Limitations of Eddy Current Testing in a Fast Reactor Environment

Transcription

1 Limitations of Eddy Current Testing in a Fast Reactor Environment Tao Wu and John R. Bowler Department of Electrical and Computer Engineering, Iowa tate University, Ames IA 5 Abstract. The feasibility of using eddy current probes for detecting flaws in fast nuclear reactor structures has been investigated with the aim of detecting defects immersed in electrically conductive coolant including under liquid sodium during standby. For the inspections to be viable, there is a need to use an encapsulated sensor system that can be move into position with the aid of visualization tools. The initial objective being to locate the surface to be investigated using, for eample, a combination of electromagnetic sensors and sonar. Here we focus on one feature of the task in which eddy current probe impedance variations due to interaction with the eternal surface of a tube are evaluated in order to monitor the probe location and orientation during inspection. INTRODUCTION Test Conditions There is a requirement to inspect 4th generation sodium cooled fast reactors at a standby temperature of 5 o C with the conductive coolant present in a liquid state. Because liquid sodium is opaque and has an electrical conductivity roughly ten times that of the steels typically used for nuclear reactor structures, there are severe limitations in carrying out an eddy current inspection. The challenges arise because the effect of the sodium reduces the eddy current flaw signal by attenuating the electromagnetic field between the probe and the workpiece and by providing a conductive path between crack faces. A further challenge is the need to use sensors in guiding the probe module to locations where it can perform inspections of the structure. An important task in approaching these challenges is to establish a baseline performance for the eddy current detection of flaws under controlled conditions in air and when immersed in conductive fluid. The initial performance assessments are being carried out using electric discharge machine notches on a variety of plate and tube specimens made from steels used in fast reactors. It is envisaged that an encapsulated sensor module immersed in the coolant can be guided to the point of inspection with the aid of sonar to visualize the surrounding structure. We can use numerical models to evaluate eddy current probe signals due to immersed flaws but also to evaluate eddy current probe interactions with unflawed structure to navigate an inspection module. In the final approach to the surface to be inspected, eddy current sensors can potentially be used at low frequency for short range adjustments of the module to locate it in a position where the inspection for possible defects can take place. In order to carry out the positioning phase of an inspection, one needs interpret probe signals at an arbitrary position and orientation with respect to structural components. A knowledge of the eddy current signals and their variation with probe position can then be used, for eample, to locate the surface of a tube and deduce the position of the tube ais prior to and during the inspection. Once this has been done, changes in the signals due to flaws can then be properly evaluated. To evaluate the overall feasibility of under coolant inspection using eddy currents requires a combination of eperiment to establish flaw signals and noise levels for a range of defects together with estimates of detection probabilities. The limited data set can be enhanced by the use of models to predict signals and synthesize detection probabilities based on noise estimates obtained eperimentally. Models available include those based on integral equation methods [,, 3] and boundary element schemes which have been validated against eperiment.

2 z FIGURE. The ais of eternal coil shown is in a plane whose normal is in the direction of a pipe ais. The ais of the coil and tube do not intersect. With reference to the case where they do intersect, the coil has undergone an azimuthal rotation. Transverse field components One effective way of analyzing probetube interactions when the a circular probe coil has an arbitrary orientation is to use spherical coordinates in representing the field and relate these to the coordinate system of the tube analytically [4]. An alternative method is to seek a relationship between the probe signals and tube position using two related cylindrical polar coordinate systems, referred to as the global system; that of a tube, and the local system; that of the source of the probe field, then compute the interaction using appropriate coordinate transforms. This is done here first for a circular filament in free space and then for a coil in each case using a scalar representation of the electromagnetic field. Further steps are then needed to model the performance of the probe encapsulated in the module by etending the analysis in local coordinates. However, this aspect of the problem will not be considered here. Consider therefore a quasistatic timeharmonic electromagnetic field varying as the real part of ep( ıωt). In the absence magnetic sources and assuming linear material properties, the magnetic field which in this case has zero divergence is represented by a transverse electric (TE) potential ψ, and a transverse magnetic (TM) potential ψ, defined with respect to the unit vector a z which is the direction of the global coordinate ais: H = (a z ψ ) + k (a z ψ ), () where k = ıωµσ, σ being the electric conductivity of the material. In a conductive region the scalar potentials are solution of the Helmholtz equation: ( + k )ψ i = i =,. () In air and any nonconducting region, the transverse electric potential satisfies the Laplace equation and Equation () reduces to H = ψ z. (3) which is not valid in a region of an electric current since it implies that H = in conflict with Ampère s law. But the coil field in a current free region can be deduced by introducing a discontinuity in the TE potential. This is done first for the field of a circular filament, Figure. FIELD THEORY Filament Field By integrating the field of a circular filament we can determine the field of any aially symmetric coil on the ais and if the coil has a rectangular cross section, the integration is straightforward. We shall use local coordinates, Fig.,

