Thermal Properties of Functionally Graded Fibre Material

Transcription

1 Ja Tua, Elżbea Radaszewska Depae of Techcal Mechacs ad Ifoacs, Lodz Uvesy of Techology Żeoskego 6, 9-94 Łódź, Polad E-al: Theal Popees of Fucoally Gaded Fbe Maeal DOI:.564/ Absac I hs pape he poble of odellg gaded aeals he fo of a fbe copose wh vayg fbe daee s cosdeed. The a of odellg was o deee he co ad acoscopc heal popees of hs ype of aeal, whch he aveage heal coducvy elao o fbe sauao chages was calculaed a ay po of fbe FGM, ad he he effecve heal coducvy of a whole laye of he aeal was deeed. To do ha, a u cell of he aeal of gve sucue was solaed ad he oedesoal hea flux passg hough was cosdeed. As a effec of he vesgao, he pocedue of effecve heal coducves calculao was peseed ad llusaed wh a uecal exaple. Addoally he dscee ad couous appoach o he effecve heal coducves calculaos wee aalysed ad copaed. Key wods: fucoally gaded aeal, fbe coposes, gaded aeals odelg, effecve heal coducvy. Ioduco Hea asfe pobles ae wdely cosdeed ay aeas of egeeg. Fo exaple, he hua eed fo heal cofo leads o cosucg pope buldg aeals [, ] ad pas of clohg [3, 4]; kowledge of he behavou of sucues ude a appled heal load ca lead o defcao of he aeal o sucual feaues [5, 6]; he aufacug pocess ca eque scly desged heal codos o pefo a echologcal pocess coecly [7, 8]; he opeag codos of ache pas ca foce egees o ceae a aeal sucue hghly essa o heal sesses [9]. I ay aeas of egeeg, heal sesses play a poa ole. I soe egeeg sucues lke sple ad coplcaed elees of aches, ca eges, ube blades, aeospace sucues ad eegy coveso syses, whch wok a hgh ad o-ufo epeaue felds, he a poace s o desg heal essa sucues. The ecessy of desgg aeal fo a loge lfee ples he applcao of ew echologcally advaced aeals. I addo o well-kow ad cooly used classc aeals of ufo echacal ad heal popees, fucoally gaded aeals ae also ulsed. Fucoally gaded aeals (FGM) ae a ew class of copose aeals whee he coposo of copoes geeaes couous ad sooh gadao of popees of a copose. Mulphase coposes wh a couously vayg volue of facos ae chaacesed by soohly chagg echacal ad heal popees. The cocep of fucoally gaded aeals was poposed he ealy 98s by aeals scess Japa as a eas of pepag heal bae aeals []. The ew geeao of fucoally gaded fbe aeals has dyac, effecve heal popees ad he volue faco of he aeals chages gadely. The o-hoogeeous, vaable cosucues of hese aeals cause couously gaded acoscopc popees, such as he heal coducvy, specfc hea, ass desy ad elasc odulus. These aeals have bee developed as supe-essa aeals ode o decease heal sesses ad cease he effec of poeco fo hea [, ]. Accodg o aufacug echques, FGM ay exhb ohoopc o asoopc aeal popees due o paccal egeeg equees. Apa fo he a advaage of good hea essace, hey ae chaacesed by vey low desy, esulg lghwegh sucues wh vey good heo-echacal popees ad sall opeag coss [3, 4]. Thee ae dffee kds of fabcao pocesses fo poducg fucoally gaded aeals. Each of he aufacug pocesses of gade aeals s adaped o he ype, shape ad sze of aeal, as well as o he value of he gade of aeal vaaos o he cosucue of gade copoes [5]. Fucoally gaded aeals ae always aufacued by xg wo dffee aeal phases, fo exaple eal ad ceac o fbes ad es. The cocep s o ake a copose aeal by vayg he cosucue fo oe aeal o aohe wh a specfc gade. Ths eables he aeal o have vayg, desged popees. The easoably low heal sesses allow us o ceae