COMPUTATIONAL AND EXPERIMENTAL ESTIMATION OF BOUNDARY CONDITIONS FOR A FLAT SPECIMEN

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1 COPUTATIOAL AD EXPERIETAL ESTIATIO OF BOUDARY CODITIOS FOR A FLAT SPECIE S. Abboud*, E. Artoukhe**, H. Rad* * LERPS, EA 171, Ittut Polytechque de Sévea B.P Belort Cedex, Frace Poe : (33) , Fax : (33) , E-mal : aïd.abboud@utbm.r ** Ittut de Gée Eergétque,, Aveue Jea oul, 9 Belort cedex, Frace Poe : (33) , Fax : (33) , E-mal : artyukh@ge.uv-comte.r ABSTRACT The obectve o the propoed tudy to aalyze umercally ad expermetally traet heatg o a lat pecme. The aaly baed o the oedmeoal heat coducto model. The ma problem o the aaly to etmate the heat lux aborbed by a pecme. Temperature evoluto meaured de the pecme are ued to olve the vere heat coducto problem, two method are ued : the teratve regularzato method () ad the equetal ucto peccato method (). The te derece method utlzed or olvg the drect problem. Thee two method are rt vered ad compared or oe layer pecme by ug mulated umercally data. The luece o the dmeole tme tep ( αfo umber) aalyzed ad comparo reult are demotrated. The the method are appled to aalyze expermetal data obtaed wth the ue o a thermal cyclg devce. OECLATURE C pecc heat umber o approxmato parameter T umber o tme tep umber o temperature eor T, T a, T temperature, ambet, tal temperature Q etvty coecet at tme t ad poto X e thcke o the pecme meaured temperature at poto X h heat traer coecet t, t tme, al tme q, qexact heat lux dety, exact heat lux adot varable γ temperature varato ± dety thermal coductvty thermal duvty etmated error or crtero 1 αt αfo αq WIR tme tep dmeole tme tep umber heat lux tep wth teratve regularzato KEYWORDS Ivere problem, regularzato, heat lux, expermetal aaly, umercal mulato, etmato. ITRODUCTIO The ll-poed vere problem o etmatg the urace heat lux rom traet temperature htore meaured a heat coductg old cotatly o a great teret durg three lat decade. A lterature revew ad a preetato o deret method gve, or example, Tkhoov ad Are (1977), Beck et al. (1985), Heel (1991), uro (1993), Alaov et al. (1995). Deret applcato o varou method are preeted, partcular, Zabara et al. (1993) et Delauay et al. (1996). I th paper, reult o a expermetal ad umercal aaly are reported, the goal o whch to etmate the heat lux aborbed by a lat pecme cooled at the back de ad ulated at t lateral urace. We ue the teratve regularzato method () (Alvaov et al., 1995) ad the ucto peccato method () (Beck et al., 1985) to olve th vere heat coducto problem. The rt umercal algorthm baed o the mmzato o the redual uctoal whch the tegrated derece betwee temperature htore meaured ad thoe calculated by olvg the drect problem. The cougate gradet method ued to olve the vere problem. The redual uctoal gradet computed by olvg the adot problem ad the optmal decet parameter calculated by olvg the problem or temperature varato. The heat lux evoluto approxmated by cubc B-ple (Alvaov et al 1987). The ecod umercal algorthm baed o the mmzato o the dcrete leat quare crtero by takg to accout a ew uture tme tep. The te derece method utlzed or olvg the drect problem. Thee two method are rt vered ad

