National Seminar & Exhibition on Non-Destructive Evaluation, NDE 014, Pune, December 4-6, 014 (NDE-India 014) Vol.0 No.6 (June 015) - The e-journal of Nondestructive Testing - ISSN 1435-4934 www.ndt.net/?id=1788 Electro-Thermal Modeling of Thermal Wave Imaging of Carbon Fibre Composites Krishnendu Chatterjee 1,a and Suneet Tuli 1,b 1 Centre for Applied Research in Electronics. Indian Institute of Technology Delhi, India a krishnendu.chatterjee@gmail.com b suneet@care.iitd.ac.in Abstract. To model heat flow inside a material, as often required by thermal NDE, the test piece is divided into finite elements, each of which obeys the basic law of heat conduction and retention. The size of these elements plays a crucial role in determining the speed and accuracy of the simulation. This paper proposes an approach towards the optimization of slice thickness using mathematical analogies between heat and current flow. The capabilities of this optimized model is demonstrated with thermal wave imaging of carbon fibre composites. A test piece with artificially created circular defects at various depths is modeled. Structural anisotropy and the effect of paint layer, which is often applied for better contrast, are also discussed. Introduction Electro-thermal modeling is the study of the thermal behavior of an object by exploiting the problem s mathematical similarities to that of an electrical network of resistors and capacitors. It uses the laws of heat transfer, charge transfer, heat retention, and charge retention, to predict the thermal evolution of a body. Fig. 1 and Table 1 summarize the theory behind the model. Fig. 1b shows a slab of thicknessl and surface area A subjected to a constant temperature difference of T across the surface, whereas Fig. 1a shows an equivalent RC-network subjected to a constant voltage difference V. Table 1: Analogy between electrical and thermal models Electrical model Thermal model Analogy Law of resistor Current (I) Heat flow (H cnd ) Voltage difference ( V ) Temperature difference ( T ) l/ka I = (1/ ) V H cnd = (ka/l) T Law of capacitor Change in voltage (δv ) Change in temperature (δt ) Charge transferred (δq) Heat transferred (δh) C e Alρc P δq = C e δv δh = (Alρc P ) δt Optimization of slice thickness In any finite element modeling, element size plays a crucial role in determining the speed and accuracy of the solution. Too big an element size causes erroneous results, while too small an element size slows down the calculation speed. It is thus necessary to optimize the element size so that the
l V Heat m Mass ρ Density c P Specific heat A Area I in C e I out T 1 Hot Cold T T (a) (b) Fig. 1: Analogy between electrical and thermal quantities: Electro-thermal modeling. (a) Single node RC-network. (b) One dimensional heat flow through a rectangular slab. calculation runs at maximum possible speed without causing any significant error. Fig. shows the one dimensional electro-thermal model which physically represents a semi-infinite bar like material. The values of and C e are calculated from the thermal conductivity (k), density (ρ), and specific C e χ net C e χ net χ net Fig. : 1D infinite RC network and its equivalent representation for analytical impedance calculation. heat (c P ) of the material. Since heating is applied on the surface of a body, which is analogous to a current being applied at the outermost element of the network, its temperature response, which is analogous to voltage, can be calculated by multiplying the applied current with the reactance of the network (χ net ). If χ net is complex, the voltage response exhibits a phase shift with respect to the excitation current. This phase shift is analogous to the phase lag between periodic heating excitation and resultant oscillating surface temperature of a body. To find χ net, the outer most node is isolated. Being an infinite series of RC, this isolation does not affect the reactance of the remaining network, and hence it can
0 Argument of network impedance (Degree) -5-10 -15-0 -5-30 -35-40 -45 0.01 0.1 1 10 100 Normalized slice thickness Fig. 3: Optimization of slice parameters in finite element modeling: Phase shift vs. (l/µ). be replaced withχ net. This leads to the following equation. ( )] χ net = [χ Re R e + C +χ net ( ) R χ e net = + χ C (1) where χ C = i/c e ω = the reactance of the capacitor, and ω is the angular excitation frequency. It can be shown that, if the material is progressively sliced into finer elements, the angle of the reactance tends towards 45 o [1]. Fig. 3 plots this result. For all practical purposes, 10% of thermal diffusion length (µ) can be taken to be the upper limit of slice thickness. For more accurate result,l=1% ofµ might be the right choice. However, it considerably increases the simulation time. Modeling a CFRP test piece The simulator