7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Sensing Characteristics of Electrostatic Sensor Arrays for gassolid flow measurement
Chuanlong Xu Heming Gao Jian Li Shimin Wang Yiqian Xu
Thermal Energy Engineering Research Institute, Southeast University, Nan~ing, 210096, P.R.China.
Email: chuanlongxu~seu.edu.cn smwang~seu.edu.cn
Keywords: Frequency response characteristics: Spatial filtering effect: Electrostatic Sensor Arrays: Gassolid twophase
flows: Particle charging
Abstract
In recent years, great advance has been made on electrostatic sensing technique for gassolid flow measurement. However,
reports on sensing mechanism and characteristics of the electrostatic sensor arrays were very few. Electrostatic sensor array is
one of key part of electrostatic tomography system. The geometric sizes of the arrays, particle distribution and velocity
distribution have important effects on its spatial sensitivity, spatial filtering effect and temporal frequency response
characteristics. In this paper the charge induced on the electrode of the arrays with different geometric sizes from a single
particle having a unity charge was modelled mathematically, and the 3dimensional electrostatic field generated by the point
charge in the sensing zone of the arrays was solved using finite element method. The effects of geometric and material
parameters of the sensor arrays including the length and angel of the electrode, the thickness, length and permittivity of the
dielectric pipe, the radius of metal screening on the sensing field of the electrode were investigated systematically, and further
a calculation model for the nondimensional sensitivity of the arrays was suggested based on Gaussian function and its spatial
filtering fundamental theory and spatial filtering effect were also analyzed quantitatively. The temporal frequency response
characteristics of the sensor arrays were also derived. The experimental results on a gravityfed conveyor were presented. The
theoretical and experimental results obtained demonstrate that the sensor arrays acts as a lowpass filter in the spatial
frequency domain and its spatial frequency characteristics are closely related to the space position of the charged particle and
the length of the electrode. The measurement system, including the sensor arrays and signal conditioning circuit, acts as a band
pass filter, and the space position, the length of the electrode and particle velocity have important effects on the temporal
frequency characteristics of the measurement system.
Introduction
Measurement of Gassolid flow parameters such as velocity,
concentration distribution, flow rate and regime is an
important research direction in the multiphase flow field,
and also is one of the urgent problems to be solved in
theoretical and experimental study of gassolid flow.
However, due to the random interface effect and velocity
slip between the gassolid two phases in space and time, the
gassolid flow parameters measurement is much more
complex than the singlephase flow. For example, the
properties and spatial distribution of solid particles, and
flow characteristics affect the measurement accuracy of the
instruments for gassolid flow in pneumatic conveying
system (Y.Yan et al 1995). So far, based on different
principles many noncontact measurement methods
including electrical method (capacitance, electrostatic),
attenuation (optical, ray), process tomography and NMR
have been researched and developed, which have their own
characteristics and application scope (Y.Yan 1996).
Electrostatic technique is one of electrical methods. When
particles are transported pneumatically in the pipelines,
charges will be generated on the particles due to the
following effects: 1) Frictional contact charging between
particles and pipe wall; 2) Collisions between particles: 3)
Friction between the particles and gas flow. Particle
charging in pneumatic pipeline is a comprehensive result of
gassolid twophase flow dynamics such as the flow
characteristics, the regime and the energy exchange,
material and configuration of the pneumatic conveyor and
the particle properties (particle size and shape, work
functions, impurities, particle surface roughness, moisture
content of particle, volume resistivity, permittivity etc)
(H.MASUDA et al 1976/19777, Shuji et al 2003, D.I.
ArmourCh!elu and S. R. Woodhead 2002, Y. Li et al 2002),
and thus the particle charging contains rich flow
information on particles movement, individual particle sizes,
collisions between particles and pipe wall, clusters or
agglomerates (temporary or persisting) of particles and
equipmentdominated structures such as "stratified stream"
overall particle flow patterns. However, because of the
gassolid flow randomness and the complexity of particle
charging mechanism, it is difficult to predict the charge
carried by the moving particles theoretically, and yet the
charge can be detected by an electrostatic sensor. When the
charged particles pass through the sensor, the
quasielectrostatic field generated by the moving charged
particles fluctuates which leads to the continuous variation
of the induced charge on the sensor, and thus the parameters
such as particle velocity, concentration and mass flow rate
can be obtained using the electrostatic sensor combined
with suitable signal analyzing and processing techniques (Y.
Yan et al 1995, C. Xu et al 2008, J. Ma and Y. Yan 2000, J.
B. Gajewski 1996, J. B. Gajewski 2008, J. Q. Zhang et al
2003, C. Xu et al 2009). Recently, passive electrostatic
technique has been widely applied in particle flow
parameters measurement because of its simple structure and
being a noncontact, highly sensitive, lowcost and safe
sensor that is suitable for the harsh industrial environment.
Among the developed electrostatic sensors, the
mathematical model of the ring sensor has been established,
and the sensing characteristics such as the spatial sensitivity
distribution, spatial filtering effect and frequency bandwidth
has been studied in depth, which provides a theoretical basis
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
defined spatial weight function in the sensing zone of the
arrays. Therefore the electrode of the sensor arrays acts as a
spatial filter in the spatial domain (C.Xu et al 2007). The
size of the electrode affects the scale, temporal frequency
characteristics and bandwidth of the output signal.
Therefore it is very important for researching the sensing
characteristics of the electrostatic sensor arrays to reveal its
sensing mechanism, to explain the nature of the output
signal and to evaluate the performance of the measurement
system.
