CHEMICAL ENGINEERING TRANSACTIONS VOL. 61, 2017 A publication of The Italian Association of Chemical Engineering Online at www.aidic.it/cet Guest Editors: Petar S Varbanov, Rongxin Su, Hon Loong Lam, Xia Liu, Jiří J Klemeš Copyright © 2017, AIDIC Servizi S.r.l. ISBN 978-88-95608-51-8; ISSN 2283-9216 Modeling of the Distribution of Electric Discharge Between Metal Balls in the Aqueous Solution Igor S. Nadezhdin*,a, Aleksey G. Goryunova, Flavio Manentib aNational Research Tomsk Polytechnic University, Institute of Physics and Technology, Department of Electronics and Automation of Nuclear Plants, Tomsk 634050, Russian Federation bPolitecnico di Milano, Dept. di Chimica, Materiali e Ingegneria Chimica „Giulio Natta“, Piazza Leonardo da Vinci 32, 20133 Milano, Italy kun9@list.ru In this work it is planned to apply the MPC approach for the optimization of energy consumption of electroerosion process for water purification. For this purpose is necessary to develop a mathematical model of the water purification process. Distribution of electrical discharges between the metal balls in contaminated water is one of the stages of electroerosion process for water purification. In this paper a mathematical model of the distribution of electrical discharges between the metal balls in the aqueous solution is presented. Method of probabilistic cellular automata was used to develop a mathematical model, since the distribution of electrical discharges is a stochastic character. The modeling results were compared with experimental data. 1. Introduction Nowadays, the tasks of creation and upgrading of technologies and methods for the purification of air and water from pollution are of scientifically interest. CO2 emissions in the atmosphere are increasing due to the large use of natural gas and coal for the production of electric energy. Reducing CO2 content in the atmosphere is an actual problem. Azmi et al. (2016) proposed a combined process of adsorption and gas hydrate formation as an alternative approach for the separation of carbon dioxide (CO2) content from gas stream. The objective of this research was to study adsorption isotherms of the CO2 onto the synthesized calcium oxide (CaO) via a static volumetric method at 2 °C and at different amount of water ratio. Zarogiannis et al. (2016) studied a systematic approach for the preliminary screening of binary amine mixtures as CO2 capture candidates considering several important properties as selection criteria. A study of the population of any country with quality classification of drinking water and the treatments used for industrial wastewater is also an urgent task. Today a large number of water purification methods are used: reverse osmosis, reagent coagulation, aeration, sedimentation, distillation, etc. Each of the foregoing methods have different advantages and disadvantages. Lutchmiah et al. (2014) reviewed problems and prospects the use of reverse osmosis membranes for wastewater treatment. The main disadvantages dealing with these methods are: a large consumption of reagents, the need for periodic replacement of membranes, high costs of reagents and membranes, large areas needed for equipment installation and the ineffectiveness to clean water sources by toxic substances, like arsenic, and dissolved salts. In the scientific literature, several articles studied different optimization approaches and the development of automatic control systems. In the study of Manenti et al. (2015) an optimization work on the reverse osmosis module with recycle is developed using MPC approach. However, the proposed control and optimization system cannot eliminate the disadvantages inherent to these methods. Therefore, it is necessary to develop a new energy-saving and resource-efficient methods for water purification. One of these methods, based on the use of electric energy, is the water purification by electroerosion process (EDM process) of metal balls. The electroerosion process of metal workpieces has been known for more than 70 years, but the use this method for water purification is a novelty. This new process of electroerosion for water purification has several advantages: it is based on cheap raw materials (metal balls) and characterized by low energy consumption. Since this is a new and poorly known method, it is necessary the study of a suitable DOI: 10.3303/CET1761087 Please cite this article as: Nadezhdin I.S., Goryunov A.G., Manenti F., 2017, Modeling of the distribution of electric discharge between metal balls in the aqueous solution, Chemical Engineering Transactions, 61, 535-540 DOI:10.3303/CET1761087 535 mathematical model for the optimization and the development of a control system. The MPC approach for development of control systems was applied. The mathematical model of the process is necessary for modeling, optimization and development of control systems based on the predictive model. There are papers devoted to the modeling electrical discharge machining (EDM) of metal products by using machines, as well as papers devoted to the modeling individual stages of the electric erosion process. Wuyi Ming et al. (2014) proposed, for electrical discharge machining process, a hybrid intelligent process model, based on finite-element method and Gaussian process regression. In order to predict material removal rate and surface roughness a model of single-spark EDM process has been done based on finite-element method, considering: the latent heat, the variable heat distribution coefficient of cathode and the plasma flushing efficiency. However, these models are not suitable for the simulation of electroerosion process for water purification, because there is a difference in the distribution of the discharge energy, and consequently in the products erosion formation. The aim of this paper is to develop a mathematical model of distribution of electrical discharges between the metal balls in the aqueous solution. 