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ABSTRACT
Laminar natural convection has been numerically investigated from two differentially heated horizontal cylinders in a square enclosure filled with power-law fluids. Two basic configurations, namely, vertical- and diagonal-alignment of the cylinders at various locations have been considered. The coupled continuity, momentum and energy equations have been solved numerically to elucidate the effect of the Grashof number (102–104), Prandtl number (0.7–100) and power-law index (0.2–2) for a range of symmetric and asymmetric locations of the cylinders. The velocity and temperature fields are visualized in terms of streamlines, isothermal contours and plots of the local and average Nusselt number for different positions of the cylinders. The occurrence of the power-law index in the definitions of the Grashof and Prandtl numbers accentuates the interplay between the viscous, inertial and buoyancy forces thereby leading to nonlinearity in the observed trends. The presence of the dead zones coupled with the dominance of conduction under certain conditions strongly influences the overall heat transfer. All else being equal, it is possible to improve heat transfer for asymmetric positioning of the cylinders, especially at high values of the Prandtl number and Grashof number in shear-thinning fluids. A predictive correlation has been developed thereby enabling the estimation of the heat transfer coefficient in a new application in terms of the geometric and kinematic parameters.
Acknowledgments
Raj P. Chhabra would like to gratefully acknowledge the award of the JC Bose Fellowship to him (SB/S2/JCB-06/2014) by the Department of Science & Technology, New Delhi, India for the period 2015–2020.
Nomenclature
| a | = | characteristic length (D for cylinder; L for enclosure), m |
| A | = | surface area for heat transfer, m2 |
| Br | = | brinkman number |
| CP | = | thermal heat capacity of fluid, J.kg−1.K−1 |
| D | = | diameter of cylinder, m |
| g | = | acceleration due to gravity, m.s−2 |
| Gr | = | Grashof number, dimensionless |
| h | = | local heat transfer coefficient, W.m−2.K−1 |
| k | = | thermal conductivity of fluid, W.m−1.K−1 |
| L | = | side length of the square enclosure, m |
| m | = | consistency index, Pa.sn |
| n | = | power-law index, dimensionless |
| ns | = | unit normal vector to the surface, dimensionless |
| Nu | = | local Nusselt number, dimensionless |
| = | surface-averaged Nusselt number, dimensionless | |
| Nu∞ | = | Nusselt number in the limit of pure conduction, dimensionless |
| P | = | pressure, dimensionless |
| Pr | = | Prandtl number, dimensionless |
| R | = | radius of the cylinder, m |
| Ra | = | Rayleigh number, dimensionless |
| Re | = | Reynolds number, dimensionless |
| s | = | distance along the surface of enclosure, m |
| S | = | diagonal of the square enclosure, m |
| T | = | temperature of fluid, K |
| TC | = | cold fluid temperature, K |
| TH | = | hot cylinder temperature, K |
| ΔT | = | temperature difference (= TH − TC), K |
| UCi | = | reference velocity, m.s−1 |
| = | viscous scale of the characteristic velocity, m.s−1 | |
| U, V | = | x- and y- components of velocity, dimensionless |
| x, y, z | = | cartesian coordinates, m |
| X, Y | = | cartesian coordinates, dimensionless |
| Greek symbols | ||
| α | = | thermal diffusivity of fluid, m2.s−1 |
| β | = | coefficient of volume expansion, K−1 |
| δ | = | horizontal displacement from the vertical (or horizontal) center-line of the enclosure to the center of cylinder, dimensionless |
| ϵ | = | displacement from the center of the square enclosure of the center of the upper cold cylinder along the diagonal of the square enclosure, dimensionless |
| = | rate of strain tensor, s−1 | |
| Ω | = | location parameter, dimensionless |
| ρ | = | density of the fluid, kg.m−3 |
| ρc | = | density of the fluid at reference temperature (TC), kg.m−3 |
| Θ | = | temperature, dimensionless |
| θ | = | position on the surface of the cylinder, degree |
| τ | = | extra stress tensor, Pa |
| Ψ | = | stream function, m2.s−1 |
| Subscripts | ||
| cylinder | = | surface of single circular cylinder |
| enclosure | = | surface of square enclosure |
| lower | = | surface of lower hot cylinder |
| upper | = | surface of upper cold cylinder |
| c | = | at temperature TC |
Additional information
Notes on contributors
Lubhani Mishra
Lubhani Mishra is a Ph.D. student in the Department of Chemical Engineering at the Indian Institute of Technology Kanpur, India. She obtained her Bachelor's degree in 2013 from UICET, Panjab University, Chandigarh, India where she investigated the process of manufacture of styrene from ethylbenzene as a part of her undergraduate research project. Her doctoral research is focused on the computational fluid dynamics and primarily deals with the role of non-Newtonian rheology on transport phenomena in complex geometries.
Raj P. Chhabra
Raj P. Chhabra is a Professor of Chemical Engineering at the Indian Institute of Technology, Kanpur, India. He received his Ph.D. from Monash University, Melbourne, Australia in 1980. The primary focus of his research has been in the field of non-Newtonian fluid mechanics including multi-phase flows in pipes, particulate flows and bluff-body flows. He has published extensively in this area and has authored or coauthored three books in the broad field of non-Newtonian fluid mechanics and applied rheology. He serves on the editorial boards of Journal of Non-Newtonian Fluid Mechanics (Elsevier), Heat Transfer-Asian Research (Wiley) and Industrial Crops & Products (Elsevier). He is a Fellow of the Indian National Academy of Engineering and of the Indian National Science Academy. He has been a visiting professor at several universities in Australia, USA, Canada, France, Poland, Japan and South Africa.