Prandtl Number Formula
The Prandtl number relates momentum diffusivity to thermal diffusivity.
It determines whether heat transfer or momentum transfer dominates in a fluid.
The Formula
The Prandtl number (Pr) is a dimensionless parameter that characterizes the relative thickness of the velocity boundary layer to the thermal boundary layer. Named after German physicist Ludwig Prandtl (1875–1953), it appears in virtually every convective heat transfer correlation. It depends only on fluid properties, not on flow conditions.
Variables
| Symbol | Meaning | Unit |
|---|---|---|
| Pr | Prandtl number | dimensionless |
| μ | Dynamic viscosity | Pa·s |
| Cp | Specific heat at constant pressure | J/(kg·K) |
| k | Thermal conductivity | W/(m·K) |
| ν = μ/ρ | Kinematic viscosity (momentum diffusivity) | m²/s |
| α = k/(ρCp) | Thermal diffusivity | m²/s |
Prandtl number ranges and physical meaning:
- Pr << 1 (liquid metals, e.g. mercury Pr ≈ 0.025): heat diffuses much faster than momentum. Thermal boundary layer is much thicker than velocity boundary layer.
- Pr ≈ 1 (gases, e.g. air Pr ≈ 0.71): heat and momentum diffuse at similar rates.
- Pr >> 1 (viscous oils, e.g. engine oil Pr ≈ 1000): momentum diffuses much faster than heat. Velocity boundary layer is much thicker than thermal boundary layer.
Example 1 — Air at 25°C
Calculate the Prandtl number for air at 25°C: μ = 1.849 × 10−5 Pa·s, Cp = 1006 J/(kg·K), k = 0.02551 W/(m·K).
Pr = μCp/k = (1.849 × 10−5 × 1006) / 0.02551
= 1.860 × 10−2 / 0.02551
Pr ≈ 0.729 — air has nearly equal thermal and momentum diffusivity, making it "easy" for heat transfer calculations.
Example 2 — In Heat Transfer Correlations
The Dittus-Boelter equation for forced convection in a pipe: Nu = 0.023 Re^0.8 Pr^n (n=0.4 for heating, 0.3 for cooling). For Re=10,000, Pr=7 (water): find Nu.
Nu = 0.023 × (10000)^0.8 × (7)^0.4
= 0.023 × 1585 × 2.179
Nu ≈ 79.3 — Nusselt number for water flow. Higher Pr means significantly better heat transfer than air (Pr≈0.7) at the same Reynolds number.
When to Use It
The Prandtl number is used when:
- Computing the Nusselt number for convective heat transfer correlations
- Selecting cooling fluids for heat exchangers — water (Pr≈7) is much better than air (Pr≈0.7)
- Designing liquid metal cooling for nuclear reactors (very low Pr)
- Analyzing natural convection in air and other fluids
- Comparing thermal performance of different fluids at the same flow rate