polykin.flow.prv

area_relief_2phase_subcooled

area_relief_2phase_subcooled(
    Q: float,
    P1: float,
    P2: float,
    Ps: float,
    rho1: float,
    rho9: float,
    Kd: float = 0.65,
    Kb: float = 1.0,
    Kc: float = 1.0,
    Kv: float = 1.0,
) -> PRVResult

Calculate the required effective discharge area of a pressure relief device for subcooled liquid flow using the omega method.

The calculation is done according to the API standard 520, Appendix C.2.3.

This method can also be used for liquids that are saturated as they enter the relief device, but no condensable vapor or noncondensable gas should be present at the inlet.

References

• Sizing, Selection, and Installation of Pressure-relieving Devices in Refineries: Part I—Sizing and Selection, API Standard 520, 10th ed., 2020.

PARAMETER	DESCRIPTION
Q	Required relieving volume flow rate (L/min). TYPE: float
P1	Upstream relieving pressure, absolute (bara). TYPE: float
P2	Downstream back pressure, absolute (bara). TYPE: float
Ps	Saturation (bubble) pressure at upstream relieving temperature (bara). TYPE: float
rho1	Liquid density at upstream relieving conditions (m³/kg). TYPE: float
rho9	Overall density evaluated at 90% of the saturation pressure Ps (m³/kg). The flash calculation should be carried out isentropically, but an isenthalpic (adiabatic) flash is sufficient for low-vapor-content mixtures far from the thermodynamic critical point. TYPE: float
Kd	Effective discharge coefficient. For a preliminary size estimation, use Kd=0.65 for subcooled liquids, and Kd=0.85 for saturated liquids. TYPE: float DEFAULT: 0.65
Kb	Backpressure correction factor for balanced bellows valves. The value can be obtained from the manufacturer or estimated with the help of Figure C.3 of the API standard 520. TYPE: float DEFAULT: 1.0
Kc	Combination correction factor for installations with a rupture disk upstream of the PRV. Use Kc=1.0 if there is no rupture disk, and Kc=0.9 if there is a rupture disk. TYPE: float DEFAULT: 1.0
Kv	Viscosity correction factor. Use Kv=1.0 if the liquid phase has a viscosity less or equal than 0.1 Pa·s. TYPE: float DEFAULT: 1.0

RETURNS	DESCRIPTION
PRVResult	Dataclass containing the results of the calculation.

Examples:

Estimate the required discharge area of a pressure relief device handling a subcooled liquid, according to the API standard 520. The required flow is 378.5 L/min, the upstream relieving pressure is 20.733 bara, the downstream back pressure is 1.703 bara, the saturation pressure at the relieving temperature is 7.419 bara, the density of the liquid is 511.3 kg/m³ and the density of the two-phase mixture evaluated at 90% of the saturation pressure is 262.7 kg/m³.

>>> from polykin.flow import area_relief_2phase_subcooled
>>> area_relief_2phase_subcooled(Q=378.5, P1=20.733, P2=1.703, Ps=7.419, 
...                              rho1=511.3, rho9=262.7)
PRVResult(Pcf=nan bara, critical_flow=True, A=1.35e+02 mm²)

Source code in src/polykin/flow/prv.py

def area_relief_2phase_subcooled(Q: float,
                                 P1: float,
                                 P2: float,
                                 Ps: float,
                                 rho1: float,
                                 rho9: float,
                                 Kd: float = 0.65,
                                 Kb: float = 1.0,
                                 Kc: float = 1.0,
                                 Kv: float = 1.0
                                 ) -> PRVResult:
    r"""Calculate the required effective discharge area of a pressure relief
    device for subcooled liquid flow using the omega method.

    The calculation is done according to the API standard 520, Appendix C.2.3.

    This method can also be used for liquids that are saturated as they enter
    the relief device, but no condensable vapor or noncondensable gas should be
    present at the inlet.

