polykin.transport.diffusion

uptake_constc_sheet

uptake_constc_sheet(Fo: float) -> float

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

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:

\[ \frac{\bar{C}-C_0}{C_s -C_0} = 1 - \frac{8}{\pi^2} \sum_{n=0}^{\infty}\frac{1}{(2n+1)^2} \exp\left[-\frac{(2n+1)^2 \pi^2}{4}Fo\right] \]

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

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

RETURNS	DESCRIPTION
float	Fractional mass uptake.

See also

• profile_constc_sheet: related method to determine the concentration profile.
• uptake_convection_sheet: related method for surface convection boundary condition.

Examples:

Determine the fractional mass uptake after 500 seconds for a polymer solution film with a thickness of 0.2 mm and a diffusion coefficient of 1e-11 m²/s.

>>> from polykin.transport import uptake_constc_sheet
>>> t = 5e2   # s
>>> a = 2e-4  # m
>>> D = 1e-11 # m²/s
>>> Fo = D*t/a**2
>>> uptake_constc_sheet(Fo)
0.39892798988456807

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

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

    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:

    $$ \frac{\bar{C}-C_0}{C_s -C_0} = 
        1 - \frac{8}{\pi^2} \sum_{n=0}^{\infty}\frac{1}{(2n+1)^2}
        \exp\left[-\frac{(2n+1)^2 \pi^2}{4}Fo\right] $$

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

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

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

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

    Examples
    --------
    Determine the fractional mass uptake after 500 seconds for a polymer solution
    film with a thickness of 0.2 mm and a diffusion coefficient of 1e-11 m²/s.
    >>> from polykin.transport import uptake_constc_sheet
    >>> t = 5e2   # s
    >>> a = 2e-4  # m
    >>> D = 1e-11 # m²/s
    >>> Fo = D*t/a**2
    >>> uptake_constc_sheet(Fo)
    0.39892798988456807
    """
    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((-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 = -(Fo*pi**2)/4
        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