Analysis of an Approach for Detecting Arc Positions During Vacuum Arc Remelting Based on Magnetic ﬂux Density Measurements

Analysis of an approach for detecting arc positions during vacuum arc remelting based on magnetic ﬂux density measurements

Miguel F. Soler Graduate Research Assistant School of Mechanical, Industrial, and Manufacturing Engineering Oregon State University Corvallis, OR 97331-6001 Email: [email protected]

Kyle E. Niemeyer∗ Assistant Professor School of Mechanical, Industrial, and Manufacturing Engineering Oregon State University Corvallis, OR 97331-6001 Email: [email protected]

Vacuum arc remelting (VAR) is a melting process for the current into the system, in a vacuum environment. The result production of homogeneous ingots, achieved by applying a is a high-quality metal ingot that exhibits increased homo- direct current to create electrical arcs between the input elec- geneity and decreased defects. The high-quality metals pro- trode and the resultant ingot. Arc behavior drives quality of duced by VAR are typically used for high-performance ap- the end product, but no methodology is currently used in VAR plications such as aerospace systems [1]. VAR is often used furnaces at large scale to track arcs in real time. An arc po- on Ni- and Ti-based alloys [2–5]. sition sensing (APS) technology was recently developed as Figure1 depicts a VAR furnace cross section. The cur- a methodology to predict arc locations using magnetic field rent applied to the system forms electrical arcs between the values measured by sensors. This system couples finite el- melted ingot and the input consumable electrode. Since no ement analysis of VAR furnace magnetostatics with direct ingot exists at the start of the process, common practices in- magnetic field measurements to predict arc locations. How- clude the addition of small metal pieces to the bottom of the ever, the published APS approach did not consider the effect crucible to form an arc. These arcs begin the melting process of various practical issues that could affect the magnetic field of the electrode, which then transfers mass to the bottom of distribution and thus arc location predictions. In this paper, the crucible due to gravity. This mass solidifies at the bot- we studied how altering assumptions made in the finite el- tom of the ingot as the arcs and heat transfer take place at the ement model affect arc location predictions. These include electrode-ingot gap, which travels up the crucible as more the vertical position of the sensor relative to the electrode- mass is transferred from the electrode to the ingot. Arcs can ingot gap, a varying electrode-ingot gap size, ingot shrink- simultaneously form in multiple positions; presently, opera- age, and the use of multiple sensors rather than a single sen- tors can neither visualize nor control the formation of arcs. A sor. Among the parameters studied, only vertical distance water-cooled jacket prevents the copper crucible from melt- between arc and sensor locations causes large sources of er- ing. At the top of the melted ingot a liquid pool of the mate- ror, and should be considered further when applying an APS rial exists. The characteristics of this melt pool have a large system. However, averaging the predicted locations from impact on final quality of the ingot [4–6]. four evenly spaced sensors helps reduce this error to no more Arc behavior drives the remelting process and de- than 16 % for a sensor position varying from 0.508 m below termines ingot quality, but arc positions are challenging and above the electrode-ingot gap height. to quantify due to the VAR system geometry and high- temperature environment. Currently, video cameras directed down the annular gap between the electrode and crucible give 1 Introduction operators qualitative information of arc behavior, as well as Vacuum arc remelting, or VAR, is the metallurgical pro- side arc detection, however these systems cannot track in- cess of remelting metal ingots with the application of a direct stantaneous arc formation and motion. A robust arc detection and tracking system would give insight into the material properties of the final ingot. A common approach for de- ∗Address all correspondence to this author. to soluteingots partitioning [3]. Wang of et the al. alloying [9] developed elements. a Recenttwo-dimensional ax- [3] resultsisymmetric have indicated model both to study through arc characteristics modeling and under different experimentally[4] that significant Fe macrosegregation occursaxial during magnetic the melting fields of using Ti-10V-2Fe-3Al, the commercially and this available soft- macrosegregationware FLUENT. changes Their as model a function focused of the on total magnetohydrody- currentnamics entering and the plasma ingot. Thebehavior modeling in the study electrode-ingot assumed gap, as- an axisymmetric and Gaussian distribution of the arc, whethersuming this thatis valid plasma is one consisted of the aims of onlyof the electrons current and ions and work.its flow can be described with a hydrodynamic approxima- Perhapstion. Theythe defect concluded of most concern that the with effect Ti-6Al-4V of current is density dis- so called hard-alpha interstitial inclusions, a type 1 low- densitytribution, inclusion arc (LDI), distribution, because was these significant inclusions can for VAR because becomeit directly crack initiation correlates sites to the leading heat to flux premature density at the anode. fatigueWoodside failure. Theet al. fact [2, that12] hard-alpha used the multiphysics material can finite-element have a similar melting point and similar density as comparedmodeling to the (FEM) alloy makes software it di COMSOLfficult to remove to simulatevia the magne- VAR.tostatics The term ofhard-alpha a VAR furnace. actually They refers assumed to material an axisymmetric overfurnace, a range homogeneous in the Ti-N phase material diagram, properties, sometimes and a single non- referreddiffuse to as arc nitrides, in a three-dimensional and within this range, model. there Model is results were considerable variation in the fracture toughness.[5] Ti-N inclusionused ‘‘survival’’ to develop times a relationship in a VAR between melt pool measured as a magnetic functionfield of readings particle at size a notional and density Hall have sensor been position mod- and arc loca- [6] eled.tionsAs [ mentioned,2]. Nair et the al. [ arc10] current used the drives FEM the software fluid Opera3d to motion. An understanding of the fluid dynamics in the poolstudy is critical the to use predicting of magnetic the ability source of VAR tomography to reduce to understand thesearc defects. behavior[7] Therefore, in VAR knowing systems. the VARThey arc modeled distri- electrostatics butionwhile is in assuming turn critical homogeneous to making accurate material predictions properties, and in- Fig. 1—Cross section of the VAR furnace. Sketch is courtesy of ATI of the dissipation of hard alpha inclusions within the FigureAllvac, 1: Cross with a modification section of to show VAR instrumentation. furnace. Image taken from VARcluded melt pool. both Producing single and ingots double free non-diffuse of high-density arcs. Nair et al. Woodside et al. [2]. inclusionsconcluded (HDI), that a arc type locations 1 defect can