Applying Fourier transformation to Eq. (14), the spatial frequency-response characteristic of a ring-shaped electrode to a unit impulse of net charge carried by particles is as follows:
8. Spatial frequency response of a ring-shaped electrostatic pulverised fuel meter In practice, the electrode must be connected to measuring equipment or to a pre-amplifier. An ac-coupled (differential) circuit is usually designed to detect the root mean square rms value of the ac signal components on the electrode, corresponding to the fluctuation in the net charge carried by particles. In this paper, the combination of the electrode and the electronics comprise the PF meter. A simplified representation of an ac-coupled circuit is shown in Fig. 5. Here Q represents the charge induced on the electrode; the equivalent resistance Rn and capacitance Cn of the electrode to the earthed conveying pipe are also considered. The input resistance and capacitance of the pre-amplifier are ignored.
A general transfer function of ac-coupled pre-amplifier is as follows:
Taking the net charge carried by the solids as the input and Uo as the output of the pulverised fuel meter, the transfer function T°$Þ of the meter is the product of both the electrode and the pre amplifier transfer functions. The meter's frequency spatial sensitivity therefore can be expressed as Tr°$Þ ¼ Hr°$Þ _ P°$Þ, i.e.,
The values of Rn and Cn of a given electrode depend on its geometry and the insulation between the pipe wall and the electrode. Assuming that the electronics are designed to detect charge variations on the electrode that correspond only to fluctuations in the net particle charge, the power spectral function of the meter output is as follows:
where QPNETr°$Þ is the frequency transfer function of the net charge of the roping stream of particles at radial position r and qPNETr°$Þ is the fluctuation in the net charge. The ''flow noise'' is usually regarded as a band-limited white noise, as is the fluctuation in net charge qPNETr°$Þ. Fig. 6 shows the predicted power spectrum of the meter at a velocity of 25 m/s based on Eq. (19), where the fluctuation in net charge is assumed to be band-limited white noise having a constant power density spectrum over the effective frequency range. Typical parameters for the circuit of Fig. 5 are the
following: Rn ¼ 1MO, Cn ¼ 1.47 nF, Rd ¼ 0:1MO, and Cd ¼ 1mF.
9. Spatial sensitivity expressed in the time domain
According to Parseval's formula [15], a stochastic process is related to its power spectral density function. From Eq. (19), it thus follows that
where U0rms is the rms PF meter output, and oT is the effective upper cut-off frequency.
If jqP NETr°$Þj2 is the white noise of value A0 up to frequency oT, then the spatial sensitivity of the ring-shaped, pulverised fuel meter can be calculated using the following equation:
10. Experimental validation
The experiments described in this section were carried out at the laboratories of Casella CRE Energy, Stoke Orchard, Cheltenham in the United Kingdom. A14 inch (355.6 cm) diameter electrostatic meter was installed in a test facility, as depicted in Fig. 8. The roping air-solids flowing in parallel with the pipeline were provided using a 1 inch (2.54 cm) diameter jet connected to the solids dispenser. A Venturi tube was located beneath the dispenser, where compressed air and solids were mixed and conveyed to the test section. The jet position in the conveyor could be adjusted both horizontally and vertically, so that the roping flow could be injected at different radii. The flow rates of the solids and conveying air mass were monitored by computer and logged into files. Their values and their fast Fourier transforms (FFT) were calculated using National Instrument's Labview software. The sampling rate of data collection was 10kHz, and 2048 data points were collected for each measured parameter. The solids ''roping'' flow rate was about 300 kg/h, and the conveying airmass flow rate in the jet was about 60 kg/h. The velocity of the ''roping'' flow was kept at about 25 m/s with the help of a draft fan at the end of the rig. Both theoretical analysis and experimental results were normalised by setting their respective maximum values to unity. The experimental results can therefore be compared to those predicted from Eqs. (19) and (21), because the net charge carried by the particles and its fluctuation level are assumed to be proportional to the solids concentration.
