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Article

Numerical Study on Hydroelastic Responses of Submersible High-Density Polyethylene Circular Seaweed Platforms Held by Single-Point Mooring System and Buoys

1
School of Civil Engineering, The University of Queensland, St. Lucia, QLD 4072, Australia
2
Blue Economy Cooperative Research Centre, Launceston, TAS 7248, Australia
3
The Climate Foundation, Woodford, QLD 4514, Australia
*
Author to whom correspondence should be addressed.
J. Mar. Sci. Eng. 2024, 12(8), 1437; https://doi.org/10.3390/jmse12081437
Submission received: 4 July 2024 / Revised: 13 August 2024 / Accepted: 16 August 2024 / Published: 20 August 2024
(This article belongs to the Special Issue Hydroelastic Behaviour of Floating Offshore Structures)

Abstract

:
This paper investigates the hydroelastic behavior of submersible circular seaweed platforms under wave action. The circular platform comprises circular collars constructed from high-density polyethylene (HDPE) pipes and seaweed grow-out lines arranged in the radial direction. The HDPE pipes may be filled with air, seawater, or pressurized seawater. The platform is kept in place by using a single-point mooring system and buoys. The platform may be lowered to over a hundred meters below the water surface to allow the seaweed to be soaked in cold nutrient-rich waters during the night and be raised to the surface for photosynthesis during the day. Also, the platform will be submerged during storms to avoid strong surface waves. The submergence is achieved by using a service vessel and surface buoys that secure the submerged platform with ropes. The hydroelastic analysis of the platform is performed using AquaSim software (v. 2.18), which has been developed specifically for hydroelastic analysis of aquaculture infrastructure. It is found that the hydroelastic response of the seaweed platform can be reduced by filling HDPE pipes with seawater and through the installation of seaweed grow-out lines. It is also found that the compressive stresses in HDPE pipes can be reduced by pressurizing the filled seawater, thereby keeping them below the allowable compressive stresses for severe sea states where pipes with unpressurized seawater show excessive compressive stresses.

1. Introduction

Seaweed has long been used in the culinary traditions of Japan, Korea, and China, dating back to prehistoric times. As early as 600 BC, the Chinese philosopher Sze Teu extolled the virtues of certain algae, deeming them fit for esteemed guests, including royalty. Across these nations, a diverse array of seaweed species, totaling 21 in Japan alone, have been integral to everyday cuisine for centuries. Notably, by 1973, seaweed consumption surged, with households averaging 3.5 kg annually—a remarkable 20% increase over the preceding decade, underscoring its enduring popularity and nutritional importance [1,2]. Nowadays, seaweed finds applications in many industries, such as animal feed, pharmaceuticals, industrial extracts, cosmetics, fertilizers or biostimulants, and biofuels [3].
Most of seaweed production is from cultivation. For example, according to the 2019 data [4], farmed seaweeds contributed to over 96% of the global seaweed production, while the small remaining production was from wild collection. Traditionally, seaweed farming takes place in nearshore waters; but this farming approach faces multiple challenges, including competition with coastal developments, rising temperatures, and environmental risks such as competing with other nearshore marine organisms for light and nutrients [3]. To overcome these challenges, moving seaweed farming offshore is suggested as a viable strategy [4]. Offshore locations offer advantages such as a vast open sea for large-scale seaweed cultivation, and seaweed cultivation infrastructure can be integrated with other-purpose offshore structures for fish farming and renewable energy production for increased sustainability and cost reductions [4,5].
Developments of infrastructure for offshore seaweed cultivation have received significant global interest, but previously proposed infrastructure designs have yet to be commercialized. The first development was recorded in the 1970s in the United States, with the proposal and trial test of an inverted umbrella-shaped structure [6]. Some other notable developments include a core guiding ring accompanied by connected support ropes and cultivation lines in 2002; a seaweed carrier in 2010; the BioArchitecture Lab (BAL) system in 2010; the Macroalgal Cultivation Rig in 2010–2019; and the system comprising multiple tension-leg platforms in 2016 [7]. Currently, extensive research and development efforts are dedicated to improving the designs of seaweed cultivation infrastructure, focusing on enhancing structural safety and cost-effectiveness. The MARINER and UNITED [8,9] initiatives represent significant ongoing research programs aimed at advancing seaweed cultivation infrastructure. These programs investigate diverse concepts, including the integration of a single-point mooring system, submerging seaweed infrastructure, and optimizing the configuration of buoy-line-anchor systems to enhance overall structural performance. Another interesting development is the use of inflatable SeaStruct beams for holding negative buoyant seaweed as proposed by Impact9 [10].
In previously proposed designs for offshore seaweed cultivation, the infrastructure has been made of steel or comprises a system of buoys, lines, and anchors. Nguyen et al. [11] recently investigated the utilization of circular submersible high-density polyethylene (HDPE) structures for seaweed farming. The key advantages of employing such structures include the following: (i) HDPE’s excellent resistance to corrosion and UV light, high durability in harsh marine conditions, and effective antibiofouling properties [12,13]; (ii) the construction of HDPE ring structures for seaweed cultivation can be easily accomplished through existing fabrication methods in the fish farming industry; (iii) integration of seaweed cultivation infrastructure into existing fish farms is feasible due to the similarities in their designs. This integrated system has the potential to enhance the sustainability of aquaculture as seaweed can absorb excess nitrogen and phosphorus from fish waste [4]. The Climate Foundation Australia is presently conducting trials on their seaweed cultivation platform in Cebu waters in the Philippines (refer to Figure 1) to demonstrate the feasibility of using circular HDPE pipe structures for seaweed cultivation in exposed sites.
This paper aims to investigate the effects of filling HDPE pipes with seawater and pressurized seawater with the view to improve the hydroelastic responses and compressive strength of The Climate Foundation’s circular HDPE platforms. AquaSim software (v. 2.18) will be adopted for the hydroelastic analysis of the flexible HDPE platforms under wave action.

