#
Evaluation of Landweber Coupled Least Square Support Vector Regression Algorithm for Electrical Capacitance Tomography for LN_{2}–VN_{2} Flow

^{*}

## Abstract

**:**

_{2}-VN

_{2}). The inversion images based on the conventional LBP and Landweber algorithms are also presented for comparison. The benefits and drawbacks of the developed algorithms are revealed and discussed, according to the results. It is demonstrated that the correlated coefficients of the images based on the developed algorithm reach more than 0.88 and a maximum of 0.975. In addition, the minimum void fraction error of the algorithm is reduced to 0.534%, which indicates the significant optimization of the LSSVR coupled method over the Landweber algorithm.

## 1. Introduction

_{2}/VN

_{2}(1.4337/1.0021) is much lower [19]. Therefore, the micro-capacitance acquisition circuit is required to have a higher sensitivity and anti-noise interference ability. In our previous studies, the potential feasibility of cryogenic ECT was evaluated [20] by conducting a substitutive ambient temperature experiment based on the pair of materials with a dielectric constant ratio close to that of cryogenic fluids. In addition, the reconstruction algorithm was especially improved to reconstruct the two-phase images for better quality. The potential application of electrical capacitance volume tomography (ECVT) to cryogenic cases was also evaluated by us, using numerical experiments [21], in which the essential roles of the shifted plane and the axial guard electrode play in ECVT were verified. Recently, Hunt et al. [22] conducted a pioneering cryogenic experiment to measure the density of the LN

_{2}-VN

_{2}flow based on the eight-electrode ECT system. The ECT sensor in their alternative experiment can provide images of non-conducting inclusions in the flow, representing gas bubbles at room temperature. Although the final LN

_{2}-VN

_{2}phase distribution reconstruction image was not obtained, their work was still constructive and provided evidence for the application of ECT in cryogenic fluids. Sun et al. [23] conducted a real-time cross-sectional holdup imaging experiment of high-pressure gas–liquid carbon dioxide (CO

_{2}) flow using the ECT method. The accuracy of real-time measurement based on the ECT system was acceptable; it was calculated by three widely used algorithms and evaluated with the images captured at the sight window. The convenience of the Calderon algorithm for the CO

_{2}two-phase image reconstruction was also summarized. Tian et al. [24] compared the performance of deep neural network (DNN) modification to the classical linear algorithms (DNN-EC) and the DNN directly for image reconstruction (DNN-C). The generalization ability of these methods was validated in the numerical experiment, and the feasibility of these models for the cryogenic application was evaluated by a cryogenic experiment. The results showed that the DNN-C is a better solution than the DNN-EC, based on error capacitance. Gao [25] introduced the transformer technique to the U-net convolutional neural network (CNN). The trained CNN was applied to the numerical experiment and achieved excellent and stable results. The model was also used to reconstruct the LN

_{2}-VN

_{2}flow in the cryogenic experiment, with clear and accurate images.

^{1}-norm regularization (TV L

^{1}-norm) [29] algorithms. Many efforts have also been made to improve the algorithms’ efficiencies. Xie et al. [27] demonstrated the significantly increased imaging speed of the back projection algorithm in determining the oil concentration. Yang et al. [28] proposed a new image inversion algorithm based on the modified Landweber iteration algorithm by applying a regularization, which showed a faster convergence speed and enhanced immunity to noise. Liu et al. [30] optimized the iterative step length of the Landweber algorithm based on minimizing the norm of the capacitance error vector. Soleimani and Lionheart [29] employed regularization techniques to overcome the ill-posedness, which proved the advantage of the TV regularization algorithm over the image reconstruction of different inclusions. To enhance the performance of the reconstruction image, Wang et al. [31] proposed an adaptive cell refinement approach based on the TV method to preserve the edge. Li and Yang [19] proposed a nonlinear iteration algorithm, in which the sensitivity matrix was updated rather than fixed in the iterative process. The results indicated that the relative capacitance residual and the image error decrease more quickly than in the fixed sensitivity matrix cases. Guo et al. [11] developed a machine-learning method by avoiding the postprocessing steps, which increased the imaging speed. The training samples were collected in different flow patterns through high-throughput experiments. Xie et al. [20] proposed an approach of the least squares support vector regression (LSSVR) coupled linear algorithm, reducing the error by 68%, at most, via fitting the correlation between the capacitance vector and the linearization error.

