Equilibrium behavior, model

One of the simplest cases of phase behavior modeling is that of soHd—fluid equilibria for crystalline soHds, in which the solubility of the fluid in the sohd phase is negligible. Thermodynamic models are based on the principle that the fugacities (escaping tendencies) of component are equal for all phases at equilibrium under constant temperature and pressure (51). The soHd-phase fugacity,, can be represented by the following expression at temperature T ... 

Unfortunately, the analysis of chemical absorption is far more complex than physical absorption. The vapor-liquid equilibrium behavior cannot be approximated by Henry s Law or any of the methods described in Chapter 4. Also, different chemical compounds in the gas mixture can become involved in competing reactions. This means that simple methods like the Kremser equation no longer apply and complex simulation software is required to model chemical absorption systems such as the absorption of H2S and C02 in monoethanolamine. This is outside the scope of this text. 

Two classes of mathematical models have been developed those which are specific and attempt to describe the transport and degradation of a chemical in a particular situation and those which are general or "evaluative" and attempt to generally classify the behavior of chemicals in a hypothetical environment. The type of modeling discussed here, equilibrium partitioning models, fall into the latter category. Such models attempt, with a minimum of information, to predict expected environmental distribution patterns of a compound and thereby identify which environmental compartments will be of primary concern. 

In the above subsection it was demonstrated that the inclusion of electrostatic interactions into the pressure-area-temperature equation of state provides a better fit to the observed equilibrium behavior than the model with two-dimensional neutral gas. Considering this fact, we would like to devote our attention now to the character of the lipid film under the dynamical, nonequilibrium conditions. In the following we shall describe the dynamical behavior of the phospholipid(l,2-dipalmitoyl-3-sn-phosphatidylethanolamines DPPE) thin films in the course of the compression and expansion cycles at air/water interface. 

The non-random two-liquid segment activity coefficient model is a recent development of Chen and Song at Aspen Technology, Inc., [1], It is derived from the polymer NRTL model of Chen [26], which in turn is developed from the original NRTL model of Renon and Prausznitz [27]. The NRTL-SAC model is proposed in support of pharmaceutical and fine chemicals process and product design, for the qualitative tasks of solvent selection and the first approximation of phase equilibrium behavior in vapour liquid and liquid systems, where dissolved or solid phase pharmaceutical solutes are present. The application of NRTL-SAC is demonstrated here with a case study on the active pharmaceutical intermediate Cimetidine, and the design of a suitable crystallization process. 

A mechanism is determined from these data by choosing one which is consistent with the overall equilibrium behavior and which correctly matches the rate relationships derived for the postulated mechanism e.g., assuming the bimolecular adsorption/desorption reaction mechanism, as given in Equation 1, and using the kinetic model described above, the following relationship between xp and reactant and product concentrations can be derived (see Appendix C) ... 

The reaction is reversible and therefore the products should be removed from the reaction zone to improve conversion. The process was catalyzed by a commercially available poly(styrene-divinyl benzene) support, which played the dual role of catalyst and selective sorbent. The affinity of this resin was the highest for water, followed by ethanol, acetic acid, and finally ethyl acetate. The mathematical analysis was based on an equilibrium dispersive model where mass transfer resistances were neglected. Although many experiments were performed at different fed compositions, we will focus here on the one exhibiting the most complex behavior see Fig. 5. 

Processes in which chemical reaction and phase equilibria are simultaneously of significance present a considerable challenge to the thermodynamicist. The challenge is both to develop models which are suitable to describe the mixtures and to find computational procedures which permit analysis of equilibrium behavior. 

A simplified equilibrium extraction model (Fig. 6) was presented by Dordick and colleagues [188] to explain the resolution behavior of glycoproteins in affinity based reverse micellar extraction and separation (ARMES). Their system for the study includes soybean peroxidase (SBP, MW 37 KDa, pi 4.1) and aj-acid glycoprotein (AGP, MW 43 KDa, pi 3.7) as glycoprotein solutes, concanavalin A (ConA) as the affinity ligand in AOT/isooctane RMs. The separation factor (a) for the separation of SBP from AGP can be given by... 

We have shown that the iGLE, interpreted as a nonstationary ( irreversible ) GLE, satisfies the correct equilibrium behavior in quasi-equilibrium limits as well as more generally in illustrative models. ... 

The work that was performed in this set of experiments was an extension of work performed by Inoue and Kaufman (7). In the previous work, the migration of strontium in glauconite was modeled using conditions of local equilibrium for flows up to 6.3 kilometers per year (72 cm/hr). The differences between the predicted and experimental results in the experiments performed by Inoue and Kaufman may be due to the existence of non-equilibrium behavior. 

