Applying Eyring s Model to Times to Breakdown of Insulating Fluid

Transcription

1 Ieaoal Joual of Pefomably Egeeg, Vol. 8, No. 3, May 22, pp RAMS Cosulas Ped Ida Applyg Eyg s Model o Tmes o Beakdow of Isulag Flud DANIEL I. DE SOUZA JR. ad R. ROCHA Flumese Fed. Uvesy, Cvl Egeeg Dep., Gad. Pogam, Neó, RJ, Bazl ad Noh Flumese Sae Uvesy, Idusal Egeeg Dep., Campus, RJ, Bazl (Receved o Ocobe 28, 2 ad evsed o Febuay 2, 22) Absac: I hs pape he es pupose wll have wo objecves: Fs wll be o vefy f mes o beakdow of sulag flud bewee elecodes ecoded a hee dffee volages have a expoeal dsbuo as pedced by heoy. Secod wll be o assess whehe o o he acceleaed model poposed by Eyg wll be able o aslae esuls fo he shape ad scale paamees of a udelyg Ivese Webull model, obaed ude wo acceleag usg codos, o expeced omal usg codo esuls fo hese wo paamees. The poduc beg aalyzed s a ew ype of sulae flud, ad he acceleag faco s he volage sesses appled o he flud a wo dffee levels (3KV ad 4KV). The omal opeag volage s 25KV ad was possble o es he flud a omal volage usg codo. Boh esuls fo he wo paamees of he Ivese Webull model, obaed ude omal usg codo ad aslaed fom acceleaed usg codos o omal codos, wll be compaed o each ohe o assess he accuacy of he Eyg model whe he acceleag faco s oly he volage sess. Keywods: Eyg acceleaed model, wo-paamee vese Webull dsbuo, sequeal lfe esg; ucao mechasm, volage sesses, sulae flud. Ioduco I a pevous pape by De Souza [], we had used a combed appoach of a acceleaed lfe esg wh a sequeal lfe esg o vefy f he es esuls obaed ude acceleaed codos could be aslaed o he esuls vefed ude omal use codos wh a cea degee of accuacy. I ha case, was o possble o es he seel pas a omal usg codos. I hs sudy was possble o es he flud a omal usg codos. I suaos whe sesses ohe ha empeaue ae volved, he Eyg model [2] offes a geeal soluo o he poblem of addoal sesses. I has a heoecal devao based o chemcal eaco ae heoy ad quaum mechacs. The Eyg model s gve by: R ae T α ( E K) T + C e exp DS () Hee, R ae s he ae of eaco, E epeses he eegy of acvao of he eaco, K he gas cosa (.986 caloes pe mole), T he empeaue degees Kelv (273.6 plus he degees Cegade) a omal codo of use, S s a secod sess, C ad D ae cosas. Fom Equao () we ca oce how he fs em, whch models he effec of empeaue, compaes o he Aheus model. Excep fo he T α faco, hs em s he same as he Aheus. Theefoe, he Aheus model s successful because s a useful smplfcao of he heoecally deved Eyg model. * Coespodg auho s emal: dael.desouza@homal.com 279

