# Association equation of state for hydrogen-bonded substances.

INTRODUCTION

Various procedures have been used to improve cubic equations of state for calculating thermodynamics properties of associating fluids (Walas, 1984; Prausnitz et al., 1986; Kang, 1999; Villares et al., 2004; Folas et al., 2005). But the most common one that has been used by several authors is to introduce the general form of the compressibility factor Z in the following form (Kretschmer and Wiebe, 1954; Anderko, 1989a, 1989b, 1990, 1991, 1992; Anderko and Prausnitz, 1994; Shinta and Firoozabadi, 1995; Cho et al., 1998; Nan et al., 2001; Pires et al., 2001; Ji et al., 2004):

Z = [Z.sup.ch] + [Z.sup.ph] - 1 (1)

The physical part of the compressibility factor [Z.sup.ph] considers the molecules in their monomeric non-associating state and can be expressed by an ordinary cubic equation of state with the parameters obtained for monomeric non-associating state. Mohsen-Nia, Modarress and Mansoori (MMM) equation of state (Mohsen-Nia et al., 2003) can be used for the physical part:

[Z.sup.ph] = v + [alpha][b.sub.0]/v - [b.sub.0] - [a.sub.0]/[RT.sup.3/2](v + N[alpha][b.sub.0] (2)

where [alpha] = 1.3191 and N = 2 and [a.sub.0] and [b.sub.0] are functions of monomeric critical constants expressed as in Mohsen-Nia et al. (2003).

[a.sub.0] = 0.47312 [R.sup.2][T'.sub.c.sup.2.5]/[P'.sub.c][[1 + (0.32 + 0.64[omega]')(1 - [square root of [T'.sub.r]])].sup.2] (3)

[b.sub.0] = 0.04616 [RT'.sub.c]/[P'.sub.c][[1 + [n.sub.1](1 - [square root of [T'.sub.r]]) + [n.sub.2][(1 - [T'.sub.r.sup.0.75])]].sup.2] (4)

In Equations (3)and (4) [T'.sub.r] = T/[T'.sub.c] where [T'.sub.c] and [P'.sub.c] are respectively monomeric critical temperature and pressure and [n.sub.1] and [n.sub.2] are given by the following equations:

[n.sub.1] = 3.270572 - 6.4127[omega]' + 10.6821[[omega]'.sup.2] (5)

[n.sub.2] = 1.72192 + 3.85288[omega]' - 7.202286[[omega]'.sup.2] (6)

where [omega]' is the monomeric acentric factor.

The chemical part is defined as the ratio of the true number of moles in associated state to the number of moles in the monomeric state. According to this definition the inverse of [Z.sup.ch] represents the mean association number [chi] (Shinta and Firoozabadi, 1995). The well-established and widely used form of [Z.sup.ch] is:

[Z.sup.(ch)] = n/[n.sub.0] = 1/[chi] (7)

where [n.sub.0] is the number of moles of monomers in non-associating state, and n is the number of all species in the system in their actual, monomeric and associated state. In the previous models various forms of [Z.sup.ch] have been deduced (Kretschmer and Wiebe, 1954; Anderko, 1989a, 1989b, 1990, 1991, 1992; Anderko and Prausnitz, 1994; Shinta and Firoozabadi, 1995; Cho et al., 1998; Nan et al., 2001; Pires et al., 2001; Ji et al., 2004). To derive an equation for [Z.sup.ch], we have to obtain n and [n.sub.0] from an association model. The infinite linear association model as described in the following section is used for obtaining n and [n.sub.0]. First [Z.sup.ch] is obtained for a pure associating fluid and then it is extended to an associating fluid mixture.

Pure Associating Fluid

The associating reaction will be represented by the following reaction:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

where [K.sub.y] is the equilibrium constant of the reaction expressed on mole fractions and is assumed to be independent of the association number i. Therefore, the number of moles of i-mer [n.sub.Ai] can be obtained as:

[n.sub.Ai] = [([K.sub.y]/n).sup.i-1] [[n.sub.A1.sup.i] (9)

The total number of moles in associating fluid n is presented as:

n = [[infinity].summation over (i=1)] [n.sub.Ai] = [[infinity].summation over (i=1)] [([K.sub.y]/n).sup.i-1] [n.sub.A1.sup.i] (10)

and the total number of moles n0 in the fluid, if all were in monomeric state, would be presented by the following equation:

[n.sub.0] = [[infinity].summation over (i=1)] [in.sub.Ai] = [[infinity].summation over (i=1)]i [([K.sub.y]/n).sup.i-1] [n.sub.A1.sup.i] (11)

