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Search Results (399)

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Keywords = hybrid nanofluid

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14 pages, 3534 KiB  
Article
Photothermal Conversion Performance of Fe3O4/ATO Hybrid Nanofluid for Direct Absorption Solar Collector
by Jeonggyun Ham, Hyemin Kim and Honghyun Cho
Energies 2024, 17(20), 5059; https://doi.org/10.3390/en17205059 - 11 Oct 2024
Viewed by 390
Abstract
In order to enhance the efficiency of direct absorption solar collectors, this study carried out an experimental analysis about the optical and photothermal conversion performance of Fe3O4, ATO (Antimony-doped tin oxide), and Fe3O4/ATO nanofluids with [...] Read more.
In order to enhance the efficiency of direct absorption solar collectors, this study carried out an experimental analysis about the optical and photothermal conversion performance of Fe3O4, ATO (Antimony-doped tin oxide), and Fe3O4/ATO nanofluids with a total concentration of 0.1 wt%. According to the results of the experiments, Fe3O4 nanofluid outperforms ATO nanofluid in terms of optical absorption; nevertheless, at wavelengths shorter than 600 nm, it also shows significant scattering reflection. The solar-weighted absorption coefficient of Fe3O4/ATO nanofluid rose from 0.863 (mFe3O4/mTotal = 0.2) to 0.932 (mFe3O4/mTotal = 0.8) when the optical path length increased from 0.01 m to 0.06 m. Moreover, the Fe3O4/ATO hybrid nanofluid achieved a photothermal conversion efficiency of 0.932 when the mass ratio of Fe3O4 to total mass was 0.2, surpassing the efficiencies of 0.892 and 0.898 recorded for 0.1 wt% ATO and Fe3O4 nanofluids, respectively. When present together, the opposing optical characteristics of Fe3O4 and ATO boost photothermal conversion performance, which is anticipated to raise the efficiency of direct absorption solar collectors. Full article
(This article belongs to the Special Issue Advances in Solar Thermal Energy Harvesting, Storage and Conversion)
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Figure 1
<p>Manufactured nanofluids: (<b>a</b>) 0.1 wt% Fe<sub>3</sub>O<sub>4</sub> NF, (<b>b</b>) 0.1 wt% ATO NF, (<b>c</b>) Fe<sub>3</sub>O<sub>4</sub>/ATO nanofluids.</p>
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<p>TEM images: (<b>a</b>) Fe<sub>3</sub>O<sub>4</sub>; (<b>b</b>) ATO; (<b>c</b>) Fe<sub>3</sub>O<sub>4</sub>/ATO and nanoparticle diameter distribution; (<b>d</b>) Fe<sub>3</sub>O<sub>4</sub>; (<b>e</b>) ATO; (<b>f</b>) Fe<sub>3</sub>O<sub>4</sub>/ATO nanofluid.</p>
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<p>Experimental setup of photothermal conversion experiment.</p>
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<p>Optical absorbance and reflectance of Fe<sub>3</sub>O<sub>4</sub>, ATO, and Fe<sub>3</sub>O<sub>4</sub>/ATO NFs. (<b>a</b>) Absorbance of Fe<sub>3</sub>O<sub>4</sub>/ATO nanofluid. (<b>b</b>) Reflectance of Fe<sub>3</sub>O<sub>4</sub>/ATO nanofluid.</p>
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<p>Solar-weighted absorption coefficient according to the optical length.</p>
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<p>Increased temperature of Fe<sub>3</sub>O<sub>4</sub>/ATO NFs according to lighting exposure time.</p>
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<p>Total energy portion of water, 0.1 wt% Fe<sub>3</sub>O<sub>4</sub> NF, 0.1 wt% ATO NF, and 0.1 wt% Fe<sub>3</sub>O<sub>4</sub>/ATO nanofluid according to light exposure time.</p>
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<p>(<b>a</b>) Rising temperature and (<b>b</b>) local receiving efficiency at t = 9000 s.</p>
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<p>Comparison of photothermal conversion efficiency and solar-weighted absorption efficiency.</p>
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25 pages, 3863 KiB  
Article
Hybrid CNC–MXene Nanolubricant for Tribological Application: Characterization, Prediction, and Optimization of Thermophysical Properties Evaluation
by Sakinah Muhamad Hisham, Norazlianie Sazali, Kumaran Kadirgama, Devarajan Ramasamy, Mohd Kamal Kamarulzaman, Lingenthiran Samylingam, Navid Aslfattahi and Chee Kuang Kok
Processes 2024, 12(10), 2146; https://doi.org/10.3390/pr12102146 - 2 Oct 2024
Viewed by 432
Abstract
In the present work, hybrid Cellulose Nanocrystal–MXene (CNC–MXene) nanolubricants were prepared via a two-step method and investigated as potential heat-transfer hybrid nanofluids for the first time. CNC–MXene nanolubricants were synthesized via a two-step method by varying the weight percentage of CNC–MXene nanoparticles (ranging [...] Read more.
In the present work, hybrid Cellulose Nanocrystal–MXene (CNC–MXene) nanolubricants were prepared via a two-step method and investigated as potential heat-transfer hybrid nanofluids for the first time. CNC–MXene nanolubricants were synthesized via a two-step method by varying the weight percentage of CNC–MXene nanoparticles (ranging from 0.01 to 0.05 wt%) and characterized using Fourier-Transform Infrared Spectroscopy and TGA (Thermogravimetric Analysis). Response surface methodology (RSM) was used in conjunction with the miscellaneous design model to identify prediction models for the thermophysical properties of the hybrid CNC–MXene nanolubricant. Minitab 18 statistical analysis software and Response Surface Methodology (RSM) based on Central Composite Design (CCD) were utilized to generate an empirical mathematical model investigating the effect of concentration and temperature. The analysis of variance (ANOVA) results indicated significant contributions from the type of nanolubricant (p < 0.001) and the quadratic effect of temperature (p < 0.001), highlighting non-linear interactions that affect viscosity and thermal conductivity. The findings showed that the predicted values closely matched the experimental results, with a percentage of absolute error below 9%, confirming the reliability of the optimization models. Additionally, the models could predict more than 85% of the nanolubricant output variations, indicating high model accuracy. The optimization analysis identified optimal conditions for maximizing both dynamic viscosity and thermal conductivity. The predicted optimal values (17.0685 for dynamic viscosity and 0.3317 for thermal conductivity) were achieved at 30 °C and a 0.01% concentration, with a composite desirability of 1. The findings of the percentage of absolute error (POAE) reveal that the model can precisely predict the optimum experimental parameters. This study contributes to the growing field of advanced nanolubricants by providing insights into the synergistic effects of CNC and MXene in enhancing thermophysical properties. The developed models and optimization techniques offer valuable tools for tailoring nanolubricant formulations to specific tribological applications, potentially leading to improved efficiency and durability in various industrial settings. Full article
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<p>FTIR spectrum for CNC–MXene at different concentration (0.01%, 0.03%, and 0.05%).</p>
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<p>FTIR graph analysis for all concentrations of different types of nanolubricant.</p>
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<p>TGA graph of CNC–MXene at different concentrations.</p>
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<p>TGA graph analysis for varying concentrations (0.01%, 0.03%, and 0.005%) of CNC, MXene, MXene–CNC nanolubricants.</p>
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<p>Surface plot for thermal conductivity.</p>
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<p>Contour plot for thermal conductivity.</p>
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<p>Surface plot for dynamic viscosity.</p>
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<p>Contour plot for dynamic viscosity.</p>
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<p>Comparison of the experimental and predicted model for dynamic viscosity.</p>
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<p>Comparison of the experimental and predicted model of the thermal conductivity.</p>
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<p>Optimization plot.</p>
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18 pages, 1287 KiB  
Article
Performance Assessment of Flat Plate Solar Collector Using Simple and Hybrid Carbon Nanofluids at Low Thermal Capacity
by José Michael Cruz, Sandra Angélica Crepaldi, Geydy Luz Gutiérrez-Urueta, José de Jesús Rubio, Alejandro Zacarías, Cuauhtémoc Jiménez, Guerlin Romage, José Alfredo Jiménez, Abel López and Ricardo Balcazar
Appl. Sci. 2024, 14(19), 8732; https://doi.org/10.3390/app14198732 - 27 Sep 2024
Viewed by 544
Abstract
Installation of flat solar collectors (FSCs) has been increasing due to the zero cost of renewable energy. However, the performance of this equipment is limited by the area, the material and the thermophysical properties of the working fluid. To improve the properties of [...] Read more.
