Fast Numerical Optimization of Electrode Geometry in a Two-Electrode Electric Resistance Furnace Using a Surrogate Criterion Derived Exclusively from an Electromagnetic Submodel

Abstract

The Joule heat generated by current flow between electrodes in a resistance furnace not only melts and heats the charge but also induces mixing of the molten material. Increased mixing promotes improved chemical and temperature uniformity within the bath. This paper presents a novel approach to effectively optimizing electrode geometry in resistance furnaces. The method relies on a surrogate criterion derived exclusively from an electromagnetic submodel, which governs the process hydrodynamics. This criterion is based on the location of the Joule heat generation center in the bath. Its idea is to lower this center as much as possible while keeping it close to the vertical bath axis. Owing to this, the best conditions for the development of natural convection were obtained. The developed methodology was demonstrated through an application to a two-electrode furnace. The results showed that the influence of forced MHD convection is negligible in this furnace (with a Lorentz force of only about 0.0015 N/kg). The validation of the optimized geometry, derived using solely the electromagnetic submodel, was carried out using a full process model, including time-consuming hydrodynamic calculations. The proposed optimization methodology enabled a 10-fold increase in the average mixing velocity (from 0.0008 to 0.0084 m/s). The main significance of the presented study is the introduction of a surrogate criterion that allows for a multiple reduction in the time of numerical optimization of the mixing intensity in electrode resistance furnaces in comparison to the standard solution based on the flow velocity criterion determined from the hydrodynamic model.

1. Introduction

Electric furnaces with graphite electrodes are widely used in the metallurgical industry. In non-ferrous metallurgy, these furnaces are mainly used for the recovery of copper from waste slags. These slags are generated as a result of copper smelting from a copper concentrate in a flash furnace, where some of the copper is oxidized and ends up in the slag. Slag cleaning is driven by ever-increasing restrictions related to further slag processing and financial losses. For the process of recovering copper from the oxide phase, resistance electric furnaces with two or three graphite electrodes are used, in which there is no electric arc [1].

Figure 1 shows schematic diagrams of two types of electric furnaces, along with a description of the device’s main components. The primary structural element of electric furnaces is a steel shell (gray components in the model), which protects the refractory bricks. The refractory bricks (the orange component) isolate the molten slag from the surroundings and minimize heat loss to the environment. The slag is indicated as a yellow component in the model. The center of Joule heat generation within the slag (as a result of the current flow between the electrodes) is marked with the symbol “x”. It can be seen that regardless of the number of electrodes, the Joule heat release center is located between them at their end, so the thermal phenomena, and especially the mechanism of natural convection related to the place of heat generation, are similar in these two types of furnaces [2,3].

The optimal methodology for investigating industrial copper slag reduction installations would ideally involve conducting experiments directly within the operational environment. However, this approach encounters significant obstacles due to the prohibitive costs associated with such endeavors and the inherent limitations in comprehensively measuring all pertinent process parameters. Consequently, computer simulations emerge as a compelling and pragmatic research alternative, offering the capacity to effectively optimize processes. An illustrative example of such an approach is presented in the publication [4], which provides a comprehensive overview of the applications of numerical modeling within the domain of electric furnace design. In order to ensure that the outcomes derived from computer simulations accurately reflect the actual process dynamics, it is imperative to meticulously define boundary conditions, initial conditions, and material specifications that most closely approximate the real-world scenario. This complex issue was addressed by the author of the publication [5].

An intriguing approach to simplify the complexity of electric furnace models was put forth by Sheng et al. [6,7]. They proposed a method utilizing a constant electrical potential as the input parameter instead of alternating current (AC), thereby enabling a reduction in computational costs. A similar approach based on electric potential was used in the paper [8]. Of course, this means that phenomena related to the alternating electromagnetic field, such as conversion forced by Lorentz forces, are not taken into account in the model. However, as shown by simulation results in this paper, the influence of magnetohydrodynamic phenomena can be negligibly small.

