Porous medium approach for aerodynamic characterization of fog water collector mesh

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(1)PONTIFICIA UNIVERSIDAD CATÓLICA DE CHILE ESCUELA DE INGENIERÍA. POROUS MEDIUM APPROACH FOR AERODYNAMIC CHARACTERIZATION OF FOG WATER COLLECTOR MESH. DANIEL ANTONIO MONTOYA CONTRERAS. Thesis submitted to the Office of Research and Graduate Studies in partial fulfillment of the requirements for the degree of Master of Science in Engineering. Advisor: WOLFRAM JAHN VON ARNSWALDT. Santiago de Chile, September 2018 c MMXVIII, DANIEL A NTONIO M ONTOYA C ONTRERAS.

(2) PONTIFICIA UNIVERSIDAD CATÓLICA DE CHILE ESCUELA DE INGENIERÍA. POROUS MEDIUM APPROACH FOR AERODYNAMIC CHARACTERIZATION OF FOG WATER COLLECTOR MESH. DANIEL ANTONIO MONTOYA CONTRERAS. Members of the Committee: WOLFRAM JAHN VON ARNSWALDT FRANCISCO IGNACIO SUÁREZ POCH JUAN DE DIOS RIVERA AGUERO ANDRÉS RODRIGO GUESALAGA MEISSNER Thesis submitted to the Office of Research and Graduate Studies in partial fulfillment of the requirements for the degree of Master of Science in Engineering. Santiago de Chile, September 2018 c MMXVIII, DANIEL A NTONIO M ONTOYA C ONTRERAS.

(3) Gratefully to my parents, siblings, friends and my beloved girlfriend.

(4) ACKNOWLEDGEMENTS. I would like to thank Wolfram Jahn for his support and guidance during my studies. Also thanks to Juan de Dios Rivera, for his absolute support in the development of this research. Special thanks to my parents Roberto and Elizabeth, my siblings Pablo, Aylin and Carolina, because with family support anything is possible. Finally, infinite thanks to Constanza, who supported me in the most difficult moments of this Thesis. And of course, thanks to Moraga family for always believe in me.. iv.

(5) TABLE OF CONTENTS. ACKNOWLEDGEMENTS. iv. LIST OF FIGURES. vii. LIST OF TABLES. x. ABSTRACT. xi. RESUMEN. xii. 1.. ARTICLE INTRODUCTORY BACKGROUND. 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1. Objectives and hypothesis . . . . . . . . . . . . . . . . . . . . . . .. 3. Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4. 1.2.1.. Fog characterization . . . . . . . . . . . . . . . . . . . . . . . . . .. 4. 1.2.2.. Aerodynamic characterization . . . . . . . . . . . . . . . . . . . . .. 5. 1.2.3.. Fog modeling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 7. Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 8. 1.3.1.. Open Foam for simulation . . . . . . . . . . . . . . . . . . . . . . .. 9. 1.3.2.. Realistic mesh simulation at confined flow condition . . . . . . . . .. 10. 1.3.3.. Porous medium simulation at confined flow condition . . . . . . . . .. 11. 1.3.4.. Porous medium and realistic mesh comparison at open flow condition .. 12. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 14. 1.4.1.. Pressure drop coefficient results . . . . . . . . . . . . . . . . . . . .. 15. 1.4.2.. Porous medium approach input. . . . . . . . . . . . . . . . . . . . .. 16. 1.4.3.. Run time comparison . . . . . . . . . . . . . . . . . . . . . . . . . .. 17. 1.4.4.. open flow results . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 17. Conclusions and future work . . . . . . . . . . . . . . . . . . . . . . . .. 18. 1.1.. 1.1.1. 1.2.. 1.3.. 1.4.. 1.5.. v.

(6) 2.. Validation of Porous Media model using CFD to characterize the aerodynamics of fog water collectors. 22. 2.1.. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 22. 2.2.. Theoretical framework . . . . . . . . . . . . . . . . . . . . . . . . . . .. 28. 2.2.1.. Porous medium equation and Forchheimer Equation . . . . . . . . . .. 30. 2.2.2.. Experimental data . . . . . . . . . . . . . . . . . . . . . . . . . . .. 32. 2.2.3.. Numerical Simulation . . . . . . . . . . . . . . . . . . . . . . . . .. 33. CFD Modeling of FWC in the wind tunnel . . . . . . . . . . . . . . . . .. 36. 2.3.1.. Numerical setup . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 37. 2.3.2.. Results of realistic approach in confined flow and discussion . . . . .. 40. 2.3.3.. Porous medium model and simulations-results in confined flow and. 2.3.. 2.4.. discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 43. FWC model at open flow . . . . . . . . . . . . . . . . . . . . . . . . . .. 45. Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . .. 46. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 50. 2.4.1. 2.5.. REFERENCES. 51. vi.

(7) LIST OF FIGURES. 1.1. Examples of the shapes of different fog collectors. a) Large fog collector; b) 3-D fog collector. Taken from (Brown & Bhushan, 2016) . . . . . . . . . . .. 1.2. 1. Examples of different mesh used for fog collectors. a) Raschel mesh; b) stainless steel and poly-yarn mesh; c) prototype polymaterial. Taken from (Klemm et al., 2012) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1.3. 2. The distribution of liquid water content in front of the mesh as a function of droplet diameter. Taken from (Schemenauer & Joe, 1989) . . . . . . . . . .. 4. 1.4. Superposition scheme. Taken from (Rivera, 2011) . . . . . . . . . . . . . .. 5. 1.5. Porous medium approach. A simulation of airflow through a mesh screen with the filament details (left). A porous slab with porous medium properties of the mesh screen (right). Taken from (Teitel, 2010) . . . . . . . . . . . . . . . .. 1.6. 8. Filaments dimensions in millimeters on simulated mesh screen (left). Wind tunnel dimensions in millimeters (right). . . . . . . . . . . . . . . . . . . .. 10. 1.7. Dimensions of the control volume to analyze the open flow case. . . . . . . .. 13. 1.8. Pressure drop as function of inlet velocity in the wind tunnel, from results of the realistic mesh approach (•), simulation of the porous medium approach (4). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1.9. 14. C0 as a function of the Reynolds number, from the results of the realistic mesh approach (•), simulation of the porous medium approach (4). . . . . . . . .. 15. 1.10 Flow field for realistic mesh approach simulation at open flow condition. . . .. 18. 1.11 Flow field for Porous medium approach simulation at open flow condition. . .. 19. vii.

(8) 1.12 Superposition of isobars from realistic mesh and porous medium simulations. Red contour lines correspond to the realistic mesh simulations, while blue lines correspond to the porous medium approach. . . . . . . . . . . . . . . . . .. 19. 1.13 Velocity field near from the realistic mesh (left) and near from porous slab of 40 mm thick (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 20. 1.14 Horizontal velocity field error between the realistic mesh and porous medium approach at open flow condition. . . . . . . . . . . . . . . . . . . . . . . .. 20. 2.1. Schematic drawing of a FWC system. . . . . . . . . . . . . . . . . . . . . .. 22. 2.2. Map of locations where fog collection has a high potential for success. (Domen, Stringfellow, Camarillo, & Gulati, 2014) . . . . . . . . . . . . . . . . . . .. 23. 2.3. Geometry of a Raschel 35 mesh. . . . . . . . . . . . . . . . . . . . . . . .. 27. 2.4. Logarithmic plot of friction factor versus permeability Reynolds number taken from (Nield & Bejan, 2013). The fK = 0.55 asymptote changes depending on the specific porous medium.. 2.5. . . . . . . . . . . . . . . . . . . . . . . . . .. 33. C0 as function of solidity for knitted ribbon filaments type. (Echevarrı́a Johnson, 2015) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 34. 2.6. Testing framework assembled on the wind tunnel. . . . . . . . . . . . . . .. 37. 2.7. Wind tunnel realistic mesh setup. . . . . . . . . . . . . . . . . . . . . . . .. 38. 2.8. Boundary layer on the contour of mesh fibers. . . . . . . . . . . . . . . . .. 39. 2.9. Porous slab on wind tunnel. . . . . . . . . . . . . . . . . . . . . . . . . . .. 39. 2.10 Pressure drop as a function of the inlet velocity into the wind tunnel, from a realistic mesh approach simulation (•) and experimental data (4). . . . . . .. 41. 2.11 Friction factor versus permeability Reynolds number, for simulation results. .. 42. viii.

