## 2 Materials and Methods
### Design of Experiments  
The parameters for the continuous reaction systems (MX-V and T-V) were varied according to a full factorial design, which allows a subsequent regression analysis. According to a 2\({}^{3}\) experimental design, eight experimental points and three middle points were provided. The molar ethanol/oil ratio (x\({}_{1}\)), the volume flow rate (x\({}_{2}\)) and the amount of the KOH catalyst (x\({}_{3}\)) were chosen for the investigation, while the time on stream (4 min) and the temperature (50 \({}^{\circ}\)C) were kept constant. Based on previous works the initial parameter values were selected [8; 10]. The experimental plan for the full factorial design can be seen in Table 1.  
The volume flow rate in the system with the T-reactor was varied between 1.79 and 3.99 mL/min, while all other parameters were the same as with the MX-mixer (Table 1). The experimental plan was examined with respect to the adequacy of the model approach according to [PERSON] [11] and the significance of the investigated parameters. Based on the obtained experimental results, a linear regression equation, which shows the parameter impact on the reaction yield of FAEE, can be formulated:  
\[\mathrm{y}=\mathrm{f}(\mathrm{x}_{\mathrm{k}})=\mathrm{b}_{0}+\mathrm{b}_{1} \mathrm{x}_{1}+\mathrm{b}_{2}\mathrm{x}_{2}+\mathrm{b}_{3}\mathrm{x}_{3} \tag{4}\]  
Figure 1: Experimental setup for continuous flow experiments.  
Figure 2: Structure of mixers and dwell section: (**a**) MX-mixer; (**b**) T-mixer; (**c**) dwell section V.  
For better operation, the parameters \(x_{k}\) were transformed into dimensionless values \(\vartheta_{k}\) which assumed value +1 or -1 and formed the plan matrix of the experimental design. A normalized, linear regression approach can be set up from the plan matrix:  
\[\mathrm{y}=\mathrm{f}(\vartheta_{k})=\mathrm{a}_{0}+\mathrm{a}_{1}\cdot \vartheta_{1}+\mathrm{a}_{2}\cdot\vartheta_{2}+\mathrm{a}_{3}\cdot\vartheta_{3} \tag{5}\]  
The regression equation includes experimental errors as well as errors introduced due to the model choice and is to be confirmed by statistical tests. For this purpose, the model dispersion \(D_{m}\) was determined from the sum of squared residuals (SSR) and the degrees of freedom of the model \(FG_{1}\). The sum of squared errors (SSE) was calculated using the mean value of the experimental values in the middle point of the plan matrix. Taking into account the degrees of freedom of the experiment \(FG_{2}\), the experimental dispersion \(D_{E}\) was calculated from the SSE. A quotient of the model and experimental dispersion provides information about the adequacy of the regression approach [12]. It has to be smaller than the value of the F-distribution (\(F_{\alpha},FG_{1},FG_{2}\)) at a significance level of the test \(\alpha=5\%\).  
Based on the model adequacy, the dimensionless coefficients of the regression equation were considered for their significance. The dimensionless coefficients (\(\vartheta_{\mathrm{i}}\)ethanol-oil-ratio, \(\vartheta_{2}\)flow rate, \(\vartheta_{\mathrm{i}}\)mass percentage of catalyst) have to exceed a value \(\Delta\mathrm{a}_{k}=t_{\alpha,\ FG}\sqrt{s_{a}^{2}}\). with \(t_{\alpha,\ FG}\)--value for the Student's t-distribution and \(s_{a}^{2}\)--dispersion of model coefficients. The significance levels \(\Delta\mathrm{a}_{k}=6.3\) for the system with T-mixer and 3.3 for the system with the MX-mixer.  
The new experimental parameters for the optimization were determined via dimensionless, significant coefficients of the regression equation. According to the Box-Wilson method, the objective function is optimized in the direction of the highest increase [12]. For this purpose, a new search step size was defined for the parameter with the largest coefficient of the regression equation. The new step size was the double starting step size for the most significant parameters of the experimental plan. The new values for the parameters with significant coefficients were calculated and optimal values were applied to the experiments.