## Abstract

The investigation of innovative macroalgal cultivation is important and needed to optimize farming operations, increase biomass production, reduce the impact on the ecosystem, and lower system and operational costs. However, most macroalgal farming systems (MFSs) are stationary, which need to occupy a substantial coastal area, require extensive investment in farm infrastructure, and cost high fertilizer and anchoring expenses. This study aims to model, analyze, and support a novel binary species free-floating longline macroalgal cultivation concept. The expected outcomes could provide a basis for the design and application of the novel MFS to improve biomass production, decrease costs, and reduce the impact on the local ecosystem. In this paper, Saccharina latissima and Nereocystis luetkeana were modeled and validated, and coupled with longline to simulate the binary species MFS free float in various growth periods and associated locations along the US west coast. The numerical predictions indicated the possibility of failure on the longline and breakage at the kelp holdfasts is low. However, the large forces due to an instantaneous change in dynamic loads caused by loss of hydrostatic buoyancy when the longline stretches out of the water would damage the kelps. Buoy-longline contact interactions could damage the buoy, resulting in the loss of the system by sinking. Furthermore, the kelp-longline and kelp-kelp entanglements could potentially cause kelp damage.

## 1 Introduction

Marine macroalgae (commonly known as seaweed) in the open ocean is a clean and valuable resource because it takes advantage of natural nutrients and ocean water, without significantly altering the physical environment, and may enhance natural habitats [1,2]. Harvested macroalgae is widely used in food production including animal feed and fertilizer [3]. In recent years, it was deemed an important source of biofuel to contribute to the global energy demands [4,5]. However, the current state of macroalgae culture is not capable to achieve the scale, efficiency, and production costs necessary to support a seaweed-to-fuels industry [6]. An increasing amount of research has focused on improving the macroalgae farming system (MFS), including species, cultivating methods, and system configurations to optimize farming operations, increase biomass production, and lower system and operational costs. In offshore macroalgal farming, the maximum hydrodynamic load effects under high sea states are often identified as a major cause of MFS failures such as plant damage and mass loss [7,8]. A comprehensive study to understand hydrodynamic load effects (or fluid macroalgae interaction) and to develop advanced techniques to optimize farming operations and lower system costs is needed and timely.

Macroalgal cultivation has been conventionally achieved using an anchored longline culture technique in the open sea [9,10]. The application of the longline system is responsible for more than 95% of global macroalgae production [11]. However, the traditional stationary longline farming system usually occupies a substantial coastal area and requires extensive investment in farm infrastructure. Moreover, the application of fertilizer to stationary seaweed farming systems is difficult given the dynamic nature of the ocean [11]. Fertilizers applied to seaweeds in the ocean will rapidly disperse, making seaweed culture costly, and creating the potential of polluting the coastal environment. Furthermore, a sufficiently heavy and expensive mooring system is often needed to provide sufficient anchoring capability to avoid excessive drifting of the structure in the rough sea [12,13]. In comparison, an unmoored longline MFS based on the completely free-floating concept avoids occupying coastal space “permanently”. Free-floating along nutrient-rich ocean currents could reduce the large cost of fertilizer and eliminate potential pollution of the ocean environment. The high cost of anchoring could be also removed without losing farm structures and macroalgae from large current and wave forces during storms. To the authors’ knowledge, the only free-floating MFS application is taking place with *Sargassum* rafts in the northern Caribbean Basin and the Atlantic Ocean [14]. It appears that research and investment to date in free-floating longline MFS are insufficient and should be studied more extensively.

An essential component to achieving large-scale, economically viable macroalgal production is determining the species and cultivated method best fitted to MFS. One of the most widely cultivated seaweed applied to the longline farming system commercially is *Saccharina latissima* (sugar kelp) [9,15,16]. Thorough studies and research on hydrodynamic effects and coefficients of sugar kelp were performed through laboratory experiments [17–20], field tests [21,22], and numerical modeling [23–25]. The dynamics of sugar kelp blades in waves can be modeled using the cable model by representing the structure as a cantilever beam discretized as blade segments [26]. Another species, *Nereocystis luetkeana* (bull kelp), has not been commercially cultivated, but the potential for commercial utilization has been recognized [27]. To date, most studies on hydrodynamic effects and coefficients are conducted using experimental methods [27–30]. The numerical model of bull kelp is simplified as a point element (*pneumatocyst*) tethered by a flexible rope (stipe), and hydrodynamic force due to fluid-kelp interaction was calculated using the Morison equation [31–34]. To maximize biomass production, together with sugar kelp and bull kelp, a binary species cultivation method could be applied. In detail, sugar kelp and bull kelp could be grown simultaneously in a single line to maximize nutrient utilization and achieve a significant increase in biomass [35]. Bull kelp growth forms a surface canopy, while sugar kelp forms a non-floating understory. These two species occur in nature in this double canopy configuration. Obviously, the hydrodynamic effects on each species have been well studied. However, the study on binary cultivation of sugar kelp and bull kelp seems lacking, and extensive research is needed. Moreover, compared to the physical experimental method, a numerical study is preferred to support the novel MFS concept because it can offer advantages including (1) efficiency in both simulation time and cost, and (2) capability for obtaining parameters and results (e.g., forces, velocities, displacements, etc.) in the whole system domain.

