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polykin.thermo.eos¤

GasEoS ¤

Abstract base class for gas equations of state.

A gas EoS is defined in terms of a compressibility factor function having pressure and temperature as independent variables:

\[ Z = Z(T,P) \]

All other thermodynamic properties are derived from \(Z\).

To implement a specific gas EoS, subclasses must:

  • Implement the Z method.
  • Preferably override the gR, P, and phi methods for efficiency.
Source code in src/polykin/thermo/eos/base.py 
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class GasEoS(EoS):
    r"""Abstract base class for gas equations of state.

    A gas EoS is defined in terms of a compressibility factor function having
    pressure and temperature as independent variables:

    $$ Z = Z(T,P) $$

    All other thermodynamic properties are derived from $Z$.

    To implement a specific gas EoS, subclasses must:

    * Implement the `Z` method.
    * Preferably override the `gR`, `P`, and `phi` methods for efficiency.
    """

    @abstractmethod
    def Z(self, T: Number, P: Number, y: FloatVector) -> float:
        r"""Calculate the compressibility factor of the fluid.

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

        Parameters
        ----------
        T : Number
            Temperature [K].
        P : Number
            Pressure [Pa].
        y : FloatVector (N)
            Mole fractions of all components [mol/mol].

        Returns
        -------
        float
            Compressibility factor.
        """

    def gR(self, T: Number, P: float, y: FloatVector) -> float:
        r"""Calculate the molar residual Gibbs energy of the fluid.

        $$ g^R = R T \int_0^P (Z-1)\frac{d P}{P} $$

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

        Returns
        -------
        float
            Molar residual Gibbs energy [J/mol].
        """
        res = integrate.quad(lambda P_: (self.Z(T, P_, y) - 1) / P_, eps, P)

        return R * T * res[0]

    def P(self, T: float, v: float, y: FloatVector) -> float:
        r"""Calculate the pressure of the fluid.

        $$ P = \frac{Z(T,P,y) R T}{v} $$

        Parameters
        ----------
        T : float
            Temperature [K].
        v : float
            Molar volume [m³/mol].
        y : FloatVector (N)
            Mole fractions of all components [mol/mol].

        Returns
        -------
        float
            Pressure [Pa].
        """
        sol = root_newton(
            lambda P_: P_ * v / (R * T) - self.Z(T, P_, y),
            x0=R * T / v,
        )

        if not sol.success:
            raise RootSolverError(
                f"Could not solve for pressure in `GasEoS.P` method.\n{sol.message}"
            )

        return sol.x

    def phi(self, T: float, P: float, y: FloatVector) -> FloatVector:
        r"""Calculate the fugacity coefficients of all components.

        $$ \ln \hat{\phi}_i = \frac{1}{RT}
           \left( \frac{\partial (n g^R)}{\partial n_i} \right)_{T,P,n_j} $$

        Parameters
        ----------
        T : float
            Temperature [K].
        P : float
            Pressure [Pa].
        y : FloatVector (N)
            Mole fractions of all components [mol/mol].

        Returns
        -------
        FloatVector (N)
            Fugacity coefficients of all components.
        """

        def GR(n: np.ndarray):
            """Total residual Gibbs energy."""
            nT = n.sum()
            return nT * self.gR(T, P, n / nT)

        dGRdn = jacobian_forward(GR, y)

        return exp(dGRdn / (R * T))

    def f(self, T: float, P: float, y: FloatVector) -> FloatVector:
        r"""Calculate the fugacity of all components.

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

        Parameters
        ----------
        T : float
            Temperature [K].
        P : float
            Pressure [Pa].
        y : FloatVector (N)
            Mole fractions of all components [mol/mol].

        Returns
        -------
        FloatVector (N)
            Fugacities of all components [Pa].
        """
        return self.phi(T, P, y) * y * P

    def v(self, T: float, P: float, y: FloatVector) -> float:
        r"""Calculate the molar volume the fluid.

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

        Parameters
        ----------
        T : float
            Temperature [K].
        P : float
            Pressure [Pa].
        y : FloatVector (N)
            Mole fractions of all components [mol/mol].

