A Computationally Efficient Approach to Applying the SAFT Equation for CO₂ + H₂O Phase Equilibrium Calculations

Abstract

The statistical associating fluid theory equation of state (EoS) is employed in a time efficient way for the correlation and prediction of vapor–liquid equilibrium of the CO₂ + H₂O binary system for the temperature (10–100 °C) and pressure (1–600 bar) ranges suitable for simulation of CO₂ geologic sequestration. The effective number of segments and energy parameter are correlated with the reduced temperature. Simple mixing rules are applied to obtain binary interaction parameters. Assigning a fixed H₂O composition in the mixing rule makes the phase equilibrium calculations relatively fast compared to other EoS’s. The results obtained by the model used were found to be in satisfactory agreement with the literature data.

Appendices

Appendix

This section provides necessary functions to calculate chemical potential μ _i , compressibility factor Z, and fugacity coefficient $\varphi_{i}$. First, μ _i , which is a derivative of $A^{\rm {res}}$ with respect to the mole number of component i at constant T, volume $V$, and non-i components, is shown.

Expression for Chemical Potential

We discuss $\mu_{i}$ in the order used to present the corresponding Helmohltz energy terms, i.e., hard-sphere, chain, dispersion, and association terms. The expression is:

$$ \mu_{i}^{\rm {res}} = \mu_{i}^{\rm {hs}} + \mu_{i}^{\rm {chain}} + \mu_{i}^{\rm {disp}} + \mu_{i}^{\rm {assoc}} . $$

(25)

The hard-sphere contribution to the chemical potential, $\mu_{i}^{\rm {hs}}$, can be expressed as:

$$ \begin{gathered} \frac{{\mu_{i}^{\rm {hs}} }}{RT} = \frac{{m_{i} d_{i}^{3} \zeta_{0} + 3m_{i} d_{i}^{2} \zeta_{1} + 3m_{i} d_{i} \zeta_{2} - 3m_{i} d_{i}^{2} \zeta_{2}^{2} /\zeta_{3}^{2} + 2m_{i} d_{i}^{3} \zeta_{2}^{3} /\zeta_{3}^{3} - m_{i} d_{i}^{3} \zeta_{2}^{3} /\zeta_{3}^{2} }}{{1 - \zeta_{3} }} \hfill \\ { + }\frac{{3m_{i} d_{i}^{3} \zeta_{1} \zeta_{2} + 3m_{i} d_{i}^{2} \zeta_{2}^{2} /\zeta_{3}^{2} - m_{i} d_{i}^{3} \zeta_{2}^{3} /\zeta_{3}^{2} - 2m_{i} d_{i}^{3} \zeta_{2}^{3} /\zeta_{3}^{3} }}{{\left( {1 - \zeta } \right)^{2} }} + \frac{{2m_{i} d_{i}^{3} \zeta_{2}^{3} /\zeta_{3}^{2} }}{{\left( {1 - \zeta_{3} } \right)^{3} }} \hfill \\ { + }\frac{{(3m_{i} d_{i}^{2} \zeta_{2}^{2} \zeta_{3} - 2m_{i} d_{i}^{3} \zeta_{2}^{3} )\ln (1 - \zeta_{3} )}}{{\zeta_{3}^{3} }} - m_{i} \ln (1 - \zeta_{3} ). \hfill \\ \end{gathered} $$

(26)

$\mu_{i}^{\rm {chain}}$, the chain contribution to the chemical potential, is expressed as:

$$ \frac{{\mu_{i}^{\rm {chain}} }}{RT} = (1 - m_{i} )\ln [g_{i}^{\rm {hs}} (d_{i} )] + \rho \sum\limits_{i = 1}^{N} {\frac{{x_{i} (1 - m_{i} )}}{{g_{i}^{\rm {hs}} (d_{i} )}}\left[ {\frac{{\partial g_{i}^{\rm {hs}} }}{{\partial \rho_{i} }}} \right]_{{T,V,\rho_{j \ne i} }} } , $$

(27)

where

$$ \left[ {\frac{{\partial g_{i}^{\rm {hs}} (d_{i} )}}{{\partial \rho_{i} }}} \right]_{{T,V,\rho_{j \ne i} }} = \frac{\pi }{6}m_{i} d_{i}^{2} N_{\rm {AV}} \left[ {\frac{{d_{i} }}{{(1 - \zeta_{3} )^{2} }} + \frac{{3d_{i} (1 - \zeta_{3} + 2d_{i} \zeta_{2} )}}{{2(1 - \zeta )^{3} }} + \frac{{d_{i}^{2} (2\zeta_{2} - 2\zeta_{2} \zeta_{3} + 3d_{i} \zeta_{2}^{2} )}}{{2(1 - \zeta_{3} )^{4} }}} \right]. $$

(28)