Installation of flat solar collectors (FSCs) has been increasing due to the zero cost of renewable energy. However, the performance of this equipment is limited by the area, the material and the thermophysical properties of the working fluid. To improve the properties of the fluid, metal and metal oxide nanoparticles have mainly been used. This paper presents the performance assessment of the FSCs using simple and hybrid carbon nanofluids of low thermal capacity. Energy and mass balance modeling was performed for this study. A parametric analysis was conducted to examine the impact of key variables on the performance of the solar collectors using simple graphite and fullerene nanofluids, as well as hybrid metal–oxide–carbon nanofluids. From the results of heat transfer in FSCs, using graphite and fullerene nanofluids, it can be concluded that adding these nanoparticles improves the convection coefficient by 40% and 30%, respectively, with 10% nanoparticles. The graphite and fullerene nanoparticles can enhance the efficiency of FSCs by 2% and 1.5% more than base fluid. As the decrease in efficiency using fullerene with magnesium oxide is less than 0.2%, fullerene hybrid nanofluids could still be used in FSCs. Full article
(This article belongs to the Section Energy Science and Technology)
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<p>Physical model of the system: (<b>a</b>) FSC system; (<b>b</b>) sectional view of FSC.</p>
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<p>Physical model of the system: (<b>a</b>) FSC cross-section view; (<b>b</b>) 3D thermodynamic system of FSC.</p>
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<p>Flowchart followed in the simulation.</p>
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<p>Useful heat and efficiency of the FSC with respect to the mass flow of the base fluid.</p>
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<p>Density, viscosity and specific heat capacity as a function of volume fraction for the graphite and fullerene nanofluids.</p>
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<p>(<b>a</b>) TC and (<b>b</b>) thermal diffusivity as a function of volume fraction for the graphite and fullerene nanofluids.</p>
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<p>The nanofluid’s (<b>a</b>) Reynolds and (<b>b</b>) Prandtl number with respect to volume fraction for graphite and fullerene nanofluids.</p>
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<p>Nusselt number and internal convection heat transfer coefficient with respect to volume fraction for graphite and fullerene nanofluids.</p>
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<p>Useful heat with respect to the volume fraction for metal oxide and carbon nanofluids.</p>
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<p>Efficiency with respect to the volume fraction for metal oxide and carbon nanofluids.</p>
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<p>Useful heat of hybrid carbon nanofluids with respect to volume fraction.</p>
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<p>Efficiency of hybrid carbon nanofluids with respect to volume fraction.</p>
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22 pages, 9448 KiB  
Article
Numerical Analysis of Natural Convection in an Annular Cavity Filled with Hybrid Nanofluids under Magnetic Field
by Souad Benkherbache, Salah Amroune, Ahmed Belaadi, Said Zergane and Chouki Farsi
Energies 2024, 17(18), 4671; https://doi.org/10.3390/en17184671 - 19 Sep 2024
Viewed by 448
Abstract
This paper presents a numerical study of natural convection in an annular cavity filled with a hybrid nanofluid under the influence of a magnetic field. This study is significant for applications requiring enhanced thermal management, such as in heat exchangers, electronics cooling, and [...] Read more.
This paper presents a numerical study of natural convection in an annular cavity filled with a hybrid nanofluid under the influence of a magnetic field. This study is significant for applications requiring enhanced thermal management, such as in heat exchangers, electronics cooling, and energy systems. The inner cylinder, equipped with fins and subjected to uniform volumetric heat generation, contrasts with the adiabatic outer cylinder. This study aims to investigate how different nanoparticle combinations (Fe3O4 with Cu, Ag, and Al2O3) and varying Hartmann and Rayleigh numbers impact heat transfer efficiency. The finite volume method is employed to solve the governing equations, with simulations conducted using Fluent 6.3.26. Parameters such as volume fraction (ϕ2 = 0.001, 0.004, 0.006), Hartmann number (0 ≤ Ha ≤ 100), Rayleigh number (3 × 103 ≤ Ra ≤ 2.4 × 104), and fin number (N = 0, 2, 4, 6, 8) are analyzed. Streamlines, isotherms, and induced magnetic field contours are utilized to assess flow structure and heat transfer. The results reveal that increasing the Rayleigh number and magnetic field enhances heat transfer, while the presence of fins, especially at N = 2, may inhibit convection currents and reduce heat transfer efficiency. These findings provide valuable insights into optimizing nanofluid-based cooling systems and highlight the trade-offs in incorporating fins in thermal management designs. Full article
(This article belongs to the Section J: Thermal Management)
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<p>Schematic of the annular cavity.</p>
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<p>Mesh of the annulus: (<b>a</b>) annulus without fins, (<b>b</b>) annulus with N = 2 fins, (<b>c</b>) annulus with N = 4 fins.</p>
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<p>Comparison of the isotherms between the present study and the experimental study carried out by Grigull and Hauf [<a href="#B42-energies-17-04671" class="html-bibr">42</a>].</p>
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<p>Comparison of the Nusselt number between the present study and results obtained by Santra et al. [<a href="#B43-energies-17-04671" class="html-bibr">43</a>].</p>
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<p>Variation of the stream function (m<sup>2</sup>/s), Temperature (K) of velocities (m/s) with Hartmann number for Ag nanoparticles, ϕ<sub>2</sub> = 0.004.</p>
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<p>Variation of the stream function (m<sup>2</sup>/s), Temperature (K) of velocities (m/s) with Hartmann number for Cu nanoparticles, ϕ<sub>2</sub> = 0.004.</p>
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<p>Variation of the stream function (m<sup>2</sup>/s), Temperature (K) of velocities (m/s) with Hartmann number for Al<sub>2</sub>O<sub>3</sub> nanoparticles, ϕ<sub>2</sub> = 0.004.</p>
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<p>Variation of the Temperature (K) in the center line of the cavity with Hartmann number for, ϕ<sub>2</sub> = 0.004, (<b>a</b>) Fe<sub>3</sub>O<sub>4</sub>-Cu, (<b>b</b>) Fe<sub>3</sub>O<sub>4</sub>-Ag, (<b>c</b>) Fe<sub>3</sub>O<sub>4</sub>-Al<sub>2</sub>O<sub>3</sub>.</p>
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<p>Variation of the induced magnetic field B (T) with the Hartmann number for the mixture Fe<sub>3</sub>O<sub>4</sub>-Cu, ϕ<sub>2</sub> = 0.004.</p>
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<p>Variation of the induced magnetic field B (T) with the Hartmann number for the mixture Fe<sub>3</sub>O<sub>4</sub>-Ag, ϕ<sub>2</sub> = 0.004.</p>
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<p>Variation of the induced magnetic field B (T) with the Hartmann number for the mixture Fe<sub>3</sub>O<sub>4</sub>-Al<sub>2</sub>O<sub>3</sub>, ϕ<sub>2</sub> = 0.004.</p>
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<p>Variation of the profiles of induced magnetic field B (T) in the center line of the cavity at different Hartmann number for ϕ<sub>2</sub> = 0.004 and for the three mixtures: (<b>a</b>) Fe<sub>3</sub>O<sub>4</sub>-Cu, (<b>b</b>) Fe<sub>3</sub>O<sub>4</sub>-Al<sub>2</sub>O<sub>3</sub>, (<b>c</b>) Fe<sub>3</sub>O<sub>4</sub>-Ag.</p>
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<p>Variation in the Nusselt number with the Hartmann number and nanoparticles volume fraction (<b>a</b>) ϕ<sub>2</sub> = 0.001, (<b>b</b>) ϕ<sub>2</sub> = 0.004, (<b>c</b>) ϕ<sub>2</sub> = 0.006.</p>
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<p>Variation of the thermal conductivity ratio with Ha, for ϕ<sub>2</sub> = 0.001, 0.004 and 0.006.</p>
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<p>Variation of the average Nusselt number Nu<sub>avg</sub> with the nanoparticles concentration ϕ<sub>2</sub>.</p>
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<p>Variation of the average Nusselt number Nu<sub>avg</sub> with the Rayleigh number for N = 2, ϕ<sub>2</sub> = 0.004 and Ha = 35.</p>
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<p>Effect of the fin number on temperature, velocity and magnetic field Ra = 1.2 × 10<sup>4</sup>, Fe<sub>3</sub>O<sub>4</sub>-Cu nanohybrid at ϕ<sub>2</sub> = 0.004, Ha = 55.</p>
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13 pages, 1792 KiB  
Article
Parametric Optimization of Entropy Generation in Hybrid Nanofluid in Contracting/Expanding Channel by Means of Analysis of Variance and Response Surface Methodology
by Ahmad Zeeshan, Rahmat Ellahi, Muhammad Anas Rafique, Sadiq M. Sait and Nasir Shehzad
Inventions 2024, 9(5), 92; https://doi.org/10.3390/inventions9050092 - 27 Aug 2024
Cited by 1 | Viewed by 533
Abstract
This study aims to propose a central composite design (CCD) combined with response surface methodology (RSM) to create a statistical experimental design. A new parametric optimization of entropy generation is presented. The flow behavior of magnetohydrodynamic hybrid nanofluid (HNF) flow through two flat [...] Read more.
