CFD-based thermal-hydraulic analysis of double-pipe heat exchangers equipped with structured metal-foam inserts

2.1. Geometric model description

To investigate the influence of metal-foam configurations on thermal-hydraulic performance for a counter flow DPHE was conducted using a 3D numerical model. The heat exchanger consists of a copper inner tube that carries hot water and a PVC outer tube that allows cold water to flow in the opposite direction. The heat exchanger measures 1 m in total length and comprises an inner tube with an outer diameter of 25. 4 mm and an inner diameter of 21. 4 mm, encased within an outer tube with an outer diameter ranging from 76 mm to 80 mm. Hot water enters the inner copper tube at an intake temperature of 75 °C with a constant volumetric flow rate of 3 L/min, while cold water flows into the annular zone at an input temperature of 30 °C with five specific flow rates: 1, 3, 5, 7, and 9 L/min. The heat exchanger functions in a counter-flow configuration, with the outside surface of the PVC pipe considered adiabatic. The geometric configurations of the nine representatives of DPHE, both with and without metal-foam inserts, are summarized in Table 2 and depicted in Fig. 1. The first configuration pertains to the baseline double-pipe heat exchanger (DPHE) devoid of any foam insertion. The second design fills the annular space with open-cell metal foam. The third arrangement arranges a ring-shaped foam design in the direction of flow. The fourth arrangement comprises three longitudinal foam strips arranged 120° apart around the annulus. The final five variations (Cases 5-9) utilize interrupted-strip topologies with 5, 7, 9, 11, and 13 axial cuts, respectively, while preserving the same total foam volumes and masses as the ring-shaped foam arrangement. The metal foam utilized in all upgraded designs is copper-based, distinguished by a porosity ($\epsilon$) of 0. 9 and an average pore density of 40 PPI, while. The thermophysical properties of the materials used are presented in Table 3. This design provides an excellent thermal conductivity pathway and a large surface area that promotes convective heat transfer. Each foam insert was placed concentrically within the annular section of the composite tube to ensure uniform flow distribution and consistent heat transfer along the tube walls. The diverse configurations were numerically simulated to comprehensively examine the impact of foam topology and cold-side Reynolds numbers on the Nusselt number, friction factor, and overall performance evaluation factor (PEF).

Fig. 1Schematic of the nine metal-foam configurations

a) Case 1(smooth)

b) Case 2(Fully filled)

c) Case 3 (Circular-shaped baffles)

d) Case 4 (Cont.-3-1 fins)

e) Case 5 (Int.-3-5 fins)

f) Case 6 (Int.-3-7 fins)

g) Case 7 (Int.-3-9 fins)

h) Case 8 (Int.-3-11 fins)

i) Case 9 (Int.-3-13 fins)

2.2. Governing equations and modeling assumptions

The flow problem for the incompressible and steady turbulent flow for both the fluid and porous phases in the DPHE was solved numerically. The governing equations (consisting of continuity, momentum, and energy equations) were solved by discretization using the SST $k$-$\omega$ turbulence model, conjugate heat transfer analysis, and the Darcy-Forchheimer approach. The governing equation general form is illustrated below [22, 23].

Continuity equation:

Momentum Equation:

Energy equation:

where ($\overrightarrow{V}$) is the velocity vector, ($\rho$) is the fluid density, ($P$) is the pressure, ($T$) is the temperature, ($C_P)$ is the specific heat, and ($K$) is the thermal conductivity.

For the case of copper foam materials, the governing equations for fluid flow and heat transport are given by the Forchheimer-Brinkman extension of Darcy's law using LTNE assumption. Hence, the equation that represents mass balance in 3D incompressible flow is written as [27]:

The porosity, ($\epsilon$), is defined as the ratio of the connected void volume to the total volume of the porous medium:

For steady state and uniform flow density, the governing equation becomes the standard divergence-free condition for superficial velocity which is represented by ∇·$\overrightarrow{\text{V}}$ = 0.

