Skin and Proximity Effects in Electrodes and Furnace Shells
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Abstract
A review of two-dimensional (2D) analytical models of skin and proximity effects in large industrial furnaces with three electrodes arranged in an equilateral triangle is given. The models cover three different cases: one electrode only, three electrodes where two are approximated by line currents, and induced shell currents where all electrodes are approximated by line currents. The first two models show how the skin and proximity effects depend on electrode material properties and size, and the distance between the electrodes. The third model shows how the strength of the induced shell currents will depend on electrode position and furnace size. These models are compared to numerical studies including distributed electrodes and shell currents. The analytical models are accurate when induced shell currents can be disregarded. However, strong shell currents may have a significant impact on the current distribution within the electrodes. This electrode-shell proximity effect competes with the electrode-electrode proximity effect. Finally, the 2D models have been compared with three-dimensional (3D) case studies of large industrial furnaces. In 3D, the shell currents are significantly smaller than what are predicted by the 2D models, but they are sufficiently strong to cause a significant correction of the electrode current density.
Introduction
In many metal-producing units, the energy needed for the primary reactions are delivered by electric currents through large electrodes. Examples of such units are slag resistance furnaces and electric arc furnaces, used in the production of steel, ferroalloys, calcium carbide and silicon.[1,2] Typically, three-phase alternating current (AC) circuits using three or more electrodes are used, operating at the grid frequency of 50 or 60 Hz. Correct and stable operation of the electrodes is of crucial importance for successful and cost-effective operation of the furnace.[3,4] Detailed understanding of current densities, thermal conditions and mechanical stresses is needed to address electrode problems such as electrode breakages and electrode consumption.[5, 6, 7] Also, for designing the furnace and for understanding its process, the current-carrying capabilities and sizing of the electrode play an important role.[8, 9, 10]
The electrodes are made of electrically conductive graphite or baked carbon material, and their main function is to carry the electric currents needed for powering the chemical reactions of the process. Although direct current (DC) furnaces can be found in the industry, most units use AC current, and in AC electrodes, there is considerable concentration of currents close to the surface; the well-known skin effect.[2] The skin effect is caused by eddy currents that are induced by the changing magnetic field produced by the current. The current density distribution in an electrode with skin effect is found by solving Maxwell’s equations, and for the typical case with vertical currents and cylinder geometry, the current density is axisymmetric and given in terms of Bessel functions.[11,12] In addition to the skin effect, there are also considerable proximity effects caused by eddy currents opposing the magnetic field produced by the currents in the neighboring electrodes. The proximity effects typically yield an asymmetric current density in the electrodes.[13, 14, 15, 16, 17, 18] Although these effects are significant and well-known,[19, 20, 21, 22, 23] they are often neglected in electrode models which typically concentrate on a single electrode.[7,24, 25, 26]
Most industrial furnaces have a furnace shell. Its function is to carry the weight of the reactants, products and lining materials in the furnace, and it is typically made of structural steel. However, such materials are both highly conductive and ferromagnetic, and strong eddy currents are induced in the furnace shell. A theoretical and experimental study of this phenomenon, can be found in Reference 27. When present, these shell currents may also modify the currents of the electrode, effectively creating electrode-shell proximity effects.
In this work, mathematical models of skin and proximity effects in three-phase AC furnaces with the typical triangular electrode arrangement are studied. Analytical models of skin and proximity effects found in the literature are reviewed and supplemented with numerical simulations of more representative 2D geometries. In particular, the electrode-shell proximity effects are included in this study. The strength of the skin and proximity effects mainly depend upon electrical material properties and important geometrical sizes such as the furnace diameter, electrode diameter and electrode-electrode distance. The main goal is to study how these electromagnetic effects depend on the geometrical sizes. Also, both the analytical models and the 2D numerical models are compared with numerical simulations of industrial furnaces in 3D.
In Section II, the basics of the models are reviewed. The numerical models are described in Section III. Section IV presents the numerical results including discussions and comparisons with the models of Section II. The conclusions are given in Section V.
Review of Mathematical Models
In this section, three analytical models are reviewed. The first model describes the skin effect in a single electrode. The second model describes both skin and proximity effects in electrodes assuming the typical triangular electrode configuration used in three-phase AC furnaces. The third model describes the induced currents in the furnace steel shell.
