Stochastic Heat Transfer in Fins and Transient Cooling Using Polynomial Chaos and Wick Products

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

Department of Mechanical Engineering, University of Washington, Seattle, WA 98195-2600emery@u.washington.edu

D. Bardot

Department of Mechanical Engineering, University of Washington, Seattle, WA 98195-2600

J. Heat Transfer 129(9), 1127-1133 (Sep 19, 2006) (7 pages) doi:10.1115/1.2739586 History: Received March 22, 2006; Revised September 19, 2006

Stochastic heat transfer problems are often solved using a perturbation approach that yields estimates of mean values and standard deviations for properties and boundary conditions that are random variables. Methods based on polynomial chaos and Wick products can be used when the randomness is a random field or white noise to describe specific realizations and to determine the statistics of the response. Polynomial chaos is best suited for problems in which the properties are strongly correlated, while the Wick product approach is most effective for variables containing white noise components. A transient lumped capacitance cooling problem and a one-dimensional fin are analyzed by both methods to demonstrate their usefulness.

Copyright © 2007 by American Society of Mechanical Engineers
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Figure 1

(a) Measured temporal variation of ambient air temperature. (b) Measured surface heat flux, during cooling compared to predicted.

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Figure 2

Comparison of σ(Θ) with respect to τ for C being a random variable and a stochastic process for α=0.2

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Figure 3

Representation of f(Θ) from Eq. 1 by first- and second-order expansions (Eq. 6) for α=0.2

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Figure 4

Convergence of the polynomial chaos solution to the exact value for C being a random variable

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Figure 5

Illustrating the convergence of the K-L approach to solving Eq. 2 for white noise

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Figure 6

Convergence of the Wick solution to Eq. 1 for white noise

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Figure 7

Typical realizations obtained with the Monte Carlo solution to Eq. 2 for C having a white noise component, α=0.1

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Figure 8

Standard deviations of Φ(ζ) for 10% noise compared to 5000 MC simulations




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