**6. Heat Transfer Analysis**

For heat transport, the turbulent kinetic viscosity plays a major role, as can be seen from Equation (4). Especially near the wall, the heat transfer is dominated by the thermal diffusivity, while it can be neglected in the main flow. In the main flow, the thermal diffusivity has a negligible contribution compared to the turbulent transport. In order to be able to neglect the turbulent Prandtl number in the modeling approach, the ratio of molecular to turbulent viscosity is examined in Figure 9. From the results, it can be seen that the turbulent kinetic viscosity *ν<sup>t</sup>* near the wall area is approaching zero, since the resolution of the mesh is very high here. This relationship can be explained by the following equation which is used for the calculation of *νt*:

$$\boldsymbol{\nu}\_{t} = (\mathbb{C}\_{\mathbb{S}} \boldsymbol{\Lambda})^{2} \mathbf{OP}. \tag{9}$$

where Δ is the cell width, *CS* is a model constant and **OP** is the LES model operator. The strong increase at *y*/*r* ≈ 0.05 can be explained by the quadratic relationship of the cell width. Since the cells in this range become larger, the value of *ν<sup>t</sup>* increases as can be concluded using Equation (9).

**Figure 9.** Representation of the turbulent viscosity in relation to the molecular viscosity.

In Figure 10, the measured Nusselt number is compared with the results of the simulation. The relationship between the Nusselt number of the examined structure and a smooth pipe is displayed. This ration is defined as *EF* with:

$$EF = \text{Nu} / \text{Nu}\_{\text{S}\prime} \tag{10}$$

where *NuS* is calculated with the Dittus–Boelter correlation [27]:

$$Nu\_S = 0.023 \cdot Re^{0.8} \cdot Pr^{0.4} \tag{11}$$

Measurements can only be made for the area between the ribs, and a detailed explanation can be found in Virgillio et al. [21]. The grey area corresponds to the tolerance range of the measurement errors. For a better overview, this representation is used instead of error bars. The comparison shows a very good agreement between the simulation and the measurement, except for the area directly before and after the rib. Investigations by Campet et al. [19] were able to demonstrate that, analogous to the results of this research, the deviation between simulation and experiment are significantly higher at these points. It can therefore be assumed that a measurement of the correct values for the heat transfer at these points is critical.

**Figure 10.** Comparison between measurement and simulation of the Nusselt number in relation to the smooth pipe between the ribs; the grey area shows the maximum and minimum values of the measurement.

As previously proven in Mayo et al. [17], there is a local maximum for the Nusselt number between two ribs along the axial flow direction in the pipe, from the point where the flow is reattached. This can be determined by evaluating the wall shear stress *τ<sup>x</sup>* in the flow direction. In Figure 11 , *τ<sup>x</sup>* is displayed in the range *x*/*e* = 2 and 8, and the reattachment point is located at the point where the value of *τ<sup>x</sup>* changes its sign. It can thus be determined at *x*/*e* ≈ 5.05. The local maximum of EF between the ribs, shown in Figure 12, is located in the same area. This is caused by the cold fluid transported from the bulk flow in wall direction, thus resulting in a higher temperature difference between wall and fluid. This increase is linear in relation to the Nusselt number. After the reattachment of the flow at *x*/*e* ≈ 5.05, the Nusselt number becomes smaller again.

**Figure 11.** Representation of *τx* to determine the reattachment point of the flow after detachment at the rib.

**Figure 12.** Display of the Nusselt number in the range from *x*/*e* = 2 to *x*/*e* = 8 to show the influence of the reattachment point of the flow on the Nusselt number.

For RIB-1, a reattachment point of the flow at *x*/*e* ≈ 4.25 has been determined in Virgilio et al. [20]. The author points out that the reattachment point of the flow for RIB-2 is located further upstream than with RIB-1, which can be confirmed by the results shown here. The authors state that the interruptions of the helix prevent the flow from a blockage

in comparison to the geometry in RIB-1. Using Figure 13, it can be illustrated with the streamlines that the fluid can also flow around the interrupted rib.

**Figure 13.** Streamlines of the magnitude velocity, over an interrupted rib.

The analysis of EF in Figure 14 demonstrates that an increase in efficiency compared to the smooth pipe is achieved almost everywhere, with the exception of the region highlighted in orange. In contrast to the RIB-1 with a continuous corrugation, the interruption and reduction at the sides results in reduced global heat transfer. Because the area in front of the centre of the rib before the flow separates has the highest heat transfer, and this area becomes smaller on both sides with this kind of geometry. The global value of EF for the measurement is 1.44 ± 0.41 for a Reynolds number of 21,160. In the simulation, the average value is 1.75 for a Reynolds number of 21,100 and is therefore still within the tolerance range of the measurement. Compared to the simulation of the RIB-1 geometry at a Reynolds number of 20,000, the value for EF decreases by 32%, showing that the interruption of the ribs reduces the heat transfer.

To compare the performance of the structured pipe with a smooth pipe, the performance evaluation factor (PEC) is used, defined as:

$$PEC = \frac{Nu/Nu\_S}{(f/f\_S)^{1/3}}\tag{12}$$

*Nu*, *f* , *NuS* and *fS* are the Nusselt number and the friction factor of the investigated geometry and the ones of a smooth pipe (index *S*). To calculate the friction factor, the correlation of Gnielinski [28] is applied. If the value of the PEC factor is higher than 1, the improvement of the heat transfer is greater than the increased costs due to the pressure loss. A PEC factor of 1.53 is determined for the pipe examined here. Thus, it can be shown that the additional heat transfer is higher than the additional loss due to the additionally required pump power.

**Figure 14.** Representation of the ratio of Nusselt number to Nusselt number smooth pipe over a complete rib.
