**2. Problem Statement and Boundary Conditions**

Figure 1 schematically shows the tube with twisted tape. The height of the proposed twisted tape (*H*) is 95% of the tube diameter (*D* = 20 mm) with a thickness (*t*) of 0.4 mm. Four different twisted tape pitches (*P*) are investigated, as shown in Figure 1a. To study the position of truncated twisted tape, three different locations, i.e., entrance, center and exit, are examined, as shown in Figure 1b. Note that the length of the tube is 400 mm. Water enters the tube at 300 K with uniform velocity, and a symmetrical constant heat flux of 5000 W/m<sup>2</sup> is applied on the walls, while the twisted tape walls are thermally insulated. A pressure outlet condition is also employed for the tube outlet [44–46].

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**Figure 1.** Schematic of the proposed system for (**a**) different twisted tape pitches and (**b**) different twisted tape positions. **Figure 1.** Schematic of the proposed system for (**a**) different twisted tape pitches and (**b**) different twisted tape positions.

**Figure 1.** Schematic of the proposed system for (**a**) different twisted tape pitches and (**b**) different

**Figure 1.** Schematic of the proposed system for (**a**) different twisted tape pitches and (**b**) different

**Figure 1.** Schematic of the proposed system for (**a**) different twisted tape pitches and (**b**) different

#### **Figure 1.** Schematic of the proposed system for (**a**) different twisted tape pitches and (**b**) different twisted tape positions. **3. Governing Equations 3. Governing Equations**  twisted tape positions. twisted tape positions. twisted tape positions.

**3. Governing Equations** 

**3. Governing Equations** 

**3. Governing Equations** 

**3. Governing Equations**  This research investigates a steady laminar flow of incompressible nanofluid, neglecting the effects of radiation and viscosity losses. The governing equations are defined as follows: This research investigates a steady laminar flow of incompressible nanofluid, neglecting the effects of radiation and viscosity losses. The governing equations are defined as follows: **3. Governing Equations 3. Governing Equations 3. Governing Equations Figure 1.** Schematic of the proposed system for (**a**) different twisted tape pitches and (**b**) different twisted tape positions. **Figure 1.** Schematic of the proposed system for (**a**) different twisted tape pitches and (**b**) different twisted tape positions. **Figure 1.** Schematic of the proposed system for (**a**) different twisted tape pitches and (**b**) different twisted tape positions.

This research investigates a steady laminar flow of incompressible nanofluid, neglecting the effects of radiation and viscosity losses. The governing equations are defined as follows: . ൫ሬ⃗൯=0 (1) . (ρ → *V*) = 0 (1) This research investigates a steady laminar flow of incompressible nanofluid, neglecting the effects of radiation and viscosity losses. The governing equations are defined as follows: This research investigates a steady laminar flow of incompressible nanofluid, neglecting the effects of radiation and viscosity losses. The governing equations are defined as follows: This research investigates a steady laminar flow of incompressible nanofluid, neglecting the effects of radiation and viscosity losses. The governing equations are defined as follows: This research investigates a steady laminar flow of incompressible nanofluid, neglecting the This research investigates a steady laminar flow of incompressible nanofluid, neglecting the This research investigates a steady laminar flow of incompressible nanofluid, neglecting the

$$\nabla.(\rho \overrightarrow{VV}) = -\nabla p + \mu \overrightarrow{\mathcal{V}}^2 \overrightarrow{V} \tag{2}$$

$$\nabla \cdot (\rho \overrightarrow{\mathbf{V} \mathbf{C}\_p T}) = \nabla \cdot (k \nabla T) \tag{3}$$

. ൫ሬ⃗൯ = . () (3) Different parameters calculated in this study, including the average and local heat transfer coefficient and Nusselt number, friction factor and PEC are defined as follows [36,37]: . ൫ሬ⃗൯ = . () (3) Different parameters calculated in this study, including the average and local heat transfer . ൫ሬ⃗൯ = . () (3) Different parameters calculated in this study, including the average and local heat transfer . ൫ሬ⃗൯ = . () (3) Different parameters calculated in this study, including the average and local heat transfer . ൫ሬ⃗ሬ⃗൯ = − + ଶሬ⃗ (2) . ൫ሬ⃗൯ = . () (3) . ൫ሬ⃗ሬ⃗൯ = − + ଶሬ⃗ (2) . ൫ሬ⃗൯ = . () (3) . ൫ሬ⃗ሬ⃗൯ = − + ଶሬ⃗ (2) . ൫ሬ⃗൯ = . () (3) Different parameters calculated in this study, including the average and local heat transfer coefficient and Nusselt number, friction factor and PEC are defined as follows [36,37]:

coefficient and Nusselt number, friction factor and PEC are defined as follows [36,37]: <sup>=</sup> 4 (4) coefficient and Nusselt number, friction factor and PEC are defined as follows [36,37]: coefficient and Nusselt number, friction factor and PEC are defined as follows [36,37]: coefficient and Nusselt number, friction factor and PEC are defined as follows [36,37]: Different parameters calculated in this study, including the average and local heat transfer coefficient and Nusselt number, friction factor and PEC are defined as follows [36,37]: Different parameters calculated in this study, including the average and local heat transfer coefficient and Nusselt number, friction factor and PEC are defined as follows [36,37]: Different parameters calculated in this study, including the average and local heat transfer coefficient and Nusselt number, friction factor and PEC are defined as follows [36,37]: *D<sup>h</sup>* = 4*A P* (4)

$$f = \frac{2\Delta P}{\rho V^2} \frac{D\_h}{L} \tag{5}$$

$$h\_{\mathbf{x}} = \frac{q^{\prime\prime}}{T\_w - T\_b} \tag{6}$$

$$h\_{\mathbf{x}}$$

$$\mathbf{M}\_{\mathbf{U}} = \mathbf{I}\_{\mathbf{b}}$$

$$\mathbf{N}u\_{\mathbf{x}} = \frac{h\_{\mathbf{x}\mathbf{D}\_{\mathbf{h}}}}{k} \tag{7}$$

$$\mathbf{1} \quad \mathbf{C}^{\mathbf{l}}$$

(6)

