*4.1. Geometric Configuration and Boundary Conditions*

The dimensions of the numerical domain were chosen to approach and focus as much as possible on the physics of the problem of flows and heat transfer of the NFs in the PHE. Therefore, only the loop that contains the NFs' flow was considered, while heat conditions of the cold loop (where the water is flowing at a constant flow rate during all the measurements) was obtained experimentally and applied in the numerical modelling into the boundary conditions of the NFs flows. The latter helps to avoid the possible numerical errors caused by the complex design of the PHE and reduces the time costs of the simulations by using a smaller number of nodes in the mesh of the numerical domain. Moreover, only one channel was chosen from the 5 hot channels in the hot loop for the simulations to increase the accuracy of the numerical investigations, mainly for NFs based on low particles concentrations. Therefore, the channel was designed to have similar dimensions to the channel of the PHE used in the investigational setup and the numerical domain was established as a two-dimensional (2D) corrugated channel with 0.278 m length (*Lch*) and 2.4 mm distance between channel's plates (b) as it is presented in Figure 4. Moreover, the heat flux boundary conditions on the wall were found based on the measured data obtained for the cold loop, where the absorbed heat by the cold loop (*Qc*) was found by Equation (4). Thus, the heat absorbance value for one channel (*Qc*1) can be calculated (*Qc*<sup>1</sup> = *Qc*/5), considering the assumption of the current study of having equal flows and heat rates in the five channels of the cold loop, then the heat flux per unit of area on the wall of the channel can be found in Equation (11).

$$q = Q\_{\mathcal{L}} / A^{\prime} \tag{11}$$

where *A* 0 is the convection heat transfer area for one plate channel.

**Figure 4.** Schematic diagram of the numerical domain. **Figure 4.** Schematic diagram of the numerical domain.

However, the numerical investigation methodology was established considering incompressible turbulent fluid flow. The thermophysical properties of the NFs and the BF were defined and used in the model. The inlet temperature (*Tin*) was considered at 40 °C similar to the experimental measurement conditions for the hot loop. Moreover, velocity (*u*) at the channel inlet was determined based on the measurement procedures for each flow rate (0.03–0.93 L/s). However, the numerical investigation methodology was established considering incompressible turbulent fluid flow. The thermophysical properties of the NFs and the BF were defined and used in the model. The inlet temperature (*Tin*) was considered at 40 ◦C similar to the experimental measurement conditions for the hot loop. Moreover, velocity (*u*) at the channel inlet was determined based on the measurement procedures for each flow rate (0.03–0.93 L/s).

### *4.2. Governing Equations and Calculation 4.2. Governing Equations and Calculation*

In this study, the equations of Navier–Stokes and energy are used in the numerical investigation methodology as governing equations for running the simulations for the NFs flowing through the corrugated channel of PHE. The turbulent model called as Realizable K-ε found in the FLUENT-ANSYS package is adapted, where it is considered a developed form of K-ε turbulence model proposed by Shih et at. [46] applying a recent for-In this study, the equations of Navier–Stokes and energy are used in the numerical investigation methodology as governing equations for running the simulations for the NFs flowing through the corrugated channel of PHE. The turbulent model called as Realizable K-ε found in the FLUENT-ANSYS package is adapted, where it is considered a developed form of K-ε turbulence model proposed by Shih et at. [46] applying a recent formulation for eddy-viscosity.

mulation for eddy-viscosity. The dimensional governing equations for the current study conditions are as follows The dimensional governing equations for the current study conditions are as follows in Equations (12)–(14),

in Equations (12)–(14), Continuity equation: Continuity equation:

$$\nabla \cdot (\rho \mathbf{V}) = 0 \tag{12}$$

() = 0 (112

)

Momentum equation:

Momentum equation:

$$\nabla \cdot (\rho \mathbf{V} \mathbf{V}) = -\nabla P + \nabla \cdot (\mu \nabla \mathbf{V}) \tag{13}$$