2. Design and Simulations of Seaweed Platforms

Figure 2 shows The Climate Foundation’s design of a floating circular seaweed cultivation platform (of diameter 40 m) that is held in place by a single-point mooring system. The platform is constructed from two circular HDPE pipes connected to each other by 48 equally spaced HDPE brackets (with regular spacing of about 2.6 m). The HDPE pipes can be filled with air, seawater, or pressurized seawater. The HDPE platform is used to support grow-out lines for red seaweed. The grow-out lines run in the radial direction, as shown in Figure 2a. To protect the red seaweeds from strong waves and current action, they are farmed in a tube net with an initial diameter of 75 mm. The diameter of tube nets with mature seaweeds can reach up to 300 mm, as shown in Figure 3. In order to accelerate the growth of seaweed, the platform is lowered to deep water levels for nutrients and cold waters by using a system comprising a service vessel, three buoys, winches, and concentrated weights, as illustrated in Figure 2b. PV solar panels are installed on the roofs of the service vessel and buoys (as seen in Figure 1) to provide clean power.
The design parameters of the considered seaweed platform are presented in Table 1. Polypropylene ropes are adopted for mooring systems and to carry seaweed tube nets. The mooring rope diameter is 40 mm and has a breaking strength of 257 kN [14]. The rope for carrying the seaweed tube nets has a diameter of 20 mm and a breaking strength of 74.95 kN [15].
The platform is sited in a water depth of 190 m and subjected to four regular sea states:
  • Sea state S1 with wave height H = 1 m, wave period T = 3 s;
  • Sea state S2 with wave height H = 2 m, wave period T = 4 s;
  • Sea state S3 with wave height H = 3 m, wave period T = 6 s;
  • Sea state S4 with wave height H = 4 m, wave period T = 8 s.
The seaweed platforms and mooring lines are modeled using the software AquaSim (v. 2.18). Details of the AquaSim theory and validation are available from its website [16,17]. In the AquaSim simulations, the HDPE pipes and brackets were modeled using beam elements, while truss elements were employed for modeling the mooring lines and chains. The restoring effects of the service vessel and the buoys were modeled using spring elements. The seaweed tube nets are assumed to be fully filled by seaweed, and the tube nets (with seaweed inside) are modeled as solid bars with a diameter equal to that of the tube nets. The modeling of the seaweed tube net at different growth stages of seaweed is not analyzed herein, but it can be readily achieved by modeling the solid bar with an equivalent smaller diameter. Additionally, future studies will aim for more-accurate estimates of loading from waves/currents on seaweed tube nets upon obtaining actual field test results for calibration.
AquaSim estimates the wave loads on the structure using the Morison equations. The coefficients for drag and mass were set to 1 and 2, respectively, based on earlier numerical and experimental studies conducted on floating fish cages made of HDPE [18].

