## 3 Results
### Printing Resolution Effects
#### 3.4.2 Grain Size and Permeability  
In 3D printing, the element (particle, grain, or drop) size is normally constant for a single printing project. We assume that the 3 DP sample is formed by small elements layer by layer with a regular array bonded by adhesives or sintering. In that case, a mathematical model can theoretically predict the porosity and permeability of the printed samples. According to this model, the seepage channel is similar to a necklace of gems. The narrowest part is the middle part of the particles, and the widest part is the lower part of the particles. It is noted that the narrowest part controls the flow rate for a single tube, like a neck, as shown in Figure 9.  
Figure 8: Sketch of the actual flow path and the tortuosity (Black dots represent skeletons with different sizes).  
As depicted in Figure 10, the diameter of the equivalent tube can be expressed in two ways, depending on the array styles of the elements. The first arrangement yields an equivalent circular area that is equivalent to the four curved edge pores surrounded by four equal-sized particles, while the second arrangement yields an equivalent circular area that is equivalent to the three curved edge pores surrounded by three equal-sized particles. In the first arrangement, the equivalent circular area can be expressed as having a radius of \(r_{1}\), while in the second arrangement, the equivalent circular area can be expressed as having a radius of \(r_{2}\). The voids for these two arrangements are illustrated in Figure 9, which can be used to derive the theoretical porosity.  
Figure 10: Porosity of the medium with the two types of particle arrangements.  
Figure 9: (**a**) Flow channels and (**b**) the corresponding pores of two types of particle arrangements in 3D printing techniques.  
For the first type, we have:  
\[r_{1}=\sqrt{\frac{4-\pi}{\pi}}R \tag{9}\]  
\[\Phi_{1}=\frac{S_{total}-S_{particle}}{S_{total}}=\frac{2 nR\times 2 nR-n^{2} \times\pi R^{2}}{2 nR\times 2 nR}=\frac{4-\pi}{4} \tag{10}\]  
The corresponding permeability K\({}_{1}\) without considering the tortuosity is then determined, as illustrated by Equation (11).  
\[K_{1}=\Phi_{1}\frac{{r_{1}}^{2}}{8}=\frac{(4-\pi)^{2}}{32\pi}R^{2} \tag{11}\]  
For the second type, we have:  
\[r_{2}=\sqrt{\frac{2\sqrt{3}-\pi}{2\pi}}R \tag{12}\]  
\[\Phi_{2}=\frac{S_{total}-S_{particle}}{S_{total}}=\frac{2 nR\times\sqrt{3}nR-n^{ 2}\times\pi R^{2}}{2 nR\times\sqrt{3}nR}=\frac{2\sqrt{3}-\pi}{2\sqrt{3}} \tag{13}\]  
Also, the corresponding permeability without considering the tortuosity has the following form (Equation (14)):  
\[K_{2}=\Phi_{2}\frac{{r_{2}}^{2}}{8}=\frac{\left(2\sqrt{3}-\pi\right)^{2}}{32 \sqrt{3}\pi}R^{2} \tag{14}\]  
When considering the tortuosity, the actual flow paths for the two types of particle arrangements are illustrated in Figure 9. The tortuosities in these two cases are derived in the following forms:  
\[\tau_{1}=\frac{L_{e}}{L}=1 \tag{15}\]  
\[\tau_{2}=\frac{L_{e}}{L}=1.25 \tag{16}\]  
Therefore, the permeability coefficient calculation equation considering tortuosity has the following forms for the two types of particle arrangements:  
\[K_{1}=\Phi_{1}\frac{{r_{1}}^{2}}{8\tau_{1}}=\frac{(4-\pi)^{2}}{32\pi}R^{2} \tag{17}\]  
\[K_{2}=\Phi_{3}\frac{{r_{2}}^{2}}{8\tau_{2}}=\frac{\left(2\sqrt{3}-\pi\right)^{ 2}}{40\sqrt{3}\pi}R^{2} \tag{18}\]  
where the units of the permeability and the radius of the pore are cm\({}^{2}\) and cm, respectively.