Keywords

1 Introduction

Heat exchangers are widely used in multiple industrial processes [1]. For this reason, the objective to increase the heat transferred while reducing the pressure drop and minimizing the size of the thermal conversion systems has turned into a relevant milestone due to its practical purposes. Different approaches have been studied to improve the convection heat transfer of a fin and diminishing at the same time its mass such as geometric modifications [2], material reduction [3], and topological modifications [4,5,6], among others. Because of the last premise, a tendency (research area) is to manufacture windows (holes, openings) according to pre-fractal patterns, like the Sierpinski carpet. Numerical and experimental work has been carried out studying the total heat transfer, efficacy, and efficiency; results are promising, reporting an increase in the energy dissipated per unit mass.

The present work is aimed to study the convective flow on the surface of four squared-copper fins oriented vertically with gravity. Perforations were manufactured according to the first four iterations of the Sierpinski carpet, starting from iteration 0. Particle image velocimetry was used to study the velocity fields and the Schlieren technique to visualize the thermal boundary layer. As far as it is known, these kinds of studies have not been performed previously for this type of fins. The perforations are believed to perturb the convective flow, improving the heat transfer in the regions close to them.

2 Experimental Methods

The experimental setup shown in Fig. 1 was installed in an experimental area of 4.3 m length, 3.1 m width, and 3 m height. Components of the particle image velocimetry and the schlieren system were also installed in this space, all but laser. During experimentation, the ambient temperature in the room was 22.7 ℃ ± 0.4 ℃.

Four copper plates were used to manufacture the fractal fins, each 10.5 cm in height, 10 cm in width, and 0.17 cm in thickness. No modifications were made to the first one, which is considered the baseline. The other three were manufactured according to the first three iterations of the Sierpinski carpet fractal. These were named IT0 to IT3, as shown in Fig. 1. The fins were inserted 0.5 cm in a heating base to minimize temperature gradients, conformed by a trapezoidal copper bar of 10 mm and 3.15 mm bases, 10 mm in altitude, and 10 cm in length. The thermal resistance was minimized by applying thermal grease. All elements were painted matte black to have uniform emissivity; laboratory tests showed ε = 0.97. The coordinate system was fixed at the lower-left corner of the fins, which are oriented parallel to gravity. The temperature of the fins was incremented using an electrical heater of 2.8 Ω, adhered to the bottom of the heating base; it was insulated using acrylic of 0.5 cm in thickness. Electrical energy was supplied by an adjustable DC power supply (GW GPS-1850D). Details about construction of the experimental model and setup can be consulted in [7].

Six type T thermocouples (Omega TT-T-30-SLE) were instrumented. Three were used to monitor the temperature of the heating base, inserted in holes of 0.5 cm in depth and 0.25 cm in diameter manufactured at locations of Y/L = −0.03 and Z/L =  −0.25, −0.5, and −0.75. Two were used to compute de heat loss through the heating base insulation, adhered on the surfaces of the acrylic. One was used to monitor the ambient temperature of the confined space; it was installed 1.5 m from the ground level and 0.5 m far from the experimental model. The thermocouples were connected to a National Instruments data acquisition system integrated by a thermocouple module (PXIe−453) and chassis (PXIe−1073). Readings of the thermocouples were monitored and recorded in real-time using a program based on the LabVIEW software.

The convective flow around the four fins was experimentally studied using particle image velocimetry (PIV) and schlieren visualization. The PIV system was integrated by a Kiralux CMOS camera (CS135MUN) using a Navitar lens (MVL12WA) and a Spectra-Physics continuous laser (177-G0323) with a maximum output power of 400 mW. A Powell lens with 110° fan angle was used to generate the light sheet. A Dantec Dynamics fog generator (10D90P) was used to seed the particles of 5 µm diameter on average. The Stokes number is smaller than 0.1, so the particles have good tracing accuracy.

Fig. 1.
figure 1

Fins and heating base.

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The Z-type Schlieren system consisted of two parabolic mirrors of 121 cm of focal length and 15.24 cm diameter each, an adjustable light source (Dolan-Jenner DC-950), an aspheric lens of 5 cm diameter an 0.5 normal aperture, a variable slit, two flat mirrors, a razor blade as knife-edge, and a concave lens of 5 cm diameter to project the images on a squared ground glass diffuser of 15 cm per side. Images were recorded with the Kiralux CMOS camera. The components were installed so that the collimated beams were parallel to the fins. The knife-edge was positioned at the focal plane of the second mirror in vertical position; it can be moved along the direction of the bigger temperature gradients, i.e., X-axis. To minimize optical aberrations the off-axis angle was 3°.

