Document Type : Research Paper
Authors
 Mohsen Keshavarz ^{1}
 Ahad Ghaemi ^{} ^{} ^{2}
 Mansour Shirvani ^{3}
 Ebrahim Arab ^{4}
^{1} M.S. Student, School of Chemical, Petroleum, and Gas Engineering, Iran University of Science and Technology, P.O. Box 1684613114, Tehran, Iran
^{2} Associate Professor, School of Chemical Engineering, Iran University of Science and Technology, P.O. Box 1684613114, Tehran, Iran
^{3} Associate Professor , School of Chemical Engineering, Iran University of Science and Technology, P.O. Box 1684613114, Tehran, Iran
^{4} M.S. Student, School of Chemical Engineering, Iran University of Science and Technology, P.O. Box 1684613114, Tehran, Iran
Abstract
In this work, the dispersed phase holdup in a Kühni extraction column is predicted using intelligent methods and a new empirical correlation. Intelligent techniques, including multilayer perceptron and radial basis functions network are used in the prediction of the dispersed phase holdup. To design the network structure and train and test the networks, 174 sets of experimental data are used. The effects of rotor speed and the flow rates of the dispersed and continuous phases on the dispersed phase holdup are experimentally investigated, and then the artificial neural networks are designed. Performance evaluation criteria consisting of R^{2}, RMSE, and AARE are used for the models. The RBF method with R^{2}, RMSE, and AARE respectively equal to 0.9992, 0.0012, and 0.9795 is the best model. The results show that the RBF method well matches the experimental data with the lowest absolute percentage error (2.1917%). The rotor speed has the most significant effect on the dispersed phase holdup comparing to the flow rates of the continuous and dispersed phases.
Keywords
 Solvent Extraction
 Kühni Extraction Column
 Dispersed Phase Holdup
 Multilayer Perceptron
 Radial Basis Function
Main Subjects
1. Introduction
Liquidliquid extraction is an important separation process used in many industries such as petroleum, petrochemical, pharmaceutical, food, environmental, nuclear, and hydrometallurgical industries (Arab et al., 2017; Oliveira et al., 2008; Moreira et al. 2005). Various types of solvent extraction contactors have been used in liquidliquid extraction processes (Hemmati et al., 2015; TorabMostaedi et al., 2011). There are two common types of phase agitation: by rotary agitation by discs, turbines, etc., and by pulsations. Among the wide variety of equipment, a countercurrent column with a rotary agitator is more frequently utilized (Asadollahzadeh et al., 2015a, 2015b; Hemmati et al., 2015; Haunold et al., 1990). The Kühni column is one of the most important agitation column extractors. It includes a rotating shaft inside the column on which a number of impellers are mounted and separated by perforated static horizontal plates to make a number of agitation stages (Mogli and Buhlmann, 1983; Tahershamsi et al., 2016). The design and identification of the performance of liquidliquid extraction columns for achieving the effects of operating variables at scalingup stages require the careful consideration of the hydrodynamic and mass transfer parameters of the device. In a liquidliquid extraction column, the dispersed phase droplets undergo repeated coalescence and breakage, ultimately causing an equilibrium drop size distribution. The resulting fractional volumetric holdup of the dispersed phase is defined as the volume fraction of the active section of the column that is occupied by the dispersed phase (TorabMostaedi et al., 2011):

