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Thermal properties estimation of polymers using only one Active Surface

Abstract

This work describes an experimental technique for obtaining, simultaneously, the thermal diffusivity and thermal conductivity of polymer materials. This technique uses experimental data from only one of the sample surfaces. It means functions using experimental and calculated temperature are defined. An objective function representing the eigenvalue phase angle is used to determine thermal diffusivity, while a least square error function is used for the thermal conductivity estimation. The sequential unconstrained optimization technique BFGS is used to calculate the search direction. In each case the golden section method is used in a one-dimensional search, followed by a polynomial approximation. A comparison with the flash method and the guarded hot plate method gives a deviation of 2.97 % and 0.63 % for thermal diffusivity and thermal conductivity, respectively, for a Polychloroethylene (PVC) sample. An uncertainty analysis is also presented.

Parameter estimation; thermal properties measurements; heat conduction; inverse problems


Thermal properties estimation of polymers using only one Active Surface

S. M. M. de Lima e SilvaI; T. H. OngII; G. GuimarãesIII

IE-mail: metrevel@mecanica.ufu.br IIE-mail: tiong@mecanica.ufu.br IIIFederal University of Uberlândia, School of Mechanical Engineering, Avenida João Naves de Ávila, S/N, 38400-902 Uberlândia, MG. Brazil. E-mail: gguima@mecanica.ufu.br

ABSTRACT

This work describes an experimental technique for obtaining, simultaneously, the thermal diffusivity and thermal conductivity of polymer materials. This technique uses experimental data from only one of the sample surfaces. It means functions using experimental and calculated temperature are defined. An objective function representing the eigenvalue phase angle is used to determine thermal diffusivity, while a least square error function is used for the thermal conductivity estimation. The sequential unconstrained optimization technique BFGS is used to calculate the search direction. In each case the golden section method is used in a one-dimensional search, followed by a polynomial approximation. A comparison with the flash method and the guarded hot plate method gives a deviation of 2.97 % and 0.63 % for thermal diffusivity and thermal conductivity, respectively, for a Polychloroethylene (PVC) sample. An uncertainty analysis is also presented.

Keywords: Parameter estimation, thermal properties measurements, heat conduction, inverse problems

Introduction

Thermal properties estimation of a and l, is extremely important in the identification of new materials, especially in the evaluation of insulation material performance. Many experimental methods have been used for determining these properties (Parker et al., 1961; Glatzmaier and Ramirez, 1985; Huang and Yan, 1995; Guimarães et al., 1995; Dowding et al., 1996; Lima e Silva et al., 1998 and Silva Neto and Carvalho, 1999). However, the availability of these methods decreases when the properties should be estimated simultaneously and non-destructively. One way to estimate simultaneously the thermal diffusivity and the thermal conductivity is using the parameter estimation technique proposed by Beck and Arnold (1977). This method involves the minimization of an objective function composed of experimental and calculated temperatures with respect to the unknown properties. The temperature data are from the surfaces of the material and the technique can be considered non-destructive once the properties can be obtained without damaging the medium under investigation. The Flash method (Parker et al., 1961) and the method in the frequency domain developed by Guimarães et al. (1995) are some other examples of surface methods that can also be cited. The main characteristic of these works is that the experimental apparatus needs two surfaces to estimate the properties simultaneously. For example, in the Flash method one surface is used to supply the heat flux, while the temperature response is measured in the opposite surface. In the technique of Guimarães et al. (1995) temperature and heat flux data are obtained from the two surfaces. In this work, the main objective is to develop an efficient experimental technique to estimate a and l simultaneously using only one access surface. The main interest in the development of methods that use only one access surface is the application to in situ cases, like in materials with large thickness. Wall constructed in buildings or soils are some examples of applications. Some papers, like Balageas (1989), Dowding et al. (1996) and Lima e Silva et al. (1998) that uses only one surface to estimate thermal properties can be cited. While Balegeas (1989) obtain only the thermal diffusivity, Dowding et al. (1996) and Lima e Silva et al. (1998) obtain simultaneously l and rCp and a and l, respectively. The method described by Dowding et al. (1996) is not easily extended experimentally to a field application due to the practicality of instrumenting the material. More work on the design and optimization of the experiments is required to apply the method in the field. To avoid this problem, Lima e Silva et al.

(1998) developed a method that uses two different objective functions from the same experimental data. The diffusivity estimation uses a correlation function, whereas to determine the thermal conductivity a square error function is used. Both correlation and error function are defined from experimental and calculated temperatures. The independent estimation of a and l, however, are also limited for cases where the correlation function reaches sufficiently high values for the thermal diffusivity estimation. The limitation of the procedure by Lima e Silva et al. (1998) is minimized by the technique proposed here. The correlation objective function is replaced by a function based on an eigenvalue phase angle defined from experimental and theoretical temperatures in the frequency domain. The perfect correlation of these signals with no delay between them, is an exclusive function of a. So, the minimization of the phase angle provides the alternative way for thermal diffusivity estimation. The other objective function used in Lima e Silva et al. (1998), the square error function, is maintained for thermal conductivity estimation.

