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Research Papers

Inverse Analysis of Radiative Flux Maps for the Characterization of High Flux Sources

[+] Author and Article Information
Clemens Suter, Antoine Torbey Meouchi, Gaël Levêque

Laboratory of Renewable Energy
Science and Engineering,
École Polytechnique Fédérale de Lausanne,
Lausanne CH 1015, Switzerland

Sophia Haussener

Mem. ASME
Laboratory of Renewable Energy
Science and Engineering,
École Polytechnique Fédérale de Lausanne,
Lausanne CH 1015, Switzerland
e-mail: sophia.haussener@epfl.ch

1Corresponding author.

Contributed by the Solar Energy Division of ASME for publication in the JOURNAL OF SOLAR ENERGY ENGINEERING: INCLUDING WIND ENERGY AND BUILDING ENERGY CONSERVATION. Manuscript received September 7, 2018; final manuscript received November 22, 2018; published online January 8, 2019. Guest Editors: Tatsuya Kodama, Christian Sattler, Nathan Siegel, Ellen Stechel.

J. Sol. Energy Eng 141(2), 021011 (Jan 08, 2019) (13 pages) Paper No: SOL-18-1422; doi: 10.1115/1.4042227 History: Received September 07, 2018; Revised November 22, 2018

The reconstruction of the angular and spatial intensity distribution from radiative flux maps measured in high flux solar simulators (HFSS) or optical concentrators is an ill-posed inverse problem requiring special solution strategies. We aimed at providing a solution strategy for the determination of intensity distributions of arbitrarily complicated concentrating facilities. The approach consists of the inverse reconstruction of the intensities from multiple radiative flux maps recorded at various positions around the focal plane. The approach was validated by three test cases including uniform spatial, Gaussian spatial, and uniform angular distributions for which we successfully predicted the intensity for a square-shaped target with edge length of 0.5 m and for a displacement range spanning ±1.5 m at a resolution of 3.2 × 106 elements, yielding relative errors between 19.8–26.4% and 15.7–25.6% when using Tikhonov regularization and the conjugate gradient least square (CGLS) method, respectively. The latter method showed superior performance and was used at a resolution of 2.35 × 107 elements to analyze EPFL's HFSS comprising 18 lamps. The inverse solution for a single lamp retrieved from experimentally measured and simulated radiative flux maps showed peak intensities of 13.7 MW/m2/sr and 16.0 MW/m2/sr, respectively, with a relative error of 81.1%. The inverse reconstruction of the entire simulator by superimposing the single lamp intensities retrieved from simulated flux maps resulted in a maximum intensity of 18.8 MW/m2/sr with a relative error of 80%. The inverse method proved to provide reasonable intensity predictions with limited resolution of details imposed by the high gradients in the radiative flux maps.

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Figures

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Fig. 2

Schematic L-curve for solution norm versus residual norm on log-log scale: (1) Tikhonov regularization with α ranging from 0 to ∞, optimum at α*, and (2) CGLS method with iteration number ranging from 0 to ∞, optimum at it*

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Fig. 1

(a) Schematic back view of the 45 kWel HFSS at EPFL consisting of 18 lamps arranged in two concentric circles. The black point denotes the center (x =0, y =0). (b) Schematic longitudinal section of the HFSS and the target (plane) denoted by the solid line in the focal plane (z =0). The dashed-dotted line indicates the optical axes (z-axis) of the HFSS. The dashed lines show the optical axis of the individual lamps. The dotted lines depict the positions of the target off the focal plane (planes closer to HFSS for z >0, planes farer away from HFSS for z <0).

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Fig. 3

Normalized reference intensity and inversely determined intensity by Tikhonov regularization method as a function of r=x2+y2 for three test cases: (a) test case (i) of uniform distribution, (b) test case (ii) of Gaussian distribution, and (c) test case (iii) of uniform angular distribution. The reference intensity (solid lines) ranging from r =0 to 0.4 m and the inverse intensity (dots with error bars) at r =0, 0.1, 0.141, 0.2, 0.224 and 0.283 m are shown. The error bars represent the averaging over multiple coordinates (x, y) on the plane with equal r. The relative errors equal 19.8%, 20.6%, and 26.4% for the test cases (i), (ii), and (iii), respectively.

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Fig. 4

L-curve (solution norm versus residual norm) for Gaussian test case presenting: inverse solution by (a) Tikhonov regularization and (b) CGLS method. Optimum solutions for (a) α* = 0.039 and (b) it* = 270.

