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Estimation of thermophysical properties by an inverse method with experimentally determined heating region of a thin-layer heater
Infrared Physics & Technology 49 (2007) 277–280 www.elsevier.com/locate/infrared
Estimation of thermophysical properties by an inverse method with experimentally determined heating region of a thin-layer heater J. Zmywaczyk *, H. Madura, P. Koniorczyk, M. Da˛browski Military University of Technology, Kaliskiego 2, 00-908 Warszawa, Poland Available online 24 July 2006
Abstract This paper deals with the problem of estimation of thermophysical parameters by an inverse method. The thermal conductivity in radial and axial direction of a cylindrical sample and the heat capacity were simultaneously estimated using the Levenberg–Marquardt method of minimizing a mean square functional. As heat sources the thin-layer heater KHR 2/10 of diameter / = 50 mm and thickness 0.20 mm made by OMEGA as well as the Kanthal resistance wire of diameter / = 0.1 mm in a form of semicircle were simultaneously used. The main aim of using these two heaters simultaneously, both placed at one of the sample interfaces, was to generate heat fluxes in axial and additionally in radial direction. However, measurements of temperature distribution on the main surface of the thin-layer heater by using the FLIR Systems (ThermaCAM SC 3000 infrared camera) revealed a spatial heterogeneity of its temperature field, and therefore it was necessary to determine the effective heating region of that heater indispensable for solving of the coefficient inverse problem of heat conduction. 2006 Elsevier B.V. All rights reserved. Keywords: Inverse methods; Thermophysical properties; Infrared thermography
1. Introduction Knowledge of thermophysical properties of solids like thermal conductivity k, thermal diffusivity a and volumetric heat capacity C = qcp and/or emissivity e is essential particularly in modeling and design of elements of machines or devices working at extreme thermal conditions. Classical approach to determine e.g. thermal conductivity with a guarded hot-plate apparatus is a time-consuming task and usually requires a few hours per one experimental point. On the other hand the same can be performed at considerably shorter time when utilizing the inverse methods. A crucial for the method of coefficient inverse heat conduction problems (CIHCP) for the sake of accuracy and precision of estimated parameters is conformity between the theoretical model describing a given physical phenomenon and its experimental realization. The other important factors *
Corresponding author. E-mail address: [email protected] (J. Zmywaczyk).
1350-4495/$ - see front matter 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.infrared.2006.06.016
belong to precise location of sensors, a right number and arrangement of measuring points, a correct choice of time duration of experiment tf, and as exact as possible measurement of temperature Y(t) and heat flux density q(t). The results of simultaneous estimation of the orthotropic thermal conductivity kr, kz and volumetric heat capacity C for PMMA (Plexiglas) of density qPlexi = 1153.3 kg/m3 and for aerated autoclaved concrete of density qAAC = 354.9 kg/m3 were presented at the TEMPMEKO’2004 Conference in Cavtat, Croatia [5]. 2. The infrared imaging of the thin-layer heater – a few of experimental details Main part of experimental setup consists of a thin-layer heater KHR 2/10 of diameter / = 50 mm and thickness g = 0.20 mm shown in Fig. 1. The heater consists of an etched foil element of 0.10 mm thickness which is encapsulated between two layers of 0.04 mm Kapton and 0.02 mm FEP Teflon adhesive. Its
278
J. Zmywaczyk et al. / Infrared Physics & Technology 49 (2007) 277–280
3. A real temperature distribution of the heater
Fig. 1. A front of view the thin-layer heater KHR 2/10 (thickness 0.2 mm).
