Equilibrium multi-component

When oil and gas are produced simultaneously into a separator a certain amount (mass fraction) of each component (e.g. butane) will be in the vapour phase and the rest in the liquid phase. This can be described using phase diagrams (such as those described in section 4.2) which describe the behaviour of multi-component mixtures at various temperatures and pressures. However to determine how much of each component goes into the gas or liquid phase the equilibrium constants (or equilibrium vapour liquid ratios) K must be known. 

Equation (3.19) is valid for any species X. If, however, a multi-component material emits only atomic SN after attaining sputter equilibrium, X stands for elements and atoms only, and the total sputter yield Yean be written as ... 

Note that this equation holds for any component in a multi-component mixture. The integral on the right-hand side can only be evaluated if the vapor mole fraction y is known as a function of the mole fraction Xr in the still. Assuming phase equilibrium between liquid and vapor in the still, the vapor mole fraction y x ) is defined by the equilibrium curve. Agitation of the liquid in tire still and low boilup rates tend to improve the validity of this assumption. 

The framework for constructing such multi-component equilibrium models is the Gibbs phase rule. This rule is valid for a system that has reached equilibrium and it states that... 

Equilibrium data are thus necessary to estimate compositions of both extract and raffinate when the time of extraction is sufficiently long. Phase equilibria have been studied for many ternary systems and the data can be found in the open literature. However, the position of the envelope can be strongly affected by other components of the feed. Furthermore, the envelope line and the tie lines are a function of temperature. Therefore, they should be determined experimentally. The other shapes of the equilibrium line can be found in literature. Equilibria in multi-component mixtures cannot be presented in planar graphs. To deal with such systems lumping of consolutes has been done to describe the system as pseudo-ternary. This can, however, lead to considerable errors in the estimation of the composition of the phases. A more rigorous thermodynamic approach is needed to regress the experimental data on equilibria in these systems. 

Pure Solid in Contact with a Single- or Multi-Component Environment Considering the example of a Pt surface in contact and in thermodynamic equilibrium with an oxygen atmosphere, the surface free energy (5.5) becomes... 

Here va and va are the stoichiometric coefficients for the reaction. The formulation is easily extended to treat a set of coupled chemical reactions. Reactive MPC dynamics again consists of free streaming and collisions, which take place at discrete times x. We partition the system into cells in order to carry out the reactive multiparticle collisions. The partition of the multicomponent system into collision cells is shown schematically in Fig. 7. In each cell, independently of the other cells, reactive and nonreactive collisions occur at times x. The nonreactive collisions can be carried out as described earlier for multi-component systems. The reactive collisions occur by birth-death stochastic rules. Such rules can be constructed to conserve mass, momentum, and energy. This is especially useful for coupling reactions to fluid flow. The reactive collision model can also be applied to far-from-equilibrium situations, where certain species are held fixed by constraints. In this case conservation laws... 

For other cases, such as La3+ where more detail is required about the nature of the species present in solution, titration data can be computer fit to more complicated multi-equilibrium models containing Mx 1 v( OR)v forms whose stoichiometry is suggested by information gained from independent spectroscopic or kinetic techniques. One must be mindful of the pitfalls of simply fitting the potentiometric data to complex multi-component models for which there is no independent evidence for the various species. Without some evidence for the species put into the fit, the procedure simply becomes an uncritical mathematical exercise of adding and removing various real and proposed components until the goodness of fit is satisfactory. 

Stoichiometric saturation defines equilibrium between an aqueous solution and homogeneous multi-component solid of fixed composition (10). At stoichiometric saturation the composition of the solid remains fixed even though the mineral is part of a continuous compositional series. Since, in this case, the composition of the solid is invariant, the solid may be treated as a one-component phase and Equation 6 is the only equilibrium criteria applicable. Equations 1 and 2 no longer apply at stoichiometric saturation because, owing to kinetic restrictions, the solid and saturated solution compositions are not free to change in establishing an equivalence of individual component chemical potentials between solid and aqueous solution. The equilibrium constant, K(x), is defined identically for both equilibrium and stoichiometric saturation. 

The most common model for describing adsorption equilibrium in multi-component systems is the Ideal Adsorbed Solution (IAS) model, which was originally developed by Radke and Prausnitz [94]. This model relies on the assumption that the adsorbed phase forms an ideal solution and hence the name IAS model has been adopted. The following is a summary of the main equations and assumptions of this model (Eqs. 22-29). 

An equilibrium-flash calculation (using the same equations as in case A above) is made at each point in time to find the vapor and liquid flow rates and properties immediately after the pressure letdown valve (the variables with the primes F , F l, y], x j,.. . shown in Fig. 3.8). These two streams are then fed into the vapor and liquid phases. The equations describing the two phases will be similar to Eqs. (3.40) to (3.42) and (3.44) to (3.46) with the addition of (1) a multi-component vapor-liquid equilibrium equation to calculate Pi and (2) NC — 1 component continuity equations for each phase. Controller equations relating 1 to Fi and P to F complete the model. 

Equilibrium sedimentation technique working with a multi-component solvent forming a density gradient in a centrifugal field... 

