Miguel Soler
email: solerm@oregonstate.edu

Kyle Kyle

Niemeyer Niemeyer

E Niemeyer

Kyle E Niemeyer
email: kyle.niemeyer@oregonstate.edu

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

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Introduction

Vacuum arc remelting, or VAR, is the metallurgical process of remelting metal ingots with the application of a direct current into the system, in a vacuum environment. The result is a high-quality metal ingot that exhibits increased homogeneity and decreased defects. The high-quality metals produced by VAR are typically used for high-performance applications such as aerospace rotating systems. VAR is often used on Ni-and Ti-based alloys [START_REF] Woodside | Arc distribution during the vacuum arc remelting of Ti-6Al-4V[END_REF][START_REF] Pericleous | A multiscale 3D model of the vacuum arc remelting process[END_REF][START_REF] Zhao | Numerical simulation of fluid flow caused by buoyancy forces during vacuum arc remelting process[END_REF][START_REF] Yang | Numerical simulation of temperature distribution and heat transfer during solidification of titanium alloy ingots in vacuum arc remelting process[END_REF].

Figure 1 depicts a VAR furnace cross section. The current applied to the system forms electrical arcs between the melted ingot and the input consumable electrode. Since no ingot exists at the start of the process, common practices include the addition of small metal pieces to the bottom of the crucible to form an arc. These arcs begin the melting process of the electrode, which then transfers mass to the bottom of the crucible due to gravity. This mass solidifies at the bottom of the ingot as the arcs and heat transfer take place at the electrode-ingot gap, which travels up the crucible as more mass is transferred from the electrode to the ingot. Arcs can simultaneously form in multiple positions; presently, operators can neither visualize nor control the formation of arcs. A water-cooled jacket prevents the copper crucible from melting. At the top of the melted ingot a liquid pool of the material exists. The characteristics of this melt pool have a large impact on final quality of the ingot [START_REF] Zhao | Numerical simulation of fluid flow caused by buoyancy forces during vacuum arc remelting process[END_REF][START_REF] Yang | Numerical simulation of temperature distribution and heat transfer during solidification of titanium alloy ingots in vacuum arc remelting process[END_REF][START_REF] Beaman | Modeling of the vacuum arc remelting process for estimation and control of the liquid pool profile[END_REF].

melting rates are much higher than utilized during nickel alloy melting. Thus, the molten pool is quite deep, sometimes being described as having a ''soda can'' shape. This means the localized heat flux from the arcs will have less impact on the solidification front as compared to nickel melting because of the difference in distances from arc to solidification front. However, arc constrictions toward the side wall can potentially lead to shelf remelting, causing material to fall into the pool, thus the commonly used term ''fall-in.'' This material can remain intact in the melt pool and can be a source for point defects as its composition will often differ from the nominal alloy composition. An example is a type 2 defect characterized as having too much primary alpha phase due to an elevated concentration of aluminum, which is the alpha stabilizer. This leads to a region having slightly elevated hardness as compared to the alloy, and this can be accompanied by a neighboring region having depleted aluminum and lower hardness.

Perhaps a less obvious effect of arc distribution is the effect of the magnetohydrodynamic stirring of the molten metal. In a coaxial furnace design, the electrical current from the arcs enters the melt pool and then largely flows in a radial direction, exiting the ingot near the top of the ingot before moving up the crucible (description in terms of electron flow). The high currents associated with titanium alloy melting create a vigorous convection pattern with a downward flow at the axis of the ingot. This tends to homogenize melt pool temperatures and results in a steep temperature gradient at the solidification front. The convection also mixes the material. The net result can be macrosegregation due to solute partitioning of the alloying elements. Recen results have indicated both through modeling [3] and experimentally [4] that significant Fe macrosegregation occurs during the melting of Ti-10V-2Fe-3Al, and thi macrosegregation changes as a function of the tota current entering the ingot. The modeling study assumed an axisymmetric and Gaussian distribution of the arc whether this is valid is one of the aims of the curren work.

