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NRTL ¤

NRTL multicomponent activity coefficient model.

This model is based on the following molar excess Gibbs energy expression:

\[ \frac{g^{E}}{RT} = \sum_i x_i \frac{\displaystyle\sum_j x_j \tau_{ji} G_{ji}} {\displaystyle\sum_j x_j G_{ji}} \]

where \(x_i\) are the mole fractions, \(\tau_{ij}\) are the interaction parameters, \(\alpha_{ij}\) are the non-randomness parameters, and \(G_{ij}=\exp(-\alpha_{ij} \tau_{ij})\).

In this particular implementation, the model parameters are allowed to depend on temperature according to the following empirical relationship (as done in Aspen Plus):

\[\begin{aligned} \tau_{ij} &= a_{ij} + b_{ij}/T + e_{ij} \ln{T} + f_{ij} T \\ \alpha_{ij} &= c_{ij} + d_{ij}(T - 273.15) \end{aligned}\]

Moreover, \(\tau_{ij}\neq\tau_{ji}\), \(\tau_{ii}=0\), and \(\alpha_{ij}=\alpha_{ji}\).

References

  • Renon, H. and Prausnitz, J.M. (1968), Local compositions in thermodynamic excess functions for liquid mixtures. AIChE J., 14: 135-144.
PARAMETER DESCRIPTION
N

Number of components.

TYPE: int

a

Matrix of interaction parameters, by default 0.

TYPE: FloatSquareMatrix(N, N) | None DEFAULT: None

b

Matrix of interaction parameters, by default 0. Unit = K.

TYPE: FloatSquareMatrix(N, N) | None DEFAULT: None

c

Matrix of interaction parameters, by default 0.3. Only the upper triangle must be supplied.

TYPE: FloatSquareMatrix(N, N) | None DEFAULT: None

d

Matrix of interaction parameters, by default 0. Only the upper triangle must be supplied. Unit = 1/K.

TYPE: FloatSquareMatrix(N, N) | None DEFAULT: None

e

Matrix of interaction parameters, by default 0.

TYPE: FloatSquareMatrix(N, N) | None DEFAULT: None

f

Matrix of interaction parameters, by default 0.

TYPE: FloatSquareMatrix(N, N) | None DEFAULT: None

name

Name.

TYPE: str DEFAULT: ''

See also
  • NRTL_gamma: related activity coefficient method.
Source code in src/polykin/thermo/acm/nrtl.py 
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class NRTL(SmallSpeciesActivityModel):
    r"""[NRTL](https://en.wikipedia.org/wiki/Non-random_two-liquid_model)
    multicomponent activity coefficient model.

    This model is based on the following molar excess Gibbs energy
    expression:

    $$ \frac{g^{E}}{RT} = 
        \sum_i x_i \frac{\displaystyle\sum_j x_j \tau_{ji} G_{ji}}
        {\displaystyle\sum_j x_j G_{ji}} $$

    where $x_i$ are the mole fractions, $\tau_{ij}$ are the interaction
    parameters, $\alpha_{ij}$ are the non-randomness parameters, and
    $G_{ij}=\exp(-\alpha_{ij} \tau_{ij})$. 

    In this particular implementation, the model parameters are allowed to
    depend on temperature according to the following empirical relationship
    (as done in Aspen Plus):

    \begin{aligned}
    \tau_{ij} &= a_{ij} + b_{ij}/T + e_{ij} \ln{T} + f_{ij} T \\
    \alpha_{ij} &= c_{ij} + d_{ij}(T - 273.15)
    \end{aligned}

    Moreover, $\tau_{ij}\neq\tau_{ji}$, $\tau_{ii}=0$, and
    $\alpha_{ij}=\alpha_{ji}$.

    **References**

    *   Renon, H. and Prausnitz, J.M. (1968), Local compositions in
        thermodynamic excess functions for liquid mixtures. AIChE J.,
        14: 135-144.

