Conservative Averaging and Finite Difference Methods for Transient Heat Conduction in 3D Fuse

Transcription

1 Ramods Vums Adrs Buks Coservatve Averagg ad Fte Dfferece Metods for Traset Heat Coducto 3D Fuse RAIMONDS VIUMS ANDRIS BUIKIS Isttute of Matematcs ad Computer Scece Uversty of atva Raa uv 9 Rga V459 ATVIA ramodsvums@uv uks@atetv ttp://wwwav/scetsts/ukstm Astract: Tree-dmesoa matematca mode of te automotve fuse s cosdered ts paper Itay parta dffereta equatos of te traset eat coducto are gve to descre eat-up process te fuse Coservatve averagg metod s used to ota aaytca appromato of tese equatos y te system of tree ordary dffereta equatos Fte dfferece sceme s gve f coservatve averagg procedure s stopped oe step efore e after D proem of parta dffereta equatos s otaed Key-Words: Heat coducto Quas-ear Traset process Tree-dmesoa Aaytca reducto Coservatve averagg Fte dfferece sceme Itroducto Usuay matematca modeg of te fuse s mpemeted y makg oe dmesoa assumptos []-[4] I ts paper we use orga metod of coservatve averagg to trasform ta 3D statemet of te proem to te statemet of ew type tat cossts of tree ordary dffereta equatos Appromate aaytca 3D souto s otaae from te souto of te trasformed proem Coservatve averagg metod s teoretcay we fouded for ear parta dffereta equatos [6]-[] Here (as [3] [4] we vestgate quas-ear proem Fg Geometry of te Mode We start wt geometrc assumptos of te typca car fuse (Fg ad Fg Fg3 Because of te symmetry t s eoug to use oy te saded part of te mode (Fg4 ad Fg5 Fg Eampe of automotve fuse wt ad wtout pastc se We seemgy stragte out te fuse ad use geometry of te mode as sow Fg3 Fg4 ISSN: Issue Voume 3 Jauary 8

2 Ramods Vums Adrs Buks Fg5 We gve ref descrpto of te metod te et capter furter foow matematca statemet of ts proem ad usage of te coservatve averagg 3 Sort Descrpto of Coservatve Averagg Metod Coservatve averagg metod was deveoped as appromate aaytca ad umerca metod for sovg parta dffereta equatos wt pecewse cotuous coeffcets Te usage of ts metod for separate reatvey t su-doma or for sudoma wt arge eat coducto coeffcet eads to reducto of te doma wc te souto must e foud Metod ca e apped for severa su-domas smutaeousy To appy ts metod for a su-domas of te ayered meda a speca type of te spe s costructed: te tegra averaged vaues terpoatg paraoc spe Usage of ts spe aows dmsg te dmesos of ta proem per oe It s mportat tat te orga R proem wt dscotuous coeffcets trasforms to proem wt cotuous coeffcets R a cases More detaed descrpto of te metod s gve papers [6]-[3] But o cocrete steady-state eat coducto eampe ma dea of te metod s gve ere et us assume tat we ave doma D tat cossts of two su-domas (rectages ad G (Fg6 y G G G Fg6 G {( y ( y ( } G {( y ( y ( } D G G {( y y ( } Ojectve s to fd fucto (cotuous doma G ad fucto (cotuous doma G tat fufs foowg equatos: a dffereta equatos: U U k ( k F y y y ( U U k ( k F y y y ( cojugato codtos at : U (3 k U U U k (4 c oudary codtos: U (5 U (6 U (7 y U k ( U Θ (8 y y We requre tat a dervates of te equatos ( ( are cotuous correspodg su-domas Souto of ts matematca proem ca e treated as temperature two ayer meda for eat trasfer process We assume tat a coeffcets are costat ere Temperature depedat coeffcets are cosdered te et capter were coservatve averagg s apped for te mode of te fuse et us assume tat doma G s t -drecto or t s made y matera tat as reatvey etter eat coductvty ta te oter oe (or ot codtos take pace We ca ovousy assume tat temperature s amost costat -drecto te If ts assumpto s ot te case we ca assume tat dstruto of te temperature dffers from some oter curve oy sgty e poyoma or fucto of te epoeta eavor Terefore te frst tg s to uderstad wc doma ad wc drecto te eavor of te ukow fucto s predctae Before we coose specfc represetato we troduce tegra averaged vaue fucto over cose terva If we take doma G ad terva [ ] te te defto of ts fucto s ISSN: Issue Voume 3 Jauary 8

