In summary, the unexpected chirality behaviour of the accounted materials clearly pointed to the important fact that chiral properties can be created and amplified at the supramolecular level in the solid state and finely tuned by mild reactions in the solid-liquid interface [51–53].

Important research is being directed to the design of molecular systems able to display the strong luminescence of lanthanide metals [53]. To accomplish it with efficiency, two stringent conditions have to be met: (i) a suitable organic chromophore should absorb light and efficiently transfer energy to the metal (*antenna effect*), (ii) the coordination sphere of the metal should be free of water molecules because the OH oscillators easily quench metal emission. The first condition has been achieved either by simple acid-base intercalation or by topotactic exchange of suitable chromophores in the organic-inorganic lattice. The second, by using the pillared materials with polyethylenoxa columns described above, constituted an excellent spider web to enshroud the oxygenphylic lanthanide metals and isolate them within the solid matrix. Additionally, the chromophores have been provided with chirality. The resulting materials exhibited circularly polarized luminescence (CPL) of lanthanides, an elusive property in the solid state which could find important applications in the technology of optical handling of information.

### **4.3. Gas and vapour storage**

and O2

example.

P(OH)2

130 Phosphoric Acid Industry - Problems and Solutions

interface, has never been observed before [44].

**4.2. Dissymmetry and luminescence signalling**

acidic groups of the layers are broken. Therefore, the number of water molecules

in the interlayer region is increased and the polyethylenoxide chains are forced to align perpendicular to the layers. This odd supramolecular behaviour in the solid state, where the porosity of a layered material is abruptly increased in response to a simple acid-base reaction in the solid-liquid

**Figure 7.** Idealized molecular model of **γ-ZrP** exchanged with pentaethylene glycol diphosphonate as a representative

Molecular modelling of the *polar/polar* and *polar/non-polar* materials (see previous section) in the extended arrangement, that is, when the amine was present, suggested that in order to fit the largest achieved interlayer distances, the polyethylenoxa columns had to be helicoidally arranged. Polyethylene glycols are actually known to attain that conformation in aqueous solution. Yet, what makes a notable difference in the previously discussed materials is that the polyethylenoxa columns are covalently attached by both ends to the inorganic layers, thus conferring to the whole scaffold a supramolecular dimension. In principle, should the appropriate conditions be established, the random P/M helicity of the chains could be driven towards homochirality. Notably, this homochirality has been revealed by optical rotation measurements. To achieve that, we performed the following simple experiment.

Dispersions of small amounts in water/acetone of either native **γ-ZrP** or exchanged at 25% with hexaethylene glycol diphosphonate (n = 6; *cf*. molecular formula in previous section) displayed no sizeable optical rotation as expected. However, the material pillared with hexaethylene glycol diphosphonate intercalated with (+)-phenethylamine rendered a relatively large value of optical rotation. Surprisingly, when the enantiomerically pure amine was smoothly

Hydrogen storage is attained by *physisorption* in porous matrices [4, 54] among other procedures. The attachment of polyphenyl or polyphenylethynyl diphosphonates to either of the ZrP phases (cf. **Figure 6** as an example) led to materials with slit-like ultramicropores of different length. Further exchange reactions allowed for the attachment of polar or non-polar groups at both ends of the pores. This set of materials was tested for hydrogen storage. The best results were obtained with the material named **αT60Li**, α-ZrP with 60% of terphenyldiphosphonates [6], the remainder phosphates bearing Li+ as counterions. At 800 Torr and 77K, 1.7% w/w of hydrogen could be stored and the Department of Energy goal for 2010 (45 g of hydrogen per L of material) was thus attained below 2 atm. at 77K.

### **4.4. Building DSSCs**

Phosphonate derivatives of the RuII(bpy)3 complex were attached to γ-ZrP and placed side-byside to different electron acceptors ranging from relatively simple viologens to more elaborated fullerene derivatives [55]. The first important observation was that the phosphorescence emission of the RuII(bpy)3 complex was heavily quenched by the presence of the electron-acceptor species, indicating that the pursued electron transfer took place. More recently, flash photolysis revealed some of these materials to have outstanding properties in that the separation of charges is long lived and the initially thought inert inorganic layer resulted in semiconducting, thus driving the separated electrons and/or holes along the material [56]. Additional results showed that these powders can be arranged as solar cells with promising efficiency [57].

### **4.5. Molecular confinement of drugs and dyes**

Confinement of drugs into molecular matrices might be a good solution to their slow release at particular locations on the body [55]. Preliminary studies show that the erythromycin family antibiotics can intercalate into ZrP phases. The measurement of the interlayer distance versus the stoichiometry of the reactants shows that mono- and bilayers of erythromycin derivatives, as an example, may be enclosed between the inorganic lamellae.

On its part, dyes are an integral part of many technologies. For example, Malachite green (MG) is used in Gram's stain which is a technique used to classify bacteria. Also, it is used for anti-fungal purposes. However, many dyes are toxic and carcinogenic and have to be handled with special care if serious environmental contamination is to be avoided. Purification of water and air resources that contain even traces of dyes is an important technological challenge and has attracted wide attention. Dye-containing coloured water is of no use but appropriately bleached solutions may still be used for washing, cooling, irrigation and cleaning purposes. Materials able to efficiently entrap dyes are thus quite desirable. **ZrP** phases serve this purpose as well (**Figure 8**) [8, 9].

These preliminary results are a good omen regarding the high prospects of using ZrP phases in the uptake of different drugs and dyes like MG or its next-of-kin methyl- and crystal violet. They also show the importance of ZrP and its surfactant composites in the field of intercalation and inclusion chemistry of bioactive organic compounds within layered inorganic and organic-inorganic materials.

### **4.6. Metal uptake**

One measure of water quality is its 'hardness' which is defined by its Mg2+ and Ca2+ ion contents. The synthesis and design of materials towards efficiently entrapping Ca2+ are of great industrial importance because hard water, on being heated, precipitates calcium carbonate which clogs boilers and pipes. Moreover, Ca2+ reacts with soap and produces an insoluble product (soap scum). This scum is abrasive and may weaken clothes fibres.

groups at both ends of the pores. This set of materials was tested for hydrogen storage. The best results were obtained with the material named **αT60Li**, α-ZrP with 60% of terphenyl-

77K, 1.7% w/w of hydrogen could be stored and the Department of Energy goal for 2010 (45 g

side to different electron acceptors ranging from relatively simple viologens to more elaborated fullerene derivatives [55]. The first important observation was that the phosphorescence emis-

species, indicating that the pursued electron transfer took place. More recently, flash photolysis revealed some of these materials to have outstanding properties in that the separation of charges is long lived and the initially thought inert inorganic layer resulted in semiconducting, thus driving the separated electrons and/or holes along the material [56]. Additional results showed that these powders can be arranged as solar cells with promising efficiency [57].

Confinement of drugs into molecular matrices might be a good solution to their slow release at particular locations on the body [55]. Preliminary studies show that the erythromycin family antibiotics can intercalate into ZrP phases. The measurement of the interlayer distance versus the stoichiometry of the reactants shows that mono- and bilayers of erythromycin

On its part, dyes are an integral part of many technologies. For example, Malachite green (MG) is used in Gram's stain which is a technique used to classify bacteria. Also, it is used for anti-fungal purposes. However, many dyes are toxic and carcinogenic and have to be handled with special care if serious environmental contamination is to be avoided. Purification of water and air resources that contain even traces of dyes is an important technological challenge and has attracted wide attention. Dye-containing coloured water is of no use but appropriately bleached solutions may still be used for washing, cooling, irrigation and cleaning purposes. Materials able to efficiently entrap dyes are thus quite desirable. **ZrP** phases serve

These preliminary results are a good omen regarding the high prospects of using ZrP phases in the uptake of different drugs and dyes like MG or its next-of-kin methyl- and crystal violet. They also show the importance of ZrP and its surfactant composites in the field of intercalation and inclusion chemistry of bioactive organic compounds within layered inorganic and

One measure of water quality is its 'hardness' which is defined by its Mg2+ and Ca2+ ion contents. The synthesis and design of materials towards efficiently entrapping Ca2+ are of great

derivatives, as an example, may be enclosed between the inorganic lamellae.

as counterions. At 800 Torr and

complex were attached to γ-ZrP and placed side-by-

complex was heavily quenched by the presence of the electron-acceptor

diphosphonates [6], the remainder phosphates bearing Li+

**4.4. Building DSSCs**

sion of the RuII(bpy)3

Phosphonate derivatives of the RuII(bpy)3

132 Phosphoric Acid Industry - Problems and Solutions

**4.5. Molecular confinement of drugs and dyes**

this purpose as well (**Figure 8**) [8, 9].

organic-inorganic materials.

**4.6. Metal uptake**

of hydrogen per L of material) was thus attained below 2 atm. at 77K.

λ-ZrP and its new ethylenediaminetetraacetic acid (EDTA)-functionalized derivative (λ-ZrPH2 Y) strongly complexes Ca2+. The results are given in **Figure 9** [10].

**Figure 8.** Possible arrangement of MG intercalated in **γ-ZrP** complying with the observed interlayer distance measured by XRD.

**Figure 9.** Kinetic curves of Ca2+ uptake from tap water samples (T.W.). (A) The remaining concentration of Ca2+ (ppm) in TW samples after treatment with λ-solids. (B) % Ca2+ uptake by λ-ZrP and λ-ZrPH2 Y. (C) Ca2+ uptake capacities of λ-ZrP and λ-ZrPH2 Y (mg g−1).

