metals Article
Electrical Resistivity Measurement of Carbon Anodes Using the Van der Pauw Method Geoffroy Rouget 1,2 , Hicham Chaouki 2 , Donald Picard 2 , Donald Ziegler 3 and Houshang Alamdari 1,2, * 1 2 3
*
Department of Mining, Metallurgical and Materials Engineering, Laval University, Quebec, QC G1V0A6, Canada;
[email protected] NSERC/Alcoa Industrial Research Chair MACE and Aluminum Research Center, Laval University, Quebec, QC G1V0A6, Canada;
[email protected] (H.C.);
[email protected] (D.P.) Alcoa Primary Metals, Alcoa Technical Centre, 859 White Cloud Road, New Kensington, PA 15068, USA;
[email protected] Correspondence:
[email protected]; Tel.: +1-418-656-7666
Received: 27 July 2017; Accepted: 8 September 2017; Published: 13 September 2017
Abstract: The electrical resistivity of carbon anodes is an important parameter in the overall efficiency of the aluminum smelting process. The aim of this work is to explore the Van der Pauw (VdP) method as an alternative technique to the standard method, which is commonly used in the aluminum industry, in order to characterize the electrical resistivity of carbon anodes and to assess the accuracy of the method. For this purpose, a cylindrical core is extracted from the top of the anodes. The electrical resistivity of the core samples is measured according to the ISO 11713 standard method. This method consists of applying a 1 A current along the revolution axis of the sample, and then measuring the voltage drop on its side, along the same direction. Theoretically, this technique appears to be satisfying, but cracks in the sample that are generated either during the anode production or while coring the sample may induce high variations in the measured signal. The VdP method, as presented in 1958 by L.J. Van der Pauw, enables the electrical resistivity of any plain sample with an arbitrary shape and low thickness to be measured, even in the presence of cracks. In this work, measurements were performed using both the standard method and the Van der Pauw method, on both flawless and cracked samples. Results provided by the VdP method appeared to be more reliable and repeatable. Furthermore, numerical simulations using the finite element method (FEM) were performed in order to assess the effect of the presence of cracks and their thicknesses on the accuracy of the VdP method. Keywords: carbon anodes; aluminum smelters; electrical resistivity; Van der Pauw
1. Introduction The Hall–Héroult process [1,2] is used to perform the electrolysis of alumina. To this end, an electrical current is applied through the electrolytic bath between carbon anodes and cathodes. The carbon anodes are used to provide carbon for the chemical reaction, as well as the electrical current necessary for the electrolysis. To ensure the proper functioning of the electrolysis cells, carbon anodes are subject to quality controls such as chemical reactivity [3,4] or mechanical properties such as strength [5] or density [6] during their production. The quality control is performed on core samples, and usually extracted from the top of the anodes beside the stub holes. This location may lead to core samples having some structural flaws, especially at their bottom [7–9]. Core sampling itself may also induce cracks in the samples [10]. The electrical resistivity characterization of anode cores is achieved using the ISO 11713 standard method. This method is merely an adaptation of ASTM B193-02, which is used as a test method for the resistivity of electrical conductor materials, although this latter method requires a sample with neither cracks nor visible defects. Cracks located in the transverse axis (placed Metals 2017, 7, 369; doi:10.3390/met7090369
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normal to the revolution axis of the sample) may most probably induce overestimated values on the measured electrical resistivity, which would not necessarily be representative of the electrical resistivity of the whole anodic block. To overcome this drawback, the Van der Pauw (VdP) method, developed to measure the electrical resistivity of samples with various shapes, represents a potential solution [11,12]. Koon studied the effect of size and place of contacts using the VdP method [13]. De Vries and Wieck targeted the current distribution in certain shapes of samples [14]. Rietveld et al. investigated the accuracy of the VdP method on several shapes of samples [15]. Kasl C. and Hoch M.J.R. [16] proposed a study using the VdP method on circular and cylindrical samples. This study shows that samples with a thickness smaller than their diameter give an accurate value of electrical resistivity. In addition, as exposed by Kasl and Hoch contacts placed along the edge of the sample are preferred to those placed on the top of the sample. Using the VdP method enables the use of samples much smaller than those used with ISO 11713 Standard. Therefore, by reducing the size of the sample, the risk of the presence of flaws reduces. In this study, the reliability of the VdP method for the electrical characterization of carbon anodes was investigated and compared with the standard method, which is currently used in the aluminum industry. Finally, defects were intentionally introduced to the anode cores, and their electrical resistivity was measured using both methods in order to assess their sensitivity to the presence of defects. The first part of this work consists of the comparison of both standard and VdP methods to confirm the reliability of the use of the VdP method for carbon anodes. In the second part, the electrical resistivity of industrial core samples is measured, and the results are compared with the values obtained by the industrial practice. For the third part of this study, another set of experiments is performed on the same batch of anode cores, as used in the preliminary experiment, in order to measure the effect of defect orientation on the measured electrical resistivity. Finally, finite element simulations were performed in order to verify the accuracy of the previously performed experiments, and to validate the use of the VdP method for cracked anode cores. 2. Experimental For both methods performed in the laboratory, the 1 ampere current is provided by a DC Power Supply GW-INSTEK, GPS-3030D, (GW-INSTEK, Taipei, Taiwan). Current and voltage are measured with an Agilent 34461A 61/2 Digit Multimeter (Agilent Technologies, Santa Clara, CA, USA). 2.1. Standard Method In the case of the standard method, the sample was maintained between two steel plates, both placed on springs to apply 3 MPa pressure, which is required by the standard. Voltage drop was measured with two thin pins mounted on spring, in order to apply the same pressure on each spot. These pins are located on a handle, keeping the same distance apart. The experimental setup is presented schematically in Figure 1. Measurements according to the industrial practice were performed using the R&D Carbon RDC 150 for Specific Electrical Resistance apparatus (R&D Carbon, Sierre, Switzerland). To calculate the electrical resistivity of anode core using the ISO 11713, the following equation is used: VS ρ= (1) I L where ρ is the electrical resistivity of the material, V is the voltage drop measured, I is the intensity of the current applied, S is the cross-section of the sample, and L is the length between the voltage probes, as shown in the Figure 1.
