Prandtl Number Calculator — Pr = μ × cp / k
The Prandtl number (Pr) is a dimensionless ratio of momentum diffusivity to thermal diffusivity — a key parameter in convective heat transfer. Calculate it from fluid properties μ, cp, k, or from ν and α, and see which fluid class your result falls into.
Calculation mode
Pa·s
J/(kg·K)
W/(m·K)
Ratio of momentum diffusivity to thermal diffusivity
- 1
Numerator: μ × cp
0.000018 × 1,005 = 0.018191Product of dynamic viscosity and specific heat capacity. - 2
Pr = (μ × cp) ÷ k
0.018191 ÷ 0.0257 = 0.7078
How does this calculator work?
Pr = μ × cp / k (or ν / α). For air (20 °C): Pr ≈ 0.71; water (20 °C): Pr ≈ 7; engine oil: Pr ≈ 100–10,000; liquid mercury: Pr ≈ 0.02. Pr < 1 means heat diffuses faster than momentum; Pr > 1 means momentum diffuses faster.
Formula
How this is calculated
The Prandtl number is defined as Pr = ν / α, where ν = μ/ρ is the kinematic (momentum) diffusivity and α = k/(ρ × cp) is the thermal diffusivity. Substituting: Pr = (μ/ρ) / (k/(ρ × cp)) = μ × cp / k. The density ρ cancels, so Pr can be computed from just three readily tabulated properties: dynamic viscosity μ, isobaric specific heat cp, and thermal conductivity k.
Pr tells you which physical process diffuses faster: momentum (viscosity) or heat (thermal conduction). When Pr < 1, heat diffuses faster than momentum — characteristic of liquid metals such as mercury (Pr ≈ 0.02) and sodium (Pr ≈ 0.006), where very high thermal conductivity drives rapid heat spreading. When Pr ≈ 0.7, diffusivities are comparable — typical of gases (air at 20 °C: Pr ≈ 0.71). When Pr ≫ 1, momentum diffuses faster than heat — as in water (Pr ≈ 7 at 20 °C) or engine oil (Pr ≈ 100–10,000).
In engineering, Pr appears in correlations for forced and natural convection: the Nusselt number Nu is typically expressed as a function of Reynolds and Prandtl numbers for forced convection (e.g. Nu = C × Re^a × Pr^b), or Grashof and Prandtl numbers for natural convection. Prandtl number values are strongly temperature-dependent — liquid water, for example, drops from Pr ≈ 13 at 0 °C to Pr ≈ 1.7 at 100 °C as viscosity decreases with temperature.
Frequently asked questions
It means momentum diffuses about 7 times faster than heat in water. The thermal boundary layer grows more slowly than the velocity boundary layer. In practice, water is an efficient coolant not because its Pr is extreme, but because it combines moderately high Pr with high volumetric heat capacity — the product of density and cp.
Liquid metals have very high thermal conductivity (k is large) because conduction electrons carry heat efficiently — the same mechanism that makes metals electrically conductive. This large k drives α = k/(ρ × cp) up and Pr = ν/α down to values like 0.003–0.03, far below gases.
It appears in empirical Nusselt number correlations. For turbulent pipe flow, the Dittus-Boelter equation is Nu = 0.023 × Re^0.8 × Pr^n (n = 0.4 for heating, 0.3 for cooling). For natural convection, correlations use the Rayleigh number Ra = Gr × Pr. Without Pr, these correlations cannot correctly account for the fluid's heat transfer characteristics.
Also known as
TG we-Calculate Editorial Team. (2026). Prandtl Number Calculator — Pr = μ × cp / k [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/prandtl-number-calculator
TG we-Calculate Editorial Team. "Prandtl Number Calculator — Pr = μ × cp / k." TG we-Calculate. 2026. https://we-calculate.com/calculator/prandtl-number-calculator.
TG we-Calculate Editorial Team, "Prandtl Number Calculator — Pr = μ × cp / k," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/prandtl-number-calculator
@misc{wecalculate_prandtl_number_calculator, title = {Prandtl Number Calculator — Pr = μ × cp / k}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/prandtl-number-calculator}}, year = {2026}, note = {TG we-Calculate} }
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