Prandtl Number Calculator — Momentum vs Thermal Diffusivity
Calculate the Prandtl number from viscosity, specific heat, and thermal conductivity. Fluid classification for gases, water, oils, and liquid metals included.
Prandtl Number
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The Formula
The Prandtl number is the ratio of momentum diffusivity (kinematic viscosity ν) to thermal diffusivity (α). It describes the relative thickness of the velocity and thermal boundary layers. Pr < 1: heat diffuses faster than momentum (liquid metals). Pr ≈ 0.7: typical for gases. Pr > 1: momentum diffuses faster (water, oils).
Variable Definitions
Dynamic Viscosity
Resistance to flow (Pa·s or kg/m·s). Higher μ means thicker, more resistant fluid. Water: 0.001, Air: 1.8×10⁻⁵, Honey: ~10 Pa·s.
Specific Heat Capacity
Energy required to raise 1 kg of fluid by 1 K (J/kg·K). Water has an unusually high cp (4184), making it an excellent coolant.
Thermal Conductivity
Rate of heat conduction through the fluid (W/m·K). Metals conduct heat well (high k); gases and insulators do not (low k).
How to Use This Calculator
- 1
Enter the dynamic viscosity (μ) in Pa·s.
- 2
Enter the specific heat capacity (cp) in J/kg·K.
- 3
Enter the thermal conductivity (k) in W/m·K.
- 4
The Prandtl number tells you whether heat or momentum diffuses faster.
Quick Reference
| From | To |
|---|---|
| Pr ≈ 0.004–0.03 | Liquid metals (sodium, mercury, NaK) |
| Pr ≈ 0.7 | Air and most gases |
| Pr ≈ 1–2 | Water at moderate temps |
| Pr ≈ 7 | Water at 20°C |
| Pr ≈ 50–500 | Light engine oils |
| Pr ≈ 1,000–10,000 | Heavy oils, glycerol |
| Pr > 100,000 | Silicone oils, polymers |
Common Applications
- Heat exchanger design — Pr appears in virtually every forced-convection Nusselt number correlation (Nu = C·Re^m·Pr^n).
- Boundary layer analysis — Pr tells you whether the thermal boundary layer is thicker (Pr < 1) or thinner (Pr > 1) than the velocity boundary layer.
- CFD simulations — Pr is a required fluid property input for conjugate heat transfer models in ANSYS Fluent, OpenFOAM, and COMSOL.
- Coolant selection — high Pr fluids (water) maintain thin thermal boundary layers; low Pr fluids (liquid metals) diffuse heat rapidly.
- Aerodynamic heating — Pr ≈ 0.7 for air determines the relationship between skin friction and heat transfer at supersonic speeds.
This prandtl number covers fluid dynamics and heat transfer. Use the worked examples to verify your understanding and bookmark for quick reference.
Pro Tips
Bookmark this calculator for quick reference — these calculations are frequently needed in engineering workflows.
Verify results against standard handbook values before applying to critical design decisions.
Use the worked examples to confirm your understanding of the underlying formulas.
Understanding the Concept
The Prandtl number (Pr), named after German physicist Ludwig Prandtl (1875–1953), is a dimensionless number that characterizes the relative effectiveness of momentum and energy transport in a fluid. Prandtl introduced the concept of the boundary layer in 1904, revolutionizing fluid dynamics. Pr < 1 means heat diffuses faster than momentum — the thermal boundary layer is thicker than the velocity boundary layer. This occurs in liquid metals (Pr ≈ 0.01), making them excellent coolants for nuclear reactors. Pr ≈ 0.7 for air and most diatomic gases — this near-unity value means the velocity and thermal boundary layers have similar thickness, which simplifies many heat transfer calculations. Pr > 1 means momentum diffuses faster — the velocity boundary layer extends beyond the thermal layer. Water at room temperature has Pr ≈ 7. Oils have Pr > 100, meaning the thermal boundary layer is extremely thin compared to the velocity layer. The Prandtl number is purely a fluid property — it depends only on the fluid state (temperature, pressure), not on flow geometry or velocity. This makes it one of the most useful dimensionless numbers: look up Pr for your fluid at the operating temperature, and it immediately tells you how heat and momentum will interact.
Worked Examples
You are designing a shell-and-tube heat exchanger using water at 60°C. μ = 0.000467 Pa·s, cp = 4184 J/kg·K, k = 0.654 W/m·K.
0.000467
4184
0.654
Result:
Insight: Pr = (0.000467 × 4184) / 0.654 = 2.99. At 60°C, water has Pr ≈ 3, which is lower than at room temperature (Pr ≈ 7). The thermal boundary layer is thicker at this temperature, which slightly reduces the convective heat transfer coefficient compared to cold-water flow. Use this Pr value in your Nusselt correlation for accurate heat exchanger sizing.
Air at 300°C flows over a turbine blade. μ = 2.93×10⁻⁵ Pa·s, cp = 1044 J/kg·K, k = 0.044 W/m·K.
0.0000293
1044
0.044
Result:
Insight: Pr = (2.93×10⁻⁵ × 1044) / 0.044 = 0.695. Even at 300°C, air Pr remains near 0.7. This remarkable constancy across a wide temperature range is characteristic of diatomic gases and simplifies gas turbine cooling design considerably.
Limitations
- This calculator computes Pr from bulk fluid properties. It does not account for temperature-dependent property variations within the thermal boundary layer. For fluids near their critical point, property variations can be extreme and this simple calculation may be insufficient. The category interpretation assumes standard engineering contexts at moderate temperatures and pressures.
Frequently Asked Questions
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