Perhaps the defect of most concern with Ti-6Al-4V i so called hard-alpha interstitial inclusions, a type 1 low density inclusion (LDI), because these inclusions can become crack initiation sites leading to prematur fatigue failure. The fact that hard-alpha material can have a similar melting point and similar density a compared to the alloy makes it difficult to remove vi VAR. The term hard-alpha actually refers to materia over a range in the Ti-N phase diagram, sometime referred to as nitrides, and within this range, there i considerable variation in the fracture toughness. [5] Ti-N inclusion ''survival'' times in a VAR melt pool as function of particle size and density have been mod eled. [6] As mentioned, the arc current drives the fluid motion. An understanding of the fluid dynamics in th pool is critical to predicting the ability of VAR to reduc these defects. [7] Therefore, knowing the VAR arc distri bution is in turn critical to making accurate prediction of the dissipation of hard alpha inclusions within th VAR melt pool. Producing ingots free of high-densit inclusions (HDI), a type 1 defect consisting of refractory element such as tungsten, is also importan but is expected to be less dependent on arc distribution This is because these inclusions tend to rapidly sink to the bottom of the melt pool, so changes in the arc driven fluid dynamics are less important.

It is also possible that the arc distribution impacts th physical structure of the ingot sidewall surface. This i significant because the sidewall integrity and grain structure can in turn affect subsequent forging opera tions and product yield. Multiple techniques are used within industry to improve ingot surface quality, bu determining the effect of the arc distribution on th ingot surface has been difficult.

Arc distribution is relevant to the quality of an material melted via VAR. The focus on this paper is on Ti-6Al-4V not because this alloy deserves the mos attention but rather because of circumstantial conve nience in terms of the experiments. The techniqu described is applicable to other VAR operations, but i should be noted that the reported arc motion and distribution results may be specific to the furnac and procedures followed by ATI Albany Operation (Albany, OR).

A. The Vacuum Arc

A vacuum arc is more accurately called a metal vapo plasma arc. The VAR arc is sustained by vaporization and ionization of the electrode material, rather than an ambient gas. In VAR, the two critical components fo the arc are the metal vapor plasma and the cathode spot The cathode spot emanates the bulk of the electrica Arc behavior drives the remelting process and determines ingot quality, but arc positions are challenging 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 operators some indication of arc behavior, mainly for the detection of side arcing, but these cannot track instantaneous arc formation and motion. A robust arc detection and tracking system would give insight into the material properties of the final ingot. The most common approach for detailed study of VAR furnaces is numerical modeling [START_REF] Woodside | Arc distribution during the vacuum arc remelting of Ti-6Al-4V[END_REF][START_REF] Pericleous | A multiscale 3D model of the vacuum arc remelting process[END_REF][START_REF] Zhao | Numerical simulation of fluid flow caused by buoyancy forces during vacuum arc remelting process[END_REF][START_REF] Yang | Numerical simulation of temperature distribution and heat transfer during solidification of titanium alloy ingots in vacuum arc remelting process[END_REF][START_REF] Beaman | Modeling of the vacuum arc remelting process for estimation and control of the liquid pool profile[END_REF][START_REF] Gartling | Finite element simulation of vacuum arc remelting[END_REF][START_REF] Reiter | Numerical simulation of the VAR process with calcosoft R -2D and its validation[END_REF][START_REF] Wang | Numerical simulation of vacuum arc under different axial magnetic fields[END_REF]. The application of large currents through the system results in a strong magnetic field surrounding the furnace, on which several studies focused [START_REF] Woodside | Arc distribution during the vacuum arc remelting of Ti-6Al-4V[END_REF][START_REF] Wang | Numerical simulation of vacuum arc under different axial magnetic fields[END_REF][START_REF] Nair | An analysis of the use of magnetic source tomography to measure the spatial distribution of electric current during vacuum arc remelting[END_REF]. The arc position in the ingot-electrode gap is a key parameter that affects the magnetic field. Arcs concentrate the electrical current passing through the system and impacts the distribution of the magnetic field.

