SOME ASPECTS OF MEASUREMENT ERRORS ANALYSIS IN ELECTROMAGNETIC FLOW METERS FOR OPEN CHANNELS

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1 OME APECT OF MEAUREMENT ERROR ANALYI IN ELECTROMAGNETIC FLOW METER FOR OPEN CHANNEL Adrzej Michalski () Wiesław Piotrowski () () Warsaw Uiversity of Techoloy, Istitute of the Theory of Electrical Eieeri ad Electrical Measuremets, ul.koszykowa 75, WARZAWA, POLAND, FAX:(48) -6996, ami@iem.pw.edu.pl () Military Uiversity of Techoloy, Faculty of Electroics, Istitute of Electroics Fudametals, ul. Kaliskieo, WARZAWA, POLAND, FAX: (48) , WPiotrowski@wel.wat.waw.pl Abstract - This paper deals with some problems of measuremet error aalysis for flow measuremet by electromaetic flow meter. Two mai roups of errors are discussed. The first roup of errors is caused by various flow parameters which differ from the flow characteristics duri calibratio. The authors preset a ew method of errors estimati, called the movi stream method that allows to estimate o error of a flow meter duri the desi stae much more precisely tha classical methods. The secod roup of errors is caused by a variatio of liquid level i a flow chael duri flow meters operatio. Authors propose a selecti a appropriate liquid level as a parameter i desii the exciti coil of electromaetic flow meter as a best way of that error miimizatio. ome results of umerical simulatio are preseted.. INTRODUCTION The electromaetic techique of flow measuremets imposes a artificial maetic field i a coductive fluid flow ad the measures the resulti potetial differece across the flow (Fi.). measuremet electrode flow chael Z exciti coil Fi.. Idea of electromaetic flow meter The maetic flux desity vector, B, has to be perpedicular to the fluid s velocity vector, ϑ. The movi fluid i the flow chael acts as the coductor ad the iduced voltae betwee the electrodes, which are placed o both sides of a flow chael, is proportioal to the mea velocity of the flow i the area betwee the electrodes. The measured voltae ca be described as follows: Y X U= W ϑdv () V where: W is the weiht vector, ( W = B J, where J is the virtual curret desity vector, B is the maetic flux desity vector), U is the voltae across the electrodes, V is the flow volume, ϑ is the velocity vector of the liquid. If the excitatio coil of the flow meter eerates a field such that rot ( B J ) = rot( W ) = 0, () over the etire domai V [], the the voltae betwee the electrodes would be proportioal to the mea velocity of uspecified flow,. This kid of flow meters is cosidered ϑ m ideal [,4,6]. The mai oal of desier is to mould the excited coil i a such way to fulfils the formula (), but techical limitatio allows to meet this requiremet oly approximately [,3,6]. Resulti measuremet error cosists of a two mai parts: error which is caused by various flow parameters which differ from the flow characteristics duri calibratio ad error caused by liquid level variatio. From the poit of view of the desi procedures it is possible to precisely estimate the value of the first error ad effectively miimize the value of the secod error.. ERROR CAUED BY VARIOU FLOW PARAMETER - METHODOLOGY OF ETIMATION Classical methods for estimati the error caused by various flow parameters are based o determii the distributio of the weiht vector. [3,6] To determie the distributio of a weiht vector, first of all, we must determie the distributio of the virtual curret desity vector ad distributio of the maetic flux desity vector as well. Accordi to the Bevir method, the distributio of a virtual curret desity vector i the measuremet zoe is equivalet to the distributio of a hypothetical uitary curret i the area betwee the electrodes []. Because of umerical calculatios, the distributio of J ca be easier foud as a div radϕ = 0 for solutio of the Laplace equatio ( )

