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Aspen HYSYSThermodynamics COMInterface

Reference Guide



Version Number: V7.2July 2010

Copyright (c) 1981-2010 by Aspen Technology, Inc. All rights reserved.

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Technical Support

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

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



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iii

Table of Contents

Technical Support................................................... 13

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

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

1 Introducing Aspen HYSYS Thermodynamics COMInterface1-1

1.1 Introduction .................................................... 1-2

2 Thermodynamic Principles ...................................2-1

2.1 Introduction .................................................... 2-32.2 Chemical Potential & Fugacity ............................ 2-6

2.3 Chemical Potential for Ideal Gas......................... 2-7

2.4 Chemical Potential & Fugacity for a Real Gas........ 2-8

2.5 Fugacity & Activity Coefficients .......................... 2-9

2.6 Henry’s Law ...................................................2-12

2.7 Gibbs-Duhem Equation ....................................2-16

2.8 Association in Vapour Phase - Ideal Gas .............2-20

2.9 Equilibrium Calculations ...................................2-24

2.10 Basic Models for VLE & LLE ...............................2-26

2.11 Phase Stability................................................2-33

2.12 Enthalpy/Cp Departure Functions ......................2-38

3 Thermodynamic Calculation Models...................... 3-1

3.1 Equations of State............................................ 3-2

3.2 Activity Models ...............................................3-98

3.3 Chao-Seader Model .......................................3-191

3.4 Grayson-Streed Model ...................................3-192

4 Physical Property Calculation Methods ................ 4-1

4.1 Cavett Method ................................................. 4-24.2 Rackett Method................................................ 4-8



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



Introducing Aspen HYSYS Thermodynamics COM Interface 1-1

1-1

1 Introducing Aspen HYSYS

Thermodynamics COM Interface

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



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.



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.



1-4 Introduction

1-4



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 Henry’s Law................................................................................. 15

2.6.1 Non-Condensable Components................................................. 172.6.2 Estimation of Henry’s 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



2-2

2-2

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

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



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 π )

i = Component (from 1 to nc)

μi 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

μi jdn i

j

i 1=

nc

∑ j 1=

π

∑+

j 1=

π

∑ –

j 1=

π

∑=

μi j

ni∂

∂U

⎝ ⎠

⎜ ⎟⎛ ⎞

S V nk 1≠ j

, ,=



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 π( 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 ( π - 1)( nc + 2) independentequations.

(2.5)

(2.6)

(2.7)

dS S d j

j 1=

π

∑ 0= =

dV V d j

j 1=

π

∑ 0= =

dn i j

0= j 1=

π

∑ i 1, ..., nc=




2-5

The variables can be eliminated in the following way:

(2.8)

(2.9)

(2.10)

dS 1

S d j

j 2=

π

∑ – =

dV 1 V d j

j 2=

π

∑ – =

dn i1 dn i

j

j 2=

π

∑=



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 π Thermal Equilibrium - no heat flux across phases (2.14)

P 1 = P 2 =...=P π

Mechanical Equilibrium - no phase displacement (2.15)

μi1 = μi

2 =...=μi

πMass 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

μi j μi1 – ( )dn i j

i 1=

nc

∑ j 1>

π

∑+ j 1>

π

∑ – j 1>

π

∑=

U ∂S ∂------- 0= U ∂

V ∂------- 0= U ∂n i∂------- 0= U 2∂

S 2∂--------- 0=

T 1 T j= j 2, ..., π=




2-7

equations, or area affects are not being considered.



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.




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 μ i n

id

i 1=

nc

∑+ +=

μ i n i∂∂G⎝ ⎠⎛ ⎞

T P n k 1≠, ,=

dG i S idT – V idP +=



2-10 Chemical Potential for Ideal Gas

2-10

where:

Now assuming the system to be at constant temperature:

(2.20)

(2.21)

G iG∂n i∂-------⎝ ⎠

⎛ ⎞

T P n k 1≠, ,=

d μi dG i V idP = =




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 fluid’s equation of state—from those models, the fugacity is calculated.

(2.22)

(2.23)

(2.24)

μi C i RT f iln+=

f i1

f i2 … f i

π= = =

P ∂∂ f iln( )⎝ ⎠⎛ ⎞

T

V i RT -------=



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 =




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)

φ 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 i∂∂V ⎝ ⎠⎛ ⎞

T P n k i≠ j, ,

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

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

RT ------- V i

RT P ------- –

⎝ ⎠⎛ ⎞

0

P

∫ dP =

φi f i

Px i--------=



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,=

γi f i

L

f i L pu re,

xi

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




2-15

2.6 Henry’s 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 1→lim γi

xi 1→lim 1= =

f i L

f ire f xi

------------ xi 0→lim γi

∗ xi 0→lim 1= =



2-16 Henry’s 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 γi* it can be shown that:

where: H ij is called Henry’s Constant of component i in solvent j.

Therefore, the Henry’s constant is the standard state fugacityfor a non-condensable component in a specific solvent. Usuallythe Henry’s constant is a rather strong function of temperature,but a weak function of the pressure and composition. Theextension of Henry’s 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 = Henry’s 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 0→lim 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∞




2-17

2.6.1 Non-Condensable

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

A mixing rule for the Henry’s 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 Henry’s 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

∑-----------------------------------------------=



2-18 Henry’s Law

2-18

2.6.2 Estimation of Henry’s

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)

Δ g Δ g I Δ g

II +=

Δ g I

RT f i

L pu re,

f iG

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

Δ g II RT γi xiln=

f iG γi xi f i

L pu re,=

Δ g 0=

RT γiln νi L δ j δi – ( )2φ j

2=





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 γiln 0=

i 1=

nc

∑

γi P ref

γi P V i

RT ------- P d

P

P ref

∫⎝ ⎠⎜ ⎟⎜ ⎟⎛ ⎞

exp=

f i L γi

P xi f i

re f = or f i

L γi P ref

xi f ire f V i

RT ------- P d

P ref

P

∫⎝ ⎠⎜ ⎟⎜ ⎟⎛ ⎞

exp=




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 φi sa t V i

RT ------- P d

P ivap

P ref

∫⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞

exp=

f i L

P iva p φi

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

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

P ivap

P ref

∫exp=



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,φi

sat = 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 γi xi P i

va p=

f ire f P i

va p=

f i L γi xi P i

va p φi sa t V i

RT -------⎝ ⎠⎛ ⎞ P d

P ivap

P

∫exp=

f i L γi xi P i

va p φi sa t V i

RT -------⎝ ⎠⎛ ⎞ P d

P ivap

P

∫exp=




2-23

integration of Equation (2.60) leads to:

(2.61)

(2.62)

f i L γi xi P i

va p φi sa t V i P P i

va p

– ( ) RT

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

f ire f P i

va p φi sa t V i P P i

va p – ( ) RT

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



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 γi xi P i

va p φi sa t =

f ire f P i

va p φi sa t =

2 A A2↔

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




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 A

va p---------------------------------- e 2 e – ( )

4 P Ava p 1 e – ( )2

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

e 1

1 4 KP Ava p

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



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 solved

for:

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 γ A x A P m°=

w A nm M m 2 nd M m+=

n A

nm

2 nd

+=




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

γ2 x2 P 2va p

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

f dimerizing L P

P P d +----------------⎝ ⎠⎛ ⎞ P m° 1 2 KP m+( )=



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 p

non dimer iz ing – =

K i yi

xi----=




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 φi

V yi P =

f i L φi

L xi P =

K iφi

L

φiV ------=

f Li φ Li xi P γi xi f

re f i= =

φi L γi f i

re f

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



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).








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.




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




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).

Henry’s LawFor systems containing non-condensable components, you cansupply Henry’s law information via the extended Henry’s 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




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 = Henry’s 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 - Henry’s Law for theuse of Henry’s Law.

H ijln A B

T

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




2-37

Only components listed in the table will be treated as Henry’sLaw components. If the program does not contain pre-fittedHenry’s 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 …= =



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 γilni 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 –

∑=




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



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: β is the phase fraction

Therefore, ( G2 , x i ), ( GI , x i I ) and ( GII , x i

II ) are co-linear pointsand you can calculate G2 = βGI + (1- β )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)

β xi x i

I –

xi II xi

I – ----------------= and β 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 γilni 1=

nc

∑= =





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




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 °

∞

∫=




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 – =

∞→




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 – – =




2-46

you were to call I an ideal gas property:

(2.110) I mi x xi I ii 1=

nc

∑=




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: ΔH 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 phase—Ideal 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 Δ H mi x

L+

i 1=

nc

∑=

H mi xV H mi x

L Δ H V Δ H P V Δ H mi x

V + + +=

It is assumed that H iL at

the reference temperatureis zero.

Δ H mi xV

Δ H P V




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 ofVapourization

pressure correction to bringthe vapour to saturation

pressure tocompress theliquid



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-Plöcker................................................................ 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



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.



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




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 -------=






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∑=

φiln 0=




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 – ( )+------------------------------------------------- – =




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 cα=

a c 0.45724 R2T c

2

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

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

α 1 κ 1 T r 0.5

– ( )+=

κ 0.37464 1.5422 ω 0.26992 ω2

– +=

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.




3-9


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 -------=






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= –




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

∫ – = –




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)

φiln 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 φi yi P =




3-14

Property Class Name and Applicable Phase

PR Cv (isochoric)The following relation calculates the isochoric heat capacity.


Mixing RulesThe mixing rules available for the PR EOS state are shown

below.


COTHPRLnFugacity Class Vapour and Liquid

(3.22)


COTHPRCv Class Vapour and Liquid

(3.23)

(3.24)

C v C p

T ∂ P ∂T ------⎝ ⎠⎛ ⎞2

V

∂ P ∂V ------⎝ ⎠⎛ ⎞

T

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

a x i x ja ij( ) j 1=

nc

∑i 1=

nc

∑=

b b i xii 1=

nc

∑=




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, ξij, 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 ξij a ci a cj α iα j=

α i 1 κi – ( ) 1 T ri0.5 – ( )=

a ci0.45724 R

2T ci

2

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

b i0.07780 RT ci

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

κi 0.37464 1.54226 ωi 0.26992 ωi2 – += ωi 0.49<

ξij 1 Aij – B ij T C ij T 2+ +=

ξij 1 Aij – B ij T C ijT

-------+ +=




3-16

Mixing Rule 3The 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, ξij, which isdefined as

Mixing Rule 4The definition of terms a and b are the same for all Mixing

Rules . The only difference between the mixing rules is thetemperature dependent binary interaction parameter, ξij, which isdefined as:


Mixing Rule 5The definition of terms a and b are the same for all MixingRules . The only difference between mixing rules is thetemperature dependent binary interaction parameter, ξij, which isdefined as:


(3.32)

(3.33)

(3.34)

ξij 1 xi Aij B ij C ij T 2+ +( ) – x j A ji B ji T C ji T 2+ +( ) – =

ξij 1 xi Aij B ij T C ijT

-------+ +⎝ ⎠⎛ ⎞ – x j A ji B ji

C jiT

-------+ +⎝ ⎠⎛ ⎞ – =

ξij 1 A ij B ij T C ij T

2+ +( ) A ji B ji T C ij T

2+ +( )

xi Aij B ij T C ij T 2+ +( ) x j A ji B ji T C ji T 2+ +( )+---------------------------------------------------------------------------------------------------------------- – =




3-17

Mixing Rule 6The definition of terms a and b are the same for all MixingRules . The only difference between mixing rules is thetemperature dependent binary interaction parameter, ξij, which isdefined as:


3.1.3 HysysPR Equation ofState

The HysysPR EOS is similar to the PR EOS with severalenhancements to the original PR equation. It extends its rangeof applicability and better represents the VLE of complexsystems. The HysysPR equation is represented by:

where:

(3.35)

(3.36)

(3.37)

ξij 1 A ij B ij T

C ijT

-------+ +⎝ ⎠⎛ ⎞ A ji B ji T

C ijT

-------+ +⎝ ⎠⎛ ⎞

xi Aij B ij T C ijT

-------+ +⎝ ⎠⎛ ⎞ x j A ji B ji T

C jiT

-------+ +⎝ ⎠⎛ ⎞+

----------------------------------------------------------------------------------------------------- – =

P RT V b – ------------ a

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

a a cα=

a c 0.45724 R2T c

2

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

b 0.077480 RT c P c---------=




3-18

The functional dependency of the “a” term is shown in thefollowing relation as Soave:

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


(3.38)

CalculationMethod

ApplicablePhase Property Class Name


COTH_HYSYS_ZFactor Class


COTH_HYSYS_Volume Class


COTH_HYSYS_PREnthalpy Class


COTH_HYSYS_Entropy Class

Isobaric heatcapacity

Vapour andLiquid

COTH_HYSYS_Cp Class


Vapour andLiquid

COTH_HYSYS_LnFugacityCoeffClass

Fugacity calculation Vapour and

Liquid

COTH_HYSYS_LnFugacity Class

Isochoric heatcapacity

Vapour andLiquid

COTH_HYSYS_Cv Class

α 1 S 1 T r 0.5 – ( )+=

S 0.37464 1.5422 ω 0.26992 ω2 – +=




3-19

HysysPR Z FactorThe compressibility factor, Z, is calculated as the root for thefollowing equation:


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


(3.39)

(3.40)

(3.41)

(3.42)


COTH_HYSYS_Volume Class Vapour and Liquid

The compressibility factor, Z, is calculated using HysysPR ZFactor . For consistency, the HysysPR molar volume alwayscalls the HysysPR Z Factor for the calculation of Z.

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

A aP

R2T

2------------=

B bP RT -------=

V ZRT P

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






3-21

HysysPR EntropyThe following relation calculates the entropy.



(3.44)


COTH_HYSYS_Entropy Class Vapour and Liquid

The volume, V, is calculated using HysysPR Molar Volume .For consistency, the HysysPR Entropy always calls theHysysPR Volume for the calculation of V.

S S IG R V b – RT ------------⎝ ⎠⎛ ⎞ 1

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

V b 1 2 – ( )+----------------------------------⎝ ⎠⎛ ⎞da

dT ------ln – ln= –




3-22

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

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


HysysPR Fugacity CoefficientThe following relation calculates the fugacity coefficient.

(3.45)


COTH_HYSYS_Cp Class Vapour and Liquid

(3.46)

(3.47)

(3.48)

C p C p IG T

∂2 P

∂T 2---------⎝ ⎠⎜ ⎟⎛ ⎞

V

V RT

∂V ∂T ------⎝ ⎠⎛ ⎞

P

2

∂V ∂ P ------⎝ ⎠⎛ ⎞

T

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

∞

V

∫ – = –

φi

ln V b – ( ) b

V 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---------=




3-23


HysysPR FugacityThe following relation calculates the fugacity for a specificphase.


HysysPR Cv (isochoric)The following relation calculates the isochoric heat capacity.


COTH_HYSYS_LnFugacityCoeffClass

Vapour and Liquid

The volume, V, is calculated using HysysPR Molar Volume .For consistency, the HysysPR Fugacity Coefficient alwayscalls the HysysPR Volume for the calculation of V. Theparameters a and b are calculated from the Mixing Rules .

(3.49)


COTH_HYSYS_LnFugacityClass

Vapour and Liquid

(3.50)

f i φi yi P =

C v C p

T ∂ P ∂T ------⎝ ⎠⎛ ⎞2

V

∂ P ∂V ------⎝ ⎠⎛ ⎞

T

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




3-24


Mixing RulesThe mixing rules available for the HysysPR EOS state are shown


COTH_HYSYS_Cv Class Vapour and Liquid




3-25

below.

where: κij = asymmetric binary interaction parameter

(3.51)

(3.52)

(3.53)

(3.54)

(3.55)

(3.56)

(3.57)


nc

∑i 1=

nc

∑=

b b i xii 1=

nc

∑=

a ij 1 k ij – ( ) a ci a cj α iα j=

α i 1 κi – ( ) 1 T ri0.5

– ( )=

a ci

0.45724 R2T ci2

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

b i0.07780 RT ci

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

κi

0.37464 1.54226 ωi 0.26992 ωi2 – +

0.37964 1.48503 ωi 0.16442 ωi2

– 0.016666 ωi3

+ +⎩⎪⎨⎪⎧

=ωi 0.49<

ωi 0.49≥




3-26

3.1.4 Peng-Robinson Stryjek-

VeraThe Peng-Robinson 10Stryjek-Vera PRSV, 1986) equation of stateis a two-fold modification of the PR equation of state thatextends the application of the original PR method for highly non-ideal systems. It has been shown to match vapour pressurescurves of pure components and mixtures more accurately,especially at low vapour pressures.

It has been extended to handle non-ideal systems providingresults similar to those obtained using excess Gibbs energyfunctions like the Wilson, NRTL or UNIQUAC equations.

The PRSV equation of state is defined as:

where:

(3.58)

(3.59)

P RT V b – ------------ a

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

a a cα=

a c 0.45724 R2T c

2

P c

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

b 0.077480 RT c P c---------=




3-27

One of the proposed modifications to the PR equation of state byStryjek and Vera was an expanded alpha, " α", term that becamea function of acentricity and an empirical parameter, κi, used for

fitting pure component vapour pressures.

where: κ1 = Characteristic pure component parameter

ωi = Acentric factor

The adjustable κ1

parameter allows for a much better fit of thepure component vapour pressure curves. This parameter hasbeen regressed against the pure component vapour pressure forall library components.

For hypocomponents that have been generated to represent oilfractions, the κ1 term for each hypocomponent will beautomatically regressed against the Lee-Kesler vapour pressurecurves. For individual user-added hypothetical components, κ1 terms can either be entered or they will automatically beregressed against the Lee-Kesler, Gomez-Thodos or Reidelcorrelations.

The second modification consists of a new set of mixing rules formixtures. To apply the PRSV EOS to mixtures, mixing rules arerequired for the “ a” and “ b” terms in Equation (3.46) . Refer tothe Mixing Rules section for the set of mixing rules applicable.

(3.60)

α i 1 κi 1 T r 0.5 – ( )+[ ]

2=

κi κ0 i κ1 1 T ri0.5+( ) 0.7 T ri – ( )+=

κ0 i 0.378893 1.4897153 ωi 0.17131848 ωi2

– 0.0196554 ωi3

+ +=




3-28

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


Calculation MethodApplicablePhase Property Class Name


COTHPRSVZFactor Class


COTHPRSVVolume Class


COTHPRSVEnthalpy Class


COTHPRSVEntropy Class

Isobaric heat capacity Vapour andLiquid

COTHPRSVCp Class


Vapour andLiquid

COTHPRSVLnFugacityCoeffClass


COTHPRSVLnFugacity Class


COTHPRSVCv Class


COTHPRSVab_1 Class


COTHPRSVab_2 Class


COTHPRSVab_3 Class


COTHPRSVab_4 Class


COTHPRSVab_5 Class


COTHPRSVab_6 Class




3-29

PRSV Z FactorThe compressibility factor, Z, is calculated as the root for thefollowing equation:


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


(3.61)

(3.62)

(3.63)

(3.64)


COTHPRSVVolume Class Vapour and Liquid

The compressibility factor, Z, is calculated using PRSV ZFactor . For consistency, the PRSV molar volume always callsthe PRSV Z factor for the calculation of Z.

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

A aP

R2T

2------------=

B bP RT -------=

V ZRT P

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




3-30

PRSV EnthalpyThe following relation calculates the enthalpy



(3.65)


COTHPRSVEnthalpy Class Vapour and Liquid

The volume, V, is calculated using PRSV Molar Volume . Forconsistency, the PRSV Enthalpy always calls the PRSVVolume for the calculation of V.

H H IG PV RT a dadT ------⎝ ⎠⎛ ⎞T –

⎝ ⎠⎛ ⎞ – 1

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

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




3-31

PRSV EntropyThe following relation calculates the entropy.



(3.66)


COTHPRSVEntropy Class Vapour and Liquid

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

S S IG

R V b – RT ------------⎝ ⎠⎛ ⎞ 1

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

V b 1 2 – ( )+----------------------------------⎝ ⎠⎛ ⎞da

dT ------ln – ln= –




3-32

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


PRSV Fugacity CoefficientThe following relation calculates the fugacity Coefficient.

(3.67)


COTHPRSVCp Class Vapour and Liquid

(3.68)

(3.69)

(3.70)

C p C p IG T

∂2 P

∂T 2---------⎝ ⎠⎜ ⎟⎛ ⎞

V

V RT

∂V ∂T ------⎝ ⎠⎛ ⎞

P

2

∂V ∂ P ------⎝ ⎠⎛ ⎞

T

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

∞

V

∫ – = –

φiln V b – ( ) bV b – ------------ a

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

V b 1 2 – ( )+----------------------------------⎝ ⎠⎛ ⎞ 1 – a

a--- b

b---++

⎝ ⎠⎛ ⎞ln+ +ln – =

a ∂n

2a

∂n------------=

b ∂nb

∂n---------=




3-33


PRSV FugacityThe following relation calculates the fugacity for a specificphase.


PRSV Cv (isochoric)The following relation calculates the isochoric heat capacity.


COTHPRSVLnFugacityCoeffClass

Vapour and Liquid

The volume, V, is calculated using PRSV Molar Volume . Forconsistency, the PRSV Fugacity Coefficient always calls thePRSV Volume for the calculation of V. The parameters a andb are calculated from the Mixing Rules .

(3.71)


COTHPRSVLnFugacity Class Vapour and Liquid

(3.72)

f i φi yi P =

C v C p

T ∂ P ∂T ------⎝ ⎠⎛ ⎞2

V

∂ P ∂V ------⎝ ⎠⎛ ⎞

T

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




3-34


Mixing RulesThe mixing rules available for the PRSV equation are shownbelow.


COTHPRSVCv Class Vapour and Liquid

(3.73)

(3.74)

(3.75)

(3.76)

(3.77)

(3.78)

a x i x j a ij( ) j 1=

nc

∑i 1=

nc

∑=

b b i xii 1=

nc

∑=

a ij a ii a jj( )0.5 ξij=

α i 1 κi – ( ) 1 T ri0.5

– ( )=

a i0.45724 R

2T ci

2

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

b i0.07780 RT ci

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






3-36






(3.81)

(3.82)

(3.83)

ξij 1 Aij – B ij T C ij T 2

+ +=


-------+ +=

ξij 1 xi Aij B ij C ij T 2

+ +( ) – x j A ji B ji T C ji T 2

+ +( ) – =




3-37





Mixing Rule 6The 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, ξij:


(3.84)

(3.85)

(3.86)

ξij 1 x i Aij B ij T C ijT

-------+ +⎝ ⎠⎛ ⎞ – x j A ji B ji

C jiT

-------+ +⎝ ⎠⎛ ⎞ – =

ξij 1 A ij B ij T C ij T 2+ +( ) A ji B ji T C ij T 2+ +( )


ξij 1 A ij B ij T

C ijT

-------+ +⎝ ⎠⎛ ⎞ A ji B ji T

C ijT

-------+ +⎝ ⎠⎛ ⎞

xi Aij B ij T C ijT

-------+ +⎝ ⎠⎛ ⎞ x j A ji B ji T

C jiT

-------+ +⎝ ⎠⎛ ⎞+

----------------------------------------------------------------------------------------------------- – =




3-38

3.1.5 Soave-Redlich-Kwong

Equation of StateWilson (1965, 1966) noted that the main drawback of the RKequation of state was its inability of accurately reproducing thevapour pressures of pure component constituents of a givenmixture. He proposed a modification to the RK equation of stateusing the acentricity as a correlating parameter, but thisapproach was widely ignored until 1972, when 11Soave (1972)proposed a modification of the RK equation of this form:

The “a” term was fitted to reproduce the vapour pressure ofhydrocarbons using the acentric factor as a correlatingparameter. This led to the following development:

Empirical modifications for the “ a” term for specific substanceslike hydrogen were proposed by 12Graboski and Daubert (1976),and different, substance specific forms for the “ a” term withseveral adjusted parameters are proposed up to the present,varying from 1 to 3 adjustable parameters. The SRK equation ofstate can represent the behaviour of hydrocarbon systems forseparation operations with accuracy. Since, it is readilyconverted into computer code, its usage has been intense in thelast twenty years. Other derived thermodynamic properties, like

enthalpies and entropies, are reasonably accurate forengineering work, and the SRK equation has wide acceptance inthe engineering community today.

(3.87)

(3.88)

P RT V b – ------------

a T T c ω, ,( )V V b+( )--------------------------- – =

P RT V b – ------------

a cαV V b+( )--------------------- – =

a c Ωa R2T c

2

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

α 1 S 1 T r 0.5 – ( )+=

S 0.480 1.574 ω 0.176 ω2 – +=




3-39

To apply the SRK EOS to mixtures, mixing rules are required forthe “ a” and “ b” terms in Equation (3.270) .

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




COTHSRKZFactor Class


COTHSRKVolume Class

Enthalpy Vapour and

Liquid

COTHSRKEnthalpy Class


COTHSRKEntropy Class

Isobar ic heat capacity Vapour andLiquid

COTHSRKCp Class


Vapour andLiquid

COTHSRKLnFugacityCoeffClass


COTHSRKLnFugacity Class


COTHSRKCv Class


COTHSRKab_1 Class

Mixing Rule 2 Vapour and

Liquid

COTHSRKab_2 Class


COTHSRKab_3 Class


COTHSRKab_4 Class


COTHSRKab_5 Class


COTHSRKab_6 Class

Refer to the MixingRules section for theapplicable set of mixing






3-41


SRK EnthalpyThe following relation calculates the enthalpy.




COTHSRKVolume Class Vapour and Liquid

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

(3.93)


COTHSRKEnthalpy Class Vapour and Liquid

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

H H IG PV RT – 1b--- a T

∂a∂T ------ –

⎝ ⎠⎛ ⎞ V

V b+-------------ln+= –




3-42

SRK EntropyThe following relation calculates the entropy.

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


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

(3.94)


COTHSRKEntropy Class Vapour and Liquid

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

(3.95)

S S IG

R V b – RT ------------⎝ ⎠⎛ ⎞ 1

b--- ∂a

∂T ------⎝ ⎠⎛ ⎞ V b+

V -------------⎝ ⎠⎛ ⎞ln – ln= –

C p C p IG

T ∂2 P ∂T

2---------⎝ ⎠⎜ ⎟⎛ ⎞

V

V RT

∂V

∂T ------⎝ ⎠⎛ ⎞

P

2

∂V ∂ P ------⎝ ⎠⎛ ⎞

T

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

∞

V

∫ – = –




3-43


SRK Fugacity CoefficientThe following relation calculates the fugacity coefficient.



COTHSRKCp Class Vapour and Liquid

(3.96)

(3.97)

(3.98)


COTHSRKLnFugacityCoeffClass

Vapour and Liquid

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

φiln V b – ( ) bV b – ------------ a

RT b---------- b

b--- a

a--- – 1 –

⎝ ⎠⎛ ⎞ V b+

V -------------⎝ ⎠⎛ ⎞ln+ +ln=

a ∂n

2

a∂n------------=

b ∂nb

∂n i---------=




3-44

SRK FugacityThe following relation calculates the fugacity for a specificphase.


