Influence of position of an anvil beam on vibrations
of collecting electrodes
Andrzej Nowak, Paweł Nowak, Jan Awrejcewicz
Abstract: The paper presents experimentally verified computational model of the
collecting electrodes in electrostatic precipitator (ESP). Discretisation of the whole
system, both the beams and the electrodes, is carried out by means of the modified
rigid finite element method (RFEM). The RFEM allows easy to reflect mass and
geometrical features of the considered. Moreover, the method is convenient for the
introduction of additional concentrated masses which reflects elements such as the
joints that fasten the electrodes to the suspension beam, distance-marking bushes,
riveted or screw joints, etc... The system of nine electrodes 16 m long and several
various positions of an anvil beam in this system are analyzed. Position of the anvil
beam in the system has essential influence on tangent and normal accelerations at
different points of the plates, and thus on the effectiveness of the dust removal
process. The aim of the simulations is to find optimal position of an anvil beam,
which guaranties the maximum amplitudes of vibrations and their proper distribution
in the plates.
1.
Introduction
Particulate matter emitted by power and industrial plants is a polydisperse particulate – it constitutes a
group of particles of different size, from submicron particles to particles of a diameter of several tens
μm [1]. Particle size distribution is affected by both the pollution generated by the same source and
the efficiency of the used dust collector [2]. Modern electrostatic precipitators, thanks to their high
total dust extraction efficiency, reaching 99.9%, are used for a thorough purification of gases from
dust particles. A scheme of a single-sectional electrostatic precipitator is demonstrated in Fig. 1.
Figure 1. Scheme of a single-sectional electrostatic precipitator.
391
The basic elements of currently built ESP are: a set of emitting electrodes 2, a set of collecting
electrodes 3, electric power system 1 and housing with supply 4 and discharge 5 conduits for the gas
from the dust collector. The dust is discharged using channel 6.
The interior of a dry ESP is a complex steel structure. It consists of multiple, repetitive
subassemblies, called sections of collecting electrodes (Fig. 2a). Sectioning of the ESP is applied in
the case of dust extraction of large volumes of gas. A single section is formed by up to several tens of
collecting electrodes, separated alternately with sets of discharge electrodes (Fig. 2b).
a)
b)
Figure 2. Scheme of: a three-sectional dry ESP, b) a section of collecting electrodes.
The effective operation of an ESP is largely dependent on, among others, the structure and
proper selection of the profile of collecting electrodes [3]. One of the profiles, often used in domestic
structures, is presented in Fig. 9b. The systems of collecting electrodes are connected with the sets of
rappers, responsible for the removal of dust accumulated on these electrodes. In the case of ESP with
a gravitational system of rappers, each section has its own drive shaft of rappers. The number of
rappers corresponds to the number of the systems of collecting electrodes in the section. An important
element of a rapping system, in terms of vibrations induced in the system of collecting electrodes, is a
beater hanging on the arm rotated by a drive shaft (Fig. 3). The removal of the particulate matter
collected on the electrodes occurs as a result of excitation of vibrations of accelerations allowing for
the effective removal of particles coagulated on their surfaces. The analysis of the phenomena
accompanying the operation of rapping devices, the improvement of reliability of the operation of
rapping systems and the achievement of more efficient removal of particulates from collecting
electrodes have been under investigation for many years [4, 5]. Initially, the studies were based on
physical models. For example [6] presents a physical model of a dry ESP, which served to search for
optimal parameters of rapping, expressed by the thickness of the particulate layer, the value of the
generated acceleration only in 2 control points of collecting electrodes. However, in the last decade,
392
instead of building physical models of ESP, specialized, numerically advanced models are created,
which serve to perform simulations of a whole device [7] or only its selected subassemblies [8].
Figure 3. Rapping system.
The earlier works of the authors [9-12] present models allowing for a simulation of vibrations of
electrodes, induced by an impulse force. The models were obtained using: the finite element method
[12], the slab strip method [9] and the rigid finite element method [10, 11]. In the above-mentioned
publications much space is devoted to the validation of the models – by comparing the results of
measurements and computer simulations. The adopted compatibility criteria demonstrated that
practically all the applied methods give a satisfactory consistency of the results in the range of peak
values of accelerations. It also applies to the RFEM, used in the current study.
Based on the models verified with the use of measurements, an attempt was made to use the
developed computer software to determine the effect of geometric and structural parameters of the
electrode-rapper system on the values of accelerations. The results of the conducted analyses may be
useful to engineers and designers of ESP.
2.
