A dynamic thermal identification method applied to condutor ... - IEM

514 V. L. Borges et al.model can be associated with the dynamic model given by equation (2). In this case,using the same procedure described in the one-dimensional (1D) case in the previouswork [10] the equivalent thermal model can be obtained as the convolution productin the frequency domain. 1 ð f Þ 2 ð f Þ¼G þ 12 ð f Þ 1ð f Þ ð6Þwhere the variable f indicates that Fourier transform was applied to the variables , ,and G þ 12. A comparison of equation (6) with equation (1) givesG þ 12 ðt Þ ¼hðt Þ ¼ ½ Gðx 1, y 1 , z 1 , t Þ Gðx 2 , y 2 , z 2 , t ÞŠ ð7ÞIt can be observed that as i (t) e i (t) are obtained by discrete measurements, Fouriertransforms can be performed numerically by using the Cooley–Tukey algorithms(Discrete Fast Fourier Transform) for these data [7]. Therefore, the equivalent thermalsystem to the dynamic system can be represented by:Zð f Þ¼hð f Þ¼ 1ðfÞ 2 ðfÞðfÞ 1¼ Yð f ÞXð f Þð8Þwhere the function Z( f ), also called impedance generalized, is equivalent to theresponse in frequency H( f ) defined by equation (2). Observing equations (4) it can beconcluded that the frequency response H( f ) is strongly dependent of the thermalproperties, which means:Hð f Þ¼ 1ð f Þ 2 ð f Þ¼ function ð, Þð9Þ 1 ð f ÞIt should be observed that the transformed impedance in the f–x plane is a complexvariable which in a polar form can be written byZð f Þ¼Hð f Þ¼Hð f Þ ej’ð f Þð10Þwhere jHj and ’ represent, respectively the modulus and the phase factor of H. Thephase factor can be written by’ð f Þ¼arctan g ½Im Hð f Þ=Re H ð f ÞŠ ð11Þwhere Re H( f ) and Im H( f ) are the real and imaginary parts of H.The phase of frequency response H( f ) and the time evolution of superficialtemperatures, T 1 (t), and T 2 (t) are the experimental data used for estimation of thermaldiffusivity and thermal conductivity respectively.2.2. Thermal diffusivity estimation: frequency domainThe fact that the phase factor is just a function of the thermal diffusivity is the greatconvenience of working in the frequency domain. The basic idea here is the observationthat the delay between the experimental and theoretical temperature is an exclusivefunction of . Therefore, the minimization of an objective function, S p , based on the