Laminar natural convection in power-law liquids from a heated semi-circular cylinder with its flat side oriented downward Anurag Kumar Tiwari, R.P. Chhabra ⇑ Department of Chemical Engineering, Indian Institute of Technology Kanpur, Kanpur 208016, India article info Article history: Received 12 September 2012 Received in revised form 21 October 2012 Accepted 14 November 2012 Keywords: Semi-circular cylinder Natural convection Power-law fluid Grashof number Nusselt number Prandtl number abstract In this work, laminar free convection heat transfer from a heated semi-circular cylinder, with its flat base facing downward, submerged in quiescent power-law fluids has been studied. The coupled momentum and energy equations have been solved numerically in the following ranges of dimensionless parameters: Grashof number (10 6 Gr 6 10 5 ), Prandtl number (0:72 6 Pr 6 100) and power-law index (0:2 6 n 6 1:8). The detailed flow and temperature fields in the vicinity of the heated object are visualized in terms of the streamline and isotherm contours respectively. The rate of heat transfer is described in terms of the local Nusselt number variation along the surface of the cylinder together with its average value over the above-mentioned ranges of parameters. As expected, the value of the local Nusselt number increases from the front stagnation point up to the sharp corner and then decreases all the way up to the rear stagnation point. Broadly, over the range of conditions spanned here, the flow remains attached to the surface of the cylinder. The average Nusselt number increases with both the Grashof and Prandtl numbers and it decreases with the increasing power-law index. Furthermore, shear-thinning behavior (n < 1) enhances heat transfer whereas shear-thickening behavior (n > 1) impedes it with reference to that in Newtonian fluids otherwise under identical conditions. The numerical results obtained herein are correlated using a composite parameter consistent with the boundary layer analysis and these are contrasted with the results for the other two-dimensional shapes thereby suggesting the general useful- ness of the composite parameter. Ó 2012 Elsevier Ltd. All rights reserved. 1. Introduction In many process engineering applications, heating and cooling of variously shaped objects are encountered in a range of industrial settings. Typical examples include heating/cooling of particulate suspensions consisting of pulp and paper fibers, thermal treatment of foodstuffs, reheating of processed foods, melting of polymeric pellets and chemical reactions carried out in multiphase reactors like packed and fluidized beds, slurry reactors, etc. [1–3]. In princi- ple, whenever a temperature gradient exists between different parts of an equipment and/or in different regions in the fluid med- ium itself, heat transfer occurs in the mixed convection regime, i.e., both force and natural convection mechanisms contribute to the overall rate of heat transfer in varying proportions. Evidently, when the imposed flow (force convection) is weak, i.e., at low Rey- nolds numbers, free convection induced by the temperature dependent fluid density contributes significantly to the overall heat transfer. Of course, the latter contribution progressively diminishes with the increasing value of the Reynolds number. Be- sides, when the fluid medium is severely confined, natural convec- tion contributes significantly, e.g., in the reheating of canned foodstuff like soups with suspended particles in them or in the freezing of fruit yoghurts, other packaged ready to eat multi-phase foods, etc. [1,2]. It is frequently required to heat or cool such pro- cess streams in engineering applications and this calculation re- quires reliable values of heat transfer coefficient. In addition, a systematic study of free convection heat transfer from variously shaped objects constitutes an important class of problems within the domain of transport phenomena [4,5]. Notwithstanding the intricate role played by the complex geometries encountered in real engineering applications, experience has shown that the study of free convection from model shapes like sphere, circular and elliptic cylinder, etc. has provided useful insights into the underly- ing physical processes. Consequently, over the past 50 years or so, a large body of knowledge has accrued on the momentum and heat transfer phenomena from idealized shapes like an isolated sphere, circular cylinder, elliptic cylinder, square cylinder, semi-circular cylinder, etc. in Newtonian fluids like air and water, e.g., see Refs. [4–7]. Suffice it to say here that based on a combination of approx- imate analyses, numerical and experimental results, it is now pos- sible to predict the value of heat transfer coefficient for the aforementioned simple shapes immersed in Newtonian fluids in a new application in the free convection regime [4,5,7]. 0017-9310/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2012.11.051 ⇑ Corresponding author. Tel.: +91 512 2597393; fax: +91 512 2590104. E-mail address: chhabra@iitk.ac.in (R.P. Chhabra). International Journal of Heat and Mass Transfer 58 (2013) 553–567 Contents lists available at SciVerse ScienceDirect International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt