at SciVerse ScienceDirect

Journal of Electrostatics 70 (2012) 327e332

Contents lists available

Journal of Electrostatics

journal homepage: www.elsevier .com/locate/elstat

Vibration of collecting electrodes in electrostatic precipitators e Modelling,measurements and simulation tests

Andrzej Nowak*

University of Bielsko-Biała, ul. Willowa 2, 43-309 Bielsko-Biała, Poland

a r t i c l e i n f o

Article history:Received 8 January 2012Received in revised form4 April 2012Accepted 11 April 2012Available online 27 April 2012

Keywords:PrecipitatorParticle dislodgingFEMNumerical modelExperimental validationVibration analysis

* Tel./fax: þ48 33 8279289.E-mail address: a.nowak@ath.eu.

0304-3886/$ e see front matter � 2012 Elsevier B.V.doi:10.1016/j.elstat.2012.04.004

a b s t r a c t

The paper presents a numerical model for simulating the vibration of collecting electrodes in an elec-trostatic precipitator. The method of finite elements (FEM) was used to describe the shell elements of thecollecting electrodes. The remaining elements of the system were modelled with the application of therigid finite elements method (RFEM). The results of measuring validation and testing calculations arediscussed.

� 2012 Elsevier B.V. All rights reserved.

1. Introduction

The effectivework of an electrostatic precipitator depends on theconditions of gasflow, the generated electricfield and the geometricparameters of the electrostatic precipitator (ESP) [1]. The geometricparameters of the ESP are of essential importance to the remainingtwo factors, i.e. the arrangement of electrode systems in the sectionsentails conditions of gas flows [2], while the type of profile of col-lecting the electrodes influences both the gas flow parameters anddistribution of the electric field potential [3]. The efficiency of ESPalso depends on the efficiency of the periodic cleaning of the col-lecting electrodes [4,5]. Removing the dust collected on the elec-trodes is achieved by bringing them to vibrationswith accelerationsthat allow for an effective separation of the dust coagulated on theirsurface. To achieve this, gravity-operated rapping systems [6] orelectro-vibrators [7] are typically used. There are also individualsolutions, i.e. the so-called acoustic tubes, which are used to excitevibrations of the electrodes [8]. In the case of the gravity-operatedrapping systems, excitation of vibrations is achieved by the axialstriking of a beater on the anvil beam, in which an intensive stress-wave is generated. Both the electrodes’ geometrical features and theforce impact have an essential influence on tangent and normal

All rights reserved.

accelerations at different points of the electrodes, and thus on theeffectiveness of the dust removal process.

There is extensive literature on the analysis of the phenomena ofcharging and capturing dust particles on the surfaces of collectingelectrodes. However, the question remains as to how to lead theelectrodes’ cleaning process so that it does not reduce the effec-tiveness of the ESP? Some attempts to answer this question havebeen presented [9,10]. However, these were experimental studies.The disadvantage of this type of work is the analysis of accelera-tions of electrodes in already existing systems so that, inmost cases,the results lead to conclusions that this type of cleaning processcould be difficult or even impossible to implement in existingstructures because of the design and/or for economic reasons.Computer simulations do not have this defect.

Chapter 2 presents a computational model which is proposedfor simulation of vibrations of collecting electrodes caused by theirstriking during the process of shaking off dust. It has been assumedthat an average class PC should be enough to do all of thecomputations, and that the real-time processing should not exceedseveral dozen minutes. Two methods were used to model thesystem under consideration: the deformable finite elementsmethod (FEM) and the rigid finite elements method (RFEM). Themethod of validating the model is also presented briefly in thechapter. Within the process of validation, conformity of results ofown calculations was compared with the results of measurements

A. Nowak / Journal of Electrostatics 70 (2012) 327e332328

conducted on a special test stand. The quantitative analysis of theresults was carried out by means of two different indicators [11]:verifiability FAC2 and conformity qε. The object on whichmeasurements of acceleration were carried out was equipped withSIGMA VI-type electrodes (Fig. 1b). The effect of the collectingelectrodes’ shape as well as various positions of the anvil beam onthe vibrations is discussed in Chapter 3.

