**2. Materials and Methods**

#### *2.1. 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 ±*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*(*xp*,*yp*) 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 *HLU*, *HRU*, *HLD*, *HRD* in Equations (2)–(5), respectively.

$$H\_y = H\_{LII} + H\_{LD} + H\_{RII} + H\_{RD} \tag{1}$$

$$H\_{\rm LLI} = \frac{+m}{4\pi\mu} \int\_{-\frac{\Phi}{2} - W}^{\frac{\Phi}{2}} \frac{y\_p}{\left\{ \left( x\_p - \mu \right)^2 + \left( y\_p \right)^2 \right\}^{\frac{3}{2}}} d\mu \tag{2}$$

$$H\_{RLI} = \frac{-m}{4\pi\mu} \int\_{\frac{\mathcal{D}}{2}}^{\frac{\mathcal{D}}{2} + W} \frac{y\_p}{\left\{ \left( x\_p - u \right)^2 + \left( y\_p \right)^2 \right\}^{\frac{3}{2}}} d\mu \tag{3}$$

$$H\_{\rm LD} = \frac{-m}{4\pi\mu} \int\_{-\frac{R}{2} - W}^{0} \frac{(y\_p + h)}{\left\{ \left( x\_p - u \right)^2 + \left( y\_p + h \right)^2 \right\}^{\frac{3}{2}}} du\tag{4}$$

$$H\_{RD} = \frac{+m}{4\pi\mu} \int\_0^{\frac{D}{2} + W} \frac{\left(y\_p + h + d\right)}{\left\{\left(x\_p - u\right)^2 + \left(y\_p + h + d\right)^2\right\}^{\frac{2}{2}}} du\tag{5}$$

**Figure 1.** 2D dipole model of magnetic flux leakage testing system for wall thinning of the boiler water-cooled tube.

Figures 2 and 3 show the result of calculating *Hy* for the depth of the defect *d* in the range of 0~3 mm and position *xp* in the range of −15–15 mm using Equations (1)–(5); where *<sup>m</sup>* was assumed as 2<sup>π</sup> × <sup>10</sup>−<sup>4</sup> [H/m], relative magnetic permeability of the ferromagnetic material *μ* = 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 *Hy* 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 *Hy* with the defect's depth at a different position on the *x*-axis. It shows a less sensitivity of the *Hy* 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 2.** Simulation result with magnetic dipole model for different depth *d*.

**Figure 3.** (**a**) Relationship between the depth of damage *d* and the vertical component of magnetic flux intensity *Hy* and (**b**) the normalized *Hy* at different positions from the yoke's center in the *x*-axis.

From the results in Figure 3, it is possible to estimate the relationship between the *Hy* and *d* by a quadratic equation, as shown in Equation (6). Here, α1, α2, and α<sup>3</sup> are constants. On the other hand, when using the Hall sensor, the magnetic flux density in the vertical direction can be measured by the Hall sensor output voltage *VH*, as expressed in Equation (7) (relative permeability of the air is assumed to 1):

$$H\_y = \mathfrak{a}\_1 (d - \mathfrak{a}\_2)^2 + \mathfrak{a}\_3 \tag{6}$$

$$V\_H = kIB\cos\theta = kIH\_y\tag{7}$$

where *VH*, *k*, *B*, *I*, *θ* denote the Hall voltage, the Hall constant, the magnetic flux density incident on the Hall sensor, the Hall sensor input current, and the direction angle of the magnetic flux density perpendicularly incident on the Hall sensor. The Hall voltage *VH* by Equation (7) is linearly proportional to *Hy*, the magnetic flux density in the vertical direction. On the other hand, if the Hall constant (*k*) and the Hall sensor input current (*I*) are constant, and Equation (6) is substituted into Equation (7), it is expressed as Equations (8) and (9). That is, by measuring the magnetic flux density in the vertical direction, the depth of the defect can be quantitatively evaluated, where *c*<sup>1</sup> and *c*<sup>2</sup> are constants:

