# Center-to-Center Distance’s Effect between Vertical Square Tubes of a Horizontal Array on Natural Convection Heat Transfer

^{1}

^{2}

^{*}

## Abstract

**:**

^{2}. Results show that at small S/D, the Nusselt number of any tube in the array is lower than that of the single tube up to a specific S/D and then increases as the ratio increases. Empirical correlations are obtained for each tube in the array at different S/D using the modified Rayleigh numbers only. General correlations using S/D as a parameter are obtained for each tube, and an overall general correlation using both S/D and the tube number (n) as parameters is obtained. The difference between the predicted and experimental Nusselt numbers is in the reasonable range even at high Rayleigh numbers.

## 1. Introduction

## 2. Experimental Setup

#### 2.1. Heat Transfer Analyses

_{c}, q

_{r}, q

_{b}, A

_{s}, and A

_{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) ≈ 0.72. The heat dissipated by conduction through the Bakelite ends and by radiation is obtained from:

_{ib}, T

_{ob}, δ, k

_{b}are the inside and outside surface temperatures of the Bakelite ends, thickness, and the thermal conductivity of the Bakelite, respectively. Furthermore, $\mathsf{\epsilon},\text{}\mathsf{\sigma},{F}_{1\infty}$ 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

#### 2.2. Experimental Uncertainty

^{2}, respectively. The Wattmeter manual provides the accuracy of the voltage as 0.5% of reading ±2 counts with a 0.7 resolution of 0.1 V and 0.7% of reading ± 5 counts + 1 mA with a resolution of 1 mA for the current. It should be noted that at each heat flux, 40 temperature measurement scans were obtained using the data acquisition system, and the average was taken. The method suggested by Kline and McClintock [17] and Moffat [18] was followed in calculating the uncertainty of the results. Table 1 shows the maximum uncertainties of the calculated results.

## 3. Results and Discussion

^{2}of tube number one and the other tubes at different S/D ratios compared to that of a single tube.

_{x}as S/D increases for different Rayleigh numbers compared to that of a single tube (at S/D = 0). These results show that the percent of degradation increases as the Ra

_{x}increases, as shown for S/D = 1.75; however, Ra

_{x}has a low effect on Nu

_{x}at large S/D = 4.25. Table 5 shows the percentage of degradation or enhancement over the single tube for the three tubes in the array at different S/D as shown in Figure 12a–c.

^{2}are shown in Table 6. Figure 13a–c shows differences between the predicted and the experimental Nusselt numbers with S/D as a parameter using a correlation number (13). The maximum deviation, which appears at high Rayleigh numbers, is shown in Table 6. Figure 14 shows a maximum difference of 17.9% between the predicted and the experimental Nusselt numbers using both S/D and the tube’s number (n) as parameters. The overall correlation for the array is obtained as:

## 4. Conclusions

_{x}decreases sharply and then starts to increase until it equals that of the single tube. If the S/D increases, the effect of buoyancy force overcomes the effect of accumulated boundary layers, which leads to an enhancement in Nu

_{x}over that of the single tubes. If S/D increases more, it would be expected that Nu

_{x}reached that of the single tubes. Table 5 shows the percent of degradation and enhancement in Nu

_{x}compared to that of the single tube. Two empirical correlations are obtained for Nu

_{x}versus $R{a}_{x}^{*}$ only, for Nu

_{x}, $R{a}_{x}^{*}$, and S/D as a parameter for each tube in the array. A more general correlation is obtained for all tubes in the array using both S/D and the tube’s number (n) as parameters (Equation (14)). These correlations are necessary for any engineering applications related to such kinds of tubes, which could be used in heat exchangers.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

Ab | Bakelite cross-section area |

As | the Tube outside surface area |

C | Constant |

D | Tube’s equivalent hydraulic diameter |

F | Shape factor |

g | Gravitational acceleration |

h | Heat transfer coefficient |

k | Thermal conductivity |

L | Tube length |

Nu | Nusselt number |

n | Tube number |

qb | Conduction heat flux through the Bakelite insulation capped ends |

qc | Convection heat flux |

qr | Radiation heat flux |

$R{a}_{x}^{*}$ | Local modified Rayleigh number |

S | Center-to-center distance |

T | Temperature |

x | Local length |

Greek symbols | |

α | Thermal diffusivity |

β | Coefficient of thermal expansion |

δ | Bakelite thickness |

ε | Surface emissivity |

θ | Arithmetic mean temperature |

ϑ | Kinematic viscosity |

σ | Planks constant |

Subscripts | |

s | Single tube |

x | axial or circumference averaged |

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**Figure 2.**Specification of each tube; (

**a**) cross-section showing the internal heating element at the center and (

**b**) thermocouple locations along each tube.

**Figure 3.**Local circumference average normalized temperatures along the vertical surface of the tube at different heat fluxes.

