Temperature Profile Optimization in a New Robotic Local Induction Heat Treatment Process

TEMPERATURE PROFILE OPTIMIZATION IN A NEW ROBOTIC LOCAL INDUCTION HEAT TREATMENT PROCESS

Mathieu Gendron1, Éric Boudreault2, Bruce Hazel2, Henri Champliaud1, Xuan-Tan Pham1

1École de technologie supérieure, 1100 rue Notre-Dame Ouest, Montréal, QC J4K 2T9, Canada 2Institut de recherche d’Hydro-Québec, 1800 boulevard Lionel Boulet, Varennes, QC J3X 1S1,

Canada

Keywords: Induction heating modeling, Optimization, Finite element method

Abstract Post-weld heat treatment (PWHT) restores mechanical properties and relieves internal stress present in hydraulic turbine runners assembled by welding. For many aging turbines, PWHT is needed to ensure high-quality repairs. It is impossible, however, to dismantle large runners or other such steel equipment and take it to a furnace. A new robotic process was thus developed to perform local induction heat treatment in situ. Heat is generated by moving an induction coil back and forth over a specific area. The critical factor in this process is to maintain a uniform temperature profile within a precise range. A method combining finite element analysis and optimization is proposed to set heating parameters. Through a series of stationary and transient thermal analyses, the heating path and parameters are optimized to achieve a uniform temperature profile. The method was tested by performing robotic PWHT in the laboratory. Results show good agreement between simulations and measurements.

Introduction A hydraulic turbine runner is an assembly of different cast and machined parts welded

together. It is well known that welding generates high internal stresses, reduced yield strain and has a negative impact on the fatigue life of many steel alloys. To restore initial properties (and reduced residual stresses), manufacturers place the assembly in a huge furnace to perform heat treatment.

Boudreault et al. developed a new robotic process for in situ heat treatment on large steel components. They used a coil-type portable induction heating source mounted on the Scompi robotic arm [1] to perform similar post-weld heat treatment (PWHT) on site. This technique provides the opportunity to perform safe crack repairs on aging equipment. Recent work on robotic induction heating is presented by Fisk [2] and Flextrol [3].

Some heat treatments, such as those performed on CA6NM stainless steel, require that the temperature be kept within a very narrow range. As suggested by Robichaud [4] and Song et al. [5] heat treatments on CA6NM stainless steel are performed by manufacturers at between 610°C and 630°C. To do this, Boudreault et al. [1] proposed moving the pancake coil back and forth as fast as the manipulator permits. Good results were obtained with this method but the heat-treated zone was small compared to the path of the coil.

To improve the system, this paper presents a new strategy to extend the heat-treated zone and make the temperature profile more uniform. First, finite element analyses (FEA) are used to optimize pancake coil geometry. Second, path parameters are related to coil geometry to compute the optimal distance between back-and-forth paths and longitudinal power modulation.

To simplify the study, heat treatment duration is considered assumed long enough to be close to the steady-state condition. Given this assumption, optimizations are performed in steady

Materials Science and Technology (MS&T) 2013October 27-31, 2013, Montreal, Quebec, Canada

state and validated in transient state with FEA and test measurements. The difference between FEA results and mean temperatures from testing is compensated with a feedback loop. Godin et al. [6] present an application of this method.

Finite Element Formulation

Finite element thermal analysis is formulated based on the approach in Cook [7]. Solving the heat equation in which every parameter is temperature-dependent would require excessive computing capacity. Boudreault et al. [1] assert that the solver must be fast enough to perform faster-than-real-time simulation. To achieve this, assumptions make the system linear on one time step:

1. First, thermal properties are constant enough over a single time step.

2. The radiation heat loss 𝑒 (W/m2) is computed with Eq. 1. Second, the time step is assumed small enough to make Tn+1 equal to Tn in hrad coefficient computing.

𝑒 = ℎ𝑟𝑎𝑑(𝑇𝑛+1 − 𝑇𝑓𝑙) and ℎ𝑟𝑎𝑑 = 𝜀𝜎(𝑇𝑛2 + 𝑇𝑓𝑙2)(𝑇𝑛 + 𝑇𝑓𝑙) (1) where Tfl is the air temperature, hrad the radiation coefficient, ε the emissivity and σ the Stefan-Boltzmann constant. The formulation used is given by Eq. 2 and 3.