temperature range of work is from 200 to +200 C. The thin-layer heater and the Kanthal wire were supplied at the same time by the two DC PPS 2017 power supplies. Typical input voltage for the heater during experiment was equal 20.0 V or 60 V and the current was equal 0.041 A or 0.128 A, respectively (heating time 3 s). The effective diameter of the heater was initially estimated to be 45.0 mm so a half of the heat flux density (because of an axial symmetry of experimental setup) amounted to 257.8 W/m2 or 2414.4 W/m2, respectively. As an example comparison of the measured temperature histories by means of the K-type (NiCr–NiAl) thermocouples with their hot junctions located at the central part of the sample PMMA – Th1 as well as between the heaters – Th2, both situated on the sample surface, and the calculated temperature histories at the same location which were resulting from the solution of the boundary-value problem of heat transfer where the thermophysical parameters of the sample material were estimated using the Levenberg–Marquardt iterative procedure applied to minimize the mean square functional is shown in Fig. 2. Before doing numerical calculation it was assumed that the heat flux density generated by the thin-layer heater is uniformly distributed in a circle of diameter / = 45.0 mm.
In order to image temperature field of the heater during its heating and cooling time the ThermaCAM SC 3000 infrared camera – FLIR Systems was used. With its realtime 14 bit digital output storage and analysis, high thermal sensitivity 0.02 C and high speed data acquisition (upto 700 Hz), 50 Hz real-time digital recording and evaluation of thermal images it makes up an excellent investigative tool to analysis of the high-frequency temperature fields. In Fig. 3 there is depicted temperature distribution of the heater just after the end of heating time th (th = 4.5 s, U = 60 V, I = 0.128 A). The initial temperature T0 of the heater was equal +20 C, relative humidity of air amounted to 60%, the measurement distance between the lens of ‘close-up’ type and the target was equal 0.2 m and there was used the spectral range 8–9 lm. (Before imaging one of the interface of the heater was sprinkled by a thin-layer of a black lacquer of emissivity e = 0.94). The maximum temperature T = 89.4 C was observed in the middle part of the KHR 2/10 heater. It is evident that the temperature distribution of the heater is non-uniform and the effective diameter of the heating area does not equal 50.0 mm, i.e. a standard diameter of the heater. In addition to this, there is a rectangular region
Fig. 3. Temperature distribution of the thin-layer heater KHR 2/10.
Fig. 2. Temperature histories for Plexiglas specimen at locations (a)-Th1 and (b)-Th2 (heating time 3 s).
J. Zmywaczyk et al. / Infrared Physics & Technology 49 (2007) 277–280
~u ¼ ½k r ; k z ; CT ; RT ¼ ½Tð~uÞ Y; T T oT ð~uÞ Xð~uÞ ¼ ; X ¼ diagðXT XÞ; o~u
Temperature (°C)
31.0 29.5 28.0
25.0 23.5
J ðuT Þ ¼ 2.5
5.0
7.5
10.0
12.5
15.0
ð7Þ
NP X Nt X 2 T ðP i ; tn ; ~uT Þ Y i ðtn ; uT Þ ; i¼1
17.5
Time (s) Fig. 4. Maximal (upper) and minimal (bottom) temperature histories of the KHR 2/10 heater (U = 20.0 V, I = 0.043 A, th = 4.5 s).