$$V\_H = kI\{a\_1(d+a\_2)^2 + a\_3\}\tag{8}$$

$$d = \sqrt{c\_1 V\_H + c\_2} - a\_2 \tag{9}$$

Figure 4 shows the block diagram of the signal processing for a single Hall sensor element. The output voltage of the Hall sensor *VH* was low-pass filtered (LPF) to remove the high-frequency noise signal. The first stage amplifier was used to gain the signal before transferring to the main signal processing circuits. The LPFs and first stage amplifier were attached near to the Hall sensor in the sensor probe. The second stage amplifier has a controllable gain, which was adjusted by the software in the PC. The signal was then converted to digital via A/D converter and real-time display/process in the PC. The proposed inspection system uses multiple Hall sensors; thus, the number of LPFs, first amplifiers and second amplifiers are the same as the number of Hall sensors for simultaneous signal processing.

**Figure 4.** Signal processing block diagram of the magnetic flux leakage testing system. The block diagram is for a single Hall sensor.

Figure 5 shows the proposed magnetic leakage testing system to inspect the corrosion in the water-cooled tube wall. A magnetic sensor array and magnetizer were manufactured to fit with the water-cooled tube's surface, as shown in the left and middle drawing. The magnetic sensor array was placed at the middle of magnetizer poles for measuring the distribution of vertical magnetic flux, as discussed in the previous paragraphs. There were three wheels (a front and two rears wheels) used to maintain the lift-off between the sensor and the tube and help scan the tube easily.

**Figure 5.** Configurations of the magnetic flux leakage testing system.

#### *2.2. Flexible Ultrasonic Testing*

Ultrasonic probe usually requires a supplement of a coupling material for transmitting ultrasonic wave from the transducer to the test specimen. It complicates the inspection system and is waste of coupling material. In addition, the test specimen surface should be flat enough to maintain a positive lift-off (non-contact) for protecting the collision of the transducer with the test specimen. It is hard for the inspection of near-surface defects in the water-cooled tube because the changes of the tube wall could make an unpredictable lift-off that could lead to the collision and break the transducer. Therefore, we propose using a flexible transducer that the lift-off could be varied and not require using the coupling material [23]. At the head of a normal transducer, we attached a flexible membrane that was water-filled. The membrane has a sphere shape after filing the water and maintains contact with the tube even though the lift-off can vary. Also, the ultrasound wave can still propagate from the transducer to the water membrane and come to the test specimen.

A sample flexible transducer is shown in Figure 6a. The transducer has a spring that keeps the contact between the membrane with the test specimen during the scan. The received time-domain signal of the transducer, which is A-scan signal (*u*(*t*)), is shown in Figure 6b. For a better signal-to-noise ratio, the spectrogram of the A-scan signal was processed (*S*(*τ*,*f*)) and extracted only the signal (*SA*(*τ*)) at the center frequency of the transducer (*fc*), as shown in Figure 6c,d. The spectrogram (*S*(*τ*,*f*)) and extracted signals (*SA*(*τ*)) are calculated as expressed in Equations (10) and (11), respectively; where, *h* is a sliding Gaussian window. The extracted signal (*SA*(*τ*)) was then stacked to form the B-scan signal while scanning the transducer along with the test specimen.

$$S(\tau, f) = \left| \int\_{-\infty}^{\infty} u(t)h(t-\tau)e^{-j2\pi ft}dt \right|^2\tag{10}$$

$$S\_A(\tau) = S(\tau, f)|\_{f=f\_c} \tag{11}$$

**Figure 6.** (**a**) A single flexible ultrasonic transducer measuring a thickness of a specimen, (**b**) its time-domain signal, (**c**) spectrogram signal, (**d**) the cross-section view signal of the spectrogram, and (**e**) the stacked cross-section view signal (B-scan).

