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Biomimetic crossflow filtration with wave minimal surface geometry for particulate biochar water treatment

  • Mason Anderson ,

    Contributed equally to this work with: Mason Anderson, Vibhav Durgesh, Martin Baker, Paulo Yu, Gregory Möller

    Roles Formal analysis, Investigation, Methodology, Software, Validation, Visualization, Writing – original draft, Writing – review & editing

    Affiliation Department of Chemical and Biological Engineering, University of Idaho, Moscow, Idaho, United States of America

  • Vibhav Durgesh ,

    Contributed equally to this work with: Mason Anderson, Vibhav Durgesh, Martin Baker, Paulo Yu, Gregory Möller

    Roles Formal analysis, Methodology, Supervision, Visualization, Writing – review & editing

    Affiliation Department of Mechanical Engineering, University of Idaho, Moscow, Idaho, United States of America

  • Martin Baker ,

    Contributed equally to this work with: Mason Anderson, Vibhav Durgesh, Martin Baker, Paulo Yu, Gregory Möller

    Roles Conceptualization, Data curation, Formal analysis, Funding acquisition, Investigation, Methodology, Software, Validation, Visualization, Writing – review & editing

    Affiliation Department of Soil and Water Systems, University of Idaho, Moscow, Idaho, United States of America

  • Paulo Yu ,

    Contributed equally to this work with: Mason Anderson, Vibhav Durgesh, Martin Baker, Paulo Yu, Gregory Möller

    Roles Data curation, Formal analysis, Methodology, Software, Validation, Visualization, Writing – original draft, Writing – review & editing

    Affiliation Department of Soil and Water Systems, University of Idaho, Moscow, Idaho, United States of America

  • Gregory Möller

    Contributed equally to this work with: Mason Anderson, Vibhav Durgesh, Martin Baker, Paulo Yu, Gregory Möller

    Roles Conceptualization, Data curation, Funding acquisition, Investigation, Methodology, Project administration, Resources, Supervision, Writing – review & editing

    Affiliations Department of Chemical and Biological Engineering, University of Idaho, Moscow, Idaho, United States of America, Department of Soil and Water Systems, University of Idaho, Moscow, Idaho, United States of America


Wave minimal surfaces (WMSs) are mathematically defined structures that are commonly observed in nature. Their unique properties have allowed researchers to harness their potential for engineering applications. Since WMSs can be represented by mathematical equations, the geometry can be parametrized and studied using computational fluid dynamics (CFD) for particle separation. Low energy particle separation in water treatment can yield low-carbon footprint technology approaches such as biochar water treatment where removal and recovery of adsorbed N and P on biochar can address water pollution, climate change and food security. The objective of this work was to demonstrate the capability of WMS as a crossflow filtration system to remove particulates in water. For this purpose we used CFD to optimize WMS geometry and studied the performance of the 3D-Printed (3DP) optimized WMS using experimental fluid dynamics (EFD) in a water tunnel. CFD studies quantified planar vorticity, fluid filtrate flux, and particle behavior of WMS. For inflow velocities of 0.2–0.4 m/s, CFD results showed that a reverse wave filter design with convex shape leading-edge, angle of incidence of 90o, and maximum width of n = 1.0 captured 15–25% of upstream velocity at the filter port. CFD analysis showed more than 95% separation efficiency at velocities and pressures of 0.2–0.32 m/s and 5–35 kPa, respectively. Particle Image Velocimetry (PIV) was used for EFD fluid flow measurements with an optimized wave minimal surface (OMWS). Comparison of OMWS CFD and PIV velocity fields showed good agreement with a root-mean-square error of less than 10%. Particle size analysis showed that the 3DP OMWS could filter particle sizes ranging from 1–30 μm with at least 50% particle count reduction in the filtrate. Thus, we successfully demonstrated a novel framework for analyzing a crossflow water filtration system from conceptual design to initial benchtop experiments using iterative CFD, 3DP, and EFD.


