Optimum design of complementary transient experiments for estimating thermal properties

Transcription

1 Opiu desig of copleeary rasie experies for esiaig heral properies Jaes V. Beck*, Filippo de Moe, Doald E. Aos *Depare of Mechaical Egieerig, Michiga Sae Uiversiy, USA Depare of Idusrial ad Iforaio Egieerig ad Ecooics, Uiversiy of Aquila, Ialy Sadia Naioal aboraories, USA Absrac. I he desig of rasie heral experies for esiaig he heral coduciviy, k, ad volueric hea capaciy, C, he scaled sesiiviy coefficies are uilized. These coefficies should be large ad ucorrelaed. They are larges whe he heaed surface eperaure is held a he axiu value. By usig copleeary experies, he sesiiviy coefficies ca be ade ore ucorrelaed ha if a sigle experie is used. The resul of he wo cobied experies is ha he wo paraeers are foud wih he greaes accuracy. They are also of equal accuracy, which is geerally o he case. Acually his proble is uique o oly because of irisic soluios bewee he sall ad large ie soluios bu i he exchagig of he roles i he scaled sesiiviy coefficies for k ad C. I fac, he k-scaled sesiiviy coefficie fucio of he XB0T0 proble is equal o he C- scaled sesiiviy fucio for he XB0T0 case.. Iroducio Theral properies eed o be easured usig rasie ehods for several reasos. Oe of which is ha biological aerials chage durig heaig. Aoher is rasie ehods are eeded o deerie boh siulaeously. May geoeries ca be uilized, icludig oe-diesioal ad wodiesioal recagular ad also D ad D radial ad axial. Various boudary codiios ca also be iposed icludig eperaure ad hea flux, which are called he firs ad secod kids. Tiedepede boudary codiios are also possible. Aoher ajor cosideraio is he locaio of he sesors ad he duraio of he experie. Clearly ay aspecs of he proposed experies are possible. I his work, oly esiaio of wo paraeers, heral coduciviy k ad volueric hea capaciy C, is discussed. Their esiaio requires ha he k- ad C- scaled sesiiviy coefficies are uilized. These coefficies should be large ad ucorrelaed. By usig copleeary experies, he sesiiviy coefficies ca be ade ore ucorrelaed ha if a sigle experie is used. I oher words, hey allow us o iiize he area or hyper-volue of he cofidece regio [, Chaper 8]. Also, he cofidece regio is a iiu whe he deeria of he relaed arix for wo paraeers is a axiu. I paricular, he sesiiviy coefficies are copued for he heaed surface of he XB(XB0T0)0T0 proble, where B(XB0T0) idicaes ha he ie-depede hea flux applied o he boudary x = 0 is he oe coig fro he surface hea flux of he XB0T0 proble

2 (which has zero eperaure a x = ad is iiially a zero eperaure). A copleeary proble is o apply he hea flux correspodig o he XB0T0 proble (which has zero hea flux a x = ) o he proble XB(XB0T0)0T0. The heaed surface scaled sesiiviy coefficies for hese wo probles are copleeary. The k-scaled sesiiviy coefficie fucio of he XB0T0 proble is equal o he C-scaled sesiiviy fucio for he XB0T0 proble. This sae reversal occurs for he C sesiiviies. The resul of he wo cobied experies is ha he wo paraeers are foud wih he greaes accuracy. They are also of equal accuracy, which is geerally o he case. Whe copuig he sesiiviy fucios, he sall ie soluios are cosidered for he XB0T0 ad XB0T0 probles ad he relaioship idicaed above is deosraed. The whole ie regio soluio is cosidered oo. A advaage of he early ie soluios is ha hey are aheaically sipler ad he ie rage of validiy is larger ha oe would expec. Boh he early ie ad whole ie soluios are helpful for several reasos. Oe is he copuaioal advaage for early ies of he oe ad he efficiecy of oher soluio for oderae large ie. Aoher reaso is ha havig wo differe soluios for he sae proble provides a idicaio of irisic verificaio. Acually his proble is uique o oly because of irisic verificaio bewee he sall ad large ie soluios bu i he exchagig of he roles i he scaled sesiiviy coefficies for k ad C.. XB(XB0T0)0T0 rasie experie A diesioal aheaical saee for he XB(XB0T0)0T0 proble is T x = T α ( 0< x < ; > 0 ) (a) k T x (0,) = q XB0T0 (0,) ( > 0 ) (b) T(, ) = 0 ( > 0 ) (c) T( x,0) = 0 ( 0< x < ) (d) where q XB0T0 (0,) is he surface hea flux of he XB0T0 case. The soluio is discussed i ex subsecios for early ies ad for he whole ie regio... Sei-ifiie body soluio. Oe-er approxiaio For early ies, less ha he so-called deviaio ie defied as [] () d = 0. Aα ( x) ( A =,3,...,5 ), () he eperaure ad hea flux of he XB0T0 proble are he sae (wih errors less ha 0 A ) as he X0BT0 case. The soluios are T XB0T0 (x,) T X0BT0 (x,) = erfc x 4α ( d () ) (3a)

