomimic the behaviour of the process substances at the operatingconditions

Aspen HYSYS Thermodynamics COM Interface is a completethermodynamics package that encompasses property methods,flash calculations, property databases, and property estimation.The package is fully componentized, and therefore fullyextensible to the level of detail that allows the user to utilize,supplement, or replace any of the components. The objective ofthis package is to improve the engineering workflow byproviding an open structure that can be used in many differentsoftware applications and obtain consistent results.


11/358

Introducing Aspen HYSYS Thermodynamics COM

1-3

The main benefit of Aspen HYSYS Thermodynamics COMInterface is delivered via consistent and rigorousthermodynamic calculations across engineering applications.

Aspen HYSYS Thermodynamics COM Interface enables theprovision of specialized thermodynamic capabilities to theHYSYS Environment and to other third party applicationsincluding internal legacy tools. It also allows the user to supportdevelopment of internal thermo capabilities. Aspen HYSYSThermodynamics COM Interface is written to specifically supportthermodynamics.

The Aspen HYSYS Thermodynamics COM Interface referenceguide details information on relevant equations, models, and thethermodynamic calculation engine. The calculation engine

encompasses a wide variety of thermodynamic propertycalculations, flash methods, and databases used in the AspenHYSYS Thermodynamics COM Interface framework.


12/358

1-4 Introduction

1-4


13/358

Thermodynamic Principles 2-1

2-1

2 ThermodynamicPrinciples

2.1 Introduction................................................................................... 3

2.2 Chemical Potential & Fugacity........................................................ 8

2.3 Chemical Potential for Ideal Gas .................................................... 9

2.4 Chemical Potential & Fugacity for a Real Gas ............................... 11

2.5 Fugacity & Activity Coefficients.................................................... 12

2.6 Henrys Law................................................................................. 15

2.6.1 Non-Condensable Components................................................. 172.6.2 Estimation of Henrys constants................................................ 18

2.7 Gibbs-Duhem Equation ................................................................ 19

2.7.1 Simplifications on Liquid Fugacity using Activity Coeff.................. 21

2.8 Association in Vapour Phase - Ideal Gas ...................................... 242.9 Equilibrium Calculations............................................................... 28

2.10 Basic Models for VLE & LLE ........................................................ 30

2.10.1 Symmetric Phase Representation............................................ 302.10.2 Asymmetric Phase Representation .......................................... 302.10.3 Interaction Parameters .......................................................... 312.10.4 Selecting Property Methods.................................................... 322.10.5 Vapour Phase Options for Activity Models................................. 35

2.11 Phase Stability ........................................................................... 37

2.11.1 Gibbs Free Energy for Binary Systems..................................... 38


14/358

2-2

2-2

2.12 Enthalpy/Cp Departure Functions...............................................42

2.12.1 Alternative Formulation for Low Pressure Systems .....................47


15/358

Thermodynamic Principles

2-3

2.1 IntroductionTo determine the actual state of a mixture defined by itscomponents and two intensive variables (usually pressure andtemperature), a unique set of conditions and equations definingequilibrium is required.

Consider a closed, multi-component and multi-phase systemwhose phases are in thermal, mechanical, and mass transferequilibrium. At this state, the internal energy of the system is ata minimum, and any variation in U at constant entropy andvolume vanishes ( 1Prausnitz et al, 1986):

The total differential for the internal energy is:

where: j = Phase (from 1 to )

i = Component (from 1 to nc)

i j = Chemical potential of component i in phase j, defined as

(2.1)

(2.2)

(2.3)

(2.4)

dU TdS PdV =

dU ( )S V , 0=

dU T j S d j

P j

V d j

i jdn i

j

i 1=

nc

j 1=

+ j 1=

j 1=

=

i j

ni

U

S V nk 1 j

, ,=


16/358

2-4 Introduction

2-4

Since the system is closed, the differentials in number of moles,volume and entropy are not all independent, but are insteadconstrained as follows:

Therefore, a system of equations with ( nc +2) variables and nc + 2 constraints ( Equations (2.5) , (2.6) and (2.7) ) is defined.The constraints can be used to eliminate some variables andreduce the system to a set of ( - 1)( nc + 2) independentequations.

(2.5)

(2.6)

(2.7)

dS S d j

j 1=

0= =

dV V d j

j 1=

0= =

dn i j

0= j 1=

i 1, ..., nc=


17/358


2-5

The variables can be eliminated in the following way:

(2.8)

(2.9)

(2.10)

dS 1

S d j

j 2=

=

dV 1 V d j

j 2=

=

dn i1 dn i

j

j 2=

=


18/358

2-6 Introduction

2-6

The result is as follows:

where: the differentials on the right side of Equation (2.11) areindependent.

Setting all of the independent variables constant except one, atequilibrium you have:

Therefore:

Repeating the same argument for all of the independentvariables, the general conditions necessary for thermodynamicequilibrium between heterogeneous phases are established (forall i ):

From now on, it is assumed that Equations (2.14) and (2.15) are always satisfied. The equilibrium condition established inEquation (2.16) will be discussed in more detail. Note that thedescription of equilibrium according to Equations (2.13) ,(2.14) , (2.15) , and (2.16) is at best incomplete, since otherintensive variables could be important in the process beinganalysed. For example, the electric or magnetic fields in the

(2.11)

(2.12)

(2.13)

T 1 = T 2 =...=T

Thermal Equilibrium - no heat flux across phases (2.14)

P 1 = P 2 =...=P

Mechanical Equilibrium - no phase displacement (2.15)

i1 = i2 =...=i

Mass Transfer Equilibrium - no mass transfer forcomponent i between phases (2.16)

dU T j T 1 ( ) S d j

P j P 1 ( ) V d j

i j i1 ( )dn i ji 1=

nc

j 1>

+ j 1>

j 1>

=

U S ------- 0=

U V ------- 0=

U n i

------- 0= U 2S 2

--------- 0=

T 1 T j= j 2, ..., =


19/358


2-7

equations, or area affects are not being considered.


20/358

2-8 Chemical Potential & Fugacity

2-8

Nevertheless, Equations (2.13) , (2.14) , (2.15) and (2.16) are important in chemical engineering thermodynamiccalculations, and will be assumed to always apply. Provided that

a reasonable physical model is available for the propertycalculations, virtually all chemical engineering problems thatinvolve phase equilibria can be represented by the aboveequations and constraints.

The following will relate the chemical potential in Equation(2.16) with measurable system properties.

2.2 Chemical Potential &

FugacityThe concept of chemical potential was introduced by J. WillardGibbs to explain phase and chemical equilibria. Since chemicalpotential cannot be directly related with any directly measuredproperty, G.N. Lewis introduced the concept of fugacity in 1902.Using a series of elegant transformations, Lewis found a way tochange the representation using chemical potential byrepresenting the equilibrium conditions with an equivalentproperty directly related to composition, temperature andpressure. He named this property "fugacity." It can be seen as a"thermodynamic pressure" or, in simpler terms, the effective

partial pressure that one compound exerts in a mixture.


