Produção Nacional
Revisado por pares
Computational modelling approach for the optimisation of a pulsed electric field system for liquid foods
2018; Institution of Engineering and Technology; Volume: 13; Issue: 3 Linguagem: Inglês
10.1049/iet-smt.2018.5311
ISSN1751-8830
AutoresEduardo J. Araújo, Ivan J. S. Lopes, J.A. Ramírez,
Tópico(s)Radiation Effects and Dosimetry
ResumoIET Science, Measurement & TechnologyVolume 13, Issue 3 p. 337-345 Research ArticleFree Access Computational modelling approach for the optimisation of a pulsed electric field system for liquid foods Eduardo J. Araujo, Corresponding Author Eduardo J. Araujo edu_jose0701@yahoo.com.br Graduate Program in Electrical Engineering, Federal University of Minas Gerais, Av. Antônio Carlos 6627, 31270-901 Belo Horizonte, BrazilSearch for more papers by this authorIvan J. S. Lopes, Ivan J. S. Lopes Department of Electrical Engineering, Federal University of Minas Gerais, Av. Antônio Carlos 6627, 31270-901 Belo Horizonte, BrazilSearch for more papers by this authorJaime A. Ramirez, Jaime A. Ramirez Department of Electrical Engineering, Federal University of Minas Gerais, Av. Antônio Carlos 6627, 31270-901 Belo Horizonte, BrazilSearch for more papers by this author Eduardo J. Araujo, Corresponding Author Eduardo J. Araujo edu_jose0701@yahoo.com.br Graduate Program in Electrical Engineering, Federal University of Minas Gerais, Av. Antônio Carlos 6627, 31270-901 Belo Horizonte, BrazilSearch for more papers by this authorIvan J. S. Lopes, Ivan J. S. Lopes Department of Electrical Engineering, Federal University of Minas Gerais, Av. Antônio Carlos 6627, 31270-901 Belo Horizonte, BrazilSearch for more papers by this authorJaime A. Ramirez, Jaime A. Ramirez Department of Electrical Engineering, Federal University of Minas Gerais, Av. Antônio Carlos 6627, 31270-901 Belo Horizonte, BrazilSearch for more papers by this author First published: 01 May 2019 https://doi.org/10.1049/iet-smt.2018.5311Citations: 1AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract The microbial inactivation and specific energy of a pulsed electric field treatment system are dependent on the electric field distribution and treatment time since they have opposite behaviour in relation to these parameters, featuring a problem of multi-objective optimisation. This study proposes a computational methodology capable of providing Pareto optimal solutions for these two objectives, using a coupled electrical-thermal model, solved by COMSOL, which has been integrated to a multi-objective algorithm NSGA-II implemented in MatLab. The simulations were run for a computational design of experiment with the following variables: applied voltage, treatment time and the internal electrode radius (three levels for each one). In the post-processing analysis, the Pareto curves were plotted for two typical microorganisms of grape juice: E. coli and S. aureus, providing a set of solutions in terms of the log of the survival rate versus the specific energy. The methodology enables the decision maker to select the best solution from the Pareto curves as a function of a required microbial inactivation and energy features. 1 Introduction Pulsed electric field (PEF) is an alternative technique for the treatment of liquid foods, applicable in industrial treatment of fruit juice, beverages and dairy products [1], being capable of inactivating microorganisms while maintaining physical, chemical and nutritional characteristics of food products [2]. Considering that there is no significant loss of flavour and taste [3], PEF is more attractive treatment for fresh foods than conventional thermal method [4]. It is widely agreed that PEF has a great potential as an alternative technology for thermal pasteurisation [5]. The PEF method involves the application of high voltage electric pulses producing high electric field levels (15–80 kV/cm) for a short time (1–100 μs) [6, 7]. On the microbial cell, the electric field may lead to irreversible damage of the cell membrane and end in cellular death [8]. The efficiency of the PEF treatment depends on the process parameters such as electric field strength and treatment time, microorganism parameters, such as type, size and growth phase, and physical properties of the liquid [9]. A treatment chamber is a key component of the system and has a significant influence on the electric field distribution [10]. It can be either static or continuous flow and the design of a continuous chamber is a gradual development from static [11]. One of the goals for designing a treatment chamber is to provide a high and uniform electric field distribution, without dielectric breakdown [12]. An electric field analysis using finite-element method associated with an optimisation technique was used in [13] for coaxial and parallel plate treatment chambers in order to obtain high-intensity electric