Multicomponent model

Thirdly, the multicomponent model was applied to the case of crystallization of a random A-B copolymer by Helfand and Lauritzen [127]. Their main result is that the composition of, 4 s and B s in the crystal is determined by kinetic, rather than equilibrium considerations the inclusion of excess B increases with growth rate. 

Although the pure component LRC coefficients, as applied in the multicomponent model of equation (4), generally gave an excellent fit of the breakthrough loadings, there was one series of runs (the low concentration C02/air/5A system) for which the pure component LRC s did not provide a satisfactory fit of the data. For these cases, the A3 and Xq constants for the pure component CO2/5A LRC correlation were modified to provide a better fit of the breakthrough loadings. 

Deeper insight into the consequences of counterion condensation is gained by an effective monomer-monomer and counterion-counterion potential, respectively. The idea is to reduce the multicomponent system (macromolecules + counterions) to effective one-component systems (macromolecules or counterions, respectively). We define the simplified model in such a way that the effective potential between the counterions or monomers, respectively, of the new system yields exactly the same correlation function (gcc, gmm) as found in the multicomponent case at the same density. Starting from the correlation function gcc -respectively gmm-of the multi-component model we calculate an effective direct correlation function cefy via the one-component Ornstein-Zernike equation. An effective potential is then obtained from the RLWC closures of the one- and multicomponent models [24]. For low and moderate densities the effective potential is well approximated by... 

The protection of ecosystems, upon which our health and lives depend (J ), requires that we understand natural processes and develop the capability to predict the effect of changes, such as the addition of pollutants, on these ecosystems. The prediction of trace-element behavior in ecosystems requires a multicomponent model by which one can 1) calculate aqueous speciation of the trace elements among both natural organic and inorganic ligands ... 

While ultimately the objective of our project will be the theoretical analysis of the behavior of complex mixtures, at this point we only can correlate our results with previously developed multicomponent models. With the objective of predicting the total amount adsorbed and the adsorbed phase composition from no more than pure-component data, only a few specially selected systems have been used in previous experimental programs to verify the various models. Hence, these relationships, which usually were derived from theoretical considerations, have not been tested sufficiently yet. Thus, they all can be classified as empirical models. 

In general, the diffusive mass flux is composed of diffusion due to concentration gradients (chemical potential gradients), diffusion due to thermal effects (Soret diffusion) and diffusion due to pressure and external forces. It is possible to include the full multicomponent model for concentration gradient driven diffusion (Taylor and Krishna, 1993 Bird, 1998). In most cases, in the absence of external forces, it is... 

In this study we restrict our consideration by a class of ionic liquids that can be properly described based on the classical multicomponent models of charged and neutral particles. The simplest nontrivial example is a binary mixture of positive and negative particles disposed in a medium with dielectric constant e that is widely used for the description of molten salts [4-6], More complicated cases can be related to ionic solutions being neutral multicomponent systems formed by a solute of positive and negative ions immersed in a neutral solvent. This kind of systems widely varies in complexity [7], ranging from electrolyte solutions where cations and anions have a comparable size and charge, to highly asymmetric macromolecular ionic liquids in which macroions (polymers, micelles, proteins, etc) and microscopic counterions coexist. Thus, the importance of this system in many theoretical and applied fields is out of any doubt. 

A multicomponent model is often used to describe spherical micelles or globular proteins in solution. In this case the ions are treated as charged hard spheres immersed in a solvent of dielectric constant uy r. In this way the micellar solution is depicted as an electrolyte where the ions grossly differ in size and charge. The solvent averaged potential in this case is given by (2aab = aa+crb) the equation,... 

In the last figure to this section, Fig. 6, we present new MC data (symbols) for zp = —60 and a concentration of added +1 —1 electrolyte equal to 0.005 mol dm-3 as a function of the macroion concentration. In this case the increase of the probability of finding two counterions in contact is over 100 times No convergent IET result could be obtained for multicomponent model. 

To examine the potential of this new approach, we analyze the experimental data for the osmotic pressure of bovine serum albumin (BSA) in 0.15 mol dm-3 sodium chloride [112] and human serum albumin (HSA) solution in 0.1 molx dm-3 phosphate buffer [111]. According to a previous experimental and theoretical study [111] the two solutions differ substantially in the degree of protein association. The theoretically determined osmotic coefficient can be fitted to the experimental results to obtain the fraction of dimers in the solution. The results of our analysis are presented in Figs. 11 and 12. The protein molecular weights used in these calculations were 69,000 g/mol for BSA and 66,700 g/mol for HSA. The hard-sphere diameter of spherical proteins was assumed to be 6.0 nm. For the case of the multicomponent model, the ions of the low-molecular weight +1 — 1 electrolyte were modelled as charged hard spheres with diameter 0.4 nm. 

Figure 11. Experimental osmotic coefficient data (X) [41, 112] for bovine serum albumin (BSA) and the corresponding multicomponent model results (full line) at pH = 5.4. For pH = 7.3, experimental data are denoted by (+) and one-component model results by the dashed line.

