The Pennsylvania State University. The Graduate School THERMAL TRANSIENT ANALYSIS WITH NONLINEAR THERMOPHYSICAL

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1 he Pennsylvania Sae Universiy he Graduae School he Deparmen of Engineering Science and Mechanics HERMAL RANSIEN ANALYSIS WIH NONLINEAR HERMOPHYSICAL PROPERIES: INVERSE AND DIREC CONSIDERAIONS A hesis in Engineering Mechanics by David Richard Engels 9 David Richard Engels Submied in Parial Fulfillmen of he Requiremens for he Degree of Maser of Science May 9

2 ii he hesis of David Richard Engels was reviewed and approved* by he following: Alber E. Segall Professor of Deparmen of Engineering Science and Mechanics hesis Co-Advisor Corina Drapaca Assisan Professor of Deparmen Engineering Science and Mechanics hesis Co-Advisor Bernhard R. imann Schell Professor of Deparmen of Engineering Science and Mechanics Judih A. odd Professor of Deparmen of Engineering Science and Mechanics P.B. Breneman Deparmen Head Deparmen of Engineering Science and Mechanics *Signaures are on file in he Graduae School

3 ABSRAC In many engineering applicaions, hermal shock is an issue ha requires serious aenion. When a maerial eperiences a large change in emperaure, is mechanical properies can change dramaically. Furhermore, any sudden changes in emperaure can adversely deform and sress he maerial such ha failure occurs. I is imporan o be able o monior sudden changes in emperaure in order o avoid hese failures. Ofen imes in sudies of hermal shock, he boundary condiion a he surface is no known, and he direc soluion is no longer useful. However, he emperaure mus sill be calculaed. hese ypes of siuaions require an inverse mehod. As is name implies, i is he opposie of he direc mehod. Insead, he hermal response a some remoe locaion is used o solve for he boundary condiion a he surface. In his hesis, he inverse hea conducion problem was analyzed where he hermal conduciviy was assumed o depend on emperaure. he inverse mehod was performed on a semi-infinie slab wih an asympoic eponenial hermal surface loading wih boh increasing and decreasing conduciviy. A generalized direc soluion was devised and analyzed. he emperaure was calculaed symbolically in erms of unknown coefficiens. A hermal response was measured a a remoe locaion, and he daa was fi o he unknown coefficiens using he leas-squares mehod. A finie elemen simulaion was run as confirmaion. he inverse problem showed ecellen agreemen compared o he finie elemen model. In addiion o solving for surface emperaure, he inverse mehod can also be used o solve for hermophysical properies such as hermal conduciviy. he conduciviy can be calculaed using he inverse problem provided ha he coefficiens of he conduciviy equaion are a leas on he order of one. he same process was performed for he finie slab; however, due o complicaions in compuaion, he sofware used was unable o perform he inverse mehod. However, he mehod is sill concepually valid and could be done wih a more argeed approach. herefore, only he direc problem was analyzed for he finie slab. Using a aylor series approimaion, good agreemen was achieved beween he analyical and finie elemen models. Afer several seconds, his approimaion would become very crude, as he analyical model would sharply diverge from he finie elemen model. I was proven ha an inegral wihin he soluion can be modeled as a consan several seconds ino he ransien analysis; his prevened he divergence ha was caused by he aylor series approimaion. he agreemen beween boh he semi-infinie and finie slab models and heir finie elemen counerpars improved as he hermal wave reached he poin of ineres more quickly. Given he versailiy of he polynomials used in he cases presened, he mehod appears well suied for monoonic or mildly-oscillaory hermal scenarios provided he analysis is resriced o he ime inerval used o deermine he polynomial. In addiion, he mehod can be adaped for any ranscendenal funcion, no necessarily polynomials, including periodic funcions. iii

4 ABLE OF CONENS iv LIS OF FIGURES... v LIS OF ABLES... vii Chaper 1 Inroducion... 1 Chaper Lieraure Review Consan hermophysical Properies emperaure Dependen hermophysical Properies Goodman s Hea Balance Inegral Chaper 3 he Inverse Problem he Direc Problem versus he Inverse Problem An Algorihm for he Inverse Problem Using he Inegral Mehod... 1 Chaper 4 Resuls and Daa Semi-Infinie Slab Verificaion of he Analyical and Numerical Models Considering Consan hermophysical Properies Comparing he Analyical and Finie Elemen Models Where he Conduciviy Has a Linear Dependence on emperaure Inverse Analysis Involving a Cubic Variaion in hermal Conduciviy Using he Inverse Mehod o Solve for hermophyiscal Properies... 4 Chaper 5 Resuls and Daa Finie Slab Calculaing emperaure for he Finie Slab Using he Inegral Mehod hermally hin Models hermally hick Models... 5 Chaper 6 Conclusion Chaper 7 Limiaions and Fuure Work Bibliography... 7 Appendi 1 Inpu File for ANSYS Appendi Mahemaica File for he Semi-Infinie Slab Appendi 3 Mahemaica File for he Finie Slab... 8

5 v LIS OF FIGURES Figure 1-1: his figure shows he hermal deformaion on he ouer surface of an alumina ube caused by hermal shock.... Figure 3-1: In he direc problem of a semi-infinie solid, picured here, he boundary condiion a = is applied o find he emperaure a any locaion as a funcion of ime Figure 3-: In he inverse case of a semi-infinie solid, picured here, he emperaure a some remoe locaion, such as (b,), is used o deermine he unknown hermal boundary condiion a he surface.... Figure 4-1: A simplified version of he finie elemen mesh is presened Figure 4-: A plo of non dimensional emperaure versus non dimensional ime is presened for he semi-infinie slab a =.3 cm comparing all hree mehods. ma= 1K o = K, and κ=1.66 cm /s... 9 Figure 4-4: A plo of non dimensional emperaure versus ime is presened for he semi-infinie slab a = 4 cm comparing he Goodman and eac soluions. ma = 1K, o=k, and κ=1.66 cm /s Figure 4-5: A plo of non dimensional emperaure versus ime where conduciviy increased linearly for a semi-infinie slab a =.3 cm. ma=1k and o=k, and he iniial diffusiviy κ =1.66cm /s Figure 4-6: A plo of non dimensional emperaure versus ime where conduciviy decreased linearly for a semi-infinie slab a =.3 cm. ma=1k and o=k, and he iniial diffusiviy κ =1.66cm /s Figure 4-7: A plo of non dimensional emperaure versus ime where conduciviy increased wih a cubic variaion for a semi-infinie slab a =.7 cm. ma=1k and o=k, and he iniial diffusiviy κ =1.66cm /s Figure 4-8: A plo of non dimensional emperaure versus ime where conduciviy decreased wih a cubic variaion for a semi-infinie slab a =.7 cm. ma=1k and o=k, and he iniial diffusiviy κ =1.66cm /s Figure 4-9: A comparison of he hermal conduciviy and he inverse predicion Figure 5-1: A finie slab of lengh L wih an insulaed back boundary is shown Figure 5-: A plo of non dimensional emperaure versus ime where conduciviy increased wih a cubic variaion for a finie slab a L = 1 cm. ma=1k and o=k, and he iniial diffusiviy κ =.85cm /s.... 5