3 Ramods Vums Adrs Buks u ( y U ( y d (9 I our case fucto represets averaged temperature terva [ ] o gve e y Net we seect fucto tat w appromate our ukow fucto cose doma ad segmet It soud descre partcuar pysca stuato It meas etter vew of te stuato we ave more approprate fucto we ca coose For eampe et us use epoeta appromato -drecto Geera form of te fucto U te s ( ( ( ( U y a y y e c y e ( Represetato of te fucto U cotas ukow fuctos ay ( y ( cy ( Tey are otaed suc way tat tey fuf codtos o te oudares ad tegra equaty (9 Practcay yperoc fuctos coud e used stead of epoet I our case etter form of te epoeta appromato s U ( y a( y ( y cos s( ( cy ( s cos( Takg to accout dervatve of ts fucto U y ( s cy ( cos ( ad oudary codto o te order (5 we ota tat cy ( If we appy tegra (9 to te formua ( we ota: ay ( u( y Cojugato codto (4 o te secod oudary of te doma G gves ukow fucto : k U y ( s( After ukow fuctos are foud we ca rewrte appromato of te fucto U foowg form: U ( y u ( y k U (3 cos s( s( Te et step of te coservatve averagg metod s tegrato of te ma dffereta equato ( over te terva [ ] : u U U k k F( y d y y et us take a ook at te frst added: U k d k U k U k U We used codtos (4 (5 o te orders of te doma G to make aaytca trasformatos ere O te oter ad we ca use represetato (3 ad t aso eads to te same resut Order of te tegrato ad dervato s swapped for te oter dervatves te tegra Itegra formua (9 s used after tat: U k d y y d du k Ud k y y dy dy Cosequety dffereta equato for te ukow averaged vaue fucto u ( y s d du k U k ( dy dy f y (4 were f ( y s averaged vaue of te source fucto: f ( y: F ( y d We ave trasformed ta proem to te ew oe Dffereta equato for te fucto U ( y te doma G remas te same ( Te secod dffereta equato (4 s for te averaged vaue fucto u ( y terva y [ ] y G Fg7 If we take to accout represetato of te averaged fucto (3 te cojugato codto (3 gves codto etwee fuctos u ad U : ISSN: Issue Voume 3 Jauary 8