SRK Cv (isochoric)The following relation calculates the isochoric heat capacity.


(3.99)


COTHSRKLnFugacity Class Vapour and Liquid

(3.100)


COTHSRKCv Class Vapour and Liquid

f i φi yi P =

C v C p

T ∂ P ∂T ------⎝ ⎠⎛ ⎞2

V

∂ P ∂V ------⎝ ⎠⎛ ⎞

T

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




3-45

Mixing RulesThe mixing rules available for the SRK EOS state are shownbelow.

(3.101)

(3.102)

(3.103)

(3.104)

(3.105)

(3.106)

(3.107)


nc

∑i 1=

nc

∑=

b b i xii 1=

nc

∑=

a ij ξij a ci a cj α iα j=

α i 1 κij – 1 T ri0.5 – ( )=

a ci0.42748 R

2T ci

2

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

b i0.08664 RT ci

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

κi 0.48 1.574 ωi 0.176 ωi2 – +=




3-46





Mixing Rule 3The 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, ξij, which isdefined as :

(3.108)

(3.109)

(3.110)


+ +=


-------+ +=

ξij 1 xi Aij B ij C ij T 2

+ +( ) – x j A ji B ji T C ji T 2

+ +( ) – =




3-47



(3.111)ξij 1 x i Aij B ij T C ijT

-------+ +⎝ ⎠⎛ ⎞ – x j A ji B ji

C jiT

-------+ +⎝ ⎠⎛ ⎞ – =






3-49

3.1.6 Redlich-Kwong Equation

of StateIn 1949, Redlich and Kwong proposed a modification of the vander Waals equation where the universal critical compressibilitywas reduced to a more reasonable number (i.e., 0.3333). Thismodification, known as the Redlich-Kwong (RK) equation ofstate, was very successful, and for the first time, a simple cubicequation of state would be used for engineering calculationswith acceptable accuracy. Previous equations used forengineering calculations were modifications of the virialequation of state, notably the Beatie-Bridgeman and theBenedict-Webb-Rubin (BWR).

These other equations, although capable of accuratelyrepresenting the behaviour of pure fluids, had many adjustableconstants to be determined through empirical fitting of PVTproperties, and received limited use. On the other hand, the RKequation required only T c and P c , and (fortunately) theprinciples of corresponding states using T c and P c applies withfair accuracy for simple hydrocarbon systems. This combinationof simplicity and relative accuracy made the RK equation ofstate a very useful tool for engineering calculations inhydrocarbon systems. The Redlich-Kwong equation of state isrepresented by the following equation:

(3.114) P RT V b – ------------ a

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

T ------- – =




3-50

and the reduced form is represented by:

Although simple systems approximately obey the correspondingstates law as expressed by the RK equation, furtherimprovements were required, especially when using theequation to predict the vapour pressure of pure substances. Itwas noted by several researchers, notably Pitzer, that thecorresponding states principle could be extended by the use of athird corresponding state parameter, in addition to T c and P c .The two most widely used third parameters are the criticalcompressibility ( Z c ) and the acentric factor ( ω). The acentricfactor has a special appeal for equations of state based on thevan der Waals ideas, since it is related to the lack of sphericity ofa given substance. Pitzer defined the acentric factor as:

In this way, one may consider developing an equation of stateusing T c, P c, and ω as correlating parameters.

To apply the RK EOS to mixtures, mixing rules are required forthe “ a” and “ b” terms in Equation (3.64) . Refer to the MixingRules section for the set of mixing rules applicable.

(3.115)

(3.116)

P r 3T r

V r 3Ωb – ----------------------9Ωa

T r 0.5

V r V r 3 Ωb+( )------------------------------------------- – =

Ωa 0.42748=

Ωb 0.08664=

a Ωa R2 T c

2.5

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

b Ωb RT c

P c-----=

Pitzer's definition is basedon an empirical study inwhich it was verified thatnoble gases have areduced pressure of about0.1 at T r = 0.7.

ω 1 – P r log – = when T r 0.7=




3-51

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




COTHRKZFactor Class


COTHRKVolume Class


COTHRKEnthalpy Class


COTHRKEntropy Class

Isobar ic heat capacity Vapour andLiquid

COTHRKCp Class


Vapour andLiquid

COTHRKLnFugacityCoeffClass


COTHRKLnFugacity Class


COTHRKCv Class


COTHRKab_1 Class


COTHRKab_2 Class


COTHRKab_3 Class


COTHRKab_4 Class


COTHRKab_5 Class


COTHRKab_6 Class




3-52

RK Z FactorThe compressibility factor is calculated as the root for the

following equation:


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

(3.117)

(3.118)

(3.119)

(3.120)

Z 3

Z 2

– Z A B – B2

– ( ) AB – + 0=

A aP

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

B bP RT -------=

V ZRT P

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




3-53


RK EnthalpyThe following relation calculates the enthalpy.




COTHRKVolume Class Vapour and Liquid

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

(3.121)


COTHRKEnthalpy Class Vapour and Liquid

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

H H IG

PV RT – 1b--- a T

∂a∂T ------ –

⎝ ⎠⎛ ⎞ V

V b+-------------ln+= –




3-54

RK EntropyThe following relation calculates the entropy.



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


(3.122)


COTHRKEntropy Class Vapour and Liquid

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

(3.123)


COTHRKCp Class Vapour and Liquid

S S IG R V b – RT ------------⎝ ⎠⎛ ⎞ 1

b--- ∂a

∂T ------⎝ ⎠⎛ ⎞ V b+

V -------------⎝ ⎠⎛ ⎞ln – ln= –

C p C p IG T

∂2 P

∂T 2---------⎝ ⎠⎜ ⎟⎛ ⎞

V

V RT

∂V ∂T ------⎝ ⎠⎛ ⎞

P

2

∂V ∂ P ------⎝ ⎠⎛ ⎞

T

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

∞

V

∫ – = –




3-55

RK Fugacity CoefficientThe following relation calculates the fugacity coefficient.


RK FugacityThe following relation calculates the fugacity for a specificphase.


(3.124)

(3.125)

(3.126)


COTHRKLnFugacityCoeff Class Vapour and Liquid

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

(3.127)


COTHRKLnFugacity Class Vapour and Liquid

φiln V b – ( ) bV b – ------------ a

RT b---------- b

b--- a

a--- – 1 –

⎝ ⎠⎛ ⎞ V b+

V -------------⎝ ⎠⎛ ⎞ln+ +ln=

a ∂n

2a

∂n------------=

b ∂nb

∂n i---------=

f i φi yi P =




3-56

RK Cv (isochoric)The following relation calculates the isochoric heat capacity.


Mixing RulesThe mixing rules available for the RK EOS state are shownbelow.

(3.128)


COTHRKCv Class Vapour and Liquid

(3.129)

(3.130)

(3.131)

(3.132)

C v C p

T ∂ P ∂T ------⎝ ⎠⎛ ⎞

V

∂ P ∂V ------⎝ ⎠⎛ ⎞

T

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


nc

∑i 1=

nc

∑=

b b i xii 1=

nc

∑=

a ij ξij a ia j=

a i 0.42748 R2

T ci

2.5

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




3-57


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




(3.133)

(3.134)

(3.135)

(3.136)

b i

0.08664 RT ci

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

ξij 1 Aij – B ij T C ij T 2+ +=


-------+ +=





3-58







(3.137)

(3.138)

(3.139)


-------+ +⎝ ⎠⎛ ⎞ – x j A ji B ji

C jiT

-------+ +⎝ ⎠⎛ ⎞ – =



ξij 1 A ij B ij T

C ijT

-------+ +⎝ ⎠⎛ ⎞ A ji B ji T

C ijT

-------+ +⎝ ⎠⎛ ⎞

xi Aij B ij T C ijT

-------+ +⎝ ⎠⎛ ⎞ x j A ji B ji T

C jiT

-------+ +⎝ ⎠⎛ ⎞+

----------------------------------------------------------------------------------------------------- – =




3-59

3.1.7 Zudkevitch-Joffee

Equation of StateThe 13Zudkevitch-Joffee (ZJ, 1970) model is a modification of theRedlich- Kwong equation of state. This model has beenenhanced for better prediction of vapour-liquid equilibria forhydrocarbon systems, and systems containing Hydrogen. Themajor advantage of this model over previous versions of the RKequation is the improved capability of predicting pure compoundvapour pressure and the simplification of the method fordetermining the required coefficients for the equation.

Enthalpy calculations for this model are performed using the

Lee-Kesler method.

The Zudkevitch-Joffe EOS is represented by the followingequation:

To apply the ZJ EOS to mixtures, mixing rules are required forthe “ a” and “ b” terms in Equation (3.84) . Refer to the MixingRules section for the set of mixing rules applicable.

(3.140) P RT V b – ------------ a

V V b+( )--------------------- – =




3-60

Property MethodsCalculation methods for ZJ EOS are shown in the following table.


ZJ Z FactorThe compressibility factor is calculated as the root for thefollowing equation:



Z Factor Vapour and Liquid COTHZJZFactor Class

Molar Volume Vapour and Liquid COTHZJVolume Class

Enthalpy Vapour and Liquid COTHZJEnthalpy Class

Entropy Vapour and Liquid COTHZJEntropy Class

Isobaric heat capaci ty Vapour and Liquid COTHZJCp Class


Vapour and Liquid COTHZJLnFugacityCoeffClass

Fugacity calculation Vapour and Liquid COTHZJLnFugacity Class

Isochoric heat capacity Vapour and Liquid COTHZJCv Class

Mixing Rule 1 Vapour and Liquid COTHZJab_1 ClassMixing Rule 2 Vapour and Liquid COTHZJab_2 Class

Mixing Rule 3 Vapour and Liquid COTHZJab_3 Class




(3.141)

(3.142)

(3.143)

Z 3 Z 2 – Z A B – B2 – ( ) AB – + 0=

A aP

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

B bP RT -------=




3-61

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


ZJ EnthalpyThe following relation calculates the enthalpy.


(3.144)


COTHZJVolume Class Vapour and Liquid

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

(3.145)

V ZRT P

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

H H IG PV RT – 1b--- a T ∂a

∂T ------ –

⎝ ⎠⎛ ⎞ V

V b+-------------ln+= –




3-62


ZJ EntropyThe following relation calculates the entropy.




COTHLeeKeslerEnthalpyClass

Vapour and Liquid

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

(3.146)


COTHLeeKeslerEntropyClass Vapour and Liquid

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

S S IG

R V b – RT ------------⎝ ⎠⎛ ⎞ 1

b--- ∂a

∂T ------⎝ ⎠⎛ ⎞ V b+

V -------------⎝ ⎠⎛ ⎞ln – ln= –




3-63

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


ZJ Fugacity CoefficientThe following relation calculates the fugacity coefficient:

(3.147)


COTHLeeKeslerCp Class Vapour and Liquid

(3.148)

(3.149)

(3.150)

C p C p IG T

∂2 P

∂T 2---------⎝ ⎠⎜ ⎟⎛ ⎞

V

V RT

∂V ∂T ------⎝ ⎠⎛ ⎞

P

2

∂V ∂ P ------⎝ ⎠⎛ ⎞

T

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

∞

V

∫ – = –

φiln V b – ( ) bV b – ------------ a

RT b---------- b

b--- a

a--- – 1 –

⎝ ⎠⎛ ⎞ V b+

V -------------⎝ ⎠⎛ ⎞ln+ +ln=

a ∂n2a

∂n------------=

b ∂nb

∂n i---------=




3-64


ZJ FugacityThe following relation calculates the fugacity for a specific

phase.


ZJ Cv (isochoric)The following relation calculates the isochoric heat capacity.


COTHZJLnFugacityCoeff Class Vapour and Liquid

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

(3.151)


COTHZJLnFugacity Class Vapour and Liquid

(3.152)

f i φi yi P =

C v C p

T ∂ P ∂T ------⎝ ⎠⎛ ⎞2

V

∂ P ∂V ------⎝ ⎠⎛ ⎞

T

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




3-65


Mixing RulesThe mixing rules available for the ZJ EOS state are shown below.

(for T r < 0.9) ( 41Soave, 1986)

With M 1 and M 2 determined at 0.9 T c to match the value and


COTHZJCv Class Vapour and Liquid

(3.153)

(3.154)

(3.155)

(3.156)

(3.157)

(3.158)

a x i x j a ij( ) j 1=

nc

∑i 1=

nc

∑=

b b i x ii 1=

nc

∑=

a ij ξij a ia jα iα j=

α i sub c – ri t ical 1 D k

P r T r ----- 10ln – ln –

k 1+2------------

D k P r T r ----- 10ln – ln –

k 1 –

k 3=

10

∑+

k 1=

2

∑+=

P r P i sa t P ci ⁄ =

α su pe r cr it ic al – ln 2 M 1 1 T r M 2 – ( )=




3-66

slope of the vapour pressure curve ( 14Mathias, 1983):



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, ξij, which is

(3.159)

(3.160)

(3.161)

(3.162)

(3.163)

(3.164)

M 1 M 212---

d αdT r --------⎝ ⎠

⎛ ⎞

0.9 T c – =

M 2 M 1 1 –

M 1----------------=

a ci

0.42748 R2T ci2

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

b i

0.08664 RT ci

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

κi 0.48 1.574 ωi 0.176 ωi2 – +=


+ +=




3-67

defined as:


(3.165)ξij 1 Aij – B ij T C ijT -------+ +=




3-68





where: Aij , B

ij , and C

ij are asymmetric binary interaction parameters

(3.166)

(3.167)

(3.168)



-------+ +⎝ ⎠⎛ ⎞ – x j A ji B ji

C jiT

-------+ +⎝ ⎠⎛ ⎞ – =








3-70

The KD equation of state is similar to the SRK equation of state,with the following modifications:

• Inclusion of a second energy parameter. The a i ’

secondary energy parameter is a function of thehydrocarbon structure expressed as a group factor G i .The G i factor is assumed to be zero for all non-hydrocarbons, including water.

• Different alpha function for water ( 16Kabadi and Danner,1985).

The interaction parameters between water and hydrocarbonwere generalized by Twu and Bluck, based on the k ij valuesgiven by Kabadi and Danner:

where: Watson is the hydrocarbon characterization factor, definedas:

The group factors G i are expressed as a perturbation fromnormal alcane values as generalized by 17Twu and Bluck (1988):

(3.171)

(3.172)

(3.173)

(3.174)

(3.175)

(3.176)

(3.177)

k iw

0.315 Watson 10.5<0.3325 – 0.061667 Watson+ 10.5 Watson 13.5≤≤

0.5 Watson 13.5>⎩⎪⎨

⎪⎧

=

WatsonT b3

SG----------=

Gln G ° 1 2 f +1 2 f – --------------⎝ ⎠⎛ ⎞

2ln=

f f 1 SG f 2 SG 2Δ+Δ=

f 1 C 1 C 2 T bln ⁄ R( )+=

f 2 C 3 C 4 T bln ⁄ R( )+=

SGΔ e5 SG ° SG – ( )1 – =




3-71

The alcane group factor Go is calculated as:

To apply the KD EOS to mixtures, mixing rules are required forthe “ a” and “ b” terms in Equation (3.170) . Refer to theMixing Rules section for the applicable set of mixing rules.

(3.178)

(3.179)

(3.180)

(3.181)

Coefficients

a 1 = 0.405040

a 6 = -0.958481

a 2 = 1.99638 c 1 = -0.178530

a 3 = 34.9349 c 2 = 1.41110

a 4 = 0.507059

c 3 = 0.237806

a 5 = 1.2589 c 4 = -1.97726

G° 1.358 – 426 1.358 – ----------------------------⎝ ⎠⎛ ⎞

a 5 1a 4-----

N gv a 6 F °+

N gv F ° – ---------------------------⎝ ⎠⎛ ⎞ln=

N gv1 a 6e

a 4 – +

1 e a 4 – – -------------------------=

F ° 1 a 3e

a 1 – +

1 ea 1 –

– ------------------------ 1 e

a 1τ – –

1 a 3ea 1 – τ

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

τ T b 200.99 –

2000 200.99 – ----------------------------------⎝ ⎠⎛ ⎞

a 2

=

(3.182)

(3.183)

(3.184)

SG ° 0.843593 0.128624 β – 3.36159 β3 – 13749.5 β12 – =

β 1T bT c----- – =

T b

c°------- 0.533272 0.191017 3 – ×10 T b 0.779681 7 – ×10 T b2 0.284376 10 – ×10 T b

3 – 95.9468T b

100---------⎝ ⎠⎛ ⎞

13 – + + +=






3-73

KD Z FactorThe compressibility factor is calculated as the root for thefollowing equation:


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


(3.185)

(3.186)

(3.187)

(3.188)


COTHKDVolume Class Vapour and Liquid

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

Z 3 Z 2 – Z A B – B2 – ( ) AB – + 0=

A aP

R2T

2------------=

B bP RT -------=

V ZRT P

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




3-74

KD EnthalpyThe following relation calculates the enthalpy.



KD EntropyThe following relation calculates the entropy.



(3.189)


COTHKDEnthalpy Class Vapour and Liquid

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

(3.190)


COTHKDEntropy Class Vapour and Liquid

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

H H IG PV RT – 1b--- a T

∂a∂T ------ –

⎝ ⎠⎛ ⎞ V

V b+-------------ln+= –

S S IG

R V b –

RT ------------⎝ ⎠

⎛ ⎞ 1b---

∂a∂T ------⎝ ⎠⎛ ⎞ V b+

V -------------⎝ ⎠

⎛ ⎞

ln – ln= –




3-75

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


KD Fugacity CoefficientThe following relation calculates the Fugacity Coefficient:

(3.191)


COTHKDCp Class Vapour and Liquid

(3.192)

(3.193)

(3.194)

C p C p IG

T ∂2

P

∂T 2---------⎝ ⎠⎜ ⎟⎛ ⎞

V

V RT

∂V ∂T ------⎝ ⎠⎛ ⎞

P

2

∂V ∂ P ------⎝ ⎠⎛ ⎞

T

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

∞

V

∫ – = –

φiln V b – ( ) bV b – ------------ a

RT b---------- b

b--- a

a--- – 1 –

⎝ ⎠⎛ ⎞ V b+

V -------------⎝ ⎠⎛ ⎞ln+ +ln=

a ∂n2a

∂n

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

b ∂nb

∂n i---------=




3-76


KD FugacityThe following relation calculates the fugacity for a specificphase.



COTHKDLnFugacityCoeff Class Vapour and Liquid

The volume, V, is calculated using KD Molar Volume . Forconsistency, the KD Fugacity Coefficient always calls the KDVolume for the calculation of V.

(3.195)


COTHKDLnFugacity Class Vapour and Liquid

f i φi yi P =




3-77

KD Cv (isochoric)The following relation calculates the isochoric heat capacity.


(3.196)


COTHKDCv Class Vapour and Liquid

C v C p

T ∂ P ∂T ------⎝ ⎠⎛ ⎞2

V

∂ P ∂V ------⎝ ⎠⎛ ⎞

T

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






3-79






(3.205)

(3.206)

(3.207)


+ +=


-------+ +=





3-80







(3.208)

(3.209)

(3.210)


-------+ +⎝ ⎠⎛ ⎞ – x j A ji B ji

C jiT

-------+ +⎝ ⎠⎛ ⎞ – =



ξij 1 A ij B ij T

C ijT

-------+ +⎝ ⎠⎛ ⎞ A ji B ji T

C ijT

-------+ +⎝ ⎠⎛ ⎞

xi Aij B ij T C ijT

-------+ +⎝ ⎠⎛ ⎞ x j A ji B ji T

C jiT

-------+ +⎝ ⎠⎛ ⎞+

----------------------------------------------------------------------------------------------------- – =




3-81

3.1.9 The Virial Equation of

StateThe Virial equation of state has theoretical importance since itcan be derived from rigorous statistical mechanical arguments.It is represented as an infinite sum of power series in theinverse of the molar volume:

where: B is the second virial coefficient, C the third, etc.

The above equation may be rewritten as a series in molardensity:

and pressure:

The last format is not widely used since it gives an inferiorrepresentation of Z over a range of densities or pressures ( 6Reid,Prausnitz and Poling, 1987). It is clear that B can be calculatedas:

(3.211)

(3.212)

(3.213)

(3.214)

(3.215)

The term Virial comesfrom the Latin vis (force)and refers to theinteraction forces between2, 3 or more molecules.

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

V --- C

V 2------ D

V 3------ …+ + + += =

Z 1 Bρ C ρ2 D ρ3 …+ + + +=

Z 1 B' P C ' P 2

D' P 3 …+ + + +=

Z 1 Bρ C ρ2 D ρ3 …+ + + +=

ρ∂∂ Z ⎝ ⎠⎛ ⎞

T B 2 C ρ 3 D ρ2 …+ + +=




3-82

and taking the limit where ρ -> 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 ρ∂∂ Z

⎝ ⎠⎛ ⎞

T ρ 0→lim=

C ρ2

2

∂∂ Z

⎝ ⎠⎜ ⎟⎛ ⎞

T ρ 0→lim= D

ρ3

3

∂∂ Z

⎝ ⎠⎜ ⎟⎛ ⎞

T ρ 0→lim=

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

V ---+= =

B B 0( ) ω B 1( )+=




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 μR 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( )

ω B

1( )

B

2( )

+ +=

μ R

10 5μ2 P cT 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 ,( )+ +=




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 μij

*'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 ijΔT ij∗

-----------⎝ ⎠⎜ ⎟⎛ ⎞

exp=

B ij D

chemical ,( ) b ij0 E ij 1

1500 ηij

T -------------------⎝ ⎠⎛ ⎞exp –

⎝ ⎠⎛ ⎞=




3-85

where:

For pure components:

(3.227)

1

T ij*'

------ 1

T ij*

------ 1.6 ωij – =

T ij* T

εij k ⁄ ( )-----------------=

b ij0 1.26184 σij

3= cm 3 gmol ⁄ ( )

μij*' μij

*= if μij* 0.04<

μij*' 0= if 0.04 μij

*≤ 0.25<

μij*' μij

* 0.25 – = if μij* 0.25≥

A ij 0.3 – 0.05 μij* – =

H ijΔ 1.99 0.2 μij*2+=

μij* 7243.8 μiμ j

εij

k -----⎝ ⎠⎛ ⎞σij

3---------------------------=

E ij ηij650

εij

k -----⎝ ⎠⎛ ⎞ 300+

-------------------------- 4.27 –

⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞

⎩ ⎭⎪ ⎪⎨ ⎬⎪ ⎪⎧ ⎫

exp= if ηij 4.5<

E ij ηij42800

εij

k -----⎝ ⎠⎛ ⎞ 22400+

-------------------------------- 4.27 –

⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞

⎩ ⎭⎪ ⎪⎨ ⎬⎪ ⎪⎧ ⎫

exp= if ηij 4.5≥

ωi 0.006026 R Di 0.02096 R Di2 0.001366 R Di

3 – +=

εij

k -----

εij

k -----⎝ ⎠⎛ ⎞′ 1 ξC 1 1 ξ 1

C 12

------+⎝ ⎠⎛ ⎞ –

⎝ ⎠⎛ ⎞ –

⎝ ⎠⎛ ⎞=

σi σi' 1 ξC 2+( )1 3 ⁄ =

εi

k ----⎝ ⎠⎛ ⎞′ T c i, 0.748 0.91 ωi 0.4

ηi

2 20 ωi+--------------------- – +

⎝ ⎠⎛ ⎞=




3-86

and

For the cross parameters:

(3.228)

(3.229)

(3.230)

σi' 2.44 ωi – ( ) 1.0133T c i,

P c i,---------

⎝ ⎠⎛ ⎞

1 3 ⁄ =

ξ 0= if μi 1.45<

or

ξ 1.7941

7×10 μi4

2.8821.882 ωi

0.03 ωi+---------------------- –

⎝ ⎠⎛ ⎞T c i, σi'

6 εi

k ----⎝ ⎠⎛ ⎞′

-----------------------------------------------------------------------------------

⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞

if μi 1.45≥=

C 116 400 ωi+

10 400 ωi+--------------------------- an d C 2

310 400 ωi+---------------------------==

ωij12--- ωi ω j+( )=

εij

k -----⎝ ⎠⎛ ⎞ εij

k -----⎝ ⎠⎛ ⎞′ 1 ξ′C 1′+( )=

σij σij′ 1 ξ – ′C 2′( )=

εij

k -----⎝ ⎠⎛ ⎞′ 0.7

εii

k -----⎝ ⎠⎛ ⎞ ε jj

k -----⎝ ⎠⎛ ⎞

12---

0.61

εii k ⁄ ------------ 1ε jj k ⁄ ------------+

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




3-87

Thus, Hayden-O'Connell models the behaviour of a mixturesubject to physical (polarity) and chemical (associative andsolvation) forces as a function of T c , P c , RD (radius of gyration),μ (dipole moment) and two empirical constants that describe the"chemical" behaviour of the gas:

This is discussed in more detail in the next section.

Mixing RulesFor a multi-component mixture, it can be shown that B mix isrigorously calculated by:

and the fugacity coefficient for a component i in the mixture

(3.231)

(3.232)

(3.233)

σij σii σ jj( )12---

=

ξ′u i

2 ε jjk -----⎝ ⎠⎛ ⎞

2 3 ⁄ σ jj

4

εij

k -----⎝ ⎠⎛ ⎞′ σij

6----------------------------------= if μi 2 and μ j≥ 0=

ξ′u2 εii

k -----⎝ ⎠⎛ ⎞2σii

4

εij

k -----⎝ ⎠⎛ ⎞′ σ′

ij6

---------------------------= if μ j 2 and μi≥ 0=

ξ′ 0 for all other values of μi and μ j=

C 1′ 16 400 ωij+

10 400 ωij+---------------------------- - an d C 2

′ 310 400 ωij+---------------------------- -==

ηii association pa ra meter =

ηij solvation pa ra meter =

Bmi x yi y j B ij j∑

i∑=




3-88

comes from:

(3.234)φiln 2 yi B ij Bmi x – j∑⎝ ⎠⎜ ⎟

⎛ ⎞ P RT -------=




3-89

Vapour Phase Chemical Associationusing the Virial EquationAlthough it was suggested many years ago that the non-idealityin mixtures could be explained by pseudo-chemical reactionsand formation of complexes, there is evidence that this is trueonly in a few special cases. Of special practical importance aremixtures which contain carboxylic acids. Carboxylic acids tend todimerize through strong hydrogen bonding.

This is not limited to carboxylic acids alone; hydrofluoric acidforms polymers (usually hexamers) and the hydrogen bondingcan happen with dissimilar molecules.

Usually, hydrogen bonding between similar molecules is calledassociation, while bonding between dissimilar molecules iscalled solvation.

