polykin.hmt.correlations

Nu_sphere_free

Nu_sphere_free(Ra: float, Pr: float) -> float

Calculate the Nusselt number for free convection on a sphere.

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

\[ \overline{Nu} = 2 + \frac{0.589 Ra^{1/4}}{[1 + (0.469/Pr)^{9/16}]^{4/9}} \]

\[\begin{bmatrix} Ra \lesssim 10^{11} \\ Pr \ge 0.7 \\ \end{bmatrix}\]

where \(Ra\) is the Rayleigh number and \(Pr\) is the Prandtl number. The properties are to be evaluated at the film temperature.

References

• Churchill, S.W, "Free convection around immersed bodies", in Heat Exchange Design Handbook, Section 2.5.7, Hemisphere Publishing, New York, 1983.
• Incropera, Frank P., and David P. De Witt. "Fundamentals of heat and mass transfer", 4th edition, 1996, p. 502.

PARAMETER	DESCRIPTION
Ra	Rayleigh number based on sphere diameter. TYPE: float
Pr	Prandtl number. TYPE: float

RETURNS	DESCRIPTION
float	Nusselt number.

See Also

• Nu_sphere: related method for forced convection.

Examples:

Estimate the external heat transfer coefficient for a 50 mm sphere with a surface temperature of 330 K immersed in water with a bulk temperature of 290 K.

>>> from polykin.hmt import Nu_sphere_free
>>> rho = 1.0e3   # kg/m³
>>> mu = 0.70e-3  # Pa.s
>>> cp = 4.2e3    # J/kg/K
>>> k = 0.63      # W/m/K
>>> beta = 362e-6 # 1/K
>>> D = 50e-3     # m
>>> g = 9.81      # m/s²
>>> Pr = cp*mu/k
>>> Gr = g*beta*(330-290)*D**3/(mu/rho)**2
>>> Ra = Gr*Pr
>>> Nu = Nu_sphere_free(Ra, Pr)
>>> h = Nu*k/D
>>> print(f"h={h:.1e} W/m²·K")
h=7.8e+02 W/m²·K

Source code in src/polykin/hmt/correlations.py

def Nu_sphere_free(Ra: float, Pr: float) -> float:
    r"""Calculate the Nusselt number for free convection on a sphere.

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

    $$ \overline{Nu} = 2 + \frac{0.589 Ra^{1/4}}{[1 + (0.469/Pr)^{9/16}]^{4/9}} $$

    \begin{bmatrix}
    Ra \lesssim 10^{11} \\
    Pr \ge  0.7 \\
    \end{bmatrix}

    where $Ra$ is the Rayleigh number and $Pr$ is the Prandtl number. The
    properties are to be evaluated at the film temperature.

    **References**

    * Churchill, S.W, "Free convection around immersed bodies", in Heat Exchange
      Design Handbook, Section 2.5.7, Hemisphere Publishing, New York, 1983.
    * Incropera, Frank P., and David P. De Witt. "Fundamentals of heat and
      mass transfer", 4th edition, 1996, p. 502.

    Parameters
    ----------
    Ra : float
        Rayleigh number based on sphere diameter.
    Pr : float
        Prandtl number.

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

    See Also
    --------
    - [`Nu_sphere`](Nu_sphere.md): related method for forced convection.

    Examples
    --------
    Estimate the external heat transfer coefficient for a 50 mm sphere with a
    surface temperature of 330 K immersed in water with a bulk temperature of
    290 K.
    >>> from polykin.hmt import Nu_sphere_free
    >>> rho = 1.0e3   # kg/m³
    >>> mu = 0.70e-3  # Pa.s
    >>> cp = 4.2e3    # J/kg/K
    >>> k = 0.63      # W/m/K
    >>> beta = 362e-6 # 1/K
    >>> D = 50e-3     # m
    >>> g = 9.81      # m/s²
    >>> Pr = cp*mu/k
    >>> Gr = g*beta*(330-290)*D**3/(mu/rho)**2
    >>> Ra = Gr*Pr
    >>> Nu = Nu_sphere_free(Ra, Pr)
    >>> h = Nu*k/D
    >>> print(f"h={h:.1e} W/m²·K")
    h=7.8e+02 W/m²·K
    """
    check_range_warn(Ra, 0, 1e11, "Ra")
    check_range_warn(Pr, 0.7, inf, "Pr")

    return 2 + 0.589 * Ra ** (1 / 4) / (1 + (0.469 / Pr) ** (9 / 16)) ** (4 / 9)