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Fig. 5

Relative error as a function of number of targets for the three test cases and the CGLS solution

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Fig. 6

Solution strategy for inverse analysis of EPFL's HFSS: The Monte Carlo ray-tracing simulations provide reference intensities for validation and simulated radiative flux maps, serving as model input yielding inverse intensities from simulated radiative flux maps; experimental setup provides measured radiative flux maps, serving as model input yielding inverse intensities from measured radiative flux maps

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Fig. 11

L-curve (solution norm versus residual norm) for inverse solution for lamp 3 for Nx = Ny = 20, (a) Nz = 11, Nθ = Nφ = 8 and it* = 74, (b) Nz = 11, Nθ = Nφ = 32 and it* = 72, (c) Nz = 26, Nθ = Nφ = 32 and it* = 81, and (d) Nz = 51, Nθ = Nφ = 32 and it* = 86. Input radiative flux maps were based on Monte Carlo ray-tracing simulations.

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Fig. 12

Normalized inversely determined intensity I0 of lamp 3 as a function of solid angle at focal plane for Nx = Ny = 20, Nz = 51, Nθ = Nφ = 32 at (a) x =2 cm, y =0 cm and I0,max = 5.9 MW/m2/sr, and (b) x =0 cm, y =2 cm and I0,max = 7.1 MW/m2/sr, both for ε = 0.811 and it* = 86. Input radiative flux maps were based on Monte Carlo ray-tracing simulations.

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Fig. 13

(a) Normalized inversely determined intensity as a function of solid angle for lamp 3 at focal plane in the center (x =0 cm, y =0 cm) with I0,max = 13.7 MW/m2/sr, and (b) L-curve (solution norm versus residual norm) for lamp 3 for Nx = Ny = 20, Nz = 51, and Nθ = Nφ = 32 with it* = 30. Input radiative flux maps were based on experimental radiative flux measurements.

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Fig. 7

Normalized radiative flux map of lamp 3 as a function of x and y at focal plane for Nx = Ny= 20. (a) Simulated radiative flux map with qmax = 0.93 MW/m2 and (b) experimentally measured radiative flux map with qmax = 0.96 MW/m2.

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Fig. 8

Normalized reference intensity I0 of lamp 3 as a function of solid angle at focal plane in the center (x =0 cm, y =0 cm) for Nx = Ny= 20, (a) Nθ = Nφ = 8 and I0,max = 21.1 MW/m2/sr, and (b) Nθ = Nφ = 32 and I0,max = 33.1 MW/m2/sr. The intensity was retrieved from Monte Carlo ray-tracing simulations.

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Fig. 9

Normalized reference intensity I0 of lamp 3 as a function of solid angle at focal plane off-center for Nx = Ny = 20, Nθ = Nφ = 32 at (a) x =2 cm, y =0 cm and I0,max = 28.8 MW/m2/sr, and (b) x =0 cm, y =2 cm and I0,max = 30.5 MW/m2/sr. The intensity was retrieved from Monte Carlo ray-tracing simulations.

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Fig. 10

Normalized inversely determined intensity I0 of lamp 3 as a function of solid angle at focal plane in the center (x =0 cm, y =0 cm) for Nx = Ny = 20, (a) Nz = 11, Nθ = Nφ = 8 and I0,max = 15.4 MW/m2/sr, ε = 0.760 and it* = 74, (b) Nz = 11, Nθ = Nφ = 32 and I0,max = 15.9 MW/m2/sr, ε = 0.813 and it* = 72, (c) Nz = 26, Nθ = Nφ = 32 and I0,max = 16.2 MW/m2/sr, ε = 0.813 and it* = 81, and (d) Nz = 51, Nθ = Nφ = 32 and I0,max = 16.0 MW/m2/sr, ε = 0.811 and it* = 86. Input radiative flux maps were based on Monte Carlo ray-tracing simulations.

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Fig. 14

Normalized intensity I0 as a function of solid angle for entire HFSS at focal plane in the center (x =0 cm, y =0 cm) for Nx = Ny = 20 and Nθ = 90, Nφ = 192. (a) Reference intensity retrieved from Monte Carlo ray-tracing simulations with I0,max = 35.2 MW/m2/sr. (b) Inversely determined intensity retrieved from radiative flux maps based on Monte Carlo ray-tracing simulations with I0,max = 18.8 MW/m2/sr. The inverse intensity was reconstructed by superposition of solutions for each single lamp.

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