## 3 Results and Discussion
### ES Technique  
Since the S technique ignores the fraction of mass transfer resistance due to fluid-to-solid film diffusion, the overall resistance is assumed to be equal to the intraparticle diffusion resistance in the method. As a result, the value of \(D_{s,sim}\) is smaller than that of \(D_{s,rig}\), as indicated in Table 1. Therefore, the ES technique is proposed in this study to minimize the estimation error. The relative ratio of diffusivities (RRD, \(D_{s,sim}/D_{s,rig}\)) between the S technique and R technique is evaluated under various conditions in this study. The results are presented in Figure 6 with the relative ratio of diffusivities (RRD) against the time ratio (TR, \((t_{0.2}/t_{0.8})_{exp}/(T_{0.2}/T_{0.8})_{sim}\)). As indicated in Figure 6, the RRD is a function of the time ratio (TR). For example, when the TR is larger than 0.6, the RRD is about 0.8, which means that there is less than a 20% difference between the \(D_{s,sim}\) and \(D_{s,rig}\). A biquadratic polynomial equation was obtained using data points, as shown in Figure 6. The correlation between the RRD and TR can be further defined in the following biquadratic polynomial equation:  
\[\text{RRD}=-3.7784\ ^{*}\ \text{TR}^{4}+11.392\ ^{*}\ \text{TR}^{3}-13.063\ ^{*}\ \text{TR}^{2}+7.2592\ ^{*}\ \text{TR}-0.8163 \tag{17}\]  
Since the RRD is the relative ratio of diffusivities between the S technique and R technique, it is used as a correction factor in the ES technique to obtain a much more accurate diffusivity value. The diffusivity value estimated by the ES technique uses Equation (17).  
\[D_{s,eff}=D_{s,sim}/\text{RRD}=D_{s,sim}/(D_{s,sim}/D_{s,rig})=D_{s,sim}/D_{s, sim}\ ^{*}\ D_{s,rig}=D_{s,rig} \tag{18}\]  
The procedures of the ES technique are shown in Figure 7. An additional two steps (Steps 6 and 7) are added in the ES technique (Figure 7) compared with the S technique (Figure 4). The time ratio (TR) \((t_{0.2}/t_{0.8})_{exp}/(T_{0.2}/T_{0.8})_{sim}\) is determined in Step 6. For  
Figure 5: Simulation results of simplified diffusion technique (-) and rigorous diffusion technique (\(\bullet\)) for Bi = 40, 10, and 4, respectively.  
example, \((t_{0.2}/t_{0.8})_{exp}/(T_{0.2}/T_{0.8})_{sim}=26.91/34.88=0.7713\). As seen in Figure 5, or as determined by using Equation (17), the RRD is about 0.906. As shown in Equation (17), \(D_{s.eff}=2.72\times 10^{-8}\,(\rm{cm}^{2}/\rm{s})/0.906=3.00\times 10^{-8}\,(\rm{cm}^{2}/ \rm{s})\).  
Three cases of different scenarios are presented in Table 1 with Biot numbers 40, 10, and 4, which correspond to negligible, medium, and significant fluid film resistance cases. As shown in Table 1, the relative estimated error of \(D_{s,sim}\) is significantly increased with a decreasing Biot number. For example, compared to the diffusivity (\(D_{s,sim}\)) results obtained using the R technique, the estimation errors obtained using the S technique are about 9.33, 36.67, and 64.00% for Biot numbers 40, 10, and 4, respectively. The S technique is only valid in cases when the contribution of fluid film resistance is negligible. In contrast, the estimated errors for those same Biot numbers are about 0.00, 3.33, and 3.33% for the ES technique, which suggests that the ES technique can be applied to a wide range of Biot numbers. In conclusion, the estimation errors of the ES technique are significantly smaller than those of the S technique.  
In addition, physical parameters such as the viscosity of the liquid might significantly contribute to the description of phenomena occurring in these systems. Since this study did not have exact information regarding the impact of the viscosity of the liquid, we used Biot numbers, which encompass these factors. Future research will be performed to investigate the impact of the viscosity of the liquid in the CMBR system.  
Figure 6: Correlation between relative ratio of diffusivities (RRD) and time ratio (TR).