3 z A R O Q B P FIGURE. Polar zais of the global coordinate system is normal to the yplane of the filament ais. for epressing the electric field of a circular filament in air as E = ıωµ (a ψ ) (4) where a is the aial direction. The origin of the local Cartesian coordinate system is at Q Fig., which is to be the center of the coil. Note that we treat a special case where the coil ais lies in a plane whose normal is the same as the zais of the global coordinate system. In general, however one can tilt the yplane, which we call a polar tilt. Here we simply deal with an azimuthal rotation. The general case is dealt with in a separate article [5]. By taking the curl of Equation (4) and using the induction law, one infers the epression for the magnetic field of a filament in air using local coordinates and can write the coil field whose TE potential is ψ in the same form. Thus for the filament we write H = ψ and for the coil H = ψ. (5) Comparing the second of the relationships with Equation (3) one notes that the TE potential in the global cylindrical polar coordinates can easily be determined from that in the local Cartesian coordinate. ψ z = ψ. (6) Continuity of the tangential electric field means that ψ is continuous at a circular disk bounding the filament. At a filament in the plane = we introduce a discontinuity at the disk related to the filament current, [ ] ψ = I, (7) = The discontinuity conceived as a single layer potential can be viewed as the source of the field. This is in contrast to the view taken in using Ampére s law where the source is the filament current or when using the magnetic scalar potential when it is a magnetic dipole layer [6]. Because it is a solution of the Laplace equation, the filament potential can be epressed in terms of the global coordinate system as ψ (r) = I 4π m= e ımφ D m (υ) I m ( υ ρ) K m ( υ ρ) eıυz dυ ρ < α ρ > α, (8) where the functions, D m (υ), are to be determined for the filament centered at R, Fig.. We consider here the case where the normal to the plane of the filament ais is the direction of the ais of the global coordinate system, shown here as the vertical direction. In the more general case, to be dealt with in future work, there is what we refer to as a polar tilt where the ais is tilted out of the horizontal plane. The solution domain, Fig. 3, is divided into three regions by two concentric circular cylinders, the radii of which, α and α, are the limits of the alternative solutions in (8). We only need the solution for ρ < α and ρ > α. The

4 α α A O Q R B FIGURE 3. olution limits are α and α. The filament lies between A and B with its ais in the direction. r r ϕ α θ ϕ FIGURE 4. Triangle used to specify the addition theorem used for a coordinate transform. potential of a filament carrying a time harmonic current I is given by a relationship derived using Green s second theorem for a single layer potential: ψ = I 4π R d, (9) where the integration is over the circular disk bounded by the filament. To evaluate Equation (9) and produce an epression that allows us to identify D m (υ) we note that [7] R = π m= e ım(φ φ ) I m ( υ ρ < )K m ( υ ρ > )e ıυ(z z ) dυ, () where ρ > is the greater and ρ < the lesser of ρ and ρ. Note that (9) and the dependence R = / r r implies that, ψ = ψ () where the Cartesian coordinate system is defined with the direction in the direction of the coil ais and hence the direction of the ais of any filament comprising the coil. In fact it is convenient to put the origin of this coordinate system at the coil center. Equation () simplifies the integration with respect to in finding the coil field. ubstituting equation () into Eq. (9) gives ψ = I 4π m= e ımφ I m ( υ ρ) K m ( υ ρ) eıυz K m ( υ ρ ) I m ( υ ρ ) e ı(mφ +υz ) d dυ ρ < α ρ > α () Comparing with Eq. (8) we find that D m (υ) = K m ( υ ρ ) I m ( υ ρ ) e ı(mφ +υz ) d ρ < α ρ > α (3)