The threedimensional mathematical model of the induced
charge on the electrostatic sensor arrays due to a single
charged particle with a unity charge was established in this
paper and the electrostatic field in the sensor array was
calculated by finite element method. Further the effect of
geometric and material parameters of the sensor arrays on
its spatial sensitivity distribution was investigated and a
calculation model for the nondimensional sensitivity of the
arrays was suggested based on Gaussian function. On the
basis of the spatial sensitivity, the spatial filtering theory
was introduced and the spatial filtering effect was also
analyzed quantitatively, and further the temporal frequency
response function of the sensor arrays was derived
theoretically.
Mathematical model of the electrostatic sensor
arrays
Construction and circuit of the electrostatic sensor
arrays
The schematic diagram of the electrostatic sensor arrays is
shown in Fig. 1. The arrays mainly consists of 8 arc
electrodes with good conduction, 2 grounded guard rings, a
grounded mental screening to resist electromagnetic
interference, and a dielectric pipe to isolate the electrodes,
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 R; and the outer radius by Rz, the
length of the dielectric pipe by I and its relative permittivity
by e,, the length of the electrode by W and its electrode
angle by 8, and space between the guard ring and the
electrodes by 1,.
for evaluating the performance and optimal design of the
sensor (Y. Yan et al 1995, J. B. Gajewski 2000, J.Krabicka
and Y.Yan 2008, L. Peng et al 2008, A Nesterov et al 2007,
J. Zhang and J. Coulthard 2005, C. Xu et al 2007). At
present, the ring sensors have had wide industrial
applications in gassolid flow parameters and flow
characteristics measurement. However, they were limited to
the correlation velocity measurement of pneumatically
conveyed particles, while the measurements of particle
concentration and velocity distribution in gassolid flow
system can not be realized by the ring sensors. This reason
is that the induced signal on the sensor is a certain result or
resultant action of the electrostatic induction of all (positive
and negative) charges in the sensing zone at a given instant,
and thus local information of the particle flow in the cross
section of pipeline can not be obtained. To solve the
problem, on the one hand the particle charging mechanism
should be further studied deeply and on the other hand it is
very necessary to research and develop the novel
electrostatic sensors. Electrostatic Tomograply (EST), a
passive tomography for gassolid flow developed in recent
years, uses the electrostatic sensor arrays to measure particle
charge distribution on the cross section of the pipeline, and
hence the particle flow parameters such as the concentration
distribution, velocity distribution and mass flow rate can be
obtained (Mlasashi Machida and Brian Scarlett 2005, R. G
Green et al 1997, R. G Green, M. F. Rahmat and M. Henry
1997, J. zhang 2009). But the sensing mechanism of the
electrostatic sensor arrays has not been understood clearly.
The geometric and material parameters of the electrostatic
sensor arrays including the length and angel of the electrode.
the thickness, length and permittivity of the dielectric pipe
and the radius of metal screening have important effect on
its sensing characteristics (spatial sensitivity distribution,
spatial filtering effect and frequency characteristics). Spatial
sensitivity reflects the differences of the induced charge on
the electrode when a unity point charge is differently
positioned in the sensing zone of the sensor, which is caused
by the nonlinear sensing mechanism and structural
imperfections of the electrostatic sensor arrays. It is very
challenging to use the electrostatic sensor arrays for the
measurement of solid particle concentration and velocity
distribution due to its inhomogeneous spatial sensitivity
distribution. Take an electrostatic tomography system for
example, even with particles distributed uniformly in the
sensing zone, the gray value distribution of the
reconstructed image is not uniform. Similarly, even if the
solid particles have the same plwsical and chemical
properties, and flow conditions are similar in the pipeline,
but the particle distribution in the sensing zone is different,
the output signal of the sensor arrays is not the same.
Therefore to reduce the effect of flow pattern on
measurement accuracy, a basic requirement for electrostatic
sensor arrays is to have a higher and uniform sensitivity
field. In addition, because the particle velocity and
concentration distribution over the crosssection of the
pipeline exist and the electrode of the electrostatic sensor
arrays acts as a "body sensor", it is necessary to determine
and clarify the effective sensitive zone of each electrode.
When the charged particles move along the axis of the
pipeline and pass through the electrostatic sensor arrays
with a defined geometric shape and size, the "electrostatic
flow noise" produced by them will be averaged with the
Guardring electrode
Dielectriepipe ~
10ental screening
Electrodes
r~
(a) crosssecti
(b) 3dimensional
ion
Fig. 1 Schematic diagram of the electrostatic sensor arrays:
(a) crosssection (b) 3dimensional
The electrostatic sensor arrays have to be connected to a
preamplifier to amplify the induced charge on the electrodes.
The electrode is simplified as an equivalent circuit,
schematically shown in Fig. 2. According to Kirchhoff's
current law, the following expression applies
dq(t) du, (t) u, (t)
dt dt R ()
where u,(t) is the input voltage of the preamplifier, R= (Re
R I ,ii .+R,): C= Ce C,OC,; Ce and Re are the equivalent
capacitance and insulation resistance of the electrode'
respectively: C, and R, are the equivalent input capacitance
and input resistance of the preamplifier: C, is the tray
capacitance of the cable, and q(t) is the induced charge on
the electrode.
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Generally, the frequency of the output signal of the
electrode is less than 2000 Hz. The condition sRC<<1 is
easy to be fulfilled with the reasonable choice of the
equivalent capacitance C, and hence the interface circuit
was resistive, namely case (1).
