Prandtl number

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.^[1] That is, the Prandtl number is given as:

\mathrm {Pr} ={\frac {\nu }{\alpha }}={\frac {\mbox{viscous diffusion rate}}{\mbox{thermal diffusion rate}}}={\frac {\mu /\rho }{k/c_{p}\rho }}={\frac {c_{p}\mu }{k}}

where:

$\nu$ : momentum diffusivity (kinematic viscosity), $\nu =\mu /\rho$ , (SI units: m²/s)
$\alpha$ : thermal diffusivity, $\alpha =k/(\rho c_{p})$ , (SI units: m²/s)
$\mu$ : dynamic viscosity, (SI units: Pa s = N s/m²)
$k$ : thermal conductivity, (SI units: W/m-K)
$c_{p}$ : specific heat, (SI units: J/kg-K)
$\rho$ : density, (SI units: kg/m³).

Note that whereas the Reynolds number and Grashof number are subscripted with a length scale variable, the Prandtl number contains no such length scale in its definition and is dependent only on the fluid and the fluid state. As such, the Prandtl number is often found in property tables alongside other properties such as viscosity and thermal conductivity.

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:

0.003 for molten potassium at 975 K^[1]
around 0.015 for mercury
around 0.16-0.7 for mixtures of noble gases or noble gases with hydrogen
0.63 for oxygen^[1]
0.065 for molten lithium at 975 K^[1]
around 0.7-0.8 for air and many other gases
1.38 for gaseous ammonia^[1]
between 4 and 5 for R-12 refrigerant
around 7 for water (At 20 °C)
13.4 and 7.2 for seawater (At 0 °C and 20 °C respectively)
50 for n-butanol^[1]
between 100 and 40,000 for engine oil
1000 for glycerol^[1]
10,000 for polymer melts^[1]
around 1×10²⁵ for Earth's mantle.

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, for engine oil, convection is very effective in transferring energy from an area in comparison to pure conduction, so momentum diffusivity is dominant.

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). This means that for liquid metals the thickness of the thermal boundary layer is much bigger than the velocity boundary layer.

The mass transfer analog of the Prandtl number is the Schmidt number.

References

1 2 3 4 5 6 7 8 9 Coulson, J. M.; Richardson, J. F. (1999). Chemical Engineering Volume 1 (6th ed.). Elsevier. ISBN 978-0-7506-4444-0.

White, F. M. (2006). Viscous Fluid Flow (3rd. ed.). New York: McGraw-Hill. ISBN 0-07-240231-8.

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Prandtl number

See also

References