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Figure 1. Schematic view of the setting used for the measurement of electrical resistivity of carbon
Figure 1. Schematic view of the setting used for the measurement of electrical resistivity of carbon Figure 1. Schematic the setting used for the measurement of electrical resistivity of carbon anodes accordingview to theof ISO 11713 standard. anodes according to the ISO 11713 standard. anodes according to the ISO 11713 standard. 2.2. Van der Pauw Method
2.2. Van Van der der Pauw Pauw Method Method 2.2. For the Van der Pauw method, since it was decided to take advantage of the cylindrical symmetry, the sample holder was made of adecided self-centering whichofisthe designed for lathes. For the der Pauw method, since it was to takechuck, advantage cylindrical symmetry, For theVan Van der Pauw method, since it was decided to take advantage of the cylindrical Currentholder suppliers and voltage probes were made of copper bars, with a V-shape facing the sample the sample was made of a self-centering chuck, which is designed for lathes. Current suppliers symmetry, the sample holder was made of a self-centering chuck, which is designed for lathes. edge, in order to minimize the contact area, as represented in Figure 2. and voltage probesand were madeprobes of copper with a V-shape facing samplefacing edge, the in order to Current suppliers voltage werebars, made of copper bars, with the a V-shape sample minimize the contact area, as represented in Figure 2. edge, in order to minimize the contact area, as represented in Figure 2.
Figure 2. Picture of the Van der Pauw setting for laboratory measurement. The sample is placed at the center, between the four V-shaped copper probes.
Figure 2. Picture of the Van der Pauw setting for laboratory measurement. The sample is placed at Figure 2. Picture of the Van der Pauw setting for laboratory measurement. The sample is placed at the the center, between the four V-shaped copper probes. center, between the four V-shaped copper probes.
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Each copper bar is mounted with springs on plastic jaws that slide on the chuck’s body (Figure 2). The plastic jaws are used to ensure a good electrical insulation between each copper bar and the chuck. The scroll ring turns to adjust the opening of the probes. All of the probes are related to move at the same time and with the same distance, in order to keep the sample always centered. As presented by Van der Pauw method [11,12], the electrical resistivity, the resistance, and the thickness of the sample between two contiguous sets of points of measurement on its edge must be related by the following equation: e
−π · dL ρ R AB,DC
+e
−π · Lρ R BC,AD
=1
(2)
where L is the thickness of the sample (m); RAB,DC is the measured electrical resistance when the current is injected between probes A and B, and the voltage drop is measured between D and C (µΩ); RBC,AD is the resistance measured when the current is injected between probes B and C, and the voltage drop is measured between A and D (µΩ); and ρ is the electrical resistivity (µΩ·m). For the standard method performed in the laboratory, four measurements were performed around each sample. The average value and the standard deviation were calculated. For the VdP method, eight measurements were performed: four in a first position, and four after rotating the sample by 45◦ . Considering the top view of the setting, which is presented in Figure 2 and the Equation (2), it can be seen that a pair of two contiguous measurements are required to obtain a single value of electrical resistivity. Due to this requirement, four values of electrical resistivity would be obtained after the eight measurements. The average electrical resistivity and standard deviation were obtained after the four electrical resistivities were calculated. To avoid positioning errors, the sample remained in the sample holder the entire time for each set of four measurements, and only the wires from the copper bars were changed. 2.3. Samples Preparation A first series of experiments allowed comparing the ISO 11713 standard method and the VdP method. To this end, eight samples, named 1 to 8, and each with a diameter of 5 cm and a length of 10 cm, were cored from the bottom of a single anode block. The electrical resistivity of each sample was measured using both techniques. This part was done to compare the reproducibility of the measurement, and assumed that the samples would have very close resistivity, as they were taken from the same anode block. To emphasize the comparison between the two techniques, a second series of experiments was carried out. Six industrial samples were tested, using the VdP method and the ISO 11713 standard method. All of the samples had their electrical resistivity measured previously using routine industrial practice (R&D Carbon method). Three samples were intact (in appearance), named Samples A, B and C, and the three other presented noticeable defects (broken into two pieces), named D, E and F. The intact samples were tested using the standard method at the university laboratory in order to compare the results with those obtained by the industrial laboratory. After the measurement using the standard method, three slices were cut out of the samples between the locations of the voltage probes. The electrical resistivity of each slice was measured using the VdP method. The position of the slices in an anode core is presented in Figure 3. The three samples containing defects were directly cut in three slices, as their condition would not allow measuring their resistivity using the standard method. Figure 4 shows the condition of the broken anode core after sampling for measurement by the VdP method.
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Figure Anode core shapes for for measuringelectrical electricalresistivity resistivityusing using the standard method Figure3. 3. the standard method (A)(A) andand the Figure 3. Anode Anode core coreshapes shapes formeasuring measuring electrical resistivity using the standard method (A) and the Van der Pauw method (B). Van der Pauw method (B). the Van der Pauw method (B).
Figure 4. Broken sample prepared for measurement using the Van der Pauw method. The failure was Figure 4. Broken sample prepared der Pauw Pauw method. method. The Figureat 4. the Broken prepared for for measurement measurement using using the the Van Van der The failure failure was was circled rightsample of the sample. circled at the right of the sample. circled at the right of the sample.