The determination of individual binary equilibrium diagrams usually only involves the characterisation of a limited number of phases, and it is possible to obtain some experimental thermodynamic data on each of these phases. However, when handling multi-component systems or/and metastable conditions there is a need to characterise the Gibbs energy of many phases, some of which may be metastable over much of the composition space. 

One of the earliest examples of Gibbs energy minimisation applied to a multi-component system was by White et al. (1958) who considered the chemical equilibrium in an ideal gas mixture of O, H and N with the species H, H2, HjO, N, N2, NH, NO, O, O2 and OH being present. The problem here is to find the most stable mixture of species. The Gibbs energy of the mixture was defined using Eq. (9.1) and defining the chemical potential of species i as... 

CALPHAD calculation. To overcome this difficulty they combined a semi-empirical model from Reddy and Blander (1987, 1989) with an equilibrium CALPHAD calculation for the multi-component oxide system Si02-Al203-Ti02-CaO-MgO-MnO-FeO. The approach can be summarised as follows. 

The DICTRA programme is based on a numerical solution of multi-component diffusion equations assuming that thermodynamic equilibrium is locally maintained at phase interfaces. Essentially the programme is broken down into four modules which involve (1) the solution of the diffusion equations, (2) the calculation of... 

A variety of other thermodynamic functions may be evaluated from this. For example, the chemical potential — the quantity equalized in equilibrium calculations — of species / in a multi-component system is given by... 

In order to determine the products of decomposition for equilibrium reactions the Kistiakowsky-Wilson or the Springall Roberts rules can be applied as a starting point. From the products of decomposition the heat and temperature of explosion can then be calculated. The temperature of explosion can then be used to calculate the products of decomposition. In practice, this process is repeated many times until there is agreement between the answers obtained. Equilibria of complex reactions and of multi-component systems are today obtained by computer however, the ability to use tabulated data is useful in predicting the direction and extent of the reaction. 

An advantage of the Wilson equation is that it involves only two parameters per binary and may be extended, without further information, to estimate multi-component phase equilibrium behavior. 

A multi-component gas solubility model and a multi-component surface adsorption model are generally required to estimate the monomer concentration at active sites. If the latter equilibrium can be neglected then the gas-solubility in the suspending agent determines the monomer concentration near the active site, which changes significantly with temperature and pressure. 

Up to this point we have dealt with the thermodynamics of planar boundaries. Let us add several relations for curved interfaces. First, we have to establish an equivalent to the Gibbs-Thomson equation which holds for curved external surfaces in a multi-component system. For incoherent (fluid-like) interfaces, this can be done by considering Figure 10-5. From the equilibrium condition at constant P and T, one has... 

For preparative applications, the expensive and configurationally unstable donor 128 can be simply prepared in situ by the action of ribose 5-phosphate isomerase (EC 5.3.1.6) on D-ribose 5-phosphate (39). This technique was applied to the stereoselective synthesis of d-[1-13C] fructose 6-phosphate 38 from [13C] formaldehyde [376,377] which also included a second enzymatic isomerization of the D-arafrino-3-hexulose 6-phosphate intermediate 129 into the more stable 2-hexulose derivative 38. Notable are the conflicting demands for high substrate levels (necessary to shift the fully reversible multi-component equilibrium) versus the notorious enzyme inactivation that occurs at higher formaldehyde concentrations. 

There have been few studies reported in the literature in the area of multi-component adsorption and desorption rate modeling (1, 2,3., 4,5. These have generally employed simplified modeling approaches, and the model predictions have provided qualitative comparisons to the experimental data. The purpose of this study is to develop a comprehensive model for multi-component adsorption kinetics based on the following mechanistic process (1) film diffusion of each species from the fluid phase to the solid surface (2) adsorption on the surface from the solute mixture and (3) diffusion of the individual solute species into the interior of the particle. The model is general in that diffusion rates in both fluid and solid phases are considered, and no restrictions are made regarding adsorption equilibrium relationships. However, diffusional flows due to solute-solute interactions are assumed to be zero in both fluid and solid phases. 

To perform a further detailed process calculation on this multi-component absorption and stripping process, vapor liquid equilibrium data for methane, hydrogen, and carbon monoxide is... 

After a discussion of the fundamental concepts in Section II, we present, in Section III, an approach to the lineshape theory of dynamic NMR spectra which comprises the most general case, namely that of a multi-component system where various intra- and inter-molecular exchange processes take place. We believe that a fully correct NMR theory of such an equilibrium has not been put forward yet. Section IV is concerned with the methods of simulation and analysis of complicated dynamic spectra. In Section V, we present our views on solving the numerous practical problems which usually appear upon the application of the theory to the analysis of dynamic spectra. 

For safety reasons, the cycling operation is interrupted during the weekend. After an interruption, always happening at the end of a discharge phase, several cycles are necessary to join the continuous curve representative of the evolution of the behaviour of the tank. This phenomenon tends to demonstrate the importance of the kinetics of the multi-component adsorption equilibrium in the case of a complex mixture of gas. 

---