2 28 Dael I. De Souza J. ad R. Rocha The acceleag faco AF 2/ fo he Eyg model (o he ao of he specfc aes of eaco R 2 /R ), a wo dffee sess empeaues, T 2 ad T, ad a wo dffee sess volages, V ad V 2, wll be gve by: α ( E K) T2 + C ( E K) V2 + D A T e e AF 2/ 2, o ye: α ( E K) T + C ( E K) V + D A T e e α 2 E exp K T AF 2/ E T exp T T (2) 2 K V V 2 Sce he esg empeaues a he wo acceleag esg codos ae he same ad he oly dffee acceleag sess s he volage, Equao (2) becomes: E AF 2/ exp (3) K V V 2 Applyg aual logahm o boh sdes of Equao (3) ad afe some algebac mapulao, we wll oba: E l ( AF 2 / ) (4) K V V 2 Fom Equao (4) we ca esmae he em E/K by esg a wo dffee sess volages ad compug he acceleao faco o he bass of he fed dsbuos. E l ( AF ) The; 2/ (5) K V V2 The acceleao faco AF 2/ wll be gve by he elaoshp / 2, wh epeseg a scale paamee o a pecele a a sess level coespodg o V. Oce he em E/K s deemed, he acceleao faco AF 2/ o be appled a he omal sess volage s obaed fom Equao (4) by eplacg he sess volage V wh he sess volage a omal codo of use V. The: E AF 2 / exp (6) K V V 2 2. The Acceleag Codo De Souza [3] has show ha ude a lea acceleao assumpo, f a heepaamee Ivese Webull model epeses he lfe dsbuo a oe sess level, a hee-paamee Ivese Webull model also epeses he lfe dsbuo a ay ohe sess level. The same easog apples o he wo-paamee Ivese Webull model. We wll be assumg a lea acceleao codo. The Ivese Webull model has bee used Bayesa elably esmao o epese he fomao avalable abou he shape paamee of a udelyg Webull samplg dsbuo, De Souza ad Lambeso [4]. I has also bee used elably esmao of elecoc poducs whee seems o have a bee aswe o he accuacy poblem peseed by he Webull model, as show by De Souza [3]. I happes ha whe he shape paamee of he Webull model s geae ha seve, he Webull cuve becomes hghly poed, esulg some compuaoal dffculy (accuacy)

3 Applyg Eyg s Model o Tmes o Beakdow of Isulag Flud 28 calculag he compoe s chaacescs of ees values. Also whe he shape paamee value s close o lowe ha oe, he Webull model wll aga exhb some compuaoal dffcules. I geeal, he scale paamee ca be esmaed by usg wo dffee sess levels (empeaue o volages o cycles o mles, ec.), ad he aos wll povde he desed value fo he acceleao faco AF. So, we wll have: AF (7) a Aga, based o he pape by De Souza [3], fo he wo-paamee Ivese Webull model he cumulave dsbuo fuco a omal esg codo F ( ) fo a cea esg me, wll be gve by: F ( ) F a exp AF (8) AF AF Equao (8) ells us ha, ude a lea acceleao assumpo, f a wo-paamee Ivese Webull model epeses he lfe dsbuo a oe sess level, a wo-paamee Ivese Webull model also epeses he lfe dsbuo a ay ohe sess level. The shape paamee emas he same whle he acceleaed scale paamee s mulpled by he acceleao faco. The equal shape paamee s a ecessay mahemacal cosequece of he ohe wo assumpos, ha s, assumg a lea acceleao model ad assumg a wo-paamee Ivese Webull samplg dsbuo. If dffee sess levels yeld daa wh vey dffee shape paamees, he ehe he wo-paamee Webull samplg dsbuo s he wog model fo he daa o we do o have a lea acceleao codo. 3. Maxmum Lkelhood Esmao fo he Two-Paamee Ivese Webull Model fo Cesoed Type II Daa (Falue Cesoed) The lkelhood fuco L(;) fo he shape ad scale paamees of a Ivese Webull samplg dsbuo fo cesoed Type II daa (falue cesoed) wh 2 wll be gve by: L ( ;)! f ( ) [ F( )] k! f ( ) [ R( )] ; > (9) Hee, 2 epeses he falue mes; dcaes he oal umbe of falues; s he sample sze; F( ) chaacezes he cumulave fuco fo he las falue me ; R( ) epeses he elably fuco fo he las falue me ; f( ) ae he desy fucos fo a cea he falue me ad,2,,. Wh f ( ) + L ( ; )! R exp, we wll have + ( ) e ( ) e () exp ad ( ) ( )