3. Hydroelastic Responses of Unpressurized HDPE Platforms in Floating Condition

Consider the floating seaweed platform described in Section 2. We study hydroelastic responses of the platform for three cases:
  • Case 1: HDPE pipes filled with air and without seaweed lines;
  • Case 2: HDPE pipes filled with seawater and without seaweed lines;
  • Case 3: HDPE pipes filled with seawater and with seaweed lines.
The case of the platform without seaweed lines illustrates the platform configuration before the installation of young seaweed lines and after harvesting mature seaweed; whereas the HDPE platform with seaweed lines illustrates the platform configuration after the installation of young seaweed lines and before seaweed harvesting.
Figure 4 shows the locations where the von Mises stress and horizontal/vertical displacements will be determined. Figure 5 presents the amplitudes of horizontal displacements of HDPE platforms in the floating condition and in the four sea states. It can be seen from Figure 5 for the simulation cases 1 and 2 that HDPE pipes with seawater can help reduce the horizontal displacements when compared to their counterpart pipes with air. The horizontal displacement can be reduced by up to 70% in sea states S1 and S2, and 45% in sea state S4. In sea state S3, the horizontal displacements over the entire platform still can be reduced significantly owing to the filled water, although the displacement reduction is minimal at the aft of the platform, as seen in Figure 5c. Figure 5 also shows that the horizontal displacements are well controlled when the seaweed lines are installed. For the platform with seaweed lines (i.e., Case 3), the horizontal displacement amplitudes are the smallest within the three simulation cases. When compared to the case of pipes filled with air and no seaweed lines (i.e., Case 1), the horizontal displacements in Case 3 can be reduced by 85% in S1, 75% in S2, 45% in S3, and 70% in S4. In addition, the platform with seaweed lines shows the most-regular horizontal displacements, with amplitudes not changed significantly over the entire platform, as seen in the other two simulation cases. This indicates the high effectiveness of seaweed lines in controlling the platform’s horizontal displacements.
Figure 6 presents the amplitudes of vertical displacements of HDPE platforms in the floating condition and in the four sea states. It can be seen from Figure 6 that the vertical displacements in S1 and S2 can increase/decrease slightly when the HDPE pipes are filled with seawater (i.e., Case 2) instead of air (i.e., Case 1). In S3 and S4, the vertical displacements of the platform with filled seawater are considerably higher than those of the platform with filled air. In S3, the platform with filled seawater shows a 30% increase in vertical displacements. In S4, the increase in vertical displacements is about 15%. The increase in vertical displacements may result from the increased inertial forces due to the additional weight of filled water. Figure 6 also shows that in the four examined sea states, the inclusion of seaweed lines to the platform results in reducing the vertical displacements of the platform by up to 23% as compared to the case of pipes filled with seawater but no seaweed lines. This reduction may be attributed to the restraint created by seaweed lines connecting different portions of the seaweed platform.
Figure 7 presents the maximum von Mises stress in HDPE platforms in the floating condition and in the four sea states. It can be seen from Figure 7 for the simulation Cases 1 and 2 that the von Mises stresses can increase/decrease when the HDPE pipes are filled with seawater (i.e., Case 2) instead of air (i.e., Case 1). However, once the seaweed lines are installed, the von Mises stresses decrease significantly as the lines stiffened the circular HDPE pipes from deformation, thereby reducing the stresses. When compared to the results of Case 2, the platform with seaweed lines shows a stress reduction of 38% in S1, 21% in S2, 57% in S3, and 45% in S4. For the platform with seaweed lines, the von Mises stresses are the highest for sea states S1 and S2, where the maximum stresses are about 9 to 9.5 MPa. The maximum von Mises stress is lower in the other sea states, i.e., up to 6.5 MPa in S3 and 4.5 MPa in S4. These results can be explained by the incident wave steepness H/λ, where H is the wave height and λ is the wavelength. The wave steepness is H/λ = 0.07 for S1, 0.08 for S2, 0.05 for S3, and 0.04 for S4. It can be seen that sea states S1 and S2 have the highest wave steepness, which explains why the platform has the higher maximum stresses.
Figure 8 shows the top views of deformed platform shapes in the four sea states and when the pipes are filled with air (Case 1), seawater (Case 2), or seawater with the presence of seaweed lines (Case 3). For Case 1, the deformations are seen to significantly increase from sea state S1 to sea state S4, with the deformations becoming excessive in S3 and S4. Thus, the use of the platform filled with air should be limited to lower sea states, e.g., S1 and S2. For Case 2, where the HDPE pipes are filled with seawater, the deformations are significantly reduced as compared to Case 1. However, they still increase significantly from sea state S1 to S4. In sea states S3 and S4, the platform experiences an oval shape, with largest displacements at the fore and aft of the platform. Note that apart from filling pipes with seawater, some other solutions with which to reduce the platform deformations include increasing the pipe’s cross-section size and using a mooring system that can keep the platform in place more firmly at multiple locations.
Figure 8 also shows that the deformations of the platform can be reduced significantly by having seaweed lines. Note that in sea states S3 and S4, the platform with seaweed lines remains in an original circular shape because the wave steepnesses are less than the those associated with sea states S1 and S2. This finding shows that seaweed lines do not just carry the seaweed tube nets but also play an important role in enhancing the stiffness of the HDPE rings against deformation. When installing the seaweed line, it should be kept taut to restrain the deformation of HDPE platforms. Both the lines and their connections to the platform should be designed to withstand the maximum line tensions.
Figure 9 presents the maximum line tensions in the four sea states at the nine locations at which the seaweed lines are connected. The line tension can reach nearly 3 kN, which is relatively small compared to the breaking strength of 74.95 kN of the line, as provided in Section 2. The lines connecting to the platform’s fore and the platform’s aft show the largest tension forces within the seaweed line system.

