System of mass balance equations

In a matrix form, the system of mass balance equation constraints (component matrix) reads... 

The systems of mass balance equations with the proper isotherm equations are integrated numerically to obtain the concentration profiles at the column outlet. 

The mass balance equation was derived and studied first by Wicke [3] and later and independently by Wilson [4] and DeVault [5], under different forms. In the derivation of this equation several assumptions are made. For the sake of clarity these assumptions will merely be stated in the text of Section 2.1.1. They are discussed separately in Section 2.1.2. As we show, their consequences are minor and do not limit significantly the range of validity of the system of mass balance equations finally derived. 

The band profiles will be obtained as the solution of the relevant system of mass balance equations (Eq. 2.2), completed with a relationship between each stationary phase concentration, Csj, and the mobile phase concentrations, C (Eq. 2.4 or 2.5), and with the proper set of initial (Eq. 2.6) and boundary conditions (Eqs. 2.8 to 2.15). There is an equation of each type for each component of the feed and of the mobile phase, except for the weak solvent in the mobile phase. However, the mass balance equations of the additives or strong solvents, whose retention factors... 

It has been shown by Giddings [67], Van Deemter et al. [29] and Haarhoff and Van der Linde [68] (see Section 2.2.6) that when the mass transfer kinetics are fast but not infinitely fast, the system of mass balance equations (Eq. 2.2) and kinetic equations (Eq. 2.5) can be replaced by the following equation ... 

Jacob et al. used the method of characteristics to discuss the general properties of the system of mass balance equations in multicomponent preparative gas chromatography (GC) [34-36], assuming either a linear or a nonlinear isotherm. The GC problem is more complicated than the HPLC one because the gas mobile phase is much more compressible than a solution and the mobile phase velocity is very different inside and outside a high concentration band because the partial molar volumes of compounds are much larger in the gas mobile phase than in the condensed stationary phase (the sorption effect). They showed that the method of characteristics appHes to multicomponent systems as well as to single component... 

Figure 11.3 Numerical solution of the system of mass balance equation for two components. Total concentration profile. Competitive Langmuir isotherm, O] = 12, U2 = 24, j = q 2 = 100. Phase ratio, f = 0.25 efficiency N = 2000 loading factor, 80%.

As described, PBPK models are systems of mass balance equations describing the mass or concentration of chemicals at various loci within the tissues. Most often the loci have been at the level of the whole organ. However, the mass can be described at as fine a resolution as is necessary and as knowledge allows. For example, a PBPK model can describe the concentration at the cellular or subcellular level (Blancato and Bischoff 1993). Ultimately, the chemical input into pharmacodynamic and systems level models could easily be the output of PBPK models at whatever the necessary resolution. For example, if a perturbation in a specific cell is directly related to a concentration or mass of chemical at that cell, the PBPK model would be expanded to describe that mass or concentration. This though raises... 

Consider in particular the system of mass balance equations (Chapter 3). Then the reduced incidence matrix C of graph G in (3.1.6) is partitioned... 

In a system of mass balance equations, the equation (9.6.7)j can be formulated as the mass balance of the reduced graph G, hence the residual r (9.6.8a) represents a vector of node imbalances. Then also an analysis of residuals makes sense see Remark to Section 9.4. 

FIGURE 9 System of mass balance equations describing the flow of metabolites through the urea cycle of hepato-cytes. Measurable fluxes are labeled with a star. 

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