field and spatially uniform distribution. The analytical expression of the electric field intensity between the two electrodes for the coaxial chamber used in this study is given by [11] (1) where E is the electric field intensity [], V is the applied voltage [], r is the radius at which electric field is measured [], is the external electrode radius [] and is the internal electrode radius []. Although PEF is intended to be a non-thermal technique, a temperature rise is present due to the electric current flowing in the liquid food (ohmic heating) [14]. The temperature rise is dependent on the electric conductivity, electric field strength, treatment time and the liquid thermal properties. The temperature range of the liquid must be, preferably, in the range of 10–45°C to preserve its properties, in particular nutritional features, flavour and taste [15]. Specific energy is an important parameter to be considered in a PEF system. It is calculated as a function of the treatment chamber volume, liquid conductivity, electric field strength and treatment time [16]. Depending on the product, treatment chamber geometry and processing parameters, the specific energy requirements vary in a broad range from 50 to several hundreds of kJ/kg [17]. The specific energy determines the extent of temperature rise of the food and is one of the major cost factors of the technology [18]. Therefore, it is important to keep this parameter as low as possible [10]. Considering that the microbial inactivation depends mainly on electric field strength and treatment time, an increment in the microbial inactivation due to an increase in the electric field strength or treatment time leads, inevitably, to an increase in the specific energy. Thus, microbial inactivation and specific energy have conflicting behaviour with respect to electric field strength and treatment time, which can be understood as a multi-objective optimisation problem. Some studies associate the microbial inactivation with specific energy for different electric field and/or temperature distribution in order to analyse the treatment efficiency. The inactivation of E. coli in Ringer's solution in relation to specific energy at different treatment temperatures for electric fields of 16 and 20 kV/cm was investigated by Toepfl et al. [19]. The microbial inactivation as function of treatment time and specific energy for three different microorganisms of grape juice was analysed by Huang et al. [14]. Previous modelling studies on the PEF technology have coupled electric-thermal analysis and some of them include also the fluid flow. The influence of temperature on microbial inactivation was investigated in a static parallel electrode treatment chamber with tempered electrodes using a coupled electric-thermal analysis by Saldaña et al. [20]. An interactive algorithm able to optimise geometric shape of a co-linear electrodes arrangement based on electric field analysis and a performance indicator was presented by Knoerzer et al. [21]. Huang et al. performed multiphysics simulations in order to compare colinear and coaxial treatment chambers. The numerical simulations indicated a better performance of the coaxial arrangement in relation to electric field and temperature distribution [22]. Jaeger et al. carried out studies aimed at improving the electric field distribution, flow profile and temperature distribution through the insertion of stainless steel and polypropylene grids in a co-linear treatment chamber using numerical simulations in COMSOL™ [23]. In none of the previous works, a multi-objective approach integrated to a coupled electrical-thermal analysis has been used. In an earlier study presented by the authors [24], a 3D finite-element algorithm implemented in MATLAB™ was used for numerical calculation of the electric field in three geometries (cylindrical needles, parallel plates and coaxial arrangements) with a generic liquid containing E. coli. The coaxial geometry showed better set of optimal solutions in comparison with other arrangements, but the thermal analysis was not performed. This paper main contribution is to develop an improved computational methodology for the optimisation of the treatment chamber considering microbial inactivation and energy efficiency as conflicting objectives. A numerical methodology based on electric-thermal coupled study and multi-objective optimisation for a simultaneous analysis of microbial inactivation and specific energy is capable of providing optimised solutions for the treatment chamber and process variables in terms of a Pareto set, which gives the engineer a broader range of options. The computational methodology is applied to a coaxial geometry for two typical microorganisms of grape juice: S. aureus and E. coli. The computational simulations are performed using MATLAB™, COMSOL™ and the interface Livelink™. A NSGA-II (non-dominated sorting genetic algorithm) multi-objective algorithm is used to obtain the optimal curves for microbial inactivation and specific energy for each microorganism. This paper is organised as follows. In the next section, the models and methods are discussed, including the treatment chamber and material properties, modelling of the microbial inactivation, governing equations of the physical processes (electrical and temperature calculation) and the computational modelling involving the coupling of COMSOL and MatLab. In Section 3, the results are presented and discussed. It includes 3D pictures of the electric field and temperature distribution within the treatment chamber and Pareto curves of the specific energy versus the log of survival rate for the S. aureus and E. coli. Finally, Section 4 concludes the work. 2 Models and methods 2.1 Treatment chamber and material properties A static treatment chamber was considered in this study. As presented in Fig. 1, it consists of a coaxial arrangement with two stainless steel electrodes. Two toroidal rings also made of stainless steel on the top and bottom of internal electrode associated with two PTFE (Teflon™) insulators were modelled in order to reduce the electric field strength near to the internal electrode borders. The geometry has also a filling port at the top insulator for injection of the liquid into the treatment chamber. The effective volume of the treated liquid varies from 22.9 to 28.2 ml, depending on the internal electrode radius, which is defined as a design variable, ranging from 0.5 to 1.0 cm. Fig 1Open in figure viewerPowerPoint Cross-section view of the treatment chamber with coaxial electrodes arrangement The electrodes and toroidal rings were modelled based on the material properties from COMSOL™ database for stainless steel: thermal conductivity (k) = 44.5 W/m K, density () = 7850 kg/m3 and heat capacity () = 475 J/kg K. The insulators (top and bottom) made of PTFE were also modelled following COMSOL™ database, with the following properties: k = 0.24 W/m K, and . The following thermo-physical properties of grape juice: thermal conductivity, thermal capacity and density, as a function of temperature, were taken from the literature [25-27]. The electrical conductivity was determined experimentally using a conductivity meter (CD-850 Conductivity meter, Instrutherm Ltd, São Paulo, Brazil) in grape juice samples (Maguary grape juice, produced in Araguari, Minas Gerais, Brazil). The data obtained as a function of temperature (20–50°C) were adjusted by regression using statistics toolbox of MATLAB™R2010. The electrical conductivity obtained is (R2>0.99) (2) where is the electrical conductivity dependent on temperature [] and T is the grape juice temperature [°C]. 2.2 Modelling microbial inactivation The design of effective PEF treatment systems involves the development of mathematical models to predict microbial inactivation. The microbial inactivation is evaluated by calculating the log reduction of the survival ratio (S), which represents the ratio between the number of microorganisms that survived to the treatment and the initial number of microorganisms. The implementation of kinetic inactivation into numerical models, which are able to provide electric field, flow velocity and temperature distribution inside the treatment chamber, is an inevitable trend that is essential for the development of PEF technology [28]. Some kinetic models have been proposed in the literature to describe microbial inactivation, such as Hulsheger, Peleg and Weibull distributions [16]. The Weibull distribution has been applied to describe survival curves of thermal and PEF-treated foods, according to (3). This model covers more than cycles reduction of microbial inactivation with more accuracy than other prediction models (Hulsheger and Peleg models), which are capable of covering few cycles reduction [17]. The Weibull model estimates the microbial inactivation through two parameters that are called scale parameter () and the shape parameter (). The scale parameter, with dimension of time, represents a characteristic time at which the survival function . The shape parameter represents the form of the curve: corresponds to a concave upward curve, corresponds to a concave downward curve and corresponds to a linear curve. Therefore (3) where S is the survival ratio and t is the treatment time . The parameter can be written as a function of the electric field strength using secondary models. These models were obtained from experimental data regression. Equation (4) represents a general form of third-order polynomial, which was the best fit of experimental data reported in [14] for S. aureus and E. coli in grape juice (4) where , , , are coefficients of the polynomial regression and E is the electric field strength []. Tertiary models were obtained by substituting the