is calculated iteratively using the Levenberg–Marquardt method [1,3] T
½ðXðsÞ Þ XðsÞ þ lðsÞ XðsÞ uðsþ1Þ T
of area 96.42 mm with a clearly lower temperature so the effective heating area is equal to 1122.20 mm2 which corresponds to the effective diameter of the first region of the highest temperature and equals 37.8 mm. The second, third, fourth and the fifth region has an average temperature equals 75, 65, 55 and 40 C, respectively. In the next figure (Fig. 4) temperature histories are taken at the two fixed points of a local maximal and minimal temperature – (see Fig. 3). In Fig. 4 it is visible that the maximal temperature difference is about 1.4 C and such a difference was kept during cooling. 4. Problem formulation and the method of its solution The inverse problem can be formulated as follows: find the orthotropic thermal conductivity kr, kz and the heat capacity C which minimize the mean square functional J ðk r ; k z ; CÞ ¼ eT e
ð1Þ
where the residual vector e stands for the difference between the calculated temperature Ti(t) and the measured temperature Yi(t) taken at the same point Pi and time t = tn. The calculated temperatures Ti(t) are the solution of governing equation oT ¼ div½KðT ÞgradðT Þ; ot
K ¼ ½k r 0; 0 k z
ð8Þ
n¼1
T
¼ ½ðXðsÞ Þ XðsÞ þ lðsÞ XðsÞ uðsÞ ðXðsÞ Þ RT ;
2
qcp ðT Þ
ð6Þ
the sought vector u(s+1) = u which minimize the mean square functional J(uT) defined as
26.5
22.0 0.0
279
ð9Þ
where l is the damping factor, s ¼ 0; 1; . . . ; s (s – the final number of iteration), X is the matrix of sensitivity coefficients which elements can be calculated using the following numerical scheme: X i;j ¼
T i ½uj ð1 þ eÞ T i ½uj ð1 eÞ ; 2euj
ð10Þ
where e is a small number (e = 104). Because the vector Y of measuring temperatures involves measurement errors which are assumed to fulfill the statistical assumptions given by Beck [1,3] so it is advisable to enclose statistical uncertainty of the estimated parameters. The standard deviation of estimated parameters ru may be calculated from the following relationship: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ruj ¼ rY ½XT Xj;j ; j ¼ 1; 2; 3; ð11Þ where rY is the standard deviation of the measured temperatures. Thus the confidence interval at the (1a) = 0.99 confidence level for the jth estimated parameter can be expressed as
ð2Þ
(K stands for 2 · 2 matrix of thermal conductivity) subject to the initial condition T jt¼0 ¼ T 0 and the boundary conditions oT oT k z ¼ qðr; tÞ; ¼ 0; oz C1 oz C3 oT oT ¼ 0; k r ¼ hðT T 0 Þ; oz C2 or C4
ð3Þ
ð4Þ ð5Þ
where Ci is the ith boundary of the sample. The finite volume method [4] was applied to solve the boundary-value problem Eqs. (2)–(5) enabling to obtain the calculated temperature Ti(tn). In a matrix form, after introducing the following quantities:
Fig. 5. A view of the main specimen surface with marked heaters and location of one of the thermocouple hot junction (Th2).
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J. Zmywaczyk et al. / Infrared Physics & Technology 49 (2007) 277–280
Table 1 Results of estimation of thermophysical parameters for PMMA of density q = 1153.3 (kg m3) at mean temperature T = 23.5 C Parameter
kr (W m1 K1)
kz (W m1 K1)
qcp (J m3 K1)
Inverse method (own) Jurkowski et al. [2] Complementary measurements (own)
0.2003 ± 0.0158 – –
0.2002 ± 0.0108 0.2017 ± 0.0053 0.1931
1153.3 Æ (1411.75 ± 6.89) 1182 Æ (1380 ± 27.6) 1153.3 Æ (1452.9)
uj 2:576ruj 6 uj 6 uj þ 2:576ruj
j ¼ 1; 2; 3
ð12aÞ
or when (1a) = 0.95 then uj 1:96ruj 6 uj 6 uj þ 1:96ruj
j ¼ 1; 2; 3:
ð12bÞ
5. The results of parameter estimation for plexiglas In Fig. 5 it is shown a face surface of the specimen with the both heaters and with marked location of one of the thermocouple hot junction (Th2). Within the temperature range from 10 to +70 C the thermal conductivity and the specific heat for Plexiglas were determined using a separated measuring stands such as: a guarded hot-plate apparatus and the DSC Perkin– Elmer Pyris 1 micro-calorimeter so as to compare the estimated and the measured results (complementary measurements). In the above mentioned temperature range it was obtained experimentally kðT Þ ¼ 0:1851 þ 3:383 104 T ;
Systems) and by using of thermocouples of K-type have confirmed that the heater generates spatially heterogeneous temperature field. Knowing such temperature distribution of the heater is essential to make possible to determine its effective diameter and consequently its effective heat flux density which in turn is necessary to estimate the thermophysical parameters of the sample materials by the inverse method. The results listed in Table 1 showed that there was not observed anisotropy with respect to the thermal conductivity of the investigated material (PMMA) and that the inverse method could be treated as an investigative tool making up alternative for classic measuring methods. Acknowledgements The authors would like to express their gratitude to the FLIR company for hire the ThermaCAM SC 3000 infrared camera.
