Numerical Study of Heated Tube Arrays in the Laminar Free Convection Heat Transfer

Abstract

Laminar free convection heat transfer from a heated cylinder and tube arrays is studied numerically to obtain the local and average Nusselt numbers. To verify the numerical simulations, the Nusselt numbers for a single cylinder were compared to other authors for the Rayleigh numbers of 10³ and 10⁴. Furthermore, the vertically arranged heated tube arrays 4 × 1 and 4 × 2 with the tube ratio spacing S_V/D = 2 were considered, and obtained average Nusselt numbers were compared to the existing correlating equations. A good agreement of the average Nusselt numbers for the single cylinder and the bottom tube of the 4 × 1 tube array is proved. On the other hand, the bottom tubes of the 4 × 2 tube array affect each other, and the Nusselt numbers have a different course compared to the single cylinder. The temperature fields for the tube array 4 × 4 in basic, concave, and convex configurations are studied, and new correlating equations were determined. The simulations were done for the Rayleigh numbers in the range of 1.3 × 10⁴ to 3.7 × 10⁴ with a tube ratio spacing S/D of 2, 2.5, and 3. On the basis of the results, the average Nusselt numbers increase with the Rayleigh numbers and tube spacing increasing. The average Nusselt number and total heat flux density for the convex configuration increase compared to the base one; on the other hand, the average Nusselt number decreases for the concave one. The results are applicable to the tube heaters constructional design in order to heat the ambient air effectively.

1. Introduction

Laminar free convection heat transfer from heated tubes to air has been used in various technical applications such as heat exchangers, electronic components, heat storage equipment, and waste heat recovery systems. The advantage of free convection heat transfer is a creation of thermal comfort without a draft and dust fragments swirling in the space. This energy-saving way of heat transfer is reliable and the operation without fans is cost-effective and noise reducing. On the other hand, heat transfer at free convection is lower; therefore, a sufficiently large heat transfer surface and its suitable shape and arrangement is needed to fulfil effective heat transfer. Owing to this, the research of a chimney effect over the heat transfer surfaces is justified and required.

Several authors have dealt with the numerical calculations of heat transfer from a heated cylinder or tubes array at free convection. Churchill and Chu [1] developed an empirical expression for the average Nusselt number Nu_av of a horizontal cylinder at presented Rayleigh numbers Ra and Prandtl numbers Pr. The temperature fields and airflows around the horizontal cylinder at free convection were researched by Morgan [2]. Kuehn and Goldstein [3] studied laminar free convection heat transfer from the horizontal cylinder by solving the Navier–Stokes and energy equations using an elliptical numerical procedure for 10⁰ ≤ Ra ≤ 10⁷.

The effect of vertical spacing on free convection heat transfer for a pair of heated horizontal tubes arranged one above the other was studied by Sparrow and Niethammer [4]. They investigated how the heat transfer characteristics of a top cylinder are affected by a bottom one, while changing Ra in the range of 2 × 10⁴–2 × 10⁵ and the tube spacing from 2 to 9 cylinder diameters. It was found that the Nusselt numbers of the top cylinder are strongly affected by the tube spacing. A decrease of the Nusselt numbers occurs at small spacing, an enhancement prevails at higher one. Regarding the temperature differences, their effect on the Nusselt numbers is higher at small tube spacing, unlike the higher one. Saitoh et al. [5] presented bench-mark solutions with accuracy to at least three decimal places for Ra = 10³ and 10⁴, where Nu_φ, Nu_av, isotherms, streamlines, and vorticities were obtained. The laminar free convection around an array of two isothermal tubes arranged above each other for Ra in the range of 10² to 10⁴ was studied numerically by Chouikh et al. [6]. Decreased Nu at close spacing and enhanced Nu at a large one occurred at the upper tube. Within the same tube spacing, heat transfer at the upper tube increases with Ra. Herraez and Belda [7] used the holographic interferometry method for the temperature field visualization around the tubes of different diameters when changing the surface temperature. Furthermore, the functions of an exponential form were defined, and Nu numbers were calculated for the range of Gr·Pr = 1.2 × 10³–1.6 × 10⁵.

Corcione [8] numerically studied a steady laminar free convection from horizontal isothermal tubes set in a vertical array. In this research, a computer code based on the SIMPLE-C algorithm was developed and simulations for the arrays of 2–6 tubes with the center-to-center distance from 2 to more than 50 tube diameters were performed for Ra in the range between 5 × 10² and 5 × 10⁵. Moreover, the heat transfer correlating equations for individual tubes in the array and for the whole tube array were proposed. Furthermore, a steady laminar free convection from a pair of vertical tube arrays for Ra in the range of 10² and 10⁴ was numerical studied by Corcione [9]. The pairs of tube arrays consisted of 1–4 tubes with the center-to-center horizontal and vertical spacing from 1.4 to 24 and from 2 to 12 tube diameters, respectively.

