Santiago Pindado *, Javier Cubas, Felix Sorribes-Palmer

Transcription

1 On the harmnic analysis f cup anemmeter rtatin speed: A principle t mnitr perfrmance and maintenance status f rtating meterlgical sensrs Santiag Pindad *, Javier Cubas, Felix Srribes-Palmer Institut Universitari de Micrgravedad "Ignaci Da Riva" (IDR/UPM), Universidad Plitecnica de Madrid, ETSI Aernautics, Pza. del Cardenal Cisne Madrid 280, Spain ARTICLE INFO ABSTRACT Article histry: Received August 204 Received in revised frm 2 March 205 Accepted 29 May 205 Available nline 8 June 205 Keywrds: Cup anemmeter Anemmeter calibratin Cup aerdynamics Analytical mdel Furier analysis Anmaly detectin The calibratin results f ne anemmeter equipped with several rtrs, varying their size, were analyzed. In each case, the 30-pulses pert turn utput signal f the anemmeter was studied using Furier series decmpsitin and crrelated with the anemmeter factr (i.e., the anemmeter transfer functin). Als, a 3-cup analytical mdel was crrelated t the data resulting frm the wind tunnel measurements. Results indicate gd crrelatin between the pst-prcessed utput signal and the wrking cnditin f the cup anemmeter. This crrelatin was als reflected in the results frm the prpsed analytical mdel. With the present wrk the pssibility f remtely checking cup anemmeter status, indicating the presence f anmalies and, therefre, a decrease n the wind sensr reliability is revealed.. Intrductin Tday, the use f wind speed anemmeters is very cmmn, their applicatins having spread frm sectrs such as meterlgy r wind energy t thers where the effect f the wind can be imprtant (mving bridges in civil engineering, big cranes, etc.). Nevertheless, the wind energy industry still can be cnsidered as the biggest cnsumer f anemmeters all ver the wrld. Apart frm the fact that the wind pwer is prprtinal t the third pwer f the wind speed, which underlines the imprtance f having the mst accurate instruments [], the wind energy sectr is extremely cncerned with tw aspects that, despite technlgical advances such as LIDAR and SODAR [2-5], require the use f anemmeters: wind energy prductin frecast n the field, and wind turbine perfrmance cntrl [6]. Within the past decades the wind energy sectr has been penly supprted by gvernments (Germany, Denmark, Spain...), cncerned abut clean energies and a reductin f their dependence n fssil fuels [7]. In additin, new strng players in this industry like China, U.S.A., Brazil r India are nw being very active, with high figures in terms f installed wind pwer and grwing rates. Accrding t these facts it seems reasnable t assume that the mentined massive demand f anemmeters frm this sectr will be maintained in the incming years, if nt increased. Amng the different instruments devted t measure the wind speed, the cup anemmeter remains tday the mst used in the wind energy sectr [8], as it is inexpensive when cmpared t ther devices (e.g., snic anemmeters), shws linear respnse in the nrmal wind speed range (frm 4 m s _ t 6 m s _ ) [9], and is capable t perate in quite extreme weather cnditins [0]. Glbal Wind Energy Cuncil; US Energy Infrmatin Administratin.

2 The present paper is part f the IDR/UPM Institute's research framewrk n cup anemmeter perfrmance, which includes a review f large series f calibratins perfrmed n cmmercial anemmeters [8], the analysis f aging and the climatic effects n anemmeter transfer functin [,2], the experimental study f the cups and rtr shapes n anemmeter respnse [3-5], and the review and imprvement f analytical methds t study anemmeter perfrmance in rder t have a better understanding f the physics regarding its rtatin (aerdynamics, frictin, etc.) [8,3-6]... Mdeling the cup anemmeter perfrmance in steady cnditins.50% \ \ \ \ \,\» \ K \ i\ \ \ \ V \ N Frm the wind industry pint f view, the cup anemmeter perfrmance is based n the transfer functin: V = A-f B, 0) where V is the wind speed, / is the anemmeter's rtatin frequency utput, and A (slpe) and B (ffset) are the calibratin cefficients. This linear equatin, 2 defined in the first quarter f the 20th century fr the Rbinsn-type anemmeter 3 [7], crrelates the wind speed and the anemmeter's utput frequency [8], and must be defined by means f a calibratin prcess [9-22]. In rder t have an expressin with a clearer physical meaning, the transfer functin can be rewritten in terms f the anemmeter's rtatin frequency, f r, instead f the utput frequency, /, intrducing in the expressin the number f pulses per revlutin given by the anemmeter, JV P : V = A-N p -f r B = A r -f r B. (2) The abve equatin als allws a direct cmparisn between different anemmeters [8], and between analytical mdels and experimental results [23,24]. The number f pulses, N p, is different depending n the anemmeter's inner system fr translating the rtatin int electric pulses. Magnet-based systems give -3 pulses per revlutin, whereas ptelectrnics-based systems nrmally give higher pulse rates per revlutin, frm 6 t 44 [8]. T analyze the behavir f cup anemmeter, analytical mdels have been prpsed by ther authrs in the past. These mdels are develped frm the fllwing expressin [9]: /(?^^ (3) where / is the mment f inertia f the rtr, QA is the aerdynamic trque, and Qj is the frictinal trque that depends n the air temperature, T, and the rtatin speed, 2 Sme authrs claim that a nn-linear expressin shuld be used instead f a linear ne, especially at lw wind speeds [0]. 3 The cup anemmeter invented by Rbinsn in the XIX century [66-68] had fur cups/arms instead f three, which is the present standardized cnfiguratin thanks t the wrk f Pattersn [3 ], wh fund that the "3-cup anemmeter is decidedly superir t the 4-cup" due t a quicker and mre unifrm respnse, and a higher aerdynamic trque prduced by the cups [69-7]. In 924 the 3-cup anemmeter was adpted as a standard fr meterlgy in the United States f America and Canada [69] B&J-I; CN (alpha) * V JJ ^V^» B&J-II; CN (alpha) "^*tp^ S. B&J-II; CN (theta) 0 B&J-II; CN (alpha), furier -h -.. a- B&J-II; CN (alpha), furier 6-h [ ] Fig.. Nrmal aerdynamic frce cefficient, c N, f the Brevrt & Jyner Type-I (hemispherical) and Type-II (cnical) cups [28], pltted as a functin f the wind directin with respect t the cup, a. The cefficient regarding Type II cup is als pltted as a functin f the rtr's rtatin angle, 9 (calculated with expressin (6) fr an anemmeter factr K = 3.5). See in the sketch the variables invlved in the rtatin f an anemmeter cup: nrmal aerdynamic frce n the cup, N, wind speed, V, relative wind speed t the cup, V r, rtr rtatin angle, 9, rtr rtatinal speed, m, and wind directin with respect t the cup, a. The -harmnic term Furier series apprximatin (expressin (9)) t the Type-II cup has been pltted, tgether with the 6-harmnic terms Furier series apprximatin [4]. c (frm [25]: Qf = B 0 (T) B^T)c B 2 {T)c 2, where cefficients B 0. Bi, and B 2 are negative 4 ). The frictinal trque, Qf, can be neglected as it is nrmally very small in cmparisn t the aerdynamic trque [26,27]. The aerdynamic trque, QA, can be derived frm the aerdynamic frces n the rtr cups, which are nrmally measured in a wind tunnel in "static" cnfiguratin, that is, measuring the frces n an islated and fixed cup immersed in a cnstant wind speed air flw and withut cnsidering any rtatinal speed [3,4,28]. The aerdynamic trque due t ne single cup is expressed as a functin f the aerdynamic nrmal-t-the-cup frce cefficient measured in a wind tunnel, c N, the wind speed, V, the rtatin speed, c, the cup radius, R c, the cup center rtatin radius, R rc, and the air density, p. See in Fig. the nrmal-t-the-cup frce cefficient, CJV, measured n hemispherical and cnical cups in "static" cnfiguratin, as a functin f the wind angle with respect t the cup, a, [28]. As far as the authrs' knwledge ges, the first analytical mdel was develped by Schrenk [29] in 929. This 2-cup psitins mdel relates the afrementined aerdynamic trque, QA, t the rtatin speed, c, the wind speed, 4 The frictin trque, Qf, in expressin (3) has a negative sign in the updated versin f reference [9]. Therefre, cefficients B 0, Bi and B 2 f the frictin trque expressin will be psitive, if this is taken int accunt.

