Geometrical models

A porous medium consists of a very complex solid matrix making it almost impossible to describe the geometry with an exact technique. However, it is often possible to describe the porous medium as a continuum, where the hydraulic resistance generated in each pore is averaged to a mean pressure drop of the entire medium. Even though the arrangement of pores is random, it is possible to analyse a small portion of it and generalise the results for the complete geometry. Numerical studies have used two different approaches to tackle this issue. The first is the macroscopic approach. This methodology utilises volume-averaged equations to describe the overall flow characteristics. The other methodology is the microscopic approach. Here the researcher uses a specific geometry resembling the porous medium (Krishnan et al. 2006). The validity and advantages of both techniques are still under discussion.

For the microscopic approach, there are two major techniques to characterise a porous medium, i.e. 2D and 3D modelling. Even though the 2D approach seems to overlook some

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of the important features of a porous medium, interesting results have been obtained via this method. The simplest way to build the porous media is by arranging a group of struts, which are used to represent the porous media as shown in Figure 2-12. Porosity can be changed by simply changing the size of the strut or horizontal distance between strut centres (Saito and de Lemos 2005).

Figure 2-12 2D Arrangement of cylinders and computational box containing one cylinder (left) and streamline plot of flow (right), (Zhu et al. 2014)

For instance, Kundu et al. (2014) used a periodic array of square cylinders to simulate turbulent flow through porous media. In their case, porosity was varied by changing the size of the fixed struts. It was reported that the macroscopic turbulent kinetic energy and dissipation rate increase with decrease in porosity. This approach was used in a similar way by Pedras and de Lemos (2001). In their study they used an array of elliptic rods to analyse turbulence in a porous medium. The dissipation rate and kinetic energy reported were in good agreement with data from previous studies. Yet, these are not the only models that exist to study porous metals. Another example is the use of a random pattern of cylinders as seen in Figure 2-13. In this case, porosity is achieved by changing the cylinder radius only. The number of struts remains fixed and they do not overlap.

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Figure 2-13 2D representation of a fibrous material generated by a Monte Carlo procedure accounting for 200 cylinders (Yazdchi and Luding 2011)

A different approach is using representative 3D models. Such an approach is perfectly capable of representing the real material, but simple enough so it does not require too much time to compute. Arrangements of groups of spheres or different 3D structures have been used. Different techniques are used in the literature to model a porous structure. Some examples of the geometries found in the literature are presented in Table 2-5. It is important to note that most of the geometries found for numerical modelling are related to open-cell structures and one of the main features when deciding for the correct model is porosity.

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Table 2-5 Current geometry creation models found in the literature

Reference Geometrical model Porous medium Porosity

(Hellström and Lundström 2006)

2D array of aligned parallel cylinders - 40%, 60%, 70%

(Papathanasiou et al. 2001)

2D array of squares and hexagons

Fibrous porous media

(30% − 60%)

(Machado, 2012)

2D model with random squares distribution.

Micro power plants

88.5% , 95%

(Karimian and Straatman 2008)

3D model based on interconnected cubes with a spherical hollow space

Graphitic foam

75% 80%, 85%, 90%

(Jiang and Lu 2006)

3D model with a limited number of identical particles as the porous media

Sintered porous plate channels

40.2%, 44.4%, 46.3%

(P. de Carvalho et al., 2015) 3D digital representations were generated using microcomputer tomography scans for open cell metal

foams of different grades.

Open-cell metal foams 88.92%, 84.65%, 88.62% (Iasiello et al., 2014) 89.4% , 90% (Krishnan et al. 2006)

3D model obtained from a solid cube and the subtraction of 5 equally sized overlapping spheres located in a BCC

pattern.

Open-cell foam structure

> 94%

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Table 2-5 (Continuation) Current geometry creation models found in the literature

Reference Geometrical model Porous medium Porosity

(Bai and Chung 2011)

3D single sphere- centred tetrakaidecahedron Open-cell metal foam 5 𝑝𝑝𝑖, 10 𝑝𝑝𝑖, 20 𝑝𝑝𝑖, 40 𝑝𝑝𝑖 𝑝𝑝𝑖 = 𝑝𝑜𝑟𝑒𝑠 𝑝𝑒𝑟 𝑖𝑛𝑐 (Duggirala et al., 2008)

3D model obtained from a solid cube and the subtraction of 5 equally sized spheres located in a

BCC pattern. In this structure the spheres do not overlap and there are small cylindrical channels crossing the structure that work

as the micro-channels. Activated carbon powder entrapped in a matrix of cylindrical fibres 41%, 47%, (Yang et al., 2013)

3D periodic arrays of cubic, spherical or ellipsoidal particles

- (60% − 75%)

Lately, tetrakaidecahedron has been used to model the geometry (Bai and Chung 2011). This 3D geometry comes as a result of what is better known as Kelvin’s conjecture. The best filling arrangement that has minimal surface area of similar cells of equal volume is polyhedrons consisting of eight hexagonal and six quadrilateral faces. In this case, the isoperimetric quotient of the polyhedron is close to 0.753367.

