A hybrid model of capsule-typed electromagnetic acoustic transducers for inspection of tubular conductors

Abstract

In-service tubular conductors are prone to such anomalies as the wall-thinning defect and hidden corrosion which pose a severe threat to structural integrity. In this paper, a capsule-typed electromagnetic acoustic transducer is proposed for inspection of tubular conductors such as small-diameter pipes subject to external corrosion. An efficient hybrid model of Electromagnetic Acoustic Transduction (EMAT) is established in an effort to reveal the underlying physical phenomena and simulate the resulting responses to external wall-thinning defects in tubular conductors. The model integrates a fast analytical model computing the electromagnetic field and resulting Lorentz force with Finite Element Modeling (FEM) for simulation of ultrasonic field. The established model is corroborated via FEM and experiments, and beneficial to the optimization of the proposed capsule-typed electromagnetic acoustic transducer for non-destructive evaluation of tubular conductors.

Keywords

Electromagnetic non-destructive evaluation electromagnetic acoustic transduction tubular conductor analytical modeling hybrid model

1. Introduction

It is highly demanded that tubular conductors such as small-diameter nonmagnetic pipes subject to anomalies involving the external corrosion occurring in the outer surface of the pipe are noninvasively inspected and evaluated before catastrophic accident occurs. Complementary to other Non-destructive Evaluation (NDE) techniques, Electromagnetic Acoustic Transduction (EMAT) has been found to be advantageous in terms of noncontact inspection without couplants, high-speed evaluation of structural integrity, etc [1]. Nevertheless, EMAT for inspection of small-diameter nonmagnetic pipes currently takes time because the single EMAT probe has to scan over the pipe internal surface axially as well as circumferentially [2]. Even though the array of EMAT probes [3] has been proposed in a bid to enhance the inspection efficiency, the issue still remains. The reasoning lies in the fact that the circumferential distribution of the electromagnetic field involving the bias magnetic field generated by permanent magnets, dynamic field and eddy currents induced by induction coils within the pipe is non-uniform, which can hardly make the EMAT inspection free of the probe circumferential scanning. In light of this, a capsule-typed EMAT probe for efficient inspection of small-diameter nonmagnetic pipes is proposed in this paper. The probe configuration is portrayed in Fig. 1. As can be seen from Fig. 1 that the proposed EMAT probe consists of: (1) a pair of permanent magnets (with the same poles facing to each other) for generation of the bias magnetic field; and (2) a bobbin coil (deployed in between the magnets) which induces dynamic electromagnetic field within the pipe, and captures signals regarding the reflected ultrasonic wave producing the Electromotive Force (EMF) in the coil. With this probe configuration, the circumferential distribution of the electromagnetic field in the pipe is uniform, which could enhance the inspection efficiency.

In order to reveal the underlying physical phenomena and particularly predict the signals of the proposed EMAT probe, simulations are demanded. Currently, simulations regarding EMAT inspection of conductive structures are usually implemented and preferred by using Finite Element Modeling (FEM) [1–4] and analytical modeling [5,6]. In addition to one of the electromagnetic phenomena of EMAT, i.e. the Lorentz force imposed on both nonmagnetic and ferromagnetic materials, the other two phenomena, which are more complex and include the magnetization force as well as magnetostriction, have been taken into account in simulation models particularly regarding EMAT inspection of ferromagnetic conductors [3,6–8]. This is of great interest and has caught much attention in the research regarding EMAT simulations. For the proposed EMAT probe, since the inspection target is a small-diameter nonmagnetic pipe, the Lorentz force is the essential source generating the ultrasonic field. In order for fast computation of the Lorentz force and EMAT signals i.e. EMFs induced in the bobbin coil, the Extended Truncated Region Eigenfunction Expansion (ETREE) modeling [9,10] is adopted for formulation of field quantities of electromagnetics whilst the dimensions of the magnet pair and bobbin coil are taken into account. FEM is utilized for the simulation of the ultrasonic field in the pipe. This gives rise to a hybrid model of EMAT inspection of tubular conductors by using the proposed capsule-typed EMAT probe.

