Outline
Comptes Rendus

Semi-empirical correlation of solid solute solubility in supercritical carbon dioxide: Comparative study and proposition of a novel density-based model
Comptes Rendus. Chimie, Volume 21 (2018) no. 5, pp. 494-513.

Abstract

A comprehensive data set on experimental solubility of 210 solid solutes in supercritical CO2 counting 5550 data points has been used for comparison of the correlation performance of 21 empirical models. On the basis of the comparison results a new eight-parameter density-based model has been proposed. The comparison shows that the three-parameter models are the least accurate. The results also show that models that relate the logarithm of the solubility to the logarithm of solvent density and temperature are more accurate than models that include the pressure. When comparing the overall correlating performance in terms of average absolute relative deviation the proposed model is by far the best with an average absolute relative deviation lying in the range 0.17–81.99% and an average value of 8.88%.

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DOI: 10.1016/j.crci.2018.02.006
Keywords: Solubility, Supercritical CO2, Modeling, Density-based models

Aicha Belghait 1; Cherif Si-Moussa 1; Maamar Laidi 1; Salah Hanini 1

1 Laboratory of Biomaterials and Transport Phenomena (LBMPT), Faculty of Science and Technology, University Yahia Fares of Medea, Algeria
@article{CRCHIM_2018__21_5_494_0,
     author = {Aicha Belghait and Cherif Si-Moussa and Maamar Laidi and Salah Hanini},
     title = {Semi-empirical correlation of solid solute solubility in supercritical carbon dioxide: {Comparative} study and proposition of a novel density-based model},
     journal = {Comptes Rendus. Chimie},
     pages = {494--513},
     publisher = {Elsevier},
     volume = {21},
     number = {5},
     year = {2018},
     doi = {10.1016/j.crci.2018.02.006},
     language = {en},
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%0 Journal Article
%A Aicha Belghait
%A Cherif Si-Moussa
%A Maamar Laidi
%A Salah Hanini
%T Semi-empirical correlation of solid solute solubility in supercritical carbon dioxide: Comparative study and proposition of a novel density-based model
%J Comptes Rendus. Chimie
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Aicha Belghait; Cherif Si-Moussa; Maamar Laidi; Salah Hanini. Semi-empirical correlation of solid solute solubility in supercritical carbon dioxide: Comparative study and proposition of a novel density-based model. Comptes Rendus. Chimie, Volume 21 (2018) no. 5, pp. 494-513. doi : 10.1016/j.crci.2018.02.006. https://comptes-rendus.academie-sciences.fr/chimie/articles/10.1016/j.crci.2018.02.006/

Version originale du texte intégral

1 Introduction

Supercritical fluid (SCF) technologies have been gaining an increasing attention in process engineering during the past decades due to its various advantages. Carbon dioxide is the most commonly used SCF because it is environmentally benign, nontoxic, nonflammable, cheap, and readily available. As a green solvent, supercritical carbon dioxide (scCO2) finds applications in many conventional and novel emerging technologies such as dry dyeing processing techniques, which operate without wastewater emissions, efficient environmentally friendly and benign new green biotechnological processes, and so forth [1,2]. The accurate knowledge of the phase behavior of the systems involved and, in particular, of the solubility of the solid solutes in high-pressure scCO2 is crucial to the design, optimization, and operation of such processes. The solid solutes of interest, such as dyes or pharmaceuticals, are generally high molecular weight complex molecules and in their majority low volatile solids. This implies that very often their experimental physical properties, such as critical properties and sublimation pressure, are not readily available in the literature. The experimental measurement of the solubility of such compounds in scCO2 is laborious and costly. To avoid expensive and tedious experiments, flexible and robust predictive models are the ideal targets. In the absence of such an ideal predictive model, however, simple and accurate enough correlative models are very useful in engineering applications such as used in commercial process simulators. This is the reason why both types of modeling have been and still remain a goal for many research works.

Three major modeling approaches have been used to model the solubility of a solid solute in an SCF for predictive or correlative purposes: equations of state (EoSs) and activity coefficient models, computational intelligence techniques, and density-based semiempirical equations. It should be mentioned that the three approaches rely more or less on experimental data and, therefore, their use outside the range of experimental data from which the models have been obtained is not safe.

EoSs are the straightforward and obvious methods as the solubility is a phase equilibrium problem. Simple cubic EoS with various mixing rules has been extensively used for calculation of solid solute solubility in scCO2 (to cite just few, see, for example, Refs. [3–12]). Detailed review of the use of cubic EoSs in this context has been given by Yazdizadeh et al. [10]. The use of cubic EoSs requires critical properties, acentric factor to compute fugacity coefficient, which is needed along with the sublimation vapor pressure, and the molar volume of the solid solute to compute the solubility. In the absence of experimental values of these solid solute parameters, they are estimated using group contribution (GC) methods. Sublimation vapor pressure, which is very low, and experimental value, unavailable for many solids, may also be considered as an adjusted parameter to be fitted to solubility data as used in Refs. [8,13,14]. The accuracy of the correlation or prediction using the cubic EoS depends greatly on the method used to estimate these solid solute properties [3,4,11]. The effect of mixing rules has been investigated by Yazdizadeh et al. [9,10] in two comparative studies. The first one [9] concerned 52 solid solutes and compared two cubic EoSs combined with four mixing rules and showed a global superiority of Esmaeilzadeh–Roshanfekr EoS [15] combined with Wong–Sandler mixing rule [16] and a global average absolute deviation of 9%. In the second one [10] 23 compounds were considered, five cubic EoSs combined with four mixing rules and the results showed a global best performance of a modified Esmaeilzadeh–Roshanfekr EoS [17] with Wong–Sandler mixing rule [16] and a global average absolute deviation of 7%. Sang et al. [8] considered three cubic EoSs combined with five mixing rules where they used a modification of the calculation procedure by introducing some approximations. They proposed two- and three-adjustable-parameter models for the calculation of the fugacity coefficient. Accordingly, the only solid solute parameter required was the molar volume. They showed that the three-adjustable-parameter model gave a global average absolute deviation of 5.5% for the same data set used in Ref. [10].

Despite their success for fluid mixtures, theoretical–sound state of the art statistical thermodynamic models have not been used for modeling complex molecules such as pharmaceuticals. Statistical associating fluid theory (SAFT)-type models use fluid-specific parameters that usually are fitted to pure fluid experimental data. In common fluids, experimental vapor pressures and liquid densities are typically used. However, for the majority of pharmaceutical complex chemicals such extended experimental data do not exist [18]. Most SAFT-type models require five parameters for each pure associating compound (three for nonassociation ones). Three parameters for nonassociating fluids: the segment number, the interaction energy, and the hard core segment diameter. Two other parameters for associating fluids are the association energy and the affective association volume. These parameters are calculated using GC methods, such as proposed by Tihic et al. [19,20] or estimated from vapor pressure and liquid-density data over extended temperature ranges [21,22]. In the absence of such data for specific compounds, such as polymers and pharmaceuticals, GC methods, such as GC-SAFT [23,24], GC-SAFT variable range (SAFT-VR) [25–32], and GC-SAFT-γ [33–35], are used for the estimation of required properties. As regards modeling the solubility of complex solid molecules in scCO2 using noncubic EoS only few works are reported. This is due to the lack of sufficient experimental data as this family of EoSs uses fluid-specific parameters that usually are fitted to pure fluid experimental data [18]. Tsivintzelis et al. [18] have used nonrandom hydrogen bonding theory to predict the solubility of pharmaceuticals in SCFs. SAFT models have also been used to predict or correlate the solubility of solids in SCFs [36–38].

Activity coefficient models, such as universal functional activity coefficient, the regular solution theory-based models, and more sophisticated models, such as conductor-like screening model (COSMO), cannot be used for high-pressure phase equilibrium calculations without being coupled with another model, namely, an EoS [18]. COSMO-based approach has been used to predict solubility of solids in scCO2 [39–42]. The only experimental data required for solutes are the melting temperature and the melting enthalpy of the solid solute.

Computational intelligence techniques such as artificial neural networks, adaptive neurofuzzy inference system, and least square support vector machine are considered as powerful modeling tools that can map complex highly nonlinear input/output relationships of any systems. This ability has made them suited for a wide range of engineering applications. Applications of such techniques to correlate solubility of various solid solutes in scCO2 have been reported: Refs. [43–51] for artificial neural networks, Refs. [52,53] for least square support vector machine, and Ref. [54] for adaptive neurofuzzy inference system.

The review of the EoS and computational intelligence approaches in modeling the solubility of solid solutes in supercritical solvents reveals that both approaches, whether used for predictive or correlative purposes, with the exception of the COSMO model, rely on experimental data of phase equilibrium and the physical properties of pure solid solute. As regards the EoS approach, phase equilibrium data are required for fitting at least one binary interaction parameter used by mixing rules, fitting other required pure solute properties, or for the validation of models. Regarding computational intelligence methods, experimental data are necessary for training and validation of the model. Pure component properties for complex multifunctional solid solutes, such as pharmaceuticals, are not always available and need to be estimated by GC methods.

Semiempirical modeling is used as a simpler alternative insofar as it does not require any solute property, although there are few models that include some solute parameters such as temperature and enthalpy of fusion and infinite dilute activity coefficient [55], solubility parameter, and molar volume of the solute [56]. In a very recent article [57], we have presented an extensive review of semiempirical models. Comparative studies [58–64] on the performance of semiempirical models show there is no model that presents the best correlation for all the experimental data of all solutes. In the present work, it is intended to carry a larger scale comparative study that includes all semiempirical models on a more comprehensive data set. It also aims for the improvement of the quality of density-based correlation of the solubility of solid solutes in scCO2.

2 Modeling solid–SCF phase equilibrium

The fugacities of a solid solute in the solid and fluid phases are related by the following fundamental isofugacity relationship:

f2S=f2SCF(1)
in which f designates the fugacity and superscripts S and SCF designate solid and SCF phases, respectively.

The fugacity of solid solute in an SCF phase can be written as the product of the solute mole fraction or solubility (y2), the fugacity coefficient (ϕ2), and the equilibrium pressure P as follows:

f2SCF=y2ϕ2P(2)
The fugacity of the solid phase is usually considered as a pure phase and can expressed as
f2s=P2subϕ2satexp[V2sRT(PP2sub)](3)
where P2sub, ϕ2sat, and V2s are the sublimation pressure, the saturated fugacity coefficient, and the molar volume of the solid, respectively. T is the temperature, P is the pressure, and R is the universal gas constant (8.314 J mol−1 K−1).

Because of sublimation pressures, ϕ2sat is equal to 1 for most solutes that do not have a strong tendency of association. Furthermore, because of the incompressibility of solids, values of V2s can be assumed to be pressure independent.

Hence, y2 in SCF reduces to

y2=P2subP1ϕ2SCFexp[V2sRT(PP2sub)](4)
Note that ϕ2SCF is a function of y2, T, P, and ZSCF (the compressibility coefficient of the supercritical phase), itself a function of y2, T, and P:
ϕ2SCF=ϕ(T,P,ZSCF,y2)(5)
ZSCF=Z(T,P,y2)(6)

Eqs. 5 and 6 are the basis for calculating the solubility of a solid solute in an SCF by an EoS. The quality of predictions or correlations depends on the EoS and mixing rules to be used. Calculations are very sensitive to the accuracy of the required solute properties [3,4,38].

Taking the logarithm of Eq. 4 gives

lny2=lnP2sublnPlnϕ2SCF+V2sRPTV2sP2subR1T(7)

The volumetric properties of the solvent are governed by the equation of state of a real gas:

P=ZRTρ1(8)

The solubility of the solute at a given T and P can be related to the density and the coefficient of compressibility of the pure solvent by replacing the pressure in Eq. 7:

lny2=lnP2subln(ZRTρ1)lnϕ2SCF+V2sZρ1V2sP2subR1T(9)

All of the density-based models that have been proposed can be considered as a simple empirical approximation to the right-hand side of Eq. 9, which generally does not require any solute property. The next section presents a brief review of density-based model for correlation of solid solubility data.