The third and last series of experiments was done in order to identify the effect of noticeable The third and last series of experiments was done in order to identify the effect of noticeable defects in third the sample on series the measured electricalwas resistivity. defects were intentionally in The and last of experiments done inThe order to identify the effect of created noticeable defects in the sample on the measured electrical resistivity. The defects were intentionally created in the sample. Samples were machined to create cracks placed in radial and in transversal positions, as defects in the sample on the measured electrical resistivity. The defects were intentionally created the sample. Samples were machined to create cracks placed in radial and in transversal positions, as shown in the Figure 5(A-2,B-2). The samples used in this experiment wereand taken the same anode in the sample. Samples were machined to create cracks placed in radial in from transversal positions, shown in the Figure 5(A-2,B-2). The samples used in this experiment were taken from the same anode as those used for the first set of measurements. as shown in the Figure 5(A-2,B-2). The samples used in this experiment were taken from the same as those used for the first set of measurements. The were performed on samples containing defects before cutting, using the anode as measurements those used for the first set of measurements. The measurements were performed on samples containing defects before cutting, using the standard and then afterperformed cutting, using the VdP method. Finally, numerical simulations were The method, measurements were on samples containing defects before cutting, using the standard method, and then after cutting, using the VdP method. Finally, numerical simulations were carried out to verify the predictive capabilities of the Van der Pauw method. To this end, an electrical standard method, and then after cutting, using the VdP method. Finally, numerical simulations were carried out to verify the predictive capabilities of the Van der Pauw method. To this end, an electrical model developed the Abaqus software [17]. these simulations, carriedwas out to verify the in predictive capabilities of the VanInder Pauw method. To the this anode end, anelectrical electrical model was developed in the Abaqus software [17]. In these simulations, the anode electrical resistivity is prescribed. The electrical resistances of Equation (2) are estimated. Then, the electrical model was developed in the Abaqus software [17]. In these simulations, the anode electrical resistivity resistivity is prescribed. The electrical resistances of Equation (2) are estimated. Then, the electrical resistivity corresponding to the VdP model is calculated, byestimated. solving Equation (2),electrical and comparing it is prescribed. The electrical resistances of Equation (2) are Then, the resistivity resistivity corresponding to the VdP model is calculated, by solving Equation (2), and comparing it with the prescribed value. by considering samples with different configurations, it corresponding to the VdPMoreover, model is calculated, by solving Equation (2), andcrack comparing it with the with the prescribed value. Moreover, by considering samples with different crack configurations, it isprescribed possible to evaluate the effect of these cracks on the accuracy of the VdP method. value. Moreover, by considering samples with different crack configurations, it is possible is possible to evaluate the effect of these cracks on the accuracy of the VdP method. to evaluate the effect of these cracks on the accuracy of the VdP method.
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Figure 5. 5. Schematic Schematic view view of of the the cuts cuts performed performed on on sliced sliced samples, samples, along revolution axis Figure along the the revolution axis (z-axis) (z-axis) (A-1,A-2) and normal to the revolution axis (B-1,B-2). (A-1,A-2) and normal to the revolution axis (B-1,B-2).
3. Results 3. Results All of the values presented in the experimental part are the resistivity values relative to the All of the values presented in the experimental part are the resistivity values relative to the average of the measurements performed on the first eight samples using the ISO 11713 standard average of the measurements performed on the first eight samples using the ISO 11713 standard method, in which the average resistivity measured was of 52.5 μΩ∙m. The results presented are thus method, in which the average resistivity measured was of 52.5 µΩ·m. The results presented are thus a relative electrical resistivity, and a negative value refers to a measured electrical resistivity smaller a relative electrical resistivity, and a negative value refers to a measured electrical resistivity smaller than the average value obtained via standard method. than the average value obtained via standard method. 3.1. Comparison Standard and and Van Van de de Pauw Pauw Methods Methods 3.1. Comparison of of Standard The graph graph presented presented in in Figure Figure 66 shows shows the the values values of of the the relative relative electrical electrical resistivity resistivity of of these these The samples. The baseline of this graph is the average value of eight measurements obtained by the ISO samples. The baseline of this graph is the average value of eight measurements obtained by the ISO 11713 standard standardmethod. method.The Thestandard standarddeviation deviation measurements is presented as percentage of 11713 ofof thethe measurements is presented as percentage of the the absolute measured value, and indicated on each value. The data points marked as “average” on absolute measured value, and indicated on each value. The data points marked as “average” on the the graph represent the average value ofeight all eight measurements performed by two techniques, graph represent the average value of all measurements performed by two techniques, thusthus the the relative forISO the11713 ISO standard 11713 standard is zero. Theshow results that thedeviations standard relative valuevalue for the methodmethod is zero. The results thatshow the standard deviations of the VdP method are much than those obtained bymethod. the standard can of the VdP method are much smaller than smaller those obtained by the standard It canmethod. be notedItfrom be noted from the ninth set of columns, named “Average”, that the average value obtained using the ninth set of columns, named “Average”, that the average value obtained using the VdP methodthe is VdP method is very close to the one obtained using the standard method, while the standard very close to the one obtained using the standard method, while the standard deviation of the former deviation is half that of thefirst standard method. This first set ofthe measurements showed is half thatofofthe theformer standard method. This set of measurements showed high accuracy of the the high accuracy of the VdP method for characterizing resistivity of the anode core. VdP method for characterizing the electrical resistivity ofthe theelectrical anode core.
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5 Relative resistivity (μΩ∙m)
+/- 1.56 %
Std-UL
4 3
VdP
+/- 2.26 %
2
+/- 1.13 % +/- 0.15 %
+/- 0.89 % +/- 1.93 %
1
+/- 0.14 % +/- 2.30 %
0 +/- 7.72 %
-1 -2
+/- 0.55 %
+/- 0.76 %
+/- 2.47 %
+/- 4.86 % +/- 4.72 %
+/- 0.07 %
+/- 0.11 % +/- 4.72 %
-3 -4
+/- 0.87 %
-5 1
2
3
4
5
6
7
8
Average
Sample Number
Figure 6. Comparison of electrical resistivity using standard Pauw methods Figure 6. Comparison of electrical resistivity using thethe standard andand VanVan derder Pauw methods forfor carbon carbon anode cores. anode cores.
3.2. Validation of Van der Pauw Method for Intact Samples
3.2. Validation of Van der Pauw Method for Intact Samples
In the second set of measurements, the three intact industrial samples (A, B and C) were tested.
