Simplified Model Predicts Binder Behavior in Sand Mold Printing

: Binder jetting is a crucial process in additive manufacturing (AM) and is widely used in sand mold casting. This study explores the challenges of simulating binder droplets in ANSYS Fluent, including complexity and computational time. To overcome these challenges, we propose a geometric approach that models the binder droplet as a circular shape instead of an actual droplet. Additionally, the dynamic mesh feature is employed to transform the initial boundary condition into a wall condition at a speciﬁed time interval ( ∆ t). This simpliﬁed approach eliminates the need to simulate actual droplets, leading to signiﬁcant computational resource and time savings. By adopting this geometric approach, we can accurately predict the diffusion and penetration behavior of binder droplets with varying materials and volumes in porous media with different porosities. Through data analysis, it was found that the main variables affecting the diffusion diameter and penetration depth are binder volume and porosity. The successful implementation of this simpliﬁed model enables researchers and engineers to expedite the simulation of binder behavior, facilitating process optimization and enhancing the understanding of binder jetting technology in the ﬁeld of additive manufacturing.


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 E. Sachs 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 eff ) and penetration depth (P 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, H. Miyanaji et al. proposed a mathematical model that used capillary pressure to explore (horizontal) diffusion and (vertical) penetration with Ti-6Al-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]. Løvoll 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 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 eff and P 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]. Yiwei Han'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]. Raj R, Krishna S.V.V's team has explored the shear patterns and viscosity relationship of materials extruded from a 3D printing nozzle using Computational Fluid Dynamics (CFD) [26]. López Martínez J.A's 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. Sadeghi 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]. M. Nazari 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]. Hongxin Deng et al. designed cylinders of different sizes to investigate the effect of different porosities on the permeation mechanism [34]. Xiangyu Gao conducted similar research by distributing cylinders of different sizes. However, his results differed from the previous simulation analysis, which utilized a single layer. Gao's multilayer simulation analysis corresponded more closely to actual printing conditions [35]. The difference in materials and amount used affects the final experimental result.
Due to the small size of the binder droplets and the porous material, it is often necessary to cut the mesh very finely for calculation. This situation causes difficulties for researchers using CFD, which requires more core calculations to save time on simulation analysis. Therefore, researchers often simplify the simulation process using a two-dimensional rather than a three-dimensional structure [36]. This geometric simplification of the porous media must be compared with actual printing to ensure the accuracy of the simulation [37,38]. This study investigates the diffusion and penetration of a single binder droplet in porous media.

Simplified Model
The simulation of binder droplets in ANSYS Fluent presents challenges in complexity and computational time. To address these challenges, we propose a geometric approach where the binder droplet is modeled as a circular shape rather than an actual droplet, and the dynamic mesh feature is utilized to transform the initial boundary condition into a wall at a specified time ∆t. This method eliminates the need to simulate actual droplets, resulting in significant savings in computational resources and time. By adopting this simplified approach, we can efficiently predict the diffusion and penetration behavior of binder droplets with different materials and volumes in porous media of varying porosities. The successful implementation of this simplified model empowers researchers and engineers to expedite the simulation of binder behavior, thereby facilitating process optimization and advancing the understanding of binder jetting technology in the additive manufacturing field. Figure 1 represents a schematic diagram of this study. We used a micropipette to inject different volumes of binder into the powder bed to carry out the penetration experiments and used the simplified model of ANSYS Fluent for simulation analysis. Afterward, the experimental results and analysis results were compared and verified.

