CONCEPTOS BÁSICOS EN ADOBE FLASH

As discussed in Chapter 3, the pore size distribution can be computed by simulating mercury porosimetry and the accuracy can be up to 99.9% [173], but to do this the model needs to be represented by a set of cubic or spherical volumes in a process referred to as voxelisation [173,185]. The voxelisation of a model can lead to a large memory requirement to store the diameter and coordinates of sphere centres which limit its application. For example, a cell (10mm × 10mm × 10mm) in a VS was voxelised by the code provided by Yue [185] and the file for the voxelised cell was 1.01GB. Following trials, the voxelisation methodology proposed by Yue was difficult to implement because of computer memory usage. It proved impossible to voxelise this sample using a standard specification desktop (Intel core ™2 Duo CPU [email protected] with 4GB memory) and therefore the code was redeveloped as described below. 30 40 50 60 70 80 90 100 15 25 35 45 55 65 75 85 95 105 P or osit y, 𝜌 r, (% )

Percentage of randomisation, rand, (%) Measured porosity

The bounding box of this porous structure was determined using equations ( 4-1 ) to ( 4-6 ). As seen in Figure 4-43, this bounding box was meshed with a specified gap termed ‘resolution’ defined by equation ( 4-35 ). The intersections between the grid lines are nodes which are located either in the inside region or the outside region which is partitioned by the tessellated wall of the pore.

( 4-35 )

is the length of a bounding box

is the width of a bounding box

Is the height of a bounding box

Figure 4-43 A gridded bounding box of a pore and illustrated nodes located in or out of pore. The porosity of a single pore is measured by using a number of spheres. Taking a single pore as an example, a sphere, whose centre is located at an inside node, is grown with a specified step size, until it touches the wall (represented by triangles) of the pore using the faster intersection test method for a sphere with a triangle detailed in Chapter 2. This is principally different from the previous method which is based on the intersection test method for a sphere and sphere. It results in an intensive computing time for a large number of sphere and sphere intersection test.

Wall of pore Node A (out of pore)

Node B (inside of pore)

Bounding box of pore

As shown in Figure 4-44, once this is the case, it stops growing. The maximum sphere diameter at this node is recorded with the node coordinates. By repeating this process, a set of spheres with their centre coordinates is obtained. The total volume of the spheres, not including any overlap can be determined by using the VTK volume function (Chapter 3). This volume is the measured void space of the porous structure and the porosity can be obtained based on the definition of porosity. By decreasing the mesh size, the computed porosity is closer to the actual porosity of pore and the process is repeated. The process is a similar way to the mercury porosimetry (detailed in Chapter3) which approaches the actual porosity by increasing the pressure forcing the mercury to intrude into the smaller pores. As the pressure increases, the more mercury intrudes into void space and the measured volume of mercury is closer to the actual void space. However decreasing the mesh gap results in an increase in computing time and computer memory usage. Therefore, in the next stage, an equation ( 4-36 ) is given to stop the computing procedure when the difference between the current compute porosity and previous porosity is limited to a pre-defined value, termed stop value.

Figure 4-44 Schematic of the maximum sphere at the node P.

( 4-36 )

is set prior to computing procedure

Node P

Then a set of porosities was obtained as well as their corresponding smallest sphere diameters. A cumulative histogram is then constructed, where represents the probability of finding a point in the model space with a pore size greater than or equal to . The pore size distribution is the negative of the differential coefficient of with respect to :

( 4-37 )

In order to verify this method, an arbitrary porous structure (cube 10mm × 10mm × 10mm) with specified pore size distribution was generated as shown in Figure 4-45 and its file size was only 1MB. By using the revised method, the computed pore size distribution was achieved as shown in Figure 4-46 by the use of standard personal computer.

Figure 4-45 Illustration of a cube with specified pore size distribution.

No intruded sphere in pore

Pore totally filled by intruded sphere

Pore partially filled by intruded sphere

X Z

Figure 4-46 Comparison of the generated pore size distribution and computed pore size distribution.

Further work was carried out to fully understand the above results. The computed pores and arbitrary porous structure are visualised in Figure 4-47. This shows that the small pores are not intruded by the spheres and large pores are not totally filled with intruded spheres and this results in a reduced accuracy when the pore size is too small or big as in Figure 4-46. In order to solve the above issue, a higher resolution is required to ensure the smaller pores are intruded and the large pores are fully filled with spheres.

Figure 4-47 Visualisation of pores in cube and intruded spheres. 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0 0.5 1 1.5 2 P or e Siz e D is trib u tion , pd Pore Size, D, (mm) Generated_Pore size distribution

Computed_Pore size distribution

Boundary of porous material

Spherical pore with randomised distribution of size and position

X Z

The computing error is expressed by an equation ( 4-38 ) and when it is smaller than the defined error, the process is stopped.

Alternatively, the error can be expressed by ( 4-39 ) and the porosity of a porous model can be measured by some commercial software such as Magics. The computed porosity is related to the resolution. For a given resolution, when the error is equal to zero which indicates that all the void space is intruded and further processing is not needed.