Particle filtration from water has been broadly explored as it has diverse applications encompassing biological, environmental, and industrial areas [17]. The science and engineering community has worked extensively in this area to address different aspects of the subject, such as mechanisms involved in solid-liquid interactions [8, 9], improved engineering designs to enhance filtration processes and systems [2, 3, 10] and understanding biological processes such as feeding mechanisms of animals [1117]. Recently, researchers have explored new technological systems that are inspired by nature [11, 13, 1821]. In this regard, the current study aims to develop a framework in utilizing computational, experimental fluid dynamics and 3D printing approaches to investigate the potential of minimal surfaces for particulate water treatment.

Novel particle filtration technologies, with lower carbon equivalent (CO2e) footprints, can concomitantly address a range of engineering grand challenges [22] and sustainable development goals [23]. Systems thinking in the design stage has shown that life cycle impacts are an increasingly important criteria during the technology innovation process [24, 25]. Addressing the global challenges of clean water, food security and the carbon sequestration required to mitigate climate change, granular and powdered biochar particulates have received considerable attention in adsorptive phosphate and nitrogen removal and recovery from water [2628]. To this end, our laboratory’s ongoing work in the field of biochar water treatment, especially where the biochar particles can be used to remove and recover phosphorus from water, has demonstrated its potential as an emerging water treatment approach [2831]. Removing and recovering phosphorus from water addresses the nutrient pollution linked to harmful algae blooms [32, 33] while also recovering non-renewable phosphorus, a vital nutrient for agriculture [34]. However, efficiently capturing back biochar particles with adsorbed phosphorus from the water stream after dosing with traditional dead-end filtration systems requires a considerable energy and pressure gradient, resulting in a significant CO2e footprint that can cancel the carbon sequestration benefits of biochar [35].

In nature, energy-efficient filter feeding mechanisms can yield an evolutionary advantage, such as is observed in Earth’s largest creature, the balaenid blue whales [36]. To design an efficient and non-clogging filter, researchers have looked at feeding mechanisms used by marine animals as the possible inspiration for particle filtration. Some of these initial studies have shown the unique design of the filter feeders, e.g., bivalves [16], American paddlefish [11], balaenid whales [12, 14, 15, 17], and manta rays [20, 21]. Wright et al. [16] investigated different filter-feeding bivalve mollusks and their ability to filter out particles as small as bacteria. They found that the bivalves were able to filter bacteria with great efficiency due to the designs of their cilia. Goldbogen et al. [12] and other researchers [14, 15, 17] have studied balaenid whales and their energy efficient feeding strategy despite their gigantic size. Divi et al. [20] studied manta rays’ feeding mechanisms and found that these fishes have wing-like structures to intercept particles (i.e., planktons) that ricochet away from the filter pores. Brooks et al. [11] investigated American paddlefishes’ feeding mechanisms and found that vortices generated in their branchial arches provide a novel mechanism for the suspension, concentration, and transport of particles in the suspension-feeding fish. These studies have shown filter feeding mechanisms have complex designs and even more complex fluid flows that allow for efficiently filtrating out nutrients.

Researchers have used biomimetic designs in several applications ranging from ion-mediated membranes [37, 38], oil-water emulsions filtration [39], and crossflow filtration [13, 19, 40]. Sanderson et al. [13] found that the vortices formed between the gills during crossflow enabled the fishes to capture particles without clogging. The continued efforts helped develop a new filter design such as a helical-shaped filter that transported the particles at the downstream end of the filter [40]. Similarly, Liao et al. [39] prepared a filter mesh inspired by armors and scales of shellfish, which effectively prevented them from marine oil contamination. The results of their mesh design have excellent superhydrophilic and underwater superoleophobic properties suitable for gravity-driven oil-water separations. In another similar study, Clark and San Miguel [19] used a lobe filtration design from a manta ray’s filter-feeding mechanism [20]. They proposed a scaled microfluidic device and showed that the device was capable of filtering particles on the order of 10 μm.

