Modified electrical capacitance tomography linear inversion algorithms for cryogenic fluids two-phase flow measurement

doi:10.11949/0438-1157.20190313

Abstract

Abstract:

Based on the dielectric properties of liquid and vapor phases of cryogenic fluids, the modified linear inversion algorithms of electrical capacitance tomography (ECT) are obtained by introducing a correcting deviation into the linear approximate equation of the traditional capacitance-permittivity distribution function. The 8-electrode capacitance tomography sensor was used as the geometric model to simulate the two-phase flow in the liquid oxygen-oxygen tube. The original linear algorithm, the modified linear algorithm and the total variation 1 norm regularity were compared under four different flow patterns. The inversion results of four different flow patterns of original linear algorithms, modified linear algorithms and non-linear algorithms represented by total variation L ¹-norm regularization (TV L ¹-norm) are compared. The correlation coefficient and image error are taken to measure the quality of the inversion results. The results of numerical experiments show that compared with the original linear algorithms, the quality of the inversion results is greatly improved by the modified linear inversion algorithm. The TV L ¹-norm takes more than 10 times as much time as the modified linear inversion algorithm. And when 3% measurement noise is applied, the non-linear algorithm shows strong sensitivity to measurement noise. The modified linear algorithms have better anti-noise ability and accuracy of inversion results.

Key words: two-phase flow, measurement, cryogenic fluids, tomography, linear inversion algorithm

摘要：

针对低温流体液、气两相的介电特性，通过向传统的电容-介电分布函数关系的线性近似方程中引入修正偏量，得到了修正的电容层析成像线性反演算法。以8电极电容层析成像传感器作为几何模型，对液氧-氧气管内两相流进行了模拟，对比了4种不同流型下原始线性算法、修正的线性算法及以全变分1范数正则化为代表的非线性算法的反演结果。采用相关性系数及图像误差定量衡量了反演结果的质量，结果表明，修正线性反演算法显著提升了反演结果的质量，全变分1范数正则化的求解耗时是修正线性算法的10倍以上。当赋以3%测量噪声时，非线性算法表现出对噪声极强的敏感性，修正线性算法具备更好的抗噪能力和反演结果准确性。

关键词: 两相流, 测量, 低温流体, 成像, 线性反演算法

CLC Number:

O 359.1

Huangjun XIE, Hong CHEN, Xu GAO, Xudong ZHENG, Xiaobin ZHANG. Modified electrical capacitance tomography linear inversion algorithms for cryogenic fluids two-phase flow measurement[J]. CIESC Journal, 2019, 70(9): 3320-3328.

谢黄骏, 陈虹, 高旭, 郑旭东, 张小斌. 应用于低温流体两相流测量的修正电容层析成像线性反演算法[J]. 化工学报, 2019, 70(9): 3320-3328.

Figures/Tables 13

Fig.1 Geometric structure of eight-electrode ECT sensor

Fig.2 Mesh scheme of computational domain for inversion problem

Table 1 Relative permittivity of different fluids at 1 atm（101325 Pa）[28]

流体介质	相对介电常数
水/空气 (300 K温度下)	77.747/1.0005
液氮/氮气(77 K温度下)	1.4337/1.0021
液氧/氧气 (90 K温度下)	1.4877/1.0016
液化/气态甲烷 (112 K温度下)	1.6299/1.0020

Fig.3 Pre-set phase distribution

Fig.4 Variation of Rxy with regularization parameter

Fig.5 Variation of IE (%) with regularization parameter

Fig.6 Comparisons of inversion results calculated by original linear algorithms, modified linear algorithms and TV L 1-norm algorithm

Fig.7 Rxy of inversion results

Fig.8 IE of inversion results

Table 2 Computational time of inversion algorithms/s

Algorithms	Case 1	Case 2	Case 3	Case 4
TR	0.395	0.388	0.398	0.399
MTR	0.420	0.402	0.427	0.400
ITR	0.613	0.605	0.649	0.614
MITR	0.680	0.704	0.734	0.728
SIRT	0.487	0.425	0.525	0.411
MSIRT	0.570	0.549	0.580	0.515
TV L ¹-norm	7.999	7.954	7.925	7.978

Fig.9 Comparisons of inversion results calculated by original linear algorithms, modified linear algorithms and TV L 1-norm algorithm under noise interference

Fig.10 Comparisons of Rxy for inversion results of noise-contaminated and noise-free

Fig.11 Comparisons of IE for inversion results of noise-contaminated and noise-free

References 32

1	杨道业, 施源, 徐锌锋 . 基于双截面ECT的气/固两相流参数检测系统[J]. 仪器仪表学报, 2013, 34(9): 1968-1974.
	Yang D Y , Shi Y , Xu X F . Parameter measurement system for gas/solid two-phase flow based on twin-plane electrical capacitance tomography [J]. Chinese Journal of Science Instrument, 2013, 34(9): 1968-1974.
2	Yang W Q . Sensors and instrumentation for monitoring and control of multi-phase separation [J]. Measurement and Control, 2006, 39(6): 178-184.
3	罗琴, 赵银峰, 叶茂, 等 . 电容层析成像在气-固流化床测量中的应用[J]. 化工学报, 2014, 65(7): 2504-2512.
	Luo Q , Zhao Y F , Ye M , et al . Application of electrical capacitance tomography for gas-solid fluidized bed measurement [J]. CIESC Journal, 2014, 65(7): 2504-2512.
4	Xu Z , Jiang Y D , Wang B L , et al . Sensitivity distribution of CCERT sensor under different excitation patterns [J]. IEEE Access, 2017, 5: 14830-14836.
5	No H C , Jeong J H . Flooding correlation based on the concept of hyperbolicity breaking in a vertical annular flow[J]. Nuclear Engineering & Design, 1996, 166(2): 249-258.
6	Wongwises S , Pipathattakul M . Flow pattern, pressure drop and void fraction of two-phase gas-liquid flow in an inclined narrow annular channel [J]. Experimental Thermal & Fluid Science, 2006, 30(4): 345-354.
7	Zhang X B , Zhu J K , Wu Z , et al . Performance prediction of structured packing column for cryogenic air separation with hybrid model [J]. Chinese Journal of Chemical Engineering, 2014, 22(8): 930-936.
8	Chen J Y , Wang Y C , Zhang W , et al . Capacitance-based liquid holdup measurement of cryogenic two-phase flow in a nearly-horizontal tube [J]. Cryogenics, 2017, 84: 69-75.
9	Wang H X , Tang L , Cao Z . An image reconstruction algorithm based on total variation with adaptive mesh refinement for ECT [J]. Flow Measurement and Instrumentation, 2007, 18(5/6): 262-267.
10	Wu T T , Lange K . Coordinate descent algorithms for lasso penalized regression [J]. The Annals of Applied Statistics, 2008, 2(1): 224-244.
11	Xie C G , Huang S M , Hoyle B S , et al . Electrical capacitance tomography for flow imaging: system model for development of image reconstruction algorithms and design of primary sensors [J]. Circuits Devices & Systems IEE Proceedings G, 1992, 139(1): 89-98.
12	Peng L H , Merkus H , Scarlett B . Using regularization methods for image reconstruction of electrical capacitance tomography [J]. Particle & Particle Systems Characterization, 2000, 17(3): 96-104.
13	Lionheart W R B . Reconstruction algorithms for permittivity and conductivity imaging [C]//Proceedings of the 2nd World Congress on Industrial Process Tomography. 2001: 4-11.
14	Yan H , Liu C , Gao J . Electrical capacitance tomography image reconstruction based on singular value decomposition[C]// World Congress on Intelligent Control & Automation. IEEE, 2004: 3783-3786.
15	Yang W Q , Peng L H . Image reconstruction algorithms for electrical capacitance tomography [J]. Measurement Science & Technology, 2003, 14: R1-R13.
