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polykin.thermo.eos¤

PengRobinson ¤

Peng-Robinson equation of state.

This EOS is based on the following \(P(v,T)\) relationship:

\[ P = \frac{RT}{v - b_m} -\frac{a_m}{v^2 + 2 v b_m - b_m^2} \]

where \(P\) is the pressure, \(T\) is the temperature, \(v\) is the molar volume, \(a_m(T,y)\) and \(b_m(y)\) are the mixture EOS parameters, and \(y\) is the vector of mole fractions.

For a single component, the parameters \(a\) and \(b\) are given by:

\[\begin{aligned} a &= 0.45724 \frac{R^2 T_{c}^2}{P_{c}} [1 + f_\omega(1 - T_{r}^{1/2})]^2 \\ f_\omega &= 0.37464 + 1.54226\omega - 0.26992\omega^2 \\ b &= 0.07780\frac{R T_{c}}{P_{c}} \end{aligned}\]

where \(T_c\) is the critical temperature, \(P_c\) is the critical pressure, and \(T_r = T/T_c\) is the reduced temperature.

References

  • RC Reid, JM Prausniz, and BE Poling. The properties of gases & liquids 4th edition, 1986, p. 37, 40, 80, 82.
PARAMETER DESCRIPTION
Tc

Critical temperatures of all components. Unit = K.

TYPE: float | FloatVectorLike

Pc

Critical pressures of all components. Unit = Pa.

TYPE: float | FloatVectorLike

w

Acentric factors of all components.

TYPE: float | FloatVectorLike

k

Binary interaction parameter matrix.

TYPE: FloatSquareMatrix | None DEFAULT: None

Source code in src/polykin/thermo/eos/cubic.py 
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class PengRobinson(Cubic):
    r"""[Peng-Robinson](https://en.wikipedia.org/wiki/Cubic_equations_of_state#Peng%E2%80%93Robinson_equation_of_state)
    equation of state.

    This EOS is based on the following $P(v,T)$ relationship:

    $$ P = \frac{RT}{v - b_m} -\frac{a_m}{v^2 + 2 v b_m - b_m^2} $$

    where $P$ is the pressure, $T$ is the temperature, $v$ is the molar
    volume, $a_m(T,y)$ and $b_m(y)$ are the mixture EOS parameters, and
    $y$ is the vector of mole fractions.

    For a single component, the parameters $a$ and $b$ are given by:

    \begin{aligned}
    a &= 0.45724 \frac{R^2 T_{c}^2}{P_{c}} [1 + f_\omega(1 - T_{r}^{1/2})]^2 \\
    f_\omega &= 0.37464 + 1.54226\omega - 0.26992\omega^2 \\
    b &= 0.07780\frac{R T_{c}}{P_{c}}
    \end{aligned}

    where $T_c$ is the critical temperature, $P_c$ is the critical pressure,
    and $T_r = T/T_c$ is the reduced temperature.

    **References**

    *   RC Reid, JM Prausniz, and BE Poling. The properties of gases &
        liquids 4th edition, 1986, p. 37, 40, 80, 82.

    Parameters
    ----------
    Tc : float | FloatVectorLike
        Critical temperatures of all components. Unit = K.
    Pc : float | FloatVectorLike
        Critical pressures of all components. Unit = Pa.
    w : float | FloatVectorLike
        Acentric factors of all components.
    k : FloatSquareMatrix | None
        Binary interaction parameter matrix.
    """
    _u = 2.
    _w = -1.
    _Ωa = 0.45724
    _Ωb = 0.07780

    def __init__(self,
                 Tc: Union[float, FloatVectorLike],
                 Pc: Union[float, FloatVectorLike],
                 w: Union[float, FloatVectorLike],
                 k: Optional[FloatSquareMatrix] = None
                 ) -> None:

        super().__init__(Tc, Pc, w, k)

    def _alpha(self, T: float) -> FloatVector:
        w = self.w
        Tr = T/self.Tc
        fw = 0.37464 + 1.54226*w - 0.26992*w**2
        return (1. + fw*(1. - sqrt(Tr)))**2

Bm ¤

Bm(T: float, y: FloatVector) -> float

Calculate the second virial coefficient of the mixture.

\[ B_m = b_m - \frac{a_m}{R T} \]

References

  • RC Reid, JM Prausniz, and BE Poling. The properties of gases & liquids 4th edition, 1986, p. 82.
PARAMETER DESCRIPTION
T

Temperature. Unit = K.

