Skip to content

polykin.transport.hmt¤

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.transport 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/transport/hmt.py
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
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.transport 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)