Basics of Boltzmann statistical averaging

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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. For example, temperature is a statistical measure of the kinetic energy of the individual molecules, and the pressure of a gas in a container is the result of many molecules’ collisions with the wall. For some engineering problems, the macroscopic properties of the fluid as well as the microscopic properties are required for design or analysis.

Most numerical codes are based on the Navier-Stokes equations, which treats a fluid as a continuous field. It is well known that a fluid is made of a discrete number of particles or molecules. Since the number of molecules is extremely large (Avogadro’s number = 6.022×1023 atoms/mole) for almost all practically sized systems, it may never be computationally viable to track each particle and its interactions with other particles. The number of molecules in a given region and the molecular interaction are described through the fluid’s density and transport coefficients (i.e., viscosity) in the continuous model. Modeling the individual molecules for a small system over a small period of time has been achieved by molecular dynamic simulations (MDS). The computational requirements needed in these simulations can be greatly reduced if the degrees of freedom of the system are reduced. Also, instead of considering individual molecules, groups of molecules can be considered. The degrees of freedom can be reduced by restricting the movement of the molecules to a lattice. A lattice is simply a predefined direction in which a molecule can move.

From this standpoint the independent variables are space, velocity and time, while the dependent variable is a molecular distribution function for species i, {f_i}\left( {{\mathbf{x}},{\mathbf{c}},t} \right). The Boltzmann equation relates the distribution function at \left( {{\mathbf{x}},{\mathbf{c}},t} \right) to the distribution function at ({\mathbf{x}} + \Delta {\mathbf{x}}, {\mathbf{c}} + \Delta {\mathbf{c}}, t + \Delta t). The location in space is x, and the particle velocity is c. It is important to note that the particle velocity is directly related to the mass average velocity, V, that is used throughout this book. This distribution function can be related to the Navier-Stokes equations as well as other transport equations; these relationships give insight to the origin of transport coefficients such as viscosity. A detailed presentation of Boltzmann statistical averaging including the discussion of Lattice Boltzmann model for both single and multiphase systems can be found in Faghri and Zhang (2006).

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.

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