_{2}-VN

_{2}two-phase flow was carried out to further evaluate the reconstructed images of the three algorithms for the eight-electrode ECT. By comparing the cryogenic imaging results of the proposed algorithm with the LBP algorithm and the original Landweber algorithm, it was proved that the proposed algorithm can improve the imaging performance.

## 2. ECT Cryogenic Experimental System

_{2}and another tube for releasing the VN

_{2}. Two additional axial shield rings are placed at both ends of the electrodes. Around the electrodes, there exists an electromagnetic shield to avoid the interference of the environmental magnetic field. Both the external and axial shields can reduce the parasitic effect. The material of all the electrodes and shields is copper. The whole cryogenic sensor is placed in a square chamber made of acrylic boards, as shown in Figure 2. The chamber is filled with static nitrogen to replace the original wet air to avoid frosting on the surface of the cryogenic pipe. The nitrogen also serves to reduce the heat leak due to the smaller thermal conductivity compared with the wet air. The design greatly simplifies the facility structure which should be in the complex vacuum insulation, while keeping it visual and maintaining an acceptable heat-leakage rate.

_{2}and VN

_{2}successively to calibrate the sensor. The stratified flows with different liquid levels were obtained when the LN

_{2}slowly evaporated in the pipe, which was filled with LN

_{2}at the beginning. The phase interface can be considered as the steady state because the change of liquid level caused by the evaporation is slow enough. The real phase distributions were recorded by a camera and the capacitance values between different electrodes were collected by the acquisition circuit.

## 3. Image Reconstruction Approach

#### 3.1. Forward Problem of ECT

_{2}-VN

_{2}and the experimental pipe, there is no electronic conduction between the outer environment and the test section. The governing equation of the potential in the computational domain can be obtained as in [32]:

#### 3.2. Calculation of Sensitivity Field

_{2}-VN

_{2}cases is shown in Figure 5. The full 28 sensitivities can be determined from the presented four cases by the relative position of the electrode pair. Since the electrodes of the sensor are centrally symmetric, the 28 sensitive fields can be regarded as a transformation of these four sensitivity fields obtained by rotating 45, 90, 135, or 180 degrees around the center of the measurement area.

#### 3.3. The Inverse Problem of ECT

#### 3.3.1. Linear Back-Projection (LBP) Algorithm

#### 3.3.2. Landweber Iterative Algorithm

#### 3.3.3. Landweber Coupled LSSVR Algorithm

**E**, the nonlinear error vector

**Y**can be obtained using the LSSVR fitting equation:

## 4. Experiment and Analysis

#### 4.1. The Tentative Experiments

_{2}-VN

_{2}pair are chosen as the working fluids, as listed in Table 3. The eight-electrode sensor here is a quartz glass tube with an outer diameter of 60 mm and a thickness of 2 mm. Each electrode attached to the pipe has a length of 90 mm and a circumferential coverage rate of 80%.

#### 4.2. Cryogenic Experiments

_{2}inside the pipe is not intense and the LN

_{2}level can be kept stable and observed without intensive boiling; therefore, the reconstruction images can reflect the real distribution of cryogenic flow.

_{2}-VN

_{2}two-phase flow based on the ECT experimental facility are shown in Figure 8, where the red part represents the liquid phase and the remaining part represents the gas phase. The left column shows the real images of the experimental pipe with LN

_{2}inside. An extra black line is added to the figure to clarify the position of the liquid level. The second column shows the real distribution mapped to the computational domain according to the liquid level measurement. The other columns in the figure provide the image reconstruction results based on the LBP, Landweber, and Landweber coupled LSSVR algorithm, respectively.

