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Research Papers: Conduction

Combining Integral Transforms and Bayesian Inference in the Simultaneous Identification of Variable Thermal Conductivity and Thermal Capacity in Heterogeneous Media

[+] Author and Article Information
Carolina P. Naveira-Cotta, Helcio R. B. Orlande

LTTC—Laboratory of Transmission and Technology of Heat, Mechanical Engineering Department – Escola Politécnica & COPPE,  Universidade Federal do Rio de Janeiro, UFRJ, Cx. Postal 68503—Cidade Universitária, 21945-970 Rio de Janeiro, RJ, Brasilcotta@mecanica.coppe.ufrj.br

Renato M. Cotta1

LTTC—Laboratory of Transmission and Technology of Heat, Mechanical Engineering Department – Escola Politécnica & COPPE,  Universidade Federal do Rio de Janeiro, UFRJ, Cx. Postal 68503—Cidade Universitária, 21945-970 Rio de Janeiro, RJ, Brasilcotta@mecanica.coppe.ufrj.br

1

Corresponding author.

J. Heat Transfer 133(11), 111301 (Aug 31, 2011) (10 pages) doi:10.1115/1.4004010 History: Received December 21, 2009; Revised April 11, 2011; Published August 31, 2011; Online August 31, 2011

This work presents the combined use of the integral transform method, for the direct problem solution, and of Bayesian inference, for the inverse problem analysis, in the simultaneous estimation of spatially variable thermal conductivity and thermal capacity for one-dimensional heat conduction within heterogeneous media. The direct problem solution is analytically obtained via integral transforms and the related eigenvalue problem is solved by the generalized integral transform technique (GITT), offering a fast, precise, and robust solution for the transient temperature field. The inverse problem analysis employs a Markov chain Monte Carlo (MCMC) method, through the implementation of the Metropolis-Hastings sampling algorithm. Instead of seeking the functions estimation in the form of local values for the thermal conductivity and capacity, an alternative approach is employed based on the eigenfunction expansion of the thermophysical properties themselves. Then, the unknown parameters become the corresponding series coefficients for the properties eigenfunction expansions. Simulated temperatures obtained via integral transforms are used in the inverse analysis, for a prescribed concentration distribution of the dispersed phase in a heterogeneous media such as particle filled composites. Available correlations for the thermal conductivity and theory of mixtures relations for the thermal capacity are employed to produce the simulated results with high precision in the direct problem solution, while eigenfunction expansions with reduced number of terms are employed in the inverse analysis itself, in order to avoid the inverse crime. Gaussian distributions were used as priors for the parameter estimation procedure. In addition, simulated results with different randomly generated errors were employed in order to test the inverse analysis robustness.

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Copyright © 2011 by American Society of Mechanical Engineers
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Figures

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Figure 1

Schematic representation of experimental setup for thermophysical properties determination, employed in the direct/inverse problem solutions

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Figure 2

Convergence behavior of the thermal conductivity and heat capacity expansions [original function-solid line, expansions-dashed (four terms), dot-dashed (seven terms), dotted (ten terms) lines]: (a) k(x) Nk  = 4, 7, and 10 and (b) w(x) Nw =4, 7, and 10

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Figure 3

Behavior of priors: (a) filler concentration distribution with 20% standard deviation and resulting function (solid) and expansion (dashed) of (b) thermal capacity and (c) thermal conductivity

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Figure 4

(ad)—Estimated functions in case 1: (a) k(x)—exact (solid thin), exact expanded (solid thick), and estimated (dashed); (b) k(x)—exact (solid thin), confidence bounds (solid thick), and estimated (dashed); (c) w(x)—exact (solid thin), exact expanded (solid thick), and estimated (dashed); (d) w(x)—exact (solid thin), confidence bounds (solid thick), and estimated (dashed)

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Figure 5

(ad)—Estimated functions in case 2: (a) k(x)—exact (solid thin), exact expanded (solid thick), and estimated (dashed); (b) k(x)—exact (solid thin), confidence bounds (solid thick), and estimated (dashed); (c) w(x)–exact (solid thin), exact expanded (solid thick), and estimated (dashed); (d) w(x)—exact (solid thin), confidence bounds (solid thick), and estimated (dashed)

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Figure 6

(ad)—Estimated functions in case 3: (a) k(x)—exact (solid thin), exact expanded (solid thick), and estimated (dashed); (b) k(x)—exact (solid thin), confidence bounds (solid thick), and estimated (dashed); (c) w(x)—exact (solid thin), exact expanded (solid thick), and estimated (dashed); (d) w(x)—exact (solid thin), confidence bounds (solid thick), and estimated (dashed)

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