**2. External Corrosion Detection Sensor**

The corrosion detection sensor consists of a semicircular shaped plastic curved beam (shown in Figure 1a), attached to a dog-bone-shaped metal component (see Figure 1b), with exact material property as of the pipeline, and an optical fiber with FBG sensors. Each FBG sensor will be glued to a semicircular shaped plastic curved beam at point A, as shown in Figure 1a.

**Figure 1.** (**a**) External corrosion detection sensor. (**b**) Dog-bone-shaped metal.

As shown in Figure 2, the outer diameter of the semicircular shaped plastic curved beam, do, is chosen to be larger than the length Lo by design. When the semicircular shaped plastic curved beam is radially compressed inwards and inserted to the two holes of the dog-bone-shaped metal, the metal and the plastic curved beam at point A are placed in tension.

**Figure 2.** (**a**) Semicircular plastic component. (**b**) Metal similar to pipeline.

This tension is also felt by the FBG sensor as it is glued to the semicircular shaped plastic curved beam at point A. Once tension is observed at point A by the interrogator, the interrogator is zeroed. As the dog-bone-shaped metal corrodes, resulting in failure of the dog-bone-shaped metal, the tension at point A is relieved; thus, a signal is picked up by the interrogator. Figure 3 shows how the sensor of Figure 1 will be implemented in the field.

**Figure 3.** An O&G pipeline with external corrosion detection sensors, spaced every few meters. (**a**) Pipeline with an optical fiber and corrosion detection sensors, (**b**) Corrosion detection sensor.

In the field, multiple optical fibers with FBG sensors will be placed on each O&G pipeline. The fiber optic cables with corrosion sensors are attached to the pipeline by simple zip ties, straps, or large hose clamps, see Figure 3. Neither the sacrificial steel material nor the semicircular plastic part shall be rigidly attached to the pipe. Our proposed sensor sits very near the pipeline but not rigidly attached to the pipeline, basically acting as an environmental corrosivity sensor near the pipeline. When the sensor fails due to corrosion, most likely the pipeline may be already corroded since the corrosion sensor is very near to the pipeline. When corrosion is severe at any pipeline location, the dog-bone-shaped metal corrodes at that location and the prestressed semicircular plastic curved beam will break the dog-bone-shaped metal in two pieces, thus a signal is detected by the interrogator at the control room. When a sensor fails at any particular location, the pipeline inspector will visit the pipeline at that location. He or she will conduct a visual inspection first. If corrosion is observed on the pipeline, he or she will use the inspection techniques such as ultrasound technology or eddy current probe and other techniques to further assess the severity of the pipeline corrosion. If repairs are needed, the pipe will be repaired at that location. As the dimension of the dog-bone-shaped metal is known, and, following the ultrasound inspection, the thickness of pipeline corrosion damage is also known, the time to failure is also known, and an average corrosion rate can be calculated. If no corrosion is observed on the pipeline, only the dog-bone-shaped metal is replaced till the next sensor failure.

Note that as the plastic curved beam can be in the deformed state for a long periods of time, seeing extreme ambient temperatures (high and low temperatures) and humidity conditions, the plastic material needs to exhibit no permanent set or creep and should be able to withstand the temperature and humidity condition of the ambient environment. Also, the plastic material needs to have UV resistance, and its mechanical properties should remain constant with aging.

The number of FBG sensors that one can incorporate within a single fiber depends on the wavelength range of operation of each sensor and the total available wavelength range of the interrogator. FBG strain sensors are often given a 4 nm range. Most commercially available interrogators provide a measurement range of 60 to 80 nm. At 80 nm wavelength range, one can only incorporate 20 sensors per fiber. A new interrogator with 160 nm wavelength range and 16 channels has been recently made available to the public. Assuming 4 nm wavelength range of operation for each FBG sensor, and using this new interrogator, at least 40 FBG sensors can be implemented in a single fiber. With a sixteen-channel interrogator, at least 640 FBG sensors can be implemented per pipeline without using any optical switches. Most flowlines are 1 to 4 km long. Assuming a 4 km long oil or gas pipeline (flowline), without using any optical switches, one can monitor corrosion of O&G pipelines every 6.25 m. With the use of optical switches, we can further reduce the distance between the corrosion sensors. If a 3 nm range is used for each FBG sensor, then the total number of sensors that can be used on each pipeline will be 848, meaning every 5 m we can place an external corrosion detection sensor.

Crevice corrosion refers to the localized attack on a metal surface at, or immediately adjacent to, the gap or crevice between two joining surfaces. To eliminate the crevice corrosion between the dog-bone-shaped metal and the plastic curved beam, the ends of the dog-bone-shaped metal can be coated with a thin film of plastic or some anticorrosion coatings. By coating the ends, we force the corrosion to only occur at the center of the dog-bone-shaped metal; thus, eliminating crevice corrosion. Crevice corrosion between the pipeline and the sensor needs to be also avoided. That is why our sensor is not attached to the pipeline but only attached to the optical fibers.