Mir et al. [START_REF] Mir | Thermal behaviour of the consumable electrode in the vacuum arc remelting process[END_REF] studied the thermal behavior of the consumable electrode using infrared cameras, focusing on heat transfer characteristics. However, their technique required alteration of a furnace and revealed little insight on arc behavior. Woodside et al. [START_REF] Woodside | Arc distribution during the vacuum arc remelting of Ti-6Al-4V[END_REF][START_REF] Woodside | A measurement system for determining the positions of arcs during Vacuum arc remelting[END_REF] used the multiphysics finite-element modeling (FEM) software COMSOL to simulate the magnetostatics of a VAR furnace. They assumed an axisymmetric furnace, homogeneous material properties, and a single non-diffuse arc in a three-dimensional model. Model results were used to develop a relationship between measured magnetic field readings at a notional Hall sensor position and arc locations [START_REF] Woodside | Arc distribution during the vacuum arc remelting of Ti-6Al-4V[END_REF]. Zhao et al. [START_REF] Zhao | Numerical simulation of fluid flow caused by buoyancy forces during vacuum arc remelting process[END_REF] used the two-dimensional FEM software ANSYS to study fluid dynamics in the molten pool. The model assumed that only buoyant forces act on the melt pool, the melt pool exhibits turbulent flow properties, chemical reactions are negligible, and material properties depend only on temperature. They did not consider the effects of arc location on the melt pool, but mentioned it as a source of interest. Nair et al. [START_REF] Nair | An analysis of the use of magnetic source tomography to measure the spatial distribution of electric current during vacuum arc remelting[END_REF] used the FEM software Opera3d to study the use of magnetic source tomography to understand arc behavior in VAR systems. They modeled electrostatics while assuming homogeneous material properties, and included both single and double non-diffuse arcs. Nair et al. concluded that arc locations can be predicted based on measurements of magnetic flux density outside the furnace. Wang et al. [START_REF] Wang | Numerical simulation of vacuum arc under different axial magnetic fields[END_REF] developed a two-dimensional axisymmetric model to study arc characteristics under different axial magnetic fields using the commercially available software FLUENT. Their model focused on magnetohydrodynamics and plasma behavior in the electrode-ingot gap, assuming that plasma consisted of only electrons and ions and its flow can be described with a hydrodynamic approximation. They concluded that the effect of current density distribution, arc distribution, was significant for VAR because it directly correlates to the heat flux density at the anode. Pericleous et al. [START_REF] Pericleous | A multiscale 3D model of the vacuum arc remelting process[END_REF] developed a three-dimensional transient multiscale model that incorporated a macro-level FEM-based computational fluid dynamics model coupled with a microscale solidification model to study VAR processes. They found that arc location and characteristics drive the occurrence of "freckles" and "white spots," two key defects in the quality of VAR ingots [START_REF] Pericleous | A multiscale 3D model of the vacuum arc remelting process[END_REF].

Gartling et al. [START_REF] Gartling | Finite element simulation of vacuum arc remelting[END_REF] created a numerical model of the VAR process that delivered qualitatively accurate results. They emphasize one of the parameters that needed to be addressed are the characteristics of melt pool stirring due to electromagnetic, and therefore arc, characteristics. Reiter et al. [START_REF] Reiter | Numerical simulation of the VAR process with calcosoft R -2D and its validation[END_REF] simulated heat transfer in VAR ingots during the melting process, where the coupling of an electromagnetic mathematical model was essential for accurate results. According to literature, arc locations and characteristics directly affect ingot characteristics. [START_REF] Woodside | Arc distribution during the vacuum arc remelting of Ti-6Al-4V[END_REF][START_REF] Pericleous | A multiscale 3D model of the vacuum arc remelting process[END_REF][START_REF] Zhao | Numerical simulation of fluid flow caused by buoyancy forces during vacuum arc remelting process[END_REF][START_REF] Gartling | Finite element simulation of vacuum arc remelting[END_REF][START_REF] Reiter | Numerical simulation of the VAR process with calcosoft R -2D and its validation[END_REF][START_REF] Wang | Numerical simulation of vacuum arc under different axial magnetic fields[END_REF][START_REF] Nair | An analysis of the use of magnetic source tomography to measure the spatial distribution of electric current during vacuum arc remelting[END_REF] Various studies showed that arc locations can be predicted accurately using magnetic flux density measurements around VAR furnaces combined with accurate numerical models. [START_REF] Woodside | Arc distribution during the vacuum arc remelting of Ti-6Al-4V[END_REF][START_REF] Nair | An analysis of the use of magnetic source tomography to measure the spatial distribution of electric current during vacuum arc remelting[END_REF] However, these methodologies made assumptions and simplifications that should be validated further to encourage their application in industry.