3 q XB0T0 (x,) q X0BT0 (x,) = T k x 0 πα e 4α ( () d ) (3b) A he boudary surface x = 0, he hea flux is q XB0T0 (0,) k πα = D πφ / ( 0.4 / ( Aα ) ) (4) where he group D = k / (uis of W - ) is cosidered kow whe he surface hea flux applies o he saple durig experies. Sice α is cosidered kow oo, le i be replaced by φ. The surface eperaure for he XB(XB0T0)0T0 proble ca accuraely be replaced a early ies (less ha he deviaio ie) wih he surface eperaure for he XB(X0BT0)0T0 case. The laer ca be calculaed usig Gree s fucios [3] where a furher approxiaio wih errors less ha 0 A ca be perfored for co-ies ( τ ) 0. ( x) / ( Aα ), ha is, G X (x,0, τ ) G X0 (x,0, τ ) = πα( τ ) exp x (5) 4α( τ ) Therefore, T X B( X B0)0 (0,) T X 0 B( X 0 B) (0,) = α k = D kπ α φ dτ = D τ ( τ ) φ q X 0 B (0, τ )G X 0 (0,0, τ )dτ kc (6) ha is valid wih errors less ha 0 A for 0.4 / ( Aα ). The k- ad C-scaled sesiiviy coefficies a he heaed surface x = 0 ca be obaied by direc differeiaio as k T X B( X B0)0 k (0,) k T X 0 B( X 0 B) k (0,) = D φ kc = (7a) C T X B( X B0)0 (0,) C T X 0 B( X 0 B) (0,) = D φ kc = (7b) Noice ha hese wo sesiiviy coefficies are equal ad fucios of kc. For his case usig he X0BT0 hea flux ad he X0 Gree s fucio, he scaled sesiiviy coefficies are boh equal o he egaive of he eperaure rise divided by. Therefore, heir su is equal o he egaive of he eperaure rise a he heaed surface. I a diesioless for, wih errors less ha 0 A hey are

4 k T X k (0, ) = k T X B( X B0)0 (0, ) k ( 0.4 / A ) (8a) C T X (0, ) = C T X B( X B0)0 (0, ) ( 0.4 / A ) (8b) where = α /. Also, he subscrip X ca be used as a shor oaio... Fiie body soluio for shor ies. Three-ers approxiaio Use he soluio for he eperaure give o page 30, Eq. (6) of Carslaw ad Jaeger [4] o ge, + x ( + ) x TXB 0T0( x, ) = T 0 erfc erfc = 0 4α 4α (9a) The hea flux obaied usig Eq. (9a) is q X B0 (x,) = k πα A x = 0, his equaio gives q exp + x ( +) x 4α + exp 4α (9b) =0 0 XB0T0(0, ) exp exp = 0 ( ) + kt = + (0) πα α α For ies less ha aoher deviaio ie give by () d = 0. Aα ( + x) ( A =,3,...,5 ), (a) which is loger ha he previous oe defied by Eq. (), he exac soluios Eqs. (9a), (9b) ad (0) ca be copued wih oly wo ers wih errors less ha 0 A. However, as for he locaio x = 0 which is here of ieres i resuls i d () = d (), a loger deviaio ie is cosidered. I is defied as (3) d = 0. Aα (4 x) ( A =,3,...,5 ), (b) ad he exac soluios lised before ca be copued wih oly hree ers wih errors less ha 0 A for d (3). The superscrips () ad (3) idicae he uber of ers required, ha are wo ad hree, respecively. I paricular, he firs hree ers for he surface hea flux defied by Eq. (0) are

