Research Papers: Conduction

Predicting Thermal System Performance and Estimating Parameters for Systems Burdened With Uncertainties and Noise Using Hierarchical Bayesian Inference

[+] Author and Article Information
A. F. Emery

Department of Mechanical Engineering,
University of Washington,
Seattle, WA 98195
e-mail: emery@u.washington.edu

D. Bardot

Medical Device Innovation Consortium,
1550 Utica Avenue South, Suite 725,
St. Louis Park, MN 55416
e-mail: dbardot@mdic.org

1Corresponding author.

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received June 14, 2012; final manuscript received August 14, 2013; published online November 15, 2013. Assoc. Editor: Oronzio Manca.

J. Heat Transfer 136(3), 031301 (Nov 15, 2013) (9 pages) Paper No: HT-12-1287; doi: 10.1115/1.4025640 History: Received June 14, 2012; Revised August 14, 2013

The precision of estimates of system performance and of parameters that affect the performance is often based upon the standard deviation obtained from the usual equation for the propagation of variances derived from a Taylor series expansion. With ever increasing computing power it is now possible to utilize the Bayesian hierarchical approach to yield improved estimates of the precision. Although quite popular in the statistical community, the Bayesian approach has not been widely used in the heat transfer and fluid mechanics communities because of its complexity and subjectivity. The paper develops the necessary equations and applies them to two typical heat transfer problems, measurement of conductivity with heat losses and heat transfer from a fin. Because of the heat loss the probability distribution of the conductivity is far from Gaussian. Using this conductivity distribution for the fin gives a very long tailed distribution for the heat transfer from the fin.

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Fig. 1

Illustrating the mode, mean and 95% high density region (credible interval) for z

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Fig. 2

Schematic of conductivity measuring experiment

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Fig. 3

(a) The posterior f(k∧) for no losses, (b) the posterior f(k∧) for losses, 0≤FL≤0.1, (c) f(k∧) from α for σ(α) = 10% and σ(ρcp)=20 %

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Fig. 4

Distributions of Qw N = normal distribution, U = uniform

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Fig. 5

Distributions of Qe N = normal distribution, U = uniform

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Fig. 6

(a) Accuracy of sampling the distribution of k∧ for the fin, (b) 1000 samples of k∧ compared to Fig. 3(b), (c) 10,000 samples of k∧ compared to Fig. 3(b)

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Fig. 7

(a) f(Qw) based on 10,000 MC samples (b) f(Qw) based on 10,000 MCMC samples

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Fig. 8

Comparison of exact and Gauss quadrature results hL and hE normally distributed as a function of for different distributions of k∧ (for k-N, the curves are coincident) U is for σ(k) = 30%

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Fig. 9

Changing nature of f(Qw) for uniform f(hL) and f(k) as the variation in hL approaches that of k, σ(k) = 10%



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