2. Methodology

The subject of modelling consists of a single set of collectingelectrodes made up of a suspension beam, on which there are pcollecting electrodes (usually not more than 10), buckled at thebottomwith an anvil (Fig. 1a). The force impulse F(t) was measuredand assumed to be as shown in Fig. 1c.

2.1. Computational model

This section contains the basic considerations that allow todetermine thematrices and vectors from the equations of motion ofa subsystem of collecting electrodes and of the whole systemwhenthe finite elements method (FEM) [12] is applied. The equations ofmotion were derived from Lagrange equations of second order. Theway of discretisation of the beams modelled with the rigid finiteelements method (RFEM) was presented thoroughly in [13]. Theconsiderations that allow to define the stiffness matrix are pre-sented in detail in [14], where a different hybrid method wasapplied. Therefore, only the main, novel features of the method arepresented below. A coordinate system with axes, directed as inFig. 2, is connected with k electrode. A single strip of the electrode(Fig. 2) has a constant width and thickness. The energy of the elasticstrain of an element with the number (k, j, i) (Fig. 2) is independentfrom i (in view of the division into elements with constant lengthDxk in the direction x) and from angle bk;j. A rectangular four-nodeelement (where the indexes are: k e plate, j e strip, i e element ofstrip j), as presented in Fig. 2, is defined by the following values:

q0k;j;i;s ¼

hu0k;j;i;s;v

0k;j;i;s;w

0k;j;i;s;4

0k;j;i;s;q

0k;j;i;s;j

0k;j;i;s

iTfor s ¼ 1;2;3;4;

(1)

Fig. 1. Schematic view of a collecting electrode system (a), SIGMA VI profile (b) and (c) meaforce impulse F(t).

where u0k;j;i;s, v0k;j;i;s, w

0k;j;i;s are displacements of node (k, j, i, s) in the

direction of axis x0k;j;i, y0k;j;i, z

0k;j;i, respectively, and 40

k;j;i;s, q0k;j;i;s, j

0k;j;i;s

are rotations in node (k, j, i, s) around the axis parallel to x0k;j;i, y0k;j;i

and z0k;j;i, respectively.In further considerations, during which the potential energy of

the elastic strain of the element and its kinetic energy weredescribed, indexes (k, j, i) were omitted to simplify the notation.Therefore, the nodal displacements are described by the followingvectors:

q0s ¼ �u0s; v

0s;w

0s;4

0s; q

0s;j

0s�T

for s ¼ 1;2;3;4: (2)

It is assumed that the shield state is described by displacementsu0s; v0s and rotation angles j0

s, whereas displacement field u0; v0 isdescribed by the functions:

u0ðx0;y0; tÞ ¼ au1 þ au2x0 þ au3y

0 þ au4x0y0 þ au5ðy0Þ2þau6x

0ðy0Þ2; ð3:1Þ

v0ðx0;y0; tÞ ¼ av1 þ av2x0 þ av3y

0 þ av4x0y0 þ av5ðx0Þ2þav6ðx0Þ2y0; (3.2)

and j0is described by the relations [15]:

j0 ¼ 12

�� vu0

vy0þ vv0

vx0

�: (3.3)

It is also assumed that the plate state is described by deflectionangles w

0and 4

0, q

0, which are described by the relations:

w0ðx0; y0; tÞ ¼ aðwÞ1 þ aðwÞ

2 x0 þ aðwÞ3 y0 þ aðwÞ

4 ðx0Þ2þaðwÞ5 x0y0

þ aðwÞ6 ðy0Þ2þaðwÞ

7 ðx0Þ3þaðwÞ8 ðx0Þ2y0

þ aðwÞ9 x0ðy0Þ2þaðwÞ

10 ðy0Þ3þaðwÞ11 ðx0Þ3y0 þ aðwÞ

12 x0ðy0Þ3;(4.1)

40 ¼ vw0

vy0; (4.2)

q0 ¼ �vw0

vx0: (4.3)

sured force impulse F(t). Vibrations are generated as the system’s response for a single

Fig. 2. Plate strip j with width bk, j and inclination angle bk;j towards axis y: { } e system of global coordinates, {k} e coordinate system connected with electrode k and with axesparallel to the axis of system { }.