**Figure 4.**Steady-state temperature profiles at the outer surface of the tubes at three different supplied powers.

**Figure 5.**Temperature measurements in the middle of the tube at different heat fluxes at the three circumference surfaces: front (F), left (L), and right (R); (

**a**) 108.0 W/m

^{2}, (

**b**) 221.0 W/m

^{2}, and (

**c**) 341.0 W/m

^{2}.

**Figure 6.**Thermocouple calibration against a platinum resistance thermometer using samples of seven thermocouples.

**Figure 8.**The effect of a small center-to-center distance ratio on the natural convection heat transfer from the left tube in an array of three tubes; (

**a**) S/D = 1.75 and (

**b**) S/D = 2.75.

**Figure 9.**Local Nusselt numbers versus the modified Rayleigh numbers for different S/D ratios of the left tube in the array of three tubes compared to that of the single tube.

**Figure 10.**Local Nusselt numbers versus the modified Rayleigh numbers for different S/D ratios corresponding to the middle and right tube compared to that of a single tube; number 2 in (

**a**), and number 3 in (

**b**).

**Figure 11.**Local Nusselt numbers versus S/D ratios for different modified Rayleigh numbers; tube number 1 in (

**a**), tube number 2 in (

**b**), and tube number 3 in (

**c**).

**Figure 12.**The percentage of degradation and enhancement of Nu

_{x}versus S/D for different Rayleigh numbers for the different tubes in the array; (

**a**) tube number 1, (

**b**) tube number 2, and (

**c**) tube number 3.

**Figure 13.**Differences between the predicted and the experimental Nusselt numbers using S/D as a parameter; (

**a**) tube number 1, (

**b**) tube number 2, and (

**c**) tube number 3.

**Figure 14.**Differences between the predicted and the experimental Nusselt numbers using S/D and tube number (n) as parameters for the array.

Quantity | Uncertainty (±%) |
---|---|

Electrical input power | 3.4 |

${q}_{r}$ | 9.0 |

${q}_{c}$ | 4.7 |

${h}_{x}$ | 4.8 |

$N{u}_{x}$ | 4.8 |

$R{a}_{x}^{*}$ | 4.7 |

Heat Flux = 108.0 W/m^{2} | |||||

Temperature, °C | |||||

Left Surface (L) | Front Surface (F) | Right Surface (R) | Mean Value | Standard Deviation ${S}_{x}=\sqrt{\frac{{{\displaystyle \sum}}_{i=1}^{n}{\left({x}_{i}-\overline{x}\right)}^{2}}{n-1}}$ | |

Tube No. 1 | 48.51 | 50.84 | 47.73 | 49.03 | 1.62 |

Tube No. 2 | 49.55 | 48.90 | 48.39 | 48.95 | 0.58 |

Tube No. 3 | 47.40 | 47.21 | 49.15 | 47.92 | 1.07 |

Overall mean | 48.63 | ||||

Overall standard deviation | 1.14 | ||||

Overall relative standard deviation percent = standard deviation × 100/mean | 2.34 | ||||

Heat flux = 221.0 W/m^{2} | |||||

Temperature, °C | |||||

Left surface (L) | Front surface (F) | Right surface (R) | Mean value | Standard deviation ${S}_{x}=\sqrt{\frac{{{\displaystyle \sum}}_{i=1}^{n}{\left({x}_{i}-\overline{x}\right)}^{2}}{n-1}}$ | |

Tube No. 1 | 66.29 | 71.18 | 65.46 | 67.64 | 3.09 |

Tube No. 2 | 68.35 | 67.15 | 67.02 | 67.51 | 0.73 |

Tube No. 3 | 63.80 | 63.95 | 68.09 | 65.28 | 2.43 |

Overall mean | 66.81 | ||||

Overall standard deviation | 2.31 | ||||

Overall relative standard deviation percent = standard deviation × 100/mean | 3.46 | ||||

Heat flux = 341.0 W/m^{2} | |||||

Temperature, °C | |||||

Left surface (L) | Front surface (F) | Right surface (R) | Mean value | Standard deviation ${S}_{x}=\sqrt{\frac{{{\displaystyle \sum}}_{i=1}^{n}{\left({x}_{i}-\overline{x}\right)}^{2}}{n-1}}$ | |

Tube No. 1 | 79.73 | 84.10 | 76.06 | 79.96 | 4.03 |

Tube No. 2 | 80.06 | 76.61 | 76.75 | 77.81 | 1.95 |

Tube No. 3 | 73.26 | 75.03 | 80.07 | 76.12 | 3.56 |

Overall mean | 77.96 | ||||

Overall standard deviation | 3.30 | ||||

Overall relative standard deviation percent = standard deviation × 100/mean | 4.23 |

**Table 3.**Recorded temperatures of the seven thermocouples against the platinum resistance thermometer at boiling state.