Our analysis indicates no self-association of protein molecules for BSA solutions [112] at pH = 5.4 and 7.3 (Fig. 11). The fraction of dimers giving good agreement with experiment in this case is zero this holds true for both the one-component [41] and the multicomponent model. The results obtained by the two theoretical models for pH = 5.4, where experimental results are denoted by (x), practically coincide. Experimental data for pH=7.3, are denoted by (+) and one-component model results by the dashed line. In this case no IET results since the multicomponent model could be obtained for concentrations above 330 g/dm-3 and therefore only one-component calculations are shown. 

Vlachy, V., and Prausnitz, J.M. Donnan equilibrium - hypernetted-chain study of one-component and multicomponent models for aqueous polyelectrolyte solutions. Journal of Physical Chemistry, 1992, 96, No. 15, p. 6465-6469. 

Analysis of Molecular Weight Distribution Using Multicomponent Models... 

Step 1 Compute stage efficiencies from multicomponent model. 

The applicability of the multicomponent mass diffusion models to chemical reactor engineering is assessed in the following section. Emphasis is placed on the first principles in the derivation of the governing flux equations, the physical interpretations of the terms in the resulting models, the consistency with Pick s first law for binary systems, the relationships between the molar and mass based fluxes, and the consistent use of these multicomponent models describing non-ideal gas and liquid systems. 

A qualified question is then whether or not the multicomponent models are really worthwhile in reactor simulations, considering the accuracy reflected by the flow, kinetics and equilibrium model parts involved. For the present multiphase flow simulations, the accuracy reflected by the flow part of the model is still limited so an extended binary approach like the Wilke model sufEce in many practical cases. This is most likely the case for most single phase simulations as well. However, for diffusion dominated problems multicomponent diffusion of concentrated ideal gases, i.e., for the cases where we cannot confidently designate one of the species as a solvent, the accuracy of the diffusive fluxes may be significantly improved using the Maxwell-Stefan approach compared to the approximate binary Fickian fluxes. The Wilke model might still be an option and is frequently used for catalyst pellet analysis. 

In this work, we have analyzed the phase behavior of various freeze-dried mixtures of DPPE, DPPC, and cholesterol and have examined the effects of trehalose addition to these liposomes. Generally, dehydration leads to increase in transition temperature of the phospholipids and also to phase separation. Addition of trehalose, however, can prevent the increase in transition temperature and phase separation freeze-dried DPPC-cholesterol liposomes exhibit only one transition and their retention capability increases by more than 40%. Further studies on the phase separation and stability of multicomponent model membranes will be required to understand better its relation to the survival of cells to freeze-drying procedures. 

The equilibrium ratio [Equation (12.1)] involves physical equilibrium between phases. The system involved may be binary or multicomponent and ideal or nonideal, according to the terminology used in solution thermodynamics. Methods for correlating or predicting equilibria are based on an application of thermodynamics to each phase and to the solutions of the components within each phase. The techniques are described in standard reference works such as Walas and Reid et al. For multicomponent systems, the usual approach is to model the equilibria for each of the binary pairs and then combine the binary models in special ways to obtain the multicomponent model. 

Thus the classical evolution scheme of (Ca-, Na-rich) smectite-illite/smectite-(K-rich) illite-(K-rich) muscovite should be replaced by a more complex multiphase/ multicomponent model, in which K-rich smectite, Na-, NH4- and Ca-rich illite and white mica also play significant roles. 

A multicomponent model of adsorption can be reduced to three main components first, H2S, second, VOC with average properties of found species, and third, H2O. Moreover, H2O adsorption can be described as quasi-equilibrium due to the high concenbation of H2O and the long operation time of the carbon bed. Rectangular type isotherms are used to model the equilibrium of VOC and H2S. The amount of H2S adsorbed depends on available space left after VOC adsorption and it can be described bj equation ... 

Isotherm models have to be divided into single-component and multicomponent models. Because most of the multicomponent models are directly derived from single-component models, the latter will be presented first. 

Sun SS, Chumlea WC, Heimsfield SB et al. 2003. Development of bioelectrical impedance analysis prediction equations for body composition with the use of a multicomponent model for use in epidemiological surveys. Am Clin Nutr 77,331-340. 

The diad and G-centered triad intensities measured for the whole polymers are reported in Table I. As expected, all three samples fitted poorly to one-conq)onent 1 order Markovian (or Bemoullian) models. The mean deviations in all three cases are larger than 1%. Instead, the data fitted reasonably well to two-component 1 order Markovian models. This finding is consistent with the latest NMR studies on alginates (16,17) which also indicated that alginates NMR data should be fitted not to one-con onent models, but to multicomponent models. 

Despite the increasing popularity of LBM in simulating complex fluid systems, one should also be aware of the limitations of this novel approach. At present, high Mach number flows in aerodynamics are still difficult for LBM, and a consistent thermohydrodynamic scheme is absent. For multiphase/multicomponent models, the interfacial thickness is usually large and the density ratio across the interface can only be small when compared with real fluids. Nevertheless, advances and applications of this method during the past two decades have proven its potential in computational physics, including computational microfiuidics. 

In the multicomponent model, relative pressure is defined as x = P/Pd, where is the dew point pressure. The average compressibility ratio Zav is the ratio of the compressibility of the vapor at pressure P to the vapor compressibility at the dew point P - As the dew point is approached, Zav approaches unity, and so in the neighborhood of the dew point may be taken as unity. 

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