21/358


2-9

2.3 Chemical Potential for

Ideal GasYou start by finding an equivalent to Equation (2.5) whichallows us to work with a better set of independent variables,namely pressure and temperature. This brings us to the Gibbsfree energy, which is expressed as a function of P and T :

where:

The chemical potential is the partial molar Gibbs free energy,since partial molar properties are defined at constant P and T .Note that the chemical potential is not the partial molar internalenergy, enthalpy or Helmholtz energy. Since a partial molarproperty is used, the following holds:

(2.17)

(2.18)

(2.19)

dG SdT Vd P i n

id

i 1=

nc

+ +=

i n iG

T P n k 1, ,=

dG i S idT V idP +=


22/358

2-10 Chemical Potential for Ideal Gas

2-10

where:

Now assuming the system to be at constant temperature:

(2.20)

(2.21)

G iGn i-------

T P n k 1, ,=

d i dG i V idP = =


23/358


2-11

2.4 Chemical Potential &

Fugacity for a Real GasAlthough Equation (2.21) has only limited interest, i ts basicform can still be used. Pressure, P , can be replaced by anotherthermodynamic quantity which represents a real gas. Thisquantity is called fugacity, and it is a function of pressure,temperature and composition:

It is interesting to note that the combination of Equations(2.22) and (2.16) results in a simple set of equations for themulti-phase, multi-component phase equilibria:

Assuming again that the system is at constant temperature,Equations (2.21) and (2.22) can be combined, resulting in aworking definition for fugacity:

In principle, if the behaviour of the partial molar volume isknown, the fugacity can be computed, and the phase equilibriais defined. In reality, the engineering solution to this problemlies in the creation of models for the fluids equation of statefrom those models, the fugacity is calculated.

(2.22)

(2.23)

(2.24)

i C i RT f iln+=

f i1

f i2 f i

= = =

P f iln( ) T

V i RT -------=


24/358

2-12 Fugacity & Activity Coefficients

2-12

2.5 Fugacity & Activity

CoefficientsWriting the fugacity expressions for a real and ideal gas:

Subtracting and rearranging Equation (2.26) from Equation

(2.25) yields:

You integrate from 0 to P, noting that the behaviour of any realgas approaches the behaviour of an ideal gas at sufficiently lowpressures (the limit of f/P as P 0 = 1):

Using the definition of compressibility factor ( PV = ZRT ),Equation (2.28) can be expressed in a more familiar format:

(2.25)

(2.26)

(2.27)

(2.28)

(2.29)

RT d f ln Vd P =

RTd P ln V ideal

dP =

RTd f P ---ln V V ideal ( )dP =

f P ---ln V

RT ------- V

RT -------

ideal

0

P

dP =

f P ---ln Z 1 ( )

P -----------------

0

P

dP =


25/358


2-13

The ratio f/P measures the deviation of a real gas from ideal gasbehaviour, and is called the fugacity coefficient:

These results are easily generalized for multi-componentmixtures:

The partial molar compressibility factor is calculated:

substituting Equation (2.32) into Equation (2.31) andrearranging:

The quantity f i / Px i measures the deviation behaviour ofcomponent i in a mixture as a real gas from the behaviour of anideal gas, and is called the fugacity coefficient of component i inthe mixture:

(2.30)

(2.31)

(2.32)

(2.33)

(2.34)

f P ---=

f i Px i--------ln

Z i 1 ( ) P

------------------0

P

dP =

Z i n i Z

T P n k i j, ,

P RT -------

n iV

T P n k i j, ,

PV i RT ---------= = =

f i Px i--------ln 1

RT ------- V i

RT P -------

0

P

dP =

i f i

Px i--------=


26/358

2-14 Fugacity & Activity Coefficients

2-14

For mixtures in the liquid state, an ideal mixing condition can bedefined. Usually this is done using the Lewis-Randall concept ofideal solution, in which an ideal solution is defined as:

where: f i ,pure refers to the fugacity of pure component i in the vapour

or liquid phase, at the mixture pressure andtemperature.

The definition used by Lewis and Randall defines an idealsolution, not the ideal gas behaviour for the fugacities.Therefore, the fugacities of each pure component may be givenby an accurate equation of state, while the mixture assumesthat different molecules do not interact. Although very fewmixtures actually obey ideal solution behaviour, approximateequilibrium charts (nomographs) using the Lewis-Randall rulewere calculated in the 1940s and 50s, and were successfullyused in the design of hydrocarbon distillation towers.

Generalizing Equation (2.36) for an arbitrary standard state,the activity coefficient for component i can written as:

It is important to properly define the normalization condition(the way in which ideal solution behaviour is defined (i.e., whenthe activity coefficient approaches one), so that supercriticalcomponents are handled correctly, and the Gibbs-Duhemequation is satisfied.

(2.35)

(2.36)

(2.37)

f iV yi f i

V pure,=

f i L xi f i

L pu re,=

i f i

L

f i L pu re,

xi-------------------- -=


27/358


2-15

2.6 Henrys LawThe normalized condition is the state where the activitycoefficient is equal to 1. For ordinary mixtures of condensablecomponents (i.e., components at a temperature below thecritical temperature), the normalization condition is defined as( 2Prausnitz et al, 1980):

However, the definition does not apply for components that

cannot exist as pure liquids at the conditions of the system.Sometimes, for components like carbon dioxide at near ambientconditions, a reasonably correct hypothetical liquid fugacity canbe extrapolated. But for components like hydrogen andnitrogen, this extrapolated liquid behaviour has little physicalsignificance.

For solutions of light gases in condensable solvents, a differentnormalization convention can be defined than the (other thanthe one in Equation (2.38) ):

(2.38)

(2.39)

f i L

f i L pu re, xi

-------------------- - xi 1lim i

xi 1lim 1= =

f i L

f ire f xi

------------ xi 0lim i

xi 0lim 1= =


28/358

2-16 Henrys Law

2-16

This equation implies that the fugacity of component i in a multi-component mixture approaches the product of the mole fractionand standard state fugacity in very dilute solutions of

component i. Using the definition of i* it can be shown that:

where: H ij is called Henrys Constant of component i in solvent j.

Therefore, the Henrys constant is the standard state fugacityfor a non-condensable component in a specific solvent. Usuallythe Henrys constant is a rather strong function of temperature,but a weak function of the pressure and composition. Theextension of Henrys law into more concentrated solutions andat higher pressures is represented by the Kritchevsky-Ilinskayaequation:

where: P j sat = Solvent saturation pressure at mixture temperature

H ij sat = Henrys law calculated at the saturation pressure of

the solvent

A ij = Margules interaction parameter for molecularinteractions between the solute and solvent

= Infinite dilution partial molar volume of solute i insolvent j

(2.40)

(2.41)

f ire f f i

L

xi----

xi 0lim H ij= =

H ijln H ij P j

Sat A ij RT ------- x j

2 1 ( ) V i

P P j

sa t ( )

RT --------------------------------+ +ln=

V i


29/358


2-17

2.6.1 Non-Condensable

ComponentsNon-condensable components are treated using Henrysconstants as defined by Equation (2.40) . The temperaturedependency of the Henrys law for a binary pair ij is representedby an Antoine-type of equation with four parameters per binarypair:

A mixing rule for the Henrys constant of a non-condensablecomponent in a mixture of condensable components must bedefined. There are several alternatives, but the followingformulation works reasonably well:

(2.42)

(2.43)

H ij

ln Aij

Bij

T ------ C

ij T ln D

ijT + + +=

The Henrys constant ofcomponent i in a multi-component mixture isestimated neglecting thesolvent-solventinteractions.

