Vindforsk report Project /V-238

Transcription

1 Repot Vindfosk 3988-/V-38 Vindfosk epot Pojet 3988-/V-38 Ny teknik fö isning indkftsing New Tehnologies fo de-iing Wind Tuines Ls Bååth nd Hns Löfgen Hlmstd Uniesity phone: 46 () Vindfosk: 8/ : /54

2 Repot Vindfosk 3988-/V-38 Inde Smmnfttning...3 Summy...4 Intodution...5 Wte...6. Moleul stutue nd popeties...6. Eletomgneti popeties of wte Genel Wte Vpo Liquid Wte Ie Pesumptions Flow studies D onetion simultions CFD esults fo the wing tip, egion I CFD esults fo egion II nd III Conetie losses oe the wing due to gs flow Conetie losses due to doplets Doplet flow Detemintion of the D ptile flow ginst ylinde Detemintion of the ptile flow ginst ylinde D stgntion point nlysis fo it ginst ylinde LWC lose to the stgntion point of flow ginst ylinde The inisid i flow in the iinity to thin ifoil Detemintion of the itil lue of fo the flow ginst thin ifoil Tehnologies Het doplets efoe hit wing Het wte on wing Conetion Eletomgneti heting Melt ie on wing Wing fom nd sufe Fom Sufe nno-stutue Disussion Conlusion Refeenes...54 Vindfosk: 8/ : /54

3 Repot Vindfosk 3988-/V-38 Smmnfttning Denn ppot pesente esulttet en föstudie om teknike fö isning indkftek. Rppoten pesente oh diskute möjlig metode oh teknike fö tt ntingen äm ttendopp till öe fyspunkten, elle smält is som h ildts på ingen. Polemtiken fö ing på indkftek skilje sig mknt fån nedisning flygplnsing i tt: () ing på indkftek tilling ll sin tid i den delen tmosfäen dä isken fö nedisning ä som stöst; oh () hstigheten fö ingen mot luft ie med stånd fån ottionsentum medn den ä konstnt öe ingen på ett flygpln. Fomen på ingen på ett indkftek ie okså fån toppen in till entum fö tt kompense fö itionen elti hstighet mot luften. Rppoten konentes på isildning inom tempetuintellet - C C oh doppstolek - μm. Nedisning ske äen id myket läge tempetue, men då ske toligen isildningen diekt fån ttenång. Vi d följnde slutstse fån å studie: - Fomen på ingen, speiellt id kontktytn mot gsflödet, kn h etydelse fö nedisning. - Nno-stuktuen ytn på ingen kn toligen konstues så tt ttendoppn få en miniml kontktyt mot ingen. Vå föstudie is dessutom: - Mikoågo ä lltfö ineffekti fö tt äm ent tten elle smält is. Teknike fö diekt stålning mikoågo mot tten elle is på ing ö således inte ide utekls. - Millimeteågo ä tilläkligt effekti, men genetionen ågo på så hög fekense ä toligen lltfö ineffekti fö tt dett sk en möjlig äg fmåt. - Infött ljus ä myket effektit fö tt äm ttendopp elle smält is oh ö undesöks ide. - Vämeledning ä okså effektit oh ö utekls. En oust oh effekti metod kn tt äm ingytn med mikoågo så tt kontkten mot den m ytn smälte isen. Vå föstudie is tt polemtiken med undiknde isildning på, elle isning, indkftsek inte h sitt s i en end teknik. Fomen på ingen oh stuktuen på dess yt kn spel en iktig oll i föhållnden fö isildning. Båd dess ile kn ehö ies eoende på ltitud oh tmosfäiskt klimt. Ytstuktuen måste toligen okså ie öe ingytn, åde längs med ingen oh täs, fö tt optime fö de lokl föhållnden. Dessutom kn smältning is medelst ämning ingytn en iktig et möjlighet fö tt undik effektföluste. Me foskning ä nödändig, men i smmnftt tt det stöst intesset just nu ä tt stude flödet dopp öe ingen som funktion täsnittsytns fom oh kontkten melln ingytn som funktion ytstuktuen (t.e. Lotus effekten). Denn ppot ä esulttet ett föstudiepojekt. Vi ämn nu fotsätt med ett djupe foskningspojekt som konentes på fomen oh ytstuktuen enligt d som fmkommit å nlys oh å dtosimuleing. Vindfosk: 8/ : 3/54

4 Repot Vindfosk 3988-/V-38 Summy This is pilot study to inestigte iing on wings of wind powe tuines. In this epot we pesent nd disuss ious wys nd mens to eithe het wte doplets o melt ie when fomed on the wings of wind tuines. The sitution is diffeent fom iing on wings of iplnes in tht () the wings of wind tuines spend ll of thei time in the tmosphee whee the isk of iing is highest nd () the speed of wing to i ies oe the wing whee it is onstnt fo n iplne. The fom of the wind tuine wings lso ies fom tip to ente, to ompenste fo the ying eltie i speed. We he onentted on iing onditions t tempetues - C C nd doplet sizes of - μm. Iing ous lso t muh lowe tempetues, ut this will poly e euse of diet feezing of wte pou to ie. This is pesently outside the sope of ou pilot pojet epot. We onlude tht - The fom of the wing, espeilly on the ontt e my e uil to the iing polem. - Also the nno-meti stutue of the wing sufe n poly e designed so tht the wte doplets he minimized ontt e to the wing. Ou pilot inestigtion lso suggests the following: - Miowes e muh too ineffiient to het wte o melt ie. Diet miowe deies should theefoe not e deeloped. Indiet heting with miowes is possile. - Millimete wes e suffiiently effiient, ut the genetion is most poly too ineffiient to e of ny ptil use. - Infed wes e ey effiient to het wte nd melt ie nd should e inestigted. - Het ondution is lso effiient nd should e pusued. Using miowes to het the wing sufe whih then ondut het to the wte/ie is ey effiient nd oust method. Ou pe-study suggests tht the solution to oid iing o de-ie wings of wind tuines most poly is not one single tehnology. The fom nd sufe stutue of the wings ply impotnt ole fo iing onditions. Both iles he to e modified depending on the ltitude nd tmosphei limte. The sufe stutue lso hs to e designed to y oe the wing, oth long nd oss to e optimized fo the men onditions t the site. In ddition, heting of the impt e, o t lest the possiility to het this, my e impotnt to oid loss of enegy output due to ie. Futhe eseh is equied. We stongly suggest inestigting the wte doplet flow oe the wing s funtion of the oss setion fom, nd the ontt with the wing sufe s funtion of the sufe stutue (e.g. Lotus effet). The pesent epot is the esult of pe-study pojet. We will now ontinue with deepe pojet whih will onentte on the fom nd sufe stutue suggestions whih esults fom ou nlysis nd flow simultions. Vindfosk: 8/ : 4/54