Another illustrative instance of employing computational numerical modeling to facilitate a comprehensive analysis of metallurgical slag reduction processes within electric furnaces is presented in the study [2]. The paper showcases the outcomes of numerical simulations pertaining to flow and heat transfer within a cylindrical electric furnace dedicated to the processing of metallurgical slags. The results derived from the numerical analyses were subsequently validated by juxtaposing the relationship between slag temperature and electrical potential with empirical data procured from the steelworks. The potential for enhanced slag reduction and copper recovery within crossed electric and magnetic fields is also elucidated in the study [9]. This work encompasses an analysis of the physicochemical phenomena inherent to the copper slag reduction process, leveraging the capabilities of numerical modeling. Ritchie et al. [10] investigated the influence of electrode immersion depth and furnace operating power on the smelting of platinum group metals (PGMs) from ores. These analyses indicated significant differences in melt temperature with shallow electrode immersion and a uniform temperature distribution in the case of deep electrode immersion and high power. Analyses that take into account Joule heating are presented in refs. [11,12,13], where the influence of electrode shape and immersion depth on the heating of an electric furnace in ferronickel processing is analyzed. They demonstrate the influence of the slag layer height on the amount of Joule heat generated.

The literature presents many applications of the finite element method (FEM) in single-criterion optimization. Research [14] including the analysis of the effect of selected process parameters on the temperature field for the Fused Filament Fabrication’s optimization was performed. In the work [14], the optimization criterion was determined by a FEM model that calculates the temperature field for three input parameters, such as scanning speed, molding chamber temperature, and nozzle temperature. For that matter, the FEM was used to simulate the temperature field distribution. Another instance is presented in the research [15], where the FEM was used to calculate the stress distribution for various geometries to optimize the topology of aircraft seats. The publication [16] considers using the FEM to optimize magnetic field strength in Electromagnetic Forming (EMF), where the influence of parameters such as current density, gap, and coil size is analyzed. Another study [17] presented the optimization of the geometry of thermoelectric generators, where FEM was used to analyze the temperature distribution. In addition to the presented single-criterion optimization examples, the FEM is also used in multi-criteria optimization. For instance, the publication [18] considered the problem involving a Switched Reluctance Motor (SRM) where both current and torque are optimized by using a Pareto front as well as the FEM method, among others.

The literature also provides examples of using the FEM to directly determine criteria for optimization algorithms. Refs. [19,20] presents an evolutionary algorithm that iteratively uses the FEM to determine the short-circuit impedance and the magnetic field distribution for conductor size optimization. The authors of [21] presented a way to interconnect the independent mesh mode, eXtended Finite Element Method (XFEM), with multi-objective optimization based on the evolution algorithms for electromagnetic problems. The work [21] proposed the solution where the genetic algorithm generates the input parameters for the XFEM model and receives a response with the optimization criterion value. The entire process takes place in a loop until the stop condition of the algorithm is met. The work [22] used the FEM with a gradient-based optimization algorithm. A mesh is generated with given input parameters, where the simulation result is passed to the optimization algorithm, which calculates new input parameters for the local minimum found, for which the mesh is readjusted. Next, the publication [23] proposed the optimization method for electrode placement in radiofrequency ablation, RFA. In this method, the optimization algorithm iteratively generates the new electrode position, which is given to the FEM model, which calculates the new criterion value, which is the Arrhenius survival fraction.

This paper is about one of the design elements of an electrode resistance furnace, which is the geometry of its electrodes. The modification of the furnace design in this respect is technically very simple to carry out, and as the research has shown, it allows for a significant improvement of an important process parameter, which is the intensity of mixing of the processed bath (e.g., slag).

Joule heat, which is generated by the flow of electric current between electrodes in a resistance furnace, plays a critical role in the melting and heating of the charge. However, its impact extends beyond these primary functions. The heat also induces the mixing of the molten material within the furnace, a factor that is crucial for the overall process efficiency. This increased mixing contributes to a more homogeneous distribution of temperature and chemical composition within the molten bath, leading to improved results of the technological process.

This paper introduces a novel methodology aimed at optimizing the geometry of electrodes in resistance furnaces. The proposed approach is distinctive in that it leverages a surrogate criterion based solely on the electromagnetic submodel. The criterion is intimately connected to the process hydrodynamics within the furnace. The idea has already been used in a process where magnetohydrodynamic phenomena played a leading role [24,25]. In the analyzed process, natural convection plays a leading role, but the principle of omitting the hydrodynamic model in optimization is similar.

The effectiveness of the developed methodology is demonstrated through its application to a resistance furnace equipped with two electrodes. The results highlight the potential of this approach to enhance furnace performance by improving the mixing of the molten material, thereby offering a significant advancement in the optimization of resistance furnace operations.

