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Radiative imaging of combustion flame in furnace of power plant plays an increasingly important role in combustion diagnosis. The flame radiation image taken by a charge-coupled device (CCD) camera can reconstruct three-dimensional flame temperature distribution in the furnace. CCD cameras are used for capturing the flame images to obtain the line-of-sight radiation intensities. The temperature reconstruction matrix equation is a seriously pathological equation. Thus the temperature field reconstruction problem is an ill-posed problem. The two algorithms (Tikhonov regularization and truncated singular value decomposition (TSVD)) for solving the temperature field reconstruction are introduced. The size of the numerical simulation system is 10 m × 10 m × 10 m, which is divided into 10 × 10 × 10 volume elements in the three dimensions. Each volume element is a unit cube. Generalized cross-validation (GCV) is used to select the correct regularization parameter. The measured data are simulated by adding different random errors to the exact solution of the direct problem. The reconstructed temperature deviation is calculated by the two algorithms separately. When the measuring errors are 0.05 and 0.10, the reconstruction errors based on Tikhonov are respectively 19.3% and 7.0%, less than those based on TSVD. When the measuring errors are 0, 0.01, 0.03 and 0.07, the differences between the two kinds of errors are all less than 3%. Both the algorithms can reconstruct the correct temperature field. The times required to reconstruct the temperature field by the two algorithms are compared and their effects of the maximum temperature are also compared. When the measuring errors are 0, 0.01, 0.03, 0.05, 0.07 and 0.1, the reconstruction times based on Tikhonov are respectively-0.0917,-0.049, 0.161, 0.002, 0.135 and 0.091 s, shorter than the reconstruction times based on TSVD. There is singular value decomposition (SVD) in TSVD. And this process takes more than 2 s. If the problem is more complicated, SVD takes much more time. The errors of the maximum reconstruction temperature under Tikhonov are smaller. And the position of the maximum reconstruction temperature under Tikhonov is near the position of the exact maximum temperature in space. The maximum reconstruction temperature under TSVD is not so good as that under Tikhonov. Preliminary results indicate that the Tikhonov-based reconstruction is slightly better than the TSVD-based reconstruction, especially in reconstruction error, reconstruction time, and effects of the maximum temperature.
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Keywords:
- temperature reconstruction /
- Tikhonov regularization /
- truncated singular value decomposition /
- random
[1] Zhou H C 2005 Furnace Flame Visual Inspection Principle and Technology (Beijing: Science Press) p2 (in Chinese) [周怀春 2005 炉内火焰可视化检测原理与技术(北京: 科学出版社) 第2页]
[2] Liu D, Wang F, Yan J H 2008 Int. J. Heat Mass Transfer 51 3434
[3] Wang F, Huang Q X, Liu D 2008 Energy Fuels 22 1731
[4] Smart J, Lu G, Yan Y 2010 Combust. Flame 157 1132
[5] Yan Y, Lu G, Colechin M 2002 Fuel 81 647
[6] Huang Q X, Wang F, Yan J H 2013 Opt. Commun. 292 2530
[7] Feng Y X, Huang Q X Liang J H 2012 Acta Phys. Sin. 61 134702 (in Chinese) [冯云霄, 黄群星, 梁军辉 2012 61 134702]
[8] Han S D, Zhou H C, Sheng F 2000 Proc. CSEE 20 6771 (in Chinese) [韩曙东, 周怀春, 盛锋 2000 中国电机工程学报 20 6771]
