Data analysis for performance index of plate heat exchanger filled by ionanofluid-oil: parallel versus counterflow | Scientific Reports

Scientific Reports volume 15, Article number: 4688 (2025) Cite this article

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Plate heat exchangers ensure their continued relevance and adaptability in meeting modern energy, environmental, and industrial demands. Innovations in this area significantly improve energy efficiency, reduce environmental impact, and enhance industrial processes. However, the ionanofluid significantly enhanced the thermal conductivity and heat transfer coefficient compared to conventional fluids. This study focuses on a three-chambered parallel plate heat exchanger (PHE) using cold ionanofluid (a mixture of graphene as nanoparticle and 1-ethyl-3-methylimidazolium thiocyanate ([C2mim][SCN]) as ionic liquid) in the top and bottom channels and hot oil in the middle channel. It aims to determine the most effective configuration for performance index (η) and energy efficiency, given the unique properties of ionanofluid. The governing equations of Navier–Stokes and energy balance are solved numerically using the finite element method. The impact of different Reynolds numbers (Re) and solid concentrations (ϕ) of nanoparticles on the fluid velocity and temperature fields is observed, and various related parameters are calculated. A comprehensive data analysis is conducted using the surface response method, including ANOVA, sensitivity analysis, and optimization testing. The counter and parallel flow designs achieve approximately 76.23% and 70.07% thermal enhancement at Re = 1, respectively. The optimal performance index occurs at (Re = 1, ϕ = 0.025), chosen to maximize predicted η (= 33972.3) and actual η (= 34020.03). The results indicate that the counterflow configuration consistently outperforms the parallel flow configuration regarding heat transfer efficiency. The counterflow configuration exhibited superior overall performance and provided a more uniform temperature distribution across the PHE. This research offers important insights into the application of ionanofluids in heat exchangers by comparing both parallel and counterflow setups. The study demonstrates high R2 values for the response function, highlighting its significance, and suggests practical implications for the optimization of heat exchanger designs in industrial settings.

Heat transfer analysis progresses through new technologies, cutting-edge applications, and creative ideas. Our lives are made more accessible by advancements in this age of cutting-edge technologies, but we also constantly encounter new difficulties. Thus, we must keep improving upon the concepts we now have. This study’s goal aligns with this contemporary necessity.

Heat exchangers are widely utilized in various industries, including electrical and mechanical, as well as in a wide range of commercial, industrial, and residential machinery. Heat exchanger design and planning must be done carefully to increase energy efficiency and reduce operating expenses. In recent years, there has been an increase in interest in using ionanofluids and their enhanced capacity for heat transmission in various applications, including electronic cooling, heat exchangers, heat pipes, and radiators1. The working fluids of the three-chambered counterflow heat exchanger discussed in this research are ionanofluid and oil. Ionanofluids, or nanofluids with an ionic liquid as the base fluid, have garnered attention because of their unique thermal properties and potential to improve heat transfer performance2,3,4. The thin steel plates that divide the channels function as efficient heat transfer surfaces while preserving structural integrity5,6.

Hong et al.7 evaluated fluid flow and heat transfer characteristics, looked into the effect of strip-fin size on performance, and offered insights for design optimization in offset strip-fin microchannel heat sinks for microelectronic cooling. Matkowska et al.8 demonstrated measurements of pure ionic liquids’ pressure, density, and temperature. Discussion of the effects of cation alkyl chain length and anion character on the reported parameters was included. Said et al.9 investigated using a SWCNTs-water nanofluid to increase the thermal efficiency of a flat plate solar collector. The results showed increased energy and exergy efficiency compared to water, indicating the possibility of SWCNTs as a functional fluid for improving solar collector performance. Chereches et al.10 highlighted the need to thoroughly understand ionanofluids’ thermophysical characteristics, viscosity’s impact on convective flow, and the fluids’ potential as promising heat transfer materials. França et al.11 examined the thermal characteristics of ionic liquids and ionanofluids, noting the difficulties in predictive modeling and emphasizing the possibility of effective heat transfer with improved conductivity.

Chereches et al.10 emphasized the significance of comprehending thermophysical features and the possibility of improved convection due to increased thermal conductivity and viscosity fluctuation. This showed the promise of ionanofluid as a promising heat transfer fluid. Oster et al.12 examined ionanofluids made of nanoparticles and trihexyl(tetradecyl)phosphonium-based ionic liquids. They evaluated their thermophysical characteristics and heat exchange effectiveness, compared to conventional heat transfer fluids. Bakthavatchalam et al.13 investigated the characteristics, preparation techniques, and advantages of real-world applications of nanofluids, ionanofluids, and nanofluid-assisted devices, intending to improve heat transmission. Napitupulu et al.14 examined the efficiency of a pass shell and tube heat exchanger with one shell and two tubes in cooling oil, showing a notable 32% drop in oil temperature. Derakhshanpour et al.15 examined ribbed microchannel designs’ hydraulic and thermal performance with various corner configurations, emphasizing how adding filleted ribs and corners significantly improved overall performance and the heat transfer coefficient.

Bakthavatchalam et al.16 examined using Ionanofluids, more precisely MWCNT-based Ionanofluids, to improve the cooling and heat transfer capabilities of a liquid-cooled microchannel heat sink. Results showed improved cooling, a lower thermal resistance, and an increased heat transfer coefficient compared to pure propylene glycol. Wang et al.17 suggested a novel double-layered microchannel heat sink design that takes into account variables, including pressure drop, coolant velocity, and mixing. It features porous vertical ribs and wavy microchannels, exhibiting higher cooling capacity than previous designs. G. Rao et al.18 Thermal ionanofluids were used to compute computationally examine laminar natural convection heat transfer. This work demonstrated the potential of ionanofluids as advantageous heat transfer fluids because of their increased thermal conductivity. The simulation results are validated by agreement with data from the literature. Haseˇci´c et al.19 emphasized the importance of considering temperature-dependent thermophysical parameters because presuming constant values significantly impacts heat transfer efficiency. Józwiak et al.20 examined the heat transfer capabilities and financial feasibility of ionanofluids made of ionic liquid and carbon nanotubes, emphasizing the latter’s potential for effective energy conversion procedures and environmentally friendly energy systems.

