Balances macroscopic

Material Balances Whenever mass-transfer applications involve equipment of specific dimensions, flux equations alone are inadequate to assess results. A material balance or continuity equation must also be used. When the geometiy is simple, macroscopic balances suffice. The following equation is an overall mass balance for such a unit having bulk-flow ports and ports or interfaces through which diffusive flux can occur ... 

S. Blenk, H. Ehrentraut, W. Muschik. Statistical foundation of macroscopic balances for liquid crystals in alignment tensor formulation. Physica A 77 119-138, 1991. 

The above equations can be used to predict the output 1/ in the effluent from a continuous-flow gas-liquid contractor. Let the flow rate of phase a be F. Then, a macroscopic balance at a steady state leads to the equation ... 

The objectives of this book are twofold (1) for the student, to show how the fundamental principles underlying the behavior of fluids (with emphasis on one-dimensional macroscopic balances) can be applied in an organized and systematic manner to the solution of practical engineering problems, and (2) for the practicing engineer, to provide a ready reference of current information and basic methods for the analysis of a variety of problems encountered in practical engineering situations. 

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... 

R. B. Bird, R. K. Prud homme, and M. Gottlieb, Extrudate Swell as Analyzed by Macroscopic Balances, The University of Wisconsin, Rheology Research Center Report RRC-35, 1975. 

Balances. The measurements of the O2 and CO2 concentrations in the gas stream at the entrance and exit of the bioreactor can be utilized by making a macroscopic balance over the bioreactor. Thus, if R is the total rate of growth (in gr biomas/cm3 s), V the reactor volume and yq2 yc02 t ie biomass yields with respect to O2... 

Other authors have also used approximate methods to solve the radiation problem. Li Puma and Yue (2003) used a thin film slurry model which does not include scattering effects. More recently, Li Puma et al. (2004), Brucato et al. (2006), and Li Puma and Brucato (2007) have used six flux models for different geometries. Salaices et al. (2001, 2002) used a model which allows for an adequate evaluation of the absorbed radiation in terms of macroscopic balances, based on radiometric measurements. They measured separately total transmitted radiation and nonscattered transmitted radiation, modeling the decay of both radiative fluxes with concentration by exponential fimctions. 

Macroscopic balance model. On the tube (feed) side ... 

As mentioned earlier, most membrane reactor models are based on isothermal macroscopic balances in the axial direction and do not solve the transport equations for the membrane/support matrix. They all account for the effects of membrane permeation through the use of some common relevant parameters (as a permeation term) in the transport equations for both the feed and permeate sides. Those parameters are to be determined experimentally. The above approach, of course, is feasible only when the membrane (or membrane/support) is not catalytic. 

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... 

Consider a well-stirred batch reactor of constant fluid volume V in which the reactions occurring are homogeneous. As the system for our macroscopic balances, we choose the fluid in the reactor volume V then the inflows Wn and outflows Wi2 vanish and no mass-transfer surface 0 is required. The species inventories are expressed in terms of fluid concentrations as... 

Consider the steady operation of a tubular reactor in which only homogeneous reactions occur. The state of the fluid now depends on the downstream distance z. Our chosen system for the macroscopic balances consists of the reactor contents between z and z + Az. Equation (3.1-4) then gives... 

The macroscopic balance ignores all the detail within a system and consequently results in a balance about the entire system. Only time remains as an independent variable in the balance. The dependent variables, such as concentration and temperature, are not functions of position but represent overall averages throughout the entire volume of the system. In effect, the system is assumed to be sufficiently well mixed so that the output concentrations and temperatures are equivalent to the concentrations and temperatures inside the system. 

As mentioned before, in the macroscopic balance the independent variable is time. In mathematics, when the quantity x varies with time, we consider dx to be the change in x that occurs during time dt if the process continues through the interval beginning at time t. Let us say that t is the independent variable and x is the dependent one. In solving problems you can use one or a combination of Eqs. (6.6) (6.8) directly, or alternatively you can proceed, as shown in some of the examples, to set... 

Stephanopoulos, G. Application of macroscopic balances and bioenergetics of growth to the on-line identification of biological reactors. Ann. N. Y. Acad. Sci. 1986, 469 (Biochem. Eng. 4), 332-349. 

As introduced in the previous section, class and sectional methods are based on a discretization of the internal coordinate so that the GPBE becomes a set of macroscopic balances in state space. Indeed, the fineness of the discretization will be dictated by the accuracy needed in the approximation of the integrals and derivative terms appearing in the GPBE. As has already been anticipated, the methods differ according to the number of internal coordinates used in the description and depend on the nature of the internal coordinates. Therefore, in what follows, we will discuss separately the univariate, bivariate, and multivariate PBE, and the use of these methods for the solution of the KE. 

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