Ashjaee and Yousefi [10] experimentally investigated the free convection heat transfer from the horizontal tubes arranged above each other and the tubes shifted in a horizontal direction. The tube spacing varied from 2 to 5 tube diameter in a vertical direction and from 0 to 2 tube diameter in a horizontal direction for the inclined array between Ra = 10³ and 3 × 10³. The Mach–Zehnder interferometer was used to visualize the temperature fields. It was found out that the location of each tube affects the flow and heat transfer on the other tubes in the array. Heo and Chung [11] numerically investigated the natural convection heat transfer of two staggered cylinders for laminar flows. They used the Ansys Fluent software to examine the effect of varying the Prandtl number and the vertical and horizontal pitch-to-diameter ratios for Ra of 1.5 × 10⁸. The heat transfer rates of the upper cylinder were affected by thermal plumes from the lower cylinders, and lower cylinders were not affected by the upper cylinders. When the vertical pitch is very small, the upper cylinders are affected.

Cernecky et al. [12] visualized the temperature fields around the two heated tubes arranged above each other at the surface temperature of 323 K. The local heat transfer coefficients were determined from the holographic interferogram images of the temperature fields and the results were compared with numerical simulations. New criterion equations for calculating Nu of electrically heated tubes at free convection heat transfer were presented by Malcho et al. [13]. The vertical center-to-center distance of the tubes was gradually changed from 20 to 100 mm with a step of 20 mm. The surface temperatures of the tubes were 30, 45, 60, 75, 90, and 105 °C at a fluid temperature of 20 °C. Lu et al. [14] numerically investigated heat transfer from the vertical tube arrays (2–10 horizontal tubes) in molten salts at Ra = 2 × 10³–5 × 10⁵. It was found out that the tube spacing affects the average heat transfer rate around the whole tube array. The research resulted in a determination of the heat transfer dimensionless correlating equations for any individual tube in two vertically aligned horizontal tubes.

Kitamura et al. [15] experimentally investigated the free convection heat transfer around a vertical row of 10 heated tubes with the diameters of 8.4, 14.4, and 20.4 mm at modified Ra in the range of 5 × 10² to 10⁵. The thermal plumes arising from the upstream tubes remained laminar throughout the rows when the gaps between the tubes were smaller than 20.6 mm. When the gaps were higher than 30.6 mm, the thermal plumes began to sway and undergo the turbulent transition on the halfway of the rows. Cernecky et al. [16] visualized the temperature fields in a set of four tubes arranged above each other at horizontal shift of 1/4 and 1/2 of the tube diameter by the holographic interferometry. Subsequently, the local heat transfer coefficients around the tubes were evaluated. The spacing between the tubes in vertical direction has a significant effect on the heat transfer parameters compared with the horizontal spacing of the tube centers. The heat transfer parameters on the other tubes in the direction of free convection flow varied depending on the tube position and geometry of the array. Ma et al. [17] studied free convection for a single cylinder by the Computational Fluid Dynamics with laminar and different turbulent models. Moreover, a thermal chimney was studied, and the effect of horizontal spacing arrangement of tubes on temperature and velocity fields was clarified.

The mentioned results were compared with those obtained from our simulations and are presented in this paper. First, a numerical simulation for a single cylinder at Ra = 10³ and 10⁴ was performed, and the local Nusselt numbers Nu_φ were compared with other authors. Subsequently, the results of Nu_av for the tube arrays 4 × 1 and 4 × 2 are presented according to arrangement in Figure 1. The main part of the presented contribution are the results of Nu_φ and Nu_av for the tube array 4 × 4 at basic, concave, and convex configurations. The simulations were performed for Ra in the range of 1.3 × 10⁴ to 3.7 × 10⁴ and for vertical and horizontal spacing between the tubes S/D = 2, 2.5 and 3. New correlating equations for the Nusselt numbers and the temperature fields for the tube arrays were created. The goal of this paper is to compare the base configuration of the tube arrays 4 × 4 (Figure 2a) with the concave (Figure 2b) and convex one (Figure 2c) and to increase the Nu_array and q_array, respectively, by means of a suitable configuration. Furthermore, the goal is to determine new correlating equations for the 4 × 4 tube array of basic, concave, and convex configurations with subsequent evaluation of the tubes’ arrangement and spacing effects on Nu_array. The results are applicable to the design of heating equipment where free convection is used such as the tube heaters with a hot water circuit and electric heaters which are suitable for bathroom and toilet heating.

