## 3 Results and Discussion
### Numerical Simulations of Supercritical Debinding  
The numerical simulations were run using the Comsol Multiphysics(r) finite-element software platform version 5.2 developed by Comsol Inc. (Burlington, MA, USA) in 2015. The main objective was to use [PERSON]'s model and the identified coefficient to confront the experimental and numerical results of the weight loss during the PEG extraction by supercritical CO\({}_{2}\) in an injected component. The chemical module transfer of the software species was used to simulate the mass transport and chemical reactions in various environments. The transport interface of the diluted species was also used to estimate the concentration of an aqueous solution in a solvent. When working with the numerical solution of any time-dependent problem, a very costly part is assembling the finite-element matrix at each time step [35]. In this study, the following simplifying hypotheses are assumed for the physical model:  
* the component is continuous;
* the soluble polymer (PEG) is distributed homogeneously in the component;
* PEG is the only soluble binder;
* the weight loss is only due to the dissolution and elimination of the soluble binder;
* there is no interaction between the other binder constituents (PP, SA) contained in the component; and,
* there is no temperature variation during the supercritical debinding. The solution to the problem requires initial boundary conditions. Initially, the concentration \(C_{i}\), which represents the average molar concentration of PEG in the injected components, is equal to 1.3 \(\times\) 10\({}^{-2}\) mol/m\({}^{3}\). To simplify the numerical results' interpretation, the relative value of the concentration, which includes 100% PEG initially in the component, was used. The diffusion coefficient identified previously was used (\(D\) = 7.46 \(\times\) 10\({}^{-6}\) cm\({}^{2}\)/s). During the debinding by the supercritical fluid (CO\({}_{2}\)), two boundary conditions existed. One was the impermeable surface, which represents the end of the cylindrical component (support). The outer surfaces, which are exposed to the supercritical CO\({}_{2}\), have a concentration \(C_{o}\) equal to zero:  
* \(\frac{\partial C}{\partial x}=\frac{\partial C}{\partial y}=\frac{\partial C}{ \partial z}=0\) for \(x\) = 0 at \(t\geq 0\);
* \(C=C_{o}=0\) for outer surfaces at \(t=0\);
* \(C=C_{i}=100\%\) at \(t=0\). A cylindrical component with length \(L=15\) mm and diameter \(d\) = 10 mm, as shown in Figure 5, was modelled in three dimensions (3D) by the finite-elements method (FEM) and simulated using the various modules described previously. The debinding time retained for this simulation was four hours, like the experimental debinding. The 3D domain was discretized using an automatic mesh generator with 77,645 linear tetrahedral elements with four nodes. A convergence test, which is not detailed in this study, validated the mesh quality. An explicit time-stepping scheme was used for the time-dependent solver algorithm. The solution is given each minute until the entire process reaches \(t=t_{f}=4\) h.  
Figure 5: Description of three-dimensional geometry used for finite-elements method (FEM) simulation with boundary conditions. Numerical simulations were carried out on the PEG distribution during the debinding step using supercritical CO\({}_{2}\) on an injected cylindrical component with a powder-volume loading equal to 60%, for various debinding kinetics in terms of temperature and time parameters. As mentioned above, the binder system is composed of PP, SA, and PEG, which is the soluble binder. Figure 6 shows the remaining PEG distribution in the cylindrical component during the debinding by supercritical CO\({}_{2}\), according to a multi-slice view from the initial time (\(t=0\)) to the final stage (\(t=4\) h). It is evident that the binder remains from the component's center to its outer surfaces during the debinding. The obtained results show an isotropic diffusion, and at the final stage the binder is present in the component's center.