2. Field formulation

Since EMAT is closely related to electromagnetics and structural mechanics, therefore the hybrid model is established in an effort to solve quantities regarding electromagnetic and ultrasonic fields. It is noteworthy that: (1) the electromagnetic problems involving the bias magnetic field from permanent magnets, dynamic electromagnetic field (transient magnetic field and eddy currents) in the pipe, Lorentz force imposed on particles of the pipe and EMF signals from the bobbin coil are solved via ETREE; and (2) FEM is utilized to simulate the ultrasonic field whose input and output are the Lorentz force and particle velocity, respectively.

Fig. 1.

The capsule-typed EMAT probe for inspection of a tubular conductor: (a) schematic illustration; (b) picture of the probe used in experiments.

2.1. Bias magnetic field

The bias magnetic field is closely related to the permanent magnet which can be analytically modeled via the amperian current approach [11]. For a cylindrical magnet shown in Fig. 1(a), it is analogous to a thin coil with a surface current I _pm distributing over the magnet lateral and flowing in 𝜙 direction. The surface current density J _pm is thus written as J _pm = N _pm I _pm∕(z ₂ − z ₁), where N _pm is the number of turns of the thin coil. The modeling treatment holds under the condition that J _pm = M ₀, where M ₀ denotes the magnetization of the magnet. The polarization of the magnet is thus equal to 𝜇₀ J _pm, where 𝜇₀ denotes the permeability of vacuum. It can also be seen from Fig. 1 that the bias magnetic field of the proposed EMAT probe is generated by a pair of cylindrical magnets with the same poles facing to each other. For such case, z-component of the bias magnetic field $B_{z}^{(bias)}$ at the boundary z = 0 vanishes. Consequently, the magnet pair of the proposed EMAT probe is modeled using the amperian current approach in conjunction with the odd parity configuration [12].

Based on ETREE [13], the closed-form expression of the bias magnetic field $\vec{B}^{(bias)}$ at an arbitrary position in Regions II, III and IV is formulated as: $\begin{eqnarray}\displaystyle \vec{B}^{(bias)}(r,z) & = & \displaystyle \frac{2{\mu}_{0}r_{pm}J_{pm}}{h}\mathop{\sum }_{i=1}^{\infty }[\cos (a_{i}z_{2})-\cos (a_{i}z_{1})]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \times ∼[\cos (a_{i}z)K_{1}(a_{i}r)\vec{r}_{0}+\sin (a_{i}z)K_{0}(a_{i}r)\vec{z}_{0}]I_{1}(a_{i}r_{pm})\end{eqnarray}$ (1) where, $\vec{r}_{0}$ and $\vec{z}_{0}$ are unit vectors. a _i = iπ∕h. I _n and K _n are the modified Bessel functions of first and second kinds respectively.

