**1. Introduction**

The pipe, as a flow channel, is one of the simplest and most commonly used in power engineering and thermal devices. It is widely applied in heat exchangers and to transport the medium in installations. The phenomena occurring in it are related to both fluid mechanics and thermal processes; therefore, the key issue is to design the geometry in such a way as to maximize heat transfer while limiting a negative effect of flow resistance.

There are many methods to intensify the heat transfer process in the pipe. Among others, intensifying inserts of various shapes and other flow turbulence devices are widely used. Wijayanta et al. [1] conducted a numerical study to evaluate the thermal hydraulic performance of a turbulent flow inside a tube equipped with a square-cut twisted tape and a classical twisted tape insert. In the range of Reynolds number from 8000 to 18,000 under investigation, the tube with a square-cut twisted tape had the highest values of heat transfer rate, friction factor, and thermal performance factor for the twisted ratio *y*/*W* = 2.7. In [2], Wijayanta et al. investigated the thermal performance of a tube heat exchanger with punched delta-winglet vortex generators. The numerical studies for *Re* = 9100–17,400 and the attack angles of 30◦, 50◦, and 70◦ allowed one to determine values of the Nusselt number, friction factor, and thermal performance factor, which increased with an increasing value

**Citation:** Jasi ´nski, P.B.; Kowalczyk, M.J.; Romaniak, A.; Warwas, B.; Obidowski, D.; Gutkowski, A. Investigation of Thermal-Flow Characteristics of Pipes with Helical Micro-Fins of Variable Height. *Energies* **2021**, *14*, 2048. https:// doi.org/10.3390/en14082048

Academic Editor: Pouyan Talebizadeh Sardari

Received: 12 March 2021 Accepted: 2 April 2021 Published: 7 April 2021

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of attack angle. Jasi ´nski [3–5] conducted experimental and numerical investigations of a flow in the circular tube with ball turbulators. The investigations for different diameters of the balls, different distances between them, and *Re* = 10,000–300,000 were carried out. The results showed that the highest increase in the Nusselt number and the friction factor was observed for the insert ball with largest diameters; however, the highest values of thermal performance factor were observed for the smallest balls. Arjmandi et al. [6] conducted a numerical investigation of applying twisted tape turbulator and Al2O3/water nanofluid in double pipe heat exchanger. The studies of heat transfer coefficient and pressure drop were carried out for different pitches ratios 0.09–0.18, angles 0–30◦, and Reynolds numbers in the range 5000–20,000. According to obtained results, the optimization accomplished by the response surface methodology was performed, thanks to which the optimal vortex generator geometry was created.

Another method to intensify heat transfer is to apply nano-liquids as a modification of the working fluid. Patil et al. [7] made a review and presented investigations on synthesis, thermo-physical properties, and a heat transfer mechanism of nanofluids. Shajahan et al. [8] carried out experimental studies with a combination of nanofluids and inserts under the conditions of laminar flow. The results allowed for the determination of highest values of the Nusselt number, friction factor, and thermal performance factor according to different twist ratios of inserts and various types of nanofluids. Kristiawan et al. [9] investigated numerically an influence of micro-fins and TiO2/water nanofluids on thermo-hydraulic performance. The highest *PEC* (Performance Evaluation Criteria) achieved the squared mini-channel with micro-fins and the nanoparticle concentration of 0.01 vol.%. Asirvatham et al. [10] presented the results of experimental studies of convective heat transfer with a low volume fraction of the CuO/water nanofluid. According to the gained experimental data, a correlation for the Nusselt number was evolved.

The third method of heat transfer intensification is surface finning, which is considered in this article. The problem of similar geometry has been already addressed widely in the scientific literature. In most cases, researchers perform experimental investigations, but an increasing number of publications based on combined numerical and experimental tests is observed. For example, Mann and Eckels [11], in order to improve heat transfer and reduce a pressure drop, optimized the following geometrical pipe parameters: the number of fins, their height and helical angle, for a Reynolds number ranging from 30,000 to 60,000. They used the ANSYS Fluent software and the NSGA II algorithm in the study. After comparing the numerical results with the experimental results, the researchers concluded that for helical angles above 45◦, the relationship between heat transfer and geometry was chaotic rather than ordered. Wen-Tao et al. [12] conducted an experiment for a developed turbulent flow in the Reynolds number range from 10,000 to 100,000, for 16 different pipes with internal grooves. They compared the experimental results with the analytical method using the Gnielinski equation, which was then extended by the Nusselt number. The proposed extended equation yielded a good agreement of the results, where the deviation for 93% of the data was within ±20%.

Li et al. [13] used PIV (Particle Image Velocimetry) technology in the experimental investigations. They determined a relationship between flow characteristics and heat transfer for the geometry of a micro-finned rectangular duct. They observed an influence of the formation of coherent structures in the fluid on an increase in the heat transfer coefficient. Additionally, for low Reynolds numbers, characteristic of the laminar and transitional flow, they observed a lack of vortices filling the spaces between the fins, which was equivalent to the deterioration of heat transfer. Guo-Dong et al. [14] compared the experimental results for a plain pipe with an internally finned pipe using the (HMiM) BF4 medium (1-hexyl-3-methyl-imidazolium-tetrafluoroborate). Based on the obtained results, they determined empirical coefficients to calculate a friction coefficient and a heat transfer coefficient for a laminar flow (*Re* = 60–600). Brognaux et al. [15] developed the characteristics of a single-phase flow through the pipe with two types of internal fins. They analyzed an influence of geometry on the Prandtl number and determined a dependence

of the Reynolds number on the friction coefficient. On the basis of experimental results, they observed that the Prandtl number exponent agreed well with other correlations for the geometry with two-dimensional roughness.