An accurate representation of the phase equilibrium behavior is required to design or simulate any separation process. Equilibrium data for salt-free systems are usually correlated by one of a number of possible equations, such as those of Wilson, Van Laar, Margules, Redlich-Kister, etc. These correlations can then be used in the appropriate process model. It has become common to utilize parameters from such correlations to obtain insight into the fundamentals underlying the behavior of solutions and to predict the behavior of other solutions. This has been particularly true of the Wilson equation, which is shown below for a binary system. 

Liquid-Phase Models. Theoretical models of the liquid state are not as well established as those for gases consequently, the development of general equations for the description of liquid-phase equilibrium behavior is not far advanced. Cubic equations of state give a qualitative description of liquid-phase equilibrium behavior, but do not generally yield good quantitative results (3). For engineering calculations, equations and estimation techniques developed specifically for liquids must normally be used. 

Peters and Luthy (1993, 1994) performed a detailed analysis of the equilibrium behavior of solvent coal tar water mixtures in work that was complementary to column studies performed by Roy, et. al. (1995). Peters and Luthy successfully modeled ternary phase diagrams of coal tar/n-butylamine/water systems. In addition, Peters and Luthy identified n-butylamine as the leading solvent for coal tar extraction. Pennell and Abriola (1993) report the solubilization of residual dodecane in Ottawa sand using a nonionic surfactant, polyoxyethylene sorbitan monooleate, which achieved a 5 order of magnitude increase over the aqueous solubility, but is still 7 times less than the equilibrium batch solubility with the same surfactant system. 

Experimental results are presented for high pressure phase equilibria in the binary systems carbon dioxide - acetone and carbon dioxide - ethanol and the ternary system carbon dioxide - acetone - water at 313 and 333 K and pressures between 20 and 150 bar. A high pressure optical cell with external recirculation and sampling of all phases was used for the experimental measurements. The ternary system exhibits an extensive three-phase equilibrium region with an upper and lower critical solution pressure at both temperatures. A modified cubic equation of a state with a non-quadratic mixing rule was successfully used to model the experimental data. The phase equilibrium behavior of the system is favorable for extraction of acetone from dilute aqueous solutions using supercritical carbon dioxide. 

A model based on a modified mixing rule for the Peng-Robinson equation of state was able to reproduce quantitatively all features of the observed phase equilibrium behavior, with model parameters determined from binary data only. The use of such models may substantially facilitate the task of process design and optimization for separations that utilize supercritical fluids. 

Although the multicomponent Langmuir equations account qualitatively for competitive adsorption of the mixture components, few real systems conform quantitatively to this simple model. For example, in real systems the separation factor is generally concentration dependent, and azeotrope formation (a = 1.0) and selectivity reversal (a varying from less than 1.0 to more than 1.0 over the composition range) are relatively common. Such behavior may limit the product purity attainable in a particular adsorption separation. It is sometimes possible to avoid such problems by introducing an additional component into the system which will modify the equilibrium behavior and eliminate the selectivity reversal. 

Thermodynamic non-idealities are taken into account while calculating necessary physical properties such as densities, viscosities, and diffusion coefficients. In addition, non-ideal phase equilibrium behavior is accounted for. In this respect, the Elec-trolyte-NRTL model (see Section 9.4.1) is used and supplied with the relevant parameters from Ref. [50]. The mass transport properties of the packing are described via the correlations from Refs. [59, 61]. This allows the mass transfer coefficients, specific contact area, hold-up and pressure drop as functions of physical properties and hydrodynamic conditions inside the column to be determined. 

This chapter describes in a step-by-step manner a generic strategy that we have been using for the development of various industrial processes. The chapter begins with a discussion of the overall workflow for process development, followed by a description of the way in which a process is synthesized, which relies heavily on the use of phase diagrams. The deviation from equilibrium behavior is accounted for using a model-based approach. This is illustrated with an example on the asymmetric transformation of an enantiomer. 

A useful literature relating to polypeptide and protein adsorption kinetics and equilibrium behavior in finite bath systems for both affinity and ion-ex-change HPLC sorbents is now available160,169,171-174,228,234 319 323 402"405 and various mathematical models have been developed, incorporating data on the adsorption behavior of proteins in a finite bath.8,160 167-169 171-174 400 403-405 406 One such model, the so-called combined-batch adsorption model (BAMcomb), initially developed for nonporous particles, takes into account the dynamic adsorption behavior of polypeptides and proteins in a finite bath. Due to the absence of pore diffusion, analytical solutions for nonporous HPLC sorbents can be readily developed using this model and its two simplified cases, and the effects of both surface interaction and film mass transfer can be independently addressed. Based on this knowledge, extension of the BAMcomb approach to porous sorbents in bath systems, and subsequently to packed-, expanded-, and fluidized-bed systems, can then be achieved. 