        Returns
        -------
        float
            Molar volume of the fluid [m³/mol].
        """
        return self.Z(T, P, y) * R * T / P

    def beta(self, T: float, P: float, y: FloatVector) -> float:
        r"""Calculate the thermal expansion coefficient.

        $$ \beta \equiv
         \frac{1}{v} \left( \frac{\partial v}{\partial T} \right)_{P,y_i}
         = \frac{1}{T}
         + \frac{1}{Z} \left( \frac{\partial Z}{\partial T} \right)_{P,y_i} $$

        Parameters
        ----------
        T : float
            Temperature [K].
        P : float
            Pressure [Pa].
        y : FloatVector (N)
            Mole fractions of all components [mol/mol].

        Returns
        -------
        float
            Thermal expansion coefficient, $\beta$ [K⁻¹].
        """
        dZdT, Z = derivative_centered(lambda T_: self.Z(T_, P, y), T)

        return 1 / T + dZdT / Z

    def kappa(self, T: float, P: float, y: FloatVector) -> float:
        r"""Calculate the isothermal compressibility coefficient.

        $$ \kappa \equiv
        - \frac{1}{v} \left( \frac{\partial v}{\partial P} \right)_{T,y_i}
        = \frac{1}{P}
        - \frac{1}{Z} \left( \frac{\partial Z}{\partial P} \right)_{T,y_i} $$

        Parameters
        ----------
        T : float
            Temperature [K].
        P : float
            Pressure [Pa].
        y : FloatVector (N)
            Mole fractions of all components [mol/mol].

        Returns
        -------
        float
            Isothermal compressibility coefficient, $\kappa$ [Pa⁻¹].
        """
        dZdP, Z = derivative_complex(lambda P_: self.Z(T, P_, y), P)

        return 1 / P - dZdP / Z

    def aR(self, T: float, P: float, y: FloatVector) -> float:
        r"""Calculate the molar residual Helmholtz energy of the fluid.

        $$ a^R = g^R - R T (Z - 1) $$

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

        Returns
        -------
        float
            Molar residual Helmholtz energy [J/mol].
        """
        return self.gR(T, P, y) - R * T * (self.Z(T, P, y) - 1.0)

    def hR(self, T: float, P: float, y: FloatVector) -> float:
        r"""Calculate the molar residual enthalpy of the fluid.

        $$ h^R = g^R + T s^R $$

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

        Returns
        -------
        float
            Molar residual enthalpy [J/mol].
        """
        return self.gR(T, P, y) + T * self.sR(T, P, y)

    def sR(self, T: float, P: float, y: FloatVector) -> float:
        r"""Calculate the molar residual entropy of the fluid.

        $$ s^R = -\left( \frac{\partial g^R}{\partial T} \right)_{P,y_i} $$

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

        Returns
        -------
        float
            Molar residual entropy [J/(mol·K)].
        """
        return -1 * derivative_centered(lambda T_: self.gR(T_, P, y), T)[0]

    def vR(self, T: float, P: float, y: FloatVector) -> float:
        r"""Calculate the molar residual volume of the fluid.

        $$ v^R = \left(Z - 1 \right) \frac{R T}{P} $$

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

        Returns
        -------
        float
            Molar residual volume [m³/mol].
        """
        return R * T / P * (self.Z(T, P, y) - 1.0)

DA ¤

DA(T: float, V: float, n: FloatVector, v0: float) -> float

Calculate the departure of Helmholtz energy.

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

Temperature [K].

TYPE: float

V

Volume [m³].

TYPE: float

n

Mole amounts of all components [mol].

TYPE: FloatVector(N)

v0

Molar volume in reference state [m³/mol].

TYPE: float

RETURNS DESCRIPTION
float

Helmholtz energy departure, \(A - A^{\circ}\) [J].
Source code in src/polykin/thermo/eos/base.py 
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def DA(
    self,
    T: float,
    V: float,
    n: FloatVector,
    v0: float,
) -> float:
    r"""Calculate the departure of Helmholtz energy.

    $$ A(T,V,n) - A^{\circ}(T,V,n)$$

    Parameters
    ----------
    T : float
        Temperature [K].
    V : float
        Volume [m³].
    n : FloatVector (N)
        Mole amounts of all components [mol].
    v0 : float
        Molar volume in reference state [m³/mol].