2 hypothetical electric potetial ϕ with a Dirichlet boudary coditios o measuremets electrodes (ϕ e =, ϕ e = 0) [6]. If the aalysed problem is liear, tha the distributio of J ca be described, as follows: J = αradϕ, (3) where α is a coefficiet described by formula (4). The value of the coefficiet α depeds o the actual shape of the flow chael ad the coductivity of the chael bed ad baks. Coefficiet α ca be described by α = ϕ d s e (4), where is the coductivity of the liquid, e is the area of m leth of the electrode, ϕ d s is the real value of the e electrode curret. For two dimesioal calculatios the total hypothetical curret will be equal A/m of electrode s leth. Oce you determie the distributio of J the you ca bei desii the excitatio coil. The distributio of the maetic flux desity ca be calculated (i most cases usi fiite elemet aalysis ad a variable metric method []) duri the desi of the crosssectio of the excitatio coil i the electromaetic flowmeter. The velocity vector, ϑ, has oly oe compoet, ϑ z, with the appropriate istallatio of the primary trasducer for most atural ad artificial flow chaels, (Fi.). Cosequetly, the three-dimesioal problem ca be aalysed i a two-dimesioal co-ordiate system ad the formula () ca be rewritte as follows: J B J B =w y x x y (5) I formula (5), the costat w should be chose to obtai a measurable value of the output sial () ad yet be sufficietly small to maitai reasoable coil dimesios ad supply power. The coil shape must miimize the followi objective fuctio to obtai uiformity i the distributio of w for a ive chael cofiuratio: [ w ( JxBy 0 JyBx) ] ds 0 ( x,i y,i y,i x,i ) [ ] w J B J B i (6) F = w0 w0 where: is the area of a cross-sectio of aalysed chael, w o is the desired value of flow sial (for ϑ ds = ), is the umber of fiite elemets i which the area was divided ito i (Fiite Elemets Method). The objective fuctio eerally describes the ohomoeeity of the distributio of W. The fial value of the objective fuctio ca be used to estimate, duri the desi procedure, the expected error for a electromaetic flowmeter caused by the variable characteristics of the flow. The ew approach to determii the error of a primary trasducer is based o a movi stream method derived by the authors. The movi stream method allows the desier to check the sesitivity of the trasducer s desi for various velocity distributios i the flow chael. Therefore, it is possible to estimate the error of a flowmeter for the actual coditios i the field. The value of objective fuctio (4) is determied maily by the o-uiform distributio of the vector product B J. This o-uiformity reaches maximum values ear the electrodes ad the chael s walls []. Whe cosideri the real velocity profile (e..ϑ = 0 ear the walls), it does ot seem reasoable to use the fial value of the objective fuctio i estimati the error. These results will always be quite differet from the actual values of measuremet error. Geerati velocity profiles similar to real oes i the chael s cross-sectio provides the basis for determii the flowmeter s respose to variable profiles of velocity. These profiles should provide ull boudary coditios (ϑ = 0 ) at the walls of the chael. The fuctio that describes the velocity profile depeds o eometric parameters of the chael ad is defied by the followi equatio: ϑ( x, y) f( x, y) h( x, y) = (7) where π ( a x,y a p,q x > p the f x,y = cos ( a( p,q ) if a x,y x < p the f x,y = si a( p,q) π. c x,y y > q the h x, y = si c( p,q) π π ( c x,y c p,q y < q the h x, y = cos ( c( p,q ) The variables a(x,y) ad c(x,y) are defied as follows: y, cxy (, ) =. d x axy (, ) = w d y d w w + Parameters w, w ad d describe the width of the chael bottom, the width of the water s surface, ad the depth of the chael respectively. You may determie the poit M of maximum flow velocity for the ive distributio by setti the parameters p ad q (coordiates i a two-dimesioal model of the flow chael). You ca chae the simulated velocity profile by chai the coordiates of poit M, but the averae speed i a measuremet zoe always remais costat. If the flow is ot caused by the force of ravity, virtually ay velocity profile is valid. The rae of p ad q is limited oly by the eometry of the chael. Fiure shows