2 compared betwee them or oe layer pecme by ug mulated umercally data. The luece o the dmeole tme tep ( αfo Ζ αt / e ) aalyzed. The the method are appled to aalyze expermetal data obtaed wth the ue o a thermal cyclg devce. A decrpto o expermetal etup ad rt reult o the expermetal data proceg are reported. IVERSE PROBLE FORULATIO The pecme heated by a heat lux o ukow dety at the actve urace ad cooled by a orced covecto low at the oppote urace. The ollowg hypothee have bee take to accout: thermophycal properte are uppoed to be cotat, heat traer oe-dmeoal. heat traer coecet cotat at the cooled urace. Uder thee codto, the heat traer proce the pecme ca be decrbed by the ollowg ytem o equato : T T Ζ, Ψ x Ψ e, Ψ t t (1) t T (, ϑ Ζ q( () T ( e, ϑ Ζ hξt ( e, ϑ T a ζ (3) T ( x,) Ζ T, x e (4) I the model (1)-(4), the heat lux dety q ( ukow. To get addtoal ormato about the temperature dtrbuto the pecme, temperature htore are meaured the pecme at a certa umber o pot wth coordate x Ζ X, Ζ 1,,..., : Tmea ( X, Ζ (, Ζ 1,,..., (5) Th ormato, together wth the model (1)-(4), ued to olve the vere problem. EXPERIETAL DEVICE The expermetal tudy wa realzed o a thermal cyclg devce wth a oe-layer pecme (gray cat ro). The heatg devce ue a heat ource o a relatvely weak heat lux (a hotwd). The coolg lud ued the preurzed water crculatg chael. Two eor have bee corporated the pecme to record the temperature evoluto whe the pecme ubected to deret type o thermal cyclg. I order to lmt lateral heat loe, the ample coated wth a ceramc re. A luxmeter (heat lux eor), placed at the back urace o the pecme, allow oe to etmate the lux dety appled o the ormer. The expermetal devce compoed o the ollowg elemet (gure 1) : heat ource ytem : hot ar wth the temperature meaured by the thermocouple TC, pecme wtch the temperature htore are meaured by TC 1 ( X 1 = 1.6 mm) ad TC ( X = 7.8 mm) repectvely, Heat Flux TC Ceramc re ( TC 6, TC 7, TC 8 ) e l Gray cat ro TC 1 TC x Ceramc coat Outlet o lud ( TC 3 ) Ilet o lud ( TC 4 ) Outlet o lud ( TC 5 ) Heat lux eor ( TC 9, TC 1 ) =.37m Fgure 1. Smpled cheme o expermetal devce water coolg ytem : oe put ( TC 4 ), two output Iulatg ceramc rg wth three thermocouple ( TC 3,TC 5 ). ( TC 6, TC 7, TC 8 ) located at deret poto. The purpoe o th rg to lmt thermal loe radal

3 drecto order to coder the heat traer a moodmeoal. heat lux eor coted o aluma coatg (1 mm thcke ad 5mm radu) depoted o the back ace o the ample. Temperature ( TC 9, TC 1 ) de th coatg are meaured at two locato eparated by.5.1 mm. umercal mulato how that the tme repoe o the heat lux eor very weak ad the temperature prole de the ytem lear. The type o the thermocouple TC ( Ζ,...,8) K ad E or Ζ 9, 1. ITERATIVE REGULARIZATIO ETHOD () To buld a computatoal algorthm, we ue the varatoal ormulato o the vere problem o teret. The problem to d uch ukow ucto q ( or whch temperature htore computed rom the mathematcal model (1) to (4) at the eor locato are cloe to meaured htore. That lead to the problem o mmzg the redual uctoal : t J ( q) Ζ ξt ( X, t; q) ϑ ( ζ dt Ζ1 (6) where T ( X, t; q), Ζ 1,,...,, are temperature htore computed at the eor locato wth the heat lux dety q ( gve. Ukow ucto parametrzed the orm o a cubc B-ple q( Ζ p ( (7) Ζ1 m m where p m, m Ζ 1,,..., m (, m Ζ 1,,...,, are ukow parameter,, are gve ba cubc B-ple. The umber o approxmato parameter uually xed a pror. A a reult, the vere problem reduced to the etmato o a vector o parameter T p Ζ [ p1, p,..., p ]. Ukow ucto codered a a elemet o the ucto pace L [, t ] o parametrzed ucto. We ue the ucotraed cougate gradet method o optmzato. The redual uctoal gradet a well a the decet drecto L pace have the orm : J q = Ζ D( Ζ m Ζ1 m Ζ1 g m m ( (8) d m m ( (9) So, the gradet characterzed by the vector T g Ζ [ g1, g,..., g ] ad the decet drecto by the T vector d Ζ d, d,..., ]. It eay to how that the [ 1 d redual uctoal mmzato wth repect to dered parametrzed ucto reduced to thoe wth repect to ukow parameter. The ucceve mprovemet o dered parameter are cotructed a ollow : p Η1 Ζ p Η ƒ d, Ζ,1,... (1) where the umber o the terato uder way, ƒ the decet parameter, p a tal gue or the vector o ukow