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Chemistry » Inorganic Chemistry » "Phosphoric Acid Industry - Problems and Solutions", book edited by Michael Schorr Wiener and Benjamin Valdez, ISBN 978-953-51-3354-4, Print ISBN 978-953-51-3353-7, Published: July 12, 2017 under CC BY 3.0 license. © The Author(s).

Chapter 7

Purification of Phosphoric Acid by Liquid‐Liquid Equilibrium

By Khatereh Bahrpaima
DOI: 10.5772/67926

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Binodal curves for (water + phosphoric acid + propylene carbonate) mixture at two different temperatures and atmospheric pressure: () T = 298.2 K, and () T = 318.2 K [6].
Figure 1. Binodal curves for (water + phosphoric acid + propylene carbonate) mixture at two different temperatures and atmospheric pressure: () T = 298.2 K, and () T = 318.2 K [6].
Tie lines of the system water + phosphoric acid + MIAK at two different temperatures and atmospheric pressure: () T = 308.2 K, and (), T = 318.2 K [9].
Figure 2. Tie lines of the system water + phosphoric acid + MIAK at two different temperatures and atmospheric pressure: () T = 308.2 K, and (), T = 318.2 K [9].
Distribution coefficients D2 of phosphoric acid as a function of the mass percent W21 of acid in aqueous phase for NRTL LLE data of {water + phosphoric acid + mixed-solvent (DCM + DCE)} system at temperature T = 298.15 K: (25% DCM + 75% DCE); , (50% DCM + 50% DCE); , (75% DCM + 25% DCE);  [11].
Figure 3. Distribution coefficients D2 of phosphoric acid as a function of the mass percent W21 of acid in aqueous phase for NRTL LLE data of {water + phosphoric acid + mixed-solvent (DCM + DCE)} system at temperature T = 298.15 K: (25% DCM + 75% DCE); , (50% DCM + 50% DCE); , (75% DCM + 25% DCE); [11].
The effects of solvents with various functional groups on separation factor S for {water + phosphoric acid + solvent} system at mentioned temperature: (methylcyclohexane, 308.2 K [24]), ; (ethyl benzene, 298.2 K [25]), ; (isoamyl acetate, 308.2 K [9]), ; (n-pantane, 308.2 K [23]), ; (1-octanol, 298.2 K [16]), ; bis(2-methylpropyl) ether, 298.2 K [22]), .
Figure 4. The effects of solvents with various functional groups on separation factor S for {water + phosphoric acid + solvent} system at mentioned temperature: (methylcyclohexane, 308.2 K [24]), ; (ethyl benzene, 298.2 K [25]), ; (isoamyl acetate, 308.2 K [9]), ; (n-pantane, 308.2 K [23]), ; (1-octanol, 298.2 K [16]), ; bis(2-methylpropyl) ether, 298.2 K [22]), .

Purification of Phosphoric Acid by Liquid‐Liquid Equilibrium

Khatereh Bahrpaima
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Various ternary and quaternary liquid‐liquid phase equilibrium data for water + phosphoric acid + solvent(s) have been reported. Salting‐out, solvent, and temperature effects on the binodal curve and the tie lines have been highlighted and the capability of solvents with different functional groups to extract phosphoric acid from water has been compared. Studying of influence of magnetic, electromagnetic, and ultrasonic fields on the separation factors and distribution coefficients of aqueous phosphoric acid mixtures has been proposed. Moreover, a summary of the optimized binary interaction values, which resulted from non‐random two‐liquid (NRTL) and universal quasi‐chemical (UNIQUAC) thermodynamic models using genetic algorithm (GA), bee algorithm (BA), and simulated annealing (SA), has been presented. Group method of data handling (GMDH) and linear solvation energy relationship (LSER) methods for the correlation of experimental liquid‐liquid equilibrium (LLE) data have been used.