(1) 
The dispersed phase holdup is one of the most important hydrodynamic parameters in extraction columns since it affects both the flooding specification and the mass transfer characteristic of the column. Also, the dispersed phase holdup is used to calculate the interfacial area and the residence time of droplets (Kumar and Hatland, 1995). The dispersed phase holdup is necessary for calculating the slip velocity between the phases in extraction columns. There are two methods for calculating the dispersed phase holdup, including explicit and implicit methods. In the implicit method, the slip velocity is used to achieve the dispersed phase holdup. The available proposed correlations are either approximate or usable in a specified range of operation conditions. There are a lot of experimental studies performed on various scales of the Kühni extraction column to determine the hydrodynamic and mass transfer parameters of the column. Kumar et al. (1986) investigated the change of the dispersed phase holdup and flooding point for a waterorthoxylene system in the Kühni extraction column. They understood that the variations of the dispersed phase holdup along the column was not uniform and that the effects of the speed of rotation and the flow rates of the continuous (Q_{c}) and dispersed (Q_{d}) phases on the dispersed phase holdup were almost similar, while the maximum of the dispersed phase holdup occurred almost in the middle stages of the column. Hufnagl et al. (1991) provided a differential model for mass transfer from the continues to the dispersed phase in a tolueneacetonewater system in the Kühni extraction column (Hufnagl et al., 1991). Oliveira et al. (2008) evaluated the effects of operational parameters on the dispersed phase holdup for a waterExxsol D80 system (Oliveria et al., 2008). Florian et al. (2012) studied the separation of toluene from heptane. They used population balance modeling to simulate a pilotscale Kühni extraction column (Florian et al., 2012). Arthur and Mansur (2013) used a combination of a dynamic model with the droplet population balance equations to simulate the Kühni extraction column (Neto and Mansur, 2013). Some of the correlations proposed for predicting the dispersed phase holdup in different types of extractor columns are summarized in Table 1.
Mechanical mixing devices or pulsing mechanisms are used for agitation in the liquidliquid extraction columns to improve mass transfer rate, which increases the complexity of the process modeling in such columns (Florian et al., 2012). The modeling and simulation of the extraction devices are necessary for determining the affecting parameters and their impact on the performance of equipment. The model of the process involves a wide set of equations, and solving them simultaneously is difficult and timeconsuming (Gomes et al., 2006; Rode et al., 2013; Attarakih et al., 2015). On the other hand, the numerical methods of artificial neural networks (ANN) have found a wide interest in the modeling of process systems in recent years.
Table 1
Summary of correlations for the calculation of the dispersed phase holdup in different contactors.
Type of column 
Correlation 
Reference 
RDC 
Haunold et al., 1990 

RDC 
TorabMostaedi et al., 2011 

Kühni 
Haunold et al., 1990 

Kühni 
Oliveira et al., 2088 

ORC 
Sharker et al., 1985 
In this work, two different methods of ANN are used to predict the dispersed phase holdup of the Kühni extraction column. The data are from the two liquidliquid systems, namely toluenewater and nbutyl acetatewater systems, with different physical properties in the condition of no mass transfer. The experiments were conducted to determine the effects of operating parameters such as the rotor speed and the flow rates of the continuous and dispersed phases on the dispersed phase holdup. A set of 174 experimental data were used in the artificial neural networks, from which 93 data were from toluenewater system and 81 data from nbutyl acetatewater system. The inputs were the speed of rotation and the flow rates of the dispersed and continuous phases, and the output was the dispersed phase holdup.
The purpose of the current paper is to develop artificial intelligencebased models for predicting the dispersed phase holdup in the Kühni extraction column. The comparison of the ability of empirical correlation and ANN models to predict anonymous data can be considered as the innovation of this paper. Due to the complexity of the design of extraction columns and the effects of multiple parameters, using ANNsbased artificial intelligence can obviate the need for experimental data to some extent by reducing the cost of the experiments.
2. Experimental
There are numerous methods such as local displacement, pressure drops, and sampling methods for measuring the dispersed phase holdup in the extraction columns (Chen et al., 2002). The experiments were carried out using a pilotscale Kühni extraction column. The active portion of the column was made of a transparent plastic material resistant to the solvent used in the experiments (Arab et al., 2017). The column consisted of 10 stages, each stage being separated by the perforated plates, each of which includes 36 holes with a diameter of 5.7 mm (Figure 1). The aqueous phase is introduced at the top of the column flowing downward, and the dispersed phase flows countercurrently. Agitation at each stage is achieved with a sixblade turbine agitator having a diameter of 50 mm with accurate speed control. In all the experiments, water is used as the continuous phase and the organic solvent as the dispersed phase. All the experiments were carried out far from flooding conditions. All the physical properties of the liquid systems reported in Table 2 are constant.
Table 2
Physical properties of the tested systems (Arab et al., 2017).
Physical property 
Toluenewater 
nButyl acetatewater 
998.2 
997.8 