_{2}level in the pipe, can be captured in all the reconstructed images. The results indicate that the ECT system can image the cryogenic capacitance data with considerable accuracy, and the three reconstruction algorithms all can reconstruct the phase distribution.

_{2}, which only fuzzes the boundary between the gas and liquid phases and is not capable of changing the geometric shape of the dividing line.

## 5. Conclusions

_{2}-VN

_{2}flow. The alternative experiment showed the potential of applying the projected LBP algorithm, the projected Landweber iteration algorithm, and the projected Landweber iteration coupled LSSVR algorithm to the inversion imaging of the two-phase flow with a relative dielectric permittivity ratio close to 1.6. In the cryogenic experiment, the interface of the different LN

_{2}levels of the stratified flow was detected and imaged. The Landweber coupled LSSVR algorithm presented the best performance on image reconstruction, which reached the smallest VF error of 0.554% and highest CC of 0.975. While the projected LBP algorithm achieved a relatively fuzzy image and acceptable image reconstruction quality, the Landweber iteration algorithm reached the highest accuracy in VF, with an error of 4.509%, although it produced distortion at the interface and the edge of the computational domain. The LSSVR method can improve the performance of the Landweber algorithm. Strengthening the advantage of the Landweber algorithm, the modified algorithm presents a clearer phase interphase. However, it cannot eliminate the artifacts at the boundary of the computational region—considering that it inherited from the Landweber algorithm the artifacts brought by the Landweber algorithm, such artifacts are inevitable. The reason is that the LSSVR coupled algorithm is a fusion-driven method—its modification is based on the error capacitance. The error capacitance is obtained from the calculation result from the linear algorithm, and the reconstructed distribution vector’s dimensionality is reduced by the sensitivity matrix. This downsampling process has lost some information; thus, as an additional correction, the modification results will not deviate too far from the original algorithm. However, the error capacitance term is optimized by LSSVR based on the trained samples, and the improvement can be observed, which is reflected by the quality evaluation from CCs, IEs, and VF errors.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

a | Linear offset vector of LSSVR | $\widehat{\mathit{S}}$ | Normalized sensitive field matrix |

b | Coefficient vector of LSSVR | ${\widehat{\mathit{S}}}_{\mathrm{i},\mathrm{j}}$ | Normalized sensitive field between the measuring electrode i and j |

C | Capacitance value | $U$ | Excitation voltage |

$\mathit{E}$ | Normalized vector of dielectric permittivity distribution | $\mathit{Y}$ | Residual error of normalized capacitance |

$\mathit{K}\left(.,.\right)$ | Kernal function | Greek symbol | |

M | Total number of effective capacitance values | ||

N | Total number of grid cells | $\alpha $ | Iteration step size |

$\overrightarrow{n}$ | Unit normal vector at the wall | $\Gamma $ | Electrode position |

$u$ | Electric potential distribution | $\mathit{\epsilon}$ | Dielectric permittivity distribution |

P(.) | Projection operation | $\mathit{\lambda}$ | Normalized measured capacitance vector |

Q | Quantity of electric charge | $\Omega $ | Computational domain |

${\mathit{S}}_{\mathrm{i},\mathrm{j}}$ | Sensitive field between the measuring electrode i and j | $\partial \Omega \backslash \Gamma $ | Boundary of the computational domain which is not covered by the electrode area |

