Assessing the feasibility of applying ERT for the evaluation of electrical conductivity of green carbon anode

Assessing the Feasibility of Applying ERT for the

Evaluation of Electrical Conductivity of Green Carbon

Anode

Mémoire

Somaiieh Yousefi

Maîtrise en génie des matériaux et de la métallurgie

Maître ès Sciences (M. Sc.)

Québec, Canada

© Somaiieh Yousefi, 2017

Assessing the Feasibility of Applying ERT for the

Evaluation of Electrical Conductivity of Green Carbon

Anode

Mémoire

Somaiieh Yousefi

Sous la direction de:

iii

Résumé

La qualité de l'anode, particulièrement du point de vue de la résistivité électrique, est primordiale pour l'industrie de l'aluminium. Plus la résistivité électrique est faible, moins l'énergie serait perdue lors de l’opération. Le but du présent projet est donc d'évaluer la qualité de l'anode de ce point de vue. L’anode est composée de trois phases; particules de coke, matrice liant (pitch) et des porosités ou des fissures. La résistivité électrique de l'anode est fortement affectée par la taille (la distribution de taille) et la forme (la distribution de forme) de ces phases. Par conséquent, il est essentiel de comprendre les mécanismes de conduction de l'anode et de mettre en évidence l'effet de la microstructure de l'anode sur sa résistivité électrique.

Dans la présente étude, nous avons essayé de créer une carte de résistivité électrique (ou conductivité) de l'anode et de la corréler avec la répartition de différentes phases. La possibilité d'utiliser de la tomographie par résistance électriques (ERT) a été évaluée en tant qu’une méthode pour cartographier la distribution de la résistivité électrique. La carte est une image de la distribution de la résistance électrique en 2-D des tranches d’anode. ERT est un processus d'estimation de la résistivité à partir des mesures de tension sur un domaine d'intérêt. Le procédé ERT consiste à mettre une série d’électrodes sur la surface de la pièce, puis injecter le courant dans une paire d’électrode et mesurer la tension des autres électrodes. Le processus continue jusqu'à ce que chaque électrode soit considérée une fois comme électrode d'injection. Alors que les tensions mesurées fournissent une matrice des mesures, le potentiel électrique à l'intérieur du matériau est calculé en utilisant la méthode des éléments finis (FEM). En comparant les tensions mesurées et calculées et en minimisant l'erreur des algorithmes de reconstruction, l'image de la conductivité est obtenue.

Les images électriques par ERT ont été comparées avec des images obtenues par la microscopie électronique à balayage (MEB) et la microscopie de la Fluorescence de Rayons X (XRF). La comparaison a montré une certaine corrélation entre la répartition des phases dans l'anode de carbone et sa carte de la distribution de la résistance électrique.

iv

Abstract

The quality of anode especially from electrical resistivity aspect is important to the aluminum industry. The higher the current can pass through the anode by the means of lower electrical resistivity, the lower the energy would be lost. The purpose of present project is to evaluate the quality of anode from the electrical properties point of view. Anode consists of three phases; coke particles, binder matrix (pitch) and porosities or cracks. The electrical resistivity of anode is highly affected by the size (size distribution) and shape (shape distribution) of these phases. Therefore, it is essential to understand the conduction mechanisms of anode and to reveal the effect of anode microstructure on its electrical resistivity.

In the present study, we attempted to create a map of electrical resistivity (or conductivity) of anode regarding the distribution of the phases. Feasibility of using Electrical Resistance Tomography (ERT) was assessed as a method for mapping the electrical resistivity distribution in carbon anode. The map is a computed image of the distribution of electrical resistance in 2-D slice through a conducting region. ERT is a process of estimating from voltage measurements at the domain of interest. ERT method involves with putting electrodes at the boundary, injecting the current to each pair and measuring the voltage from the remaining ones until each electrode once considered as the injecting electrode. While the measured voltages provide a matrix of measurements, the electrical potential inside the material would be calculated using the Finite Element Method (FEM). By comparing the measured and calculated voltages and minimizing the error and utilizing reconstruction algorithms, the conductivity image of the desired surface will be obtained.

In the final analysis, the electrical images by ERT were evaluated using SEM microscope and XRF analysis. The comparison suggested a good correlation between the electrical images and the distribution of the phases in carbon anode.

v

Table of Contents

Résumé ... iii

Abstract ... iv

Table of Contents ... v

List of Figures ... vii

Acknowledgements ... ix

Chapter 1. Introduction ... 1

1.1. Aluminum production ... 1

1.2. Anode properties respecting raw materials ... 3

1.3. The electrical resistivity of anode ... 4

1.4. Four point measurement ... 4

1.5. The effect of raw materials and fabricating process on the electrical resistivity of anode ... 5

1.6. Computed Tomography ... 7

1.7. Problem statement ... 7

1.8. Conclusion ... 8

Chapter 2. Literature Review... 9

2.1. Electrical Impedance Tomography ... 9

2.2. Using ERT (EIT) for characterizing materials ... 11

2.3. ERT Fundamentals ... 16

2.3.1. The forward problem ... 18

2.3.2. Inverse problem ... 19

2.4. EIT for simple systems ... 21

2.5. ERT system and measured data set ... 23

2.5.1. Reconstruction of γ in a 4-channel EIT system: ... 24

2.6. Hypothesis ... 28

2.7. Objectives ... 28

2.8. Summary ... 28

Chapter 3. Materials and methods ... 29

3.1. Methodology ... 29

3.2. Finite Element Simulation ... 30

3.2.1. Defining material ... 30

3.2.2. Assigning the properties ... 30

3.2.3. Meshing ... 31

3.2.4. Applying boundary conditions ... 32

3.2.5. Running ... 33

3.2.6. Output of the FEM system of a bed of coke with a pitch particle in the middle ... 33

3.2.7. Defining material ... 34

3.2.8. Output of the FEM system of a bed of pitch with a coke particle in the middle ... 35

3.2.9. Characterization methods ... 36

3.3. Experiment with ERT equipment ... 37

3.3.1. ERT Setup ... 38

Chapter 4. ERT Experimental Results ... 40

4.1. Quarter 1 ... 45

4.2. Quarter 2 ... 48

vi

4.4. Quarter 4 ... 56

4.5. ERT-XRF Image Comparisons ... 63

4.6. Conclusion ... 69

4.7. Recommendations for Future work ... 69

Chapter 5. Appendix ... 75

5.1. Mathematical setting ... 75

vii

List of Figures

Figure 1 Hall-Héroult reduction cell in prebaked electrolytic cell ... 2

Figure 2 Schematic of a set up for determining the specific electrical resistivity of anode[15] ... 5

Figure 3 Correlation between baked density and specific electrical resistivity [15] ... 6

Figure 4 Specific electrical resistivity versus mixing time [15] ... 7

Figure 5 Specific electrical resistivity versus vibration time [15] ... 7

Figure 6 EIT (ERT) setup; A. ERT operating system [26], B. Computer setup [25]. ... 9

Figure 7 Binary image of a block of soil. A network of cracks can be detected… [56] ... 11

Figure 8 A. Photographs of the three level surfaces B. Interpreted …[57]... 12

Figure 9 Top view of the specimen A. Configuration setup of ERT …[58] ... 13

Figure 10 Illustration of non-linear low-pass filtering for regularization in 2D EIT [59] ... 14

Figure 11 Cylindrical simulated model A. 3D view of the specimen …[60]. ... 15

Figure 12 The images reconstructed from the cylinder model calculated …[60] ... 16

Figure 13 Current flux near a. active and b. passive electrode [61] ... 17

Figure 14 A very EIT system …[77] ... 22

Figure 15 A 4-channel EIT system …[77] ... 25

Figure 16 An example of complex potentials …[77] ... 26

Figure 17 The process of minimization… [77] ... 27

Figure 18 Defining material ... 30

Figure19 Assigning material properties ... 31

Figure 20 Meshing ... 31

Figure 21 Assigning boundary conditions ... 32

Figure 22 Applying boundary conditions ... 33

Figure 23 Electric potential distribution with a particle of pitch…... 34

Figure 24 Defining material ... 34

Figure 25 Assigning material properties ... 35

Figure 26 Electric potential distribution of a coke particle in a bed of pitch…... 36

Figure 27 ERT setup fabricated in Ryerson University… ... 37

Figure 28 Full electrode scan… ... 39

Figure 29 Reconstructed image of the first ring of the sample with 96% relative density… ... 41

Figure 30 Reconstructed image of the first ring of the sample with 96% relative density… ... 42

Figure 31 Reconstructed image of the second ring of the sample with 96% relative density… .. 43

Figure 32 Reconstructed image of the third ring of the sample with 96% relative density…... 44