H i mixture,ln

H ij x

jV

c j,

3---

ln j 1 j i,=

nc

x jV c j,

23---

j 1 j i,=

nc

-----------------------------------------------=


30/358

2-18 Henrys Law

2-18

2.6.2 Estimation of Henrys

constantsA rigorous estimation of gas solubilities in condensable solventsdepends on the existence of a rigorous theory of solutions,which currently does not exist. On the other hand,corresponding states and regular solution theory give us acorrelative tool which allows us to estimate gas solubilities. Theuse of regular solution theory assumes that there is no volumechange on mixing. Therefore consider a process in which thepure gas, i , is condensed to a liquid-like state, corresponding tothe partial molar volume of the gas in the solvent. At this point,

liquid gas is dissolved in the solvent (Prausnitz et al, 1986):

Since the gas dissolved in the liquid is in equilibrium with thegas in the gas phase:

and therefore:

Using regular solution theory to estimate the activity coefficientof the gas in the solvent:

(2.44)

(2.45)

(2.46)

(2.47)

(2.48)

(2.49)

g g I g II +=

g I RT f i

L pu re,

f iG

----------------ln=

g II RT i xiln=

f iG i xi f i

L pu re,=

g 0=

RT iln i L j i ( )

2 j2=


31/358


2-19

and finally the expression for the Henrys constant is:

Since regular solution theory assumes that the activitycoefficient is proportional to the inverse of temperature, theterm i L( j - i ) j 2 is temperature independent, and anyconvenient temperature (usually 25 oC) can be used for thecalculation of iL, j L, i, and j . Note also that as a firstapproximation, j is very close to 1, and Equation (2.50) simplifies to:

This is the equation used when estimating Henrys constants.The Henrys constants are calculated constants at 270.0, 300.0,330.0, 360.0, 390.0, 420.0, 450.0 K and fits the results usingEquation (2.42) , for each non-condensable/condensable pairpresent in the fluid package.

2.7 Gibbs-DuhemEquation

At constant temperature and pressure, the Gibbs-Duhem

(2.50)

(2.51)

The interaction between two non-condensable components

are not taken into account.

H ij f iG

xi----- f i

L pu re, i L

j i ( )2

j2

RT ----------------------------------exp= =

H ij f i

G

xi----- f i

L pu re, i L i j ( )

2

RT ----------------------------exp= =


32/358

2-20 Gibbs-Duhem Equation

2-20

equation states that:

This equation applies to condensable and non-condensablecomponents and only when integrating the Gibbs-Duhemequation should normalization conditions be considered. A more

general form of the Gibbs-Duhem is also available, which isapplicable to non-isothermal and non-isobaric cases. Theseforms are difficult to integrate, and do little to help in thedefinition of the standard state fugacity.

If the liquid composition of a binary mixture was varied from x i = 0 to x i = 1 at constant temperature, the equilibrium pressurewould change. Therefore, if the integrated form of Equation(2.52) is used to correlate isothermal activity coefficients, all ofthe activity coefficients will have to be corrected to someconstant reference pressure. This is easily done if thedependency of fugacity on pressure is known:

Now if the fugacity equation is written using activity coefficients:

The definition of the standard state fugacity now comes directly

(2.52)

(2.53)

(2.54)

xid iln 0=i 1=

nc

i P ref i

P V i RT ------- P d

P

P ref

exp=

f i L i

P xi f i

re f = or f i

L i P ref

xi f ire f V i

RT ------- P d

P ref

P

exp=


33/358


2-21

from the Gibbs-Duhem equation using the normalizationcondition for a condensable component; i.e., f i

ref is the fugacityof component i in the pure state at the mixture temperature and

reference pressure preference. The standard state fugacity canbe conveniently represented as a departure from the saturatedconditions:

Combining Equations (2.54) and (2.55) :

This equation is the basis for practically all low to moderatepressure engineering equilibrium calculations using activitycoefficients. The exponential correction in Equations (2.54) and (2.55) is often called the Poynting correction , and takesinto account the fact that the liquid is at a different pressurethan the saturation pressure. The Poynting correction at low tomoderate pressures is very close to unity.

2.7.1 Simplifications on LiquidFugacity using ActivityCoeff

There are many traditional simplifications involving Equation(2.56) which are currently used in engineering applications.

(2.55)

(2.56)

f ire f P i

va p i sa t V i

RT ------- P d

P ivap

P ref

exp=

f i L

P iva p i

sa t V i RT -------

V i RT -------+ P d

P ivap

P ref

exp=


34/358

2-22 Gibbs-Duhem Equation

2-22

Ideal GasWhen ideal gas behaviour is assumed, this usually implies thatthe Poynting correction is dropped. Also, since the gas is ideal,isat = 1:

Low Pressures & Conditions Awayfrom the Critical PointFor conditions away from the critical point and at low tomoderate pressures, the activity coefficients are virtuallyindependent of pressure. For these conditions, it is common toset P ref = P i

vap giving us the familiar equation:

It is common to assume that the partial molar volume isapproximately the same as the molar volume of the pure liquid i at P and T , and equation simplifies even further:

Since fluids are usually incompressible at conditions removedfrom the critical point, V i can be considered constant and the

(2.57)

(2.58)

(2.59)

(2.60)

f i L i xi P i

va p=

f ire f P i

va p=

f i L i xi P i

va p i sa t V i

RT ------- P d

P ivap

P

exp=

f i L i xi P i

va p i sa t V i

RT ------- P d

P ivap

P

exp=


35/358


2-23

integration of Equation (2.60) leads to:

(2.61)

(2.62)

f i L i xi P i

va p i sa t V i P P iva p

( ) RT

-------------------------------exp=

f ire f P i

va p i sa t V i P P i

va p ( ) RT

-------------------------------exp=


36/358

2-24 Association in Vapour Phase - Ideal Gas

2-24

This is the equation used when taking into account vapour phasenon-ideality. Sometimes, Equation (2.60) is simplified evenfurther, assuming that the Poynting correction is equal to 1:

Equations (2.57) , (2.60) and (2.61) form the basis used toname several of the activity coefficient based propertypackages.

