Natural convection of power-law fluids in rectangular cross-sectional cylindrical annular enclosures with differentially heated vertical walls

Abstract

Laminar natural convection of power-law fluids in rectangular cross-sectional cylindrical annular enclosures with differentially heated vertical walls has been numerically investigated in this analysis for both constant wall temperature (CWT) and constant wall heat flux (CWHF) boundary conditions for the active walls. Two-dimensional axisymmetric steady-state numerical simulations have been conducted for a range of values of normalized internal radius 0.125 ≤ ri/L ≤ 16 (where ri internal cylinder radius, L is the difference between outer and inner cylinder radius), aspect ratio 0.125 ≤ AR ≤ 8 (AR = H/L, where H is the enclosure height), power-law index (i.e. 0.6 ≤ n ≤ 1.8) and nominal Rayleigh number (i.e. Ra = 103 − 106) for a single representative nominal value of Prandtl number: Pr = 103. It has been found that the mean Nusselt number based on the inner periphery of the annular space Nui = hiL/k (where hi is the mean heat transfer coefficient on the inner periphery of the annular space and k is the thermal conductivity) increases with increasing Ra due to the strengthening of buoyancy forces. By contrast, Nui increases with decreasing n due to the weakening of viscous resistance. The mean Nusselt number Nui decreases with increasing ri/L before approaching the mean Nusselt number for a rectangular enclosure in the limit of ri/L→ ∞. By contrast Nui normalized by the corresponding Nusselt number for pure conduction (i.e. Nui/Nucond) increases with increasing ri/L. The ratio of convection to conduction strength increases with increasing ri/L, since the Nusselt number for pure conductive transport Nuconddecreases with increasing ri/L for cylindrical annular enclosures (i.e. Nucond = (L/ri)/(ln(1 + L/ri)). Additionally, it has been found that Nui exhibits a non-monotonic variation with increasing AR for a given set of values of Ra,Pr,ri/L for shear thinning (n<1), Newtonian (n= 1) and shear thickening (n> 1) fluids in the CWT configuration, whereas Nui increase monotonically with increasing AR in the CWHF configuration irrespective of the value of n. The completion between the strengthening of thermal convection and the weakening of conductive thermal transport with increasing AR is responsible for the non-monotonic variation of Nui with AR in the CWT configuration. A detailed scaling analysis is utilized to explain the effects of normalized radius, aspect ratio, nominal Rayleigh and Prandtl numbers, power-law index on Nui for natural convection of power-law fluids within rectangular cross-sectional cylindrical annular enclosures. Finally, new correlations have been proposed for Nui for both CWT and CWHF boundary conditions, which have been shown to provide satisfactorily predictions of Nui for the range of the parameters considered in this analysis. © 2017, Begell House Inc. All Rights Reserved.