El-Maghlany et al.21 examined an ionanofluid’s behavior during natural convection, stressing the effect of alumina nanoparticles on heat transfer and entropy formation and indicating a possible utility for these materials in forced convection applications. Zheng et al.22 examined how a cone-column combination heat sink performs better in heat transfer than a circular microchannel. It finds that the combined heat sink has a better flow state, a lower base temperature, and a higher average Nusselt number, which makes it a viable option for cooling high-power electronic components. Huminic et al.23 investigated the potential for improving the thermal performance of ionanofluids, specifically silicon carbide, and graphene dispersed in an ionic liquid, by demonstrating their capacity for heat transfer under laminar flow circumstances. Zhu et al.24 found that improving the sidewall geometry of the grooves in microchannel heat sinks enhances performance overall, with the water-droplet-shaped grooves showing minimal pressure drop and favorable heat transfer enhancement. Very recently, Sohail et al.25,26 performed a time-dependent numerical analysis for thermal performance enhancement using hybrid/tetra hybrid nanofluid due to a radially stretched surface/ spinning sphere. Gurrampati et al.27,28 used ternary composite nanofluid/Casson-Maxwell nanofluids due to a porous stretching surface/cylinder considering non-Fourier law to observe the effects of different pertinent parameters. Fadodun et al.29,30 conducted a numerical study to explore the hydrothermal performance and the rates of entropy generation of a hybrid nanofluid composed of reduced graphene oxide (rGO) and cobalt oxide (CO3O4) suspended in water (H2O). This investigation focused on the behavior of the nanofluid as it flowed through corrugated-converging pipes and wavy channels, utilizing a discrete phase model to provide a detailed analysis of the fluid dynamics and heat transfer characteristics in these complex geometries.

To the best of the authors’ knowledge, following all the literature cited above, we can come to the following conclusion,

Very little research has been conducted using nanofluid for flat plate heat exchangers.

Data analysis for performance index, thermal efficiency, and average heat transfer for heat exchangers is not available in the literature.

The combination of ionic liquid with a single nanoparticle and hot oil has not been previously utilized in heat exchanger design for counterflow analysis.

The comparative study for parallel and counter flow, considering ionanofluid-oil, has not been conducted previously.

In this study, we looked into the following:

In our investigation, we utilize ionic liquid (IL) namely 1-ethyl-3-methylimidazolium thiocyanate ([C2mim][SCN]) as the base fluid and graphene (G) as a nanoparticle;

We focus primarily on analyzing counter flow and comparing fluid flow and heat transfer phenomena for flat plate heat exchangers in both parallel and counter flow configurations;

Our analysis includes calculating heat transfer rates, pressure drop, fanning friction, synergy number, thermal enhancement efficiency, and thermal performance index;

Additionally, we conduct a comprehensive data analysis using the surface response method, which involved ANOVA, sensitivity analysis, and optimization testing for the obtained data.

Overall, we can conclude that the study aims to improve our understanding of heat transfer in three-chambered counterflow heat exchangers. It will provide valuable data for optimizing construction and operation parameters. The study will also consider the effects of ionanofluids on heat transfer efficiency and energy consumption when assessing the potential benefits of using them in heat exchanger systems. Lastly, the study aims to determine the viability and performance of a three-chambered counterflow heat exchanger that utilizes ionanofluid and oil, to enhance heat exchanger technology.

The physical model consists of a parallel plate heat exchanger with three channels, each with a uniform thickness of 0.1 mm. The channels measure 20.2 mm in width and 14.1 mm in height. The model is constructed by stacking three blocks on each other and sandwiching two thin steel plates between them. The steel plates are 0.2 mm thick and the same size as the blocks. The top and bottom channels of the flow configuration are used for the cold ionanofluid, and the middle channel is used for the hot oil, as shown in (Fig. 1).

Model geometry for counter flow.

Figure 2 displays the geometry of the parallel flow heat exchanger. Graphene (G) nanoparticles with thermophysical properties defined in Table 1 make the ionanofluid in the analysis. The ionanofluid also has the ionic liquid 1-ethyl-3-methylimidazolium thiocyanate ([C2mim][SCN]), with an inlet velocity of 0.005 m/s and a temperature of 295 K, the cold ionanofluid enters the heat exchanger. The oil used in the study has an inlet velocity of 0.004 m/s and a solid concentration of 1% when it enters the heat exchanger. At a temperature of 325 K, the hot oil flows through the middle.

Model geometry for parallel flow.

The system of governing equations along with boundary conditions following the Boussinesq approximation is given as31,32,33,34,35,36,37:

At the top and bottom channels: inlet velocity, \(u= {u}_{in }\text{m}/\text{s}, v=0, w=0\), inlet temperature T = 295 K.

At the middle hot channel: inlet velocity, \(u= -0.004 \text{m}/\text{s}, v=0, w=0\), inlet temperature, T = 325 K.

All other solid walls: \(u=v=0\).

The Temperature of the top and bottom surfaces of the model are maintained at periodic conditions (temperature offset \(\Delta T=0\) K, so that the source and destination temperatures are equal).

Both vertical surfaces of the solid channel are kept insulated.

All the solid–fluid interfaces: \({T}_{inf}={T}_{s}, {k}_{inf}\frac{\partial {T}_{inf}}{\partial n}={k}_{s}\frac{\partial {T}_{s}}{\partial n}\).

At all outlets: \(P=0\).

In the current research work, we have collected quantitative data from the model using the following expression after numerical simulation:

Reynolds number,

Hydraulic diameter,

Prandtl number,

The average Nusselt number at hot fluid inlet surface,

Total pressure drop31,

The apparent fanning friction factor,

The field synergy number,

Total pumping power37,

Volumetric flow rate6,

Thermal effectivenes37,

Performance index (Hydraulic)6,

Thermal performance index6,

where, \({N}_{ch}\) is number of channel, \({q}_{max}=m{C}_{p}\left({T}_{hi}-{T}_{ci}\right),\) \({\text{and } q}_{actual}={{(m{C}_{p})}_{h}(T}_{hi}-{T}_{ho})={{(m{C}_{p})}_{c}(T}_{ci}-{T}_{co}),\) are the maximum and actual heat transfer, respectively.

Tables 1 and 2 show the thermo-physical properties of all fluids, materials, and effective properties of ionanofluid, respectively.

For numerical simulations, we utilized numerical analysis to generate a mesh for the parallel plate heat exchanger38,39,40,41. The constructed mesh aligns cellular elements with the geometry, typically resulting in rectangular or square elements. This mesh type is well-suited for the regular geometry of the channels in the parallel plate heat exchanger. In our numerical computations, we employed the finite element method (FEM) with the Galerkin weighted residual technique. The thermophysical properties of the base fluid and solids are provided in Tables 1, 2, respectively. The software COMSOL Multiphysics has been utilized for numerical computations.

The mesh used in this study for the parallel heat exchanger is depicted in (Fig. 3a,b). The domain’s solution has been divided into non-uniform finite element meshes. The border elements consist of 6-node triangles and 10-node tetrahedrons, as shown in (Fig. 3a). In Fig. 3b, five different grids are applied at Re = 2, \(\phi =0.01\). The total number of mesh elements comprises 74163 (extra coarse), 147888 (coarser), 356098 (coarse), 635651 (normal), and 1538952 (fine), respectively. The computational times are 150, 280, 800, 2375, and 5494 s, respectively, and the corresponding Nusselt numbers are 0.5861, 0.5941, 0.6110, 0.6151, and 0.61520. We noticed that the computation time increased significantly, but when we switched to using normal meshes, the heat transfer rate increased only marginally, which had an insignificant effect. So, we used a normal mesh for heat exchange for the numerical simulation to balance computational cost and accuracy. The continuity equation is inherently satisfied for high values of the specified constraint. In this framework, the velocity components (\(U, V\)) and temperature (\(\theta\)) are expanded using a basis set. Furthermore, the condition for convergence criteria is chosen as \(\left|{\psi }^{m+1}-{\psi }^{m}\right|\le 1{0}^{-5}\), where \(m\) and \(\psi\) are the iteration numbers and the functions of dependent variables, respectively. The pressure (\(P\)) is subsequently eliminated through the application of a specific constraint. The non-linear residual equations are solved using the Newton–Raphson method to determine the coefficients of the expansions.

(a) Mesh generation, (b) zoom view, and (c) grid test for Re = 2, \(\phi =0.01\).

Model validation is a vital component of numerical investigations. In this study, we benchmark our numerical results against those presented by Jia et al.31, which examine counterflow parallel heat exchangers utilizing cold water and hot oil as fluids, with steel as the solid material. As depicted in (Fig. 4a,b), there is remarkable agreement between our findings and those of Jia et al.31 (Figs. 4, 5) regarding both the surface temperature of the computational domain and the slice plot in the y-z plane along the channel length. These validations enhance our confidence in the numerical code, enabling us to pursue the objectives of the current investigation.

Code validation of the present study with Jia et al.31 in (a) surface temperature of the computational domain and (b) temperature field in the y-z cut along the channel length.

Code validation of the present study with Jia et al.42 in (a) average fluid temperatures in hot and cold channels and (b) heat transfer effectiveness versus channel length.

According to the quantitative analysis for code validation conducted by Jia et al.42, as shown in Fig. 5, the average fluid temperatures in the hot (solid red) and cold (solid blue) channels are examined along the ratio of (y/L) where L = 0.2 m is the channel length in (Fig. 5a). A strong agreement is observed between the results for the cold (dotted blue) and hot (dotted purple) channels from the current study and those presented by Jia et al.42. In Fig. 5b, it is evident that the heat transfer effectiveness reported by Jia et al.42 (solid blue) closely aligns with that of the present study (dotted red), which supports the validation of the model.

The analysis of the parallel plate heat exchanger model yielded promising results. This included the examination of surface velocity, surface temperature, streamlines, isothermal lines, efficient heat transfer rate, pressure drop, fanning fraction, synergy number, and pumping power relating to flow and heat transfer phenomena. For numerical computation, the relevant parameters are as follows:

The pertinent parameters for the analysis are considered as \(1\le Re\le 60\), 0.001 \(\le \phi \le 0.025\). However, for cold ionanofluids, the corresponding inlet velocity (\({u}_{in}\)) values are 0.001, 0.0017, 0.0042, 0.0085, 0.017, 0.034, and 0.051 m/s, corresponding to Reynolds numbers of 1, 2, 5, 10, 20, 40, and 60, respectively. In the contour plot of streamlines, the black solid lines represent the streamlines, while the color maps depict the surface plot of the velocity profile. Similarly, the solid black lines represent the isotherms for the isothermal lines, and the color maps represent the contour surface.

Figure 6a–c depict the surface temperature, surface velocity of the three-dimensional computational region, and temperature along the cut field in the Y-Z plane, respectively. It’s evident that hot oil enters from the central channel at a uniform temperature of 325 K and is gradually cooled along the channel length. On the other hand, cold water enters from the top and bottom channels at a temperature of 295 K and is subsequently heated along the channel length. Eleven sectional views are taken for the y-z plane, demonstrating the heat transfer phenomena for cold ionanofluid and hot oil.

(a) Surface velocity, (b) surface temperature, and (c) temperature fields in the y-z cut-planes along channel length for Re = 2, \(\phi =0.01\).