TYPE: float

y

Mole fractions of all components. Unit = mol/mol.

TYPE: FloatVector

RETURNS DESCRIPTION
float

Mixture second virial coefficient, \(B_m\). Unit = m³/mol.
Source code in src/polykin/thermo/eos/cubic.py 
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def Bm(self,
       T: float,
       y: FloatVector
       ) -> float:
    r"""Calculate the second virial coefficient of the mixture.

    $$ B_m = b_m - \frac{a_m}{R T} $$

    **References**

    *   RC Reid, JM Prausniz, and BE Poling. The properties of gases &
        liquids 4th edition, 1986, p. 82.

    Parameters
    ----------
    T : float
        Temperature. Unit = K.
    y : FloatVector
        Mole fractions of all components. Unit = mol/mol.

    Returns
    -------
    float
        Mixture second virial coefficient, $B_m$. Unit = m³/mol.
    """
    return self.bm(y) - self.am(T, y)/(R*T)

DA ¤

DA(T, V, n, v0)

Calculate the departure of Helmholtz energy.

\[ A(T,V,n) - A^{\circ}(T,V,n)\]
PARAMETER DESCRIPTION
T

Temperature. Unit = K.

TYPE: float

V

Volume. Unit = m³.

TYPE: float

n

Mole amounts of all components. Unit = mol.

TYPE: FloatVector

v0

Molar volume in reference state. Unit = m³/mol.

TYPE: float

RETURNS DESCRIPTION
FloatVector

Helmholtz energy departure, \(A - A^{\circ}\). Unit = J.
Source code in src/polykin/thermo/eos/cubic.py 
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def DA(self, T, V, n, v0):
    nT = n.sum()
    y = n/nT
    u = self._u
    w = self._w
    d = sqrt(u**2 - 4*w)
    am = self.am(T, y)
    bm = self.bm(y)

    return nT*am/(bm*d)*log((2*V + nT*bm*(u - d))/(2*V + nT*bm*(u + d))) \
        - nT*R*T*log((V - nT*bm)/(nT*v0))

P ¤

P(T: float, v: float, y: FloatVector) -> float

Calculate the pressure of the fluid.

PARAMETER DESCRIPTION
T

Temperature. Unit = K.

TYPE: float

v

Molar volume. Unit = m³/mol.

TYPE: float

y

Mole fractions of all components. Unit = mol/mol.

TYPE: FloatVector

RETURNS DESCRIPTION
float

Pressure. Unit = Pa.
Source code in src/polykin/thermo/eos/cubic.py 
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def P(self,
      T: float,
      v: float,
      y: FloatVector
      ) -> float:
    r"""Calculate the pressure of the fluid.

    Parameters
    ----------
    T : float
        Temperature. Unit = K.
    v : float
        Molar volume. Unit = m³/mol.
    y : FloatVector
        Mole fractions of all components. Unit = mol/mol.

    Returns
    -------
    float
        Pressure. Unit = Pa.
    """
    am = self.am(T, y)
    bm = self.bm(y)
    u = self._u
    w = self._w
    return R*T/(v - bm) - am/(v**2 + u*v*bm + w*bm**2)

Z ¤

Z(T: float, P: float, y: FloatVector) -> FloatVector

Calculate the compressibility factors of the coexisting phases a fluid.

The calculation is handled by Z_cubic_roots.

PARAMETER DESCRIPTION
T

Temperature. Unit = K.

