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Table of Contents Introduction .................................................................................................................................................. 2

Cubic Equations of State ............................................................................................................................... 2

Peng-Robinson (PR) Variants .................................................................................................................... 4

PR76 ...................................................................................................................................................... 4

PR78 ...................................................................................................................................................... 5

PR76 – Boston Mathias and PR78 – Boston Mathias ............................................................................ 5

Soave-Redlich-Kwong (SRK) Variants ........................................................................................................ 6

SRK ........................................................................................................................................................ 6

SRK – Boston Mathias ........................................................................................................................... 7

Predictive Models ..................................................................................................................................... 7

PPR78 .................................................................................................................................................... 8

SRK-JP .................................................................................................................................................... 8

Fugacity Coefficients and Enthalpy Calculations in EOS Models .............................................................. 8

Enthalpy Calculations in EOS Models ........................................................................................................ 9

Calculation of the Specific Liquid Molar Volume ...................................................................................... 9

Activity Model (NRTL) ................................................................................................................................. 10

Calculation of Activity Coefficients ......................................................................................................... 10

Liquid Mixture Enthalpy .......................................................................................................................... 10

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Advanced Thermo Package Equations and Definitions

Introduction Mimic v3.7.0 contains the addition of an Advanced Thermo Package. The functionality of the Advanced Thermo Package can broadly be classified as:

Cubic Equation of State (Cubic EOS) Model Activity Model (namely, NRTL)

In this document, we review the substituent equations and models making up these two classes of model. This document is not intended as a replacement for a textbook on thermodynamics. Instead, it is intended to provide a convenient source of equations and terminology for users interested in tweaking Binary Interactions Parameters (BIPs or Binaries, interchangeably). The user is encouraged to refer to the referenced source material for a more thorough understanding.

The Cubic Equation of State models are based on the cubic van der Waals type equation of state (EOS). The Advanced Thermo Package includes the most widely used EOS’s of the van der Waals type: Peng–Robinson (PR) and Soave–Redlich–Kwong (SRK) EOS. These equations utilize empirical binary interaction parameters (BIPs) for each binary pair in a mixture.

The Mimic EOS package contains two methods of providing BIPs for each of the PR and SRK EOS models:

Constant values of BIPs, defined by approximation of the experimental data for each binary pair in a mixture, are used in PR, PR78, and SRK models. The Mimic property database provides some of the possible interactions.

Temperature dependent values of BIPs, calculated by a group contribution method for the binary pair in a mixture, are used in the Predictive models PPR78 and PSRK-JP.

These parameters are generally denoted as in literature (note that = and = 0).

The additional features offered by the activity model are represented by the NRTL (Non Random Two Liquids) model. The NRTL equation is applied for calculation of the liquid phase activity coefficients and excess enthalpy ℎ ( , , ) (enthalpy of mixing). The vapor phase model in this case is an ideal gas.

Cubic Equations of State The Cubic Equations of State models available in the Advanced Thermo Package can be seen as descending from the Peng-Robinson and Soave-Redlich-Kwong EOS’s. These formulations have three major components:

1. Equation Form 2. Attractive parameter ( ( )) 3. Alpha function ( ( , ), an empirical parameter which improves the vapor pressure curve

representation and modifies ( ) 4. Covolume parameter ( ) (sometimes called “repulsive paremter”)

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Both the attractive and covolume parameters are calculated from critical properties of the fluid. All of the Cubic EOS models available in the Advanced Thermo Package derive from either SRK or PR. The available Cubic EOS models are:

Peng-Robinson (PR) Variants o Peng-Robinson (PR76) o Peng-Robinson (PR76 - Boston Mathias Alpha) o Peng-Robinson (PR78) o Peng-Robinson (PR78 - Boston Mathias Alpha) o Predictive Peng-Robinson (PPR78)

Soave-Redlich-Kwong (SRK) Variants o Soave-Redlich-Kwong (SRK) o Soave-Redlich-Kwong (SRK - Boston Mathias Alpha) o Predictive Soave-Redlich-Kwong by Jaubert and Privat (SRK-JP)

When using an EOS model, first and values are calculated for pure components in the system. When applying Cubic EOS models to a mixture, mixing rules are used to calculate the values of a and b as:

= ( ) = ∗ ∗ ∗ 1 −

Equation 1 - Mixing of a

= ( ) = ∗

Equation 2 - Mixing of b

where zi represents the mole fraction of component “i” in the mixture and Nc is the number of components in the mixture. kij values, are the binary interaction parameters (BIPs) characterizing molecular interactions between molecules “i” and “j”.