consisting be predicted of a based on mea- melting rates are much higher than utilized during nickel refractorysurements element of suchmagnetic as tungsten, flux density is also outside important the furnace with alloy melting. Thus, the molten pool is quite deep, but is expected to be less dependent on arc distribution. sometimes being described as having a ‘‘soda can’’ Thissufficient is because accuracy these inclusions under tendthe right to rapidly circumstances. sink to According tailedshape. study This of VAR means furnaces the localized is numerical heat flux from modeling the arcs [2–9].the bottomto literature, of the melt arc pool, locations so changes and characteristics in the arc driven directly affect The applicationwill have less of impactlarge currents on the through solidification the system front as resultsfluidingot dynamics characteristics. are less important. [2–4,7–10] Various studies showed that compared to nickel melting because of the difference in It is also possible that the arc distribution impacts the in a strongdistances magnetic from arc field to solidification surrounding front. the furnace,However, on arc whichphysicalarc locations structure of can the be ingot predicted sidewall accurately surface. This using is magnetic flux severalconstrictions studies focused toward the [2, side9, 10 wall]. can The potentially arc position lead to in thesignificantdensity because measurements the sidewall around integrity VARfurnaces and grain combined with ingot-electrodeshelf remelting, gap causing is a key material parameter to fall that into affects the pool, the mag-structureaccurate can innumerical turn affect models. subsequent [2, 10 forging] However, opera- these method- thus the commonly used term ‘‘fall-in.’’ This material tions and product yield. Multiple techniques are used neticcan field. remain Arcs intact concentrate in the melt the pool electrical and can be current a source passingwithinologies industry made to assumptionsimprove ingot and surface simplifications quality, but that should be throughfor point the system defects as and its impact composition the distribution will often differ of from the mag-determiningexamined the further effect of to encouragethe arc distribution their application on the in industry. neticthe field. nominal alloy composition. An example is a type 2 ingot surfaceThe has purpose been di offfi thiscult. study is to model the magnetostatics defect characterized as having too much primary alpha Arc distribution is relevant to the quality of any Mirphase et due al. [ to11 an] studied elevated the concentration thermal behavior of aluminum, of the con-materialof the melted VARvia processVAR. The in differentfocus on this scenarios, paper is onwhile evaluating sumablewhich electrode is the alpha using stabilizer. infrared This cameras, leads focusing to a region on heatTi-6Al-4Vthe impact not because of previously this alloy made deserves assumptions, the most to determine the transferhaving characteristics. slightly elevated However, hardness their as compared technique tothe requiredattentionpotential but rathererrors becauseof arc locations of circumstantial predicted conve- by the Arc Position alloy, and this can be accompanied by a neighboring nience in terms of the experiments. The technique alterationregion of having a furnace depleted and aluminum revealed and little lower insight hardness. on arc be-describedSensing is applicable (APS) system to other of VAR Woodside operations, and but King it [12]. Under- havior.Perhaps Zhao et a al.less [ obvious4] used e theffect two-dimensional of arc distribution isFEM the soft-shouldstanding be noted the thatbehavior the reported of the APS arctechnology motion and due to chang- waree ANSYSffect of the to study magnetohydrodynamic fluid dynamicsin stirring the molten of the pool.distributioning parameters results may could be lead specific to further to the validation furnace or improve- molten metal. In a coaxial furnace design, the electrical andment procedures [2, 10 followed]. We used by aATI multiphysics Albany Operations FEM simulation soft- The modelcurrent assumed from the that arcs only enters buoyant the melt forces pool act and on then the melt(Albany, OR). pool,largely the melt flows pool in a exhibits radial direction, turbulent exiting flow the properties, ingot near chem- ware to study the system. First, in Section2 we describe the top of the ingot before moving up the crucible ical reactions are negligible, and material properties dependA. Thethe Vacuum methodology Arc and approach used to establish a working (description in terms of electron flow). The high currents model of the system, and discuss the methodology of the arc onlyassociated on temperature. with titanium They alloy did melting not consider create a thevigorous effects of A vacuum arc is more accurately called a metal vapor arc locationconvection on pattern the melt with pool, a downward but mentioned flow at the it axis as a of sourceplasmalocation arc. The prediction VAR arc equations. is sustained Next, by vaporization in Section3, we inspect of interest.the ingot. Gartling This tends et al. to [homogenize7] created melt a numerical pool temper- model ofandseveral ionization factors of the that electrode may material, impact the rather accuracy than an of arc location the VARatures process and results that in delivered a steep temperature qualitatively gradient accurate at the results.ambientpredictions, gas. In VAR, including the two the critical vertical components distance for between the sen- solidification front. The convection also mixes the the arc are the metal vapor plasma and the cathode spot. Theymaterial. emphasize The one net of result the parameterscan be macrosegregation that needed dueto be ad-Thesor cathode and arc,spot the emanates size of the the bulk electrode-ingot of the electrical gap, the effects dressed are the characteristics of melt pool stirring due to of ingot shrinkage, and the use of multiple sensors. Finally, electromagnetic,METALLURGICAL and AND therefore MATERIALS arc, TRANSACTIONS characteristics. B Reiter et in Section4 we summarize VOLUME 44B, our FEBRUARY results into 2013—155 primary conclu- al. [8] simulated heat transfer in VAR ingots during the melt- sions, and make some recommendations of best practices for ing process, where the coupling of an electromagnetic math- using—and further developing—the APS technology. ematical model was essential for accurate results. Pericleous et al. [3] developed a three-dimensional transient multi-scale model that incorporated a macro-level FEM-based computa- 2 Methodology tional fluid dynamics model coupled with a microscale solid- In this section, we describe our approach to modeling ification model to study VAR processes. They found that arc the VAR furnace and predicting arc locations based on model location and characteristics drive the occurrence of “freck- results. We used the multiphysics FEM software COMSOL les” and “white spots,” two key defects in the quality of VAR Multiphysics [13] to simulate a simplified VAR furnace, with arcs located in different locations in the electrode-ingot gap. Our model only considered magnetostatics, based on the Input Current steady-state Ampere’s Law and current conservation equa- Ground tion:

−1 −1 Ram J = ∇ × (µ0 µr B) − σv × B (1) B = ∇ × A (2) ∇ · J = 0 (3) Crucible

where J is the current vector, µ0 is the permeability of a vac- Vacuum uum, µr is the relative permeability of the material, B is the chamber magnetic ﬂux density vector, σ is the electrical conductivity, v is the particle velocity, and A is the magnetic vector potential. Electrode

Table 1: Geometric speciﬁcations of modeled VAR furnace Arc Component Radius (m) Height (m)

Electrode 0.381 1.000 Ingot 0.432 1.057 Crucible (outer) 0.472 4.000 Ingot Crucible (inner) 0.432 4.000 Furnace shell Furnace shell (outer) 0.640 4.000 Furnace shell (inner) 0.472 4.000 Arc 0.010 0.0254

We modeled a simplified axisymmetric VAR furnace, Figure 2: Diagram of the geometry employed in the model based on the geometry of Woodside et al. [2]. Table1 lists the (not to scale). geometric specifications, and Fig.2 shows the geometry of the VAR furnace studied. The electrode and ingot were both assumed to be titanium, with an electrical conductivity of as Nair and Ward [2, 10]. The top of the copper crucible 7.407 × 105 S/m and relative permeability of 7.9585 × 10−7. was assigned a current source of 35,000 A, and the system The surrounding crucible was selected as copper (electrical was grounded at the ram that feeds the electrode. Domain conductivity of 5.998 × 107 S/m and relative permeability of boundary conditions were set to mimic an infinite domain. near 1), and the outer shell as steel (electrical conductiv- The entire domain used a mesh consisting of free tetrahedral ity of 4.032 × 106 S/m, and relative permeability of near 1). cells, that was automatically determined by COMSOL. We The annular space between the electrode and crucible and assigned the mesh size near the electrode-ingot gap as “fine” the electrode-ingot gap were modeled as near-vacuums, with settings, with a minimum element size of 0.0016 m. The rest an electrical conductivity of essentially zero (1 × 10−15 S/m), of the furnace geometry was set to “medium” settings, with and relative permeability of exactly 1. A small cylinder a minimum element size of 0.08 m. The outer boundary was connecting the ingot and electrode represented the arc, with set to “coarse” mesh settings, with a minimum element size all current forced through that small location; the arc was of 0.224 m. Mesh settings resulted in a total element count assigned an electrical conductivity 20 orders of magnitude of approximately 170,000. Further refining the grid changed larger than its surroundings. The size of the cylinder has solutions to within 5 % of the results published here, so we minimal effects on the magnetostatics of the scenario if its selected the aforementioned settings as a tradeoff between position and current remain constant. [10]. Another simpli- acceptable accuracy and time-to-solution. fication applied to the model is the assumption that princi- Next, we simulated sensor readings from one location. ples of superposition can be used in order to take into ac- We chose a point in the three-dimensional space of the do- count the effects of multiple arcs. [2]. The application of main to represent the sensor location; a physical sensor itself superposition was utilized by both Woodside et al. as well was not modeled, as its presence should not affect the magnetic field distribution. We performed parametric sweeps of of the furnace, d0 is the input radius of an arc, from the cen- the arc location to calculate flux density changes at the sen- ter of the furnace, θi is the angle from the sensor to the arc sor location. (Swept parameters in COMSOL include r0 and location, di is the length from the sensor to the arc location, θ0, where r0 is the radial position of the arc from the center rs is the distance from the sensor to the center of the furnace, of the furnace and θ0 is the angular position of the arc with and fi is the fraction of the total current associated with the respect to the x-axis.) Figure3 shows the magnetic flux den- arc. sity (T) as a function of arc position for a sensor located at (0 m,−0.64 m).