Kopanidis et al. (2010) used this structure (Figure 2-14) to simulate flow and heat transfer at pore scale level. They implemented the tetrakaidecahedron and the resulting values were compared with experimental data available in open-cell foams.

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Figure 2-14 Two tetrakaidecahedron united (Kelvin cells) (Boomsma et al. 2003)

Their results showed the influence of entrance effects on pressure drop and heat transfer coefficient to be significant. This technique was also used by Boomsma et al. (2003). They created a periodic unit volume by joining 8 cells, with the unit cells representing the pores and interstices created by joining other cells to the model. Their numerical simulation results were compared to experiments resulting in good agreement between both techniques.

However, this kind of technique is suitable for structures with high porosity. Besides, the construction of a structure made of several number of this type of cells requires a huge amount of computational resources.

In a similar approach, Dukhan and Suleiman (2013) used Kelvin’s cells as a single hollow structure for porosities greater than 90%. In their approach the vertices are used as ligaments which create an assembly similar to the ligaments within open-cell metal foam. The technique used to create this structure is simple. First, a solid cube is subjected to a Boolean operation where eight spheres are subtracted from the corners as shown in Figure 2-15. The centre of each sphere is located in one of the cube’s vertices and the radius of the spheres is not larger than half the length of the cube’s side.

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Figure 2-15 First Boolean operation for Kelvin's cell creation (Krishnan et al. 2006)

Later, another six spheres are subtracted from the resultant structure. These spheres are located on the planar faces of the remaining body. The distance between the centre of these spheres and the centre of the solid body, as well as the radius of all fourteen spheres are in direct relationship with the desired porosity (Dukhan and Suleiman 2013). The overlapping section will create small spherical caps that will be withdrawn. This geometry modelling is shown in Figure 2-16. This construction is similar to a FCC crystalline structure arrangement.

Figure 2-16 Second Boolean operation for Kelvin's cell creation (Dukhan and Suleiman 2013)

The last step is to create a hollow body from the remaining structure. This is easily achieved by subtracting a single sphere which is located in the centre of the structure. It is important to note that this last sphere will have the same radius as the previous ones. The resultant structure has shown to have minimum volume ratio compared to other structures (Krishnan et al. 2006).

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Figure 2-17 Hollow tetrakaidecahedron with a porosity of 𝟗𝟐. 𝟖% (Dukhan and Suleiman 2013)

Figure 2-17 is the resultant structure, showing a porosity close to 93%. The porosity is determined by considering the original cube as the total space and the remaining structure as the solid matrix. The volume of the remaining structure accounts for overlapping spheres and spherical cap reductions. The porosity can be calculated using the following expression (Dukhan and Suleiman 2013):

𝜀 = 1 𝑎𝑐3 [8 3𝜋𝑅 3_{− 2𝜋 (𝑅 −}𝑎𝑐 2) 2 (2𝑅) −16 3 𝜋 (𝑅 − 𝑎𝑐√3 4 ) 2 (2𝑅 +𝑎𝑐√3 4 )] (2.58)

where 𝑎 is the centre to centre distance between the central sphere and the polar sphere and 𝑅 is the radius of any sphere. This technique is suitable for high porosity porous metals.

Xu and Jiang (2008) numerically analysed fluid flow and heat transfer in low porosity porous metals. They used the Simple Cubic (SC) crystal structure as a base model and air as a working fluid. Instead of using the particles as pores, they used the remaining body as the porous structure. With this they achieved porosities closed to 40%. Their results showed that the friction factor calculated for the non-slip flow regime agreed well with the correlation for normal scale porous media.

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The use of X-ray computer tomography to characterize foam structures is a common practice (Lefebvre et al. 2008). Jaganathan et al. (2008) presented a CFD analysis using 3D images from a real fibrous medium to analyse its permeability. They compared their model with an analytical technique. Their results were consistent and in good agreement with previous findings. Scanned 3D geometries seem like the proper approach to tackle the porous media problem by providing a structure close enough to the real material. Nevertheless, this kind of scan generates geometries that require a large amount of computational resources for meshing and simulation processes.