Measurement of thermal radiative and conductive properties of semitransparent materials using a photothermal crenel method

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Journal of Quantitative Spectroscopy & Radiative Transfer 109 (2008) 620–635 www.elsevier.com/locate/jqsrt

Measurement of thermal radiative and conductive properties of semitransparent materials using a photothermal crenel method Zied Cheheb, Fethi Albouchi, Sassi Ben Nasrallah Ecole Nationale d’Inge´nieurs de Monastir, Laboratoire d’Etudes des Syste`mes Thermiques et Energe´tiques, Avenue Ibn Eljazzar, 5019 Monastir, Tunisia Received 1 March 2007; received in revised form 24 June 2007; accepted 8 August 2007

Abstract This paper deals with a theoretical and an experimental study allowing the measurement of the radiative and the conductive properties of semitransparent materials. The method consists of applying a crenel heat flux on the front face of a semitransparent sample and recording the temperature at the rear face using an open thermocouple junction. Parameter identification is performed by the minimization of the ordinary least-squares function comparing the measured and the calculated temperatures. This later is obtained from the thermal model describing the heat transfer by conduction and radiation in the medium. This model is built by the thermal quadrupole formalism. Measurements are reported on commercial glasses and plexiglass samples, and the used iterative algorithm is based on the Gauss–Newton method. r 2007 Elsevier Ltd. All rights reserved. Keywords: Semitransparent materials; Radiative and conductive transfer; Quadrupole method; Crenel heating excitation; Inverse problem

0. Introduction The semitransparent materials such as glasses, foams and granular materials are usually employed in various industrial sectors such as building, agronomy, energy and environment. In order to study the heat transfer in these materials, the radiative heat transfer must be taken into account and the coupled radiative–conductive heat transfer must be studied. Towards this objective, many works have been accomplished and both analytical and numerical formulations have been developed [1–6]. To carry on these studies, radiative and conductive properties of the studied materials must be determined. In the literature, the thermal characterization of materials, especially the determination of the thermal diffusivity, was the object of many papers, and different materials were studied [7,8]. In these studies, radiative parameters were fixed and only conductive characteristics were estimated. However, the estimation of both radiative and conductive properties was the subject of few works [9,10]. These radiative properties can be obtained from theoretical formulations or from transmittance and reflectance measurements. Corresponding author. Tel.: +216 95 56 31 27; fax: +216 73 50 05 14.

E-mail address: [email protected] (Z. Cheheb). 0022-4073/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.jqsrt.2007.08.005

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Nomenclature Ai, Bi, a Bi C cp e h I J k ss M N nr Npl p Qc qr Qt Rr r S t T T0 tc Tc Tm Tr X z Z

Ci, Di matrix transfer coefficients of model i thermal diffusivity (m2 s1) Biot number quadratique criterium heat capacity (J kg1 K1) sample thickness (m) heat transfer coefficient (W m2 K1) radiative intensity (W m2 str1) Hessian matrix absorption coefficient (m1) scattering coefficient (m1) global radiative parameter global radiative parameter refractive index Planck number Laplace parameter density of the heat flux (W m2) radiative heat flux (W m2) crenel excitation (W m2) dimensionless radiative resistance reflectivity surface (m2) time (s) temperature (K) reference temperature (K) heating time (s) calculated temperature (K) measured temperature (K) transmittance sensitivity coefficient space parameter (m) reduced sensitivity coefficient (K)

Greek symbols s¯ b be ei Z y l r sn t0 tr

Stephan–Boltzmann constant (W m2 K4) estimated parameter/s extinction coefficient (m1) emissivity of surface I measurement noise Laplace temperature (K s) thermal conductivity (W m1 K1) density (kg m3) standard deviation of the measurement noise optical thickness transmissivity