2.1 Density-based model for correlation of solid solubility data

In a recent article [57], we have presented an extensive review of density-based empirical models. Chronologies of their development, the experimental findings, and theoretical explanations of some of the model parameters have been presented. Table 1 shows semiempirical models used to correlate the experimental solubility data that have been proposed by different authors in the last four decades with the aim of improving the quality of correlation of new experimental data. Models that require solute properties, which are not readily available for compounds of interest such as pharmaceuticals and dyes, have not been considered in this study. In Table 1, y2 is the solubility of the solid solute in mole fraction, ρ1 is the density of the SCF (in kilograms per cubic meter), T is the temperature (in Kelvin), P is the pressure (in bar), ai represents the characteristic parameters determined by fitting experimental data of the solute/solvent considered. For the Sparks et al. [73] model, solubility is expressed in a dimensionless form, whereas in the Bain et al. [64] model the solute solubility, c2, is in kilograms per cubic meter. In both models c2 is related to y2 by Eq. 24a, in which Mw1 and Mw2 are the molecular weights of the solvent and the solute, respectively. It is worth mentioning that some of the models do not relate the solubility explicitly to the density of the SCF solvent. The model proposed in our previous work [57] is based on the following facts: (1) models that relate the solubility of the solid solute to temperature, pressure, and the density of the SCF violate the phase rule, because the volumetric properties (P, T, and ρ1) of scCO2 are governed by the phase rule, hence only two variables need to be fixed; (2) just as the density expansion of the virial EoS is more accurate than the pressure expansion at increased pressures [80], relating the solubility to density at isothermal condition instead of pressure would lead, a priori, to better correlation.

Table 1

Density-based models for the correlation of solid solutes in SCFs.

ModelEquationEquation no.Reference
Chrastillny2=a0+a1lnρ1+a2T(10)[65]
Adachi and Lulny2=a0+(a1+a2ρ1+a3ρ12)lnρ1+a4T(11)[66]
del Valle and Aguileralny2=a0+a1lnρ1+a2T+a3T2(12)[67]
Kumar and Johnstonlny2=a0+a1ρ1+a2T(13)[68]
Bartle et al.lny2PPref=a0+a1(ρ1ρ1ref)+a2T(14)[69]
Gordillo et al.lny2=a0+a1P+a2P2+a3PT+a4T+a5T2(15)[70]
Mendez-Santiago and TejaTlny2P=a0+a1ρ1+a2T(16)[71]
Sung and Shimlny2=(a0+a1T)lnρ1+a2T+a3(17)[72]
Jouyban et al.lny2=a0+a1P+a2P2+a3PT+a4TP+a5lnρ1(18)[58]
Sparks et al.c2=ρr,1(a0+a1ρr,1+a2ρr,12)exp(b0+b1Tr+b2Tr2)
with c2=c2ρc,1;ρr,1=ρ1ρc,1;Tr=TTc,1
(19)[73]
Garlapati et Madraslny2=a0lnρ1+a1T+a2(20)[74]
Garlapati et Madraslny2=a0+(a1+a2ρ1)lnρ1+a3T+a4ln(ρ1T)(21)[75]
Jafari Nedjad et al.lny2=a0+a1P2+a2T2+a3lnρ1(22)[76]
Ch and Madrasy2=(PP)(k1)exp(a0T+a1ρ1+a2)(23)[77]
Bian et al.c2=ρ1(a0+a1ρ1+a2/lnT)exp(a3+a4ρ1T+a5)c2=ρ1Mw2y2Mw1(1y2)(24)
(24a)
[59]
Keshmiri et al.lny2=a0+a1T+a2P2+(a3+a4T)lnρ1(25)[78]
Ali Akbar Amooeylny2=a0+a1/ρ1+a2/ρ12+a3lnT+a4(lnT)21+a5/ρ1+a6lnT+a7(lnT)2+a8(lnT)3(26)[60]
Hozhabr et al.lny2=a0+a1T+a2ρ1Ta3lnP(27)[79]
Khansary et al.lny2=a0T+a1P+a2P2T+(a3+a4P)lnρ1(28)[61]
Bian et al.lny2=a0+a1T+a2ρ1T+(a3+a4ρ1)lnρ1(29)[64]
Si-Moussa et al.lny2=a0+a1ρ1+a2ρ12+a3ρ1T+a4Tρ1+a5lnρ1(30)[57]

2.2 Proposed model

On the basis of the two arguments mentioned by Si-Moussa et al. [57], the proposed model is derived first by a modification of Gordillo et al. [70] model where the pressure is replaced by the density of the solvent, which leads to the following equation:

lny2=a0+a1ρ1+a2ρ12+a3ρ1T+a4T+a5T2(31)

As the linear relationship between ln(y2) and ln ρ1 in isothermal conditions, as well as the inverse of temperature effect on ln(y2), has been considered in many models that demonstrated acceptable correlation performance in previous comparative studies [57–64], these two terms (a6lnρ1+a7/T) are added to Eq. 31. The proposed model is therefore

lny2=a0+a1ρ1+a2ρ12+a3ρ1T+a4T+a5T2+a6lnρ1+a7/T(32)

The eight parameters a0a7 are obtained from regression of experimental data.

3 Results and discussion

In the present study, a data set of 210 solid solutes counting 5550 data points of experimental solubility in scCO2, compiled from the literature published in the last 25 years, is used for comparison of correlation performance of 22 empirical models. Table 2 displays the range of experimental temperature, pressure, solubility, Ni number of experimental points, and source of the 210 selected binary solute–scCO2 systems.

Table 2

Sources and ranges of solubility data of solid solutes in scCO2.