5 Repot Vindfosk 3988-/V-38 Intodution Wind powe is one of the fstest gowing industies in Sweden, nd in the wold, of tody. Wind powe is seen s len genetion of eletil powe nd new tes on geen house gs emissions will mke it ompetitie soue of enegy. Lge wind powe pks e plnned in Sweden to meet the mitious plns. Espeilly the nothen mountin egions, the ostl se es nd the inne high plteu lndspes nd suoundings he geneted get inteests fo inestos. In genel, ll es of Sweden do sometimes duing the winte enounte times whee iing my ou. When wm i lifts fom the ostl ses onto the highe inlnd es, it ings with it sustntil mounts of wte po. The wte po then ondenses to liquid wte dop-lets when the i is ooled t highe ltitudes. Suh dop-lets n in su-zeo tempetues eithe feeze to snow o hil, o sty liquid s supe-ooled dop-lets. Supe-ooled dop-lets will dietly feeze to fom ie when they enounte mteil to whih it my gie off enegy, suh s the wings of wind tuine. Fomtion of ie on wind tuine wings is theefoe not limited to the f noth, ut my ou on suh southen sites s Bi whee tempetues my eh just elow zeo degees Celsius. Iing is ey muh Euopen polem. A nume of ppes nd epots he disussed the ouene of iing. The estimtes nge fom -7 dys pe ye fo low-lnd Sweden to moe thn 3 dys pe ye fo the high ltitude mountin nges. We suggest studying the EU epot y Lkso et l. [ef ] fo moe detils. Een few dys loss of enegy genetion is signifint fto fo lge wind pks sine ALL of the tuines within the pk would e ffeted. Also, sine this loss will e duing winte time when the spot pie on enegy is t its pek, the eonomi loss would e signifint. Enegimyndigheten (STEM) in Sweden hs gien gnt to the Hlmstd Uniesity to inestigte news wys to eithe oid iing o de-ie wings of wind tuines. Peious pojets of STEM he inestigted using miowes fo de-iing, e.g. Senson et l. [ef ] nd Bth et l. [ef 3]. In this epot we he tken moe omplete iew on the polem. Vindfosk: 8/ : 5/54

6 Repot Vindfosk 3988-/V-38 Wte. Moleul stutue nd popeties Wte is one of the most undnt moleules on eth nd lso one of the most impotnt fo life on this plnet. Wte is fomed y single Oygen tom ound with two Hydogen toms. The Oygen shes two eletons with the two Hydogen toms s olent ond. The esulting moleule of wte hs 3 nulei (O, H, nd H) nd eletons whih esult in 39 oodintes nd thus 39 degees of feedom. Figue The wte moleule (left) nd the Hydogen onding (ight) The si fom of wte mkes it ipol moleule in tht one end will he pedominntly negtie hge (Oygen end) nd the othe positie hge (Hydogen end). Wte moleules theefoe my fom ond with eh othe o othe ipol moleules s Hydogen ond. - Cohesion efes to tttion to othe wte moleules. - Adhesion efes to tttion to othe moleul speies. Cohesion is ey stle nd uses the sufe tension whih holds doplets togethe nd llows mtte to flot on wte sufe. Adhesion mkes wte doplets nd ie stik to metl nd othe mteil. The dhesie foe of the hydogen ond is ey stong, s is demonstted y how hd it my e to len the windseen fom ie in the winte. Wte, like ll mteils, eists in silly thee foms: - Gs fom, wte po, whee the moleules e llowed to moe moe o less t ndom in Bownin motion. - Solid mtte, ie, whee the moleules e ound y Hydogen onding into ystl stutue. - Liquid, wte, whih is n intemedite phse whee the moleules e loosely ound ut n moe eltie eh othe. The thee foms he quite diffeent popeties, lso s to thei flow popeties nd intetion with wind tuine wing. Vindfosk: 8/ : 6/54

7 Repot Vindfosk 3988-/V-38 Figue The wte moleul stutue s solid (ie) [ef 4]. Wte fom Speifi het (kj/kg/kelin) Ltent het (kj/kg) Vpo.996 Liquid Ie Tle Het pities nd ltent hets fo wte. Tle shows the speifi hets nd ltent hets fo wte in its thee foms. Note tht wte hs the highest speifi het nd theefoe equies moe enegy to het thn eithe ie o po. Note lso tht ie hs the highest ltent het nd theefoe equies muh moe enegy to melt thn to oil wte. It equies out the sme mount of enegy to melt kg of ie s to het kg of wte y 8 degees. Figue 3 Phse digm of wte. Stndd tmosphei pessue t se leel is.35 kp (denoted y lue line). The phse digm of wte [ef 5] is shown in figue 3 oe. The tmosphei stndd pessue of.35 kp is indited y hoizontl lue line. Note tht t this pessue the tnsfomtion fom ie to liquid (melting point) is 73 Kelin nd fom liquid to po (oiling point) t 373 Kelin. Note lso the tiple point t 73.6 Kelin nd kp whee wte eists in its thee foms simultneously. It should e noted tht the feezing of liquid wte into ie equies tht the wte moleules n gie off t lest 334 kj pe kg of enegy to the suoundings. This my not e possile een t low tmosphei tempetues nd liquid wte my theefoe eist s supe-ooled doplets. Vindfosk: 8/ : 7/54

8 Repot Vindfosk 3988-/V-38 Figue 4 Dew point of wte t se leel s funtion of i tempetue [ef 6]. The dew point is the tempetue whee po ondenses to liquid wte. This is dependent on the tempetue of the mient i nd the eltie wte po ontent (humidity). The eltie humidity is the eisting humidity diided y totl mount of po tht the tmosphee my hold t tht tempetue. A eltie humidity of moe thn % theefoe mens tht wte eist lso in liquid fom in the tmosphee (s dop-lets). The dew point is shown in Figue 4 oe s funtion of mient i tempetue. Red the figue s follows: Fo n i tempetue of C go to tht on the - is. Go to the line epesenting the eisting eltie humidity nd ed the de point on the y-is. The dew point is hee the tempetue to whih the i hs to e loweed to ondenste wte s liquid. The tmosphee t se leel pessue hs the pity of holding etin mount of wte s po whih is dependent of tempetue. Figue 5 elow shows the stution point of wte po s funtion of tempetue. Note tht the tmosphee my hold sustntil mount of wte een well elow the feezing point nd tht zeo wte ontent is hieed s low s ound -4 C. Figue 5 Peipittion of wte in i t se leel s funtion of i tempetue [ef 7]. Vindfosk: 8/ : 8/54