    Parameters
    ----------
    N : int
        Number of components.
    a : FloatSquareMatrix (N,N) | None
        Matrix of interaction parameters, by default 0.
    b : FloatSquareMatrix (N,N) | None
        Matrix of interaction parameters, by default 0. Unit = K.
    c : FloatSquareMatrix (N,N) | None
        Matrix of interaction parameters, by default 0.3. Only the upper
        triangle must be supplied.
    d : FloatSquareMatrix (N,N) | None
        Matrix of interaction parameters, by default 0. Only the upper triangle
        must be supplied. Unit = 1/K.
    e : FloatSquareMatrix (N,N) | None
        Matrix of interaction parameters, by default 0.
    f : FloatSquareMatrix (N,N) | None
        Matrix of interaction parameters, by default 0.
    name: str
        Name.

    See also
    --------
    * [`NRTL_gamma`](NRTL_gamma.md): related activity coefficient method.

    """

    _a: FloatSquareMatrix
    _b: FloatSquareMatrix
    _c: FloatSquareMatrix
    _d: FloatSquareMatrix
    _e: FloatSquareMatrix
    _f: FloatSquareMatrix

    def __init__(self,
                 N: int,
                 a: Optional[FloatSquareMatrix] = None,
                 b: Optional[FloatSquareMatrix] = None,
                 c: Optional[FloatSquareMatrix] = None,
                 d: Optional[FloatSquareMatrix] = None,
                 e: Optional[FloatSquareMatrix] = None,
                 f: Optional[FloatSquareMatrix] = None,
                 name: str = ''
                 ) -> None:

        # Set default values
        if a is None:
            a = np.zeros((N, N))
        if b is None:
            b = np.zeros((N, N))
        if c is None:
            c = np.full((N, N), 0.3)
        if d is None:
            d = np.zeros((N, N))
        if e is None:
            e = np.zeros((N, N))
        if f is None:
            f = np.zeros((N, N))

        # Check shapes
        for array in [a, b, c, d, e, f]:
            if array.shape != (N, N):
                raise ShapeError(
                    f"The shape of matrix {array} is invalid: {array.shape}.")

        # Check bounds (same as Aspen Plus)
        check_bounds(a, -1e2, 1e2, 'a')
        check_bounds(b, -3e4, 3e4, 'b')
        check_bounds(c, 0., 1., 'c')
        check_bounds(d, -0.02, 0.02, 'd')

        # Ensure tau_ii=0
        for array in [a, b, e, f]:
            np.fill_diagonal(array, 0.)

        # Ensure alpha_ij=alpha_ji
        for array in [c, d]:
            np.fill_diagonal(array, 0.)
            enforce_symmetry(array)

        super().__init__(N, name)
        self._a = a
        self._b = b
        self._c = c
        self._d = d
        self._e = e
        self._f = f

    @functools.cache
    def alpha(self,
              T: float
              ) -> FloatSquareMatrix:
        r"""Compute matrix of non-randomness parameters.

        $$ \alpha_{ij} = c_{ij} + d_{ij}(T - 273.15) $$

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

        Returns
        -------
        FloatSquareMatrix (N,N)
            Non-randomness parameters.
        """
        return self._c + self._d*(T - 273.15)

    @functools.cache
    def tau(self,
            T: float
            ) -> FloatSquareMatrix:
        r"""Compute the matrix of interaction parameters.

        $$ \tau_{ij} = a_{ij} + b_{ij}/T + e_{ij} \ln{T} + f_{ij} T $$

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

        Returns
        -------
        FloatSquareMatrix (N,N)
            Interaction parameters.
        """
        return self._a + self._b/T + self._e*log(T) + self._f*T

    def gE(self, T: float, x: FloatVector) -> float:
        tau = self.tau(T)
        alpha = self.alpha(T)
        G = exp(-alpha*tau)
        A = dot(x, tau*G)
        B = dot(x, G)
        return R*T*dot(x, A/B)

    def gamma(self, T: float, x: FloatVector) -> FloatVector:
        return NRTL_gamma(x, self.tau(T), self.alpha(T))

Dgmix ¤

Dgmix(T: float, x: FloatVector) -> float

Molar Gibbs energy of mixing, \(\Delta g_{mix}\).

\[ \Delta g_{mix} = \Delta h_{mix} -T \Delta s_{mix} \]
PARAMETER DESCRIPTION
T

Temperature. Unit = K.

