Analytical and numerical analysis and simulation of heat transfer in electrical conductors and fuses

Transcription

1 Uvestät de Budesweh Mühe Fakultät fü Elektotehk ud Ifomatostehk Aaltal ad umeal aalss ad smulato of heat tasfe eletal odutos ad fuses Audus Ilgevus Vostzede des Pomotosausshusses: Pof. D.-Ig. K. Lades. Behtestatte: Pof. D.-Ig. H.-D. Leß. Behtestatte: Pof. Habl. D. R. Cegs 3. Behtestatte: Pof. D.-Ig. H. Dalhau ag de Püfug: Mt de Pomoto elagte akademshe Gad: Dokto-Igeeu (D.-Ig. eubbeg, de. ovembe 4 De Dssetato wude am be de Uvestät de Budesweh Mühe egeeht.

2 Cotets Lst of smbols Itoduto. Obetves of uet stud. Methodolog of uet eseah....3 Setf ovelt....4 Reseah appoval ad publatos... Phsal models of odutos ad the heat tasfe equatos. Ovevew.... Geomet of phsal models.3 Cosevatve fom of the heat tasfe equatos Flat ables Roud wes Elet fuses....4 Phsal mateal ostats....5 Detemato of heat tasfe oeffets Coveto oeffet fo the log hozotal lde.5. Coveto oeffet fo hozotal plate Exat mathematal expessos of the phsal ostats of a Radato.6 Bouda odtos.. 3 Aaltal aalss of heat tasfe a stead state 3. Calulato of the themo-eletal haatests of flat ables. 3.. Vetal heat tasfe wth tempeatue-depedet oeffets. 3.. Vetal heat tasfe wth tempeatue-depedet oeffets 3..3 Statoa soluto of vetal heat tasfe equato 3. Calulato of the themo-eletal haatests of oud wes Radal heat tasfe wth tempeatue-depedet oeffets Radal heat tasfe wth tempeatue-depedet oeffets Statoa soluto of adal heat tasfe equato Calulato of the themo-eletal haatests of fuses Axal heat tasfe wth tempeatue-depedet oeffets Axal heat tasfe wth tempeatue-depedet oeffets Avalahe effet metall oduto Statoa soluto fo axal heat tasfe equato umeal alulato of tempeatue behavou a taset state 4. Ovevew of the umeal methods used heat tasfe omputato Fudametals of the fte volume method (FVM o-lea heat tasfe model of eletal odutos Appoxmato of heat tasfe equatos b FVM Flat elet able Roud elet we Elet fuse umeal mplemetato of bouda odtos Flat elet able

3 Cotets Roud elet we Elet fuse Soluto of the equato sstem b ewto-raphso method Flat elet able Roud elet we Elet fuse Bas osdeatos of the expemet ad expemetal setup 5. Bas osdeatos of the expemet 5.. Det uet esstae vesus tempeatue measuemet Det uet vesus voltage measuemet Expemetal setup Detemato of the able oduto tempeatue oeffet. 5.. Detemato of the able oduto tempeatue Measug poess ad paamete aqusto Detemato of the able oduto tempeatue oeffet Detemato of the able oduto tempeatue Mathematal model valdato ad tepolato of the umeal esults 6. Mathematal model valdato. 6. Itepolato of the umeal esults to edue heat tasfe equatos Calulatos of the heat tasfe a mult-we budle 7. Coodate tasfomato of mult-we budle geomet Calulato of the heat tasfe the eal mult-we budle Summa ad outlook 8. Summa 8. Colusos Suggestos fo futue eseah. Appedx A Heat tasfe equatos fo elet odutos A. Heat tasfe equatos fo flat elet able A. Heat tasfe equatos fo oud elet we. A.3 Heat tasfe equatos fo a elet fuse elemet B umeal algothm applato fo heat tasfe smulato B. umeal heat tasfe smulato ad tepolato of the esults.. B. Calulato of themo-elet haatests b the polomal futos C Softwae fo measuemet data aqusto C. Algothm despto ad measuemet pogam C. Measuemet esults.. Bblogaph 35 Akolegmet

4 Lst of smbols A sufae aea, m A f oss seto aea of flat able, m A fu oss seto aea of the fuse, m a,b,,d polomal oeffets of polomal futo ( Eq..,.3 a,b, temedate vaables t-dagoal matx b wdth of flat able, m d thkes of flat able,m D mult-we budle damete, m E elet feld stegth, V/m E elet feld stegth at efeee tempeatue, V/m F fllg fato of mult-we budle f fllg fato of elet we oduto G heat odutae mult-we budle, W/mK G Gashof umbe g gavtatoal aeleato, m/s spatal dex umeal alulato I elet uet, A I omal elet uet of elet wes o ables, A J elet uet dest, A/m K K d,k, K,K, K 3 umbe of tme steps umeal alulato algothm temedate vaable of heat oveto equato (seto L haatest legth of the fuse elemet o able, m L legth of the we, m umbe of odes the umeal sheme veto sze the umeal algothm o umbe of tme ostats u usselt umbe P elet powe pe ut legth, W/m o pemete, m P Padtl umbe q heat flux, W/m q heat flux aused b the oveto, W/m q heat flux aused b the adato, W/m q v ate of eeg geeato pe ut volume, W/m 3 R ohm esstae, Ω Ra Ralegh umbe, lde adus, m,φ,z ldal oodates S thkess of sulato of mult-we budle, m tempeatue, C tempeatue dffeee, K ev evomet tempeatue, C s sufae tempeatue of the oduto, C absolute tempeatue, K t tme, s t g heatg-up tme, s u pemete of fuse elemet, m

5 v Lst of smbols x,,z etagula oodates, m W eeg ate, W W st stoed eeg the sold, W W out eeg eteg the sold, W W t eeg geeated the sold b the Joule losses, W W out eeg ate dsspated b the sold, W Geek Lettes α oveall heat oveto oeffet, W/m K α heat oveto oeffet, W/m K α adato oeffet α ρ,α lea tempeatue oeffet of oppe esstae, /K ß volumet themal expaso oeffet, /K ß ρ,ß squae tempeatue oeffet of oppe esstae, /K χ legth ostat, m Laplae opeato ε emssvt? heat odutvt oeffet, W/mK ν kemat vsost, m /s ρ dest, kg/m 3 ρ el spef esstvt, Ωm ρ spef esstvt at efeee tempeatue C, Ωm σ Stefa-Bolzma ostat φ azmuthal agle, ad γ spef heat apat pe volume, J/m 3 K τ tme ostat, s τ g heatg-up tme ostat, s τ tme ostat at I, s Subspts ave ev el f fu g v Supespts aveage oveto evomet elet flat able fuse heatg-up tme otato spatal odes otato umeal algothm adato, oud we volume fee-steam odtos * absolute tempeatue K tme dex the umeal algothm 4 tempeatue of fouth ode

6 CHAPER IRODUCIO hemo-eletal vestgatos of eletal odutos (wes, ables, fuses have bee desbed a geat vaet of applatos ad gaed easg atteto b a umbe of eseah woks [,,3,4]. he mao pat of these woks was devoted to the aalss of heat tasfe eletal odutos fo hgh voltage powe dstbuto sstems. Howeve, toda, powe suppl moble sstems lke aafts, shps o as have to be osdeed due to weght esttos. he ma dffeee betwee powe les ad wes fo moble applatos s the legth, whh does ot exeeds 8 m.e. the as. hs auses hghe uet dest that leads hghe voltage dop. oda, the mode moble vehles eletal ad eleto equpmet s of geat mpotae. Eletos s used fo the applatos lke eletomehaal dves (sevomotos, pumps as well as fo a odtoes ad safet equpmet. I the futue eve safet tal sstems the as mght be eplaed b so-alled x-b-we teholog [5,6], whee steeg, bakg, shftg ad thottle s pefomed b eletos. he eletos eplaes the mehaal sstems due to the followg easos: - to ease passege omfot, - to edue the weght of a vehle whle easg the e spae, - to ease safet, - to edue fuel osumpto ad osts Se, the powe osumes ae dstbuted ove the whole vehle, the powe must be delveed to the osumes b eletal wes. Wth easg umbe of osumes, the amout of wes ad the we sze ses also. Se the spae moble sstems s lmted ad weght s alwas beg edued, we oduto szes must be kept as small as possble. heefoe, t s eessa to vestgate heat tasfe eletal odutos ode to be able to alulate optmal oduto oss-seto fo log lastg load. hs fomato a be obtaed fom the uet-tempeatue ( stead state haatest of eah we. It s also mpotat to osde uet-tme ( taset-state haatest of wes vesus fuses. hs fomato s mpotat fo the fuse desg, whose uet-tme haatest should math we uet-tme haatest ode to potet the we elable agast oveload ad shot-ut uets. he ma developmet the feld of heat tasfe omputato elet powe ables was made b the wok of ehe ad MGath [6] publshed 957. Late, thee wee a umbe of publatos publshed as IEEE tasatos. I 997 based o IEEE tasatos Geoge J. Ades publshed the fst book [7], whh s the ol devoted solel to the fudametal theo ad pate of omputg the maxmum uet a powe able

7 Chapte. Itoduto a a wthout oveheatg. Almost all efeees to setf atles ad books of heat tasfe aalss elet ables ae summazed ths book. Howeve, lteatue [7] s ol devoted to the heat tasfe omputatos fo tasmsso, dstbuto, ad dustal applatos. he poblem dealg wth moble sstems, s ot oveed b the book. he ma dffeee betwee the elet ables used dustal applatos ad moble sstems s that the latte have geeall shote legths ad muh hghe opeatg tempeatue ages. he fst attempt to develop a theo of heat tasfe alulato elet odutos fo moble applatos was made b. Shulz [8]. I hs dssetato, the stead-state heat tasfe equatos of elet odutos have bee solved aaltall wth some smplfatos. hs s suffet to elaboate tede. Fo moe pese alulatos, howeve, umeal methods should be appled. I addto to ths, thee s also a eed fo the mathematal elatoshps of themoeletal haatests fo ompute aded desg pogam. he peset avalable ompute smulato pogams fo heat tasfe lke CableCad o Ass [9,] ae too omplex, use pue umeal methods equg spef kowledge, ad ae ot spea l- zed fo heat tasfe alulato elet ables ad fuses. O the ota, the mplemetato of a smple mathematal model to a ompute pogam, would allow the developmet of a ve tme-effet able desg tool. All ths shows, that thee s a equemet to vestgate the heat tasfe eletal odutos ad to develop effet algothms fo the alulato of the themoeletal haatests. I ths stud, effet algothms meas, that all haatests of odutos should be desbed b smple mathematal futos. Oe of the possble was to solve ths poblem s to ombe aaltal ad umeal aalss methods.. Obetves of uet stud he am of the peset eseah s to aalse heat tasfe of oe-dmesoal elet oduto models ad to develop a smplfed alulato methodolog of themoeletal haatests fo ompute aded elet able desg algothms. I ode to aheve ths goal the followg poblems must be solved: o eate oe-dmesoal mathematal model of elet odutos fo alulato of themo-eletal haatests of eletal ables ad fuses; o aalze stead-state heat tasfe b solvg patal dffeetal equatos aaltall; o alulate stead / taset state haatests usg a oe-dmesoal umeal model; o vef the obtaed umeal model b expemetal data; o develop a smplfed alulato methodolog of elet oduto haatests b fttg eale obtaed umeal esults wth polomal futos.

8 . Methodolog of uet eseah 3. Methodolog of uet eseah hs eseah wok pesets a oe-dmesoal (-D aaltal ad umeal model to smulate a heat tasfe flat ables, ldal wes ad elet fuses sepaatel. Heat dsspato due to fee oveto ad adato to a Heat dsspato due to fee oveto ad adato to a Coppe we PVC sulato a b Heat dsspato due to fee oveto ad adato to a Heat dfuso due to heat oduto to the we Heat dfuso due to heat oduto to the we Heat dsspato due to fee oveto ad adato to a Fg.. Heat dsspato b fee oveto elet oduto models: a - flat able, b - oud we, - fuse I ths stud both appoahes.e. aaltal ad umeal, ae used fo the aalss of heat tasfe. Aaltal solutos wee used to obta stead-state tempeatues fo leased oduto models. he leasato was doe although o-lea heat tasfe models would be appeable. he expemetal data have show that leased model have qute a good ageemet wth expemetal data. umeal model was appled fo taset-state tempeatue alulatos osdeg a o-lea heat tasfe model. he heat tasfe eletal sstems (ables ad fuses (Fgue. s obvousl of two - o thee -dmesoal atue (-D o 3-D. he heat tasfe ous due to the heat

9 4 Chapte. Itoduto dffuso fom fuse to the we o to the fuse holde; also the heat s dsspated fom the sufaes of odutos to the ambet due to tempeatue dffeees. Howeve, due to the omplext of the umeal model ad lage tme sale of heat tasfe poesses the ables t s ot omputatoall effet to use thee-dmesoal models to smulate heat tasfe eletal sstems. he CPU tme fo smulatg the same phsal sstem usg two- o thee -dmesoal models s sgfatl loge tha equed b a smplfed -D model. I eatg a mathematal model of flat ables (Fgue.a we egad heat tasfe ol deto (see thee-dmesoal dawg whle sde effets ae eglgble. Bouda odtos ae smmetal ad ovetve-adatve. Hee, oveto s assumed ufoed ad lama. Flat able has sulato/oduto/sulato lae sequee, whee the sulato s PolVlChlode (PVC ad the oduto s oppe. Isulato lae s desbed b heat odutvt, spef heat apat ad heat dsspato oeffet. Coduto lae s heated wth ufom volumet heat, geeated b eletal uet. I the ase of ldal wes (Fgue.b, the 3-D poblem s edued to -D egadg ol adal heat tasfe ad as fte legth of the we. he same mateal popetes ad bouda odtos appl as fo flat able. he fuse model (. a also be osdeed as a ldal oduto, ol wth fte legth ad wthout sulato. he model s also edued to a -D model egletg adal heat tasfe, beause the fuse elemet has ve hgh heat odutvt. Se the fuse elemet has fte legth, axal heat tasfe s modelled wth pesbed tempeatues o the boudaes (,t ad (L,t. hese tempeatues ae kow fom we tempeatues detemed eale. Due to the o-lea behavou of mateal popetes wth espet to the tempeatue, a umeal algothm had to be appled. A fte volume (FV method was used to appoxmate patal devatves of heat tasfe equato. he obtaed sstem of olea algeba equatos was solved b teatve ewto-raphso method ode to fd odal ukows of tempeatues the odutos. he fal step of ths wok was the evaluato of umeal smulato esults b the polomal fttg poedue usg the least squae (LS algothm. A umbe of mathematal methods have bee poposed [,,,3,4] fo the aalss of heat tasfe eletal odutos. Usuall these methods ae pue-aaltal o umeal. Aaltal methods ae eas to hadle, phsall meagful but of lmted applato fo omplated models (o-lea, o-homogeous ad bouda odtos. A umeal appoah eables us to mplemet moe ealst bouda odtos, whh a be appled to omplated geometes. I ode to udestad phsal meag of the esults eeved fom the umeal smulato, alulato esults have to be desbed b smple mathematal equatos wth as small a umbe of ukow vaables as possble. heefoe, themo-eletal haatests of eletal odutos ae aalsed b polomal o logathmal futos. he seod easo of devato of smplfed equatos s to mplemet these fomulas to ompute tool, whee a ve good tme-effe a be aheved.

10 .3. Setf ovelt 5.3 Setf ovelt he speal setf otbuto of ths wok s the patula wa to ombe aalt - al ad umeal methods to alulate the themal behavou of eletal odutos. he poposed algothm s based o the followg steps:. Aaltal devato of the heat tasfe equatos.. Aaltal soluto of the obtaed dffeetal equatos wth mal tempeatue depedet o lea depedet phsal ostats. 3. Smplfato of the obtaed aaltal soluto to edue the umbe of vaables. 4. umeal appoxmato of the heat tasfe equatos wth o- lea tempeatue depedet phsal ostats. 5. Model valdato of the umeal esults b expemetal data. 6. Itepolato (fttg of the eeved umeal esults wth the smplfed equatos deved fom the aaltal soluto of the heat tasfe equatos. 7. Evaluato of the esults to eeve a lmted amout of depedet ostats (e.g. tempeatue to desbe the themal-eletal haatests wth suffet aua. I ths stud, fo the fst tme, a methodolog of heat tasfe aalss elet sstems fo moble applatos has bee fomulated. It s show that t s possble to desbe ma themo-eletal haatests b smplfed quas-aaltal futos, whh ae vald fo oe patula oduto tpe. Obtaed themo-eletal haatests of eletal odutos ae: - themo eletal haatest (I : ( I I a I b - heatg-up tme haatest t g (I : t I I I ( I > I τ l I (. (. - tme ostat haatest τ(i :

11 6 Chapte. Itoduto τ τ.5 I I d I (.3 Havg the elatoshp betwee the oduto tempeatue ad eletal uet (Eq.., voltage dop the oduto a be alulated as followg: - voltage dop pe legth haatest E(I : hee: Iρ E A Iρ( α ρ β ρ ( A oduto tempeatue dffeee agast evomet K I uet A I omal uet A a,b,,d ostats t heatg up tme s τ omal tme ostat s τ uet depedet tme ostat s τ I tme ostat at zeo uet s E voltage dop pe legth V/m ρ spef esstae (esstvt Ωm ρ spef esstae at efeee tempeatue (e.g. C Ωm α ρ lea tempeatue oeffet of the spef esstae /K β ρ squae tempeatue oeffet of the spef esstae /K A oduto oss setoal aea m (.4 I ths wok a algothm s poposed to desbe themo-eletal haatests wth the smplfed equatos (see above.-.4, whh wee obtaed fom aaltal ad umeal models. hs algothm s suted fo mplemetato the ompute aded able desg pogam. Based o the poposed algothm to alulate themo-eletal haatests a ompute pogam to desg eletal sstems as has bee wtte [5]..4 Reseah appoval ad publatos Ceated methodolog ad algothms, whh have bee developed to alulate themoeletal haatests of eletal ables fo a applatos wee mplemeted b able haess maufatue Leo Bodetzssteme GmbH ad DamleChsle AG. he bas ahevemets of peset eseah have bee peseted at the followg teatoal ofeees:

12 .4. Reseah appoval ad publatos 7 - he 7 th Iteatoal Cofeee Eletos 3 Kauas, Lthuaa, 3; - he 8 th Iteatoal Cofeee Mathematal Modellg ad Aalss aka, Lthuaa, 3 he otet of the dssetato ludes thee setf publatos: the two papes ae publshed the oual Mathematal Modellg ad Aalss ad oe publato Eletos ad Eletal Egeeg. Both ouals ae edted Lthuaa b a teatoal edtoal boad.

13 CHAPER PHYSICAL MODELS OF CODUCORS AD HEIR HEA RASFER EQUAIOS. Ovevew Befoe the dsusso of the theoetal model, a shot gude wll be peseted at fst. hs gudae s teded to show osel what steps the heat tasfe equatos ae gog to be developed. It wll also be dsussed how these equatos ae solved fo able atg poblems. Afte a shot toduto to the model geomet, heat tasfe equatos of dffeet model geometes wll be deved. hese equatos desbe the tempeatue behavou eletal odutos ad fuses. As a ext step, the heat oveto ad adato oeffets wll be detemed. he heat ovetve oeffet s peseted fo ldal ad hozotal sufaes. Beause of ts oleat, ths oeffet wll have to be leazed fo the late aaltal aalss of the heat equato. Followg ths, the ma phsal mateal paametes of the heat equato wll be osdeed. Beause of ts o-leat (e.g. heat odutvt ad eletal esstae ealt, eta smplfatos have to be todued. It wll be show that these smplfatos a be toleated fo the themal aalss of the eletal oduto ad do ot estt the valdt of the smplfed themal oduto model the tempeatue age of teest. Fall, equed bouda odtos wll be todued. he have to be leazed ode to mplemet them to a aaltal soluto of the heat equato. Wth these pepaatos, t wll be possble to vestgate the themo-eletal haatests of odutos ad alulate the atgs.