Energy equation:

(16)):

$$\nabla \cdot \left( \rho \mathbb{C}\_p \mathbf{V} T \right) = \nabla \cdot \left( k \nabla T \right) \tag{14}$$

Energy equation: ൫ ൯ = () (14) where represents the vector of velocity, is the pressure, is the density of the fluid where *V* represents the vector of velocity, *P* is the pressure, *ρ* is the density of the fluid and *µ* is the viscosity of the fluid. Moreover, the kinetic energy of turbulence (*K*) and the dissipation rate (*ε*) are attained by the following transport equations (Equations (15) and (16)):

$$\nabla \cdot (\rho \mathbf{V} \mathbf{K}) = \nabla \cdot \left[ \left( \mu + \frac{\mu\_l}{\sigma\_\mathbf{K}} \right) \nabla \mathbf{K} \right] + \mathbf{G}\_\mathbf{K} - \rho \varepsilon \tag{15}$$

$$\nabla \cdot (\rho \mathbf{V} \varepsilon) = \nabla \cdot \left[ \left( \mu + \frac{\mu\_t}{\sigma\_\varepsilon} \right) \nabla \varepsilon \right] + \rho \mathbf{C}\_{1\varepsilon} \mathbf{S} \varepsilon - \rho \mathbf{C}\_{2\varepsilon} \frac{\varepsilon^2}{\mathbf{K} + \sqrt{v\varepsilon}} \tag{16}$$

 () =൬ + ௧ ఌ ൰ ൨ + ଵఌ − ଶఌ ଶ + √ (16) In these equations, symbolizes the generation of turbulence kinetic energy term for the gradients in velocities. ଶఌ and ଵఌ are equation' constants. and ఌ represent In these equations, *G<sup>k</sup>* symbolizes the generation of turbulence kinetic energy term for the gradients in velocities. *C*2*<sup>ε</sup>* and *C*1*<sup>ε</sup>* are equation' constants. *σ<sup>K</sup>* and *σ<sup>ε</sup>* represent the turbulent Prandtl numbers for *K* and *ε*, respectively. *S* is the average strain rate [44]. Moreover, the value of the eddy viscosity of the used Realizable model is not constant, and it is calculated from the following Equation (17):

$$
\mu\_l = \rho \cdot \mathbb{C}\_{\mu} \cdot \frac{K^2}{\varepsilon} \tag{17}
$$

and it is calculated from the following Equation (17): where *C*<sup>µ</sup> is a factor associated with the eddy viscosity. However, the model constants (C2*<sup>ε</sup>* , *σK*, *σ<sup>ε</sup>* , *A*<sup>0</sup> and *As*) have been formed to ensure the good performance of the numerical

for the turbulent flow conditions. The mentioned constants of the numerical model are provided as: particle concentrations (0.01, 0.05, 0.1 and 0.15 and 0.2 vol.%) flowing through the corrugated channel of the hot loop in the PHE in the similar working conditions (flow rates,

௧ = ∙ µ ∙

where µ is a factor associated with the eddy viscosity. However, the model constants (C2*ε*, , σε, and ௦) have been formed to ensure the good performance of the numerical for the turbulent flow conditions. The mentioned constants of the numerical model

However, the simulations were carried out for the BFs and the NFs for the various

<sup>ଶ</sup>

(17)

*Nanomaterials* **2022**, *12*, x FOR PEER REVIEW 8 of 15

$$C\_{2\varepsilon} = 1.90, \ A\_0 = 4.040, \ \sigma\_\mathcal{K} = 1.00, \ \sigma\_\varepsilon = 1.20 \text{ and } A\_\mathcal{s} = \sqrt{6} \cos \varphi.$$

