Macroscopic Mass, Energy, and Momentum Balances

DEVELOPMENT OF MACROSCOPIC MASS, ENERGY, AND MOMENTUM BALANCES 

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For the student, this is a basic text for a first-level course in process engineering fluid mechanics, which emphasizes the systematic application of fundamental principles (e.g., macroscopic mass, energy, and momentum balances and economics) to the analysis of a variety of fluid problems of a practical nature. Methods of analysis of many of these operations have been taken from the recent technical literature, and have not previously been available in textbooks. This book includes numerous problems that illustrate these applications at the end of each chapter. 

Macroscopic mass, energy, and momentum balances provide the simplest starting point for reactor modeling. These equations give little spatial detail, but provide a first approximation to the performance of chemical reactors. This section builds on Chapter 22 of Bird, Stewart, and Llghtfoot (2002). A table of notation is given at the end of the current chapter. 

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 discussion overall or macroscopic mass balances were made because we wish to describe these balances from outside the enclosure. In this section on overall mass balances, some of the equations presented may have seemed quite obvious. However, the purpose was to develop the methods which should be helpful in the next sections. Overall balances will also be made on energy and momentum in the next sections. These overall balances do not tell us the details of what happens inside. However, in Section 2.9 a shell momentum balance will be made to obtain these details, which will give us the velocity distribution and pressure drop. To further study these details of the processes occurring inside the enclosure, differential balances rather than shell balances can be written and these are discussed in other later Sections 3.6 to 3.9 on differential equations of continuity and momentum transfer. Sections 5.6 and 5.7 on differential equations of energy change and boundary-layer flow, and Section 7.5B on differential equations of continuity for a binary mixture. 

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. 

Bird RB (1957) The equations of change and the macroscopic mass, momentum, and energy balances. Chem Eng Sci 6 123-181... 

Furthermore, appropriate energy space averaging over Eqs. (12), derived through two-term approximation from the Boltzmann equation, yields the consistent macroscopic balance equations of the electrons. In particular, the particle and power balance can be derived from the first equation of system (12) and the momentum balance equation, normalized on the electron mass can be derived from the second equation of (12). These balance equations are... 

While the Navier-Stokes equations and Bernoulli s equation, along with the macroscopic mass and linear momentum balances, were clearly available prior to 1888, it seems likely that the general macroscopic mechanical energy balance was not. Nevertheless, the fixed control volume... 

As if in answer to Bliss editorial. Prof. Bird (1957) provided a detailed derivation of the macroscopic balances for mass, momentum and mechanical energy. These were presented for an arbitrary moving control volume, and thus captured in a rigorous manner all of the details that had been added by persuasion over the previous years. One might think that this would have ended the confusion about the macroscopic mechanical energy balance, but this has not been the case and a precise understanding of this result seems to be the exception rather than the rule. 

Three types of theoretical approaches can be used for modeling the gas-particles flows in the pneumatic dryers, namely Two-Fluid Theory [1], Eulerian-Granular [2] and the Discrete Element Method [3]. Traditionally the Two-Fluid Theory was used to model dilute phase flow. In this theory, the solid phase is being considering as a pseudo-fluid. It is assumed that both phases are occupying every point of the computational domain with its own volume fraction. Thus, macroscopic balance equations of mass, momentum and energy for both the gas and the solid... 

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