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The tunable nature of the solubility of various compounds, including molecules of pharmaceutical and biological interest, in supercritical fluids (SCFs) makes SCF extraction technology attractive for many separation and purification processes. Among the different influencing parameters, the most important one in the supercritical based processes is the knowledge of solubility of model solute. But, experimental measurement of the solubility of all pharmaceuticals in wide ranges of temperature and pressure is not only cost effective but also impossible in some cases. Regarding this fact, during the past decades, several approaches are proposed to model the solubility of the compounds in the supercritical fluids especially carbon dioxide. In this way, in the current investigation, two different approaches including five semi-empirical density based correlations (Mendez-Santiago and Teja (MST), Bartle et al ., Chrastil, Kumar and Johnston (KJ) and Hezave et al .) and Peng-Robinson equation of state are used to find if it is possible to correlate the solubility of cetirizine with acceptable deviation as a function of temperature and pressure. The results reveal that among the examined approaches Hezave and Lashkarbolooki model leads to better overall correlative capability with average absolute relative deviation of 5.04% although Peng-Robinson EoS leads to lower AARD % of 3.85 % in 338 K isotherm.

Supercritical fluid (SCF) technologies have been gaining an increasing attention in different chemical processes during the past decades due to its applicable advantages. In this way, supercritical fluid (SCF) technology has been continuously developed for the processing of food, pharmaceuticals, and polymeric and specialty chemicals [

SCFs are advantageous since they introduce desired properties of both liquids (good solvating strength) and gases (good diffusivity) make them a good alternative for conventional solvents. Among the different possible solvents for supercritical fluid technologies, carbon dioxide is most commonly used due to its mild critical pressure and temperature which make it a good candidate for processing of thermal labile compounds. In addition, carbon dioxide is non-toxic, green, available, cheap and non-toxic solvent. Also, since carbon dioxide introduces a mild critical condition, after expansion which occurs mostly into the ambient condition, it is vaporized from the matrix of the process material which remains no contamination. Due to this unique advantages, one of the most common applications of SCF technologies can be considered in pharmaceutical engineering for different applications such as drug delivery [

Among the different parameters affecting the efficiency of the supercritical based technologies, solubility of pharmaceutical compounds is the major criterion for choosing appropriate process. In this way, several literatures reported the experimental measured solubility of different compounds as a function of temperature and pressure [

As a way out, during the past decades, different predictive approaches are proposed by different researchers to correlate the solubility of different compounds as a function of temperature and pressure as simply as possible [

Due to several limitations of EoS models, researchers proposed a new generation of predictive methods called semi-empirical density based correlations which used only the density of pure SCF and operational temperatures and pressures [_{2}-solutes using these semi-empirical equations. For example, Hojjati et al. [

Regarding these progresses, it seems possible to use predictive tools for correlating solubility data and reduce the experimental measurement which consequently decreases the operational costs. In this way, in the current investigation, two different approaches including equation of sate (Peng-Robinson model) and semi-empirical density based correlations (including Mendez-Santiago and Teja (MST), Bartle et al., Chrastil, Kumar and Johnston (KJ) and Hezave and Lashkarbolooki) are used to compare their capability for correlating the solubility of cetirizine in supercritical carbon dioxide in wide range of temperature (308 - 338 K) and pressure (160 - 400 bar).

The measured solubility data were correlated using four semi-empirical density-based correlations namely MST, KJ, Bartle et al. and Chrastil models. A brief description of the used correlations is given as follows. In the first stage, the Bartle et al. model [

where a, b and c are fitting parameters, P^{ref}, y and ρ are the reference pressure of 1 bar, the solute solubility and the density of carbon dioxide at a specific pressure and temperature modified by subtracting 700 kg∙m^{−3} considering as the reference density, ρ_{ref}, respectively.

The second utilized model was Mendez Santiago and Teja (MST) which is one of the most popular one due to its accuracy and simplicity. In brief, MST has presented an empirical model based on the theory of infinitely dilute solutions [

where T is the temperature, y_{2 }is the solubility of the compound in terms of mole fraction, p is the pressure,

Since the sublimation pressures are not often available, the proposed model was combined with a Clausius-Clapeyron type expression for the sublimation pressure and a semi-empirical relation, with three adjustable parameters, for the solid solubility was derived:

The adjustable parameters of the MST correlation including a, b, and c obtained by performing a simple graphical data fitting to

Finally, solubility data were correlated using Kumar and Johnston method (KJ) and Chrastil model (see Equations (4) and (5)).

where y is the solubility of the solid, a, b and c are the fitting parameters and

where, a, b and c are the fitting parameters and S is the solute solubility in_{sublimation}= −Rb).

Among the above mentioned models, Chrastil method which is one of the oldest semi-empirical density-based correlations proposed in 1980s. This correlation was based on this assumption that the solute molecules surrounded by c molecules of a solvent form a solute-solvent complex. In other words, in the Chrastil model (Equation (5)), the fitting parameter c indicates the number of solvent molecules surrounding the solute molecule.

Finally, the correlation proposed by the authors [

The proposed correlation was considered as a function of temperature, pressure and density of pure SC-CO_{2} (see Equation (6)) to not only into account the nature of the involved compounds but also to account the effects of system pressure and temperature on the solubility of the pharmaceutical in SC-CO_{2} [

The optimization of the coefficient of the proposed correlation was obtained using a genetic algorithm. In the recent years, several computer-based methods for simulating and modeling of the different chemical phenomena have been proposed and examined [

As aforementioned, besides the semi-empirical correlations, Peng-Robinson EoS was used to model the measured solubility data. Regarding this purpose, the Peng-Robinson EoS coupled with conventional van der Waals mixing rule was used to evaluate the fugacity coefficient of solid in compressed fluid phase.