This study aims to propose a central composite design (CCD) combined with response surface methodology (RSM) to create a statistical experimental design. A new parametric optimization of entropy generation is presented. The flow behavior of magnetohydrodynamic hybrid nanofluid (HNF) flow through two flat contracting expanding plates of channel alongside radiative heat transmission was considered. The lower fixed plate was externally heated whereas the upper porous plate was cooled by injecting a coolant fluid with a uniform velocity inside the channel. The resulting equations were solved by the Homotopic Analysis Method using MATHEMATICA 10 and Minitab 17.1. The design consists of several input factors, namely a magnetic field parameter (M), radiation parameter (N) and group parameter (Br/A1). To obtain the values of flow response parameters, numerical experiments were used. Variables, especially the entropy generation (Ne), were considered for each combination of design. The resulting RSM empirical model obtained a high coefficient of determination, reaching 99.97% for the entropy generation number (Ne). These values show an excellent fit of the model to the data. Full article
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<p>Flow geometry of the problem.</p>
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<p>Normal probability graph of the total energy of <math display="inline"><semantics> <mrow> <mi mathvariant="normal">N</mi> <mi mathvariant="normal">e</mi> </mrow> </semantics></math>.</p>
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<p>Histogram of the empirical correlation of <math display="inline"><semantics> <mrow> <mi mathvariant="normal">N</mi> <mi mathvariant="normal">e</mi> </mrow> </semantics></math>.</p>
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<p>Observation order against standardized residual plot of <math display="inline"><semantics> <mrow> <mi mathvariant="normal">N</mi> <mi mathvariant="normal">e</mi> </mrow> </semantics></math>.</p>
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<p>Sensitivity analysis at <math display="inline"><semantics> <mrow> <mi mathvariant="normal">C</mi> </mrow> </semantics></math> at the central level and <math display="inline"><semantics> <mrow> <mi mathvariant="normal">A</mi> </mrow> </semantics></math> at the lower level for <math display="inline"><semantics> <mrow> <mi mathvariant="normal">N</mi> <mi mathvariant="normal">e</mi> </mrow> </semantics></math>.</p>
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<p>Sensitivity analysis at <math display="inline"><semantics> <mrow> <mi mathvariant="normal">C</mi> </mrow> </semantics></math> at the central level and <math display="inline"><semantics> <mrow> <mi mathvariant="normal">A</mi> </mrow> </semantics></math> at the central level for <math display="inline"><semantics> <mrow> <mi mathvariant="normal">N</mi> <mi mathvariant="normal">e</mi> </mrow> </semantics></math>.</p>
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<p>Sensitivity analysis at <math display="inline"><semantics> <mrow> <mi mathvariant="normal">C</mi> </mrow> </semantics></math> at the central level and <math display="inline"><semantics> <mrow> <mi mathvariant="normal">A</mi> </mrow> </semantics></math> at the upper level for <math display="inline"><semantics> <mrow> <mi mathvariant="normal">N</mi> <mi mathvariant="normal">e</mi> </mrow> </semantics></math>.</p>
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20 pages, 9374 KiB  
Article
A Numerical Investigation of Activation Energy Impact on MHD Water-Based Fe3O4 and CoFe2O4 Flow between the Rotating Cone and Expanding Disc
by Kandavkovi Mallikarjuna Nihaal, Ulavathi Shettar Mahabaleshwar, Nedunchezhian Swaminathan, David Laroze and Igor V. Shevchuk
Mathematics 2024, 12(16), 2530; https://doi.org/10.3390/math12162530 - 16 Aug 2024
Viewed by 593
Abstract
Hybrid nanofluids have caught the attention of scholars and investigators in the present technological period due to their improved thermophysical features and the desire to boost heat transfer rates compared to those of conventional fluids. The present paper is mainly concerned with heat [...] Read more.
Hybrid nanofluids have caught the attention of scholars and investigators in the present technological period due to their improved thermophysical features and the desire to boost heat transfer rates compared to those of conventional fluids. The present paper is mainly concerned with heat transmission in cone-disk geometry in the presence of a magnetic field, activation energy, and non-uniform heat absorption/generation. In this work, the cone-disk (CD) apparatus is considered to have a rotating cone (RC) and a stretching disk, along with iron oxide and cobalt ferrite-based hybrid nanofluid. Appropriate similarity transformations are employed to change the physically modeled equations into ordinary differential equations (ODEs). Heat transfer rates at both surfaces are estimated by implementing a modified energy equation with non-uniform heat absorption/generation. The outcomes illustrated that the inclusion of such physical streamwise heat conduction variables in the energy equation has a significant impact on the well-known conclusions of heat transfer rates. To understand flow profile behavior, we have resorted to the RKF-45 method and the shooting method, which are illustrated using graphs. The findings provide conclusive evidence that wall stretching alters the flow, heat, and mass profile characteristics within the conical gap. The wall deformation caused by disk stretching was found to have a potential impact of modifying the centripetal/centrifugal flow characteristics of the disk, increasing the flow velocity and swirling angles. A rise in activation energy leads to an improved concentration field. Full article
(This article belongs to the Special Issue Computational Fluid Dynamics with Applications)
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<p>Schematic diagram.</p>
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<p>Flow chart of RKF-45.</p>
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<p>Effect of <math display="inline"><semantics> <mi>M</mi> </semantics></math> over <math display="inline"><semantics> <mi>f</mi> </semantics></math> (Blue—0.5, Green—1.0, Orange—1.5, Red—2.0).</p>
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<p>Effect of <math display="inline"><semantics> <mi>M</mi> </semantics></math> over <math display="inline"><semantics> <mi>g</mi> </semantics></math> (Blue—0.5, Green—1.0, Orange—1.5, Red—2.0).</p>
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<p>Effect of <math display="inline"><semantics> <mi>M</mi> </semantics></math> over <math display="inline"><semantics> <mi>θ</mi> </semantics></math> (Blue—0.5, Green—1.0, Orange—1.5, Red—2.0).</p>
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<p>Impact of <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>Re</mi> </mrow> <mo>Ω</mo> </msub> </mrow> </semantics></math> on <math display="inline"><semantics> <mi>f</mi> </semantics></math> (Blue—1, Green—3, Orange—5, Red—7).</p>
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<p>Impact of <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>Re</mi> </mrow> <mo>Ω</mo> </msub> </mrow> </semantics></math> on <math display="inline"><semantics> <mi>g</mi> </semantics></math> (Blue—1, Green—3, Orange—5, Red—7).</p>
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<p>Impact of <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>Re</mi> </mrow> <mo>Ω</mo> </msub> </mrow> </semantics></math> on <math display="inline"><semantics> <mi>θ</mi> </semantics></math> (Blue—1, Green—3, Orange—5, Red—7).</p>
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<p>Influence of <math display="inline"><semantics> <mrow> <msup> <mi>A</mi> <mo>∗</mo> </msup> </mrow> </semantics></math> over <math display="inline"><semantics> <mi>θ</mi> </semantics></math> (Blue—0.3, Green—0.5, Orange—0.7, Red—0.9).</p>
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<p>Influence of <math display="inline"><semantics> <mrow> <msup> <mi>B</mi> <mo>∗</mo> </msup> </mrow> </semantics></math> over <math display="inline"><semantics> <mi>θ</mi> </semantics></math> (Blue—0.3, Green—0.7, Orange—0.9, Red—1.1).</p>
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<p>Fluctuation of <math display="inline"><semantics> <mi>f</mi> </semantics></math> for distinct values of <math display="inline"><semantics> <mrow> <msup> <mi>S</mi> <mo>∗</mo> </msup> </mrow> </semantics></math> (Blue—1, Green—3, Orange—5, Red—7).</p>
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<p>Fluctuations of <math display="inline"><semantics> <mi>g</mi> </semantics></math> for distinct values of <math display="inline"><semantics> <mrow> <msup> <mi>S</mi> <mo>∗</mo> </msup> </mrow> </semantics></math> (Blue—1, Green—3, Orange—5, Red—7).</p>
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<p>Fluctuations of <math display="inline"><semantics> <mi>θ</mi> </semantics></math> for distinct values of <math display="inline"><semantics> <mrow> <msup> <mi>S</mi> <mo>∗</mo> </msup> </mrow> </semantics></math> (Blue—1, Green—3, Orange—5, Red—7).</p>