865.2 
882.3 

0.96 
1.04 

0.75 
0.76 

36.10 
14.10 
Two PENTAX centrifugal pumps were used to move the light and heavy phase. A 1.1 kW electric motor (MOTOGEN Company, Iran) was used to rotate the extraction column. An optical sensor was used to control the interface between two phases at the top of the column. Four metal tanks made of stainless steel with a volume of about 85 liters were built to hold and collect the light and heavy phases. Duo to the experimental limits, in these experiments the ratio of the mixer diameter to the column diameter is constant (Arab et al., 2017).
Figure 1
A schematic of the Kühni extraction column (Arab et al., 2017).
The liquid systems studied are toluene–water and nbutyl acetate–water. These systems have been recommended by the European Federation of Chemical Engineering (EFCE) as official test systems for extraction investigations. During the experiments, the effects of rotor speed and the flow rates of the dispersed and continuous phases on the dispersed phase holdup were studied. A rotor speed in the range of 90 to 270 rpm and the flow rates of the dispersed and continuous phases in the range of 18 to 42 lit/hr. were applied. In each experiment, one of the parameters was changed while the others were kept constant. Totally, 93 data sets of toluenewater and 81 data sets of nbutyl acetatewater were obtained and used in the modeling procedures. The range of the inputs and the details of the information on the simulation are tabulated in Table 3 (Arab et al., 2017).
Table 3
Details of the collected information on both systems used in the experiments.
System 
Inputs parameter 
Ranges 
The mean of the parameter 
Standard deviation 
Toluenewater 
Rotor speed (rpm) 
90270 
154.1940 
46.0404 
Flow rate of the dispersed phase, Q_{d} (l/hr.) 
1842 
26.9355 
6.1996 

Flow rate of the continuous phase, Q_{c} (l/hr.) 
1842 
26.9355 
6.1996 

nButyl acetatewater 
Rotor speed (rpm) 
90210 
144.0740 
39.6640 
Flow rate of the dispersed phase, Q_{d} (l/hr.) 
1840 
26.4200 
4.8190 