4. Hydroelastic Responses of Unpressurized HDPE Platforms in Submerged Conditions

The proposed seaweed platform may be lowered to a hundred meters below the water surface to allow the seaweed to be soaked in cold nutrient-rich waters. This section aims to study the hydroelastic responses of the HDPE platform in submerged conditions. In the analysis, the platform’s pipes are filled with seawater.
Figure 10 shows the horizontal displacement amplitudes of the seaweed platform at different submergence depths in sea states S2 and S4. The simulations were conducted with cylindrical buoys (of height 2 m and diameter 2 m). It can be seen that the horizontal displacement is reduced significantly when the platform is submerged. By submerging the platform 10 m below the water surface, the horizontal displacements decrease by up to 50% in sea state S2 and 57% in sea state S4. In sea state S2, the horizontal displacements further decrease when the submergence depth increases. At 100 m submergence depth, the horizontal displacement is only about 25% of the displacement at the free surface. In sea state S4, larger displacement reductions can be seen where the horizontal displacement at 100 m submergence depth is only about 10% of the displacements at the free surface. The significant reductions in the platform’s horizontal displacements when submerged are expected because the dynamic wave pressure is largest at the free surface and decreases exponentially with increasing water depth. For example, for an 8 s wave period and at a site with a water depth of 190 m, the dynamic wave pressure decreases by 70% from the surface wave pressure when submerged at 20 m and by 95% at 50 m below the water surface [19].
Figure 11 shows the vertical displacement amplitudes of the seaweed platform at different submergence depths in sea states S2 and S4. It can be seen that the vertical displacement is also reduced significantly when the platform is submerged. By submerging the platform 10 m below the water surface, the vertical displacements decrease by up to 50% in sea state S2 and 30% in sea state S4. In sea state S2, the vertical displacements decrease only slightly when the submergence depth further increases. At 100 m submergence depth, the vertical displacement is still about 85% of the displacement at 10 m submergence depth. The vertical displacements at 25–100 m submergence depths are almost the same. The motions of the platform with submergence depths ≥ 25 m should be primarily driven by the motions of the surface buoys. This is because these depths are larger than half of the wavelength λ ≈ 25 m in sea state S2, where the water motions are only minimal. In sea state S4, the vertical displacements decrease significantly as the submergence depth increases up to 50 m. When the submergence depth increases beyond 50 m, the change in vertical displacements is not considerable. This is because such submergence depths are larger than half of the wavelength λ ≈ 100 m in sea state S4.
Figure 12 shows the maximum von Mises stresses of the seaweed platform at different submergence depths. In sea state S2, Figure 12a shows that the von Mises stress in the middle portion of the platform can be reduced by 15% when the platform is submerged. However, there are stress concentrations at the fore and aft of the platform that are connected to the surface buoys/vessel via ropes. At these locations, the maximum stresses can reach the maximum stress level seen in the floating platform. Figure 12a also shows that the von Mises stresses are changed only slightly for different submergence depths, which may result from the corresponding relatively small changes in displacements as seen in Figure 10a and Figure 11a.
In sea state S4, Figure 12b shows that the maximum von Mises stress can increase when the platform is submerged. The largest stresses are observed at the fore and aft of the platform that are connected to the surface buoy and service vessel. This indicates stress concentrations at those locations. However, the maximum stress is below 8.8 MPa. Figure 12b also shows that the largest stress is seen for the submergence depth of 10 m, and the stress fluctuated slightly when the submergence depth further increases to 100 m.
Figure 13 shows the side and top views of the deformed platform at the free surface and different submergence depths in sea state S2. It can be seen that the floating platform is subjected to significant bending due to surface waves. The bending deformations are reduced considerably when the platform is submerged 10 m below the water surface. The platform’s responses are changed only slightly when the submergence depth increases to 100 m. This is because when the submergence depths are larger than half of the wavelength, the platform’s responses are driven primarily by the motions of the surface vessel and buoys.
Figure 14 shows the side and top views of the deformed platform at the free surface and different submergence depths in sea state S4. It can be seen that the floating platform experiences minimal bending, with rigid body motions. This is expected because the length of the platform is about 40 m, whereas the wavelength in sea state S4 is 100 m. Owing to minimal bending, the stress in the floating platform is below 5 MPa. When the platform is submerged 10 m below the water surface, more significant bending is observed. The bending is primarily due to the combination of wave action and the motions of surface vessels/buoys that cause concentrated loads on the platform. The von Mises stress for the platform at the 10 m submergence depth is higher than the stress in the platform at the free surface, primarily due to the stress concentrations at the platform’s portions with connections to the surface vessel/buoys. When the submergence depth increases from 10 m to 25 m, the platform’s deformations are reduced. This may be because of the reductions in wave action with respect to the water depth. Correspondingly, the von Mises stresses are reduced, with less significant stress concentrations. As the submergence depth increases further to 100 m, the platform is almost planar. This may be because for such deep elevations, the water particle motions are minimal, and the platform’s responses are mainly affected by the motions of the surface vessel/buoys.