(6)

(8)

(**b**)

(**b**)

(6)

(6)

(8)

(8)

(8)

$$Nu\_{avg} = \frac{1}{l} \int\_{0}^{l} Nu\_{x} dx \tag{8}$$

$$Nu\_{x} / Nu\_{x}$$

$$\text{PEC} = \frac{\text{Nu} / \text{Nu}\_0}{\left(f / f\_0\right)^{1/3}} \tag{9}$$

(/)ଵ/ଷ (9)

(/)ଵ/ଷ (9)

(/)ଵ/ଷ (9)

= /

= /

= /

### **4. Numerical Procedure 4. Numerical Procedure**

The commercial ANSYS Fluent computational fluid dynamics (CFD) code is employed to perform the simulation and solve the equations [47–49]. The velocity–pressure coupling is resolved using the coupled algorithm, and the convection terms are discretized using the second-order upwind scheme. The convergence criteria of 10−<sup>6</sup> are also selected. The commercial ANSYS Fluent computational fluid dynamics (CFD) code is employed to perform the simulation and solve the equations [47–49]. The velocity–pressure coupling is resolved using the coupled algorithm, and the convection terms are discretized using the second-order upwind scheme. The convergence criteria of 10−6 are also selected.

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### *4.1. Grid Study 4.1. Grid Study*

Figure 2 illustrates the computational mesh with finer mesh generated in the near-wall region due to velocity and temperature boundary layers and high gradients of variables. A fully structured mesh is created as shown using ANSYS meshing to have a high degree of quality and faster convergence in CFD FLUENT code, which also results in less computational time. Figure 2 illustrates the computational mesh with finer mesh generated in the near-wall region due to velocity and temperature boundary layers and high gradients of variables. A fully structured mesh is created as shown using ANSYS meshing to have a high degree of quality and faster convergence in CFD FLUENT code, which also results in less computational time.

**Figure 2.** The meshing of the computational domain. **Figure 2.** The meshing of the computational domain.

Different grid numbers are examined for grid independency analysis in the case of fully filled twisted tape with a pitch of L/4 and a Reynolds number of 1000. The average Nusselt number is selected as the selection criterion [50] presented in Table 1. As listed, case 3 is chosen for all the simulations, since using finer meshing leads to relative errors of less than one percent. Different grid numbers are examined for grid independency analysis in the case of fully filled twisted tape with a pitch of L/4 and a Reynolds number of 1000. The average Nusselt number is selected as the selection criterion [50] presented in Table 1. As listed, case 3 is chosen for all thesimulations, since using finer meshing leads to relative errors of less than one percent.


**Table 1.** Grid independence analysis. **Case Number of Elements Nusselt Number Error (%)** 

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**Table 1.** Grid independence analysis.

#### *4.2. Validation* To obtain a reliable result, the experimental results of Qi et al. [42] for the average Nusselt

To obtain a reliable result, the experimental results of Qi et al. [42] for the average Nusselt number for laminar fluid flow in a heat exchanger using stationary twisted tape are used. They experimentally examined pure water and nanofluid flow for different Reynolds numbers for a twisted tape length of 1600 mm and a pitch size of 100 mm with a width and thickness of 16 and 2 mm, respectively. Figure 3 displays the average Nusselt number for the cases of pure water and stationary twisted tapes for different Reynolds numbers. As shown, the results are in excellent agreement with the experimental data of Qi et al. [42], where the maximum difference is less than 2%. number for laminar fluid flow in a heat exchanger using stationary twisted tape are used. They experimentally examined pure water and nanofluid flow for different Reynolds numbers for a twisted tape length of 1600 mm and a pitch size of 100 mm with a width and thickness of 16 and 2 mm, respectively. Figure 3 displays the average Nusselt number for the cases of pure water and stationary twisted tapes for different Reynolds numbers. As shown, the results are in excellent agreement with the experimental data of Qi et al. [42], where the maximum difference is less than 2%.

**Figure 3.** Validation of the present numerical results with the experimental data of Qi et al. [42]. **Figure 3.** Validation of the present numerical results with the experimental data of Qi et al. [42].

#### **5. Results and Discussion 5. Results and Discussion**

Numerical simulations are performed at four pitch values () of L, L/2, L/3 and L/4, four Reynolds numbers () of 250, 500, 750 and 1000, three twisted tape truncation (λ) percentages of 25, 50 and 75% and three positions of twisted tape at the entrance, center and exit of the tube, which are investigated in the following: Numerical simulations are performed at four pitch values (*P*) of L, L/2, L/3 and L/4, four Reynolds numbers (*Re*) of 250, 500, 750 and 1000, three twisted tape truncation (λ) percentages of 25, 50 and 75% and three positions of twisted tape at the entrance, center and exit of the tube, which are investigated in the following:

### *5.1. Effect of Twisted Tape Pitch 5.1. E*ff*ect of Twisted Tape Pitch*

tube where cooling is required.

In the first step, the effect of twisted tape pitch on the hydrothermal characteristics of the tube is examined and analyzed. Figure 4 represents the local Nusselt number () throughout the tube length for the plain tube (PT) and four twisted tape pitch values at = 250 while the twisted tape is fully fitted in the tube (no truncation). Using the twisted tape and decreasing its pitch magnitude noticeably augments the local along the tube length. The reason is that the twisted tape inserts create secondary flow as a result of flow swirling, which consequently improves the flow mixing, disturbs the thermal boundary layer and increases the heat transfer rate [51]. In other words, the twisted tape redirects the colder core fluid with a better cooling capacity to the heated walls of the In the first step, the effect of twisted tape pitch on the hydrothermal characteristics of the tube is examined and analyzed. Figure 4 represents the local Nusselt number (*Nu*) throughout the tube length for the plain tube (PT) and four twisted tape pitch values at *Re* = 250 while the twisted tape is fully fitted in the tube (no truncation). Using the twisted tape and decreasing its pitch magnitude noticeably augments the local *Nu* along the tube length. The reason is that the twisted tape inserts create secondary flow as a result of flow swirling, which consequently improves the flow mixing, disturbs the thermal boundary layer and increases the heat transfer rate [51]. In other words, the twisted tape redirects the colder core fluid with a better cooling capacity to the heated walls of the tube where cooling is required.