Magnetic Field Sensor

#10-3 Furnace Shell 0.4 3.5

0.3 3 Crucible

0.2 ] T [ 2.5 m

r rs o Electrode ] 0.1 N m y

[ r t B i s

n 0 2 n d o e 0

i θ t 0 D i Arc s x di u o t l B i

p θ F - c y 1.5 -0.1 i t e n g a M -0.2 1

-0.3 Br 0.5

-0.4 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 Bt x-position [m] Figure 4: Overhead cross-sectional diagram of a VAR fur- Figure 3: Magnetic ﬂux density norm with respect to arc nace, shown with four two-axis magnetic ﬁeld sensors and location in Cartesian coordinates. the geometry of the variables for one sensor.

Arc position predictions can now be examined. As de- Figure4 shows a top-down cross-section of the VARfur- termined by Woodside et al. [2] arc position determination nace modeled, including sensor locations. The furnace coef- can be achieved through the application of the Biot–Savart ficients m and m depend on the geometry and configuration law. The Biot–Savart law with a magnetostatic derivation of t r of individual furnaces [2]. The input angle for the COMSOL the Maxwell–Ampere law, using the relation between super- model and the angle θ are different measurements. Using the imposed line sources of current, and magnetic flux density Biot–Savart equations, a nonlinear regression was used to de- vector B at a location r is given by termine the unknown furnace coefficients mt and mr. Once these were determined, the single-line current versions of the µ Z dI0 × rˆ Biot–Savart equations were solved for d and θ (according to B(r) = 0 I , (4) i i 4π krk2 Fig.4) with input or measured magnetic flux density components. A vector reference frame rotation and translation is done to transform the magnetic field values from the refer- where dI0 is an element of the length along the total current, ence of the center of the furnace to each sensor location. The r is the vector from the source to the point, and rˆ is the unit equations take the form vector of r. This equation can be used to find the components of the magnetic field: Imrmt di = r , and (7) 2 2 2 2 I mr mt 2IBt mt mr 2 2 2 2 fi sinθi 1 2 + r + Br mt + Bt mr B = m I − , and (5) rs s t t ∑ d r i s −B d θ = cos−1 r i . (8) − fi cosθi i m I Br = mrI ∑ . (6) r di These equations represent the basis of the APS technol- where Bt and Br are the tangential and radial components of ogy [2]; using a given geometry and FEM-based furnace co- the magnetic flux density, mt is the tangential furnace coeffi- efficients, the arc can be located using measurements of mag- cient, mr is the radial furnace coefficient, I is the line current, netic flux density and the current through the system. Per- θ0 is the input angle in the model of an arc, from the center forming a parametric sweep of arc locations in the model where x, y are the exact positions andx ˆ,y ˆ are the predicted 0.4 positions. Arc location predictions Exact locations 0.3

0.2 3 Results and discussion

0.1

] In this section, we describe the results of our studies on m [ n

o the impact of various factors on the accuracy of arc location i

t 0 i s o

p predictions. We considered the effects on arc location predic- - y -0.1 tions of vertical sensor position (i.e., relative vertical distance between the sensor and arc), electrode-ingot gap size, ingot -0.2 shrinkage, and using the average of predictions from multi-

-0.3 ple sensors. All error calculations were based on the difference between the known position speciﬁed in the COMSOL -0.4 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 model and the location predicted using Eqs. (7) and (8). x-position [m]

Figure 5: Predicted arc locations compared to exact locations 3.1 Effect of vertical sensor position (calculated using furnace coefficients m = 8.15 × 10−8 N/A2, r The original APS studies of Woodside et al. [2, 12] con- m = 4.98 × 10−8 N/A2). t sidered a single sensor located in the plane of the electrode- ingot gap, where theoretically the most accurate results would be achieved due to the line-current source assumption. results in magnetic flux density components at the sensor In reality, due to the continuous movement of the electrode location. This array of values is equivalent to experimen- and growth of the ingot, the sensor would be in that plane tally measured sensor magnetic flux density values. Figure5 only for a small amount of time relative to the the entire pro- shows the predicted arc locations in contrast to the exact lo- cess. In addition, Ingots used for industrial VAR applications cations. Table2 compares the furnace coefficients found here can be several meters in length amplifying any error caused with those of Woodside et al. [2]; our implementation pre- by deviation from the electrode-ingot gap. A solution to this dicts arc locations within five percent of the published val- problem is the application of multiple rings of sensor in- ues. This discrepancy likely resulted from slight differences stalled in intervals that lie within acceptable error. We there- in geometry or solver setup (we developed our model based fore want to quantify the potential error induced by a vertical on published descriptions), differing COMSOL versions, or separation between sensor and arc, in order to decide how a different selection of input arc locations to calculate the frequently or far apart sensors should be placed along a fur- furnace coefficients. nace. To determine the error of vertical distance between the arc and sensor, we varied the vertical sensor position away from the plane of the gap in multiples of the electrode-ingot Table 2: Comparison of furnace coefficients from Woodside gap height: ±0, 1, 2, 3, 5, 7, 10, 15, and 20. et al. [2] with those determined here