The volume averaged momentum equation for the porous copper is obtained based on the Forchheimer-Brinkman approach, which considers both the Darcy resistance term and Forchheimer inertial term. In addition to this, buoyancy effects are considered in the equation with the help of Boussinesq’s approximation [32]:

where (𝐾) is the permeability of the porous medium and ($C_1$) is the inertial loss coefficient. For open-cell copper foams, these quantities are correlated to the pore and fiber scales as:

The thermal behavior of the saturated porous region is described using an LTNE formulation. The energy equation for the fluid phase flowing through the foam can be expressed as [32]:

where $T_f$ and $T_s$ are the fluid and solid (foam matrix) temperatures, respectively; $k_f$ is the effective thermal conductivity of the fluid, $a_{sf}$is the specific interfacial area density between solid and fluid, and $h_{sf}$ is the local interfacial heat-transfer coefficient.

The interfacial area density and local heat transfer coefficient were found by the following equations:

Dispersion thermal conductivity was found as follows:

Solid matrix:

The local convective heat transfer at the porous-fluid interface between the fluid in the fluid area and the edge of foam ligaments may be represented as follows:

where $k_s$is solid thermal conductivity.

The Reynolds number $Re$ employed in defining the flow regime is:

where $\rho$ represents the fluid density, $D_{eq}$ denotes the hydraulic diameter signifies the characteristic velocity, and $\mu$ indicates the dynamic viscosity. The hydraulic diameter is determined by:

where $A$ is the cross-sectional flow area and $p_w$ is the wetted perimeter.

The current model considers the problem of resolving wall gradients while maintaining numerical stability in the free-stream zone. This model combines the advantages of the $k$-$\omega$ turbulence model near the surface with the $k$-$\epsilon$ turbulence model far from the walls, making it suitable for modeling enhanced heat transfer internal turbulent flows. Thermo-hydraulic behavior of the designs under study was analyzed based on three nondimensional criteria: Nusselt number ($Nu$), friction factor ($f$), and performance evaluation factor (PEF). These values were determined in accordance with the formulas provided in the literature [11]:

For carrying out the numerical calculation and maintaining a reasonable level of accuracy, the following assumptions were made:

– The flow is taken as steady, 3D, turbulent, and incompressible [22].

– The thermal properties of water were considered constant during all simulations.

– Heat transfer from the system to its surroundings was not considered by making the outer surface of the channel adiabatic.

– Complete thermal interaction was considered between the surfaces of the copper foam and the tube.

– Thermal non-equilibrium between the fluid and solid phases was also considered in the porous media.

4.1. Numerical setup

The numerical simulations were carried out at atmospheric pressure. To improve numerical stability and increase convergence rate, double precision calculation and six cores of parallel computing were used. Gravity was considered in the negative $Y$-direction to avoid any buoyancy effects. Energy equations were also enabled to couple the heat and flow fields. SST $k$-$\omega$ turbulence model was chosen due to its high computational performance in dealing with near-wall flows and its ability to be used in mixed convection problems in double pipe heat exchangers. Local Thermal Non-Equilibrium (LTNE) approach was applied to the regions filled with copper foam, which allows treating the fluid and solid phases separately in order to provide a realistic description of heat transfer from the working fluid to the copper foam matrix. Porous media was modeled using additional momentum sink sources that represent viscous resistance and inertial loss of the foam, which are proportional to the reciprocal of permeability and friction factor, respectively. Additionally, the interfacial area density ($a_{sf}$) and interfacial heat transfer coefficient ($h_{sf}$) were taken into account in order to simulate the thermal interaction between solid and fluid phases. Local value of interfacial heat transfer coefficient ($h_{sf}$) was provided using User Defined Function (UDF) that allows changing it depending on local flow velocity according to the correlation proposed in ANSYS Fluent Users Guide.