5. Effects of Pressurization of Filled Water on Responses of HDPE Platform

Under cyclical sinusoidal wave action, the stress in the HDPE pipes alternates between tensile and compressive. It is important to ensure that both tensile and compressive stresses are below their allowable levels. For HDPE pipes, the recommended allowable compressive stress is 8 MPa, while the allowable tensile stress is 23 MPa [20,21]; noting that HDPE pipes perform better when they are in tension. This feature reflects the actual behaviors of HDPE materials, where HDPE pipes tend to be damaged on the compression side instead of on the tension side. The damage on the compression side of HDPE pipes used for fish cages can be seen in previous numerical and experimental studies [22,23].
First, consider the platform with unpressurized filled water, as in Section 3, in a floating condition and in sea states S1 and S2. Figure 15 shows the maximum normal compressive and tensile stresses at nine locations in the HDPE ring. It can be seen that the tensile and compressive stresses at the same locations are almost the same in magnitude. However, while the tensile stress is well below its allowable level, the compressive stress may be larger than the allowable compressive stress at some locations. This result indicates the need to reduce the compressive stress in order to minimize the damage probability. This can be achieved by increasing the HDPE pipe size, limiting the use of existing platform configuration to lower sea states, or imposing a prestressed tension to the pipe, which can be achieved by filling the pipe with pressurized sea water.
As the AquaSim software is unable to account for the pressure of the filled water, we need to adopt an indirect modeling approach to impose the internal pressure. In this indirect approach, we first calculate the stress in the HDPE pipes caused by the internal pressure. Next, we need to determine the stress in the HDPE pipes due to waves using the AquaSim software. Total stress will be the combination of the stresses due to internal pressure and due to waves.
The stresses caused by internal pressure only can be calculated using the analytical formulae provided in Appendix A. These formulae have been developed and validated for toroidal shells.
With regard to calculating the stress in pressurized HDPE pipes due to waves, we utilize the AquaSim software, but we need to calibrate the material’s Young modulus to account for the effect of pressurization on the HDPE pipe’s rigidity (if considerable). To check if such an effect is considerable, we conduct free vibration analyses via the ANSYS software (v. 2023 R1) for the pressurized HDPE pipes. Consider a half circular HDPE ring with two fixed end supports. The ring has a diameter of 40 m, and the pipe’s cross-section has a diameter of 225 mm and SDR 17. The pipe’s mass density is modified so that it is equivalent to the case where the pipe is filled with water. The interior surface of the pipe is subjected to an internal pressure of 1 MPa. Figure 16 presents the mode shapes and vibration frequencies for the first three modes of unpressurized and 1 MPa pressurized pipes. It can be seen that the pressurization has a negligible effect on the mode shapes and vibration frequencies, indicating that the flexural rigidity of the pipes with and without pressurized water is about the same. Based on this finding, we will model the pressurized HDPE platform as an unpressurized HDPE platform under wave action in AquaSim. The normal stress obtained from AquaSim will later be combined with the stress from the analytical formulae in Appendix A to account for the effect of the internal pressure on the stress state of the pipe.
Figure 17 shows the normal stresses in the floating platform where the HDPE pipes are filled with 1 MPa pressurized water in seas state S1 and S2. The presented stresses are those combined from the stress due to internal pressure (calculated by using the analytical formulae) and the stress due to waves only (determined via the AquaSim software). Figure 17 shows that by pressurizing the filled seawater at 1 MPa, the tensile stress increases while the compressive stress is reduced by 3.8 Mpa, which is caused by presence of 1 MPa internal pressure. The resulting compressive and tensile stresses due to waves and pressurization are kept within the allowable limits, thereby reducing the possibility of pipe damage.
Figure 18 presents the allowable stresses for the HDPE pipes under wave action with respect to filled seawater pressure. These allowable stresses are those caused by waves only, without combination with the stress caused by the internal pressure. After combining these allowable stresses with the stress caused by the internal pressure, the resulting allowable stresses are 8 MPa for compressive stress and 23 MPa for tensile stress, which are determined from the HDPE materials. It can be seen from Figure 18 that when pressurized, the HDPE pipe can take more compressive stress (i.e., up to 50%). Under wave action, as the compressive and tensile stresses have almost the same magnitude (e.g., see Figure 15), the bearing capacity of the HDPE pipes tends to be determined by the compressive stress because its allowable level has a smaller magnitude. Thus, as the allowable compressive stress increases with respect to filled seawater pressure, the bearing capacity of the pipes increases correspondingly.
Figure 19a shows the normal stress caused by the internal pressure for different SDR pipes with respect to filled seawater pressure. Note that for the same SDR, the normal circumferential stress caused by the internal pressure is the same for different pipes’ outer diameters, as seen in the formulae in Appendix A. Figure 19b shows the corresponding allowable stress due to waves only for different SDR pipes. It can be seen that through pressurizing the filled seawater, the pipe can take compressive stress up to 12 MPa, representing an increase of 50% when compared to the allowable compressive stress of 8 MPa for unpressurized pipes.