The goal of this work is to model, analyze, and support a novel offshore binary species sensor-equipped carbon-fiber longline MFS. The MFS is seeded and free floats with nutrient-rich ocean currents southward along the US west coast. The unmoored buoys, supporting lines, longlines, and kelps were modeled, and the maximum hydrodynamic load effects on the system under high sea states were examined in different growth periods and associated locations. A dynamic response analysis was performed on the longline and the kelp to identify critical issues and potential failure modes of the system. The expected outcomes could provide a scientific basis for the design and application of the novel free-floating longline system and binary species cultivation method to improve biomass production and lower system and operational costs. The methodology and parameters to model the *Saccharina latissima*, *Nereocystis luetkeana*, and free-floating longline are presented in Sec. 2. Numerical model developments and validations are presented in Sec. 3. The simulation of integrated *Saccharina latissima*, *Nereocystis luetkeana*, and free-floating longline, analysis of the predictive results, and potential failure modes of the kelps and system are presented in Sec. 4. Finally, conclusions are presented in Sec. 5.

## 2 Methodology

A generic physical model of the MFS with both sugar kelp and bull kelp being cultivated on a 5 km, high-strength, extremely durable, recycled carbon fiber longline is introduced in this section (see Fig. 1). The mathematical model includes governing equations and solution procedures based on simplified physical models and assumptions derived.

### 2.1 Physical Models and Simplifications.

In a typical physical farm system, *Saccharina latissima* (sugar kelp) is grown very densely with up to hundreds of plants per meter on a single longline. In such conditions, it is not possible to model each sugar kelp plant individually. However, the sugar kelp is grown (or attached) to the longline in an evenly spaced cluster as a consequence of the seeding process [20]. Therefore, each cluster of sugar kelp can be modeled using a flexible line with specific parameters [25]. Because the total mass of the kelp can be approximated based on the growth period, we can calculate an effective diameter using sugar kelp density of 1092 kg/m^{3} [22], resulting in net buoyancy forces of the clusters consistent with the physical system. However, this model might slightly change the hydrodynamic properties due to the simplifications of the shape of the clusters. Hence, the determination of optimal drag coefficient and effective drag diameter equivalent to the physical model to minimize such induced approximation errors is essential. The optimal drag coefficient can be identified using numerical simulations. Unlike the numerical clusters model, the physical model of (Fig. 5 in Ref. [20]) sugar kelps is dense and does not have clear spatial separation. Therefore, to better calculate the drag force, the effective drag diameter may (and often) differ from the cluster (or flexible line) diameter. With the assumption that the sugar kelps will be evenly planted on the longline, the effective drag diameter can be calculated using the length of the longline divided by the total number of clusters. The cluster model (or line object) is considered elastic, with bending and tensile Young’s modulus of 4.9 × 10^{6} Pa and 4.7 × 10^{6} Pa, respectively [22]. In this modeling method, the length of the sugar kelp is assumed to be uniform, and kelp-kelp interaction is not considered.

*Nereocystis luetkeana* (bull kelp) is easily recognized by the long stipe extending from the holdfast attached to the sea bottom (see Fig. 2(a)). Up to 64 laminar blades are attached to the floating gas-filled hollow bulb [29–31]. The physical model can be simplified as an elastic line structure (stipe) with a small floater attached at the top (see Fig. 2(b)). The distributed forces acting on the pneumatocyst and blades as well as the mass are assumed to act at the small floater. This model has been widely used in the analysis of the hydrodynamic properties of bull kelp [31,33]. In general, allometric relationships (Eq. (18) in Ref. [31]) are applied to approximate the character of interests including overall mass, blade area, minimum equivalent stipe radius, and net buoyancy using stipe length. The bull kelp plant length and blade area increase until the pneumatocyst approaches the water surface. In the numerical bull kelp plant model (see Fig. 2(b)), the minimum equivalent stipe diameter of the physical plant is applied as the line structure diameter and the model of the stipe is solid and homogeneous (constant diameter along the stipe with proper material property adjustments). The general drag coefficient for a slender structure, 1.2, is used in calculating the drag force on the stipe. The stipe stiffness is 1.2 × 10^{7} Pa [33]. The density of the bull kelp material is 1090 kg/m^{3} [29]. The mass and volume of the floater (*pneumatocyst*) attached at the top of the line can be calculated based on the net buoyancy and mass. The blade area is used as the drag area of the floater in computing the drag force. The added mass coefficient of 3.0 is suggested [36].

A high-stiffness recyclable carbon-fiber (rCF) macroalgae longline is selected to avoid entangling ocean animals and self-entanglement during the free drift. The physical properties of the longline include a diameter of 0.015 m, dry mass of 0.19 kg/m, bending stiffness of 1.088 × 10^{5} N·m^{2}, and axial stiffness of 8.241 × 10^{6} N. The normal and tangential drag coefficients are 1.2 and 0.008, respectively, with an added mass coefficient of 1.