TYPE: float

x

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

TYPE: FloatVector(N)

RETURNS DESCRIPTION
float

Molar Gibbs energy of mixing. Unit = J/mol.
Source code in src/polykin/thermo/acm/base.py 
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def Dgmix(self,
          T: float,
          x: FloatVector
          ) -> float:
    r"""Molar Gibbs energy of mixing, $\Delta g_{mix}$.

    $$ \Delta g_{mix} = \Delta h_{mix} -T \Delta s_{mix} $$

    Parameters
    ----------
    T : float
        Temperature. Unit = K.
    x : FloatVector (N)
        Mole fractions of all components. Unit = mol/mol.

    Returns
    -------
    float
        Molar Gibbs energy of mixing. Unit = J/mol.
    """
    return self.gE(T, x) - T*self._Dsmix_ideal(T, x)

Dhmix ¤

Dhmix(T: float, x: FloatVector) -> float

Molar enthalpy of mixing, \(\Delta h_{mix}\).

\[ \Delta h_{mix} = h^{E} \]
PARAMETER DESCRIPTION
T

Temperature. Unit = K.

TYPE: float

x

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

TYPE: FloatVector(N)

RETURNS DESCRIPTION
float

Molar enthalpy of mixing. Unit = J/mol.
Source code in src/polykin/thermo/acm/base.py 
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def Dhmix(self,
          T: float,
          x: FloatVector
          ) -> float:
    r"""Molar enthalpy of mixing, $\Delta h_{mix}$.

    $$ \Delta h_{mix} = h^{E} $$

    Parameters
    ----------
    T : float
        Temperature. Unit = K.
    x : FloatVector (N)
        Mole fractions of all components. Unit = mol/mol.

    Returns
    -------
    float
        Molar enthalpy of mixing. Unit = J/mol.
    """
    return self.hE(T, x)

Dsmix ¤

Dsmix(T: float, x: FloatVector) -> float

Molar entropy of mixing, \(\Delta s_{mix}\).

\[ \Delta s_{mix} = s^{E} - R \sum_i {x_i \ln{x_i}} \]
PARAMETER DESCRIPTION
T

Temperature. Unit = K.

TYPE: float

x

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

TYPE: FloatVector(N)

RETURNS DESCRIPTION
float

Molar entropy of mixing. Unit = J/(mol·K).
Source code in src/polykin/thermo/acm/base.py 
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def Dsmix(self,
          T: float,
          x: FloatVector
          ) -> float:
    r"""Molar entropy of mixing, $\Delta s_{mix}$.

    $$ \Delta s_{mix} = s^{E} - R \sum_i {x_i \ln{x_i}} $$

    Parameters
    ----------
    T : float
        Temperature. Unit = K.
    x : FloatVector (N)
        Mole fractions of all components. Unit = mol/mol.

    Returns
    -------
    float
        Molar entropy of mixing. Unit = J/(mol·K).
    """
    return self.sE(T, x) + self._Dsmix_ideal(T, x)

N property ¤

N: int

Number of components.

activity ¤

activity(T: float, x: FloatVector) -> FloatVector

Activities, \(a_i\).

\[ a_i = x_i \gamma_i \]
PARAMETER DESCRIPTION
T

Temperature. Unit = K.

TYPE: float

x

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

TYPE: FloatVector(N)

RETURNS DESCRIPTION
FloatVector(N)

Activities of all components.
Source code in src/polykin/thermo/acm/base.py 
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def activity(self,
             T: float,
             x: FloatVector
             ) -> FloatVector:
    r"""Activities, $a_i$.

    $$ a_i = x_i \gamma_i $$

    Parameters
    ----------
    T : float
        Temperature. Unit = K.
    x : FloatVector (N)
        Mole fractions of all components. Unit = mol/mol.

    Returns
    -------
    FloatVector (N)
        Activities of all components.
    """
    return x*self.gamma(T, x)

alpha cached ¤

alpha(T: float) -> FloatSquareMatrix

Compute matrix of non-randomness parameters.

\[ \alpha_{ij} = c_{ij} + d_{ij}(T - 273.15) \]
PARAMETER DESCRIPTION
T

Temperature. Unit = K.

