Numerical investigation of convection in tubes with aluminium and carbon steel fins: evaluating the assumption of convective heat transfer coefficient as that for the tube without fins and relating physical processes with the optimum spacing between fins

Finned tubes exist in diverse geometric configurations, usually the convective heat transfer coefficient is unknown and approximated as that for a geometry similar to the actual one. This paper presents a numerical investigation about the convective heat transfer in a horizontal finned tube. Ten geometric configurations were considered, which differed on the distance between fins, and two materials were analysed for the fins: aluminium and carbon steel. Eleven different values were considered for the temperature difference between the base of the tube and air. Therefore, a total of one hundred and ten different conditions were studied, for each material. The flow regime was laminar. The analysis showed that approximating the convective heat transfer of the finned tube as that for a tube without fins, for which correlations are available in the literature, can lead to significant errors. The maximum difference verified was for the spacing between fins equal to 2 mm, for which the convective heat transfer coefficient was about seven times lower than that for the tube without fins. For spacings equal to 6 mm and 8 mm, associated to the maximum heat transfer rate, the convective heat transfer coefficient for the tube without fins was about 30% to 50% higher than that for the finned tube. The common assumption of uniform fins surface temperature was also evaluated for the two considered materials the fins were made of. The results showed that for the steel fins the approximation leads to an error of about 10%, while for aluminium it is only about 3%. Results that allow a better understanding of the physical phenomena related to the occurrence of the optimum spacing between fins, which maximize the heat transfer, are also presented and analysed.


Introduction
Enhancement of heat transfer is of great interest for the industry and can provide considerably energy savings. A simple manner of enhancing heat transfer is by using fins, which are employed in a large variety of applications. Thus, the procedures for calculating heat transfer in finned surfaces, necessary to design fins, are available in many textbooks, some examples are [1][2][3]. In such procedures, the convective heat transfer coefficient is generally assumed as constant along the fins surfaces and commonly estimated as that of the surface without fins. However, the inclusion of fins modifies the flow field, which is expected to become considerably more complex than that which would occur without the fins.
With the goal of determining the correct convective heat transfer coefficient, experiments have been accomplished in a series of research works in the past century and recent years. These works include fins under different flow conditions, as natural [4,5] and forced convection [6], laminar [7] and turbulent flows [5,6] and fins with different geometry regarding cross-sectional shape, as rectangular fins [5,[8][9][10][11], annular fins [6,11], pin fins [4,[12][13][14], and others. Natural convection around finned tubes occurs in many engineering applications [2], thus the research about heat transfer in finned tubes is of great interest. Kayansayan and Karabacak [15] analysed the convective heat transfer in 15 different geometric configurations of horizontal finned tubes and verified that the Nusselt number was always lower when fins were attached on the tubes as compared to the case without fins. Comparing the results obtained for different geometric configurations, the authors identified the distance between fins which leads to the maximum heat transfer, for the considered cases. Hahne and Zhus [16] investigated the influence of the fins diameter in cases in which there was an imposed heat rate, which was varied from 10 to 60 W. Yildiz and Yüncü [17] studied the dependence of the convective heat transfer coefficient with respect to fins diameters, spacing between fins and difference of temperature between the base of the fins and ambient air, and presented a procedure to determine the optimum spacing between fins.
In addition to experimental approaches, numerical simulations are also used to study the heat transfer in finned tubes. Chen and Hsu [18] used experimental procedures and the numerical technique of finite differences to investigate diffusion and convection in fins 2 mm thick, with diameter equal to 99 mm, attached to a tube with diameter equal to 27 mm, constituted by steel AISI 304. They analysed the effect of the spacing between fins and verified that when the distance was approximately greater than 20 mm, the convective heat transfer coefficient and efficiency became invariable. Yaghoubi and Mahdavi [19] also used both experimental and numerical techniques to investigate finned tubes. However, tubes and fins were constituted by aluminium, and solely one geometric configuration was considered: fins with diameter equal to 56 mm, thickness of 0.4 mm and distant 2 mm one each other. Three different temperatures were considered for the base of the fins and air, totalling nine temperature combinations. A correlation for the Nusselt number was proposed. Kumar et al. [20] numerically investigated the natural convection in a tube and considered the effects of the spacing to fin diameter ratio and difference of temperature between the base surface and ambient air. They verified that the optimum spacing was 8-10 mm and that the convective heat transfer coefficient was higher for the tube without fins than when fins were attached on the tube. Kumar et al. [20] validated their simulations by comparing some of the obtained results with the experimental results presented by Yaghoubi and Mahdavi [19]. Chen et al. [21] also investigated numerically and experimentally finned tubes with various spacing between fins, employing CFD. Inverse methods were used to determine the temperature field and convective heat transfer coefficient and compared results obtained for laminar flow and with the turbulence models RNG k − and zero-equations to others obtained experimentally. Chen et al. [22] analysed the effect of the computational domain on the results for cases similar to that approached by Chen et al. [21]. Chen et al. [23] studied experimentally and numerically the laminar/turbulent mixed flow regime in finned tubes considering different diameters and spacings between fins.