2.8 Association in VapourPhase - Ideal GasFor some types of mixtures (especially carboxylic acids), there isa strong tendency for association in the vapour phase, wherethe associating component can dimerize, forming a reasonablystable associated component. Effectively, a simple chemicalreaction in the vapour phase takes place, and even at modestpressures a strong deviation from the vapour phase behaviourpredicted by the ideal gas law may be observed. This happens

because an additional component is present in the mixture(Walas, 1985).

where: A is the associating component in the mixture (assumedbinary for simplicity).

the equilibrium constant for the chemical reaction can be writtenas:

(2.63)

(2.64)

(2.65)

(2.66)

f i L i xi P i

va p i sa t =

f ire f P i

va p i sa t =

2 A A2

K A2[ ] A[ ]2-----------=


37/358


2-25

Assuming that the species in the vapour phase behave like idealgases:

where: P d is the dimer partial pressure

P m is the monomer partial pressure

At equilibrium, the molar concentrations of monomer and dimerare:

where: e is the extent of dimerization

The expression for the dimerization extent in terms of theequilibrium constant can be written as follows:

Solving for e the following:

(2.67)

(2.68)

(2.69)

(2.70)

(2.71)

K P d [ ]

P m[ ]2

--------------=

ym

2 2 e

2 e ------------=

yd e

2 e -----------=

K P d

P m2

------- P A

va p yd

P Ava p ym( )

2------------------------- e 2 e ( )

2 2 e ( )2 P Ava p---------------------------------- e 2 e ( )

4 P Ava p 1 e ( )2

----------------------------------= = = =

e 1

1 4 KP Ava p

+----------------------------=


38/358

2-26 Association in Vapour Phase - Ideal Gas

2-26

The vapour pressure of the associating substance at a giventemperature is the sum of the monomer and dimer partialpressures:

The hypothetical monomer vapour pressure P om can be solvedfor:

The partial pressure of the monomer can be written as afunction of a hypothetical monomer vapour pressure and theactivity coefficient of the associated substance in the liquidmixture:

Note that in the working equations the mole fraction of dimer isnot included. The associating substance is used when calculatingthe number of moles in the vapour phase:

where: w A = Weight of associating substance

n m , n d = Number of moles of monomer and dimer

M m = Molecular weight of monomer

Dividing by M m :

(2.72)

(2.73)

(2.74)

(2.75)

(2.76)

P Ava p P m P d + P m K P m[ ]

2+= =

P m 1 4 KP A

va p+ 1

2 K -----------------------------------------=

P m A x A P m=

w A nm M m 2 nd M m+=

n A

nm

2 nd

+=


39/358


2-27

Since there are n t total number of moles in the vapour phase,the mass balance is expressed as:

where: the index 2 represents the non-associating component in themixture.

Since it is assumed that the components in the mixture behavelike an ideal gas:

where: P A is the total pressure using Equation (2.77) .

Knowing that:

You have:

The usage of Equations (2.80) and (2.81) can be easilyaccomodated by defining a new standard state fugacity forsystems with dimerization:

(2.77)

(2.78)

(2.79)

(2.80)

(2.81)

(2.82)

xm 2 xd x2+ + 1=

P A P m 2 P d P 2+ +=

P P m P d P 2+ +=

y A P m 2 P d +

P m 2 P d P 2+ +-----------------------------------

P m 2 P d +

P P d +-----------------------= =

y2 P 2

P m 2 P d P 2+ +-----------------------------------

P 2 P P d +----------------

2 x2 P 2va p

P P d +----------------------= = =

f dimerizing L P

P P d +---------------- P m 1 2 KP m+( )=


40/358

2-28 Equilibrium Calculations

2-28

Several property packages in DISTIL support ideal gasdimerization. The standard nomenclature is:

[Activity Coefficient Model] + [Dimer] = Property PackageName

For example, NRTL-Dimer is the property package which usesNRTL for the activity coefficient calculations and the carboxylicacid dimer assuming ideal gas phase behaviour. The followingcarboxylic acids are supported:

Formic Acid Acetic Acid Acrylic Acid Propionic Acid Butyric Acid IsoButyric Acid Valeric Acid Hexanoic Acid

2.9 EquilibriumCalculations

When performing flash calculations, K-values are usuallycalculated. K-values are defined as follows:

where: y i is the composition of one phase (usually the vapour)

x i is the composition of another phase (usually the liquid)

(2.83)

(2.84)

f n o n d imerizing L P

P P d +---------------- P va pnon dimer iz ing =

K i yi xi----=


41/358


2-29

When using equations of state to represent the vapour andliquid behaviour, you have:

and therefore:

Activity coefficient based models can easily be expressed in thisformat:

and therefore:

where the standard state reference fugacity is calculated byEquations (2.58) , (2.62) or (2.64) depending on the desiredproperty package.

(2.85)

(2.86)

(2.87)

(2.88)

(2.89)

f iV i

V yi P =

f i L i

L xi P =

K ii

L

iV ------=

f Li Li xi P i xi f re f i= =

i L i f i

re f

P ------------=


42/358

2-30 Basic Models for VLE & LLE

2-30

2.10 Basic Models for VLE

& LLE2.10.1 Symmetric Phase

RepresentationSymmetric phase representation is the use of only onethermodynamic model to represent the behaviour of the vapourand liquid phases. Examples are the Peng-Robinson and SRKmodels.

The advantages of symmetric phase representation are asfollows:

Consistent representation for both liquid and vapourphases.

Other thermodynamic properties like enthalpies,entropies and densities can be readily obtained.

The disadvantages of symmetric phase representation are asfollows:

It is not always accurate enough to represent thebehaviour of the liquid and vapour phase for polarcomponents. Unless empirical modifications are made on

the equations, the representation of the vapourpressures for polar components is not accurate. The simple composition dependence usually shown by

standard cubic equations of state is inadequate torepresent the phase behaviour of polar mixtures.

2.10.2 Asymmetric PhaseRepresentation

Asymmetric phase representation is the use of one model torepresent the behaviour of the vapour phase and a separatemodel to represent the behaviour of the liquid phase (such asIdeal Gas/UNIQUAC, or RK/Van Laar).


43/358


2-31

The advantages of asymmetric phase representation are:

The vapour pressure representation is limited only by theaccuracy of the vapour pressure correlation.

There are more adjustable parameters to represent theliquid mixture behaviour. There is the possibility of reasonable liquid-liquid

representation.

The disadvantages of asymmetric phase representation are:

The necessity of an arbitrary reference state. There are potential problems representing supercritical

components. There are problems when approaching the mixture

critical point. Enthalpies, entropies and densities must be computed

using a separate model.

2.10.3 Interaction ParametersThe phase behaviour of mixtures is generally not wellrepresented using only pure component properties. Whenworking with either the symmetric or asymmetric approach, itwill often be necessary to use some experimental data to "help"the semi-theoretical equations represent reality. If you are usingan equation of state, this experimental parameter is usuallycalled "k ij " , and is commonly used to correct the quadraticmixture term in cubic equations of state, roughly representing

the energies of interaction between components present in themixture.

If you are using an activity model, the experimental parametersare usually called a ij and a ji . Several different approachescreate different equations with different interpretations of whatinteraction parameters are. As a rough comparison, theMargules and Van Laar equations are polynomial expansions ofthe Gibbs free energy of mixture, and the Wilson, NRTL andUNIQUAC methods are statistical mechanics equations based onthe Local Composition Concept.


44/358


2-32

2.10.4 Selecting Property

MethodsThe various property packages available allow you to predictproperties of mixtures ranging from well defined lighthydrocarbon systems to highly non-ideal (non-electrolyte)chemical systems. Enhanced equations of state (PR and PRSV)are provided for rigorous treatment of hydrocarbon systems andactivity coefficient models for chemical systems. All of theseequations have their own inherent limitations and you areencouraged to become more familiar with the application ofeach equation. This section contains a description of eachproperty package as well as listings of referenced literature.

For oil, gas and petrochemical applications, the Peng-RobinsonEOS (PR) is generally the recommended property package. Theenhancements to this equation of state enable it to be accuratefor a variety of systems over a wide range of conditions. Itrigorously solves any single, two-phase or three-phase systemwith a high degree of efficiency and reliability, and is applicable

over a wide range of conditions, as shown in the following table.