### 2.2 Governing Equation and Solving Procedure.

A fully nonlinear time domain finite-element analysis software, OrcaFlex [37], is employed to perform a computational structure dynamic (CSD) analysis. A lumped-mass element is applied to simplify the mathematical formulation and allow quick and efficient development of the software to include additional force terms (e.g., hydrodynamic force) and constraints (e.g., joint connection) on the system. The equilibrium configuration of the system considering the hydrostatic problem of the system initially at rest in the ocean without wave and current is obtained. The results are applied as the initial condition for the following dynamic analysis. Then waves and current are gradually ramped up linearly until a stochastic equilibrium is achieved. Numerical prediction data over a specific simulation period after stochastic stationarity are achieved and collected for analysis and comparison. Compressibility of water and viscous effects are not the main concerns in this study and were neglected in simulations and subsequent analyses.

*t*is the simulation time, $x$ is the structure displacement vector, and $x\u02d9$, $x\xa8$ represent the first- and second-time derivative of the displacement which are velocity and acceleration vectors. Equation (1) is a local equation of motion for each free body and each line node. Solving each of these equations of motion merely requires the inversion of a three-by-three mass matrix. Loads including weight, buoyancy, hydrodynamic drag and added mass effects, tension, shear, and bending acting on each free body or node are calculated separately on the right-hand side.

*C*

_{m}is the inertia coefficient for the object,

*ρ*is the fluid density,

*V*is the volume of the object in fluid, $uf$ is the velocity of the fluid, and $ub$ is the velocity of the object so $u\u02d9f$ and $u\u02d9b$ represent the acceleration of the fluid and acceleration of the body respectively,

*C*

_{a}is the added mass coefficient,

*C*

_{d}is the drag coefficient, and

*A*is the projected area.

Linear wave theory [38] is used to calculate the velocities on the objects and nodes from given wave profiles, and power law method [37] is applied to compute ocean current speed. Under regular wave conditions, the wave height, period, and direction are specified. Irregular waves are obtained using linear superposition of regular waves of distinct amplitudes and periods characterized in the frequency domain by a wave spectrum. In this study, a JONSWAP spectrum [39] is employed to generate irregular waves specified in terms of significant wave height and peak period.

A generalized-*α* implicit time integration scheme is used [40]. At the start of each time-step, the positions and orientations of all objects in the model are known from previous calculations. Then the forces and moments including weight, buoyancy, hydrodynamic loads, tension and shear, bending, and torque and forces applied by links acting on each free body and node are calculated. The equation of motion is solved implicitly (at the end of the time-step) by an iterative solution method for best stability and accuracy.

## 3 Numerical Modeling and Validation

In this section, the sugar kelp cluster model and bull kelp plant model are developed and validated by comparison to experimental data from the literature. The line structures (sugar kelp cluster, bull kelp stipe, supporting line, longline), small volume structure (buoy, weight), and linker, are not discussed in this section because those models have been well applied and validated [37].

### 3.1 Sugar Kelp Clusters Numerical Modeling and Validation.

In this section, the sugar kelp cluster model is developed and validated by comparison to the results from an experiment performed by Endresen et al. [20]. To be consistent with the available experimental data, the numerical model developed contains a 3 m rope with cultivated sugar kelp under different current velocities ranging from 0.25 m/s to 0.85 m/s. The velocities and forces values presented in the reference are average values for each test run over a time interval. The ropes are mounted perpendicular to the experienced water current. Seawater property (*ρ* = 1022.1 kg/m^{3} recommended by [20]) was applied in the simulation to be consistent in the experiment. The submerged depth of all the ropes is 1.1 m. The water depth of the experimental facility is 14 m. The sugar kelp cluster planted to the rope is evenly spaced and the spacing is 0.2 m (five clusters per meter). The measurements of drag forces on the rope for groups n3 and n4 (see Ref. [20], page 5) are selected to validate the simulation results because the plants in those groups are evenly distributed which satisfied the assumption of the mathematical model. The parameters used in the model are listed in Table 1. Different drag coefficients for the sugar kelp cluster are examined and the drag coefficient of 0.45 is confirmed to be applicable. The drag forces computed using the drag coefficient of 0.45 under different current velocities from the simulations for two groups (n3 and n4) are compared to the experimental data (see Fig. 3). The results obtained from numerical simulations are consistent with the experimental measurements for both groups. The total drag force increases with increasing towing velocities. The averaged percentage differences of the drag forces between the experimental data and numerical results are 10.42% in group n3 and 13.69% in group n4, which the accuracy of the results computed from the CSD numerical model is acceptable from an engineering design perspective which generally employed a large design/safety factor.

