Library / Dimensional analysis and similarity / DIM-13
Dimensional analysis and similarity
Péclet number
Ratio of advective transport to diffusive transport over a length L.
- Ref
- DIM-13
- Rev
- v1
- Field
- Dimensional analysis and similarity
- Result
- Pe
Pe = UL/Γ compares transport by the flow with transport by diffusion over a length L. Use the thermal diffusivity α for heat, which gives Pe = Re Pr, or the binary mass diffusivity DAB for a species, which gives Pe = Re Sc. The same calculation covers both. When Pe is much greater than 1, advection dominates and the heat or species is swept along with the flow; when it is much less than 1, diffusion dominates and the distribution spreads out in place.
Ratio of advective transport to diffusive transport over a length L. Use the thermal diffusivity for heat or the binary mass diffusivity for a species, and the same calculation covers both.
Calculator
- Advection time, L/U0.1 s
- Diffusion time, L²/Γ1,000 s
- Length at which Pe = 10.001 mm
- Pe with the flow speed halved5,000
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- Pe = 10,000: advection is far stronger than diffusion over this length, so the heat or species is carried along with the flow and diffusion matters only in thin layers.
- The transport time by diffusion, L²/Γ = 1,000 s, is 10,000 times the flow time L/U = 0.1 s: that ratio is the Péclet number.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| Γ Diffusivity (α for heat, D_AB for a species) | 1e-4 to 1,000 mm²/s | 1 | 1e7 | 10,000,000 | +10% | −9.09 % |
| U Flow speed | 1e-4 to 100 m/s | 10 | 1e7 | 10,000,000 | +10% | +10 % |
| L Characteristic length | 0.01 to 10,000 mm | 10 | 1e7 | 10,000,000 | +10% | +10 % |
Sweep of Flow speed
| Flow speed (m/s) | Péclet number |
|---|---|
| 1e-4 | 10 |
| 5.623e-4 | 56.23 |
| 0.003162 | 316.2 |
| 0.01778 | 1,778 |
| 0.1 | 10,000 |
| 0.5623 | 56,230 |
| 3.162 | 316,200 |
| 17.78 | 1,778,000 |
| 100 | 1e7 |
Sweep one variable
Variables and how each one contributes
Péclet number
What the equation gives you. Everything else feeds it.
Flow speed
The flow speed, in m/s. Raises the Péclet number in direct proportion. Raising it by 10 % (to 0.11) moves the result by +10 %, from 10,000 to 11,000.
Characteristic length
The characteristic length over which transport is judged, in mm. Raises the Péclet number in direct proportion. Raising it by 10 % (to 11) moves the result by +10 %, from 10,000 to 11,000.
Diffusivity (α for heat, D_AB for a species)
The diffusivity, in mm²/s: the thermal diffusivity α for heat (water about 0.14, air about 20) or the mass diffusivity DAB for a species (a dissolved solute in water about 0.002). Lowers the Péclet number in inverse proportion: a slow diffuser is carried further by the flow. Raising it by 10 % (to 0.11) moves the result by −9.09 %, from 10,000 to 9,091.
Worked example with the values above
- Convert: L = 10 mm = 0.01 m, Γ = 0.1 mm²/s = 1×10−7 m²/s.
- Advection rate: UL = 0.1 × 0.01 = 0.001 m²/s.
- Pe = 0.001 / 1×10−7 = 10,000.
Assumptions and limits
- A single characteristic speed and length, and a constant diffusivity.
- The diffusivity matches what is being transported: α for heat and DAB for a species.
Common mistakes
- Using the kinematic viscosity as the diffusivity: that gives the Reynolds number.
- Entering a diffusivity in m²/s instead of mm²/s.
- Applying a heat Péclet number to a species problem, or the reverse.
Statistics and uncertainty
The result is a product of powers, so to first order its relative error is the root sum of squares of each input’s relative error multiplied by its exponent (U × 1, L × 1, Γ × −1). The diffusivity is usually the least certain: mass diffusivities especially vary with temperature, concentration and the solute, so use a value measured or tabulated for the actual system.
Where it is used
- Deciding whether heat or a solute in a pipe, reactor, heat exchanger or porous medium is carried mainly by the flow or by diffusion.
- Choosing between an advection-dominated and a diffusion-dominated model, and the numerical scheme for it.
Maths you need
- Thermal diffusivity: how fast temperature differences spread by conduction
- Mass diffusivity: how fast a species spreads by diffusion