polykin.transport.flow

vt_sphere

vt_sphere(
    D: 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:

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

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

Tip

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

References

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

PARAMETER	DESCRIPTION
D	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.transport 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/transport/flow.py

def vt_sphere(D: 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:

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

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

    !!! tip

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

    **References**

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

    Parameters
    ----------
    D : 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.transport 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):
        Re = rho*vt*D/mu
        Cd = Cd_sphere(Re)
        return vt - sqrt(4*D*g*(rhop - rho)/(3*Cd*rho))

    sol = fzero_newton(fnc, x0=1.)

    return sol.x