3 the examples of velocity profiles for half of a rectaular flow chael that measures m deep by m wide. Fi.. Example of velocity profiles eerated for a rectaular flow chael. The fuctio defied i (7) ca eerate a velocity profile very similar to the real oe reardless of the drivi force, ravity or artificial. For a ideal flow meter, the voltae betwee the measuremet electrodes ive by () should be costat for all velocity profiles havi the same averae value. The estimated error i a flow meter caused by variatio i the flow ca be defied as the stadard deviatio of a mea value of electrode potetial versus the mea value of electrode potetial obtaied for a series of differet simulated velocity profiles, where the averae velocity remaied costat: η = m= u um m= u 00% (8) where: is the umber of simulated velocity profiles, u is the mea value of electrode potetial calculated for a series of differet velocity profiles, u m is the calculated electrode potetial for m-th velocity profile. everal flow chaels were aalyzed for the expected error by simulati the trasducer behavior uder differet velocity profiles. Table shows results for rectaular (Rect.) ad trapezoid (Trap.) chaels with various water-tobed coductivity ratios, w / ( - [/m]). Tab. Results of umerical calculatios for various chael cross-sectios ad coductivity ratios. Chael Coductivity ratio [/m] Objective fuctio defied by (6) Error defied by (8) Rect. 0./ Rect. 0.0/ Trap. 0./ Trap. 0.0/ ixty differet velocity profiles were aalyzed for each chael, all havi the same mea velocity of m/s. The error was calculated accordi to formula (8). The movi stream method shows apparet correlatio betwee the error ad the fial value of the objective fuctio (compare the first ad secod rows or the third ad fourth rows i table ). We could ot estimate the real value of the error by cosideri the fial value of the objective fuctio aloe. 3. ERROR CAUED BY LIQIUD LEVEL VARIATION - METHODOLOGY OF MINIMIZATION Classical sythesis of a cross-sectioal shape of the excitatio coil takes ito cosideratio full filli flow chael [,6]. Full filli of a flow meters cross-sectio ca be easily achieved i case of closed chaels, whereas i ope chaels such coditios practically ever exist. o the iitial coditios for desi procedure were quite differet from reality. Duri the umerical simulatios the chaes of a voltae sial across the electrodes as a fuctio of chael s filli were moitored with costat mea velocity of the liquid. Calculatios were carried out for a coil desied for fully filled chael. The results idicated sial chaes related to the level of the liquid eve thouh theoretically voltae across the electrodes is oly a fuctio of the mea velocity. This effect is caused by o-uiform distributio of a weiht vector itroduced by variable filli. It ca be clearly see that the acquired shape of the exciti coil is ot optimal for applicatios where liquid level chaes are expected ad trasfer fuctio of such a flow meter is ambiuous. Miimizatio of errors itroduced by chai level of the liquid is coducted maily by socalled dry calibratio [3] procedures whe the level is costatly moitored ad appropriate correctio factors are applied. But it turs out, that for extreme level chaes ad small liquid-to-roud coductivity ratios those coefficiets exceed value of oe, maki this whole procedure quite useless. Authors suest to choose such level of the liquid duri calculatios of the shape of the coil, for which the ifluece of level chaes o voltae sial would be miimal. Assumptio of certai level of filli duri the desii process affects the results. Hitherto existi results of works, i which classical approach was used, showed, that full filli of chael does ot resolve problem uivocally. Choices of level of filli based o averae oe year's filli or radom selectio also are ot best because of ecessity of measuremet of hih states. Dimiishi of filli accepted to calculatios i approximate maer will ot permit fully to verify received results which is caused by o-liear trasfer fuctio of a flow meter. To poit out the filli which could be accepted to calculatios, over a doze of coils were projected, each for differet filli. Every oe of them assured uiform distributio of weiht vector oly at filli for which it was projected. For every case a differet coversio characteristic (U= f(h)) was obtaied. To measure this characteristic, a series of computer simulatio was coducted ad voltae across the electrodes was calculated. The flow velocity was set at m/s ad filli was chaed from 0% to 00% of maximal value. O the basis of those results a optimal coil could be chose. The results, however, could ot be compared directly because of differet fillis. The proposed method of compariso was a ormalizatio obtaied by multiplyi a electrode potetial by a followi coefficiet (9):

