## 4 Conclusions  
This study analyzed the coefficient of performance (_COP_) of a thermoacoustic device based on five dimensionless parameters, using linear thermoacoustic theory. Employing the design of experiments (DOE) and statistical tools like ANOVA enabled a thorough evaluation of the contributions and interactions of these parameters, which is crucial for optimizing device performance. The key findings were as follows:  
* A _COP_ of 1.76 was achieved for \(X_{s}\) = 0.05 m, \(\Delta T_{m}\) = 5 K, \(B\) = 0.25, \(L_{s}\) = 0.05 m, and \(D\) = 1.
* We note the importance of selecting the stack position \(x_{n}\) to enhance the _COP_, as positioning it too far from the pressure antinode reduced efficiency. Interactions between parameters were also significant. For instance, Factor D influenced how Factor B affected the _COP_, emphasizing the need to consider these interactions in the design.
* The different effects of Factor B on the _COP_ at varying levels of Factor D were due to their interaction. Factor D modified the sensitivity of the system to changes in Factor B, leading to different performance outcomes, depending on the level of Factor D. This highlights the importance of considering interactions between factors in the design and optimization of the device.
* The full factorial design allows for a systematic variation of all factors at all levels, providing comprehensive quantitative and qualitative insights into the main parameters and their interactions, it facilitates easy interpretation, due to the clear separation of effects, and it enables a systematic exploration of the parameter space, which may allow engineers to study the effects of each factor on performance metrics like the _COP_, acoustic power, and cooling capacity.
* Regarding optimization performance, the DOE not only helps to find the optimal parameter settings that maximize or minimize a desired response, such as maximizing the _COP_, but it also allows for the fine-tuning of the thermoacoustic engine design, leading to better performance and efficiency.
* Using the insights gained from the DOE, the design process can not only be streamlined and a more efficient prototype developed with a higher probability of meeting the desired performance targets, but it also helps in reducing the number of prototype iterations needed, saving time and resources.
* Statistical analyses, such as analysis of variance (ANOVA), can be used to determine the significance and sensibility of each factor and its interactions on the _COP_ responses. This is useful in identifying robust designs that are less sensitive to variations in manufacturing or operating conditions.  
A systematic exploration of parameter space will be proposed, to analyze the interval where the _COP_ response value is 2.5. Finally, the performance relative to Carnot COPR vs the _COP_ values obtained in Table 6 will be analyzed in future work, to find out the parameter differences, applying the proposed DOE methodology.  
Data curation, C.A.E.V.; formal analysis, H.P.F. and C.A.E.V.; investigation, H.P.F. and C.A.E.V.; methodology, H.P.F.; project administration, H.P.F. and C.A.E.V.; resources, H.P.F. and C.A.E.V.; software, H.P.F. and C.A.E.V.; supervision, C.A.E.V.; validation, H.P.F.; visualization, H.P.F. and C.A.E.V.; writing--original draft, H.P.F.; writing--review and editing, C.A.E.V. All authors have read and agreed to the published version of the manuscript.  
This work was supported by CONAHCYT project 322615 (LaNITeF).  
Not applicable.  
The raw data supporting the conclusions of this article will be made available by the authors on request.  
**Acknowledgments:** The authors would like to thank the \"Laboratorio Nacional de Investigacion en Tecnologias del Frio--LaNITef\". [PERSON]. [PERSON] would like to thank the CONAHCYT's \"Investigadoras e investigadores por Mexico\" program.  
**Conflicts of Interest:** The authors declare no conflicts of interest.