Péclet number

In continuum mechanics, the Péclet number ( $Pe$ , 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 ( $Re \times Sc$ ). In the context of the thermal fluids, the thermal Péclet number is equivalent to the product of the Reynolds number and the Prandtl number ( $Re \times Pr$ ).

The Péclet number is defined as:

Plan view: For $Pe_{L}\rightarrow 0$ , advection is negligible, and diffusion dominates mass transport.

\mathrm {Pe} ={\dfrac {\mbox{advective transport rate}}{\mbox{diffusive transport rate}}}

For mass transfer, it is defined as:

\mathrm {Pe} _{L}={\frac {Lu}{D}}=\mathrm {Re} _{L}\,\mathrm {Sc}

Plan view: For $Pe_{L}=1$ , diffusion and advection occur over equal times, and both have a not negligible influence on mass transport.

Such ratio can also be re-written in terms of times, as a ratio between the characteristic temporal intervals of the system:

\mathrm {Pe} _{L}={\frac {u/L}{D/L^{2}}}={\frac {L^{2}/D}{L/u}}={\frac {\mbox{diffusion time}}{\mbox{advection time}}}

For $\mathrm {Pe_{L}} \gg 1$ 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.

Plan view: For $Pe_{L}\rightarrow \infty$ , diffusion is negligible, and advection dominates mass transport.

For heat transfer, the Péclet number is defined as:

\mathrm {Pe} _{L}={\frac {Lu}{\alpha }}=\mathrm {Re} _{L}\,\mathrm {Pr} .

where $L$ is the characteristic length, $u$ the local flow velocity, $D$ the mass diffusion coefficient, $Re$ the Reynolds number, $Sc$ the Schmidt number, $Pr$ the Prandtl number, and $α$ the thermal diffusivity,

\alpha ={\frac {k}{\rho c_{p}}}

where $k$ is the thermal conductivity, $ρ$ the density, and $c p$ 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.^[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 $Br$ , 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]

References

^ Patankar, Suhas V. (1980). Numerical Heat Transfer and Fluid Flow. New York: McGraw-Hill. p. 102. ISBN 0-89116-522-3.
^ Promoted by S. G. Mason in publications from circa 1977 onward, and adopted by a number of others.^[who?]
^ Gommes, Cedric; Tharakan, Joe (2020). "The Péclet number of a casino: Diffusion and convection in a gambling context". American Journal of Physics. 88 (6): 439. Bibcode:2020AmJPh..88..439G. doi:10.1119/10.0000957. S2CID 219432227.

[1] Patankar, Suhas V. (1980). Numerical Heat Transfer and Fluid Flow. New York: McGraw-Hill. p. 102. ISBN 0-89116-522-3.

[2] Promoted by S. G. Mason in publications from circa 1977 onward, and adopted by a number of others.^[who?]

[3] Gommes, Cedric; Tharakan, Joe (2020). "The Péclet number of a casino: Diffusion and convection in a gambling context". American Journal of Physics. 88 (6): 439. Bibcode:2020AmJPh..88..439G. doi:10.1119/10.0000957. S2CID 219432227.

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See also

References