6 Figure 5-3: A plo of non dimensional emperaure versus ime where conduciviy decreased wih a cubic variaion for a finie slab a L = 1 cm. ma=1k and o=k, and he iniial diffusiviy κ =.85cm /s Figure 5-4: he value of he inegral in Eq was evaluaed wih a numerical echnique and wih a aylor Series Approimaion for he case of L=1 cm Figure 5-5: he value of he inegral in Eq was evaluaed wih a numerical echnique and wih a aylor Series Approimaion for he case of L=5 cm Figure 5-6: A plo of non dimensional emperaure versus ime where conduciviy increased wih a cubic variaion for a finie slab a L = 3 cm. ma=1k and o=k, and he iniial diffusiviy κ =.85cm /s Figure 5-7: A plo of non dimensional emperaure versus ime where conduciviy increased wih a cubic variaion for a finie slab a L = 3 cm. ma=1k and o=k, and he iniial diffusiviy κ =.85cm /s Figure 5-8: A plo of non dimensional emperaure versus ime where conduciviy increased wih a cubic variaion for a finie slab a L = 4 cm. ma=1k and o=k, and he iniial diffusiviy κ =.85cm /s Figure 5-9: A plo of non dimensional emperaure versus ime where conduciviy increased wih a cubic variaion for a finie slab a L = 5 cm. ma=1k and o=k, and he iniial diffusiviy κ =.85cm /s Figure 5-1: A plo of non dimensional emperaure versus ime where conduciviy decreased wih a cubic variaion for a finie slab a L = cm. ma=1k and o=k, and he iniial diffusiviy κ =.85cm /s Figure 5-11: A plo of non dimensional emperaure versus ime where conduciviy decreased wih a cubic variaion for a finie slab a L = 3 cm. ma=1k and o=k, and he iniial diffusiviy κ =.85cm /s vi

7 vii LIS OF ABLES able 3-1: Calculaions of Eq. 3.4 a =.7 cm... 4

8 1 Chaper 1 Inroducion In many engineering applicaions, hermal shock is an issue ha requires serious aenion. When a maerial eperiences a large change in emperaure, is mechanical properies can change dramaically. Furhermore, any sudden changes in emperaure can adversely deform and sress he maerial such ha failure occurs. Figure 1-1 shows he consequences of hermal shock on an alumina ube. he ube underwen inernal radial heaing on he inner diameer, and a localized (and severe) eernal heaing and quenching caused he failure shown. I is imporan o be able o closely monior such sudden changes in emperaure in cerain maerials in order o avoid he consequences shown in Figure 1-1. When solving a boundary value problem in hea conducion, one mus ypically solve Fourier s hea conducion wih a given se of parameers. hen one will solve for he emperaure of he sysem as a funcion of space and ime. In addiion, hermoelasic quaniies such as hermal sress or srain can be calculaed a any locaion and insan in ime. his process is ypically is referred o as he direc mehod. In oher words, a boundary condiion is imposed on he surface, and a ransien response can be measured somewhere else.

9 Figure 1-1: his figure shows he hermal deformaion on he ouer surface of an alumina ube caused by hermal shock.

10 3 Ofen imes in sudies of hermal shock, he boundary condiion a he surface is no known, and he direc soluion is no longer useful. However, he emperaure mus sill be calculaed. hese ypes of siuaions require an inverse mehod. As is name implies, i is he opposie of he direc mehod. Insead, he hermal response a some remoe locaion is used o solve for he boundary condiion a he surface. Many algorihms have been formulaed for he Inverse Hea Conducion Problem (IHCP). However, mos of hem approimae hermophysical properies as being consan. A very high emperaures, hese properies ofen depend on emperaure. By definiion, hermal shock means ha here are large emperaure changes in a shor of amoun of ime; herefore, many hermal shock scenarios deal wih high or low emperaure condiions where properies are emperaure dependen. In hese cases, he approimaion of independen hermophysical properies leads o error. herefore, i would be advanageous o formulae an inverse mehod ha accouns for properies ha change wih emperaure. his hesis inends o devise a mehod ha considers all of hese facors. Chaper presens a review of he lieraure, while Chaper 3 illusraes he concep of he inverse mehod. Chaper 4 shows he resuls and daa for he semi-infinie slab model; Chaper 5 shows he resuls for he finie slab model. he conclusion is given in Chaper 6, and he paper finishes wih fuure work suggesions in Chaper 7.

11 Chaper Lieraure Review In he lieraure review conduced, all auhors analyzed hea conducion problems, where he emperaure was found using Fourier s Equaion: ρc = [ k( ) ].1 In Eq..1, ρ is densiy, c is specific hea, k() is he emperaure dependen hermal conduciviy, is ime, and is he gradien operaor. In addiion, i is assumed ha for his model (and for all proceeding models) ha here is no inernal hea generaion. As shown by he lieraure o be discussed, some solved his equaion using he direc mehod, applying known boundary condiions o find a emperaure as a funcion of space and ime, while ohers aemped o go he Inverse Hea Conducion Problem (IHCP) roue by measuring a hermal response a a remoe locaion and deermining he hermal surface boundary condiion. However, he naure, difficuly, and ulimaely, he accuracy of he resuling soluion was dicaed o a large degree by how he hermophysical properies were handled.

12 5.1 Consan hermophysical Properies In 1996, Blanc and Raynaud [1] devised one of many mehods o solve he IHCP. An unknown inernal hea flu was compued by relying on hermal srain and emperaure measuremens; previously, only emperaure measuremens were used o solve for he flu. A sequenial procedure was developed using he sensiiviy sysem and solving for he sensiiviy coefficiens of he srain o he hea flu. he mehod was used on several benchmark es cases wih eperimenal validaion. However, in his mehod, here are numerous hermal boundary condiions, iniial condiions, and mechanical boundary condiions ha are necessary, including no change in emperaure in he radial direcion of an annular cylinder. Ulimaely, hese assumpions resric he usabiliy of his mehod ino real-world engineering applicaions. In 3, Chen, e al. [] applied a hybrid numerical algorihm, which combined Laplace ransforms and a finie difference mehod o solve he inverse hea conducion problem. he mehod relied on several consiuive equaions in hea ransfer applicaions; however, here are a combined seven boundary and iniial condiions ha are assumed in order o perform he numerical calculaions. he Laplace ransforms successfully eliminae he ime-dependen erms and are consequenly combined wih various finie difference equaions for he hybrid mehod; he resuling predicions give a srong agreemen wih a numerical eample. Once again, he seven differen boundary

13 6 and iniial condiions provide srong resricions o he applicabiliy of his mehod for engineering applicaions. In 5, Yvonne, e al. [3] devised ye anoher scheme o solve he IHCP in relaion o calculaing he hea flu in a ool used in orhogonal cuing. his hea flu disribuion was approimaed hrough a piecewise consan funcion. he average flu was calculaed hrough finie elemen modeling and he Newon-Raphson mehod. Respecable accuracy was achieved when compared o eperimenal emperaure values. Once again, here were wo iniial condiions ha he iniial hea flu and he iniial convecive hea ransfer coefficien be known a priori. As previously eplained, hese boundary condiions limi how easily he inverse mehod can be used. In 198, Grysa, e al. [4] found an inverse mehod ha relied on hermal sress heory as opposed o he IHCP. he researchers used Laplace ransform echniques for heir mehod and verified he resuls wih a numerical eample. Using his mehod, eiher emperaure or hea flu a he boundary of a body can be calculaed wih a known emperaure a anoher locaion. In 6, Ishizaka, e al. [5] devised an inverse mehod where he objecive was o minimize he hoop sresses on he ouer surface of a pipe in order o maimize he life of power plans. In he cied paper, he auhors used a hin-walled approimaion for he ube and assumed ha he ouer surface was insulaed for heir calculaions. he emperaure disribuion and hermal hoop sress disribuion in he radial direcion were hen calculaed. Subsequenly, a mehod was devised o minimize he ransien hermal hoop sress. his analyical mehod can be useful, bu once again, only in cerain siuaions

14 7 More recenly, Segall [6] devised a linear inverse mehod for he hea conducion problem. In he inverse mehod, a generalized direc soluion for emperaure deermined by Duhamel s convoluion inegral was relaed o srain. he measured srain hisory was hen fi o a polynomial, and a leas-squares approach was employed o solve for he unknown coefficiens in he convoluion inegral. he resuling coefficiens were hen subsiued back ino he original polynomial ha in urn became he direc soluion. he conceps of he generalized direc soluion and leas-squares fi o coefficiens will be used in he models presened in his hesis. All of he previously menioned mehods are innovaive and quie applicable under cerain condiions. However, if he hermophysical properies of he maerial change wih emperaure, hese models become severely limied. herefore, i is necessary o devise a model ha accouns for hermophysical properies ha vary wih emperaure.

15 8. emperaure Dependen hermophysical Properies Vujanovic [7] creaed an opimal linearizaion mehod for he direc problem. Densiy and specific hea are assumed o be consan, while hermal conduciviy varies wih emperaure. Saring wih he one-dimensional case of Eq..1 in Caresian coordinaes, he hea conducion equaion simplifies o ρc = (k() ). where ρ is densiy, c is specific hea, k() is he emperaure dependen hermal conduciviy, is ime, and is he spaial coordinae. he boundary condiions of an arbirary emperaure on he surface and a zero value of emperaure everywhere else were considered [7]: (, ) = (,) =.3 A new consan parameer λ is inroduced o he model, such ha: ρc = λ.4 he objecive of he mehod proposed by Vujanovic is o calculae an opimal value of λ ha can be subsiued ino Eq..4, which can be readily solved o find a emperaure soluion as a funcion of space and ime. In order o find he value, Vujanovic

16 9 implemened a mehod ha minimized he difference beween Eq.., and Eq..4 by inroducing he funcion ε, which represens he difference beween he wo equaions [7]: In order o find he bes possible value of he parameer λ, he opimal linearizaion mehod is performed [7]: where he upper and lower limis of inegraion represen general ime and space parameers ha depend on he problem being analyzed. he inegral funcion I(λ) can be simplified ino he form [7] where ( ) ( ) k d dk λ, λ,, ε =.5 ( ) = 1 1 dd, λ,, ε λ I.6 ( ) ( ) D C B λ Aλ λ I + + =.7 ( ) ( ) ( ) ( ) + = = = = dd k d dk D dd k C dd d dk B dd A.8

17 In order o find he opimal value of λ, he inegral funcion in Eq..6 can be differeniaed wih respec o λ and se equal o zero, leading o he soluion 1 B + C λ =.9 A Now known, he value of λ can be subsiued back ino Eq..4, where he emperaure soluion can be readily compued analyically. here are, however, several limiaions o he Vujanovic model. he emperaure soluion for a consan conduciviy siuaion mus be known prior o conducing he analysis; as previously discussed, his is problemaic if he hermal surface loading is unknown. Furhermore, he conduciviy is assumed o be a linear funcion of emperaure. However, his is no always a realisic profile for hermal conduciviy. Masanaiah and Muhunayagam [8] used he Vujanovic [7] mehod jus discussed o devise a modified opimal linearizaion mehod. he analysis encompassed a semiinfinie ime frame (unil he hermal layer reaches he back boundary) and a finie ime frame hereafer. he inegral mehods had a parabolic profile for he spaial emperaure disribuion. However, a parabolic profile is no always realisic; his hesis will model a cubic emperaure profile insead. Imber [9] devised a linearizaion mehod similar o he one prescribed by Vujanovic [7], ecep he mehod uses an ieraive process. A semi-infinie solid is considered where, as wih he Vujanovic model, he hermal conduciviy is assumed o have a linear dependence on emperaure. he hea conducion equaion presened in Eq.. is modified ino he form [9]