TYPE: float

P

Pressure. Unit = Pa.

TYPE: float

y

Mole fractions of all components. Unit = mol/mol.

TYPE: FloatVector

RETURNS DESCRIPTION
FloatVector

Compressibility factor of the vapor and/or liquid phases.
Source code in src/polykin/thermo/eos/cubic.py 
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def Z(self,
      T: float,
      P: float,
      y: FloatVector
      ) -> FloatVector:
    r"""Calculate the compressibility factors of the coexisting phases a
    fluid.

    The calculation is handled by
    [`Z_cubic_roots`](.#polykin.thermo.eos.cubic.Z_cubic_roots).

    Parameters
    ----------
    T : float
        Temperature. Unit = K.
    P : float
        Pressure. Unit = Pa.
    y : FloatVector
        Mole fractions of all components. Unit = mol/mol.

    Returns
    -------
    FloatVector
        Compressibility factor of the vapor and/or liquid phases.
    """
    A = self.am(T, y)*P/(R*T)**2
    B = self.bm(y)*P/(R*T)
    return Z_cubic_roots(self._u, self._w, A, B)

a cached ¤

a(T: float) -> FloatVector

Calculate the attractive parameters of the pure-components that make up the mixture.

PARAMETER DESCRIPTION
T

Temperature. Unit = K.

TYPE: float

RETURNS DESCRIPTION
FloatVector

Attractive parameters of all components, \(a_i\). Unit = J·m³.
Source code in src/polykin/thermo/eos/cubic.py 
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@functools.cache
def a(self,
      T: float
      ) -> FloatVector:
    r"""Calculate the attractive parameters of the pure-components that
    make up the mixture.

    Parameters
    ----------
    T : float
        Temperature. Unit = K.

    Returns
    -------
    FloatVector
        Attractive parameters of all components, $a_i$. Unit = J·m³.
    """
    return self._Ωa * (R*self.Tc)**2 / self.Pc * self._alpha(T)

am ¤

am(T: float, y: FloatVector) -> float

Calculate the mixture attractive parameter from the corresponding pure-component parameters.

\[ a_m = \sum_i \sum_j y_i y_j (a_i a_j)^{1/2} (1 - \bar{k}_{ij}) \]

References

  • RC Reid, JM Prausniz, and BE Poling. The properties of gases & liquids 4th edition, 1986, p. 82.
PARAMETER DESCRIPTION
T

Temperature. Unit = K.

TYPE: float

y

Mole fractions of all components. Unit = mol/mol.

TYPE: FloatVector

RETURNS DESCRIPTION
float

Mixture attractive parameter, \(a_m\). Unit = J·m³.
Source code in src/polykin/thermo/eos/cubic.py 
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def am(self,
       T: float,
       y: FloatVector
       ) -> float:
    r"""Calculate the mixture attractive parameter from the corresponding
    pure-component parameters.

    $$ a_m = \sum_i \sum_j y_i y_j (a_i a_j)^{1/2} (1 - \bar{k}_{ij}) $$

    **References**

    *   RC Reid, JM Prausniz, and BE Poling. The properties of gases &
        liquids 4th edition, 1986, p. 82.

    Parameters
    ----------
    T : float
        Temperature. Unit = K.
    y : FloatVector
        Mole fractions of all components. Unit = mol/mol.

    Returns
    -------
    float
        Mixture attractive parameter, $a_m$. Unit = J·m³.
    """
    return geometric_interaction_mixing(y, self.a(T), self.k)

b cached property ¤

b: FloatVector

Calculate the repulsive parameters of the pure-components that make up the mixture.

RETURNS DESCRIPTION
FloatVector

Repulsive parameters of all components, \(b_i\). Unit = m³/mol.

bm ¤

bm(y: FloatVector) -> float

Calculate the mixture repulsive parameter from the corresponding pure-component parameters.

\[ b_m = \sum_i y_i b_i \]

References

  • RC Reid, JM Prausniz, and BE Poling. The properties of gases & liquids 4th edition, 1986, p. 82.
PARAMETER DESCRIPTION
y

Mole fractions of all components. Unit = mol/mol.