Pure component physical property data for the five species in our simulation of the HDA process were obtained from Chemical Engineering (1975) (liquid densities, heat capacities, vapor pressures, etc.). Vapor-liquid equilibrium behavior was assumed to be ideal. Much of the flowsheet and equipment design information was extracted from Douglas (1988). We have also determined certain design and control variables (e.g., column feed locations, temperature control trays, overhead receiver and column base liquid holdups.) that are not specified by Douglas. Tables 10.1 to 10.4 contain data for selected process streams. These data come from our TMODS dynamic simulation and not from a commercial steady-state simulation package. The corresponding stream numbers are shown in Fig. 10.1. In our simulation, the stabilizer column is modeled as a component splitter and tank. A heater is used to raise the temperature of the liquid feed stream to the product column. Table 10.5 presents equipment data and Table 10.6 compiles the heat transfer rates within process equipment. 

Nonlinear optimization techniques have been applied to determine isotherm parameters. It is well known (Ncibi, 2008) that the use of linear expressions, obtained by transformation of nonlinear one, distorts the experimental error by creating an inherent error estimation problem. In fact, the linear analysis method assumes that (i) the scatter of points follows a Gaussian distribution and (ii) the error distribution is the same at every value of the equilibrium liquid-phase concentration. Such behavior is not exhibited by equilibrium isotherm models since they have nonlinear shape for this reason the error distribution gets altered after transforming the data... 

Even at steady state, efficiencies vary from component to component and with position in a column. Thus, if the column is not at steady state, then efficiencies also must vary with time as a result of changes to flow rates and composition inside the column. Thus, equilibrium-stage models with efficiencies should not be used to model the dynamic behavior of distillation and absorption columns. Nonequilibrium models for studying column dynamics are described hy, e.g., Kooijman and Taylor [AlChE 41, 1852 (1995)], Baur et al. [Chem. 

The sensitivity to the band tail slope accoimts for the change in defect density with deposition conditions and annealing. Low deposition temperatures, below the usual 200-250 C, result in an increased defect density, reaching above 10 cm" with room temperature deposition. The band tails are much broader at the low deposition temperatures, so that Eq. (6.50) predicts the higher defect density. When the low deposition temperature material is annealed, the defect density is reduced. In Section 2.3.4, this behavior was taken as evidence that the material was not initially in equilibrium. The model... 

Dynamic Performance. Most models do not attempt to separate the equilibrium behavior trom the mass-transfer behavior. Rather they treat adsorption as one dynamic process with an overall dynamic response of the adsorbent bed to the feed stream. Although numerical solutions can be attempted for the rigorous partial differential equations, simplifying assumptions are often made to yield more manageable calculating techniques. 

We have utilized a chemical model in this investigation to interpret the equilibrium behavior of iron in acid mine waters. A successful correlation between calculated and measured Eh values has been found, using WATEQ2, the computerized ion association model. This correlation supports the basic assumption of homogeneous solution equilibrium in these waters and simultaneously corroborates both the validity of the aqueous model and the quantitative interpretation of Eh measurements in these waters. This interpretation makes it possible to calculate the distribution of iron... 

The nonlinear one-zone models can be generalized by the introduction of explicitly stochastic terms for any of the rates. The easiest, and physically most interesting, term to introduce is that of time-variable infall. In this case, Ferrini et al. have shown that there are analytic solutions possible for the simple two-component gas model which agree well with both the equilibrium behavior predicted by the linearized models and the deterministic systems. 

In this section we have assumed that the adsorption isotherm of an adsorbate is unaffected by the presence of constituents other than the adsorbate in the fluid mixture. If such ideaiir> is assumed for the Langmuir isotherm developed in the previous example, you could use the derived expression for any gaseous system containing carbon tetrachloride and the same activated carbon. In reality, however, the presence of other solutes that have an affinity for the carbon surface alters the CCI4 equilibrium behavior. An accurate system representation would require data or models for the complete multicomponent mixture. 

Comins H. N. and McMurtrie R. E. (1993) Long-term response of nutrient-limited forests to CO2 enrichment—equilibrium behavior of plant-soil models. Ecol. Appl. 3(4), 666-681. 

The single-molecule nature of the gene allows the gene in steady state to behave much differently from what the macroscopic laws postulate. Even the law of mass action is modified as seen in Fig. 28.4. Again, most models of gene regulation have assumed macroscopic equilibrium behavior and would obey the law of mass action but this simple picture breaks down due to singlemolecule effects. 

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