1 Introduction

In current years, the attention in green technology has led to the extension of a novel class of highly tunable and unique components that are called ionic liquids (IL). The ILs are the components that have revolutionized the corresponding researches and chemical industries. They are investigated as green chemical components that play a very significant role as a solvent to decrease applying of harmful, toxic and hazardous chemical components for the environment (Nishan et al., 2021). The main reason of increased considers on ILs is the aim of researchers to look for an appropriate alternative for volatile conventional organic solvents among the industries. Indeed, volatile conventional organic solvents are main environmental pollution resource in chemical industries (El shafiee et al., 2021). Although, all ILs are not completely considered as green solvents, some of ILs are even considerably toxic. Ionic liquids have involved a significant interest throughout the last three decades in the industrial and academic fields due to their unique properties (Agafonov et al., 2020; Omar et al., 2021). ILs are usually composed of inorganic or organic anions and an organic cation (Lian et al., 2016). According to cations, ILs are categorized to several categories like sulfonium, amino acids, phosphonium, guanidinium, ammonium, pyridazinium, piperazinium, oxazolidinium, tetrazolium, morpholinium, uranium, thiazolium, isoquinolinium, pyrroline, piperidinium, pyrrolidinium, pyridinium and imidazolium. Moreover, nitrate, perchlorate, bromide and chloride are simple anions and bis((trifluoromethyl)sulfonyl) imide, trifluoromethanesulfonate and lactate are complex anions (Bagheri et al., 2021; Evangelista et al., 2014; Xu et al., 2009). The ionic liquids properties will depend on the special combination of the anion and cation. The significant physicochemical features of ionic liquids like high electrochemical, thermal and chemical stability, non-flammability, low melting point and low vapor pressure (non-volatility) at moderate temperatures and pressures, high solvating power for non-polar and polar compounds have made ILs the exploration subject in many applications (Mohammadzadeh et al., 2020).

Designing equipment and chemical processes that involve ILs, needs accurate data on ILs thermo-physical features. The knowledge of ILs chemical and physical features is fundamental for ILs applications. Consequently, a number of scientists have investigated the thermodynamic properties of pure ILs and binary IL-mixture systems (Mesquita et al., 2019; Bagheri and Mohebbi, 2017). Until now, most investigations related to features of IL are limited to pure ones like viscosity, heat capacity, conductivity, surface tension and density. But, the usage of IL-mixture in engineering applications needs accurate detail about thermodynamic and physical features of ILs. The volumetric features refer to be related by substance volume change, such as expansibility, compressibility, excess molar volume, partial molar volume and density. ILs density is one of the most basic ILs volumetric features that is necessary in process design and calculations of metering of IL and it is interpreting interactions of intermolecular between solvent and ILs molecular (Tao et al., 2020). It is possible to calculate the ILs density accurately using experimental methods. However, as the experimental calculations are usually difficult, time consuming and costly consequently, the experimental calculations are not continuously possible (Tao et al., 2020; Bagheri et al., 2019; Paiva et al., 2019). Indeed, to develop process based on IL from the laboratory scale to industrial applications, the sufficient theoretical models to calculate ILs thermo-physical features must be considered (Bagheri and Ghader, 2017). In point of fact, the equipment design like liquid–liquid and vapor–liquid separation, energy and material balances including liquids, tower height calculations, rebuilders and condensers and storage vessels sizing, all need accurate liquid density values. Furthermore, other ILs features like surface tension or heat capacity, are sometimes correlated by density, so the reliable values of density permit one to calculate the mentioned features by an acceptable accuracy (Lian et al., 2016).

It is difficult to calculate features of all substances especially at various temperature and pressure or expensive and toxic components that maybe cause serious injuries for environment and human health. Thus, it is required to find out a method to calculate substances features. To solve the mentioned problem several estimation method was presented to estimate pure and IL-mixture. The presented methods are different from one case to another one and such correlations are based on some adjustable parameters, some of them are based on group contribution methods, equations of state and or machine learning methods (Abumandour et al., 2020; Yang et al., 2020; Bagheri et al., 2021).