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**Figure 4.** Local along the tube length for plain tube (PT) and twisted tape inserts with P = L, L/2, L/3, L/4 at = 250. **Figure 4.** Local *Nu* along the tube length for plain tube (PT) and twisted tape inserts with P = L, L/2, L/3, L/4 at *Re* = 250. **Figure 4.** Local along the tube length for plain tube (PT) and twisted tape inserts with P = L, L/2, L/3, L/4 at = 250.

The cause of the heat transfer enhancement in Figure 4 can be seen in Figure 5, in which the streamlines colored by velocity magnitude are illustrated for PT and four twisted tape pitch values. It is visible that as the twisted tape pitch value decreases, the flow path undergoes more changes. This is because more swirl flow fronts can be seen in lower pitch values with higher radial velocity, implying stronger secondary and mixing flow. As a result, it leads to a more effective redirection of core colder fluid towards the heated wall and, consequently, more heat is transferred between the fluid and heated wall. The cause of the heat transfer enhancement in Figure 4 can be seen in Figure 5, in which the streamlines colored by velocity magnitude are illustrated for PT and four twisted tape pitch values. It is visible that as the twisted tape pitch value decreases, the flow path undergoes more changes. This is because more swirl flow fronts can be seen in lower pitch values with higher radial velocity, implying stronger secondary and mixing flow. As a result, it leads to a more effective redirection of core colder fluid towards the heated wall and, consequently, more heat is transferred between the fluid and heated wall. The cause of the heat transfer enhancement in Figure 4 can be seen in Figure 5, in which the streamlines colored by velocity magnitude are illustrated for PT and four twisted tape pitch values. It is visible that as the twisted tape pitch value decreases, the flow path undergoes more changes. This is because more swirl flow fronts can be seen in lower pitch values with higher radial velocity, implying stronger secondary and flow. As a result, it leads to a more effective redirection of core colder fluid towards the heated wall and, consequently, more heat is transferred between the fluid and heated wall.

**Figure 5.** Streamlines colored by velocity magnitude for PT and twisted tape inserts with P = L, L/2, L/3, L/4 at = 250. **Figure 5.** Streamlines colored by velocity magnitude for PT and twisted tape inserts with P = L, L/2, L/3, L/4 at = 250. **Figure 5.** Streamlines colored by velocity magnitude for PT and twisted tape inserts with P = L, L/2, L/3, L/4 at *Re* = 250.

Figure 6 shows how decreasing the twisted tape pitch magnitude affects the cooling of the heated wall at = 250. The temperature distribution on the heated wall implies that the secondary and mixing flow intensity affect the heated wall. In PT, the uniform enhancement of temperature along the tube length is visible, showing the thermal boundary layer development without any disturbance. On the contrary, as the twisted tape is inserted in the tube, the change in temperature distribution on the heated wall is visible. The temperature value on the heated wall decreases along the tube length and strengthens as the pitch value reduces. In higher pitch values, some hotspot regions are visible on the heated wall temperature contour; however, these regions decline at lower pitch values, resulting in better cooling performance of the system. Figure 6 shows how decreasing the twisted tape pitch magnitude affects the cooling of the heated wall at = 250. The temperature distribution on the heated wall implies that the secondary and mixing flow intensity affect the heated wall. In PT, the uniform enhancement of temperature along the tube length is visible, showing the thermal boundary layer development without any disturbance. On the contrary, as the twisted tape is inserted in the tube, the change in temperature distribution on the heated wall is visible. The temperature value on the heated wall decreases along the tube length and strengthens as the pitch value reduces. In higher pitch values, some hotspot regions are visible on the heated wall temperature contour; however, these regions decline at lower pitch values, resulting in better cooling performance of the system. Figure 6 shows how decreasing the twisted tape pitch magnitude affects the cooling of the heated wall at *Re* = 250. The temperature distribution on the heated wall implies that the secondary and mixing flow intensity affect the heated wall. In PT, the uniform enhancement of temperature along the tube length is visible, showing the thermal boundary layer development without any disturbance. On the contrary, as the twisted tape is inserted in the tube, the change in temperature distribution on the heated wall is visible. The temperature value on the heated wall decreases along the tube length and strengthens as the pitch value reduces. In higher pitch values, some hotspot regions are visible on the heated wall temperature contour; however, these regions decline at lower pitch values, resulting in better cooling performance of the system.

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**Figure 6.** Temperature contours on the heated wall for PT and twisted tape inserts with P = L, L/2, L/3, L/4 at = 250. **Figure 6.** Temperature contours on the heated wall for PT and twisted tape inserts with P = L, L/2, L/3, L/4 at *Re* = 250. **Figure 6.** Temperature contours on the heated wall for PT and twisted tape inserts with P = L, L/2, L/3, L/4 at = 250.

To better understand the temperature distribution, four different cross-sections are defined along the tube length and are displayed in Figure 7. In the following, different parameters, such as temperature and velocity, are illustrated in them. To better understand the temperature distribution, four different cross-sections are defined along the tube length and are displayed in Figure 7. In the following, different parameters, such as temperature and velocity, are illustrated in them. To better understand the temperature distribution, four different cross-sections are defined along the tube length and are displayed in Figure 7. In the following, different parameters, such as temperature and velocity, are illustrated in them.

**Figure 7.** Generated cross-sectional surfaces throughout the tube length for post-processing purposes. **Figure 7.** Generated cross-sectional surfaces throughout the tube length for post-processing purposes. **Figure 7.** Generated cross-sectional surfaces throughout the tube length for post-processing purposes.