TYPE: float

RETURNS DESCRIPTION
FloatSquareMatrix(N, N)

Non-randomness parameters.
Source code in src/polykin/thermo/acm/nrtl.py 
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@functools.cache
def alpha(self,
          T: float
          ) -> FloatSquareMatrix:
    r"""Compute matrix of non-randomness parameters.

    $$ \alpha_{ij} = c_{ij} + d_{ij}(T - 273.15) $$

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

    Returns
    -------
    FloatSquareMatrix (N,N)
        Non-randomness parameters.
    """
    return self._c + self._d*(T - 273.15)

gE ¤

gE(T: float, x: FloatVector) -> float

Molar excess Gibbs energy, \(g^{E}\).

PARAMETER DESCRIPTION
T

Temperature. Unit = K.

TYPE: float

x

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

TYPE: FloatVector(N)

RETURNS DESCRIPTION
float

Molar excess Gibbs energy. Unit = J/mol.
Source code in src/polykin/thermo/acm/nrtl.py 
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def gE(self, T: float, x: FloatVector) -> float:
    tau = self.tau(T)
    alpha = self.alpha(T)
    G = exp(-alpha*tau)
    A = dot(x, tau*G)
    B = dot(x, G)
    return R*T*dot(x, A/B)

gamma ¤

gamma(T: float, x: FloatVector) -> FloatVector

Activity coefficients based on mole fraction, \(\gamma_i\).

PARAMETER DESCRIPTION
T

Temperature. Unit = K.

TYPE: float

x

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

TYPE: FloatVector(N)

RETURNS DESCRIPTION
FloatVector(N)

Activity coefficients of all components.
Source code in src/polykin/thermo/acm/nrtl.py 
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def gamma(self, T: float, x: FloatVector) -> FloatVector:
    return NRTL_gamma(x, self.tau(T), self.alpha(T))

hE ¤

hE(T: float, x: FloatVector) -> float

Molar excess enthalpy, \(h^{E}\).

\[ h^{E} = g^{E} + T s^{E} \]
PARAMETER DESCRIPTION
T

Temperature. Unit = K.

TYPE: float

x

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

TYPE: FloatVector(N)

RETURNS DESCRIPTION
float

Molar excess enthalpy. Unit = J/mol.
Source code in src/polykin/thermo/acm/base.py 
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def hE(self,
       T: float,
       x: FloatVector
       ) -> float:
    r"""Molar excess enthalpy, $h^{E}$.

    $$ h^{E} = g^{E} + T s^{E} $$

    Parameters
    ----------
    T : float
        Temperature. Unit = K.
    x : FloatVector (N)
        Mole fractions of all components. Unit = mol/mol.

    Returns
    -------
    float
        Molar excess enthalpy. Unit = J/mol.
    """
    return self.gE(T, x) + T*self.sE(T, x)

sE ¤

sE(T: float, x: FloatVector) -> float

Molar excess entropy, \(s^{E}\).

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

Temperature. Unit = K.

TYPE: float

x

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

TYPE: FloatVector(N)

RETURNS DESCRIPTION
float

Molar excess entropy. Unit = J/(mol·K).
Source code in src/polykin/thermo/acm/base.py 
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def sE(self,
       T: float,
       x: FloatVector
       ) -> float:
    r"""Molar excess entropy, $s^{E}$.

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

    Parameters
    ----------
    T : float
        Temperature. Unit = K.
    x : FloatVector (N)
        Mole fractions of all components. Unit = mol/mol.

    Returns
    -------
    float
        Molar excess entropy. Unit = J/(mol·K).
    """
    return -1*derivative_complex(lambda t: self.gE(t, x), T)[0]

tau cached ¤

tau(T: float) -> FloatSquareMatrix

Compute the matrix of interaction parameters.

\[ \tau_{ij} = a_{ij} + b_{ij}/T + e_{ij} \ln{T} + f_{ij} T \]
PARAMETER DESCRIPTION
T

Temperature. Unit = K.

TYPE: float

RETURNS DESCRIPTION
FloatSquareMatrix(N, N)

Interaction parameters.
Source code in src/polykin/thermo/acm/nrtl.py 
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@functools.cache
def tau(self,
        T: float
        ) -> FloatSquareMatrix:
    r"""Compute the matrix of interaction parameters.

    $$ \tau_{ij} = a_{ij} + b_{ij}/T + e_{ij} \ln{T} + f_{ij} T $$

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

    Returns
    -------
    FloatSquareMatrix (N,N)
        Interaction parameters.
    """
    return self._a + self._b/T + self._e*log(T) + self._f*T