Cooling by Free Convection at High Rayleigh Number of Cylinders Positioned Above a Plane

Free convection cooling of isothermal circular cylinders positioned above a horizontal plane is investigated numerically, using a commercial Computational Fluid Dynamics (CFD) software package. Computation is performed for high Rayleigh number, in the range 10 − 10. Chien’s turbulence model of low-Reynolds-number K-ε is used, with Prandtl number of 0.707, corresponding to air near standard conditions. Influence of the underlying plane on heat transfer from the cylinders' surface is examined. As the gap between the plane and cylinders is narrowed, a pattern can be seen whereby heat transfer reaches a minimum that moves closer to the cylinder surface with higher Rayleigh number. The plane’s thermal condition, adiabatic versus isothermal, produces no significant difference in the heat transfer for the present range of gap ratio, in contrast to laminar case. Nomenclature A surface area per unit length of the whole cylinder cp specific heat at constant pressure D cylinder’s diameter g gravitational acceleration (9.81 m/s) Gr Grashof number = g β D(Ts − T∞) / ν h gap between cylinder and the plane haverage average heat transfer coefficient = Q / [A (Ts − T∞)] k (molecular) thermal conductivity K turbulent kinetic energy Nuaverage average Nusselt number = haverage D / k p pressure Pr (molecular) Prandtl number = μ cp / k = ν / α Prt turbulent Prandtl number = μ t cp / k t = ν t / α t Q heat transfer rate per unit length of whole cylinder surface Ra Rayleigh number = g β D (Ts − T∞) / (ν α) T temperature Ts temperature of cylinder’s surface (387.52 K) T∞ ambient temperature (300 K) u velocity in the x-direction uτ friction velocity = (τw / ρ) v velocity in the y-direction x coordinate in the horizontal direction y coordinate in the vertical direction (increasing upward) y non-dimensional distance from closest wall = δ / (ν / uτ) δ distance from closest wall yplane y-coordinate of the horizontal plane α (molecular) thermal diffusivity = k / (ρ cp) β thermal expansion coefficient ε turbulent kinetic energy’s dissipation rate μ (molecular) viscosity ν (molecular) kinematic viscosity = μ / ρ ρ density τw wall shear stress Subscript t: turbulent Introduction This paper reports on heat transfer by free-convection at high Rayleigh number from a heated, isothermal horizontal circular cylinder positioned above a horizontal plane. This work is an extension to the turbulent regime of a previous one [7]. Circular cylinder is a very common geometry, and objects of this shape abound. Examples are fluid-carrying pipes, electrical wires, etc. Heat transfer from circular cylinders has been investigated by many authors, especially the situation of an isothermal horizontal cylinder exchanging heat with its surroundings in a totally unobstructed free convection regime. For this configuration, the empirical correlations of Morgan [19] who had reviewed a large body of literature, and of Churchill and Chu [4] have been particularly well accepted [8,17]. This unobstructed free convection around an isothermal, horizontal circular cylinder has also been investigated computationally by other authors [5,14-15,18,22]. The common situation when a cylinder is positioned close to a plane has also been considered. However, focus has been on the case of the heated cylinder being positioned between vertical walls [6,11,16,21], or below a ceiling [1,12,15]. The case of the cylinder having its surface temperature higher than its surroundings’ and positioned above a plane (or, equivalently, the cylinder having surface temperature lower than its surroundings’ and positioned below a plane) seems to have only been treated experimentally by Jones and Masson [9], and computationally for isothermal underlying plane and low Grashof numbers (Gr ≤ 8000) by Müller and co-workers [10,23] (whose heat transfer results are, however, too low in comparison with correlations of, for example, [4]). In this work, the situation of a heated cylinder positioned above a horizontal plane is investigated using computational method for Rayleigh number in the range of 10 − 10, using a fluid with Prandtl number 0.707 (corresponding to Gr = 1.41×10 − 1.41×10). Both cases of isothermal and adiabatic plane will be considered. Modelling and Computation The physical model is depicted in Figure 1. A horizontal circular cylinder of diameter D is positioned above a solid horizontal plane, with gap h between the two items. The cylinder’s surface is assumed to have a uniform temperature Ts, while the surrounding fluid has the constant ambient temperature T∞, with Ts > T∞. Here T∞ is fixed at 300 K, and Ts at 387.52 K. Note that this situation is also equivalent to when the cylinder is positioned below the plane, but with Ts < T∞. The underlying plane is assumed to either be of the same temperature as the ambient fluid’s (300 K), or be insulated. Free convection would result from temperature difference between the cylinder’s surface and the surrounding fluid, and is the subject of this study. Attention will however be given to the total heat transfer rate from the cylinder’s surface, which is characterised by an average Nusselt number.