Applications to Nuclear Power

Abstract

The nuclear-power industry faces the serious challenge of convincing a skeptical public and regulatory agencies that it can operate safely and efficiently. Nondestructive evaluation (NDE) plays a significant role in this task, and computer modeling is playing a significant role in NDE. The industry now realizes the value of using such modeling to replace expensive experimental tests, as well as to design equipment, and interpret results. Eddy-currents have a traditional place in the inspection of steam generator tubing, and the industry seeks improved tools for such inspections.

Appendix

1.1 Modeling Probes + Cables

In order to characterize a probe for the purpose of modeling probe-flaw responses and validating benchmark tests, it is necessary to consider the effects of real conditions under which the ideal probe operates. Consider a probe that is to be operated as a differential-bobbin within a 70/30 Cu-Ni tube with no defects. The inner radius of the tube is 0.2475 in and the outer radius is 0.3125 in. The tube’s conductivity is 2. 61 ×10⁶ S/m, and its relative magnetic permeability is unity. Almost nothing is known about the probe except some external dimensions.

Data taken for a single probe of the differential-pair are given in Table 18.11.

Table 18.11 Frequency response of a probe in freespace and within a tube

Referring to the air measurements, we note that the resistance of the probe varies with frequency, and the reactance is not proportional to frequency. This suggests that there is a frequency-sensitive two-port network that connects the probe to the impedance analyzer. It could be something as simple as a shunt self-capacitance of the coil, but in this case it is a 25-foot (7.62 m), 5/16-inch coaxial cable. Cables of this length (or more) are used to connect the probe to the measuring instrument when inspecting heat-exchanger tubes.

The problem, now, is to characterize a probe coil whose parameters are not known, and to characterize the cable that connects the coil to the impedance analyzer. This can be done quite simply, though a bit tediously, with VIC-3D ^®. Trial and error on the turns and inner- and outer-radii of the coil suggest that a good fit to the measured data is given by: N _turns = 55, IR = 0.19 in. and OR = 0.21 in. The height of the coil is known to be 0.055 in. When we run VIC-3D ^® for this probe in freespace, we get an inductance of 49. 55 μH, which is close to, but smaller than, the inductance measured at 10 kHz. The freespace resistance of the model coil is, of course, 0 $\Omega$, so we manually enter a resistance of $R_{0} = 4.5\,\Omega$ into the VIC-3D ^® file to see what that gives. We choose this value because it agrees with the measured resistance at the lowest frequency.

Cable effects can be accounted for by defining four parameters: characteristic impedance, capacitance per unit length, attenuation in dB/m and length. This follows from transmission-line theory [98]. Let Z _L be the load (terminating) impedance of a coaxial cable, and Z _in be the driving-point (input) impedance of the loaded cable. Then

$$\displaystyle\begin{array}{rcl} Z_{\mathrm{in}}& =& Z_{0}\frac{Z_{L} + Z_{0}\tanh (j\beta l +\alpha l)} {Z_{0} + Z_{L}\tanh (j\beta l +\alpha l)} \\ Z_{0}& =& \sqrt{(}L/C) =\mathrm{ characteristic\ impedance\ of\ the\ cable} \\ \beta & =& \frac{\omega } {v_{p}} \\ v_{p}& =& \frac{1} {\sqrt{(}LC)} \approx 2 \times 1{0}^{8}\ \mathrm{m/s}. {}\end{array}$$

(18.2)

L and C are, respectively, inductance and capacitance per unit length of the cable. The value for phase velocity, v _P , is a reasonable approximation for typical transmission lines.

Typical values for coaxial cables that are used in eddy-current NDE are 50 Ω for the characteristic impedance, and 100 pF/m for the distributed capacitance. Our problem calls for a cable length of 7.62 m, which leaves only the attenuation to be determined. Attenuation, α, is frequency dependent, but VIC-3D ^® assumes it to be constant, so it becomes necessary to run several tests to determine the frequency response of the cable.

The results for tests of the probe+cable in freespace are shown in Table 18.12. The diagonal entries in this table agree with the corresponding freespace results of Table 18.11 within 2 %. This indicates that the attenuation varies with frequency as shown in the left-hand column. This variation agrees with theory, in that attenuation always increases with frequency.

Table 18.12 Freespace impedance response of the coil + cable

When modeling the probe within the tube, we use the same cable parameters as above, except that we use α = 0. 0 only for f = 10⁴ Hz, α = 0. 005 only for f = 5 ×10⁴, etc. The results for the model calculations for the coil within the tube are given in Table 18.13. The differences between these values and those shown in Table 18.11 are 3 %, or less, except for $X_{1{0}^{4}}$ and $R_{1{0}^{5}}$.

Table 18.13 In-tube impedance response of the coil + cable

It is difficult to determine an outer radius of a many-turn bobbin coil when it is within a tube, because the windings on the outer layer of the coil nearest the tube will not lie smoothly in that layer, as shown in Fig. 18.26. This gives the appearance of an uneven spacing between the outer radius of the coil and the inner tube wall. This is not as serious a problem when the coil is in the “pancake” aspect.

Fig. 18.26

The nonuniform distribution of turns within a typical real coil

1.2 A “Layered” Model of Corrosion Pits

One possible scheme for determining the shape of a corrosion pit that is assumed to have a fixed morphology would be to use the layered-pit model shown in Fig. 18.27.

Fig. 18.27

A layered model of the ID pit shown in Fig. 18.19

The objective is to determine the conductivity of each layer, and from that result infer the size, and perhaps shape, of the pit. We assume that the pit has a circular cross-section, as before, so that the only parameter that defines the model is the radius of the anomalous region. If the layer has a conductivity of 0S/m, then clearly that layer is entirely filled by the pit, whereas if the layer has a conductivity of 1. 4 ×10⁶S/m, then the layer is entirely free of the pit, being host material, only. Anything in between requires further consideration, as we show next.

The unknowns in the inversion process are $\sigma _{1},\ldots,\sigma _{4}$, the conductivities of layers 1–4, with 1 corresponding to the inner wall and 4 to the outer wall, as well as the diameter of the anomalous region. Once the conductivity of a layer has been estimated by NLSE, we apply a simple volume-fraction computation to determine the relative volume of cavity to host material within that layer. For example, suppose layer L has a conductivity of 1. 135 ×10⁶ S/m. Then to see how much of the Lth layer is occupied by the cavity, we compute the volume-fraction of the cavity: $\mathrm{VF}_{L} = 1 -\frac{1.135} {1.40} = 0.1893$. Thus, the cavity occupies less than 19 % of the Lth layer.