Reactive Zinc Extraction in MRDC Column; Case Study by Mathematical Modelling and Applying the Droplet Size Distribution with Forward Mixing Approach

In this survey, the reactive mass transfer data are determined for zinc extraction from chloride solution using D 2 EHPA in the MRDC extraction column. The numerical analysis for evaluating the column performance is applied to describe mass balance equations. Four mathematical models (backflow, forward mixing, plug flow, and axial dispersion) are investigated to compute the mass transfer coefficients of the dispersed phase. The solvent extraction experiments showed that the optimum zinc transport efficiency in rotor speed of 410 rpm in this column is equal to 98.85% and 99.85 for extraction and stripping stages, respectively. The model's achievement is compared with the solvent extraction data and a significant validity is obtained by coupling the forward mixing approach. The mathematical modeling expresses that the coefficients of axial dispersion and backflow based on the continuous phase increase by an increase in the rotor speed and inlet continuous phase rate. While these coefficients reduce at a higher inlet dispersed phase rate. The FMM method is preferred to predict the reactive mass transfer rate in the MRDC column due to the lowest relative deviation. The experimental study and mathematical modeling in this report provide beneficial information about the metallurgical industry to design solvent extraction equipment. phase mathematical modeling reactive transfer and continuous phase this transfer forward mixing model and for


Introduction
Zinc extraction from the industrial residue is substantial because of usages in electrical equipment and the chemical industries such as cosmetics, rubber, pharmaceuticals, and batteries 1,2 . Therefore, different methods have been investigated for the recovery of zinc ions from effluents, such as solvent extraction 3 , ion exchange 4 , and precipitation 5 . The several benefits especially less expensive setup, more product purity, and wide selectivity are the features of the solvent extraction method 6 . In recent decades, different acidic extractants such as D2EHPA 7 , PC-88A 8 , Cyanex272 9 , Cyanex302 10 , and Cyanex921 11 have been commonly suggested for Zn (II) extraction from sulfate medium. The selective extraction of zinc from manganese and cadmium in the hydrochloric acid media using D2EHPA has been investigated by Jafari and coworkers 12 . By considering the impact of contact time, equilibrium pH, extractant concentration, O/A ratio, and temperature on the solvent extraction process, they investigated the required theoretical stages for zinc extraction as well as the thermodynamic behavior of chemical reaction. Keshavarz Alamdari and co-workers studied the separation of zinc and cadmium from the synthetic sulfate leach solution by the synergistic effect of MEHPA and D2EHPA 13 . The experimental work on the zinc extraction from hydrochloric acid solution by coupling the effects of Cyanex 272 and N1923 was reported by Jia and co-workers, which compared with the zinc (II) extraction, the insignificant impact of synergism solvent extraction on the cadmium (II) was observed 14 . A.M.I. Ali and co-workers performed the solvent extraction experiments for extracting the zinc ions with the Cyanex272 in the mixer-settler unit 15 . The feasibility of the proposed flow sheet for the elimination of zinc from industrial waste solutions was also conducted in a 15 stage-horizontal mixer-settler unit.
Several mass transfer devices have been developed to perform solvent extraction tests 16 . The amount of turbulence in the liquid-liquid systems and the interfacial area is directly proportional by enhancing the efficiency of the reactive extraction rate 17 . Mechanically rotating columns are extremely applied in the hydrometallurgy, pharmacy, and nuclear industry. Among various types of these columns, the RDC column achieves acceptable extraction efficiency because of its high throughput. The MRDC column with the perforated configuration is one of the upgraded versions of the RDC column, which improves the distribution of the droplets and mass transfer performance 18,19 .
The column internal diameter and the total height of the extractor are the main parameters in order to evaluate, design, and scale up the extraction columns. Hydrodynamic characteristics comprising droplets behavior, holdup of the dispersed phase, slip velocity, characteristic velocity, and flooding points are vital to determine the column internal diameter 20,21 .
Knowledge of mass transfer evaluation is of essential significance for obtaining the column height, and theoretical models are used for calculating this function 22 . In recent years, the complication and importance of mass transfer rate were led to conduct several studies on mass transfer performance in the RDC columns [23][24][25][26] . The experimental and numerical studies on the evaluation of mass transfer data in the RDC columns centralizes on the physical systems, and simplified equations were utilized for interpreting the column performance without consideration of the chemical resistance to mass transfer 27-29 . The published correlations were also compared with the experimental results for investigating the mass transfer rate, which these investigations are still far from satisfactory to describe the reactive and non-reactive systems 30, 31 .
Optimum situations and used operating parameters determination is established to design the solvent extraction process by evaluating the concentration profile curves. The relation of overall mass transfer rates with the physical properties and operational variables can be calculated by applying the theoretical models such as plug flow, forward mixing, backflow, and axial diffusion 32, 33 . As a result, creating theoretical models can be acknowledged as one of the foremost influential and time-saving methodologies for assessing column performance.
Literature survey indicates that the PFM approach is used to compute the mass transfer predictions in the mixer-settlers units by assuming the ideal plug flow 34 . The non-uniformity of drops along the extraction column leads to deflection from the ideal conditions in which this phenomenon causes remarkable errors in the calculation steps. For this reason, the modified plug flow model is presented as backflow and axial dispersion models in terms of non-ideal state. The axial mixing coefficients based on the dispersed phase and continuous phase (Ed and Ec) are used at the axial dispersion model to convert some deviation in the plug flow model to high efficiency. When the active height of the mass transfer device divides by the tiny lengths in terms of the specific volumetric parts, the backflow mathematical model can be applied for predicting the reactive mass transfer rate. The backflow from the present stage to the prior stage is formulated by axial mixing in the mentioned model, which represented as α and β for continuous and dispersed phases, respectively. In order to approach the actual extraction conditions, the drop behavior and terminal velocity of dispersed phase droplets are considered in the forward mixing model, and consequently, the forward mixing model (FMM) is better than the other mentioned models.
Several investigations on the MRDC column with the perforated structure have been reported under physical systems in the literature 35-37 . But, scanty investigations have been observed for chemical reaction conditions in the mentioned column 30, 38 . However, for the first time, the solvent extraction technique has been utilized to remove zinc ions from the chloride leach 6 solution in the MRDC column. The mathematical modeling study in this extractor to evaluate the mass transfer performance is the new findings and novelty of this research in the field of heavy metals extraction. The analysis of reactive mass transfer rate based on the dispersed phase is developed by associating four theoretical mass transfer models and hydrodynamic parameters for zinc transport in the extraction and stripping stages.