3 F i =\pv? {9\2W)S c c N {9 \20 ) F 3 =\pv? (024O )S c c N (0 2 ) ^> V Fi=ipr?{)sM0) Fig. 2. Sketch f the 3-cup analytical mdel cnfiguratin. V, and the aerdynamic nrmal cefficient, c N, measured at tw wind speed angles, that is, at tw psitins f the cups, a = 0 (c D = CJV(0) in Fig. ) and a = 80 (c D2 = c N (80) in Fig. ). This mdel has been widely used t study the cup anemmeter perfrmance [6,30]. Althugh this 2-cup analytical mdel has crrelated well t experimental calibratin results frm different cmmercial anemmeters [8], it is fair t say that it has certain limitatins, derived frm the fact that the behavir f the rtr is based n tw psitins f the cups nly. T have a better apprach t the prblem, the 3-cup analytical mdel was develped integrating the aerdynamic nrmal frce n all three cups in an entire rtatin [3,23,24]. The starting pint f this 3-cup mdel is the mentined expressin (3), taking int accunt, as said, the aerdynamic trque prduced by each cup f the anemmeter rtr, see Fig. 2. If frictin is als left ut f Eq. (3), the fllwing expressin can be derived then fr the rtr mvement: dc ~d7 = 2 ^ps cr rc V 2 (8)c N (i(8)) -ps c R rc V 2 (8-2O )c N (a(02o ))- psc R rc V 2 (8-24O )c N (a(0 24O )), where V r is the wind speed relative t the cups, c N is the aerdynamic nrmal frce cefficient, a is the lcal wind directin with respect t the cups, 8 is the angle f the rtr with respect t a reference line (see sketch in Fig. ), and S c is the frnt area f the cups (S c = nr 2 ). Wind speed V r, relative t the cup at rtr angle 8 with respect t the reference line, is expressed as: V r {8) = yv 2 {0)R rc ) 2-2V)R rc cs{8), (5) whereas the wind directin with respect t the cup, a, can be derived, as a functin f the rtr's psitin angle, 8, frm the fllwing expressin [3]: tan(a) (4) Ksin(0) : Kcs(0)-l : (6) where K is the anemmeter factr, defined as the rati between the wind speed, V, and the speed f the cup center averaged in ne cmplete rtatin, a>r rc [9,3]: K = V <DR A r f r B 2nf r R rc A r 2nR7c!-( ) In the present wrk the anemmeter factr is cnsidered leaving aside the ffset f the transfer functin, B, as it is nrmally very small when cmpared t the wind speed: 5 A r K = 2nR (8)... Mdeling the aerdynamic frces n the cups The aerdynamic nrmal frce cefficient, c N, can be simplified taking int accunt nly the tw first cefficients f the Furier series expansin [4]: c N (a) = c 0 Ci cs(a). (9) Als, the related-t-the-cup wind angle, a, was quite accurately related t the rtr's rtatin angle, 8, with the expressin: cs(a) =ri 0 ri cs(8) rj 2 cs (ff) 2 rj 3 cs (fff, (0) where the cefficients t] 0, rji, t] 2, and t] 3 are expressed as a functin f anemmeter factr K: - K V = m VIK 2 ' I \/lk 2 ' <7i = V^ K 2 % K 2 -\ VTTF (7) (ii) Taking int accunt expressins (9) and (0), Eq. (4) can be slved fr the statinary state, averaging its value alng ne turn f the rtr. Then, a direct relatinship between the anemmeter factr and cefficients c 0 and c x is derived: 5 The ffset wind speed, B, f the transfer functins related t mst f Class anemmeters is nrmally lwer than 0.25 m s _ and negligible [8]. Hwever, the effect f this cnstant shuld be taken int accunt fr calculatins at lw wind speeds [5].

4 0= ( F)( cci (^H)-t(^!^ (2) Using expressins (), the abve equatin was finally rewritten in terms f the anemmeter factr: -(-?)H^) c : / K 3K 2-4\ This equatin gives, as a functin f the Furier cefficients rati Ci/c 0 (that nly depends n the aerdynamics f the cup), the anemmeter factr, K, and then, fr each wind speed, V, the averaged value f the rtatinal speed f the rtr, c, can be btained frm expressin (7). This 3-cup mdel has been successfully crrelated with experimental results [4]...2. Mdeling the rtr's respnse Additinally, if the attentin is fcused n the left term f Eq. (4), we shuld bear in mind that this term is nt equal t zer, althugh its average value is. Under a perfectly cnstant, hrizntal and unifrm wind speed, and cnsidering the statinary cnditin, the rtatinal speed crrespnding t a 3-cup rtr can be decmpsed alng ne turn int a cnstant term, c 0. plus ne harmnic term that crrespnds t a frequency three times bigger than the ne related t the mentined cnstant term, 3c 0. and its multiples, 6c 0. 9c 0. 2c 0 -.[5]: a(t) = c ^c3 sin(3nct<p 3 ). 0 4 ) n=l Obviusly, fr a cup anemmeter in a real situatin, that is, taking int accunt sme degree f turbulence, wind nn-unifrmities, the wake created dwnstream the anemmeter's bdy, etc., ther harmnic terms nt multiple f 3 shuld be included in the abve equatin as a result f the Furier expansin: c(t) = c 0 (Q\ sin(c 0 t cp ) <D 2 sin(2c 0 t cp 2 ) c 3 sin(3c 0 t cp 3 )... CO = ) 0 ^2) n sin(n) 0 t cp n ). (5) n=l Nevertheless, taking int accunt results measured n damaged anemmeters [32], expressin (5) can be quite accurately simplified by taking nly the cnstant term, c 0. and the first and third harmnic terms f the series: c(t) = c 0 (Q\ sin(c 0 t <pj) c 3 sin(3c 0 t cp 3 ), (6) as the third harmnic term is the mst imprtant due t the cup anemmeter shape (which prduces three acceleratins and three deceleratins f the rtr in ne turn), while the first harmnic reflects the anmalies that affect the nrmal rtatin f the anemmeter's rtr..2. Understanding f the rtatinal speed harmnic terms The aim f the present wrk is t have a better understanding f the effect f the first and the third harmnic terms, m^ and c 3, n cup anemmeter perfrmance, cntinuing the experimental research already initiated in [5]. The analytical relatinship between this third harmnic term and the cefficients c 0 and c x (frm the simplified expressin f the aerdynamic nrmal frce cefficient, c N, see Eq. (9)), is als analyzed fr bth statinary and transient cases. Besides, the first harmnic term is studied as a way t analyze the evlutin f cup anemmeter perfrmance degradatin. The idea behind this research is the attempt t establish sme parameter related t the anemmeter wear and tear in rder t ptimize the maintenance prcesses, as arund 30% f mast-munted anemmeters return fr recalibratin far frm nrmal peratinal cnditins [33]. Sme effrt has been dne t mnitr the status f anemmeters, as their wrking cnditin when perating in the field must be checked thrugh frequent calibratins [34]. Calibratin-nthe-field prcedures have been studied as a cst-effective slutin in rder t reduce anemmeter maintenance and recalibratin [35,36]. This prblem is a majr cncern in the industry, and as a result several patents and inventins have been develped. The prcedure behind sme f them is based n frcing the anemmeter's rtatin and then crrelating its respnse with the crrespnding input [37-4], whereas ther patents are based n the crrelatin between the measured wind speed by the anemmeter and a different parameter related t the wind speed, e.g., the pwer curve f a wind turbine [42-45]. Anmaly detectin is quite an imprtant prblem within many industrial, technlgical and research areas, with three classical appraches [46]: Case-based reasning, based n a list f rules derived frm wide human experience. Mdel-based diagnsis, in which the decisin criteria are based n mathematical mdels assciated t physical attributes. Sme parameters with physical meaning are then mnitred as indicatrs f the pssible degradatin f the system. Nn-parametric mdels such us neural netwrk-based methds. This apprach des nt require expert knwledge. Hwever, a huge amunt f training data needs t be pst-prcessed in rder t validate the mdel. The present wrk shuld be cnsidered then an effrt t identify a new parameter, the first harmnic term f the rtatin speed, t be used in anemmeter cnditin diagnsis. Fcusing n the cup anemmeter as a mechanism, it is lgical t assume that any malfunctin f the anemmeter bearings system, r any defect prduced n the rtr gemetry (dirt accumulatin, brken parts...), shuld be translated int nise added t the utput signal. Apart frm the fact that this nise means a change f the transfer functin cnstants and, cnsequently, a degradatin f anemmeter perfrmance (lwer rtatin frequency at same wind velcity), it can be assumed that it is frmed by a number f signal perturbatins that are