Metal uptake reached the equilibrium after ca. 24 h, where the maximum Ca2+ uptake is obtained (32.0 and 40.4 mg Ca2+/g λ-phase, respectively), corresponding to ca. 0.8 and 1.0 mmol Ca2+/g λ-phase and % Ca2+ uptake of 60 and 74% (**Figure 9**B), respectively.

These positive results have stimulated the testing of λ-ZrP-EDTA material for the removal of toxic metals such as Cu2+ and Ni2+ from water. The results are summarized in **Figures 10** and **11** [58].

Equilibrium was reached for Cu2+ and Ni2+ after ca. 48 h, where the maximum uptake is obtained (72.0 and 93.5 mg M2+/g λ-ZrPH2 Y, respectively). These uptake values correspond to ca. 4.58 and 5.49 mmol M2+/g λ-ZrPH2 Y and % M2+ uptake of 58.0 and 76.6 %, respectively. This shows that λ-ZrPH2 Y has higher Ni2+ uptake capacity as compared to Cu2+.

These experiments showed that EDTA was successfully incorporated inside the interlayer gallery of λ-ZrP by means of topotactic carboxylate/Cl ligand exchange reaction. The resulting

**Figure 10.** Kinetic curves of Cu2+ uptake from aqueous solutions: The remaining concentration of Cu2+ (ppm) in samples after treatment with λ-ZrPH2 Y (*top left*). % Cu2+ uptake by λ-ZrPH2 Y (*top right*). Cu2+ uptake capacities of λ-ZrPH2 Y (mg g−1) (*bottom*).

Phosphate-Based Organic-Inorganic Hybrid Materials: Design, Construction and Technological Applications http://dx.doi.org/10.5772/68140 135

**Figure 11.** Kinetic curves of Ni2+ uptake from aqueous solutions (initial conc. 1375 ppm): The remaining concentration of Ni2+ (ppm) in samples after treatment with λ-ZrPH2 Y (*top left*). % Ni2+ uptake by λ-ZrPH2 Y (*top right*). Ni2+ uptake capacities of λ-ZrPH2 Y (mg g-1) (*bottom*).

composite phase, λ-ZrPH2 Y, exhibited high potential for the extraction of various metal ions from water samples.

### **5. Conclusion and prospects**

Metal uptake reached the equilibrium after ca. 24 h, where the maximum Ca2+ uptake is obtained (32.0 and 40.4 mg Ca2+/g λ-phase, respectively), corresponding to ca. 0.8 and 1.0

These positive results have stimulated the testing of λ-ZrP-EDTA material for the removal of toxic metals such as Cu2+ and Ni2+ from water. The results are summarized in **Figures 10**

Equilibrium was reached for Cu2+ and Ni2+ after ca. 48 h, where the maximum uptake is

Y has higher Ni2+ uptake capacity as compared to Cu2+. These experiments showed that EDTA was successfully incorporated inside the interlayer gallery of λ-ZrP by means of topotactic carboxylate/Cl ligand exchange reaction. The resulting

% Cu2+ uptake

0 10 20 30 40 50 60 70 80

Time (h)

**Figure 10.** Kinetic curves of Cu2+ uptake from aqueous solutions: The remaining concentration of Cu2+ (ppm) in samples

Y (*top left*). % Cu2+ uptake by λ-ZrPH2

 pH 3.3 pH 3.76 pH 4.24 pH 4.8 pH 5.41 pH 6.17

Y, respectively). These uptake values correspond to

0 10 20 30 40 50 60 70 80

 pH 3.30 pH 3.76 pH 4.24 pH 4.80 pH 5.41 pH 6.17

Y (mg

Time (h)

Y (*top right*). Cu2+ uptake capacities of λ-ZrPH2

 pH 3.30 pH 3.76 pH 4.24 pH 4.80 pH 5.41 pH 6.17

Y and % M2+ uptake of 58.0 and 76.6 %, respectively. This

mmol Ca2+/g λ-phase and % Ca2+ uptake of 60 and 74% (**Figure 9**B), respectively.

and **11** [58].

shows that λ-ZrPH2

after treatment with λ-ZrPH2

g−1) (*bottom*).

Remaining Cu2+ (ppm)

obtained (72.0 and 93.5 mg M2+/g λ-ZrPH2

Untreated sample (1375 ppm)

0 10 20 30 40 50 60 70 80

Time (h)

mg Cu2+/g lambda-ZrP-EDT

A

ca. 4.58 and 5.49 mmol M2+/g λ-ZrPH2

134 Phosphoric Acid Industry - Problems and Solutions

Phosphoric acid chemistry has limitless features. In this article, we have gathered a collection of eloquent experimental facts where the primordial phosphoric acid, combined with Zr metal and other chemicals, led to the building of inorganic-organic 2D and 3D structures that can play a crucial role in numerous research areas. The number of rather complex composites that can be figured out from the presented rationale is inexhaustible and only limited by the imagination and chemical needs of the researcher. Moreover, the confinement of organic molecules within the ZrP frameworks would confer them new properties at the supramolecular level and give rise to many surprises driven by the high level of *serendipity* that was and will be found in this field.

