Mass balance equations equilibrium calculations

If he selects the still pressure (which for a binary system will determine the vapour-liquid-equilibrium relationship) and one outlet stream flow-rate, then the outlet compositions can be calculated by simultaneous solution of the mass balance and equilibrium relationships (equations). A graphical method for the simultaneous solution is given in Volume 2, Chapter 11. 

Fortunately, few of these variables are truly independent. Geochemists have developed a variety of numerical schemes to solve for equilibrium in multicomponent systems, each of which features a reduction in the number of independent variables carried through the calculation. The schemes are alike in that each solves sets of mass action and mass balance equations. They vary, however, in their choices of thermodynamic components and independent variables, and how effectively the number of independent variables has been reduced. 

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 combined effects of a +2.9%o equilibrium fractionation between Fe(III)aq and Fe(II)jq, and a +2.0%o fractionation between Fe(Ill)aq and ferrihydrite upon precipitation is illustrated in Figure 10. Although a k ki ratio of 5 appears to fit the fractionations measured by Bullen et al. (2001) (Fig. 10), we note that at these relatively low kjk ratios. Equation (16) cannot be used, but instead the calculations are made using an incremental approach and simple isotope mass-balance equations (e.g., Eqn. 12). As the kjki ratio increases to 20, Equation (16) may... 

In the case of the flash calculations, different algorithms and schemes can be adopted the case of an isothermal, or PT flash will be considered. This term normally refers to any calculation of the amounts and compositions of the vapour and the liquid phase (V, L, y,-, xh respectively) making up a two-phase system in equilibrium at known T, P, and overall composition. In this case, one needs to satisfy relation for the equality of fugacities (eq. 2.3-1) and also the mass balance equations (based on 1 mole feed with N components of mole fraction z,) ... 

In order to calculate the equilibrium partitioning of a substance within the model world, a mass balance exercise is conducted. Thus, for a system comprising simply air, water, and soil, the mass balance equation would be... 

First, pseudocomponents determined by the quadrature method may well be unrealistic ones for instance, if the label x is (proportional to) the number of carbon atoms, pseudocomponents may well correspond to noninteger jcjc values. This may be aesthetically unpleasant, but it does not represent a real problem. More seriously, the appropriate pseudocomponents obviously depend on the composition and, hence, in a repeated calculation such as is required in a distillation tower, pseudocomponents will need to be different at each step. This puts out of tilt the mass balance equations that are coupled to the equilibrium ones, and, even if this problem could be circumvented (as, at least in principle, it can), the procedure would certainly not be applicable to existing software for distillation column calculations. 

We can now calculate the surface acidity equilibrium constants (equations 20 and 21). There are five species, =FeOH, =FeOH, =FeO", H , OH", that are interrelated by the two acidity mass law constants (equations 20 and 21), by the ion product of water Kw = [H l [OH"]) and two mass balance equations ... 

Peak profiles can be calculated with a proper column model, the differential mass balance equation of the compound(s), the adsorption isotherm, the mass transfer kinetics of the compound(s) and the boundary and initial conditions [13], When a suitable column model has been chosen, the proper parameters (isotherm and mass transfer parameters and experimental conditions) are entered into the calculations. The results from these calculations can have great predictive value [13, 114], The most important of the column models are the ideal model , the equilibrium-dispersive (ED) model , the... 

In these models, the mass balance equation (Eq. 2.2) is combined vHth a kinetic equation (Eq. 2.5), relating the rate of variation of the concentration of each component in the stationary phase to its concentrations in both phases and to the equilibrium concentration in the stationary phase [80-93]. Although in principle kinetic models are more exact than the equilibrium-dispersive model, the difference between the individual band profiles calculated using the equilibrium-dispersive model or the linear driving force model, for example, is negligible when the rate constants are not very small i.e., when the column efficiency exceeds a few him-dred theoretical plates), as shown in Chapter 14 (Section 14.2). 

For mass balance reasons, Cbfl = Cafi. The problem is to find a relationship between the isotherm parameters and the parameters and of the f-th harmonic. In the calculations made to relate the equilibrium isotherm and the response of the system (Eq. 3.102), the equilibrium-dispersive model is used (Chapter 2, Section 2.2.2) and the mass balance equation is integrated with the Danck-werts boundary conditions (Chapter 2, Section 2.1.4.3) and with the initial conditions C = Ca,o, q = qiCa,o). 

