**3. Results**

After performing numerical simulations, the correctness of the obtained results was assessed by analyzing the distribution of selected physical quantities. In Figure 8, results from the numerical simulations are shown. The contours of temperature and vectors of velocity distributions are presented for the cases in Figure 5, respectively.

**Figure 8.** Results from the numerical simulations: (**a**) contour of temperature for *H* = 0.05 mm; (**b**) vectors of velocity (tangential projection) for *H* = 0.05 mm; (**c**) contour of temperature for *H* = 0.25 mm; (**d**) vectors of velocity (tangential projection) for *H* = 0.25 mm; (**e**) contour of temperature for *H* = 0.40 mm; (**f**) vectors of velocity (tangential projection) for *H* = 0.40 mm.

#### *3.1. Data Processing*

Based on the experimental data obtained from the examination of the industrial pipe TECTUBE fin 12736CV50/65D, the numerical model was validated and verified. The most important parameters in terms of the performance of the analyzed pipes are the friction factor and the heat transfer coefficient, as expressed by the Nusselt number. To create such characteristics, it is necessary to define basic flow parameters such as velocity, temperature, and pressure drop for each pipe geometry, which were obtained as a result of the computer simulations.

In numerical calculations, the pressure gradient described by Equation (1) was used to force the flow:

$$
\text{grad } p = \frac{\Delta p}{L} = f \cdot \frac{\mathfrak{u}\_{av}^2 \cdot \rho}{2 \cdot d}. \tag{1}
$$

For each tested geometry, the friction factor was calculated using the Darcy–Weisbach Equation (2), which is a modification of Equation (1):

$$f = \frac{2 \cdot \Delta p \cdot d}{\rho \cdot u\_{\text{av}}^2 \cdot L}. \tag{2}$$

The theoretical value of the friction factor for a plain pipe, as the reference level for the numerical results, was calculated from the Blasius correlation (3):

$$f\_{plain} = 0.3164 \cdot Re^{-0.25}.\tag{3}$$

Similarly, for the plain pipe, the Nusselt number was calculated from the well-known Dittus–Boelter [33,34] Equation (4):

$$Nu\_{plain} = 0.023 \cdot Re^{0.8} \cdot Pr^{0.4}.\tag{4}$$

For the investigated cases of a finned tube, the formula (5) was used to calculate the Nusselt number:

$$Nu = \frac{h \cdot d}{k}.\tag{5}$$

The heat transfer coefficient *h* used in Equation (5) was determined from the formula for heat flux (6), from the obtained results of the numerical tests.

$$h = \frac{q}{T\_{wall} - T\_{bulk}}\tag{6}$$

#### *3.2. Friction Factor*

The results of numerical simulations, in the form of the characteristics of the friction factor *f*(*Re*), are shown in Figure 9. The graph also features a curve for a smooth pipe, which was calculated from the Blassius correlation (3), to show the reference level.

**Figure 9.** Results from the numerical simulations for various fin heights in the tube—*f* vs. *Re*.

It is difficult to find any simple approximation function that would express the variation in the height of micro-fins with a mathematical formula. For this reason, the friction factor was approximated separately for each pipe geometry, using an exponential thirdorder decay function (7), which makes the best fitting of the research results. The calculated correlation coefficients of the function are given in Table 4.

$$f = \ y\_0 + A\_1 \cdot \exp\left(\frac{R\varepsilon}{t\_1}\right) + A\_2 \cdot \exp\left(\frac{R\varepsilon}{t\_2}\right) + A\_3 \cdot \exp\left(\frac{R\varepsilon}{t\_3}\right) \tag{7}$$


**Table 4.** Fitting parameters for Equation (7). For all geometries coefficient, *y*<sup>0</sup> = 0.0208.

For the micro-fins of height *H* = 0.05 mm, *H* = 0.25 mm, and *H* = 0.40 mm (minimal, medium, and maximal micro-fin height, respectively) the curves were calculated for several *ε* values from Table 2, using the empirical formula (8) given by Swamee and Jain [35].