A. Nowak / Journal of Electrostatics 70 (2012) 327e332 329

Factors aðuÞ1 � aðuÞ6 ; aðvÞ1 � aðvÞ6 and aðwÞ1 � aðwÞ

12 can be determinedfrom respective boundary conditions [13].

The energy of elastic strain of the shell element in Fig. 2 is thesum of:

E ¼ EðtÞ þ EðpÞ; (5)

where E(t) is the energy of the shield state and E(p) is the energy ofthe plate state. After necessary transformations [13], the energy ofelastic strain of element (k, j, i) may be presented in the followingform:

Ek;j;i ¼12

X4l¼1

X4s¼1

q0Tk;j;i;lC0k;j;i;l;sq

0k;j;i;s: (6)

where q0k;j;i;s ¼ q0s; q0s is specified in eq. (2), C 0k;j;i;s are square,

symmetric stiffness matrices of the element with dimensions of24 � 24, as explained in [14].

The kinetic energy of the element under consideration isdefined differently than in the hybrid method [14]. As in the case offormulae for energy of elastic strain, the expression for the kineticenergy of element (k, j, i) may be presented in the following form:

Tk;j;i ¼12

X4l¼1

X4s¼1

_q0Tk;j;i;lM0k;j;i;l;s _q

0k;j;i;s (7)

whereM0k;j;i;l;s are matrices with dimensions of 6 � 6 with constant

elements. In the computational model, the nodal displacements inthe global coordinate system (Fig. 2) are taken as generalisedcoordinates. Thus, formulae (6) and (7), which describe the kineticand potential energy of the elastic strain of element (k, j, i), may bepresented in the following forms:

Ek;j;i ¼12

X4l¼1

X4s¼1

qTk;j;i;lCk;j;i;l;sqk;j;i;s (8)

Tk;j;i ¼12

X4l¼1

X4s¼1

_qTk;j;i;lMk;j;i;l;s _qk;j;i;s (9)

where Mk;j;i;l;s ¼ Rk;jM0k;j;i;l;sR

Tk;j; Ck;j;i;l;s ¼ Rk;jC

0k;j;i;l;sR

Tk;j; and

M0k;j;i;l;s and C 0

k;j;i;l;s, as described in eqs. (6) and (7), are the mass andstiffness matrices of the elements expressed in local coordinate

systems {k, j, i}0;Rk;j ¼"Rk;j 00 Rk;j

#;Rk;j ¼

"1 0 00 cosbk;j �sinbk;j0 sinbk;j cosbk;j

#;

qk;j;i;s ¼ ½uk;j;i;s; vk;j;i;s;wk;j;i;s;4k;j;i;s; qk;j;i;s;jk;j;i;s�T and is the vector ofdisplacement of the nodes expressed in the global coordinatesystem.

In this paper the geometrical and physical linear systems areconsidered (spring vibrations around the static equilibrium). Thisleads to the following equations of the system’s motion:

M€qþ Cq ¼ �G þ Q ; (10)

where Mn�n is the mass matrix, Cn�n is the stiffness matrix, qn�1 isthe vector of generalised coordinates.

Gn�1 ¼ vVg=vq, Vg is the potential energy of gravity forces,Qn�1 ¼ ½Q1 . Qn �T is the vector of generalised forces and n isthe number of degrees of the system’s freedom. The equation (10)were integrated with the Newmark method with a constant inte-gration step.