The PR equation of state has been enhanced to yield accuratephase equilibrium calculations for systems ranging from lowtemperature cryogenic systems to high temperature, highpressure reservoir systems. The same equation of statesatisfactorily predicts component distributions for heavy oilsystems, aqueous glycol and methanol systems, and acid gas/sour water systems.

The range of applicability in many cases is more indicative ofthe availability of good data rather than on the actuallimitations of the Equation of State.

Method Temperature, F Pressure, psia

PR > -456 (-271C)

< 15,000 (100,000kPa)

SRK > -225 (-143C)

< 5,000 (35,000kPa)


45/358


2-33

Although the Soave-Redlich-Kwong (SRK) equation will alsoprovide comparable results to the PR in many cases, it has beenobserved that its range of application is significantly more

limited and this method is not as reliable for non-ideal systems.For example, it should not be used for systems with methanol orglycols.

As an alternative, the PRSV equation of state should beconsidered. It can handle the same systems as the PR equationwith equivalent, or better accuracy, plus it is more suitable forhandling non-ideal systems.

The advantage of the PRSV equation is that not only does ithave the potential to more accurately predict the phasebehaviour of hydrocarbon systems, particularly for systemscomposed of dissimilar components, but it can also be extendedto handle non-ideal systems with accuracies that rival traditionalactivity coefficient models. The only compromise is increasedcomputational time and an additional interaction parameterwhich is required for the equation.

The PR and PRSV equations of state can be used to performrigorous three-phase flash calculations for aqueous systemscontaining water, methanol or glycols, as well as systemscontaining other hydrocarbons or non-hydrocarbons in thesecond liquid phase. The same is true for SRK, but only foraqueous systems.

The PR can also be used for crude systems, which havetraditionally been modeled with dual model thermodynamicpackages (an activity model representing the liquid phasebehaviour, and an equation of state or the ideal gas law for thevapour phase properties). These earlier models become lessaccurate for systems with large amounts of light ends or whenapproaching critical regions. Also, the dual model system leadsto internal inconsistencies. The proprietary enhancements to thePR and SRK methods allow these Equations of State to correctlyrepresent vacuum conditions and heavy components (a problemwith traditional EOS methods), and handle the light-end

components and high-pressure systems.


46/358


2-34

The table below lists some typical systems and therecommended correlations. However, when in doubt of theaccuracy or application of one of the property packages, call

Technical Support. They will try to either provide you withadditional validation material or give the best estimate of itsaccuracy.

The Property Package methods are divided into eight basiccategories, as shown in the following table. Listed with each ofthe property methods are the available methods for VLE andEnthalpy/Entropy calculations.

Type of System Recommended PropertyMethod

TEG Dehydration PR

Cryogenic Gas Processing PR, PRSV

Air Separation PR, PRSV

Reservoir Systems PR, PR Options

Highly Polar and non-hydrocarbonsystems

Activity Models, PRSV

Hydrocarbon systems where H 2 Osolubility in HC is important Kabadi Danner

Property Method VLE Calculation Enthalpy/EntropyCalculation

Equations of State

PR PR PR

SRK SRK SRK

Equation of State Options

PRSV PRSV PRSV

Kabadi Danner Kabadi Danner SRK

RK-Zudekevitch-Joffee RK-Zudekevitch-Joffee RK-Zudekevitch-Joffee

Activity Models

Liquid

Margules Margules Cavett

Van Laar Van Laar Cavett

Wilson Wilson Cavett

NRTL NRTL Cavett

UNIQUAC UNIQUAC Cavett

Chien Null Chien Null CavettVapour

Ideal Gas Ideal Gas Ideal Gas


47/358


2-35

2.10.5 Vapour Phase Optionsfor Activity Models

There are several models available for calculating the vapourphase in conjunction with the selected activity model. Thechoice will depend on specific considerations of your system.However, in cases when you are operating at moderatepressures (less than 5 atm), choosing Ideal Gas should besatisfactory.

IdealThe ideal gas law will be used to model the vapour phase. Thismodel is appropriate for low pressures and for a vapour phasewith little intermolecular interaction.

Peng Robinson and SRKThese two options have been provided to allow for betterrepresentation of unit operations (such as compressor loops).

Henrys LawFor systems containing non-condensable components, you cansupply Henrys law information via the extended Henrys lawequations.

Ideal Gas/Dimer Ideal Gas/Dimer Ideal Gas

RK RK RKPeng Robinson Peng Robinson Peng Robinson

Virial Virial Virial

Property Method VLE Calculation Enthalpy/EntropyCalculation


48/358


2-36

The program considers the following components as non-condensable:

This information is used to model dilute solute/solventinteractions. Non-condensable components are defined as thosecomponents that have critical temperatures below the systemtemperature you are modeling.

The equation has the following form:

where: i = Solute or non-condensable component

j = Solvent or condensable component

H ij = Henrys constant between i and j in kPa, Temperature indegrees K

A = A coefficient entered as a ij in the parameter matrix

B = B coefficient entered as a ji in the parameter matrix

C = C coefficient entered as b ij in the parameter matrix

D = D coefficient entered as b ji in the parameter matrix

T = temperature in degrees K

Component Simulation NameC1 Methane

C2 Ethane

C2= Ethylene

C2# Acetylene

H2 Hydrogen

He Helium

Argon Argon

N2 Nitrogen

O2 Oxygen

NitricOxide Nitric Oxide

CO Carbon Monoxide

CO2 Carbon DioxideH2S Hydrogen Sulfide

(2.90)

Refer to Section 2.6.1 -Non-CondensableComponents and Section2.6 - Henrys Law for theuse of Henrys Law.

H ijln A B

T

--- C T ( ) DT +ln+ +=


49/358


2-37

Only components listed in the table will be treated as HenrysLaw components. If the program does not contain pre-fittedHenrys coefficients, it will estimate the missing coefficients. To

supply your own coefficients, you must enter them directly intothe a ij and b ij matrices according to Equation (2.90) .

No interaction between "non-condensable" component pairs istaken into account in the VLE calculations.

2.11 Phase StabilitySo far, the equality of fugacities on the phases for eachindividual component has been used as the criteria for phaseequilibria. Although the equality of fugacities is a necessarycriteria, it is not sufficient to ensure that the system is atequilibrium. A necessary and sufficient criteria forthermodynamic equilibrium is that the fugacities of theindividual components are the same and the Gibbs Free Energyof the system is at its minimum.

Mathematically:

and Gsystem

= minimum .

The problem of phase stability is not a trivial one, since thenumber of phases that constitute a system is not known initially,and the creation (or deletion) of new phases during the searchfor the minimum is a blend of physics, empiricism and art.

(2.91) f i I f i

II f i II I = =


50/358

2-38 Phase Stability

2-38

2.11.1 Gibbs Free Energy for

Binary SystemsAccording to the definitions, the excess Gibbs energy is givenby:

From the previous discussion on partial molar properties,; thus, if you find a condition such that :

is smaller than:

where: np = number of phases

(2.92)

(2.93)

(2.94)

G E

G G ID

RT x i ilni 1=

nc

RT x i f i

xi f ire f

-------------ln= = =

E xiG E

i=

G E xi P G i

P E ,

i

nc

j 1=

np

=

G E xi P G i

P E ,

i

nc

j 1=

np 1

=


51/358


2-39

The former condition is more stable than the latter one. If GE fortwo phases is smaller than GE for one phase, then a solutionwith two phases is more stable than the solution with one. This

is represented graphically as shown in the following figures.