2. Mathematical Formulation

Free convection heat transfer is conducted between the heated cylinder surface and ambient environment (air) and between the heated tube array and ambient air, respectively. In the mathematical formulation, the uniform and constant temperature of the cylinder surface (tube array surfaces) T_s is considered, and the ambient air is taken with the temperature T_f without the interfering airflow. The air, flown at free convection, is considered as steady, two-dimensional (x, y), laminar, and incompressible. The properties of air are shown in

Table 1. The properties of air.

Property and Unit	Values
Density ρ (kg/m3)	1.196
Specific heat c (J/kgK)	piecewise polynomial
Thermal conductivity λ (W/mK)	−2.48 × 10−8 · T2 + 8.92 × 10−5 · T + 1.12 × 10−3
Dynamic viscosity μ (kg/ms)	−3.76 × 10−11 · T2 + 6.95 × 10−8 · T + 1.12 × 10−6

.

The mass (continuity equation), momentum equations, and the energy equation for incompressible and compressible flows are the following:

$$\frac{\partial \left(\rho u\right)}{\partial x}+\frac{\partial \left(\rho v\right)}{\partial y}=0,$$

(1)

Momentum equation in x direction:

$$\rho \left(u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}\right)=\mu \left(\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2}\right)-\frac{\partial p}{\partial x},$$

(2)

Momentum equation in y direction:

$$\rho \left(u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}\right)=\mu \left(\frac{{\partial }^{2}v}{\partial x^2}+\frac{{\partial }^{2}v}{\partial y^2}\right)-\frac{\partial p}{\partial y}+\rho g\beta \left(T_s-T_f\right),$$

(3)

Energy equation:

$$\left(u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}\right)=\alpha \left(\frac{{\partial }^{2}T}{\partial x^2}+\frac{{\partial }^{2}T}{\partial y^2}\right).$$

(4)

The local Nusselt numbers Nu_φ of any cylinder are calculated as [8]:

$$N{u}_{\phi }=\frac{qD}{{\lambda }_f\left(T_s-T_f\right)}=-{\frac{\partial T}{\partial r}|}_{r=0.5}.$$

(5)

The average Nusselt numbers Nu_av of any cylinder are calculated as [8]:

$$N{u}_{av}=\frac{Q}{{\lambda }_f\pi \left(T_s-T_f\right)}=-\frac{1}{\pi }{{\displaystyle \int }}_0^{\pi }{\frac{\partial T}{\partial r}|}_{r=0.5}d\phi .$$

(6)

The average Nusselt number of the whole array Nu_array is obtained as an arithmetic average value of Nu_av from the individual tubes in the array [8]:

$$N{u}_{array}=\frac{1}{N}{\displaystyle \sum }_{i=1}^{N}N{u}_{av}.$$

(7)

3. Arrangement of Investigated Tube Arrays and Boundary Conditions

The numerical simulations of temperature fields and calculation of the local and average Nusselt numbers were performed in the Ansys Fluent software. The following boundary conditions are applied:

(a) at the right symmetry line B-C (symmetry):

$$\frac{\partial V}{\partial X}=0;U=0;\frac{\partial T}{\partial X}=0,$$

(8)

(b) on the cylinder surfaces (wall):

$$U=0;V=0;T=1,$$

(9)

(c) at the bottom boundary line A-B (velocity inlet):

$$\frac{\partial V}{\partial Y}=0;U=0;T=0ifV\ge 0,$$

(10)

(d) at the left boundary line A-D (pressure outlet):

$$V=0;\frac{\partial U}{\partial X}=0;\frac{\partial T}{\partial X}=0ifU\ge 0,$$

(11)

(e) at the top boundary line C-D (pressure outlet):

$$\frac{\partial V}{\partial Y}=0;U=0;\frac{\partial T}{\partial Y}=0ifV\ge 0.$$

(12)

Inlet boundary: velocity inlet and constant temperature of 295 K. Outlet boundary, left wall: pressure outlet = 0 gauge pressure and constant temperature of 295 K. Right wall: symmetry. Tube walls: constant temperature from 313 to 373 K (Figure 1 and Figure 2).

A preliminary study of the mesh grid size was carried out where quadrilateral elements were used. As shown in

Table 2. Mesh independence test for single cylinder.

No. of Elements Around the Half Tube	Average Nu	No. of Elements Perpendicular to the Tube	Average Nu
36	4.749	10	4.747
50	4.759	15	4.721
60	4.721	20	4.710
80	4.719	30	4.700
90	4.710	45	4.695

, five different numbers of elements were created around the half tube where the average Nusselt numbers of the single cylinder arrangement were being evaluated. On the basis of this study, 50 elements around the half tube are sufficient from the accuracy point of view. Owing to having more information about the local Nusselt numbers, 90 elements around the half tube (2° angle increment) were used despite the computational time increase. Furthermore, the element length, perpendicular to the tube, was studied. Table 2 shows the numbers of elements on the length of 8 mm perpendicular to the tube with 90 elements around the half tube. It is clear that 15 is a sufficient number of elements. Considering the elements quality (aspect ratio, angles, etc.), 20 elements were chosen. The average element length size close to the tube is 0.3 mm.

