Equilibrium function

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. 

Many environmental applications involve dilute systems whose equilibrium functions can be linearized over the operating range to yield... 

A second equation is needed to determine the surface tension as a function of axial position. We adopt the quasistatic assumption that a is a unique equilibrium function of the surface excess concentration, T, even during dynamic events (17). A surface species continuity balance dictates how T varies along the interface. Upon neglect of surface diffusion and for h <1, the steady state form of this balance is... 

In the linearized LB equation Eq. (7), the ensemble averaged effect of the particle-particle collision is now represented by a relaxation of the distribution function ft to the equilibrium function /fq, where the matrix Ly does not necessarily have to correspond to an existing set of collision rules. The question now arises if L can be simplified even further to the form Ly = aSy, so that the LB equation takes the form (with x — -St/a)... 

At first sight this does not seem possible because the specific form of L, for instance Eq. (8), which followed from the requirement of symmetry. However, it turns out that b, c, and d can be put to zero, if an equilibrium function ffeq with different weights is used for the different directions, so that the lack of symmetry can be remedied (Qian et al., 1992 Ladd, 1994 Sued, 2001). Specifically, the/fq should take the form... 

Altmann, R. S. and Buffle, J. (1988). The use of differential equilibrium functions for the interpretation of metal binding in complex ligand systems its relation to site occupation and site affinity distributions, Geochim. Cosmochim. Acta, 52, 1505-1519. 

Abate G, Masini JC. Complexation of Cd(II) and Pb(II) with humic acids studies by anodic stripping voltammetry using differential equilibrium functions and discrete site models. Org. Geochem. 2002 33 1171-1182. 

Tighilet B, Lacour M. (1995). Pharmacological activity of the Ginkgo biloba extract (EGb 761) on equilibrium function recovery in the unilateral vestibular neurectomized cat. J Vestib Res. 5(3) 187-200. 

The knowledge of the complete set of functions plus that of the intcrmoleculuf forces would permit the compulation of all equilibrium properties of the liquid, and indeed if the inlcrmolecular forces are the suni of pair forces, only a knowledge of />. ai all P. 7 values is necessary. An adequate although numerically difficult, theory of the transport properties also exists, using the equilibrium functions. /> . At present, only qualitative success is obtained in the completely a priori use of the equations. 

It is important to emphasize the fact that the process of adsorption itself changes the equilibrium function p(E) by facilitating the formation of sectors with high heats of adsorption. 

When we assume equilibrium stages, we write Yj = F(Xj) for an equilibrium function F(Xj) that depends on the variable Xj only. 

By using an equilibrium function F of this linear form in equation (6.86), we obtain... 

For strongly asymmetric junctions (e.g. rR distribution function remains close to the equilibrium function in the right lead fR(c — efR), thus essentially simplifying the solution. While for symmetric junctions the distribution function has the double-step form and is very different from the equilibrium one. 

The procedures employed to evaluate the coefficients 5) and Qt and the explicit asymptotic forms for Q and S2 are given in Section III.B.8.a note representation (4.133) for the equilibrium function /0. 

The wave and pulse patterns of nonreactive separation processes, as well as the integrated reaction separation processes illustrated above, can be easily predicted with some simple graphical procedures derived from Eqs. (4) and (5). The behavior crucially depends on the equilibrium function y(x) in the nonreactive case, and on the transformed equilibrium function Y(X) in the reactive case. In addition to phase equilibrium, the latter also includes chemical equilibrium. An explicit calculation of the transformed equilibrium function and its derivatives is only possible in special cases. However, in Ref. [13] a numerical calculation procedure is given, which applies to any number of components, any number of reactions, and any type of phase and reaction equilibrium. 

For nonconvex equilibrium functions, combined wave solutions are possible which consist in parts of spreading waves and in others of constant pattern waves. In the equilibrium diagram combined wave solutions represent the convex hull of the equilibrium line, as illustrated in Fig. 5.5(c). 

New phenomena compared to nonreactive Langmuir systems are the same as in the binary case - that is, the existence of combined waves due to the occurrence of inflection points of the equilibrium functions y(x) or Y(X) and limitations on feasible product composition due to adsorptivity reversal similar to azeotropic distillation. Nonreactive examples for the latter were treated in Refs. [6 - 8], reactive examples will be discussed in the next section. 

Fig. 5.7. Transformed equilibrium function of a reactive distillation process with a reaction of type 2 A B + C and a mixture with constant relative volatilities (schematic)...

In summary, we find that the transformed equilibrium function provides useful insight not only into the steady-state behavior but also in to the transient behavior. This was illustrated here with a very simple example. However, an application to more complicated examples is straightforward, as shown in Ref. [13]. 

T, and other thermodynamic (equilibrium) functions can be defined and used even though equilibrium does not exist. 

The following step of the Onsager method involves the determination of the equilibrium function f(u) which minimizes functional (2.2). Unfortunately, the direct minimization leads to an integral equation, which cannot be solved in a standard way (see, however, Refs.29,30)), So, in Ref.75 the variational method was used, the trial function being chosen in the form... 

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