continuum mechanics Continuum mechanics is a branch of mechanics that deals with the deformation of and transmission of forces through materials modeled as a ''continuous medium'' (also called a ''continuum'') rather than as discrete particles. Continuum mec ...

, the Péclet number (, after Jean Claude Eugène Péclet) is a class of

dimensionless number Dimensionless quantities, or quantities of dimension one, are quantities implicitly defined in a manner that prevents their aggregation into unit of measurement, units of measurement. ISBN 978-92-822-2272-0. Typically expressed as ratios that a ...

s relevant in the study of

transport phenomena In engineering, physics, and chemistry, the study of transport phenomena concerns the exchange of mass, energy, charge, momentum and angular momentum between observed and studied systems. While it draws from fields as diverse as continuum mec ...

in a continuum. It is defined to be the ratio of the rate of

advection In the fields of physics, engineering, and earth sciences, advection is the transport of a substance or quantity by bulk motion of a fluid. The properties of that substance are carried with it. Generally the majority of the advected substance is a ...

of a

physical quantity A physical quantity (or simply quantity) is a property of a material or system that can be Quantification (science), quantified by measurement. A physical quantity can be expressed as a ''value'', which is the algebraic multiplication of a ''nu ...

by the flow to the rate of

diffusion Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in Gibbs free energy or chemical p ...

of the same quantity driven by an appropriate

gradient In vector calculus, the gradient of a scalar-valued differentiable function f of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p gives the direction and the rate of fastest increase. The g ...

. In the context of species or

mass transfer Mass transfer is the net movement of mass from one location (usually meaning stream, phase, fraction, or component) to another. Mass transfer occurs in many processes, such as absorption, evaporation, drying, precipitation, membrane filtra ...

, the Péclet number is the product of the

Reynolds number In fluid dynamics, the Reynolds number () is a dimensionless quantity that helps predict fluid flow patterns in different situations by measuring the ratio between Inertia, inertial and viscous forces. At low Reynolds numbers, flows tend to ...

and the

Schmidt number In fluid dynamics, the Schmidt number (denoted ) of a fluid is a dimensionless number defined as the ratio of momentum diffusivity (kinematic viscosity) and mass diffusivity, and it is used to characterize fluid flows in which there are simultan ...

(). In the context of the

thermal fluids Thermofluids is a branch of science and engineering encompassing four intersecting fields: *Heat transfer *Thermodynamics *Fluid mechanics *Combustion The term is a combination of "thermo", referring to heat, and "fluids", which refers to liquids ...

, the thermal Péclet number is equivalent to the product of the Reynolds number and the

Prandtl number The Prandtl number (Pr) or Prandtl group is a dimensionless number, named after the German physicist Ludwig Prandtl, defined as the ratio of momentum diffusivity to thermal diffusivity. The Prandtl number is given as:where: * \nu : momentum d ...

().

The Péclet number is defined as :

\mathrm = \dfrac.

For mass transfer, it is defined as :

\mathrm_L = \frac = \mathrm_L \, \mathrm,

where is the

characteristic length In physics, a characteristic length is an important dimension that defines the scale of a physical system. Often, such a length is used as an input to a formula in order to predict some characteristics of the system, and it is usually required by ...

, the local

flow velocity In continuum mechanics the flow velocity in fluid dynamics, also macroscopic velocity in statistical mechanics, or drift velocity in electromagnetism, is a vector field used to mathematically describe the motion of a continuum. The length of the f ...

, the mass diffusion coefficient, the Reynolds number, 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: :

\mathrm_L = \frac = \frac = \frac.

For

\mathrm \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. Pe greater 1

For

heat transfer Heat transfer is a discipline of thermal engineering that concerns the generation, use, conversion, and exchange of thermal energy (heat) between physical systems. Heat transfer is classified into various mechanisms, such as thermal conduction, ...

, the Péclet number is defined as :

\mathrm_L = \frac = \mathrm_L \, \mathrm,

where the Prandtl number, and the

thermal diffusivity In thermodynamics, thermal diffusivity is the thermal conductivity divided by density and specific heat capacity at constant pressure. It is a measure of the rate of heat transfer inside a material and has SI, SI units of m2/s. It is an intensive ...

, :

\alpha = \frac,

where is the

thermal conductivity The thermal conductivity of a material is a measure of its ability to heat conduction, conduct heat. It is commonly denoted by k, \lambda, or \kappa and is measured in W·m−1·K−1. Heat transfer occurs at a lower rate in materials of low ...

, the

density Density (volumetric mass density or specific mass) is the ratio of a substance's mass to its volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' (or ''d'') can also be u ...

, and the

specific heat capacity In thermodynamics, the specific heat capacity (symbol ) of a substance is the amount of heat that must be added to one unit of mass of the substance in order to cause an increase of one unit in temperature. It is also referred to as massic heat ...

. 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. 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 , in honour of Howard Brenner. 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.

References

{{DEFAULTSORT:Peclet Number Convection Dimensionless numbers of fluid mechanics Dimensionless numbers of thermodynamics Fluid dynamics Heat conduction

See also

References