Mass balance, definition

Since residue cur ves do not by definition cross separatrices, the distillate and bottoms compositions must be in the same distillation region with the mass balance line intersecting a residue cur ve in two places. Mass balance lines for mixing and for other separations not involving vapor-hquid equihbria, such as extraction and decantation, are of course not hmited by distiUation boundaries. 

The boundary conditions normally associated with Equation (9.14) are known as the Danckwerts or closed boundary conditions. They are obtained from mass balances across the inlet and outlet of the reactor. We suppose that the piping to and from the reactor is small and has a high Re. Thus, if we were to apply the axial dispersion model to the inlet and outlet streams, we would find = 0, which is the definition of a closed system. See... 

Figure 3 Possible blood circulation connections in a physiologically based pharmacokinetic model. (A) Venous return incorporated into lung mass balance equation (B) separate venous blood compartment. See text for definition of symbols.

Definitions for the variables and constants appearing in eqns. 1 and 2 are given in the nomenclature section at the end of this paper. The first of these equations represents a mass balance around the reactor, assuming that it operates in a differential manner. The second equation is a species balance written for the catalyst surface. The rate of elementary reaction j is represented by rj, and v j is the stoichiometric coefficient for component i in reaction j. The relationship of rj to the reactant partial pressures and surface species coverages are given by expressions of the form... 

Attempts to define operationally the rate of reaction in terms of certain derivatives with respect to time (r) are generally unnecessarily restrictive, since they relate primarily to closed static systems, and some relate to reacting systems for which the stoichiometry must be explicitly known in the form of one chemical equation in each case. For example, a IUPAC Commission (Mils, 1988) recommends that a species-independent rate of reaction be defined by r = (l/v,V)(dn,/dO, where vt and nf are, respectively, the stoichiometric coefficient in the chemical equation corresponding to the reaction, and the number of moles of species i in volume V. However, for a flow system at steady-state, this definition is inappropriate, and a corresponding expression requires a particular application of the mass-balance equation (see Chapter 2). Similar points of view about rate have been expressed by Dixon (1970) and by Cassano (1980). 

As depicted in Figure 2.3, electrons are transferred from the oxidation step to the reduction step of the redox reaction. The number of electrons exchanged is the fundamental basis for establishing the stoichiometry of the redox process. This fact is crucial when establishing a mass balance, as will be done by modeling sewer processes (cf. Chapters 5 and 6). The OX value is, by definition, a key element in determination of this number. 

VAN AKEN et al. 0) and EDWARDS et al. (2) made clear that two sets of fundamental parameters are useful in describing vapor-liquid equilibria of volatile weak electrolytes, (1) the dissociation constant(s) K of acids, bases and water, and (2) the Henry s constants H of undissociated volatile molecules. A thermodynamic model can be built incorporating the definitions of these parameters and appropriate equations for mass balance and electric neutrality. It is complete if deviations to ideality are taken into account. The basic framework developped by EDWARDS, NEWMAN and PRAUSNITZ (2) (table 1) was used by authors who worked on volatile electrolyte systems the difference among their models are in the choice of parameters and in the representation of deviations to ideality. 

Hydrocarbon-degrading microorganisms are ubiquitous in most ecosystems [32,117] however, it is often very difficult to prove that transformation, degradation, and mineralization actually occur in the environment because it is difficult to distinguish contributions from biotic and abiotic processes under uncontrolled conditions in the natural environment [338]. In contrast, laboratory assays can provide definitive evidence for microbial degradation, and sterilized samples can be used to determine abiotic losses. Thus, contributions from microbial degradation can be differentiated from abiotic loss by a mass balance... 