4 Ramods Vums Adrs Buks ( ta( U u U k (5 Ts equato togeter wt te equato (4 coud e cosdered as o-cassca oudary codtos for te order doma G Boudary codtos we y ad y for te fucto are smar as for te fucto U : du u (6 dy y du k y ( u Θ dy (7 y To e accurate tese equates are otaed after tegra s apped to oudary codtos (7 (8 Trasformatos are smar to tose tat were doe for te ma dffereta equato ( Boudary codtos (6 (7 (8 rema te same for te fucto U ( y of te doma G We ave trasformed orga proem ad reduced dmeso of oe doma Usuay t s mposse to fd aaytca souto Te oy coce s to sove t umercay It takes ess computer power to cacuate suc matematca proem ecause of reduced dmeso It s posse to recostruct temperature dstruto U ( y doma G from te represetato (3 after fuctos u ( y ad U ( y are cacuated Note tat t s posse to get vaue at ay pot of a terva - ot oy some dscrete pots as t woud e after appyg fte dfferece metod to ta proem Matematca proem s reduced y oe dmeso for woe system f coservatve averagg metod s apped for te doma G -drecto over terva [ ] Averagg procedure ca e apped cotuousy severa drectos reducg dmesos of te proem oe y oe Numerca cacuatos are aso reduced y order et us ook ack to te orga proem (-(8 For ot domas coservatve averagg coud e apped y-drecto at frst (Fg8 Fg8 Boudary codtos at y ad y trasfer to te ew dffereta equatos te If costat appromato y-drecto s used U( y u( y U( y u( y ( te dffereta equatos for t wre wt covecto s otaed: d du y k ( u d d Θ f (8 f ( F( y Remag oudary ad cojugato codtos actuay stay te same: du du (9 d d du du u u k k ( d d We gave ma steps of te coservatve averagg metod as a summary of ts secto Frst coose fucto of te appromato Secod tegrate ma dffereta equato Trd use oudary ad cojugato codtos 4 Orga Proem ad ts Appromato y Coservatve Averagg Metod 4 Matematca Statemet of te Ita Proem We cotue wt accurate formuato of te treedmesoa matematca mode of te traset eat coducto proem for fuse Fg9 et us treat ma doma (Fg9 as two coected su-domas G ad G : G {( y [ ] y [ ] [ ] } G {( y [ ] y [ ] [ H] } If temperature doma G s deoted as fucto U ( y t te dffereta equato for te eat trasfer s U U γu k k t y y ( U k F( y t U ( y G t> ISSN: Issue Voume 3 Jauary 8

5 Ramods Vums Adrs Buks F Source fucto (eat produced y eectrca curret ca e appromated wt ear fucto: F( y t U B( α ( U Ur ( ρref I ρref I were B B H Parameter ρ ref s resstvty of te matera at te referece temperature Ur α s temperature coeffcet at te same referece temperature; I eectrca curret Heat coductvty k ad eat capacty (per voume γ deped o temperature Besdes ma equatos ( we add symmetry codtos: U U U y y (3 U U U y y (4 ad eat ecage codtos o outer surfaces U k y ( U Θ y y U k y ( U Θ y y (5 U k ( U Θ U k ( U Θ H (6 U k ( U Θ [ H] (7 were ΘΘt ( s temperature of te evromet ut y are eat covecto coeffcets for surfaces correspodg drecto tat aso deped o temperature We aso add cojugato codtos e cotuty of te temperature ad eat fues etwee ot parts of te fuse: U U U U (8 y [ ] [ ] Fay we add ta codtos: U U U cost (9 t t 4 Coservatve Averagg y-drecto We troduce te tegra average vaue of te fuctos U ( y t te y-drecto: V( t U( y t dy (3 I pras frsty te tckess s very sma comparso wt te wdt of te fuse Secody matera of te fuse (meta as g eat coductvty coeffcet Tese features aow us to use te smpest form of te coservatve averagg metod appromato y te costat Detaed procedure of te aaytca trasformatos s gve prevous secto Sorty we tegrate ma equato ( over te segmet y [ ] ad te we use oudary codtos (5 ad ear represetato of te source fucto ( Fay we take to accout tegra equaty (3 ad ota: V V ( γv k k t (3 y ( V Θ B ( α ( V U r Because of te earty te addtoa oudary codtos (BC of ew proem are te same as te statemet of te orga proem (-(9: V V V (3 V k ( V Θ (33 V k ( V Θ H V k ( V Θ (34 [ H] We aso add cojugato codtos at [ ] : V V V V (35 ad ta codtos: V V U cost (36 t t 43 Coservatve Averagg -drecto As te et step we w make coservatve averagg te -drecto We defe oe averaged vaue fucto over doma G ad two separate fuctos for te doma G te frst for terva ( ad te secod for terva ISSN: Issue Voume 3 Jauary 8