4.2. Validation

To verify the accuracy and predictive performance of the current numerical model, the latter was validated against experimental data obtained from Issam et al. [21]. As mentioned above, the geometry and thermo-hydraulic features of the experimental apparatus used in the reference work resemble the ones of the present work and therefore provide a basis for comparison with the predicted results. In terms of verification, the comparison between the experimentally measured and numerically calculated mean Nusselt number ($Nu$) and friction factor ($f$) at $Re=$ 16,300 < $Re<$ 32,500 for the baseline smooth pipe geometry was performed (as shown in Figs. 4-5). Good agreement was observed between the numerical calculations and experimental data in terms of both mean $Nu$ and $f$. The difference in the values of the mean Nusselt number did not exceed ±5 %, while for the friction factor, this difference was limited to ± 7 % (respectively). Both discrepancies are acceptable for turbulent flows around wall-bounded surfaces. Other smaller discrepancies between the numerical data and experiments may occur because of the assumptions related to the turbulence-model coefficients used and the roughness of surfaces, diffusion numerical error, development of the inlet velocity profile, etc. Therefore, it is safe to conclude that the current numerical approach allows predicting adequately the flow and heat transfer properties of smooth DPHEs, which implies the use of such a numerical model for studying the inserts based on a copper foam.

Fig. 3Validation of the friction factor (f) outcome

Fig. 4Validation of the average Nusselt number (Nu) results

5.1. Influence of baffle geometry and arrangement on heat transfer

It is essential to establish the association between the volumetric flow rates and the corresponding Reynolds numbers advance of presenting the simulation results. This step enables a consistent interpretation of thermo-hydraulic response under different flow conditions. The relationship between various volumetric flow rates (1, 3, 5, 7, and 9 L/min) and a Reynolds number range of 205 to 1845 is clarified in Fig. 5. The linear correlation between flow velocity and fluid momentum, as can be seen from the numerical simulations, adequately maintained the boundary conditions. Furthermore, the thermophysical properties of water were nearly constant within the examined temperature range, as corroborated by this relationship.

The effect of the Reynolds number ($Re$) on the average Nusselt number ($Nu$) across the various configurations examined is illustrated in Fig. 6. The Nusselt number increases consistently with the Reynolds number, attributable to improvements in convective heat transfer resulting from increased flow rates and reduced boundary layer thickness. When the Reynolds numbers were 205 and 1845, the Nusselt number in the smooth DPHE rose from 12 to 38. Such a slight increase indicates the limited convection occurring without flow interference, where conductive heat transfer across the thermal boundary layer prevails. The use of metal foam significantly improved the coefficient of heat transfer within the DPHE. The fully filled foam model achieved the highest Nusselt number values, ranging from approximately 184 at low Reynolds numbers to nearly 291 at higher $Re$. This signifies approximately fifteen and eight times the enhancements, respectively, as compared to the smooth tube heat exchanger at the greatest and lowest Reynolds numbers. The observed improvement stems from the foam's substantial surface area and open pore structure, which facilitates fluid mixing and disrupts the boundary layer. Circular-shaped baffles exhibit lower Nusselt number values than the fully filled model, with values ranging from 65 to 126 at Reynolds numbers of 205 and 1845. Accordingly, the improvement of the Nusselt number or (heat transfer coefficient) was nearly five and three times greater, respectively, relative to the smooth tube. The continuous 3-1 fin and the interrupted foam 3-(5, 7, 9, 11, and 13) fins achieved the most balanced performance, with the int-3-13 fins model reaching a peak Nusselt number of approximately 158 at $Re=$ 1845. In contrast, the lowest Nusselt number of 23 was recorded in the continuous-3-1 fin at $Re=$ 205. Finally, the thermal performance of the DPHE was assessed in the subsequent sequence: filled, int-3-(13, 9, 11, 7, 5) fins, con-3-1 fin, circular baffles, and smooth tube. Among all configurations, the interrupted 3-13 fins yielded the highest Nusselt number, achieving approximately a 315 % and 275 % enhancement over the smooth tube at the high and low Reynolds numbers, respectively. The simulation data confirms that the design configuration plays a critical role in promoting local mixing and enhancing heat transfer. Furthermore, the frequency of interruptions directly influences the convective behavior. Augmenting foam sections typically improves recirculation and disrupts the boundary layer; However, severe obstruction may counteract these advantages. This study elucidates the suboptimal performance of the circular-shaped baffles, which, while generating significant disturbances, also create prolonged stagnant zones behind each baffle.