The governing Equations (1)–(4) considering the boundary conditions (8)–(12) were solved by the finite volume method using the Ansys Fluent software. Both transient and steady analyses were compared with each other from an accuracy point of view. The comparison was performed on the single cylinder simulation. While the transient analysis was done with variable air properties, the steady analysis was done with both variable and constant air properties. Regarding the variable properties, thermal conductivity and viscosity were considered as polynomial functions of temperature while specific heat was considered as a piecewise-polynomial function of temperature.

On the basis of negligible differences shown in Figure 3, the steady pressure-based analyses with the variable air properties were performed for the other arrangements. The Boussinesq approximation was used as a computational model and the pressure–velocity coupling was handled by the SIMPLE-C algorithm. For a momentum and energy gradient, the Quick and Second order upwind schemes were used. The solution was considered to be fully converged when the residuals of continuity, x-velocity, y-velocity, and energy met the convergence criterion 10⁻⁶. Finally, Nu_φ were exported and subsequently evaluated.

4. Results and Discussion

4.1. Results of the Single Cylinder

Laminar free convection heat transfer from the single isothermal circular cylinder with a diameter of 20 mm was studied (Figure 1a). The simulations were performed for Ra = 10³ and Ra = 10⁴. For a comparison, numerical simulations for the steady analysis with constant and variable material properties were performed (Figure 3).

The results of Corcione [8] were compared with ours. The discrepancies of Nu_φ values for Ra = 10³ are in the range of 0.37–5.90% for the steady state at constant air properties, and 0.14% to 6.11% for the steady state at variable air properties. The discrepancies of Nu_φ values for Ra = 10⁴ are in the range of 0.23–3.14% for the steady state at constant air properties and 0.33–3.07% for the steady state at variable air properties.

The present numerical simulations quantitatively match the data of the numerical simulations by Corcione [8], Saitoh et al. [5], and Wang et al. [18] for the single cylinder shown in Figure 4. The present numerical simulations were performed at the steady state analysis and variable air properties, described in Section 3.

In comparison with Saitoh et al. [5], the discrepancies of Nu_φ values are in the range of 0.02–6.70% for Ra = 10³, and 0.36–3.00% for Ra = 10⁴, respectively. In comparison with Wang et al. [18], the discrepancies of Nu_φ values are in the range of 1.05–7.84% for Ra = 10³ and 0–5.13% for Ra = 10⁴, respectively.

The numerical simulations for the single cylinder were performed to create a suitable computational model which was verified with other authors. On the basis of this model, the other simulations for the tube arrays 4 × 1, 4 × 2, and 4 × 4 were performed.

4.2. Results of the Tube Array 4 × 1a

We have studied the laminar free convection heat transfer from the vertical array of four isothermal circular tubes with a diameter of 20 mm. The simulations were performed for the tubes at a center-to-center dimensionless vertical distance of S_V/D = 2 (Figure 1b) at Ra = 10³ and Ra = 10⁴. The courses of the local Nusselt numbers Nu_φ for the individual tubes in the array are shown in Figure 5a,b for the angles of φ = 0° to 180°. Owing to the model symmetry, the courses of Nu_φ are identical for the right side as well.

As it is shown in Figure 5, the highest values of Nu_φ are achieved on the tube I circumference. For Ra = 10⁴, Nu_φ increases for all tubes I–IV in comparison with Ra = 10³. A significant increase of Nu_φ on the downstream area of the tube II (φ = 0°–70°) is shown in comparison with Ra = 10³ (Figure 5b). Due to the higher temperature difference between the tube surface and ambient air at Ra = 10⁴, heat transfer to the tube II is affected by the heat flux from the tube I, and therefore Nu_φ increases. Simultaneously, the tubes III and IV are affected by the heat flux of previous tubes. When comparing the same positions of the tubes at Ra = 10³ and 10⁴ and the angles of φ = 0°–180°, Nu_φ increases for Ra = 10⁴ in the range of 35–61% for the tube I, 30–171% for the tube II, 29–158% for the tube III, and 29–121% for the tube IV.

