polykin.transport.hmt

Nu_cylinder

Nu_cylinder(Re: float, Pr: float) -> float

Calculate the Nusselt number for cross flow over a circular cylinder.

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

\[ \overline{Nu} = 0.3 + \frac{0.62 Re^{1/2} Pr^{1/3}} {\left[1 + (0.4/Pr)^{2/3}\right]^{1/4}} \left[1 + \left(\frac{Re}{282 \times 10^3}\right)^{5/8}\right]^{4/5} \]

\[ \left[ Re Pr > 0.2 \right] \]

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

References

• Churchill, S. W., and Bernstein, M. "A Correlating Equation for Forced Convection From Gases and Liquids to a Circular Cylinder in Crossflow", ASME. J. Heat Transfer. May 1977; 99(2): 300.
• Incropera, Frank P., and David P. De Witt. "Fundamentals of heat and mass transfer", 4th edition, 1996, p. 370.

PARAMETER	DESCRIPTION
Re	Reynolds number based on cylinder diameter. TYPE: float
Pr	Prandtl number. TYPE: float

RETURNS	DESCRIPTION
float	Nusselt number.

See also

• Nu_cylinder_bank: specific method for a bank of tubes.
• Nu_cylinder_free: related method for free convection.

Examples:

Estimate the external heat transfer coefficient for water flowing at 2 m/s across a DN25 pipe.

>>> from polykin.transport import Nu_cylinder
>>> rho = 1e3   # kg/m³
>>> mu = 1e-3   # Pa.s
>>> cp = 4.2e3  # J/kg/K
>>> k = 0.6     # W/m/K
>>> v = 2.      # m/s
>>> D = 33.7e-3 # m (OD from pipe chart)
>>> Re = rho*v*D/mu
>>> Pr = cp*mu/k
>>> Nu = Nu_cylinder(Re, Pr)
>>> h = Nu*k/D
>>> print(f"h={h:.1e} W/m².K")
h=7.0e+03 W/m².K

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

def Nu_cylinder(Re: float, Pr: float) -> float:
    r"""Calculate the Nusselt number for cross flow over a circular cylinder.

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

    $$ \overline{Nu} = 0.3 + \frac{0.62 Re^{1/2} Pr^{1/3}}
    {\left[1 + (0.4/Pr)^{2/3}\right]^{1/4}} 
    \left[1 + \left(\frac{Re}{282 \times 10^3}\right)^{5/8}\right]^{4/5} $$

    $$ \left[  Re Pr > 0.2 \right] $$

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

    **References**

    * Churchill, S. W., and Bernstein, M. "A Correlating Equation for Forced
      Convection From Gases and Liquids to a Circular Cylinder in Crossflow",
      ASME. J. Heat Transfer. May 1977; 99(2): 300.
    * Incropera, Frank P., and David P. De Witt. "Fundamentals of heat and
      mass transfer", 4th edition, 1996, p. 370.

    Parameters
    ----------
    Re : float
        Reynolds number based on cylinder diameter.
    Pr : float
        Prandtl number.

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

    See also
    --------
    - [`Nu_cylinder_bank`](Nu_cylinder_bank.md): specific method for a bank of
      tubes.
    - [`Nu_cylinder_free`](Nu_cylinder_free.md): related method for free
      convection.

    Examples
    --------
    Estimate the external heat transfer coefficient for water flowing at 2 m/s
    across a DN25 pipe.
    >>> from polykin.transport import Nu_cylinder
    >>> rho = 1e3   # kg/m³
    >>> mu = 1e-3   # Pa.s
    >>> cp = 4.2e3  # J/kg/K
    >>> k = 0.6     # W/m/K
    >>> v = 2.      # m/s
    >>> D = 33.7e-3 # m (OD from pipe chart)
    >>> Re = rho*v*D/mu
    >>> Pr = cp*mu/k
    >>> Nu = Nu_cylinder(Re, Pr)
    >>> h = Nu*k/D
    >>> print(f"h={h:.1e} W/m².K")
    h=7.0e+03 W/m².K
    """
    check_range_warn(Re*Pr, 0.2, inf, 'Re*Pr')

    return 0.3 + 0.62*Re**(1/2)*Pr**(1/3)/(1 + (0.4/Pr)**(2/3))**(1/4) \
        * (1 + (Re/282e3)**(5/8))**(4/5)

Graphical Illustration