**1. Introduction**

Uranium fuels more than 400 nuclear reactors worldwide and provides over 13% of the world's electricity. While uranium is among the most abundant elements found in natural crustal rock, it is seldom sufficiently concentrated to be economically recoverable. The uranium ore in the ground has remained as the single most important conventional uranium resource. Based on current consumption rates, the known uranium ore resources that can be mined at current costs are estimated to be sufficient to produce fuel for about a century. Although at low concentrations, the world oceans hold the largest reserves of uranium. In fact, extracting metals (e.g., Na, Mg, and K) from seawater has been commercialized for a long time [1]. The possibility of recovery of seawater uranium by ion-exchange resins was studied shortly after World War II, but was not economically viable compared to exploitation of known uranium ores on land [2]. While extracting seawater uranium is not yet commercially viable, it serves as a "backstop" to the conventional uranium resources and provides an essentially unlimited (~6500 years) supply of uranium [3]. Driven by the rapid growth of global energy demand in recent decades, interest in extracting uranium from seawater for nuclear energy has been renewed. With recent advances in seawater uranium

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extraction technology, extracting uranium from seawater could become economically feasible especially when the extraction devices are deployed at large scales of several hundred km2 [4].

Sugo *et al.* [5] introduced the braided adsorbent farm technology that is potentially feasible for large-scale uranium extraction from seawater. The fibers are braided around a low-density core to result in positively-buoyant braids approximately 60 meters in length. The material is carried to the deployment site and moored to the ocean floor with anchor chains. The proposed design calls for deployment of over a million long braided moorings, 60 m in height over an area of about 680 km2 . The submerged farm closely resembles a kelp forest, which is known to exert a substantial drag on coastal currents [6]. Hence, there is concern that the large scale deployment of adsorbent farms could result in potential impact to the hydrodynamic flow field in an oceanic setting.

In this study, a kelp-type structure module was incorporated into the Finite Volume Coastal Ocean Model (FVCOM) to simulate the retardation effect of a farm of uranium extraction devices on the flow field. The kelp-type structure module is based on the classic momentum sink approach that approximates the blockage effect of structures on flows as additional drag force in the momentum equations. This paper summarizes the kelp-type module development and validation processes.

#### **2. Methodology**

#### *2.1. Kelp-Type Structure Module Development*

A number of modeling studies have been carried out to investigate the hydrodynamic effects of underwater structures, including aquaculture farms, vegetation canopies, as well as wind and tidal energy farms. For instance, Grant and Bacher [7] developed a two-dimensional (2-D) finite element circulation model for Sungo Bay, China to study the effect of bivalve culture structure on flows. The drag exerted by the culture drop ropes was parameterized as additional form drag in the hydrodynamic model, which predicted a 54% reduction in current speed in the midst of the culture area. By approximating the shellfish farm drag as additional bottom friction in a 2-D hydrodynamic model, Plew [8] studied the shellfish farm-induced changes to tidal circulation in an embayment in New Zealand, and found that the current speeds were reduced inside most farms. Struve *et al.* [9] studied the influence of model mangrove trees on the hydrodynamics in a flume through both flume experiments and 2-D depth-integrated numerical modeling. The model results compared very well with experiment measurements when the resistance created by mangroves was modeled as an additional drag force. Hence, in this study, a similar momentum sink approach was adopted for the kelp-type structure module to assess the hydrodynamic impact of seawater uranium extraction devices. Specifically, the additional resistance force on flow caused by a single uranium adsorbent braid or kelp frond is defined as:

$$
\pi = \frac{1}{2} \rho \mathcal{C}\_d A |\vec{u}| \,\vec{u} \tag{1}
$$

where IJ additional resistance force by uranium adsorbent braid (N), ȡ = seawater density (kg/m3 ), *Cd* = drag coefficient of the equivalent (cylindrical) braid or kelp structure, *A* = flow-facing area of

