Equilibrium Constraints

The estimated true values must satisfy the appropriate equilibrium constraints. For points 1 through L, there are two constraints given by Equation (2-4) one each for components 1 and 2. For points L+1 through M the same equilibrium relations apply however, now they apply to components 2 and 3. The constraints for the tie-line points, M+1 through N, are given by Equation (2-6), applied to each of the three components. 

As is well known, we can consider the ensemble of many molecules of water either at equilibrium conditions or not. To start with, we shall describe our result within the equilibrium constraint, even if we realize that temperature gradients, velocity gradients, density, and concentration gradients are characterizations nearly essential to describe anything which is in the liquid state. The traditional approaches to equilibrium statistics are Monte Carlo< and molecular dynamics. Some of the results are discussed in the following (The details can be found in the references cited). 

Bird, D.K. and Helgeson, H.C. (1981) Chemical interaction of aqueous solution with epidote-feldspar mineral assemblages in geologic systems II, Equilibrium constraints in metamorphic/geothermal processes. Am. J. Set, 281, 576-614. 

The addition of Equations 8 and 10, subject to equilibrium constraints, yields... 

The variable / depends on the particular species chosen as a reference substance. In general, the initial mole numbers of the reactants do not constitute simple stoichiometric ratios, and the number of moles of product that may be formed is limited by the amount of one of the reactants present in the system. If the extent of reaction is not limited by thermodynamic equilibrium constraints, this limiting reagent is the one that determines the maximum possible value of the extent of reaction ( max). We should refer our fractional conversions to this stoichiometrically limiting reactant if / is to lie between zero and unity. Consequently, the treatment used in subsequent chapters will define fractional conversions in terms of the limiting reactant. 

Appendix 4A. The Magnitudes of Equilibrium Constraints in Cell Systems. 183... 

These levels are illustrated in Figure 1.1. Levels (1) and (2) are domains of kinetics in the sense that attention is focused on reaction (rate, mechanism, etc.), perhaps in conjunction with other rate processes, subject to stoichiometric and equilibrium constraints. At the other extreme, level (3) is the domain of CRE, because, in general, it is at this level that sufficient information about overall behavior is required to make decisions about reactors for, say, commercial production. Notwithstanding these comments, it is possible under certain ideal conditions at level (3) to make the required decisions based on information available only at level (1), or at levels (1) and (2) combined. The concepts relating to these ideal conditions are introduced in Chapter 2, and are used in subsequent chapters dealing with CRE. 

Figure 21.8 shows schematically the linear energy equation or operating line added to lines shown in Figure 21.7, both for an exothermic reaction, Figure 21.8(a), and for an endothermic reaction, Figure 21.8(b). Both cases illustrate the limits imposed by the equilibrium constraint for reversible reactions. 

Phase changes are typically associated with the evaporation of any suspended liquid phase in an aerosol release. As air is mixed with an aerosol, equilibrium constraints cause additional evaporation of the liquid phase which reduces the temperature of the liquid phase (and the vapor phase if thermal equilibrium is maintained). 

Vapor-liquid equilibria data are often correlated using two adjustable parameters per binary mixture. In many cases, multicomponent vapor-liquid equilibria can be predicted using only binary parameters. For low pressures, the equilibrium constraint is... 

The product composition from the fuel reformer generally consists of 35 to 40% H2 and 6 to 10% CO balanced with H20, C02, and N2 [39], The CO is further reduced to 2 to 3% by HT WGS and then down to <0.5% CO with LT WGS. It is not possible to reduce the concentration of CO down to a few ppm with LT WGS because of equilibrium constraints. This has to be done with preferential oxidation of CO in the last step. However, owing to the development of high-temperature PEMFC... 

As shown by Moore (1984 126), the velocity is only equal to the multiplier under the assumption that the period of production and the period dining which the multiplier is completed are the same as the period during which a given volume of money circulates. Under this assumption, the magnitude of spill-over effects, associated with an initial money outlay on investment through the multiplier process, provides an equilibrium constraint on the size of the velocity.8... 

In the above MINLP model, the potential recovery is the objective function (Eqn. 26) subject to other property and phase equilibrium constraints (Eqn s. 27-40). The decisions that need to be made are the identities of the solvent and anti-solvent (binary variables) and the composition of solvent and anti-solvent... 

