polykin.flow.friction

vt_sphere

vt_sphere(
    Dp: float, rhop: float, rho: float, mu: float
) -> float

Calculate the terminal velocity of an isolated sphere in laminar or turbulent flow.

In both laminar and turbulent flow, the terminal velocity of an isolated sphere is given by:

\[ v_t = \sqrt{\frac{4 D_p g (\rho_p - \rho)}{3 C_d \rho}} \]

where \(C_d(Re_p)\) is the drag coefficient. This implementation uses the drag correlation proposed by Turton and Levenspiel.

Tip

In laminar flow, \(v_t \propto D_p^2\), while in turbulent flow, \(v_t \propto D_p^{1/2}\).

References

• Turton, R., and O. Levenspiel. "A short note on the drag correlation for spheres", Powder technology 47.1 (1986): 83-86.

PARAMETER	DESCRIPTION
Dp	Particle diameter [m]. TYPE: float
rhop	Particle density [kg/m³]. TYPE: float
rho	Fluid density [kg/m³]. TYPE: float
mu	Fluid viscosity [Pa·s]. TYPE: float

RETURNS	DESCRIPTION
float	Terminal velocity [m/s].

See also

• vt_Stokes: specific method for laminar flow.
• Cd_sphere: related method to estimate the drag coefficient.

Examples:

Calculate the terminal velocity of a 1 mm styrene droplet in air.

>>> from polykin.flow import vt_sphere
>>> vt = vt_sphere(1e-3, 910., 1.2, 1.6e-5)
>>> print(f"vt = {vt:.1f} m/s")
vt = 3.8 m/s

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

def vt_sphere(
    Dp: float,
    rhop: float,
    rho: float,
    mu: float
) -> float:
    r"""Calculate the terminal velocity of an isolated sphere in laminar or
    turbulent flow.

    In both laminar and turbulent flow, the terminal velocity of an isolated
    sphere is given by:

    $$ v_t = \sqrt{\frac{4 D_p g (\rho_p - \rho)}{3 C_d \rho}} $$

    where $C_d(Re_p)$ is the drag coefficient. This implementation uses the drag
    correlation proposed by Turton and Levenspiel.

    !!! tip

        In laminar flow, $v_t \propto D_p^2$, while in turbulent flow, 
        $v_t \propto D_p^{1/2}$.

    **References**

    * Turton, R., and O. Levenspiel. "A short note on the drag correlation for
      spheres", Powder technology 47.1 (1986): 83-86.

    Parameters
    ----------
    Dp : float
        Particle diameter [m].
    rhop : float
        Particle density [kg/m³].
    rho : float
        Fluid density [kg/m³].
    mu : float
        Fluid viscosity [Pa·s].

    Returns
    -------
    float
        Terminal velocity [m/s].

    See also
    --------
    * [`vt_Stokes`](vt_Stokes.md): specific method for laminar flow.
    * [`Cd_sphere`](Cd_sphere.md): related method to estimate the drag
      coefficient.

    Examples
    --------
    Calculate the terminal velocity of a 1 mm styrene droplet in air.
    >>> from polykin.flow import vt_sphere
    >>> vt = vt_sphere(1e-3, 910., 1.2, 1.6e-5)
    >>> print(f"vt = {vt:.1f} m/s")
    vt = 3.8 m/s
    """

    def fnc(vt):
        Rep = rho*vt*Dp/mu
        Cd = Cd_sphere(Rep)
        return vt - sqrt(4*Dp*g*(rhop - rho)/(3*Cd*rho))

    sol = root_newton(fnc, x0=1.0)

    if sol.success:
        vt = sol.x
    else:
        raise RootSolverError("vt_sphere: could not converge to a solution.")

    return vt