Péclet number: Difference between revisions
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{{short description|Ratio of a fluid's advective and diffusive transport rates}} | {{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 | 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 continuous environment. 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.]] | [[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 | The Péclet number is defined as | ||
<math display="block">\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 non-negligible influence on mass transport.]] | [[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 non-negligible influence on mass transport.]] | ||
For mass transfer, it is defined as | For mass transfer, it is defined as | ||
<math display="block">\mathrm{Pe}_L = \frac{L u}{D} = \mathrm{Re}_L \, \mathrm{Sc},</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. | 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. | ||
Such ratio can also be re-written in terms of times, as a ratio between the characteristic temporal intervals of the system: | Such ratio can also be re-written in terms of times, as a ratio between the characteristic temporal intervals of the system: | ||
<math display="block">\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. | 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. | ||
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For [[heat transfer]], the Péclet number is defined as | For [[heat transfer]], the Péclet number is defined as | ||
<math display="block">\mathrm{Pe}_L = \frac{L u}{\alpha} = \mathrm{Re}_L \, \mathrm{Pr},</math> | |||
where {{math|Pr}} the Prandtl number, and {{mvar|α}} the [[thermal diffusivity]], | where {{math|Pr}} the Prandtl number, and {{mvar|α}} the [[thermal diffusivity]], | ||
<math display="block">\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]]. | where {{mvar|k}} is the [[thermal conductivity]], {{mvar|ρ}} the [[density]], and {{mvar|c{{sub|p}}}} the [[specific heat capacity]]. | ||
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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> | 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|url-access=subscription }}</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|url-access=subscription }}</ref> | ||
==See also== | ==See also== | ||
Latest revision as of 20:13, 21 October 2025
In continuum mechanics, the Péclet number (Template:Math, after Jean Claude Eugène Péclet) is a class of dimensionless numbers relevant in the study of transport phenomena in a continuous environment. 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 gradient. In the context of species or mass transfer, the Péclet number is the product of the Reynolds number and the Schmidt number (Template:Math). In the context of the thermal fluids, the thermal Péclet number is equivalent to the product of the Reynolds number and the Prandtl number (Template:Math).
The Péclet number is defined as
For mass transfer, it is defined as
where Template:Mvar is the characteristic length, Template:Mvar the local flow velocity, Template:Mvar the mass diffusion coefficient, Template:Math the Reynolds number, Template:Math the Schmidt number.
Such ratio can also be re-written in terms of times, as a ratio between the characteristic temporal intervals of the system:
For 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.
For heat transfer, the Péclet number is defined as where Template:Math the Prandtl number, and Template:Mvar the thermal diffusivity, where Template:Mvar is the thermal conductivity, Template:Mvar the density, and Template:Mvar 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.[1]
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 Template:Math, in honour of Howard Brenner.[2]
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.[3]