The ratio of the average Nusselt number for the i^th cylinder and single cylinder $N{u}_{iav}/N{u}_{0av}$ for a vertical column of four horizontal cylinders at Ra = 10⁴ is shown in Figure 6a. The presented values were compared with Razzaghpanah and Sarunac [19] on the basis of Table 3 and Figure 9 [19] and Corcione [8] on the basis of Equation (17) [8]. The average Nu for the first cylinder in a column is the same as for the single cylinder. The relative error of the other presented tubes for the ratio $N{u}_{iav}/N{u}_{0av}$, compared with Razzaghpanah and Sarunac [19], is up to 5.2%. Figure 6b compares the presented average Nusselt numbers of the i^th cylinder with Equation (22) of the authors Razzaghpanah and Sarunac [19] and Equations (11)–(13) of the authors Kitamura et al. [15].

Corcione [8] obtained numerical results for the average Nusselt number of the tube array Nu_array which may be correlated to the Rayleigh number Ra, the tube ratio spacing S_V/D, and the number of the tubes in the array N:

$$\begin{gathered} N{u}_{array}=R{a}^{0.235}\left\{0.292ln\left[{\left(\frac{S_V}{D}\right)}^{0.4}\times N^{-0.2}\right]+0.447\right\}, \\ 2\le N\le 6;5\times {10}^2\le Ra\le 5\times {10}^5, \\ {S}_V/D\le 10-\log \left(Ra\right), \end{gathered}$$

(13)

with the percent standard deviation of error E_SD = 2.25% and error E in the range of -4.79% to +5.27%. The values Nu_array obtained by our simulations lie out of the error range according to Corcione [8] by 4.9% for Ra = 10³ and 3.7% for Ra = 10⁴, respectively.

4.3. Results of the Tube Array 4 × 2

We have studied the laminar free convection heat transfer from the two vertical arrays of four isothermal circular tubes with a diameter of 20 mm. The simulations were performed for the center-to-center dimensionless vertical distance S_V/D = 2 and horizontal distance S_H/D = 2 (Figure 1c) at Ra = 10³ and Ra = 10⁴.

The courses of the local Nusselt numbers Nu_φ for the individual tubes in the array 4 × 2 are shown in Figure 7. The courses of Nu_φ are valid for the left tube array; the right tube array has mirrored Nu_φ courses. As shown in Figure 7a, the highest values of Nu_φ for Ra = 10³ are achieved on the tube I in the range of 1.167–3.615. For Ra = 10⁴, the local Nusselt numbers on the tube I are in the range of 1.518–6.118, which is an increase of 30–69% in comparison with Ra = 10³. When comparing the same positions of the tubes at Ra = 10³ and 10⁴ and the angles of φ = 0°–360°, Nu_φ increase for Ra = 10⁴ in the range of 30–70% for the tube I, 48–183% for the tube II, 55–245% for the tube III, and 48–265% for the tube IV.

Corcione [9] obtained numerical results for the average Nusselt number of the double tube array Nu_array which may be correlated to the Rayleigh number Ra, the tube ratio spacing S_V/D and S_H/D, and the number of the tubes in each vertical tube array of the pair N:

$$\begin{gathered} N{u}_{array}=0.43R{a}^{0.235}{\left(S_H/D\right)}^{0.14}{\left(S_V/D\right)}^{0.2}\times N^{-0.1}, \\ 2\le N\le 4;2.4-0.2\log {\left(Ra\right)}_{}\le S_H/D<5, \\ 2\le S_V/D<5;{10}^2\le Ra\le {10}^4, \end{gathered}$$

(14)

with the percent standard deviation of error E_SD = 2.12% and error E in the range of -5.32% to +5.67%. The values Nu_array obtained by our simulations lie out of the error range according to Corcione [9] by 1.2% for Ra = 10³. For Ra = 10⁴, our values lie in the error range according to Corcione [9].

The values of average Nusselt numbers Nu_av for the single cylinder and tube arrays 4 × 1 and 4 × 2 are given in

Table 3. Average Nusselt numbers Nu_av for the single cylinder and tube arrays 4 × 1, 4 × 2 at Ra = 10³ and Ra = 10⁴.

	Ra = 103				Ra = 104
Single Cylinder	2.906				4.710
Tube Array 4 × 1	I	II	III	IV	I	II	III	IV
	2.875	1.974	1.692	1.645	4.722	3.654	3.044	2.827
Tube Array 4 × 2 (Left Column)	I	II	III	IV	I	II	III	IV
	2.946	2.248	1.928	1.861	4.816	4.344	4.015	3.824

for Ra = 10³ and Ra = 10⁴. The values of Nu_av increase with Ra, specifically from 62% to 108% at Ra = 10⁴ against Ra = 10³ for individual tubes within the same arrangement. For the tube array 4 × 1, the values of Nu_av decrease from the tube I to the tube IV. For the tube array 4 × 2, the columns affect each other, where the values of Nu_av are higher from 2% to 35% on each tube compared with the tube array 4 × 1.

The temperature fields around the single cylinder and created thermal chimney at Ra = 10³ and 10⁴ are shown in Figure 8a,b. The temperature fields of the tube arrays 4 × 1 and 4 × 2 together with created thermal chimneys at Ra = 10³ and 10⁴ are shown in Figure 8c–f.

As the thermal chimney develops upwards, its temperature decreases due to heat transfer to the cold ambient air. A density difference between the thermal chimney and ambient air induces a buoyancy force; near the heated cylinder, the buoyancy force is higher.

In Figure 8c,d, it can be seen a gradual increase of the thermal boundary layer thickness in a vertical direction of the tube array 4 × 1. The widening of the thermal boundary layer causes the diminishing of the temperature gradient on the surfaces and the reduction of the local and average Nusselt numbers Nu_φ and Nu_av, respectively. In Figure 8e–f, a mutual effect of two tube columns in the tube array 4 × 2 can be seen, where an in-draft heat occurs between the first and second columns and a common thermal chimney is achieved over the tube array. For Ra = 10⁴, a thinner thermal boundary layer around the tube circumferences can be observed. Owing to this, the values of Nu_φ and Nu_av increase compared with Ra = 10³.

The comparison of the local Nusselt numbers Nu_φ of the single cylinder with the bottom tube of the tube array 4 × 1 and with the bottom left tube of the tube array 4 × 2 is shown in Figure 9 for Ra = 10³ and 10⁴. The courses of Nu_φ for the tube array 4 × 2 are valid for the left tube column; the right tube column has mirrored Nu_φ courses. As shown, the values Nu_φ for the bottom tube of the 4 × 1 array are similar to the values Nu_φ for the single cylinder at Ra = 10³ and 10⁴. When comparing the bottom tube of the 4 × 2 array and the single cylinder, there are different courses due to the right tube column effect.

4.4. Results of the Tube Array 4 × 4 (Base, Concave, Convex)

The laminar free convection heat transfer from the four vertical arrays of four isothermal circular tubes with the diameter of 20 mm has been studied. Moreover, the effect of the tube ratio spacing on the Nusselt numbers had been investigated for different Ra before. The tube spacing in the vertical direction S_V and horizontal one S_H increased in the same way; therefore, general tube ratio spacing S/D is defined. The simulations were performed for the S/D ratios of 2, 2.5, and 3 (Figure 2) for the tube array 4 × 4 at three different configurations (Base—Figure 2a, Concave—Figure 2b, Convex—Figure 2c). The courses of the Nu_φ values in the tube array 4 × 4 for base, concave, and convex configurations at Ra = 1.3 × 10⁴ and 3.7 × 10⁴ with a S/D ratio of 2 are shown in Figure 10. The courses for the same parameters and the S/D ratio of 2.5 are shown in Figure 11 and for the S/D ratio of 3 in Figure 12. When increasing the Rayleigh number, the convection increases in strength. Furthermore, the Nu_av and Nu_array values increase with the S/D ratio and Ra increasing.

The ratio of presented Nu for the in-line bundle of horizontal cylinders and the single cylinder for Ra = 10⁴ was compared with Razzaghpanah and Sarunac [20] on the basis of Figure 10 [20]. For the ratio S_L/D and S_T/D = 2, the values Nu_array = 4.726 and Nu_o = 4.710 were obtained. The mentioned ratios Nu_array/Nu_o = 1.003 and Nu_o/Nu_array = 0.997 are close to those obtained by Razzaghpanah and Sarunac [20] (1.073 and 0.900, respectively).

Subsequently, the average Nusselt numbers of the whole array Nu_array for three different configurations (base, concave, convex) with the tube ratio spacing S/D of 2, 2.5, and 3 were compared for Ra in the range of 1.3 × 10⁴ to 3.7 × 10⁴ (Figure 13). On the basis of the results, the Nu_array value increases with the increase of Ra and tube spacing. The highest values Nu_array are achieved at the convex configuration. The discrepancies of Nu_array between the concave and convex configurations are up to 3.7% depending on Ra. Owing to negligible error ranges for the correlating equations, these are interchangeable with similar discrepancies. The Nu_array values for the base configuration lie between the values for the concave and convex ones for all considered Ra.

The temperature fields for the tube arrays 4 × 4 with the tube ratio spacing S/D of 2 obtained by the numerical simulations are shown in Figure 14. The temperature fields for the tube array 4 × 4 with the tube ratio spacing S/D of 2.5 and 3 are shown in Figure 15 and Figure 16, respectively. For the middle columns of the tubes (V to VIII–Figure 2), it is clear that the thermal plume from the lower tubes affects the local and average Nusselt numbers of the upper tubes. The upper tubes create a quasi-forced convection which tends to a heat transfer increase from the upper tubes. It causes a lower temperature difference between the tube surface and the incoming air, which tends to a heat transfer decrease at the tubes downstream that is of a major importance at close spacing.

