Macroscopic Energy Balances

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. 

Differential momentum, mechanical-energy, or total-energy balances can be written for each phase in a two-phase flowing mixture for certain flow patterns, e.g., annular, in which each phase is continuous. For flow patterns where this is not the case, e.g., plug flow, the equivalent expressions can usually be written with sufficient accuracy as macroscopic balances. These equations can be formulated in a perfectly general way, or with various limitations imposed on them. Most investigations of two-phase flow are carried out with definite limits on the system, and therefore the balances will be given for the commonest conditions encountered experimentally. 

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... 

The last term is the rate of viscous energy dissipation to internal energy, Ev = jv <5 dV, also called the rate of viscous losses. These losses are the origin of frictional pressure drop in fluid flow. Whitaker and Bird, Stewart, and Lightfoot provide expressions for the dissipation function <5 for Newtonian fluids in terms of the local velocity gradients. However, when using macroscopic balance equations the local velocity field within the control volume is usually unknown. For such... 

Integration of Eq. 2.9-11 leads to the macroscopic mechanical energy balance equation, the steady-state version of which is the famous Bernoulli equation. Next we subtract Eq. 2.9-11 from Eq. 2.9-10 to obtain the differential thermal energy-balance... 

The macroscopic mechanical energy balance is obtained by integrating Eq. 2.9-11. 

The most important relationship in designing flow systems is the macroscopic mechanical-energy balance, or Bernoulli s equation. Not only is it required for calculating the pump work, but it is also used to derive formulas for sizing valves and flow meters. Bird, et al. [6] derived this equation by integrating the microscopic mechanical-energy balance over the volume of the system. The balance is given by... 

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 mechanical energy balance is not a fundamental principle rather, it is a corollary (Bird 1957 Bird. Stewart, and Lightfoot 2002) of the equation of motion. For constant fluid density, the macroscopic mechanical energy balance takes the form... 

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 any particular region of space (Fig. 2.2-1) the macroscopic balances express the fact (hat the time rate of change of mass, species, momentum, or energy within the system is equal to the sum of the net flow across the boundaries of the region and the rate of generation within the region. 

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... 

The advantage of this approach over kinematically extended continua is twofold first, the additional parameter is abstract and does not require special assumptions of the local kinematics as it is necessary in micropolar continua, for example. As a consequence, the corresponding boundary conditions can be formulated more flexibly. Second, the coupling between the macroscopic balance of momentum and the additional balance of equilibrated forces is obtained consti-tutively, via the free Helmholtz energy function and not via the free energy function in combination with measures of deformation assumed a priori. Therefore, such an approach allows for size effects in both ways - either smaller is stiffer" or smaller is weaker depending on the boundary conditions. 

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