Abstract

The paper discusses a simple multi-variable optimization problem: the bifurcation of a branch of a pipe of circular cross-section with a given initial radius r0 and delivering a given mass flow rate m0.The optimization is performed using an objective function that prescribes the minimization of the entropy generation rate due — in this simple case — only to viscous flow effects within the tubes. Several fundamental simplifying assumptions are made to reduce the problem to a multi-variable optimization in three independent variables: the aspect ratio of the domain served by the flow, the diameter ratio of the primary and secondary branches, and the length of the secondary branch (the location of both the “source” of the fluid and the “sink”, i.e., the place of desired delivery of the fluid, being a datum).It is shown that the solution is strongly dependent both on the aspect ratio and on the diameter ratio, and that the "optimal" configurations display some resemblance to the branching patterns observed in natural structures. The study poses a challenge both to Designers and to Natural Scientists: are the optima suggested by the present procedure compatible with the structures currently used in heat exchangers and flow devices? Are they compatible with the structures observed in nature? No final answer is provided in this preliminary study, but a possible "falsification" procedure is outlined in the conclusions.

Highlights

  • The scope of this paper is a simple analytical examination of the functional dependence of the entropy generation rate in a bifurcated flow on the geometric parameters of the bifurcation

  • Equation (9) was solved analytically for different aspect ratios ar = H/L and different bifurcation lengths, under three physically meaningful situations: (1) The Reynolds number remains constant over the entire fluid path: Re0 = Re1. This defines a diameter ratio δ = r1/r0 = 0.5; (2) The velocity remains constant over the entire fluid path: U0 = U1

  • This defines a diameter ratio δ = r1/r0 = 0.707; (3) The volume occupied by the fluid in the unsplit portion is equal to that occupied by the two bifurcated branches

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Summary

Introduction

The scope of this paper is a simple analytical examination of the functional dependence of the entropy generation rate in a bifurcated flow on the geometric parameters of the bifurcation. Let us focus our attention on the second question: the conventional approach followed by heat transfer practitioners is that of devising a set of similar (or modular) structures (each member being a properly constructed series of bifurcations) and to check a posteriori which member of the family attains the best performance under a pre-assigned set of design constraints This heuristic approach may be very successful in practice [1,2,3], but its final result (the “optimal” structure) strongly and unavoidably depends on the initial choice of the geometrical features the “family” must possess: in other words, the final result essentially depends on the ingenuity and insight of the designer. The two “measures” (the pressure drop and the viscous entropy generation rate) may be numerically equivalent, in the sense that the minimum of the former coincides with the minimum of the latter, but the insight provided by an entropic analysis is much deeper, if only for the fact that it can be assessed locally [5,6] and immediately applied to design modifications

The standard derivation of the velocity profile in plane Poiseuille flow
Alternative derivation of the velocity profile of plane Poiseuille flow
The Entropy Generation Rate in a Simple Bifurcation
Results and Discussion
A more realistic case with added bifurcation losses
Conclusions

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