Jasi ´nski [16] used numerical simulations to model a flow in the internally finned pipe for various helical angles of the micro-fins in the range of 0–90◦. Using the EGM (Entropy Generation Minimization) method to assess the flow, he showed that for each geometry, the minimum entropy was generated for the range *Re* = 60,000–90,000. Tang and Li [17] developed a correlation of the friction coefficient for data from various experiments. The data include old and new refrigerants that were used in internally finned tube heat exchangers. Jensen and Vlakancic [18] compared the experimental results for different geometries of internally fined pipes and compared them with theoretical relationships. They developed an empirical correlation to calculate the friction coefficient and the Nusselt number. The developed criteria differed for high fins and micro-fins, which allowed for the appropriate use of the resulting correlations depending on the geometry of the pipe. Dastmalchi et al. [19] optimized the geometry of micro-finned tubes using numerical methods to increase heat transfer and minimize flow resistance. The research concerned double pipe heat exchangers during the turbulent flow. In the study, they changed the geometric parameters of the pipe, i.e., its internal diameter, number of fins, fin height and their helix angle, for various Reynolds numbers from 3000 to 100,000. Among other findings, the authors noted that the optimal micro-fin height increases with an increasing Reynolds number for all internal pipe diameters considered. In addition, Dastmalchi et al. [20] conducted oil flow tests with low Reynolds numbers, ranging from 100 to 1000, in a tube with internal micro-fins. As before, they investigated an influence of the tube geometry on heat transfer efficiency and pressure drop during flow. The sample results for *Re* = 1000 show the maximal heat transfer increase of 44%, but also a 69% increase in the friction coefficient, for the flow through micro-finned tubes compared to smooth tube flow.

Filho and Jabardo [21] investigated an influence of the geometry of three selected types of pipes: smooth, micro-finned, and herringbone on the thermo-hydraulic flow characteristics with the use of refrigerants. On the basis of the conducted research, they found that the thermal efficiency was the best for the herringbone pipe; however, it had the highest pressure drop. Raj et al. [22] investigated the properties of various types of internally and externally finned pipes for two fluids: water and 46% glycol solution. They showed that the Prandtl number had a large impact on the intensification of heat transfer, and at the same time, it was dependent on the temperature of fluids.

The present paper attempts to investigate the problem of thermal efficiency of several pipe geometries with micro-fins. By changing the height of inner fins, the researchers searched for the highest values of the *PEC* coefficient while maintaining the same pumping power. The tests were performed using numerical simulations, whereas for one pipe geometry, experimental tests were carried out as well. The paper presents in detail and discusses an influence of the height of micro-fins on the friction factor, heat transfer, and thermal efficiency of pipes, as well as a correlation of these parameters with mathematical functions. The main goal of this article, which provides the novelty, was the discovery of the best geometry of a pipe in terms of thermal efficiency by determining the highest value of PEC coefficient.

#### **2. Materials and Methods**

#### *2.1. Experimental Stand*

Before the numerical simulations were carried out, one of the pipe geometries was tested on the experimental stand by Zawadzki et al. [23], and the results obtained from the experiment were used to validate the numerical model. The object of the investigations was an internal finned pipe, produced industrially by KME Germany AG & Co. KG under the working name "TECTUBE fin 12736CV50/65D" and with the dimensions shown in Figure 1, which is used for the production of both single-phase and two-phase heat exchangers, e.g., evaporators in the refrigeration equipment.

**Figure 1.** Scheme and dimensions of the micro-finned tube "TECTUBE fin 12736CV50/65D" produced by KME Germany AG & Co. KG company.

A schematic diagram of the experimental stand on which the test was performed is presented in Figure 2. The list of used devices in the experimental stand is introduced in Table 1. It consists of cold and hot water circulations, a data acquisition system, and a test section, which is a double pipe heat exchanger. During the measurements, constant temperatures in the circulation of cold and hot water were maintained at the inlets to the exchanger. In order to obtain thermal-flow characteristics, the water flow rate was gradually changed within the range of Reynolds numbers 10,000–60,000.

**Figure 2.** Experimental stand scheme for measuring micro-finned tubes [23].


**Table 1.** List of content used in experimental stand.

In Figure 3, the results of the experiment and numerical tests for the micro-finned tube from the previous studies are shown. This procedure also verified the two most popular turbulence models—the classic k-ε and the SST k-ω (Shear Stress Transport). It can be clearly seen that the best fit data and a good agreement with the experimental values for both pressure drop and heat transfer were obtained with the SST k-ω turbulence model. This is the reason why that model was used for further numerical calculations presented in this paper.

**Figure 3.** Comparison between results of the simulation and the experiment for the micro-fin tube and the SST and k-ε turbulence model: (**a**) mass flow vs. *grad p*, (**b**) *Nu* vs. *Re* [16,24].
