ABSTRACT
ABSTRACT
Experimental investigation is reported on natural convection heat transfer from the outer surface of a vertical array of horizontal square tubes in air. Five tubes equally spaced are used with cross section 0.02 × 0.02 m2. The tubes are subject to constant heat flux boundary condition using internal constant heat flux heating elements in the range 46–510 W/m2. Experiment is done for arrays of 2–5 square tubes and for four center-to-center separation distance to hydraulic diameter ratios. Study is concentrated on the effect of tube location in the array and on the geometry of the array. Results show that the downstream tubes exhibit reduced Nusselt numbers than that of a single tube for small center-to-center separation ratio of 2.5. This reduction depends on the location of the tube in the array and the number of tubes in each array. Results also show that as the ratio increases, enhancement in heat transfer over that of a single tube is observed and critical ratio is obtained at a specified value of the modified Rayleigh number for the upper (downward) tubes in each array. Local circumference averaged correlations are proposed for the upper tubes in each array and for any other individual tube in each array geometry. An overall general averaged correlation is also reported for each tube in the array.
Introduction
Natural convection heat transfer from arrays of horizontal cylinders has many engineering applications. Such applications are extended fins heat exchangers, heaters for high viscosity fuel oils (Corpe [1]), space heating, cooling or heating of liquids in process plants and in cooling of electronic devices. Eckert and Soehngen [2] was the first to introduce an experimental study of natural convection heat transfer induced by a vertical array of three horizontal cylinders. They observed that the heat transfer from the lowest cylinder is the same as that of a single cylinder but the other downstream cylinders decrease with elevation in the array. Experimental natural convection studies of vertical array of wires or cylinders heated by uniform heat flux have been reported by Lieberman and Gebhart [3] and Marsters [4], respectively. Both studies agreed that the Nusselt number of the lowest cylinder was unaffected by the downstream cylinders and it was identical to that for a single cylinder. However, the upper downstream cylinders observed to degrade the Nusselt number at close center-to-center spacing and to enhance it at large spacing. Similar experiment was done by Sparrow and Niethammer [5] for two-cylinder array. The distance between the two cylinders was changed between two to nine diameter and it was found that degradation of the Nusselt number of the upper cylinder occurred at small separations and enhancement at large ones if compared to the lower cylinder. It was also found that the maximum enhancement observed at a separation distance range of seven to nine cylinder diameters. Experimental free convection heat transfer from a vertical array consisting of horizontal cylinders was conducted by Tokura et al. [6] for Grashof numbers in the range between 4 × 104 and 4 × 105. They reported that the averaged convection heat transfer coefficient of the lower cylinder in the array was almost the same as a single horizontal cylinder, except when the spacing is less than the diameter of the cylinder. They also mentioned that the maximum averaged convection heat transfer coefficient of the downstream cylinders reached maximum values when the spacing was larger than five times the cylinder diameter. Numerical study of free convection from a pair of vertically separated horizontal circular cylinders was reported by Park and Chang [7] for Rayleigh numbers in the range between 104 to 105 and for spacing/diameter ratio between 2 and 4. Their study confirmed that the upper cylinder was influenced by the upstream cylinders and the lower cylinder was uninfluenced by the upper ones. Sadeghipour and Asheghi [8] have reported an experimental study for natural convection heat transfer from horizontal isothermal cylinders in vertical arrays of two to eight at low Rayleigh numbers of 500, 600, and 700. Their results showed that the maximum enhancements of Nusselt number could be reached at a distance to diameter ratio of 15 and if the separation distance between the cylinders increased more than this ratio then no more heat transfer improvement could be obtained from the array. Numerical study of natural convection from an array of two cylinders was presented by Chouikh et al. [9] for Rayleigh numbers rang 100 to 10,000. Their results showed that the upper cylinder degraded the Nusselt number of a single cylinder at close spacing whereas enhanced it at large spacing, and for the same spacing the heat transfer from the upper cylinder increased with Rayleigh number. Correlating equations were proposed numerically for natural convection from a horizontal circular array of 2-6 cylinders by Corcione [10]. Experimental study on free convection heat transfer from isothermal cylinders in vertical array was studied for Rayleigh numbers range of 103 to 3 × 103 in free space by Ashjaee and Yousefi [11]. In their study, the cylinder spacing was varied from 2 to 5 cylinder diameter and heat transfer correlations were developed for any individual cylinder in the array and for the whole arrays. Similar experiment was done to study the same parameters but for elliptic cylinders in a vertical array where new correlations were obtained by Yousefi and Ashjaee [12]. A pair of vertically aligned horizontal cylinders in water was investigated by Reymond et al. [13] for Rayleigh numbers 2 × 106, 4 × 106 and 6 × 106 and for a range of cylinder spacing of 1.5, 2, and 3 diameters. It was observed that the plume rising from the lower cylinder oscillated out of phase with the flow around the upper cylinder increasing the mixing and the result was enhanced heat transfer. Experimental investigation on natural convection at high Rayleigh numbers from a pair of evenly spaced cylinders in water was reported by Grafsrønningen and Jensen [14] for five different center to center separation distances. Their results showed that at small separation distance of the upper cylinder, the increase in averaged Nusselt number was about 6% and for large one was about 15% to 40%. The effect of dissimilar cylinder spacing between horizontal cylinders in vertical arrays of two or three cylinders was investigated by Grafsrønningen and Jensen [15]. Their results have shown that the Nusselt number on the middle cylinder, in a three-cylinder array, increased compared to the lower cylinder and that of the upper cylinder increased too with not much difference than that of the middle cylinder. Ali [16] has investigated experimentally the steady state natural convection from the outer surface of a single horizontal rectangular or square cylinder in air. He developed a correlation for a transition regime local Nusselt numbers as a function of the modified Rayleigh numbers which is cited here for comparison: 
(1)
Ashjaee et al. [17] investigated the effect of Rayleigh numbers and cylinder spacing from the adiabatic ceiling on both the local and averaged Nusselt numbers around the cylinder numerically. Laminar natural convection heat transfer to Bingham plastic fluids, from two differentially heated isothermal cylinders confined in a square enclosure, was investigated numerically by Baranwal and Chhabra [18]. They obtained simple correlations for the prediction of the average Nusselt number and the limiting Bingham number. Kalendar et al. [19] have reported a natural convective heat transfer from the exposed top surface of an inclined isothermal cylinder, with a circular cross section, mounted on a flat adiabatic base plate numerically. They developed empirical correlations for the heat transfer rates from the top exposed surface of the cylinder. The effect of Prandtl number on laminar natural convection heat transfer in a square enclosure consisting of two differentially heated cylinders were studied numerically by Baranwal and Chhabra [20]. They found that the surface-averaged Nusselt number has a positive dependence on Grashof and Prandtl numbers for a fixed location of the two cylinders and a correlation has been developed as a function of Rayleigh numbers and geometric parameters. Karimi et al. [21] reported a numerical study about steady-state mixed convection around two heated horizontal cylinders in a square 2-D enclosure. Their results showed that both heat transfer rates from the heated cylinders and the dimensionless fluid temperature in the enclosure increase with increasing Richardson number and the cylinder diameter.
