VAR modeling applied to process control

Vacuum arc remelting (VAR) is a process used throughout the specialty metals industry to control casting of segregation-sensitive and reactive metal alloy ingots. Of particular importance in the former group are nickel-base superalloys such as Alloy 718, commonly used for forged rotating parts in the hot stages of jet aircraft engines and increasingly in land-based turbines for power generation. Reactive metal alloys include titanium, zirconium and uranium alloys, such as Ti-6Al-4V, which is used for a variety of aerospace applications. VAR also is used to produce various grades of forgeable stainless steel ingots.

Fig 1 Schematic depiction of the vacuum arc remelting process

In the VAR process, a cylindrically shaped alloy electrode is loaded into a water-cooled, copper crucible of a VAR furnace. The furnace is evacuated, and a dc arc is struck between the electrode (cathode) and some start material (e.g. metal chips) at the bottom of the crucible (anode). The arc heats both the start material and the electrode tip, eventually melting both. As the electrode tip is melted away, molten metal drips off and an ingot forms in the copper crucible. Because the crucible diameter is larger than the electrode diameter, the electrode must be translated downward toward the anode pool to keep the mean distance (electrode gap) between the electrode tip and pool surface constant. The objective of VAR is to produce an ingot free of macrosegregation, porosity, shrinkage cavities and other defects associated with uncontrolled solidification during casting. Figure 1 shows a schematic depiction of the VAR process.

Maintaining control of electrode gap is of primary importance in the VAR control problem. Large variations in electrode gap cause variations in the partitioning of arc power between the molten ingot pool surface and the mold wall, which creates transients in the solidification mushy zone thermal gradient, a condition conducive to the formation of melt related defects [1]. It also causes the ingot sidewall quality to vary and destabilizes the ingot-mold contact boundary. The majority of industrial VAR controllers control electrode gap using a closed loop feedback system to adjust electrode drive speed to maintain a drip-short (momentary arc interruption caused by metal drips bridging the electrode gap and contacting the ingot pool surface) frequency [2] or voltage setpoint. This method is relatively effective, problems arising only in response to process upsets that affect arc voltage or the drip-short mechanism.

Fig 2 Schematic depiction of the nonlinear VAR controller layout; Fig 3 Drip-short frequency and electrode gap data during gap control exercise. Effects of physical filtering are evident in the estimated gap data.

Electrode melt rate is also a vitally important control variable. Melt rate variations also produce transients in the solidification mushy zone and cause variations in ingot growth rate. These types of melt transients have been linked to freckle formation [3] in nickel-base superalloys and solidification white spot formation in Alloy 718 [4]. Effective melt rate control is a more challenging problem than gap control. VAR electrode melting dynamics are described by Bertram and Zanner [5], who assumed uniform electrode tip heating and successfully modeled the melting dynamics for a 0.42 m (17 in.) diameter Alloy 718 electrode, a size commonly used in industry. Their simulations demonstrated that approximately one hour is required to reestablish steady-state melting after a step in current in this melting system, because that is the time required for the temperature gradient in the electrode tip to adjust to the new power level. This indicates that the melt rate obtained for a given power level depends on the thermal history of the electrode. Thus, to successfully control melt rate on a time scale that is fast relative to thermal diffusion in the electrode, it is advantageous to have a controller that "understands" these melting dynamics.

This paper describes a method of dynamic VAR process control capable of simultaneously controlling melt rate and electrode gap (U.S. Patent # 6,115,404; 2002). The control method performs well even in regions of the process space characterized by highly transient conditions [6]. The method is designed around an optimal process estimator (or Kalman filter) that incorporates an accurate, low-order model of the process dynamics. A full treatment of the mathematical development underlying the controller is reported in Ref. [7].

The dynamic VAR problem

A low-order process model that can be inverted to find the control inputs is required to design a closed-loop VAR control system (see Ref. 7). Results for melting dynamics can be summarized by two nonlinear differential equations where D is the thermal boundary layer (distance from the tip along the electrode axis to where the temperature falls to e-1 of melt temperature), •S is electrode burn-off rate, ar is room-temperature thermal diffusivity, µ is melting efficiency, VC is the cathode fall voltage [8], I is current, G is electrode gap, Ae is the cross sectional area of the electrode tip, hm is the volume specific enthalpy at melt temperature and R1 and RG are resistance parameters associated with the circuit and electrode gap, respectively [9]. CDD, CDp, CSD and CSp are dimensionless constants for a given material [7]. Their values depend on the latent heat of fusion and sensible superheat, the melt temperature specific enthalpy, and the room and melt temperature thermal conductivities.

Other dynamic equations required to completely specify the dynamic VAR problem are represented in the equation where U is electrode drive velocity and rliq is the density of the molten metal dripping from the electrode at superheat temperature. The area ratio parameter is defined as 1-Ae/AI, where AI is the cross-sectional area of the ingot.

Equations (1), (2), (3) and (5) depend on total power, PT = [VCI + (RI + RGG)I2]. However, only a fraction of the total power (as determined by the process efficiency) goes into melting. Specifying efficiency as part of the dynamic problem is an essential feature of the controller design because it provides a convenient variable to track heat conduction disturbances in the electrode (due to a crack or void, for example). It is treated as a disturbance variable because one has little control over its value. Its time derivative is set to zero in the dynamic problem because its value should not change during normal steady-state melting. This does not mean that estimates of µ generated by the process estimator will never change. If this were true, there would be no point in specifying µ as a process variable. The estimator produces an estimate of µ consistent with the state of the process as defined by all the dynamic state equations. Generally, this is only true if the dynamic system as specified satisfies the requirements of observability.