Figure 8 demonstrates the cross-sectional temperature contours on the surfaces shown in Figure 7 for PT and four twisted tape pitch values at = 250. The PT temperature contours show the normal development of the thermal boundary layer throughout the tube, leading to a great drop in heat transfer along the tube. On the other hand, the thermal boundary layer disturbance is intensified and gets thinner as the twisted tape is inserted in the tube, which is more effective for a lower pitch in heat transfer between the fluid and heated wall. Besides, fewer hotspot regions causing a reduction in heat transfer enhancement is visible at lower pitch values of the twisted tape, proving the cooling process improvement. Another point that can be noticed in this figure is that the presence of twisted tape and lowering its pitch value redirects the core colder fluid to the vicinity of the hot wall. The twisted tape causes more efficient heat dissipation from the wall and, consequently, more heat is transferred from the wall to the fluid. It should be noted that the contours are almost symmetrical, related to the center of the tube in all cases. Figure 8 demonstrates the cross-sectional temperature contours on the surfaces shown in Figure 7 for PT and four twisted tape pitch values at = 250. The PT temperature contours show the normal development of the thermal boundary layer throughout the tube, leading to a great drop in heat transfer along the tube. On the other hand, the thermal boundary layer disturbance is intensified and gets thinner as the twisted tape is inserted in the tube, which is more effective for a lower pitch in heat transfer between the fluid and heated wall. Besides, fewer hotspot regions causing a reduction in heat transfer enhancement is visible at lower pitch values of the twisted tape, proving the cooling process improvement. Another point that can be noticed in this figure is that the presence of twisted tape and lowering its pitch value redirects the core colder fluid to the vicinity of the hot wall. The twisted tape causes more efficient heat dissipation from the wall and, consequently, more heat is transferred from the wall to the fluid. It should be noted that the contours are almost symmetrical, related to the center of the tube in all cases. Figure 8 demonstrates the cross-sectional temperature contours on the surfaces shown in Figure 7 for PT and four twisted tape pitch values at *Re* = 250. The PT temperature contours show the normal development of the thermal boundary layer throughout the tube, leading to a great drop in heat transfer along the tube. On the other hand, the thermal boundary layer disturbance is intensified and gets thinner as the twisted tape is inserted in the tube, which is more effective for a lower pitch in heat transfer between the fluid and heated wall. Besides, fewer hotspot regions causing a reduction in heat transfer enhancement is visible at lower pitch values of the twisted tape, proving the cooling process improvement. Another point that can be noticed in this figure is that the presence of twisted tape and lowering its pitch value redirects the core colder fluid to the vicinity of the hot wall. The twisted tape causes more efficient heat dissipation from the wall and, consequently, more heat is transferred from the wall to the fluid. It should be noted that the contours are almost symmetrical, related to the center of the tube in all cases.

tape plane.

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**Figure 8.** Cross-sectional temperature contours for PT and twisted tape inserts with P = L, L/2, L/3, L/4 at = 250. **Figure 8.** Cross-sectional temperature contours for PT and twisted tape inserts with P = L, L/2, L/3, L/4 at *Re* = 250.

Figure 9 displays the cross-sectional velocity contours on the surfaces shown in Figure 7 for PT and four twisted tape pitch values at = 250 . The velocity contours imply the intensity of secondary and mixing flow in the presence of twisted tape inserts. In other words, the higher velocity of the fluid near the heated wall shows a stronger secondary flow, and, as a result, a higher fluid momentum near the wall and a better cooling process could be achieved. It is visible in this figure that inserting twisted tape with a pitch value of L results in a high-velocity region of the fluid, which is strengthened as the twisted tape pitch value decreases, resulting in the stronger secondary flow observed in Figure 5. It should be noted that the contours are almost symmetrical, related to the center of the tube in all cases; however, for the case with P = L, it is almost symmetrical related to the twisted Figure 9 displays the cross-sectional velocity contours on the surfaces shown in Figure 7 for PT and four twisted tape pitch values at *Re* = 250. The velocity contours imply the intensity of secondary and mixing flow in the presence of twisted tape inserts. In other words, the higher velocity of the fluid near the heated wall shows a stronger secondary flow, and, as a result, a higher fluid momentum near the wall and a better cooling process could be achieved. It is visible in this figure that inserting twisted tape with a pitch value of L results in a high-velocity region of the fluid, which is strengthened as the twisted tape pitch value decreases, resulting in the stronger secondary flow observed in Figure 5. It should be noted that the contours are almost symmetrical, related to the center of the tube in all cases; however, for the case with P = L, it is almost symmetrical related to the twisted tape plane.

**Figure 9.** Cross-sectional velocity contours for PT and twisted tape inserts with P = L, L/2, L/3, L/4 at = 250. **Figure 9.** Cross-sectional velocity contours for PT and twisted tape inserts with P = L, L/2, L/3, L/4 at *Re* = 250.

To better quantify the heat transfer modification in the presence of twisted tape inserts with different pitch values, Figure 10 is provided for various . increases as changes due to the more effective advection phenomenon and fluid momentum in higher fluid velocities. As seen in this figure, as an example, using twisted tape with pitch values of L, L/2, L/3 and L/4 increases the average by about 26.87, 55.03, 86.59 and 151.42% at = 1000 compared with the PT, respectively. To better quantify the heat transfer modification in the presence of twisted tape inserts with different pitch values, Figure 10 is provided for various *Re*. *Nu* increases as *Re* changes due to the more effective advection phenomenon and fluid momentum in higher fluid velocities. As seen in this figure, as an example, using twisted tape with pitch values of L, L/2, L/3 and L/4 increases the average *Nu* by about 26.87, 55.03, 86.59 and 151.42% at *Re* = 1000 compared with the PT, respectively.

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**Figure 10.** Variations of average ratio with respect to PT at various pitch values and . **Figure 10.** Variations of average *Nu* ratio with respect to PT at various pitch values and *Re*. **Figure 10.** Variations of average ratio with respect to PT at various pitch values and .