(9) 2.12 Pressure drop as function of inlet velocity into wind tunnel, from results of porous medium approach simulation with porous properties obtained from realistic approach (Simulation a) (•), simulation with porous properties obtained from experimental data (Simulation b) (♦), and pressure drop from experimental data (4). . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 43. 2.13 Flow field for the realistic mesh approach simulation at open flow condition. .. 46. 2.14 Flow field for the porous medium approach simulation at open flow condition. 47 2.15 Superposition of isobars from realistic mesh and porous medium simulations. Red contour lines correspond to the realistic mesh simulations, while blue lines correspond to the porous medium approach . . . . . . . . . . . . . . . . . .. 47. 2.16 Velocity field from a simulation with a realistic mesh (a) and from simulation with a porous slab 40 mm thick (b). . . . . . . . . . . . . . . . . . . . . . .. 48. 2.17 Horizontal velocity field error between the realistic mesh and porous medium approach at open flow condition. . . . . . . . . . . . . . . . . . . . . . . .. 49. 2.18 Horizontal velocity profile superposition between the realistic mesh and porous medium approach at open flow condition. The wake is recognizable between -400 mm and 400 mm. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. ix. 49.

(10) LIST OF TABLES. 1.1. Air properties used for simulations. (Mills, 1999) . . . . . . . . . . . . . . .. 1.2. Average C0 obtained by experimental results, realistic mesh approach and. 10. porous medium approach . . . . . . . . . . . . . . . . . . . . . . . . . . .. 16. 1.3. Coefficients obtained from Realistic mesh approach. . . . . . . . . . . . . .. 16. 1.4. Execution time (seconds) for realistic mesh (tr ) and porous medium (tp ) approach, for meshing and solver. . . . . . . . . . . . . . . . . . . . . . . .. 17. 2.1. Coefficients obtained from Realistic mesh approach. . . . . . . . . . . . . .. 41. 2.2. C0 coefficients obtained from porous medium approach. . . . . . . . . . . .. 44. 2.3. Execution time (seconds) for realistic mesh (tr ) and porous medium (tp ) approach, for meshing and solver. . . . . . . . . . . . . . . . . . . . . . . .. x. 45.

(11) ABSTRACT. Harvesting water from coastal fog is a promising alternative for regions of water scarcity such as northern Chile. The fog water collector (FWC) is a mesh screen usually installed perpendicular to the main wind direction, intercepting the air flow and extracting the water droplets contained in it. In order to maximize the water output, many FWC have to be located in an optimized manner. For optimizing the layout, computational simulations that consume large amounts of computational resources are necessary. The aim of the present study is to develop a computationally cheap methodology of simulating flow through FWCs. For this a porous medium model was validated in order to adequately represent the aerodynamics of FWC at open flow conditions. Computational simulations of airflow were carried out through a mesh screen at confined flow and compared to simulations of flow through a porous medium at confined flow. The results were then extrapolated to air flow through a mesh screen at open flow conditions. The pressure drop coefficients of both simulations at confined flow were compared, yielding difference of less than 4%. For the open flow simulations, the velocity fields for the simulation with detailed mesh screen and the porous medium approach were compared, and it was determined that outside of the wake the difference between both simulations does not exceed 5%, while within the wake the error is close to 30%. The flow field is generally similar with only minor differences in the pressure distribution. This indicates that the model of porous medium is a valid methodology to simulate the flow through a FWC for both, confined flow and open flow conditions at a much smaller computational cost.. xi.

(12) RESUMEN. La colección de agua de niebla es una alternativa prometedora en regiones con escasez de agua, como en el norte de Chile. El atrapanieblas es una malla instalada perpendicularmente a la dirección del viento, interceptando el flujo aire y extrayendo las gotas de agua que contiene. Para maximizar la cantidad de agua colectada, se deben instalar muchos atrapanieblas de forma optimizada. Para optimizar la disposición espacial de los atrapanieblas se requieren simulaciones que utilizan una gran cantidad de recursos computacionales. El objetivo del presente estudio es desarrollar un modelo de bajo costo computacional para simular el flujo de viento a través de atrapanieblas. Para esto se validó un modelo de medio poroso que permite representar adecuadamente la aerodinámica de un atrapaniebla en condiciones de flujo externo. Se realizaron simulaciones de flujo de viento a través de una malla en flujo confinado y se compararon con simulaciones de viento a través de un medio poroso en flujo confinado. Posteriormente, los resultados se extrapolaron a flujo de viento a través de una malla en condiciones de flujo abierto. Se compararon los coeficientes de caı́da de presión C0 de ambas simulaciones en flujo confinado, la diferencia entre ambos resultados menor a un 4%. Para las simulaciones en flujo abierto se comparó el campo de velocidad para la malla y el medio poroso y se determinó que fuera de la estela la diferencia entre ambas simulaciones no supera el 5%, mientras que dentro de la estela el error es cercano al 30%. El campo de flujo es similar para ambas simulaciones, solo con pequeñas diferencias en la distribución de presión. Se concluye que el modelo de medio poroso es una metodologı́a válida y de bajo costo computacional para simular el flujo a través de una malla de atrapaniebla, tanto en flujo confinado como en flujo abierto.. xii.

(13) 1. ARTICLE INTRODUCTORY BACKGROUND 1.1. Introduction Water scarcity affects 2.7 billion people worldwide at least a month in a year, which corresponds to 37% of the world population (WWF, 2017). It is mainly caused by water pollution, population growth, human and industrial waste, and the indiscriminate and irresponsible use of water for agriculture. Lack of water has led to use every source of this precious resource. One type of water source can be found in the air: fog is an important fount of freshwater for many coastal regions (Cereceda, Hernández, Leiva, & Rivera, 2014). Fog is defined as water vapor condensed into small water droplets suspended in air, either on continental or oceanic surfaces (Da Franca & Dos Anjos, 1998). Fog harvesting occurs when water droplets in the airflow collide with an object so that it can be collected into a container as a source of freshwater. Fog collector projects are usually located in arid and semi-arid climates, due to the presence of fog events and the lack of a more direct freshwater source, such as rain or groundwater (Klemm et al., 2012).. Figure 1.1. Examples of the shapes of different fog collectors. a) Large fog collector; b) 3-D fog collector. Taken from (Brown & Bhushan, 2016). 1.

(14) A fog water collector (FWC) is a vertical mesh screen usually installed against wind direction to intercept the water droplets transported by the air. The water agglomerates within the fibers of the mesh, then drains because of gravity into a container at the bottom of the collector. Finally, it flows into a tank to be stored. The shape of fog collectors can be flat or three dimensional, as shown in Fig. 1.1. The material used for the mesh is usually a knitted polymer with different fiber widths and knitting patterns, see Fig. 1.2.. Figure 1.2. Examples of different mesh used for fog collectors. a) Raschel mesh; b) stainless steel and poly-yarn mesh; c) prototype polymaterial. Taken from (Klemm et al., 2012) Parameters to decide if a specific site is feasible to install a fog collector are the water quality, wind velocity and liquid water content of the fog. The water quality obtained from fog collectors helps to determine if it is suitable for human consumption. Different studies have demonstrated that water obtained by fog collectors in many sites around the world met the World Health Organization (WHO) standards for drinking water (Klemm et al., 2012). The liquid water content (LWC) corresponds to the amount of condensed water transported by air measured in kilograms of water per cubic meter of air. Montecinos et al. (2018) conclude that its value must be equal or higher than 0.045 gm−3 for establishing water collection, because no water collection is registered when LWC is below that number. Wind velocity for most sites are between 1 ms−1 and 16 ms−1 (de la Jara, 2012). The amount of collected water obtained by a FWC is described in terms of the average collection rate measured in liter per square meter per day (LDM). Collection rate varies on 2.