2 2 mr (N/A ) mt (N/A )

80 FEM results 8.15 × 10−8 4.98 × 10−8 70 Woodside et al. 9 × 10−8 4.9 × 10−8 60 ] % [

r 50 o r r E

t 40 n

The remaining sections of the paper describe the perfor- e c r e mance of the arc position sensing approach as various model P 30 parameters are varied or assumptions are relaxed. This per- 20 formance is described in terms of percent error in the predicted arc locations with respect to the known arc locations, 10 normalized by the ingot radius (0.432 m). The error is de- 0 termined as the difference between two position vectors, the -20 -15 -10 -7 -5 -3 -2 -1 0 1 2 3 5 7 10 15 20 Sensor Distance from Ingot Top as Factor of Gap Height(hg = 0.0254[m])(in z-axis) predicted and exact locations of the arc: Figure 6: Statistical error distribution of predicted arc loca- p(x − xˆ)2 + (y − yˆ)2 tions with respect to vertical sensor position using constant error = ∗ 100 (9) furnace coefficients. radiusingot Figure6 shows a box-and-whisker statistical distribution all sensor locations, compared to using constant coefficients. of the percent error in predicted arc position with respect to Looking at four points in more detail, for a sensor located vertical sensor position using a single sensor; clearly, the er- 5× above and below the gap, median error for variable co- ror increases both as the sensor moves above and below the efficients increased by 5.43 % and decreased by 0.46 % re- gap. The red line inside the rectangles represents the me- spectively, compared to their constant coefficient counter- dian, the blue boxes span the 25th and 75th percentiles of the parts. For a sensor located 10× above and below the gap, data, and the black whiskers span the maximum and min- median error decreased by 0.31 % and increased by 2.92 %, imum values not defined as outliers. Outliers—defined as respectively. In real world systems the true vertical posi- values greater than three times the standard deviation—are tion of the electrode-ingot gap is unknown, however reason- represented by red plus-sign markers. Interestingly, the trend able estimates can be made using data from the melt such as in increasing error is asymmetric, with error increasing more weight and size of the ingot, time, and ram position. Includ- rapidly for sensor locations above the gap. The asymmetry ing this step in the algorithm could result in higher accuracy observed in error is most likely caused by the asymmetry of as shown in the results, however it would require real-world the current loop in the system. The reasons for the error be- testing and experimentation to confirm. ing lower for sensors below the electrode-ingot gap is not While percent error at each location offers some infor- clear. mation about measurement accuracy, examining the actual predicted locations and how they change can give insight on trends. Figure8 shows arc location predictions as the sensor location moves from the electrode-ingot gap plane up- 80 wards along the furnace wall, for vertical sensor locations

70 of 0, 0.0762, 0.254, and 0.508 m (or 0, 3, 10, and 20 hg). As the sensor position moves away from the electrode-ingot gap, 60

] the arc location predictions cluster together near the center of % [

r 50 the furnace. We hypothesize that this results from the current o r r

E density concentrating inwards inside the electrode as it tra-

t 40 n

e verses through the electrode and into the smaller-radius ram. c r e

P 30 The equations used to locate arc positions are two dimensional, using tangential and radial magnetic flux values in the 20 plane of the sensor’s vertical position. The magnetic flux val- 10 ues measured by sensors positioned away from the electrode- ingot gap plane are small; for example, a sensor positioned 0 -20 -15 -10 -7 -5 -3 -2 -1 0 1 2 3 5 7 10 15 20 at the electrode-ingot gap plane, a maximum of 3.3 × 10−3 T Sensor Distance from Ingot Top as Factor of Gap Height(hg = 0.0254[m])(in z-axis) in the radial direction and 3.5 × 10−3 T in the tangential direction are observed. On the other hand, a sensor located Figure 7: Error distribution of predicted arc locations with 0.5 m above the electrode-ingot gap measures a maximum respect to vertical sensor position using adaptive furnace co- of 2.5 × 10−4 T in the radial direction and 7.5 × 10−4 T in the efficients. tangential direction. In Eqs.7 and8 used to calculate position, magnetic flux components appear in the denominator— so the order-of-magnitude smaller values at the higher sensor For the results shown in Fig.6, arc locations were pre- location result in predicted locations further away. However, dicted assuming constant furnace coefficients determined us- Fig.8 shows that the location predictions shift further away ing data for a sensor placed at the plane of the electrode-ingot and also cluster around the center of the furnace. This sup- gap. We can also recalculate coefficients for each sensor lo- ports the hypothesis that the electrical current funnels as it cation, and investigate whether this practice improves results. traverses the electrode, and the magnetic flux density values This procedure is identical to that of the normal coefficient being measured at sensor locations away from the gap is the calculation, but uses magnetic flux density measurements at current density in the electrode for that plane. This clustering the various vertical sensor locations (rather than in the plane behavior would impact results by indicating the presence of of the gap). Figure7 shows the distribution of error in pre- arcs near the center of the furnace, when they might actually dicted arc locations using furnace coefficients recalculated occur near the edges of the electrode and ingot. for each sensor location. The varying furnace coefficients aided in suppressing outliers, and reducing maximum error. For the sensor position of 20× hg, using varying coefficients 3.2 Effect of gap size reduced maximum error by more than 30 %, and decreased All previous calculations assumed a constant electrode- median error by approximately 5 %. Median error shows a ingot gap. In theory, the gap size should remain approxi- slight increase for all vertical sensor positions; however, us- mately constant as the ram raises the electrode based on its ing varying furnace coefficients reduced the overall distri- melting and solidification rate; in reality, the gap size is con- bution of error. The application of varying furnace coeffi- stantly changing slightly throughout this process. This vari- cients decreased maximum error by 9.27 % on average for ation could introduce non-negligible errors into the predic- 1 1

