The Prandtl number (Pr) or Prandtl group is a dimensionless number, named after the German physicist Ludwig Prandtl, defined as the ratio of momentum diffusivity to thermal diffusivity.
For most gases over a wide range of temperature and pressure, Pr is approximately constant.
Therefore, it can be used to determine the thermal conductivity of gases at high temperatures, where it is difficult to measure experimentally due to the formation of convection currents.
[1] Typical values for Pr are: For air with a pressure of 1 bar, the Prandtl numbers in the temperature range between −100 °C and +500 °C can be calculated using the formula given below.
The Prandtl numbers for water (1 bar) can be determined in the temperature range between 0 °C and 90 °C using the formula given below.
Small values of the Prandtl number, Pr ≪ 1, means the thermal diffusivity dominates.
Whereas with large values, Pr ≫ 1, the momentum diffusivity dominates the behavior.
For example, the listed value for liquid mercury indicates that the heat conduction is more significant compared to convection, so thermal diffusivity is dominant.
However, engine oil with its high viscosity and low heat conductivity, has a higher momentum diffusivity as compared to thermal diffusivity.
[3] The Prandtl numbers of gases are about 1, which indicates that both momentum and heat dissipate through the fluid at about the same rate.
Heat diffuses very quickly in liquid metals (Pr ≪ 1) and very slowly in oils (Pr ≫ 1) relative to momentum.
Consequently thermal boundary layer is much thicker for liquid metals and much thinner for oils relative to the velocity boundary layer.
In heat transfer problems, the Prandtl number controls the relative thickness of the momentum and thermal boundary layers.
When Pr is small, it means that the heat diffuses quickly compared to the velocity (momentum).
In laminar boundary layers, the ratio of the thermal to momentum boundary layer thickness over a flat plate is well approximated by[4] where
For incompressible flow over a flat plate, the two Nusselt number correlations are asymptotically correct:[4] where
These two asymptotic solutions can be blended together using the concept of the Norm (mathematics):[4]