polykin.transport.diffusion

profile_sphere

profile_sphere(
    t: float, r: float, a: float, D: float
) -> float

Concentration profile for transient diffusion in a sphere.

For a sphere of radius \(a\), where the concentration is initially \(C_0\) everywhere, and the surface concentration is maintained at \(C_s\), the normalized concentration is:

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

where \(r\) is the radial distance from the center of the sphere, \(t\) is the time, and \(D\) is the diffusion coefficient.

References

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

PARAMETER	DESCRIPTION
t	Time (s). TYPE: float
r	Radial distance from center of sphere (m). TYPE: float
a	Radius of sphere (m). TYPE: float
D	Diffusion coefficient (m²/s). TYPE: float

RETURNS	DESCRIPTION
float	Normalized concentration.

See also

• uptake_sphere: related method to determine the mass uptake.

Examples:

Determine the fractional concentration change after 100 s at the center of a polymer sphere with a radius of 0.2 mm and a diffusivity of 1e-10 m²/s.

>>> from polykin.transport.diffusion import profile_sphere
>>> profile_sphere(t=100., r=0., a=0.2e-3, D=1e-10)
0.8304935009764247

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

def profile_sphere(t: float,
                   r: float,
                   a: float,
                   D: float
                   ) -> float:
    r"""Concentration profile for transient diffusion in a sphere. 

    For a sphere of radius $a$, where the concentration is initially $C_0$
    everywhere, and the surface concentration is maintained at $C_s$, the
    normalized concentration is:

    $$ \frac{C - C_0}{C_s - C_0} =
        1 + \frac{2 a}{\pi r} \sum_{n=1}^\infty \frac{(-1)^n}{n} 
        \exp \left(-\frac{D n^2 \pi^2 t}{a^2} \right)
        \sin \left(\frac{n \pi r}{a} \right) $$

    where $r$ is the radial distance from the center of the sphere, $t$ is the
    time, and $D$ is the diffusion coefficient.

    **References**

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

    Parameters
    ----------
    t : float
        Time (s).
    r : float
        Radial distance from center of sphere (m).
    a : float
        Radius of sphere (m).
    D : float
        Diffusion coefficient (m²/s).

    Returns
    -------
    float
        Normalized concentration.

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

    Examples
    --------
    Determine the fractional concentration change after 100 s at the center of
    a polymer sphere with a radius of 0.2 mm and a diffusivity of 1e-10 m²/s.
    >>> from polykin.transport.diffusion import profile_sphere
    >>> profile_sphere(t=100., r=0., a=0.2e-3, D=1e-10)
    0.8304935009764247
    """
    N = 4  # Number of terms in series expansion (sufficient for convergence)

    A = 2*sqrt(D*t)/a
    B = -D*pi**2*t/a**2
    C = r/a
    if abs(C) < eps:
        # Solution for particular case r->0
        S = sum((-1 if n % 2 else 1) * exp(B*n**2) for n in range(1, N))
        result = 1 + 2*S
    else:
        if A < 1.:
            # Solution for small times
            S = sum(erfc(((2*n + 1) - C)/A) - erfc(((2*n + 1) + C)/A)
                    for n in range(0, N))
            result = S/C
        else:
            # Solution for normal times
            S = sum((-1 if n % 2 else 1) / n * exp(B*n**2) * sin(C*pi*n)
                    for n in range(1, N))
            result = 1 + (2/(pi*C))*S

    return result

Graphical Illustration

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