Figure 33 Images of the first ring, A. ERT image, B. Surface image… ... 45

Figure 34 SEM image of the green area in quarter 1 of the ERT image… ... 46

Figure 35 EDS analysis of some areas in the quarter, A. Zone A…... 47

Figure 36 Porosities at the top of quarter 1 (zone Y1) ... 48

Figure 37 Porosity in quarter 2 (zone Y2) ... 49

Figure 38 EDS analysis, A. Zone A, B. Zone A1... 50

Figure 39 Several porosities in quarter 2 (zone Y2) ... 50

Figure 40 EDS diagrams, A. Zone A1, B. Zone A2, C. Zone A3, D. Zone B ... 51

Figure 41 SEM image of a large coke particle in quarter 3 (zone Z3) ... 52

Figure 42 EDS diagrams, A. Zone A1, B. Zone A2, C. Zone A3, D. Zone A4, E. Zone A5 ... 53

Figure 43 SEM image of a coke particle in quarter 3 (zone Z3) ... 54

viii

Figure 45 SEM image, random area in quarter 3 ... 55

Figure 46 EDS analysis of the third quarter, A. Zone A… ... 56

Figure 47 SEM image of a porosity in quarter 4 (zone Y4) ... 57

Figure 48 EDS analysis in quarter 4, A. Zone A, B. Zone B, C. Zone C, D. Zone D ... 58

Figure 49 SEM image, porosity in quarter 4 (zone Y4) ... 59

Figure 50 SEM image, porosity in quarter 4 (zone Y4) ... 59

Figure 51 SEM image, porosity in quarter 4 (zone Y4) ... 60

Figure 52 SEM image, porosity in quarter 4 (zone Y4) ... 60

Figure 53 EDS analysis in quarter 4, A. Zone A, B. Zone B, C. Zone C… ... 61

Figure 54 SEM image, random area in quarter 4 ... 62

Figure 55 EDS analysis, random area in quarter 4 ... 63

Figure 56 ERT image ... 64

Figure 57 XRF image, four numbered regions… ... 65

ix

Acknowledgements

I would like to express my acknowledgements to Prof. H. Alamdari and Prof. M. Fafard for their interests and guidance for this project.

I also sincerely appreciate the warm welcoming and the supports of Prof. F. Ein-Mozaffari in Ryerson University for the permission to use the ERT equipment in his laboratory and his team of research including Mr. Ali Hemmati and A. Kazemzadeh for designing the special setup for our anode lab sample.

I also would like to express my appreciations to Dr. K. Azari for his generous rendering information as well as his anode samples.

Last but not least, I would like to present my appreciations to my parents who have always supported me for taking steps in my life.

1

&KDSWHU ,QWURGXFWLRQ

 $OXPLQXPQH[WWRVWHHOLVWKHPRVWZLGHO\XVHGPHWDOLQWKHZRUOG2ZLQJWRLWVSURSHUWLHV DQG SHUIRUPDQFHEHVLGHLWVSURGXFWLRQFRVWVDOXPLQXPLVSUHIHUDEOHLQWKHPDMRULW\RIDSSOLFDWLRQV7KH ZLGHUDQJHRIDOXPLQXPDSSOLFDWLRQLVGXHWRLWVJRRGVWUHQJWKWRZHLJKWUDWLRJRRGHOHFWULFDODQG WKHUPDOFRQGXFWLYLW\UHVLVWDQFHWRFRUURVLRQUHF\FODELOLW\IRUPDELOLW\FRDWLQJDELOLW\DSSHDUDQFH OLJKWZHLJKWUHVLVWDQFHWRZHDWKHULQJWKHIDFLOLW\RILWVIDEULFDWLRQDQGQRQWR[LFLW\LQFRQWDFWZLWK ZLGHUDQJHRIIRRGV>@

$OXPLQXP LV QHYHU IRXQG LQ WKH QDWXUH LQ PHWDOOLF IRUP GXH WR LWV KLJK UHDFWLYLW\ ZLWK DLU ,W FRQVWUXFWVDERXWRIWKHFUXVWRIHDUWKLQGLIIHUHQWPLQHUDOV>@3URGXFWLRQRIDOXPLQXP LQYROYHV ZLWK WKH UHGXFWLRQ RI LWV R[LGH E\ HOHFWURO\VLV RI GLVVROYHG DOXPLQD ܣ݈_ଶܱ_ଷ LQ PROWHQ FU\ROLWH ܰܽ_ଷܣ݈ܨ_଺ LQ DSSUR[LPDWHO\ Ԩ 7KLV SURFHVV LV FDOOHG +DOO+pURXOW 7KH SURFHVV LV FRQGXFWHG LQ ODUJH UHFWDQJXODU FHOOV ZLWK FDUERQ FDWKRGH EORFNV DQG LQVXODWLQJ EULFNV +LJK HOHFWULFDOFXUUHQWLVSDVVHGWKURXJKDQRGHVFDUERQEORFNVWKHPROWHQDOXPLQXPILQDOO\LVFROOHFWHG DWWKHFDWKRGHVORFDWHGLQWKHERWWRPRIWKHFHOOV>@)LJ ,Q+DOO+pURXOWSURFHVVWKHUHDFWLRQHTXDWLRQLVDVEHOORZ ʹܣ݈_ଶܱ_ଷ൅ ͵ܥ ՜ Ͷܣ݈ ൅ ͵ܥܱ_ଶ  ,WLVLPSRUWDQWIRUFDUERQDQRGHVWRKDYHWKHVHSURSHUWLHV x +LJKFKHPLFDOSXULW\ x +LJKHOHFWULFDOFRQGXFWLYLW\ORZUHVLVWLYLW\ x +LJKWKHUPDOVKRFNUHVLVWDQFH x +LJKPHFKDQLFDOVWUHQJWKDQGKRPRJHQHLW\ x /RZUHDFWLYLW\WRZDUGVFDUERQGLR[LGHܥܱ_ଶDQGDLU :LWKLQWKH+DOO+pURXOWSURFHVVFDUERQDQRGHVDUHFRQVLGHUHGDVDSDUWRIHOHFWULFDOFLUFXLW&DUERQ LVFRQVXPHGGXULQJHOHFWURO\VLVDQGVKRXOGEHUHSODFHGHYHU\WRZHHNVGHSHQGLQJRQWKHVL]HRI WKH DQRGH DQG LWV FXUUHQW GHQVLW\ 7KH FKDQJLQJ WLPH LV ZKHQ WKH WKUHHTXDUWHU RI WKH DQRGH LV FRQVXPHG7KHUHPDLQGHULVWKHµEXWW¶ZKLFKLVWKHQFOHDQHGRIHOHFWURO\WHFUXVKHGDQGUHF\FOHGDV DSDUWRIQHZHOHFWURGHV7KHFRVWRIHOHFWULFLW\XVHGLQWKLVSURFHVVKDVDODUJHFRQWULEXWLRQRIWKH ZKROHDOXPLQXPSURGXFWLRQFRVWV

2

Figure 1 Hall-Héroult reduction cell in prebaked electrolytic cell

There are some reactions in the cell which do not contribute in the metal reduction. Air burning is one of these reactions that result in excess consumption. This reaction is the oxidation of carbon when the prebaked anode is immersed in electrolyte and a gradient temperature of 400 to 700 ℃ is established along the block [6]. If the anode is not completely covered by the mixture of bath/alumina, the reaction between the surface of aluminum and the atmospheric oxygen takes place (air burn)(Eq. 1.2)

O₂+ C → CO₂ (1.2)

Anode consumption in the whole aluminum production process is involved with the term “current efficiency”. The reaction of aluminum with carbon dioxide cause current efficiency loss (Eq. 1.3).