2.2. Dynamic electromagnetic field

The dynamic electromagnetic field, which results from the bobbin coil driven by a transient excitation current in an arbitrary waveform and is independent of the magnet pair, involves the transient magnetic field and induced eddy currents within the nonmagnetic pipe (with the conductivity of σ₁ and relative permeability 𝜇₁, 𝜇₁ = 1), i.e., Region III in Fig. 1. Referring to [14], the closed-form expressions of the transient magnetic field $\vec{B}^{(dynamic)}$ and eddy current density $\vec{J}_{ec}$ at an arbitrary location in Regions III are written as: $\begin{eqnarray}\displaystyle \vec{B}^{(dynamic)}(r,z) & = & \displaystyle \displaystyle \frac{{\mu}_{0}N}{hd_{1}z_{c}(r_{2}-r_{1})}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \times ∼\mathop{\sum }_{n=1}^{\infty }\sin ({\kappa}_{n}z_{c})[{\kappa}_{n}\sin ({\kappa}_{n}z){\eta}_{1}\vec{r}_{0}+{\lambda}_{n}\cos ({\kappa}_{n}z){\eta}_{2}\vec{z}_{0}]{\chi}({\kappa}_{n}r_{1},{\kappa}_{n}r_{2})\displaystyle \frac{1}{{\kappa}_{n}^{3}}\end{eqnarray}$ (2) $\begin{eqnarray}\displaystyle J_{ec}(r,z) & = & \displaystyle \displaystyle \frac{-{\sigma}_{1}{\mu}_{0}N}{hd_{1}z_{c}(r_{2}-r_{1})}\displaystyle \mathop{\sum }_{n=1}^{\infty }\cos ({\kappa}_{n}z)\sin ({\kappa}_{n}z_{c}){\chi}({\kappa}_{n}r_{1},{\kappa}_{n}r_{2})\frac{{\eta}_{3}}{{\kappa}_{n}^{3}}.\end{eqnarray}$ (3) In Eqs (2) and (3), 𝜅_n = (2n − 1)π∕(2h). ${\lambda}_{n}=\sqrt{{\kappa}_{n}^{2}+j{\omega}{\mu}_{0}{\sigma}_{1}}$ , where, ω denotes the angular frequency of each harmonic within the excitation current. N stands for the number of turns of the bobbin coil. 𝜒(𝜅_n r ₁, 𝜅_n r ₂) is the coil coefficient which can be readily computed by referring to [13 ]. The other terms include: $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}l@{}}\displaystyle {\eta}_{1}=\frac{1}{{\pi}}\int _{-\infty }^{+\infty }\frac{k_{2}K_{1}({\lambda}_{n}r)-k_{4}I_{1}({\lambda}_{n}r)}{k_{1}k_{2}-k_{3}k_{4}}I({\omega})e^{j{\omega}t}d{\omega};\\[12.0pt] \displaystyle {\eta}_{2}=\displaystyle \frac{1}{{\pi}}\int _{-\infty }^{+\infty }\frac{k_{4}I_{0}({\lambda}_{n}r)+k_{2}K_{0}({\lambda}_{n}r)}{k_{3}k_{4}-k_{1}k_{2}}I({\omega})e^{j{\omega}t}d{\omega}\\[12.0pt] \displaystyle {\eta}_{3}=\frac{1}{{\pi}}\int _{-\infty }^{+\infty }j{\omega}\frac{k_{2}K_{1}({\lambda}_{n}r)-k_{4}I_{1}({\lambda}_{n}r)}{k_{1}k_{2}-k_{3}k_{4}}I({\omega})e^{j{\omega}t}d{\omega}\end{array}\right. & & \displaystyle\end{eqnarray}$ (4) where, I (ω) denotes the harmonics within the excitation current. k _n (n = 1, 2, 3 and 4) is written as: $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}l@{}}\displaystyle k_{1}={\kappa}_{n}I_{0}({\kappa}_{n}d_{1})K_{1}({\lambda}_{n}d_{1})+{\lambda}_{n}I_{1}({\kappa}_{n}d_{1})K_{0}({\lambda}_{n}d_{1});\\[7.0pt] k_{2}={\kappa}_{n}I_{1}({\lambda}_{n}d_{2})K_{0}({\kappa}_{n}d_{2})+{\lambda}_{n}I_{0}\left({\lambda}_{n}d_{2}\right)K_{1}\left({\kappa}_{n}d_{2}\right)\\[7.0pt] \displaystyle k_{3}={\kappa}_{n}I_{0}({\kappa}_{n}d_{1})I_{1}({\lambda}_{n}d_{1})-{\lambda}_{n}I_{1}({\kappa}_{n}d_{1})I_{0}({\lambda}_{n}d_{1});\\[7.0pt] \displaystyle k_{4}={\kappa}_{n}K_{1}({\lambda}_{n}d_{2})K_{0}({\kappa}_{n}d_{2})-{\lambda}_{n}K_{0}({\lambda}_{n}d_{2})K_{1}({\kappa}_{n}d_{2}).\end{array}\right. & & \displaystyle\end{eqnarray}$ (5) It is noted that Eq. (4) can be evaluated efficiently and numerically via Inverse Fast Fourier Transform (IFFT) [9].