Skin Effect
The resulting expressions for the geometry of Figure 1 are quoted in Appendix A.2. Most importantly, the expression for the AC resistance factor \(R_{{\rm AC}}/R_{{\rm DC}}\) is given in Eq. [A7], and the expression for the root-mean-squared current density \(J_{z,\text{rms}}\) is given in Eq. [A12]. Note that the current density expression is axisymmetric, i.e., the only spatial dependence is with the radial coordinate r. Also note that when \(J_{z,\text{rms}}\) is normalized by the DC current density \(J_{z,\text{DC}} = I_{{\rm rms}}/(\pi R^2),\) it only depends on two nondimensional numbers, namely the nondimensional radius r/R and the ratio \(R/\delta ,\) which can be thought of as inverse nondimensional skin depth. In other words, the shape of the current density depends only on \(R/\delta .\) The strength of the skin effect is controlled by \(R/\delta .\) When \(R/\delta \ll 1,\) the skin effect is negligible, and the current density is uniform. When \(R/\delta \sim 1,\) the current starts to accumulate near the surface of the electrode, and when \(R/\delta \gg 1,\) the current flows only within the skin of the electrode. The importance of \(R/\delta \) is also seen when studying the resistance of the electrode (see Eq. [A7]). The AC resistance factor \(R_{{\rm AC}}/R_{{\rm DC}},\) which quantifies the increased resistance associated with skin effect in AC electrodes, depends only on \(R/\delta .\)
Electrode-Electrode Proximity Effect
The most important expressions are the root-mean-squared current density in Eq. [A15] and the AC resistance factor in Eq. [A20]. In these expressions, the electrode-electrode proximity effects are included as a series expansion in orders of the nondimensional number R/d, which is the ratio between the electrode radius and the electrode-electrode distance. For any practical application, the series converges with few terms, and the formula can be implemented efficiently.
Together with \(R/\delta ,\)R/d is important for determining the strength of the proximity effects. In the limit when \(R/d \rightarrow 0,\) Eqs. [A15] and [A20] reduce to the skin effect expressions of Eqs. [A12] and [A7]. With increasing values of R/d, the importance of the proximity effect terms increases. Note that these terms also depend on the azimuthal angle \(\theta ,\) and the current density is no longer axisymmetric. The proximity effect yields a shift of the current density towards the leading electrode.[13, 14, 15, 16, 17, 18,20,22]
Properties for Some Industrial Three-Phase Furnaces with Søderberg Electrodes
ID | Process | \(\sigma \) (\(\hbox{Sm}^{-1}\)) | R (m) | d (m) | \(R_{\rm f}\)(m) | \(\delta\)(m) | \(R/\delta \)(–) | R/d (–) | \(d/(\sqrt{3}R_{\rm f})\) (–) | Reference |
---|---|---|---|---|---|---|---|---|---|---|
A | ferromanganese | \(5.0\times 10^4\) | 0.95 | 4.9 | 7.0 | 0.32 | 3.0 | 0.19 | 0.40 | |
B | ferrosilicon | \(5.0\times 10^4\) | 0.90 | 4.0 | 5.5 | 0.32 | 2.8 | 0.23 | 0.42 | |
C | ferrochrome | \(2.8\times 10^4\) | 0.78 | 3.8 | 5.1 | 0.43 | 1.8 | 0.21 | 0.43 |
The AC Resistance Factor and the Proximity Effect Factor for the Example Electrodes of Table I
ID | \({R_{\rm AC}}/{R_{\rm DC}}\) (–) | \({R_{\rm AC}}/{R_{\rm AC,s}}\) (–) |
---|---|---|
A | 1.91 | 1.08 |
B | 1.86 | 1.11 |
C | 1.28 | 1.07 |
Accurate calculation of the skin and proximity effects in the electrodes have important practical consequences for furnace operation and, in particular, electrode operation. Electrode conditions and operating procedures are optimized using electrode models, in which the calculation of temperatures and thermal stresses in the electrodes are based upon accurate estimates of the Joule heating inside the electrode.[7] As shown here, the models should include proximity effects in order to be sufficiently accurate. Also, \(R_{{\rm AC}}/R_{{\rm DC}}\) goes directly into typical expressions for electrode load factors and the current-carrying capacities of electrodes.[5,8,10] In these expressions, the electrode load factor increases with \(R_{{\rm AC}}/R_{{\rm DC}}.\) For example, decreasing proximity effects by increasing the electrode-electrode distance d will increase the current-carrying capacity of an electrode. Consequently, accurate estimates of electrode skin and proximity effects are needed for determining key parameters related to operation and furnace design.
Shell Currents
Relatively simple expressions for the shell currents induced by the three-phase electrode currents can be found in the literature.[27,29,30] In these works, the electrodes are modelled as wires, and the cylindrical shell is assumed to be so thick that all fields decay within. The latter assumption is good when the shell skin depth is much smaller than its thickness. For typical steel with \(\sigma = 6 \times 10^6\,\hbox{Sm}^{-1}\) and \(\mu = \mu_{\rm r} \mu_0,\) with relative permeability \(\mu_{\rm r} = 200,\) the skin depth \(\delta_{{\rm sh}} = {2}\,\hbox{mm}\) at \(f = {50}\,\hbox{Hz}.\) Hence, the approximation is good for industrial furnaces with shell thicknesses that typically are an order of magnitude larger than the skin depth.