By a straightforward algebraic manipulation Equations (10) and (11), can be simplified as:

n = [n.sub.A1](1 + [K.sub.y]) (12)

[n.sup.0] = [n.sub.A1][(1 + [K.sub.y]).sup.2] (13)

Substituting Equations (12) and (13) in Equation (7) results in:

[Z.sup.ch] = 1/(1 + [K.sub.y]) (14)

The mole fraction based association constant [K.sub.y] is related to activity based association constant K by the following equation (Walas, 1984):

K = [K.sub.y] ([P.sup.o]/P)[[[phi].sub.i]/([[phi].sub.i]-1[[phi].sub.1]] (15)

where [P.sup.o] is the standard pressure, P is the total pressure and [[phi].sub.i], [[phi].sub.i-1] and [[phi].sub.1] are respectively fugacity coefficients of [A.sub.i], [A.sub.i-1] and [A.sub.1]. We can calculate the ratio of fugacity coefficients in Equation (15) by using the following equation for species j in the associative mixture, where j represents species [A.sub.i], [A.sub.i-1] and [A.sub.1] (Hu et al., 1984).

1n([[phi].sub.j]) = [[mu].sup.r.sub.j]/(RT) - 1N(Z) (16)

In Equation (16) the residual chemical potential [[mu].sup.r.sub.j] is defined as the chemical potential of a substance in the mixture minus its chemical potential in the ideal state.

For chemical reaction represented by Equation (8) the following equation holds:

[[mu].sup.r.sub.i] - [[mu].sup.r.sub.i-1] - [[mu].sup.r.sub.1] = 0 (17)

where [[mu].sup.r.sub.i], [[mu].sup.r.sub.i-1] and [[mu].sup.r.sub.1] are respectively residual chemical potentials of [A.sub.i], [A.sub.i-1] and [A.sub.1]. By substituting for residual chemical potential from Equation (16) in Equation (17) and Z = Pv/RT the following equation will be obtained (Anderko, 1990):

[K.sub.y] = (K/[P.sub.o])(RT/v) (18)

where v = V/[n.sub.0] is molar volume of monomeric substance. On substituting for [K.sub.y] from Equation (18) in Equation (14) [Z.sup.ch] will be in the following form:

[Z.sub.ch] = 1/[1 + [K.sub.y] = (K/[P.sub.o])(RT/v)](19)

On the other hand, the activity based association constant K can be expressed as (Anderko, 1990):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)

where [DELTA][H.sup.o]([T.sub.0]) is the standard enthalpy change and [DELTA][S.sup.o]([T.sub.0]) is the standard entropy change for association reaction at [T.sub.0] = 273.15 K and [DELTA][C.sub.p.sup.o] is the standard heat capacity change of association reaction that is assumed to be independent of temperature.

It is notable that for pure component the chemical contribution to the second virial coefficient should converge to the following limit (Prausnitz et al., 1986):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (21)

where the equilibrium constant of dimerization; K' = K/[P.sup.o] and [P.sup.o] is the standard pressure. This limit is satisfied by Equation (19)

By substituting for K from Equation (20) in Equation (19), an equation for [Z.sup.ch] can be derived. [Z.sup.ch] from this equation and [Z.sup.ph] from Equation (2) are substituted in Equation (1) and then a new associating equation of state will be obtained. This equation has six adjustable parameters, namely [T'.sub.c], [P'.sub.c], [omega]', [DELTA][H.sup.o]([T.sub.0]), [DELTA][C.sub.p.sup.o] and [DELTA][S.sup.o]([T.sub.0]). These parameters can be evaluated by fitting the vapour pressure and saturation liquid density of pure associating fluid from the triple point temperature to the critical point temperature.

An Associating Fluid in a Mixture with Non-Associating Fluids

For a mixture the physical part of compressibility factor has the same form as Equation (2), except the parameters [a.sub.0] and [b.sub.0] are expressed by the following mixing rules:

[a.sub.0] = [N+1.summation over (i=1)] [N+1.summation over (l=1)] [x.sub.i][x.sub.l] [([a.sub.i][a.sub.l]).sup.1/2](1 - [k.sub.il]) (22)

[b.sub.0] = [N+1.summation over (l=1)] [x.sub.i][b.sub.i] (23)

where N+1 represents the number of components in the mixture where N components are non-associating and one component is associating. It is worth noting that for associating component, in Equations (22) and (23), the parameters should be indicated [a.sub.0i] and [b.sub.0i] but for simplifying the equations [a.sub.i] and [b.sub.i] were used.