0.9 0.9

0.8 0.8

] 0.7 ] 0.7 m [ m [ n o n i o t 0.6 i i t

0.6 s i o s o p - p - y 0.5 y 0.5

0.4 0.4

0.3 0.3

0.2 0.2 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 x-position [m] x-position [m]

(a) Arc location prediction with a sensor at the electrode-ingot gap (b) Arc location prediction with a sensor 0.0762 m above the (0 m above the electrode-ingot gap). electrode-ingot gap.

1 1

0.9 0.9

0.8 0.8 ] 0.7 ] 0.7 m [ m [ n n o i o i t 0.6 i t 0.6 i s s o o p p - - y 0.5 y 0.5

0.4 0.4

0.3 0.3

0.2 0.2 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 x-position [m] x-position [m]

(c) Arc location prediction with a sensor 0.254 m above the (d) Arc location prediction with a sensor 0.508 m above the electrode-ingot gap. electrode-ingot gap.

Figure 8: Arc location prediction trends for varying, single sensor position in the positive z direction with the origin at the sensor location (0,0.64 m) using constant furnace coefficients. tions of arc locations, and therefore we studied the effect of 3.3 Effect of ingot shrinkage gap size on the accuracy of arc location predictions. Previ- The physical characteristics of the ingot differ drasti- ous studies set the gap height to a constant 0.0254 m [2, 12]; cally from top to bottom during the melting process. At the here, we varied the height between 0.5–2.5 times the baseline top, a molten pool of liquid metal circulates on top of the soft, value, or specifically 0.0127, 0.0254, 0.0381, and 0.0635 m. hot metal solidifying near the sides and bottom. As the VAR Gap sizes considered in the literature include 0.01 m [6, 10] process continues and the ingot grows, the metal cools and for smaller radius ingots/electrodes, although Zanner stud- contracts. This causes the metal to shrink and pull away from ied the effect of gap sizes ranging 0.006–0.05 m [14] on melt the crucible, reducing the electrical contact surface area with rate. the crucible; this behavior is known as ingot shrinkage. The section of the material that remains in contact with the crucible wall is called the contact zone. Since shrinkage changes Figure9 shows arc location predictions with varying gap the surface area of the ingot that contacts the crucible wall— height for a single sensor location. The error in arc location and thus the area where electrical current passes—it could al- prediction exhibits little sensitivity to gap height; between ter the current path and thus the magnetic field distribution, the smallest and largest gap heights, the median error differs potentially affecting predictions of arc location. We stud- by less than 1.5 % and the maximum error by less than 5 %. ied the effects of shrinkage by applying grounded boundary nent does not play a role in the current APS approach based on horizontal (i.e., x and y) components.

20 3.4 Effect of using multiple sensors Thus far, we only used magnetic ﬁeld measurements at ] 15 %

[ one sensor location for calculations to determine arc loca- r o

r tions. Both our results and those from Woodside et al. [2, 12] r E

t show that the error increases as the arc moves away from the

n 10 e c

r sensor. Therefore, for an axisymmetric system, we hypoth- e P esize that the results from multiple sensors can be averaged