    **References**

    * Sizing, Selection, and Installation of Pressure-relieving Devices in
      Refineries: Part I—Sizing and Selection, API Standard 520, 10th ed., 2020.

    Parameters
    ----------
    Q : float
        Required relieving volume flow rate (L/min).
    P1 : float
        Upstream relieving pressure, absolute (bara).
    P2 : float
        Downstream back pressure, absolute (bara).
    Ps : float
        Saturation (bubble) pressure at upstream relieving temperature (bara).
    rho1 : float
        Liquid density at upstream relieving conditions (m³/kg).
    rho9 : float
        Overall density evaluated at 90% of the saturation pressure `Ps` (m³/kg). 
        The flash calculation should be carried out isentropically, but an 
        isenthalpic (adiabatic) flash is sufficient for low-vapor-content mixtures
        far from the thermodynamic critical point.
    Kd : float
        Effective discharge coefficient. For a preliminary size estimation, use
        `Kd=0.65` for subcooled liquids, and `Kd=0.85` for saturated liquids.
    Kb : float
        Backpressure correction factor for balanced bellows valves. The value
        can be obtained from the manufacturer or estimated with the help of 
        Figure C.3 of the API standard 520.
    Kc : float
        Combination correction factor for installations with a rupture disk
        upstream of the PRV. Use `Kc=1.0` if there is no rupture disk, and
        `Kc=0.9` if there is a rupture disk.
    Kv : float
        Viscosity correction factor. Use `Kv=1.0` if the liquid phase has a
        viscosity less or equal than 0.1 Pa·s.

    Returns
    -------
    PRVResult
        Dataclass containing the results of the calculation.

    Examples
    --------
    Estimate the required discharge area of a pressure relief device handling a
    subcooled liquid, according to the API standard 520. The required flow is
    378.5 L/min, the upstream relieving pressure is 20.733 bara, the downstream
    back pressure is 1.703 bara, the saturation pressure at the relieving 
    temperature is 7.419 bara, the density of the liquid is 511.3 kg/m³ and the
    density of the two-phase mixture evaluated at 90% of the saturation pressure
    is 262.7 kg/m³.
    >>> from polykin.flow import area_relief_2phase_subcooled
    >>> area_relief_2phase_subcooled(Q=378.5, P1=20.733, P2=1.703, Ps=7.419, 
    ...                              rho1=511.3, rho9=262.7)
    PRVResult(Pcf=nan bara, critical_flow=True, A=1.35e+02 mm²)
    """

    # Convert pressures from bar to Pa
    P1 *= 1e5
    P2 *= 1e5
    Ps *= 1e5

    # Omega parameter for saturated liquid
    ws = 9*(rho1/rho9 - 1)

    # Transition saturation pressure ratio
    ηst = 2*ws/(1 + 2*ws)

    # High/low subcooling region
    ηs = Ps/P1
    ηa = P2/P1
    high_subcooling = (ηs < ηst)

    # Mass flux (kg/s.m²)
    if high_subcooling:
        critical_flow = (P2 <= Ps)
        P = Ps if critical_flow else P2
        Pcf = nan
        G = 1.414*sqrt(rho1*(P1 - P))
    else:
        ηc_ = ηs*(2*ws/(2*ws - 1))*(1 - sqrt(1 - 1/ηs*((2*ws - 1)/(2*ws))))
        ηc = ηc_ if ηs > ηst else ηs
        Pcf = P1 * ηc
        critical_flow = (P2 <= Pcf)
        η = ηc if critical_flow else ηa
        G = sqrt(2*(1 - ηs) + 2*(ws*ηs*log(ηs/η) - (ws - 1)*(ηs - η))) * \
            sqrt(P1*rho1)/(ws*(ηs/η - 1) + 1)

    # Area (mm²)
    A = 16.67*Q*rho1/(Kd*Kb*Kc*Kv*G)

    return PRVResult(Pcf/1e5, critical_flow, A)