### **Acknowledgements**

This article has been written within the collaborative research work activities between the research groups of Prof. H. Ahendawi (*Al-Azhar University – Gaza, Palestine*) and Prof. E. Brunet (*Autonomous University of Madrid, Spain*). It should be mentioned that parts of this chapter are reproduced from the authors' previous publications [12].

### **Author details**

Hussein M. H. Alhendawi1 \*, Ernesto Brunet2 and Elena Rodríguez-Payán2

\*Address all correspondence to<EMAIL_ADDRESS>
1 Department of Chemistry, Faculty of Science, Al-Azhar University-Gaza, Gaza, Palestine

2 Department of Organic Chemistry, Faculty of Sciences, Universidad Autónoma de Madrid, Madrid, Spain

### **References**

[7] Brunet E, Alhendawi H, Alonso M, Cerro C, Jiménez L, Juanes O, de la Mata MJ, Salvador A, Victoria M, Rodríguez-Payán E, Rodríguez-Ubis JC. The Use of Laminar Inorganic Salts to Make Organic Molecules Display New Properties at the Supramolecular Level in the Solid State. J. Phys.: Conf. Ser. 2010; **232**:012017. DOI: 10.1088/1742-6596/232/1/012017

**Acknowledgements**

136 Phosphoric Acid Industry - Problems and Solutions

**Author details**

Madrid, Spain

**References**

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10.1039/b817317f

reproduced from the authors' previous publications [12].

\*, Ernesto Brunet2

\*Address all correspondence to<EMAIL_ADDRESS>
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1 Department of Chemistry, Faculty of Science, Al-Azhar University-Gaza, Gaza, Palestine

2 Department of Organic Chemistry, Faculty of Sciences, Universidad Autónoma de Madrid,

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## **Purification of Phosphoric Acid by Liquid‐Liquid Equilibrium**

Khatereh Bahrpaima

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/67926

### Abstract

Various ternary and quaternary liquid-liquid phase equilibrium data for water þ phosphoric acid þ 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

© 2017 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 [2–4].

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

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, con-

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

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

2. Phase diagram of multicomponent liquid mixtures: measurement and

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

similar boiling points, high boiling points, and temperature-sensitive components [2].

centration, addition of additives, and so on, have been investigated [2–4].

144 Phosphoric Acid Industry - Problems and Solutions

commercially produced through liquid-liquid equilibrium process.

visualization

Figure 1. Binodal curves for (water þ phosphoric acid þ propylene carbonate) mixture at two different temperatures and atmospheric pressure: ( ) T ¼ 298.2 K, and ( ) T ¼ 318.2 K [6].

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

Figure 2. Tie lines of the system water þ phosphoric acid þ MIAK at two different temperatures and atmospheric pressure: ( ) T ¼ 308.2 K, and ( ), T ¼ 318.2 K [9].

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 þ phosphoric acid þ methyl isoamyl ketone (MIAK)) [9] and (water þ phosphoric acid þ 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).

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

### 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 ¼ 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 þ phosphoric acid þ mixed-solvent (dichloromethane þ 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.

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 þ phosphoric acid þ methyl isoamyl ketone (MIAK)) [9] and (water þ phosphoric acid þ 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

Aqueous phase (raffinate) mass percent Organic phase (extract) mass percent D1 D2 S

70.48 15.00 14.52 6.14 1.40 92.46 0.087 0.093 1.071 66.82 16.39 16.79 6.49 1.67 91.84 0.097 0.101 1.049 58.19 19.78 22.03 6.60 2.01 91.39 0.113 0.102 0.896 50.89 23.28 25.83 6.78 2.92 90.30 0.133 0.125 0.941 45.74 24.33 29.93 6.80 3.31 89.89 0.149 0.136 0.915

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 ¼ 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 þ phosphoric acid þ mixed-solvent (dichloromethane þ 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

Table 1. Calculated tie line data, distribution coefficient, and separation factor for the ternary system of water (1) þ

(see Figures 1 and 2).

146 Phosphoric Acid Industry - Problems and Solutions

GMDH is used to predict the tie lines.

\*

3. Separation factor and distribution coefficients

phosphoric acid (2) þ DCM (3), according to the GMDH\* results [10].

W11 W21 W31 W13 W23 W33

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); , (50% DCM + 50% DCE); , (75% DCM + 25% DCE); [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 [12–27]. 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.

