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polykin.transport.diffusion¤

profile_sheet ¤

profile_sheet(
    t: float, x: float, a: float, D: float
) -> float

Concentration profile 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 normalized concentration is:

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

where \(x\) is the distance from the center of the sheet, \(t\) 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. 47.
PARAMETER DESCRIPTION
t

Time (s).

TYPE: float

x

Distance from the center or sealed face (m).

TYPE: float

a

Half thickness of sheet (m).

TYPE: float

D

Diffusion coefficient (m²/s).

TYPE: float

RETURNS DESCRIPTION
float

Fractional accomplished concentration change.
See also
  • uptake_sheet: 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.diffusion import profile_sheet
>>> profile_sheet(t=1e2, x=0., a=0.2e-3, D=1e-10)
0.3145542331096479
Source code in src/polykin/transport/diffusion.py 
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def profile_sheet(t: float,
                  x: float,
                  a: float,
                  D: float
                  ) -> float:
    r"""Concentration profile 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 normalized concentration is:

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

    where $x$ is the distance from the center of the sheet, $t$ 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. 47.

    Parameters
    ----------
    t : float
        Time (s).
    x : float
        Distance from the center or sealed face (m).
    a : float
        Half thickness of sheet (m).
    D : float
        Diffusion coefficient (m²/s).

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

    See also
    --------
    * [`uptake_sheet`](uptake_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.diffusion import profile_sheet
    >>> profile_sheet(t=1e2, x=0., a=0.2e-3, D=1e-10)
    0.3145542331096479
    """
    N = 4  # Number of terms in series expansion (sufficient for convergence)

    A = 2*sqrt(D*t)/a
    if A < 1.:
        # Solution for small times
        S = sum((-1 if n % 2 else 1) * (erfc(((2*n + 1) - x/a)/A) + erfc(((2*n + 1) + x/a)/A))
                for n in range(0, N))
        result = S
    else:
        # Solution for normal times
        B = -D*pi**2*t/(4*a**2)
        C = pi*x/(2*a)
        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 \(D t / a^2\).

profile_sheet