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<p>Fluctuations of <math display="inline"><semantics> <mi>χ</mi> </semantics></math> for distinct values of <math display="inline"><semantics> <mrow> <msup> <mi>S</mi> <mo>∗</mo> </msup> </mrow> </semantics></math> (Blue—1, Green—3, Orange—5, Red—7).</p>
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<p>Impact on <math display="inline"><semantics> <mi>χ</mi> </semantics></math> (Blue—0.8, Green—1.4, Orange—2.0, Red—2.6).</p>
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<p>Impact on <math display="inline"><semantics> <mi>χ</mi> </semantics></math> (Blue—0.5, Green—1.3, Orange—2.1, Red—2.9).</p>
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<p>Solid volume fraction effect over <math display="inline"><semantics> <mi>f</mi> </semantics></math>.</p>
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<p>Solid volume fraction effect on <math display="inline"><semantics> <mi>θ</mi> </semantics></math>.</p>
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<p>Variations in <math display="inline"><semantics> <mrow> <mi>N</mi> <msub> <mi>u</mi> <mi>d</mi> </msub> </mrow> </semantics></math> for rising <math display="inline"><semantics> <mi>ϕ</mi> </semantics></math> and <math display="inline"><semantics> <mrow> <msup> <mi>A</mi> <mo>∗</mo> </msup> </mrow> </semantics></math> values.</p>
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<p>Variations in <math display="inline"><semantics> <mrow> <mi>N</mi> <msub> <mi>u</mi> <mi>c</mi> </msub> </mrow> </semantics></math> for rising <math display="inline"><semantics> <mi>ϕ</mi> </semantics></math> and <math display="inline"><semantics> <mrow> <msup> <mi>A</mi> <mo>∗</mo> </msup> </mrow> </semantics></math> values.</p>
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<p>Variations of <math display="inline"><semantics> <mrow> <mi>N</mi> <msub> <mi>u</mi> <mi>d</mi> </msub> </mrow> </semantics></math> for rising <math display="inline"><semantics> <mi>Μ</mi> </semantics></math> and <math display="inline"><semantics> <mi>ϕ</mi> </semantics></math> values (Blue—1.5, Red—2.0, Orange—2.5).</p>
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<p>Variations of <math display="inline"><semantics> <mrow> <mi>N</mi> <msub> <mi>u</mi> <mi>c</mi> </msub> </mrow> </semantics></math> for rising <math display="inline"><semantics> <mi>M</mi> </semantics></math> and <math display="inline"><semantics> <mi>ϕ</mi> </semantics></math> values (Blue—1.5, Red—2.0, Orange—2.5).</p>
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38 pages, 11862 KiB  
Article
A COMSOL-Based Numerical Simulation of Heat Transfer in a Hybrid Nanofluid Flow at the Stagnant Point across a Stretching/Shrinking Sheet: Implementation for Understanding and Improving Solar Systems
by Ahmad Ayyad Alharbi and Ali Rashash R. Alzahrani
Mathematics 2024, 12(16), 2493; https://doi.org/10.3390/math12162493 - 13 Aug 2024
Viewed by 723
Abstract
The present study investigates hybrid nanofluid (HNF) behavior at the stagnation point near a stretching/shrinking sheet using the Tiwari and Das model. The governing equations were transformed into a boundary layer flow model and simulated using COMSOL Multiphysics 6.0. This research examines flow [...] Read more.
The present study investigates hybrid nanofluid (HNF) behavior at the stagnation point near a stretching/shrinking sheet using the Tiwari and Das model. The governing equations were transformed into a boundary layer flow model and simulated using COMSOL Multiphysics 6.0. This research examines flow characteristics, temperature profiles, and distributions by varying parameters: stretching/shrinking (λ, −2 to 2), slip flow (δ, 0 to 1 m), suction (γ, 0 to 1), and similarity variables (η, 0 to 5). The HNF comprised equal ratios of copper and alumina with total concentrations ranging from 0.01 to 0.1. The results showed that velocity profiles increased with distance from the stagnation point, escalated in shrinking cases, and decayed in stretching cases. Increased suction consistently reduced velocity profiles. Temperature distribution was slightly slower in shrinking compared to stretching cases, with expansion along the sheet directly proportional to η estimates but controllable through suction adjustments. The findings were applied to enhance photovoltaic thermal (PV/T) system performance. Stretching sheets proved crucial for improving electricity production efficiency. Non-slip wall conditions and increased copper volume fractions in the presence of suction effects led to notable improvements in electrical efficiency. The maximum average efficiency was achieved when γ = 0.4, λ = 2, δ = 0.7, and ϕ2 = 0.01, which was of about 10%. The present numerical work also aligned well with the experimental results when evaluating the thermal efficiency of conventional fluids. These insights contribute to optimizing PV/T system parameters and advancing solar energy conversion technology, with potential implications for broader applications in the field. Full article
(This article belongs to the Section Computational and Applied Mathematics)
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Figure 1

Figure 1
<p>Schematic diagram of the flow channel (<b>a</b>) stretching sheet and (<b>b</b>) shrinking sheet [<a href="#B46-mathematics-12-02493" class="html-bibr">46</a>].</p>
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<p>Schematic diagram of the flow channel (<b>a</b>) stretching sheet and (<b>b</b>) shrinking sheet [<a href="#B46-mathematics-12-02493" class="html-bibr">46</a>].</p>
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<p>A sketch of the geometry while it was processed in COMSOL Multiphysics 6.0.</p>
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<p>Schematic diagram of the finite element meshing procedure.</p>
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<p>The mesh independence study for Equation (8) by fixing the estimations <math display="inline"><semantics> <mi>γ</mi> </semantics></math> = 1, <math display="inline"><semantics> <mi>λ</mi> </semantics></math> = 2, <math display="inline"><semantics> <mrow> <msub> <mi>ϕ</mi> <mn>2</mn> </msub> </mrow> </semantics></math> = 0.1, and <math display="inline"><semantics> <mi>δ</mi> </semantics></math> = 1.</p>
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<p>A comparison of the present numerical work with the previous work performed by Yashkun et al. [<a href="#B46-mathematics-12-02493" class="html-bibr">46</a>] when <math display="inline"><semantics> <mrow> <msub> <mi>ϕ</mi> <mn>2</mn> </msub> </mrow> </semantics></math> = 0.1, <math display="inline"><semantics> <mrow> <msub> <mi>ϕ</mi> <mn>2</mn> </msub> </mrow> </semantics></math> = 0, <math display="inline"><semantics> <mi>λ</mi> </semantics></math> = −2, and <math display="inline"><semantics> <mi>δ</mi> </semantics></math> = 1.</p>
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<p>Velocity outlines for different estimations of <math display="inline"><semantics> <mi>λ</mi> </semantics></math> when <math display="inline"><semantics> <mrow> <msub> <mi>ϕ</mi> <mn>2</mn> </msub> </mrow> </semantics></math> = 0.1, <math display="inline"><semantics> <mi>δ</mi> </semantics></math> = 0, (<b>a</b>) <math display="inline"><semantics> <mi>γ</mi> </semantics></math> = 0, and (<b>b</b>) <math display="inline"><semantics> <mi>γ</mi> </semantics></math> = 1.</p>
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<p>Velocity outlines for all estimations of <math display="inline"><semantics> <mi>δ</mi> </semantics></math>, when <math display="inline"><semantics> <mrow> <msub> <mi>ϕ</mi> <mn>2</mn> </msub> </mrow> </semantics></math> = 0.1, <math display="inline"><semantics> <mi>γ</mi> </semantics></math> = 0, (<b>a</b>) <math display="inline"><semantics> <mi>λ</mi> </semantics></math> = −2, (<b>b</b>) <math display="inline"><semantics> <mi>λ</mi> </semantics></math> = 0, and (<b>c</b>) <math display="inline"><semantics> <mi>λ</mi> </semantics></math> = 2.</p>
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<p>Velocity outlines for all estimations of <math display="inline"><semantics> <mi>δ</mi> </semantics></math>, when <math display="inline"><semantics> <mrow> <msub> <mi>ϕ</mi> <mn>2</mn> </msub> </mrow> </semantics></math> = 0.1, <math display="inline"><semantics> <mi>γ</mi> </semantics></math> = 0, (<b>a</b>) <math display="inline"><semantics> <mi>λ</mi> </semantics></math> = −2, (<b>b</b>) <math display="inline"><semantics> <mi>λ</mi> </semantics></math> = 0, and (<b>c</b>) <math display="inline"><semantics> <mi>λ</mi> </semantics></math> = 2.</p>
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<p>Velocity outlines for all estimations of <math display="inline"><semantics> <mrow> <msub> <mi>ϕ</mi> <mn>2</mn> </msub> </mrow> </semantics></math> when <math display="inline"><semantics> <mi>δ</mi> </semantics></math> = 1, <math display="inline"><semantics> <mi>γ</mi> </semantics></math> = 0, (<b>a</b>) <math display="inline"><semantics> <mi>λ</mi> </semantics></math> = −2, (<b>b</b>) <math display="inline"><semantics> <mi>λ</mi> </semantics></math> = 0, and (<b>c</b>) <math display="inline"><semantics> <mi>λ</mi> </semantics></math> = 2.</p>