In addition, the presence of dissolved salt changes the phase equilibrium behavior of a mixture significantly [28–31]. 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 þ phosphoric acid þ solvent þ salt). In particular, as can be seen in Table 2, maximum values of separation factor for (water þ phosphoric acid þ DCM þ 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 þ liquid) equilibrium data of

Table 2. The maximum values of separation factors for the (water þ phosphoric acid þ solvent) ternary systems.

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

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]), .

the mixtures with a mixed solvent has become the turning point topic of the chemistry investigations in recent years [11, 35–42]. As an illustration, Figure 3 compares distribution coefficients of phosphoric acid for three quaternary systems of (water þ phosphoric acid þ mixed solvent (25% DCM þ 75% DCE) or (50% DCM þ 50% DCE) or (75% DCM þ 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 þ 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 ¼ 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 [43–48]. 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:

$$\log \mathbf{S} = \log \mathbf{S}\_0 + s(\mathbf{x}\pi^\* + d\mathbf{x}\delta) + a\mathbf{x}\alpha + b\mathbf{x}\beta \tag{1}$$

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, EN T , 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. EN T , 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

$$\log \mathbb{S} = \log \mathbb{S}\_0 + a \cdot E\_T^N + b \cdot \left(\frac{\varepsilon - 1}{2\varepsilon + 1}\right) + c \cdot \left(\frac{n^2 - 1}{2n^2 + 1}\right) + d \cdot \left(\frac{T}{298.2}\right) \tag{2}$$

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 twoliquid (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, 16–19, 24–26], 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

$$\frac{G^{\ominus}}{\rm RT} = \mathbf{x}\_1 \mathbf{x}\_2 \left( \frac{\tau\_{21} G\_{21}}{\mathbf{x}\_1 + \mathbf{x}\_2 G\_{21}} + \frac{\tau\_{12} G\_{12}}{\mathbf{x}\_2 + \mathbf{x}\_1 G\_{12}} \right) \tag{3}$$

where

$$\mathbf{G}\_{12} = \exp\left(-a\_{12}\tau\_{12}\right);\;\mathbf{G}\_{21} = \exp\left(-a\_{21}\tau\_{21}\right) \tag{4}$$

$$\pi\_{12} = \frac{\mathcal{g}\_{12} - \mathcal{g}\_{22}}{RT} \; ; \; \pi\_{21} = \frac{\mathcal{g}\_{21} - \mathcal{g}\_{11}}{RT} \tag{5}$$

The significance of gij is an energy parameter characteristic of the i-j 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 ¼ 0.3. From Eq. (3), the activity coefficients are

$$\ln \mathbf{\nu}\_1 = \mathbf{x}\_2^2 \left[ \mathbf{\tau}\_{21} \left( \frac{\mathbf{G}\_{21}}{\mathbf{x}\_1 + \mathbf{x}\_2 \mathbf{G}\_{21}} \right)^2 + \frac{\mathbf{\tau}\_{12} \mathbf{G}\_{12}}{\left( \mathbf{x}\_2 + \mathbf{x}\_1 \mathbf{G}\_{12} \right)^2} \right] \tag{6}$$

$$\ln \mathbf{\gamma}\_2 = \mathbf{x}\_1^2 \left[ \tau\_{12} \left( \frac{\mathbf{G}\_{12}}{\mathbf{x}\_2 + \mathbf{x}\_1 \mathbf{G}\_{12}} \right)^2 + \frac{\tau\_{21} \mathbf{G}\_{21}}{\left( \mathbf{x}\_1 + \mathbf{x}\_2 \mathbf{G}\_{21} \right)^2} \right] \tag{7}$$

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

$$\frac{G^E}{RT} = \sum\_{i=1}^m \mathbf{x}\_i \frac{\sum\_{j=1}^m \mathbf{r}\_{jl} G\_{ji} \mathbf{x}\_j}{\sum\_{l=1}^m G\_{li} \mathbf{x}\_1} \tag{8}$$

where

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 parame-

Katritzky [51] introduced a multiparameter polarity scale with combination of Riechardt's sol-

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

literature [49]. A modified form of the Katritzky equation may be used, consisting of the addition

2ε þ 1

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 refer-

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 twoliquid (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, 16–19, 24–26], while a group contribution approach (UNIFAC) [54]

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

> τ21G21 x1 þ x2G21

τ12 ¼ g12 � g22

þ

RT ; τ21 ¼ g21 � g11

τ12G12 x2 þ x1G12

G12 ¼ exp ð�α12τ12Þ; G21 ¼ exp ð�α21τ21Þ ð4Þ

RT ð5Þ

T , the dielectric constant, ε, and the index of refraction, n, functions. The Katritzky

þ c : n2 � 1 2n2 þ 1

T , ε, and n values of solvents are also present in

þ d : T

298:2

ð2Þ

ð3Þ

ters (α, β, and π\*) of the solvents are obtained from literature [49].