parameter gve a pror. The vector d computed a ollow : ϑ1 d Ζ ϑg Η ϒ d, Ζ,1,..., (11) () (ϑ1 ) () Εg ϑ g,g Φ ϒ, Ζ where Ζ ϒ (1) () g The realzato o the teratve procedure (1) baed o computg the vector g at each terato. Th vector determed by the relatohp or the redual uctoal varato : J ( q, q) Ζ ( J, p) Ζ ( J, q) (13) p R q where J p the redual uctoal gradet pace o approxmato parameter ad J q the gradet L pace o parametrzed ucto, (, ) L R the calar product. By ug the parametrc orm (7), t ca be how that (Alvaov et al., 1987): J p Ζ Gg (14) where G the Gram' matrx or ba ucto : G Ζ G m Ζ (, m ) L,, m 1,,..., (15), Ζ Th matrx ymmetrc ad potve dete. The Choleky decompoto ued to olve the ytem (14). The mot eectve method or calculatg the gradet J p R pace baed o troducg a adot problem. The ollowg expreo or the gradet compoet ca be derved : t J Ζ (, ( dt, m Ζ 1,,..., (16) pm m where ( x, the oluto o the ollowg adot problem (Alvaov et al., 1995): ϑ Ζ, Ψ x Ψ e, Ψ t t (17) t 3

4 (, ϑ Ζ (18) X, Ω x ( Η1 ξ ( X, x ζ γ T( X, Ω (, Ζ1,,..., ϑ1 (19) ( X, Ζ 1( X,, Ζ 1,,..., () Η ϑ Ζ h ( e, (1) ( x, t ) Ζ, Ψ x Ψ e () A lear approxmato ued to etmate the parameter ƒ (Alvaov et al., 1995) : t ξ T ( X, ϑ ( X, γ( X, dt Ζ1 ƒ Ζ ϑ (3) t Ζ1 ξ ζ γ( X, where γ ( x, the oluto o the ollowg boudaryvalue problem : γ γ Ζ, Ψ x Ψ e, Ψ t t (4) t γ(, ϑ Ζ D( (5) γ( e, ϑ Ζ hγ( e, (6) ( x,) Ζ, Ψ x Ψ e (7) where D ( the decet drecto L pace o parametrzed ucto (9). To obta table oluto o the vere problem uder coderato, the teratve regularzato ued (Alvaov et al 1995). The ma dea to termate the teratve procedure wth the redual crtero : * J ( q ) (8) where the total (tegrated) meauremet error deed by : t Ζ ( dt (9) t Ζ1 ( ) a etmate o the tme-depedet tadard devato or the th temperature htory meaured. Th procedure gve the mot table oluto. The umber * o the lat terato the regularzato parameter o the method. ζ dt It eceary to ote that the umber o approxmato parameter hould be correctly choe or the dered ucto. Th umber ha to be choe o that the redual crtero would be vered. I th cae, the lexblty o the oluto good eough or ug the teratve regularzato method. Oe terato o the umercal algorthm compoed by the ollowg tep : oluto o the drect problem ad computato o the redual uctoal, vercato o the redual crtero, oluto o the adot problem ad computato o the redual uctoal gradet L pace, computato o the decet drecto, oluto o the problem or temperature varato ad computato o the optmal decet parameter, calculato o the heat lux approxmato. FUCTIO SPECIFICATIO ETHOD () To etmate traet heat lux, the um o quare ucto (Beck et al. 1985) : J r Η1 Η1 ( q ) [ T ( X, tη ; q ) ϑ ( tη Ζ1 Ζ1 Ζ )] (3) Η1 mmzed wth repect q. The rt ubcrpt reer to pace where the eor umber ad the ecod to tme where r the uture tme. The temporal tablzato, or the ucto peccato method, obtaed through the temporary aympto that lux cotat over the r uture temperature : Η1 Η Ηr Η1 q Ζ q Ζ... Ζ q Ζ q Η αq (31) Η1 the the mmzato o the um J ( q ) : J ( q αq Η1 Η 1 ) Ζ lead to the expreo o αq Η1 : r Η ξ Ζ1 Ζ1 Η1 ξ ζ Η Q ζ (3) Q T ( X, tη ; q ) ϑ ( tη ) Η1 Ζ1 Ζ1 αq Ζ (33) Η Η r Η1 where Q Ζ T / q the etvty coecet at tme t Η ad eor locato X. The etvty coecet are computed umercally by ug the ollowg relato Q Η [ T Ζ Η ( q Η q q ) ϑ T Η1 Η where a mall coecet. ( q )] (34) 4