18 11 where C, and a are consans o be deermined and represens a rial emperaure ha saisfies he boundary condiions. As wih he Vujanovic model, he opimal linearizaion mehod is performed, leading o he funcion I(a) [9]: he funcion I(a) is differeniaed wih respec o a and se equal o zero, resuling in he opimal value of he parameer a [9]: he value of C is deermined such ha he parameer is se equal o he iniial hermal conduciviy value k i. If necessary, his process can be ieraed unil a saisfacory soluion is compued. If ieraed a second ime, he hea conducion would become [9]: he inegral funcion I(a), would be modified ino he following form [9]: C a ρc + =.1 ( ) ( ) = dd C a k a I.11 = dd dd C k a C C a ρc + + =.13 ( ) ( ) = 1 1 dd C C a k a I.14

19 1 Once again, he funcion I(a) is differeniaed wih respec o a and se equal o zero for he opimal value of he parameer a [9] his ieraion process can be conduced a oal of N imes, where and here are several issues wih his mehod. Once again, he hermal conduciviy is assumed o be a linear funcion of emperaure. In addiion, if he surface emperaure is unknown, here is no clear guideline of deciding when o sop he ieraive process. herefore, his model would be problemaic for inverse analyses. Sucec and Hedge [1] creaed a finie difference mehod for he direc case. hey analyzed a finie slab wih where conduciviy also had a linear dependence on emperaure. heir analysis included a convecive boundary condiion on he surface as opposed o emperaure or hea flu, which is more commonly chosen for hese kinds of = 1 1 dd dd C C k a.15 = + = N n n a ρc.16 = = N n n dd dd k a.17

20 13 problems. he finie difference equaions were eplici, ecep a he boundary. Implici equaions were implemened a he surface for sabiliy purposes. he resuls of a sample problem compared well wih a Heisler Char. Numerical mehods such as he one proposed by Sucec and Hedge can be very accurae and versaile. However, i is beneficial o devise an analyical mehod ha can be used o validae numerical resuls. Cai and Zhang [11] have employed a separaion of variables echnique o solve he IHCP. he soluion o he hea conducion equaion was separaed ino a ransien soluion and a seady sae soluion (only dependen on spaial coordinaes). he seady sae soluion was separaed ino individual componens, such ha each componen has only one spaial componen dependence (e.g., X(), Y(y), and Z(z) for Caresian coordinaes). Various eplici soluions were given for one and wo dimensional cases for boh Caresian and cylindrical coordinaes [1]. Hilon [13] analyzed hermal gradiens as hey perain o hermoelasic sresses wih boh a hick walled cylinder and radial disk analyzed. Several case sudies were conduced, encompassing differen hermal gradiens. In addiion, elasic properies were assumed o be emperaure-dependen. hese properies reduced he maimum sress in he differen geomeries as well as increased he maimum srain. he hin plae approimaion was made for he circular disk. he validiy of he analysis did no depend grealy on wheher or no he properies were emperaure dependen, as hey only relied on he magniude of he hermal gradien.

21 14 Muki and Sernberg [14] also analyzed hermal sress wih emperaure dependen properies. wo es cases were sudied; he firs encompassed a slab ha was finie in he direcion of hea conducion, bu infinie in he wo ransverse direcions. he second case analyzed a sphere wih a symmeric hermal loading. Each siuaion had is own individual se of displacemen consrains. Several numerical analyses were conduced, and various hermoelasic consiuive relaions were saisfied.

22 15.3 Goodman s Hea Balance Inegral While emperaure dependen properies have been considered (as previously oulined), he aforemenioned models were no sufficienly general o allow for an inverse analysis. Goodman [15] solved he problem iniially posed by Yang [16] of a semiinfinie slab wih zero emperaure undergoing an arbirary sep change in emperaure a he surface a =. Uilizing Eq.., Goodman considered he following boundary condiions: (,) = (,) = (, ) =,, s ( ) > >.18 where s is he arbirary surface emperaure. he Goodman soluion o his problem involves he following change of variable: his ransformaion simplifies he hea conducion equaion o v = ρc d.19 v v = κ(v). where κ(v) is he hermal diffusiviy of he maerial, which is dependen on he new variable v. he boundary condiions in Eq..18 become:

23 16 v(,) = v(, ) = v(,) = v s.1 A his poin, a new variable θ is inroduced in he model, which is defined as: where he upper limi of inegraion δ is he hermal layer or deph of peneraion. By definiion, a remoe locaion on he slab can eperience a change in emperaure only if he hermal wave has reached ha poin, i.e. δ θ = vd. v(δ( ) =.3 In addiion, here is no flu in he maerial unil he hermal wave has reached he poin of ineres: v ( δ, ) =.4 Furhermore, he emperaure a a remoe locaion is no changing wih respec o ime unil he hermal wave reaches he poin of ineres, meaning ha [15] v ( δ, ) =.5 he parial derivaive on he righ-hand side of Eq.. can be evaluaed wih respec o and equaed o he condiion given in Eq..5, providing he following relaionship: ( v) κ v v ( δ, ) + κ( v) ( δ, ) =.6

24 he firs erm in Eq..6 is equal o zero due o he condiion provided in Eq..4; hence, one can uilize he so-called smoohing condiion 17 v ( δ, ) =.7 Given he condiions provided, he Goodman soluion o Eq..1 and Eq..18 is of an assumed form: he definiion of δ is given by: (, ) 3 = s ( ) 1.8 δ he propery κ s is he hermal diffusiviy a he surface. herefore, if he emperaure s () and diffusiviy are known a he surface, he hermal layer can be calculaed by Eq..9, and he emperaure of he slab can be calculaed as funcion of space and ime in Eq..8. his inegral mehod will be used o solve problems of ineres in he proceeding chapers. 1/ 6 δ = κss ( ) d s ( ).9

25 Chaper 3 he Inverse Problem 3.1 he Direc Problem versus he Inverse Problem In many hermal scenarios, a surface boundary condiion is imposed, leading o a emperaure soluion as a funcion of space and ime. However, as previously discussed, here are ofen cases ha require an inverse analysis where a hermal response is deermined a a remoe locaion, and i is desirable o deermine he hermal boundary condiion imposed on he surface. his concep is illusraed for he semi-infinie slab in Figure 3-1 and Figure 3-, where Figure 3-1 shows he direc case and Figure 3- shows he inverse case. In he direc case, a known hermal boundary condiion a he surface is imposed in order o find a emperaure soluion (,) as a funcion of space and ime. Once his is known, he ransien hermal response can be deermined a a remoe locaion such as =b, as shown in Figure 3-1. In he inverse case, he ransien hermal response is measured a =b and subsequenly used o deermine he unknown hermal boundary condiion (,) a he surface, as shown in Figure 3-.