    Returns
    -------
    float
        Helmholtz energy departure, $A - A^{\circ}$ [J].
    """

N property ¤

N: int

Number of components.

P ¤

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

Calculate the pressure of the fluid.

\[ P = \frac{Z(T,P,y) R T}{v} \]
PARAMETER DESCRIPTION
T

Temperature [K].

TYPE: float

v

Molar volume [m³/mol].

TYPE: float

y

Mole fractions of all components [mol/mol].

TYPE: FloatVector(N)

RETURNS DESCRIPTION
float

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

    $$ P = \frac{Z(T,P,y) R T}{v} $$

    Parameters
    ----------
    T : float
        Temperature [K].
    v : float
        Molar volume [m³/mol].
    y : FloatVector (N)
        Mole fractions of all components [mol/mol].

    Returns
    -------
    float
        Pressure [Pa].
    """
    sol = root_newton(
        lambda P_: P_ * v / (R * T) - self.Z(T, P_, y),
        x0=R * T / v,
    )

    if not sol.success:
        raise RootSolverError(
            f"Could not solve for pressure in `GasEoS.P` method.\n{sol.message}"
        )

    return sol.x

Z abstractmethod ¤

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

Calculate the compressibility factor of the fluid.

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

Temperature [K].

TYPE: Number

P

Pressure [Pa].

TYPE: Number

y

Mole fractions of all components [mol/mol].

TYPE: FloatVector(N)

RETURNS DESCRIPTION
float

Compressibility factor.
Source code in src/polykin/thermo/eos/base.py 
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@abstractmethod
def Z(self, T: Number, P: Number, y: FloatVector) -> float:
    r"""Calculate the compressibility factor of the fluid.

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

    Parameters
    ----------
    T : Number
        Temperature [K].
    P : Number
        Pressure [Pa].
    y : FloatVector (N)
        Mole fractions of all components [mol/mol].

    Returns
    -------
    float
        Compressibility factor.
    """

aR ¤

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

Calculate the molar residual Helmholtz energy of the fluid.

\[ a^R = g^R - R T (Z - 1) \]
PARAMETER DESCRIPTION
T

Temperature [K].

TYPE: float

P

Pressure [Pa].

TYPE: float

y

Mole fractions of all components [mol/mol].

TYPE: FloatVector

RETURNS DESCRIPTION
float

Molar residual Helmholtz energy [J/mol].
Source code in src/polykin/thermo/eos/base.py 
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def aR(self, T: float, P: float, y: FloatVector) -> float:
    r"""Calculate the molar residual Helmholtz energy of the fluid.

    $$ a^R = g^R - R T (Z - 1) $$

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

    Returns
    -------
    float
        Molar residual Helmholtz energy [J/mol].
    """
    return self.gR(T, P, y) - R * T * (self.Z(T, P, y) - 1.0)

beta ¤

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

Calculate the thermal expansion coefficient.

\[ \beta \equiv \frac{1}{v} \left( \frac{\partial v}{\partial T} \right)_{P,y_i} = \frac{1}{T} + \frac{1}{Z} \left( \frac{\partial Z}{\partial T} \right)_{P,y_i} \]
PARAMETER DESCRIPTION
T

Temperature [K].

TYPE: float

P

Pressure [Pa].

TYPE: float

y

Mole fractions of all components [mol/mol].

TYPE: FloatVector(N)

RETURNS DESCRIPTION
float

Thermal expansion coefficient, \(\beta\) [K⁻¹].
Source code in src/polykin/thermo/eos/base.py 
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def beta(self, T: float, P: float, y: FloatVector) -> float:
    r"""Calculate the thermal expansion coefficient.

    $$ \beta \equiv
     \frac{1}{v} \left( \frac{\partial v}{\partial T} \right)_{P,y_i}
     = \frac{1}{T}
     + \frac{1}{Z} \left( \frac{\partial Z}{\partial T} \right)_{P,y_i} $$

    Parameters
    ----------
    T : float
        Temperature [K].
    P : float
        Pressure [Pa].
    y : FloatVector (N)
        Mole fractions of all components [mol/mol].

