## 4 Conclusions  
The ES technique using a completely mixed batch reactor (CMBR) is proposed in this study for the determination of diffusivity. This technique overcomes the main limitation of the S technique, which is only valid when the contribution of fluid film resistance is negligible. The relative ratio of diffusivity (RRD) is introduced as a correction factor in the ES technique to reduce the estimation error of diffusivity. The ES technique can be used under a wide range of conditions, including under conditions when both intra-particle and fluid film resistances are significant. In conclusion, the proposed ES technique would be useful for field applications to determine the surface diffusivity for an aqueous adsorption system using the completely mixed batch reactor (CMBR) approach.  
Figure 7: Procedures for efficient simplified diffusion technique.  
**Author Contributions:** Conceptualization, Y.K. and Y.S.; methodology, H.-J.F., E.F. and N.S.; software, E.F.; validation, Y.K. and Y.S.; investigation, N.S. and Y.K.; data curation, Y.K.; writing--original draft preparation, Y.K., H.-J.F., Y.S. and E.F.; writing--review and editing, H.-J.F. and E.F.; supervision, H.-J.F. and E.F. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This research received no external funding.  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** The data presented in this study are available on request from the corresponding author. The data are not publicly available due to privacy.  
**Conflicts of Interest:** Author [PERSON] was employed by the company Techno Media Lab, Inc. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.  
**Nomenclature**  
Bi Biot number, Bi = \(\left(\frac{k_{F}p_{T}}{\zeta_{S}p_{S}}\right)\)  
\(q_{P}\) Surface area based on solid particle [cm\({}^{2}\)/cm\({}^{3}\)]  
\(c_{0}\) Concentration within vessel at time, 0 [mmol/cm\({}^{3}\)]  
\(c_{e}\) Equilibrium concentration at time, infinity [mmol/cm\({}^{3}\)]  
\(c_{s}\) Concentration at interface between solid and fluid-to-solid film [mmol/cm\({}^{3}\)]  
\(C_{s}\) Dimensions \(c_{s}\), \(C_{S}=\left(\frac{c_{s}}{\epsilon_{0}}\right)\)  
\(c_{t}\) Concentration within vessel at time, \(t\) [mmol/cm\({}^{3}\)]  
\(C_{t}\) Dimensions \(c_{t}\), \(C_{t}=\left(\frac{c_{t}}{\epsilon_{0}}\right)\)  
\(D_{s}\) Effective surface diffusivity (based on solid concentration difference) [cm\({}^{2}\)/s]  
\(D_{s,trig}\) Effective surface diffusivity estimated using R technique [cm\({}^{2}\)/s]  
\(D_{s,sim}\) Effective surface diffusivity estimated using S technique [cm\({}^{2}\)/s]  
\(D_{s,eff}\) Effective surface diffusivity estimated using ES technique [cm\({}^{2}\)/s]  
\(k\) Freundlich constant [(mmol/g) (cm\({}^{3}\)/mmol)n]  
\(k_{F}\) liquid film mass transfer coefficient [cm\({}^{2}\)/s]  
\(m\) Weight of adsorbent [g]  
\(1/n\) Freundlich exponent [-]  
\(q_{0}\) Amount adsorbed in equilibrium with fluid concentration, \(c_{0}\) [mmol/g]  
\(q_{int}\) Initial amount adsorbed at \(t=0\) [mmol/g]  
\(q_{m}\) Amount adsorbed at \(r=r\) [mmol/g]  
\(Q_{m}\) Dimensionless \(q_{m}\), \(Q_{m}=\left(\frac{q_{m}}{q_{0}}\right)\)  
\(q_{t}\) Average amount adsorbed within a solid particle at time, \(t\) [mmol/g]  
\(Q_{t}\) Dimensionless \(q_{t}\), \(Q_{t}=\left(\frac{q_{t}}{q_{0}}\right)\)  
\(r\) Internal radial length (length from solid center) [cm]  
\(r_{p}\) Particle radius [cm]  
\(R\) Dimensionless internal radial length, \(R=\left(\frac{r}{r_{p}}\right)\)  
\(t\) Time [s]  
\(T\) Dimensionless time, \(T=\left(\frac{D_{s}}{r_{p}^{2}}\right)t\)  
\(V\) Volume of vessel [cm\({}^{3}\)]