Group | Plants (–) | Mass (kg) | Length (m) | Clusters (–) | Mass/Cluster (kg) | Equivalent cluster diameter (m) | Equivalent drag diameter (m) |
---|---|---|---|---|---|---|---|

n3 | 615 | 8.34 | 0.597 | 16 | 0.5213 | 0.0319 | 0.2 |

n4 | 546 | 9.09 | 0.686 | 16 | 0.5681 | 0.0311 | 0.2 |

Group | Plants (–) | Mass (kg) | Length (m) | Clusters (–) | Mass/Cluster (kg) | Equivalent cluster diameter (m) | Equivalent drag diameter (m) |
---|---|---|---|---|---|---|---|

n3 | 615 | 8.34 | 0.597 | 16 | 0.5213 | 0.0319 | 0.2 |

n4 | 546 | 9.09 | 0.686 | 16 | 0.5681 | 0.0311 | 0.2 |

### 3.2 Bull Kelp Plant Numerical Modeling and Validation.

The single bull kelp plant model is developed and validated in this section by field measurements data from Ref. [31] with the tensile forces imposed on bull kelp stipe measured using a force transducer in the field. The device was deployed off the shore at Hopkins Marine Station, Pacific Grove, CA. The tensile force was recorded for 30 min. The wave profile is not present in the paper. However, the wave conditions during the test were characterized by a significant wave height of 0.8 m and a peak period of approximately 16 s, combined with a high amplitude surfbeat with a peak period of approximately 200 s. The current at this site is small and can be neglected. To simulate such wave conditions, the JOWSWAP spectrum is used to develop the irregular waves with 0.8 m significant wave height and 16 s peak period. Then a slow modulation method [41] is applied to modify the irregular waves and combined with a high amplitude surfbeat with a peak period of approximately 200 s. The modified irregular waves are used as input for the simulation. The water depth at this site is approximately 10 m and a constant water depth is assumed in the simulation. The total simulation time is 1600 s with 0.5 s time-step to ensure enough data can be obtained to analyze and validate the model. The morphological characteristics of the bull kelp plant tested in the field are listed in Table 2. The bull kelp plant model is built based on the characteristic parameters (see Fig. 2(b)).

Length (m) | 6.57 |

Mass (kg) | 8.00 |

Net buoyancy (N) | 12.51 |

Equivalent minimum stipe radius (m) | 0.0058 |

Blade area (m^{2}) | 4.159 |

Length (m) | 6.57 |

Mass (kg) | 8.00 |

Net buoyancy (N) | 12.51 |

Equivalent minimum stipe radius (m) | 0.0058 |

Blade area (m^{2}) | 4.159 |

The bull kelp plant model is validated by comparison to the mean measured stipe tension and the overall distribution of stipe tensions recorded in the field (see Fig. 4). The time series of stipe tension cannot be used for comparison because there are no measured wave profiles available from the field. The mean measured tension on stipe is 9.99 ± 3.20 N (mean ± SD, *n* = 3112). The mean calculated stipe tension from the simulation is 9.96 ± 2.39 N (mean ± SD, *n* = 3200). The numerical model predicts the mean stipe tension to be very close to the mean value from the field measurement (within 0.03 N). In both the field measurement and numerical prediction, the mean tension is less than the plant’s net buoyancy (12.51 N). Therefore, unlike the sugar kelp, the effect of the net buoyancy force on the bull kelp plant can be significant. The overall distribution of stipe tensions computed from the simulation is shown in Fig. 4 with the distribution recorded in the field. Since the physical wave profile is unavailable, the waves developed based on the characterized parameters may not capture all the large waves (possibly highly nonlinear ones). Therefore, the predicted fraction of large tensions on stipe is less than the measurements from the field.

## 4 Simulation, Analysis, and Discussions

The free-floating binary species longline macroalgae farming system models with different locations and growth periods are developed in this section. In the field, the system will be seeded and released in mid-March from WA, then drift freely over a 90 days growth period and be harvested in CA in late June. The trajectory of the floating system is determined and simulation locations are selected based on growth periods and ocean assessment data availability. Then the waves and current measurements data are collected and characterized at the selected locations. Case studies are performed to analyze: (1) the hydrodynamic load effects on the free-floating binary species longline MFS in different kelps growth periods under high sea states and (2) the dynamic response of the system to identify critical issues and potential failure modes in different growth periods.

### 4.1 Trajectory Determination and Simulation Locations Selection.

A predicted free-floating MFS trajectory is achieved from a trajectory model conducted by Pacific Northwest National Laboratory (PNNL) [42]. The seeding and release location (45.6, −128.28) is in WA and is marked in Fig. 5. Each grid cell is assigned a color that represents the number of days and warmer colors show fewer days. The six “black squares” represent the locations we pre-selected to perform the simulations for the following reasons: (1) the six locations distributed along the free-floating MFS drift trajectory and include different kelps growth periods from the initial stage (top black square location) to fully grown stage (bottom black square location) and (2) the environmental data can easily be collected because these locations are all near the National Data Buoy Center (NDBC) stations and the data can be freely downloaded from the NDBC website.

### 4.2 Waves and Current Characterization.

The wave data, including significant wave heights and dominant periods, are collected at six NDBC stations. The significant wave height (meters) is calculated as the average of the highest one-third of all the wave heights during the 20 min sampling period, and the dominant wave period (seconds) is the period with the maximum wave energy. Unlike the traditional stationary offshore structure with a design service life of 50 years, the free-floating MFS is only deployed in the ocean for about 90 days (from March to June). Therefore, the typical 50-year or 100-year return period is not applied to determine the extreme hydrodynamic load in this study. Instead, we collect ten years (from 2010 to 2019) of wave historical data including significant wave heights and dominant periods from March 15 to June 25 in each selected NDBC station with a hypothetical deployment date of March 15 and harvest date of June 25. The maximum significant wave height and associated dominant period in each NDBC station are identified and applied to predict the high sea states wave profiles using the JONSWAP spectrum. The maximum significant wave height and dominant period in each station are listed in Table 3.