4 282 Dael I. De Souza J. ad R. Rocha The log lkelhood fuco wll be gve by: L ( )! l + ( ) l + ( ) l ( ) + ( ) l ( ) () To fd he values of ad ha maxmzes he log lkelhood fuco, we ake he ad devaves ad make hem equal o zeo. The, we wll have: d dl ( ) (2) d dl + ( ) l ( ) l l ( ) l (3) Fom Equao (2) we oba: ( ) + (4) Noce ha, whe, Equao (4) educes o he maxmum lkelhood esmao fo he vese expoeal dsbuo. Usg Equao (4) fo Equao (3) ad applyg some algeba, Equao (3) educes o: ( ) l + ( ) ( ) ( ) ( ) + + l l (5) Equao (5) mus be solved eavely. 4. The Sequeal Lfe Tesg The wo-paamee Ivese Webull dsbuo has a shape paamee whch specfes he shape of he dsbuo, ad a scale paamee, whch epeses he chaacesc lfe of he dsbuo. Boh paamees ae posve. The Ivese Webull desy fuco s gve by: ( ) f + exp ; ;, > (6) The hypohess esg suaos wee gve by Kapu ad Lambeso [5] ad by De Souza [6]:. Fo he scale paamee : H : ; H : < The pobably of accepg he ull hypohess H wll be se a (-α) f. Now, f whee <, he he pobably of accepg H wll be se a a low level γ. 2. Fo he shape paamee : H : ; H : < The pobably of accepg H wll be se aga a (-α) f. Now, f, whee <, he he pobably of accepg H wll also be se a a low level γ.

5 Applyg Eyg s Model o Tmes o Beakdow of Isulag Flud 283 Hee, H epeses he ull hypohess; H s he aleave hypohess; α dcaes he Type I (poduce) eo; γ s he Type II (cosume) eo; ad ae he paamees of he ull hypohess H ; ad ae he paamees of he aleave hypohess H. The developme of a sequeal es uses he lkelhood ao (LR) gve by he followg elaoshp descbed by Kapu ad Lambeso [5]: LR L ; /L ;. The sequeal pobably ao (SPR) wll be gve by: SPR L ; /L ;. Based o he pape by De Souza [3], fo he wo-paamee Ivese Webull model he (SPR) wll be: l ( γ) l < W < α l ( ) α + l (7) γ W + ( ) ( ) ( ) l ( ) (8) The coue ego wll become A< SPR < B, whee A γ /(-α) ad also B (- γ)/α. We wll accep he ull hypohess H f SPR B ad we wll ejec H f SPR A. Now, f A <SPR< B, we wll ake oe moe obsevao. 5. Expeced Sample Sze of a Sequeal Lfe Tesg Accodg o Mood ad Gaybll [7], a appoxmae expesso fo he expeced sample sze E() of a sequeal lfe esg wll be gve by: E E ( ) ( W * ) (9) E( w) f ( ;, ) w l (2) f ( ;o, o ) E ( W * ) P(,) l A + [ P(,) ] l B (2) Accodg o De Souza [3], fo he wo-paamee Ivese Webull samplg dsbuo, we wll have: E ( w) l + ( + ) El ( ) ( + ) El ( ) + (22) E ( ) E ( ) To fd he E[l()] some umecal egao pocedue (Smpso s /3 ule hs wok) wll have o be used. The soluo of each compoe of Equao (22) ca be foud De Souza [3]. 6. Example The daa s fom Nelso [8]. Table () below shows daa o me o beakdow of a sulag flud ecoded a hee dffee volages (4KV, 3KV ad 25 KV) o 2 asfomes. The opeag omal volage s 25KV. The es pupose wll have wo objecves: The fs pupose wll be o vefy f mes o beakdow of sulag flud