The hydrogen bonding process can be observed as a chemicalreaction:

where: i and j are monomer molecules and ij is the complex formedby hydrogen bonding

The following may be written to describe the chemical reaction:

where: Z is the true mole fraction of the species in equilibrium

is the fugacity coefficient of the true species

P is the system pressure

k ij is the reaction equilibrium constant

(3.235)

(3.236)

i j ij↔+

k ij f ij f i f j------

Z ij φ#

ij

Z i Z jφi#φ#

j P ---------------------------= =

φ#




3-90

If y i is defined as the mole fraction of component i in the vapourphase, disregarding dimerization, it can be shown that:

where: denotes the apparent fugacity coefficient of component i

If it is assumed that the vapour solution behaves like an idealsolution (Lewis), the following may be written:

where: B i F is the contribution to the second virial coefficient from

physical forces

If the Lewis ideal solution is carried all the way:

The chemical equilibrium constant is found from the relation:

where: Bij

D is the contribution of dimerization to the second virialcoefficient

(3.237)

(3.238)

(3.239)

(3.240)

(3.241)

φ#i Z i φi yi= or φi

φi# Z i

yi----------=

φi

φln i# B i P

RT ----------=

k ijφij Z ij P

φi Z i P φ j Z j P ----------------------------=

k ij Z ij

Z i Z j---------- 1

P ---

Bij F P

RT -------

⎝ ⎠⎛ ⎞exp

B ii F P

RT -------

⎝ ⎠⎛ ⎞ B jj F P

RT -------

⎝ ⎠⎛ ⎞expexp------------------------------------------------------------×=

k ij B ij

D – 2 δij – ( ) RT

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

δij0 i j≠1 i j=⎩⎨⎧

=




3-91

Therefore:

The calculation of the fugacity coefficient for species i and j isaccomplished by solving the previous chemical equilibriumconstant equation combined with the restriction that the sum ofZ i , Z j and Z ij is equal to 1.

Application of the Virial EquationThe equation enables you to better model vapour phasefugacities of systems displaying strong vapour phaseinteractions. Typically this occurs in systems containingcarboxylic acids, or compounds that have the tendency to formstable hydrogen bonds in the vapour phase. In these cases, thefugacity coefficient shows large deviations from ideality, even atlow or moderate pressures.

The regression module contains temperature dependentcoefficients for carboxylic acids. You can overwrite these bychanging the Association (ij) or Solvation (ii) coefficients fromthe default values.

If the virial coefficients need to be calculated, the softwarecontains correlations utilizing the following pure componentproperties:

• critical temperature• critical pressure• dipole moment• mean radius of gyration• association parameter• association parameter for each binary pair

(3.242)k ij

Z ij Z i Z j---------- 1

P ---

B ij F P

RT -------⎝ ⎠⎛ ⎞exp

B ii F P

RT -------

⎝ ⎠⎛ ⎞ B jj

F P RT -------

⎝ ⎠⎛ ⎞expexp

------------------------------------------------------------×=

B ij

D – 2 δij – ( ) RT

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




3-92

The equation is restricted to systems where the density ismoderate, typically less than one-half the critical density. TheVirial equation used is valid for the following range:

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


(3.243)


Molar Volume Vapour COTHVirial_Volume Class

Enthalpy Vapour COTHVirial_Enthalpy Class

Entropy Vapour COTHVirial_Entropy ClassIsobaric heat capacity Vapour COTHVirial_Cp Class


Vapour COTHVirial_LnFugacityCoeffClass

Fugacity calculation Vapour COTHVirial_LnFugacity Class

Density Vapour COTHVirial_Density Class

Isochoric HeatCapacity

Vapour COTHVirial_Cv Class

Gibbs Energy Vapour COTHVirial_GibbsEnergy Class

Helmholtz Energy Vapour COTHVirial_HelmholtzEnergyClass

Z Factor Vapour COTHVirial_ZFactor Class

P T 2---

yi P c ii 1=

m

∑

yiT c ii 1=

m

∑---------------------≤






3-94


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


Virial Fugacity CoefficientThe following relation calculates the fugacity coefficient:



COTHVirial_Entropy Class Vapour

(3.247)


COTHVirial_Cp Class Vapour

(3.248)


COTHVirial_LnFugacityCoeffClass

Vapour

C p C p ° – T T 2

2

∂∂ P ⎝ ⎠⎜ ⎟⎛ ⎞

V d

∞

V

∫ T T ∂∂ P ⎝ ⎠⎛ ⎞

V

2

T ∂∂ P ⎝ ⎠⎛ ⎞

T

----------------- – R – =

φiln 2 yi B ij Bmi x – j∑

⎝ ⎠⎜ ⎟⎛ ⎞ P

RT -------=




3-95

Virial FugacityThe following relation calculates the fugacity for a specificphase.


Virial DensityThe following relation calculates the molar density for a specificphase.


Virial Cv (isochoric)The following relation calculates the isochoric heat capacity.

(3.249)


COTHVirial_LnFugacityClass

Vapour and Liquid

(3.250)


COTHVirial_Density Class Vapour and Liquid

(3.251)

f i φi yi P =

ρ P ZRT -----------=

C v

C v° – T

T 2

2

∂

∂ P

⎝ ⎠

⎜ ⎟⎛ ⎞

V d

∞

V

∫=




3-96


Virial Gibbs EnergyThe following relation calculates the Gibbs energy.


Virial Helmholtz EnergyThe following relation calculates the Helmholtz energy.



COTHVirial_Cv Class Vapour and Liquid

(3.252)


COTHVirial_GibbsEnergyClass

Vapour

(3.253)


COTHVirial_HelmholtzEnergyClass

Vapour

G A RT Z 1 – ( )+=

A Ao RT V V B – -------------ln RT V

V o------ln – = –




3-97

Virial Z FactorThe following relation calculates the Z Factor.


3.1.10 Lee-Kesler Equation ofState

The 50Lee-Kesler (LK, 1975) method is an effort to extend themethod originally proposed by Pitzer to temperatures lower than0.8 T r . Lee and Kesler expanded Pitzer's method expressing thecompressibility factor as:

where: Z o = the compressibility factor of a simple fluid

Z r = the compressibility factor of a reference fluid

They chose the reduced form of the BWR EOS to represent bothZ o and Z r :

(3.254)


COTHVirial_ZFactor Class Vapour

(3.255)

(3.256)

Z 1 BV ---+=

Z Z ° ω

ωr

------ Z r Z ° – ( )+=

Z 1 B

V r ----- C

V r 2

------ D

V r 5

------ D

T r 3V r

3------------ β γ

V r 2

------ – ⎝ ⎠⎜ ⎟⎛ ⎞

e

γV r

2------⎝ ⎠⎛ ⎞ –

+ + + +=




3-98

where:

The constants in these equations were determined usingexperimental compressibility and enthalpy data. Two sets ofconstants, one for the simple fluid ( ωo = 0) and one for thereference fluid ( ωr=0.3978, n-C 8 ) were determined.

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


CalculationMethod Applicable Phase Property Class Name

Enthalpy Vapour and Liquid COTHLeeKeslerEnthalpy Class

Entropy Vapour and Liquid COTHLeeKeslerEntropy Class

Isobaric heat

capacity

Vapour and Liquid COTHLeeKeslerCp Class

V r VP c RT c---------=

B b 1b2

T r ----- – b3

T r 2

----- – b4

T r 4

----- – =

C c 1c2

T r ----- –

c3

T r 3

-----+=

D d 1d 2T r -----+=






3-100

where:


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


(3.261)

(3.262)


COTHLeeKeslerEntropyClass

Vapour and Liquid

The values of T c and V c are calculated from the Mixing Rules .

(3.263)


COTHLeeKeslerCp Class Vapour and Liquid

T r

T

T c-----=

V r V V c-----=

C p C p IG

T ∂2 P

∂T 2

---------

⎝ ⎠

⎜ ⎟⎛ ⎞

V

V RT

∂V ∂T ------⎝ ⎠⎛ ⎞

P

2

∂V ∂ P ------⎝ ⎠⎛ ⎞T

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

∞

V

∫ – = –




3-101

Mixing RulesFor mixtures, the Critical properties for the LK EOS state aredefined as follows.

3.1.11 Lee-Kesler-PlöckerThe Lee-Kesler-Plöcker equation is an accurate general methodfor non-polar substances and mixtures. 3Plöcker et al, applied

the Lee-Kesler equation to mixtures, which itself was modifiedfrom the BWR equation.

The compressibility factors are determined as follows:

(3.264)

(3.265)

ω xiωii 1=

N

∑=

z c i0.2905 0.0851 ωi – =

V c i

Z c i RT c i

P c i

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

V c18--- xi x j V c i

13---

V c j

13---

+⎝ ⎠⎜ ⎟⎛ ⎞

3

j 1=

N

∑i 1=

N

∑=

T c1

8V c--------- xi x j V c i

13--- V c j

13---+

⎝ ⎠⎜ ⎟⎛ ⎞

3

T c iT c j

( )0.5

j 1=

N

∑i 1=

N

∑=

P c 0.2905 0.085 ω – ( ) RT c

V c---------=

The Lee-Kesler-Plöckerequation does not use theCOSTALD correlation incomputing liquid density.This may result indifferences whencomparing results

z z o( ) ω

ω r ( )--------- z r ( )

z o( )

– ( )+=

z pv RT -------

p r vr

T r ---------- z T r vr Ak , ,( )= = =




3-102

where:

Mixing rules for pseudocritical properties are as follows:

where:

(3.266)

(3.267)

z 1 B

vr

---- C

vr 2

----- D

vr 5

----- C 4T r

3vr 2

----------- β γvr

2-----+

γ –

vr 2

-----exp+ + + +=

vr p cv

RT c---------=

C c 1

c2

T r ----- –

c3

T r 2

-----+=

ω o( )0=

B b 1b2

T r ----- –

b3

T r 2

----- – b4

T r 3

----- – =

D d 1d 2

T r ----- – =

ω r ( )0.3978=

T cm1

V cmη---------

⎝ ⎠⎜ ⎟⎛ ⎞

xi x j vc ij

j∑

i∑=

T c ijT c i

T c j( )1 2 ⁄

= T c iiT c i

= T c jjT c j

=

vcm xi x jvc ij

j∑

i∑= vc ij

18--- vc i

1 3 ⁄ vc j

1 3 ⁄ +( )

3=

vc i z c i

RT c i

pc i

----------= z c i0.2905 0.085 ωi – =

p cm z cm

RT cm

vcm

------------= z cm0.2905 0.085 ωm – =

ωm xiωii

∑=




3-103

3.2 Activity ModelsAlthough equation of state models have proven to be veryreliable in predicting properties of most hydrocarbon-basedfluids over a large range of operating conditions, theirapplication has been limited to primarily non-polar or slightlypolar components. Polar or non-ideal chemical systems havetraditionally been handled using dual model approaches. In thisapproach, an equation of state is used for predicting the vapourfugacity coefficients (normally ideal gas or the Redlich-Kwong,Peng-Robinson or SRK equations of state) and an activitycoefficient model is used for the liquid phase. Although there isconsiderable research being conducted to extend equation ofstate applications into the chemical arena (e.g., the PRSVequation), the state of the art of property predictions forchemical systems is still governed mainly by activity models.

Activity models are much more empirical in nature whencompared to the property predictions in the hydrocarbonindustry. For this reason, they cannot be used as reliably as theequations of state for generalized application or extrapolatedinto untested operating conditions. Their adjustable parametersshould be fitted against a representative sample of experimentaldata and their application should be limited to moderatepressures. Consequently, caution should be exercised whenselecting these models for your simulation.

The phase separation or equilibrium ratio K i for component i (defined in terms of the vapour phase fugacity coefficient andthe liquid phase activity coefficient), is calculated from thefollowing expression:

where: γi = Liquid phase activity coefficient of component i

f i o= Standard state fugacity of component i

P = System pressure

f i = Vapour phase fugacity coefficient of component i

(3.268)

Activity models generatethe best results when theyare applied in theoperating region in whichthe interaction parameterswere generated.

K i yi

xi----

γi f i° P φi----------= =



3-104 Activity Models

3-104

Although for ideal solutions the activity coefficient is unity, formost chemical (non-ideal) systems this approximation isincorrect. Dissimilar chemicals normally exhibit not only large

deviations from an ideal solution, but the deviation is also foundto be a strong function of the composition. To account for thisnon-ideality, activity models were developed to predict theactivity coefficients of the components in the liquid phase. Thederived correlations were based on the excess Gibbs energyfunction, which is defined as the observed Gibbs energy of amixture in excess of what it would be if the solution behavedideally, at the same temperature and pressure.

For a multi-component mixture consisting of n i moles ofcomponent i , the total excess Gibbs free energy is representedby the following expression:

where: γi is the activity coefficient for component i

The individual activity coefficients for any system can beobtained from a derived expression for excess Gibbs energyfunction coupled with the Gibbs-Duhem equation. The earlymodels (Margules, van Laar) provide an empiricalrepresentation of the excess function that limits their

application. The newer models such as Wilson, NRTL andUNIQUAC use the local composition concept and provide animprovement in their general application and reliability. All ofthese models involve the concept of binary interactionparameters and require that they be fitted to experimental data.

Since the Margules and van Laar models are less complex thanthe Wilson, NRTL and UNIQUAC models, they require less CPUtime for solving flash calculations. However, these are older andmore empirically based models and generally give poorer resultsfor strongly non-ideal mixtures such as alcohol-hydrocarbonsystems, particularly for dilute regions.

(3.269)G E

RT n i γiln( )∑=




3-105

The following table briefly summarizes recommended models fordifferent applications.

Vapour phase non-ideality can be taken into account for eachactivity model by selecting the Redlich-Kwong, Peng-Robinsonor SRK equations of state as the vapour phase model. When oneof the equations of state is used for the vapour phase, thestandard form of the Poynting correction factor is always usedfor liquid phase correction.

The binary parameters required for the activity models havebeen regressed based on the VLE data collected from DECHEMA,Chemistry Data Series. There are over 16,000 fitted binary pairsin the library. The structures of all library components applicable

for the UNIFAC VLE estimation have been stored. The Poyntingcorrection for the liquid phase is ignored if ideal solutionbehaviour is assumed.

Application Margules van Laar Wilson NRTL UNIQUACBinary Systems A A A A A

multi-componentSystems

LA LA A A A

Azeotropic Systems A A A A A

Liquid-Liquid Equilibria A A N/A A A

Dilute Systems ? ? A A A

Self-Associating Systems ? ? A A A

Polymers N/A N/A N/A N/A A

Extrapolation ? ? G G G

A = Applicable; N/A = Not Applicable;? = Questionable; G = Good; LA = LimitedApplication

All the binary parameters stored in the properties libraryhave been regressed using an ideal gas model for the vapourphase.




3-106

If you are using the built-in binary parameters, the ideal gasmodel should be used. All activity models, with the exception ofthe Wilson equation, automatically calculate three phases given

the correct set of energy parameters. The vapour pressuresused in the calculation of the standard state fugacity are basedon the pure component library coefficients using the modifiedform of the Antoine equation.

3.2.1 Ideal Solution ModelThe ideal solution model is the simplest activity model thatignores all non-idealities in a liquid solution. Although this modelis very simple, it is incapable of representing complex systemssuch as those with azeotropes.

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

The calculation methods from the table are described in the

following sections.

The internally stored binary parameters have NOT beenregressed against three-phase equilibrium data.

CalculationMethod


Activity coefficient Liquid COTHIdealSolLnActivityCoeffClass

Fugacity coefficient Liquid COTHIdealSolLnFugacityCoeffClass

Fugacity Liquid COTHIdealSolLnFugacity Class

Activity coefficientdifferential wrttemperature

Liquid COTHIdealSolLnActivityCoeffDTClass

Enthalpy Liquid COTHIdealSolEnthalpy Class

Gibbs energy Liquid COTHIdealSolGibbsEnergy Class




3-107

Ideal Solution Ln Activity CoefficientThis method calculates the activity coefficient of components, i ,using the Ideal Solution model. The extended, multi-componentform of the Ideal Solution is shown in the following relation:

where: γi = activity coefficient of component i


Ideal Solution Ln FugacityCoefficientThis method calculates the fugacity coefficient of componentsusing the Ideal Solution activity model. The fugacity coefficientof component i , φi , is calculated from the following relation.

where: γi = 1

P = pressure

f i = standard state fugacity

(3.270)


COTHIdealSolLnActivityCoeff Class Liquid

(3.271)

γiln 0=

φln i ln f i

st d

P --------⎝ ⎠⎜ ⎟⎛ ⎞

=






3-109

Ideal Solution Activity CoefficientDifferential wrt TemperatureThis method calculates the activity coefficient differential wrt totemperature using the Ideal Solution model from the followingrelation.


Ideal Solution Gibbs EnergyThis method calculates the Gibbs free energy using the IdealSolution activity model from the following relation.

where: x i = mole fraction of component i

Gi = Gibbs energy of component i

(3.273)


COTHIdealSolLnActivityCoeffDT Class Liquid

(3.274)

∂ γiln∂T

------------ 0=

G xiG i RT x i xiln

i

n

∑+

i

n

∑=




3-110


Ideal Solution EnthalpyThis method calculates the enthalpy using the Ideal Solutionactivity model from the following relation.

where: x i = mole fraction of component i

H i = enthalpy of component i



COTHIdealSolGibbsEnergy Class Liquid

(3.275)


COTHIdealSolEnthalpy Class Liquid

H x i H ii

n

∑=






3-112

Regular Solution Ln ActivityCoefficientThis method calculates the activity coefficient of components, i ,using the Regular Solution model as shown in the expressionbelow.


V i = liquid molar volume of component i

δi = solubility parameter of component i


(3.276)

(3.277)


COTHRegSolLnActivityCoeff Class Liquid

γi

V i RT ------- δi ϕ jδi

j∑ –

2=ln

ϕ j

x jV j

xk V k

k

∑------------------=




3-113

Regular Solution Ln FugacityCoefficientThis method calculates the fugacity coefficient of componentsusing the Regular Solution activity model. The fugacitycoefficient of component i , φi , is calculated from the followingrelation.


P = pressuref i

std = standard state fugacity


Regular Solution Ln FugacityThis method calculates the fugacity of components using theRegular Solution activity model. The fugacity of component i , f i ,is calculated from the following relation.

(3.278)


COTHRegSolLnFugacityCoeff Class Liquid

The term, ln γi , in the above equation is exclusivelycalculated using the Regular Solution Ln Activity Coefficient .

For the standard fugacity, f i std , refer to Section 5.4 -Standard State Fugacity .

(3.279)

φln i ln γi f i

st d

P --------

⎝ ⎠⎜ ⎟⎛ ⎞

=

ln f i

ln γi x

i f

i

st d ( )=




3-114


f istd = standard state fugacity

x i

= mole fraction of component i


Regular Solution Activity CoefficientDifferential wrt TemperatureThis method calculates the activity coefficient differential wrt totemperature using the Regular Solution model from thefollowing relation.



COTHRegSolLnFugacity Class Liquid

The term, ln γi , in the above equation is exclusively calculatedusing the Regular Solution Ln Activity Coefficient . For the

standard fugacity, f i std

, refer to Section 5.4 - Standard StateFugacity .

(3.280)


COTHVanLaarLnActivityCoeffDT Class Liquid

d γilndT

------------




3-115

Regular Solution Excess GibbsEnergyThis method calculates the excess Gibbs energy using theRegular Solution activity model from the following relation.


x i = mole fraction of component i

T = temperatureR = universal gas constant


(3.281)


COTHRegSolLnActivityCoeffDT Class Liquid

The term, ln γi , in the above equation is exclusivelycalculated using the Regular Solution Ln Activity Coefficient .

G E RT x i γiln

i

n

∑=




3-116

3.2.3 van Laar ModelIn the Van Laar (

2

Prausnitz et al., 1986) activity model, it isassumed that, if two pure liquids are mixed at constant pressureand temperature, no volume expansion or contraction wouldhappen ( V E = 0 ) and that the entropy of mixing would be zero.Thus the following relation:

simplifies to:

To calculate the Gibbs free energy of mixing, the simple VanLaar thermodynamic cycle is shown below:

(3.282)

(3.283)

Figure 3.2

G E U E PV E TS E – +=

G E H E U E ==

P r e s s u r e

Pure Liquid

Liquid Mixtu re

Vap ourize eac h liquid dropping system P to a ve ry low va lue (Ideal Gas)

Co mp ress Vap our Mixt ure

Idea l Gas Mix Ideal Cases




3-117

Since U is a point function, the value of U E is:

The expression for ΔU I is:

The following is true:

Therefore:

In the van Laar model, it is assumed that the volumetricproperties of the pure fluids could be represented by the van derWaals equation. This leads to:

(3.284)

(3.285)

(3.286)

(3.287)

(3.288)

U E U I

U II

U II I

+ +=

U Δ I V ∂∂U ⎝ ⎠⎛ ⎞

T T

T ∂∂ P ⎝ ⎠⎛ ⎞

V P – = =

The expression

can be derived fromfundamentalthermodynamicrelationships.

V ∂∂U ⎝ ⎠⎛ ⎞

T T

T ∂∂ P ⎝ ⎠⎛ ⎞

V P – =

T ∂∂ P

⎝ ⎠⎛ ⎞

V T ∂∂V

⎝ ⎠⎛ ⎞

– P V ∂∂ P

⎝ ⎠⎛ ⎞

T P – =

T ∂∂ P ⎝ ⎠⎛ ⎞

V T ∂∂V ⎝ ⎠⎛ ⎞ –

P P ∂∂V ⎝ ⎠⎛ ⎞

T ⁄ =

V ∂∂U ⎝ ⎠⎛ ⎞

T P T

T ∂∂V ⎝ ⎠⎛ ⎞

P

P ∂∂V ⎝ ⎠⎛ ⎞

T

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

⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞

– =

V ∂∂U ⎝ ⎠⎛ ⎞

T

a

V 2------=




3-118

Assuming that there are x 1 moles of component 1 and x 2 ofcomponent 2 and x 1 + x 2 = 1 mole of mixture:

thus:

and:

Using the van der Waals equation:

and for a real fluid well below its critical point, should be alarge negative number (since liquids exhibit low compressibility)and consequently:

Therefore,

(3.289)

(3.290)

(3.291)

(3.292)

(3.293)

(3.294)

x1 U id

U – ( )1a 1 x1

V 2----------- V d

V 1 L

∞

∫ a 1 x1

V 1 L

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

x2 U id

U – ( )2a 1 x1

V 2

----------- V d

V 2 L

∞

∫ a 2 x2

V 1 L

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

U I Δ x1 U id

U – ( )1 x2 U id

U – ( )2+=

U I Δ a 1 x1

V 2 L

-----------a 2 x2

V 1 L

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

V ∂∂ P ⎝ ⎠⎛ ⎞T

RT

V b – ( )2-------------------- – 2a

V 3------+=

V ∂∂ P ⎝ ⎠⎛ ⎞

T

V b 0≅ – or V b≅

U I Δ a 1 x1

b1-----------

a 2 x2

b2-----------+=




3-119

It follows that:

And since two ideal gases are being mixed,

Again, it is assumed that the van der Waals equation applies.

Using the simple mixing rules for the van der Waals equation:

Finally, after some manipulation:

and:

(3.295)

(3.296)

(3.297)

(3.298)

(3.299)

(3.300)

U II Δ 0=

U II I Δ a mi x

bmi x---------- – =

a mi x xi x j a ia j∑∑ x12a 1 x2

2a 2 2 x1 x2 a 1a 2+ += =

bmi x x ii 1=

nc

∑ b i x1b1 x2b2+= =

G E x1 x2b1b2

x1b1 x2b2+----------------------------

a 1

b1---------

a 2

b2--------- –

⎝ ⎠⎜ ⎟⎛ ⎞

2

=

γ1ln A

1 A B--- x1 x2-----+

2---------------------------=

γ2ln B

1 B A--- x2 x1-----+

2---------------------------=




3-120

where:

Two important features that are evident from the activitycoefficient equations are that the log of the activity coefficient isproportional to the inverse of the absolute temperature, andthat the activity coefficient of a component in a mixture isalways greater than one. The quantitative agreement of the vanLaar equation is not good, mainly due to the use of the van derWaals equation to represent the behaviour of the condensedphase, and the poor mixing rules for the mixture.

If one uses the van Laar equation to correlate experimental data(regarding the A and B parameters as purely empirical), goodresults are obtained even for highly non-ideal systems. Onewell-known exception is when one uses the van Laar equation tocorrelate data for self-associating mixtures like alcohol-hydrocarbon.

Application of the van Laar EquationThe van Laar equation was the first Gibbs excess energyrepresentation with physical significance. The van Laar equationis a modified form of that described in "Phase Equilibrium inProcess Design" by Null. This equation fits many systems quitewell, particularly for LLE component distributions. It can be usedfor systems that exhibit positive or negative deviations fromRaoult's Law, however, it cannot predict maximas or minimas inthe activity coefficient. Therefore, it generally performs poorlyfor systems with halogenated hydrocarbons and alcohols. Due tothe empirical nature of the equation, caution should beexercised in analyzing multi-component systems. It also has atendency to predict two liquid phases when they do not exist.

(3.301) A

b1

RT -------a

1b1

---------a

2b2

--------- – ⎝ ⎠⎜ ⎟

⎛ ⎞=

Bb2

RT -------

a 1

b1---------

a 2

b2--------- –

⎝ ⎠⎜ ⎟⎛ ⎞

=

Ethanol: T c=513.9 K

Pc=6147 kPa

a=1252.5 l 2 /gmol 2

b=0.087 l 2 /gmol 2

Water: T c=647.3 K

Pc=22120 kPa

a=552.2 l 2 /gmol 2

b=0.030 l 2 /gmol 2

System: T = 25 C

Aij = 4.976




3-121

The van Laar equation has some advantages over the otheractivity models in that it requires less CPU time and canrepresent limited miscibility as well as three-phase equilibrium.

Property MethodsA quick reference of calculation methods is shown in the tablebelow for the van Laar model.



Activity coefficient Liquid COTHVanLaarLnActivityCoeff Class

Fugacity coefficient Liquid COTHVanLaarLnFugacityCoeffClass

Fugacity Liquid COTHVanLaarLnFugacity Class


Liquid COTHVanLaarLnActivityCoeffDTClass

Excess Gibbs Liquid COTHVanLaarExcessGibbsEnergyClass

Excess enthalpy Liquid COTHVanLaarExcessEnthalpy Class

Enthalpy Liquid COTHVanLaarEnthalpy Class

Gibbs energy Liquid COTHVanLaarGibbsEnergy Class

The Van Laar equation alsoperforms poorly for dilutesystems and cannotrepresent many commonsystems, such as alcohol-hydrocarbon mixtures,with acceptable accuracy.