6. Conclusions

This paper presents the hydroelastic responses of circular seaweed platforms on the water surface and at various submergence depths under wave action. Considered in this study are circular HDPE pipes filled with (a) air, (b) seawater, and (c) pressurized (1 MPa) seawater. The hydroelastic analysis was conducted using AquaSim software (v. 2.18).
It is found that filling HDPE pipes with seawater helps in reducing the hydroelastic responses of the platform. Moreover, the seaweed grow-out lines are important in providing additional stiffness to the circular HDPE pipes against deformation. Furthermore, by pressurizing (say, by 1 MPa) the filled waters in HDPE pipes, the pipes’ compressive bearing capacity can be increased by up to 50%.
Future studies will involve physical model and field tests to calibrate and validate the AquaSim numerical model, which will then be used to optimize the design of the circular seaweed platform.

Author Contributions

Pressurization conceptualization, B.v.H; analysis conceptualization, H.-P.N. and C.-M.W.; methodology, H.-P.N. and C.-M.W.; formal analysis, H.-P.N., C.-M.W. and B.v.H.; writing—original draft preparation, H.-P.N. and C.H.; writing—review and editing, C.-M.W., B.v.H. and C.H.; funding acquisition, C.-M.W. and B.v.H. All authors have read and agreed to the published version of the manuscript.

Funding

The authors acknowledge the financial support of the Blue Economy Cooperative Research Centre, established and supported under the Australian Government’s Cooperative Research Centres Program, grant number CRC-20180101.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

The data presented in this study are available from the corresponding author on reasonable request.

Acknowledgments

The authors would like to thank Aquastructures AS for granting access to their amazing AquaSim software (https://aquasim.no).

Conflicts of Interest

Brian von Herzen was employed by The Climate Foundation. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Appendix A