5 repeated each turn f the rtr. S, in principle, any degradatin r change f the anemmeter's wrking cnditin shuld be translated int an increase f the first harmnic term f Eq. (5). This kind f analysis based n the harmnic respnse has been traditinally used fr wind turbine cnditin mnitring. These machines, that invlve several rtating systems (gear, bearings, shaft), include different sensrs and transducers (psitin, velcity, and acceleratin) used t mnitr the maintenance status thrugh Furier analysis f vibratins spectra [47-50]. In additin, it shuld be pinted ut that fault detectin in mechanical units such us cup anemmeters is usually carried ut by perfrmance mnitring, this prcess requiring large time-series prcessing [5]. Different techniques have been develped t reduce the cmplexity f wrking with that large time-series. The nrmal way t perfrm this reductin is t characterize the time-series by a spectral decmpsitin (ther characterizatins as wavelet decmpsitin r singular value decmpsitin are als pssible) [52-54]. The harmnic terms f the cup anemmeter rtatin speed defined in the present wrk can be cnsidered equivalent t the spectral decmpsitin. In this cntext, the PHM 20 Data Challenge Cmpetitin shuld be mentined, as it was a very interesting prpsal t analyze the prblem f cup anemmeter status based n the measurements recrd. In this cmpetitin, a set f measurements (mean, standard deviatin, maximum and minimum wind speed), taken by several paired anemmeters installed at different heights alng a vertical mast were analyzed in rder t study the anemmeters' wrking cnditin. Different slutins were btained by researchers wh tk part in the afrementined cmpetitin, the mst relevant being based n direct cmparisn f signals frm tw different cup anemmeters nce prperly filtered [55], the crrelatin f the differences in measured wind speed frm tw anemmeters with a Weibull distributin [56], and the use f a neural netwrk mdel [57]. The last apprach, i.e., the use f neural netwrk mdels, seems t be a quite accurate tl fr cup anemmeter perfrmance analysis, as it btained the highest scre in the PHM 20 Data Challenge Cmpetitin, and it has als been used with gd results t cmpensate anemmeter verspeeding in real-time measurements [58]. This wrk is rganized as fllws. In Sectin 2 the testing methdlgy and results pst-prcessing carried ut in the present research are described. The results are discussed in Sectin 3, which is divided in three subsectins. The first subsectin is devted t a first analysis f the testing results, whereas the secnd and the third subsectins include the develpment f an analytical methdlgy based n harmnic decmpsitin, t study respectively the cup anemmeter wrking under ptimal cnditins and under the effect f a peridical perturbatin. Finally, cnclusins are summarized in Sectin Testing cnfiguratin and results pst-prcessing The characteristics f the 32 different rtrs analyzed in the present wrk are included in Table. 26 rtrs were equipped with cnical cups (90 cne-angle, 4 with cup radius: R c = 20 mm, cup center rtatin radius varying frm R rc = 30 mm t R rc = mm; 5 with cup radius: R c = 25 mm, cup center rtatin radius varying frm R rc = mm t R rc = 00 mm; 6 with cup radius: R c = 30 mm, cup center rtatin radius varying frm R rc = mm t R rc = 20 mm; 5 with cup radius: R c = 35 mm, cup center rtatin radius varying frm R rc = 50 mm t R rc = 20 mm; and 6 with cup radius: R c = mm, cup center rtatin radius varying frm R rc = mm t R rc = mm). 3 rtrs were equipped with elliptical cups (frnt surface equal t the cnical cups: S c = mm 2 and R rc =mm). And finally, 3 rtrs were equipped with prus cups (frnt surface, including the empty area, equal t the cnical cups: S c = mm 2, cup radius: R c = 25 mm, truncated shape with hle diameter h = 9 mm, h = 9 mm and h = 24 mm, and R rc = mm). See in Fig. 3 a sketch f the rtr cups tested. A secnd testing rund, devted t analyze the effect f rtr nn-symmetries n the anemmeters' respnse, was carried ut using the h-9/ rtr as a base cnfiguratin. This rtr was calibrated 4 mre times, ne repeating the calibratin f the first testing rund, and the ther three replacing ne f the cups with a cup frm the h-09/, h-24/ and c-25/ rtrs (see Fig. 4). The cups were made f ABS plastic using a 3D printer, and the arm n each cup was made f aluminum tubing 5 mm in diameter. In Table, the weight, W, and the mment f inertia, /, crrespnding t each rtr have been als included. The mment f inertia was calculated fr each case adding the cntributins f the head f the rtr, the arms and the cups. The mment f inertia f the cups was calculated with the CAD prgram used fr their design. In Fig. 5 the calculated mments f inertia are pltted as a functin f the squared cup radius multiplied by the squared cup center rtatin radius, R 2 c R 2 rc - The experimentally measured mments f inertia f several rtrs frm cmmercial anemmeters [25,59-6] have been als included in the graph (see als in Table 2 the mass and gemetrical characteristics f these rtrs). It can be bserved in the afrementined figure that the calculated mments f inertia regarding the rtrs tested seem t be quite similar t the nes experimentally measured. The mment f inertia has prven t be f great imprtance in relatin t the anemmeters perfrmance. Anemmeters with damaged rtrs nt nly can have different aerdynamic behavir [62], a different perfrmance can be shwn due t changes n the mment f inertia if ne cup is missing r damaged [32,62]. Due t this reasn it is relevant t highlight that the testing rtrs have bth similar gemetry and mment f inertia when cmpared t cmmercial anemmeters. The testing campaign whse results are analyzed in the present wrk was carried ut using a Climatrnics anemmeter, which gives 30 squared pulses per turn. In Fig. 6, a plt f the utput signal recrded in ne turn f this anemmeter equipped with the h-24/ rtr is shwn. The crrespnding nn-dimensinal rtatin speed, c/c. is als included in the figure. It was calculated

6 Table Gemetrical, weight and inertia characteristics f the rtrs tested: cup center rtatin radius, R rc, frnt area f the cups, S c, cup radius (cnical and prus cups), Re, rati f cup radius t the cups' center rtatin radius (cnical cups), RclR rc, weight f the rtr, W, mment f inertia f the rtr, /, hle diameter f prus cups, h, and semi-majr and semi-minr axes, a and b, f elliptical cups. See als Fig. 3. Rtr R c (mm) S c (mm 2 R rc (mm) RcIRrc W(kg) J(kgm 2 ; Cnical cups c-20/30 c-20/ c-20/50 c-20/ c-25/ c-25/50 c-25/ c-25/80 c-25/00 c-30/ c-30/50 c-30/ c-30/80 c-30/00 c-30/20 c-35/50 c-35/ c-35/80 c-35/00 c-35/20 c-/50 c-/ c-/80 c-/00 c-/20 c-/ x 0~ 2.78 x x 0~ 2.87 x 0~ x x 0~ x 0~ x x 0~ x 0~ x 0~ x 0~ x 0~ x 0~ x 0~ x 0~ 2 5. x 0~ x 0~ x 0~ 2 5. x 0~ x 0~ x 0~ x 0~ x 0~ x x 0~ x 0~ 6. x x 0~ x 0~ x x 0~ x 0~ 5. x x 0~ x 0~ x 0~ 5.08 x 0~ 4.85 x 0~ x 0~ x 0~ 4.33 x 0~ 4.80 x 0~ x 0~ x 0~ x 0~ 4.89 x 0~ x 0~ x 0~ x 0~ x x 0~ 3 Rtr a (mm) b (mm) S c (mm 2 R rc (mm) W(kg) /(kgm 2 ) Elliptical cups a-27/ a-30/ a-35/ ) 2.48) 2.24) 0~ i- i x x x 0-5 Rtr R c (mm) S c (mm 2 R rc (mm) h (mm) W(kg) /(kgm 2 ) Prus cups h-9/ h-9/ h-24/ x x x x x x 0-5 averaging grups f 30 pulses (ne revlutin), frm the recrded data at every pint (that is, at every wind flw velcity) f each perfrmed calibratin. The calibratins were carried ut at the IDR/UPM Institute, in the S4 wind tunnel. This facility is an pen-circuit wind tunnel with a clsed test sectin measuring 0.9 by 0.9 m. It is served by fur 7.5 kw fans with a flw unifrmity under 0.2% in the testing area. Mre details cncerning the facility and the calibratin prcess are included in reference [8], The uncertainty levels f the calibratins perfrmed at the S4 calibratin wind tunnel are specified fllwing the ISO/IEC 7025 standard [63], this levels being 0. m s _ fr wind speeds frm 4 ms - t loms -, and O.OlVms- fr wind speeds, V, frm 0 m s _ t 23 m s _. The calibratins studied in the present wrk were perfrmed fllwing the MEASNET [20,2] recmmendatins (ver 3 pints and frm 4 ms- t 6 ms- wind speed). In each pint (wind speed) f every calibratin perfrmed, the anemmeter's utput was sampled during 20 s at 0,000 Hz. In Table 3 the results f the calibratins that were carried ut are included. The first nine harmnic terms f the Furier expansin (expressin (5)) were calculated in every case (each rtr tested and every wind speed during its calibratin) upn the nn-dimensinal rtatin speed that, as said, was btained averaging the data frm several revlutins (i.e., grups f 30 pulses). See in Fig. 7 (left) the nndimensinal first and third harmnic terms (c /c 0 and c 3 /c 0. respectively), calculated at every pint f the calibratin perfrmed t the rtr h-9/. In rder t characterize the rtr perfrmance, averaged values f the nn-dimensinal first and the third harmnic terms, ci and m 3, were calculated with data frm the 3 pints f the calibratin prcedure: CO} = _y-tt>3 C0 3 = (7) 3-^c 0 3^CO 0 tgether with the crrespnding standard deviatins, ^ and er 3, calculated using the general prcedure:

7 Cnical cups Prus cups Elliptical cups Fig. 3. Sketch f the cups and rtr gemetries tested. Dimensins in mm. See als Table. Fig. 4. Pictures f ne f the nn-axially symmetrical rtrs tested (2 cups h-9/; cupc-25/). Rtr islated (left); and installed n the Climatrnics anemmeter (right). " = \ y-il3 { w± IT* ) 3- <T3 = \ T 3 /ffi3 <» )3 The abve parameters are als indicated in Fig. 7 (left). The first nine averaged nn-dimensinal harmnic terms (8) calculated with the prcedure abve described fr rtrs c-25/, h-09/, h-9/, and h-24/ are included in

8 0.00 : : ThiesClima NRG Maximum v R C 2 RJ Thies Clima *L-, A^ Thies Clima A QU Ris P2546 J Vaisala WAA5 ctr Instruments A00LK M Fig. 5. Mment f inertia, J, f the cnical cup rtrs tested as a functin f the squared cup radius multiplied by the squared cup center rtatin radius, R c 2 R rc 2 - The results crrespnd t rtrs equipped with R c = 20 (rhmbi), 25 (circles), 30 (squares), 35 (triangles), and (crsses) mm cups (see Table ). Gray circles crrespnd t different cmmercial anemmeter rtrs (see Table 2). Fig. 7 (right). As expected, the highest figures crrespnd t the third, sixth and ninth harmnic terms, tgether with the first ne, which is very similar in the fur cases included in the figure. 3. Results and discussin 3.. Experimental results In Table 3 and Figs. 8 and 9, the results crrespndent t the calibratins perfrmed n the cnical-cup rtrs are included. In Fig. 8 the anemmeter factr, K, calculated leaving aside the calibratin ffset (see expressin (8)), is shwn as a functin f the rati f cup radius t the cup center rtatin radius, r r = R c lr rc - On the ther hand, the graphs cntained in Fig. 9 shw respectively the first and the third averaged harmnic terms f the rtatin, CO] and 0 3, (just called first and third harmnic terms hereinafter), and the crrespndent nrmalized standard deviatins, a-i/cbi and "3/ct> 3, as a functin f the rati f cup radius t the cup center rtatin radius, r r = RjR rc. The data crrespnding t calibratins perfrmed t 3 Class- new and nt calibrated befre anemmeters (Thies Clima 4.335; Vectr Instruments A00 L2; and Vaisala WAA 5), have been als included in the graphs. The results frm a calibratin perfrmed t the anemmeter used in the present testing campaigns, the Climatrnics anemmeter, equipped with its riginal rtr are als included in the afrementined figure graphs. The effect f the rati f cup radius t the cup center rtatin radius, r r, n the anemmeter factr, K, has been analyzed and largely discussed [3-6,24,65]. Nevertheless, it shuld be pinted ut that, leaving aside primary gemetrical parameters f the rtr such as the cup radius, R c, and the cup center rtatin radius, R rc, sme ther shape parameters as the cne angle f the cups, the cup edges (beaded r nt, sharp r blunt...), the anemmeter "neck" diameter... seem t have sme relevance n anemmeter perfrmance. It can be bserved in Fig. 8 that, in cmparisn with the cnical cup rtrs manufactured fr the present testing campaign, the cmmercial anemmeters tested and the Climatrnics anemmeter equipped with the riginal rtr have lwer values f K, being mre efficient in terms f transfrming the wind speed int rtatinal speed (i.e. fr the same wind speed, V, higher values f the rtatinal speed, c, are reached). The nly exceptin, the Vectr Instruments A00 L2 anemmeter, has a similar perfrmance in terms f anemmeter factr t the manufactured rtrs. Cncerning the first harmnic term, CO], higher values f this parameter are clearly shwn by the Climatrnics anemmeter (cnsidering all rtrs tested, included the riginal ne), when cmpared t the nes crrespnding t the 3 Class- new anemmeters tested. As already indicated in the intrductin f the present wrk, the first harmnic term is related t all physical effects that are repeated with a perid f ne turn. These effects are caused by bth frictin and aerdynamic frces n the rtr. If nw the results crrespnding t the third harmnic term, c 3, which is directly related t the aerdynamic frces n each cup [5], are examined, it can be bserved that n big differences are shwn amng the data Table 2 Gemetrical characteristics f different cmmercial anemmeter rtrs cup center rtatin radius, R rc frnt area f the cups, S c cup radius, R a rati f cup radius t the cup center rtatin radius (cnical cups), RJRrc, and mment f inertia f the rtr, /. Anemmeter Ris0 P2546 Thies Clima Thies Clima Thies Clima Vaisala WAA5 NRG Maximum Vectr Instruments A00LK Rc (mm) S c (mm 2 ) R rc (mm) RJRrc W(kg) 5.8 x 0-2 [59] 9.6 x 0-2 [59] x 0-2 [59] - - J(kgm 2 ) 9.74 x 0-5 [25,59].0 x 0-4 [64] 9.92 x 0-5 [6] 8.87 x 0-4 [25,59] 2.93 x 0-5 [25] 2.89 x 0-4 [6] 6.4 x 0-5 [6] 5.2 x 0-5 [59].0 x 0-4 [6] 9.2 x 0-5 [59] 4. x 0-5 [6]