Despite the extensive research effort in using bio-inspired designs, there are several challenges in designing filters using bio-inspired geometries. One such challenge is the complex geometry and fabricating such models can be time and cost-intensive. Therefore, this study focuses on mathematically defined minimal surface geometry for filter design. The minimal surface geometry is commonly observed in nature, and the minimal surface geometries are one of the most energy-efficient designs for the given constraints. Another benefit of using mathematically defined minimal surface geometry design is that it allows for parametric studies and can be fabricated using 3D printing and advanced manufacturing technology.

Minimal surfaces are a widely researched area in mathematics literature [4145], and are commonly observed in several natural patterns, designs, and fluid flow patterns. Different definitions arise when describing minimal surfaces, and a few are described here. The term minimal surface stems from the idea that a surface is minimal if and only if its mean curvature is zero [45]. Another definition of a minimal surface is the surface if and only if every point has a neighborhood with the least area relative to its boundary [42]. Minimal surfaces can be observed in nature and architecture from the structure of the cerebellum [46], nucleic acid folding in viruses [47], and structural buildings [48, 49]. Minimal surfaces have also been used in particle filtration, such as investigations performed by Sreedhar et al. [50], Thomas et al. [51], and Fu et al. [52] on triply periodic minimal surfaces.

Several researchers have attempted to provide different filtration approaches with varying success. However, there is a lack of comprehensive studies of filter designs that are mathematically defined and incorporate a framework to design filters from concept to a laboratory experiment. Therefore, this work aims to perform a feasibility study utilizing a bio-inspired filtration mechanism and a wave minimal surface (WMS) [44, 53]. For this study, a WMS was selected as it is mathematically defined, allowing for parametrization of the design space for systematic testing.

There are three main objectives for this study. First, computational fluid dynamics (CFD) parametric studies of the WMS were conducted to identify their vortex generation capabilities for particle filtration. Further CFD studies were conducted on the model to configure its increased mass flow rate at the filter pore, leading to the optimized wave minimal surface (OMWS) design. Second, a comparison of CFD studies with experimental fluid dynamics (EFD) measurements was performed for the OMWS. Lastly, the performance of the OMWS design was quantified in lab-scale experiments to assess its filtration capabilities. The initial results from this investigation showed that the OMWS has potential use in particle filtration for a certain range of velocities and particle diameters.

Materials and methods

The approach used for the iterative CFD and EFD studies can be seen in Fig 1. The flow chart describes the steps used in modeling, fabricating, and testing of the wave minimal surface. The initial conception of the WMS can be seen in “1” where parametric CFD studies were performed to test and modify the geometry. The selected geometry modification “2” was fabricated where laboratory experiment were conducted to validate the simulations. The obtained simulation and experimental results would then be used for further studies in a bench scale implementation. Details of critical steps in the design process of “1” and “2” can be found in following sections.

Fig 1. Process diagram of computational fluid dynamics (CFD) and experimental fluid dynamics (EFD) used in this study.

The three major benchmarks of the process include 1) defining the biomimetic geometry, 2) optimizing that geometry by CFD, and 3) testing the geometry with EFD to advance full scale implementation.

Geometry generation and parametrization

The WMS used for this study can be defined by a set of parametrized functions in Cartesian coordinates as (1) (2) (3) where m, n, o, and p are functional parameters, and u and v are grid points for mesh evaluation. Here, we note the variable n which impacts the maximum width of the WMS. This was one of the variables considered in optimizing the filtering ability of the WMS.

The WMS was first visualized and designed using CAD software (Rhinoceros 3D with Grasshopper plugin [43, 54]) and the general schematic of the WMS is shown in Fig 2. The figure highlights the critical features of the geometry. Before CFD and EFD studies, the WMS was analyzed for structural rigidity identifying the appropriate location of the particle filtering outlet port. This was done by first creating surfaces using the outer boundaries of the WMS shown in Fig 2. These surfaces result in providing structural stability to the design. Next, a filter pore was designed at the mid-plane of the geometry to increase the fluid velocity at the lip. This pore diameter was set to 1.35mm throughout the CFD studies.

