# On the Svelteness as an Engineering Tool in Constructal Design: A Critical Review
## 3 Svelteness as a Tool  
Following the approach of [PERSON] and [PERSON] [1], [PERSON] and [PERSON] [23] assessed the role of Sv in the configuration depicted in Figure 2 of a central inlet channel and several outlets aimed at improving the cooling of a micro-mold insert. Figure 3 depicts the direction of flow evolution towards greater performance. In this work, the configuration designated as _umbrella_ had the following relation between the pressure loss ratio and a functional relation depending on Sv and Re:  
Figure 1: Geometry and related Sv function of the criteria to neglect or consider local pressure losses. \[\frac{\Delta p_{local}}{\Delta p_{distr}}=\frac{\Lambda(n,\alpha)}{64\ \psi(n,\alpha)}\bigg{(}\frac{\text{Re}}{\text{Sv}^{3/2}}\bigg{)} \tag{3}\]  
where \(\Lambda\) and \(\psi\) are functions of the number of return channels (\(n\)) and the angle between entry and return channels (\(\alpha\)). \[\Lambda(n,\alpha)=\frac{1+n^{-2/3}-2\cos(\alpha)n^{-1/3}}{1+n^{1/3}\sec( \alpha)} \tag{4}\]  
\[\psi(n,\alpha)=\sqrt{\frac{\frac{\pi}{4}\big{(}1+n^{1/3}\sec(\alpha)\big{)}}{ \bigg{[}\tan(\alpha)\Big{[}n\Big{(}\sin(\big{(}\frac{\pi}{n}\big{)}+\frac{\cot( \alpha)}{2}\Big{)}\Big{]}^{1/2}\bigg{]}^{3}}} \tag{5}\]  
Applying the criterion considering local pressure losses, at most, 10% of distributed pressure losses due to friction, \(\Delta p_{local}/\Delta p_{distr}<0.1\) Equation (3), allows establishing a minimum Sv value that complies with the pressure loss criterion as:  
\[\text{Sv}_{min}=\frac{1}{16}\bigg{(}10\ \frac{\Lambda(n,\alpha)}{\psi(n,\alpha)} \text{Re}\bigg{)}^{2/3} \tag{6}\]  
Another example of such a relation is the study of the impact of Svelteness, the Reynolds number, and the bifurcation angle in the pressure drop of micro-channel tree branches by [PERSON] et al. [10]. The authors highlighted that flow and thermal distribution uniformity increase with Sv. [PERSON] et al. [24] showed that an appropriate configuration of a solar chimney would allow scaling the design. In this case, they introduced Svelteness to explore the effect of the solar chimney design on the significance of local losses and defined it as  
\[\text{Sv}=\frac{R+H}{(\pi R^{2}h+\pi D^{2}H/4)^{1/3}} \tag{7}\]  
where \(H\) and \(D\) are the tower height and diameter and \(R\) and \(h\) are the roof radius and height (see Figure 4). The external length scale is the distance covered by the flow. In contrast, the internal length scale is the internal flow space of the system. In an analysis varying \(H\) and  
Figure 3: Evolution of the _umbrella_ geometry with growing divergence angle. Figure 2: Schematic example of the _umbrella_ geometry. \(R\) from a reference solar chimney and comparing local pressure losses with friction losses induced by the acceleration of airflow inside the chimney, [PERSON] et al. [24] concluded that local losses are negligible when \(\mathrm{Sv}>6\) and that Sv decreases for larger roof radius \(R\). Since Sv guides the designer in the evolutionary direction of the flow configuration, comparing Sveltenesses allows for a helpful interpretation of the technological evolutionary direction of configurations. [PERSON] [25] illustrated this scenario by comparing a design with two different options for storing energy with a phase-change material in a storage tank with a vertical pipe (Figure 5). The configurations alternate the location of the phase-change material and the thermal fluid, changing the direction of energy flow. Comparing the Sv of each case leads to the following relation:  
\[\mathrm{Sv}^{A} =\frac{H}{\left(\frac{\pi}{4}d^{2}H\right)^{1/3}} \tag{8a}\] \[\mathrm{Sv}^{B} =\frac{H}{\left[\frac{\pi}{4}(D^{2}-d^{2})H\right]^{1/3}}\] (8b) \[\frac{\Delta p_{local}}{\Delta p_{distr}}=\frac{\Lambda(n,\alpha)}{64\ \psi(n,\alpha)}\bigg{(}\frac{\text{Re}}{\text{Sv}^{3/2}}\bigg{)} \tag{3}\]0  
where \(D\) is the storage tank diameter, \(d\) is the vertical pipe diameter, and \(H\) is the storage tank length.