Prandtl Number Calculator

The Prandtl number (Pr) is a dimensionless number that compares how fast momentum diffuses through a fluid to how fast heat diffuses through it: Pr = μ·c_p/k, where μ is dynamic viscosity, c_p specific heat and k thermal conductivity. Worked example (Mode 1): water at 20 °C has μ = 0.001002 Pa·s, c_p = 4182 J/(kg·K), k = 0.598 W/(m·K), giving Pr = 0.001002 × 4182 / 0.598 = 7.01.

The defining formula is Pr = μ·c_p/k. This calculator solves it in all four directions: Pr = μ·c_p/k, μ = Pr·k/c_p, c_p = Pr·k/μ, k = μ·c_p/Pr. Inverse example (Mode 1): for Pr = 7.01, c_p = 4182 J/(kg·K), k = 0.598 W/(m·K), viscosity μ = 7.01 × 0.598 / 4182 = 0.0010024 Pa·s — recovering the water value above.

Air at 20 °C: 0.705. Water at 20 °C: 7.01 (at 100 °C it falls to 1.75). Mercury: 0.0246 and liquid sodium: 0.0098 — both far below 1. Engine oil at 20 °C: about 5631; glycerin at 20 °C: about 12,672 — far above 1. Gases cluster near 0.7–1, ordinary liquids near 1–10, and oils reach thousands.

Pr << 1 (liquid metals: mercury 0.0246, sodium 0.0098) means heat diffuses far faster than momentum, so the thermal boundary layer is much thicker than the velocity boundary layer. For mercury the laminar estimate δ/δ_t ≈ Pr^(1/3) gives 0.29, i.e. the thermal layer is about 3.44× thicker than the velocity layer.

Pr >> 1 (engine oil 5631, glycerin 12,672) means momentum diffuses far faster than heat, so the thermal boundary layer is very thin compared with the velocity boundary layer. For engine oil at 20 °C, δ/δ_t ≈ 5631^(1/3) = 17.79: the velocity layer is roughly 18× thicker than the thermal layer, and heat transfer is dominated by steep wall temperature gradients.

Yes — strongly, mostly through viscosity. Water drops from Pr = 7.01 at 20 °C to Pr = 1.75 at 100 °C because its viscosity falls ~3.5×. Engine oil is even more dramatic: Pr ≈ 5631 at 20 °C vs Pr ≈ 274 at 100 °C. Always evaluate Pr at the actual fluid temperature — Mode 2 of this calculator interpolates the fluid table for you.

Pr appears in almost every forced-convection correlation. The most famous is Dittus–Boelter for turbulent pipe flow: Nu = 0.023·Re^0.8·Pr^n (n = 0.4 heating, 0.3 cooling). Boundary with the sibling tool: the Nusselt Number Calculator implements those correlations and takes Pr as an input — this tool is the one that computes Pr itself.

Yes: the equivalent form is Pr = ν/α, with ν = μ/ρ (momentum diffusivity) and α = k/(ρ·c_p) (thermal diffusivity). Example — water at 20 °C: ν = 1.004×10^-6 m²/s, α = 1.433×10^-7 m²/s, so Pr = 1.004×10^-6 / 1.433×10^-7 = 7.01, identical to μ·c_p/k. Use whichever form matches the data you have.

Any consistent set works — Pr is dimensionless, so the units cancel. In SI: μ in Pa·s, c_p in J/(kg·K), k in W/(m·K). Imperial combos work too, e.g. μ in lb/(ft·s), c_p in BTU/(lb·°F), k in BTU/(hr·ft·°F). Mode 1 converts each property to SI before dividing, so you can mix units freely. The classic pitfall is entering viscosity in cP (1 cP = 1 mPa·s = 0.001 Pa·s) while the formula expects Pa·s — off by 1000×.

All three are dimensionless, but each answers a different question. The Reynolds number Re describes the flow regime (laminar vs turbulent) from velocity and size. The Prandtl number Pr describes the fluid's own heat-vs-momentum diffusivity, independent of the flow. The Nusselt number Nu is the result — how much convection enhances heat transfer — and correlations predict it from the other two: Nu = f(Re, Pr). This tool computes Pr; the sibling Nusselt Number Calculator computes Nu from Re and Pr.