These combined parameters are then used to calculate a polynomial in Z of the form:

+ ∗ + ∗ + = 0

This polynomial is then solved for the vapor and liquid phases (cubic form of the Equation of State for Zv and ZL, respectively)

The resulting roots of the EOS are then used in the calculations for:

fugacity coefficients for liquid and vapor phases, ɸLi and ɸV

i , respectively K-values for VLE of the mixture, Ki = φL

i /φVi

Enthalpy departure and overall enthalpy for both phases

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Peng-Robinson (PR) Variants All Peng-Robinson variants take the form of

= ( – )

−( )

( + ) + ( − )

Equation 3 - Peng-Robinson Type EOS

The variants of Peng-Robinson are:

Peng-Robinson (PR76) Peng-Robinson (PR78) Peng-Robinson (PR76 - Boston Mathias) Peng-Robinson (PR78 - Boston Mathias) Predictive Peng-Robinson (PPR78)

Details of the predictive form are covered in the Predictive Models section.

PR76 For a pure component “i”:

= 0.0777960739 ∗R ∗ Tc,i

,

Equation 4 - PR76 b Parameter

= 0.37464 + 1.54226 − 0.26992

Equation 5- PR76 Alpha mi

( ) = 1 + 1 −,

Equation 6 - Standard Alpha Function

( ) = ( ) = 0.457235529 ∗2

,2

,∗ ( )

Equation 7- PR76 a Parameter

where T is the temperature, Tc,i - critical temperature, Pc,i - critical pressure of the component, R- ideal gas constant, and ωi is the acentric factor.

These mixed terms (as calculated from the foregoing and Equation 1 and Equation 2) are used in the polynomial:

− (1 − ) ∗ + ( − 3 − 2 ) ∗ − ( − − ) = 0

Equation 8 - PR Polynomial Form in Z

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where

=

Equation 9 - Cubic EOS B

=( )

Equation 10 - Cubic EOS A

For more information, see:

Peng-Robinson 76 - D.-Y. Peng and D. Robinson, Ind. Eng. Chem. Fundam., vol. 15, pp. 59-64, 1976

PR78 PR78 is an improved version of PR76. The only difference from PR76 is the calculation of mi for the heavy hydrocarbons depending on . Equation 5 is modified to give the piecewise function:

=0.37464 + 1.54226 − 0.26992 2, ≤ 0.491

0.379642 + 1.48503 ∗ − 0.164423 ∗ + 0.016666 ∗ , > 0.491

Equation 11 - PR78 Modification of mi

For more information, see:

D. Robinson and D.-Y. Peng, Gas Proc. Assoc., Research Report 28, 1978.

PR76 – Boston Mathias and PR78 – Boston Mathias The attraction term for a pure component “i" (see Equation 7) is usually presented as the product of two terms,

= , ∗ ( )

Equation 12 - General ai Parameter

where ( ) is called an alpha function. The standard alpha function is shown as Equation 6. For supercritical components, it is advantageous to use the modified alpha function of Boston and Mathias of the form:

( ) = [exp 1 − ,

Equation 13 - Boston-Mathias Alpha Function

with

= 1 +2

Equation 14 - Boston-Mahias di

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= 1 −1

Equation 15 - Boston-Mathias ci

Here, mi is calculated according to the corresponding equation version (Equation 4 and Equation 11 for PR76 and PR78, respectively).

For Peng-Robinson, , is calculated as:

= 0.457235529 ∗2

,2

,

Equation 16 - Peng-Robinson aci

which is merely Equation 7 before multiplying by ( ).

Soave-Redlich-Kwong (SRK) Variants SRK is the other popular Cubic EOS provided by the Advanced Thermo Package. It was the predecessor of PR that uses the same two parameters – namely, a (the attraction parameter) and b (the covolume parameter). The Equation Form of SRK is:

= ( – )

−( )

( + )

Equation 17 - SRK Type EOS

SRK For the SRK model, the parameters a and b are defined similarly to PR as:

= = 0.08664 ∗∗ ,

,

Equation 18 - SRK b Parameter

= 0.480 + 1.574 ∗ − 0.176 ∗

Equation 19 - SRK Alpha mi

( ) = ( ) = 0.42748 ∗2

,2

,∗ ( )

Equation 20 - SRK a Parameter

where T is the temperature, Tc,i - critical temperature, Pc,i - critical pressure of the component, R- ideal gas constant, and ωi is the acentric factor. The standard alpha function (Equation 6) is used in this equation as well.