To obtain [Z.sup.ch] for the mixture of one associating component A and N non-associating components [B.sub.j], Equation (12) for (n) the total number of moles of the mixture will be presented as:

n = [n.sub.A1]/[1 - ([K.sub.y]/n)[n.sub.A1]] + [N.summation over (j=1)] [n.sub.Bj] (24)

and Equation (13) for ([n.sub.0]) the total number of moles of monomers A and [B.sub.j] will be presented as:

[n.sub.0] = [n.sub.A1]/[[1 - ([K.sub.y]/n)[n.sub.A1]].sup.2] + [N.summation over (j=1)] [n.sub.Bj] (25)

where [n.sub.A1] represents the number of moles of associating component existing in monomeric state, [n.sub.Bj] is the number of moles of [B.sub.j] the non-associating component.

By linear association model, the number of moles of associating component is presented as:

[n.sub.A] = [[infinity].summation over [i=1][in.sub.Ai] = [[infinity].summation over [i=1] i [([K.sub.y/n).sup.i-1] [n.sub.A1.sup.i] (26)

Then by using Equations (25) and (26) the mole fraction of associating component xA will be obtained as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (27)

By using Equations (24), (25) and (27) the following expression for n/[n.sub.0], the ratio of the number of moles in associating state to the number of moles in monomeric state, will be obtained:

n/[n.sub.0] = [x.sub.A][1- ([K.sub.y]/n)[n.sub.A1]] + [N.summation over (j=1)[x.sub.bj] = 1 - [K.sub.y][x.sub.A] ([n.sub.A1]/n) (28)

where from Equation (24) [n.sub.A1]/n is given as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (29)

Then from Equations (28), (29) and Equation (18) for [K.sub.y], the chemical part of compressibility factor [Z.sup.ch] = n/[n.sub.0] will be derived in the following form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (30)

where K is given by Equation (20).

In order to simplify Equation (30) the term 4(K/[P.sup.o]) (RT/v)[x.sub.A] (1 - [x.sub.A]) can be neglected since for most associating fluids it is smaller than unity. Taking water as an example the experimental association constant is K < < 1 in temperature range (400 - 500 K) (Burke et al., 1993), considering the ideal gas ([P.sup.o]v/RT) = 1 and the maximum value of [x.sub.A](1-[x.sub.A]) = 0.25, it is seen that neglecting this term is a valid assumption. Therefore, Equation (30) is simplified as:

[Z.sup.ch] = [1 + (K/[P.sup.o]) (RT/v)(1 - [x.sub.A])] / [1 + (K/[P.sup.o]) (RT/v)] (31)

This simplification can be verified by considering that in Equations (30) and (31) [Z.sup.ch] is a function of mole fraction of associated species [x.sub.A] and [K.sub.y] association constant on mole fraction; [K.sub.y] = (K/Po)(RT/v). By plotting the absolute difference of [Z.sup.ch], calculated by Equations (30) and (31) for various values of [x.sub.A] ([x.sub.A] = 0.01-0.9), versus [K.sub.y] = (K/[P.sup.o])(RT/v); it is seen from Figure 1 that the difference at its maximum near [K.sub.y] = 4 does not exceed 0.13 and for the other values of [K.sub.y] is much smaller.

CALCULATIONS

Pure Associating Component

The saturation pressure and liquid density of water were obtained from Sonntag et al. (2003). For ethanol and butanol these data were obtained from Ambrose and Sprake (1970) and Perry et al. (1984). The saturation pressure and saturation liquid molar volume for each temperature were calculated by using fugacity equality at vapour-liquid equilibrium for pure substances:

[[phi].sup.v] = [[phi].sup.l] (32)

The fugacity of pure substance can be calculated via the following well-known equation (Edmister and Lee, 1983):

ln [phi] = Z - 1 - ln Z + 1/RT [[integral].sup.v.sub.[infinity]] (RT/v - P)dv (33)

The compressibility factor Z in the above equation was obtained by substituting Equations (2), (19) and (20) in Equation (1). The six unknown parameters [T'.sub.c], [P'.sub.c], [omega]', [DELTA][H.sup.o]([T.sub.0]), [DELTA][C.sub.p.sub.o] and [DELTA][S.sup.o]([T.sub.0]) in Equation (20) were calculated by using non-linear regression analysis of Nelder-Mead (Bazaraa et al., 1993) for minimizing the average absolute deviation [(AAD).sub.v] in saturation liquid molar volume:

[(AAD).sub.v] = 1/N [N.summation over (i=1)]|[v.sup.exp.sub.i] - [v.sup.cal.sub.i]| (34)

and the average absolute deviation [(AAD).sub.p] in saturation pressure:

[(AAD).sub.p] = 1/N [N.summation over (i=1)]|[p.sup.exp.sub.i] - [p.sup.cal.sub.i]| (35)

where N is the number of data point, [v.sup.exp.sub.i] and [p.sup.exp.sub.i] exp are respectively the liquid molar volume and saturation pressure and [v.sup.cal.sub.i] and [p.sup.cal.sub.i] are their calculated values.