14 Chapte. Phsal models of odutos ad the heat tasfe equatos. Geomet of phsal models O the bass of eletal odutos, thee dffeet models wll be osdeed: flat sulated able, oud sulated we, ad eletal fuse. hese thee dffeet tpes of odutos ove the ma pat of powe suppl sstem ma applatos. I the flat able model, the tem able s used beause t has moe tha oe we. All models ae oe dmesoal sstems, beause the othe dmesos all ases vash due to lage dffeee betwee oss-setos (fo a oud we o fuse o thkess (fo flat able ad legth of the odutos. A. he flat able model (Fg..,a s edued to oe-dmesoal heat oduto, wheeb spatal devatves wth espet to x ad z ae egleted: ( x (... z (.... he eduto of the model s possble beause of fte legth of the able L ad muh bgge wdth b ompaed to the thkess d. Due to lateal smmet of ths model, t s suffet to aalse the uppe pat of the flat able ol. he model ossts of thee laes ad a be exteded depedg o the flat able stutue. Fom bottom to top the fgue (.,a we have: Polvlhlode (PVC sulato Metall odutos (pue oppe Polvlhlode (PVC sulato Fo the sake of smplt, the odutos (the mddle lae ae osdeed as a homogeeous oduto lae. B. I oud we model (Fg.., b all spatal devatves of the heat equato vash wth espet to x ad ϕ: ( x (... ϕ (.... he heat oduto the axal deto s egleted, beause omall the legth of the we s muh lage tha ts aea, theefoe, the bouda effets a be egleted. he agula dmeso ϕ s also egleted due to otatoal smmet of the oduto ad sulato lae. he whole model ossts of two laes ad a be exteded to moe laes, depedg o the we ostuto. I ths model, we have: Metall oduto (98% oppe

15 . Geomet of phsal models Polvlhlode (PVC sulato (x,,z,t~(,t ~ PVC sulato z b d Coppe oduto x d a

16 Chapte. Phsal models of odutos ad the heat tasfe equatos x q v (,x,t~(,t ~ Volumet heat geeato PVC sulato s Coppe we,effet s s ( od s ev Evomet Isulato Coduto Isulato ev m Evomet b

17 .3 Cosevatve fom of the heat tasfe equatos 3 We Fuse holde Fuse holde elemet Fuse elemet elemet We Max.empeatue max empeatue x empeatue We Fuse holde elemet Fuse elemet Fuse holde elemet We x Fg.. Model geometes ad heat oduto paametes: a flat able, b oud we, elet fuse he metall oduto s assumed homogeeous ad a pefet lde. I ealt, the oe of we s made of a umbe of sgle odutos wth small a gaps betwee. If sgle odutos ae aaged smmetall, the the we has a hexagoal shape. C. he eletal fuse model s oe dmesoal (Fg.., wth the heat oduto ol alog the x axs. he heat oduto deto s ot osdeed beause of ve hgh heat odutvt of oppe ompaed to the heat oveto fom the sufae. he shape of the fuse model x deto s o-homogeeous. he whole model ossts of oe lae oppe, bas o a othe allo..3 Cosevatve fom of the heat tasfe equatos I ode to alulate heat dsspato (heat oduto, oveto ad adato, the elevat heat tasfe equatos have to be solved. hese equatos defe the elatoshp betwee the heat geeated b eletal uet metall oduto, ad the tempeatue dstbuto wth the we o able (oduto ad sulato ad ts suoudgs.

18 4 Chapte. Phsal models of odutos ad the heat tasfe equatos he aalss of heat tasfe s goveed b the law of osevato of eeg. We wll fomulate ths law o a eeg ate bass; whh meas, that at a stat, thee must be a balae betwee all powe ates, as measued Joules ( Ws. he eeg osevato law a be wtte followg fom: W W t W W (. et st out whee: W et s the ate of eeg eteg the eletal oduto. hs eeg ma be geeated b othe ables o wes loated the vt of othe ables o b sola eeg, W t s the ate of heat geeated teall b Joule losses, W st s the ate of eeg stoed wth the able, W out s the ate of eeg whh s dsspated b oduto, oveto, ad adato. he flow ad outflow tems W et ad W out ae sufae pheomea, ad these ates ae popotoal to the sufae aea. he themal eeg geeato ate W t s assoated wth the ate of oveso of eletal eeg to themal eeg ad s popotoal to the volume. he eeg stoage s also a volumet pheomea, but t s smpl assoated wth a ease (W st > o deease (W st < the eeg of able. Ude stead-state odtos, thee s, of ouse, o hage eeg stoage (W st. A detaled devato of the heat tasfe equato s gve Appedx A. Fom the Equato (., (see also Appedx A geeal fom of the heat tasfe equato osevatve fom Catesa (. ad ldal (.a oodates s obtaed as follows []: λ λ λ qv γρ (. x x z z t x λ x λ λ φ φ q V γρ t (.a hee:? heat odutvt W/mK q V volumet heat geeato W/m 3 he osevatve fom s a fom of heat oduto equato whee spae depedat themal odutvt o othe oeffets emas oseved wth dffeet meda of mateals. he osevatve fom s gve as λ x x γ ad the oosevatve fom as λ t x γ λ x x

19 .3 Cosevatve fom of the heat tasfe equatos 5 γ spef heat apat W/kgK ρ dest kg/m 3 he heat equatos (.,.a ae the bass fo futue heat tasfe aalss eletal odutos..3. Flat ables he heat tasfe equato (. fo flat able (Fg.., a, whh s deved ( Appedx A. s smplfed fo oe-dmeso as follows: (, t (, t λ (, γ (, ρ qv (, (.3 t As metoed the Chapte.3, ths model t s osdeed mddle smmet (Fg... hs assumpto s allowed beause heat oveto ad adato fom top sde of the able sufae has almost the same heat dsspato ate as fom the bottom sde of the able. It s mpotat to emphasze, that the fee oveto a stuato s osdeed. he able s plaed hozotal the a. I ode to smplf the model, the metall oduto s teated as a homogeeous bod aoss the able wdth d (see Fg..,a. Hee, the heat odutvt oeffet λ s spae depedat, due to dffeet mateal laes the we. he spef heat apat tem γ s a o-lea futo of tempeatue fo oppe ad PVC sulato. he heat geeato b eletal uet s expessed as q v tem ad s alled volumet spef heat flux. It s a lea futo of tempeatue metall oduto ad vashes PVC sulato. Isulato Metall oduto x Fg.. Flat able model wth homogeeous metall oduto

20 6 Chapte. Phsal models of odutos ad the heat tasfe equatos Hee, the equato (.3, volumet heat flux s expessed as: q V dl I dq dr I el ρj dv A dl ρ A dl I ρ A [ α ( ] (.4 hee ρ el spef esstae of the metall oduto gve b [ α ( ] ρ ρ Om, el ρ spef esstae of the oduto at C tempeatue α oppe tempeatue oeffet at C /K (α /K l legth of the able m J uet dest A/m I deotes uet though the we A A aea of metall oduto m..3. Roud wes Heat tasfe oud wes s detemed, pple, b the same equato as (.3, heat tasfe adal deto must also be osdeed. he geeal fom of heat equato ldal oodates s: λ(, γ (, ρ qv (, t λ(, φ φ λ(, x x (.5 akg to aout the model smplfatos gve eale (see Fg.,b, the heat equato s edued to the oe-dmesoal fom (see also Appedx A.: (, t (, t λ (, γ (, ρ qv (, (.6 t he tempeatue pofle flat ables ad oud wes show Fgue (.,a,b ude assumpto, that the tempeatue gadet a metall oduto s ve small due to ts ve hgh heat odutvt. I the sulato the tempeatue gadet s muh lage. he ma tempeatue dop, howeve, s betwee the we sufae ad evomet. hs tempeatue dop s aused b oveto ad desbed b heat oveto oeff-

21 .4 Phsal mateal ostats 7 et α. heefoe, hee t s ve mpotat to deteme ths oeffet oetl. hs poblem wll be dsussed the seto Elet fuses he followg dffeetal equato fo the heat tasfe the fuse elemet s gve (Appedx, A.3: ( x, t A( x λ x x ( x, t γ( ρa( x A( x q t 4 4 [ α ( α ( ( x, t ] V ( ev u (.7 hee: A oss seto aea of the fuse elemet m α, α oveto ad adato oeffets espetvel u umfeee m ( x, t K ev Aodg to the model (Fg..,, the heat tasfe should be aalsed ol the x deto, beause of the shot legths of the fuse meltg elemet. he mathematal model of fuse elemet should alulate meltg tempeatue of the fuse. Hee, adal heat oduto a be egleted due to hgh heat odutvt of the fuse mateal. I equato (.8 the heat flux q V s deved the same wa as equato (.5. I addto to ths, the equato s vald also fo a vaable oss setoal aea..4 Phsal mateal ostats Heat tasfe equato gve seto.3 depeds o the spef esstae, heat odutvt ad the heat apat of the oduto mateal. All thee values ae tempeatue depedet, howeve the values ae ol kow fo eta tempeatues. I ode to tepolate betwee these gve values, a lea o squae futo has to be used to desbe the elatoshp. hs estmato s ve mpotat ode to model the heat tasfe qualtatvel. Dffeet alulato peso tea ae defed fo the aaltal appoah ad fo the umeal appoah. Fo the aaltal appoah t s eessa to have tempeatue depedet o lea depedet ostats. he umeal appoah of the heat tasfe model allows moe pese tempeatue alulato the odutos. Hee, o-lea futos a be mplemeted fo the despto of the mateal ostats.

22 8 Chapte. Phsal models of odutos ad the heat tasfe equatos he followg dagams show the exat gaphal ad umeal oeffets of the spef esstae, ρ, of oppe, of the heat odutvt,?, of pue oppe ad PVC, ad of the spef heat apat, γ, of pue oppe ad PVC [6]. he tempeatue age the dagams s ve wde, although ths wok ol tempeatue up to C has bee osdeed. he easo of ths hgh tempeatue age the hats s to show the ovevew how the oeffets behave wth wde tempeatue age. Lea ad o-lea appoxmato has bee made usg the avalable data. 5,5E-8 5,E-8 Cu Spef esstvt ρ Ωm 4,5E-8 4,E-8 3,5E-8 3,E-8,5E-8,E-8,5E Absolute tempeatue K a 45 hemal odutvt λ W/mK Cu Absolute tempeatue K b

23 .4 Phsal mateal ostats 9 Spef heat γ J/kgK Cu Absolute tempeatue K Fg..3 Values of: (a spef esstae, (b themal odutvt ad ( spef heat apat of pue oppe Heat odutvt ad spef heat apat values of PVC: ame of mateal DI ode empeatue C 5 hemal heat odutvt? W/Km Polvlhlode PVC Spef heat apat γ J/kgK Polvlhlode PVC ab.. Values of themal odutvt ad heat apat of PVC Appoxmato of the tempeatue depedet oppe ad PVC mateal oeffets: a Spef esstae of oppe : ρ [ α ( ( ] ρ β ρ ρ hee: ρ spef esstae at C, ρ Om α ρ lea tempeatue oeffet, α ρ /K ß ρ squae tempeatue oeffet. ß ρ /K tempeatue of the oduto C

24 Chapte. Phsal models of odutos ad the heat tasfe equatos efeee tempeatue. I ths stud efeee tempeatue odes wth evomet tempeatue ev. b spef heat apat of oppe g: γ γ, C α γ hee: γ heat apat at C efeee tempeatue, γ 38 J/kgK α γ appoxmated lea tempeatue oeffet of heat apat /K α γ.7 /K spef heat apat of PVC g: γ γ C α γ βγ hee: γ heat apat at C efeee tempeatue, γ 9 J/kgK α γ appoxmated lea tempeatue oeffet of heat apat /K α γ.3 /K ß γ appoxmated squae tempeatue oeffet of heat apat /K ß γ.74 /K.5 Detemato of heat tasfe oeffets he heat tasfe fom the sufae s goveed b oveto ad adato. hs effet a be desbed b the oespodg oveto ad adato heat tasfe oeffets. Both deped o the sufae ad evomet tempeatues. Coveto takes plae betwee the bouda sufae ad a heat taspot b a flud (e.g. a moto at a dffeet tempeatue. Radato ous b eletomaget wave heat exhage betwee the sufae ad ts suoudg evomet sepaated b a. I ths wok the ovetve heat tasfe oeffet of lama flow has to be examed fo the followg two dffeet model geometes: - hozotal lde sufaes - hozotal plate sufaes he esult of ths examato leads to two dffeet heat tasfe oeffets vald fo oud ad fo plate sufaes. he oveto ad adato oeffet appeas the bouda odtos of the heat tasfe equatos fo the eletal oduto models. At the lowe tempeatues, whh ae tpal fo elet able applatos, oveto s the bas heat dsspato ompoet (a. 9%.

25 .5 Detemato of heat tasfe oeffets I ths wok, the heat tasfe eletal odutos s omputed b a aaltal alulato (of the heat oduto equatos the stead state egme ad b a umeal algothm a taset state egme. heefoe, the oveto ad adato oeffets fo the aaltal soluto has to be leazed ad to be peseted a appoxmated fom ode to obta smple but suffetl auate equatos of the oveto ad adato oeffets. Fo the umeal algothm the oeffets wll be deved a o-lea fom se both ae o-lea (tempeatue depedet..5. Coveto oeffet fo the log hozotal ldes he mal appled oud geomet has bee studed extesvel. Ma oelatos exst betwee the dffeet alulato methods. he lteatue [] pesets smple algothms fo the alulato of ovetve oeffets of the ldes. hs wok fo l- lows the poedue poposed b [7], whee ma appoahes of the vaous poedues ae summased. he equatos of ths poedue wee valdated b the expemetal data the dploma wok [8]. All otatos of phsal ostats ad mateal popetes wll be used fom the woks [7, 8]. I geeal, the heat dsspato b oveto s defed as: q ( α s (.8 hee: s sufae tempeatue of the sold C, ev 73.5 the absolute tempeatue of the flud K. he oveto oeffet α a be alulated as follows: λ α u (.9 d hee: λ heat oduto of a W/m K, u usselt umbe d damete of lde m. he usselt umbe fo a hozotal lde aodg to Wämeatlas (Heat asfe Atlas [7] s expessed b: 6.387Ra u.75 8 ( P I ths equato the Ralegh umbe Ra s alulated as:

26 Chapte. Phsal models of odutos ad the heat tasfe equatos Ra G P (. Hee: P G Padtl umbe (see ab.. ad Gashof umbe defed b the followg equato: G 3 gd β ( v, (. hee: g gavtatoal aeleato m/s, ß volumet themal expaso oeffet /K, ν kemat vsost (m /s. he ß oeffet fo deal gas wth ustfable eo a be osdeed as: β (.3 whee ev the absolute tempeatue of the flud ( K he mateal ostats λ, ν ad P of a ae take fom Heat asfe Atlas [7]. hese ostats ae depedet o the aveage tempeatue ave : ave ( s ev (.4 hee s s tempeatue of the sufae of lde ( C ad ev evomet tempeatue ( C. Wth the equatos (.9 ad (., the oveto oeffet α s wtte as follows: 6 λ.387ra α.75 8 (.5 d P Replag the equato (.5 the Ralegh umbe Ra, the Padl umbe P ad heat odutvt λ leads to the followg fom, whh s ol damete d ad tempeatue dffeee depedat: α Kd K d ( 6 (.6

27 .5 Detemato of heat tasfe oeffets 3 whee: K d. 75λ, (.7 ad K.387λ.559 P P gβ v 6 (.8 he phsal ostats of a.e. (heat odutvt λ, kemat vsost ν ad the Padtl umbe P a be foud the lteatue [7]. Fo the volumet themal expaso oeffet ß, a s osdeed as a deal gas. Fo efeee, evomet tempeatue s take. I the table. K d ad K values fo a tempeatue age fom to 4 C ae gve. Sufae tempeatue e mpe atue Heat odut v- t l -3 Kemat vsost -6 m /s Padtl umbe ave W/mK P C C Aveage: ab.. Phsal ostats of a fo tempeatue fom to 4 C K d K he aveaged fom of the ovetve oeffet fo tempeatue age fom to 4 C s followg: α d ( 6 (.9.5. Coveto oeffet fo hozotal plates Fo the applato fo flat ables the fee oveto of hozotal plates has bee osdeed as well. Fo ths geomet, we have to dstgush betwee the oveto fom the top sde of the plate sufae ad the bottom sde.

28 4 Chapte. Phsal models of odutos ad the heat tasfe equatos he oveto oeffet α s alulated smla to equato (.9: λ α u (. l hee l s haatest legth, whh s defed as: A l, P whee A ad P ae the plate sufae ad pemete, espetvel. A. he usselt umbe fo the uppe sde of hozotal plate aodg to Wämeatlas (Heat asfe Atlas [7] s expessed b: a. Fo lama flow: u Ra P, (. hee:.3 4 Ra 7. P b. Fo tubulet flow: u Ra P (. hee:.3 4 Ra 7. P B. he usselt umbe fo the lowe sde of a hozotal plate has the followg fom (ol lama oveto:

29 .5 Detemato of heat tasfe oeffets u Ra P (.3 hee.49 P < Ra < All the equatos (.,.3 ad usselt umbes gve (.,.,.3 ae seted to equato (.. hs leads to the followg fom of the oveto oeffets: A. Uppe sde a. Lama flow: hee: K α K 8 3 l ( 5 5 (.4 P gβ λ P (.5 v b. ubulet flow: hee: K α ( 3 K l 3 (.6 P.3.5 gβ λ P (.7 v B. Lowe sde (lama flow ol: hee: α K l ( 5 (.8

30 6 Chapte. Phsal models of odutos ad the heat tasfe equatos K P.3.6 gβ λ P (.9 v.5.3 Exat mathematal expessos of the phsal ostats of a he phsal ostats of a deped ve muh o tempeatue. hese futos ae of hghe polomal ode, whh wee obtaed b fttg of the gve esults the Wämeatlas [7]. Wth these futos, a ve hgh aua of oveto oeffet a be aheved ad the futo a easl be mplemeted to the ompute pogam. Hee, the wde tempeatue age s used ode to expad the valdt age of tempeatue depedet ostats the ompute pogam. A. Heat odutvt a l ( ave : empeatue age fo the fttg poedue: C ave C Obtaed polomal futo b fttg: λ( ave ave ave ave ave (.3 4

31 .5 Detemato of heat tasfe oeffets Data aodg to Heat Atlas [7] Empal futo (Polom 4.Gade Heat odutvt of a λ -3 Wm - K empeatue ave C Fg..4 Heat odutvt of a as a futo of tempeatue at ostat pessue P 5 Pa B. Kemat vsost ( ave : empeatue age fo the fttg poedue: C ave C Obtaed polomal futo b fttg: ν ( ave ave.47 ave ave (.3 4 ave

32 8 Chapte. Phsal models of odutos ad the heat tasfe equatos Data aodg to Heat Atlas [7] Empal futo (Polom 4. Gade Kemat vsost ν -7 m s empeatue ave C Fg..5 Kemat vsost of a as a futo of tempeatue at os tat pessue P 5 Pa C. Padtl umbe P( ave empeatue age fo the fttg poedue: 5 C ave 65 C Obtaed polomal futo b fttg: P( ave ave ave ave ( ave

33 .5 Detemato of heat tasfe oeffets Data aodg to Heat Atlas [7] Empal fukto (Polom 4.Gade fo empeatue age - C < ave < 65 C Padtl-umbe P C 65 C empeatue ave C Fg..6 Padtl-umbe of a as a futo of tempeatue at ostat pessue P 5 Pa.5.4 Radato I ode to desbe heat tasfe b the themal adato eletal odutos, the exhage of adato eeg betwee the sulated oduto sufae ad the ftel lage evomet s osdeed. It ma ou ot ol fom sold sufaes but also fom lquds ad gases []. he eeg of the adato s taspoted b eletomaget waves (o alteatvel, photos. Whle the tasfe of eeg b oduto o oveto eques the pesee of a mateal medum, adato does ot. I fat, adato tasfe ous most effetl a vauum. he omplete eletomaget spetum s show Fgue.7. he shot wavelegth gamma as, X as ad ultavolet (UV adato ae pmal of teest to the hgh eeg phsst ad ulea egee, whle the log wavelegth mowaves ad ado waves ae of oe to the eletal egees. It s the temedate poto of the spetum, whh exteds fom appoxmatel. to µm. It ludes a pat of the UV ad all of the vsble faed (IR, that s alled themal adato ad belogs to heat tasfe.