The primary contribution of the presented study lies in the development of a novel surrogate criterion, which significantly enhances the efficiency of numerical optimization for determining mixing intensity within electrode resistance furnaces. This criterion allows for a substantial reduction in the computation time required for optimization, achieving a level of time efficiency that is considerably greater than that of the conventional approach, which relies on the flow velocity criterion derived from a hydrodynamic model.

2. Mathematical Model of the Process

To optimize any technological process using numerical methods, it is first necessary to create a representative mathematical model that reproduces all the phenomena influencing the characteristics being improved.

The electrode resistance furnace process involves a complex interplay of current flow, electromagnetic, thermal, and hydrodynamic phenomena. Consequently, an accurate model of the electrode furnace must incorporate coupled electromagnetic, thermal, and fluid flow fields. The considered optimization concerns the period after melting the material and after the stabilization of the process. Owing to this, it is possible to assume a simplification in the form of constant and uniform electrical conductivity for the bath. This allows for the use of a computationally effective one-way coupling between the electromagnetic and hydrodynamic models (Figure 2).

2.1. Electromagnetic Submodel

The electromagnetic problem was solved using finite element harmonic analysis at a frequency of 50 Hz. For the numerical domain, the magnetic vector potential equation was applied in the following form:

$$\nabla \times \left({\displaystyle \frac{1}{\mu }}\nabla \times \mathbf{A}\right)+j\omega \sigma \mathbf{A}={\mathbf{J}}_{\mathbf{s}}$$

(1)

where $\mu$ is the magnetic permeability, $\sigma$ is the electric conductivity, $\omega$ is the angular frequency, and ${\mathbf{J}}_{\mathbf{s}}$ is the current density source.

Knowing the distribution of the magnetic vector potential $\mathbf{A}$, both the magnetic induction $\mathbf{B}$ and the eddy current density $\mathbf{J}$ can be determined as follows:

$$\mathbf{B}=\nabla \times \mathbf{A},$$

(2)

$$\mathbf{J}=-j\omega \sigma \mathbf{A}.$$

(3)

With the current density, one can obtain the volume density of the Joule heat power generated in the bath processed in the furnace:

$$q_{\mathrm{JH}}={\displaystyle \frac{1}{2}}{\displaystyle \frac{{\left|\mathbf{J}\right|}^2}{\sigma }}.$$

(4)

This quantity is of key importance for the analyzed process because, in addition to heating the molten material, it causes natural convection, the leading mechanism of stirring and homogenizing the bath. The second mechanism, which also mixes the bath to a very small extent (as the later results will show), is convection forced by Lorentz forces:

$${\mathbf{f}}_{\mathrm{LF}}={\displaystyle \frac{1}{2}}Re\left(\mathbf{J}\times {\mathbf{B}}^{*}\right)$$

(5)

where ${\mathbf{B}}^{*}$ is the complex conjugate of $\mathbf{B}$.

The spatial distributions of Joule heat sources and Lorentz force densities calculated in the EM model are transferred to the hydrodynamic submodel via one-way coupling (Figure 2).

Figure 3 shows the geometry of the EM submodel used for the example of a basic furnace. At the boundary of the computational domain, the zero value of the magnetic vector potential $\mathbf{A}$ was assumed. On the electrode terminators, the current forcing was assumed with a value selected depending on the electrode geometry and ensuring constant power supplied to the device for comparison purposes.

The geometry and mesh of the electromagnetic submodel were generated in Gmsh software 4.13.1 [26]. The electromagnetic calculations were carried out based on the GetDP software 3.5.0. integrated with Gmsh [27]. The advantage of using the Gmsh and GetDP packages is that they can be fully controlled by scripts, which significantly facilitates their use in optimization systems.

2.2. Hydrodynamic Submodel

The distributions of Joule heat and Lorentz forces allow us to calculate in the hydrodynamic submodel the target criterion related to the homogenization of the processed bath—the average flow velocity. In the submodel, a steady simulation was carried out, determining the above-mentioned value at the moment of the stabilization of the process in the furnace.

The steady state for an incompressible fluid and laminar flow is described by the following equation:

$$\rho \nabla \left(\mathbf{v}\mathbf{v}\right)=-\nabla p+\mu {\nabla }^2\mathbf{v}+\rho \mathbf{g}+{\mathbf{f}}_{\mathrm{LF}}+{\mathbf{f}}_{buo}$$

(6)

where $\rho$ is the fluid density, $\mathbf{v}$ is the velocity, p is the pressure, $\mu$ is the dynamic viscosity, $\mathbf{g}$ is the gravitational acceleration vector, and ${\mathbf{f}}_{buo}$ is the buoyancy force.