## References

- Sakamoto, Y.; Kobayashi, H.; Naruo, Y.; Takesaki, Y.; Nakajima, Y.; Furuichi, A.; Tsujimura, H.; Kabayama, K.; Sato, T. Investigation of the Void Fraction–quality Correlations for Two-phase Hydrogen Flow Based on the Capacitive Void Fraction Measurement. Int. J. Hydrogen Energy
**2019**, 44, 18483–18495. [Google Scholar] [CrossRef] - Yang, Z.Q.; Chen, G.F.; Zhuang, X.R.; Song, Q.L.; Deng, Z.; Shen, J.; Gong, M.Q. A New Flow Pattern Map for Flow Boiling of R1234ze(E) in a Horizontal Tube. Int. J. Multiph. Flow
**2018**, 98, 24–35. [Google Scholar] [CrossRef] - Forte, G.; Clark, P.; Yan, Z.; Stitt, E.H.; Marigo, M. Using a Freeman Ft4 Rheometer and Electrical Capacitance Tomography to Assess Powder Blending. Powder Technol.
**2018**, 337, 25–35. [Google Scholar] [CrossRef] - Chen, J.Y.; Wang, Y.C.; Zhang, W.; Qiu, L.M.; Zhang, X.B. Capacitance-based Liquid Holdup Measurement of Cryogenic Two-phase Flow in a Nearly-horizontal Tube. Cryogenics
**2017**, 84, 69–75. [Google Scholar] [CrossRef] - Khalil, A.; Mcintosh, G.; Boom, R. Experimental Measurement of Void Fraction in Cryogenic Two Phase Upward Flow. Cryogenics
**1981**, 21, 411–414. [Google Scholar] [CrossRef] - Filippov, Y.P.; Kovrizhnykh, A.M.; Miklayev, V.M.; Sukhanova, A.K. Metrological Systems for Monitoring Two-phase Cryogenic Flows. Cryogenics
**2000**, 40, 279–285. [Google Scholar] [CrossRef] - Filippov, Y.P. How to Measure Void Fraction of Two-phase Cryogenic Flows. Cryogenics
**2001**, 41, 327–334. [Google Scholar] [CrossRef] - Filippov, Y.P.; Kakorin, I.; Kovrizhnykh, A.M. New Solutions to Produce a Cryogenic Void Fraction Sensor of Round Cross-section and Its Applications. Cryogenics
**2013**, 57, 55–62. [Google Scholar] [CrossRef] - Harada, K.; Murakami, M.; Ishii, T. PIV Measurements for Flow Pattern Void Fraction in Cavitating Flows of He II and He I. Cryogenics
**2006**, 46, 648–657. [Google Scholar] [CrossRef] - Che, H.Q.; Ye, J.M.; Tu, Q.Y.; Yang, W.Q.; Wang, H.G. Investigation of Coating Process in Wurster Fluidised Bed Using Electrical Capacitance Tomography. Chem. Eng. Res. Des.
**2018**, 132, 1180–1192. [Google Scholar] [CrossRef] - Guo, Q.; Ye, M.; Yang, W.Q.; Liu, Z. A Machine Learning Approach for Electrical Capacitance Tomography Measurement of Gas–solid Fluidized Beds. Aiche J.
**2019**, 65, e16583. [Google Scholar] [CrossRef] - Mohamad, E.J.; Rahim, R.A.; Rahiman, M.H.F.; Ameran, H.L.M.; Muji, S.Z.M.; Marwah, O.M.F. Measurement and Analysis of Water/oil Multiphase Flow Using Electrical Capacitance Tomography Sensor. Flow Meas. Instrum.
**2016**, 47, 62–70. [Google Scholar] [CrossRef] - Yang, W.Q.; Stott, A.L.; Beck, M.S.; Xie, C.G. Development of Capacitance Tomographic Imaging Systems for Oil Pipeline Measurements. Rev. Sci. Instrum.
