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PHASE EQUILIBRIA Amyn S. Teja The equilibrium relationship for any component i in an equilibrium stage is defined in terms of the distribution coefficient or K-value yi Ki (1) xi The more volatile components of a mixture will have higher K-values, whereas the less volatile components will have lower K-values. In distillation, the efficiency of separation of two components is often compared via a quantity called the relative volatilityij Ki y / xi ij i (2) Kj yj / xj A relative volatility close to unity means that the separation of the two components is likely to be difficult, whereas a relative volatility much greater or much less than unity means that few equilibrium stages are likely to be needed for separation. When the relative volatility has a value of one, yi = xi and separation is no longer feasible. A relative volatility of one also signifies the existence of an azeotrope or a critical point. THERMODYNAMIC FRAMEWORK IDEAL SYSTEMS For the simplest case of an ideal gas mixture in equilibrium with an ideal liquid solution: xi Pi sat y i P (3) and, therefore, yi Pi sat Ki (4) xi P and Pi sat ij (5) P jsat The relative volatility of a system that obeys Raoult’s law is thus a ratio of two vapor pressures and is a function only of the temperature. A feature of this system is that the P- x behavior (or the bubble curve) is linear and given by: 1 P P sat x1 P sat P2sat 1 (6) Only a small number of systems containing chemically similar components obey Raoult’s law, and then only at low pressures (< 1 MPa). As a consequence, K-values can be predicted from pure component data only for such mixtures. The majority of real systems are nonideal. Non-ideal systems Non-ideal behavior can be described using two approaches – the activity coefficient approach and the equation of state approach. Activity coefficient approach At low pressures (< 1 MPa), we can write ˆ i xi Pisat iV yi P (7) ˆ Often iV ~ 1.0 for vapor phases at moderate pressures. Hence: i xi Pi sat y i P (8) and K i i Pi sat / P (9) 2 Vapor pressures at subcritical temperatures may be obtained from experimental data using equations such as the Antoine equation. Activity coefficients may be obtained from excess Gibbs energy gE models as described below. Equation of state approach The equation of state approach results in ˆ ˆ iV yi iL xi (10) and ˆ ˆ K i iL / iV (11) The calculation of K-values is therefore reduced to the calculation of fugacity coefficients. CALCULATION OF FUGACITY COEFFICIENTS Calculation of the fugacity coefficient requires knowledge of the P-V-T-x behavior of the system. This information is obtained from an equation of state. Two representative types of equation of state will be discussed below – a volume-explicit virial equation and a pressure-explicit cubic equation. Volume explicit virial equation Virial equations of state are infinite-series expansions of the compressibility Z as a function either of the density or pressure. The pressure series may be written as: PV Z 1 B P C P 2 ...... RT (12) BP (C B 2 ) 2 1 P ......... RT ( RT 2 ) 3 where B is the second virial coefficient, C the third virial coefficient and so on. Typically, the two-term truncated form of the virial equation is used for gases at low pressures: BP Z 1 BP 1 (13) RT which can be rearranged in the volume-explicit form RT V B (14) P The truncated virial equation is only applicable to gases at densities that are less than about half the critical density. Mixture second virial coefficient is given by: B yi y j Bij (15) i j where Bii is the second virial coefficient of component i and Bij is a cross second virial coefficient. This leads to: ˆ 1 ln iV Bii y k yl (2 ki kl ) (16) 2 where kl = 2 Bkl – Bkk - Bll. The fugacity coefficient of any component in the vapor phase can thus be calculated if the second virial coefficients of the pure components and the cross second virial coefficients are available. Since the truncated virial equation is only applicable to gases at low to moderate pressures, fugacity coefficients calculated using Equation (16) are generally employed only when Equation (7) is used to calculate vapor- liquid equilibria. 