## 4 Results and Discussion
### Results for EIS
#### 4.1.3 Creating Transfer Function Models from EIS Data and Comparative Analysis

Table 2: Transfer function models tested. Figure 7: Simulink program used to convert from rectangular coordinates to complex coordinates. The output data was used to create an frd model object. The second model (Model 2) tested was a second-order system as given by Equation (15). It has no zeroes and two poles. \[\frac{Y(s)}{X(s)}=\frac{A_{0}}{B_{2}s^{2}+B_{1}s+B_{0}} \tag{15}\]  
The third model (Model 3) tested was with two poles and one zero in the form of Equation (16). \[\frac{Y(s)}{X(s)}=\frac{A_{1}s+A_{0}}{B_{2}s^{2}+B_{1}s+B_{0}} \tag{16}\]  
Table 3 shows the percentage fit to the data estimation (tfest fit %) of the transfer function fitted using Model 1, Model 2, and Model 3 for different working conditions, i.e., for different concentrations of sucrose solutions, and different effective ethanol to water ratios. The fit per cent is a quantitative assessment of the estimation of normalized root mean squared Error (NRMSE) measure of how well the response of the model fits the estimation data, expressed as the percentage fit = 100(1 - NRMSE). The fit percentage for Model 1 and Model 2 was higher at 55% _w/w_ and 66.67% _w/w_ solutions compared to the solutions with lower concentrations of 33% _w/w_ and 44% _w/w_. However, the goodness of fit does not increase monotonically from Model 1 to Model 2. This issue was resolved in Model 3 by adding one zero to the second-order transfer function, which results in a good agreement between the experimental data and the response of the model with a minimum of 90.9 % fit for all the solutions. Figure 8a-d shows the bode plot comparison of the fit of Model 1, Model 2, and Model 3 transfer functions to the acquired experimental data for the solution with sucrose concentration 33 % w/w at various ethanol to water ratios. The highest fit percentage of the Model 3 can be observed graphically using bode representation. Figure 9a,b shows the change in model parameters: steady-state gain (K) and time constant (\(\tau\)), for various sucrose concentrations at increasing ethanol to water ratios evaluated using the first-order transfer function. It confirms that the considered system is nonlinear. However, for a constant working condition, a simplified linear model in the form of the transfer function can be applied. An increase in the gain constant K and a decrease in system time constant \(\tau\) was observed as the ethanol to water ratio increased (Table 3), Model 1 could be used as a simple prediction tool with a limited accuracy. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{ \begin{tabular}{c} **Solution** \\ **Index** \\ \end{tabular}  } & \multirow{2}{*}{ \begin{tabular}{c} **Effective Ethanol** \\ **to Water Ratio** \\ \end{tabular}  } & \multirow{2}{*}{\ \begin{tabular}{c} **Different Concentrations of** \\ **Sucrose Solutions (\% _w/w_)** \\ \end{tabular}  } & \multirow{2}{*}{ \begin{tabular}{c} **Effective Sucrose/Ethanol** \\ **Ratio in 100 mL Solution** \\ \end{tabular}  } & \multicolumn{2}{c}{**Trest Fit \%**} \\ \cline{5-7}  & & & 33 & - & 55.2 & 67.8 & 96.43 \\
2 & & 44 & - & 55.56 & 68.48 & 95.84 \\
3 & & 55 & - & 62.31 & 76.26 & 93.42 \\
4 & & 66.67 & - & 88.98 & 80.1  
Figure 8: _Cont._  
Figure 10a-d shows the trend for the four coefficients A\({}_{1}\), A\({}_{0}\), B\({}_{1}\), and B\({}_{0}\) of the Model 3, and B\({}_{2}\) = 1. The increase in A\({}_{1}\) was observed as the ethanol to water ratio is increased, whereas a decrease was observed in the coefficients A\({}_{0}\), B\({}_{0}\), and B\({}_{1}\). The decrease in coefficients was greater for the lower concentration of ternary solutions of 33% _w/w_ and 44% _w/w_ as compared to the solutions with higher concentrations. The analysis of the model parameters shows a clear dependence between the concentration of the sucrose constituent component with EIS measurements. Accordingly, at a given sucrose concentration, the model parameters can be utilised to correspond to specific ethanol to water ratio present in the crystallizer.