# On the Svelteness as an Engineering Tool in Constructal Design: A Critical Review
## 2 Svelteness as a Design Criterion  
In fluid flow, Svelteness has been used to show how the architecture of bifurcations impacts and relates to the pressure drop, showing how distributed losses dominate over local losses when Sv surpasses a certain threshold [1; 10]. Although not explicitly declared, these results assume a value for the Svelteness and neglect local pressure losses over distributed losses in fluid flows. Higher values of Svelteness usually translate to better usage of the available space for minimizing the flow resistance while producing more complex flow architectures. However, complexity is a byproduct of Constructal Design, and its maximization is not a goal. In this sense, two main criteria are used in the literature for assessing Svelteness. [PERSON] and [PERSON] [1] introduced the first criterion using a sudden expansion in a pipe to demonstrate the effect of Sv on the relevance of local pressure losses relative to distributed friction losses (see Figure 1a). The interpretation of the criterion, widely used in the literature, considers that Sv \(>10\) is enough to neglect local losses. This means the external length scale should be, at least, one order of magnitude larger than the internal length scale to validate this consideration: \(\mathcal{O}(\mathcal{L}_{e})>10\ \mathcal{O}(\mathcal{L}_{i})\). Although the criterion is bound to a precise geometry simplification, it is common to consider it a simplification with no correlation with the geometry of the scenario in which it was defined. However, the criterion defined by [PERSON] and [PERSON] [1] is not strictly Sv \(>10\), independently of the configuration, but a result of analyzing a sudden expansion, and using this value as universal can lead to a misuse of the criterion. Mistaking a value order by an order of magnitude can undermine the universal validity of such a criterion and should require further attention when used in Constructal Design, as shown in the following example. In another approach, [PERSON] et al. [21] investigated the effect of local losses on the optimized architecture of a T-shaped flow from a central point to a circular area of radius \(R\). What are the conditions, and how do local losses affect the optimized geometry? The authors' approach to Svelteness (Sv) originates in Slenderness (Sl), an important factor for the friction effects and proportional to local losses. The Slenderness of a channel is the ratio between its length and diameter, \(\text{Sl}=L_{i}/D_{i}\). Through normalization of both the length (\(\tilde{L}_{i}=L_{i}/R\), with \(R\) as the external length scale) and diameter (\(\tilde{D}_{i}=D_{i}/V^{1/3}\), with \(V^{1/3}\) as the internal length scale), [PERSON] et al. [21] considered Svelteness as:  
\[\text{Sv}=\text{Sl}\,\frac{\tilde{D}_{i}}{\tilde{L}_{i}} \tag{2}\]  
Analyzing the T-shaped flow in laminar and turbulent conditions, with smooth and rough ducts for turbulent flow, [PERSON] et al. [21] showed that the parametric domain that neglects junction losses is governed by Sv, and for domains where \(\text{Sv}^{2}<10\), junction losses have an essential effect on the optimized geometry. By analogy, the \(\text{Sv}^{2}<10\) criterion extends the initial version (Sv \(>10\)) to \(\text{Sv}>\sqrt{10}=3.16\), implying \(\mathcal{O}(\text{Sv})>1\), instead of 10. Still, the criterion does not consider its validity for a specific design. Both criteria (Sv \(>10\) or \(\mathcal{O}(\text{Sv})>1\)) originate from the design intention that \(\Delta p_{local}\ll\Delta p_{distr}\), but one should assess its impact on the flow configuration. A reasonable threshold to assume local losses are irrelevant compared to distributed losses is to consider the former at least one order of magnitude smaller: \(\Delta p_{local}/\Delta p_{distr}<0.1\). The conservation of mass and momentum applied to the flow architecture allows relating the ratio between local and distributed head losses with Sv.