34 3 Chapte. Phsal models of odutos ad th e heat tasfe equatos Vsble Volet Blue Gee Yellow Red Gamma as X as Ultavolet Ifaed Mowave hemal adato λ, µm Fg..7 Spetum of eletomaget adato he maxmum flux (W/m at whh adato ma be emtted fom a sufae s gve b the Stefa-Boltzma law: q σ (.33 4 s whee S s the absolute tempeatue (K of the sufae ad σ s the Stefa-Boltzma 8 4 ostat ( σ 5.67 W / m K. Suh a sufae s alled a deal adato o blak bod. he heat flux emtted b a eal sufae s less tha that of the deal adato ad s gve b q εσ (.34 4 s whee ε s a adatve popet of the sufae alled the emssvt. hs popet dates how effetl the sufae emts ompaed to a deal adato. he ate of heat exhage betwee the able sufae ad ts suoudgs, expessed pe ut aea of the sufae, s: q 4 4 ( εσ (.35 s ev I ode make t ompatble wth heat oveto, t s oveet to expess the adato heat exhage the fom: q ( α (.36 s ev

35 .6 Bouda odtos 3 whee fom Equato (.35 the adato heat tasfe oeffet α s: α εσ ( ( s ev s ev (.37 Hee we have modelled the adato the same wa as oveto. I ths sese we have leased the adato ate equato, makg the heat ate popotoal to a tempeatue dffeee athe tha to the dffeee betwee two tempeatues to the fouth powe. ote, howeve, that α depeds stogl o tempeatue, whle the tempeatue depedee of the oveto heat tasfe oeffet α s geeall weak. Se the fee oveto ad adato tasfe ous smultaeousl, the oveto ad adato has to be added. he the total ate of heat tasfe fom the sufae s as follows: 4 4 q q q α ( εσ( (.38 s ev s ev he total heat tasfe b oveto ad adato expessed as the heat tasfe oeffet α s: α α α α εσ ( ( s ev s ev (.39.6 Bouda odtos I ode to have a uque soluto of the PDE (patal dffeetal equato, bouda ad tal odtos have to be spefed as show below. I ase of dffeetal equatos fo the eletal fuse, pesbed bouda odtos ae used. PDE s of flat ad oud eletal ables wll have smmet ad o-lea ovetve-adatve bouda odtos.. Flat eletal able - tal odto (, ( (.4 ev

36 3 Chapte. Phsal models of odutos ad the heat tasfe equatos - bouda odtos (, t lm λ, λ α 4 4 ( l, ( εσ(. ev ev (.4. Roud eletal we - tal odto (, ( (.4 ev - bouda odtos (, t lm λ, λ α 4 4 ( d, ( εσ(. ev ev ( Eletal fuse - tal odto ( x, ( x (.44 ev - bouda odtos (, t ( t ( x, t ( t (.45 he bouda ad tal odtos equatos ( ae geeall vald ad mplemeted to the umeal algothm of heat tasfe alulatos. I the aaltal aalss of heat tasfe (Chapte 3, some addtoal bouda od - tos wll be used to solve the PDE of flat ables ad oud wes. Hee we have to alulate wth the ostat heat tasfe oeffet ad do ot take to aout the olea pheomea of adato. he followg addtoal bouda odtos appl fo a flat eletal able:

37 .6 Bouda odtos 33 d d d d EI ( d b λ α λ s ( s ev, (.46 I ase of ldal we: d d d d EI π λ α λ s s, ( ev (.47

38 CHAPER 3 AALYICAL AALYSIS OF HEA RASFER I A SEADY SAE I the peedg Chapte, a defto of heat tasfe equatos fo the stud of aaltal ad umeal heat tasfe omputato was gve. he obetve of those equatos s to deteme the tempeatue feld dffeet kds of eletal odutos whee heat oduto, oveto/adato ad eeg geeato takes plae. Dffeet bouda odtos wee also gve fo the solutos of those equatos. he am of the peset hapte s to obta exat aaltal solutos a stead-state egme. Beause of the leazato of dffeetal equatos, some dffeee betwee umeal ad aaltal esults wll ou, but these msmathes a be aepted ma stuatos. It s alwas oveet to have a smple aaltal soluto f a stead state s equed. he followg assumptos ae made to smplf the patal dffeetal equatos: a stead-state odtos, b oe-dmesoal oduto, ostat o lea mateal popetes, d ufom volumet heat geeato, e ostat heat tasfe oeffet. 3. Calulato of the themo-eletal haatests of flat ables 3.. Vetal heat tasfe wth tempeatue-depedet oeffets Fo pue vetal heat tasfe flat ables equato (.6, Chapte wll be used: λ ( γ ( ρ qv (, (.6 t Cosdeg assumptos fo the heat equato made befoe we get the followg equato:

39 36 Chapte 3. Aaltal aalss of heat tasfe a stead state (, t EI γρ (, t λa λ t (3. o, (, t (, t C D t (3. EI I hee: C ρ λ A λa ; γρ D. λ 3.. Vetal heat tasfe wth tempeatue-depedet oeffets Cosdeg spef esstae ρ ad eletal feld stegth E depedee o tempeatue: [ α ( (, t ] ρ ( ρ ρ ev (3.3 [ ( (, t ] E( E α ρ ev (3.4 hee: α ρ lea tempeatue oeffet of esstae /K ρ spef esstae at efeee tempeatue C E feld stegth at efeee tempeatue C he, equato (3. obtas ths fom: ( t IE λa, γρ [ ( ] (, t α, t ρ λ t ( t, α ρ IE IE λa (, t λa γρ λ t (, t (3.5 o, (, t B (, t C D t (, t (3.6

40 3. Calulato of the themo-eletal haatests of flat ables 37 Se equato (3.6 the tem wth B s tempeatue depedat, stead state a ol be eahed f addtoal odtos ae satsfed. he eesst of suh a odto ases fom the fat that the spef esstae ρ eases wth tempeatue. he soluto of tempeatue hage tme a be peseted the followg fom: d ( t, g ( s t π (3.6b d hs esult s eogsed as a Foue se-sees expaso of the abta futo (, fo whh the ostat ampltudes g ae gve b: g d d ( t ( s π d. (3.6 d d Ol, f B < π the soluto of stead state tempeatue exst. d hee: α E I B ρ ; λa EI C ; λa γ C D ; ˆ λ B α ρ ;, t ev ( Statoa soluto of vetal heat tasfe equato he statoa soluto wll be obtaed fo etagula ables, amel, flat ables, whee the able wdth b s muh lage tha the thkess d. hs soluto desbes the tempeatue patte a metall oduto of a flat able ad ts sulato vetal deto. hee dffeet ases of eletal oduto ae osdeed, fo whh a statoa soluto of the heat equato s aheved: A Cable wthout sulato ad tempeatue-depedet spef esstae ρ ( B, Dhlet bouda odtos; B Cable wthout sulato ad tempeatue-depedet spef esstae ρ ( B, smmet ad ovetve bouda odtos; C Cable wth sulato ad tempeatue-depedet spef esstae ρ ( B, smmet ad ovetve bouda odtos. Case A. Cable wthout sulato ad tempeatue-depedet spef esstae ρ ( B.

41 38 Chapte 3. Aaltal aalss of heat tasfe a stead state he heat equato (3.6 fo stead-state smplfes to: ( B ( C (3.7 he geeal soluto of equato (3.7 s: ' C ( s B os B (3.8 B hee, tegato ostats C ad C ' B ev C. I ode to get a tempeatue pofle, bouda odtos fo the equato (3.8 have to be appled. he tempeatues ae fxed at the bouda (Dhlet odtos at the bottom sde of the flat able (-d/ ad uppe sde - (d/. Fo -d/: d d d s B B os ˆ (3.9a Fo d/: d d d B B os ˆ s (3.9b hs leads to the tegato ostats, : (3.a d s B ˆ (3.b d os B he seto of tegato ostats to the geeal soluto (3.8 gves the followg tempeatue dstbuto the flat able: ˆ s B os B ˆ d d s B os B ( (3.

42 3. Calulato of the themo-eletal haatests of flat ables 39 hee: d thkess of the able, - bouda tempeatue at -d/, -bouda tempeatue at d/. Fo equato (3. a be smplfed: ˆ ˆ d os B ( os B (3. Case B. Cable wthout sulato ad tempeatue-depedet spef esstae ρ ( B I ase the tempeatue depedee of the spef esstae a be egleted, the equato (3. smplfes to: (, t C (3.3 he geeal soluto of ths equato (3.3 s: C ( (3.4 whee ad ae tegato ostats. Smmet ad ovetve bouda odtos (Fg.3. ae appled: λ s d d ( d b α ( d b( d ev d -d x q v (, t lm λ d Fg. 3. Bouda odtos osdeg tempeatue gadet oduto ol of flat able

43 4 Chapte 3. Aaltal aalss of heat tasfe a stead state at : ; ad (3.5 at d : λ ( d b α ( d b ( d α ( λ d ev ev, o. (3.6 he elatoshp (3.6 s developed b applg a sufae eeg balae. Hee the heat tasfe oeffet s osdeed ostat. Substtutg the appopate ate equatos (3.3, 3.4, 3.5 ad 3.6 tempeatue pofle the oduto of flat able s obtaed: C EI ( d d ev d α ( d b (3.7 C EI ( d d ev d α ( d b hee: C Case C. Cable wth sulato ad tempeatue-depedet spef esstae ρ ( B Fo the alulato of the tempeatue dstbuto a sulated flat able (ase C, the bouda odtos should be appled to the bodes of the sulato (see Fg.3.. Due to hgh themal odutvt of the oduto ompaed to the sulato, the tempeatue gadet the metall oduto a be assumed to be zeo. Applg as oveall eeg balae law to the flat able model, we obta followg bouda od - tos: d d ( d b α ( d b( d x -d -d λ λs ( d b EI q d v s d d Fg. 3. Bouda odtos osdeg tempeatue gadet the sulato aloe of flat able d d d ev

44 3. Calulato of the themo-eletal haatests of flat ables 4 at d : λ d d ( d b EI s d, o d d d ( d EI bλ s ; (3.8 at d : λ ( d b α ( d b(, o ( s d d d ev d d α λ d s ev (3.9 he equato (3.7 a be wtte as follows ( (3. whh b tegato beomes: (3. whee s a tegato ostat. akg to aout the lmt odto (3.8 the ostat s: λ EI ( d b s he tempeatue s of the oute sufae of the sulato, aodg to (3.9 s gve b: s ev α EI ( d b (3. he tempeatue pofle the sulato bod a be detemed b tegatg the equato (3.: ( s λ s EI ( d b (3.3

45 4 Chapte 3. Aaltal aalss of heat tasfe a stead state empeatue at the e sde of sulato, whh also meas tempeatue of metall oduto s gve wth d : s λ s EI ( d b d (3.4 o, expessed as a futo of evomet tempeatue ev : ev α EI ( d b λ ( d b s EI ( Calulato of themo-eletal haatests of oud wes 3.. Radal heat tasfe wth tempeatue-depedet oeffets Fo adal heat tasfe we osde fte legth ldal we thus egletg ed effets. hs assumpto s easoable f the ato of lde legth L ad lde adus s L/. he geeal heat equato fo adal sstem s: (, t EI γ (, t λa λ t (3.6 o, (, t (, t C D t (3.7 EI I hee: C ρ λ A λa ; γ D λ 3.. Heat tasfe equatos wth tempeatue-depedet oeffets Hee the spef esstae depedee o tempeatue wll be osdeed: [ α ( (, t ] ρ ( ρ ρ (3.8

46 3. Calulato of themo eletal haatests of oud wes 43 he eletal feld stegth E hages wth tempeatue as followg: [ ( (, t ] E( E α ρ (3.9 hee: α ρ - lea tempeatue oeffet of esstae ρ spef esstae at efeee tempeatue E feld stegth at efeee tempeatue he the equato (3.7 obtas the followg fom: (, t IE λa γ [ ( ] (, t α, t ρ λ t (, t IE (, t IE γ (, t α ρ (3.3 λa λa λ t o, (, t B (, t (, t C D t (3.3 hee: α E I B ρ ; λa EI C ; λa γ D. λ 3..3 Statoa soluto of adal heat tasfe equato Befoe solvg the equatos, a shot explaato of the applatos shall be gve whee the solutos ae applable. Aga, fst the heat equato wll be solved fo the aked we.e. ldal we wthout sulato (Fg. 3.a. I ths ase, tempeatue dstbuto ous ol the metall oduto. Seodl, the heat equato wll be appled to the oud we wth sulato (Fg. 3.b. Hee the tempeatue dstbuto wll be alulated whlst the sulato lae whle tempeatue gadet of the metall oduto s assumed to be zeo. Fo stead state ad ostat mateal popetes, the heat tasfe equato edues to B : ( C (3.3

47 44 Chapte 3. Aaltal aalss of heat tasfe a stead state Isulato Coduto Coduto ( ( e s - e - - a b Fg. 3. empeatue dstbuto a plae of ldal we: a we wthout sulato; b eletal we wth sulato Sepaatg vaables ad assumg ufom heat geeato, the equato a be tegated to obta: ( C (3.33 Repeatg the poedue, the geeal soluto fo the tempeatue dstbuto beomes: C ( l ( o obta tegato ostats ad we appl the followg bouda odtos: ( at :, ad at : ( ; he fst odto esults fom the smmet of the lde. I the ete of the lde, the tempeatue gadet must be zeo. Usg the seod bouda odto at wth the equato (3.34 we obta: C 4 (3.35 he tempeatue dstbuto s theefoe: C ( 4 (3.36

48 3. Calulato of themo eletal haatests of oud wes 45 o elate the sufae tempeatue,, to the evomet tempeatue ev, a oveall eeg balae equato leads to the esult: EI π L π o ( α π L ev EI ev (3.37 π α hee: L legth of ldal we m he, the tempeatue dstbuto the metall oduto osdeg heat dsspato fom the sufae b oveto: C EI ( 4 π α ev (3.38 I ode to deteme tempeatue a sulated ldal we (Fg.3.b, we use the same heat equato (3.3 but dffeet bouda odtos shall be osdeed: a fom the eeg balae equato fo the followg equato a be wtte: π λs EI ; b egletg adato, fo the bouda odto s as followg: α λ s π α π( ev, o ( ev. λ s Repeatg the same poedue as the soluto of Eq. (3.7 fo a o-sulated we, we obta the followg soluto fo tempeatue pofle the sulato of ldal we: EI EI ( l ev (3.39 πλs Ł ł π α Equato (3.39 eables us to ompute the tempeatue pofle the sulato. I the metall oduto, the tempeatue gadet s osdeed to be zeo. hs assumpto

49 46 Chapte 3. Aaltal aalss of heat tasfe a stead state s easoable, beause the heat odutvt of a metall oduto s ve hgh, ompaed wth the sulato heat odutvt. empeatue of metall oduto at fom Eq. (3.39 s theefoe: EI EI l πλs Ł ł π α ev ( Calulato of the themo-eletal haatests of eletal fuses 3.3. Axal heat tasfe wth tempeatue depedet oeffets Fo the aaltal aalss of axal heat tasfe we wll use smla equato to (Eq..8, Chapte ad todue tempeatue-depedet oeffets. he the equato has the fom: ( x, t αu EI ( x, t x λa λa γ λ ( x, t t (3.4 hee: αu EI γ B ; C ; D. λa λa λ he equato (3.4 a be ewtte followg: ( x, t B ( x, t C D x t (3.4 Let us desbe the oeffet phsal meag of equato (3.4. hese oeffets do ot deped o tempeatue. Coeffet B a be wtte the followg fom: α u B (3.43 λ A χ

50 3.3 Calulato of themo eletal haatests of eletal fuses 47 hee χ - s the legth ostat, e.g. the vesed squae oot of oeffet B : λa χ (3.44 B αu χ a be osdeed as a legth dea a futo of tempeatue, whh eases f the ato A / u eases. Coeffet C s alled empeatue feld gadet K/m. he oeffet meas the ato of the volumet geeated heat EJ the fuse ad the heat odutvt oeffet? : EI ρj C (3.45 λa λ We todue the asmptot tempeatue temˆ. Asmptot tem a be udestood as a fal tempeatue of fte legth we afte stead state. empeatue ˆ the fuse wll ot be aheved f the fuse has a ve shot legth. I a ase ˆ wll ot be eahed taset state. he fomula of ˆ s the followg: C EI ˆ χ C B αu (3.46 Coeffet D a be alled epoal tempeatue odutvt o epoal heat taspot velot ad s desbed as the quotet of heat apat ad heat odutvt: γ D (3.47 λ 3.3. Axal heat tasfe wth tempeatue-depedet oeffets I ths seto we wll osde tempeatue depedat spef eletal esstae of oppe o bass. Spef esstae, ρ, fo tempeatue hage fom to 8 C a be alulated as follows: [ α ( ( x, t ] ρ ( ρ ρ ev (3.48 he feld stegth, E, hages wth espet to tempeatue as: [ ( ( x, t ] E( E α ρ ev (3.49

51 48 Chapte 3. Aaltal aalss of heat tasfe a stead state hee: α ρ lea tempeatue oeffet of esstae ρ spef esstae at efeee tempeatue E feld stegth at efeee tempeatue Cosdeg Eq. (3.49, equato (3.4 takes the followg fom: [ ], (, ( (, (, ( t t x A t x IE t x A u x t x ev λ γ λ α λ α ρ (, (, (, ( t t x A IE t x A IE u x t x ev λ γ λ λ α α ρ (3.5 o, (, (, ( t D C t x B x t x ev whee the oeffets have the followg meag: IE u A B ρ α α λ χ (3.5 ev IE u EI C B C ˆ α ρ α χ ( Avalahe effet metall oduto A oduto wth a postve tempeatue oeffet α ρ shows a so-alled avalahe effet, whee due to too lage eeg geeato, the equlbum, geeated betwee eeg the fuse ad dsspated heat to ambet a ot be aheved. hs s vald fo the legth ostat as well as fo the fal tempeatue of a we wth fte legth. Beause of ths effet, tempeatue ses otuousl ad the legth ostat χ A ad fal tempeatue A ˆ beomes fte f the followg odtos ae satsfed: A I IE u ρ α α α ρ ρ (3.53

52 3.3 Calulato of themo eletal haatests of eletal fuses 49 he: χ A, ˆ A Avalahe uet a be alulated ths wa: I A αua α ρ ρ ( Statoa soluto fo axal heat tasfe equato he soluto of the equato fo axal heat tasfe gves the tempeatue dstbuto the x deto. Fo stead-state we appl bouda odtos gve equato (.46, Chapte: ( ( x l (.46 he, the equato (3.4 of axal heat tasfe smplfes to: ( x B( x C x (3.55 he geeal soluto of Eq. (3.55 s: C x e B ' x x ( e B (3.56 B hee: B C ', lea tempeatue depedet heat dsspato oeffet to ambet /m B ev C tempeatue depedet heat geeato b eletal uet /m tegato ostats, whh the bouda odtos ae set C I ode to fd, we todue bouda values at x ad x l. he, the tempeatue dstbuto the fuse s as followg:

53 5 Chapte 3. Aaltal aalss of heat tasfe a stead state ( ( e e e e e e e e x Bx Bl Bl l Bl l x B Bl l Bl Bl ˆ ˆ ˆ ( ( (3.57 hee: l legth of fuse m (x tempeatue dstbuto alog the fuse C the bouda tempeatue of the fuse at x C l the bouda tempeatue of the fuse at x l C Replag B b χ, equato (3.7 a be wtte as: e e e e e e e e x x l l l l l x l l l l ˆ ˆ ˆ ( χ χ χ χ χ χ χ χ (3.58 Equato 3.58 gves the tempeatue dstbuto a fuse elemet wth the fte legth ad fxed bouda tempeatues. Fom the aaltal soluto the avalahe effet (Eq. 3.5 a be obseved, the legth ostat χ s obtaed (Eq.3.5, ad hpothetal tempeatue ˆ a be deved (Eq. 3.5.

54 CHAPER 4 UMERICAL CALCULAIO OF EMPERAURE BEHAVIOUR I A RASIE SAE 4. Ovevew of the umeal methods used heat tasfe omputato he heat tasfe of a sulated eletal we s desbed b a o-lea ad ohomogeeous patal dffeetal equato. A uque aaltal soluto s ol feasble fo dealsed ad smple odtos. Fo patal ases, t s equed to mplemet umeal methods. Avalable aaltal ad expemetal esults ae of osdeable mpotae vefg the aua ad valdt of umeal esults. he lmtatos of a aaltal soluto ase fom the followg popetes of eletal wes: eletal odutvt s seod ode tempeatue depedet heat odutvt λ of sulato s at least lea tempeatue depedet heat oveto ad adato α s at least thd ode tempeatue depedat he umeal methods allow ot ol bette epesetato of the mutual heatg effets, but also pemt moe auate modellg of the boudaes (e.g. a ovetoadato bouda to the evomet. A umeal soluto s obtaed fom dsetsato of the patal dffeetal equato. hee ae fou dstt steams of umeal soluto tehques: Fte Dffeees (FD Fte Elemet (FE Spetal Method (SM Fte Volume Method (FV he fou metoed methods dffe mal the appoxmato of the vaables ad the dsetsato poesses.

55 5 Chapte 4 umeal alulato of tempeatue behavou a taset state egme. Fte dffeee methods (FD I ths method the patal devatos of equatos ae appoxmated b a tuated alo sees. hs method s patulal appopate fo a equdstat Catesa mesh. alo sees expaso of a futo f(x about a pot x the fowad (.e. postve x ad bakwad (.e., egatve x detos ae gve, espetvel, b: 3 3 df d f ( x d f ( x f ( x x f ( x x... 3 dx dx! dx 3! 3 3 df d f ( x d f ( x f ( x x f ( x x... 3 dx dx! dx 3! hese two expessos fom the bass fo developg dffeee appoxmatos fo the fst ode devatve df/dx about x. Reaagg the expessos, the fowad ad bakwad fte dffeee appoxmatos fo the fst ode devatve, espetvel, beome: df dx df dx f ( x x f ( x ( x x (fowad f ( x f ( x x ( x x (bakwad Moe about FD method heat tasfe a be foud the elevat lteatue [9].. Fte elemet method (FE hs method ogated fom the stutual aalss as a esult of ma eas of eseah, mal betwee 94 ad 96. I ths method the poblem doma s deall subdvded to a olleto of small egos of fte dmesos, alled fte elemets. he elemets a -D ase have ethe a tagula o quadlateal fom (Fgue 4.,a ad a be etlea o uved. Afte subdvso of the doma, the soluto of the dsete poblem s assumed to have pesbed fom. hs epesetato of the soluto s stogl lked to the geomet dvso of sub domas ad haatesed b the pesbed odal values of the mesh. Fo heat tasfe the eletal wes, the dsete soluto wth FE a be ostuted as follows:. A fte umbe of pots the soluto ego s detfed. hese pots ae alled odal pots o odes.. he value of tempeatue at eah ode s deoted as a vaable whh has to be detemed.