    Returns
    -------
    float
        Thermal expansion coefficient, $\beta$ [K⁻¹].
    """
    dZdT, Z = derivative_centered(lambda T_: self.Z(T_, P, y), T)

    return 1 / T + dZdT / Z

f ¤

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

Calculate the fugacity of all components.

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

Temperature [K].

TYPE: float

P

Pressure [Pa].

TYPE: float

y

Mole fractions of all components [mol/mol].

TYPE: FloatVector(N)

RETURNS DESCRIPTION
FloatVector(N)

Fugacities of all components [Pa].
Source code in src/polykin/thermo/eos/base.py 
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def f(self, T: float, P: float, y: FloatVector) -> FloatVector:
    r"""Calculate the fugacity of all components.

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

    Parameters
    ----------
    T : float
        Temperature [K].
    P : float
        Pressure [Pa].
    y : FloatVector (N)
        Mole fractions of all components [mol/mol].

    Returns
    -------
    FloatVector (N)
        Fugacities of all components [Pa].
    """
    return self.phi(T, P, y) * y * P

gR ¤

gR(T: Number, P: float, y: FloatVector) -> float

Calculate the molar residual Gibbs energy of the fluid.

\[ g^R = R T \int_0^P (Z-1)\frac{d P}{P} \]
PARAMETER DESCRIPTION
T

Temperature [K].

TYPE: Number

P

Pressure [Pa].

TYPE: float

y

Mole fractions of all components [mol/mol].

TYPE: FloatVector

RETURNS DESCRIPTION
float

Molar residual Gibbs energy [J/mol].
Source code in src/polykin/thermo/eos/base.py 
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def gR(self, T: Number, P: float, y: FloatVector) -> float:
    r"""Calculate the molar residual Gibbs energy of the fluid.

    $$ g^R = R T \int_0^P (Z-1)\frac{d P}{P} $$

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

    Returns
    -------
    float
        Molar residual Gibbs energy [J/mol].
    """
    res = integrate.quad(lambda P_: (self.Z(T, P_, y) - 1) / P_, eps, P)

    return R * T * res[0]

hR ¤

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

Calculate the molar residual enthalpy of the fluid.

\[ h^R = g^R + T s^R \]
PARAMETER DESCRIPTION
T

Temperature [K].

TYPE: float

P

Pressure [Pa].

TYPE: float

y

Mole fractions of all components [mol/mol].

TYPE: FloatVector

RETURNS DESCRIPTION
float

Molar residual enthalpy [J/mol].
Source code in src/polykin/thermo/eos/base.py 
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def hR(self, T: float, P: float, y: FloatVector) -> float:
    r"""Calculate the molar residual enthalpy of the fluid.

    $$ h^R = g^R + T s^R $$

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

    Returns
    -------
    float
        Molar residual enthalpy [J/mol].
    """
    return self.gR(T, P, y) + T * self.sR(T, P, y)

kappa ¤

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

Calculate the isothermal compressibility coefficient.

\[ \kappa \equiv - \frac{1}{v} \left( \frac{\partial v}{\partial P} \right)_{T,y_i} = \frac{1}{P} - \frac{1}{Z} \left( \frac{\partial Z}{\partial P} \right)_{T,y_i} \]
PARAMETER DESCRIPTION
T

Temperature [K].

TYPE: float

P

Pressure [Pa].

TYPE: float

y

Mole fractions of all components [mol/mol].

TYPE: FloatVector(N)

RETURNS DESCRIPTION
float

Isothermal compressibility coefficient, \(\kappa\) [Pa⁻¹].
Source code in src/polykin/thermo/eos/base.py 
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def kappa(self, T: float, P: float, y: FloatVector) -> float:
    r"""Calculate the isothermal compressibility coefficient.

    $$ \kappa \equiv
    - \frac{1}{v} \left( \frac{\partial v}{\partial P} \right)_{T,y_i}
    = \frac{1}{P}
    - \frac{1}{Z} \left( \frac{\partial Z}{\partial P} \right)_{T,y_i} $$

    Parameters
    ----------
    T : float
        Temperature [K].
    P : float
        Pressure [Pa].
    y : FloatVector (N)
        Mole fractions of all components [mol/mol].