Procedures to calculate heat transfer for fins design are available in textbooks, as Bejan [24] and Bergman et al. [25]. Simplifications are adopted in the calculations, such as assuming the convective heat transfer coefficient equal to that which would occur without the fins, which can be calculated from correlations available in the literature. However, inclusion of fins changes the flow and therefore the convective heat transfer coefficient. This paper investigates the effect of assuming the convective heat transfer coefficient of a finned tube as that which would occur without fins. An study about the optimum spacing between fins which leads to the maximum heat rate is also presented. Although some research works investigated the optimum spacing between fins [8,17,26,27], the present paper shows how the spacing between fins influences flow and thermal boundary layers, and the manner it affects the convective heat transfer coefficient. In addition, the three-dimensional heat diffusion in fins made of two different materials is computed to investigate how variations on the temperature of fins surfaces affect the general heat transfer. The dimensions of tube and fins are the same as considered by Yaghoubi and Mahdavi [19], that is, the tube diameter is 25.4 mm, and the fins diameter and thickness are, respectively, 56 mm and 0.4 mm. However, differently than in the experimental work accomplished by Yaghoubi and Mahdavi [19], now different spacings between fins are considered, namely, 2, 6, 8, 10, 12 15, 20, 25 and 30 mm. Two materials are considered for the fins: aluminium and carbon steel. Eleven values were considered for the temperature difference between tube and fins, ΔT : and 30 • C , for so much, five temperatures were considered for the tube surface ( 3 • C , 8 • C , 12 • C , 15 • C and 19 • C ), and others three for the ambient air ( 22 • C , 27 • C and 33 • C ). Thus, with the nine different spacing between fins and eleven temperature combinations, this research considers ninetynine different conditions for each material.

Geometry and computational domain
As previously mentioned, the considered tube has diameter d = 25.4 mm , while the fins have diameter D = 56 mm and thickness t = 0.4 mm . The following spacings between the fins, S, were considered: 2, 4, 6 ,8 10, 12, 15, 20, 25 and 30 mm. It was assumed that the external surface of the tube has uniform temperature; however, temperature is allowed to vary throughout the fins. Although finned tubes use to have several fins attached on it, the flow and heat transfer process occurring between two of the fins tend to repeat along all finned tube, and there exists a vertical symmetry plane containing the tube axis. Taking into account these considerations, only two quarter-fins and the external surface of a segment of half tube between fins were considered in the simulations. This geometry is shown in Fig. 1.
The simulations were performed using the ANSYS-CFX 13 software, which requires that solid and fluid domains be defined. The rectangular box with dimensions 7D × 3D × (S + t) , shown in Fig. 1, contains both the fluid and solid domains. The solid domain is constituted by the two quarter-fins, while the fluid domain is the space inside the box except the solid domain and the inner region of the tube. The boundaries where the boundary conditions are applied, as will be described in Sect. 3, are: the external surface of the considered segment of tube; the front wall, which does not include the fins and inner region of the tube; the back wall; and the lateral walls.

Equations and solution procedure
The simulations involved heat diffusion inside the fins and natural convection around the finned tube. The flow was assumed as incompressible and remained laminar in the considered domain and the steady-state solution was obtained.
Previous simulations considering unsteady conditions were accomplished for fins with diameter equal to 10 mm, and difference of temperature between the base of the fins and air equal to 3 K, 14 K and 30 K. The time steps were set such that the Courant number, Eq. (1), was lower than 5.
where u is the velocity component in the x direction, Δt is time step and Δx is the length of the mesh in the x direction.
The steady state was reached in 160 s of simulation solving the unsteady equations. The results for the heat rate were compared to that obtained using the steady formulation, differences were approximately 0.5%.
