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Cubic Equations of State

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Thermophysical Properties of Heavy Petroleum Fluids

Part of the book series: Petroleum Engineering ((PEEN))

Abstract

In the previous Chapter, the thermodynamic potentials and the equilibrium criteria were obtained as a function of the fugacity and activity coefficients. In turn, the fugacity coefficient was expressed as a function of the two sets of measurable variables. The former consists of temperature, pressure and composition, while the latter consists of temperature, volume and composition. In either case, an equation of state is needed in order to obtain practical expressions. In this manuscript, two equations of state will be outlined, emphasizing applications to hydrocarbon mixtures in general and heavy petroleum fluids in particular.

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Abbreviations

CEoS:

Cubic Equation of State

GCVOL:

Group-Contribution-Volume

HR:

Huron Vidal

MHV1:

Modified Huron-Vidal of order one

MHV2:

Modified Huron-Vidal of order two

NRTL:

Non-Random-Two-Liquid

PSRK:

Predictive SRK equation of state

UMR:

Universal mixing rule

UNIFAC:

Universal quasi-chemical functional group activity coefficient

VTPR:

Volume translated Peng Robinson

\( \left( {Z_{1} } \right) \) :

Roots for vapor phase (largest)

\( \left( {Z_{2} } \right) \) :

Roots for liquid phase (smallest)

a :

Attractive forces between molecules

\( \tilde{A}^{E} \) :

Helmholtz free energy molar excess

\( A,B \) :

Dimensionless numbers (Eqs. (3.4–3.6))

\( a_{i}^{{\prime }} \) :

Partial derivatives (Eq. (3.33))

\( A_{mc} \) :

CEoS parameters

\( a_{mn} \) :

Interaction parameters for PSRK

b :

Co-volume

B :

Dimensionless co-volume

\( b_{i}^{{\prime }} \) :

Partial derivatives (Eq. (3.34))

\( b_{mn} \) :

Interaction parameters for PSRK

C :

Components

\( c_{1} ,c_{2} ,c_{3} \) :

Coefficients (Eq. (3.3))

\( c_{mn} \) :

Interaction parameters for PSRK

D :

Discriminant

\( E_{ij} \) :

Binary interaction parameter Peneloux and Abdoul

F :

Function

\( f_{i} \) :

Fugacity

\( f_{il} \) :

Occurrence fraction of functional group

\( k_{0} ,k_{1} ,k_{2} ,k_{4} \) :

Constants (Eq. (3.48))

\( k_{ij} \) :

Binary interaction parameters

M :

Molecular weight

Q :

Numbers of the formula of Cardiano

q :

Function (Eq. (3.73))

\( q_{1} \) :

MHV1 parameter

\( q_{i} \) :

Pure-component parameter

\( Q_{k} \) :

Relative group area

\( r_{i} \) :

Pure-component parameter

\( R_{k} \) :

Relative group volume

S :

Real root

s :

Volume translation of component

\( S_{j} \) :

Specific gravity

T :

Real root

\( T_{bj} \) :

Normal boiling temperature

U :

Numbers of the formula of Cardiano

\( u,v \) :

Parameters of Cubic Equations of State

\( x_{i} \) :

Molar fraction

\( X_{m} \) :

Group mole fraction

\( \alpha \) :

CEoS parameters

\( \alpha_{f} \) :

Empirical dimensionless function

\( \alpha_{ij} , g_{ij} , g_{jj} \) :

Binary interaction parameters (Eq. (3.62))

\( \beta \) :

CEoS parameters

\( \gamma_{i} \) :

Activity coefficient

\( \gamma_{i}^{comb} \) :

Combinatorial contribution

\( \gamma_{i}^{FH} \) :

Flory-Huggins contribution

\( \gamma_{i}^{resi} \) :

Residual contribution

\( \gamma_{i}^{SG} \) :

Staverman-Guggenheim contribution

\( \varGamma_{k} \) :

Activity coefficient corresponding to the functional group k

\( \delta_{i} \) :

Relationship beteewn \( a_{i} \), \( b_{i} \)

\( \epsilon \) :

CEoS-dependent parameter

\( \epsilon^{*} \) :

Parameter (Eq. (3.84))

\( \eta \) :

Packing fraction

\( \eta_{mc} \) :

CEoS parameters

\( \varTheta_{i} \) :

Molecular surface fraction

\( \theta_{m} \) :

Area fraction

\( \rho \) :

Density

\( \sigma \) :

Molecular diameter

\( \upsilon_{ki} \) :

Number of times (occurrence)

\( \varphi \) :

CEoS-variable

\( \varPhi_{i} \) :

Molecular volume fraction

\( \phi_{i} \) :

Fugacity coefficient

\( \chi_{kl} , \psi_{kl} \) :

Contributions for the binary interaction

\( \varPsi_{mn} \) :

Binary main group interaction parameters

\( \omega \) :

Acentric factor

\( \varOmega_{a} , \varOmega_{b} \) :

CEoS-dependent coefficients

\( \sim \) :

Molar

0:

Pure component

\( \infty \) :

Infinite

\( C_{1i} , C_{2i} \) :

Coefficients of function of the molecular weight and acentric factor

\( d_{jk} \) :

Correlation parameters

E :

Excess property

Exp :

Experimental value

ext :

Extrapolated

ig :

Ideal gas

is :

Ideal solution

l :

Liquid

R :

Real

R :

Residual

resi :

Residual (Eq. (3.108))

v :

Vapor

\( \iota_{i} , \lambda_{j} , \chi_{j} \) :

Parameters of GCVOL

\( \hat{\rho } \) :

Mass density

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Authors and Affiliations

  1. Instituto Mexicano del Petróleo, Gerencia de Ingeniería de Yacimientos, Mexico City, Mexico

    Prof. Bernardo Carreón-Calderón

  2. Instituto Mexicano del Petróleo, Gerencia de Ingeniería de Yacimientos, Mexico City, Mexico

    Dr. Verónica Uribe-Vargas

  3. Instituto de Ciencias Aplicadas y Tecnología, Universidad Nacional Autónoma de México, Mexico City, Mexico

    Dr. Juan Pablo Aguayo

Authors
  1. Prof. Bernardo Carreón-Calderón
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  2. Dr. Verónica Uribe-Vargas
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  3. Dr. Juan Pablo Aguayo
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Correspondence to Bernardo Carreón-Calderón .

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Carreón-Calderón, B., Uribe-Vargas, V., Aguayo, J. (2021). Cubic Equations of State. In: Thermophysical Properties of Heavy Petroleum Fluids. Petroleum Engineering. Springer, Cham. https://doi.org/10.1007/978-3-030-58831-1_3

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