HYDROSTATIC HEAD CORRECTION
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1 XVI IMKO World oress Measureme - Suppors Scece - Improves Tecoloy - Proecs vrome... ad Provdes mployme - Now ad e Fuure Vea, AUSTIA,, Sepember 5-8 YDOSTATI AD OTION W. Kolacza Seco for e, Area, Ale, Poomery, Sac Volume Budesam für c- ud Vermessuswese BV Arlasse 5, A-6 We, Ausra Absrac: Te volume of a cyldrcal sorae ak w a vercal axs caes - besdes of e fluece of oer quaes - w e ydrosac pressure appled by e lqud sored e ak. Te formula of M. K. Sayad [] descrbes e fluece of e lqud ead o e expaso of sorae aks bes av e adcap a ca be evaluaed dvdually for eac ak oly by meas of a umercal proram because e explc soluo works oly for a ak coss of a sle course. Te auor red o mee wo requremes: o ve a formula for a explc soluo for aks coss of several courses ad o ve a formula wc leads o a resul as accurae as a oe obaed by umercal soluo accord o e formula of Sayad. Bo codos are fulflled by e formula derved. Keywords: sorae aks, volume, measureme uceray INTODUTION Te volume of sorae aks caes - besdes of e fluece of oer quaes - w e ydrosac pressure appled by e lqud sored e ak. Te resul orders of maude are o o be eleced we assess e uceray of e calbrao measureme of e sorae ak, parcular, f e calbrao s carred ou by e opcal meod a empy ak. O oe ad e paper preseed sall elp o mmze e compoe of e uceray of measureme caused by e expaso of e ak, o e oer ad e calculao of e correco for expaso sall be carred ou easly. I aaloy o Sayad s formula [], equao 5 ca be see ow e auor developed s formulae: e wole formula of Sayad cosss of pars wc ca be erpreed as BASI MOD, XPANSION OF T BASI MOD w reard o e vercal sress ad e XPANSION TO T GNA SOUTION w reard o e fluece of e boom of e sorae ak. So e auors effors wll eveually lead o soluos vald for ay ve umber of courses. BASI MOD Te assumpo of e basc model s a e lqud ead leads o deformao of e sell a orzoal layer. A maemacal apparaus sall be foud descrb e aeal sress of e sell e case of a sle course model ad laer w a model o e assumpo of may courses.. ase of sle course e us cosder a ak form of a crcular cylder w vercal axs ad a radus, lqud e ad plae ckess. Te relave cae of e radus w radal pressure P s ve by equao = were s You s modulus of elascy. Te radal pressure P e e, provded e ak s flled up o e e, s a fuco of. Terefore P = were s e desy of e lqud sored e ak, s e desy of e medum dur e calbrao procedure ar, waer, ec. ud s e accelerao due o ravy. A ak form of a crcular cylder w vercal axs deformed by e ydrosac pressure ca be cosdered as a roaoal body ad e cae of e volume for e fll e caused by e
2 XVI IMKO World oress Measureme - Suppors Scece - Improves Tecoloy - Proecs vrome... ad Provdes mployme - Now ad e Fuure Vea, AUSTIA,, Sepember 5-8 expaso of e ak ca be calculaed accord o a equao, wc a expresso for accord o s subsued = V d emove e expresso from w e brackes ad pu as a cosa fro of e eral. Te square e bomal fuco e eral. If you elec e quadrac erms of, e absolue value of e error resul ereby s usually 5-4. As resul of s ad furer more raslaos lke a subsuo for P accord o you wll fd a las d Solv e eral you wll e For e relave volume of expaso /V apples 4 5 V 6. ase of may courses Te assumpo a a sorae ak as e same radus ad e same plae ckess for e wole e s o realsc; sorae aks cyldrcal sape w vercal axs are as a rule bul up by several cyldrcal secos w radus r, e of seco ad plae ckess. For e calculao of e correco of e expaso equao 4 e r-ad sde e sum of e correced voluma of e specfc secos, a are releva for e overall correco, s o be pu. Tus s = r d were o = ad =. Solv e erals leads o r = Because of = + ad = - - we e 7 8 = + r 9 Subsu r = ad = for all ad = equao 9 leads o equao 5. XPANDD MOD Te formulae for e calculao of e correco for e expaso meoed above are based o a pyscal-maemacal model a cosders e aeal sress of e sell by e radal pressure P a orzoal plae due o e ydrosac pressure, caused by e lqud ead bu o e sress e vercal dreco resul from. Te correcos of e expaso calculaed accord o ese formulae are subjec o a error wc corbues o e measureme uceray wle calbra e sorae ak. If s measureme uceray sall be mmzed, e model for e calculao of e correcos for e expaso as o be modfed a approprae way. A approac s e model used