26 19 Surface emperaure (,) known (b,) unknown Figure 3-1: In he direc problem of a semi-infinie solid, picured here, he boundary condiion a = is applied o find he emperaure a any locaion as a funcion of ime.

27 Surface emperaure (,) unknown (b,) known Figure 3-: In he inverse case of a semi-infinie solid, picured here, he emperaure a some remoe locaion, such as (b,), is used o deermine he unknown hermal boundary condiion a he surface.

28 1 3. An Algorihm for he Inverse Problem Using he Inegral Mehod In he inverse problem of a semi-infinie slab, he boundary condiion a he surface, = is no known. If one firs considers he direc problem wih he unknown boundary condiion aking he form of a polynomial: N 3 n () = a + a + a +... = a 3.1 s 1 3 n= 1 n an alernaive mehod eiss o erac he surface eciaion. he hermal diffusiviy, a funcion of emperaure (and subsequenly a funcion of ime) can be inegraed wih his generalized emperaure in Eq..9 o find he hermal layer in erms of he coefficiens a n. he resuling hermal layer can hen be subsiued ino Eq..8, giving a generalized emperaure in erms of he coefficiens a any spaial locaion and insan in ime. If a ransien hermal response can be measured a some remoe locaion such as = b, his daa can be correlaed o he generalized emperaure a = b, compued in Eq..8 by a leas-squares fi. Once his has been done, one can solve for he coefficiens ha can be subsiued back ino Eq. 3.1, hus providing he hermal boundary condiion. A simplified eample will illusrae his concep. Suppose one analyzes a semiinfinie solid wih a consan diffusiviy of cm /s. he surface loading is chosen o be a linear funcion of ime:

29 s = 4 3. he choice of hese parameers is no necessarily realisic, bu hey will help illusrae he concep of how he inverse mehod is performed. If one subsiues hese values in Eq..8 and Eq..9, one will obain he following resuls: (, ) ( ) 3 δ = 3.3 Wih his equaion for emperaure, one can calculae he emperaure a any locaion. In his eample, =.7 cm was chosen. he emperaure a his locaion was calculaed a 7 differen insances in ime. he resuls of hese calculaions can be seen in able 3-1. Equaions 3.3 and 3.4 calculae he direc problem, as he surface emperaure s () is known. For he inverse problem, he eac same process is performed. his may seem problemaic, as he surface emperaure is no known for he inverse siuaion. However, by applying Eq. 3.1, wih he coefficiens ye o be deermined, one can perform he direc calculaions wih a generalized surface emperaure. Using Eq. 3.1 and choosing N=1 for convenience, one has he generalized equaion where he coefficien a is o be solved for. he emperaure for all space and ime can be calculaed via Eq..8 and Eq..9 symbolically: δ inverse 3 = ( 16 4) + ( ) a inverse ( ) s, = 3.5 ( a ) 3 a = 3.6 a

30 inverse (, ) = a a ( a ) Now ha he emperaure in Eq. 3.7 has been calculaed in erms of a, i can be fi o he daa in able 3-1. While i may be emping o solve Eq. 3.7 direcly wih Eq. 3.4, more complicaed problems would make his mehod mahemaically difficul. Insead, one mus fi he daa in able 3-1 o inverse (.7cm, ) via a leas squares deerminaion. his will work no only for his simplified case, bu also more complicaed rials. Wih his eample, one can readily employ he leas squares mehod and obain a value of a = 4, which of course is he desired resul. he eample jus presened provides insigh ino how he inverse calculaions are performed in he proceeding chapers. In hese problems, he diffusiviy will have a cubic dependence on he surface emperaure, which will be a fifh order polynomial wih respec o ime. he equaions for he hermal layer and emperaure are unwieldy and are herefore no shown eplicily in his paper; insead, he resuls will be presened graphically. a 3 3

31 4 able 3-1: Calculaions of Eq. 3.4 a =.7 cm ime (seconds) emperaure (K)

32 Chaper 4 Resuls and Daa Semi-Infinie Slab In he resuls o be discussed, a numerical model was devised in ANSYS o verify he analyical soluions (used in Mahemaica) ha were developed. For he finie elemen model, hermal elemens were fi ino a recangular se of nodes. he reader mus be warned ha is essenial o use hermal elemens and no srucural elemens; oherwise, hermal ransien daa will be unaainable. Furhermore, he lengh in he conducing direcion mus be much smaller han he lengh in he ransverse direcion o ensure ha he analysis is essenially one-dimensional in space. However, for he semiinfinie solid, a relaively large number of elemens is also required in he conducing direcion o simulae an infinie dimension. A simplified version of he mesh creaed for he finie elemen model is shown in Figure 4-1. he mesh presened is a very simplified model shown for illusraive purposes. More deails on he finie elemen model including he inpu files used for he hesis can be found in Appendi 1.

33 6 Figure 4-1: A simplified version of he finie elemen mesh is presened. In order o verify boh of he analyical and numerical soluions, each model was compared o an already known soluion. Once each model showed good agreemen wih his soluion, hey could be used in fuure consideraions wih confidence. As in [15], he iniial verificaion problem chosen was a semi-infinie slab wih consan conduciviy properies and a sep change in emperaure. his problem has an eac soluion of he form (, ) s = Erfc( ) κ 4.1 where Erfc represens he complemenary error funcion. Wih he eac soluion already known, he analyical and numerical soluions were compared o Eq I mus be noed ha i is cerainly unrealisic o creae a slab of ruly infinie lengh in a finie elemen program. However, his issue can be avoided if he analysis is resriced o he region of ime in which he hermal wave has no reached

34 7 he back boundary in he conducing direcion. In he analysis conduced, he emperaure on he back boundary was measured; hroughou he span of ime chosen, he back boundary showed no noiceable changes in emperaure, hus validaing he slab as being infinie in he direcion of conducion for he given imeframe.