In order to minimize these objective functions the sequential unconstrained optimization technique Broydon-Fletcher-Goldfarb-Shanno (BFGS) is used to calculate the search direction (Vanderplaats, 1995). In the unidirectional search the golden section method is followed by a polynomial approximation with the purpose of reaching better confidence and precision (Vanderplaats, 1984).

In fact, the method proposed in this work represents an alternate form of simultaneous estimation of a and l for materials of low thermal conductivity, can be used for a large range of thicknesses, and adapted for in situ application.

Nomenclature

Greek Symbols

Subscripts

Experimental Aspects

Figure 1 shows the experimental apparatus used to determine the thermal properties a and l of polymers materials. At time t = 0, the sample is in thermal equilibrium at T0. At this time, it is then submitted to a unidirectional and uniform heat flux on its upper surface (access surface). Two polymers samples were used, one of Polymethylmethacrylate (Perspex) and the other of Polychloroethylene (PVC), both with thickness of 50 mm and both with lateral dimensions of 305 x 305 mm.


The thermal diffusivity and thermal conductivity are, then, calculated from the mathematical model using the acquired data for the heat flux and temperature. Heat is supplied by a 22 W electrical resistance heater, covered with silicone rubber, with lateral dimensions of 305 x 305 mm and thickness 1.4 mm. A heat flux transducer with lateral dimensions of 50 x 50 mm, thickness 0.1 mm, and constant time less than 10 ms measures the heat flux input (Leclerq & Thery, 1983). The transducer is based on the thermopile conception of multiple thermoelectric junction (made by electrolytic deposition) on a thin conductor sheet. The electromotive force (emf) measured is proportional to the heat flux crossing the measured area. The evolution of temperature with time at the upper frontal boundary surface is measured using a surface thermocouple (type k). The silicone rubber for thermal contact has lateral dimensions of 305 x 305 mm and thickness 3 mm. The signals of temperature and heat flux are acquired by a data acquisition system HP Series 75000 with voltmeter E1326B controlled by a Pentium II personal computer.

Theory of the Method

Temperature Model

The temperature distribution in a finite one-dimensional, constant-property body which has an initial temperature distribution T0, is described by

subjected to the following boundary conditions:

and initial condition

where

f1(t) is the imposed heat flux and q1(t) is the upper surface temperature evolution as shown in Fig. 2.


The solution to the problem given by Eqs. (1)–(4) can be obtained through the use of Green's function (Lima e Silva, 1995). In this case q1 can be written as

where bm are the eigenvalues defined by bm = mp/ L, with m = 1, 2... It can be seen that the evolution f1(t) must be known to perform the integral analytically in Eq. (6). However, for experimental cases f1(t) assumes discrete data. In this sense, an approximation by polynomial or other function can be difficult to perform and a great number of intervals would be necessary to obtain a perfect curve fit. For this case, the curve adjustment with various types of functions represents a time consuming process and gives to rise complexity in the numerical and analytical integration. Besides, when the direct integration is possible, the difficulties of convergence of series are common. In this work the integrals are performed numerically using a technique based on the convolution theorem and the well-known numerical algorithm Discrete Fast Fourier Transform to solve the heat conduction problem without direct integration (Lima e Silva et al., 1998).

Eigenvalue Phase Angle Objective Function

The present work uses the input/output system idea to determine a. In this system, the input and output are given by the experimental and estimated temperature signals, respectively. Thus, once the true values of the thermal properties are known, under ideal conditions the temperature calculated by the theoretical model should be very close to the experimental temperatures. In other words, if a cross correlation between the experimental and calculated temperature is defined, its magnitude must go to one and its delay must go to zero. In this case, the behavior of the cross-correlation tends to the autocorrelation. This behavior can be analyzed in both time or frequency domain. Due to the signal process characteristic of the temperature, the frequency domain is used. The basic idea here is the observation that the delay between the experimental and theoretical temperature is an exclusive function of a. The correct value of the thermal diffusivity minimizes the delay in the time domain, and leaves the phase angle as zero in the frequency domain. This minimization is the way to determine the thermal diffusivity. This procedure is shown in Fig. 3


The output y(t) of the system, shown in Fig. 3, can be computed by the convolution integral

where h(t) = 0 for t < 0 and x(t-t) is the input with a time delay t. This is a general property of linear, physically realizable systems with constant parameters (Bendat and Piersol, 1986).

