Nonlinear-Equilibrium

In the highly nonlinear equilibrium situations characteristic of liquid separations, the use of priori initial estimates of phase compositions that are not very close to the true compositions of these phases can lead to divergence of iterative computations or to spurious convergence upon feed composition. 

Now, we discuss briefly the situation when one or both of the adiabatic electronic states has/have nonlinear equilibrium geometry. In Figures 6 and 7 we show two characteristic examples, the state of BH2 and NH2, respectively. The BH2 potential curves are the result of ab initio calculations of the present authors [33,34], and those for NH2 are taken from [25]. 

Srinivas, B. K. and El-Halwagi, M. M. (1994), Synthesis of reactive mass-exchange networks with general nonlinear equilibrium functions. AlChE J., 40(3), 463-472. 

Figure 8 illustrates the performance of a system with three equilibrium stages in the absorber and six in the stripper. The actual steam requirement is 147 moles/mole SO2 (41.3 kg/kg). The use of a finite number of stages increases the steam requirement a factor of 2.5 from the case of infinite stages with a nonlinear equilibrium. 

As mentioned earlier, to solve explicitly for the temperature T2 and the product composition, one must consider a, mass balance equations, (/j, a) nonlinear equilibrium equations, and an energy equation in which one of the unknowns T2 is not even explicitly present. Since numerical procedures are used to solve the problem on computers, the thermodynamic functions are represented in terms of power series with respect to temperature. 

The introductory information presented above allows us to outline a certain hierarchy of electro-diffusional phenomena and levels of description that form the backbone of this monograph. At the bottom, simplest level of this hierarchy lie the nonlinear equilibrium effects to be treated in Chapter 2. The entire treatment here will be based upon the Poisson-Boltzmann equation. 

Three-degree-of-freedom model fitted to experimental data for the overall band structure by Schinke et al. [159] (nonlinear equilibrium geometry). 

The driving force for transport within the zeolite crystals appears to be the gradient of chemical potential rather than the concentration gradient, and, for systems with a nonlinear equilibrium isotherm, the diffusivity is therefore concentration dependent (6-8). 

Unfortunately the first simplifying assumption of a linear equilibrium relation in the mass-balance model is not very accurate for practical chemical/biological systems. Therefore we will also present numerical solutions for linear high-dimensional systems with nonlinear equilibrium relations. A model that accounts for mass transfer in each tray will be... 

Multistage Absorption with a Nonlinear Equilibrium Relation... 

This gives rise to a set of nonlinear equations that must be solved numerically. This more realistic and more accurate model of the nonlinear equilibrium case is much more interesting. 

All other nonlinear equilibrium relations Y = F(X) can be treated analogously. 

This difference is due to the different characteristics of linear and nonlinear equilibrium relations. 

Nonequilibrium Multistages with Nonlinear Equilibrium Relations... 

In this chapter we have presented multistage systems with special emphasis on absorption processes. We have studied multitray countercurrent absorption towers with equilibrium trays for both cases when the equilibrium relation is linear and when it is nonlinear. This study was accompanied by MATLAB codes that can solve either of the cases numerically. We have also introduced cases where the trays are not efficient enough to be treated as equilibrium stages. Using the rate of mass transfer RMT in this case, we have shown how the equilibrium case is the limit of the nonequilibrium cases when the rate of mass transfer becomes high. Both the linear and the nonlinear equilibrium relation were used to investigate the nonequi-librium case. We have developed MATLAB programs for the nonequilibrium cases as well. 

Rate-limited sorption can also be modeled assuming a kinetic rate expression coupled with a nonlinear equilibrium expression. If we assume a Freundlich isotherm and a first-order rate expression, we can use the following equation to model sorption kinetics [21] ... 

In this chapter we present a general-purpose transport model of the multireaction type. The model was successfully used to predict the adsorption as well as transport of several heavy metals in soils (Selim, 1992 Hinz and Selim, 1994 Selim and Amacher, 2001). Multireaction models are empirical and include linear and nonlinear equilibrium and reversible and irreversible retention reactions. A major feature of... 

FIGURE 12.5 Experimental and simulated (solid and dashed curves) BTCs of S04 effluent concentrations from the Bs horizon (column Bs-I, input S04 (C0) of 0.005 M). Simulations are for a range of n values where a nonlinear-equilibrium model was assumed. 

Parameter Estimates, Standard Errors, Root Mean Square Errors, and Correlation Coefficients for Linear and Nonlinear Equilibrium Models for Various Columns... 

The calculation of temperatures and equilibrium compositions of gas mixtures involves simultaneous solution of linear (material balance) and nonlinear (equilibrium) algebraic equations. Therefore, it is necessary to resort to various approximate procedures classified by Carter and Altman (Cl) as (1) trial and error methods (2) iterative methods (3) graphical methods and use of published tables and (4) punched-card or machine methods. Numerical solutions involve a four-step sequence described by Penner (P4). 

It is always found that M + L N, whence M (generally nonlinear) equilibrium equations and L (linear) atom-conservation equations determine the equilibrium composition (that is, may be solved... 

The importance of linear chromatography comes from the fact that almost all analytical applications of chromatography are carried out xmder such experimental conditions that the sample size is small, the mobile phase concentrations low, and thus, the equilibrixim isotherm linear. The development in the late 1960s and early 1970s of highly sensitive, on-line detectors, with detection limits in the low ppb range or lower, permits the use of very small samples in most analyses. In such cases the concentrations of the sample components are very low, the equilibrium isotherms are practically linear, the band profiles are symmetrical (phenomena other than nonlinear equilibrium behavior may take place see Section 6.6), and the bands of the different sample components are independent of each other. Qualitative and quantitative analyses are based on this linear model. We must note, however, that the assumption of a linear isotherm is nearly always approximate. It may often be a reasonable approximation, but the cases in which the isotherm is truly linear remain exceptional. Most often, when the sample size is small, the effects of a nonlinear isotherm (e.g., the dependence of the retention time on the sample size, the peak asymmetry) are only smaller than what the precision of the experiments permits us to detect, or simply smaller than what we are ready to tolerate in order to benefit from entertaining a simple model. 

For fixed beds, simplified methods of calculation are available when the equilibrium can be considered highly favorable, highly unfavorable, or linear. The general case of nonlinear equilibrium has been solved for only a few types of equilibrium equations (Section II, A, 3). The discussion immediately following (Section II, A, 2) gives a wide selection of equilibrium relationships which can be used empirically in cases allowing the algebraic calculation of column performance such cases are identified by italicized type descriptions. 

The additivity of tba individual resistances is dependant on tbe linearity of the flux expressions and of the equilibrium relationship. For nonlinear equilibrium relationships Eq. (2,4-IOa) and (2,4-lOb) can still ha used provided m is recognized to he a function of the interfacia] composition. Overall coefficients are often employed in the analysis of fluid-fluid mass transfer operations despite their complex dependence on the hydrodynamics, geometry and compositions of the two phases, In some instances the overall coefficients cen bs predicted from correlations for the individual coefficients for each phase provided the conditions in the apparatus are comparable to those for which the correlation was developed. 

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