hgh heal essace aeals whch ca wok chagg evoeal codos. The os cooly used gade aeals ade of fbe coposes ae hose whch fbes have a cosa daee, bu he sauao ca be dffee. Gaded aeals ca be also ceaed fo egulaly aaged fbes of vaable daee. Glass ad cabo fbes ae he os popula classes of fllg fbes. Glass fbes ae cooly used due o he low cos ad excelle popees. Cabo fbes ae oe of he os poa classes of fllg fbes because of he popees, such as hgh sffess, hgh esle segh, low wegh, hgh epeaue oleace ad low heal expaso. These feaues ake he vey popula ay egeeg applcaos. Howeve, cabo fbes ae elavely expesve copaso o fllg fbes, such as glass o plasc fbes [6, 7]. By coollg he desy of he aagee of fbes o he daee, s possble 68 Tua J, Radaszewska E. Theal Popees of Fucoally Gaded Fbe Maeal. FIBRES & TEXTILES Ease Euope 6; 4, 4(8): DOI:.564/

2 o fluece he duably of soe sucual pas. A exaple of such usage of gade aeal ca be a bea sucue ade of copose aeal flled wh fbes of cosa daee bu wh dffee sauao of he ax o bea flled wh fbes of vaable daee. I hs case he efocg fbes ca be oe desely aaged pas oe dsa fo he eual axs. Pope aagee of fllg fbes eables bee usage of he copoes, whch eas a oe expesve ad usually oe essa faco ca have geae sauao a pos of hghe sesses. Geeally, hs ype of cosuco aeal ca be used as sucual elees of buldgs, such as beas, plaes o shell elees I addo o he echacal popees of fbe FGM, s heal popees ca also be poa whe cosdeg heal cofo sde buldgs. I hs pape he effecve heal coducvy of a laye of gve hckess bul fo fbe FGM elao o vayg fbe sauao was deeed. To do ha, he aveage heal coducvy a a abay po of he sucue was cosdeed. I he vesgaos was assued ha log paallelly-placed fbes of vaable daee ae aaged a egula hexagoal cosucue. Wh espec o he eal, dscee sucue of fbe FGM (bul wh ay layes of copose aeal), he heal behavou of a dscee aeal sucue was copaed wh ha of he gade aeal cosdeed as a couu. Poble foulao Le us cosde wo-desoal seady sae hea asfe fo a body bul wh ohoopc fbe FGM aeal. To descbe he behavou of he body a each po, we ca use ypcal elaos descbed by a hea equao ad Foue s law [8]: dvq + f, q - e FIBRES & TEXTILES Ease Euope 6, Vol. 4, 4(8) T Ω () Whee, q ad f deoe he hea flux esy ad hea souce, especvely, e - he ax of aveage heal coducvy coeffces (followg fo popees of he ax ad fbe aeals), ad Τ deoes he gade of he epeaue feld. The a poble fo FGM s he kowledge of elees of he ax e a each po of he body doa. I he Caesa coodae syse fo ohoopc axes of he aeal, we have o kow elees ex ad ey of he a dagoal of he e ax. Addoally o be able o solve he poble, we have o kow pope bouday codos chaacesc fo evoeal codos of he sucue aalysed descbg, fo exaple, he epeaue, hea flux esy o coveco pope pas of he bouday (Fgue ). The codos eoed ca be we he fo: T T o G T, q q q o G q, () q h(t - T ) o G c whee, T & q deoe pescbed values of he epeaue ad hea flux, h he coveco coeffce, T - he evoeal epeaue, ad s he oal u veco of he bouday le a a chose po. I ay paccal applcaos we should kow he effecve heal coducvy fo a elee of gve hckess elao o fbe sauao chages, whch chaaceses he oal heal essace of a laye bul fo fbe FGM. I hs case we ae eesed, fac, oedesoal hea flux (as show Fgue ) of pescbed epeaue T ad effecve epeaue T, fo sace. Kowledge of hs coeffce gves us he possbly o cosde he heal popees of fbe FGM as cosa elao o fbe sauao chages, ad leads o he