5 q X B0 (0,) k πα + e α + e α = D + e πφ / φ (.6 / ( Aα ) ) () where D ad φ are he sae cosas appearig i Eq. (4). The hree-er approxiae Gree s fucio for he X case a x = x = 0 wih errors less ha 0 A for co-ies ( τ ) 0.4 / ( Aα ) is [3] α( τ) GX (0,0, τ ) e πα( τ ) (3) The soluio for he surface eperaure of he XB(XB0T0)0T0 proble for sall diesioless ies ca be foud usig Gree s fucios ad is aheaically give by T XB(XB0T0)0T0 (0,) = α k C = D kc q X B0 (0,τ )G X (0,0, τ )dτ D + e φτ πφτ / e πα( τ ) α ( τ e ) + e φτ π φτ ( τ ) dτ α ( τ ) dτ (4) where he cross ers were dropped. Soe eeded iegrals are give i Maheaica. For coveiece, oe of hese iegrals is e ατ dτ = τ ( τ ) e α ( τ ) dτ = πerfc τ ( τ ) α (5) Usig hese iegrals, Eq. (4) produces T XB(XB0T0)0T0 (0,) D φkc C erfc k + erfc φ (6) ha is valid for.6 / ( Aα ) wih errors less ha 0 A. The heral coduciviy scaled sesiiviy coefficie obaied by differeiaig Eq. (6) is

6 k T XB(XB0T0)0T0 (0,) k T D (0,) XB(XB0T0)0T0 φkc = D / k C πφ / e k = T C C π k e k πφ / e C k (7a) where φ is equal o k/c. Siilarly, he volueric hea capaciy sesiiviy coefficie is C T XB(XB0T0)0T0 (0,) T 0 4 πφ / e C k (7b) Noice ha boh of he are valid for.6 / (α A) wih errors less ha 0 A. Also, he su of hese wo scaled sesiiviy coefficies is equal o he egaive of he eperaure rise a he heaed surface. I a diesioless for, hey are k T X k (0, ) = k T XB(XB0T0)0T0 k C T X (0, ) = C T XB(XB0T0)0T0 (0, ) (0, ) + 4e / π 4e / π (.6 / A ) (8a) (.6 / A ) (8b) where = α / = φ /. Also, a early ies (less ha he deviaio ie 0.4 / A ), Eqs. (8a) ad (8b) ca be used..3. Fiie body soluio for large ies. Exac soluio A soluio valid for all > 0 is ( ) T XB0T0 (x,) = x si β x / = β e β α (9a) where β = π are he eigevalues. The soluio for he hea flux is q X B0 (x,) = kt 0 + cos β x = e β α (9b) The hea flux give by Eq. (9b) a x = 0 is q X B0 (0,) = k + e β = α = D + = φ e β (0)

7 where D ad φ are he sae cosas appearig i Eqs. (4) ad (). For sall ies, he hea flux a x = 0 eds o ifiiy ad goes o a cosa for very large ies. Usig he hea flux as beig kow, he surface eperaure is copued usig large-coie Gree s fucios [3], T XB(XB0T0)0T0 (0,) = α k = D C q X B0 (0,τ )G X (0,0, τ )dτ + e β φτ e η α ( τ ) dτ, η = ( / )π = = () The firs par of his iegral is = e η α ( τ ) dτ α = η = =/ = e η α η α e = = η α η () The secod iegral i Eq. () is φτ e β dτ = = = e η α ( τ ) = = = = e η α = α e η e (β φ η α ) τ e β φ β φ η α dτ (3) Cobiig he wo iegrals i Eq. () gives he surface eperaure as T XB(XB0T0)0T0 (0,) = D k e = η α η + = = e η α e β φ β Cφ η k (4) where he double suaio ca be expaded o e η α e β φ = β Cφ η k Cφ = = = e η α = (5) β η α / φ k β φ / α η = e β φ = The suaios wihou he expoeials coverge slowly. However, he followig relaios (foud usig Maheaica) ca be eployed o reove his covergece proble, = = ( ) η α / φco η α / φ = = β η α / φ ( π) η α / φ η α / φ (6a)