Upon this survey, one can conclude that correlations for natural convection from the outer surface of vertical arrays of horizontal noncircular tubes such as squares are not available. Therefore, more studies are still needed to explain the behavior of natural convection heat transfer from such arrays to be useful for engineering applications, which motivates the present study. This experimental study uses five horizontal square tubes aligned in a vertical array of two, three, four or five tubes. Four tubes center to center distance to hydraulic diameter ratio S/Dh are used. Local and averaged natural heat transfer analyses are obtained to determine the Nusselt numbers with the modified Rayleigh numbers in a correlated form for the upper tube in each array and also for any tube in different arrays.
Experimental setup
Figure 1(a) shows a schematic of the vertical array of five horizontal square tubes. Each tube has a cross section of 0.02 × 0.02 m2 with inserted heating element (H) of outside diameter 0.0066 m passing through the center of each tube (Figure 1(b)). The tubes were made of steel (polished mild steel) with 1 m length and uniform thickness of 0.002 m. The tubes ends were caped with Bakelite insulating material of thermal conductivity 0.15 W/mK [22] to reduce the axial conduction. The thickness of each Bakelite end is 2.06 cm. The inside tube was filled with sand to insure lateral uniform conduction heat transfer from the heating element up to the outer surface of the tube (Figure 1(b)). The surface temperature was measured at nine points in the longitudinal direction of each tube 0.1 m apart at three different surfaces (lower, upper, and either side surface), as seen in Figure 1(c). Therefore, 27 total number of calibrated self-adhesive type K thermocouples (Chromel/Alumel, 0.3 second time response with flattened bead) were stuck on each square tube. Two more were stuck on the outer surface of the Bakelite end plates; one at each plate and two thermocouples (0.01 in. or 0.0025 m diameter, one at each plate) were inserted through the Bakelite thickness and leveled with its inside surface at each tube. The ambient air temperature was measured by one more thermocouple. The tubes were oriented horizontally as seen in Figure 1(a) in an array form with equal center to center distance ratio S/Dh in the vertical direction using two vertical bars stands in a room 4.4 × 3.0 m2 with no windows, any air conditioning system or ventilation openings, it has only one entrance which was covered by a thick curtain to minimize any possible forced convection of air. The experiment was done for four S/Dh ratios of 2.5, 5.0, 7.5, and 10.0 and special attention was given to two more ratios of 12.5 and 15. Thermocouples were connected to data acquisition systems. 156-channels were used and connected to a computer where the measured surface temperatures were stored for further analysis. Voltage regulator was used to evenly control the input electrical power to the heating element (H) of each tube of the array. The power consumed by each tube was measured by a Wattmeter and assumed uniformly distributed along its length. The heat flux per unit surface area of the tube was calculated by dividing the consumed power (after deducting the heat lost by both axial conduction through the Bakelite end plates and radiation) over the tube outer surface area. The input power increased such that the maximum tube surface temperature did not exceed 100°C. Tube surface temperature measurements were taken after 2 hours of setting where steady state temperature should have been reached as seen in Figure 2 for two different heat fluxes. It should be mentioned that the surface temperature measurements were taken at different convection heat fluxes in the range 46–510 W/m2. The procedure outlined above was used to generate natural convection heat transfer data in air (Prandtl number ≈ 0.72).
The Effect of Square Tube Location in a Vertical Array of Square Tubes on Natural Convection Heat Transfer
Published online:
13 September 2017Figure 1. Schematic of the experimental system; (a) vertical array of square tubes, (b) tube cross section, (c) thermocouple locations and (d) tubes’ sequential order in the array.
Figure 1. Schematic of the experimental system; (a) vertical array of square tubes, (b) tube cross section, (c) thermocouple locations and (d) tubes’ sequential order in the array.
The Effect of Square Tube Location in a Vertical Array of Square Tubes on Natural Convection Heat Transfer
Published online:
13 September 2017
Figure 2. Tube surface temperature profiles for two different heat fluxes.