Meanwhile, Fig. 7a–c depict streamlines, isothermal lines, pressure contours, and fluid temperature variations along the length of the heat exchanger for the top and bottom channels at Re = 2, and \(\phi =0.01\). According to the heat transfer analysis, the temperature of the cold ionanofluid changes from 295 to 304.2 K which confirm the temperature difference of 9.2 K. The streamlined profile shows flow directions and patterns in the cold channels, while the isotherm lines indicate temperature distribution, highlighting areas with varying temperatures. Furthermore, the pressure drop analysis reveals a favorable pressure drop across the heat exchanger.

(a) Streamline, (b) isotherm lines, and (c) pressure contour plot at the cold channels at Re = 2, \(\phi =0.01.\)

By Fig. 8a–c, the streamline, isotherm line, and pressure contour plots from the hot channel are depicted at Re = 2, and \(\phi =0.01\). Along the channel length, the temperature of the hot oil decreases from 325 to 312.34 K. Additionally, a gradual decrease in pressure is observed from the inlet to the outlet along the flow path. The streamlines plots of the heat exchanger’s hot channel illustrate the flow patterns and directions. At the same time, the temperature distribution is represented by the isotherm lines, showing changes in temperature along the hot channel.

(a) Streamline at the hot channel, (b) isotherm lines at the hot channel, and (c) pressure contour plot at the hot channel Re = 2, \(\phi =0.01.\)

Figure 9a–c show the impact of of Re with \(\phi =1\%\) on the velocity magnitude, temperature, and pressure of the computational domain. As Re increases, the velocity magnitude increases due to the heightened fluid momentum. This leads to a rise in temperature along the channel length, with counterflow exhibiting a more even distribution due to effective heat transfer. Additionally, counterflow, which has a longer flow route and faster velocity, experiences a greater decrease in pressure as the channel length increases.

Effect of Re on the model (1) with \(\phi =1\%\) on (a) velocity magnitude along y-z cut plane, (b) surface temperature, and (c) isosurface pressure.

Figure 10a–c represent the effect of \(\phi\) when Re = 2 on streamlines, isothermal lines and pressure profile for the studied domain.

Effect of \(\phi\) with Re = 2 on (a) streamlines, (b) isothermal lines, and (c) surface pressure of the system.

When \(\phi\) (volume fraction of nanoparticles) increases, the streamlines pattern within the flow domain exhibits a more concentrated behavior toward the center, suggesting that the flow becomes more laminar. This is likely due to the increased viscosity and the enhanced thermal properties of the ionanofluid, which contains a higher concentration of nanoparticles. Concurrently, the isotherm lines, representing areas of equal temperature, appear closer with the increase of \(\phi\). This phenomenon indicates a faster rate of heat transfer, attributed to the enhanced thermal conductivity and specific heat capacity of the ionanofluid as the concentration of nanoparticles increases. However, this increase in \(\phi\) also leads to a higher viscosity in the ionanofluid, which results in a higher pressure drop across the flow. This is because the fluid resistance to flow increases with viscosity, necessitating more pumping power to maintain the flow rate, thus affecting the efficiency and operating costs of the heat exchanger system has a higher viscosity, the pressure profile indicates an increase in pressure drops with increasing \(\phi\).

The graph in Fig. 11a illustrates the relationship between Reynolds number (Re) and Nusselt number (Nu) in the heat exchanger.The Nu is calculated using Eq. (10). As the Reynolds number increases, Nusselt number steadily rises, indicating improved convective heat transfer. Nusselt numbers range from 1.89 at Re = 60 to 0.324 at Re = 1, indicating that higher flow rates or velocities enhance heat transfer efficiency. The increasing trend in Nusselt number suggests that the ionanofluid-oil combination holds potential for energy-efficient heat transfer systems, promising more effective heat exchange between the fluids. In Fig. 11b, the graph displays the Nusselt number values for a specific Reynolds number (Re = 2) for varying \(\phi\) in the computational domain of the study. As the flow rate, indicated by the dimensionless parameter \(\phi\), increases from 0.001 to 0.025, the Nusselt number also increases progressively from 0.595 to 0.65. This suggests that higher convective heat transfer occurs in the heat exchanger at higher flow rates. These findings indicate that combining ionanofluid and oil can enhance thermal performance in heat transfer systems, resulting in lower energy consumption.

Nusselt number of the model (1) for (a) Re, and (b) \(\phi\).

The relationship between the Reynolds number (Re) and pressure drop for the ionanofluid-oil counter flow parallel plate heat exchanger is illustrated in Fig. 12 using Eq. (12). The graph shows that as the Reynolds number increases, there is a noticeable increase in pressure drop. For instance, at Re = 1, the pressure drop is 89.08 Pa, while at Re = 60, it is 5900.24 Pa. This highlights the importance of considering the impact of Reynolds number on pressure drop when designing and operating heat exchangers to ensure effective and efficient heat transfer.

Pressure drop against Re with \(\phi =1\%\).

The fanning friction factor (or the fanning factor) is a dimensionless number used in fluid dynamics to characterize the pressure drop in a pipe due to fluid flow. The correlation between Reynolds number (Re) and fanning friction (f) in the system being investigated is represented in (Fig. 13). The fanning friction values demonstrate a decreasing trend as the Reynolds number increases. For instance, the fanning friction is 3632.92 at Re = 1, and decreases to 66.84 at Re = 60. This is described by Eq. (12). As the inlet velocity increases, the Fanning friction factor (f) decreases. It is well known that the Fanning friction factor in laminar flow is inversely proportional to the Reynolds number. As the Reynolds number (Re) increases within the laminar regime, the Fanning friction factor decreases sharply. This shows that the resistance to flow, or fanning friction, diminishes as the fluid flow rate increases. These results underscore the influence of Reynolds number on flow characteristics and suggest that higher Reynolds numbers result in reduced system friction losses.

Fanning friction for \(\phi =1\%\).