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Superscripts  * + 0

backward direction dimensionless forward direction black body

Subscripts 1 2 a c p r

front face rear face absorbing conductive participating radiative

Theoretical formulations are interested in the determination of the radiative properties of opaque surfaces [1] and porous media. Different models are presented in the literature for granular and fibrous materials [11–14]. Experimental determination of the radiative properties of semitransparent materials can be performed through transmittance and reflectance measurements. The experimental devices use monochromators [15,16], and Fourier transform infrared (FTIR) spectrometers [17,18]. The use of FTIR spectrometers improves the measurement precision and the acquisition time. In the preceding works [19,20], we presented photothermal methods for semitransparent powders’ characterization using crenel flux excitation. In these works, we have been interested only in the conduction heat transfer. In the present article, we study the possibility of the simultaneous experimental measurement of the conductive and the radiative properties of semitransparent solids at ambient temperature using a crenel flux excitation and an inverse method. Identification of the conductive and the radiative properties is based on the minimization of the ordinary least squares (OLS) by comparing the measured and the calculated temperature given by direct model. In this work, we present in the first part, the thermal models built with the quadrupole formalism for participating, purely absorbing and purely scattering semitransparent materials. In the second part, we present the sensitivity analysis and the identification procedure: initially, we study the identification of the thermo-physical parameters with simulated signals. Then, we present the identification using experimental measurements for commercial glass and plexiglass samples. Finally, we discuss the experimental results and the feasibility of the identification technique. 1. Theoretical formulation 1.1. Energy equation The sample is constituted by a plane parallel, gray and isotropic semitransparent medium. Black boundaries are considered (e1 ¼ e2 ¼ 1) in order to increase the radiative heat transfer within the material. The sample is initially assumed at uniform temperature T0. The heat transfer on the two faces with the surrounding environment is taken into account and it is represented by two heat transfer coefficients h1 and h2, while the lateral heat losses are neglected. Seeing that we work at ambient temperature, we suppose that the heat transfer coefficients on the two faces are the same (h1 ¼ h2 ¼ h). The front face of the sample is uniformly subjected to a constant crenel heat flux Qt(t): ( ) Qc 0ptptc Qt ¼ , (1) 0 tXtc where tc is the duration of the applied heating.

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The rear face temperature can be obtained by solving the one-dimensional energy equation, and dimensionless variables are introduced to simplify the calculations. The dimensionless time and space parameters are, respectively, defined as t* ¼ at/e2 and z* ¼ z/e. The dimensionless temperature is given by y ¼ T   T 0 ¼

T  T0 , ðQc =rcp eÞ

where (Qc/rcpe) is the adiabatic temperature. In the same way, radiative dimensionless variables are introduced such as the optical thickness t0 ¼ (k+ss)  e, and the Planck number N pl ¼ lk=4n2r sT ¯ 30 , where nr is the refractive index and s¯ is the Stephan–Boltzmann constant. The dimensionless intensity and the radiative flux are, respectively, defined by I  ¼ pI= 4n2r sT ¯ 40 and qr ¼ qr =4n2r sT ¯ 40 . Using these dimensionless variables, the energy balance equation can be written as qy q2 y t0 T 0 qqr ¼  ; qt qz2 N pl qz

0oz o1.

(2)

1.2. Quadrupole formulation In this part, we are interested in the development of the thermal models using the thermal quadrupole formalism. Two models are presented for two different media: a model for participating medium and a model for absorbing emitting medium. For the rest of this study, dimensionless quantities will be considered and the superscript ‘*’ will be omitted. 1.2.1. Participating medium In this case, a linear heat transfer is assumed. This assumption remains valid as far as the input energy is small and the temperature rise at the rear face is not important. We assume the two-flux approximation [2,21,22]. The radiative transfer is then expressed by the two coupled differential equations:   qI þ P y 4 þ  1þ þ MI ¼ NI þ , (3) 4 T0 qz   qI  P y 4 1þ  MI  ¼ NI þ  , (4) 4 T0 qz where M, N and P are global radiative parameters defined by M ¼ (2k+ss)  e, N ¼ ss  e and P ¼ 2ke. The radiative flux vector is given by [2,22] Z z P P qr ðzÞ ¼  2CC 1 enz þ 2CC 2 enz þ ðenz  enz Þ þ yðz0 Þ expðnðz  z0 ÞÞdz0 4n T0 0  Z z þ yðz0 Þ expðnðz0  zÞÞdz0 , ð5Þ 0