Physical properties
In this study, densities of the continuous and the dispersed phases have calculated by the pycnometer methodology. The viscosities of both fluids were also determined via DVI-Prime viscometer. The interfacial tensions in extraction and stripping stages were obtained with a Krüss tensiometer. Literature survey expresses that the measurement of physical properties especially 9 interfacial tension is approximately invalid due to uncertainty existence along the column 39 . As a result, the mean values of Zn (II) concentration in the continuous and dispersed phases at the inlet and outlet of the MRDC column were obtained to remove the measurement errors in this phenomenon. Table 1 lists the physical properties of dispersed and continuous phases for both mass transfer directions. Table 1 3

.3. Measurement of hydrodynamic parameters
The aqueous phase containing zinc chloride solution was first pumped into the main section of the extractor, and then the mass transfer device was filled by the dispersed phase for performing the solvent extraction process. By considering the digital camera (Nikon D5000), droplets size has been determined by a photographic approach in which ImageJ software has been applied to interpret the image analysis. Photograph analysis method using this software is completely described in Ref 40 . The following relation is presented to obtain the equivalent diameter of the non-spherical droplets, where d1 and d2 indicate the major and minor axes, respectively: The average droplet size has then calculated as follows: In this analysis, the inlet and outlet valves of the MRDC column were rapidly closed while the steady-state situations were established. The continuous and dispersed phases were separated over time from each other and then the holdup values were calculated by the following equation: The relative speed of dispersed phase drops within the column is described as the slip velocity.
This hydrodynamic variable is defined as the linear velocity summation of both fluids during the operational situations that are given as below: The interfacial area was also measured for Zn (II) reaction in the extraction and stripping stages by the following equation:

MRDC column experiments
The solvent extraction behavior for zinc extraction in both mass transfer directions has been