Figure 2.1

Figure 2.2

xi0.5

1

G 1

xi

0.50

dG 1

dx


52/358

2-40 Phase Stability

2-40

If you have a binary mixture of bulk composition x i , the GibbsFree Energy of the mixture will be G1 = G i x i + G j x j . If youconsider that two phases can exist, the mass and energy

balance tells us that:

where: is the phase fraction

Therefore, ( G2 , x i ), ( GI , x i

I ) and ( GII , x i II ) are co-linear points

and you can calculate G2 = GI + (1- )G II .

where:

Thus, the conditions for phase splitting can be mathematicallyexpressed as requiring that the function G1 has a localmaximum and is convex. This is conveniently expressed usingderivatives:

If you use

(2.95)

(2.96)

(2.97)

(2.98)

xi x i

I

xi II xi

I ----------------= and G

2 G I

G II G I --------------------=

G I G I xi I x j

I P T , , ,( )= G II G II xi II x j

II P T , , ,( )=

xi

G 1

T P ,

0= an d x

i

2

2

G 1

T P ,

0=

G E G G ID RT x i ilni 1=

nc

= =


53/358


54/358

2-42 Enthalpy/Cp Departure Functions

2-42

2.12 Enthalpy/Cp

Departure FunctionsLet Prop be any thermodynamic property. If you define thedifference of Prop-Prop o to be the residual of that property (itsvalue minus the value it would have at a reference state) andcall this reference state the ideal gas at the system temperatureand composition, you have:

where: P is an arbitrary pressure, usually 1 atm.

If you have an equation of state defined in terms of V and T (explicit in pressure), the Helmholtz free energy is the mostconvenient to work with ( dA = -SdT -PdV ).

(2.102)

Figure 2.3

P V RT = or V RT

P -------=

P r e s s u r e

I d e a l G a s

Enthalpy

A

B

C D

Isobar 1

Isobar 2

I s o t h e r m 2 I s o t h e

r m 1


55/358


2-43

At constant temperature you have dA = -PdV and if youintegrate from the reference state, you have:

You can divide the integral into two parts:

The second integral refers to the ideal gas, and can beimmediately evaluated:

It is interesting to note that A-Ao

for an ideal gas is not zero. The A-A o

(2.103)

(2.104)

(2.105)

A A P V d V

V

=

A A P V d

V

P V d V

=

P RT V

-------= and P V d V

RT V ------- V d V

=


56/358


2-44

term can be rearranged by adding and subtracting and thefinal result is:

(Notice that (P-RT/V) goes to zero when the limit V isapproached).

(2.106)

RT V

------- V d

V

A A P RT V

------- V d

V

RT V V ------ln =


57/358


2-45

The other energy-related thermodynamic functions areimmediately derived from the Helmholtz energy:

By the definition of C p , you have:

and integrating at constant temperature you have:

A more complete table of thermodynamic relations and a veryconvenient derivation for cubic equations of state is given by6Reid, Prausnitz and Poling (1987). The only missing derivationsare the ideal gas properties. Recalling the previous section, if

(2.107)

(2.108)

(2.109)

S S T

A A ( )V T P

V

RV --- V R V

V ------ln+d

V

= =

H H A A ( ) T S S ( ) RT Z 1 ( )+ +=

C p T H

P = an d

P C p

T T

T 2

2

V

P

=

dC p T T 2

2

V

P dP =

C p C p T T

2

2

V

P

P d

P

P

=

or

C p C p T T

2

2

P

V

V d

V

T P

T ------

V

2

P T ------

T

------------------- R =


58/358


2-46

you were to call I an ideal gas property:

(2.110) I mi x xi I ii 1=

nc

=


59/358


2-47

2.12.1 Alternative Formulation

for Low Pressure SystemsFor chemical systems, where the non-idealities in the liquidphase may be severe even at relatively low pressures, alternateformulations for the thermal properties are as follows:

The vapour properties can be calculated as:

where: H V is the enthalpy of vapourization of the mixture at thesystem pressure

Usually the term is ignored (although it can be computed

in a fairly straight forward way for systems where association inthe vapour phase is important ( 2Prausnitz et al., (1980)).

The term is the contribution to the enthalpy due tocompression or expansion, and is zero for an ideal gas. Thecalculation of this term depends on what model was selected forthe vapour phaseIdeal Gas, Redlich Kwong or Virial.

(2.111)

(2.112)

H i L Cp i T d

T ref ,

T

= and H L x i H i L H mi x L+i 1=

nc

=

H mi xV H mi x

L H V H P V H mi x

V + + +=

It is assumed that H iL at

the reference temperatureis zero.

H mi xV

H P V


60/358


2-48

All contribution to the enthalpy at constant temperature can besummarized as follows ( 7Henley and Seader, 1981):

Figure 2.4

A

B

C

D

T TcCritical Temperature

P = 0 ( I d

e a l G a s

V a p o u r

a t Z e r o P r e

s s u r e

P = S y s t e m

P

M o l a r E n t h a l p y H

Absolute Temperature T

{Heat ofVapourizationpressure correction to bringthe vapour to saturation

pressure tocompress theliquid


61/358

Thermodynamic Calculation Models 3-1

3-1

3 ThermodynamicCalculation Models

3.1 Equations of State.......................................................................... 2

3.1.1 Ideal Gas Equation of State ....................................................... 33.1.2 Peng-Robinson Equation of State................................................73.1.3 HysysPR Equation of State....................................................... 173.1.4 Peng-Robinson Stryjek-Vera..................................................... 253.1.5 Soave-Redlich-Kwong Equation of State .................................... 363.1.6 Redlich-Kwong Equation of State .............................................. 463.1.7 Zudkevitch-Joffee Equation of State.......................................... 563.1.8 Kabadi-Danner Equation of State.............................................. 653.1.9 The Virial Equation of State ..................................................... 773.1.10 Lee-Kesler Equation of State .................................................. 923.1.11 Lee-Kesler-Plcker................................................................ 96

3.2 Activity Models............................................................................. 98

3.2.1 Ideal Solution Model ..............................................................1013.2.2 Regular Solution Model ..........................................................1063.2.3 van Laar Model .....................................................................1113.2.4 Margules Model.....................................................................1233.2.5 Wilson Model ........................................................................1303.2.6 NRTL Model ..........................................................................1413.2.7 HypNRTL Model.....................................................................1543.2.8 The General NRTL Model ........................................................1553.2.9 HYSYS - General NRTL ...........................................................1573.2.10 UNIQUAC Model ..................................................................1583.2.11 UNIFAC Model .....................................................................1703.2.12 Chien-Null Model .................................................................182

3.3 Chao-Seader Model .................................................................... 191

3.4 Grayson-Streed Model................................................................ 192


62/358

3-2 Equations of State

3-2

3.1 Equations of StateThe program currently offers the enhanced Peng-Robinson (PR),and Soave-Redlich-Kwong (SRK) equations of state. In addition,several methods are offered which are modifications of theseproperty packages, including PRSV, Zudkevitch Joffee andKabadi Danner. Of these, the Peng-Robinson equation of statesupports the widest range of operating conditions and thegreatest variety of systems. The Peng-Robinson and Soave-Redlich-Kwong equations of state (EOS) generate all requiredequilibrium and thermodynamic properties directly. Although theforms of these EOS methods are common with other commercialsimulators, they have been significantly enhanced to extendtheir range of applicability.

