What Is the Prandtl Number?
This Prandtl number calculator helps you determine this dimensionless number for different fluids. The Prandtl number (Pr) is a function of a fluid's viscosity and thermal diffusivity. This dimensionless entity relates momentum transport to heat transport. The value of the Prandtl number tells us how heat will diffuse relative to momentum — in other words, whether heat conduction is more dominant than convection for a given fluid. Prandtl number plays a vital role in heat transfer via fluids, a phenomenon used extensively in boundary-layer flows.
Consider a fluid — say water — flowing over a hot flat plate. To describe the rate at which energy or heat is transferred by conduction or convection, we use the Prandtl number. Put simply, the Prandtl number is the ratio of momentum diffusivity to thermal diffusivity:
Pr = momentum diffusivity / thermal diffusivity = ν / α
- Momentum diffusivity (ν): the diffusion of momentum between particles in a fluid, also known as the kinematic viscosity. It equals the dynamic viscosity μ divided by the fluid density ρ: ν = μ / ρ. Its units are area per time — m²/s or ft²/s.
- Thermal diffusivity (α): how fast heat travels from one location to another. It depends on the thermal conductivity k, the specific heat at constant pressure cp, and the density ρ: α = k / (ρ · cp). Its units are also area per time — m²/s or ft²/s.
How to Calculate the Prandtl Number
Because the densities in ν and α cancel out, the Prandtl number can be written purely in terms of the fluid's transport properties — which is exactly what this calculator uses:
Pr = ν / α = (μ / ρ) / (k / (ρ · cp)) = μ · cp / k
where:
- Pr — Prandtl number (dimensionless)
- μ — Dynamic (absolute) viscosity — Pa·s (metric) or lb/(ft·s) (American)
- cp — Specific heat at constant pressure — J/(kg·K) or BTU/(lb·°F)
- k — Thermal conductivity — W/(m·K) or BTU/(hr·ft·°F)
The Prandtl number is dimensionless: when consistent units are used, all physical units cancel, leaving a pure number. This calculator supports both metric (SI) and American (imperial) unit systems and converts your inputs automatically.
Example: Using the Prandtl Number Calculator
Let's find the Prandtl number of water at 20 °C (68 °F), using its standard transport properties:
- μ = 0.001002 Pa·s
- cp = 4182 J/(kg·K)
- k = 0.598 W/(m·K)
Pr = (0.001002 × 4182) / 0.598 ≈ 7.0
A Prandtl number of about 7 means momentum diffuses roughly seven times faster than heat in water, so the velocity boundary layer is thicker than the thermal boundary layer.
Prandtl Number of Different Fluids
The Prandtl number spans an enormous range — from roughly 0.004 for liquid metals to tens of thousands for heavy oils. Typical values include:
| Fluid | Prandtl number (Pr) |
|---|---|
| Liquid sodium (~100 °C) | 0.011 |
| Mercury (20 °C) | 0.025 |
| Air (20 °C) | 0.71 |
| Carbon dioxide (gas) | 0.77 |
| Water (100 °C) | 1.75 |
| Water (20 °C) | 7.0 |
| Engine oil (20 °C) | ~10 400 |
| Glycerin (20 °C) | ~12 500 |
The Prandtl number of air is approximately 0.71 across a wide range of temperatures because, for gases, both the viscosity and thermal conductivity scale similarly with temperature.
Physical Significance of the Prandtl Number
The Prandtl number compares how quickly momentum and heat diffuse through a fluid, which governs the relative thickness of the velocity and thermal boundary layers:
- Pr < 1 — heat diffuses faster than momentum. The thermal boundary layer is thicker than the velocity boundary layer. Typical of gases and especially liquid metals, where conduction is very effective.
- Pr ≈ 1 — momentum and heat diffuse at comparable rates; the two boundary layers grow together. Most gases (e.g., air) sit close to this value.
- Pr > 1 — momentum diffuses faster than heat. The velocity boundary layer is thicker than the thermal one. Typical of water and oils.
A useful rule of thumb is that the ratio of the velocity to thermal boundary-layer thickness scales as δ / δt ≈ Pr1/3. This is why the Prandtl number appears in countless convective heat-transfer correlations (such as the Nusselt-number relations) used to design heat exchangers, cooling systems, and aerodynamic surfaces.
Frequently Asked Questions
What is the Prandtl number?
The Prandtl number is a dimensionless number equal to the ratio of momentum diffusivity (kinematic viscosity) to thermal diffusivity, Pr = ν / α = μ·cp/k. It indicates whether momentum or heat diffuses faster in a fluid.
How do you calculate the Prandtl number of air?
Use air's properties at the temperature of interest. At 20 °C, μ ≈ 1.825 × 10⁻⁵ Pa·s, cp ≈ 1005 J/(kg·K), and k ≈ 0.0257 W/(m·K), giving Pr = (1.825 × 10⁻⁵ × 1005) / 0.0257 ≈ 0.71.
Is the Prandtl number dimensionless?
Yes. As long as consistent units are used (all SI or all imperial), the units cancel and the Prandtl number is a pure number. This calculator handles both metric (SI) and American (imperial) unit systems.
Why is the Prandtl number of liquid metals so small?
Liquid metals such as mercury and sodium have very high thermal conductivity, so heat diffuses much faster than momentum. This gives them a Prandtl number well below 1 — which is why they are excellent coolants in applications like nuclear reactors.
What does a high Prandtl number mean?
A high Prandtl number (Pr ≫ 1), typical of oils, means momentum diffuses far faster than heat. The thermal boundary layer is much thinner than the velocity boundary layer, so heat transfer is concentrated very close to the surface.