View source for Péclet number

{{short description|Ratio of a fluid's advective and diffusive transport rates}}

In [[continuum mechanics]], the '''Péclet number''' ({{math|'''Pe'''}}, after [[Jean Claude Eugène Péclet]]) is a class of [[dimensionless number]]s relevant in the study of [[transport phenomena]] in a continuum. It is defined to be the ratio of the rate of [[advection]] of a [[physical quantity]] by the flow to the rate of [[diffusion]] of the same quantity driven by an appropriate [[Potential gradient|gradient]]. In the context of species or [[mass transfer]], the Péclet number is the product of the [[Reynolds number]] and the [[Schmidt number]] ({{math|Re × Sc}}). In the context of the [[thermal fluids]], the thermal Péclet number is equivalent to the product of the Reynolds number and the [[Prandtl number]] ({{math|Re × Pr}}).

[[File:Pe less 1.gif|thumb|Plan view: For <math>Pe_L \to 0</math>, advection is negligible, and diffusion dominates mass transport.]]
The Péclet number is defined as
: <math>\mathrm{Pe} = \dfrac{\text{advective transport rate}}{\text{diffusive transport rate}}.</math>

[[File:Pe equal 1.png|thumb|Plan view: For <math>Pe_L = 1</math>, diffusion and advection occur over equal times, and both have a not negligible influence on mass transport.]]
For mass transfer, it is defined as
: <math>\mathrm{Pe}_L = \frac{L u}{D} = \mathrm{Re}_L \, \mathrm{Sc}.</math>

Such ratio can also be re-written in terms of times, as a ratio between the characteristic temporal intervals of the system:
: <math>\mathrm{Pe}_L = \frac{u/L}{D/L^2} = \frac{L^2/D}{L/u} = \frac{\text{diffusion time}}{\text{advection time}}.</math>

For <math>\mathrm{Pe_L} \gg 1</math> the diffusion happens in a much longer time compared to the advection, and therefore the latter of the two phenomena predominates in the mass transport. 
[[File:Pe greater 1.png|thumb|Plan view: For <math>Pe_L \rightarrow \infty</math>, diffusion is negligible, and advection dominates mass transport.]]

For [[heat transfer]], the Péclet number is defined as
: <math>\mathrm{Pe}_L = \frac{L u}{\alpha} = \mathrm{Re}_L \, \mathrm{Pr},</math>
where {{mvar|L}} is the [[characteristic length]], {{mvar|u}} the local [[flow velocity]], {{mvar|D}} the [[Fick's law|mass diffusion coefficient]], {{math|Re}} the Reynolds number, {{math|Sc}} the Schmidt number, {{math|Pr}} the Prandtl number, and {{mvar|α}} the [[thermal diffusivity]],
: <math>\alpha = \frac{k}{\rho c_p},</math>
where {{mvar|k}} is the [[thermal conductivity]], {{mvar|ρ}} the [[density]], and {{mvar|c{{sub|p}}}} the [[specific heat capacity]].

In engineering applications the Péclet number is often very large. In such situations, the dependency of the flow upon ''downstream'' locations is diminished, and variables in the flow tend to become "one-way" properties. Thus, when modelling certain situations with high Péclet numbers, simpler computational models can be adopted.<ref>{{cite book |last=Patankar |first=Suhas V. |year=1980 |title=Numerical Heat Transfer and Fluid Flow |location=New York |publisher=McGraw-Hill |isbn=0-89116-522-3 |page=102 }}</ref>

A flow will often have different Péclet numbers for heat and mass. This can lead to the phenomenon of [[double diffusive convection]].

In the context of particulate motion the Péclet number has also been called '''Brenner number''', with symbol {{math|'''Br'''}}, in honour of [[Howard Brenner]].<ref>Promoted by S. G. Mason in publications from ''circa'' 1977 onward, and adopted by a number of others.{{Who|date=July 2016}}</ref>

The Péclet number also finds applications beyond transport phenomena, as a general measure for the relative importance of the random fluctuations and of the systematic average behavior in mesoscopic systems.<ref>{{cite journal |last1=Gommes |first1=Cedric |last2=Tharakan |first2=Joe |title=The Péclet number of a casino: Diffusion and convection in a gambling context |journal=American Journal of Physics |date=2020 |volume=88 |issue=6 |page=439 |doi=10.1119/10.0000957 |bibcode=2020AmJPh..88..439G |s2cid=219432227 |url=https://aapt.scitation.org/doi/abs/10.1119/10.0000957}}</ref>

==See also==
* [[Nusselt number]]

==References==
{{Reflist}}

{{NonDimFluMech}}
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{{DEFAULTSORT:Peclet Number}}
[[Category:Convection]]
[[Category:Dimensionless numbers of fluid mechanics]]
[[Category:Dimensionless numbers of thermodynamics]]
[[Category:Fluid dynamics]]
[[Category:Heat conduction]]