No.Solid soluteMw (g/mol)T (K)P (MPa)y2 × 104NiReference
1(S)-Boc-piperazine285.3900308.20–328.2090.10–205.0013.70–65.2021[81]
21,8-Dihydroxy-9-anthrone226.2274308.00–348.00101.00–355.001.33–4.8754[82]
310-Undecenoic acid184.2800308.00–333.00100.00–180.004.00–174.0018[83]
41-Chloro-2,4-dinitrobenzene202.5520308.00–313.0095.00–145.0017.80–61.9012[84]
51-Hydroxy-2,4-dimethyl-9-anthrone238.2812308.00–348.00101.00–355.000.24–1.0654[82]
61-Hydroxy-2-ethyl-9-anthrone238.2812308.00–348.00101.00–355.001.08–9.7854[82]
71-Hydroxy-2-methyl-9-anthrone224.2546308.00–348.00101.00–355.000.87–7.0554[82]
81-Hydroxythioxanthone228.2660308.00–348.00121.00–354.000.99–4.2445[85]
92,2′-Oxybis(N,N-dibutylacetamide)356.5432313.00–333.0088.00–131.0015.20–86.0015[86]
102,2′-Oxybis(N,N-diethylacetamide)240.2988313.00–333.0087.00–133.0022.30–1171.0015[86]
112,2′-Oxybis(N,N-dihexylacetamide)468.7558313.00–333.0090.00–164.0010.60–53.6015[86]
122,3-Dimethylaniline121.1800313.00–333.0080.00–200.003.30–95.2018[87]
132,4-Dinitrophenol184.1100308.00–348.00121.60–483.7012.40–140.0041[88]
142,5-Dinitrophenol184.1100308.00–348.00121.60–486.4019.10–94.4045[88]
152-Methoxybenzoic acid152.1500308.20–328.2099.80–246.9090.00–1110.0023[89]
162-Naphthol144.1700308.10–328.10100.50–300.002.49–12.3018[90]
172-Phenyl-4H-1,3-benzoxazin-one223.2270308.15–328.18651.46–951.270.80–4.5026[91]
182-Propenamide71.0800308.20–323.2090.00–400.000.12–1.6328[92]
192-Trifluoromethylbenzoic acid190.1190308.20–323.2093.40–226.0013.70–85.5021[93]
203-Acetylpyridine121.1400313.15–343.15100.00–260.0069.00–1619.0036[94]
213-Aminobenzoic acid137.1360308.00–328.00100.00–210.000.02–0.0715[95]
223-Nitrotoluene137.1360313.00–333.0080.00–200.001.50–59.2018[87]
233-Trifluoromethylbenzoic acid190.1190308.20–323.2094.10–225.4055.10–469.0021[93]
244-Aminoantipyrine203.2400308.20–328.20100.50–220.401.16–3.6721[96]
254-Aminosalicylic acid153.1350308.00–328.00110.00–210.000.04–0.3615[97]
264-Dimethylaminoantipyrine231.2940308.20–328.20100.30–220.305.25–25.5021[96]
274-Hydroxybenzaldehyde122.1200308.00–328.00110.00–210.000.07–1.3815[98]
284-Trifluoromethylbenzoic acid190.1190308.20–323.2096.80–224.401.06–4.5621[93]
295-Fluorouracil130.0770308.15–328.15125.00–250.000.04–0.1518[91]
305-Hydroxymethylfurfural126.1100314.10–343.2097.40–195.805.06–23.8422[99]
31Acenaphthene154.2080308.00–338.00121.60–354.6023.95–47.3736[100]
32Acetylferrocene228.0680308.00–348.0077.00–221.002.50–79.2018[101]
33All-trans retinoic acid300.4351308.20–318.20102.00–303.000.01–0.1815[102]
34Amical-48 (diiodomethyl p-tolyl sulfone)422.0200318.00–338.00100.00–300.000.06–1.1518[103]
35Ammonium benzoate139.1520308.00–328.00110.00–210.000.02–0.0515[104]
36Amoxicillin365.4040308.15–338.15160.00–400.000.10–72.3028[105]
37Anastrozole293.3700308.00–348.00122.00–355.000.04–3.8045[106]
38Anthracene178.2290308.10–328.10100.00–300.000.42–1.1315[90]
39Antipyrine188.2260308.20–328.20100.40–220.402.09–7.1721[96]
40Aspirin180.1570308.15–328.15120.00–250.000.63–3.4724[107]
41Atorvastatin558.6400308.00–348.00121.60–354.600.17–14.4645[108]
42Atropine289.3690308.00–348.00122.00–355.000.60–16.7045[109]
43Azadirachtin720.7140308.15–333.15100.00–260.000.01–0.1554[110]
44Azelaic acid188.2210313.15–333.15100.00–300.000.01–0.0814[111]
45Azobenzene182.2212308.20–318.2091.00–253.0036.40–98.4010[112]
46Azodicarbonamide116.0788308.15–328.15100.00–300.000.09–0.2026[91]
47Beclomethasone dipropionate521.0420338.00–358.00213.00–385.000.07–0.3421[113]
48Benzamide121.1366308.00–328.00110.00–210.000.02–0.1615[104]
49Benzenesulfonamide172.2050308.15–328.15110.00–210.000.65–2.9015[114]
50Benzocaine165.1891308.00–348.00122.00–355.0010.30–121.2040[115]
51Benzoin212.2400308.15–328.15111.30–244.300.41–4.1019[116]
52Bisacodyl361.4000308.00–348.00122.00–355.000.36–5.8339[117]
53Budesonide430.5340338.00–358.00213.00–385.000.06–0.2921[113]
54Caffeine194.1900313.00–353.00199.00–349.002.83–11.3024[118]
55Cannabinol310.4299314.00–334.00130.00–202.001.26–2.1734[119]
56Capecitabine359.3501308.00–348.0015.20–35.400.08–1.5940[120]
57Carbamazepine236.2686308.00–348.00122.00–355.000.11–0.9439[109]
58Carvedilol406.4000308.00–338.00160.00–400.000.11–50.1028[121]
59Cefixime trihydrate507.5000308.00–328.00183.00–335.000.002–0.00318[122]
60Celecoxib381.3700323.20–343.20150.00–300.000.02–0.1518[123]
61Cetirizine388.8880308.15–338.15160.00–400.000.11–49.2028[124]
62Chlormezanone273.7000308.20–328.20119.30–239.900.05–5.4521[125]
63Chlorothalonil265.9100318.00–338.00100.00–300.000.13–2.6823[103]
64Chlorpheniramine maleate390.8600308.00–338.00160.00–400.000.15–4.2624[126]
65Cholesterol386.6500313.15–333.15100.00–250.000.02–1.4524[127]
66Cholesteryl acetate428.6900308.15–328.1590.00–240.000.044–10.4024[127]
67Cholesteryl benzoate490.7600308.15–328.15120.00–270.000.05–0.5320[127]
68Cholesteryl butyrate456.7400308.15–328.15100.00–240.000.22–8.9320[127]
69Cinnarizine368.5000308.15–328.15139.70–240.000.03–2.0521[128]
70Climbazole292.7600313.20–333.20105.50–398.906.20–48.8024[129]
71Clobetasol propionate466.9700308.00–348.00122.00–355.000.004–0.03544[130]
72Clofenamic acid247.6800313.00–333.00120.00–360.000.04–0.3524[131]
73Clofibric acid214.6000308.20–328.20100.10–220.202.57–8.5621[132]
74Clotrimazole344.8370308.00–348.00122.00–355.000.04–1.0645[133]
75Clozapine326.8230318.00–348.00121.60–354.600.04–0.4227[134]
76Codeine299.3642308.00–348.00122.00–355.001.00–12.3045[109]
77Corosolic acid472.7000308.15–333.1580.00–300.000.01–743.0040[135]
78Cortisone acetate402.4800308.15–373.1582.40–226.500.005–0.2330[136]
79Cyproheptadine278.4000308.00–338.00160.00–400.000.81–30.9028[137]
80Cyproterone acetate416.9380308–348122.00–355.000.13–2.6140[138]
81Desoxycorticosterone acetate372.4970308–348122.00–355.000.01–1.3945[130]
82Dexamethasone392.4610308.15–328.15151.10–348.300.013–0.02830[139]
83Diazepam284.7400308.00–348.00122.00–355.001.80–11.1045[109]
84Dichlone227.0400313.00–333.0070.70–325.800.07–2.5123[140]
85Diclofenac acid331.3400308.15–338.15120.00–400.000.40–19.8032[141]
86Diflunisal250.1980308.20–328.2093.00–246.000.02–0.0821[142]
87Diheptyl-2,2′-oxidiacetate330.4596313.00–333.0087.00–141.0021.99–81.6615[143]
88Dinonyl-2,2′-oxidiacetate386.5659313.00–333.0091.00–142.0014.86–67.1015[143]
89Dipentadecyl-2,2′-oxidiacetate554.8848313.00–333.00110.00–193.005.39–14.0315[143]
90Dipentyl-2,2′-oxidiacetate274.3532313.00–333.0085.00–125.0022.31–125.0515[143]
91Ditridecyl-2,2′-oxydiacetate498.7790313.00–333.0099.00–156.005.57–18.2215[143]
92Diundecyl-2,2′-oxidiacetate442.6722313.00–333.0092.00–145.007.44–24.3415[143]
93Diuron233.0900308.20–328.20100.00–200.000.009–3.1819[144]
94Docetaxel807.8792308.00–338.00152.00–354.000.37–5.0332[120]
95Dutasteride528.5000308.00–348.00122.00–355.000.001–1.6045[145]
96Epicatechin290.2680313.15–343.15120.00–260.000.007–0.1860[146]
97Ergosterol396.6500318.15–333.15120.00–240.000.03–0.3015[147]
98Ethanamide59.0700308.20–323.2090.00–400.002.38–31.6230[92]
99Ethylene glycol62.0680313.15–353.1570.00–190.0083.00–429.0035[148]
100Exemestane296.4000308.00–348.00122.00–355.000.13–18.7645[106]
101Fenofibrate360.8310308.20–328.20100.10–220.2018.30–66.6021[132]
102Ferrocene186.0310308.00–348.0080.00–244.008.90–31.2022[101]
103Ferulic acid194.1840301.50–333.50120.00–280.000.02–0.1218[149]
104Finasteride372.5000308.00–348.00122.00–355.000.83–3.3345[145]
105Fluoranthene202.2506308.15–328.1586.00–249.001.13–9.3028[150]
106Fluoxetine hydrochloride345.7900308.15–338.15160.00–400.000.35–8.1228[151]
107Flurbiprofen244.2609303.15–323.1589.00–245.000.22–1.9727[152]
108Flutamide276.1000308.00–348.00122.00–355.000.05–5.0945[145]
109Fluvastatin411.4700308.00–348.00121.60–354.600.22–6.0145[108]
110Gabapentin171.2400308.00–338.00160.00–400.002.31–73.6028[153]
111Gemfibrozil250.3400308.20–328.20100.10–220.200.29–41.9021[132]
112Geraniol154.2500308.00–333.00100.00–180.0027.00–252.0018[83]
113Geranyl butyrate224.3400308.15–333.15100.00–180.0022.00–250.0018[83]
114Hinokitiol164.2000313.20–333.20101.40–358.405.91–24.9032[154]
115Hydrocortisone362.4600308.15–373.1599.00–226.500.0005–0.007326[136]
116Ibuprofen206.2808308.15–318.1580.00–220.000.53–58.4029[155]
117Imipramine HCl316.8700313.50–323.50300.00–500.000.05–0.1010[156]
118Iodopropynyl butylcarbamate281.0900308.15–333.1587.60–341.5012.00–47.0027[157]
119Isoniazid137.1393308.00–313.00130.00–185.000.005–0.0618[158]
120Juglone174.1528308.20–328.2092.00–244.000.20–15.9018[159]
121Ketoconazole531.4310308.00–348.00122.00–355.000.005–1.7545[133]
122Ketoprofen254.2806313.15–328.1590.00–250.000.03–1.8815[160]
123Lamotrigine256.0910318.00–338.00121.60–354.600.004–0.04627[134]
124Letrozole285.3100308.00–348.00122.00–355.000.01–0.8345[106]
125Levonorgestrel384.5000308.00–338.0012.20–35.500.004–0.0336[161]
126Levulinic acid116.1152313.00–342.4094.10–182.704.91–90.0034[162]
127Lovastatin404.5500308.00–348.00121.60–354.600.11–1.1445[108]
128Mandelic acid152.1500308.15–328.15101.00–230.600.27–29.0421[116]
129m-Dinitrobenzene168.1100308.00–328.0095.00–145.001.90–55.4018[84]
130Medroxyprogesterone acetate386.2460308.00–348.00122.00–355.000.40–4.1340[138]
131Mefenamic acid241.2900308.15–338.15160.00–400.000.83–59.4028[163]
132Megestrol acetate312.5000308.00–338.0012.20–35.500.03–8.7036[161]
133Meloxicam sodium salt373.3800303.00–323.00149.00–255.000.04–0.1315[164]
134Menadione172.1800313.00–333.0097.20–306.707.50–10.0618[140]
135Metaxalone221.3000308.20–328.20119.50–240.000.04–1.4821[125]
136Methimazole114.1700308.00–348.00122.00–355.000.54–18.9940[165]
137Methocarbamol241.2000308.20–328.20119.90–220.500.61–4.8721[125]
138Methyl gallate184.1500313.00–333.00100.00–500.000.001–0.04227[165]
139Methyl salicylate152.1473343.15–423.1590.00–310.005.10–753.3044[167]
140Methylparaben152.1473308.00–348.00122.00–355.001.08–12.1340[117]
141Metronidazole benzoate275.7600308.00–348.00122.00–355.000.70–45.5040[115]
142m-Nitrophenol139.1100308.00–348.00121.60–486.403.94–44.5049[88]
143N-(4-Ethoxyphenyl)ethanamide179.2160308.00–328.0090.00–190.000.09–0.4316[77]
144Nabumetone228.3000308.20–328.20100.00–220.000.39–26.8021[168]
145Naproxene230.2592313.10–333.1089.60–193.100.02–0.3218[169]
146Nicotinamide122.1246313.15–373.1555.00–305.000.10–31.4024[170]
147Nicotinic acid123.1100313.15–373.1545.00–302.000.003–0.10422[168]
148Nifedipine346.3350333.15–373.15126.00–296.000.05–0.7129[171]
149Niflumic acid282.2200313.20–353.20190.00–310.000.07–0.2121[123]
150Nimodipine418.4403313.00–333.00100.00–250.000.006–0.42321[87]
151Nitrendipine360.3600308.00–328.0080.00–200.000.01–0.6315[172]
152Norfloxacin319.3308313.15–323.1599.40–303.300.01–0.2415[173]
153Ofloxacin361.3675318.15–323.15101.50–300.600.004–0.01310[173]
154O-Nitrobenzoic acid167.1200308.00–328.00100.00–210.000.06–0.3615[174]
155O-Tolidine212.2900308.00–328.00110.00–210.000.05–0.4815[175]
156Oxymatrine264.3700308.15–328.15110.40–210.700.007–0.02518[176]
157Oxymetholone332.4770308.00–328.00121.00–305.000.16–1.4920[122]
158Paclitaxel853.9061308.15–328.15100.00–275.000.012–0.06221[91]
159p-Dimethylaminoazobenzene225.2890308.20–318.2091.00–253.004.37–12.5010[112]
160Penicillin G334.3901313.15–333.15100.00–350.000.04–0.6318[70]
161Penicillin V350.3800314.85–334.8579.90–280.503.24–4.2624[177]
162Pentoxifylline278.3000308.15–328.15119.80–24000.30–13.7021[128]
163Phenazopyridine213.2385308.00–348.00122.00–355.000.44–20.2145[165]
164Phenol94.1112333.15–363.15100.00–350.0011.40–906.4033[178]
165Phenylbutazone317.4458308.20–328.20100.00–220.000.20–26.5021[168]
166Phenylephrine hydrochloride203.6660308.15–338.15160.00–400.001.01–28.9028[179]
167p-Hydroxyazobenzene198.2206308.20–318.2091.00–253.0025.70–63.6010[112]
168Picric acid229.1100308.00–348.00121.60–486.400.22–26.5050[88]
169Pindolol248.3210298.00–318.0080.00–275.000.66–1.9230[180]
170Piracetam142.2000308.15–328.15120.00–240.000.08–0.3721[128]
171Piroxicam331.3460308.15–338.15160.00–400.000.12–5.1228[181]
172p-Methylbenzenesulfonic acid172.2020318.00–328.0080.00–210.000.20–0.7415[182]
173p-Nitroaniline183.1300308.00–328.00110.00–210.000.06–0.4315[183]
174p-Nitrophenol139.1100308.00–348.00121.60–486.402.23–19.7049[88]
175Prednisolone360.4400308.15–373.1582.40–226.500.00004–0.0009928[136]
176Prednisone358.4300308.15–373.1582.40–226.500.0001–0.005828[136]
177Propranolol259.3434308.00–348.00122.00–355.003.58–239.6145[165]
178Propyl 4-hydroxybenzoate180.2010308.15–328.1594.10–220.200.19–6.1221[116]
179Propyl p-hydroxybenzoate180.2000308.15–328.1580.00–230.000.038–6.17518[184]
180Protocatechualdehyde138.1200313.00–333.00100.00–500.000.004–0.04624[166]
181Protocatechuic acid154.1200313.00–333.00100.00–500.000.0004–0.025924[166]
182p-Toluenesulfonamide171.2200308.15–328.1580.00–210.000.05–0.5115[185]
183Pyrocatechol110.1106333.15–348.15100.00–350.001.22–38.4722[178]
184rac-Boc-piperazine285.3900308.20–328.20102.00–204.001.24–5.9019[81]
185Racemic paroxetine329.3650308.20–328.2088.30–242.000.73–11.8424[186]
186Resorcinol110.1100308.15–338.15120.00–400.001.10–9.7332[187]
187Rosuvastatin481.5380308.00–348.00121.60–354.600.23–2.4445[108]
188Salicylamide137.1360308.20–328.20100.00–220.000.09–2.1021[167]
189Salicylic acid138.1200308.15–318.1592.60–157.900.97–3.9820[188]
190Simvastatine418.5662308.00–348.00121.60–354.600.02–5.3545[108]
191Spironolactone416.5730308.00–338.00160.00–400.000.63–50.1028[189]
192Sulindac356.4100308.15–338.15160.00–400.000.37–86.9028[190]
193Syringic acid198.1700313.00–333.00100.00–500.000.003–0.12727[191]
194TCMTB [2-(thiocyanomethylthio)benzothiazole]238.3600323.00–338.00100.00–300.000.02–13.9012[103]
195Tebuconazole307.8200323.00–338.00100.00–300.000.04–18.5712[103]
196Testosterone288.4200308.15–373.1582.40–226.500.07–0.9030[136]
197Tetramethylpyrazine136.2000318.00–338.00100.00–300.00100.00–1310.0015[192]
198Theobromine180.1700313.00–353.00193.00–344.000.09–0.4723[118]
199Theophylline180.1700313.00–353.00199–349.000.10–0.3324[118]
200Thymidine242.2286308.15–328.15100–300.000.01–0.0825[91]
201Thymol150.2176308.15–323.1578.00–250.008.37–147.0017[193]
202Tolfenamic acid461.7100313–333120.00–360.000.01–0.1824[131]
203Triclocarban315.5800313.20–333.20109.30–389.600.90–8.7024[129]
204Triphenylene228.2879308.15–328.1585.00–252.000.01–0.4228[150]
205Triphenylmethyl chloride278.7754308.15–338.15150.00–400.002.93–2718[194]
206Triphenyltin chloride385.4500308.15–328.15150.00–300.002.71–10.9612[192]
207Uracil112.0868308.15–338.15150.00–400.000.000769–0.003423[195]
208Vanillic acid168.1400313–33385.00–500.000.0143–0.60728[191]
209Xanthohumol354.3960328.15–358.15350.00–950.000.012–0.1715[196]
210Zopiclone388.8080313–333100.00–250.000.015–0.21921[87]

The optimization procedure, for fitting the model parameters, reduces the percentage average absolute relative deviation (AARD %) and is defined as

AARD(%)=100Ni=1N|y2expy2caly2exp|(33)

Initial approximations of model parameters are first obtained by genetic algorithm function (ga MATLAB function) through several runs to avoid convergence to local minimum. These approximations are then used as initial guess for least-square fitting with Levenberg–Marquardt algorithm as minimization option (lsqcurvefit MATLAB function).