4 Pressure explicit cubic equation of state Cubic equations of state express the pressure as a cubic function of the molar volume, and their origin stems from the van der Waals equation, which was the first cubic equation of state to qualitatively represent both vapor and liquid phases. Several hundred modifications of the van der Waals equation have been reported in the literature. An example of a recent modification that is better able to represent P-V-T-x data for both vapour and liquid mixtures is the equation of state proposed by Patel and Teja in 1982. This equation may be written as: RT a P 2 (17) v b v bv cv bc where R 2Tc2 a a (18) Pc RTc b b (19) Pc RTc c c (20) Pc 1 m 1 T / Tc 2 (21) a 3 c 31 2 c b b 1 3 c 2 2 (22) c 1 3 c (23) ζc = Pcvc / RTc (24) and b is the smallest positive root of : 5 3 2 3 c b 3 c b 3 0 b 2 2 c (26) In the above equations, the subscript c denotes a value at the critical point. Note that by setting the parameter c=0 in Equation (17), the Patel-Teja equation reduces to the Redlich-Kwong-Soave equation of state; and by setting c=b, it reduces to the Peng- Robinson equation of state. Both the Redlich-Kwong-Soave and the Peng-Robinson equation are widely used in process design calculations. For nonpolar fluids, ζc and m are calculated from the following relationships in terms of the acentric factor : c 0.329032 0.076799 0.0211947 2 (27) m 0.452413 1.30982 0.2959372 (28) A knowledge of Pc, Tc and is therefore sufficient to calculate the parameters of the equation of state. Alternatively, ζc and m may be calculated from experimental values of the vapor pressure and liquid density of the substance. The parameters a, b and c for a mixture can be calculated using the following mixing rules: a zi z j a ij (29) i j b z i bi (30) i c z i ci (31) i where zi can be xi or yi and a ij 1 k ij a i a j . The kij are binary interaction parameters that are usually obtained by fitting experimental VLE data. The fugacity coefficient can be written as follows: 6 ˆ b Pv Pv b a bi b z j a ij ln 1 v 2 ln i i 1 ln 1 (32) b RT RT v bRT b a j Equation (32) can be used to calculate both the vapor and liquid phase fugacity coefficients. In the case of the vapor phase, zi = yi and v is the vapour molar volume, whereas for the liquid phase zi = xi and v is the liquid molar volume. The ratio of the two fugacity coefficients yields the K-value at the conditions of interest. CALCULATION OF ACTIVITY COEFFICIENTS Activity coefficients i are generally calculated by differentiation of the excess Gibbs energy gE A number of expressions have been proposed for gE as a function of composition. Some of the more popular of these are outlined below. Margules equation The Margules equation is one of the simplest expressions for the molar excess Gibbs Energy. For a binary solution, gE x1 x 2 A21 x1 A12 x 2 (33) RT where A12 and A21 are binary parameters dependent on temperature, but not on the composition. The Margules activity coefficients in a binary mixture are given by ln 1 A12 2 A21 A12 x1 x 2 2 (34) ln 2 A21 2 A12 A21 x 2 x12 (35) A12 and A21 are generally obtained by fitting VLE data. Note that the value of the activity coefficient of each component tends to one as the mole fraction of that component goes to one. This behavior is inherent in all gE models. The Margules equation works well for 7 binary systems in which the two components are very similar in size, shape and chemical nature. Wilson equation The Margules equation is based on the assumption that the ratio of species 1 to species 2 molecules surrounding any molecule is the same as the ratio of the mole fractions of species 1 and 2. A different class of gE models has been proposed based on the assumption that, around each molecule, there is a local composition that is different from the bulk composition. The Wilson equation is such a local composition model and the Wilson excess Gibbs energy has the following form for a binary system: gE x1 ln x1 12 x 2 x 2 ln x 2 21 x1 (36) RT where 12 and 21 are parameters specific to the binary pair. These parameters are defined in terms of the molar liquid volume vi of the pure component i, and the energies of interaction ij between the molecules i and j as follows vj ii ij exp ij (37) vi RT The expressions for the liquid activity coefficients are: 12 21 ln 1 ln x1 12 x2 x2 x x x x (38) 1 12 2 2 21 1 21 12 ln 2 ln x 2 21 x1 x1 x x x x (39) 2 21 1 1 12 2 8 The Wilson equation has two parameters and (or equivalently, - and - ) and is able to correlate VLE data for a wide variety of miscible systems, including those containing polar or associating components in nonpolar solvents. However, the equation is incapable of predicting liquid-liquid immiscibility in a system. For multicomponent mixtures, the Wilson equation is written as follows: m m x ln k ln x j kj 1 m i ik (40) j 1 i 1 x j ij j 1 Note that only binary parameters ij are required to evaluate activity coefficients in multicomponent systems. These parameters are obtained by fitting VLE data for the binary pairs, and many of these parameters have been tabulated in the DECHEMA data books. Moreover, because a temperature dependence is included in Equation (37), the same binary parameters may be used over a range of