9 tit Fig. 6. Climatrnics anemmeter placed in the S4 wind tunnel f the IDR/UPM Institute, t carry ut a calibratin equipped with a prus-cups rtr (left). Anemmeter utput signal, V 0 utput, measured in ne turn (right-tp). Relative-t-the-average rtatinal speed, mj( 0, calculated in that turn (right-bttm). Tis the perid f the anemmeter's rtatin [5]. related t the calibratins perfrmed t the Climatrnics anemmeter and the 3 ther perfrmed t the new Class- anemmeters. Based n this fact, it can be assumed that the differences between the Climatrnics anemmeter and the new nes regarding the first harmnic term are directly related t the frictin effects which, taking int accunt the large uptime f this anemmeter [], can be attributed t a certain level f degradatin regarding the bearings system. Results indicate the same trend f bth the third harmnic, 0 3, and the anemmeter factr, K, that is, lwer values f parameter r r imply greater values f K and c 3, the pattern being similarly linear. This crrelatin between the anemmeter factr and the third harmnic indicates higher rtatinal speeds f rtrs with lwer third harmnic values. This particular behavir has been checked with the rtrs with prus and elliptical cups. In Fig. 0 the anemmeter cnstant, K, and the third and first harmnic terms, m 3 and ct)], regarding these rtrs have been pltted as a functin f the eccentricity, e = J - (b/a) 2, in the case f the elliptical cups, and as a functin f the rati f the hle diameter t the cup diameter, h\2r c, in the case f the prus cups. In the graphs f this figure the results f c-25/ rtr have been included, as it represents the "zer" case in terms f eccentricity and hle diameter (i.e., e = 0 and hj2r c = 0), fr the elliptical and prus cup rtrs respectively. It shuld be underlined that all these rtrs are characterized by the same cups center rtatin radius, R rc = mm, and cups with the same frnt area, S c = mm. Fr bth cases, elliptical and prus cups, the same pattern regarding the anemmeter factr, K, and the third harmnic term, c 3, which is in tune with the cmmented result related t the cnical cups, can be bserved in Fig. 0. Finally, very similar values f the first harmnic term, ct>i, are shwn in all these cases, indicating a much greater crrelatin f this parameter with the mechanical behavir f the anemmeter than with the rtr aerdynamics. Additinally, sme cnclusins can be extracted frm the testing results in relatin t the effect f the mment f inertia, /, n the anemmeter perfrmance (see Fig. ). N effect f this parameter has been bserved n the first harmnic term, CO]. On the ther hand, if bth the anemmeter factr, K, and the third harmnic term, 0 3, are pltted as a functin f the mment f inertia (see Fig. ), it can be clearly bserved the effect f the size f the cups and the mment f the inertia. If the value f the rtr's mment f inertia is cnstant, bigger cups (i.e., larger values f the cup radius, R c ) lgically prduce lwer values f bth the anemmeter factr, K, and the third harmnic term, c 3, as the aerdynamic frces are increased. Reversely, fr cnstant cups size (i.e., maintaining cnstant the aerdynamic frces), higher values f the rtr's mment f inertia result in higher values f the anemmeter factr, K, and the third harmnic term, m Mdeling the cup anemmeter steady behavir under ptimal wrking cnditins (i.e., anemmeter nt affected by degradatin) The equatin that defines the 3-cup mdel (4), can be expressed as: "d7 =/(0) /(0 20 ) /(0 2 ) where (9)

10 M M S? S- re 5 w n 7? C 3' a re S- nt can cr metric acc e II M I -i ^ n < II NJ ^ n 5 t t t UJ : n n n n '. t3 t3 "O "d ID ID ID ID JJ ~CTi ~CTi ~CTi ~CTi *< 3 n n n n n n n n n n n n n n n n n n n n n n n n n n n n n p i ^^^^^^UJUJUiuiUJUJUJUJUJUJUJt O O O O O O ^ ^ W W ^ O O O O O O ^ W ^ U i U i O O O O ~. ^ ^ ^ ^ I S T n ^ ^ l ^ ^ ^ ^ ^ l S T n ^ ^ l ^ ^ ^ l S T n ^ T u Ct 4 ^ t t t ~ - 2 U 3 "> n 3 2 e expr frmu nt Eqs OJ re r-~, "en t > 3" S re 3 re D. ^ CLV^ >< S.» 2 Cb s s 2 K) H?a n i' --I in 4^ ~ ^ en -i Ui -J e OJ 3 3 S --I m U) en --I ID O Ui 4^ 4^ Ui J* bl c en 4^ t O in I*J en in t t! ' H J. H J. O O O O H J. 0 0 tccn4^ujccn cncncncnuiui-j-jc u u i m i c u ^ u w O H J O O O O O O O O O O O O ' w c m ^ w w c i ^ w w u i i i - J - J C T I C O C O C O C O I D I D I D I D I D H J H J I IDC0C0U)CTilDtO-J-J toujidujuictictiuictictiujcticoco' m - J O ) O H i H i w j \ H i i w ) O 3 I O J CO W ID 4^ U) --J --J U) Hi Hi Ln ID en t l*j en i ' in t en --j in en en --j U) U) to HJ I si HI ai b ui t b i b 4 ^ H J t c n b 4 ^ H J c c n b 4 ^ l D l D O l D - J C T - t O t 0 4 ^ H J H J t HJHJOCOUlHJtOHJCOUlCTlHJ O l M O M O O O l U i H i H i O O O ^ 4^OC7<CTi--JCTilDHJC0tOUJLn UJtOtOHJHJUJtOtOHJHJOtOtOHJ O H J. H J. p P c Ln t LD en uuiuiuiuicasiuiuim I D O U i C 0 C 0 L n i D C 0 I D C 0 4 ^ U i U J C n O C 0 O 4 ^ H J L n t O t O - J C 0 O - - J H J H J OIDLn4^COtOCOIDOCT-HJ4^COCO 3 D. 9 ff rt. re re sr 3 w p p 4^ 4^ Ln Ln t p p ^ ui Ln u 3 HJ. 4^ HJ. ^ Ui Ui Ji vi 0 W HI ^ U) ID 4^ t. to to ^-L HJ U) ID 3 ^L Ui t ID ID to U) vi t L t t HJ I-J I-J 4^ CO U) en en U) HI w ID ai M L k ) H j ^ j ^ J 4 ^ L k ) H j ^ J L n L k ) k ) t 4 ^ L t U J 4 ^ t O C 0 U i l D l D C 0 L n 4 ^ l D l D U J l D 4 ^ O C T i O H J U J s j H J i - J i - J U J O O t O 4 ^ C 0 C T i U i C T i 4 ^ t O - J - J O H J U J O CTi O ID U) Ui HJ -JCTiU)LnCOLnUiOtOOUiCOUJ4^COCTilD4^tOlD to Ui to to to 4^ HJC0tO4^HJU)UJUJUJCTiHJLnC0LnCTiOHJC0C0CTi OJ OJ 3 OJ re "~~ XJ 3 e OJ < re si < 3 ID ID ID ID ID ID ID ID CO CO U} (fl ID ID ID ID ID ID ID ID ID ID CO CO ID ID ID ID ID ID ID ID ID ID ID ID ID CO CO ID P P P P P P P P P P P P P ^ P P P P P P P P P P P P LDLDLDLDLDLDLDLDLDLDLDLDLDOIDLDLDLDLDLDLDLDLDLDLDLD I D I D I D I D I D I D I D I D I D I D I D I D I D O I D I D I D I D I D I D I D I D I D I D I D I D I D I D I D I D I D I D I D I D I D I D I D I D I D O l D I D I D I D I D I D I D I D I D I D I D I D I D I D I D I D I D I D I D I D I D I D I D I D I D O l D I D I D I D I D I D I D I D I D I D I D I D LnCTiCTiCOIDIDIDIDCOIDIDIDCOOCOIDUilDIDIDIDIDIDIDIDLn W U) Ui 8i r r r - to U) H-l ^ HJ. en en 8i r r r - t t U) re a n ' n OJ 3 u H-. 2 T PJ,, NJ NJ -' t t ui t U) ID ID 4^ (D W UD -J to HJ Ln b 4^ r ' r ' r ' r ' P P r ' r ' r ' P P r ' r ' r ' P P P r ' r ' LnLnLb^LnLn4^HjlD0i^LnLiDC0Cn--jLni s l J ^ s j s l M C O D U i ^ U i ^ a i ^ J l D M ^ C a B p p ^ ^ p p LD en 4^ b c Ln to U) U) ID 4^ HJ