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where C ¼ ðM  NÞ=ðM þ NÞ, n2 ¼ M 2  N 2 and C1, C2 are two constants which depend on the radiative boundaries conditions [2,22]. Using the linear heat transfer assumption, we write   y 4 y 1þ 1þ4 . (6) T0 T0 The differential approximation [2,22] allows one to write q2 qr P qy . ¼ n 2 qr þ 2 2 T 0 qz qz

(7)

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The Laplace transform is applied to Eq. (3) and the temperature rise is obtained by solving the following equation:   2¯ d4 y¯ 2Pt0 2 d y (8)  pþ þn þ pn2 y¯ ¼ 0. dz4 dz2 N pl Using the thermal quadrupole method [2,22], we can write ! ! ! ¯ ¯ Ap Bp yð1Þ yð0Þ , ¼ ¯ ¯ C p Dp fð1Þ fð0Þ

(9)

¯ ¯ where yð0Þ and fð0Þ are the Laplace transform of the temperature and the flux on the front face and ¯yð1Þ and fð1Þ ¯ are those on the rear face. The parameters Ap, Bp, Cp, and Dp are the coefficients of the heat transfer matrix. 1.2.2. Absorbing emitting medium In the case of an absorbing emitting medium, the reduced radiative flux is given by 1 qr ðzÞ ¼ I þ ð0Þet0 z  I  ð1Þet0 ð1zÞ  ðet0 z  et0 ð1zÞ Þ 4 Z Z t0 z 0 t0 1 0 0 yðz Þ expðt0 ðz  z ÞÞdz0  yðz Þ expðt0 ðz0  zÞÞdz0 , þ T0 0 T0 z

ð10Þ

where t0 has been changed to t ¼ (3/2)t0 due to the kernel substitution [22]. The dimensionless intensities I+(0) and I(1) are given by the radiative boundary conditions [2,22]. In this case, Eq. (8) can be written as   2¯ d4 y¯ t2 2 d y (11)  pþ2 þt þ pt2 y¯ ¼ 0, dz4 dz2 N with N ¼ (3/2)Npl This Eq. (11) is ! ¯ yð0Þ ¼ ¯ fð0Þ

(due to the kernel substitution). solved using the thermal quadrupole formulation. The system is then represented by ! ! ¯ Aa Ba yð1Þ . (12) ¯ C a Da fð1Þ

For the two models, the heat losses are expressed by the Biot number, defined as Bi ¼

he . l

In this case, the system is represented by ! !  !   ¯ ¯ Ai Bi yð1Þ yð0Þ 1 0 1 0 . ¼ ¯ ¯ C i Di fð1Þ fð0Þ Bi 1 Bi 1

(13)

(14)

In the Laplace space, the rear face temperature is given by ¯ yð1Þ ¼

¯ fð0Þ , BiðAi þ Di þ Bi  BiÞ þ C i

(15)

¯ 0 is the Laplace transform of the crenel excitation and it is given by where f Q ¯ fð0Þ ¼ c ½1  expðtc pÞ, p

(16)

where Q(c) is the density of the crenel heating flux (Fig. 1). ¯ using The temperature in the usual space T(t) for the two models is obtained by the inverse transform of yð1Þ the numerical algorithm proposed by Gover–Stehfest [22].

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Q (t)

Qc

Measured temperature

h

h

tc

t

0

e

z

Fig. 1. Principle of the crenel method.