2 𝐴𝑙 + 3𝐶𝑂₂ → 𝐴𝑙₂𝑂₃+ 3𝐶𝑂 (1.3)

The Current Efficiency (CE) is significantly affected by the electrical resistivity of carbon anode. One of the factors which can greatly affect the CE is the current loss of different sources:

 Electronic conduction due to dissolved metal,

 Short-circuit between the anode and cathode,

 Parallel current path through the top crust [7],

3

7KHFRQFHQWUDWLRQRIWKHPHWDOLQFU\ROLWHLVORZDQGWKLVGLVVROYHGPHWDOGLR[LGLVHGEHIRUHUHDFKLQJ WKHDQRGH7KXVWKHHOHFWURQLFFRQGXFWLRQRIWKHEDWKFDQEHQHJOHFWHG6KRUWFLUFXLWEHWZHHQDQRGH DQG FDWKRGH LV WKH RWKHU FRPPRQ SUREOHP HVSHFLDOO\ LQ 6RGHUEHUJ FHOOV &XUUHQW OHDNDJH WKURXJK WKH WRS FUXVW DV WKH RWKHU IDFWRU FDQ EH FRQVLGHUHG QHJOLJLEOH GXH WR WKH KLJKHU UHVLVWLYLW\ LQ WKH VROLGSKDVHHJRKPFPDWԨDQG‘ŠǤ …DWԨ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ܥܱଶDQGDLU>@7KHVHIDFWRUVDUHKLJKO\

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

$FFRUGLQJWRWKHSDVWVWXGLHVLWKDVEHHQVKRZQWKDWWKHSUHVHQFHRISRURVLWLHVLQFRNHSDUWLFOHVDQG DQRGHPD\EHDQLPSRUWDQWIHDWXUHLQPHFKDQLFDOSURSHUWLHVDQGHOHFWULFDOUHVLVWLYLW\RIWKHDQRGH > @ 3URGXFWLRQ VWHSV VXFK DV YDULDEOH PL[LQJ PHWKRGV EDNLQJ FRQGLWLRQV DQG YLEURIRUPLQJ ZRXOGDIIHFWWKHSRURVLWLHVDQGFRQVHTXHQWO\WKHHOHFWULFDOSURSHUWLHVRIWKHDQRGHV>@

4

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 7KH DGYDQWDJH RI WKLV PHWKRG LV LWV VLPSOH VHWXS +RZHYHU LW FRXOG QRW UHODWH WKH HOHFWULFDO UHVLVWLYLW\GLVWULEXWLRQWRWKHGLVWULEXWLRQRIWKHSKDVHVLQDQRGH  7KHVSHFLILFHOHFWULFDOUHVLVWLYLW\DWDFRQVWDQWFXUUHQWLVGHWHUPLQHGE\PHDVXULQJWKHYROWDJHGURS RYHUDF\OLQGULFDOVDPSOHXVLQJDIRXUSRLQWPHDVXUHPHQWHTXLSPHQW)LJ>@  ߩ ൌܷ௔௕Ǥ ܣ ܮǤ ܫ    :KHUH ߩLVWKHVSHFLILFHOHFWULFDOUHVLVWLYLW\LQȳǤ ݉ 8DELVYROWDJHGURSRYHUWKHPHDVXUHGVDPSOHLQݒ $LVWKHFURVVVHFWLRQLQ݉ଶ_ /LVWKHGLVWDQFHEHWZHHQSRWHQWLDOFRQWDFWVLQP ,LVWKHHOHFWULFDOFXUUHQWWKURXJKWKHVDPSOH $VLWFDQEHLQIHUUHGIURP(TWKLVHTXDWLRQSUREHVWKHHOHFWULFSRWHQWLDOEHWZHHQWZRSRLQWV KDYLQJ WKH YDOXH RI WKH LQMHFWHG FXUUHQW ZKLFK LV D PHDVXUHPHQW UDWKHU WKDQ SURYLGLQJ WKH LQIRUPDWLRQRIWKHGLVWULEXWLRQRIWKHWKUHHSKDVHV7KHUHIRUHWKLVPHWKRGZRXOGQRWEHVXIILFLHQW IRUPDSSLQJWKHFRQGXFWLYLW\GLVWULEXWLRQ

 

5  )LJXUH6FKHPDWLFRIDVHWXSIRUGHWHUPLQLQJWKHVSHFLILFHOHFWULFDOUHVLVWLYLW\RIDQRGH>@  ,QRUGHUWRNQRZWKHGHSHQGHQF\RIDQRGHHOHFWULFDOUHVLVWLYLW\WRGLIIHUHQWLQYROYHGSDUDPHWHUVRQH VKRXOGILUVWKDYHDNQRZOHGJHRQWKHUDZPDWHULDOVDQGDOVRWKHPDQXIDFWXULQJSURFHVV >@+HUH VRPHRIWKHPRVWLPSRUWDQWIHDWXUHVDUHEULHIO\PHQWLRQHG

&RNH SDUWLFOHV KDYH WKUHH GLIIHUHQW W\SHV DFFRUGLQJ WR WKHLU WH[WXUH DQLVRWURSLF LVRWURSLF DQG VSRQJHFRNH>@5HJDUGLQJWKHUHVHDUFKE\0HGHNDQG:HLVKDXSWRYDLQQHHGOHFRNHHOHFWURGHV ZLWK LQFUHDVLQJ LQ SRURVLW\ WKH HOHFWULFDO UHVLVWLYLW\ LQFUHDVHG DQG WKH VWUHQJWK GHFUHDVHG >@ )XUWKHUPRUH LW KDV EHHQ VKRZQ WKDW LQFUHDVH LQ WKH GHQVLW\ RI WKH EDNHG DQRGH ZRXOG FDXVH D GHFUHDVHLQWKHHOHFWULFDOUHVLVWLYLW\RIDQRGH)LJ>@

7KHRWKHUDIIHFWLQJIDFWRUWRZDUGVHOHFWULFDOUHVLVWLYLW\LVFDOFLQLQJSURFHVV ,WKDVEHHQIRXQGWKDW WKH HOHFWULFDO UHVLVWLYLW\ RI WKH FRNHV GHFUHDVHV OLQHDUO\ ZLWK WKH FDOFLQDWLRQ WHPSHUDWXUH >@ &DOFLQLQJLVDKHDWWUHDWPHQWWRԨLQZKLFKWKHDPRUSKRXVSKDVHJUDGXDOO\FKDQJHVWRJUDSKLWH VKHHWV,QJUDSKLWHVKHHWVݏ݌ଶIUHHHOHFWURQVDVVLVWHDVLHUFXUUHQWIORZDQGDVWKHUHVXOWWKHHOHFWULFDO UHVLVWLYLW\RIFRNHGHFUHDVHVZLWKLQWKLVSURFHVV

6

Figure 3 Correlation between baked density and specific electrical resistivity [15]

Mixing is the other influencing factor for the electrical resistivity of anode. Binder is used to wet the filler surface during the mixing. The wetting process is dependent upon the characteristics of the filler and the binder [21]. Generally, the interaction of the two steps of mixing and forming is taking part in anode production. Increase in the mixing time leads to a general decrease of the specific electrical resistivity (Fig. 4). As the mixing time increases some bridges in coke form, which facilitate the electrical current and decreases the electrical resistivity of coke. Furthermore, within vibro-compacting process, in order to increase the density of coke, increasing vibration time causes closer packing that enables better electrical contacts between coke grains after baking [15]. This is seen also in Fig. 5 where the 1 minute mixed paste was mostly affected by vibration time.

During baking process, heating rate, as a factor, can affect the electrical resistivity of anode. High heating rate would result in less anode density, which can cause rising the electrical resistivity [22]. In order to reduce the effect of raw materials on the final quality of real carbon anode, Computed Tomography, is used as the diagnostic instrumentation [23].

7  )LJXUH6SHFLILFHOHFWULFDOUHVLVWLYLW\YHUVXVPL[LQJWLPH>@  )LJXUH6SHFLILFHOHFWULFDOUHVLVWLYLW\YHUVXVYLEUDWLRQWLPH>@  &RPSXWHG7RPRJUDSK\&7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

8

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

'XH WR WKH SUHVHQFH RI GLIIHUHQW SKDVHV DQG WKH SRURVLWLHV WKH HOHFWULFDO UHVLVWLYLW\ GLVWULEXWLRQ LV YDULRXV DQG DV D FRQVHTXHQFH WKH FXUUHQW GLVWULEXWLRQ LV LQKRPRJHQHRXV LQ DQRGH 7KLV LQKRPRJHQHLW\ LV QRW GHVLUDEOH IRU WKH FXUUHQW GHQVLW\ %HFDXVH LQKRPRJHQHRXV FXUUHQW SDWWHUQV UHVXOWLQJ IURP LQKRPRJHQHLWLHV RI WKH DQRGH VWUXFWXUH FDXVHV VRPH DUHDV RI PRUH GHQVH FXUUHQW SDWWHUQVWHQGLQJWREHFRQVXPHGIDVWHUDQG RWKHU UHJLRQVRIOHVVGHQVHFXUUHQWSDWWHUQV WHQGWREH OHVVFRQVXPHG

&7 LPDJHV FRXOG SURYLGH XV ZLWK WKH LQIRUPDWLRQ RI WKH VRPH FUDFNV SRURVLWLHV DQG FRNH FRQFHQWUDWLRQVLQWKHOLJKWRIDSSO\LQJ;UD\+RZHYHULQ&7LPDJHVFRNHDJJUHJDWHVDQGSLWFKDUH GHPRQVWUDWHG WKH VDPH FRORU GXH WR WKHLU HTXDOLW\ RI IXQGDPHQWDO PDWHULDO EHLQJ FDUERQ LQ ERWK SKDVHV ZKLOH WKHLU HOHFWULFDO FRQGXFWLYLW\ LV QRW FRPSDUDEOH 7KH FRQGXFWLYLW\ RI FRNH LV QHDUO\ P6FPZKLFKFRXOGEHGLIIHUHQWGHSHQGLQJRQWKHVKDSHDQGVL]HRIWKHSDUWLFOHFRPSDULQJWR WKHFRQGXFWLYLW\RISLWFKZKLFKLVDOPRVWLQVXODWLQJ