SES model of a system of collecting electrodes
The RFEM is an original Polish method, developed since the 70s of the twentieth century [13, 14]. Its
essential feature is a simple physical interpretation and numerical effectiveness – particularly in the
area of the analysis of vibrations around the position of static equilibrium. In the present study the
subject for modeling is a single set of collecting electrodes (Fig. 4) comprising a suspension beam
(upper beam), on which p electrodes are hung (usually no more than 10), braced at the bottom by the
anvil beam (bottom beam). An anvil is completed with an anvil, hit by the beater.
393
Figure 4. The system of collecting electrodes.
Based on the analysis of the shapes of collecting electrodes presented in the introduction, it can
be noted that they consist of several long slab strips, connected at certain angles. This implies the way
of their division into elements, presented below. The electrode k is associated with a coordinate
system with axes directed according to Fig. 5.
Figure 5. Strip j with width bk,j and angle of inclination Ek,j towards axis ŷ , {} – global coordinate
system, {k} – local coordinate system connected with electrode k with axes parallel to those
of system {}.
394
A single strip of an electrode has a constant width and thickness. It is assumed that the strips are
numbered from 1 to mk, while the extreme left-hand strip has number 1, and the extreme right-hand
one number mk. In the direction x the strip is divided into nk elements of the length:
'x k
lk
,
nk
(1)
where lk –length of the electrode (and thus the length of strips from 1 to mk). Therefore, the whole
collecting electrode is divided into
ne( k )
(2)
mk n k
elements. It should be noted that the elements of the strip j of the electrode k are of the same size. The
introduced division will serve to determine the energy of elastic strain of element of the electrode and
its kinetic energy, using rectangular shell elements. In the RFEM the presented division is called a
primary division. In the secondary division, primary elements are divided into four parts – rigid finite
elements (rfe), as shown in Fig. 6 and connected by means of spring-damping elements (sde).
Figure 6. Exemplary scheme of discretisation in RFEM : primary (┼) and secondary (╬) divisions.
In the present study spring-damping elements sde are placed similarly as in [16] and in contrast
to [15]. This way of sde arrangement provides the opportunity to consider the torsion of discretized
plates and eliminates singularities in the model. Upper and lower beams are discretized in a similar
way (Fig. 7).
395
a)
b)
After calculating mass parameters of rfe and stiffness coefficients of sde for plates and beams,
the equation of motion can be presented in the general form:
Mq
q Cq
(3)
f
where: M – mass matrix of constant elements, C – stiffness matrix of constant elements, q – the
vector of generalised coordinates, f=f(q, F) and F(t) – force applied to the anvil of the bottom beam
(Fig. 8).
Figure 7. Model of an impulse force inducing vibrations in the lower beam.
In equation (3) damping is omitted, as from the engineering point of view, it is important to
determine the maximum possible acceleration values. However, these are obtained for systems
without damping. It should also be considered that in the structures of real ESP the following factors
are higher than material damping: damping of the structure and damping caused by particulates
accumulated on the electrodes.
3.
Test calculations
The model presented in this study was used to simulate vibrations induced by an impulse force, the
course of which was demonstrated in Fig. 9a. In the further part the results of exemplary numerical
simulations of the vibrations of a set of electrodes of the profile presented in Fig. 9b, were
demonstrated. The analysis covered sets of 9 electrodes of a length of 16 m, differing with the
396
position of an anvil beam in the system (Fig. 9c, variants H6..H16), made of rolled sheet having a
thickness of 1,5 mm. The calculations were made for angles of application of impulse force
DF
EF
JF
0 applied to a point xF , y F , z F 0,0,0 , Fig. 8.
Figure 8. Scheme of the system for tests: a) course of the impulse force, b) analyzed profile of
electrodes, c) variants with a variable height of anvil beam position (H6.. H16).
Vibrations of electrodes have a nature of high-frequency processes. However, a direct
comparison of such courses is not very effective. For this reason, the present study compares peak
values Wmax representing indirectly analyzed courses of accelerations in the field of amplitudes,
expressed by the following dependencies:
Wmax a s ,i max a s ,i ,
(4)
0 d t dT
where: T- time of the analysis, i – number of the control point. In the above equation it was assumed
that a s ,i can be one of the following values:
397
a s ,i
therefore s
­ a x ,i
°
° a y ,i
°
®aW ,i
°a
° Q ,i
°a
¯ c ,i
– acceleration in the direction of axis x,
– acceleration in the direction of axis y,
a x2,i a 2y ,i
– tangential acceleration in plane xy,
– normal acceleration in plane xy,
a z ,i
a x2,i a 2y ,i a z2,i
– total acceleration,
^x, y, t, n, c` .