However, the simulations were carried out for the BFs and the NFs for the various particle concentrations (0.01, 0.05, 0.1 and 0.15 and 0.2 vol.%) flowing through the corrugated channel of the hot loop in the PHE in the similar working conditions (flow rates, temperature value, and thermophysical properties) of the experimental measurements. The temperature outcomes of the different samples at the outlet of the corrugated channel (*Tho*) were collected and used to calculate the corresponding CHTC at each flow. The heat removed from the hot loop (*Q<sup>h</sup>* ) is found by Equation (3). Meanwhile, the values of the temperatures of the channel in a cold loop and the heat transferred (*Tci*, *Tco*, and *Qc*) were already known from the experimental measurements. Then the mean heat transfer (*Q*) was calculated as in Equation (5). Moreover, the overall heat transfer coefficient (*U*) was found by Equations (6) and (7). Then, the CHTC for the BFs and NFs in the hot loop (*h<sup>h</sup>* ) can be determined by Equation (8). Where the *h<sup>c</sup>* is the CHTC for the water in the cold loop that was defined in the experimental investigation. () were collected and used to calculate the corresponding CHTC at each flow. The heat removed from the hot loop () is found by Equation (3). Meanwhile, the values of the temperatures of the channel in a cold loop and the heat transferred (, , and Qc) were already known from the experimental measurements. Then the mean heat transfer (Q) was calculated as in Equation (5). Moreover, the overall heat transfer coefficient (*U*) was found by Equations (6) and (7). Then, the CHTC for the BFs and NFs in the hot loop (ℎ) can be determined by Equation (8). Where the ℎ is the CHTC for the water in the cold loop that was defined in the experimental investigation. *4.3. Mesh Optimisation and Validation* The independency of the mesh was ensured by testing several meshes by increasing

### *4.3. Mesh Optimisation and Validation* the number of the nodes and comparing it with experimental measurements for water as

are provided as:

The independency of the mesh was ensured by testing several meshes by increasing the number of the nodes and comparing it with experimental measurements for water as it has a well-known hydraulic-thermal behavior. The tested meshes were 112621 (Mesh 1), 168441 (Mesh 2), 219282 (Mesh 3), and 303101 (Mesh 4): nodes were tested for several flow rates at the equivalent working conditions of the experimental investigation. The obtained numerical predictions are presented in Figure 5 for the CHTC and they show good accuracy for the mesh of 219282 nodes in comparison with the experimental measurements (Experimental data) with a deviation of around 6%. Moreover, it was found that the numerical predictions of CHTC were not changing for node numbers higher than 219282 nodes. Therefore, the mesh of 219282 nodes was selected for the intended simulations of this study to ensure the independency of the results from the mesh. A part of the mesh is presented in Figure 6, which shows the uniform and smooth structure of the mesh for accurate simulations of the turbulent flows. it has a well-known hydraulic-thermal behavior. The tested meshes were 112621 (Mesh 1), 168441 (Mesh 2), 219282 (Mesh 3), and 303101 (Mesh 4): nodes were tested for several flow rates at the equivalent working conditions of the experimental investigation. The obtained numerical predictions are presented in Figure 5 for the CHTC and they show good accuracy for the mesh of 219282 nodes in comparison with the experimental measurements (Experimental data) with a deviation of around 6%. Moreover, it was found that the numerical predictions of CHTC were not changing for node numbers higher than 219282 nodes. Therefore, the mesh of 219282 nodes was selected for the intended simulations of this study to ensure the independency of the results from the mesh. A part of the mesh is presented in Figure 6, which shows the uniform and smooth structure of the mesh for accurate simulations of the turbulent flows.

**Figure 5.** Mesh independency and validation for CHTC (*h*) of BF as a function of flow rate. **Figure 5.** Mesh independency and validation for CHTC (*h*) of BF as a function of flow rate.

**Figure 6.** A view of a section of the mesh of the numerical domain. **Figure 6.** A view of a section of the mesh of the numerical domain. **Figure 6.** A view of a section of the mesh of the numerical domain.