Fig 2. Schematic of the WMS design with CFD mesh.

(a) Front view, and (b) isometric view.

The study further focused on three geometrical features of the model to optimize the mass flow rate at the filter pore. These features were leading-edge morphology (LE), maximum width (n), and angle of fluid incidence (FI) (see Fig 2). LE was considered for modification to produce a smooth fluid contact area and to reduce momentum loss in the fluid due to initial fluid contact. Designs considered were flat, where LE was sharp at the boundary of WMS (see Fig 2a), and convex, where LE was a parabolic function such that the apex of the function was offset by 25% from the bounding edge of the WMS. The geometry width, n, was studied for structural stability, fabrication access, and vortex generation. Values selected for n were between 0.5 and 1 (see Fig 2b). Lastly, FI was considered to determine effective fluid contact to the filter pore. The values tested were 90° for standard backstep flow and 40° to maintain effective pore diameter (see Fig 2b). Note that other geometrical parameters were investigated in this study such as alternate orientations and pore locations, however, the results from these studies did not add significant impact on the overall behavior of the fluid and stability of the model. Thus, only critical geometrical features (i.e., n, FI, and LE) were presented in this study.

Computational fluid dynamics (CFD)

CFD was initially used to identify the WMS’s potential for water filtration. Parametric simulations of laminar flow over the WMS at different geometrical configurations were conducted using ANSYS Workbench (ANSYS 2021 R2 Suite) and solved using ANSYS Fluent. The flow simulations were performed to identify the OWMS geometry configuration, which maximizes the mass flow rate at the outlet port. The OWMS would be fabricated using 3D printing, and corresponding CFD results would be compared to the experiments for a few inflow velocities.

The simulations used the Reynolds Stress Model (RSM) to anticipate the vortex generation ability of the WMS. The RSM is optimized for flow cases where the geometry or the strain rate tensor is assumed to be complex with significant sections of swirling flow [55, 56]. Furthermore, the RSM was utilized for turbulence prediction. This is because the RSM modifies the two-equation turbulence model by neglecting the assumption of isotropic eddy-viscosity leading to seven additional transport functions for 3D cases. The exact form of the equations for Reynolds stress transport can be seen as [57, 58] (4) where Cij, DL,ij, Pij, and Fij are convection, molecular diffusion, stress, and system rotation tensors, respectively. Meanwhile, DT,ij, Gij, ϕij, and ij denote the turbulent diffusion, buoyancy, strain, and dissipation, respectively. Here, we note that further information about the RSM such as model assumptions can be found in [57].

The simulations used a rectangular domain as shown in Fig 3 where the dimensions replicate the test section of the water tunnel at the University of Idaho used for the experimental study (vide infra). The fluid used for the simulations was water with fluid properties determined at 20°C and 101 kPa (ρ = 998.21 kg/m3, μ = 1.00 × 10−3 Pa•s). The WMS was placed on the bottom boundary of the domain, and the inlet conditions were varied on the left boundary depending on the case study. No-slip condition was applied on the tunnel walls. Other CFD parameters pertaining to the WMS used dimensions and designed to fit the water tunnel dimensions. Furthermore, parametric CFD studies were then conducted on the WMS prior to its fabrication and validation studies. With this in mind, the simulations were set to solve the flow scenarios with at least 1000 iterations using a convergence criterion of 10−5. Convergence was achieved with roughly 400 iterations for each simulation scenario. Furthermore, convergence studies were performed using different mesh sizes and found that roughly 950,000 cells are sufficient to achieve stable results under different inflow conditions. Lastly, the simulations were run on a computer with an 18 core Intel Xeon E5-2690 processor and 80 GB of RAM. The computational time for each simulation was approximately 24–48 hours.

Experimental fluid dynamics (EFD)

Water tunnel facility.