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

9

&KDSWHU /LWHUDWXUH5HYLHZ

$FFRUGLQJWRWKHOLWHUDWXUHWKHUHDUHVRPHPHWKRGVWRLQVSHFWWKHTXDOLW\RIFDUERQDQRGHVEHIRUH XWLOL]DWLRQLQWKHHOHFWURO\VLVEDWKRIDOXPLQXPSURGXFWLRQLQFOXGLQJDFRXVWRXOWUDVRQLFWHFKQLTXHV >@ 8VLQJ DFRXVWLF LQVSHFWLRQ V\VWHP %RXEDNHU HW DO >@ WULHG WR GHWHFW WKH GHIHFWV VXFK DV FUDFNVZKLFKFRQWULEXWHVWRWKHDQLVRWURSLFSURSHUWLHVLQFDUERQDQRGH+RZHYHUWKLVPHWKRGLVQRW GLUHFWO\XVHGIRUWKHPHDQVRIHOHFWULFDOFRQGXFWLYLW\LQVSHFWLRQ

$QRWKHU PHWKRG QDPHG (OHFWULFDO ,PSHGDQFH 7RPRJUDSK\ ZKLFK LV FDSDEOH RI UHSRUWLQJ LQIRUPDWLRQ IURP WKH LQWHULRU RI WKH ERXQGDU\ RI LQWHUHVW E\ UHFRQVWUXFWLQJ LPDJHV RI WKH FRQGXFWLYLW\GLVWULEXWLRQ ,QWKHVLPSOHFDVHLWLV FDOOHG(OHFWULFDO5HVLVWDQFH 7RPRJUDSK\(57 %DVHGRQWKHNQRZOHGJHRIWKHDXWKRUWKHPHWKRGKDVQRWIRUPHUO\EHHQXVHGIRUTXDOLW\FRQWURORI FDUERQ DQRGH (57 FRXOG SURYLGH D FRUUHODWLRQ EHWZHHQ WKH GLVWULEXWLRQ RI WKH SKDVHV DQG WKH GLVWULEXWLRQRIFRQGXFWLYLW\





7KH DELOLW\ WR REWDLQ LQIRUPDWLRQ IURP LQVLGH WKH REMHFW ZKLFK LV RSWLFDOO\ LQDFFHVVLEOH SURYLGHV QXPHURXVDGYDQWDJHVLQWKHYDVWUDQJHRIVFLHQWLILFDUHDVIURPPHGLFDOGLDJQRVLVWRPRQLWRULQJWKH SURFHVVHVLQLQGXVWULDODSSOLFDWLRQVDQGJHRSK\VLFDOLQVSHFWLRQV7KHQRWLRQRIQRQLQYDVLYHLQWHULRU LQVSHFWLRQILUVWLQWURGXFHGLQPHGLFDODUHDV 7KLVLQQRYDWLYHVXFFHVVIXOPHWKRGRSHQHGQHZZD\V IRUEURDGGRPDLQVRIDSSOLFDWLRQV>@ ,Q(,7(57PHWKRGDVHWXSFRPSULVLQJDVHQVRUV\VWHPDGDWDDFTXLVLWLRQV\VWHP'$6DQGD 3&ZLWKDQDSSURSULDWHVRIWZDUHEDVHGRQ)LQLWH(OHPHQW0HWKRG)(0DQGLPDJHUHFRQVWUXFWLRQ DOJRULWKPVDUHQHHGHG)LJ   )LJXUH(,7(57VHWXS$(57RSHUDWLQJV\VWHP>@%&RPSXWHUVHWXS>@

7KH FXUUHQW SURWRFRO ZLWKLQ WKH PHDVXULQJ DQG LQMHFWLQJ HOHFWURGHV LV EDVHG RQ LQMHFWLQJ FXUUHQW XVLQJDSDLURIQHLJKERULQJHOHFWURGHVDQGPHDVXULQJWKHYROWDJHGLIIHUHQFHVIURPDOORWKHUSDLUVRI

10

electrodes (Fig. 6). This process continues until all electrodes once considered as the injecting electrode [27]. The measured data will be then processed in a known manner to yield and display on the screen; a representation of the distribution of the electrical conductivity across the cross section of material or system [28]. Using multi-plane electrical sensors provide three-dimensional description from the system process or the material.

The method Electrical Impedance Tomography (EIT), which is non-destructive low cost imaging technique, was first introduced by Barber and Brown [29], in the eighties. EIT is a member of a family of electromagnetic imaging modalities. The other methods are Electrical Capacitance Tomography (ECT)[30] , Electromagnetic Tomography (EMT) [31], and Magnetic Induction Tomography (MIT) [32]. In EIT, the objective is to estimate a set of unknown variables which represent some of the characteristics of the material, whether it is the conductivity or relative admittivity distribution, within the medium of interest in response to some boundary excitation (current) [33]. This is the basis of the inverse problems [34], which are highly ill-posed and regularization methods are generally needed to solve them [35, 36].

The application of EIT was in imaging the ventilation of the lungs with increasing the gas [37]. The feasibility of method was demonstrated by the early studies of Harris [38, 39], Holder and Temple [40]. Hahn et al. and Frerichs et al. [41, 42] have shown that the functional electrical impedance tomography (f-EIT) could be a reliable technique in monitoring the local lung ventilation under the laboratory conditions.

While Electrical Impedance Tomography covers the medical applications, Electrical Resistance Tomography (ERT), is the term which is more oriented to the industry and geophysics. Early geophysical studies are including Rush [43] offered a solution for a layer of one resistivity over-lying a semi-infinite medium with second resistivity and Stefanesco [44] et al. considered many layers of different resistivities. The widely interested ERT, has found its place in industry during the mid nineties. Publication of the books focusing on the principles of the process tomography using ERT [45] and monitoring multiphase flows [46], witnessed the popularity of this method. The applications of ERT in the chemical engineering are enormous [25]; petroleum prospection, environment protection etc. [47-50] are the other fields in which ERT has appeared perfectly. A wide range of ERT utilizations in the industry concentrate on the chemical engineering such as analysis and control of the reactors, leakage detecting in underground pipes [25], measuring solid-liquid mixtures [51], characterizing the bubble within the mixing vessels or flotation processes [52, 53], flow measurements [54] etc. However, almost none of researches attributed to the distribution of the conductivity of one phase of the material within the other phases. Using ERT Hallaji et al. [55] provided images of the distribution of moisture in cement. They were particularly interested in assessing the feasibility of using ERT in obtaining information from unstructured moisture flows when the flows are non-uniform.

In this research, the attitude of the author is to investigate the application of ERT in detecting the distribution of the phases in carbon anode, which is a multi-phase porous material. This carbon

11

PDWHULDO PD\ DOVR FRQWDLQ FUDFNV 7R DVVHVV WKH IHDVLELOLW\ RI DSSO\LQJ WKH PHWKRG (57 IRU LQYHVWLJDWLQJ WKH GLVWULEXWLRQ RI WKH SKDVHV WKURXJK REWDLQLQJ PDSV RI FRQGXFWLYLW\ GLVWULEXWLRQ &DUERQDQRGHPD\DOVRKDYHFUDFNV



6DPRXsOLDQHWDO>@WULHGWRILQGRXWWKHLQIRUPDWLRQDERXWWKHSUHVHQFHSRVLWLRQRULHQWDWLRQDQG H[WHQVLRQ RI FUDFNV IURP WKH DQDO\VLV RI DSSDUHQW UHVLVWLYLW\ REWDLQHG E\ D WKUHHGLPHQVLRQDO HOHFWULFDO VXUYH\ 7KH\ XVHG DQ DUUD\ RI TXDGULSROH HOHFWURGHV IRU WKHLU PHDVXUHPHQWV 7KH PHDVXUHPHQWVZHUHPDGHZLWKWZRGLPHQVLRQVͲιܽ݊݀ͻͲιRQDEORFNRIVRLO)LJ7KH\XVHG WZRLQGH[HVZKLFKZHUHFDOFXODWHGIURPWKHHOHFWULFDOUHVLVWLYLW\YDOXHVWRHYDOXDWHWKHGHJUHHRI VRLO LQKRPRJHQHLW\ L DQ DQLVRWURS\ LQGH[ DQG LL WKH DQJOH DUUD\ RULHQWDWLRQ 7KH DQLVRWURS\ LQGH[ SURYLGHG WKHP WKH LQIRUPDWLRQ DERXW WKH SUHVHQFH DQG WKH RULHQWDWLRQ RI WKH FUDFNV ZLWK ZLGWKVRYHUWKDQPP





)LJXUH%LQDU\LPDJHRIDEORFNRIVRLO$QHWZRUNRIFUDFNVFDQEHGHWHFWHG7KHJULGVDUHUHSUHVHQWDWLYHWRWKHHOHFWURGH SRVLWLRQV>@

6pJHU HW DO >@ XVHG (57 WR GHWHUPLQH WKH KHWHURJHQHRXV VWUXFWXUH RI VRLO DV D SRURXV PHGLD LQ RUGHUWRLPSURYHLWVFKDUDFWHUL]DWLRQDQGFRQVHTXHQWO\LWVK\GUDXOLFIXQFWLRQ,QWKHLUUHVHDUFKWKH\ DSSOLHGWZRDSSURDFKHV,QWKHILUVWRQHWKH\SXWHOHFWURGHSURILOHVLQWKUHHGLIIHUHQWGHSWKVRIWKH VRLO DQG WKH\ REWDLQHG WKH HOHFWULFDO UHVLVWLYLW\ PDS RI WKUHH OHYHOV )LJ E 2Q WKH VHFRQG DSSURDFKWKH\WRRNSKRWRJUDSKVRIWKHVDPHOHYHOV)LJ D %\FRPSDULQJWKHUHVXOWVIURPWKH WZRDSSURDFKHVWKH\FRXOGGHWHFWGLIIHUHQWSKDVHVRIVRLO

12

Figure 8 A. Photographs of the three level surfaces B. Interpreted electrical resistivity of the same three levels extracted from 3D data [57]

In the research of Yunus et al. [58] a dual-modality of electrical resistivity tomography and ultrasonic transmission tomography (UTT) for imaging a liquid-gas medium was applied. They utilized the two methods because each modality is sensitive to specific properties of the material to be imaged. They put electrodes and ultra sonic transducers around the sample (Fig. 9.A). The excitation of electrodes was conducted by injecting the ac current to some special ones. The choice of ac current was to prevent the electrode polarization. Electrical potential distributions inside the body and ultrasonic impedance measurements were performed with numerical modeling using finite element software (Fig 9.B). The medium was considered homogeneous and spherical air bubbles were introduced to the system (Fig. 9.C). Therefore, by implementing both ERT and UTT at the same time, they could detect the air bubble inside a liquid homogeneous medium.

(A)

13

Figure 9 Top view of the specimen A. Configuration setup of ERT electrodes and UTT transducers, B. Finite element meshing C. Potential distribution inside the specimen (a bubble exists on the top) [58]

Using the method EIT in the medical applications is majorly because of detecting tumors. In such applications, the quality of the reconstructed conductivity images is highly important. As it was mentioned before the reconstruction task in EIT is a highly ill-posed non-linear inverse problem [59]. There are several algorithms introduced in the literature for the image reconstruction of EIT. In medical imaging, an algorithm should be able to recover images of sharp features such as organ boundaries and also approximate ranges for realistic conductivity values. Hamilton et al. [59] utilized the low-passed filtered D-bar method together with a method so-called sinogram CGO as the methods of image reconstruction. They compared their results to the traditional ones such as current-to-voltage or Dirichlet-to-Neumann matrices.

According to Hamilton et al. [59], the nature of the ill-posedness and the inverse of EIT means that small changes in the boundary measurements can result in large changes in the interior conductivity distribution. In addition, this noise in the data amplified exponentially. Thus, regularization is required to recovery of 𝜎 from Λ𝛿𝜎.

Hamilton’s group [59] used the D-bar method for the regularization (Fig. 10). They showed that the stable computation of non-linear Fourier transform is only possible in a disc at the center of the frequency domain. After low-pass filtering with the cut-off frequency R, the good part would be extracted. In D-bar method the transform domain has two parts; stable part (|𝑘| < 𝑅) and unknown unstable part (|𝑘| ≥ 𝑅), which means that high frequency part of data is eliminated. The high frequency part of data belongs to boundaries.

In EIT the fact is that information about these high-frequencies of the un-known conductivities are missing. An applicable assumption is to consider the conductivity as piecewise constant with clear boundaries between homogeneous regions. With this assumption and benefiting from modern imaging science, recovering images has become possible with several options.

14

Figure 10 Illustration of non-linear low-pass filtering for regularization in 2D EIT [59] C. simulated heart-and-lungs phantom. Voltage-to-current matrix 𝚲𝜹𝝈 the orange square, which can be used for non-linear Fourier transform. A. presence

of noise causes in numerical instabilities (white areas in (A)). D. the reconstruction of this noisy data is un-stable and in-accurate. B. low-pass filtering transform. E. Approximate reconstruction after inverse transform

Another image recovery research has been performed by Javaherian et al. [60]. They developed an algorithm for sparse recovery with the application in tumours detection in the body according to their research, regularization algorithms (quadratic regularized solvers) for large size 3D EIT encountered problems due to the large number of unknown parameters. Thus, they applied conjugate gradient methods. Commonly employed in 3D EIT, the predicted conjugate gradient (PCG) is a quadratic solver. Using PCG, which is a random method, they roughly estimated the sparse of unknown conductivity filed inside their specimen. As this prediction technique is a random projection, it contains mistakes. Then they employed finite element method with the application of compressive sensing [60](CS), to the solution already calculated with PCG.

They assumed K as the maximum level of sparsity in the conductivity field. The finite element modeling was done with K-sampled sparsity elements with the CS solution as the initial guess. The conductivity value over the remaining elements was set to background.

In the numerical part they considered the normalized dimension of 1 in radius and 3 in height. Their mesh generation method provided 57024 elements. They simulated 32 electrodes on the CEM model with the contact impedance of 100 Ω per electrode. The background conductivity was considered 1 𝑆𝑚−1_{. Then they put two inclusions of different conductivities in separate locations}

(Fig. 11). A 20 dB white Gaussian noise was also added to the calculations. (C)

(A)

(B)

(E) (D)

15

Figure 11 Cylindrical simulated model A. 3D view of the specimen B. The lateral view C. Top view [60].

With the inclusions is illustrated in Fig. 11. The results of this model are presented in Fig 12. The PCG calculations accordingly, have been done with different termination thresholds of 𝜀1 and 𝜀2

(Fig. 18. A and B). 𝜀2 stands for the optimized solution threshold among all iterations. They used

𝜀2 as the benchmark to evaluate PCG and CS-based calculations. They also used a standard

algorithm namely SpaRSA [60] (Fig. 12.C) and provided extra information regarding this algorithm. Their finite element model sampled by CS is showed in Fig. 12.d.

With comparing the images in Fig. 12.b, c and e with Fig. 11, they concluded that CS-based calculations lead in more accurate results than customary application of the reconstruction algorithms. This demonstrates the importance of having a literally accurate approximation of the conductivity distribution inside the specimen. Without this first guess, finite element simulation is not possible. In addition having an inaccurate algorithm would provide us with poor image reconstruction.

(C) (B)

16  )LJXUH7KHLPDJHVUHFRQVWUXFWHGIURPWKHF\OLQGHUPRGHOFDOFXODWHGE\$3&*ZLWKFDOFXODWLRQWHUPLQDWLRQDWࢿ૚% 3&*DWࢿ૛&WKHVWDQGDUG6SD56$'WKHILQLWHHOHPHQWVDPSOHGE\&6(WKHLPDJHUHFRQVWUXFWHGE\&6EDVHG6SD56$ >@  7KHFXUUHQWSDWWHUQVDSSOLHGLQ(57DUHLQORZIUHTXHQFLHV>@0RUHYR[HOVDUHLQYROYHGIRUWKH HOHFWULFDOGLUHFWFXUUHQWLQ(57FRPSDULQJZLWK&RPSXWHG7RPRJUDSK\&7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7KHPHDVXUHPHQWRIYROWDJHLVIURPSDVVLYHHOHFWURGHV)LJ%

%DVLFDOO\PHDVXULQJWKHDPSOLWXGHDQGWKHSKDVHDQJOHRIWKHYROWDJHFDQUHVXOWLQWKHLPDJHVRIWKH HOHFWULFDO SHUPLWWLYLW\ DQG FRQGXFWLYLW\ LQVLGH WKH YROXPH >@ $OWKRXJK WKH HOHFWULF ILHOG LV UHTXLUHGDOWHUQDWLQJFXUUHQWSDWWHUQVDUHSUHIHUDEOHWRDYRLGSRODUL]DWLRQHIIHFWV,QWKHIUHTXHQFLHV

17

less than 100 kHz, the magnetization1 _{of current is negligible and the phase angle of current is}

small. Therefore, only the conductivity (or equivalently resistivity) is reconstructed when the phase angle is absent in the reading data and the method is simplified to electrical resistivity tomography (ERT) [62].

