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K0_Nomura ¤

K0_Nomura(
    Dw: float,
    Dp: float,
    q: float,
    dp: float,
    *,
    b: float = 10.0
) -> float

Monomeric radical desorption coefficient according to Nomura's two-film mass-transfer model.

The monomeric radical desorption coefficient is given by:

\[ K_0 = \frac{6}{d_p} \left[ \left( \frac{b D_p}{d_p} \right)^{-1} + \left( \frac{2 D_w}{q d_p} \right)^{-1} \right]^{-1} \]

where \(D_w\) is the diffusivity of the monomeric radical in the aqueous phase, \(D_p\) is the diffusivity of the monomeric radical in the particle phase, \(q\) is the partition coefficient of the monomeric radical between the particle and aqueous phases, \(d_p\) is the particle diameter, and \(b\) is a parameter that characterizes the particle-side mass transfer coefficient.

References

  • Nomura, M. and Harada, M. (1981), Rate coefficient for radical desorption in emulsion polymerization. J. Appl. Polym. Sci., 26: 17-26.
PARAMETER DESCRIPTION
Dw

Diffusion coefficient of the monomeric radical in the aqueous phase [m²/s].

TYPE: float

Dp

Diffusion coefficient of the monomeric radical in the particle phase [m²/s].

TYPE: float

q

Partition coefficient of the monomeric radical (particle/aqueous).

TYPE: float

dp

Particle diameter [m].

TYPE: float

b

Proportionality constant for the particle-side mass transfer coefficient. Literature values vary substantially: Ugelstad and Hansen (1976) use b=2, Nomura and Harada (1981) use b=1/3, Asua et al. (1989) use b=1, and Hernandez and Tauer (2008) used b=10. The default value is b=10, corresponding to the linear driving force approximation for intraparticle diffusion. This choice is adopted here as the most physically consistent representation of particle-side mass transfer.

TYPE: float DEFAULT: 10.0

RETURNS DESCRIPTION
float

Monomeric radical desorption coefficient [s⁻¹].
See Also

Examples:

Evaluate the monomeric radical desorption coefficient for a system with a diffusivity of 1e-9 m²/s in the aqueous phase, a diffusivity of 1e-10 m²/s in the particle phase, a partition coefficient of 30, and a particle diameter of 200 nm.

>>> from polykin.kinetics import K0_Nomura
>>> K0 = K0_Nomura(Dw=1e-9, Dp=1e-10, q=30, dp=200e-9)
>>> print(f"K0 = {K0:.2e} s⁻¹")
K0 = 9.38e+03 s⁻¹
Source code in src/polykin/kinetics/emulsion/desorption.py 
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def K0_Nomura(
    Dw: float,
    Dp: float,
    q: float,
    dp: float,
    *,
    b: float = 10.0,
) -> float:
    r"""Monomeric radical desorption coefficient according to Nomura's two-film
    mass-transfer model.

    The monomeric radical desorption coefficient is given by:

    $$ K_0 = \frac{6}{d_p} \left[
             \left( \frac{b D_p}{d_p} \right)^{-1} +
             \left( \frac{2 D_w}{q d_p} \right)^{-1} \right]^{-1} $$

    where $D_w$ is the diffusivity of the monomeric radical in the aqueous phase, $D_p$ is
    the diffusivity of the monomeric radical in the particle phase, $q$ is the partition
    coefficient of the monomeric radical between the particle and aqueous phases, $d_p$
    is the particle diameter, and $b$ is a parameter that characterizes the particle-side
    mass transfer coefficient.

    **References**

    *   Nomura, M. and Harada, M. (1981), Rate coefficient for radical desorption in
        emulsion polymerization. J. Appl. Polym. Sci., 26: 17-26.

    Parameters
    ----------
    Dw : float
        Diffusion coefficient of the monomeric radical in the aqueous phase [m²/s].
    Dp : float
        Diffusion coefficient of the monomeric radical in the particle phase [m²/s].
    q : float
        Partition coefficient of the monomeric radical (particle/aqueous).
    dp : float
        Particle diameter [m].
    b : float
        Proportionality constant for the particle-side mass transfer coefficient.
        Literature values vary substantially: Ugelstad and Hansen (1976) use b=2,
        Nomura and Harada (1981) use b=1/3, Asua et al. (1989) use b=1, and Hernandez and
        Tauer (2008) used b=10. The default value is b=10, corresponding to the linear
        driving force approximation for intraparticle diffusion. This choice is adopted
        here as the most physically consistent representation of particle-side mass
        transfer.

    Returns
    -------
    float
        Monomeric radical desorption coefficient [s⁻¹].

    See Also
    --------
    * [`kdesorption_Asua`](kdesorption_Asua.md): Related desorption coefficient method.

    Examples
    --------
    Evaluate the monomeric radical desorption coefficient for a system with a diffusivity
    of 1e-9 m²/s in the aqueous phase, a diffusivity of 1e-10 m²/s in the particle phase,
    a partition coefficient of 30, and a particle diameter of 200 nm.
    >>> from polykin.kinetics import K0_Nomura
    >>> K0 = K0_Nomura(Dw=1e-9, Dp=1e-10, q=30, dp=200e-9)
    >>> print(f"K0 = {K0:.2e} s⁻¹")
    K0 = 9.38e+03 s⁻¹
    """
    return (12 * b * Dw * Dp) / (dp**2 * (2 * Dw + b * q * Dp))