4 k = (9) opt w0 where: opt - the area of a filled cross sectio of the flow chael for which the exciti coil was optimized, w 0 - the desired value of coefficiet w. After applyi this ormalizatio, a set of -elemet data vectors was obtaied for each coil. The defiitio of ormalizatio procedure is so costructed, that for ideal coil a uitary vector is expected. For such a ideal coil the measuremet sial depeds oly o the mea velocity of the flow. With the uity vector as referece, it is possible to fid the coil which trasfer fuctio (represeted by its specific data vector) is closest to the ideal oe. The problem of classificatio ca be resolved by choosi a appropriate represetatio of the sial treated as a vector i space. I that case the represetatio is a ormalized electrode potetial. For all data are i a form of vectors, it is atural to use metric spaces as a most coveiet tool of classificatio. Every coil is represeted by a -elemet sequece of real umbers. Let s cosider a set R of all such sequeces. I this set a metric ca be itroduced i may ways. For prelimiary classificatio some basic metrics were chose. Those were the followi fuctioals: d ( x, y = x i y i ; d = x i ( x, y) y i ; ) = d3 ( x, y) x i y i d 4 ( x, y) = x i y i ; { x y } p p ; d x, y x i y 6 = i d ( x, y) = mi (5) 5 i i, where: d (x,y) d 6 (x,y) the value of metric, x i sequece or vector of desired values (uity vector), y i sequece or vector of measured data (y i =k U i ).,5,5 0,5 0 0,05 0,045 0,04 0,035 0,03 0,05 d (x,y) w = 0,7 0,8 0,9 0,99 d (x,y) 6 w =00 0,7 0,8 0,9 0,99 Fi.3. The relatios betwee the metric value ad relative water level (h) take to desi process for metric d ad coductivity ratio w/ =, for metric d 6 ad coductivity ratio w/ =00. The value of metrics is a quatitative value of quality for each represetatio. For cosidered metrics, the better h h solutio is for the smaller values of the fuctioal. This meas that ive represetatio approaches optimal expected value. Lare values of fuctioal mea that the coil has to be rejected. The relatios betwee the metric value ad relative water level take to desi process is show o Fi. 3 ad Fi. 4. 0,07 0,03 0,009 0,005 0,00 0, 0,5 0, 0,05 0 d (x,y) 5 w = 0,6 0,7 0,8 0,9 0,99 h d (x,y) 5 =00 0,7 0,8 0,9 0,99 h Fi.4. The relatios betwee the metric value ad relative water level (h) take to desi process, for metric d 5 ad coductivity ratio w/ =, for metric d 5 ad coductivity ratio w/ =00. The fial decisio which represetatio to choose is ot always clear. It is caused by eerally lare values of fuctioal for fillis lower the 40%. Those values ifluece the fial result ad probably oly the reater fillis should be cosidered (Fi.3,4). By aalysi the characteristics d(x,y) it ca be see, that there is always a optimal coil, closest to the ideal oe. This coil is optimized for a specific filli, which should be take as a starti poit for a optimizatio process. 8,00E-0 7,80E-0 7,60E-0 7,40E-0 7,0E-0 7,00E-0 6,80E-0 U[V] w h Fi. 5 The relatio betwee flow sial ad chaels filli for two flow meters, with 00% filli take to the desi procedures, - with optimized chael s filli take to the desi procedures, both flow meters ware desied for isulated flow chael. It should be oticed that the filli, for which a fuctioal reaches its miimum depeds o may evirometal