As can be seen, the equation assumes constant fluid density while natural convection is implemented via the Boussinesq model:

$${\mathbf{f}}_{buo}=-{\rho }_0\beta (T-T_0)\mathbf{g}$$

(7)

where T is the temperature, ${\rho }_0$ is the density at the reference temperature $T_0$, and $\beta$ is the thermal expansion coefficient.

To determine temperature distribution and heat transfer within the furnace, the following energy equation is solved:

$$\rho c\mathbf{v}·\nabla T=\nabla ·(k\nabla T)+q_{\mathrm{JH}}$$

(8)

where k is the thermal conductivity and c is the specific heat capacity. Figure 4 shows the geometry of the HD submodel based on the basic furnace. Convective heat transfer was applied to the external walls of the domain, with the heat transfer coefficient ($h_{\mathrm{ext}}$) of 25 W/m²·K and the ambient temperature ($T_{\mathrm{ext}}$) of 21.85 °C. Heat transfer was defined using the following formula:

$$\dot{Q}=h(T_{ext}-T)$$

(9)

where h is the heat transfer coefficient and $T_{\mathrm{ext}}$ is the external ambient temperature.

On the upper surface of the slag, the mixed heat exchange condition was implemented, incorporating both convection and radiation. The parameters for this condition were the heat transfer coefficient of 25 W/m²·K, the ambient temperature of 21.85 °C, and an external emissivity of 0.8. Mixed heat transfer was defined using equation [28]:

$$\dot{Q}=h(T_{ext}-T)+ϵ\sigma (T_{ext}^4-T^4)$$

(10)

where $ϵ$ is the emissivity of the slag surface and $\sigma$ is the Stefan–Boltzmann constant. For the free surface of the slag, no friction was assumed:

$${\displaystyle \frac{\partial v_{\left|\right|}}{\partial \mathbf{n}}}=0$$

(11)

where $v_{\left|\right|}$ is the velocity component parallel to the wall and $\mathbf{n}$ is the vector normal to the wall.

Geometry for hydrodynamic submodel generated in Ansys SpaceClaim 22R2 software [29]. Mesh was generated in Ansys Fluent Meshing 22R2 software [30]. Hydrodynamic calculations were carried out based on the Ansys Fluent 22R2 [28]. Data obtained from calculations performed in GetDP were interpolated onto the mesh for the hydrodynamic model. Data on the value of the Lorentz force and Joule heat were added to the model using Fluent User Defined Functions.

3. Optimization System

Ensuring the chemical and thermal homogeneity of the processed bath is of crucial importance for the processes carried out in electrode furnaces. When developing the furnace optimization system discussed in the paper, the authors assumed that the target optimization criterion that would well describe this feature of the furnace would be the average bath flow velocity. Of course, as intuition suggests, increasing the flowing current intensity will intensify the phenomena of natural and forced convection affecting this quantity. However, the optimization problem described in this paper concerns the modification of the furnace design (more precisely, the geometry of electrodes), assuming an unchanged electrical power supplied to the furnace.

The standard, full optimization loop (Figure 5, the green path) for such a task shown in the figure includes, in the first step, the creation of an electromagnetic submodel of the analyzed furnace solution for a given point of the solution space (provided in the form of parameters).

Once the electromagnetic submodel has performed its calculations, the distributions of the Joule heat power density and Lorentz force fields are passed to the hydrodynamic model. On this basis, the HD model calculates the velocity field in a steady simulation. The spatial velocity distribution is subjected to a simple analysis providing a target criterion: the average bath velocity. This value is passed to the optimization algorithm, which modifies the parameters starting the next optimization cycle. However, this relatively simple solution is hindered by the substantial computational time required for hydrodynamic calculations, which can exceed several hours for the 3D process under consideration. Even when restricting the optimization to local algorithms, several hundred iterations are typically needed to find an optimal solution. A simple calculation tells us that optimizing the furnace with this approach will take months.

The solution to this problem proposed in this paper is to use a surrogate criterion whose value can only be determined based on the electromagnetic submodel. Since the computation time in such a model is several dozen seconds, a radical reduction of the entire optimization time is achieved. Such an accelerated optimization loop is shown in Figure 5.

