Research Papers: Conduction

The Effect of Spatially Correlated Roughness and Boundary Conditions on the Conduction of Heat Through a Slab

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

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

H. Dillon, A. M. Mescher

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

J. Heat Transfer 132(5), 051301 (Mar 08, 2010) (11 pages) doi:10.1115/1.4000445 History: Received February 02, 2009; Revised October 01, 2009; Published March 08, 2010; Online March 08, 2010

The nominally one-dimensional conduction of heat through a slab becomes two dimensional when one of the surfaces is rough or when the boundary conditions are spatially nonuniform. This paper develops the stochastic equations for a slab whose surface roughness or convective boundary condition is spatially correlated with correlation lengths ranging from 0 (white noise) to a length long in comparison to the slab thickness. The effect is described in terms of the standard deviation and the resulting spatial correlation of the heat flux as a function of depth into the slab. In contrast to the expectation that the effect is monotonic with respect to the correlation length, it is shown that the effect is maximized at an intermediate correlation length. It is also shown that roughness or a random convective heat transfer coefficient have essentially the same effects on the conducted heat, but that the combination results in a much deeper penetration than does each effect individually. In contrast to the usual methods of solving stochastic problems, both the case of a rough edge and a smooth edge with stochastic convective heat transfer coefficients can only be treated with reasonable computational expense by using direct Monte Carlo simulations.

Copyright © 2010 by American Society of Mechanical Engineers
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Figure 1

Characteristic air and fiber diameter histories: (a) laminar, (b) oscillatory, and (c) chaotic flow regimes (note the different scales)

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Figure 2

Schematic of a slab with a rough surface

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Figure 3

Contours of qx(x,y) for W=1, ΔT=100, and k=1

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Figure 4

Number of vectors N using Eq. 4 needed to achieve 99.9% of the true variance as a function of the correlation length L/H (the number approaches infinity as L→0 (white noise))

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Figure 5

Principal component analysis for L/H=0.2

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Figure 6

An element to represent a rough edge

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Figure 7

Variation of terms in the conductance matrix, Eq. 7: (a) kx(2,3) and (b) ky(3,4)

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Figure 8

(a) Convergence for L=0.075. For meshes with degrees of freedom ranging from 1000 to 68,000. (b) Effect of the number of edge elements on the value of σ(qx) at x=0.1W and y=0.5H.

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Figure 9

Contours of the standard deviations of the heat fluxes for an edge roughness of 2% and L=0.05W: (a) σ(qx) and (b) σ(qy)

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Figure 10

(a) Effect of L on the behavior of qx for an edge roughness of 2%. (b) Simplified grid for quadrilateral elements for the rough edge.

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Figure 11

(a) behavior of qx and qy and (b) correlation length

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Figure 12

Comparing the effects of a rough edge and a stochastic convective heat transfer coefficient

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Figure 13

Comparing the case of h(0)=10 and h(0)=h(W)=10

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Figure 14

Depth of penetration at which σ(qx)=σ(qx(for L=∞))

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Figure 15

Effect of L/W on σ(qx) for a rough edge alone and combined with a stochastic heat transfer coefficient

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Figure 16

Correlation depth L for a rough edge and for a stochastic heat transfer coefficient when L/W=0.125 at x=0

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Figure 17

Probability density distributions at x=0.1W for a rough edge (a) and a stochastic heat transfer coefficient (b) when L/W=0.125 at x=0




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