35 8 4.1 Verificaion of he Analyical and Numerical Models Considering Consan hermophysical Properies wo arbirary dephs were chosen in he slab, and he emperaure changes from he finie elemen analysis were compared o he eac and Goodman soluions given in Eq. 4.1 and Eq..8, respecively. he resuls of his analysis are given in Figure 4-, Figure 4-3, and Figure 4-4. hese figures, as well as all figures presened in his hesis, will plo non dimensional values for emperaure and ime. In Figure 4-, a =.3 cm, agreemen was ecellen among all hree mehods. here is lile o no deviaion from he eac soluion compared o eiher of he oher wo. In Figure 4-3, a deph of = 4 cm was analyzed. In his figure, he finie elemen analysis also shows ecellen agreemen wih he eac soluion. However, he Goodman soluion does deviae noiceably in his graph. In Figure 4-4, he Goodman soluion was compared o he eac soluion, bu over a greaer period of ime. he figure shows some iniial deviaion in he graph, bu as he emperaure reaches an asympoic value, he wo graphs converge o he same value. herefore, i can be concluded ha he Goodman soluion may have some iniial deviaion, bu beer agreemen laer. he hree figures discussed show ha he finie elemen model has ecellen agreemen wih he eac soluion. As a resul, he finie elemen model can be used wih confidence in fuure analyses, as he problems become more complicaed. he Goodman soluion does show some deviaion for a shor amoun of ime a greaer dephs in he slab. However, his deviaion lass for a relaively shor ime such ha agreemen improves as he emperaures reach heir asympoic values.

36 (, ) ma Finie Elemen Eac Goodman Iniial Diffusiviy κ = 1.66 cm /s Figure 4-: A plo of non dimensional emperaure versus non dimensional ime is presened for he semi-infinie slab a =.3 cm comparing all hree mehods. ma= 1K o = K, and κ=1.66 cm /s κ

37 3.35 (, ) ma Finie Elemen Goodman Eac κ Figure 4-3: A plo of non dimensional emperaure versus non dimensional ime is presened for he semi-infinie slab a = 4 cm comparing all hree mehods. ma = 1K o = K, and κ=1.66 cm /s

38 (, ) ma.4.3. ANSYS Eac Goodman Iniial Diffusiviy κ =.85cm /s κ Figure 4-4: A plo of non dimensional emperaure versus ime is presened for he semiinfinie slab a = 4 cm comparing he Goodman and eac soluions. ma = 1K, o=k, and κ=1.66 cm /s

39 3 4. Comparing he Analyical and Finie Elemen Models Where he Conduciviy Has a Linear Dependence on emperaure Afer analyzing he condiions wih consan properies, he case of varying conduciviy was analyzed. he hermal conduciviy is now of he linear form: k() = α + α1 4. In he equaion, α is he hermal conduciviy ha was used in he consan conduciviy analysis. he wo values of α 1 used in he analyses were.5 and -.5 so ha he analysis could include boh increasing and decreasing conduciviies wih increasing emperaure. In addiion, he boundary condiion was differen for his analysis. Insead of a sep change, he ofen encounered asympoic loading on he inernal surface of s =V(1-e -.5 ) fi o a polynomial was employed for he comparisons. Once again, an arbirary deph of =.3 cm was chosen for boh analyses. Figure 4-5 shows he analysis wih increasing hermal conduciviy while Figure 4-6 shows he analysis wih decreasing hermal conduciviy. For hese figures, as well as all figures peraining o varying conduciviy, a reference diffusiviy value mus be chosen for he non dimensional ime ais since he diffusiviy will also change wih emperaure. he iniial diffusiviy a = was chosen for all of he varying conduciviy plos and is denoed by κ. Figure 4-5 and Figure 4-6 show ecellen agreemen beween he

40 (, ) ma Finie Elemen Goodman Figure 4-5: A plo of non dimensional emperaure versus ime where conduciviy increased linearly for a semi-infinie slab a =.3 cm. ma=1k and o=k, and he iniial diffusiviy κ =1.66cm /s. κ

41 (, ) ma Finie Elemen Goodman κ Figure 4-6: A plo of non dimensional emperaure versus ime where conduciviy decreased linearly for a semi-infinie slab a =.3 cm. ma=1k and o=k, and he iniial diffusiviy κ =1.66cm /s.

42 35 Goodman and finie elemen models. Figure 4-5 does show a slighly beer agreemen beween he wo mehods, bu his is o be epeced since an increasing hermal conduciviy leads o an increasing hermal diffusiviy and a hermally hinner maerial. As previously discussed, he Goodman model will deviae slighly from he finie elemen model unil an asympoic value is reached. By definiion, as he hermal diffusiviy increases, he hermal wave passes hrough he slab a a greaer rae. As he hermal wave passes hrough he slab more quickly, his asympoic value is reached more quickly, hus decreasing he ime span in which here is a noiceable deviaion beween he wo models. For he case of decreasing hermal conduciviy, he reverse is rue; here is a smaller diffusiviy, meaning he hermal wave passes hrough he slab less quickly. Hence, i akes longer for he asympoic value o be reached, giving a greaer ime span in which here is deviaion beween he wo models. However, even in his scenario, ecellen agreemen was sill achieved.

43 Inverse Analysis Involving a Cubic Variaion in hermal Conduciviy Afer considering hermal conduciviy possessing a linear dependence, he analysis of conduciviy having a cubic variaion was considered. As wih he linear variaion, he asympoic loading on he inernal surface of s =V(1-e -.5 ) fi o a polynomial was employed. wo es cases were considered; in he firs case, he hermal conduciviy increased wih increasing emperaure, while he second case considered hermal conduciviy decreasing wih increasing emperaure. In each siuaion sudied, V was se equal o one, and he conduciviy, in general, was described by he polynomial: k() = α α1 + α α3 where all of he α coefficiens are nonzero. In addiion, he maerial was given he following hemophysical properies: ρ=3. kg/cm 3 and c =.376 J/kg-K. A generalized surface emperaure described in Eq. 3.1 was used in each case. his emperaure soluion was deermined using Eq..8 and Eq..9 A finie elemen model was creaed ha had idenical hermophysical properies o he ones previously menioned. he slab was sufficienly long in he conducing direcion such ha he hermal wave did no reach he back boundary during he inerval of sudy. As a resul, his slab could be modeled as semi-infinie. An asympoic eponenial was hermally loaded on he surface, a finie elemen simulaion was conduced, and he hermal response a =.7 cm was deermined. In he case of increasing conduciviy, he conduciviy followed he form of Eq. 4.3, wih he following coefficiens: α=., α1=.3546, α=.3361, α3 =.39.