In order to verify whether the flow was laminar, the Rayleigh number ( Ra D ) is calculated with Eq. (2), to evaluate the flow around the fins, and the Grashof ( Gr y t ) is calculated with Eq. (3), to evaluate the plume which forms below the tube.
where D is the diameter of the fins, g is the gravitational acceleration, is the kinetic viscosity, y t is the distance from the tube, and is the thermal expansion coefficient, which in the case of an ideal gas is given by 1 / T.
The maximum Rayleigh was about 2.5 × 10 5 , while Grashof remained lower than 1.92 × 10 5 in the entire space bellow the tube, characterizing laminar flow.
With these considerations, no model for turbulence was necessary, and density varied only due to the temperature gradient, accordingly the full buoyancy model. In this case, it is necessary to solve the following form of mass conservation, momentum and energy equations.
Mass conservation: where is density, and u, v and w are the velocity components, respectively, in the x, y and z coordinate directions. Momentum:

�� ⃗
V is the velocity vector, is dynamic viscosity, g is acceleration of gravity, which acts only in the z direction and B is the buoyancy term given by where ref is the reference density, which is evaluated considering no temperature gradient. Energy: where T is temperature, k is the thermal conductivity and c p is the specific heat.
It is worth to observe that for the diffusion solution �� ⃗ V = 0 , therefore the momentum equations are not used, and the energy equation reduces to Two different materials were considered for the fins: aluminium and carbon steel, while the fluid domain is air, assumed as ideal gas. The thermal boundary condition for the tube surface was prescribed temperature T W , and five different surface temperatures were considered, The flow boundary condition for the tube surface was no slip condition, �� ⃗ V w = 0 . Open boundary was assumed for both bottom and top surfaces, with pressure equal to 1atm, and prescribed temperatures were assumed for the air entering and exiting the computational domain, T Bottom = T Top = T air . Three different temperatures were considered for the air, T air = 22 • C , 27 • C and 33 • C . Simmetry conditions were assumed to the front, back and lateral walls. Thus, Equations from (4) to (10) were solved by the finite volume method using the ANSYS-CFX. The upwind scheme was used to treat the advective terms appearing in the equations. For coupling velocity and pressure, it was used the ANSYS-CFX default Rhie-Chow algorithm. The adopted convergence criterium was the root mean square (RMS) of 10 −6 . The grid convergence index [28] was employed to ensure mesh independent solutions, and tests were preformed for three different meshes with varied levels of refinement. The computational mesh used for the solutions presented in this work is shown in Fig. 2. It contains approximately 8.9x10 5 nodes and is nonuniform with higher resolution in the region close to the tube surface, which contains the fins. The elements orthogonality varies from 0.65 to 1.

Model validation
The model for the numerical solution was validated by comparing some simulated results with experimental results presented in the literature and results obtained by largely used correlations for computing the convective heat transfer coefficient. The average heat transfer coefficient is computed from the heat rate obtained by the simulation as where q is the heat rate in the tube surface, including the parts without and with fins, which is provided by the simulation; A f and A t are, respectively, the area of the fins surface  [28] and the area of the surface of the tube without fins; T ∞ is the temperature far from the tube, which was set as equal to the temperature of the fluid entering the domain by the top surface, and T w is the temperature of the tube surface.
In order to validate the numerical model, results were obtained for parallel circular plates for which the experimental correlations presented by Tsubouchi and Massuda [29] are available. Figure 3 shows the Nusselt number computed with Eq. (12) in function of the Rayleigh number, computed with Eq. (13): Three different plante diameters, D, and four different spacings between plates, S, were considered. The fins thickness is 2 mm for all cases. As can be verified, the simulated results agreed with that obtained by the experimental correlations. Figure 4 presents results for the convective heat transfer coefficient in function of the difference of temperature between tube surface and air, ΔT = (T ∞ − T w ) . Two geometries are considered: tube without fins and tube with fins distant 2 mm one each other. In the case of the tube without fins, the simulated results were compared with results obtained using correlations presented by Churchil and Chu [30] and Morgan [31], while for the finned tube, the results obtained by simulation were compared with the experimental results presented by Yaghoubi and Mahdavi [19]. In both cases, the maximum differences were of the order of 10%.