Experimental analysis
The uniformly generated heat inside each tube dissipates by convection and radiation from the outer surface in addition to heat lost by conduction through the Bakelite end plates as 
(2) q″ and qr, are the fraction of the heat flux dissipated from the tube surface by convection and radiation, respectively. The heat flux lost by radiation qr and by axial conduction through the Bakelite end plates qbk can be calculated, respectively, as 
(3) 
(4)
In Eq. (4); TiB and TOB are the measured inside and outside surface temperature of the Bakelite end plates, respectively, and kbk and δ present the Bakelite thermal conductivity and thickness, respectively. It should be noted that qr is estimated using the total overall averaged surface temperature 
at each heat flux of the tube and ϵ is the surface emissivity of the tube and it is estimated as 0.27 for polished mild steel [23]. Measurements show that, the fraction of radiated heat transfer is about 25.74% of the total input power while the axial conduction heat lost through the Bakelite end plates is 1.6% at most. Each tube loses heat by radiation either to the surrounding air or to its adjacent tubes. Exchange of thermal radiation between similar tubes is negligibly small, therefore it is assumed that neighboring tubes are in radiant balance [4]. Consequently, the net radiant exchange is with the room walls. Depending on its location in the array, the tube has either one or two tubes in its neighborhood. The shape factor between two long parallel tubes is given as [23]: 
(5) where S is the spacing between centers of the adjacent tubes and Dh is the hydraulic diameter of the tube. Therefore, the shape factor between a tube and the ambient, F1∞ is given as in the second of Eq. (5) where, α = 1 for the top and bottom tubes and α = 2 for the others. For the current square tube array; the hydraulic diameter is used which turned out to be the side length of the tube cross-section (Dh = 0.02 m).
Axial (circumference averaged) convection heat transfer coefficient
In this case, the surface temperature at any station x in the longitudinal direction for each constant heat flux is determined as 
(6) where j is the thermocouple number in the circumference direction at any station x along the surface of the tube. The arithmetic mean surface temperature is calculated along the axial direction for each heat flux as 
(7)
Therefore, for each heat flux there are 9 Tx longitudinal temperature measurements for each tube in the array. Consequently, once the electrical input power to each tube is measured, qr and qbk from Eqs. (3) and (4) and q″ from Eq. (2), then the axial (circumference averaged) convection heat transfer coefficient hx can be calculated for each tube. 
(8)
Hence, the non-dimensional Nusselt and the modified Rayleigh numbers are obtained as 
(9)
All physical properties are evaluated at the axial circumference averaged mean temperature θx for each q″.
Total overall averaged convection heat transfer coefficient
Following our previous study for single horizontal square tube [16] in air where the circumference averaged convection heat transfer coefficient hx is first evaluated at each station x as in Eq. (8) and then the overall longitudinal averaged 
is obtained as (excluding the end effects as we will see next at the result section) 
(10)
Therefore, each heat flux q″ is presented by only one overall averaged convection heat transfer coefficient on contrary to the local case, where q″ is presented by seven hx along the longitudinal direction given by Eq. (8). All circumference averaged physical properties are first obtained at θx, and then the overall averaged properties are obtained the same way following Eq. (10). The non-dimensional overall averaged Nusselt and the modified Rayleigh numbers are defined using the characteristic length Dh as 
(11) and the total averaged convection heat transfer at each heat flux is given by 
(12)
Experimental uncertainty
The experimental uncertainty is estimated in the calculated results on the basis of uncertainties in the primary measurements. It should be noted that, some of the experiments had to be repeated in order to check the calculated results and the general trends of the data to be sure that the experiment is going as planned. The error in measuring the temperature, estimating the emissivity and in calculating the surface area is ± 0.5 °C, ± 0.02, and ± 0.003 m2, respectively. The accuracy in measuring the voltage is taken from the manual of the Wattmeter as 0.5% of reading ± 2 counts with a resolution of 0.1 V and the corresponding one for the current is 0.7% of reading ± 5 counts + 1 mA with a resolution of 1 mA. At each heat flux, 40 scans of the temperature measurement were made by the data acquisition system for each channel and the mathematical averaged of those scans was obtained. It should be noted that the uncertainty in the calculated result was estimated using the method recommended by Kline and McClintock [24] and Moffat [25] and a computer program was written to do that. Table 1 shows the maximum itemized uncertainties of the calculated result.
The Effect of Square Tube Location in a Vertical Array of Square Tubes on Natural Convection Heat Transfer
Published online:
13 September 2017Table 1. The maximum percentage uncertainties of various quantities.
Results and discussion
Natural convection heat transfer experiment from a single horizontal tube is obtained first to compare the current result with those in the literature to have confidence in our results. Therefore, Figure 3 shows a comparison between the Nusselt numbers and the modified Rayleigh numbers for a heated horizontal square tube in air. This comparison shows that 91% of the present experimental data lie within a bandwidth of +15% and/or –10% of correlation (1) of Ali [16]. These differences could be attributed to that the experiment of [16] was done such that air is trapped inside the tube surrounding the heater in the middle. However, in the current experiment sand is used to fill the gap inside the tube to insure uniform heat conduction transfer between the heater and the outer surface of the horizontal tube. Figure 4(a) shows some samples of the local circumference averaged temperatures for the upper tube of a five-tube array normalized by the ambient temperature for some selected heat fluxes. It can be seen that the temperature slightly increases up to almost the middle of the tube (x/L = 0.6) and then decreases. Those slight temperature drops near the ends could be attributed to the thinning of the boundary layer due to the edge effect as observed previously for horizontal plate [26]. This change in the temperature profile occurs between the dimensionless axial distances x/L of 0.2 and 0.8 where the end effect is clear beyond those two limits where the temperature profiles have sharp increase or decrease, respectively. These sharp changes are shown in Figure 4(a) on the left or right of the vertical dashed lines at x/L of 0.2 and 0.8, respectively. Therefore, the test section of all tubes is chosen between these two limits. This figure shows also that as the heat flux increases the temperature increases as expected. The corresponding local circumference averaged convection heat transfer coefficients (hx) are presented in Figure 4(b) for the same heat fluxes used in Figure 4(a). As mentioned earlier, the end effect is clear such that the convection heat transfer coefficients have sharp decrease or increase at both limits, respectively. However, by inspection it can be seen that the convection heat transfer coefficient decreases up to x/L = 0.6 and then increases up to x/L = 0.8. The vertical dashed line at x/L = 0.6 separates these two regions.
