Multicomponent relations

The analogue of the Clapeyron equation for multicomponent systems can be derived by a complex procedure of systematically eliminating the various chemical potentials, but an alternative derivation uses the Maxwell relation (A2.1.41)... 

Equations (2.15) or (2.16) are the so-called Stefan-Maxwell relations for multicomponent diffusion, and we have seen that they are an almost obvious generalization of the corresponding result (2.13) for two components, once the right hand side of this has been identified physically as an inter-molecular momentum transfer rate. In the case of two components equation (2.16) degenerates to... 

Though illustrated here by the Scott and Dullien flux relations, this is an example of a general principle which is often overlooked namely, an isobaric set of flux relations cannot, in general, be used to represent diffusion in the presence of chemical reactions. The reason for this is the existence of a relation between the species fluxes in isobaric systems (the Graham relation in the case of a binary mixture, or its extension (6.2) for multicomponent mixtures) which is inconsistent with the demands of stoichiometry. If the fluxes are to meet the constraints of stoichiometry, the pressure gradient must be left free to adjust itself accordingly. We shall return to this point in more detail in Chapter 11. 

We see, then, that pressure gradients must necessarily exist in catalyst pellets to free the fluxes from the constraints Imposed by Graham s relation (11,42), or Its generalization = 0 in multicomponent systems. Without this freedom the fluxes are unable to adjust to the demands... 

There are a number of physical properties of copolymers that are directly related to the multicomponent character of these materials and which an... 

In real-life appHcations, many other failure mechanisms are present and this type of curve is not necessarily obtained. For example, in a multicomponents system the quaUty related failures do not necessarily all drop out early but might be phased out over a longer period of time. 

Other chemometrics methods to improve caUbration have been advanced. The method of partial least squares has been usehil in multicomponent cahbration (48—51). In this approach the concentrations are related to latent variables in the block of observed instmment responses. Thus PLS regression can solve the colinearity problem and provide all of the advantages discussed earlier. Principal components analysis coupled with multiple regression, often called Principal Component Regression (PCR), is another cahbration approach that has been compared and contrasted to PLS (52—54). Cahbration problems can also be approached using the Kalman filter as discussed (43). 

From the definition of a partial molar quantity and some thermodynamic substitutions involving exact differentials, it is possible to derive the simple, yet powerful, Duhem data testing relation (2,3,18). Stated in words, the Duhem equation is a mole-fraction-weighted summation of the partial derivatives of a set of partial molar quantities, with respect to the composition of one of the components (2,3). For example, in an / -component system, there are n partial molar quantities, Af, representing any extensive molar property. At a specified temperature and pressure, only n — 1) of these properties are independent. Many experiments, however, measure quantities for every chemical in a multicomponent system. It is this redundance in reported data that makes thermodynamic consistency tests possible. 

UNIQUAC is significant because it provides a means to estimate multicomponent interactions using no more than binary interaction experimental data, bond angles, and bond distances. There is an implicit assumption that the combinatorial portion of the model, ie, the size and shape effects, can be averaged over a molecule and that these can be directly related to molecular surface area and volume. This assumption can be found in many QSAR methods and probably makes a significant contribution to the generally low accuracy of many QSAR prediction techniques. 

The general XT E problem involves a multicomponent system of N constituent species for which the independent variables are T, P, N — 1 liquid-phase mole fractions, and N — 1 vapor-phase mole fractions. (Note that Xi = 1 and y = 1, where x, and y, represent liquid and vapor mole fractions respectively.) Thus there are 2N independent variables, and application of the phase rule shows that exactly N of these variables must be fixed to estabhsh the intensive state of the system. This means that once N variables have been specified, the remaining N variables can be determined by siiTUiltaneous solution of the N equihbrium relations ... 

Problem Solving Methods Most, if not aU, problems or applications that involve mass transfer can be approached by a systematic-course of action. In the simplest cases, the unknown quantities are obvious. In more complex (e.g., iTmlticomponent, multiphase, multidimensional, nonisothermal, and/or transient) systems, it is more subtle to resolve the known and unknown quantities. For example, in multicomponent systems, one must know the fluxes of the components before predicting their effective diffusivities and vice versa. More will be said about that dilemma later. Once the known and unknown quantities are resolved, however, a combination of conservation equations, definitions, empirical relations, and properties are apphed to arrive at an answer. Figure 5-24 is a flowchart that illustrates the primary types of information and their relationships, and it apphes to many mass-transfer problems. 

Lejfler-Cullinan They extended their binaiy relation to an arbi-traiy multicomponent mixture, as follows ... 