Materials
The materials we used will be introduced in this study. The material properties must be provided for simulation software analysis and are also key elements for judging the permeability mechanism. The binder (as shown in Table 1) used in this study is a furan resin. This resin, with a density of 2.34 g/cm 3 , can be used at room temperature. We found that the viscosity of the furan resin was 14.55 cp (measured every 30 s at room temperature, using a total of six measurements to obtain the average). The contact angle measuring instrument was used to measure the average surface tension of the five sets of sand powders, with a result of 40.23 mN/m. We tested three different sand materials commonly used in additive manufacturing, namely SiO2, Al2O3, and ZrO2. Furthermore, we repeated the experiment six times to obtain the average density and divided the silica sand into three groups of samples (A-1, A-2, A-3) according to particle size, which were tested with a Bruker Micro CT skyscan 1272 to obtain the average particle size and porosity. Bettersize's Bettersizer 2600 (wet) was used for particle size analysis to analyze effective particle sizes D10, D50, and D90. Finally, we used an OSA60 produced by NBSI, Ltd. for contact angle measurement (please refer to Table 2 for particle descriptions).

Materials
The materials we used will be introduced in this study. The material properties must be provided for simulation software analysis and are also key elements for judging the permeability mechanism. The binder (as shown in Table 1) used in this study is a furan resin. This resin, with a density of 2.34 g/cm 3 , can be used at room temperature. We found that the viscosity of the furan resin was 14.55 cp (measured every 30 s at room temperature, using a total of six measurements to obtain the average). The contact angle measuring instrument was used to measure the average surface tension of the five sets of sand powders, with a result of 40.23 mN/m. We tested three different sand materials commonly used in additive manufacturing, namely SiO 2 , Al 2 O 3 , and ZrO 2 . Furthermore, we repeated the experiment six times to obtain the average density and divided the silica sand into three groups of samples (A-1, A-2, A-3) according to particle size, which were tested with a Bruker Micro CT skyscan 1272 to obtain the average particle size and porosity. Bettersize's Bettersizer 2600 (wet) was used for particle size analysis to analyze effective particle sizes D 10 , D 50 , and D 90 . Finally, we used an OSA60 produced by NBSI, Ltd. for contact angle measurement (please refer to Table 2 for particle descriptions).

Solution and Solver in Simulation
ANSYS Fluent is software used for the numerical analysis of fluid mechanics using the Finite Volume Method (FVM) to solve differential equations. We used two-phase VOF modeling to calculate the liquid/gas interface. The diffusion and penetration of the binder in the powder bed needed to be solved with a transient solution. Since the binder volume was tiny, the mesh and initial time also had to be very small (please refer to Table 3 for parameters).

Geometry and Mesh
This study obtained the particle size and porosity of the powders used in the experiment via CT scanning, then set the geometry in the model to 10 mm (w) × 10 mm (l) × 5 mm (h), as shown in Figure 2a for grid processing. Doing so saves analysis and simulation time and prevents the mesh from breaking or failing to simulate the actual situation smoothly. Since the binder volume was between 0.1-2.5 µL, we set grid smoothing to medium, limited minimum size and maximum size to between 5.0 −8 and 4.096 −4 m, and used the tetrahedron method (most suitable for fluid dynamics simulation) to process the grid, as shown in Figure 2b. The final mesh metrics were below 0.9 (average and stable mesh quality). We then input the parameters needed for the simulation, material properties such as liquid and powder density, viscosity, contact angle, and surface tension, as shown in Tables 1 and 2.

Setup
This simulation used VOF for two-phase flow (liquid/gas) analysis, set interface modeling to sharp, and selected Implicit Body Force. The material used was a furan resin with a density of 2.34 g/cm 3 , viscosity of 14.55 cp, and surface tension of 40.23 mN/m, as shown in Table 1. The other phase was set as air. For the parameters of the solid materials, such as Cell Zone Conditions, we set porosity (shown in Table 2). Inlet velocity was 1 m/s and treated as a liquid-gas mixture. Except for the bottom wall, which was set to Stationary Wall and No Slip Shear Condition, the interfaces were all set to allow fluid to flow in and out (as shown in Figure 2c) because the binder penetrates to the bottom and diffuses in other directions after entering the powder bed. The sand surfaces allow liquids and gases to pass through. To convert the binder volume shown in Figure 1a, we checked the Dynamic Mesh option in the software interface section to generate an event. At Δt 1.0 −4 s, we reset the boundary conditions to Wall to control the binder volume flowing into the powder bed to equal a single spherical droplet (as in Figure 1b).
The binder drops deposited from the micropipette in Figure 1a can be regarded as spheres driven by the waveform and voltage of the piezoelectric nozzle [39]. We converted their volumes into cylinders with different radii, as shown in Figure 1b. The binder volume radii are shown in Table 4. The radius can be used to obtain the corresponding height of the cylinder under the condition of a fixed flow rate.