The solubility of a pure solid (component 2) in a supercritical fluid was estimated using the classical expression:

where, _{2}

In addition, the l_{ij} and k_{ij} parameters were optimized using a differential evolution technique using MATLAB software. Differential evolution (DE) is a method optimizes a problem by iteratively trying to improve a candidate solution with regard to a given measure of quality. In brief, differential evolution is capable of handling non-differentiable, nonlinear and multimodal objective functions. It has been also used to train neural networks having real and constrained integer weights. This kind of solver works iteratively in which at each iteration, called a generation, new vectors are generated by the combination of vectors randomly chosen from the current population (mutation). The resultant vectors are then mixed with a predetermined target vector. This operation is called recombination and produces the trial vector. Finally, the trial vector is accepted for the next generation if and only if it yields a reduction in the value of the objective function. This last operator is referred to as a selection. This procedure continued until the desired error obtained.

In the current investigation, the solubility of cetirizine measured by the co-authors is gathered from the previously published literature [

In the first stage of the current investigation, the solubility of cetirizine was correlated using Peng-Robinson EoS using VDW1 mixing rules as previously described. The two adjustable parameters (k_{ij} and l_{ij}) were optimized using the DE method using the following objective function:

where _{2} such as critical pressure (17.50 bar) and temperature (1025.1 K) were estimated using Joback and Reid method [

A closer examination in the listed results in

A closer examination of the listed results in

Also, comparing the results obtained for Peng-Robinson EoS using two different mixing rules including VDW1 (see

In the second stage, five different semi-empirical density based correlations were used to find if it is possible to correlate the solubility if cetirizine with better accuracy compared with the results obtained by Peng-Robin- son EoS correlative approach (see

Besides the accurate correlative capability of the examined semi-empirical correlations, it is possible to extrapolate the solubility of cetirizine since the self-consistency test of these correlations leads to a straight line which means it is extrapolative capability of these correlations.

In details, as demonstrated in

Temperature (K) | k_{ij} | l_{ij} | AARD (%) |
---|---|---|---|

308 | 0.196833 | 0.099997881 | 50.78 |

318 | 0.152875 | 0.099999220 | 46.50 |

328 | 0.104759 | 0.099998463 | 33.76 |

338 | 0.051526 | 0.099997864 | 3.86 |

Temperature (K) | k_{ij} | l_{ij} | AARD (%) |
---|---|---|---|

308 | −0.037 | −0.1 | 29.3 |

318 | −0.091 | −0.1 | 23.6 |

328 | −0.158 | −0.1 | 9.25 |

338 | −0.233 | −0.1 | 10.15 |

Model | Correlation | Constant | |||
---|---|---|---|---|---|

a | b | c | d | ||

Chrastil | −80.71 | −13,330.5 | 18.18 | ----- | |

Kumar and Johnstone | 15.40 | −13,276 | 0.93 | ------ | |

Mendez-Santiago-Teja | −22,696 | 46.09 | 8.34 | ------ | |

Bartle | 42.6 | −15,695.36 | 0.026 | ------ | |

Hezave and Lashakrbolooki | 43.3262 | −15,986.08 | 0.0266 | 5.44 × 10^{−9} |

This observed trend can probably relate to the following fact that at the higher pressures which the density of compressed CO_{2} amounts to a value of 972.3 kg∙m^{−3} which is not so far away from water at ambient conditions can lead to a phenomenon called “squeezing out”. In other words, at higher pressure and liquid-like densities, the effect of “squeezing out”, that is a retrograde solubility, can often be observed leads to this fact that the density-based models do no longer work willingly in that regions. This phenomenon previously was reported by Kurnik and Reid [

Finally, based on the obtained results it can be concluded that among the examined approaches, semi-empir- ical density based correlations are well able to satisfactorily correlate the solubility of cetirizine. In addition, among the examined semi-empirical density based correlations correlation proposed by Hezave and Lashakrbolooki correlate the solubility of cetirizine with acceptable level of accuracy with AARD % of 5.03%. At last, the worth mentioning point is that the semi-empirical density based correlations are not only able to well correlate the solubility of cetirizine in wide range of temperature and pressure but also they are able to extrapolate the solubility of compounds in supercritical carbon dioxide only as a function of temperature, pressure ans density of supercritical carbon dioxide which introduce them as an efficient tool for solubility interpolation and extrapolation.

In the current investigation, two different approaches were used and compared to find if it was possible to correlate the solubility of cetirizine as a function of temperature and pressure. In this way, one EoS namely Peng-Ro- binson Eos and five semi empirical density based correlations namely Mendez-Santiago and Teja (MST), Bartle et al., Chrastil, Kumar and Johnston (KJ) and Hezave and Lashkarbolooki were used and the obtained results were compared. The obtained result revealed that among the different predictive methods the semi-empirical density based correlations not only were easier to be used but also led to more accurate results compared with the Peng- Robinson EoS. In addition, the results of self-consistency test revealed that the semi-empirical density based correlations not only were able to correlate the solubility of cetirizine but also were able to extrapolate the solubility of the cetirizine. Finally, the results illustrated that among the different semi-empirical density based correlations, Hezave and Lashkarbolooki model led to more accurate results with minimum AARD % of 5.04%.

Finally, based on the obtained results one can conclude that although these predictive methods suffer from inaccuracies in some cases, it is possible to propose correlations which are able correlate the solubility of compounds in the supercritical carbon dioxide as a function of temperature and pressure with acceptable level of accuracy.

Mostafa Lashkarbolooki,Ali Zeinolabedini Hezave, (2015) Comparison between Modeling of Cetirizine Solubility Using Different Approaches: Semi-Empirical Density Based Correlations vs. Peng-Robinson EoS. Open Access Library Journal,02,1-16. doi: 10.4236/oalib.1101715