44 37 Boh he analyical and finie elemen model were given hese coefficiens. he surface emperaure was imposed from Eq. 3.1, where N was chosen o be 5. A generalized emperaure was found via he Goodman soluion, and hen fi o he daa measured from he finie elemen model. Figure 4-7 shows a measured response a =.7 cm. From his response, he inverse predicion was made and compared o he surface loading. he figure shows ha he inverse predicion and he surface loading are in ecellen agreemen. In he case of decreasing conduciviy, he conduciviy polynomial followed he form of Eq. 4.3, wih he following coefficiens: α =., α 1 =-.6353, α =-.941, α 3 =.759. Anoher finie simulaion was run wih hese coefficiens. Oherwise, he process performed in he firs case was repeaed. Figure 4-8 also shows a measured response a =.7 cm. he inverse predicion from his response was compared o he surface loading, where eac agreemen was again achieved. Wih a fifh order polynomial for surface emperaure and a hird order polynomial for hermal diffusiviy, he eplici soluion o Eq..8 is eremely lenghy and herefore will no be shown. he graphs presened in Figure 4-7 and Figure 4-8 represen he resuls for his polynomial. he key poin o emphasize is ha hey mach he imposed asympoic surface loading, hus validaing he mehod. Given he versailiy of he polynomials used, he mehod appears well suied for monoonic or mildly-oscillaory hermal scenarios provided he analysis is resriced o he ime inerval used o deermine he polynomial. In addiion, he mehod can be adaped for any ranscendenal funcion, no necessarily polynomials, including periodic funcions.

45 (, ) ma Surface Loading Inverse Predicion Iniial Diffusiviy κ = 1.66 cm /s.3 Measured Response (=.7cm) κ Figure 4-7: A plo of non dimensional emperaure versus ime where conduciviy increased wih a cubic variaion for a semi-infinie slab a =.7 cm. ma=1k and o=k, and he iniial diffusiviy κ =1.66cm /s.

46 (, ) ma Surface Loading Inverse Predicion Measured Response (=.7cm) Iniial Diffusiviy κ = 1.66 cm /s κ Figure 4-8: A plo of non dimensional emperaure versus ime where conduciviy decreased wih a cubic variaion for a semi-infinie slab a =.7 cm. ma=1k and o=k, and he iniial diffusiviy κ =1.66cm /s.

47 4 4.4 Using he Inverse Mehod o Solve for hermophyiscal Properies In addiion o unknown surface emperaure hisories, hermophysical properies can also be calculaed via he inverse problem. his can be beneficial o calculae properies such as hermal conduciviy or diffusiviy. Finding hese properies and heir behavior wih respec o emperaure would be relaively simple using he inverse mehod presened. In fac, he algorihm is quie similar o he inverse problem calculaing surface emperaure. However, in his problem, he surface emperaure mus be known, and he conduciviy/diffusiviy polynomial is deermined. If a known hermal eciaion is imposed on he surface, he hermal response could be measured on a some locaion. he algorihm presened in Chaper 3 would be conduced, ecep s () is imposed and herefore known. he conduciviy would follow Eq. 4.3, ecep ha α, α 1, α, and α 3 are all unknown and herefore calculaed symbolically. he emperaure ha is calculaed is now in erms of he α coefficiens in Eq. 4.3 as opposed o he a coefficiens in Eq he measured hermal response can be fi o he emperaure equaion, like in previous eamples, via a leas-squares deerminaion. A es case was run in order o validae his mehod. Firs, a direc problem was analyzed, where all boundary condiions and properies were known. he hermal conduciviy followed Eq. 4.3 wih he following coefficiens: α =.13, α 1 =1, α =,

48 41 α 3 =3. Ne, he inverse problem was run, where he hermal conduciviy for Eq. 4.3 was calculaed wih unknown coefficiens o be solved for. Figure 4-9 shows resuls of hese wo cases, and he figure shows ecellen agreemen beween he conduciviy polynomial and inverse predicion. 6 5 Conduciviy W cmk 4 3 Conduciviy Inverse Predicion emperaure (K) Figure 4-9: A comparison of he hermal conduciviy and he inverse predicion. I mus be noed ha his case is limied o coefficiens of Eq. 4.3 whose values are no close o zero when using Mahemaica for he leas squares fiing. his influenced he choice of coefficien values for he conduciviy equaion. In running various rials wih differen coefficiens, he mehod only worked if α 1, α, and α 3 were a leas on he order of one. If hey were less han ha, he leas-squares mehod did no

49 4 produce real values. he concep of fiing he daa and solving for he coefficiens of ineres is sill valid. However, wih smaller values of coefficiens, he leas-squres used by Mahemaica breaks down. If an improved algorihm could fi he daa o he coefficiens, hen i would be advised o use his mehod insead of he leas-squares fi. In addiion, he following calculaion was made for he semi-infinie slab. If he hermal response was measured a he back boundary of a finie slab, his approimaion would no be valid for imes greaer han ha required for he hermal wave o reach he back boundary. Hence, for his ype of analysis, he emperaure mus be measured a an inernal locaion, and only for he imes up unil he hermal wave reaches he back boundary. For a hermally hin maerial, here perhaps would no be sufficien ime o acquire he necessary daa o perform he inverse mehod. Hence, he models based on a semi-infinie solid do have significan limiaions. herefore, i is definiely desirable o perform he inverse calculaions on a finie slab for surface emperaure as well as diffusiviy.

50 43 Chaper 5 Resuls and Daa Finie Slab In his chaper, he more realisic and pracical finie slab of lengh L is analyzed. An illusraion of his model is given in Figure 5-1. As wih he semi-infinie solid model, he lengh in he ransverse direcion is significanly longer han he lengh in he conducing direcion, which ensures ha he analysis is one-dimensional in space. he finie elemen model, verified wih he eac soluion in Chaper 4, was used again (afer sligh modificaions o reflec a finie hickness) as validaion for he analyical soluion derived. he resuls of Chaper 4 enabled he auhor o use he model wih confidence. Wih a finie slab, he Goodman soluion given by Eq..9 for emperaure is no longer valid afer he hermal wave reaches he back boundary. As a resul, a new model mus be devised. Forunaely, he inegral mehod used for he semi-infinie slab model can also be used for he finie slab model wih some adjusmens, including he back boundary of he slab, =L, modeled as a perfec insulaor.