3-122

van Laar Ln Activity CoefficientThis method calculates the activity coefficient of components, i ,using the van Laar activity model. The extended, multi-component form of the van Laar equation is shown in thefollowing relation:



where: T = temperature (K)

n = total number of components

(3.302)

(3.303)

(3.304)

E i = -4.0 if A i Bi < 0.0, otherwise 0.0

(3.305)

γiln Ai 1.0 z i – [ ]21.0 E i z i+( )=

A i x ja ij b ij T +( )1.0 xi – ( )---------------------------

j 1=

n

∑=

B i x ja ji b ji T +( )1.0 xi – ( )---------------------------

j 1=

n∑=

z i A i xi

A i xi B i 1.0 xi – ( )+[ ]------------------------------------------------ -=




3-123

a ij = non-temperature-dependent energy parameter betweencomponents i and j

b ij = temperature-dependent energy parameter between

components i and j [1/K]

a ji = non-temperature-dependent energy parameter betweencomponents j and i

b ji = temperature-dependent energy parameter betweencomponents j and i [1/K]


van Laar Ln Fugacity CoefficientThis method calculates the fugacity coefficient of componentsusing the van Laar activity model. The fugacity coefficient ofcomponent i , φi , is calculated from the following relation.


P = pressure




COTHVanLaarLnActivityCoeff Class Liquid

(3.306)


COTHVanLaarLnFugacityCoeff Class Liquid

The four adjustableparameters for the VanLaar equation are the a ij,a ji

, bij

, and b ji

terms. Theequation will use storedparameter values storedor any user-supplied valuefor further fitting theequation to a given set ofdata.

φln i ln γi f i

st d

P --------

⎝ ⎠⎜ ⎟⎛ ⎞

=




3-124

The term, ln γi , in the above equation is exclusivelycalculated using the van Laar Ln Activity Coefficient . For thestandard fugacity, f i

std , refer to Section 5.4 - Standard StateFugacity .




3-125

van Laar Ln FugacityThis method calculates the fugacity of components using thevan Laar activity model. The fugacity of component i , f i , iscalculated from the following relation.


f istd = standard state fugacity



(3.307)


COTHVanLaarLnFugacity Class Liquid

The term, ln γi , in the above equation is exclusively calculatedusing the van Laar Ln Activity Coefficient . For the standardfugacity, f i

std , refer to Section 5.4 - Standard State Fugacity .

ln f i ln γi xi f i st d ( )=




3-126

van Laar Activity CoefficientDifferential wrt TemperatureThis method calculates the activity coefficient differential wrt totemperature using the van Laar model from the followingrelation.

where:


(3.308)


COTHVanLaarLnActivityCoeffDT Class Liquid

d γiln

dT ------------ 1 z i – ( )2 1 E i zi+( )

dA i

dT -------- 2 Ai 1 z i – ( ) 1 Ez i+( )

dz idT ------- – A 1 z i – ( )2 E i

dz idT -------+=

dA i

dT --------

x jb ij

1 xi – -------------

j 1=

n

∑=

dB i

dT --------

x jb ji

1 xi

– -------------

j 1=

n

∑=

dZ idT --------

xi 1 xi – ( ) dA i

dT -------- B i

dB i

dT -------- A i –

⎝ ⎠⎛ ⎞

A i xi B i 1 xi – ( )+[ ]2-------------------------------------------------------------- -=




3-127

van Laar Excess Gibbs EnergyThis method calculates the excess Gibbs energy using the vanLaar activity model from the following relation.




van Laar Gibbs EnergyThis method calculates the Gibbs free energy using the van Laaractivity model from the following relation.

where: G E = excess Gibbs energy



(3.309)


COTHVanLaarExcessGibbsEnergyClass

Liquid

The term, ln γi , in the above equation is exclusivelycalculated using the van Laar Ln Activity Coefficient .

(3.310)

G E RT x i γiln

i

n

∑=

G xiG i RT x i xi G E +ln

i

n

∑+

i

n

∑=






3-129

van Laar EnthalpyThis method calculates the enthalpy using the van Laar activitymodel from the following relation.

where: ΗΕ= excess enthalpy




3.2.4 Margules ModelThe Margules equation was the first Gibbs excess energyrepresentation developed. The equation does not have anytheoretical basis, but is useful for quick estimates and datainterpolation. The software has an extended multi-componentMargules equation with up to four adjustable parameters perbinary.

The four adjustable parameters for the Margules equation arethe a ij and a ji (temperature independent) and the b ij and b ji terms (temperature dependent). The equation will use stored

parameter values or any user-supplied value for further fittingthe equation to a given set of data.

(3.312)


COTHVanLaarEnthalpy Class Liquid

The term, H E, in the above equation is exclusively calculatedusing the van Laar Ln Activity Coefficient .

H x i H i H E +

i

n

∑=

This equation should notbe used for extrapolationbeyond the range overwhich the energyparameters have beenfitted.




3-130

Property MethodsA quick reference of calculation methods is shown in the tablebelow for the Margules property model.


Margules Ln Activity CoefficientThis method calculates the activity coefficient for components, i ,using the Margules activity model from the following relation:


Activity Coefficient Liquid COTHMargulesLnActivityCoeffClass


Liquid COTHMargulesLnFugacityCoeffClass

Fugacity calculation Liquid COTHMargulesLnFugacity Class


Liquid COTHMargulesLnActivityCoeffDTClass

Excess Gibbs Liquid COTHMargulesExcessGibbsEnergyClass

Excess enthalpy Liquid COTHMargulesExcessEnthalpyClass

Enthalpy Liquid COTHMargulesEnthalpy Class

Gibbs energy Liquid COTHMargulesGibbsEnergy Class

(3.313)γiln 1.0 xi – [ ]2 A i 2 xi Bi Ai – ( )+[ ]=




3-131

where: γi = activity Coefficient of component i


where: T = temperature (K)

n = total number of components

a ij = non-temperature-dependent energy parameter betweencomponents i and j

b ij = temperature-dependent energy parameter betweencomponents i and j [1/K]

a ji = non-temperature-dependent energy parameter betweencomponents j and i

b ji = temperature-dependent energy parameter betweencomponents j and i [1/K]

(3.314)

(3.315)

A i x ja ij b ij T +( )1.0 xi – ( )---------------------------

j 1=

n

∑=

B i x ja ji b ji T +( )1.0 xi – ( )---------------------------

j 1=

n

∑=




3-132


Margules Ln Fugacity CoefficientThis method calculates the fugacity coefficient of componentsusing the Margules activity model. The fugacity coefficient ofcomponent i , φi , is calculated from the following relation.




Margules FugacityThis method calculates the fugacity logarithm of componentsusing Margules activity model. The fugacity of component i , f i , is


COTHMargulesLnActivityCoeffClass

Liquid

(3.316)


COTHMargulesLnFugacityCoeffClass

Liquid

The term, ln γi , in the above equation is exclusively calculatedusing the Margules Ln Activity Coefficient . For the standardfugacity, f i


φln i ln γi f i

st d

P --------⎝ ⎠⎜ ⎟

⎛ ⎞

=




3-133

calculated from the following relation.


f istd = Standard state fugacity



Margules Activity CoefficientDifferential wrt TemperatureThis method calculates the activity coefficient wrt totemperature from the following relation.

(3.317)


COTHMargulesLnFugacity Class Liquid

The term, ln γi , in the above equation is exclusively calculatedusing the Margules Ln Activity Coefficient . For the standardfugacity, f i


(3.318)

ln f i

ln γi x

i f

i

st d ( )=

∂ γiln∂T

------------




3-134


Margules Excess Gibbs EnergyThis method calculates the excess Gibbs energy using theMargules activity model from the following relation.




Margules Gibbs EnergyThis method calculates the Gibbs free energy using the Margulesactivity model from the following relation.


COTHMargulesLnActivityCoeffDTClass

Liquid

(3.319)


COTHMargulesExcessGibbsEnergyClass

Liquid

The term, ln γi , in the above equation is exclusivelycalculated using the Margules Ln Activity Coefficient .

(3.320)

G E

RT x i γiln

i

n

∑=


i

n

∑+

i

n

∑=






3-136

Margules Excess EnthalpyThis method calculates the excess enthalpy using the Margulesactivity model from the following relation.




Margules EnthalpyThis method calculates the enthalpy using the Margules activitymodel from the following relation.


x i = mole fraction of component i H i = enthalpy of component i

(3.321)


COTHMargulesExcessEnthalpy Class Liquid

The term, , in the above equation is exclusivelycalculated using the Margules Activity Coefficient Differentialwrt Temperature .

(3.322)

H E RT 2 xid γiln

dT ------------

i

n

∑ – =

d γiln

dT ------------

H x i H i H E +

i

n

∑=




3-137


3.2.5 Wilson ModelThe 20Wilson (1964) equation is based on the Flory-Hugginstheory, assuming that intermolecular interactions are negligible.

First, imagine that the liquid mixture can be magnified to a pointwhere molecules of type 1 and type 2 in a binary mixture can bevisualized. Consider molecules of type 1, and determine theratio of the probability of finding a molecule of type 2 over theprobability of finding a molecule of type 1 in the surrounding ofthis particular molecule of type 1.

Wilson proposed that:

The parameters a 21 and a 11 are related to the potential energiesof the 1-1 and 1-2 pairs of molecules. Similarly, to see what ishappening in the region of a specific molecule of type 2, youhave:


COTHMargulesEnthalpy Class Liquid

The term, H E, in the above equation is exclusively calculatedusing the Margules Excess Enthalpy .

(3.323)

(3.324)

x21

x11-------

x2a 21

RT ------- –

⎝ ⎠⎛ ⎞exp

x1

a11 RT ------- – ⎝ ⎠⎛ ⎞exp

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

x12

x22-------

x1a 12

RT ------- –

⎝ ⎠⎛ ⎞exp

x2a 22

RT ------- –

⎝ ⎠⎛ ⎞exp

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




3-138

Wilson defined the local volume fractions based on the twoequations above, using the pure component molar volumes asweights:

When the above relations for φ are substituted into the Flory-Huggins equation:

where:

and:

The Wilson equation, although fundamentally empirical,provides a fair description of how real liquid systems behave.Also, it is a powerful framework for regression and extension ofexperimental data. Of primary importance, the Wilson equationcan be extended to multi-component mixtures without the useof simplifications (as in the case of van Laar and Margules) orternary or higher parameters. In other words, if one has the λij -

λii parameters for all binaries in a multi-component mixture, theWilson equation can be used to model the multi-componentbehaviour.

(3.325)

(3.326)

(3.327)

(3.328)

φ1

V 1 x11

V 1 x11 V 2 x21+----------------------------------= φ2

V 2 x22

V 1 x12 V 2 x22+----------------------------------=

φi is the volume fraction ofcomponent i.

G E

RT ------- xi

φi

xi----⎝ ⎠⎛ ⎞ln∑=

G E

RT ------- x1 x1 Λ12 x2+( ) x2 x2 Λ21 x1+( )ln – ln – =

Λ12

V 2V 1------

λ12

RT -------- –

⎝ ⎠⎛ ⎞exp=

Λ21V 1V 2------

λ21

RT -------- –

⎝ ⎠⎛ ⎞exp=

γ1ln x1 Λ12 x2+( ) x2

Λ12

x1 Λ12 x2+

--------------------------Λ21

x2 Λ21 x1+

-------------------------- – +ln – =

γ2ln x2 Λ21 x1+( ) x1

Λ12

x1 Λ12 x2+--------------------------

Λ21

x2 Λ21 x1+-------------------------- – +ln – =




3-139

This is very important, since multi-component data are ratherscarce and tedious to collect and correlate. In the same way thatthe CS correlation opened the doors for VLE modeling of fairly

complex hydrocarbon systems, the Wilson equation enabled thesystematic modeling of fairly complex non-ideal systems.However, one still has to measure the VLE behaviour to obtainthe binary parameters. Only in very specific situations can theparameters be generalized ( 30Orye and Prausnitz, 1965).

Perhaps more importantly, the Wilson equation can not predictphase splitting, thus it cannot be used for LLE calculations. Anempirical additional parameter proposed by Wilson to accountfor phase splitting did not find wide acceptance, since it cannotbe easily extended for multi-component mixtures. An interestingmodification of the Wilson equation to account for phasesplitting is the one by Tsuboka and Katayama, as described inthe 21Walas (1985).

To extend the applicability of the Wilson equation

It is modeled as a simple linear function of temperature:

(3.329)

(3.330)

a ij Λij Λ ji – =

a ij b ij c ij T +=




3-140

Application of Wilson EquationThe Wilson equation was the first activity coefficient equation

that used the local composition model to derive the excessGibbs energy expression. It offers a thermodynamicallyconsistent approach to predicting multi-component behaviourfrom regressed binary equilibrium data. Experience also showsthat the Wilson equation can be extrapolated with reasonableconfidence to other operating regions with the same set ofregressed energy parameters.

Although the Wilson equation is more complex and requiresmore CPU time than either the van Laar or Margules equations,it can represent almost all non-ideal liquid solutionssatisfactorily, except electrolytes and solutions exhibiting limitedmiscibility (LLE or VLLE). It provides an excellent prediction ofternary equilibrium using parameters regressed from binarydata only. The Wilson equation will give similar results as theMargules and van Laar equations for weak non-ideal systems,but consistently outperforms them for increasingly non-idealsystems.

The Wilson equation used in this program requires two to fouradjustable parameters per binary. The four adjustableparameters for the Wilson equation are the a ij and a ji (temperature independent) terms, and the b ij and b ji terms(temperature dependent). Depending upon the availableinformation, the temperature dependent parameters may be setto zero. Although the Wilson equation contains terms fortemperature dependency, caution should be exercised whenextrapolating.

Setting all four parameters to zero does not reduce thebinary to an ideal solution, but maintains a small effect dueto molecular size differences represented by the ratio ofmolar volumes.




3-141

Property MethodsA quick reference of calculation methods is shown in the tablebelow for the Wilson property model.


Wilson Ln Activity CoefficientThis method calculates the activity coefficient for components, i ,using the Wilson activity model from the following relation.

CalculationMethod


Activity Coefficient Liquid COTHWilsonLnActivityCoeff Class


Liquid COTHWilsonLnFugacityCoeff Class

Fugacity calculation Liquid COTHWilsonLnFugacity Class


Liquid COTHWilsonLnActivityCoeffDTClass

Excess Gibbs Liquid COTHWilsonExcessGibbsEnergyClass

Excess enthalpy Liquid COTHWilsonExcessEnthalpy Class

Enthalpy Liquid COTHWilsonEnthalpy Class

Gibbs energy Liquid COTHWilsonGibbsEnergy Class

(3.331)γiln 1.0 x j Λij j 1=

n

∑ln – xk Λki

xk Λkj j 1=

n

∑-----------------------

k 1=

n

∑ – =




3-142

where: γi = Activity coefficient of component i

x i = Mole fraction of component i

T = Temperature (K)

n = Total number of components

a ij = Non-temperature dependent energy parameter betweencomponents i and j (cal/gmol)

b ij = Temperature dependent energy parameter betweencomponents i and j (cal/gmol-K)

V i = Molar volume of pure liquid component i in m3/kgmol(litres/gmol)

Property Class Name and Applicable PhasesProperty Class Name Applicable Phase

COTHWilsonLnActivityCoeff Class Liquid

This method uses the Henry’s convention for non-condensable components.

ΛijV jV i-----

a ij b ij T +( ) RT

--------------------------- – exp=




3-143

Wilson Fugacity CoefficientThis method calculates the fugacity coefficient of componentsusing the Wilson activity model. The fugacity coefficient ofcomponent i , φi , is calculated from the following relation.


P = Pressure

f i = Standard state fugacity


(3.332)


COTHWilsonLnFugacityCoeff Class Liquid

The term, ln γi , in the above equation is exclusivelycalculated using the Wilson Ln Activity Coefficient . For thestandard fugacity, f i


φln i ln γi f i

st d

P --------

⎝ ⎠⎜ ⎟⎛ ⎞

=




3-144

Wilson FugacityThis method calculates the fugacity of components using theWilson activity model. The fugacity of component i , f i , iscalculated from the following relation.





Wilson Activity CoefficientDifferential wrt TemperatureThis method calculates the activity coefficient wrt totemperature from the following relation.

(3.333)


COTHWilsonLnFugacity Class Liquid

The term, ln γi , in the above equation is exclusivelycalculated using the Wilson Ln Activity Coefficient . For thestandard fugacity, f i


(3.334)


d γiln

dT ------------

x jd Λij

dT ---------------

j 1=

n

∑

x jΛij

j 1=

n

∑

------------------------- –

xk

d Λki

dT ----------- x jΛkj

j 1=

n

∑⎝ ⎠⎜ ⎟⎜ ⎟⎛ ⎞

xk Λij x j

d Λkj

dT -----------

j 1=

n

∑⎝ ⎠⎜ ⎟⎜ ⎟⎛ ⎞

–

x jΛkj

j 1=

n

∑⎝ ⎠⎜ ⎟⎜ ⎟

⎛ ⎞2

------------------------------------------------------------------------------------------------

k 1=

n

∑ – =




3-145


Wilson Excess Gibbs EnergyThis method calculates the excess Gibbs energy using theWilson activity model from the following relation.



T = temperature

R = universal gas constant



COTHWilsonLnActivityCoeffDTClass

Liquid

(3.335)


COTHWilsonExcessGibbsEnergy Class Liquid

The term, ln γi , in the above equation is exclusivelycalculated using the Wilson Ln Activity Coefficient .

G E

RT x i γiln

i

n

∑=




3-146

Wilson Gibbs EnergyThis method calculates the Gibbs free energy using the Wilsonactivity model from the following relation.





(3.336)


COTHWilsonGibbsEnergy Class Liquid

The term, G E, in the above equation is exclusively calculatedusing the Wilson Excess Gibbs Energy .


i

n

∑+

i

n

∑=






3-148


3.2.6 NRTL ModelThe Wilson equation is very successful in the representation ofVLE behaviour of completely miscible systems, but is not

theoretically capable of predicting VLE and LLE. 22Renon andPrausnitz (1968) developed the Non-Random Two-LiquidEquation (NRTL). In developing the NRTL, they used the quasi-chemical theory of Guggenheim and the two-liquid theory fromScott. To take into account the "structure" of the liquidgenerated by the electrostatic force fields of individualmolecules, the local composition expression suggested byWilson is modified:

where: α12 = is a parameter which characterizes the non-randomnessof the mixture.

x = is mole fraction of component

g = is free energies for mixture


COTHWilsonEnthalpy Class Liquid

The term, H E, in the above equation is exclusively calculatedusing the Wilson Excess Enthalpy .

(3.339)

(3.340)

x21

x11

------- x2

x1

-----

α12 g 21 –

RT --------------------⎝ ⎠⎛ ⎞exp

α12 g 11 – RT

--------------------⎝ ⎠⎛ ⎞exp

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

x21

x22-------

x1

x2-----

α12 g 12 –

RT --------------------⎝ ⎠⎛ ⎞exp

α12 g 22 –

RT --------------------⎝ ⎠⎛ ⎞exp

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




3-149

The local model fractions are restricted by material balance to

x 12 + x 22 = 1 and x 21 + x 11 = 1. If the ratios and are

multiplied:

When the material balance equations are substituted:

Scotts Two Liquid Theory

The quasi-chemical theory of Guggenheim with the non-randomassumption can be written as:

where: Z = is the coordination number

ω = is the energy of interaction between pairs

(3.341)

(3.342)

Figure 3.3

(3.343)

x21

x11-------

x12

x22-------

x21

x11-------

x12

x22-------× α12

2 g 12 g 11 – g 22 – ( ) RT

------------------------------------------- – ⎝ ⎠⎛ ⎞exp=

1 x21 – ( ) 1 x12 – ( ) α122 g 12 g 11 – g 22 – ( )

RT ------------------------------------------- –

⎝ ⎠⎛ ⎞exp x21 x12=

P r e s s u r e

Pure Liq uid

Liquid Mixtu re

Vap ourize eac h liquid dropping system P to a ve ry low va lue (Ideal Gas)

Co mp ress Vap our

Mixt ure

Idea l Gas Mix Ideal Cases

1 x21 – ( ) 1 x12 – ( ) 1 Z ---

2ω12 ω11 – ω22 – ( ) RT

----------------------------------------------- – ⎝ ⎠⎛ ⎞exp x21 x12=




3-150

x = is mole fraction of components

This gives a physical interpretation of the αij parameter. Sincethe coordination number represents the number of neighbourmolecules a given molecule may have, the usual value issomewhere between 6 and 12, giving an α value in the order ofpositive 0.1 to 0.3. The significance of α is somewhat ambiguouswhen its value is greater than 0.3, where a hypothetical fluidmixture in which a molecule with very few neighbours shouldexist. The following equations for the local compositions exist:

and

Renon and Prausnitz used the above equations in the two-liquidtheory of Scott. Scott assumed that a liquid mixture can beidealized as a set of cells, in which there are cells with moleculesof type 1 and type 2 in the centre. "For cells with molecules oftype 1 in the centre, the residual Gibbs free energy (the Gibbsfree energy when compared with that of an ideal gas at thesame temperature, pressure and composition) is the sum of allthe residual Gibbs free energies for two body interactionsexperienced by centre molecule of type 1" ( 22Renon andPrausnitz, 1968). Thus:

(3.344)

(3.345)

(3.346)

x21

x2 α12

g 21 g 11 – ( ) RT

-------------------------- – exp

x1 x+ 2 α12 g 21 g 11 – ( )

RT -------------------------- – exp

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

x12

x1 α12 g 12 g 22 – ( )

RT -------------------------- – exp

x1 x+ 2 α12 g 12 g 22 – ( )

RT -------------------------- – exp

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

g 1( )

x11 g 11 x21 g 21+=

g pu re1( )

g 11=




3-151

A molecule of type 2 in the centre can be:

The Gibbs excess energy is the sum of the changes wheremolecules of type 1 from a cell of pure component 1 aretransferred into the centre of a cell of liquid 2; the samereasoning applies for molecule 2.

Consequently:

substituting and finally:

where: g E is the excess Gibbs free energy

g is Gibbs free energy for interaction between components

and the activity coefficients are:

(3.347)

(3.348)

(3.349)

(3.350)

(3.351)

g 2( )

x22 g 22 x12 g 12+= g pu re

2( ) g 22=

g E x1

g 1( )

g pu re

1( ) – ( ) x

2 g 2

( ) g

pu re

2( ) – ( )+=

g E x1 x21 g 21 g 11 – ( ) x2 x12 g 12 g 22 – ( )+=

γ1ln x22 τ21

2α12 τ21 – ( )exp

x1 x2 α12 τ21 – ( )exp+[ ]2---------------------------------------------------------- τ12

2α12 τ12 – ( )exp

x2 x1 α12 τ12 – ( )exp+[ ]2----------------------------------------------------------+

⎝ ⎠⎜ ⎟⎛ ⎞

=

γ2ln x12 τ12

2α12 τ12 – ( )exp

x2 x1 α12 τ12 – ( )exp+[ ]2---------------------------------------------------------- τ21

2α12 τ21 – ( )exp

x1 x2 α12 τ21 – ( )exp+[ ]2----------------------------------------------------------+

⎝ ⎠⎜ ⎟⎛ ⎞

=




3-152

where:

The NRTL equation offers little advantage over Wilson forsystems that are completely miscible. On the other hand, theNRTL equation can be used for systems that will phase split.When the g ij - g ji parameters are temperature dependent, theNRTL equation is very flexible and can be used to model a widevariety of chemical systems. Although the αij term has aphysical meaning and 22Renon and Prausnitz (1968) suggested aseries of rules to fix its value depending on the mixture type, itis better treated as an empirical parameter to be determinedthrough regression of experimental data. That is, if there isenough data to justify the use of 3 parameters.

The NRTL equation is an extension of the original Wilsonequation. It uses statistical mechanics and the liquid cell theoryto represent the liquid structure. These concepts, combined withWilson's local composition model, produce an equation capableof representing VLE, LLE and VLLE phase behaviour. Like theWilson equation, the NRTL is thermodynamically consistent andcan be applied to ternary and higher order systems usingparameters regressed from binary equilibrium data. It has anaccuracy comparable to the Wilson equation for VLE systems.

The NRTL combines the advantages of the Wilson and van Laarequations, and, like the van Laar equation, it is not extremelyCPU intensive and can represent LLE quite well. It is importantto note that because of the mathematical structure of the NRTLequation, it can produce erroneous multiple miscibility gaps.Unlike the van Laar equation, NRTL can be used for dilute

systems and hydrocarbon-alcohol mixtures, although it may notbe as good for alcohol-hydrocarbon systems as the Wilsonequation.

(3.352)

τ12

g 12 g 22 –

RT ---------------------=

τ21 g 21 g 11 –

RT ---------------------=

g 12 α12 τ12 – ( )exp=

g 21 α12 τ21 – ( )exp=




3-153

Property MethodsA quick reference of calculation methods is shown in the tablebelow for the NRTL property model.



Activity Coefficient Liquid COTHNRTLLnActivityCoeff Class


Liquid COTHNRTLLnFugacityCoeffClass

Fugacity calculation Liquid COTHNRTLLnFugacity Class

Activity coefficientdifferential wrt temperature

Liquid COTHNRTLLnActivityCoeffDTClass

NRTL temperaturedependent binaryinteraction parameters

Liquid COTHNRTLTempDep Class

Excess Gibbs Liquid COTHNRTLExcessGibbsEnergyClass

Excess enthalpy Liquid COTHNRTLExcessEnthalpy Class

Enthalpy Liquid COTHNRTLEnthalpy Class

Gibbs energy Liquid COTHNRTLGibbsEnergy Class




3-154

NRTL Ln Activity CoefficientThis method calculates the activity coefficient for components, i ,

using the NRTL activity model from the following relation:




τij = Temperature-dependent energy parameter betweencomponents i and j (cal/gmol-K)


(3.353)


COTHNRTLLnActivityCoeff Class Liquid

This method uses Henry’s convention for non-condensablecomponents.

The values G ij and τij are calculated from the temperaturedependent binary interaction parameters.

γiln

τ ji x jG ji j 1=

n

∑

xk G ki

k 1=

n

∑----------------------------

x jG ij

xk Gkj------------- τij

τmi xmG mim 1=

n

∑

xk G kj

k 1=

n

∑------------------------------------ –

⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞

j 1=

n

∑+=




3-155

NRTL Fugacity CoefficientThis method calculates the fugacity coefficient of componentsusing the NRTL activity model. The fugacity coefficient ofcomponent i , φi , is calculated from the following relation.


P = Pressure



NRTL FugacityThis method calculates the fugacity of components using theNRTL activity model. The fugacity of component, f i , is calculatedfrom the following relation.




(3.354)


COTHNRTLLnFugacityCoeff Class Liquid

The term, ln γi , in the above equation is exclusively calculatedusing the NRTL Ln Activity Coefficient . For the standardfugacity, f i


(3.355)

φln i ln γi f i

st d

P --------

⎝ ⎠⎜ ⎟⎛ ⎞

=





3-156


NRTL Activity Coefficient Differentialwrt TemperatureThis method analytically calculates the differential activitycoefficient with respect to temperature from the followingrelation.



