# Experimental and Numerical Study of Vacuum Resin Infusion for Thin-Walled Composite Parts
## 3 Model of the Resin Infusion: Problem Statement
The Comsol FE system, in which the simulation was performed, uses the splitting of the fourth-order Equation (4) into two second-order partial differential equations (PDE):  
\[\frac{\partial\phi}{\partial t}+\mathbf{u}\cdot\
abla\phi=\
abla\cdot\frac{ \gamma\lambda}{\varepsilon^{2}}\
abla\psi \tag{6}\]  
\[\psi=-\
abla\cdot\varepsilon^{2}\
abla\phi+\left(\phi^{2}-1\right)\phi \tag{7}\]  
The local filling of the micro volumes of the preform \(V_{r}\) can be expressed through the phase field variable \(\phi\) by the relation:  
\[V_{r}=(\phi+1)/2 \tag{8}\]  
so that a completely filled volume (\(V_{r}=1\)) corresponds to a value \(\phi=1\), an empty volume (\(V_{r}=0\)) corresponds to a value \(\phi=-1\). The boundary conditions of the phase field equation are shown in Table 1, while the used values of all equation's parameters are present in Table 2. \begin{table}
\begin{tabular}{c c} \hline \hline
**Boundary** & **Boundary Condition** \\ \hline \hline \multicolumn{1}{c}{Resin injection gate:} & \(\phi=1\) \\ \hline \multicolumn{1}{c}{Wetted walls boundary conditions, where \(\mathbf{n}\) is a unit vector normal to the preform boundaries, and the contact angle \(\theta_{w}\) was accepted as \(\pi\)/2:} & \(\left\{\begin{array}{c}\mathbf{n}\cdot\frac{\gamma\lambda}{\varepsilon^{2}} \
abla\psi=0;\\ \mathbf{n}\cdot\varepsilon^{2}\
abla\phi=\varepsilon^{2}\cos(\theta_{w}) \big{|}\
abla\phi\big{|}=0.\end{array}\right.\) ;} \\ \hline \multicolumn{1}{c}{Vacuum vent:} & \(\phi=-1\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The boundary conditions of the phase field equation. The initial condition for the variable \(\phi\) defined everywhere in the volume of preform is \(\phi=-1\). The second equation of the coupled problem describing the dynamics of the viscous resin should be an equation connecting the field of the resin's driving pressures with the velocity field of the medium. The Richards equation [49] representing fluid motion in unsaturated soils is adopted as:  
\[\partial\big{(}V_{\phi}S\big{)}/\partial t-\
abla\cdot\left(\frac{K\cdot k_{r}} {\mu}(\
abla p)\right)=0 \tag{9}\]  
which models the behavior of saturation \(S\) as a function of pressure \(p\), permeability and porosity \(V_{\phi}\). Permeability in Equation (9) is the product of constant permeability \(K\) and relative permeability \(k_{r}\in[0;1]\). The saturation \(S\) in Equation (9) is an analog of the filling, i.e., resin volume fraction \(V_{r}\). To correctly take into account the void preform's compacting and the resulting change in its permeability at the vacuum infusion processes, the following models were most widely used [6; 34]: compacting model--changes in the reinforcing volume fraction \(V_{f}\) (see Figure 9):  
\[V_{f}=V_{f0}\cdot P_{comp}^{B} \tag{10}\]  
where the compacting pressure \(P_{comp}\) is the difference between atmospheric pressure and the pressure in the region inside the preform, \(V_{f0}\) is the reinforcing fiber volume fraction at the zero compacting pressure and the Kozeny-Karman model for the permeability \(K\) of the porous medium:  
\[K=k\cdot\left(1-V_{f}\right)^{3}/{V_{f}}^{2} \tag{11}\]  
where the Kozeny constant \(k\), determined experimentally, reflects the structure of the porous medium, i.e., depends on the geometry of the reinforcement. Combining Equations (10) and (11), it is easy to obtain the dependence of permeability on compacting pressure. Using the relation between porosity \(V_{\phi}\) and \(V_{f}\), the dependence of permeability can also be expressed in terms of the porosity  
\[K=K_{0}\cdot V_{\phi}^{3}/\left(1-V_{\phi}\right)^{2} \tag{12}\]  
These models have been used in our study; their graphical representations are present in Figures 9 and 10 together with the experimental data used for these models' identification. In Equation (9), the value of the saturation can be expressed as some function of pressure that activates the movement of the liquid [49].