The hydrodynamic model selected in this study is the finite volume coastal ocean model FVCOM developed by Chen *et al.* [10]. As a three-dimensional (3-D) unstructured-grid coastal ocean model, FVCOM is capable of simulating water surface elevation, velocity, temperature, salinity, sediment, and water quality constituents. The unstructured grid and finite volume approach employed in the model provides geometric flexibility and mass conservation that is well suited to simulate hydrodynamic transport at various spatial scales within a large model domain. For computational efficiency, a mode splitting scheme is used to solve the momentum equations. FVCOM has been extensively used by the estuarine and coastal modeling community to study a variety of scientific and engineering problems in estuaries and coastal oceans [11–13]. The numerical aspects and detailed formulations of FVCOM have been presented in Chen *et al.* [10,14] and many other FVCOM publications, thus they will not be elaborated here except for the portions in the momentum governing equations that were modified to include the momentum sink induced by underwater structures.

The modified FVCOM momentum equations in the horizontal directions have the following general form [13]:

$$\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} + w \frac{\partial u}{\partial z} - fv = -\frac{1}{\rho\_o} \frac{\partial p}{\partial x} + \frac{\partial}{\partial z} \left( K\_m \frac{\partial u}{\partial z} \right) + F\_\ddagger - F\_\ddagger^M \tag{2}$$

$$\frac{\partial \upsilon}{\partial t} + u \frac{\partial \upsilon}{\partial x} + \upsilon \frac{\partial \upsilon}{\partial y} + w \frac{\partial \upsilon}{\partial z} + fu = -\frac{1}{\rho\_o} \frac{\partial p}{\partial y} + \frac{\partial}{\partial z} \left( K\_m \frac{\partial \upsilon}{\partial z} \right) + F\_y - F\_y^M \tag{3}$$

where (*x*, *y*, *z*) are the east, north, and vertical axes in the Cartesian coordinates; (*u*, *v*, *w*) are the three velocity components in the *x*, *y*, and *z* directions, respectively; (*Fx*, *Fv*) are the horizontal momentum diffusivity terms in the *x* and *y* directions, respectively; *Km* is the vertical eddy viscosity coefficient; U is water density; *p* is pressure; and *f* is the Coriolis parameter. ܨ௫ ெ and ܨ௬ ெ are the momentum sink term (m/s2 ) induced by the uranium adsorbent device that was added to the original FVCOM governing equations [10,14], and is defined as the following general form:

$$
\overrightarrow{F^M} = \frac{1}{2} \frac{NC\_d A}{V\_c} \|\overrightarrow{u}\|\ \overrightarrow{u} \tag{4}
$$

where *Vc* = momentum control volume in which the adsorbent device is deployed (m3 ), *N* = the number of adsorbent braids deployed within the same momentum control volume, and the rest terms were defined previously in Equation (1).

FVCOM solves the momentum equations using the finite-volume method and sigma-stretched coordinate transformation in the vertical direction. Assuming one adsorbent braid may occupy multiple ı-layers and is located within a single momentum control element (*i.e.*, the model grid size is much larger than the width of adsorbent braid), the integrated form of Equations (2) and (3) for the 3-D internal mode can be written as:

$$\frac{\partial (A\_e \Delta\_\sigma Du)}{\partial t} + R\_u - f A\_e \Delta\_\sigma Dv = -\frac{1}{2} N \left[ \mathcal{C}\_d A\_\sigma \sqrt{u^2 + v^2} u \right] \tag{5}$$

$$\frac{\partial \langle A\_{\epsilon} \Delta\_{\sigma} D \upsilon \rangle}{\partial t} + R\_{\upsilon} + f A\_{\epsilon} \Delta\_{\sigma} Du = -\frac{1}{2} N \left[ \mathcal{C}\_{d} A\_{\sigma} \sqrt{u^{2} + \upsilon^{2}} \upsilon \right] \tag{6}$$

where *Ae* = triangular element surface area (m2 ), *ǻıD* = *ı*-layer thickness (m), *Ru* and *Rv* = all the remaining momentum terms including advection, diffusion, and pressure gradient. The right hand side of Equations (5) and (6) represents the volumetric momentum sink rate (m4 /s2 ) contributed by the adsorbent braid or kelp frond defined in Equation (1), and *Aı* = flow-facing area of braid adsorbent within the *ı*-layer.