56 4. Ovevew of the umeal methods used heat tasfe omputato he soluto ego s dvded to a fte umbe of subegos alled ele mets. hese elemets oet ommo odes, ad olletvel appoxmate the shape of the ego. 4. empeatue s appoxmated ove eah elemet b a polomal expesso that s defed usg odal values of the tempeatue (see Fg. 4., b: P Aω Bω Cω m whee ω, ω, ω m ae the aea oodates defed as Fg. 4., b. hese aea oodates uquel defe the posto of a pot P sde the tagle m. A dffeet polomal s defed fo eah elemet, but the elemet polomals ae seleted suh a wa that otut s mataed alog the elemet boudaes. he odal values ae omputed so that the povde the best appoxmato possble to the tue tempeatue dstbuto. hs seleto s a omplshed b mmsg some quatt assoated wth the phsal pob lem o b usg Galek s method [9], whh deal wth the dffeetal equa tos detl. he soluto veto of the algeba equatos gves the equed odal tempeatues. he aswe s the kow thoughout the soluto ego. Moe about FE method a be foud lteatue []. m P ω ω ω m x a b Fg. 4. agula o quadlateal fte elemets of a two-dmesoal doma (a ad aea oodates (b 3. Spetal method Spetal methods appoxmate the ukows b meas of tuated Foue sees o sees of Chebshev polomals. Ulke the fte dffeee o fte elemet appoah the appoxmatos ae ot loal but vald thoughout the ete omputatoal doma. he ukows the goveg equato ae eplaed b the tuated sees. he osta that leads to the algeba equatos fo the oeffets of the Foue o Chebshev sees s povded b a weghted esduals oept smla to the fte elemet method o b makg the appoxmate futo ode wth the exat soluto at a

57 54 Chapte 4 umeal alulato of tempeatue behavou a taset state egme umbe of gd pots. Futhe fomato o ths spealsed method a be foud [] 4. he fte volume method he fte volume method was ogall developed as a speal fte dffeee fomulato [, 3]. I FV method, the patal devato of equatos s ot detl appoxmated lke FD appoah. Istead, the equatos ae tegated ove a otol volume V, whh s defed b odes of gds o the mesh: V dv ( λ gad dv q dv γ V v V dv t he volume tegal tems wll be eplaed b sufae tegals usg the Gauss fomula. Fo a veto a ths theoem states: adv V dv. a da A hese sufae tegals defe the ovetve ad dffusve fluxes though the sufaes. Due to the tegato ove the volume, the method s full osevatve. hs s a mpotat popet of FV method. hs lea elatoshp betwee the umeal algothm ad the udelg phsal osevato pple foms oe of the ma attatos of the fte volume method ad makes ts oept muh smple to udestad b egees tha fte elemet ad spetal methods. I fat, 4 eas ago Lax ad Wedoff poved mathematall that osevatve umeal methods, f oveget, do ovege to oet soluto of the equato. hs stud shall be solel oeed wth ths most well establshed ad thooughl valdated geeal-pupose omputatoal flud dam (CFD ad heat tasfe tehque. heefoe the method s dsussed moe detaled. 4. Fudametals of the fte volume method he bas laws of heat tasfe ae the osevato equatos, whh ae statemets that expess the osevato of: mass, mometum, ad eeg a volume losed b ts sufae.

58 4. Fudametals of the fte volume method 55 (x x x - B / - B x Fgue 4. Oe-dmesoal Fte Volume Mesh Ceta equemets ae eessa to ovet these laws to patal dffeetal equatos. hese equemets aot alwas be guaateed. I a ase whee a dsotut ous, a auate epesetato of the osevato laws s mpotat. I othe wods, t s of bg mpotae that these osevato equatos ae auatel epeseted the tegal fom. he most atual method to aomplsh ths s obvousl to dstse the tegal fom of the equatos but ot the dffeetal fom. hs s the bass of fte volume (FV method. I two dmesoal ases the feld o doma s subdvded the same wa as the fte elemet method, amel, to a set of o-ovelappg ells that ove the whole doma o whh the equatos ae appled. O eah ell the osevato laws ae appled to deteme the flow vaables some dsete pots of the ells, alled odes, whh ae tpal loatos of the ells suh ell-ete (ell eteed mesh o ellvetes (ell vetex mesh (Fgue 4.3. Obvousl, thee s osdeable feedom the hoe of the ell shapes. he a be tagula, quadlateal et. ad geeate a stutued o ustutued mesh. Due to ths ustutued fom, ve omplex geometes a be hadled wth ease. hs s leal a mpotat advatage of the method. Addtoall the soluto of the equato of the ell s ot stogl lked to the geomet epesetato of the doma. hs s aothe mpotat advatage of the fte volume method otast to the fte elemet method.

59 56 Chapte 4 umeal alulato of tempeatue behavou a taset state egme (a (b Fgue 4.3 wo-dmesoal stutued Fte Volume Mesh sstem: (a Cell Ceted mesh (b Cell Vetex mesh umeal popetes of dsetsato shemes. It s omall dstgushed betwee tme ad spatal dsetsato of otuum equato. he spatal dsetsato a be appled o dffeet foms of gds suh as Catesa, o-othogoal, stutued ad ustutued. me dsetsato s usuall doe b FD sheme, whh a be explt o mplt. omall, the explt method s used fo stog ustead flows o whe tme gadet s ve bg. Whe a det omputato of the depedet vaables a be made tems of kow quattes, the omputato s sad to be explt. I otast, whe the depedet vaables ae defed b oupled sets of equatos, ad ethe a matx o teatve tehque s eeded to obta the soluto, the umeal method s sad to be mplt. he hoe of whethe a mplt vesus explt method should be used depeds ultmatel o the goal of the omputato. he osequees of usg both methods have to do wth umeal stablt ad umeal aua. Usg explt methods we a aheve equed aua tme wth moe omputatoal effot tha mplt method. Although, explt methods ae smple to mplemet mathematall, the ae almost all ases ol odtoall stable. Implt fomulato s sad to be alwas uod - toall stable. A soluto fo the ukows at oe tme level ma be obtaed fo a sze of tme step. I omputatoal heat tasfe, the goveg equatos ae olea. Ude these odtos mpltl fomulated equatos ae almost alwas solved usg teatve tehques. Se heat tasfe eletal odutos has o stog ustead flows as well as osdeg effe ad stablt of mplt method,

60 4.3 o-lea heat tasfe model of eletal odutos 57 wheeve possble mplt methods ae used ths wok but explt method a be appled optoall. Good udestadg of the umeal soluto algothm s ual. hee mathematal oepts ae useful detemg the suess o othewse of suh algothms: ovegee, osste ad stablt. Covegee s the popet of a umeal method to podue a soluto, whh appoahes the exat soluto as the gd spag, otol volume sze o elemet s edued to zeo ( lm (, exat soluto of patal dffeetal equato, mesh soluto of fte dffeee equato. Cosstet umeal shemes podue sstems of algeba equatos, whh a be demostated to be equvalet to the ogal goveg equatos, as the gd spag teds to zeo. Stablt s assoated wth dampg of eos as the umeal method poeeds. If a tehque s ot stable eve oud off eos the tal data a ause eat osllatos ad dvegee. Covegee s usuall ve dffult to establsh theoetall ad pate Lax s equvalee theoem s used [4], whh states that fo lea poblems a eessa ad suffet odto fo ovegee s that the method s both osstet ad stable. I heat tasfe alulatos ths theoem s of lmted use se we stated that the goveg equatos ae o-lea. I suh poblems osste ad stablt ae eessa odtos fo ovegee, but ot suffet. 4.3 o-lea heat tasfe model of eletal odutos It has alead bee metoed that usg the umeal methods the dffeetal ad the tegal equatos a be tasfomed to dsete algeba equatos. Based o the metoed easos the pevous seto (4., the FV method has bee hose fo the dsetsato of PDE. Equato (., Chapte λ λ λ qv γ (4. x x z z t a be ewtte the tegal fom. We tegate Equato (. ove a small fxed volume V: dv( λ gad dv qvdv γ dv (4. t V V V

61 58 Chapte 4 umeal alulato of tempeatue behavou a taset state egme he volume tegal ove the dvegee of heat flux veto s tasfomed to a sufae tegal b meas of the dvegee theoem. he Equato (4. beomes ( λ gad ds qvdv γ dv (4.a t S V V whee S s the sufae aea of the fte volume. Se, gad (4.b the, the equato (4.a gves: S λ ds qvdv dv γ (4. t V V hee: V small fte volume; outwad daw omal ut veto devatve alog the outwad daw omal to the sufae of the otol volume. Equato (4.a epesets the pple of osevato of eeg ove fte volume V. It states that the ate of eeg eteg the otol volume though ts bouda sufae S plus the ate of eeg geeated the volume elemet s equal to the ate of ease of stoed eeg the otol volume. Futhemoe, se fluxes ae oseved taspot betwee the otol volumes, the osevato pple s also satsfed fo a assembl of fte volumes. hat s, the umeal soluto wll satsf both the loal ad global osevato popetes, hee the fomulato gve b Equato (4. s full osevatve Appoxmato of heat tasfe equatos b FVM Flat elet able We osde the taset state dffuso wth ovetve-adatve bouda odto of a flat eletal able a oe-dmesoal doma defed Fgue 4.4.

62 4.3 o-lea heat tasfe model of eletal odutos 59 A (fee oveto ad adato ev α, (x,,z,t~(,t ~ z b d PVC sulato?(,γ(,α( Coppe oduto?(,γ( x λ α 4 4 ( l ( εσ(, ev ev d q v ( (, t lmλ Fgue 4.4 he phsal model of flat eletal able heated b eletal uet It s assumed that the dmesos the x- ad z- detos ae so lage that tempeatue gadets ae sgfat the - deto ol. he used gd s show Fgue 4.5: / - / - s Fgue 4.5 he fte volume gd of flat able he goveg equato s: λ ( γ (, qv (, (4.3 t

63 6 Chapte 4 umeal alulato of tempeatue behavou a taset state egme he Equato (4.3 models the heat oduto a eletal we, whh has dffeet mateals amel a oduto (hee: oppe ad a sulato (hee: PVC PolVl- Chlode wth dffeet heat odutvtes λ. Moeove, thee s o heat geeato q v we (PVC sulato. heefoe the equato has dsotuous oeffets λ ad q v : < < < < ; ;,, ;, s v s v q q λ λ λ λ (4.4 he tegal fom of the goveg equato fo teo odes of flat able gves: V V V q v dv dv t dv, ( ( γ λ (4.5 Applg ths tegal fom to the fte volume V [-.5;.5] (Fgue 4.5 the Equato (4.5 a be ewtte wheeb the volume elemet dv s eplaed b the sufae elemet d (usg Gauss theoem: , (, ( ( v d q d t d γ λ (4.6 Itegato of the Equato (4.6 ove [-.5;.5] leads to: v q t, (, ( ( (.5.5 γ λ λ (4.7 Replag patal devatves spae b etal dffeees ad devatve tme b bakwad dffeee, the Equato (4.7 takes the fom: z q t, ( ( ( ( ( γ λ λ (4.8 he patal devatves Eq. (4.8 spae ae of seod ode aua ad tme of fst ode (,t.

64 4.3 o-lea heat tasfe model of eletal odutos 6 Equato (4.8 s solved mpltl tme. I ths stud sem-mplt sheme s used, so the tme step a ot be hose too lage. he spae step ad tme step t a be omputed b these expessos: d d s.5 ad t t (4.8a K hee: t tme eeded to eah stead state some ode, K umbe of tme steps eeded to eah stead state egme. he esult of ths umeal soluto gves tempeatue dstbuto the metall oduto ad sulato of the flat able Roud elet we Cldal eletal we of fte legth s gve (Fgue 4.6

65 6 Chapte 4 umeal alulato of tempeatue behavou a taset state egme A (fee oveto ad adato ev, α x q v (,x,t~(,t ~ Volumet heat geeato s PVC sulato?(,γ(,α( Coppe we?(,γ(,effet s Fgue 4.6 he model of heat oduto oud eletal we adus of the metall oduto (oppe, s adus of the sulato (PVC,,effet effetve adus of the oduto (pue oppe wthout a gaps betwee sgle odutos I the heat tasfe model of ldal we we osde heat oduto the oss seto ad eglet the oduto alog the we, assumg, that the ed effets of the we have o fluee to the amed alulato esults. hs appoxmato s easoable fo L/ s >, whee L legth of the we. he umeal sheme s show Fgue 4.7:

66 4.3 o-lea heat tasfe model of eletal odutos 63 / - / - s Fgue 4.7 he fte volume gd of oud wes he goveg equato s: d λ ( γ (, qv (, (4.9 d t Equato (4.9 has dsotuous oeffets λ ad P: λ λ, qv λ λs, qv ; ; <, < < s ; (4. Afte multplato of the Equato (4.9 b, the tegal fom of the goveg equato fo teo odes of ldal we beomes: d λ qv (, dv (4. d t ( dv γ (, dv V V V Applg ths tegal fom to the fte volume V [-.5;.5] (Fgue 4.7 we a ewte the Equato (4. as followg: λ ( d γ(, d qv(, d (4. t.5 Itegatg ove [-.5;.5] we get:.5 λ (4.3.5 (.5λ( γ(, qv(,.5.5 t

67 64 Chapte 4 umeal alulato of tempeatue behavou a taset state egme Replag patal devatves spae b etal dffeees ad devatve tme b bakwad dffeee, the Equato (4.3 takes the fom: / λ / / λ/ γ ( ( / / t ( q (4.4 Equato (4.4 s solved sem-mpltl tme. he spae step ad tme step t a be omputed b these expessos: s.5 ad t t (4.5 K hee: s t K the adus of sulated we, umbe of odes, the tme eeded to eah stead state some ode, a umbe of tme steps eeded to eah stead state egme Elet fuse Hee, the poblem deals wth the axal heat tasfe alulato a oppe (o bass sold o hollow lde (fuse meltg elemet pototpe wth fte legth (Fgue 4.8. Axal tempeatue dstbuto the fuse meltg elemet must be omputed ode to obta the maxmum tempeatue (meltg tempeatue of ths elemet. I the mathematal model of heat tasfe the fuse, the o- homogeous geomet, heat oveto ad adato s osdeed though the sufae of the fuse elemet ad pesbed tempeatue bouda odtos ae appled. he tempeatue gadet the adal deto s egleted, beause heat odutvt oeffet of oppe (o bass s ve lage ad gves a ostat tempeatue dstbuto fo the model geomet osdeed hee. hs model eduto edues omputatoal effots osdeabl.

68 4.3 o-lea heat tasfe model of eletal odutos 65 d3 A (fee oveto ad adato ev, α d (,t (L,t d d d (x,,z,t~(x,t ~ We Fuse holde elemet Fuse elemet Fuse holde elemet We x Fgue 4.8 Phsal model of the fuse meltg elemet used fo umeal omputato he heat tasfe equato of the eeg osevato law fo a -D poblem the axal deto a be expessed as: A( x λ x x γ(, A( x t 4 4 [ α ( α ( ] A( x q v ( ev u (4.6 hee: α oveto heat tasfe oeffet W/m K, tempeatue dffeee betwee the tempeatue o the sufae ad evomet tempeatue ev K, α εσ adato heat tasfe oeffet W/m K 4, u umfeee of the fuse elemet m, A aea (depeds o the geomet of the fuse elemet m. Followg the same poedue as pevous setos ( we tegate Eq. (4.6 ove [-.5;.5] (Fg Afte tegato equato elds: λ A( x d dx.5 γ(, A( x x λ A( x d dt d dx.5 A( x q ( x v 4 4 [ α ( α ( ] ev u (4.7 Replag the devatves b etal dffeees spae ad devatve tme b bakwad dffeee, the Equato (4.7 gves:

69 66 Chapte 4 umeal alulato of tempeatue behavou a taset state egme λ.5 A x 4 4 [ α ( ( α (( ] γ.5.5 Ax t ( ( ev λ.5a x.5.5 A q x ev ( u (4.8 he spae step x ad tme step t s omputed b: L x ad t t (4.9 K hee: L legth of the fuse elemet, umbe of odes whh the tempeatue s alulated, tme eeded to eah stead state some x ode, K umbe of tme steps eeded to eah stead state egme. x/ x x - / - x x Fgue 4.9 he fte volume gd of fuse umeal mplemetato of bouda odtos Flat elet able he mathematal model of heat tasfe the flat able ossts of PDE ad two bouda odtos. heefoe, the heat tasfe poblem must be osdeed as a tal-bouda value poblem. Hee we have to deal wth mxed-tpe bouda od - tos, whh osst of eumma tpe bouda odtos (seod kd ad ovetve-adatve lmt odtos (thd kd. Se the flat able model s a smmetal sstem (see Fg. 4.4, we a use smmet bouda odto at. he able s plaed a a hozotal posto ad affeted ol b lama fee oveto ad adato to the evomet. Due eglgble dffeee betwee the heat oveto o the uppe ad lowe sdes of the able, ths smmet assumpto s oet.

70 4.3 o-lea heat tasfe model of eletal odutos 67 I ths seto we show how to appoxmate ths tpe of bouda odto ode to have the same ode of aua as the goveg equato. Ital ad bouda odto s gve as: [ ] ( ( ( ( (,, (, (,, (,, ( ( lm,, (, ( 4 * 4 * t t t d t t t ev ev ev εσ α λ λ (4. hee * ev s the absolute tempeatue of evomet K. he fst bouda odto, whh s deved fom the smmet of the phsal model, has to be seted to the ma equato (4.5 ad tegated ove ego [ ;.5 ]: d q d t d.5.5.5, (, ( ( γ λ (4.a Followg the same poedue as the devato of the dsete fom of the goveg equato (4.8, the followg dsete fom of the fst bouda odto s obtaed: ( ( q t γ λ (4.b I ode to desbe heat oduto though the bouda of able sulato ad evomet, we have to tegate Equato (4.3 ove the ego [ -.5 ; ]: d q d t d ( γ λ (4. Itegatg Equato (4. ove [ -.5 ; ] ad osdeg ovetve adatve pheomea at the bouda of sulato lae the od we get heat oduto equato fo the aea [ -.5 ; ]:

71 68 Chapte 4 umeal alulato of tempeatue behavou a taset state egme ( ( q t ev ev, ( ( ( ( ( * 4 * γ λ β α ( Roud elet we he heat oduto equato of ldal elet we has the same tpe of bouda odtos as flat able (seto 4.3..: smmet ad ovetve-adatve bouda odto. he we s plaed a a hozotal posto ad affeted b lama fee oveto ad adato to the evomet. Ital ad bouda odtos ae gve as followg: [ ] ( ( ( ( (,,, ( lm.5,, (, ( 4 * 4 * t d t ev ev ev εσ α λ λ (4.3 Hee, the fst bouda odto, whh s also deved fom the smmet of the phsal model, has to be seted to the ma equato (4.9 ad tegated ove ego [ ;.5 ]: d q d t d.5.5.5, (, ( ( γ λ (4.3a Followg the same poedue as the devato of the dsete fom of the goveg equato (4., the followg dsete fom of the fst bouda odto s obtaed: ( ( q t γ λ (4.3b I ode to appoxmate the heat flux at, we have to tegate Equato (4.9 ove ego [ -.5 ; ] ad set the seod bouda odto of equato sstem (4.3: d q d t t d.5.5.5, (, ( ( ( γ γ λ (4.3

72 4.3 o-lea heat tasfe model of eletal odutos 69 Itegatg Equato (4.3 ove [ -.5 ; ] ad osdeg ovetve adatve pheomea at the bouda of sulato lae pot, we get the heat oduto equato fo [ -.5 ; ] aea: ( α ( ev q εσ (, 4 * ( ( * 4.5λ.5 γ ev (.5 t ( (4.3d Elet fuse Fo axal heat tasfe the fuse elemet we osde Dhlet (pesbed tempeatue bouda odtos. he tempeatue o both fuse holdes should be equal to the maxmal pemssble tempeatue of the eletal we. heefoe, tempeatue of the eletal we a be assged to the boudaes of the fuse holdes. Ital bouda odtos: [, x ] ( x, ev ( x, x (, t µ ( t, t, (4.4 ( x, t µ ( t, t, Soluto of the equato sstem b ewto-raphso method Flat elet able Setos 4.3. ad 4.3. have show how to appoxmate dffeetal equatos b the fte volume appoah. hus, the heat taspot poblem eletal able, whh s goveed b a sgle dffeetal equato ad bouda odtos a be appoxmated b a sstem of algeba equatos. It s ve mpotat to udestad what methods ae best applable to solve these sstems of algeba equatos. If the umbe of equatos to be solved s lage ad the equatos ae o-lea, oe eeds to exame the atue of the esultg sstem of equatos. Fom setos 4.3. ad 4.3. t a be see, that we have to solve o-lea sstem of equatos beause heat apat γ of oppe s

