Prandtl Number Calculator
Compute Prandtl number Pr = μCp/k from fluid properties.
Regime classifier for liquid metals, gases, liquids, and oils, with reference values.
The Prandtl number is a fluid property, not a flow property. It depends only on what the fluid is, not on how fast it is moving or what shape it is moving through. That makes it different from the Reynolds number (which depends on flow speed and geometry) and the Nusselt number (which depends on both). Pr tells you, for a given fluid, whether momentum or heat diffuses faster.
The formula:
Pr = μ·Cp / k = ν / α
Where μ is dynamic viscosity (Pa·s), Cp is specific heat at constant pressure (J/kg·K), k is thermal conductivity (W/m·K), ν is kinematic viscosity (m²/s), and α is thermal diffusivity (m²/s). The two forms are algebraically identical; the second form, the ratio of momentum diffusivity to thermal diffusivity, is the physically most illuminating way to read it.
What Pr means physically:
Imagine a hot plate immersed in a moving fluid. The fluid develops two boundary layers near the plate. The velocity boundary layer is where viscosity slows the fluid down. The thermal boundary layer is where the fluid heats up. These two layers are usually different thicknesses, and the ratio of their thicknesses is roughly √Pr.
- Pr ≈ 1: the layers are about the same thickness. Gases (air, nitrogen) fall here. Momentum and heat diffuse at the same rate.
- Pr » 1: thermal boundary layer is much thinner than the velocity layer. Heat doesn’t penetrate far. Oils and viscous fluids. Heat transfer is dominated by what happens very close to the wall.
- Pr « 1: thermal boundary layer is much thicker than the velocity layer. Heat penetrates well beyond the slow-moving fluid near the wall. Liquid metals (sodium, mercury, lead in nuclear coolant loops).
Named after Ludwig Prandtl:
Prandtl introduced boundary-layer theory in 1904 in a famous 10-minute conference talk at the International Congress of Mathematicians in Heidelberg. The theory was so radical that audience members reportedly thought it could not be right. By the 1940s, every aerospace and heat-transfer engineer was using it; Prandtl is now considered the founder of modern fluid mechanics. The dimensionless number bearing his name appeared in his student Wilhelm Nusselt’s 1915 work, and the two have been paired in heat-transfer correlations ever since.
Reference values at room temperature (20-25 °C):
| Fluid | Pr | Note |
|---|---|---|
| Mercury (liquid) | 0.025 | Liquid metal; thermal diffusivity dominates |
| Sodium (liquid, hot) | 0.005 | Nuclear reactor coolant |
| Hydrogen gas | 0.69 | Light gas, Pr near 1 |
| Air | 0.71 | Most engineering examples |
| Helium | 0.70 | Similar to air |
| Steam (superheated) | 1.0 | Slightly higher than air |
| Water (20 °C) | 7.0 | Standard reference |
| Water (60 °C) | 3.0 | Pr drops with temperature |
| Water (100 °C) | 1.75 | At boiling, much closer to air |
| Glycerin | 12,500 | Highly viscous, slow heat transfer |
| Engine oil (cold) | 10,000-50,000 | Why engines warm up slowly |
| SAE 50 oil (20 °C) | 6,400 | Lubrication oils |
| Liquid sodium (200 °C) | 0.007 | Sodium-cooled fast reactors |
Notice how dramatically Pr changes with temperature for liquids. Water at 100 °C has Pr ≈ 1.75; at 20 °C, Pr ≈ 7. This is why heat-exchanger design uses Pr at the mean fluid temperature, not at one fixed reference.
Why Pr matters for design:
In Dittus-Boelter (Nu = 0.023·Re^0.8·Pr^n), Pr appears with exponent 0.3 (cooling) or 0.4 (heating). Doubling Pr increases Nu by roughly 25%, holding Re fixed. So switching from air-cooling (Pr ≈ 0.7) to water-cooling (Pr ≈ 7) at the same flow rate doesn’t multiply heat transfer by 10× — it multiplies by about 2.5×. But the higher k of water (0.6 vs 0.026) also raises h directly. The combined effect: water-cooling delivers ~30× the heat transfer of air-cooling at similar Re. This is why CPU water-cooling exists.
Worked example, water at 25 °C:
μ = 8.9 × 10⁻⁴ Pa·s, Cp = 4180 J/kg·K, k = 0.61 W/m·K.
Pr = (8.9 × 10⁻⁴ × 4180) / 0.61 = 3.72 / 0.61 = 6.10
Close to the textbook value of ~7 for water at 20 °C. Small differences come from exact temperature, dissolved gases, and which property tables you trust.
Worked example, engine oil at 20 °C:
μ ≈ 0.8 Pa·s (huge!), Cp ≈ 1900 J/kg·K, k ≈ 0.145 W/m·K.
Pr = (0.8 × 1900) / 0.145 = 1520 / 0.145 = 10,483
The high Pr means a very thin thermal boundary layer relative to the velocity layer — heat sits stubbornly at the wall, the fluid only an inch away barely notices. This is exactly why cold engine oil takes so long to warm up; it doesn’t conduct heat into its bulk efficiently.
Pr and the Schmidt number:
The mass-transfer analog of Pr is the Schmidt number Sc = ν / D, where D is molecular diffusivity. The Lewis number is Le = α / D = Sc / Pr. These three appear together in coupled heat-and-mass transfer problems (drying, evaporative cooling, fuel combustion).
Independence of Pr from flow:
The most important property of Pr is what it is NOT a function of: velocity, length scale, density, gravity. Get the temperature right, look up the fluid in a property table, and Pr is set. This is what makes Pr a useful catalog parameter — you can tabulate it for every common fluid at every common temperature, and that table is universal across all flow problems.