The dispersion contribution to the chemical potential, $\mu_{i}^{\rm {disp}}$ is expressed as:

$$ \frac{{\mu_{i}^{\rm {disp}} }}{RT} = \frac{{m_{i} A_{1}^{\rm {disp}} }}{{T_{\rm {R}} }} + \frac{{m_{i} A_{2}^{\rm {disp}} }}{{T_{\rm {R}}^{2} }} + \frac{{\rho_{s} }}{{T_{\rm {R}} }}\left[ {\frac{{\partial A_{1}^{\rm {disp}} }}{{\partial \rho_{i} }}} \right]_{{T,V,\rho_{j \ne i} }} - \left( {\frac{{\rho_{s} A_{1}^{\rm {disp}} }}{{T_{\rm {R}}^{2} }} + 2\frac{{\rho_{s} A_{2}^{\rm {disp}} }}{{T_{\rm {R}}^{3} }}} \right)\left[ {\frac{{\partial T_{\rm {R}} }}{{\partial \rho_{i} }}} \right]_{{T,V,\rho_{j \ne i} }} , $$

(29)

where

$$ \left[ {\frac{{\partial A_{1}^{\rm {disp}} }}{{\partial \rho_{i} }}} \right]_{{T,V,\rho_{j \ne i} }} = \frac{{m_{i} d^{3} }}{\sqrt 2 }( - 8.5959 - 9.0848\rho_{\rm {R}} - 6.3807\rho_{\rm {R}}^{2} + 41.141\rho_{\rm {R}}^{3} ), $$

(30)

$$ \left[ {\frac{{\partial A_{2}^{\rm {disp}} }}{{\partial \rho_{i} }}} \right]_{{T,V,\rho_{j \ne i} }} = \frac{{m_{i} d_{i}^{3} }}{\sqrt 2 }\left( { - 1.9075 + 19.9449\rho_{\rm {R}} - 66.648\rho_{\rm {R}}^{2} + 63.616\rho_{\rm {R}}^{3} } \right), $$

(31)

$$ \left[ {\frac{{\partial T_{\rm {R}} }}{{\partial \rho_{i} }}} \right]_{{T,V,\rho_{j \ne i} }} = \frac{{2m_{i} \rho \left( {\varepsilon_{x} \sum\limits_{l = 1}^{N} {x_{l} m_{l} \rho_{il}^{3} - \sum\limits_{l = 1}^{N} {x_{l} m_{l} \rho_{il}^{3} \varepsilon_{il} } } } \right)}}{{\beta \varepsilon_{x}^{2} \sigma_{x}^{3} \rho_{s}^{2} }}. $$

(32)

$\mu_{i}^{\rm {assoc}}$, the association contribution to the chemical potential can be expressed as:

$$ \frac{{\mu_{i}^{\rm {assoc}} }}{RT} = \left\{ {\sum\limits_{{A_{i} }} {\left( {\ln X^{{A_{i} }} - .5X^{{A_{i} }} } \right) + .5M_{i} + \sum\limits_{j} {x_{j} \rho \sum\limits_{{A_{j} }} {\left[ {\left( {\frac{{\partial X^{{A_{j} }} }}{{\partial \rho_{i} }}} \right)_{{T,\rho_{j \ne i} }} \left( {\frac{1}{{X^{{A_{j} }} }} - .5} \right)} \right]} } } } \right\}, $$

(33)

where

$$ \begin{gathered} \left( {\frac{{\partial X^{{A_{j} }} }}{{\partial \rho_{j} }}} \right)_{{T,\rho_{j \ne i} }} = - \left( {X^{{A_{j} }} } \right)^{2} N_{\rm {AV}} \left\{ {\sum\limits_{{B_{i} }} {X^{{B_{i} }} } \Delta^{{A_{j} B_{i} }} + \rho \sum\limits_{k} {\sum\limits_{{B_{k} }} {X_{k} \left[ {\Delta^{{A_{j} B_{kl} }} \left( {\frac{{\partial X^{{B_{k} }} }}{{\partial \rho_{i} }}} \right)_{{T,\rho_{l \ne i} }} } \right.} } + } \right. \hfill \\ \, \left. {X^{{B_{k} }} \left. {\left( {\frac{{\partial \Delta^{{A_{j} B_{k} }} }}{{\partial \rho_{i} }}} \right)_{{T,\rho_{l \ne i} }} } \right]} \right\}. \hfill \\ \end{gathered} $$

(34)

$\left( {\frac{{\partial X^{{B_{k} }} }}{{\partial \rho_{i} }}} \right)_{{T,\rho_{l \ne i} }}$ and so forth can be found similarly following Eq. 34. Now the expression for $\left( {\frac{{\partial \Delta^{{A_{j} B_{k} }} }}{{\partial \rho_{i} }}} \right)_{{T,\rho_{l \ne i} }}$is given by:

$$ \left( {\frac{{\partial \varDelta^{{A_{j} B_{k} }} }}{{\partial \rho_{i} }}} \right)_{{T,\rho_{l \ne i} }} = d_{jk}^{3} \left[ {\frac{{\partial g_{jk} (d_{jk} )^{hs} }}{{\partial \rho_{i} }}} \right]_{{T,\rho_{l \ne i} }} \left[ {\exp \left( {\varepsilon^{{A_{j} B_{k} }} /kT} \right) - 1} \right]\kappa^{{A_{j} B_{k} }} , $$