of a dimensionless term as T/298.2 that characterizes temperature effect

T þ b : ε � 1

are regressed using experimental data. EN

150 Phosphoric Acid Industry - Problems and Solutions

logS ¼ logS0 þ a : EN

ences [24] are temperature sensitive.

may be utilized to predict the LLE mixtures.

mixtures, and for the binary mixtures the excess Gibbs energy is

GE RT ¼ x1x2

4. LLE correlation

4.1. NRTL equation

where

vent polarity, EN

$$
\tau\_{i\dot{\prime}} = \frac{\mathcal{g}\_{i\dot{\prime}} - \mathcal{g}\_{\dot{\prime}\dot{\prime}}}{\mathcal{RT}}, \qquad \tau\_{\dot{\prime}\dot{\prime}} = \frac{\mathcal{g}\_{\dot{\prime}\dot{\prime}} - \mathcal{g}\_{\dot{\prime}\dot{\prime}}}{\mathcal{RT}} \tag{9}
$$

$$\mathcal{G}\_{\vec{\mu}} = \exp\left(-\alpha\_{\vec{\mu}}\tau\_{\vec{\mu}}\right) \quad (\alpha\_{\vec{\mu}} = \alpha\_{\vec{\eta}}) \tag{10}$$

The activity coefficient for any component i is given by

$$\ln \mathbf{\gamma}\_{i} = \frac{\sum\_{j=1}^{m} \tau\_{ji} \mathbf{G}\_{ji} \mathbf{x}\_{j}}{\sum\_{l=1}^{m} \mathbf{G}\_{li} \mathbf{x}\_{l}} + \sum\_{j=1}^{m} \left[ \frac{\mathbf{x}\_{j} \mathbf{G}\_{ij}}{\sum\_{l=1}^{m} \mathbf{G}\_{lj} \mathbf{x}\_{l}} \left( \tau\_{ij} - \frac{\sum\_{r=1}^{m} \mathbf{x}\_{r} \tau\_{rj} \mathbf{G}\_{rj}}{\sum\_{l=1}^{m} \mathbf{G}\_{lj} \mathbf{x}\_{l}} \right) \right] \tag{11}$$

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 wellknown 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

Table 3. Correlated LLE results from the NRTL (α12 ¼ 0.3) model using SA; the corresponding binary interaction parameters, Aij and Aji, for ternary system water (1) þ phosphoric acid (2) þ DCE (3) þ 10 wt% CaCl2 at temperature T ¼ 298.15 K.

Table 4. NRTL (α12 ¼ 0.3) interaction parameters, Aij and Aji, using BA for quaternary LLE system water (1) þ phosphoric acid (2) þ (50% DCM (3) þ 50% DCE (4)) at temperature T ¼ 298.15 K.

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 þ phosphoric acid þ mixed-solvent (DCM þ DCE)) system at temperature T ¼ 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 þ 50% DCE) [61].

### 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 quasichemical 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

$$\frac{G^E}{RT} = \left(\frac{G^E}{RT}\right)\_{\text{combined}} + \left(\frac{G^E}{RT}\right)\_{\text{residual}}\tag{12}$$

For a binary mixture,

$$\left(\frac{G^{\mathbb{E}}}{\mathbb{RT}}\right)\_{\text{combined}} = x\_1 \ln \frac{\Phi\_1}{x\_1} + x\_2 \ln \frac{\Phi\_2}{x\_2} + \frac{z}{2} \left(x\_1 q\_1 \ln \frac{\theta\_1}{\Phi\_1} + x\_2 q\_2 \ln \frac{\theta\_2}{\Phi\_2}\right) \tag{13}$$

$$\left(\frac{G^E}{\text{RT}}\right)\_{\text{residual}} = -\mathbf{x}\_1 q\_1' \ln\left(\theta\_1' + \theta\_2' \tau\_{21}\right) - \mathbf{x}\_2 q\_2' \ln\left(\theta\_2' + \theta\_1' \tau\_{12}\right) \tag{14}$$

where the coordination number z is set equal to 10. Segment fraction, Ф, and area fraction, θ and θ0 , are given by

$$\Phi\_1 = \frac{\mathbf{x}\_1 r\_1}{\mathbf{x}\_1 r\_1 + \mathbf{x}\_2 r\_2}, \qquad \Phi\_2 = \frac{\mathbf{x}\_2 r\_2}{\mathbf{x}\_1 r\_1 + \mathbf{x}\_2 r\_2} \tag{15}$$

$$
\theta\_1 = \frac{\mathbf{x}\_1 \mathbf{q}\_1}{\mathbf{x}\_1 \mathbf{q}\_1 + \mathbf{x}\_2 \mathbf{q}\_2}, \qquad \theta\_2 = \frac{\mathbf{x}\_2 \mathbf{q}\_2}{\mathbf{x}\_1 \mathbf{q}\_1 + \mathbf{x}\_2 \mathbf{q}\_2} \tag{16}
$$

$$
\theta'\_1 = \frac{\mathbf{x}\_1 \mathbf{q}'\_1}{\mathbf{x}\_1 \mathbf{q}'\_1 + \mathbf{x}\_2 \mathbf{q}'\_2}, \qquad \theta'\_2 = \frac{\mathbf{x}\_2 \mathbf{q}'\_2}{\mathbf{x}\_1 \mathbf{q}'\_1 + \mathbf{x}\_2 \mathbf{q}'\_2} \tag{17}
$$

where

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 þ phosphoric acid þ mixed-solvent (DCM þ DCE)) system at temperature T ¼ 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