The water tunnel experiments for this study were performed in the flow visualization water tunnel facility (Model 501 Water Tunnel, Engineering Laboratory Design, Inc., Lake City, Minnesota, USA) in the Integrated Research and Innovation Center (IRIC) at the University of Idaho. The water tunnel shown in Fig 4 has a cross-sectional test area and length of 15.24 cm by 15.24 cm and 45.72 cm (6 in. × 6 in. × 18 in.), respectively. The test section is made of clear, abrasion-resistant acrylic, which allows optical access to the flow field for visualization measurements. The water tunnel has a capacity of approximately 0.246 m3 (65 gallons), and the fluid used for the experiments was water with fluid properties determined at 20#x00B0;C and 101 kPa (ρ = 998.21 kg/m3, μ = 1.00 × 10−3 Pa•s). The water tunnel also featured a variable frequency drive to precisely control inflow velocities of 0.014 m/s-0.409 m/s in the test section. To ensure uniform flow in the test section, the settling length upstream of the contraction in the tunnel is fitted with a precision tubular cell, plastic, honeycomb section, and three stainless steel screens with 60% porosity.

Fig 4. Water tunnel experimental setup using PIV for velocity field measurements.

Laser and sheet optics are not shown in the figure.

Particle Image Velocimetry (PIV)

Velocity measurements were performed on the WMS model using a 2D PIV system [59]. Fig 4 shows a picture of the experimental setup with the WMS model in the water tunnel facility. The model was attached to a flat surface made of Plexiglass and held by adapters that were clamped to the test section frame. The filtered water was collected through the outlet of the model using clear plastic tubing. The PIV system setup had a dual frame 8MP charge-coupled device (CCD) camera and a 200mJ Nd:YAG double pulsed laser which could pulse at 15Hz. The laser sheet thickness used in the experiments was 1mm created using sheet optics. The flow in the water tunnel was then seeded with silver coated, neutrally buoyant, polyamide particles that were 20 μm in diameter. (Polyamide Particles, LaVision Inc., Ypsilanti, Michigan, USA). The mean diameter and density were of 20 μm, ρ = 1.03 g/cm3, respectively. The PIV measurement plane shown in the figure was calibrated before experiments. The calibration yielded a field of view of 110 mm × 80 mm in the test region with a spatial resolution of 0.063 mm/pixel. For each velocity field measurement, 500 image pairs were acquired, and these PIV images were then processed to determine the velocity fields using DaVis software. The post-processing settings used cross-correlation, multi-pass analyses with 50% overlap, and a final interrogation window of 32 pixels × 32 pixels. The resulting velocity fields contained 9000 vectors which were then used for analyses. Lastly, the PIV experiments were performed using flow velocities of 0.200, 0.300, and 0.400 m/s. The experiments used a camera frame rate of 13Hz with time between image pairs to be 1000–2000 μs based on flow velocity. The averaged velocity fields of each flow scenario were then used to compare with the CFD scenarios and presented in this study.

Particle size analyzer

For particle filtration in a lab-scale scenario, a particle size analyzer (Model 780 AccuSizer, Particle Sizing Systems, Inc., Santa Barbara, California, USA) was used to measure the particle size distribution of the fluid samples upstream and at the outlet port of the OWMS. The calibrated particle-size analyzer used a light blockage principle could detect particles with particle diameters ranging from 0.50–400 μm.Furthermore, the particle size analyzer has a particle sensitivity of 10 ppt, size accuracy of 2% and count accuracy of 10%, and detection limit of up to 35,000 particles/s. The particle analyzer was set to sample up to 50 mL of fluid. For each test scenario, samples were taken in triplicates. The particle size analyzer was set to collect three fluid samples per container (~15 mL), and the averaged results are presented in this study. Furthermore, the equipment tubes were flushed using deionized water before testing to avoid contamination from earlier experiments.