51 44 = L Insulaed Back boundary (=L) Figure 5-1: A finie slab of lengh L wih an insulaed back boundary is shown.

52 Calculaing emperaure for he Finie Slab Using he Inegral Mehod As previously discussed, an insulaed back boundary condiion was imposed, where he back boundary is = L. he surface ( = ) was given he same asympoic eponenial hermal loading from Chaper 4. Using he inegral approach, he emperaure of his hermal model is assumed o be of he form [17] he quaniy η() mus be calculaed in order o eplicily define he emperaure. As in he semi-infinie slab model, he quaniy v is he same variable defined in Eq..19. Once again, he variable θ is inroduced o he model: In his analysis, he upper limi of inegraion is =L insead of he hermal layer δ. If Eq. 5.1 is subsiued ino Eq. 5., he inegral, merely a polynomial funcion of, can be readily evaluaed ino he form Furhermore, if Eq.. is inegraed from = o =L, he following equaion will be obained: 3 1 v (, ) = vs( ) 1 + η( ) η( ) 5.1 L 3 L L (, ) θ = v d 5. 5 θ = vs ( ) L + Lη( ) dθ v v = κ( v) ( L,) κ( v) (, ) 5.4 d

53 A parial derivaive wih respec o of he polynomial in Eq. 5.1 can be easily differeniaed a = and =L, simplifying Eq. 5.4 ino he form where he subscrip s denoes quaniies a he surface. he value for θ given by Eq. 5.3 can be differeniaed wih respec o ime and equaed o Eq. 5.5, resuling in he relaion d d v dθ d which can be differeniaed wih respec o ime, resuling in he following form: ( ) vsη = κs 5.5 L 5η 1 ( ) v η( ) s s 1 + = κs 5.6 L 46 dv d s ( ) 5v dη( ) 5η s κsvs = η d L By grouping all of he η() erms on one side, Eq. 5.7 can be algebraically manipulaed ino he following form: Eq. 5.8 is a linear, nonhomogeneous differenial equaion ha can be readily solved analyically. o do so, i mus firs be simplified ino he following form: where and dη d ( ) ( ) 1κ( ) ( ) 1 dv s 1 1 dvs + η( ) + = 5.8 vs( ) d 5L 5 vs( ) d dη d ( ) ( ) ( ) f ( ) f ( ) + η = s f 1 + vs( ) d ( ) 1κ( ) 1 dv = 5.1 5L

54 Eq. 5.9 can be muliplied by he inegraing facor Ep[F()], where he soluion o Eq. 5.9 hen becomes where c 1 is a consan of inegraion. he inegral in Eq is inracable using direc inegraion; herefore, a aylor series approimaion was employed. he value for η() can hen be subsiued ino Eq. 5.1 for he emperaure soluion. Since he densiy ρ and he specific hea c are assumed o be consan, he soluion given by Eq. 5.1 can be simplified o ( ) where he quaniy η() is now defined in Eq he consan of inegraion mus be solved using a boundary condiion. he definiion of he hermal layer δ() in Eq..9 can be used. In spie of he fac ha his definiion works for a semi-infinie slab, he reader mus be reminded ha unil he hermal wave reaches he back boundary, he semi-infinie soluion is valid. As a resul, he value of ime a which he hermal wave reaches he back boundary (which will be called L ) can be found by solving he equaion f ( ) ( ) 1 1 dvs = v d ( ) f ( ) s F = 1 d 5.1 F F ( ) = + F() η c e e ( ) e f d (, ) = s ( ) 1 + η( ) η( ) 5.14 L 3 L δ( L ) = L

55 48 he value for L can hen be subsiued ino Eq If he iniial emperaure (,) in Eq..18 is defined as i, hen he consan c 1 can be found by solving he equaion L i s ( L, ) = = 1 η ( ) Once he consan c 1 has been deermined, he emperaure has been compleely solved as a funcion of space and ime. As wih he semi-infinie slab analysis, he asympoic loading on he inernal surface of s =V(1-e -.5 ) fi o a polynomial was employed. wo more cases were considered; one case analyzed increasing hermal conduciviy wih increasing emperaure, while he second case considered decreasing conduciviy wih increasing emperaure. In each case, V was se equal o one, and he conduciviy, in general, was described by he polynomial in Eq. 4.3, where all of he α coefficiens are nonzero. he maerial was given he following hemophysical properies: ρ=3. kg/cm 3 and c =.376 J/(kgK). A finie elemen model was creaed ha had idenical hermophysical properies o he ones previously menioned. Since he slab now has a finie lengh, he model mus have a nonzero hermal response on he back boundary oherwise, he finie slab analysis is no required. An asympoic eponenial was hermally loaded on he surface, and a finie elemen simulaion was conduced. he hermal response a = L was measured because in many engineering applicaions, emperaure, or some oher hermal response, is measured a he back boundary and no a an inernal locaion.

56 49 5. hermally hin Models In he increasing conduciviy case, he conduciviy followed he form of Eq. 4.3, wih he following coefficiens: α =.13, α 1 =.897, α =.778, α 3 = Boh he analyical and finie elemen model were given hese coefficiens. A generalized emperaure was found via Eq. 5.1 and fi o he daa measured from he finie elemen model. Figure 5- shows a measured response a =L, where L = 1 cm. he figure shows ha boh models are in good agreemen. In he case of decreasing conduciviy, he values for conduciviy were se o Eq. 4.3 wih he following coefficiens: α=.13, α1=-.937, α=-.5873, α3= All oher hermophysical properies were he same as previous case sudies. A finie elemen simulaion was run as confirmaion, wih he same parameers as he analyical case. Figure 5-3 shows he measured response a = L, where L = 1cm. he figure shows relaively good agreemen beween he wo models, bu no as close as he increasing conduciviy case. Again, given he versailiy of he polynomials used, he mehod appears well suied for monoonic or mildly-oscillaory hermal scenarios provided he analysis is resriced o he ime inerval used o deermine he polynomial. In addiion, he mehod can be adaped for any ranscendenal funcion, no necessarily polynomials, including periodic ones.