What gets carried versus what spreads
Pe compares transport by the flow (advection) with spreading by molecular motion (diffusion). Over a length L with a temperature difference ΔT, advection carries roughly ρcpUΔT/L per unit volume and conduction roughly kΔT/L². Their ratio is UL/(k/ρcp) = UL/α, the coefficient that appears on the diffusion term of the dimensionless energy equation.
As times: the flow crosses L in L/U, diffusion needs about L²/α, and their ratio is Pe. At Pe = 100 the flow crosses L a hundred times before heat diffuses across it once. For 20 °C water at 0.1 m/s and L = 10 mm, the transit time is 0.1 s and the diffusion time about 697 s.
With the same U, L and properties, Peh = Re × Pr; with a species diffusivity D in place of α, Pem = Re × Sc. The calculator takes α as its third input, so it returns the thermal Pe unless you enter D.
Choosing the diffusivity and the length
v is a representative speed (m/s), L the comparison length (m) and α = k/(ρcp) (m²/s); vL and α share units, so Pe has none. In a pipe, the diameter suits diffusion across the section and the pipe length suits spreading along the axis. Pe scales with L, so always state which L you used. In a 20 mm pipe, taking 2 m of length as L makes Pe 100 times the diameter-based value; both are correct answers to different questions.
| Fluid and diffusing quantity | Diffusivity [m²/s] | Pr or Sc | Pe at vL = 1×10⁻³ m²/s |
|---|---|---|---|
| Water 20 °C, heat | α = 1.435 × 10⁻⁷ | Pr 6.981 | 6,968 |
| Water 60 °C, heat | α = 1.583 × 10⁻⁷ | Pr 3.005 | 6,318 |
| Air 20 °C, 1 atm, heat | α = 2.142 × 10⁻⁵ | Pr ≈ 0.71 | 46.7 |
| Dissolved oxygen in water 20 °C | D = 1.97 × 10⁻⁹ | Sc ≈ 509 | 507,614 |
Water α uses k, ρ and cp from IAEA Table 3.3 (saturated liquid, taken as low-pressure water). Air α uses k = 0.0258 W/(m·K) and cp = 1000 J/(kg·K) from IAEA Table 3.1 (98.1 kPa) with density scaled to 1 atm as an ideal gas, 1.2043 kg/m³. The oxygen value is Cornell's Wilke–Chang estimate (about ±10%). At equal vL, oxygen in water has over ten thousand times the Pe of heat in air.
Open this calculatorExamples: water pipe, dissolved oxygen, groundwater column
Example 1. Thermal Péclet number for 20 °C water in a pipe
Given: an inner diameter of 20 mm and a mean velocity of 0.1 m/s are assumed operating conditions. Properties are the 20 °C values from IAEA Table 3.3: ρ = 998.2 kg/m³, μ = 1.0000×10⁻³ Pa·s, k = 0.5995 W/(m·K), cp = 4185 J/(kg·K), Pr = 6.981.
Unit conversion: L = 20 mm = 0.02 m. α = 0.5995 ÷ (998.2 × 4185) = 1.435 × 10⁻⁷ m²/s.
Calculation: Pe = 0.1 × 0.02 ÷ 1.435 × 10⁻⁷ = 13,937. As a check, Re = 998.2 × 0.1 × 0.02 ÷ 0.001 = 1,996.4, and multiplying by Pr gives 1,996.4 × 6.981 = 13,937, the same value.
Interpretation: the flow is laminar, yet advection dominates heat transport. The common laminar thermal entry-length estimate 0.05 × Re × Pr × D gives 0.05 × 13,937 × 0.02 = 13.9 m, so in a tube a few metres long the fully developed Nu = 3.66 underestimates heat transfer.
Example 2. Mass Péclet number for dissolved oxygen in the same pipe
Given: the same pipe and assumed velocity as Example 1, with the oxygen diffusivity in 20 °C water D = 1.97 × 10⁻⁹ m²/s (Cornell, Wilke–Chang estimate).
Calculation: Pem = 0.1 × 0.02 ÷ 1.97 × 10⁻⁹ = 1.02 × 10⁶. Since Sc = ν/D = 1.0020×10⁻⁶ ÷ 1.97×10⁻⁹ ≈ 509, Re × Sc gives the same magnitude. The ratio to the thermal Pe is α/D = 72.8.
As a diffusion distance: 1 m takes 1 ÷ 0.1 = 10 s. Heat spreads about √(αt) = √(1.435×10⁻⁷ × 10) ≈ 1.20 mm, oxygen only √(Dt) ≈ 0.140 mm; both are small against the 10 mm radius, and the concentration layer stays much thinner.
Interpretation: the same flow gives Pe values about 73 times apart. Using the thermal Pe for mass transfer overstates diffusion. That is also why mass-transfer-limited devices, such as an oxygen-consuming biofilm or an electrode at the wall, need Sc in place of Pr in heat-transfer-style correlations.
