Thermal characterization of an insulating material through a tri-layer transient method

Transcription

1 Advaced Sprig School «Thermal Measuremets & Iverse techiques», Domaie de Fraço, Biarritz, March T6 Thermal characterizatio of a isulatig material through a tri-layer trasiet method V. Félix, Y. Jaot, A. Degiovai, V. Schick LEMTA, Uiversité de Lorraie, CNRS UMR 7563,, aveue de la Forêt de Haye, TSA VANDOEUVRE Frace Abstract. The three layers trasiet method is dedicated to the thermal properties measuremet of isulatig materials. The three layers experimetal device (brass/sample/brass) ad the priciple of the measuremet based o a pulsed method will be first preseted. The three dimesioal modellig of the system will be developed ad used for a sesitivity aalysis. The estimatio method will be described ad its applicatio to simulated oisy measuremets realized with COMSOL will be preseted. Durig the workshop, several experimets will be carried o differet materials ad the experimetal temperature records will be used to estimate the thermal properties of the tested samples. Some improvemets to the iitial model such as takig ito accout a parallel or series thermal resistace will be discussed. 6. Itroductio The three layers method [] has bee specially developed for thermal coductivity measuremet of small samples of low-desity isulatig materials sice the existig methods are ot suited to this type of materials. The cotact trasiet methods usig plae or liear heatig elemet: hot disk [-3], hot wire [4], hot plate [5], hot strip [6-7] caot measure precisely thermal coductivity of low desity isulatig materials for the followig reasos: - The thermal capacity ad the thermal resistace of the heatig elemet (ofte heterogeeous ad made of a metal wire iserted betwee two plastic films) is ot kow with precisio ad is ofte take ito accout by a simplified model. - The sesitivity of the measured temperature to the thermal capacity of the heatig elemet is very high if the thermal capacity of the isulatig material is low (case of a low desity material) - The thermal coductivity of the heatig elemet is higher tha the coductivity of the isulatig material. The logitudial heat trasfer (parallel to the cotact surface betwee the heat source ad the sample) i the heatig elemet that is ot take ito accout i the models may be ad lead to estimatio errors. - The Flash method [8] is difficult to use for the followig reasos: - The isulatig materials are ofte semi-trasparet to the radiatios of the Flash lamp, - It is very difficult to measure precisely a surface temperature o a low desity material, - The heat trasfer o the heated face is ofte very differet of the heat trasfer o the other faces (very importat temperature differeces). To avoid the first two disadvatages, the sample may be iserted betwee two heat coductig plates; a device based o this priciple has already bee used for the liquids [9]. For very low desity materials, oe ca show that the sesitivity of the uheated face temperature to the thermal coductivity is highly correlated to the sesitivity to the covective losses ad that the sesitivity to the thermal diffusivity is low. The aim of this work was to develop a ew method suited to the thermal coductivity (ad evetually diffusivity) measuremet of low ad very low desity isulatig materials. 0