Figure 13 Current flux near a. active and b. passive electrode [61]

EIT (ERT) in the general sense is an inverse problem. To solve the inverse problem, one needs to solve the forward problem first. In the forward solution the objective is estimating the electric field inside the volume using assuming regarding the assumed conductivities using Finite Element method, for which Dirichlet and Neumann boundary conditions are required. In the inverse solution though, the approach is the process of estimating the conductivity using the boundary measurements (image reconstruction). The process of reconstructing images is an ill-posed inverse problem. EIT calculations are complex. The reason for its difficulty lies in the nature of the ill-posed problem. In order to know the ill-posed nature of EIT, one should formerly know about the definition of the posed problems. According to Hadamard a mathematical model for a physical problem is well-posed only when the three following criteria would be met:

1- a solution consistent with the observations exists,

2- the solution is unique,

3- and the solution depends continuously on the observations [62]

The process of recovering unknown conductivity from boundary data is severely ill-posed. Because the third criterion is not met in EIT problem [61]. This means that relatively large changes in the conductivity distribution could not be detected from the boundary voltage measurements. Furthermore, EIT is highly sensitive to the noise. Even small noise in the sensing electrodes at the boundary may result in unstable or in-accurate image.

1 _{When electrical current passes through a component, a magnetic filed is produced. This magnetic} filed is normal in the current direction and circular around the conductive.

18

The first and the second criteria are not the troubles because the solution exists. For the uniqueness of the problem we shall refer to the Calderon problem, which is a conductivity inverse boundary problem. It shows that a complete relationship between the boundary voltage and current can determine the conductivity uniquely [63]. The uniqueness of the solution has been met under variety of assumptions [64]. This problem is also crucially non-linear.

2.3.1. The forward problem

In the forward problem, the objective is to estimate the potential distribution inside the material and at the surfaces of a conductive volume and subsequently assemble the boundary observations. The solution for this problem can be either analytical, using the equations or numerical using the finite element method [62].

Only for simple geometries or conductivities and homogeneous materials, the forward problem can be solved analytically. However for more complicated geometries and high contrast conductivities such as multiphase materials numerical methods should be applied.

The physical model proposed for EIT should respect the boundary geometry and the structural properties of the volume. Once the model is created, mathematical equations which relate the physical conditions in the boundaries and the electric filed inside the volume are required [62]. Furthermore, the boundary electrodes should have proper mathematical model because the excitation and the measurements are being done through the electrodes.

In order to understand the forward solution for EIT, the mathematical background of EIT and the Maxwell’s equations are required. More information about the mathematical part and the equations are presented in appendix.

2.3.1.1. Solving the forward problem using finite element method

In order to solve the inverse problem, the forward problem should be solved in advance. In the forward solution, voltages are calculated regarding the assumed conductivities. The interior electric fields are used for the Jacobian calculations. For inhomogeneous media like multiphase materials, numerical methods are utilized for Jacobian calculations. In this method, the conductivity and the domain should be discretized. The discretized areas are in the form of polyhedra which called elements. Usually, the transformation from continuous domain to discrete ones are achieved by numerical integral methods such as Galerkin method [62]. The transformation start from Partial Differential Equations (PDE) to integral form using Galerkin test functions. After that, the integral form is discretized using FEM. The integration on the discretized form could be exact or numerical under the condition that the exact integral does not exist.

In order to have a better understanding of the EIT fundamentals, basic equations are introduced in the appendix. The reader is kindly asked to see the appendix for more information.

19 2.3.1.2. Jacobian calculations

In EIT, a complete matrix of partial derivatives of voltages with respect to conductivity parameters is called the Jacobian matrix. In medical applications it is called sensitivity map. In optimizing methods, calculating the derivative of voltage regarding to the conductivities is necessary. There are some Quasi-Newton methods in which Jacobian is updated from the forward solution. This has been used in geophysics [65].

In order to calculate the Jacobian matrix, the conductivity should be discretized. The most simple way to discretize the conductivity is to consider that as piecewise constants on polyhedral domains such as voxels [65]. Therefore, voxels can be regarded as our elements in the FEM calculations of Jacobian matrix.

In the case of non-linear parameterization of the conductivity, for instance the shapes in anatomical model, the Jacobian can be calculated using the chain rule with respect to the new conductivity parameters [61].

2.3.2. Inverse problem

Up until now, the forward solution with employing the finite element method gives us a discrete approximation of the electric potential from the domain for a given conductivity distribution. The forward solution model should be able to predict the initial electric filed for the given conductivity [65]. The inverse solution tends to recovery 𝜎 from Λ𝛿𝜎. For this purpose, the norm of discrepancy

between the computed and the measured voltages on the electrodes should be minimized [60]. In the other words, the conductivity field would be updated with minimization of the residuals [66]. Starting from a rough matrix of admittivity (𝛾) (or in the simple case conductivity (𝜎)), measuring the electrical potentials on the boundary (from electrodes) and calculating the potentials inside the anode, these two matrices of the electrical potential values are compared together in order to have the minimum difference between these values. In the next step, if the deviation is acceptable, the calculations would be terminated. However, if the discrepancy is not yet minimized, the calculations would be continued until reaching a new matrix of 𝛾, for which this difference is minimum. Supposedly, the final matrix of admittivity (or conductivity) would best describe the conductivity distribution of the material. Image reconstruction algorithms then applied to the measurement to produce images [67].

2.3.2.1. Regularization

Image reconstruction is the process of invoking the matrix of conductivity (admittivity) using special algorithms taking into account the calculated and measured electric potentials. Basically, EIT and tomography are not the same terms. The fact is that it is not possible to reconstruct the images slice by slice [65]. The reason lies in the continuous distribution of current inside the material. However, there is a relationship between EIT and tomographic images, which is reconstruction algorithm.

20

The EIT problem is non-linear thus it should be linearized and the output linear problem is ill-posed so it should be regularized [65]. The regularized algorithms are suitable for linearized inverse problems and divided into two categories; methods that use single regularization like Tikhonov [68] or the methods which use iterative regularization [69]. The non-linear algorithms are able to high-contrast inhomogenities. For the complex geometries or large high-contrasts between conductivities, the linear approximation is not capable for the image reconstructions. Therefore, the procedure should be applied iteratively. To do the process in the iterative way, first the assumed conductivity 𝜎 represented by parameter s. The simplest way is to consider the conductivity as the sum of basic functions of a set of voxels [65].

Typically, image reconstruction in EIT engages both forward and inverse solutions [60]. In the reconstruction part the potential, which is calculated by the forward solution is utilized. According to what previously mentioned, so far, for a given conductivity distribution, the forward model is able to give us the potential inside the body; let us call that F. On the other hand V is the measured voltages around the boundary, which V=F(s). As the goal is to fit the voltage measurements 𝑉𝑚𝑒𝑎𝑠,

the most straight and simple approach is to minimize the sum of square errors:

‖𝑉_{𝑚𝑒𝑎𝑠}− 𝐹(𝑠)‖2 (2.1)

which is called output least square approach. Usually, the raw least square is not common and weighted least square is used instead [65]. For a unique solution, one must add a function which contains additional information about conductivity. An example would be considering G(s) as a penalty for highly oscillatory conductivities and doing the minimization:

𝑓(𝑠) = ‖𝑉𝑚𝑒𝑎𝑠− 𝐹(𝑠)‖2+ 𝐺(𝑠) (2.2)

A typical simple choice for G(s) [70] could be

𝐺(𝑠) = 𝛼2_{‖𝐿(𝑠 − 𝑠}

𝑟𝑒𝑓)‖

2 _(2.3)

in which L is a matrix approximation of some partial differential operators and 𝑠𝑟𝑒𝑓 is a reference

conductivity. There are also other smooth options for G [71]. 2.3.2.2. Linearizing the problem

Consider a simple case in which F(s) is replaced by a linear approximation

𝐹(𝑠0) + 𝐽(𝑠 − 𝑠0) (2.4)

where J is the matrix of Jacobian of F calculated at some initial conductivities 𝑠0. We shall define

𝛿𝑠 = 𝑠 − 𝑠₀ and 𝛿𝑉 = 𝑉_{𝑚𝑒𝑎𝑠}− 𝐹(𝑠₀). The solution for the linearized regularization problem (a quadratic minimization problem for f) is given by

𝛿𝑠 = (𝐽∗𝐽 + 𝛼2𝐿∗𝐿)−1(𝐽∗𝛿𝑉 + 𝛼2𝐿∗𝐿(𝑠𝑟𝑒𝑓− 𝑠0)) (2.5)

or any equivalent form of this equation [72]. While there are many other regularizations are possible for ill-posed problems this Tikhonov regularization has some benefits [65].

