polykin.thermo.eos

Z_cubic_roots

Z_cubic_roots(
    u: float, w: float, A: float, B: float
) -> FloatVector

Find the compressibility factor roots of a cubic EOS.

\[\begin{gathered} Z^3 + c_2 Z^2 + c_1 Z + c_0 = 0 \\ c_2 = -(1 + B - u B) \\ c_1 = A + w B^2 - u B - u B^2 \\ c_0 = -(A B + w B^2 + w B^3) \\ A = \frac{a_m P}{R^2 T^2} \\ B = \frac{b_m P}{R T} \end{gathered}\]

Equation	\(u\)	\(w\)
Redlich-Kwong	1	0
Soave-Redlich-Kwong	1	0
Peng-Robinson	2	-1

References

• RC Reid, JM Prausniz, and BE Poling. The properties of gases & liquids 4th edition, 1986, p. 42.

PARAMETER	DESCRIPTION
u	Parameter of polynomial equation. TYPE: float
w	Parameter of polynomial equation. TYPE: float
A	Parameter of polynomial equation. TYPE: float
B	Parameter of polynomial equation. TYPE: float

RETURNS	DESCRIPTION
FloatVector	Compressibility factor(s) of the coexisting phases. If there are two phases, the first result is the lowest value (liquid).

Source code in src/polykin/thermo/eos/cubic.py

def Z_cubic_roots(
    u: float,
    w: float,
    A: float,
    B: float
) -> FloatVector:
    r"""Find the compressibility factor roots of a cubic EOS.

    \begin{gathered}
        Z^3 + c_2 Z^2 + c_1 Z + c_0 = 0 \\
        c_2 = -(1 + B - u B) \\
        c_1 = A + w B^2 - u B - u B^2 \\
        c_0 = -(A B + w B^2 + w B^3) \\
        A = \frac{a_m P}{R^2 T^2} \\
        B = \frac{b_m P}{R T}
    \end{gathered}

    | Equation            | $u$ | $w$ |
    |---------------------|:---:|:---:|
    | Redlich-Kwong       |  1  |  0  |
    | Soave-Redlich-Kwong |  1  |  0  |
    | Peng-Robinson       |  2  | -1  |

    **References**

    *   RC Reid, JM Prausniz, and BE Poling. The properties of gases &
        liquids 4th edition, 1986, p. 42.

    Parameters
    ----------
    u : float
        Parameter of polynomial equation.
    w : float
        Parameter of polynomial equation.
    A : float
        Parameter of polynomial equation.
    B : float
        Parameter of polynomial equation.

    Returns
    -------
    FloatVector
        Compressibility factor(s) of the coexisting phases. If there are two
        phases, the first result is the lowest value (liquid).
    """
    c3 = 1.0
    c2 = -(1 + B - u*B)
    c1 = (A + w*B**2 - u*B - u*B**2)
    c0 = -(A*B + w*B**2 + w*B**3)

    roots = np.roots((c3, c2, c1, c0))
    roots = [x.real for x in roots if (abs(x.imag) < eps and x.real > B)]

    Z = [min(roots)]
    if len(roots) > 1:
        Z.append(max(roots))

    return np.array(Z)