Fluke PRT (°C) | TC1 (°C) | TC2 (°C) | TC3 (°C) | TC4 (°C) | TC5 (°C) | TC6 (°C) | TC7 (°C) |
---|---|---|---|---|---|---|---|

97.65 | 97.49 | 97.48 | 97.49 | 97.40 | 97.35 | 97.36 | 97.27 |

**Table 4.**Constants A, B, and R

^{2}appear in the correlation (12) at different S/D ratios for the horizontal array of three vertical tubes compared to that of the single tube.

Tube Number 1, (①, 2, 3) | |||

S/D | A | B | R^{2}, % |

Single tube | 0.299 | 0.240 | 98.3 |

1.75 | 0.364 | 0.225 | 99.1 |

2.75 | 0.346 | 0.231 | 97.8 |

3.25 | 0.385 | 0.227 | 96.9 |

3.75 | 0.336 | 0.235 | 97.9 |

4.25 | 0.375 | 0.237 | 98.1 |

Tube number 2, (1, ②, 3) | |||

Single tube | 0.343 | 0.235 | 99.0 |

1.75 | 0.467 | 0.213 | 99.3 |

2.75 | 0.327 | 0.233 | 98.0 |

3.25 | 0.400 | 0.225 | 97.9 |

3.75 | 0.435 | 0.224 | 98.7 |

4.25 | 0.409 | 0.232 | 99.0 |

Tube number 3, (1, 2, ③) | |||

Single tube | 0.165 | 0.267 | 97.4 |

1.75 | 0.280 | 0.238 | 99.1 |

2.75 | 0.204 | 0.255 | 98.2 |

3.25 | 0.201 | 0.255 | 97.5 |

3.75 | 0.185 | 0.262 | 96.2 |

4.25 | 0.182 | 0.267 | 98.1 |

**Table 5.**Percent of degradation or enhancement in Nusselt numbers over that of a single tube at different Rayleigh numbers (Nu

_{x}– Nu

_{xs}) × 100/Nus.

Tube Number 1, (①, 2, 3) | ||||

S/D | Ra = 5 × 10^{11} | Ra = 1 × 10^{11} | Ra = 1 × 10^{10} | Ra = 1 × 10^{9} |

Single tube | 0 | 0 | 0 | 0 |

1.75 | −17.74 | −15.78 | 12.90 | −9.92 |

2.75 | −8.06 | −6.78 | −4.94 | −3.05 |

3.25 | −7.91 | −6.05 | −3.31 | −0.51 |

3.75 | −0.70 | 0.05 | 1.11 | 2.19 |

4.25 | 16.05 | 16.60 | 17.37 | 18.15 |

Tube number 2, (1, ②, 3) | ||||

S/D | Ra = 5 × 10^{11} | Ra = 1 × 10^{11} | Ra = 1 × 10^{10} | Ra = 1 × 10^{9} |

Single tube | 0 | 0 | 0 | 0 |

1.75 | −23.92 | −21.23 | −17.21 | −13.00 |

2.75 | −10.64 | −10.29 | −9.81 | −9.31 |

3.25 | −10.43 | −9.00 | −6.93 | −4.79 |

3.75 | −6.12 | −4.40 | −1.91 | 0.65 |

4.25 | 11.00 | 11.47 | 12.16 | 12.86 |

Tube number 3, (1, 2, ③) | ||||

S/D | Ra = 5 × 10^{11} | Ra = 1 × 10^{11} | Ra = 1 × 10^{10} | Ra = 1 × 10^{9} |

Single tube | 0 | 0 | 0 | 0 |

1.75 | −21.67 | −17.97 | −12.37 | −6.40 |

2.75 | −10.36 | −8.63 | −6.08 | −3.47 |

3.25 | −12.01 | −10.29 | −7.77 | −5.18 |

3.75 | −2.01 | −1.22 | −0.07 | 1.10 |

4.25 | 9.23 | 9.31 | 9.43 | 9.56 |

**Table 6.**Constants appearing in Equation (13) with the corresponding R

^{2}and the maximum deviation appear in Figure 13 according to each tube in the array.

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**MDPI and ACS Style**

Alsuhaibani, Z.; Ali, M.; Saleh, N.S.
Center-to-Center Distance’s Effect between Vertical Square Tubes of a Horizontal Array on Natural Convection Heat Transfer. *Appl. Sci.* **2023**, *13*, 6345.
https://doi.org/10.3390/app13106345

**AMA Style**

Alsuhaibani Z, Ali M, Saleh NS.
Center-to-Center Distance’s Effect between Vertical Square Tubes of a Horizontal Array on Natural Convection Heat Transfer. *Applied Sciences*. 2023; 13(10):6345.
https://doi.org/10.3390/app13106345

**Chicago/Turabian Style**

Alsuhaibani, Zeyad, Mohamed Ali, and Nader S. Saleh.
2023. "Center-to-Center Distance’s Effect between Vertical Square Tubes of a Horizontal Array on Natural Convection Heat Transfer" *Applied Sciences* 13, no. 10: 6345.
https://doi.org/10.3390/app13106345