# Averaging approaches

### From Thermal-FluidsPedia

The objectives of the various averaging methods are twofold: (1) to define the average properties for the multiphase system and correlate the experimental data, and (2) to obtain solvable governing equations that can be used to predict the macroscopic properties of the multiphase system. This chapter will address the application of averaging methods to the governing equations.

Based on the physical concepts used to formulate multiphase transport phenomena, the averaging methods can be classified into three major groups: (1) *Eulerian averaging*, (2) *Lagrangian averaging*, and (3) *Molecular statistical averaging*.

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## Volume Averaging

Eulerian averaging is the most important and widely-used method of averaging, because it is consistent with the control volume analysis that we used to develop the governing equations in the preceding section. It is also applicable to the most common techniques of experimental observations. Eulerian averaging is based on time-space description of physical phenomena. In the Eulerian description, changes in the various dependent variables, such as velocity, temperature, and pressure, are expressed as functions of time and space coordinates, which are considered to be independent variables. One can average these independent variables over both space and time. The integral operations associated with these averages smooth out the local spatial or instant variations of the properties within the domain of integration.

*See Main Article* Volume Averaging

## Lagrangian Averaging

Lagrangian averaging is directly related to the Lagrangian description of a system, which requires tracking the motion of each individual fluid particle. Therefore, Lagrangian averaging is a very useful tool when the dynamics of individual particles are of interest. To obtain Lagrangian time averaging, it is necessary to follow a specific particle and observe its behavior for a certain time interval. Then, the behavior of this particle is averaged over the time interval.

*See Main Article* Lagrangian Averaging

## Boltzmann Statistical Averaging

When the collective mechanics of a large number of particles is of interest, molecular statistical averaging may be employed. This relies on the concept of particle number density, which is the number of particles per unit volume. For a system with a large number of particles, the behavior of each individual particle is random because random collisions occur. To describe the behavior of each particle, it is necessary to track the motion resulting from each collision – an impractical and often unnecessary task. Although the behavior of each particle is random, the collection of particles may demonstrate some statistical behaviors that are different from those of the individual particles. When the number of molecules involved in the averaging process is large enough, the statistical average value becomes independent of the number of molecules involved. The statistical average value of the microscopic properties for a large number of molecules is related to the macroscopic properties of the system.

*See Main Article* Boltzmann Statistical Averaging

## References

Faghri, A., and Zhang, Y., 2006, *Transport Phenomena in Multiphase Systems*, Elsevier, Burlington, MA

Faghri, A., Zhang, Y., and Howell, J. R., 2010, *Advanced Heat and Mass Transfer*, Global Digital Press, Columbia, MO.