The shapes of the corresponding curves are similar in these two figures, except that the peak positions and the geometrical means of the curves in Fig. 9 are both shifted downward in frequency. This may have been caused by the jet vibration which would have contributed low-frequency components to the flow stream. The solids moving adjacent to the pipe-wall contribute higher frequency components and have higher level overall response, because the sensing length of the electrode along the pipeline decreases with increased relative radial position (r/R) [13]. The overall shape of the measured spatial response shown in Fig. 10 is similar to the calculated response shown in Fig. 7. In practice, it is difficult to obtain reliable experimental spatial response results near the pipe wall where the roping flow diverges. In practice, it was very difficult to keep the roping flow perfectly constant, because the flow rate of the conveying air, as well as the pressure at the Venturi tube,
changed randomly. Thus, although the curve in Fig. 10 is comprised of about 2000 data points, it is not smooth
2. Mathematical model and finite element model of electrostatic inductive sensor
2.1. Fundamental theory and mathematical modeling of the sensor
The charging process of pneumatically transported particles is quite complex. The magnitude and sign of charged particle is not only dependent on the physical and chemical properties of the particle including particle shape, particle size, roughness degree, relative moisture content, and work function, etc., but also dependent on the material and configuration of the pipe and transportation conditions
in the pipeline. Up to now, most research work has been based on experiments to understand particle electrification [18,19]. When powder particles are conveyed pneumatically in the pipeline the typical examples of electrostatic charging problems are an inadequate flowability and ease to cohesiveness and adhesiveness of powder particles, which may result in the particle sedimentation in the inner pipe wall. The most serious is that the local air will be broken down if the electrostatic charge of powder particles accumulates too much, which may cause electrostatic discharge, fire, and even explosion.
So it is very necessary to carry out in-depth research on the charging mechanism of particles and to generate better tools for prevention and even making full use of the charging behavior of powder particles [20]. An electrostatic inductive sensor, which is based on particle electrification, is used to measure the net charge of particles and particle flow parameters. A simple schematic diagram of an
electrostatic inductive probe is illustrated in Fig. 1. The probe mainly consists of a ring-shaped electrode with good conduction, a grounded shield screening to resist electromagnetic interference, and a dielectric pipe to isolate the electrode, the pneumatic transport pipeline and particles. As shown in Fig. 1, the radius of the screen is denoted by R3, the inner radius of the pipeline by R1 and the outer
radius by R2, the length of the dielectric pipe by l and its relative permittivity by eri, and the axial length of the electrode by We. Gajewski [21] and Yan [2], respectively, proposed the mathematic models of an electrostatic sensor when a point
charge moves along the pipeline. In Gajewski's model, only when the charged particles move along the central axis of the sensor, the electrical potential of the electrode was studied. And the effects of particle position and the geometric structure of the probe on the output signal of the sensor were not pointed out. Hence, the spatial sensitivity of the sensor cannot be obtained from the model. In Yan's model, the electrostatic field formed by a point charge within the pipeline was treated as a free electrostatic field in infinite space, therefore the induced charge on the electrode can be calculated. In fact, the presence of the electrode and the pipeline will cause the interaction between the induced charge and the electrostatic field generated by the point charge, and thereby influencing the induced charge on the electrode. Therefore, Yan's model only gives some qualitative information about
the sensing characteristics of the sensor. The electric field formed by the point charge interacts
with the induced charge on the electrode, allowing the sensing electrode to reach an electrostatic equilibrium. This process is so short °10E _ 17sÞ [22] that the interaction between the moving point charge and the electrode could be treated as an electrostatic field. Additionally, compared with electrostatic effects, electromagnetic effects due to small fluctuations of charged particles flow are estimated to be negligible. When a point charge is positioned in the sensing zone of an electrostatic inductive probe the electrostatic field can be described using the following Poisson equation and boundary conditions:
where f is the electric potential, r is the body charge density, e is the dielectric permittivity, Gp, Gs, Ge are the boundaries of the pipeline, the shield, and the electrode, respectively, cons denotes that the electrode is an equivalent potential body. On the electrode, the induced charge density s can be
written as follows:
where D is the electric displacement vector near the inner wall of the electrode, E is the electrostatic field intensity, e0 is the permittivity of air. Therefore, the induced charge on the electrode with surface area S can be calculated using the following formula:
If the structural parameters of the probe and the position of the unity charge are already known, the numerical solution of the above mentioned mathematical model can be obtained using finite element method (FEM), and the induced charge can be calculated using Eq. (3).