Setup
This simulation used VOF for two-phase flow (liquid/gas) analysis, set interface modeling to sharp, and selected Implicit Body Force. The material used was a furan resin with a density of 2.34 g/cm 3 , viscosity of 14.55 cp, and surface tension of 40.23 mN/m, as shown in Table 1. The other phase was set as air. For the parameters of the solid materials, such as Cell Zone Conditions, we set porosity (shown in Table 2). Inlet velocity was 1 m/s and treated as a liquid-gas mixture. Except for the bottom wall, which was set to Stationary Wall and No Slip Shear Condition, the interfaces were all set to allow fluid to flow in and out (as shown in Figure 2c) because the binder penetrates to the bottom and diffuses in other directions after entering the powder bed. The sand surfaces allow liquids and gases to pass through. To convert the binder volume shown in Figure 1a, we checked the Dynamic Mesh option in the software interface section to generate an event. At ∆t 1.0 −4 s, we reset the boundary conditions to Wall to control the binder volume flowing into the powder bed to equal a single spherical droplet (as in Figure 1b).
The binder drops deposited from the micropipette in Figure 1a can be regarded as spheres driven by the waveform and voltage of the piezoelectric nozzle [39]. We converted their volumes into cylinders with different radii, as shown in Figure 1b. The binder volume radii are shown in Table 4. The radius can be used to obtain the corresponding height of the cylinder under the condition of a fixed flow rate.

Solution Method
Finally, since the porous medium is composed of particles of different sizes and shapes, the differences in the particles caused the mesh to have a larger inclination angle, so we selected the Pressure-Implicit with Splitting of Operators (PISO) algorithm to obtain the solution. This is a velocity-pressure algorithm based on the extension of SIMPLE because it does not return to the initial momentum equation after each calculation of the velocity field but rather directly updates the internal operation matrix. The pressure field obtained at each step is more accurate and converges faster, so the overall calculation process is decreased, and the operating speed is faster. This method is suitable for calculating transient incompressible fluids. For the Gradient setting, we selected green-gauss node-based. As the name suggests, this method is based on Node-Based calculations. This algorithm consumes more computer resources but can obtain higher accuracy. It is suitable for small volumes and uses Calculation of Fluid Mechanics with Triangular Mesh.
We selected PRESTO! for pressure, which is more suitable for the significant pressure changes found in VOF modeling in porous media, and is also the default option in ANSYS Fluent. For the momentum equation, we chose Second Order Upwind, appropriate for use with smaller triangular meshes. Finally, we ran 2000 Number of Time Steps with a time step size of 1.0 −12 s to obtain the solution (Table 3).
In this study, SiO 2 (A-1, A-2, A-3), Al 2 O 3 , and ZrO 2 sand powders, as shown in Figure 2d, commonly used in 3D printing, were used to simulate and analyze binder volumes of 0.1, 0.5, 1.0, 1.5, 2.0, and 2.5 µL. The silica sand was divided into three different particle sizes. Five sets of simulations and experimental comparisons were carried out.

Experiment
As shown in Figure 1a, for the binder printing we placed the carrier on the dropper holder and pushed the microdroplet through the air pressure device so that the binder droplet was ejected at the same speed as the simulated speed (1 m/s). The distance from the dropper to the sand surface was 6 mm. However, in the experiments in this study, the distance from the sand surface had little effect on the D eff . Then we used the Dino-Lite Premier AM4113T5X microscope with Media Cybernetics Image-Pro 10 software to observe and obtain the average D eff of the six groups of samples through image analysis. Since penetration depth P inf cannot be observed with the microscope and image processing, we did CT scans of the six groups of samples with a Bruker Micro CT skyscan 1272 and calculated the P inf of the six groups of samples. To do so, we put the powder into a container and made a preliminary scan. Afterward, we dropped the binder into the powder bed (the powder in the container) and waited for 30 min, then performed a second scan and compared the relative positions of the powders between the two to calculate the penetration depth.