Furthermore, the resulted profiles of the velocity and temperature fields, at the entrance and end of the channel of the PHE, are presented in Figure 7 which shows good consistency with the fluid flow characteristics under the PHE problem conditions. It can be noticed from Figure 7 that the values of the temperature at the entrance of the channel starts high and it becomes lower in the outlet area of the channel with lower values of temperature near the upper wall. Moreover, the velocity profile shows the boundary layer of the flow where there are lower values of velocity near the walls which make closely a parabolic profile of the velocity in the channel. Furthermore, the resulted profiles of the velocity and temperature fields, at the entrance and end of the channel of the PHE, are presented in Figure 7 which shows good consistency with the fluid flow characteristics under the PHE problem conditions. It can be noticed from Figure 7 that the values of the temperature at the entrance of the channel starts high and it becomes lower in the outlet area of the channel with lower values of temperature near the upper wall. Moreover, the velocity profile shows the boundary layer of the flow where there are lower values of velocity near the walls which make closely a parabolic profile of the velocity in the channel. Furthermore, the resulted profiles of the velocity and temperature fields, at the entrance and end of the channel of the PHE, are presented in Figure 7 which shows good consistency with the fluid flow characteristics under the PHE problem conditions. It can be noticed from Figure 7 that the values of the temperature at the entrance of the channel starts high and it becomes lower in the outlet area of the channel with lower values of temperature near the upper wall. Moreover, the velocity profile shows the boundary layer of the flow where there are lower values of velocity near the walls which make closely a parabolic profile of the velocity in the channel.

**Figure 7.** The numerical section of study (test) with the temperature and velocity profiles at the entrance and outlet of the channel. **Figure 7.** The numerical section of study (test) with the temperature and velocity profiles at the entrance and outlet of the channel. **Figure 7.** The numerical section of study (test) with the temperature and velocity profiles at the entrance and outlet of the channel.

#### **5. Heat Transfer Results and Discussion 5. Heat Transfer Results and Discussion 5. Heat Transfer Results and Discussion**

The experimental and numerical results of the heat transfer performance of two types of NFs (Al2O3 and TiO2 NFs) flowing through the hot loop of the CPHE are presented in Figure 8. The heat transfer enhancements of the NFs are determined in comparison with the corresponding BFs. The outcomes show good improvement with the increase in loading of nanoparticles for all flow rates (Figure 8), and better thermal performance for Al2O3 compared to TiO2 NFs for a deviation of 9% at 2.0 vol.% particles that decreases with reducing the concentration of the particles. The maximum average enhancement of about 24.6% is observed for the highest particle concentration (0.2 vol.%) through the experimental measurements (Figure 8) of Al2O3 NF. It was anticipated as the enhanced thermal conductivity of nanofluids was found to increase with increasing the amount of nanoparticles into the BF. Moreover, it is due to the fact that Al2O3 NFs exhibit higher thermal conductivity compared to TiO2 NFs (as presented in Figure 1). The experimental and numerical results of the heat transfer performance of two types of NFs (Al2O3 and TiO2 NFs) flowing through the hot loop of the CPHE are presented in Figure 8. The heat transfer enhancements of the NFs are determined in comparison with the corresponding BFs. The outcomes show good improvement with the increase in loading of nanoparticles for all flow rates (Figure 8), and better thermal performance for Al2O3 compared to TiO2 NFs for a deviation of 9% at 2.0 vol.% particles that decreases with reducing the concentration of the particles. The maximum average enhancement of about 24.6% is observed for the highest particle concentration (0.2 vol.%) through the experimental measurements (Figure 8) of Al2O3 NF. It was anticipated as the enhanced thermal conductivity of nanofluids was found to increase with increasing the amount of nanoparticles into the BF. Moreover, it is due to the fact that Al2O3 NFs exhibit higher thermal conductivity compared to TiO2 NFs (as presented in Figure 1). The experimental and numerical results of the heat transfer performance of two types of NFs (Al2O<sup>3</sup> and TiO<sup>2</sup> NFs) flowing through the hot loop of the CPHE are presented in Figure 8. The heat transfer enhancements of the NFs are determined in comparison with the corresponding BFs. The outcomes show good improvement with the increase in loading of nanoparticles for all flow rates (Figure 8), and better thermal performance for Al2O<sup>3</sup> compared to TiO<sup>2</sup> NFs for a deviation of 9% at 2.0 vol.% particles that decreases with reducing the concentration of the particles. The maximum average enhancement of about 24.6% is observed for the highest particle concentration (0.2 vol.%) through the experimental measurements (Figure 8) of Al2O<sup>3</sup> NF. It was anticipated as the enhanced thermal conductivity of nanofluids was found to increase with increasing the amount of nanoparticles into the BF. Moreover, it is due to the fact that Al2O<sup>3</sup> NFs exhibit higher thermal conductivity compared to TiO<sup>2</sup> NFs (as presented in Figure 1).