NDBC station | Maximum significant wave height (m) | Dominant period (s) | Maximum current speed (m/s) | Water depth (m) |
---|---|---|---|---|

1 (46089) | 9.64 | 12.12 | 0.76 | 2293 |

2 (46050) | 8.91 | 14.81 | 0.91 | 160 |

3 (46015) | 8.84 | 12.90 | 0.97 | 446 |

4 (46022) | 8.03 | 16.00 | 0.91 | 505 |

5 (46013) | 7.65 | 17.39 | 0.89 | 123 |

6 (46028) | 7.08 | 17.39 | 0.72 | 1128 |

NDBC station | Maximum significant wave height (m) | Dominant period (s) | Maximum current speed (m/s) | Water depth (m) |
---|---|---|---|---|

1 (46089) | 9.64 | 12.12 | 0.76 | 2293 |

2 (46050) | 8.91 | 14.81 | 0.91 | 160 |

3 (46015) | 8.84 | 12.90 | 0.97 | 446 |

4 (46022) | 8.03 | 16.00 | 0.91 | 505 |

5 (46013) | 7.65 | 17.39 | 0.89 | 123 |

6 (46028) | 7.08 | 17.39 | 0.72 | 1128 |

Unlike the wave data from six NDBC stations, current information for an entire 10-year period is unavailable. The one- or two-year data are not sufficiently long to generate reliable statistical current parameters and determine the maximum current speed. Therefore, the ocean current statistics data along the west coast are produced based on HYCOM + NCODA Global 1/12 deg Reanalysis provided by PNNL. The maximum current speeds in six selected locations are gathered and listed in Table 3. In the numerical simulations, the maximum current speed will be applied in each case (location) to represent extreme current activities and assumed constant in the water domain. The water depth in each location is determined based on the value provided on the NDBC website and listed in Table 3. Because the water depth is large, the effect of the seabed profile on waves and structure is small and can be negligible. Therefore, the seabed is assumed as flat and the water depth is constant in each location.

### 4.3 Case Studies: Binary Species Free-Floating Longline Macroalgal Farming System Modeling.

Four cases of free-floating longline MFS in different growth periods and associated locations are modeled and studied. Case 1 (see Fig. 6(a)) in location NDBC 46089 with a 1–10-day growth period, is considered as the initial stage where the kelps are so small that their presence is negligible in any simulation. In the system design, the length of each longline is 5 km. In this case, a 1 km longline is selected as a representative model to balance the real situation and computational efficiency. The number of buoys and specifications are determined based on the calculation of the maximum buoyancy force needed at the critical growth configurations over the kelp’s growth period. The submerged depth of the longline is determined as 5 m, considering the sunlight, temperature, nutrient, etc. [42] so the length of supporting lines is 5 m. In addition, optimal number and size of weighted disks are determined by the simulations, especially for the case where the bull kelp has not reached the water surface. In such a situation, the proper weighted disks can hold the longline in position. Otherwise, the upward buoyancy force of the bull kelp will pull the longline up and the bull kelp may cease to grow when they reach the sea surface in early stage. The parameters of the buoys, longline, supporting lines, and weighted disks used in the numerical model are listed in Table 4. An 11–30-day growth period model in NDBC 46050 is considered as case 2 (see Fig. 6(b)). In this case, the bull kelp would grow to 3.8 m approximated using an average growth rate [43] and 20 days growth period. The sugar kelp may grow to 0.3 m with a biomass of 4.6 kg/m in 20 days [44]. A 31–50-day growth period model is considered as case 3 (see Fig. 6(c)). Two locations, NDBC 46015 and NDBC 46022 are situated during this growth period. The NDBC 46015 location with larger maximum significant wave height and current is selected as the simulation site to analyze the maximum hydrodynamic load condition under high sea states. In this case, as the longline submerged depth is 5 m, the bull kelp will reach the sea surface and cease to grow in length. The sugar kelp length can be approximated as 0.58 m with 8.8 kg/m mass based on the growth rate. Similarly, a 51–90-day growth period model in NDBC 46013 is considered as case 4 (see Fig. 6(d)). The bull kelp has reached the sea surface but the blade area continues to increase. However, the erosion of the blade tip begins to slow the overall growth of the blade area, especially under high-speed current. Ultimately, an equilibrium between growth and erosion is reached in about 50 days in growing season [27]. Therefore, we assume the parameters of the bull kelp can still be obtained by the allometric relationships based on stipe length. The sugar kelp can grow to 0.85 m with 13 kg/m in a 51–90-day growth period [45]. The simulation cases, simulation site information, growth periods, and kelp model descriptions are summarized and listed in Table 5.