In of measurements, theanode threeplant, intact industrial samples (A, B in and were tested. Allthe the second samplesset originated from the same Plant 1. The results obtained theC) industrial All the samplesare originated the same plant, Plant Thefour-point results obtained in theof industrial laboratory labeled asfrom “Indus”. Then, anode those obtained using1.the measurements the standard method are labeled as STD-UL. The three slices cut out of each anode core, between theof the laboratory are labeled as “Indus”. Then, those obtained using the four-point measurements locations of the voltage sensors, in order to use the VdP method were labeled VDP-1, VDP-2 and standard method are labeled as STD-UL. The three slices cut out of each anode core, between the VDP-3, theinmean electrical of each sample was calculated using the and locations of respectively. the voltage Finally, sensors, order to use resistivity the VdP method were labeled VDP-1, VDP-2 VdP method, and labeled as VDP-AVG. The results of the measurements are presented in Figure 7. VDP-3, respectively. Finally, the mean electrical resistivity of each sample was calculated using the It can be seen that the electrical resistivity measured at the industrial laboratory and in our laboratory VdP method, and labeled as VDP-AVG. The results of the measurements are presented in Figure 7. are very close when using the standard method. We notice that standard deviation for tests carried It canout be in seen the electrical resistivity the that industrial laboratory were not measured provided. at the industrial laboratory and in our laboratory are very close when using the Cstandard method. We notice thatblocks standard deviation for by tests The samples A, B and were cored from different anode that were provided thecarried same out in theanode industrial laboratory were not provided. plant. This means they share the same raw materials and anode-forming process. However, The A, B and C were from different anode blocks that weremethod, provided byatthe they samples show a strong difference in cored electrical resistivity measured by the standard both thesame industrial laboratory at the university (Indus STD-ULand respectively). On the other hand, the anode plant. This meansand they share the same rawand materials anode-forming process. However, the in VdP method resistivity show that the electricalby resistivity values for the three they results show aobtained strong through difference electrical measured the standard method, both at samples are very close to each other. In addition, the standard deviations were smaller compared the industrial laboratory and at the university (Indus and STD-UL respectively). On the other with those provided by the standard method, which suggested that the VdP method could be a more hand, the results obtained through the VdP method show that the electrical resistivity values for accurate technique of measurement. The drastic change in results measured using the standard the three samples are very close to each other. In addition, the standard deviations were smaller method, both by the industrial partner and at the university, and the VdP method can be explained. compared those provided thethe standard which thatthe the VdP method could When with the standard method isby used, current method, flows through thesuggested sample along axis of revolution be a (Figure more accurate technique of measurement. The drastic change in results measured 5(A-1)), but the voltage drop is measured only on the edge of the sample. During theusing core the standard method, both by the industrialand partner at the containing university,their and the method can be sampling of the anode, microcracks coarseand particles ownVdP defects can be explained. When the standard method is used, current flows through thearea sample along the axis of produced, which may strongly influence thethe current flow at the specific of measurement. However, these5(A-1)), defects probably cannot be seeniswith the naked eye.on Onthe theedge otherof hand, when using revolution (Figure but the voltage drop measured only the sample. During the VdP method, the current flows normally to the axis of revolution (Figure 5(B-1,B-2)), parallel to can the core sampling of the anode, microcracks and coarse particles containing their own defects the cross-section of the sample. This implies that the defects generated during the machining on any be produced, which may strongly influence the current flow at the specific area of measurement. of the edges of the sample, may not have a strong influence on the measurements performed. It is However, these defects probably cannot be seen with the naked eye. On the other hand, when using suggested by the author that the measurement using the standard method occurs at the surface, while the VdP method, the current flows normally to the axis of revolution (Figure 5(B-1,B-2)), parallel to the when using the VdP method, the measurement is over the volume. cross-section of the sample. This implies that the defects generated during the machining on any of the edges of the sample, may not have a strong influence on the measurements performed. It is suggested by the author that the measurement using the standard method occurs at the surface, while when using the VdP method, the measurement is over the volume.
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Relative electrical resistivity (μΩ∙m)
Sample A
Sample B
+/- 0.43 %
0 -0.5
+/- 0.09 %
Sample C
+/- 0.76 %
+/- 0.18 %
+/- 0.70 %
N/C
-1 -1.5 -2 -2.5 -3
+/- 0.06 %
N/C
+/- 8.68 %
-3.5
+/- 1.99 %
-4 -4.5
+/- 0.50 %
+/- 0.42 % N/C
+/- 0.25 % +/- 5.71 %
-5
Indus
STD-UL
+/- 0.43 %
+/- 0.56 %
+/- 0.07 %
VDP-1
VDP-2
VDP-3
VDP AVG
Figure 7. Comparison of standard and Van der Pauw (VdP) methods for intact samples cored from Figure 7. Comparison of standard and Van der Pauw (VdP) methods for intact samples cored from the industrial laboratory. “Indus” and “STD-UL” are the measurements provided respectively by the the industrial laboratory. “Indus” and “STD-UL” are the measurements provided respectively by the industrial laboratory and those achieved using standard method in our laboratory. “VDP (1-3)” are industrial laboratory and those achieved using standard method in our laboratory. “VDP (1-3)” are measurements performed on slices 1, 2 and 3 of each sample (VdP method); VDP-AVG is the average measurements performed on slices 1, 2 and 3 of each sample (VdP method); VDP-AVG is the average ofthe thethree threecombined combinedmeasurements. measurements. of
3.3. Validation of Van der Pauw Method for Broken Samples 3.3. Validation of Van der Pauw Method for Broken Samples Another set of tests was performed on three samples provided by the industrial partner, and Another set of tests was performed on three samples provided by the industrial partner, containing noticeable flaws, as shown in Figure 4. Samples D and E originated from the same anode and containing noticeable flaws, as shown in Figure 4. Samples D and E originated from the same plant (Plant 2), while sample F came from a different anode plant (Plant 3). These samples were anode plant (Plant 2), while sample F came from a different anode plant (Plant 3). These samples already broken upon reception. This condition prevented the performance of any measurement using were already broken upon reception. This condition prevented the performance of any measurement the standard method. In this case, only the measurements provided by the industrial partner using the standard method. In this case, only the measurements provided by the industrial partner (standard) and VdP methods are compared. The anode core pieces were long enough to cut three (standard) and VdP methods are compared. The anode core pieces were long enough to cut three slices out of each sample. Figure 8 presents the results provided by the industrial laboratory and those slices out of each sample. Figure 8 presents the results provided by the industrial laboratory and those obtained using the VdP method. Similar to the previous set of samples, the industrial laboratory did obtained using the VdP method. Similar to the previous set of samples, the industrial laboratory did not provide the standard deviation for the measurements. not provide the standard deviation for the measurements. It can be seen that there are strong differences between samples D and E when the standard It can be seen that there are strong differences between samples D and E when the standard method was used in the industrial laboratory. However, the VdP method showed that they exhibit method was used in the industrial laboratory. However, the VdP method showed that they exhibit very close relative electrical resistivities (−4.57 μΩ∙m for D and −4.91 μΩ∙m for E). The two samples very close relative electrical resistivities (−4.57 µΩ·m for D and −4.91 µΩ·m for E). The two samples D D and E showed a similar electrical resistivity using the VdP method, which might be because the and E showed a similar electrical resistivity using the VdP method, which might be because the same same raw materials and anode-forming process were used in the anode plant. For sample F, the raw materials and anode-forming process were used in the anode plant. For sample F, the standard standard and VdP methods lead to the same results. and VdP methods lead to the same results.
