Geometry-Based Entropy Generation Minimization in Laminar Internal Convective Micro-Flow

Abstract

Abstract In this theoretical study, fully developed forced convective laminar water flow is considered in circular micro-tubes, for the constant wall heat flux boundary condition. The change in entropy generation rate ( Δ S ˙ gen \Delta {\dot{S}_{\mathrm{gen}}} ) for N micro-tubes (each of diameter D N {D_{\mathrm{N}}} ) relative to a reference tube (of 1 mm diameter) was investigated towards the micro-scale, for different tube length (l). A given total heat flow rate is to be removed using a fixed total mass flow rate through N tubes. Hence, the wall heat flux for one of the N tubes decreases towards the micro-scale, which is “thermal under-loading”. For given l, Δ S ˙ gen \Delta {\dot{S}_{\mathrm{gen}}} due to fluid conduction decreases and Δ S ˙ gen \Delta {\dot{S}_{\mathrm{gen}}} due to fluid friction increases towards the micro-scale. There exists an optimum D N {D_{\mathrm{N}}} ( = D N , opt ={D_{\mathrm{N},\mathrm{opt}}} ) at which the change in sum-total S ˙ gen {\dot{S}_{\mathrm{gen}}} ( Δ S ˙ gen , tot \Delta {\dot{S}_{\mathrm{gen},\mathrm{tot}}} ) is minimum; where D N , opt {D_{\mathrm{N},\mathrm{opt}}} decreases with decreasing l. For given l, cooling capacity of the heat sink increases towards the micro-scale. A general criterion for minimization of Δ S ˙ gen , tot \Delta {\dot{S}_{\mathrm{gen},\mathrm{tot}}} is obtained in terms of Reynolds number, Brinkman number, and dimensionless l.