3.1. Surrogate Optimization Criterion

The concept of the proposed substitute criterion is based on the analysis of the main mechanism of molten bath mixing, which is natural convection. It is based on the decrease in fluid density as the temperature increases. The warmer, less dense fluid is displaced upwards by the cooler, more dense fluid. The idea is to lower the location of the Joule heat generation center to the maximum. This allows the heated liquid to flow the longest possible distance to the surface of the bath, which allows for an increase in its velocity. The first variant of the criterion analyzed in the base was based only on this quantity:

$$C_1={\displaystyle \frac{{\sum }_{i=1}^N{q}_{JH,i}{z}_i{V}_i}{{\sum }_{i=1}^N{q}_{JH,i}{V}_i}}$$

(12)

where N is the number of cells in the liquid bath area, $q_{JH,i}$ is the Joule volume power density in the i-th cell, $z_i$ is the vertical coordinate of the i-th cell’s centroid, and $V_i$ is the volume of this cell. As can be seen, the calculation of this surrogate criterion requires only data from the electromagnetic model.

Qualitative analysis of the phenomena associated with natural convection seemed to indicate that it is beneficial to place the Joule heat generation center as close as possible to the vertical axis passing through the center of the bath for the intensification of the bath mixing. Therefore, the research also included such a modification of the optimization goal. This can be achieved by including this second goal in the second version $C_2$ of the criterion that takes into account the distance D of the Joule heat generation center from the mentioned central axis:

$$D=\sqrt{{\left({\displaystyle \frac{{\sum }_{i=1}^N{q}_{JH,i}{x}_i{V}_i}{{\sum }_{i=1}^N{q}_{JH,i}{V}_i}}\right)}^2+{\left({\displaystyle \frac{{\sum }_{i=1}^N{q}_{JH,i}{y}_i{V}_i}{{\sum }_{i=1}^N{q}_{JH,i}{V}_i}}\right)}^2},$$

(13)

$$C_2=C_1+aD$$

(14)

where $x_i,y_i$ are horizontal coordinates of the i-th cell. The a coefficient in the criterion decides how strongly the system tends to place the heat generation center in the bath axis. Since this is a secondary goal, a value of a equal to 0.5 was assumed in the study. Of course, assuming zero a means reducing criterion $C_2$ to criterion $C_1$. During the research, an attempt was also made to use the opposite of this criterion, where the a coefficient had a negative value (−0.5). This means that the optimization algorithm strives to move the Joule heat generation center as far from the bath axis as possible.

3.2. Optimization Parameters

The optimization parameters for the two-electrode furnace are shown in

Table 1. Optimization parameters for a 2-electrode furnace.

Parameter
Immersion of electrode 1
Immersion of electrode 2
Deflection X of electrode 1
Deflection X of electrode 2
Deflection Y of electrode 1
Deflection Y of electrode 2
Position X of electrode 1
Position X of electrode 2
Position Y of electrode 1
Position Y of electrode 2

. As can be seen, only the position and orientation of the electrodes are subject to optimization. The same and constant diameter of each electrode was assumed.

3.3. Optimization Constraints

The optimization constraints were based on a set of intuitive geometric dependencies that can be easily determined from the solution parameters:

• Distances of electrodes from the bottom wall $D_{e1,btw}$, $D_{e2,btw}$, $D_{e3,btw}$ greater than the assumed margin,
• Distances of electrodes from the left wall $D_{e1,lw}$, $D_{e2,lw}$, $D_{e3,lw}$ greater than the margin,
• Distances of electrodes from the right wall $D_{e1,rw}$, $D_{e2,rw}$, $D_{e3,rw}$ greater than the margin,
• Distances of electrodes from the front wall $D_{e1,fw}$, $D_{e2,fw}$, $D_{e3,fw}$ greater than the margin,
• Distances of electrodes from the back wall $D_{e1,bw}$, $D_{e2,bw}$, $D_{e3,bw}$ greater than the margin,
• Immersion depths $D_{e1}$, $D_{e2}$, $D_{e2}$ greater than the margin.