2 Advaced Sprig School «Thermal Measuremets & Iverse techiques», Domaie de Fraço, Biarritz, March Priciple ad experimetal device The experimetal device icludes a cylidrical sample (R = cm, e = 5 to 0mm) of the material to be characterized iserted betwee two brass discs with a thickess e b = 0,4mm ad the same radius (cf. figure ). Two type K thermocouple with wire diameter 0,05mm are welded o the exteral face of each brass disc by the techique of the separated cotact (with a distace of 5mm betwee the two wires). The lower disc is i direct cotact with a plae circular heatig elemet havig the same diameter ad set o a isulatig material. A pressure is applied o the uheated brass disc by four PVC (chose for its low thermal coductivity) tips with a very low cotact surface area. The upper surface of the uheated brass disc exchages with the ambiet air by atural covectio ad radiatio. A heat flux is applied durig a few secods to the heatig elemet ad the temperatures T b (t) ad T b (t) of the brass discs are recorded. A three-dimesioal model associated to a iverse method is the used to estimate the thermal coductivity ad the thermal diffusivity of the isulatig material iserted betwee the two brass discs. The heat flux is produced by a plae-heatig elemet durig a few secods istead of beig produced by a flash lamp (device iitially tested) durig a few millisecods for the followig reasos: - The temperature icreasig of the heated face must ot be too fast to be compatible with the thermocouple respose time. - A uiform pressure may be easily applied o the brass discs through the plae heatig elemet. - A light part of the flash may reach the lateral surfaces by reflectio, this disadvatage is avoided with a plae heatig elemet. h Brass T b (t) Brass h 3 Sample T b (t) Heatig elemet Isulatig material Figure. Experimetal device 6.3 Model ad estimatio method Assumptios: - The temperatures T b ad T b i the brass discs are uiform - The thermal cotact resistaces (typically 0-4 m.k.w - ) betwee the sample ad the brass discs are egligible i compariso with the thermal resistace of the sample (greater tha 5 x 0 - m.k.w - for the tested samples). 0

3 Advaced Sprig School «Thermal Measuremets & Iverse techiques», Domaie de Fraço, Biarritz, March As a first step, the followig case is cosidered: a uique sample with heat trasfer by covectio with the ambiet air o all its faces receives a direct ad short heatig o oe face (o brass discs as represeted i figure ). Settig T( r,z,t ) = T( r,z,t ) Te The equatio of heat becomes: ( r,z,t) T ( r,z,t) T ( r,z,t) T ( r,z,t) T + + = r a Iitial ad boudary coditios may be writte as: t (6.) T ( r, 0,t) = 0 λ = h T ( r,,t) φ ( t) (6.) z 0 0 T ( r,e,t ) = e λ = h T ( r,e,t ) (6.3) z T (0,z,t) r = 0 = 0 (6.4) T ( R,z,t ) = R λ = h T ( R,e,t ) (6.5) r 3 t = 0 T( r,z, 0) = 0 (6.6) The Laplace trasform applied to relatio () with L[ T( r,z,t )] θ( r,z, p ) θ( r,z, p) θ ( r,z, p) θ( r,z, p) + r The Laplace trasforms of the boudary coditios are: + = leads to: p = a θ ( r, 0,t) λ = hθ ( r, 0,t) Φ 0 ( t) θ( r,z, p) (6.7) (6.8) 03

4 Advaced Sprig School «Thermal Measuremets & Iverse techiques», Domaie de Fraço, Biarritz, March Settig: θ ( r,z, p) = R( r, p ) Z( z, p ) Oe obtais: θ ( r,e,t ) λ = hθ ( r,e,t ) (6.9) θ ( 0,z,t ) = 0 (6.0) θ ( R,z,t ) λ = h3 θ ( R,e,t ) (6.) θ ( r,z, 0 ) = 0 (6.) Where: A ( r,z, p) A J0 ( αr) { β ch[ γ ( e z) ] + H sh[ γ ( e z) ]} (6.3) θ = = Φ0( p) = λ (6.4) ω ω + J ω β + H H sh β + β H + H ch β H 3 ( ) [( ) ( ) ( ) ( )] e ω is solutio of: ω J ( ω) = H3 J 0( ω ), β = p e e e h + ω, e h H =, H =, a R λ λ h3 R H3 = λ The mea temperatures for z = 0 ad z = e may be calculated by itegratio of relatio (3) betwee r = 0 ad r = R: ( ) [ ( ) ( )] ( ) 4 Φ0 p βch β + H sh β θ moy 0, p = λ (6.5) = ω + ω β + H H sh β + β H + H ch β H 3 [( ) ( ) ( ) ( )] e ( ) ( ) 4 Φ0 p β θ moy e, p = λ (6.6) = ω + ω β + H H sh β + β H + H ch β H 3 [( ) ( ) ( ) ( )] The ext step is the study of the three layers device represeted i figure 3. The relatios (), (4), (5) ad (6) remai valid i this case. The local thermal balace at radius r o the heated brass disc (supposed at uiform temperature T b (t)) is: T R b T ( r, ) π R e ρ c = φ ( t) π R h π R T h π R e T + π r dr λ e 0 (6.7) b b t 0 b 3 e b 0 s 04