Using the convolution theorem (Bendat and Piersol, 1986), the frequency response function can be identified in the frequency domain by

where Y(ƒ) and X(ƒ) are the output and input, respectively, in the transformed f plane. Their values are found by application of the finite Fourier transform of the data y(t) and x(t), respectively (Bendat and Piersol, 1986). The Fourier transforms are performed numerically by using the Cooley-Tukey algorithms (Discrete Fast Fourier Transform) (Brigham, 1988). Multiplying Eq. (8) by the complex conjugate X(ƒ),

where Sxy is the cross-spectral density of x(t) and y(t), and Sxx is the autoespectral density of x(t). Equation (9) is more suitable to calculate H(ƒ) due to the more stable behavior of the spectral density, with ƒ. In polar form, H(ƒ) can be written as

where the modulus of H(ƒ) is

and the phase angle y(f) is

By observing Eq. (10) we note that if the phase angle, y(ƒ), goes to 0 (zero), then y(ƒ) goes to the value of |H(f)|. In this case, the output autoespectral density, Syy(ƒ), is nearly equal to the input autoespectral density Sxx(ƒ) (Eq. 11) once the theoretical and experimental temperatures tends to have the same behavior. As previously mentioned, the minimization of the square phase angle is used to adjust the a parameter once the phase angle behavior is only a function of this property. However, since the model used here to calculate the frequency response is not analytical, the unique dependence of a in the phase angle can be shown through an analysis of the square function of phase angle defined by

with respect to the variations of the values of a and l in a large region.

At this point, it should be noted that for the phase angle calculus, the cross-spectral density must be obtained as shown in Eq. (12). It can also be noted that two different terms define the cross-spectral density, a theoretical temperature, q1, obtained from performing the integral Eq. (6) and the experimental temperature data, qe1. In this analysis a Polychloroethylene sample (PVC), was chosen to obtain the experimental temperature. In this case the reference values of the thermal diffusivity and conductivity obtained by two different techniques: Flash method (Degiovanni and Laurent, 1986) and the guarded hot plate method (Ayaichia, 1992) was used. These values were obtained in the CETHIL laboratory (Centre de Thermique, Lyon-France) and are used here for comparison with the estimated values presented in this work (Lima e Silva, 2000). It means that even if we are interested in just simulating the behavior of the phase angle for different values of thermal properties, we must first specify the sample. Otherwise we can not obtain the experimental temperature, simulated or real.

Hence, using the reference values of a = 1.28 x 10-07 m2/s and l = 0.157 W/m.K, the eigenvalue phase angle objective function, Sy, can then be calculated for different values of a and l.

Figure 4 presents the evolution of Sy with variation of a. It can be noted the existence of an optimum value of a that minimizes Sy.


In order to guarantee that the thermal conductivity does not affect the Sy, this function is also calculated for different values of l (in the region of 0.1 W/m.K < l < 0.3 W/m.K). Figure 5 shows that any changes in the thermal conductivity will not affect the value of Sy. This confirms the choice of independent estimation of thermal diffusivity via phase angle minimization.


As already mentioned the thermal diffusivity could be estimated using the correlation principles in the time domain, specifically by considering an objective function that is defined by a correlation that by its term is sensitive to the temperature decay at the front surface. This procedure can be seen in the paper of Lima e Silva et al. (1998). The main disadvantage of this technique is the low sensitivity that the objective function has with the thermal diffusivity. Figure 6 shows a comparison between the correlation function, Sh, (Lima e Silva et al., 1988) and the function, Sy, an function of a. It can be shown that for the same region of a, the angle phase function, proposed here, is approximately 3 times more sensitive than the correlation function proposed by Lima e Silva et al. (1988).


Least Square Error Function

Once the thermal diffusivity value is obtained, the only parameter yet to be estimated in Eq. (6) is the thermal conductivity. This parameter can be estimated using the minimization of the error function defined by

where j represents the discrete time measurements.

Using the previous experiments, a new data treatment is employed to visualize the evolution of Smq with l (Fig. 7). The best value of l to minimize the function Smq is well characterized. This figure is presented using a logarithmic scale.


Uncertainty Analysis

There are inherent bias errors due to the limitations of the theoretical model and the uncertainty in the experimental values. In fact, the sample was supposed to be homogeneous and the heat flux to be unidirectional. In addition, numerical errors produced when calculating the distributions and using the fast Fourier transform must be taken into account.

A 2D transient model was formulated to simulate the influence of lateral heat losses on q (0,y). It can be shown that these heat losses are negligible for values of the ratio y/D smaller than 0.6, where D is the half-width of the sample and y is the distance from the axis of symmetry of the sample, taken along its upper surface. The sample lateral dimensions are 305 x 305 mm. The lateral dimensions of the heat flux transducers are 50 x 50 mm. When compared to the lateral dimensions of the sample, this gives a ratio y/D = 0.167, significantly lower than 0.6. The temperature evolution at the point (x,y) = (0,0) at the upper surface of the sample was calculated from 0 to 5000 s using the 2D transient model. The results were compared with the 1D solution for validation. The influence of lateral heat losses was smaller than 1.5 %.