Some optimization algorithms have constraints built into their structure. However, the vast majority of algorithms, including those used in the presented research and a large group of popular population algorithms, do not have this mechanism directly implemented. The solution to this constraint is to include the constraint in the form of a penalty function in the optimization criterion. In the discussed solution, the penalty function for each constraint described by the geometric dependence $D_{\gamma }$ is described by the following equation:

$$P_{\gamma }\left(D_{\gamma }\right)=\left\{\begin{array}{cc}{(M-D_{\gamma })}^b\hfill & \mathrm{if}{D}_{\gamma }<M\hfill \\ 0\hfill & \mathrm{if}{D}_{\gamma }\ge M\hfill \end{array}\right.$$

(15)

where M is the margin and b the exponent of the strength of the penalty, which is equal to 2 in this study.

If a violation of geometric dependencies has occurred, then the total criterion $C_{T,i}$ returns the sum penalty of functions and the maximum theoretically possible value of the criterion. Electromagnetic calculations are not performed for this defective geometry.

$$C_{T,i}=\left\{\begin{array}{cc}{C}_{max,i}+{\displaystyle \sum _{\gamma }}{P}_{\gamma }\hfill & \mathrm{if}{\displaystyle \sum _{\gamma }}{P}_{\gamma }>0\hfill \\ C_i\hfill & \mathrm{if}{\displaystyle \sum _{\gamma }}{P}_{\gamma }=0\hfill \end{array}\right.$$

(16)

where $C_{max,i}$ is the highest possible value of the criterion in the allowable parameter space. For criterion $C_1$, it is a value 0 for the location of the heat generation center at the upper surface of the bath. For criterion $C_2$, this is the value of the criterion for the location of the center in one of the upper corners of the bath (for a positive coefficient a).

4. Results

The verification of the developed methodology for a rapid optimization of the resistance furnace was carried out on a laboratory furnace used in the decoppering process at the Institute of Non-Ferrous Metals. The dimensions of the slag chamber and the material properties are shown in

Table 2. Material properties, dimensions of the slag, and supply parameters.

Slag
Density [kg · ${\mathrm{m}}^{-3}$]	3600
Viscosity [kg · ${\mathrm{m}}^{-1}$ · ${\mathrm{s}}^{-1}$]	0.1
Thermal expansion	10.3 · ${10}^{-6}$
Electrical conductivity [S · ${\mathrm{m}}^{-1}$]	80
Thermal conductivity [W · ${\mathrm{m}}^{-1}$ · ${\mathrm{K}}^{-1}$]	0.178
Graphite electrode
Density [kg · ${\mathrm{m}}^{-3}$]	2830
Electrical conductivity [S · ${\mathrm{m}}^{-1}$)]	112500
Thermal conductivity [W · ${\mathrm{m}}^{-1}$ · ${\mathrm{K}}^{-1}$]	100
Dimensions of the computational domain
Slag	0.54 × 0.27 × 0.18 [m]
Electrode diameter	0.045 [m]
Supply parameters
Frequency	50 [Hz]
Power	18 [kW]

. Material properties were taken from the literature [31,32]. The power supply parameters were determined for the above-mentioned laboratory furnace. Unfortunately, there is currently no technical method available to measure the average velocity of a molten slag bath at temperatures exceeding 1000 °C. The only possibility of direct validation, whether the applied surrogate criterion really represents the target, utilitarian goal of improving the homogenizing mixing intensity, is to perform a hydrodynamic simulation. With suitably carefully defined boundary conditions and material properties, the provided results, at least in a comparative scope, allow us to confirm the correctness of the proposed optimization methodology.

4.1. Variant 0—Basic Construction of the Furnace

The basic design model of the electric furnace was created based on the algorithm for calculating electric furnaces with graphite electrodes proposed by the authors [33]. The subsequent step entails determining the dimensions of the electric furnace’s working space. These dimensions are established based on the following formulas:

$$\begin{array}{c}\hfill B\approx 6D\\ \hfill L\approx 12D\\ \hfill l\approx 3D\end{array}$$

(17)

where D—diameter of the graphite electrode; B—width of the slag bath; L—length of the slag bath; l—electrode spacing.

In the literature [33], it is assumed that the electrodes should be immersed to half the depth of the bath as shown in the illustration below. The basic design of the furnace is shown in Figure 6.

Figure 7 shows the power density distribution of the Joule heat released in the bath for 18 kW of electrical energy supplied to the furnace. As expected, a fully symmetric distribution of Joule heat power density was obtained with respect to the vertical longitudinal and transverse planes of the bath. For this electrode arrangement, the Joule heat generation center determined according to Formula (12) was located 0.0662 m below the bath surface.

