DENG Fang-fang, ZHANG Jian-yong, CHENG Rui-xue,ZHOU Han-chang, WANG Gao， YAN Bing

(1. Science and Technology on Electronic Test & Measurement Laboratory, North University of China, Taiyuan 030051, China;2. School of Information and Communication Engineering, North University of China, Taiyuan 030051, China;3. Teesside University, Middlesbrough TS1 3BA, UK)

0 Introduction

In the gas-solid two-phase flow, the particles are naturally charged due to the friction between particles and pipe wall and collision among particles[1-3]. The circular electrostatic sensor utilizes this phenomenon to measure particle flow[4-6].

As shown in Fig.1, the sensor described in this paper uses a circular electrode structure.

Fig.1 Structure of ring electrostatic sensor

The inner surface of the electrode is exposed to the flowing medium and embedded flush with the inner pipe wall.

In order to analyze the response of the sensor on particle flow, the first step is to establish the response of the sensor to a single charged particle, which is known as “spatial sensitivity”. This model is established based on simulation results by the finite element method (FEM)[7], however it has not been optimized and validated.

1 Model establishment

In order to improve the above model, the simulation is repeated, the simulation model is shown in Fig.2. In the simulation, assume a fixed ratio of electrode width (W) to the radius (R) of the pipe is 1/5, the diameter of the pipe is 100 mm. The pipe and electrode are made of stainless steel with the dielectric constant of 1 and the conductivity of 1.1×106; the material of the insulating layer is set as rubber with the dielectric constant of 3 and the conductivity of 1×10-5. The metal pipes and shielding are grounded. The charged particle is assumed as a quartz glass with negligible volume, with the dielectric constant of 3.78 and the conductivity of 0.

Fig.2 Simulation model

The amount of charge applied to the particle is a constant. The space of the medium is set to be vacuum. As the metal pipe can extend indefinitely, the boundary of the pipe is set as the balloon boundary, which is a common method for solving Maxwell 3D problem. A total of 7 333 tetrahedral meshes are used and 1 071 points are calculated, the specific graphic division is shown in Fig.3.

Fig.3 Mesh diagram

Based on the above settings, by changing the spatial location of the charged particle, the variation of induced charge on the electrode against axial locationxis obtained as shown in Fig.4 at a fixed radial positionr=0 mm, and more results can be seen in Fig.5 for other radial positions.

From Figs.4 and 5, it is clear that for a given particle to carry charge, when the charged particle moves along thexaxis at a fixed radial positionR, the relationship betweenQandxon the electrode is similar to that of bell shape curve, which is symmetrical tox=0.

By observing the curves in Figs.4 and 5, the relationship between the induced chargeQand the axial distancexis expressed as a Gaussian function[7-8]

Q=Ae-Kx2,

(1)

Q=Be-Cx2+De-Ex2,

(2)

whereQrepresents the ratio of the induced charge on the electrode to the charge carried by the particle;xis the distance from the position of the point charge in the axial direction to the center of the electrode, mm; for a given width,AandK(mm-2) are two constants, which are functions of radial positionr.

Fig.4 Simulation results at different r/R

Fig.5 Simulation results at different W/R

Although the double Gaussian approach is more accurate[9-10], in this paper, the model improvement and validation is carried out on the simplicity of the single Gauss equation.

2 Model improvement

Eq.(1) represents an overall mathematic representation. For a given radial position, parametersAandKare regarded as constants, and it is not necessary that the simulation results at each point exactly fit to Eq.(1). Based on the least square method, however the sum of the square of the differences between the simulation results and the results generated with this equation can be used to find the best fit parametersAandKunder the least square criteria, that is

(3)

(4)

(5)

Under the above criteria, forW/R=1/5, atr=0 mm,A=0.378 9 andK=0.006 9, the expression of the fitted curve is

Q=0.378 9e-0.006 9x2.

