Prandtl Number Calculator

Pr = μc_p/k — ratio of momentum diffusivity to thermal diffusivity. A pure fluid property that governs the relative thickness of velocity and thermal boundary layers.

Heat Transfer · Dimensionless Number

Inputs

Solve for:

Fluid preset (fills all properties):

Dynamic viscosity μ (Pa·s)

Specific heat capacity cp (J/(kg·K))

Thermal conductivity k (W/(m·K))

Also compute ν (kinematic viscosity) and α (thermal diffusivity) — enter density ρ

Theory

Definition (three equivalent forms):

Pr = μ × c_p / k [from measurable fluid properties]

Pr = ν / α [kinematic viscosity / thermal diffusivity]

Pr = δ_v / δ_T [velocity BL thickness / thermal BL thickness] (Pr^(1/3) scaling)

Physical meaning:

Pr tells you which boundary layer is thicker. For Pr = 1, velocity and thermal boundary layers grow at the same rate. For Pr > 1 (oils), the velocity boundary layer is thicker — momentum diffuses faster than heat. For Pr < 1 (liquid metals), heat conducts faster than momentum diffuses.

Typical Pr values:

Fluid	Pr	Regime
Liquid metals (Na, Hg)	0.003 – 0.03	Thermal diffusion dominates
Air (20 °C)	0.71	Gases — Pr ≈ 0.7
Water (20 °C)	7.0	Moderate convection
Water (60 °C)	3.0	Moderate convection
Ethylene glycol	~ 150	Viscous liquid
Engine oil (20 °C)	~ 2500	Momentum diffusion dominates

Prandtl number in heat transfer correlations:

Dittus-Boelter (turbulent pipe): Nu = 0.023 Re⁰·⁸ Prⁿ — valid 0.6 < Pr < 160
Flat plate laminar: Nu = 0.664 Re^½ Pr^⅓ — valid Pr > 0.6
Churchill-Chu (natural convection): Nu depends on Ra = Gr × Pr
At Pr → 0 (liquid metals): separate correlations needed (e.g. Seban-Shimazaki)

Pr vs Biot (Bi) vs Nusselt (Nu):

Pr = μ c_p / k pure fluid property — no geometry, no flow

Nu = h L / k_f convection intensity — depends on flow and geometry

Bi = h L / k_s solid internal resistance — uses solid conductivity