9 Repot Vindfosk 3988-/V-38. Eletomgneti popeties of wte.. Genel Atoms s well s moleules e ound togethe with eletomgneti foe. An etenl eletomgneti field will theefoe ffet mtte in wys whih e dependent on the fequeny of the field nd the omposition of the mteil. All moleules nd toms he speifi size nd will ffet eh othe nd how they espond to the etenl field y how f pt they e (density) nd how fst they moe eltie eh othe (tempetue). The esponse to n etenl eletomgneti field depends on its fequeny. At low fequenies, few Hz to few GHz, the field will loose enegy y moing hged ptiles suh s eletons o ions. If the mtte ontins fee eletons, then eletoni ondutiity will e the pedominnt loss, e.g. in metls. If no fee eletons e ille then ioni ondutiity my eist. This equies ions whih e fee to moe in the gs o solution, e.g. ions in slt wte. At highe fequenies in the dio miowe mm we su mm we nds, GHz to 5 THz, the eletomgneti field ffets moleule y hnging its ottionl is. This is quntized enegy jump etween two ottionl leels nd n e eithe soption (dding enegy to the moleule) o emission (diting enegy fom the moleule). At highe fequenies in the inf ed nd, the moleule my gin o loose enegy in tnsitions etween diffeent itionl leels. At optil nd to UV fequenies the tnsitions e on the eleton leel etween diffeent eleton shells. Asoption (nd emission) of the eletomgneti we in mtte is usully desied y omple pmete lled the di-eleti onstnt o pemittiity The imginy pt epesents the esponse of the mteil to n etenl eletomgneti field while the el pt epesents the intenl field. The pemittiity is fequeny dependent s shown in figue 6 elow. Figue 6 The di-eleti onstnt (el (ε ) nd imginy (ε ) pts). When n eletomgneti we psses though medium it will e ffeted to hnge fom hing then intensity I to outgoing I s shown in figue 7 elow. Vindfosk: 8/ : 9/54

10 Repot Vindfosk 3988-/V-38 Inoming field V Outgoing field V Figue 7 The hnge in eletomgneti field when pssing though medium. The field omple oltge is hnged s: l V V e i π υ n Δt whee υ is the fequeny, t is the time of flight in uum of the we though the mteil of thikness l nd n is the omple eftie inde of the mteil. The omple inde of eftion is onneted to the pemittiity s: n n i κ n ε ε ε κ ε ε ε The el pt, n, is the dely of the signl though the medium, while the imginy pt, κ, is the soption within the medium. The hnge of intensity (sque of the omple oltge) of the eletomgneti field n now e witten s: I I e α l whee α is the soption oeffiient in pts pe length nd l is the thikness of the medium. The soption oeffiient n e witten s: π κ ν α 4 whee is the speed of light in uum. The soption within medium is theefoe dependent on the intenl field of the medium, how this n ouples to the etenl field nd the thikness though whih the etenl field penettes. The intensity, o powe, soed in the mteil n now e witten s: ΔP I( e α l ) Vindfosk: 8/ : /54

11 Repot Vindfosk 3988-/V-38.. Wte Vpo Figue 8 The soption oeffiient in wte po s funtion of fequeny. Spetum is lulted fom Hitn [ef 8] dt. Moleules moe feely in gs nd the tnsitions e puely quntized. Figue 8 oe shows the ottionl tnsitions in the miowe nge of wte po. Eh spetl line epesents tnsition etween two quntum sttes, whee the lowest t.35 GHz is the soption tnsition fom ottionl stte to. Note tht the soption oeffiient is ey low nd tht ey long olumn of wte po is equied to so ny signifint mount of intensity in this fequeny nge. The width of the lines is depending on the pessue nd tempetue. The highe the pessue nd/o tempetue, the lose the moleules will eh eh othe nd the moe thei speifi intenl field will ffet eh othe nd the wide the lines will e. Pessue odening is eident in the figue oe whih is lulted wte lines t tmosphei pessue nd t 7 C. We onlude fom this tht wte po is not good soe of enegy t miowe fequenies. Vindfosk: 8/ : /54

12 Repot Vindfosk 3988-/V Liquid Wte Figue 9 The soption oeffiient of liquid wte. Dt e fom Segelstein [ef 9]. Wte in liquid fom hs ey diffeent esponse to eletomgneti dition. Figue 9 oe shows the soption oeffiient of liquid wte fom shotwe UV to the long we miowe. The yellow e indites the optil pt of the spetum. Note tht liquid wte is tnspent t optil nd dio fequenies ut moe o less opque t the othe pts of the spetum. Pue liquid wte does not ontin ny fee eletons nd no ions nd is theefoe not good onduto. Potons my hnge ples etween wte moleules nd tnspot some hge, ut this is ey smll effet. Asoption in liquid wte is theefoe lso minly di-eleti used y tnsitions etween quntized enegy sttes. The moleules in liquid wte e suffiiently lose to ffet eh othe nd the spetl lines seen in po is tnsfomed into wide nds whih dds togethe to wide nd spetum s seen in figue 9. The soption t miowe fequenies is still ey low, of the ode of.- m -. It is fist in the infed egion tht soption eomes sustntil - m -. Figue Comple pemittiity of liquid wte. Left pnel shows pue wte. Aows indite inesing tempetue. Right pnel shows sline wte. Aows indite inesed slinity Figue oe shows the omple pemittiity of wte po t miowe fequenies [ef ]. The fequeny of stndd miowe oens is shown s thik lk line. The soption oeffiient of pue wte nges etween.- m - t.5 GHz mking it poo soe. The sitution hnges dmtilly if slt is dded to the wte. The dipol stutue of the wte moleules esults in tht the ion onding of the slt moleule is split nd ions e fomed. These my moe feely in the sline wte nd ion ondutiity n e quite sustntil. The figue oe shows the pemittiity of wte fo diffeent slinity. Note tht the soption oeffiient ineses signifintly t the low fequeny (longe welength) of the spetum with inesing slinity. Sline wte is theefoe muh ette soe t.45 GHz thn pue wte. Howee, the liquid wte elent to wind powe tuine wings nd iing is epeted to e lmost pue wte ontining no ioni slt. Vindfosk: 8/ : /54

13 Repot Vindfosk 3988-/V Ie Figue The di-eleti pemittiity of wte (left pnel) nd ie (ight pnel) [ef ]. Ie is the solid phse of wte. This is ystl, whee the wte moleules e loked into positions y the stong Hydogen ond. The stength of the ond is demonstted y the ey high ltent het of ie. It equies 334 kj/kg to melt ie into liquid wte, while it equies out the sme mount, o 48 kj/kg, to het liquid wte fom C to the oiling point of C. Figue oe shows ompison etween the pemittiity of liquid wte nd ie [ef ]. Note tht ie hs di-eleti pemittiity whih is out times lowe then liquid wte t miowe fequenies. At.45 GHz the soption o-effiient of ie is in the ode of.-. m -. Ie is theefoe ey poo soe t miowe fequenies. Figue The sone of wte in the infed egion. Figue shows the sone of wte in its thee foms within the infed nd [ef ]. The we nume is /λ (we length in m) is fequently used to define spetum in the infed nge. The spetum shown is etween.5 μm to 3.5 μm. Ie is y f the est soe t infed welengths with wte po s the lowest. Vindfosk: 8/ : 3/54