Results
Given the large number of samples and experimental items, it was not feasible to provide all of them. Thus, we presented the selected simulation and experimental results in Figures 3 and 4. In Figure 3, we used the conversion method to drop binder volumes of 0.1-2.5 µL into porosities of 7.6, 16.6, and 36.6%, which cover the various powder parameters used. In Figure 4, we dropped a binder volume of 1.0 µL into porosities of 7.6, 16.6, 36.3, and 40.7%. Using CT scans, we obtained the penetration depth and diffusion diameter and obtained several insightful results depicted in Figure 4. Ultimately, all the simulation and experimental data were organized into graphs for clear representation in Figure 5.

Diffusion Comparison in the Simulation and Experiment
At first, the simulation results were very close to the experimental results regarding D eff . The most significant difference was that the geometry of the actual printing results was messier, as in Figure 3a,b, rather than a perfect circle. The simulation resulted in an entirely perfect circle, as in (Figure 3a). This result is typical because the simulation does not simulate every particle but uses the overall porosity method. In actual conditions, the particle size of the powder is not fixed, which leads to differences in the flow of the binder between the particles, which in turn affects the geometry of the D eff . This becomes more significant when there is greater porosity or larger particle size. If the particle size and porosity are small, the D eff correlates closely with the simulation result and presents a perfect circle, as in Figure 3b. When particle size and porosity decrease, D eff is closer to a circle.
In our simulation analysis, we used the porous media setting in ANSYS Fluent to define the porosity. This allows us to add resistance in the geometry to simulate the pressure drop experienced by the fluid in a porous medium. This idealized scenario provides a convenient and easily understandable simulation result. However, in the experimental process, the powder bed consists of numerous randomly distributed particles, resulting in varying porosity within each unit volume. Additionally, the particle sizes are not completely uniform, adding to the unknown factors in the powder bed (such as the adsorption capacity of the binder by the powder). Moreover, during binder falling, the velocity may not be strictly vertical due to operator manipulation or external influences. This can introduce inertial effects and lead to different diffusion geometries once the binder enters the powder bed. Figure 3b indicates that the geometric shape becomes more perfect as the binder volume decreases. Image analysis software was utilized to analyze the diffusion diameter in the experiment, which is shown in the partially enlarged view in Figure 3b. Figure 5b compares the D eff data in the simulation analysis and experiment. When the porosity is low, for example, 7.6-16.6%, the measured D eff is relatively consistent. However, once porosity reaches 36.3%, the sample points gradually disperse; when porosity reaches 38.2 and 40.7%, the sample points tend to be slightly concentrated. As shown in Figure 5b, D eff gradually increases as the binder volume increases, and the result presented is a quadratic curve. There is no significant difference in the diffusion diameter obtained after the binder volume reaches 2.0 and 2.5 µL with a porosity of 36.3%. The above phenomenon is attributed to the excellent absorption of binder by dry sand. When the porosity is low, the dry sand has a higher surface area ratio and has the largest absorption capacity for binders. Therefore, the rate of increase is relatively high at 16.6%. However, at 7.6%, despite having better absorption force, the limited space between particles to accommodate the binder affects the absorption speed of the binder. Conversely, porosity greater than 36.3% has a high binder capacity but poor absorption. The diffusion of the binder is affected by the inertial effect caused by the velocity at the initial absorption. This phenomenon also occurs in the infiltration phenomenon, but the gravity and the inertial effect of the binder ejection affect the infiltration phenomenon.
Furthermore, our lab determines the penetration depth and diffusion diameter by comparing CT scans taken before and after binder deposition. By analyzing the positional changes in the powder bed caused by the displacement resulting from binder infiltration, we can determine the achieved penetration depth and diffusion diameter. The accuracy of this method depends on the CT scan resolution and filtering mechanisms we employed. If the positional change in a particle between pre-and post-deposition scans exceeds 30% of its diameter, we consider it as having experienced displacement (to save processing time and filter out errors caused by the CT scan process). This approach is effective for both small and large particles in assessing penetration depth and diffusion diameter. Therefore, it is possible to have particles that have come into contact with the binder but have not experienced any displacement. We believe that if the binder only adheres to the powder without significant bonding strength between the particles, it may not withstand the impacts and loads during the casting process. Hence, we applied empirical formulas such as the Washburn equation and soil water retention curve to exclude non-conforming areas. Based on our lab's experience, we determined that the effective area accounts for approximately 75% of the overall area. The data distribution presented in Figure 5 is the result after being adjusted.