The Effect of Square Tube Location in a Vertical Array of Square Tubes on Natural Convection Heat Transfer
Published online:
13 September 2017
Figure 3. Comparison with Ali [16] for single square horizontal tube in air. Dashed lines present the error bandwidth and the solid line presents the correlation of Ref. [16].
The Effect of Square Tube Location in a Vertical Array of Square Tubes on Natural Convection Heat Transfer
Published online:
13 September 2017Figure 4. Natural convection profiles for different heat fluxes; (a) temperatures and (b) local circumference averaged convection heat transfer coefficients.
Figure 4. Natural convection profiles for different heat fluxes; (a) temperatures and (b) local circumference averaged convection heat transfer coefficients.
Local circumference averaged results
In this section, the local circumference averaged data are presented using the tube longitudinal axial coordinate as a characteristic length in both Nusselt and the modified Rayleigh numbers (Eq. (9)).
The upper (downstream) tube in the array
The natural convection heat transfer from the upper tube in different arrays of tubes is presented in this section compared to that of a single tube in air. Figure 5 shows the Nusselt numbers of the upper tube in a five-tube array of equally spaced S/Dh = 2.5 (Figure 1(a or d)) compared to that of single tube as a function of the modified Rayleigh numbers. It is clear that the effect of the upper tube geometric position in the array is to degrade the heat transfer by about 12% of the single tube alone. It should be noted that the convection heat transfer from such an array has a competition between the developed velocity of the rising plume from all the bottom tubes and that of the accumulated thermal boundary layer. The natural convection velocity of the plume tends to enhance the convection heat transfer coefficient from the upper tube. On the other hand, this plume increases the thermal boundary layer around the tube, which tends to degrade the heat transfer from such a tube. This competition depends on the center to center distance ratio S/Dh between the tubes in different arrays, for small distance ratio the effect of thermal boundary layer dominates and as the distance ratio increases this degradation decreases and for large distance ratio it enhances the convection heat transfer over that of a single tube. The fitted correlation of the upper tube alone is shown in Figure 5 as dashed line whereas that in the array is shown as solid line and given, respectively for S/Dh = 2.5 as: 
(13) 
(14) where the coefficient of determination (R2) are 99.3% and 99.6%, respectively. Figures 6(a–c)Table 2show Nusselt numbers of the upper (downstream) tube in four, three and two-tube array, respectively, for S/Dh = 2.5. The degradation in Nusselt numbers from that of the single tube is 11%, 9%, and 7%, respectively. Correlations of those top tubes in different arrays are obtained with those of the single tube alone of the form 
(15)
The Effect of Square Tube Location in a Vertical Array of Square Tubes on Natural Convection Heat Transfer
Published online:
13 September 2017
Figure 5. Nusselt numbers versus the modified Rayleigh numbers showing the effect of the upper tube in an array of five tubes compared to that of a single tube, dashed and solid lines, respectively, present the fitting correlations given by Eqs. (13) and (14) for S/Dh = 2.5.
The Effect of Square Tube Location in a Vertical Array of Square Tubes on Natural Convection Heat Transfer
Published online:
13 September 2017Figure 6. Nusselt numbers versus the modified Rayleigh numbers showing the effect of the upper (downstream) tube in different arrays for S/Dh = 2.5 compared to that of a single tube. Dashed and solid lines present the fitting correlations given by Eq. (15) (see Table 2 for details); (a) four-tube array, (b) three-tube array and (c) two-tube array.
Figure 6. Nusselt numbers versus the modified Rayleigh numbers showing the effect of the upper (downstream) tube in different arrays for S/Dh = 2.5 compared to that of a single tube. Dashed and solid lines present the fitting correlations given by Eq. (15) (see Table 2 for details); (a) four-tube array, (b) three-tube array and (c) two-tube array.
The above constants a, b, and R2 for the upper tubes in different arrays are shown in Table 2. By inspection, the exponent b of (Ra*x)b in Eq. (14) and those shown in Table 2 of other tubes are in the range ≤ 0.25 which indicates that the mode of heat transfer is almost laminar since the laminar natural convection heat transfer is characterized by an exponent b ≅ 1/5 and with the other constant a ≤ 0.669 [26–29]. Figure 7 shows the correlations of the upper tube in different arrays compared to that of a single tube. It is clear that as the number of upstream tubes increases the degradation of heat transfer increases. The general correlation, for S/Dh = 2.5, covers each upper (downstream) tube in different arrays is obtained as: 
(16)
The Effect of Square Tube Location in a Vertical Array of Square Tubes on Natural Convection Heat Transfer
Published online:
13 September 2017Table 2. Correlation coefficients for the top tubes in different arrays given by Eq. (15) for S/Dh = 2.5 and for modified Rayleigh number range 3 × 108 < Ra*x < 7 × 1011 compared to that of a single tube.
The Effect of Square Tube Location in a Vertical Array of Square Tubes on Natural Convection Heat Transfer
Published online:
13 September 2017Figure 7. Nusselt numbers versus the modified Rayleigh numbers showing the upper (downstream) tube's correlations of different equally spaced arrays (S/Dh = 2.5) compared to that of the single tube, the general correlation for any upper tube is given by Eq. (16).
