NU_RA_PROBERT

This function calculates the Nusselt number for free convection between idealized horizontal plates using the Probert correlation. The model uses the Rayleigh number \mathrm{Ra} = \mathrm{Pr}\,\mathrm{Gr} and applies different power-law expressions by regime.

In the laminar range, heat transfer follows a quarter-power scaling, while higher Rayleigh values use a one-third-power turbulent-style relation. If buoyancy is not assisting or the Rayleigh number is below onset, the correlation returns the conduction limit.

\mathrm{Nu} = \begin{cases} 0.208\,\mathrm{Ra}^{0.25}, & 1708 < \mathrm{Ra} \le 2.2\times 10^4 \\ 0.092\,\mathrm{Ra}^{1/3}, & \mathrm{Ra} > 2.2\times 10^4 \end{cases}

Excel Usage

=NU_RA_PROBERT(Pr, Gr, buoyancy)
  • 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) (-).

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

Example 1: Probert correlation at high Rayleigh

Inputs:

Pr Gr buoyancy
5.54 321000000 true

Excel formula:

=NU_RA_PROBERT(5.54, 321000000, TRUE)

Expected output:

111.462

Example 2: Probert correlation in laminar regime

Inputs:

Pr Gr buoyancy
0.7 20000 true

Excel formula:

=NU_RA_PROBERT(0.7, 20000, TRUE)

Expected output:

2.26254

Example 3: Probert correlation in turbulent regime

Inputs:

Pr Gr buoyancy
0.7 10000000 true

Excel formula:

=NU_RA_PROBERT(0.7, 10000000, TRUE)

Expected output:

17.599

Example 4: Probert correlation with buoyancy opposed

Inputs:

Pr Gr buoyancy
0.9 1000000 false

Excel formula:

=NU_RA_PROBERT(0.9, 1000000, FALSE)

Expected output:

1

Python Code

Show Code 
from ht.conv_free_enclosed import Nu_Nusselt_Rayleigh_Probert as ht_Nu_Nusselt_Rayleigh_Probert

def Nu_Ra_Probert(Pr, Gr, buoyancy=True):
    """
    Calculate the Nusselt number between infinite horizontal plates using the Probert 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.

    Returns:
        float: Nusselt number between plates, or error message if invalid.
    """
    try:
        return ht_Nu_Nusselt_Rayleigh_Probert(Pr=Pr, Gr=Gr, buoyancy=buoyancy)
    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) (-).