**Figure 8.** Enhancement of average heat transfer coefficient of DW-based Al2O3 and TiO2 NFs obtained experimentally and numerically as a function of nanoparticle concentration. **Figure 8.** Enhancement of average heat transfer coefficient of DW-based Al2O<sup>3</sup> and TiO<sup>2</sup> NFs obtained experimentally and numerically as a function of nanoparticle concentration.

Therefore, the numerical findings (Num. in Figure 8) and experimental measurements (Exp. in Figure 8) confirm that the dispersion of nanoparticles characterized by superior thermal conduction such as Al2O3 and TiO2 particles with a conventional heat transfer fluid (e.g., DW in this study) enhances the heat transfer performance, due to the greater thermal conductivity of the NFs. As presented in Figure 8, an increase in nanoparticles' concentrations led to higher enhancement value in CHTC. It should be also noted that the temperature and velocity profiles and their boundary layer developments into the flow regime are influenced by the advanced properties of the NFs performing better heat transfer between the two loops of the PHE, especially in the conditions of the current study where the NFs are operating in the hot loop under high temperature that leads to having higher thermal conduction characteristic and lower viscosity values of the NFs. The latter explains the excellent heat transfer improvement that reached around 24.6% for 0.2 vol.% of Al2O3 and 15.3% for 0.2 vol.% of TiO2. On other hand, the random movement of the nanoparticle in the turbulent flow regime and the possible migration of the nanoparticles into the flow inside the PHE (as appeared in the experimental investigations) can cause further development in the heat transfer rates [47]. These mentioned factors can also be the reasons for the higher enhancements of the CHTCs for Al2O3 and TiO2 NFs than the enhancements of their thermal conductivity (e.g., Figure 1). Several relevant studies have also highlighted that the improved convection heat transfer for the laminar flow of Al2O3 NFs through a horizontal tube mainly due to the migration phenomena of nanoparticles into the flow [48]. The latter is proved by the current study when the experimental results were compared with the numerical results where the impact of the nanoparticles' movements and the migration phenomena are not considered for the numerical approach (nanofluids are simulated as single-phase fluids) but it exists in the experimental investi-Therefore, the numerical findings (Num. in Figure 8) and experimental measurements (Exp. in Figure 8) confirm that the dispersion of nanoparticles characterized by superior thermal conduction such as Al2O<sup>3</sup> and TiO<sup>2</sup> particles with a conventional heat transfer fluid (e.g., DW in this study) enhances the heat transfer performance, due to the greater thermal conductivity of the NFs. As presented in Figure 8, an increase in nanoparticles' concentrations led to higher enhancement value in CHTC. It should be also noted that the temperature and velocity profiles and their boundary layer developments into the flow regime are influenced by the advanced properties of the NFs performing better heat transfer between the two loops of the PHE, especially in the conditions of the current study where the NFs are operating in the hot loop under high temperature that leads to having higher thermal conduction characteristic and lower viscosity values of the NFs. The latter explains the excellent heat transfer improvement that reached around 24.6% for 0.2 vol.% of Al2O<sup>3</sup> and 15.3% for 0.2 vol.% of TiO2. On other hand, the random movement of the nanoparticle in the turbulent flow regime and the possible migration of the nanoparticles into the flow inside the PHE (as appeared in the experimental investigations) can cause further development in the heat transfer rates [47]. These mentioned factors can also be the reasons for the higher enhancements of the CHTCs for Al2O<sup>3</sup> and TiO<sup>2</sup> NFs than the enhancements of their thermal conductivity (e.g., Figure 1). Several relevant studies have also highlighted that the improved convection heat transfer for the laminar flow of Al2O<sup>3</sup> NFs through a horizontal tube mainly due to the migration phenomena of nanoparticles into the flow [48]. The latter is proved by the current study when the experimental results were compared with the numerical results where the impact of the nanoparticles' movements and the migration phenomena are not considered for the numerical approach (nanofluids are simulated as single-phase fluids) but it exists in the experimental investigation. However, the advanced heat transfer performance of Al2O<sup>3</sup> NFs for different types of heat exchangers was widely reported in the literature [12,49,50] as well as for TiO<sup>2</sup> NFs [51,52] in agreement with the finding of the current study. Furthermore, Tiwari et al. [42] found in their experimental investigation of CeO2, Al2O3, TiO2, and SiO<sup>2</sup> NFs for gasketed PHE that there was heat transfer boost for all the particles' types with maximum values for CeO<sup>2</sup> NFs, whereas the Al2O<sup>3</sup> NFs showed better heat transfer enhancement than TiO<sup>2</sup> NFs. Nevertheless, the findings and discussion can vary among different researchers even for the same NF type because of many parameters related to the concentration and morphology of the particles,

