Prandtl Number Calculator - Pr = nu / alpha

A prandtl number calculator computes the Prandtl number Pr = nu / alpha, the textbook dimensionless group that compares momentum diffusivity to thermal diffusivity in a flowing fluid, so a student or engineer can read the heat-transfer behaviour of a fluid from its transport properties.

• • Heat-transfer homework and exam problems: Compute Pr from dynamic viscosity, specific heat, thermal conductivity, and density when a textbook gives mu, c_p, k, and rho for air, water, oil, or mercury.
• • Thermal boundary layer analysis: Read Pr alongside the Reynolds number to estimate the relative thickness of the velocity and thermal boundary layers in forced-convection flow.
• • Convective heat-transfer correlations: Plug Pr into the Dittus-Boelter or Sieder-Tate correlations when sizing a heat exchanger or a coolant loop.
• • Comparing fluids at a glance: Switch between air, water (20 C and 100 C), engine oil, glycerin, and mercury presets to see how Pr spans about 0.025 for mercury up to a few thousand for heavy oils.

The result panel never carries a unit because Pr is dimensionless, and the same form doubles as a thermal diffusivity calculator because alpha = k / (rho c_p) is computed and surfaced next to Pr.

When the heat-transfer problem also needs the inertial-to-viscous force ratio that drives the boundary-layer thickness, the Reynolds Number Calculator computes Re from the same fluid-property set this calculator already takes.

The prandtl number calculator applies Pr = nu / alpha to the fluid properties the user enters, with automatic conversion of the four-property form Pr = mu * c_p / k and an optional two-diffusivity path when both diffusivities are supplied.

• mu: Dynamic viscosity in pascal-seconds (Pa.s). Air at 20 C: 1.81e-5 Pa.s. Water at 20 C: 1.002e-3 Pa.s.
• c_p: Specific heat at constant pressure in J/(kg.K). Air: 1005. Water at 20 C: 4182.
• k: Thermal conductivity in W/(m.K). Air at 20 C: 0.0257. Water at 20 C: 0.598.
• rho: Density in kg/m^3. Air at 20 C: 1.204. Water at 20 C: 998.2.
• nu: Kinematic viscosity in m^2/s; equals mu / rho.
• alpha: Thermal diffusivity in m^2/s; equals k / (rho * c_p).

When the user supplies only mu, c_p, k, and rho, the calculator first forms nu = mu / rho and alpha = k / (rho c_p) and then divides to get Pr. When both nu and alpha are typed in directly, the calculator uses Pr = nu / alpha on the values as given.

According to Wikipedia - Prandtl Number, the Prandtl number is defined as Pr = nu / alpha = mu * c_p / k and is dimensionless, with air at 20 C near 0.71 and liquid metals well below 0.1.

According to the Engineering Toolbox air properties reference, dry air at 20 C has dynamic viscosity 1.81e-5 Pa.s, specific heat 1005 J/(kg.K), thermal conductivity 0.0257 W/(m.K), and density 1.204 kg/m^3, giving Pr ∼ 0.71 from mu * c_p / k.

When the steady-state conduction side of the same problem needs a thermal resistance or a heat rate, the Heat Transfer Conduction Calculator takes the thermal conductivity k this calculator already reads and returns the conductive heat flow for a chosen geometry.

Four ideas make the result panel easier to read: the dimensionless-group definition, the connection between nu and alpha, the boundary-layer interpretation, and the low-versus-high classification.

Because the result is dimensionless, the calculator compares a room-temperature gas with a room-temperature liquid on the same axis - mercury at Pr ∼0.025 sits about 300x lower than water at Pr ∼7.

When the dynamic viscosity mu needs to be re-expressed in poise, centipoise, or another cgs unit before it is fed into Pr, the Poise Stokes Converter handles the unit conversion without losing precision.

Pick a fluid preset, confirm the four transport properties, and read Pr from the result panel.

1. 1 Choose a fluid preset: Select air, water (20 C or 100 C), engine oil, glycerin, or mercury from the preset list. The four transport properties fill in automatically and can be edited on the spot.
2. 2 Or stay on Custom: Choose Custom fluid to type mu, c_p, k, and rho directly when the fluid of interest is not in the preset list.
3. 3 Confirm or edit the transport properties: Inspect mu in Pa.s, c_p in J/(kg.K), k in W/(m.K), and rho in kg/m^3. Edit any value when a tabulated source supersedes the preset.
4. 4 Optionally supply nu and alpha directly: Type kinematic viscosity nu and thermal diffusivity alpha in m^2/s when those values are already tabulated. The calculator then uses Pr = nu / alpha directly.
5. 5 Read the Prandtl number: Read the dimensionless Prandtl number in the result panel, alongside the computed nu and alpha in m^2/s and the regime label.
6. 6 Pair with Reynolds number: Use Pr together with the Reynolds number from a flow problem to size the thermal boundary layer or evaluate a Nu = f(Re, Pr) correlation.

Pick Water (20 C) from the preset list. The calculator fills mu = 1.002e-3, c_p = 4182, k = 0.598, rho = 998.2 and returns Pr ∼ 7.01, which is the textbook value for water at room temperature and the input you would feed into the Dittus-Boelter correlation.

When the rho value on the form needs to match a non-standard temperature, pressure, or humidity, the Air Density Calculator computes the same density from the ideal-gas or moist-air branch.

The prandtl number calculator gives a single dimensionless result from either the four-property form or the two-diffusivity form.

• • Two input paths, one result: Accepts either the four-property form (mu, c_p, k, rho) or the two-diffusivity form (nu, alpha) so the user picks whichever inputs are already on hand.
• • Preset fluid library: Covers air, water (20 C and 100 C), engine oil, glycerin, and mercury - the same set transport-phenomena textbooks use to anchor the Prandtl-number scale.
• • Side-by-side nu and alpha readouts: Surfaces kinematic viscosity and thermal diffusivity in m^2/s next to Pr, doubling the form as a thermal diffusivity calculator for unsteady conduction.
• • Regime label at a glance: Tags the result as low (liquid metals), moderate (gases), or high (oils and water) so a non-expert can read the boundary-layer behaviour without a fluid mechanics textbook.
• • Traceable inputs: Shows mu, c_p, k, and rho in SI base units so each Pr can be re-derived from the same numbers if the result needs to be checked.

When the broader fluid-mechanics problem also needs the pressure, velocity, and elevation head along a streamline, the Bernoulli Equation Calculator reads the same fluid properties this calculator surfaces and returns the energy balance along the flow path.

Three transport properties drive Pr in the four-property form Pr = mu * c_p / k, and the density rho hides in both diffusivities but cancels in the ratio.

• • The four-property form assumes the four transport properties are evaluated at the same temperature and pressure. Fluids with strong temperature dependence (oils, liquid metals) need a film temperature.
• • The dimensionless-group definition is exact for Newtonian fluids only. Non-Newtonian fluids need a generalized Prandtl number using an apparent viscosity, which the four-property form here does not cover.

The calculator rejects zero or negative values for mu, c_p, k, and rho. When both nu and alpha are supplied, they are used directly; otherwise the calculator falls back to the four-property form.

According to Engineering Toolbox - Air Prandtl Number, the Prandtl number Pr is a dimensionless number approximating the ratio of momentum diffusivity (kinematic viscosity) to thermal diffusivity, and is often used in heat transfer and free and forced convection calculations.

When the gas preset on this form needs a different temperature or pressure to back out a new rho, c_p, or k, the Ideal Gas Calculator computes the new molar density and mole fraction from P, V, n, and T.