Mathematical model of the electrostatic sensor arrays
A charged cylinder in pipe is infinitesimal compared with
the sensor arrays, and thus it can be expressed as a point
charge. When the point charge passes through the electrodes,
the charge and voltage will be generated on the electrodes
due to electrostatic induction. The electric field formed by
the point charge interacts with the induced charge on the
electrodes, allowing the sensing electrode to reach an
electrostatic equilibrium. This process is so short (10E17s)
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 the
charged particles flow are negligible. When a point charge is
positioned in the sensing zone of an electrostatic sensor
array, the electric field can be described using the Poisson
equation and boundary conditions. Electrostatic sensor
arrays are followed by different interface circuits and the
boundary conditions is different (seen in section 2.1). When
the interface circuit is resistive, the electrodes and mental
screening of the sensor arrays meet Dirichlet boundary
condition. Therefore the mathematical model can be
described as follows:
V (E(x, y, z)V#(x, y, z)) = p(x, y, z)
#(x, y, z) cXYZ=>sr = 0
(8)
#(x, y, z) cx,y,z>sr,,r ,rg =0
E, = 0
Where is electric potential: p is body charge density: E
is dielectric permittivity: Fp, Es, Eg, Fe are the boundaries
of the pipeline, the shield, guard electrodes and the
electrodes, respectively: E is the electric field strength at
the infinite distance.
According to the electrostatic field theory, if the distribution
of the relative permittivity e(x, y, z), charge density
distribution p(x, y, z) and boundary conditions are known
within the sensing zone of the electrostatic sensor arrays, the
only electric potential distribution can be determined based
on the above equation (8), and thus the induced charge q on
the electrodes with surface area S can be calculated using
the following formula:
q2 V #ds (9)
The construction of the electrostatic sensor arrays is
threedimensional and complicated. At present, it is difficult
to derive analytical solution of the mathematical model, but
finite element method can be used to solve the model. A
variety of electromagnetic field calculation software
developed in recent years makes it possible to calculate
complex electrostatic field. In the paper, Ansoft (Maxwell
3D Field Simulator) was applied to calculating induced
charge on the electrodes of the sensor arrays.
Finite element model of the electrostatic sensor arrays
Considering that the electrostatic sensor arrays is
Cd R{r CI R~ C
R
n'" >( 4 qm
Fig. 2 Equivalent circuit of the electrode of the electrostatic
sensor arrays
If the initial condition q(0) is zero, applying the Fourier
transform to Eq. (1) gives the following equation
/0})1+ juRCjw (
where U,0w) is the Fourier transform of u,(t), Qgw) is the
Fourier transform of q(t), and xv is the angular frequency.
Three cases are discussed under the following different
interface circuits:
1) IjwRC<<1, Equation (2) canbe simplified as
U, (jw)= jwRQ(jW) (3)
Then u,(t) is
dq(t)
U, (t) =R (4)
Equation (4) shows that the input voltage of the interface
circuit u,(t) is proportional to the induced current, and thus
the interface circuit is resistive.
2) IjwRC>>1, Equation (5) can be derived
g q/ U (5)
Then u,(t) is
q(t)
u, (t) = (6)
Equation (6) shows that u,(t) is proportional to the induced
charge on the electrode and is inversely proportional to the
equivalent capacitance, so the interface circuit is capacitive.
3) If [IjvRC is comparable with 1, we have
Ul~C RC R
Equation (7) shows that u,(t) is related to the induced charge
on the electrode and its integration with time so the circuit
is resistive and capacitive. C is a constant.
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
The sensitivity distribution of the electrostatic sensor
arrays
8 independent values of the induced charge can be obtained
in the 8electrode electrostatic sensor arrays, and each has
their own the sensing zone. But there is only 1 type of the
sensing field distribution because of the axial symmetry of
the electrostatic sensor arrays. So it just needs to calculate
the sensitivity distribution of one electrode, and the
sensitive field distribution of other electrodes of the arrays
can be obtained by rotating method.
0.35
axisvmmetric and threedimensional finite element
simulation calculation is timeconsuming, 5 flow
streamlines (denoted by a, b, c, d, e) shown in Fig. 3 were
selected to represent the axial sensitivity distribution of the
arrays. Because the sensor arrays will be applied to
densephase pneumatic conveying systems of pulverized
coal under highpressure, in the numerical simulations the
inner radius of the pipeline R; is 5 nun, and the positions of
the point charge over the cross section of the pipe are
corresponded to the radial coordinates r =4.5 nun, 3 nun, O
nun and the angles 2, 0. Fig. 4 shows the finite element
mesh of the electrostatic sensor arrays, infinitesimal charged
cylinder and dielectric pipe. When the point charge is in a
position, the induced charge values on the electrodes can be
produced. For example, for the electrode 1, when the point
charge is located in space poison d, the induced charge on
the electrode 1 can be obtained through the calculating the
electrostatic field, and the induced charge values on the
other seven electrodes (28) also can be got. Due to the axial
symmetry of the electrostatic sensor arrays, the 7 values are
equivalent to the ones corresponding to the electrode 1 in
the seven different space positions of the same circle
(dld7). Similarly, when the point charge is located in b, the
calculated value on the electrode 2 is equivalent to the value
on the electrode 1 in the point position f:
3 2
4 + +
5 ~8 Y
Fig. 3 Typical streamlines passing through the electrostatic
sensor arrays
Fig. 4 Finite model of the electrostatic sensor arrays
Spatial sensitivity field of the electrostatic sensor
arrays
The spatial sensitivity of an electrostatic sensor arrays can
be defined as the induced charge on the electrode when a
unity point charge is differently positioned at space
coordinate (x, y, z) in the sensing zone of the arrays.
Therefore the spatial sensitivity can be represented by the
following dimensionless parameter:
s(x, y, z) =  (10)
Q
where q is the induced charge on the electrode and Q is the
point charge.