6 284 Dael I. De Souza J. ad R. Rocha bewee elecodes ecoded a hee dffee volages have a expoeal dsbuo as pedced by heoy. The secod pupose wll be o assess whehe o o he acceleaed model poposed by Eyg wll be able o aslae esuls fo he shape ad scale paamees of a udelyg Ivese Webull model, obaed ude wo acceleag usg codos, o expeced omal usg codo esuls fo hese wo paamees. The es was ucaed a 3KV afe 5,75 secods ad 25KV afe 4,29 secods. The fs hee falue mes a 4KV wee dscaded fo beg o sgfca (vey small). The sample sze of each volage was 2. Table : Tmes o beakdow of sulag flud wh cesog Tme o beakdow secods; KV Klovols. Volages 4KV 3KV 25KV Tme/sec 5. 2,5. Tme/sec 34. 4,56. Tme/sec 87. 2,553. Tme/sec ,29. Tme/sec 2.,448. * Tme/sec 25.,468. * Tme/sec 46. 2,29. * Tme/sec 56. 2,932. * Tme/sec 68. 4,38. * Tme/sec 9. 5,75. * Tme/sec 323. * * Tme/sec 47. * * Usg he maxmum lkelhood esmao appoach fo he scale ad shape paamees of he udelyg Ivese Webull samplg dsbuo fo cesoed Type II daa (falue cesoed), we oba he followg values fo hese paamees: Volage Shape Paamee Scale Paamee 4KV (9 Iems) KV ( Iems) KV (4 ems).35 4,436.7 Sce he values of he shape paamees fo he hee volages ae elavely close (.56 fo 4KV wh he aalyss of e falue mes;.69 fo 3KV wh he obsevao of e falue mes ad.35 fo 25KV wh he speco of oly fou falue mes), we ca assume a lea acceleao codo. Sce hese values fo he shape paamees ae o close o oe, we ca say ha he expoeal dsbuo ca o epese well he mes o beakdow of sulag flud bewee elecodes, as pedced by heoy. Now, usg he esul of he scale paamee of he udelyg Ivese Webull model obaed a 4KV ad 3KV sesses, we wll esmae he value of hs paamee ude omal volage usg codo (25KV). The, we wll compae he esmaed esul wh he oe obaed a omal esg codos. We wa o assess whehe o o he acceleaed model poposed by Eyg wll be able o aslae esmaed esuls fo he shape ad scale paamees of a udelyg Ivese Webull model o expeced omal usg codo esuls fo he sulag flud beg aalyzed. Theefoe, usg Equaos () o (7), we wll have he acceleao faco fo he scale paamee AF 2/. Ulzg Equao (7) we wll oba: AF Usg ow Equao (5), we ca esmae he em E/K. The:

7 Applyg Eyg s Model o Tmes o Beakdow of Isulag Flud 285 E l ( AF ) l( 8.69) [( 3) ( 4) ] 2/ K V V2 Applyg Equao (6), he acceleao faco fo he scale paamee o be appled a he omal sess volage AF 2/ wll be: E AF 2 / exp exp K V V Theefoe, he scale paamee of he compoe a omal opeag sess volage s esmaed o be: AF 2/ ,23.48 secods The peceage dffeece bewee he calculaed value of (4,436.7 secods) obaed wh he speco of oly fou falue mes ad he esmaed value of (4,23.48 secods) wll be: % Dffeece( calculaed esmaed) 4, , % The, we ca see ha he acceleaed model poposed by Eyg wll be able o aslae wh a cea degee of pecso, esuls fo he shape ad scale paamees of a udelyg Ivese Webull model, obaed ude wo acceleag usg codos, o expeced omal usg codo esuls fo hese wo paamees. To evaluae he accuacy (sgfcace) of he wo-paamee values esmaed ude omal codos fo he udelyg Ivese Webull model we wll employ, o he expeced omal falue mes, a sequeal lfe esg usg a ucao mechasm developed by De Souza [3]. These expeced omal falue mes wll be acqued by mulplyg he welve falue mes obaed ude acceleaed esg codos a 4 KV gve by Table (), by he acceleag faco AF of Table (2) shows hese expeced omal falue mes. Table 2: Tmes o beakdow of sulag flud wh cesog Tme o beakdow secods; KV Klovols. Volages 4KV Expeced Nomal Tme/sec Tme/sec 2. 2, Tme/sec 25. 4, Tme/sec 46. 8, Tme/sec 56., Tme/sec 68. 3, Tme/sec 9. 2,34.66 Tme/sec ,33.62 Tme/sec 47. 8,47.58 I was decded ha he value of α was.5 ad γ was.. I hs example, he followg values fo he aleave ad ull paamees wee chose: aleave scale paamee 4,8. secods ad aleave shape paamee.9; ull scale paamee 4,44. secods ad ull shape paamee.7. Now elecg P(,) o be., we ca calculae he expeced sample sze E() of hs sequeal lfe esg ude aalyss. Applyg Equaos (9) o (22), we wll have:

8 286 Dael I. De Souza J. ad R. Rocha E ( w) l + ( ) E[ l( )] E + E ( ) ( ) E , ( w) Now, wh A γ /(-α); B (-γ)/α; α.5; γ. ad also P(,)., we wll have: E ( W * ) P(,) l A + [ P(,) ] l B P E ( ) (,) l A + [ P(,) ] l B ems E( w).6399 So, we could make a decso abou accepg o ejecg he ull hypohess H afe he aalyss of obsevao umbe 5. Usg ow Equaos (7) ad (8) ad he e falue mes obaed ude acceleaed codos a 4 KV gve by Table (2), mulpled by he acceleag faco AF of , we calculae he sequeal lfe esg lms ,8 (.) l l , ,8 (.5) l + l ,44. W ( ) ( ) l The, we ge: < W < (23) The pocedue s defed by he followg ules:. If W , we wll accep H. 2. If W , we wll ejec H. 3. If < W < , we wll ake oe moe obsevao. Table (3) shows he esuls of hs es fo he Webull model case. + ( ) ( ) Table 3: Resuls fo he 34 KV Case Two-Paamee Samplg Ivese Webull Model U Numbe Lowe Lm Uppe Lm Value of W I hs case, eve afe he obsevao of 9 mes o falue, was o possble o make he decso o accep o ejec he ull hypohess H. Sce we could make a decso

9 Applyg Eyg s Model o Tmes o Beakdow of Isulag Flud 287 abou accepg o ejecg he ull hypohess H afe he aalyss of obsevao umbe 5, we wll oduce a pocedue fo ealy ucao. 7. A Pocedue fo Ealy Tucao Accodg o Kapu ad Lambeso [2], whe he ucao po s eached, a le paog he sequeal gaph ca be daw as show Fgue (). Ths le s daw hough he og of he gaph paallel o he accepace ad ejeco les. The decso o accep o ejec H smply depeds o whch sde of he le he fal oucome les. V A L U E O F W ACCEPT Ho REJECT Ho NUMBER OF ITEMS TRUNCATION POINT Fgue : Sequeal es gaph fo he wo-paamee Ivese Webull model Obvously hs pocedue chages he levels of α ad γ of he ogal es; howeve, accodg o Kapu ad Lambeso [5], he chage s slgh f he ucao po s o oo small (less ha hee). Fgue () above shows he sequeal es gaph developed fo hs example. As we ca see Fgue (), he ull hypohess H should be acceped sce he fal obsevao (obsevao umbe 5) les o he sde of he le elaed o he accepace of H. 8. Coclusos I hs pape we appled he Eyg`s model o mes o beakdow of sulag flud bewee elecodes. To esmae he paamees of he Ivese Webull model we used a maxmum lkelhood appoach fo cesoed falue daa, wh he lfe-esg emag a he mome he ucao po was eached. To evaluae he accuacy (sgfcace) of he wo-paamee values esmaed ude omal codos fo he udelyg Ivese Webull model we employed, o he expeced omal falue mes, a sequeal lfe esg usg a ucao mechasm developed by De Souza [3]. The shape paamee emaed he same whle he acceleaed scale paamee ad he acceleaed mmum lfe paamee wee mulpled by he acceleao faco. The equal shape paamee s a ecessay mahemacal cosequece of he ohe wo assumpos; ha s, assumg a lea acceleao model ad a wo-paamee Ivese Webull samplg dsbuo. If dffee sess levels yeld daa wh vey dffee shape paamees, he ehe he wopaamee Ivese Webull samplg dsbuo s he wog model fo he daa o we do o have a lea acceleao codo. Sce he obaed values of he shape paamees fo he hee volages ae elavely close (.56 fo 4KV wh he aalyss of e falue mes;.69 fo 3KV wh he obsevao of e falue mes ad.35 fo 25KV wh he speco of oly fou falue mes), we ca assume a lea acceleao codo. Sce hese values fo he shape paamees ae o close o oe, we ca say ha he expoeal dsbuo ca o