2.3. Lorentz force

Based on Eqs (1), (2) and (3), the formulation of closed-form expressions of Lorentz forces (F _s and F _d resulting from interactions of the eddy current with the bias magnetic field and transient magnetic field, respectively) imposed on the particle at an arbitrary position in Region III is straightforward. By using the identity $\vec{F}=\vec{J}_{ec}\times \vec{B}$ , we come across: $\begin{eqnarray}\displaystyle \vec{F}_{s}(r,z) & = & \displaystyle \displaystyle \frac{2r_{pm}J_{pm}{\sigma}_{1}N{\mu}_{0}^{2}}{h^{2}d_{1}z_{c}(r_{2}-r_{1})}\left\{-\left[\mathop{\sum }_{n=1}^{\infty }P_{n}\times \mathop{\sum }_{i=1}^{\infty }Q_{i}\sin \left(a_{i}z\right)K_{0}\left(a_{i}r\right)\right]\vec{r}_{0}\right.\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \left.+\left[\mathop{\sum }_{n=1}^{\infty }P_{n}\times \mathop{\sum }_{i=1}^{\infty }Q_{i}\cos \left(a_{i}z\right)K_{1}\left(a_{i}r\right)\right]\vec{z}_{0}\right\}\end{eqnarray}$ (6) $\begin{eqnarray}\displaystyle \vec{F}_{d}(r,z) & = & \displaystyle {\sigma}_{1}\left[\displaystyle \frac{{\mu}_{0}N}{hd_{1}z_{c}(r_{2}-r_{1})}\right]^{2}\left\{-\left[\mathop{\sum }_{n=1}^{\infty }P_{n}\times \mathop{\sum }_{n=1}^{\infty }W_{n}{\lambda}_{n}\cos ({\kappa}_{n}z){\eta}_{2}\right]\vec{r}_{0}\right.\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \left.+\left[\mathop{\sum }_{n=1}^{\infty }P_{n}\times \mathop{\sum }_{n=1}^{\infty }W_{n}{\kappa}_{n}\sin ({\kappa}_{n}z){\eta}_{1}\right]\vec{z}_{0}\right\}\end{eqnarray}$ (7) where, $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}l@{}}P_{n}={\eta}_{3}\cos ({\kappa}_{n}z)\sin ({\kappa}_{n}z_{c}){\chi}({\kappa}_{n}r_{1},{\kappa}_{n}r_{2})/{\kappa}_{n}^{3}\\[5.0pt] W_{n}=\sin ({\kappa}_{n}z_{c}){\chi}({\kappa}_{n}r_{1},{\kappa}_{n}r_{2})/{\kappa}_{n}^{3}\\[5.0pt] Q_{i}=[\cos (a_{i}z_{2})-\cos (a_{i}z_{1})]I_{1}(a_{i}r_{pm}).\end{array}\right. & & \displaystyle\end{eqnarray}$ (8)

2.4. EMF signals

The ultrasonic wave reflected from the external surface of the pipe vibrates each particle on the inner surface of the pipe with the velocity of $\vec{v}$ , which induces the so-called velocity-induced eddy current in the presence of the bias magnetic field. This is constituted by the identity $J_{v}={\sigma}_{1}(\vec{v}\times \vec{B}^{(bias)})$ , where J _v denotes the density of the velocity-induced eddy current. The velocity-induced eddy current subsequently gives rise to the electric field within the area covering Regions I and II. Based on ETREE, the intensity of the electric field is formulated as: $\begin{eqnarray}\displaystyle E(r,z)=-\displaystyle \frac{2{\mu}_{0}d_{1}}{h}\frac{\partial J_{v}}{\partial t}\mathop{\sum }_{i=1}^{\infty }\sin (a_{i}z)I_{1}(a_{i}r)\sin (a_{i}z_{0})K_{1}(a_{i}d_{1}) & & \displaystyle\end{eqnarray}$ (9) where, z ₀ denote the axial coordinate of a particle. The electric field induces EMF in the bobbin coil, and the EMF signal which results from the vibration of a particle over the pipe inner surface is expressed as: $\begin{eqnarray}\displaystyle {\xi}(t)=\frac{4{\pi}{\mu}_{0}d_{1}}{h}\displaystyle \frac{\partial J_{v}}{\partial t}\mathop{\sum }_{i=1}^{\infty }[\cos (a_{i}z_{2})-\cos (a_{i}z_{1})]\sin (a_{i}z_{0})K_{1}(a_{i}r_{0}){\chi}(a_{i}r_{1},a_{i}r_{2})\frac{1}{a_{i}^{3}}. & & \displaystyle\end{eqnarray}$ (10) The total EMF signal 𝜓(t) which is usually taken as the testing signal of EMAT is thus derived from the superposition of 𝜉(t) by considering all particles over the pipe inner surface.