[FIGURE 1 OMITTED]

An Associating Component in the Mixture of Non-Associating Ones

For component i in the associating mixture the following equation is true (Firoozabadi, 1999):

ln([[phi].sub.i]) = ln([[phi].sup.ph.sub.i][[Z.sup.ph]ln([[phi].sup.ch.sub.i][[Z.sup.ch] (36)

where [[phi].sub.i] is fugacity coefficient of component i in the mixture and Z is given in Equation (1). All terms in Equation (36) can be obtained by the following general equation in terms of PVT behaviour of mixture:

ln([[phi].sub.i]Z) = 1/RT [[integral].sup.v.sub.[infinity]] (RT/V - [partial derivative]P/[partial derivative][n.sub.i])dv (37)

Using [Z.sup.ch] from Equation (31) and [P.sup.ch] = [Z.sup.ch][n.sub.0]RT/V in Equation (37) the following equations will be obtained:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (38)

By using [Z.sup.ph] of MMM EOS, as given by Equation (2), and [P.sup.ph] = [Z.sup.ph][n.sub.0]RT/V in Equation (37), the following equation will be obtained:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (39)

where [alpha] = 1.3191, N = 2; [a.sub.0] and [b.sub.0] are given by Equations (22) and (23); [a.sub.i] and [b.sub.i] are given by Equations (3) and (4) and the critical constants ([T.sub.c], [P.sub.c]) of non-associating components were obtained from Perry et al. (1984), Henley and Seader (1981) and Reid et al. (1986).

RESULTS AND DISCUSSION

Table 1 reports the calculated parameters ([T'.sub.c], [P'.sub.c], [omega]', [DELTA][H.sup.o], [DELTA][C.sup.o.sub.p], [DELTA][S.sub.o]) for pure water, ethanol and butanol by using [Z.sup.ph] from MMM EOS (Mohsen-Nia et al., 2003) and [Z.sup.ch] as obtained in this work and represented by Equation (19). In this table the same parameters calculated by Anderko (1991) and Shinta and Firoozabadi (1995) for pure water are reported. Anderko proposed an empirical [Z.sup.ch] and used in his calculations along with a [Z.sup.ph] obtained from Yu and Lu (1987) EOS. Shinta and Firoozabadi (1995) also proposed an empirical [Z.sup.ch] and used in their calculations along with a [Z.sup.ph] obtained from PR EOS (Peng and Robinson, 1976). The mean association number [chi] = 1/[Z.sup.ch] can be calculated from parameters ([DELTA][H.sup.o], [DELTA][C.sup.o.sub.p], [DELTA][S.sup.o]). The calculated mean association number x for water at different temperatures by [Z.sup.ch] of this work and by those of Anderko (1991) and Shinta and Firoozabadi (1995) are reported in Table 2. The experimental values of association number obtained from both X-ray measurements (Nemethy and Scherage, 1962) and the calculated association numbers [chi] initio method (Jhon et al., 1966; Burke et al., 1993; Carey and Korenowski, 1998; Maselia, 1998; Wales and Hodges, 1998; Chaplin, 1999; Alfredsson and Hermansson, 1999; Ohtaki, 2003; Starzak and Mathlouthi, 2003) confirm [chi] = 4 for the first shell surrounding a water molecules and up to 21 for the other next nearest shells. As the results reported in Table 2 indicate the value of [chi] calculated by [Z.sup.ch] proposed in this work is close to the value of [chi] = 4 whereas by [Z.sup.ch]'s of Anderko (1991) and Shinta and Firoozabadi (1995) surprisingly much higher and unacceptable values of [chi] are calculated. Figure 2 shows the calculated molar volumes of water versus temperature from the triple point to the critical point based on the results of this work and those of Anderko (1991) and Shinta and Firoozabadi (1995). This figure shows clearly that [Z.sup.ph] and [Z.sup.ch] proposed in this work have much better agreement with experimental data. Figures 3 and 4 show the calculated and experimental values of, respectively, saturated liquid molar volumes for ethanol and 1-butanol, and saturated pressure for water, ethanol and 1-butanol. The good fit with experimental data as presented in these figures lends support to the validity and effectiveness of [Z.sup.ch] and [Z.sup.ph] proposed in this work for PVT calculations of pure associating fluids. This is confirmed by comparing the (AAD)v's with other models and regarding [(AAD).sub.p]'s of proposed model as shown in Table 1.