Keywords: phosphoric acid, LLE, optimization algorithms, GMDH, LSER

1. Introduction

The phosphoric acid due to its potential applications in various industries attracted a great deal of attention in the area of petrochemical operations, pharmaceutical productions, food industry, detergents, insecticides, and agricultural fertilizers. Generally, there are two basic methods in commercial use for the production of phosphoric acid—thermal and wet processes. The thermal process produces a pure acid with huge energy consumption, whereas the wet process is economic and practiced everywhere in the world [1]. Nevertheless, phosphoric acid produced from both methods contains a variety of impurities, which could affect quantity and the quality of the product. Therefore, for efficient removal and decreasing the impurities to below acceptable regulatory levels, not only suitable studies but also using and developing new different techniques and technologies are required. For this purpose, some of the common selective methods such as liquid‐liquid extraction, solid‐liquid extraction, crystallization, concentration, addition of additives, and so on, have been investigated [24].

Liquid‐liquid extraction or solvent extraction is now one of the most important and widely employed separation techniques in the modern process industries. It is well known due to various practical characteristics, including simplicity, rapid method development, and reasonable selectivity. Moreover, it can be used for several purposes, for instance, to separate the systems with similar boiling points, high boiling points, and temperature‐sensitive components [2].

Success of liquid‐liquid extraction and accurate understanding of the performance of that is measured through the liquid‐liquid equilibrium (LLE) data. LLE is described with the separation of components of a feed solution, containing the carrier and extract components, using an additional liquid solvent, as the mixture of feed and solvent—if enough solvent is added—forms two immiscible liquid phases. LLE is based on the differences in solubility and equilibrium distribution of these components between the two produced immiscible—or partially miscible—phases. In other words, it depends on the mass transfer of the component to be extracted from the carrier to the solvent. For having an effective extraction, the extract component should more preferably dissolve in the solvent. After settling the two phases, the raffinate and the extract phases are formed. The raffinate phase contains mostly the carrier, as well as a residue of the extracted component and solved solvent, whereas the extract phase consists of the main the solvent, a part of the extracted component, and solved carrier. Solvent extraction is a separation process aiming to purify the feed or to recover one or more components from it [5]. Nowadays, enormous amounts of acid, specifically high‐purity phosphoric acid, can be commercially produced through liquid‐liquid equilibrium process.

2. Phase diagram of multicomponent liquid mixtures: measurement and visualization

Only systems with at least three components and a miscibility gap can be used for extractions. Phase behavior of such systems at a constant temperature and pressure is conveniently represented on an equilateral triangular diagram, which its corners indicate the pure components, binary compositions are along the edges, and ternary mixtures are located inside the triangle. A common phase diagram is type I system and shown in Figure 1, where a pair of components propylene carbonate and water are partially miscible, and liquid phosphoric acid dissolves completely in propylene carbonate or water [6]. In this figure, the boundaries that each one of them separates the single‐phase region from the two‐phase region are named binodal (solubility) curves. The two‐phase region is included inside below the curved envelope. The binodal curve results are determined by cloud‐point titration method [7]. According to this method, a binary mixture of known composition was titrated with the third component at each mixture using non‐sealed glass vessels. The transition point is taken as the appearance/disappearance of turbidity in the sample and it is defined as a cloud point. The mixture temperature is regulated by a thermostatic thermometer with an accuracy of ±0.1 K. Every point on the binodal curve also has another corresponding point on the binodal curve, as these two points represent the phase equilibrium. The lines between these two equilibrium points are called tie lines. The tie lines are usually not parallel and their slopes can increase and decrease drastically. Tie line experiments were carried out in a jacketed 150 ml glass cell. The biphasic mixture with known compositions was placed in the extraction cell and was vigorously agitated by a magnetic stirrer for 4 h, and then left to settle for minimum 4 h for phase separation. After separation, samples of both phases were transparent and were carefully weighed and analyzed to determine their compositions. Then, the concentrations of the acid in both the phases were obtained by potentiometric NaOH titration [6]. Besides, the water content of the organic phase was measured by the Karl‐Fisher method [6] and the water contained in the aqueous layer was determined using refractive index measurement method [8]. The obtained tie line data for (water + phosphoric acid + methyl isoamyl ketone (MIAK)) [9] and (water + phosphoric acid + dichloromethane (DCM)) [10] ternary systems are presented in Figure 2 and Table 1, respectively. As either the temperature or pressure is varied, the location of the binodal curve and slopes of the tie lines may change (see Figures 1 and 2).