COTHNRTLLnFugacity Class Liquid

The term, ln γi , in the above equation is exclusively calculatedusing the NRTL Ln Activity Coefficient . For the standardfugacity, f i


(3.356)

Property Class Name Applicable PhaseCOTHNRTLLnActivityCoeffDT Class Liquid

d γiln

dT ------------




3-157

Temperature Dependent BinaryInteraction ParametersThis method calculates the temperature dependent binaryinteraction parameters for the NRTL model from the followingrelation.

where:

where: a ij , b ij , c ij , d ij , e ij , = Temperature-dependent energy parameter between components i and j (cal/gmol-K)

αij = NRTL non-randomness parameters for binary interaction(note that a ij = a ji for all binaries)


(3.357)

(3.358)


COTHNRTLTempDep Class Liquid

τij a ijb ij

T ------ c ij T ln d ij T

e ij

T 2-----+ + + +

⎝ ⎠⎜ ⎟⎛ ⎞

=

G ij EXP ατ ij – ( )=

α α0 α1T +=

a ij 0 b ij 0 c ij 0 d ij 0 e ij 0τij 0=

=;=;=;=;=




3-158

NRTL Excess Gibbs EnergyThis method calculates the excess Gibbs energy using the NRTLactivity model from the following relation.




NRTL Gibbs EnergyThis method calculates the Gibbs free energy NRTL activitymodel from the following relation.




(3.359)


COTHNRTLExcessGibbsEnergy Class Liquid

The term, ln γi , in the above equation is exclusivelycalculated using the NRTL Ln Activity Coefficient .

(3.360)

G E RT x i γiln

i

n

∑=


i

n

∑+

i

n

∑=




3-159


NRTL Excess EnthalpyThis method calculates the excess enthalpy using the NRTLactivity model from the following relation.





COTHNRTLGibbsEnergy Class Liquid

The term, G E, in the above equation is exclusively calculatedusing the NRTL Gibbs Energy .

(3.361)


COTHNRTLExcessEnthalpy Class Liquid

The term, , in the above equation is exclusivelycalculated using the NRTL Activity Coefficient Differentialwrt Temperature .

H E

RT 2

xid γiln

dT ------------

i

n

∑ – =

d γiln

dT ------------




3-160

NRTL EnthalpyThis method calculates the enthalpy using the NRTL activitymodel from the following relation.





(3.362)


COTHNRTLEnthalpy Class Liquid

The term, H E, in the above equation is exclusively calculatedusing the NRTL Excess Enthalpy .

H x i H i H E +

i

n

∑=




3-161

3.2.7 HypNRTL ModelThe methods in the HypNRTL model are same as the Section3.2.6 - NRTL Model explained in the previous section. Thedifference between the models is that the HypNRTL does notoffer a flexible temperature dependence for τij. The HypNRTL isrepresented by the following relation:


T = Temperature (K)


a ij = Non-temperature-dependent energy parameter betweencomponents i and j (cal/gmol)*

b ij = Temperature-dependent energy parameter betweencomponents i and j (cal/gmol-K)*

αij = NRTL non-randomness parameters for binary interaction(note that a ij = a ji for all binaries)

(3.363)

(3.364)

G ij τij α ij – [ ]exp=

τij

a ij b ij T +

RT ----------------------=






3-163

With the General NRTL model, you can specify the format for theequations of τij and a ij to be any of the following:

Depending on the form of the equations that you choose, you

can specify values for the different component energyparameters. The General NRTL model provides radio buttons onthe Binary Coeffs tab which access the matrices for the A ij , B ij ,C ij , F ij , G ij , Alp1 ij and Alp2 ij energy parameters.

τij and αij Options

The equations options canbe viewed in the DisplayForm drop down list on theBinary Coeffs tab of theFluid Package propertyview.

τij Aij

Bij

T ------

C ij

T 2

------- F ij T G ij T ( )ln+ + + +=

α ij Alp 1 ij Alp2 ij T +=

τij

A ij B ij

T ------+

RT --------------------=

α ij Alp1 ij=

τij Aij B ij

T ------ F ij T G ij T ( )ln+ + +=


τij Aij B ij t C ijT

-------+ +=


where: T is in K and t is °C

τij Aij B ij

T ------+=

α ij Alp1 ij=




3-164

3.2.9 HYSYS - General NRTLMethod Description EquationHYSIMStdLiquidVolume

Standard LiquidVolume

HYSIMLiqDensity Density Hankinson, R.W. and Thompson, G.H., A.I.Ch.E.Journal 25, No.4, P. 653, (1979).

HYSIMLiqVolume Volume Hankinson, R.W. and Thompson, G.H., A.I.Ch.E.Journal 25, No.4, P. 653, (1979).

GenLiquid1FugCoefficient

Fugacity Coefficient

NRTLActCoeff Activity Coefficient

ActivityLiquid1Fugacity

Fugacity

CavettEnthalpy Enthalpy

CavettEntropy Entropy

CavettGibbs Gibbs Free Energy

CavettHelmholtz Helmholtz Energy

CavettInternal Internal Energy

CavettCp Cp

CavettCv Cv

V MW i

ρi--------------- xi

i 1=

nc

∑=

φi γi

f i st d

P ---------⎝ ⎠⎜ ⎟⎜ ⎟⎛ ⎞

=

γiln

τ ji x j G ji

j 1=

n

∑

xk Gki

k 1=

n

∑---------------------------------

x j G ij xk G kj--------------- τij

τmi xmGmi

m 1=

n

∑

xk Gkj

k 1=

n

∑------------------------------------------- - –

⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞

j 1=

n

∑+=

f i γi xi f i st d

=

H l

xwater H water st ea m 67

xi H i° Δ H iCavett

+⎝ ⎠⎛ ⎞

i∑+=

S l

xwater S water st ea m 67

xi S ° ΔS iCavett

+⎝ ⎠⎛ ⎞

i∑+=

G G ° A A° – ( ) RT Z 1 – ( )+ +=

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

U U ° A A° – ( ) T S S ° – ( )+ +=

Cp l xwater Cp water st ea m 67

xi Cp i° ΔCp iCavett

+⎝ ⎠⎛ ⎞

i∑+=

C v

C p

R – =




3-165

3.2.10 UNIQUAC Model23Abrams and Prausnitz (1975) derived an equation with a semi-theoretical basis like NRTL, but using only two adjustableparameters per binary pair. Their approach is heavily dependenton some statistical mechanics concepts which are outside thescope of this guide. Only a few highlights from their work will bepresented here.

"Guggenheim proposed that a liquid mixture can be seen as aset of tri-dimensional lattice sites, the volume in the immediatevicinity of a site is called a cell. Each molecule in the liquid isdivided in segments such that each segment occupies one cell"( 23Abrams and Prausnitz, 1975). Using the configurationalpartition function, it can be shown that:

where: A = Helmholtz function

n = number of moles

NRTLGe Excess Gibbs freeenergy

MRTLHe Excess enthalpy

HYSIMLiquidViscosity*

Viscosity Light Hydrocarbons (NBP<155 F) - Modified Ely &Hanley (1983)Heavy Hydrocarbons (NVP>155 F) - Twu (1984)Non-Ideal Chemicals - Modified Letsou-Stiel (see Reid,Prausnitz and Poling, 1987).

HYSIMVapourThermalK*

Thermal Conductivity Misic and Thodos; Chung et al. methods (see Reid,Prausnitz and Poling, 1987).

HYSIMSurfaceTension Surface Tension

Method Description Equation

G E

RT xi

γi

ln

i 1=

nc

∑=

H E

G E

T T ∂

∂G E

⎝ ⎠⎜ ⎟⎛ ⎞

– =

σ P c

23---

T c

1

3---

Q 1 T R – ( )a b=

(3.365) g E a E ≅

AΔn1 n 2+----------------- RT x 1 x1 x2 x2ln+ln( ) – =




3-166

x = mole fraction

In the original work of Guggenheim, he assumed that the liquidwas composed of molecules with relatively the same size; thusthe number of neighbours of type 2 to a molecule of type 1 wasa reasonable measure of the local composition.

Since Prausnitz and Abrams proposed to handle molecules ofdifferent sizes and shapes, they developed a differentmeasurement of the local composition, i.e., a local area fraction.Using this idea, coupled with some arguments based onstatistical thermodynamics, they reached the followingexpression for the Gibbs free energy:

and:

where:

q = parameter proportional to the area

(3.366)

(3.367)

(3.368)

(3.369)

(3.370)

G E

Gcombinational

E G

resdiual

E +=

Combinational refers tothe non-ideality caused bydifferences in size andshape (entropic effects).

G combinational E

x1

φ1

x1-----⎝ ⎠⎛ ⎞ x2

φ2

x2-----⎝ ⎠⎛ ⎞ Z

2--- q 1 x1

θ1

φ1-----⎝ ⎠⎛ ⎞ q2 x2

θ2

φ2-----⎝ ⎠⎛ ⎞ln+ln

⎝ ⎠⎛ ⎞+ln+ln=

G resdiual E q1 x1 θ1 θ2τ21+( )ln – q 2 x2 θ2 θ1τ12+( )ln – =

τ21

u21 u 11 –

RT --------------------- – ⎝ ⎠

⎛ ⎞exp=

τ12

u12 u 22 –

RT --------------------- –

⎝ ⎠⎛ ⎞exp=

Residual refers to non-idealities due to energeticinteractions betweenmolecules (temperature orenergy dependent).

θ1

q1 x1

q1 x1 q2 x2+----------------------------=

φ1r 1 x1

r 1 x1 r 2 x2+---------------------------=

θ2

q2 x2

q1 x1 q2 x2+----------------------------=

φ2r 2 x2

r 1 x1 r 2 x2+---------------------------=




3-167

r = parameter proportional to the volume of the individualmolecules.

And finally, the expressions for the activity coefficients are:

and ln γ2 can be found by interchanging the subscripts.

As with the Wilson and NRTL equations, the UNIQUAC equationis readily expanded for a multi-component system without the

need for ternary or higher data. Like NRTL, it is capable ofpredicting two liquid phases, but unlike NRTL, it needs only twoparameters per binary pair.

One interesting theoretical result from the UNIQUAC equation isthat it is an equation for which the entropy contributions to theGibbs free energy are separated from the temperature (energy)contributions. The idea of looking at the entropy portion basedon segments of molecules suggests that one can divide amolecule into atomic groups and compute the activity coefficientas a function of the group. This idea was explored in full by24Fredenslund et al (1975, 251977) and is implemented in the

UNIFAC method.

The UNIQUAC equation has been successfully used to predictVLE and LLE behaviour of highly non-ideal systems.

Application of UNIQUACThe UNIQUAC (UNIversal QUASI-Chemical) equation usesstatistical mechanics and the quasi-chemical theory ofGuggenhiem to represent the liquid structure. The equation iscapable of representing LLE, VLE and VLLE with accuracy

comparable to the NRTL equation, but without the need for anon-randomness factor. The UNIQUAC equation is significantlymore detailed and sophisticated than any of the other activity

(3.371)γ1ln

φ1

x1-----⎝ ⎠⎛ ⎞ Z

2--- q1

θ1

φ1-----⎝ ⎠⎛ ⎞ φ2 l 1

r 1r 2---- l 2 –

⎝ ⎠⎛ ⎞ q1 θ1 θ2τ21+( ) θ2q1

τ21

θ1 θ2τ21+-------------------------

τ12

θ2 θ1τ21+------------------------- –

⎝ ⎠⎛ ⎞+ln – +ln+ln=

l 1 Z 2--- r 1 q 1 – ( ) r 1 1 – ( ) – =




3-168

models. Its main advantage is that a good representation ofboth VLE and LLE can be obtained for a large range of non-electrolyte mixtures using only two adjustable parameters per

binary. The fitted parameters usually exhibit a smallertemperature dependence which makes them more valid forextrapolation purposes.

The UNIQUAC equation uses the concept of local composition asproposed by Wilson. Since the primary concentration variable isa surface fraction as opposed to a mole fraction, it is applicableto systems containing molecules of very different sizes andshapes, such as polymers. The UNIQUAC equation can beapplied to a wide range of mixtures containing water, alcohols,nitriles, amines, esters, ketones, aldehydes, halogenatedhydrocarbons and hydrocarbons.

This software uses the following four-parameter extended formof the UNIQUAC equation. The four adjustable parameters forthe UNIQUAC equation are the a ij and a ji terms (temperatureindependent), and the b ij and b ji terms (temperaturedependent).

The equation uses stored parameter values or any user-suppliedvalue for further fitting the equation to a given set of data.




3-169

Property MethodsA quick reference of calculation methods is shown in the tablebelow for the UNIQUAC property model.


UNIQUAC Ln Activity CoefficientThis method calculates the activity coefficient for components, i ,using the UNIQUAC activity model from the following relation.


x i = Mole fraction of component i T = Temperature (K)


Activity Coefficient Liquid COTHUNIQUACLnActivityCoeffClass


Liquid COTHUNIQUACLnFugacityCoeffClass

Fugacity calculation Liquid COTHUNIQUACLnFugacity Class


Liquid COTHUNIQUACLnActivityCoeffDTClass

Excess Gibbs Liquid COTHUNIQUACExcessGibbsEnergyClass

Excess enthalpy Liquid COTHUNIQUACExcessEnthalpyClass

Enthalpy Liquid COTHUNIQUACEnthalpy Class

Gibbs energy Liquid COTHUNIQUACGibbsEnergy Class

(3.372)γilnΦi

xi------⎝ ⎠⎛ ⎞ 0.5 Zq i

θi

Φi------⎝ ⎠⎛ ⎞ Li

Φi

xi------⎝ ⎠⎛ ⎞ L j x j q i 1.0 θ jτ ji

j 1=

n

∑ln – ⎝ ⎠⎜ ⎟⎜ ⎟⎛ ⎞

q i

θ jτ ji

θk τkj

k 1=

n

∑-----------------------

⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞

j 1=

n

∑ – +

j 1=

n

∑ – +ln+ln=




3-170


Z = 10.0 (coordination number)

a ij = Non-temperature-dependent energy parameter betweencomponents i and j (cal/gmol)

b ij = Temperature-dependent energy parameter betweencomponents i and j (cal/gmol-K)

q i = van der Waals area parameter - Aw i /( 2.5x10 9 )

A w = van der Waals area

r i = van der Waals volume parameter - Vw i /( 15.17 )

V w = van der Waals volume


(3.373)

(3.374)

(3.375)


COTHUNIQUACLnActivityCoeffClass

Liquid

L j 0.5 Z r j q j – ( ) r j – 1+=

θi

q i xi

q j x j∑----------------=

τija ij b ij T +

RT ---------------------- – exp=




3-171

UNIQUAC Fugacity CoefficientThis method calculates the fugacity coefficient of componentsusing the UNIQUAC activity model. The fugacity coefficient ofcomponent i , φi , is calculated from the following relation.




(3.376)


COTHUNIQUACLnFugacityCoeffClass

Liquid

The term, ln γi , in the above equation is exclusively calculatedusing the UNIQUAC Ln Activity Coefficient . For the standardfugacity, f i


φln i ln γi f i

st d

P --------

⎝ ⎠⎜ ⎟⎛ ⎞

=




3-172

UNIQUAC FugacityThis method calculates the fugacity of components using theUNIQUAC activity model. The fugacity of component i , f i , iscalculated from the following relation.





(3.377)


COTHUNIQUACLnFugacityClass

Liquid

The term, ln γi , in the above equation is exclusively calculatedusing the UNIQUAC Ln Activity Coefficient . For the standardfugacity, f i






3-173

UNIQUAC Activity CoefficientDifferential wrt TemperatureThis method analytically calculates the differential activitycoefficient wrt to temperature from the following relation.


(3.378)


COTHUNIQUACLnActivityCoeffDTClass

Liquid

d γiln

dT ------------




3-174

UNIQUAC Excess Gibbs EnergyThis method calculates the excess Gibbs energy using theUNIQUAC activity model from the following relation.




(3.379)


COTHUNIQUACExcessGibbsEnergyClass

Liquid

The term, ln γi , in the above equation is exclusivelycalculated using the UNIQUAC Ln Activity Coefficient .

G E RT x i γiln

i

n

∑=




3-175

UNIQUAC Gibbs EnergyThis method calculates the Gibbs free energy using theUNIQUAC activity model from the following relation.





UNIQUAC Excess EnthalpyThis method calculates the excess enthalpy using the UNIQUACactivity model from the following relation.



(3.380)


COTHUNIQUACGibbsEnergy Class Liquid

The term, G E, in the above equation is exclusively calculatedusing the UNIQUAC Excess Gibbs Energy .

(3.381)


i

n

∑+

i

n

∑=

H E

RT 2

xid γiln

dT ------------

i

n

∑ – =




3-176


UNIQUAC EnthalpyThis method calculates the enthalpy using the UNIQUAC activitymodel from the following relation.






COTHUNIQUACExcessEnthalpy Class Liquid

The term, , in the above equation is exclusivelycalculated using the UNIQUAC Activity CoefficientDifferential wrt Temperature .

(3.382)


COTHUNIQUACEnthalpy Class Liquid

The term, H E, in the above equation is exclusively calculatedusing the UNIQUAC Excess Enthalpy .

d γiln

dT ------------

H x i H i H E +

i

n

∑=




3-177

3.2.11 UNIFAC ModelFor more complex mixtures, 26Wilson and Deal (1962), and27Derr and Deal (1969), proposed a group contribution methodin which the mixture was treated as a solution of atomic groupsinstead of a solution of molecules. The concept of atomic groupactivity, although not new in chemical engineering ( 28Le Bas,1915), was shown to be applicable to the prediction of mixturebehaviour, thus increasing its utility many times. The Wilson,Deal and Derr approach was based on the athermal Flory-Huggins equation and it found acceptance, especially in Japanwhere it modified to a computer method called ASOG (AnalyticalSolution of Groups) by 29Kojima and Toguichi (1979).

In 1975, 24Fredenslund et al presented the UNIFAC (1975)method (UNIQUAC Functional Group Activity Coefficients), inwhich he used the UNIQUAC equation as the basis for the atomicgroup method. In 1977, the UNIFAC group was published in abook (1977), which included a thorough description of themethod by which the atomic group contributions werecalculated, plus the computer code which performed the activitycoefficient calculations (including fugacity coefficients using thevirial equation, vapour phase association and a distillationcolumn program). The method found wide acceptance in theengineering community and revisions are continuously beingpublished to update and extend the original group interactionparameter matrix for VLE calculations.

Figure 3.4

ethanol

ethanol

H 2 O

H 2 O OH

OH

CH 2

CH 2

CH 3 CH 3

H 2 O

H 2 O Cla ssica l View

Solution of Groups Point o f View




3-178

Also, there are specially-developed UNIFAC interactionparameter matrices for LLE calculations ( 31Magnussen et al,1981), vapour pressure estimation ( 32Jensen et al, 1981), gas

solubility estimation (33

Dahl et al, 1991) and polymer properties( 34Elbro, 1991).

The UNIFAC method has several interesting features:

• Coefficients are based on a data reduction using theDortmund Data Bank (DDB) as a source for VLE datapoints.

• Parameters are approximately independent oftemperature.

• Area and volume group parameters are readily available.• Group interaction parameters are available for many

group combinations.• The group interaction parameter matrix is being

continuously updated.• Gives reasonable predictions between 0 and 150°C, and

pressures up to a few atmospheres.• Extensive comparisons against experimental data are

available, often permitting a rough estimate of errors inthe predictions.

The original UNIFAC method also has several shortcomings thatstem from the assumptions used to make it a useful engineeringtool. Perhaps the most important one is that the group activityconcept is not correct, since the group area and volume shouldbe a function of the position in the molecule, as well as the othergroups present in the molecule. Also, 35Sandler suggested thatthe original choice of groups might not be optimal (1991a,361991b) and sometimes wrong results are predicted.

Also, the original UNIFAC VLE produces wrong LLE predictions(which is not surprising). This was remedied by 31Magnussen(1981) with the publication of interaction parameter tables forLLE calculations. This area has received considerably lessattention than the VLE, and hopefully new revisions for the LLEinteraction parameter matrix will appear.




3-179

One more interesting point is that the a mk interaction parameterterm is not, in reality, temperature independent. Thus, seriouserrors can be expected when predicting excess enthalpies.

There is work being done to extend the applicability andreliability of the UNIFAC method, especially in Denmark (1984)and Germany (1987).

The main idea is to modify the a mk term to include a temperaturedependency, in a form such as:

These refinements will probably continue for several years andUNIFAC will be continuously updated.

For more complex mixtures, 26Wilson and Deal (1962), and27Derr and Deal (1969), proposed a group contribution methodin which the mixture was treated as a solution of atomic groupsinstead of a solution of molecules. The concept of atomic groupactivity, although not new in chemical engineering ( 28Le Bas,1915), was shown to be applicable to the prediction of mixturebehaviour, thus increasing its utility many times.

(3.383)a mk a mk 0( ) a mk

1( )

T --------- a mk

2( )T ln+ +=








3-182

UNIFAC Ln Activity CoefficientThis method calculates the activity coefficient for components, i ,using the UNIFAC activity model from the following relation.

In γic is calculated in the same way as for the UNIQUAC

equation, but the residual part is calculated as follows:

where: k = functional group in the mixture

νk i = number of atomic groups of type k in molecule i

Γk = residual activity coefficient of the functional group k inthe actual mixture

Γk (i) = residual activity coefficient of the functional group k in

a mixture that contains only molecules i (this isnecessary to ensure the prediction of γi = 1 for a pureliquid)

The summation is extended over all the groups present in the

mixture. Γk is calculated in a similar manner as γiR

in theUNIQUAC equation:

(3.384)

(3.385)

(3.386)

This relation is from theUNIQUAC method γiln γi

cln γieln+=

γieln vk

i( ) Γk Γk i( )

ln – ( )lnk ∑=

Notice that normalizationis required to avoid thespurious prediction of anactivity coefficientdifferent than one for apure component liquid.

Γk ln Q k 1 θmτmk m∑⎝ ⎠⎜ ⎟⎛ ⎞ θmτmk

θnτnmn∑----------------------

⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞

m∑ – ln – =






3-184


UNIFAC Fugacity CoefficientThis method calculates the fugacity coefficient of componentsusing the UNIFAC activity model. The fugacity coefficient ofcomponent i , φi , is calculated from the following relation.


P = Pressure



COTHUNIFAC1_VLELnActivityCoeffClass

Liquid

COTHUNIFAC1_LLELnActivityCoeffClass

Liquid

The UNIFAC VLE model uses the interaction parameterswhich have been calculated from the experimental VLE data,whereas, the UNIFAC LLE uses the interaction parameterscalculated from LLE experimental data.

(3.391)φln i ln γi f i

st d

P --------

⎝ ⎠⎜ ⎟⎛ ⎞

=




3-185


UNIFAC FugacityThis method calculates the fugacity of components using theUNIFAC activity model. The fugacity of component i , f i , iscalculated from the following relation.





COTHUNIFAC1_VLELnFugacityCoeffClass

Liquid

COTHUNIFAC1_LLELnFugacityCoeffClass

Liquid

The term, ln γi , in the above equation is exclusively calculatedusing the UNIFAC Ln Activity Coefficient . For the standardfugacity, f i



(3.392)ln f i ln γi xi f i st d ( )=




3-186


UNIFAC Activity Coefficient Differential wrtTemperature

This method calculates the activity coefficient wrt totemperature from the following relation.



COTHUNIFAC1_VLELnFugacity Class Liquid

COTHUNIFAC1_LLELnFugacity Class Liquid

The term, ln γi , in the above equation is exclusively calculatedusing the UNIFAC Ln Activity Coefficient . For the standardfugacity, f i



(3.393)


COTHUNIFAC1_VLELnActivityCoeffDTClass

Liquid

COTHUNIFA1_LLECLnActivityCoeffDTClass

Liquid


d γiln

dT ------------




3-187

UNIFAC Gibbs EnergyThis method calculates the Gibbs free energy using the UNIFACactivity model from the following relation.





(3.394)


COTHUNIFAC1_VLEGibbsEnergy Class Liquid

COTHUNIFAC1_LLEGibbsEnergy Class Liquid

The term, G E, in the above equation is exclusively calculatedby the UNIQUAC Excess Gibbs Energy .



i

n

∑+

i

n

∑=




3-188

UNIFAC EnthalpyThis method calculates the enthalpy using the UNIFAC activitymodel from the following relation.





(3.395)


COTHUNIFAC1_VLEEnthalpy Class Liquid

COTHUNIFAC1_LLEEnthalpy Class Liquid

The term, H E, in the above equation is exclusively calculatedby the UNIQUAC Excess Enthalpy .


H x i H i H E +

i

n

∑=




3-189

3.2.12 Chien-Null ModelThe Chien-Null (CN) model provides a consistent framework forapplying existing activity models on a binary by binary basis. Inthis manner, the Chien-Null model allows you to select the bestactivity model for each pair in the case.

The Chien-Null model allows three sets of coefficients for eachcomponent pair, accessible via the A, B and C coefficientmatrices. Refer to the following sections for an explanation ofthe terms for each of the models.

Chien-Null FormThe Chien-Null generalized multi-component equation can beexpressed as:

Each of the parameters in this equation are defined specificallyfor each of the applicable activity methods.

Description of TermsThe Regular Solution equation uses the following:

(3.396)

(3.397)

2 Γi Lln

A j i, x j j∑⎝ ⎠⎜ ⎟⎛ ⎞

R j i, x j j∑⎝ ⎠⎜ ⎟⎛ ⎞

S j i, x j j∑⎝ ⎠⎜ ⎟⎛ ⎞

V j i, x j j∑⎝ ⎠⎜ ⎟⎛ ⎞

-------------------------------------------------------- xk

A j k , x j j∑⎝ ⎠⎜ ⎟⎛ ⎞

R j k , x j j∑⎝ ⎠⎜ ⎟⎛ ⎞

S j k , x j j∑⎝ ⎠⎜ ⎟⎛ ⎞

V j i, x j j∑⎝ ⎠⎜ ⎟⎛ ⎞

------------------------------------------------------------- ⋅k ∑+=

A i k ,

A j k , x j j∑

----------------------- R i k ,

R j k , x j j∑

-----------------------S i k ,

S j k , x j j∑

---------------------- – V i k ,

V j k , x j j∑

----------------------- – +

A i j,vi L δi δ j – ( )2

RT ----------------------------= R i j,

A i j, A j i,--------= V i j, Ri j,= S i j, R i j,=




3-190

δi is the solubility parameter in (cal/cm 3) ½ and v i L is the

saturated liquid volume in cm 3 /mol calculated from:

The van Laar , Margules and Scatchard Hamer use thefollowing:

For the van Laar, Margules and Scatchard Hamer equations:

where: T must be in K

This equation is of a different form than the original van Laarand Margules equations in HYSYS, which used an a + bT relationship. However, since HYSYS only contains a ij values, thedifference should not cause problems.

