Multilayered Media with Cylindrical Geometries
In this chapter, we describe the mathematical development of a general axisymmetric model for VIC-3D®. This model is capable of analyzing tubes with tube supports and roll-expanded transition zones. Features such as magnetite and sludge,are included, and
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Multilayered Media with Cylindrical Geometries
9.1 Introduction In this chapter, we describe the mathematical development of a general axisymR metric model for VIC-3D . This model is capable of analyzing tubes with tube supports and roll-expanded transition zones. Features such as magnetite and sludge, are included, and materials may be either ferromagnetic or nonmagnetic. The model described in this chapter will include only differential (or absolute) bobbin coils. Flaws, or anomalies, can be of three types: (1) axisymmetric (such as circumferential rings, tube supports, roll-expanded transition zones), (2) thin axially-oriented cracks, and (3) user-defined flaws, such as intergranular attack (IGA) or corrosion pits. The incident fields due to bobbin coils are computed using the axisymmetric theory developed in this chapter.
9.2 Some Typical Problems in Steam Generator Tubing There are a number of rather complicated geometries that appear in the inspection of steam generator tubing by means of eddy currents. Figures 9.1–9.3 illustrate several of them, and Fig. 9.4 illustrates a number of different flaws that must be modeled. The model must not only contend with these geometries, but it must deal with ferromagnetic bodies, as well.
9.3 Coupled Ferromagnetic Integral Equations It is possible to account for ferromagnetic effects by introducing an anomalous magnetic current, together with magnetic–magnetic, magnetic–electric, and electric–magnetic Green functions. For this problem, which is axisymmetric, and H.A. Sabbagh et al., Computational Electromagnetics and Model-Based Inversion, Scientific Computation, DOI 10.1007/978-1-4419-8429-6 9, © Springer Science+Business Media New York 2013
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9 Multilayered Media with Cylindrical Geometries
Fig. 9.1 Illustrating a steam generator tube with a tube support
Tube support
Fig. 9.2 Illustrating a roll-expanded steam generator tube Tube sheet
Fig. 9.3 Illustrating a laser-welded sleeve in a steam generator tube
Sleeve Tube
Laser weld
therefore solenoidal in the electric variables, it will be easier to use an “all electric” model for the coupled system, following the same ideas that we developed in the context of Amperian currents in Chap. 3. The electric–electric Green function is simple, and because all electric variables are solenoidal, the matrices should be well conditioned for all frequencies.
9.3 Coupled Ferromagnetic Integral Equations
Axial Flaw
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Sludge, roll expansion
Circumferential Flaw
Inter-Granular Attack
Fig. 9.4 Illustrating a variety of anomalies (“flaws”) in steam generator tubes
We start with Maxwell’s equations ∇ × E = − jω B ∇ × H = jω D + J(e) .
(9.1)
Now H = B/μ (r) = B/ μh + B/μ (r) − B/μh = B/μh − Ma , where μh is the host permeability and Ma is the anomalous magnetization vector. Thus the second of Maxwell’s equations may be written ∇ × (B/ μh ) = jω D + J(e) + ∇ × Ma ,
(9.2)
which makes clear that the Amperian current, ∇ × Ma , is an equivalent anomalous electric current that arises because of the departures of th
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