The synergy number is a concept utilized in fluid dynamics and multiphase flow to quantify the enhancement in heat transfer efficiency. It applies to systems that incorporate enhancements such as surface treatments, flow dynamics alterations, or additives. These enhancements may arise from interactions between different phases or the combined application of multiple enhancement techniques. The Fc (Synergy Number) values for the system under investigation at various Reynolds numbers (Re) are depicted in Fig. 14 using Eq. (13). As the Reynolds number increases, the Fc values exhibit a trend towards decrease. For instance, the Fc is 0.001827205 at Re = 1, decreasing to 0.000177645 at Re = 60. This illustrates that higher Reynolds numbers lead to lower Fc values, signifying reduced fluid movement and heat transfer within the system. These findings underscore the significance of accounting for flow characteristics to achieve optimal heat transfer and highlight how the Reynolds number impacts the overall performance and efficiency of the system.

Synergy number for \(\phi =1\%\).

Pumping power is the power required to push a fluid through a system, overcoming frictional losses, changes in elevation, and other resistances. It is crucial to consider when designing and operating fluid transport systems, such as pipelines and water distribution networks. The link between Reynolds number (Re) and pumping power (\({P}_{pu}\)) in the studied research is illustrated in the graph provided in (Fig. 15), which is calculated using Eq. (15). There is a noticeable increase in pumping power with an increasing Reynolds number. For instance, the pumping power is about 0.0004 W at Re = 1 and 1.5 W at Re = 60. These results suggest that higher Reynolds numbers require more pumping power to overcome fluid resistance and maintain the system’s flow rate at the required level. The data emphasizes how the Reynolds number affects the power needed for the system to operate efficiently, highlighting the crucial consideration of pumping power in the design and optimization of fluid flow operations.

Pumping power versus Re with \(\phi =1\%\).

The temperature variation along the channel length for a counterflow heat exchanger at Re = 2 and \(\phi\) = 0.01 is depicted as shown in (Fig. 16a). The hot temperature is represented by the red line on the graph, which is calculated along the middle line of the hot channel and decreases from 325 to 308.88 K as the channel length goes from 250 to 0 mm (right to left direction), resulting in an overall temperature decrease of 17.22 K. The average cold temperature, calculated along the middle line of considering top and bottom channels, is represented by the blue line showing an increasing trend over the same channel length (from left to right direction), ranging from 295 to 304.24 K, equating to a temperature increase of about 9.2 K. These graphs clearly illustrate how the average cold and hot temperatures change along the channel length, visually representing the temperature variations inside the counterflow heat exchanger.

Fluid temperature variations at different positions along the middle line of the channel for (a) counter flow, and (b) parallel flow.

However, the temperature profiles for a parallel flow heat exchanger at Re = 2 and \(\phi\) = 0.01 are depicted in (Fig. 16b). The first graph (red) represents the hot temperature, a gradual decrease from 325 to 312.34 K as the channel length from 250 to 0 mm, resulting in a temperature reduction of about 13.64 K. The second graph (blue) illustrates the average cold temperature, slightly increasing over the same channel length from 0 to 250 mm, from 295 to 304.16 K, representing a temperature rise of about 8.84 K.

When comparing this plot to the counterflow plot from earlier in (Fig. 16a), it can be seen that the hot temperature drops more significantly for the counterflow configuration compared to the parallel flow heat exchanger as in (Fig. 16b) over the channel length. In addition, the average cold temperature increases slightly in the case of counterflow compared to the parallel flow situation. These variations draw attention to how the counterflow and parallel flow heat exchanger layouts differ in their respective heat transfer properties.

When the flow \(\phi\) is at 1%, the thermal enhancement efficiency (ε) for both counterflow and parallel flow heat exchangers is illustrated in Fig. 17 as a bar chart at various Reynolds numbers (Re). Equation (17) is used to calculate the thermal efficiency. The bar chart demonstrates that the thermal enhancement efficiency decreases as the Reynolds number (Re) increases in the counterflow and parallel flow setups. Specifically, at Re = 1, the thermal enhancement efficiency of the counterflow setup is 0.77, while for parallel flow, it is 0.7. As the Reynolds number increases to 60, the efficiency of the counterflow setup decreases to 0.168, and for the parallel flow setup, it decreases to 0.15. These results indicate that higher Reynolds numbers lead to lower thermal enhancement efficiencies for counterflow and parallel-flow heat exchangers. Notably, the counterflow arrangement exhibits slightly better thermal enhancement efficiencies across the range of Reynolds numbers than the parallel flow configuration. This suggests that the Reynolds number impacts the ability of heat exchangers to improve thermal performance. In the given conditions, the counterflow arrangement may offer slightly superior thermal performance compared to the parallel flow setup.

The thermal enhancement efficiency for \(\phi =1\%\).

The data present in Fig. 18 represents the comparative hydraulic performance index (η) from Eq. (17) of parallel and counterflow heat exchangers across different Reynolds numbers (Re) in a bar chart form. The findings reveal a decline in performance index as the Reynolds number increases, irrespective of whether it’s a counterflow or parallel flow configuration. At Re = 1, the performance index for counterflow is 33465.97, whereas for parallel flow, it is 30908.57. As the Reynolds number escalates to 60, the performance index plunges to 110.78 for counterflow and 101.98 for parallel flow. These results demonstrate that higher Reynolds numbers reduce performance index values for both heat exchangers. The study meticulously analyzes and compares the performance of these two setups, underscoring the substantial influence of the Reynolds number on heat transfer efficiency, as inferred from the available dataset.

Thermal performance index for \(\phi =1\%\).

Overall, our study examined two types of data: qualitative and quantitative. The qualitative analysis encompassed velocity profiles, surface temperature, streamlines, and isothermal profiles. The quantitative analysis included the Nusselt number, pressure drop, Fanning friction factor, synergy number, thermal efficiency, and thermal performance index.