5 to improve the accuracy of the overall prediction. To test this, we averaged predicted arc locations from 2–16 evenly spaced sensors around the furnace. 0 First, we examined the trends in arc location prediction 0.0127 0.025 0.0375 0.06 for two separate sensors located at opposite sides of the fur- Electrode-Ingot Gap Height [m] nace; Figure 11 shows these (separate) predicted arc locations, compared with the exact locations. Although the sen- Figure 9: Arc location predictions with varying gap height sors predict similar locations for arcs located near the center for a single sensor location. of the furnace, near the perimeters the predicted locations exhibit a bias towards the closer sensor. Figure 12 compares exact arc locations with predictions based on the average lo- conditions to the contact zone and electrical insulation to the cation from four evenly spaced sensors. The averaging re- shrinkage gap zone, following the approach of Pericleous et sulted in an even spacial distribution of the arc positions; as al. [3]. We varied the size of the contact zone from the full Figure 13 shows, the error predicted locations is also evenly length of the ingot to 0.032 m, ranging from zero shrinkage distributed compared with that from a single sensor. This to a contact zone 3 % of the ingot height. information is useful because it can be used to develop cor- rection algorithms to predict arc locations more accurately. Based on these results, more sensors might aid in smooth- ing the error further; however, evenly distributed results do not guarantee more accurate results—in fact, outliers could 20 bias the predictions. Figure 14 shows the error distribution in predicted arc locations achieved by averaging calculations using 1, 2, 4, 8, and 16 evenly distributed sensors. The er- ] 15 % [ ror distribution contracts with the addition of sensors, but the r o r

r predictions do not improve with more than four sensors. E t

n 10 Now we can analyze whether using four sensor averag- e c r e ing of arc location predictions aid in the reduction of error P for a varying sensor height. In Section 3.1 we calculated 5 how error distribution was affected by the relative vertical position of a single sensor with respect to the electrode-ingot

0 gap. Now, we apply the same methodology while using arc

No Shrinkage 0.1 0.06 0.032 location predictions determined with four-sensor averaging. Contact Zone Height [m] The results are shown in Figure 15. The total error distribution is smaller than the single-sensor results from Section 3.1 Figure 10: Error in predicted arc locations with varying con- with a maximum error of less than 16 % at the highest and tact zone height, mimicking ingot shrinkage. lowest positions. Using four sensors also suppresses the outliers. These results were calculated using varying furnace coefficients; the coefficients were recalculated at every verti- Figure 10 shows the error distribution for predicted arc cal position. locations with increasing shrinkage, corresponding to de- creasing contact zone height. While error increases slightly as the contact zone shrinks, in general shrinkage causes an 4 Conclusions median error increase of less than 2 % in predicted arc lo- The magnetostatic characteristics of a simplified vac- cations, corresponding to a similar change in magnetic flux uum arc remelting furnace were studied using the finite el- density. However, the higher current density resulting from ement method, to determine the effects of various physical smaller contact zones could affect the z component of mag- phenomena on the resulting magnetic field. This was done netic flux density more significantly—although this compo- to analyze a system for predicting the locations of electrical 0.4 16

0.4 SP: (0,-0.64) m 0.3 14 SP: (0, 0.64) m exact 0.3 0.2 12

0.2 ] ] 0.1 % m 10 [ [ r o n r r o i 0.1 e

t 0 t i ] s n e m 8 o [ c r p n - e o i y P

t 0

i -0.1 s o

p 6 - y

-0.1 -0.2 4

-0.2 -0.3 2 -0.4 -0.3 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 x-position [m]

-0.4 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 x-position [m] (a) Percent error of arc location predictions using one sensor from the exact locations.

Figure 11: Arc location predictions for two sensors position 0.4 9 on opposite sides of the furnace: (0 m, −0.64 m) and (0 m, 0.3 0.64 m). 8 0.2 7 ] ] 0.1 % m [ [ r o n r r o i

6 e

0.4 t 0 t i s n

Predicted Locations e o c r Exact Locations p - e y -0.1 P 0.3 5

-0.2

0.2 4 -0.3

0.1 3 -0.4

] -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 m [ x-position [m] n o i

t 0 i s o p - y (b) Percent error of arc location predictions using four sensor aver- -0.1 aging from the exact locations.

-0.2 Figure 13: Comparison of error distribution and magnitude from using one sensor versus four sensor averages; error is -0.3 based on difference between predicted and exact values.

-0.4 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 x-position [m] late the furnace coefﬁcients. We then studied the effects of Figure 12: Arc location predictions using four sensors changing certain parameters or eliminating previously made around the furnace; these locations are the average of the assumptions to increase conﬁdence in this arc position sens- locations predicted by each of the four sensors. ing approach. Based on our simulations, we drew the following conclusions:

1. Error in predicted arc locations increases substantially arcs that form in the electrode-ingot gap in these furnaces. as the sensor moves vertically from the electrode-ingot Arc distribution throughout the remelting process plays a gap; for distances of 0.508 m above and below the plane strong role in the material properties of the produced ingot, of the gap, the maximum error reached around 40 % so real-time tracking of arc locations can provide a priori infor constant furnace coefficients and 25 % for adaptive dications of ingot quality. furnace coefficients. Furthermore, as the sensor moves First, we reproduced the prior results of Woodside et away from the gap, the magnetic flux density decreases al. [2] for predicting arc locations; we matched arc location and results in clustering of predicted locations near the predictions within 3.5 % of their results. This minor discrep- center of the domain. ancy likely resulted from slight differences in model setup, 2. Gap height variations do not significantly affect pre- solver version, or selection of input arc locations to calcu- dictions in arc location: varying the gap height from Overall, out of the parameters and assumptions we studied, we conclude that gap height variation and ingot shrink-

20 age do not affect arc location predictions. However, increasing vertical distance between the sensor and arc will lead to signiﬁcant errors in location prediction, and should be con-

] 15 sidered for further development of the arc position sensing % [ r

o technology. r r e t n

e 10 c r e

P Acknowledgements

5 This material is based on work supported by Oregon BEST under grant reference number “CG-SOW-2015-OSU- KW Associates”. The authors thank Paul King, Rigel Wood- 0 side, and Matt Cibula of Ampere Scientiﬁc (formerly KW 1 2 4 8 16 Associates LLC) for helpful discussions and comments on Number of Sensors this paper.

