Heat Transfer Enhancement Using Isothermal Heating Blocks of Low Height in an Annular Space Filled with Air

In this paper, we consider an annular space filled with air (Pr = 0.7) confined between two horizontal coaxial cylinders differentially heated, with a radii ratio R = 2. The space is fitted with two symmetrical heating blocks with a height h = 0.093 and a width l = 0.109 placed at the angular position φm = 0.82π. We investigate numerically the impact of different initial fields on the final solution obtained for a range of Rayleigh numbers between 10 and 10. The obtainment of a bicellular flow regime that promotes the overall heat transfer rate is ensured at different intervals of Rayleigh numbers. Each interval is characterized by specific Rayleigh numbers of the bicellular flow appearance and disappearance depending on the introduced initial fields. These initial conditions have a real influence on the computation time necessary to obtain the optimal steady regimes. Index Term— Natural convection, isothermal blocks, initial field, heat transfer, flow regimes, finite difference. Nomenclature Greek letters BCR bicellular regime α thermal diffusivity, m 2 s -1 g gravity acceleration, ms -2  thermal expansion coefficient, K h dimensionless height of blocks relative difference of heat transfer rates l dimensionless width of blocks kinematic viscosity, m 2 s -1 Nug overall Nusselt number polar angle Pr Prandtl number, Pr = α angular position of the blocks R radii ratio, r ́o / r ́i dimensionless stream function r dimensionless radial coordinate ω dimensionless vorticity r ́i inner cylinder radius, m r ́o outer cylinder radius, m Indices / Exponents Ra Rayleigh number, Ra = g(T ́i T ́o) r ́i 3 / α i inner RaIC3 initial field o outer RaBCR interval of Ra corresponding to BCR regime max upper radial facet of block RaUCR interval of Ra corresponding to UCR regime min lower radial facet of block T ́i inner cylinder temperature, K ́ dimensional T ́o outer cylinder temperature, K t dimensionless time u dimensionless radial velocity UCR unicellular regime v dimensionless angular velocity INTRODUCTION Due to the multiplicity of industrial applications where natural convection phenomenon plays a crucial role such as thermal exchangers, solar collectors, electronic components cooling and home heating, the natural convection in annular cavity has been the subject of several numerical and experimental investigations focusing on the heat transfer improvement. In this paper, we are interested to the cylindrical annular geometry filled with air as previous numerical and experimental studies have already done, which indicates the big interest that natural convection in this geometry had been given during last decades. Kuehn and Goldstein [1] carried out an experimental and theoretical investigation on natural convection where the influence of diameter ratio and Prandtl number especially for Pr = 0.7 (air) on the heat transfer and the flow was determined. In the same geometry, Powe et al. International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:16 No:02 96 166302-9494-IJMME-IJENS © April 2016 IJENS I J E N S [2] described experimentally the several flow patterns observed for different radii ratios R. In addition to that, some works showed particular interest to the influence of the injected initial profile fields on the final flow solution and the overall heat transfer rate. These initial conditions could be based on temperature fields or and velocity fields. By adequately perturbing the temperature field of pure conduction and using it as initial condition, Cheddadi et al. [3] obtained numerically with the Chorin’s method the bicellular regime for different radii ratios. For R = 2, they found the bifurcation point at Ra = 3250 and focused on the influence of the computation grid. Later on, Chung et al. [4], in a numerical investigation based on solving the governing equations from primitive variables, using initial profiles resulting from melting processes already investigated in their previous study, found it at Ra = 2819. Zhang et al. [5] showed in their experimental study the effect of the choice of the temperature distribution initial profile on the final solution obtained for a same Rayleigh number by using the holographic interferometry technique. Apart from the annular cavity filled with air, we notice a scarcity in investigations of the same cavity with the presence of heating blocks discussing the influence of initial conditions either numerically or experimentally on the overall heat transfer rate and the flow pattern found. Concerning the numerical studies, many publications described either the influence of the fins geometrical disposition or their optimal number ([6-11]). Recently, a numerical investigation on the initial conditions impact was published by Idrissi et al. [12] who presented a comparison between two types of initial conditions IC1 and IC2 by the means of flow regime and overall heat transfer for a radii ratio R = 2. From an overview of these studies mentioned above, a third type of initial conditions based on experimental methods can be presented by introducing the solutions of an established flow in order to reach a final permanent solution for the range of Rayleigh numbers between 10 3 and 10 4 . This paper presents a numerical simulation examining the influence of a new type of initial conditions in the case of two heating blocks in a cavity filled with a fluid assimilated to air, Pr = 0.7, limited by two horizontal isothermal coaxial cylinders extremely long. These blocks are isothermal and placed symmetrically to the vertical plane containing the cylinders axis on the hot wall, which is the inner cylinder of radius r ́i, with the temperature T ́i. The outer cylinder of radius r ́o is maintained at a temperature T ́o smaller than T ́i. In this work, the blocks have a dimensionless height, h = 0.093, relative to the annular space thickness (r ́o r ́i). The blocks are delimited in the angular direction by the values min and max of the polar angle calculated from the space bottom and placed in the upper part of the annular space with an average angular position m = 0.82π (Fig. 1). The radii ratio R = r ́o / r ́i and the dimensionless blocks width, l, relative to semicircle, are held constant throughout the study: R = 2 and l = 0.109.