The NRTL form for the Chien-Null uses:

The expression for the τ term under the Chien-Null incorporates

(3.398)vi L vω i, 5.7 3 T r i,+( )=

Model A i,j R i,j S i,j Vi,j

van Laar

Margules

Scatchard Hamer

γi j,∞

ln i j, A j i,-------- R i j, R i j,

2 γi j,∞

ln

1

γi j,∞

ln

γ j i,∞ln----------------⎝ ⎠⎜ ⎟

⎛ ⎞

+

------------------------------- Ai j, A j i,-------- 1 1

2 γi j,∞

ln

1γi j,

∞ln

γ j i,∞

ln----------------⎝ ⎠⎜ ⎟⎛ ⎞

+

------------------------------- Ai j, A j i,--------

vi∞

v j∞------

vi∞

v j∞------

(3.399)γi j,∞

ln a i j,b i j,T

-------- c ij T + +=

(3.400) A i j, 2τi j, V i j,= R i j, 1= V i j, c i j, – τi j,( )exp= S i j, 1= τi j, a i j,b i j,

T K ( )------------+=




3-191

the R term of NRTL into the values for a ij and b ij . As such, thevalues initialized for NRTL under Chien-Null will not be the sameas for the regular NRTL. When you select NRTL for a binary pair,

a ij will be empty (essentially equivalent to the regular NRTL b ij term), b ij will be initialized and c ij will be the α term for theoriginal NRTL, and will be assumed to be symmetric.

The General Chien-Null equation is:

In all cases:

With the exception of the Regular Solution option, all modelscan use six constants, a i,j , a j,i , b i,j , b j,i , c i,j and c j,i for eachcomponent pair. For all models, if the constants are unknownthey can be estimated from the UNIFAC VLE or LLE methods,the Insoluble option, or using Henry's Law coefficients forappropriate components. For the general Chien-Null model, thec ij values are assumed to be 1.

(3.401)

(3.402)

A i j, a i j,b i j,

T K ( )------------+= R i j, A i j, A j i,--------= V i j, C i j,= S i j, C i j,=

A i i, 0= R i i, S i i, V i i, 1= = =




3-192

Property MethodsA quick reference of calculation methods is shown in the tablebelow for the Chien-Null (CN) property model.


Chien-Null Ln Activity CoefficientRefer to Equation (3.379) to Equation (3.385) for methodson calculating the activity coefficient for components, i , usingthe CN activity model.



Activity Coefficient Liquid COTHCNLnActivityCoeff Class


Liquid COTHCNLnFugacityCoeff Class

Fugacity calculation Liquid COTHCNLnFugacity Class


Liquid COTHCNLnActivityCoeffDTClass

NRTL temperaturedependent properties

Liquid COTHNRTLTempDep Class

Excess Gibbs Liquid COTHCNExcessGibbsEnergyClass

Excess enthalpy Liquid COTHCNExcessEnthalpy Class

Enthalpy Liquid COTHCNEnthalpy Class

Gibbs energy Liquid COTHCNGibbsEnergy Class


COTHCNLnActivityCoeff Class Liquid




3-193

Chien-Null Fugacity CoefficientThis method calculates the fugacity coefficient of componentsusing the CN activity model. The fugacity coefficient ofcomponent i , φi , is calculated from the following relation.




Chien-Null FugacityThis method calculates the fugacity of components using theUNIFAC activity model. The fugacity of component i , f i , iscalculated from the following relation.




(3.403)


COTHCNLnFugacityCoeff Class Liquid

The term, ln γi , in the above equation is exclusively calculatedusing the Chien-Null Ln Activity Coefficient . For the standardfugacity, f i


(3.404)

φln i ln γi f i

st d

P --------

⎝ ⎠⎜ ⎟⎛ ⎞

=





3-194


Chien-Null Activity CoefficientDifferential wrt TemperatureThis method analytically calculates the activity coefficientdifferential wrt to temperature from the following relation.



COTHCNLnFugacity Class Liquid

The term, ln γi , in the above equation is exclusively calculatedusing the Chien-Null Ln Activity Coefficient . For the standardfugacity, f i


(3.405)


COTHCNLnActivityCoeffDT Class Liquid

∂ γiln∂T

------------




3-195

Chien-Null Excess Gibbs EnergyThis method calculates the excess Gibbs energy using the CNactivity model from the following relation.




Chien-Null Gibbs EnergyThis method calculates the Gibbs free energy using the CNactivity model from the following relation.




(3.406)


COTHCNExcessGibbsEnergy Class Liquid

The term, ln γi , in the above equation is exclusivelycalculated using the Chien-Null Ln Activity Coefficient .

(3.407)

G E RT x i γiln

i

n

∑=


i

n

∑+

i

n

∑=




3-196


Chien-Null Excess EnthalpyThis method calculates the excess enthalpy using the CN activity

model from the following relation.





COTHCNGibbsEnergy Class Liquid

The term, G E, in the above equation is exclusively calculatedusing the Chien-Null Excess Gibbs Energy .

(3.408)


COTHCNExcessEnthalpy Class Liquid

The term, , in the above equation is exclusivelycalculated using the Chien-Null Activity CoefficientDifferential wrt Temperature .

H E

RT 2

xi

d γiln

dT ------------

i

n

∑ – =

d γiln

dT ------------




3-197

Chien-Null EnthalpyThis method calculates the enthalpy using the CN activity modelfrom the following relation.





(3.409)


COTHCNEnthalpy Class Liquid

The term, H E, in the above equation is exclusively calculatedusing the Chien-Null Excess Enthalpy .

H x i H i H E +

i

n

∑=






3-199

3.4 Grayson-Streed ModelThe Grayson-Streed (GS) method is an older, semi-empiricalmethod. The GS correlation is an extension of the Chao-Seadermethod with special emphasis on hydrogen. This method hasalso been adopted by and is recommended for use in the APITechnical Data Book.


The following table gives an approximate range of applicabilityfor this method, and under what conditions it is applicable.

Grayson-StreedModel Description

Grayson-Streed Recommended for simulating heavy hydrocarbonsystems with a high hydrogen content.


COTHGraysonStreedLnFugacityCoeffClass

Liquid

COTHGraysonStreedLnFugacity Class Liquid

Method Temp. (°C) Temp. (°C) Press.(psia)

Press.(kPa)

GS 0 to 800 18 to 425 < 3,000 < 20,000

Conditions of Applicability

For all hydrocarbons (exceptCH4):

0.5 < Tr i < 1.3 and Pr mixture < 0.8

If CH4 or H2 is present: • molal average Tr < 0.93 • CH4 mole fraction < 0.3• mole fraction dissolved gases < 0.2

When predicting K values for:Paraffinic or Olefinic MixturesAromatic Mixtures

liquid phase aromatic mole fraction <0.5liquid phase aromatic mole fraction >0.5



3-200 Grayson-Streed Model

3-200

The GS correlation is recommended for simulating heavyhydrocarbon systems with a high H 2 content, such ashydrotreating units. The GS correlation can also be used for

simulating topping units and heavy ends vacuum applications.The vapour phase fugacity coefficients are calculated with theRedlich Kwong equation of state. The pure liquid fugacitycoefficients are calculated via the principle of correspondingstates. Modified acentric factors are included in the library formost components. Special functions have been incorporated forthe calculation of liquid phase fugacities for N 2 , CO 2 and H 2 S.These functions are restricted to hydrocarbon mixtures with lessthan five percent of each of the above components. As with theVapour Pressure models, H 2O is treated using a combination ofthe steam tables and the kerosene solubility charts from the APIdata book. This method of handling H 2O is not very accurate for

gas systems. Although three phase calculations are performedfor all systems, it is important to note that the aqueous phase isalways treated as pure H 2O with these correlations.




3-201



Physical Property Calculation Methods 4-1

4-1

4 Physical PropertyCalculation Methods

4.1 Cavett Method................................................................................ 2

4.2 Rackett Method .............................................................................. 8

4.3 COSTALD Method ......................................................................... 11

4.4 Viscosity ...................................................................................... 14

4.5 Thermal Conductivity ................................................................... 18

4.6 Surface Tension ........................................................................... 21

4.7 Insoluble Solids ........................................................................... 22



4-2 Cavett Method

4-2

4.1 Cavett MethodAspen HYSYS Thermodynamics COM Interface uses the three-parameter corresponding states method to represent theenthalpy of a liquid when working with the activity models.Water is the only exception which uses the 1967 formulation forsteam ( 37McClintock and Silvestri, 1967). For the Cavett method,a generalized slope for the liquid enthalpy is correlated using P c ,T c and the Cavett parameter (an empirical constant fitted tomatch the heat of vapourization at the normal boiling point).The Cavett parameter may be approximated by the criticalcompressibility factor of a component if no heat of vapourizationdata is available.

Property MethodsA quick reference of calculation methods is shown in the tablebelow for the Cavett method.


Calculation Method PhaseApplicable Property Class Name

Enthalpy Liquid COTHCavettEnthalpy Class

Entropy Liquid COTHCavettEntropy Class

Isobaric heat capacity Liquid COTHNCavettCp Class

Helmholtz energy Liquid COTHCavettHelmholtz Class

Gibbs energy Liquid COTHCavettGibbs ClassInternal energy Liquid COTHCavettInternalEnergy

Class




4-3

Cavett EnthalpyThis method calculates the liquid enthalpy using the Cavettmodel from the following relation.

where: the calculation of the change in Cavett enthalpy is shownbelow

where: i = non-aqueous components


For subcritical, non-hydrocarbon components, the change in

(4.1)

(4.2)

(4.3)

(4.4)

(4.5)

(4.6)

(4.7)

(4.8)

H l

xwater H water st ea m 67

xi H i° Δ H iCavett

+( )i∑+=

Δ H non aqueous – cavett

min Δ H icavett ( ) xi

i 1 i 1≠;=

nc

∑=

Δ H i1 T c i, a 1 a 2 1 T r i, – ( )e 1+( )=

a 1 b 1 b2χ i b 3χ i2

b4χ i3

+ + +=

a 2 b 5 b6χ i b 7χ i2 b8χ i

3+ + +=

a 9 b 9 b 10 χ i b 11 χ i2

b12 χ i3

+ + +=

e1 1 a 3 T r i, 0.1 – ( ) – =

Δ H i2 T c i, max c 1 c 2T r i,

2 c3T r i,3 c4T r i,

4 c5T r i,2 0,+ + + +( )( )=



4-4 Cavett Method

4-4

enthalpy is:

For subcritical, hydrocarbon components, the change inenthalpy is:

For supercritical components, the change in enthalpy is:

Property Class Name and Phases Applicable

Cavett EntropyThis method calculates the liquid entropy using the Cavettmodel from the following relation:

For subcritical, non-hydrocarbon components, the change inentropy is:

(4.9)

(4.10)

(4.11)

Property Class Name Phase Applicable

COTHCavettEnthalpy Class Liquid

(4.12)

(4.13)

Δ H icavett

Δ H i1

=

Δ H icavett min Δ H i

1 Δ H i2,( )=

Δ H icavett Δ H i2=

S l

xwater S water st ea m 67

xi S i° ΔS iCavett

+( )i∑+=

ΔS icavett Δ H i1

T ------------=




4-5

For subcritical, hydrocarbon components, the change in entropyis:

For supercritical components, the change in entropy is:



T = Temperature (K)


Cavett Cp (Isobaric)This method calculates the liquid isobaric heat capacity usingthe Cavett model from the following relation.


For subcritical hydrocarbons with ΔH i 1 > ΔH i

2 , the change in heatcapacity is:

(4.14)

(4.15)


COTHCavettEntropy Class Liquid

(4.16)

(4.17)

ΔS icavett mi n Δ H i

1 Δ H i2,( )

T ------------------------------------------=

ΔS icavett Δ H i

2

T ------------=

Cp l xwater Cp water st ea m 67 xi Cp i° ΔCp i

Cavett +( )i∑+=

ΔCp icavett

T r i, 2 c 2 c 5 P r i,+( ) T r i, 3 c3 T r i, 4 c4( )+( )+( )=



4-6 Cavett Method

4-6

For other subcritical components, the change in heat capacity is:

For supercritical components with ΔH i 2 equal to zero, the change

in heat capacity is:

For supercritical components with ΔH i 2 different than zero, the

change in heat capacity is:

where:


(4.18)

(4.19)

(4.20)

b1 = -67.022001 b7 = -23612.5670 c1 = 10.273695

b2 = 644.81654 b8 = 34152.870 c2 = -1.5594238

b3 = -1613.1584 b9 = 8.9994977 c3 = 0.019399

b4 = 844.13728 b10 = -78.472151 c4 = -0.03060833

b5 = -270.43935 b11 = 212.61128 c5 = -0.168872

b6 = 4944.9795 b12 = -143.59393


COTHCavettCp Class Liquid

The term, ΔH i 1 , in the above equation is exclusively

calculated using the Cavett Enthalpy .

ΔCp icavett a 1

Δ H i

1

T c i,------------ –

⎝ ⎠⎜ ⎟

⎛ ⎞a 3 1 T r i, – ( )

e1

1 T r i, – ----------------- -+⎝ ⎠⎛ ⎞log⎝ ⎠⎛ ⎞=

ΔCp icavett 0=

ΔCp icavett T r i, 2 c2 c5 P r i,+( ) T r i, 3 c3 T r i, 4 c4( )+( )+( )=




4-7

Cavett Helmholtz EnergyThis method calculates the liquid Helmholtz energy using theCavett model from the following relation.


Cavett Gibbs EnergyThis method calculates the liquid Gibbs free energy using theCavett model from the following relation.

where: H = Cavett enthalpy

S = Cavett entropy

(4.21)


COTHCavettHelmholtz Class Liquid

The term, G , in the above equation is exclusively calculatedusing the Cavett Gibbs Energy .

(4.22)

A G PV – =

G H TS – =



4-8 Rackett Method

4-8


Cavett Internal EnergyThis method calculates the liquid internal energy using theCavett model from the following relation.


4.2 Rackett MethodLiquid densities and molar volumes can be calculated bygeneralized cubic equations of state, although they are ofteninaccurate and often provide incorrect estimations. AspenHYSYS Thermodynamics COM Interface allows for alternatemethods of calculating the saturated liquid volumes includingthe Rackett Liquid Density correlations. This method wasdeveloped by Rackett (1970) and later modified by Spencer andDanner.


COTHCavettGibbs Class Liquid

The terms, H and S , in the above equation are exclusivelycalculated using the Cavett Enthalpy and Cavett Entropy ,respectively.

(4.23)


COTHCavettInternal Class Liquid

The term, H , in the above equation is exclusively calculated

using the Cavett Enthalpy .

U H PV – =

Property Packages withthis option currentlyavailable:NRTL-Ideal-ZraPeng-Robinson-RackettLiq Density




4-9

The Rackett Equation has been found to produce slightly moreaccurate estimations for chemical groups such as acetylenes,cycloparaffins, aromatics, flurocarbons, cryogenic liquids, and

sulfides.

Property MethodsA quick reference of liquid density and volume calculations areshown in the table below for the Rackett method.


Rackett Liquid VolumeThis method calculates the liquid volume using the Rackettmethod from the following relation.

where: V s = saturated liquid volume

R = ideal gas constant

T c & P c = critical constants for each compound

Z RA = Rackett compressibility factor

T r = reduced temperature, T/T c

CalculationMethod

PhaseApplicable Property Class Name

Liquid Volume Liquid COTHRackettVolumeClass

Liquid Density Liquid COTHRackettDensityClass

(4.24)V s RT c P c--------- Z RA

1 1 T r – ( )2

7---

+=



4-10 Rackett Method

4-10


Rackett Liquid DensityThis method calculates the liquid density using the Rackettmethod from the following relation.


R = ideal gas constant

T c & P c = critical constants for each compound

Z RA = Rackett compressibility factor




COTHRackettVolume Class Liquid

The Rackett Compressibility factor ( Z RA ) is a unique constantfor each compound and is usually determined fromexperimental data, however if no data is available, Z c can beused as an estimate of Z RA .

(4.25)


COTHRackettDensity Class Liquid

ρ s 1 RT c P c---------⎝ ⎠⎛ ⎞ ⁄

⎝ ⎠⎛ ⎞ Z RA

1 1 T r – ( )27---

+=




4-11

4.3 COSTALD MethodSaturated liquid volumes are obtained using a correspondingstates equation developed by 38R.W. Hankinson and G.H.Thompson which explicitly relates the liquid volume of a purecomponent to its reduced temperature and a second parametertermed the characteristic volume. This method has beenadopted as an API standard.

The pure compound parameters needed in the correspondingstates liquid density (COSTALD) calculations are taken from theoriginal tables published by Hankinson and Thompson, and theAPI data book for components contained in the HYSYS library.The parameters for hypothetical components are based on the

API gravity and the generalized Lu equation.

Although the COSTALD method was developed for saturatedliquid densities, it can be applied to sub-cooled liquid densities(i.e., at pressures greater than the vapour pressure), using theChueh and Prausnitz correction factor for compressed fluids. Itis used to predict the density for all systems whose pseudo-reduced temperature is below 1.0. Above this temperature, theequation of state compressibility factor is used to calculate theliquid density.

38R.W. Hankinson and G.H. Thompson (1979) published a new

method of correlating saturated densities of liquids and theirmixtures. This method was superior to its predecessors in that itovercame the mathematical discontinuities presented inmethods by Yen and Woods (1966) and was not limited to purecompounds. COSTALD was later successfully applied tocompressed liquids and liquid mixtures.



4-12 COSTALD Method

4-12

Property MethodsA quick reference of liquid density and volume calculations areshown in the table below for the Rackett method.


COSTALD Liquid VolumeThis method calculates the liquid volume using the COSTALDmethod for pure compounds:

CalculationMethod


Liquid Volume Liquid COTHCOSTALDVolume Class

Liquid Density Liquid COTHCOSTALDDensity Class

(4.26)

V s V ∗ ⁄ V r o( )

1 ωSR K V r δ( )

– [ ]=

V r o( )

1 Ak 1 T r – ( )k 3 ⁄

k 1=

4

∑+= 0.25 T r 0.95< <

V r δ( )

Bk T r k

k 0=

3

∑ T r 1.00001 – ( ) ⁄ = 0.25 T r 1.0< <




4-13

and for mixtures:

where: A k and B k are constantsV* = the characteristic volume

ωSRK = SRK acentric factor

T c = critical temperature for each compound



(4.27)


COTHCOSTALDVolume Class Liquid

T cm xi x j V ij∗T c ij

j∑i∑⎝ ⎠⎜ ⎟

⎛ ⎞

V m∗ ⁄ =

V m∗ 1 4 ⁄ xiV i∗

i∑ 3 xiV i∗

23---

i∑⎝ ⎠⎜ ⎟⎜ ⎟⎛ ⎞

x iV i∗13---

i∑⎝ ⎠⎜ ⎟⎜ ⎟⎛ ⎞

+=

V ij∗T c ijV i∗T c i

V j∗T c j( )

12---

=

ωSR K m x iωSR K i

i∑=



4-14 Viscosity

4-14

COSTALD Liquid DensityThis method calculates the liquid density using the COSTALDmethod from the following relation.



4.4 ViscosityThis method will automatically select the model best suited forpredicting the phase viscosities of the system under study. Themodel selected will be from one of the three available in thismethod: a modification of the NBS method ( 39Ely and Hanley),Twu's model, or a modification of the Letsou-Stiel correlation.This method will select the appropriate model using thefollowing criteria:

(4.28)


COTHCOSTALDDensity Class Liquid

The saturated liquid volume, V s , is calculated from Equations(4.26) and (4.27) .

Chemical System Vapour Phase Liquid Phase

Lt Hydrocarbons (NBP <155°F)

Mod Ely & Hanley Mod Ely & Hanley

Hvy Hydrocarbons (NBP >155°F)

Mod Ely & Hanley Twu

Non-Ideal Chemicals Mod Ely & Hanley Mod Letsou-Stiel

ρ 1V s-----=




4-15

All of the models are based on corresponding states principlesand have been modified for more reliable application. Internalvalidation showed that these models yielded the most reliable

results for the chemical systems shown. Viscosity predictions forlight hydrocarbon liquid phases and vapour phases were foundto be handled more reliably by an in-house modification of theoriginal Ely and Hanley model, heavier hydrocarbon liquids weremore effectively handled by Twu's model, and chemical systemswere more accurately handled by an in-house modification ofthe original Letsou-Stiel model.

A complete description of the original corresponding states(NBS) model used for viscosity predictions is presented by Elyand Hanley in their NBS publication. The original model hasbeen modified to eliminate the iterative procedure for

calculating the system shape factors. The generalized Leech-Leland shape factor models have been replaced by componentspecific models. This method constructs a PVT map for eachcomponent using the COSTALD for the liquid region. The shapefactors are adjusted such that the PVT map can be reproducedusing the reference fluid.

The shape factors for all the library components have alreadybeen regressed and are included in the Pure Component Library.Hypocomponent shape factors are regressed using estimatedviscosities. These viscosity estimations are functions of thehypocomponent Base Properties and Critical Properties.

Hypocomponents generated in the Oil CharacterizationEnvironment have the additional ability of having their shapefactors regressed to match kinematic or dynamic viscosityassays.

The general model employs CH 4 as a reference fluid and isapplicable to the entire range of non-polar fluid mixtures in thehydrocarbon industry. Accuracy for highly aromatic ornaphthenic crudes will be increased by supplying viscositycurves when available, since the pure component propertygenerators were developed for average crude oils. The model

also handles H 2 O and acid gases as well as quantum gases.



4-16 Viscosity

4-16

Although the modified NBS model handles these systems verywell, the Twu method was found to do a better job of predictingthe viscosities of heavier hydrocarbon liquids. The Twu model is

also based on corresponding states principles, but hasimplemented a viscosity correlation for n-alkanes as itsreference fluid instead of CH 4 . A complete description of thismodel is given in the paper entitled " 42Internally ConsistentCorrelation for Predicting Liquid Viscosities of PetroleumFractions".

For chemical systems, the modified NBS model of Ely andHanley is used for predicting vapour phase viscosities, whereasa modified form of the Letsou-Stiel model is used for predictingthe liquid viscosities. This method is also based oncorresponding states principles and was found to perform

satisfactorily for the components tested.

The shape factors contained within this methods PureComponent Library have been fit to match experimentalviscosity data over a broad operating range.

Property Class Name and Phases ApplicableProperty Class Name Phase Applicable

COTHViscosity Class Liquid and vapour




4-17

Liquid Phase Mixing Rules forViscosityThe estimates of the apparent liquid phase viscosity ofimmiscible Hydrocarbon Liquid - Aqueous mixtures arecalculated using the following "mixing rules":

• If the volume fraction of the hydrocarbon phase isgreater than or equal to 0.5, the following equation isused 51:

where: μeff = apparent viscosity

μoil = viscosity of Hydrocarbon phase

νoil = volume fraction Hydrocarbon phase

• If the volume fraction of the hydrocarbon phase is lessthan 0.33, the following equation is used 52:

where: μeff = apparent viscosity

μoil

= viscosity of Hydrocarbon phase

μH2O = viscosity of Aqueous phase

νoil = volume fraction Hydrocarbon phase

• If the volume of the hydrocarbon phase is between 0.33and 0.5, the effective viscosity for combined liquid phaseis calculated using a weighted average betweenEquation (4.29) and Equation (4.30) .

The remaining properties of the pseudo phase are calculated as

(4.29)

(4.30)

μef f μoi l e3.6 1 νoil – ( )

=

μef f 1 2.5 νoi l

μoi l 0.4 μ H 2O+

μoi l μ H 2O+------------------------------------⎝ ⎠⎜ ⎟⎛ ⎞

+ μ H 2O=



4-18 Thermal Conductivity

4-18

follows:

4.5 Thermal ConductivityAs in viscosity predictions, a number of different models andcomponent specific correlations are implemented for predictionof liquid and vapour phase thermal conductivities. The text byReid, Prausnitz and Poling 6 was used as a general guideline indetermining which model was best suited for each class ofcomponents. For hydrocarbon systems, the correspondingstates method proposed by Ely and Hanley 39 is generally used.The method requires molecular weight, acentric factor and idealheat capacity for each component. These parameters aretabulated for all library components and may either be input orcalculated for hypothetical components. It is recommended thatall of these parameters be supplied for non-hydrocarbonhypotheticals to ensure reliable thermal conductivity coefficientsand enthalpy departures.

The modifications to the method are identical to those for theviscosity calculations. Shape factors calculated in the viscosityroutines are used directly in the thermal conductivity equations.The accuracy of the method will depend on the consistency ofthe original PVT map.

The Sato-Reidel method is used for liquid phase thermalconductivity predictions of glycols and acids, the Latini et al

(4.31)

MW ef f xi MW i∑=

ρef f 1 xiρ

i----⎝ ⎠⎛ ⎞∑

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

C p eff xiC p i∑=

(molecular weight)

(mixture density)

(mixture specific heat)




4-19

method is used for esters, alcohols and light hydrocarbons in therange of C 3 -C 7 , and the Missenard and Reidel method is used forthe remaining components.

For vapour phase thermal conductivity predictions, the Misic andThodos, and Chung et al methods are used. The effect of higherpressure on thermal conductivities is taken into account by theChung et al method.

Property Class Name and Phases ApplicableProperty Class Name Phase Applicable

COTHThermCond Class Liquid and vapour



4-20 Thermal Conductivity

4-20

As with viscosity, the thermal conductivity for two liquid phasesis approximated by using empirical mixing rules for generating asingle pseudo liquid phase property. The thermal conductivity

for an immiscible binary of liquid phases is calculated by thefollowing equation 53:

where: λLmix = mixture liquid thermal conductivity at temperature T(K)

κij = liquid thermal conductivity of pure component i or j attemperature T

λL1 = liquid thermal conductivity of liquid phase 1

λL2 = liquid thermal conductivity of liquid phase 2

φ1 =

φ2 =


V i = molar volume of component i

(4.32)λ Lmixφ1

2λ L12φ1φ2λ12 φ2

2λ L2+ +=

λ Lm ix φiφ jk ij j∑

i∑=

k ij2

1 k i ⁄ ( ) 1 k j ⁄ ( )+--------------------------------------=

x1V 1

xiV ii 1=

2

∑-------------------

x2V 2

xiV ii 1=

2

∑-------------------




4-21

4.6 Surface TensionSurface tensions for hydrocarbon systems are calculated using amodified form of the Brock and Bird equation.


The equation expresses the surface tension, σ, as a function ofthe reduced and critical properties of the component. The basic

form of the equation was used to regress parameters for eachfamily of components.

where: σ = surface tension (dynes/cm 2 )

Q = 0.1207[1.0 + T BR ln P c /(1.0 - T BR )] - 0.281

T BR = reduced boiling point temperature (T b /T c )

a = parameter fitted for each chemical class

b = c 0

+ c 1

ω + c 2ω

2+ c

3 ω

3 (parameter fitted for each

chemical class, expanded as a polynomial inacentricity)

For aqueous systems, HYSYS employs a polynomial to predictthe surface tension. It is important to note that HYSYS predictsonly liquid-vapour surface tensions.


COTHSurfaceTension Class Liquid and vapour

(4.33)σ P c2 3 ⁄

T c1 3 ⁄

Q 1 T R – ( )a b×=



4-22 Insoluble Solids

4-22

4.7 Insoluble SolidsAn insoluble solid is identified from its pure compound "family"classification.

Property MethodsA quick reference of calculation methods for insoluble solids isshown in the table below.


