## 3 Results and Discussion
### Hydrolysis Kinetics
#### 3.4.3 Model Validation  
To validate the kinetic model, its predictive catalytic performance was tested with a group of experimental data. Figure 10 demonstrates that the theoretical calculation results were in very good accordance with the experimental results and the relative error was less than 3.0%. It is clear that all the experimental data fit reasonably well with the kinetic values under different inulin concentration. Meanwhile, as shown in Figure 10d, it can be observed that the average variance has a tendency to increase with the increase of reaction time under the higher inulin concentration. It is mainly because of the increased mass transfer resistance of diffusion, which is reflected in two aspects: (i) the transfer resistance of inulin from outside into pore, (ii) the release resistance of the hydrolyzed fructose from pore into solution. However, the corresponding average variances were only 7.7%. Therefore, we can conclude that the kinetic model can be used to predict the performance of the PEK-C-ILs catalyst.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Catalyst**} & \multicolumn{2}{c}{\(k\) (min\({}^{-1}\))} & \multicolumn{2}{c}{\(E_{a}\) (k/mol)} & \multirow{2}{*}{\(\ln A\)} & \multirow{2}{*}{\(R^{2}\)} \\ \cline{2-2} \cline{5-6}  & 55\({}^{\circ}\)C & & & & \\ \hline PEK-C-ILs & 0.0687 & 0.1694 & 83.21 & 27.80 & 0.9957 \\ VImalLs [13] & 0.0327 & 0.0781 & 83.68 & 27.25 & 0.9989 \\ Dilute sulfuric acid & 0.0173 & 0.0621 & 95.12 & 30.90 & 0.9802 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Comparison of hydrolysis kinetics parameters by PEK-C-ILs, VImalLs and dilute sulfuric acid.  
Figure 10: Reducing sugars yield curve as a function of process time. Experimental conditions: **(a)** Catalyst concentration 1.8%, temperature 61 \({}^{\circ}\)C, inulin concentration 15 (_w/w_)%; **(b)** Catalyst concentration 2.5%, temperature 64 \({}^{\circ}\)C, inulin concentration 20 (_w/w_)%; **(c)** Catalyst concentration 3.5%, temperature 69 \({}^{\circ}\)C, inulin concentration 25 (_w/w_)%; **(d)** Catalyst concentration 3%, temperature 74 \({}^{\circ}\)C, inulin concentration 30 (_w/w_)%.  
Response Surface Methodology (RSM) can be regarded as a collection of statistical and mathematical techniques useful for optimizing objective functions. Thus, in order to investigate the effects of two independent variables on the yield of reducing sugars, the RSM experimental design was carried out. The analysis results were shown in Figure 11. It could be seen that the relative rate of inulin hydrolysis is enhanced by increasing the temperature and catalyst concentration. Figure 11a shows the surface representation of inulin hydrolysis ratio at 4.0% PEK-C-ILs concentration over the temperature range from 300.15 K to 348.15 K and reaction time from 0 min to 80 min. And the relationship between measured inulin hydrolysis ratio, PEK-C-ILs concentration (1.0-5.0%) and reaction time (0-80 min) at temperature 338.15 K was shown in Figure 11b. Analyzing the reducing sugars yield response surface will provide the useful information about operational conditions and results and understand the property-condition relationship of PEK-C-ILs catalyst.