2.2. Finite element modeling
The axial orientation, radial orientation, and tangential orientation of the electrostatic probe are chosen as the z-axis, r-axis and y-axis of the global coordinate system, respectively. When a unity point charge is put in different positions within the infinite metal pipeline grounded, the electrostatic field formed has two different cases. The first is axially symmetrical and two-dimensional when the point charge is positioned on the central axis of the pipeline, and the second is three-dimensional when it is off the central axis. For the two-dimensional electrostatic field the numerical results can be obtained quickly and accurately using FEM while the calculation speed for three-dimensional
field is very slow. However, in the second case, it is assumed that the unity charge would be distributed evenly with the line charge density 1=°2prÞ C=m on the circle of its radial position r (seen in Fig. 2). The electrostatic field
generated by the circle line charge is axially symmetrical. The assumed axially symmetrical electrostatic field within the pipeline is different from that of the point charge, but from the point of view of the induced charge on the electrode it is the same because the induced charge is equivalent at the same axial and radial position according to the electrostatic field superposition theory and the axially symmetrical structure of the probe. Therefore, the three-dimensional problem can be reduced to a twodimensional one.
The finite element analysis software ANSYS with its friendly graphic interface, better programming structure, and graphic and interactive pre/post processing function, was adopted to calculate the numerical results of the model, which greatly reduces user's workload of modeling, finite element solving, and evaluation and analysis of numerical results. From the above analysis of the finite
element model, the electrostatic field can be treated as a two-dimensional one to be calculated wherever the point charge is positioned within the pipeline. According to the structural size, shape, and precision request PLANE121 and INFIN110, which are 2-D axisymmetric quadrangle element with eight nodes, are chosen in the finite model. Fig. 3 shows a created finite element model of the sensor when the point charge is on its central axis. The electrostatic potential of the nodes of the finite element model can be obtained, and then the electrostatic field intensity can be derived.
3. Spatial sensitivity of electrostatic inductive sensor:
The uniform spatial sensitivity distribution is required in the measurement of the particle concentration to reduce the effect of flow regime. The spatial sensitivity of a ringshaped electrostatic inductive sensor can be defined as the absolute value of the induced charge on the electrode when a unity point charge is positioned in the sensing zone of the sensor. From the finite element analysis model of the electrostatic sensor, it is known that the induced charge on the electrode is related to the axial coordinate z and the radial coordinate r of the point charge within the sensing zone and is irrespective of the tangential coordinate y. Therefore, the spatial sensitivity can be represented by the following dimensionless parameter:
where q is the induced charge on the electrode and Q is the point charge. The induced charge can be calculated using the above mentioned finite element model when the point charge is in different positions of the sensing zone. Furthermore, the spatial sensitivity is derived. The thickness of the electrode was ignored, and the other parameters are kept at R1 ¼ 50 mm, R2 ¼ 60 mm, R3 ¼ 80 mm,
eri ¼ 3:75, l ¼ 100 mm. Fig. 4 depicts the variation of the sensitivity with the axial coordinate z at different radial positions. The axial length of the electrode (WeÞ is 20 mm. It can be seen that for the different radial positions, the sensor is most sensitive at z ¼ 0, and the sensitivity decreases with the
increase of jzj. But the sensitivity is not equal to zero outside the electrode, which illustrates that the length of the sensing zone is slightly larger than that determined by the geometry of the electrode due to the edge effect. The most rapid change in the sensitivity occurs when the unity point charge moves along close to the pipe wall with a fixed moving velocity. Fig. 5 shows the variation of the sensitivity of the sensor with radial coordinate at z ¼ 0. The axial length of the electrode is 4, 10, 20, and 50 mm, respectively. It can be seen from Fig. 5 that the axial length of the electrode is an important factor influencing the spatial sensitivity of the sensor. The more homogeneous sensitivity distribution
attributes to the longer axial length of the electrode. Additionally, the sensitivity at different radial positions r, as expected, is different. The sensitivity increases with the increase of radial coordinate r, which illustrates the sensor is more sensitive to the particle adjacent to the pipe wall.
