Phase equilibrium algorithms

The phase equilibrium problem consists of two parts the phase stability calculation and the phase split calculation. For a particular total mixture composition, the phase stability calculation determines if that feed will split into two or more phases. If it is determined that multiple phases are present, then one performs the phase split calculation, assuming some specified number of phases. One must then calculate the stability of the solutions to the phase split to ascertain that the assumed number of phases was correct. The key to this procedure is performing the phase stability calculation reliably. Unfortunately, this problem—which can be formulated as an optimization problem (or the equivalent set of nonlinear equations)— frequently has multiple minima and maxima. As a result, conventional phase equilibrium algorithms may fail to converge or may converge to the wrong solution. 

As discussed above no equation of state can accurately calculate phase equilibrium for the components in 1-5 at all relevant conditions instead activity coefficient models must be used. In such models, the characteristic basic property used by the phase equilibrium algorithms is the K-value, defined as Ki=yi/xj where yi and Xi are the mole fractions in the vapour and liquid phases, respectively. 

The algorithm we used for solvent/polydisperse polymer equilibria calls for only one solvent/polymer interaction parameter. The interaction parameter (pto) i ed in the algorithm can be determined from essentially any type of ethylene/polyethylene phase equilibrium data. Cloud-point data have been used (18). while Cheng (16) and Harmony ( ) have done so from gas sorption data. 

A number of textbooks and review articles are available which provide background and more-general simulation techniques for fluids, beyond the calculations of the present chapter. In particular, the book by Frenkel and Smit [1] has comprehensive coverage of molecular simulation methods for fluids, with some emphasis on algorithms for phase-equilibrium calculations. General review articles on simulation methods and their applications - e.g., [2-6] - are also available. Sections 10.2 and 10.3 of the present chapter were adapted from [6]. The present chapter also reviews examples of the recently developed flat-histogram approaches described in Chap. 3 when applied to phase equilibria. 

The KTTS depends upon an absolute zero and one fixed point through which a straight line is projected. Because they are not ideally linear, practicable interpolation thermometers require additional fixed points to describe their individual characteristics. Thus a suitable number of fixed points, ie, temperatures at which pure substances in nature can exist in two- or three-phase equilibrium, together with specification of an interpolation instrument and appropriate algorithms, define a temperature scale. The temperature values of the fixed points are assigned values based on adjustments of data obtained by thermodynamic measurements such as gas thermometry. 

From the various possible closures, the mean spherical approximation (MSA) [189] has found particularly wide attention in phase equilibrium calculations of ionic fluids. The Percus-Yevick (PY) closure is unsatisfactory for long-range potentials [173, 187, 190]. The hypemetted chain approximation (HNC), widely used in electrolyte thermodynamics [168, 173], leads to an increasing instability of the numerical algorithm as the phase boundary is approached [191]. There seems to be no decisive relation between the location of this numerical instability and phase transition lines [192-194]. Attempts were made to extrapolate phase transition lines from results far away, where the HNC is soluble [81, 194]. 

Du, P. C., and Mansoori, G. A., Phase equilibrium computational algorithms of continuous thermodynamics. Fluid Phase Eq. 30,57 (1986). 

Robust and efficient mathematical algorithms for phase equilibrium calculations e.g., VLB, LLE, VLLE, and solid precipitation). 

In problems where the flux ratios are known (e.g., condensation and heterogeneous reacting systems where the reaction rate is controlled by diffusion) the mole fractions at the interface are not known in advance and it is necessary to solve the mass transfer rate equations simultaneously with additional equations (these may be phase equilibrium and/or reaction rate equations). For these cases it is possible to embed Algorithms 8.1 or 8.2 within another iterative procedure that solves the additional equations (as was done in Example 8.3.2). However, we suggest that a better procedure is to solve the mass transfer rate equations simultaneously with the additional equations using Newton s method. This approach will be developed below for cases where the mole fractions at both ends of the film are known. Later we will extend the method to allow straightforward solution of more complicated problems (see Examples 9.4.1, 11.5.2, 11.5.3, and others). 

The tie lines in Figures 1, 2, and 3 were calculated from the phase equilibrium relations represented by Equations 21 which upon substitution of the chemical potential models become Equations 22. There are three equations and four unknown molalities. For each tie line we therefore set a value for one of the molalities, which in our case was that of dextran in the bottom phase, and simultaneously solved Equations 22 for the remaining three molalities. The numerical algorithm used was the same one that Edmond and Ogston (6, 9) used for their model. The virial coefficients used in Equations 22 for all the calculations were predicted from the scaling expressions of Equations 19. 

In conclusion, the temperature profile in conventional distillation columns is the result of both phase equilibrium relations and enthalpy balances. In narrow-boiling mixtures, the phase equilibrium effect is generally more pronounced, while in wide-boiling mixtures, the enthalpy balances are more significant. The importance of the distinction between the two effects is twofold. First, different mathematical solution algorithms are better suited for each situation, as will be discussed in Chapter 13. Second, the understanding and prediction of column performance is enhanced when the two effects are recognized. Examples 7.1 and 7.2 illustrate the two cases. 

