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polykin.transport.diffusion¤

profile_constc_sheet ¤

profile_constc_sheet(Fo: float, xstar: float) -> float

Concentration profile 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 normalized concentration is:

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

where \(Fo = D t/a^2\) is the Fourier number, and \(x^*=x/a\) is the normalized distance from the center of the sheet.

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. 47.
PARAMETER DESCRIPTION
Fo

Fourier number, \(D t/a^2\).

TYPE: float

xstar

Normalized distance from the center or sealed face, \(r/a\).

TYPE: float

RETURNS DESCRIPTION
float

Fractional accomplished concentration change.
See also

Examples:

Determine the fractional concentration change after 100 s in a 0.2 mm-thick polymer film (diffusivity: 1e-10 m²/s) at its maximum depth.

>>> from polykin.transport import profile_constc_sheet
>>> t = 1e2    # s
>>> a = 0.2e-3 # m
>>> D = 1e-10  # m²/s
>>> x = 0.     # m
>>> Fo = D*t/a**2
>>> xstar = x/a
>>> profile_constc_sheet(Fo, xstar)
0.3145542331096478
Source code in src/polykin/transport/diffusion.py 
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def profile_constc_sheet(Fo: float, xstar: float) -> float:
    r"""Concentration profile 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 normalized concentration is:

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

    where $Fo = D t/a^2$ is the Fourier number, and $x^*=x/a$ is the normalized
    distance from the center of the sheet.

    !!! 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. 47.

    Parameters
    ----------
    Fo : float
        Fourier number, $D t/a^2$.
    xstar : float
        Normalized distance from the center or sealed face, $r/a$.

    Returns
    -------
    float
        Fractional accomplished concentration change.

    See also
    --------
    * [`uptake_constc_sheet`](uptake_constc_sheet.md): related method to
      determine the mass uptake.

    Examples
    --------
    Determine the fractional concentration change after 100 s in a 0.2 mm-thick
    polymer film (diffusivity: 1e-10 m²/s) at its maximum depth.
    >>> from polykin.transport import profile_constc_sheet
    >>> t = 1e2    # s
    >>> a = 0.2e-3 # m
    >>> D = 1e-10  # m²/s
    >>> x = 0.     # m
    >>> Fo = D*t/a**2
    >>> xstar = x/a
    >>> profile_constc_sheet(Fo, xstar)
    0.3145542331096478
    """
    N = 4  # Number of terms in series expansion (sufficient for convergence)

    if Fo == 0:
        result = 0
    elif Fo < 1/4:
        # Solution for small times
        A = 2*sqrt(Fo)
        S = sum((-1 if n % 2 else 1) * (erfc(((2*n + 1) - xstar)/A) + erfc(((2*n + 1) + xstar)/A))
                for n in range(0, N))
        result = S
    else:
        # Solution for normal times
        B = -(pi**2 / 4)*Fo
        C = (pi/2)*xstar
        S = sum((-1 if n % 2 else 1) / (2*n + 1) * exp(B*(2*n + 1)**2) * cos(C*(2*n + 1))
                for n in range(0, N))
        result = 1 - (4/pi)*S

    return result

Graphical Illustration¤

The numbers in the legend are values of \(Fo = D t / a^2\).

profile_constc_sheet