Prandtlzahlen

Prandtlzahlen, or Prandtl numbers, are dimensionless quantities used in fluid dynamics and heat transfer to characterize the relative thickness of momentum and thermal boundary layers. Named after Ludwig Prandtl, the Prandtl number denotes the ratio of momentum diffusivity to thermal diffusivity in a fluid.

The Prandtl number Pr is defined as Pr = ν/α, where ν is the kinematic viscosity (dynamic viscosity divided by density) and α is the thermal diffusivity (thermal conductivity divided by density and specific heat capacity). Equivalently, Pr can be written as Pr = μ c_p / k, relating viscosity, specific heat, and thermal conductivity. The Prandtl number thus links momentum transport to heat transport in the fluid.

Interpretation of Pr is qualitative: small Pr indicates that thermal diffusion dominates over momentum diffusion, so thermal disturbances spread quickly and the thermal boundary layer is relatively thick. Large Pr means momentum diffusion is more effective than heat diffusion, leading to thicker velocity boundary layers compared with thermal ones. In practice, Pr values span wide ranges: air around 0.7, water around 7 at room temperature, many oils can exceed 100, and liquid metals can be well below 0.1. The exact value depends on temperature and chemical composition.

Prandtl numbers are central to convective heat-transfer correlations and boundary-layer theory. They are used in forced and natural convection to relate Nusselt, Reynolds, and Prandtl numbers in empirical and semi-empirical formulas. Related dimensionless groups include the Schmidt number for mass transfer (Sc = ν/D). In non-Newtonian or multi-constituent fluids, the definition may be generalized accordingly.