11 5.00% c c3 4.00% 3.00% 2.00%.00% 0.00% ffll/fflc t_. & On,, -.. U.. <J._QO.O.O_ a, 2(7, 0 F [m/s] % 6.00% % - «j\ j\ /Al O c-25/ D h-09/ -O - h-9/ A h-24/ 2.00% - sw. A 0.00% Fig. 7. Nn-dimensinal first and third harmnic terms, c-^jc and c^jc 0, calculated at every pint f the calibratin perfrmed n the testing anemmeter equipped with h-9/ rtr (left), the average values, <Bi and <B 3, and the standard deviatins, <xi and <x 3, are als indicated in the graph; and (right) first nine averaged nn-dimensinal harmnic terms, <B[ = <Bi/<B 0 (i = t 9), calculated fr rtrs c-25/, h-09/, h-9/, and h-24/ (see als expressin (24)) ~k A, and finally, intrducing the anemmeter factr, K, in the abve expressin: / dc \ps c R rc V 2 At i)( c c ic72^2 K WAA5 8 A U O O A O Fig. 8. Anemmeter factr, K, crrespnding t calibratins perfrmed n a Climatrnics anemmeter equipped with different cnical-cup rtrs as a functin f the rati f cup radius t the cup center rtatin radius, r r = R c jr rc. The symbls stand fr the fllwing cup radius: Re = 20 mm (rhmbi), 25 mm (circles), 30 mm (squares), 35 mm (triangles), and mm (crsses). Gray circles crrespnd t results frm calibratins perfrmed t new 3 Class- anemmeters (Thies Clima 4.335; Vectr Instruments A00 L2; Vaisala WAA 5), and the Climatrnics anemmeter equipped with its riginal rtr. / dm ps c K rc ~df = (y 2 (c»k rc ) 2 ) UO C ] J/ 0 -C ] J/ 2 J CORrcV Ci^j^Ci^ ((V 2 (c»k rc ) 2 ) ]c,n 3 - mrr^cn^j cs(30), (23) ^>= -h>)!»«(24) If the statinary cnditin f the anemmeter is cnsidered, and taking int accunt nly the third harmnic term f the rtatinal wind speed (Eq. (6)), the abve equatin can be rewritten as: /3c 0 c 3 ~ys c R rc v, 2 sin(3c 0 t<p 3 ) C ci (n A I7I3 IC 2 Cl 7^V3-^V2 cs (38). IC (25) The abve expressin can be decmpsed int tw equatins. The first ne: 0 = CCi(J7 0 2»?2 -rci\n,^3 (26) has been already mentined in Sectin as Eq. (3). As said, it relates the anemmeter factr, K, t the Furier cefficients rati Oi/c 0, which, as mentined, nly depends n cup aerdynamics [4]. This expressin gives then the average rtatin speed f the anemmeter rtr, c 0. as a functin f the wind speed, the cups center rtatin radius,

12 .80% H.50%.20% S>! 0.90% 0 A \ A i,.50%.20% ft>3 0.90% CL *\ A00L2 WAA 5 Q ^ p TH % 0.% O a n 0.30% 0.30% 0.00% % % 6.00% A00L2 6.00% % 8.00% 4.00% O WAA 5 TH a,cl O A ~t~ 0.00% % % % % - * > An 0 a 0 CL ta O 0 TH4.335 WAA 5 O A00L t Fig. 9. First and third averaged harmnic terms f the rtatin, <Bi and <B 3, and crrespndent nrmalized standard deviatins, <Ti/<Bi and <T 3 /<B 3, as a functin f the rati f cup radius t the cup center rtatin radius, r, = RJRrc- The results crrespnd t calibratins perfrmed n a Climatrnics anemmeter equipped with R c = 20 (rhmbi), 25 (circles), 30 (squares), 35 (triangles), and (crsses) mm cup rtrs. Gray circles crrespnd t results frm calibratins perfrmed t three new Class- anemmeters (Thies Clima 4.335; Vectr Instruments AlOO L2; Vaisala WAA 5), and the Climatrnics anemmeter equipped with its riginal rtr. R rc, and the aerdynamic characteristics f the cup. The linear fitting: K = C' (27) is prved t be a gd apprximatin t Eq. (26) within the range frm Oi/c 0 = 2.25 (/<=3) t Oi/c 0, = 6.74 (J<=0), where the factrs regarding mst cmmercial anemmeters lie [4]. On the ther hand, the secnd equatin: IO) 0 O) 3 \pscrrcv 2 4)3 - v r\2 K' (28) can be derived t an expressin frmed by dimensinless terms: 0)3 where = >3= «(K 2 l)»/3-2mfc Cirj^, 0 = pr5 rc (29) (30) is an inertial parameter that represents the rati between the inertia f a mass f air prprtinal t anemmeter rtr dimensins, and the rtr's inertia. Taking int accunt the relatinship between the anemmeter factr, K, and the cefficients rati, c^c, stated in Eqs. (26) and (27), it can be assumed that the third term f the equatin abve depends als n the cefficients rati c^c, a reasnable fitting being: (^ )^-2^ ( - \c 0 0.5, (3) fr the bracket between /<=.8 and /<=0. The abve expressin is interesting as it reveals the existence f a theretical minimum value f the third harmnic term, i.e. c 3 /c 0 ~ 0, at a certain value f the cefficients rati arund Oi/c 0 = It shuld be underlined that the mentined cefficients rati nly depends n the cup aerdynamics (see Fig. and expressin (9)). Finally, a direct relatinship between the nn-dimensinal third harmnic term, c 3 /c 0. and the frce cefficient Oi is bserved, this result being expected as this cefficient is related t the

