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Research Papers: Conduction

The Use of Inverse Heat Conduction Models for Estimation of Transient Surface Heat Flux in Electroslag Remelting

[+] Author and Article Information
Alex Plotkowski

Purdue Center for Metal Casting Research,
School of Materials Engineering,
Purdue University,
West Lafayette, IN 47907
e-mail: aplotkow@purdue.edu

Matthew John M. Krane

Purdue Center for Metal Casting Research,
School of Materials Engineering,
Purdue University,
West Lafayette, IN 47907
e-mail: krane@purdue.edu

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received June 10, 2014; final manuscript received October 30, 2014; published online December 2, 2014. Assoc. Editor: Wilson K. S. Chiu.

J. Heat Transfer 137(3), 031301 (Mar 01, 2015) (9 pages) Paper No: HT-14-1396; doi: 10.1115/1.4029038 History: Received June 10, 2014; Revised October 30, 2014; Online December 02, 2014

Three inverse heat conduction models were evaluated for their ability to predict the transient heat flux at the interior surface of the copper mold in the electroslag remelting (ESR) process for use in validating numerical ESR simulations and real-time control systems. The models were evaluated numerically using a simple one-dimensional (1D) test case and a 2D pseudo-ESR test case as a function of the thermocouple locations and sample frequency. The sensitivity of the models to measurement errors was then tested by applying random error to the numerically calculated temperature fields prior to the application of the inverse models. This error caused large fluctuations in the results of the inverse models, but these could be mitigated by implementing a simple Savitzky–Golay filter for data smoothing. Finally, the three inverse methods were applied to a fully transient ESR simulation to demonstrate their applicability to the industrial process. Based on these results, the authors recommend that the 2D control volume method described here be applied to industrial ESR trials.

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References

Figures

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Fig. 4

Results of Stolz and 1D control volume inverse models for 1D test case proposed by Beck [18]

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Fig. 3

Schematic of 1D test case for inverse heat conduction models, after Beck [18]

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Fig. 2

Thermocouple configurations for the three inverse methods with the applicable dimensions labeled: (a) the Stolz method, (b) 1D control volume method, and (c) 2D control volume method

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Fig. 1

Schematic of the ESR process [1]

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Fig. 5

The 2D test case uses predictions from an ESR simulation to construct an approximation of the heat flux into the mold. This heat flux distribution is applied as a moving boundary condition in a separate transient conduction model to predict internal mold temperatures. These temperatures are used by the inverse models to predict the wall heat flux, and the methods are evaluated by comparing these results to the input flux.

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Fig. 6

Results for a 4-cm-mold thickness with sensor depth of 1 cm and sample rate of 1 Hz

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Fig. 7

Comparison of the average error for the three inverse methods as a function of the sensor depth for an inverse sample rate of 1 Hz. Axial spacing for the 2D control volume method was held constant at 2 cm.

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Fig. 8

Comparison of the average error for the Stolz and 1D control volume methods for 4-cm and 8-cm thick molds as a function of sensor depth for an inverse sample rate of 1 Hz

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Fig. 9

Comparison of the average error for the three inverse methods as function of the inverse sample frequency for a sensor depth of 1 cm. Axial spacing for the 2D control volume method was held constant at 2 cm.

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Fig. 10

The average error of the 2D inverse method as a function of the thermocouple spacing in the axial direction for a sensor depth of 1 cm and a sample rate of 1 Hz

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Fig. 11

Results of the Stolz algorithm with added measurement error and two levels of data smoothing using a Savitzky–Golay filter with 11 points, and 25 points. Sensor depth is 1 cm and sample rate is 1.0 Hz.

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Fig. 12

Results of the 1D control volume method with added measurement error and two levels of data smoothing using a Savitzky–Golay filter with 11 points and 25 points. Sensor depth is 1 cm and sample rate is 1.0 Hz.

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Fig. 13

Results of the 2D control volume method with added measurement error and two levels of data smoothing using a Savitzky–Golay filter with 11 points and 25 points. Sensor depth is 1 cm, sample rate is 1.0 Hz, and axial thermocouple spacing is 2 cm.

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Fig. 14

Comparison of the effect of the magnitude of the unsmoothed measurement error on the inverse model results for (a) a standard deviation of 1.1 °C and (b) a standard deviation of 0.5 °C

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Fig. 15

Schematic of the numerical domains for the ESR model. At the beginning of the simulation the initial height of the ingot, H = 17 cm, and grows as the electrode melts.

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Fig. 16

Graph with results of inverse models from ESR simulation

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