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<p>Velocity outlines for all estimations of <math display="inline"><semantics> <mrow> <msub> <mi>ϕ</mi> <mn>2</mn> </msub> </mrow> </semantics></math> when <math display="inline"><semantics> <mi>δ</mi> </semantics></math> = 1, <math display="inline"><semantics> <mi>γ</mi> </semantics></math> = 0, (<b>a</b>) <math display="inline"><semantics> <mi>λ</mi> </semantics></math> = −2, (<b>b</b>) <math display="inline"><semantics> <mi>λ</mi> </semantics></math> = 0, and (<b>c</b>) <math display="inline"><semantics> <mi>λ</mi> </semantics></math> = 2.</p>
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<p>Temperature outlines for all estimations of <math display="inline"><semantics> <mi>λ</mi> </semantics></math> when <math display="inline"><semantics> <mi>γ</mi> </semantics></math> = 0, <math display="inline"><semantics> <mrow> <msub> <mi>ϕ</mi> <mn>2</mn> </msub> </mrow> </semantics></math> = 0.1, (<b>a</b>) <math display="inline"><semantics> <mi>δ</mi> </semantics></math> = 0, (<b>b</b>) <math display="inline"><semantics> <mi>δ</mi> </semantics></math> = 0.7, and (<b>c</b>) <math display="inline"><semantics> <mi>δ</mi> </semantics></math> = 1.</p>
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<p>Temperature outlines against the <math display="inline"><semantics> <mi>η</mi> </semantics></math> for all estimations of <math display="inline"><semantics> <mi>δ</mi> </semantics></math> when <math display="inline"><semantics> <mi>γ</mi> </semantics></math> = 0, <math display="inline"><semantics> <mrow> <msub> <mi>ϕ</mi> <mn>2</mn> </msub> </mrow> </semantics></math> = 0.1, (<b>a</b>) <math display="inline"><semantics> <mi>λ</mi> </semantics></math> = −2, and (<b>b</b>) <math display="inline"><semantics> <mi>λ</mi> </semantics></math> = 2.</p>
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<p>Temperature outlines for all estimations of <math display="inline"><semantics> <mrow> <msub> <mi>ϕ</mi> <mn>2</mn> </msub> </mrow> </semantics></math> when <math display="inline"><semantics> <mi>δ</mi> </semantics></math> = 1, <math display="inline"><semantics> <mi>γ</mi> </semantics></math> = 0, (<b>a</b>) <math display="inline"><semantics> <mi>λ</mi> </semantics></math> = −2, (<b>b</b>) <math display="inline"><semantics> <mi>λ</mi> </semantics></math> = 0, and (<b>c</b>) <math display="inline"><semantics> <mi>λ</mi> </semantics></math> = 2.</p>
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<p>Temperature outlines for all estimations of <math display="inline"><semantics> <mrow> <msub> <mi>ϕ</mi> <mn>2</mn> </msub> </mrow> </semantics></math> when <math display="inline"><semantics> <mi>δ</mi> </semantics></math> = 1, <math display="inline"><semantics> <mi>γ</mi> </semantics></math> = 0, (<b>a</b>) <math display="inline"><semantics> <mi>λ</mi> </semantics></math> = −2, (<b>b</b>) <math display="inline"><semantics> <mi>λ</mi> </semantics></math> = 0, and (<b>c</b>) <math display="inline"><semantics> <mi>λ</mi> </semantics></math> = 2.</p>
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<p>Temperature distribution vs. <math display="inline"><semantics> <mi>η</mi> </semantics></math> at different estimations <math display="inline"><semantics> <mi>λ</mi> </semantics></math>, <math display="inline"><semantics> <mi>δ</mi> </semantics></math> = 1, <math display="inline"><semantics> <mrow> <msub> <mi>ϕ</mi> <mn>2</mn> </msub> </mrow> </semantics></math> = 0.1, (<b>a</b>) <math display="inline"><semantics> <mi>γ</mi> </semantics></math> = 0, and (<b>b</b>) <math display="inline"><semantics> <mi>γ</mi> </semantics></math> = 1.</p>
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<p>Temperature distribution vs. <math display="inline"><semantics> <mi>η</mi> </semantics></math> for all estimations of <math display="inline"><semantics> <mi>δ</mi> </semantics></math> when <math display="inline"><semantics> <mi>γ</mi> </semantics></math> = 0, <math display="inline"><semantics> <mrow> <msub> <mi>ϕ</mi> <mn>2</mn> </msub> </mrow> </semantics></math> = 0.1, (<b>a</b>) <math display="inline"><semantics> <mi>λ</mi> </semantics></math> =−2, (<b>b</b>) <math display="inline"><semantics> <mi>λ</mi> </semantics></math> = 0, and (<b>c</b>) <math display="inline"><semantics> <mi>λ</mi> </semantics></math> = 2.</p>
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<p>Temperature distribution vs. <math display="inline"><semantics> <mi>η</mi> </semantics></math> for all estimations of <math display="inline"><semantics> <mi>δ</mi> </semantics></math> when <math display="inline"><semantics> <mi>γ</mi> </semantics></math> = 0, <math display="inline"><semantics> <mrow> <msub> <mi>ϕ</mi> <mn>2</mn> </msub> </mrow> </semantics></math> = 0.1, (<b>a</b>) <math display="inline"><semantics> <mi>λ</mi> </semantics></math> =−2, (<b>b</b>) <math display="inline"><semantics> <mi>λ</mi> </semantics></math> = 0, and (<b>c</b>) <math display="inline"><semantics> <mi>λ</mi> </semantics></math> = 2.</p>
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<p>Temperature distribution vs. <math display="inline"><semantics> <mi>η</mi> </semantics></math> for all <math display="inline"><semantics> <mrow> <msub> <mi>ϕ</mi> <mn>2</mn> </msub> </mrow> </semantics></math> when <math display="inline"><semantics> <mi>δ</mi> </semantics></math> = 1, <math display="inline"><semantics> <mi>γ</mi> </semantics></math> = 1, (<b>a</b>) <math display="inline"><semantics> <mi>λ</mi> </semantics></math> = −2, (<b>b</b>) <math display="inline"><semantics> <mi>λ</mi> </semantics></math> = 0, and (<b>c</b>) <math display="inline"><semantics> <mi>λ</mi> </semantics></math> = 2.</p>
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<p>Temperature distribution vs. <math display="inline"><semantics> <mi>η</mi> </semantics></math> for all <math display="inline"><semantics> <mrow> <msub> <mi>ϕ</mi> <mn>2</mn> </msub> </mrow> </semantics></math> when <math display="inline"><semantics> <mi>δ</mi> </semantics></math> = 1, <math display="inline"><semantics> <mi>γ</mi> </semantics></math> = 1, (<b>a</b>) <math display="inline"><semantics> <mi>λ</mi> </semantics></math> = −2, (<b>b</b>) <math display="inline"><semantics> <mi>λ</mi> </semantics></math> = 0, and (<b>c</b>) <math display="inline"><semantics> <mi>λ</mi> </semantics></math> = 2.</p>
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<p>The comparison with [<a href="#B48-mathematics-12-02493" class="html-bibr">48</a>] of the percentage change in the temperature due to ambient temperature.</p>
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<p>The pattern of electrical efficiency of PV/T vs. <math display="inline"><semantics> <mi>η</mi> </semantics></math> for all estimations of stretching and shrining parameters.</p>
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<p>The pattern of electrical efficiency of PV/T vs. <math display="inline"><semantics> <mi>η</mi> </semantics></math> for all estimations of stretching and shrining parameters.</p>
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<p>The pattern of electrical efficiency vs. <math display="inline"><semantics> <mi>η</mi> </semantics></math> for all estimations of the suction parameter <math display="inline"><semantics> <mi>γ</mi> </semantics></math>.</p>
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<p>The pattern of electrical efficiency vs. <math display="inline"><semantics> <mi>η</mi> </semantics></math> for all estimations of slip flow parameters.</p>
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<p>The pattern of electrical efficiency vs. <math display="inline"><semantics> <mi>η</mi> </semantics></math> for all estimations of slip flow parameters.</p>
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<p>The pattern of electrical efficiency vs. <math display="inline"><semantics> <mi>η</mi> </semantics></math> for all estimations of the volume fraction of copper.</p>
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<p>The pattern of electrical efficiency vs. <math display="inline"><semantics> <mi>η</mi> </semantics></math> for all estimations of the volume fraction of copper.</p>
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<p>A comparison of the thermal efficiency of the PV/T system with the experimental work [<a href="#B53-mathematics-12-02493" class="html-bibr">53</a>].</p>
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19 pages, 5575 KiB  
Article
Impact of Navier’s Slip and MHD on a Hybrid Nanofluid Flow over a Porous Stretching/Shrinking Sheet with Heat Transfer
by Thippaiah Maranna, Gadhigeppa Myacher Sachin, Ulavathi Shettar Mahabaleshwar, Laura M. Pérez and Igor V. Shevchuk
Fluids 2024, 9(8), 180; https://doi.org/10.3390/fluids9080180 - 10 Aug 2024
Cited by 1 | Viewed by 866
Abstract
The main objective of this study is to explore the inventive conception of the magnetohydrodynamic flow of a hybrid nanofluid over-porous stretching/shrinking sheet with the effect of radiation and mass suction/injection. The hybrid nanofluid advances both the manufactured nanofluid of the current region [...] Read more.
