The Prandtl number (Pr) is a dimensionless number, named for the German fluid dynamicist Ludwig Prandtl. It is defined as the ratio of momentum diffusivity to thermal diffusivity. The Prandtl number is a ratio of physical properties that may be arranged in several ways

::<math> \mathrm{Pr} = \frac{\nu}{\alpha} = \frac{\mbox{momentum diffusivity{\mbox{thermal diffusivity = \frac{\mu / \rho}{k / (c_p \rho)} = \frac{c_p \mu}{k} </math>

where the symbols are as follow.

  • <math>\nu</math> : kinematic viscosity (momentum diffusivity), <math>\nu = \mu/\rho</math> [m<sup>2</sup>/s]
  • <math>\alpha</math> : thermal diffusivity, <math>\alpha = k/(\rho c_p)</math> [m<sup>2</sup>/s]
  • <math>\mu</math> : dynamic viscosity [Pa s = N s/m<sup>2</sup>]
  • <math>k</math> : thermal conductivity [W/(m·K)]
  • <math>c_p</math> : specific heat [J/(kg·K)]
  • <math>\rho</math> : density [kg/m<sup>3</sup>].

The Prandtl number is a property of the fluid. Unlike the Reynolds number and Grashof number, it does not change with length scale or other conditions of the flow field.

The mass transfer analog of the Prandtl number is the Schmidt number. The ratio of the Schmidt number to the Prandtl number is called the Lewis number.

Experimental values

The Prandtl number is often given in fluid property tables alongside other properties such as viscosity and thermal conductivity.

For most gases, Pr is approximately constant over a wide range of temperature and pressure. The temperature is to be used in the unit degree Celsius. The deviations are a maximum of 0.1% from the literature values.

::<math>\mathrm{Pr}_\text{air} = \frac{10^9}{1.1 \cdot \vartheta^3-1200 \cdot \vartheta^2 + 322000 \cdot \vartheta + 1.393 \cdot 10^9}</math>,

where <math>\vartheta </math> is the temperature in Celsius.

The Prandtl numbers for water (1 bar) can be determined in the temperature range between 0&nbsp;°C and 90&nbsp;°C using the formula given below. The Prandtl numbers of gases are about 1, which indicates that momentum and heat diffuse at about the same rate.

In a laminar boundary layer on a flat plate, the ratio of the thermal to momentum boundary layer thickness is well approximated by