Mass | Volume | Diameter | Bending stiffness | C_{d} | C_{a} | |
---|---|---|---|---|---|---|

Buoy (B1800MB) | 5.9 kg | 0.105 m^{3} | 0.94 m | — | 1.2 | 1 |

Weighed disk | 12 kg | 0.0019 m^{3} | 0.075 m | — | 1.2 | 1 |

Longline | 0.19 kg/m | — | 0.012 m | 1.09 × 10^{5} Nm^{2} | 1.2 | 1 |

Supporting line | 0.43 kg/m | — | 0.021 m | 38.168 Nm^{2} | 1.2 | 1 |

Mass | Volume | Diameter | Bending stiffness | C_{d} | C_{a} | |
---|---|---|---|---|---|---|

Buoy (B1800MB) | 5.9 kg | 0.105 m^{3} | 0.94 m | — | 1.2 | 1 |

Weighed disk | 12 kg | 0.0019 m^{3} | 0.075 m | — | 1.2 | 1 |

Longline | 0.19 kg/m | — | 0.012 m | 1.09 × 10^{5} Nm^{2} | 1.2 | 1 |

Supporting line | 0.43 kg/m | — | 0.021 m | 38.168 Nm^{2} | 1.2 | 1 |

Simulation case | Location | Growth period (days) | Numerical model notes |
---|---|---|---|

1 | 1 (NDBC 46089) | 1–10 | 1 km longline model (initial stage, kelps can be neglected) |

2 | 2 (NDBC 46050) | 11–30 | Bull kelps length is 3.8 m Sugar kelp length 0.3 m with mass of 4.6 kg/m |

3 | 3 (NDBC 46015) | 31–50 | Bull kelps reach the sea surface Sugar kelp length 0.58 m with mass of 8.8 kg/m |

4 (NDBC 46022) | |||

4 | 5 (NDBC 46013) | 51–90 | Bull kelps reach the sea surface Sugar kelp length 0.85 m with mass of 13 kg/m |

6 (NDBC 46028) |

Simulation case | Location | Growth period (days) | Numerical model notes |
---|---|---|---|

1 | 1 (NDBC 46089) | 1–10 | 1 km longline model (initial stage, kelps can be neglected) |

2 | 2 (NDBC 46050) | 11–30 | Bull kelps length is 3.8 m Sugar kelp length 0.3 m with mass of 4.6 kg/m |

3 | 3 (NDBC 46015) | 31–50 | Bull kelps reach the sea surface Sugar kelp length 0.58 m with mass of 8.8 kg/m |

4 (NDBC 46022) | |||

4 | 5 (NDBC 46013) | 51–90 | Bull kelps reach the sea surface Sugar kelp length 0.85 m with mass of 13 kg/m |

6 (NDBC 46028) |

Note: For two locations that have the same growth period, the location that has the larger maximum significant wave height is selected as the simulation location and are highlighted in bold.

In addition, a judicious wave input selection method is developed which uses a relatively short period of time wave profile as input to decrease the simulation time significantly without sacrificing computational accuracy. In detail, we first created a 1 h irregular wave profile based on the maximum significant wave height and dominant period using JONSWAP spectrum (see Figs. 7–10(a)). Second, we applied the zero up-crossing method to obtain discrete values of wave height and corresponding wave period. In linear wave theory, it is straightforward to show the probability distribution function of the wave height follows the Rayleigh distribution [38]. Therefore, we evaluated the wave heights by comparison to the Rayleigh distribution for low sea states (see Figs. 7–10(b)). However, in high sea states, waves are nonlinear, and extreme wave height distribution predicted by the Rayleigh model (based on linear wave theory) results in significant errors [38], and a parameter-based Weibull distribution has been demonstrated to provide more accurate prediction for large wave heights [46]. The evaluations of the dataset based on both model distributions are shown in Figs. 7–10(c). The low sea state’s wave heights from 1 h wave profile prediction are fitted by Rayleigh distribution and the high sea state’s wave heights are fitted by Weibull distribution. Thus, we can conclude that the wave predictions can be used to represent the real ocean wave in all four cases. Then, we selected a 300 s long wave profile containing the maximum wave height in the 1 h waves as input to simulate the maximum hydrodynamic load conditions (see Figs. 7–10(a) and 10(d)). Finally, we evaluated the relatively small wave heights from the 300 s waves by comparison to Rayleigh distribution (Figs. 7–10(e)) and the relatively large wave heights by Weibull distribution (Figs. 7–10(f)). Combining the two evaluations, the close matching of small wave heights to Rayleigh distribution and large wave heights to Weibull distribution in all four cases, demonstrate that the selected 300 s waves applied as the input wave profile in each simulation can achieve reliable results. In each selected wave profile, we started the wave train with zero wave elevation, and in each simulation, an 8 s ramp-up is set to avoid the effects of the initial conditions.

### 4.4 Numerical Results and Discussions.

In this section, the numerical predictions for four cases are analyzed and the potential failure modes are determined. For case 1, the global behavior of the longline is studied and for cases 2–4, the local properties of the short interval longline and kelps are analyzed. Because in this study, the objective is to study the maximum hydrodynamic load effects on the MFS, the potential biotic effects are not considered. We assume all kelp samples are healthy and have no biotic damages during the free-floating period. Hence, in each case (or location), the plant number remains the same.