Table 3 shows the results of parameter estimation (ai) of the model proposed in this work (Eq. 32) for the 210 systems considered. Parameters of the other models can be accessed in the Supplementary material.

Table 3

Estimated model parameters (ai) of the model proposed in this work (Eq. 32) for the 210 solid solutes considered.

No.a0a1a2a3a4a5a6a7
1−105.070.083−4.48 × 10−56.59 × 10−51.32−2.08 × 10−3−24.58−1.85
2−1531.82−0.021−3.45 × 10−79.44 × 10−54.50−4.41 × 10−3−2.92175,549.35
3−4084.36−0.0601.46 × 10−51.38 × 10−412.65−1.32 × 10−22.92437,823.79
4−0.970.073−6.70 × 10−54.56 × 10−42.89−5.14 × 10−3−80.05−0.44
58205.31−0.016−5.75 × 10−61.10 × 10−4−25.392.61 × 10−2−4.19−879,263.63
6−1982.72−0.0131.33 × 10−75.77 × 10−55.95−5.97 × 10−3−0.40217,476.69
7573.760.010−7.18 × 10−64.24 × 10−5−1.912.14 × 10−3−5.20−54,779.81
8−546.60−0.0122.81 × 10−63.79 × 10−51.59−1.58 × 10−30.8258,878.77
9374.660.219−1.10 × 10−4−2.37 × 10−5−1.191.91 × 10−3−43.891.06
10−525.74−0.3572.08 × 10−4−1.38 × 10−40.15−9.85 × 10−5101.71−5.43
11−382.66−0.1639.82 × 10−5−1.12 × 10−40.64−8.50 × 10−453.66−3.04
12−1.66−0.0061.44 × 10−69.60 × 10−6−0.062.26 × 10−52.162.23
1311,328.330.018−2.33 × 10−62.17 × 10−7−34.563.52 × 10−2−5.74−1,234,124.05
14−392.250.001−6.50 × 10−67.90 × 10−51.32−1.41 × 10−3−8.2646,727.41
15−85.82−0.0611.76 × 10−55.13 × 10−5−0.071.37 × 10−417.55−7.53
16−83.090.050−1.15 × 10−5−6.50 × 10−50.53−6.75 × 10−4−5.92−10.79
17−885.78−0.6892.25 × 10−47.63 × 10−5−2.303.56 × 10−3244.70−4.01
18−262.350.001−2.34 × 10−62.89 × 10−51.63−2.56 × 10−3−1.91−1.69
191.84−0.0441.76 × 10−5−1.18 × 10−5−0.781.35 × 10−319.532.52
20−251.12−0.0641.38 × 10−51.28 × 10−40.53−4.15 × 10−45.6630,058.65
21−95.390.0029.40 × 10−7−3.53 × 10−60.43−5.89 × 10−40.480.15
22−11.28−0.0071.23 × 10−61.35 × 10−5−0.02−2.87 × 10−52.622.48
23133.120.0213.05 × 10−6−1.31 × 10−4−1.662.97 × 10−315.94−15.54
24−74.340.058−1.49 × 10−5−8.87 × 10−50.31−3.33 × 10−4−2.110.85
25−25.71−0.0195.32 × 10−62.56 × 10−5−0.183.73 × 10−46.141.78
26−360.010.190−1.32 × 10−5−5.80 × 10−41.24−1.19 × 10−312.320.63
27−141.29−0.0231.67 × 10−5−6.03 × 10−50.098.10 × 10−517.82−0.78
28598.29−0.0802.73 × 10−54.44 × 10−5−4.587.28 × 10−322.116.32
29−2460.28−1.2033.79 × 10−4−6.65 × 10−5−0.841.47 × 10−3494.35−24.73
30−692.41−0.0512.41 × 10−51.05 × 10−51.96−2.04 × 10−314.3063,063.17
31−9800.68−0.00023.53 × 10−6−3.46 × 10−530.61−3.19 × 10−26.551,033,864.84
32−13,205.450.023−1.31 × 10−55.01 × 10−540.60−4.14 × 10−2−9.421,437,033.73
33−236.85−0.0049.77 × 10−6−5.92 × 10−50.87−1.17 × 10−311.850.60
34−156.11−0.0171.04 × 10−5−1.48 × 10−50.58−7.95 × 10−47.760.72
35−190.060.113−1.17 × 10−5−2.68 × 10−40.91−1.04 × 10−3−3.410.83
361954.650.088−2.45 × 10−5−5.31 × 10−5−5.826.02 × 10−3−9.43−223,043.77
37−2792.520.025−9.65 × 10−61.15 × 10−58.51−8.53 × 10−3−4.84303,725.89
3843.350.011−4.59 × 10−67.94 × 10−6−0.315.27 × 10−4−2.202.05
39−115.460.057−9.23 × 10−6−1.33 × 10−40.39−4.02 × 10−42.5621.70
40145.930.185−5.94 × 10−5−5.96 × 10−50.55−6.99 × 10−4−52.95−1.75
417678.04−0.0751.73 × 10−51.27 × 10−4−23.472.37 × 10−29.28−847,146.30
42852.150.030−1.00 × 10−51.22 × 10−5−2.552.63 × 10−3−4.10−98,485.91
43−1645.58−0.0044.24 × 10−6−1.65 × 10−55.13−5.37 × 10−34.70165,008.13
44−1.95−0.1383.39 × 10−52.11 × 10−4−0.506.00 × 10−418.213.11
450.930.0121.17 × 10−6−4.92 × 10−5−0.235.31 × 10−42.41−0.27
46−716.84−0.4271.50 × 10−4−8.76 × 10−5−1.171.95 × 10−3171.93−5.92
4711,214.795.746−1.90 × 10−37.50 × 10−5−0.324.38 × 10−4−2179.3893.76
48−227.100.121−1.59 × 10−5−2.37 × 10−41.37−1.72 × 10−3−10.71−1.82
49107.14−0.006−1.81 × 10−67.24 × 10−5−0.629.96 × 10−4−4.784.22
50−2958.54−0.0617.14 × 10−61.47 × 10−48.72−8.72 × 10−34.93328,517.55
51−26.690.098−2.40 × 10−5−1.34 × 10−40.25−1.56 × 10−4−10.956.24
52−164.84−0.0681.10 × 10−51.71 × 10−40.22−8.52 × 10−53.1324,939.84
531060.660.514−1.82 × 10−48.90 × 10−5−0.354.70 × 10−4−198.278.60
5434.680.004−3.01 × 10−64.46 × 10−5−0.111.64 × 10−4−5.715.29
55−1339.30−0.0545.01 × 10−5−2.49 × 10−46.55−9.78 × 10−346.62−18.29
561164.460.181−5.51 × 10−5−4.82 × 10−5−2.692.71 × 10−3−51.34−110,426.70
573303.720.214−7.27 × 10−52.94 × 10−5−9.049.37 × 10−3−79.39−314,535.29
58−19,001.50−0.3081.25 × 10−4−1.47 × 10−457.79−6.04 × 10−2130.401,924,912.82
59−53.000.038−9.72 × 10−6−2.77 × 10−50.52−7.32 × 10−4−10.48−0.54
60−51.010.025−5.36 × 10−6−3.85 × 10−50.047.42 × 10−52.27−1.51
61−1138.320.108−3.25 × 10−51.40 × 10−53.90−3.89 × 10−3−30.78132,743.12
6250.190.107−3.55 × 10−5−2.14 × 10−50.19−1.32 × 10−4−24.43−16.81
6312.490.012−1.32 × 10−57.90 × 10−50.11−1.88 × 10−4−8.57−1.40
641562.77−0.1414.12 × 10−51.04 × 10−4−5.295.21 × 10−341.52−202,400.70
659228.66−0.009−2.13 × 10−65.99 × 10−5−28.642.96 × 10−20.64−995,613.08
66119.60−0.0709.87 × 10−61.75 × 10−4−1.031.53 × 10−37.483.78
67−237.860.042−6.03 × 10−6−1.26 × 10−40.69−8.09 × 10−413.81−1.24
68−40.730.124−3.66 × 10−5−7.46 × 10−50.88−1.20 × 10−3−27.01−2.77
69671.210.654−1.88 × 10−4−3.30 × 10−40.89−7.66 × 10−4−180.492.74
70101.910.110−3.27 × 10−5−6.30 × 10−5−0.051.91 × 10−4−24.391.91
71−2234.78−0.0451.50 × 10−61.93 × 10−46.75−6.79 × 10−3−9.01257,254.20
72−81.150.044−1.07 × 10−5−4.84 × 10−50.41−4.81 × 10−4−4.05−2.43
73−584.030.199−2.44 × 10−5−5.37 × 10−42.84−3.80 × 10−37.58−3.85
74246.330.033−1.02 × 10−5−7.38 × 10−6−0.727.64 × 10−4−3.25−32,096.23
7550.19−0.1853.64 × 10−53.55 × 10−4−0.565.21 × 10−414.702.15
76−2563.28−0.0781.94 × 10−51.34 × 10−47.57−7.63 × 10−38.38279,924.96
7751,417.64−0.049−2.31 × 10−52.93 × 10−4−160.071.66 × 10−13.75−5,507,449.07
78−11.31−0.0321.59 × 10−52.13 × 10−5−0.213.21 × 10−46.311.26
79−717.560.144−4.74 × 10−57.99 × 10−53.17−3.27 × 10−3−61.25106,770.33
80452.85−0.0852.19 × 10−51.64 × 10−4−1.551.55 × 10−34.99−46,538.55
81−350.16−0.0263.00 × 10−68.53 × 10−50.83−6.91 × 10−41.5742,383.68
822576.44−0.0691.52 × 10−51.10 × 10−4−8.499.05 × 10−39.10−269,462.56
83145.56−0.0407.81 × 10−61.04 × 10−4−0.525.54 × 10−4−0.73−12,431.30
84232.96−0.0721.18 × 10−51.50 × 10−4−1.602.34 × 10−36.508.38
853580.56−0.0367.25 × 10−67.43 × 10−5−11.121.14 × 10−26.73−395,352.20
86240.50−0.055−2.89 × 10−72.07 × 10−4−1.392.02 × 10−3−3.655.82
87−243.80−0.1798.83 × 10−51.77 × 10−50.11−1.94 × 10−446.11−1.30
88−659.43−0.1718.83 × 10−5−2.60 × 10−40.89−1.12 × 10−394.34−8.06
89−2624.87−1.2014.12 × 10−4−2.41 × 10−40.101.46 × 10−4498.22−22.53
90−312.90−0.0673.99 × 10−5−2.59 × 10−51.22−1.83 × 10−321.33−3.88
911293.790.663−2.86 × 10−42.29 × 10−4−1.111.57 × 10−3−226.1730.25
92−7198.49−5.0972.27 × 10−3−1.55 × 10−40.60−7.51 × 10−41461.03−74.64
93476.02−0.2953.43 × 10−56.00 × 10−4−3.444.37 × 10−334.00−132.83
943749.410.069−3.24 × 10−51.39 × 10−4−11.021.14 × 10−2−45.12−374,414.07
95860.660.065−1.99 × 10−5−2.68 × 10−5−2.462.48 × 10−3−9.44−98,727.09
96387.570.000−3.17 × 10−72.78 × 10−6−1.191.21 × 10−33.01−52,764.44
97−40.90−0.0622.84 × 10−62.00 × 10−40.27−5.48 × 10−4−0.863.73
98−65.300.002−4.13 × 10−63.36 × 10−50.40−6.11 × 10−4−1.965.24
99−902.14−0.0314.74 × 10−68.00 × 10−52.59−2.52 × 10−30.81103,437.48
1002766.570.054−8.54 × 10−6−8.46 × 10−5−7.927.68 × 10−3−3.01−327,864.36
101−551.600.281−4.77 × 10−5−6.35 × 10−42.91−3.81 × 10−3−3.79−12.88
102334.39−0.0441.49 × 10−54.95 × 10−5−1.181.25 × 10−36.75−39,319.98