�1∆𝑡

[𝐶(𝑇𝑛)] + 𝛽({𝐾(𝑇𝑛)} + {𝐻(𝑇𝑛)})� {𝑇}𝑛+1 = {𝐵(𝑇𝑛)} + 𝛽��𝑅𝑄(𝑇𝑛+1)�+ {𝑅ℎ(𝑇𝑛)}� (2)

{𝐵(𝑇𝑛)} = �1∆𝑡

[𝐶(𝑇𝑛)] − (1 − 𝛽)([𝐾(𝑇𝑛)] + [𝐻(𝑇𝑛)]� {𝑇}𝑛 + (1 − 𝛽)��𝑅𝑄(𝑇𝑛)� + {𝑅ℎ(𝑇𝑛)}� (3)

Δt is the time step, n the time step index, {T} the temperature vector, {RQ} the heat input vector, {Rh} the heat transfer vector, [H] the heat lost matrix, [K] the conductivity matrix and [C] the specific heat matrix. For stability, factor β in the Newmark Beta scheme is set to 0.5.

Path Modeling Paths are defined with a sequence of connected cubic arcs, with continuity of second derivative. The method is based on the approached detailed by Hazel et al. [8]. In summary, each arc is defined by an initial and final position/velocity couple (P0, P1 and V0, V1) at an initial and final time t0 and t1. For each cubic arc segment, position P and velocity V can be interpolated at time t with the general Eq. 4. The parametric coordinate u is related to time using 𝑢 = (𝑡 −𝑡0)/∆𝑡, where ∆𝑡 = 𝑡1 − 𝑡0 and 0 ≤ 𝑢 ≤ 1. 𝑃 = 𝑃𝑜𝑒1(𝑢) + 𝑃1𝑒2(𝑢) + ∆𝑡𝑉𝑜𝑒3(𝑢) + ∆𝑡𝑉1𝑒4(𝑢) (4)

where: 𝑒1(𝑢) = 2𝑢3 − 3𝑢2 + 1, 𝑒2(𝑢) = −2𝑢3 + 3𝑢2

𝑒3(𝑢) = 𝑢3 − 2𝑢2 + 𝑢, 𝑒4(𝑢) = 𝑢3 − 𝑢2 (5)

An example definition of a cubic arc is provided in Figure 1 (right). Figure 1 (left) illustrates the path use to generate the temperature profile. Compared to the path used by Boudreault et al. [1], an interpass distance e between back-and-forth paths is included to improve lateral temperature profile control (along the Y-axis).

Figure 1. Coil trajectory (left) and cubic arc definition and 𝑑𝑡𝑖(𝑥𝑖, 𝑦𝑖) calculation (right)

Power density approximation

As exposed in [1], computing the temperature profile with sufficient precision using the real displacement of the coil is impossible in real-time simulations on a laptop. Given an adequate coil velocity, the heat injected over the entire coil path could be modeled by an average source. For a given finite element eli, the power density 𝑞𝑖 is assumed proportional to the time interval 𝑑𝑡𝑖 that the coil spends over the element, relative to the time dttot the coil takes to cover one back-and-forth path (Eq. 6). Based on [1], the coil power density distribution is assumed constant within a specific annular heating volume.

𝑞𝑖(𝑥𝑖 ,𝑦𝑖) = 𝑄𝑜𝑢𝑡𝑑𝑡𝑖(𝑥𝑖, 𝑦𝑖)𝑑𝑡𝑡𝑜𝑡

𝑎𝑛𝑑 𝑄𝑜𝑢𝑡 = 𝑄𝑖𝑛𝑒𝑒𝑒 (6)

where Qin is the input power and eff the system efficiency. For example, in Figure 1 (right), for a coil radius R, 𝑑𝑡𝑖(𝑥𝑖,𝑦𝑖) for 𝑒𝑙𝑖(𝑥𝑖,𝑦𝑖) is the time difference between instant tA and tB. Parametric positions at these times, uA and uB, are located at the intersection of a circle of radius R, centered in (𝑥𝑖,𝑦𝑖), and the cubic arc. As a general formulation, a given intersection in parametric coordinate uk is found using the cubic arc Eq. 8 (from Eq. 4) and the circle Eq. 7.

