NDT.net Aug 2003, Vol. 8 No.08 |
Classical thermal non destructive testing is a technique for obtaining surface temperature profile in a structure. It can reveals the presence of a flaw by observation of the temperature distribution profile anomaly which the flaw produces on the surface of the structure. In fact, a flaw inside the structure will generally alter the heat flow through the structure due to the difference in its heat transfer properties and those of the unflawed structure. If the heat flow pattern is sufficiently altered, a difference of temperature on the structure in the unflawed and the flawed regions will be observed. In consequence, a difference in thermal transfer function, defined in frequency domain like the report of temperature or flux variations at the input and the variations of the same quantity at the output of the structure, between the areas with and without flaw will be observed too. The frequency analysis of this thermal transfer function permits to characterize the structure to be inspected.
The simulations realized according to the position of the defect in the structure and the frequency treat several situations depending on the geometry of the structure in question. These results are compared to those obtained by numerical method of volumes of control. The two methods applied are complementary in the analysis and the characterization of a defect in a given structure. We limit our study, in this paper, to a one-dimensional heat conduction assumption.
Key words:
NDT, thermal transfer function, numerical method.
The thermal non destructive testing is a method of inspection and analysis of the various structures homogeneity by the acquisition of information at the possible system accesses, and the establishment of a relation between this information and the imperfections that they contain. The presence of a defect in a given structure, generally, changes thermal flow value compared to a structure devoid of any defect. This is due to the thermophysical characteristics difference between the structure and the defect. If this difference is large, the variation of the surface temperature profile will be significant and can be detected. The analysis of this temperature difference and its connection to possible anomalies in the structures was largely discussed in literature [ 1,2,3 ]. In the preceding articles [ 4,5 ] we introduced and showed the interest of the thermal impedance function concept for plane materials inspection used in civil engineering. The study was primarily centered on the effect of the defect position . In this article, we propose to study, in addition to the defect position, the effect of its dimensions and thermophysical properties. For this, we use the transfer function concept whose advantage is to represent an independent formulation of the thermal excitation nature, and to be better adapted to the complex signals cases i.e. measurements in-situ [ 4 ], and then the numerical method of control volumes (method of Patankar) [ 6]. This method, allows the evaluation of the output flow of the homogeneous or multi-layer structures according to the position and thickness. The assumption of a one-dimensional conduction permits the independent evaluation of temperatures or transfer function in the areas with or without defect. This assumption is valid when the transversal dimensions are rather large compared to the wall thickness. The inclusion defect can be an unspecified material, whose characteristics are different from the wall material in question, as an insulating layer, a blade of air...
2.1 Used model
The used model, shown in figure 1, is represented by an homogeneous plane wall of thickness l, conductivity k_{1}, effusivity b_{1} and diffusivity a_{1}, whose output temperature at (x=l) is supposed to be constant. It contains a defect of thickness x_{2}, placed at x_{1} from the input face (x=0) and x_{3} from the output one (x=l), perpendicularly to the heat propagation direction. The defect is characterized by the conductivity k_{2}, effusivity b_{2} and diffusivity a_{2. }The whole wall-defect is supposed to be in initial thermal balance at temperature T_{0}.
The structure can be decomposed into two areas I and II. The area I represents the homogeneous part, and area II the part containing the defect. Taking into account this difference between the two areas, the transfer process of heat will be developed differently, which will generate a different surface temperature profiles, or transfer function in flux, from one area to another.
Fig 1: Plane wall containing a defect. |
Area I
The temperature evolution in area I of thickness l, is solution of the Fourier’s equation (1) to which we associate the Fourier‘s law (2)
(1) |
(2) |
the boundary (3), (4) and initial (5) conditions,
(3) |
(4) |
(5) |
Area II
The Fourier's equation (6) and Fourier's Law (7)
(6) |
(7) |
the continuity of the temperature (8) and flow (9) at the interfaces of layers 1-2 and 2-3
(8) |
continuity of flow
(9) |
boundary conditions at the input (10), and the output (11) and initial condition (12),
(10) |
(11) |
(12) |
2.2 Method of volumes of control
The numerical method used is that of volumes of control for a fixed and non regular grid [ 6 ]. It consists in cutting out the model of figure 1 in a group of small volumes of controls, on which are integrated the preceding equations of heat transfer.