Simulations of vibrations for the above-mentioned variants were conducted at the following
calculation and material parameters:
x
x
force impulse: the course as shown in Fig. 9a,
integration step: hc 1 u 106 s,
x
density of discretisation of electrode strips: nk
x
simulation time: T
x
Young’s modulus: E
x
Poisson number: Q
400,
2
1 u 10 s,
2,06 u 1011 Pa,
0,3.
Peak values Wmax a s of the tangential and normal component and total acceleration were compared.
100 control levels, spaced at 0.16 m were assumed. At each level acceleration values were determined
in 225 points. No numbers were assigned to the points.
3.1. Analysis of the shift of the anvil beam
The study of the impact of the position of the anvil beam on the distribution of accelerations in the
system of electrodes was limited to the analysis of 11 cases (Fig.8c). In the first case the beam was
located at the bottom edge of electrodes (16 m below the suspension beam), and thus in the position
applied most often in the ESP produced in Poland. In successive cases the beam was “raised” 1 meter
up as compared to a previous case. For clarity, the presentation of the maps of peak values was
limited to total accelerations. Fig. 9 presents a map of peak values Wmax ac of total accelerations of
control points of the system, obtained during the simulation. Based on the presented results it can be
observed that the shift of the anvil beam towards the upper beam allows for a more even distribution
of peak values Wmax ac throughout the system.
The quantitative assessment of the vibrations in the system was performed by comparing mean
peak values Wmax,H ac of control levels (Fig. 10) and mean peak levels Wmax a s of the system of
electrodes (Fig. 11). The analysis of the values Wmax,H a s from Fig. 11 helps to observe that the
position H8 of the anvil beam allows for a more even distribution of peak values of accelerations in
the system.
398
a)
b)
c)
Figure 9. Peak values Wmax ac of the total acceleration at time t
1 u 10 2 s after the application of
the impulse force; the position of the anvil beam: a) H16, b) H11, c) H8.
Wmax,H(a c) [ms-2]
0
2000
4000
6000
8000
10000
12000
0
2
4
x [m]
6
H6
H7
8
H8
H9
H10
10
H11
12
H12
H13
14
H14
H15
16
H16
Figure 10. Impact of the position of the anvil beam on mean peak values Wmax,H ac in the following
variants: H6..H16 of the position of the anvil beam.
399
On the other hand, in the position H11 mean peak values of: Wmax an - normal component,
Wmax at - tangential component and Wmax ac - total acceleration are in this case the highest
(Fig. 11).
[ms-2]
4000
3876
3673
3194
2672
3437
3005
3254
2788
2307
3955
3500
3382
3334
2801 2881
3000
2500
2748
2000
3512
2602
1500
2980
2937
2391 2485
1000
2426
500
0
2422
2343 2466 2227
2239
2176
2118
2031 2052
1929
1928
aw
at
an
an
at
H6
H7
H8
H9
H10 H11 H12 H13 H14 H15 H16
Position of the anvil beam
aw
Figure 11. Impact of the position of the anvil beam on mean peak values Wmax a s of the electrode
system in the following variants: H= 16 m .. H=6 m of the position of the beam,
where s in {n, t, c}.
4.
Conclusions
In summary, it should be concluded that the process of inducing vibrations and their propagation in
the electrode system is a resultant of many factors. It depends not only on the size of impulse force,
but also on physical, geometrical and structural parameters of all the components of the system. The
experiments presented in the present study imply the following remarks:
1. Treating the normal component of acceleration as the sole criterion for the selection of
collecting electrodes and the beater of the rapper is not correct and may lead to incorrect conclusions.
2. The values of total acceleration cannot also determine the selection of a profile of electrodes as
the sole factor. Only the combined use of indicators of the assessment of peak values of all the
components of acceleration allows for the selection of the optimal variant.
400
3. The position (height) of anvil beam installation in the system of electrodes may affect the size
and distribution of generated vibrations, as well as the time of their propagation. As shown in the
study, the “classical” position of the rod in the system is not the most favorable.
4. The shift of rapping rod towards the upper beam does not cause a significant increase in the
acceleration values, but it can contribute to their more even distribution.
The results of test calculations demonstrate that intuition of engineers responsible for the design
of electrodes was generally correct. The location of the anvil beam in a position other than the
“classic” one would require significant changes in current structures, encountering a number of
technical difficulties.
References
[1] Jaworek, A., Krupa, A., Adamiak, K. Removal of submicrometer dust particles by a charged
spherical collector, Proceedings: 6th International Conference on Electrostatic Precipitation (ICESP
V), June 18–21, 1996, Birmingham, AL., USA,Washington, DC USA, (1996).
