# Simplified Model Predicts Binder Behavior in Sand Mold Printing
## 5 Conclusions  
This study proposes a simplified model that converts the binder from droplet-shaped to cylindrical-shaped through unit volume transformation. Using transient simulation analysis in ANSYS Fluent, we convert the velocity at the boundary into a wall boundary condition through \(\Delta t\) settings to achieve volume transformation. This method effectively saves simulation analysis time and reduces the difficulty of setup. We applied this research to predict the penetration depth and diffusion diameter through experimental design. Different volumes of binders (0.1, 0.5, 1.0, 1.5, 2.0, and 2.5 \(\mu\)L) were deposited onto the powder bed using a micropipette, and CT scans were conducted to determine the penetration depth and diffusion diameter. We found that the porosity is related to the penetration depth and diffusion diameter. In the volume range of 0.1-2.0 \(\mu\)L, as the binder volume increases, both the penetration depth and diffusion diameter increase. However, the difference between the 2.0 and 2.5 \(\mu\)L volumes is small, indicating that the penetration behavior is not linearly distributed. Conversely, the diffusion diameter shows a linear distribution. Additionally, as the porosity increases, both the penetration depth and diffusion diameter increase due to larger gaps between particles, allowing more binder to flow toward the bottom under inertia and gravity. Conversely, if the porosity decreases, the penetration depth and diffusion diameter also decrease. Through the proposed simplified model in this study, we observed that the experimental results exhibit the same trend as the simulation results. Based on the experience and results obtained in this study, the porosity also affects the overall absorb force between the powder and binder. This absorb force leads to different penetration depths and diffusion diameters, with smaller porosity resulting in greater adhesive force. However, if the porosity is too small, the space between particles becomes insufficient, which also affects the flow of the binder between particles. More binder is bonded between particles, resulting in higher strength of the printed product. We further applied the model to actual printing using a physical machine. When using a piezoelectric printhead in the machine, the volume of a single binder is smaller than when using a micro pump. It is difficult to measure the penetration of these tiny binders in real-time through microscopy or high-speed cameras. Therefore, we predicted the penetration depth and diffusion diameter using the simplified model. The results still matched the printing outcomes of the machine, with even smaller errors compared to using the micro-pump with higher binder volumes. This indicates that in the future, with the pursuit of high precision in binder jetting, using less binder and lower porosity sand will become a trend. The proposed simplified model in this study is better suited to meet future needs. The simplified model proposed in this study is intended to quickly predict the penetration depth and diffusion diameter, providing users with a basic understanding of the dimensional accuracy of the final product. Therefore, it omits many complex model constructions and theoretical calculations. The theoretical foundation of penetration behavior lies in porous media, particle behavior, and fluid mechanics. In the increasingly popular field of binder jetting, a further understanding of penetration behavior is essential. We have summarized it in the following six points:  
1. The study proposes a simplified model that converts the binder from droplet-shaped to cylindrical-shaped through unit volume transformation, saving simulation analysis time and reducing setup difficulty;
2. Experimental design and CT scans were conducted to predict the penetration depth and diffusion diameter of binders with different volumes deposited onto the powder bed;
3. The porosity of the powder bed affects the penetration depth and diffusion diameter, with higher binder volumes leading to increased values;
4.