# 3D Reconstruction of a Single Bubble in Transparent Media Using Three Orthographic Digital Images
## 5 Verification by Gas Injection Test  
To further verify the accuracy of the method in this paper, a gas injection test was conducted using the experimental apparatus in Section 2. During this experiment, the bubble images were taken by three cameras, the gas flow rate was set as 1 sccm, and the gas pressure was recorded by pressure sensor sequentially. As we can see in Figure 15, the pressure increased steadily until it met a peak, and then dropped ephemerally. After that, it maintains a stable horizontal level. The bubble appeared soon after the pressure reached peak and it grew during the period of pressure drop and the stable pressure level. Images at four different time nodes (_A-D_) were stitched together to visualize the bubble growth. The experimental bubble volumes were calculated by following steps based on [PERSON] (1873). The initial condition (point \(O_{0}\)) could be written as:  
\[\left(P_{0}+\frac{n_{0}^{2}a}{V_{pipe}^{2}}\right)\left(V_{pipe}-n_{0}b\right) =n_{0}RT \tag{18}\]  
where, \(P_{0}\) is the initial pipe pressure, and the \(n_{0}\) is initial mole numbers inside the pipe, \(V_{pipe}\) is the pipe volume. The Van der Waals parameters \(a=0.1408\) Pa \(\cdot\) m\({}^{6}\) \(\cdot\) mol\({}^{-2}\), \(b=3.912\times 10^{-5}\) m\({}^{5}\) \(\cdot\) mol\({}^{-1}\), and the  
Figure 14: Sample NO.1 used for 3 test cases, along with their projections. Figure 15: Variation of gas pressure during experiment. ideal gas constant \(R=8.31441\pm 0.00026\) J/(mol\(\cdot\)K). The thermodynamic temperature \(K\) inside the laboratory was 288 K. For a moment (point \(O\)) where the bubble had not generated:  
\[\left(P^{\prime}+\frac{\left(n_{0}+n^{\prime}\right)^{2}a}{V_{pipe}^{2}}\right) \left(V_{pipe}-\left(n_{0}+n^{\prime}\right)b\right)=\left(n_{0}+n^{\prime} \right)RT \tag{19}\]  
By combining Equations (18) and (19), both \(P_{0}\) and \(n_{0}\) were solved. For the period of pressure drop (point \(O_{1}\) to \(O_{2}\)) in Figure 15:  
\[\left(P_{1}+\frac{\left(n_{0}+n_{2}\right)^{2}a}{V_{pipe}^{2}}\right)\left(V_ {pipe}-\left(n_{0}+n_{2}\right)b\right)=\left(P_{2}+\frac{\left(n_{0}+n_{2} \right)^{2}a}{\left(V_{pipe}+V_{b2}\right)^{2}}\right)\left(V_{pipe}+V_{b2}- \left(n_{0}+n_{2}\right)b\right) \tag{20}\]  
where, \(P_{1}\) is the beginning pressure of pressure drop, and \(P_{2}\) is the ending pressure of pressure drop, \(V_{b2}\) is the calculated bubble volume at the point \(O_{2}\), \(n_{2}\) is the mole numbers at the point \(O_{2}\):  
\[n_{2}=ql_{2}/22.4 \tag{21}\]  
Then, the bubble volume (\(V_{b}\)) after \(O_{2}\) are:  
\[V_{b}=q(t_{x}-t_{2})+V_{b2} \tag{22}\]  
where, \(t_{x}\) is the experimental time. A series of experiment pictures (corresponding to points \(A\)--\(D\) respectively in Figure 15) were taken based on the setup, some of which were shown as example in Figure 16. To further testify the accuracy of the method proposed above, they were reconstructed in Figure 17. It can be seen that the bubbles developed vertically and the projections for horizontal direction were analogue. The bubble shape was close to cone at the beginning, and then developed to ellipsoid quickly. The tip of the cusp emerged as bubble volume increased, and eventually evolved into shape of inverted teardrop. Figure 14:0 Growth of N2 bubbles injected in transparent soil. Figure 14:1 The reconstructed bubble shapes from an experimental data using the method in this paper. The bubble volume measured by the methods in this paper, in [PERSON] et al. [21] and in [PERSON] et al. [20], are shown in Figure 14:2 The construction results of the method in this paper fits well with the theoretical results and the MRE of bubble volume later \(O_{2}\), which was 1.79%. Similarly, the construction results of the method in [PERSON] et al. [21] also fits the theoretical results perfectly, and the MRE of bubble volume later \(O_{2}\) was 1.74%. The construction results of the method in [PERSON] et al. [20] fits the theoretical results at the beginning; however, the error arose as time went by.