1038621.92−0.0192.17 × 10−5−2.28 × 10−4−28.042.97 × 10−248.90−949,369.22
104−425.91−0.0561.89 × 10−62.17 × 10−41.26−1.22 × 10−3−9.5961,284.65
105−76.070.038−1.10 × 10−5−5.30 × 10−50.26−2.77 × 10−40.4336.77
1060.16−0.2016.48 × 10−52.67 × 10−5−1.201.18 × 10−378.68−51,364.38
107−5.170.046−1.06 × 10−5−1.26 × 10−50.31−3.39 × 10−4−14.060.60
108−677.14−0.1041.55 × 10−52.35 × 10−41.65−1.54 × 10−37.8782,605.10
109915.94−0.016−3.56 × 10−61.17 × 10−4−2.832.94 × 10−3−5.88−94,980.76
110−1663.480.211−5.86 × 10−5−1.22 × 10−45.99−6.14 × 10−3−48.99190,361.87
111−288.61−0.0402.79 × 10−5−1.50 × 10−40.31−2.32 × 10−438.92−14.73
11213,211.97−0.0341.98 × 10−61.31 × 10−4−41.694.38 × 10−2−2.97−1,390,552.83
113−3716.29−0.1344.82 × 10−51.37 × 10−411.07−1.13 × 10−220.92394,533.29
114−194.120.0061.38 × 10−7−6.99 × 10−50.52−6.85 × 10−415.690.17
115−434.950.0043.54 × 10−5−2.46 × 10−41.53−1.94 × 10−325.132.49
116−160.030.048−2.34 × 10−52.49 × 10−51.15−1.73 × 10−3−9.81−48.21
117195.607.771−2.08 × 10−3−1.36 × 10−3104.33−1.62 × 10−1−3225.354.15
118127.43−0.021−9.84 × 10−78.65 × 10−5−0.771.13 × 10−3−1.233.22
119−980.73−0.1371.67 × 10−4−1.48 × 10−3−4.861.04 × 10−2276.44−2.50
12041.320.018−1.92 × 10−6−4.23 × 10−5−0.508.82 × 10−43.08−1.55
121−1936.350.101−2.90 × 10−5−5.30 × 10−56.44−6.69 × 10−3−20.54204,158.22
1220.970.071−3.74 × 10−58.39 × 10−50.65−1.02 × 10−3−24.951.66
123−48.070.033−1.03 × 10−5−1.79 × 10−50.28−3.53 × 10−4−5.040.94
124−2999.330.053−2.60 × 10−55.89 × 10−59.47−9.67 × 10−3−18.88330,889.01
125−407.180.0082.20 × 10−6−1.37 × 10−51.18−1.12 × 10−3−0.7041,782.09
1261452.110.004−1.72 × 10−64.59 × 10−6−4.464.53 × 10−31.03−161,529.34
1271609.020.021−8.32 × 10−6−5.68 × 10−6−4.874.93 × 10−3−2.69−178,903.05
12810.680.013−2.88 × 10−62.54 × 10−5−0.143.60 × 10−4−3.43−1.12
129−249.41−0.0191.79 × 10−5−4.63 × 10−51.12−1.67 × 10−311.04−1.74
130−5601.28−0.0296.06 × 10−69.34 × 10−516.72−1.66 × 10−2−2.06625,762.42
1311943.21−0.0155.00 × 10−64.98 × 10−5−6.146.38 × 10−34.35−215,444.10
1321973.71−0.0721.95 × 10−51.05 × 10−4−6.085.97 × 10−312.18−228,276.88
133−2087.93−1.2493.60 × 10−44.06 × 10−4−0.931.02 × 10−3439.09−21.03
134−25.36−0.0061.14 × 10−61.30 × 10−50.10−1.51 × 10−40.581.44
135433.520.069−4.48 × 10−52.43 × 10−4−1.362.03 × 10−3−45.503.74
136−1476.05−0.1082.31 × 10−52.02 × 10−44.18−4.20 × 10−310.03164,342.67
137−49.110.187−4.62 × 10−5−1.93 × 10−40.99−1.16 × 10−3−33.993.39
138−124.360.0174.46 × 10−6−8.00 × 10−50.27−2.10 × 10−47.257.88
139−161.270.0137.31 × 10−6−3.18 × 10−50.37−2.63 × 10−40.7118,861.41
140439.34−0.0131.26 × 10−65.72 × 10−5−1.431.55 × 10−3−1.92−45,085.94
1411837.590.007−5.92 × 10−65.63 × 10−5−5.856.26 × 10−3−6.70−187,579.63
142−964.63−0.0509.53 × 10−61.01 × 10−42.86−2.97 × 10−35.23101,856.34
143154.020.059−2.34 × 10−52.10 × 10−5−0.498.21 × 10−4−19.175.30
144−61.04−0.0601.33 × 10−57.55 × 10−5−0.192.95 × 10−415.81−0.34
145−91.130.0101.53 × 10−6−3.05 × 10−50.30−3.54 × 10−42.701.25
146−2044.48−0.0671.48 × 10−51.41 × 10−45.85−5.69 × 10−33.98233,327.78
147384.04−0.0195.06 × 10−64.45 × 10−5−1.221.24 × 10−30.80−44,257.62
14810.31−0.0791.93 × 10−51.40 × 10−4−0.303.69 × 10−47.801.62
149−79.29−0.0521.24 × 10−56.04 × 10−50.01−3.90 × 10−512.951.50
150−13.36−0.0691.49 × 10−51.09 × 10−4−0.385.65 × 10−4 × 10−412.48−3.63
151−77.13−0.0248.64 × 10−62.73 × 10−50.16−1.76 × 10−46.26−0.45
152186.97−0.257−2.02 × 10−51.06 × 10−30.45−1.71 × 10−3−33.223.37
1530.340.0035.40 × 10−5−4.32 × 10−4−1.843.62 × 10−341.421.20
154−32.810.016−3.91 × 10−6−1.36 × 10−5−0.011.89 × 10−4−0.04−2.69
155−3.390.017−5.48 × 10−6−8.53 × 10−6−0.133.20 × 10−4−0.941.02
156−249.260.078−9.95 × 10−6−1.58 × 10−41.46−2.01 × 10−3−5.99−1.82
157−9.69−0.0373.71 × 10−69.18 × 10−5−0.172.25 × 10−45.572.35
158−1835.62−0.8452.97 × 10−4−1.29 × 10−40.71−8.42 × 10−4329.42−11.27
1590.310.049−3.71 × 10−6−1.24 × 10−4−0.185.30 × 10−4−1.42−1.50
16044.26−0.0811.29 × 10−52.07 × 10−4−0.282.97 × 10−41.102.75
161−94.41−0.0241.21 × 10−54.80 × 10−60.31−4.18 × 10−46.19−0.93
162−12.300.079−2.34 × 10−5−5.22 × 10−50.18−9.26 × 10−5−11.5916.92
1631429.40−0.0165.10 × 10−78.54 × 10−5−4.414.54 × 10−3−3.38−152,612.74
164−36.560.021−4.22 × 10−6−1.76 × 10−50.16−1.78 × 10−4−1.57−2.21
165−69.00−0.0561.37 × 10−55.96 × 10−5−0.182.93 × 10−416.86−8.38
1665830.12−0.2065.82 × 10−51.24 × 10−4−19.231.99 × 10−263.96−658,885.28
167−0.99−0.0134.42 × 10−65.06 × 10−6−0.244.69 × 10−44.59−1.05
168−2923.03−0.0983.07 × 10−57.40 × 10−58.53−8.70 × 10−324.20305,416.57
169112.44−0.0401.13 × 10−58.74 × 10−5−0.771.17 × 10−31.003.11
170106.560.105−2.97 × 10−5−7.50 × 10−5−0.154.01 × 10−4−23.363.67
171−13,659.770.031−2.01 × 10−51.97 × 10−443.02−4.46 × 10−2−39.271,487,919.98
172−18.24−0.0111.87 × 10−62.54 × 10−50.002.29 × 10−51.314.05
173−42.820.044−1.03 × 10−5−5.78 × 10−50.20−1.71 × 10−4−4.443.12
174−32.24−0.0164.67 × 10−62.82 × 10−50.003.89 × 10−53.71−1.83
175−100.38−0.0583.57 × 10−57.10 × 10−70.05−1.74 × 10−514.020.55
1762.14−0.0863.24 × 10−51.38 × 10−4−0.385.56 × 10−48.87−1.92
1771346.38−0.0982.06 × 10−52.00 × 10−4−4.414.60 × 10−36.60−141,805.22
17823.810.022−6.93 × 10−69.39 × 10−6−0.153.36 × 10−4−5.20−0.23
179−160.90−0.1473.83 × 10−52.34 × 10−40.50−8.99 × 10−417.55−2.76
180−11.19−0.0089.67 × 10−6−4.01 × 10−5−0.499.44 × 10−410.360.69
181−184.93−0.0951.65 × 10−51.87 × 10−40.68−1.16 × 10−313.2144.17
182−78.04−0.0055.66 × 10−6−2.25 × 10−50.23−2.75 × 10−44.42−1.27
1830.940.034−8.40 × 10−6−3.84 × 10−5−0.031.21 × 10−4−3.12−0.20
184−16.280.065−3.08 × 10−53.83 × 10−50.57−8.45 × 10−4−19.3111.93
18550.56−0.0375.90 × 10−68.97 × 10−5−0.497.47 × 10−43.630.36
186−1572.060.028−8.59 × 10−61.77 × 10−55.10−5.30 × 10−3−12.28170,879.91
187472.290.045−1.67 × 10−56.58 × 10−6−1.351.42 × 10−3−9.97−49,967.76
188−35.35−0.0449.82 × 10−65.59 × 10−5−0.233.75 × 10−411.370.70
189−0.040.045−1.20 × 10−5−3.13 × 10−50.13−7.83 × 10−5−9.230.14
190−1016.150.041−1.71 × 10−53.20 × 10−53.20−3.22 × 10−3−11.13113,622.34
191−9888.680.148−5.24 × 10−58.59 × 10−532.16−3.36 × 10−2−68.811,082,525.80
192−1468.070.055−2.61 × 10−51.18 × 10−44.82−4.76 × 10−3−30.13175,156.62
193−415.39−0.1224.34 × 10−51.38 × 10−51.04−1.53 × 10−343.6120.11
1942.140.071−4.37 × 10−59.66 × 10−50.55−8.69 × 10−4−22.632.17
1950.060.022−2.85 × 10−51.67 × 10−40.45−7.23 × 10−4−18.02−0.04
19611.47−0.0701.73 × 10−51.34 × 10−4−0.324.28 × 10−46.920.70
197−18.14−0.0113.92 × 10−69.96 × 10−6−0.113.03 × 10−43.196.16
198−303.35−0.1846.24 × 10−56.66 × 10−50.16−2.57 × 10−453.32−4.75
199−141.35−0.2486.43 × 10−52.52 × 10−4−0.535.40 × 10−450.60−0.51
200−1602.81−0.5201.86 × 10−4−9.83 × 10−53.36−5.11 × 10−3203.24−11.97
201718.88−0.0188.52 × 10−6−1.12 × 10−6−4.847.68 × 10−37.376.19
202−38.32−0.006−2.87 × 10−65.73 × 10−50.17−2.20 × 10−4−2.081.94
203−44.520.123−3.11 × 10−5−1.02 × 10−40.88−1.17 × 10−3−26.523.52
204−127.080.0091.63 × 10−6−6.44 × 10−50.23−2.16 × 10−410.890.53
205−1.500.105−2.81 × 10−5−9.46 × 10−50.34−3.21 × 10−4−18.43−0.20
20644.33−0.1342.24 × 10−52.29 × 10−4−0.971.28 × 10−324.08−5.17
207−1972.880.128−2.30 × 10−5−1.91 × 10−48.10−1.00 × 10−2−15.63145,287.58
208−22.21−0.0313.99 × 10−68.16 × 10−5−0.032.56 × 10−52.852.61
209605.760.371−1.06 × 10−4−6.44 × 10−60.95−1.28 × 10−3−152.586.90
210−170.010.0059.12 × 10−6−8.17 × 10−50.57−7.11 × 10−48.98−0.19