Penetration Comparison in the Simulation and Experiment
We found that the penetration depth increased with decreasing porosity, as shown in the curves in Figures 3c and 4. In addition, the diffusion diameter also increased with increasing porosity, as observed in Figure 5b. More specifically, when porosity was low, the diffusion diameter values were relatively consistent; when porosity increased beyond 36.3%, the diffusion diameter values gradually became more dispersed. Moreover, we observed that the relationship between the diffusion diameter and binder volume followed a quadratic curve, as depicted in Figure 5b. Overall, these findings suggest that binder volume and porosity play critical roles in determining penetration depth and diffusion diameter, which can help enhance our understanding of the infiltration phenomenon.
We determined the penetration of the binder in the powder bed using CT scans and compared the results with the simulation results. It can be seen that P inf is related to porosity and particle size. As the binder volume increases, P inf becomes more significant. However, if the porosity is too great, the binder will continue to infiltrate downward until resistance with the particle surfaces balances the force of penetration. Conversely, if the particle size range and porosity are small, the binder is more likely to be retained between the particles [40][41][42], decreasing the P inf of the binder. When the porosity exceeds a certain value, the permeation shape gradually becomes cylindrical, which is most obvious when porosity exceeds 36.3%, according to the simulation results in this paper. Figure 4 shows that the affected particles were displaced after being combined with the binder. Moreover, it presents a somewhat spherical penetration range.
Moreover, at a porosity of 40.7%, the cross-sectional geometry is not clearly visible due to the difficulty in capturing displacement changes between particles under CT scans at higher porosities. As a result, the difference is not easily discernible from a side view. However, as depicted in Figure 4b,f, at lower porosities, the side view shows that the permeation geometry is between semicircular and cylindrical. The larger the porosity, the less resistance the particles exhibit to the binder. Gravity and inertia also have a more significant influence on the binder, resulting in a deeper penetration depth and vice versa. This trend aligns with the simulation result presented in Figure 3c and can be confirmed by the Washburn equation [43]. Ultimately, we present the penetration depth data results in Figure 5. Figure 5 shows all samples and analysis results in a statistical chart of P inf and D eff data for six different binder volumes in order of increasing porosity. The red line segments and sample points represent the simulation results, and the black line segments and sample points represent the experimental results. When porosity is low, the P inf presents a quadratic curve distribution and generally ranges between 500-1500 µm. However, as porosity increases, the overall diffusion depth increases significantly, and the trend line tends to close to a straight line. The result is that the binder cannot be retained between the particles. It infiltrates downwards, driven by inertia and gravity [37,38]. If porosity is low, the surface force between the particles increases, which increases the resistance of the particles to infiltration, resulting in a decrease in P inf (but with little effect on D eff ), thereby increasing the precision of the mold after molding. The results obtained by the simplified model proposed in this experiment are the same as the quadratic curve trend resulting from the experiments. If binder volume and porosity are used as the main variables, the P inf and D eff results are almost identical at lower porosities. However, when the porosity exceeds 35%, the P inf prediction results of the simplified model produce significant errors.

