## 1 Introduction  
Natural convection heat transfer from vertical heated objects such as surfaces and cylinders is of great importance; it is relevant to a wide range of engineering applications such as cooling of process piping, cooling of electronic devices, HVAC systems, radiators, oil cooling systems, nuclear reactors, passive heat exchangers, etc. The local natural convection correlation from vertical constant heat flux surfaces was proposed by [PERSON] and [PERSON] [1] as:  
\[Nu_{x}=0.59\;(Ra_{x}^{*})^{0.22},\;1.0\times 10^{11}\;\leq\;Ra_{x}^{*}\;\leq 1.0 \times 10^{16} \tag{1}\]  
[PERSON] [2] has studied natural convection heat transfer experimentally from vertical uniform surface heat flux tubes with rectangular cross-sections in the air. His empirical correlation for the transition regime was given by  
\[Nu_{x}=0.14\;(Ra_{x}^{*})^{0.269},\;5.0\times 10^{9}\;\leq\;Ra_{x}^{*}\;\leq 1.5\times 10^{16} \tag{2}\]  
Cylinders, in general, can either have a circular or noncircular cross-section; the non-circular one includes polygonal cross-sections such as square, rectangular, triangular, etc. The natural convection heat transfer from a single vertical cylinder differs from that of a similar cylinder placed in an array due to the effect of other cylinders. [PERSON] and [PERSON] [3] have experimentally studied the natural convection heat transfer from an array of uniformly heated vertical cylinders in mercury. An equilateral triangular pattern of cylinderswas arranged and the data were focused on the center cylinder. Their results indicated a strong natural convection dependence on cylinder spacing but with less dependence on heat flux and the circumference position. [PERSON] et al. [4] experimentally studied the natural convection heat transfer from a line of horizontal cylinders in the air. The effect of cylinder spacing on heat transfer was reported and empirical correlations for the heat transfer were proposed. [PERSON] [5] numerically investigated the free convection heat transfer from a vertical array of equally spaced horizontal cylinders in the air. Correlations were developed for any individual cylinder in the array and for the whole tube array. [PERSON] and [PERSON] [6] experimentally studied natural convection heat transfer from a vertical array of horizontal isothermal elliptic cylinders. It was found that the heat transfer from any cylinder depended on the position of the cylinder in the array, the Rayleigh number, and the separation distance. Empirical correlations were obtained for each cylinder in the array and for the whole array. [PERSON] and [PERSON] [7] have experimentally investigated the natural convection heat transfer from vertical and inclined arrays of horizontal cylinders in the air. Correlations were developed for each cylinder in the vertical and inclined arrays and for the whole arrays. [PERSON] et al. [8] have experimentally studied free convection heat transfer from a vertical array of two horizontal cylinders. It was observed that the heat transfer from the lower cylinder was unaffected by the presence of the upper cylinder; however, the upper cylinder was affected by the rising plume from the lower cylinder. [PERSON] and [PERSON] [9] have experimentally investigated the natural convection heat transfer from a pair of vertical arrays consisting of equally spaced uniformly heated cylinders in the air. It was found that the Nusselt number of each cylinder either enhanced or degraded, compared to that of a single cylinder, based on the location of the cylinder in the array, the geometry of the array, and the Rayleigh numbers. [PERSON] et al. [10] have experimentally investigated the free convection from an array of three vertical tubes at different orientations filled with hot mineral oil. Their results showed a temperature increase along the tubes for all inclinations. [PERSON] et al. [11] have numerically studied the natural convection heat transfer from two attached cylinders forming a vertical array in the air.