Figure 11 shows the variations of the friction coefficient ratio with respect to PT for different pitch values and . This figure shows that applying twisted tape enhances the friction coefficient ratio due to the added surface area and flow blockage [52]. Furthermore, higher values for the friction coefficient ratio are observed in lower pitch values due to the creation of the more intense secondary flow shown in Figure 5; however, when the twisted tape pitch is equal to L, its enhancement is insignificant due to the creation of very weak secondary flow in the tube. Moreover, this figure reveals that the friction coefficient ratio increases as changes because more intense swirling flow. Figure 11 shows the variations of the friction coefficient ratio with respect to PT for different pitch values and *Re*. This figure shows that applying twisted tape enhances the friction coefficient ratio due to the added surface area and flow blockage [52]. Furthermore, higher values for the friction coefficient ratio are observed in lower pitch values due to the creation of the more intense secondary flow shown in Figure 5; however, when the twisted tape pitch is equal to L, its enhancement is insignificant due to the creation of very weak secondary flow in the tube. Moreover, this figure reveals that the friction coefficient ratio increases as *Re* changes because more intense swirling flow. Figure 11 shows the variations of the friction coefficient ratio with respect to PT for different pitch values and . This figure shows that applying twisted tape enhances the friction coefficient ratio due to the added surface area and flow blockage [52]. Furthermore, higher values for the friction coefficient ratio are observed in lower pitch values due to the creation of the more intense secondary flow shown in Figure 5; however, when the twisted tape pitch is equal to L, its enhancement is insignificant due to the creation of very weak secondary flow in the tube. Moreover, this figure reveals that the friction coefficient ratio increases as changes because more intense swirling flow.

**Figure 11.** Variations of friction coefficient ratio with respect to PT at various pitch values and . **Figure 11.** Variations of friction coefficient ratio with respect to PT at various pitch values and . **Figure 11.** Variations of friction coefficient ratio with respect to PT at various pitch values and *Re*.

#### *5.2. Effect of Truncated Twisted Tape Position and Percentage 5.2. Effect of Truncated Twisted Tape Position and Percentage 5.2. E*ff*ect of Truncated Twisted Tape Position and Percentage*

So far, it has been shown that the twisted tape with a pitch of L/4 results in the best thermal performance of the system. Therefore, the following simulations are performed for this pitch value. To show the effect of twisted tape truncation percentage on local throughout the tube length for different twisted tape positions, Figure 12 is provided at = 250. In Figure 12a, the twisted tape is embedded at the entrance of the tube with different values for λ. It is visible that as the flow enters the twisted tape at the tube inlet, the thermal boundary layer is disturbed and, due to the creation of secondary flow, the local increases; however, as the flow passes the twisted tape, the thermal boundary layer starts to develop normally, and as a result, it tends to develop towards the local So far, it has been shown that the twisted tape with a pitch of L/4 results in the best thermal performance of the system. Therefore, the following simulations are performed for this pitch value. To show the effect of twisted tape truncation percentage on local throughout the tube length for different twisted tape positions, Figure 12 is provided at = 250. In Figure 12a, the twisted tape is embedded at the entrance of the tube with different values for λ. It is visible that as the flow enters the twisted tape at the tube inlet, the thermal boundary layer is disturbed and, due to the creation of secondary flow, the local increases; however, as the flow passes the twisted tape, the thermal boundary layer starts to develop normally, and as a result, it tends to develop towards the local So far, it has been shown that the twisted tape with a pitch of L/4 results in the best thermal performance of the system. Therefore, the following simulations are performed for this pitch value. To show the effect of twisted tape truncation percentage on local *Nu* throughout the tube length for different twisted tape positions, Figure 12 is provided at *Re* = 250. In Figure 12a, the twisted tape is embedded at the entrance of the tube with different values for λ. It is visible that as the flow enters the twisted tape at the tube inlet, the thermal boundary layer is disturbed and, due to the creation of secondary flow, the local *Nu* increases; however, as the flow passes the twisted tape, the thermal boundary layer starts to develop normally, and as a result, it tends to develop towards the local *Nu*

tape is inserted at the tube entrance.

curve of PT. Consequently, lower values of λ cause a higher heat transfer rate when the twisted tape is inserted at the tube entrance. in at the twisted tape entrance, the local decreases, and its curve tends to reach the value of the fully fitted twisted tape.

thermal boundary layer disturbance and stronger mixing flow. In this case, after a sudden increase

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curve of PT. Consequently, lower values of λ cause a higher heat transfer rate when the twisted

In Figure 12b, the twisted tape is embedded at the center of the tube with different values for λ. In this figure, the local curve starts to grow as the fluid reaches the twisted tape due to the thermal boundary layer disturbance. It is visible that the local at the truncated cases surpasses the fully fitted twisted tape, and a higher maximum value for is visible for higher values of λ. As the flow passes the twisted tape, the thermal boundary layer starts to grow and becomes fully developed.

**Figure 12.** Local along the tube length for different λ values at three twisted tape positions for (**a**) entrance, (**b**) center, (**c**) exit and P = L/4. **Figure 12.** Local *Nu* along the tube length for different λ values at three twisted tape positions for (**a**) entrance, (**b**) center, (**c**) exit and P = L/4.

Figure 13 displays the streamlines colored by velocity magnitude in different twisted tape truncation percentage values inserted at the tube entrance for = 250. In the truncated cases, the flow path swirls to the end of tube length, but the secondary flow intensity is reduced as the fluid In Figure 12b, the twisted tape is embedded at the center of the tube with different values for λ. In this figure, the local *Nu* curve starts to grow as the fluid reaches the twisted tape due to the thermal boundary layer disturbance. It is visible that the local *Nu* at the truncated cases surpasses the fully fitted twisted tape, and a higher maximum value for *Nu* is visible for higher values of λ. As the flow passes the twisted tape, the thermal boundary layer starts to grow and becomes fully developed.

In Figure 12c, the twisted tape is embedded at the exit of the tube with different values for λ. This figure also shows the local *Nu* enhancement as the fluid enters the twisted tape as a result of thermal boundary layer disturbance and stronger mixing flow. In this case, after a sudden increase in *Nu* at the twisted tape entrance, the local *Nu* decreases, and its curve tends to reach the value of the fully fitted twisted tape.