73 7 Chapte 4 umeal alulato of tempeatue behavou a taset state egme seod ode tempeatue depedet ad the adato bouda odto s fouth ode tempeatue depedet. Othe o-leat poblems lke eletal esstae olea behavo ad heat odutvt o-lea tempeatue depedee a be egleted se these pheomea have a small fluee o omputatoal esults f the tempeatue does ot exeed about 5 C. he obetve of ths seto s to llustate how to solve a o-lea sstem of algeba equatos obtaed fom the goveg sgle dffeetal equato ad ts bouda odtos ode to deteme ukow tempeatue vaables Se we have to deal wth a oe-dmesoal heat tasfe poblem, the Gauss elmato method [9] a be futhe smplfed b takg advatage of the zeos of the tdagoal oeffet matx. hs modfed poedue, geeall efeed to as homas Algothm, s a extemel effet method fo solvg a lage umbe of suh equatos [9]. Usg ths algothm, the umbe of bas athmet opeatos fo solvg a td - agoal set s of the ode, otast to O( 3 opeatos equed fo solvg wth Gauss Elmato. heefoe, ot ol ae the omputato tmes muh shote, but the oud off eos ae also sgfatl edued. I the Equato sstem (4.5 a ewto-raphso teato Method used to lease equatos. he ewto-raphso method s a algothm fo fdg the oots of sstems of olea algeba equatos b teato. If a good tal guess s made, ewto- Raphso teato poess oveges extemel fast. Iteatos ae temated whe the omputed hages the values of P P beome less tha some spefed quatt ε.. ( ( (... 4( ( ;... ; ; ; ev ev a q f P P a b a q f P b P P a b a q f P b P P a b q f P b P εσ α εσ α εσ α (4.5 hee: P ukow tempeatue vaables; tall guessed values. empeatue vaables P ae foud b followg expessos:

74 4.3 o-lea heat tasfe model of eletal odutos 7..., ; beta alpha alpha a beta a f alpha a beta a f beta alpha a b alpha f beta b alpha (4.6 hee:.5.5 b λ ; t γ λ ; t EJ q.5 γ, ;.5.5 a λ ;.5.5 b λ ; t γ λ, t EJ q γ ; < < ;.5.5 a λ ; t γ λ ; t EJ q γ ; Roud elet we he wa of solvg the sstem of algeba equatos s aalogous to the method desbed the seto (4.3.3.:. ( ( (... 4( ( ;... ; ; ; ev ev a q f P P a b a q f P b P P a b a q f P b P P a b q f P b P εσ α εσ α εσ α (4.7 empeatue vaables P ae foud fom the Eq. (4.7, whee the oeffets a, b, ad f ae alulated b:

75 7 Chapte 4 umeal alulato of tempeatue behavou a taset state egme b λ ; t γ λ ; t EJ q.5.5 γ, ; a λ ; b λ ; t γ λ, t EJ q γ, < < ; a λ ; t γ λ ; t EJ q γ, Elet fuse I ode to alulate the tempeatue dstbuto the fuse sstem a o-lea sstem of equatos has to be solved beause of the adato tem the equato (4.6. We use, as pevous ases a ewto-raphso method to solve the poblem. he sstem of algeba equatos s gve as follows:. ;... ; ( ( ( 4( ( ; ( ( 4( ( ; µ µ εσ α εσ α εσ α εσ α εσ α εσ α a b u u u u a q P b P u u P a b u u u u a q P b P u u P a b ev ev ev ev (4.8 Hee the oeffets a, b, ad p ae alulated as follows: ; b ; 5 µ ; ; x A a λ ; x A b λ ; t x x A γ λ, t x AEJ q γ,,,,-; a ; ; 5 µ,. Results of umeal smulato usg the appoah developed ths hapte togethe wth a valdato poedue ae peseted Chapte 6. Flat eletal ables of dffeet oduto sze suppled b the ompa eholog & Iovato GmbH ad oud

76 4.3 o-lea heat tasfe model of eletal odutos 73 eletal wes suppled b the ompa Leo Bodetze GmbH wee used. Fuse elemet samples wee povded b DamleChsle AG.

77 CHAPER 5 BASIC COSIDERAIOS OF HE EXPERIME AD EXPERIMEAL SEUP hs hapte pesets the theo of the expemetal poedue used ths wok togethe wth the measuemet setup ad the data aqusto. he am of the expemets was to vestgate the eletal load apates of vaous eletal wes ad ables. Fo ths, two tpes of measuemets wee equed: a Cuet vesus voltage measuemet, b Resstae vesus tempeatue measuemet he vestgato ommeed wth the measuemet of powe dsspato as futo of tempeatue of the we sufae, alled uet vesus voltage measuemet,(a. he ext step was to valdate the lea tempeatue oeffet α, alled esstae vesus tempeatue measuemet,(b. Late the fomato fom both expemets was used to oet the theoetal model, f eessa. Fst, some fomato o how popel the measuemets have to be pefomed wll be gve. he theoetal bakgoud fo the measuemet spefato was developed extesvel dug ths eseah wok. he, a fgue detalg the expemetal set up fo both ases wll be show. he poedue of paamete aqusto usg GPIB (Geeal Pupose Itefae Bus deves ad the appopate softwae wll be also peseted. he pogamme s gve the Appedx C. 5. Bas osdeatos of the expemet 5.. Resstae vesus tempeatue measuemet he am of ths expemet s to deteme the lea tempeatue oeffet α of oppe wes ad ables. Late, ths tempeatue oeffet α s used to alulate the we ad able tempeatue fom the uet vesus voltage elatoshp obtaed fom the seod expemet: the uet vesus voltage measuemet. he eletal esstae R of the we has to be detemed fo dffeet appled tempeatue values of the we. hese tempeatues ae aheved b heatg up the able a heat hambe. Chambe tempeatue h should be eased equal steps fom evomet tempeatue ev up to max. able tempeatue. I ode to avod eos

78 76 Chapte 5. Bas osdeatos of the expemet ad expemetal setup aused b a ustead state dug the measuemet, the esstae of able should be also measued fo deeasg tempeatue values dow to ev. Dug the measuemet poess, the able s ot heated eletall. Fo the qualtatve measuemet, the followg thee odtos must be fulflled: a Fo eve esstae measuemet step, tempeatue stead-state odto must be obtaed, b he measuemet eo due to mateal estats must be estmated, hemo voltages must be avoded. A. Requemets fo a stead-state tempeatue odto Hee, the most mpotat paamete s a stead-state tme t, afte whh tempeatue dffeee betwee h ad able tempeatue beomes zeo. he tme t s flueed b a tme ostat τ, whh s dffeet fo eve we tpe. I ode to alulate steadstate tme t, oe should deteme the fato,.e. the umbe of tme ostats τ eeded to eah a stead-state: t τ (5. o alulate fato, we a osde the peso of the heatg hambe tempeatue. We a assume that the tempeatue dffeee δ L betwee the able tempeatue ad hambe tempeatue should be equal to the tempeatue eo of the hambe δ h : δ L ϑe δ h (5. Fom thee, the fato s gve b: ϑ l (5.3 δ h whee: δ L tempeatue dffeee betwee able ad hambe tempeatues K δ h tempeatue measuemet eo of the heatg hambe K τ themal tme ostat of the able s ϑ step sze of tempeatue teval K tme ostat fato t stead-state tme s Hee we osde δ h fom the tehal spefato of heatg hambe.5 K. Step sze of tempeatue teval fo esstae measuemet s ϑ K. he, the fato s alulated: l / (5.4

79 5. Bas osdeatos of the expemet 77 ths meas, that ol afte 3τ able esstae a be measued. B. Pemssble measuemet eo of able esstae. I ode to estmate pemssble measuemet eo of able esstae, we ought to use the followg fomula: [ ( β ( ] R [ α ( ( ] R( t R α β (5.5 Afte dffeetato, we obta: [ α ] d dr( t R β (5.6 Fo ß, equato (5.6 smplfes to: dr d R α (5.7 If we appl a small dffeee to the esstae δr ad tempeatue δ, we get a tempeatue eo based o efeee esstae R ( at efeee tempeatue : δr R R R R α δ (5.8 I the ase that measuemet eo of tempeatue does ot exeed a value of δ <.5 K, we get maxmal pemssble elatve esstae eo δr/r fo oppe: δr R.9%.% (5.9 he total measuemet eo ossts of measued able tempeatues δ (heatg up, δ (oolg dow ad eo of heatg hambe tempeatue δ h : δ δ δ δ h (5. Iseted the equato (5.8 we obta the followg maxmal allowed esstae eo: - lea pat: β << α

80 78 Chapte 5. Bas osdeatos of the expemet ad expemetal setup δr R 3α δ (5. h - lea ad squae pat: δr R 3( α β δ (5. h δr hee: - maxmal pemssble esstae eo. R Fo the tempeatue teval to K (o osdeg evomet tempeatue ev C, to 4 C we get: δr R ( % δh (5.3 Cosdeg the tempeatue eo of the hambe δ h.5k we obta maxmal δr pemssble esstae eo : R δr R ( %.6% (5.4 C. Measuemet eo due to themo voltages. hemo voltages ou f two dffeet metals at dffeet tempeatues ome to otat. It s tal to pefom the expemet b measug voltage dop o the able athe tha dog a fou-pole esstae measuemet. hs s obvous fom the followg example: hemo voltage fo Cu vs Cu-Fe otat s about 5 µv/k wheeas fo K t makes.5 mv. I ase of 4-pole esstae measuemet fo mm oppe we themo voltage s: at ma uet ol 7.5 µv/m, fo 3 m we - 5 µv Expemet esults ae peseted the table 5.. Hee ae evaluated lea ad squae tempeatue oeffets α 65 ad ß 65 at 65 C efeee tempeatue espetvel.

81 5. Bas osdeatos of the expemet 79 Expemet. Cable tpe α 65 /K ß 65 /K. Roud able FLYR-B.5mm Extuded PE flat able, b5mm Extuded PE flat able, b5mm PE flat able bmm PE flat able bmm PE flat able b5mm ab. 5. Lea ad squae tempeatue oeffets of oud ad flat elet ables It a be see that squae tempeatue oeffet ß 65 s egatve whle theoetal value s /K. hs pheomeo s dffult to expla. heefoe, ol the lea tempeatue oeffet α 65 was take to osdeato fo the whole model valdato.. Se, tempeatue of the ables does ot exeed 4 C, lea appoxmato of the esstae as a futo of tempeatue s suffetl pese. 5.. Cuet vesus voltage measuemet he am of the seod expemet s the detemato of the uet-voltage haatest of wes ad ables. Fom ths haatest, the elatoshp betwee we esstae ad ts tempeatue a be detemed. hs fomato s ve mpotat ode to valdate the mathematal model of wes ad ables peseted Chaptes 3 ad 4. hs model valdato s gve Chapte 6, whee expemetal uves ae ompaed wth those fom umeal alulato. he expemet s pefomed b measug the voltage dop U fo dffeet uet values I. It s advsable to ease the uet I stepwse fom to maxmal pemssble oss seto uet I max equal tevals ϑi. he same poedue should be epeated whlst edug the uet dow fom I max to. Ambet tempeatue of the expemet evomet should be kept ostat ad a foed a movemet should also be avoded. I ode to pefom qualtatve expemet almost the same equemets as (5.. have to be stated: a Fo eve voltage-uet measuemet step, stead-state odto must be obtaed. b Measuemet eos due to estat of mateals must be estmated. hemo voltages must be avoded. d A mmum equed dstae betwee uet ad voltage dop o the able otats must be kept. he fst two equemets (a ad (b have the same defto as the esstaetempeatue expemet. he thd equemet ( should be eosdeed, se the themo voltage fluee s ot tal ths ase due to hgh uet dued hghe voltage dop.

82 8 Chapte 5. Bas osdeatos of the expemet ad expemetal setup he themo voltage fo Cu vs. Cu-Fe pa s 5 µv/k, that s fo K.5 mv. Cosdeg the voltage dop mm oppe we: Fo A measuemet uet, the voltage dop leads to.75 V/m. Fo a m legth oppe we (whh s a mmum equemet to avod the fluee of the bouda tempeatue dea the voltage dop esults.75 V. hs s about 3 tmes bgge value as the themo voltage fo K. Fom ths explaato follows that themo voltages do ot sgfatl fluee to measuemet eo. he dstae betwee uet ad voltage measuemet otats of o-sulated we a be alulated aodg to equatos (3.5, Chapte 3 s gve b: λ χ α ua α Aρ I ρ (5.5 Deotg A πd ad uπd we obta: 4 πd λ λd χ 3 απ d 4α I ρ 4α α di ρ (5.6 ρ ρ We a ewte equato (5.6 the followg fom: χ λd α ρ I ρ α απ d 3 (5.7 Fo a small uet dest, the Eq. (5.7 a be smplfed as follows: λd χ, α χ λd (5.8 α hee: s the umbe of legth ostats, whh ma be appled agg fom a gve tempeatue to the able tempeatue of a fte legth wth a aeptable eo. Fo applato ths stud a fato 3, leadg to a eo of 5 %, s suffetl pese: 3 λd d / mm 3 χ.897 (5.9 α α hee: d able damete mm.

83 5. Bas osdeatos of the expemet 8 If we osde a tal heat oveto oeffet α 6 ad we damete of. mm we get: 3χ 4 mm m Followg equatos ae used to ompute able tempeatue as a futo of uet I:. Detemato of able tempeatue : [ α ( - β ( - ] R [ α ( β ( ] R( R (5. hee: α β lea tempeatue oeffet of oppe at efeee tempeatue /K squae tempeatue oeffet of oppe at efeee tempeatue /K Mateal ostats α ad β have bee detemed b the expemet desbed (5.., howeve ol lea tempeatue oeffets wll be used to alulate able tempeatue. Cable tempeatue s foud fom Eq. (5.5: R α R (5. hee: umbe of measuemets wth oe expemet. Detemato of able esstae R : Fom voltage dop measuemet o the able at uet I aodg to Ohm s law the esstae s: U R I (5. hee: I uet alulated b voltage dop o the shut U s wth ostat estae R s : U I R s s (5.3

84 8 Chapte 5. Bas osdeatos of the expemet ad expemetal setup 5. Expemetal setup 5.. Detemato of the able oduto tempeatue oeffet Fgue 5. shows expemetal setup fo able esstae / tempeatue measuemet. he able s plaed a lqud slo bath, whh s stuated a heatg hambe (pe: xo 85, ISOECH. he lqud was used as a heat tasfeg meda ode to esue ve ostat ad homogeeous tempeatue the able. he heat ha m- be has ts ow tempeatue otol, whh howeve s ot pese eough fo ths kd of expemet. heefoe, a addtoal Pt seso was used to measue the lqud ad able tempeatue. GPIB (Geeal Pupose Itefae Bus GPIB Pt Heat hambe Lqud slo bath pestalt pump Hgh peso dgtal multmete-mlommete Lqud slo Flat/oud able 4-pole esstae measuemet Fg. 5. Expemetal setup fo the able oduto esstae measuemet at dffeet tempeatues fo the detemato of the tempeatue oeffet he voltage ad the uet measuemet oeto wes wee soldeed to the able ad sealed wth tempeatue esstat sulatg epox adhesve. hs sulato s ve mpotat ode to avod shot uts the flud. Resstae measuemet was pefomed b the 8 / dgts peso dgtal multmete PREMA 64S. hs multmete as well as the heat hambe was otolled b a GPIB (Geeal Pupose Itefae Bus otolle, whh allows the automato of the whole expemet. he lqud slo s ulated b a pump, whh allows moe pese tempeatue otol. All put data

85 5. Expemetal setup 83 (otol of expemet equpmet ad output data (measuemet esults ae hadled b a LabVew pogam, whh was spefall developed fo ths pupose. 5.. Detemato of the able oduto tempeatue I fgue 5., the expemetal setup s gve to measue able voltage dop at dffeet load uets. I ths expemet, the able s plaed fee a wthout touhg athg. he laboato oom has ostat ambet tempeatue of 4 C ad o sgfat a uets. Ude these laboato odtos, a pese measuemet of the able powe dsspato s possble. GPIB (Geeal Pupose Itefae Bus GPIB Hgh peso dgtal multmete-mlommete hemoouple Pot exteso boad...3a..8v max.5a 3mV Shut m legthm I DC powe suppl ut Measued able Fg. 5. Expemetal setup fo the able oduto voltage measuemet at dffeet uets fo the detemato of the oduto tempeatue All expemetal equpmet was otolled b a ompute va GPIB tefae ad house developed LabVew softwae (Appedx C, whh eabled data aqusto. he whole expemet s full automated, avodg a tefeee of a opeato wth the expemet evomet. he voltage dop ad the uet of the powe suppl ut EA-PS 98-3 ae dated b the ut, howeve t was moe auate to use a hgh peso dgtal voltmete ad a shut (peso lass. ode to measue the voltage dop ad the load uet. he able esstae s obtaed b the dvso of the voltage dop though the load uet. Hee, t s ve mpotat to take the tempeatue dop at the ed of the able to aout. heefoe, the measug pots of the voltage o the able must have a equed mmum dstae (see Eq. 5.9 fom the uet suppl oetos. Fall,

86 84 Chapte 5. Bas osdeatos of the expemet ad expemetal setup the able tempeatue s alulated fom the able esstae wth the help of the oduto tempeatue oeffet. Fo the sake of eduda, a seod depedet method to eeve able tempeatue has bee appled b usg a kel-ostata themoouple seso, whh was attahed to the able sufae. hs seso was also used fo safet to ut off the powe suppl ut f the maxmum allowed tempeatue of the able s exeeded. hemoouple sesos measue ve pesel eve f ol ve small sufaes ae avalable. hs s ve mpotat fo small sze elet ables. It would ot be possble to measue the tempeatue of suh ables wth a Pt seso. he dsadvatage of themoouples s howeve, that a ve pese efeee tempeatue of zeo degees s equed. Fo ths pupose, a wate-e mxtue was used. 5.3 Measug poess ad paamete aqusto 5.3. Detemato of the able oduto tempeatue oeffet he measug algothm of the esstae vs. tempeatue haatest of the able s gve b P. Mak [4]. Dug ths measug poess the able esstae was measued b 4-pole measuemet ad ts esults saved the ompute. empeatue of the able was measued b Pt seso, whh had bee plaed as lose as possble to the able. he esstae of Pt was also measued b the same dgtal multmete (DMM ad late, esstae values wee oveted to the tempeatue. hs oveso s possble b the empal equato: ' ' ' ' '.45 (5.9 C C C C hee s gve b the equato: ' o, ' ' α [ W ( ] δ C C (5. δ δ 4δ W ( ' α (5. δ

87 5.3 Measug poess ad paamete aqusto 85 hee: W ( R( / R( C, R( measued esstae of Pt seso Ω, R( C.6796 Ωm, α /K, δ.4965 C Detemato of the able oduto tempeatue he measug poess ad data aqusto of the voltage vs. uet expemet had bee mplemeted Pasal laguage b. Roda [5] ad late mpoved LabVew b the autho of ths stud (see also Fg. 5.3, 5.4. he dea of the measug poess s llustated the followg table: Set the powe suppl ut to 5 VDC he use defes the step sze ad teval of uets fom to ( umbe of uets he use sets the umbe of measuemets fo a sgle uet value fom to m (m umbe of measuemets of oe uet value he use sets oe watg tme fo all the measuemets s Stead state umbe of measuemet tmes (Watg tme fo oe measuemet Reset the powe suppl to A Close the fles ab 5. Algothm of the measuemet pogam he vsualsato of the measuemet data s gve the fgues 5.5 ad 5.6. he tme sale (Fg. 5.5 llustates the measuemet poedue ad gves fomato about the taset state egme. I Fg. 5.6 the tempeatue of a sulated oppe we s depted as futo of the load uet.