Table 4. NRTL (α12 ¼ 0.3) interaction parameters, Aij and Aji, using BA for quaternary LLE system water (1) þ phosphoric

i-j AijðKÞ \* AjiðKÞ –2 7.01 62.86 –3 996.58 334.13 –3 230.64 270.65

Table 3. Correlated LLE results from the NRTL (α12 ¼ 0.3) model using SA; the corresponding binary interaction parameters, Aij and Aji, for ternary system water (1) þ phosphoric acid (2) þ DCE (3) þ 10 wt% CaCl2 at temperature T ¼ 298.15 K.

i-j AijðKÞ AjiðKÞ –2 2492.75 �419.50 –3 2303.05 1212.55 –4 7.56 �3652.94 –3 2466.50 4329.12 –4 �2316.04 �2817.74 –4 �854.74 �3413.39

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 quasichemical 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

aqueous mixture including phosphoric acid and (50% DCM þ 50% DCE) [61].

acid (2) þ (50% DCM (3) þ 50% DCE (4)) at temperature T ¼ 298.15 K.

4.2. UNIQUAC equation

\*gij (J mol�1

) ¼ Aij (K)/RT.

152 Phosphoric Acid Industry - Problems and Solutions

$$l\_{\dot{j}} = \frac{z}{2}(r\_{\dot{j}} - q\_{\dot{j}}) - (r\_{\dot{j}} - 1) \tag{18}$$

Parameters r, q, and q0 are pure component molecular structure constants depending on molecular size and external surface areas. In the original formulation, q ¼ q0 . To obtain better agreement for systems containing water or lower alcohols, q0 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 q0 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 ¼ q0 .

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

$$\tau\_{12} = \exp\left(-\frac{\Delta u\_{12}}{\text{RT}}\right) \equiv \exp\left(-\frac{a\_{12}}{T}\right) \tag{19}$$

$$\tau\_{21} = \exp\left(-\frac{\Delta u\_{21}}{\text{RT}}\right) \equiv \exp\left(-\frac{a\_{21}}{T}\right) \tag{20}$$

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.

$$\ln \eta\_1 = \ln \frac{\Phi\_1}{x\_1} + \frac{z}{2} \left. q\_1 \ln \frac{\theta\_1}{\Phi\_1} + \left. \Phi\_1 \left( l\_1 - \frac{r\_1}{r\_2} l\_2 \right) - q\_1' \ln (\theta\_1' + \theta\_2' \tau\_{21}) + \theta\_2' \left. q\_1' \left( \frac{\tau\_{21}}{\theta\_1' + \theta\_2' \tau\_{21}} - \frac{\tau\_{12}}{\theta\_2' + \theta\_1' \tau\_{12}} \right) \right) \right| \tag{21}$$

$$\ln \eta\_2 = \ln \frac{\Phi\_2}{x\_2} + \frac{z}{2} \, q\_2 \ln \frac{\theta\_2}{\Phi\_2} + \Phi\_2 \left( l\_2 - \, \frac{r\_2}{r\_1} l\_1 \right) - q\_2' \ln(\theta\_2' + \theta\_1' \tau\_{12}) + \, \theta\_1' \, q\_2' \left( \frac{\tau\_{12}}{\theta\_2' + \theta\_1' \tau\_{12}} - \frac{\tau\_{21}}{\theta\_1' + \theta\_2' \tau\_{21}} \right) \tag{22}$$

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

$$\frac{G^{\mathbb{E}}(\text{combinatorial})}{\text{RT}} = \sum\_{i=1}^{m} \mathbf{x}\_{i} \ln \frac{\boldsymbol{\Phi}\_{i}}{\boldsymbol{\chi}\_{i}} + \frac{z}{2} \sum\_{i=1}^{m} q\_{i} \mathbf{x}\_{i} \ln \frac{\boldsymbol{\Theta}\_{i}}{\boldsymbol{\Phi}\_{i}} \tag{23}$$

and

$$\frac{G^E(\text{residual})}{\text{RT}} = -\sum\_{i=1}^m q\_i' \mathbf{x}\_i \ln \left( \sum\_{j=1}^m \Theta\_j' \mathbf{r}\_{ji} \right) \tag{24}$$