CalculationMethod


MolarDensity xptInsolubleSolid

COTHSolidDensity Class

MolarVolume xptInsolubleSolid

COTHSolidVolume Class

Enthalpy xptInsolubleSolid

COTHSolidEnthalpy Class

Entropy xptInsolubleSolid

COTHSolidEntropy Class

Cp xptInsolubleSolid

COTHSolidCp Class




4-23

Insoluble Solid Molar Density


Insoluble Solid MolarVolume



COTHSolidDensity Class xptInsolubleSolid


COTHSolidVolume Class xptInsolubleSolid



4-24 Insoluble Solids

4-24

Insoluble Solid Enthalpy


Insoluble Solid Entropy


Insoluble Solid Cp



COTHSolidEnthalpy Class xptInsolubleSolid


COTHSolidEnthalpy Class xptInsolubleSolid


COTHSolidCp Class xptInsolubleSolid





5-2 Enthalpy Reference States

5-2

5.1 Enthalpy Reference

StatesAll enthalpy calculations are determined with respect to areference enthalpy which are defined in the following methods.

Property MethodsThe enthalpy reference state calculation methods are shown inthe table below.

5.1.1 Ideal Gas EnthalpyOffset

The Ideal Gas enthalpy calculates and returns an array of:

for all components.

Calculation Method Phase ApplicableProperty ClassName

Ideal Gas EnthalpyOffset

Vapour & Liquid COTHOffsetIGHClass

Enthalpy Offset Vapour & Liquid COTHOffsetH Class

(5.1) H iig offset H i+



References & Standard States

5-3


5.1.2 Enthalpy OffsetThe enthalpy offset calculates and returns an array of:

for all components.

where: H ig(25°C) = ideal gas enthalpy at 25°C.

H fig(25°C) = ideal gas enthalpy with heat of formation of thecomponent at 25°C.



COTHOffsetIGH Class Vapour & Liquid

The term, offset H i , is calculated by Section 5.1.2 - EnthalpyOffset .

(5.2)


COTHOffsetH Class Vapour & Liquid

Offset H i H iig 25 °C ( )

– H i fi g 25 °C ( )

+=



5-4 Entropy Reference States

5-4

5.2 Entropy Reference

StatesAll entropy calculations are determined with respect to areference enthalpy which are defined in the following methods.

Property MethodsThe entropy reference state calculation methods are shown inthe table below.

5.2.1 Ideal Gas Entropy OffsetThe Ideal Gas entropy calculates and returns an array of:

for all components.

Calculation Method Phase ApplicableProperty ClassName

Ideal Gas EntropyOffset

Vapour & Liquid COTHOffsetIGSClass

Entropy Offset Vapour & Liquid COTHOffsetS Class

(5.3)S iig

offset S i+




5-5


5.2.2 Entropy OffsetThe entropy offset calculates and returns an array of:

for all components.


5.3 Ideal Gas CpThe ideal gas C p calculates and returns an array containing theideal gas C p of all components.


COTHOffsetIGS Class Vapour & Liquid

The term, offset S i , is calculated by Section 5.2.2 - EntropyOffset .

(5.4)


COTHOffsetS Class Vapour & Liquid

Offset S 0=



5-6 Standard State Fugacity

5-6

5.4 Standard State

FugacityThe fugacity of component, i, in the mixture is related to itsactivity coefficient composition through the following equation.


f istd = standard state fugacity of component i


The standard state fugacity, f i std , is defined at the temperature

and pressure of the mixture. As, γi , approaches one in the limit, the standard state fugacity may be related to the vapour

pressure of component i .

where: P i

sat = vapour pressure of component i at the temperature ofthe system

φi sat =fugacity coefficient of pure component i at temperature

T and pressure P i sat

P = pressure of the system

Vi = liquid molar volume of component i at T and P

R = gas constant

T = temperature of system

The Poynting factor accounts for the effect of pressure on liquidfugacity and is represented by the exponential term in theabove equation. The correction factor generally is neglected ifthe pressure does not exceed a few atmospheres. The liquid

(5.5)

(5.6)

f i γi xi f i st d =

xi 1→

f i st d

P = i sa t φi

sa t V i RT ------- P d

P i sat

P

∫exp






5-8

5.4.1 Standard State without

Poynting CorrectionThis method calculates the standard state fugacity for allcomponents. The effects of the poynting correction and φi

sat inEquation (5.5) are neglected.

For condensible components, the standard state fugacity iscalculated as:


Notes

For non-condensible components in the presence of anycondensible components, Henry’s law is used as shown below.

In a system of all non-condensible components and nocondensible components, the standard state fugacity iscalculated as:

(5.8)


COTHIdealStdFug Class Liquid

(5.9)

(5.10)

f i st d

P = i sa t

f i st d H = i j,

f i st d P = i

sa t






5-10

5.4.3 Ideal Standard State

with Fugacity CoefficientThis method calculates the standard state fugacity for allcomponents. The effect of the fugacity coefficient, φi

sat , isincluded although the poynting factor in Equation (5.5) isneglected.

For condensible components, the standard state fugacity iscalculated as:


Notes

For non-condensible components in the presence of any

condensible components, Henry’s law is used as shown below.


The fugacity coefficient, φi sat , is calculated from the specified

vapour model.

(5.14)


COTHPhiStdFug Class Liquid

(5.15)

(5.16)

f i st d

P = i sa t φi

sa t

f i st d

H = i j, P P i sa t

– ( )V i RT ( ) ⁄ ][exp

f i st d

P = i sa t

P P i sa t

– ( )V i RT ( ) ⁄ ][exp




5-11

5.4.4 Ideal Standard State

with Fugacity Coeff &PoyntingThis method calculates the standard state fugacity for allcomponents. The effects of the fugacity coefficient, φi

sat , and thepoynting correction in Equation (5.5) are included.

For condensible components, the standard state fugacity is:


Notes

For non-condensible components in the presence of any

condensible components, Henry’s law is used as shown below.


The fugacity coefficient, φi sat , is calculated from the specified

vapour model.

(5.17)


COTHPoyntingPhiStdFug Class Liquid

(5.18)

(5.19)

f i st d

P = i sa t φi

sa t P P i

sa t – ( )V i RT ( ) ⁄ ][exp

f i st d

H = i j, P P i sa t

– ( )V i RT ( ) ⁄ ][exp

f i st d

P = i sa t

P P i sa t

– ( )V i RT ( ) ⁄ ][exp




5-12



Flash Calculations 6-1

6-1

6 Flash Calculations

6.1 Introduction................................................................................... 2

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

6.3 Vapour Fraction Flash .................................................................... 4

6.3.1 Dew Points ..............................................................................46.3.2 Bubble Points/Vapour Pressure...................................................56.3.3 Quality Points .......................................................................... 5

6.4 Flash Control Settings.................................................................... 7



6-2 Introduction

6-2

6.1 IntroductionRigorous three phase calculations are performed for allequations of state and activity models with the exception of theWilson equation, which only performs two phase vapour-liquidcalculations.

Aspen HYSYS Thermodynamics COM Interface uses internalintelligence to determine when it can perform a flash calculationon a stream, and then what type of flash calculation needs to beperformed on the stream. This is based completely on thedegrees of freedom concept. When the composition of a streamand two property variables are known, (vapour fraction,temperature, pressure, enthalpy or entropy, one of which must

be either temperature or pressure), the thermodynamic state ofthe stream is defined.

Aspen HYSYS Thermodynamics COM Interface automaticallyperforms the appropriate flash calculation when sufficientinformation is known. Depending on the known streaminformation, one of the following flashes are performed: T-P,T-VF, T-H, T-S, P-VF, P-H, or P-S.

Specified variables canonly be re-specified by youor via the Recycle Adjust , or SpreadSheet operations. They will notchange through any heator material balance





6-4 Vapour Fraction Flash

6-4

6.3 Vapour Fraction FlashVapour fraction and either temperature or pressure are theindependent variables for this type of calculation. This class ofcalculation embodies all fixed quality points including bubblepoints (vapour pressure) and dew points. To perform bubblepoint calculation on a stream of known composition, simplyspecify the Vapour Fraction of the stream as 0.0 and define thetemperature or pressure at which the calculation is desired. Fora dew point calculation, simply specify the Vapour Fraction ofthe stream as 1.0 and define the temperature or pressure atwhich the dew point calculation is desired. Like the other typesof flash calculations, no initial estimates are required.

6.3.1 Dew PointsGiven a vapour fraction specification of 1.0 and eithertemperature or pressure, the property package will calculate theother dependent variable (P or T). If temperature is the secondindependent variable, the dew point pressure is calculated.Likewise, if pressure is the independent variable, then the dewpoint temperature will be calculated. Retrograde dew points maybe calculated by specifying a vapour fraction of -1.0. It isimportant to note that a dew point that is retrograde withrespect to temperature can be normal with respect to pressureand vice versa.

The vapour fraction is always shown in terms of the totalnumber of moles. For instance, the vapour fraction (VF)represents the fraction of vapour in the stream, while thefraction, (1.0 - VF), represents all other phases in the stream(i.e. a single liquid, 2 liquids, a liquid and a solid).

All of the solids will appearin the liquid phase.



Flash Calculations 6-5

6-5

6.3.2 Bubble Points/Vapour

PressureA vapour fraction specification of 0.0 defines a bubble pointcalculation. Given this specification and either temperature orpressure, the flash will calculate the unknown T or P variable. Aswith the dew point calculation, if the temperature is known, thebubble point pressure is calculated and conversely, given thepressure, the bubble point temperature is calculated. Forexample, by fixing the temperature at 100°F, the resultingbubble point pressure is the true vapour pressure at 100°F.

6.3.3 Quality PointsBubble and dew points are special cases of quality pointcalculations. Temperatures or pressures can be calculated forany vapour quality between 0.0 and 1.0 by specifying thedesired vapour fraction and the corresponding independentvariable. If HYSYS displays an error when calculating vapourfraction, then this means that the specified vapour fractiondoesn't exist under the given conditions, i.e., the specifiedpressure is above the cricondenbar, or the given temperature isto the right of the cricondentherm on a standard P-T envelope.

Enthalpy FlashGiven the enthalpy and either the temperature or pressure of astream, the property package will calculate the unknowndependent variables. Although the enthalpy of a stream cannotbe specified directly, it will often occur as the second propertyvariable as a result of energy balances around unit operationssuch as valves, heat exchangers and mixers.

If an error message appears, this may mean that an internallyset temperature or pressure bound has been encountered. Since

these bounds are set at quite large values, there is generallysome erroneous input that is directly or indirectly causing theproblem, such as an impossible heat exchange.

Vapour pressure andbubble point pressure aresynonymous.



6-6 Vapour Fraction Flash

6-6

Entropy FlashGiven the entropy and either the temperature or pressure of astream, the flash will calculate the unknown dependentvariables.

SolidsAspen HYSYS Thermodynamics COM Interface flash does notcheck for solid phase formation of pure components within theflash calculations.

Solids do not participate in vapour-liquid equilibrium (VLE)calculations. Their vapour pressure is taken as zero. However,since solids do have an enthalpy contribution, they will have aneffect on heat balance calculations. Thus, while the results of atemperature flash will be the same whether or not suchcomponents are present, an Enthalpy flash will be affected bythe presence of solids.





6-8 Flash Control Settings

6-8







Property Packages

7-3

Peng-RobinsonThis model is ideal for VLE calculations as well as calculatingliquid densities for hydrocarbon systems. However, in situationswhere highly non-ideal systems are encountered, the use ofActivity Models is recommended.

OffsetH COTHOffsetH Offset enthalpy with heat offormation.

OffsetIGS COTHOffsetIGS Ideal gas offset entropy.OffsetS COTHOffsetS Offset entropy.

XML File Name Name Description

pr_vapour Peng-Robinson Peng-Robinson Equation ofState using Mixing Rule 1 for allproperties.

Property Name Class Name Description

Enthalpy COTHPREnthalpy Peng-Robinson enthalpy.

Entropy COTHPREntropy Peng-Robinson entropy.

Cp COTHPRCp Peng-Robinson heat capacity.

LnFugacityCoeff COTHPRLnFugacityCoeff

Peng-Robinson fugacitycoefficient.

LnFugaci ty COTHPRLnFugacity Peng-Robinson fugacity.

MolarVolume COTHPRVolume Peng-Robinson molar volume.

Viscosity COTHViscosity Viscosity.

ThermalConductivity

COTHThermCond Thermal conductivi ty.

ZFactor COTHPRZFactor Peng-Robinsoncompressibility factor.

amix COTHPRab_1 Peng-Robinson amix.

IGCp COTHIdealGasCp Ideal gas heat capacity.

OffsetIGH COTHOffsetIGH Ideal gas offset enthalpy


OffsetIGS COTHOffsetIGS Ideal gas offset entropy.

OffsetS COTHOffsetS Offset entropy.




7-4 Vapour Phase Models

7-4

HysysPRThe HysysPR EOS is similar to the PR EOS with severalenhancements to the original PR equation. It extends its rangeof applicability and better represents the VLE of complexsystems.


hysyspr_vapour HysysPR HysysPR Equation of State using Mixing Rule 1 for allproperties.


Enthalpy COTHPR_HYSYS_Enthalpy

Peng-Robinson enthalpy.

Entropy COTHPR_HYSYS_Entropy

Peng-Robinson entropy.

Cp COTHPR_HYSYS_Cp Peng-Robinson heatcapacity.

LnFugacityCoeff COTHPR_HYSYS_LnFugacityCoeff


LnFugacity COTHPR_HYSYS_LnFugacity

Peng-Robinson fugacity.

MolarVolume COTHPR_HYSYS_Volume Peng-Robinson molarvolume.


ThermalConductivity COTHThermCond Thermal conductivity.




OffsetIGH COTHOffsetIGH Ideal gas offset enthalpy






Property Packages

7-5

Peng-Robinson-Stryjek-VeraThis is a two-fold modification of the PR equation of state thatextends the application of the original PR method for moderatelynon-ideal systems. It provides a better pure component vapourpressure prediction as well as a more flexible mixing rule thanPeng robinson.


prsv_vapour PRSV Peng-Robinson Stryjek-Vera using Mixing Rule 1 for allproperties.


Enthalpy COTHPRSVEnthalpy PRSV enthalpy.

Entropy COTHPRSVEntropy PRSV entropy.

Cp COTHPRSVCp PRSV heat capacity.

LnFugacityCoeff COTHPRSVLnFugacityCoeff

PRSV fugacity coefficient.

LnFugacity COTHPRSVLnFugacity PRSV fugacity.

MolarVolume COTHPRSVVolume PRSV molar volume.


ThermalConductivity

COTHThermCond Thermal conductivity.

ZFactor COTHPRSVZFactor PRSV compressibility factor.

amix COTHPRSVab_1 PRSV amix.IGCp COTHIdealGasCp Ideal gas heat capacity.

OffsetIGH COTHOffsetIGH Ideal gas Offset enthalpy.







7-6

Soave-Redlich-KwongIn many cases it provides comparable results to PR, but itsrange of application is significantly more limited. This method isnot as reliable for non-ideal systems.


srk_vapour SRK Soave-Redlich-Kwong Equation ofState using Mixing Rule 1 for allproperties.


Enthalpy COTHSRKEnthalpy SRK enthalpy.

Entropy COTHSRKEntropy SRK entropy.

Cp COTHSRKCp SRK heat capacity.

LnFugacityCoeff COTHSRKLnFugacityCoeff

SRK fugacity coefficient.

LnFugacity COTHSRKLnFugacity SRK fugacity.

MolarVolume COTHSRKVolume SRK molar volume.


ThermalConductivity


ZFactor COTHSRKZFactor SRK compressibility factor.

amix COTHSRKab_1 SRK amix.


OffsetIGH COTHOffsetIGH Ideal gas Offset enthalpy.OffsetH COTHOffsetH Offset enthalpy with heat of

formation.





Property Packages

7-7

Redlich-KwongThe Redlich-Kwong equation generally provides results similar toPeng-Robinson. Several enhancements have been made to thePR as described above which make it the preferred equation ofstate.


rk_vapour Redlich-Kwong Redlich-Kwong Equation ofState using Mixing Rule 1 for allproperties.


Enthalpy COTHRKEnthalpy RK enthalpy.

Entropy COTHRKEntropy RK entropy.

Cp COTHRKCp RK heat capacity.

LnFugacityCoeff COTHRKLnFugacityCoeff

RK fugacity coefficient.

LnFugacity COTHRKLnFugacity RK fugacity.

MolarVolume COTHRKVolume RK molar volume.


ThermalConductivity


ZFactor COTHRKZFactor SRK compressibility factor.

amix COTHRKab_1 SRK amix.

IGCp COTHIdealGasCp Ideal gas heat capacity.OffsetIGH COTHOffsetIGH Ideal gas Offset enthalpy.







7-8

Zudkevitch-JoffeeThis is a modification of the Redlich-Kwong equation of state,which reproduces the pure component vapour pressures aspredicted by the Antoine vapour pressure equation. This modelhas been enhanced for better prediction of vapour-liquidequilibrium for hydrocarbon systems, and systems containingHydrogen.


zj_vapour Zudkevitch-Joffee Zudkevitch-Joffee Equation ofState


Enthalpy COTHLeeKeslerEnthalpy

Lee-Kesler enthalpy.

Entropy COTHLeeKeslerEntropy Lee-Kesler entropy.

Cp COTHLeeKeslerCp Lee-Kesler heat capacity.

LnFugacityCoeff COTHZJLnFugacityCoeff

ZJ fugacity coefficient.

LnFugacity COTHZJLnFugacity ZJ fugacity.

MolarVolume COTHZJVolume ZJ molar volume.


ThermalConductivity


ZFactor COTHZJZFactor ZJ compressibility factor.amix COTHZJab_1 ZJ amix.








Property Packages

7-9

Kabadi-DannerThis model is a modification of the original SRK equation ofstate, enhanced to improve the vapour-liquid-liquid equilibriumcalculations for water-hydrocarbon systems, particularly indilute regions.


kd_vapour Kabadi-Danner Kabadi-Danner Equation ofState using Mixing Rule 1 for allproperties.


Enthalpy COTHKDEnthalpy KD enthalpy.

Entropy COTHKDEntropy KD entropy.

Cp COTHKDCp KD heat capacity.

LnFugacityCoeff COTHKDLnFugacityCoeff

KD fugacity coefficient.

LnFugacity COTHKDLnFugacity KD fugacity.

MolarVolume COTHKDVolume KD molar volume.


ThermalConductivity


ZFactor COTHKDZFactor KD compressibility factor.

amix COTHKDab_1 KD amix.








7-10

VirialThis model enables you to better model vapour phase fugacitiesof systems displaying strong vapour phase interactions.Typically this occurs in systems containing carboxylic acids, orcompounds that have the tendency to form stable hydrogenbonds in the vapour phase. In these cases, the fugacitycoefficient shows large deviations from ideality, even at low ormoderate pressures.


virial_vapour Virial The Virial Equation of State


Enthalpy COTHVirial_Enthalpy Virial enthalpy.

Entropy COTHVirial_Entropy Virial entropy.

Cp COTHVirial_Cp Virial heat capacity.

LnFugacityCoeff COTHVirial_LnFugacityCoeff

Virial fugacity coefficient.

LnFugacity COTHVirial_LnFugacity Virial fugacity.

MolarVolume COTHVirial_Volume Virial molar volume.


ThermalConductivity


ZFactor COTHVirial_ZFactor Virial compressibility factor.



OffsetH COTHOffsetH Offset enthalpy with heatof formation.





Property Packages

7-11

Lee-Kesler-PlöckerThis model is the most accurate general method for non-polarsubstances and mixtures.


lkp_vapour Lee-Kesler-Plöcker Lee-Kesler-Plöcker EOS usingMixing Rule 1 for all properties.


Enthalpy COTHLeeKeslerEnthalpy Lee-Kesler enthalpy.

Entropy COTHLeeKeslerEnthalpy Lee-Kesler entropy.


LnFugacityCoeff COTHLKPLnFugacityCoeff

LKP fugacity coefficient.

LnFugacity COTHLKPLnFugacity LKP fugacity.

MolarVolume COTHLKPMolarVolume LKP molar volume.


ThermalConductivity


SurfaceTension COTHSurfaceTension HYSYS surface t ension.

ZFactor COTHLKPZFactor LKP compressibility factor.



OffsetH COTHOffsetH Offset enthalpy with heat of

formation.OffsetIGS COTHOffsetIGS Ideal gas offset entropy.





7-12

Braun K10This model is strictly applicable to heavy hydrocarbon systemsat low pressures. The model employs the Braun convergencepressure method, where, given the normal boiling point of acomponent, the K-value is calculated at system temperature and10 psia (68.95 kPa).


braunk10_vapour Braun K10 Braun K10 Vapour PressureProperty Model.




Cp COTHLeeKeslerCp Lee-Kesler heatcapacity.

LnFugacityCoeff COTHIGLnFugacityCoeff Ideal gas fugacitycoefficient.

LnFugacity COTHIGLnFugacity Ideal gas fugacity.

MolarVolume COTHIGVolume Ideal gas molar volume.

MolarDensity COTHIGDensity Ideal gas molar density.


ThermalConductivity


SurfaceTension COTHSurfaceTension HYSYS surface tension.IGCp COTHIdealGasCp Ideal gas heat capacity.

OffsetIGH COTHOffsetIGH Ideal gas Offsetenthalpy.

OffsetH COTHOffsetH Offset enthalpy withheat of formation.





Property Packages

7-13

7.3 Liquid Phase ModelsThe property package information for the liquid phase models isshown in the following sections.

Ideal SolutionAssumes the volume change due to mixing is zero. This model ismore commonly used for solutions comprised of molecules nottoo different in size and of the same chemical nature.


idealsol_liquid Ideal Solution Ideal Solution Model


Enthalpy COTHCavettEnthalpy Cavett enthalpy.

Entropy COTHCavettEntropy Cavett entropy.

Cp COTHCavettCp Cavett heat capacity.

LnFugacityCoeff COTHIdealSolLnFugacityCoeff

Ideal Solution fugacitycoefficient.

LnFugaci ty COTHIdealSolLnFugaci ty Ideal solution fugaci ty.

LnActivity Coeff COTHIdealSolLnActivityCoeff Ideal solution activitycoefficient.

LnStdFugacity COTHIdealStdFug Ideal standard fugacitywith or without poyntingcorrection.

LnActivityCoeffDT COTHIdealSolLnActivityCoeff DT

Ideal solution activitycoefficient wrttemperature.

MolarDensity COTHCOSTALDDensity COSTALD molar density.

MolarVolume COTHCOSTALDVolume COSTALD molar volume.


ThermalConductivity


SurfaceTension COTHSurfaceTension Surface Tension.

Helmholtz COTHCavettHelmholtz Cavett Helmholtzenergy.

InternalEnergy COTHCavettInternalEnergy Cavett Internal energy.

GibbsEnergy COTHIdealSolGibbsEnergy Cavett Gibbs energy.



7-14 Liquid Phase Models

7-14






MolarDensity COTHSolidDensity Solid molar density.

MolarVolume COTHMolarVolume Solid molar volume.

Enthalpy COTHSolidEnthalpy Solid enthalpy.

Entropy COTHSolidEntropy Solid entropy.

Cp COTHSolidCp Solid heat capacity.




Property Packages

7-15

Regular SolutionThis model eliminates the excess entropy when a solution ismixed at constant temperature and volume. The model isrecommended for non-polar components where the moleculesdo not differ greatly in size. By the attraction of intermolecularforces, the excess Gibbs energy may be determined.


regsol_liquid Regular Solution Regular Solution Model .


Enthalpy COTHCavettEnthalpy Cavett enthalpy.Entropy COTHCavettEntropy Cavett entropy.


LnFugacityCoeff COTHRegSolLnFugacityCoef f

Regular Solutionfugacity coefficient.

LnFugacity COTHRegSolLnFugacity Regular solutionfugacity.

LnActivity Coeff COTHRegSolLnActivityCoeff Regular solution activitycoefficient.

LnStdFugacity COTHIdealStdFug Ideal standard fugacitywith or without poyntingcorrection.

LnActivityCoeffDT COTHRegSolLnActivityCoeff DT

Regular solution activitycoefficient wrttemperature.




ThermalConductivity





GibbsEnergy COTHCavettGibbs Cavett Gibbs energy.







7-16














7-18











Property Packages

7-19

MargulesThis was the first Gibbs excess energy representationdeveloped. The equation does not have any theoretical basis,but is useful for quick estimates and data interpolation.


margules_liquid Margules Two-parameter temperaturedependent Margules Model





LnFugacityCoeff COTHMargulesLnFugacityCoeff

Margules fugacitycoefficient.

LnFugacity COTHMargulesLnFugacity Margules fugacity.

LnActivity Coeff COTHMargulesLnActivityCoeff Margules activitycoefficient.

LnStdFugacity COTHMargulesStdFug Ideal standard fugacitywith or withoutpoynting correction.

LnActivityCoeffDT COTHMargulesLnActivityCoeff DT

Margules activitycoefficient wrttemperature.

MolarDensity COTHCOSTALDDensity COSTALD molar

density.MolarVolume COTHCOSTALDVolume COSTALD molar

volume.


ThermalConductivity






IGCp COTHIdealGasCp Ideal gas heatcapacity.






7-20

OffsetIGS COTHOffsetIGS Ideal gas offsetentropy.

OffsetS COTHOffsetS Offset entropy.MolarDensity COTHSolidDensity Solid molar density.








Property Packages

7-21

WilsonFirst activity coefficient equation to use the local compositionmodel to derive the Gibbs Excess energy expression. It offers athermodynamically consistent approach to predicting multi-component behaviour from regressed binary equilibrium data.However the Wilson model cannot be used for systems with twoliquid phases.


wilson_liquid Wilson Two-parameter temperaturedependent Wilson Model





LnFugacityCoeff COTHWilsonLnFugacityCoef f

Wilson fugacitycoefficient.

LnFugacity COTHWilsonLnFugacity Wilson fugacity.

LnActivity Coeff COTHWilsonLnActivityCoeff Wilson activitycoefficient.

LnStdFugacity COTHWilsonStdFug Ideal standard fugacitywith or without poyntingcorrection.

LnActivityCoeffDT COTHWilsonLnActivityCoeff DT

Wilson activity coefficientwrt temperature.




ThermalConductivity



Helmholtz COTHCavettHelmhol tz Cavett Helmholtz energy.










7-22










Property Packages

7-23

General NRTLThis variation of the NRTL model uses five parameters and ismore flexible than the NRTL model. Apply this model to systemswith a wide boiling point range between components, where yourequire simultaneous solution of VLE and LLE, and where thereexists a wide boiling point or concentration range betweencomponents.


nrtl_liquid General NRTL The General NRTL Model withfive-coefficient temperaturedependent parameters.





ActTempDep COTHNRTLTempDep HYSYS NRTL temperaturedependent properties.

LnFugacityCoeff COTHNRTLLnFugacityCoeff NRTL fugacity coefficient.

LnFugacity COTHNRTLLnFugacity NRTL fugacity.