Several semi-empirical equations were presented to calculate IL pure density such as Redlich-Kister polynomial equation, Lorentz-Lorenz equation, Tait equation and Wright equation and they have some fitting parameters (Lampreia et al., 2003; Valderrama and Zarricuetac, 2009; Geppert-Rybczyńska et al., 2010; I. Bahadur I, N. Deenadayalu, , 2011; Matkowska and Hofman, 2013; Bahadur et al., 2013; Govinda et al., 2013; Govinda et al., 2013; Singh et al., 2014; Huang et al., 2015). However, Hosseini et al. (Hosseini et al., 2013) presented the modified version of Spencer and Danner equation to predict IL-mixture density. The used data set was including 854 experimental data points and 14 binary mixtures. The obtained results indicated the performed modification had acceptable performance to predict IL-mixture density. Equations of state (EoS) and group contribution (GC) EoS are other sets of equations which were used to estimate the IL binary mixture (Abareshi et al., 2009; Wang et al., 2010; Li et al., 2011; Shen et al., 2011; Shahriari et al., 2012; Oliveira et al., 2012; Hosseini et al., 2012; Yousefi, 2012; Sheikhi-Kouhsar et al., 2015). EoSs are important tool to design in chemical engineering and supposed a developing role in the consideration of the fluid mixtures phase equilibria. However, the accuracy of some CEoSs is low to predict features of thermodynamic in a wide range of pressures and temperatures (Liu et al., 2020; Kamath et al., 2010; Ghoderao et al., 2019; Coquelet et al., 2016). The binary mixtures containing IL are complex systems, due to depend not only on molecular-molecular interaction, but also on the ion-molecular and ion-ion interactions. Cubic EoSs (CEoSs) consider only the attraction and repulsive interaction and they are unable to consider the other intermolecular forces (Panayiotou et al., 2019; Farrokh-Niae et al., 2008; Kukreja et al., 2021; Secuianu et al., 2008; Lopez-Echeverry et al., 2017). Therefore, the obtained results of IL-mixture density based on CEoSs have systematic deviation. Although some corrections like volume-translated to decrease systematic deviation was performed (Sheikhi-Kouhsar et al., 2015). The statistical associating fluid theory (SAFT) family EoS is according to perturbation theory. The SAFT family EoSs consider almost all intermolecular forces and presented the real form of intermolecular forces of components. However, computer programing of SAFT family EoSs are time consuming and need advance numerical methods (Perdomo et al., 2021; J. Gross J, G. Sadowski, , 2002; Bülow et al., 2021). Also, GC method required an accurate and perfect knowledge of the IL structure and it is time consuming and complex. Furthermore, the model fails and is considerably limited when applying large substructures as group parameters (Qiao et al., 2010; Peng et al., 2017; Chen et al., 2019). Machine learning method is another advantageous tool to predict properties pure and mixture IL and it is extensively accepted as estimation technique (Venkatraman and Alsberg, 2017). The relationship between the properties is nonlinear and the machine learning method is a significantly able algorithm to estimate certain properties like density by learning the relation between the input data (for instance temperature, pressure and critical features) and output data (like density) (Zhang et al., 2006; Yusuf et al., 2021). Also, the accuracy of the obtained results depends on the size of the training data set and the most important weakness of the machine learning method is low capability to predict the future performance of the network (generalization) (Low et al., 2020).

Density is one of the basic features of ionic liquid mixtures that would be useful in understanding intermolecular interactions between molecules of ionic liquid and solvent. The main purpose of this study is to develop a more accurate and reliable methods for estimating density of IL + solvent binary system. Consequently, the density of 130 various IL + solvent binary mixture (4626 data point) including 11 different cation-family (BuPy, C₂mim, C₃mim, C₄mim, C₆mim, C₈mim, Hmim, Mmim, Moim, OcPy and Omim) in the wide range of temperature, IL mole fraction and in the atmospheric pressure is modeled using two methods, i.e. cubic equations of state (Soave-Redlich-Kwong (SRK) EoS as two-parameter CEoS and Patel-Teja (PT) EoS as three-parameter CEoS) and six semi-empirical equations. The semi-empirical equations had fitting parameters and the they were obtained for each cation-family, separately. To obtain the fitting parameters of semi-empirical equations, the improved laplacian whale optimization algorithm (LXWOA) was applied.

2 Materials and methods

2.3

2.3 Semi-empirical equation

There is a great change of analytical expressions, which permit predicting and correlating the density of liquids. Such correlations are usually according to apply of adjustable parameters for each component (Geppert-Rybczyńska et al., 2010; Singh et al., 2014). But, the mentioned types of generalized correlations were not developed for IL-mixture systems. Several semi-empirical equations were presented to predict pure IL density. In this communication, through several presented semi-empirical equations, six semi-empirical equations were selected and the details are given as following (Elbro et al., 1991; Mchaweh et al., 2004; Gunn and Yamada, 1971; Hankinson and Thomson, 1979; Poling et al., 2001; Sandler, 2017; Kontogeorgis and Folas, 2009 Dec 1): $$\ln \frac{{M_{w}P}_c}{R\rho T}=\ln V^0+\omega \ln V^1$$ $$\ln V^0=a_0+a_1{T}_r+a_2{T}_r^2+a_3{T}_r^3+a_4{T}_r^4+a_5{T}_r^5+a_6{T}_r^6$$

(20)

$$\ln V^1=b_0+b_1{T}_r+b_2{T}_r^2+b_3{T}_r^3+b_4{T}_r^4+b_5{T}_r^5+b_6{T}_r^6$$ $$\rho =\frac{M_w}{V_c}\left(a_0+a_1{\tau }^{\frac{1}{3}}+a_2{\tau }^{\frac{2}{3}}+a_3\tau +a_4{\tau }^{\frac{4}{3}}\right)$$