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VDP
Relative electrical resistivity (μΩ∙m)
12
Indus N/C
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10
Relative electrical resistivity (μΩ∙m)
8 6
8
2
6
0
4
-2
2
-6
Indus N/C
10
4
-4
VDP
12
N/C +/- 0.59 % N/C
N/C
0 -2+/- 2.20 % N/C -4
D
+/- 0.59 %
+/- 1.67 % E
F
+/- 2.20 % +/- 1.67 % -6 Figure 8. Comparison of the standard and Van der Pauw methods for flawed samples from the Figure 8. Comparison ofDthe standard and Van der EPauw methods for flawed Fsamples from the industrial industrial laboratory. laboratory. Figure 8. Comparison of the standard and Van der Pauw methods for flawed samples from the
3.4. Validation of Van der Pauw Method for Intentionally-Generated Cracked Samples industrial laboratory. 3.4. Validation of Van der Pauw Method for Intentionally-Generated Cracked Samples Finally, measurements were conducted on the samplesCracked containing intentionally3.4. Validation of Van der Pauw Method for Intentionally-Generated Samplesnoticeable Finally, measurements were conducted on the samples containing noticeable intentionally-generated generated cracks. For all of the following experiments, the samples for the VdP method were sliced cracks. For all of the following experiments, the on samples for thecontaining VdP method were sliced to 1 cm thick Finally, measurements were conducted the samples noticeable intentionallyto 1 cm thick and 5 cm in diameter. To simplify the problem, cracks were made either along the and 5generated cm in diameter. To simplify the problem, cracks were made either along the revolution cracks. For all of the following experiments, the samples for the VdP method were sliced axis revolution axis (radial crack), or normal to it (transverse crack). to crack), 1 cm thick and 5 cm in (transverse diameter. Tocrack). simplify the problem, cracks were made either along the (radial or normal to it A radial crack was generated on a to sample before slicing. This allowed performing electrical revolution axis (radial crack), or normal it (transverse A radial crack was generated on a sample before crack). slicing. This allowed performing electrical measurements by the standard method, both before and after performing the slice cut. Then, three A radial crack was generated on a sample before slicing. This allowed the performing measurements by the standard method, both before and after performing slice cut.electrical Then, three slicesmeasurements were cut out by of the the standard sample to measure the electrical resistivity using the VdP In such method, both before and after performing the slice cut.method. Then, three slices were cut out of the sample to measure the electrical resistivity using the VdP method. In such a situation, twocut positions were used the crack towardresistivity the probes. In the oneVdP position, theIncrack slices were out of the sample to for measure the electrical using method. such was a situation, two positions were used for the crack toward the probes. In one position, the crack was a situation, twodistance positionsbetween were usedtwo for probes. the crackIn toward the probes. In one theplaced crack was placed at an equal the other position, theposition, crack was close to placed at anatequal distance between two probes. In the theother otherposition, position, the crack was placed close to placed an equal distance between two probes. In the crack was placed close to on one of the probes (Figure 9). Those options were chosen to assess the effect of position of the crack one of the probes (Figure 9). Those options were chosen to assess the effect of position of the crack on one of the probes (Figure 9). Those options were chosen to assess the effect of position of the crack on the results. the results. the results.
Figure 9. Position of the crack toward the probes: (1) crack placed between two probes at equal distance; Figure 9. Position of the crack toward the probes: (1) crack placed between two probes at equal (2) crack placed to one close of thetoprobes. distance; (2) close crack one ofthe the probes. Figure 9. Position of placed the crack toward probes: (1) crack placed between two probes at equal distance; (2) crack placed close to one of the probes.
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The tests were also performed on samples that had transversal cracks (normal to the revolution axis). In order to compare the standard and VdP methods, cracks were created before slicing. A side view of the cracked sample is shown on Figure 5(B-2). The obtained results were summarized in Table 1. As the surface and length of samples were taken into account for the electrical resistivity calculation using the standard method, the void corresponding to the crack generation had to be removed from the total sample volume. We recall that the presented values are related to the average value of the electrical resistivity measured in the first experiments, while the standard deviation is related to the real obtained values. Table 1. Electrical resistivity measurement of samples containing defects using both the standard and the Van der Pauw methods. Sample and Configuration
Relative Electrical Resistivity (µΩ·m)
Measurements Using Standard Method Sample without defect Radial Crack void: 1.7% Transversal crack void: 0.18%
−2.4 ± 12% +5 ± 12% +12.9 ± 21%
Measurements using Van der Pauw method Sample without defect Radial Crack Crack placed between two probes Radial Crack Crack placed near the probe Transverse crack
+2.5 ± 0.4% +3.2 ± 2.5% +3.3 ± 2% +4.5 ± 8%
The results presented in Table 1 clearly show that when using the standard method, a noticeable difference is measured between the sample without any cracks, and the sample containing a crack. The electrical resistivity is overestimated in the presence of cracks. Moreover, it seems that the crack orientation substantially affects the measurements provided by the standard method. The results obtained using the VdP method show that this technique is considerably less sensitive to the cracks’ size and orientation than the standard method. In fact, small differences between relative electrical resistivity measurements for different samples were obtained with the VdP method. The bigger standard deviation is obtained for the transverse crack. The thickness of the crack behaves as a layer of insulating material parallel to the sample itself. Using the VdP technique enables the electrical resistivity to be retrieved when the crack is radial, as it does not interfere with any parameter of calculation. When the crack is placed transversally, it has an influence on the thickness of the sample (parameter “d” in the Equation (2)). This causes an overestimation of the electrical resistivity. 4. Finite Element Modelling of VdP Method In order to assess the robustness of the VdP method, numerical simulations were carried out using the commercial finite element software Abaqus [17]. The purpose of these numerical simulations was to simulate the experimental set up developed for the VdP method (Figures 2 and 9). Therefore, for a given sample, with or without crack, and an assigned electrical resistivity, all electrical potentials and resistances needed for the VdP method were evaluated numerically. This enabled the estimation of the electrical resistivity of a sample through the VdP method. To this end, the Newton–Raphson scheme for nonlinear equations was used to solve Equation (2). Furthermore, using this approach, some specific cases, such as very thin cracks, which are not easily achievable in laboratory, can be simulated. In order to reproduce conditions used in the laboratory, a cylindrical sample with 1 cm thickness and a 5 cm diameter was considered. An electrical current of 1 ampere was applied on the supply connector bars. The electrical resistivity of an anode is usually around 50 µΩ·m (a conductivity of
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connector bars. The electrical resistivity of an anode is usually around 50 μΩ∙m (a conductivity of 20,000 S/m) [18]. show thethe developed CAD (Computer-Aaided Design) model, and 20,000 S/m) [18].Figures Figures1010and and1111 show developed CAD (Computer-Aaided Design) model, the mesh usedused for the in thein cases the radial betweenbetween two probes and the mesh forsimulations the simulations the of cases of the crack radialplaced crack placed two(Figures probes 10A and 11A), and the transverse crack (Figure 11A,B). (Figures 10A and 11A), and the transverse crack (Figure 11A,B).