expression of from the secondary model in (3) and using the average value of the shape parameter (), as seen in the following equation [14]: (5) 2.3 Governing equations 2.3.1 Electric field calculations The governing equation for the electric potential is based on the charge conservation principle (6) where is the electrical conductivity dependent on temperature [] and V the electric potential []. Assuming no generation of electromagnetic forces, the relation between the electric field and the electric potential is described by (7) where E is the electric field strength []. Boundary conditions for (7) were set as zero for the grounded external electrode. The internal electrode and the rings equalisers were set following the rectangular pulse applied, with initial voltage V, pulse width 10 μs and frequency 100 Hz. 2.3.2 Temperature calculations The energy balance for a pure conductive heat transfer in a liquid energy balance is given by (8) where denotes the density [], is the specific heat capacity [], T is the temperature [], k is the thermal conductivity []. The heat source Q is due to the dissipation of energy by ohmic heating effects as expressed by [29] (9) where is the temperature dependent electrical conductivity [] and is the electric field strength []. The outer wall of the treatment chamber insulators was considered thermally insulated and the external electrode walls were set as conductive, applying a heat flux boundary condition as expressed in (10) where n represents the normal direction to the boundary, is the room temperature [] and h is the heat transfer coefficient []. A heat transfer coefficient (h) of was calculated based on the Nusselt number according to (11) applied for free air convection in vertical cylinders [30] (11) where h is the heat transfer coefficient [], is the Nusselt number, k is the thermal conductivity of the air [] and L is the vertical dimension of the external electrode []. 2.4 Computational modelling The flowchart applied to this study is presented in Fig. 2. Initially, the design of the geometry is created in COMSOL™ and all parameters used to solve the electrical-thermal problem are set. Two COMSOL™ modules were coupled for modelling the system, namely, the AC/DC for electrostatics and heat transfer for temperature analysis. The mesh was automatically generated by COMSOL™ with ∼12,000 tetrahedral elements. Fig 2Open in figure viewerPowerPoint Flowchart of the computational modelling A computational design of experiment (DOE) was used for generating the results of microbial inactivation and specific energy from COMSOL™ simulations. A DOE is a series of tests in which changes are made to the inputs variables of a system in order to investigate how the inputs affect responses based on statistical analysis. DOE has been a very useful tool for the design and analysis of industrial design problems. DOE is applicable to both physical process and computer simulation models [31]. For the purpose of this work, a DOE matrix was created with the process and design variables with their respective levels, consisting in a factorial design (3 variables with 3 levels for each one). The levels are referred to as low (−), intermediate (0) and high (+). The DOE matrix combines the following variables: applied voltage (V), treatment time (t) and internal electrode radius (). The upper limits of (V) and () were specified to limit the electric field within the grape juice to 50 kV/cm in order to avoid dielectric breakdown. The lower limit of the applied voltage was specified as two-thirds of the upper limit. The upper limit of the treatment time range was specified considering a maximum value of liquid average temperature rise of 25°C for the condition of maximum electric field strength, so that the average liquid temperature does not exceed 45°C for an initial temperature of 20°C, according to the preferable range specified in [15]. Table 1 shows the variables and corresponding levels used in the DOE matrix. Table 1. Variables and levels for the DOE matrix Variable Unit Level − 0 + applied voltage (V) kV 16 20 24 treatment time (t) µs 100 200 300 internal electrode radius () cm 0.5 0.75 1.0 The internal electrode radius and rectangular pulse train were set in COMSOL™ model for each combination of DOE matrix variables. The pulse train was set based on the treatment time and applied voltage. The pulse frequency was set to 100 Hz and pulse duration to 10 μs for all combinations. The rise and fall times of each rectangular pulse were set to 0.2 μs. As the rise and fall times are essentially smaller than the pulse width, they have no significant influence on the PEF efficiency [32, 33]. Additionally, as the equivalent electric circuit of the fluid is predominantly resistive, the high-frequency components of the rectangular pulse do not have considerable influence on the model. The simulations