8 = ( β φ α) a / = (6b) βφ/ α η β φ / α Iroducig hese relaios i Eq. (5) gives e η α e β φ = β Cφ η k k = = + k e η α = η = e β φ k = e η α ( ) a β φ / α β φ / α ( ) η α / φ co η α / φ Iroducig his resul io Eq. (4) yields he surface eperaure here of ieres T XB(XB0T0)0T0 (0,) = D k = e η k η ( ) C k / (Cφ) co η k / (Cφ) η ( ) + e β φ a β Cφ / k = β Cφ / k (7) (8) As ie goes o ifiiy, Eq. (8) produces T XB(XB0T0)0T0 (0,) = D / k. Acually, he sae resul is obaied for all ies equal o or greaer ha zero for φ = α sice co( ) 0 η = ad a( β ) = 0. Equaio (8) is he desired expressio, which coverges icely. Ulike he double-suaio for, his soluio coverges wih a relaively sall ubers of ers. For a accuracy of abou 0 A, he required uber ers for eiher suaio is N = ceil π Al(0) + M = ceil π Al(0) (9) where = α / = φ /. For he exreely accurae value for A = 5 (errors less ha abou 0 5 ), he required ubers of ers for he -suaio for ies = 0.0, 0. ad 0.5 are respecively, 0, 7 ad 4. Eve he uber of 0 ers is o large; furherore, a ore efficie soluio is available for sall diesioless ies, as discussed i Subsecios. ad.. I fac, Eq. (8) is copuaioally efficie for >.6 / (α A). The sesiiviy coefficies for k ad C ca be obaied fro Eq. (8) by direc differeiaio. However, i is uch ore coplicaed ha uilizig fiie differeces which is possible because he copued eperaures ca be calculaed so precisely. A ceral fiie differece schee for copuig he diesioless scaled k- ad C-sesiiviies is

9 k T X k (0, ) T XB(XB0T0)0T0 [0,,k(+ ε),c] T XB(XB0T0)0T0 [0,,k( ε),c] ε C T X (0, ) T XB(XB0T0)0T0 [0,,k,C(+ ε)] T XB(XB0T0)0T0 [0,,k,C( ε)] ε (30a) (30b) The errors usig his equaio are abou ε so ha if ε = = 0 5, he errors would be abou 0 0. Nuerical experies wih ε values fro 0.0 o have show ha boh sall values, such as 0.0 or 0.00, ca cause sigifica errors; bu exreely sall values such as also ca. To ge a accuracy of abou 8 decial places whe copared wih he sesiiviy coefficies for k ad C obaied fro Eq. (8) by direc differeiaio, a ε value of is recoeded. Soe uerical values (o 8 digis, i.e. A = 8) for he scaled sesiiviy coefficies are give i Table. They were obaied usig Eqs. (8), (8) ad (30) wih Eq. (8) i heir ow ie rages of validiy. A plo is show i Fig. for visual proposes. Table. Diesioless k- ad C-scaled sesiiviy coefficies for he heaed surface. = α Equaio k T X = C T X k C T X = k T X k k T k + C T 0.0 (8) (8) for X (36) for X (30) wih (8) for X (45) wih (44) for X A side beefi of he uerical approxiaio give by Eq. (30) is ha of irisic verificaio. I fac, i ca also be used a early ies ad should give he sae uerical values provided by Eqs. (8) ad (8) if he equaios ad relaed copuer codes are righ. I oher words, he sesiiviy