(15) each site and is in the range of 1.5 to 81 LDM (Klemm et al., 2012), but these values have high variability depending on the season and even on a day to day basis. Studies about the cost of fog collection projects conclude that for this technology to be economically competitive, it should have a collection rate of at least 10 LDM (assuming a 24% collection efficiency), but most fog collection projects are below that number (Cereceda et al., 2014).. 1.1.1. Objectives and hypothesis The present study forms part of a long term research with the objective of obtaining an optimal layout for fog collector farms, in order to maximize the water collected by an arrangement of fog collectors placed in a given piece of land. However, modelling the flow through fine meshes is computationally expensive and would not be feasible for optimization procedures. The specific objective of this study is to find a computationally cheap model which simulates the airflow through a fog collector mesh. This would allow for iterative optimization techniques at an acceptable computational cost. In this case, the porous medium model with the Forchheimer equation is analyzed using computational fluid dynamics to simulate different scenarios. The main scientific contribution of this study is the comparison of both approaches (realistic mesh and porous medium) at open flow condition, because an analysis of this type is not found in the literature, while the comparison in wind tunnel has been studied for a woven mesh screen (Teitel, 2010). The hypothesis is that a porous medium model of a fog water collection mesh screen can be used to calculate its pressure drop coefficient and thus obtain the flow field of wind passing through and around the mesh screen. The pressure drop coefficient obtained by running the porous medium model will be compared with results obtained by experimental data and the realistic case simulation.. 3.

(16) 1.2. Literature review In this section a brief summary of the most relevant literature related to fog collection is given, along with flow simulations through mesh screen and porous medium properties of mesh screens.. 1.2.1. Fog characterization Fog water transportation and collection modeling is a multi-scale problem separated in two, macro-scale and micro-scale (de la Jara, 2012). In micro-scale analysis, water droplet diameters and its distribution are studied. On the other hand, macro-scale covers wind velocity, water content of the air and water flow rate.. Figure 1.3. The distribution of liquid water content in front of the mesh as a function of droplet diameter. Taken from (Schemenauer & Joe, 1989) Droplet diameter distribution of fog events is studied to analyze the size contribution to LWC and aerodynamic behavior. Fog is considered to have at least 0.01 gm−3 of LWC (Montecinos et al., 2018), the distribution of LWC as a function of droplet diameter (Fig. 1.3) helps to understand the droplet diameters that a FWC should harvest. Also, as droplet diameters typically have a size of up to 30 µm, it can be assumed that fog droplets follow the same trajectory of the air streamlines (de la Jara, 2012). Thus, the droplet velocities of 4.

(17) the undisturbed fog (upstream) is the same as the wind speed, and the droplet deposition on the mesh screen occurs mainly by inertial impact in a known proportion, determined by the Stokes number, St . Accordingly, the water droplet-mesh interaction can be neglected and it is enough to study the airflow behavior (Echevarrı́a Johnson, 2015).. 1.2.2. Aerodynamic characterization The aerodynamics of fog collection meshes have been studied analytically and experimentally. A simplified analytical model was developed to define the aerodynamic collection efficiency, and to find an optimal shade coefficient of a mesh screen (Rivera, 2011). Experimental measurements have focused on having data to characterize the sites where the fog collection projects are located. The maximum fraction of the undisturbed fog that can be captured by a FWC is called aerodynamic collection efficiency ηAC (Rivera, 2011). An analytical model of the airflow through a mesh screen was developed by Rivera and its analysis was based on the superposition method (valid for a potential flow). In Fig. 1.4, the real case of airflow passing through a mesh screen can be represented as the sum of air passing through a mesh screen in confined flow plus air passing around a solid plate.. Figure 1.4. Superposition scheme. Taken from (Rivera, 2011) 5.

(18) The model results in that ηAC , equation 1.1, depends on the pressure drop coefficient of the mesh C0 , drag coefficient of a non-permeable screen Cd , and on the solidity s, defined as the fraction of surface covered by the mesh that is capable of collecting droplets (Echevarrı́a Johnson, 2015).. ηAC =. s p 1 + C0 /Cd. (1.1). A theoretical optimization analysis was made by Regalado (2016), in order to find and optimal arrangement and geometry of fog collector with maximal fog water yield. The study was made by comparing the number of impacting screen at staggered arrangement, inclination of the screens, and geometry of the collector. An optimal number of screens was found, which is between 3 to 5. Rectangular flat design is preferred over cylindrical one. There exist a slight reduction of the impact efficiency with the inclination of the impacting screen. An important conclusion of the study is that the aerodynamic efficiency is limiting during the fog collection process, and it is difficult to increase. The downstream vertical velocity distribution has been measured and the wake has been characterized by tracing colored particles at the bottom into a wind tunnel (Bresci, 2002). These studies show the presence of a turbulent disturbed area in the downstream flow of the mesh screen that cannot be analyzed analytically, and a detailed computational simulation should allow to perform a better analysis of this phenomenon. Experimental measurements of the airflow through a mesh screen into a wind tunnel were made by Echevarrı́a (2015). He measured pressure drops for different types of meshes. Among the meshes studied is the Raschel 35 mesh, the most typical mesh used for fog harvesting. The most relevant result to be used in the present research is the pressure drop coefficient for the Raschel Mesh 35, with a solidity s of the mesh of 0.4806. This pressure drop coefficient C0 is 2.23.. 6.

(19) A comparative analysis of the water collected by three different types of mesh was made by Fernandez et al.(Fernandez et al., 2018). They measured the water collected by Raschel, stainless steel mesh coated with MIT-14 hydrophobic formulation and FogHaTin, at four different sites in California. MIT-14 collected more water than the other meshes, and the Raschel mesh is the less effective mesh for water collection, but it is the cheapest among those studied. Therefore, a greater fog water collection effectiveness has a higher cost and both variables should be considered for the design of fog collection projects.. 1.2.3. Fog modeling The first simulation of a mesh screen including droplet collision was made by De la Jara (2012), who simulated water droplets as Lagrangian particles and modeled the interception of those particles against cylindrical fibers as a two dimensional problem. De la Jara concludes that the collection efficiency obtained by a finite elements model has a 2% relative error, when compared to the analytical model, using the same solidity and C0 obtained from Idel’cik (1960). Other simulations on mesh screens were made by Valera (2005), who analyzed different types of meshes used for greenhouses, by comparing the pressure drop across meshes in a wind tunnel obtained by CFD simulations and experimental measurements. He also compared porous properties of the mesh obtained from both, simulations and experimental data, which are permeability, and the inertial factor. Teitel used a porous medium approach (Teitel, 2010), to model the aerodynamic behavior of the mesh screen. The method consist on simulating a porous slab in a wind tunnel as replacement of the mesh screen, see Fig. 1.5. The idea of using a porous medium is adopted in this study as a cheap computational model. Its implications on confined flow are assessed.. 7.

(20) Figure 1.5. Porous medium approach. A simulation of airflow through a mesh screen with the filament details (left). A porous slab with porous medium properties of the mesh screen (right). Taken from (Teitel, 2010) As a porous medium approach is used to model the aerodynamic of a fog collector mesh, the Forchheimer equation is used for this model, as it describes the flow of a fluid passing through a porous medium.. −. dp Y µ u + ρ 0.5 u2 = dx Kp Kp. (1.2). where p is the pressure, x the unidimensional length coordinate, µ is the dynamic viscosity of the fluid, u the fluid velocity, ρ the fluid density, Kp the permeability of the porous medium, and Y is the inertial factor or non-linear momentum loss coefficient.. 1.3. Methodology The main goal for this study is to find a computationlly cheap model that adequately represents the airflow through a mesh screen. Teitel methodology is followed, using the Forchheimer equation to model the mesh screen as a porous medium. In order to analyze the computational cost of the model and its feasibility, the research has been separated into three stages. All the simulations are based on the specific Raschel Mesh 35 with a solidity of 0.4806.. 8.

(21) The first step is to simulate the air flow through a realistic mesh screen in a wind tunnel, with full resolution of the space between filaments. Then, validate the results by comparing them with experimental measurements obtained by Echevarrı́a (2015). The results to be compared are the pressure drop coefficient C0 , Kp and Y , parameters from the Forchheimer equation. The mesh will then be replaced by a porous medium with equivalent K and Y coefficients, from the previous step, and the results will be compared with the realistic mesh simulation. The third step is to simulate both, realistic mesh and porous medium, but at open flow conditions. The flow field is compared then for both scenarios, qualitatively an quantitatively.. 1.3.1. Open Foam for simulation Simulations are run in OpenFOAM (OpenCFD, 2004), an open source computational fluid dynamics code based on C++ libraries that contains a large set of flow solvers. SimpleFoam is the solver used for this study, as it solves a version of Navier-Stokes equation (eq. (1.3) and (1.4)) correponding to incompressible, steady-state and turbulent flow, where pressure p is normalized with the density. The incompressible Navier-Stokes equations are. ∇ · ~u = 0. (1.3). ∂~u + (~u · ∇) ~u = −∇P + ν∇2~u + ~g ∂t. (1.4). P =. 9. p ρ. (1.5).