Figure Figure10. 10.CAD CAD(Computer-Aaided (Computer-AaidedDesign) Design)model modelused usedfor forthe thefinite finiteelement elementmethod method(FEM) (FEM)and andthe the VdP VdPmethod, method,(A) (A)radial radialcrack, crack,placed placedbetween betweenthe theprobes; probes;(B) (B)transverse transversecrack. crack.
SinceAbaqus Abaqus software software requires requiresan anelectrical electricalconductivity conductivity as as the the input, input, we wedecided decided to touse usethe the Since valueof of20,000 20,000S/m S/m for for simulations. simulations. Using Using the the parameters parameters mentioned mentionedabove, above,tests testswere wererun runwith witheach each value crack configuration presented previously. The electrical potential at the voltage probes was crack configuration presented previously. The electrical potential at the voltage probes was calculated calculated in the VdP in order the to estimate the sample electrical For resistivity. For and used inand the used VdP equation in equation order to estimate sample electrical resistivity. radial and radial and transversal defects, the crack was designed as 2 mm thick and 1.7 cm deep. In the case of transversal defects, the crack was designed as 2 mm thick and 1.7 cm deep. In the case of the radial the radial crack, simulations were performed with cracks placed between the probes and close to one crack, simulations were performed with cracks placed between the probes and close to one of the of the probes, to the thatexperimental in the experimental measurements. probes, similar similar to that in measurements.
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Figure 11. Mesh model used for FEM and VdP methods; (A) radial crack, placed between the probes; Figure 11. Mesh model used for FEM and VdP methods; (A) radial crack, placed between the probes; (B) transverse cracks. (B) transverse cracks.
A first simulation was run on a simple circular sample. This simulation allowed us to retrieve A first simulation runμΩ∙m on a simple circular allowed userror to retrieve an an electrical resistivity was of 49.8 using the VdP sample. method.This Thissimulation result shows that the between electrical resistivity of 49.8 µΩ · m using the VdP method. This result shows that the error between the the assigned electrical resistivity and the calculated one (using the VdP Method) is very small. After assigned electrical resistivity the calculated onerun (using the VdP Method) is very small. After the the validity of the model was and confirmed, tests were on other configurations. The obtained results validity of the model was confirmed, tests were run on other configurations. The obtained results are are stored in Tables 2 and 3. stored in Tables 2 and Table 2 shows the3.simulation results for samples containing a crack parallel to the axis of rotation Table 2 shows the simulation resultsasfor samples containing a crack parallel to the axis of rotation (Figure 5(A-2)). The crack was placed follow: (i) between two sensors at equal distance, and (ii) (Figure 5(A-2)). The crack was placed as follow: (i) between two sensors at equal distance, and (ii)being close close to one of the sensors. It is expected that, for a defect parallel to the axis of revolution to one ofto thethe sensors. is expected a defect to does the axis revolution being normal to the normal electricIt current line,that, the for volume of parallel the crack notofsubstantially affect the results. electric current line, the volume of the crack does not substantially affect the results. This assumption This assumption can be explained in that the VdP equation, Equation (2), is not related to the surface can in that the VdP equation, Equation (2), 5(A-2) is not related theposition surface of area the sample, areabe ofexplained the sample, but rather only to its height. Figure depictstothe theofcrack, which but rather only to its height. Figure 5(A-2) depicts the position of the crack, which in the case of thisa in the case of this orientation, does not interfere in the calculation. To justify the latter statement, orientation, does not interfere in the calculation. To justify the latter statement, a simulation was simulation was performed on a sample containing an infinitesimally thin crack (0.1 mm), which does performed on any a sample containing an infinitesimally thin (0.1 It mm), which does not represent not represent noticeable variation in the volume of thecrack sample. is important to note that such any noticeable variation in the volume of the sample. It is important to note that such a thin crack may a thin crack may not be visible to the naked eye. The crack was placed in the same direction, close to not be visible to the naked eye. The crack was placed in the same direction, close to one of the sensors. one of the sensors. It appears that the obtained results seem to remain stable whether the crack is It appears that obtained results seem to remain whether theiscrack is placed close to a probe, placed close tothe a probe, or centered between two, stable or even if its size highly reduced. Furthermore, or centered between two, or even if its size is highly reduced. Furthermore, the relative error between the relative error between the assigned electrical conductivity and the one retrieved using the finite the assigned electrical conductivity and the one retrieved using the finite element method (FEM) and element method (FEM) and VdP methods remains very low. VdP methods remains very low. Table 2. Electrical resistivity estimation, using the FEM and VdP methods, for a sample containing a Table Electrical resistivity estimation, using the FEM and VdP methods, for a sample containing a crack 2. parallel to the axis of revolution. crack parallel to the axis of revolution.
