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 numbers 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 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

The Péclet number is defined as

: $\backslash mathrm\{Pe\}\; =\; \backslash dfrac\{\backslash text\{advective\; transport\; rate\}\}\{\backslash text\{diffusive\; transport\; rate\}\}.$

File:Pe equal 1.png

For mass transfer, it is defined as

: $\backslash mathrm\{Pe\}\_L\; =\; \backslash frac\{L\; u\}\{D\}\; =\; \backslash mathrm\{Re\}\_L\; \backslash ,\; \backslash mathrm\{Sc\},$

where {{mvar|L}} is the characteristic length, {{mvar|u}} the local flow velocity, {{mvar|D}} the 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:

: $\backslash mathrm\{Pe\}\_L\; =\; \backslash frac\{u/L\}\{D/L^2\}\; =\; \backslash frac\{L^2/D\}\{L/u\}\; =\; \backslash frac\{\backslash text\{diffusion\; time\}\}\{\backslash text\{advection\; time\}\}.$

For $\backslash mathrm\{Pe\_L\}\; \backslash gg\; 1$ 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

For heat transfer, the Péclet number is defined as

: $\backslash mathrm\{Pe\}\_L\; =\; \backslash frac\{L\; u\}\{\backslash alpha\}\; =\; \backslash mathrm\{Re\}\_L\; \backslash ,\; \backslash mathrm\{Pr\},$

where {{math|Pr}} the Prandtl number, and {{mvar|α}} the thermal diffusivity,

: $\backslash alpha\; =\; \backslash frac\{k\}\{\backslash rho\; c\_p\},$

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.{{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 }}

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.Promoted by S. G. Mason in publications from circa 1977 onward, and adopted by a number of others.{{Who|date=July 2016}}

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.{{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 }}