Shape Parametric Numerical Analysis of Thermal-Hydraulic Performance and Fluid Flow in Circular Pipes with Perforated Plate Inserts: Impact of Plate Number, Thickness, and Hole Geometry

An important component of thermal and flow engineering is convective heat transfer in internal flow systems. Heat exchangers, cooling systems, and energy conversion devices are all directly impacted. Enhancement of heat transfer is one of the most important fields of research because interaction of fluid flow and thermal transport determines the overall efficiency in these systems. Since circular tubes are commonly used in industries, much has been done to improve their thermal capabilities [1]. Methods of enhancing heat transfer can be classified into two groups: active and passive. Whereas passive methods aim at modifying geometry of the flow, or the properties of the flow with no other energy input, active methods demand some sort of external energy input, like vibration, or electromagnetic forces. An example of this enhancement is the production of swirl flow by the use of twisted tape inserts which has been shown to enhance rate of heat transfer dramatically, but at the cost of rising pressure drop [2]. Real thermal systems using passive methods have been extensively employed due to its ease of use, low cost and simplicity. A whole host of flow-disturbing apparatus, baffles, wire coils, ribs, twisted tapes, etc. have been well explored as passive methods of augmentation. These devices improve the mixing of fluids and increase heat exchange by developing secondary flows and strengthening turbulence, as well as disturbing the thermal barrier layer [3]. Recent numerical studies of perforated tube–plate heat exchangers have also proved the potential of perforated configurations to improve convective heat transfer [4]. Unlike solid inserts, perforated devices allow some fluid flow through their structure, which enhances mixing between fluid layers and generates local jet flows. Experimental studies have also searched in heat transfer improvement by different perforated tube inserts [5]. This action modifies the parameters of the pressure drop along with an enhanced convective heat transfer. Mehta et al. [6] demonstrated that the pressure drop is also increased but slanted perforated disk inserts significantly increase the Nusselt number due to improved turbulence and fluid mixing. According to Alwatban [7] porous vertical baffles enhance the fluid mixing and heat transfer efficiency but at the expense of enhancing resistance to flow. Also, in contrast with conventional layouts, modified baffle structures enhance significantly turbulence intensity and redistribution of the flow, resulting into enhanced thermal performance under numerical simulations [3, 8]. The shape of the channel and the working fluid significantly influence the form of entropy, the resistance to flow, and the thermal performance [9]. Moreover, studies of open-cell materials and perforated plates have confirmed that flow structure and porosity has a strong influence on the properties of heat transfer and pressure drop [10]. Perforated fins are also capable of significantly enhancing thermal performance, both by enhancing the heat transfer mechanisms and by increasing the effective heat transfer area, according to experimental and numerical studies [11]. Cicek et al. [12] computational studied triangular perforated offset-strip fins and presented that the flow resistance and heat transfer enhancement were significantly affected by the geometry of the perforated fins. Recent advances in perforated geometries have been used to further demonstrate the importance of structural design in controlling thermal-hydraulic performance. By modifying the behaviour of the boundary layer, and increasing turbulence, perforated-serrated fin designs enhance heat transfer [13]. These works illustrate the significance of geometry in optimization of the heat transfer augmentation techniques. Moreover, the effects of channel geometry and nanofluids on the heat transfer efficiency have been studied numerically.These comprehensive studies leave a big gap in the literature. Most of the existing research focuses on single strategies of augmentation, e.g. twisted tapes, perforated fins or baffles, without systematically investigating the overall impact of significant geometric aspects of perforated plate inserts. In particular, the effects of whole shape, plate thickness, and plate number on overall thermal-hydraulic performance of circular tubes have not been studied in detail. Moreover, the previous studies often concentrate on enhancing heat transfer without providing a reasonable evaluation regarding the related pressure drop that is essential to realistic engineering design. The present research, therefore, aims to address these limitations by performing a rigorous numerical study of fluid flow, and heat transfer in circular pipes with perforated plate inserts. The findings of this research present design suggestions and quantitative data towards the maximization of perforated plate patterns in heat exchanger designs. Perforated structures have been investigated as operative passive heat transfer-increasing techniques in the latest investigations. A tangential perforated ring turbulator, as discussed by Chauhan et al. [14] played an important role in enhancing the thermo-hydraulic performance of double-pipe heat exchanger by improving the mixing of fluid and heat transfer between the inner and outer pipesMoreover, Aktas et al. [15] highlighted the need to balance the heat transfer enhancement and pressure-drop penalty in their experimental analysis of the thermal and hydrodynamic performance of plate-fin heat exchangers. Aoua et al. [16] investigated the influence of anisotropic parameters on flow structure and heat transfer enhancement in a partially porous channel, demonstrating that porous layer characteristics significantly affect fluid behavior, thermal distribution, and overall heat transfer performance. Although this research shows that perforated geometries can be viable enhancing strategies, the collective impacts of plate number, plate thickness, and overall geometry of a circular pipe have not been studied.