The PR and SRK packages contain enhanced binary interactionparameters for all library hydrocarbon-hydrocarbon pairs (acombination of fitted and generated interaction parameters), aswell as for most hydrocarbon-non-hydrocarbon binaries.

For non-library or hydrocarbon hypocomponents, HC-HCinteraction parameters can be generated automatically forimproved VLE property predictions.

The PR equation of state applies a functionality to some specificcomponent-component interaction parameters. Key componentsreceiving special treatment include He, H 2 , N 2 , CO 2 , H 2 S, H 2O,CH3OH, EG and TEG.

The PR or SRK EOS should not be used for non-idealchemicals such as alcohols, acids or other components.These systems are more accurately handled by the ActivityModels or the PRSV EOS.


63/358

Thermodynamic Calculation Models

3-3

3.1.1 Ideal Gas Equation of

StateTo use the fugacity coefficient approach, a functional formrelating P, V, and T is required. These functional relationshipsare called equations of state, and their development dates fromthe 17 th century when Boyle first discovered the functionalitybetween pressure and volume. The experimental resultsobtained from Boyle, Charles, Gay-Lussac, Dalton and Avogadrocan be summarized in the Ideal Gas law:

The Ideal Gas equation, while very useful in some applicationsand as a limiting case, is restricted from the practical point ofview. The primary drawbacks of the ideal gas equation stemfrom the fact that in its derivation two major simplifications areassumed:

1. The molecules do not have a physical dimension; they arepoints in a geometrical sense.

2. There are no electrostatic interactions between molecules.

PV = RT (3.1)

Figure 3.1

V

P


64/358


3-4

For further information on the derivation of the Ideal Gas lawfrom first principles, see 8Feynman (1966).

Property MethodsA quick reference of calculation methods is shown in the tablebelow for Ideal Gas.

The calculation methods from the table are described in thefollowing sections.

IG Molar VolumeThe following relation calculates the Molar Volume for a specificphase.

Property Class Name and Applicable Phases

Calculation Method ApplicablePhase Property Class Name

Molar Volume Vapour COTHIGVolume Class

Enthalpy Vapour COTHIGEnthalpy Class

Entropy Vapour COTHIGEntropy Class

Isobaric heat capacity Vapour COTHIGCp Class

Fugacity coefficientcalculation Vapour COTHIGLnFugacityCoeffClass

Fugacity calculation Vapour COTHIGLnFugacity Class

(3.2)

Property Class Name Applicable Phase

COTHIGVolume Class Vapour

Usually the Ideal Gasequation is adequate whenworking with distillation

systems withoutassociation at low

V RT P -------=


65/358


3-5

IG EnthalpyThe following relation calculates enthalpy.

where: H i IG is the pure compound ideal gas enthalpy


IG EntropyThe following relation calculates entropy.

where: S i IG is the pure compound ideal gas entropy


(3.3)


COTHIGEnthalpy Class Vapour

(3.4)


COTHIGEntropy Class Vapour

H x i H i IG=

S xiS i IG R x i xiln =


66/358


3-6

IG Cp (Heat Capacity)The following relation calculates the isobaric heat capacity.

where: Cp i IG is the pure compound ideal gas Cp


IG Fugacity CoefficientThe following relation calculates the fugacity coefficient.


(3.5)


COTHIGCp Class Vapour

(3.6)


COTHIGLnFugacityCoeffClass

Vapour

C p xiC p i IG=

iln 0=


67/358


3-7

IG FugacityThe following relation calculates the fugacity for a specificphase.


3.1.2 Peng-Robinson Equationof State

The 9Peng Robinson (1976) equation of state (EOS) is amodification of the RK equation to better represent VLEcalculations. The densities for the liquid phase in the SRK didnot accurately represent the experimental values due to a highuniversal critical compressibility factor of 0.3333. The PR is amodification of the RK equation of state which corresponds to alower critical compressibility of about 0.307 thus representingthe VLE of natural gas systems accurately. The PR equation isrepresented by:

(3.7)


COTHIGLnFugacity Class Vapour

(3.8)

f i yi P =

P RT V b ------------ a

V V b+( ) b V b ( )+------------------------------------------------- =


68/358


3-8

where:

The functional dependency of the a term is shown in thefollowing relation.

The accuracy of the PR and SRK equations of state areapproximately the same. However, the PR EOS represents thedensity of the liquid phase more accurately due to the lowercritical compressibility factor.

These equations were originally developed for pure components.To apply the PR EOS to mixtures, mixing rules are required forthe a and b terms in Equation (3.2) . Refer to the MixingRules section on the mixing rules available.

Property MethodsA quick reference of calculation methods is shown in the tablebelow for the PR EOS.

(3.9)

(3.10)


Z Factor Vapour andLiquid

COTHPRZFactor Class

Molar Volume Vapour andLiquid

COTHPRVolume Class

Enthalpy Vapour andLiquid

COTHPREnthalpy Class

Entropy Vapour andLiquid

COTHPREntropy Class

a a c=

a c 0.45724 R2T c

2

P c------------=

b 0.07780 RT c P c---------=

1 1 T r 0.5

( )+=

0.37464 1.5422 0.26992 2

+=

Equations of state ingeneral - attractive andrepulsion partsSimplest cubic EOS - vander WaalsPrinciple of correspondingstates

First successfulmodification forengineering - RKThe property that is usuallyrequired for engineeringcalculations is vapourpressure.The SRK and RK EOSpropose modificationswhich improve the vapourpressure calculations forrelatively simple gases andhydrocarbons.


69/358


3-9

The calculation methods from the table are described in thefollowing sections.

PR Z FactorThe compressibility factor, Z, is calculated as the root for the

following equation:

There are three roots for the above equation. It is consideredthat the smallest root is for the liquid phase and the largest rootis for the vapour phase. The third root has no physical meaning.

Isobaric heat capacity Vapour and

Liquid

COTHPRCp Class

Fugacity coefficientcalculation

Vapour andLiquid

COTHPRLnFugacityCoeffClass

Fugacity calculation Vapour andLiquid

COTHPRLnFugacity Class

Isochoric heat capacity Vapour andLiquid

COTHPRCv Class

Mixing Rule 1 Vapour andLiquid

COTHPRab_1 Class


COTHPRab_2 Class


COTHPRab_3 Class

Mixing Rule 4 Vapour and

Liquid

COTHPRab_4 Class


COTHPRab_5 Class


COTHPRab_6 Class

(3.11)

(3.12)

(3.13)


Z 3 1 B ( ) Z 2 Z A 3 B2 2 B ( ) AB B2 B3 ( ) + 0=

A aP

R2T 2------------=

B bP RT -------=


70/358


3-10

PR Molar VolumeThe following relation calculates the molar volume for a specificphase.