Figure 13 displays the streamlines colored by velocity magnitude in different twisted tape truncation percentage values inserted at the tube entrance for *Re* = 250. In the truncated cases, the flow path swirls to the end of tube length, but the secondary flow intensity is reduced as the fluid passes the twisted tape. As a result, truncating the twisted tape results in a reduction in heat transfer rate but less material is used, causing fewer production expenses, and also the pressure loss penalty reduces due to less flow lockage and contact area between the fluid and solid. *Symmetry* **2020**, *12*, x FOR PEER REVIEW 13 of 23 passes the twisted tape. As a result, truncating the twisted tape results in a reduction in heat transfer rate but less material is used, causing fewer production expenses, and also the pressure loss penalty reduces due to less flow lockage and contact area between the fluid and solid. *Symmetry* **2020**, *12*, x FOR PEER REVIEW 13 of 23 passes the twisted tape. As a result, truncating the twisted tape results in a reduction in heat transfer rate but less material is used, causing fewer production expenses, and also the pressure loss penalty

**Figure 13.** Streamline colored by velocity magnitude for different λ values at Re = 250 and P = L/4. **Figure 13.** Streamline colored by velocity magnitude for different λ values at Re = 250 and P = L/4. **Figure 13.** Streamline colored by velocity magnitude for different λ values at Re = 250 and P = L/4.

To show the effect of truncated twisted tape inserts and changes in the flow patch shown in Figure 13 on the temperature distribution of the heated wall, Figure 14 illustrates the temperature contours at the wall. It is visible that although there are some hotspots after the truncated twisted tape, the number of hotspots is still lower than for the PT, showing the effective cooling process even after the fluid passes the twisted tape. This figure proves the presence of secondary flow (not as intense as it is in the fully twisted tape insert) after the fluid passes the truncated twisted tape. To show the effect of truncated twisted tape inserts and changes in the flow patch shown in Figure 13 on the temperature distribution of the heated wall, Figure 14 illustrates the temperature contours at the wall. It is visible that although there are some hotspots after the truncated twisted tape, the number of hotspots is still lower than for the PT, showing the effective cooling process even after the fluid passes the twisted tape. This figure proves the presence of secondary flow (not as intense as it is in the fully twisted tape insert) after the fluid passes the truncated twisted tape. To show the effect of truncated twisted tape inserts and changes in the flow patch shown in Figure 13 on the temperature distribution of the heated wall, Figure 14 illustrates the temperature contours at the wall. It is visible that although there are some hotspots after the truncated twisted tape, the number of hotspots is still lower than for the PT, showing the effective cooling process even after the fluid passes the twisted tape. This figure proves the presence of secondary flow (not as intense as it is in the fully twisted tape insert) after the fluid passes the truncated twisted tape.

**Figure 14.** Temperature contours on the heated wall for different λ values at Re = 250 and P = L/4. **Figure 14.** Temperature contours on the heated wall for different λ values at Re = 250 and P = L/4. **Figure 14.** Temperature contours on the heated wall for different λ values at Re = 250 and P = L/4.

To clearly show the effect of twisted tape position on the local throughout the tube length at different twisted tape truncation percentage values, Figure 15 illustrates the local for three different twisted tape truncation percentage values of 25, 50 and 75% at = 250 for different positions, as shown in Figure 1b. This figure shows that when the twisted tape is at the tube entrance, no sudden increase in is visible; in contrast, there is a maximum value for along the tube length when the twisted tape is inserted at the center and exit of the tube. For all values of λ, the highest maximum throughout the tube length corresponds to the layouts where twisted tape is inserted at the tube exit, which is higher as λ changes. To clearly show the effect of twisted tape position on the local throughout the tube length at different twisted tape truncation percentage values, Figure 15 illustrates the local for three different twisted tape truncation percentage values of 25, 50 and 75% at = 250 for different positions, as shown in Figure 1b. This figure shows that when the twisted tape is at the tube entrance, no sudden increase in is visible; in contrast, there is a maximum value for along the tube length when the twisted tape is inserted at the center and exit of the tube. For all values of λ, the highest maximum throughout the tube length corresponds to the layouts where twisted tape is inserted at the tube exit, which is higher as λ changes. To clearly show the effect of twisted tape position on the local *Nu* throughout the tube length at different twisted tape truncation percentage values, Figure 15 illustrates the local *Nu* for three different twisted tape truncation percentage values of 25, 50 and 75% at *Re* = 250 for different positions, as shown in Figure 1b. This figure shows that when the twisted tape is at the tube entrance, no sudden increase in *Nu* is visible; in contrast, there is a maximum value for *Nu* along the tube length when the twisted tape is inserted at the center and exit of the tube. For all values of λ, the highest maximum *Nu* throughout the tube length corresponds to the layouts where twisted tape is inserted at the tube exit, which is higher as λ changes.

*Symmetry* **2020**, *12*, x FOR PEER REVIEW 14 of 23

**Figure 15.** Local along the tube length for different twisted tape positions at three λ values of (**a**) 25%, (**b**) 50%, (**c**) 75% and P = L/4. **Figure 15.** Local *Nu* along the tube length for different twisted tape positions at three λ values of (**a**) 25%, (**b**) 50%, (**c**) 75% and P = L/4.

Figure 16 illustrates the streamlines colored by velocity magnitude at different positions of the twisted tape in the tube at a twisted tape truncation percentage of 75% and = 250. This figure shows that when the twisted tape is embedded at the tube entrance, the whole flow patch is affected. When the twisted tape is at the tube center, half of the tube length is affected, and for the case in which twisted tape is at the tube exit, the flow path before reaching the twisted tape is similar to that Figure 16 illustrates the streamlines colored by velocity magnitude at different positions of the twisted tape in the tube at a twisted tape truncation percentage of 75% and *Re* = 250. This figure shows that when the twisted tape is embedded at the tube entrance, the whole flow patch is affected. When the twisted tape is at the tube center, half of the tube length is affected, and for the case in which twisted tape is at the tube exit, the flow path before reaching the twisted tape is similar to that of PT.

of PT. The effect of the position of the truncated twisted tape inserts on the temperature distribution of the heated wall is depicted in Figure 17. The same temperature distribution before the flow reaches the twisted tape as that of PT for the heated wall temperature distribution is also visible in this figure. Moreover, flow mixing as the fluid passes the twisted tape and the strong secondary flow in the twisted tape regions are observable in this figure.