21  ,WHUDWLYHVROXWLRQV 7KHOLQHDUDSSUR[LPDWLRQVDUHRQO\VXLWDEOHIRUVPDOOGHYLDWLRQVIURPUHIHUHQFHFRQGXFWLYLW\ݏ଴,Q PHGLFDODSSOLFDWLRQVWKHFRQGXFWLYLW\FRQWUDVWFDQEHVRODUJHEXWVWLOOLQVXFKFDVHVWKHOLQHDUL]HG PHWKRGVWRFDOFXODWHWKHFKDQJHLQDGPLWWLYLW\EHWZHHQWZRGLIIHUHQWVWDWHVRIWLPHDQGIUHTXHQF\ DUHFRPPRQ,QWKHPHGLFDODSSOLFDWLRQVWKLVPHWKRGLVFDOOHGµG\QDPLFRUGLIIHUHQFHLPDJLQJ¶,Q LQGXVWULDOSUREOHPVDQGLQSKDQWRPWDQNVWKHPHDVXUHPHQWRIKRPRJHQHRXVWDQNDVDUHIHUHQFHLV SRVVLEOH7KLVFDQEHXVHGDVWKHPHDQVRIFDOLEUDWLRQLQIRUZDUGPRGHOSDUWLFXODUO\IRUWKHFRQWDFW LPSHGDQFH HVWLPDWLRQV >@ +RZHYHU LQ WKH FDVH RI LQ YLYR PHDVXUHPHQWV WKHUH LV QR VXFK D SRVVLELOLW\ WR KDYH D UHIHUHQFH DQG DV D UHVXOW WKHUH FRXOG EH UDWKHU JUHDW GLIIHUHQFH EHWZHHQ WKH PHDVXUHGGDWDDQGWKHSUHGLFWLRQVRIWKHLQWHULRUFRQGXFWLYLW\:LWKRXWVROYLQJWKLVSUREOHPLWLV XQOLNHO\WREHDEOHWRXVHLWHUDWLYHQRQOLQHDUPHWKRGVLQYLYRXVLQJLQGLYLGXDOHOHFWURGHV>@7KXV ZLWKRXW KDYLQJ HQRXJK LQIRUPDWLRQ DERXW WKH HOHFWULFDO ILHOG GLVWULEXWLRQ GXH WR WKH FRQGXFWLYLW\ GLIIHUHQFHLQVLGHWKHERG\WKHUHVSRQVHZRXOGKDUGO\FRQYHUJH

1RQOLQHDU VROXWLRQ IRU WKLV SUREOHP PHDQV WR UHSHDW WKH SURFHVV RI FDOFXODWLQJ WKH -DFRELDQ DQG VROYLQJ D UHJXODUL]HG OLQHDU DSSUR[LPDWLRQ $ UHJXODU ZD\ WR GR WKDW LV VWDUWLQJ ZLWK I DQG PLQLPL]LQJWKDW ,IWKHIXQFWLRQ*LVEHLQJZHOOFKRVHQIZRXOGKDYHDFULWLFDOSRLQWZKLFKLVWKH PLQLPXP$WWKLV SRLQW׏݂ሺݏሻ ൌ ͲZKLFKLVHTXLYDOHQWRI KDYLQJ 1HTXDWLRQVDQG 1XQNQRZQV 7KHVROXWLRQIRUWKLVVHWRIHTXDWLRQVFDQEHREWDLQHGE\PXOWLYDULDEOH1HZWRQ5DSKVRQPHWKRG 7KH*DXVV1HZWRQDSSUR[LPDWLRQQHJOHFWVWKHVHFRQGGHULYDWLYHVRI)7LNKRQRYIRUPXODXSGDWLQJ QWKDSSUR[LPDWLRQDFFRUGLQJWRWKHFRQGXFWLYLW\SDUDPHWHUVݏ_௡>@ ݏ_௡ାଵ ൌ ݏ_௡൅ ሺܬכ_{ܬ ൅ ߙ}ଶ_ܮכ_ܮሻିଵ_ሺܬכ_ሺܸ ௠௘௔௦െ ܨሺݏ଴ሻሻ ൅ ߙଶܮכܮ൫ݏ௥௘௙െ ݏ௡൯ሻ 

ZKHUHܬ௡LV WKH -DFRELDQ HYDOXDWHG DWݏ௡ ,Q WKLV IRUPXOD VLJQV DUH LPSRUWDQW 7KLV IRUPXOD LV

H[SODLQHGSURSHUO\LQWKHWKLUGFKDSWHURIWKHUHIHUHQFH>@

,Q RUGHU WR SDUDPHWHUL]H WKH FRQGXFWLYLW\ RQH FDQ FRQVLGHU WKDW DV YR[HOV RU PRUH VSHFLILFDOO\ SLHFHZLVHFRQVWDQWVRQVPRRWKGRPDLQVDQGUHFRQVWUXFWWKHVKDSHVXVLQJ)RXULHUVHULHV>@



,QRUGHUWRKDYHDEHWWHUNQRZOHGJHRI(,7RSHUDWLRQILUVWFRQVLGHUWZRHOHFWURGHVFRQQHFWHGWRD PDWHULDODQGWKHQWKLVV\VWHPZRXOGEHH[WHQGHGWRQHOHFWURGHV>@7KLVH[DPSOHFRXOGZUDS XS WKH NQRZOHGJH KDV EHHQ REWDLQHG IURP WKH IXQGDPHQWDOV RI (,7 DV ZHOO DV WKH )LQLWH (OHPHQW VROXWLRQIRUWKDW

22

Figure 14 A very EIT system [77]

According to the Stokes’s theorem the complex potential u is defined based on the line integral

𝑢(𝑟₂) − 𝑢(𝑟1) = − ∫ 𝐸. 𝑑𝑙

𝐶𝑟1→𝑟2

(2.7)

which is approximately path independent. Where 𝐶𝑟1→𝑟2 is a curve lying in Ω with the starting

point 𝑟1 and the ending point 𝑟2. Hence, the complex potential u satisfies

−∇𝑢(𝑟) ≈ 𝐸(𝑥) 𝑖𝑛 Ω (2.8)

In order to ensure about the uniqueness of u, the reference voltage 𝑢|ℇ0=0. From ∇ × 𝐻 = 𝐽 +

𝑖𝜔𝐷 = (𝜎 + 𝑖𝜔𝜖)𝐸 (appendix), following relation would be obtained:

∇ × 𝐻 = (𝜎(𝑟, 𝜔) + 𝑖𝜔𝜖(𝑟, 𝜔))𝐸(𝑟) = −𝛾∇𝑢 (2.9)

𝜎(𝑟, 𝜔) + 𝑖𝜔𝜖(𝑟, 𝜔) = 𝛾(𝑟, 𝜔) (2.10)

since ∇. (∇ × ∇𝑢) = ∇. ∇ × 𝐻 = 0 𝑖𝑛 Ω.

Boundary conditions are generated regarding the currents injected to some electrodes. Neglecting the contact impedance at the surface ℇ0∪ ℇ1 ⊂ 𝜕Ω, the injecting current 𝐼𝑐𝑜𝑠(𝜔𝑡) into Ω through

the electrodes ℇ0 and ℇ1provide the boundary condition for 𝐽̅ and 𝐷̅. Thus, the boundary conditions

on the contacts are: ∫ 𝑛. 𝐽̅

ℰ1

𝑑𝑠 = − ∫ 𝑛. 𝐽̅

ℰ0

23

ZKHUH Q LV WKH RXWZDUG QRUPDO YHFWRU GV LV WKH VXUIDFH HOHPHQW DQGࣟ_௞LV FRUUHVSRQGLQJ WR WKH HOHFWURGHFRQWDFW7KHVHFRQGERXQGDU\FRQGLWLRQLV  ܬҧǤ ݊ ൌ Ͳ ൌ ܦഥǤ ݊RQ߲ȳ̳Ԫ_଴׫ Ԫ_ଵ   6LQFHYROWDJHVRQWKHHOHFWURGHVDUHFRQVWDQWV  ׏ ൈ ܬҧ ൌ ׏ ൈ ܦഥRQԪ଴׫ Ԫଵ   ሺߛ׏ݑሻǤ ݊ ൌ ͲRQ߲ȳ̳Ԫ଴׫ Ԫଵ   ݊ ൈ ׏ݑ ൌ ͲRQԪ_଴׫ Ԫ_ଵQHJOHFWLQJFRQWDFWLPSHGDQFH  

7KHUHIRUH WKH FRPSOH[ SRWHQWLDO DSSUR[LPDWHO\ VDWLVILHV WKH IROORZLQJ PL[HG ERXQGDU\ YDOXH SUREOHP  ە ۖ ۖ ۔ ۖ ۖ ۓ െ׏Ǥ ሺߛ׏ݑሻ ൌ Ͳ݅݊ȳ ܫ ൌ න ߛ߲ݑ ߲݊ ࣟభ ݀ݏ ൌ െ න ߛ߲ݑ ߲݊ ࣟబ ݀ݏ ߛ߲ݑ ߲݊ ൌ Ͳ߲ȳ̳Ԫ଴׫ Ԫଵ ׏ݑ ൈ ݊ȁ_ఌ଴ൌ Ͳǡݑȁ_ఌଵൌ Ͳ   7KHIROORZLQJVWHSVVKRXOGEHGRQHXVLQJ)(PHWKRGWRUHDFKWKHIRUZDUGVROXWLRQDQGGHULYHWKH PDWUL[ RI HOHFWULF SRWHQWLDOV ZKLFK ZLOO EH XWLOL]HG LQ LPDJH UHFRQVWUXFWLRQ 7KH PDWUL[ RI FRQGXFWLYLW\DGPLWWLYLW\WKHQFRXOGEHLQYRNHGRIWKHZKROHVDPSOH