2.5. Ultrasonic field

Followed by the computation of the total EMF signal, the simulation of the ultrasonic field arising from the Lorentz forces in the pipe body plays a vital role, and gives the computed results regarding the velocities of the particles distributing on the pipe inner surface. The simulation is implemented based on FEM [1,3,15]. In structural mechanics, the dynamic equilibrium equation of the entire model concerning merely the pipe can be expressed as: $\begin{eqnarray}\displaystyle [M]\{\ddot{U} \}+[C]\{\dot{U}\}+[K]\{U\}=\{F\} & & \displaystyle\end{eqnarray}$ (11) where, [M], [C], [K] and $\{F\}$ are the matrices of the coordination mass, damping, stiffness and force of each element in Region III, respectively. $\{U\}$ is the resulting displacement of each node in every element. It is noteworthy that the load matrix $\{F\}$ in Eq. (11) is efficiently computed from Eqs (6) and (7). By solving Eq. (11) via time-stepping FEM, the particle velocity $\vec{v}$ of each particle/node over the pipe inner surface is calculated, and subsequently adopted for calculation of the EMF signals which is elaborated in Section 2.4.

3. Corroboration

In a bid to verify the proposed hybrid model, FEM simulations and experiments are carried out. The parameters of the probe and nonmagnetic pipe are tabulated in Tables 1 and 2. The excitation current driving the probe is illustrated in Fig. 2. It is noted that: (1) in FEM simulations both electromagnetic and ultrasonic fields are computed via FEM without introducing the analytical modeling for calculating any field quantities; and (2) experiments are conducted using the commercial system (RITEC Advanced Measurement System RAM5000) along with the fabricated probe. The EMAT signals acquired from the proposed hybrid model, FEM simulations and experiments are presented in Fig. 3.

Table 1
Parameters of the EMAT probe

M ₀/T z ₁/mm z ₂ − z ₁ /mm r _pm/mm r ₁ /mm r ₂ /mm z _c/mm N

1.22 2.0 10.0 15.0 15.4 15.6 1.5 40

M ₀/T	z ₁/mm	z ₂ − z ₁ /mm	r _pm/mm	r ₁ /mm	r ₂ /mm	z _c/mm	N
1.22	2.0	10.0	15.0	15.4	15.6	1.5	40

Table 2

Parameters of the small-diameter nonmagnetic pipe

d ₁/mm	d ₂/mm	σ₁/MS ⋅ m⁻¹	𝜇₁	Young’s modulus/GPa	Density/Kg ⋅ m⁻³	Poisson’s ratio
16.0	21.0	34.2	1.0	71.7	2750	0.33

Fig. 2.

The excitation current driving the proposed probe.

Fig. 3.

EMAT signals obtained from the hybrid model, FEM simulation and experiment.

It can be seen from Fig. 3 that the predicted signal from the hybrid model has good agreement with those of the FEM simulation and experiment particularly regarding the first and second pulses arising from the reflected ultrasonic waves from the external surface of the conductor. It is noteworthy that the initial pulse (0 ≤ t ≤ 2 μs) within the experimental signal is automatically mitigated by the system in order to enhance the first and second pulses, the temporal span of which is normally preferred for quantitative evaluation of the sample thickness. In order for quantitative comparison in terms of the amplitudes and temporal span between the first and second pulses, after taking the absolute values of the EMAT signals the signal envelopes are obtained. The envelopes of the EMAT signals from the hybrid model and FEM are shown in Fig. 4 together with the experimental signal.