14 Repot Vindfosk 3988-/V-38 3 Pesumptions Iing on wings is well known phenomenon fom ift nd flight industy. A full disussion on the ouene of iing onditions is outside the sope of this pojet. We efe to the tutoil y NASA [ef 3] fo full oege of suh onditions. We note hee tht liquid wte doplets my he tempetue elow C, supe-ooled wte, in onditions whee moist wm i meets old font. Suh onditions my espeilly ou when moist i fom the se moes onto highe lnd es. Figue 3 elow is dopted fom NASA to demonstte tht supe-ooled wte my eist t tempetues lose to C. Figue 3 Supe-ooled doplets fom in old font. Suh supe-ooled doplets will feeze instntneously when they ome in ontt with mteil s shown in figue 4 elow. The feezing ou t the fist ttk point to the wing, i.e. the leding edge. Note tht the wing hs high speed ginst the wind nd tht ie my fom een though the i ound the wing my e oe the feezing point. Sine the wing is solid otto, the speed of the wing eltie the wind is depending on the distne fom the hu. The eltie wind speed is the tul eloity eto esulting fom the wind speed eltie the gound nd the pependiul ottion speed of the wing. This speed my e ey high, usully > 6 m/s t the tip, nd wind hill effets my theefoe use iing een t tempetues few degees oe feezing. V CLOUD OF SUPERCOOLED WATER DROPLETS Figue 4 Cloud of supe-ooled wte doplets hits ifoil. AIRFOIL The mnne in whih wte doplets ohee nd feeze to fom ie on the wing detemines the stutue of the ie. Figue 5 elow shows the usul definitions of ie: Rime ie is fomed y smll doplets whih feeze dietly upon impt nd do not he time to flow. Cle ie - is fomed y lge doplets with time to flow out oe the sufe fte the initil impt. Mied ie is fomed when supe-ooled doplets of ious sizes e intemingled. All kinds of iing, een t smll quntities, will use loss of lift nd inese dg to the wing euse of hnge in wing fom t the impt e. Vindfosk: 8/ : 4/54

15 Repot Vindfosk 3988-/V-38 Figue 5 Iing on wing: ime fost (left), le iing (ight) nd mied (middle). The mount of iing is dependent on the mount of liquid wte doplets in the i nd the tempetue. Figue 6 elow shows n old, ut still ey lid, inestigtion on liquid wte ontent (LWC) [ef 4]. We he in ou lultions used LWC etween.- g/m 3. The density of i t se leel is ssumed thoughout to e. kg/m 3. Figue 6 Liquid wte ontent (LWC) in i. TYPE Mist Dy fog Wet fog Dizzle Light in Rin DROPLET SIZE.- μm - μm -4 μm 5- μm -4 μm 5- μm Tle Doplet size fo ious types of tmosphei wte. The size of doplets diffes with type of fog nd in. Tle shows epeted itions of doplet sizes fo ious types. Dy to wet fog e the usul onditions fo supe-ooled doplets t ne, up to few hunded metes oe, gound leel. We he theefoe in ou lultions ssumed dop-let sizes of - μm. We he lso ssumed thoughout tempetue of the supe-ooled doplets of -5 C. Vindfosk: 8/ : 5/54

16 Repot Vindfosk 3988-/V-38 4 Flow studies When wing is sweeping though i with supe-ooled liquid wte doplets, n ie shell is quikly fomed t the leding edge. The gowing ie etes ough sufe whih distus the fine lmin oundy lye flow t the leding edge. The tnsition fom lmin to tuulent oundy lye flow is then moed lose to the stgntion point. This ultimtely ineses the isous fition long the sufe of the wing nd led onsequently to n inesed dg foe. The popelling foe of the wing theey deeses leding to lowe powe genetion of the wind tuine. In ode to seue the eodynmi foe lne etween the popelling pt of the lift foe nd the dg foe, the wing neessily needs to e ie fee. This n e done y keeping the wing sufe tempetue just oe the feezing point (> C). The min ojetie of this setion is theefoe to find n ppoimte mesue of the heting equiements. Figue 7 A typil wind tuine wing. The most eposed pt of the wing is of ouse the tip whee we find the highest eloities. In the fist study (setion 4..) we theefoe inestigte this egion (I).5 m elow the wing tip. At tht distne fom the wing tip we ssume the oundy effets fom the tip ote is negligile. The typil width of the wing is hee out.4 m fo 45 m long wing (inluding the hu). Two othe positions, egion II nd III (fig. 7), e onsideed in setion.3 in ode to get n oe ll mesue of the neessy het powe. The wing pofile used in the simultions is the well known NACA 63 48, see figue 8. Figue 8 NACA with the ngle of ttk α 6. A wind tuine opeting t optiml onditions hs wing tip eloity of out 6 times the wind speed. In ou studies we ssume typil wind speed of m/s. The wing tip will then e eposed to eltie eloities of out 6 m/s while the eltie eloities t egions II nd III e 4 nd m/s, espetiely. Vindfosk: 8/ : 6/54

17 Repot Vindfosk 3988-/V D onetion simultions Figue. The omputtionl gid (8 nodes). Computtionl Fluid Dynmis (CFD) is the nlysis of het nd fluid flows y mens of ompute sed simultions. It is ey poweful tool tht spns wide nge of pplitions. The solution to flow polem is defined t finite nume of positions inside the flow domin lled nodes. The numeil uy of the esults depends on the distiution of nodes (gid). Howee, it is not suffiient to he high numeil uy sine the uy of the physil models is eqully impotnt. The tuulent motion, hteisti fo most flows of engineeing impotne, pesent high degee of unetinty [ef 5]. Tuulene models used in the study of ifoil flows e well known nd show good geement with epeimentl esults fo unsepted flows. The lultion of the ompessile flow whee pefomed utilizing the ommeil CFD softwe FLUENT (.) with seond ode disetistion shemes. The tuulene is modeled using the -eq. SST k-w model (stndd hoie fo these kinds of simultions). A dw k using eqution models is thei inility to ptue the oundy lye tnsition fom lmin to tuulent flow whih leds to slight oe pedition of sufe she foes nd het flues [ef 6]. Futue studies should theefoe im to inestigte 3 nd 4 eqution models s well. In the CFD nlysis of the etenl het nd fluid flow we ssume the fee stem eloities U I 6 m / s, U II 4 m / s nd U III m / s omined with tuulene intensity of % (ms(tuulent kineti enegy)/u. In the het flow nlysis it is ey impotnt to esole the oundy lye long the wing sine the onetion polem hee is stongly oupled to the fluid flow. The oundy lye mesh then neessily needs to e ey fine, with n inne ell thikness of the ode of μm. The fee stem tempetue is set to - 5 C while the wing is keep isotheml t C. Vindfosk: 8/ : 7/54

18 Repot Vindfosk 3988-/V CFD esults fo the wing tip, egion I. In this setion we pesent the het nd fluid flow esults fo the wing tip egion I (Fee stem 6 eloity 6 m/s, Re V L / 5 ) V ν Figue 9 Veloity field t setion I. On the uppe side of the wing the eloity ineses to out 9 m/s whih is Mh.3, see figue 9. The flow is theefoe wekly ompessile whih would led to some theml effets. At low ngles of ttk we see tht the flow is niely tthed long the wing sufe. Low flow eloities e found t the stgntion point just elow the leding edge nd in the wke flow t the tiling edge. It is the eloity diffeene etween the uppe nd lowe side of the wing tht etes the powe poduing lift foe. Fo septed flows lled stll this effet disppes leding to n inesed dg foe tht ultimtely stops the enegy podution. Vindfosk: 8/ : 8/54