Flow rate of the continuous phase, Q_{c} (l/hr.) 
2036 
25.7780 
4.0000 
3. Artificial intelligencebased models
Various methods of ANN’s can be considered as a class of dynamical systems that process and develop a model from the experimental data and the knowledge or law behind the data (Du et al., 2006). They are computational tools which can be considered as computational intelligence. The origin of the method is taken from the principles of biology arranged in a mathematical description by scientists. One of the key ideas in artificial neural networks is the backpropagation algorithm created by David Rummelhart and James McClelland in 1986. The backpropagation algorithm is a common method for training AANs, which is used to minimize the cost function. The backpropagation algorithm is a supervisory learning method which requires a set of inputoutput data (Hassoun et al., 1996).
The multilayer perceptron (MLP) is one of the most famous artificial neural networks used to create nonlinear mappings. It is a kind of feedforward artificial neural network which is able to perform a nonlinear mapping with arbitrary precision by selecting the number of layers and neurons. The adjustable parameters of the network are the weights of the connections between the layers. The MLP network consists of three layers: the input layer, the hidden layer, and the output layer (Hagan and Menhaj, 1994). The training process means finding appropriate values for the connection weights between the neurons. The relation between the input, the hidden, and the output layers are established by weights “w” and biases “b”. The mathematical function, called activation function, has a variety of forms, including hyperbolic tangent, sigmoid, linear, and Gaussian functions. All of these functions are continuous and differentiable. Various learning algorithms such as the combined conjugate gradient, back propagation algorithm, the reducing gradient algorithm, the Bayesian regulating algorithm, and LevenbergMarquardt algorithm are used for training MLP models. The selection of the algorithm type affects the learning speed and the accuracy of the results (Du et al., 2006; Hassoun and Menhaj, 1994).
Radial based functions (RBF) network which were introduced by Brodhead and Lowe in 1998 for the first time is a type of forward networks with a hidden layer. With sufficient numbers of neurons in the hidden layer, they are very strong networks in approximation and able to simulate any continuous function with an acceptable degree of accuracy. With some advantages in comparison with the other artificial neural networks, including a better approximation ability, a shorter learning time, a simpler network structure, and avoiding local minimums, the RBF models are used as a reliable tool to simulate various chemical engineering processes (Vaziri and Shahsavand, 2013). This network is composed of three layers. Each layer consists of a number of nodes (neurons), and the nodes in the input layer are used only to pass the input data to the hidden layer; in fact, no calculations are performed in the input layer nodes. Each neuron in the hidden layer has two sets of parameters, namely the center parameter ( ) and the width or spread parameter ( ) associated with it. The activation function of the hidden layer is Gaussian function, and that of the output layer is a linear function. For details in this context can be found elsewhere (Dabiri et al., 2018; Mohebian et al., 2017). All of the data were normalized in the range of 1 to 1. For this purpose, Equation 2 is used:
(2) 
where, is normalized data, and and are the maximum and minimum data in the dataset respectively. There are various performance characterization criteria for evaluating the performance of regression techniques. The criteria used for this purpose include the square of the correlation coefficient “R^{2}”, root mean square error “RMSE”, and the percentage of average absolute relative error “AARE”. Their mathematical definition are given in Equations 3 to 5 (Gandhi and Joshi, 2010; Lashkarbolooki et al., 2012; Alves et al., 2012).

(3) 

(4) 