5 Oe terato o the etmated algorthm compoed by the ollowg tep : computato o temperature at the eor locato by olvg the drect problem or q ad q Η q codto. computato o the etvty coecet Q Η, computato o the lux varato computato ad tet o heat lux αq Η1, Η1 q. T ( C) T(, T(e, UERICAL SIULATIO AD COPARISO OF THE TWO ETHODS The rt expermetal tudy wa carred out wth relatvely low heatg rate. But our uture goal to aalyze teve heatg wth mall eough αfo umber. That wa a reao to compare the ececy o the aalyzed method wth deret αfo value. For our purpoe, the ma crtero wa the accuracy o etmated heat lux evoluto. Reult o uch a comparo are preeted th ecto. To mulate the umercal oluto ad to compare the method, we have uppoed the problem (1) to (4) that Ζ Ζ e Ζ1, h Ζ T Ζ. Two type o the heat lux evoluto were tuded : Step heat lux evoluto (W/m ) q ( = 1 Ψ t Ψ t ad q ( = t Ψ t Ψ t Suodal heat lux evoluto wth decreae ampltude: q ( t ) Ζ q ξ1 Η ( ζexp(-a, Ψ t t q =1 W/m, Ζ t, rd /, a =,5 or α t =,1 to,1 ad a = or α t =1. I thee codto, the α Fo umber equal to the tme tep that we have vared rom.1 to 1. I the above tet cae the meaured temperature evoluto wa mulated umercally at the back urace o the pecme wth a radom oe o 1% o the maxmal value o the temperature. For the, we have ued =51 parameter or to etmate the ukow heat lux evoluto. For the, the etvty coecet were computed by ug =.1. For each partcular cae ( α Fo ), the umber o uture tep r wa choe by realzg umercal expermet dvdually. Th umber wa creaed utl luctuato the oluto would be mmum, that there would be o urther tablzato. I gure a ad b, we how a example o temperature evoluto at x = ad x = e =1 or both tet cae aalyzed. I gure 3a, b, c ad d or the rt type o the heat lux evoluto ad gure 4 a, b, c ad d or the ecod oe repectvely, we preet a comparo betwee the two method or αfo umber varyg rom.1 to 1. Thee reult o comparo how that or α Fo >.1 the etmated a exact heat lux evoluto obtaed by the both method are a good agreemet Fgure a. Temperature evoluto at x Ζ ad x Ζ e or the rt tet cae. T ( C) 1,6 1,4 1, 1, T(, T(e, Fgure a. Temperature evoluto at x Ζ ad x Ζ e or the ecod tet cae. For umber α Fo =.1, reult etmated by rema cloe to exact heat lux but, a the αfo umber decreae, the derece betwee exact ad etmated value creaed. For the, the derece betwee exact ad etmated heat lux evoluto alo creaed but le tha that obtaed by the. Thee reult how that, or the tet cae aalyzed th paper, heat lux evoluto etmated by the are more accurate that thoe obtaed by the. Th a drect coequece o the ue o the regularzg redual crtero (8). For all codered cae, the redual value were computed by ug the ormula (9). Thee value are: αfo 6 ( 1 )

6 1, 1, 1, 8 1, , Fgure 3a. α Fo = Fgure 3e. α Fo =.1 1, Fgure 3b. α Fo =.1 Q (W/m²), 1,8 1,6 1,4 1, 1, Fgure 4a. α Fo =1 1,, 1, 8 1,8 1,6 1,4 qexact 6 4 1, 1, , 1, Fgure 3c. α Fo = Fgure 3d. α Fo =.5 4, 1,8 1,6 1,4 1, 1, Fgure 4b. α Fo = Fgure 4c. α Fo =.1 SF 6

7 q (W/m ) 1,8 1,6 1,4 1, 1, 8 6 4, 1,8 1,6 1,4 1, 1, Fgure 4d. α Fo = Fgure 4e. α Fo =.1 We ote that the comparo reult obtaed th paper der rom thoe preeted or example, Beck, A explaato that we ue aother realzato o the baed o a ple-approxmato o the ukow heat lux evoluto. I partcular, the redual uctoal gradet ot equal to zero at the al tme wth th approxmato. I term o computg tme (we ued a PC o 6 Hz), the more eectve that the, that the take le tme that the. For the oe dmeoal tet cae codered th paper the computg tme rather mall. So, the computg tme ot the bet crtero to compare the method th cae. For two ad three dmeoal problem th crtero may be much more mportat. Th queto hould be careully aalyzed. It hould be uderled that the realzed comparo o the two method wa rather retrcted. It eceary to cotue th aaly to etablh the doma where each o the method more eectve. EXPERIETAL DATA AALYSIS The expermetal tet were realzed wth a relatvely low heat lux wth a pecme o the gray cat ro ( e =.97 m, =5 W(m.K), ± =7 kg/m 3, C =67 J/(kg.K) to obta a gcat derece