polykin.transport.flow

DP_Hagen_Poiseuille

DP_Hagen_Poiseuille(
    Q: float, D: float, L: float, mu: float
) -> float

Calculate the pressure drop in a pipe using the Hagen-Poiseuille equation.

In laminar flow, the pressure drop in a circular pipe is given by:

\[ \Delta P = \frac{128 \mu Q}{\pi D^4} L \]

where \(D\) is the pipe diameter, \(L\) is the pipe length, \(Q\) is the volume flowrate, and \(\mu\) is the fluid viscosity.

PARAMETER	DESCRIPTION
Q	Volume flowrate (m³/s). TYPE: float
D	Diameter (m). TYPE: float
L	Length (m). TYPE: float
mu	Viscosity (Pa·s). TYPE: float

RETURNS	DESCRIPTION
float	Pressure drop (Pa).

Examples:

Calculate the pressure drop for a polymer solution (viscosity: 10 Pa·s) flowing at 1 L/s through 5 m of smooth pipe with a 50 mm internal diameter.

>>> from polykin.transport import DP_Hagen_Poiseuille
>>> from math import pi
>>> Q = 1e-3  # m³/s
>>> D = 50e-3 # m
>>> L = 5.    # m
>>> mu = 10.  # Pa·s
>>> rho = 1e3 # kg/m³ 
>>> v = 4*Q/(pi*D**2)
>>> Re = rho*v*D/mu
>>> print(f"Re = {Re:.1e}")
Re = 2.5e+00
>>> DP = DP_Hagen_Poiseuille(Q, D, L, mu)
>>> print(f"DP = {DP:.1e} Pa")
DP = 3.3e+05 Pa

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

def DP_Hagen_Poiseuille(Q: float,
                        D: float,
                        L: float,
                        mu: float
                        ) -> float:
    r"""Calculate the pressure drop in a pipe using the Hagen-Poiseuille equation.

    In laminar flow, the pressure drop in a circular pipe is given by:

    $$ \Delta P =  \frac{128 \mu Q}{\pi D^4} L $$

    where $D$ is the pipe diameter, $L$ is the pipe length, $Q$ is the volume
    flowrate, and $\mu$ is the fluid viscosity.

    Parameters
    ----------
    Q : float
        Volume flowrate (m³/s).
    D : float
        Diameter (m).
    L : float
        Length (m).
    mu : float
        Viscosity (Pa·s).

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

    Examples
    --------
    Calculate the pressure drop for a polymer solution (viscosity: 10 Pa·s)
    flowing at 1 L/s through 5 m of smooth pipe with a 50 mm internal diameter. 
    >>> from polykin.transport import DP_Hagen_Poiseuille
    >>> from math import pi
    >>> Q = 1e-3  # m³/s
    >>> D = 50e-3 # m
    >>> L = 5.    # m
    >>> mu = 10.  # Pa·s
    >>> rho = 1e3 # kg/m³ 
    >>> v = 4*Q/(pi*D**2)
    >>> Re = rho*v*D/mu
    >>> print(f"Re = {Re:.1e}")
    Re = 2.5e+00
    >>> DP = DP_Hagen_Poiseuille(Q, D, L, mu)
    >>> print(f"DP = {DP:.1e} Pa")
    DP = 3.3e+05 Pa
    """
    return 128*mu*Q*L/(pi*D**4)