3 XVI IMKO World oress Measureme - Suppors Scece - Improves Tecoloy - Proecs vrome... ad Provdes mployme - Now ad e Fuure Vea, AUSTIA,, Sepember 5-8 by Sayad s paper wc, o e base of ooke s law, cosders e sress of e sell all ree spaal drecos due o ydrosac pressure.. ase of a sle course For e smple case of a sorae ak w e, radus ad plae ckess a explc formula for e calculao of e relave volume of expaso /V s ve [] by equao 5, a smplfcao of s formula s ve by equao 7. For e calculao of e volume of expaso of a sorae ak assembled by several cyldrcal secos w radus r, e of e seco ad plae ckess a mplc formula s ve w equao, e soluo of wc oly could be doe by a proram for e umercal compuao of e eral separae for eac ak. Sudy [] sows a e complexy of e maemacal model rows by eac mproveme of e pyscal model very quckly ad complcaes e udersad ad e applcably. O e oer ad a smplfcao w a suffcely ood approxmao s a ad as equao 7 sows. I e follow we sall ry o coec e resuls of [] w ose of e paper preseed a way so a a suffcely ood approxmao s uaraeed despe e smplcy of e maemacal model. Te comparso of equao 7 [] ad equao 6 of e paper preseed sows a e fluece of e vercal compoe of e sress of e sell due o ydrosac pressure ca be descrbed by e expresso -µ/, were µ s e Posso rao for seel,. Tak o accou e vercal compoe 5 you e µ be e equao for e calculao of e volume of expaso. I order o derve equao e same way as was doe w equao 5, equao as o be modfed as follows: = µ If you subsue for accord o, you e = V [ µ ] d You ca remove from e erm w e brackes ad pu as a cosa fro of e eral. By squar e bomal expresso e eral ad by elma e quadrac erm ad subsu accord o you e afer reduc ad rasform correspodly µ Solv e eral leads o equao. d. ase of may courses For a sorae ak assembled by cyldrcal secos w radus r, e of e seco ad plae ckess e correco for e volume of expaso ca be calculaed cosequely accord o e follow equao: µ = r By solv e eral you e afer reduc ad rasform correspodly d 4 µ + r = 4 GNA SOUTION Bo models roduced ve a maxmum for = ; because of e sable boom of e sorae ak wc preves ay expaso e vcy of e boom of e sorae ak = apples for 5
4 XVI IMKO World oress Measureme - Suppors Scece - Improves Tecoloy - Proecs vrome... ad Provdes mployme - Now ad e Fuure Vea, AUSTIA,, Sepember 5-8 =. Terefore e expaso of e sorae ak calculaed accord o equao 5 as o be correced. As you ca see from e paper of Sayad e fluece of e boom of e sorae ak o e expaso volume of e frs course ca be descrbed by a facor k: k e + + s + + cos cos + cos = + 6 wereas s defed by e equao = r µ 4 7 w e ak fll e, er radus r ad plae ckess. Te correco facor k s dmesoless ad deermed compleely by e parameer ad erefore applcable for all sorae aks w oe course. Of course, f we are e poso o modfy equao 7 suc a way a s defed for e case of a sorae ak coss of several courses we ave foud a eeral soluo. For a easy udersad mae several courses o pled up bu pu sde by sde. For eac of ese courses equao 7 mus apply; be e e of e frs course, e e of e secod course, e e of e course: = µ 4 r 8 = = Pl up ese courses,, apples, ad by 8 we wll fd e equao = = µ 4 r 9 s Te complee equao wc akes e fluece of e boom of e sorae ak also o accou e s cos µ r + V + cos + cos = wereas fs equao 9. Subsu r = ad = for all ad = equao leads o a equao wc s equvale o e equao 5 []. 5 ONUSION Te formula derved s o more complcaed a a oe ve by Sayad [] bu s exac eou ak o accou e vercal compoe of e sress of e sell caused by radal pressure ad e fluece of e boom of e sorae ak coss of several courses. By e formula derved s possble o calculae o e oe ad e correco for e expaso of e sorae ak ad o e oer ad o mmze e compoe of e measureme uceray caused by e expaso of e sorae ak wle calbra cyldrcal sorae aks w vercal axs accordace w [], [] ad [4]. Fure sows a example of ow b e devaos of e expaso voluma ca be relaed o e respecve voluma of e sorae aks calculaed accord o formulae 9 ad 5:
5 XVI IMKO World oress Measureme - Suppors Scece - Improves Tecoloy - Proecs vrome... ad Provdes mployme - Now ad e Fuure Vea, AUSTIA,, Sepember 5-8 Fure. xample calculao see ex for deals Ts example s based o a sorae ak w radus r = 5,5 m, coss of 9 courses w e plae es =, m wereas e plae ckess vares from 5,8 mm o,4 mm 9 ; e maxmum fll e s,5 m. Ts example s a que exreme oe capacy of e sorae ak more a 8 m bu wll be elpful o esmae e maxmum relave devao e upper rae sce smaller sorae aks sow oly smaller relave devaos, of course. I fure e relave devao for ay fll e ca be see: e maxmum relave devao of e expaso voluma calculaed by equao 9 ad, respecvely, s 6, -5 ad a oe calculaed by equao 5 ad, respecvely, s 6,9-5. Terefore, case e es possble accuracy s spulaed, s very advsable o apply equao for calcula e expaso of a cyldrcal sorae ak w a vercal axs due o lqud ead. FNS [] M. K. Sayad: xpaso of rcular yldrcal Taks due o qud ead. École des mes de Pars 995 [] PTB-Prüfreel: aerbeäler Form seeder Zylder. Brauscwe/Berl 974 [] ISO Peroleum ad lqud peroleum producs - albrao of vercal cyldrcal aks Par o. 99 [4] ISO Peroleum ad lqud peroleum producs - Tak calbrao - qud meods. 98 AUTO: Dr. Wlelm KOAZIA, Seco : e, Area, Ale, Poomery, Sac Volume, Budesam für c- ud Vermessuswese BV, Arlasse 5, A-6 Vea, Ausra Poe I , Fax I , -mal: w.kolacza@bev-ec.v.a
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