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polykin.reactors.rtd¤

F_tanks_series ¤

F_tanks_series(t: float, tavg: float, N: int) -> float

Cumulative residence time distribution for a series of equal CSTRs.

\[ F(t) = 1 - e^{-N t / \bar{t}} \; \sum_{i=0}^{N-1} \frac{(N t / \bar{t})^i}{i!} \]

References

  • Levenspiel, O. "Chemical reaction engineering", 3rd ed., John Wiley & Sons, 1999, p. 327.
PARAMETER DESCRIPTION
t

Residence time.

TYPE: float

tavg

Total average residence time, \(\bar{t}\).

TYPE: float

N

Number of tanks in series.

TYPE: int

RETURNS DESCRIPTION
float

Cumulative residence time distribution.
See also
  • E_tanks_series: related method to determine the differential distribution.
Source code in src/polykin/reactors/rtd.py 
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def F_tanks_series(t: float, tavg: float, N: int) -> float:
    r"""Cumulative residence time distribution for a series of equal CSTRs.

    $$ F(t) = 1 - e^{-N t / \bar{t}} \;
        \sum_{i=0}^{N-1} \frac{(N t / \bar{t})^i}{i!}  $$

    **References**

    * Levenspiel, O. "Chemical reaction engineering", 3rd ed., John Wiley &
      Sons, 1999, p. 327.

    Parameters
    ----------
    t : float
        Residence time.
    tavg : float
        Total average residence time, $\bar{t}$.
    N : int
        Number of tanks in series.

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

    See also
    --------
    * [`E_tanks_series`](E_tanks_series.md): related method to determine the
      differential distribution.
    """
    q = t/tavg
    if q == 0:
        return 0
    elif q == inf:
        return 1
    else:
        S = sum((N*q)**i / gamma(i+1) for i in range(1, N))
        return 1 - exp(-N*q) * (1 + S)