polykin.transport.flow

fD_Colebrook

fD_Colebrook(Re: float, er: float) -> float

Calculate the Darcy friction factor using Colebrook's equation.

For turbulent flow, i.e., \(Re \gtrsim 2300\), the friction factor is given by the following implicit expression:

\[ \frac{1}{\sqrt{f}}= -2 \log \left( \frac {\epsilon/D} {3.7} + \frac {2.51} {Re \sqrt{f}} \right) \]

This equation is a historical landmark but has the disadvantage of being implicit, requiring an iterative solution.

References

• Colebrook, C F (1939). "Turbulent Flow in Pipes, with Particular Reference to the Transition Region Between the Smooth and Rough Pipe Laws", Journal of the Institution of Civil Engineers. 11 (4): 133-156.

PARAMETER	DESCRIPTION
Re	Reynolds number. TYPE: float
er	Relative pipe roughness, \(\epsilon/D\). TYPE: float

RETURNS	DESCRIPTION
float	Darcy friction factor.

See also

• fD_Haaland: alternative method.

Examples:

Calculate the friction factor for water flowing at 2 m/s through a PVC pipe with an internal diameter of 25 mm.

>>> from polykin.transport.flow import fD_Colebrook
>>> rho = 1e3 # kg/m³
>>> mu = 1e-3 # Pa·s
>>> D = 25e-3 # m
>>> v = 2.    # m/s
>>> Re = rho*v*D/mu
>>> er = 0.0015/25 # from pipe table
>>> fD = fD_Colebrook(Re, er)
>>> print(f"fD = {fD:.3f}")
fD = 0.021

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

def fD_Colebrook(Re: float, er: float) -> float:
    r"""Calculate the Darcy friction factor using Colebrook's equation.

    For turbulent flow, i.e., $Re \gtrsim 2300$, the friction factor
    is given by the following implicit expression:

    $$  \frac{1}{\sqrt{f}}= -2 \log \left( \frac {\epsilon/D} {3.7} +
        \frac {2.51} {Re \sqrt{f}} \right) $$

    This equation is a historical landmark but has the disadvantage of
    being implicit, requiring an iterative solution.

    **References**

    * Colebrook, C F (1939). "Turbulent Flow in Pipes, with Particular Reference
      to the Transition Region Between the Smooth and Rough Pipe Laws", Journal
      of the Institution of Civil Engineers. 11 (4): 133-156.

    Parameters
    ----------
    Re : float
        Reynolds number.
    er : float
        Relative pipe roughness, $\epsilon/D$.

    Returns
    -------
    float
        Darcy friction factor.

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

    Examples
    --------
    Calculate the friction factor for water flowing at 2 m/s through a PVC pipe
    with an internal diameter of 25 mm.
    >>> from polykin.transport.flow import fD_Colebrook
    >>> rho = 1e3 # kg/m³
    >>> mu = 1e-3 # Pa·s
    >>> D = 25e-3 # m
    >>> v = 2.    # m/s
    >>> Re = rho*v*D/mu
    >>> er = 0.0015/25 # from pipe table
    >>> fD = fD_Colebrook(Re, er)
    >>> print(f"fD = {fD:.3f}")
    fD = 0.021
    """

    check_range_warn(Re, 2.3e3, inf, 'Re')

    def fnc(f):
        return 2*log10(er/3.7 + 2.51/(Re*sqrt(f))) + 1/sqrt(f)

    sol = root_newton(fnc, fD_Haaland(Re, er), 1e-5)

    return sol.x

Graphical Illustration