Fig. 5The relation between Reynolds number and volumetric flow rate for the cold-water

Fig. 6Variation of the average Nusselt number with Reynolds number for the investigated DPHE configurations

5.2. Influence of baffle geometry and arrangement on friction factor

Given the extensive variation in pressure drop values observed throughout the examined configurations at the exact Reynolds numbers, Fig. 7 is divided into two segments (A and B) to facilitate more precise visualization and comparison of the results. Fig. 7(a) illustrates the relationship between the Reynolds number and the friction factor. In all examined configurations, including smooth tubes and int-3-(5,7,9,11,13) fins, the friction factor continuously declines monotonically with the Reynolds number as inertial forces increasingly overshadow viscous force effects. In the baseline heat exchanger construction with a smooth tube, the friction factor exhibited the lowest values across all evaluated configurations, decreasing from approximately 0.53 at $Re=$205 to about 0.10 at $Re=$1845.The observed trend indicates a streamlined and fully developed flow regime, in which viscous effects diminish progressively as the Reynolds number increases. In the interrupted foam configurations(int-3-5,7,9,11 and 13 fins), the friction factor was consistently low, varying from approximately 0.42 in the int-3-5 fins to about 0.44 in the int-3-13 fins at $Re=$1845.This behavior is attributed to the presence of gaps or cut regions, which reduce the continuous drag within the annulus and allow the flow to regain its momentum between successive foam blocks partially .Therefore, it ultimately resulted in a reduction in the friction factor of approximately 99.7 % relative to the fully packed configuration. In other words, the DPHE exhibits excellent hydraulic performance. The design accomplishes this decrease in flow resistance while still ensuring a significant enhancement in heat transmission. The optimal compromise between hydraulic performance and thermal enhancement was achieved using the axially interrupted 3-5 fins design. The completely filled foam arrangement has the greatest friction factor at the lowest Reynolds number, as illustrated in Fig. 7(b).The friction factor varies from approximately 693 at $Re=$205 to about 186 at $Re=$1845, indicating a nearly one order of magnitude increase relative to the circular shaped baffle arrangement inside the double-pipe heat exchanger .Similarly, the circular baffles exhibit an elevated friction factor, ranging from 104 at $Re=$205 to 18 at $Re=$1845,attributable to the partial occlusion of the flow stream .Additionally, the findings indicate that at the maximum Reynolds number examined ($Re=$1845), the fully filled configuration exhibits a friction factor approximately 90.3 % higher than that of the circular baffles. Consequently, the significant increase underscores the substantial additional hydraulic resistance introduced when the porous region is completely saturated with metal foam.

Fig. 7Variation of friction factor with Reynolds number for the investigated foam configurations

a)

b)

