Shape from Function: the exergy cost of viscous flow in bifurcated diabatic tubes

Abstract Bifurcated flows are ubiquitous in nature. Their being such a common feature of many natural “structures” has prompted a multitude of investigations in diverse scientific branches: botanists, neurophysicians, biologists and chemists have attempted to find a “structural plan” on which to build a general model of the formation and evolution of bifurcations. Not incidentally, the engineering side of the issue is also extremely interesting: from heat exchangers to pipelines to district heating networks, the existence of a “general geometric model” would much facilitate a designer’s life. The study reported in this paper is based on a straightforward application of the exergy cost theory to the development and self-sustenance of natural bifurcated structures and leads to the conclusion that the shape, connectivity and evolution of a dycotomic depend on several factors that are irreducibly case dependent. This result is of great importance for engineered bifurcations, for which the same impossibility to generalize is again demonstrated: here though, since the “design goals” are formulated as rather simple constraints (simpler than in nature!), a somewhat larger degree of (albeit always application-dependent) generality is found. The exergy cost method is valid for any virtual or real system, and assigns a “resource cost” to its products: it consists in evaluating the exergy inflows (in W) and in keeping an accurate bookkeeping of the embodied exergy (in W/kg or W/m3) into the system, to calculate an average (instantaneous or lifetime-based) exergy input. The cost is then obtained by dividing this “cumulative input” by the exergy flux of the “products” in the same time window. In natural processes, this cost is called the Exergy Footprint, because it represents the actual primary resource consumption necessary to generate the outputs. In engineered artifacts, an additional procedure can be used to internalize the externalities (Labour, Capital and Environmental Remediation cost) so that the total equivalent primary exergy consumption needed to generate a unit of “product” can be again applied as a cost indicator. The novelty of the method and of the results discussed in this paper is twofold: first, the two most popular bifurcation models (Fractal and Constructal) are critically re-evaluated to show that neither one succeeds in generating credible predictive correlations. Second, it is demonstrated that the exergy costing paradigm provides a feasible and rigorous method for identifying the optimal bifurcation geometry for practical engineering applications. To express the results in a concise sentence: it is indeed possible to accurately and rigorously predict the optimal shape of a bifurcated structure once its function is known, but at the loss of generality. Neither in nature nor in engineering sciences bifurcated flows can be optimized by a universally valid allometric rule.