Macroscopic balance equations

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... 

By "inert it means that the membrane is a separator but not a catalyst. Many membrane reactor modeling studies consider only those cases where the membrane is catalydcally inert and the catalyst is packed most often in the tubular (feed) region but sometimes in the annular (permeate) region. When it is assumed that no reaction takes place in the membrane or membrane/support matrix, the governing equations for the membrane/support matrix are usually eliminated. The overall eff ect of membrane permeation can be accounted for by a permeation term which appears in the macroscopic balance equations for both the feed and permeate sides. Thus, the diffusional gradient term... 

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. 

B. Macroscopic Balance Equations and Lumped Dissipation Frequencies. 49... 

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... 

Valuable information about the physics involved in the kinetic treatment of a specific problem can be obtained by considering the consistent macroscopic balance equations of the electrons, Eqs. (31) to (33), adapted to the specific kinetic problem. On the right side of the power and momentum balance, Eqs. (32) and (33), a difference between the corresponding gain fi-om the electric field and the total loss in collisions occurs. Gain and loss terms arise on the right side of the particle balance equation, Eq. (31), too if nonconservative electron collision processes (for instance, ionization and attachment) are additionally taken into account in the kinetic equation, Eq. (8), and thus in the equation system (12). 

For time-dependent plasmas, the macroscopic balance equations in Eqs. (31) to (33) take the simplified form (Wilhelm and Winkler, 1979 Winkler and Wuttke, 1992 Winkler, 1993)... 

In the aerosol literature, one sees frequently macroscopic balance equations for the evolution of an aerosol in space, time, and particle size or composition. As an example, the evolution of the singlet density function, n. (r,t) at position r and time t for the aerosol phase can be written... 

They arise in solving single macroscopic balance equations. Several special forms will be considered. 

The computation of every realizations of this ensemble of flow fields, even if the usual macroscopic balance equations are valid, is impossible in practice, even simply with a brute numerical method the time scales and length scales, that we know to exist within the turbulent regime, are so small with respect to the time or length scales in which we are interested, that we would need an incredible amount of computer memories and an incredible amount of computer time. In addition the computation of just one or few realizations is without interest we would not be able to perform experiments with the same initial and boundary conditions. Indeed only statistical quantities are of practical meaning in order to describe the randomness we need first mean values, then variances, correlations, and, at the best, probability density functions. 

II.1 The basic macroscopic balance equation with reactive flows are the well known Navier Stokes equations, together the continuity equation and the diffusion-reaction equations for each of the involved reactive species in the case of exothermic or endothermic reactions, an energy balance equation has to be added, but we will not emphasize this point here. 

---