5.3. Influence of baffle geometry and arrangement on performance evaluation factor (PEF)

The performance evaluation factor (PEF) was adopted as the primary comparison metric among all simulation configurations. This dimensionless index accounts for both the enhancement in heat transfer and the additional pressure losses incurred by the foam structure, as commonly defined in the literature on porous media heat exchangers [22, 23]. Fig. 8 depicts the results of the baffle-foam configuration and arrangement's impact on the PEF inside the DPHE across various Reynolds numbers. The outcomes indicate that in the fully filled foam structure, a maximum PEF of 1.38 was achieved at $Re=$205, whereas circular-shaped baffles reached a value of 0.92. The data show that the fully filled foam worked about 33 % better than the circular-shaped baffles when the flow was low. Such superiority is mainly attributed to the extensive contact area and enhanced mixing inside the fully saturated porous region. With the Reynolds number increases, the PEF for both configurations consistently declines. The persistent decline occurs because the rate of increase in pressure drop exceeds the corresponding gain in heat transfer at higher flow velocities. Additionally, it is important to demonstrate that the interrupted fins foam configurations exhibit superior overall performance compared to the fully filled, circular-shaped baffles and the continuous 3-1 fins configuration. Across the Reynolds number range of 205-1845, the peak PEF value is 2.8 at $Re=$205 in the int.-3-13 fins, while the minimum PEF value is 2.4 at the lowest Reynolds number in the int.-3-5 fins. However, the PEF for the interrupted foam does not remain constant; instead, it initially increases at low Reynolds numbers and subsequently gradually decreases as the flow rate rises. This behavior reflects the competing physical mechanisms governing the thermo-hydraulic performance. In contrast, at low Reynolds numbers, the gaps(spacing)between foam-enhanced areas are irregular. Therefore, the recurrent regeneration of the boundary layer and increased mixing in these intervening zones significantly enhance convection while maintaining moderate pressure drops. The frictional resistance in the porous regions escalates more significantly with a rise in the Reynolds number. This is attributed to the amplification of inertial forces. Notwithstanding the continuous rise in heat-transfer coefficients, the PEF progressively diminishes in this domain as the hydraulic penalty begins to offset the thermal advantages. Finally, this trend suggests that interrupted metal-foam designs offer the optimal balance between heat-transfer enhancement and pressure drop cost at low to moderate Reynolds numbers. At the same time, their relative advantage diminishes under high-inertia flow conditions due to the disproportionately increasing drag. For the porous tube at $Re<$1500, the non-monotonic) rise-drop-rise (trend in TPF can be attributed to successive porous-flow regimes. At low velocities (Darcy regime), pressure losses scale nearly linearly with velocity, while heat transfer enhancement increases relatively faster, so TPF rises. With increasing velocity, inertial effects become important (Darcy-Forchheimer), causing a sharper growth in pressure drop/friction than in $Nu$, so TPF drops. At higher flow intensity, incipient pore scale mixing/transitional effects can accelerate Nu again relative to the friction penalty, leading to a second rise in TPF [33-35].

Fig. 8Variation of the performance evaluation factor (PEF) with Reynolds number for the investigated DPHE configurations

5.4. Flow field and temperature distribution analysis

The significant thermal and hydraulic behaviors occurring in the double-pipe heat exchanger are shown under various design situations in Figs. 9-10. The simulation results of the temperature contours shown in Fig. 9 illustrate the thermal development along the flow direction for both the smooth tube (a), circular-shaped baffles (b), and the fully filled foam (c) configurations at a flow rate of 3 L/min. For the smooth tube, the temperature gradient was observed to be virtually linear and symmetric, suggesting a fully developed boundary layer with minimal thermal mixing. With the foam fully filled, the temperature gradient near the wall intensifies, and the isotherms become more compact, signifying enhanced heat transfer. This declines the outlet temperature by approximately 4-5 °C compared to the smooth channel. Conversely, the porous domains exhibit significantly non-homogeneous distributions of the local heat transfer coefficient in both the continuous (d) and interrupted fin topologies (e to i). In these configurations, the boundary layer is periodically refreshed with each reconnection, resulting in localized turbulence and increased convective mixing. This results in inhomogeneous but uniform temperature distributions, where foam portions and gaps (the spacing between fins) represent zones of elevated (hot) and lower (cold) temperatures, respectively. Consequently, the overall outcome is a 20-25 % increase in the wall-to-bulk flow temperature difference, consistent with the high Nusselt numbers explained previously.