As in most oil fields 3000 to 4000 flowlines maybe present and each pipeline may have anywhere from 600 to 800 sensors, there is a need to keep the cost of the corrosion detection sensor low. To keep the cost of the sensor low, 3D printers can be used to manufacture the plastic semicircular shaped component. The 3D printing technology to print the plastic component is very mature. As for metals, in recent years, major research has taken place with printing metal components and some 3D printing companies claim that they are able to print steel with 0.02% to 2% carbon content. The authors believe in few years' time, the technology to print all sensor components using 3D printers will be there, and the cost of the corrosion detection sensor will go down as time goes on.

#### *Sensor Design Equations*

To design the corrosion detection sensor of Figure 1 at different sizes, developing a closed form design equation for the above proposed sensor is required. Most optical fibers can only handle strains up to a limit and design equations will thus be necessary to make sure the strain of the optical fiber at the FBG sensor locations does not exceed the manufacturer specified strain limit. To verify the developed closed form design equations, ANSYS finite element software will be used to compare the analytical results with ANSYS results.

In Figure 2, as we mentioned earlier, the distance do is larger than Lo, meaning to insert the semicircular plastic curved beam to the holes of the dog-bone-shaped metal (see Figure 2b), one needs to compress the semicircular plastic curved beam radially inwards. When the semicircular plastic curved beam is compressed radially inwards and then inserted to the dog bone shaped metal piece, we get the picture of Figure 4a. The boundary conditions applied to the plastic curved beam will be as follows, see Figure 4b.

**Figure 4.** (**a**) Assembled sensor. (**b**) Boundary conditions.

In actual practice, to insert the semicircular plastic curved beam to the dog-bone-shaped metal, a radial displacement in the negative X-direction is given to point B, but to apply that displacement, a force F is required to be applied to move point B in the negative x-direction. The force F causes not only to move point B and surface BC in the negative X-direction but also in the negative Y-direction.

Force F rotates surface BC in the clock-wise direction. A counterclockwise moment M is needed to be applied to the right end of the curved beam in order to bring the slope of surface BC to zero and keep the motion of the BC-surface in the Y-direction to zero.

At any angle θ, the internal forces and moments will be as follows, see Figure 5.

**Figure 5.** Plastic semicircular curved beam internal forces and moments.

The internal forces and moments (shown in Figure 5), at any angle θ, are equal to

$$\mathbf{F}\_{\oplus} = \mathbf{F}\sin\theta \tag{1}$$

$$F\_{\mathbf{r}} = F \cos \theta \tag{2}$$

$$\mathbf{M}\_{\rm F} = \mathbf{F}\mathbf{R}\sin\Theta\tag{3}$$

The curved beam stress, at any angle θ, and at any radius r is equal to [22]

$$
\sigma = \frac{\mathbf{M}\_{\text{t}}(\mathbf{r} - \mathbf{r}\_{\text{n}})}{\mathbf{A}\mathbf{e}\mathbf{r}} - \frac{\mathbf{F}\_{\text{\Theta}}}{\mathbf{A}} \tag{4}
$$

where Mt = (MF − M). The strain, at any angle θ, and at any radius r will be equal to

$$\varepsilon = \frac{\mathbf{M}\_{\mathrm{t}}(\mathbf{r} - \mathbf{r}\_{\mathrm{n}})}{\mathbf{A}\mathbf{e}\mathbf{E}\mathbf{r}} - \frac{\mathbf{F}\_{\mathrm{\theta}}}{\mathbf{A}\mathbf{E}} \tag{5}$$

The strain limit of the FBG sensor determines the maximum deflection that can be given to point B. The Castigliano's second theorem is used to develop the closed form equations.

In Figure 5, for a curved beam with a rectangular cross section, the width is assumed to be W, and the outer and the inner radii are ro and ri, respectively. The location of neutral axis is given by Equation (6).

$$\mathbf{r}\_{\mathrm{R}} = \frac{\mathbf{A}}{\int\_{A} \frac{\mathbf{d}\Delta}{\mathbf{r}}} = \frac{\mathbf{W}(\mathbf{r}\_{\mathrm{0}} - \mathbf{r}\_{\mathrm{i}})}{\int\_{\mathbf{r}} \frac{\mathbf{W}(\mathrm{dr})}{\mathbf{r}}} = \frac{\mathbf{W}(\mathbf{r}\_{\mathrm{0}} - \mathbf{r}\_{\mathrm{i}})}{\mathbf{W} \int\_{\mathbf{r}\_{\mathrm{i}}}^{\mathbf{r}\_{\mathrm{0}}} \frac{1}{\mathbf{r}} d\mathbf{r}} = \frac{(\mathbf{r}\_{\mathrm{0}} - \mathbf{r}\_{\mathrm{i}})}{\ln \mathbf{r} |\_{\mathbf{r}\_{\mathrm{i}}}^{\mathbf{r}\_{\mathrm{0}}}} = \frac{(\mathbf{r}\_{\mathrm{0}} - \mathbf{r}\_{\mathrm{i}})}{\ln \mathbf{r}\_{\mathrm{0}} - \ln \mathbf{r}\_{\mathrm{i}}} \tag{6}$$

