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polykin.copolymerization¤

inst_copolymer_ternary ¤

inst_copolymer_ternary(
    f1: Union[float, FloatArrayLike],
    f2: Union[float, FloatArrayLike],
    r12: float,
    r21: float,
    r13: float,
    r31: float,
    r23: float,
    r32: float,
) -> tuple[
    Union[float, FloatArray],
    Union[float, FloatArray],
    Union[float, FloatArray],
]

Calculate the instantaneous copolymer composition for a ternary system.

In a ternary system, the instantaneous copolymer composition \(F_i\) is related to the monomer composition \(f_i\) by:

\[\begin{aligned} a &= \frac{f_1}{r_{21} r_{31}} + \frac{f_2}{r_{21} r_{32}} + \frac{f_3}{r_{31} r_{23}} \\ b &= f_1 + \frac{f_2}{r_{12}} + \frac{f_3}{r_{13}} \\ c &= \frac{f_1}{r_{12} r_{31}} + \frac{f_2}{r_{12} r_{32}} + \frac{f_3}{r_{13} r_{32}} \\ d &= f_2 + \frac{f_1}{r_{21}} + \frac{f_3}{r_{23}} \\ e &= \frac{f_1}{r_{13} r_{21}} + \frac{f_2}{r_{23} r_{12}} + \frac{f_3}{r_{13} r_{23}} \\ g &= f_3 + \frac{f_1}{r_{31}} + \frac{f_2}{r_{32}} \\ F_1 &= \frac{a b f_1}{a b f_1 + c d f_2 + e g f_3} \\ F_2 &= \frac{c d f_2}{a b f_1 + c d f_2 + e g f_3} \\ F_3 &= \frac{e g f_3}{a b f_1 + c d f_2 + e g f_3} \end{aligned}\]

where \(r_{ij}=k_{ii}/k_{ij}\) are the multicomponent reactivity ratios.

References

  • Kazemi, N., Duever, T.A. and Penlidis, A. (2014), Demystifying the estimation of reactivity ratios for terpolymerization systems. AIChE J., 60: 1752-1766.
PARAMETER DESCRIPTION
f1

Molar fraction of M1.

TYPE: float | FloatArrayLike

f2

Molar fraction of M2.

TYPE: float | FloatArrayLike

r12

Reactivity ratio.

TYPE: float

r21

Reactivity ratio.

TYPE: float

r13

Reactivity ratio.

TYPE: float

r31

Reactivity ratio.

TYPE: float

r23

Reactivity ratio.

TYPE: float

r32

Reactivity ratio.

TYPE: float

RETURNS DESCRIPTION
tuple[float | FloatArray, ...]

Instantaneous terpolymer composition, \((F_1, F_2, F_3)\).
See also

Examples:

>>> from polykin.copolymerization import inst_copolymer_ternary
>>> F1, F2, F3 = inst_copolymer_ternary(f1=0.5, f2=0.3, r12=0.2, r21=2.3,
...                                     r13=3.0, r31=0.9, r23=0.4, r32=1.5)
>>> print(f"F1 = {F1:.2f}; F2 = {F2:.2f}; F3 = {F3:.2f}")
F1 = 0.32; F2 = 0.41; F3 = 0.27
Source code in src/polykin/copolymerization/multicomponent.py 
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def inst_copolymer_ternary(f1: Union[float, FloatArrayLike],
                           f2: Union[float, FloatArrayLike],
                           r12: float,
                           r21: float,
                           r13: float,
                           r31: float,
                           r23: float,
                           r32: float,
                           ) -> tuple[Union[float, FloatArray],
                                      Union[float, FloatArray],
                                      Union[float, FloatArray]]:
    r"""Calculate the instantaneous copolymer composition for a ternary system.

    In a ternary system, the instantaneous copolymer composition $F_i$ is
    related to the monomer composition $f_i$ by:

    \begin{aligned}
        a &= \frac{f_1}{r_{21} r_{31}} + \frac{f_2}{r_{21} r_{32}} +
             \frac{f_3}{r_{31} r_{23}} \\
        b &= f_1 + \frac{f_2}{r_{12}} + \frac{f_3}{r_{13}} \\
        c &= \frac{f_1}{r_{12} r_{31}} + \frac{f_2}{r_{12} r_{32}} +
             \frac{f_3}{r_{13} r_{32}} \\
        d &= f_2 + \frac{f_1}{r_{21}} + \frac{f_3}{r_{23}} \\
        e &= \frac{f_1}{r_{13} r_{21}} + \frac{f_2}{r_{23} r_{12}} +
             \frac{f_3}{r_{13} r_{23}} \\
        g &= f_3 + \frac{f_1}{r_{31}} + \frac{f_2}{r_{32}} \\
        F_1 &= \frac{a b f_1}{a b f_1 + c d f_2 + e g f_3} \\
        F_2 &= \frac{c d f_2}{a b f_1 + c d f_2 + e g f_3} \\
        F_3 &= \frac{e g f_3}{a b f_1 + c d f_2 + e g f_3}
    \end{aligned}

    where $r_{ij}=k_{ii}/k_{ij}$ are the multicomponent reactivity ratios.

    **References**

    *   Kazemi, N., Duever, T.A. and Penlidis, A. (2014), Demystifying the
    estimation of reactivity ratios for terpolymerization systems. AIChE J.,
    60: 1752-1766.

    Parameters
    ----------
    f1 : float | FloatArrayLike
        Molar fraction of M1.
    f2 : float | FloatArrayLike
        Molar fraction of M2.
    r12 : float
        Reactivity ratio.
    r21 : float
        Reactivity ratio.
    r13 : float
        Reactivity ratio.
    r31 : float
        Reactivity ratio.
    r23 : float
        Reactivity ratio.
    r32 : float
        Reactivity ratio.

    Returns
    -------
    tuple[float | FloatArray, ...]
        Instantaneous terpolymer composition, $(F_1, F_2, F_3)$.

    See also
    --------
    * [`inst_copolymer_binary`](inst_copolymer_binary.md):
      specific method for binary systems.
    * [`inst_copolymer_multi`](inst_copolymer_multi.md):
      generic method for multicomponent systems.

    Examples
    --------
    >>> from polykin.copolymerization import inst_copolymer_ternary
    >>> F1, F2, F3 = inst_copolymer_ternary(f1=0.5, f2=0.3, r12=0.2, r21=2.3,
    ...                                     r13=3.0, r31=0.9, r23=0.4, r32=1.5)
    >>> print(f"F1 = {F1:.2f}; F2 = {F2:.2f}; F3 = {F3:.2f}")
    F1 = 0.32; F2 = 0.41; F3 = 0.27
    """

    f1 = np.asarray(f1)
    f2 = np.asarray(f2)

    f3 = 1. - (f1 + f2)

    a = f1/(r21*r31) + f2/(r21*r32) + f3/(r31*r23)
    b = f1 + f2/r12 + f3/r13
    c = f1/(r12*r31) + f2/(r12*r32) + f3/(r13*r32)
    d = f2 + f1/r21 + f3/r23
    e = f1/(r13*r21) + f2/(r23*r12) + f3/(r13*r23)
    g = f3 + f1/r31 + f2/r32

    denominator = f1*a*b + f2*c*d + f3*e*g

    F1 = f1*a*b/denominator
    F2 = f2*c*d/denominator
    F3 = 1. - (F1 + F2)

    return (F1, F2, F3)