The Nusselt number increases with the Re, as a higher Re signifies more vigorous fluid flow, which enhances convective heat transfer. However, pressure drops also rise with the Re, as faster fluid flow results in greater resistance due to viscous effects.

In contrast, as the Re increases, thermal efficiency, the performance index, the fanning friction factor, and the synergy number tend to decrease. The Fanning friction factor declines because, in laminar flow, it is inversely related to the Reynolds number. Similarly, the Synergy number decreases due to increasing misalignment between the velocity and temperature gradients at elevated Reynolds numbers.

Thermal efficiency diminishes because the increase in pumping power outpaces the improvement in heat transfer. Moreover, the performance index falls since heat transfer scales linearly while pumping power scales quadratically with the Re.

The response surface (RS) technique is a powerful method to study multiple variables simultaneously, using quantitative data and a well-planned experimental setup with minimal resource allocation. This technique is particularly beneficial when the response variables are affected by various factors and variables. Moreover, the analysis examines the interrelations between these parameters. Although several RS models are available, the 2nd order RS model, which includes all linear, square, and interacting variables, is generally considered adequate for estimating the response43.

The following formula defines the quadratic polynomial model:

where, y is the response function; a0 is the intercept terms; ai is the linear regression coefficient of ith factor; aii is the quadratic regression coefficient of ith factor; aij is the interaction term of i and jth factors; and xi, xj are the design parameters. Table 3 highlights a comprehensive overview of the codded (categorized) levels of input variables for CCD (Central composite design).

Equation (18) becomes

Table 4 shows the simulated configurations of the data, both in coded and actual form, using the CCD method. The statistical analysis of the quadratic polynomial correlation equations, including estimating coefficients and assessing the factors’ impact on variables (represented as coded values), is done using analytical software. The adequacy of the CFD results’ fit is evaluated using a designated coefficient of determination, and the available resources are modified accordingly. This evaluation is done by comparing the obtained p-value with a significance level of 99%, leading to the acceptance or rejection of the fitting quality.

Based on computational analysis using RS, an RS plot is a design that links two independent factors—Re and ϕ—with the response parameters (RP) Nu, ε, and η. Figure 19a,b presents contour plots and surface plots, respectively, created through RS to analyze the effects of Re and ϕ on heat transfer, effectiveness, and performance index (PI). Figure 19a shows that Nu increases as Re and ϕ increase. However, ε decreases for increasing Re, and it boosts slowly for escalating ϕ. Additionally, η devalues for rising values of Re and it escalates with elevated ϕ. The highest values of Nu, ε, and η are achieved at (Re = 60, ϕ = 0.025), (Re = 1, ϕ = 0.025), and (Re = 1, ϕ = 0.001), respectively. The surface plots in (Fig. 19b) exhibit a similar trend in response to the independent variables. The contour and surface plots demonstrate the same behavior regarding the response functions/dependent variables.

RS (a) contour plots and (b) surface plots for Nu, ε, and η.

The following Tables 5–7 present the results of a statistical analysis of the thermal–hydraulic characteristics of a heat exchanger (HE) system that uses the RS technique. The study measured the rate of thermal transport, thermal efficiency, and PI. The tables include the degrees of freedom (DOF) for the model’s distinct variables, the overall sum of squared (SS), which quantifies the collective variability resulting from multiple factors, and the mean squared (MS), which is also significant. The F-value indicates the statistical significance of the model, while the p-value measures the likelihood of the null hypothesis being valid for a particular statistical design. The p-value ranges from 0 to 1, commonly used to determine the level of statistical significance. There are three categories based on p-values to assess the incidence of statistical significance. If the p-value is less than 0.01, it is considered extremely small, and the null hypothesis is rejected due to strong evidence, indicating that the observed results are highly statistically significant. When the p-value falls within the range of 0.01 to 0.05, it is still statistically significant. However, if the p-value exceeds 0.05, it indicates inadequate proof for rejecting the null hypothesis since the observed outcome is not deemed statistically significant.

The regression model Eq. (19) for the response-dependent variable Nu is calculated from the Table 5 and is written in terms of coded values as:

Here, the p-value of ϕ2 is greater than 1%. Thus, this ϕ2 is insignificant for the regression equation of Nu. The final regression model equation for the thermal transport rate can be written as follows:

It is observed from Table 6 that the p-value of ϕ2 is 0.9513, which is greater than 0.01. Thus, the final regression model equation of the thermal efficiency can be written in terms of coded factors as:

Table 7 displays that the p-value of ϕ2 is 0.9649 > 0.01. Hence, the final regression model equation for the PI of heat exchanger (HE) in terms of coded factors is:

The residual schemes for Nu, ε, and η are depicted in (Fig. 20a,b), indicating the RS model’s exactness level. Figure 20a shows the predicted versus actual values for all three response variables, while Fig. 20b displays the normal probability. Residual plots are often used in regression analysis to assess the adequacy of the model and how well it fits. In Fig. 20a, the residual diagrams and fitted values strongly connect with the observed values and the results obtained by fitting. The residuals are distributed horizontally around the zero line, indicating that the error terms have a similar variance. Notably, none of the residuals show significant deviations from the fundamental random pattern of residuals, suggesting the absence of outliers. Figure 20a shows the residual features form almost a straight line, indicating that the regression models are well-matched with the actual outcomes. Hence, the current RS model demonstrates a satisfactory level of accuracy. Figure 20b shows that the normal probability plots corresponding to the residuals indicate a satisfactory distribution. Normal probability plots displaying the residual distributions are often used to assess the normality of the observation. The assumption of normality in the residual distribution of values for Nu, ε, and η is based on the linearity of the relationship. The planned data points closely follow a straight line, indicating that the residual profiles show a normal distribution.

Residual plots (a) predicted vs actual and (b) normal probability for Nu, ε, and η.