#### 4.4.1 Simulation Results of Case 1.

In this case, the 1 km free-floating macroalgae longline in high sea states is considered in the initial farming stage. The 1 km free-floating longline responses are very flexible and the shape turns to track the ocean surface due to wave and current activities, even for large bending stiffness because the bending stiffness is proportional to 1/*L*^{3}, where *L* is the length of the longline. The maximum tension and compression envelope along the 1 km free-floating longline are shown in Fig. 11. The positive value represents tension when the supporting line pulls the longline (see Fig. 12(a)), and the negative value means compression when the supporting line pushes the longline (see Fig. 12(b)). The rapid changes in maximum tension and compression are due to the buoy, supporting line, and weighted disk effects. The maximum tension is about 6000 N and the maximum compression is around 3900 N, which are both far below the tension/compression failure limit of 250,000 N. Therefore, the possibility of breaking or damage on the longline is very low.

#### 4.4.2 Simulation Results of Case 2, Case 3, and Case 4.

In this section, a series of numerical simulations including a free-floating binary species longline in early stage (case 2), intermediate stage (case 3), and late stages (case 4) in high sea states are considered. The objective of these simulations is to determine the local behaviors of kelps. Thus, a short interval of a 50 m longline model containing leading buoy and end buoy is separated into two spans (25 m each) with a mid-buoy at the 25 m center mark and with kelps is applied to represent the physical MFS for efficient computation. In this short-interval longline model, three buoys can cover all buoy conditions in the whole longline. Moreover, two spans with kelps can capture the effects from back span kelps onto the front span kelps, and vice versa. Thus, the 50 m longline model can be used to represent the whole longline for the study of local behavior. The maximum tension and compression envelope along the 50 m free-floating longline for all three cases are shown in Fig. 13. The relatively large tensions and compressions are concentrated in the mid area of the longline for all three growth stages for all the three different waves and current conditions. Again, the rapid changes in maximum tension and compression at the middle where the buoy is attached are due to the buoy-motion-induced tension of the supporting line and weighted disk effects.

The maximum tension of each sugar kelp cluster and bull kelp plant at the holdfast is plotted in Figs. 14 and 15. In each simulation case, 250 sugar kelp clusters and 100 bull kelp plants are defined and numbered starting from the upstream location to downstream location (*x*-axis in Figs. 14 and 15). The macroalgae attachment test in the laboratory shows both bull kelp and sugar kelp attached strongly to the longline before breakage above a force of 40–55 N. In Figs. 14 and 15, the lower bound of the breakage force of 40 N is plotted as an orange line. In all three simulation cases, all sugar kelp clusters and bull kelp plants are in the safe region even under the extreme waves and current activities. The averaged maximum tensions values of all sugar kelp clusters and bull kelp plants for each case are plotted as a red line in Figs. 14 and 15 and the values are shown below the line. The increasing value of the averaged maximum tension of sugar kelp clusters from the early stage to the late stage (see Fig. 14) is because the drag force due to the waves and current is the dominant load applied on the sugar kelp clusters. With the growing area and mass of the sugar kelps from the early stage (case 2) to the late stage (case 4), the increasing drag forces result in rising tensions in the holdfast. Slightly changing the wave characterized parameters and current will not alter the trend (e.g., the significant wave height and current speed in case 2 (8.91 m and 0.91 m/s) are larger than those in case 4 (7.65 m and 0.89 m/s) but the averaged maximum tensions value in case 2 is smaller than that in case 4). The maximum tensions at the sugar kelp cluster holdfasts in the middle of the span are relatively small compared to the tensions near the supporting lines because the bull kelps on the top of the longline behave as a damper to reduce the relative motions. However, the longline motions near the supporting lines are relatively large due to the motion of the buoys connected. These large motions may cause large tensions at the holdfast of the sugar kelp clusters. Compared to the sugar kelp clusters, the bull kelp plants always suffer more tensions at the holdfast, especially in the early growth stage (case 2) because in addition to the drag force, the net buoyancy force is also applied to the stipe due to the pneumatocyst on the top of the stipe and the effect of the net buoyancy force on the bull kelp plant is significant. The largest averaged maximum tensions at the bull kelp plant holdfasts in case 2 (see Fig. 15) well demonstrate the domination effect of net buoyancy force on the bull kelp. Compared to intermediate stage (case 3) and late stage (case 4), the early stage (case 2) bull kelp plants experience less drag force but the whole plant including the pneumatocyst is fully submerged in the water. In this situation, the bull kelp plant will suffer more net buoyancy force and produce large tension at the holdfast. In cases 3 and 4, because all the bull kelp stipes are grown and have reached the sea surface, the pneumatocysts are floating on the water surface most of the time. Therefore, the similar net buoyancy forces result in a close value of averaged maximum tensions at the bull kelp plant holdfasts in cases 3 and 4. The slightly larger averaged maximum tensions at the bull kelp plant holdfasts in case 4 compared to case 3 because of the growing blade increasing the drag force acting on the stipe or from the modeling uncertainty. The large tensions appear in case 2 due to situations where the bull kelp plants are fully submerged in the water (see Fig. 6(b)), and the significant net buoyancy forces produce large tensions at the holdfast of the plants.

