polykin.transport.diffusion

uptake_sphere

uptake_sphere(t: float, a: float, D: float) -> float

Fractional mass uptake for transient diffusion in a sphere.

For a sphere of radius \(a\), where the concentration is initially \(C_0\) everywhere, and the surface concentration is maintained at \(C_s\), the fractional mass uptake is:

\[ \frac{M_t}{M_{\infty}} = 1 - \frac{6}{\pi^2} \sum_{n=1}^{\infty}\frac{1}{n^2} \exp \left( \frac{-D n^2 \pi^2 t}{a^2} \right) \]

where \(t\) is the time, and \(D\) is the diffusion coefficient.

References

• J. Crank, "The mathematics of diffusion", Oxford University Press, 1975, p. 91.

PARAMETER	DESCRIPTION
t	Time (s). TYPE: float
a	Radius of sphere (m). TYPE: float
D	Diffusion coefficient (m²/s). TYPE: float

RETURNS	DESCRIPTION
float	Fractional mass uptake.

See also

• profile_sphere: related method to determine the concentration profile.

Examples:

Determine the fractional mass uptake after 100 seconds for a polymer sphere with a radius of 0.2 mm and a diffusion coefficient of 1e-10 m²/s.

>>> from polykin.transport.diffusion import uptake_sphere
>>> uptake_sphere(t=1e2, a=0.2e-3, D=1e-10)
0.9484368978658284

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

def uptake_sphere(t: float,
                  a: float,
                  D: float
                  ) -> float:
    r"""Fractional mass uptake for transient diffusion in a sphere. 

    For a sphere of radius $a$, where the concentration is initially $C_0$
    everywhere, and the surface concentration is maintained at $C_s$, the
    fractional mass uptake is:

    $$ \frac{M_t}{M_{\infty}} = 
        1 - \frac{6}{\pi^2} \sum_{n=1}^{\infty}\frac{1}{n^2}
        \exp \left( \frac{-D n^2 \pi^2 t}{a^2} \right) $$

    where $t$ is the time, and $D$ is the diffusion coefficient.

    **References**

    * J. Crank, "The mathematics of diffusion", Oxford University Press, 1975,
      p. 91.

    Parameters
    ----------
    t : float
        Time (s).
    a : float
        Radius of sphere (m).
    D : float
        Diffusion coefficient (m²/s).

    Returns
    -------
    float
        Fractional mass uptake.

    See also
    --------
    * [`profile_sphere`](profile_sphere.md): related method to determine the
      concentration profile.

    Examples
    --------
    Determine the fractional mass uptake after 100 seconds for a polymer sphere
    with a radius of 0.2 mm and a diffusion coefficient of 1e-10 m²/s.
    >>> from polykin.transport.diffusion import uptake_sphere
    >>> uptake_sphere(t=1e2, a=0.2e-3, D=1e-10)
    0.9484368978658284
    """
    N = 4  # Number of terms in series expansion (optimal value)

    A = sqrt(D*t)/a
    if A == 0.:
        result = 0.
    elif A < 0.5:
        # Solution for small times
        S = sum(ierfc(n/A) for n in range(1, N))
        result = 6*A * (1/sqrt(pi) + 2*S) - 3*A**2
    else:
        # Solution for normal times
        B = -D*pi**2*t/a**2
        S = sum(1/n**2 * exp(B*n**2) for n in range(1, N))
        result = 1 - (6/pi**2)*S

    return result

Graphical Illustration