The response variable measures the level of change brought about by the parameters. The analysis of SS (sum of squares) is an essential strategy in counter flow conjugate heat transfer of HE as it helps to explain how changes in multiple variables affect the learning outcome. Several factors can affect the HT (heat transfer) level at the hot inlet surface, thermal efficiency, and performance index of the HE. The SS of the output variables concerning specific operative aspects (Re, and ϕ) is calculated precisely by the partial derivatives of the response parameter (RP) task Nu, ε, and η. The partial derivatives of the RP of Eqs. (15–17) concerning the input variables are calculated as follows:

The sum of squares (SS) yield shows that when the input variable’s value increases, the output task’s value also increases. Conversely, a negative SS value indicates that raising the input variable caused a decrease in the output task’s value. These results are displayed in (Table 8). The coded value of the independent variables is used to determine the output values.

Figure 21a,b shows the standard error estimation in contour and surface plots for Nu, ε, and η. The standard error in the RS technique signifies the oscillation or hesitation around the assessed RS method. The error scheme explicitly illustrates the explicit inaccuracy of the RS model. The level of deviation of the entire process is displayed by the standard error, which can be obtained using statistical trials. The precise technique depends on the norms and features of the data.

Standard error (a) contour plots (b) surface plots for Nu, ε, and η.

Figure 22a,b displays the sensitivity study for dissimilar values of inlet velocity and solid concentration of graphene nanoparticles on the outcome reply of heat transfer, efficiency, and performance of HE. Figure 22a,b express the SS results against the coded values of Re = −1, 0, and 1 while the coded values of ϕ = −1 and 1, respectively, are kept fixed. It is observed from Fig. 22a (at coded value ϕ = −1) that the SS to Re is negative while the SS to ϕ is positive for all the response-dependent parameters Nu, ε, and η. The sensitivity of ϕ is positive, implying that growing these constraints could be crucial to an upsurge in the heat transmission level, thermal effectiveness, and PI. The SS of Re is negative, implying an upsurge in inlet velocity, lessening the heat allocation, effectiveness, and PI levels. Similarly, Fig. 22b (at coded value ϕ = 1) shows that Nu’s SS to both Re and ϕ is positive. But, the SS to Re is negative, and ϕ is positive for the other response-dependent parameters ε and η.

Sensitivity results against coded levels of Re at (a) ϕ = −1 and (b) ϕ = 1, for Nu, ε, and η.

Figure 23a,b displays the correlation analysis of Nu, ε, and η with independent variables Re, and ϕ, respectively. A strong positive correlation exists between Nu and Re with a correlation coefficient (R) = 0.916 and a weak positive correlation between Nu and ϕ with R = 0.064. However, a strong negative correlation is observed between ε and Re with R = −0.848, whereas a strong positive correlation between ε and ϕ with R = 0.009. Similarly, there is a strong negative correlation between η and Re with R = −0.809 and a strong positive correlation between η and ϕ with R = 0.005. These correlation coefficient results among the dependent and independent variables from Fig. 4 agree with the findings in (Table 6).

Correlation analysis of Nu, ε, and η with (a) Re, and (b) ϕ.

The desirability function executes the numerical ϕ optimization. The goals selected for optimizing Nu, ε, and η are the in-range level of Re and values and the maximum level of RP. Table 9 describes the optimum points, CFD, and predicted response value. Real levels of independent variables are coded according to the following equation (Mehmod et al.44):

where, Z and Z0 indicate coded and actual levels of independent variables, respectively. ΔZ represents step change, while ZC indicates the actual value at the central point.

Thirty-five (35) different solutions are found to maximize the Nu, and they contain various levels (in range) of independent variables (Re and ϕ). All 35 solutions have the maximum desirability 1. It is well known that the maximum desirability value is selected as the optimized condition. Among them, one solution point (Re = 60, ϕ = 0.025) is chosen to get the maximum predicted Nu (= 2.00058) and inserted in (Table 9). Similarly, we have nine (9) different solutions using numerical optimization techniques for maximum thermal effectiveness. However, maximum desirability is obtained at 0.999 for two solutions. Among them, we select one solution (Re = 1, ϕ = 0.025) and include it in (Table 9). To obtain maximum PI, we get twelve (12) solutions. But the maximum desirability is 0.999. The optimum solution point (Re = 1, \(\phi\) = 0.025).

Figure 24 expresses the ramp chart for statistical optimization for Nu, ε, and η. A lie detector examines the ideal combination of parameters to find the best response. The effectiveness of an investigative approach is evaluated using three different methods: numerical analysis, point prediction, and graphical analysis. The results are presented in reports and graphs, which provide the same information. Figure 24 is a design of a numerical optimization scheme for maximizing response Nu, ε, and η.

Ramp chart for statistical optimization of Nu, ε, and η.

Optimized conditions check the model’s suitability for predicting Nu, ε, and η response values. Optimized conditions are validated by performing CFD simulation under optimized conditions. Using the optimum independent variables, we get the response predicted values as Nu = 2.00058, ε = 0.770237, and η = 33972.3. On the other hand, the CFD values at optimized in-range conditions are Nu = 1.992, ε = 0.771, and η = 34020.03. The CFD response values align with predicted response values (Table 9). In addition, there is a 0.42, 0.099, and 0.14% difference between the CFD and the predicted statistical values.

We compared our current findings with a recent article by Hasan et al.6 in terms of thermal performance efficiency (ε) as shown in (Fig. 25a), and in terms of the thermal performance index (η*) as shown in (Fig. 25b). The blue line graph represents the thermal performance efficiency of Hasan et al.6, while the red-dotted graph represents our findings in (Fig. 25a). When comparing the two graphs, it is clear that, across the whole range of Reynolds numbers taken into consideration, our current results consistently beat the thermal performance efficiency stated in the study by Hasan et al.6.

Comparison of the present study with Hasan et al.6 in terms of (a) thermal enhancement efficiency (ε), and (b) performance index (η*).