#### 4.4.3 Potential Damage and Failures.

Although the maximum tensions on the longline and the holdfasts of all sugar kelp clusters and bull kelp plants are in the safe region from the simulations, several phenomena are captured through the simulations which may result in system failure or biomass loss (see Fig. 16). In the initial farming stage (case 1) simulation, two potentially damaging conditions are shown in Fig. 16(a). One is if the longline stretches out of the water (see 2 in Fig. 16(a)), the large force due to instantaneous change in dynamic loads caused by loss of hydrostatic buoyancy on young kelps may result in damage. However, the longline stretching out of the water happened instantly and the actual effect on the kelp needs to be determined experimentally.

Figure 16(a)—1 shows a potential contact between buoy and longline, and if such contact interaction is significant, the buoy may be damaged and sink into the water. The worst-case situation would be the whole system sinking into the water when the number of damaged buoys reaches the critical stage where the total net buoyancy force of the system becomes negative. To avoid potential system failure due to damage to the buoys, one could select high-strength buoys or add protections around the buoys to enable resistance against potential contact/impact forces. The maximum impact force can be determined by performing laboratory tests or using appropriate numerical simulation tools containing contact/impact and fluid-structure interaction capabilities.

Figure 16(b)—3 shows sugar kelp clusters swing-up and may flip-over in the real sea. Figure 16(b)—4 shows the crossing of the bull kelp plants. Both of these phenomena may cause potential kelp-kelp, kelp longline, and kelp-supporting line entanglements, result in damage to the kelp and biomass loss. Field tests or laboratory experiments can help determine if the swing-up/flip-over and crossing would cause entanglement and plant damage. An alternative way to analyze the potential entanglement is to use fully nonlinear fluid-structure interaction and computational fluid dynamics software to simulate waves and current interacting with multiple kelp blades. In such a model, body-to-body contact can be simulated. Entanglement is one of the main considerations for future studies using such an approach.

## 5 Concluding Remarks

An offshore binary species (*Saccharina latissima* and *Nereocystis luetkeana*) free-floating longline macroalgae farming system is presented and the maximum hydrodynamic load effects on the system under high sea states in different growth periods and associated locations are analyzed. Potential damage and failure modes are determined based on the case simulations and studies. The following conclusions can be drawn from this work:

The maximum tensions on the longline for all stages are far less than the longline breakage limit and all the maximum tensions at the holdfasts of bull kelp plants and sugar kelp clusters are below the breakage limit obtained from the laboratory tests.

The drag force due to the waves and current is the dominant load applied on the sugar kelp clusters whereas the net buoyancy force effect is more significant on the bull kelp plants.

One of the potential damages to the young kelps comes from the fact that the longline may stretch out of the water in the initial stage and the sudden change in large dynamic and hydrostatic forces on young kelps may lead to possible plant damage.

The potential significant buoy-longline contact interactions may cause damage to the buoys and result in the whole system sinking into the water. Therefore, qualified buoys are required or proper protections around the buoys are necessary to resist the critical contact/impact force.

Potential kelp-longline, kelp-supporting line, and kelp-kelp entanglements may cause kelp damage and biomass loss.

The entanglement effect is one of the main areas for potential future studies. In addition, a nonlinear ocean wave analysis based on the nonlinear Fourier transformation can be employed to determine the input waves containing maximum wave amplitudes, including interactions between nonlinear modes. Therefore, the extreme phenomenon, including the rogue wave, could be captured and would improve the simulation results significantly. Additional potential damages and failures can potentially be identified from numerical predictions under rogue waves.

## Acknowledgment

We wish to thank all the contributors in Multi-resolution, Multi-scale Modeling for Scalable Macroalgae Production team including Andrea Copping, Alicia Gorton, Jonathan Whiting, Molly Grear, Gabriel Garcia-Medina, Luca Castrucci, and Wei-cheng Wu from PNNL; Phil Wolfram, Mat Maltrud, and Steven Brus from Los Alamos National Laboratory; Kevin Haas, and Martin Jang from Georgia Institute of Technology. We also thank the contributors in NOMAD team including Scott Edmundson, Song Gao from PNNL; Jason Quinn and Jonah Greene from Colorado State University; Jascha Gulden from Reliance Laboratories; and Geoff Wood from Composite Recycling Technology Center. We are grateful for the support and guidance from U.S. Department of Energy.

## Funding Data

The U.S. Department of Energy (DOE), Advanced Research Projects Agency-Energy (ARPA-E) Macroalgae Research Inspiring Novel Energy Resources (MARINER) program, Award No. 17/CJ000/09/01 and 17/CJ000/09/02.

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

## Nomenclature

*l*=length of the bull kelp stipe

- L =
length of the free-floating longline

*A*=drag area

*V*=volume of the object in fluid

- $C$ =
system damping matrix

- $F$ =
the external load vector

- $K$ =
system stiffness matrix

- $M$ =
system mass matrix including structural mass and hydrodynamic mass

*C*_{a}=added mass coefficient

*C*_{d}=drag coefficient

*C*_{m}=inertia coefficient for the object

- $uf$, $u\u02d9f$ =
velocity and acceleration of the fluid

- $ub$, $u\u02d9b$ =
velocity and acceleration of the structure

- $x$, $x\u02d9$, $x\xa8$ =
structure displacement, velocity, and acceleration vector

*ρ*=fluid density

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