LnActivity Coeff COTHNRTLLnActivityCoeff NRTL activity coefficient.

LnStdFugacity COTHNRTLStdFug Ideal standard fugacitywith or without poyntingcorrection.

LnActivityCoeffDT COTHNRTLLnActivityCoeff DT

NRTL activity coefficientwrt temperature.




ThermalConductivity



Helmholtz COTHCavettHelmholtz Cavett Helmholtz energy.

InternalEnergy COTHCavettInternalEnergy

Cavett Internal energy.






7-24











Property Packages

7-25

UNIQUACThis model uses statistical mechanics and the quasi-chemicaltheory of Guggenheim to represent the liquid structure. Theequation is capable of representing LLE, VLE, and VLLE withaccuracy comparable to the NRTL equation, but without theneed for a non-randomness factor.


uniquac_liquid UNIQUAC UNIQUAC Model with two-coefficient temperaturedependent parameters.





LnFugacityCoeff COTHUNIQUACLnFugacityCoef f

UNIQUAC fugacitycoefficient.

LnFugacity COTHUNIQUACLnFugacity UNIQUAC fugacity.

LnActivity Coeff COTHUNIQUACLnActivityCoeff UNIQUAC activitycoefficient.

LnStdFugacity COTHIdeallStdFug Ideal standardfugacity with orwithout poyntingcorrection.

LnActivityCoeffDT COTHUNIQUACLnActivityCoeff DT

UNIQUAC activitycoefficient wrttemperature.

MolarDensity COTHCOSTALDDensity COSTALD molardensity.

MolarVolume COTHCOSTALDVolume COSTALD molarvolume.


ThermalConductivity




InternalEnergy COTHCavettInternalEnergy Cavett Internalenergy.





7-26














Property Packages

7-27

Chien-NullThis model provides consistent framework for applying existingActivity Models on a binary by binary basis. It allows you toselect the best Activity Model for each pair in your case.


cn_liquid Chien-Null Three-parameter temperaturedependent Chien-Null Model .





LnFugacityCoeff COTHCNLnFugacityCoeff CN fugacity coefficient.

LnFugacity COTHCNLnFugacity CN fugacity.

LnActivity Coeff COTHCNLnActivityCoeff CN activity coefficient.

LnActivityCoeffDT COTHCNLnActivityCoeff DT

CN activity coefficient wrttemperature.

LnStdFugacity COTHIdealStdFug Ideal standard fugacity withor without poyntingcorrection.

ActTempDep COTHCNTempDep HYSYS CN temperaturedependent properties.

MolarDensity COTHCOSTALDDensity COSTALD molar d ensity.



ThermalConductivity



Helmholtz COTHCavettHelmholtz Cavett Helmholtz energy.

InternalEnergy COTHCavettInternalEnergy

Cavett Internal energy.










7-28








Property Packages

7-29

AntoineThis model is applicable for low pressure systems that behaveideally.


antoine_liquid Antoine UNIQUAC activity model withtwo-coefficient temperaturedependent parameters.





LnFugacityCoeff COTHAntoineLnFugacityCoeff

Antoine fugacitycoefficient.

LnFugacity COTHAntoineLnFugacity Antoine fugacity.

LnActivity Coeff COTHAntoineLnActivityCoeff

Antoine activitycoefficient.



Viscosity COTHViscosity HYSYS Viscosity.

ThermalConductivity

COTHThermCond HYSYS Thermalconductivity.















7-30

Braun K10This model is strictly applicable to heavy hydrocarbon systemsat low pressures. The model employs the Braun convergencepressure method, where, given the normal boiling point of acomponent, the K-value is calculated at system temperature and10 psia (68.95 kPa).


braunk10_liquid Braun K10 Braun K10 Vapour PressureProperty Model.





LnFugacityCoeff COTHBraunK10LnFugacityCoeff

Braun K10 fugacitycoefficient.

LnFugacity COTHBraunK10LnFugacity Braun K10 fugacity.

LnActivity Coeff COTHBraunK10LnActivityCoeff

Braun K10 molarvolume.

MolarDensity COTHCOSTALDDensity Costald molar density.

MolarVolume COTHCOSTALDVolume Costald molar volume.

Viscosity COTHViscosity HYSYS viscosity.

ThermalConductivity COTHThermCond HYSYS thermalconductivity.

SurfaceTension COTHSurfaceTension HYSYS surface tension.













Property Packages

7-31

Esso TabularThis model is strictly applicable to hydrocarbon systems at lowpressures. The model employs a modification of the Maxwell-Bonnel vapour pressure model.


essotabular_liquid

Esso Tabular Esso Tabular vapour PressureProperty Model.





LnFugacityCoeff COTHEssoLnFugacityCoeff

Esso fugacity coefficient.

LnFugacity COTHEssoLnFugacity Esso fugacity.

LnActivity Coeff COTHEssoLnActivityCoef f

Esso activity coefficient.

MolarDensity COTHCOSTALDDensity Costald molar density.

MolarVolume COTHCOSTALDVolume Costald molar volume.


ThermalConductivity

COTHThermCond HYSYS thermalconductivity.














7-32

Chao-SeaderThis method for heavy hydrocarbons, where the pressure is lessthan 10342 kPa (1500 psia), and temperatures range between -17.78 and 260°C (0-500°F).


cs_liquid Chao-Seader Chao-Seader Model is a semi-empirical property method





LnFugacityCoeff COTHChaoSeaderLnFugacityCoeff

Chao-Seader fugacitycoefficient.

LnFugacity COTHChaoSeaderLnFugacity Chao-Seaderfugacity.

MolarVolume COTHRKVolume Redlich-Kwong molarvolume.

ZFactor COTHRKZFactor Redlich-Kwongcompressibility factor.

amix COTHRKab_1 Redlich-Kwong EOSamix.


ThermalConductivity


SurfaceTension COTHSurfaceTension HYSYS surfacetension.











Property Packages

7-33







7-34

Grayson-StreedThis model is recommended for simulating heavy hydrocarbonsystems with a high hydrogen content.


gs_liquid Grayson-Streed Grayson-Streed Model is asemi-empirical propertymethod.





LnFugacityCoeff COTHGraysonStreedLnFugacityCoeff

Grayson-Streedfugacity coefficient.

LnFugacity COTHGraysonStreedLnFugacity

Grayson-Streedfugacity.

MolarVolume COTHRKVolume Redlich-Kwong molarvolume.

ZFactor COTHRKZFactor Redlich-Kwongcompressibility factor.

amix COTHRKab_1 Redlich-Kwong EOSamix.


ThermalConductivity COTHThermCond HYSYS thermalconductivity.













Property Packages

7-35

HysysPRThe HysysPR EOS is similar to the PR EOS with severalenhancements to the original PR equation. It extends the rangeof applicability and better represents the VLE of complexsystems.


hysyspr_liquid HysysPR Peng-Robinson EOS using MixingRule 1 for all properties.


Enthalpy COTHPR_HYSYS_Enthalpy Peng-Robinsonenthalpy.

Entropy COTHPR_HYSYS_Entropy Peng-Robinsonentropy.

Cp COTHPR_HYSYS_Cp Peng-Robinson heatcapacity.

LnFugacityCoeff COTHPR_HYSYS_LnFugacityCoeff

Peng-Robinsonfugacity coefficient.

LnFugacity COTHPR_HYSYS_LnFugacity Peng-Robinsonfugacity.

MolarVolume COTHPR_HYSYS_Volume Peng-Robinson molarvolume.


ThermalConductiv

ity

COTHThermCond HYSYS thermal

conductivity.ZFactor COTHPRZFactor Peng-Robinson

compressibilityfactor.







MolarDensity COTHSolidDensity Solid molar density.MolarVolume COTHMolarVolume Solid molar volume.





7-36






Property Packages

7-37

Kabadi-DannerThis model is a modification of the original SRK equation ofstate, enhanced to improve the vapour-liquid-liquid equilibriumcalculations for water-hydrocarbon systems, particularly indilute regions.


kd_liquid Kabadi-Danner Kabadi-Danner EOS using MixingRule 1 for all properties.


Enthalpy COTHKDEnthalpy Kabadi-Danner enthalpy.

Entropy COTHKDEntropy Kabadi-Danner entropy.

Cp COTHKDCp Kabadi-Danner heatcapacity.

LnFugacityCoeff COTHKDLnFugacityCoeff

Kabadi-Danner fugacitycoefficient.

LnFugacity COTHKDLnFugacity Kabadi-Danner fugacity.

MolarVolume COTHKDVolume Kabadi-Danner molarvolume.


ThermalConductivity

COTHThermCond HYSYS thermal conductivi ty.


ZFactor COTHKDZFactor Kabadi-Dannercompressibility factor.

amix COTHKDab_1 Kabadi-Danner amix.














7-38

Peng-RobinsonThis model is ideal for VLE calculations as well as calculatingliquid densities for hydrocarbon systems. However, in situationswhere highly non-ideal systems are encountered, the use ofActivity Models is recommended.


pr_liquid Peng-Robinson Peng-Robinson EOS using MixingRule 1 for all properties.


Enthalpy COTHPREnthalpy Peng-Robinson enthalpy.

Entropy COTHPREntropy Peng-Robinson entropy.

Cp COTHPRCp Peng-Robinson heat capacity.

LnFugacityCoeff COTHPRLnFugacityCoeff


LnFugacity COTHPRLnFugacity Peng-Robinson fugacity.

MolarVolume COTHPRVolume Peng-Robinson molarvolume.


ThermalConductivity

COTHThermCond HYSYS thermal conductivity.
















Property Packages

7-39

Peng-Robinson-Stryjek-VeraThis is a two-fold modification of the PR equation of state thatextends the application of the original PR method for moderatelynon-ideal systems. It provides a better pure component vapourpressure prediction as well as a more flexible Mixing Rule thanPeng robinson.


prsv_liquid PRSV Peng-Robinson-Stryjek-VeraEOS using Mixing Rule 1 for allproperties.


Enthalpy COTHPRSVEnthalpy PRSV enthalpy.

Entropy COTHPRSVEntropy PRSV entropy.

Cp COTHPRSVCp PRSV heat capacity.

LnFugacityCoeff COTHPRSVLnFugacityCoeff

PRSV fugacity coefficient.

LnFugacity COTHPRSVLnFugacity PRSV fugacity.

MolarVolume COTHPRSVVolume PRSV molar volume.


ThermalConductivity



ZFactor COTHPRSVZFactor PRSV compressibilityfactor.

amix COTHPRSVab_1 PRSV amix.














7-40

Soave-Redlich-KwongIn many cases it provides comparable results to PR, but itsrange of application is significantly more limited. This method isnot as reliable for non-ideal systems.


srk_liquid SRK Soave-Redlich-Kwong EOS usingMixing Rule 1 for all properties.


Enthalpy COTHSRKEnthalpy SRK enthalpy.

Entropy COTHSRKEntropy SRK entropy.

Cp COTHSRKCp SRK heat capacity.

LnFugacityCoeff COTHSRKLnFugacityCoeff

SRK fugacity coefficient.

LnFugacity COTHSRKLnFugacity SRK fugacity.

MolarVolume COTHSRKVolume SRK molar volume.


ThermalConductivity



ZFactor COTHSRKZFactor SRK compressibility factor.

amix COTHSRKab_1 SRK amix.













Property Packages

7-41

VirialThis model enables you to better model vapour phase fugacitiesof systems displaying strong vapour phase interactions.Typically this occurs in systems containing carboxylic acids, orcompounds that have the tendency to form stable hydrogenbonds in the vapour phase. In these cases, the fugacitycoefficient shows large deviations from ideality, even at low ormoderate pressures.


virial_liquid Virial Virial Equation of State.


LnFugacityCoeff COTHPR_LnFugacityCoeff Peng-Robinson fugacitycoefficient.

LnFugacity COTHPR_LnFugacity Peng-Robinson fugacity.

LnStdFugacity COTHIdealStdFug Ideal standard fugacity.

MolarVolume COTHSolidVolume Molar solid volume.


ThermalConductivity



ZFactor COTHPR_ZFactor Peng-Robinsoncompressibility factor.

Enthalpy COTHPR_Enthalpy Peng-Robinson enthalpy.

Enthalpy COTHSolidEnthalpy Insoluble solid enthalpy.

Entropy COTHPR_Entropy Peng-Robinson entropy.

Entropy COTHSolidEntropy Insoluble solid entropy.

Cp COTHPR_Cp Peng-Robinson heatcapacity.

Cp COTHSolidCp Insoluble solid heatcapacity.










7-42

Zudkevitch-JoffeeThis is a modification of the Redlich-Kwong equation of state,which reproduces the pure component vapour pressures aspredicted by the Antoine vapour pressure equation. This modelhas been enhanced for better prediction of vapour-liquidequilibrium for hydrocarbon systems, and systems containingHydrogen.


zj_liquid Zudkevitch-Joffee Zudkevitch-Joffee Equation ofState.


Enthalpy COTHLeeKeslerEnthalpy

Lee-Kesler enthalpy.

Entropy COTHLeeKeslerEntropy

Lee-Kesler entropy.


LnFugacityCoeff COTHZJLnFugacityCoeff

Zudkevitch-Joffee fugacitycoefficient.

LnFugacity COTHZJLnFugacity Zudkevitch-Joffee fugaci ty.

MolarVolume COTHZJVolume Zudkevitch-Joffee molarvolume.


ThermalConductivity

COTHThermCond HYSYS thermal conductivi ty.


ZFactor COTHZJZFactor Zudkevitch-Joffeecompressibility factor.

amix COTHZJab_1 Zudkevitch-Joffee amix.







MolarVolume COTHMolarVolume Solid molar volume.Enthalpy COTHSolidEnthalpy Solid enthalpy.



Property Packages

7-43







7-44

Lee-Kesler-PlöckerThis model is the most accurate general method for non-polarsubstances and mixtures.


lkp_liquid Lee-Kesler-Plöcker Lee-Kesler-Plöcker EOS usingMixing Rule 1 for all properties.



Entropy COTHLeeKeslerEnthalpy Lee-Kesler entropy.


LnFugacityCoeff COTHLKPLnFugacityCoef f

LKP fugacity coefficient.

LnFugacity COTHLKPLnFugacity LKP fugacity.

MolarVolume COTHLKPMolarVolume LKP molar volume.


ThermalConductivity



ZFactor COTHLKPZFactor LKP compressibility factor.



OffsetH COTHOffsetH Offset enthalpy with heat of

formation.OffsetIGS COTHOffsetIGS Ideal gas offset entropy.











8-2 Introduction

8-2

8.1 IntroductionThe utility commands are a set of tools, which interact with aprocess by providing additional information or analysis ofstreams or operations. In HYSYS, utilities become a permanentpart of the Flowsheet and are calculated automatically whenappropriate.

8.2 Envelope UtilityCurrently there are two utilities in HYSYS that are directlyrelated to Aspen HYSYS Thermodynamics COM Interface:

• HYSYS Two-Phase Envelope Utility• Aspen HYSYS Thermodynamics COM Interface Three-PhaseEnvelope Utility

They can be accessed through the Envelope utility in HYSYS.Refer to the Envelope Utility section in Chapter 14 in theOperations Guide for more information.



References

9-1

9 References 1 Prausnitz, J.M.; Lichtenthaler, R.N., and de Azeuedo, E.G. “Molecular

Thermodynamics of Fluid Phase Equilibria”, 2nd Ed. Prentice Hall,Inc. (1986).

2 Prausnitz, J.M.; Anderson, T.; Grens, E.; Eckert, C.; Hsieh, R.; andO'Connell, J.P. “Computer Calculations for Multi-ComponentVapour-Liquid and Liquid-Liquid Equilibria” Prentice-Hall Inc.(1980).

3 Modell, M. and Reid, R.D., “Thermodynamics and its Applications”,2nd Ed., Prentice-Hall, Inc. (1983).

4 Michelsen, M.L., “The Isothermal Flash Problem. Part I. Stability, PartII. Phase Split Calculation, Fluid Phase Equilibria”, 9 1-19; 21-40.(1982).

5 Gautam, R. and Seider, J.D., “Computation of Phase and ChemicalEquilibrium. I. Local and Constrained Minima in Gibbs Free Energy;II. Phase Splitting, III. Electrolytic Solutions.”, AIChE J. 24, 991-1015. (1979).

6 Reid, J.C.; Prausnitz, J.M. and Poling, B.E. “The Properties of Gasesand Liquid” McGraw-Hill Inc. (1987).

7 Henley, E.J.; Seader, J.D., “Equilibrium-Stage Separation Operationsin Chemical Engineering”, John Wiley and Sons. (1981).

8 Feynman, R.P., Leighton, R.B., and Sands, M., “The Feyman Lectureson Physics” Addison-Wesley Publishing Company. (1966).

9 Peng, D.Y. and Robinson, D.B. “A New Two Constant Equation ofState” Ind. Eng. Chem. Fundamen. 15, 59-64. (1976).

10 Stryjek, R. and Vera, J.H. “PRSV: An Improved Peng-RobinsonEquation of State for Pure components and Mixtures” The CanadianJournal of Chemical Eng. 64. (1986).

11 Soave, G. “Equilibrium Constants from a Modified Redlich-KwongEquation of State”. Chem. Eng. Sci. 27, 1197-1203. (1972).

12 Graboski, M.S. and Daubert, T.E., “A Modified Soave Equation of Statefor Phase Equilibrium Calculations. 3. Systems Containing

Hydrogen” Ind. Eng. Chem. Fundamen. 15, 59-64. (1976). 13 Zudkevitch, D. and Joffee, J., Correlation and Prediction of Vapor-

Liquid Equilibria with the Redlich Kwong Equation of State, AIChE



9-2

9-2

J.; 16, 112-119. (1970). 14 Mathias, P.M., “Versatile Phase Equilibrium Equation of State”, Ind.

Eng. Chem. Process Des. Dev. 22, 385-391. (1983).

15 Mathias, P.M. and Copeman, T.W. “Extension of the Peng Robinson ofstate to Complex Mixtures: Evaluations of the Various Forms of theLocal Composition Concept”. (1983).

16 Kabadi, V.N.; Danner, R.P., “A Modified Soave Redlich Kwong Equationof State for Water-Hydrocarbon Phase Equilibria”, Ind. Eng. Chem.process Des. Dev., 24, 537-541. (1985).

17 Twu, C.H. and Bluck, D., “An Extension of Modified Soave-Redlich-Kwong Equation of State to Water-Petroleum Fraction Systems”,Paper presented at the AIChE Meeting. (1988).

18 Tsonopoulos, C. AIChE Journal 20, 263. (1974). 19 Hayden, J.G. and O'Connell, J.P. “A Generalized Method for Predicting

Second Virial Coefficients” Ind. Eng. Chem. Process Des. Dev. 14,209-216. (1975).

20 Wilson, G.M. “Vapour-Liquid Equilibrium XI: A New Expression for theExcess Free Energy of Mixing” J. Am. Chem Soc. 86, 127-130.(1964).

21 Walas, S.M. “Phase Equilibria in Chemical Engineering” ButterworthPublishers. (1985).

22 Renon, H. and Prausnitz, J.M. “Local Compositions in ThermodynamicExcess Functions for Liquid Mixtures” AIChE Journal 14, 135-144.(1968).

23 Abrams, D.S. and Prausnitz, J.M., “Statistical Thermodynamics ofLiquid Mixtures: A New Expression for the Excess Gibbs Energy ofPartly of Completely Miscible Systems” AIChE Journal 21, 116-128.(1975).

24 Fredenslund, A. Jones, R.L. and Prausnitz, J.M. “Group ContributionEstimations of Activity Coefficients in non-ideal Liquid Mixtures”AIChE Journal 21, 1086-1098. (1975).

25 Fredenslund, A.; Gmehling, J. and Rasmussen, P. “Vapour-LiquidEquilibria using UNIFAC” Elsevier. (1977).

26 Wilson, G.M. and Deal, C.H. “Activity Coefficients and MolecularStructure” Ind. Eng. Chem. Fundamen. 1, 20-33. (1962).

27 Derr, E.L. and Deal, C.H., Instn. Chem. Eng. Symp. Ser. No. 32, Inst.Chem. Engr. London 3, 40-51. (1969).

28 Le Bas, G. “The Molecular Volumes of Liquid Chemical Compounds”Longmans, Green and Co., Inc. New York. (1915).



References

9-3

29 Kojima, K. and Tochigi, K. “Prediction of Vapour-Liquid Equilibriausing ASOG” Elsevier. (1979).

30 Orye, R.V. and Prausnitz, J.M. “Multi-Component Equilibria with the

Wilson Equation” Ind. Eng. Chem. 57, 18-26. (1965). 31 Magnussen, T.; Rasmussen, P. and Fredenslund, A. “UNIFAC

Parameter Table for Prediction of Liquid-Liquid Equilibria” Ind. Eng.Chem. Process Des. Dev. 20, 331-339. (1981).

32 Jensen, T.; Fredenslund, A. and Rasmussen, “Pure ComponentVapour-Pressures using UNIFAC Group Contribution” Ind. Eng.Chem. Fundamen. 20, 239-246. (1981).

33 Dahl, Soren, Fredenslund, A. and Rasmussen, P., “The MHV2 Model: AUNIFAC Based Equation of State Model for Prediction of GasSolubility and Vapour-Liquid Equilibria at Low and High Pressures”Ind. Eng. Chem. Res. 30, 1936-1945. (1991).

34 “Group Contribution Method for the Prediction of Liquid Densities as a

Function of Temperature for Solvents, Oligomers and Polymers”,Elbro, H.S., Fredenslund, A. and Rasmussen, P., Ind. Eng. Chem.Res. 30, 2576-2586. (1991).

35 W.H., H.S. and S.I. Sandler, “Use of ab Initio Quantum MechanicsCalculations in Group Contribution Methods. 1. Theory and theBasis for Group Identifications” Ind. Eng. Chem. Res. 30, 881-889.(1991).

36 W.H., H.S., and S.I. Sandler, “Use of ab Initio Quantum MechanicsCalculations in Group Contribution Methods. 2. Test of New Groupsin UNIFAC” Ind. Eng. Chem. Res. 30, 889-897. (1991).

37 McClintock, R.B.; Silvestri, G.J., “Formulations and IterativeProcedures for the Calculation of Properties of Steam”, TheAmerican Society of Mechanical Engineers, New York. (1967).

38 Hankinson, R.W. and Thompson, G.H., AIChE J., 25, 653. (1979). 39 Ely, J.F. and Hanley, H.J.M., “A Computer Program for the Prediction

of Viscosity and Thermal Conductivity in Hydrocarbon Mixtures”,NBS Technical Note 1039. (1983).

40 Hildebrand, J.H., Prausnitz, J.M. and Scott, R.L “Regular and RelatedSolutions”, Van Nostrand Reinhold Co., New York. (1970).

41 Soave, G., Direct Calculation of Pure-Component Vapour Pressurethrough Cubic Equations of State, Fluid Phase Equilibria, 31, 203-207. (1986).

42 Twu, C.H., I.E.C. Proc. Des. & Dev. 24, 1287. (1985). 43 Twu, C.H., “An Internally Consistent Correlation for Predicting the





I-1

IndexA

Activity Coefficient Modelsvapour phase options 2-31

Activity Coefficients 2-9See individual activity modelsActivity Models 3-98

See individual Activity modelsAsymmetric Phase Representation 2-26

B

Bubble Point 6-5BWR Equation 3-96

C

carboxylic acid 2-24Cavett Method 4-2Chao Seader 3-191

semi-empirical method 3-191Chao-Seader Model 3-191Chemical Potential

ideal gas 2-7real gas 2-8

Chien-Null Model 3-182property classes 3-185property methods 3-185

COSTALD Method 4-11Cp 2-38

D

Departure FunctionsEnthalpy 2-38

Dew Point 6-4Dimerization 2-21

E

Enthalpy Flash 6-5Enthalpy Reference States 5-2Entropy Flash 6-6Entropy Reference States 5-4Equations of State

See also individual equations of stateEquilibrium Calculations 2-24Equilibrium calculations 2-24

F

FlashT-P Flash 6-3vapour fraction 6-3 – 6-4

Flash Calculationstemperature-pressure (TP) 6-2

Flash calculations 2-24Fugacity 2-8

ideal gas 2-18simplifications 2-18

G

General NRTL Model 3-155Gibbs Free Energy 2-34Gibbs-Duhem Equation 2-16Grayson Streed 3-192

semi-empirical method 3-192Grayson-Streed Model 3-192

H

Henry’s Law 2-12, 2-31estimation of constants 2-15

HypNRTL Model 3-154HysysPR Equation of State 3-17

mixing rules 3-24property classes 3-18property methods 3-18

I

Ideal Gas Cp 5-5Ideal Gas Equation of State 3-3

property classes 3-4property methods 3-4

Ideal Gas Law 2-31Ideal Solution Activity Model 3-101

property classes 3-101property methods 3-101Insoluble Solids 4-22Interaction Parameters 2-27Internal Energy 2-3

K

Kabadi-Danner Equation of State 3-65mixing rules 3-74property classes 3-68property methods 3-68

K-values 2-24

L

Lee-Kesler Equation of State 3-92mixing rules 3-96property classes 3-93



I-2

I-2

property methods 3-93Lee-Kesler-Plocker Equation 3-96Liquid Phase Models 7-13

MMargules Model 3-123


N

Non-Condensable Components 2-14NRTL Model 3-141

property classes 3-146, 3-155property methods 3-146, 3-155

P

Peng Robinson Equation of State


Peng-Robinson Equation 2-31Peng-Robinson Equation of State 3-7Peng-Robinson Stryjek-Vera Equation of State3-2335


Phase Stability 2-33Property Packages 7-1

recommended 2-30selecting 2-28

Q

Quality Pressure 6-5

R

Rackett Method 4-8Redlich-Kwong Equation of State 3-46


Regular Solution Activity Model 3-106property classes 3-106property methods 3-106

S

Scott's Two Liquid Theory 3-142Soave-Redlich-Kwong Equation 2-31

Soave-Redlich-Kwong Equation of State 3-36mixing rules 3-43property classes 3-37property methods 3-37

Solids 6-6Standard State Fugacity 5-6Surface Tension 4-21Symmetric Phase Representation 2-26

T

Thermal Conductivity 4-18T-P Flash Calculation 6-3

U

UNIFAC Model 3-170property classes 3-174property methods 3-174

UNIQUAC Equation 3-158application 3-160

UNIQUAC Model 3-158property classes 3-162property methods 3-162

V

Van Laar Equationapplication 3-115

Van Laar Model 3-111property classes 3-116property methods 3-116

Vapour Phase Models 7-2Vapour Pressure 6-5

Virial Equation 3-86calculating second virial coefficient 3-78vapour phase chemical association 3-84

Virial Equation of State 3-77mixing rules 3-83property classes 3-87property methods 3-87

Viscosity 4-14liquid phase mixing rules 4-17

W

Wilson Equationapplication 3-133

Wilson Model 3-130




I-3

I-3

Z

Zudkevitch-Joffee Equation of State 3-56mixing rules 3-62


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