(13) Ra S = g T air − T w S 3 Figure 5 shows results for the convective heat transfer coefficient in function of the difference of temperature between tube surface and air, ΔT = (T air − T w ) , for different spacings between fins, S, and for the tube without fins. As can be noticed, the convective heat transfer coefficient increases considerably from the case of spacing equal to 2 mm to the case without fins. Comparing the results for these two  conditions, one can notice that the convective heat transfer coefficient for the tube without fins is about seven times higher than that for spacing of 2 mm. Therefore, using the convective heat transfer coefficient obtained by correlations for tubes without fins, as that provided by the correlations presented by Churchil and Chu [30] and Morgan [31], can lead to significant errors. Also, it can be observed that for spacings equal to 25 and 30 mm, the convective heat transfer coefficient has similar magnitude. An usual assumption to calculate the heat transfer in finned tubes is that the convective heat transfer coefficient is equal to the obtained for the tube without fins. Observing the results for the average convective heat transfer coefficient, shown in Fig. 5, one can see that the convective heat transfer coefficient for the tube without fins is notably higher than that for spacings equal to 6 and 8 mm, associated to the maximum heat transfer. For the lowest temperature difference, it was approximately 50% higher, while for the maximum temperature difference, it was about 30% higher. As expected, for the minimum spacing of 2 mm the difference is higher, being the convective heat transfer coefficient for the tube without fins seven times the one associated to the minimum spacing. Thus, the results show that the assumption can lead to significant errors.

Results
The evolution of the convective coefficient with respect to S can be seen in Fig. 6, for different ΔT . The convective coefficient increases considerably until S = 8 mm , and barely varies after S = 10 mm . The results presented in Fig. 6 are for the aluminium fins. The same trend was verified in the case of steel fins, with the difference that the convective coefficient was somewhat smaller, presenting a The convective coefficient does not vary significantly for S approximately larger than 10 mm because the influence of the fins to the total heat transfer becomes small as compared to the relatively large segment of tube without fins. In Fig. 7, the relative contribution of the fins and segment of tube without fins to the total heat rate can be observed, this figure shows the heat rate of the tube segment without fins, q t , Fig. 7a; the fins heat rate, q f , Fig. 7b; and the total heat rate, q = q f + q t , Fig. 5c, for the case of aluminium. As expected, q t increases with S, since there is an increase on the heat exchange area, A t . Differently, the area of the fins do not increase with S. Thus, the fins heat rate increases until about S = 10 mm , and after that remains approximately constant, as shown in Fig. 7b. The initial rise of the heat rate with S is due to changes on the flow. However, for large values of S, the flow around each fin is not significantly influenced by the opposite fin. For ΔT ≤ 21 • C a slight decrease on the heat rate can be noticed. This behaviour was also verified by Petraci [6] and Yldiz and Yücü [17]. For S = 2 mm , the heat transfer hate is low, suggesting that in this case diffusion in the air dominates the heat transfer. Figure 8a presents the velocity distribution in a line which goes from one fin to the other, located at half height of the fins, as depicted in the figure. The velocities for S = 2 mm are low, reinforcing the hypotheses of diffusion dominant regime. For S < 10 mm , the maximum velocity occurs at the half distance between the fins, indicating that the boundary layers which form from each fin occupy the entire region between fins, while for S ≥ 10 mm , the maximum velocity occurs in two locations closer to the fins. The location of the maximum velocity for spacing greater than 20 mm is in agreement with the analytical solution for plane plate, as presented by Bejan [24]. Figure 8b presents an non-dimensional temperature given by T adm = T∕T w , for the same line considered in Fig. 8a. The thermal boundary layers occupy the entire space between fins for S ≤ 20 mm . For spacings greater than 20 mm, the temperature remains constant in a central region, out of the boundary layers, and decreases close to the fins. Comparing the results of Fig. 8a, b, one can conclude that the thermal boundary layer is larger than the fluid-dynamic boundary layer, which was expected, since the Prandtl number is about 0.7, less than 1.
One of the advantages of the numerical simulation is that it allows to see details regarding the physical processes. Figure 9 presents the velocity filed for the case of ΔT = 30 K , and aluminium fins. For S = 2 mm , the velocities are quite low remaining below 0.04 m/s. Thus, in this The velocity field is important in the convective terms appearing in the energy equations. Figure 10 shows a comparison between the diffusive and advective terms of the energy equation, for the case of ΔT = 30 K , and aluminium fins. For S = 2 mm the diffusion terms are considerably more important than the convective terms, reinforcing that in this case diffusion dominates the heat transfer process. As the distance between fins is increased, the advective terms gain importance, but diffusion remains dominant close to the fins.