was deployed to simulate circular defects in carbon fibre reinforced plastic (CFRP) and the results are verified experimentally. Fig. 4 shows a drawing of the sample which was used for experimental verification of the simulator. The test piece was periodically heated with two 1 kw tungsten-halogen flood lamps. IR screens, consisting of glass tanks containing water (30 mm of water and 5 mm glass on each side), were used to remove the IR radiation emitted by the flood lamps that interferes with the lock-in tests. For this experimental work five shallowest 6 mm diameter holes were considered. They have been marked by a rectangle in Fig. 4. They were chosen because the blind frequency effect was most prominent in their phase images, a phenomenon where a defect becomes invisible in a certain phase image. Since CFRP exhibits anisotropy, the bulk thermal conductivity parallel (k ) and perpendicular (k ) to the fibre are entered in the simulation[]. All simulation input parameters are tabulated in Table. To simulate a 6 mm wide cylindrical defect, at depths ranging from 0.5 mm to 1.5 mm from the surface, a 10 mm tall, 30 mm wide cylindrical CFRP body was considered. The cylindrical body was placed in such a way that the Z-axis coincided with the cylinder s axis, and its bottom surface rested
Fig. 4: A drawing of the CFRP test piece used for comparison studies (all dimensions are in mm). Table : Parameters used in CFRP simulation. Body parameters Diameter Thickness Material K (lateral) K (transverse) Density Specific heat Defect parameters Diameter Depth 30 mm 10 mm CFRP 4.3 W m 1 K 1 0.8 W m 1 K 1 1600 kg m 3 100 J kg 1 K 1 6 mm 0.5 mm to 1.5 mm in steps of 0.5 mm Paint parameters Thickness 30 µm Diffusivity 10 7 m s 1 Effusivity 3000 J K 1 m s 1/ Simulation parameters Excitation Frequency Sampling frequency Duration 16.66 mhz, 50 mhz, 83.33 mhz 10 Hz 60 s on the XY-plane. A 6 mm wide flat bottomed back drilled hole was then created at the centre to depict the defect. The depth of the hole was varied, such that the defect depth ranged from 0.5 mm to 1.5 mm in steps of 0.5 mm. In addition, the top surface of the body was also painted with a 30 µm thick layer of black paint. To speed up the simulation, the problem was reduced to a -dimensional simulation problem by exploiting the cylindrical symmetry as shown in Fig. 5.
Z Defect Defect Y in out in + out Paint layer Z z CFRP X Y Rz e Cylindrical defect Dangling resistors are placed inwards The paint layer is sliced into a finer mesh than the body, because of the low thermal diffusivity of paint. This causes multiple paint layer terminal resistors to be connected to a single resistor in the body. Fig. 5: Modeling of a cylindrical defect in CFRP sample using cylindrical coordinate system.
Results Fig. 6 shows the simulated and experimentally observed phase images, at 16.6 mhz, 50 mhz, and 83.3 mhz, side by side. It also shows the blind frequency effect [3, 4], as the defects colour change (a) (b) (c) (d) Fig. 6: Comparison of simulated and experimental phase images of defects in CFRP test piece. (a) Simulated phase images of CFRP sample at 16.7 mhz (left), 50.0 mhz (middle), and 83.3 mhz (right). (b)-(d) Experimentally observed phase images respectively at 16.6 mhz, 50 mhz, and 83.3 mhz. from white to black with the increasing excitation frequency. This is in agreement with the experimentally observed phase images of similar defects (Fig. 6b-6d). Conclusion A thermal simulator based on electro-thermal model has been designed, implemented and demonstrated. The optimum element size of the test piece is found to be 10% of thermal diffusion length. This produces 45 phase shift of the surface temperature oscillation with respect to excitation heat flux. The simulator is deployed to simulate circular defects in CFRP. The defects diameter were chosen to be 6 mm which produces blind frequency effect in the frequency range of 0 mhz to 80 mhz. This phenomenon is predicted by the simulator showing its validity and usefulness. References [1] K. Chatterjee and S. Tuli, Prediction of blind frequency in lock-in thermography using eletro-thermal model based numerical simulation, Journal of Applied Physics, no. doi: 10.1063/1.488480. [] R. Rolfes and U. Hammerschmidt, Transverse thermal conductivity of cfrp laminates: A numerical and experimental validation of approximation formulae, Composite Science and Technology, vol. 54, pp. 45 54, 1995. [3] W. Bai and B. S. Wong, Evaluation of defects in composite plates under convective environments using lock-in thermography, Measurement Science and Technology, vol. 1, pp. 14 150, 001. [4] C. Wallbrink, S. A. Wade, and R. Jones, The effect of size on the quantitative estimation of defect depth in steel structures using lock-in thermography, Journal of Applied Physics, vol. 101, pp. 104907 8, 007.