19 Repot Vindfosk 3988-/V-38 Figue Tempetue field t setion I. (Fee stem temp. 68 K ( 5 C) nd wing temp. 73 K ( C). It is inteesting to note tht ompessile effets etes inesed i tempetues of out degees t the stgntion point nd tempetue dop of out 3 degees oe the uppe pt of the wing. These effets e esily undestood y onsideing the stedy-stte enegy eqution h V onst long stemline Fo n idel gs the enthlpy h is gien y dh dt Δh ΔT whee p kj / kgk fo i. Hene, the tempetue dop long stemline is p ( V V p p ΔT ) () Fo the flow on the uppe side of the wing eq.() onfims tempetue dop of.5 K while we find tempetue inese t the stgntion point of.8 K. We my then onlude tht if it is found tht the uppe pt of the wing is not suseptile to iing it eomes impotnt to isolte this side in ode to deese the het flu fom n intenlly heted wing. Vindfosk: 8/ : 9/54

20 Repot Vindfosk 3988-/V-38 Figue Het flu t setion I. (Fee stem tempetue 5 C nd onstnt wing tempetue C) The highest het flu is found t the leding edge n long the uppe side of the wing. This is due to the thin oundy lyes eted y the eleted i flow ound the leding edge nd the tempetue dop indued y ompessile effets. Integting the lol het flu in figue the totl het onetion pe unit length of the wing t setion I is found to e q kon I, 7W / m The estimted Newtonin het tnsfe oeffiient pe unit length [ W mk] though whee T s T 5 K qi, luft q, luft H( Ts T ) H I ( T T ). H I / is detemined 4 W I / s mk Vindfosk: 8/ : /54

21 Repot Vindfosk 3988-/V-38 Figue Appoimte het powe equied to keep the outemost mete of the wing t onstnt tempetue of 73 K ( C) fo diffeent fee stem i tempetues. T Vindfosk: 8/ : /54

22 Repot Vindfosk 3988-/V CFD esults fo egion II nd III. In this setion we pesent the esults fom the CFD simultions fo the D flow t oss setion II nd III. The min inteest hee is to get glimpse of how the typil flow hte hnges long the wing nd to e le to get ough mesue of the oe ll onetie het flu. Figue 3 Contous of eloity mgnitudes. In figue 3 we see the sme flow ptten ound the wing. This is euse the Reynolds 6 numere U L / ν 5 is the sme t oth positions. The mimum Mh numes found on the uppe pt of the leding edge e hee M II. (wekly ompessile) nd M. (ppoimtely inompessile). Hene, the i flow eomes ompessile somewhee etween position II nd III. We my then ssume to find non-unifom tempetue field t II (not t III) Vindfosk: 8/ : /54

23 Repot Vindfosk 3988-/V-38 Figue 4 Contous of tempetue. Utilizing the stedy stte enegy eq. we find tempetue dop of out -.4 K long the uppe side of the wing nd stgntion tempetue inese of.8 K fo se II. Fo se III the themodynmi hnge in tempetue is only few pts of degee K nd onsequently ppoimtely inompessile. Oe ll, it should e pointed out tht ompessile effets eists nd e impotnt in the study of iing of wind tuine wings. Fo the study of puely fluid dynmil popeties ompessile effets n sfely e negleted fo Mh numes <.3 (Boussinesq ppoimtion [ef 7]). Howee, fo themodynmil popeties the enegy eqution is found to e oupled to the momentum eqution fo the oute hlf of the wing. Vindfosk: 8/ : 3/54

24 Repot Vindfosk 3988-/V Conetie losses oe the wing due to gs flow Finlly we onlude these simultions with mesue of the oe ll onetie het flow oe the oute 3 m of 45 m long the wing. The wing is ssumed to opete t optiml onditions, mening tht the tip eloity is out 6 times the wind speed. The wing tempetue is held t C while the fee stem i tempetue is -5 C. At positions I, II nd III we found the lol onetie het flow pe unit length to e q q q I, i II, i III, i 7 W / m 6 W / m 7 W / m Using the men lues multiplying with the distne etween the positions (5 m) we get the oe ll het flow Q i 5 kw fo the oute 3 m of 45 mete long wing (inluding the hu). Vindfosk: 8/ : 4/54

25 Repot Vindfosk 3988-/V Conetie losses due to doplets The min ojetie fo heting the wing is to pelude iing when the wing is sujet to flow of supe-ooled liquid doplets. It is theefoe inteesting to gie n uppe estimte of the het loss due to the doplets lone. Figue 5 Polem, set-up. In the lultion of the ooling effet of the doplets we ssume tht ll doplets within the dshed lines will hit the pojeted e of the wing. This is lely n oe pedition sine some doplets will e swept wy due to isous dg foes indued y the i defleted flow in whih they e emedded. The ooling effet due to supe-ooled liquid wte doplets my then lulted s: 45m ΔQ mc & ΔT U ( ) L( ) sinα LWC C p, H O 5m whee U I, 6 U ( ) [m/s], L 45 The wing width L s funtion of the distne fom the hu is gien y Speifi dt: yields 45 L( ).4 (3.4) [m] 3 α 6 3 LWC. g/ m C 4kJ / kgk p, H O ΔT 5 C p ΔTd Δ Q 37 W fo the oute 3 m of 45 m long wing (inluding the hu). Vindfosk: 8/ : 5/54

26 Repot Vindfosk 3988-/V Doplet flow In this study we like to dw ttention to some nlytil esults onening the flow of smll wte doplets following fee i stem ginst solid odies. The min ojetie is to inestigte the itil onditions fo whih the doplets e swept ound the ody without hitting the sufe. This sitution is impotnt sine it ultimtely peent iing. We will ll this sitution impt fee ptile flow. Knowing the mehnism ehind this phenomenon might yield n indiet tehnique fo keeping the wing ie fee o less suseptile to iing, in ontst to diet tehniques like heting. Two odies e studied fo this pupose: the iul ylinde nd the infinitely thin ifoil. Both ses n e seen s eteme ses of typil wind tuine wing. The ylindil se is lso of inteest in the field of mpping tmosphei ie lods whee ylinde with dimete of 3 mm is used. The flow of smll liquid doplets in i is goened y the siene of multiphse flows. In multiphse flows the phses e defined s n identified lss of mteil tht hs ptiul inetil esponse to nd intetion with the flow in whih it is immesed. Fo the typil doplet flow ginst wind tuine wings the doplets intets with the i flow without ffeting it. This getly simplifies the nlytil tetment. In this study, whih is puely nlytil, we onside the ptile flow fom n Eulein pespetie, mening tht we see the ptile flow s field in ontst to the Lggin iew. In futue studies, the uthos like to inestigte moe glol hteistis of the doplet flows ginst typil wing pofiles y mens of CFD simultions. Suh studies will pesent detiled infomtion of how the doplets will distiute oe the wing sufe, neessy fo n effetie de-iing sttegy. Vindfosk: 8/ : 6/54