Figure 7. Nusselt numbers versus the modified Rayleigh numbers showing the upper (downstream) tube's correlations of different equally spaced arrays (S/Dh = 2.5) compared to that of the single tube, the general correlation for any upper tube is given by Eq. (16).
Where, N is the total number of tubes in each array (2, 3, 4 or 5). In other words, if N = 4, Eq. (16) corresponds to the upper tube in an array consists of four equally spaced tubes with center to center distance ratio S/Dh = 2.5. The coefficient of determination (R2) of Eq. (16) is 95.4%.
Upstream tubes in the array
Does the existence of any downstream tubes affect the upstream tubes? In order to answer this question Figures 8 and 9 are constructed. Figure 8 shows the effect of changing the tube location in different arrays, as being at the top (downstream) tube in a four-tube array or the second tube from the top in a five-tube array. By inspection, it can be seen that (at this S/Dh = 2.5 center to center distance ratio) at Ra*x > 3 × 1010 the two correlations are identical which means that the downstream tube has no effect on the upstream test tube. However, as the Ra*x decreases, very small differences are observed and their maximum occurs at Ra*x = 3 × 108 as 4% enhancement in Nusselt number due to the existence of one more tube at the top of the test tube. Nevertheless, this small enhancement is within the experimental uncertainty percentage error in Nusselt numbers as mentioned earlier. Therefore, it can be concluded that at S/Dh = 2.5 the tubes in the arrays are affected only by the upstream tubes and are independent of those downstream. This observation is obtained for all other S/Dh ratios. Figure 8 shows also the Nusselt number profile for the same tube if it existed as a single tube in air for comparison and the solid and dashed lines present the fitting curves through the data. It should be noted that this finding for S/Dh = 2.5 agrees very well with the scenario obtained by Corcione [10] where he has reported for circular cylinder array problem that it is a one-way coordinate problem for S/D > 2 where the cylinder in the array interact only with the upstream ones and independent of the downstream cylinders. Figure 9(a, b) shows another similar example of that where the tubes in the array of S/Dh = 2.5 is unaffected by the tubes over them but only affected by the tubes beneath them. Figure 9(a) shows the Nusselt numbers profiles for the tube being at the top in two-tube array or at number four in a five-tube array compared to its single tube profile. Figure 9(b) shows the correlations with no symbols for more clarification. The area on the right of the vertical line (Ra*x > 2 × 1010) shows no differences however at Ra*x = 5 × 108, the maximum enhancement is 3.9% which is within the experimental uncertainty as mentioned earlier. It should be noted that the upstream tube (lowest one) in any array is unaffected by the number of downstream tubes and it behaves as a single tube. Figure 10 shows that for the bottom of a five-tube array compared to that of a single tube. Beyond Ra*x > 3 × 1010 (vertical line) the two correlations are identical however, at Ra*x = 7 × 108 a difference of 4.5% which is in the range of the experimental uncertainty is observed. The percentage degradation in Nusselt numbers compared to that of a single tube at different modified Rayleigh numbers and for a five-tube array is presented in Figure 11 for S/Dh = 2.5, where Y/Dh is the dimensionless vertical distance measured from the center of the lowest tube as seen in Figure 1(d). It is clear that as the position of the tube increases downstream the degradation percentage increases whereas these degradations get small as the modified Rayleigh number increases for each tube in the array. It should be noted that, dashed lines showing in Figure 11 are connected only to make it easier for the reader to visualize the percentage of degradation from that of a single tube alone. The overall general correlation for each tube, in any different array, is obtained for S/Dh = 2.5 as: 
(17)
The Effect of Square Tube Location in a Vertical Array of Square Tubes on Natural Convection Heat Transfer
Published online:
13 September 2017Figure 8. Nusselt numbers versus the modified Rayleigh numbers showing that the upstream tubes in the array is almost unaffected by the downstream tubes (see Figures 1(a) and (d). for the tubes’ sequential order). Dashed and solid lines present the fitting correlations through the data.
Figure 8. Nusselt numbers versus the modified Rayleigh numbers showing that the upstream tubes in the array is almost unaffected by the downstream tubes (see Figures 1(a) and (d). for the tubes’ sequential order). Dashed and solid lines present the fitting correlations through the data.
The Effect of Square Tube Location in a Vertical Array of Square Tubes on Natural Convection Heat Transfer
Published online:
13 September 2017Figure 9. Nusselt numbers versus the modified Rayleigh numbers showing the upstream tube geometric position in the array is almost unaffected by the downstream tubes (S/Dh = 2.5). Dashed and solid lines present the fitting correlations through the data; (a) fitting correlations with added symbols and (b) no symbols for clarification.
Figure 9. Nusselt numbers versus the modified Rayleigh numbers showing the upstream tube geometric position in the array is almost unaffected by the downstream tubes (S/Dh = 2.5). Dashed and solid lines present the fitting correlations through the data; (a) fitting correlations with added symbols and (b) no symbols for clarification.
The Effect of Square Tube Location in a Vertical Array of Square Tubes on Natural Convection Heat Transfer
Published online:
13 September 2017Figure 10. Nusselt numbers versus the modified Rayleigh numbers showing the lower tube in the array is almost unaffected by the number of downstream tubes compared to that of a single tube. Dashed and solid lines present the fitting correlations through the data.
Figure 10. Nusselt numbers versus the modified Rayleigh numbers showing the lower tube in the array is almost unaffected by the number of downstream tubes compared to that of a single tube. Dashed and solid lines present the fitting correlations through the data.
The Effect of Square Tube Location in a Vertical Array of Square Tubes on Natural Convection Heat Transfer
Published online:
13 September 2017Figure 11. The effect of tube position in the array on the percentage degradation of Nusselt numbers compared to that of a single tube at different modified Rayleigh numbers for S/Dh = 2.5. Numbers next to the symbols present the tubes’ sequential order as seen in Figures 1(a) and (d).
