polykin.thermo.acm

UNIQUAC_gamma

UNIQUAC_gamma(
    x: FloatVector,
    q: FloatVector,
    r: FloatVector,
    tau: FloatSquareMatrix,
) -> FloatVector

Calculate the activity coefficients of a multicomponent mixture according to the UNIQUAC model.

\[\begin{aligned} \ln{\gamma_i} &= \ln{\gamma_i^R} + \ln{\gamma_i^C} \\ \ln{\gamma_i^R} &= q_i \left(1 - \ln{s_i} - \sum_j \theta_j\frac{\tau_{ij}}{s_j} \right) \\ \ln{\gamma_i^C} &= 1 - J_i + \ln{J_i} - 5 q_i \left(1 - \frac{J_i}{L_i} + \ln{\frac{J_i}{L_i}} \right) \end{aligned}\]

with:

\[\begin{aligned} J_i &=\frac{r_i}{\sum_j x_j r_j} \\ L_i &=\frac{q_i}{\sum_j x_j q_j} \\ \theta_i &= x_i L_i \\ s_i &= \sum_j \theta_j \tau_{ji} \end{aligned}\]

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

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.
• JM Smith, HC Van Ness, MM Abbott. Introduction to chemical engineering thermodynamics, 5th edition, 1996, p. 740-741.

PARAMETER	DESCRIPTION
x	Mole fractions of all components. Unit = mol/mol. TYPE: FloatVector(N)
q	Relative surface areas of all components. TYPE: FloatVector(N)
r	Relative volumes of all components. TYPE: FloatVector(N)
tau	Interaction parameters, \(\tau_{ij}\). It is expected (but not checked) that \(\tau_{ii}=1\). TYPE: FloatSquareMatrix(N, N)

RETURNS	DESCRIPTION
FloatVector(N)	Activity coefficients of all components.

See also

• UNIQUAC: related class.

Source code in src/polykin/thermo/acm/uniquac.py

def UNIQUAC_gamma(x: FloatVector,
                  q: FloatVector,
                  r: FloatVector,
                  tau: FloatSquareMatrix
                  ) -> FloatVector:
    r"""Calculate the activity coefficients of a multicomponent mixture
    according to the UNIQUAC model.

    \begin{aligned}
    \ln{\gamma_i} &= \ln{\gamma_i^R} + \ln{\gamma_i^C} \\
    \ln{\gamma_i^R} &= q_i \left(1 - \ln{s_i} - 
                       \sum_j \theta_j\frac{\tau_{ij}}{s_j} \right) \\
    \ln{\gamma_i^C} &= 1 - J_i + \ln{J_i} - 5 q_i \left(1 - \frac{J_i}{L_i} +
                      \ln{\frac{J_i}{L_i}} \right)
    \end{aligned}

    with:

    \begin{aligned}
    J_i &=\frac{r_i}{\sum_j x_j r_j} \\
    L_i &=\frac{q_i}{\sum_j x_j q_j} \\
    \theta_i &= x_i L_i \\
    s_i &= \sum_j \theta_j \tau_{ji}
    \end{aligned}

    where $x_i$ are the mole fractions, $q_i$ and $r_i$ denote the
    pure-component parameters, and $\tau_{ij}$ are the interaction parameters.

    **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.
    *   JM Smith, HC Van Ness, MM Abbott. Introduction to chemical engineering
        thermodynamics, 5th edition, 1996, p. 740-741.

    Parameters
    ----------
    x : FloatVector (N)
        Mole fractions of all components. Unit = mol/mol.
    q : FloatVector (N)
        Relative surface areas of all components.
    r : FloatVector (N)
        Relative volumes of all components. 
    tau : FloatSquareMatrix (N,N)
        Interaction parameters, $\tau_{ij}$. It is expected (but not checked)
        that $\tau_{ii}=1$.

    Returns
    -------
    FloatVector (N)
        Activity coefficients of all components.

    See also
    --------
    * [`UNIQUAC`](UNIQUAC.md): related class.

    """

    J = r/dot(x, r)
    L = q/dot(x, q)
    theta = x*L
    s = dot(theta, tau)

    return J*exp(1 - J + q*(1 - log(s) - dot(tau, theta/s)
                            - 5*(1 - J/L + log(J/L))))