(21)

$$\tau =1-\frac{T_r}{{\left[1+\left(0.48+1.574\omega -0.176{\omega }^2\right)\left(1-\sqrt{T_r}\right)\right]}^2}$$ $$\rho =\frac{P_c{M}_w}{{R{T}_{c}V}_1\left(1-\omega V_2\right)\left(a_0+a_1\omega \right)}$$ $$0.2<T_r<0.8:V_1=b_0+b_1{T}_r+b_2{T}_r^2+b_3{T}_r^3+b_4{T}_r^4$$ $$0.8<T_r<1:V_1=b_0+b_1\sqrt{{1-T}_r}\log \left({1-T}_r\right)+b_2\left({1-T}_r\right)++b_3{{(1-T}_r)}^2$$

(22)

$$V_2=c_0+c_1{T}_r+c_2{T}_r^2$$ $$\rho =\frac{M_w}{V_c\left[V^0+\left(1-\frac{\omega \left(b_0+b_1{T}_r+b_2{T}_r^2+b_3{T}_r^3\right)}{T_r+b_4}\right)\right]}$$

(23)

$$V^0=a_0+a_1{{(1-T}_r)}^{\frac{1}{3}}+a_2{{(1-T}_r)}^{\frac{2}{3}}+a_3{(1-T}_r)+a_4{{(1-T}_r)}^{\frac{4}{3}}$$

(24)

$$\rho =\left(\sum _i\frac{P_{\mathit{ci}}{M}_{\mathit{wi}}}{R{T_{\mathit{ci}}x}_i}\right){\left(a_0+a_1\omega +a_2{\omega }^2+a_3{\omega }^4\right)}^{-\left(1+{\left(1-T_r\right)}^{\frac{2}{7}}\right)}$$

Also, the Nasrifar-Moshfeghian (NM) equation was used to predict the density of binary IL + solvent. The equation is expressed by (Nasrifar and Moshfeghian, 1998; Rabari et al., 2014; Mathias and Copeman, 1983):

(25)

$$\rho =\frac{M_w}{V_c}{\rho }_0\left[1+\delta {\left(f(T_r)-1\right)}^{\frac{1}{3}}\right]$$

(26)

$${\rho }_0=1+1.1688{\left(1-\frac{T_r}{f(T_r)}\right)}^{\frac{1}{3}}+1.8177{\left(1-\frac{T_r}{f(T_r)}\right)}^{\frac{2}{3}}-2.65811.1688\left(1-\frac{T_r}{f\left(T_r\right)}\right)+2.1613{\left(1-\frac{T_r}{f(T_r)}\right)}^{\frac{4}{3}}$$ $$f\left(T_r\right)={\left[1+c_0\left(1-\sqrt{T_r}\right)+c_1{\left(1-\sqrt{T_r}\right)}^2+c_2{\left(1-\sqrt{T_r}\right)}^3\right]}^2,T_r<1$$

(27)

$$f\left(T_r\right)={\left[1+c_0\left(1-\sqrt{T_r}\right)\right]}^2,T_r>1$$

As mentioned, the presented semi-empirical equations, for first time, were used for pure liquids (except Eq. (24)). Subsequently, to develop them to IL-mixture systems the mixing rules must be applied. There are five thermodynamic properties in equations (20) - (27) i.e. acentric factor, critical temperature, critical pressure, critical volume and molecular weight. In this study, various mixing rules were considered. The details of all mixing rules and the final mixing rules that were applied, are provided in Table 3.

Table 3 The details of all mixing rules for thermodynamic properties (Poling et al., 2001; Sandler, 2017; Kontogeorgis and Folas, 2009 Dec 1; Nasrifar and Moshfeghian, 1998).

