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Chinese Physics B
Chin. Phys. B, 2020, Vol. 29(11): 110202    DOI: 10.1088/1674-1056/aba608
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An efficient inverse approach for reconstructing time- and space-dependent heat flux of participating medium
Shuang-Cheng Sun(孙双成)1,2, †, Guang-Jun Wang(王广军)1,2, and Hong Chen(陈红)1,2$
1 School of Energy and Power Engineering, Chongqing University, Chongqing 400044, China
2 Key Laboratory of Low-grade Energy Utilization Technologies and Systems, Ministry of Education, Chongqing University, Chongqing 400044, China
Abstract  

The decentralized fuzzy inference method (DFIM) is employed as an optimization technique to reconstruct time- and space-dependent heat flux of two-dimensional (2D) participating medium. The forward coupled radiative and conductive heat transfer problem is solved by a combination of finite volume method and discrete ordinate method. The reconstruction task is formulated as an inverse problem, and the DFIM is used to reconstruct the unknown heat flux. No prior information on the heat flux distribution is required for the inverse analysis. All retrieval results illustrate that the time- and space-dependent heat flux of participating medium can be exactly recovered by the DFIM. The present method is proved to be more efficient and accurate than other optimization techniques. The effects of heat flux form, initial guess, medium property, and measurement error on reconstruction results are investigated. Simulated results indicate that the DFIM is robust to reconstruct different kinds of heat fluxes even with noisy data.
Keywords:  decentralized fuzzy inference      surface heat flux reconstruction      inverse heat transfer problem      participating medium  
Received:  04 May 2020      Revised:  26 May 2020      Accepted manuscript online:  15 July 2020
Fund: the Natural Science Foundation of Chongqing (CSTC, Grant No. 2019JCYJ-MSXMX0441).
Corresponding Authors:  †Corresponding author. E-mail: scsun@cqu.edu.cn   

Cite this article: 

Shuang-Cheng Sun(孙双成), Guang-Jun Wang(王广军), and Hong Chen(陈红)$ An efficient inverse approach for reconstructing time- and space-dependent heat flux of participating medium 2020 Chin. Phys. B 29 110202

Fig. 1.  

Schematic diagram of CRCHT in 2D participating medium.
Parameter Lx/m Ly/m λ/W⋅m−1⋅K−1 κa/m−1 κs/m−1 n εw Φ
Value 1.0 1.0 2.268, 22.68, 226.8 1.0 0.0 1.0 1.0 1.0
Table 1.  

Parameters in the verification case.
Fig. 2.  

Verification of the FVM–DOM solution for solving CRCHT problem in 2D participating medium.
Fig. 3.  

Membership functions of (a) fuzzy input set and (b) fuzzy output set.
em,n NB NM NS ZO PS PM PB
um,n PB PM PS ZO NS NM NB
Table 2.  

Fuzzy rules of DFIM.
Fig. 4.  

Fuzzy inference system of DFIM.
Fig. 5.  

Flowchart of DFIM for reconstructing time- and space-dependent heat flux.
Fig. 6.  

Reconstruction results of heat flux Q1 using DFIM. (a) Exact distribution, (b) reconstructed results, and (c) relative error distribution.
Optimization technique Average relative error/% Maximum relative error/% Iteration number Computational time/h
DFIM 0.0002 0.0092 46 0.3
SQP 0.0003 0.0229 306 86.2
CGM 0.0010 0.1341 698 153.3
L-M 0.0015 0.2056 717 172.9
PSO 44.8900 287.7838 10000 23.8
GA 59.2025 392.2087 10000 36.1
Table 3.  

Reconstruction results of time- and space-dependent heat flux using different optimization techniques.
Fig. 7.  

Retrieval results of heat flux Q2. (a) Exact distribution, and (b) reconstruction result.
Fig. 8.  

Retrieval results of heat flux Q3. (a) Exact distribution, and (b) reconstruction result.
Fig. 9.  

Retrieval results of heat flux Q4. (a) Exact distribution, and (b) reconstruction result.
Fig. 10.  

Retrieval results of heat flux Q5. (a) Exact distribution, and (b) reconstruction result.
Heat flux Average relative error/% Maximum relative error/% Iteration number Computational time/s
Q1 0.0002 0.0092 46 921.1
Q2 0.0002 0.0099 46 922.7
Q3 0.0001 0.0028 45 919.8
Q4 0.0001 0.0030 46 922.3
Q5 0.0001 0.0075 46 923.9
Table 4.  

Reconstruction results of different kinds of heat fluxes.
Initial guess/W⋅m−2 Average relative error/10−6 Computational time/s
Q1 Q2 Q3 Q4 Q5 Q1 Q2 Q3 Q4 Q5
0 1.40 1.97 1.07 0.92 0.79 923.2 923.5 921.0 923.0 921.5
5000 1.38 1.96 1.06 0.93 0.83 921.5 926.4 923.4 921.6 919.4
10000 1.50 1.92 1.05 0.92 0.77 921.1 922.7 919.8 922.3 923.9
30000 1.53 1.90 1.03 0.91 0.83 923.0 921.3 920.1 925.6 921.6
50000 1.99 1.93 1.02 0.91 0.81 922.4 925.4 920.4 922.9 924.1
100000 2.21 2.00 1.16 1.05 0.88 921.6 924.3 922.6 921.7 923.8
Table 5.  

Reconstruction results for different initial guesses.
ρ/kg⋅m−3 cp/J⋅kg−1⋅ K−1 λ/W⋅m−1⋅ K−1 κa/m−1 κs/m−1 n
Ceramic 5600 456 1.4 300 12000 1.8
Glass 2219 840 1.025 1244 0 1.4
Table 6.  

Parameter of ceramic and glass.[1]
Fig. 11.  

Reconstruction results of Q3 in ceramic and glass industries. (a) Exact and reconstructed heat fluxes, (b) relative error distribution of ceramic, and (c) relative error distribution of glass.
Measurement error Average relative error/% Maximum relative error/% Computational time/s
0.01 0.0203 0.1560 923.1
0.05 0.1062 0.6897 921.9
0.1 0.2115 1.3735 922.1
0.5 1.0208 6.9042 919.6
1.0 2.0916 12.2036 921.5
Table 7.  

Retrieval relative errors for different measurement errors.
Fig. 12.  

Reconstruction results of heat flux Q3 for different measurement errors. (a) σ = 0.01, (b) σ = 0.05, (c) σ = 0.1, (d) σ = 0.5, and (e) σ = 1.0.
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