## 2 Experimental Setup
### Heat Transfer Analyses  
Natural convection heat flux from each tube is determined following the equation  
\[\text{Electrical input power}=\text{A}_{\text{s}}\ (q_{c}+q_{r})+(\text{A}_{\text{b}}\ q_{b}) \tag{3}\]  
where \(q_{c},q_{r},q_{b}\), \(\text{A}_{\text{s}}\), and \(\text{A}_{\text{b}}\) are the convection heat flux, radiation heat flux, conduction heat flux through the Bakelite insulation capped ends, tube outside surface area, and the Bakelite cross-section area, respectively. The \(q_{c}\) is in the range of 77-360 W/m\({}^{2}\) and the surrounding medium is air with a Prandtl number (Pr) \(\approx\) 0.72. The heat dissipated by conduction through the Bakelite ends and by radiation is obtained from:  
\[q_{b}=\text{k}_{\text{b}}\ \frac{T_{ib}-T_{ob}}{\delta} \tag{4}\]  
\[q_{r}=\varepsilon\ \sigma\ F_{1\infty}\left(\overline{T}^{4}-T_{\infty}^{4}\ \right) \tag{5}\]  
where \(T_{ib}\), \(T_{ob}\), \(\delta\), \(\text{k}_{\text{b}}\) are the inside and outside surface temperatures of the Bakelite ends, thickness, and the thermal conductivity of the Bakelite, respectively. Furthermore, \(\varepsilon,\ \sigma,\ F_{1\infty}\) stand for the surface emissivity of the tube (0.27 for polished mild steel [15]), the Planks constant, and the shape factor, respectively. It should be noted that \(q_{r}\) is calculated at the overall average surface temperature \(\overline{T}\) corresponding to each heat flux. Following [15], the shape factor between two long parallel tubes can be obtained from  
\[F_{12}=\frac{1}{\pi}\left[sin^{-1}\ \left(\frac{D}{S}\right)+\left(\left( \frac{S}{D}\right)^{2}-1\right)^{0.5}-\left(\frac{S}{D}\right)\right] \tag{6}\]  
\[F_{1\infty}=1-\alpha\ F_{12} \tag{7}\]  
where \(F_{1\infty}\) is the shape factor between each tube and the ambient and \(\alpha=1\) for the right and left tube and 2 for the middle one. It should be mentioned that exchanging radiation between similar tubes is negligibly small; therefore, it will be assumed that neighboring tubes are in radiant balance [16]. Consequently, the net radiant exchange is with the room walls. The circumference average surface temperature at any axial distance along the tube and for each uniform heat flux can be calculated from  
\[T_{x}=\sum_{j=1}^{3}\ \frac{T_{x,j}}{3} \tag{8}\]  
where \(j\) stands for the thermocouple number in the circumference direction at each axial distance along the tube. The physical properties are calculated at the arithmetic mean temperature as  
\[\theta_{a}=0.5\ (T_{x}+T_{\infty}),\ x\ =\ 1,2,\ldots.,9 \tag{9}\]  
Following the above temperature distribution, then there are nine \(T_{x}\), which corresponds to each heat flux for each tube in the array. The axial circumference averaged local heat transfer coefficient \(h_{x}\) can be calculated for each tube as  
\[h_{x}=\frac{q_{c}}{T_{x}-T_{\infty}} \tag{10}\]  
It should be noted that for each tube there are nine axial values of the circumference averaged local heat transfer coefficient \(h_{x}\) corresponding to each heat flux. Consequently, the non-dimensional Nusselt and the modified Rayleigh numbers can be determined as  
\[Nu_{x}=\frac{h_{x}\ x}{k},\ RA_{x}^{*}=\frac{\ g\ g\ q_{c}\ x^{4}}{\theta\ k\ \alpha} \tag{11}\]It should be noticed that the local Nusselt number Nux and the local modified Rayleigh number \(Ra_{x}^{*}\) are used throughout this study and sometimes the superscript * and the subscript x are dropped out just for simplicity.