2.3 Concept of thermal transfer function
By application of Fourier’s transform of equations (1), (2), (3) and (4) we will find a linear relation, well known [ 5 ], between the output variable, and the input ones for the homogeneous part (area I),
(13) |
where q(0,w) and q(l,w) are the temperature variation Fourier’s transform at x=0 and x=1 et the flow variation Fourier’s transform at x=0 and x=1, and where
(14) |
in the case of three-layers systems in perfect contacts, identical relations are obtained, then we show that the transfer functions in flux of areas I (15) and II (16) are given by
(15) |
(16) |
(17) |
obtained when the variations in temperature at the output surface can be neglected.
The comparison of transfer function in flux, or the output flux, of the area II (area with defect), to those of the area I (area without defect) permits the detection of a possible characteristic anomaly of the presence of the defect.
3.1 Effect of the position and thickness on the transfer function in flux and on the output flux:
We have simulated the transfer function in module and phase of a plane structure of concrete containing a defect materialized by a plate of polystyrene as indicated in figure (1).
3.1.1 Effect of the defect position:
The figures (2-a) and (2-b) respectively represent the module and the phase of the transfer function in flow for three positions.
Fig 2: Effect of the defect position on (a) the module, and (b) the phase. |
The curves show that the presence of the resistive defect plays a role of filtering of the spectral components. Indeed, more the defect is far from the entry of material, more the filtering is accentuated, that means the components high frequencies are amortized before reaching the output face of material.
In order to highlight this effect of position, we have recorded the noted frequencies F_{P}, corresponding to the beginning of the steady state statement. The table (1) recapitulates the obtained results from calculation. However the effect of the defect position on the phase remains negligible.
The same phenomenon occurs in temporal evolution of the output flow.
Defect position (mm) | Homogeneous | 2 | 6 |
Fp (Hz) | 5,8.10^{-5 } | 2,9.10^{-5 } | 1,3.10^{-5 } |
Table 1: |
These results permit to choose the frequencies of excitations which will cross the studied structure without being completely amortized in it, in order to inspect a given material. The value of these frequencies depend on dimension and physical nature of material in question.
For the same configuration, the responses of the system in temporal domain are represented in the figure (3) where the curve (a) corresponds to an homogeneous structures, and the curves (2) and (3) to a structure containing a defect, placed at 2 and 6mm from the entry, respectively. The curves show that more the defect is placed far from the entry of the structure more we have a delay in the establishment of the steady regime. The system behaves as if its time constant takes a great value.
Fig 3: Effect of the position defect on the output flow. |
These results can be represented by the Fourier's time in the table (2).
Defect position (mm) | Homogeneous | 2 | 6 |
Fourier's time (s) | 434 | 966 | 2128 |
Table 2: |
3.1.2 Effect of thickness of defect:
The figures (4-a) and (4-b) respectively represent the module and the phase of the transfer function in flow for various thicknesses.
Fig 4: Effect of the defect thickness on (a) the module, and (b) the phase. |
Fig 5: Effect of the thickness defect on the output flow. |
Equally, in order to highlight the effect of thickness, we recorded the noted frequencies Fp as previously. The table below recapitulates the obtained results.
However the effect of the defect thickness on the phase remains negligible.
The same phenomenon occurs in temporal evolution of the output flow.
Thickness defect (mm) | Homogeneous | 1 | 2 |
Fp (Hz) | 5,8.10^{-5 } | 1,5.10^{-5 } | 8.10^{-6 } |
Table 3: |
Thickness defect (mm) | Homogeneous | 1 | 2 |
Fourier's time (s) | 392 | 1834 | 3234 |
Table 4: |
The results presented in the frequency or the time domain show the interest of the analysis of the thermal signals of the systems for a non destructive characterization. Simulations realized show clearly the influence of different parameters related to the inclusion defect. A defect is easily detected when it is close to the inspection face, or has a thickness rather significant. A defect placed towards the input face will be difficult to detect particularly when we cannot reverse the inspection face of material. This analysis can be used in the establishment of the adequate TNDT equipment parameters. The used numerical method is actually developing in our laboratory to tackle a two or three dimensional defect.
a: Diffusivity m^{2}.s^{-1}
b: Effusivity j.k^{-1}.m^{-2}.s^{-1/2}
k: Thermal conductivity w.m^{-1}.k^{-1}
w: Pulsation rad.s^{-1}
c: mass heat j.kg^{-1}.k^{-1}
r: Volumetric mass kg.m^{-3}
l: wave length m
q: Temperature variation k
f: Flow density w.m^{-2}
H_{f}: Thermal impedance k.m^{2}.w^{-1}
x: Thickness
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