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

Table 3 reports the calculated mole fractions of liquid and vapour phases for the feed compositions of mixtures ([H.sub.2]O/C[H.sub.4]/ C[O.sub.2]/[H.sub.2]S) at three temperatures and pressures. The experimental results for this mixture were obtained by Huang et al. (1984). It is worth noting that, the calculations for this mixtures were done for the interaction parameters, [k.sub.il] = 0 (as shown in Equation (19)). The AAD reported in Table 3 are defined as:

[(AAD).sub.x] = 1/N [N.summation over (i=1)][absolute value of [x.sup.exp.sub.i] - [x.sup.cal.sub.i]] (40)

[(AAD).sub.y] = 1/N [N.summation over (i=1)][absolute value of [y.sup.exp.sub.i] - [y.sup.cal.sub.i]| (41)

Table 4 reports the (Error%) in K-values for quaternary system of ([H.sub.2]O/C[H.sub.4]/C[O.sub.2]/[H.sub.2]S) at 310.92 K and is compared with those reported in Shinta and Firoozabadi (1995). As it is seen from Table 4, the AEOS proposed in this work has better accuracy than that of Shinta and Firoozabadi (1995). At higher temperatures and in some cases the AEOS of Shinta and Firoozabadi (1995) may have better accuracy. However, bearing in mind that the AEOS of Shinta and Firoozabadi (1995) is an empirical equation proposed only for water, whereas AEOS of this work a general analytic AEOS derived on the basis on linear association model that can be applied to the all associating fluids without limitation, its overall performance is quite justifiable.

[FIGURE 5 OMITTED]

[FIGURE 6 OMITTED]

[FIGURE 7 OMITTED]

Figure 5 shows experimental and calculated mole fraction of water in [H.sub.2]O/n[C.sub.10] mixture as P-x,y diagram at temperatures, 573.15 K and 593.15 K, where x is mole fraction of water in liquid phase and y is mole fraction of water in vapour phase. [k.sub.il] for this binary mixture is 0.57. Figure 6 shows the results of VLE calculations for binary mixture of, ethanol/C6, as P-x,y diagram at 473.15 K. x and y are mole fraction of ethanol, respectively, in liquid and vapour phases. For this mixture [k.sub.il] = 0.23. Figure 7 is T-x,y diagram for binary mixture, ethanol/ toluene, at 1 atm with [k.sub.il] = 0.13 for T > 362 K and [k.sub.il] = 0.11 at T < 362 K. x and y are ethanol mole fraction in respective liquid and vapour phases. Figure 8 shows y-P and x-P for 1-butanol/C[O.sub.2] mixture, at 324.16 K and 333.58 K. y is mole fraction of 1-butanol in vapour and x is mole fraction of it in liquid phase. [k.sub.il] = 0.17 at 324.16 K and at 333.58 K, [k.sub.il] = 0.13. Figure 9 shows T-x,y diagram for 1-butanol/[C.sub.2][H.sub.2][Cl.sub.2] at 40 and 66.6 kpa with [k.sub.il] = 0.08. x and y are mole fraction of 1-butanol, respectively, in liquid and vapour phase. Figure 10 shows P-x,y diagram for 1-butanol/benzene mixture at T = 313.15 K and with [k.sub.il] = 0.05. Similarly x and y are liquid and vapour composition of 1-butanol, respectively.

CONCLUSION

[Z.sup.ph] and [Z.sup.ch] for associating pure and mixture fluids proposed in this work were used to calculate PVT behaviour of pure water, ethanol and 1-butanol and mixtures ([H.sub.2]O/C[H.sub.4]/C[O.sub.2]/[H.sub.2]S), (water/[nC1.sub.0]), (ethanol/n[C.sub.6]), (ethanol/toluene), (1-butanol/ C[O.sub.2]), (1-butanol/dichloroethane), and (1-butanol/benzene). The parameters [T'.sup.c,], [P'.sup.c], [omega]', [DELTA][H.sup.o], [DELTA][C.sup.o], [DELTA][S.sup.o] for associating fluids used in this work were evaluated. The mean associating number for pure water was evaluated from three calculated parameters ([DELTA][H.sup.o], [DELTA][C.sup.o], [DELTA][S.sup.o]). The mean associating number was in agreement with those reported in literature obtained from experimental data of pure water. The calculations for mixture of an associating component with non-associating components by using [Z.sup.ph] and [Z.sup.ch] also were in agreement with experimental data. Therefore, it was concluded that [Z.sup.ph] and [Z.sup.ch] proposed in this work have better capability to treat the PVT behaviour of pure associating and mixtures of one associating component with non-associating ones.