As we consider pipes with internal pressure, the following stress formulae and maximum allowable stresses are briefly reviewed. Consider a circular HDPE pipe with internal pressure p, as shown in Figure A1. The meridional stress σm and the circumferential stress σc in the pipe (defined in Figure A1) are given by [24]
σ m = p d 4 t 2 r + d 2 sin θ r + d 2 sin θ ,
σ c = p d 4 t ,
where d is the cross-section inner diameter of pipes and t is the wall thickness.
The upper bound of meridional stress is at point M (as indicated in Figure A1) and is given by
σ m = p d 4 t 2 r d 2 r d 2 > p d 2 t ,
whereas the lower bound of meridional stress is given by
σ m = p d 4 t 2 r + d 2 r + d 2 < p d 2 t .
Figure A1. Stresses in circular pipe under internal pressure.
Figure A1. Stresses in circular pipe under internal pressure.
Jmse 12 01437 g0a1
Figure A2 presents the maximum allowable internal pressure for pipes with different standard ratios (SDR) [25]. Pipes with smaller SDRs (or larger wall thickness) can take a higher internal pressure. The maximum allowable internal pressure for HDPE pipes is between 0.4 MPa to 2.5 MPa.
Figure A2. Maximum allowable internal pressure with respect to standard dimension ratios.
Figure A2. Maximum allowable internal pressure with respect to standard dimension ratios.
Jmse 12 01437 g0a2
Figure A3 presents the upper and lower bounds of meridional stresses and circumferential stress in pipes with (a) 225 mm cross-section diameter, SDR 17, and r = 20.25 m; (b) 160 mm cross-section diameter, SDR 17, and r = 20 m. The difference between the upper and lower bounds of meridional stresses is minimal, resulting from a large ring diameter when compared to the pipe cross-section diameter. The meridional stresses are almost twice the circumferential stress. In addition, the stresses for the 225 mm pipe are almost identical to those for the 160 mm pipe. This is expected as the pipes’ SDR is the same.
Figure A3. Upper and lower bounds of meridional stresses and circumferential stresses in pipes with different internal pressures: (a) 225 mm cross-section diameter, SDR 17, and r = 20.25 m; (b) 160 mm cross-section diameter, SDR 17, and r = 20 m.
Figure A3. Upper and lower bounds of meridional stresses and circumferential stresses in pipes with different internal pressures: (a) 225 mm cross-section diameter, SDR 17, and r = 20.25 m; (b) 160 mm cross-section diameter, SDR 17, and r = 20 m.
Jmse 12 01437 g0a3