Example 3. Groundwater sand column with a Darcy velocity
Given: column length L = 0.3 m, Darcy velocity u = 2×10⁻⁶ m/s, porosity φ = 0.4 and hydrodynamic dispersion coefficient Dh = 1×10⁻⁸ m²/s, all assumed for illustration. The definition follows Penn State CE 574: Pe = Lu/(φDh).
Calculation: the advection time is τa = φL/u = 0.4 × 0.3 ÷ 2×10⁻⁶ = 60,000 s (about 16.7 hours), and the dispersion time is τd = L²/Dh = 0.09 ÷ 1×10⁻⁸ = 9,000,000 s (about 104 days). Pe = τd/τa = 150.
Using the calculator: Darcy velocity is per total cross-section, so enter the pore velocity u/φ = 5×10⁻⁶ m/s as v and Dh as α to get 150. Entering u gives a Pe smaller by φ, here 60.
Interpretation: Penn State notes that larger Pe means stronger advection and sharper breakthrough fronts. At 150 the front clearly moves with the flow, yet dispersive spreading remains measurable.
What different Pe ranges tell you
- Pe ≪ 1: diffusion outruns the flow; the field is close to the pure conduction or diffusion solution.
- Pe around 1: neither advection nor diffusion can be dropped.
- Pe ≫ 1: diffusion survives only in thin layers at walls or fronts. MIT 3.185 derives δT/x = 3.6/√Pe for a uniform velocity, so at Pe = 10,000 δT is about 3.6% of x.
The same notes list Pe as the test of whether a continuous-flow reactor behaves as plug flow or as a well-mixed tank (CSTR): large axial Pe means plug flow, small means strong axial mixing. Cutoffs such as 1, 10 or 100 depend on geometry and boundary conditions and are not universal.
Where Péclet numbers are easy to misread
- Turbulence: eddy mixing can exceed molecular diffusion by orders of magnitude, so a molecular Pe says nothing about it; wall heat transfer still comes from Nu.
- Porous media: mixing up Darcy and pore velocity, or diffusivity and dispersion coefficient, shifts Pe by a few times to orders of magnitude.
- Local versus global: boundary layers use a local Pex from the leading edge; a whole device uses its length. Do not compare the two directly.
- Numerical grids: Penn State mentions L taken as the grid spacing. A large cell Pe can make central differencing oscillate, prompting upwinding or refinement; it measures discretisation quality, not physics.
Common questions about using the Péclet number
For pipe flow, should L be the diameter or the pipe length?
Use the diameter for how fast heat evens out across the radius, and the pipe length for whether axial conduction is negligible. Heat-exchanger correlations use diameter-based Re and Pr, so compare on that basis.
Re × Pr and vL/α give slightly different answers. Why?
They are equal by definition; the gap is property rounding. Example 1 gives 13,937 both ways, but air's tabulated Pr 0.71 recomputes from μcp/k as 0.705, nearly 1% off. Compute Pr from one consistent property set.
In a groundwater problem, can I put the Darcy velocity straight into the v field?
Under Penn State's Pe = Lu/(φDh), enter u/φ. In Example 3, entering u at porosity 0.4 gives 60 instead of 150. Some texts define u as pore velocity already, so check. Putting a molecular diffusivity where a dispersion coefficient belongs is another common slip that inflates Pe.
Is the Péclet number still meaningful in turbulent flow?
A molecular Pe still compares fluids and flows but not actual mixing. Turbulent mixing needs an effective diffusivity from an eddy viscosity, which depends on the turbulence model.
Above what Pe can diffusion be neglected?
No single value. At large Pe diffusion may be negligible in the bulk, but it controls transfer in the wall layer; even at 13,937 in Example 1 heat reaches the wall by conduction.
Related dimensionless numbers
These numbers multiply into Pe or are read beside it.
- Prandtl number (Pr): dividing the thermal Pe by Re leaves Pr, so checking both quickly reveals a wrong property input.
- Schmidt number (Sc): the mass Pe equals Re × Sc, and with Sc in the hundreds to thousands for liquids it ends up far above the thermal Pe.
- Nusselt number (Nu): Pe only compares transport inside the fluid; the heat actually delivered at a wall comes from Nu and the heat-transfer coefficient.
References
- Penn State CE 574: 6.6 Dimensionless Numbers. Porous-media Pe = Lu/(φDh) with Darcy velocity; grid-spacing L; sharper fronts at large Pe
- Michigan Technological University: CM3120 Module 2 Lecture IV. Pe = Re × Pr and Pe = Re × Sc as convective-to-diffusive rate ratios
- MIT OpenCourseWare (3.185): Transport Phenomena in Materials Engineering. Derivation of δT/x = 3.6/√Pe, Pe as the plug-flow versus CSTR criterion
- CFD Direct: Notes on CFD: Scale Similarity. Pe in the dimensionless energy equation and Pe = Re × Pr
- IAEA: Thermophysical Properties of Materials for Nuclear Engineering. Tables 3.1 (dry air) and 3.3 (saturated water)
- Cornell University: Calculating the Oxygen Diffusion Coefficient in Water. Oxygen diffusivity in 20 °C water, 1.97×10⁻⁹ m²/s (Wilke–Chang estimate)