TYPE: FloatVector

RETURNS DESCRIPTION
float

Mixture repulsive parameter, \(b_m\). Unit = m³/mol.
Source code in src/polykin/thermo/eos/cubic.py 
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def bm(self,
       y: FloatVector
       ) -> float:
    r"""Calculate the mixture repulsive parameter from the corresponding
    pure-component parameters.

    $$ b_m = \sum_i y_i b_i $$

    **References**

    *   RC Reid, JM Prausniz, and BE Poling. The properties of gases &
        liquids 4th edition, 1986, p. 82.

    Parameters
    ----------
    y : FloatVector
        Mole fractions of all components. Unit = mol/mol.

    Returns
    -------
    float
        Mixture repulsive parameter, $b_m$. Unit = m³/mol.
    """
    return dot(y, self.b)

fV ¤

fV(T: float, P: float, y: FloatVector) -> FloatVector

Calculate the fugacity of all components in the vapor phase.

\[ \hat{f}_i = \hat{\phi}_i y_i P \]

\(\hat{f}_i\) is the fugacity in the vapor phase, \(\hat{\phi}_i(T,P,y)\) is the fugacity coefficient, \(P\) is the pressure, and \(y_i\) is the mole fraction in the vapor phase.

PARAMETER DESCRIPTION
T

Temperature. Unit = K.

TYPE: float

P

Pressure. Unit = Pa.

TYPE: float

y

Mole fractions of all components. Unit = mol/mol.

TYPE: FloatVector

RETURNS DESCRIPTION
FloatVector

Fugacity coefficients of all components.
Source code in src/polykin/thermo/eos/base.py 
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def fV(self,
        T: float,
        P: float,
        y: FloatVector
       ) -> FloatVector:
    r"""Calculate the fugacity of all components in the vapor phase.

    $$ \hat{f}_i = \hat{\phi}_i y_i P $$

    $\hat{f}_i$ is the fugacity in the vapor phase, $\hat{\phi}_i(T,P,y)$
    is the fugacity coefficient, $P$ is the pressure, and $y_i$ is the mole
    fraction in the vapor phase.

    Parameters
    ----------
    T : float
        Temperature. Unit = K.
    P : float
        Pressure. Unit = Pa.
    y : FloatVector
        Mole fractions of all components. Unit = mol/mol.

    Returns
    -------
    FloatVector
        Fugacity coefficients of all components.
    """
    return self.phiV(T, P, y)*y*P

phiV ¤

phiV(T: float, P: float, y: FloatVector) -> FloatVector

Calculate the fugacity coefficients of all components in the vapor phase.

\[\begin{aligned} \ln \hat{\phi}_i &= \frac{b_i}{b_m}(Z-1)-\ln(Z-B^*) +\frac{A^*}{B^*\sqrt{u^2-4w}}\left(\frac{b_i}{b_m}-\delta_i \right)\ln{\frac{2Z+B^*(u+\sqrt{u^2-4w})}{2Z+B^*(u-\sqrt{u^2-4w})}} \\ \delta_i &= \frac{2a_i^{1/2}}{a_m}\sum_j y_j a_j^{1/2}(1-\bar{k}_{ij}) \end{aligned}\]

References

  • RC Reid, JM Prausniz, and BE Poling. The properties of gases & liquids 4th edition, 1986, p. 145.
PARAMETER DESCRIPTION
T

Temperature. Unit = K.

TYPE: float

P

Pressure. Unit = Pa.

TYPE: float

y

Mole fractions of all components. Unit = mol/mol.