**1995**, 66, 4326–4332. [Google Scholar] [CrossRef] - Ortiz-alemán, C.; Martin, R. Inversion of Electrical Capacitance Tomography Data By Simulated Annealing: Application to Real Two-phase Gas–oil Flow Imaging. Flow Meas. Instrum.
**2005**, 16, 157–162. [Google Scholar] [CrossRef] - Ismail, I.; Gamio, J.; Bukhari, S.; Yang, W.Q. Tomography for Multi-phase Flow Measurement in the Oil Industry. Flow Meas. Instrum.
**2005**, 16, 145–155. [Google Scholar] [CrossRef] - Dyakowski, T.; Edwards, R.B.; Xie, C.G.; Williams, R.A. Application of Capacitance Tomography to Gas-solid Flows. Chem. Eng. Sci.
**1997**, 52, 2099–2110. [Google Scholar] [CrossRef] - Cui, Z.Q.; Yang, C.Y.; Sun, B.Y.; Wang, H.X. Liquid Film Thickness Estimation Using Electrical Capacitance Tomography. Meas. Sci. Rev.
**2014**, 14, 8–15. [Google Scholar] [CrossRef] - Román, A.; Cronin, J.; Ervin, J.; Byrd, L. Measurement of the Void Fraction and Maximum Dry Angle Using Electrical Capacitance Tomography Applied to a 7 mm Tube with R-134a. Int. J. Refrig.
**2018**, 95, 122–132. [Google Scholar] [CrossRef] - Xie, H.J.; Chen, H.; Gao, X.; Zheng, X.D.; Zhi, X.Q.; Zhang, X.B. Theoretical Analysis of Fuzzy Least Squares Support Vector Regression Method for Void Fraction Measurement of Two-phase Flow by Multi-electrode Capacitance Sensor. Cryogenics
**2019**, 103, 102969. [Google Scholar] [CrossRef] - Xie, H.J.; Xia, T.; Tian, Z.N.; Zheng, X.D.; Zhang, X.B. A Least Squares Support Vector Regression Coupled Linear Reconstruction Algorithm for ECT. Flow Meas. Instrum.
**2021**, 77, 101874. [Google Scholar] [CrossRef] - Xia, T.; Xie, H.J.; Wei, A.B.; Zhou, R.; Qiu, L.M.; Zhang, X.B. Preliminary Study on Three-dimensional Imaging of Cryogenic Two-phase Flow Based on Electrical Capacitance Volume Tomography. Cryogenics
**2020**, 110, 103127. [Google Scholar] [CrossRef] - Hunt, A.; Rusli, I.; Schakel, M.; Kenbar, A. High-speed Density Measurement for Lng and Other Cryogenic Fluids Using Electrical Capacitance Tomography. Cryogenics
**2021**, 113, 103207. [Google Scholar] [CrossRef] - Sun, S.J.; Zhang, W.B.; Sun, J.T.; Cao, Z.; Xu, L.J.; Yan, Y. Real-Time Imaging and Holdup Measurement of Carbon Dioxide Under CCS Conditions Using Electrical Capacitance Tomography. IEEE Sens. J.
**2018**, 18, 7551–7559. [Google Scholar] [CrossRef] - Tian, Z.; Gao, X.; Qiu, L.; Zhang, X. Experimental imaging and algorithm optimization based on deep neural network for electrical capacitance tomography for LN2-VN2 flow. Cryogenics
**2022**, 127, 103568. [Google Scholar] [CrossRef] - Gao, X.; Tian, Z.; Qiu, L.; Zhang, X. A hybrid deep learning model for ECT image reconstruction of cryogenic fluids. Flow Meas. Instrum.
**2022**, 87, 102228. [Google Scholar] - Xie, C.G.; Huang, S.M.; Hoyle, B.S.; Thorn, R.; Lenn, C.; Snowden, D.; Beck, M.S. Electrical capacitance tomography for flow imaging system model for development of image reconstruction algorithms and design of primary sensors. Proc. G Circuits Devices Syst.