Axial coordinate (mm)
Fig.5 Variation of the sensitivity with the axial coordinate :
for different streamlines (W=10nun, 840, Rz=10nun,
RI=5nun, e,=3.4, R3=15 nun, I =45 nun, 1, =1 nun)
Fig.5 depicts the variation of the sensitivity with the axial
COordinate z for different streamlines. Fig. 6 shows the
sensitivity distribution of the sensor arrays at the central
cross section (z=0). It can be seen from Fig.5 and Fig. 6 that
the sensitivity of each electrode of the sensor arrays is
threedimensional, and is inhomogeneous both in the cross
section and the axial direction of the pipe. At a fixed cross
sectional position, the electrode is most sensitive at z=0, and
the sensitivity decreases with the increase of z. 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 electrode. The reason is that that when a point
charge moves nearby the pipe wall of the electrode, the
solid angle relative to the electrode is larger, and thus the
inducted charge on the electrode is greater. With the axial
coordinate z increasing, the sensitivity on the nonaxis
changes quicker than the one on the axis. Fig. 6 shows the
sensitivity distribution at the central cross section (z=0) of
the electrostatic sensor arrays. It can be seen that the sensing
zone of the electrode 1 is a fanshaped one, which covers
the space determined by the electrodes 1, 2 and 8. However
it should be noted that on the other streamlines a, Jf g, the
sensitivity of the electrode 1 is so low that the charge on the
electrode 1 induced by the particle charge can be negligible.
The electrode 1 is sensitive in the electrode angle region
nearby the pipewall while the sensitivity is very low on the
center and outside the electrode angle. The electrode of the
sensor arrays acts as a body sensor. Its sensing zone
includes not only the space determined by their own
electrode angels, but also half of space of the adjacent
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
electrodes. Therefore the crosstalk between the adjacent
electrodes exists. When designing the electrostatic sensor
arrays, it is very necessary to minimize the crosstalk and to
clarify their own sensing zones.
mm, 1,=1 mm)
~0.3
S0.2
5
0.4
,0.3'
'0.2
c
0>
5
0
y/(mm)
0
x/(mm)
5 5
(a) W=5 mm
y/(mm) 5 5~ x/(mm)
Fig. 6 Sensitivity distribution at the central cross section
(z=0)(W=10mm, 8=40, R2=10mm, RI=5mm, er3.4, R3=15
mm, I =45 mm, 1, =1 mm)
Actually the axial length (0 and the opening angle (0) of
the electrode, the thickness (R2RI), length (1) and
permittivity (e,) of the dielectric pipe, the inner diameter
(R3) Of the mental screening all affect the sensitivity
distribution of the electrode. These factors will be
systematically discussed in the following sections.
Effect of the length of the electrode
For the electrodes with different lengths, the axial sensitivity
distribution of the streamlines a, d, e is investigated. Fig. 7
shows the variation of the sensitivity with the axial
coordinate z for the electrodes with different lengths (0 of
5 mm,10 mm and 15 mm. Fig. 8 shows the sensitivity
distribution at the central cross section (z=0) for the
electrodes with different lengths. From Fig. 6, Fig. 7 and Fig.
8, it can be seen that whether long or short electrode, the
sensitivity nearby the pipe wall is significantly higher than
other space zone. The longer the electrode length, the higher
the sensitivity in the corresponding crosssection and axial
position, and the larger the sensing space determined by the
electrode.
0 35
0 3 *d5
031
0 25 10
Ae10
r~0 2 I AL al5
+dl5
0 15 ee5
01
0 05
0.4
,0.3
S0.2
co 0.1
0:
5
'`
o
y/(mm)
0
5 5 ""'
(b) FF 15 mm
Fig. 8 Sensitivity distribution at the central cross section
(z=0) for the electrodes with different lengths(W=5 mm, 10
mm, 15 mm) (840, R2=10 mm, RI=5mm, er3.4, R3=15 mm,
l=45 mm, 1,=1 mm)
Effect of the electrode angle
For the electrodes with different angles, the axial sensitivity
distribution of the streamlines a, d, e is investigated. Fig. 9
shows the variation of the sensitivity with the axial
coordinate z for the electrodes with different angles (0) of
200, 300and 400. Fig. 10 shows the sensitivity distribution at
the central cross section (z=0) for the electrodes with angles.
From Fig. 6, Fig. 9 and Fig. 10, it can be concluded that the
larger the electrode angle, the higher the sensitivity
COrresponding to the axial space and cross sectional
pOSitions, and the larger the sensing zone determined by the
electrode on the crosssection. But the sensing space in the
axial direction is almost consistent.
Axlal coordinate (mm)
Fig.7 Variation of the sensitivity with the axial coordinate z
for electrodes with different lengths (W=5 mm, 10 mm, 15
mm) (840, R2=10 mm, RI=5mm, er3.4, R3=15 mm, I =45
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
and Fig. 12, it can be concluded that, the sensitivity
corresponding to axial and on the central crosssectional
positions increases and hence the sensing space with the
dielectric pipe length increasing. This may be explained by
the electrostatic field distribution within the sensing zone of
the arrays. Increasing the dielectric pipe length causes the
electric fluxline issued by the point charge to concentrate to
the electrodes, so the sensitivity increase. But it is noted that
the spatial sensitivity tends to a constant with the increase of
the dielectric pipe length.
