During mine filling, the caking in the pipeline and the waste rock in the filling slurry may cause serious safety accidents such as pipe blocking or explosion. Therefore, the visualization of the inner mine filling of the solid–liquid two-phase flow in the pipeline is important. This paper proposes a method based on capacitance tomography for the visualization of the solid–liquid distribution on the section of a filling pipe. A feedback network is used for electrical capacitance tomography reconstruction. This reconstruction method uses radial basis function neural network fitting to determine the relationship between the capacitance vector and medium distribution error. In the reconstruction process, the error in the linear back projection is removed; thus, the reconstruction problem becomes an accurate linear problem. The simulation results show that the reconstruction accuracy of this algorithm is better than that of many traditional algorithms; furthermore, the reconstructed image artifacts are fewer, and the phase distribution boundary is clearer. This method can help determine the location and size of the caking and waste rock in the cross section of the pipeline more accurately and has great application prospects in the visualization of filling pipelines in mines.

In the process of mining, cemented tailing filling technology is often adopted to reduce environmental pollution and solve problems such as surface subsidence [

ECT, a nondestructive testing method, is an effective means to monitor multiphase flow. As the permittivity of the medium in the measured pipeline may vary, ECT can calculate the permittivity distribution from the measured capacitance through the reconstruction algorithm, which can reflect the corresponding phase distribution [

In this study, an ECT reconstruction feedback network based on an RBFNN was established, and the center selection of the RBFNN was realized using the orthogonal least squares (OLS) algorithm. There are major errors in the calculation results of linear reconstruction algorithms. Using a feedback reconstruction network, the errors were removed, and accurate imaging results were obtained.

In

In this study, an ECT system was investigated and verified through simulation experiments. The ECT system is illustrated in

The 12-electrode ECT system was studied, and there were 66 independent measured capacitance values.

In the capacitance tomography system, there is a nonlinear relationship between the capacitance C between the plates and the dielectric constant

Because the number of pixels required for imaging is much larger than the number of measured capacitance values, the dimension of g is much larger than that of C. The number of calculations required for sensitivity S will increase with an increase in the precision of the sensitive field partition. Because the condition number of S is large, the disturbance with a small measured capacitance value will cause a large change in the dielectric constant [

To address these problems, this paper proposes the use of an RBFNN to establish the nonlinear relationship between capacitance and dielectric constant values and to select the center of the RBFNN using the OLS method.

In this study, the LBP reconstruction algorithm was used to solve the ECT inverse problem, and an RBFNN was used to realize the nonlinear mapping between the approximation measurement capacitance and LBP reconstruction error. An OLS was used to optimize the RBFNN. Thus, the ECT feedback network was implemented. This section introduces the overall framework of the ECT feedback network based on the RBFNN, the principle of OLS-RBF, and the generation of the training set.

In the LBP algorithm, the solution of the dielectric constant value is expressed as follows:

Hypothesis

which can be obtained through

If the nonlinear relationship between the measured capacitance and the reconstruction error is known, the error generated by

An RBFNN based on the OLS algorithm was established to realize the nonlinear mapping between the measured capacitance and the reconstruction error. The feedback ECT reconstruction method based on the RBFNN proposed in this paper is shown in

An RBFNN has a feedforward structure with a single hidden layer, and the application of its kernel function can map low-dimensional linearly indivisible data to high-dimensional space, making it linearly separable in a high-dimensional space [

The RBFNN takes neurons as nodes and the radial basis function as the excitation function. The activation function of the hidden layer node responds locally to the input. When the input is close to the central range of the base function, the hidden layer node will produce a large output; when the input is far from the central point, the output decays exponentially. The hidden layer neurons obtain the output layer by linear addition.

RBFs are non-negative real-valued functions whose values depend on the input distance from the central point and are radially symmetric. The Gaussian function is expressed as follows:

When using an RBFNN, the most important factor is selecting the center point. In this study, the OLS algorithm was used to select the center and construct the RBFNN. RBF network can be viewed as the following regression model:

OLS transforms the P set into orthogonal basis vectors, and then obtains the effect of each basis vector on the output,

The gram-Schmidt method can be used to obtain the value of the least squares estimate

COMSOL Multiphysics finite element software was used to establish four distributions of ECT models: single-core, two-core, three-core, and stratified. When modeling different samples in COMSOL, different parameters were used to describe their characteristics.

Each distribution sample was simulated by COMSOL to obtain 5,000 datasets, with a total of 20,000 data sets. Each dataset consists of a measured capacitance vector C and a dielectric constant value vector g. C and g are the input and expected output of the network, respectively. To accelerate the convergence speed of the network training, before using the data set to train the network, each group of measured capacitance values needs to be normalized. The normalization formula for C is as follows:

To verify the effectiveness of the proposed reconstruction method, two criteria of correlation coefficient (CC) and image error (IE) are used to quantitatively evaluate the reconstruction accuracy:

Our method | ART | Tikhonov | Landweber | |||||
---|---|---|---|---|---|---|---|---|

IE (%) | CC | IE (%) | CC | IE (%) | CC | IE (%) | CC | |

1 | 3.46 | 0.9956 | 1.30 | 0.9887 | 14.19 | 0.9431 | 22.52 | 0.8861 |

2 | 2.71 | 0.9908 | 4.59 | 0.9623 | 21.46 | 0.9264 | 26.64 | 0.8600 |

3 | 4.38 | 0.9871 | 6.81 | 0.9315 | 24.36 | 0.9387 | 28.87 | 0.7773 |

4 | 3.12 | 0.9957 | 8.24 | 0.8917 | 12.13 | 0.9347 | 2.08 | 0.9463 |

5 | 2.06 | 0.9987 | 4.25 | 0.9725 | 5.09 | 0.9822 | 2.78 | 0.9837 |

To further study the anti-noise performance of the algorithm, the capacitance data are added to the noise and then inputted to the feedback reconstruction network for calculation. The signal-to-noise ratio (SNR) is expressed as follows:

Noise was added to the capacitor vector, as shown in

A feedback reconstruction network based on an RBFNN was proposed for mine filling pipeline visualization and to reduce the error caused by nonlinearity in the ECT reconstruction process. After the calculation error caused by the linear algorithm is removed from the model, the ECT reconstruction problem is transformed from nonlinear to linear inverse. In this study, typical two-phase flow data samples were used to train the RBFNN for predicting the reconstruction error, and the LBP algorithm was used to complete the image reconstruction. In the simulation results, for the data without noise and the data interfered by noise, the proposed reconstruction method has a high reconstruction accuracy, fewer imaging artifacts, and a clear phase distribution boundary. It can effectively judge the caking and blockage in the pipeline, which has an important application prospect. The feedback reconstruction network greatly reduces the error of the linear model of ECT. It can be combined with a more complex reconstruction algorithm to further improve the accuracy of the reconstruction algorithm, providing a theoretical basis for the visual detection of mine filling pipelines.

I would like to acknowledge Professor, Ningbo Jing, for inspiring my interest in the development of innovative technologies