## 2 Materials and Methods  
### Measurement of Magnetic Flux Density  
Figure 1 shows the simplified 2D dipole model of the thickness changing on the boiler water-cooled wall tube due to corrosion [21,22]. A U-type magnetizer is placed on the surface of the membrane. The width, distance between poles, and height of the magnetizer are expressed as \(W\), \(D\), and \(H\), respectively. The corrosion depth and length are \(d\) and \(D/2+W\), respectively. The distance between the magnetizer and the specimen, i.e., lift-off, is expressed as \(h\). Then, the lift-off at the corrosion is \(h+d\). In the dipole model, magnetic charges \(\pm m\) per unit area are assumed to be distributed along the length of the magnetizer poles, membrane length, and corrosion length according to the assumption in the dipole model [21,22]. The magnetic flux density in the \(y\)-axis direction at the position of \(P(x_{p}\),\(y_{p})\) is the summary of the magnetic field produced from the magnetic charges, as expressed in Equation (1). The vertical magnetic field from the left magnetizer pole, right magnetizer pole, no-corrosion specimen length, and corrosion specimen length are expressed as \(H_{\text{LU}}\), \(H_{\text{RU}}\), \(H_{\text{LD}}\), \(H_{\text{RD}}\) in Equations (2)-(5), respectively. \[H_{y}=H_{\text{LU}}+H_{\text{LD}}+H_{\text{RU}}+H_{\text{RD}} \tag{1}\]  
\[H_{\text{LU}}=\frac{+m}{4\pi\mu}\int_{-\frac{D}{2}-W}^{\frac{D}{2}}\frac{y_{p} }{\left\{\left(x_{p}-u\right)^{2}+\left(y_{p}\right)^{2}\right\}^{\frac{3}{2} }}du \tag{2}\]  
\[H_{\text{RU}}=\frac{-m}{4\pi\mu}\int_{\frac{D}{2}}^{\frac{D}{2}+W}\frac{y_{p} }{\left\{\left(x_{p}-u\right)^{2}+\left(y_{p}\right)^{2}\right\}^{\frac{3}{2} }}du \tag{3}\]  
\[H_{\text{LD}}=\frac{-m}{4\pi\mu}\int_{-\frac{D}{2}-W}^{0}\frac{\left(y_{p}+h \right)}{\left\{\left(x_{p}-u\right)^{2}+\left(y_{p}+h\right)^{2}\right\}^{ \frac{3}{2}}}du \tag{4}\]  
\[H_{\text{RD}}=\frac{+m}{4\pi\mu}\int_{0}^{\frac{D}{2}+W}\frac{\left(y_{p}+h+d \right)}{\left\{\left(x_{p}-u\right)^{2}+\left(y_{p}+h+d\right)^{2}\right\}^{ \frac{3}{2}}}du \tag{5}\]Figures 2 and 3 show the result of calculating \(H_{y}\) for the depth of the defect \(d\) in the range of 0\(\sim\)3 mm and position \(x_{p}\) in the range of \(-\)15-15 mm using Equations (1)-(5); where \(m\) was assumed as \(2\pi\times 10^{-4}\) [H/m], relative magnetic permeability of the ferromagnetic material \(\mu\) = 500, the lift-off \(h\) = 1 mm, and the width (\(D\)) and width (\(W\)) of the magnetic poles were assumed to be 10 mm. Large changes of the magnetic flux intensity on the defect size with different depths are shown in Figure 2. It is noted that the \(H_{y}\) has small changes at the center position of the magnetizer and increases as closer to the pole of the magnetizer. Thus, it should not position the magnetic sensor at the center of the magnetizer. Furthermore, Figure 3 shows the relationship between the \(H_{y}\) with the defect's depth at a different position on the \(x\)-axis. It shows a less sensitivity of the \(H_{y}\) to the defect's depth when placing the sensor at the center of the magnetizer (\(x\) = 0), and a similar-high sensitivity when the sensor is at 2-5 mm from the magnetizer's center. However, the closer to the magnetizer's pole, the higher the intensity of the magnetic flux that could saturate the magnetic sensor. Therefore, the results suggest positioning the sensor at a distance of 2 mm where the sensitivity to the defect's depth is high, and the magnetic flux density is hard to saturate the magnetic sensor. Figure 1: 2D dipole model of magnetic flux leakage testing system for wall thinning of the boiler water-cooled tube. Figure 2: Simulation result with magnetic dipole model for different depth \(d\). From the results in Figure 3, it is possible to estimate the relationship between the \(H_{y}\) and \(d\) by a quadratic equation, as shown in Equation (6). Here, \(\alpha_{1}\), \(\alpha_{2}\), and \(\alpha_{3}\) are constants.