27 Repot Vindfosk 3988-/V Detemintion of the D ptile flow ginst ylinde Knowing the ptile flow ginst the stgntion point of wind tuine wing is essentil fo the undestnding of the iing polem. In this setion we inestigte the ptile flow ound iul ylinde. The stgntion flow ginst iul ylinde is muh simple thn the flow ginst the leding edge of wing ut is ssumed to he the sme physil mehnisms. We will mke the polem dimensionless in ode to identify impotnt pmetes nd ty to gie them physil eplntion. The ptile flow ginst iul ylinde epesents n ppoimtion to the flow ginst the leding edge of typil wind tuine wing. The physil mehnism fo IFPF is hee mde s le s possile. Let us stt y onsideing the undelying ssumptions: Figue 6 Foe lne. Figue 7 Ai flow field ginst ylinde. In the following we ssume tht smll doplets with the dimete φ e swept y fee stem with the U eloity f wy fom the ylinde. Close to the ylinde the stgntion pessue kes nd deflets the i eloity u ound the ylinde. The pessue gdient is howee not ffeting the motion of the doplets. The doplets feel only thei own ineti long with the isous she foes t the fee sufe due to the fition ginst the suounding i flow. The foe lne fo spheil doplet is m F D () whee m is the mss of the doplet gien y nd its eletion 6 3 m ρ H O π φ () ssuming tht the flow is stedy-stte F F ( ) ( ) sphee with the dimete φ is ( doplet eloity) (3) D D. The isous dg foe F D fo Vindfosk: 8/ : 7/54

28 Repot Vindfosk 3988-/V-38 ( u F ) φ μ π i D 3 (4) The inisid i eloity field u ound ylinde with the dius R is gien y os R U u (5) sin R U u (6) [ef 7].Intoduing the dimensionless pmetes D U R U U luft O H μ φ ρ 9 ; ; ; u u (7) into the foe lne we get (8) u whee the dimensionless inisid i flow is witten os u (9) sin u () nd the dimensionless eloity gdient ˆ ˆ () The foe lne in eh dietion is u ( -led) () ˆ u (ˆ -led) (3) Asymptoti oundy ondition s ˆ sin ˆ os U. (4) A genel solution to eq. () nd (3) hs not een found. Insted we pesent the symptoti solutions lid fo smll nd lge. Vindfosk: 8/ : 8/54

29 Repot Vindfosk 3988-/V-38 We now pefom n symptoti nlysis fo <<. Assume solution of the fom ( ) O ( ) O (5) Insetion of eq. (5) into () nd (3) we find the dimensionless doplet eloity os os 3 O ) (6) sin sin O 3 ) (7) ( ( Figue 8 Doplet fee egion fo smll. The physil eplntion to the doplet fee egion in the iinity of the ylinde (eept t the stgntion point) is tht the doplets e eing eleted y the ngul i eloity efoe hitting the sufe. The ngul eloity of the doplets tully eomes tht lge tht the distne to the ylinde sufe ineses. Hene, we my sy tht the inisid i stem ts s entifugl septo. The effet of the isous oundy lye is not studied ut it is the utho s elief tht it will t to deese the ngul eletion leding to doplet impt lose to the stgntion point. The isous oundy lye thikness t the stgntion point is gien y ν luft δ. 4 (8) k whee ( ) u ( ) k (9) Vindfosk: 8/ : 9/54

30 Repot Vindfosk 3988-/V-38 In the limit eq. (5) yields U k () R Inseted into (8) the oundy lye thikness is gien y ν luft R δ.4 () U Fo R 3 m, U 6 m / s 5 nd ν ( 5 C).3 m s isδ.4 mm. luft / Asymptoti solutions fo >>. Assume the solution O ( ) O ( ) () Insetion of (8) into () nd (3) yields os O( ) (3) sin O( ) (4) Fo the se of lge lues of the pmete the doplets pss ight onto the ylinde lmost without ny defletion. The mss flow is miml t the stgntion point nd deeses with the ngle, see figue 9. The ition of the liquid wte ontent LWC [ kg/m ontinuity eqution, i.e. Fo smll we ssume the solution ( ) LWC LWC 3 ] n e deied onsideing the stedy-stte LWC (5) LWC LWC O( ) fo << (6) tht yields to O() LWC (7) sine (inompessile flow). This mens tht LWC onstnt long stemlines. LWC Sine ll stemlines omes fom unifom distiution we onlude tht s long s << LWC is onstnt thoughout the flow. Howee, thee eist doplet fee egion lose to the ylinde sufe fom eq. (6), see lso figue (8). Ou solution is theey Vindfosk: 8/ : 3/54

31 Repot Vindfosk 3988-/V-38 LWC LWC O() Fo lge we ssume the solution << (8) LWC LWC O( ) >> (9) Insetion of eq. (9) into eq. (5) gies in the sme wy onstnt LWC though eq. (7), i.e. LWC LWC O( ) >> (3) LWC n theey e onsideed onstnt fo smll nd lge. Fo of ode O() numeil studies fo the ition of LWC e needed. Figue 9 Typil mss flows pe unit sufe e fo lge. In this nlysis we only inestigted smll nd lge lues of the pmete. But sine the flow ehio hnges fom eing swept ound to pssing dietly onto the ylindil sufe we onlude tht thee hs to e itil lue of O () fo tht mks the hnge in solutions. In the net setion we seek this itil lue. Vindfosk: 8/ : 3/54

32 Repot Vindfosk 3988-/V-38 Figue 3 s funtion of the doplet dimete fo ylinde dimete of the dimension s leding edge t the top of wind tuine wing. Vindfosk: 8/ : 3/54

33 Repot Vindfosk 3988-/V Detemintion of the ptile flow ginst ylinde The symptoti solutions fo smll nd lge lues of the pmete indite two diffeent ptile flow ehios. Fo ptile flows with smll eey doplet is swept ound the ylinde without hitting the sufe. In ft we found tht doplet fee egion is fomed lose to the ylinde. This is due to the ngul eletion of the inisid flow tht thows the ptiles futhe out k into the stem. Fo flows with lge the ptiles tels moe dietly ginst the ylinde sine they e found to e lgely unffeted y the suounding i flow. In this se, the ineti of the ptiles is lge thn the isous dg. The question we hee like to nswe is theefoe: At whih ppe? it will the hnge in flow ptten Figue 3 Stgntion stemline. Let us study the ptile flow long the stgntion stemline nd gue tht the hnge to solutions will fist ppe s non-zeo eloity t the stgntion point. The one dimensionl polem long the stgntion stemline is gien y d d u () whee is the ptile eloity nd u (...)... () ( ) is the inisid i flow (positie dietion towds the stgntion point). Fo ( ) leding to solutions of the type it is... (3) Insetion of eq. () nd (3) in () yields (4) O(): ± 8 (5) Knowing tht u s the oet oot must e lim. The limit poess of (5) gies Vindfosk: 8/ : 33/54