[𝑥𝑘 𝑦𝑘] �1/𝑅2 00 1/𝑅2

� �𝑥𝑘𝑦𝑘� + [−2𝑥𝑖 −2𝑦𝑖] �

𝑥𝑘𝑦𝑘� + 𝑥𝑖2 + 𝑦𝑖2 − 1 = 0 (7)

�𝑥𝑘𝑦𝑘� = �

𝑃𝑥0𝑃𝑦0

� 𝑒1(𝑢𝑘) + �𝑃𝑥1𝑃𝑦1

� 𝑒2(𝑢𝑘) + ∆𝑡 ��𝑉𝑥0𝑉𝑦0

� 𝑒3(𝑢𝑘) + �𝑉𝑥1𝑉𝑦1

� 𝑒4(𝑢𝑘)� (8)

P0[Px0, Py0]

P1[Px1, Py1]

V0[Vx0, Vy0]

V1[Vx1, Vy1]

eli(xi, yi)

Rint Pancake coil

uA(xA, yA)

uB(xB, yB)

Putting Eq. 8 in Eq. 7, a sixth order polynomial equation, the function of uk, is obtained. In fact, all intersection coordinates are found to be the real roots, between 0 and 1, of this polynomial equation. Once the parametric positions uk are known, it is easy to related them to time, as seen previously for Eq. 4. Taking into account the lack of heating in the middle of the coil, the inside radius time interval for each element is subtract from the outside radius time interval. Thus, roots are found for both the external radius Rout and the internal radius Rint. Validation The heat input power distribution model is validated in the steady state. The temperature profile is assumed nearly constant over time after 2 hours. Experiments are performed on a 305 x 267 x 64 mm UNS41500 steel plate. Temperatures are measured with thermocouples and an infrared camera. Figure 2 shows results for an e = 15 mm path. The measurements shown are average temperatures over the last 10 back-and-forth cycles.

Figure 2. Comparison between simulations and measurements for e = 15 mm

In the zone directly heated by the coil (HZ), the deviation between thermocouples and computed temperature is generally less than 7°C at 640°C, which is a relative deviation of about 1.1%. Similar results are found for other configuration. This numerical power density model offers a good approximation of the temperature profile.

Path Planning The objective is to keep the temperature in a specific area as uniform as possible and as close as possible to a target temperature Ttarget. This is achieved when the root-mean-square temperature deviation at every selected node i in this area is minimized (Eq. 9).

min 𝜑(𝜀) =1𝑗��T�𝑄𝑜𝑢𝑡,𝑅𝑒𝑥𝑡,𝑅𝑖𝑛𝑡, 𝑒�i

− T𝑡𝑎𝑟𝑔𝑒𝑡�2

The temperature at a specific node i, T(𝑄𝑜𝑢𝑡,𝑅𝑒𝑥𝑡,𝑅𝑖𝑛𝑡, 𝑒)i, depends on the output power, and on coil and path parameters. This temperature is obtained by solving Eq. 3. A conjugated gradient algorithm [9] is combined to the thermal finite element solver to optimize these parameters. To simplify the problem, coil geometry and interpass distance are optimized separately. Moreover,

-130-110 -90 -70 -50 -30 -10 10 30 50 70 90 110 130

Position y (mm)

Lateral temperature profile

T. measured T. computedT. Camera |Deviation|

-150-125-100 -75 -50 -25 0 25 50 75 100 125 150

Position x (mm)

Longitudinal temperature profile

T. measured T. computedT. Camera |Deviation|

assuming a constant temperature profile along the X-axis, only the node on a single row following Y-axis is selected. The optimization area is a rectangular surface defined by a width A and a length B, following the X-axis and Y-axis. Coil Geometry Optimization The coil’s internal and external radii, Rint and Rext are optimized for a straight back-and-forth path, without interpass distance. For this configuration, the lateral temperature profile is only impacted by the coil geometry. Analysis of the lateral temperature profile shows that the smaller the internal radius, the more uniform the temperature profile. Thus, Rint is set to the tubing’s minimum bending radius, 13 mm. Figure 3 shows the lateral temperature profile corresponding to the optimal outside radius obtained on different optimization area widths.