Test conditions

The CFD simulations allowed for parametric study by systematically changing inflow velocity, geometrical characterization, and particle filtration. The CFD studies can be categorized into three different simulation sets and were summarized in Table 1. The first set of CFD simulations (i.e., S1R01-S1R06) aimed to identify the orientation of the WMS with respect to the incoming flow that increased its vortex generation ability. This was set to a forward-facing WMS and the reverse-facing WMS (RWMS). The second set of CFD simulations (i.e., S2R01-S2R360) was conducted to determine the optimum configuration of the WMS that best increases its mass flow rate. Finally, the last set of simulations (i.e., S3R01-S360) was performed to determine the filtration capability of the OMWS at different inflow velocities, pressures, and particle diameters. The results from the third set of simulations were used to compare with experimental results performed at inflow velocities of 0.2 and 0.4 m/s. These inflow velocities were the mid to high range of the water tunnel capability and were selected as they yielded expected vortex dynamics from the CFD studies.

Table 1. Summary of CFD simulations performed on the WMS.

WMS = Wave Minimal Surface, RWMS = Reverse Wave Minimal Surface, OMWS = Optimized Wave Minimal Surface.


Vorticity analysis

Earlier studies [10, 60] have shown that the vortex generated by the crossflow filters plays a crucial role in the antifouling/anticlogging behavior of these filters. Therefore, this study evaluated the different shapes of the WMS for the vortex generation capabilities. For this set of simulations (see Table 1, S1R01-S1R06), the WMS was evaluated in forward and reverse orientations facing the flow which range from 0.03–0.3 m/s. Fig 5a–5d show the averaged vorticity in the x-direction obtained from the CFD simulations at different upstream conditions. These figures show the peak vorticity values in the x-direction as a fraction of the respective geometry to provide insight into their vortex generation abilities. The results in the figure showed the increased averaged x-vorticities of WMS and RWMS geometry for the first row of the simulation (see Fig 5a and 5b). With the flow going through the second row of the array configuration of the geometries (see Fig 5c and 5d), there was a clear reduction in the vorticity values due to the wake from the first row of their respective geometries. The figures indicate that RWMS provided increased vorticity values where there was an increase in the shear layer generated from this orientation. Furthermore, eddies were generated from the free stream due to the orientation, which may be used for filtration purposes. Thus, the RWMS orientation was selected for further CFD studies to enhance its geometrical characteristics for particle filtration.

Fig 5. Average vorticity in the x-direction for different orientations.

(a) First row at 0.03 m/s, (b) first row at 0.3 m/s, (c) second row at 0.03 m/s, and (d) second row at 0.3 m/s. Raw data can be found in S1 Data.

Optimization of WMS

The next set of simulations was analyzed to determine the WMS configuration that provided the increased mass flow of the filtrate at the filter pore (see Table 1, S2R01-S2R360). The impact of WMS configuration on the flow rate at the outlet port was studied to determine the amount of filtered fluid that can be captured using this design. Hence, for each WMS configuration, different suction pressures from 0 kPa to 35 kPa were applied at the outlet port of the model. Different upstream velocities from 0.20–0.40 m/s were also applied for each simulation scenario. With these in mind, the results for one of the configurations can be seen in Fig 6a. The flow rate at the outlet port was evaluated as a percentage of the mass flow from upstream of the model. The results in Fig 6a illustrate the flow rate capability of the WMS for the convex shape n = 1 and FI = 90o configuration. Here we note that this configuration was denoted as OMWS as it provided the most mass flow rate percentage in all the studies performed. The results show the increase in mass flow rate at the outlet port is proportional to the increase in suction pressure. The flow rate also showed that the OMWS captured 15–25% of the inlet condition. This can be attributed to the different configurations of the model where the interaction of the incoming fluid to the geometry changes. Further systematic studies need to be performed to identify the mechanisms for this behavior. Nevertheless, for all the other configurations studied here, the flow rate reduced significantly compared to the results shown in Fig 6a and thus are not discussed here.

Fig 6.

(a) Mass flow rate at the outlet port at different upstream velocities and suction pressures for the convex, n = 1, and FI = 90o configuration. (b) Particle efficiency at different inflow conditions. Raw data for (a) can be found in Table A in S2 Data while raw data for (b) can be found in Table B in S2 Data.

