polykin.transport.diffusion

uptake_constc_sphere

uptake_constc_sphere(Fo: float) -> float

Fractional mass uptake for transient diffusion in a sphere subjected to a constant surface concentration boundary condition.

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:

\[ \frac{\bar{C}-C_0}{C_s -C_0} = 1 - \frac{6}{\pi^2}\sum_{n=1}^{\infty}\frac{1}{n^2} \exp (-n^2 \pi^2 Fo) \]

where \(Fo = D t/a^2\) is the Fourier number.

References

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

PARAMETER	DESCRIPTION
Fo	Fourier number, \(D t/a^2\). TYPE: float

RETURNS	DESCRIPTION
float	Fractional mass uptake.

See also

• profile_constc_sphere: related method to determine the concentration profile.
• uptake_convection_sphere: related method for surface convection boundary condition.

Examples:

Determine the fractional mass uptake after 100 seconds for a polymer sphere with a radius of 0.1 mm and a diffusion coefficient of 1e-11 m²/s.

>>> from polykin.transport import uptake_constc_sphere
>>> t = 1e2   # s
>>> a = 1e-4  # m
>>> D = 1e-11 # m²/s
>>> Fo = D*t/a**2
>>> uptake_constc_sphere(Fo)
0.7704787380259631

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

def uptake_constc_sphere(Fo: float) -> float:
    r"""Fractional mass uptake for transient diffusion in a sphere subjected
    to a constant surface concentration boundary condition. 

    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:

    $$ \frac{\bar{C}-C_0}{C_s -C_0} =
    1 - \frac{6}{\pi^2}\sum_{n=1}^{\infty}\frac{1}{n^2} \exp (-n^2 \pi^2 Fo) $$

    where $Fo = D t/a^2$ is the Fourier number.

    **References**

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

    Parameters
    ----------
    Fo : float
        Fourier number, $D t/a^2$.

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

    See also
    --------
    * [`profile_constc_sphere`](profile_constc_sphere.md): related method to
      determine the concentration profile.
    * [`uptake_convection_sphere`](uptake_convection_sphere.md): related method
      for surface convection boundary condition.

    Examples
    --------
    Determine the fractional mass uptake after 100 seconds for a polymer sphere
    with a radius of 0.1 mm and a diffusion coefficient of 1e-11 m²/s.
    >>> from polykin.transport import uptake_constc_sphere
    >>> t = 1e2   # s
    >>> a = 1e-4  # m
    >>> D = 1e-11 # m²/s
    >>> Fo = D*t/a**2
    >>> uptake_constc_sphere(Fo)
    0.7704787380259631
    """
    N = 4  # Number of terms in series expansion (optimal value)

    if Fo == 0:
        result = 0
    elif Fo < 1/4:
        # Solution for small times
        A = sqrt(Fo)
        S = sum(ierfc(n/A) for n in range(1, N))
        result = 6*A * (1/sqrt(pi) + 2*S) - 3*Fo
    else:
        # Solution for normal times
        B = -Fo*pi**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