10 coefficies ca be calculaed i wo differe ad idepede ways ad he sae uerical values should be obaied. For exaple, he copued scaled sesiiviies for sall ies go o -0.5, which is kow o be correc fro wha was obaied i Subsecios. ad.. This is oe idicaio of irisic verificaio. 0 Scaled sesiiviy coefficies for k ad C C T/ k T/ k XB80T0 case Diesioless ieἀ/ Figure. Diesioless scaled sesiiviy coeffices for k ad C for he heaed surface locaio. 3. XB(XB0T0)0T0 rasie experie A diesioal aheaical saee for he XB(XB0T0)0T0 proble is T x = T α ( 0< x < ; > 0 ) (3a) k T x (0,) = q XB0T0 (0,) ( > 0 ) (3b) k T x (,) = 0 ( > 0 ) (3c) T( x,0) = 0 ( 0< x < ) (3d) where q XB0T0 (0,) is he surface hea flux of he XB0T0 case. The soluio is discussed below. 3.. Sei-ifiie body soluio. Oe-er approxiaio For early ies, less ha he so-called deviaio ie defied by Eq. (), he eperaure ad hea flux of he XB0T0 proble are he sae (wih errors less ha 0 A ) as he X0BT0 case reaed i Subsecio..

11 Therefore, he surface eperaure for he XB(XB0T0)0T0 proble ca accuraely be replaced a early ies (less ha he deviaio ie) wih he surface eperaure for he XB(X0BT0)0T0 case. The laer ca be calculaed usig Gree s fucios where a furher approxiaio ca be perfored for co-ies ( τ ) 0. ( x) / ( Aα ) wih errors less ha 0 A, ha is, G X (x,0, τ ) G X0 (x,0, τ ) defied by Eq. (5). I follows ha for shor ies he sesiiviy coefficies are sill give by Eqs. (7a) ad (7b) ad, i diesioless for, by Eqs. (8a) ad (8b) where he subscrip X has o be replaced wih X. 3.. Fiie body soluio for shor ies. Three-ers approxiaio A soluio ha is paricularly efficie for he shor ies is give i Carslaw ad Jaeger [4] (page 309, Eq. (3)) + x ( + ) x TXB 0T0( x, ) = T 0 ( ) erfc + erfc = 0 4α 4α (3a) Usig his equaio ad akig jus he firs hree ers, he hea flux a he heaed surface is approxiaely (wih errors less ha 0 A for.6 / (α A) ) q X B0 (0,) k πα e α D e α πφ (3b) where he sybol φ is used o idicae ha he heral diffusiviy α is kow; also, he group D = k /. The Gree s fucio a x = 0 is α( τ) GX (0,0, τ ) + e πα( τ ) (33) The eperaure soluio a x = 0 for he XB(XB0T0)0T0 proble is T XB(XB0T0)0T0 (0,) = α k q X B0T0 (0,τ )G X (0,0, τ )dτ α D φτ e + e k πφτ πα( τ ) D e φτ + e α ( τ ) π kc φτ ( τ ) dτ α ( τ ) τ (34) where he cross ers were dropped. Perforig he iegraio ers produces

12 T X B( X B0)0 (0,) D φkc C + erfc k erfc φ (35) which is he sae as Eq. (6) excep for he + ad sigs iside he brackes. The scaled sesiiviy coefficies i diesioless for are k T X k (0, ) = k T XB(XB0T0)0T0 k C T X (0, ) = C T XB(XB0T0)0T0 (0, ) (0, ) 4e / π + 4e / π (36a) (36b) which are he sae as Eq. (8) excep for he + ad sigs iside he brackes. Therefore, he k-scaled sesiiviy coefficie of he XB(XB0T0)0T0 proble is equal o he C-scaled sesiiviy fucio for he XB(XB0T0)0T0 case. This sae reversal occurs for he C sesiiviies Fiie body soluio for large ies. Exac soluio The soluio for he XB0T0 proble is ( η x ) η α si / TXB0T0 ( x, ) = T0 e, η = ( / ) π (37a) = η The hea flux is give by q XB0T0 (x,) = k = x cos η e η α (37b) A x = 0, his hea flux is XB0T0 η = φ q (0, ) = D e (38) where D ad φ are he sae cosas appearig i Eq. (3b). The eperaure a x = 0 for his hea flux assued kow ca be derived usig Gree s fucios as T XB(XB0T0)0T0 (0,) = α k = D C q X B0T0 (0,τ )G X (0,0, τ )dτ = e η φτ + α ( τ ) e β = dτ, β = π (39)