[9] Zhou H C, Han S D, Sheng F 2003 J. Chin. Soc. Power Eng. 23 2154 (in Chinese) [周怀春, 韩曙东, 盛锋 2003 动力工程学报 23 2154]
[10] Lou C, Zhou H C 2007 Proc. CSEE 27 5256 (in Chinese) [娄春, 周怀春 2007 中国电机工程学报 27 5256]
[11] Cheng Q, Zhang X Y, Wang Z C 2014 Heat Transfer Eng. 35 770
[12] Huang Q X, Liu D, Wang F 2007 Acta Phys. Sin. 56 6742 (in Chinese) [黄群星, 刘冬, 王飞 2007 56 6742]
[13] Liu D, Wang F, Huang Q X 2007 Proc. CSEE 9 7277 (in Chinese) [刘冬, 王飞, 黄群星 2007 中国电机工程学报 9 7277]
[14] Phillips D L 1962 J. Assoc. Comput. Mach. 9 8497
[15] Tikhonov A N 1963 Soviet Math. Dokl. 4 1035
[16] Tang Y, Prieur C, Girard A 2015 Automatica 57 110
[17] Rajan M P, Reddy G D 2015 Appl. Math. Comput. 259 412
[18] Hansen P C 1990 J. Sci. Stat. Comput. 11 503
[19] Shea J D, Veen B, Hagness S C 2012 Trans. Biomed. Eng. 59 936
[20] Wu Z M, Bian S F, Xiang C B 2013 Math. Probl. Eng. 2013 161834
[21] Zhou H C, Han S D, Sheng F 2002 J. Quant. Spectrosc. Radiat. Transfer 72 361
[22] Wang F, Liu D, Cen K F 2008 J. Quant. Spectrosc. Radiat. Transfer 109 2171
[23] Hansen P C 2007 Regularization Tools version 4.0 for Matlab 7.3 manual (e-book) pp65-66
[24] Mottershead J E, Friswell M I, Ahmadian H 1998 16th International Modal Analysis Conference Santa Barbara, CA, February 2-5, 1998
[25] Hansen P C 1992 SIAM Rev. 34 561
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[1] Zhou H C 2005 Furnace Flame Visual Inspection Principle and Technology (Beijing: Science Press) p2 (in Chinese) [周怀春 2005 炉内火焰可视化检测原理与技术(北京: 科学出版社) 第2页]
[2] Liu D, Wang F, Yan J H 2008 Int. J. Heat Mass Transfer 51 3434
[3] Wang F, Huang Q X, Liu D 2008 Energy Fuels 22 1731
[4] Smart J, Lu G, Yan Y 2010 Combust. Flame 157 1132
[5] Yan Y, Lu G, Colechin M 2002 Fuel 81 647
[6] Huang Q X, Wang F, Yan J H 2013 Opt. Commun. 292 2530
[7] Feng Y X, Huang Q X Liang J H 2012 Acta Phys. Sin. 61 134702 (in Chinese) [冯云霄, 黄群星, 梁军辉 2012 61 134702]
[8] Han S D, Zhou H C, Sheng F 2000 Proc. CSEE 20 6771 (in Chinese) [韩曙东, 周怀春, 盛锋 2000 中国电机工程学报 20 6771]
[9] Zhou H C, Han S D, Sheng F 2003 J. Chin. Soc. Power Eng. 23 2154 (in Chinese) [周怀春, 韩曙东, 盛锋 2003 动力工程学报 23 2154]
[10] Lou C, Zhou H C 2007 Proc. CSEE 27 5256 (in Chinese) [娄春, 周怀春 2007 中国电机工程学报 27 5256]
[11] Cheng Q, Zhang X Y, Wang Z C 2014 Heat Transfer Eng. 35 770
[12] Huang Q X, Liu D, Wang F 2007 Acta Phys. Sin. 56 6742 (in Chinese) [黄群星, 刘冬, 王飞 2007 56 6742]
[13] Liu D, Wang F, Huang Q X 2007 Proc. CSEE 9 7277 (in Chinese) [刘冬, 王飞, 黄群星 2007 中国电机工程学报 9 7277]
[14] Phillips D L 1962 J. Assoc. Comput. Mach. 9 8497
[15] Tikhonov A N 1963 Soviet Math. Dokl. 4 1035
[16] Tang Y, Prieur C, Girard A 2015 Automatica 57 110
[17] Rajan M P, Reddy G D 2015 Appl. Math. Comput. 259 412
[18] Hansen P C 1990 J. Sci. Stat. Comput. 11 503
[19] Shea J D, Veen B, Hagness S C 2012 Trans. Biomed. Eng. 59 936
[20] Wu Z M, Bian S F, Xiang C B 2013 Math. Probl. Eng. 2013 161834
[21] Zhou H C, Han S D, Sheng F 2002 J. Quant. Spectrosc. Radiat. Transfer 72 361
[22] Wang F, Liu D, Cen K F 2008 J. Quant. Spectrosc. Radiat. Transfer 109 2171
[23] Hansen P C 2007 Regularization Tools version 4.0 for Matlab 7.3 manual (e-book) pp65-66
[24] Mottershead J E, Friswell M I, Ahmadian H 1998 16th International Modal Analysis Conference Santa Barbara, CA, February 2-5, 1998
[25] Hansen P C 1992 SIAM Rev. 34 561
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