Property	Tested mixing rule	Final mixing rule
$M_w$	$M_w={\sum }_i{x}_i{M}_{\mathit{wi}}$	$M_w={\sum }_i{x}_i{M}_{\mathit{wi}}$
$T_c$	$T_c=\frac{T_{\mathit{ci}}+T_{\mathit{cj}}}{2}{T}_c={\sum }_i{x}_i{T}_{\mathit{ci}}$ $T_c={\sum }_i{\sum }_j{x}_i{x}_j{T}_{\mathit{cij}}$ , $T_{\mathit{cij}}=\sqrt{T_{\mathit{ci}}{T}_{\mathit{cj}}}$ $T_c=\frac{1}{{\sum }_i\frac{x_i}{T_{\mathit{ci}}}}$	$T_c={\sum }_i{\sum }_j{x}_i{x}_j{T}_{\mathit{cij}}$
$V_c$	$V_c={\left(\frac{\sqrt[3]{V_{c1}}+\sqrt[3]{V_{c2}}}{2}\right)}^3{V}_c={\sum }_i{x}_i{V}_{\mathit{ci}}$ $V_c={\sum }_i{\sum }_j{x}_i{x}_j{V}_{\mathit{cij}}$ , $V_{\mathit{cij}}={\left(\frac{\sqrt[3]{V_{c1}}+\sqrt[3]{V_{c2}}}{2}\right)}^3$	$V_c={\sum }_i{x}_i{V}_{\mathit{ci}}$
$P_c$	$P_c=\frac{P_{\mathit{ci}}+P_{\mathit{cj}}}{2}{P}_c={\sum }_i{x}_i{P}_{\mathit{ci}}{P}_c=\frac{1}{{\sum }_i\frac{x_i}{P_{\mathit{ci}}}}$ $P_c={\sum }_i{\sum }_j{x}_i{x}_j{P}_{\mathit{cij}}$ , $P_{\mathit{cij}}=\sqrt{P_{\mathit{ci}}{P}_{\mathit{cj}}}$ $P_c=\frac{Z_{\mathit{cij}}R{T}_{\mathit{cij}}}{V_{\mathit{cij}}}$ , $Z_{\mathit{cij}}=\frac{Z_{\mathit{ci}}+Z_{\mathit{cj}}}{2}$ , $T_{\mathit{cij}}=\sqrt{T_{\mathit{ci}}{T}_{\mathit{cj}}}$ , $V_{\mathit{cij}}={\left(\frac{\sqrt[3]{V_{c1}}+\sqrt[3]{V_{c2}}}{2}\right)}^3$	$P_c={\sum }_i{x}_i{P}_{\mathit{ci}}$
$\omega$	$\omega =\frac{{\omega }_i+{\omega }_j}{2}\omega =\sqrt{{\omega }_i{\omega }_j}\omega ={\sum }_i{x}_i{\omega }_i$	$\omega =\sqrt{{\omega }_i{\omega }_j}$

2.3.1

2.3.1 Pre-processing of the train and test

The AI learning approaches used to estimate the target parameters applying input parameters are widely nonlinear and depend mainly on the learning database (Memarzadeh et al., 2020; Ali et al., 2018; Nguyen-Huy et al., 2018). In cases where we are dealing with large-scale data processing, dimensionality reduction and subset selection methods can be necessary and useful tools in model development (Xu et al., 2019; Riahi-Madvar and Seifi, 2018). Important points that are essential in designing the precise prediction model for the AI model include the selection of appropriate input and data clustering techniques due to the diversity of the data set (Riahi-Madvar and Seifi, 2018; Loey et al., 2021; Mueller and Massaron, 2021). In order to achieve reliable data subsets for training and testing, the subset selection of maximum dissimilarity method (SSMD) was used to select training and testing subsets through a random manipulation of these data sets. Selecting the most appropriate train sets in the proposed preprocessing strategy is the most important task in SSMD-based approaches for data selection, which extremely depends on the expansion of the data (Riahi-Madvar and Seifi, 2018). In the SSMD method, data selection is not focused only on a specific area. Although these data are very different from each other, they can still show the statistical features of the original dataset (Lajiness and Watson, 2008). The data must first be normalized. Thus, the process of normalization and then de-normalization of the initial data set is performed by Eq. (28), respectively.

(28)

$$y_n=\frac{y_i-y_{\mathit{min}}}{y_{\mathit{max}}-y_{\mathit{min}}}\to y_i=y_n\left(y_{\mathit{max}}-y_{\mathit{min}}\right)+y_{\mathit{min}}$$

The Kennard-Stone (KS) algorithm is used to describe the SSMD in which a subset of N points in j dimensional space is selected (Kennard and Stone, 1969). The candidates of subset for training can be defined by the following matrix Y:

(29)

$$y=\left(\begin{array}{ccc}{y}_{11}& \cdots & y_{j1}\\ ⋮& \ddots & ⋮\\ y_{1N}& \cdots & y_{\mathit{jN}}\end{array}\right)$$

where Y represents a data set as Y = (y₁; y₂; · · · ; y_j) that contains j factor and a set of N points in j dimensional space defined by j factors (Kennard and Stone, 1969). The selection of design points is done sequentially (Riahi-Madvar et al., 2019). The goal in each step is to select points that are evenly spaced along the object area at each step of the algorithm. The first point is selected close to the average of the dataset and the second point is chosen so that it has the greatest distance from the first point. Eventually, the third point is selected so that it has the longest distance from the aforementioned two points. The training subcomponent involves these aspects of data points, in addition, the test subset includes the remaining data points (Wu et al., 1996). The subsets are selected by maximizing the minimum distance between the train dataset and the rest of the data points in the original dataset. The statistical features of the subsets obtained from the SSMD method are depicted in Table 4. As it can be seen from Table 4; all parameters have approximately the same distributions over train and test datasets, which is in accordance with the equal scale of standard deviation, kurtosis, skewness. Moreover, a wide range of $\rho$ is observed from the longitudinal dispersion data in both training and testing stages.

