#### 3.1. Factors and Response Variable Selection

The thermoacoustic efficiency ${\eta}_{ta}$ entails a trade-off between usable acoustic power $\text{}{\dot{E}}_{4}$, and power that is spent in the system feedback $\text{}{\dot{E}}_{2}$. It is therefore a good indicator of the performance of the device and, in the present work, used to study the response to the variation of the studied factors. The impact of these factors on the response variable will be measured and analyzed, in order to maximize the thermoacoustic efficiency.

In this study, the seven parameters of

Table 3 have been selected to describe the configuration of TA-SLiCE. The range for each of the factors is delimited by the high and low levels. It is usually accepted [

21] that the range of analysis to study the influence of factors on the response is reasonably established to 25% of the extreme range. The extreme range introduces constructive restrictions in the model, since TA-SLiCE design values are considered to set its boundaries, according to experiments carried out in the lab.

If a full factorial design, with all the combinations of each factor in the two levels proposed, was to be carried out, 2^{7} = 128 experiments or simulations would be required. Therefore, a factorial design to reduce the high amount of simulations would be very beneficial.

#### 3.2. Dimensional Reduction (${2}_{IV}^{7-3}$ FFD)

In order to identify the number of significant factors, an FFD is made. From now on, for greater simplicity, letters will be assigned to the factors as shown in

Table 4. An RStudio script has been developed, which detects the significant factors, adjusting the linear model to the response values. The R code used also allows the visualization of the results by contour and Pareto plots.

This highly fractional factorial design

${2}_{\mathrm{IV}}^{7-3}$ needs to run 2

^{4} = 16 simulations to predict the effects of one factor in the levels of the other factors and to obtain conclusions that are valid in the range of −1 to +1. The code “+1” represents the highest level and “−1” the lowest level corresponding to each factor. This design of resolution IV provides solid and robust predictions, since the confusion of the main effects will only occur with interactions of three factors and above, of little significance [

22].

To obtain the confusion pattern proposed in

Table 4, 16 numerical simulations have been performed in DeltaEC. The first four columns contain a complete factorial design for four factors A, B, C, and D with all the combinations of each factor in orthogonal code. The next three columns correspond to the factors E, F, and G. They are generated by assigning values to those factors using the relations E = ABC, F = ABD, and G = ACD. The response variables of the last column

$y={\eta}_{ta}$ are calculated by combining the high and low levels of the factors in real notation during DeltaEC simulations.

The Pareto diagrams in

Figure 5, derived from the generated FFD, show the main effects, as well as the interaction effects and their direction (positive or negative impact).

All the Pareto plots in

Figure 5 visually indicate the significance of factors B and C; C has a negative effect in the response whereas B has a positive one. Factors A and E have little significance, and factors F, D, and G are negligible coefficients. Usually model terms are selected or rejected based on the significant probability value (p-value within 5% significance level) [

22]. Factors F, D, and G are not significant, and their variation can be ignored for the response because their p-values are greater than 0.05. In plots

Figure 5b,d factors F, D, and G are no longer considered. In

Figure 5b, the interactions of the factors are included, while in

Figure 5d only the main effects are visualized.

In order to make the model sufficiently precise and robust, it must include the interactions. Therefore, the relevance of factors A and E is set, based on the interpretation of the Pareto diagram

Figure 5b. Of all the possible and numerous interactions between two factors, diagram

Figure 5b identifies the BC, AC, and AB interactions, thus indicating the importance of these three interactions over all other. From the results it is clearly inferred that the BC and AC interactions are significant. This fact indicates that factor B modifies the effect of factor C, and, although to a lesser extent, factor A modifies the effect of factor C. This dependence implies that the factor A of the model cannot be discarded although the graphs show that it has narrow relevance. However, the E factor and the AB interaction can be definitely neglected in the model.

The impact of the parameters on the thermoacoustic efficiency of TA-SLiCE, in the range determined by the limits −1 and +1 for each factor, can be summarized in the following concluding remarks:

The compliance inner diameter (B = ${D}_{C}$) and inertance inner diameter (C = ${D}_{L}$) are critical parameters for thermoacoustic efficiency. The inertance inner diameter has a reducing effect on the thermoacoustic efficiency, while the compliance inner diameter has an increasing effect;

The inner diameter of inertance and compliance parameters have significant and positive interaction effects (BC);

The operating frequency (A = $f$) has a positive and relatively smaller impact than factors B and C on thermoacoustic efficiency;

The operating frequency and inertance inner diameter parameters have positive and mild interaction effects (AC);

The lengths of the inertance (D = ${L}_{L}$), the compliance (E = ${L}_{C}$), the thermal buffer tube (G = G = ${L}_{TBT}$), and the hot heat exchanger (F = ${L}_{HHX}$) have a negligible effect on thermoacoustic efficiency;

The operating frequency and compliance inner diameter parameters have negligible interaction effects (AB). Similarly, all the other interactions are negligible.

Contour plots measure the differences between the effects of one factor on the response, at different levels of another. To interpret the effects of interaction between two factors derived from the generated FFD, the contour plots, shown in

Figure 6 are used.

In

Figure 6, the trend and relative magnitude of the interaction effects on thermoacoustic efficiency are visualized. The contour lines of those plots show values of the response of the model according to the level (color scale), while in the ordinate and abscissa, the levels of the two-factor interaction are shown. The four points delimit the range at levels −1 and +1 for each factor in what is called a cube plot. Plot

Figure 6a shows contour lines practically parallel within the cube plot, which is interpreted as a weak interaction between factors A and B. If the lines lose the parallelism to a greater or lesser degree, the curvature of the contour lines is interpreted as evidence of interaction between the factors, so that the greater the curvature, the greater the degree of interaction. According to this reasoning, the contour plot

Figure 6b shows a medium interaction between factors A and C, and an interaction of factors B and C somewhat stronger in the plot

Figure 6c. This information indicates that factor B modifies the effect of factor C. Similarly, factor A modifies the effect of factor C. For the rest of interactions, where AB is included, no evidence of interaction has been found.

The plots displayed above are the upshot of the least squares model derived from the

${2}_{IV}^{7-3}$ FFD, in terms of the relevant input parameters for thermoacoustic efficiency:

The quality verification of the regression model in Equation (5) sets R^{2} = 0.9743, which means that the model can explain about 97.43% of the variance in the response. Therefore, the model is acceptable for testing of statistical significance.

In the next section, the effects of the three significant factors (A, B, C) and interactions (AC, BC) will be further evaluated by RSM. The factors and interactions that have insignificant effects on the response will thus not be considered in what follows. D, E, F, and G factors will be established at their midpoint value “0” and will not vary for further simulations.