Each of these dissociation reactions also specifies a definite equilibrium concentration of each product at a given temperature consequently, the reactions are written as equilibrium reactions. In the calculation of the heat of reaction of low-temperature combustion experiments the products could be specified from the chemical stoichiometry but with dissociation, the specification of the product concentrations becomes much more complex and the s in the flame temperature equation [Eq. (1.11)] are as unknown as the flame temperature itself. In order to solve the equation for the n s and T2, it is apparent that one needs more than mass balance equations. The necessary equations are found in the equilibrium relationships that exist among the product composition in the equilibrium system. 

The system for which we want to write a total continuity equation is all the liquid phase in the tank. We call this a macroscopic system, as opposed to a microscopic system, since it is of definite and finite size. The mass balance is around the whole tank, not just a small, differential element inside the tank. 

Figure 2 Rogers system definition of the mass-balance method [2]...

Based on an overall mass balance, the mathematical relationships can be defined between the in- and outflows of the conversion system. First some of the most important definitions and assumptions of the conceptual model for the conversion system are presented on which the mathematical model is based. 

Figure 59 The system definition and the system boundary for the mass-balance, based on three-step model.

We obtain the same result by just noting that this difference when divided by dz is simply the definition of a derivative.] Both At and dz can be canceled in each term so the mass balance on species j becomes... 

Therefore, Eq, (4.27) together with the boundary conditions of Eqs. (4.28-4.31) provide a definition of the problem of pressure P(x, y, t) with an unknown boundary Xo(y, t) provided that function q(t) specifying the pressure at the exit from a point gat into a cavity is known. In practice, function P0(t) is known having determined the mass balance function q(t), the final formulation takes the form ... 

The assessment of degradation in pharmaceutical products involves two aspects of analytical measurement. First, a selective analytical method must be available for accurate assay of the parent drug compound, in order to correctly measure any loss. Second, methodology should be in place for quantification of the degradation products formed. Ideally, when degradation occurs, the measured amount of parent drug lost should correlate well with the measured increase in degradation products. This correlation is referred to as mass balance )- More recently, the International Conference on Harmonization (ICH) has provided a definition of mass balance material balance as follows ... 

The steady-state volume of distribution is 12 1/kg. Donepezil hydrochloride is approximately 96% bound to human plasma proteins. The distribution of donepezil hydrochloride in various body tissues has not been definitively studied. However, in a mass balance study conducted in healthy male volunteers, 240 h after the administration of a single 5 mg dose of 14C-labeled donepezil hydrochloride, approximately 28% of the label remained unrecovered. This suggests that donepezil and/or its metabolites may persist in the body for more than 10 days. 

The inherent chemical complexity of DOM presents many challenges to understanding the role of DOM in C and N cycling and other processes in aquatic ecosystems. The measurement of trace organic moieties in major fractions of DOM, such as fulvic acids, can provide valuable data for understanding sources and biogeochemical pathways. In field studies, multiple lines of evidence can be critical for definitive interpretation of results. The tracer approaches outlined in this chapter should be used in conjunction with mass balance and flux measurements, for example. 

The boundary conditions normally associated with Equation (9.14) are known as the Danckwerts or closed boundary conditions. They are obtained from mass balances across the inlet and outlet of the reactor. We suppose that the piping to and from the reactor is small and has a high Re. Thus, if we were to apply the axial dispersion model to the inlet and outlet streams, we would find Din = Dout = 0, which is the definition of a closed system. See Figure 9.8. The flux in the inlet pipe is due solely to convection and has magnitude Qi ain. The flux just inside the reactor at location z = 0+ has two components. One component, Qina(0+), is due to convection. The other component, —DAc[da/dz 0+, is due to diffusion (albeit eddy diffusion) from the relatively high concentrations at the inlet toward the lower concentrations within the reactor. The inflow to the plane at z = 0 must be matched by material leaving the plane at z = 0+ since no reaction occurs in a region that has no volume. Thus,... 

For process analysis including energy balances, the same procedure defined for mass balances is followed. The only change is that the source of equations, described in item 3 of the procedure, now includes the energy balance equation as well as the definition of different energy terms. Again, this procedure will be illustrated by example. 

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