The main objective of this study is to explore the inventive conception of the magnetohydrodynamic flow of a hybrid nanofluid over-porous stretching/shrinking sheet with the effect of radiation and mass suction/injection. The hybrid nanofluid advances both the manufactured nanofluid of the current region and the base fluid. For the current investigation, hybrid nanofluids comprising two different kinds of nanoparticles, aluminium oxide and ferrofluid, contained in water as a base fluid, are considered. A collection of highly nonlinear partial differential equations is used to model the whole physical problem. These equations are then transformed into highly nonlinear ordinary differential equations using an appropriate similarity technique. The transformed differential equations are nonlinear, and thus it is difficult to analytically solve considering temperature increases. Then, the outcome is described in incomplete gamma function form. The considered physical parameters namely, magnetic field, Inverse Darcy number, velocity slip, suction/injection, temperature jump effects on velocity, temperature, skin friction and Nusselt number profiles are reviewed using plots. The results reveal that magnetic field, and Inverse Darcy number values increase as the momentum boundary layer decreases. Moreover, higher values of heat sources and thermal radiation enhance the thermal boundary layer. The present problem has various applications in manufacturing and technological devices such as cooling systems, condensers, microelectronics, digital cooling, car radiators, nuclear power stations, nano-drag shipments, automobile production, and tumour treatments. Full article
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<p>Diagrammatic representation of fluid flow.</p>
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<p>The impact of velocity slip parameter on skin friction.</p>
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<p>The impact of magnetic field on skin friction.</p>
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<p>The impact of mass suction/injection on the radial velocity profile.</p>
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<p>The impact of mass suction/injection on the velocity profile.</p>
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<p>An impact of magnetic field on velocity profile.</p>
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<p>An impact of inverse Darcy number on velocity profile.</p>
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<p>The impact of velocity slip on velocity profile.</p>
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<p>The impact of mass suction/injection on temperature distribution.</p>
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<p>The impact of magnetic field on temperature distribution.</p>
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<p>The impact of inverse Darcy number on temperature distribution.</p>
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<p>The impact of velocity slip parameter on temperature distribution.</p>
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<p>The impact of temperature increase on temperature distribution.</p>
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<p>The impact of radiation on temperature distribution.</p>
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<p>The impact of magnetic field on Nusselt number.</p>
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<p>The impact of velocity slip on Nusselt number.</p>
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16 pages, 1038 KiB  
Article
Examining Hybrid Nanofluid Flow Dynamics in the Conical Gap between a Rotating Disk and Cone Surface: An Artificial Neural Network Approach
by Julien Moussa H. Barakat, Zaher Al Barakeh and Raymond Ghandour
Appl. Syst. Innov. 2024, 7(4), 63; https://doi.org/10.3390/asi7040063 - 22 Jul 2024
Cited by 1 | Viewed by 1033
Abstract
To comprehend the thermal regulation within the conical gap between a disk and a cone (TRHNF-DC) for hybrid nanofluid flow, this research introduces a novel application of computationally intelligent heuristics utilizing backpropagated Levenberg–Marquardt neural networks (LM-NNs). A unique hybrid nanoliquid comprising aluminum oxide, [...] Read more.
To comprehend the thermal regulation within the conical gap between a disk and a cone (TRHNF-DC) for hybrid nanofluid flow, this research introduces a novel application of computationally intelligent heuristics utilizing backpropagated Levenberg–Marquardt neural networks (LM-NNs). A unique hybrid nanoliquid comprising aluminum oxide, Al2O3, nanoparticles and copper, Cu, nanoparticles is specifically addressed. Through the application of similarity transformations, the mathematical model formulated in terms of partial differential equations (PDEs) is converted into ordinary differential equations (ODEs). The BVP4C method is employed to generate a dataset encompassing various TRHNF-DC scenarios by varying magnetic parameters and nanoparticles. Subsequently, the intelligent LM-NN solver is trained, tested, and validated to ascertain the TRHNF-DC solution under diverse conditions. The accuracy of the LM-NN approach in solving the TRHNF-DC model is verified through different analyses, demonstrating a high level of accuracy, with discrepancies ranging from 1010 to 108 when compared with standard solutions. The efficacy of the framework is further underscored by the close agreement of recommended outcomes with reference solutions, thereby validating its integrity. Full article
(This article belongs to the Section Artificial Intelligence)
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<p>Rotating disk and cone.</p>
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<p>Neural network diagram for TRHNF-DC.</p>
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<p>Plots for solving <span class="html-italic">M</span> for <math display="inline"><semantics> <mrow> <msup> <mi>f</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mi>ζ</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>.</p>
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<p>(<b>a</b>,<b>c</b>): Plots of <math display="inline"><semantics> <mrow> <mi>R</mi> <msub> <mi>e</mi> <mi>ω</mi> </msub> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>R</mi> <msub> <mi>e</mi> <mo>Ω</mo> </msub> </mrow> </semantics></math> for <math display="inline"><semantics> <mrow> <mi>G</mi> <mo>(</mo> <mi>ζ</mi> <mo>)</mo> </mrow> </semantics></math>. (<b>b</b>,<b>d</b>): Plots for solving <span class="html-italic">M</span> for <math display="inline"><semantics> <mrow> <mi>θ</mi> <mo>(</mo> <mi>ζ</mi> <mo>)</mo> </mrow> </semantics></math>.</p>
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<p>(<b>a</b>,<b>c</b>): Plots of <math display="inline"><semantics> <mrow> <mi>R</mi> <msub> <mi>e</mi> <mi>ω</mi> </msub> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>R</mi> <msub> <mi>e</mi> <mo>Ω</mo> </msub> </mrow> </semantics></math> for <math display="inline"><semantics> <mrow> <mi>G</mi> <mo>(</mo> <mi>ζ</mi> <mo>)</mo> </mrow> </semantics></math>. (<b>b</b>,<b>d</b>): Plots for solving <span class="html-italic">M</span> for <math display="inline"><semantics> <mrow> <mi>θ</mi> <mo>(</mo> <mi>ζ</mi> <mo>)</mo> </mrow> </semantics></math>.</p>