**1992**, 139, 89–98. [Google Scholar] [CrossRef] - Yang, W.Q.; Spink, D.M.; York, T.A.; McCann, H. An Image-reconstruction Algorithm Based on Landweber’s Iteration Method for Electrical-capacitance Tomography. Meas. Sci. Technol.
**1999**, 10, 1065–1069. [Google Scholar] [CrossRef] - Soleimani, M.; Lionheart, W.R.B. Nonlinear Image Reconstruction for Electrical Capacitance Tomography Using Experimental Data. Meas. Sci. Technol.
**2005**, 16, 1987–1996. [Google Scholar] [CrossRef] - Liu, S.; Fu, L.; Yang, W.Q. Optimization of an Iterative Image Reconstruction Algorithm for Electrical Capacitance Tomography. Meas. Technol.
**1999**, 10, 1970–1980. [Google Scholar] [CrossRef] - Wang, H.X.; Tang, L.; Cao, Z. An Image Reconstruction Algorithm Based on Total Variation with Adaptive Mesh Refinement for ECT. Flow Meas. Instrum.
**2007**, 18, 262–267. [Google Scholar] [CrossRef] - Li, Y.; Yang, W.Q. Image Reconstruction by Nonlinear Landweber Iteration for Complicated Distributions. Meas. Sci. Technol.
**2008**, 19, 94014. [Google Scholar] [CrossRef] - Yang, W.Q.; Liu, S. Electrical Capacitance Tomography with Square Sensor. Electron. Lett.
**1999**, 35, 295–296. [Google Scholar] [CrossRef] - Xie, H.J.; Yu, L.; Zhou, R.; Qiu, L.M.; Zhang, X.B. Preliminary Evaluation of Cryogenic Two-phase Flow Imaging Using Electrical Capacitance Tomography. Cryogenics
**2017**, 86, 97–105. [Google Scholar] [CrossRef] - Cui, Z.Q.; Wang, Q.; Xue, Q.; Fan, W.R.; Zhang, L.L.; Cao, Z.; Sun, B.Y.; Wang, H.X.; Yang, W.Q. A Review on Image Reconstruction Algorithms for Electrical Capacitance/resistance Tomography. Sens. Rev.
**2016**, 36, 429–445. [Google Scholar] [CrossRef] - Isaksen, Ø. A Review of Reconstruction Techniques for Capacitance Tomography. Meas. Sci. Technol.
**1996**, 7, 325–337. [Google Scholar] [CrossRef] - Liao, A.M.; Zhou, Q.Y. Application of ECT and Relative Change Ratio of Capacitances in Probing Anomalous Objects in Water. Flow Meas. Instrum.
**2015**, 45, 7–17. [Google Scholar] [CrossRef] - Yang, W.Q.; Peng, L.H. Image Reconstruction Algorithms for Electrical Capacitance Tomography. Meas. Sci. Technol.
**2003**, 14, R1–R13. [Google Scholar] [CrossRef] - Peng, L.H.; Merkus, H.; Scarlett, B. Using Regularization Methods for Image Reconstruction of Electrical Capacitance Tomography. Part. Part. Syst. Charact.
**2000**, 17, 96–104. [Google Scholar] [CrossRef] - Yang, Y.J.; Peng, L.H. Data Pattern with ECT Sensor and Its Impact on Image Reconstruction. IEEE Sens. J.
**2013**, 13, 1582–1593. [Google Scholar] [CrossRef] - Smits, G.F.; Jordaan, E.M. Improved SVM regression using mixtures of kernels. In Proceedings of the International Joint Conference on Neural Networks, IEEE Xplore, Honolulu, HI, USA, 12–17 May 2002; pp. 2785–2790. [Google Scholar]