Table 4 The summary statistics of the parameters in train, test and over all of the sets.

	Parameter	Mean	Mode	SD	Min	Median	Maximum	Skewness	Kurtosis
Total data set	ρ X1 T Mw1 Tc1 ω1 Vc1 Mw2 Tc2 ω2 Vc2	1.110 0.459 305.470 236.916 862.300 0.712 689.000 59.080 589.171 0.4997 195.260	1.0836 0.200 298.150 236.30 968.100 0.8140 676.80 18.020 647.100 0.345 55.900	0.121 0.298 15.583 51.811 162.090 0.209 146.162 56.734 83.601 0.178 186.530	0.772 0.0009 278.150 107.110 571.300 0.308 298.240 18.020 506.550 0.198 55.900	1.120 0.406 303.150 236.300 898.800 0.814 676.800 46.070 562.050 0.564 1670	1.398 0.997 353.15 648.867 1376.1 1.408 1983.5 350 837.15 1.106 1175	−0.481 0.220 0.687 0.636 0.054 −0.226 0.988 3.552 0.942 0.477 3.790	3.000 1.739 3.241 10.208 1.759 2.264 13.758 18.205 3.302 2.667 20.000
Testing data set	ρ X1 T Mw1 Tc1 ω1 Vc1 Mw2 Tc2 ω2 Vc2	1.1025 0.453 306.686 238.119 851.230 0.7174 696.919 64.049 590.359 0.526 213.477	1.083 0.200 298.150 236.300 968.100 0.814 676.800 18.020 647.100 0.345 55.900	0.114 0.280 15.593 54.294 164.862 0.214 154.383 64.633 85.869 0.183 214.080	0.772 0.0009 278.150 107.110 571.300 0.308 298.240 18.020 506.550 0.198 55.900	1.108 0.409 303.150 240.000 802.145 0.814 701.300 46.070 562.050 0.564 167.000	1.398 0.997 353.15 648.867 1376.100 1.408 1983.500 350.000 837.150 1.106 1175.000	−0.350 0.213 0.687 0.364 0.280 −0.142 0.670 3.220 1.024 0.374 3.356	3.131 1.852 3.224 7.896 1.777 2.258 10.302 14.377 3.490 2.649 15.254
Training data set	ρ X1 T Mw1 Tc1 ω1 Vc1 Mw2 Tc2 ω2 Vc2	1.129 0.474 302.654 234.107 888.174 0.699 670.528 47.478 586.395 0.436 152.741	1.029 0.242 298.150 236.300 968.100 0.814 676.800 18.020 647.100 0.345 55.900	0.134 0.335 15.195 45.385 152.386 0.198 122.960 28.090 78.008 0.149 80.346	0.780 0.0009 278.150 107.111 571.30 0.325 298.24 18.02 506.55 0.198 55.900	1.167 0.400 298.150 236.300 968.100 0.814 676.800 41.050 560.000 0.345 167.000	1.395 0.990 353.150 648.867 13761.000 1.357 1983.500 134.178 802.050 0.870 422.000	−0.796 0.187 0.714 1.623 −0.527 −0.501 2.187 0.760 0.655 0.527 0.404	2.989 1.481 3.288 20.296 2.167 2.153 32.166 2.565 2.424 2.106 2.442

Note: The subscript 1 is related to IL and the subscript 2 is related to solvent.

2.3.2

2.3.2 LXWOA theories and formulation

To determine the coefficient of the semi-empirical equation using experimental data of IL-mixture, optimization algorithms must be used. Some of the algorithms that used to obtain the coefficients of semi-empirical equations are genetic algorithm, artificial bee colony algorithm, imperialist competitive algorithm, particle swarm optimization, and simulated annealing (Singh, 2019). The whale optimization algorithm is a nature-inspired metaheuristic optimization algorithm, which is inspired by the search behavior of humpback whales. The main difference between LXWOA and WOA is the improved convergence of the algorithm. The LXWOA algorithm is a combination of the WOA and Laplace crossover algorithms, which was introduced by Singh for the first time (Singh, 2019). The whale optimization algorithm is a novel nature-inspired meta-heuristic optimization method, which was presented by Mirjalili and Lewia (Mirjalili and Lewis, 2016). The WOA method is inspired by the social behavior and the bubble-net hunting strategy of humpback whales in which the bubble-net hunting behavior is used to prey hunting. One of the advantages of the WOA method is that it does not need a large amount of initial information. This feature makes the method have a small number of adjustment parameters that can be easily tuned for a great variety of applications. The LXWOA algorithm consists of the following steps: In the first step, an initial population of size N_p is generated randomly from the data set. In the second step, the WOA method is first followed by the LXWOA in each iteration, then two agents are chosen (Singh, 2019). The best agent is determined by the first agent and the second agent is randomly chosen from the present population. In the third step, laplace crossover operators are used to create two offspring by combining the best and randomly selected agents. The fitness of each offspring over the worst agent in the present population is evaluated one by one. If offspring has better fitness over the worst agent, it is then replaced by the worst particle. The best search agent is then updated and iteration is increased. Finally, this process is followed by the algorithm till termination criteria is satisfied (Liakos et al., 2018).