(22) where P is the normalized pressure and air properties used from table 1.1. The turbulence model applied was SST k − ω RANS, since this model is practical for the fibers of the mesh that act as walls along the entire cross-section area. Table 1.1. Air properties used for simulations. (Mills, 1999) Symbol Value Unit Property ◦ C Temperature T 20 Density ρ 1.204 kgm−3 −5 Dynamic viscosity µ 1.811× 10 Pa s −5 ν 1.504× 10 m2 s Kinematic viscosity. 1.3.2. Realistic mesh simulation at confined flow condition The simulation process begins by creating CAD files of the mesh screen with a width of 0.3 mm, from geometrical details given by Holmes (2015), and a wind tunnel, both are shown in Fig. 1.6.. Figure 1.6. Filaments dimensions in millimeters on simulated mesh screen (left). Wind tunnel dimensions in millimeters (right). The mesh screen is placed in the middle of the wind tunnel, covering all the crosssection area. A refinement region is defined around the mesh screen for the numerical grid. Boundary conditions are defined for each surface, no-slip condition for the lateral faces of the wind tunnel and filament surface, fixed velocity for the wind tunnel inlet and zero pressure for the outlet. The simulations are run for different inlet velocity ranging from 0.1 ms−1 to 8 ms−1 .. 10.

(23) The post-processing includes the average inlet pressure, which corresponds to the pressure drop ∆P through the wind tunnel. From eq. (1.6) it is possible to obtain the pressure drop coefficient for each inlet velocity (Kundu, Cohen, & Dowling, 2012).. C0 =. ∆p (1/2)ρu2. (1.6). Further, it is possible to relate inlet velocity to pressure drop. From that, the Forchheimer coefficients, Kp and Y , can be obtained by using a quadratic equation to represent the data,. ∆p = a1 u2 + a2 u.. (1.7). Then, Kp and Y values are obtained from a discrete version of eq. (1.2).. ∆xµ a2. (1.8). a1 K 0.5 ∆xρ. (1.9). K=. Y =. In the context of the current study ∆x refers to the thickness of the mesh screen.. 1.3.3. Porous medium simulation at confined flow condition The porous medium approach is described in this section. The method consist of replacing the mesh screen by a porous slab in the wind tunnel. The porous slab thickness does not necessarily have to be the same as that of the mesh. Even more, it is useful to have greater thickness, to have a coarse computational mesh and this to reduce the simulation cost. However, bigger thickness values can lead to errors in the simulation, so that an. 11.

(24) appropriate distance between the end of the block and the end of the wind tunnel should be considered. The simulation setup is the same as the one used for the realistic mesh simulation, but with a porous slab instead of the mesh screen. For this case, a block of 1600 mm2 with 10 mm thickness is placed at the center of the wind tunnel, in the same position where the mesh screen was located. The input of the model are porous properties obtained from realistic mesh simulations. As the porous region has a different thickness than the mesh, porous properties Kp and Y are not directly used as inputs. The modified values for those properties are defined as Kp0 and Y 0 and depend on the thickness ratio,. ∆x0 Kp Kp0 = ∆x 0.5 ∆x 0 Y = Y ∆x0. (1.10) (1.11). where ∆x0 is the thickness of the porous slab. Within the porous cells, the Forchheimer equation is added as a source term to the momentum equation of the simpleFoam solver. Once the model is run, the pressure drop versus inlet velocity curve can be plotted to analyze its difference compared to the curve obtained from the realistic mesh simulation. The average pressure drop coefficient can also be compared.. 1.3.4. Porous medium and realistic mesh comparison at open flow condition The final goal of this research is to analyze the porous medium approach as a cheap computational model to be used instead of the realistic mesh approach. Thus, it is necessary to test the model under realistic conditions of open flow.. 12.

(25) The porous slab simulation at open flow condition is compared to the realistic mesh simulation in the same situation. Therefore, it will be compared to the flow field around the mesh and around the porous region, for a given upstream velocity, perpendicular to the screen. The control volume used to analyze airflow through the mesh is shown in Fig. 1.7. The center of the mesh of 400 mm x 400 mm is placed at the origin of the coordinate axis, the bounding box coordinates are from (-400, -800, -800) mm to (1600, 800, 800) mm. A refinement region is defined around the mesh, to have the necessary resolution to capture the filaments of the mesh.. Figure 1.7. Dimensions of the control volume to analyze the open flow case. Boundary condition for the bounding box are fixed inlet velocity at the upstream and zero pressure for other surfaces, non-slip is defined in the surfaces of the mesh screen. The simulation is run for two models of the mesh screen. One is for the detailed mesh screen, i.e. the realistic mesh approach at open flow condition. The other is for the porous slab, which is called the porous medium approach at open flow condition.. 13.

(26) Once the simulation for both cases is run, a qualitative and quantitative comparison is made. Quantitative comparison is oriented to determine the difference between velocity flow fields for both cases. Qualitative analysis is made to compare isobars around the mesh for both cases.. 1.4. Results A realistic mesh approach and a porous medium approach were simulated for two conditions: at confined flow and at open flow. A pressure drop versus inlet velocity curve is plotted for the realistic mesh simulation and for the porous medium simulation. Fig. 1.8 shows this curve. It can be seen that both curves are very close to each other, which means that the simulated flow fields are similar.. Figure 1.8. Pressure drop as function of inlet velocity in the wind tunnel, from results of the realistic mesh approach (•), simulation of the porous medium approach (4).. 14.

(27) 1.4.1. Pressure drop coefficient results From the results obtained at confined flow condition, the pressure drop coefficient can be obtained for each inlet velocity, for both cases analyzed. Also, the Reynolds number considering the fiber width, Ref , can be calculated. A C0 versus Reynolds number is plotted to analyze its behavior.. Figure 1.9. C0 as a function of the Reynolds number, from the results of the realistic mesh approach (•), simulation of the porous medium approach (4). In Fig. 1.9, it can be seen that for high Reynolds numbers, C0 becomes constant. In the realistic mesh simulation, C0 slightly increases with Ref but for the porous medium simulation C0 slightly decreases. Despite the variation of C0 with Ref the average pressure drop coefficients obtained for realistic mesh and porous medium approach have a 4% difference between them. The average C0 can also be compared with the one obtained experimentally. The error shown in Table 1.2 is relative to experimental value obtained by Echevarrı́a (2015).. 15.

(28) Table 1.2. Average C0 obtained by experimental results, realistic mesh approach and porous medium approach C0 Error Echevarrı́a 2.230 Realistic mesh approach 2.254 1.1% Porous medium approach 2.167 2.8% By analyzing the relative error for C0 obtained from simulations, it can be concluded that the porous medium approach is a valid method for representing the aerodynamic characteristics of a Raschel 35 mesh.. 1.4.2. Porous medium approach input The input parameters for porous medium approach are Kp and Y , both obtained by fitting a quadratic equation to the realistic medium approach curve of the pressure drop (eq. (1.12)). The fitting equation for the normalized pressure drop is eq. (1.13).. ∆p = 1.3597u2 + 0.0065u. (1.12). ∆P = 1.1429u2 + 0.0053u. (1.13). In Table 1.3 the values obtained from the quadratic fitting are both shown: pressure drop and normalized pressure drop. Table 1.3. Coefficients obtained from Realistic mesh approach. C0 Kp · 10−7 Pressure drop 2.25 7.990 Normalized pressure drop 1.87 0.102. 16. Kp0 · 10−5 Y0 Y (∆x0 = 10mm) (∆x0 = 10mm) 3.365 2.663 19.426 3.204 3.417 18.496.

