The basic mass balances

The reactants may be brought into the reactor by a number of different pipes for since we are assuming that there is good internal mixing, there is no need to premix the feed qf always refers to the total volume feed rate. 

The first two terms on the right-hand side are due to advection, the third to reaction. If qf — q and, therefore, T is constant, we may divide through by q and get 

A similar balance on the number of moles of B in the tank would give a similar equation. 

The differences between the two equations are (1) there is no term in 5 since we assumed that the feed was pure A (2) the sign of the reaction term is changed since the reaction uses up A but forms B. 

Here then are two equations for the two concentrations a and 5, but since there is only one reaction we might suspect that only one equation should be needed. This is true under certain circumstances. Suppose, for example, that at time t = 0 the concentrations in the tank were a = and b — and that the total concentration, a + o, were equal to the total feed concentration, Of. Then writing c = a + 5 and adding Eqs. (7.1.3) and (7.1.4) would give 

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When one of the above three resistances is predominant and the other two are negligible, the basic mass balance differential equation can be integrated and gives expressions for the relation between the extent of reaction and time, both being experimentally observable. Typical results... 

The change of mass (or moles) of oxygen isotope within the small cube with volume dV can alternatively be calculated from the change of mass, dm, of the system. Mass gain or loss of can be calculated by multiplying the concentration of by the volume of the system, dV. Hence we obtain for the basic mass balance equation ... 

Note that the concentration gain of an isotope in the system is equal to the negative of the change in flux of this isotope. The dV terms can be dropped to yield the basic mass balance equation for stable isotope transport (or any other element) ... 

Let us start by formulating the basic mass balance in spherical coordinates. The spherical cousin of equation (7-2) is... 

Bartlett et al. (2004) have modeled COj in naturally ventilated classrooms occupied by children. The only air exchange present in naturally ventilated classrooms is provided by infiltration and exfiltration, mostly through open doors and windows. Bartlett et al. (2004) gave the basic mass balance equation for COj as... 

The basic mass balance equation for drying can be expressed as in the following equation ... 

Ffowever, from the basic mass balance equation (6.4.67),... 

We resort to the basic mass balance equation to obtain the average distillate composition used in the case of constant reflux mode. 

The component mass balance equation, combined with the reactor energy balance equation and the kinetic rate equation, provide the basic model for the ideal plug-flow tubular reactor. 

To date, in-bed filtration was more or less a black box as any measurement within the bed was virtually impossible. The design of such filters was based on models that could be validated only by integral measurements. However, with the MRI method even the (slow) dynamics of the filtration process can be determined. The binary gated data obtained by standard MRI methods are sufficient for the quantitative description of the system. With spatially resolved measurements the applicability of basic mass balances based on improved models can be shown in detail. 

These equations can be solved simultaneously with the material balance equations to obtain x[, x, xf and x1,1. For a multicomponent system, the liquid-liquid equilibrium is illustrated in Figure 4.7. The mass balance is basically the same as that for vapor-liquid equilibrium, but is written for two-liquid phases. Liquid I in the liquid-liquid equilibrium corresponds with the vapor in vapor-liquid equilibrium and Liquid II corresponds with the liquid in vapor-liquid equilibrium. The corresponding mass balance is given by the equivalent to Equation 4.55 ... 

Biomass and substrate must be separately described to establish a concept for classification of wastewater directed toward a description of the microbial processes. For several reasons, e.g., to allow widespread application and to observe a basic mass balance, the organic matter expressed in terms of COD is a central parameter for wastewater quality. According to the concepts used in the active sludge models, the classification of wastewater in a sewer network can also be subdivided as outlined in Figure 3.1 (Henze et al., 1987, 1995a, 2000). A direct interaction between sewer and treatment plant processes is therefore within reach. 

The DO mass balance in a sewer can be established at different levels. Basically, it is either expressed by the conceptual model, i.e., Equation (5.9) that is the same as column number 5 in the matrix, Table 5.3, or simply stated as follows (Parkhurst and Pomeroy, 1972 Jensen and Hvitved-Jacobsen, 1991 Matos and de Sousa, 1991, 1996) ... 

Since the basic model generated by the Program Builder Block looks similar in many ways to the previously illustrated model, we have not shown it here. One difference in this case is that we have an expression for solid-liquid equilibria which is written from the last mass balance expression entered by the user and is then written by the Program Builder as ... 

Rather than proceed by trying to read a reaction factor fA from Fig. 4.3, it is better to set out the basic material balance for mass transfer and reaction as below. Locating the position of 0 on Fig. 4.3 does however confirm that reaction will be occurring in the main bulk of the liquid and that an agitated tank is a suitable type of reactor. 