The outer column’s air flow (I to IV–Figure 2) is affected by the inner tube columns and the heat is directed to the central part of the tube array with a subsequent thermal plume creation. It may be seen that the Rayleigh number increases the “chimney effect”. As the temperature fields around the tubes do not affect each other significantly, the boundary layer thickness decreases and Nu_av increases with the higher ratio S/D. The effect of the tube spacing on the temperature field shapes is obvious in Figure 14, Figure 15 and Figure 16. On the basis of the results, we can state that the higher tube spacing, the more effective heat transfer.

The effect of the S/D value and the tube array configuration for Ra = 1.3 × 10⁴ and 3.7 × 10⁴ is obvious in

Table 4. Average Nusselt numbers for different ratios S/D and configurations of 4 × 4 tube arrays at Ra = 1.3 × 10⁴.

S/D	Config.	Average Nusselt Numbers, Nuav (-)								Nuarray (-)	qarray (W/m2)
		I	II	III	IV	V	VI	VII	VIII
2	Base	5.273	5.169	5.163	4.996	5.970	4.519	4.212	4.339	4.955	118.371
	Concave	5.198	5.118	5.136	5.003	6.081	4.294	4.114	4.194	4.892	116.869
	Convex	5.302	5.225	5.278	4.968	5.819	4.901	4.398	4.672	5.070	121.117
2.5	Base	5.279	5.162	5.129	4.909	6.018	5.255	5.108	5.471	5.291	126.381
	Concave	5.238	5.145	5.114	4.945	6.111	5.018	5.023	5.342	5.242	125.201
	Convex	5.287	5.219	5.219	4.869	5.837	5.556	5.216	5.629	5.354	127.875
3	Base	5.256	5.161	5.095	4.820	5.984	5.690	5.672	6.057	5.467	130.550
	Concave	5.244	5.152	5.082	4.859	6.077	5.501	5.668	5.971	5.444	130.014
	Convex	5.264	5.208	5.168	4.793	5.795	5.863	5.686	6.122	5.487	131.049

and

Table 5. Average Nusselt numbers for different ratios S/D and configurations of 4 × 4 tube arrays at Ra = 3.7 × 10⁴.

S/D	Config.	Average Nusselt Numbers, Nuav (-)								Nuarray (-)	qarray (W/m2)
		I	II	III	IV	V	VI	VII	VIII
2	Base	6.485	6.412	6.432	6.206	7.444	5.998	5.869	6.695	6.443	768.405
	Concave	6.392	6.376	6.396	6.237	7.556	5.664	5.690	6.462	6.346	756.880
	Convex	6.522	6.477	6.544	6.098	7.229	6.436	5.979	6.895	6.523	777.818
2.5	Base	6.466	6.343	6.350	6.056	7.452	6.792	6.633	7.626	6.715	800.298
	Concave	6.429	6.330	6.335	6.122	7.572	6.481	6.510	7.526	6.663	794.168
	Convex	6.496	6.438	6.459	5.984	7.226	7.093	6.815	7.549	6.757	805.471
3	Base	6.427	6.332	6.304	5.957	7.393	7.195	7.127	8.141	6.860	817.236
	Concave	6.422	6.321	6.296	6.019	7.512	6.956	7.049	8.108	6.835	814.315
	Convex	6.484	6.441	6.419	5.949	7.177	7.376	7.252	7.994	6.887	820.744

. The Nu_av value for the tubes I–VIII, the Nu_array value for whole tube arrays, and the q_array value were evaluated for all configurations and S/D ratios. When increasing the S/D ratio, also Nu_av, Nu_array, and q_array increase, where the tube array configuration has a crucial effect.

As it is shown in Table 4, Nu_array and q_array values decrease by 1.27% at the concave configuration compared with the base one for S/D = 2 and Ra = 1.3 × 10⁴. On the contrary, these representative values increase by 2.32% at the convex configuration compared with the base one for the same parameters. Additionally, they increase by 3.64% at the convex configuration compared with the concave one. For S/D = 2.5, the same comparison is carried out, where Nu_array and q_array decrease by 0.93% at the concave configuration and increase by 1.19% at the convex configuration compared with the base one. When comparing the convex configuration with the concave one, Nu_array and q_array increase by 2.14%. For S/D = 3, Nu_array and q_array decrease by 0.42% at the concave configuration and increase by 0.37% at the convex configuration compared with the base one. When comparing the convex configuration with the concave one, Nu_array and q_array increase by 0.79%.