(29) 1.4.3. Run time comparison Computing time can be divided in two aspects, numerical start-up time (i.e. time to compute the numerical grid and to apply boundary and initial conditions at all cells) and solver execution time. These times are presented in Table 1.4, for the porous medium approach and realistic mesh screen approach for confined flow. Table 1.4. Execution time (seconds) for realistic mesh (tr ) and porous medium (tp ) approach, for meshing and solver. tr , s tp , s Rate tr /tp Meshing 38 243 545 70 242 089 226 186 Solver Start-up time of the numerical simulation for the realistic mesh approach takes up to 70 times longer than in the case of the porous medium approach. Execution time for the detailed mesh approach is 186 times larger than for the porous medium approach. Thia is a considerable time reduction and the aim to have a cheaper computationally costing model is fulfilled.. 1.4.4. open flow results The realistic mesh and porous medium approach simulations were run for a single inlet velocity u = 2 ms−1 . The comparison of both simulations was carried out from a cut plane, located at the center of the mesh. Velocity field and and isobars were plotted in the symmetry plane X-Y, for both cases, see Fig. 1.10 and 1.11. Figure 1.12 shows the superposition of realistic mesh and porous medium flow field. Isobars of the porous medium are similar in shape and are very close to the correlative isobar of the realistic mesh. This demonstrates that the flow resultant from both scenarios is similar. The velocity field can also be compared qualitatively, in Fig. 1.13. Both velocity flow fields are shown for realistic and porous medium simulations. It can be seen than these 17.

(30) Figure 1.10. Flow field for realistic mesh approach simulation at open flow condition. have the same shape far from the intercept medium, and present larger differences close to the medium. This means that the flow, when passing across a mesh screen, can be modeled by a porous medium, but that significant differences arise close to the mesh. However, for optimization purpose, the near field to the mesh is not relevant. The quantitative analysis is made through a relative error field. Its results are plotted in Fig. 1.14. When analyzing a 5 % error between realistic mesh and porous medium approach for open flow, it can be noted that only on the downstream flow passing through the mesh is larger than 5%. The rest of the the flow field is almost equal in both scenarios.. 1.5. Conclusions and future work In this study the airflow through a fog collector mesh screen is simulated using a porous medium approach, where the filaments of the mesh are replaced by a porous slab and porous media equations are used to model the pressure drop.. 18.

(31) Figure 1.11. Flow field for Porous medium approach simulation at open flow condition.. Figure 1.12. Superposition of isobars from realistic mesh and porous medium simulations. Red contour lines correspond to the realistic mesh simulations, while blue lines correspond to the porous medium approach. The porous medium approach at confined conditions is validated by experimental measurements and realistic mesh simulations. The pressure drop coefficient C0 obtained. 19.

(32) Figure 1.13. Velocity field near from the realistic mesh (left) and near from porous slab of 40 mm thick (right).. Figure 1.14. Horizontal velocity field error between the realistic mesh and porous medium approach at open flow condition. through the porous medium approach is similar to experimental results (2.8% error) and realistic mesh approach (1% error). The porous medium approach results in a cheaper computational cost model when compared to simulations with the filament detail mesh screen. It approach is 186 times faster for the case analyzed and the results only differ in up to 1%.. 20.

(33) The porous medium approach at external conditions is validated by realistic mesh simulations. The error obtained between realistic and porous medium approach is less than 5% outside the wake. However, it should still be validated against experimental measurements using the Particle Image Visualization (PIV) technique, or a high density arrangement of anemometers. Future work should focus on the applicability of the porous medium model on external conditions. A first step is to measure experimental values for external conditions to have a better uncertainty analysis of the method. To have a more reliable droplet impact model it is necessary to simulate the droplet impact on a Raschel 35 Mesh. The results have to be compared with experimental measurements. An optimization model has to be implemented to find an optimal layout for a fog collector farm. An optimization algorithm has to be chosen so that it is compatible with the physics of this problem. Furthermore, a fog model should be added to the optimization problem with a complete simulation with the all the variables that affect the fog water collection.. 21.

(34) 2. VALIDATION OF POROUS MEDIA MODEL USING CFD TO CHARACTERIZE THE AERODYNAMICS OF FOG WATER COLLECTORS 2.1. Introduction Ensuring sufficient drinking water supply is a major challenge in many regions around the world, especially in deserts and arid zones. According to the World Health Organization, water scarcity affects all continents, and for 40% of the world’s population the situation is getting worse “due to population growth, urban development and use of water for industrial and domestic purpose” (Organization, 2007). Several technologies have been developed to obtain water in places where it is not possible to obtain it directly from rivers, lakes, dams or groundwater. One such technology, fog water collection (FWC) or fog harvesting (Gischler, 1991; Echevarrı́a Johnson, 2015; Rivera, 2011; Cereceda et al., 2014), consists of extracting condensed water from humid air by conveniently placing fine meshes to intercept the air flow above ground level. Water droplets in the airflow collide with the filaments of the mesh as the air flows through it, and are thus captured, and accumulate in a receptacle at the bottom of the collector. Details of this can be seen in Fig. 2.1.. Figure 2.1. Schematic drawing of a FWC system. 22.

(35) This technology is suited for regions with a persistent fog formation only, which typically occurs at the tropical, temperate and arid regions, mainly in the coast and its nearby islands around the globe (California, West Africa, Chile) (Schemenauer, Cereceda, & Osses, 2003). Fig. 2.2 shows a map with locations where fog water collection is, in principle, viable.. Figure 2.2. Map of locations where fog collection has a high potential for success. (Domen et al., 2014) Fog water collection has been extensively studied in order to improve its performance. Many authors have investigated different topics related to fog harvesting: Fog resources such as fog formation physics (Bruijnzeel, Eugster, & Burkard, 2005), fog as sustainable fresh water resource (European Community Directorate General XII, 1995), development and location of fog events around the world (Klemm et al., 2012), fog water collection related to fog events (Marzol, 2008) and variability of fog water as a consequence of the meteorological condition for Atacama Desert (Cereceda, Larrain, Osses, Farı́as, & Egaña, 2008; del Rı́o, Osses, Wolf, Garcı́a, & Siegmund, 2016; Osses et al., 2016). De la Jara et al. studied the collision mechanics of water droplets, describing their transport in air flow and showing that most of water drops for fog harvesting follow air streamlines of the undisturbed flow until some of the droplets collide with the fibers of the mesh screen, mainly by inertial impact (de la Jara, 2012), Echevarrı́a analyzes the collection medium, experimentally studying aerodynamic parameters of fog collector mesh 23.

(36) screens (Echevarrı́a Johnson, 2015), and Park et al. looked at the fiber design for optimal fog harvesting (Park, Chhatre, Srinivasan, Cohen, & McKinley, 2013). The effect of surface coating is studied by Rajaram et al. (Rajaram, Heng, Oza, & Luo, 2016), and several authors show that the best filament shape is cylindric and that they should be as thin as possible (Schemenauer & Joe, 1989; Rivera, 2011). Schemenauer et al. quantify the amount of water collected by different FWC systems (Schemenauer, Cereceda, & Carvajal, 1987), while other authors focus on the collection efficiency (Holmes et al., 2015; Rivera, 2011; Schemenauer & Joe, 1989), studying an analytical model of the aerodynamics of fog collector and the relevance of the correct characterization of screen meshes by type of filament and shade coefficient. Finally, the socio-economical impact has been studied, and its high cost established because it would exceed the market price (Cereceda et al., 2014; LeBoeuf & de la Jara, 2014). These last studies conclude that it is still necessary to improve the technology of FWC to make it commercially competitive. Fog collectors have evolved since; dimensions, type of mesh used, even orientation to wind direction have changed (Cereceda et al., 2014). Holmes, Rivera and de la Jara (Holmes et al., 2015) describe the Large Fog Collector (LFC) as the most used topology for large scale purpose of water collection, such as agriculture and water supply to small villages. The LFC consists of a double layer mesh screen of 48 m2 (Schemenauer & Joe, 1989), usually sited in arrays with one screen standing next to its neighbor. This commonly used layout results from purely practical reasons, and does not respond to an effort of optimizing the water output. The average water collection rate (measured usually in liters per day per square meter– LDM) is the main parameter to consider when analyzing the economical performance of fog collectors as a source of fresh water (LeBoeuf & de la Jara, 2014). LDM depends on the time integration of the liquid water flux of the location and on collection efficiency. As Lebouf and de la Jara mention, the threshold for large-scale fog collection to be economically competitive is of 10 LDM (LeBoeuf & de la Jara, 2014). They specify the minimum. 24.