    Returns
    -------
    float
        Isothermal compressibility coefficient, $\kappa$ [Pa⁻¹].
    """
    dZdP, Z = derivative_complex(lambda P_: self.Z(T, P_, y), P)

    return 1 / P - dZdP / Z

phi ¤

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

Calculate the fugacity coefficients of all components.

\[ \ln \hat{\phi}_i = \frac{1}{RT} \left( \frac{\partial (n g^R)}{\partial n_i} \right)_{T,P,n_j} \]
PARAMETER DESCRIPTION
T

Temperature [K].

TYPE: float

P

Pressure [Pa].

TYPE: float

y

Mole fractions of all components [mol/mol].

TYPE: FloatVector(N)

RETURNS DESCRIPTION
FloatVector(N)

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

    $$ \ln \hat{\phi}_i = \frac{1}{RT}
       \left( \frac{\partial (n g^R)}{\partial n_i} \right)_{T,P,n_j} $$

    Parameters
    ----------
    T : float
        Temperature [K].
    P : float
        Pressure [Pa].
    y : FloatVector (N)
        Mole fractions of all components [mol/mol].

    Returns
    -------
    FloatVector (N)
        Fugacity coefficients of all components.
    """

    def GR(n: np.ndarray):
        """Total residual Gibbs energy."""
        nT = n.sum()
        return nT * self.gR(T, P, n / nT)

    dGRdn = jacobian_forward(GR, y)

    return exp(dGRdn / (R * T))

sR ¤

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

Calculate the molar residual entropy of the fluid.

\[ s^R = -\left( \frac{\partial g^R}{\partial T} \right)_{P,y_i} \]
PARAMETER DESCRIPTION
T

Temperature [K].

TYPE: float

P

Pressure [Pa].

TYPE: float

y

Mole fractions of all components [mol/mol].

TYPE: FloatVector

RETURNS DESCRIPTION
float

Molar residual entropy [J/(mol·K)].
Source code in src/polykin/thermo/eos/base.py 
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def sR(self, T: float, P: float, y: FloatVector) -> float:
    r"""Calculate the molar residual entropy of the fluid.

    $$ s^R = -\left( \frac{\partial g^R}{\partial T} \right)_{P,y_i} $$

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

    Returns
    -------
    float
        Molar residual entropy [J/(mol·K)].
    """
    return -1 * derivative_centered(lambda T_: self.gR(T_, P, y), T)[0]

v ¤

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

Calculate the molar volume the fluid.

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

Temperature [K].

TYPE: float

P

Pressure [Pa].

TYPE: float

y

Mole fractions of all components [mol/mol].

TYPE: FloatVector(N)

RETURNS DESCRIPTION
float

Molar volume of the fluid [m³/mol].
Source code in src/polykin/thermo/eos/base.py 
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def v(self, T: float, P: float, y: FloatVector) -> float:
    r"""Calculate the molar volume the fluid.

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

    Parameters
    ----------
    T : float
        Temperature [K].
    P : float
        Pressure [Pa].
    y : FloatVector (N)
        Mole fractions of all components [mol/mol].

    Returns
    -------
    float
        Molar volume of the fluid [m³/mol].
    """
    return self.Z(T, P, y) * R * T / P

vR ¤

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

Calculate the molar residual volume of the fluid.

\[ v^R = \left(Z - 1 \right) \frac{R T}{P} \]
PARAMETER DESCRIPTION
T

Temperature [K].

TYPE: float

P

Pressure [Pa].

TYPE: float

y

Mole fractions of all components [mol/mol].

TYPE: FloatVector

RETURNS DESCRIPTION
float

Molar residual volume [m³/mol].
Source code in src/polykin/thermo/eos/base.py 
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def vR(self, T: float, P: float, y: FloatVector) -> float:
    r"""Calculate the molar residual volume of the fluid.

    $$ v^R = \left(Z - 1 \right) \frac{R T}{P} $$

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

    Returns
    -------
    float
        Molar residual volume [m³/mol].
    """
    return R * T / P * (self.Z(T, P, y) - 1.0)