Prandtl Number Calculator
The Prandtl number is a dimensionless quantity used in fluid mechanics and heat transfer to characterize the ratio of momentum diffusivity to thermal diffusivity in a fluid. The formula Pr = cp mu / k uses the specific heat at constant pressure (cp), the dynamic viscosity (mu), and the thermal conductivity (k). A Prandtl number near 1 (such as air, Pr = 0.71) indicates that heat and momentum transfer at similar rates. Pr is a key parameter in convective heat transfer correlations used in engineering design of heat exchangers, cooling systems, and fluid flow over surfaces.
Prandtl number formula
Pr = cp × mu / k
cp is specific heat at constant pressure (J/(kg K)), mu is dynamic viscosity (Pa s = kg/(m s)), k is thermal conductivity (W/(m K)). Thermal diffusivity alpha = k / (rho cp), where rho is density. Pr = kinematic viscosity / thermal diffusivity = nu / alpha.
Prandtl number interpretation
- Pr much less than 1 (liquid metals, e.g. mercury Pr = 0.02): thermal diffusivity dominates; temperature equilibrates faster than velocity profile.
- Pr approximately 1 (gases such as air Pr = 0.71): velocity and thermal boundary layers are similar in thickness.
- Pr much greater than 1 (oils, Pr = 1,000+): momentum diffusivity dominates; thick velocity boundary layer, thin thermal boundary layer.
Frequently asked questions
What is the Prandtl number?
The Prandtl number (Pr) is a dimensionless number that characterizes the relative thickness of the velocity and thermal boundary layers in a fluid. Pr = cp mu / k, where cp is specific heat, mu is dynamic viscosity, and k is thermal conductivity.
What does the Prandtl number represent physically?
Pr = momentum diffusivity / thermal diffusivity. Pr less than 1 means heat diffuses faster than momentum (liquid metals). Pr greater than 1 means momentum diffuses faster than heat (most liquids and gases). Water at 20 degrees C has Pr approximately 7.
What are typical Prandtl numbers?
Air at 20 degrees C: Pr approximately 0.71. Water at 20 degrees C: Pr approximately 7.0. Engine oil: Pr approximately 1,000-100,000. Mercury: Pr approximately 0.023. Liquid sodium (coolant): Pr approximately 0.005.
How is the Prandtl number used?
Pr appears in heat transfer correlations such as the Dittus-Boelter equation for forced convection in pipes (Nu = 0.023 Re^0.8 Pr^0.4) and in natural convection. It determines whether the thermal or velocity boundary layer is thicker.
What units should I use?
Specific heat cp in J/(kg K), dynamic viscosity mu in Pa s (= kg/(m s)), and thermal conductivity k in W/(m K). The Prandtl number is dimensionless; units cancel: (J/(kg K)) x (kg/(m s)) / (W/(m K)) = (J/s) / W = 1.
Official sources
- NIST Chemistry WebBook: Thermophysical Properties of Fluid Systems.
- ASHRAE: ASHRAE Handbook of Fundamentals.
Reviewed by the CalculatorHub team, edited by James Graham, 15 June 2026. See our methodology.