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: