polykin.transport.diffusion

diffusivity_composite

diffusivity_composite(
    Dd: float, Dc: float, fd: float, sphericity: float = 1
) -> float

Calculate the effective diffusivity of a composite medium containing a dispersed particle phase.

The effective diffusivity \(D\) is calculated using a generalization of Maxwell's analytical solution for spherical particles:

\[ \frac{D - D_c}{D + x D_c} = \phi_d \frac{D_d - D_c}{D_d + x D_c} \]

with \(x = 3/s - 1\). Here, \(D_d\) is the diffusivity of the dispersed phase, \(D_c\) is the diffusivity of the continuous phase, \(\phi_d\) is the volume fraction of the dispersed phase, and \(s\) is the sphericity of the dispersed particles.

References

• J. Crank, "The mathematics of diffusion", Oxford University Press, 1975, p. 271.

PARAMETER	DESCRIPTION
Dd	Diffusity of the dispersed phase. TYPE: float
Dc	Diffusity of the continuous phase. TYPE: float
fd	Volume fraction of the dispersed phase. TYPE: float
sphericity	Sphericity of the particles. Ratio of the surface area of a sphere of volume equal to that of the particle, to the surface area of the particle. TYPE: float DEFAULT: 1

RETURNS	DESCRIPTION
float	Effective diffusivity of the composite medium.

Examples:

Determine the effective diffusivity of a composite medium containing 5 vol% of spherical particles with a diffusivity of 1e-10 m²/s. The diffusivity of the continuous phase is 1e-11 m²/s.

>>> from polykin.transport import diffusivity_composite
>>> diffusivity_composite(Dd=1e-10, Dc=1e-11, fd=0.05)
1.1168831168831167e-11

Source code in src/polykin/transport/diffusion.py

def diffusivity_composite(Dd: float,
                          Dc: float,
                          fd: float,
                          sphericity: float = 1) -> float:
    r"""Calculate the effective diffusivity of a composite medium containing a 
    dispersed particle phase.

    The effective diffusivity $D$ is calculated using a generalization of
    Maxwell's analytical solution for spherical particles:

    $$ \frac{D - D_c}{D + x D_c} = \phi_d \frac{D_d - D_c}{D_d + x D_c} $$

    with $x = 3/s - 1$. Here, $D_d$ is the diffusivity of the dispersed phase, 
    $D_c$ is the diffusivity of the continuous phase, $\phi_d$ is the volume
    fraction of the dispersed phase, and $s$ is the sphericity of the dispersed
    particles.

    **References**

    * J. Crank, "The mathematics of diffusion", Oxford University Press, 1975,
      p. 271.

    Parameters
    ----------
    Dd : float
        Diffusity of the dispersed phase.
    Dc : float
        Diffusity of the continuous phase.
    fd : float
        Volume fraction of the dispersed phase.
    sphericity : float
        Sphericity of the particles. Ratio of the surface area of a sphere of 
        volume equal to that of the particle, to the surface area of the
        particle.

    Returns
    -------
    float
        Effective diffusivity of the composite medium.

    Examples
    --------
    Determine the effective diffusivity of a composite medium containing 5 vol%
    of spherical particles with a diffusivity of 1e-10 m²/s. The diffusivity
    of the continuous phase is 1e-11 m²/s.
    >>> from polykin.transport import diffusivity_composite
    >>> diffusivity_composite(Dd=1e-10, Dc=1e-11, fd=0.05)
    1.1168831168831167e-11
    """
    x = 3/sphericity - 1
    Y = (Dd - Dc)/(Dd + x*Dc)
    return Dc*(1 + fd*Y*x)/(1 - fd*Y)