These mixed terms (as calculated from the foregoing and Equation 1 and Equation 2) are used in the polynomial:

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− + ( − + ) ∗ − = 0

Equation 21 - SRK Polynomial Form in Z

where again

= =( )

For more information, see:

G. Soave, Chem. Eng. Sci., vol. 27, pp. 1197-1203, 1972

SRK – Boston Mathias Similarly to the Boston Mathias modifications for Peng-Robinson (as explained in the section PR76 – Boston Mathias and PR78 – Boston Mathias), the SRK – Boston Mathias model modifies the alpha function to use Equation 13 through Equation 15 with mi from Equation 19.

As before, , is taken as ( ) (Equation 20) before multiplying by ( ). That is to say that:

, = 0.42748 ∗2

,2

,

Equation 22 - SRK aci

is used in place of Equation 16.

Predictive Models Predictive models allow for updating as a function of temperature based on a decomposition of the components in the mixture. The Advanced Thermo Package introduces two predictive methods based on the work of Jaubert and colleagues (from 2004-2015).

The main features of these predictive models (both based on PPR) are as follows:

PPR78 is based on a group-contribution method (GCM) allowing estimation of the temperature-dependent BIPs kij (T) for Peng-Robinson equation of state.

The proposed group-contribution method (GCM) estimates the temperature-dependent kij of the PR EOS working with classical mixing rules (linear on b and quadratic on a).

The proposed model can predict any phase equilibrium of mixtures containing CO2 and light alkanes, “permanent gases” (O2, N2, H2, H2S), and H2O.

The binary interaction parameter kij (T) between two components i and j depends only on the temperature (T) and on the pure components data (critical temperature (TCi,TCj), critical pressure (PCi, PCj) and acentric factor (ωi, ωj)).

The two predictive Cubic EOS models provided by the Advanced Thermo Package are:

Predictive Peng-Robinson 78 (PPR78) Predictive Soave-Redlich-Kwong by Jaubert and Privat (SRK-JP)

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PPR78 PPR78 calculates BIPs ( values) using the following expression:

( ) =

−12 ∑ ∑ − − ∗

298.15−

( )−

( )

2 ( ) ( )

Equation 23 - Predictive Peng-Robinson kij Calculation

where

ai(T) - temperature dependent function (alpha function) of the equation of state

Akl, Bkl - group-interaction parameters facilitating the calculation of the binary interaction parameters kij.

αik is the fraction of molecule “i” occupied by group “k”(occurrence of group k in molecule i divided by the total number of groups present in molecule “i”).

Ng is the number of different groups defined by the method.

For more information, see:

J. Jaubert and F. Mutelet, J. Fluid., vol. 224, pp. 285-304, 2004

SRK-JP SRK-JP is a predictive version of SRK developed using PPR78 as a basis. The expression for the SRK-JP is similar to Equation 23. Instead, it uses the values of ( ) and from SRK (see the section titled SRK) and uses a conversion term for the values of the group interaction parameters as discussed in:

J. Jaubert and R. Privat, J. Fluid., vol 295, pp. 26-37, 2010.

Fugacity Coefficients and Enthalpy Calculations in EOS Models Fugacity coefficient of the component “i” in the mixture for liquid and vapor phases, φL

i and φVi,

respectively, are used in EOS models in calculations of K-values for VLE of the mixture, Ki = φLi /φV

i .

For the PR equation, the expression for the fugacity coefficient is:

ln( ) = − ln( − ) + ( − 1) ∗ −2 .

12 1 − −

∗ ln+ (2 . + 1)

− (2 . − 1)

Equation 24 - Fugacity Coefficient for PR

For the SRK equation, the expression for the fugacity coefficient is:

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ln( ) = − ln( − ) + ( − 1) ∗ −1

2 1 − − ln 1 +

Equation 25 - Fugacity Coefficient for SRK

Here, A and B are given by Equation 9 and Equation 10. Z is the solution of the EOS for the vapor and liquid phases (Zv and Zl, respectively). zj is a component “i" mole fraction in the phase.

Enthalpy Calculations in EOS Models The enthalpy of the pure component or mixture ( ( , )) is calculated as the sum of the ideal gas enthalpy ( ∗( , )) and the enthalpy departure . is the correction of enthalpy of a fluid with composition zi at temperature T and pressure P with respect to the enthalpy of an ideal gas ∗ at the same temperature, pressure, and composition. That is:

( , ) = ∗( , ) + ( , )

The departure function for the Peng-Robinson EOS is:

( , ) = ( − 1) +

( ) − ( )

2√2

+ 1 + √2

+ 1 − √2

Equation 26 - Enthalpy Departure for PR

The departure function for the SRK EOS is:

( , ) = ( − 1) +1

∗( )

− ( ) 1 +

Equation 27 - Enthalpy Departure for SRK

The values ( ) and ( )

in Equation 26 and Equation 27 depend on the type of alpha function applied.

