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Research Papers: Heat and Mass Transfer

Iterative Multiscale Approach for Heat Conduction With Radiation Problem in Porous Materials

[+] Author and Article Information
Ronen Haymes

The Energy Engineering Division,
Ben-Gurion University of the Negev,
Beer-Sheva 84105, Israel
e-mail: ronenhay@post.bgu.ac.il

Erez Gal

Department of Structural Engineering,
Ben-Gurion University of the Negev,
Beer-Sheva 84105, Israel
e-mail: erezgal@bgu.ac.il

1Corresponding author.

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received December 15, 2017; final manuscript received February 13, 2018; published online April 11, 2018. Assoc. Editor: Laurent Pilon.

J. Heat Transfer 140(8), 082002 (Apr 11, 2018) (17 pages) Paper No: HT-17-1752; doi: 10.1115/1.4039420 History: Received December 15, 2017; Revised February 13, 2018

This paper describes a thermal homogenization approach to the application of a multiscale formulation for heat conduction with radiation problems in a porous material. The suggested formulation enables to evaluate the effective macroscopic thermal conductivity, based on the microscopic properties such as porosity, and can also provide the microscopic radiosity heat flux, based on the macroscopic temperature gradient field. This is done by scaling up and down between the microscopic and macroscopic models according to the suggested methodology. The proposed methodology involves a new iterative upscaling procedure, which uses heat conduction at macroscopic problem and heat transfer by conduction and radiation at microscopic problem. This reduces the required computational time, while maintaining the required level of accuracy. The suggested multiscale formulation has been verified by comparing its results with reference finite element (FE) solutions of porous (filled with air) materials examples; the results shows excellent agreement (up to 5% discrepancy) with reference solutions. The efficiency of the suggested formulation was demonstrated by solving a full-scale engineering transient problem.

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Figures

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Fig. 1

Periodic porous materials: (a) domain Ωξ and (b) unit cell Θ*

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Fig. 2

Application of the suggested thermal multiscales analysis for steady-state heat conduction with radiation problem

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Fig. 3

Porous material structure (left), unit cell model (middle), and macroscopic structure (right), with their boundary conditions

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Fig. 4

Physical properties of the air: (a) thermal conductivity, (b) specific heat capacity, and (c) density, as a function of temperature

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Fig. 5

Visualization images for influence functions (H) of the heat fluxes, for the two first two guesses in the: (a) x direction, (b) y direction, and (c) z direction, for unit cell 1 and 2

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Fig. 6

Application of the suggested iterative thermal homogenization for steady-state heat conduction with radiation problem in the first example

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Fig. 7

Temperature (°C) distribution results: (a) reference model, (b) macroscopic model, (c) middle surface of the reference model, and (d) middle surface of the macroscopic model

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Fig. 8

Radiosity (j) (W/m2) results: (a) first row of the reference model, (b) unit cell 1 model, (c) second row of the reference model, and (d) unit cell 2 model

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Fig. 9

porous material structure (left), unit cell models (middle), and macroscopic structure (right), together with their boundary conditions

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Fig. 10

Visualization images for influence functions (H) of the heat fluxes, for the first two guesses in the: (a) x direction, (b) y direction, and (c) z direction, for unit cell 2

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Fig. 11

Application of the suggested iterative thermal homogenization for steady-state heat conduction with radiation problem in the second example

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Fig. 12

Temperature (°C) distribution results: (a) reference model, (b) macroscopic model, (c) middle surface of the reference model, and (d) middle surface of the macroscopic model

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Fig. 13

Radiosity (j) (W/m2) results: (a) first row of the reference model, (b) unit cell 1 model, (c) second row of the reference model, and (d) unit cell 2 model

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Fig. 14

Macromodel (size 3 × 2 m), which represents the FE model of the engineering problem (without details), and the porous material unit cell of the wall (size 300 × 200 × 200 mm, with air spaces, diameter ∅150 mm), with their boundary conditions

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Fig. 15

Heat fluxes in August and January in the Middle East

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Fig. 16

Ambient temperatures in August and January in the Middle East

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Fig. 17

Macroscopic effective thermal conductivity (kỹ) as a function of time (24 h)

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Fig. 18

Microscopic radiosity heat flux (j) (W/m2) results compared to those obtained by the reference solution, at (a) 8:00, (b) 12:00, and (c) 18:00 during January

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Fig. 19

Microscopic radiosity heat flux (j) (W/m2) results compared to those obtained by the reference solution, at (a) 8:00, (b) 12:00, and (c) 18:00 during August

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