Figure 11. The effect of tube position in the array on the percentage degradation of Nusselt numbers compared to that of a single tube at different modified Rayleigh numbers for S/Dh = 2.5. Numbers next to the symbols present the tubes’ sequential order as seen in Figures 1(a) and (d).
Where N and M stand for number of tubes in the array (2, 3, 4, or 5) and the sequence number of the tube in the array (1, 2, 3, or 4), respectively as seen in Figures 1(a) and (d). The coefficient of determination (R2) for Eq. (17) is 95.2%. The position of the tube in the array is highly affected by increasing S/Dh ratio. Figure 12(a–c) shows the Nusselt numbers percentage of enhancement over that of single tube versus the tube position in a five-tube array for different S/Dh = 5, 7.5, and 10, respectively at four different modified Rayleigh numbers. Figure 12(a–c) shows that the percentage of enhancement depends strongly on the tube position in the array as Y/Dh increases for fixed S/Dh. Furthermore, increasing S/Dh for the same tube in the array enhances also the heat transfer from such tube as seen, for example, for tube number four (+) in Figures 12(a–c). It is clear that as the ratio S/Dh increases beyond 2.5, the degradations shown in Figure 11 have been recovered to enhancement over that of a single tube and this enhancement increases as the ratio S/Dh increases. These enhancements could be attributed to the effect of rising plume velocity that has managed to overcome that of the accumulated thermal boundary layer as explained earlier. Dashed lines in Figure 12 connecting the data points are showing only to make it easier for the reader to visualize the percentage of enhancement from that of a single tube alone. Table 3 shows the percentage of enhancement for each tube (Y/Dh) over that of a single tube for a five-tube array at S/Dh = 5, 7.5 and 10. The overall general correlation covering the position of each tube in different arrays for S/Dh = 5, 7.5 and 10 is obtained as 
(18) where N and M as defined earlier and the coefficient of determination (R2) is 94.7%.
The Effect of Square Tube Location in a Vertical Array of Square Tubes on Natural Convection Heat Transfer
Published online:
13 September 2017Figure 12. The effect of tube position in a five-tube array on the enhancement percentage of Nusselt numbers compared to that of single tube at different modified Rayleigh numbers for different S/Dh: (a) S/Dh = 5, (b) S/Dh = 7.5 and (c) S/Dh = 10. Numbers next to the symbols present the tubes’ sequential order as seen in Figures 1(a) and (d).
Figure 12. The effect of tube position in a five-tube array on the enhancement percentage of Nusselt numbers compared to that of single tube at different modified Rayleigh numbers for different S/Dh: (a) S/Dh = 5, (b) S/Dh = 7.5 and (c) S/Dh = 10. Numbers next to the symbols present the tubes’ sequential order as seen in Figures 1(a) and (d).
The Effect of Square Tube Location in a Vertical Array of Square Tubes on Natural Convection Heat Transfer
Published online:
13 September 2017Table 3. Percentage of heat transfer enhancement for each tube (Y/Dh) for a five-tube array (N = 5) at S/Dh = 5, 7.5, and 10.
Overall averaged results
Figure 13 shows the overall averaged results for a five-tube array for S/Dh = 2.5. As in the local results, the upstream tube (number 5 in Figure 13) is unaffected by the downstream tubes and its 
is almost equal to that of a single tube similar to the result of [6] for circular cylinder array. On the other hand, the average Nusselt number of any downstream tube decreases as the tube geometric position increases away from the bottom one towards downstream as seen in the figure. Fitting correlation is obtained for each tube in the array in the form: 
(19) where a, b, and the coefficient of determination (R2) are given in Table 4. The overall general correlation covering all tubes in all different arrays is obtained for (S/Dh = 2.5) as: 
(20) with R2 = 94.2%. The degradation ratio of Nusselt numbers is shown in Figure 14 for different modified Rayleigh numbers for all tubes in a five-tube array. It is worth noting that the fourth tube from the top is not affected by the change in the modified Rayleigh number because the strength of the natural convection velocity developed only by one lower tube however, as the number of tubes increases downstream the effect of the modified Rayleigh number become remarkable. Dashed lines in Figure 14 connecting the data points are showing only to make it easier for the reader to visualize the percentage of degradation from that of a single tube alone. The position of the tube in the array is highly affected by increasing S/Dh ratio as seen for the local results. Figure 15(a–c) shows that effect for S/Dh = 5, 7.5, and 10, respectively, on the enhancement of Nusselt numbers over that of a single tube for each tube (at different Y/Dh) in a five-tube array. It is clear that as the ratio increases beyond 2.5, the degradations shown in Figure 14 have been changed to enhancement over that of a single tube and this enhancement increases as the ratio S/Dh increases. This enhancement could be attributed to the effect of rising plume velocity overcomes that of the accumulated thermal boundary layer as the ratio increases as explained earlier. Dashed lines in Figure 15 connecting the data points are showing only to make it easier for the reader to visualize the percentage of enhancement from that of a single tube alone. The overall general correlation covering the position of each tube in different arrays for S/Dh = 5.0, 7.5, and 10 is obtained as: 
(21) with R2 = 90.4%. As mentioned in the local case, the geometric position of each tube in the array is affected by the S/Dh ratio. The effect of S/Dh on the upper downstream tube in different arrays is shown in Figure 16 at Ra*x = 6.91 × 104 as a pe-rcentage of that of a single tube. It is clear that for fixed number of tubes (N) in the array the percentage of enhancement in convection heat transfer coefficient increases as the space between tubes increases, which means that the effect of rising plume velocity overcomes that of the accumulated thermal boundary layer as, explained earlier. On the other hand, as the space increases more the plume velocity gets weak and the plume dispersed where it has a little effect on the upper (downstream) tube. Therefore, Figure 16 shows that a critical spacing distance does exist where the enhancement in convection heat transfer coefficient reaches its maximum and then degrades. The vertical line connects the locus of the critical spacing distance. This figure shows also that for small spacing (S/Dh = 2.5) as the number of tubes in the array decreases the degradation from that of the single tube gets small. However, for large spacing (S/Dh ≥ 5) as the number of tubes in the array increases the percentage of enhancement increases too. The overall correlation covering the upper (downstream) tube in the arrays for S/Dh = 2.5 is given by 
(22) where, N is the number of tubes in each array (2, 3, 4, or 5). In other words, if N = 4, Eq. (22) corresponds to the upper tube in an array consists of four equally spaced tubes with center to center distance ratio S/Dh = 2.5. The coefficient of determination (R2) for Eq. (22) is 94.3%. It should be noted that for other S/Dh ratios, Eq. (21) should be used with M = 1.