Figure 3. Lateral temperature profile for different optimization zone widths (A)

The coil size must be small enough to ensure manoeuvrability on complex surfaces and large enough to cover a zone at least 25.4 mm wide (average groove width for crack repair). An external radius of 37 mm meets these conditions and also generates the most uniform lateral temperature profile (smallest 𝜑(𝜀) value). Optimization of Distance between Back-and-Forth Paths (e) The widest uniform temperature profile is obtained by optimizing e over a 50.8 mm wide zone. The optimal interpass distance e found for this area is 70 mm. Moreover, other analyses show that for zones wider than 25.4 mm, the global minimum for Eq. 9 is always about twice the outside radius. As show in figure 4, for a target of 620 ±10 °C, the interpass configuration almost doubled the heat treated width. Moreover, due to a highest power requirement, the treated depth is also doubled.

0 0.01 0.02 0.03 0.04 0.05 610

Position y (m)

A = 25.4 mm; φ(ε) = 0,39; Rext = 37 mm A = 38.1 mm; φ(ε) = 2,94; Rext = 39 mm A = 50.8 mm; φ(ε) = 3,36; Rext = 57 mm A = 76.2 mm; φ(ε) = 3,74; Rext = 67 mm A = 101.6 mm; φ(ε) = 5,34; Rext = 74 mm

A = 25.4 mm

Figure 4. Lateral profile uniformization

Longitudinal Temperature Profile Optimization The previous configuration with the interpass distance set to 70 mm causes an overheated zone near where the path ends. To avoid theses hot spots, output power must be reduced in the turn-around section. Since the path is straight and symmetric, power only needs to be adjusted along the longitudinal axis. For each element, the output power 𝑄𝑜𝑢𝑡 in (6) is then expressed as follows:

𝑄𝑜𝑢𝑡(𝑥𝑖,𝑦𝑖) = 𝑄𝑖𝑛 ∗ 𝑒𝑒𝑒𝑖(𝑥𝑖, 𝑦𝑖) = 𝑄𝑖𝑛 ∗1

(𝑢𝐵 − 𝑢𝐴)∗ � 𝑒𝑒𝑒(𝑢)𝑑𝑢

𝑢𝐵

𝑢𝐴 (10)

As illustrated in Figure 5 (left), the efficiency function eff(u) is characterized by a series of effn point, linked by linear functions. All points are optimized to minimized Eq. 9, for nodes selected along the X-axis.

Figure 5. Height optimization schema (left) and optimal power/height profile (right)

Due to induction power supply bandwidth constraints, instead of varying the system input power, power is adjusted by changing the distance between the coil and the plate. An empirical relationship is used to related output power and height. For the present voltage operating range, good results are obtained assuming linear behavior. Figure 6 (left) compares the numerical

0.00 0.01 0.02 0.03 610 612 614 616 618 620 622 624 626 628 630

Position y (m)

E = 0 mm, P = 1987 W E = 70 mm, P = 3037 W

0123456789

-0.10 -0.05 0.00 0.05 0.10 0.0

Position x (m)

Qout(x) and h(x)

Qout(x) h(x)

eli(xi, yi)

eff1 eff3 effn eff2

≈ 33 mm

≈ 6.5 mm

≈ 18 mm

≈ 3 mm

Hot spot

longitudinal (X-axis) profile with and without power adjustment. Figure 5 (right) compares the optimal power profile 𝑄𝑜𝑢𝑡(𝑥) found with FEA, and its corresponding height.

Figure 6. Longitudinal (X-axis) uniformization (left) and PWHT experimental setup (right)

Application: Post-Weld Heat Treatment

PWHT heat treatment is performed on UNS S41500 steel plates. To reproduce a crack repair, a groove is made and filled with 410 NiMo filler metal. The aim is to keep the weld zone (WZ) at 620±10°C for 60 minutes. Details are presented by Godin et al. [6]. The numerical method described in this paper is used to set parameters. Those parameters are used as feed forward into a PID feedback loop. The simulated profiles are verified on a test plate instrumented with thermocouples along the X- and Y-axes, before the real treatment. Figure 6 (right) shows the experimental setup and WZ position. Figure 7 compares measured and computed temperatures. The target surface temperature is set to the upper tolerance (630°C) to increase the heat-treated depth.

Figure 7. Longitudinal (X-axis) and lateral (Y-axis) temperature profiles

The maximum temperature deviation is about 16°C near the hot spot. Otherwise, the difference between the numerical and measured temperature is less than 10°C or 1.6%. This error can be attributed to thermocouple position error and numerical assumptions.