The purpose of this study is to model the magnetostatics of the VAR process in different scenarios, while evaluating the impact of previously made assumptions, to determine the potential errors of arc locations predicted by the Arc Position Sensing (APS) system of Woodside and King [START_REF] Woodside | A measurement system for determining the positions of arcs during Vacuum arc remelting[END_REF]. Understanding the behavior of the APS technology due to changing parameters could lead to further validation or improvement [START_REF] Woodside | Arc distribution during the vacuum arc remelting of Ti-6Al-4V[END_REF][START_REF] Nair | An analysis of the use of magnetic source tomography to measure the spatial distribution of electric current during vacuum arc remelting[END_REF]. We used a multiphysics FEM simulation software to study the system. First, in Section 2 we describe the methodology and approach used to establish a working model of the system, and discuss the methodology of the arc location prediction equations. Next, in Section 3, we inspect several factors that may impact the accuracy of arc location predictions, including the vertical distance between the sensor and arc, the size of the electrode-ingot gap, the effects of ingot shrinkage, and the use of multiple sensors. Finally, in Section 4 we summarize our results into primary conclusions, and make some recommendations of best practices for using-and further developing-the APS technology.

Methodology

In this section, we describe our approach to modeling the VAR furnace and predicting arc locations based on model results. We used the multiphysics FEM software COMSOL Multiphysics [START_REF]COMSOL Multiphysics[END_REF] 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 steady-state Ampere's Law and current conservation equation:

J = ∇ × (µ -1 0 µ -1 r B) -σv × B (1) B = ∇ × A (2)
∇ • J = 0 ( 3
)
where J is the current vector, µ 0 is the permeability of a vacuum, µ r is the relative permeability of the material, B is the magnetic flux density vector, σ is the electrical conductivity, v is the particle velocity, and A is the magnetic vector potential. We modeled a simplified axisymmetric VAR furnace, based on the geometry of Woodside et al. [START_REF] Woodside | Arc distribution during the vacuum arc remelting of Ti-6Al-4V[END_REF]. Table 1 lists the 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 7.407 × 10 5 S/m. The surrounding crucible was selected as copper (electrical conductivity of 5.998 × 10 7 S/m) and the outer shell as steel (electrical conductivity of 4.032 × 10 6 S/m). The annular space between the electrode and crucible and the electrode-ingot gap were modeled as near-vacuums, with an electrical conductivity of essentially zero (1 × 10 -15 S/m). A small cylinder connecting the ingot and electrode represented the arc, with all current forced through that small location; the arc was assigned an electrical conductivity 20 orders of magnitude larger than its surroundings. The top of the copper crucible was assigned a current source of 35,000 A, and the system was grounded at the ram that feeds the electrode. Domain boundary conditions were set to mimic an infinite domain.

The entire domain used a mesh consisting of free tetrahedral cells, that was automatically determined by COMSOL. We assigned the mesh size near the electrode-ingot gap as "fine" settings, with a minimum element size of 0.0016 m. The rest of the furnace geometry was set to "medium" settings, with a minimum element size of 0.08 m. The outer boundary was set to "coarse" mesh settings, with a minimum element size of 0.224 m. Mesh settings resulted in a total element count of approximately 170,000. Further refining the grid changed solutions to within 5 % of the results published here, so we selected the aforementioned settings as a tradeoff between acceptable accuracy and time-to-solution.

Next, we simulated sensor readings from one location. We chose a point in the three-dimensional space of the domain to represent the sensor location; a physical sensor itself was not modeled, as its presence should not affect the magnetic field distribution. We performed parametric sweeps of the arc location to calculate flux density changes at the sensor location. (Swept parameters in COMSOL include r 0 and θ 0 , where r 0 is the radial position of the arc from the center of the furnace and θ 0 is the angular position of the arc with respect to the x-axis.) Figure 3 shows the magnetic flux density (T ) as a function of arc position for a sensor located at (0 m,-0.64 m).