5 Advaced Sprig School «Thermal Measuremets & Iverse techiques», Domaie de Fraço, Biarritz, March z Brass e + e e h T b (t) Brass h 3 Sample 0 -e R r T b (t) h Figure 3. Experimetal schema of a three layers device The itegratio of this local balace betwee r = 0 ad r = R with a Laplace trasformatio leads to: θmoy ( e, p) h3e λ = Φ0( t) ρb cb e p + h + θb (6.8) This equatio is similar to relatio (8) cosiderig the mea temperature for a give value of z istead of the local temperature at (r,z) ad replacig h by a corrected coefficiet: h c R h3e = ρ b cb e p + h + (6.9) R The thermal balace of the uheated disc leads similarly to a corrected coefficiet: h c h3e = ρ b cb e p + h + (6.0) R By cosiderig the mea temperature at z istead of the local temperature at (r,z), this boudary coditios are the same as for a uique sample. So that for the three layers system brass/sample/brass, the expressios (8) ad (9) of the mea temperatures respectively at z = 0 ad z = e for a uique sample remais valid if h is replaced by h c ad h by h c i relatio (7). The trasfer fuctio H(p) of the system may be writte as: Ad: T b H θ ( e, p) ( p) = (6.) θ ( 0, p) ( t) = T ( t) L [ H( p) ] (6.) b φ 0 (t) These two relatios are true whatever the boudary coditio o the heated brass disc is, particularly if this disc exchage heat by coductio with a isulatig material (as described i figure ) rather tha by covectio with air. The priciple of the method is to estimate the trasfer fuctio H(p) by estimatig the values of the three parameters a, λ ad h = h = h 3 that miimize the sum of the quadratic errors betwee the experimetal 05

6 Advaced Sprig School «Thermal Measuremets & Iverse techiques», Domaie de Fraço, Biarritz, March values of T b (t) ad those calculated by relatio (5) with experimetal values of T b (t). The umber of parameters to be estimated is the same that i the classical flash method (heat flux desity φ, thermal diffusivity a ad covectio heat trasfer coefficiet h. The miimizatio is realized by usig the Leveberg-Marquart method. Oe of the advatage of the method is that it is ot sesitive to the heat trasfer o the heated face; i the case of a covective heat trasfer with importat variatios of temperature o this face, the hypothesis of a costat covectio coefficiet beig the same o the heated ad o the uheated face is ot totally true. It will be show that this model approximatio ca lead to errors i the estimated parameters particularly i the case of a measuremet realized o a isulatig material where the heated face temperature may reach several decades degrees ad where the resistace to the exteral trasfer (covectio) is of the same order of magitude that the resistace to the iteral trasfer (coductio). Compared to the limits (described i itroductio) of the classical cotact methods ad of the Flash method for the thermal coductivity measuremet of low desity material, the proposed method has the followig advatages: - The temperature measuremets are more precise sice they are doe o a heavy coductive material (brass), - The thermal capacity of the two discs (homogeeous) is kow precisely ad take ito accout, - The logitudial heat trasfers i the discs are take ito accout without approximatio i the model (boudary coditio of uiform temperature). 6.4 Sesitivity aalysis The reduced sesitivities of the trasfer fuctio H(p) have bee calculated ad represeted i figure 4 for a three layers system with the followig characteristics : - Brass discs: e b = 0.4mm, D = 40 mm, λ =00.04 W.m -.K -, a =3.4 x 0-5 m.s - ; - Samples: o Isulatig material: λ = 0.0 W.m -.K -, a = 4 x 0-6 m.s -, e = 5mm et e = 0mm o Polystyree: λ = W.m -.K -, a = 8 x 0-7 m.s -, e = 0mm o Cellular cocrete: λ = 0.5 W.m -.K -, a = 5 x 0-7 m.s -, e = 0mm It ca be oticed that: - For a give material, if the sample thickess decreases the the sesitivity to the thermal diffusivity icreases ad the sesitivity to the thermal coductivity decreases. - For a give sample thickess, if the thermal capacity of the sample icreases the the sesitivity to the thermal diffusivity icreases - The sesitivity to the thermal diffusivity is quite low for very low desity materials so that the thermal diffusivity will ot be measurable with this method. Nevertheless, this sesitivity is high eough to estimate the thermal diffusivity of isulatig material with higher desity such as polystyree or cellular cocrete. The sesitivity to the thermal coductivity is high i each case ad is ot correlated with the sesitivities to the covectio coefficiet ad to the thermal diffusivity. The thermal coductivity is thus estimable by this method for all type of isulatig materials. 06