Runs were carried out for power inputs of 4.7 W, 9.0 W and 13.7 W at the base of the fins after losses. The experimental space was closed 24 h before experimentation to minimize external perturbations. The activities started introducing the seeding particles to the area and turning-on the power supply. At least two hours were necessary to fins reach steady conditions. One hundred photographs were taken for PIV analysis and 10 for schlieren visualization.

For each case, PIV measurements were taken in positions parallel to the XY-plane at Z/L =  −0.166, −0.375 and −0.5. The open-source PIVlab software was used to compute the velocity fields [8]. Cross-correlation in two steps was used to analyze the images; interrogation areas of 64 × 64 pixels and 32 × 32 pixels were used in the first and second steps, respectively. The one hundred velocity fields were averaged to obtain a resultant velocity field. A velocity uncertainty of 4.4 × 10−3 m/s was estimated. For schlieren visualization, the knife-edge was adjusted to block 40% of the light spot to improve the contrast of the images and the edge detection of the thermal boundary layer. The technique averages the temperature fluctuations on the surface of the fin along the path light, i.e., the Z-axis.

Fig. 2.
figure 2

Experimental results for IT0 and 4.7 W, measurement plane Z/L =  −0.5. Left: streamlines, Center: Schlieren visualization. Right: velocity profiles of v.

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3 Results

In Fig. 2 the results of IT0 are presented for 4.7 W at Z/L =  −0.5. An upward flow parallel to the fin can be observed on the streamline plot. It skirts the experimental model from the heating base to the trailing edge of the fin, where the flow bends left to occupy the zone behind the fin. Additionally, a secondary flow is induced by drag far from the fin. The velocity profiles of the v component depict that the thickness of the hydrodynamic boundary layer reaches X/L = 0.2. For all profiles, the velocity peak is reached at X/L = 0.043. As the flow travels on the surface of the fin, its velocity increases to 0.12 m/s at Y/L = 5/6. On the other side, the velocity of the secondary flow is 1.3 × 10−2 m/s. With respect to the thermal boundary layer, it has a maximum thickness of 9 mm and covers the whole surface of the fin. The bigger temperature gradients are presented on the surface of the fin from Y/L = 0.2 to Y/L = 0.8. The black lines in plots represent the area occupied by the fin. In the contour plots, bigger temperature gradients are colored in yellow tones and the smaller ones in blue.

In Fig. 3 are presented the velocity profiles of v for the three supplied powers, for middle plane Z/L = 0.5, at Y/L = 1/2. Increasing this parameter, the velocity of the hydrodynamical boundary layer increases as can be seen in v the velocity profiles for 4.7 W and 9.0 W. But for 13.7 W only the secondary flow is affected, reaching 5.7 × 10−2 m/s. Additionally, the base temperature was incremented for all fins, as seen in Table 1. On the other hand, for all fins the thickness of the thermal boundary layer increases as higher the supplied power, and its extension has the same behavior. For example, the thickness of IT0 and 13.7 W was 10 mm, and it extended far from Y/L = 1.2, i.e., is out of the visualization area of the schlieren system.

Fig. 3.
figure 3

Effect of the supplied power and the fractal iteration. Left: velocity profiles of v, at Y/L = 1/2. Center and Right: Streamlines and schlieren visualization, for IT2 and 9.0 W.

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Referring to the effect of the Sierpinski fractal iteration, the higher the iteration, the fewer the heat transfer area of the fins. Consequently, for the same power the temperature of the base of the fins increases, as shown in Table 1. The decrease in the area diminishes the buoyancy forces, so the velocity in the hydrodynamic boundary layer decreases too. Furthermore, the thermal boundary layer shrinks in thickness and extension; for example, IT0 and IT2 in Figs. 2 and 3 have the same thickness, although the power in IT2 is bigger. The perforations manufactured to the fins affect the convective flow, generating a rippled flow as shown by the streamlines in Fig. 3.

Table 1. Temperature of the base of the fins.
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4 Conclusions

In the current work, the velocity fields and the temperature gradients on the surface of four pre-fractal fins were measured and visualized. Each fin was analyzed for three different power supplies on the heating base, obtaining the following conclusions:

  • For larger supplied power, the faster the speed of the hydrodynamic boundary layer. At the same time, the thickness of the thermal and hydrodynamic boundary layer increases.

  • Due to drag effects, a secondary flow is generated far from the fin, and for a higher power, the interaction between this flow and the hydrodynamical boundary layer was more evident.

  • The perforations affect the thermal and hydrodynamical boundary layer, and its effect can be identified as a waving movement of the flow.

  • The higher the fractal iteration, the lower the heat transfer area of the fins; consequently, the temperature of its base incremented.