ARTICLE IN PRESS
Consequently, Eq. (8) can be rewritten as follows:
where F denotes the Fourier transform, S°wzÞ denotes the spatial filtering transfer function of the electrostatic inductive sensor. jS°wzÞj denotes the spatial amplitude frequency spectrum of the electrostatic sensor at the given radial position. The spatial frequency properties at different radial positions are shown in Fig. 7. The structural parameters of the probe are referred to in Section 3. It can be seen that the electrostatic inductive sensor acts as a low-pass filter in spatial frequency domain. Namely, only the signal with a lower spatial frequency can pass through the probe. Additionally, the smaller the radial coordinate r is, the narrower the spatial frequency bandwidth is. Fig. 8 depicts the comparison of the spatial filtering property of the electrostatic sensor with the electrodes' axial length at 20 and 10 mm, respectively. It shows that wherever the central axis of the probe or near pipe wall is, the shorter the axial length of the electrode, and the wider the spatial frequency bandwidth is relatively. And the amplitude increases with the increase of the axial length in the working spatial frequency range, which illustrates that there is a larger energy transfer with the longer axial length.
5. Temporal frequency response characteristics of electrostatic inductive sensor:
If the radial position r is fixed and the moving particle is seen as a unit point charge the input signal of electrostatic inductive sensor can be substituted by the unit impulse signal d°z þ vtÞ. d denotes Dirac function. Then the unit impulse response function °h°tÞÞ of the probe can be expressed as follows:
where SS°f zÞ is the spatial Fourier transform of s°zÞ and equal to S°2pf zÞ. Hence the Fourier transform of h°tÞ can be used to represent the temporal frequency response function of the probe. The frequency response property H°f Þ can be expressed as follows:
Fig. 9 shows the temporal frequency response characteristics H°f Þ of the probe. As can be seen from Fig. 9 the temporal frequency characteristics are similar to its spatial frequency property and also act as a low-pass filter. And the effects of the radial position r and the axial length of the electrode on the temporal frequency response characteristics are similar to those in the spatial frequency domain. Fig. 10 depicts the effects of particle velocity on the temporal frequency response characteristics of the probe. It can be seen that the quicker the particle velocity is, the wider the temporal frequency bandwidth of the probe, but the smaller the amplitude of the temporal frequency characteristics is relatively. The probe has to be connected to a preamplifier to amplify the induced charge signal. So the frequency response characteristics of the measurement system should be dependent on both the temporal filtering property of the probe and the dynamic property of the preamplifier. The electrostatic inductive sensor is simplified as an equivalent circuit shown in Fig.11. Where q°tÞ is the induced charge on the electrode, Ce, Re are the equivalent capacitance and insulation resistance of the electrode, respectively, Ci, Ri are the equivalent input capacitance and input resistance of the signal conditioning circuit, respectively, and ui°tÞ is the input voltage of the preamplifier. Therefore, the transfer function of the signal conditioning circuit can be expressed as follows:
where R ¼ °Re _ RiÞ=°Re þ RiÞ, C ¼ Ce þ Ci, Ui°sÞ denotes the Laplace transform of ui°tÞ and Q°sÞ is the Laplace transform of q°tÞ. Obviously, the signal conditioning circuit is a high-pass filter.
Generally, the upper frequency of the output signal of an electrodynamic sensor is very low at less than 2000 Hz. If the condition jsRCj51 can be satisfied and the initial induced charge equals zero, the output signal of electrostatic sensor can be expressed as follows
From Eq. (13), it can be seen that the output signal of the sensor is the change of the induced charge on the electrode with time. Therefore the temporal frequency response characteristics of the sensor, including the probe and the signal conditioning circuit, can be expressed as
6. Experiment results and discussion:
The schematic diagram of the experimental apparatus of the dynamic property of an electrostatic inductive sensor is shown in Fig. 12. Particles are supplied from the funnel into a PVC pipe (inner diameter 10 mm) and then conveyed towards the probe by gravity. The distance between the
funnel and the center of the electrode can be adjusted upwards or downward to change the free fall velocity of the particles passing through the probe. For example, if the distance is h, the free fall velocity of the particles can be approximated using Eq. °2ghÞ1=2. Different radial positions can be controlled by the horizontal adjustment of the exit of the PVC pipe. The measurement system consists of the probe, a preamplifier, and a PC-based data acquisition system. The output signal of the probe is amplified by the preamplifier, then is sampled, saved, and analyzed by the PC. The sampling rate is 5000 Hz and the length of data logged is 5120 points at a time. The mathematical model of the electrostatic inductive sensor is based on a point charge, but the particles conveyed pneumatically in the pipeline have a special shape and size distribution range in the defined industrial process. Hence, during the experiments, the point charge was substituted with a charged sphere particle. From the theoretical analysis of an electrostatic inductive sensor, we know that there are many factors influencing its dynamic performance. To determine their effects on the dynamic characteristics and
evaluate the mathematical model of the electrostatic sensor, a series of experiments were carried out. Because it is very difficult to control the charge carried by the particle, all the experimental results were normalized to make them comparable. The structural parameters of the probe are referred to in Section 3, and except for special illustration, the length of the electrode is 10 mm.