The thermal efficiency is calculated from Eq. (16). The latest results demonstrate a greater thermal performance efficiency of 0.71 and 0.69 at lower Reynolds numbers (Re = 30, and Re = 40) than Hasan et al.’s findings of 0.66 and 0.65, respectively. This trend continues as the Reynolds number increases, with our results consistently outperforming those documented by Hasan et al.6.

At Re = 200, our thermal performance efficiency (ε) is 0.3, while Hasan et al.6 obtained 0.32. Additionally, our efficiency at Re = 500 is 0.18, compared to Hasan et al.’s6 0.21. This pattern holds across all investigated Reynolds numbers.

Overall, our results consistently exhibit higher thermal performance efficiency compared to the data provided by Hasan et al.6, clearly indicating the superiority of the current study. This suggests that the system under study demonstrates better heat transfer and thermal performance efficiency due to our model or experimental setup.

The information in the text contrasts our latest findings with those of a recent study by Hasan et al. in terms of the thermal performance index (η*) (1/Pa) in (Fig. 25b) which is calculated using Eq. (17a). Our results are shown as a red dotted line graph, while a blue solid line graph represents Hasan et al.’s thermal performance index.

Upon examining Fig. 25b, it is evident that our current results consistently surpass the thermal performance index published in Hasan et al.'s paper for all Reynolds numbers considered. The current results indicate higher thermal performance index values of 0.0056 and 0.0054, respectively, at lower Reynolds numbers (e.g., Re = 30, and Re = 40) than Hasan et al.’s 0.0053 and 0.005. This trend continues as the Reynolds number increases, with our findings consistently outperforming the thermal performance index of Hasan et al.

For instance, our thermal performance index at Re = 200 is 0.0003, while Hasan et al.6 obtained 0.0005. Our performance index is 0.00009 at Re = 500, but Hasan et al. achieved 0.00018. This pattern applies to all investigated Reynolds numbers.

At Re = 30, the current study’s thermal performance efficiency (ε) of 10.61% and thermal performance index (η*) (1/Pa) of 6.38% were noticeably higher. The employment of an ionanofluid with improved thermal properties, improved heat exchanger design, and the application of more precise numerical techniques are all credited with this improvement.

Since our results consistently demonstrate greater thermal performance index values than the data published by Hasan et al.6, the comparison favors our current study. This indicates that the system under study has a better heat transfer performance index due to our model or experimental setup.

This study explored the use of ionanofluids in three-chambered counterflow and parallel flow plate heat exchangers. The results show that the counterflow design is more efficient and offers better thermal performance than the parallel flow configuration.

In the counterflow design, the Nusselt number increases by 483.33% when the Reynolds number rises from 1 to 60 at a solid concentration of ϕ = 0.01. It also rises by about 9.24% with solid concentration changes from 0.001 to 0.025 at Re = 2.

Total pressure drop increases by 6523.52% as Re changes from 1 to 60, while higher inlet velocity reduces the field synergy number by 928.57% and the apparent fanning friction factor by 5335.40%.

Approximately 76.23 and 70.07% thermal enhancement are achieved by the counter, and parallel flow design, respectively at Re = 1.

The counter flow design consistently has a higher thermal performance index about 8.27% than the parallel flow configuration at Re = 1. This index takes into account both heat transfer and pumping power.

Both configurations show increased cold and hot temperatures along the channel length. At Re = 2 and ϕ = 0.01, the counter flow configuration shows a 3.13% rise in freezing temperature and a 4.86% decrease in hot temperature. In comparison, the parallel flow configuration showed a 3.10% increase in cold temperature and a 3.66% decrease in hot temperature.

The optimized Nu is obtained at a solution point (Re = 60, ϕ = 0.025), which is chosen to get the maximum predicted Nu (= 2.00058) and CFD Nu (= 1.992).

The maximum thermal effectiveness is achieved at (Re = 1, ϕ = 0.025), selected to obtain the highest predicted ε (= 0.770237) and CFD ε (= 0.771).

The optimal performance index occurs at (Re = 1, ϕ = 0.025) chosen to maximize predicted η (= 33,972.3) and CFD η (= 34,020.03).

This research achieves an enhanced heat transfer rate using an ionanofluid with a higher convective heat transfer coefficient and thermal conductivity. The improved heat exchanger design, featuring thin steel plates and a three-chamber layout, facilitates better heat transfer. More accurate numerical techniques also provide precise assessments of heat transfer performance. The findings underscore the advantages of ionanofluids and advanced heat exchanger designs for improved thermal efficiency compared to traditional methods, potentially leading to smaller heat exchangers and reduced energy consumption. Notably, the counterflow design surpasses the parallel flow configuration in thermal performance, making it ideal for applications requiring high heat transfer rates.

The data set used and/or analysed during the current study available from the corresponding author on reasonable request.

Surface area (m2)

Specific heat (Jkg−1 K−1)

Hydraulic diameter (m)

Size of nanoparticle (nm)

Apparent friction factor

Field synergy number

Graphene

Channel height (m)

Thermal conductivity (Wm−1 K−1)

Boltzmann constant

Channel length (m)

Shape factor of nanoparticle

Number of channel

Mean nusselt number

Prandtl number

Pressure drop (Pa)

Pressure (kgms−2)

Volumetric flow rate (m3/s)

Reynolds number

Temperature (K)

Co-ordinates velocity (ms−1)

Inlet velocity (ms−1)

Channel width (m)

Thermal diffusivity (m2s−1)

Viscosity dynamical (Nsm−2)

Thermal enhancement efficiency

Density (kgm−3)

Performance index

Solid concentration

Cold

Channel

Hot

Ionic liquid

Ionanofluid

Pump

Reference

Solid

Wall

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Ishrat Zahan, Rehena Nasrin & Nusrat Jahan Jakia

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Zahan, I., Nasrin, R. & Jakia, N.J. Data analysis for performance index of plate heat exchanger filled by ionanofluid-oil: parallel versus counterflow. Sci Rep 15, 4688 (2025). https://doi.org/10.1038/s41598-025-88851-2

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