[FIGURE 8 OMITTED]

[FIGURE 9 OMITTED]

[FIGURE 10 OMITTED]

APPENDIX A

For a mixture of one associating component [A.sub.i] (i species, I = 1 monomer and I = 2 dimer, etc.) and N non-associating components [B.sub.j] (j number of components), the total number of moles (n) of the mixture can be presented as:

n = [summation over [i=1][n.sub.Ai] + [summation over (j=1)][n.sub.Bj] = n/[K.sub.y] [[infinity].summation over (i=0)][([K.sub.y]/n[n.sub.A1]).sup.i]-1] + [summation over (j=1)][n.sub.Bj] (A1)

where [K.sub.y], the association constant, has been defined in Equation (18). The total number of moles of monomers [n.sub.0] will be presented as:

[n.sub.0] = [summation over [i=1][in.sub.Ai] + [summation over [j=1][n.sub.Bj] = n/[K.sub.y] [[infinity].summation over (i=0)]i[([K.sub.y]/n[n.sub.A1]).sup.i]-1] + [summation over [j=1][n.sub.Bj] (A2)

where the right-hand side of the above equations can be obtained by substituting for [n.sub.Ai] from Equation (9).The above equations can be simplified by using the following mathematical equalities:

[[infinity].summation over (i=0)][x.sup.i] = 1/1-x and [[infinity].summation over (i=0)][ix.sup.i] x/[(1 - x).sup.2] for x < 1 (A3)

The final equations will be in the following form:

N = [n.sub.A1]/[1 - [K.sub.y]/n)[n.sub.A1]] + [N.summation over (j=1)][n.sub.Bj] (A4)

and

[N.sub.0] = [n.sub.A1]/[[1 - [K.sub.y]/n)[n.sub.A1]].sup.2] + [N.summation over (j=1)][n.sub.Bj] (A4) (A5)

Equations (A4) and (A5) are the same as Equations (24) and (25).

From Equations (A4) and (A5) it is obtained:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A6)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. By comparing Equation (A6) with Equation (27), the following equation will be derived:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A7)

This equation is the same as Equation (28). [n.sub.A1]/n in the above equation can be substituting from Equation (24) or (A4) in the form of Equation (29) and after rearrangement:

(1 + [K.sub.y]) [(n/[n.sub.0]).sup.2] - (2[K.sub.y](1 - [x.sub.A]) + 1](n/[n.sub.0]) + [K.sub.y](1 - ([x.sub.A].sup.2] = 0 (A8)

By solving the above equation for n/[n.sub.0] (or [Z.sup.ch]) and substituting for [K.sub.y] from Equation (18) we will have:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A9)

This is Equation (30). For 4(K/[P.sup.o])(RT/v)[x.sub.A](1 - [x.sub.A]) < < 1, this equation will be reduced to Equation (31).

APPENDIX B

For chemical contribution, Equation (37) can be written as:

Ln([[phi].sup.ch.sub.i][Z.sup.ch]) = 1/RT [[integral].sup.v.sub.[infinity]] (RT/V - [partial derivative][p.sup.ch]/[partial derivative][n.sub.i])dV (B1)

By using Equations (31) [P.sup.ch] can be presented as:

[P.sup.ch] = [n.sub.0]RT/V - (K/P.sub.o])[n.sub.0][n.sub.A][(RT).sup.2]/V / V + (K/P.sub.o])[n.sub.0]RT (B2)

Therefore, for component i in the mixture the following equations can be obtained as:

(a) for non-associating components i [not equal to] A,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (B3)

for i [not equal to] A,

(b) for associating component I = A,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (B4)

By substituting Equations (B3) and (B4) in Equation (B1), Equation (38) will be derived.