Fig. 9Temperature distribution contours along the heat exchanger and the outlet section

a) Smooth

b) Fully filled

c) Circular-shaped baffles

d) nt.-3-1 fins

e) Int.-3-5 fins

f) Int.-3-7 fins

g) Int.-3-13 fins

The mechanisms responsible for the observed thermal enhancement can be interpreted using the velocity contours shown in Fig. 10. In the smooth tube (a), the velocity profile remains nearly parabolic and fully developed, indicating a stable and symmetric flow field. However, the incorporation of porous regions (b-i) leads to a clear disruption of this symmetry. The existence of interrupted (e-i) foam blocks compels the fluid to accelerate around their edges, generating localized secondary flows and minor recirculation regions at the upstream and downstream interfaces of each portion. These flow disturbances intensify fluid mixing and repeatedly regenerate the thermal boundary layer, both of which contribute significantly to the enhancement of convective heat transfer. Although the formation of vortices increases flow resistance, leading to a higher friction factor, the thermal gains are much greater than the hydraulic penalties. This balance is reflected in the increased values of the performance evaluation factor (PEF) obtained for the interrupted designs.

As shown in Fig. 11, the velocity-vector field in the vicinity of the interrupted foam inserts provides direct support for the proposed heat-transfer enhancement mechanism. When the flow approaches the leading edges of the foam segments, the local flow passage is constricted, resulting in a clear acceleration of the fluid through the adjacent open regions. The enlarged views further reveal low-velocity wake regions immediately downstream of the interrupted sections, together with local recirculating motion and vector deflection near the insert surfaces. These features indicate repeated flow separation and reattachment around the foam interruptions, which continuously disturb the developing hydrodynamic and thermal boundary layers.

This repeated interruption of the boundary layer promotes its regeneration along the flow direction and enhances fluid exchange between the near-wall region and the core flow. As a result, the interrupted configuration produces stronger local mixing and more effective renewal of the fluid adjacent to the heated surface, thereby improving convective heat transfer. Therefore, the thermal enhancement observed in the present design can be attributed not only to the presence of the porous insert itself, but also to the combined effects of local flow acceleration, wake formation, recirculation, and repeated boundary-layer disruption/regeneration evidenced by the velocity vectors.

Fig. 10Velocity contour

a) Smooth

b) Fully filled

c) Circular-shaped baffles

d) Cont.-3-1 fins

e) Int.-3-5 fins

f) Int.-3-7 fins

g) Int.-3-9 fins

h) Int.-3-11 fins

i) Int.-3-13 fins

5.5. Comparative performance assessment of DPHE

Fig. 12-13 demonstrate the comparative performance of the friction factor, Nusselt number, and performance evaluation factor (PEF) across various designs and flow rates. These characteristics indicate the simultaneous effect of design geometry and flow rate on the global thermo-hydraulic performance in DPHE. Fig. 12 shows a significant decrease in the friction factor as the water flow rate increases, indicating the rising dominance of inertial effects over viscous effects at greater Reynolds numbers. In all examined cases, the lowest thermal resistance is achieved at a flow rate of 5 L/min, while the highest resistance is observed at 1 L/min.

Fig. 11Velocity vectors around the interrupted foam inserts at 3 L/min, showing local recirculation and boundary-layer regeneration

This rise at lower flow rates is primarily attributed to the formation of a thicker thermal boundary layer, accompanied by an extended fluid residence time in the DPHE, subsequently prolonging the period of heat transfer. Also, the con.-3-1fin, and int.-3-(5,7,9,11,13) fins show greater friction factors than the reference case as a result of the additional blockage of flow. However, the slight increase in friction suggests a manageable hydraulic penalty, at least in the laminar-to-transition range.

Fig. 12Effect of water flow rate on the friction factor (1, 3 and 5 L/min)

Fig. 13Effect of water flow rate on the Nusselt number (1, 3 and 5 L/min)

The impact of the volumetric flow rate on the Nusselt number in a DPHE with straight interrupted porous fins is shown in Fig. 13. The results indicated that the Nusselt number grows considerably as the flow rate rises for all configurations, due to enhanced convective mixing and increased wall heat transfer coefficients. The average Nu for the enhanced int.-3-(5,7,9,11,13) fins is approximately three to four times that of the plain tube (smooth DPHE) at a flow rate of 5 L/min. The porous channel, including int.-3-(9,13) fins, shows the most significant improvement of the Nusselt number, due to the repeated regeneration of the boundary layer and localized flow acceleration within the porous channels. Additionally, these configurations preserve a more pronounced temperature gradient between the wall and the fluid throughout the length of the exchanger.

