# Flood Detection in Steel Tubes Using Guided Wave Energy Leakage

## Abstract

**:**

## 1. Introduction

## 2. Leaky Guided Waves

#### Dispersion Curves and Mode Shapes

_{ph}is the phase velocity. The index M can be zero or an integer and defines how the fields generated by the guided wave modes vary with the angular coordinate θ in a cylinder cross-section. By solving Equation (1) for a known frequency, using the software DISPERSE, the dispersion curves for all three vibrational modes could be obtained. Dispersion curves for a steel tube of 2 mm in thickness and 270 mm in diameter flooded by water are plotted in Figure 1.

_{r}, is given by:

_{r}stands for the radial component of the velocity field of the mode, T

_{rr}represents the normal traction component, evaluated at the surface of the tube in contact with the liquid, and the asterisk denotes the complex conjugate [50]. Therefore, the radial component of the exciting modes predominantly influences the amount of leakage and its amplitude loss. In this application, it is desirable to excite guided wave modes that exhibit a certain amount of leakage into the surrounding liquid since this parameter is the basis to determine if there is a flooded member. Taking this into consideration, a frequency excitation of 40 kHz was selected. According to the software DISPERSE mode shapes, the radial power flow (P

_{w}R

_{r}) energy and the radial displacement (U

_{r}) from the inner wall of the tube to its center abated at 127 mm and 106 mm, respectively, for the F(1,1) mode, compared to the L(0,1) mode, which continued leaking energy up to its center, as illustrated in Figure 2.

## 3. Wavelet Energy Entropy Signal Processing

_{j}

_{,k}are known as wavelet coefficients, j is the resolution level, and k are each of the elements in the time series. The DWT yields discrete blocks of wavelet coefficients with several elements that reduce as j increases [53]. The concept of wavelet frames determines the viability of the wavelet coefficients to represent a full signal. A frame consists of a family of wavelet functions, where the energy of the wavelet coefficients resides within a bounded range of the energy of the original signal [54], given by:

_{j}

_{,k}is the wavelet coefficients. If A = B = 1, wavelets are orthonormal, and the original signal can be completely reconstructed by:

_{2}M. The concept of energy is linked with the notions derived from the Fourier theory, where the spectral energy density of a signal x(t) is E(f) = |X(f)|

^{2}, where X(f) is the Fourier transform of x(t). Then, since orthonormal wavelets, by definition, have finite energy, it is possible to measure the energy of each resolution level (j) by [55]:

_{j}denotes the probability distribution of energy at each resolution level j, then Σp

_{j}= 1. Shannon entropy measures the average uncertainty of a random variable. Hence, wavelet energy entropy using p

_{j}is a tool that permits the analysis of the amount of disorder in any distribution [56]. Entropy can be defined by:

_{j}is the probability distribution of the energy at each resolution level. Since the WT guarantees the accurate decomposition of the original signal into components of different frequencies that allow one to study each component separately, hence, the energy of wavelet coefficients at a particular frequency can be analyzed using the wavelet energy entropy to detect events that modify the energy distribution of the signal.

## 4. FMD-Guided Wave System

^{TM}, a proprietary DSP-based module, which includes signal conditioning, a 54 dB gain instrumentation amplifier, and a USB interface communication to display information in a Personal Computer, shown in Figure 5.

_{ave}(k), where N is the number of averages, using the typical expression given by:

## 5. Experiment Setup

## 6. Results

#### 6.1. Off-Line FMD via Wavelet Energy Entropy

_{j}) and the total energy of the signal (E

_{tot}), as indicated in Equations (7) and (8), and obtain the probability distribution of energy at each resolution level (p

_{j}), as shown in Equation (9). Entropy is a quantitative criterion for analyzing probability and the amount of disorder in any distribution; therefore, wavelet energy entropy using p

_{j}displays the amount of disorder of any distribution, as shown in Equation (10), and allows one to detect events that modify the energy distribution of a signal. Figure 9 shows the wavelet energy entropy calculated for the experiment setup for the three conditions of the tube, dry and flooded.

#### 6.2. Real-Time FMD via Time–Space between Pulses

## 7. Conclusions and Further Work

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Phase velocity dispersion curve of a steel tube 2 mm × 270 mm diameter mm loaded with water.

**Figure 2.**Radial power flow (P

_{w}R

_{r}) energy and the radial displacement (U

_{r}) at 40 kHz. (

**a**) Flexural F(1,1) mode; (

**b**) longitudinal L(0,1) mode.

**Figure 4.**(

**a**) PPM transmitter main components; (

**b**) actual PPM transmitter design; (

**c**) PPM pulse width; (

**d**) PPM communication scheme.

**Figure 6.**Experiment setup using a hollow steel pipe, 1.5 m × 270 mm × 2 mm, a PPM transmitter, and a DSP instrumentation receiver connected to a PC via USB port.

**Figure 8.**Wavelet coefficients and their spectrogram for a received guided wave pulse centered at 40 kHz using the experiment setup with the tube dry and the tube flooded at 5% and 10% of its height.

**Figure 10.**A received guided wave pulse: (

**a**) dry tube, (

**b**) tube loaded with 5% water, and (

**c**) tube loaded with 10% water.

**Figure 11.**Received guided wave pulses amplified and filtered; (

**left**), dry steel tube, TSBP of 30 ms; (

**right**), steel tube flooded at 10%, TSBP of 40 ms.

**Figure 12.**Guided wave analog pulses converted to digital pulses to estimate the TSBP; (

**upper**), for a steel tube in dry conditions; (

**lower**), for a steel tube flooded at 5% of its height.

**Figure 13.**(

**a**) Normalized FMD threshold in percentage for the calculated number of TSBP; (

**b**) displayed results in a human–machine interface in ASCII standard.

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**MDPI and ACS Style**

Mijarez, R. Flood Detection in Steel Tubes Using Guided Wave Energy Leakage. *Sensors* **2023**, *23*, 1334.
https://doi.org/10.3390/s23031334

**AMA Style**

Mijarez R. Flood Detection in Steel Tubes Using Guided Wave Energy Leakage. *Sensors*. 2023; 23(3):1334.
https://doi.org/10.3390/s23031334

**Chicago/Turabian Style**

Mijarez, Rito. 2023. "Flood Detection in Steel Tubes Using Guided Wave Energy Leakage" *Sensors* 23, no. 3: 1334.
https://doi.org/10.3390/s23031334