3.1 Permeability mechanism
Totally 747 original images including projections, dark field and flat field images were captured in one cycle. However, they only show side elevation instead of directly reflecting the inner structure of this porous ceramic, as shown in Fig. 5(a). So it needs to transfer the original images to slice images through ReNamer, P3 and P3B software.
These slice images can visually show cross-section views. The cross section (Z = 3.068 mm) of ceramic sample after failure is selected to display the cracks in Fig. 5(b), while the rendered models are chosen from the ceramic sample before loading in Fig. 5(d), (e) and (f). Gray values of pixels in binary images are strictly divided into two gray values for presenting information more distinct. White region represents the matrix of porous ceramic, and black gray value exhibits the pore structures information (Fig. 5(c)). The image processing from slice images to binary images is achieved by Image J software. The image analysis for the greyscale pixel intensity value of ceramic matrix and pore structure is carried out using this software. There is a marked difference between the greyscale value of the ceramic matrix pixels and the pore structure pixels. Then the threshold has been selected according to the image segmentation [44–46] in order to transform to black and white ones. Three-dimensional reconstruction of this porous ceramic is achieved through these slice images with Avizo software in this paper. Each voxel represents a physical volume of ~ 3.25 × 3.25 × 3.25 µm3. Random part of this porous ceramic (diameter of 337.9 µm, height of 737 µm) has been selected to show the pore structures and permeability process, as shown in Fig. 5(d), (e) and (f). It has a uniform distribution for pores inside the porous ceramic (see Fig. 5(d)). White region represents the matrix and blue region is regarded as pores inside the porous ceramic in 5(d).
Pore characteristics are extracted from three-dimensional model of porous ceramic (see Fig. 5(e)). Figure 5(f) shows the state of permeability test of porous ceramic, and three-dimensional streamlines obtained by using Avizo software are used to simulate the pore fluid flow based on the Darcy’s law and the Navier-stokes equation. Streamlines with different colors represent flow rates in this permeability simulation process, it also indicates uniformity of permeability during the permeability test. The red streamlines denote faster flow rate, while blue streamlines show slower flow rate. According to mass conservation of incompressible fluid, the smaller area of flow passing can contribute to a faster flow rate, so the streamlines can visually display the porosity and size of pores. Note that where streamlines are concentrated, pores are also more concentrated, which indicates that these streamlines can also be used to qualitatively and visually identify the pore zones information inside the porous ceramic. As we all know, fluid usually escapes the region with bigger flow resistance, and prefers to flow through the pores structures with smaller flow resistance, this phenomenon may cause heat transfer deterioration in the transpiration cooling. On the basis of streamlines, porosity uniformity of porous ceramic can still be further improved by optimizing the fabrication technology even though the porosity is basically uniform. The streamlines video associated with this permeability process can be found in the supplementary material.
3.2 Influence of Loads on permeability
Figure 6 shows the stress-strain curve and permeability-strain curve under the loads of F = 0 MPa, F = 5.28 MPa, F = 10 MPa and failure (F = 12.4 MPa), which respectively corresponds to A, B, C, D in stress-strain curve and a, b, c, d in permeability-strain curve.
Due to mechanical transmission of the miniature loading device, there are two stress relaxation stages of BB’ and CC’ where the scan imaging and permeability testing are carried out, respectively. The strains of ceramic sample in these two stages are virtually remained unchanged, so they can’t affect the experimental results and the height of the sample has little variation. During the experiment, A point is the original state that has no loading and no permeability test, in this state, the x-ray imaging, permeability test and x-ray imaging are successively carried out in turn. AB is first loading, B’C is second loading, C’D is last loading until the sample has been broken. The x-ray imaging, permeability test and x-ray imaging are successively carried out after each loading. For permeability-strain curve, as the strain increases, the permeability rate has a tendency to decrease for ab, and a slight rise for bc, the last a sharp rise for cd.
3.3 Permeability variation mechanism
Permeability rate is mainly determined by the inner pore structures of the sample. 3D rending of porous ceramic with pores and matrix (blue represents matrix, yellow is pore) has been built in Fig. 7, it has obvious changes for pores under different loading stages, while the pores of sample before and after permeability test have little noticeable change under the same load.
The visual cross-section views of a sample under different loads are exhibited in Fig. 8. For clarity, cross-section views are chosen from six different positions (Z1 = 0.64 mm, Z2 = 1.31 mm, Z3 = 1.97 mm, Z4 = 2.64 mm, Z5 = 2.97 mm and Z6 = 3.30 mm) of sample, as clearly shown in Fig. 8(a). The porosities of these six positions under different loads were also illustrated Fig. 8(b), it is obvious that the porosities at different positions firstly decrease and then increase with the increasing of loads. Similarly, the slice images of these six positions can also reflect this tendency (see Fig. 8(c)). The black areas of these six cross-section positions after compression of 5 MPa are smaller than those before loading, which indicates that pores inside the sample become smaller under this load. This trend corresponds to the permeability rate of sample after compression of 5 MPa. As the stress increases to 10 MPa, the porosity and the permeability rate have some increase. The stress continues to be increased, it is clear that the large cracks begin to appear for some cross-section views like Z2 = 1.311 mm, Z3 = 1.971 mm, Z4 = 2.636 mm, Z5 = 2.969 mm and Z6 = 3.301 mm. These cracks provide the bigger channels for deionized water, which causes a sharp rise in the permeability rate of the sample. The cross-section views of each position before and after permeability have been displayed, respectively. Compared with cross-section views before permeability test, the pore structures of these cross-section views after permeability test have little noticeable change. It indicates that the water pressure (0.08 MPa) inside sample almost doesn’t impact on the pore structures at this stage.
Figure 9 displays the porosity distribution along the thickness (from Z7 = 0.256 mm to Z8 = 3.424 mm) of porous ceramic with different loads. Note that it is verified that porosity of C/SiC ceramic without any loads ranges from 10.6–15.1% along the thickness, namely uniformity > 95.4%. In these curves, the porosity includes volume fraction of pores and cracks caused by loading. The changing trends of porosity of porous ceramic with different loads are consistent with those of permeability, which indicates cracks can also be the channels for deionized water in permeability tests. To investigate the variation of porosity of different cross-sections in porous ceramics under different loads, respectively, the dispersion degree of porosity of different cross-sections has been estimated according to the discrete variance formula accepted widely by researchers:
where D represents the dispersion of porosity of different cross-sections in the porous ceramic, E is mathematical expectation of the porosity. The dispersion degree under loads of 0 MPa, 5.28 MPa, and 10 MPa are 0.94, 1.1, 0.61, separately. Note that dispersion degree after failure is 8.1. It distinctly indicates pores along the thickness of porous ceramic without large cracks are uniform, which shows that the uniform distribution of pores can provide reasonable channels for deionized water in the transpiration cooling.
The three-dimensional structure model of porous ceramic (the diameter of 4 mm, and the height of 737 µm) has been restructured with Avizo software in order to see the pore structures clearly, pores have been uniformly distributed in the porous ceramic (see Fig. 10(a)). After maximum loading, the pore structures including pores and cracks are displayed in Fig. 10(b), blue region represents pores and red part shows cracks. Compared to porous ceramic before loading in Fig. 10(a), it is clearly seen that large cracks have randomly appeared in the porous ceramic, and these cracks can also be the channels for deionized water in the experiment, so the permeability has a sharp increase in cd in Fig. 6. The porosity formed from cracks is very non uniform along the thickness direction of porous ceramic (see Fig. 9). Therefore, these pore channels of porous ceramic after achieving maximum loading cannot control accurately and uniformly water flow.