# On the Svelteness as an Engineering Tool in Constructal Design: A Critical Review
## 4 The Role of Length Scales  
As previously defined, Sv is the ratio of internal and external length scales (Equation (1)). Although the flow volume is well established as the ground for the internal length scale, there is still some divergence related to the external length scale. The common use of Sv focuses on two definitions for the external length scale: \(L_{c}\), the characteristic length occupied by an entire branch (Equation (9a)); and \(A^{1/2}\), based on the total area containing the flow configuration (Equation (9b)). What are the implications of considering one versus the other, and what are their limitations? \[\text{Sv}_{L} =\frac{L_{c}}{V^{1/3}} \tag{9a}\] \[\text{Sv}_{A} =\frac{A^{1/2}}{V^{1/3}} \tag{9b}\]  
Consider two different architectures, serpentine (Figure 6a) and canopy-to-canopy (Figure 6b), where both layouts result in a set of parallel channels in a Z-type arrangement. The serpentine layout results from equally spaced U-turn channels, creating a set of horizontal parallel segments from bending a single tube of fixed diameter. The canopy-to-canopy layout consists of two horizontal manifolds interconnected with vertical parallel channels. [PERSON] et al. [26] designed these two configurations in a set of different aspect ratios (\(\xi=W/L\)) with a fixed area and volume, \(A=1.35\) m\({}^{2}\) and \(V=5.64\times 10^{-4}\) m\({}^{3}\). The authors considered two types of canopy-to-canopy arrangements. One of them kept the tube diameter ratio (\(\phi=D_{1}/D_{2}\)) constant with \(\phi=1\) (\(D_{1}=D_{2}\)), and the other varied \(\phi\) (\(D_{1}\
eq D_{2}\)); however, the focus will be on the latter. Figure 5: Thermal energy storage system with phase-change material inside a vertical pipe (**A**) or with the thermal fluid inside the vertical pipe (**B**). [PERSON] et al. [26] explored the application of the configurations on suspended radiant ceiling panels for cooling purposes. The authors assessed the performance by comparing the panel's cooling capacity (\(q_{cool}\)\([W]\)) to the pumping power required (\(\dot{W}\)). Cooling capacity depends on the panel's temperature distribution. Therefore, the best configuration should produce the most homogeneous temperature distribution. On the other hand, pumping power is affected by pressure losses. However, since the authors disregarded local losses compared to distributed losses, keeping the flow total length path small is advantageous. According to [PERSON] et al. [26], Figure 7 shows that parallel channels outperform serpentine one, with an optimum configuration for the aspect ratio of \(\xi=1.05\) with different diameters because this is the aspect ratio with the lowest flow resistance, as demonstrated in Appendix A. By analyzing Sv in these configurations according to Equations (9a) and (9b), we obtain a constant Sv for any case that uses the total area \(A\), and this makes any analysis of Sv pointless. The definition in Equation (9a) seems the correct option, but defining \(\mathcal{L}_{e}\) is not straightforward. Furthermore, by restricting both the area and volume, we limit the freedom to morph the system to increase its complexity. In that setting, any configuration with a given volume in a given area will have the same Sv, thus eliminating any insight relating the evolution of flow architecture with the system's performance. Figure 6: (**a**) Serpentine and (**b**) canopy-to-canopy configurations investigated in [PERSON] et al. [26]. Figure 7: Cooling-capacity-to-pumping-power ratio as a function of the aspect ratio. Data retrieved from [26]. [PERSON] et al. [26] introduced a different setting for the external length scale, as \(L_{p}\times\tilde{\xi}\), i.e., the product between the flow path of the system and its aspect ratio. The flow path is the length from the inlet to the outlet, therefore directly related to the pressure drop. \[\text{Sv}_{M}=\frac{L_{p}\ \tilde{\xi}}{V^{1/3}} \tag{10}\]  
For the serpentine configuration, if the number of tubes is \(n=W/S-1\), from the lowest aspect ratio investigated by [PERSON] et al.