The principles and algorithms for calculating fluid-phase equilibria are discussed in many textbooks [36 0]. Here, we focus on methods and data requirements for calculating the component fugacities in a phase as a function of temperature, pressure, and composition this is the key element in all phase-equilibrium calculations. 

With both staged equipment and differential contactors, availability oradequate phase-equilibrium models and rate expressions would allow application of existing correlations and simulation algorithm). For example. knowledge of metal-extraction kinetics in terms of interfacial species concentrations conld be combined with correlations of film mass transfer coefficients in a particular type of equipment to obtain the inlerfacial flux as a fuuction of bulk concentrations. Correlations or separate measurements of inierfacial area and an estimate of dispersion characteristics would allow calculation of extraction performance as a... 

Although the same enclosures are used for the component-material balances in the formulation of the Almost Band Algorithm as were used in the formulation of the 2N Newton-Raphson method, it is convenient in this case (because of the form of the phase equilibrium relationships) to include Tr+l, Ts, vr+lt,, and usl in the set of independent variables. Thus, in the formulation of the Almost Band Algorithm, the following choice of independent variables is made. 

In this chapter, we describe an algorithm for predicting feasible splits for continuous single-feed RD that is not limited by the number of reactions or components. The method described here uses minimal information to determine the feasibility of reactive columns phase equilibrium between the components in the mixture, a reaction rate model, and feed state specification. This is based on a bifurcation analysis of the fixed points for a co-current flash cascade model. Unstable nodes ( light species ) and stable nodes ( heavy species ) in the flash cascade model are candidate distillate and bottom products, respectively, from a RD column. Therefore, we focus our attention on those splits that are equivalent to the direct and indirect sharp splits in non-RD. One of the products in these sharp splits will be a pure component, an azeotrope, or a kinetic pinch point the other product will be in material balance with the first. 

Problems at the end of the chapter give you opportunities to explore other algorithms. More technical discussions of these and other algorithms for phase-equilibrium calculations can be found in the book by Prausnitz et al. [3] and in the papers by Michelsen [4-6]. 

The central portion of the algorithm in Figure 11.6 exactly parallels the standard Rachford-Rice procedure. First, we use (11.1.27)-(11.1.29) to compute the mole fractions for all phases, then we compute all fugacity coefficients and all activity coefficients. With those quantities we can obtain new estimates for the Cs and Ks from the phase-equilibrium relations (11.1.15) and (11.1.24). Now we use (11.1.31) and (11.1.32) to calculate values for the Rachford-Rice functions, Fj and F2, and test for convergence. If our convergence criteria are not met at iteration k, then we use the Newton-Raphson method to estimate the unknown L and V at the next iteration (fc + 1). 

A principal advantage of this algorithm is that it applies to any number of components C > 3, though in every case we solve only fhe two equations (11.1.31) and (11.1.32). However, this method fails for binary mixtures. To see why, note that for binaries in three-phase equilibrium, (11.1.23) requires us to specify values for T = 3 variables. We then have five equations that can be solved for five unknowns. The five equations are four phase-equilibrium relations (11.1.15) and (11.1.24) plus the one Rachford-Rice function (11.1.31). In the Rachford-Rice approach, the five unknowns would be i/p plus the fractions L and V. However, L and V appear in only one... 

Coupled phase-reaction equilibrium problems not only raise no new thermodynamic issues, but they also raise few new computational issues. By building on the phase and reaction-equilibrium algorithms presented earlier in this chapter, we can devise an elementary algorithm. Reaction-equilibrium problems typically start with known values for T, P, and initial mole numbers N° in a phase-equilibrium context, these variables identify an T problem, such as an isothermal flash calculation. Therefore we can combine the Rachford-Rice method with the reaction-equilibrium calculation given in 11.2 an example is provided in Figure 11.8 for a vapor-liquid situation. This is a traditional way for attacking multiphase-multireaction problems [21, 22] ... 

Phases and compositions at 358 K, 1.0133 bar. With a slight increase in temperature, from 355 K to 358 K, the algorithm in Figure 11.8 finds no two-phase equilibrium the system is a one-phase gas. Otherwise, with all temperature-independent parameters the same as in the previous calculation, and with the standard Gibbs energies of formation from Table 11.8, we now find the following equilibrium compositions ... 

Use the one-phase multireaction algorithm in Figure 11.6 to determine the extent to which formation of tetramers of acetic acid affect the fractional conversion during esterification of ethanol. That is, repeat the vapor-phase calculation at 358 K, 1.0133 bar illustrated in the last part of 11.3.3, but now include not only dimers but also tetramers. (Spectroscopic evidence suggests that formation of trimers is unfavored [32].) Sebastiani and Lacquaniti give the equilibrium constant for formation of tetramers as [32]... 

---