Figure 1.

Binodal curves for (water + phosphoric acid + propylene carbonate) mixture at two different temperatures and atmospheric pressure: (media/UF8.png) T = 298.2 K, and (media/UF10.png) T = 318.2 K [6].


Figure 2.

Tie lines of the system water + phosphoric acid + MIAK at two different temperatures and atmospheric pressure: (media/UF8.png) T = 308.2 K, and (media/UF5.png), T = 318.2 K [9].

Aqueous phase (raffinate) mass percentOrganic phase (extract) mass percentD1D2S

Table 1.

Calculated tie line data, distribution coefficient, and separation factor for the ternary system of water (1) + phosphoric acid (2) + DCM (3), according to the GMDH* results [10].

[i] - *GMDH is used to predict the tie lines.

3. Separation factor and distribution coefficients

In order to evaluate the extracting capability of the solvent for the separation of components from feed solutions with liquid‐liquid extraction, the separation factor, S, is calculated. The separation factor is defined as the ratio of distribution coefficients of extract component, 2, to carrier one, 1, S = D2/D1. Additionally, distribution coefficient is the ratio of concentrations of a component in a mixture of two immiscible phases at equilibrium. This ratio is therefore a measure of the difference in the component solubilities in these two phases (see Table 1). Depending on the system, the distribution coefficient and separation factor can be a function of temperature, the concentration of chemical species in the system, and a large number of other parameters. Figure 3 gives the distribution coefficient as a function of the mass percent of phosphoric acid in aqueous phase for water + phosphoric acid + mixed‐solvent (dichloromethane + 1,2‐dichloroethane (DCE)) system at different ratios of mixed solvent [11]. Generally, if the separation factor is greater than one (S > 1) for the systems investigated, it means that the component can be extracted by the proposed solvent—for example, according to the reported S for studied system in Table 1, DCM can be considered as a possible candidate for the recovery of the aqueous phosphoric acid solutions.


Figure 3.

Distribution coefficients D2 of phosphoric acid as a function of the mass percent W21 of acid in aqueous phase for NRTL LLE data of {water + phosphoric acid + mixed-solvent (DCM + DCE)} system at temperature T = 298.15 K: (25% DCM + 75% DCE); media/UF1.png, (50% DCM + 50% DCE); media/UF6.png, (75% DCM + 25% DCE); media/UF3.png [11].

In recent years, significant investigations on LLE measurements and the extraction of phosphoric acid from aqueous solutions have been carried out, while many solvents have mainly been tested, in order to understand and provide further information for accurate interpretation of phase equilibria of the different multicomponent mixtures with phosphoric acid [1227]. As a detailed evaluation of extraction abilities of the different solvents reported in the literature, including alcohols, esters, ethers, ketones, and hydrocarbons to extract phosphoric acid from aqueous solutions, a comparative study is listed in Table 2. Figure 3 also highlights the definitive effects of different chemical structures present in solvents on recovering of the acid. As it is drawn, results show the following order of effectiveness of solvent for systems containing phosphoric acid: methylcyclohexane > ethyl benzene > isoamyl acetate > n‐pentane > 1‐octanol > bis(2‐methylpropyl) ether. From the obtained experimental results, it can be concluded that methylcyclohexane is the most appropriate solvent for the separation of a mixture of water and phosphoric acid. It is also apparent from Figure 4 that bis(2‐methylpropyl) ether is a less favorable solvating agent for phosphoric acid.


Figure 4.