It is seen that the fugacity of non-associating components in the mixture depends on the association constant (K) of the associating component. This is an interesting result, which has been obtained analytically in this work by using the linear association model.
```NOMENCLATURE

a EOS attractive parameter, Pa.[m.sup.6]/
[mol.sup.2]
[a.sub.0] monomeric EOS attractive parameter,
Pa.[m.sup.6]/[mol.sup.2]
A associating component
AEOS association equation of state
b EOS co-volume parameter, [m.sup.3]/mol
[b.sub.0] monomeric EOS co-volume parameter,
[m.sup.3]/mol
B non-associating component; second virial
coefficient, [m.sup.3]/mol
[B.sup.ch] chemical part of second virial coefficient,
[m.sup.3]/mol
Cal calculated results
[DELTA][C.sub.p.sup.o] standard heat capacity of association,
J/mol K
EOS equation of state
Exp experimental results
f fugacity, Pa
[DELTA][H.sup.o] standard enthalpy of association at [T.sub.0]
= 273.15K, J/mol
[k.sub.il] binary interaction parameter between
component i and l
K activity based association constant
K' pressure based association constant,
[Pa.sup.-1]
[K.sub.y] mole fraction based association constant
MMM EOS Mohsen-Nia, Modarress and Mansoori
equation of state
n number of all species in the system in their
actual, monomeric and associated state
[n.sub.0] number of moles of monomers in
non-associating state
[n.sub.1], [n.sub.2] EOS parameter
[n.sub.A] number of moles of associating component
[n.sub.A1] number of moles of monomer of associating
component
[n.sub.Ai] number of moles of i-mer of associating
component
[n.sub.Bj] number of moles of jth non-associating
component
N number of non-associating component, number
of data point, EOS parameter
P pressure, Pa
[P.sup.ch] chemical part of pressure equal to [Z.sup.ch]
RT/v, Pa
[P.sub.c] critical pressure, Pa
[P.sup.'c] monomeric critical pressure, Pa
[P.sup.o] standard pressure, Pa
[P.sup.i] saturation pressure of component i, Pa
R universal gas constant, Pa.[m.sup.3]/mol.K
[DELTA][S.sup.o] standard entropy of association at [T.sub.0]
= 273.15K, J/mol K
T temperature, K
[T.sub.0] standard temperature, K
[T.sub.c] critical temperature, K
[T'.sub.c] monomeric critical temperature, K
[T'.sub.r] monomeric reduced temperature
v molar volume, [m.sup.3]/mol
V total volume, [m.sup.3]
[x.sub.A] mole fraction of associating component,
[n.sub.A]/[n.sub.0]
[x.sub.Bj] mole fraction of jth non-associating
component, [n.sub.Bj]/[n.sub.0]
[x.sub.i] mole fraction of component i in liquid phase
[y.sub.i] mole fraction of component i in vapour phase
Z compressibility factor
[Z.sup.ch] chemical part of the compressibility factor,
n/[n.sub.0]
[Z.sup.ch.sub.eq.(30)] chemical part of the compressibility factor
calculated using Equation (30)
[Z.sup.ch.sub.eq.(31)] chemical part of the compressibility factor
calculated using Equation (31)
[Z.sup.ph] physical part of the compressibility factor

Greek Symbols

[alpha] EOS parameter
[micro] chemical potential
[micro].sup.r.sub.j] chemical potential of a substance j in the
mixture minus its chemical potential in the
ideal state
[rho] molar density, mol/[m.sup.3]
[phi] fugacity coefficient
[chi] mean associating number, [n.sub.0]/n
[omega] acentric factor
[omega]' monomeric acentric factor

Subscripts

A associating component
c critical property
i i-mer spice of associating component, index for
component
j, k indices for component
0 monomeric state
P saturation pressure
r reduced parameter
v saturation liquid molar volume
x mole fraction of liquid phase
y mole fraction of vapour phase

Superscripts

Cal calculated value
Ch chemical part
exp experimental value
l liquid phase
o standard condition
Ph physical part
r residual property
v vapour phase
```

Manuscript received January 29, 2006; revised manuscript received June 28, 2006; accepted for publication September 13, 2006.