(35)

where

$$ \begin{gathered} \left[ {\frac{{\partial g_{jk} (d_{jk} )^{\rm {hs}} }}{{\partial \rho_{i} }}} \right]_{{T,\rho_{l \ne i} }} = \frac{{\pi N_{\rm {AV}} }}{6}m_{i} \left\{ {\frac{{d_{i}^{3} }}{{(1 - \zeta_{3} )^{2} }}} \right. + 3\frac{{d_{j} d_{k} }}{{d_{j} + d_{k} }}\left[ {\frac{{d_{i}^{2} }}{{(1 - \zeta_{3} )^{2} }} + \left. {\frac{{2d_{i}^{3} \zeta_{2} }}{{(1 - \zeta_{3} )^{3} }}} \right]} \right. \hfill \\ { + 2}\left( {\frac{{d_{j} d_{k} }}{{d_{j} + d_{k} }}} \right)^{2} \left. {\left[ {\frac{{2d_{i}^{2} \zeta_{2} }}{{(1 - \zeta_{3} )^{3} }} + \frac{{3d_{i}^{3} \zeta_{2}^{2} }}{{(1 - \zeta_{3} )^{4} }}} \right]} \right\}. \hfill \\ \end{gathered} $$

(36)

Expression for Compressibility Factor Z

The compressibility factor is calculated from the Helmholtz energy, A, through:

$$ Z = \rho \left[ {\frac{\partial (A/RT)}{\partial \rho }} \right]_{T,N} = 1 + Z^{\rm {hs}} + Z^{\rm {disp}} + Z^{\rm {chain}} + Z^{\rm {assoc}} , $$

(37)

where:

$$ \begin{gathered} Z^{\rm {hs}} = \frac{{\sum\limits_{i = 1}^{N} {x_{i} m_{i} } }}{{\zeta_{0} }}\left[{\frac{{6\zeta_{1} \zeta_{2} - \zeta_{2}^{3} /\zeta_{3}^{3} - \zeta_{2}^{3} /\zeta_{3} + \zeta_{0} \zeta_{3} }}{{1 - \zeta_{3} }}} \right. \hfill \\ +\, \frac{{6\zeta_{1} \zeta_{2} \zeta_{3} - 3\zeta_{1} \zeta_{2} - 2\zeta_{2}^{3} /\zeta_{3} + 2\zeta_{2}^{3} /\zeta_{3}^{2} }}{{(1 - \zeta_{3} )^{2} }} \hfill \\ + \, \left. {\frac{{3\zeta_{2}^{3} /\zeta_{3} - \zeta_{2}^{3} /\zeta_{3}^{2} }}{{(1 - \zeta_{3} )^{3} }}} \right], \hfill \\ \end{gathered} $$

(38)

$$ Z^{\text{disp}} = \sum\limits_{i = 1}^{N} {x_{i} m_{i} \left( {Z_{1}^{\text{disp}} /T_{\rm {R}} + Z_{2}^{\text{disp}} /T_{\rm {R}}^{2} } \right)} , $$

(39)

and

$$ Z_{1}^{\rm {disp}} = \rho_{\rm {R}} ( - 8.5959 - 9.0848\rho_{\rm {R}} - 6.3807\rho_{\rm {R}}^{2} + 41.1416\rho_{\rm {R}}^{3} ), $$

(40)

$$ Z_{2}^{\rm {disp}} = \rho_{\rm {R}} ( - 1.9075 + 19.9449\rho_{\rm {R}} - 66.648\rho_{\rm {R}}^{2} + 63.616\rho_{\rm {R}}^{3} ), $$

(41)

$$ Z^{\rm {chain}} = \sum\limits_{i = 1}^{N} {\frac{{x_{i} (1 - m_{i} )}}{{g_{i}^{\rm {hs}} (d_{i} )}}\left[ {\frac{{\zeta_{3} + 1.5d_{i} \zeta_{2} }}{{(1 - \zeta_{3} )^{2} }} + \frac{{3d_{i} \zeta_{2} \zeta_{3} + d_{i}^{2} \zeta_{2}^{2} }}{{(1 - \zeta_{3} )^{3} }} + \frac{{1.5d_{i}^{2} \zeta_{2}^{2} \zeta_{3} }}{{(1 - \zeta_{3} )^{4} }}} \right]} , $$

(42)

$$ Z^{\rm {assoc}} = \frac{1}{RT}\left[ {\sum\limits_{i} {x_{i} \mu_{i}^{\rm {assoc}} - A^{\rm {assoc}} } } \right]. $$

(43)

Here $\mu_{i}^{\rm {assoc}}$and $A^{\rm {assoc}}$are presented by Eqs. 33 and 20, respectively.

Expression for Fugacity Coefficient

The relation between $\varphi_{i}$(fugacity coefficient) and the residual chemical potential of component i is as follows:

$$ \ln \varphi_{i} = \frac{{\mu_{i}^{\rm {res}} }}{RT} - \ln Z. $$

(44)

$\mu_{i}^{\rm {res}}$ and Z can be obtained from Eqs. 25 and 37, respectively.