Department of Physics, School of Science and Technology, University of Evora, Evora, Portugal
Tree flow networks are ubiquitous in nature and abound in engineered systems. A parent tube branching into two daughter tubes is the main building block of these networks. These branched tubes should be designed to provide easier access to flow under different size constraints. Optimal tree networks follow a homothetic scaling where the sizes of tubes have the same ratios between successive generations. In this study, different approaches aiming at optimal design of bifurcating tubes are presented and compared. The cross-sectional area of the tubes is obtained using two methods, based on Lagrange multipliers with a size constraint to respect, and including the size limitations directly into the function to optimize via chain rule. The optimal length of the tubes is obtained based both on the equipartition of forces/resistances and on the equal thermodynamic distance. These methods can be understood as a way of connecting entropy generation and the size of branching tubes. This study shows that applying the Lagrange Multiplier Method and applying the chain rule with constraint provides the same result. A similar result is obtained when the equipartition of forces/resistances and equal thermodynamic distance design methods are applied. These results are valid for different size constraints. In summary, our paper provides a comprehensive comparison of the different methods for a better choice, and is intended to provide insights into tree networks of tubes of any shape under different size constraints, for design and analysis.