Discussion
The reason for this is that there are still many uncertainties in permeability processes under conditions of high porosity [40]. As binder jetting technology improves, studies and equipment are moving toward smaller particle sizes, which means that low porosity is becoming a trend in high-precision, high-strength binder jetting. Therefore, the simplified model proposed in this study still has a high reference value.
Regarding penetration depth, the curves of 2.0 and 2.5 µL demonstrate a gentle slope because of the extremely high initial absorption force in the dry sand state, which results in rapid penetration. However, as the dry sand is transformed into wet sand, the contact angle differs in the dry and wet states, as depicted in Table 5. Consequently, the capillary force and surface tension are influenced, leading to a slower penetration rate, and the binder and hardening agent begin to react, increasing the binder's viscosity. As a result, the penetration depths at 2.0 and 2.5 µL did not differ significantly. By observing the samples under the microscope, we found that while the penetration depth was not significantly different, more binder remained between the powders at 2.5 µL than at 2.0 µL [38]. The penetration rate of the binder in dry and wet sand is influenced by the contact angle. A larger contact angle results in stronger adhesion and faster binder penetration into the powder. Conversely, a smaller contact angle leads to slower penetration. Therefore, we can understand that when the sand is in a dry state, at the moment we introduce the binder, the powder bed begins to transition from dry to wet, causing the contact angle to decrease. Additionally, the internal voids between particles are gradually filled with the binder, resulting in minimal differences between the results of 2.0 and 2.5 µL binder volumes. Furthermore, this phenomenon can be discussed using the Washburn equation for capillary infiltration [43]. The infiltration velocity decreases exponentially as the state of the sand changes from dry to wet, implying that the penetration rate of the binder into the powder slows down. Subsequent binders cannot push out or push down the initially introduced binder, resulting in differences in the final penetration depth and diffusion diameter. However, our simplified model does not consider the variation in contact angle (transition between dry and wet sand) and assumes the powder bed remains in a dry state throughout the process. This is also why the simulation results in Figure 5 are consistently higher than the experimental results. To verify that this simplified model applies to current binder jetting technology, we conducted a series of tests, as shown in Figure 6. Figure 6a,b shows powders mixed with A-1 and A-2 at a ratio of 30:70 wt %, and the porosity becomes 7.2%. We obtained D eff and P inf with binder volumes of 0.1, 1.0, and 2.0 µL. The model results correlated with the experimental results. We then decreased the size of the droplets to replicate the piezoelectric nozzles used in existing machines. We measured the D eff and P inf of binder droplets printed by a piezoelectric nozzle (Fujifilm SG1024, Fujifilm, Valhalla, NY, USA). As the droplets were extremely small and difficult to measure and photograph, we only adjusted the binder volume to three separate volumes (16.2, 26.2, and 35.8 nL) by controlling the waveform [44] of the piezoelectric nozzle to verify the simplified model. (Figure 6c,d) shows A-1 samples with a porosity of 7.6% printed with these three binder volumes. Figure 6e,f shows A-2 samples with a porosity of 36.3%. The results indicate that although the simplified ANSYS Fluent model still correlates with the experimental results, the range of D eff and P inf obtained was still large due to the large porosity. More experimental verification is needed. However, compared with the simulation results, smaller binder volumes result in more accurate prediction by the model, and the trend is still consistent. This shows that when using small binder volumes, if porosity and binder volume are used for simulation, the permeability mechanism can be easily deduced, as shown in Figure 6.

Conclusions
This study proposes a simplified model that converts the binder from droplet-shaped to cylindrical-shaped through unit volume transformation. Using transient simulation

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 ∆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 µL) 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 µL, as the binder volume increases, both the penetration depth and diffusion diameter increase. However, the difference between the 2.0 and 2.5 µL 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.
The proposed simplified model aligns with the experimental results and exhibits the same trends; 5.
The model was applied to actual printing using a physical machine, and the results matched the printing outcomes with small errors;

6.
Using less binder and lower porosity sand will be a future trend for achieving higher precision in binder jetting.