ð13aÞ 3
cp ðT Þ ¼ 1339:750 þ 4:7814T þ 1:4060 10 T 2 :
ð13bÞ
Analysis of temperature distribution on the surface of the KHR 2/10 heater using the ThermaCAM SC 3000 infrared camera has confirmed an earlier estimated and assumed value of its effective diameter to be equal to / = 45.0 mm. The results of 2D parameter estimations for Plexiglas at a mean temperature T = 23.5 C and the complementary results are listed in Table 1. 6. Conclusions The results of investigations the temperature distribution of the thin-layer heater KHR 2/10 made by OMEGA using the ThermaCAM SC 3000 infrared camera (FLIR
References [1] J.V. Beck, K.J. Arnold, Parameter Estimation in Engineering and Science, first ed., John Wiley, New York, 1977. [2] T. Jurkowski, Y. Jarny, D. Delaunay, Estimation of thermal conductivity of thermoplastics under moulding conditions: an apparatus and an inverse algorithm, Int. J. Heat Mass Transf. 40 (17) (1997) 4169– 4181. ¨ zisik, H.R.B. Orlande, Inverse Heat Transfer Fundamentals [3] M.N. O and Applications, Taylor&Francis, New York, 2000. [4] S.V. Patankar, Numerical Heat Transfer and Fluid Flow, Hemisphere, Washington, DC, 1980. [5] J. Zmywaczyk, P. Koniorczyk, J. Terpiłowski, G. Zapotoczna-Sytek, Simultaneous estimation of thermal conductivity and heat capacity – 2D formulation, in: Proceedings of the 9th International Symposium on Temperature and Thermal Measurements in Industry and Science, 2005, pp. 1349–1355.
Estimation of thermophysical properties by an inverse method with experimentally determined heating region of a thin-layer heater J. Zmywaczyk *, H. Madura, P. Koniorczyk, M. Da˛browski Military University of Technology, Kaliskiego 2, 00-908 Warszawa, Poland Available online 24 July 2006
Abstract This paper deals with the problem of estimation of thermophysical parameters by an inverse method. The thermal conductivity in radial and axial direction of a cylindrical sample and the heat capacity were simultaneously estimated using the Levenberg–Marquardt method of minimizing a mean square functional. As heat sources the thin-layer heater KHR 2/10 of diameter / = 50 mm and thickness 0.20 mm made by OMEGA as well as the Kanthal resistance wire of diameter / = 0.1 mm in a form of semicircle were simultaneously used. The main aim of using these two heaters simultaneously, both placed at one of the sample interfaces, was to generate heat fluxes in axial and additionally in radial direction. However, measurements of temperature distribution on the main surface of the thin-layer heater by using the FLIR Systems (ThermaCAM SC 3000 infrared camera) revealed a spatial heterogeneity of its temperature field, and therefore it was necessary to determine the effective heating region of that heater indispensable for solving of the coefficient inverse problem of heat conduction. 2006 Elsevier B.V. All rights reserved. Keywords: Inverse methods; Thermophysical properties; Infrared thermography
1. Introduction Knowledge of thermophysical properties of solids like thermal conductivity k, thermal diffusivity a and volumetric heat capacity C = qcp and/or emissivity e is essential particularly in modeling and design of elements of machines or devices working at extreme thermal conditions. Classical approach to determine e.g. thermal conductivity with a guarded hot-plate apparatus is a time-consuming task and usually requires a few hours per one experimental point. On the other hand the same can be performed at considerably shorter time when utilizing the inverse methods. A crucial for the method of coefficient inverse heat conduction problems (CIHCP) for the sake of accuracy and precision of estimated parameters is conformity between the theoretical model describing a given physical phenomenon and its experimental realization. The other important factors *
Corresponding author. E-mail address: [email protected] (J. Zmywaczyk).
1350-4495/$ - see front matter 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.infrared.2006.06.016
belong to precise location of sensors, a right number and arrangement of measuring points, a correct choice of time duration of experiment tf, and as exact as possible measurement of temperature Y(t) and heat flux density q(t). The results of simultaneous estimation of the orthotropic thermal conductivity kr, kz and volumetric heat capacity C for PMMA (Plexiglas) of density qPlexi = 1153.3 kg/m3 and for aerated autoclaved concrete of density qAAC = 354.9 kg/m3 were presented at the TEMPMEKO’2004 Conference in Cavtat, Croatia [5]. 2. The infrared imaging of the thin-layer heater – a few of experimental details Main part of experimental setup consists of a thin-layer heater KHR 2/10 of diameter / = 50 mm and thickness g = 0.20 mm shown in Fig. 1. The heater consists of an etched foil element of 0.10 mm thickness which is encapsulated between two layers of 0.04 mm Kapton and 0.02 mm FEP Teflon adhesive. Its
278
J. Zmywaczyk et al. / Infrared Physics & Technology 49 (2007) 277–280
3. A real temperature distribution of the heater
Fig. 1. A front of view the thin-layer heater KHR 2/10 (thickness 0.2 mm).
temperature range of work is from 200 to +200 C. The thin-layer heater and the Kanthal wire were supplied at the same time by the two DC PPS 2017 power supplies. Typical input voltage for the heater during experiment was equal 20.0 V or 60 V and the current was equal 0.041 A or 0.128 A, respectively (heating time 3 s). The effective diameter of the heater was initially estimated to be 45.0 mm so a half of the heat flux density (because of an axial symmetry of experimental setup) amounted to 257.8 W/m2 or 2414.4 W/m2, respectively. As an example comparison of the measured temperature histories by means of the K-type (NiCr–NiAl) thermocouples with their hot junctions located at the central part of the sample PMMA – Th1 as well as between the heaters – Th2, both situated on the sample surface, and the calculated temperature histories at the same location which were resulting from the solution of the boundary-value problem of heat transfer where the thermophysical parameters of the sample material were estimated using the Levenberg–Marquardt iterative procedure applied to minimize the mean square functional is shown in Fig. 2. Before doing numerical calculation it was assumed that the heat flux density generated by the thin-layer heater is uniformly distributed in a circle of diameter / = 45.0 mm.
In order to image temperature field of the heater during its heating and cooling time the ThermaCAM SC 3000 infrared camera – FLIR Systems was used. With its realtime 14 bit digital output storage and analysis, high thermal sensitivity 0.02 C and high speed data acquisition (upto 700 Hz), 50 Hz real-time digital recording and evaluation of thermal images it makes up an excellent investigative tool to analysis of the high-frequency temperature fields. In Fig. 3 there is depicted temperature distribution of the heater just after the end of heating time th (th = 4.5 s, U = 60 V, I = 0.128 A). The initial temperature T0 of the heater was equal +20 C, relative humidity of air amounted to 60%, the measurement distance between the lens of ‘close-up’ type and the target was equal 0.2 m and there was used the spectral range 8–9 lm. (Before imaging one of the interface of the heater was sprinkled by a thin-layer of a black lacquer of emissivity e = 0.94). The maximum temperature T = 89.4 C was observed in the middle part of the KHR 2/10 heater. It is evident that the temperature distribution of the heater is non-uniform and the effective diameter of the heating area does not equal 50.0 mm, i.e. a standard diameter of the heater. In addition to this, there is a rectangular region
Fig. 3. Temperature distribution of the thin-layer heater KHR 2/10.
Fig. 2. Temperature histories for Plexiglas specimen at locations (a)-Th1 and (b)-Th2 (heating time 3 s).
J. Zmywaczyk et al. / Infrared Physics & Technology 49 (2007) 277–280
~u ¼ ½k r ; k z ; CT ; RT ¼ ½Tð~uÞ Y; T T oT ð~uÞ Xð~uÞ ¼ ; X ¼ diagðXT XÞ; o~u
Temperature (°C)
31.0 29.5 28.0
25.0 23.5
J ðuT Þ ¼ 2.5
5.0
7.5
10.0
12.5
15.0
ð7Þ
NP X Nt X 2 T ðP i ; tn ; ~uT Þ Y i ðtn ; uT Þ ; i¼1
17.5
Time (s) Fig. 4. Maximal (upper) and minimal (bottom) temperature histories of the KHR 2/10 heater (U = 20.0 V, I = 0.043 A, th = 4.5 s).
is calculated iteratively using the Levenberg–Marquardt method [1,3] T
½ðXðsÞ Þ XðsÞ þ lðsÞ XðsÞ uðsþ1Þ T
of area 96.42 mm with a clearly lower temperature so the effective heating area is equal to 1122.20 mm2 which corresponds to the effective diameter of the first region of the highest temperature and equals 37.8 mm. The second, third, fourth and the fifth region has an average temperature equals 75, 65, 55 and 40 C, respectively. In the next figure (Fig. 4) temperature histories are taken at the two fixed points of a local maximal and minimal temperature – (see Fig. 3). In Fig. 4 it is visible that the maximal temperature difference is about 1.4 C and such a difference was kept during cooling. 4. Problem formulation and the method of its solution The inverse problem can be formulated as follows: find the orthotropic thermal conductivity kr, kz and the heat capacity C which minimize the mean square functional J ðk r ; k z ; CÞ ¼ eT e
ð1Þ
where the residual vector e stands for the difference between the calculated temperature Ti(t) and the measured temperature Yi(t) taken at the same point Pi and time t = tn. The calculated temperatures Ti(t) are the solution of governing equation oT ¼ div½KðT ÞgradðT Þ; ot
K ¼ ½k r 0; 0 k z
ð8Þ
n¼1
T
¼ ½ðXðsÞ Þ XðsÞ þ lðsÞ XðsÞ uðsÞ ðXðsÞ Þ RT ;
2
qcp ðT Þ
ð6Þ
the sought vector u(s+1) = u which minimize the mean square functional J(uT) defined as
26.5
22.0 0.0
279
ð9Þ
where l is the damping factor, s ¼ 0; 1; . . . ; s (s – the final number of iteration), X is the matrix of sensitivity coefficients which elements can be calculated using the following numerical scheme: X i;j ¼
T i ½uj ð1 þ eÞ T i ½uj ð1 eÞ ; 2euj
ð10Þ
where e is a small number (e = 104). Because the vector Y of measuring temperatures involves measurement errors which are assumed to fulfill the statistical assumptions given by Beck [1,3] so it is advisable to enclose statistical uncertainty of the estimated parameters. The standard deviation of estimated parameters ru may be calculated from the following relationship: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ruj ¼ rY ½XT Xj;j ; j ¼ 1; 2; 3; ð11Þ where rY is the standard deviation of the measured temperatures. Thus the confidence interval at the (1a) = 0.99 confidence level for the jth estimated parameter can be expressed as
ð2Þ
(K stands for 2 · 2 matrix of thermal conductivity) subject to the initial condition T jt¼0 ¼ T 0 and the boundary conditions oT oT k z ¼ qðr; tÞ; ¼ 0; oz C1 oz C3 oT oT ¼ 0; k r ¼ hðT T 0 Þ; oz C2 or C4
ð3Þ
ð4Þ ð5Þ
where Ci is the ith boundary of the sample. The finite volume method [4] was applied to solve the boundary-value problem Eqs. (2)–(5) enabling to obtain the calculated temperature Ti(tn). In a matrix form, after introducing the following quantities:
Fig. 5. A view of the main specimen surface with marked heaters and location of one of the thermocouple hot junction (Th2).