For predictive PR and SRK (PPR78 and PSRK-JP, respectively), these values also depend upon the form of the binary interaction parameter (see Equation 23). In the predictive formulations, ( ) itself becomes a function of temperature and must be accounted for when taking the derivative.

Calculation of the Specific Liquid Molar Volume The Advanced Thermo Package implements several options for calculation of the specific molar liquid volume (i.e., density). The following options are available:

Standard Mimic calculation (Core Thermo) is based on the component molar liquid densities, supplied from the components properties database.

Rackett method. This method requires compound critical temperatures, pressures, molecular weights, and Rackett parameters (for which the values of critical compressibility are used if unknown)

Rackett modified for mixtures. The Rackett modified equation applies special mixing rules for critical mixture temperature T_mc, using the Prausnitz BIP correlations

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Yen-Woods method. The mixture critical temperature, volume, and compressibility are calculated with the "normal" mixing rules. The correlation expression has two branches depending on the values of the critical compressibility Zc.

Directly from Selected EOS using ZL value (the results might be poor, particularly for SRK model).

Activity Model (NRTL) The activity models are represented in the Advanced Thermo Package by the NRTL (Non Random Two Liquids) model. The NRTL equation is applied for calculation of the liquid phase activity coefficients and excess enthalpy ( , , ) (enthalpy of mixing).

Calculation of Activity Coefficients The liquid phase activity coefficients γi are calculated by NRTL Model (as originally created by Renon) according to the following equation:

ln( ) =∑ =1

∑ =1

+∑ =1

∗ −∑ =1

∑ =1=1

Equation 28 - NRTL Activity Coefficient

where

=∆

∗

Equation 29 - NRTL τij

Δ = + ∗

Equation 30 - NRTL Interaction Δgij

= (− ∗ )

Equation 31 - NRTL Interaction Gij

Note that = but ≠ and = .

Liquid Mixture Enthalpy The liquid mixture enthalpy is calculated from the component enthalpies with

H(T,P)L =∑

where xi is the mole fraction of component “i" in the liquid phase.

The component enthalpy can be expressed as the sum of the ideal contribution , and an excess enthalpy :

= , +

, = , − ΔH

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The component excess enthalpy is computed using the derivative of the activity coefficient calculated for the liquid phase (Equation 28) as:

= − ( )

Note that all values in the foregoing are calculated at the system temperature T and pressure P if applicable.

For more information, see:

H. Renon and J.M. Prausnitz, AIChE J. 14 (1968) 135.

Concluding Remarks This document provides an overview of the equations and methods used in the Mimic Advanced Thermo Package. Inquiring users are encouraged to read the following sources:

Peng-Robinson 76 - D.-Y. Peng and D. Robinson, Ind. Eng. Chem. Fundam., vol. 15, pp. 59-64, 1976

Peng-Robinson 78 - D. Robinson and D.-Y. Peng, Gas Proc. Assoc., Research Report 28, 1978. Soave-Redlich-Kwong - G. Soave, Chem. Eng. Sci., vol. 27, pp. 1197-1203, 1972 Predictive Peng-Robinson 78 - J. Jaubert and F. Mutelet, J. Fluid., vol. 224, pp. 285-304, 2004 Soave-Redlich-Kwong - Jaubert-Privat - J. Jaubert and R. Privat, J. Fluid., vol 295, pp. 26-37, 2010. NRTL - H. Renon and J.M. Prausnitz, AIChE J. 14 (1968) 135. Thermophysical Properties of Fluids – M. Assael et al., Imperial College Press, 1996. Introduction to Chemical Engineering Thermodynamics – J.M. Smith et al., McGraw Hill, 7e,

2005. Chemical Thermodynamics for Process Simulation – J. Gmehling et al., Wiley-VCH, 2012. Phase Equilibria in Chemical Engineering – S.M. Walas, Butterworth, 1985. Thermophysical Properties of Chemicals and Hydrocarbons – C.L. Yaws, Elsevier, 2e, 2014. The Yaws Handbook of Vapor Pressure – C.L. Yaws, Elsevier, 2e, 2015.