5 A O η ϕ' ρ ' ' Q T R η ϕ U B P FIGURE 5. Integration over the filament disk, between A and B, is carried out in two parts using an addition theorem whose variables are related by the cosine rule applied to OT and OU. In order to carry out the integration over the filament disk, we transform (3) using an addition theorem for Bessel functions in which the arguments are related by the cosine rule, r = α + r αr cos η, Fig. 4. Rather than writing it in the form of Watson treatise [8] we use instead the eternal angle φ = π η to get K ν (r) I ν (r) e ıνφ = n ( ) n K ν n (α) I ν n (α) I n (r )e ınφ (4) Viewed as a coordinate transform, this represents the relationship between cylindrical polar coordinates r, φ and r, φ defined with respect to their origins at O and respectively, Fig. 3. Note that we have reversed the sign of n compare with Watson s epression, but this make no difference to the summation result. In applying the transformation, we identify the terms illustrated in Fig. 4 with those associated with the similar triangle OT in Fig. 5 as follows r = υ ρ, r = υ ρ and α = υ ( a + /sin φ ), (5) where = QR a = OQ and R is the center of the filament, Fig. 5. By integrating the filament field with respect to one can determine the field of a coil field centered at Q. The addition theorem is applied twice, since the integration over the filament disk is divided into two parts at a zdirected line through. First we use coordinate transforms by using the addition theorem with reference to OT, and also with reference to OU. For a filament of radius s, with R = cot φ, the integration over the filament disk gives where D m (, s, υ) = = υ n= ( ) ( ) n K m n υ a + υ s + cot φ ( sin φ) I m n υ a + υ e ınφ s (ρ cot φ ) I n ( υ ρ ) sin φ e ıυz dz dρ s (ρ cot φ ) s cot φ s (ρ + cot φ ) I n ( υ ρ ) e ıυz dz dρ s (ρ + cot φ ) n= +e ın(π φ ) ( ) n K m n ( υ a + υ sin φ ) I m n ( υ a + υ sin φ ) F n(r, c) = e ınφ [ F ( n υ s, υ ) cot φ + ( ) n F n ( υ s, υ )] ρ < α cot φ (6) ρ > α r a I n (ξ c) sin r ξ dξ. (7) and the prime refers to the derivative with respect to the first argument of the function. For the case where φ = π/, sin φ =., cot φ =. and (6) becomes D m (υ) = ( ) n K m n ( υ a + υ ) υ I m n ( υ a + υ cos(nπ/)f ) n ( υ s ρ < α, ) (8) ρ > α n= which gives the source coefficients for the case of a filament whose ais is normal to that of the tube, hence by integration it can be used to give the source coefficient for the corresponding coil.