(6)

The similarity degree between two curves is evaluated using a correlation coefficient. The correlation coefficientγis defined as

(7)

The correlation coefficient of single Gauss isγs=0.953 2， and double Gauss isγd=0.996 2. It is shown that the improved model has a strong correlation with the simulation results. And the correlation coefficients can be used to show that the accuracy of the double Gauss is higher. With the same methodology, the improved parametersAandKat eachrare listed in Table 1.

It is clear that for a givenW/R,AandKare the functions of radial positionr. It means an overall equation can be used to analytically express the relationship between the induced charges on the electrode for the point charge at any location in the sensing volume. It can be seen from the correlation coefficient that the values obtained at the central axis positionr/R=0 are higher. This work has been carried out in this investigation, however due to the limited space, the detailed analysis is not included in this paper.

Table 1 Parameter A and K at different r/R

3 Experimental verification and analysis

The purpose of experiments is to validate the improved model. The experiment set-up is shown in Fig.6.

Fig.6 Experiment set-up

The sensor has a radius of 50 mm and electrode width of 10 mm. The particle is a plastic bullet with diameter of 1 mm. The bullet is shot using an air gun and treated before each shot with the same procedure. The sensor is installed in the horizontal for convenience of alignment and safety consideration. A 2D coordinate paper is fixed on the target side to determine the radial position of the travelling bullet. The instantaneous signal from the electrode is recorded with the time base as shown in Fig.7.

The signal in Fig.7 is considered as a “dynamic signal”, indicating the signal at different radial positions when the given electrode width isW/R=1/5, i.e., the first derivative of Fig.4. It must be explained that in Fig.7, the horizontal coordinate is time, which can be converted intoxcoordinate as in Fig.4 with a constant speed of the bullet.

As seen from Fig.7, the peaks of the signal are almost symmetrical with opposite polarity. With the increase ofr/Rvalue, the amplitude of the detection signal increases as the bullet passing near the pipe wall.

Fig.7 Selected experiment results

For Fig.4, if the first order derivative is calculated against equivalent time for the velocity of 15 m/s, which is the bullet’s velocity before it enters into the pipe section in the experiment, the left part of differentiation of the curve would look like a curve with solid line in Fig.7. Scatter indicates the result of the experiment. In order to compare the experimental results with simulation results, the results are normalized in amplitude. The details can be seen from Fig.8, where the horizontal coordinate is axial positionx.

Fig.8 Model validation without correction

However the error caused by the setting must be considered. At first, due to the air resistance, the speed of the bullet decreases as it passing the sensing zone. Secondly as the bullet travelled following the trajectory curve, in the actual experiments, there is bias from the given radial position during the travelling of the bullet.

The experimental curve are modified based on the bullet trajectory and velocity changes. The validation in Fig.9 shows that if the background noise is removed or smoothed, the experimental curve is very close to the optimized model represented by the solid line based on the simulation results, so the model has been verified.

According to the principle of electrostatic field superposition, in the study of multi particles, for given electrode widthW/R=1/5, when the particles move along the axis of the center ofr/R=0, the signal of the induced charge on the electrode is shown in Fig.10.

Fig.9 Model validation with correction

Fig.10 Multi particles signal

The correlation coefficients between the measured results and the simulation results and the improved results are compared and analyzed. The correlation coefficients areγ1=0.974 2 andγ2=0.993 5 respectively. It can be seen that the improved model has good effect on the study of multi particles.

4 Conclusion

This paper presents a model improvement and validation of a circular electrostatic sensor. The FEM is used to solve a set of the Poisson equation based on the electrostatic induction theory. The numeric iteration is adopted to solve a non-linear problem and calculate the parameters of the single Gaussian equation under the least square criteria. The experiments are carried out in an electrostatic meter, the results are corrected based on the trajectory differential equation. The results confirm the improved model is feasible and practical. This investigation has laid the foundation for improved measurement through spatial sensitivity weighting.

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