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nbar_Li_Brooks ¤

nbar_Li_Brooks(alpha: float, m: float) -> float

Average number of radicals per particle according to the Li-Brooks approximation.

\[ \bar{n} = \frac{2 \alpha}{m + \sqrt{m^2 + \frac{8 \alpha \left( 2 \alpha + m \right)}{2 \alpha + m + 1}}} \]

This formula agrees well with the exact Stockmayer-O'Toole solution, with a maximum deviation of about 4%.

References

  • Li B-G, Brooks BW. Prediction of the average number of radicals per particle for emulsion polymerization. J Polym Sci, Part A: Polym Chem 1993;31:2397-402.
PARAMETER DESCRIPTION
alpha

Dimensionless entry frequency.

TYPE: float

m

Dimensionless desorption frequency.

TYPE: float

RETURNS DESCRIPTION
float

Average number of radicals per particle.
Source code in src/polykin/emulsion/nbar.py 
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def nbar_Li_Brooks(alpha: float, m: float) -> float:
    r"""Average number of radicals per particle according to the Li-Brooks
    approximation.

    $$ \bar{n} = \frac{2 \alpha}{m + \sqrt{m^2 + 
        \frac{8 \alpha \left( 2 \alpha + m \right)}{2 \alpha + m + 1}}} $$

    This formula agrees well with the exact Stockmayer-O'Toole solution, 
    with a maximum deviation of about 4%.

    **References**

    *   Li B-G, Brooks BW. Prediction of the average number of radicals per
        particle for emulsion polymerization. J Polym Sci, Part A: Polym Chem
        1993;31:2397-402.

    Parameters
    ----------
    alpha : float
        Dimensionless entry frequency.
    m : float
        Dimensionless desorption frequency.

    Returns
    -------
    float
        Average number of radicals per particle.
    """
    return 2*alpha/(m + sqrt(m**2 + 8*alpha*(2*alpha + m)/(2*alpha + m + 1)))