polykin.transport.flow

DP_GL_Lockhart_Martinelli

DP_GL_Lockhart_Martinelli(
    mdotL: float,
    mdotG: float,
    D: float,
    L: float,
    rhoL: float,
    rhoG: float,
    muL: float,
    muG: float,
    er: float,
) -> float

Calculate the pressure drop due to friction in two-phase liquid-gas flow through a horizontal pipe using the Lockhart-Martinelli correlation.

The pressure drop due to friction in a two-phase flow is estimated by:

\[ (\Delta P)_{TP} = (\Delta P)_{L} \phi^2_{L} \]

where \((\Delta P)_{L}\) is the pressure drop if the liquid phase were alone, and \(\phi_{L}\) is the liquid-phase multiplier. The latter is given by:

\[ \phi_{L} = 1 + \frac{C}{X} + \frac{1}{X^2} \]

where \(X=\sqrt{(\Delta P)_L/(\Delta P)_G}\) is the Lockhart-Martinelli parameter and \(C\) is a coefficient that varies based on the flow regimes of the liquid and gas phases.

References

• Walas, S. M., "Chemical Process Equipment: Selection and Design", Singapore: Butterworths, 1988.

PARAMETER	DESCRIPTION
mdotL	Mass flow rate of liquid (kg/s). TYPE: float
mdotG	Mass flow rate of gas (kg/s). TYPE: float
D	Diameter (m). TYPE: float
L	Length (m). TYPE: float
rhoL	Density of liquid (kg/m³). TYPE: float
rhoG	Density of gas (kg/m³). TYPE: float
muL	Viscosity of liquid (Pa·s). TYPE: float
muG	Viscosity of gas (Pa·s). TYPE: float
er	Relative pipe roughness, \(\epsilon/D\). Only required for turbulent flow. TYPE: float

RETURNS	DESCRIPTION
float	Pressure drop (Pa).

See also

• DP_GL_Mueller_Bonn: alternative method.

Examples:

Calculate the pressure gradient due to friction in a 80 mm inner diameter pipe with 2 kg/s of liquid and 1 kg/s of gas. The liquid and gas have densities of 1000 and 1 kg/m³, respectively, and viscosities of 1e-3 and 2e-5 Pa·s, respectively.

>>> from polykin.transport import DP_GL_Lockhart_Martinelli
>>> mdotL = 2.0 # kg/s
>>> mdotG = 1.0 # kg/s 
>>> D = 80e-3   # m
>>> L = 1.0     # m
>>> rhoL = 1e3  # kg/m³
>>> rhoG = 1e0  # kg/m³
>>> muL = 1e-3  # Pa·s
>>> muG = 2e-5  # Pa·s
>>> er = 0.0    
>>> DP = DP_GL_Lockhart_Martinelli(mdotL, mdotG, D, L, rhoL, rhoG, muL, muG, er)
>>> print(f"DP = {DP:.1e} Pa/m")
DP = 8.2e+03 Pa/m

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

def DP_GL_Lockhart_Martinelli(mdotL: float,
                              mdotG: float,
                              D: float,
                              L: float,
                              rhoL: float,
                              rhoG: float,
                              muL: float,
                              muG: float,
                              er: float
                              ) -> float:
    r"""Calculate the pressure drop due to friction in two-phase liquid-gas
    flow through a horizontal pipe using the Lockhart-Martinelli correlation.

    The pressure drop due to friction in a two-phase flow is estimated by:

    $$ (\Delta P)_{TP} = (\Delta P)_{L} \phi^2_{L} $$

    where $(\Delta P)_{L}$ is the pressure drop if the liquid phase were alone,
    and $\phi_{L}$ is the liquid-phase multiplier. The latter is given by:

    $$ \phi_{L} = 1 + \frac{C}{X} + \frac{1}{X^2} $$

    where $X=\sqrt{(\Delta P)_L/(\Delta P)_G}$ is the Lockhart-Martinelli
    parameter and $C$ is a coefficient that varies based on the flow regimes of
    the liquid and gas phases.

    **References**

    * Walas, S. M., "Chemical Process Equipment: Selection and Design",
      Singapore: Butterworths, 1988.

    Parameters
    ----------
    mdotL : float
        Mass flow rate of liquid (kg/s).
    mdotG : float
        Mass flow rate of gas (kg/s).
    D : float
        Diameter (m).
    L : float
        Length (m).
    rhoL : float
        Density of liquid (kg/m³).
    rhoG : float
        Density of gas (kg/m³).
    muL : float
        Viscosity of liquid (Pa·s).
    muG : float
        Viscosity of gas (Pa·s).
    er : float
        Relative pipe roughness, $\epsilon/D$. Only required for turbulent flow.

    Returns
    -------
    float
        Pressure drop (Pa).

    See also
    --------
    * [`DP_GL_Mueller_Bonn`](DP_GL_Mueller_Bonn.md): alternative method.

    Examples
    --------
    Calculate the pressure gradient due to friction in a 80 mm inner diameter
    pipe with 2 kg/s of liquid and 1 kg/s of gas. The liquid and gas have
    densities of 1000 and 1 kg/m³, respectively, and viscosities of 1e-3 and
    2e-5 Pa·s, respectively. 
    >>> from polykin.transport import DP_GL_Lockhart_Martinelli
    >>> mdotL = 2.0 # kg/s
    >>> mdotG = 1.0 # kg/s 
    >>> D = 80e-3   # m
    >>> L = 1.0     # m
    >>> rhoL = 1e3  # kg/m³
    >>> rhoG = 1e0  # kg/m³
    >>> muL = 1e-3  # Pa·s
    >>> muG = 2e-5  # Pa·s
    >>> er = 0.0    
    >>> DP = DP_GL_Lockhart_Martinelli(mdotL, mdotG, D, L, rhoL, rhoG, muL, muG, er)
    >>> print(f"DP = {DP:.1e} Pa/m")
    DP = 8.2e+03 Pa/m
    """

    # Pressure gradient if liquid were alone
    A = (pi/4)*D**2
    if mdotL > 0.0:
        vL = mdotL/(A*rhoL)
        ReL = rhoL*vL*D/muL
        fL = 64/ReL if ReL < 2.3e3 else fD_Haaland(ReL, er)
        dPL = DP_Darcy_Weisbach(vL, D, 1.0, rhoL, fL)
    else:
        dPL = 0.0

    # Pressure gradient if gas were alone
    if mdotG > 0.0:
        vG = mdotG/(A*rhoG)
        ReG = rhoG*vG*D/muG
        fG = 64/ReG if ReG < 2.3e3 else fD_Haaland(ReG, er)
        dPG = DP_Darcy_Weisbach(vG, D, 1.0, rhoG, fG)
    else:
        dPG = 0.0

    # Two-phase pressure drop
    if not dPG:
        dP = dPL
    elif not dPL:
        dP = dPG
    else:
        X = sqrt(dPL/dPG)
        if ReL > 1e3:
            C = 20.0 if ReG > 1e3 else 10.0
        else:
            C = 12.0 if ReG > 1e3 else 5.0
        YL = 1 + C/X + 1/X**2
        dP = YL*dPL
    DP = dP*L

    return DP