The AARDs (%) of each model equation with each solid compound are shown in Table 4 where the minimal AARD for the correlation of the solubility data of each binary system is shown in bold. It is clear that empirical models, considered in this study, as a whole correlate quite satisfactorily the data set with a minimal AARD (0.17%) obtained by the proposed model (Eq. 32) for component 45 (azobenzene) and a maximum AARD (56.47%) obtained by Jouyban et al. model (Eq. 18) for component 77 (corosolic acid). The average of minimal AARDs is 6.87%. Therefore, 0.17% represents the minimum of minimums and 56.47% represents the maximum of minimums and 6.87% represents the average of the minimums for the data set considered. It is worth mentioning that the value of 56.77% was obtained for corosolic acid for which the experimental data present presumably a misprint, which results in excessive AARD.

Table 4

AARD (%) for 210 compounds calculated for each of the 22 studied correlations.

No.(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22)(23)(24)(25)(26)(27)(28)(29)(30)(32)
121.089.9921.0828.6521.0517.3523.1813.7312.966.2021.1414.7418.4212.899.8513.7710.0813.88015.3111.698.626.34
219.3410.7019.3411.8019.0910.4718.5918.478.4010.6319.3110.6412.1711.416.7012.557.0012.00711.839.568.676.64
324.359.4824.3513.9620.219.4119.5321.0812.559.0224.369.4612.249.598.4912.179.189.769.458.917.457.58
45.382.375.384.077.000.786.834.943.602.375.383.703.913.901.573.406.953.9734.021.661.161.17
546.6543.3046.6544.9543.9545.8043.0946.4243.3740.9046.5343.4645.4043.3841.5945.8938.0643.20246.7644.8142.9439.98
613.307.7413.308.7910.2814.139.9212.437.377.2813.267.8310.707.675.9811.166.338.2299.458.407.675.84
717.1212.7817.1213.5014.9517.3713.7616.2310.779.5217.0312.0015.0012.9611.2116.319.7312.79714.9813.5711.419.56
89.443.619.443.6210.747.429.477.463.273.469.403.435.313.592.915.202.864.1375.533.663.392.99
99.169.099.169.028.539.099.058.423.213.149.023.298.488.563.264.086.579.0783.518.097.563.06
106.236.006.237.147.005.987.385.964.565.786.215.995.757.095.086.044.707.0706.165.224.134.30
1112.348.8012.3410.5910.629.879.969.977.698.8912.389.3212.2510.489.019.719.079.94610.708.878.607.99
123.633.393.6310.8110.4211.8911.723.195.683.183.643.383.389.743.213.083.3811.11510.823.204.783.11
1333.3524.0033.3626.7031.1227.3129.5829.7626.6723.9933.2724.1926.9727.3124.0326.3724.3527.20027.9923.9823.6618.10
1418.4013.5118.4014.3518.0311.5917.5418.5412.4613.2218.4614.1016.8514.3611.6117.0612.1815.41215.2811.7810.8611.62
153.542.873.543.786.6312.406.603.544.372.993.512.882.793.572.603.033.063.6182.953.142.752.59
166.464.296.466.218.0613.416.695.344.874.226.475.326.404.804.205.389.944.4385.854.963.903.41
1731.9123.5531.9130.9832.8213.2933.2228.6621.3215.0131.8917.1122.9525.6417.5522.6826.0324.54123.8123.8721.6614.21
1813.159.3913.159.6719.0615.3018.3911.978.457.1813.167.9612.009.347.2811.219.099.80010.169.288.667.37
197.595.137.5914.3110.3817.5910.447.228.414.407.585.777.0010.495.937.314.7810.6288.516.155.104.27
2014.689.6714.689.7513.998.4614.6114.895.5610.4414.6710.217.659.065.325.935.4810.1797.657.744.944.55
215.952.655.952.6110.036.589.674.334.602.225.952.223.842.661.973.306.912.9283.022.692.742.16
223.573.103.5713.7911.4914.1813.102.326.042.913.603.053.2012.012.442.373.1113.86511.002.533.142.38
2319.9212.8819.9232.2925.3231.2926.3919.4817.786.5719.906.6614.9421.436.1315.8310.4420.85016.398.037.346.26
249.606.339.6017.4311.4218.0212.338.847.136.389.597.447.499.867.627.597.9510.1968.847.826.736.65
253.562.423.564.934.7911.213.942.394.151.563.501.661.882.781.732.266.662.7023.532.291.921.58
2621.9223.6921.9226.6024.1931.6323.6021.9723.1724.0421.8923.7922.5125.7721.4724.0117.4325.40824.1521.9422.0521.59
276.503.526.506.828.0716.537.474.177.702.536.464.914.326.563.404.187.406.4315.543.391.891.99
2816.6215.1816.6220.0123.9911.4823.9616.598.537.2016.5810.2516.1518.8710.9616.0810.6119.11215.7315.6814.907.23
298.728.228.728.529.025.208.768.206.745.278.706.048.018.805.767.696.908.7678.177.836.724.53
308.622.668.624.978.633.888.137.022.522.378.614.194.652.994.313.192.702.9244.034.282.562.25
3112.9612.7912.9614.0116.1314.3916.2312.4615.3411.5013.0211.7213.9713.6712.3312.4611.8314.15713.1212.4712.438.95
3224.8724.4824.8726.1925.2919.6826.9524.2920.2824.5624.7623.9723.7725.9822.9924.5923.3928.37423.5523.8624.0517.03
337.585.817.5812.659.9027.249.857.1111.425.347.656.087.239.716.046.8011.189.7425.036.856.534.89
349.177.109.1710.6911.0114.0811.008.149.786.639.198.448.9510.558.268.129.2910.6128.588.307.816.20
3510.075.4310.076.3912.749.1511.473.605.605.3110.066.157.115.512.813.535.525.2405.743.834.022.83
3636.6436.4236.6439.7949.2917.3650.0836.5415.6936.4236.5936.6125.8932.0636.4120.9244.9434.39731.2936.4237.0435.88
3711.017.2211.029.517.2517.266.739.8110.156.2510.837.388.856.627.419.267.056.72110.067.656.974.49
385.743.935.745.579.1012.387.994.584.453.455.733.874.983.874.084.499.003.6035.254.303.433.49
398.126.158.1216.6810.6517.6311.228.046.916.028.096.016.119.735.506.039.449.5918.765.555.775.31
405.153.005.145.665.575.825.165.355.652.635.194.715.984.924.455.396.215.1515.444.653.312.28
4118.8817.6618.8717.5618.2816.4519.9018.0419.4516.9519.0216.9820.4117.5515.3816.6814.4418.97516.8015.1816.2412.56
4212.974.0412.9710.583.7130.568.6410.7011.943.9812.865.9010.133.645.449.094.626.65510.645.513.843.89
434.592.674.595.058.5310.278.033.885.252.684.603.363.264.393.432.607.724.4413.703.433.052.88
4424.2821.5124.2822.4424.7327.8224.7524.5717.6921.5424.2821.8618.3522.4319.8216.8127.2522.98915.8020.2217.0416.82
452.171.162.173.3011.929.6911.330.695.771.162.171.952.002.530.320.740.392.3471.510.590.700.17
4615.8810.6115.8815.1017.157.6717.1715.899.858.4315.8711.1211.8712.989.9310.9813.8612.94510.5312.3410.167.01
47100.9235.72100.9295.2396.50111.0598.32118.9892.9336.51100.8893.4498.7185.4781.8293.80111.9084.77095.8396.7635.2035.02
4813.825.4813.828.0915.356.3114.006.244.325.5713.825.459.335.842.905.9310.425.5103.653.984.612.36
4912.575.5412.576.9911.687.3010.618.385.285.0312.555.258.166.334.697.2514.316.1747.925.745.324.75
5011.8211.9911.8215.3112.2215.1513.1012.008.3311.1711.6911.1310.6812.226.2611.056.3513.22512.5511.169.674.87
516.525.556.5210.556.4410.266.706.534.364.796.495.154.665.504.773.874.784.4126.134.762.803.10
5215.9710.8515.9712.5214.3110.6813.8215.947.179.9915.8712.2312.8212.446.1012.966.5813.40415.0611.029.166.60
5312.5710.1112.5712.3514.168.0514.0712.508.559.8012.5512.079.4210.3311.868.6913.0010.4510.0412.299.839.39
544.953.434.963.455.663.393.553.202.882.924.712.953.073.442.863.202.903.055.423.242.902.85
5531.4126.9931.4031.7132.636.0632.7223.2924.876.0231.428.7832.4632.1111.8321.955.9431.9231.6223.3319.874.96
5610.495.8710.499.668.5911.5310.3811.6611.715.6810.649.0912.668.888.5611.0719.5810.4511.448.576.955.44
5719.5613.0819.5616.2119.3416.0618.9119.3915.0412.2219.4215.8316.9517.1715.4317.6322.4717.0418.0215.7811.8512.10
5858.5342.3358.5254.4360.7835.8659.6950.9640.2938.1558.6941.0749.8039.1840.3546.9238.7038.5543.5745.1942.9033.70
592.392.332.392.373.660.643.462.383.010.702.400.702.882.360.612.110.762.391.811.952.220.61
602.262.252.265.082.296.212.172.182.871.802.191.801.762.291.441.751.331.952.931.741.421.35
617.425.367.427.645.0914.906.416.218.124.177.065.005.575.074.625.8429.174.7911.185.434.114.21
625.794.125.797.753.9316.125.225.614.764.105.755.145.473.944.795.6117.124.547.344.974.134.15
6323.9815.7123.9818.5819.1120.9919.6923.4619.4516.6323.9719.1921.8717.9317.7621.7721.7018.8919.0617.6114.4814.89
649.198.839.188.689.767.5510.469.0812.856.719.526.6211.718.836.086.917.279.676.516.547.356.29
657.104.977.107.916.1419.746.506.784.915.127.095.285.385.434.224.745.636.045.234.414.454.34
6612.137.1912.1310.789.0932.559.9611.914.727.0912.168.9910.658.165.389.094.899.549.446.876.314.54
676.806.136.808.556.9310.446.886.748.466.196.846.496.396.975.796.225.726.716.665.835.945.81
686.615.866.618.666.8518.697.626.624.735.176.704.767.076.445.066.577.247.456.075.846.535.50
696.805.356.817.545.848.215.975.934.804.756.655.896.225.855.975.6313.695.298.356.044.123.88
705.254.145.258.204.0611.643.545.273.723.315.223.833.184.073.372.835.962.755.513.372.392.30
7121.2614.3821.2615.6920.5710.7520.0521.4312.0214.6121.2315.8215.3515.929.5914.8010.8917.5816.7112.4810.958.92
725.314.995.318.695.4916.454.785.086.765.015.305.325.175.415.184.979.325.185.995.044.804.92
7347.1635.8147.1654.0783.4320.5982.0847.1824.0835.7347.2136.6837.8438.6321.3634.7728.0838.9533.4023.6626.8719.63
748.542.838.5411.472.8126.767.017.388.632.718.474.767.552.904.587.0325.415.508.344.543.712.65
7528.2824.0928.2824.2925.6418.1426.2330.7217.7724.0228.3024.8225.4325.7917.1322.7238.3427.7725.8618.7515.0218.06
7617.9112.6717.9112.1517.5911.6116.5515.8510.6912.7217.8712.2613.6912.6310.7214.2824.6413.0313.3112.4312.6812.26
7792.2860.8592.28714.91630.01454.94855.59113.3056.4756.8292.4976.4866.92356.62102.8689.8585.24669.0869.05116.9471.3681.99
7829.6014.4629.6024.8029.5813.3529.6024.6623.8412.1429.3317.3226.7121.8712.7121.6612.7021.2337.5321.3115.4010.73
7910.307.2310.308.569.009.019.9210.456.937.1910.288.4610.198.447.608.9024.799.919.937.517.107.11
8027.7111.0427.7118.4827.524.5326.3224.2610.2510.2427.7310.6910.2411.887.6710.3218.1912.319.708.625.554.49
8113.848.2213.859.918.1316.128.1911.455.826.9813.687.649.447.395.799.885.688.2812.398.727.905.62
828.857.728.858.4416.915.7316.778.854.397.738.858.296.728.217.506.488.468.215.897.537.427.00
8316.667.4716.6610.8519.323.8518.3014.795.007.4016.667.548.009.143.427.9319.149.448.114.583.532.96
8421.9020.2921.9027.9730.7023.7431.6514.7416.5121.2421.8921.1321.9928.179.3215.1214.5729.8524.1917.9310.5612.75
859.546.609.547.469.6820.2610.179.5613.195.409.615.318.116.704.225.814.298.615.844.245.014.06
8619.7412.4719.7414.0917.0211.6017.0919.4810.0513.0219.7313.2915.4914.2510.6015.5425.1414.5014.5211.2011.059.87
873.703.573.704.614.638.755.383.693.773.693.693.803.704.623.643.723.694.723.723.623.543.68
886.686.076.687.177.059.387.366.706.276.316.696.456.417.404.466.184.217.536.295.515.784.52
896.573.936.577.266.773.487.435.485.593.336.563.516.616.243.465.473.567.544.243.672.682.55
907.567.557.567.758.036.528.157.531.381.477.641.457.997.271.452.372.217.123.217.577.951.22
914.834.734.835.575.113.575.544.921.602.404.732.803.424.462.693.393.435.563.294.173.892.11
928.134.458.138.498.505.118.688.168.233.878.146.708.218.616.337.786.088.718.076.954.633.54
93112.3186.65112.31122.17114.93119.25114.5799.13106.4687.74112.3187.10108.04114.7471.7697.9240.72115.27119.7987.1187.2779.83
948.456.148.468.478.646.107.908.476.345.918.398.348.618.096.408.336.708.189.097.635.835.48
9511.233.9811.2314.035.7834.2411.5210.7010.804.0811.287.2311.155.456.7210.016.148.5810.266.945.053.92
960.930.930.936.893.436.245.090.957.020.920.920.944.623.070.930.951.794.380.880.952.000.85
977.196.947.197.997.615.568.106.863.496.137.206.036.357.342.423.822.808.164.972.452.482.43
986.265.286.268.7012.0517.4611.366.395.895.056.266.066.434.945.386.434.975.506.435.534.554.62
9914.1613.4414.1613.5317.914.8719.8212.566.4113.2614.1713.028.3210.686.156.415.8310.7710.896.866.754.96
10022.5318.1622.5218.7120.4422.8721.8922.2826.3810.7522.8410.8526.1118.8010.1619.9127.6721.1317.7117.0520.608.63
10169.5445.5069.5481.04122.6726.85120.7769.5330.1945.4469.6149.0652.6654.4525.6647.1838.4955.0547.6328.2732.7424.10
10210.454.9310.4511.3213.947.7614.6210.796.344.5910.439.486.2911.606.966.397.9013.047.899.612.852.72