where segment fraction Ф and area fractions θ and θ0 are given by

$$\begin{aligned} \boldsymbol{\Phi}\_{i} = \frac{\mathbf{x}\_{i} r\_{i}}{\sum \mathbf{x}\_{\uparrow} r\_{j}}, \qquad \boldsymbol{\Theta}\_{i} = \frac{q\_{i} \mathbf{x}\_{i}}{\sum\_{j=1}^{m} q\_{j} \mathbf{x}\_{j}}, \qquad \boldsymbol{\Theta}\_{i}^{\prime} = \frac{q\_{i}^{\prime} \mathbf{x}\_{i}}{\sum\_{j=1}^{m} q\_{j}^{\prime} \mathbf{x}\_{j}} \end{aligned} \tag{25}$$

The activity coefficient for any component i is given by

$$\ln \eta\_i = \ln \frac{\Phi\_i}{\mathbf{x}\_i} + \frac{z}{2} q\_i \ln \frac{\Theta\_i}{\Phi\_i} + l\_i - \frac{\Phi\_i}{\mathbf{x}\_i} \sum\_{j=1}^m \mathbf{x}\_j l\_j - q\_i' \ln \left( \sum\_{j=1}^m \Theta\_j' \tau\_{ji} \right) + q\_i' - q\_i' \sum\_{j=1}^m \frac{\Theta\_j' \tau\_{ji}}{\sum\_{k=1}^m \Theta\_k' \tau\_{kj}} \tag{26}$$

\* ri and qi are the UN1QUAC structural parameters.

Table 5. Optimized UNIQUAC binary interaction energy parameters, uij and ujiðJ mol�1 Þ, for water (1) þ phosphoric acid (2) þ butyl acetate (3) ternary system at T ¼ 308.2 K [12].

The molecular interaction-based model of UNIQUAC successfully has correlated many LLE data of ternary and quaternary systems [10, 15–17, 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].

### 4.3. GMDH-type neural network

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.

> 0 2 q0 1

0 1 q0 2

1

i ¼ q0 i xi Xm j¼1 q0 j xj

> 1 A þ q0

i � q0 i Xm j¼1

τ21

τ12

� τ12 θ 0 2 þ θ 0 1τ12

� τ21 θ 0 1 þ θ 0 2τ21

ð21Þ

ð22Þ

ð23Þ

ð25Þ

ð26Þ

!

!

A ð24Þ

θ0 j τij

Xm k¼1 θ0 kτkj

Þ

q1 ¼ 1. 400, q2 ¼ 4.000, q3 ¼ 4.196

Þ, for water (1) þ phosphoric

θ 0 1 þ θ 0 2τ21

θ 0 2 þ θ 0 1τ12

� q0 1 lnðθ 0 1 þ θ 0 2τ21Þ þ θ

� q0 2 lnðθ 0 2 þ θ 0 1τ12Þ þ θ

For a multicomponent system, the UNIQUAC equation for the molar excess Gibbs energy is

i¼1

i¼1 q0 i

xjlj � q0 i ln Xm j¼1 θ0 j τji

xi Xm j¼1 qj xj

xiln Φi xi þ z 2 Xm i¼1 qi xiln θi Φi

xiln Xm

0 @

j¼1 θ0 j τji

, θ0

Þ ujiðJ mol�1

0 @

þ Ф1 l1 � r1

þ Ф2 l2 � r2

GEðcombinatorialÞ

GEðresidualÞ

where segment fraction Ф and area fractions θ and θ0 are given by

þ li � Φi xi Xm j¼1

1–2 3589.99 �2835.91 1–3 909.55 4182.19 2–3 3043.92 �4435.44

r1 ¼ 0.920, r2 ¼ 3.000, r3 ¼ 4.827 \*

Table 5. Optimized UNIQUAC binary interaction energy parameters, uij and ujiðJ mol�1

Φi ¼ xiri X j xjrj

The activity coefficient for any component i is given by

lnγi ¼ ln Φi

xi þ z 2 qi ln θi Φi

i-j uijðJ mol�1

ri and qi are the UN1QUAC structural parameters.

acid (2) þ butyl acetate (3) ternary system at T ¼ 308.2 K [12].

RT ¼ Xm

RT ¼ �Xm

, θi ¼ qi

r2 l2 � �

r1 l1 � �

lnγ1 ¼ ln Φ1

lnγ2 ¼ ln Φ2

and

\*

\*

x1 þ z 2 q1ln θ1 Ф1

154 Phosphoric Acid Industry - Problems and Solutions

x2 þ z 2 q2ln θ2 Ф2

given by the sum of

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 þ phosphoric acid þ 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.

### Acknowledgements

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

### Author details

Khatereh Bahrpaima

Address all correspondence to<EMAIL_ADDRESS>
Department of Chemistry, Firoozabad Branch, Islamic Azad University, Firoozabad, Iran

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