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TECHNICAL PAPERS: Conduction

Combined Parameter and Function Estimation With Application to Thermal Conductivity and Surface Heat Flux

[+] Author and Article Information
Tahar Loulou

Centre de Recherche, Université de Bretagne Sud, LET2E, Rue Saint Maude, F-56321 Lorient Cedex, Francetahar.loulou@univ-ubs.fr

J. Heat Transfer 129(10), 1309-1320 (Feb 09, 2007) (12 pages) doi:10.1115/1.2755064 History: Received December 01, 2005; Revised February 09, 2007

This is a numerical and experimental study on the combined parameter and function estimation. The determination of thermal conductivity and the surface heat flux is an illustration of combined estimation of one parameter and one function by means of the conjugate gradient method with vectorial descent parameter. The experimental example developed herein uses one set of good data obtained by Beck and Arnold (1977, Parameter Estimation in Engineering and Science, Wiley, New York). For this case, two measured temperatures in the solid are used to illustrate combined estimation. The unknown boundary condition and thermal conductivity of this solid were satisfactorily reconstructed and a good enough comparison is demonstrated between the known and estimated unknowns. The temperature data of Beck and Arnold are found to be excellent. Also, it is shown that the developed approach is general, stable, powerful, and able to process a wide variety of heat transfer problems where a simultaneous estimation is unavoidable.

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Copyright © 2007 by American Society of Mechanical Engineers
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Figures

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Figure 1

Configuration C1. Recovered surface heat flux for the four combinations of sensors: Tc1‐Tc5,…,Tc4‐Tc8. To alleviate the presentation of all heat flux profiles, the results are shifted one by one toward the top.

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Figure 2

Temperature residuals of the obtained results with the four combinations of sensors: Tc1‐Tc5, Tc2‐Tc6, Tc3‐Tc7, and Tc4‐Tc8 and in the case of configuration C1

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Figure 3

Recovered surface heat flux by using all measurements and resulting temperature residuals in the case of configuration C1. All measurements mean that temperatures were averaged on each side of the solid.

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Figure 4

Measured and calculated temperatures for averaged measurement example in the case of configuration C1

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Figure 5

Recovered heat surface flux with and without regularization in the case of configuration C1 obtained with sensor combination Tc4‐Tc8

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Figure 6

Sequential estimation of thermal conductivity with averaged measurements in the case of configuration C1

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Figure 7

Recovered surface heat flux for different experimental times in the case of configuration C1 and using average combination of measurements. The displayed profiles are shifted up one by one to get a better graphical presentation.

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Figure 8

Temperature residuals for different experimental times in the case of configuration C1 and using average combination of measurements. For the sake of clarity, residuals are shifted up one by one.

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Figure 9

Recovered surface heat flux and temperature residuals by using parametric form and averaged measurements

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Figure 10

Measured and recomputed temperatures when the parametric form of q(t) is used

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