polykin.transport.diffusion

profile_constc_sphere

profile_constc_sphere(Fo: float, rstar: float) -> float

Concentration profile for transient diffusion in a sphere subjected to a constant surface concentration boundary condition.

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:

\[ \frac{C - C_0}{C_s - C_0} = 1 + \frac{2}{\pi r^*} \sum_{n=1}^\infty \frac{(-1)^n}{n} \exp(- n^2 \pi^2 Fo) \sin (n \pi r^*) \]

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

References

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

PARAMETER	DESCRIPTION
Fo	Fourier number, \(D t/a^2\). TYPE: float
rstar	Normalized radial distance from the center of the sphere, \(r/a\). TYPE: float

RETURNS	DESCRIPTION
float	Normalized concentration.

See also

• uptake_constc_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 import profile_constc_sphere
>>> t = 1e2    # s
>>> a = 0.2e-3 # m
>>> r = 0      # m
>>> D = 1e-10  # m²/s
>>> Fo = D*t/a**2
>>> rstar = r/a
>>> profile_constc_sphere(Fo, rstar)
0.8304935009764246

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

def profile_constc_sphere(Fo: float, rstar: float) -> float:
    r"""Concentration profile for transient diffusion in a sphere subjected
    to a constant surface concentration boundary condition. 

    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:

    $$ \frac{C - C_0}{C_s - C_0} =
        1 + \frac{2}{\pi r^*} \sum_{n=1}^\infty \frac{(-1)^n}{n} 
        \exp(- n^2 \pi^2 Fo) \sin (n \pi r^*) $$

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

    **References**

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

    Parameters
    ----------
    Fo : float
        Fourier number, $D t/a^2$.
    rstar : float
        Normalized radial distance from the center of the sphere, $r/a$.

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

    See also
    --------
    * [`uptake_constc_sphere`](uptake_constc_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 import profile_constc_sphere
    >>> t = 1e2    # s
    >>> a = 0.2e-3 # m
    >>> r = 0      # m
    >>> D = 1e-10  # m²/s
    >>> Fo = D*t/a**2
    >>> rstar = r/a
    >>> profile_constc_sphere(Fo, rstar)
    0.8304935009764246
    """
    N = 4  # Number of terms in series expansion (sufficient for convergence)

    if abs(rstar) < eps:
        # Solution for particular case r->0
        B = -(pi**2)*Fo
        S = sum((-1 if n % 2 else 1) * exp(B*n**2) for n in range(1, N))
        result = 1 + 2*S
    else:
        if Fo == 0:
            result = 0
        elif Fo < 1/4:
            # Solution for small times
            A = 2*sqrt(Fo)
            S = sum(erfc(((2*n + 1) - rstar)/A) - erfc(((2*n + 1) + rstar)/A)
                    for n in range(0, N))
            result = S/rstar
        else:
            # Solution for normal times
            B = -(pi**2)*Fo
            S = sum((-1 if n % 2 else 1) / n * exp(B*n**2) * sin(rstar*pi*n)
                    for n in range(1, N))
            result = 1 + (2/(pi*rstar))*S

    return result

Graphical Illustration

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