*Symmetry* **2020**, *12*, x FOR PEER REVIEW 15 of 23

*Symmetry* **2020**, *12*, x FOR PEER REVIEW 15 of 23

**Figure 16.** Streamline colored by velocity magnitude for different twisted tape positions at Re = 250, P = L/4 and λ = 0.75. **Figure 16.** Streamline colored by velocity magnitude for different twisted tape positions at Re = 250, P = L/4 and λ = 0.75. the twisted tape as that of PT for the heated wall temperature distribution is also visible in this figure. Moreover, flow mixing as the fluid passes the twisted tape and the strong secondary flow in the

twisted tape regions are observable in this figure.

**Figure 17.** Temperature contours on the heated wall for different twisted tape positions at Re = 250, P = L/4 and λ = 0.75. **Figure 17.** Temperature contours on the heated wall for different twisted tape positions at Re = 250, P = L/4 and λ = 0.75.

**Figure 17.** Temperature contours on the heated wall for different twisted tape positions at Re = 250, P = L/4 and λ = 0.75. To quantify the heat transfer rate for different truncation values and positions of the twisted tape, Figure 18 is provided to show the variations of the ratio compared with PT at various λ and Re values, different twisted tape positions and a pitch of L/4. For all twisted tape positions and , higher values of λ result in lower and heat transfer due to the fact that the secondary and mixing To quantify the heat transfer rate for different truncation values and positions of the twisted tape, Figure 18 is provided to show the variations of the ratio compared with PT at various λ and Re values, different twisted tape positions and a pitch of L/4. For all twisted tape positions and , higher values of λ result in lower and heat transfer due to the fact that the secondary and mixing flows in a tube fully fitted with twisted tape are much stronger than for a tube equipped with a truncated twisted tape, as shown in Figure 14. As seen in this figure, using twisted tape with a pitch value of L/4 at the entrance of the tube and λ values of 0, 25, 50 and 75% increase the average by about 71.26, 68.50, 57.59 and 37.34% at = 250 and 151.42, 133.99, 109.52 and 71.43% at = 1000 in comparison with the PT, respectively. To quantify the heat transfer rate for different truncation values and positions of the twisted tape, Figure 18 is provided to show the variations of the *Nu* ratio compared with PT at various λ and Re values, different twisted tape positions and a pitch of L/4. For all twisted tape positions and *Re*, higher values of λ result in lower *Nu* and heat transfer due to the fact that the secondary and mixing flows in a tube fully fitted with twisted tape are much stronger than for a tube equipped with a truncated twisted tape, as shown in Figure 14. As seen in this figure, using twisted tape with a pitch value of L/4 at the entrance of the tube and λ values of 0, 25, 50 and 75% increase the average *Nu* by about 71.26, 68.50, 57.59 and 37.34% at *Re* = 250 and 151.42, 133.99, 109.52 and 71.43% at *Re* = 1000 in comparison with the PT, respectively.

flows in a tube fully fitted with twisted tape are much stronger than for a tube equipped with a truncated twisted tape, as shown in Figure 14. As seen in this figure, using twisted tape with a pitch value of L/4 at the entrance of the tube and λ values of 0, 25, 50 and 75% increase the average by about 71.26, 68.50, 57.59 and 37.34% at = 250 and 151.42, 133.99, 109.52 and 71.43% at = 1000 in comparison with the PT, respectively. Considering the position of the twisted tape, for all values of λ at Re of 1000, when the truncated twisted tape is placed at the tube entrance, a higher *Nu* is obtained, followed by the cases with twisted tape at the tube center, and the lowest *Nu* correspond to the cases with twisted tape at the tube exit. This is due to the fact that as the fluid passes the twisted tape, the flow is still affected by the twisted tape and swirl flow is visible to the end of the tube, as shown in Figure 16. Thus, for the case of twisted tape inserts in the tube entrance, more of the tube length experiences swirl flow, and as a result, the cooling process improves.

**Figure 18.** Variations of average ratio with respect to PT at different twisted tape positions and λ values for Re = 1000. **Figure 18.** Variations of average *Nu* ratio with respect to PT at different twisted tape positions and λ values for Re = 1000.

Considering the position of the twisted tape, for all values of λ at Re of 1000, when the truncated twisted tape is placed at the tube entrance, a higher is obtained, followed by the cases with twisted tape at the tube center, and the lowest correspond to the cases with twisted tape at the tube exit. This is due to the fact that as the fluid passes the twisted tape, the flow is still affected by the twisted tape and swirl flow is visible to the end of the tube, as shown in Figure 16. Thus, for the case of twisted tape inserts in the tube entrance, more of the tube length experiences swirl flow, and as a result, the cooling process improves. Figure 19 represents the variations of friction coefficient ratio with respect to PT at various λ for *Re* of 1000, different twisted tape positions and a pitch of L/4. It is visible that for all twisted tape positions, as λ increases, the friction coefficient ratio is reduced due to the lower level of solid material Figure 19 represents the variations of friction coefficient ratio with respect to PT at various λ for *Re* of 1000, different twisted tape positions and a pitch of L/4. It is visible that for all twisted tape positions, as λ increases, the friction coefficient ratio is reduced due to the lower level of solid material and, as a result, less flow blockage and smaller contact area between the fluid flow and solid. Furthermore, for all values of λ, there are almost similar values for the friction coefficient ratio for the cases where twisted tape is placed at the tube entrance and center; however, lower values for the friction coefficient ratio are observable when the twisted tape is embedded at the tube exit. The cause of this scenario can be attributed to the fact that when the twisted tape is at the tube exit, no swirl flow is generated before the fluid reaches the twisted tape. *Symmetry* **2020**, *12*, x FOR PEER REVIEW 17 of 23

**Figure 19.** Variations of friction coefficient ratio with respect to PT at different twisted tape positions and λ values for Re = 1000. **Figure 19.** Variations of friction coefficient ratio with respect to PT at different twisted tape positions and λ values for Re = 1000.

As discussed above, the application of twisted tape inserts in the captured cases increases both the heat transfer rate as a desirable outcome and the friction coefficient as an undesirable result. Therefore, there is an interplay between the benefits of using twisted tape in heat transfer enhancement and their side effects in forcing more pumping power into the system. To analyze this issue, the dimensionless PEC number introduced in Equation (9) is discussed here. In fact, this

increases, the sensitivity of PEC to the λ value is reduced.