were performed in the time domain. The electric field and temperature inside the treatment chamber were numerically obtained by weighted averages in order to have more accurate results, as expressed by (12) and (13), respectively. In these equations, the results of electric field and temperature for each element of the mesh from COMSOL™ were weighted by element volume (12) (13) where is the electric field strength in the each element i [], is the temperature value in the each element i [], is the volume of the element i [], is the liquid volume [], is the total number of elements in the liquid, is the weighted average of the electric field strength [] and is the weighted average temperature []. The microbial inactivation for S. aureus and E. coli, as a function of the treatment time t and the weighted average electric field in the liquid, were calculated for each combination of the DOE matrix by the tertiary models (5), applying the coefficients ( to ) and the average values of the shape parameter () reported in [14], which reflect the resistance of each microorganism to PEF treatment. These parameters are shown in Table 2. Table 2. Regression coefficients (A0–A3) and average value of the shape parameter () for the E. coli and S. aureus Parameter Microorganism E. coli S. aureus 3.31 1.77 0.166 0.6610 0.5059 The specific energy was numerically calculated by the ratio between the energy absorbed by the liquid, obtained from the sum of the energy in the elements, and the fluid mass inside the treatment chamber (14) where is the specific energy [], is the electrical conductivity dependent on temperature [], is the electric field strength in the element i [], is the volume of the element i [], t is the treatment time [], is the liquid volume inside the treatment chamber [] and is the fluid density []. The post-processing results of microbial inactivation and specific energy were adjusted by multilinear regressions using the statistics toolbox of MATLAB™ R2010 in order to obtain the objective functions. As the problem is defined in terms of two objectives, its solution is set as a solution of a multi-objective problem, and has the form of a Pareto-set [34]. A multi-objective genetic algorithm NSGA-II was applied to obtain the set of optimal solutions for this problem. The algorithm NSGA-II was initially proposed in [35] as an evolution of NSGA. In general terms, the parent and offspring populations are combined and evaluated using the fast non-dominated sorting approach, an elitist approach, and an efficient crowding mechanism [36]. Generally, the NSGA-II algorithms maintain a good spread of solutions and good convergence near the true Pareto-optimal front. A NSGA-II algorithm adapted from [37] was used to generate the Pareto optimal curves for the two microorganisms under study. In this study, the initial population was set to 100 individuals and the genetic operators crossover and mutation were defined as 0.9 and 0.5, respectively. The number of interactions was set to 50. 3 Results and discussion After modelling the coaxial geometry and parameters setting in COMSOL™, the problem was solved for each variable combination of the DOE matrix. The electric field and temperature results (weighted average and maximum values) and the post-processing data of microbial inactivation and specific energy for the S. aureus and E. coli are available in Table 3. Table 3. FEA and post-processing results Design variables FEA results Post-processing results a b (E. coli) 16 100 0.5 8.7 29.8 21.7 25.1 −0.62 −0.25 33.9 16 100 0.75 10.8 30.0 22.4 25.6 −0.64 −0.30 47.9 16 100 1.0 13.5 32.3 23.6 27.0 −0.73 −0.39 71.3 16 200 0.5 8.7 29.8 23.2 28.4 −0.87 −0.39 125.5 16 200 0.75 10.8 30.0 24.6 29.3 −0.91 −0.48 179.6 16 200 1.0 13.5 32.3 27.0 31.8 −1.04 −0.62 270.8 16 300 0.5 8.7 29.8 24.7 31.3 −1.07 −0.52 271.3 16 300 0.75 10.8 30.0 26.9 32.6 −1.12 −0.62 392.4 16 300 1.0 13.5 32.3 30.5 36.3 −1.28 −0.81 598.4 20 100 0.5 10.8 37.2 22.5 28.0 −0.64 −0.30 51.8 20 100 0.75 13.5 37.4 23.7 28.8 −0.73 −0.39 73.8 20 100 1.0 16.8 40.3 25.6 31.1 −0.96 −0.56 110.8 20 200 0.5 10.8 37.2 25.0 33.3 −0.91 −0.48 194.9 20 200 0.75 13.5 37.4 27.3 34.8 −1.04 −0.62 281.2 20 200 1.0 16.8 40.3 31.3 39.2 −1.36 −0.89 427.7 20 300 0.5 10.8 37.2 27.6 38.1 −1.12 −0.63 427.5 20 300 0.75 13.5 37.4 31.1 40.4 −1.27 −0.81 623.4 20 300 1.0 16.8 40.3 37.4 47.0 −1.67 −1.16 958.6 24 100 0.5 13.0 44.5 23.6 31.6 −0.71 −0.37 73.5 24 100 0.75 16.1 44.7 25.3 32.8 −0.90 −0.52 105.6 24 100 1.0 20.2 48.2 28.2 36.4 −1.32 −0.84 160.1 24 200 0.5 13.0 44.5 27.3 39.5 −1.01 −0.59 280.8 24 200 0.75 16.1 44.7 30.8 42.0 −1.28 −0.82 408.5 24 200 1.0 20.2 48.2 37.0 49.1 −1.88 −1.32 626.7 24 300 0.5 13.0 44.5 31.3 46.8 −1.24 −0.77 624.8 24 300 0.75 16.1 44.7 36.7 50.7 −1.57 −1.07 918.7 24 300 1.0 20.2 48.2 45.0 61.9 −2.31 −1.73 1423.8 a: maximum electric field inside the liquid. b: maximum temperature inside the liquid. 