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Figure 1. Overhead drone photo of 1000 m2 seaweed cultivation infrastructure with single-point mooring system tested in the Philippines in 2023. Photo courtesy of Eric Smith at The Climate Foundation.
Figure 1. Overhead drone photo of 1000 m2 seaweed cultivation infrastructure with single-point mooring system tested in the Philippines in 2023. Photo courtesy of Eric Smith at The Climate Foundation.
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Figure 2. The Climate Foundation’s design of the test submersible seaweed platform: (a) plan view; (b) side view in submerged state.
Figure 2. The Climate Foundation’s design of the test submersible seaweed platform: (a) plan view; (b) side view in submerged state.
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Figure 3. (a) Initial tube nets with Kappaphycus alvarezii. (b) Tube nets at harvest. Photo courtesy of The Climate Foundation.
Figure 3. (a) Initial tube nets with Kappaphycus alvarezii. (b) Tube nets at harvest. Photo courtesy of The Climate Foundation.
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Figure 4. Locations where stresses and displacements are determined.
Figure 4. Locations where stresses and displacements are determined.
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Figure 5. Horizontal displacement amplitude in four sea states: (a) S1; (b) S2; (c) S3; (d) S4.
Figure 5. Horizontal displacement amplitude in four sea states: (a) S1; (b) S2; (c) S3; (d) S4.
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Figure 6. Vertical displacement amplitude in four sea states: (a) S1; (b) S2; (c) S3; (d) S4.
Figure 6. Vertical displacement amplitude in four sea states: (a) S1; (b) S2; (c) S3; (d) S4.
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Figure 7. Maximum von Mises stress in four sea states: (a) S1; (b) S2; (c) S3; (d) S4.
Figure 7. Maximum von Mises stress in four sea states: (a) S1; (b) S2; (c) S3; (d) S4.
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Figure 8. Top views of deformed platforms with HDPE pipes filled with air, seawater, and with filled water and seaweed lines. The deformed shapes are taken at the time showing the largest horizontal displacements.
Figure 8. Top views of deformed platforms with HDPE pipes filled with air, seawater, and with filled water and seaweed lines. The deformed shapes are taken at the time showing the largest horizontal displacements.
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Figure 9. Maximum tensions in seaweed lines at the connections to HDPE ring.
Figure 9. Maximum tensions in seaweed lines at the connections to HDPE ring.
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Figure 10. Horizontal displacement amplitude of platform at different submergence depths in sea state: (a) S2; (b) S4.
Figure 10. Horizontal displacement amplitude of platform at different submergence depths in sea state: (a) S2; (b) S4.
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Figure 11. Vertical displacement amplitude of platform at different submergence depths in sea state: (a) S2; (b) S4.
Figure 11. Vertical displacement amplitude of platform at different submergence depths in sea state: (a) S2; (b) S4.
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Figure 12. Maximum von Mises stresses in platform at different submergence depths in sea state: (a) S2; (b) S4.
Figure 12. Maximum von Mises stresses in platform at different submergence depths in sea state: (a) S2; (b) S4.
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Figure 13. Deformed platform in floating and submerged conditions in sea state S2 with buoy size 2 m × 2 m: floating state; 25 m submergence; 50 m submergence; 100 m submergence.
Figure 13. Deformed platform in floating and submerged conditions in sea state S2 with buoy size 2 m × 2 m: floating state; 25 m submergence; 50 m submergence; 100 m submergence.
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Figure 14. Deformed platform in floating and submerged conditions in sea state S4 with buoy size 2 m × 2 m: floating state; 25 m submergence; 50 m submergence; 100 m submergence.
Figure 14. Deformed platform in floating and submerged conditions in sea state S4 with buoy size 2 m × 2 m: floating state; 25 m submergence; 50 m submergence; 100 m submergence.
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Figure 15. Normal stresses in HDPE pipe with unpressurized seawater in floating condition: (a) sea state S1; (b) sea state S2.
Figure 15. Normal stresses in HDPE pipe with unpressurized seawater in floating condition: (a) sea state S1; (b) sea state S2.
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Figure 16. First three non-zero natural frequencies and mode shapes for pipes filled with seawater and 1 MPa pressurized seawater.
Figure 16. First three non-zero natural frequencies and mode shapes for pipes filled with seawater and 1 MPa pressurized seawater.
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Figure 17. Normal (i.e., circumferential) stress for the floating HDPE platform filled with 1 MPa pressurized water: (a) sea state S1; (b) sea state S2.
Figure 17. Normal (i.e., circumferential) stress for the floating HDPE platform filled with 1 MPa pressurized water: (a) sea state S1; (b) sea state S2.
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Figure 18. Allowable tensile and compressive stresses due to waves only with respect to filled seawater pressure.
Figure 18. Allowable tensile and compressive stresses due to waves only with respect to filled seawater pressure.
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Figure 19. (a) Normal stress caused by internal pressure. (b) Allowable compressive stress caused by waves only with different levels of internal pressure and pipe SDR.
Figure 19. (a) Normal stress caused by internal pressure. (b) Allowable compressive stress caused by waves only with different levels of internal pressure and pipe SDR.
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Table 1. Design parameters of seaweed platform.
Table 1. Design parameters of seaweed platform.
ComponentsDescriptionMagnitudeUnit
HDPE pipes, standard dimension ratio (SDR) = 17Diameter of outer ring40.5m
Diameter of inner ring40m
Cross-section diameter of outer ring0.225m
Cross-section diameter of inner ring0.16m
Young’s modulus900MPa
Poisson’s ratio0.4-
Mass density958kg/m3
Pressure of filled seawater0 to 1.2MPa
Polypropylene ropes;
anchor chains
Mooring rope diameter0.04m
Seaweed line (or rope) diameter0.02m
Rope Young’s modulus1500MPa
Rope mass density910kg/m3
Anchor chain diameter0.032m
Chain Young’s modulus54.4GPa
Anchor chain mass16kg/m
Cylindrical buoyBuoy diameter2m
Buoy height2m
Tube netsDiameter of tube nets7.5cm
Spacing between tube nets 1.2m
Submergence depth0 to 100m
Mass density of seaweed1030kg/m3
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Nguyen, H.-P.; Huang, C.; von Herzen, B.; Wang, C.-M. Numerical Study on Hydroelastic Responses of Submersible High-Density Polyethylene Circular Seaweed Platforms Held by Single-Point Mooring System and Buoys. J. Mar. Sci. Eng. 2024, 12, 1437. https://doi.org/10.3390/jmse12081437

AMA Style

Nguyen H-P, Huang C, von Herzen B, Wang C-M. Numerical Study on Hydroelastic Responses of Submersible High-Density Polyethylene Circular Seaweed Platforms Held by Single-Point Mooring System and Buoys. Journal of Marine Science and Engineering. 2024; 12(8):1437. https://doi.org/10.3390/jmse12081437

Chicago/Turabian Style

Nguyen, Huu-Phu, Chenxuan Huang, Brian von Herzen, and Chien-Ming Wang. 2024. "Numerical Study on Hydroelastic Responses of Submersible High-Density Polyethylene Circular Seaweed Platforms Held by Single-Point Mooring System and Buoys" Journal of Marine Science and Engineering 12, no. 8: 1437. https://doi.org/10.3390/jmse12081437

APA Style

Nguyen, H. -P., Huang, C., von Herzen, B., & Wang, C. -M. (2024). Numerical Study on Hydroelastic Responses of Submersible High-Density Polyethylene Circular Seaweed Platforms Held by Single-Point Mooring System and Buoys. Journal of Marine Science and Engineering, 12(8), 1437. https://doi.org/10.3390/jmse12081437

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