Estimation of droplet size distribution by using maximum entropy programming and population balance equations in pulsed disc-doughnut column
Introduction
The size distribution of the dispersed phase droplets is the significant measuring of interfacial area for operational conditions, including mass transfer with simultaneous chemical reactions (Asadollahzadeh et al., 2021c, d; Shakib et al., 2021). In the design of extraction equipment, the extractor’s height and diameter must be determined for a desired mass transfer and admissible stream rates in all phases (Asadollahzadeh et al., 2021b; Saremi et al., 2021; Weber et al., 2019). Drop size significantly affects the holdup of the dispersed phase and the maximum throughput. Moreover, the interfacial area of mass transfer can be resolved, utilizing drop size and holdup together (Asadollahzadeh et al., 2017b; Asadollahzadeh et al., 2021a, e). The equipment utilized for such operation is extensively characterized relying upon the strategy for dispersion of the phases, as gravity extractors, mechanically agitated (Asadollahzadeh et al., 2016a; b; c; 2017a; Shakib et al., 2020a; Torab-Mostaedi et al., 2017) and pulse extractors (Amokrane et al., 2016; Shakib et al., 2020b). The determination of an appropriate extractor includes consideration of the scope of conflicting prerequisites (Jahya et al., 2009; Torkaman et al., 2017a). Morello and Poffenberger introduced a helpful study of extractor types (Morello and Poffenberger, 1950). Pratt extended it in an extractor selection chart giving numerical ratings for each range (Pratt, 1954). Then, it was updated more recently by Pratt and Stevens (1991).
Pulsed equipment, having a place with the class of pulse agitated extractor, finds comprehensive utilization in solvent extraction, especially in treating toxic materials (Sen et al., 2018; Torab-Mostaedi et al., 2011a; Torkaman et al., 2017b, 2015). The nonattendance of mechanical moving parts inside the column and the facility of creating pulsing remotely away from the column are two explicit points of interest of the pulsed column (Torab-Mostaedi et al., 2011b). The pulsed disc-doughnut column (PDDC) has recently been introduced in various metallurgical applications worldwide, contrasted to mixer-settlers, which are habitually utilized in the mining industry (Torab-Mostaedi et al., 2012). This column is attractive from safety and economic standpoints, specifically its effortlessness of configuration, less space utilization, higher throughput, and no internal moving parts (Torab-Mostaedi et al., 2011c). Li et al. illustrated the new pulsed disc-doughnut column with Tenova kinetics internals. It was observed that this column was performed under lower holdup and larger Sauter-mean drop size compared to the standard column (Li et al., 2018). Performance evaluation of this column in pulsed and non-pulsed conditions showed that the agitation intensity plays a vital role in achieving the desired level in the column (Wang et al., 2016). A chemical system containing nitric acid in the aqueous phase and a solvent of tributyl phosphate in the organic phase was used to investigate the behavior of the PDDC column. The results showed that the PDDC column performance is better in this condition than the pulsed sieve plate column (Sarkar et al., 2019).
The modeling of this column and other liquid–liquid extraction columns is possible based on the population balance of droplets and CFD simulation. Many factors need to be considered in modeling and simulation analysis, such as coalescence and droplet breakage, different time of droplet size movement, and reactive mass transfer processes (Bart et al., 2020; Hlawitschk et al., 2016). The optimization of modeling and CFD simulation has been done based on various methods such as one primary and one secondary particle method (Attarakih et al., 2013), integral formulation of the population balance equation (Attarakih, 2013), simplified volume averaged models (Buffo et al., 2016), and differential maximum entropy method (Attarakih and Bart, 2014). In the simulation studies of PDDC column with droplet balance and computational fluid dynamic simulation, the coefficients of improved breakage and coalescence kernels were proposed (Amokrane et al., 2016).
Since many parameters are effective in the simulation of the extraction column, new techniques in the modeling process, such as the maximum entropy technique are suggested.
This manuscript depicts an examination of the measurements drop size distribution across a pulsed disc-doughnut column as a function of the amplitude of pulsing, frequency of pulsing, and inlet flow rates with three various liquid feeds (standard feed systems). A comparison of maximum entropy models and droplet population balance in this column has been investigated to evaluate the droplet size distribution. So far, no research has been published on studying two models for the prediction droplet size distribution in the PDDC column. It is the research innovation that, in the following sections, these models are described.
Section snippets
Modeling of size distribution of the dispersed phase droplets
The dispersed phase droplets' size distribution in the solvent extraction process is one of the imperative parameters needed for the fundamental examination of extractors. More data about the size distribution is crucial for liquid feed systems' design and performance. This paper uses the maximum entropy strategy (Asadollahzadeh et al., 2015; Stecker, 2008) and the population balance approach (Maluta et al., 2021; Tu et al., 2021; Yu et al., 2020a, b) to get the size distribution in a pulsed
Materials
The dispersed phases in this study were toluene/n-butyl acetate/n-butanol with 99.5% purity. The continuous phase was distilled water. The data of physical properties are shown in Table 1.
Equipment
The pulsed disc-doughnut column (Fig. 2) consists of a 74 cm height with an inner diameter of 7.6 cm. In the main column section, thirty pairs of discs-doughnuts were designed alternately, 1 cm space, 2 cm compartment height. The discs ∼0.67 cm ID, and the donut apertures ∼0.36 cm, showed the open free areas
Results and discussion
This section investigates the impact of the operational parameters, including inlet feed phase flow rates and pulse intensity.
Conclusion
A progression of experiments has been performed to examine the impact of operational parameters and physical properties on the drop size distributions in a pulsed disc-doughnut column. The drop size distributions anticipated from the statistical methodology are compared with the experimental information. The empirical correlations are proposed to portray Lagrange multipliers in the maximum entropy function regarding operational variables and physical properties of the feed streams. The maximum
Authors’ contributions
Rezvan Torkamn analyzed and interpreted the data. Mehdi Asadollahzadeh performed visualization of data and were major contributors in writing the manuscript. Meisam Torab-Mostaedi both gave helpful advices on the frame and schematic of this research. All authors read and approved the final manuscript.
Funding
This work was not supported by any funding.
Data availability
The datasets used and/or analyzed during the current study are available from the corresponding author on reasonable request.
Competing interests
The authors declare that they have no competing interests.
Ethics approval and consent to participate
Not applicable.
Consent for publication
Not applicable.
Declaration of Competing Interest
The authors report no declarations of interest.
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2022, Progress in Nuclear EnergyCitation Excerpt :The design of the PDDC column involves that the chemical engineering expertise to determine the main dimension of the column and the mechanical engineering expertise to design the column and its pulsed section (Lo and Schweitzer, 1979). The volumetric flow rates, drop size distribution, dispersion phase hold-up, and hydrodynamic velocities are all needed to determine the column's diameter and height (Asadollahzadeh et al., 2012, 2015, 2016a, 2016b, 2017a, 2017b, 2021a, 2021b, 2022; Torab-Mostaedi et al., 2017; Torkaman et al., 2017a; Ghaemi et al., 2021; Shakib et al., 2021, 2022; Hemmati et al., 2022). The dispersion phase hold-up amount is required to compute slip velocity and interfacial zone per unit volume.