A three-dimensional steady state CFD model was used to investigate the convective heat transfer and fluid flow of a smooth circular pipe and perforated plate-inserted pipes. The numerical simulations were performed with ANSYS Fluent. To determine the influence of plate number on thermal-hydraulic performance, three perforated plates, as well as one and two, were incorporated into the computational domain. Six plate thicknesses (5, 8, 10, 15, 18, and 20 mm) were considered to determine the geometric effects in a systematic manner. Moreover, two hole geometries, circular and rectangular holes with the same porosity, were studied as demonstrated in Figures 1 and 2. To tackle the laminar and transitional flow regimes of compact heat exchanger applications, the Reynolds number at the intake was varied between 500 and 2000. The Reynolds number range selected (500 to 2000) is the range of laminar and transitional flow conditions that often occur in energy recovery devices, cooling channels, chemical process pipework, electronic cooling systems and small heat exchangers. These operating conditions are well suited to a number of industrial applications as a result of maintaining reasonable pumping power requirements whilst maintaining consistent flow characteristics. Engineers interested in enhancing heat transmission while minimizing pressure drop penalties can be interested in the thermal-hydraulic performance of inserts in the form of perforated plates in this Reynolds number range and gain valuable design information from the analysis. The fluid under study was water, which was believed to have a constant thermophysical behaviour, be incompressible, and Newtonian fluid. This is a common assumption in a study of the same sort and valid in the temperature range considered. It was decided to model the flow with the laminar viscous model due to the diameter of the pipes used in the study and very low Reynolds numbers. The governing equations of mass, momentum, and energy conservation were solved by a pressure-based solver. Numerical accuracy was enhanced through second-order spatial discretization algorithms, and the SIMPLE algorithm was applied in the pressure velocity coupling. The effects of heat transfer were represented by switching on the energy equation. The following are the set of boundary conditions: The constant pressure outlet condition was imposed at the outlet, and a uniform velocity profile at the inlet with the assigned Reynolds number was imposed. A steady heat flux situation was imposed on the wall of the pipes and the perforated plates. The no-slip boundary conditions were applied to all solid surfaces. To ensure convergence and numerical stability, appropriate under-relaxation factors, i.e., 0.3 of pressure, 0.7 of momentum, and 1.0 of density, body forces, and energy, were used. Convergence was achieved when the measured outlet temperature and pressure dropped to constant values and the residuals of the continuity, momentum, and energy equations reduced to values less than 10 ^-1. A grid independence analysis was done to ensure that the numerical results are not dependent on the mesh size. The reliability of the existing simulations was also ensured through the validation of the numerical model with the reported findings of Taskesen et al. [9] that showed that there was a good agreement with the findings within the acceptable error margin This numerical framework, in comparison to the previous studies that examined these factors independently, allows a comprehensive evaluation of the combined impacts of the plate number, plate thickness, and hole geometry on the enhancement of heat transfer as well as the pressure drop and provides more insight into the mechanism of the flow.

Figure 1. The circular shape of plate's holes

Figure 2. The rectangular shape of the plate's holes

3.1 Boundary conditions

Measurement of the length and diameter of the pipe is according to Taskesen et al. [9]. Figure 3 indicates the geometry (without a plate, one plate, two plates, and three plates). All the boundary conditions are presented in Table 1.

Figure 3. Geometry model: a- pipe without plate b-pipe with one plate c- pipe with two plates d- pipe with three plates

Table 1. Parameters of the model

3.2 Mesh generation

ANSYS Fluent 2020R2 was used to make the mesh using the meshing model. In Figures 4 and 5, a tetrahedron mesh having a body size of 0.001 m was used. The size of the plate face sphere of influence and the sphere radius depend on the thickness of the plate; plate thicknesses of 5, 8, and 10 mm have a sphere radius of 0.02 m; plate thicknesses of 15, 18, and 20 mm have a sphere radius of 0.03 m.

Figure 4. Mesh of pipe with three plates

Figure 5. Magnifier photo for mesh of one plate

3.3 Grid convergence study

A grid independence study was conducted to ensure that the numerical results are independent of mesh size and to determine the optimal mesh resolution. Four different mesh sizes, ranging from 189,717 to 20,465,749 elements, were tested. The outlet temperature and pressure drop were monitored as key parameters to evaluate mesh sensitivity. The relative error between successive meshes was calculated using the following equation:

Error% = $\left|\frac{\emptyset_{\text {fine}}-\emptyset_{\text {coarse}}}{\emptyset_{\text {fine}}}\right| * 100$     (1)

where, ф represents the monitored parameter (e.g., outlet temperature or pressure drop), and the subscript "fine" refers to the finer mesh, while "coarse" refers to the compared mesh. The results of the grid independence analysis are presented in Table 2. The difference in the pressure drops across the finest and medium mesh was found to be approximately 2.44%, which is within acceptable numerical tolerance, and the change in outlet temperature was found to be less than 0.03%. The mesh consisting of 2,596,253 elements was selected to be used in all the simulations due to it offering an appropriate trade-off between the computational cost and the solution accuracy.