0.35
*a20
Sd20
EI e20
*a30
+ d30
0.3
0.25
S0.2
S0.15
0.1
0.05
0.35
0.3
0.25
0.15
10 0
Axial coordinate /(mm)
Fig.9 Variation of the sensitivity with the axial coordinate z
for the electrodes of different angles (0=200, 300
400)(FV=10 mm, R2=10 mm, RI=5mm, ei=3.4, R3=15 mm, I
= 45 mm 1, =1 m m)
5 05
Axial coordinate (mm)
0.3
. 0.2
0 U
Fig.11 Variation of the sensitivity with the axial coordinate z
for different dielectric pipe lengths (l=15 mm, 30 mm, 45
mm) (W=10 mm, 840, R2=10 mm, RI=5mm, ei=3.4, R3=15
mm, 1, =1 mm)
0.4,
0
y /(mm)
0
x /(mm)
5 5
(a) 0200
>x 0.3
S0.2
0.1
0.3
S0.2
S0.1
5
0
y/(mm)
0
x/(mm)
5 5
(a) l=15 mm
0
m(/y m)
5 5 x /(mm)
(a) 0300
Fig. 10 Sensitivity distribution at the central cross section
(z=0) for the electrodes of different angles (0=200, 300,
400)(W=10 mm, R2=10 mm, RI=5mm, e,3.4, R3=15 mm, I
= 45 mm 1, =1 mm )
Effect of the length of the dielectric pipe
For the dielectric pipes with different lengths, the axial
sensitivity distribution of the streamlines a, d, e is
investigated. Fig. 11 shows the variation of the sensitivity
with the axial coordinate z for the different dielectric pipe
lengths of (1) of 15 mm, 30 mm and 45 mm. Fig. 12 shows
the sensitivity distribution at the central cross section (z=0)
for the different dielectric pipe lengths. From Fig. 6, Fig. 11
0
x/(mm)
y/(mm)
5 5
(b) l=30 mm
Fig. 12 Sensitivity distribution at the central cross section
(z=0) for different dielectric pipe lengths (l=15 mm, 30 mm,
45 mm) (W=10 mm, 840, R2=10 mm, RI=5mm, e,3.4,
R3= 15 mm, 1, =1 mm)
Effect of the permittivity of the dielectric pipe
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
(b) e,=8
Fig. 14 Sensitivity distribution at the central cross section
(z=0) for different relative permittivity (e,=1.5, 3.4, 8)
( W= 10 mm, l= 45 mm, 840, R2= 10 mm, Ry= 5mm, R3= 15
mm, 1,=1 mm)
Effect of the thickness of the dielectric pipe
The dielectric distribution in sensing space of the sensor
arrays affects the electrostatic field, so the thickness (R2R,)
of the dielectric pipe will have an impact on the sensitivity
distribution of the electrostatic sensor arrays. The sensitivity
distribution of the streamlines a, d, e was analyzed when the
values ofR2RI is respectively 3 mm, 5 mm and 6.5 mm. It is
noted that the thickness is changed by changing the inner
diameter Rlof the dielectric pipe to ensure the electrode size
be constant. Fig.15 shows the variation of the sensitivity
with the axial coordinate z for different dielectric pipe
thicknesses. Fig. 16 shows the sensitivity distribution at the
central cross section (z=0) for different dielectric pipe
thicknesses. From Fig. 6, Fig. 15 and Fig. 16, it can be seen
that the sensitivity increases with the increase of the
thickness of the dielectric pipe at fixed crosssectional
position (x, y), especially nearby the electrode. Obviously
the higher the sensitivity nearby the electrode is, the thinner
the thickness of the dielectric pipe is. Therefore when
designing the sensor arrays, it is better to choose the thinner
dielectric pipe.
0.4
as
41d5
0.3
= a3
~ p  d3
.9 9 e3
S0.2~ =~ Q1~03
E $I ~k~t1 a6.5
(I) A \t d6.5
0.1
Axial coordinate
Fig.15 Variation of the sensitivity with the axial coordinate z
for different dielectric pipe thicknesses (R2RI=3 mm, 5 mm,
6.5 mm)(W=10 mm, l=45 mm, 8=40, 8,=3.4, R2=10 mm,
R3=15 mm, =1, mm)
For the dielectric pipes with different permittivity, the axial
sensitivity distribution of the streamlines a, d, e is
investigated. Fig. 13 shows the variation of the sensitivity
with the axial coordinate z for the different relative
permittivity of (e,) of 1.5, 3.4, and 8. Fig. 14 shows the
sensitivity distribution at the central cross section (z=0) for
the different relative permittivity. From Fig. 6, Fig. 13 and
Fig. 14, the sensitivity, corresponding to the axial and
central crosssectional positions, increases with the increase
of the permittivity of the dielectric pipe. With an initial
increase in dielectric pipe permittivity, the sensitivity fast
increases, then increases slow, and finally tends to be a
constant. Therefore, increasing the permittivity of the
dielectric pipe leads to the increase of the sensitivity, but do
not have effect on the size of the sensing space.
0.35 
ed1.5
0.3 +ls
a3.4
0.25 +34
~ ~e3.4
>0.2 cas
lM /~7\ +d8
0.15 ~ p/ saL vea
0.1
0.05
0 1 01020
Axial coordinate /(mm)
Fig.13 Variation of the sensitivity with the axial coordinate z
for different relative permittivity (e,=1.5, 3.4, 8) (W=10 mm,
l=45 mm, 840, R2=10 mm, R y=5mm, R3=15 mm, 1, =1 mm)
0.4
,0.3
S0.2
0 0.1
0
5
I~
y/(mm)
5 5
(a) e,=1.5
x/(mm)
0.4
0.3
S0.2
i~0.1
,x 0.3
: 0.2
5
0
5 x/(mm)
y/(mm) 5
(a) R2RI=3 mm
y/(mm)
0
x/(mm)
5 5
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
0.3
:f 0.2
" 0.1
0
5 5
0
5 5 x (mm)
x/mm)
y /(mm)
(b) R,RI=6.5 mm
Fig.16 Sensitivity distribution at the central cross section
(z=0) for different dielectric pipe thicknesses (RzRI=3 mm,
5 mm, 6.5 mm)(W=10 mm, l=45 mm, 840, e,=3.4, Rz=10
mm, R3= 15 mm, 1, =1 mm)
Effect of the radius of the mental screening
Electric fluxline formed by the charged particles in pipe
terminate at the metal pipewall, the electrodes and mental
screening. Does the plwsical construction of the metal
screening have effect on the sensitivity field of the
electrostatic sensor arrays? The mental screening
construction is mainly dependant on its radius R3 and length,
and the axial length is equal to the length of the dielectric
pipe (1). Fig. 17 shows the variation of the sensitivity of the
streamlines a, d, e with the axial coordinate z for the
different radiuses of (R3) Of 15 mm, 20 mm and 30 mm.