In Figure 10, an influence of the pipe geometry on a value of the friction factor *f*(*H*) for several Reynolds numbers is shown. The individual curves show the friction factor with respect to micro-fins height *H* for specific Reynolds numbers, i.e., for the same flow parameters.

**Figure 10.** Results from the numerical simulations of the friction factor for various micro-fin heights in the tube—*f* vs. *H* (for a specific value of Reynolds Number).

An analysis of the obtained results allowed us to observe a large deviation from the characteristics of rough pipes presented in the Moody diagram [36]. In the case of pipes, the relative roughness is defined as a ratio of the height of unevenness to the diameter, and for the tested geometries, it is closely related to the height of the micro-fins. For the presented channels, it is 0.004–0.033, which was presented in Table 2.

$$f = \frac{0.25}{\left(\log\left(\frac{\xi}{37} + \frac{5.74}{\log^{10}}\right)\right)^2} \tag{8}$$

#### *3.3. Heat Transfer*

In Figure 11, thermal characteristics for all tested geometries are shown, in the form of the *Nu*(*Re*) function, and one curve for a plain pipe, calculated from (4) as the reference level.

**Figure 11.** Results from the numerical simulations for various fin heights in the tube—*Nu* vs. *Re*.

Between the functions of Nusselt numbers shown in Figure 11, there is no simple geometric dependence (similar to the friction factor); nevertheless, certain mathematical functions can be adjusted to these data.

For the correlation of the Nusselt number function for the studied cases, the most suitable formula is the exponential function (9):

$$N\mu = A \cdot R e^B \cdot Pr^{0.4}.\tag{9}$$

In Table 5, values of the *A* and *B* coefficients for each fin height are shown.

**Table 5.** Fitting parameters for Equation (9).


#### *3.4. PEC (Performance Evaluation Criteria)*

The *PEC* (Performance Evaluation Criteria) thermal efficiency rating for the tested pipes was calculated from Equation (10). This parameter combines the Nusselt number and the friction factor obtained from the tests of channels with geometries other than the smooth pipe, and the *Nu* and *f* numbers calculated for a smooth pipe with the same Reynolds number. The *PEC* is an indicator of how the heat transfer will increase in the tested channel relative to a plain pipe for the same pumping power. *PEC* values above 1 indicate a greater impact of intensification of the heat transfer in the pipe than flow resistance, while values below 1 indicate greater flow resistance in relation to the benefit obtained from intensifying the heat transfer for the tested geometry (Figure 12).

$$PEC = \frac{\frac{Nu}{Nu\_{plair}}}{\left(\frac{f}{f\_{plair}}\right)^{\frac{1}{3}}} \tag{10}$$

**Figure 12.** PEC (Performance Evaluation Criteria) coefficient of effectiveness for each tested model— *PEC* vs. *Re*.

#### **4. Discussion**

A validation of the numerical model with the experimental data from the tests for a pipe with the fin height of *H* = 0.25 mm was presented. When analyzing the results, the discrepancies between the experiment and the obtained numerical results found are as follows: for the friction factor—maximum 7%; for a Nusselt number—maximum 12%, as indicated in Figure 3.

Using the Blasius formula determining the friction factor, a comparison of the obtained numerical results for different micro-fin heights with the plain pipe was made (Figure 9). For each case of micro-finned pipes, the obtained values are greater than the values for a smooth pipe, which indicates the physical correctness of the results obtained. For *Re* = 10,000–25,000, the lowest values of the friction factor are achieved by the pipe with micro-fins of the height of *H* = 0.05 mm; whereas the highest values for the height of micro-fins are *H* = 0.30 mm. All curves are rather regular and straight lines on a logarithmic plot. For Reynolds numbers above 25,000, the lowest values of the friction factor are achieved by tubes with micro-fins of the height of *H* = 0.10 mm and *H* = 0.15 mm; in turn, the highest values are achieved by two geometries for pipes with micro-fins of the height of *H* = 0.30 mm and *H* = 0.35 mm. In this range, functions change their character, and it is difficult to find a regularity in their position.