Arc position predictions can now be examined. As determined by Woodside et al. [START_REF] Woodside | Arc distribution during the vacuum arc remelting of Ti-6Al-4V[END_REF] arc position determination can be achieved through the application of the Biot-Savart law. The Biot-Savart law with a magnetostatic derivation of the Maxwell-Ampere law, using the relation between superimposed line sources of current, and magnetic flux density vector B

B(r) = µ 0 4π I dI × r r 2 , (4)
where dI is an element of the length along the total current, r is the vector from the source to the point, and r is the unit vector of r. This equation can be used to find the components of the magnetic field:

B t = m t I ∑ f i sin θ i d i - 1 r s , and (5)
B r = m r I ∑ -f i cos θ i d i . ( 6
)
where B t and B r are the tangential and radical components of the magnetic flux density, m t is the tangential furnace coefficient, m r is the radial furnace coefficient, I is the line current, θ 0 is the input angle in the model of an arc, from the center of the furnace, d 0 is the input radius of an arc, from the center of the furnace, θ i is the angle from the sensor to the arc location, d i is the length from the sensor to the arc location, r s is the distance from the sensor to the center of the furnace, and f i is the fraction of the total current associated with the arc.

Figure 4 shows a top-down cross-section of the VAR furnace modeled, including sensor locations. The furnace coefficients m t and m r depend on the geometry and configuration of individual furnaces [START_REF] Woodside | Arc distribution during the vacuum arc remelting of Ti-6Al-4V[END_REF]. The input angle for the COMSOL model and the angle θ are different measurements. Using the Biot-Savart equations, a nonlinear regression was used to determine the unknown furnace coefficients m t and m r . Once these were determined, the single-line current versions of the Biot-Savart equations were solved for d i and θ i (according to 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 reference of the center of the furnace to each sensor location. The equations take the form

d i = Im r m t I 2 m 2 r m 2 t r 2 s + 2IB t m t m 2 r r s + B 2 r m 2 t + B 2 t m 2 r
, and

θ i = cos -1 -B r d i m r I . ( (7)
) 8
These equations represent the basis of the APS technology [START_REF] Woodside | Arc distribution during the vacuum arc remelting of Ti-6Al-4V[END_REF]; using a given geometry and FEM-based furnace coefficients, the arc can be located using measurements of magnetic flux density and the current through the system. Performing a parametric sweep of arc locations in the model results in magnetic flux density components at the sensor location. This array of values is equivalent to experimentally measured sensor magnetic flux density values. Figure 5 shows the predicted arc locations in contrast to the exact locations. Table 2 compares the furnace coefficients found here with those of Woodside et al. [START_REF] Woodside | Arc distribution during the vacuum arc remelting of Ti-6Al-4V[END_REF]; our implementation predicts arc locations within five percent of the published values. This discrepancy likely resulted from slight differences in geometry or solver setup (we developed our model based on published descriptions), differing COMSOL versions, or a different selection of input arc locations to calculate the furnace coefficients. The remaining sections of the paper describe the performance of the arc position sensing approach as various model parameters are varied or assumptions are relaxed. This performance is described in terms of relative error in the predicted arc locations with respect to the known arc locations, normalized by the ingot radius (0.432 m).

Results and discussion

In this section, we describe the results of our studies on the impact of various factors on the accuracy of arc location predictions. We considered the effects on arc location predictions of vertical sensor position (i.e., relative vertical distance between the sensor and arc), electrode-ingot gap size, ingot shrinkage, and using the average of predictions from multiple sensors. All error calculations were based on the difference between the known position specified in the COMSOL model and the location predicted using Eqs. ( 7) and (8).

Effect of vertical sensor position

The original APS studies of Woodside et al. [START_REF] Woodside | Arc distribution during the vacuum arc remelting of Ti-6Al-4V[END_REF][START_REF] Woodside | A measurement system for determining the positions of arcs during Vacuum arc remelting[END_REF] considered 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. In reality, due to the continuous movement of the electrode and growth of the ingot, the sensor would be in that plane only for a small amount of time relative to the the entire process. We therefore want to quantify the potential error induced by a vertical separation between sensor and arc, in order to decide how frequently or far apart sensors should be placed along a furnace. 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 gap height: ±0, 1, 2, 3, 5, 7, 10, 15, and 20.