Figure 14: Percent error statistical distribution for 1, 2, 4, 8, and 16 sensors. References [1] Yu, K. O., 2002. Modelling for Casting and Solidiﬁca- tion Processing. Dekker.

16 [2] Woodside, C. R., King, P. E., and Nordlund, C., 2013. “Arc distribution during the vacuum arc remelting of 14 Ti-6Al-4V”. Metall and Materi Trans B, 44(1), Feb.,

12 pp. 154–165.

] [3] Pericleous, K., Djambazov, G., Ward, M., Yuan, L., and % [ 10 r

o Lee, P. D., 2013. “A multiscale 3D model of the vac- r r

E 8 uum arc remelting process”. Metall and Mat Trans A, t n e

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e 6 P [4] Zhao, X.-h., Li, J.-s., Yang, Z.-j., Kou, H.-c., Hu, R., 4 and Zhou, L., 2011. “Numerical simulation of fluid flow caused by buoyancy forces during vacuum arc 2 remelting process”. J. Shanghai Jiaotong Univ. (Sci.), 0 16(3), June, pp. 272–276. -20 -15 -10 -7 -5 -3 -2 -1 0 1 2 3 5 7 10 15 20 Sensor Distance from Ingot Top as Factor of Gap Height(hg = 0.0254[m])(in z-axis) [5] Yang, Z.-j., Zhao, X.-h., Kou, H.-c., Li, J.-s., Hu, R., and Zhuo, L., 2010. “Numerical simulation of temperature distribution and heat transfer during solidification Figure 15: Error distribution of predicted arc locations with of titanium alloy ingots in vacuum arc remelting pro- respect to vertical sensor position using the average results cess”. Transactions of Nonferrous Metals Society of from four sensors. China, 20(10), Oct., pp. 1957–1962. [6] Beaman, J. J., Felipe Lopez, L., and Williamson, R. L., 2014. “Modeling of the vacuum arc remelting process 0.0128 m to 0.063 m resulted in the median error chang- for estimation and control of the liquid pool profile”. J. ing by 0.55 % and the maximum error by 1.65 %. Dyn. Sys., Meas., Control, 136(3), May, pp. 031007– 3. Ingot shrinkage does not affect predictions in arc lo- 11. cation; varying the ingot contact zone from the entire [7] Gartling, D. K., and Sackinger, P. A., 1997. “Finite length down to 0.032 m kept the maximum error around element simulation of vacuum arc remelting”. Int. J. 10 %, and did not change the median error of around Numer. Methods Fluids, 24(12), June, pp. 1271–1289. 2 percent. [8] Reiter, G., Maronnier, V., Sommitsch, C., Gäu- 4. Averaging the measurements of four evenly spaced sen- mann, M., Schützenhöfer, W., and Schneider, R., sors around the furnace reduces prediction error distri- 2003. “Numerical simulation of the VAR process with bution by 12.64 % compared to a single sensor; there is calcosoft®-2D and its validation”. In LMPC 2003, little to no improvement on location predictions with the pp. 77–86. implementation of more than four sensors. The total er- [9] Wang, L., Jia, S., Shi, Z., and Rong, M., 2005. “Nu- ror distribution reduces by 0.85 % comparing the imple- merical simulation of vacuum arc under different axial mentation of four sensors with sixteen. The application magnetic fields”. J. Phys. D: Appl. Phys., 38(7), Mar., of four sensor averaging aids in the suppression of error pp. 1034–1041. when taking into account vertical sensor position. [10] Nair, B. G., and Ward, R. M., 2009. “An analysis of the use of magnetic source tomography to measure the spatial distribution of electric current during vacuum arc remelting”. Meas. Sci. Technol., 20(4), Feb., pp. 045701–12. [11] Mir, H. E., Jardy, A., Bellot, J.-P., Chapelle, P., Lasalmonie, D., and Senevat, J., 2010. “Thermal be- haviour of the consumable electrode in the vacuum arc remelting process”. Journal of Materials Processing Technology, 210(3), Feb., pp. 564–572. [12] Woodside, C. R., and King, P. E., 2010. “A measurement system for determining the positions of arcs during Vacuum arc remelting”. In 2010 IEEE Instrumenta- tion & Measurement Technology Conference Proceed- ings, IEEE, pp. 452–457. [13] COMSOL Inc., 2013. COMSOL Multiphysics. 4.3b, May. [14] Zanner, F. J., Adasczik, C., O’Brien, T., and Bertram, L. A., 1984. “Observations of melt rate as a function of arc power, CO pressure, and electrode gap during vacuum consumable arc remelting of Inconel 718”. Metal- lurgical Transactions B, 15(1), pp. 117–125.