# Simplified Model Predicts Binder Behavior in Sand Mold Printing
## 1 Introduction  
Binder jetting is one of the essential processes of additive manufacturing (AM), the most widely used technology in sand mold casting [1; 2; 3; 4]. The technology was first demonstrated by [PERSON] et al. in 1990 [5]. Because binder jetting is now used in the production of multiple different materials, such as TEG-DMA [6], ceramics [7], alumina [8], and silica sand [9], it has become a frequent subject of scientific research. One issue with binder jetting is that the flow of the binder in the powder bed affects diffusion diameter (D\({}_{\text{eff}}\)) and penetration depth (P\({}_{\text{inf}}\) [infiltrate]). Due to gravity and the capillary action of the sand particles, the binder tends to permeate down to the bottom of the sand mold during printing, which affects the dimensional error of the sand mold in the vertical direction [9; 10; 11; 12]. In one study of this phenomenon, [PERSON] et al. proposed a mathematical model that used capillary pressure to explore (horizontal) diffusion and (vertical) penetration with Ti-6 Al-4V and 420 stainless steel powder. They then created a phase diagram based on the ratios of binder liquid viscosity, air, and capillary number to explain the compact front versus the finger-like pattern of the permeation volume [13; 14]. [PERSON] et al. explored the influence of viscous force, capillary force, and gravity on vertical penetration. They used Bond and Capillary numbers to predict penetration and confirm the correlation between saturation, binder P\({}_{\text{inf}}\), and capillary pressure [15]. The results showed that binder volume is the direct cause of dimensional error in sand molds in both the horizontal and vertical directions and that using a mix of particle sizes has the effect of reducing D\({}_{\text{eff}}\) and P\({}_{\text{inf}}\)[16]. ANSYS Fluent is a leading computational fluid dynamics (CFD) software that simulates fluid flow and heat transfer in various engineering applications, including automotive,aerospace, energy [17; 18; 19], chemical [20], and process engineering [21; 22]. Many researchers have turned to using Computational Fluid Dynamics (CFD) to simulate the behavior of 3D printing [23; 24]. [PERSON]'s team has investigated the penetration and diffusion behavior of droplets impacting the surface of a powder bed using Finite Element Analysis (FEA) and predicted their mechanisms [25]. [PERSON], [PERSON] team has explored the shear patterns and viscosity relationship of materials extruded from a 3D printing nozzle using Computational Fluid Dynamics (CFD) [26]. [PERSON] team has also investigated the penetration and diffusion mechanisms of binder in paper-based 3D printing using ANSYS Fluent [27]. In additive manufacturing, CFD is primarily used in Laser Powder Bed Fusion (L-OBF) and Fused Deposition Modeling (FDM) at present [28; 29; 30; 31]. Both methods heat the binder jetting mold, so temperature variables must be considered in the simulation process. Thus, CFD obtains more accurate simulation results. [PERSON] et al. determined the CFD parameters used in binder jetting and researched the influence of the Reynolds number, the Weber number, porosity, and contact angle on permeation kinetics [32]. [PERSON] proposed the influence of the hydrophilicity or hydrophobicity of a porous surface on permeation using a 2D model, established the hydrophobic parameters for the porous medium, and attempted to predict the permeation mechanism of the binder fluid in the porous medium [33]. [PERSON] et al. designed cylinders of different sizes to investigate the effect of different porosities on the permeation mechanism [34]. [PERSON] conducted similar research by distributing cylinders of different sizes. However, his results differed from the previous simulation analysis, which utilized a single layer. [PERSON]'s multilayer simulation analysis corresponded more closely to actual printing conditions [35]. The difference in materials and amount used affects the final experimental result.