λ (%) 0 25 50 75

Entrance Center Exit

PEC 0.94 0.96 0.98 1.00 1.02 1.04 1.06 1.08 1.10 1.12

viewpoint of energy-saving potential. Generally, higher values of PEC imply superior energy saving. Figure 20 illustrates the PEC parameter for all cases investigated in this study for Re of 1000. It is visible that decreasing the twisted tape pitch value from Figure 20a–d enhances the PEC number, showing the fact that applying twisted tape with a lower pitch value is efficient from the viewpoint of both heat transfer enhancement and energy saving. It is also visible that as the twisted tape pitch

(**a**) (**b**)

PEC 0.90 0.95 1.00 1.05 1.10 1.15 1.20 1.25 1.30

> λ (%) 0 25 50 75

Entrance Center Exit

**6. Conclusions** 

reduction.

Canada.

and λ values for Re = 1000.

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

3.0

f/f0

As discussed above, the application of twisted tape inserts in the captured cases increases both the heat transfer rate as a desirable outcome and the friction coefficient as an undesirable result. Therefore, there is an interplay between the benefits of using twisted tape in heat transfer enhancement and their side effects in forcing more pumping power into the system. To analyze this issue, the dimensionless PEC number introduced in Equation (9) is discussed here. In fact, this number is used to evaluate the practical use of any modified heat transfer technique from the viewpoint of energy-saving potential. Generally, higher values of PEC imply superior energy saving. Figure 20 illustrates the PEC parameter for all cases investigated in this study for Re of 1000. It is visible that decreasing the twisted tape pitch value from Figure 20a–d enhances the PEC number, showing the fact that applying twisted tape with a lower pitch value is efficient from the viewpoint of both heat transfer enhancement and energy saving. It is also visible that as the twisted tape pitch increases, the sensitivity of PEC to the λ value is reduced. the heat transfer rate as a desirable outcome and the friction coefficient as an undesirable result. Therefore, there is an interplay between the benefits of using twisted tape in heat transfer enhancement and their side effects in forcing more pumping power into the system. To analyze this issue, the dimensionless PEC number introduced in Equation (9) is discussed here. In fact, this number is used to evaluate the practical use of any modified heat transfer technique from the viewpoint of energy-saving potential. Generally, higher values of PEC imply superior energy saving. Figure 20 illustrates the PEC parameter for all cases investigated in this study for Re of 1000. It is visible that decreasing the twisted tape pitch value from Figure 20a–d enhances the PEC number, showing the fact that applying twisted tape with a lower pitch value is efficient from the viewpoint of both heat transfer enhancement and energy saving. It is also visible that as the twisted tape pitch increases, the sensitivity of PEC to the λ value is reduced.

**Figure 19.** Variations of friction coefficient ratio with respect to PT at different twisted tape positions

λ (%) 0 25 50 75

Entrance Center Exit

As discussed above, the application of twisted tape inserts in the captured cases increases both

*Symmetry* **2020**, *12*, x FOR PEER REVIEW 17 of 23

**Figure 20.** Variations of PEC number at different twisted tape positions, λ and P values. (**a**) P = L, (**b**) P = L/2, (**c**) P = L/3 and (**d**) P = L/4 for Re = 1000. **Figure 20.** Variations of PEC number at different twisted tape positions, λ and P values. (**a**) P = L, (**b**) P = L/2, (**c**) P = L/3 and (**d**) P = L/4 for Re = 1000.

Furthermore, for all λ values, placing the twisted tape at the tube entrance leads to higher PEC magnitudes. Therefore, for P = L, L/2, L/3 and L/4, the optimum cases from the viewpoint of energy saving are twisted tapes with λ = 75, 50, 50 and 0%, for which the related PEC numbers at = 1000 are almost equal to 1.08, 1.24, 1.4 and 1.76, respectively. In addition, PEC numbers in all cases are tabulated in Appendix A of this paper. Furthermore, for all λ values, placing the twisted tape at the tube entrance leads to higher PEC magnitudes. Therefore, for P = L, L/2, L/3 and L/4, the optimum cases from the viewpoint of energy saving are twisted tapes with λ = 75, 50, 50 and 0%, for which the related PEC numbers at *Re* = 1000 are almost equal to 1.08, 1.24, 1.4 and 1.76, respectively. In addition, PEC numbers in all cases are tabulated in Appendix A of this paper.

pressure drop penalty, the PEC number was calculated. The obtained results indicated that using the twisted tape and reducing its pitch value increases the Nusselt number, friction coefficient and PEC number due to the generation of efficient secondary and mixing flow. The average Nusselt number increased by about 151.42 for a Reynolds number of 1000 in the case of fully fitted twisted tape at a pitch value of L/4. It was also found that increasing the twisted tape truncation percentage reduced both heat transfer and pressure drop in comparison with the fully fitted twisted tape case. Moreover, the best position for the truncated twisted tapes was at the tube entrance to reach the highest thermal performance. Ultimately, it was concluded that for P = L, L/2, L/3 and L/4, the optimum cases from the viewpoint of energy saving are twisted tapes with λ = 75, 50, 50 and 0%, for which the related PEC numbers at a Reynolds number of 1000 are almost equal to 1.08, 1.24, 1.4 and 1.76, respectively. The finding of this research provides a framework for researchers working in this area toward higher performance based on energy saving according to both heat transfer enhancement and pressure drop

**Author Contributions:** R.M. and P.T. performed and designed the numerical simulations; M.G., H.A., R.M., H.M.A. and W.Y. analyzed the data and performed the discussion; M.G., H.A., R.M., P.T., H.M.A. and W.Y.

**Funding:** The APC was funded by Dr Wahiba Yaïci from CanmetENERGY Research Centre, Natural Resources

wrote the paper. All authors have read and agreed to the published version of the manuscript.

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

This paper investigated the laminar convection heat transfer in a tube equipped with twisted tape inserts. A parametric study was conducted to evaluate the impact of key design variables, including the twisted tape truncation percentage, pitch value, position in the tube and Reynolds