13 0.0% 0.0 / 2 9.0% 9.0% 0 8.0% 7.0%? 8.0% 7.0% K 6.0% 5.0% 6.0% 5.0% 4.0% 4.0% 3.0% 2.0% ) ((P w 2.0% < - _-D- ^ r\,-, 4.0%?^r ~ e ; h/2r c e ; h/2r c e ; h/2r c J ) Fig. 0. Anemmeter cnstant, K, and nn-dimensinal third and first harmnic terms, <B 3 and <Bi, regarding calibratins perfrmed t the anemmeter equipped with elliptical (squares) and prus (circles) cup rtrs, as a functin f the eccentricity, e = J\ - (b/af, and the rati f the hle diameter t the frnt cup area, hj2r c, respectively. 3.0% K O Q O DA4. DA O n O a A / [kg m 2 ;.5 - m n a A c,.4 / [kg m i- ; / [kg m 2 l Fig.. Anemmeter cnstant, K, and nn-dimensinal third and first harmnic terms, <B 3 and <Bi, regarding calibratins perfrmed t the anemmeter equipped with cnical cup rtrs, in relatin t the rtr's mment f inertia, J. The results crrespnd t calibratins perfrmed n a Climatrnics anemmeter equipped with R c = 20 (rhmbi), 25 (circles), 30 (squares), 35 (triangles), and (crsses) mm cup rtrs. aerdynamic nrmal frce scillatin n each cup in ne rtatin. In Table 4 cefficients c 0 and Oi frm the Furier series crrespndent t the nrmal-t-the-cup frce cefficient, c N (Eq. (9)), are included fr the cnical (represented by the c-25/ rtr), prus and elliptical cups. These cefficients were all experimentally measured with the nly exceptin f the h-09/ rtr cups [4]. In this particular case the cefficients were interplated frm the ther testing results. The anemmeter factr, K, measured (expressin (8)) and calculated (expressin (27)) is als included, tgether with the measured (Table 3) and calculated (Eq. (29)) third harmnic term, m 3. As already stated, the calculated anemmeter factr fits reasnably well with experimental results [3,4], althugh it underestimates the average rtatin speed, c 0. up t a 22% in the wrst result f the studied cases. On the ther hand, significant differences are bserved between the calculated and the measured third harmnic terms. These differences can be explained as the measured third harmnic term values are arund 2% f the average rtatin speed (a maximum f 7.56% was measured in the extreme case represented by the h-24/ rtr), which is quite small cmpared t the deviatins fund regarding the average rtatin speed, m 0. It seems reasnable t assume larger deviatins in the calculated secndary term (i.e., the third harmnic term, c 3 ), if the deviatin frm the primary ne (the average rtatin speed, c 0 ) is greater than the measured value f

14 Table 4 Aerdynamic cup cefficients c 0 and C] frm the Furier series crrespndent t the nrmal-t-the-cup frce cefficient, c N (Eq. (9)), calculated frm the cnical (represented by the c-25/ rtr), prus and elliptical cups wind tunnel measurements [4]. The cefficients c 0 and C] crrespnding t the h-09/ cups were interplated as n testing results were available. The anemmeter factr, K, and third harmnic term, <B 3, calculated upn the afrementined cefficients and based n the measurements (Table 3) are als included in the table. Rtr cups c-25/ h-09/ h-9/ h-24/ a-27/ a-30/ a-35/ Cup cefficients C C, Ci/C Calculated K ffi>3 (%) Measured K ffi>3 (%) the secndary term. In rder t analyze this deviatin frm the testing results, the third harmnic term f the prus and elliptical cup rtrs was related t the rtr frmed by cnical cups (i.e., c-25/ rtr), the results being included in Fig. 2. As said in the previus sectin, the c-25/ rtr represents the "zer" case in terms f eccentricity and hle diameter fr the elliptical and prus cup rtrs. The analytical methd underestimates the effect f the changes n the cups n the cmpared third harmnic term, althugh the tendencies seem t be smehw reflected in the graph with the exceptin f the h-24/ rtr. Taking int accunt the reductin in terms f aerdynamic frces n the cups due t its high prsity (see Table 4), the deviatin f the h-24/ rtr culd be attributed, in principle, t a higher level f aerdynamic interactin between the rtr cups and the stream frmed by the anemmeter "neck" and the cups arms (which als have in this case a larger relative cntributin t the aerdynamic mment). Mrever, the measured third harmnic terms, ct>3, frm the cnical cup rtrs tested are pltted as a functin f inertial parameter <P multiplied by the squared rati f cup radius t the cup center rtatin radius, r r 2, in Prus cups; calc Prus cups; exp. Elliptical cups; calc Elliptical cups; exp. 2.00% -.50% -.00% % % - Ajy y^ y = x J? 2 = XD d>r Fig. 3. Measured third harmnic term, <B 3, f the cnical cup rtrs as a functin f the inertia parameter, <P, multiplied by the squared rati f the cup diameter t the cup center rtatin radius, r r. The symbls stand fr the fllwing cup radius: i? c = 20mm (rhmbi), 25 mm (circles), 30 mm (squares), 35 mm (triangles), and mm (crsses). Fig. 3. It can be bserved that the results indicate a linear tendency just like the present mdel des with expressin (29). Further imprvements shuld be made t this analytical mdel in rder t have a clser apprximatin t testing results with regard the third harmnic term. In this sense, recent results indicate that the aerdynamic frce n the rtr cups is nt unifrm due t the rtatinal mvement. The implementatin f this effect culd prbably increase the accuracy f the 3-cup analytical mdel as it has been dne with the 2-cup analytical mdel [6] Mdeling the cup anemmeter behavir under the effect f a peridical perturbatin As previusly mentined in Sectin 2, different nn-symmetrical rtrs were manufactured t study the effect f a peridical perturbatin n the anemmeter perfrmance. The perturbatin n the rtr dynamics is then intrduced by mdifying the aerdynamic behavir f ne cup. This case can ccur if ne f the cups is damaged as a result f the nrmal degradatin causes f the anemmeter (i.e., frst and snw accumulatin [33]). If the 3-cup analytical mdel is cnsidered t analyze the aerdynamics f a rtr frmed by tw cups f nrmal-t-the-cup aerdynamic frce cefficient c^a), and a third cup f affected cefficient c N *(a), the equatin f the mvement can be rewritten as: }(k =^. t(e) f{g 2QO) f{g 24QO) = /(<?) f(0 20 ) f(0 2 ) f'(0) (32) where f(6), /(0 2O ) and /(0 24O ) are defined by expressin (20) and: Fig. 2. Third harmnic term f the prus and elliptical cup rtrs related t the third harmnic related t the cnical cup rtr (c-25/ rtr), <B3/<B 3 c _ 2 5/6. respectively pltted as a functin f the nndimensinal hle diameter, hj2r c, and the eccentricity, e, f the cups. f'(8)=jps c R r c(v 2 (cr rc ) 2 2)R rc V x cs(0))ac N (0). with, taking int accunt Eqs. (9) and (0): (33)

