polykin.thermo.flash

residual_Rachford_Rice

residual_Rachford_Rice(
    beta: float,
    K: FloatVector,
    z: FloatVector,
    derivative: bool = False,
) -> tuple[float, ...]

Rachford-Rice flash residual function and its derivative.

The residual function is defined as:

\[ F = \sum_i \frac{z_i(K_i - 1)}{1 + \beta(K_i - 1)} \]

and the derivative with respect to \(\beta\) is:

\[ \frac{\partial F}{\partial \beta} = -\sum_i \frac{z_i(K_i - 1)^2}{(1 + \beta(K_i - 1))^2} \]

where \(K\) is the vector of K-values, \(z\) is the vector of feed mole fractions, and \(\beta\) is the vapor phase fraction.

References

• Rachford, H.H., and J.D. Rice. "Procedure for Use of Electronic Digital Computers in Calculating Flash Vaporization Hydrocarbon Equilibrium", J. Pet. Technol. 4 (1952): 19-3.

PARAMETER	DESCRIPTION
beta	Vapor phase fraction (mol/mol). TYPE: float
K	K-values. TYPE: FloatVector(N)
z	Feed mole fractions (mol/mol). TYPE: FloatVector(N)
derivative	Flag specifying if the derivative should be returned. TYPE: bool DEFAULT: False

RETURNS	DESCRIPTION
tuple[float, ...]	Tuple with residual and derivative, (F, dF).

See also

• solve_Rachford_Rice: related method to solve the equation.

Source code in src/polykin/thermo/flash/vle.py

def residual_Rachford_Rice(
    beta: float,
    K: FloatVector,
    z: FloatVector,
    derivative: bool = False
) -> tuple[float, ...]:
    r"""Rachford-Rice flash residual function and its derivative.

    The residual function is defined as:

    $$ F = \sum_i \frac{z_i(K_i - 1)}{1 + \beta(K_i - 1)} $$

    and the derivative with respect to $\beta$ is:

    $$ \frac{\partial F}{\partial \beta} =
       -\sum_i \frac{z_i(K_i - 1)^2}{(1 + \beta(K_i - 1))^2} $$

    where $K$ is the vector of K-values, $z$ is the vector of feed mole
    fractions, and $\beta$ is the vapor phase fraction.

    **References**

    *  Rachford, H.H., and J.D. Rice. "Procedure for Use of Electronic Digital
       Computers in Calculating Flash Vaporization Hydrocarbon Equilibrium", 
       J. Pet. Technol. 4 (1952): 19-3.

    Parameters
    ----------
    beta : float
        Vapor phase fraction (mol/mol).
    K : FloatVector(N)
        K-values.
    z : FloatVector(N)
        Feed mole fractions (mol/mol).
    derivative : bool
        Flag specifying if the derivative should be returned.

    Returns
    -------
    tuple[float, ...]
        Tuple with residual and derivative, `(F, dF)`.

    See also
    --------
    * [`solve_Rachford_Rice`](solve_Rachford_Rice.md): related method to solve
      the equation. 
    """

    F = np.sum(z*(K - 1)/(1 + beta*(K - 1)))

    if not derivative:
        return (F,)
    else:
        dF = -np.sum(z*(K - 1)**2/(1 + beta*(K - 1))**2)
        return (F, dF)