Prandtl number

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

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It is defined as:

where:

• ν is the kinematic viscosity, ν = μ / ρ [Pa s m³ kg^-1].
• α is the thermal diffusivity, α = k / (ρ c_p) [m² S^-1].
• μ [Pa s] , k[J s^-1 m^-1 K^-1], ρ[kg m^-3], C_p[J kg^-1 K^-1]

Typical values for Pr are:

• around 0.7 for air and many other gases,
• 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 materials 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.

See also

• Turbulent Prandtl number

References

• Viscous Fluid Flow, F. M. White, McGraw-Hill, 3rd. Ed, 2006