0.00 0.02 0.04 0.06 0.08 0.10610

Position x (m)

Eff = 0.5

Eff = eff(x)

ΔT ≈

02468101214161820

0.00 0.02 0.04 0.06 0.08 0.10 Tem

Position x (m)

02468101214161820

0 0.02 0.04 0.06 0.08 Tem

Position y (m)

Pancake coil

6 dof robotic arm

Weld zone

Conclusion The experimental results prove that the numerical optimization method does make the temperature profile more uniform over an extended heat-treated zone. First, pancake coil geometry optimization shows that a 13 mm inside radius and 37 mm outside radius generate the most uniform profile across a 25.4 mm wide zone for a straight back-and-forth path (e = 0 mm). Second, an optimal interpass distance of 69 mm is found. Including this interpass in the back-and-forth paths doubles the heat-treated width and depth. Third, output power is optimize along the longitudinal axis to achieve a more uniform temperature profile. Lastly, the numerical method is shown to be effective in generating path and heating parameters for laboratory PWTH. These results are obtained using an optimized distance for the back-and-forth path. Optimization is performed using an averaging source model implemented in an in-house FEA software. Based on measured temperature profiles, the power density approximation represents reality. Further work will be done to control and optimize the orientation of the pancake coil. In addition, to demonstrate that the method can be applied to complex shapes like that of a hydraulic turbine runner, experiments will be performed on a plate of complex shape.

Acknowledgment The authors would like to thank who contributed to the project, and specifically Alexandre Lapointe, Clément Choquet for their help with the experiments. They also thank the NSERC.

References

1. Boudreault, E., Hazel, B., Godin, S., Côté, J., A new robotic process for in situ heat treatment on large steel components, in ASME Power conference 2013: Boston, Massachusetts, USA. 2. Fisk, M., Lundbäck, A., Simulation and validation of repair welding and heat treatment of an alloy 718 plate. Finite elements in Analysis and Design, Vol. 58, 2012. 3. Ruffini, R.T., Nemkov, V. S., Goldstein, R. C., Influence of magnetic flux Controllers on Induction Heating Systems, Computer Simulation and Practice, ASM. 2001. 4. Robichaud, P., Stabilité de l'austénite résiduelle de l'acier inoxydable 415 soumis à la fatigue oligocyclique, in Mechanical Engineering, École de technologie supérieur: Montréal, 2007. 5. Song, Y.Y., Li, X. Y., Rong, L. J., Li, Y. Y., The influence of tempering temperature on the reversed austenite formation and tensile properties in Fe-13%Cr-4%Ni-Mo low carbon martensitic stainless steels. Materials Science & Engineering, vol. 528: p. 4075-4079, 2011. 6. Godin, S., Boudreault, E, Hazel, B., On-site post weld heat treatment of weld made of 410 NiMo steel, in MS&T COM 2013: Montreal, Canada, 2013. 7. Cook, R.D., Concepts and applications of finite element analysis, Wiley, 2011. 8. Hazel, B., Côté, J., Laroche Y., Mongenot, P., A portable, multiprocess, track-based robot for in situ work on hydropower equipment. Journal of field robotics, vol. 29: p. 69-101, 2012. 9. Galassi, M. and B. Gough, GNU Scientific Library: Reference Manual, Network Theory Limited, 2009.

Temperature Profile Optimization in a New Robotic Local Induction Heat Treatment Process

Documents

DEPARTMENT PROFILE

Robotic Radiosurgery - Volume 1: Index

Video System in Robotic Applications

A bio-robotic approach

Class Profile

Alliance Profile

Robotic Software Integration Using MARIE

PDE-based robust robotic navigation

Robotic Automation

Subsurface robotic exploration for geomorphology

Robotic Wireless Sensor Networks - Unpaywall

Induction Program

Mathematical Induction and Induction in Mathematics

Robotic cardiac surgery: overview

Adaptive Homing--Robotic Exploration Tours

Robotic Machining White Paper Project

Robotic Fettling Project

Vision Based Robotic Gesture Recognition

In ovo delivery of Newcastle disease virus conjugated hybrid calcium phosphate nanoparticle and to study the cytokine profile induction

Prequalification Profile