Original Electrical Conductivity Electrical Conductivity Estimated Using Conductivity Estimated Using FEM and ErrorError Original Conductivity (S/m) FEM and VdP Methods (S/m) (S/m) VdP Methods (S/m) 2-mm wide crack, parallel to the axis of 20,000 20,316 +1.58% revolution, equal distance to sensorsto the axis 2-mm wide crack, parallel wide crack, parallel to the axis of of 2-mm revolution, equal distance to 20,316 +1.58% 20,000 20,000 20,260 +1.30% revolution, close to one sensor sensors 0.1-mm thin crack, parallel to the axis of 20,000 20,253 +1.37% revolution, close to one sensor 2-mm wide crack, parallel to the axis 20,000 20,260 +1.30% of revolution, close to one sensor In a second approach, thetocrack was placed on a plane normal to the axis of rotation (Figure 5(B-2)). 0.1-mm thin crack, parallel the axis 20,000 20,253 +1.37% Inofthis case, the void with the real height of the sample and most probably affects the results. revolution, close interferes to one sensor To justify this assumption, two cases were investigated using the FEM and VdP methods. The crack In a second approach, crack placed on awhich plane created normal to the axis of rotation was initially 2 mm wide inthe a 10 mmwas thick sample, a non-negligible void(Figure within 5(Bthe 2)). In this case, the void interferes with the real height of the sample and most probably affects the results. To justify this assumption, two cases were investigated using the FEM and VdP methods. The Crack Type Position Crack Type and and Position
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crack was initially 2 mm wide in a 10 mm thick sample, which created a non-negligible void within the material. Referring to Figure 5(B-2) and Equation (2), one can easily state that the void created by material. Referring toprobably Figure 5(B-2) can easily state that created this crack would most affectand the Equation calculation(2), of one electrical resistivity. Afterthe thevoid 2 mm crack,by a this crack would most probably affect the calculation of electrical resistivity. After the 2 mm crack, 0.1 mm crack was placed in the same spot to compare the results. Table 3 shows the results of both a 0.1 mm crack placed theauthors, same spot compare the when results.the Table 3 shows the results both simulations. As was expected byinthe the to error obtained crack is macroscopic is of bigger simulations. As expected by the authors, the error obtained when the crack is macroscopic is bigger than when the crack in infinitesimally thin. In the case of the thin crack placed normal to the axis of than whenthe theerror crackcalculated in infinitesimally thin. In the case ofthan the that thin in crack placed normal to theplaced axis of revolution, tended to be even smaller the case of the thin crack revolution, the same error calculated tended presented to be even after smaller than that in the corroborate case of the thin placed parallel to the axis. The results both simulations the crack hypothesis parallel to the same axis. The results presented after both simulations corroborate the hypothesis initially made by the authors; i.e., when the crack is placed normal to the axis of revolution, namely initially to made the authors; i.e.,thickness when the of crack placed normal axis of revolution, namely parallel the by current lines, the the iscrack will have to anthe influence on the measured parallel to the current lines, the thickness of the crack will have an influence on the measured electrical electrical resistivity. Moreover, the thickness of the crack will induce a non-negligible volume of void resistivity. Moreover, thickness of the crack will inducethe a non-negligible volume of void with a with a higher electricalthe resistivity, after taking into account calculation of the electrical resistivity higher electrical resistivity, after takingin into the electrical calculation of the electrical resistivity of the sample. This leads to an increase theaccount calculated resistivity or, as seen in Tableof3,the a sample. This leads to an increase in the calculated electrical resistivity or, as seen in Table a decrease decrease in electrical conductivity. The bigger the crack, the stronger the error will be3, observed. in electrical conductivity. Thelines bigger the crack, the stronger the error beof observed. Figure 12Figure shows Figure 12 shows the current flowing through the sample in thewill case the radial crack. the current lines sample in theprobes case ofare theplaced radial on crack. Figure 12A 12A represents theflowing currentthrough flowing the when the current the same side of represents the crack; the current flowing when the current probes are placed on the same side of the crack; meanwhile, meanwhile, Figure 12B shows the current lines when the current probes are placed on both sides of Figure 12B shows the current lines when the current probes are placed on both sides of the crack. the crack. In both cases, it can be noticed that the electrical current flows in a plan perpendicular to In both cases, it can be noticed that the electrical current flows in a plan perpendicular to the revolution the revolution axis. These results are in agreement with authors’ hypothesis about the electrical axis. These current flow.results are in agreement with authors’ hypothesis about the electrical current flow.
Figure 12. Cont.
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Figure 12. Representation of the current line on a sample containing radial crack; (A) probes placed Figure 12. Representation of the current line on a sample containing radial crack; (A) probes placed on on the same side of the crack; (B) probes placed on both sides of the crack. the same side of the crack; (B) probes placed on both sides of the crack. Table 3. Electrical resistivity of a sample containing a crack, normal to the axis of rotation, calculated Table 3. Electrical resistivity of a sample containing a crack, normal to the axis of rotation, calculated using the FEM and VdP methods. using the FEM and VdP methods.
Electrical Conductivity Estimated Original Electrical Conductivity Estimated Original Conductivity (S/m)Using FEM and VdP Methods Error Error Using FEM and VdP Methods (S/m) Conductivity (S/m) (S/m) 2-mm wide crack, normal to the axis of revolution 20,000 19,362 3.19% 0.1-mm thin crack, normal to the axis of revolution 20,000 20,215 +1.07% 2-mm wide crack, normal 20,000 19,362 3.19% to the axis of revolution 5. Detection Defects 0.1-mm of thin crack, Using normalVdP Method 20,000 20,215 +1.07% theprevious axis of revolution Intothe section, it was shown that the electrical resistivity could be easily retrieved using Type and Position Position CrackCrack Type and
the VdP method, even if the sample contains structural flaws. Referring to Equation (2), the VdP 5. Detection Defects Using VdP Method method requiresof the measurement of two contiguous resistances, RAB,DC and RBC,AD , in order to calculateIn the electrical resistivity of the material. samples have a cylindrical and are the previous section, it was shown thatThe the electrical resistivity could begeometry, easily retrieved using supposed to be homogeneous in composition. In a crack-free sample, the resistances R and AB,DC the VdP method, even if the sample contains structural flaws. Referring to Equation (2), the VdP RBC,AD present very close In the case of samples containing cracks along axis of revolution, method requires the values. measurement of two contiguous resistances, RAB,DCthe and RBC,AD , in order to the calculate cylindrical symmetry is lost, and only the axis of symmetry remains along the crack. Thus, the electrical resistivity of the material. The samples have a cylindrical geometry, and are the resistances R and R show a significant difference. In the case of cracks normal to the axis AB,DC BC,AD supposed to be homogeneous in composition. In a crack-free sample, the resistances RAB,DC and RBC,AD of revolution, such a difference between and RBC,AD is not cracks observed. This beofexplained by the present very close values. In the caseRAB,DC of samples containing along thecan axis revolution, the orientation the crack is toward the current flow. According to theremains geometryalong of thethe system, can the cylindrical of symmetry lost, and only the axis of symmetry crack.one Thus, stateresistances that the current flows mostly parallel to the faces of the sample; as a result, defects oriented in RAB,DC and RBC,AD show a significant difference. In the case of cracks normal to the the axis of same direction as theacurrent flows will have a lower than those placed in a different direction. revolution, such difference between RAB,DC and Reffect BC,AD is not observed. This can be explained by the For simplicity, RBC,AD arethe respectively expressed as R1 to and hereafter.ofTable 4 showsone the can AB,DC orientationRof the and crack toward current flow. According theR2,geometry the system, results obtained after the simulations. It reveals that for a crack placed in a radial direction, parallel to in state that the current flows mostly parallel to the faces of the sample; as a result, defects oriented the axis of revolution, a strong difference between the two contiguous resistances can be observed, the same direction as the current flows will have a lower effect than those placed in a different regardless of the or R the position of the crack toward the sensors. direction. Forthickness simplicity, AB,DC and RBC,AD are respectively expressed as R1 and R2, hereafter. Table 4
shows the results obtained after the simulations. It reveals that for a crack placed in a radial direction, parallel to the axis of revolution, a strong difference between the two contiguous resistances can be observed, regardless of the thickness or the position of the crack toward the sensors.