splfcao of ypcal paccal calculaos. Aveage heal coducves fo fbe FGM Fbe FGM s a xue of ax ad fbes. To deee he aveage heal coducvy of hs ype of aeal we should kow how o deee he heal essace of copose aeal whe hea flux passes hough he copoes aaged paallel o a seal ae, ad how o ake o accou he shape of usos. The fbe aagee, coss seco shapes ad fbe sauao he ax play a sgfca ole fo heal popees of he whole aeal. Paallel ad seal aagee of copoes The goal ow s o deee he effecve heal coducves fo oe-desoal hea flow fo coposes bul wh ay layes of dffee, heally soopc aeals. The poble s well Fgue. FGM sucue subjeced o sevce loadg. Fgue. Oe-desoal hea flow hough FGM sucue. kow ad was solved ay yeas ago, bu s awaeess s eally poa fo fuhe vesgaos caed ou hs pape. Le us cosde seady sae hea asfe hough a ullaye sucue of u legh, bul wh layes of dffee hckesses ad heal coducves, as show Fgue 3.a (see page 7). Fs we cosde he oe-desoal hea flow pepedcula o he laye deco, Fgue 3.b. Takg o accou ha he flux coduced hough layes s he sae each laye of he dscee sucue ad should be he sae hoogesed aeal, we ca we: (3) whee, j deoes he epeaue jup bewee -h ad j-h sufaces, ad sx - he effecve heal coducvy of he copose he y deco. Nog ha (4) we ca eaage elaoshps (3) o oba he effecve heal coducvy ad we he fo: sy 69

3 a) b) c) Fgue 3. a) sadwch sucue, b) hea flow hough elees coeced sees, c) hea flow hough elees coeced paallel. (5) I he case of oe-desoal hea flow paallel o he layes (Fgue 3.c), he oal flux passg hough he sucue s equal o he su of fluxes passg hough all layes, ad should be equal o he flux passg hough he hoogesed aeal. Cosequely assug a addoally cosa epeaue a each bouday, we ca we: ΔT ΔT sx (6) o fally oba he effecve heal coducvy he y deco he fo: sy sx (7) Aveage heal coducves a a abay po of fbe FGM Le us ake o accou a epeaable sucue of fucoally gaded aeal flled wh log fbes of chagg daee ad aaged a hexagoal sucue, as show Fgue 4. The a goal s o deee he aveage heal coducvy coeffce of FGM he y deco. To do ha, we have o defe s value a each po of he sucue. To deee he aveage heal coducvy coeffce a a abay po, we solae he epeaable u cell, show he Fgue 4, ad we coec o he local coodae syse ξ η. Wh espec o syecy of he cell, we cosde oly oe-foh of he cell (Fgue 4). Because of he dffee possble daees of fllg fbes, we ca obseve wo dffee aagees of he elee cosdeed. I he fs case (Fgue 5.a) we ca obseve he ovelappg of fbes ( < + ), ad he secod case (Fg- ue 5.b) a ax aeal laye exss bewee fbes ( > + ). I he fs case, sla o wha was doe [9], he aveage heal coducvy coeffce a he y (cf. Fgue 4) coodae po ca be calculaed as a paallel coeco of hee pas of he cell: f, ff & f (Fgue 5.a). Usg he pope oao (7) chaacesc fo ou u cell, we ca we: y ( y) (8) f ( ) + ff ( + ) + f( ) whee, f, ff, f deoe he effecve heal coducvy of f, ff & f pas, especvely. All effecve heal coducves eoed Equao 8 ca be calculaed a sla way. Le us cosde he pocedue of calculao of he ff coeffce. To calculae, we ca we he oal hea balace fo he fesal seco dξ, show Fgue 6, o sply use Equao 5 applyg he pope oao. Adapg Equao 5, we ca we: Fgue 4. Hexagoal epeaable aagee of fbes FGM ad ue cell. 7 Fgue 5. The oe foh of he u cell; a) wh ovelappg fbes, < +,b) wh o ovelappg fbes, > +. FIBRES & TEXTILES Ease Euope 6, Vol. 4, 4(8)

