Table of Contents
What Is the Prandtl Number?
The Prandtl number (Pr) is a dimensionless number that represents the ratio of momentum diffusivity (kinematic viscosity) to thermal diffusivity. Named after the German physicist Ludwig Prandtl, it characterizes the relative thickness of the velocity and thermal boundary layers in a fluid flow. A fluid with a low Prandtl number conducts heat much faster than it transports momentum, meaning the thermal boundary layer is thicker than the velocity boundary layer.
When Pr is much less than 1, heat diffuses quickly compared to momentum. This is typical of liquid metals like mercury (Pr around 0.025) where thermal conduction dominates. When Pr is much greater than 1, momentum diffusivity dominates over thermal diffusivity, as seen in viscous oils (Pr can exceed 1000). For gases, the Prandtl number is typically close to 0.7, meaning their velocity and thermal boundary layers have similar thicknesses.
Formula & Derivation
Where μ is dynamic viscosity (Pa·s), Cp is specific heat capacity at constant pressure (J/kg·K), k is thermal conductivity (W/m·K), ν = μ/ρ is kinematic viscosity (m²/s), and α = k/(ρ·Cp) is thermal diffusivity (m²/s). The Prandtl number connects the velocity field to the temperature field in convection problems.
In the Nusselt number correlations for forced and natural convection, the Prandtl number appears as a key parameter. For example, the Dittus-Boelter equation for turbulent pipe flow is Nu = 0.023 Re0.8 Prn, where n depends on heating or cooling conditions.
Prandtl Numbers of Common Fluids
| Fluid | Temperature | Pr |
|---|---|---|
| Liquid Mercury | 20 °C | 0.025 |
| Liquid Sodium | 100 °C | 0.004 |
| Air | 20 °C | 0.71 |
| Water | 20 °C | 6.99 |
| Water | 100 °C | 1.75 |
| Engine Oil | 20 °C | ~2000 |
| Glycerin | 20 °C | ~12,500 |
Applications in Engineering
The Prandtl number is crucial for heat exchanger design, cooling system optimization, and thermal management of electronics. In nuclear reactor design, liquid metal coolants with very low Prandtl numbers require special correlations because the thermal boundary layer extends far beyond the velocity boundary layer. In polymer processing, very high Prandtl numbers mean viscous dissipation effects become important, and heat penetrates slowly into the thick, slow-moving fluid.
- Pr < 0.1: Liquid metals -- thermal boundary layer much thicker than velocity boundary layer
- Pr ≈ 0.7: Gases -- both boundary layers have similar thickness
- Pr ≈ 1–10: Water and light liquids -- velocity boundary layer slightly thicker
- Pr > 100: Heavy oils -- velocity boundary layer much thicker than thermal boundary layer
Frequently Asked Questions
What does a Prandtl number of 1 mean?
A Prandtl number of 1 means that momentum and thermal diffusivities are equal, so the velocity and thermal boundary layers grow at the same rate. The temperature and velocity profiles have the same shape (Reynolds analogy). In practice, many gases have Pr near 0.7, which is close enough that the Reynolds analogy is often applied with a correction factor.
Why is the Prandtl number important for CFD simulations?
In computational fluid dynamics, the Prandtl number determines the mesh resolution needed near walls. For high-Pr fluids, the thermal boundary layer is very thin and requires an extremely fine mesh to resolve the steep temperature gradients near the wall. For low-Pr fluids like liquid metals, standard wall functions developed for gases fail, and specialized turbulence models are required.
Does the Prandtl number depend on temperature?
Yes, because viscosity, specific heat, and thermal conductivity all vary with temperature. For water, Pr drops from about 13 at 0 degrees Celsius to about 1.75 at 100 degrees Celsius because viscosity decreases rapidly with temperature while thermal conductivity changes less dramatically.