All cases of practical importance in liquid chromatography deal with the separation of multicomponent feed mixtures. As shown in Chapter 2, the combination of the mass balance equations for the components of the feed, their isotherm equations, and a chromatography model that accounts for the kinetics of mass transfer between the two phases of the system permits the calculation of the individual band profiles of these compounds. To address this problem, we need first to understand, measure, and model the equilibrium isotherms of multicomponent mixtures. These equilibria are more complex than single-component ones, due to the competition between the different components for interaction with the stationary phase, a phenomenon that is imderstood but not yet predictable. We observe that the adsorption isotherms of the different compounds that are simultaneously present in a solution are almost always neither linear nor independent. In a finite-concentration solution, the amount of a component adsorbed at equilib-... 

The equilibrium-dispersive model had been discussed and studied in the literature long before the formulation of the ideal model. Bohart and Adams [2] derived the equation of the model as early as 1920, but it does not seem that they attempted any calculations based on this model. Wicke [3,4] derived the mass balance equation of the model in 1939 and discussed its application to gas chromatography on activated charcoal. In this chapter, we describe the equilibrium-dispersive model, its historical development, the inherent assumptions, the input parameters required, the methods used for the calculation of solutions, and their characteristic features. In addition, some approximate analytical solutions of the equilibrium-dispersive model are presented. 

Calculate the concentration of all species present in a solution containing 1.00 M and 0.010 M Cd(N03)2 at 25°C. Because the solution is strongly acidic, Cd-OH complexing need not be considered. Solving the problem requires the simultaneous solution of equilibrium constant expressions and mass-balance equations involving the aqueous species. We are given stepwise formation (equilibrium) constant expressions for the Cd-Cl complexes that can be reformatted to give... 

Although the solubilities of quartz, its polymorphs, and amorphous silica are fixed and pH-in-dependent in most natural waters, the dissociation of silicic acid at alkaline pH s leads to substantial increases in their solubilities above pH 9 to 10. The following calculations show how we can predict this effect. First, the solubility of any silica solid must equal the sum of the concentrations of all species of silica in solution at equilibrium. This summation is given by the mass-balance equation... 

Numerical Methods and Data Structure. Both EQ3NR and EQ6 make extensive use of a combined method, using a "continued fraction" based "optimizer" algorithm, followed by the Newton-Raphson method, to make equilibrium calculations. The method uses a set of master or "basis" species to reduce the number of iteration variables. Mass action equations for the non-basis species are substituted into mass balance equations, each of which corresponds to a basis species. 

Note the difference between this method of calculation and the one used in the previous illustration. There we did vapor-liquid equilibrium calculations only for the conditions needed, and then solved the mass balance equations analytically. In this illustration we first had to do vapor-liquid equilibrium calculations for all compositions (to construct the. t- v diagram), and then for this binary mixture we were able to do all further calculations graphically. As shown in the following discussion, this makes it easier to consider other reflux ratios than the one u.sed in this illustration. 

The predictions of three-phase equilibria considered so far were done as two separate two-phase calculations. Although applicable to the examples here, such a procedure cannot easily be followed in a three-phase flash calculation in which the temperature or pressure of a mixture of two or more components is changed so that three phases are formed. In this case the equilibrium relations and mass balance equations for all three phases must be solved simultaneously to find the compositions of the three coexisting phases. It is left to you (Problem 11.3-7) to develop the algorithm for such a calculation. 

These speciation diagrams are calculated from the equilibrium constant for formation of each species plus mass balance equations. In this section we describe the use of equilibrium constants in modeling the speciation in a natural water. 

However, one should be aware of the following three important issues when geochemical models are applied to determine the solid-phase control of soil solutions (1) Solid-phase chemistry is based on the assumption of equilibrium. Therefore, soil solutions to be tested should be close to a steady-state condition (i.e., the condition in which little change has occurred in the major ions involved). (2) The mass-balance equation for each dissolved species should contain all possible solution species to ensure accurate calculation of the free concentration of the dissolved species. Omission of any significant solution species from the mass-balance equation will cause overestimation of the free concentration of the dissolved species. (3) Variations occur in the equilibrium constants for solution species and solid phases. All these factors could lead to misinterpretation of solid-phase equilibria in soil solutions. 

For a number of nonlinear and competitive isotherm models analytical solutions of the mass balance equations can be provided for only one strongly simplified column model. This is the ideal model of chromatography, which considers just convection and neglects all mass transfer processes (Section 6.2.3). Using the method of characteristics within the elegant equilibrium theory, analytical expressions were derived capable to calculate single elution profiles for single components and mixtures (Helfferich and Klein, 1970 Helfferich and Carr 1993 Helfferich and Whitley 1996 Helfferich 1997 Rhee, Aris, and Amundson, 1970 ... 

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