Caractérisation dans le plan de matériaux isolants par thermographie infrarouge et méthode convolutive
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1 Journée SFT du Jeud 3 Jun 3 Caractérsaton Température / Température Caractérsaton dans le plan de matérau solants par termogrape nfrarouge et métode convolutve B. REMY, A. DEGIOVAI & D. MAIET B.Rémy, A.Degovann and D.Mallet " Measurement of te In-Plane Termal Dffusvty of Materals by Infrared Termograpy " (DOI:.7/s z Internatonal Journal of Termopyscs - Vol.6, n (5, pp EMTA - UMR 7563 (CRS Unversté de orrane ESEM avenue de la Forêt de Haye TSA VADOEUVRE CEDEX Emal : benjamn.remy@unv-lorrane.fr Journée SFT du Jeud 3 Jun 3
2 Te n-plane dffusvty measurement tecnques Soluton n te case of a sem-nfnte medum submtted to a eat step stmulaton : φ e ( t T, Assumptons: - Sem-nfnte medum - One drectonal eat transfer - Insulated T (, t φ at ep erfc λ π 4at 4at 4at a: termal dffusvty λ: termal conductvty Metod : poston and dfferent tmes (Steere, 966 & Harmaty, 964 T T (,t (, t erfc erfc at at Anoter tecnque: at e,5τ t m m m ( τ t ln( τ t m Steere, 967 Square pulse (duraton τ
3 Te n-plane dffusvty measurement tecnques Metod : dfferent postons at a gven tme (Katayama, 969 T πe ep erfc T 4at 4at 4at Drawbacks: - Heat Flu Dstrbuton must be perfectly known - Heatng element s deal (no capacty and n perfect contact wt te sample - Medum s assumed to be perfectly nsulated - D Heat transfer ( pt Metod : aplace transform (Kavanpour & Beck, 977 θ ( p ( T ( t T ( t ep dt θ T ( p a T t θ a p θ ( p ( p a ϕ ep λ p a, ln θ θ (, p (, p ( Te soluton s ndependent of te n-tme eat flu dstrbuton p a
4 Te n-plane dffusvty measurement tecnques Metod : Fn s metod (Hadsaroyo, 99 Unknown eat flu (, t T ( t T, ( e + S e. T S λ T t θ p a S λ ( T T + θ et a e l ln θ (, p (, p θ p S ( + a λ Unknown eat flu T (, t T (, t e Bot dffusvty a and eat losses are taken nto account (sem-nfnte medum s assumed l Interest of te nfrared camera: - on-necessary n space unform stmulaton - D transfer n te case of nsulatng materals
5 O J D D J J A J O Te Fn s metod n Transtory Regme To measure λ, te sample s stmulated n by a non necessary unform and constant eat flu ϕ(y,t or temperature step T (y,t. Te averaged temperature s ten consdered: O 6 6 T (, t T (, t T et 6 Heat Transfer Equaton : ( e + T T T e λ a t Boundares Condtons : n T λ ϕ ( t or T T ( t Tet n T λ ϕ ( t T or ( et T t T e << or y Fn s Appromaton Intal Condton : T at t
6 Teoretcal Model: Soluton n te aplace doman Te soluton s obtaned n te aplace doman (ntegral transform F ( pt ( p ( f ( t f ( t ep Te Quadrupole formulaton allows to lnearly lnks te aplace transforms of te nner and outer temperatures and flues: T Φ ϕ θ ( ( dt θ Φ A C θ Φ θ Φ e [ M ] [ M ] B D θ Φ e θ Φe A B C D sn λk cos ( k ( k λk ( k sn ( k p a Two reference temperature profles are cosen n and sn θ (, p θ( p F (, p + θ ( p F (, p wt: ( ( α( F, p sn( α( and F (, p ( α p a + eλ sn sn ( α( ( α(
7 Teoretcal Model: Soluton n te Tme doman In te tme doman, te soluton can be wrtten as a sum of two convoluton products: - ( t T ( t f (, t + T ( t f ( t wt: f (, t ( F ( p T,,, measured by nfrared camera f and f are functons of te unknown parameters: a and H..5 *//8 *//4 */3/8 *// *//8 *//4 */3/8 *//..5 * * * * f ( f (,t,t t * at/ t * at/
8 Drect Model: Test Case Test Case: square sample 4mm and emm a 5. 7 m s. λ W. m. K W. m. K Temperature step n Temperature * θ (, p In-tme Varatons: θ T θ ( p ( T T p et (, p cos( α( T p cos( α( et Insulated n * * Profle T ( lm p. θ ( p p In-space Varatons: cos cos ( α ( ( α ( T*(T-T et /(T -T et * *. * *.4 *.6 *.8 *. T*(T-T et /(T -T et t*. t*.78 t*.3 t*.6 t*. t * Steady-State Profle t * at/ * /
9 Inverse Model: Parameter Estmaton Ordnary east Squares Metod: T te ( t β T T te te ( t, β, ( t, β, Tte k, M M (, t, β S n t ( Tte ( t Tep ( t ( n + profles and:, β β T ep ( t T T T ep ep ep ( t, M ( k, t ( M, t a H wt: Unknown parameters k β j k n (, K k,n S β j ( t, β nt Tte te β β j ( T ( t, T ( t ( j ep X j β j T te β ( t β, j Senstvty Coeffcent Ideal case on-deal case T ep ( t T ( t, β + ε ( t te E( ε and ( ε V σ b β ˆ β + t t ( X X X ε( t ( t, β T (, t, β + ε ( t f (, t + ( t f ( t T, ep k, te k k ε ε ε k V E( ˆ β β t ( β σ ( X X ˆ b σ b Var Cov ( a Cov( a, H ( a, H Var( H σ σ Var( a σ σ Var( H a b. H b.
10 Optmal coce of te reference profles statc profles: movng profles: T * (T-T /(T -T et et X. a X H t * at/ T * (T-T /(T -T et et Var ( a Var ( H Cov ( a, H ρ ( a, H X a X H t * at/ 3 Drect Model General Model k k k k + atural & Optmal Coce: and
11 Effect of a non-unform eat transfer coeffcent T*(T-T /(T -T et et Sold : Dased : (.3/4*(/ / t * at/ In-space varyng eat transfer coeffcent wt: ( ( d (FEM Software - FlePDE
12 Effect of a non-unform eat transfer coeffcent a 4.98e-7 H 4 a 4.98e-7 H 5456 T * (T-T /(T -T et et ,96W. m. K T * (T-T /(T -T et et ,4w. m. K C Ste ( t * at/ t * at/ omnal values: 7 a 5. m. s and H 5 λ W. m. K W. m. K
13 Epermental Results Termograms (lne source stmulaton: Temperature profles and termograms: Estmaton: profle over
14 Conclusons An n-plane dffusvty estmaton metod by convoluton metod for low conductve materals as been presented. It s a very smple and sem-analytcal metod tat allows us usng two reference measured temperatures as boundary condtons To measure te n-plane dffusvty of low conductve materals Watever te In-Space and In-Tme stmulaton eat flu, And eat transfer coeffcent varatons are. Tank you for your attenton
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