The Effect of Square Tube Location in a Vertical Array of Square Tubes on Natural Convection Heat Transfer
Published online:
13 September 2017Figure 13. Nusselt numbers versus the modified Rayleigh numbers showing the overall averaged results for a five-tube array (S/Dh = 2.5). Numbers next to the symbols present the tubes’ sequential order as seen in Figures 1(a) and (d) and dashed and solid lines present the fitting correlations through the data (see Eq. (19) and Table 4 for details).
Figure 13. Nusselt numbers versus the modified Rayleigh numbers showing the overall averaged results for a five-tube array (S/Dh = 2.5). Numbers next to the symbols present the tubes’ sequential order as seen in Figures 1(a) and (d) and dashed and solid lines present the fitting correlations through the data (see Eq. (19) and Table 4 for details).
The Effect of Square Tube Location in a Vertical Array of Square Tubes on Natural Convection Heat Transfer
Published online:
13 September 2017The Effect of Square Tube Location in a Vertical Array of Square Tubes on Natural Convection Heat Transfer
Published online:
13 September 2017Figure 14. The effect of tube position in the array on the degradation ratio of Nusselt numbers compared to that of a single tube at different modified Rayleigh numbers (Overall averaged results) for S/Dh = 2.5. Numbers next to the symbols present the tubes’ sequential order as seen in Figures 1(a) and (d).
Figure 14. The effect of tube position in the array on the degradation ratio of Nusselt numbers compared to that of a single tube at different modified Rayleigh numbers (Overall averaged results) for S/Dh = 2.5. Numbers next to the symbols present the tubes’ sequential order as seen in Figures 1(a) and (d).
The Effect of Square Tube Location in a Vertical Array of Square Tubes on Natural Convection Heat Transfer
Published online:
13 September 2017Figure 15. The effect of tube position in a five-tube array on the enhancement ratio of Nusselt numbers compared to that of a single tube at different modified Rayleigh numbers (Overall averaged results) for different S/Dh; (a) S/Dh = 5, (b) S/Dh = 7.5 and (c) S/Dh = 10. Numbers next to the symbols present the tubes’ sequential order as seen in Figures 1(a) and (d).
Figure 15. The effect of tube position in a five-tube array on the enhancement ratio of Nusselt numbers compared to that of a single tube at different modified Rayleigh numbers (Overall averaged results) for different S/Dh; (a) S/Dh = 5, (b) S/Dh = 7.5 and (c) S/Dh = 10. Numbers next to the symbols present the tubes’ sequential order as seen in Figures 1(a) and (d).
The Effect of Square Tube Location in a Vertical Array of Square Tubes on Natural Convection Heat Transfer
Published online:
13 September 2017Figure 16. The effect of upper (downstream) tube position in the array compared to that of a single tube at Ra*D = 6.91 × 104 for different S/Dh ratio.
Figure 16. The effect of upper (downstream) tube position in the array compared to that of a single tube at Ra*D = 6.91 × 104 for different S/Dh ratio.
Comparison with the previous results
Most of the previous literature, cited earlier on horizontal circular cylinders in vertical array, have used overall averaged Nusselt numbers with the Rayleigh numbers in their correlations. Corcione [10] has developed an averaged correlation for the Nusselt number for any circular cylinder (Mi) in the array (excluding the bottom one) as a function of Y/D. His equation was correlated with Rayleigh number as 
(23)
Therefore, the current experimental data is re-analyzed for the third tube (M = 3) in an array of 5 tubes with center to center distance 5.0 cm (S/Dh = 2.5) based on overall averaged results using the regular Rayleigh numbers not the modified one. The correlation obtained for that tube at Y/Dh = 5 is 
(24) with R2 = 94.4%. Figure 17 shows a comparison between Eq. (24) of the current experimental results and the numerical correlation of [10], Eq. (23) for Mi = 3 and Y/D = 5. In spite of the fact that the comparison is not one to one, Figure 17 gives a quantitative agreement of the profile trend of Nusselt numbers with the Rayleigh numbers for both arrays.
The Effect of Square Tube Location in a Vertical Array of Square Tubes on Natural Convection Heat Transfer
Published online:
13 September 2017
Conclusions
The effect of square horizontal tube location in a vertical array of 2, 3, 4, or 5 tubes on natural convection is investigated. Local and average results show that the lowermost tube is uninfluenced by the downstream tubes and its Nusselt number is substantially identical to that of single tube in free space up to the closest investigated ratio S/Dh of 2.5. It is also shown that, any tube in the array is influenced only by the upstream tubes and unaffected by the downstream ones for the range of S/Dh studied. Furthermore, the top (downstream) tube in each array always degrade the Nusselt number the most than that of a single tube followed by other tubes upstream towards the lower tube at small S/Dh = 2.5. This degradation in general depends on the tube center to center distance ratio S/Dh in the array. As S/Dh increases the degradation is recovered and changed to enhancement in Nusselt numbers. Critical S/Dh is obtained for the upper (downstream) tube in each array. This enhancement depends on the number of tubes in the array as well as on the S/Dh ratio. Local and averaged correlations are obtained for each tube in different arrays in terms of Nusselt numbers and the modified Rayleigh numbers for S/Dh = 2.5 where the degradation occurs (Eqs. (17) for local and (20) for average). Furthermore, general local and averaged correlations, where enhancement in heat transfer occurs, are obtained by Eqs. (18) and (21), respectively for S/Dh = 5, 7.5, and 10.
