NDT.net Aug 2003, Vol. 8 No.08 |

Laboratoire d'Energétique et de Traitement du signal

U.F.R Instrumentation et Traitement du signal

Faculté des Sciences, 24000 El Jadida, Maroc

Corresponding author. E-mail :Sahnoun@ucd.ac.ma

S. Belattar,

Laboratoire d'Energétique et de Traitement du signal

U.F.R Instrumentation et Traitement du signal

Faculté des Sciences, 24000 El Jadida, Maroc

Email: ssahnoun@hotmail.com

Classical thermal non-destructive testing is a technique for obtaining surface temperature profiles on a structure, and subsequently relating this information to some imperfection within the structure. The steady temperature responses are analysed assuming a two dimentional heat flow conduction. The main aim of the authors is to present a technique for characterizing the presence of a circular defect in any plan wall by use of a numerical model. The results were analysed taking into account the most influencial parameters such as:postion and physical nature of the defect.

The processes of heat transfer seem to pervade all aspects of our life. These processes that occur, for example, in engineering equipment, in the heating and air conditioning of building, play a vital role that can be observed in a great variety of practical situations. In all these cases, predictions offer economic benefits because no experimental study can be expected to measure the distributions of all variables over the entire domain. For this reason, even when an experiment is performed, there is great value in obtaining a companion computer solution to supplement the experimental information.

Among the different methods of predictions, the numerical solution based on finite differences method seem quite promising. Composite materials are of increasing interest to the industry and their performance characteristics are desirable. Often, an internal anomaly within these structures modifies strongly their physical properties. Thermal non-destructive testing is able to reveal the presence of a defect without damaging the specimen [1-3]. It employs heated surface temperature variations caused by delaminations, cracks, voids, corroded regions, etc. [4]. Numerical analysis is useful to consider different defect geometry and determine their detectability without the expense of making and testing the corresponding specimens [5,6]. The planar size, depth, and adimensional thermal resistance of the detected defects are the parameters of interest. The two-dimensional model offers more flexibility for the defect geometry, so the thermal response from a defect with finite width and thickness can be obtained and studied.

In this paper, a numerical analysis is applied to simulated thermal responses in order to obtain the description of the inclusion-type defect in a flat panel. A two-dimensional finite difference model in Cartesian coordinates is used. The simulation process involves two sequential steps: solving the two dimensional transient heat diffusion equation for a sample with a circular defect, then estimating the defect location and size from the surface spatial and thermal distributions, calculated from the simulation. Thermal properties of the inclusions are varied to represent the defects of different nature.

The heat equation is a direct result of the gradient-driven flux law expressed by the Fourier law and of the conservation principle implied by the First Law of Thermodynamics.

The governing differential equation for unsteady two- dimensional conduction in homogenous material without thermal source is :

(1) |

where the quantities rc and k stand for the heat capacity per unit volume and conductivity of the material respectively.

The task is to find the temperature distributions at all subsequent time. The strength of a numerical method lies in the replacement of the differential equation by a set of algebraic equations, which are easier to solve. The solution obtained by the control-volume method will always give a perfect overall balance of energy for the whole calculation domain. It consists of subdividing the entire calculation domain into a number of control volumes over which the heat conduction equation is integrated [7].

For numerical simulation the initial and boundary condition must be defined as well as the material of the sample, material of the defect and sample geometry. The sample shape is rectangular with width x_{0}=4cm and length y_{0}=6cm. A uniform heat flow F_{0}is imposed on the front face of the sample while the rear face is maintained at constant temperature T_{0}. Thus, the initial and boundary conditions are

(2) |

(3) |

(4) |

(5) |

The aim of the research is to analyze the possibility of obtaining the description of internal defect structure by plotting steady isothermal lines. The defects have a circular shape (diameter d=1cm), placed at different distances (x_{1}=5, 15 and 25 cm) from the front surface. The coordinate y1 if the defect center is taken constant (Fig.1). The defect is considered resistive when its thermal conductivity is greater than that of the material above it. On the opposite, it is called capacitive when its conductivity is weaker than that of the reference material.

