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Research Papers: Forced Convection

Estimation of Unknown Boundary Heat Flux in Laminar Circular Pipe Flow Using Functional Optimization Approach: Effects of Reynolds Number

[+] Author and Article Information
Peng Ding

School of Energy and Power Engineering, Xi’an Jiaotong University, Xi’an 710049, China

Wen-Quan Tao

School of Energy and Power Engineering, Xi’an Jiaotong University, Xi’an 710049, Chinawqtao@mail.xjtu.edu.cn

J. Heat Transfer 131(2), 021701 (Dec 12, 2008) (9 pages) doi:10.1115/1.3013825 History: Received September 16, 2007; Revised September 12, 2008; Published December 12, 2008

An inverse forced convection problem was studied in this paper. The unknown space-dependent heat flux at the outer boundary of a circular pipe was identified from the temperature measurements within the flow using the algorithm based on an improved conjugate gradient method, which is a combination of the modified inverse algorithm proposed by Ozisik (Huang and Ozisik, 1992, “Inverse Problem of Determining Unknown Wall Heat Flux in Laminar Flow Through a Parallel Plate Duct  ,” Numer. Heat Transfer, Part A21, pp. 2615–2618) and the general inverse algorithm based on the conjugate gradient method. The effects of the convection intensity, the number of thermocouples, the location of the thermocouples, and the measurement error on the performance of the modified inverse algorithm method and the improved inverse algorithm were studied thoroughly through three examples. It is shown that the improved inverse algorithm can greatly improve the solution accuracy in the entire computation domain. The accuracy and stability of both the modified inverse algorithm method and the improved inverse algorithm are strongly influenced by the Reynolds number and the shape of the unknown heat flux. Those functions, which contain more high-frequency components of Fourier series, are more sensitive to the increase in the Reynolds number.

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Copyright © 2009 by American Society of Mechanical Engineers
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Figures

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Figure 1

Schematic view of the pipe system and position of the thermocouples. The velocity profile is fully developed and the inlet temperature is constant.

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Figure 2

Three test cases considered in this study. (a), (b), and (c) correspond to case (a), case (b), and case (c), respectively.

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Figure 3

Computation results for case (a) and case (b) with Rmea=0.6R0, σ=0.0 and 800 measurement points, and Re=100. (a) and (b) correspond to the convergence characteristic of the end point value Q(L) when using the modified inverse algorithm, (c) and (d) correspond to the wall heat flux Q(X) identified by the improved inverse algorithm.

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Figure 4

The effects of the number of thermocouples on the accuracy of estimation for case (c), Re=100. (a) and (b) correspond to the wall heat flux Q(X) identified by the improved inverse algorithm.

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Figure 5

Computation results for case (a) with measurement errors of σ=0.1 and σ=0.19 and Re=100. (a) corresponds to the wall heat flux Q(X) identified by the improved inverse algorithm with σ=0.1, (b) corresponds to the wall heat flux Q(X) identified by the improved inverse algorithm with σ=0.19.

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Figure 6

Computation results for cases (a) and (b) with Rmea=0.6R0, σ=0.0 and 800 measurement points, and Re=1000. (a) and (b) correspond to the convergence characteristic of the end point value Q(L) when using the modified conjugate gradient method. (c) and (d) correspond to the wall heat flux Q(X) identified by the improved inverse algorithm.

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Figure 7

Some computation results for the case of Re=2000. (a) corresponds to the wall heat flux Q(X) identified by the improved inverse algorithm. (b) corresponds to the wall heat flux Q(X) identified by the improved inverse algorithm with σ=0.19.

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