The structural constraints used in the first case study namely, Eqn s 27,28 and 29 are used again. The melting point, boiling point and flash point, are used as constraints for both solvent and anti-solvent. Since the solvent needs to have high solubility for solute and the anti-solvent needs to have low solubility for the solute limits of 17 <8 < 19 and 5 > 30 (Eqn s. 33 and 37) are placed on the solubility parameters of solvent and anti-solvents respectively. Eqn.38 gives the necessary condition for phase stability (Bernard et al., 1967), which needs to be satisfied for the solvent-anti solvent pairs to be miscible with each other. Eqn. 39 gives the solid-liquid equilibrium constraint. 

In both situations the interaction of the medium inside the pore with the pore wall (1) is increased (2) or changed which affect the transport and separation properties (surface diffusion, multilayer adsorption) and/or help overcome equilibrium constraints in membrane reactors. Membrane modifications can be performed by depositing material in the internal pore structure from liquids (impregnation, adsorption) or gases. Several modification possibilities are schematically shown in Figure 2.3. Some results obtained by Burggraaf, Keizer and coworkers are summarized in Table 2.7. Composite structures on a scale of 1-5 nm were obtained. 

The equilibrium constraint of Equation (13.3) can be met only if = P, which is the condition for mechanical equilibrium. (We will discuss several special cases to which this requirement does not apply.) Or, to put the argument differently, if the pressures of two phases are different, the phase with the higher pressure will spontaneously expand and the phase with the lower pressure will spontaneously contract, with a decrease in A, until the pressures are equal. Thus, for p phases, p — independent relationships among the pressures of the phases can be written as follows ... 

As discussed earlier, the equilibrium constraint of low vapor pressure is particularly important for stability of the organic aerosols. Organic aerosol formation requires accumulation of condensable species in excess to their gas-phase saturation concentrations. In turn, examination of vapor-pressure for various oxygenated... 

From a practical viewpoint they suffer from serious product inhibition and/or equilibrium constraints. One way of avoiding the latter issue is to couple the transamination step to a second enzymatic step which removes the coproduct In the example shown in Scheme 6.16 acetophenone undergoes TA-catalyzed transamination using L-alanine as the amine donor. The unfavorable equilibrium and... 

In designing a wall-cooled tubular reactor, we want to operate such that the trajectory stays near the maximum rate for all temperatures. Thus for an exothermic reversible reaction the temperature should increase initially while the conversion is low and decrease as the conversion increases to stay away from the equilibrium constraint. One can easily program a computer to compute conversion and T versus t to attain a desired conversion for rninimum T in a PFTR. These curves are shown in Figure 5-17 for the three situations. 

To attain conversions higher than attainable fi om equilibrium constraints in a single-phase process and... 

It is intuitively clear that the structure of resultant (25) should reflect the equilibrium constraint (30). Let us call the binomial... 

The methane decomposition reaction is severely constrained by equilibrium. A few studies have also been undertaken to circumvent the equilibrium constraints.Otsuka and coworkers used the addition of CaNis to Ni/Si02 for cheating equilibrium. The physical mixture of CaNis and Ni/Si02 showed greater than equilibrium methane (decomposition) conversion due to the hydrogen absorption property of CaNis. 

The model satisfied microscopic reversibility with the equilibrium constraints determined a priori from free energy data. 

The chemical equilibrium assumption often results in modeling predictions similar to those obtained assuming infinitely fast reaction, at least for overall aspects of practical systems such as combustion. However, the increased computational complexity of the chemical equilibrium approach is often justified, since the restrictions that the equilibrium constraint places on the reaction system are accounted for. The fractional conversion of reactants to products at chemical equilibrium typically depends strongly on temperature. For an exothermic reaction system, complete conversion to products is favored thermodynamically at low temperatures, while at high temperatures the equilibrium may shift toward reactants. The restrictions that equilibrium place on the reaction system are obviously not accounted for by the fast chemistry approximation. 

Once cooled, the effluent from the process separates into a liquid-water phase and a gaseous phase, the latter containing mainly carbon dioxide along with oxygen which was in excess of the stoichiometric requirements, and nitrogen. The separation is carried out in multiple stages in order to minimize erosion of valves, as well as to maximise the separation due to phase-equilibrium constraints. 