15 Ac N (0)=c N *(0)-c N (0) = Uc 0 *-c 0 ) (c *-c )U 0 2'?2jJ (c *-c )fj7 5 J7 3 Jcs(0) 2(ci*-c ))7 2 cs(2e)-(c *-c ))73Cs(3e) = (Ac 0 Ad U 2^2)) Ad U 4I3J cs(<9) -Ad^2 C0S ( 2e ) -7 Ad ^3 cs(36). (34) Then, using the same mathematical prcedure described in Sectin 3. the fllwing expressin can be derived: / dc \ps c R rc V 2 df = (( ^)( C Cl(,?0^2) ) where 4 Cl l^4^3 'LlsiH-rWii-cs&O) G(Ac;Aci;/f: /. G{Ac 0 ;Ac ;K;8)=-n- 2 -\ /. [Ac 0 Ac l \n 0 -r^2 (35) ^^{2Ac 0 Ac (2f] 0 -f] 2 ) )cs(<9) cht^ f A d>? 2 cs(20) 6 V K 2 _ ;Ad()?i (? 3 )cs(20) "3r T2( F) Acl '?3COS(3e) - 5 ^ Ad»7 2 cs(3<9) - ^AdJ/3 cs(4<9). (36) Taking int accunt the abve expressins, the steady state equatin is nw defined by the fllwing equatin: \\ =[\' A)[ K CO C A ^ 2 n2 )) ~K Cl [^ 4f?3 -^r( ( (. 3 VV (^2 K J(Ac Ac (/7 0 -/7 2 j j --Ac (/ 7l -i7 3 4*0) _ r = VV J?) V C ',(,/^27 ^ V 4 f?3 ))' where, bviusly: c' 0 = c 0 -Ac 0 ; C\ = Ci - Ad (37) (38) As said in the previus sectins, expressin (37) gives a value f the anemmeter factr as a functin f the rati c\/c' Q, which can be apprximated by the fllwing expressin: Cn C'J Cj \Ac a d- CilAci,C 'd Ad 3d~ Ad 3c 0 (39) Cncerning the first t the furth harmnic terms, they can be derived frm expressins (35) and (36): a 0" r 2 _ c 0" c 3 c 0" /7D - 2 ^ VAd /it =C02 = U /7D (K 2 )^^3) V^m) Ad C<, () (V l^-k^^ad rf<, ((/C 2 l)i/7 3 -/C )72 )c ((^ l)^ 3^2)Ad r 2 <P, ^ = C 4 = (32 ic(7 3Ad r 2^, (4) (42) (43) In Table 5 the results calculated with this mdel are included tgether with the testing results. In Fig. 4, the measured and calculated anemmeter factr, K, and the first, secnd and third, ct>i, c 2, and c 3, are pltted as a functin f percentage variatin f the c 0 aerdynamic cefficient regarding the perturbed cup f the nn-symmetrical rtr, Ac 0 /c 0. This parameter seems t be a gd chice t represent the change f nrmal-t-the-cup aerdynamic frce n the cup, bearing in mind that a cmpletely flat cup which is unable t create aerdynamic mment n the rtr is characterized by c 0 = 0 due t its symmetry. The averaged rtatinal speed, c«. characterized by the anemmeter factr, K, is, as expected, affected by the change f the aerdynamic frce n ne f the cups, the factr being lwer (i.e., higher rtatinal speeds) fr higher values f the afrementined aerdynamic frce. Als, the results indicate a gd crrelatin between the studied analytical mdel and the testing results. With regard t the first harmnic term, CO], the testing results clearly shw higher values fr the nn-symmetrical rtrs, with again gd crrelatin with the data resulting frm the analytical mdel. The secnd and the third harmnic terms, c 2, and c 3, experimentally measured shw a different pattern in cmparisn with the first ne, their values being inversely prprtinal with the increment f aerdynamic frce n the affected cup. With respect t the third harmnic term it shuld be underlined that, althugh it decreases with the increase f the aerdynamic frce n ne cup f the rtr, it cannt be taken as an increase f the rtatin speed unifrmity due t the higher values f the first harmnic term, ci. In this case the crrelatin f the results frm the analytical methd with the testing results is wrse than the

16 Table 5 Anemmeter factr, K, and the first fur harmnic terms, c lt 6) 2,6) 3 and <B 4, analytically calculated and measured with the Climatrnics anemmeter equipped with nn-symmetrical rtrs. The descriptin f the rtrs (i.e., the cups frming them) and their mment f inertia, /, are included in the table. See als Fig. 4 and Table 3. Rtr cups (2) h-9/ (l)c-25/ (2) h-9/ (l)h-09/ (3) h-9/ (2) h-9/ (l)h-24/ J(kgm 2 ) 5.68 x 0~ x 0~ x 0~ x 0~ 5 Calculated K a, (%) ffl 2 (%) <M%) ffl 4 (%) 7.8 x 0~ x 0~ x 0~ 4 Measurec K <M%) ffl 2 (%) ffl 3 (%) ffl 4 (%) % 6% K 4-»! 4% 2 - Calculated Experimental.5% 0-00% -50% 0% 50% 00% Ac 0 /c 0 0% -00% -50% 0% 50% 00% 5% Ac 0 /c 0.2% 4% 0.9% 3% 0.6% 0.3% 0.0% -00% -50% 0% 50% 00% Ac 0 /c 0 2%- % 0% -00% - Calculated - Experimental - =C^^X>- -50% 0% Acjc, 50% 00% Fig. 4. Results with regard t the perfrmance analysis (experimental and calculated with the prpsed analytical mdel) f the Climatrnics cup anemmeter equipped with nn-symmetrical rtrs (seefig. 4). Variatin f the anemmeter factr, K, and the first, secnd and third, <Bi,<B 2, and <B 3,as a functin f the percentage variatin f the aerdynamic frces n the perturbed cup f the nn-symmetrical rtr, Ac 0 /c 0 (see expressins (9) and (34)). ne described fr the anemmeter factr and the first harmnic term. Mre precisely, it seems that in the case f the secnd harmnic term, c 2. there is sme level f crrelatin measured in the case f a decrease in the aerdynamic frce n the affected cup, whereas n crrelatin between the calculatins and the testing results is bserved in the case f the third harmnic term, m Cnclusins In the present wrk, the harmnic terms f cup anemmeter rtatin speed, c, and their effects n anemmeter perfrmance (n steady wind) are experimentally and analytically studied, with quite gd agreement amng results frm bth methdlgies. The effect f mechanical perturbatins n the anemmeter perfrmance has been als analyzed with nn-symmetrical rtrs. The mst relevant cnclusins resulting frm this wrk are: Anemmeters wrking in nrmal cnditins (i.e., nt affected my any mechanical perturbatin) shw a high crrelatin level between their aerdynamic efficiency (i.e., the capability f transfrming the incming wind

17 speed int rtatinal speed) and the rtatin unifrmity characterized by the third harmnic term, m 3. Therefre, a mre unifrm rtatin f the anemmeter rtr results in higher averaged rtatin speed, m 0. Als, the afrementined third harmnic term, c 3, seems t be mre unifrm within the calibratin speed range (frm 4 m s t 6 m s _ ) fr certain rtr sizes, with values f the rati f cup radius t the cup center rtatin radius, r r = R c jr rc, frm r r = 0.4 t r r = 0.5. The 3-cup analytical mdel, even being accurate enugh t analyze the effect f the rtr shape n the anemmeter factr K (i.e., the averaged rtatin speed, c 0 ), needs t be further imprved t accurately reprduce the third harmnic term f the rtatin speed, c 3. Nevertheless, the mdel was able t crrectly indicate sme aspects related t this harmnic term such us the linear relatinship with the rtr inertia parameter <P = pr-rc/l One f the mst prmising imprvements t this 3-cup analytical mdel culd be t cnsider the pressure distributin n the cups nt cnstant, as sme recent results indicate higher aerdynamic pressure cefficient at the area f the cups clser t the rtatin axis [6]. Experimental results clearly indicate higher values f the first harmnic term, CO], in calibratins carried ut n an ld anemmeter (affected by a certain degree f wear and tear) equipped with different rtrs, than the nes related t calibratins perfrmed n new anemmeters. Anemmeters affected by an aerdynamic perturbatin n ne single cup f the rtr shwed, as expected, higher values f the first harmnic term, CO]. Als, the testing results indicated a gd crrelatin amng the perturbatin and its effect n anemmeter aerdynamic perfrmance, that is, an increase f the aerdynamic nrmal-t-the-cup frce cefficient in ne f the rtr cups was translated int higher values f rtr aerdynamic efficiency (as said, lwer value f anemmeter factr, /<), whereas a decrease f the afrementined cefficient resulted in the ppsite effect. The calculatins dne using the 3-cup analytical mdel accurately reprduced the effect f the afrementined aerdynamic perturbatin n ne single cup f the rtr. Althugh mre testing campaigns fcusing specifically n the crrelatin between the first harmnic term, ci, and the anemmeter degradatin due t wear and tear shuld be carried ut, this prmising result suggests a new parameter as a key variable in the cnditin mnitring/diagnsis f cup anemmeters. Acknwledgments The authrs are indebted t Prf. Angel Sanz Andres fr his encuraging supprt regarding the research prgram n cup anemmeters perfrmance. 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