34 Repot Vindfosk 3988-/V-38 ± lim 8 ± ( 4) lim ( ) ( ) fom whih we onlude tht ou sought oot is 8 (6) The ptile eloity in the iinity of the stgntion point is theey gien y 8... fo. (7) 8 This solution loses mening t / 8 whih is the itil lue we e looking fo. it Fo > we my ssume the solution 8... (8) Inseted into () gie us no infomtion out how depends on ut we find tht. (9) Figue 3 Ptile eloities fo diffeent (numeil solutions). d ( ); s d Numeil solution of the diffeentil eqution lso show tht / 8 is the oet solution nd how the solutions ehe oe nd unde the itil lue. The net step is now to study the two dimensionl ptile flow in the iinity of the stgntion point in ode to undestnd the deelopment of the ptile fee egion. it Vindfosk: 8/ : 34/54

35 Repot Vindfosk 3988-/V D stgntion point nlysis fo it ginst ylinde In this setion we like to inestigte the shpe of the ptile fee egion in the iinity of the stgntion point. The petution nlysis is sed on powe seies of the noml nd tngentil oodintes. In this nlysis we study the ptile flow equtions in the iinity of the stgntion point, i.e. fo nd. The eqution system, defined in setion., is fo smll nd gien y ( ) ( ) () ( ) ( ) ( ) 6 3 () whee we mde the ile sustitution. A onsequene of the hypothesis mde in the nlysis in setion 4.. is tht thee eist doplet fee egion lose to the ylindil sufe fo smll. Mthemtilly this oesponds to n outflow oundy ondition fo. Assume: is n een funtion of out onst s ) ( O (3) is n odd funtion of out fo ( ) O (4) Insetion of (3) nd (4) in () gies ( ) ( )( ) ( ) ( ) ( ) ( ) Soling fo the diffeent odes O 8 ) : ( (5) Note tht we get 8 it just s epeted! The flow field fo 8 > theey neessily needs nothe ssumption. ( ) ( ) ) : ( O Vindfosk: 8/ : 35/54

36 Repot Vindfosk 3988-/V-38 Insetion of fom (7) yields ( ) (6) Figue 33 The onstnts nd s funtion of. Inseting (3) nd (4) in () gies ( ) ( ) ) )( )( ( ) )( )( ( ( ) ( ) ( ) ) ( Soling fo the diffeent odes yields ( ) O 8 : (7) ( ) ( ) : O (8) Vindfosk: 8/ : 36/54

37 Repot Vindfosk 3988-/V-38 Figue 34 Constnts nd s funtion of. The eloity field in the stgntion point is theey ( ) (9) () An inteesting question is now: Wht is the shpe of the inne stemline of the ptile flow in the stgntion egion? Let us nswe this question y setting in eqution (9). This etes the eqution () Vindfosk: 8/ : 37/54

38 Repot Vindfosk 3988-/V-38 Figue 35 Shpe of the inne stemline lose to the stgntion point fo diffeent. Fom figue 35 we see tht the inne stemline lifts fom the ylindil sufe with inesing. At the itil lue / 8 the inne stemline mkes the widest pth ound the ylinde. Fo > / 8 this flow phenomenon disppes s the inne stemline ollpse ginst the ylinde t the stgntion point. The net question we like to inestigte is how the liquid wte ontent LWC hnge in the egion lose to the stgntion point. Vindfosk: 8/ : 38/54

39 Repot Vindfosk 3988-/V LWC lose to the stgntion point of flow ginst ylinde. To get feeling of the ptile flow ginst the stgntion point we hee like to study the distiution of the liquid wte ontent LWC long the stgntion stemline. We ssume to find n inesing LWC the lose we get to the stgntion point. Figue 36 Stgntion stemlines. We will hee inestigte the LWC lose to the stgntion point. The stedy-stte ontinuity eqution is ( LWC ) () Along the stgntion stemline eq.() eomes whee ( LWC) LWC () ( ). (3) The ptile eloity field lose to the stgntion point << is... (4)... (5) found in setion.3. The diegene in the iinity of the stgntion point eomes (( )...) (...) (...) (...) (6) Inseted into () gies ( LWC) α LWC LWC onst s (7) whee ( see setion.3) 8 nd 8 yields Vindfosk: 8/ : 39/54

40 Repot Vindfosk 3988-/V-38 8 α (8) 8 The onstnt in (7) n e detemined though the limiting poess lim LWC LWC whee limα LWC onst. lim α onst. The LWC in the stgntion point egion then eomes LWC LWC s whee α 8 α. (9) 8 Hene, LWC ineses dmtilly the lose we ome the stgntion point. Figue 37 LWC s funtion of the distne to the stgntion point (eq.(9)). Fo /8 LWC ineses singully ginst the wll. This is due to the deesing ptile eloity. Fo lge thee will not e the sme inese in LWC, insted will the doplets fom liquid 3 wte lye t the sufe with LWC ρ H kg m. o / Vindfosk: 8/ : 4/54

41 Repot Vindfosk 3988-/V The inisid i flow in the iinity to thin ifoil The net geomety tht we like to inestigte is the thin ifoil. It will simulte the impotnt mehnism of lift whih is stongly oupled to the iultion phenomenon. In this setion we deie the neessy stgntion point flow needed in ode to find the itil pmete. Figue 38 The flow ginst thin ifoil. In this nlysis we like to deie the iflow in the iinity of the stgntion point. This solution will then e used in the sme mnne s fo the ylindil se in ode to find the itil lue of whih mks the limit fo non-hitting ptile flows. The dimensionless inisid i flow ginst the thin ifoil in figue 38 is [ef 8]. z u i osα sinα with z iy () z whee the wing oupies the intel nd y. We e hee going to inestigte the flow lose to the ing, i.e. y <<. Stt y studying the tio within the sque oot whih is z z ( z)( z) z z zz ( z)( z) z z zz y iy ( ) y Letting y we get y i nπ ( O y z z y i ( ) y ) i e n,,... Tking the oot of this genetes two solutions z z ± i y () The eloity field on the lowe side of the wing () is gien y y u i osα sinα i sinα (3) ( ) The stgntion point s is detemined y setting the eloities equl to zeo Vindfosk: 8/ : 4/54

42 Repot Vindfosk 3988-/V-38 s osα sinα s osα s (4) Hene, long the stgntion stemline the etil eloity ies s y y <, y << (5) 3 4sin α osα Knowing the etil eloity lose to the stgntion point we my now utilize the sme sttegy fo detemining the itil lue of s fo the ylindil se. This deition is pesented in the net setion. Vindfosk: 8/ : 4/54

43 Repot Vindfosk 3988-/V Detemintion of the itil lue of fo the flow ginst thin ifoil In this setion we inestigte the ptile flow ginst thin ifoil nd espeilly seek the itil lue of the pmete. The ylinde nd the thin ifoil mks the oute limits of el wing nd will theefoe gie us ough mesues of wht to epet. The min question hee is how the itil lue of the pmete hnges with espet to the ngle of ttk. Figue 39 The flow ginst thin ifoil. The dimensionless equtions fo stedy-stte ptile flows e u () If we sle the nd y oodintes with hlf the wing spn (L / ) we get the dimensionless pmete ρ H Oφ U () 9 μ i L in odne with the definition in setion.. The dimensionless i eloity long the stgnting stemline is (see setion.5) y u yˆ y << (3) 3 4sin α osα Fo doplets whih it we ssume the etil eloity pofile y yˆ y << (4) The y-dietion of eqution () the eomes y y y 3 4sin α osα (5) o (6) 3 4sin α osα Vindfosk: 8/ : 43/54