10331.2031.3831.2131.6462.676.8361.6729.357.3628.2031.1228.8916.4812.2727.7216.6928.0212.1911.8427.7227.7326.54
10424.3716.4324.3721.0225.9714.8325.6324.5615.2816.3524.3718.8520.9020.8512.0119.0023.9421.3621.2213.1411.5712.03
10510.968.6010.9623.9816.1336.6017.0610.559.678.1810.937.899.1613.437.889.167.9113.3610.867.897.818.37
10610.589.6710.5710.8911.3413.2911.2410.5613.639.1510.6110.2411.3110.7910.1310.4510.5011.109.4810.459.829.14
10710.255.1510.266.7612.947.6111.897.455.855.1610.245.396.895.775.366.1911.325.815.975.405.235.16
10819.8619.7719.8621.7620.0921.0421.2320.6313.3919.7719.8319.6519.3319.9513.1115.9015.3221.4118.8417.5415.6312.57
10915.788.2815.789.8910.8113.9711.6515.326.878.1315.7310.0912.049.454.4011.424.0011.3212.196.283.713.02
11010.8010.3110.8013.2010.6615.7611.1310.5411.7710.2110.7910.8210.6510.7310.4910.1916.3010.1812.5810.459.619.27
11117.5310.6717.5327.7321.5730.2322.8416.3115.4211.5717.5011.7913.8518.3811.1113.989.3118.3816.4810.8611.3511.39
11216.348.0616.349.5014.2711.6213.5513.5012.068.1916.328.5912.979.398.0211.638.289.2012.168.608.137.73
11325.928.7325.9220.2427.148.0825.9320.497.778.6725.9016.319.048.8515.378.999.748.849.3516.507.366.49
1148.974.328.9711.265.9712.236.048.855.974.298.974.304.485.543.214.283.835.003.543.593.823.00
11553.1340.6953.1366.3674.0446.8486.9551.9440.3134.3553.2252.9652.8655.8748.5849.3428.2359.1755.6752.7444.2620.88
11610.818.4010.8122.3711.8452.8613.3611.2111.358.5310.8111.9010.439.6711.6910.6310.4710.649.7311.866.228.09
11717.6112.9517.6117.6317.377.2517.3917.547.2012.9517.6112.999.4116.8211.367.3717.5116.567.648.157.677.68
1187.116.307.116.6010.0610.3110.296.945.855.107.096.306.686.532.696.395.426.866.416.015.513.46
1198.658.338.658.358.805.738.596.716.516.918.636.898.248.305.215.706.888.216.735.535.375.29
1206.896.696.8915.606.8535.967.304.937.184.456.814.576.456.934.804.514.227.407.984.614.083.96
12120.959.3220.9413.2613.2422.3614.3819.7918.407.6121.1311.9121.9912.5911.5318.8943.6014.1014.6712.6812.206.54
12210.958.4910.9515.8711.5323.7113.828.976.208.3510.9510.8410.9210.388.389.317.9912.519.868.386.186.25
12358.092.9758.0937.8368.6110.9068.5047.777.323.4258.0911.6830.7320.306.6024.6042.7720.3723.4710.794.312.98
12419.3814.8619.3821.5016.4420.9119.1119.2518.7515.2319.5119.8722.0616.3418.2719.8417.0718.4919.5018.3313.9613.53
12515.009.6415.0011.2617.9612.7216.6011.6211.0210.0314.989.9910.689.6010.089.5114.959.5810.529.659.549.98
126114.2319.48114.2398.66131.3516.82131.2455.5516.7519.82114.2224.3740.1625.3022.6936.9067.1925.2619.9222.5418.7519.27
1275.334.085.339.664.4612.185.505.057.904.025.404.956.324.544.824.884.815.504.254.954.293.22
12818.773.9018.774.738.309.757.1913.224.493.4118.763.5810.184.623.367.5736.504.807.754.113.563.18
1298.434.788.437.9410.218.2910.547.944.042.378.446.184.707.486.294.829.908.134.977.123.972.29
13035.0024.1335.0027.3733.8724.0532.0833.0626.2125.7434.9525.6925.0524.5822.4927.7744.1324.8726.2124.9723.6421.68
1316.913.616.913.895.3310.634.435.425.963.746.743.823.853.553.493.8918.984.276.983.623.423.37
13224.6022.3824.6022.2326.0224.6125.6824.5123.8819.6224.7320.0421.8122.4020.8019.9919.0723.0618.9621.3121.1019.39
1335.734.295.735.986.455.846.225.724.954.205.725.715.355.985.734.895.396.065.745.783.202.93
1341.511.321.511.2911.811.5211.651.531.150.751.510.771.391.290.831.000.671.411.3522.590.600.56
13527.1022.9627.1022.2627.8913.8127.1425.6011.6823.6927.0523.9320.9820.0525.5722.1845.2021.3919.9312.2422.8925.49
13618.9814.0518.9813.7418.6211.3318.0318.889.0814.2818.9614.1213.8913.8210.1012.4310.3415.1113.494.219.468.99
1375.584.015.584.795.569.604.474.305.964.025.584.364.964.094.124.2810.543.795.136.423.822.96
13811.248.1711.2412.209.9842.349.026.7016.238.2111.258.159.739.116.386.0013.778.807.6312.274.926.19
13928.7716.4528.7718.7515.7915.7512.0123.5927.0814.1128.6114.4226.3513.9411.7410.9618.3112.5517.525.8411.6910.17
14011.745.8511.756.7913.636.1412.358.935.025.1011.675.157.146.333.727.2413.256.387.4714.855.213.82
14120.8815.5520.8916.1317.7314.0716.0515.1714.439.3020.639.9615.5616.139.6214.8010.4015.2819.487.1812.868.07
14212.0111.0612.0111.4314.1412.1614.4512.129.2210.3012.0610.2311.4310.917.188.037.0412.7411.255.356.496.54
1438.374.578.375.668.864.037.916.885.063.098.344.516.565.694.146.379.905.477.443.823.853.19
14410.722.8910.7223.7917.6121.1819.546.209.022.9410.713.727.1110.813.475.5910.0810.855.683.741.751.39
1456.533.836.534.215.4310.725.315.454.943.816.544.085.454.253.673.843.464.144.3434.973.773.52
14667.4467.5067.4487.83114.1429.99135.3736.1844.0966.2967.5765.4056.4053.5024.3127.5645.4948.5677.1319.3725.2327.36
14727.6021.4127.6021.5031.6015.0035.2525.3916.0221.1627.6021.2817.5719.0114.1118.8319.6219.6126.2814.4917.6815.24
14819.0716.1119.0717.0818.3411.5421.4719.1410.0016.3619.0716.3613.9717.7011.1513.1511.7719.9717.141.0512.1610.69
1492.601.912.604.452.872.524.461.173.211.182.831.174.672.751.040.930.874.341.877.680.860.86
1509.307.389.3018.7714.6718.2316.557.328.457.389.288.088.9214.116.486.657.2915.6910.148.704.805.00
1519.187.799.1819.4718.8824.7320.809.039.117.679.179.388.2718.178.447.559.2120.029.6813.598.256.67
15233.2829.3233.2833.0637.4113.0237.7929.9713.4229.6633.2931.3329.2232.5012.8719.6634.8232.8427.339.0014.9311.62
15315.098.8515.0913.4123.316.2923.1111.9410.508.8315.099.1711.018.768.437.8012.638.928.423.776.644.99
1545.923.665.926.625.1116.064.714.326.062.855.863.074.073.323.524.1011.773.715.233.583.122.89
1553.993.403.995.974.9611.374.033.545.022.853.952.943.193.383.123.556.403.354.357.242.972.90
1569.937.319.937.8012.684.8412.178.335.286.199.936.398.427.236.028.096.777.274.813.007.595.61
1575.233.835.238.824.288.255.683.083.873.835.243.744.503.362.542.922.523.733.7910.852.832.59
15818.718.6318.7118.5022.249.0328.8312.679.308.6518.7012.1317.169.217.9211.0411.789.8910.082.418.478.07
1597.353.687.354.2515.088.9314.182.975.433.707.354.235.474.022.133.124.863.812.7116.322.802.05
16038.8823.0038.8828.6841.2714.4040.3336.8211.0422.3038.8822.6324.4222.4316.4924.0538.4922.7619.199.1115.5715.95
16110.613.7610.6110.7316.7813.2817.6010.585.423.6910.619.076.989.809.137.108.8010.158.306.054.973.67
1626.145.696.1510.106.1915.756.986.096.545.706.115.995.905.996.045.7512.965.956.617.555.615.54
16316.387.4316.388.7514.2110.2513.2114.307.207.4216.348.3110.928.786.5911.0825.159.3210.896.146.676.56
16418.985.2018.985.816.0915.706.7615.7911.905.2318.975.9012.735.245.8810.347.635.998.174.636.235.38
16511.002.9911.0027.1220.5523.6122.816.838.873.0511.004.447.1712.223.766.113.3711.886.8111.672.632.39
16612.1210.8112.1311.6013.2113.1612.3211.6613.1410.7512.0711.3511.0011.7711.2011.2511.3911.7411.571.9510.419.51
1672.031.162.034.7810.8110.3310.212.045.471.172.031.982.003.151.182.031.693.211.8513.851.191.10
16815.8510.7115.8513.1119.3621.4418.8815.7612.2110.8515.8413.8910.1313.0413.7111.3120.5014.2711.938.859.1910.18
16933.628.6433.6224.0144.6410.9343.7724.9714.088.3833.618.2118.0912.499.1916.3631.3812.4513.763.977.677.32
1705.274.275.274.586.785.345.713.943.373.055.203.234.594.443.243.664.254.045.2813.602.752.54
17117.8116.0317.8116.3617.3015.3318.0917.8015.1714.3017.9514.0416.8116.5013.6014.4214.1417.8414.402.8713.8611.27
1724.294.034.295.6511.056.6111.373.783.323.794.304.053.725.632.732.913.026.253.763.403.092.84
1733.953.493.955.485.738.034.513.313.383.663.953.684.033.363.423.276.573.324.104.483.423.13
1748.954.598.954.8712.1511.5110.617.546.254.528.944.495.234.713.835.1615.155.435.5830.383.783.73
17557.1819.6057.1838.2844.1322.2638.1142.7438.0417.9756.8329.7551.8136.2625.2629.6612.3031.4055.7735.2919.0319.18
17666.4527.0466.4637.0242.4729.8743.7055.1536.8126.6465.8633.8639.7631.6124.7122.9119.8128.4848.3113.2221.0417.40
17724.8915.8324.8917.4424.389.5323.1023.167.4715.8224.8415.7414.5116.3110.7314.0528.9917.2715.174.1911.299.58
17811.823.4111.824.127.239.595.627.534.592.8811.803.937.233.884.265.4119.623.696.9526.293.152.99
17941.2430.4041.2445.0851.8830.9753.7630.3630.1928.5941.2539.9531.2341.8323.8020.3232.0644.2931.3813.9914.1817.33
18017.4913.5417.4917.6916.2044.0815.6114.3914.3412.1117.4413.1213.5515.5412.6413.6322.6715.1314.8214.3614.2411.81
18120.4719.8420.4626.3722.6640.3225.4720.3916.8520.3820.5119.0719.4722.6914.4817.5514.9125.8019.084.0811.6014.44
1824.093.054.0910.9811.2315.9111.893.954.553.104.094.193.959.003.823.803.899.266.047.444.912.42
18312.176.5812.179.135.8514.535.7210.4410.346.4512.178.3411.155.867.149.948.705.728.5211.196.576.48
18412.019.5012.0112.6210.7214.0811.1112.0911.439.0612.0011.2611.9610.7211.0912.1211.1011.0812.024.059.288.87
1856.804.056.805.528.6212.888.006.223.253.916.793.924.314.893.024.6314.085.454.765.123.652.73
18623.813.5923.8117.0235.832.7235.3215.275.694.0523.805.5512.725.775.1111.9920.265.823.657.003.423.77
1878.653.718.659.733.8618.775.249.127.693.188.687.208.893.976.928.846.365.039.833.573.823.44
1887.722.757.7216.6311.3516.4012.715.407.692.697.713.556.059.073.435.012.749.254.313.611.781.95
1895.193.615.193.646.725.776.454.313.703.615.193.643.863.643.413.836.153.693.879.593.633.67
19011.806.5711.8012.766.3423.069.1311.319.386.3811.719.7311.116.378.9811.297.628.2712.958.415.925.98
19113.3812.0113.3713.5013.797.1914.8011.8420.048.3913.539.8418.7913.367.6111.419.8814.1913.759.229.685.35
1929.748.449.7510.838.5412.759.329.7012.697.429.528.849.198.556.749.4613.118.4210.9719.766.706.40
19324.4412.0324.4474.0748.02152.9254.0719.1727.7412.2424.4719.6722.1231.7220.5419.4318.0135.3518.2478.6712.2410.86
194183.6433.67183.6496.44113.4632.24114.62170.7222.2839.08183.6479.50132.3478.2741.57132.97297.0778.4486.6746.8714.2828.91
195159.8021.42159.8080.7197.1718.8296.91145.8121.7224.47159.8055.34103.1455.5821.28103.87236.2355.8654.3233.4413.3916.15
19634.9131.7234.9154.2562.0223.7777.2133.7523.1530.8134.7535.0130.0149.4523.5025.1128.3853.1843.963.3724.3520.97
1973.602.903.605.528.388.338.813.5910.542.303.592.742.035.012.832.991.595.412.904.613.632.13
1986.754.486.756.048.983.478.226.184.413.756.754.434.873.504.433.834.083.543.285.263.173.21
1995.335.345.335.395.884.555.715.274.195.325.335.335.445.364.244.824.715.535.0923.634.613.28
20027.2123.0227.2126.9928.2312.1728.2427.2322.8913.8827.2414.5923.3124.5214.5022.4121.1725.5522.1118.3522.8812.94
20118.5418.3618.5420.1919.6730.8419.6818.5517.3012.7518.5113.1518.1018.9017.9318.2214.3318.7118.153.7018.2312.73
2025.414.535.416.126.6611.236.575.407.914.485.424.715.104.503.704.313.685.144.815.343.603.58
2037.185.837.186.488.9512.157.065.373.565.827.195.886.835.784.715.174.795.446.059.445.153.94
20411.308.9511.3018.8713.9123.1915.1010.7312.178.8611.309.1010.4013.839.0710.019.4714.6811.815.649.229.28
2055.755.395.758.856.2011.365.765.786.715.815.825.866.105.875.355.405.275.746.255.035.605.27
2066.804.636.808.284.385.015.125.152.564.526.814.775.824.063.435.294.374.246.4744.534.763.44
20750.2149.8650.2151.1651.5810.3453.0547.5755.9110.4450.6310.1553.5051.167.6144.9712.4052.4044.255.9644.557.86
20813.859.4013.859.3118.2226.1419.0913.338.269.4013.878.739.999.304.908.406.2710.867.9410.985.615.96
20912.8111.0312.8013.3812.903.4013.6810.9313.443.7313.044.5114.1512.524.5610.634.5213.727.755.4811.923.77
2106.715.966.719.0510.6213.0310.145.455.325.956.716.566.548.125.285.456.277.907.260.835.774.24
Min.0.930.930.931.292.290.642.170.691.150.700.920.701.391.290.320.740.391.410.880.590.600.17
Max.183.6486.65183.64714.91630.01454.94855.59170.72106.4687.74183.6493.44132.34356.62102.86132.97297.07669.08119.79116.9487.2781.99
Overall AARD18.8011.9418.8022.2723.7419.9425.7916.9612.6310.8718.7912.6915.0216.1110.6713.5215.3718.9014.5312.5610.398.88

