polykin.transport.heat

Nu_sphere

Nu_sphere(Re: float, Pr: float, mur: float) -> float

Calculate the Nusselt number for an isolated sphere.

The average Nusselt number \(\overline{Nu}=\bar{h}D/k\) is estimated by the following expression:

\[ \overline{Nu} = 2 + ( 0.4 Re^{1/2} + 0.06 Re^{2/3} ) Pr^{0.4} \left(\frac{\mu}{\mu_s}\right)^{1/4} \]

\[\begin{bmatrix} 3.5 < Re < 7.6 \times 10^{4} \\ 0.71 < Pr < 380 \\ 1.0 < (\mu/\mu_s) 3.2 \end{bmatrix}\]

where \(Re\) is the sphere Reynolds number, \(Pr\) is the Prandtl number, \(\mu\) is the bulk viscosity, and \(\mu_s\) is the surface viscosity. All properties are to be evaluated at the bulk temperature, except \(\mu_s\).

References

• Whitaker, S. (1972), "Forced convection heat transfer correlations for flow in pipes, past flat plates, single cylinders, single spheres, and for flow in packed beds and tube bundles", AIChE J., 18: 361-371.
• Incropera, Frank P., and David P. De Witt. "Fundamentals of heat and mass transfer", 4th edition, 1996, p. 374.

PARAMETER	DESCRIPTION
Re	Sphere Reynolds number. TYPE: float
Pr	Prandtl number. TYPE: float
mur	Ratio of bulk viscosity to surface viscosity, \(\mu/\mu_s\). TYPE: float

RETURNS	DESCRIPTION
float	Nusselt number.

See also

• Nu_drop: specific method for drops.

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

def Nu_sphere(Re: float, Pr: float, mur: float) -> float:
    r"""Calculate the Nusselt number for an isolated sphere.

    The average Nusselt number $\overline{Nu}=\bar{h}D/k$ is estimated by the
    following expression:

    $$ \overline{Nu} = 2 + ( 0.4 Re^{1/2} + 0.06 Re^{2/3} ) Pr^{0.4}
                           \left(\frac{\mu}{\mu_s}\right)^{1/4} $$

    \begin{bmatrix}
    3.5 < Re < 7.6 \times 10^{4} \\
    0.71 < Pr < 380 \\
    1.0 < (\mu/\mu_s) 3.2
    \end{bmatrix}

    where $Re$ is the sphere Reynolds number, $Pr$ is the Prandtl number, 
    $\mu$ is the bulk viscosity, and $\mu_s$ is the surface viscosity. All 
    properties are to be evaluated at the bulk temperature, except $\mu_s$.

    **References**

    * Whitaker, S. (1972), "Forced convection heat transfer correlations for
      flow in pipes, past flat plates, single cylinders, single spheres, and
      for flow in packed beds and tube bundles", AIChE J., 18: 361-371.
    * Incropera, Frank P., and David P. De Witt. "Fundamentals of heat and
      mass transfer", 4th edition, 1996, p. 374.

    Parameters
    ----------
    Re : float
        Sphere Reynolds number.
    Pr : float
        Prandtl number.
    mur : float
        Ratio of bulk viscosity to surface viscosity, $\mu/\mu_s$.

    Returns
    -------
    float
        Nusselt number.

    See also
    --------
    * [`Nu_drop`](Nu_drop.md): specific method for drops.
    """
    check_range_warn(Re, 3.5, 7.6e4, 'Re')
    check_range_warn(Pr, 0.71, 380, 'Pr')
    check_range_warn(mur, 1.0, 3.2, 'mur')

    return 2 + (0.4*Re**(1/2) + 0.06*Re**(2/3))*Pr**0.4*(mur)**(1/4)

Graphical Illustration