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Table 4. Difference for two contiguous resistances for a sample containing a crack parallel to the axis of rotation, calculated using the FEM method. Original Conductivity (S/m)
Electrical Resistance Measured Using Finite Element Method
Difference
2-mm wide crack, parallel to the axis of revolution, equal distance to sensors
20,000
R1 = 0.7 mΩ R2 = 1.74 mΩ
59.8%
2-mm wide crack, parallel to the axis of revolution, close to one sensor
20,000
R1 = 0.52 mΩ R2 = 1.99 mΩ
73.9%
0.1-mm thin crack, parallel to the axis of revolution, close to one sensor
20,000
R1 = 0.553 mΩ R2 = 1.91 mΩ
71.5%
Crack Type and Position
On the other hand, Table 5 presents the results obtained in the case of a crack placed normally to the axis of revolution of the sample. This means that the crack shares the direction of the electrical current lines. In this case, referring to Equation (2) and considering the Figure 5(B-2), the parameter “d” is obtained by subtracting the thickness of the crack from the thickness of the sample. Therefore, the thickness of the crack implicitly influences the measured values when the thickness of the crack is not negligible compared with that of the sample. In this configuration, the effect is limited to a diminution of the actual volume of matter through which the current flows. Results presented in Table 5 show that the difference between contiguous probes is very low when compared with the case of a crack normal to the electrical field lines. The difference between the results in the two sections of Table 5 is probably because when a crack is of a non-negligible size, the lack of material to conduct the current, which acts as a much more resistive part, will lead to higher resistance values and therefore a lower electrical conductivity. Consequently, the resistances measured for a 2 mm thick crack are slightly higher than that measured for a 0.1 mm thin crack. Those results are in a good adequacy with the above-mentioned hypothesis. Table 5. Difference for two contiguous resistances for a sample containing a crack normal to the axis of rotation, calculated using the FEM method. Crack Type and Position
Original Conductivity (S/m)
Electrical Resistance Measured Using Finite Element Method
Difference
2 mm wide crack, normal to the axis of revolution
20,000
R1 = 1.1 mΩ R2 = 1.18 mΩ
6.8%
0.1 mm thin crack, normal to the axis of revolution
20,000
R1 = 1.086 mΩ R2 = 1.0969 mΩ
0.99%
The difference of values between R1 and R2 can indicate very precisely the presence of cracks, specifically if they are placed parallel to the axis of revolution. According to the obtained results, it seems that the Van der Pauw method, when coupled with FEM, helps detect the presence of defects in an anode core. Depending on the type, size, and orientation of the defects, the detection can be either good or totally inadequate. This method is complementary to the standard method that is usually used for the characterization of the anode core, and could provide better defect detection. 6. Conclusions In this work, the electrical resistivity of baked carbon anodes used in the aluminum industry was studied using both the ISO 11713 standard method and the Van der Pauw method. Comparison side by side was achieved by experimental analysis; then, numerical simulation was used to emphasize the previously obtained results. The results obtained experimentally show that on industrially produced anode cores, the Van der Pauw method appears to be more accurate and less sensitive to macro and micro structural defects in the material. As the cracks in anode cores are not necessarily representative of the microstructure of the whole anode, the knowledge of the electrical resistivity of the material without the effect of crack, which can be considered as intrinsic electrical resistivity of the material, gives better information on the
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electrical properties of the anode block, i.e., inhomogeneity, paste formulation effect, etc. This method shows a good repeatability in the measurement of the electrical resistivity of samples. Numerical simulations using the finite element method showed that this method is perfectly suitable for the measurement of the electrical resistivity of samples containing defects or flaws such as macroscopic cracks, in different orientations. It has to be noted that depending on the size and orientation of the defect toward the sensors, the defect may induce an error in the calculation of electrical resistivity. Also, the Van der Pauw method, complementary with the standard methods ISO 11713, can reveal the presence of defects in the material that cannot be seen by the naked eye. The Van der Pauw method is very simple and easy to use, and would be easy to automate for industrial purposes. Acknowledgments: The authors gratefully acknowledge the financial support provided by Alcoa Inc., the Natural Sciences and Engineering Research Council of Canada, and the Centre Québécois de Recherche et de Développement de l’Aluminium. A part of the research presented in this article was financed by the Fonds de recherche du Québec-Nature et Technologies by the intermediary of the Aluminium Research Centre–REGAL. Author Contributions: Geoffroy Rouget and Houshang Alamdari conceived and designed the experiments; Geoffroy Rouget and Hicham Chaouki performed the experiments and numerical simulations; Donald Picard and Donald Ziegler contributed in data analysis; Geoffroy Rouget wrote the paper and all co-authors commented/corrected it. Conflicts of Interest: The authors declare no conflict of interest.
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Kasl, C.; Hoch, M.J.R. Effect of sample thickness on the van der Pauw technique for resistivity measurements. Rev. Sci. Instrum. 2005, 76, 033907. [CrossRef] Abaqus Analysis User’s Guide; Version 6; Dassault Systèmes Simulia Corp.: Providence, RI, USA, 2014. Mannweiler, U.; Keller, F. The design of a new anode technology for the aluminium industry. J. Miner. Met. Mater. Soc. 1994, 48, 15–21. [CrossRef] © 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