Filtration of OWMS

Multiphase CFD simulations were performed to assess the WMS models’ particle filtration efficiency for a range of upstream velocities, outlet pressure, and particle diameters (see Table 1, S3R01-S3R60). For each simulation, 5000 particles of constant diameter and density were released at the upstream location of the model. The density of the particles (ρp) was simulated to be similar to water (i.e., ρp = 1 g/mL). Furthermore, the particles were released at x/n = 5.0 upstream to allow the particles to follow the incoming streamline before the model. The filtration efficiency was then determined by the ratio of particles captured at the outlet port and the number of particles per unit length from the upstream location. The filtering efficiency results are shown in Fig 6b. The results show a significantly high filtration efficiency for the OMWS, with a reduction of filtering capability with the decrease in inflow velocity. The decrease in the filtering capability could be attributed to particle inertia as the particles flow around the model. Furthermore, the OMWS was observed to be the most efficient filter for the studied flow conditions. The results also suggest that particle diameters greater than 50 μm can be filtered, and thus particle diameters play a role in the filtration efficiency of the current design.

Here, we note that the results obtained in the CFD studies could be an artifact particle density and where the particles were released with respect to the model. Further, controlled CFD studies need to be performed for validation of the observed conditions. One such study could be the impact of different particle counts upstream of the model. The number of particles used in the current manuscript was chosen primarily to optimize the computational time and resources. Nevertheless, the trends observed in the results point to the capability of the OMWS to filter a certain range of particles at high efficiency. To complement this initial observation, experimental studies were performed on the 3D-printed OMWS using Stratasys F370 3D printer with build size of 355 × 254 × 355 mm (14 × 10 × 14 in.) and layer resolution of 0.0127 cm. The model was 3D-Printed (3DP) using black-ABS filament material.

Flow field validation

The averaged velocity field result from the CFD of the OMWS model was compared with experimental data using PIV for validation purposes. Fig 7a and 7b show the CFD and EFD average velocity contour fields for 0.20 m/s inflow velocity, while Fig 7c shows the root-mean square error between the CFD and EFD. There is a good agreement between the two flow fields, and the root-mean-square error was estimated to be less than 10%. As observed in the figure, the fluid accelerates over the model, and there is a large region of slow-moving recirculating fluid in the wake of the model. It can be seen from the figure that the upstream flow slowed down near the stagnation point on the OMWS model. The acceleration over the model could be attributed to the design of the OMWS providing a converging duct effect. A shear layer region was observed near the downstream side of the OMWS as slow-moving fluid was present in this area. The CFD results were validated with EFD measurements to show that the CFD model approach and geometry optimizations performed could be justified. Results were also obtained for 0.35 m/s and 0.40 m/s scenarios and yielded good agreement between CFD and EFD and were not presented in the manuscript for brevity.

Fig 7. Flow field comparison for upstream velocity of 0.2 m/s.

(a) CFD, (b) EFD, and (c) RMS error. Flowfield data for (a) can be found in S3 Data while flow field data for (b) can be found in S4 Data. Uncertainty in (b) can be found in S5 Data.

OMWS lab-scale experiments

The CFD simulations demonstrated the potential of the OMWS to perform water filtration. To support this observation, the OMWS was 3DP and tested for its particle filtration capabilities in a lab-scale scenario. These experiments were performed for inflow velocities of 0.07 m/s-0.40 m/s. The fluid was sampled upstream of the model and outlet port of the model. The collected samples were analyzed using the particle size analyzer to assess the size and density of particles in these sampled fluids. The filtration effectiveness of the OMWS was quantified by the ratio of counted particles in the outport port of the model (filtrate) and upstream fluid.