6.1. Verification of the mathematical model of electrostatic inductive sensor:
From the mathematical model established above, the induced charge on the electrode can be calculated using FEM. In principle, the induced charge can be obtained from the integral of the output signal of the electrostatic sensor during the experiments, but it is very difficult to accurately control the charge carried by the particle percent experiment. Therefore, the normalized values of theoretical induced current and experimental current of the sensor were compared to verify the mathematical model. The length of the electrode is 20 mm. Fig. 13 shows the comparison between the normalized theoretical and experimental currents at r ¼ 0 and 40 mm, respectively. The conveyed material is a glass particle with a diameter of 4mm and the free fall velocity of the particle is about 4.95 m/s °°2ghÞ1=2 ¼ °2 _ 9:8 _ 1:25Þ1=2Þ. It can be seen that the trend of the theoretical output currents of the electrostatic sensor is consistent with the experimental currents. But the range of the theoretical currents in the time domain is less than that of the experimental currents. This may be due to the existence of particle shape and size, which lead to the longer stay time of the particle in the
sensing zone of the sensor.
6.2. Effects of radial positions on the temporal frequency characteristics
Fig. 14 shows the output voltage of the electrostatic inductive sensor for a glass particle with a diameter of 4mm dropped through the axis and the radial position °r ¼ 40mmÞ of the sensing electrode, respectively. The time recorded is 1 s and the free fall velocity of the particle is 4.95 m/s. One can see that the polarity of the output voltages changes from positive to negative when the
charged particle passes through the electrode, from which we can judge that the particle carries a negative charge. Compared to the output voltage on the axis of the electrode, the voltage range near the pipe wall is relatively narrower in the time domain, which leads to the wider frequency range in the temporal frequency domain. Fig. 15 illustrates the FFT results of the output voltages (Fig. 14), which represents the effect of the radial position on the temporal frequency response characteristics of the sensor. It is obvious that the electrostatic inductive sensor, including the probe and signal conditioning circuit, acts as a band-pass filter in the temporal frequency domain. And the temporal frequency bandwidth is wider near the pipe wall, which corresponds to the theoretical temporal frequency characteristics of the electrostatic sensor. But it can also be found that there exists the difference between the theoretical and experimental temporal frequency characteristics of the measurement system. The theoretical frequency bandwidth is wider than the experimental value, and the experimental peak frequency shifts to the left. This may be due to the particles having shape and size, yet the theoretical frequency characteristics are based on point charge.
6.3. Effect of particle velocity on the temporal frequency characteristics
Fig. 16 illustrates the effect of the velocity on the temporal frequency characteristics of the electrostatic sensor. The free fall velocities of the particle are 3.83 and 4.95 m/s, respectively. One can see that with the increase of particle velocity, the peak frequency at the maximum amplitude moves rightward and the temporal frequency bandwidth widens.
6.4. Effect of the axial length of the electrode on the temporal frequency characteristics
Fig. 17 shows the effect of the axial length of the electrode on the temporal frequency characteristics of the sensor. Particle velocity is about 4.95 m/s, and the axial lengths of the electrode are 10 and 20 mm, respectively. It can be seen that wherever on the central axis of the electrode or near pipe wall, the wider temporal frequency bandwidth of the measurement system attributes to the shorter electrode, which corresponds to theoretical prediction.
6.5. Effects of particle size on the temporal frequency characteristics
According to Sections 6.1 and 6.2, the existence of particle shape and size causes the discrepancy between the frequency response characteristics of the sensor. The moving velocity of particles is approximately 4.95 m/s. The effect of particle size on the temporal frequency characteristics of the sensor is shown in Fig. 18. Wherever on the central axis or near the pipe wall, the bandwidth for
the larger particle passing through the electrode is relatively narrower. This may be due to the larger particle being within the sensing region of the electrode for a longer time at the same velocity, which leads to the lower change and longer period of the output signal of the sensor.