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<p>(<b>Up-left</b>): Impact of <span class="html-italic">M</span> on <math display="inline"><semantics> <mrow> <mi>F</mi> <mo>(</mo> <mi>ζ</mi> <mo>)</mo> </mrow> </semantics></math>. (<b>Up-right:</b>) Impact of <math display="inline"><semantics> <msub> <mi>ϕ</mi> <mrow> <mi>A</mi> <msub> <mi>l</mi> <mn>2</mn> </msub> <msub> <mi>O</mi> <mn>3</mn> </msub> </mrow> </msub> </semantics></math> on <math display="inline"><semantics> <mrow> <mi>F</mi> <mo>(</mo> <mi>ζ</mi> <mo>)</mo> </mrow> </semantics></math>. (<b>Down-left</b>): Impact of <math display="inline"><semantics> <msub> <mi>ϕ</mi> <mrow> <mi>C</mi> <mi>u</mi> </mrow> </msub> </semantics></math> on <math display="inline"><semantics> <mrow> <mi>F</mi> <mo>(</mo> <mi>ζ</mi> <mo>)</mo> </mrow> </semantics></math>. (<b>Down-right</b>): Influence of <span class="html-italic">M</span> on <math display="inline"><semantics> <mrow> <mi>G</mi> <mo>(</mo> <mi>ζ</mi> <mo>)</mo> </mrow> </semantics></math>.</p>
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<p>(<b>Up-left</b>): Influence of <span class="html-italic">M</span> on <math display="inline"><semantics> <mrow> <mi>G</mi> <mo>(</mo> <mi>ζ</mi> <mo>)</mo> </mrow> </semantics></math>. (<b>Up-right</b>): Influence of <math display="inline"><semantics> <msub> <mi>ϕ</mi> <mrow> <mi>A</mi> <msub> <mi>l</mi> <mn>2</mn> </msub> <msub> <mi>O</mi> <mn>3</mn> </msub> </mrow> </msub> </semantics></math> on <math display="inline"><semantics> <mrow> <mi>G</mi> <mo>(</mo> <mi>ζ</mi> <mo>)</mo> </mrow> </semantics></math>. (<b>Middle-left</b>): Case (1) influence over <math display="inline"><semantics> <mrow> <mi>G</mi> <mo>(</mo> <mi>ζ</mi> <mo>)</mo> </mrow> </semantics></math>. (<b>Middle-right</b>): Case (2) influence over <math display="inline"><semantics> <mrow> <mi>G</mi> <mo>(</mo> <mi>ζ</mi> <mo>)</mo> </mrow> </semantics></math>. (<b>Down-left</b>): Case (3) influence on <math display="inline"><semantics> <mrow> <mi>G</mi> <mo>(</mo> <mi>ζ</mi> <mo>)</mo> </mrow> </semantics></math>. (<b>Down-right</b>): Case (3) influence on <math display="inline"><semantics> <mrow> <mi>G</mi> <mo>(</mo> <mi>ζ</mi> <mo>)</mo> </mrow> </semantics></math>.</p>
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<p>(<b>Up-left</b>): Influence of <span class="html-italic">M</span> on <math display="inline"><semantics> <mrow> <mi>θ</mi> <mo>(</mo> <mi>ζ</mi> <mo>)</mo> </mrow> </semantics></math>. (<b>Up-right</b>): Influence of <math display="inline"><semantics> <msub> <mi>ϕ</mi> <mrow> <mi>A</mi> <msub> <mi>l</mi> <mn>2</mn> </msub> <msub> <mi>O</mi> <mn>3</mn> </msub> </mrow> </msub> </semantics></math> on <math display="inline"><semantics> <mrow> <mi>θ</mi> <mo>(</mo> <mi>ζ</mi> <mo>)</mo> </mrow> </semantics></math>. (<b>Down-left</b>): Influence of <math display="inline"><semantics> <msub> <mi>ϕ</mi> <mrow> <mi>C</mi> <mi>u</mi> </mrow> </msub> </semantics></math> on <math display="inline"><semantics> <mrow> <mi>θ</mi> <mo>(</mo> <mi>ζ</mi> <mo>)</mo> </mrow> </semantics></math>. (<b>Down-right</b>): Influence of <math display="inline"><semantics> <mrow> <mi>P</mi> <mi>r</mi> </mrow> </semantics></math> on <math display="inline"><semantics> <mrow> <mi>θ</mi> <mo>(</mo> <mi>ζ</mi> <mo>)</mo> </mrow> </semantics></math>.</p>
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14 pages, 2795 KiB  
Article
Hybrid Nanofluid Flow over a Shrinking Rotating Disk: Response Surface Methodology
by Rusya Iryanti Yahaya, Norihan Md Arifin, Ioan Pop, Fadzilah Md Ali and Siti Suzilliana Putri Mohamed Isa
Computation 2024, 12(7), 141; https://doi.org/10.3390/computation12070141 - 10 Jul 2024
Viewed by 680
Abstract
For efficient heating and cooling applications, minimum wall shear stress and maximum heat transfer rate are desired. The current study optimized the local skin friction coefficient and Nusselt number in Al2O3-Cu/water hybrid nanofluid flow over a permeable shrinking rotating [...] Read more.
For efficient heating and cooling applications, minimum wall shear stress and maximum heat transfer rate are desired. The current study optimized the local skin friction coefficient and Nusselt number in Al2O3-Cu/water hybrid nanofluid flow over a permeable shrinking rotating disk. First, the governing equations and boundary conditions are solved numerically using the bvp4c solver in MATLAB. Von Kármán’s transformations are used to reduce the partial differential equations into solvable non-linear ordinary differential equations. The augmentation of the mass transfer parameter is found to reduce the local skin friction coefficient and Nusselt number. Higher values of these physical quantities of interest are observed in the injection case than in the suction case. Meanwhile, the increase in the magnitude of the shrinking parameter improved and reduced the local skin friction coefficient and Nusselt number, respectively. Then, response surface methodology (RSM) is conducted to understand the interactive impacts of the controlling parameters in optimizing the physical quantities of interest. With a desirability of 66%, the local skin friction coefficient and Nusselt number are optimized at 1.528780016 and 0.888353037 when the shrinking parameter (λ) and mass transfer parameter (S) are −0.8 and −0.6, respectively. Full article
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Figure 1

Figure 1
<p>Schematic diagram of the flow problem.</p>
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<p>Flowchart of the numerical and statistical procedures.</p>
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<p>The profiles of (<b>a</b>) radial velocity, (<b>b</b>) tangential velocity, (<b>c</b>) axial velocity, and (<b>d</b>) temperature with various values of <math display="inline"><semantics> <mrow> <mi>λ</mi> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>S</mi> </mrow> </semantics></math>.</p>
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<p>The profiles of (<b>a</b>) radial velocity, (<b>b</b>) tangential velocity, (<b>c</b>) axial velocity, and (<b>d</b>) temperature with various values of <math display="inline"><semantics> <mrow> <mi>λ</mi> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>S</mi> </mrow> </semantics></math>.</p>
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<p>The profiles of (<b>a</b>) local skin friction coefficient and (<b>b</b>) local Nusselt number with various values of <math display="inline"><semantics> <mrow> <mi>λ</mi> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>S</mi> </mrow> </semantics></math>.</p>
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<p>Contour and surface plots for Response 1.</p>
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<p>Contour and surface plots for Response 2.</p>
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32 pages, 10224 KiB  
Article
Economic and Exergy Analysis of TiO2 + SiO2 Ethylene-Glycol-Based Hybrid Nanofluid in Plate Heat Exchange System of Solar Installation
by Sylwia Wciślik and Dawid Taler
Energies 2024, 17(13), 3107; https://doi.org/10.3390/en17133107 - 24 Jun 2024
Viewed by 593
Abstract
This paper concerns an economic and exergetic efficiency analysis of a plate heat exchanger placed in a solar installation with TiO2:SiO2/DI:EG nanofluid. This device separates the primary circuit—with the solar fluid—and the secondary circuit—in which domestic hot water flows [...] Read more.