6 Ramods Vums Adrs Buks ( H ecause of dfferet codtos o te e : W ( t V ( t d ( W ( t V ( t d ( (37 W ( t V ( t d ( H I ts case we use epoeta appromato te foowg form: V( t W( t p( t (38 cos s ( V( t W( t p( t (39 cos s ( Equates (38 (39 are cose suc way tat tey fuf tegra equates (37 (coservato of te eat eergy ad BC (3 at ad We use cojugato codtos (35 to fd ukow fuctos p p ad afterwards ota fuctos V V : V( t W( t CW( t W( t ( e (4 cos s ( C V( t W( t CW( t W( t (4 cos s ( ( We fd fucto p ad represetato of fucto V terva ( H from epresso (39 y meas of BC (34: V ( t W ( t C W ( t Θ( t ( ( (4 cos s ( ( H e C k( e Dscotuty for te temperature fed coud appear o te e Ts kd of dscotutes was cosdered papers [3] [4] We tegrate dffereta equatos (3 o te frst step of averagg order to ota equatos for te secod step We use represetatos (4 (4 (4 of te fuctos V V ad tegra equates (37 to make appromate aaytca reducto of D system to D system of parta dffereta equatos W D ( γ W k ( WW t (43 y ( W Θ B ( α ( W U r W D ( γ W k ( W W t (44 y ( W Θ B ( α ( W U r W ( γ W ( k D W Θ t (45 y ( W Θ B ( α ( W U r Ck s( D Cks( D As we metoed te troducto we cosder te quas-ear proem ere Ts proem sgfcaty dffers from te proem cosdered our papers [] [] We ave made te averagg procedure over te su-doma wt ear dffereta equato te earer statemet of te proem Terefore we w epa deeper te averagg procedure for te eft ad sde of te equato (43 (procedure for te equatos (44 (45 ca e reaed te same way I ts paper we use etapy form of te eat equato (see eg [5] capter 7 Ts form s sustatay more sutae for te use of te mea vaue teorem: [ ( V V] d t γ γ ( V Vod t [ ] t γ W γ γ( V V V( t It s posse to coose te mea vaue more or ess freey We propose to use te correspodg mdde pot e or averaged temperature: γ γ( W γ γ( W γ γ( W Aga oudary ad ta codtos are te same as te orga proem ecause of te earty: W W (46 ISSN: Issue Voume 3 Jauary 8

7 Ramods Vums Adrs Buks W ( Θ k W W ( Θ k W t t t H (47 W W W U cost (48 We aso ask for cotuty of te averaged temperature ad fues o te e Tat gves addtoa cojugato codtos: W W W W (49 44 Coservatve Averagg -drecto Fay we w make coservatve averagg procedure te -drecto We troduce tree ew fuctos for ts purpose: u( t W( t d u ( t W ( t d H u ( t W ( t d H (5 We use epoeta appromato te form W( t u( t q(cos t s ( W( t u( t q(cos t s ( (5 W( t u( t q(cos t s ( H q3( t s cos ( H We fuf te tegra equates (coservato of te eat eergy (5 ad te symmetry codtos (46 at y ts represetato Usg BC (47 ad cojugato codtos (49 we ca fd four ukow parameters te represetato (5 Ts gves: W( t u e( uθ cos s ( Wt ( u (5 e( u u e( uθ cos s( W( t u e3( u u e4( uθ cos s( H e5( u u e6( uθ s cos ( H u u ( t u u ( t u u ( t e are: were costats e e ke ( ( ( ( e e k e H e 7 e e ( H ( e e e 7 ( ( ( e ( H( e k( e ( e( H e 3 7 e ( H e ( e ( e ( H e 4 7 e ( H ( e k( e ( e e 5 7 e ( H ( e ( e e e ( ( e k e H ( ( H ( e ( e3( e( H After tegrato of equatos (43-(45 we fay ota system of ordary dffereta equatos: d D ( γ u ( uu E( u Θ dt (53 B α u U ( ( r d D y ( γ u ( u u ( uθ dt E e u u e u Θ B u U ( ( ( α ( r d ( ˆ γ u E( uu dt y E3 D ( u Θ ( α ( r B u U Here costats E E E E3 ad coeffcets γ : y e E e k s( E k( e3s( e5(cos( E H ( (54 (55 ISSN: Issue Voume 3 Jauary 8