The thermal contact resistance between the sensor and the sample was simulated by Guimarães et al. (1995). The authors showed that there is no temperature gradient in the heat transducer area. In addition, the influence of the thermal contact resistance for non metallic materials suitable for this kind of measurement is negligible (Guimarães et al., 1995).

The procedure used to estimate the error in the thermal properties determination is by calculus of the uncertainty in the measured variables (heat flux and temperature) and the calculus of numerical Fourier transform used to obtain the theoretical temperature, which is calculated using the experimental heat flux. Since the heat flux indirectly affects the uncertainty propagation calculus, once both objective functions (angle phase and least square) used the theoretical model temperature. Besides, as the determination of a is done in the frequency domain, the uncertainty in the use of the fast Fourier transform in the eigenvalue phase angle function is also calculated. The functions Sy and Smq are dependent on theoretical and experimental temperature signals. The uncertainty in Sy and Smq is obtained by considering the uncertainty in the experimental temperature, in the theoretical temperature and in the discrete fast Fourier transform algorithm used to calculate the spectral density. The hypothesis of linear propagation was used (Taylor, 1988). An analysis of propagation errors (Lima e Silva, 2000) shows that the uncertainty in the original data propagates in a conservative way, and the total uncertainty was found to be

with a confidence equal to 99.87 %.

Results and Discussion

Fifty independent runs for Polychloroethylene (PVC) and forty independent runs for Polymethylmethacrylate (Perspex) were realized. For both samples 4096 points were taken where the time intervals, Dt, were 1 s for Perspex and 0.8793 s for PVC. The time duration of heating, th, was approximately 150 s with a heat pulse generated by a 40 V (dc) supply. Figures (8 and 9 ) present, respectively, the heat flux and temperature signals in the upper surface of the PVC sample. As the heat flux and temperature curves for the Perspex sample have the same form of the PVC, these curves are presented for PVC sample only.


Table 1 presents the value estimated for a with 99.87 % confidence interval and standard deviation, s, of 0.0166 x 10-07 m2/s. Table 2 presents the value estimated of l also with 99.87 % confidence interval and standard deviation, s, of 0.00172 W/m.K. The sequential unconstrained optimization technique BFGS (Broydon-Fletcher-Goldfarb-Shanno) is used to minimize the objective functions (Eqs. 13 and 14). In addition, the golden section method is used in the one-dimensional search, followed by a cubic polynomial interpolation (Vanderplaats, 1984). The package Design optimization Tools (DOT) (Vanderplaats, 1995) is used to apply the optimization technique. Table 3 presents a summary of the simultaneous estimation of a and l for the PVC sample. In this table, the value obtained for a by using the Flash method (Degiovanni & Laurent, 1986) and the value obtained for l by using the guarded hot plate method (Ayaichia, 1992) are also presented.

Table 4 presents a summary of the simultaneous estimation of a and l for the Perspex sample. Similarly to the PVC sample, the sequential unconstrained optimization technique BFGS is used for Perspex. For this sample the reference values used for a and l were obtained by Miller & Kotlar (1993) and NPL (1991), respectively.

It can be observed that the results present good agreement with the respective reference values. The deviations for the PVC sample were less than 3 % and 1 % for the thermal diffusivity and thermal conductivity, respectively (Table 3). It can also be observed that these deviations increases for the Perspex sample (Tables 3 and 4). However, it must be cited that the two sample had their experiments made in two different ways. It means, that each sample used a different data acquisition system. For the Perspex sample the signals of heat flux and temperature are acquired by a data acquisition card (PCL-818) controlled by a 486-DX personnal computer, while for the PVC sample a more accurate data acquistion system HP (Series B 75000) was used. The random data uncertainty in the temperature measurement with the two systems can be cited just for a comparison. If the data acquistion card PCL-818 is used, the temperature uncertainty is on the range of ± 0.25 K, while the with the data acquistion system HP it would be expected a uncertainty on the range of ± 0.10 K.

A comparison between a and l for PVC with the reference values obtained by the Flash and guarded hot plate method validates the use of the method of eigenvalue phase angle in the frequency domain and the least square error in time domain.

Acknowledgements

The authors express their acknowledgment to Professors Martin Raynaud and Michel Laurent for their contribution. The authors would also like to acknowledge the financial support of Fapemig, CNPq and Capes.

Paper accepted December, 2002. Technical Editor: Atila P.Silva Freire

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  • Publication Dates

    • Publication in this collection
      18 Mar 2004
    • Date of issue
      Mar 2003

    History

    • Received
      Dec 2002
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