Prandtl number

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The Prandtl number is a dimensionless number approximating the ratio of momentum diffusivity (kinematic viscosity) and thermal diffusivity. It is named after the German physicist Ludwig Prandtl.

It is defined as:

$\mathit{Pr} = \frac{\nu}{\alpha} = \frac{\mbox{viscous diffusion rate}}{\mbox{thermal diffusion rate}} = \frac{c_p \mu}{k}$

where:

$ν$ : kinematic viscosity, $ν = μ / ρ$ , (SI units : m²/s)
$α$ : thermal diffusivity, $α = k / (ρ c p)$ , (SI units : m²/s)
$μ$ : viscosity, (SI units : Pa s)
k : thermal conductivity, (SI units : W/(m K) )
c_p : specific heat, (SI units : J/(kg K) )
$ρ$ : density, (SI units : kg/m³ )

Typical values for Pr are:

around 0.7-0.8 for air and many other gases,
around 0.16-0.7 for mixtures of noble gases or noble gases with hydrogen
around 7 for water
around 7×10²¹ for Earth's mantle
between 100 and 40,000 for engine oil,
between 4 and 5 for R-12 refrigerant
around 0.015 for mercury

For mercury, heat conduction is very effective compared to convection: thermal diffusivity is dominant. For engine oil, convection is very effective in transferring energy from an area, compared to pure conduction: 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 very 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.

[edit] See also

[edit] References

Viscous Fluid Flow, F. M. White, McGraw-Hill, 3rd. Ed, 2006
Effects of Prandtl number and a new instability mode in a plane thermal plume. R. Lakkaraju, M Alam. Journal of Fluid Mechanics, vol. 592, 221-231 (2007)