The area ratio parameter is usually thought of as a system constant, which is only approximately true. It may change during the course of melting due to voids in the electrode or the use of tapered electrodes and molds. Like µ, it too is considered a disturbance variable.

Finally, this formulation assumes that process voltage is adequately described by V = VC +(RI + RGG)I. However, process upsets sometimes affect voltage, causing deviations away from the model behavior. A common example is day-to-day variation in crucible coating thickness when processing large titanium alloy electrodes. It is common practice to coat the crucible (or sometimes the electrode) with salt or titania to facilitate arc control and improve sidewall quality in the casting. Some of the coating material is vaporized into the arc, changing the arc plasma conductivity and, therefore, arc voltage. Voltage bias, Vb, is an upset variable used to account for this voltage change.

Fig 4 (near right) Plots of load cell data, five minute “average” melt rate derived from the load cell data, and melt rate reference during a melt rate step exercise; Fig 5 (far right) Plots of electrode ram position, melt rate reference, and melt rate estimated from the average ram velocity demonstrating feedback melt rate control without electrode weight data

VAR control problem

The above equations in combination with process inputs and output measurements are used to design a VAR process estimator that produces accurate, unbiased estimates of process state variables. The estimates are used as feedback variables for process control. The nonlinear controller development is similar to that described in [10]. The control problem was formulated using state-space methodology and assumes that the plant formulation meets the criteria of controllability and observability, as described in references on control theory and application, such as [11]. The model function includes all the dynamics specified in Equations (1 to 8). Five measurements typically made using the VAR estimator for control are electrode gap, position and weight, and process current and voltage. The calculation requires specification of a linear dynamic model, as well as process and measurement uncertainty terms. Gain elements are chosen to minimize the steady-state covariance of the estimation error given the input and measurement noise covariance matrices.

Figure 2 shows a nonlinear feedback control system for the VAR process. The reference inputs to the controller are melt rate and electrode gap. Functions fP and fI are obtained by solving Equation (5) for current. Function fG is obtained using Equations (3) and (5) to find ram velocity. Note that the estimated gap is fed back to the ram velocity through the feedback gain KG to ensure that the controller drives the system to the gap reference. Without this feedback, one could only guarantee that the controller would hold a gap consistent with the requirements of the melt rate due to uncertainty in the initial state of the process. The measured process outputs are each marked with an asterisk in the figure.

Some test results

Gap control was exercised during an industrial VAR test melting 0.42 m diameter Alloy 718 electrode into 0.51 m (20 in.) diameter ingot. Figure 3 shows three electrode gap steps of 1.0 to 0.8 cm, 0.8 to 0.6 cm and 0.6 to 1.0 cm. The objective of the exercise was to test the controller's ability to run at very tight gaps-average drip-short frequency in the range of 15-20 s-1. The figure shows that control was maintained at the smallest gap and the steps are relatively sharp. Note also that the noise in the estimated gap is much smaller than that in the "measured" gap, evidence that the Kalman filter is doing its job. That is, the filter "knows" the process dynamics and automatically places boundaries on the gap data reflecting the possible range of fluctuations determined by the system physics. This is very important because gap data obtained from voltage or drip-short based measurement models are typically very noisy; they require a great deal of averaging before a control decision can be made.

Figure 4 shows a melt rate control exercise carried out during the same VAR melt as the gap exercises in Fig. 3. The melt rate was ramped from 60 to 90 g/s at a linear rate of 3.0 g/s/min, held for a little more than 40 min, then ramped back to 60 g/s at a rate of 15.0 g/s/min. It is noteworthy that there is no significant overshoot or undershoot of the target melt rates at the end of the ramps, indicating that the melting dynamics are correctly described by the model. Regression of the load cell data in the region of the 90 g/s hold gave a melt rate of 91.5 g/s with a regression coefficient of 0.9999. Values for the ramp rates were calculated from second-order coefficients found by fitting the appropriate regions of the load-cell data with second-order polynomials. The results for the two ramps are 2.7 g/s/min (R = 0.998) and 16 g/s/min (R = 0.96). The agreement is considered very good relative to the performance of conventional melt rate controllers based solely on electrode weight feedback.

The controller commands involve estimates of µ, G and D, but not M. Thus, M may be removed from the state vector, and the estimator equations reformulated without an electrode-weight measurement. This provides a means of feedback control for melt rate that does not require a measurement of electrode weight. An electrode weight measurement does lead to improved performance, but is not necessary. This is important because many older VAR furnaces in use throughout the specialty metals industry are not equipped with load-cell transducers to measure electrode mass. Figure 5 shows electrode position data, melt rate reference, and melt rate derived from position data for VAR of 0.36 m (14 in.) diameter Ti-6Al-4V electrode into 0.46 m (18 in.) diameter ingot. The data were acquired from a furnace not equipped with a means of measuring electrode mass. The duration of the initial ramp from 60 to 90 g/s was eight minutes, and the final ramp from 90 g/s to 60 g/s was five minutes. Melt rate may be related to ram velocity assuming a constant electrode gap. The relationship is represented in the equation where _ is an experimental correction term for the area ratio parameter found to be ~0.94 for this application. The position data were smoothed for one minute before being used to calculate melt rate. Although this method produces rather noisy results, reasonable melt rate control is maintained throughout the melt although some slight overshoot is apparent between ~8.9 and ~9.1 hours.

Conclusions

This work demonstrates a VAR process control method that achieves highly responsive melt rate control under normal melting conditions with minimal fluctuations in power, even under conditions where the electrode temperature distribution has been driven away from, or has not yet achieved, steady state. The method allows extremely dynamic melt rate control when required, such as during start-up or power cutback. Process efficiency is effective as an electrode temperature distribution disturbance variable when controlling the VAR process. Good electrode gap control is established and maintained. IH

Acknowledgment

References

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