2.3.2.1

2.3.2.1 Whale optimization algorithm

Searching for a prey, encircling prey, and spiral bubble-net feeding maneuver are two main phases that are used by humpback whales to prey hunting. In the searching for a prey (the first phase), a random search is done to explore the prey depending on the position of each other (exploration phase). In the encircling prey and spiral bubble-net attacking (the second phase), the location of prey is perceived by the humpback whales after encircling them and the spiral updating position are implemented (exploitation phase). Since the optimal solution is not known in the a priori search space, the best current candidate solution is considered by the WOA algorithm as the target prey or close to the optimum target point. The WOA algorithm has received a lot of attention in recent years due to its advantages such as simple principle, simple operation, easy implementation, few adjustment parameters, and strong robustness, and many valuable research results have been obtained in this field. Moreover, the research results show that the WOA algorithm significantly outperformed other optimization algorithms, such as differential evolution and gravitational search in terms of solution accuracy and algorithm stability. Therefore, according to the above explanations, this algorithm was used in the present study. After the position of the best search agent is defined by the algorithm, the position of the other search agents will be updated towards the best search agent. The mathematical model of WOA is provided as follows: let N_P be the population size and M_itr be MaxIter  (Mirjalili and Lewis, 2016; Liakos et al., 2018; Kotsiantis et al., 2006; Dutton and Conroy, 1997; Maxwell et al., 2018). The method consists of two adjustable parameters a and b, which the value of the parameter a linearly decreases in the range 2 to 0 through iterations and the value of the parameter b reduces from −1 to −2. Next, the coefficients A_i, C_i and a random number l_i are calculated for agent i by the following equations (Singh, 2019):

(30)

$$A_i=2a{r}_1-a,C_i=2r_2,l_i=\left(b-1\right)r_3+1$$

where r₁, r₂ and r₃ represent three random numbers that are uniformly distributed on the interval from 0 to 1. Next, a random number p_r is generated in (0; 1). If 0.5 ≥ p_r and |A_i| ≥ 1, the d^th particle of the next position ( $x_i^d(t+1)$ ) is updated by the i^th agent through the $x_{\mathit{rand}}^d(t)$ agent as follows (Singh, 2019):

(31)

$$D_i^d=\left|C_i{x}_{\mathit{rand}}^t\left(t\right)-x_i^d(t)\right|,x_i^d\left(t+1\right)=x_{\mathit{rand}}^d\left(t\right)-A_i{D}_i^d$$

where $x_{\mathit{rand}}^d\left(t\right)$ indicates a random position of i^th agent in d^th dimension at time t. Now, If p_r < 0.5 and |A_i| < 1, the d^th particle of next position ( $x_i^d\left(t+1\right)$ ) is updated by the i^th agent through the $x_{\mathit{best}}^d$ agent as follows (Singh, 2019):

(32)

$$D_i^d=\left|C_i{x}_{\mathit{best}}^t\left(t\right)-x_i^d(t)\right|,x_i^d\left(t+1\right)=x_{\mathit{bset}}^d\left(t\right)-A_i{D}_i^d$$

Next, if p_r ≥ 0.5 the following spiral model is generated by evaluating the distance between the whale postion (x_i) and prey postion (x_best (t)) in d^th dimension $D_i^d=\left|C_i{x}_{\mathit{best}}^t\left(t\right)-x_i^d(t)\right|$ at iteration t to imitate the helix-shaped movement of humpback whales (Singh, 2019):

(33)

$$x_i^d\left(t+1\right)=D_i^d{e}^{b{l}_i}\cos \left(2\pi l_i\right)+x_{\mathit{bset}}^d\left(t\right)$$

The Laplace crossover (LX) is proposed by Deep and Thakur for the first time in which two off-springs i.e. $y_1=\left(y_1^1,y_1^2,\cdots \cdots ,y_1^m\right)$ and $y_2=\left(y_2^1,y_2^2,\cdots \cdots ,y_2^m\right)$ are generated from a pair of parents i.e. $x_1=\left(x_1^1,x_1^2,\cdots \cdots ,x_1^m\right)$ and $x_2=\left(x_2^1,x_2^2,\cdots \cdots ,x_2^m\right)$ in such a way that both of the off-spring have a symmetric position over the parents position. Next, two uniformly distributed random numbers u_i 2 [0; 1] and v_i 2 [0; 1] are firstly generated (Mirjalili and Lewis, 2016; Kotsiantis et al., 2006). Then, a random number l_i is generated by simply inverting the distribution function of Laplace distribution as follows:

(34)

$$l_i=\left\{\begin{array}{c}p-qln\left(u_i\right)v_i\le 0.5\\ p+qln\left(u_i\right)v_i>0.5\end{array}\right.$$

where p ∊ R and q > 0 are called the location parameter and the scale parameter, respectively.

The off-springs are generated as follows:

(35)

$$y_1^i=x_1^i+l_i\left|x_1^i-x_2^i\right|,y_1^i=x_2^i+l_i\left|x_1^i-x_2^i\right|$$

If for some of i the created off-spring are outside the search space i.e. $y^i<y_{\mathit{low}}^i$ or $y^i>y_{\mathit{up}}^i$ , a random number is then selected from $\left[y_{\mathit{low}}^i,y_{\mathit{up}}^i\right]$ for yⁱ as the generated off-spring. For smaller values of q, LX produces the off-spring close to the parents and for greater values of q; LX produces the off-springs away from the parents. For fixed values of p and q, LX dispenses off-springs according to the parents dispersion (Mirjalili and Lewis, 2016; Liakos et al., 2018). An equation for ρ is firstly defined based on previous literature to run the algorithm. The parameters obtained from the equation are then used as variables for the LXWOA algorithm. Moreover, a suitable objective function will be defined for the implementation of the algorithm in the following section. In this regards, at the end of each iteration, the best value of $x_{\mathit{best}}^d$ must be found by the algorithm through changing the values of the variables. Eventually, the optimal solution rapidly converges towards a local optimum after a couple hundred times of running the algorithm. Eventually, after running the program a couple of hundred times, the convergence towards the optimal solution is achieved by the algorithm. This means that the hunting location has been identified and surrounded by the humpback whale. The flowchart of the implied algorithm is noticed in Fig. 1.

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Fig. 1

The algorithm of LXWOA method.

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The main contribution of the proposed LXWOA algorithm is addressed in this subsection. After performing repeated runs of the optimization algorithm, it was found that it is not feasible to provide an equation that covers all the data. This creates some difficulty; first of all, the algorithm will not converge properly. The second problem is that the error value R₂ will increase because the algorithm has insisted on synchronizing and fitting itself overall data despite the high errors. Thus, a solution must be provided to prevent this problem at the various stages of the learning process. This means that one should try to converge the proposed model to less error data as much as possible. In this regard, the cumulative distribution function (CDF) function of the errors is defined as an objective function for the LXWOA optimization algorithm. Therefore, the objective function is considered as a coefficient (γ = 0.8) of CDF for the error defined as follows: $$Error\left(i\right)=y_{\mathit{Predictedvalue}}\left(i\right)-y_{\mathit{Observedvalue}}\left(i\right)$$

(36)

$$\mathit{Fitness}=\gamma \left(\sum \left|{\mathit{CDF}}_{\mathit{errors}}\right|\right)$$

Indeed, some of the data generated by the LXWOA model may differ significantly from the actual data. Therefore, there is a lot of error in this data. This portion of the data may contain up to 10 to 20 percent of total data per run of the optimizer program. Therefore, if we want to present the cost function in the optimization algorithm in such a way that it puts pressure on all data to reduce the total error, it causes the model to move away from convergence in data with less error at the cost of achieving a lower error. In this regard, we try to reduce the model pressure on high errors each time the optimization process is performed. For this purpose, the CDF of the errors is first obtained. Part of the errors (with a coefficient (0.8)) is then considered as a criterion for the cost function.

3 Results and discussion

No CEoS can prepare exact descriptions of behavior of real-fluid in all interest regions. Both SRK EoS and PT EoS consider only the attractive and repulsive intermolecular force between molecules. However, they have fundamental challenge to pure and mixture liquid density and the modification has been made to improve CEoS capability using modifying the repulsive and attractive terms (Sheikhi-Kouhsar et al., 2015). In this study, in the first step, the accuracy of SRK EoS and PT EoS was investigated. For example, Fig. 2 indicates the density behavior of C₆mimBr + EGMME at x_IL = 0.3990 and x_IL = 0.7003 and C₆mimPF₆ + EGMME at x_IL = 0.2006 and x_IL = 0.8019 and at atmospheric pressure based on SRK EoS and PT EoS. Both CEoSs could not predict the mixture density with satisfactory accuracy. To decrease the deviation, the binary interaction parameter (k_ij) was used for both of them; however, the effect of interaction parameter on improving the modeling results was insignificant.

