Chaotic heat transfer enhancement in rotating eccentric annular-flow systems

Abstract

Thermal Taylor dispersion theory for time-periodic systems was used to study the extent of chaotic laminar heat transfer enhancement and axial thermal dispersion occurring during combined transverse and axial annular flow between two nonconcentric circular cylinders undergoing alternate rotations. A local Newton’s “law of cooling” heat transfer boundary condition was used on the outer cylinder, whereas the inner cylinder was supposed insulated. The effective heat transfer coefficient H̄* describing the global rate of heat loss from the system (differing in general from the true microscale Newton’s law heat transfer coefficient h on the outer cylinder) was calculated as a function of the system parameters, thereby serving to quantify the extent of chaotic heat transfer enhancement. The axial thermal Taylor dispersivity provided an independent measure of the effects of chaotic mixing, as too did the axial thermal velocity. Calculations were performed for three different cases: (i) concentric cylinder rotation (for which case the resulting circular transverse flow has no effect upon the effective transport properties); (ii) nonconcentric counter-rotating circular cylinders, each undergoing a steady rotation, thereby creating a time-independent transverse flow field; (iii) nonconcentric counter- and co-rotating circular cylinders, each undergoing time-periodic alternate rotation while the other remains at rest. A “regular” enhancement of the heat transfer rate over the concentric cylinder case was observed in case (ii), arising from the presence of a secondary-flow recirculation region. Enhancement due to chaotic advection was observed in case (iii) [about 50% more than that of case (ii) and more than double that of case (i), all other things being equal]. Concomitant values of the axial thermal Taylor dispersivity and axial thermal velocity confirmed the existence of enhanced transverse transport due to chaotic advection. It was observed that the functional dependence of the enhanced heat transfer rate upon the system parameters does not consistently display the same trends as are qualitatively suggested by the “degree of chaoticity” of the comparable Poincaré plots. This observation signals the need for caution in simply assuming that the greater the degree of chaotic “mixing” implicit in the Poincaré plot the greater will be the corresponding global transport rate. By simple redefinition of the symbols used in the present paper, our energy transport results may be re-interpreted so as to apply to the case of reactive-species transport involving a first-order irreversible chemical reaction occurring on the outer-cylinder surface; explicitly, the Nusselt number quantifying the local heat transfer coefficient rate is simply replaced by a comparable Damköhler number quantifying the local kinetics of the surface reaction.