Figure 6 shows a box-and-whisker statistical distribution of the percent error in predicted arc position with respect to vertical sensor position using a single sensor; clearly, the error increases both as the sensor moves above and below the gap. The red line inside the rectangles represents the median, the blue boxes span the 25th and 75th percentiles of the data, and the black whiskers span the maximum and minimum values not defined as outliers. Outliers-defined as values greater than three times the standard deviation-are represented by red plus-sign markers. Interestingly, the trend in increasing error is asymmetric, with error increasing more rapidly for sensor locations above the gap. The asymmetry observed in error is most likely caused by the asymmetry of the current loop in the system. The reasons for the error being lower for sensors below the electrode-ingot gap is not clear.

For the results shown in Fig. 6, arc locations were predicted assuming constant furnace coefficients determined using data for a sensor placed at the plane of the electrode-ingot gap. We can also recalculate coefficients for each sensor location, and investigate whether this practice improves results. This procedure is identical to that of the normal coefficient calculation, but uses magnetic flux density measurements at the various vertical sensor locations (rather than in the plane of the gap). Figure 7 shows the distribution of error in predicted arc locations using furnace coefficients recalculated for each sensor location. The varying furnace coefficients aided in suppressing outliers, and reducing maximum error. For the sensor position of 20× h g , using varying coefficients reduced maximum error by more than 30 %, and decreased median error by approximately 5 %. Median error shows a slight increase for all vertical sensor positions; however, using varying furnace coefficients reduced the overall distribution of error. The application of varying furnace coefficients decreased maximum error by 9.27 % on average for all sensor locations, compared to using constant coefficients. Looking at four points in more detail, for a sensor located 5× above and below the gap, median error for variable coefficients increased by 5. 7: Error distribution of predicted arc locations with respect to vertical sensor position using adaptive furnace coefficients respectively, compared to their constant coefficient counterparts. For a sensor located 10× above and below the gap, median error decreased by 0.31 % and increased by 2.92 %, respectively.

While percent error at each location offers some information about measurement accuracy, examining the actual predicted locations and how they change can give insight on trends. Figure 8 shows arc location predictions as the sensor location moves from the electrode-ingot gap plane upwards along the furnace wall, for vertical sensor locations of 0, 0.0762, 0.254, and 0.508 m (or 0, 3, 10, and 20 h g ). As the sensor position moves away from the electrode-ingot gap, the arc location predictions cluster together near the center of the furnace. We hypothesize that this results from the current density concentrating inwards inside the electrode as it traverses through the electrode and into the smaller-radius ram. The equations used to locate arc positions are two dimensional, using tangential and radial magnetic flux values in the plane of the sensor's vertical position. The magnetic flux values measured by sensors positioned away from the electrode-ingot gap plane are small; for example, a sensor positioned at the electrode-ingot gap plane, a maximum of 3.3 × 10 -3 T in the radial direction and 3.5 × 10 -3 T in the tangential direction are observed. On the other hand, a sensor located 0.5 m above the electrode-ingot gap measures a maximum of 2.5 × 10 -4 T in the radial direction and 7.5 × 10 -4 T in the tangential direction. In Eqs. 7 and 8 used to calculate position, magnetic flux components appear in the denominator-so the order-of-magnitude smaller values at the higher sensor location result in predicted locations further away. However, Fig. 8 shows that the location predictions shift further away and also cluster around the center of the furnace. This supports the hypothesis that the electrical current funnels as it traverses the electrode, and the magnetic flux density values being measured at sensor locations away from the gap is the current density in the electrode for that plane. This clustering behavior would impact results by indicating the presence of arcs near the center of the furnace, when they might actually occur near the edges of the electrode and ingot.