2. Process description The electroerosion process for water purification is only one stage of technological process of water purification from contaminants. The main destination of this method is that convert soluble salts into insoluble precipitate that can be easily removed from water. The scheme of water purification is shown in Figure 1. Figure 1: Possible technological scheme of the water purification plant Technological scheme (see Figure 1) includes a tank with feed water, electrical discharge (electro erosion) of water purification plant, high pressure pump, membrane module for filtration of erosion products from water. The membrane module may be replaced by a settler tank in which the electroerosion products can be deposited. It will allow reduce the cost of water purification, but will worsen the degree of water purification. In this paper was discussed processes in the electrical discharge in water purification plant. The electrical discharge equipment of water purification plant consists of electrical pulse generator (PG) and the tank-reactor Two electrodes are located in the reactor-tank and connected to the PG. The interelectrode gap, in the tank- reactor, is filled with metal balls and purified water. Then, electrical impulses of short duration passed through the layer of metal balls. Electrical discharges arise between the balls when the electrical impulses pass. These discharges are characterized by high energy. As a result, electrical erosion process occurs on the surface of metal balls. The separated erosion products are highly dispersed particles of metal. The size of the dispersed particles is about 1 - 100 nm. Nadezhdin et al. (2016) presented the mathematical model and the mechanism of formation of electroerosion holes and electroerosion products. The erosion products were oxidized by water. As a result, metal hydroxides and oxides were formed and were active coagulants. The formed metal hydroxides and oxides efficiently adsorbe impurities contained in water and form insoluble salts that can precipitated. Nadezhdin et al. (2016) presented the chemical reactions that describe chemical processes occurring in the tank-reactor and a mathematical model able to describe chemical reactions kinetic of electroerosion process for water purification. The optimum power of electrical pulses can be determined. The maximum amount of erosion products is formed as result of these impulses. This will optimize the energy consumption of electroerosion process for water purification. The task of increasing the electrical discharges between balls ("tracks" discharges) as result of one electrical impulse arises. The mathematical model of the distribution process of electrical discharges between the metal balls into aqueous solution is planned to be applied to optimize. 536 3. Mathematical modeling of the distribution of "tracks" discharges 3.1 Application of probabilistic cellular automata for modeling of "tracks" discharges Place initiation and trajectory of the distribution of electrical discharges between the metal balls in the interelectrode gap of the electrical discharge have a stochastic character. Method of probabilistic cellular automata was used to develop a mathematical model of distribution of electrical discharges between the metal balls ("tracks" discharges). The metal balls are located in layer a certain height in the plant, depending by the backfilling mass. As can see in Figure 2, one ball is adjacent to 6 balls from its layer and to 3 from above and below layer. Thus, one ball in the plant may have up to 12 of adjacent balls. Figure 2: Location of metal balls in tank of the electrical discharge of water purification plant Considering each layer, the balls separately it can be seen that the distribution of balls satisfies of the hexagonal lattice cellular automata, which shown in Figure 3а. The metal ball is located in each cell of the hexagonal lattice (see Figure 3a). The hexagonal lattice is undergoing certain transformations for computer implementation (see Figure 3b). The cells of lattice can take on two values during modelling. The first value indicates that the discharge has passed through the cell (the ball), the second value indicates that the cell (the ball) was not exposed to electric discharge. a) the initial hexagonal lattice; b) lattice, which being implemented in a computer; c) scheme for determining the position of the ball in the cell Figure 3: Location of metal balls in the layer of the backfilling The interelectrode gap is represented as a series-parallel circuit of the resistors. The metal balls and aqueous solution filling the space between the electrodes are represented as a circuit of resistors. Between