Many additional consistency tests can be derived from phase equilibrium constraints. From thermodynamics, the activity coefficient is known to be the fundamental basis of many properties and parameters of engineering interest. Therefore, data for such quantities as Henry s constant, octanol—water partition coefficient, aqueous solubility, and solubility of water in chemicals are related to solution activity coefficients and other properties through fundamental equilibrium relationships (10,23,24). Accurate, consistent data should be expected to satisfy these and other thermodynamic requirements. Furthermore, equilibrium models may permit a missing property value to be calculated from those values that are known (2). 

The large effective heat capacity of the liquid-solid slurry absorbent enables relatively small slurry flows to absorb the carbon dioxide heat of condensation with only modest absorber temperature rise. This contrasts with other acid gas removal processes in which solvent flows to the carbon dioxide absorber are considerably larger than flows determined by vapor-liquid equilibrium constraints. Large flows are required to provide sensible heat capacity for the large absorber heat effects. Small slurry absorbent flows permit smaller tower diameters because allowable vapor velocities generally increase with reduced liquid loading (8). 

While some reconciliation models only have material balance relationships, more meaningful reconciliation results are obtained with models that include material balances, heat balances, equilibrium constraints (both in the separation and reaction domains), rate relationships (heat transfer, mass transfer, momentum transfer, and kinetics), as well as equipment-specific relationships. In other words, one should include more than just material balance constraints when reconciling a model. Heat balance, kinetics, transport relationships—if needed for the... 

We are considering an irreversible reaction in this book. For reversible reactions, Luyben2 demonstrated the opposite effect of recycle flowrate on production rate, specifically, increasing recycle flowrate increases production rate. With reversible reactions, the impact of temperature is less important than the impact of concentrations because of the reaction equilibrium constraints. 

Thorstenson and Plummer (1977), in an elegant theoretical discussion (see section on The Fundamental Problems), discussed the equilibrium criteria applicable to a system composed of a two-component solid that is a member of a binary solid solution and an aqueous phase, depending on whether the solid reacts with fixed or variable composition. Because of kinetic restrictions, a solid may react with a fixed composition, even though it is a member of a continuous solid solution. Thorstenson and Plummer refer to equilibrium between such a solid and an aqueous phase as stoichiometric saturation. Because the solid reacts with fixed composition (reacts congruently), the chemical potentials of individual components cannot be equated between phases the solid reacts thermodynamically as a one-component phase. The variance of the system is reduced from two to one and, according to Thorstenson and Plummer, the only equilibrium constraint is IAP g. calcite = Keq(x>- where Keq(x) is the equilibrium constant for the solid, a function of... 

This result can be transformed further under the assumption that AIC2O4 equilibrates much more rapidly with Al3+ and C2O4 than does A1F2 with F" (otherwise it would not be useful to consider displacement of C2Ol by F, but rather the converse). Then the equilibrium constraint... 

Dfaz-Cruz, J.M., Agullo, J., Dfaz-Cruz, M.S., Arino, C., Esteban, M., and Tauler, R., Implementation of a chemical equilibrium constraint in the multivariate curve resolution of voltammograms from systems with successive metal complexes, Analyst, 126, 371-377, 2001. 

Experiments were executed in an autoclave at temperature between 130 and 180 °C, with alcohol/acid ratios between 1/9 to 27/1, as well as sulfated zirconia catalyst concentration up to 5 wt%. The experimental conditions preserved the chemical equilibrium constraint. Details are given elsewhere [2]. Two contributions in forming the reaction rate can be distinguished enhancement due to the solid catalyst and an autocatalysis effect by the fatty acid. Consequently, the following expression can be formulated for the overall reaction rate ... 

Another limitation is dictated by equilibrium constraints for the reforming Reactions (1) and (2), i.e., they do not reach completion. As a result, the reformate stream contains unconverted methane, carbon monoxide and carbon dioxide. Typically, at lower temperatures (600°C) the amount of carbon monoxide in the reformate gas is reduced to 2%, however the amount of methane increases to 28%. At high temperatures (900°C), the amount of methane in the reformate is reduced to 2%, however the amount of carbon monoxide increases to 20%. In UMR reaction equilibrium is shifted favorably by the carbon dioxide sorbent. 

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