The effects of solvents with various functional groups on separation factor S for {water + phosphoric acid + solvent} system at mentioned temperature: (methylcyclohexane, 308.2 K [24]), media/UF1.png; (ethyl benzene, 298.2 K [25]), media/UF2.png; (isoamyl acetate, 308.2 K [9]), media/UF3.png; (n-pantane, 308.2 K [23]), media/UF4.png; (1-octanol, 298.2 K [16]), media/UF11.png; bis(2-methylpropyl) ether, 298.2 K [22]), media/UF9.png.

SystemSolvent typeT/KSRef.
Water + phosphoric acid + 1‐butanolAlcohol308.21.65[12]
Water + phosphoric acid + 2‐methyl‐2‐butanolAlcohol298.21.82[13]
Water + phosphoric acid + isoamy1 alcoholAlcohol308.21.82[14]
Water + phosphoric acid + 2‐ethyl‐1‐hexanolAlcohol298.21.46[15]
Water + phosphoric acid + 1‐octanolAlcohol298.26.35[16]
Water + phosphoric acid + 1‐nanonolAlcohol298.23.54[17]
Water + phosphoric acid + 1‐decanolAlcohol303.23.80[18]
Water + phosphoric acid + 1‐dodecanolAlcohol303.24.30[18]
Water + phosphoric acid + cyclohexanolAlcohol, cyclic alcohol308.21.57[14]
Water + phosphoric acid + 1‐dodecanethiolThiol, aliphatic thiol303.218.20[18]
Water + phosphoric acid + butyl acetateEster308.25.78[12]
Water + phosphoric acid + isobutyl acetateEster298.27.46[13]
Water + phosphoric acid + isoamyl acetateEster298.219.70[9]
Water + phosphoric acid + hexyl acetateEster298.27.00[19]
Water + phosphoric acid + cyclohexyl acetateEster298.26.20[19]
Water + phosphoric acid + propylene carbonateEster298.22.82[6]
Water + phosphoric acid + tributyl phosphateEster298.23.40[20]
Water + phosphoric acid + diisopropyl etherEther293.21.89[21]
Water + phosphoric acid + bis(2‐methylpropyl) etherEther298.23.76[22]
Water + phosphoric acid + n‐pentaneHydrocarbon308.215.08[23]
Water + phosphoric acid + 2‐methylpentaneHydrocarbon308.23.36[23]
Water + phosphoric acid + n‐hexaneHydrocarbon308.24.60[23]
Water + phosphoric acid + cyclohexaneHydrocarbon, cycloalkane308.215.30[24]
Water + phosphoric acid + methyl cyclohexaneHydrocarbon, cycloalkane308.243.10[24]
Water + phosphoric acid + tolueneHydrocarbon, aromatic hydrocarbon308.226.50[24]
Water + phosphoric acid + ethyl benzeneHydrocarbon, aromatic hydrocarbon298.232.70[25]
Water + phosphoric acid + isopropyl benzeneHydrocarbon, aromatic hydrocarbon298.223.70[25]
Water + phosphoric acid + dichloromethane (DCM)Halogenated aliphatic hydrocarbon298.21.04[26]
Water + phosphoric acid + DCM + 10 wt% NaClHalogenated aliphatic hydrocarbon + salt298.22.20[26]
Water + phosphoric acid + DCM + 10 wt% CaCl2Halogenated aliphatic hydrocarbon + salt298.21.39[26]
Water + phosphoric acid + 1,2‐dichloroethane (DCE)Halogenated aliphatic hydrocarbon298.21.10[26]
Water + phosphoric acid + DCE + 10 wt% NaClHalogenated aliphatic hydrocarbon + salt298.22.35[26]
Water + phosphoric acid + DCE + 10 wt% CaCl2Halogenated aliphatic hydrocarbon + salt298.22.13[26]
Water + phosphoric acid + methyl isobutyl ketoneKetone308.21.26[14]
Water + phosphoric acid + methyl isoamyl ketoneKetone298.211.00[9]
Water + phosphoric acid + methyl ethyl ketoneKetone308.20.91[27]

Table 2.

The maximum values of separation factors for the (water + phosphoric acid + solvent) ternary systems.