Fig. 4.

Signal envelopes of the EMAT signals obtained from the hybrid model and FEM simulation along with the experimental signal (2 μs ≤ t ≤ 8 μs).

It is observed from Fig. 4 that compared with FEM, the extrema of the pulses of the predicted signal via the hybrid model have better agreement with those of the experimental signal. Further analysis reveals that: (1) the maximum relative error of the predicted pulse extremum from the hybrid model against the experimental value is less than 8.7% whilst it is 11.3% for FEM; and (2) the predicted temporal span between the two pulses from the hybrid model (3.23 μs) has smaller discrepancy against the experimental result (3.22 μs) than that from FEM (3.25 μs). The cross comparison indicates that for the proposed capsule-typed probe, simulations based on the hybrid model give better prediction regarding the EMAT signals as well as their features for evaluation of tubular conductors than FEM. In regard to the computational time, it takes the hybrid model 873s to predict the EMAT signal whist FEM consumes 1750 s. In consideration of the computational accuracy and efficiency, it is noteworthy that the hybrid model is superior to FEM. The reasoning lies in the fact that the analytical modeling i.e. ETREE included in the hybrid model for is independent of meshes, and enhances the accuracy and efficiency in calculation of the electromagnetic quantities which are paramount in EMAT simulations.

From Fig. 4, it is also noticeable that comparing results from the hybrid model and those of experiments there are discrepancies. This could be caused by the factors which are barely taken into account in simulations and include: (1) small eccentricity when deploying the capsule-typed probe in the tubular sample; (2) filters and other electronics employed in the commercial EMAT system; and (3) extraneous noise during the course of the experiment. Nevertheless, the comparison of results from the hybrid model with those of experiments implies that the established model has considerably higher computational efficiency without much loss in accuracy. This is beneficial to the optimization of the proposed capsule-typed EMAT probe for non-destructive evaluation of tubular conductors.

4. Conclusion

In this paper, a hybrid model regarding the proposed capsule-typed EMAT probe for inspection of tubular conductors is established. The model integrates the analytical modeling i.e. ETREE for efficient computation of electromagnetic quantities and FEM for calculation of ultrasonic field. Particularly in modeling of electromagnetics, the closed-form expressions of the electromagnetic field, resulting Lorentz force and EMF signals are formulated. The hybrid model is subsequently corroborated by FEM and experiments. From cross comparison regarding EMF signals acquired from the hybrid model, FEM and experiments, it can be observed that thanks to ETREE, the established hybrid model is advantageous over FEM whose computational efficiency and accuracy are highly dependent on meshes and their qualities especially in simulations of electromagnetic field. Simulations based on the established hybrid model will benefit: (1) optimization of the proposed capsule-typed EMAT probe; (2) investigation of physical phenomena underlying the inspection; and (3) quantitative evaluation of tubular conductors prone to external corrosion.

Footnotes

Acknowledgements

The authors would like to thank the Natural Science Foundation of China (Grant No. 51477127, 51777149), National Magnetic Confinement Fusion Program of China (Grant No. 2013GB113005), Fundamental Research Funds for the Central Universities of China (Grant No. XJJ2018027) and Natural Science Basic Research Plan in Shaanxi Province of China (Program No. 2016JM5075).

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A hybrid model of capsule-typed electromagnetic acoustic transducers for inspection of tubular conductors

Abstract

Keywords

1. Introduction

2. Field formulation

Table 1 Parameters of the EMAT probe M 0/T z 1/mm z 2 − z 1 /mm r pm /mm r 1 /mm r 2 /mm z c/mm N 1.22 2.0 10.0 15.0 15.4 15.6 1.5 40

Footnotes

Acknowledgements

References

Table 1
Parameters of the EMAT probe

M ₀/T z ₁/mm z ₂ − z ₁ /mm r _pm/mm r ₁ /mm r ₂ /mm z _c/mm N

1.22 2.0 10.0 15.0 15.4 15.6 1.5 40