In Eq. [A23] is an expression for the total induced current that can be associated with the shell section nearest electrode 1. This expression reveals that the nondimensional number \(d/(\sqrt{3}R_{\rm f})\) is important for determining the strength of the induced shell currents. The distance from the furnace center to the electrode center is equal to \(d/\sqrt{3},\) so \(d/(\sqrt{3}R_{\rm f})\) is the ratio between the electrode center radius and the shell radius of the furnace. When this number is small, the induced currents of the shell are small. This is reasonable, since this limit corresponds to the situation where the three electrodes are close and the shell is far away. The three-phase electrodes can then be considered as a single conductor with zero net current, and consequently, there will not be any induced currents in the shell. In the opposite limit, when \(d/(\sqrt{3}R_{\rm f}) \rightarrow 1,\)\(I_{{\rm sh},1} \rightarrow -I.\) That is, the total current in the shell section is equal to the current of the electrode, but with opposite phase. This corresponds to three separate, non-interacting electrode-wall systems. Note that this limit is not realistic in a furnace since it would imply that the electrodes are line currents.
Numerical Model Description
The numerical calculations in this work were performed by using the Magnetic Field interface in the AC/DC module of the COMSOL Multiphysics software.[31,32] In this interface, Maxwell’s equations in Eq. [A2] are solved by a formulation in terms of the magnetic vector potential \({\mathbf{A}}.\) The electromagnetic fields are then given by \({\mathbf{B}} = \nabla \times {\mathbf{A}}\) and \({\mathbf{E}} = -\partial {\mathbf{A}}/\partial t.\) That is, the electric potential is not solved for except within the electrodes where the coil domain feature was used.[32] The equations are discretized using the finite-element method with quadratic element order. To avoid influence from the boundaries of the simulation domain, the electrodes and furnace shell are surrounded by dielectric material with a radius of \(10R_{\rm f}\) and with infinite element domains towards the outer boundary of the simulation domain.[31] At the outer boundary, the magnetic insulation boundary condition was used. In most of the simulation domain, triangular mesh elements were used, with high mesh density in the electrodes in order to resolve the gradients caused by the skin and proximity effects. For the thin steel shell with very small skin depth, thin quadrilateral boundary layer mesh elements were used in order to resolve the large gradients close to the surface. In total, there were typically around \(1 \times 10^5\) mesh elements for each calculation.
In industrial furnaces, the furnace shell is typically made of magnetic steel. In this model, the steel shell has been modelled in a simplified manner, as a linear magnetic material with a constant \(\mu_{{\rm r}} = {200}.\) Moreover, \(f = {50}\,\hbox{Hz},\)\(R = {1}\,\hbox{m}\) and \(I_{{\rm rms}}={115}\,\hbox{kA}\) and steel shell thickness \(t_{{\rm sh}}={25}\,\hbox{mm}.\) Since the model is relatively simple and the solution typically converges in less than a minute, many calculations were performed in order to map out the parameter space spanned by the nondimensional numbers of the system. This was done by varying parameters \(\sigma ,\)d, \(R_{\rm f}\) and the steel shell electrical conductivity \(\sigma_{{\rm sh}}.\)
Results and Discussion
Triangular Electrode Configuration
We have studied the current and power distribution for the triangular electrode configuration numerically. Compared with the analytical expressions of Section II–B, the numerical model is more accurate since all electrode currents are distributed throughout the electrode cross section. In total, results for 480 different parameter combinations have been calculated numerically. This was done by choosing 20 different values of \(\sigma \) and 24 different values of d such that all combinations of \(R/\delta \in \{{0.25}, {0.50}, \ldots , {5.00}\}\) and \(R/d \in \{{0.015}, {0.030}, \ldots , {0.360}\}\) were studied.
The same comparison between the numerically and analytically calculated values can also be performed for the relative power density variation, and the result is presented in Supplementary Figure S-4. The results show that the difference between the numerically and analytically calculated values are \(\sim \)1 pct.
The results presented in this section (Figures 9 to 11) show that deviations are small between the numerical model results and the analytical expressions presented in Section II–B. The differences found can be attributed to the line current approximation of neighboring electrode currents, as assumed in Section II–B. In reality, there is a spatial extent of the electrodes, and thus, currents that are close to other electrodes are more important for proximity effects. Also, because of proximity effects, currents will, on average, be displaced with respect to the electrode center. In most cases, this means that proximity effects are slightly more pronounced than when calculated by the expressions of Section II–B.