Fig. 14 shows the influence of flow rate on the thermal-hydraulic performance of all examined configurations with and without metal foam, as determined by the Performance Evaluation Factor. The results proves that the PEF declines with rising flow rate for fully filled and circular-shaped baffles in the DPHE (cases 2 and 3). The rationale for this decline is that the growing of the friction factor exceeds the improvement in heat transfer. Consequently, when the flow rate increases, the friction factor rises more rapidly, resulting in a significant decline in PEF. In contrast, for configurations using straight, interrupted foam fin geometries, the optimal PEF occurs at a flow rate of 3 L/min. As a result, an advantageous balance between heat transfer enhancement and pressure drop is achieved. The PEF peaks for int.-3-(7,13) fins, reaching around 2.8-2.77 at 3 L/min, indicating that the enhancement in heat transfer surpasses the pressure drop. Overall, int.-3-7 fins provide the most favorable balance between thermal performance and pressure across the considered range, achieving the optimum geometry.

Fig. 14Effect of flow rate on the performance evaluation factor (PEF) (1, 3 and 5 L/min)

The enhancement ratios obtained in the present study compare favorably with those reported in earlier investigations on metal-foam-enhanced DPHEs, although direct quantitative comparison should be treated cautiously because of differences in geometry, operating conditions, and performance definitions. In the current work, the fully filled foam configuration produced the highest thermal enhancement, with the average Nusselt number increasing by up to about 15 times relative to the smooth baseline, while the preferred interrupted configuration, Int.-3-7, provided the most favorable thermo-hydraulic balance. This level of improvement is higher than the Nusselt-number enhancement of about 117-146 % reported for customized copper-foam inserts by Zuhair et al. [22, 23], and also exceeds the 184 % enhancement reported by Nati et al. [24, 25] for perforated metal-foam topologies combined with CuO-water nanofluid. Moreover, the overall thermo-hydraulic behavior of the interrupted layouts in the present study is consistent with the experimental findings of Aljubury et al. [21], who reported Nusselt-number improvements of about 275-500 % over the clear tube for metal-foam twisted tapes, and with the PEC values reported by Nati et al. [24, 25], where the best perforated configuration reached about 2.14, compared with approximately 2.8 in the present work.

5.6. Correlation of TPF

A data-driven nonlinear model is formulated to represent the correlation between the PEF and Reynold’s number ($Re$), number of fins ($N$), Nusselt’s number ($Nu$), and friction factor ($f$) as in Eq. (24):

where $a_i$, $i=$ 1, 2, 3, 4 referred to the correlation model coefficients whereas$$refers to the error term. The experimental readings are firstly splitted into two samples, namely training and testing. The training sample is selected to form 70 % of the entire data while the remaining 30 % is kept unseen by the model as testing sample. The cross validation process, with four folds, is used to shuffle the training and testing for each time of training model to enhance the reliability of the model. At each training time, the optimum st of coefficient are selected based on the Levenberg-Marquardt algorithm. The model accuracy is evaluated using the mean absolute error percentage. Table 6 presents the optimum set of the model’s coefficients and the percentage mean absolute error between the measured and predicted TPF values, where ${PEF}_{exp}$, ${PEF}_p$ refer to the measured and estimated values of PEF respectively. Using cross validation of four folds, it is shown that the model has successfully predicted the PEF in terms of Reynold’s number ($Re$), number of fins ($N$), Nusselt’s number ($Nu$), and friction factor ($f$). The mean of the percentage error is very small which reflect the model accuracy.