As shown in Table 5, Nu_array and q_array values decrease by 1.51% at the concave configuration compared with the base one for S/D = 2 and Ra = 3.7 × 10⁴. These representative values increase by 1.24% at the convex configuration compared with the base one. Additionally, they increase by 2.79% at the convex configuration compared with the concave one. For S/D = 2.5, Nu_array and q_array decrease by 0.77% at the concave configuration and increase by 0.63% at the convex configuration compared with the base one. When comparing the convex configuration with the concave one, Nu_array and q_array increase by 1.41%. For S/D = 3, Nu_array and q_array decrease by 0.36% at the concave configuration and increase by 0.39% at the convex configuration compared with the base one. When comparing the convex configuration with the concave one, Nu_array and q_array increase by 0.76%. It is obvious that the increase of the S/D ratio and Ra values causes a decrease of Nu_array and q_array percentage discrepancies between the individual configurations.

New correlating equations for Nu_array were created for all Ra values in the range of 1.3 × 10⁴ to 3.7 × 10⁴ as follows:

1.

The base configuration:

$$N{u}_{array}=0.475R{a}^{0.2351}{\left(S/D\right)}^{0.1907},$$

(15)

with the percent standard deviation of error E_SD = 0.70%, and error E ranged from -0.94% to +1.37%,

2.

The concave configuration:

$$N{u}_{array}=0.463R{a}^{0.2348}{\left(S/D\right)}^{0.2132},$$

(16)

with the percent standard deviation of error E_SD = 0.65%, and error E ranged from -0.90% to +1.37%,

3.

The convex configuration:

$$N{u}_{array}=0.520R{a}^{0.2303}{\left(S/D\right)}^{0.1555},$$

(17)

with the percent standard deviation of error E_SD = 0.54%, and error E ranged from -0.84% to +1.14%.

The correctness of the correlating equations is shown in Figure 17, where the values of Nu_array from the correlating equations and numerical simulations were compared.

Considering the correlating equations for the individual tubes in the array, there is an effort to create the equation containing the tube position in numerical order from bottom to top. For the Nusselt numbers of individual tubes situated in the outer columns, it is possible to assume an approximation by a linear function with the maximum discrepancy of 3% for specific tube position, spacing, configuration, and Ra. However, the monotony is not uniform throughout the tube’s interval. The correlating equation indicates a decreasing of Nu_av with increasing the tube position. On the contrary, in general, Nu_av for the tube 3 is higher than the value for the tube 2. This statement is valid for all configurations, tube spacing and Ra. The Nusselt numbers of the tubes for the inner columns do not have the same course character when changing the parameters. On the basis of the previous facts, the correlating equations for Nu_av are not created.

5. Conclusions

The paper has reported on the results of the numerical investigation of laminar free convection heat transfer from various tube configurations. The single cylinder and the tube arrays of 4 × 1 and 4 × 2 arrangements at Ra = 10³ and 10⁴ were used for computational model verification. The main part of the research focused on the tube arrays in the configuration of 4 × 4 (base, concave, convex) at Ra in the range of 1.3 × 10⁴ to 3.7 × 10⁴. From the obtained local and average Nusselt numbers and temperature fields, the following conclusions can be drawn:

• The local Nusselt numbers Nu_φ for the single circular cylinder at Ra = 10³ and 10⁴ were confirmed.
• The average Nusselt numbers Nu_av for the tube array 4 × 1 lie out of the error range, according to the Corcione correlating Equation [8], by 4.9% for Ra = 10³ and 3.7% for Ra = 10⁴, respectively.
• The average Nusselt numbers Nu_av for the tube array 4 × 2 lie out of the error range, according to the Corcione correlating Equation [9], by 1.2% for Ra = 10³ and lie in the error range for Ra = 10⁴.
• There is a good agreement between the Nu_φ of the single cylinder and bottom tube of the tube array 4 × 1 at Ra = 10³ and 10⁴.
• The left bottom tube for the tube array 4 × 2 is affected by the right tube column; therefore, the Nu_φ values have different courses for Ra = 10³ and 10⁴.
• The Nu_av values increase with the S/D ratio and Ra increasing for the tube array 4 × 4 at the base, concave, and convex configurations.
• The highest values of Nu_av for the tube array 4 × 4 are noticed at the convex configuration which has discrepancies up to 3.7% in comparison with the concave one depending on Ra in the range of 1.3 × 10⁴ to 3.7 × 10⁴. The value of Nu_av for the base configuration lies between the convex and concave ones for all Ra.
• The concave configuration achieves a decrease of Nu_array and q_array compared with the base one. The convex configuration achieves an increase of these representative values compared with the base one.
• The convex configuration achieves an increase of Nu_array and q_array compared with the concave one for all ratios S/D and Ra. When increasing the ratio S/D and Ra, a decrease of Nu_array and q_array percentage discrepancies between the individual configurations is reached.
• The new correlating equations for Nu_array calculation concerning the tube array 4 × 4 were defined. The equations have negligible error ranges and are interchangeable with each other.