polykin.reactors.rtd

F_dispersion_model

F_dispersion_model(
    t: float, tavg: float, Pe: float
) -> float

Cumulative residence time distribution for the dispersed plug flow model (also known as dispersion model).

Only approximate analytical solutions are available for this model. For small deviations from plug flow, i.e. when \(Pe > 10^2\), the distribution is approximated by:

\[ F(t) = \frac{1}{2}\left[ 1 + \mathrm{erf} \left( \frac{\sqrt{Pe}}{2}(\theta-1)\right) \right] \]

Otherwise, when \(Pe \le 10^2\), the distribution is approximated by the numerical integral of the so-called open-open solution:

\[ F(t) = \frac{1}{\bar{t}} \sqrt{\frac{Pe}{4\pi}} \int_0^{\theta} \frac{1}{\sqrt{\theta'}} \exp\left[-\frac{Pe}{4}\frac{(1-\theta')^2}{\theta'}\right] d\theta' \]

where \(\theta = t/\bar{t}\), and \(Pe = (v L)/D\).

PARAMETER	DESCRIPTION
t	Residence time. TYPE: float
tavg	Average residence time, \(\bar{t}\). TYPE: float
Pe	Péclet number, \((v L)/D\). TYPE: float

RETURNS	DESCRIPTION
float	Cumulative residence time distribution.

See also

• E_dispersion_model: related method to determine the differential distribution.

Source code in src/polykin/reactors/rtd.py

def F_dispersion_model(t: float, tavg: float, Pe: float) -> float:
    r"""Cumulative residence time distribution for the dispersed plug flow
    model (also known as dispersion model).

    Only approximate analytical solutions are available for this model. For
    small deviations from plug flow, i.e. when $Pe > 10^2$, the distribution is
    approximated by:

    $$ F(t) = \frac{1}{2}\left[ 1 +
        \mathrm{erf} \left( \frac{\sqrt{Pe}}{2}(\theta-1)\right) \right]  $$

    Otherwise, when $Pe \le 10^2$, the distribution is approximated by the
    numerical integral of the so-called open-open solution:

    $$ F(t) = \frac{1}{\bar{t}} \sqrt{\frac{Pe}{4\pi}}
        \int_0^{\theta} \frac{1}{\sqrt{\theta'}}
        \exp\left[-\frac{Pe}{4}\frac{(1-\theta')^2}{\theta'}\right] d\theta' $$

    where $\theta = t/\bar{t}$, and $Pe = (v L)/D$.

    Parameters
    ----------
    t : float
        Residence time.
    tavg : float
        Average residence time, $\bar{t}$.
    Pe : float
        Péclet number, $(v L)/D$.

    Returns
    -------
    float
        Cumulative residence time distribution.

    See also
    --------
    * [`E_dispersion_model`](E_dispersion_model.md): related method to determine
      the differential distribution.
    """
    q = t/tavg
    if q == 0:
        return 0
    elif q == inf:
        return 1
    else:
        if Pe > 1e2:
            return 0.5*(1 + erf(sqrt(Pe)/2*(q - 1)))
        else:
            result, _ = integrate.quad(
                E_dispersion_model,
                a=0.,
                b=t,
                args=(tavg, Pe),
                epsabs=1e-5)
            return result