44 Repot Vindfosk 3988-/V-38 Soling fo yields 3 sin α osα (7) The itil lue of is then gien y it sin 3 α osα (8) 3 α fo smll ttk ngles α <.3 ( α < 5 ) it fo whih ou ssumption eomes inlid nd the doplets get non-zeo stgntion point eloity (i.e. hitting the sufe). Figue 4 Citil lue of s funtion of the ngle of ttk. The itil lue of fo the thin ifoil is edily seen to e muh less thn tht fo the iul ylinde without iultion. We theefoe ssume tht the itil fo the typil wind tuine wing is 3 somewhee etween /8. α it Vindfosk: 8/ : 44/54

45 Repot Vindfosk 3988-/V-38 5 Tehnologies In the following hpte we disuss diffeent tehnologies to oid iing. Ou disussions entes on heting the wte doplets to oe feezing tempetue efoe they hit the wing o on the wing (potie) nd melting of ie when it hs fomed on the wing (e-tie). 5. Het doplets efoe hit wing The wte doplets my e heted to oe feezing efoe they tully hit the wing. This is ontinuous poess when onditions fo iing he oued. The powe equied to ise the tempetue of doplets y T degees Kelin n e witten s: ΔT V q C p LWC kw/m Δs A whee C p is the speifi het of wte in kj/g/k; LWC is the liquid wte ontent in g/m 3 ; s is the ess time in seonds to the doplets; V is the olume in m 3 swept y the wing nd A is the tget wing e in m. Assuming m of tget e nd g/m 3 of liquid wte ontent will esult in olumn length of peipitted wte of μm. Thus, if the doplets he dimete > μm, then the sum e of ll doplets is less thn the totl e oeed nd signl powe will e lost due to lk of em filling s: Λ f A dop A whee A dop is the e of eh dop. The equied powe n now e witten s: q ΔT Δs V A C p LWC Λ f This n e simplified s: ΔT q C p ρ H O φ 3 Δs whee ρ HO is the density of liquid wte ( kg/m3) nd φ is the dimete of the doplets. Note tht this eqution is independent on the liquid wte ontents, mening tht if we het one dop we will het ll within the olume. The doplets n e heted efoe they stuk the wind eithe y emitting hot i in the fowd dietion of the wing. We he ssumed this to e impope sine it would indeed wok ginst the flow. We theefoe suggest tht the only ptil men to het the doplets is y emitting eletomgneti dition fom the font of the wings. The equied powe to het the wte with miowe o infed would e: C q p ΔT LWC Δs V A α φ ( e ) whee α is the soption oeffiient of liquid wte s peiously disussed. Assuming tht we mount tnsmitte on the font of the wings with opening ngle of fowd the wing dietion of motion nd width of dm oe 45m wings. Futhemoe, the 3 wings e ssumed to sweep t wing tip speed of 6 m/s nd the doplets e μm in dimete nd he Vindfosk: 8/ : 45/54

46 Repot Vindfosk 3988-/V-38 tempetue of -5 C. The powe loss to the wte doplets (effiieny) nd the equied ontinuous powe is shown in tle 3 elow fo some welength nds in the miowe to infed nds. Welength Effiieny Powe 3 mm.3 % 7 MW 3 mm 5 %.4 MW 3 mm 7 % 5 kw μm 63 % kw Tle 3 Powe equiement to het wte efoe it hits the wings. 5. Het wte on wing The wte my e heted lso y onetie mens one the wte doplets e in ontt with the wing. Suh heting ould e to het the wing sufe mteil eithe y hot i inside the wing o eleti heting with low fequeny eleti field. In this pt we will disuss onetie heting s well s heting the wte with miowe o infed dition. 5.. Conetion Ou simultions show the oe ll onetie het flow oe the oute 3 m of the wing. At positions I, II nd III we found the lol onetie het flow pe unit length to e q q q I, i II, i III, i 7 W / m 6 W / m 7 W / m Using the men lues multiplying with the distne etween the positions (5 m) we get the oe ll het flow Q i 5 kw fo the oute 3 m of 45 mete long wing (inluding the hu). We he lso shown tht the onetie flow due to doplets n e ppoimted s Δ Q 37 W fo the oute 3 m of 45 m long wing (inluding the hu). Hene, we find tht ΔQ << Q i, fom whih we onlude tht heting the wing in ode to keep the supe ooled liquid doplets fom feezing is ey pimitie method, whee lmost ll of the het powe is lost into thin i. The effiieny of this poess is less thn.7 %! Een if the LWC is 3 highe, let s sy g/ m the effiieny only ineses to 7 % ut is still ey low. 5.. Eletomgneti heting One the wte doplets hit the font wing e they will fom thin wte lye. This lye will e efuished y doplets t speed of U m/s. The powe equied to het m thik lye of wte n e lulted s: q C p LWC U ΔT A α ( e ) Vindfosk: 8/ : 46/54

47 Repot Vindfosk 3988-/V-38 Assuming lye thikness of. mm oe wing stip of m of 3 times 45 m wings nd tempetue of -5 C we get: Welength Powe 3 mm 5.3 MW 3 mm 368 kw 3 mm 9 kw μm 6 kw Tle 4 Powe equiement to het wte on wings. 5.3 Melt ie on wing Heting of the wte equies ontinuous powe emission duing iing onditions. One the ie hs fomed it n e melted with speifi mount of enegy lulted s: E L ρ ie ie A whee L ie is the ltent het of 334 kj/kg; ρ ie is the density of ie (96 kg/m 3 ); is the ie lye thikness nd A is the e. Assuming m width on the wing font nd 3 wings of 45 m length we get n enegy equiement of -.3 kwh fo mm of ie nd - 3 kwh fo mm of ie It my e possile to het only the inne mm lye of ie in ode to dismount the ie. The het n e injeted to the ie with miowes o infed emission. In this se we get the following enegy onsumption: Welength Effiieny Enegy 3 mm % MWh 3 mm 9 %. MWh 3 mm 73 % 5 kwh μm % 3 kwh Tle 5 Powe equiement to melt mm lye of ie on wings. Note tht the effiieny t miowes is so low tht it does not mtte if thee is mm lye o mm lye, most of the signl will e lost nyhow. In the infed nge, the soption is suffiiently high to estit the melting zone to few mm only, thus minimize the enegy equiement to the theoetil leel. Conetion n lso e used to melt the ie. Hee the wing hs to e heted nd tnsfe the equied mount of enegy to the ie. This is possile, ut epensie sine the full wing will e het sink whih is ooled y the steming ool i s in 5.. oe. Vindfosk: 8/ : 47/54