The volume integral of the Lorentz force norm acting on the bath with mass 93.4543 kg was only 0.1041 N, which means that magnetohydrodynamic phenomena, although they were included in the model (6), have a negligible influence on the hydrodynamics of the process in this furnace. Figure 8 shows the velocity distribution obtained for the basic furnace geometry determined. The average flow velocity of the bath was 0.00077 m/s.

4.2. Variant 1—Optimization Without Forcing the Heat Generation Center in the Bath Axis

The first optimization variant was carried out based on the basic criterion $C_1$ (12). This gave the optimizer complete freedom to choose the geometry to minimize the center of Joule heat generation. The optimization was carried out based on three local optimization algorithms: Nelder–Mead [34], Powell [35], and COBYLA [36]. The advantage of these popular algorithms is that there is no need to calculate the gradient and a suboptimal solution can be obtained in a limited number of iterations. In order to reduce the tendency of these local algorithms to get stuck in local minima, the multi-start method with N = 20 was used in the optimization. The criterion minimization curves for these three algorithms for the best starts are shown in Figure 9. The lowest value of the criterion $C_1$ was obtained for the Powell algorithm, and the solution from this algorithm (Figure 10) was submitted for validation in the hydrodynamic model.

Figure 11 shows the power density distribution of Joule heat released in the bath for 18 kW of electrical energy supplied to the furnace. For obvious reasons, this time, an asymmetric distribution was obtained. The Joule heat generation center is significantly lowered compared to the base variant and is located at a depth of −0.0996 m.

The volume integral of the norm of the Lorentz force acting on the bath will again be only 0.1617 N, which again means that magnetohydrodynamic phenomena have a negligible influence on the hydrodynamics of the process in this furnace. Changing the electrode geometry significantly affects the velocity field in the bath. Figure 12 shows the velocity distribution obtained for the variant 1 geometry. The average flow velocity for this variant is significantly higher compared to variant 0 and is 0.00782 m/s. A higher flow velocity means better conditions for homogenizing the processed material.

4.3. Variant 2: Optimization with Forcing the Heat Generation Center in the Bath Axis

The next experiment concerned the optimization aimed not only at lowering the center of Joule heat generation, but also at locating this center close to the vertical axis of the bath. In this variant, a modified version of the criterion (14) was used, in which the distance of the Joule heat generation center from the bath axis was taken into account with a weight of 0.5. The three local optimization algorithms were used again. Figure 13 shows the optimization progress. The best solution was again provided by Powell’s algorithm. For this best solution (Figure 14), the Joule heat generation center location was z = −0.0929, and its distance from the bath axis was 0.0011.

Figure 15 shows the power density distribution of the Joule heat released in the bath for 18 kW of electrical energy supplied to the furnace. It is clearly seen that the main part of the Joule heat is generated near the vertical axis of the bath (according to the distance from this axis quoted above).

The volume integral of the Lorentz force module acting on the bath with a mass of 92.4846 kg was again only 0.1601. Figure 16 shows the velocity distribution obtained for the variant 2 geometry determined. The placement of the Joule heat generation center allowed for the formation of distinct vortices on the sides of this center. As a result, the average flow velocity in the bath reached 0.00845 m/s.

4.4. Variant 3: Optimization with Forcing the Heat Generation Center as Far as Possible from the Bath Axis

The results obtained for variant 2 determined the need to conduct another test. In this optimization variant, the value of coefficient a in the criterion was changed to −0.5. This caused the optimization system, in addition to lowering the Joule heat release center, to move this center away from the bath axis as much as possible, increasing the asymmetry of the system. Figure 17 shows the system determined based again on the most efficient Powell algorithm (Figure 18).

Figure 19 shows the clearly asymmetrical power density distribution of the Joule heat released in the bath for 18 kW of power supplied to the furnace. The Joule heat generation center is located at a depth of −0.0918 m. The use of a negative coefficient a in the criterion (14) caused the distance between the Joule heat generation center and the vertical axis of the bath to be equal to 0.15421 m. This means a strong asymmetry in the natural convection mechanism.

The volume integral of the Lorentz force norm acting on the bath was 0.1370 N, which again confirms that magnetohydrodynamic phenomena in this process are negligible. Figure 20 shows the velocity distribution obtained for the variant 3 geometry determined. The introduction of strong asymmetry worsened the intensity of bath mixing. The average flow velocity of the bath was 0.00593 m/s.

