# Experimental and Numerical Study of Vacuum Resin Infusion for Thin-Walled Composite Parts
## 3 Model of the Resin Infusion: Problem Statement
In the work [50], on the base of the results of [PERSON]'s research [51], it was shown that the saturation dependence on the pressure can be satisfactorily described by the function:  
\[S(p)=2/\pi\cdot a\tan(\alpha p) \tag{13}\]  
where a variable \(\alpha\) is determined experimentally, which was implemented in [49]. Satisfactory agreement with the experimental data was achieved for the value \(\alpha=1/p_{ref}\), where the reference pressure \begin{table}
\begin{tabular}{c c} \hline
**Parameter Name** & **Parameter Value** \\ \hline Interface thickness for the variable \(\phi\) & \(\epsilon=h_{\max}/2\ =\ 0.0025\) m \\ \hline The maximum size of the finite elements at the used & \(h_{max}=0.005\) m \\ FE meshing & \\ \hline Surface tension coefficient & \(\sigma=1\) N/m \\ \hline The mobility setting parameter & \(\chi=\frac{\sqrt{2}}{3}\frac{u_{max}}{\sigma}\frac{h_{max}}{\varepsilon}=0.001\) m-s/kg \\ \hline Estimated maximum resin velocity & \(u_{max}=0.001\) m/s \\ \hline \end{tabular}
\end{table} 
Table 2: The values of the phase field equation’s parameters, which were accepted at the finite element (FE) model implementation. is taken equal to the current atmospheric pressure. Thus, after performing time differentiation, Equation (9) is transformed to the form:  
\[V_{\phi}\frac{2}{\pi}\frac{\alpha}{1+(ap)^{2}}\frac{\partial p}{\partial t}- \sqrt{\left(\frac{K}{\mu}\
abla p\right)}=0 \tag{14}\]  
It supplements the phase field equation, is connected with it through porosity and allows us to establish a relationship between pressure and fluid flow rate using Relation (1) of Darcy's law, which includes the pressure value determined from Equation (14). The pressure field in the preform, determined from Equation (14), is used to calculate the compacting pressure \(P_{comp}(\mathbf{r})\) acting externally on the porous structure of the preform  
\[P_{comp}(\mathbf{r})=p_{atm}-p(\mathbf{r}) \tag{15}\]  
and, therefore, changing its porosity and fiber volume fraction according to Equation (10). Equation (14) must be valid in the entire volume of the preform, both in filled and in empty areas. Consequently, the values of permeability and viscosity should vary in accordance with the resin filling \(V_{r}\) and porosity \(V_{\phi}\). The resin permeability dependence on the fiber volume fraction is described by  
Figure 10: The dependence of the porous medium permeability \(K\) on the reinforcement volume fraction \(V_{f}\): experimental data borrowed from the article [6] and their approximation by the model in Equation (11) with the parameter \(k=2.41\times 10^{-12}\). Figure 9: The dependence of the reinforcement volume fraction \(V_{f}\) on the compacting pressure: experimental data borrowed from the article [6] and their approximation by the model in Equation (10) with the parameters: \(V_{f0}=0.21\); \(B=0.093\). Equation (11). The study of the air permeability along and transverse of the hexagonally arranged fabrics with different fiber volume fractions suggest the following relations for these permeabilities [52]  
\[K_{long}^{hex}=\frac{8\rho^{2}}{c_{hex}}\cdot\frac{\left(1-V_{f}\right)^{3}}{V_{ f}{}^{2}};\quad K_{trans}^{hex}=C_{hex}r^{2}\bigg{(}\sqrt{V_{f}^{\max}/V_{f}-1} \bigg{)}^{5/2} \tag{16}\]  
where these models parameters are: \(c_{hex}=53\), \(C_{hex}=16/\big{(}9\pi\sqrt{6}\big{)}\), \(V_{f}^{\max}=\pi/2\sqrt{3}\) and \(r\approx 3\) um is a fiber radius. The validity of the relations (16) is confirmed by numerous experimental data for different CFRP and GFRP (Glass Fiber Reinforced Plastic) fabrics, e.g., [52; 53]. These dependencies are present in Figure 11. A comparison of Figures 10 and 11 demonstrates a very close coincidence of the nature of permeability for liquid and air; therefore, the dependence of permeability for both contacting phases is assumed to be the same, and it was calculated by the Formula (11). However, the viscosities of the liquid resin and air differ several orders of magnitude.