7 Advaced Sprig School «Thermal Measuremets & Iverse techiques», Domaie de Fraço, Biarritz, March Estimatio from umerical simulatios The temperatures i a three layers device have bee simulated with COMSOL. The followig data has bee cosidered: - Sample: diameter = 35mm, thickess = 5.6mm, thermal properties: λ = 0.0 W.m -.K -, ρc = 5000 J.m - 3.K -, a = 4 x 0-6 m.s - - Metallic discs (copper): diameter = 35mm, thickess = 0.4 mm, thermal properties: λ = W.m -.K -, ρ = 8940 kg.m -3, c = J.kg -.K -. First, the simulatio of the temperatures has show that the hypothesis of uiform temperature i the two metallic discs is valid. The, the simulated temperatures T b (t) ad T b (t) have bee cosidered as experimetal data ad a estimatio parameter has bee applied accordig to the two followig methods:. The uheated face temperature T b (t) is the oly experimetal data cosidered as i the classical flash method. A iversio method is used to estimate the heat flux desity φ 0, the thermal diffusivity a, the thermal coductivity λ ad the covectio coefficiet h that miimize the sum of the quadratic errors betwee the experimetal curve ad the simulated curve T b (t) calculated by relatio (9) with the corrected covectio coefficiets give by relatios () ad (3).. The temperature T b (t) of the heated face is cosidered as iput experimetal data of the system both with the temperature T b (t). A iversio method is used to estimate the thermal diffusivity a, 07

8 Advaced Sprig School «Thermal Measuremets & Iverse techiques», Domaie de Fraço, Biarritz, March the thermal coductivity λ ad the covectio coefficiet h that miimize the sum of the quadratic errors betwee the experimetal curve ad the simulated curve T b (t) calculated by relatio (5) of the uheated face temperature. The results of these two estimatios are reported i table where t s is the upper boud of the estimatio time iterval. Table. Results of the estimatios realized by two differet methods from temperatures Estimatio method simulated with COMSOL h = h = h 3 = 0 W.m -.K - t s = 00s t s = 00s t s = 300s 0 6 a 0 3 λ h 0 6 a 0 3 λ h 0 6 a 0 3 λ h h = 5 W.m -.K - ; h = h 3 = 5 W.m -.K - t s = 00s t s = 00s t s = 300s Estimatio method 0 6 a 0 3 λ h 0 6 a 0 3 λ h 0 6 a 0 3 λ h h = 5 W.m -.K - ; h = h 3 = 5 W.m -.K - ; dt b = dt b = 0.0 C Estimatio method t s = 00s t s = 00s t s = 300s 0 6 a 0 3 λ h 0 6 a 0 3 λ h 0 6 a 0 3 λ h Estimatio method h = h 3 = 0 W.m -.K - ; h = 5 W.m -.K - t s = 00s t s = 00s t s = 300s 0 6 a 0 3 λ h 0 6 a 0 3 λ h 0 6 a 0 3 λ h It ca be oticed that: - The method, based o the 3D model ad cosiderig that the oly kow temperature is T b (t), does ot lead to a precise estimatio of λ if the covectio coefficiet h o the heated face is differet of the coefficiets h ad h 3 o the other faces. - The method gives i all cases a precise estimatio of the thermal coductivity, eve whe a radom oise with a amplitude of 0.0 C has bee added to the temperatures simulated with COMSOL. The estimatio of the thermal diffusivity becomes less precise if the measuremet oise icreases. - A simulatio based o method with h = 5 W.m -.K - ad h = h 3 = 0 W.m -.K - leads to a error of 3.5% o the estimated value of λ ad of % o the estimated value of a. A extreme case has bee cosidered (h 3 = h ) so that the errors with experimetal data will be less importat. 08