In addition, the presence of dissolved salt changes the phase equilibrium behavior of a mixture significantly [2831]. Recently, Govindarajan and Sabarathinam [32], Mohsen‐Nia et al. [33], and Santos et al. [34] have investigated effects of some inorganic salts such as Na2SO4, ZnSO4, (NH4)2SO4, NaCl, KCl, KBr, and NaNO3 on the ternary liquid‐liquid equilibria data. Similarly, by considering the importance of the salting effect on the extraction of phosphoric acid from aqueous mixtures, it is worthwhile to study LLE of mixtures of (water + phosphoric acid + solvent + salt). In particular, as can be seen in Table 2, maximum values of separation factor for (water + phosphoric acid + DCM + 10 wt% NaCl and 10 wt% CaCl2) solutions are improved compared with those in the absence of added salts [26].

Although solvents with high separation are considered as potential candidates to carry out the extraction, due to the economic, environmental, and technical factors, the suitable solvent at optimum conditions is proposed to be chosen for the extraction of phosphoric acid from aqueous mixtures. Meanwhile, it seems that the measurement of the (liquid + liquid) equilibrium data of the mixtures with a mixed solvent has become the turning point topic of the chemistry investigations in recent years [11, 3542]. As an illustration, Figure 3 compares distribution coefficients of phosphoric acid for three quaternary systems of (water + phosphoric acid + mixed solvent (25% DCM + 75% DCE) or (50% DCM + 50% DCE) or (75% DCM + 25% DCE)) [11]. The results confirm the different effects of the dichloromethane as co‐solvent on 1,2‐dichloroethane for the extraction of phosphoric acid from aqueous mixtures. The comparison between the experimental separation factors obtained for the mixed solvent (DCM + DCE) with various ratios also indicates that mixed solvent (25% DCM, 75% DCE) has a higher separation factor than the other mixed solvents at T = 298.15 K, which means that the extraction of phosphoric acid by mixed solvent (25% DCM, 75% DCE) is more suitable.

Besides, it is anticipated that the magnetic, electromagnetic, and ultrasonic fields as external factors can affect LLE data by influencing solvents properties such as polarity and permittivity, especially for polar solvents. Therefore, the application of them to the phase separation in the solvent extraction process has been considered from the technical and economical viewpoints. For studying this purpose, the effect of the applied different fields on phase behavior of several ternary systems has been obtained and reported [4348]. There are a number of effective potentials for future research to evaluate the effects of the magnetic, electromagnetic, and ultrasonic fields on the extraction of phosphoric acid from aqueous mixtures.

The linear solvation energy relationship (LSER) models [49] such as Kamlet (Eq. (1)) and Katritzky (Eq. (2)) equations [50, 51] were used to correlate separation factor for several ternary LLE systems with phosphoric acid, reporting in references [16, 19, 24]. Kamlet LSER model with the solvatochromic parameters is defined according to the following equation:

logS=logS0+ s(xπ*+dxδ)+axα+bxβ

where π* is a measure of solvent dipolarity/polarizability, δ is a discontinuous polarizability correction term, α is a measure of the solvent hydrogen‐bond donor acidity, and β is a measure of the hydrogen‐bond acceptor basicity. The coefficients S0, s, d, a, and b include the properties of solute and are derived from regression. The values of Kamlet‐Taft solvatochromic parameters (α, β, and π*) of the solvents are obtained from literature [49].

Katritzky [51] introduced a multiparameter polarity scale with combination of Riechardt's solvent polarity, ETN, the dielectric constant, ε, and the index of refraction, n, functions. The Katritzky equation (Eq. (2)) allows one to estimate independent description of solvent dipolarity, polarizability, and specific interactions (such as hydrogen bonding, π‐π interaction). The coefficients, a, b, c, and d measure the relative susceptibilities of log (S) to the indicated solvent parameters and are regressed using experimental data. ETN, ε, and n values of solvents are also present in literature [49]. A modified form of the Katritzky equation may be used, consisting of the addition of a dimensionless term as T/298.2 that characterizes temperature effect

logS=logS0+a . ETN+b . (ε12ε+1)+c . (n212n2+1)+d . (T298.2)

The LSER model values showed a good regression of the experimental data for all investigated systems. The temperature susceptibility coefficient, d, shows that all the systems in references [24] are temperature sensitive.