Effect of gap size

All previous calculations assumed a constant electrode-ingot gap. In theory, the gap size should remain approximately constant as the ram raises the electrode based on its melting and solidification rate; in reality, the gap size is constantly changing slightly throughout this process. This variation could introduce non-negligible errors into the predictions of arc locations, and therefore we studied the effect of gap size on the accuracy of arc location predictions. Previous studies set the gap height to a constant 0.0254 m [START_REF] Woodside | Arc distribution during the vacuum arc remelting of Ti-6Al-4V[END_REF][START_REF] Woodside | A measurement system for determining the positions of arcs during Vacuum arc remelting[END_REF]; here, we varied the height between 0.5-2.5 times the baseline value, or specifically 0.0127, 0.0254, 0.0381, and 0.0635 m. Gap sizes considered in the literature include 0.01 m [START_REF] Beaman | Modeling of the vacuum arc remelting process for estimation and control of the liquid pool profile[END_REF][START_REF] Nair | An analysis of the use of magnetic source tomography to measure the spatial distribution of electric current during vacuum arc remelting[END_REF] for smaller radius ingots/ electrodes, although Zanner studied the effect of gap sizes ranging 0.006-0.05 m [START_REF] Zanner | Observations of melt rate as a function of arc power, CO pressure, and electrode gap during vacuum consumable arc remelting of Inconel 718[END_REF] on melt rate. Figure 9 shows arc location predictions with varying gap height for a single sensor location. The error in arc location prediction exhibits little sensitivity to gap height; between the smallest and largest gap heights, the median error differs by less than 1.5 % and the maximum error by less than 5 %.

Electrode-Ingot Gap Height [m]

Effect of ingot shrinkage

The physical characteristics of the ingot differ drastically from top to bottom during the melting process. At the top, a molten pool of liquid metal circulates on top of the soft, hot metal solidifying near the sides and bottom. As the VAR process continues and the ingot grows, the metal cools and contracts. This causes the metal to shrink and pull away from the crucible, reducing the electrical contact surface area with 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 the surface area of the ingot that contacts the crucible wall-and thus the area where electrical current passes-it could alter the current path and thus the magnetic field distribution, potentially affecting predictions of arc location. We studied the effects of shrinkage by applying grounded boundary conditions to the contact zone and electrical insulation to the shrinkage gap zone, following the approach of Pericleous et al. [START_REF] Pericleous | A multiscale 3D model of the vacuum arc remelting process[END_REF]. We varied the size of the contact zone from the full length of the ingot to 0.032 m, ranging from zero shrinkage to a contact zone 3 % of the ingot height.

Figure 10 shows the error distribution for predicted arc locations with increasing shrinkage, corresponding to decreasing contact zone height. While error increases slightly as the contact zone shrinks, in general shrinkage causes an median error increase of less than 2 % in predicted arc locations, corresponding to a similar change in magnetic flux density. However, the higher current density resulting from smaller contact zones could affect the z component of magnetic flux density more significantly-although this component does not play a role in the current APS approach based on horizontal (i.e., x and y) components.

Effect of using multiple sensors

Thus far, we only used magnetic field measurements at one sensor location for calculations to determine arc locations. Both our results and those from Woodside et al. [START_REF] Woodside | Arc distribution during the vacuum arc remelting of Ti-6Al-4V[END_REF][START_REF] Woodside | A measurement system for determining the positions of arcs during Vacuum arc remelting[END_REF] show that the error increases as the arc moves away from the sensor. Therefore, for an axisymmetric system, we hypothesize that the results from multiple sensors can be averaged 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. First, we examined the trends in arc location prediction for two separate sensors located at opposite sides of the furnace; Figure 11 shows these (separate) predicted arc locations, compared with the exact locations. Although the sensors predict similar locations for arcs located near the center 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 location from four evenly spaced sensors. The averaging resulted in an even spacial distribution of the arc positions; as Figure 13 shows, the error predicted locations is also evenly distributed compared with that from a single sensor. This information is useful because it can be used to develop correction algorithms to predict arc locations more accurately. Based on these results, more sensors might aid in smoothing the error further; however, evenly distributed results do not guarantee more accurate results-in fact, outliers could 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 error distribution contracts with the addition of sensors, but the predictions do not improve with more than four sensors. Now we can analyze whether using four sensor averaging of arc location predictions aid in the reduction of error for a varying sensor height. In Section 3.1 we calculated how error distribution was affected by the relative vertical position of a single sensor with respect to the electrode-ingot gap. Now, we apply the same methodology while using arc location predictions determined with four-sensor averaging. The results are shown in Figure 15. The total error distribution is smaller than the single-sensor results from Section 3.1 with a maximum error of less than 16 % at the highest and lowest positions. Using four sensors also suppresses the outliers. These results were calculated using varying furnace coefficients; the coefficients were recalculated at every vertical position.

