Prandtl Number Calculator: Pr from Viscosity, Heat Capacity and Conductivity
Enter a fluid's dynamic viscosity, specific heat capacity, and thermal conductivity to calculate its Prandtl number instantly. Pick a fluid preset to auto-fill typical values, or enter your own. The "show your work" panel walks through every step of the calculation, a gauge shows where your fluid sits on the momentum-to-thermal diffusivity spectrum, and a reference table lists Pr values for a dozen common fluids.
Formula
Worked example
For water at 20 °C: μ = 1.002 × 10⁻³ Pa·s, Cₚ = 4182 J/(kg·K), k = 0.598 W/(m·K). Pr = (1.002 × 10⁻³ × 4182) / 0.598 = 4.190 / 0.598 ≈ 7.01.
What is the Prandtl number?
The Prandtl number (Pr) is a dimensionless quantity in fluid mechanics and heat transfer named after the German physicist Ludwig Prandtl. It expresses the ratio of momentum diffusivity (kinematic viscosity, nu) to thermal diffusivity (alpha): Pr = nu / alpha = mu * Cp / k. When Pr is much less than 1, heat spreads through the fluid far faster than momentum, a situation found in liquid metals. When Pr equals roughly 0.7, momentum and thermal diffusion are nearly equal, which is the case for most gases. When Pr is much greater than 1, as in oils, viscous momentum diffusion dominates and heat transfer becomes relatively inefficient. The number determines the relative thickness of the velocity and thermal boundary layers, which is fundamental to predicting heat transfer coefficients in forced and natural convection.
The formula and its two forms
The primary form is Pr = (mu × Cp) / k, where mu is dynamic viscosity in Pa·s (or kg/(m·s)), Cp is the isobaric specific heat capacity in J/(kg·K), and k is the thermal conductivity in W/(m·K). An equivalent form uses the kinematic quantities: Pr = nu / alpha, where nu = mu / rho is the kinematic viscosity (m²/s) and alpha = k / (rho × Cp) is the thermal diffusivity (m²/s). Both forms give the same result. The primary form is more practical in most tabulated data because viscosity, heat capacity, and conductivity are all commonly measured and listed for fluids. All three properties are temperature-dependent, so Pr varies with temperature, especially strongly for liquids like water and oils.
Prandtl number ranges and their physical meaning
Liquid metals (mercury, sodium, lead-bismuth alloys) have Pr between about 0.003 and 0.03. Their high thermal conductivity means the thermal boundary layer is far thicker than the velocity boundary layer, which is why liquid metals are used as coolants in high-flux applications such as fast-breeder nuclear reactors. Gases (air, nitrogen, carbon dioxide, steam) cluster between 0.65 and 0.9. For air the famous value is about 0.71, close enough to 1 that the Reynolds analogy between heat and momentum transfer is often applied. Water at 20 °C has Pr near 7; at 0 °C it rises to about 13.6 and at 100 °C it drops to about 1.76, so water-cooled systems must account for this temperature dependence. Transformer and engine oils reach Pr of several thousand at room temperature, meaning their thermal boundary layers are extremely thin and wall temperature corrections are critical in design. This calculator accepts any combination of mu, Cp, and k, so it works for any fluid including exotic process fluids.
Reverse solving and engineering applications
In experimental or inverse heat transfer problems you may know the Prandtl number and two of the three primary properties, and need to back-calculate the third. Use the "Solve for" selector to switch to any of the three reverse modes. For example, if you have measured the Prandtl number of an unknown fluid and its specific heat and thermal conductivity, the calculator will return its dynamic viscosity directly. In design work, the Prandtl number feeds directly into Nusselt number correlations for forced convection in pipes (Dittus-Boelter, Gnielinski), external flow over flat plates (Pohlhausen), natural convection (Churchill-Chu), and two-phase flow regimes. A higher Pr generally means a lower convection heat transfer coefficient for the same geometry and flow rate, which is why engineers prefer water over oil as a cooling medium when both are feasible.
Prandtl numbers for common fluids at approximately 20 °C
| Fluid | Pr (approx.) | Regime | Typical application |
|---|---|---|---|
| Mercury | 0.025 | Liquid metal | Thermometers, nuclear reactor coolant |
| Sodium (liquid) | 0.011 | Liquid metal | Fast-breeder reactor coolant |
| Hydrogen (gas) | 0.68 | Gas | Fuel cells, cryogenic cooling |
| Air | 0.71 | Gas | HVAC, turbines, aerospace |
| Carbon dioxide | 0.75 | Gas | Supercritical cooling, fire suppression |
| Water | 6.9 | Low-Pr liquid | Heat exchangers, boilers, cooling towers |
| Seawater | 7.2 | Low-Pr liquid | Ocean thermal energy, desalination |
| Ethanol | 16 | Moderate liquid | Cooling loops, chemical processing |
| Ethylene glycol | 151 | High-Pr liquid | Engine antifreeze, coolant loops |
| Engine oil (20) | ~10,400 | Very high-Pr | Lubrication, hydraulic systems |
| Glycerol | ~11,600 | Very high-Pr | Pharmaceuticals, food processing |
Values sourced from Incropera's "Fundamentals of Heat and Mass Transfer", 7th edition, Appendix A.
Frequently asked questions
What does a Prandtl number greater than 1 mean?
Pr greater than 1 means the fluid transfers momentum more readily than it transfers heat. In boundary layer terms, the velocity boundary layer is thicker than the thermal boundary layer. Most liquids fall in this category. The higher the Pr, the more viscous forces dominate thermal conduction, and the harder it becomes to transfer heat from a hot surface to the bulk fluid. Engine oils with Pr in the thousands are much harder to cool through than water with Pr around 7.
Why do gases have Prandtl numbers close to 0.7?
For most simple gases, kinetic theory predicts Pr = 4 gamma / (9 gamma - 5), where gamma is the heat capacity ratio. For diatomic gases like air (gamma ≈ 1.4), this gives about 0.74, very close to the measured value. This proximity of Pr to 1 for gases is why the Reynolds analogy - which assumes Pr = 1 - is a useful approximation in gas-phase convection problems.
What is the Prandtl number of water?
Water's Prandtl number is strongly temperature-dependent: about 13.6 at 0 °C, 6.9 at 20 °C, 4.3 at 40 °C, 2.99 at 60 °C, and 1.76 at 100 °C. The drop occurs because viscosity decreases much faster with temperature than thermal conductivity. For any heat exchanger design involving water, you should use the Pr value at your actual operating temperature rather than a generic room-temperature value.
How is the Prandtl number used in Nusselt number correlations?
In forced convection correlations, the Prandtl number appears as a power-law factor. For turbulent pipe flow the Dittus-Boelter equation is Nu = 0.023 Re^0.8 Pr^n, where n = 0.4 for heating and 0.3 for cooling. The more accurate Gnielinski correlation uses Nu = (f/8)(Re - 1000)Pr / [1 + 12.7 sqrt(f/8) (Pr^(2/3) - 1)]. Both are valid for 0.6 < Pr < 160. For very high or very low Pr, specialized correlations must be used.
What is the Prandtl number of air at room temperature?
Air at 20 °C and standard pressure has a Prandtl number of approximately 0.713. It is nearly constant across a wide temperature range: 0.714 at 0 °C, 0.707 at 100 °C, and 0.683 at 500 °C. This near-constancy makes air-side calculations relatively simple compared to liquid-side calculations where Pr changes strongly with temperature.
Can the Prandtl number be less than 1?
Yes. Liquid metals have very low Prandtl numbers because their thermal conductivity is exceptionally high. Mercury has Pr around 0.025, liquid sodium around 0.011, and lead-bismuth eutectic around 0.015. In these fluids, thermal energy diffuses far faster than momentum, and conventional Nusselt number correlations (derived assuming Pr greater than or equal to about 0.6) do not apply. Liquid-metal specific correlations must be used instead.