Open-loop behavior of multicomponent distillation may be studied by solving modifications of the multicomponent equations of Distefano [Am. Inst. Chem. Eng. J., 14, 190 (1968)] as presented in the subsection Batch Distillation. One frequent modification is to include an equation, such as the Francis weir formula, to relate liquid holdup on a tray to liquid flow rate leaving the tray. Applications to azeotropic-distillation towers are particularly interesting because, as discussed by and ihustrated in the Following example from Prokopalds and Seider... 

McCormick [97] presents a correlation for Gilliland s chart relating reflux, minimum reflux, number of stages, and minimum stages for multicomponent distillation. Selecting a multiplier for actual reflux over minimum reflux is important for any design. Depending on the com-... 

For multicomponent systems, the relation of the system can be expressed using the relative volatility ... 

Multicomponent distillations are more complicated than binary systems due primarily to the actual or potential involvement or interaction of one or more components of the multicomponent system on other components of the mixture. These interactions may be in the form of vapor-liquid equilibriums such as azeotrope formation, or chemical reaction, etc., any of which may affect the activity relations, and hence deviations from ideal relationships. For example, some systems are known to have two azeotrope combinations in the distillation column. Sometimes these, one or all, can be broken or changed in the vapor pressure relationships by addition of a third chemical or hydrocarbon. 

Although this method has not found as much wide acceptance when referenced to use by designers or controversial discussion in the literature, it nevertheless allows a direct approximate solution of the average multicomponent system with accuracy of 1-8% average. If the key components are less than 10% of the feed, the accuracy is probably considerably less than indicated. If a split key system is considered, Scheibel reports fair accuracy when the split components going overhead are estimated and combined with the light key, the badance considered with the heavy key in the L/D relation. 

A family of interesting polycychc systems 106 related to pyrrolidines was obtained in a one-pot double intermolecular 1,3-dipolar cycloaddition, irradiating derivatives of o-allyl-sahcylaldehydes with microwaves in toluene for 10 min in presence of the TEA salt of glycine esters [71]. A very similar approach was previously proposed by Bashiardes and co-workers to obtain a one-pot multicomponent synthesis of benzopyrano-pyrrolidines 107 and pyrrole products 108 (Scheme 37). The latter cycloadducts were obtained when o-propargylic benzaldehydes were utihzed instead of o-allyhc benzalde-hydes, followed by in situ oxidation [72]. 

Diffusion of ions can be observed in multicomponent systems where concentration gradients can arise. In individnal melts, self-diffnsion of ions can be studied with the aid of radiotracers. Whereas the mobilities of ions are lower in melts, the diffusion coefficients are of the same order of magnitude as in aqueous solutions (i.e., about 10 cmVs). Thus, for melts the Nemst relation (4.6) is not applicable. This can be explained in terms of an appreciable contribntion of ion pairs to diffusional transport since these pairs are nncharged, they do not carry cnrrent, so that values of ionic mobility calculated from diffusion coefficients will be high. 

Table 5.4-3 summarizes the design equations and analytical relations between concentration, C/(, and batch time, t, or residence time, t, for a homogeneous reaction A —> products with simple reaction kinetics (Van Santen etal., 1999). Balance equations for multicomponent homogeneous systems for any reaction network and for gas-liquid and gas-liquid-solid systems are presented in Tables 5.4-7 and 5.4.8 at the end of Section 5.4.3. 

Although NMR spectrometers of operating frequencies > 400 MHz are cosdy and require specialist technical support staff, the technique provides a broad picture of the chemical modifications arising from the reactions of free radicals or related oxidants in complex, multicomponent systems such as intact biofluids, tissue sample... 

At this point we introduce the formal notation, which is commonly used in literature, and which is further used throughout this chapter. In the new notation we replace the parameter vector b in the calibration example by a vector x, which is called the state vector. In the multicomponent kinetic system the state vector x contains the concentrations of the compounds in the reaction mixture at a given time. Thus x is the vector which is estimated by the filter. The response of the measurement device, e.g., the absorbance at a given wavelength, is denoted by z. The absorbtivities at a given wavelength which relate the measured absorbance to the concentrations of the compounds in the mixture, or the design matrix in the calibration experiment (x in eq. (41.3)) are denoted by h. ... 

Equilibrium data correlations can be extremely complex, especially when related to non-ideal multicomponent mixtures, and in order to handle such real life complex simulations, a commercial dynamic simulator with access to a physical property data-base often becomes essential. The approach in this text, is based, however, on the basic concepts of ideal behaviour, as expressed by Henry s law for gas absorption, the use of constant relative volatility values for distillation and constant distribution coeficients for solvent extraction. These have the advantage that they normally enable an explicit method of solution and avoid the more cumbersome iterative types of procedure, which would otherwise be required. Simulation examples in which more complex forms of equilibria are employed are STEAM and BUBBLE. 

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