9 Advaced Sprig School «Thermal Measuremets & Iverse techiques», Domaie de Fraço, Biarritz, March The effect of a error o the supposed kow value of the thermal capacity ρc mp of the two metallic plates has also bee ivestigated. It was foud that a error of x% o ρc mp leads to the same error o the estimated value of λ but has o sigificat ifluece o the estimated value of a, for x < 5%. It has bee verified that a error of 5% o the thermal coductivity of the metallic plates has o ifluece o the estimated values. 6.6 Experimetal study A first experimet will be carried o a sample of polyethylee foam with the followig characteristics: e = 3mm, D = 40mm. The estimated values will be compared to the followig oes estimated by use of other methods: ρc = J.m -3 (measured by DSC) ; λ = 0.04 W.m -.K - (measured by the cetred hot plate method [9]). Other experimets will be carried o with the followig materials ad/or processed: - Spaceloft (super-isulator) with e = mm - PVC: ρc =.40x0 6 J.m -3 (measured by DSC) ; λ = 0.84 W.m -.K - (measured by the tiy hot plate method [0]). The aalysis of the residuals may lead i some cases to use a modified model takig ito accout a parallel or a series thermal resistace. 6.7 Refereces [] Jaot Y, Degiovai A, Payet G 009 Thermal coductivity measuremet of isulatig materials with a three layers device Iteratioal Joural of Heat ad Mass Trasfer [] Gustafsso SE 005 Trasiet plae source techiques for thermal coductivity ad thermal diffusivity measuremets of solid materials Rev. Sci. Istrum [3] He Y 005 Rapid thermal coductivity measuremet with a hot disk sesor. Part. Theoretical cosideratios Thermochimica Acta [4] Nagazaka Y, Nagashima A 98 Simultaeous measuremet of the thermal coductivity ad the thermal diffusivity of liquids by the trasiet hot-wire method Rev. Sci. Istrum [5] Zhag X, Degiovai A 993 Mesure de l effusivité thermique de matériaux solides et homogèes par ue méthode de «sode» plae Joural de Physique III [6] Hammerschmidt U 003 A ew pulse hot strip sesor for measurig thermal coductivity ad thermal diffusivity of solids Iteratioal Joural of Thermophysics [7] Jaot Y, Meukam P 004 Simplified estimatio method for the determiatio of thermal effusivity ad thermal coductivity with a low cost hot strip Measuremet Sciece ad Techology [8] Degiovai A, Lauret A 986 Ue ouvelle techique d idetificatio de la diffusivité thermique pour la méthode flash Revue de Physique Appliquée [9] Rémy B, Degiovai A 005 Parameters estimatio ad measuremet of thermophysical properties of liquids Iteratioal Joural of Heat ad Mass Trasfer [0] Jaot Y, Felix V, Degiovai A 00 A cetered hot plate method for measuremet of thermal properties of thi isulatig materials Measuremet Sciece ad Techology. [] Jaot Y, Rémy B, Degiovai A 00 Measuremet of thermal coductivity ad cotact resistace through a tiy hot-plate experimet High Temperatures High Pressures