Triangular Electrode Configuration with Shell
In this section, results are shown for the general model with a triangular electrode configuration and a shell with finite thickness. This model has been investigated by changing four parameters: \(\sigma ,\)\(\sigma_{{\rm sh}},\)d and \(R_{\rm f}.\) These have been varied such that the nondimensional numbers have been studied in intervals \(R/\delta \in [{0.25}, {5}],\)\(R/d \in [{0.015}, {0.45}],\)\(d/(\sqrt{3}R_{\rm f}) \in [{0.10}, {0.90}]\) and \(t_{{\rm sh}}/\delta_{{\rm sh}} \in [{0.1}, {30.0}].\) By assuming that electrodes should not overlap with each other, nor the furnace shell, there are constraints that must be obeyed, \(R/d<0.5\) and \(d/(\sqrt{3}R_{\rm f})< (1+\sqrt{3}R/d)^{-1}.\)
Shell currents
Electrode currents
Similar to Section IV–A, we study the current and power distribution of the electrodes, but now including the interactions with the shell currents.
Comparison with 3D Results
The 2D models investigated in this work represent the horizontal cross sections of three-phase AC furnaces with triangular electrode configuration. Models without the furnace shell represent cross sections in regions above the furnace edge, and models with the furnace shell represent cross sections below the furnace edge. However, for the 2D models to be quantitatively accurate, the vertical length of the regions must be large compared with the horizontal extent of the system, and this is not the case for 3D industrial furnaces. Particularly, when it comes to results for induced currents in the steel shell, we expect that the 2D results are quantitatively different compared with 3D. The reason for this is that the induced shell currents are expected to be smaller in 3D, because of the finite height of the steel shell.
Comparison of Model Results for the Induced Shell Currents \(\left| I_{{\rm sh},1}/I\right| ,\) for Furnaces A and B of Table I
Comparison of Model Results of the Power Density Variation \(\Delta p_{R}/\overline{p_{R}},\) for Different Electrode Cross Sections of Furnaces A and B of Table I
Conclusions
2D models describing the skin effect, electrode-electrode proximity effects and currents in the furnace shell have been reviewed. The models reveal how these effects depend on furnace and electrode sizes, as well as the location of the electrodes. In a large furnace, there will be considerable skin and proximity effects in the electrodes.
The nondimensional number \(R/\delta \) quantifies the strength of induction effects in the electrodes. When this number is small, the current density in the electrode is uniform, and when this number is large, the current flows only close to the periphery. For electrodes in large industrial furnaces, \(R/\delta \) is larger than 1, and there is a considerable accumulation of current close to the surface of the electrode.
For the electrode-electrode proximity effect, the nondimensional number R/d, which is the ratio between the electrode radius and the electrode distance, is also important. With increasing value of R/d, large proximity effects can be noticed, with higher current densities towards the leading electrode.
The electrode currents will also induce currents in the furnace shell. These currents can be large, given that the ratio between the thickness and the skin depth of the furnace shell, \(t_{{\rm sh}}/\delta_{{\rm sh}} > 1.\) Where a magnetic steel shell is used, \(t_{{\rm sh}}/\delta_{{\rm sh}} \gg 1,\) and large currents will be induced in a thin layer on the inside of the furnace shell. The strength of these shell currents is determined by the ratio between the ”electrode circle radius” and the furnace radius, \(d/(\sqrt{3}R_{\rm f}).\)
The analytical expressions have been compared with numerical models that account for the finite extents of the electrodes and the furnace shell. When shell currents are neglected, the analytical expressions are accurate. However, the analytical models do not account for the electromagnetic interactions between the shell and the electrodes. The numerical simulations show that strong shell currents will have a significant impact on the current distribution within the electrodes. This is the electrode-shell proximity effect, and it effectively competes with the electrode-electrode proximity effect. In industrial furnaces, both effects need to be considered. The outcome of the competition depends, to a large degree, on the value of \(d/(\sqrt{3}R_{\rm f}).\)
The 2D numerical simulations have been compared to 3D case studies. For industrial furnaces, the shell height is comparable to its diameter, and the region with vertical shell currents should therefore be relatively small. Hence, the shell currents are not properly approximated by 2D models. The simulations show that the shell currents are significantly smaller than predicted by the 2D models, but they are sufficiently strong to cause a significant reduction in the electrode-electrode proximity effects. 3D simulations are therefore recommended to gain proper insight into the current distribution within the electrodes in large industrial furnaces.
Footnotes
Notes
Acknowledgments
This paper is published as part of the project Electrical Conditions and their Process Interactions in High Temperature Metallurgical Reactors (ElMet), with financial support from The Research Council of Norway and the companies Elkem and Eramet Norway.
Supplementary material
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