88 86 Chapte 5. Bas osdeatos of the expemet ad expemetal setup Fg. 5.3 Expemetal setup of flat elet able (FFC Fg. 5.4 Expemetal setup of oud elet able

89 5.3 Measug poess ad paamete aqusto 87 aset state able tempeatue C aset state of able sufae tempeatue Reahed stead state Measuemet tme t s Fg. 5.5 Expemetal taset tempeatue -tme haatest of eletal able

90 88 Chapte 5. Bas osdeatos of the expemet ad expemetal setup Measued stead state able tempeatue C Measuemet uet I A Fg. 5.6 Expemetal stead-state tempeatue -uet haatest of eletal able

91 CHAPER 6 MAHEMAICAL MODEL VALIDAIO AD IERPOLAIO OF HE UMERICAL RESULS hs hapte stats wth the valdato of the umeal smulato b the measuemet esults. Seto 6. pesets a least-squae tepolato of the valdated esults. he theoetal model s detemed b the themal odutvt of sulatg mateals, the tempeatue oeffet of oppe esstae ad the oveto ad adato oeffets. he umeal esults ae ftted b polomal o logathm futos, whee the oeffets of those futos ae obtaed b applg the least-squae tehque. 6. Mathematal model valdato Befoe statg the ompaso of the theoetal (umeal esults wth the expemetal oes, the qualt of the model equatos ad the evaluato poedue fo fttg them to the expemetal data should be heked. he qualt of deved heat equatos a be udestood as the eduto of a 3-D model to a -D model. Dug expemetal wok, the model eduto to -D poblem tued out to be suffet wth a eglgble eo betwee the theoetal ad the phsal model. I addto, bouda odtos a be osdeed as the qualt fato. Implemeted lmt odtos must be as lose to ealst odtos as possble. I ou model, the bouda odtos epeset lama fee oveto to the a whle the effets aused b tubulet oveto wee egleted. I ealt, the expemets wee pefomed a evomet whee the foed oveto was mmsed. I addto, oveto aused b tubulee was eglgble. he fst step was to the valdate lea tempeatue oeffet of the oduto esstae α ρ, se ths paamete s mpotat fo the oduto eletal esstae behavou the theoetal model. he bas mateal fo the poduto of staded odutos fo automotve wes s oxge oppe aodg to DI 45, pat 4. he DI desgato of ths oduto tpe s E-CU 58 F. Se ths mateal does ot osst of pue oppe, ad the maufatug poess mght have haged the popetes, t

92 9 Chapte 6. Mathematal model valdato ad tepolato of the umeal esults was eessa to measue ad valdate tempeatue oeffet of oppe esstae aga. Below the measuemet esults of oduto esstae as a futo of tempeatue ae gve:.8 Resstae R oms Measuemet data Polomal ft of measuemet data. Refeee tempeatue - C R.753[.4d] empeatue delta K Fg. 6. Measued esstae R of oud we fo tempeatue age fom to K. Measued tempeatue oeffet of oppe esstae s a.4 /K (based o C Resstae R oms Measuemet data Polomal ft of measuemet data. Refeee tempeatue - 65 C R.887[.336d] empeatue delta K Fg. 6. Measued esstae R of oud we fo a tempeatue age fom to 6 K. Measued tempeatue oeffet of oppe esstae s a.36 /K (based o 65 C

93 6.. Mathematal model valdato 9 Fom the measuemet esults (Fg.6. t a be see that α ρ has a value of.4 /K stead of.38 /K as gve the lteatue fo pue oppe mateal. Fgue 6. shows a tempeatue oeffet of.36 /K (based o 65 C, whh oespods the value of.4 /K (based o C. he value of.36 /K s also useful, se all ou alulatos ae based o the evomet tempeatue of 65 C. Below ae equatos fo the oveso of the tempeatue oeffet α ρ to a efeee tempeatue. Fo the devato, the followg futos of the eletal esstae R( though the pots ad ae osdeed: R( R R( R ( α β ( α β (6. hee: he am s to obta a elatoshp betwee α ad α. Fst ad seod ode devatves of Eq. (6. ae: ( ' R ( R α β (6.a ( ' R ( R α β (6.b '' R ( β R (6. '' R ( β R (6.d R βr β (6.e α R α R β R ( (6.f R R R( βr( α (6.g he the esult s the followg: R [ ( ( ] R β α (6. α β( ( β ( α (6.3 α

94 9 Chapte 6. Mathematal model valdato ad tepolato of the umeal esults β (6.4 ( β ( α β Se the o-lea pat of the tempeatue oeffet ß s ve small, fo tempeatue up to 4 C ß a be assumed zeo (ß. I that ase, the equatos ae smplfed as follows: R [ ( ] R α (6.5 α α α ( (6.6 he ext step of the model valdato s to estmate the heat odutvt oeffet? of the sulato mateal. It should be oted that dug ths eseah wok the heat odutvt oeffet was ot vestgated expemetall. Hee, ode to save vestgato tme ad osts, the oeffet? was obtaed b fttg of the tempeatue measuemet esults wth theoetal oes. Of ouse, t would have bee possble to use the oeffet fom the lteatue, howeve the avalable values wee ol gve a ve aow tempeatue age. heefoe, the? oeffet was vaed ode to ft to the expemetal uves of the oduto sufae tempeatue. Below the measued oduto sufae tempeatue of oud ad flat eletal ables ae ompaed wth the omputed oes. empeatue C Cable wdth: 5 mm Coduto wdth: 5 mm Lae sequee: (PE: µm Coppe: 47 µm (PE: µm Expemet umeal omputato Cuet I A Fg. 6.3 Measued tempeatue of a flat able oduto (oduto wdth 5 mm as futo of the elet load

95 6.. Mathematal model valdato 93 empeatue C Cable wdth: 74 mm Coduto wdth: 74 mm Lae sequee: (PE: µm Coppe: 6 µm (PE: µm Expemet umeal omputato Cuet I A Fg.6. 4 Measued tempeatue of a flat able oduto (oduto wdth 74 mm as futo of the elet load Fg Flat able (FFC examples

96 94 Chapte 6. Mathematal model valdato ad tepolato of the umeal esults empeatue C Coduto oss-seto:.5 mm Coduto: oppe Cu58 F Isulato: (PE Expemet umeal omputato (lambda.58 umeal omputato (lambda Cuet I A Fg Measued tempeatue of a oud we oduto (oss seto.5mm as futo of the elet load empeatue C Coduto oss-seto: 35 mm Coduto: oppe Cu58 F Isulato: (PVC Expemet umeal omputato Cuet I A Fg Measued tempeatue of a oud we oduto (oss seto 35mm as futo of the elet load

97 6.. Mathematal model valdato 95 Fg Roud able examples 8 6 Expemet umeal omputato 8 Amp empeatue C Amp 4 Amp Amp Amp 8 Amp 6 Amp 4 Amp,,,4,6,8, Legth m Fg. 6.9 Measued tempeatue of a fuse elemet as futo of the ele t load

98 96 Chapte 6. Mathematal model valdato ad tepolato of the umeal esults Fg. 6. Fuse elemet pototpe wth tempeatue sesos (themoouples. Fuse elemet (hollow lde dmesos: e damete 6mm, oute damete 8mm. Mateal tpe of the fuse elemet: Bas 58 (CuZ39Pb3. Fall, the heat ovetve ad adatve oeffets had to be valdated. Dug sepaate expemetal wok [8] a umbe of measuemets of vaous oud wes was aed out ode to valdate the empal heat oveto ad adato fomulas, whh wee peseted the seto (.5. he esults of ths expemetal wok have show that the evaluato poedue of oveto ad adato oeffets s oet. Se, t has alead bee desbed how the model was valdated, ow the obtaed esults a be studed. he followg thee models, whh dffe b the geomet ad tpe of mateals ae peseted:. Fgues 6.3 ad 6.4 show the flat able oduto tempeatue as futo of the eletal uet. hese esults ae obtaed fo a able, whh s plaed fee a, a hozotal posto. he tempeatue s take afte the stead state has bee eahed. Both, umeal smulato ad measuemet data, gves a ve good ageemet.. Fgues 6.6 ad 6.7 show the oud we oduto tempeatue as futo of the eletal uet. B alteg of the odutvt oeffet λ of PE (Polethlee fgue 6.6, a bette ageemet of both uves a be aheved. 3. Fgue 6.9 shows the tempeatue dstbuto alog the fuse elemet. he legth of the whole sstem was about mete, se the fluee of the fuse holdes ad otats must be osdeed. Also eletal we, whh s poteted b ths fuse had to be take to osdeato too. he tempeatue dstbuto s peseted fo dffeet eletal uet values. he alulated uves all fgues math wth the measued data a

99 6.. Itepolato of the umeal esults to edue heat tasfe equatos 97 eptabl well. At the boudaes of the fuse holdes ad otats, a lage eo s peset. hs a be explaed b the lmted pefomae of the expemet setup. empeatue of the fuse sstem was measued ol b thee themoouple sesos. Late, thee measuemet pots wee tepolated. Howeve, hgh aua of tempeatue gadet o the boudaes s ot of pma mpotae. Ol the fuse elemet maxmal tempeatue s of teest, beause ths tempeatue auses the equed teupto of the fuse-meltg elemet. Dsussed esults show the fat that the appoxmato made fo the devatves of the equatos (Chapte :.4,.6,.8 as well as the valdato of the phsal model ostats ae applable ude the expemetal odtos, whh ae teested ths otext. 6. Itepolato of the umeal esults to edue heat tasfe equatos houghout the ete stud, the heat tasfe aalss algothm was deved usg aaltal / umeal methods. hs algothm allows the detemato of the themoeletal haatest of eletal odutos both a stead- ad taset-state egmes. he poposed appoah povdes ve good aua betwee theo ad expemetal esults, s applable fo dffeet oduto geometes, ad a be exteded fom a -D to a -D poblem. Howeve, ve ofte, umeal smulato of heat tasfe eques a lot of omputato tme that s ot aeptable f the umeal smulato oute has to be tegated to aothe omplex smulato sstem. hs leads fall to a stuato, whee the whole pefomae of a omplex smulato sstem beomes ve poo. Aothe dsadvatage of pue umeal smulato of heat tasfe poblem s that ve ofte a eal lfe, the alulatos have to be doe ve qukl ad a smple mae heefoe, ou teso ths stud s ot ol to peset full-developed umeal models but also to develop smple, wth the mmum umbe of phsal ostats, aaltal equatos, whh desbe best the stead-state ad taset-state heat tasfe egmes a tpe of odutos. hese equatos should have the advatage of podug a maageable elatoshp havg ol two o thee ostats, whh a be obtaed easl b the least-squae (LS algothm. Fall, havg those smple equatos, a opeato a pefom the alulatos a eas wa. I ths seto we wll peset polomal ad logathmal equatos of themoeletal haatests ad show how to appl the LS algothm [8,9] fo the alulato of the smplfed-equato oeffets. It s ve mpotat to pedt oetl the oet equatos wth a mmum umbe of ukow ostats. Basall, two tpe of futos ae of teest: polomal egesso ad logathmal futos. Wth these two futos, all mpotat themo-eletal haatests a be desbed. he followg futos ae osdeed, whose phsal meag wll be gve late: - themo-eletal haatest (I (Fg.6.

100 98 Chapte 6. Mathematal model valdato ad tepolato of the umeal esults - heatg-up tme haatest tg(i (Fg.6. - tme ostat haatest τ(i (Fg elet-feld stegth haatest E(I (Fg.6.4 It s woth to emphasse that these fou futos ae vald fo a kd of oduto (flat ables, fuses o able budles, whee heat geeato b eletal uet takes plae. Fo the llustato of the LS algothm, a oud sulated we of FLRY-B -.5mm tpe s used. Hee the maxmal fal tempeatue of the we s 5 C ad evomet tempeatue 65 C. 6 empeatue delta K umeal omputato Polomal ft a.8 b.469 d a*i b* I Cuet I A Fg. 6.. hemo-eletal haatest D(I of oud sulated we (FLRY-B,.5mm obtaed fom umeal alulato ad appoxmated b polomal futo

101 6.. Itepolato of the umeal esults to edue heat tasfe equatos 99 Heatg up tme tg se umeal omputato Logathmal ft tau I I Cuet I A Fg. 6.. Heatg-up tme tg(i haatest of oud sulated we (FLRY-B,.5mm obtaed fom umeal alulato ad appoxmated b logathmal futo me ostat tau se umeal omputato Polomal ft tau d.935 tau tau - * I.5 d*i Cuet I A Fg me ostat t(i haatest of oud sulated we (FLRY-B,.5mm obtaed fom umeal alulato ad appoxmated b polomal futo

102 Chapte 6. Mathematal model valdato ad tepolato of the umeal esults Eletal mfeld Steght E V/m Cuet I A Fg. 6.4 Elet feld stegth E(I haatest of oud sulated we (FLRY-B,.5mm obtaed fom Eq. (6. he uves peseted fgues 6., 6., 6.3 ad 6.4 a be desbed b the followg equatos: - themo-eletal haatest (hee I : - heatg-up tme haatest: ( I I a I b I (6.7 - tme ostat haatest: t g I I ( I > I τ g l I (6.8 τ τ - eletal feld stegth haatest:.5 I d I (6.9 Iρ E A Iρ( α ρ β ρ ( A (6.

103 6.. Itepolato of the umeal esults to edue heat tasfe equatos I equato (6. tempeatue dffeee s alulated b Eq. (6.7 he oeffets of the equatos (6.7, 6.8, 6.9 a, b, I, τ g, τ,, d, ae vald ol fo oe spef tpe of we. If aothe we tpe has to be vestgated, the oeffets have to be e-omputed. he Least-Squae Method (LS a be used to obta these oeffets. Applg LS method ad solvg the lea sstem of the equatos leads to the followg equed oeffets: - fal tempeatue pe uet oeffet a: ł Ł - - I I I I I I I a (6. - fal tempeatue pe uet squae oeffet b: ł Ł - - I I I I I I I b (6. - heatg-up tme ostat τ g ł Ł - - g I I I I I I t l l τ (6.3 - tme ostat pe squae oot uet oeffet

104 Chapte 6. Mathematal model valdato ad tepolato of the umeal esults τ I I - τ I I - I I τ I 4.5 I I - I Ł ł I τ (6.4 - tme ostat pe squae uet oeffet d hee: τ I I - τ I I - I I τ I I τ d.5 4 I - I I Ł ł (6.5, ; whee s the umbe of alulatg pots the we haatests; I the eletal uet values, whh ae used the haatests of Fg tempeatue values, whh ae used the haatest of Fg. 6. t heatg up tme values, whh ae used the haatest of Fg. 6. τ tme ostat values, whh ae used the haatest of Fg. 6.3 he elatoshp peseted Fg. 6. s a stead-state load haatest of elet ables. Hee the tempeatue epesets the stead-state fo a patula load uet value. omall, ths haatest eds wth the maxmum allowed tempeatue of the we o able afte fte tme. I Fg. 6. heatg up tme s gve as a futo of the load uet. hs elatoshp s well kow fom the fuse tme-uet haatest ad t makes sese to appl the same haatest to a elet we o able. he heatg up tme s the tme to eah maxmal pemssble tempeatue wth a uet geate tha the omal load. Fo example, Fg C degees ae gve as maxmal pemssble tempeatue. Havg avalable the tme-uet haatests fo both: wes ad fuses, t s possble to model the geomet of the fuse to math the heatg-up tme futo of the we. Fall, havg a fuse wth suh a haatest, t s possble to potet the we wth good aua. Fg. 6.3 gves the tme ostat as a futo of load uet. hs uve gves the possblt to obta a τ ostat as a futo of ol oe vaable; the uet. It s also staghtfowad to ompute the taset state aaltall, havg τ as the kow paamete. Fg. 6.4 epesets the elet feld stegth depedee o load uet. Hee, the o-lea uve behavou s due to o-lea elet esstae depedee o tempeatue.

105 CHAPER 7 CALCULAIO OF HE HEA RASFER I A MULI-WIRE BUDLE Cota to pevous haptes, whee the heat tasfe was modelled fo a sgle elet able, ths hapte possble methods to alulate the heat oduto a mult-able budle wll be show. he ma effot to solve ths poblem s devoted to the lea oodate tasfomato ode to smplf the model geomet ad to the detemato of a aveaged heat odutvt oeffet of the mult-able mateal meda. hs hapte deals wth a oe-dmesoal adal stead state heat oduto poblem, whee the heat tasfe equato s solved aaltall. I aalog to a sgle sulated oduto (see Fg. 7.a, the mult-sulated able oduto (see Fg. 7.b s osdeed as a sulated mxed oduto. 7. Coodate tasfomato of the mult-we budle geomet he alulato of heat tasfe a mult-able budle belogs to the heat oduto poblems fo asotop mult-laeed meda [8]. he heat odutvt oeffet has bee detemed usg osevatve aveagg method fo laeed meda [78], whee heat odutvt of sgle ables ae tasfomed to a ommo mxed popet o mxed heat odutvt oeffet. he algothm of tempeatue detemato usg a osevatve aveagg method a be foud Chapte 8, (3 he alulato method osdes a asotop mateal that s homogeeous ad has ostat themo-phsal popetes. It also osdes adal smmet,.e. o agula tempeatue gadet / φ. he, the goveg patal dffeetal equato fo the heat oduto poblem a ldal oodate sstem beomes: ( λ qv ( (7. hee:? q v themal odutvt oeffet, tempeatue feld, volumet heat geeato.

106 4 Chapte 7. Calulato of th e heat tasfe a mult-we budle hs leads to a tempeatue dop the able budle, whh s obtaed a smla wa as Chapte 3 (aaltal aalss of heat tasfe ldal wes: p π α I 4 S l ( D S λ D λ L (7.a hee:? I,? L heat odutvt of sulato ad of mxed oduto W/mK, D damete of the able wthout sulato m, I the uet A, p p desbes the eletal powe pe legth EI,.e. a sum of all sgle wes W/m, S thkess of sulato m. I a mathematal sese, Eq. (7. s tasfomed b the lea oodate tasfomato as show fgue 7.. I a phsal sese, the goveg equato (7. of a asotop heat oduto poblem s oveted to a equvalet sotop poblem b eplag dffeet mateal oeffets b mxed mateal popetes. hs tasfomato has the followg haatests: a t s lea ad otuous, b a asotop poblem s oveted to a sotop poblem afte tasfomato, thee s o steth ad the otato adal deto, d o gaps o ovelaps ae geeated alog the tefae, e o sldg ad msmathes ou alog the tefae. hese featues offe advatages dealg wth staght boudaes ad tefaes the mult-laeed sstem. I ths stud the oveso of oud sulated wes to a squae oes wth the same oduto, sulato ad a oss seto s poposed (see Fg. 7.. a b Fg. 7. Isulated sgle (a ad sulated mult-we oduto (b

107 7.. Coodate tasfomato of the mult-we budle geomet 5 a b Fg. 7. asfomato of sulated oud odutos (a to squaes of same ae a (b he squae stutue a be ow alulated easl as a themal seal-paallel swthed model of smlal oveed aeas. I ode to smplf the alulato, the omplete sold mateal was sepaated fom the a ad ombed thee laes. he fluee of the lowe heat odutvt of a wll be osdeed late wth a so-alled fllg fato. a b Fg. 7.3 Detemato of the mxed aea odutvt: (a assembl, (b ut Fo log (ompaed to the thkess wes, the whole mateal a be teated twodmesoall wth a so-alled aea odutvt?l ad whh s popotoal to the kow heat odutvt?. he heat odutae G s gve b

108 6 Chapte 7. Calulato of th e heat tasfe a mult-we budle G λlx ad fo eah lae gve Fg.3 oe obtas: G λl, l, G λ a G λ l, b a b a G 3 λ 3 l (7. b Fall, a Mxed Mateal Equato of the followg fom s obtaed: a λ λ (7.3 λ b b λ a whee the heat odutae ae eplaed b the heat odutvtes. hee:? adal mxed heat odutvt,? heat odutvt of the oduto,? heat odutvt of the we sulato (see Fg.7., a edge of heat oduto mateal (?, mm, b edge of mxed heat oduto mateal ad (? mm. he ext step s to deteme the elatoshp of b/a whh s alulated fom the aea a of the mateals ad (whh s the oss seto of the oduto ad the solato. Se the oduto ossts of sgle we ves wth a gaps betwee, the eal oduto oss seto a has stll to be multpled wth the so-alled fllg fato f, whh s the elatoshp of the eal odutg oss seto to the oss seto to be detemed b ts measued damete: A a f. he oss seto of the oduto ad the solato togethe esults : A f A b (7.4 fom whh fall b/a a be alulated, wheeb the aeas A ad A ma be eplaed b the sums of the damete squaes of the oduto d ad of the we ves δ: b a fa A f d δ f Σ (7.5

109 7.. Coodate tasfomato of the mult-we budle geomet 7 he model desbed b the equato 7.3 s ol vald fo mateals wth smla heat odutvt. If oe pat dvets as fa as a ompaed to oppe, the model s o loge applable. I ths ase, the assumpto s made, that the oduto ossts of two dffeet odutg mateals, whh ae swthed paallel as follows: G G a G b. Fg. 7.4 Volume hage due to empt spaes, aodae wth the fllg fato f o F I aodae wth Fg. 7.4 the heat odutvtes a be alulated as follows: G λ, G a λa, G b b b λ a (7.6 b hee:? mxed heat odutvt,? a,? b heat odutvt of aea a ad b, espetvel, a, b vtual legth of all odutos ad of mxed oduto, vtual legth /b of b. Replag a squae elemet b two etagula elemets wth dffeet heat odutvt, (see Fg. 7.4 leads to the mxed heat odutvt: b λ λa λb. b b Usg the fllg fato: f b

110 8 Chapte 7. Calulato of th e heat tasfe a mult-we budle fo the a betwee ves a we ad F b b fo the a betwee wes a able gves the thg equatos: λ λ f λ ( f λ f, λ λ F λ ( F λ a b a a b a F (7.7 Assumg the applato of the fst model (Eq. 7.3 s moe suted fo sulated odutos ad the seod model (Eq. 7.7 s moe suted fo o-sulated odutos wth gaps betwee, the two equatos a be ombed as follows. I ths ase, the heat odutvt? of the mxed mateal wthout a a be eplaed b the heat odutvt?f of the mxed mateal wth a, wheeb F s the sulated we fllg fato. Assumg, that the oduto mateal has muh hghe heat odutvt tha the a betwee, the equato ma be smplfed eve futhe, e.g. fo? f>>? F oe obtas: λ λ F λf f λ f Σ λ f Σ f Σ f Σ ( Calulato of tempeatue dstbuto the eal mult-we budle Cosdeg the followg able budle stutue, (see ptue 7.6: We tpe Coss seto umbe of sgle wes (ves Damete of the sgle we (ve We (ve damete umbe of wes the budle δ m mm mm mm FLRY-A,35 7,6,8 FLRY-A,5 9,9, FLRY-A,75 9,3, FLRY-A,5 9,3,7 5 FLRY-B,5 5,6, 5 FLRY-B 4 56,3,75 ab 7.. Phsal data of the able budle

111 7.. Calulato of tempeatue dstbuto the eal mult-we budle 9 Fst, alulatg the fll fato f of the we: δ f (7.9 hee:? we damete mm Fom the oduto ad we ve squaed sum data, a oss seto quotet Σ of able budle s alulated: Σ d m m d δ δ (7. Se, expemet data ae avalable of the able budle peseted the 7. table, the budle fll fato of able budle F empall a be alulated: ( 4 S U U I l F S π ρ π Σ (7. Fall, havg all equed fomato to alulate adal aveaged heat oduto oeffet?, Eq. (7.8 a be used to obta ths oeffet: Σ Σ f f F λ λ (7.8

112 Chapte 7. Calulato of the heat tasfe a mult-we budle 9 8 Measued tempeatue Calulated tempeatue 7 empeatue K Cuet A Fg. 7.5 Expemetal vesus theoetal esults of the able budle. Evomet te m- peatue 3 C I Fg. (7.5 the tempeatue depedee o load uet of the peseted able budle (see ab. 7. s gve. Hee the expemetal esults ae ompaed wth the esults obtaed usg the ew deved aveaged heat odutvt oeffet?. hee s some eo betwee two uves, whh a ot be full explaed et. It seems that the appled model fo the budle does ot ompletel desbe the ealt eve ase. he easo fo ths obsevato mght be geometal dffeees betwee the ealt ad the model. Aothe easo ould ome though a dffeee betwee the eal ad the alulated heat adato fom the budle sufae. he elevat emssve oeffet s ot kow pese eough fo ths applato. All these obsevatos ae a aea of futhe osdeato f eessa.

