Mixed convection from a heated sphere in power-law fluids N. Nirmalkar, R.P. Chhabra n Department of Chemical Engineering, Indian Institute of Technology, Kanpur, UP 208016, India HIGHLIGHTS c Effect of Richardson number on heat transfer from a sphere is studied. c Aiding-buoyancy flow enhances heat transfer. c Shear-thinning fluid behaviour promotes heat transfer. c Nusselt number shows positive dependence on Grashof and Prandtl numbers. article info Article history: Received 26 July 2012 Received in revised form 21 November 2012 Accepted 21 November 2012 Available online 29 November 2012 Keywords: Convective transport Non-Newtonian fluids Richardson number Heat transfer Prandtl number Transport processes abstract In this work, the steady, laminar, mixed convection heat transfer from a heated sphere immersed in power-law fluids has been investigated in the so-called aiding-buoyancy configuration. The momentum and thermal energy equations have been solved numerically over the following ranges of conditions: Richardson number, 0 rRi r2, power-law index, 0:2 rn r2, sphere Reynolds number, 1 rRe r100 and the generalized Prandtl number, 1 rPr r100. The detailed kinematics of the flow and temperature fields are visualized in terms of the streamline (stream trace) and isotherm contours in the close proximity of the heated sphere. Further insights are provided in terms of the distribution of pressure coefficient and local Nusselt number along the surface of the heated sphere over wide ranges of conditions from weak to strong free convection flow. Finally, the overall macroscopic characteristics are reported in terms of the individual and total drag coefficients and the surface average Nusselt number as functions of the pertinent dimensionless parameters. The drag coefficient is seen to increase monotonically with the Richardson and Prandtl numbers at low Reynolds numbers and the type of dependence changes at a critical Richardson number. Similarly, the average Nusselt number shows a positive dependence on the Reynolds number, Prandtl number and Richardson number. Broadly, shear- thinning viscosity (n o1) promotes heat transfer over and above that in Newtonian fluids otherwise under identical conditions and, as expected, shear-thickening behaviour (n 41) somewhat impedes it. Finally, the present numerical results have been correlated in terms of the modified Reynolds and Prandtl numbers thereby enabling interpolation of the present results for the intermediate values of the governing parameters. & 2012 Elsevier Ltd. All rights reserved. 1. Introduction Heat transfer from a single sphere immersed in moving or quiescent non-Newtonian fluids denotes an idealization of several industrially important processes. Typical examples include fixed and fluidized bed reactors, three-phase sparged reactors, production and processing of suspensions encountered in food, mineral, pharma- ceutical, personal care products related settings, etc. (e.g., see Awuah and Ramaswamy, 1996; Coulson and Richardson, 2002; Sastry and Cornelius, 2002; Ramaswamy and Zareifard, 2003; Meng and Ramaswamy, 2007). Depending upon the strength of the externally imposed flow vis-a-vis that of the buoyancy-induced flow, it is possible to discern three distinct regimes of heat transfer, namely, forced convection, free convection and mixed convection. In order to ascertain the relative importance of these two mechanisms, the so-called Richardson number (Ri), defined as the ratio of the Grashof number (Gr) to the square of the Reynolds number (Re), i.e., Ri ¼ Gr/Re 2 is employed, to delineate the main mode of heat transfer. When the externally imposed flow is much stronger than that induced by the temperature-dependent density of fluid, i.e., Re bGr, i.e., Ri-0 denotes the pure forced convection regime. On the other hand, at the other extreme, when Re 5Gr, the main mode of heat transfer is free convection characterized by large values of the Richardson number, Ri-1. For intermediate values of the Richardson number, O(1), heat transfer occurs in the so-called mixed regime. Thus, for instance, at Ri ¼ 1, the strength of the buoyancy- induced flow is comparable to that of the externally imposed flow and therefore the contributions of the both, free and forced Contents lists available at SciVerse ScienceDirect journal homepage: www.elsevier.com/locate/ces Chemical Engineering Science 0009-2509/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ces.2012.11.031 n Corresponding author. Tel.: þ91 512 2597393; fax: þ91 512 2590104. E-mail address: chhabra@iitk.ac.in (R.P. Chhabra). Chemical Engineering Science 89 (2013) 49–71