Considering the dependence of the friction factor in relation to the fin height for different Reynolds numbers, its value decreases with an increase in the Reynolds number for each flow channel. On the basis of Figure 10, apart from minor irregularities in the characteristics, one can notice their quite clear trend. For the micro-fins height *H* = 0.30–0.35 mm, a clear maximum can be seen for all the characteristics, which means that these pipe geometries give the highest flow resistance. On the other hand, for the height of approximately *H* = 0.15 mm, one can observe a "slight" minimum of these curves and a decrease in the value for the highest height of micro-fins *H* = 0.40 mm. The decrease in the value of the friction factor for *H* = 0.40 mm is probably due to low thickness of the fin compared to other dimensions (Figure 5). At the same time, a small contact area with the main turbulent core, where the highest flow velocities occur, exerts also an influence on a decrease in the friction factor.

Each tested tube had a different relative roughness related to the height of the microfins. In Figure 9, a complete discrepancy between the positions of the curves obtained from the numerical simulations and those calculated theoretically on the basis of the well-known formula (8) was shown for the same relative roughness. One can notice that it is not possible to calculate the friction factors from Equation (8) for the tested geometries as the model derives significantly different values than the ones obtained in the tests. Therefore, one of the fundamental conclusions resulting from the numerical tests carried out is a lack of correlation of the friction factor between the theoretically calculated (for irregular roughness) and the one obtained from the tests (for the same roughness but with regular shapes). The same fact was recognized by Wang et al. in their research [37].

When analyzing the obtained results of the heat transfer intensity for the geometries under investigation, several phenomena can be observed. The presented results show an irregular order of the Nusselt number characteristics for various fluid flow rates (Figure 11). For Reynolds numbers above 20,000, pipes with micro-fins having the height of *H* = 0.20 mm and higher achieve significantly larger values of the Nusselt number than for the plain pipe, compared to the cases with micro-fins below *H* = 0.20 mm, for which the characteristics are very similar to those of the smooth pipe. In the entire range of Reynolds numbers, the highest values of Nusselt numbers are achieved by pipes with micro-fin heights equal to *H* = 0.30 mm and *H* = 0.35 mm, and the same pipes for which the highest friction factor was observed. The irregular position of these characteristics indicates a significant influence of turbulences in the vicinity of the laminar boundary layer and the size of the heat transfer surface related to the height of the micro-fins.

As can be seen in Figure 12, in the range of low Reynolds numbers up to approximately 25,000, the *PEC* value of less than 1 was observed for all geometries. It means that using these pipes in this flow range is less efficient than using the regular plain pipe. For Reynolds numbers higher than 25,000, all characteristics are higher than 1, and it is within this range that the use of such pipes is justified. The highest *PEC* values, up to 1.25, are achieved by tubes with the micro-fin height of 0.30 and 0.35 mm for Reynolds numbers above 50,000. A characteristic feature of these geometries is a virtually constant value of this coefficient in the given Reynolds number range. Therefore, these micro-fins heights can be considered the most optimal for thermal-flow applications among all numerically tested in this work.

#### **5. Conclusions**

Based on the numerical investigations presented, the most important conclusions of this work can be drawn as follows:


**Author Contributions:** Conceptualization, P.B.J., M.J.K., A.R., B.W., and D.O.; methodology, D.O.; validation, M.J.K., A.R., and B.W.; investigation, M.J.K., A.R., and B.W.; writing—original draft preparation, M.J.K., A.R., and B.W.; writing—review and editing, P.B.J. and A.G.; supervision, P.B.J. and A.G.; All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** This article has been completed while the second, third, and fourth authors were Doctoral Candidates in the Interdisciplinary Doctoral School at the Lodz University of Technology, Poland. We would like to thank Malgorzata Jozwik for her significant linguistic help during the preparation of the manuscript.

**Conflicts of Interest:** The authors declare no conflict of interest.

## **Nomenclature**


#### **References**