Fig.18 shows the sensitivity distribution at the central cross
section (z=0) for different mental screening radiuses.
Whether in the axial coordinate or over the cross section of
the pipe, the sensitivity distribution is almost consistent,
which illustrates that the size of the mental screening has no
effect on the sensitivity distribution of the sensor arrays.
0.35
(a) R3=20 mm
0.3
: 0.2
" 0.1
0:
5
QI
0
Y /(mm)
5 5 x (mm)
(b) R3=10 mm
Fig.18 Sensitivity distribution at the central cross section
(z=0) for different mental screening radiuses (R3=15 mm,
20mm, 30mm) (W=10 mm, l=45 mm, 840, e,=3.4, Rz=10
inm, RI=5mm, g1, mm)
Dimensionless model of the Sensitivity of Electrostatic
Sensor arrays
In the above section, the sensitivity distribution is
determined under the specific geometric sizes of the
electrostatic sensor arrays, how can the sensitivity model be
extended to other arrays with the similar geometric
construction in order to improve their universal? Similarity
theory provides the basis for solving the above problem.
The plwsical variables related to the similarity of the
electrostatic field distribution include the geometric
construction, the permittivity of the dielectric pipe and the
dielectric distribution. The similarity of the geometric
construction of the sensor arrays means that the angle
determined by any two line segments is equivalent and the
corresponding line segments stay the same ratio in the
arrays. The permittivity of the dielectric pipe is equal in the
similar models. The similarity of the dielectric distribution
means the space positions of the point charges within the
sensor arrays are similar. If the mentioned plwsical variables
are similar, then the sensitivity of the extended model
remains the same with the original model.
Under the similar conditions, the sensitivity distribution of
the electrostatic sensor arrays depends on electrode angle
(0), dimensionless length of the electrode (FF7R2),
dimensionless thickness (R2RI) Rz and dimensionless
length of the dielectric pipe (1 R,), dimensionless radius of
the mental screening (R3 R,), dimensionless space between
guard and the electrode (18 R,). Fig. 19 shows the
0.25
>0.2
S0.15
0.1
20 10 0 10 20
Axial coordinate /(mm)
Fig.17 Variation of the sensitivity with the axial coordinate:
for different mental screening radiuses (R3=15 mm, 20mm,
30mm) (W=10 mm, l=45 mm, 8=40, e,=3.4, Rz=10 mm,
RI=5mm, lg=1 mm)
comparison of the sensitivity and relative error of two
electrostatic sensor arrays with the same dimensionless
parameters (seen in Table 1). It can be seen that the
sensitivity distribution of the two arrays is consistent and the
relative errors are within +10%, which confirms that the
sensitivity is related to the nondimensional parameters and
is independent of the actual sizes of the electrostatic sensor
arrays. Therefore, similarity theory expands the universal of
the sensitivity distribution model of the electrostatic sensor
arrays.
Table 1 similar geometric condition
(FF7R,) (0) (R,RI)/R, (1R) (R3/ R,) 1/,
modell 1)/1() 4() (1()5)/1() 45/1() 15/1() 1/1()
model 2 15/15 4() (157.5)/15 67.5/15 22.5/15 1.5/15
Table 2 Fitting coefficients A, B and accuracy
Radial position r (mm) .4 B Rsquare
(), ()) ().7384 ().9()46 ().9991
((), ().3) ().1732 ().7931 ().9991
((), ().45) ().2522 ().7348 ().9981
(().1()3, .282) ().1867 ().7867 ().9984
(().154, (.423) ().3()28 ().7()56 ().9983
Spatial filtering effect and frequency
characteristics of the electrostatic sensor arrays
The charge carried by the particles in pipe is the function of
the space coordinates (x, y, z) and time (t), that is the charge
density distribution p=p(x, y, z, t). Due to the randomness of
the particle flow and complexity of the particle charging
mechanism in pipe, p is a random function related to time
and space coordinate (x, y, z, t), which can be regarded as
the superposition of a mean and an independent, limited
bandwidth Gaussian distribution function. So the charge
distribution produced by the discrete particles in the
gassolid flow process is also known as "electrostatic flow
noise" (J. B. Gajewski 1996, J. B. Gajewski 2008). When
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
(r I7~ ~
s(z,,) =A(x,,,y,,)e (11)
where 4 and B are the fitting coeffcients, which are related
to the geometric size of the sensor arrays and the
nondimensional space position (xo, Yo) of the point charge.
Fig. 20 shows the fitting curves of the sensitivity
distribution along the dimensionless coordinate zo at
different streamlines (xo, yo). At each streamline (xo, yo), the
coeffcients and fitting accuracy are listed in table 2. At the
typical streamlines, the Rsquare values of the fitting curves
are better than 0.99, which shows good consistency of the
Gaussian function based fitting curve and finite element
calculation results. Therefore the sensitivity distribution of
the electrostatic sensor arrays can be represented by a series
of coeffcients (4, B). The fitting curve gives a general
formula of the similarity of the similar sensor arrays and
improves the universal of the sensitivity distribution model
of these arrays.