4 ff ξ ( ξ) (9) ( CD + AB) + BC f AD I hs way we oba he subsug heal coducvy coeffce fo elee dξ as a fuco of he pope leghs of he elee cosdeed as well he heal coducves of ax ad fbe aeals f. Nog ha CD ξ, BA ( ) ξ ad AD CD + BC + AB 3, elao (9) ca be ewe he fo: ) ff ξ ( ξ FIBRES & TEXTILES Ease Euope 6, Vol. 4, 4(8) () 3 3 ξ + ( ξ ) + f Kowg ha he oal hea flux passg hough he pa of he cell cosdeed s he sae eal ad hoogesed aeal, we ca we Equao : ΔT ΔT ff ξ dξ ff AD AD ( + ),() whee, s he jup epeaue bewee pos A ad D of he u cell, whch s assued o be cosa alog he ξ axs he cell cosdeed. Cosequely we ca we: ff + ff ξ dξ () The ssg coeffces (8) ca be obaed a sla way. Cosequely subsug he heal coducvy he fesal seco dξ fo pas f ad f ca be expessed as follows: f ξ f ξ 3 ( ξ ) f ( ξ ) f ( ξ ) 3 + ( ξ ) (3) ad effecve heal coducves equed have he fo: f f ξdξ, f f ξ dξ (4) Geeally he egals Equao ad Equao 4 have o be calculaed a uecal ae. I he secod case (Fgue 5.b), he aveage heal coducvy coeffce ca be calculaed slaly ad ca be we he fo: ( y) y f + ( ) + f (5) Whee, f ad f ae deeed as follows: f f ξdξ, f f ξ dξ (6) The aveage heal coducvy coeffce fo he whole cell (a a abay po of FGM fo a y coodae) s easly deeed fo Equao 5 he seal coeco of he uppe ad lowe oe fouh of cell pas (cf. Fgue 4): ) + y yj ay( y (7) yyj whee, ad j ae equal o o accodg o he chaace of he uppe ad lowe oe-fouh of he cell. If ecessay, we ca oba he aveage heal coducvy ax fo a abay y coodae he sae ae. I s woh og ha wh espec o he fbe aeal aagee, coeffces ax ad ay fo he chose y ae cosa alog he x deco. Effecve heal coducvy fo a laye of fbe FGM Havg deeed he aveage heal coducvy coeffces a each po of he aeal, we ca deee he effecve heal coducvy coeffce of he whole laye. Le us assue ha he laye of he aeal has hckess (as show Fgue ) ad we kow fuco ey (y). Accodg o Foue s law, he esy of hea flux he y deco s equal o: dt qy ay y) dy ( (8) Sepaag he vaables ad egag boh sdes wh ls y, we oba: T T qy, (9) dy ay whee, T ad T deoe he epeaues o he uppe ad lowe suface of he laye, especvely. Copag he hea flux passg he y deco hough seco dx eal ad hoogesed aeal, we ca we: Fgue 6. The oe-foh of he u cell wh sgled fesal seco dy. T T T T dx ey dx, () aydy whee, ey deoes he effecve heal coducvy he y deco of he FGM laye aalysed, whch ca be easly deeed as: ey () dy ay The expesso Equao ca be used whe we ea he aeal sucue as couous. The FGM aeal sucue s fac dscee. Cosequely sead of Equao, kowg he eal sucue of FGM coeced wh he aufacug pocess (daees of fbes, he ube of fbe layes, ec.), we ca also use Equao 5 o deee he effecve heal coducvy equed. Nuecal exaples The a goal of hs seco s o llusae he appoach o effecve heal coducvy calculaos poposed ad ex o copae he dscee ad couous appoach fo he calculao. Nuecal exaple of calculao of he effecve heal coducvy fo a laye of FGM To show he ably of he ehod of effecve heal coducvy calculao poposed (accodg o Equao ), a uecal exaple was ealsed. I was assued ha he adus of fbes s kow a ay po of he aeal ad ca by descbed by he fuco R(y). I ou case hs fuco was assued he lea fo: R( y) y + () whee, ad deoe he adus of fbe a pos y ad y - whch eas o exeal boudaes of laa of hckess. Calculaos wee pefoed fo chagg ao / assug he axal possble sze (cf. Fgue 6). I pacula, was assued:.5, 7

5 Subsue coducvy, W/(K) Fgue 7. Fgue 8. Subsue coducvy, W/(K) / Nube of laye Relave eo, % Fgue 9. Nube of laye Fgue 7. The effecve heal coducvy vs. chages of fbes adus alog laa hckess. Fgue 8. The effecve heal coducves fo fbe FGM. Fgue 9. The elave eo of calculao of he effecve heal coducvy obaed as a esul of dscee appoach vs. couous appoach. ube of layes, W/(K), ad f 5 W/(K). A plo of he effecve heal coducvy obaed s show Fgue 7. Fo geoecal vesgaos of he u cell poposed, we ca see ha he axal sauao of fbe aeal he ax (fo / ad ) s equal o p/( 3).969, whch allows o ceae a copose wh effecve heal coducvy appoxaely wo es lowe ha fo pue fbe aeal. Copaso of dscee ad couous appoaches o calculao of effecve heal coducvy coeffces Eve f we kow he eal dscee sucue of a fbe FGM, we ea as couous aeal. The queso s how good hs appoxao s ad whe we ca ea a dscee aeal as couous. The queso s sla o ha whe we ea fbe coposes as ufo hoogesed aeal fogeg abou he e sucue. I he case of fbe FGM cosdeed, he effecve heal coducvy coeffce ca be obaed usg he dscee (cf. Equao 5) o couous appoach (cf. 7 Equao ). The queso ow s how ay layes ae eeded o ea a dscee sucue as a couous oe. To exae hs, we ca do uecal ess usg Equao 5 o Equao, whee he effecve heal coducvy coeffce a a abay po s calculaed he sae ae usg Equao 7. I he case of he dscee appoach, we have o calculae he effecve heal coducvy coeffce each laye accodg o Equao 5. I he case of he couous appoach, we have o calculae, a uecal ae, he egal appeag Equao. I hs case we foge abou he eal aeal sucue, bu we ae eesed he heal coducvy coeffce a soe pos accodg o he ehod of uecal calculaos chose ou case Gauss pos. The sucue of he u cell (he daees of fbes) esuls fo he assued fuco R(y) of fbe daee vaably, ad ca be descbed a ay po ad o oly eal fbe posos. Ths appoach ceaes vual (o exsg) fbes whch ae used he calculao pocess. Nuecal ess wee caed ou fo laa coposed of 3 o layes. Each laye had a cosa hckess esulg fo he assued deso.5. Addoally was assued Equao ha ad. Ths assupo causes a chage fbe sauao he ax fo he axu o u possble value. Dug uecal egao Equao he ube of Gauss pos was assued o be equal o, whch eas ha he egal Equao was calculaed as a esul of calculao y a wo pos, ad fuco couse y (y) was appoxaed wh a hd ode polyoal. Plos of he effecve heal coducves vesus he ube of layes fo he dscee ad couous appoaches ae show Fgue 8. The elave dffeece bewee he coeffces obaed usg boh appoaches s depced Fgue 9. The couous appoach gves a eally sall elave eo copaso o he dscee oe. Fo a ube of layes equal o 3 he eo s salle ha.5%, ad fo a aeal bul fo oe ha 5 layes s always salle ha.%. The geae he ube of Gauss pos used uecal calculaos, he bee he accuacy ha ca be acheved; FIBRES & TEXTILES Ease Euope 6, Vol. 4, 4(8)

6 howeve, sees o o be ecessay wh espec o sasfacoy pecso of calculaos fo a ube of Gauss pos equal o. Cocluso eak I he pape, a sple way of aveage heal coducvy calculao of FGM flled wh log paallely-aaged fbes was peseed. Nex he effecve heal coducvy fo a laye of fbe FGM was deeed he gade deco of fbe daee chages. Nuecal calculaos wee caed ou fo a aeal of lea vaably of fbe daee, assug dffee velocy of fbe adus chages acoss he hckess of he aeal saple. Addoally was show ha he case of heal popees, laa cossg of hee o oe layes ca be eaed as couous wh good accuacy. Cosequely hee s o eed o cosde all layes of laa because we ca hoogese he oe sucue wh he pescbed fuco defg he vaably of he fbe daee, ad he ea as couous. The effecveess of he poposed ehod