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UNIQUAC ¤

UNIQUAC multicomponent activity coefficient model.

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

\[\begin{aligned} g^E &= g^R + g^C \\ \frac{g^R}{R T} &= -\sum_i q_i x_i \ln{\left ( \sum_j \theta_j \tau_{ji} \right )} \\ \frac{g^C}{R T} &= \sum_i x_i \ln{\frac{\Phi_i}{x_i}} + 5\sum_i q_ix_i \ln{\frac{\theta_i}{\Phi_i}} \end{aligned}\]

with:

\[\begin{aligned} \Phi_i =\frac{x_i r_i}{\sum_j x_j r_j} \\ \theta_i =\frac{x_i q_i}{\sum_j x_j q_j} \end{aligned}\]

where \(x_i\) are the mole fractions, \(q_i\) (a relative surface) and \(r_i\) (a relative volume) denote the pure-component parameters, and \(\tau_{ij}\) are the interaction parameters.

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

\[ \tau_{ij} = \exp( a_{ij} + b_{ij}/T + c_{ij} \ln{T} + d_{ij} T ) \]

Moreover, \(\tau_{ij} \neq \tau_{ji}\) and \(\tau_{ii}=1\).

References

  • Abrams, D.S. and Prausnitz, J.M. (1975), Statistical thermodynamics of liquid mixtures: A new expression for the excess Gibbs energy of partly or completely miscible systems. AIChE J., 21: 116-128.
PARAMETER DESCRIPTION
N

Number of components.

TYPE: int

q

Relative surface areas of all components.

TYPE: FloatVectorLike(N)

r

Relative volumes of all components.

TYPE: FloatVectorLike(N)

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.

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

d

Matrix of interaction parameters, by default 0.

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

name

Name.

TYPE: str DEFAULT: ''

See also
Source code in src/polykin/thermo/acm/uniquac.py 
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class UNIQUAC(SmallSpeciesActivityModel):
    r"""[UNIQUAC](https://en.wikipedia.org/wiki/UNIQUAC) multicomponent
    activity coefficient model.

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

    \begin{aligned}
    g^E &= g^R + g^C \\
    \frac{g^R}{R T} &= -\sum_i q_i x_i \ln{\left ( \sum_j \theta_j \tau_{ji} \right )} \\
    \frac{g^C}{R T} &= \sum_i x_i \ln{\frac{\Phi_i}{x_i}} + 5\sum_i q_ix_i \ln{\frac{\theta_i}{\Phi_i}}
    \end{aligned}

    with:

    \begin{aligned}
    \Phi_i =\frac{x_i r_i}{\sum_j x_j r_j} \\
    \theta_i =\frac{x_i q_i}{\sum_j x_j q_j}
    \end{aligned}

    where $x_i$ are the mole fractions, $q_i$ (a relative surface) and $r_i$
    (a relative volume) denote the pure-component parameters, and $\tau_{ij}$
    are the interaction parameters. 

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

    $$ \tau_{ij} = \exp( a_{ij} + b_{ij}/T + c_{ij} \ln{T} + d_{ij} T ) $$

    Moreover, $\tau_{ij} \neq \tau_{ji}$ and $\tau_{ii}=1$.

    **References**

    *   Abrams, D.S. and Prausnitz, J.M. (1975), Statistical thermodynamics of
        liquid mixtures: A new expression for the excess Gibbs energy of partly
        or completely miscible systems. AIChE J., 21: 116-128.

    Parameters
    ----------
    N : int
        Number of components.
    q : FloatVectorLike (N)
        Relative surface areas of all components.
    r : FloatVectorLike (N)
        Relative volumes of all 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.
    d : FloatSquareMatrix (N,N) | None
        Matrix of interaction parameters, by default 0.
    name : str
        Name.

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

    """

    _q: FloatVector
    _r: FloatVector
    _a: FloatSquareMatrix
    _b: FloatSquareMatrix
    _c: FloatSquareMatrix
    _d: FloatSquareMatrix

    def __init__(self,
                 N: int,
                 q: FloatVectorLike,
                 r: FloatVectorLike,
                 a: Optional[FloatSquareMatrix] = None,
                 b: Optional[FloatSquareMatrix] = None,
                 c: Optional[FloatSquareMatrix] = None,
                 d: 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.zeros((N, N))
        if d is None:
            d = np.zeros((N, N))

        # Check shapes -> move to func
        q = np.asarray(q)
        r = np.asarray(r)
        for vector in [q, r]:
            if vector.shape != (N,):
                raise ShapeError(
                    f"The shape of vector {vector} is invalid: {vector.shape}.")
        for array in [a, b, c, d]:
            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, -50., 50., 'a')
        check_bounds(b, -1.5e4, 1.5e4, 'b')

        # Ensure tau_ii=1
        for array in [a, b, c, d]:
            np.fill_diagonal(array, 0.)

        super().__init__(N, name)
        self._q = q
        self._r = r
        self._a = a
        self._b = b
        self._c = c
        self._d = d

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

        $$ \tau_{ij} = \exp( a_{ij} + b_{ij}/T + c_{ij} \ln{T} + d_{ij} T ) $$

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

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

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

        r = self._r
        q = self._q
        tau = self.tau(T)

        phi = x*r/dot(x, r)
        theta = x*q/dot(x, q)

        p = x > 0.
        gC = np.sum(x[p]*(log(phi[p]/x[p]) + 5*q[p]*log(theta[p]/phi[p])))
        gR = -np.sum(q[p]*x[p]*log(dot(theta, tau)[p]))

        return R*T*(gC + gR)

    def gamma(self,
              T: float,
              x: FloatVector
              ) -> FloatVector:
        return UNIQUAC_gamma(x, self._q, self._r, self.tau(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)

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/uniquac.py 
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def gE(self,
       T: float,
       x: FloatVector
       ) -> float:

    r = self._r
    q = self._q
    tau = self.tau(T)

    phi = x*r/dot(x, r)
    theta = x*q/dot(x, q)

    p = x > 0.
    gC = np.sum(x[p]*(log(phi[p]/x[p]) + 5*q[p]*log(theta[p]/phi[p])))
    gR = -np.sum(q[p]*x[p]*log(dot(theta, tau)[p]))

    return R*T*(gC + gR)

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/uniquac.py 
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def gamma(self,
          T: float,
          x: FloatVector
          ) -> FloatVector:
    return UNIQUAC_gamma(x, self._q, self._r, self.tau(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.

\[ \tau_{ij} = \exp( a_{ij} + b_{ij}/T + c_{ij} \ln{T} + d_{ij} T ) \]
PARAMETER DESCRIPTION
T

Temperature. Unit = K.

TYPE: float

RETURNS DESCRIPTION
FloatSquareMatrix(N, N)

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

    $$ \tau_{ij} = \exp( a_{ij} + b_{ij}/T + c_{ij} \ln{T} + d_{ij} T ) $$

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

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