8 Ramods Vums Adrs Buks E k( e4s( e6(cos( ( H ( W t ( W t ( W t 3 γ γ ˆ ˆ ( γ γ ( γ γ ( ˆ ( H or γ γ ( u( t γ γ ( u( t ˆ γ γ ( u ( t Ts system of tree ordary dffereta equatos must e suppemeted wt ta codtos: u u U (56 t t 45 Smpfed Averaged System of Ordary Dffereta Equatos Te ma goa of ts matematca mode s to predct tme efore metg of te matera te test su-doma G caused y admsse strog curret Accordg to epresso ( desty of te eectrca curret s H / tmes gger ts su-doma Ts reaso aows us to propose aoter mode esdes te frst oe As te secod step of te averagg we use te smpest appromato te -drecto appromato y costat We troduce averaged vaues: w( t V( t d (57 H w( t V( t d H We assume tat temperature s costat -drecto ecause t cages oy sgty comparso wt -drecto: w( t V( t (58 w( t V( t Itegrato of te dffereta equatos ( mmedatey gves system of two D parta dffereta equatos: w ( γ ( w w k t y ( w Θ B( α ( w Ur (59 w ( γ ( w w k t y ( wθ B( α ( wur H Boudary codtos rema te same: w w (6 Te secod cojugato codto cages sustatay ecause of covectve eat osses over [ H] : te surface { } w w w w ( ( k Hk H w Θ (6 Itegrato of te oudary codto ad cojugato codtos was made to ota prevous equato By te way suc type of te secod cojugato codto was used paper [4] Te ta codtos rema te same: w w U (6 t t As te ast step we w appy te coservatve averagg metod -drecto We w use epoeta appromato as te form used earer: ( w( t u( t p(cos t s (63 w( t u( t p(cos t s ( We ave troduced te average tegra vaues aga: u( t w( t d (64 u( t w( t d We ota parameters p ( t p ( t of te represetatos (63 from te cojugato codtos (6: p( t e[ u( t u( t ] p( t g( u u g( u Θ p( t e g3 e g( u u g( uθ p ( t e g3 Here H g k ( e g k ( e g ( H g3 g g g Fay we tegrate parta dffereta equatos (59 ad ota system of ordary dffereta equatos: ISSN: Issue Voume 3 Jauary 8

9 Ramods Vums Adrs Buks d G ( γ u ( g g ( u u g( u Θ dt y ( u Θ B( α ( u Ur (65 d G ( γ u g( u u g ( u dt Θ y ( uθ B( α ( uur H (66 Here γ γ w ( t or γ γ u ( t ( ( ( ( ( γ γ w ( t or γ γ u ( t k e G g3 It remas to add te ta codtos for te competeess of te fu statemet of te -D proem: u u U (67 t t 5 Fte Dfferece Metod for D Proem To mprove te accuracy of te coservatve averagg metod we ca use te fte dfferece metod for te umerca appromato of te system of two D eat equatos (59 5 Te Statemet of te D Proem Te system of two quas-ear D eat equatos as te form: w ( γ ( w w k f( t w t (68 w ( γ ( w w k f( t w t Here source fuctos f( tw ca pay te roe of te eat sources or eat sks depedg o vaues of te frst or te secod term epressos (69: f ( t w B α w U ( ( r y Θ f ( t w B w U ( w ( α ( r H ( w y Θ (69 Boudary codtos are take as te geeraato of te omogeeous oudary codtos (4: w w q( t q( t (7 Cojugato codtos dffer from te dea terma cotact codtos as we as from o-dea terma cotact codtos: w w w k (7 w Hk ( H ( w Θ Ita codtos are o-omogeeous: w w U ( (7 t t From matematca pot of vew mportat s a fact tat te fuctos f( tw (as we as fucto γ ( w fuf foowg estmatos: f j( t wj γ M j M wj wj Tese costrats guaratee te uqueess of te souto of te proem (68-(7 5 Te Costructo of te Fte Dfferece Sceme Te fte dfferece metod for eat trasfer proems are we epaed terature eg [8] Te fte dfferece souto of te D proem w e deoted as vj wj( t We w use uform tme step: t τ N ; te space step w e pece-wse costat: Δ ; Δ ( I Δ I I Δ We appromate eat coductvty term te foowg way (temporary we w omt te otato of te tme-eve: Λ v [ a( v v] I (73 Here we ave used tradtoa otatos eg [8]: v v v v v v Δ Δ were Δ Δ < < ; Δ Δ < < I For coeffcet av ( (73 severa equvaet ISSN: Issue Voume 3 Jauary 8