2.2. Validation of the model

The presented FEM model was extended by introducing RFEMmodels of suspension and anvil beams [13], and implemented intoVibroESP calculation software. Validation was performed bycomparing the results of own numerical simulations withmeasurements performed on a test stand built by a producer ofelectrostatic precipitators (Fig. 3a).

The system of electrodes shown in Fig. 3b was consideredduring model validation. The electrodes were 16 m long and1.5 mm thick. A simulation was carried out with: time ofanalysis T ¼ 1 � 10�2 s, integration step hc ¼ 1 � 10�6 s and thediscretisation density of electrodes defined by 400 elementsalong axis x (Fig. 2). The configuration of checkpoints corre-sponds to the arrangement of acceleration sensors on the teststand. The equipment used consists of a 16-channel recorderTEAC lx110, a portable computer with LX Navi and FlexProsoftware, 5 triaxial vibration ICP sensors and an ICP force sensor.The signals were recorded with a sampling rate of 24 kHz (perchannel) as the response of the system for a single force impulseF(t) applied to the anvil (Fig. 3b).

The vibration of electrodes changes very rapidly. A directcomparison of such processes is not very effective, or is evenunfeasible. That iswhy thepeak valueWMax and rootemeanesquarevalue WRMS were used to evaluate signals in the domain of ampli-tudes; the values were expressed by the following relationships:

Fig. 3. Test stand (a) and positions of accelerometers (b) at control levels IeV. A single control level is a set of checkpoints with equal values of coordinate x.

A. Nowak / Journal of Electrostatics 70 (2012) 327e332330

WMax; s; s_; i ¼ max

0�t�T

��as; s_;i��

WRMS; s; s_; i ¼

1T

24ZT

0

a2s; s

_;idt

35

12

9>>>=>>>;; (11)

where T is the time of analysis, s_is the index that takes the value n

_

if signal as; s_;i was determined as the result of numerical calculationsor p

_if it was received as the result of measurements, and i is the

number of checkpoints. In the above formulae it was assumed thatas; s_;i may have one of the values below:

as; s_;i ¼

8>>>>>><>>>>>>:

ax; s_;i �acceleration in the direction of axis x;ay; s_;i �acceleration in the direction of axis y;

as; s_;i ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2x; s

_;iþ a2

y; s_;i

q�tangential acceleration in plane xy;

an; s

_;i ¼ az; s_;i �normal acceleration to plane xy;

ac; s_;i ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2x; s

_;iþ a2

y; s_;iþ a2

z; s_;i

q�total acceleration;

where s ¼ {x, y, s, y, c}. Quantitative analysis of the results is carriedout by means of two different indicators: FAC2 and qε. These areused, among others, to evaluate the forecasting models. In [16] it isassumed that the accuracy of the model is acceptable whenFAC2 � 0.5 and q

ε� 0.66. Verifiability indicator FAC2 was used as

error estimation in the process of validation:

FAC2g ¼ 1np

Xnp

i¼1

Nfi whereNf

i ¼8<:1 for

12�W

g;si;n_

�asi;n

_

�W

g;si;p_

�asi;p

_

�< 2

0 otherwise

(12)

where Wg; si; s

_ and Wg; si; p

_ are calculated in accordance with eq.

(11), g ¼ {Max, RMS}, and np is the number of checkpoints. Theconformity ratio q(as) is defined as:

qg ¼ 1np

Xnp

i¼1

Nqi where

Nqi ¼

8<:1 for

��Wg; si; n

_

�asi; n

_

��Wg; si; p

_

�asi;p

_

���W

g; si; p_

�asi;p

_

� < ε

0 otherwise

; ð13Þ

where ε is themaximumrelative error assumed. Itwas assumed thatthe relativeerror is equal to theaccuracyofmeasurements, so ε¼0.4.