5 parameters. The most importat of them is roud-to-liquid coductivity ratio. Fi. 5 presets the relatio betwee flow sial ad chael s filli for two flow meters, - with optimized chael s filli take to the desi procedures with 00% filli take to the desi procedures, both flow meters ware desied for isulated flow chael. All calculated metrics, for that flow chael, poited out the optimal filli at 0.9 h max. The testi procedure was curried out for several differet flow chaels. The obtaied fial results allows to summarise, that applyi the proposed procedure to selecti a appropriate liquid level take to the desi procedures, could effectively miimize (about 0% less) the sesitivity of the flow meter to various liquid level. As a fial testi routie, authors have decided to check the sesitivity of the improved flow meter for various velocity distributio (variable flow parameters). The results of testi are preseted i Table. Tab. Error eerated by variable flow parameters for full ad optimal filli for flow chael m m. Coductivity ratio [-] Liquid level Error defied by (8) 80 (optimal) 5, ,04 80 (optimal) 3, , (optimal) 7, , (optimal) 5, , (optimal) 6, ,43 Desii the excitatio coil of a electromaetic flow meter accordi to the ew procedures of choosi iitial values, leads additioally to 50% reductio of error defied by (8). iificat improvemet ca be obtaied especially for roud - to - liquid coductivity ratios close to. For isulated chaels the error reductio reaches 30%. 4. CONCLUION From the poit of view of the desier ad ed user of the flow meters is very importat to be able to precisely estimate the value of the error caused by various flow parameters as well as effectively miimize the sesitivity of the flow meter to various liquid level i the flow chael. The most importat is that all estimatios could be doe o the desi stae. The classical method of error, caused by various flow parameters, estimatio, would ive a error almost 0 times hiher tha the value obtaied usi the movi stream method. (ee Table.) This is because the objective fuctio icludes a ouiform distributio of the weiht vector i the measuremet zoe. The movi stream method miimizes this ouiformity by setti the real velocity profile (e.. ϑ = 0 ear the walls) existi i the field. Cosequetly, the estimated error is much more closer to the actual value foud i the field. The reductio of error over the classical method s shows how much the ouiformity of the distributio of the weiht vector ca be decreased by applyi real distributios of the flow velocity. All simulatio carried out for a wide roup of a flow chaels cofirm the depedece betwee flow sial ad liquid level i a flow chael. Primarily that depedece was miimized usi dry calibratio procedure. Preseted approach, based o applyi a metric space to selecti a appropriate liquid level take to desi procedures, allows to effectively miimize this depedece. The most importat is that all calculatio ad simulatio could be doe o the desi stae. Geerally we ca say that preseted procedure allows to miimize the sesitivity of the flow meter to various liquid level. Additioally we ca observe the serious improvemet i measuremet accuracy. The value of measuremet error caused by various flow parameters has bee decreasi about 40% i compariso to the flow meter desied for fully filli flow chael. The biest improvemet of measuremet accuracy ca be obtaied whe roud-to liquid coductivity ratio is aroud oe. REFERENCE. M.K. Bevir, The Theory of Electromaetic flowmeter, Joural Fluid Mech., No 43, A.Michalski, J.tarzyński,.Wiceciak, Optimal desi of the coils of a electromaetic flowmeter, IEEE Trasactio o Maetics, vol. 34, No 5, EPTEMBER 998, pp A.Michalski, "Dry calibratio procedure of electromaetic flowmeter for ope chaels, Trasactios o Istrumetatio ad Measuremet, vol. 49, o, April, 000 pp M.K. Bevir, Lo iduced voltae electromaetic flowmeters ad the effect of velocity profile Quart. Jour. Mech. Ad Applied Math., vol XXIV, Pt pp E.Luta, J.Halttue, The effect of velocity profile o electromaetic flow measuremet, esors ad Actuators, o pp A.Michalski,. Wiceciak, J. tarzyński, Electromaetic flowmeters for ope chaels two dimesioal approach to desi procedures, IEEE esors Joural, vol. o, July 00.

LECTURE 17: Linear Discriminant Functions

LECTURE 17: Linear Discriminant Functions LECURE 7: Liear Discrimiat Fuctios Perceptro leari Miimum squared error (MSE) solutio Least-mea squares (LMS) rule Ho-Kashyap procedure Itroductio to Patter Aalysis Ricardo Gutierrez-Osua exas A&M Uiversity