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Mohsen Vafaie-Sefti (1*), Hamid Modarress (2), Majid Emami Meibodi (1) and Seyyed Ali Mousavi-Dehghani (1)

(1.) Department of Chemical Engineering, Tarbiat Modarres University, Tehran, Iran

(2.) Chemical Engineering Faculty, Amir Kabir University of Technology, Tehran, Iran
```Table 1. The parameters for water ethanol and 1-butanol
evaluated by three association equation of state and the
average absolute deviation in saturation volume and pressure

Shinta and
(1991) (1995)

Parameters Water Water

[T.sub.c] [k] 334 391
[P.sub.c] [Mpa] 19.0491 23.305
[omega] 0.224 0.04
[DELTA][H.sup.o][KJ/mol] -29.138 -26.641
[DELTA][C.sub.p.sup.o][J/mol.K] 0 0
[DELTA][S.sup.o][J/mol.K] -105.71 -93.49

This work

Parameters Water Ethanol 1-butanol

[T.sub.c] [k] 645.8 521.76 548.66
[P.sub.c] [Mpa] 22.934 5.88 4.3
[omega] 0.55 0.55 0.41
[DELTA][H.sup.o][KJ/mol] -20.958 -1.31 -16.51
[DELTA][C.sub.p.sup.o][J/mol.K] -28.9 0 -25.88
[DELTA][S.sup.o][J/mol.K] -114 -66.97 -89.26

Table 2. Calculated association number [chi] for water by three
association equations of state. Experimental values: [chi] = 4,
(Nemethy and Scheraga, 1962) and its maximum [chi] = 21, (Jhon
et al., 1966; Burke et al., 1993; Carey and Korenowski, 1998;
Maselia, 1998; Wales and Hodges, 1998; Chaplin, 1999;
Alfredsson and Hermansson, 1999; Ohtaki, 2003; Starzak and
Mathlouthi, 2003).

[chi], (Folas [chi], [chi],
T[K] et al., 2005) (Anderko, 1994) This work

273.15 226401 276 14

313.15 9801 67 6

353.15 857 23 3

413.15 56 7 2

473.15 9 3 1

513.15 5 2 1

613.15 2 1 1

Table 3. Feed, calculated and experimental * mole fractions of
equilibrium liquid and vapour and their average absolute
deviation (AAD) for quaternary mixture of [H.sub.2]O,
C[H.sub.4], C[O.sub.2], [H.sub.2]S at different temperatures

Liquid
P(MPa) T[K] Component Feed Exp. Cal.

16.93 310.92 [H.sub.2]O 0.5002 0.9777 0.9928
C[H.sub.4] 0.1492 0.00099 0.001
C[O.sub.2] 0.3006 0.0154 0.0048
[H.sub.2]S 0.0497 0.00608 0.0015

17.17 380.37 [H.sub.2]O 0.4997 0.9834 0.9807
C[H.sub.4] 0.1496 0.00 0.002
C[O.sub.2] 0.3009 0.0113 0.0112
[H.sub.2]S 0.0498 0.00473 0.0061

17.31 449.82 [H.sub.2]O 0.4946 0.9827 0.9644
C[H.sub.4] 0.15 0.0011 0.0033
C[O.sub.2] 0.3015 0.0114 0.018
[H.sub.2]S 0.0499 0.00478 0.0103

Vapour
P(MPa) T[K] Component Exp. Cal.

16.93 310.92 [H.sub.2]O 0.00199 0.0014
C[H.sub.4] 0.3021 0.2999
C[O.sub.2] 0.5963 0.6002
[H.sub.2]S 0.0996 0.0985

17.17 380.37 [H.sub.2]O 0.0179 0.0074
C[H.sub.4] 0.2935 0.3007
C[O.sub.2] 0.5916 0.5973
[H.sub.2]S 0.097 0.0945

17.31 449.82 [H.sub.2]O 0.0848 0.0577
C[H.sub.4] 0.2762 0.2864
C[O.sub.2] 0.552 0.5651
[H.sub.2]S 0.087 0.0868

* the experimental mole fractions are from Huang et al. (1984)

Table 4. Calculated error percent in K-values (K = y/x) for
quaternary system [H.sub.2]O/C[H.sub.4]/C[O.sub.2]/[H.sub.2]S
by the proposed AEOS in this work (*) and those reported in
Shinta and Firoozabadi (1995) (**) at T=310.92 K. The binary
interaction coefficient for [H.sub.2]O/C[O.sub.2] is
[k.sub.ij] = -0.0682 and for [H.sub.2]O/[H.sub.2]S is
[k.sub.ij] = -0.0668 and for other binaries [k.sub.ij] = 0

component [x.sub.CAL] [y.sub.CAL] Error% * Error% **

0.9776 0.0018 -8.3 -47.7
[H.sub.2]O
0.0011 0.3044 -7.97 -20.17
C[H.sub.4]
0.0155 0.5983 -0.048 -2.17
C[O.sub.2]
0.0058 0.0955 0.0095 6.439
[H.sub.2]S

Error % = 100 x [K.sub.cal] - [K.sub.exp]/[K.sub.exp]
```