Conservation principles microscopic

Macroscopic and Microscopic Balances Three postulates, regarded as laws of physics, are fundamental in fluid mechanics. These are conservation of mass, conservation of momentum, and con-servation of energy. In addition, two other postulates, conservation of moment of momentum (angular momentum) and the entropy inequality (second law of thermodynamics) have occasional use. The conservation principles may be applied either to material systems or to control volumes in space. Most often, control volumes are used. The control volumes may be either of finite or differential size, resulting in either algebraic or differential consei vation equations, respectively. These are often called macroscopic and microscopic balance equations. 

Microscopic Balance Equations Partial differential balance equations express the conservation principles at a point in space. Equations for mass, momentum, totaf energy, and mechanical energy may be found in Whitaker (ibid.). Bird, Stewart, and Lightfoot (Transport Phenomena, Wiley, New York, 1960), and Slattery (Momentum, Heat and Mass Transfer in Continua, 2d ed., Krieger, Huntington, N.Y., 1981), for example. These references also present the equations in other useful coordinate systems besides the cartesian system. The coordinate systems are fixed in inertial reference frames. The two most used equations, for mass and momentum, are presented here. 

The engineering science of transport phenomena as formulated by Bird, Stewart, and Lightfoot (1) deals with the transfer of momentum, energy, and mass, and provides the tools for solving problems involving fluid flow, heat transfer, and diffusion. It is founded on the great principles of conservation of mass, momentum (Newton s second law), and energy (the first law of thermodynamics).1 These conservation principles can be expressed in mathematical equations in either macroscopic form or microscopic form. 

The principle of a lattice gas is to reproduce macroscopic behavior by modeling the underlying microscopic dynamics. In order to successfully predict the macro-level behavior of a fluid from micro-level rules, three requirements must be satisfied. First, the number of particles must be conserved and, in most cases, so is the particle momentum. States of all the cells in the neighborhood depend on the states of all the others, but neighborhoods do not overlap. This makes application of conservation laws simple because if they apply to one neighborhood they apply to the whole lattice. 

As discussed in Chapter 1, the basic principles that apply to the analysis and solution of flow problems include the conservation of mass, energy, and momentum in addition to appropriate transport relations for these conserved quantities. For flow problems, these conservation laws are applied to a system, which is defined as any clearly specified region or volume of fluid with either macroscopic or microscopic dimensions (this is also sometimes referred to as a control volume ), as illustrated in Fig. 5-1. The general conservation law is... 

The flow behavior of fluids is governed by the basic laws for conservation of mass, energy, and momentum coupled with appropriate expressions for the irreversible rate processes (e.g., friction loss) as a function of fluid properties, flow conditions, geometry, etc. These conservation laws can be expressed in terms of microscopic or point values of the variables, or in terms of macroscopic or integrated average values of these quantities. In principle, the macroscopic balances can be derived by integration of the microscopic balances. However, unless the local microscopic details of the flow field are required, it is often easier and more convenient to start with the macroscopic balance equations. 

In this chapter we are concerned with developing the equations of energy conservation to be used in the thermodynamic analysis of systems of pure substances. (The thermodynamics of mixtures is more complicated and will be considered in later chapters.) To emphasize both the generality of these equations and the lack of detail necessary, we write these energy balance equations for a general black-box system. For contrast, and also because a more detailed description will be.useful in Chapter 4, the rudiments of the more detailed microscopic description are provided in the final, optional section of this chapter. This microscopic description is not central to our development of thermodynamic principles, is suitable only for advanced students, and may be omitted. 

This section opened with an example of the macroscopic theory which is based, of course, on the conservation laws. The "mesoscopic" description (a term due to VAN KAMPEN [2.93) permits knowledge not only of the average behavior of an aerosol but also of its stochastic behavior through so-called master equations. However, this mesoscopic level of description may require (in complex systems) some physical assumptions as to the transition probabilities between states describing the system. Finally, the microscopic approach attempts to develop the theory of an aerosol from "first principles"—that is, through study of the dynamics of molecular motion in a suitable phase space. Master equations and macroscopic theory appear from the microscopic theory by the reduction of the complete dynamical description of the system in a suitable phase space to small subsets of chosen variables. 

Sason Not a problem I am going to teach balancing equations using microscopic constitution The principle of conservation of the number of atoms on the two sides of the chemical equation. 

In this section the statistical theorems or mathematical tools needed to understand the Boltzmann equation in itself, and the mathematical operations performed developing the macroscopic conservation equations starting out from the microscopic Boltzmann equation, are presented. Introductory it is stressed that a heuristic theory, which resembles the work of Boltzmann [10] and the standard kinetic theory literature, is adopted in this section and the subsequent sections deriving the Boltzmann equation. Irrespective, the notation and concepts presented in Sect. 2.2 are often referred, or even redefined in a less formal wrapping, thus the underlying elements of classical mechanics are prescience of outmost importance understanding the true principles of kinetic theory. 

Finally, it is quite impressive to recognize that at the very base of LCD technology, there lie such frmdamental physical principles as symmetries and broken symmetries, conservation laws, and the microscopic reversibility. 

The conservation of momentum principle is a common approach to a system composed of an arbitrary (differential) cubical volume within any flow field. By accounting for convection of momentum throughout the surface, all possible stress components on any and all surfaces, and any other forces (e.g. gravity), a general microscopic form of momentum equation can be derived, as shown in Eq. 3.9, which is valid at all points within any fluid. 

To some, stoichiometry is no more exciting than the law of conservation of mass, but make no mistake—stoichiometry is important. Chemists use stoichiometric principles routinely to plan experiments, analyze their results, and make predictions, all of which contribute to making new discoveries and expanding our knowledge of the microscopic world of atoms, molecules, and ions. 

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