The mesh with 20,465,749 elements was numerically the most accurate, but was slightly better than the mesh with 2,596,253 elements. The outlet-temperature deviation remained at less than 0.03%, and the difference between the two meshes was 2.44% in terms of the pressure drop. The other way around, however, caused a significant increase in the simulation time and processing cost. The mesh with 2,596,253 elements was selected as the best grid because it is mesh independent and has much fewer processing requirements.

Table 2. Validation of mesh

3.4 The governing equations

The governing equations in three-dimensional, laminar, steady-state, incompressible flow with conjugate heat transfer have been solved using the finite volume method. It was believed that the working fluid was characterized by constant thermophysical parameters and was Newtonian. Such assumptions are valid in the case of moderate temperature variations and the range of Reynolds numbers (500–2000). Heat conduction was considered in the solid plates, and conservation laws of mass, momentum, and energy were applied to the fluid region [6, 18].

Continuity equation

$\nabla \cdot u=0$     (2)

Energy equation (fluid region)

$\rho c_p(u . \nabla T)=k_f \nabla^2 T$     (3)

Energy equation (plate region)

$k_s \nabla^2 T s=0$     (4)

Momentum equation

$\rho(u . \nabla) u=-\nabla p+\mu \nabla^2 u$     (5)

Heat flux continuity at the fluid–solid interface

$k f\left(\frac{\delta T f}{\delta n}\right)=k s\left(\frac{\delta T s}{\delta n \delta}\right)$     (6)

At the fluid–solid interface, both temperature continuity and heat flux continuity were enforced to accurately model conjugate heat transfer between the fluid and perforated plates, which is consistent with previous studies on perforated structures and heat transfer enhancement [10].

Performance Evaluation Criterion (PEC)

The overall thermal-hydraulic performance was evaluated using the PEC, defined as [4, 14, 15]:

$P E C=\frac{\left(N u_{\text {modified}} / N u_{\text {smooth pipe}}\right)}{\left(f_{\text {modified}} / f_{\text {smooth pipe}}\right)^{1 / 3}}$     (7)

The finite volume method of ANSYS Fluent was employed to discretize and numerically solve the above equations. The flow field, temperature gradient, and the properties of enhancement of heat transfer and pressure drop in pipes that have in them inserts of perforated plates can all be correctly predicted courtesy of this formulation.

3.5 Validation of the model

Taskesen et al. [9] conducted a numerical study of the heat transport as well as the flow properties of water and nanofluids in pipes of different shapes. As observed in Figure 6 and Table 3, the validation results show there is a good agreement between the current numerical results and the data reported by Taskesen et al. [9]. The present numerical model was checked by comparing the change in Nusselt number with Reynolds number to the results that were reported by Taskesen et al. [9]. The largest variance of about 2.56% confirmed the accuracy and reliability of the current model, which is within an acceptable range of numerical simulations.

Table 3. Percentage error validation

Figure 6. Model validation

This work explored the effects of plate number, plate thickness, and hole form on fluid flow and heat transfer in a circular pipe with perforated plates through the use of three-dimensional CFD. The results showed that increasing the number of plates enhances the efficiency of heat transfer. As an example, the Nusselt number changed by approximately 7–8% when the number of plates increased by one to three at Re = 1500. But, since the plates give more resistance to the flow, this enhancement comes with a significant increase in pressure drop. In the range studied (520 mm), the effect of plate thickness on heat transmission was observed to be very small, with only slight variations in the Nusselt number. But the larger the thickness, the larger the pressure losses, which negatively impact the overall thermal-hydraulic performance. The geometry of the holes is also very important in determining system performance. Due to the more intense local flow disturbances, rectangular holes generally yield a slightly higher rate of heat transfer. However, this is associated with increased pressure drop that might reduce the overall efficiency. Plate configuration and flow conditions both affect ideal performance, according to the PEC study. When the heat transfer enhancement exceeded the pressure penalty (low to moderate Reynolds numbers), the maximum PEC values were observed. There was also a decrease in PEC due to higher friction losses in multi-plate layouts, but the best performance was achieved in a single-plate system with round holes. In general, it can be concluded that the ideal design requires a trade-off between pressure drop and heat transfer improvement, where the thickness of plates and the shape of holes play secondary roles in the optimal design within the studied range, and the number of plates and hole shape are the most significant factors.