4. LLE correlation

Various thermodynamic models with different abilities can be used to accurately describe the multicomponent LLE systems. Aided by these activity coefficient equations and a parameter fit, it is possible to reproduce the measurement and control the data. The non‐random two‐liquid (NRTL) [52] and the universal quasi‐chemical (UNIQUAC) [53] methods have been successfully applied for the correlation of many ternary and quaternary liquid‐liquid phase equilibrium solutions [12, 1619, 2426], while a group contribution approach (UNIFAC) [54] may be utilized to predict the LLE mixtures.

4.1. NRTL equation

The basic idea in NRTL equation follows from the concept of local composition which was used by Renon and Prausnitz [52]. Renon's equation is applicable to partially miscible liquid mixtures, and for the binary mixtures the excess Gibbs energy is



G12=exp(α12τ12); G21=exp(α21τ21)
τ12=g12g22RT ; τ21=g21g11RT

The significance of gij is an energy parameter characteristic of the ij interaction. Parameter α12 is related to the nonrandomness in the mixture; when α12 is zero, the mixture is completely random and Eq. (1) reduces to the two‐suffix Margules equation [55]. The NRTL equation contains three parameters, but reduction of experimental data for a large number of binary systems indicates that α12 varies from about 0.20 to 0.47; when experimental data are scarce, the value of α12 can often be set arbitrarily; a typical choice is α12 = 0.3. From Eq. (3), the activity coefficients are


For strongly nonideal mixtures and especially for partially immiscible systems [56], the NRTL equation often provides a good representation of experimental data if care is exercised in data reduction to obtain the adjustable parameters.

The equations discussed are readily to as many components as desired without any additional assumptions and without introducing any constants other than those obtained from binary data. For a solution of m components, the NRTL equation is




The activity coefficient for any component i is given by


Eqs. (8) and (11) contain only parameters obtained from binary data. The experimental tie line LLE data of multicomponent mixtures can be correlated using the NRTL model within well‐known simulation software such as Aspen Plus. The quality of the correlation is measured by the root‐mean‐square deviation (RMSD) [16]. Many NRTL LLE values for the ternary and quaternary systems containing water, phosphoric acid, and solvent have appeared in the available literature [11, 16, 17]. For example, in Figure 3, the calculated NRTL data have been used to plot and show the variation of distribution coefficients of acid for (water + phosphoric acid + mixed‐solvent (DCM + DCE)) system at temperature T = 298.15 K. Furthermore, several optimization algorithms such as genetic algorithm (GA) [57], bee algorithm (BA) [58], and simulated annealing (SA) [59] can be applied to predict the binary interaction parameters. As can be seen in Table 3, the SA and NRTL models have been used to estimate the optimized binary interaction parameters for an aqueous ternary system containing phosphoric acid [60]. Table 4 reports 12 NRTL binary interaction parameters obtained using BA for the quaternary aqueous mixture including phosphoric acid and (50% DCM + 50% DCE) [61].

ijAij(K) *Aji(K)

Table 3.

Correlated LLE results from the NRTL (α12 = 0.3) model using SA; the corresponding binary interaction parameters, Aij and Aji, for ternary system water (1) + phosphoric acid (2) + DCE (3) + 10 wt% CaCl2 at temperature T = 298.15 K.

[i] - *gij (J mol−1) = Aij (K)/RT.


Table 4.

NRTL (α12 = 0.3) interaction parameters, Aij and Aji, using BA for quaternary LLE system water (1) + phosphoric acid (2) + (50% DCM (3) + 50% DCE (4)) at temperature T = 298.15 K.