gation. However, the advanced heat transfer performance of Al2O3 NFs for different types of heat exchangers was widely reported in the literature [12,49,50] as well as for TiO2 NFs

found in their experimental investigation of CeO2, Al2O3, TiO2, and SiO2 NFs for gasketed PHE that there was heat transfer boost for all the particles' types with maximum values for CeO2 NFs, whereas the Al2O3 NFs showed better heat transfer enhancement than TiO2 NFs. Nevertheless, the findings and discussion can vary among different researchers even for the same NF type because of many parameters related to the concentration and morphology of the particles, BF, flow rate, the operation temperature value, and the method

On the other hand, the differences between the numerical and the experimental results

are shown in Figure 9. The numerical results show relatively lower heat transfer

of preparing the NFs and the type of heat exchanger and its dimensions.

BF, flow rate, the operation temperature value, and the method of preparing the NFs and the type of heat exchanger and its dimensions. *Nanomaterials* **2022**, *12*, x FOR PEER REVIEW 11 of 15

> On the other hand, the differences between the numerical and the experimental results are shown in Figure 9. The numerical results show relatively lower heat transfer enhancements in comparison with experimental data with a deviation between 1.0% and 3.3% for TiO<sup>2</sup> NFs and a deviation between 1.6% and 7.2% for Al2O<sup>3</sup> NFs. enhancements in comparison with experimental data with a deviation between 1.0% and 3.3% for TiO2 NFs and a deviation between 1.6% and 7.2% for Al2O3 NFs.

**Figure 9.** The deviation between experimental (Exp) and numerical (Num) findings of the enhancement of CHTC (h%) as a function of flow rate of (**a**) Al2O3 and (**b**) TiO2 NFs. **Figure 9.** The deviation between experimental (Exp) and numerical (Num) findings of the enhancement of CHTC (h%) as a function of flow rate of (**a**) Al2O<sup>3</sup> and (**b**) TiO<sup>2</sup> NFs.

