polykin.thermo.acm

NRTL_gamma

NRTL_gamma(
    x: FloatVector,
    tau: FloatSquareMatrix,
    alpha: FloatSquareMatrix,
) -> FloatVector

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

\[ \ln{\gamma_i} = \frac{\displaystyle\sum_{k}{x_{k}\tau_{ki}G_{ki}}} {\displaystyle\sum_{k}{x_{k}G_{ki}}} +\sum_{j}{\frac{x_{j}G_{ij}}{\displaystyle\sum_{k}{x_{k}G_{kj}}}} {\left ({\tau_{ij}-\frac{\displaystyle\sum_{k}{x_{k}\tau_{kj}G_{kj}}} {\displaystyle\sum_{k}{x_{k}G_{kj}}}}\right )} \]

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

References

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

PARAMETER	DESCRIPTION
x	Mole fractions of all components. Unit = mol/mol. TYPE: FloatVector(N)
tau	Interaction parameters, \(\tau_{ij}\). It is expected (but not checked) that \(\tau_{ii}=0\). TYPE: FloatSquareMatrix(N, N)
alpha	Non-randomness parameters, \(\alpha_{ij}\). It is expected (but not checked) that \(\alpha_{ij}=\alpha_{ji}\). TYPE: FloatSquareMatrix(N, N)

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

See also

• NRTL: related class.

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

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

    $$ \ln{\gamma_i} = \frac{\displaystyle\sum_{k}{x_{k}\tau_{ki}G_{ki}}}
    {\displaystyle\sum_{k}{x_{k}G_{ki}}}
    +\sum_{j}{\frac{x_{j}G_{ij}}{\displaystyle\sum_{k}{x_{k}G_{kj}}}}
    {\left ({\tau_{ij}-\frac{\displaystyle\sum_{k}{x_{k}\tau_{kj}G_{kj}}}
    {\displaystyle\sum_{k}{x_{k}G_{kj}}}}\right )} $$

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

    **References**

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

    Parameters
    ----------
    x : FloatVector (N)
        Mole fractions of all components. Unit = mol/mol.
    tau : FloatSquareMatrix (N,N)
        Interaction parameters, $\tau_{ij}$. It is expected (but not checked)
        that $\tau_{ii}=0$.
    alpha : FloatSquareMatrix (N,N)
        Non-randomness parameters, $\alpha_{ij}$. It is expected (but not
        checked) that $\alpha_{ij}=\alpha_{ji}$.

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

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

    """

    G = exp(-alpha*tau)

    # N = x.size
    # A = np.empty(N)
    # B = np.empty(N)
    # for i in range(N):
    #     A[i] = np.sum(x*tau[:, i]*G[:, i])
    #     B[i] = np.sum(x*G[:, i])
    # C = np.zeros(N)
    # for j in range(N):
    #     C += x[j]*G[:, j]/B[j]*(tau[:, j] - A[j]/B[j])

    A = dot(x, tau*G)
    B = dot(x, G)
    C = dot(tau*G, x/B) - dot(G, x*A/B**2)

    return exp(A/B + C)