Abstract

The traditional methods of thermal non destructive control are based on the analysis of the temperature profiles of surface, which are responses to a thermal excitations at the output of the studied system. Generally, these methods require an excitation of form well defined in order to execute the inspection. In this article, we present a method of thermal non- destructive analysis of thermal response of material containing a plan defect in the frequency domain, not necessitating any specific form of the input signal. This method is based on the concept of thermal input impedance, which may describes the dynamic behaviour of any system when the thermal limit conditions are known. The investigations by simulations implemented in this work, in the assumption of one dimensional condition, concern particularly the distribution of the thermal input impedance versus the frequency of excitation signal, and the defect position in the structure in question.

Keywords:
Diffusivity; Effusivity; Thermal impedance; Temperature; Heat flow; Non- destructive testing

1- Introduction

Thermal non-destructive testing is a method of inspection and analysis of homogeneity of various structures by simple acquisition of information at the possible access of system considered [1]. These thermal methods are very well adapted to the analysis of thermal transfer of structure containing defects (defect in thermal meaning) i.e. defects which will modify the thermal transfer process because of their presence [2]. To this effect, the introduction of the concept of thermal quadripole concept already well-known proves very interesting since it permits the introduction of the thermal input impedance quantity characteristic of the transfer process [3-7].

The study of the functional relation flux-temperature on the input face of a one- dimensional structure permits to introduce the concept of thermal impedance. This quantity characterizes entirely the system if the boundary conditions are isothermal or adiabatic type. In order to understand non-destructive testing in frequency domain we present in this paper simulations of input impedance of multilayer materials, where the second layer is assumed to be a defect in the structure.

2- Theoretical results

Let us consider a plane wall of thikness L, in which is inserted a layer of different thermophysical nature, of thikness e, representing a defect. The present study will relate to two types of defects, a resistive defect characterized by a great thermal resistance and a capacitive one characterized by a low resistance. The configuration considered is schematized by the figure below.

Fig 1: The wall containing a defect.

The studied wall consists of three layers:

-the region (I) which represents the first layer of the structure and located between the plans x=0 and x=l₁ is characterized by the parameters:

(1)

- the region (II) which represents the defect and located between the plans x=l₁ and x=l₂ is characterized by the parameters:

(2)

- the region (III) located between the plans x=l₂ and x=L is characterized by the parameters:

(3)

In the case of a resistive defect, we will choose the concrete as material of reference in which is inserted a layer of polystyrene. In the case of capacitive defect, we will take as basic material the Plexiglass in which is inserted a layer of copper.

For each type of defect, resistive or capacitive one, we have considered three different configurations depending on the defect position in material. Initially the defect was placed at 1mm from the entry, then at 5mm from the entry (in the middle) and finally at 1mm from the output face.

2-1- Thermal input impedance
The theory of the quadripole, well known, shows that [ 4 -5 ] for a three layer wall whose the output temperature is maintained constant, the expression of the thermal impedance is given by :

(4)

where

(5)

For our configuration Zc₁=Zc₃ (the same material)

3- Results of simulations and discussions

From the preceding theoretical results we carried out simulations of the input impedance by taking into account several configurations related to the geometrical and thermophysical characteristics of the studied structure

3-1- Space and frequency analysis
3-1-1- Resistive defect
The figure (2) represents the distribution of input impedance in module and phase within a material containing a resistive defect for spectral components ranging from 10^-6Hz to 1 Hz (frequency is taken as a parameter).

a. Position of defect : l1=1mm

b. Position of defect :l1=5mm

Position of defect : l1=8mm

Fig 2: Space and frequency evolution in module and in phase of the input impedance.

The space analysis shows that the presence of defect is highlighted by a discontinuity of the evolution of the input impedance distribution in the material for the three configurations and all the spectral components.

For the very low frequencies ranging from 10^-6 Hz to 10^-5 Hz, the module of the input impedance , when the plan of the observation is located at the entry, has the same value for the three configurations. This value decreases when the frequency increases. And when we are close to the entry of defect this value increases because of the difference of the effusivities which increase the temperature when the value of the flow is fixed.

Conversely, flow density decreases when the temperature is fixed.

Concerning the curves of phases, we find the same observations made on the curves of modules.

3-1-2- Capacitive defect
The figure (3) represents the distribution in module and phase of the input impedance within a material containing a capacitive defect for spectral components ranging from 10^-6 Hz to 1 Hz (the frequency is taken as a parameter).

The space analysis also shows a discontinuity of the input impedance evolution .

For very low frequencies ranging from 10^-6 Hz to 10^-4 Hz. At the entry of the system, we find a constant value of the input impedance module for the three configurations. This value is equal to the sum of resistances of the two layers surrounding a capacitive defect. More one approaches from the defect, more the value of the input impedance in module decreases. This is due to the difference of the effusivities which will bring a reduction in the temperature on the entry when the flow density is imposed, and conversely.

Concerning the curves of phases, we find the same observations made on the curves of modules.

a. Position of defect : l1=1mm

b. Position of defect : l1=5mm

Position of defect : l1=8mm

Fig 3: Space and frequency evolution in module and in phase of the input impedance.

4- Conclusion

This study shows the interest of the frequency analysis applied to the thermal non destructive testing. It shows the effect of the defect nature and its position in the considered structure on the heat transfer . Currently, the studies in our laboratory seek the applying of this method of analysis to configurations with two dimensional heat transfer.

Bibliography

1. J. Williams, S. Mansouri, S. Lee, J. N.D.T. (1980) 113-118.
2. A. Obbadi, S. Belattar, M. Beihaqi, A. Tmiri, accepted for the (ASPNDEM), 7-10 auguest (2001) Quebec, Canada.
3. S. Belattar, B. Duthoit, P. Thery, Congres Inter , Hanover, (USA), 1990.
4. H. S. Carslaw, J. C. Jaeger, 2^nd Ed. (1959) (Oxford: Clarendon).
5. S. Belattar, Thèse de doctorat, juillet 1992, Faculté des Sciences II Casablanca (Maroc).
6. A.Tmiri, S. Sahnoun, S. Belattar, S.F.T., Nantes (2001), Elsevier.
7. S. Belattar, A. Hamdoun, B. Duthoit, P. Thery, J. of Physics E. Sci. Instruments 2 (1991) 553-562
8. A. Tmiri, S. Belattar, A. Elouahlouli, Troisème Conférence Intérnationale sur les Mathématiques Appliquées et les Sciences de l'Ingénieur 23-25 oct. (2000) Casablanca.

Nomenclature

a: Diffusivity	m2s-1
b: Effusivity	j.k-1.m-2.s-1/2
w: Pulsation	rad.s-1
F: Flow density	w.m-2
Ze: Thermal impedance	k.m2 .w-1
Zc: Characteristic impedance	k.m2.w-1

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