10 Ramods Vums Adrs Buks epressos te sese of te order of appromato O ( ca e apped eg Δ v v av ( k (74 or k( Δ v k( v av ( (75 Now we ca propose two-step predctor-correctortype fte dfferece sceme for te dffereta equatos (68 (t s mportat to sow te tme eve ere ut otato j of te su-segmet may e omtted: v v γ ( v [ a( v v] f( t v τ v v γ ( v (76 τ [ av ( v ] f( tv ; < < < < I I ( Now we w pay speca atteto to ota appromato of te oudary codtos (7 ad cojugatos codtos (7 (7 wt te same order of appromato O( Δ as te fte dfferece equatos (76 To guaraty te secod order of te appromato we empoy te dea of use of ma dffereta equato o te order [9] We start wt Tayor seres epasos for te fuctos v j as dfferetae fuctos of argumets t : v kv ± kv ±Δk v 3 k O( Δ Δ We draw te reader s atteto to foowg uace: te eat coductvty coeffcet k s take te fed pot Ts assumpto aows us to rewrte te ast formua te form: v kv ± kv ±Δ k (77 Δ v 3 k O( Δ Net two equates foow from (77: v v v Δ v k k k O( Δ Δ v v v v k k Δ k O( Δ Δ Te assumpto tat fuctos vt ( fuf dffereta equatos (68 gves foowg epressos for te frst dervatves: v v v k k Δ Δ ( γ v f t v O Δ t v v k k Δ Δ t ( γ ( ( v v f t v O Δ ( ( It remas to use oudary codtos (69 ad accordace wt dfferece sceme (76 we ota secod order fte dfferece appromato of ot oudary codtos We ave foowg dfferece equatos for te predctor step: Δ v v γ ( v a( v( v τ Δ f ( t v q ( t (77 Δ v I v I γ ( v I a( v I( v I τ Δ f ( t v I q ( t Te dfferece equatos for te corrector step are: Δ v v γ ( v a( v ( v τ Δ f ( t v q ( t (78 Δ v I v I γ ( v I a( v I ( v I τ Δ f ( I t v I q ( t We make smar costructo of te secod order appromato o te order etwee ot parts ( te pot Here we eed to e carefuy wt otato ad use dfferet dees for dffereces to te eft (ad rgt from te order pot Te secod cojugato codto (7 ca e rewrtte foowg equvaet form: ISSN: Issue Voume 3 Jauary 8

