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Result
Your inputs are calculated by your browser on this page; the operator does not receive or store them.
Reading the result as a ratio of two times
The result is the diffusion time L²/α divided by the transit time L/v. For the 60 °C water in Example 1 below, the flow crosses the 15 mm diameter at 0.5 m/s in 0.03 s, while heat needs a time of order 1,421 s to diffuse across the same diameter. Their ratio is Pe = 47,378. Both times are scales built on the chosen L, not predictions of how long the fluid takes to heat up.
The result text only says that a larger Pe means relatively stronger advection. It does not split the result into named regimes, as the Reynolds calculator does, because Pe has no cutoff that holds for every problem.
Variables and SI units
| Symbol | Meaning | SI unit | Example |
|---|---|---|---|
| v | Representative speed along the chosen length | m/s | 0.01 |
| L | Length over which advection and diffusion are compared | m | 0.1 |
| α | α = k/(ρcₚ) | m²/s | 1e-7 |
The “Fill example values” button enters v = 0.01 m/s, L = 0.1 m and α = 1×10⁻⁷ m²/s and returns Pe = 10,000. These values only demonstrate the input format and do not describe a particular device.
When α is not given directly
- If a property table lists only k, ρ and cₚ, compute α = k/(ρcₚ). The 60 °C saturated-water values in IAEA Table 3.3 give 0.6510 ÷ (983.2 × 4183) = 1.583 × 10⁻⁷ m²/s.
- The fields accept exponent notation, so type 1.583e-7. Data in mm²/s convert to m²/s by multiplying by 10⁻⁶.
- If you already have Re and Pr, compare their product with this result to catch input mistakes. A gap above about 1% usually means mixed property sources or units.
- Pick v and L to match the direction you care about: the diameter for diffusion across a pipe, or the pipe length for transport along it.
Using it as a mass Péclet number
Enter the molecular diffusivity D of a species (m²/s) in the α field and the calculator returns vL/D, the mass Péclet number, which equals Re × Sc. Read “thermal diffusion” in the explanation as “mass diffusion”. Oxygen in 20 °C water has D = 1.97 × 10⁻⁹ m²/s (Cornell, Wilke–Chang estimate), about 73 times smaller than α for water at the same temperature, so the mass Pe in the same flow is larger by that factor.
Calculator input examples
Example 1. Heat-exchanger tube carrying 60 °C water
Conditions: a 15 mm inner diameter and 0.5 m/s velocity are assumed operating conditions. Properties are the 60 °C saturated-water values in IAEA Table 3.3 (ρ = 983.2 kg/m³, μ = 4.677×10⁻⁴ Pa·s, k = 0.6510 W/(m·K), cₚ = 4183 J/(kg·K), Pr = 3.005), used as an approximation for low-pressure liquid water.
Calculator input: v = 0.5, L = 0.015, α = 1.583e-7 gives Pe = 47,378.
Cross-check: Re = 983.2 × 0.5 × 0.015 ÷ 4.677×10⁻⁴ = 15,767, and multiplying by Pr = 3.005 gives 47,378 again.
Reading it: the flow is turbulent by Re, so this Pe is a comparison based on molecular diffusion. Wall heat transfer has to come from a turbulent Nusselt-number correlation, not from Pe.
Example 2. Pe versus Re in a 20 °C air duct
Conditions: a 0.2 m characteristic length and 2 m/s air speed are assumed. Air properties are k = 0.0258 W/(m·K) and cₚ = 1000 J/(kg·K) from IAEA Table 3.1, with the table's 98.1 kPa density scaled to 1 atm as an ideal gas, 1.2043 kg/m³. That gives α = 0.0258 ÷ (1.2043 × 1000) = 2.142 × 10⁻⁵ m²/s.
Calculator input: v = 2, L = 0.2, α = 2.142e-5 gives Pe = 18,674.
Cross-check: with the 1 atm kinematic viscosity 1.5104×10⁻⁵ m²/s, Re = 2 × 0.2 ÷ 1.5104×10⁻⁵ = 26,483. Using the same properties, Pr = ν/α = 0.705, and Re × Pr ≈ 1.87 × 10⁴, matching the calculator.
Reading it: air has a Pr a little below 1, so Pe comes out slightly below Re. For water, Pe is 3 to 7 times Re at the same Re, so changing the fluid also changes how thick the thermal layer is relative to the velocity layer.
Example 3. Cell Péclet number from a CFD grid spacing
Conditions: 20 °C water (α = 1.435 × 10⁻⁷ m²/s, computed from IAEA Table 3.3) flows at 0.1 m/s through a region meshed at an assumed 0.5 mm spacing. The Penn State page notes that the grid spacing is sometimes used as the length in Pe.
Calculator input: v = 0.1, L = 0.0005, α = 1.435e-7 gives a cell Pe of 348. On the same grid, dissolved oxygen (D = 1.97 × 10⁻⁹ m²/s) has a cell Pe of 2.54 × 10⁴.
Reading it: the textbook condition for central differencing to give a non-oscillating solution of the steady 1D advection-diffusion equation is a cell Pe of 2 or less. Meeting it here would need a spacing of 2α/v = 2.87 μm, so practical solvers use upwind-type schemes and check the effect of their numerical diffusion through a grid-dependence study.
Interpreting the size
Around Pe ≈ 1 both advection and diffusion matter, but no single number is a universal boundary. In turbulent flow, mixing by eddies can be far stronger than molecular diffusion, so a Pe built on molecular α should not be read as the actual temperature field or heat-transfer rate. For porous-media data, enter the pore velocity (Darcy velocity divided by porosity) as v to match the Penn State definition.
Questions about the Péclet calculator
I entered a diffusivity D in the α field and the result text still talks about advection. Is the number right?
Yes. The calculator simply divides vL by the third input, so entering D gives the mass Péclet number directly. Read the diffusion in the result text as mass diffusion.
How do I enter very small values such as 1e-7?
The fields accept exponent notation, so type 1e-7 or 1.435e-7. Writing 0.0000001 makes it easy to miscount zeros.
For air, Pe comes out close to Re. Is that expected?
Yes. With the same v and L, Pe/Re = ν/α = Pr, and Pr for 20 °C air is about 0.71. In Example 2, Pe = 18,674 is about 30% below Re = 26,483.
Can I use this calculator to check the cell Péclet number of a CFD mesh?
Enter the grid spacing as L. Keep in mind that the result guides the choice of discretisation scheme, not the physics, so do not mix it with the advection-to-diffusion ratio of the whole device.
Related guides and calculators
Guide
Calculation notes
- Cross-check Re, Pr and Pe for pipe flow: follows one input set step by step and matches this calculator's output against the product of the Reynolds and Prandtl results.
- Thermal and mass diffusion: the distinct roles of Pr, Sc, Pe and Nu: shows how a Pe from this page turns into a boundary-layer thickness or a diffusion distance.
Related calculators
- Prandtl number (Pr): get Pr from k, μ and cₚ, then multiply by Re to check the result here.
- Schmidt number (Sc): with a diffusivity D, compute Sc first and compare Re × Sc with the mass Pe from this page.
- Nusselt number (Nu): switch to it when you need the heat actually exchanged at a wall.
References
- IAEA: Thermophysical Properties of Materials for Nuclear Engineering. Water (Table 3.3) for Examples 1 and 3, air (Table 3.1) for Example 2
- Michigan Technological University: CM3120 Module 2 Lecture IV. Pe = Re × Pr and Pe = Re × Sc used in the cross-checks
- Penn State CE 574: 6.6 Dimensionless Numbers. Porous-media Pe definition and grid spacing as the length scale
- Cornell University: Calculating the Oxygen Diffusion Coefficient in Water. Oxygen diffusivity in 20 °C water