*3.4. Mathematical Modeling*

The values of out-of-plane permeability, both saturated and unsaturated, of the three different preforms have been further validated by comparison with the results of predictive models. As recently reported by Karaki et al. [18,61], among the several mathematical models available in the literature for estimating the permeability of unidirectional reinforcements, the most accurate ones, able to fit the experimental data of out-of-plane permeability of unidirectional yarns, are those proposed by Gebart [62] and Berdichevsky and Cai [63].

Gebart [62] derived an analytical model to predict the unidirectional permeability starting from Navier-Stokes equation, considering two fibers arrangements, a quadratic array and a hexagonal array:

$$\mathbf{K}\_3 = \mathbf{C}\_1 \left( \sqrt{\frac{\mathbf{V}\_{f\text{max}}}{\mathbf{V}\_f}} - 1 \right)^{\frac{5}{2}} \mathbf{r}\_f^2 \tag{10}$$

where V<sup>f</sup> is the fiber volume fraction, r<sup>f</sup> the fiber radius and the constants C<sup>1</sup> and Vfmax depend on the fiber packing arrangement according to the values reported in Table 3.


**Table 3.** Gebart model: constant values for quadratic and hexagonal fiber arrangement.

Berdichesvky and Cai [63] derived an unified empirical model where Stokes flow and Darcy flow are considered at different regions and the gap between neighboring fibers governs the flow resistance:

$$\mathbf{K}\_3 = 0.229 \mathbf{r}^2 \left(\frac{1.814}{\mathbf{V}\_\mathbf{a}} - 1\right) \left(\frac{\left(1 - \sqrt{\frac{\mathbf{V}\_t}{\mathbf{V}\_\mathbf{a}}}\right)}{\sqrt{\frac{\mathbf{V}\_t}{\mathbf{V}\_\mathbf{a}}}}\right)^{2.5} \tag{11}$$

where V<sup>f</sup> is the fiber volume fraction and V<sup>a</sup> is equal to 0.9069 and 0.7854 in case of hexagonal or quadratic fiber arrangement, respectively.

Preform B

Preform C

Unsaturated

Saturated

Unsaturated

Saturated

Models predictions and the experimental results on the unsaturated and saturated permeabilities are compared in Figure 8, where only the best fit among modeling and experimental data are reported. A nonlinear fit of the K<sup>3</sup> values as a function of fiber volume fraction has been performed using Equations (10) and (11), in order to obtain the fiber radius r<sup>f</sup> , whose values are reported in Table 4. The quadratic fiber arrangement is not able to properly model the experimental values, as inferred by the high differences reported in Table 4, since the obtained fiber radius is higher than the nominal value, which is equal to 3.35 µm for Preform A and 2.5 µm for Preforms B and C. Therefore, the fit obtained with the quadratic fiber arrangement with both the models are not reported in Figure 8 for the sake of clarity. *Materials* **2020**, *13*, x FOR PEER REVIEW 13 of 17 Therefore, the fit obtained with the quadratic fiber arrangement with both the models are not reported in Figure 8 for the sake of clarity.

**Figure 8.** Mathematical modeling of unsaturated (**a**–**c**) and saturated permeability (**d**–**f**) values: (**a**,**d**) preform A; (**b**,**e**) preform B; (**c**,**f**) preform C. **Figure 8.** Mathematical modeling of unsaturated (**a**–**c**) and saturated permeability (**d**–**f**) values: (**a**,**d**) preform A; (**b**,**e**) preform B; (**c**,**f**) preform C.

**Table 4.** Unsaturated out-of-plane fiber radius obtained from the mathematical modeling of the experimental K3 values and percentage difference from the nominal fiber radius. **Material Out-Of-Plane Permeability Model Fiber Radius from Model Best Fit, r<sup>f</sup> (µm) Nominal Fiber Radius rfn (µm) Difference [(rfn − rf)/rf] × 100 (%)**  Gebart-hexagonal 4.57 3.45 +32 On the other hand, the best fit is always obtained adopting the hexagonal array and Berdichevsky and Cai model, with differences from the nominal fiber radius lower than 15%. Only the best fit of saturated K<sup>3</sup> of preform A, with all models, leads always to fiber radii much larger than the nominal ones. From Figure 8 it can be observed good agreement among the unsaturated K<sup>3</sup> values calculated by ultrasonic measurement and the model prediction. This can be considered a further validation of the proposed measurement method, based on ultrasonic wave propagation.

Preform A Unsaturated Gebart-quadratic 5.45 3.45 +58 Berdichevsky-hexagonal 3.23 3.45 −6.8 Berdichevsky-quadratic 4.80 3.45 +39 Saturated Gebart-hexagonal 7.09 3.45 +105 Gebart-quadratic 8.41 3.45 +144 It should be also underlined that the schematization in hexagonal and quadratic array is often ideal since the fiber distribution is often irregular and falls in between hexagonal and quadratic arrangement. Fiber clustering, in real preforms, makes reasonable the almost always higher radii obtained from fitting compared to the nominal ones.

Berdichevsky-hexagonal 4.99 3.45 +45 Berdichevsky-quadratic 7.39 3.45 +114

Gebart-hexagonal 2.89 2.5 +16 Gebart-quadratic 4.04 2.5 +62 Berdichevsky-hexagonal 2.20 2.5 −12 Berdichevsky-quadratic 3.73 2.5 +49

Gebart-hexagonal 3.32 2.5 +33 Gebart-quadratic 4.69 2.5 +88 Berdichevsky-hexagonal 2.53 2.5 +1.2 Berdichevsky-quadratic 4.43 2.5 +77

Gebart-hexagonal 2.81 2.5 +12 Gebart-quadratic 3.87 2.5 +55

Berdichevsky-quadratic 3.65 2.5 +46

Gebart-hexagonal 3.09 2.5 +24 Gebart-quadratic 4.29 2.5 +72 Berdichevsky-hexagonal 2.35 2.5 −6 Berdichevsky-quadratic 4.05 2.5 +62


**Table 4.** Unsaturated out-of-plane fiber radius obtained from the mathematical modeling of the experimental K<sup>3</sup> values and percentage difference from the nominal fiber radius.