11 Ramods Vums Adrs Buks w χ w ( k k λ w Θ (79 ( H χ λ H H We appromate te eft ad sde fu of equato (79 for te predctor stage as foow: Δ v v χ γ( v f( t v τ χav ( ( v J We ota smar epresso for te rgt ad sde: av ( ( v λ( v Θ Δ v v γ ( v f( t v J τ Takg to accout te frst cojugato codto (7 (cotuty: v v fay we ave at te order: J J or suc equato for te predctor stage: Δ v v χ γ( v f( t v τ χav ( ( v av ( ( v (8 λ( v Θ Δ v v γ ( v f( t v τ We ave foowg equato for te corrector stage: Δ v v χ γ( v f( t v τ χav ( ( v av ( ( v (8 λ( v Θ Δ v v γ ( v f( t v τ Te dfferece equatos (77-(8 togeter wt sef evdet ta codtos v U ( (8 v U ( I are compete dfferece sceme of te secod order of appromato Automotve fuse of te oma curret 5A (Fg s take as a sampe Fg 5A Fuse wtout se Geometry s trasferred to our matematca mode (Fg9 Oe egt of te fuse s cosdered ecause of te symmetry Notatos of te dmesos are as paragraps 4 ad 5: 3mm 7mm mm 9mm H 8mm Dmesos are otaed y measurg te fuse Fuse s made of c Propertes of te matera deped o temperature Vaues are kow at some referece temperatures Spe s costructed from tem It s satsfactory to use ear spe (Fg Fg Fg Heat coductvty of c Fg Heat capacty of c (per voume Net fgures sow oter parameters Fg3 sows eat covecto coeffcet y ad Sod e s for te test part of te fuse ( ; das e s for te terva ( 6 Numerca Eampes 5A Fuse ISSN: Issue Voume 3 Jauary 8

12 Ramods Vums Adrs Buks Fg3 Heat covecto to te ar Net fgures sow eat produced y eectrca curret te ter part of te fuse ad te ades F at dfferet curret vaues F % of Rated Curret M Ma 6 % 4 s s 35 % s 7 s % s 6 s 35 % 6 s 8 s Tae Mama temperature s reaced te mdde of te fuse Tme to reac metg temperature s cacuated ad compared amog a tree matematca modes (Tae Fg % of Ratg D PDE-s [s] 3 ODE-s [s] ODE-s [s] 6 % % % % Tae Fg4 Heat produced y eectrca curret F Fg5 Heat produced y eectrca curret F It s vse from te fgures tat eat producto te ter part s more ta tmes arger ta te oter part of te fuse Numerca cacuatos are doe 3 dfferet ways Frst souto s otaed from te system of 3 ODE-s (53-(55 Secod cacuatos are doe from te system of ODE-s (65-(66 Trd D matematca proem tat cossts of PDE-s (68 ad addtoa codtos (7-(7 s soved y appyg dfferece sceme from te secto 5 Resuts are compared atogeter ad wt stadard DIN tat defe tme terva of te urout of te fuses (Tae Fg6 5A Fuse tme-curret curve Das es are tme mts from te DIN stadard Numerca resuts are approprate f curret s greater te % of te ratg Heat gve away y radato ad coducto over ades pays greater roe f te curret s cose to oma vaue ad rea fuse reakg tme s arger Ts s ot cosdered ts partcuar mode ut coud e added to orga 3D matematca mode Coservatve averagg coud e apped te same maer For eampe f we take to accout radato addtoa term soud e added to te oudary codtos (5-(7: U 4 4 k y( U Θ εσ ( U Θ (83 y y U 4 4 k ( U Θ εσ ( U Θ (84 U 4 4 k ( UΘ εσ ( U Θ H ISSN: Issue Voume 3 Jauary 8

13 Ramods Vums Adrs Buks U k ( U Θ [ H] 4 4 ( U εσ Θ [ H] (85 3D temperature dstruto coud e recostructed womever averagg s coused It s eoug to sow temperature oy ( pae ecause we ave assumpto aout costat temperature dstruto over y-dmeso Fg7 sows temperature recostructo after system of ODE-s s soved case of A (% curret s apped to te fuse Fg9 Net fgure (Fg cotas temperature dstruto o te e [ ] for a appromatos used Fg7 Smar grapc (Fg8 coud e otaed for te souto of 3 ODE-s y formuas (4-(4 (5 D PDE ODE-s 3 ODE-s Fg It takes aout oe mute o moder desktop computer to cacuate partcuar eampe at gve curret Cacuato of ODE-s s eve qucker It s more effcet to cacuate averaged matematca proems rater te fu 3D proems Fg8 Souto s dscotuous ecause dscotuous appromato fucto V ( t were used ((4- (4 atoug averaged vaues w ( t are more precse ta prevous case Souto of D PDE-s soud e used f temperature dstruto te fuse s aso mportat ad ot oy fuse reakg tme Fg9 sows temperature o te e [ ] 7 Cocusos We ave appromated 3D proem ad reduced ts souto to te souto of te tme-depedet oear system of two or tree ordary dffereta equatos Reducto was reaed two dfferet ways y dfferet assumptos Bot systems ave smar structure ut dfferet coeffcets Te systems of ordary dffereta equatos are sovae wt stadard tecques Appromate aaytca 3D souto coud e easy otaed from te souto of te trasformed proem afterwards Ackowedgemets: Researc was supported y Europea Soca Fud ad Couc of Sceces of atva (grat 555 Speca taks to Prof H-D ess from Uversty of te Budeswer Muc for usefu cosutatos aout pysca ackgroud of te automotve fuses ISSN: Issue Voume 3 Jauary 8

