Equilibria numerical solution

For noncoustaut diffusivity, a numerical solution of the conseiwa-tion equations is generally required. In molecular sieve zeohtes, when equilibrium is described by the Langmuir isotherm, the concentration dependence of the intracrystalline diffusivity can often be approximated by Eq. (16-72). The relevant rate equation is ... 

Displacement Development A complete prediction of displacement chromatography accounting for rate factors requires a numerical solution since the adsorption equilibrium is nonlinear and intrinsically competitive. When the column efficiency is high, however, useful predictious can be obtained with the local equilibrium theoiy (see Fixed Bed Transitions ). 

FIG. 18 Scaling plot of L t) from the numerical solution of the rate equations for quenches to different equilibrium states [64] from an initial exponential MWD with L = 2. The final mean lengths are given in the legend. The inset shows the original L t) vs t data. Here = (0.33Lqo). ... 

The orders of reaction, U , ivith respect to A, B and AB are obtained from the rate expression by differentiation as in Eq. (11). In the rare case that we have a complete numerical solution of the kinetics, as explained in Section 2.10.3, we can find the reaction orders numerically. Here we assume that the quasi-equilibrium approximation is valid, ivhich enables us to derive an analytical expression for the rate as in Eq. (161) and to calculate the reaction orders as ... 

Eq. (122) represents a set of algebraic constraints for the vector of species concentrations expressing the fact that the fast reactions are in equilibrium. The introduction of constraints reduces the number of degrees of freedom of the problem, which now exclusively lie in the subspace of slow reactions. In such a way the fast degrees of freedom have been eliminated, and the problem is now much better suited for numerical solution methods. It has been shown that, depending on the specific problem to be solved, the use of simplified kinetic models allows one to reduce the computational time by two to three orders of magnitude [161],... 

While offering a more inherently realistic method of solution, however, the technique may cause some additional problems in the numerical solution, since high values of Kl can lead to increased stiffness in the differential equations. Thus in using this technique, a compromise between the approach to equilibrium and the speed of numerical solution may have to be adopted. Continuous single-stage extraction is treated in the simulation example EQEX. Reaction with integrated extraction is demonstrated in simulation example REXT. 

Because the more complicated model that required numerical solution still neglected important effects, we chose to use a simple analytical model for convenience. We chose Oddson s because its major features had been verified by Huggenberger (15, 16) for lindane, one of the compounds in our study. Oddson included the kinetics of adsorption by assuming that the rate of adsorption is proportional to the difference between the amount that has already adsorbed and the equilibrium value ... 

The other variables in Equations 3.32-3.34 are either known values, such as the equilibrium constants K and reaction coefficients v, or, in the case of the activity coefficients y, yj and activities aw, a, values that can be considered to be known. In practice, the model updates the activity coefficients and activities during the numerical solution so that their values have been accurately determined by the time the iterative procedure is complete. 

Treatment of the full six-step kinetic scheme above with the SSH leads to very cumbersome expressions for Cg, Cgj, etc., such that it would be better to use a numerical solution. These can be simplified greatly to lead to a rate law in relatively simple form, if we assume (1) the first four steps are at equilibrium, and (2) kri = kr2 ... 

The additional number of differential equations and increased complexities of the equilibrium relationships may also be compounded by computational problems caused by widely differing magnitudes in the equibbrium constants for the various components. As discussed in Section 3.3.2, it is shown that this can lead to widely differing values in the equation time constants and hence to stiffness problems for the numerical solution. 

This again shows that the component balance may thus have different time constants, which depend on the relative magnitudes of the equilibrium constants Ki which again can lead to problems of numerical solution due to equation stiffness. 

When equilibrium between the fluid and the solid cannot be assumed, it may still be possible to obtain analytical solutions for beds operating non-isothermally. In general, however, it will be necessary to look for numerical solutions. This problem has been summarised by Ruth vex 16. ... 

The numerical solution of the theory provides us with the complete picture of the interface, namely the functions X /(z), p (z), Jt(z), P a),fc z),f., mps (z) and s(z). These fundions can also be used to calculate the equilibrium eledrochemistry. To do this we solve the theory for increasing eledrode potentials and calculate the redox and... 

A unique solution for the equilibrium concentrations of each ion is obtained by fixing the temperature and chloride concentration. The resulting atmospheric level of CO2 can also be calculated. An example of the numerical solution to this multicomponent equilibrium concentration calculation is shown in Table 21.10. The predicted major ion concentrations are close to the observed values. Nevertheless, this model is not widely accepted as realistic because little evidence has been found for the establishment of equilibria between seawater and the solid phases. In feet, concentration gradients in the bottom and pore waters suggest that equilibrium is not being attained (Figure 21.2). This model is also not able to predict chloride concentrations because the major sedimentary component (halite) is nowhere near saturation with respect to average seawater. 

In Figure 1 we compare our numerical solutions with the molecular dynamics computer simulations of Thompson, et al. (7). In this comparison we use liquid and vapor densities obtained from the simulation studies. In the next section we obtain the required boundary values by approximate evaluation of vapor-liquid equilibrium for a small system. 

The DICTRA programme is based on a numerical solution of multi-component diffusion equations assuming that thermodynamic equilibrium is locally maintained at phase interfaces. Essentially the programme is broken down into four modules which involve (1) the solution of the diffusion equations, (2) the calculation of... 

In eq. (18), Mi t) and Mzit) are the magnetizations at time t for sites 1 and 2 respectively, M oc) is the equilibrium value of the magnetization in either site, R is the spin-lattice relaxation rate (= 1/Ti) for the two sites, which are assumed to be equal, and k is the exchange rate. For a larger system, numerical solutions are readily available with standard eigenvalue methods [42-44]. 

Electron attachment rates have been measured for numerous solutes. Many of these studies were limited to three solvents cyclohexane, 2,2,4-trimethylpentane, and tetrame-thylsilane (TMS), and those rates are discussed here. What to expect in other liquids can be inferred from these results. Considerable insight has been gained into certain reactions. Equilibrium reactions of electrons are particularly interesting since they provide information not only on energy levels, as mentioned above, but also on the partial molar volume of trapped electrons. This has led to a better understanding of the mechanism of electron transport. 

The radial functions in Eq.(ll) are numerical solutions of monodimensional Schrodinger equations, in which the potential corresponds to the three-dimensional one in which all degrees of freedom are frozen at their equilibrium values except that radial coordinate under consideration[40, 31]. 

The structure of the present subsection is as follows. First, the governing b.v.p. is formulated and reduced, following the scheme of 4.2, to a system of algebraic equations. Then, two important limit cases are discussed counterion selectivity near equilibrium and selectivity at high concentration polarization. Finally, we present and discuss the results of a numerical solution of the above algebraic system for the intermediate range of deviations from equilibrium. 

Before turning to the appropriate numerical solution, we discuss next the two important limiting cases equilibrium membrane selectivity and selectivity very far from equilibrium, near the saturation of the voltage against current curve. 

We answer this complicated question by considering stoichiometry and then by using a spreadsheet to find a numerical solution. First, let s evaluate the reaction quotient to find out in which direction the reaction must proceed to reach equilibrium ... 

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