(37) collection efficiency for different scenarios of number of large fog collectors and cost associated. It is important to consider that the average of collection rate for most locations is about 5 LDM, which indicates that an enhancement of the fog harvesting methods is necessary in order to reach the threshold and become economically competitive. Significant progress has been made in several aspects of fog collection, e.g. optimal design of the mesh (Regalado & Ritter, 2016), use of surface coating (Park et al., 2013; Rajaram et al., 2016), position of the mesh (Holmes et al., 2015), solidity and type of mesh (Echevarrı́a Johnson, 2015; Fernandez et al., 2018; Rivera, 2011). However, the layout optimization of fog collector farms has not been addressed so far. Layout optimization has been extensively used for wind farms, heat exchanger configuration, cooling system design and many others, and it is suitable to be used in any system which is able to get a greater flow access as mentioned in constructal law (which explains the tendency to an optimal geometry for several natural flow systems) (Bejan, 2016). The optimization procedure relies on an adequate mathematical representation of the underlying physics (a model) that acts as a constraint to the optimization problem. The numerical implementation of that model needs to be solved at an acceptable computational cost. There are two main groups of numerical optimization techniques: gradient based optimization and gradient free optimization. As indicated by the name, gradient based optimization uses information of rates of change with respect to the parameters of the problem in order to find a (local) optimum, while gradient free optimization relies on holistic approaches such as swarm behavior or evolutionary forces. Advantages and limitations of these methods can be found in the literature (Nocedal & Wright, 2006). Regardless of the approach, all of those optimization techniques ultimately require the repetitive solving of the constraining physics (such as the flow around wind turbine blades, or through fog collectors). Depending on the goal and technique used, tens to thousands of scenarios must be simulated.. 25.

(38) The wind farm optimization problem (Samorani, 2013) is one of the most relevant examples in terms of optimal spacing, as significant parallels can be drawn to fog collection farms. This problem has the objective of reducing the wake effect of the wind turbines, i.e. the effect of the wake produced by one wind turbine on the performance of other turbines downstream. The wake effect could potentially result in losses for the system, and the prevention of such losses is essential to maximize power production. In order to improve water collection of LFGs, the layout of fog collectors farms must be optimized, so that the impact of individual fog collectors on their neighbor is minimized. The optimization consists of maximizing the water collected by an arrangement of fog collectors placed in a given piece of land. It has been shown that the collection efficiency can be related to the flow dynamics of the air passing through the mesh (de la Jara, 2012). Thus, for the purpose of optimal fog collector farm layout, the water droplet-mesh interaction can be neglected and it is enough to study the air flow through the mesh screen. However, it has to be studied the effect of the water collection by a LFC on downstream collectors. The main objective of this study is to find a computationally cheap model which adequately simulates the air flow through a fog collector mesh. This would allow for iterative optimization techniques at an acceptable computational cost. This study will focus on a typical mesh used for fog harvesting (knitted ribbon filament type meshes, commercially known as Raschel mesh 35). The shape of the mesh fabric is shown in Fig. 2.3. For solving the air flow through and around the fog collector, a computational fluid dynamics (CFD) solver is used. Since the flow around the filaments of the collector mesh requires a significant refinement of the numerical grid (the space between filaments is of the order of millimeters), substantial computing times and use of memory resources are to be expected.. 26.

(39) (a) Photograph of a Raschel 35 mesh screen (www.marienberg.cl).. (b) Dimensions in mm of mesh screen used by Holmes (Holmes et al., 2015).. Figure 2.3. Geometry of a Raschel 35 mesh. With the objective of reducing computational resources, a porous medium approach will be used to model the pressure drop across the Raschel mesh. The methodology proposed by Teitel (Teitel, 2010) describes the use of a porous medium instead of a woven screen, when air flow passes through a wind tunnel. This approach allows to dispense with the detail of the mesh screen and replace it by porosity properties of the medium, resulting in a much lower computational cost. In order to replace the detailed simulation of flow through the mesh by a porous medium model, it is necessary to assess the capabilities of this approach of representing the aerodynamics of fog collectors at open flow conditions. This will be done in three steps: (i) Validate a realistic mesh approach simulation at confined flow. The aerodynamic characterization of Raschel mesh screen has been studied by Echevarrı́a (Echevarrı́a Johnson, 2015), by testing air flow through the mesh in a wind tunnel and then obtaining the pressure drop coefficient and the porous medium properties. The realistic mesh approach consists of simulating the experimental configuration with geometric details of the mesh and tunnel with accurate computational methods. Results obtained from the simulations are compared to the. 27.

(40) experimental tests in the wind tunnel. The realistic mesh approach is validated if its results are significantly similar to wind tunnel measurements. (ii) Validate a porous medium approach simulation at confined flow. Porous properties of the mesh screen can be obtained from pressure drop recordings through the mesh screen as function of air velocity. The pressure-velocity curve can be plotted from wind tunnel testing and/or from realistic mesh approach simulations. The porous medium approach consists of replacing the mesh screen in the wind tunnel by a porous slab with the same porous properties as the mesh. This approach is validated if pressure drop across the porous slab is significantly similar to that obtained from wind tunnel measurements. (iii) Validate a porous medium approach in a scenario with open flow. For this step two approaches are simulated: On one hand the air flow through the mesh is simulated with detailed geometry (realistic mesh approach), and on the other hand the air flow through a porous slab. In absence of experimental measurements at open flow, results from the validated realistic mesh approach can be used. To validate the approach the fluid flow field obtained from simulations for two approaches at open flow conditions are compared. If both flow fields are significantly similar, this porous medium approach for open flow conditions can be validated, and could thus be used for optimization calculations at large scale.. 2.2. Theoretical framework Water collection modeling in the context of layout optimization is a multi-scale problem and three scales can be distinguished. The small scale, less than a few millimeters, the mesh filaments and water droplets are part of this scale. Medium scale, between millimeters and tens of meters, where the fog collectors are. Large, between several tens and hundreds of meters, and fog collector farms belong to this group. Many authors (e.g. (de la Jara, 2012; Echevarrı́a Johnson, 2015)) have shown that fog water droplets follow a very similar trajectory as the air that carries them, and the 28.

(41) impact with an obstacle depends on the Stokes number (defined as the ratio between the characteristic stopping time of the particle and the characteristic flow time around the collector (Israel & Rosner, 1982)), St . If St > 1, droplets inertia causes the impact with the obstacle, because of its migration across streamlines. If St < 1, water droplets follows the streamlines and does not impact the object. Each scale is associated to a range of the Stokes number, as the average droplet diameter is close to 15 µm. For small scale (St > 1), for the medium scale (1 > St > 10−4 ), and for the large one (St < 10−4 ). Schemenauer and Joe (Schemenauer & Joe, 1989) defined the collection efficiency ηcoll 00 of a LFC in terms of the water flow rate collected per unit area ẇcoll , of the unperturbed. flow velocity u0 and of the liquid content of the air LWC (kilograms of water per cubic meter of air) (Schemenauer & Joe, 1989). ηcoll =. 00 ẇcoll u0 · LW C. (2.1). The overall collection efficiency depends on several other efficiencies, as suggested by equation (2.2).. ηcoll = ηAC ηd ηdr. (2.2). The aerodynamic efficiency, ηAC , proposed by Rivera (Rivera, 2011),“represents the portion of droplets in the unperturbed fog that would collide with the mesh” (Rivera, 2011). The deposition efficiency ηd represents the rate of water drops that remain deposited on the mesh filaments, without being returned back into the air flow. This phenomenon is determined by the Stokes number St , and its expression is given in eq. (2.3). The draining efficiency ηdr is defined as the fraction of water drops that actually fall down into the receptacle (Park et al., 2013), and its value commonly assumed as unity for lack of information.. ηd =. St St + π/2 29. (2.3).