Moreover, the deviation, in most cases, increases with the decrease in flow rates and the rise in particles concentrations (Figure 9). The latter refers to the existence of some factors in experimental investigations responsible for extra heat transfer enhancement than the ones in numerical investigations. Those factors are mainly related to the nanoparticle's movements in the flow which is not considered in the numerical investigation methodology. However, at higher flow rates the hydraulic impact of the flow on convection heat transfer performance becomes higher than the impact of the thermal conductivity increases in the NFs which led to a slightly lower impact of the nanoparticle's movements on heat transfer. On other hand, the increase in the deviation between the numerical and experimental data refers to the increase in the impact of nanoparticles' movements with increasing the concentration due to their influence on hydraulic and thermal boundary layer development leading to higher heat transfer levels. However, some previous studies have conducted both numerical and experimental examinations on the NFs through PHE, such as a study by Pantzali et al. [53] for CuO NFs in a miniature PHE and they mentioned better heat transfer levels at low flow rates reaching the overall heat transfer enhancement of about 29.41%. Their numerical results were in good agreement with the experimental results, demonstrating CFD as a reliable tool for investigating NFs in PHE. Moreover, Bhattad et al. [54] have numerically and empirically examined the behavior of hybrid NF (Al2O3 + MWCNT/water) in PHE and reported an increase in CHTC of 39.16%. Their numerical predictions were in good agreement with the experimental results with a smaller deviation. The latter conclusion is also agreed with the findings of a numerical study by Tiwari et al. [31] using CeO2 and Al2O3 NFs in PHE. Therefore, the numerical results of the current study showed significant advantages to define the parameters that cannot be determined when only the experimental methods are used. The parameters such as the nanoparticles movements and their impact on the fluid flow and heat transfer through the channels of PHE can't be defined by the traditional experimental in-Moreover, the deviation, in most cases, increases with the decrease in flow rates and the rise in particles concentrations (Figure 9). The latter refers to the existence of some factors in experimental investigations responsible for extra heat transfer enhancement than the ones in numerical investigations. Those factors are mainly related to the nanoparticle's movements in the flow which is not considered in the numerical investigation methodology. However, at higher flow rates the hydraulic impact of the flow on convection heat transfer performance becomes higher than the impact of the thermal conductivity increases in the NFs which led to a slightly lower impact of the nanoparticle's movements on heat transfer. On other hand, the increase in the deviation between the numerical and experimental data refers to the increase in the impact of nanoparticles' movements with increasing the concentration due to their influence on hydraulic and thermal boundary layer development leading to higher heat transfer levels. However, some previous studies have conducted both numerical and experimental examinations on the NFs through PHE, such as a study by Pantzali et al. [53] for CuO NFs in a miniature PHE and they mentioned better heat transfer levels at low flow rates reaching the overall heat transfer enhancement of about 29.41%. Their numerical results were in good agreement with the experimental results, demonstrating CFD as a reliable tool for investigating NFs in PHE. Moreover, Bhattad et al. [54] have numerically and empirically examined the behavior of hybrid NF (Al2O<sup>3</sup> + MWCNT/water) in PHE and reported an increase in CHTC of 39.16%. Their numerical predictions were in good agreement with the experimental results with a smaller deviation. The latter conclusion is also agreed with the findings of a numerical study by Tiwari et al. [31] using CeO<sup>2</sup> and Al2O<sup>3</sup> NFs in PHE. Therefore, the numerical results of the current study showed significant advantages to define the parameters that cannot be determined when only the experimental methods are used. The parameters such as the nanoparticles movements and their impact on the fluid flow and heat transfer through the channels of PHE can't be defined by the traditional experimental investigation methods. The numerical results of the current study allowed to isolate (comparing with

vestigation methods. The numerical results of the current study allowed to isolate (comparing with the experimental data of heat transfer) those important parameters that are

In this study, numerical simulations and experimental measurements on the flow and heat transfer performance of Al2O3 and TiO2 NFs in a compact PHE are carried out.

**6. Conclusions** 

the experimental data of heat transfer) those important parameters that are considered responsible for the extra enhancement of the heat transfer.