[2] Brocilo, D., Podlinski, J., Chang, J. S., Mizeraczyk, J., Findlay, R.D. Electrode geometry effects
on the collection efficiency of submicron and ultra–fine dust particles in spikeplate electrostatic
precipitators. Journal of Physics: Conference series 142, 012032 (2008), pp. 1–6.
[3] Lind, L., Bojsen, E.M. Experience with baffle–free collecting plates in an electrostatic
precipitator, Proceedings: 5th International Conference on Electrostatic Precipitation (ICESP V),
April 5–8, 1993, Washington, DC USA (1993), pp. 36.1–14.
[4] Talaie, M.R. Mathematical modeling of wire–duct single–stage electrostatic precipitators,
Journal of Hazardous Materials 124, 1–3 (2005), pp. 44–52.
[5] Yang, X.F., Kang, Y.M., Zhong, K. Effects of geometric parameters and electric indexes on the
performance of laboratory–scale electrostatic precipitators, Journal of Hazardous Materials 169, 1–3
(2009), pp. 941–947.
[6] Lee, J.–K., Ku, J.–H., Lee, J.–E., Kim, S.–C., Ahn, Y.–C., Shin, J.–H., Choung, S.–H. An
experimental study of electrostatic precipitator plate rapping and reentrainment, Proceedings, ICESP
VII, Sep. 20–25, 1998, Kyongju, Korea (1998).
[7] Franci,s S.L., Bäck, A., Johansson, P. Reduction of Rapping Losses to Improve ESP
Performance, Proceedings, ICESP XI, October 20–25, 2008, Hangzhou, China (2008), pp. 45–49.
[8] [8] Caraman, N., Bacchiega, G., Gallimberti, I., Arrondel, V., Hamlil M. Development of an
industrial model of rapping–effect on the collecting efficiency, Proceedings, ICESP X, June 25–29,
2006, Cairns, Australia (2006).
[9] Adamiec-Wójcik, I., Nowak, A., Wojciech, S. Comparison of methods for vibration analysis of
electrostatic precipitators. Acta Mech. Sinica 1, 27 (2011), pp. 72–79.
[10] Adamiec-Wójcik, I. Modelling of Systems of Collecting Electrodes of Electrostatic Precipitators
by means of the Rigid Finite Element Method. Archive of Mechanical Engineering 58, 1 (2011), pp.
27–47. DOI: 10.2478/v10180-011-0002-x.
[11] Adamiec-Wójcik, I., Awrejcewicz, J., Nowak, A., Wojciech, S. Vibration Analysis of Collecting
Electrodes by means of the Hybrid Finite Element Method. Mathematical Problems in Engineering
2014, Article ID 832918, 19 pages (2014). doi:10.1155/2014/832918.
401
[12] Nowak, A. Modelling and measurements of vibrations of collecting electrodes of dry
electrostatic precipitators, (in Polish Modelowanie i pomiary drgań elektrod osadczych elektrofiltrów
suchych). Rozprawy Naukowe 35, Wydawnictwo Naukowe Akademii Techniczno-Humanistycznej,
Bielsko-Biała, 2011.
[13] Kruszewski, J., Gawroński, W., Wittbrodt, E., Najbar, F., Grabowski, S. Rigid finite method (in
Polish: Metoda sztywnych elementów skończonych), Arkady, Warszawa, 1975.
[14] Gawroński, W., Kruszewski, J., Ostachowicz, W., Tarnowski, J., Wittbrodt, E. The finie element
metod in Dynamics of structures (in Polish Metoda elementów skończonych w dynamice konstrukcji).
Arkady, Warszawa, 1984.
[15] Adamiec-Wójcik, I., Wojciech, S. Application of a rigid finite element method in dynamic
analysis of plane manipulators. Mechanism and Machine Theory 28, 3 (1993), pp 327–334.
[16] Abramowicz, M. Modelowanie drgań przestrzennych i identyfikacja parametrów dyskretnych
modeli stalowo-betonowych belek zespolonych, Ph.D. thesis, West Pomeranian University of
Technology, Szczecin, 2014, Berczyński, S.
Jan Awrejcewicz, Professor: Technical University of Łódź , Department of Automation,
Biomechanics and Mechatronics, 1/15
Stefanowskiego
Street, 90-924 Łódź, Poland,
([email protected]).
Andrzej Nowak, Associate Professor: University of Bielsko-Biala, Department of Transport and
Informatics, Willowa 2, 43-300 Bielsko-Biala, Poland ([email protected]).
Paweł Nowak, M.Sc.: University of Bielsko-Biala, Department of Transport and Informatics,
Willowa 2, 43-300 Bielsko-Biala, Poland. The author gave a presentation of this paper during one of
the conference sessions ([email protected]).
402