(42) Rivera (Rivera, 2011) showed that ηAC depends only on the solidity s of the mesh screen, and its pressure drop coefficient. ηAC =. s p 1 + C0 /Cd. (2.4). The solidity of the mesh screen “represents the fraction of surface covered by the mesh that is capable of collecting droplets”, and can be accurately obtained by an image analysis method, described in detail by Echevarrı́a (Echevarrı́a Johnson, 2015). In eq. (2.4) the constant Cd is the drag coefficient for a non-porous plate and its value depends mainly on the aspect ratio of the surface and does not change significantly (White, 2011). The constant C0 is the pressure drop coefficient which depends on the solidity, the Reynolds number of the flow and type of mesh screen. This aerodynamic analysis allows for focusing on a single parameter, C0 , in order to validate computational simulation for a given mesh screen, as the solidity depends on the surface covered ratio, Cd is relatively constant, and only C0 varies with the geometry of the filaments and the thickness of the mesh screen. Therefore, it is only necessary to study the air flow passing through and around a specific mesh screen in order to model the trajectory of water droplets. In consequence, knowing the liquid water content of the upstream air and the collection efficiency would suffice to determine the volume of water collected by a single mesh screen. 00 The net water flow rate collected per unit area, ẇcoll , is the objective function to be. maximized in a layout optimization exercise of LFC farms. Through eqns. (2.1), (2.2) and (2.4), this flow rate is linked to the pressure drop coefficient across the mesh screen. This, assuming a known LWC downstream the mesh screen.. 2.2.1. Porous medium equation and Forchheimer Equation A porous medium can be defined as a “material consisting of a solid matrix with an interconnected void” (Nield & Bejan, 2013). In that context, a Raschel mesh screen can be considered a porous medium and thus be described by its porous medium properties. The 30.

(43) Forchheimer equation describes the flow through a porous medium, considering viscous and inertial terms (Nield & Bejan, 2013),. −. µ Y dp = u + ρ 0.5 u2 dx Kp Kp. (2.5). where p is the pressure, x the uni-dimensional length coordinate, µ is the dynamic viscosity of the fluid, u the fluid velocity, ρ the fluid density, Kp the permeability of the porous medium, and Y is called the inertial factor, representing a non-linear momentum loss coefficient. The Reynolds number for porous materials is commonly defined with the characteristic length as the void space between solid walls (space between filaments in case of the Raschel mesh). It is important to note that for Rep > 150 the viscous term in equation (2.5) is negligible (Miguel, Van De Braak, & Bot, 1997), resulting in the so called Forchheimer regime. For air flow passing through the mesh used in this study, the Forchheimer regime corresponds to wind speeds of the order of 0.4 ms−1 . When Rep < 150, the porous regime is modified and the linear term becomes important. When the quadratic term is negligible (i.e. for Rep < 150), the porous momentum equation depends only on the permeability. The regime in this region is called the Darcy regime (Nield & Bejan, 2013). A confined flow through a porous medium can also be described as a dimensionless term from Bernoulli’s equation (Kundu et al., 2012),. C0 ≡. ∆p , (1/2)ρu2. (2.6). where C0 is defined as the pressure drop coefficient, ∆p is the pressure drop across the mesh and u correspond to the upstream velocity. Equation (2.6) applies only when Bernoulli’s conditions are fulfilled, i.e. for steady, incompressible and inviscid flow along a streamline(Kundu et al., 2012). Flow through a porous medium within a wind tunnel is such a case, so it applies to this study. 31.

(44) When eq. (2.5) is discretized using a finite difference approach, and the viscous term neglected, a relationship between the porous medium parameters, Kp and Y , and the pressure drop coefficient can be found,. Y 1 0.5 ∆x = C0 . 2 Kp. (2.7). Here ∆x is the thickness of the porous medium. Note that equation (2.7) is independent of density. Thus, both porous medium parameters and pressure drop coefficient depend only on the geometry of the medium for the mentioned conditions. However, the parameters Y and Kp cannot be obtained independently from eq. (2.7). The different flow regimes through porous medium are shown in Fig. (2.4), where the following variable definitions are used, ReK =. uKp 0.5 ρ , µ. fk =. ∆p Kp 0.5 . ∆x ρu2. The horizontal asymptote—for high ReK —corresponds to the Forchheimer regime, while the oblique asymptote—for low ReK —corresponds to the Darcy regime. The shape of the curve in Fig. 2.4 is similar for all porous materials, with changes only in the value of the horizontal asymptote.. 2.2.2. Experimental data The pressure drop of air flow as it passes through different types of meshes was analyzed by Echevarrı́a for several meshes commonly used in LFC (Echevarrı́a Johnson, 2015). He found a correlation between the pressure drop coefficient (C0 ) and the solidity of different kinds of meshes. For a Raschel ribbon filament type the correlation for the pressure drop coefficient is as follows,. C0 = 0.0744e6.85s . 32. (2.8).

(45) Figure 2.4. Logarithmic plot of friction factor versus permeability Reynolds number taken from (Nield & Bejan, 2013). The fK = 0.55 asymptote changes depending on the specific porous medium. For the specific mesh type used in this study, the Raschel Mesh 35 with s=0.4806, the pressure drop coefficient is C0 = 2.23. 2.2.3. Numerical Simulation Air flow (i.e. wind) is governed by the Navier-Stokes equations (Kundu et al., 2012), and due to the low velocities and relatively small temperature differences, density variations in the flow can generally be neglected. This allow for an incompressible approximation. The incompressible Navier-Stokes equations can then be solved numerically using an adequate discretization technique. The numerical resolution of the Navier-Stokes equations is known as Computational Fluid Dynamics (CFD), and is an active field of research as well as a powerful technique for the analysis of fluid flow systems (Versteeg. 33.

(46) Figure 2.5. C0 as function of solidity for knitted ribbon filaments type. (Echevarrı́a Johnson, 2015) & Malalasekera, 2007). The incompressible Navier-Stokes equations together with the continuity constraint are summarized as follows:. ∇ · ~u = 0 ∂~u 1 ~ + (~u · ∇) ~u = − ∇p + ν∇2~u + S ∂t ρ µ ν= , ρ. (2.9) (2.10) (2.11). ~ accounts for external body forces such as where ν is the kinematic viscosity and S gravity. Any particular scenario of slow fluid flow is described by these equations if properly defined through adequate boundary and initial conditions, that describe the geometry and flow characteristics at openings and walls. 34.

(47) While laminar flow can be solved directly using CFD, in many natural phenomena (including wind) turbulent flow will predominate. Turbulence is a complex phenomenon that occurs over a large range of spatial scales (spanning several orders of magnitude). This makes it complicate to simulate numerically, as the required grid size for a direct numerical simulation (DNS) would be prohibitively small for even medium scale simulated scenarios. The main effect of turbulence on flow behavior corresponds to energy losses due to turbulent energy dissipation at a microscopic scale. In order to take into account theses energy losses, several modeling approaches have been developed (Versteeg & Malalasekera, 2007). They can be gathered into two groups: Reynolds Averaged Navier-Stokes (RANS) modeling and Large Eddy Simulation (LES). In general terms RANS approaches the problem by averaging the flow properties in time, while LES applies a spatial filter and uses a model to account for sub-grid energy dissipation (Versteeg & Malalasekera, 2007). The RANS modeling approach is generally simpler and requires less computational resources. In the context of the current analysis (optimization of fog collector placement in order to minimize wake effects) RANS is a more convenient choice due to the lower computational cost. RANS results in a set of additional transport equations that must be solved in parallel to the main flow equations. This additional computational cost is offset by the possibility of using a relatively coarse computational grid. The choice of how exactly the energy dissipation is modeled, i.e. which additional transport equations is solved, depends on the particular approach taken. Spalart-Allmaras developed a model consisting of a single additional transport equation for kinematic eddy viscosity (SPALART & ALLMARAS, 1992); The popular κ- model was developed by Launder and Spalding (Launder & Spalding, 1974), and consists of adding two transport equations, turbulence kinetic energy κ and the rate of dissipation of turbulent kinetic energy (rate of viscous dissipation) . It yields particular good results for flow far away from walls, but has shown to present problems. 35.