It is observed from the ultrasonic transducer signal that there are multiple peaks. The first peaks are the reflected wave from the specimen surface. It has a delay time of about 4 μs (*t*1, *u*1), which is the propagation time from within the probe membrane. This delay time could be varied due to the flexibility of the membrane (lift-off). Thus, it is necessary to eliminate this delay time by shifting the signal with an amount of time −*t*1. In addition, there are four peaks (*t*2, *u*2), (*t*3, *u*3), (*t*4, *u*4), (*t*5, *u*5) next to the specimen surface peak (*t*1, *u*1), which correspond to the repetitions from the bottom surface of the specimen. The time intervals of these four peaks are the same and can be used to calculate the specimen thickness, as shown in Equation (12); where *v* is the speed of the ultrasound in the specimen.

$$d = \left(t\mathfrak{z} - t\_1\right) \times \frac{\upsilon}{2} = \left(t\mathfrak{z} - t\mathfrak{z}\right) \times \frac{\upsilon}{2} = \dots = \left(t\mathfrak{z} - t\_4\right) \times \frac{\upsilon}{2} \tag{12}$$

Figure 7 is a schematic of the flexible ultrasonic probe (FUP) for quantitatively measuring the specimen thickness. The FUP is an array of multiple transducers (i.e., 6) arranged for covering the tube wall, welding lines, and specimen membrane area. The FUP could be alternated the magnetizer and magnetic sensor array modules in Figure 5.

**Figure 7.** Schematic of the flexible ultrasonic probe (FUP) for measuring the water-cooled wall thicknesses.

#### **3. Experiment and Results**

*3.1. Specimen*

Figure 8 shows the shape and location of damages on a specimen. A total of four watercooled tubes (SA210C) with inner and outer diameters of 47.3 to 51.3 mm and 63.5 mm, respectively, one-sided (t11–t14) and double-sided artificial damages (t41–44) simulated for wears were produced on Tube-1 and Tube-4, respectively. In Tube-2, slit-type artificial damages (t21–t28) with the same width of 7.0 mm, depth of 0.9 mm, and lengths from 20 to 100 mm were produced. In Tube-3, slit-type artificial damages (t31–t38) having the same width of 7 mm and length of 60 mm and different depths from 0.3 to 3.1 mm were produced. The detailed location and size are as shown in Tables 1 and 2.

**Figure 8.** Specimen with different shape and size of artificial damages.


**Table 1.** Specification of artificial taper-type wear on the Tube-1 and Tube-4.

**Table 2.** Specification of artificial slit-type damages on the Tube-2 and Tube-3.


The tubes were welded with a 6.0 mm thick membrane. There six slit-type artificial damages (w11, w12, w13, w31, w31, w33) on the two Membrane −1 and −2. The damages have the same width of 7.0 mm, different lengths from 20 to 80 mm, and different depths from 0.3 to 2.4 mm, as shown in Table 3. On the four welding lines (Welds 1, 2, 3, and 4), there are ten slit-type artificial damages (w11–w42) with the same width of 7.0 mm, different lengths from 30 to 100 mm, and different depths from 0.6 to 3.0 mm, as shown in Table 4. Totally, there are 40 artificial damages produced on the tubes, membranes, and welding lines. The picture of the specimen with damages is shown in Figure 9.



**Table 4.** Specification of slit-type damages on the Weld-1~4.

**Figure 9.** Sample specimens with four water-cooled tubes and artificial damages.

#### *3.2. Inspection System*

Figure 10 shows the prototype of the inspection system. In the magnetic leakage testing (MFLT) module, the magnetizer has a pole distance of 15 mm and has manufactured the profile following the tube and membrane surfaces. It maintains about 1.0 mm of distance above the specimen surface by the support of the three wheels. The magnetizer has 100 turns of copper wire and supplied by a current of about 200 mA to produce a magnetic field into the specimen. There are 48 Hall sensors arrayed at an interval of 2.5 mm on a curve following the tube and membrane surfaces. The MFLT probe scanned the specimen with steps of 4.0 mm. In the FUP, there are 6 flexible ultrasound transducers having a center frequency of 5 MHz. The MFLT module, including the magnetizer and magnetic sensor array, can be exchanged with the FUP module. The measured signal can be processed and displayed in real-time in a LabVIEW software on a notebook.

**Figure 10.** The prototype of the inspection system for water-cooled tubes.
