polykin.reactors.rtd

E_dispersion_model

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

Differential 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:

\[ E(t) = \frac{1}{\bar{t}} \sqrt{\frac{Pe}{4\pi}} \exp\left[-\frac{Pe}{4}(1-\theta)^2\right] \]

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

\[ E(t) = \frac{1}{\bar{t}} \sqrt{\frac{Pe}{4\pi\theta}} \exp\left[-\frac{Pe}{4}\frac{(1-\theta)^2}{\theta}\right] \]

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

References

• Levenspiel, O. "Chemical reaction engineering", 3rd ed., John Wiley & Sons, 1999, p. 296-301.

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	Differential residence time distribution.

See also

• F_dispersion_model: related method to determine the cumulative distribution.

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

def E_dispersion_model(t: float, tavg: float, Pe: float) -> float:
    r"""Differential 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:

    $$ E(t) = \frac{1}{\bar{t}} \sqrt{\frac{Pe}{4\pi}}
              \exp\left[-\frac{Pe}{4}(1-\theta)^2\right]  $$

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

    $$ E(t) = \frac{1}{\bar{t}} \sqrt{\frac{Pe}{4\pi\theta}}
              \exp\left[-\frac{Pe}{4}\frac{(1-\theta)^2}{\theta}\right]  $$

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

    **References**

    * Levenspiel, O. "Chemical reaction engineering", 3rd ed., John Wiley &
      Sons, 1999, p. 296-301.

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

    Returns
    -------
    float
        Differential residence time distribution.

    See also
    --------
    * [`F_dispersion_model`](F_dispersion_model.md): related method to determine
      the cumulative distribution.
    """
    q = t/tavg
    if q == 0 or q == inf:
        return 0
    else:
        if Pe > 1e2:
            return sqrt(Pe/pi)/(2*tavg) * exp(-Pe/4*(1 - q)**2)
        else:
            return sqrt(Pe/(pi*q))/(2*tavg) * exp(-Pe/(4*q)*(1 - q)**2)