continuum mechanics Continuum mechanics is a branch of mechanics that deals with the mechanical behavior of materials modeled as a continuous mass rather than as discrete particles. The French mathematician Augustin-Louis Cauchy was the first to formulate such mo ...

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

dimensionless number A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a quantity to which no physical dimension is assigned, with a corresponding SI unit of measurement of one (or 1) ...

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 mecha ...

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

advection In the field 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 al ...

of a

physical quantity A physical quantity is a physical property of a material or system that can be quantified by measurement. A physical quantity can be expressed as a ''value'', which is the algebraic multiplication of a ' Numerical value ' and a ' Unit '. For exam ...

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 ...

of the same quantity driven by an appropriate

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

. 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 filtration ...

, the Péclet number is the product of the

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

and the

Schmidt number Schmidt number (Sc) 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 simultaneous momentum and mass diffusion convec ...

(). In the context of the thermal fluids, 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: : \mathrm = \frac = \fr ...

(). The Péclet number is defined as: :

\mathrm = \dfrac

For mass transfer, it is defined as: :

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

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 convection, and therefore the latter of the two phenomena predominates in the mass transport. 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 conducti ...

, the Péclet number 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, the Prandtl number, and the

thermal diffusivity In heat transfer analysis, thermal diffusivity is the thermal conductivity divided by density and specific heat capacity at constant pressure. It measures the rate of transfer of heat of a material from the hot end to the cold end. It has the SI ...

, :

\alpha = \frac

where is the

thermal conductivity The thermal conductivity of a material is a measure of its ability to conduct heat. It is commonly denoted by k, \lambda, or \kappa. Heat transfer occurs at a lower rate in materials of low thermal conductivity than in materials of high thermal ...

, the

density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematicall ...

, and the

specific heat capacity In thermodynamics, the specific heat capacity (symbol ) of a substance is the heat capacity of a sample of the substance divided by the mass of the sample, also sometimes referred to as massic heat capacity. Informally, it is the amount of heat t ...

. 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 Howard Brenner (16 March 1929 – 17 February 2014) was a professor emeritus of chemical engineering at Massachusetts Institute of Technology. His research profoundly influenced the field of fluid dynamics, and his research contribution to funda ...

. 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