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# Hybrid Method to Calculate Direct Exchange Areas Using the Finite Volume Method and Midpoint Intergration

[+] Author and Article Information
Weixue Tian

Department of Mechanical Engineering,  University of Connecticut, Storrs, CT 06269-3139

Wilson K. Chiu

Department of Mechanical Engineering,  University of Connecticut, Storrs, CT 06269-3139wchiu@engr.uconn.edu

J. Heat Transfer 127(8), 911-917 (Jan 27, 2005) (7 pages) doi:10.1115/1.1929786 History: Received March 29, 2004; Revised January 27, 2005

## Abstract

This paper presents a hybrid method to calculate direct exchange areas for an infinitely long black-walled rectangular enclosure. The hybrid method combines the finite volume method (FVM) with the midpoint integration scheme. Direct numerical integration of direct exchange areas for adjacent and overlapping zones is difficult because of singularities in the integrand. Therefore, direct exchange areas of adjacent and overlapping zones are calculated using the FVM. Direct exchange areas of nonadjacent zones are calculated by the efficient midpoint integration scheme. Thus, direct exchange areas in an infinitely long enclosure can be obtained with both efficiency and accuracy. Volume-volume direct exchange areas of zones with various aspect ratios and optical thickness have been calculated and compared to exact solutions, and satisfactory results are found.

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## Figures

Figure 1

Schematic of volume i and volume j zone in an infinite rectangular duct

Figure 2

Schematic of an infinitely small volume i to volume j zone

Figure 3

Change of η and Δτ with τij for 5% error

Figure 4

Schematic of adjacent zones for direct exchange areas calculation using the FVM

Figure 5

Enclosure with 10×10 zones to calculate direct exchange areas to zone 0

Figure 6

Midpoint integration and its error for τL=τH=1 and Δτ=0.1: (a) comparison of direct exchange areas by midpoint integration and exact solution; and (b) relative error by midpoint integration

Figure 7

Finite volume method and its error for τL=τH=1 and Δτ=0.1: (a) comparison of direct exchange areas by FVM and exact solution; and (bb) relative error by FVM

Figure 9

Change of maximum error with varying aspect ratio, 10×10 zones, τL=0.5 and τH varying from 0.5 to 2.5: (a) maximum relative error of nonadjacent direct exchange areas by midpoint integration; and (b) maximum relative error of adjacent direct exchange areas by FVM

Figure 8

Change in maximum error with optical thickness, 10×10 zones and τL=τH varying from 0.1 to 9: (a) maximum relative error of nonadjacent direct exchange areas by midpoint integration; and (b) maximum relative error of adjacent direct exchange areas by FVM

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