3. Results and discussion

Fig. 4 presents a combination of the values of indicators FAC2-g (as) and qg(as). The model showed high values of the verifiabilityindicator FAC2Max (as) e about 4% of checkpoints were not withinthe acceptable range, whereas in the case of FAC2RMS (ac)e only onecheckpoint was outside the range.

The comparison of the values of qg(as) looks slightly worse. Inthis case the differences in values fluctuate between 4% and 25%.

Fig. 5. Different geometrical parameters: shapes of electrodes (a) and positions of the anvil beam in the system (b). In all cases the electrodes were 16 m long and 1.5 mm thick. Theforce impulse was assumed to be as in Fig. 1c.

Fig. 6. Results of numerical simulations for different: profiles of electrodes (a, b) and positions of the anvil beam in the system (c, d). In Fig. 6a,c the average peak values of normalacceleration are compared. Tangential components of acceleration are shown in Fig. 6b,d.

Fig. 4. Indicators FAC2g (as) and qg(as).

A. Nowak / Journal of Electrostatics 70 (2012) 327e332 331

A. Nowak / Journal of Electrostatics 70 (2012) 327e332332

Such differences may result from themeasurements as well as fromsimplifications assumed in the computational model.

The numerical simulation enabled us to analyse the influence ofthe system’s different geometrical parameters. There was a total of100 control levels; and the distance between each control level was0.16 m. At each level the accelerations were computed in 225checkpoints. The checkpoints were not numbered e for the testingcalculations theywere identified in the global coordinate system { }only by their coordinates. The average calculation time for theactual time of 10 ms was about 70 min.

Firstly, different shapes of the profile of electrodes were exam-ined (Fig. 5a). It can be seen that the wavier the profile, the higherthe normal acceleration of the electrode (Fig. 6a), and thus perhapsthe higher efficiency of the dust removal process. The flat elec-trodes reached the highest tangential acceleration at all controllevels (PLATE F - Fig. 6b), so, if this were the only criterion forselection, that profile would be the best.

However, the ESP is designed and the electrode profile isselected, so an important criterion is to obtain a maximum collect-ing surface, but the flat electrodes do not provide this. They also donotmeet the criterion bywhich the value of the normal componentof acceleration at any point in the systemmay not be less than 100 g(g e acceleration due to gravity) [10] e in case of flat electrodes, inthe numerical model, these values are equal to zero (Fig. 6a).

Different positions of the anvil beam (Fig. 5b) have also beenexamined e moving it upwards did not cause the accelerations toincrease, however, they were more evenly distributed (Fig. 6c,d).The position of the anvil beam in the system affected both the sizeand distribution of the vibration generated, also during the time oftheir propagation. For these reasons the central position of thebeam in the system seems to be the best. On the other hand, placingthe anvil beam horizontally halfway along the length of the elec-trodes could require substantial changes in the existing construc-tions, thus several technical difficulties might be encountered.

4. Conclusions

The results of validation of the model allow to state that themodel correctly reproduces the character of dynamic phenomenathat appear in the system of collecting electrodes during thevibrations that are generated in them by the impulse of forcecoming from the hammer of the rapping system. The experimentalinvestigations demonstrate some of the model’s imperfections; yetthe differences between the results of calculations and measure-ments are within the range of values accepted in engineeringpractice. Moreover, the results are characterised by precisioncomparable to the precision achieved in commercial models, yetreached at considerably smaller computational costs [13]. Thecomputer implementation of the model, as presented in this paper,found application in the design office of one of the Polish producersof electrostatic precipitators.

The overall conclusion is that the process of vibration excitationand wave propagation in the system of electrodes is the result of

many factors. This process depends not only on the impact force,but also on the physical parameters, geometry and construction ofall the elements that make up this system [10]. In this respect themodel presented in this paper is an important novelty since usingthe testing calculations can predict properties of the future struc-ture as early as at the stage of its design.

Acknowledgements

The paper is part of the project N R03 0035 04 financed by theNational Centre for Research and Development, Poland.

References

[1] T. Yamamoto, Effects of turbulence and electrohydrodynamics on the perfor-mance of electrostatic precipitators, Journal of Electrostatics 22 (1989) 11e22.

[2] E. Wittbrodt, I. Adamiec-Wójcik, S. Wojciech, Dynamics of Flexible MultibodySystems Rigid Finite Element Method, Springer-Verlag, Berlin Heidelberg,2006.

[3] Z. Long, Q. Yao, Q. Song, S. Li, A second-order accurate finite volume methodfor the computation of electrical conditions inside a wire-plate electrostaticprecipitator on unstructured meshes, Journal of Electrostatics 67 (4) (2009)597e604.

[4] S.L. Francis, A. Bäck, P. Johansson, Reduction of Rapping Losses to Improve ESPPerformance Proceedings ICESP XI, Hangzhou, China (2008), http://www.isesp.org/ICESP%20XI%20PAPERS/pdfs/045Reduction%20of%20Rapping%20Losses%20to%20Improve%20ESP%20Performance.pdf.

[5] M. Sarna, Self Exploring ESP Rapping Optimisation System Proceedings ICESPVI, Budapest, Hungary (1996), http://www.isesp.org/ICESP%20VI%20PAPERS/VI-ICESP%20P080.pdf.

[6] A. Nowak, S. Wojciech, Optimisation and experimental verification of a dust-removal beater for the electrodes of electrostatic precipitators, Computers andStructures 82 (22) (2004) 1785e1792.

[7] S.H. Kim, K.W. Lee, Experimental study of electrostatic precipitator perfor-mance and comparison with existing theoretical prediction models, Journal ofElectrostatics 48 (1999) 3e25.

[8] L. Buhl, S. Drue, L. Lind, Field Testing of Acoustical Cleaning of ElectrostaticPrecipitators Proceedings ICESP VI, Budapest, Hungary (1996), http://www.isesp.org/ICESP%20VI%20PAPERS/VI-ICESP%20P071.pdf.

[9] S.H. Kim, K.W. Lee, Influence of Contaminated Discharging Electrode andCollection Plates on Particle Collection Characteristics of ESP ProceedingsICESP VII, Kyongju, Korea (1998), http://www.isesp.org/ICESP%20VII%20PAPERS/VII-ICESP%20P111.pdf.

[10] A. Strchlow,M. Schmoch,Comparisonof theTechniques forElectrodeRapping inElectrostatic Precipitators (2001) Proceedings ICESP VIII, Birmingham, AL., USA.

[11] R. Britter, M. Schatzmann (Eds.), Background and Justification Documentto Support the Model Evaluation Guidance and Protocol, COST Office, Brussels,2007 COST Action 732, http://www.mi.uni-hamburg.de/fileadmin/files/forschung/techmet/cost/cost_732/pdf/BackgroundDocument_1-5-2007_www.pdf.

[12] O.C. Zienkiewicz, R.L. Taylor, J.Z. Zhu, The Finite Element Method: Its Basis andFundamentals, sixth ed., reprint, Amsterdam: Elsevier Butterworth Heine-mann, 2006.

[13] A. Nowak, Modelling and Measurements of Vibrations of Collecting Electrodesin Dry Electrostatic Precipitators (In Polish) (2011) Research Monograph(habilitation), Bielsko-Bia1a, Poland.

[14] I. Adamiec-Wójcik, Modelling of systems of collecting electrodes of electro-static precipitators by means of the rigid finite element method, The Archiveof Mechanical Engineering LVIII (1) (2011) 27e47.

[15] M. Huang, Z. Zhao, C. Shen, An effective planar triangular element with dril-ling rotation, Finite Elements in Analysis and Design 46 (2010) 1031e1036.

[16] J.C. Chang, S.R. Hanna, Technical Descriptions and User’s Guide for the BOOTStatistical Model Evaluation Software Package, Version 2.0 (2005).http://www.harmo.org/kit/Download/BOOT_UG.pdf.