5. Discussion

Table 3. Summary of electrode geometry variants.

	Variant 0	Variant 1	Variant 2	Variant 3
Vertical position of JHC [m]	−0.0662	−0.0996	−0.0950	−0.0918
Distance of JHC from axis [m]	0.0000	0.0850	0.0011	0.15421
Average velocity [m/s]	0.0008	0.00782	0.00845	0.00593
Maximum velocity [m/s]	0.01067	0.04000	0.04185	0.04635
Total Lorentz force [N/kg]	0.00136	0.00175	0.00173	0.00148

shows a comparison of various global measures for the analyzed variants of electrode geometry. For all variants, it can be seen that the action of the Lorenz force (expressed by the integral of its norm divided by the mass of the bath) is negligible, which is consistent with the literature [6,7,8]. This confirms the assumption that magnetohydrodynamic phenomena can be omitted in the optimization procedure. All optimized variants aimed to lower the location of the Joule heat generation center. This was achieved by bringing the electrode tips as close as possible to the bottom of the bath and simultaneously orienting them obliquely. Because the lower ends of both electrodes are closest to each other, the electric potential gradient between them is the greatest, and it is between these ends that the main electric current flows. The Joule heat generation center is always located above because part of the current also flows between the higher parts of the electrodes. However, the location of the heat generation center was only a surrogate criterion, which could be determined only from the electromagnetic submodel. The target, usable criterion, the value of which constitutes the validation of the developed methodology, is the average flow velocity of the bath. This value expresses the intensity of the desired homogenizing mixing of the bath. Table 3 clearly shows that lowering the Joule heat center is always beneficial and always results in a dramatic increase in the mixing rate relative to the design suggested by the literature (variant 0). Even when the only surrogate criterion is the vertical position of the heat generation center (variant 1), it results in an 877% increase in the mixing intensity. Additional enforcement of horizontal symmetry of the system (the best, variant 2) allows for an additional increase in the bath mixing intensity by 8%. Placing the bath mixing mechanism (Joule heat causing natural convection) close to its vertical axis allows for the vortices in the bath to fully develop. Conversely, maximizing the horizontal asymmetry of the system (variant 3) results in lower mixing performance compared to the variant without symmetry control (variant 1).

6. Conclusions

This paper presents a new methodology for the rapid optimization of electrode resistance furnaces, the aim of which is to increase the intensity of bath mixing and, thus, improve its homogeneity. Omitting the hydrodynamic model from the optimization cycle allowed us to shorten its duration by two orders of magnitude. Owing to this, it was possible to optimize the electrode furnace using local algorithms, as part of the presented research. The proposed surrogate criterion was based solely on Joule heat analysis due to the negligible influence of convection forced by Lorentz forces (only about 0.0015 N/kg). Based on the accelerated optimization loop, various variants of the furnace design solutions were determined, depending on the adopted definitions of the criteria and the optimization parameter space. Hydrodynamic simulations were performed on the basis of these solutions, and the target criterion value was obtained, i.e., the average bath flow velocity. All variants obtained using the proposed optimization methodology showed a significant increase in the mixing intensity of the processed bath compared with solutions based on geometric relationships given in the literature (at least a 631% improvement). The best variant, determined for the substitute criterion, maximized the lowering of the Joule heat generation center while forcing its location closer to the system axis. This increased the mixing intensity by 956% compared to the base variant.

Using a surrogate criterion drastically reduced the criterion determination time (from 6 h to 3 min on a computer with an AMD EPYC 7302, 16 Core, 3.00 GHz processor), allowing for the methodology to be extended toward global optimization algorithms (e.g., PSO, GA). This could increase the chances of finding a global optimum and further improve the bath mixing intensity.

The developed methodology was applied to a two-electrode furnace as an example. However, based on the diagram shown in Figure 1, it can be seen that the location of the heat release center in the three-electrode furnace also depends on the geometry of the electrodes, especially the position of their lower ends. As shown in the article, controlling the position of the Joule heat release center allows for increasing the intensity of bath mixing. Regardless of whether this heat is generated as a result of a single-phase current flow between two electrodes or a three-phase current flow between three electrodes, the convection-driving mechanism remains the same. Therefore, it seems that the proposed surrogate criterion can also be used in the optimization of three-electrode furnaces. This is a general, conceptual analysis, but it encourages further research involving three-electrode furnaces.