Results and discussion
The dispersed phase holdup, droplets behavior, and slip velocity are essential to study the mass transfer characteristics in the continuous extraction columns. Prediction of the mass transfer data using the mentioned models has been examined to interpret the column performance under chemical reaction conditions. In this research, each class's mass transfer coefficients have been determined with the technique of fitting concentration profiles. Also, the concentration profiles of organic and aqueous phases have been evaluated step-by-step through the column length. The quantities of the continuous and dispersed phases in these models are represented by X and Y, respectively. The experimental data for the holdup of the dispersed phase, average droplet diameter, slip velocity, zinc transport efficiency, and the values of AARE for four theoretical models are listed in Tables 2 and 3 for the extraction and stripping stage, respectively. Table 2   Table 3 The poor predictive capability of PFM is referred to the fact that the MRDC column performance for reactive extraction systems does seriously deviate from ideal plug conditions. In these conditions, the modified models by considering the non-ideality state, namely the axial dispersion and backflow mathematical models have been utilized to reduce some errors. When the reactive mass transfer rates are calculated by these models, the quantities of the absolute average relative error for X and Y parameters decrease. As can be seen, the relative deviation error of mass transfer coefficients is highly decreased in both mass transfer directions by considering the forward mixing model due to adding the impact of droplet size distribution. At constant inlet phase rates, Fig.2 shows the rotor speed impacts on the holdup values, mean drop size, and slip velocity in the extraction and stripping stages. The break-up of the large droplets into smaller ones has occurred at higher rotor speed due to increasing the shear forces.
Consequently, this phenomenon leads to enhancement of residence time of drops along the extraction column, which the holdup values enhance with an increase in the rotor speed. The slip velocity decreases with an enhancement in the disc speed based on the decrement in the mean drop size and dispersed phase at higher rotor speeds. Dispersed phase holdup in the extraction stage is higher than the stripping stage at similar operating situations. The resistance to the mass transfer direction and the interfacial tension deviation leads to appear this event. It was also found that the slip velocity and average droplet size in the extraction system are lower than in the stripping system. The hydrodynamic parameters for reactive systems against the continuous phase rate are indicated in Fig.3.     The experimental results and numerical modeling based on the forward mixing approach indicated that the relative deviations in the calculation of concentration profiles are acceptable owing to the perforated structure of discs in this column. As a result, the profile changes using the curve fitting process and forward mixing model are shown in Fig.6. It was observed that the predictive of actual data with this model based on adding the effects of the droplet size distribution were near enough to the regression line.     In this study, the forward mixing model has been used for the numerical solution of mass transfer coefficients owing to the best performance among other mathematical models (see in Table 4).
The numerical results to interpret the reactive mass transfer rate as a function of rotor speed, continuous and dispersed phases flow rate are represented in Fig.11. It was found that the rotor speed is one of the critical operating variables in the MRDC column for reactive systems. This parameter has a significant effect on the dispersed phase holdup, mean drop size, interfacial area, and slip velocity in which the surface area increment for performing the chemical reaction causes the extraction system in this column to move towards the maximum separation process.
Although the rigid state creation for solvent phase droplets at high rotor speeds causes a decrease in the overall mass transfer coefficients, increasing the required surface area provides high efficiency for the overall mass transfer rates. As mentioned, no remarkable effect on the dispersed phase and mean drop size was observed with changes in the flow rate of the dispersed and continuous phases. Nonetheless, the impact of holdup quantities on the required area is more than the average drop size. Consequently, the interfacial area for zinc (II) chemical reaction in the extraction and stripping stages slightly increases with an enhancement in the flow rate of both phases. The mass transfer coefficients improve due to inner circulation within the drops.
Thus, the mass transfer performance is increased at higher values of Qd and Qc. According to this figure, the output results indicate that the effect of rotor speed and dispersed phase rate on the reactive mass transfer data is more significant than the continuous phase rate.  Table 4 4

. Conclusion
In this study, the MRDC column performance for Zn (II) extraction from chloride solution has been examined. The operational conditions of the mentioned column were reviewed by changing the rotor speed and the flow rate of the inlet phases. The desirability achieves for zinc transport efficiency in both mass transfer directions with an increase in the operating parameters.