Conclusions

The magnetostatic characteristics of a simplified vacuum arc remelting furnace where studied using the finite element method, in order to determine the effects of various physical phenomena on the resulting magnetic field. This was done to validate a system for predicting the locations of electrical arcs that form in the electrode-ingot gap in these furnaces. Arc distribution throughout the remelting process plays a strong role in the material properties of the produced ingot, so real-time tracking of arc locations can provide a priori indications of ingot quality.

First, we reproduced the prior results of Woodside et al. [START_REF] Woodside | Arc distribution during the vacuum arc remelting of Ti-6Al-4V[END_REF] for predicting arc locations; we matched arc location predictions within 3.5 % of their results. This minor discrepancy likely resulted from slight differences in model setup, solver version, or selection of input arc locations to calculate the furnace coefficients. We then studied the effects of changing certain parameters or eliminating previously made assumptions to validate this overall approach for the arc position sensing technology. Based on our simulations, we drew the following conclusions:

1. Error in predicted arc locations increases substantially as the sensor moves vertically from the electrode-ingot gap;

for distances of 0.508 m above and below the plane of the gap, the maximum error reached around 40 % for constant furnace coefficients and 25 % for adaptive furnace coefficients. Furthermore, as the sensor moves away from the gap, the magnetic flux density decreases and results in clustering of predicted locations near the center of the domain. Overall, out of the parameters and assumptions we studied, we conclude that gap height variation and ingot shrinkage will 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 considered for further development of the arc position sensing technology.

Fig. 1 -

 1 Fig. 1-Cross section of the VAR furnace. Sketch is courtesy of ATI Allvac, with a modification to show instrumentation.

Fig. 1 :

 1 Fig.1: Cross section of VAR furnace. Image taken from Woodside et al.[START_REF] Woodside | Arc distribution during the vacuum arc remelting of Ti-6Al-4V[END_REF]

Fig. 2 :

 2 Fig. 2: Diagram of the geometry employed in the model (not to scale)

Fig. 3 :

 3 Fig. 3: Magnetic flux density norm with respect to arc location in Cartesian coordinates

Fig. 4 :Fig. 5 :

 45 Fig. 4: Overhead cross-sectional diagram of a VAR furnace, shown with four two-axis magnetic field sensors and the geometry of the variables for one sensor

Fig. 6 :

 6 Fig.6: Statistical error distribution of predicted arc locations with respect to vertical sensor position using constant furnace coefficients

Fig. 8 :

 8 Fig.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

Fig. 9 :

 9 Fig. 9: Arc location predictions with varying gap height for a single sensor location

Fig. 10 :

 10 Fig. 10: Error in predicted arc locations with varying contact zone height, mimicking ingot shrinkage

Fig. 11 :Fig. 12 :Fig. 13 :Fig. 14 :

 11121314 Fig. 11: Arc location predictions for two sensors position on opposite sides of the furnace: (0 m, -0.64 m) and (0 m, 0.64 m)

SensorFig. 15 :

 15 Fig. 15: Error distribution of predicted arc locations with respect to vertical sensor position using the average results from four sensors

Table 1 :

 1 Geometric specifications of modeled VAR furnace

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

Table 2 :

 2 Comparison of furnace coefficients from Woodside et al.[START_REF] Woodside | Arc distribution during the vacuum arc remelting of Ti-6Al-4V[END_REF] with those determined here

		m r (N/A 2 )	m t (N/A 2 )
	FEM results	8.15 × 10 -8 4.98 × 10 -8
	Woodside et al. 9 × 10 -8	4.9 × 10 -8

Acknowledgements

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 Woodside, and Matt Cibula of KW Associates LLC for helpful discussions and comments on this paper.