113 7.. Calulato of tempeatue dstbuto the eal mult-we budle Fg. 7.6 Expemetal setup of mult-we budle Despte of all these small eos, the esults of the alulatos ae moe tha suffet fo patal applatos.

114 CHAPER 8 SUMMARY AD COCLUSIOS 8. Summa Heat tasfe ad tempeatue dstbuto eletal ables ad fuses have bee studed aaltall ad umeall the peset eseah wok. Obtaed esults fom umeal smulato ae tepolated b the Least-Squae method that led to smple polomal ad logathmal futos of the ma themo-eletal haatests. he geometes of phsal models ad appopate heat tasfe equatos ae peseted Chapte. Aaltal solutos of the heat equatos fo dffeet oduto tpes have bee obtaed Chapte 3. A umeal appoah based o a fte volume method has bee studed Chapte 4. A ew omputatoal algothm to ompute taset state themo-eletal haatests of eletal ables was also desbed Chapte 4. he expemetal setup ad the exeuto poedue of the expemets s explaed Chapte 5. he aheved omputatoal esults of the developed mathematal model wee vefed b laboato expemets. Itepolato usg Leastsquaes tehque of umeal esults ode to edue the amout of umeal data s gve Chapte 6. Fall, a ew appoah to alulate heat oduto oeffet of multlae able budles s peseted Chapte 7. he ultmate goal of ths wok s to develop a methodolog of the aalss of heat tasfe eletal ables ad fuses. he ma esult of ths aalss s: tempeatue dstbuto the odutos ad taset state themo-eletal haatests. o obta these stead- / taset-state haatests a umeal algothm had to be emploed, se most mateals ae tempeatue depedat. Aothe easo fo a use of umeal algothm s that the heat tasfe alulato s muh ease to pefom b a umeal appoah stead of applg aaltal Foue sees solutos. A oe-dmesoal model fo dffeet kd s of elet odutos s peseted seto.. he phsal models of thee dffeet tpes of odutos: flat able, oud we ad fuse meltg elemet ae take as phsal examples fo mathematal modellg, smulato ad aalss of heat tasfe. Whe eatg the mathematal models (seto.3 the heat tasfe equatos wee edued to oe-dmesoal poblems (Eq..3,.6,.7. I ode to have a lea udestadg about the modellg of the heat tasfe elet odutos the ete seod hapte s devoted to the devato of a mathematal model. heefoe, setos.4,.5 desbe the aalss of heat oduto ad themal oveto espetvel. Se, oe has to deal wth a tal-bouda

115 4 Chapte 8. Summa ad olusos value poblem, bouda odtos ae desbed seto.6. Hee ae the fst kd (Dhlet ad mxed tpe (smmet ad ovetve-adatve lmt odtos ae peseted. Addtoall, umeal mplemetato of the same kd of bouda odtos s gve seto Afte eato ad pepaato of the mathematal model of elet odutos the fo l- lowg Chaptes 3 ad 4 gve detaled aalss of aaltal ad umeal alulato poedues of tempeatue dstbuto the odutos. I the aaltal aalss (Chapte 3, exat solutos of stead state heat tasfe equatos wee obtaed. hese solutos ae gve fo tempeatue depedet mateal oeffets as well as fo tempeatue depedet oeffets. Dffeet bouda odtos (smmet o deved fom eeg balae equato wee mplemeted fo aaltal expessos. A ve mpotat popet was obtaed fom the heat equato wth tempeatue depedat oeffets. hs popet s alled avalahe effet ad a be desbed b the followg equato: f, α ρ ρi αu α ρ IE (3.53 A the, χ, ˆ. A A he ext step of heat tasfe aalss the elet odutos was to develop a umeal model to ompute stead- ad taset-state tempeatue behavou the odutos. he umeal model was eated usg a fte volume method. he tegal fom of the heat equato was dsetsed usg etal dffeees spae ad a bakwad dffeee sheme tme. hus, seod ode aua sheme spae ad fst ode aua tme was aheved. Dsetsato of the equatos tme wee made mpltl ode to aheve uodtoal stablt ad ease omputatoal effe. All fte volume shemes wee developed o stutued gds, howeve ths method s staghtfowad applable o ustutued gds too. Sstems of algeba equatos wee solved b the teatve ewto-raphso method, whh has fast ovegee f a sutable tal guess s made. As omputatoal esults, followg haatests wee obtaed (Fg. 6., , Chapte 6: a oduto sufae tempeatue as a futo of load uet, b heatg up tme as a futo of load uet, tme ostat as a futo of load uet, d elet feld stegth as a futo of load uet (osdeg o-lea elet esstae depedee o tempeatue e tempeatue dstbuto the fuse elemet.

116 8. Summa 5 Chapte 5 detals how the theoetal model of elet ables ad fuses has bee tested. Fo ths pupose, two dffeet expemets (seto 5.. ad 5.. wee made. he fst expemet deals wth the detemato of the tempeatue oeffet of oppe esstae. hs oeffet s a mpotat paamete fo the detemato of the able esstae depedee o tempeatue. hee dffeet ables wee plaed a heated lqud slo bath. he able oduto esstae hage due to tempeatue was measued. eessa theo was developed ad desbed the seto 5... he expemetal set up s desbed seto 5.. he seod expemet was desged to measue the able powe dsspato. Hee, all avalable wes ad ables as well as fuses ad able budles wee oeted to the det uet powe suppl soue ad loaded wth the powe fom to the maxmal allowed value. he whole expemet was otolled b the softwae developed fo ths pupose. he eessa theo of ths expemet ad expemetal setup ae gve the setos 5.. ad 5. espetvel. he measug poess ad the paamete aqusto poedue was desbed the seto 5.3. Chapte 6 desbes the valdato of the mathematal model of elet odutos ad the tepolato poedue of the umeal esults b smple polomal futos. he valdato of the model was based o estmatg the paametes, whh most fluee the heat tasfe elet ables. heefoe, osdeable atteto was pad to the valdato of the heat oveto oeffet, α, ad the tempeatue oeffet of oppe esstae, α ρ. Fom seto 6. oe ould see that the mathematal model was popel deved ad appoxmated, se haatests peseted fgues 6.3, 6.4, 6.6, 6.7, 6.9 have good ageemet wth expemetal data. I seto 6. a algothm was developed to tepolate umeal esults. All themoeletal haatests wee tepolated b smple polomal futos usg the Least-Squae method. hese futos ae ve useful fo patal alulato poblems of elet ables. Also, f suh equatos ae mplemeted the ompute pogamme to alulate themal pefomae of elet ables, the pogamme gves ve good omputatoal effe tme. he polomal futos ae gve Eq. ( he fal hapte oludes ths stud wth a ewl developed appoah of heat tasfe alulato the mult-we able budle. hs hapte apples the pples, whh have bee deved fo sgle odutos to mult-wes. Ol afte detaled eseah of heat tasfe models fo sgle wes ad ables was t possble to deve the methodolog fo the able budle. he ke of ths methodolog s to alulate aveaged heat odutvt oeffet of mult- we laes. heefoe, t was poposed to use the lea oodate tasfomato ode to smplf able budle geomet. A aveagg method fo laeed meda was the used to deteme a mxed o aveaged heat odutvt oeffet. he able budle model s gve Fg. 7. ad ts tasfomato to squae fames of the same aea Fg. 7.. Developed mxed mateal equatos fo themal odutvt oeffet s gve b the Eq I the followg seto 7. the ompaso of the esults was gve. he wee obtaed wth the alulated mxed heat

117 6 Chapte 8. Summa ad olusos odutvt oeffet ad ompaed wth measuemets fom the able budle. Fom the fgue 7.5, oe ould see that the developed method to alulate the aveaged heat odutvt oeffet gves suffet aua fo patal applatos. 8. Colusos Fou dffeet oe-dmesoal aaltal ad umeal models wee suessfull developed ths eseah wok, whh ae able to smulate heat tasfe a kd of elet odutos suh as flat ables, oud elet wes, elet fuses o mult-we able budles. he ew aaltal-umeal appoah, whh has bee poposed ths wok, allows aalss of the themo-eletal haatests of elet odutos. Fom ths stud, seveal othe olusos ae as follows: Aaltal solutos fo stead-state tempeatue dstbuto a sgle oduto wee obtaed. hese solutos ae vald fo tempeatue depedet mateal ostats as well as tempeatue depedat mateal ostats. A avalahe effet has bee obtaed fom these aaltal solutos (seto It was also obseved that taset heat tasfe equatos eque ve omplated tehques whh ae tme ad spae depedet ode to obta exat solutos. heefoe t s ot woth makg too muh effot to solve taset heat tasfe equatos. Istead, a umeal appoah should be used. A umeal model as developed to smulate heat tasfe the odutos, whh s based o the fte volume method. hs smulates heat tasfe ve pesel due to the sheme of seod ode spae. Also mplt shemes tme poved to have bette omputatoal pefomae tha explt oes f tme aua s ot of pma mpotae. Expemets delveed qualtatve data ad eablg a estmate of the qualt of the mathematal model. A ew method to alulate heat tasfe the mult-we able budle was eated. Ogal equatos of heat odutvt oeffet wee used b tasfomg the omplated we budle geomet to moe smple squaes. Equatos of mxed mateal popetes podue aveaged heat odutvt oeffets fo a able budle, whh osst of mateals as oppe, PVC ad a. A aaltal-umeal appoah was developed to smulate heat tasfe elet odutos. hs was mplemeted to a ompute aded desg pogam to optmse themal pefomae of the ables. he developed softwae allows aalss of themo-elet popetes of both wes ad fuses. hs aalss s ve mpotat fo fuse maufatug poesses ode to obta bette tme-uet haatests fo we ad able poteto agast oveload ad shot-ut uets. I the futue, ompaes

118 8.. Colusos 7 ould desg fuse mateals, usg the developed softwae. he esult would be ve aow tme-uet haatest of a fuse (lose to we tme-uet haatest. Fall, wes ad ables a be loaded wth % load whle beg poteted b the fuses a elable wa. he ma esults of ths stud wee peseted thee teatoal ofeees (Lthuaa, 3 ad publshed thee teatoal ouals. Lst of publatos:. A. Ilgevus, H.D. Less. hemal Aalss of Eletal Wes b Fte Volume Method. Eletos ad Eletal Egeeg.. 4 (46, Kauas, 3.. A. Ilgevus, H.D. Less. Calulato of the Heat asfe Cldal Wes ad Fuses b Implt Fte Volume Method. Mathematal Modellg ad Aalss, 8(3, 7-7, H.D.Less, A. Ilgevus. Aaltal vesus umeal Solutos of Phsal Poblems. he Beefts of ts Combato. Mathematal Modellg ad Aalss, 8(4, 9-3, Suggestos fo futue eseah I the peset eseah a oe-dmesoal model fo dffeet geometes of phsal models have bee developed. Howeve a two-dmesoal model would be appeable fo heat tasfe smulato omplated fuse elemet geometes ad mult-we budles. Also, a mesh geeato tool fo Catesa ad adal oodates s desable.

119 8 Chapte 8. Summa ad olusos Fg. 8.. Mult-we tee of o-boad eletal sstem Moe eseah wok should be udetake developg mathematal models of able budles o eve able tees (see Fg. 8.. Heat odutvt sde the budle has to be estmated b futhe expemetal woks ode to valdate the umeal model. Studes should also be made to aseta whh umeal method s moe appopate fo the model fo ables budle ad able tees: fte volume o fte elemet method.

120 APPEDIX A HEA RASFER EQUAIOS FOR ELECRIC CODUCORS A. Heat tasfe equatos fo flat elet able he am of ths appedx s to gve the devato of heat tasfe equatos fom the phsal pot of vew. It wll be show how to obta oe-dmesoal taset state heat equatos fo dffeet heat tasfe detos eletal odutos. I ode to deve the heat equato fo vetal heat tasfe applable fo heat tasfe a flat able, two assumptos have to be fomulated:. he heat flux q s a veto. hs veto s omal to the oss seto aea. Moe geeall, the heat flows a sotop meda wth some heat odutvt? agast the veto of tempeatue gadet: q λ gad (A.. he hage of heat flux dv q s a veto. hs veto desbes the heat flux hage spae pe volume ad tme ad s popotoal to the heat apat γ ad to the ate of tempeatue hage: dvq γ t (A. o, dv( λ gad γ (A.3 t Fo? ost. ad se dv gad UU, dffeetal equato fo a meda s of the followg fom: γ λ t (A.4 hee:? Heat apat Ws/m 3 K? Heat odutvt W/mK? Opeato /m q Heat flux W/m empeatue K ode C he opeato? s gve b: fo Catesa oodates:

121 Appedx A. Heat tasfe equatos fo eletal odutos x z fo ldal oodates: x ϕ dp Fg. A. Vetal heat tasfe the flat able Aodg to the ptue A, elet powe P e s gve b: dp e O E J d (A.5 I z- deto the heat powe P z a be egleted: P z (A.6 I - deto the heat powe though the able sufae s as follows: d(, t P λ O (A.7 d I the able aumulated heat eeg Q: dq γ O (, td (A.8 hee: O a l s the sufae aea m Fom the eeg balae equato, P e s gve b:

122 A. Heat tasfe equatos fo oud elet we dq Pe P o dt dq dpe dp d (A.9 Ł dt ł B setg the equatos A.5, A.7 ad A.8 to the equato A.9, we obta a dffeetal equato fo taset tempeatue dstbuto the flat able: OE J (, t d (, t d λ O γ Od d (A. d dt dvdg b O d: (, t (, t λ E J γ (A. t o, (, t E J λ γ λ (, t t (A. A. Heat tasfe equatos fo oud elet we Fo adal heat tasfe, dffeetal equatos wth ldal oodates have to be used. I fgue A. heat oduto s gve. Fg. A. Radal heat tasfe the oud we Elet powe P e s gve b: dp e O E J d (A.3

123 Appedx A. Heat tasfe equatos fo eletal odutos whee: O l u p l - the sufae dp e p l E J d (A.3a Radal heat powe of the we P s gve b: d (, t d (, t P λ O π λ l (A.4 d d d(, t d (, t dp π λ l d π λ l d d I the we aumulated heat eeg Q: (, t d π γ l d ( td d Q γ l u d, (A.5 Fom the eeg balae equato, fo adal heat oduto, the powe P s gve b: dq dq Pe P o dpe dp d (A.6 dt dt Afte seto of the equatos A.3a, A.4 ad A.5 to the equato A.6, the dffeetal equato fo taset tempeatue dstbuto the ldal we follows: (, t d (, t π l E J d π λ l d (, t π l λ π l γ d (A.7 d dt dvded b π l d we have, (, t (, t (, t λ λ E J γ (A.7a t o, (, t (, t E J λ γ λ (, t t (A.8

124 A3. Heat tasfe equatos fo elet fuse elemet 3 A.3 Heat tasfe equatos fo a elet fuse elemet Hee we osde axal heat tasfe a ldal o flat bod elet fuse wth ostat oss seto. he axal heat oduto a ldal fuse elemet s gve ptue A.3. Fg. A.3 Axal heat tasfe the fuse Aodg to Fg. A3, elet powe P e s gve b: dp e A E J dx (A.9 Radal heat oduto P though the sufae s: dp a u (x,t dx (A. Axal heat oduto P x alog the fuse s:: d ( x, t P x λ A (A. dx I the we aumulated heat eeg Q s: dq γ A ( x, tdx (A. Fom the eeg balae equato, P s gve b:

125 4 Appedx A. Heat tasfe equatos fo eletal odutos dq dq Pe P Px o dpe dp dpx d (A.3 dt dt Repeatg the same poedue fo the fuse elemet as was appled fo flat ad ldal ables. Equatos A.9, A., A. ad A. ae seted to the equato A.3. Fall the dffeetal equato fo taset tempeatue dstbuto the fuse elemet leads to: ( x, t d ( x, t AE J dx α u ( x, t dx λ A γ Ad dx (A.4 dx dt dvded b A dx, ( x, t d α u ( x, t λ ( x, t E J γ d (A.4a dx A dt o, ( x, t x α u λ A E J λ γ λ ( x, t t ( x, t (A.5

126 APPEDIX B UMERICAL ALGORIHM APPLICAIO FOR HEA RASFER SIMULAIO B. umeal heat tasfe smulato ad tepolato of the esults he am of ths appedx s to llustate the patal mplemetato of a umeal algothm (Chapte 4 ode to smulate the heat tasfe elet odutos ad the mplemetato of the tepolato algothm (Chapte 6. I the fgue B. the gaphal tefae to put pe-poessg data fo the umeal smulato of heat tasfe ldal wes s gve. Hee the use has the possblt to hoose a geomet of the we ldal oodates, evomet tempeatue ad to appl dffeet tpes of mateals. he bouda odtos ae, howeve, fxed, ad expose fee oveto ad adato to the a. As postpoessg fomato, umeal epesetato a ASCII fle (see Fg. B.. Itepolato esults ae saved the ba fle (see Fg. B.3 Fg. B. Wdow of pe -poessg fomato fo the heat tasfe smulato pogam ldal wes

127 6 Appedx B. umeal algothm applato fo heat tasfe smulato Fg. B. Smulato data of a ldal we wth mm oss seto. Hee: I load uet amps, oduto sufae tempeatue C, E eletal feld stegth V/m, tau tme ostat afte 5 se, tg heatg up tme to eah 9 C se. B. Calulato of themo-elet haatests b the polomal futos I ode to be able to wok wth a tepolated mathematal model o wth smplfed polomal futos, aothe pogamme was eated (see Fg. B4. Hee, usg equatos ( gve seto 6., themo-eletal haatests a be alulated.

128 7 B.. Calulato of themo-eletal haatests b the polomal futos Fg. B.3 Ba fle, whee all tepolato oeffets ae saved (A, B, C, D, au.

129 8 Appedx B. umeal algothm applato fo heat tasfe smulato Fg. B.4 Pogamme, whh alulates themo-eletal haatests of ldal we b smplfed polomal futos.

130 APPEDIX C SOFWARE FOR MEASUREME DAA ACQUISIIO C. Algothm despto ad measuemet pogam I the seto 5.3 two dffeet measuemet poedues of elet ables to valdate the mathematal model wee explaed. I ths appedx, a lose look at the measuemet softwae wll be gve. Both expemets wee u b the same tpe of softwae, mplemeted the LabVew 6. evomet. Fo the explaato ths hapte, ol the seod expemet softwae wll be peseted. hs pogam otols the powe suppl of det uet ad dgtal multmete, to whh the measuemet sesos wee attahed. he otol of measuemet equpmet s mplemeted b a GPIB otolle. he pogam has the followg stutue: - set the voltage to 5V; - ope the fles eeded to save the esults ad the omputatos; - eset the multmete; - wte to the fles the date ad the tme; - ompute R 65 ode to do ths: appl a amp uet fo 6 seods, ead the voltage dops o the able ad o the shut, ead the evomet tempeatue ad fall ompute t; - fo all uet values (fo to umbe of uets do: o ompute the peset uet ad appl t to the able; o fo umbe of measuemets 4 to umbe of measuemets do: set the multmete sale to DC voltage age; ead the voltage dop o the shut ad wat fo 4 seods also save the value a vaable; ead the voltage dop o the able ad wat fo 4 seods also save the value a vaable; ead the seso tempeatue - ead atuall a voltage dop ad the ompute the tempeatue wth the followg fomula: s * VoltageDop *. Remak: ths themoouple seso s a safet seso, whh uts off the powe suppl f the oduto tempeatue exeeds maxmal allowed value. ead the evomet tempeatue ad save the value a vaable fo late use. based o the saved values ompute the able esstae, the able tempeatue, the able powe. wte the esults the fle dedated to omputed values. - wte to both fles the date ad the tme (the measuemet has fshed at ths pot; - eset both powe suppl ad multmete; - lose both fles; he pogam tefae s show the ptue C..