Acknowledgments
The authors extend their appreciation to the Deanship of Scientific Research at King Saud University for funding this work through the research group project No RGP-080.
Nomenclature
| As | tube surface area, m2 |
| Abk | end plate cross section area, m2 |
| D | diameter of circular cylinder as a characteristic length (m) |
| Dh | hydraulic diameter equal to the square of cross section side length, m |
| EIP | electrical input power, W |
| F12 | shape factor between two long parallel cylinders |
| F1∞ | shape factor between any tube and the ambient |
| g | acceleration due to gravity, m/s2 |
| H | heating element |
| h | convection heat transfer coefficient, W m-2 K-1 |
| k | thermal conductivity, W m-1 K-1 |
| L | tube length, (= 1m) |
| M | ordinal number of the ith tube in the array, 1 ≤ M ≤ 4 |
| N | number of tubes in the array, 2 ≤ N ≤ 5 |
| Nu | Nusselt number, h Dh / k or h x / k |
| q″ | convection heat flux, W/m2 |
| qr | radiation heat flux, W/m2 |
| qbk | heat lost by conduction through the Bakelite end plates, W/m2 |
| Qc | averaged convection heat transfer rate, W |
| R2 | coefficient of determination (correlation coefficient) |
| Ra* | modified Rayleigh number, gβ q″x4 ν− 1α− 1 k− 1 or g β q″D4h − 1− 1k− 1 |
| Ra | Rayleigh number, gβ ΔTD3 ν− 1α− 1 |
| S | distance of tube enter to the next tube center, m |
| S/Dh | distance ratio |
| T | temperature, K |
| TiB | inside surface temperature of the Bakelite end plate, K |
| ToB | outside surface temperature of the Bakelite end plate, K |
| x | axial or axial distance, m |
| Y/Dh | non-dimensional coordinate |
Greek symbols
| α | thermal diffusivity, m2 s− 1 |
| β | coefficient for thermal expansion, K-1 |
| δ | Bakelite thickness, m |
| ϵ | emissivity |
| θ | arithmetic mean temperature defined by Eq. (7), K |
| ν | kinematics viscosity, m2 s− 1 |
| σ | Stefan- Boltzmann constant, (= 5.67 × 10−8, W m−2 K−4) |
Subscripts
| bk | Bakelite |
| Dh | characteristic length, m |
| j | indices in the perimeter direction ranging from 1 to 3 |
| x | indices in the axial direction ranging from 1 to 9 |
| x | characteristic length, m |
| ∞ | ambient condition |
Superscripts
| − | averaged value |
| Quantity | Uncertainty (±%) |
|---|---|
| EIP | 2.5 |
| qr | 9.8 |
| q″ | 4.4 |
| hx | 5.7 |
| Nux | 5.7 |
| Ra*x | 5.4 |
 | 6.2 |
 | 4.2 |
 | 10.2 |
| Number of tubes in the array, equally spaced at S/Dh = 2.5. | ||||||||||||
| Single tube | 4 tubes | 3 tubes | 2 tubes | |||||||||
| Fig. # | a | b | R2% | a | b | R2% | a | b | R2% | a | b | R2% |
| 6(a) | 0.413 | 0.235 | 99.3 | 0.302 | 0.242 | 99.2 | ||||||
| 6(b) | 0.259 | 0.252 | 99.3 | 0.245 | 0.251 | 99.6 | ||||||
| 6(c) | 0.366 | 0.239 | 99.1 | 0.302 | 0.244 | 99.5 | ||||||
 × 100 | ||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
 Figure 12(a) |  Figure 12(b) |  Figure 12(c) | ||||||||||
| Ra*x | 5 (+) | 10 (□) | 15 (■) | 20 (▲) | 7.5 (+) | 15 (□) | 22.5 (■) | 30 (▲) | 10 (+) | 20 (□) | 30 (■) | 40 (▲) |
| 1 × 1010 | 6.3 | 8.1 | 10.6 | 15.2 | 9.9 | 11.7 | 14.4 | 19.1 | 12.5 | 14.4 | 17.2 | 22.1 |
| 3 × 1010 | 7.3 | 9.1 | 11.6 | 16.2 | 10.9 | 12.8 | 15.5 | 20.2 | 13.6 | 15.5 | 18.3 | 23.2 |
| 1 × 1011 | 8.4 | 10.2 | 12.8 | 17.4 | 12.1 | 13.9 | 16.7 | 21.5 | 14.8 | 16.7 | 19.5 | 24.4 |
| 3 × 1011 | 9.4 | 11.2 | 13.8 | 18.5 | 13.2 | 15.1 | 17.8 | 22.6 | 15.9 | 17.8 | 20.6 | 25.6 |
| Tube number | a | b | R2% |
|---|---|---|---|
| 1 | 0.326 | 0.220 | 98.5 |
| 2 | 0.467 | 0.197 | 98.2 |
| 3 | 0.544 | 0.186 | 96.2 |
| 4 | 0.625 | 0.175 | 99.3 |
| 5 | 0.694 | 0.174 | 99.3 |