13 The firs iegral i Eq. (39) is = e η φτ dτ φ = η = =/ = e η φ η φ e = = η φ η (40a) The secod iegral i Eq. (39) is e β dτ = e β = = α ( τ ) e η φτ = α = e β = = α = e ( η φ β α ) τ dτ ( η φ β α ) e η φ β α (40b) Uilizig Eqs. (4a) ad (40b) io Eq. (39) produces T XB(XB0T0)0T0 (0,) = D C φ = φ η e η + = = e β α e η φ η φ β α (4) The double suaio coverges very slowly bu i ca be siplified o wo sigle suaios as give below, α φ β η α φ e e β η = e e = = = = = = ηφ βα ηφ βα ηφ βα α φ β a( β / ) α φ η e e = β φη αφ = = ( ) + η φ / α co η φ / α (4) where he followig algebraic ideiies (foud usig Maheaica) have bee used, ( β α φ) ( β α φ) a / a / = = (43a) ηφ βα φ β α φ β αφ = / = = =η φ β α α =η φ / α β + η φ / α co( η φ / α ) (43b) φη Uilizig Eq. (4) i Eq. (4) produces he efficie soluio of

14 ( ) T XB(XB0T0)0T0 (0,)= D k a β C + e β k / (Cφ) Cφ = β k / (Cφ) = e η φ ( ) Cφ / k co η Cφ / k η (44) For he case of φ = α, he er iside he braces is equal o sice a( ) 0 β = ad co( η ) = 0. I is sigifica o oice ha he wo expressios defied by Eqs. (8) ad (44) are irror iages i he sese ha k /( Cφ) by Cφ / k ca be ierchaged o ge ideical expressios if also φ i he expoeials is replaced by k / C. This irror iage is raher aazig. I has applicaio o he case of opial experies for esiaig k ad C. The axiu uber of ers is give by Eq. (9). Sesiiviy coefficie for he k ad C properies ca be foud usig differeces as k T X k (0, ) T XB(XB0T0)0T0 [0,,k(+ ε),c] T XB(XB0T0)0T0 [0,,k( ε),c] ε C T X (0, ) T XB(XB0T0)0T0 [0,,k,C(+ ε)] T XB(XB0T0)0T0 [0,,k,C( ε)] ε (45a) (45b) I is efficie ad accurae sice he eperaures are kow very accuraely. Nuerical values are give i Table. They were obaied usig Eqs. (8), (36) ad (45) wih Eq. (44) i heir ow ie rages of validiy. A plo is show i Fig. for visual proposes. Coclusios I was show ha, by usig copleeary experies, he sesiiviy coefficies ca be ade ore ucorrelaed ha if a sigle experie is used. Also, whe copuig he sesiiviy fucios for he heaed surface of he XB(XB0T0)0T0 ad XB(XB0T0)0T0 experies, i was oed ha he roles i he scaled sesiiviy coefficies for k ad C ca be exchaged. I addiio, he sall ad large ie soluios for heir calculaio exhibi irisic verificaio. Refereces [] J.V. Beck ad K.J. Arold, Paraeer Esiaio i Egieerig ad Sciece, Joh Wiley ad Sos, Ic., New York, 977. [] K. D. Cole, J. V. Beck, K. A. Woodbury, F. de Moe, Irisic Verificaio ad a Hea Coducio Daabase, Ieraioal Joural of Theral Scieces, Vol. 78, pp , April 04. [3] K. Cole, J.V. Beck, A. Haji-Sheikh ad B. ikouhi, Hea Coducio Usig Gree's Fucio, secod ediio, CRC Press, Baca Rao, 0. [4] H. S. Carslaw, J. C. Jaeger, Coducio of Hea i Solids, d Ediio, Oxford Uiversiy Press, New York, 959.