16	Yang W Q , Spink D M , York T A , et al . An image-reconstruction algorithm based on Landweber’s iteration method for electrical-capacitance tomography [J]. Measurement Science & Technology, 1999, 10(11): 1065.
17	Kak A C , Slaney M . Principles of Computerized Tomographic Imaging [M]. New York: IEEE Press, 1988.
18	Su B L , Zhang Y H , Peng L H , et al . The use of simultaneous iterative reconstruction technique for electrical capacitance tomography [J]. Chemical Engineering Journal, 2000, 77(1): 37-41.
19	陈夏, 胡红利, 高享想, 等 . 代数重建和同步迭代重建在电容层析成像中的比较研究[J]. 西安交通大学学报, 2011, 45(4): 25-29.
	Chen X , Hu H L , Gao X X , et al . Comparison of algebraic reconstruction technique and simultaneous iterative reconstruction technique in electrical capacitance tomography image reconstruction [J]. Journal of Xi’an Jiaotong University, 2011, 45(4): 25-29.
20	Fang W F . A nonlinear image reconstruction algorithm for electrical capacitance tomography [J]. Measurement Science and Technology, 2004, 15(10): 2124-2132.
21	Osher S , Burger M , Goldfarb D , et al . An iterative regularization method for total variation-based image restoration [J]. Multiscale Modeling & Simulation, 2005, 4(2): 460-489.
22	雷兢, 刘石, 李志宏 . 一个基于1范数的电容层析成像图像重建迭代算法[J]. 仪器仪表学报, 2008, 29(7): 1355-1358.
	Lei J , Liu S , Li Z H . Image reconstruction iteration algorithm based on 1-norm for electrical capacitance tomography [J]. Chinese Journal of Scientific Instrument, 2008, 29(7): 1355-1358.
23	王丕涛, 王化祥, 孙犇渊 . 基于l1范数的电容层析成像图像重建算法[J]. 中国电机工程学报, 2015, 35(18): 4709-4714.
	Wang P T , Wang H X , Sun B Y . l1-norm-based image reconstruction algorithm for electrical capacitance tomography [J]. Proceedings of the CSEE, 2015, 35(18): 4709-4714.
24	Yan Y , Liu L J , Xu H , et al . Image reconstruction in electrical capacitance tomography using multiple linear regression and regularization[J]. Measurement Science & Technology, 2001, 12(5): 575-581.
25	Marashdeh Q , Teixeira F L . Sensitivity matrix calculation for fast 3-D electrical capacitance tomography (ECT) of flow systems[J]. IEEE Transactions on Magnetics, 2004, 40(2): 1204-1207.
26	Gunes C , Marashdeh Q , Teixeira F L . A comparison between electrical capacitance tomography and displacement-current phase tomography [J]. IEEE Sensors Journal, 2017, 17(24): 8037-8046.
27	王彦飞 . 反演问题的计算方法及其应用[M]. 北京: 高等教育出版社, 2007.
	Wang Y F . The Calculating Methods and Applications of Inverse Problem [M]. Beijing: Science Press, 2007.
28	Rumble J . CRC Hanfbook of Chemistry and Physis [M]. 99th edition. Boca Raton: CRC Press, 2018.
29	杨晓伟, 郝志峰 . 支持向量机的算法设计与分析[M]. 北京: 科学出版社, 2013.
	Yang X W , Hao Z F . Algorithm Design and Analysis of Support Vector Machines[M]. Beijing: Science Press, 2013.
30	郭威 . CT不完全投影数据重建算法研究[D]. 长春: 吉林大学, 2011.
	Guo W . Research on reconstruction algorithms of CT with incomplete projection data [D]. Changchun: Jilin University, 2011.
31	吴丽华 . CT迭代重建算法的研究[D]. 沈阳: 东北大学, 2008.
	Wu L H . Research on iterative reconstruction algorithms of computed tomography[D]. Shenyang: Northeastern University, 2008.
32	董向元, 刘石, 阎润生, 等 . 电容层析成像中通用迭代法的研究[J]. 仪器仪表学报, 2006, 27(1): 23-25.
	Dong X Y , Liu S , Yan R S , et al . A general iterative algorithm for image reconstruction with electrical capacitance tomography [J]. Chinese Journal of Science Instrument, 2006, 27(1): 23-25.

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