10 288 Dael I. De Souza J. ad R. Rocha epese well he mes o beakdow of sulag flud bewee elecodes, as pedced by heoy. The, we compaed he esmaed omal usg codo esuls fo he wo shape ad scale paamees wh he oes obaed by esg a he omal opeag sess volage (25KV). The peceage dffeece bewee he calculaed value of obaed wh he speco of oly fou falue mes (4,436.7 secods) ad he esmaed value of (4,23.48 secods) s oly 4.84%. Theefoe, we ca assume ha he acceleaed model poposed by Eyg, (whe he acceleag faco s oly he volage sess), s able o aslae wh a cea degee of pecso, esuls fo he shape ad scale paamees of a udelyg Ivese Webull model, obaed ude wo acceleag usg codos, o expeced omal usg codo esuls fo hese wo paamees. Refeeces [] De Souza, Dael I, Foseca, D. R ad Kppe, D. A. Combed Appoach Usg Acceleaed ad a Sequeal Lfe Tesg Appled o Seel Compoes Used Hghway Ovepasses Bazl, Rsk Aalyss 2 Cofeece, WIT Tasacos o Ifomao ad Commucao Techologes, Vol. 43, pp. 5-4, Souhampo ad Boso, 2. [2] Ebelg, C. E., Relably ad Maaably Egeeg. The McGaw-Hll Compaes, Ic., pp. 328, 997. [3] De Souza, Dael I. A Maxmum Lkelhood Appoach Appled o a Acceleaed Lfe Tesg wh a Udelyg Thee-Paamee Ivese Webull Model. I Raj B. K. Rao ad Davd U Mba Eds. COMADEM 25 Codo Moog ad Dagosc Egeeg Maageme Cofeece, Uvesy Pess,. Vol., pp , Cafeld, Bedfodshe, UK., 25. [4] De Souza, Dael I. ad Lambeso, Leoad R. Bayesa Webull Relably Esmao, IIE Tasacos 27 (3), pp. 3-32, 995. [5] Kapu, Kalash ad Leoad R Lambeso. Relably Egeeg Desg, New Yok, Joh Wley & Sos, Ic., 977. [6] De Souza, Dael I. Sequeal Lfe Tesg wh a Tucao Mechasm fo a Udelyg Webull Model, Towads a Safe Wold, ESREL 2 Cofeece, Zo, Demchela ad Pcc (eds), Too, Ialy, Vol. 3, pp , Polecco D Too, 6-2 Sepembe 2. [7] Mood, A. M., ad F. A. Gaybll. Ioduco o he Theoy of Sascs, Secod Edo, New Yok, pp , 963. [8] Nelso, W. B. Appled Lfe Daa Aalyss. Joh Wley & Sos, Ic., New Yok, pp. 24, 982. Dael I. De Souza J. s a Pofesso of Lfe Tesg a he Flumese Fedeal Uvesy Neó, Bazl, whee he has augh fo he las 37 yeas. I hs capacy he seved as he Gaduae Coodao of he Cvl Egeeg Depame ad as Reseach Deco of he School of Egeeg. He s also a vsg full pofesso a he Noh Flumese Sae Uvesy Campos, Bazl. He has bee hee mes a Uvesy of Floda, Gaesvlle, FL, USA, as a eseach schola, whee he augh each me he couse Idusal Qualy Cool. He also dd some eseach a Pesylvaa Sae Uvesy, USA. He eceved hs B.S. Idusal Meallugcal Egeeg fom Flumese Fedeal Uvesy Bazl, a M.S. Opeaos Reseach fom Floda Isue of Techology, USA, ad a Ph.D. Idusal Egeeg fom Waye Sae Uvesy, MI, USA. Hs eseach eess clude lfe esg ad Webull ad Ivese Webull elably esmao. Hs publcaos have appeaed IIE Tasacos, ASQC Tasacos, Elseve Scece Poceedgs, Balkema Poceedgs, WIT Tasacos, WASET Tasacos, Comadem Poceedgs (UK), ESREL Poceedgs ad seveal Bazla jouals.