**Figure 7.**Distributions and reconstruction images by ECT of (

**a**) rice particles—air, (

**b**) mung bean particles—air, (

**c**) millet particles—air, and (

**d**) polypropylene particles—air.

Fluid Pair | Relative Dielectric Permittivity |
---|---|

Water/Air (at 300 K) | 77.747/1.0005 |

Liquid/Vapor nitrogen (at 78 K) | 1.4337/1.0021 |

Liquid/Vapor oxygen (at 90 K) | 1.4877/1.0016 |

Liquid/Vapor methane (at 112 K) | 1.6299/1.0020 |

Parameter | Value | |
---|---|---|

Landweber | Iteration number | 5 |

Landweber coupled LSSVR | Iteration number | 5 |

$\rho $ | 0.9 | |

$\mathsf{\sigma}$ | 0.1 | |

$p$ | 2 |

Fluid Pair | Relative Dielectric Permittivity |
---|---|

Polypropylene/Air | 1.6201/1.0005 |

Mung bean/Air | 5.5728/1.0005 |

Millet/Air | 5.2995/1.0005 |

Rice/Air | 6.035/1.0005 |

VF Error (%) | |||
---|---|---|---|

LBP | Landweber | Landweber Coupled LSSVR | |

Case 1a | 4.815 | 11.278 | 10.885 |

Case 2a | 3.576 | 2.821 | 0.485 |

Case 1b | 15.459 | 4.703 | 8.078 |

Case 2b | 11.509 | 8.652 | 8.961 |

Case 3b | 0.013 | 4.442 | 3.635 |

Case 1c | 35.277 | 32.297 | 13.843 |

Case 2c | 3.272 | 4.127 | 1.506 |

Case 3c | 1.891 | 5.464 | 0.695 |

Case 1d | 11.746 | 8.868 | 8.571 |

Case 2d | 4.802 | 8.570 | 5.822 |

CC | |||
---|---|---|---|

LBP | Landweber | Landweber Coupled LSSVR | |

Case 1a | 0.898 | 0.895 | 0.901 |

Case 2a | 0.803 | 0.759 | 0.835 |

Case 1b | 0.768 | 0.724 | 0.873 |

Case 2b | 0.917 | 0.904 | 0.888 |

Case 3b | 0.884 | 0.835 | 0.872 |

Case 1c | 0.901 | 0.890 | 0.808 |

Case 2c | 0.914 | 0.883 | 0.896 |

Case 3c | 0.868 | 0.772 | 0.827 |

Case 1d | 0.902 | 0.887 | 0.893 |

Case 2d | 0.882 | 0.879 | 0.895 |

IE (%) | |||
---|---|---|---|

LBP | Landweber | Landweber Coupled LSSVR | |

Case 1a | 26.681 | 26.813 | 24.225 |

Case 2a | 50.139 | 56.888 | 42.269 |

Case 1b | 25.406 | 28.744 | 14.998 |

Case 2b | 23.552 | 24.839 | 26.365 |

Case 3b | 35.202 | 42.097 | 42.785 |

Case 1c | 14.219 | 14.436 | 22.205 |

Case 2c | 24.774 | 26.699 | 26.810 |

Case 3c | 42.244 | 55.445 | 46.828 |

Case 1d | 34.418 | 35.540 | 28.145 |

Case 2d | 60.383 | 57.993 | 34.736 |

LBP | Landweber | Landweber Coupled LSSVR | ||
---|---|---|---|---|

VF error (%) | Case 1 | 12.831 | 9.979 | 8.478 |

Case 2 | 26.047 | 15.215 | 14.883 | |

Case 3 | 22.967 | 21.447 | 21.549 | |

CC | Case 1 | 0.940 | 0.963 | 0.975 |

Case 2 | 0.793 | 0.929 | 0.935 | |

Case 3 | 0.874 | 0.889 | 0.889 | |

IE (%) | Case 1 | 12.831 | 9.979 | 8.478 |

Case 2 | 26.047 | 15.215 | 14.883 | |

Case 3 | 22.967 | 21.447 | 21.549 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Tian, Z.-N.; Gao, X.-X.; Xia, T.; Zhang, X.-B.
Evaluation of Landweber Coupled Least Square Support Vector Regression Algorithm for Electrical Capacitance Tomography for LN_{2}–VN_{2} Flow. *Energies* **2023**, *16*, 7661.
https://doi.org/10.3390/en16227661

**AMA Style**

Tian Z-N, Gao X-X, Xia T, Zhang X-B.
Evaluation of Landweber Coupled Least Square Support Vector Regression Algorithm for Electrical Capacitance Tomography for LN_{2}–VN_{2} Flow. *Energies*. 2023; 16(22):7661.
https://doi.org/10.3390/en16227661

**Chicago/Turabian Style**

Tian, Ze-Nan, Xin-Xin Gao, Tao Xia, and Xiao-Bin Zhang.
2023. "Evaluation of Landweber Coupled Least Square Support Vector Regression Algorithm for Electrical Capacitance Tomography for LN_{2}–VN_{2} Flow" *Energies* 16, no. 22: 7661.
https://doi.org/10.3390/en16227661