polykin.transport.diffusion

uptake_sheet

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

Fractional mass uptake for transient diffusion in a plane sheet.

For a plane sheet of thickness \(2a\), with diffusion from both faces, where the concentration is initially \(C_0\) everywhere, and the concentration at both surfaces is maintained at \(C_s\), the fractional mass uptake is:

\[ \frac{M_t}{M_{\infty}} = 1 - \frac{8}{\pi^2} \sum_{n=0}^{\infty}\frac{1}{(2n+1)^2} \exp\left[\frac{-D(2n+1)^2 \pi^2t}{4 a^2}\right] \]

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

Tip

This equation is also applicable to a plane sheet of thickness \(a\) if one of the faces is sealed.

References

• J. Crank, "The mathematics of diffusion", Oxford University Press, 1975, p. 48.

PARAMETER	DESCRIPTION
t	Time (s). TYPE: float
a	Half thickness of sheet (m). TYPE: float
D	Diffusion coefficient (m²/s). TYPE: float

RETURNS	DESCRIPTION
float	Fractional mass uptake.

See also

• profile_sheet: related method to determine the concentration profile.

Examples:

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

>>> from polykin.transport.diffusion import uptake_sheet
>>> uptake_sheet(t=1e2, a=0.2e-3, D=1e-10)
0.562233541762138

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

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

    For a plane sheet of thickness $2a$, with diffusion from _both_ faces, where
    the concentration is initially $C_0$ everywhere, and the concentration at
    both surfaces is maintained at $C_s$, the fractional mass uptake is:

    $$ \frac{M_t}{M_{\infty}} = 
        1 - \frac{8}{\pi^2} \sum_{n=0}^{\infty}\frac{1}{(2n+1)^2}
        \exp\left[\frac{-D(2n+1)^2 \pi^2t}{4 a^2}\right] $$

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

    !!! tip

        This equation is also applicable to a plane sheet of thickness $a$ if
        one of the faces is sealed.

    **References**

    * J. Crank, "The mathematics of diffusion", Oxford University Press, 1975,
      p. 48.

    Parameters
    ----------
    t : float
        Time (s).
    a : float
        Half thickness of sheet (m).
    D : float
        Diffusion coefficient (m²/s).

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

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

    Examples
    --------
    Determine the fractional mass uptake after 100 seconds for a polymer solution
    film with a thickness of 0.2 mm and a diffusion coefficient of 1e-10 m²/s.
    >>> from polykin.transport.diffusion import uptake_sheet
    >>> uptake_sheet(t=1e2, a=0.2e-3, D=1e-10)
    0.562233541762138
    """
    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((-1 if n % 2 else 1) * ierfc(n/A) for n in range(1, N))
        result = 2*A * (1/sqrt(pi) + 2*S)
    else:
        # Solution for normal times
        B = -D*pi**2*t/(4*a**2)
        S = sum(1/(2*n + 1)**2 * exp(B*(2*n + 1)**2) for n in range(0, N-1))
        result = 1 - (8/pi**2)*S

    return result

Graphical Illustration