(48) when used for internal flow. The κ-ω approach replaces the model equation by the turbulence frequency ω=/κ, and has shown to be suitable for flow close to a wall. In this study a κ-ω Shear Stress Transport turbulence model-SST approach was used, because the fibers of the mesh act as walls along the entire cross-sectional area, and results obtained by this model showed good agreement compared to those obtained experimentally (Middelstädt & Gerstmann, 2013; Teitel, 2010).. 2.3. CFD Modeling of FWC in the wind tunnel Previous studies of air flow through mesh screens show that under certain circumstances CFD simulations can reproduce experimental measurements satisfactorily (Valera, Álvarez, & Molina, 2006; Valera et al., 2005). Nevertheless, results for a specific type of mesh are not necessarily valid for others, because of the geometry effect of the filament that modifies the flow across the entire mesh screen. It is thus necessary to verify the match of simulation results with measurements for each specific mesh. Main conclusions from studies of mesh screens simulated with CFD is that pressure drops through mesh screens can be determined by means of CFD simulations in which the realistic woven screen geometry is replaced with a porous medium (Teitel, 2010). While they use a porous medium approach to simulate the flow through a mesh in internal flow, i.e. in a flow confined by the walls of a wind tunnel, they do not extend their analysis to open flow under realistic flow conditions. The results of the internal flow are certainly encouraging, but it is important to compare the fluid flow around the mesh with the flow around the porous medium, in order to understand the applicability of using the porous medium approach to model a realistic mesh at open flow conditions. The experimental results of Echevarrı́a in a 1600 mm long wind tunnel with a crosssectional area of 400 × 400 mm were used for comparison (Echevarrı́a Johnson, 2015). In those experiments, a Raschel mesh 35 as shown in Fig. 2.3 was located at the center of the wind tunnel, covering its entire cross-sectional area (Fig. 2.6).. 36.

(49) Figure 2.6. Testing framework assembled on the wind tunnel. 2.3.1. Numerical setup OpenFoam (OpenCFD, 2004) is used to simulate the flow field in this study. For the first stage an idealization of the Raschel mesh geometry was implemented to detail using OpenFoam’s numerical grid generator, snappyHexMesh. The geometry with the realistic mesh screen implemented in the model is shown in Fig. 2.7. In order to account for near wall effects at the mesh filaments a refined boundary layer grid was added at all contours of the filaments and at the wind tunnel walls as shown in Fig. 2.8. Further away from the mesh screen the numerical resolution was reduced in order to minimize the computational effort. The inlet boundary conditions were set as fixed velocity at the inlet and zero pressure at the outlet. A non-slip velocity boundary condition was applied to the walls of the tunnel. Due to the relatively low velocities, incompressible flow was assumed, and the simpleFoam solver was used. For turbulence a SST κ − ω RANS approach was selected. The simulations were run for different inlet velocities ranging from 0.1 to 8.0 ms−1 . Pressure drop versus inlet velocity were plotted with these results, and Kp and Y were then obtained by fitting a second order polynomial curve to the data, 37.

(50) ∆p = a1 u2 + a2 u.. (2.12). From equations (2.5) and (2.12) the required constants can be identified,. ∆xµ a2 a1 K 0.5 Y = ∆xρ K=. (2.13) (2.14). In the context of the current study ∆x refers to the thickness of the mesh screen. If an homogeneous thickness is assumed, i.e. without considering the joints of the filaments of the mesh screen, ∆x is about 0.3 mm (Rajaram et al., 2016).. Figure 2.7. Wind tunnel realistic mesh setup. For simulating the flow through the Raschel 35 mesh screen using the porous medium approach, the mesh screen was replaced by a 10 mm thick porous slab in the CFD model (see Fig. 2.9). The rest of the numerical set up was maintained, and the simpleFoam solver was again used. In the porous slab section the numerical cells were assigned porous properties (porosity and the inertial factor). When a porous section is defined, the solver modifies the momentum equation in that zone by adding the Forchheimer equation as a source term. The porous slab used in the model was thicker than the actual mesh screen, so that the coefficients Kp and Y had to be replaced by modified coefficients (Kp0 and Y 0 ) in order to account for the ratio between the thicknesses,. 38.

(51) Figure 2.8. Boundary layer on the contour of mesh fibers.. Figure 2.9. Porous slab on wind tunnel.. 39.

(52) −. dp µ Y0 = 0 u + ρ 0 0.5 u2 dx Kp Kp ∆x0 Kp0 = Kp ∆x 0.5 ∆x 0 Y = Y ∆x0. (2.15) (2.16) (2.17). where ∆x0 is the thickness of the porous slab and ∆x is the thickness of the experimental mesh screen. This modification ensures that the pressure drop is the same for both cases, the porous slab and the mesh screen. The air flow is thus compared in equivalent scenarios. Therefore, the value of porosity properties of the porous slab depends on its thickness and is defined arbitrarily. In order to incur in a lower computational cost it is convenient for the porous slab dimension to be greater than the thickness of the mesh screen, since the size of the computational grid is related to the thickness of the porous cells.. 2.3.2. Results of realistic approach in confined flow and discussion The results of the first two stages of this study are presented in the following paragraphs. From the CFD simulation of the realistic mesh approach the pressure drop across the 0.3 mm thick Raschel mesh screen is obtained by post processing of the numerical simulation for a range between 0.1 and 8 ms−1 of inlet velocity. Figure 2.10 shows the simulated pressure drop ∆p as a function of the inlet velocity u together with the experimental results. The pressure drop coefficient C0 is obtained for each point from the curve of simulation results shown in Fig. 2.10. The average C0 from this realistic mesh approach is presented in Table 2.1. The parameters Kp , Y are obtained by fitting a quadratic function to the data obtained from the realistic mesh simulations by using equations (2.13) and (2.14). The modified parameters Kp0 , Y 0 can then be calculated. The same variables are obtained for. 40.

(53) Figure 2.10. Pressure drop as a function of the inlet velocity into the wind tunnel, from a realistic mesh approach simulation (•) and experimental data (4). the experimental curve, where the ∆p values where measured in the wind tunnel. Table 2.1 summarizes the results. Table 2.1. Coefficients obtained from Realistic mesh approach. C0 Kp · 10−7 Y Kp0 · 10−5 (∆x0 = 10mm) Y 0 (∆x0 = 10mm) Simulation 2.25 7.990 3.365 2.663 19.426 Experimental 2.23 2.650 1.920 0.883 11.084 For C0 , the average relative error between the value obtained by simulation and the one obtained by Echevarrı́a (Echevarrı́a Johnson, 2015) through experimental measurements is close to 1.1%. This error is small, and it can thus be assumed that the realistic mesh approach is a valid method for obtaining the pressure drop coefficient for a Raschel 35 mesh.. 41.

(54) Regarding the porous properties of the mesh screen, a ReK -fK plot can be made from simulation results, similar to Fig. 2.4. This is shown in Fig. 2.11, where most of the points lie on the asymptotic line. One simulated fK value, corresponding to the simulation for 0.1 ms−1 , escapes the asymptote. This simulated velocity corresponds to a Rep within the Forchheimer regime. Thus, the condition to neglect the viscous term of the Forchheimer equation is fulfilled from 0.5 ms−1 onwards. Not surprisingly, the difference of the C0 value obtained for 0.1 ms−1 inlet velocity is relatively large (26%) when compared to the Forchheimer regime.. Figure 2.11. Friction factor versus permeability Reynolds number, for simulation results. Using a constant C0 to describe the behavior of the air flow through the mesh screen is valid only when the viscous term of the Forchheimer equation is neglected. Wind velocities commonly used for the analysis of fog collectors are well above the critical velocity, so that the value for C0 can be assumed constant.. 42.

(55) 2.3.3. Porous medium model and simulations-results in confined flow and discussion The Permeability (Kp ) and the inertial factor (Y ) are required as input for the simulation with the porous medium approach. The adjusted parameters, Kp0 and Y 0 , are obtained from the simulations with the realistic mesh screen (Simulation a) and also from experimental data (Simulation b). The results of the simulations are presented in Fig. 2.12, for the same range of inlet velocities used in the realistic mesh approach.. Figure 2.12. Pressure drop as function of inlet velocity into wind tunnel, from results of porous medium approach simulation with porous properties obtained from realistic approach (Simulation a) (•), simulation with porous properties obtained from experimental data (Simulation b) (♦), and pressure drop from experimental data (4). The three curves in the pressure drop plot shown in Fig. 2.12 are close to each other, which suggests that the pressure drop obtained by simulations with the porous medium approach is similar to the experimentally obtained pressure drop. Additionally, it can be concluded that the simulation with a porous medium approach can be done with the parameters Kp0 and Y 0 obtained from the simulations using a realistic mesh approach or directly from experimental data. 43.