Equation (8) constitutes the basic thermodynamic equation for the calculation of the radius of the globules. Of course, explicit expressions, in terms of the radius of the globules and volume fraction, are needed fort, C and af before such a calculation can be carried out. Expressions for Af will be provided in another section of the paper, but it is difficult to derive expressions fory and C. One may, however, note that y (and also C) depends on the radius for the following two reasons (1) its value depends upon the concentrations of surfactant and cosurfactant in the bulk phases, which, because the system is closed, depend upon the amounts adsorbed on the area of the internal interface of the microemulsion (2) in addition to the above mass balance effect, there is a curvature effect on y (this point is examined later in the paper). 

FBA is attractive as a predictive tool. That is, once the model is constructed, one can predict the metabolic behavior under a number of different conditions. The basic principle underlying FBA is the steady-state conservation of mass, energy, and redox potential. A dynamic mass balance can be written arormd each metabolite (A,) within a metabolic network. Fig. 4 shows a hypothetical network with the fluxes (V) affecting a metabolite (A,). The dynamic mass balance for A,- is ... 

The species mass balance for the gas mixture in the pellet pores are written on the form (11.28). To define the alternative model versions the effective reaction term Sc = RcPcati — e) in the basic model species mass balance is substituted with a modified source term Sc being different in the various model versions ... 

Before delving into specific mathematical models, it might be helpful to look at the basic mass conservation equation, because some terms that represent adsorbent properties appear in it. The material balance equation for solute A is... 

In the first case, mass transfer and reaction occur at different locations and are necessarily sequential—what reacts at the surface must first have got there by mass transfer. Mathematically, the equations for mass transfer are the same whether a reaction occurs or not. The reaction merely determines the boundary condition at the catalyst surface. In contrast, in the second case, mass-transfer and reaction occur side by side and simultaneously in the same volume elements. Here, mass-transfer enters as a source-or-sink term in the basic material-balance equation. 

Standing of source-receptor relationships for nonreactive species in an airshed. The.se methods include the chemical mass balance (CMB) used for. source apportionment, the principal component analysis (PCA) used for source identification, and the empirical orthogonal function (EOF) method used for identification of the location and strengths of emission sources. A detailed review of all the variations of these basic methods is outside the scope of this book. For more information the reader is referred to treatments by Watson (1984), Henry et al. (1984), Cooper and Watson (1980), Watson et al. (1981), Macias and Hopke <1981), Dattner and Hopke (1982), Pace (1986), Watson et al. (1989), Gordon (1980, 1988), Stevens and Pace (1984), Hopke (1985, 1991), and Javitz et al. (1988). 

One can see immediately that this approach will be a little more detailed than the previous section, since the bubble mechanics are contained in the basic material balance. Various simplifications of equation (8-164) are possible according to the reactor type by deleting terms for a steady-state CSTR the time derivative is zero, for a batch reactor the flow rate terms are zero, etc. For the gas phase the situation is complicated by the fact that the configuration (and concentration) of bubbles can be a function of both time and position that is, the total mass of j in a given bubble, PVt,yj RT), can depend on both position z and time t. 

The homogeneous diffusion model is slightly more complex in cyUndrical coordinates relative to the model described above in rectangular coordinates. Additional complexity arises because the radial term of the Laplacian operator (V V = V ) accounts for the fact that the surface area across which radial diffusion occurs increases linearly with dimensionless coordinate r/ as one moves radially outward. Basic information for = f(t]) is obtained by integrating the dimensionless mass balance with radial diffusion and chemical reaction ... 

The basic information for molar density 4 a( ) is obtained by solving the dimensionless mass balance which includes the appropriate Hougen-Watson model ... 

The simplest model takes into account only convective transport and thermodynamics. It assumes a permanently established local equilibrium between mobile and stationary phases. This model is frequently called the ideal or basic model of chromatography (Guiochon et al., 2006). It was described first by Wicke (1939) for the elution of a single component. Subsequently, De Vault (1943) derived in more detail the corresponding mass balance. 

Adding the above two equations, we obtain the total mass balance equation (9.2-lb), which basically states that the total hold-up in both phases is governed by the net total flux contributed by the two phases. This is true irrespective of the rate of mass exchange between the two phases whether they are finite or infinite. Now back to our finite mass exchange conditions where the governing equations are (9.4-3a) and (9.4-3b). To solve these equations, we need to impose boundary conditions as well as define an initial state for the system. One boundary is at the center of the particle where we have the usual symmetry ... 

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