In order to identify the optimum spacing between fins, Fig. 11 shows the heat rate per unit length for all considered temperature differences and spacings. For ΔT ≥ 10 K , the maximum heat rate per unit length occurs for S = 6 mm , while when ΔT < 10 K , it occurs for S = 8 mm . It is worth to observe that the heat rate per unit length associated to the maximum considered spacing, S = 30 mm , is even lower than that associated to S = 2 mm , for which diffusion dominates the heat transfer regime. For S = 6 mm the heat rate for unit length is 200% times higher than for S = 30 mm, in the case of ΔT ≤ 3 K , and increases with temperature difference until 380%, in the case of ΔT = 30 K . Quite similar behaviour as this presented in Fig. 9 for the aluminium fin was verified for the steel fin, differing only on the magnitude of the heat rate per unit length, which was 7% lower in the steel case.
The maximum heat rate per unit length shown in Fig. 11 increases almost linearly with temperature difference. This can be better visualized in Fig. 12, which shows the maximum heat rate per unit length in function of ΔT , for different spacings, S. As can be observed, S for which the maximum heat rate per unit length occurs varies with ΔT . Figure 12 also shows a continuous curve, which was obtained by the following correlation fitted from the simulated results: The optimum spacings shown in Fig. 11 vary with ΔT . Figure 13 shows this variation. The figure also presentes a curve computed with the correlation given by Eq. (15), which was fitted from the results.
The correlation given by Eq. (14) provides an approximation of the maximum possible heat rate per unit length in function of the temperature difference if the finned tube is designed so that the spacing between fins is optimized, while the correlation given by Eq. (15) provides an approximation for the optimum spacing to reach this maximum heat rate per unit length based on the conditions of operation of the finned tube regarding the temperature diference between tube surface and surrounding air.   The temperature distribution inside the fins is shown in Fig. 14 for both aluminium and steel, and ΔT = 30 K , with the tube and base of the fins at 3 • C . As expected the top of the fins has higher temperature due to the contact with the hot air. Temperature gradients are lower in the aluminium fin as compared to the steel fin, since the conductivity of the aluminium is considerably higher. Also, because its lower thermal diffusivity, the steel fins reach higher temperatures, as can be seen in Fig 14. It is interesting to observe that in the region below the tube, the temperatures and gradients are low, being this region inefficient for heat transfer. In the case of S = 2 mm , the temperatures are low due to the low convective heat transfer coefficient, since the flow is confined between the fins, as shown in Fig. 9. The temperature fields for S = 8 mm and 12 mm are similar and does not changes as S increases, which is in accordance with the heat rate shown in Fig. 7b. As clearly demonstrated in Fig. 14, fins temperature presents considerable variation with radial and angular coordinates, unlike being uniform, as often assumed in heat transfer calculations for fins design. Table 1 presents the error of the result for the total heat rate obtained assuming constant surface temperature with respect to the heat rate obtained allowing temperature variation for the cases of aluminium and steel fins. In the case of aluminium, which has higher conductivity than steel, the maximum difference regarding the case of constant temperature is about 2.7%. Temperature gradients are more significant in the steel fins, since thermal conductivity is lower, and in this case the maximum difference with respect to the case of uniform temperature is almost 10%.

Conclusion
Numerical results were obtained for the heat transfer process occurring around finned tubes, considering natural convection and heat diffusion inside the fins. Two materials were considered for the fins, aluminium and steel. Simulations were accomplished for different cases, involved different temperatures and spacings between fins. For the considered cases, the optimum spacing was 6 mm when the difference of temperature between tube and air was lower than 10 K, and 8 mm for temperature differences equal or higher than 10 K. For the spacing of 2 mm, the heat transfer between fins and air occurs mostly by diffusion, since the velocities tend to be low due to the major importance of friction terms of the momentum equation as compared to inertial terms. It was demonstrated that the convective heat transfer coefficient barely varies for spacings greater than 10 mm, since for such spacing the boundary layers which form from each fin surface do not interact one with the other. The heat transfer convective coefficient computed considering the tube without fins, which is a common assumption for computing heat transfer in finned tubes, was in between approximately 30% to 50% higher than that obtained for the optimum spacing between fins, and seven times higher than that for the case of the minimum spacing of 2 mm. In addition, results for the temperature distribution in the fins demonstrate that the temperature is nonuniform, differently than assumed for usual analytical solutions of heat transfer in fins. Nonetheless, for aluminium fins, which has high thermal conductivity, the errors were low, remaining below 3%. However, for steel fins the maximum error was almost 10%.