2. Sensitivity coefficients The most used criteria to optimally design an experiment are the sensitivity coefficients. Their reduced form shows the variation qT(t, b) of the model induced by a relative variation qbj/bj of the parameter. These reduced sensitivities are defined by Z ij ðtÞ ¼ bj X ij ðti ; bÞ ¼ bj

qTðti ; bÞ . qbj

(17)

In general, these coefficients must be large and uncorrelated with each other. The relative uncertainty on the estimated parameters is obtained from the variance to covariance matrix defined as R ¼ J 1 s2n ,

(18)

where s2n is the variance of the measurement noise and J is the (m  m) reduced Hessian matrix obtained from the reduced (n  m) sensitivity coefficients matrix J ¼ Zt Z,

(19)

where m is the number of parameters and n the number of measurement points. For the sensitivity analysis, nominal values are used for all parameters and they are presented in Table 1. Figs. 2(a, b) present the parameters’ reduced sensitivities, respectively, for a participating and a purely absorbing medium. Analysis of these reduced sensitivities shows that the two models have low sensitivity for the radiative parameters M, N and nr for the participating medium, and k and nr for the purely absorbing medium. This behavior is due to the fact that at low reference temperature, the heat transfer in the medium is made essentially by conduction. We remark that the two models are very sensitive to the heat capacity, the thermal conductivity and the sample thickness, whereas they are less sensitive to the Biot number and the radiative parameters. In the case of a participating medium, it can be seen that Zl, Zrcp and Ze do not have the same form and reach their maxima at different times. So, these three parameters are uncorrelated and can be estimated simultaneously. Indeed, the sensitivity coefficients ZM and ZN are linearly dependent. Consequently, they can not be identified simultaneously. Comparing the sensitivity coefficient magnitude of thermal and radiative parameters, we can say that the radiative parameters will be estimated with a low accuracy.

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Table 1 Input parameters for simulations of Section 2 Pulse of energy Q Crenel duration tc Heat capacity rcp Thermal conductivity l Reference temperature T0 Refraction index nr Emissivities e1, e2 Biot number Bi Sample thickness e Absorption coefficient ka

5000 J m2 5s 2  106 J m3 K1 1 W m1 K1 300 K 1 1 0.01 3 mm 100 m1

Fig. 2. (a) Sensitivity coefficients for a purely scattering medium. (b) Sensitivities for an absorbing emitting medium.

3. Parameter identification method The unknown parameters are identified by the minimization of the gap between the measured and the calculated temperature given by the direct model. This gap, represented by the minimization of the leastsquares functions, is given by JðbÞ ¼

n X

ðT m;i  T c;i ðbÞÞ2 ,

(20)

i¼1

where Tm is the measured temperature, Tc is the calculated temperature, b is the parameters vector and n is the number of measurements. The solution vector bk+1 at the iteration n+1 is given by bkþ1 ¼ bk þ ½X ðbk Þt X ðbk Þ1 X ðbk Þt  ½T m  T c ðbk Þ.

(21)

4. Parameters’ estimation with simulated measurements In this study, we are interested in the case of an absorbing emitting medium. In the Laplace space, the rear face temperature is a function of several parameters: T(t) ¼ f(t, tc, rcp, l, h, e, e1, e2, nr, k), with nr the refractive index and e1 and e2, respectively, the emissivities of the front and the rear faces. The simulated measurements are obtained by adding a measurement noise to the direct model response. The added noise is characterized by its standard deviation sn which is supposed to be constant. In this section, sn is taken equal to 0.005 K. In this part, the unknown parameters are the thermal conductivity (l), the heat transfer coefficient (h), the volumetric heat capacity (rcp) and the optical thickness (t). Parameters’ supposed to be known are grouped in

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Table 2a Fixed parameters for the estimations of Section 4 5000 J m2 5s 300 K 1.3 1 3 mm

Pulse of energy Q Crenel duration tc Reference temperature T0 Refraction index nr Emissivities e1, e2 Sample thickness e

Table 2b Results of the estimation with simulated measurements

t l (W m1 K1) Bi rcp (J m3 K1)

Exact values

Estimated values

Relative uncertainty (%)

0.45 1 0.035 2  106

0.459 1.066 0.036 2.131  106

3.38 0.32 2.93 1.29

Fig. 3. (a) Measured and calculated rear face temperatures. (b) Residuals.

Table 2a. The parameters identification is performed using the Gauss–Newton algorithm and the obtained results are presented in Table 2b. We remark that all parameters are estimated with good precision and the relative errors do not exceed 3% for conductive parameters and 4% for optical thickness. In order to analyze these results, we represent in Fig. 3a the comparisons between the measured and the calculated temperatures using the estimated parameters. We note a good agreement between curves. The residuals that represent the difference between the measured and the calculated temperature are presented in Fig. 3b. We remark that the residuals are random and centered on zero. 5. Experimental study 5.1. Experimental setup The experimental setup (Fig. 4) is constituted by a heat source, an enclosure, a signal amplifier, a numerical oscilloscope and an acquisition system. The sample is horizontally placed on a holder (Fig. 5a) and the lateral surface is insulated with glass wool. The sample front face is painted with a thin black layer and submitted to a

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Lamp P.C

Glass Sample Numerical oscilloscope Holder

Thermocouple Amplifier RS-232 Fig. 4. Experimental setup.

Glass wool Sample Holder

Sample

Semiconductor tablet

copper stems

Springs

Black paint layer

P-N junction Silver layer Fig. 5. (a) Studied sample. (b) Thermocouple. (c) Studied sample and the P–N junction.

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uniform crenel heat flux delivered by a lamp of power 1200 W m2. The rear face of the sample is placed on a bismuth tellurium thermocouple with separated contact (Fig. 5b) having a high thermoelectric capacity (360 mV K–1). The signal delivered by the thermocouple is very weak, so we use an amplifier with low noise to obtain a measurable signal. Finally, the signal recorded on the numerical oscilloscope is sent to a computer via an RS-232 output. Measurements are performed on three different semitransparent samples. The characteristics of the studied samples are grouped in Table 3: two commercial glasses and plexiglass. In order to obtain the same conditions of the theoretical model (opaque, diffusely emitting and reflecting boundary), a black paint layer is applied on the two faces of the three samples. To obtain an electric conductor, we apply a thin silver layer with a diameter slightly superior to that of the thermocouple junction (Fig. 5c). 5.2. Parameters’ estimation with experimental measurements 5.2.1. Estimation of four parameters In this section, the unknown parameters are the thermal conductivity (l), the volumetric heat capacity (rcp), the Biot number (Bi) and the optical thickness (t). The emissivity of the black painted faces was measured by an infrared camera associated with a specular hemisphere [20]. Refraction indexes are obtained by ellipsometric measurements using a GESP5 SOPRA spectroscopic ellipsometer. Some parameters are fixed on nominal values presented in Tables 4a–c respectively, for the clear glass, the gray glass and the plexiglass samples.

Table 3 Studied samples properties

Diameter (mm) Thickness (mm) Color

Glass

Glass

Plexiglass

25 3 Clear

25 6 Gray

25 3 Clear

Table 4a Input parameters for the estimation for the clear glass Pulse of energy Q Crenel duration tc Reference temperature T0 Refraction index nr Emissivities e1, e2 Sample thickness e Time step

1200 W m2 3s 288 K 1.51 0.94 3 mm 0.05 s

Table 4b Input parameters for the estimation for the gray glass Pulse of energy Q Crenel duration tc Reference temperature T0 Refraction index nr Emissivities e1, e2 Sample thickness e Time step

1200 W m2 10 s 305 K 1.51 0.94 6 mm 0.1 s

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Table 4c Input parameters for the estimation for the plexiglass 1200 W m2 5s 305 K 1.3 0.94 3 mm 0.05 s

Pulse of energy Q Crenel duration tc Reference temperature T0 Refraction index nr Emissivities e1, e2 Sample thickness e Time step

Table 5a Results of estimation of four parameters for the clear glass

t l (W m1 K1) Bi rc (W m3 K1)

Estimated values

Relative uncertainty (%)

1.043 0.786 0.094 1.786  106

24.28 2.37 0.99 2.01

Table 5b Results of estimation of 4 parameters for the gray glass

t l (W m1 K1) Bi rc (W m3 K1)

Estimated values

Relative uncertainty (%)

2.178 0.895 0.136 1.975  106

14.12 2.64 1.1 2.23

Table 5c Results of estimation of 4 parameters for the plexiglass

t l (W m1 K1) Bi rc (W m3 K1)

Estimated values

Relative uncertainty (%)

2.19 0.21 0.32 2.3  106

3.61 1.41 1.51 1.05

The results of the simultaneous estimation of the four parameters for the three samples are presented in Tables 5a–c. In spite of the low sensitivity of the model for the optical thickness, the simultaneous estimation of the four unknown parameters is succesfully performed with different precisions. The thermal conductivity, the heat capacity and the Biot number are identified with good accuracies. The relative uncertainties do not exceed 3%. For plexiglass, the estimation of the radiative parameter is done with good accuracy in spite of the low reference temperature. Although, the relative uncertainty on the estimation of the optical thickness is large for glass samples, it is 14.1% for gray glass and 24.2% for the clear glass sample. These large uncertainties are due to the low sensitivity of the model to the optical thickness at low temperatures and its large sensitivity for the thermal conductivity and the volumetric heat capacity. This behavior is due to the low reference temperature. The quality of the estimation is analyzed, by comparing the experimental response and the calculated temperature using the estimated values. Figs. 6a–c present comparisons between the measurements and the

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Fig. 6. (a) Comparison between measured and calculated temperatures for the clear glass. (b) Comparison between measured and calculated temperatures for the gray glass. (c) Comparison between measured and calculated temperatures for the plexiglass.

calculated temperatures, respectively, for clear glass, gray glass and plexiglass samples. For the three cases, the curves show a good agreement. Figs. 7a–c present the residuals for the three samples. These residuals represent the difference between experimental and calculated temperatures. In the three cases, we remark that the residuals are centered on zero and they do not present any deviations or oscillations. 5.2.2. Estimation of two parameters From the qualitative analysis of the sensitivity coefficients, we have noted that the sensitivity of the model for the thermal conductivity and the volumetric heat capacity are largely higher than its sensitivity for the optical thickness and the Biot number. To check the estimation uncertainty of the optical thickness, we propose to fix the values of l and rcp at the optimal estimated values in the previous section (Tables 5), and to estimate the two other parameters (the Biot number and the optical thickness). The results of these estimations are presented in Tables 6a–c, respectively, for the clear glass, the gray glass and the plexiglass samples. We remark that the relative uncertainty on the estimated optical thickness decreases from 14.1% to 10.3% for gray glass and from 24.2% to 13.9% for clear glass. In the same way, the relative uncertainty of the Biot number decreases from 1.1% to 0.3% in the case of gray glass, from 1% to 0.3% in the case of clear glass and from 1.51% to 0.64% for plexiglass. This behavior is due to the fact that the two estimated parameters have comparable sensitivities especially at short times when the sensitivity to the optical thickness is maximum.

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0.02

0.04

0.015

0.03

0.01

0.02

0.005

0.01

Residuals

Residuals

632

0 -0.005

0 -0.01

-0.01

-0.02

-0.015

-0.03

-0.02 0

10

20 time (s)

30

40

-0.04 0

50

100

150

time (s)

0.03 0.02

Residuals

0.01 0 -0.01 -0.02 -0.03 -0.04 0

10

20

30 time (s)

40

50

60

Fig. 7. (a) Residuals between measured and calculated temperatures for the clear glass. (b) Residuals between measured and calculated temperatures for the gray glass. (c) Residuals between measured and calculated temperatures for the plexiglass. Table 6a Results of estimation of 2 parameters for clear glass

t Bi

Estimated values

Relative uncertainty (%)

0.964 0.136

13.98 0.37

Table 6b Results of estimation of 2 parameters for clear glass

t Bi

Estimated values

Relative uncertainty (%)

1.615 0.136

10.03 0.31

6. Direct measurement of the absorption coefficient In this section, the estimated method values (by the inverse method) of the radiative absorption coefficient are compared with arithmetic mean values measured by spectrometric normal transmittance measurements.

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Table 6c Results of estimation of 2 parameters for plexiglass

t Bi

Estimated values

Relative uncertainty (%)

2.15 0.37

2.8 0.64

6.1. Formulation In the case of a normal incidence, the reflectivity r is given by [16]   nr  1 2 r¼ . nr þ 1

(22)

Inside the medium, the incident beam is progressively attenuated by respective refraction on the two faces. This attenuation is due to the radiation absorption by the medium. The transmitted beam is described by the Beer’s law in the case of a normal incidence [16]: tr ¼ expðkeÞ.

(23)

The sum of all the transmitted beams is the transmittance, given by [16] Tr ¼

tr ð1  rÞ2 , 1  r2 t2r

(24)

Tr ¼

ð1  rÞ2 expðkeÞ , 1  r2 expð2keÞ

(25)

so

Eq. (25) allows the calculation of the radiative absorption coefficients using normal transmittance measurements. 6.2. Measurements The normal transmittance is measured by a Nicolet-560 FTIR spectrometer having a spectral range between 1.3 and 7.5 mm. These spectral measurements are performed by a Michelson interferometer and the samples are placed in the normal direction between the source and the detector. The measured spectral transmittances are represented in Figs. 8a–c for clear glass, gray glass and plexiglass. 6.3. Comparison of the results Since the estimated values of the radiative absorption coefficients (Section 5) are mean values (Rosseland mean absorption coefficients), we compare these values with the arithmetic mean absorption coefficient calculated from normal transmittance measurements. The comparison between these values is presented in Table 7). A good agreement between the values obtained by the two methods is remarked. The difference is 2.8% in the case of the clear glass, 10.4% for the gray glass and 11.6% for the plexiglass. 7. Conclusion This work has shown that using a crenel heating excitation and the Gauss–Newton method, the thermal and radiative parameters of a semitransparent material can be estimated with good accuracy.

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50

70 Normal transmittance (%)

Normal transmittance (%)

80

60 50 40 30 20 10 0 0

1

2

3

4

5

6

7

40 30 20 10 0 0

1

2

Normal transmittance (%)

Wave length (µm)

3 4 5 Wave length (µm)

6

7

100 90 80 70 60 50 40 30 20 10 0 0

1

2

3 4 5 Wave length (µm)

6

7

Fig. 8. (a) Normal transmittance for the clear glass. (b) Normal transmittance for the gray glass. (c) Normal transmittance for the plexiglass.

Table 7 Comparison between spectroscopic measurements and estimated values of the radiative absorption coefficients Sample

Estimated values (inverse method) (m1)

Relative uncertainty (%)

Arithmetic mean values (normal transmittance measurements) (m1)

Relative uncertainty (%)

Clear glass Gray glass Plexiglass

231.86 242.73 488.27

15.98 12.03 4.8

238.56 217.28 431.33

3.46 2.56 2.72

In the majority of work, radiative and conductive properties are separately estimated and only conductive or radiative heat transfer is considered each time. In this study, combined conductive and radiative heat transfer was taken into account. Two thermal models have been developed using the thermal quadrupole formalism. The parameters’ estimation possibility is analyzed from a sensibility study. The estimated values of the radiative parameters are compared with those obtained by spectroscopic normal transmittance measurements, and a good agreement between theses values is shown.

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