## 2 Pore-Structure Characterisation and Analysis of Permeability Data

Table 3: _Cont._  
Figure 2: **(a)** Optical image of aluminium foam characterised by monosized (L-L) pores [37], (**b**) virtually-generated image of monomodal (L-L) macroporous structure, (**c**) virtually-generated representative volume element (RVE) of monomodal macroporous structure, (**d**) X-ray computer tomography section of bimodal (L-S) macroporous structure, (**e**) virtually-generated image of bimodal (L-S) macroporous structure, (**f**) virtually-generated representative volume element (RVE) of bimodal macroporous structure, (**g**) X-ray computer tomography RVE image of Inconel 450\(\upmu\)m sample [8] (**h**) X-ray computer tomography RVE image of Inconel 1200 \(\upmu\)m sample [39] and (**i**) X-ray computer tomography RVE image of dual-porosity differential Inconel 450 and 1200 \(\upmu\)m samples [38]. In Table 2, there is evidence of increased porosity and pore openings in samples as S-S was added to L-L, further increasing the flow permeability of the porous structure and reducing the number of \"dead-end\" pores. With these, bimodal macroporous structures made by packing varied sphere sizes are expected to be more complicated, and the flow patterns within them are expected to behave differently from those of monosized pores. For example, despite capillary influence being included in the coordination number of bottleneck-type structures described in [33], the behaviour characterised by changes in predicted static viscous permeability (using models described in [30, 36, 39]) of bimodal structures were observed to underestimate their capabilities for prediction by CFD. The reason for this disagreement was that the models presented in [30, 36, 39] were developed using CFD computed permeability data for macroporous structures characterised by monosized pores and pore volume fractions between 0.59 and 0.65. Figure 3a shows a plot of reduced permeability (\(k^{x}=k_{0}/\left\lceil\sigma_{p}^{2}\right\rceil\)) against a set of multiple changes in pore-structure-related parameters (\(\left[\left[1-\varepsilon\right]^{2}/\varepsilon^{3}\right]*\left[r_{p}/r_{w }\right]\)) for combined packing conditions (monomodal and bimodal structures) emphasising an inverse relationship with near-power. Therefore, this relation can be used to provide an analytical description of permeability as a test function of parameters related to pore structure, as shown by Equation (1). \[k_{0}=\frac{\varepsilon r_{p}^{2}}{5}\left\{\frac{\varepsilon^{3}}{(1- \varepsilon^{2})}\frac{r_{w}}{r_{p}}\right\}^{\sqrt{(5.5/2)}} \tag{1}\]  
In order to make a quantitative assessment of the proposed model, experimental measurements and modelled permeability data were compared. Table 3 summarises various macroporous component parameters and permeability data (measured experimentally, computed tomography and predicted analytically). According to [33], the permeability derived from X-ray CT and measured experimentally with a variable head permeameter for the L-L structure in Figure 1a is 1.14 \(\times\) 10\({}^{-9}\) m\({}^{2}\) and 1.28 \(\pm\) 0.09 \(\times\) 10\({}^{-9}\) m\({}^{2}\), respectively. By using the expression in Equation (1), the equivalent predictive value was calculated to be 1.27 \(\times\) 10\({}^{-9}\) m\({}^{2}\), demonstrating excellent agreement between the measured and predicted values. Additionally, for bimodal pore distributed structures, CFD simulation using a 3D representative model reconstructed from 2D X-ray computed tomography slices for the \"real\" 12.5 and 25.0% volume S-S samples (Figure 2d-f) achieved permeability values of 0.65 \(\times\) 10\({}^{-9}\) m\({}^{2}\) and 1.34 \(\times\) 10\({}^{-9}\) m\({}^{2}\), respectively.