PR EnthalpyThe following relation calculates the enthalpy.

(3.14)


COTHPRVolume Class Vapour and Liquid

The compressibility factor, Z, is calculated using PR Z Factor .For consistency, the PR molar volume always calls the PR ZFactor for the calculation of Z.

(3.15)

V ZRT P

-----------=

H H IG PV RT a dadT ------ T 12 2 b

------------- V b 1 2+( )+V b 1 2 ( )+----------------------------------ln =


71/358


3-11

where: H IG is the ideal gas enthalpy calculated at temperature, T


PR EntropyThe following relation calculates the entropy.

where: S IG is the ideal gas entropy calculated at temperature, T


COTHPREnthalpy Class Vapour and Liquid

The volume, V, is calculated using PR Molar Volume . Forconsistency, the PR Enthalpy always calls the PR Volume forthe calculation of V.

(3.16)S S IG R V b RT ------------ 1

2b 2------------- V b 1 2+( )+

V b 1 2 ( )+---------------------------------- da

dT ------ln ln=


72/358


3-12


PR Cp (Heat Capacity)The following relation calculates the isobaric heat capacity.

where: Cp IG is the ideal gas heat capacity calculated at temperature,T



COTHPREntropy Class Vapour and Liquid

The volume, V, is calculated using PR Molar Volume . Forconsistency, the PR Entropy always calls the PR Volume forthe calculation of V.

(3.17)


COTHPRCp Class Vapour and Liquid

The volume, V, is calculated using PR Molar Volume . Forconsistency, the PR Entropy always calls the PR Volume forthe calculation of V.

C p C p IG

T 2 P

T 2---------

V

V RT

V T ------

P

2

V P ------

T

-------------------+ +d

V

=


73/358


3-13

PR Fugacity CoefficientThe following relation calculates the fugacity coefficient.


PR FugacityThe following relation calculates the fugacity for a specificphase.

(3.18)

(3.19)

(3.20)


COTHPRLnFugacityCoeffClass

Vapour and Liquid

The volume, V, is calculated using PR Molar Volume . Forconsistency, the PR Fugacity Coefficient always calls the PRVolume for the calculation of V. The parameters a and b arecalculated from the Mixing Rules .

(3.21)

iln V b ( ) bV b ------------ a

2 2 b------------- V b 1 2+( )+

V b 1 2 ( )+---------------------------------- 1 a

a--- b

b---++

ln+ +ln =

a n2a

n------------=

b nb

n---------=

f i i yi P =


74/358


75/358


3-15

Mixing Rule 1The definition of terms a and b are the same for all MixingRules . The only difference between the mixing rules is thetemperature dependent binary interaction parameter, ij, which isdefined as:

where: A ij , B ij , and C ij are asymmetric binary interaction parameters

Mixing Rule 2The definition of terms a and b are the same for all MixingRules . The only difference between the mixing rules is thetemperature dependent binary interaction parameter, ij, which isdefined as:

where: A ij , B ij , and C ij are asymmetric binary interaction parameters

(3.25)

(3.26)

(3.27)

(3.28)

(3.29)

(3.30)

(3.31)

a ij ij a ci a cj i j=

i 1 i ( ) 1 T ri0.5 ( )=

a ci0.45724 R

2T ci

2

P ci---------------------------------=

b i0.07780 RT ci

P ci-------------------------------=

i 0.37464 1.54226 i 0.26992 i2 += i 0.49 0, B can be expressed as:

Similarly, the following can be obtained:

This approach can easily be extended to higher terms.

It is experimentally verified that the Virial equation, whentruncated after the second Virial coefficient, gives reasonablevapour phase density predictions provided that the density issmaller than half of the critical density. The Virial EOS truncatedafter the second Virial coefficient is:

Calculating the Second VirialCoefficientThere are several ways of estimating the second virial coefficientfor pure components and mixtures. If accurate volumetric datais available, the procedure is straightforward, but tedious. Inyour applications, it is better to estimate the second virialcoefficient similar to the way in which the cubic equation of stateparameters were determined. That is, it is desired to expressthe second virial coefficient as a function of T c , P c and theacentric factor. Pitzer attempted to do this, proposing a simplecorresponding states approach:

(3.216)

(3.217)

(3.218)

(3.219)

B Z

T 0lim=

C 22

Z

T 0lim= D

33

Z

T 0lim=

Z PV RT ------- 1 B

V ---+= =

B B 0( ) B 1( )+=


143/358


3-83

where: B (0) is a simple fluid term depending only on T c

B(1) is a correction term for the real fluid, which is a functionof T c and P c

Note that this three-parameter corresponding states relationdisplays in many different forms, such as in the Soave, Peng-Robinson, Lee-Kesler and BWR-Starling equations of state.

Pitzer proposed several modifications to this simple form. Pitzerwas motivated mainly because polar fluids do not obey a simplethree-parameter corresponding states theory. 18Tsonopoulos(1974) suggested that the problem can (at least partially) besolved by the inclusion of a third term in the previousexpression:

where: B (2) is a function of T c and one (or more) empirical constants

It was found that this empirical function can sometimes begeneralized in terms of the reduced dipole moment:

where: P c is in bar and R is in debyes

The method of 19Hayden and O'Connell (1975) is used, wherethey define:

where: B ij F , non-polar = Second virial coefficient contribution from

the non-polar part due to physical interactions

Bij F , polar = Second virial coefficient contribution from the polar part due to physical interactions

(3.220)

(3.221)

(3.222)

B B

0( )

B 1( )

B

2( )

+ +=

R10 52 P c

T c--------------------- 0.9869=

B ij B ij F

B ij D

+=

B ij F

B ij F

non po la r ,( ) B ij F

pol ar ,( )+=

B ij D B ij

Dmetastable,( ) B ij

Dbound ,( ) B ij

Dchemical ,( )+ +=


144/358


3-84

Bij D , metastable = Second virial coefficient contribution due

to the formation of metastable compounds due to the"chemical" (dimerization) reaction

Bij D , bound = Second virial coefficient contribution due to theformation of chemical bonds

Bij D , chemical = Second virial coefficient contribution due to

the chemical reaction

The several contributions to the second Virial coefficient arecalculated as follows:

(3.223)

(3.224)

(3.225)

(3.226)

B ij F

non po la r , b ij0 0.94 1.47

T ij*'

---------- 0.85

T ij*'2

---------- 1.015

T ij*'3

------------- +

=

B ij F

pol ar , b ij0 ij

*'0.74 3.0

T ij*'

------- 2.1

T ij*'2

--------- 2.1

T ij*'3

---------+ + =

B ij D

metastable,( ) B ij D

bound ,( )+ b ij0 A ij

H ijT ij

-----------

exp=

B ij D

chemical ,( ) b ij0 E ij 1

1500 ijT

------------------- exp

=


145/358


3-85

where:

For pure components:

(3.227)

1

T ij*'

------ 1

T ij*

------ 1.6 ij =

T ij* T

ij k ( )-----------------=

b ij0 1.26184 ij

3= cm 3 gmol ( )

ij*' ij

*= if ij* 0.04

Documents

AspenCOMThermoV7 3 Ref