of calculao of effecve heal coducves of fbe FGM s defed by he ube of eleeay aheacal opeaos whch have o be doe o calculae pope egals. Assug uecal egao usg fve Gauss pos o calculae he aveage heal coducvy a a gve po ad wo Gauss pos o calculae he effecve heal coeffce of he whole laye of fbe FGM, we eed a lle oe he oe huded eleeay aheacal opeaos. I he eal wold of uecal calculaos, he e of execuo of hs ube of calculaos s eglgbly sall. To vefy he accuacy of hs appoach we eed paccal ess. I he case of suffce copably of paccal es esuls wh uecal calculaos, he ehod poposed should be evsed akg o accou he sochasc appoach o fbe aagees he u cell. Refeeces. Mlaka J, Šaca J. Tepeaue ad hudy pofles passve-house buldg blocks. Buld Evo 3; 6: Sog G, Paskaluk S, Sa R, Cow EM, Dale JD, Ackea M. Theal poecve pefoace of poecve clohg used fo low ada hea poeco. Tex Res J ; 8(3): Yu S, Wae JJ. Nuecal odelg of hea ad osue asfe a weaable covecve coolg syse fo hua cofo. Buld Evo 5; 93(): Koyck R. Two-desoal shape defcao fo he useady co-duco poble. Suc Muldscp O ; (3): Zucca S. Idefcao of a low-ode odel fo heal sess oog. J The Sesses 5; 8(3): Kodu VKR, Nase M, Pakala P, Vaa A. Modelg he espose of copose bea slab assebles exposed o fe. J Cos Seel Res 3; 8: Koyck R. Modelg of ase hea asfe wh bouded seas. Fbes Tex Eas Eu ; 88(5): Padue NP, Gell M, Joda EH. Theal Bae Coags fo Gas-Tube Ege Applcaos. Scece Jouals ; 96(5566): Kozu M. FGM acves Japa. Copos Pa B-Eg 997; 8(-): -4.. Shabaa YM, Noda N. Nuecal evaluao of he heoechacal effecve popees of a fucoally gaded aeal usg he hoogezao ehod. I J Solds Suc 8; 45: Y HM, Paulo GH, Bula WG, Su LZ. Effecve Theal Coducvy of Two-Phase Fucoally Gaded Paculae Coposes. J Appl Phys 5; 98(6): Gua G, Csafull D, Belouea S, Caea E. A heo-echacal aalyss of fucoally gaded beas va heachcal odellg. Copos Suc 3; 95: Loja MAR, Babosa JI, Moa Soaes CM. A sudy o he odelg of sadwch fucoally gaded paculae coposes. Copos Suc ; 94(7): Huu-Ta T, Seug-Eock K. A evew of heoes fo he odelg ad aalyss of fucoally gaded plaes ad shells 5. Copos Suc; 8: Kelly A. Cocse Ecyclopeda of Copose Maeals. Pegao, Olauj AO, Boeche SKS, Cuda TR. Theal coduco aalyss layeed fucoally gaded aeals. Cop Mae Sc ; 54: Ghoshdasda PS. Hea Tasfe. Oxfod Uvesy Pess, Des K, Radaszewska E, Tua J. Modelg of Fbe-Refoced Copose Maeal Subjeced o Theal Load. J The Sesses ; 35: Laha PK, Dashaa Y, Veugopal V. Role of buldg aeal heal cofo opcal claes A evew. Joual of Buldg Egeeg 5; 3: 4 3. Receved 5..5 Revewed..6 Isue of Texle Egeeg ad Polye Maeals The Isue of Texle Egeeg ad Polye Maeals s pa of he Faculy of Maeals ad Evoeal Sceces a he Uvesy of Belsko-Bala. The ajo ask of he sue s o coduc eseach ad develope he feld of fbes, exles ad polye coposes wh egad o aufacug, odfcao, chaacesao ad pocessg. The Isue of Texle Egeeg ad Polye Maeals has a vaey of sueao ecessay fo eseach, develope ad esg he exle ad fbe feld, wh he expese he followg scefc ehods: FTIR (cludg appg), Wde Agle X-Ray Scaeg, Sall Agle X-Ray Scaeg, SEM (Scag Eleco Mcoscopy), Theal Aalyss (DSC, TGA) Sog pac o eseach ad develope o geoexles ad geosyhecs ake he Isue Isue of Texle Egeeg ad Polye Maeals uque aog he ohe exle suos Polad. Coac: Isue of Texle Egeeg ad Polye Maeals Uvesy of Belsko-Bala Wllowa, Belsko-Bala, POLAND , e-al: p@ah.belsko.pl FIBRES & TEXTILES Ease Euope 6, Vol. 4, 4(8) 73