temperatures (although no more than about 50 K). NRTL equation The NRTL equation is also based on a local composition model for the excess Gibbs energy. However, it is applicable to miscible as well as partially miscible systems due to the inclusion of a third binary parameter in the model. The expression for the molar excess Gibbs energy is gE G 12 G12 x1 x 2 21 21 (41) RT x1 x 2 G21 x 2 x1G12 where ij and Gij are defined as Gij exp ij ij (42) 9 g ij g jj ij (43) RT gij describes the energy of interaction between component i and j and ij (= ji) is a nonrandomness parameter which is often set equal to 0.3. Thus, only two parameters ij and ji ( or equivalently, gij – gjj and gji – gii ) are required per binary pair. The activity coefficients expressions are as follows: G21 2 12 G12 ln 1 x 21 2 (44) x1 G21 x 2 2 x 2 G12 x1 2 G12 2 21G21 ln 2 x 12 2 (45) x 2 G12 x1 1 x1 G21 x 2 2 A major advantage of the NRTL equation lies in its ability to represent highly nonideal systems, particularly partially miscible systems. For multicomponent mixtures, the liquid phase activity coefficients are expressed as: m m ji G ji x j m x j Gij x r G rj rj ln i j 1 m r 1 (46) m ij m Gl 1 li xl j 1 Glj x l l 1 Glj x l l 1 As with the Wilson equation, only binary data are needed to calculate activity coefficients in multicomponent systems, and these parameters have been tabulated in the DECHEMA data books for many systems. Furthermore, because of the inclusion of the temperature in 10 Equation (43), the parameters obtained by fitting VLE data at one temperature may be used to calculate VLE at other temperatures (within about 50K). UNIQUAC equation The UNIQUAC equation expresses the molar excess Gibbs energy as a sum of a combinatorial part and residual part. g E g E (combinato rial) g E (residual) (47) The combinatorial part accounts for differences in the size and shape of the molecules, whereas the residual contribution accounts for energetic interactions. g E (combinatorial ) z x1 ln 1 x 2 ln 2 q1 x1 ln 1 q 2 x 2 ln 2 (48) RT x1 x2 2 1 2 g E ( residual ) q1 x1 ln 1 2 21 q 2 x 2 ln 2 1 12 (49) RT where x i ri i (50) x j rj j x i qi i (51) x jqj j a ji ji exp (52) RT In Equation (48), z’ is a co-ordination number (= 10 usually), i are volume fractions, and i are surface area fractions for component i. The volume and surface area parameters ri and qi can be evaluated from pure component molecular structure 11 information and are tabulated in the DECHEMA data books. Thus, there are two binary parameters aij and aji in the UNIQUAC model and these are found by fitting binary VLE data. The activity coefficient expressions become 1 z r ln 1 ln q1 ln 1 2 l1 1 l2 x1 2 1 r2 (53) 21 12 q1 ln 1 2 21 2 q1 1 2 21 2 112 2 z r ln 2 ln q 2 ln 2 1 l 2 2 l1 x2 2 2 r1 (54) 12 21 q 2 ln 2 1 12 1 q 2 2 1 12 1 2 21 where z l i ri qi ri 1 (55) 2 The UNIQUAC equation is applicable to a wide range of systems including partially miscible systems. For multicomponent systems, the UNIQUAC equation becomes z m ln i ln i qi ln i l i i xi 2 i xi x l j 1 j j m m j ij (56) qi ln j ji qi qi m j 1 j 1 k kj k 1 12 Once again, only pure component and binary data are needed to calculate the equation parameters. UNIQUAC parameters for over 6000 binary systems have been tabulated in the DECHEMA data series on vapour-liquid equilibrium. UNIFAC group contribution method When values of Margules, Wilson, NRTL, or UNIQUAC parameters are not available in the literature, or when no VLE data for the system of interest have been measured, the UNIFAC method may be used to estimate activity coefficients. The UNIFAC method is a group contribution technique for the estimation of the parameters amn of the excess Gibbs energy model. In terms of the activity coefficient: ln i ln iC ln iR (57) i z r ln i ln qi ln i i li i l j C (58) xi 2 i rj ri ki Rk (59) k qi kiQ k (60) k The group volume Rk, and the group area Qk are tabulated for a large number of groups. ki is the number of groups of k kind in molecule i. The residual contribution is expressed as follows: ln iR Q k ln k ln k( i ) (61) k ln k Qk 1 ln E k Fk (62) ln k( i ) Q k( i ) 1 ln E k( i ) Fk( i ) (63) 13 E k 1 1k 2 2 k 3 3k (64) 1 k 1 2 k 2 3 k 3 Fk (65) E1 E2 E3 X m Qm m (66) X n Qn n a mn mn exp (67) T x i mi Xm i (68) xi ki i k Since the group volume parameters Rk, and the group area parameters Qk are known, the only unknowns in the UNIFAC equations are the group interaction parameters amn and anm. Fredenslund, Ghemling, and Rasmussen have tabulated these for a large number of groups. Moreover, updated parameters are published regularly in the literature. The UNIFAC method has been successfully applied to a wide variety of binary and multicomponent systems. 14