**4. Conclusions**

**Figure 6** shows the results for this limit considering the properties reported

The helical coil illustrated in **Figure 4** is an alternative to the vertical pipe, and a geometry where the freedom to morph is larger because of the ability to change the helix diameter, number of turns and pitch angle. Alailami et al. [16] explored the morphing ability of the system in its design stage to optimize the storage of energy analyzing two scales. The timescale of heat penetrating the storage material from

temperature difference, *T t*ðÞ� *Ttf* , scaled by the initial condition, *T*ð Þ� 0 *Ttf* , with *Ttf* as the temperature of the thermal fluid. When Alailami et al. [16] simulated the

with the scaled time – *<sup>t</sup>* <sup>∗</sup> <sup>¼</sup> *<sup>t</sup>=τ<sup>c</sup>* – meaning the average temperature of the energy

experiment of this configuration to store energy in a PCM. The authors used unoptimized values for the helical coil (*ζ*≈0*:*7, *ε*≈0*:*2), from the Alailami et al. [16] point of view. However, the PCM configuration showed promising results storing 20% more energy than its equivalent mass in water. Also, the charging process in the PCM facility was slower and the authors attribute this result to the unoptimized

heat exchanger design, justifying the need of more research on this topic.

*Variation of limit timescale for the energy storaged in a PCM through annular fins distributed around a vertical*

Afterward, focusing the analysis on *<sup>t</sup>* <sup>∗</sup> <sup>¼</sup> <sup>0</sup>*:*3, and varying the helical coil diameter – *Dh* ¼ *ζD* – and pitch height – *Hh* ¼ *εH* – obtained as function of the cylinder diameter (*D*) and height (*H*), the authors reached an optimum diameter and pitch length of the helical coil, corresponding to *ζ* ¼ 0*:*6, and *ε* ¼ 0*:*3, respectively. Without using the work of Alailami et al. [16], Joseph et al. [21] performed an

*avg* <sup>¼</sup> *Tavg <sup>t</sup>* <sup>∗</sup> ð Þ�*Ttf*

*=α*; and the

*<sup>T</sup>*ð Þ� <sup>0</sup> *Ttf* – its value decreased

the boundaries of the helical coil to the cylinder diameter, *<sup>τ</sup><sup>c</sup>* <sup>¼</sup> *<sup>D</sup>*<sup>2</sup>

storage material approaches the temperature of the thermal fluid.

evolution of the scaled average temperature – *T* <sup>∗</sup>

*Heat Transfer - Design, Experimentation and Applications*

The constructal design analysis of this heat exchanger indicates diminishing returns of less than 10% for a number of fins above *n* >18, which is coherent with the numerical results presented by Ogoh and Groulx [19]. Applying the timescale defined by Eq. (30) to the experimental conditions in the work of Kamkari and Shokouhmand [20], which explore the effect of no fins with the cases of 1 and 3 fins, for the later case, *τmax* is roughly 1.3� the value measured for the total melting process. In the case of the experiments with 1 fin, the results for *τmax* are significantly larger, evidencing the role of natural convection in delaying the heat transfer

in [19].

to the PCM.

**Figure 6.**

**258**

*pipe inside a cylindrical enclosure.*

Thermal energy storage is one of the preeminent options to face the energy challenges of this century, providing a high energy saving potential and effective utilization. However, in these systems, the architecture of the heat exchangers through which energy flows, during charge and discharge, is of paramount importance. While most approaches optimize heat exchanger designs, the one presented in this chapter, based on constructal theory, follows an evolutionary design, meaning that the configuration explored at the design stage is dynamic and free to morph. It is not pre-defined, rigid, or still, but considers how it should evolve toward the greater access of the energy currents that flow through it.

Thermal energy storage systems follow two thermodynamic processes using the sensible heat of the energy storage material, or, besides the sensible heat, also the latent heat, as in Phase-Change Material (PCM). After introducing the general considerations on these systems, this chapter presents two design tools in constructal theory: the Svelteness, as a global property of any flow system, which tends to increase and evolve toward vascularization; and the scale analysis, as an expedite problem solving tool that allows obtaining relevant information of the several energetic processes involved.

Using the design tools presented, this chapter reviews and further explores the constructal theory approach in the development of heat exchangers for sensible and latent thermal energy storage configurations. The analysis evidences the explanatory potential of the constructal approach, increasing the sensibility of the engineer to the advantages of including the freedom to morph at the design stage of heat exchangers in thermal energy storage.

#### **Acknowledgements**

The author would like to acknowledge project UIDB/50022/2020 and UIDP/ 50022/2020 of ADAI for the financial support for this publication.