0.351
0.3
0.25
S0.2
S0.15
1 0 1 2 daY i
1.5 1 0.5 0 0.5 1 1.5
Non dimensional axial coordinate Nondimensional axial coordinate
(a) Comparison of axial
Fig. 20 Fitting curves of the sensitivity along dimensionless
n r\ ,i ~~~~~~cordinaR~~=., ~z=.1codnte zo at (xo, Yo) (FF7 Rz=1, 0=40, (R2RI)/ Rz =0.5, 1/
/p vr/R2=0 9N
Sr/R2=0.3
Sr/R2=0.45
1 0 1
Nondirnensional axial coordinate
(b) Relative error
Fig.19 Comparison of the sensitivity and relative error of
two electrostatic sensor arrays with the same dimensionless
parameters (W R2= 15/15, 8=40, (R2RI;) R,(157.5)/15, 1
Rz= 67.5/15, R3 Rz=22.5/15, 1, Rz =1.5/15, L Rz=120/15)
From Fig.19 (a), the nondimensional sensitivity distribution
of the electrostatic sensor arrays is very similar to Gaussian
function. Thus we used the Gaussian function to fit the
sensitivity distribution, and defined nondimensional
coordinates: .\.. \ ii', 1.. ~I ii', :.. : ii'~ The fitted equation
is as followed
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
electrostatic flow noise passes through the electrostatic
sensor arrays, the charge q(t) will be generated on each
electrode due to the electrostatic induction. According to the
definition of the sensitivity, the induced charge q(t) can be
calculated by the following equation
q(t) = JTp(xi y, z, ) s(x, yl z)dxdydz (12)
From Equation (12), the charge q(t) is an average of the
random signal p=p(x, y; z, t) with the weight function s(x, y,
z), and hence q(t) is also a complicated random signal.
During the gassolid flow process, the electrostatic flow
noise causes the change of quasielectrostatic field within
the sensing zone of the sensor arrays and thus the inducted
charge on the electrodes also fluctuate continuously. If the
charged particles move in threedimensional space (x, y; z)
with the velocity v,, v,, and v, respectively, the induced
charge q(t) on the electrodes can be rewritten as follows
q(t) = f p(x Vxt, y v,t, z v t) s(x, y, z)dxdydz (13)
Assumed that x,.=vxt, y,.=v,t, zr=v t, and then
q(t) = q(xY, yyz) (14)
= JJ p(x x,., y y,, z z,.) s(x, y, z)dxd/dz
As electrostatic f low noise p(x, y; z) is a steadystate and
ergodic stochastic process (J. Zhang and J. Coulthard 2005),
the autocorrelation function of q (x,, y, z,. is
ql(t) = pSS(x y Z v t)S (x y Z>UdxdydUZ
The autocorrelation function of q(t) is defined as
#,(z> = E[q(t + )q(t)]
=E[ p(xr, ~;y vlc,zst;vt)sulx0ylz&&z
= [#S~,ic,(xaz y r+v4t)s(z,r)*
s(a~ a4)cdvdzdadady]
(19)
(20)
where ,(Tx, z,, T) is the autocorrelation function of the
electrostatic flow noise p(x, y; z).
According to the Wiener Khintchine theorem, the power
spectrum of q(t) is
s(a~ ) )exp(j2K fz )drzdxdydzdadfd y]
According to generalized parseral theorem, Equation (22)
can be derived:
S (f) = [SSS @,Cf, f ~)exp(J r ; aj2rf fj2rf (rr r))
s K ,ff)r,(a A )exp3(j2rifr~z;~ycdrwdj~dadfdy
= [rS ,u,,iexpl9j2nll'rf r)) ( Jf SJ ). (22)
ex(p(J2ri~fr)dr ];'y'
where S*(~fx, fy; fz) is the complex conjugate of S~fxc, fy; fz),
that is, S*( ~fx, fy; fz)= S (~fx, fy: fz). Equation (22) further
indicates that since the spatial filtering effect, the frequency
component and amplitude distribution of the output signal
of the electrostatic sensor arrays is no longer electrostatic
flow noise of the measured powder, rather an average of the
noise weighted by the sensitivity function of the
electrostatic sensor arrays. Studies have shown that the
electrostatic flow noise is a white noise with the limited
bandwidth in the space. From Equation (18) and (22), the
power spectrum characteristics of the output signal of the
electrostatic sensor arrays depend entirely on its spatial
filtering characteristics. Therefore, by designing the shape
and geometry of the spatial filter appropriately, the
electrostatic sensor arrays will have a specific spatial
sensitivity, and then expected flow information may be
extracted from gassolid phase flow system.
At present, it is difficult to obtain the expressions of the
threedimensional sensitivity field distribution of the
electrostatic sensor arrays, and thus the threedimensional
spatial filtering characteristics of the arrays. To simplify the
analysis, the filtering characteristics of typical streamline
were investigated. The Fourier transform of the sensitivity
distribution function s(x, y; z) can be expressed as
S(f, f )f = F(s(x, y, z))
(23)
= ~ls(x, y, z) exp( j27r( fx S + J.y fz)drd
If the coordinate (x, y) is fixed, the spatial sensitivity s(x, :,
z) is only related to z. So the Fourier transform of the spatial
sensitivity in the spatial domain can be expressed as:
where E [...] denotes mathematical expectation.