131 3 Appedx C. Softwae fo measuemet data aqusto Fg. C. Measuemet pogamme to measue able voltage dop, esstae, elet powe ad tempeatue he measuemet of able esstae, voltage dop, elet powe ad tempeatue a be stated f all measuemet equpmet defed the pogamme s swthed o. he pogam should talse the avalable GPIB otolle, powe suppl soue, tempeatue sesos ad the dgtal multmete, whh seves as a data aqusto deve. Wheeve the hadwae s oetl talsed, the use should gve tal data to the pogam. Fst, Stom fü R box has to be flled, whee some uet value equed ode to measue old esstae R of the able. ext, the box Shutwdestad shut esstae should be defed ode to ompute exat load uet podued b the powe suppl. Alpha box defes the tempeatue oeffet of oppe esstae. hs value s equed fo the tempeatue omputato of the able. empeatu Lmt equed fo the safet easos, f the able tempeatue exeeds ths value, the powe suppl soue s ut off. M. Stom ad Max. Stom ae equed to spef the statg uet load value ad the ed uet value espetvel. ext, the step sze of measuemet teval should be gve. Fo ths, the box Stom-Itevall should be used. he measuemet tme, afte whh the stead state should be eahed s gve the box Messzet po Stom. he box Azahl Messuge meas, how ma temedate values fo eve measuemet teval should be measued ad saved the fle. hs fomato s mpotat ode to eod the taset state of able tempeatue. he measuemet poedue the begs, ad the whole expemet us automatall. I ode to obseve the measuemet data ole, the ompute wth measuemet softwae should be oeted to the Etheet etwok, ad usg Remote Desktop Coeto appl ato, the measuemet ompute a be otolled va the Etheet etwok. O the ght had sde of the pogam, ole measuemet fomato s peseted: atual load uet, atual able tempeatue obtaed b measug able voltage dop, able sufae tempeatue

132 C.. Measuemet esults 3 obtaed b themoouple seso, evomet tempeatue obtaed b Pt seso ad the old able esstae R. C. Measuemet esults he measuemet pogamme podued umeal esults, saved the ASCII tpe fle (see Fg. C.

133 3 Appedx C. Softwae fo measuemet data aqusto Measuemet me Speal Evets Measued Cable emp C Wated Cuet A A Cuet measued b PoweSuppl alulated uet Cable Powe Shut Voltage Dop Cable Voltage Dop 48 6: - Exta fle tated as: C:\CPOW\testf.log 48 6: - Evomet tempeatue:. deg C 6: - Use om Roda statet a ew Measuemet! Date: : - Settg ew uet to. A 55 33,76,36,5,39,45E-5 3,979E ,6,36,5,39,43E-5 3,945E ,47,9,3,39,46E-5 3,943E ,9,9,5,38,4E-5 3,98E :6 - Settg ew uet to. A 46 33,7 9,87 9,685,8956,936E-3,9596E-,74E ,7 9,87 9,6795,8934,9359E-3,9593E-,76E ,7 9,94 9,6795,8936,9359E-3,9595E-,76E ,3 9,94 9,6795,897,9359E-3,9576E-,74E :4 - Settg ew uet to. A 36 33,98 9,96 9,75 7,8585 3,945E-3 3,9844E-,979E ,5 9,96 9,75 7,8677 3,945E-3 3,98569E-,63E ,3 9,96 9,74 7, ,9448E-3 3,9879E-,85E ,46 9,96 9,73 7, ,9446E-3 3,993E-,34E :55 - Settg ew uet to 3. A 36 36,4 3 9,9 9,6955 7,9544 5,939E-3 6,468E-,363E ,33 3 9,98 9,6965 7, ,9393E-3 6,473E-,3637E ,38 3 9,98 9,6955 7,968 5,939E-3 6,4833E-,3678E ,46 3 9,98 9,695 7, ,939E-3 6,4995E-,3736E ,5 3 9,9 9,693 7,968 5,9386E-3 6,59E-,3795E : - Settg ew uet to 4. A , ,7435 3,5896 7,9487E-3 8,679E-,49E , ,7435 3,3397 7,9487E-3 8,5E-,47E , ,745 3,366 7,949E-3 8,76E-,399E , ,747 3,54 7,9494E-3 8,5E-,398E :8 - Settg ew uet to 5. A ,6 5 49,94 49,78 5, ,9456E-3,E,56E , ,94 49,77 5,764 9,9454E-3,E,5E ,6 5 5, 49,78 5,7553 9,9456E-3,65E,547E , ,94 49,775 5, ,9455E-3,5E,537E , ,94 49,765 5,8343 9,9453E-3,8E,558E :46 - Settg ew uet to 6. A 656 4, 6 6,4 59,785 74,584,9563E-,46E,846E ,7 6 6,4 59,78 74,589,9564E-,465E,859E ,8 6 59,96 59,78 74,5366,956E-,4675E,855E ,7 6 6,4 59,785 74,5466,9563E-,4689E,8574E :5 - Settg ew uet to 7. A ,9 7 69,98 69,767,6456,39534E-,476E,88E , ,98 69,7675,667,39535E-,4747E,9E , 7 69,98 69,768,678,39536E-,4763E,93E- 83 8:5 - Settg ew uet to 8. A , ,835 36,49,5967E-,745E,356E , ,835 36,3437,5967E-,744E,3548E , ,85 36,736,5965E-,749E,365E ,7 8 8,8 79,83 36,673,5966E-,7477E,3595E :45 - Settg ew uet to 9. A 6 5, ,95 89, ,359,79495E-,9478E,647E- 36 5,5 9 89,95 89, ,456,79495E-,9435E,654E- 46 5, ,95 89,749 74,3887,79498E-,9498E,649E- 56 5, ,95 89,748 74,3844,79496E-,943E,6496E :7 - Settg ew uet to. A 56 54,99 99,97 99,8635 9,4446,9977E-,9444E,9744E ,,4 99,865 9,7785,9973E-,9374E,967E ,4,4 99,864 9,46,9978E-,943E,97E ,,4 99,866 9,66,9973E-,9355E,9649E :9 - Settg ew uet to. A 9 58,88 9,843 69,33833,9686E-,453E,33E ,8 9,99 9, ,337,9689E-,4566E,394E ,8,6 9,84 69,737,9684E-,4546E,38E- A W V V ohm Cable Resstae Fg. C. Saved expemetal data ASCII tpe fle

134 C.. Measuemet esults 33 Cable Powe & Resstae,7 Cable powe,6 9 Cable Resstae,5 8 Polomsh (Cable powe Powe W Polomsh (Cable Resstae,4,3, Resstae Ohm 4 3,,,9,, 4, 6, 8,,, 4, 6, 8,, Cuet A Fg. C.3 Measued able powe ad esstae afte stead state egme aset state able tempeatue C Expemetal data Ftted b expoetal futo Measuemet tme t s Fg. C.4 Measued able sufae tempeatue taset state egme at the uet I 7 A

135 Bblogaph [] Haada, At. Calulato of Coduto empeatues ad Ampates of Cable Sstems Usg a Geealzed Fte Dffeee Model. IEEE asatos o Powe Delve, Vol. 6, o., 5-, Jaua 99. [] Haskew, m, A., Cawle, Rega F., Ggsb, L., L. A Algothm fo Stead- State hemal Aalss of Eletal Cables wth Radato b edued ewto- Raphso ehques. IEEE asatos o Powe Delve, Vol. 9, o., Jaua 994. [3] Hashe, B.L., Blak, W.Z. Ampat of ables sgle ope-top able tas. IEEE asatos o Powe Deleve, Vol. 9; o. 4, Otobe 994. [4] El-Kad, M.A. Calulato of the powe able ampat to vaatos of desg ad evomet paametes. IEEE tasatos o powe ad sstems. Vol. PAS-3; o.8, 43-5, 984. [5] Ra, C., Lee, G. ad Heffea, D. A Pototpe Stee-b-We ad Bake-b- We Sstem fo Eduatoal Reseah Poets. Poeeedgs U.K. Isttute of Phss - Sesos&Applatos XII. Sept. 3. [6] ehe, J.H, MGath, M.H. he Calulato of tempeatue Rse ad Load Capablt of Cable Sstems. AIEE asatos, Vol. 76, 75-77, Otobe 957. [7] Geoge J. Ades. Ratg of elet powe ables. IEEE Pess Powe Egeeg Sees. MGaw-Hll, 997. ISB [8] Shulz, homas. Gudsatzutesuhug zum empeatuvehalte elektshe Letuge ud dee Shutzelemete auf Shmelztletebass Kfz-Bodetze. Dssetato, Uvestät de Budesweh Mühe,. [9] Das, S., Kapp, R. H. ad Shmabukuo,.A. Fte Elemet Aalss - A ew ool fo Cable Desg. Po 5th Itl. We ad Cable Smposum, Pape -4, Lake Buea Vsta, FL, ov. -5,. [] Goth, C., Mülle, G. FEM fü Paktke-empeatufelde. Bad 463, Expet Velag,. Auflage, 997. [] Iopea, Fak, P., DeWtt, Davd P. Itoduto to heat tasfe. Joh Wlle & Sos, USA, 985. ISB [] Buks, A., Kals, H. he mathematal modellg of olea heat taspot th plate. Mathematal modellg ad aalss, Vol. 4, 44-5, 999. [3] Kals, H. Lass, A. Smple algothms fo alulato of the axal smmet heat taspot poblem a lde. 6(, 6-69,. [4] Buks, A., Kals, H. Calulato of eletomaget felds, foes ad tempeatue a fte lde. Mathematal modellg ad aalss, 7( -3,. [5] Zubau, O. Optmzg the we sstem pakage automoble dust. Dplom thess, Uvestät de Budesweh Mühe,. [6] slaf, Avaham. Combed popetes of odutos. Phsal see data Vol. 9, Elseve, ethelads, 98. ISB [7] VDI-Wämeatlas, 8. Auflage, Spge-Velag, Bel, ew Yok, 997. ISB [8] Wogala, aw. Ewämug elektshe Lete. Dplomabet, Uvestät de Budesweh Mühe, 999.

136 36 Bblogaph [9] Özsk, M.,. Fte dffeee methods heat tasfe. CRC Pess, I., USA, 994. ISB [] Zekew, O.C. he fte elemet method Egeeg See. ew Yok: MGaw-Hll. 97. [] Gottleb, D., Oszag, S. A. umeal Aalss of Spetal Aalss: heo ad Applatos, Soet fo Idustal ad Appled Mathemats. Phladelpha, PA, 977. [] Vesteeg, H.K, Malalsekea, W. A toduto to omputatoal flud dams. he fte volume method. Logma Goup, 995. ISB [3] Hsh, C. umeal omputato of teal ad exteal flows. Vol.: Fudametals of umeal dsetzato. ISB [4] Mak, P. Bestmmug des leae ud quadatshe empeatukoeffzete ausgewählte elektshe Lete. Studeabet, Uvestät de Budesweh Mühe,. [5] Roda,. Automatsete Messug de Wämeabletug vo elektsh behezte Lete. Studetabet, Uvestät de Budesweh Mühe,. [6] Laaste, P., Šalkauskas. K. Cuve ad sufae fttg. A toduto. Aadem pess, UK, 986. ISB [7] Cuthbet, D., Fed, S.W. Fttg equatos to data. Joh Wle & Sos, USA, 98. ISB [8] Che-Chg, Sh-We Chag. Aaltal exat solutos of heat oduto poblems fo asotop multlaeed meda. Iteatoal oual of heat ad mass tasfe 47, , 4. [9] Flethe, C.A. Computatoal Galek methods. ew Yok, Bel, Hedelbeg, oko, Spge Velag, 984. [3] Muoz-Cobo, J.L, Cobea, J.M, Chva, S. Explt fomulas fo lama atual oveto heat tasfe alog vetal ldes wth powe-law wall tempeatue dstbuto. Heat ad Mass asfe 39, 5-, 3. [3] Kas, W.M., Cawfod, M.E. Covetve Heat asfe. M Gaw Hll, 993. [3] Ades, G.J., Roz, J., Moshef, A. Advaed ompute pogams fo powe able ampat alulatos. IEEE Comput. Appl. Powe, Vol. 3, o. 3, 4-45, 996. [33] El-Kad, M.A. Calulato of the sestvt of powe able ampat to vaatos of desg ad evometal paametes. IEEE as. Powe App. Sst., Vol. PAS- 3, o. 8, 43-5, 985. [34] Flatabo,. aset heat oduto poblem powe ables solved b the fte elemet method. IEEE as. Powe App. Sst. Vol. PAS-9, 6-68, 973. [35] Haa, M.A., Chkha, A.Y., Salama, M., M., A. hemal aalss of powe ables mult-laeed sol. Pats ad. IEEE as. Powe Delve, Vol. 8, o. 3, , 993. [36] Kellow, M.A. A umeal poedue fo the alulato of tempeatue se ad ampat of udegoud ables. IEEE as. Powe App. Sst., Vol. PAS-, o. 7, , 98. [37] Mushamalwa, D., Gema,., Steffes, J.C. A -D fte elemet mesh geeato fo themal aalss of udegoud powe ables. IEEE as. Powe Delve, Vol. 3, o., 6-68, 988. [38] Cao, J.R. he oe-dmesoal heat equato. Addso-Wesle Publshg Compa, USA, 984, ISB [39] Shede, P.J. Coduto heat tasfe. Addso-Wesle,Readg, MA, 955.

137 Bblogaph 37 [4] Bumeste, L.,C. Covetve Heat asfe. d Ed., Wle, ew Yok, 995. [4] Smso, J.R. Egeeg heat tasfe. he Mamlla Pess Ltd, 975. ISB [4] Shh, e-mo. umeal popetes ad methodologes heat tasfe. Spge- Velag, Bel, 983. ISB [43] Lsek, V.D. Gd geeato methods. Spge-Velag Bel Hedelbeg ew Yok ISB [44] Hehold, L. Kabel ud Letuge fü Stakstom. el ud. Semes Velag, 4. Auflage 987. [45] Baue, H. Kaftfahtehshes ashebuh. Bosh,. Auflage-VDI-Velag, 995. ISB [46] Al-Khafa, Am Wad. umeal methods egeeg pate. Saudes College Publshg, 986. ISB [47] Moto, Keth W. umeal soluto of oveto-dffuso poblems. Chapma&Hall, 996. ISB [48] Jalua, Y. Computatoal heat tasfe. Spge-Velag, 986. ISB [49] Mülle, G. FEM fü Paktke. Expet Velag,993. ISB [5] Lax, P. D. Calulus wth applato ad omputg Vol.. Spge-Velag ISB [5] Mkowz, W.J. Hadbook of umeal heat tasfe. Joh Wle & Sos, I,988, ISB [5] Asoge, R. Iteatve soluto of olea sstem equatos. Poeedgs of a Meetg held at Obewolfah, Gema, Ja. 3-Feb. 5. Spge-Velag, 98. ISB [53] Shh, e-mo. umeal heat tasfe. Spge-Velag, 984. ISB [54] Mthell, A.R. he fte dffeee method patal dffeetal equatos. Joh Wlle & Sos, 985. ISB [55] Adeso, D. A. Computatoal flud mehas ad heat tasfe. Hemsphee publshg opoato, ew Yok, 984. ISB [56] veto, A. Itoduto to patal dffeetal equatos. Spge-Velag, 998. ISB [57] Gazha, V.G. umeal solutos fo patal dffeetal equatos. Poblems solvg usg Mathemata. CRC Pess, I., 996. ISB [58], M.U. Patal dffeetal equatos fo setst ad egees. Elseve See Publshg Co., 987, ISB [59] Stepheso, G. Patal dffeetal equatos fo setsts ad egees. Logma, 98. ISB [6] VVedesk, D. Patal dffeetal equatos wth Mathemata. Adso-Wesle, 993. ISB [6] Baeh, H. D. Heat ad Mass asfe. Spge, ew Yok, 998. [6] Bea, A. Heat asfe. Joh Wle, ew Yok, 993. [63] Iopea, F. ad DeWtt, D. Heat ad Mass asfe. Joh Wle, d Ed.,985. [64] Suaaaaa,.V. Egeeg Heat asfe. West Pub. Co., Meapols/St Paul, 995. [65] Caslaw, H.S. ad Jaege, J.C. Coduto of Heat Solds. Claedo Pess,Oxfod, UK, 959.

138 38 Bblogaph [66] Ggull, U. Heat Coduto. Spge-Velag, ew Yok; Hemsphee Pub Cop., Washgto, D.C., 984. [67] Poulkakos, D. Coduto Heat asfe. Pete Hall, Eglewood Cl_s, J, 994. [68] Kava, M. Pples of Covetve Heat asfe. Spge-Velag, ew Yok, 994. [69] Oosthuze, P.H. ad alo, D. Itoduto to Covetve Heat asfe Aalss. MGaw-Hll, 998. [7] Modest, M.F. Radatve Heat asfe. MGaw-Hll, ew Yok, 993. [7] Mes, G.E. Aaltal Methods Coduto Heat asfe. MGaw-Hll, ew Yok, 97. [7] Adams, J.A., Roges, D. F. Compute-Aded Heat asfe Aalss. MGaw-Hll, ew Yok, 973. [73] Wag, Z.J. Spetal (Fte Volume Method fo Cosevato Laws o Ustutued Gds. Joual of Computatoal Phss, 78, -5,. [74] Cegs, R. Dfeealu lgu skata spedmo metoda. Kauas, 3. ISB [75] IEEE Stadat Powe Cable Ampat ables, IEEE Std 835, 994. [76] Calulato of the Cotuous Cuet Ratg of Cables (% load fato. IEC Publato 87, 98 ad subsequet edtos. [77] A Method fo Calulatg Reduto Fatos fo Goups of Cables Fee A, Poteted fom Sola Radato. IEC Publato 4, 99. [78] Buks, A.,Ulaova,. Aaltal-umeal method fo tempeatue felds multlaeed sstem. Advaed Computatoal Methods Heat asfe, VI, Sauthappto, Bosto: WIp, ,. [79] Abou khahfe, R., Ja, Y. Detemato of heat soues ad heat tasfe oeffet fo two-dmesoal heat flow-umeal stud. It. J. Heat Mass asfe 44(7,. [8] Ca, R.X., Zhag,. Some algebaall explt aaltal solutos of ustead olea heat oduto. It. J. Heat asfe. ASME 3(6,. [8] Chag, C.Y., Ma, C.C. aset themal oduto aalss of a etagula plate wth multple sulated aks b the alteatg method. It. J. Heat Mass asfe 44(3,. [8] Che, B.S., Gu, Y.X., Gua, Z.Q., Zhag, H.W. olea taset heat oduto aalss wth pese tme tegato method. ume. Heat asfe Pat B-Fudametals 4 (3,. [83] Asllaa, F. Jeadel, G., Rohe, J.R. umeal soluto of adatve tasfe equato oupled wth olea heat oduto equato. It. J. ume. Meth. Heat Flud Flow (5 6,. [84] Kshapakas, C. K. Combed Coduto ad Radato Heat asfe a Cldal Medum. Joual of hemophss ad Heat asfe, Vol., o. 4, 65-68, 998. [85] alukda, P., Msha, S. C. aset oduto ad adato heat tasfe wth vaable themal odutvt. umeal Heat asfe, Pat A, 4: ,.

139 Akolegmet hs dssetato has bee wtte dug m atvt as a eseah egee the Isttute of Phss of Eletal Egeeg ad Ifomato eholog Fault of Uvest of the Budesweh Muh. I would lke seel to thak Pof. D. H.-D. Leß fo aeptg me to hs eseah goup ad ofdg ths top to me. Hs teest o the pogess of m eseah wok, supevso ad povded suppot had bee valuable help fo me. I wsh to expess m deep gattude to Pof. Habl. D. R. Cegs, who kdl ageed to supevse m Ph.D. dssetato umeal heat tasfe subet ad gave me a tesve ouse o the mathematal modellg top. I thak also Pof. R. Cegs fo hs otbuto as the seod evse of ths wok. Pof. Habl. D. H. Dalhau I thak fo the evso of ths wok as the thd evse. Hs emaks ad suggestos o ompute aded omputatoal tehques wee ve valuable fo me. I would lke to thak Pof. D. K. Lades fo hs otbuto as the hama of the examato ommttee ad beod t fo teestg dsussos about the eseah wok. I wsh to akowledge valuable dsussos wth the Pof. D. R. Rkevee, Pof. Habl. D. A.J. Poska, ad Pof. Habl. D. A. Smlgevus fom Vlus Gedmas eholog Uvest, Fault of Eletos dug pepaato the dssetato mauspt. I akowledge valuable suppot of Dpl. Ig. M. Glka ad Dpl.-Ig. M. Hlle fom EI 6 ad D.-Ig.. K. Wust fom LR fo povdg expesve laboato equpmet. Wthout ths equpmet would ot had bee possble to valdate the eseah theo. he o-wokes ad olleagues fom m eseah team I deepl thak fo the assstae wth the pogammg of umeal algothms ad measuemet softwae. Last but ot least, the tehal staff fom LR 6 I thak fo the help to set up the measuemet equpmet. Fall, but most deepl, I thak m paets, wthout whose ostat eouagemet, suppot, assstae, ad toleae ths dssetato would ot have bee possble.