(5) 
where, , and N are the experimental values, the average of the experimental values, the artificial neural network output, and the number of samples respectively.
4. Results and discussion
4.1. MLP model structure
Different learning algorithms, including combined conjugate gradient algorithm (Traincgb), the Bayesian algorithm (Trainrp), the reduced gradient algorithm (Traingda), the BroydenFletcherGoldfarbShanno (BFGS) algorithm (Trainbfg), and LevenbergMarquardt algorithm (Trainlm) have been used for training MLP models. In order to determine the most appropriate learning algorithm, 93 data set of the toluenewater system was used in the learning of MLP network. For this purpose, a network with one hidden layer and 8 neurons in the hidden layer was considered so that the ability of the different algorithms is checked. The results of different algorithms for the MLP model are shown in Figure 2.
According to Figure 2, the best algorithm for the simulation is LevenbergMarquardt. Various structures with different numbers of layers and neurons were studied using the LevenbergMarquardt algorithm to select the optimized structure of the network. The results of RSME and AARE evaluations are illustrated in Figures 3 and 4 for the various structures of the MLP model applied to the toluenewater system. It is noteworthy that the same investigations were also conducted on the nbutyl acetatewater. According to the evaluation criteria, the best structure for the toluenewater system includes a network with one hidden layer and 8 neurons in the hidden layer. Moreover, the best structure for the nbutyl acetatewater system is a network with one hidden layer and 6 neurons in the hidden layer.
During the calculations, the learning coefficient, the maximum number of iterations, and the momentum factor were determined as 0.01, 10000, and 0.001 respectively. Sigmoid function (tansig) was used as the activation function for all the neurons in the hidden layer. Furthermore, the linear function (purelin) was used for the output layer. The structure of the MLP model of the toluenewater system depicted in Figure 5 includes the input and output layers as well as the number of the hidden layers and neurons in the hidden layer for the network. The experimental data taken from the Kühni extraction column were divided into three distinct sets, including training, validation, and test data. From the total collection of the experimental data, 70% (65 datasets) were used for the training, 15% (14 datasets) for the validation, and 15% (14 datasets) for testing the data.
Figure 2
Evaluation criteria of different algorithms in the MLP model.
Figure 3
RMSE values of the toluenewater system in the MLP model for structures with different numbers of neurons in the first and the second layers.
Figure 4
AARE values of the toluenewater system in the MLP model for structures with different numbers of neurons in the first and the second layers.
Rotation speed 
The flow rate of the dispersed phase 
The flow rate of the continuous phase 
N2 
N8 
N7 
N6 
N5 
N4 
N3 
N1 
The dispersed phase holdup 
Input Layer 
Output Layer 
Hidden Layer 
Figure 5
The MLP model structure of the toluenewater system with 8 neurons in one hidden layer.
The values of AARE, RMSE, and R^{2} for the network of the toluenewater system were 0.8312, 0.0013, and 0.9990 respectively, and the corresponding values of the nbutyl acetatewater system were 0.8835, 0.0013, and 0.9980 respectively. The comparison between the experimental data and the MLP model predictions for the toluenewater system is given in Figure 6. Figure 6a shows the distribution of all the data and the comparison between the actual and the predicted values, and Figure 6b reveals the extent of the deviations of the data from the model predictions for various amounts of the dispersed phase values; acceptable predictions are obtained by the MLP model. Figure 6c displays the values of errors in various experiments. As shown in this figure, the 38^{th} sample has the maximum error equal to 0.0074. Figure 6d also delineates the frequency distribution of errors.
Figure 6
The values predicted by the MLP model against the experimental data for the dispersed phase holdup of the toluenewater system.
4.2. RBF model structure
Gaussian function is used to design the RBF network. The important issue of the RBF model is the spread parameter which plays an important role in improving the generalizability of the network, and it is better to select a large value for it. The magnitude of this parameter depends on the type of problem, and it must be chosen carefully. In addition, the number of neurons must not exceed the half of the number of the training data (Vaziri et al., 2013). A trial and error procedure is used to determine the appropriate number of neurons as well as the spread parameter. Figure 7 displays the variations of RMSE and AARE of the toluenewater system versus the spread parameter and the number of neurons. As can be seen, the lowest values of RMSE and AARE are related to the network with 25 neurons in the middle layer and a spread parameter equal to 22.5.
a)

b) 
Figure 7
Evaluation results of the toluenewater system based on the spread parameter and the number of neurons in the RBF model: a) RMSE and b) AARE; Minimum AARE and RMSE are 0.97948 and 0.00116 respectively.
In the same manner, a network with 22 neurons in the middle layer and a spread parameter of 19 was determined as the best fitted RBF network for the nbutyl acetatewater system. The evaluation criteria values of AARE, RMSE, and R^{2} for the RBF network adapted for the toluenewater system were 0.9795, 0.0012, and 0.9992 respectively, and the corresponding values for the RBF network adapted for the nbutyl acetatewater system were 0.8082, 0.0012, and 0.9987 respectively.
The overall values of the evaluation criteria of the two types of models and the individual evaluations of the training, validation, and testing data are given in Tables 6 and 7; generally, the RBF method represents the best results of the MLP model.
Table 6
Comparison between the evaluation criteria of the toluenewater system.
Evaluation criteria 
Data sets 
MLP model 
RBF model 
R^{2} 
Training 
0.9997 
0.9993 
Validation 
0.9970 
0.9992 

Test 
0.9986 
0.9990 

Overall data 
0.9990 
0.9992 

RMSE 
Training 
0.0006 
0.0012 
Validation 
0.0026 
0.0013 

Test 
0.0016 
0.0012 

Overall data 
0.0013 
0.0012 

AARE(%) 
Training 
0.5421 
0.8902 
Validation 
1.6527 
1.1980 

Test 
1.3518 
1.1755 

Overall data 
0.8312 
0.9795 
Table 7
Comparison between the evaluation criteria of the nbutyl acetatewater system.
Evaluation criteria 
Data sets 
MLP model 
RBF model 
R^{2} 
Training 
0.9993 
0.9994 
Validation 
0.9968 
0.9980 