If the curved beam is sectioned at an angle, θ, see Figure 5, as explained earlier, there will be two forces, Fr, and Fθ and a moment, Mt, at that section. The total strain energy of the semicircular curved beam from 0 < θ < π, can be calculated by adding four terms, shown below in Equation (7).

$$\mathbf{U} = \int \frac{\mathbf{M\_{t}^{2}}}{2\mathbf{EAe}} \mathbf{d\theta} + \int \frac{\mathbf{F\_{0}^{2}r\_{0}}}{2\mathbf{EA}} \mathbf{d\theta} + \int \frac{\mathbf{M\_{t}F\_{0}}}{\mathbf{EA}} \mathbf{d\theta} + \int \frac{\mathbf{C\_{r}^{2}r\_{r}}}{2\mathbf{A}\mathbf{G}} \mathbf{d\theta} \tag{7}$$

The first strain energy term in Equation (7) is generated by the moment Mt, the second term is due to axial force Fθ, the third term accounts for coupling energy due to Mt and axial force Fθ, and the fourth term is due to transverse shear energy due to radial force Fr [23]. The parameter C in the fourth term is the strain-energy correction factor for transverse shear, equal to 1.2 when the cross section is rectangular [24].

Using Equation (7) and conducting a lengthy mathematics, please see reference [22] for more details, we arrive to the following two important equations:

$$\mathbf{u}\_{\mathbf{x}} = \mathbf{u}\_{\mathbf{x}\mathbf{F}} - \mathbf{u}\_{\mathbf{x}\mathbf{M}} = \mathbf{F} \begin{pmatrix} \pi \mathbf{r}\_{\mathbf{c}}^2 \\ 2\mathbf{E}\mathbf{A}\mathbf{e} \end{pmatrix} - \frac{\pi \mathbf{r}\_{\mathbf{c}}}{2\mathbf{E}\mathbf{A}} \begin{pmatrix} \pi \mathbf{C} \mathbf{r}\_{\mathbf{c}} \\ 2\mathbf{A}\mathbf{G} \end{pmatrix} - \mathbf{M} \begin{pmatrix} 2\mathbf{r}\_{\mathbf{c}} \\ \mathbf{E}\mathbf{A}\mathbf{e} \end{pmatrix} + \frac{2}{\mathbf{E}\mathbf{A}} \begin{pmatrix} \mathbf{0} \\ \end{pmatrix} \tag{8}$$

$$\mathbf{u}\_{\rm Y} = \mathbf{u}\_{\rm yF} - \mathbf{u}\_{\rm yM} = \mathbf{F} \left( \frac{2\mathbf{r}\_{\rm c}^2}{\mathbf{E} \mathbf{A} \mathbf{e}} - \frac{2\mathbf{r}\_{\rm c}}{\mathbf{E} \mathbf{A}} \right) - \mathbf{M} \frac{\pi \mathbf{r}\_{\rm c}}{\mathbf{E} \mathbf{A} \mathbf{e}} \tag{9}$$

For the design of Figure 4, uy = 0, and ux is known. Knowing ux and uy, force F and moment M can be calculated from Equations (8) and (9). Knowing F and M, we can now calculate the strain ε using Equation (5). The maximum strain occurs at θ = 90 degrees and r = ro.

#### **3. Equation Validation Using Finite Elements**

To validate Equations (5), (8), and (9), ANSYS finite element (FE) software was used to model a semicircular plastic beam, made from PVC material, with following dimensions (see Table 1).


**Table 1.** Mechanical property and model geometry.

As was explained earlier in Section 2, the semicircular shaped plastic curved beam is chosen to be larger than the dog-bone-shaped metal. To insert the plastic curved beam to the metal, the beam is radially compressed inwards and inserted to the two holes of the dog-bone-shaped metal. The radial inward motion here is 1.5 mm, as shown in Table 1. The ANSYS 2D FE model is shown in Figure 6. Plane183, 2D 8-node element with "plane stress with thickness" option was used to mesh the curved beam. The thickness of the curved beam (thickness is in to the paper) is set as W = 10 mm. Both ends of the curved beam are held fixed in the "uy" direction. The left end is given ux = +0.75 mm and the right end is given ux = −0.75 mm motion, simulating 1.5 mm inward radial motion. The FE model of Figure 6 shows eight elements through the radial thickness.

**Figure 6.** ANSYS Finite Element (FE) Model.

Figure 7 shows the deformed and the undeformed shape of the semicircular curved beam due to 1.5 mm radial displacement.

**Figure 7.** The deformed and the undeformed shapes with boundary conditions.

As can be seen from Figure 8, the maximum strain occurs at r = ro, at θ = 90 degrees, and at point A (see Figure 4), and it is equal to 4487 με. The maximum strain was calculated using Equations (8), (9) and finally (5) and was found to be 4492 με. The error is only 0.1% (see Table 1). ANSYS FE stress analysis validates the derived equations. Equations (1)–(9) can now be used to design the corrosion sensor of Figure 4.

**Figure 8.** (**a**) Curved beam strain in the x-direction, (**b**) Strain in the x-direction zoomed to point A.