14 Ramods Vums Adrs Buks Refereces: [] Fuses for Automotve Appcato ttefuse Ic 8 East Nortwest Hwy Des Paes II66 USA [] Ker DQ Kraus AD Eteded Surface Heat Trasfer McGraw-H Book Compay 97 [3] Maoor M Heat Fow troug Eteded Surface Heat Ecagers Sprger-Verag: Ber ad New York 984 [4] Wood AS Tupome GE Batt MIH Heggs PJ Performace dcators for steadystate eat trasfer troug f assemes Tras ASME Joura of Heat Trasfer pp 3-36 [5] Ockedo J ao Apped Parta Dffereta Equatos Oford Uversty Press 999 [6] Buks A Aufgaesteug ud ősug eer Kasse vo Proeme der matematsce Pysk mt ctkasssce Zusatedguge Rostock Mat Kooq pp 53-6 (I Germa [7] Buks A Modeg of ftrato processes ayered porous meda y te coservatve averagg metod Dr Tess Pyscsmatematcs Kaa p (I Russa [8] Buks A Proems of matematca pyscs wt dscotuous coeffcets ad ter appcatos Rga p (I Russa upused ook [9] Vums R Estmates of appromato errors for coservatve averagg metod Master tess Rga 4 9 p (I atva [] Buks A Coservatve averagg as a appromate metod for souto of some drect ad verse eat trasfer proems Advaced Computatoa Metods Heat Trasfer IX WIT Press 6 p 3-3 [] Vums R Buks A Coservatve averagg metod for parta dffereta equatos wt dscotuous coeffcets WSEAS Trasactos o Heat ad Mass Trasfer Vo Issue 4 6 p [] Buke M Buks A Modeg of treedmesoa trasport processes asotropc ayered stratum y coservatve averagg metod WSEAS Trasactos of Heat ad Mass Trasfer 6 Issue 4 Vo p [3] Buke M Buks A System of varous matematca modes for trasport processes ayered Strata wt terayers WSEAS Trasactos o Matematcs 7 Issue 4 Vo 6 p [4] Buks A ess H-D Vums R Coservatve Averagg Metod for Cacuato of Heat Trasfer Cydrca Wre wt Isuato Matematca Modeg ad Aayss Astracts of te t Iteratoa Coferece MMA 5 5 p 48 [5] Buks A Buke M Cosed two-dmesoa souto for eat trasfer a perodca system wt a f Proceedgs of te atva Academy of Sceces Secto B Vo5 Nr5 998 pp8- [6] Buke M Smuato of steady-state eat process for te rectaguar f-cotag system Matematca Modeg ad Aayss 999 vo 4 pp [7] Mak MY Wood AS Buks A A appromate aaytca souto to a famar cojugate eat trasfer proem Iteratoa Joura of Pure ad Apped Matematcs Vo Nr 4 pp 9-7 [8] Samarsk AA Vascevc PN Computatoa Heat Trasfer Vo Matematca Modeg Jo Wey & Sos 995 [9] Samarsk AA Teory of Fte Dfferece Scemes Moscow Nauka 989 ISSN: Issue Voume 3 Jauary 8