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Fig. 2

The density behavior of (a) C₆mimBr + EGMME (system I: x_IL = 0.3990; system II: x_IL = 0.7003) and (b) C₆mimPF₆ + EGMME (system I: x_IL = 0.2006; system II: x_IL = 0.8019). The experimental data are from Pal and Kumar (Pal and Kumar, 2012).

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As another suggestion to model the IL-mixture density, application of semi-empirical equations was studied. The various semi-empirical equations were developed in this communication (see Appendix A). The semi-empirical equation parameters are determined by matching prediction density (based on semi-empirical equation) with experimental data. To model the density of IL-mixture systems based on semi-empirical equations, two methods was considered. In the first method, the coefficients of Eqs. (20) - (24) that were presented by authors (Valderrama and Zarricuetac, 2009; Hosseini et al., 2013) with all mixing rules that are provided in Table 3, was applied. The obtained results indicated presented coefficients was not appropriate idea to predict IL-mixture systems. For instance, Fig. 3 indicates the results of C₄mimHSO₄ + water at x_IL = 0.2066 and C₂mimC₂SO₄ + water x_IL = 0.8775 using Eqs. (20) - (24) with available coefficients in the literature (Valderrama and Zarricuetac, 2009; Hosseini et al., 2013). The behavior of all equations is out of experimental data. Indeed, this behavior was predictable. Due to the presented coefficients by authors (Valderrama and Zarricuetac, 2009; Hosseini et al., 2013) were obtained based on pure experimental data.

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Fig. 3

The density behavior of (a) C₄mimHSO₄ + water, x_IL = 0.2066 and (b) C₂mimC₂SO₄ + water, x_IL = 0.8775. The experimental data are from Bhattacharjee et al. (Bhattacharjee et al., 2012).

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In the second method, the coefficients of Eqs. (20) - (25) was optimized using improved laplacian whale optimization algorithm. ILs composed of various cation structures and due to the behavior of each cation-family is similar, subsequently; to present a comprehensive IL-mixture density model, the semi-empirical equation coefficients of each cation-family were obtained, separately. In order, to obtain the coefficients, 70% of experimental data of each cation-family were used (according to SSMD method) and the rest of them were applied to investigate the accuracy of presented modification. Table 5 indicates the obtained results of coefficients of each cation-family.

The statistical validation of optimized coefficients was investigated by introducing four statistical parameters of correlation factor (R²), average absolute percent deviation (AAPD), standard deviation (STD) and root mean squared error (RMSE). These parameters are formulated as following and the values of them are provided in Table 6.

For example, Fig. 4 indicates the total results of the correlation factor based on Eq. (20) and C₄mim-family with applying test data subcategories. The solid 45^° line shows the details of fitting between the experimental data and the related ones, whereas circles point show the comparison between the obtained results based on Eq. (20) and the experimental density data. The R² value is 0.9856. Indeed, closeness of the circle points to the solid line in Fig. 4 demonstrates the presented modification technique has acceptable ability to predict IL-mixture density.

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Fig. 4

Predicted versus experimental IL density mixtures for test datasets. The reported results are related to Eq. (23) and C₄mim-family.

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The six semi-empirical equations gave different results and any cation-family presented various results to predict the density of IL-mixture system at all temperatures. For example, Fig. 5 indicates the density behavior of OmimBF₄ + ethanol (x_IL = 0.247 and x_IL = 0.839) and C₄mimClO₄ + ethanol (x_IL = 0.709 and x_IL = 0.900) based on semi-empirical equations (using test data). According to Fig. 5, the semi-empirical equations could predict the mixture density with acceptable accuracy. All semi-empirical equations have similar behavior and they could predict the mixture density with acceptable accuracy.

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Fig. 5

The density behavior of (a) OmimBF₄ + ethanol, x_IL = 0.247 and x_IL = 0.839 (b) C₄mimClO₄ + ethanol, x_IL = 0.709 and x_IL = 0.900. The obtained results based on Eq. (20) with optimized coefficients. The experimental data are from Mokhtarani et al. (Mokhtarani et al., 2008).

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All semi-empirical equations, except Eq. (24), only were used to predict pure IL density up to now. For first time, we have examined the capability of these equations to predict the density of IL-mixture system at various mole fractions and temperatures. Subsequently, mixing rules had significant role in this development and they show the effect of component mole fraction. According to Table 3 various mixing rules, at least three mixing rules for each property, was investigated. The mixing rule of all thermodynamic properties except acentric factor and critical temperature, was linear. The experimental data density mixture are function of mole fraction and temperature and all semi-empirical equations are function of reduced temperature (T_r), therefore, critical temperature mixing rule have remarkable influence on accuracy of obtained results based on semi-empirical equations. Several critical temperature mixing rules have been investigated but non-linear ones indicated better results. This mixing rule is function of second power of mole fraction and also second order of radical of both IL and solvent critical temperature ( $\sqrt{T_{\mathit{ci}}{T}_{\mathit{cj}}}$ ).