Fig 1: Cross section of sample with circular defect. |

**3.1. Resistive defect**

On Figs. 2, 3 and 4, are reported the steady isothermal lines of a heated rectangular wall of concrete (k=1.7w/m.k, c=827 J/(kg.k), r =2250 kg/m^{3}) [1], containing polystyrene of circular form (k=0.041/m.k, c=1310 J/(kg.k), r =14.7 kg/m^{3}) [1], playing the role of resistive defect. For a start, one sees that these isothermal lines fit perfectly the defect form. However, in the absence of defect, the isothermal lines will be plans that are perpendicular to the thermal propagation direction defined by the heat flow vector. In the presence of the defect, the thermal flow lines (circumvent) avoid meeting the obstacle represented by the resistive defect.

Fig 2: Steady isothermal lines of a concrete flat sample with a circular resistive defect at 5 mm from the front surface. | Fig 3: Steady isothermal lines of a concrete flat sample with a circular resistive defect at 15 mm from front surface. |

In a x plan situated at the entry of the resistive defect (e.g. opposite face of the wall entry ) the temperature increases as one approaches the defect outline. On the other hand , in a x plan located at the resistive defect exit the temperature decreases as one approaches the defect outline.

The size ( diameter of the defect ) and the position of the defect act considerably on the extent and the position of the isothermal line distortion. That is a good sign for collecting some information on the size and the position of the defect in the wall.

On the other hand, the distribution of the adimensional thermal resistance versus wall coordinates x and y is represented on Fig. 5.

Fig 4: Steady isothermal lines of a concrete flat sample with a circular resistive defect at25 mm from front surface. | Fig 5: Adimentional thermal resistance of a concrete flat sample with a circular resistive defect at 5 mm from front surface. |

As x increases (in the direction of the propagation of heat) , the distribution of the adimensional thermal resistance decreases and varies on both sides of an inclined plan. This inclined plan represents the distribution of the adimensional thermal resistance of the wall without defect. One concludes that the adimensional thermal resistance seen at the front face of the defect (in the direction of the propagation of heat), is larger than that seen at the rear face.

**3.2. Capacitive defect**

The capacitive defect in question here, is simulated by the window glass (k=0.78w/m.k, c=778 J/(kg.k), r =2550 kg/m^{3}) [1], incorporated in a plexiglass wall (k=0.19w/m.k, c=1442 J/(kg.k), r =1165 kg/m^{3}) [1],. Thermal diffusivities of these materials are close. An examination of Figs. 6, 7 and 8 shows that, the isothermal lines are tangent with the defect, and in a x plan situated at the entry of the capacitive defect (e.g. opposite face of the wall entry), the temperature decreases as one approaches the defect outline.

Fig 6: Steady isothermal lines of a concrete flat sample with a circular capacitive defect at 5 mm from front surface. | Fig 7: Steady isothermal lines of a concrete flat sample with a circular capacitive defect at 15 mm from front surface. |

On the other hand , in a x plan located at the capacitive defect exit the temperature increases as one approaches the defect outline.

The heat flow lines converge towards the defect. On can say that the capacitive defect, in the opposite of the resistive defect, pumps heat from its vicinity.

In addition, the distribution of the adimensional thermal resistance (Fig. 9), is quiet different from that of the case corresponding to a resistive defect. One can say that, at the front face of the capacitive defect, the adimensional thermal resistance is weaker than that seen at the rear face.

Fig 8: Steady isothermal lines of a concrete flat sample with a circular capacitive defect at 25 mm from front surface. | Fig 9: Adimentional thermal resistance of a concrete flat sample with a circular resistive defect at 5mm from front surface. |

The numerical analysis brings invaluable information related to non-destructive testing problems. The results presented in this paper show the possibility for testing thermally materials by interpreting, graphically, the steady isothermal lines. Two typical categories of defect are considered. In presence of the resistive defect, the thermal flow lines of the wall avoid meeting the obstacle represented by this one ( resistive defect). In the opposite case, the thermal flow lines, converges towards the capacitive defect.

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