Test 
0.9989 
0.9971 

Overall data 
0.9985 
0.9987 

RMSE 
Training 
0.0008 
0.0008 
Validation 
0.0025 
0.0014 

Test 
0.0012 
0.0021 

Overall data 
0.0013 
0.0012 

AARE (%) 
Training 
0.6583 
0.5867 
Validation 
2.0406 
0.9089 

Test 
0.7963 
1.7592 

Overall data 
0.8835 
0.8082 
4.3. Effect of operating parameters
The effect of rotor speed on the dispersed phase holdup of both systems is presented in Figure 9. At constant flow rates of the dispersed and continuous phases, the dispersed phase holdup increases by raising the rotor speed of both systems. It is clear that all of the models used for simulating the data of Kühni extractor are excellent in fitting the experimental data. Actually, the droplet breakage increases but droplet size decreases with an increase in rotor speed, which results in an increase in shear forces affecting the droplets. As a result of reducing the droplet size, the relative velocity between the continuous phase and the dispersed phase (slip velocity) falls, and the dispersed phase holdup rises with an increase in the number of the droplets within the column.
A comparison between the curves in Figure 8 confirms that an increase in the interfacial tension leads to a drop in the dispersed phase holdup. That is, the dispersed phase holdup of the toluenewater system, with a higher interfacial tension, is less than that of the nbutyl acetatewater system. In fact, increasing interfacial tension raises the droplet size and consequently the slip velocity between the phases. A reduction in the residence time of the droplets corresponds to a drop in the dispersed phase holdup. By using the output predictions from the intelligent methods, one can detect the effect of rotor speed on the variations of the dispersed phase holdup at rotor speeds beyond the range of the data.
Figure 8
Effect of rotor speed on the dispersed phase holdup.
The effect of the flow rate of the dispersed phase on the dispersed phase holdup of both systems is depicted in Figures 9 and 10. The number and frequency of droplet coalescence increase at an increased flow rate of the dispersed phase, when the rotor speed and the flow rate of the continuous phase are kept constant. Thus, the dispersed phase holdup rises at a higher number of drops. Figure 11 depicts the effects of rotor speed and the flow rate of the dispersed phase on the dispersed phase holdup at two different constant flow rates of the continuous phase. As can be inferred from this figure, raising the flow rate of the dispersed phase at all values of the rotor speed and at the flow rates of the continues phase equal to 30 and 60 (l/hr.) improves the dispersed phase holdup of the toluenewater system.
Figure 9
Effect of the flow rate of the dispersed phase on the dispersed phase holdup of the toluenewater system.
Figure 10
Effect of the flow rate of the dispersed phase on the dispersed phase holdup of the nbutyl acetatewater system.
a) 
b) 
Figure 11
Variations of the dispersed phase holdup versus the rotor speed and the flow rate of the dispersed phase of the toluenewater system: a) and b) .
Figures 12 and 13 illustrate the effect of the flow rate of the continuous phase on the dispersed phase holdup. Increasing the flow rate of the continuous phase at a constant rotor speed and a constant flow rate of the dispersed phase raises the drag force between the continuous phase and the dispersed phase, which results in limitation to the droplet movement and consequentially enhances the dispersed phase holdup. Comparing Figures 9 and 10 with Figures 12 and 13 states that the effect of flow rate of the dispersed phase on the dispersed phase holdup is larger than that of the continuous phase in both systems.
Figure 12
Effect of the flow rate of the continuous phase on the dispersed phase holdup of the toluenewater system.
Figure 13
Effect of the flow rate of the continuous phase on the dispersed phase holdup of the nbutyl acetatewater system.
One of the objectives of this work is comparing the ability of different modeling methods and empirical correlations to predict the dispersed phase holdup. To this end, a set of 174 experimental data (93 data sets for the toluenewater system and 81 data sets for the nbutyl acetatewater system) are used. 150 data sets are used to produce the intelligent networks, and the remaining 24 data sets are employed to determine the empirical correlation coefficients and thus compare the performance of the models. The dispersed phase holdup is correlated with the operational conditions and the physical properties of the systems by the 150 datasets as follows:

(6) 
where, , , , , , , , , , and are the dispersed phase holdup, gravitational force, interfacial tension, the continuous phase density, the dispersed phase velocity, the continuous phase velocity, the dispersed phase viscosity, the continuous phase viscosity, rotation speed, and the density difference between the phases respectively. In Equation 6, the evaluation criteria of AARE, RMSE, and R^{2} for fitting the data are 6.6132, 0.0098, and 0.9653 respectively. Figure 14 compares the ability of the intelligent models and that of the correlation in Equation 6 to predict the dispersed phase holdup. It also reveals that the RBF model performs better than the MLP and empirical correlation in predicting the new data sets. According to Equation 5, the absolute percentage error of the empirical correlations, MLP network, and RBF network are 4.332%, 2.642%, and 2.192% respectively.
Figure 14
The actual and predicted dispersed phase holdup for 24 intact datasets: a) empirical correlations, b) MLP, and c) RBF.
5. Conclusions
In this research, the experimental investigation and simulation of the dispersed phase holdup are considered in a Kühni column. In the simulation, using intelligent techniques, a method is provided which reduces the need for the experimental data. For this purpose, two contacting systems of toluenewater and nbutyl acetatewater are investigated. The intelligent models, including MLP and RBF models are used in an optimal structure to simulate the dispersed phase holdup. Additionally, an empirical correlation is employed to predict the dispersed phase holdup as a function of the physical properties and the operation variables. Finally, the following results are obtained:
 The dispersed phase holdup increases at higher rotor speeds at a constant flow rate of the dispersed and continuous phases.
 The interfacial tension has a significant impact on the dispersed phase holdup. An increase in the interfacial tension expands the droplets, which results in a higher relative phase velocity; consequently, the dispersed phase holdup drops.
 Increasing the flow rate of the dispersed phase at a constant rotor speed and a constant flow rate of the continuous phase enhances the dispersed phase holdup.
 By increasing the flow rate of the continuous phase at a constant rotor speed and a constant flow rate of the dispersed phase, the drag force for moving the droplets rises and the speed of the droplets climbing up falls, resulting in an increase in the dispersed phase holdup.
 Among the operation parameters, the rotor speed has the greatest impact on the dispersed phase holdup.
 Among the three different modeling techniques applied to the data of Kühni extraction column, including MPL, RBF, and the empirical correlating method, the RBF modeling is proved to most accurately fit the data of the systems.
 The ability of the artificial neural network to predict anonymous data can be used to reduce the number of the relevant experiments and to lower the cost of the experiments; therefore, with fewer experiments, the desired result is achieved. In order to improve the results and predictions, using machine learning methods (support vector regression) and comparing them with ANN methods are suggested.
Nomenclature
AARE 
Average absolute relative error (%) 
D 
Column diameter (m) 
D_{s} 
Stator opening diameter (m) 
g 
Acceleration duo to gravity (m/s^{2}) 
h_{c} 
Compartment height (m) 
N 
Rotor speed (1/s) 
Q 
Flow rate of the dispersed or continuous phases (m^{3}/s) 
R 
Correlation coefficient () 
RMSE 
Root mean square error () 
V 
Superficial velocity (m/s) 
V_{s} 
Slip velocity (m/s) 
W 
Weight factor () 
Input examples (attributes) 

Target output 

Greek letters 

Density difference between the phases (kg/m^{3}) 

Dispersed phase holdup () 

Mechanical power dissipation per unit mass (w/kg) 

Interfacial tension (N/m) 

Viscosity (Pa.s) 

Density (kg/m^{3}) 

Width of radial basis function (RBF) kernel () 

Subscripts 

c 
Continuous phase 
d 
Dispersed phase 
j 
Number of neurons in hidden layer 
Superscripts 

T 
Transpose 