According to WienerKhintchine theorem, the
spectrum of q(z rs z) is expressed as
power
S (,ff ,)=Sr f,,(f sff )S(f (16)
whereS ( fx,f f ) is the power spectrum density
function of the electrostatic flow noise p(x, v; z),
S, (J J ,) is the power spectrum density function of
the spatial sensitivity function s(x, y, z) in the spatial
frequency domain. f,,4 and fz respectively denote the spatial
frequency in x, v and z directions. S, ( f,, f ,f,) and
S, (J, J, f,) are given respectively by
S ~;.f~fl)=SSSQ,(xW~y~z)exip(j2airx+fy+fz)deddz(7)
(18)
= 1~s(x, Y z)ex~p(j2(fx+ ~f4 y+ fz))dxdedz i
It can be seen from the Equation (16) that the power
spectrum of the output signal of the electrostatic sensor
arrays is the product of the power spectrum of the
electrostatic flow noise and the spatial spectrum density of
the electrostatic sensor arrays. In other words, the output
signal of the electrostatic sensor arrays is modulated by the
electrostatic sensor arrays. From the component of the
spatial frequency, the electrostatic flow noise cannot be fully
transferred to the sensing electrodes, and the electrostatic
sensor arrays acts as a low pass spatial filter.
If the particles move only along : direction, the induced
charge q(t) can then be determined by
S( f,) = F(s(z))
= are bR2~ Oap(jlnf z)dz (24)
= abR2 O~Xp((bR 72 z 2)
where F denotes Fourier transform. S@n is the spatial
filtering transfer function of the electrostatic sensor arrays.
The spatial amplitude frequency spectrum S@V) of the
sensitivity s(z) at the given position (x, y) is shown in Fig.
21. Fig.21 (a) shows the spatial frequency characteristics at
different (x, y) when the geometric and material parameters
are kept at W=10 mm, 840, R2=10 mm, RI=5mm, 8,=3.4,
R3=15 mm, I =45 mm, Ig =1 mm. It can be seen that the
electrostatic sensor arrays acts as a low pass filter in spatial
frequency domain. Namely, only the signal with a lower
spatial frequency can pass through the electrodes.
Additionally, nearby the electrode the spatial frequency
bandwidth is broader, and the amplitude is greater. Fig.
21(b) depicts the comparison of the spatial filtering
characteristics of the electrostatic sensor arrays with the
electrodes' length at 10 mm and 5 mm, respectively. It shows
that wherever on the central axis of the arrays or near pipe
wall is, the shorter the length of the electrode, and the wider
the spatial frequency bandwidth is relatively. And the
amplitude increases with the increase of the electrode length
in the working spatial frequency range, which illustrates that
there is a larger energy transfer with the longer electrode
length. The spatial filtering characteristics of the sensor
arrays are similar to the ring electrostatic sensor (C. Xu et al
2007).
x 103
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
distribution of the electrostatic sensor arrays
If the position (x, y) is fixed and the moving particle is seen
as a unit point charge the input signal p(x, y, zvzt) of
electrostatic sensor arrays can be substituted by the unit
impulse signal G(zvzt). 6 denotes Dirac function. Then the
unit impulse response function (h(t)) of the arrays can be
expressed as followed:
h(t) = Jd(z vzt)s(z)dz
= a exp
Hence the Fourier transform of h(t) can be used to represent
the temporal frequency response function of the probe. The
frequency response property H~f) can be expressed as
follows:
H (f) = Jh(t)exp( j2 7cft )dt
%, >2(26)
=fa exp bR; Xp( j27cft)dt
abR 2 Op( (bR ,if2
v_ v
The electrode of the sensor arrays has to be connected to a
preamplifier to amplify the induced charge. So the
frequency response characteristics of the measurement
system should be dependent on both the temporal filtering
property of the electrode e and the dynamic property of the
preamplifier. The resistive interface circuit of the
electrostatic sensor arrays is shown in Fig. 22, and the
frequency response characteristic H~f) of the electrostatic
sensor arrays.
abR, 2 (bR 2 f)2 (27)
U,(J) =j2nFfR exp( 2
v, v
071P
Fig.22 Interface circuit of the electrostatic sensor arrays
Fig. 23 shows the temporal frequency response
characteristics Hdf) of the electrostatic sensor arrays. Fig.
23(a) depicts the comparison of the frequency
characteristics of the electrostatic sensor arrays with the
electrodes' lengths at 10 mm and 5 mm, respectively. As can
be seen from Fig. 23(a) the sensor arrays acts as a bandpass
filter in the temporal frequency domain. The reason is that
the electrodes are lowpass filters, and yet the interface
circuit is a highpass filter. The temporal frequency
bandwidth is wider near the pipe wall and the central
frequency is higher on the central axis of the electrode.
Wherever on the central axis of the electrode or near
electrode, the wider the temporal frequency bandwidth of
the measurement system, the shorter the electrode. Fig.23
(b) shows the effect of the particle velocity on the frequency
response characteristics of the sensor arrays. The faster the
particle moves, the wider the frequency bandwidth and the
higher the central frequency.
10 100 200 300 4
Spatial frequency fz/(1/m)
(a) different cross sectional positions (x, y)
x 10 3
0 100 200 300 400
Spatial frequency fz /(1/m)
(b) different lengths of the electrode
Fig.21 Comparison of the spatial frequency spectrum
Frequency f/(Hz)
(a) Different electrode lengths
0b 500 1000 1500
Frequency f/(Hz)
(b) Different velocity
Fig.23 Temporal frequency response characteristics of
electrostatic sensor arrays
Conclusions
A mathematical model of an electrostatic sensor arrays was
established, and the spatial sensitivity, spatial filtering effect,
and temporal frequency response characteristics of the
electrostatic sensor arrays have been investigated using the
Finite Element Method and electrostatic field theory. Due to
the limited time, effects of particle velocity, radial position,
the length of the electrode, and particle size on the temporal
frequency response characteristics were not investigated
experimentally.
Acknowledgements
The authors wish to express their gratitude to the Specia
Funds for the Major State Basic Research Projects
(No.2010CB22_'ilyI' (China), to the National Natura
Science Foundation of China (No.50836003, 50906012),
and to the Science Foundation of Ministry of Education of
China for the New Teacher (No.200802861005) for funding
this research.
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