the balls there is a thin film of water, regardless of as tightly they are adjacent to each other (about 1 µm). Distribution of electrical discharges between the metal balls depends by the thickness of the water film between balls. The thinner the film of water, the smaller the resistance. The electrical discharge is distributed along the path with the least resistance. The thickness of the water film between the balls depends on the location of the balls in the plant. The balls are arranged in an arbitrary way in the apparatus. Some balls are more firmly pressed to each other. Accordingly, between them is smallest thickness of the water film. The balls are misaligned in the some direction while located in cells of the hexagonal lattice. In addition, the presence of "tracks" discharges between the potential electrode and the balls is a random process. This process is dependent on the location of the balls in the plant. Initially defined by a limiting distance between potential electrode and balls in the near to it row (lel-ball). If the distance between electrode and balls is less than the specified (lel-ball), then there is the breakdown. Thus, emerging as much of "tracks" discharges, as much as balls located at a distance of less lel- ball from the electrode. Balls location in the plant has a stochastic character. It is assumed that the center of each ball can be in some permissible region a* × b* (see Figure 3c). The dimensions of this region are defined by using the following expressions:         * * , , ball ball film ball ball film a A d A d l and b B d B d l (1) a b l1 l3 l2 l1 l2 l3 А B R a* b* Δa Δb c 537 where А, B is parameters dependent on the thickness of the water film between the balls (mm); lfilm is the thickness of the water film between the balls (mm), about 10-3 mm. Further, given by the number of possible positions of the center ball in this region, Δa and Δb are calculated by using the following Eq(2)     * * max max a b a ba and b n n (2) where max a n and max b n are the maximum number of possible positions of the center ball in the allowable region a* × b*. Placing of the center ball in the one of the possible positions is performed using a function that returns two random integers from 0 to max a n and max b n , which specify the coordinates of the center ball in region a* × b*. 3.2 The mathematical model of the distribution of "tracks" discharges The number of balls in one layer of backfilling and height of the entire layer of backfilling in the plant are necessary to understand how to determine the trajectory and the number of "tracks" discharges. Height of the entire layer of backfilling in the plant is calculated with the Eq(3)  1layer ball ball N h N (3) where Nball is the number of balls in the backfilling (pcs); 1 ball N is the number of balls in one layer of the backfilling (pcs). The number of balls in one layer of the backfilling can be found by using the Eq (4): 1 2   pl pl ball ball l w N d (4) where dball is diameter of the balls in backfilling (mm); lpl is length of the plant (mm); wpl is width of the plant (mm). The number of balls in backfilling is determined using the Eq(5)  fill ball ball m N m (5) where mfill is mass of backfilling (kg); mball is mass of one ball (kg). The mass of one ball can be determined using Eq(6)         34 3 ball ball ball ball ball m V R (6) where Vball is volume of ball (m3); ρball is density of ball (kg/m3); Rball is radius of ball (m). As said earlier, the number of arising "tracks" discharges, depends on the number balls located at a distance of less lel-ball from the electrode. Electric current flow through the least resistance path. One of the parameters of each cell is the conductivity of the film water. The conductivity of the film of water determines the current flowing between the two balls of backfilling, and it is defined by the formula Eq(7), which follows from Ohm's law:      ball ball U S I l (7) where I is current of conduction (A); U is voltage of pulse (V); S is sectional area of channel (m2); γ is electrical conductivity of the aqueous solution (Ω-1·cm-1); lball-ball is distance between the balls. The balls have a probabilistic character of the distribution in the plant, so the distance between (lball-ball) the balls is arbitrary values, thereby changing the resistance of part an electrical circuit (of water film). Since the distribution of discharges is due to potential electrode and ground electrode, then discharge has three potential areas for further spread in balls layer (see Figure 3a,b). The following conditions are checked to find the minimum distance between adjacent balls:                      1 2 1 3 2 1 2 3 3 1 3 2 & & & ball ball ball ball ball ball ball ball ball ball ball ball ball ball ball ball ball ball ball ball ball ball ball ball l l l l l l l l l l l l (8) finding the minimum distance (   mini ball ball ball ball l l ) appropriate cell changes its state and process is repeated until the ground electrode is not reached. The length of "tracks" discharge depends by electric field intensity (E) in the interelectrode gap. The electric field intensity decreases by the exponential law. The electric field intensity 538 takes a value less than value of the critical intensity (Ecrit) at a certain distance from the potential electrode. When E