280
J. Zmywaczyk et al. / Infrared Physics & Technology 49 (2007) 277–280
Table 1 Results of estimation of thermophysical parameters for PMMA of density q = 1153.3 (kg m3) at mean temperature T = 23.5 C Parameter
kr (W m1 K1)
kz (W m1 K1)
qcp (J m3 K1)
Inverse method (own) Jurkowski et al. [2] Complementary measurements (own)
0.2003 ± 0.0158 – –
0.2002 ± 0.0108 0.2017 ± 0.0053 0.1931
1153.3 Æ (1411.75 ± 6.89) 1182 Æ (1380 ± 27.6) 1153.3 Æ (1452.9)
uj 2:576ruj 6 uj 6 uj þ 2:576ruj
j ¼ 1; 2; 3
ð12aÞ
or when (1a) = 0.95 then uj 1:96ruj 6 uj 6 uj þ 1:96ruj
j ¼ 1; 2; 3:
ð12bÞ
5. The results of parameter estimation for plexiglas In Fig. 5 it is shown a face surface of the specimen with the both heaters and with marked location of one of the thermocouple hot junction (Th2). Within the temperature range from 10 to +70 C the thermal conductivity and the specific heat for Plexiglas were determined using a separated measuring stands such as: a guarded hot-plate apparatus and the DSC Perkin– Elmer Pyris 1 micro-calorimeter so as to compare the estimated and the measured results (complementary measurements). In the above mentioned temperature range it was obtained experimentally kðT Þ ¼ 0:1851 þ 3:383 104 T ;
Systems) and by using of thermocouples of K-type have confirmed that the heater generates spatially heterogeneous temperature field. Knowing such temperature distribution of the heater is essential to make possible to determine its effective diameter and consequently its effective heat flux density which in turn is necessary to estimate the thermophysical parameters of the sample materials by the inverse method. The results listed in Table 1 showed that there was not observed anisotropy with respect to the thermal conductivity of the investigated material (PMMA) and that the inverse method could be treated as an investigative tool making up alternative for classic measuring methods. Acknowledgements The authors would like to express their gratitude to the FLIR company for hire the ThermaCAM SC 3000 infrared camera.
ð13aÞ 3
cp ðT Þ ¼ 1339:750 þ 4:7814T þ 1:4060 10 T 2 :
ð13bÞ
Analysis of temperature distribution on the surface of the KHR 2/10 heater using the ThermaCAM SC 3000 infrared camera has confirmed an earlier estimated and assumed value of its effective diameter to be equal to / = 45.0 mm. The results of 2D parameter estimations for Plexiglas at a mean temperature T = 23.5 C and the complementary results are listed in Table 1. 6. Conclusions The results of investigations the temperature distribution of the thin-layer heater KHR 2/10 made by OMEGA using the ThermaCAM SC 3000 infrared camera (FLIR
References [1] J.V. Beck, K.J. Arnold, Parameter Estimation in Engineering and Science, first ed., John Wiley, New York, 1977. [2] T. Jurkowski, Y. Jarny, D. Delaunay, Estimation of thermal conductivity of thermoplastics under moulding conditions: an apparatus and an inverse algorithm, Int. J. Heat Mass Transf. 40 (17) (1997) 4169– 4181. ¨ zisik, H.R.B. Orlande, Inverse Heat Transfer Fundamentals [3] M.N. O and Applications, Taylor&Francis, New York, 2000. [4] S.V. Patankar, Numerical Heat Transfer and Fluid Flow, Hemisphere, Washington, DC, 1980. [5] J. Zmywaczyk, P. Koniorczyk, J. Terpiłowski, G. Zapotoczna-Sytek, Simultaneous estimation of thermal conductivity and heat capacity – 2D formulation, in: Proceedings of the 9th International Symposium on Temperature and Thermal Measurements in Industry and Science, 2005, pp. 1349–1355.