3.1 Electric field distribution A 3D view of the electric field distribution within the treatment chamber is presented in Fig. 3 for 24 kV applied voltage and internal electrode radius of 1.0 cm, which represents the combination that generates the higher electric field intensity. The toroidal rings and insulators were inserted in the treatment chamber in order to minimise the electric field enhancement around the top and bottom of the internal electrode. For the mentioned setting, the electric field strength peaks near to 50 kV/cm around the surface of the toroidal ring and is higher near the electrode surface, decreasing towards the external electrode, which is the expected behaviour for a coaxial arrangement. Fig 3Open in figure viewerPowerPoint 3D view of the electric field distribution for and A comparison of the electric field radial distribution at the centre of the treatment chamber (height of 1.5 cm) for the DOE values of internal electrode radii and applied voltage of 24 kV is presented in Fig. 4. The electric field strength results between the electrodes obtained numerically are in agreement with the classic expression for the analytical calculation of a coaxial arrangement shown in (1). The electric field intensity in the liquid must be high enough to reach the required level of microbial inactivation. Although the electric field peak, for of 0.5 cm, was slightly higher in comparison to the other cases, the weighted average of the electric field strength for this radius is 35.6% lower than of 1 cm and 19.2% lower than of 0.75 cm (data in Table 3). Therefore, inactivation is higher for larger internal electrode radii due to the higher electric field intensity. On the other hand, the specific energy for of 1.0 cm was 56.1% higher than for of 0.5 cm. Hence, the energy consumed by the liquid is higher for larger internal electrode radii, which is not interesting from the point of view of energy efficiency. Fig 4Open in figure viewerPowerPoint Radial distribution of the electric field strength for different internal electrode radii (height of 1.5 cm and applied voltage of 24 kV) In order to avoid dielectric breakdown and regions in which microorganisms are undertreated, the electric field distribution for a PEF treatment chamber must be as uniform as possible. Although coaxial chamber has a well-defined electric field distribution, its electric field distribution is not completely uniform. A measure of the electric field distribution uniformity in the liquid can be numerically obtained by the calculation of the coefficient of variation given by (15) (16) where CV is the coefficient of variation of the electric field distribution, is the relative standard deviation [], is the electric field strength in each element i [], is the volume of the element i [], is the liquid volume [], is the total number of elements in the liquid and is the weighted average electric field strength []. Fig. 5 shows the coefficient of variation and the weighted average electric field as a function of the internal electrode radius for an applied voltage of 24 kV. As shown, the value of decreases as the internal electrode radius increases, indicating a higher uniformity of the electric field distribution for higher values of . Considering a variation in the electrode radius from 0.5 to 1 cm, there is a significant reduction of 41.6%, indicating an improvement in the electric field uniformity towards the increase of . However, this also means an increase in the energy consumption, which has a relevant impact on the cost of a PEF system. In a similar manner, seeking a higher uniformity in the electric field by increasing the internal electrode radius affects the energy consumption of the system. Fig 5Open in figure viewerPowerPoint Coefficient of variation and the weighted average electric field as a function of the internal electrode radius (applied voltage of 24 kV) 3.2 Temperature distribution The liquid temperature during PEF is one of the critical factors affecting microbial inactivation as well the sensorial and nutritional properties of the food. Thus, the knowledge of the temperature distribution inside the PEF treatment chamber is essential for the design of an efficient application of the technology [38]. The temperature distribution simulated in the treatment chamber for a initial value of 20°C is seen in Fig. 6, considering the more critical DOE variables combination in relation to the temperature rise (, and ). The temperature rise is due to the ohmic energy dissipation, which is proportional to the square of the electric field magnitude (heat source). As shown, there are high local spots surrounding the internal electrode borders exceeding , as a consequence of high local electric field strength in these regions. These local spots may cause une
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