The new model was compared with the 21 most commonly used semiempirical models in terms of AARD (%) and the results are presented in Table 5, in which bold values represent the best performance for a group of models with the same number of adjustable parameters. Among the three parameter models, Garlapati and Madras [74] (Eq. 20) and Chrastil [65] (Eq. 10) models were found to correlate better with an average AARD of 18.79%, whereas Jafari Nedjad et al. [76] model (Eq. 22) was found to be better among four parameter models with an average AARD of 15.02%. Comparing five parameter models, Adachi and Lu [66] correlation (Eq. 11) gave the best global AARD of 11.94%. For the set of six parameter models, Si-Moussa et al. [57] model (Eq. 30) shows a slight advantage over Sparks et al. [73] (Eq. 19) and Bian et al. [59] (Eq. 24) models and a clear superiority over Jouyban et al. [58] (Eq. 18) and Gordillo et al. [70] (Eq. 15) models with an average AARD of 10.39%. To summarize, it is clear from Table 5 that the more parameters the model has the more accurate correlation performance are, except the nine-parameter model of Amooey [60] (Eq. 26) with a global AARD of 15.37%. The eight-parameter model proposed in this work shows the smallest AARD (8.88%) among all of the models considered. These findings are further illustrated in Fig. 1, which represent the correlations that best correlated 90% of systems considered. Clearly, the model proposed in this work is superior to the other models as it best correlates 87 (41%) systems of the 210 considered. The second best is the five-parameter model of Bian et al. [64] (Eq. 29) with 28 (13%) systems.

Table 5

Comparison of overall AARD according to the number of model parameters.

Number of parametersModel equationOverall AARD %Number of parametersModel equationOverall AARD %
3(10)18.805(21)12.69
3(13)22.275(25)13.52
3(14)23.745(28)14.53
3(16)25.795(29)12.56
3(20)18.796(15)19.94
4(12)18.806(18)12.63
4(17)16.966(19)10.87
4(22)15.026(24)10.67
4(23)16.116(30)10.39
4(27)18.908(32)8.88
5(11)11.949(26)15.37
Fig. 1

The number of systems best correlated by each model equation.

It should be mentioned that models that include the pressure as variable tend to give a relatively poor correlation. This is well demonstrated by the correlation results of the same data set of Jouyban et al. [58] model (Eq. 18) (12.63%) compared to Si-Moussa et al. [57] model (Eq. 30) (AARD = 10.39%). A more illustrative example, which supports this evidence, is the comparison of Gordillo et al. [70] (Eq. 15) and its modification (Eq. 31), not included in the present study, where there is a net difference between the correlation performance of the two equations (AARD of 19.94% and 11.09%, respectively).

Fig. 2 displays the scatter plot of the entire data set of the solid solute solubility calculated using the model proposed in this work (Eq. 32) versus experimental solubility (5550 experimental points). These plots are generated using the postreg function of MATLAB, trace the calculated solubility as a function of experimental solubility.

Fig. 2

Scatter plot of the entire dataset of solid solutes solubility calculated using the model proposed in this work (Eq. 32) versus experimental solubility.

In Fig. 2, the first bisector shows the exact fit between the correlated solubilities and experimental data, whereas the cross points demonstrated the real correlated solubility data by the proposed correlation versus experimental data. In detail, the closer the points to the solid line, the more accurate correlated solubility data means. On the basis of the obtained results, it can be concluded that the proposed correlation was able to correlate the solubility of solids in scCO2 with an acceptable deviation.

Figs. 3 and 4 show comparisons between calculated and experimental solubility for the best (menadione) and worst (corosolic acid) correlation results.

Fig. 3

Mole fraction solubility (log10(y2)) of menadione in scCO2 as a function of carbon dioxide density and temperature.

Fig. 4

Mole fraction solubility (log10(y2)) of corosolic acid in scCO2 as a function of carbon dioxide density and temperature.

4 Conclusions

The comparative study of semiempirical models on a fairly large database carried out in this work shows that the solubility data of solid solutes in scCO2 are better correlated by models that relate the logarithm of the solubility in the mole fraction to the density of solvent and temperature with more than five adjustable parameters. The model must include a term of the logarithm of the density, a polynomial of the density, and a term that accounts for the effect of temperature. As to the data set considered, semiempirical models correlate quite satisfactorily the data within the AARD range of 0.17–56.47% and an average value of 6.87%. However, when comparing the global correlating performance of each model, the proposed model has by far the best correlating performance with an AARD lying in the range of 0.17–81.99% and an average value of 8.88%. The second best correlation is (Eq. 30) [30] where the AARD ranges in 0.60–87.27% and an average value of 10.39%.

Acknowledgments

The authors gratefully acknowledge the Algerian Ministry of Higher Education and Scientific Research (CNEPRU Projects Nos. J0102620110007 and J0102620140015) and the University Yahia Fares of Medea.


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