&RQVLGHUDV\VWHPRI(,7ZLWKPXOWLSOHHOHFWURGHVԪ଴ǡ Ԫଵǡ ǥ ǡ Ԫ௡RQ߲ȳ:LWKWKHVHQHOHFWURGHVQ

OLQHDULQGHSHQGHQWHOHFWULFDOFXUUHQWVFRXOGEHDSSOLHG>@$VVXPHQSDLUVRIHOHFWURGHVDUHXVHG Ԫ଴Ԫ௝ M «Q WR LQMHFW WKH VLQXVRLGDO FXUUHQWܫܿ݋ݏሺ߱ݐሻܸ௝ሺݎǡ ݐሻZRXOG EH WKH FRUUHVSRQGLQJ

HOHFWULFDOSRWHQWLDOZLWKܸ௝ห_ఌ଴ ൌ ͲVXEMHFWWRWKHLQMHFWFXUUHQWܫ௝ܿ݋ݏሺ߱ݐሻ:HPHDVXUHWKHHOHFWULFDO

SRWHQWLDO ݂ሚ௝ǡ௞ሺݐሻ ൌ ܸ௝ห_ఌ௞DW DOO HOHFWURGHV ߝ௞ǡ ݇ ൌ ͳǡʹǡ ǥ ǡ ݊  7KH LQYHUVH SUREOHP LQ (,7 PHDQV

ILQGLQJߪDQG߳FRPSOH[DGPLWWLYLW\ሺߪ ൅ ݅߱߳ሻIURPܨ_ఙǡఢ 

24 𝐹_𝜎,𝜖 ≔ [ 𝑓̅₁₁(𝑡) 𝑓̅₁₂ (𝑡) … 𝑓̅_1𝑛 (𝑡) 𝑓̅₂₁(𝑡) 𝑓̅₂₂ (𝑡) … 𝑓̅_2𝑛 (𝑡) . . . . . . . . . 𝑓̅𝑛1(𝑡) 𝑓̅𝑛2 (𝑡) … 𝑓̅𝑛𝑛 (𝑡) ] ← 𝐼₁𝑐𝑜𝑠(𝜔𝑡) ← 𝐼₂𝑐𝑜𝑠(𝜔𝑡) .. . . ← 𝐼𝑛𝑐𝑜𝑠(𝜔𝑡) (2.16)

Therefore, the inverse problem is to invert the map.

For an (n+1) channel EIT system, let 𝑢𝑗 be the complex potential such that 𝑉𝑗 = 𝑅𝑒 (𝑢𝑗𝑒𝑖𝜔𝑡), where

𝑉_𝑗 is the matrix of electrical potentials. The complex potential 𝑢_𝑗 satisfies the boundary condition

{ −∇. (𝛾∇𝑢) = 0 𝑖𝑛 Ω 𝐼 = ∫ 𝛾𝜕𝑢 𝜕𝑛 ℰ1 𝑑𝑠 = − ∫ 𝛾𝜕𝑢 𝜕𝑛 ℰ0 𝑑𝑠 𝛾𝜕𝑢 𝜕𝑛 = 0 𝜕Ω\ℇ0∪ ℇ1 ∇𝑢 × 𝑛|𝜀1= 0, 𝑢|𝜀0= 0

For image reconstruction, the measured potentials at each electrode 𝜀𝑘is

𝑓𝑗(𝑘) ≔ 𝑢𝑗(𝑘) 𝑗 = 1,2, … , 𝑛

In EIT, there is a reciprocity relation which means that [77]

𝑓𝑗(𝑘) = 𝑓𝑘(𝑗) (2.17) 𝑓𝑘(𝑗) = 1 𝐼∫ 𝛾 𝜕𝑢_𝑗 𝜕𝑛 𝜕Ω 𝑢𝑘𝑑𝑠 = 1 𝐼∫ 𝛾∇𝑢𝑗. ∇𝑢𝑘𝑑𝑟 = 1 𝐼∫ 𝛾 𝜕𝑢_𝑘 𝜕𝑛 𝜕Ω 𝑢𝑗𝑑𝑠 = 𝑓𝑗(𝑘) (2.18)

r is the direction line surrounding the surface according to the Stokes theorem. The reciprocity relation implies that the matrix in (2.45) is symmetric. Thus, 𝑛+1

2 is the number of independent

data, which is the maximum number of unknown parameters of 𝛾 that can be reconstructed from the above set of data 𝐹𝛾.

2.5.1. Reconstruction of 𝜸 in a 4-channel EIT system:

As a simple example, consider a rectangular region Ω and assume that 𝜔 = 0 and 𝛾 = 𝜎 represents the conductivity which is real and isotropic [77]. A rough structure of 𝛾 = 𝜎 from the following baby Neumann-t-Dirichlet (NtD) map (table 1), taken from the measurements of the electrodes attached to the boundary of the sample, is desired.

25

Table 1 Neumann-t-Dirichlet map [77]

𝑘 = 1 𝑘 = 1 𝑘 = 1

𝑢1|ℇ𝐾 -2.0285 -1.3025 -1.0962

𝑢₂|_ℇ𝐾 -1.38068 -2.3413 -1.3633

𝑢₃|_ℇ𝐾 -1.1053 -1.3724 -2.5987

It should be noted that the reconstruction of 𝛾 with this limited number of data set 𝐹𝛾 is impossible.

The least-square methods are applied as the reconstruction technique. For a given 𝛾, let us 𝑢_𝑗𝛾 _{be the solution for 𝒫}

𝑗[𝛾]: 𝒫_𝑗[𝛾]: { −∇. (𝛾∇𝑢) = 0 𝑖𝑛 Ω 𝐼 = ∫ 𝛾𝜕𝑢 𝜕𝑛 ℰ1 𝑑𝑠 = − ∫ 𝛾𝜕𝑢 𝜕𝑛 ℰ0 𝑑𝑠 𝛾𝜕𝑢 𝜕𝑛 = 0 𝜕Ω\ℇ0∪ ℇ1 ∇𝑢 × 𝑛|_𝜀1 = 0, 𝑢|_𝜀0 = 0

Figure 15A 4-channel EIT system [77]

By solving 𝒫𝑗[𝛾] we would obtain the following simulated data 𝑢𝛾:

𝑢𝛾 = [ 𝑢₁𝛾|_ℇ1 𝑢₁𝛾|_ℇ2 𝑢₁𝛾|_ℇ3 𝑢₁𝛾| ℇ1 𝑢1 𝛾 | ℇ1 𝑢1 𝛾 | ℇ3 𝑢₁𝛾|_ℇ1 𝑢₁𝛾|_ℇ1 𝑢₁𝛾|_ℇ1]

26

This matrix is a computed voltage set with the guessed 𝛾. Fig. 16 demonstrates the complex potentials. This simulated data would be compared with the measured data:

𝑓 = [

𝑓₁|_ℇ1 𝑓₁|_ℇ2 𝑓₁|_ℇ3 𝑓₂|_ℇ1 𝑓₂|_ℇ2 𝑓₂|_ℇ3 𝑓₃|_ℇ1 𝑓₃|_ℇ3 𝑓₃|_ℇ3

]

where 𝑓𝑗|_ℇ𝑘 is the measured voltage at electrode 𝜀𝑘 subjected to the j th current.

Figure 16An example of complex potentials [77]

Then in the minimization problem 𝛾 which minimizes the misfit between simulated and measured data would be reconstructed (Fig. 12).

𝜙(𝛾) = ‖𝑓 − 𝑢‖2+ 𝜂(𝛾) = ∑ ∑ |𝑓_𝑖𝑗 − 𝑢_𝑗𝛾| 𝜀𝑘 | 3 𝑗=1 3 𝑘=1 + 𝜂(𝛾) _(2.19)

in which 𝜂(𝛾) is the regularization term.

The flowchart in Fig. 17 demonstrate the steps of minimization of the error and the reconstruction part. As it can be seen after injecting the current, the data acquisition part provides the matrix of measured voltages. On the other hand, computer program using forward solutions obtains the electric filed inside the material. By comparing the two matrices and applying the image reconstruction algorithms, the electrical images of the conductivity distributions would be obtained.

27

28



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