TYPE: FloatVector

RETURNS DESCRIPTION
FloatVector

Fugacity coefficients of all components.
Source code in src/polykin/thermo/eos/cubic.py 
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def phiV(self,
         T: float,
         P: float,
         y: FloatVector
         ) -> FloatVector:
    r"""Calculate the fugacity coefficients of all components in the vapor
    phase.

    \begin{aligned}
    \ln \hat{\phi}_i &= \frac{b_i}{b_m}(Z-1)-\ln(Z-B^*)
    +\frac{A^*}{B^*\sqrt{u^2-4w}}\left(\frac{b_i}{b_m}-\delta_i  \right)\ln{\frac{2Z+B^*(u+\sqrt{u^2-4w})}{2Z+B^*(u-\sqrt{u^2-4w})}} \\
    \delta_i &= \frac{2a_i^{1/2}}{a_m}\sum_j y_j a_j^{1/2}(1-\bar{k}_{ij})
    \end{aligned}

    **References**

    *   RC Reid, JM Prausniz, and BE Poling. The properties of gases &
        liquids 4th edition, 1986, p. 145.

    Parameters
    ----------
    T : float
        Temperature. Unit = K.
    P : float
        Pressure. Unit = Pa.
    y : FloatVector
        Mole fractions of all components. Unit = mol/mol.

    Returns
    -------
    FloatVector
        Fugacity coefficients of all components.
    """
    u = self._u
    w = self._w
    d = sqrt(u**2 - 4*w)
    a = self.a(T)
    am = self.am(T, y)
    b = self.b
    bm = self.bm(y)
    k = self.k
    A = am*P/(R*T)**2
    B = bm*P/(R*T)
    b_bm = b/bm
    Z = self.Z(T, P, y)

    if k is None:
        delta = 2*sqrt(a/am)
    else:
        delta = np.sum(y * sqrt(a) * (1 - k), axis=1)

    # get only vapor solution !!!
    z = max(Z)
    lnphi = b_bm*(z - 1) - log(z - B) + A/(B*d) * \
        (b_bm - delta)*log((2*z + B*(u + d))/(2*z + B*(u - d)))

    return exp(lnphi)

v ¤

v(T: float, P: float, y: FloatVector) -> FloatVector

Calculate the molar volumes of the coexisting phases a fluid.

\[ v = \frac{Z R T}{P} \]

where \(v\) is the molar volume, \(Z\) is the compressibility factor, \(T\) is the temperature, \(P\) is the pressure, and \(y\) is the mole fraction vector.

PARAMETER DESCRIPTION
T

Temperature. Unit = K.

TYPE: float

P

Pressure. Unit = Pa.

TYPE: float

y

Mole fractions of all components. Unit = mol/mol.

TYPE: FloatVector

RETURNS DESCRIPTION
FloatVector

Molar volume of the vapor and/or liquid phases. Unit = m³/mol.
Source code in src/polykin/thermo/eos/base.py 
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def v(self,
      T: float,
      P: float,
      y: FloatVector
      ) -> FloatVector:
    r"""Calculate the molar volumes of the coexisting phases a fluid.

    $$ v = \frac{Z R T}{P} $$

    where $v$ is the molar volume, $Z$ is the compressibility factor,
    $T$ is the temperature, $P$ is the pressure, and $y$ is the mole
    fraction vector.

    Parameters
    ----------
    T : float
        Temperature. Unit = K.
    P : float
        Pressure. Unit = Pa.
    y : FloatVector
        Mole fractions of all components. Unit = mol/mol.

    Returns
    -------
    FloatVector
        Molar volume of the vapor and/or liquid phases. Unit = m³/mol.
    """
    return self.Z(T, P, y)*R*T/P

Examples¤

Estimate the compressibility factor of a 50 mol% ethylene/nitrogen gas mixture at 300 K and 100 bar.

from polykin.thermo.eos import PengRobinson
import numpy as np

Tc = [282.4, 126.2]    # K
Pc = [50.4e5, 33.9e5]  # Pa
w = [0.089, 0.039]

eos = PengRobinson(Tc, Pc, w)
Z = eos.Z(T=300., P=100e5, y=np.array([0.5, 0.5]))

print(f"{Z[0]:.2f}")
0.79