Prandtl Number Calculator

The Prandtl number, Pr, is a dimensionless quantity that compares momentum diffusivity (kinematic viscosity) to thermal diffusivity in a fluid. It tells you whether heat or momentum spreads faster through the fluid, and it is central to every convective heat transfer correlation.

Pr = cp·μ/k, where cp is specific heat capacity in J/(kg·K), μ is dynamic viscosity in Pa·s, and k is thermal conductivity in W/(m·K). All three properties are evaluated at the same fluid temperature.

A low Prandtl number (Pr ≪ 1), typical of liquid metals like mercury or liquid sodium, means heat diffuses much faster than momentum through the fluid. Thermal boundary layers are much thicker than velocity boundary layers in these fluids.

A high Prandtl number (Pr ≫ 1), typical of oils and other viscous fluids, means momentum diffuses much faster than heat. Velocity boundary layers are much thicker than thermal boundary layers, and heat transfer is comparatively sluggish.

Air's Prandtl number of roughly 0.71 reflects that, for most common gases, momentum and heat diffuse at comparable but not identical rates. This value is remarkably stable across a wide temperature range for air and most diatomic gases.

Unused engine oil has a Prandtl number in the thousands (roughly 10,000 at room temperature) because it is highly viscous (high μ) yet a poor thermal conductor (low k), so cp·μ/k becomes very large. This is why oil cools much more slowly by convection than water at the same flow conditions.

The Dittus-Boelter correlation Nu = 0.023·Re^0.8·Pr^n uses the Prandtl number alongside Reynolds number to estimate the Nusselt number for turbulent flow in a pipe. Once Pr is known for a fluid at its operating temperature, it can be raised to the exponent n, 0.4 for heating or 0.3 for cooling, and combined with Re^0.8 to solve directly for the convective heat transfer coefficient.

Yes. All three properties in Pr = cp·μ/k, specific heat, viscosity, and thermal conductivity, change with temperature, so Pr must be evaluated at the fluid's actual operating temperature (often the average of surface and bulk temperature) for accurate results.

Water at 20°C has a Prandtl number of about 7.0, meaning momentum diffuses about seven times faster than heat. This value drops significantly as water gets hotter, down to around 1.75 near boiling, because viscosity falls sharply with temperature.

For many flows, the ratio of the velocity boundary layer thickness to the thermal boundary layer thickness scales approximately as Pr^(1/3). A high-Pr fluid like oil has a thermal boundary layer much thinner than its velocity boundary layer, while a low-Pr fluid like liquid metal has the opposite.

Yes, for extremely viscous fluids such as glycerin, silicone oils, or molten glass, Pr can reach into the tens of thousands or higher. These fluids transfer heat almost entirely by conduction near a wall because momentum diffuses so much faster than heat.