betwee the temperature htore meaured. The heat traer coecet determed, teady tate regme, by a heat balace equato at the terace pecmelud. q h Ζ Ζ 317 Ζ 88W /( m. ) T ϑt 45 ϑ 9 C α h =ΝΘΟ W/(m C), α T =ΜΚΡ= C, α q =ΘΚΤ=kW where T the temperature o the back urace o the ample, T the mea lud temperature ad q the teady heat lux etmated by the Fourer law. αt.98 q Ζ Ζ 5.3 Ζ 31.7 kw / m e.5 α T =.5= C, α =3 W(/m C),= α e =.1=mm, α q =6.45 kw. =1, T =35, α Fo =4,5 The expermetal data aaly corm that heat traer coecet h rema practcally cotat. I gure 5a ad 5b, we preet meaured ad etmated temperature at eor locato X 1 =1.6 mm, X =7.8 mm repectvely ad, gure 6, we preet the etmated heat lux evoluto. eaured ad etmated temperature htore how a good agreemet ad the derece betwee them rema le 4%. T ( C) T(X1, 1( 1 5 1, 1,5,,5 3, 3,5 Fgure 5a.Temperature htore meaured ad etmated at locato X 1 =1.6 mm, T ( C) ( T(X, 1 5 1, 1,5,,5 3, 3,5 Fgure 5b. Temperature htore meaured ad etmated at locato X =7.8 mm 7

8 q (w/m²) 5, 45, 4, 35, 3, 5,, 15, 1, 5, 5 1, 1,5,,5 3, 3,5 Fgure 6. Etmated heat lux dety. Thee reult were obtaed wth the ue o the both method uder coderato. Practcally, there o derece betwee the both oluto. Th act corm that there are doma o applcato where deret method gve practcally the ame reult (Beck et al., 1996). COCLUSIOS To etmate the heat lux evoluto rom meaured teral temperature htore, the ad wa tuded. A comparo betwee thee two method wa realzed or deret dmeole tme tep ( αfo umber) by aalyzg two tet cae. Th tudy how that or α Fo >.1, the both method gve practcally the ame reult. For α Fo <.1, the gve more accurate reult ad the error the etmated oluto are creaed much ater or the that or the the tet cae codered. The ue o the rt expermetal reult allow to obta the mlar etmated heat lux evoluto or the two method. The rt expermetal tudy wa realzed by ug a thermal cyclg devce wth a oe-layer pecme (gray cat ro). Th tudy wa carred out wth relatvely low heat lux. Two temperature htore were meaured de the pecme. The heat lux evoluto at the actve urace wa etmated wth the ue o the ad. The reult obtaed by both method are practcally detcal. The preet tudy a preparato or a expermetal tudy o thermal regme o multlayer pecme wh are typcal thermal barrer heated tevely durg hort perod o tme. Problem wth Applcato to Ivere Heat Traer Problem, Begell Houe, Ic., ew York. Beck, J. V., Blackwell, B., St. Clar, C. R., 1985, Ivere Heat Codto : Ill-Poed Problem, Wley Iterc., ew York. Beck, J. V., 1993, Comparo o the Iteratve Regularzato ad Fucto Speccato Algorthm or the Ivere Heat Coducto Problem, Zabara et al., Beck, J. V., Blackwell, B., ad Hahekh, A., 1996, Comparo o Some Ivere Heat Coducto ethod Ug Expermetal Data, Iteratoal Joural o Heat ad a Traer, Vol. 39, o. 17, pp Delauay, D., Jary,Y., ad Woodbury, K., (Ed), 1996, Proceedg o the d Iteratoal Coerece o Ivere Problem Egeerg: Theory ad Practce, Le Croc, Frace, Jue , ASE. Heel, E., 1991, Ivere Theory ad Applcato or Egeer, Pretce Hall, Eglewood Cl, ew Jerey. uro, D., 1993, The ollcato ethod ad the umercal Soluto o Ill-Poed Problem, Joh Wley ad So, Ic., ew York. Tkhoov, A., Are V., 1977, Soluto o Ill-poed Problem, Wley, ew York. Zabara,., Woodbury, K., ad Rayaud,., (Ed), 1993, Proceedg o the Frt Iteratoal Coerece o Ivere Problem Egeerg: Theory ad Practce, Palm Coat, Florda, USA, Jue , ASE. REFERECES Alaov, O.., Artyukh, E. A., ad earokomov, A. V., 1987, Sple-Approxmato o the Soluto o the Ivere Heat Coducto Problem, Takg Accout o the Smoothe o the Dered Fucto, Hgh Temperature, Vol. 5, o.5, pp Alaov, O.., Artyukh, E. A., ad Rumyatev, S. V., 1995, Extreme ethod or Solvg Ill Poed 8