NU_RA_HOLLANDS

This function computes the natural-convection Nusselt number between horizontal plates using the Hollands correlation for enclosed layers. It uses the Rayleigh number formed from Grashof and Prandtl numbers, with a critical Rayleigh threshold to account for finite-geometry effects.

With \mathrm{Ra} = \mathrm{Pr}\,\mathrm{Gr}, the model applies a buoyancy-dependent relation that transitions from conduction-dominated behavior to convection-enhanced heat transfer. For stable configurations or subcritical Rayleigh number, the result returns to the conduction limit.

\mathrm{Nu} = 1 \quad \text{for } \mathrm{Ra} < \mathrm{Ra}_c \text{ or when buoyancy is not assisting convection}

Excel Usage

=NU_RA_HOLLANDS(Pr, Gr, buoyancy, Rac)
  • Pr (float, required): Prandtl number with respect to fluid properties (-).
  • Gr (float, required): Grashof number with respect to fluid properties and plate temperature difference (-).
  • buoyancy (bool, optional, default: true): Whether convection is buoyancy assisted (hot plate) (-).
  • Rac (float, optional, default: 1708): Critical Rayleigh number (-).

Returns (float): Nusselt number between plates, or error message if invalid.

Example 1: Hollands correlation at high Rayleigh

Inputs:

Pr Gr buoyancy
5.54 321000000 true

Excel formula:

=NU_RA_HOLLANDS(5.54, 321000000, TRUE)

Expected output:

69.0267

Example 2: Hollands correlation with custom critical Rayleigh

Inputs:

Pr Gr buoyancy Rac
0.7 3210000 true 2530.5

Excel formula:

=NU_RA_HOLLANDS(0.7, 3210000, TRUE, 2530.5)

Expected output:

8.81176

Example 3: Hollands correlation with buoyancy opposed

Inputs:

Pr Gr buoyancy
0.7 1000000 false

Excel formula:

=NU_RA_HOLLANDS(0.7, 1000000, FALSE)

Expected output:

1

Example 4: Hollands correlation with insulated critical Rayleigh

Inputs:

Pr Gr buoyancy Rac
7 10000000 true 2071.0089443385655

Excel formula:

=NU_RA_HOLLANDS(7, 10000000, TRUE, 2071.0089443385655)

Expected output:

26.082

Python Code

Show Code 
from ht.conv_free_enclosed import Nu_Nusselt_Rayleigh_Hollands as ht_Nu_Nusselt_Rayleigh_Hollands

def Nu_Ra_Hollands(Pr, Gr, buoyancy=True, Rac=1708):
    """
    Calculate the Nusselt number between horizontal plates using the Hollands correlation.

    See: https://ht.readthedocs.io/en/latest/ht.conv_free_enclosed.html

    This example function is provided as-is without any representation of accuracy.

    Args:
        Pr (float): Prandtl number with respect to fluid properties (-).
        Gr (float): Grashof number with respect to fluid properties and plate temperature difference (-).
        buoyancy (bool, optional): Whether convection is buoyancy assisted (hot plate) (-). Default is True.
        Rac (float, optional): Critical Rayleigh number (-). Default is 1708.

    Returns:
        float: Nusselt number between plates, or error message if invalid.
    """
    try:
        return ht_Nu_Nusselt_Rayleigh_Hollands(Pr=Pr, Gr=Gr, buoyancy=buoyancy, Rac=Rac)
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

Prandtl number with respect to fluid properties (-).
Grashof number with respect to fluid properties and plate temperature difference (-).
Whether convection is buoyancy assisted (hot plate) (-).
Critical Rayleigh number (-).