6 O η β β ρ η Q ρ P FIGURE 6. olution limits β and β for a coil. Coil field We can epress ψ, representing the transverse electric potential, with a preferred zdirection for the coil in free space just as we have ψ for the filament in Eq. (8): ψ(r) = I 4π m= e ımφ C m (υ) I m ( υ ρ) K m ( υ ρ) eıυz dυ ρ < β ρ > β, (9) where β = and β = (a cos η + ρ ) +, are the solution limits, Fig. 6. In view of the relationship (6), and the integral relationship that determines the coil potential from the filament potential, one can deduce C m (υ) by integrating Eq. (6) with respect to and s, between the limits of the coil crosssection. We shall do this for a coil that is eternal to a tube. However, the internal coil is similar with a change of the associated Bessel function. Thus we find C m (υ) = ν ρ D m (, s, υ) D m (, s, υ) ds ıυ ρ = ıν ( ) n e ınφ [ K υ 4 m n ( υ a + υ / sin φ ) f n (υρ, υρ, υ cot φ ) n= K m n ( υ a + υ / sin φ ) f n (υρ, υρ, υ cot φ ) ] () where ν is the turns density of the coil, the factor /ıυ is a results of the integration with respect to z, as in required by (6), a trivial integral has been performed over the coil length and we define which completes the derivation of the source coefficient of a coil in free space. f n (r, r, c) = F n (r, c) F n (r, c) () CONCLUION The calculation of the interaction with a tube depends on the reflection of the the at the tube surface. The reflected field as a whole is related to the probe impedance through a reciprocity principle [9]. The procedure for evaluating the impedance changes due to the cylindrical structure and due to flaws therein is given in [] for the case of a coil interaction with the surface of a borehole. In dealing with the interaction with flaws one has a choice between

7 TABLE. Coil and inconel steam generator tube parameters for probe outside tube case Coil Inner Raids, r.59mm Coil Outer Raids, r 3.98mm Coil Thickness.44mm Number of Turns 35 Isolated DC Coil Inductance, L 465µH Tube Inner Diameter 6.64mm Tube Outer Diameter 8.99mm Conductivity (M/m).84 Relative Magnetic Permeability, µ r Liftoff, λ 3.98mm R/X η =4 o η =6 o η =9 o R/X Frequency (khz) FIGURE 7. Normalized resistance changes for a coil with different azimuthal tilt angles outside a tube treating the flaw using a hypersingular/dipole kernel but for improved accuracy the calculations can be performed using classical/monopole kernel [,, ]. The results given in Fig. 7 and 8, are for the case defined by the parameters given in Table, in which the coil is eternal to a tube. The graphs show the variation in coil impedance as a function of frequency and for different angles of coil orientation η see Fig. 6. For the case where η = 9 deg the interaction with the tube is a maimum at which orientation the ais of the coil intersects that of the tube and the impedance give a measure of the liftoff. In other words the tube signal strength is a maimum when the coil ais passes though that of the tube.

8 X/X X/X η =4 o η =6 o η =9 o 3 Frequency (khz) FIGURE 8. Normalized reactance changes for a coil with different azimuthal tilt angles outside a tube ACKNOWLEDGMENT This work was performed within the Nuclear Energy University Research Program supported by the Department of Energy on the grant RPA355: Advanced High Temperature Inspection Capabilities for mall Modular Reactors. REFERENCE J. R. Bowler,. A. Jenkins, L. D. abbagh, and H. A. abbagh, J. Appl. Phys. 7, 7 (99). J. R. Bowler, T. P. Theodoulidis, H. Xie, and Y. Ji, IEEE Trans. Mag. 48, 59 (Mar. ). 3 J. R. Bowler, T. P. Theodoulidis, and N. Poulakis, IEEE Trans. Mag. 48, 4735 (Dec. ). 4 T. P. Theodoulidis and J. Bowler, 5, (5). 5 T. Wu and J. R. Bowler, J. Phys D, Appl. Phys. to be published.. 6 J. A. tratton, Electromagnetic Theory (IEEE Antennas and Propagation ociety, 7). 7 A. Gray, G. B. Mathews, and T. M. MacRobert, A Treatise on Bessel Functions, nd ed. (MacMillan and Co., London, 95). 8 G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge university press, 995). 9 C. A. Balanis, Advanced Engineering Electromagnetics, Vol. (Wiley New York, 989). Y. Yoshida and J. R. Bowler, IEEE Trans. Mag. 36, 46 (). J. Bowler, Y. Yoshida, and N. Harfield, IEEE Trans. Mag. 33, 487 (997). N. Harfield, Y. Yoshida, and J. R. Bowler, J. Appl. Phys. 8, 49 (996).