The integrated form for the 2-D external mode of Equations (2) and (3) are expressed as:

$$\frac{\partial (A\_e D\bar{u})}{\partial t} + R\_{\bar{u}} - f A\_e D\bar{v} = -\frac{1}{2} N \sum\_{\sigma=1}^{\sigma=k} \left[ \mathcal{C}\_d A\_\sigma \sqrt{u^2 + v^2} u \right] \tag{7}$$

$$\frac{\partial \langle A\_{\epsilon} D \bar{v} \rangle}{\partial t} + R\_{\bar{v}} + f A\_{\epsilon} D \bar{u} = -\frac{1}{2} N \sum\_{\sigma=1}^{\sigma=k} \left[ \mathcal{C}\_{d} A\_{\sigma} \sqrt{u^{2} + v^{2}} v \right] \tag{8}$$

where ݑത and ݒҧ= vertically averaged velocity in the *x* and *y* directions, respectively.

#### *2.2. Module Validation*

The kelp-type structure module was validated against laboratory experiments conducted by Plew [15]. The detailed experiment configuration has been described in Plew [15], and is briefly presented here. The experiments were conducted in a 6-m long by 0.6-m wide flume (Figure 1). The structure canopies were constructed from aluminum cylinders of 9.54-mm diameter, and extended over the full width and the entire working length (4.8 m) of the flume. The velocity profiles were measured using particle tracking velocimetry (PTV) at a distance of 4 m from the flow inlet. Velocity measurements were made in two vertical planes, mid-way between cylinders and then in line with the cylinders, and were averaged horizontally in the *x*-direction over the distance (L) between cylinder rows to give an averaged vertical profiles for each plane. This enabled a vertical profile of spatially averaged velocity and turbulence statistics to be defined.

Table 1 summarizes the configuration of the four flume experiments selected for the kelp-type module validation in this study. The cylinders were suspended in the upper half of the water column in all the experiments but with different horizontal spacing/density, allowing cylinder density to increase from runs A to D.

FVCOM was configured based on the flume experiment configurations listed in Figure 1 and Table 1 to validate the kelp-type structure module. The flume was represented with an unstructured mesh consisting of 5578 elements and 2954 nodes in the horizontal plane (Figure 2a). In the vertical direction, 40 uniform sigma layers were specified. An external time-step of 0.001 second was used in all model runs. The default Mellor-Yamada 2.5 turbulence closure was used for vertical eddy viscosity and diffusivity calculations. The drag coefficient (*Cd*) of the canopy was treated as spatially uniform but its value for each validation run was calibrated based on model-data comparisons. Figure 2b shows the spatial distribution of the cylinder array in Validation Run D. The corresponding model predicted surface velocity field during the baseline condition (without cylinder array) and Run D are presented in Figure 2c,d, respectively. The presence of cylinders

significantly altered the flow field. Surface velocity was generally reduced within the cylinder canopy compared to the baseline condition.

**Figure 1.** Schematic of the experimental flume setup (adapted from Plew [15], with permission from © 2011 American Society of Civil Engineers). Cylinders were arranged in rows with a spacing of L (m) in the direction of flow, and a transverse spacing of B (m) between cylinders. Velocity measurements were taken at a distance of 4 m from the inlet. H (m) is the total water depth in the flume, hc (m) is the canopy height, and hg (m) is the distance between the canopy and the flume bed.

**Table 1.** Summary of flume experiments selected from Plew [15] for kelp-type module validation. a is the projected cylinder flow-facing area per unit volume inside the canopy, and Q is the flow rate (data obtained from Plew [15]).


**Figure 2.** (**a**) Finite Volume Coastal Ocean Model (FVCOM) model grid (in the horizontal plane) for the flume experiment. (**b**) Spatial distribution of cylinders in Run D. (**c**) Surface velocity field in the baseline condition without the cylinder array. (**d**) Surface velocity field in Run D.

#### **3. Results and Discussion**