4.2. UNIQUAC equation

A critical examination of the derivation of the NRTL equation shows that this equation is more suitable for HE than GE [53]. Abrams derived an equation that, in a sense, extends the quasi‐chemical theory of Guggenhiem [56] for nonrandom mixtures to solutions containing molecules of different size. This extension was therefore called the universal quasi‐chemical theory or, in short, UNIQUAC. The UNIQUAC equation for GE consists of two parts: a combinatorial part that attempts to describe the dominant entropic contribution and a residual part that is due primarily to intermolecular forces that are responsible for the enthalpy of mixing. The combinatorial part is determined only by the composition and by the sizes and shapes of the molecules; it requires only pure‐component data. The residual part, however, depends also on intermolecular forces; the two adjustable binary parameters, therefore, appear only in the residual part. The UNIQUAC equation is


For a binary mixture,


where the coordination number z is set equal to 10. Segment fraction, Ф, and area fraction, θ and θ′, are given by




Parameters r, q, and q′ are pure component molecular structure constants depending on molecular size and external surface areas. In the original formulation, q = q′. To obtain better agreement for systems containing water or lower alcohols, q′ values for water and alcohols were adjusted empirically by Anderson [62] to give an optimum fit to a variety of systems containing these components. For alcohols, the surface of interaction q′ is smaller than the geometric external surface q, suggesting that intermolecular attraction is dominated by the OH group (hydrogen bonding). For fluids other than water or lower alcohols, q = q′.

For each binary mixture, there are two adjustable parameters, τ12 and τ21. These, in turn, are given in terms of characteristic energies Δu12 and Δu21, by


For many cases, Eqs. (19) and (20) give the primary effect of temperature on τ12 and τ21. Characteristic energies Δu12 and Δu21 are often only weakly dependent on temperature.

lnγ1=lnΦ1x1+z2 q1lnθ1Ф1+ Ф1(l1 r1r2l2)q1 ln(θ1'+θ2'τ21)+ θ2' q1(τ21θ1'+θ2'τ21τ12θ2'+θ1'τ12)
lnγ2=lnΦ2x2+z2 q2lnθ2Ф2+Ф2(l2 r2r1l1)q2 ln(θ2'+θ1'τ12)+ θ1' q2(τ12θ2'+θ1'τ12τ21θ1'+θ2'τ21)

For a multicomponent system, the UNIQUAC equation for the molar excess Gibbs energy is given by the sum of




where segment fraction Ф and area fractions θ and θ′ are given by


The activity coefficient for any component i is given by


The molecular interaction‐based model of UNIQUAC successfully has correlated many LLE data of ternary and quaternary systems [10, 1517, 24] within Aspen Plus—for example, see Table 5; however, as previously mentioned, during the last 10 years, many LLE‐based algorithm—GA, SA, and BA—have been developed and introduced in the literature [57, 60, 61]. Recently, a study on improved binary parameters using GA for ternary mixtures: UNIQUAC model has been carried out by Hamidi [63].

ijuij(J mol1)uji(J mol1)
*r1 = 0.920, r2 = 3.000, r3 = 4.827*q1 = 1. 400, q2 = 4.000, q3 = 4.196

Table 5.

Optimized UNIQUAC binary interaction energy parameters, uij and uji(J mol1), for water (1) + phosphoric acid (2) + butyl acetate (3) ternary system at T = 308.2 K [12].

[i] - *ri and qi are the UN1QUAC structural parameters.

4.3. GMDH‐type neural network

It is observed that usually LLE measurements are successfully correlated using common models such as NRTL and UNIQUAC. However, these conventional thermodynamic methods for data prediction of complex systems are tedious and involve a certain amount of empiricism. Recently, a new prediction method, the group method of data handling, was developed to predict LLE and VLE data in order to avoid these limitations [64, 65]. Namely, the GMDH tie line data for the ternary system of water + phosphoric acid + dichloromethane have been illustrated in Table 1 [10, 26]. Furthermore, the reliability of the presented tie line results can be tested through the Othmer‐Tobias [66] and Bachman [67] correlation equations.


We gratefully acknowledge the financial support from the Research Council of Firoozabad Branch, Islamic Azad University.


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