Theoretical and numerical analysis of counter-flow parallel convective exchangers considering axial diffusion

Abstract

We perform a systematic analysis of heat transfer in a counter-current three dimensional convective exchanger, when the inlet/outlet influence is fully taken into account. The analysis, carried out for constant fluid properties, considers the various influences of the fluid/solid conductivity, the imposed convection, inlet/outlet far-field conditions, and lateral boundary conditions. Using a generalized Graetz mode decomposition which permits to consider, both transverse and longitudinal diffusion influence in the exchanger as well as in the inlets/outlets, we put forward several salient generic features of convection/conduction heat transfer. In all cases we found an optimal Péclet number for the cold or hot effectiveness. Even if, as expected, the larger the Péclet the larger the Nusselt number, high transfer performances are found to be poorly efficient and/or to necessitate non-compact elongated exchangers. Performance degradation arising at high Péclet number are found to be related to ''convective leaks" taking place within outlets. A fully developed regime occurs at large Péclet and/or for long exchangers, which is fully determined by the first eigenvalue of the generalized Graetz mode decomposition, which is an extension of classical Graetz analysis. Numerical results are found consistent with a generalized linear relation between effectiveness and the number of heat transfer units asymptotically established in the convection dominated regime. This study opens new perspectives for micro-heat exchangers where moderate convection provides the best effectiveness and compactness. This contribution is also useful for giving reference solutions to counterflow exchangers with realistic inlet/outlet boundary conditions.

Thermal significance and optimal transfer in vessels bundles is influenced by vascular density

“…The tissue/vessel system is modeled as a transversely infinite exchanger into which a pattern of periodic parallel vessels trans port heat. This choice has been made so as to gain insight into the exchanges independently of the transverse boundary condi tions as opposed to [30]. To be more precise on the terminology that we will use in the following we will refer to the exchanger region as the finite (but transversely periodic) domain within Fig.…”

“…Hence, the tissue model under study has a finite extent in the longitudinal direction, but needs boundary conditions for the heat at tubes inlets and outlets. In order to circumvent the problem of arbitrarily setting such boundary conditions at z 0 and z L e , we connect the finite tis sue exchanger to infinite inlet and outlet reservoirs connecting tubes, as in [28,30]. Hence three compartments are considered.…”

“…As in [30], we consider laminar convection diffusion arising in a fluid having constant properties. In the following we consider tubular vessels having circular sections.…”

“…On the external part of the exchanger configuration, i.e. for z ~ 0 or z ~ L, we consider homogeneous Neumann conditions as in [30] (4)…”

“…Hence, in this paper, we analyze various distinct features of heat transfer in vessels bundles, associated with thermal significance, and optimal transfer using a quasi analytic approach previously described in [29,30]. In Section 2 the set of governing equations and their dimensionless formulation are provided.…”