Fig 8 shows the filtration effectiveness of the OMWS model at different inflow velocities. The results show that the model could filter particle sizes of 1 μm-30 μm leading to at least a 50% reduction of particle counts in the filtrate sample. This was less than the observed efficiency from the CFD results (see Fig 6b), and this difference could be attributed to the simulation parameters chosen or the experimental protocols performed. However, both CFD and EFD results now point to OMWS being able to filter particles in different inflow scenarios. Although the range of particles was different for CFD and EFD studies, they both show trends that the OMWS design filter particles of certain diameters efficiently. The trend displayed by both methods shows that the OMWS exhibit certain mechanisms that allow it to filter particles, and these mechanisms are currently being studied. Here we note that for the 0.270 m/s flow scenario (not shown), there was an observed presence of higher counts of particles in the filtrate. Yao et al. [61] also observed a similar result, and they attributed it to the shear forces developed in the filter, causing a further breakdown of particles.

Fig 8. Lab-scale experiments using OMWS at different water tunnel velocities.

Raw particle size analyzer data at these inflow conditions can be found in S6S13 Data.


Computational and experimental studies were performed on a rigid WMS model to determine its potential use as a filtration design for water treatment applications. The CFD parametric studies of the WMS helped identify a configuration based on increased mass flow rate at the outlet port and vortex generations. The CFD studies yielded a reverse wave filter design with convex shape leading-edge, an angle of incidence of 90o, and maximum width integer of n = 1.0. This configuration of the WMS (i.e., OMWS) captured 15–25% of the inflow velocity at the filter port for velocities of 0.2–0.4 m/s. The flow fields around the OMWS at different upstream velocities agreed reasonably with PIV measurements. The averaged flow field results in both the CFD and PIV contour fields showed a root-mean-square error of 10%. A close match between CFD and experimental velocity flow field provided confidence in the CFD studies, particularly in the observed flow dynamics at different regions of the model. The OMWS design of the filter was computationally and experimentally studied to quantify the OMWS filtering efficiency. The computational studies showed filtering efficiencies of greater than 95% for 50–200 μm, and benchtop experiments showed efficiencies of 50% for 1–25 μm. They could be attributed to the limited number of particles used during the simulations to optimize the simulation time and computing resources, while particle clustering and degradation could occur in EFD studies. Nevertheless, it is interesting to note that despite the difference in filtering efficiency between the experiments and CFD studies, the overall trend was similar.

This investigation showed the potential of using an OMWS for particle filtration for removing biochar particles of 1–25 um diameters. The new design of OMWS filters, if implemented on a field scale, can provide a robust and efficient approach to capturing P and N. As this design utilizes the fluid vortex flow to capture the particles, the filter therefore will not get clogged with particles and does not require a large pressure gradient to filter out the particles, unlike conventional filters. However, controlled experiments and simulations must be performed to develop a field-scale filtering device. If successful, the new filter design can potentially address the nutrient pollution responsible for HABs. Lastly, this study provided a framework for studying crossflow filtration using CFD, EFD, and 3DP technology to develop lower energy separation science approaches from conceptualization to benchtop prototypes.

Supporting information

S1 Data. CFD simulation datasets of WMS for vorticity calculations.


S2 Data. Mass flow rate and particle filtration data of OMWS.


S3 Data. CFD simulation dataset of OMWS for 0.2 m/s.


S4 Data. PIV mean data of OMWS for 0.2 m/s.


S5 Data. PIV standard deviation data of OMWS for 0.2 m/s.


S6 Data. Particle size analyzer data of OMWS for 0.071 m/s upstream condition.


S7 Data. Particle size analyzer data of OMWS for 0.071 m/s filtrate condition.


S8 Data. Particle size analyzer data of OMWS for 0.138 m/s upstream condition.


S9 Data. Particle size analyzer data of OMWS for 0.138 m/s filtrate condition.


S10 Data. Particle size analyzer data of OMWS for 0.205 m/s upstream condition.


S11 Data. Particle size analyzer data of OMWS for 0.205 m/s filtrate condition.


S12 Data. Particle size analyzer data of OMWS for 0.409 m/s upstream condition.


S13 Data. Particle size analyzer data of OMWS for 0.409 m/s filtrate condition.



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