This paper concerns an economic and exergetic efficiency analysis of a plate heat exchanger placed in a solar installation with TiO2:SiO2/DI:EG nanofluid. This device separates the primary circuit—with the solar fluid—and the secondary circuit—in which domestic hot water flows (DHW). The solar fluid is TiO2:SiO2 nanofluid with a concentration in the range of 0.5–1.5%vol. and T = 60 °C. Its flow is maintained at a constant level of 3 dm3/min. The heat-receiving medium is domestic water with an initial temperature of 30 °C. This work records a DHW flow of V˙DHW,in = 3–6(12) dm3/min. In order to calculate the exergy efficiency of the system, first, the total exergy destruction, the entropy generation number Ns, and the Bejan number Be are determined. Only for a comparable solar fluid flow, DHW V˙nf=V˙DHW 3 dm3/min, and concentrations of 0 and 0.5%vol. is there no significant improvement in the exergy efficiency. In other cases, the presence of nanoparticles significantly improves the heat transfer. The TiO2:SiO2/DI:EG nanofluid is even a 13 to 26% more effective working fluid than the traditional solar fluid; at Re = 329, the exergy efficiency is ηexergy = 37.29%, with a nanoparticle concentration of 0% and ηexergy(1.5%vol.) = 50.56%; with Re = 430, ηexergy(0%) = 57.03% and ηexergy(1.5%) = 65.9%. Full article
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<p>Number of all publications registered in the Scopus database for the entry (<b>a</b>) ‘nanofluids’ + ‘heat exchangers’ from 1998 to 16 January 2024 and (<b>b</b>) ‘hybrid nanofluids’ + ‘heat exchangers’ from 1999 to 26 January 2024.</p>
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<p>Growth rate of publications registered in the Scopus database for the entry ‘nanofluids’/‘hybrid nanofluids’ + ‘heat exchangers’ from 2003 to 31 December 2023.</p>
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<p>The most common variations of plate heat exchangers (PHEs) and their corrugations: (<b>a</b>) washboard, (<b>b</b>) zigzag, (<b>c</b>) chevron or herringbone, (<b>d</b>) protrusions and depressions, (<b>e</b>) washboard with secondary corrugations, and (<b>f</b>) oblique washboard [<a href="#B8-energies-17-03107" class="html-bibr">8</a>,<a href="#B9-energies-17-03107" class="html-bibr">9</a>].</p>
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<p>Density of TiO<sub>2</sub> + SiO<sub>2</sub>/DI:EG hybrid nanofluid as a function of its concentration in comparison with literature [<a href="#B38-energies-17-03107" class="html-bibr">38</a>].</p>
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<p>Specific heat of TiO<sub>2</sub>:SiO<sub>2</sub>/DI:EG hybrid nanofluid as a function of the concentration concentration in comparison with literature [<a href="#B38-energies-17-03107" class="html-bibr">38</a>].</p>
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<p>Dynamic viscosity of TiO<sub>2</sub>:SiO<sub>2</sub>/DI:EG hybrid nanofluid versus its concentration in comparison with literature [<a href="#B38-energies-17-03107" class="html-bibr">38</a>,<a href="#B45-energies-17-03107" class="html-bibr">45</a>].</p>
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<p>Thermal conductivity of TiO<sub>2</sub>:SiO<sub>2</sub>/DI:EG hybrid nanofluid (<b>a</b>) versus its concentration in comparison with literature data and (<b>b</b>) with error analysis in comparison with literature [<a href="#B29-energies-17-03107" class="html-bibr">29</a>,<a href="#B38-energies-17-03107" class="html-bibr">38</a>].</p>
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<p>Initial conditions of the solar system and basic dimensions of a chevron-type PHE with its cross-sectional dimensions normal to the direction of troughs.</p>
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<p>Schematic test stand of a solar system with flat-plate collectors and TiO<sub>2</sub>:SiO<sub>2</sub>/DI:EG hybrid nanofluid as a working fluid: 1—flat-plate collectors, 2—pump unit, 3—a chevron-type PHE, 4—domestic hot water tank, 5—reversible valve, and 6—a pump.</p>
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<p>Analysis of a counter-flow heat exchanger.</p>
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<p>Nusselt number of (<b>a</b>) studied TiO<sub>2</sub>:SiO<sub>2</sub>/water ethylene glycol 60:40—hybrid nanofluid and (<b>b</b>,<b>c</b>) [<a href="#B36-energies-17-03107" class="html-bibr">36</a>,<a href="#B50-energies-17-03107" class="html-bibr">50</a>,<a href="#B60-energies-17-03107" class="html-bibr">60</a>] other nanofluids according to worldwide researchers versus low Reynold number.</p>
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<p>Nusselt number validation of DHW during the flow through PHE system versus Reynolds number. (<b>a</b>) own research; (<b>b</b>) in comparison with literature [<a href="#B49-energies-17-03107" class="html-bibr">49</a>].</p>
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<p>Reynolds number versus DHW flow.</p>
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<p>Friction factor calculated in present study in comparison with the literature (<b>a</b>) for deionized water and [<a href="#B50-energies-17-03107" class="html-bibr">50</a>] (<b>b</b>) nanofluids [<a href="#B50-energies-17-03107" class="html-bibr">50</a>,<a href="#B61-energies-17-03107" class="html-bibr">61</a>].</p>
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<p>Entropy generation number versus Reynolds numbers and volume concentrations of TiO<sub>2</sub>:SiO<sub>2</sub>/DI:EG hybrid nanofluid and different DHW flow from (<b>a</b>) 3 dm<sup>3</sup>/min, through (<b>b</b>) 4 dm<sup>3</sup>/min, (<b>c</b>) 5 dm<sup>3</sup>/min, (<b>d</b>) 6 dm<sup>3</sup>/min, up to (<b>e</b>) 12 dm<sup>3</sup>/min.</p>
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<p>Bejan number variation with volumetric flow of nanofluid and concentration.</p>
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<p>The exergy efficiency of PHE with working fluids of different concentration versus Reynolds number and DHW flow of (<b>a</b>) 3 dm<sup>3</sup>/min; (<b>b</b>) 4 dm<sup>3</sup>/min; (<b>c</b>) 5 dm<sup>3</sup>/min; (<b>d</b>) 6 dm<sup>3</sup>/min; (<b>e</b>) 12 dm<sup>3</sup>/min.</p>
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<p>The overall heat transfer coefficient of the hybrid nanofluid versus it % vol. concentration and DHW flow [<a href="#B14-energies-17-03107" class="html-bibr">14</a>].</p>
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<p>The heat transfer coefficient of TiO<sub>2</sub>:SiO<sub>2</sub>/DI:EG hybrid nanofluid with Reynolds number.</p>
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<p>The effectiveness of PHE versus Reynolds number and volume concentrations of TiO<sub>2</sub>:SiO<sub>2</sub>/DI:EG hybrid nanofluid and different DHW flow from (<b>a</b>) 3 dm<sup>3</sup>/min, through (<b>b</b>) 4 dm<sup>3</sup>/min, (<b>c</b>) 5 dm<sup>3</sup>/min, (<b>d</b>) 6 dm<sup>3</sup>/min, up to (<b>e</b>) 12 dm<sup>3</sup>/min.</p>
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<p>The comparison of effectiveness of PHE with available literature data [<a href="#B14-energies-17-03107" class="html-bibr">14</a>].</p>
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<p>The number of transfer units of PHE using TiO<sub>2</sub>:SiO<sub>2</sub>/DI:EG hybrid nanofluid hybrid nanofluid as a function of Reynolds number and different DHW flow from (<b>a</b>) 3 dm<sup>3</sup>/min, through (<b>b</b>) 4 dm<sup>3</sup>/min, (<b>c</b>) 5 dm<sup>3</sup>/min, (<b>d</b>) 6 dm<sup>3</sup>/min, up to (<b>e</b>) 12 dm<sup>3</sup>/min.</p>
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<p>An average number of transfer units of PHE using TiO<sub>2</sub>:SiO<sub>2</sub>/DI:EG hybrid nanofluid as a function of DHW and Reynolds number [<a href="#B2-energies-17-03107" class="html-bibr">2</a>].</p>
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<p><span class="html-italic">THPI</span> of TiO<sub>2</sub>:SiO<sub>2</sub>/DI:EG hybrid nanofluid at different <span class="html-italic">Re</span> and concentration.</p>
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<p>Heat transfer enhancement ratios of TiO<sub>2</sub>:SiO<sub>2</sub>/DI:EG hybrid nanofluid versus Reynolds number.</p>
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<p>The pumping power requirement of TiO<sub>2</sub>:SiO<sub>2</sub>/DI:EG hybrid nanofluid versus its concentration.</p>
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17 pages, 3036 KiB  
Article
Heat and Mass Transformation of Casson Hybrid Nanofluid (MoS2 + ZnO) Based on Engine Oil over a Stretched Wall with Chemical Reaction and Thermo-Diffusion Effect
by Shreedevi Madiwal and Neminath B. Naduvinamani
Lubricants 2024, 12(6), 221; https://doi.org/10.3390/lubricants12060221 - 16 Jun 2024
Viewed by 816
Abstract
This study investigates the potential of a hybrid nanofluid composed of MoS2 and ZnO nanoparticles dispersed in engine oil, aiming to enhance the properties of a lubricant’s chemical reaction with the Soret effect on a stretching sheet under the influence of an [...] Read more.
This study investigates the potential of a hybrid nanofluid composed of MoS2 and ZnO nanoparticles dispersed in engine oil, aiming to enhance the properties of a lubricant’s chemical reaction with the Soret effect on a stretching sheet under the influence of an applied magnetic field. With the growing demand for efficient lubrication systems in various industrial applications, including automotive engines, the development of novel nanofluid-based lubricants presents a promising avenue for improving engine performance and longevity. However, the synergistic effects of hybrid nanoparticles in engine oil remain relatively unexplored. The present research addresses this gap by examining the thermal conductivity, viscosity, and wear resistance of the hybrid nanofluid, shedding light on its potential as an advanced lubrication solution. Overall, the objectives of studying the hybrid nanolubricant MoS2 + ZnO with engine oil aim to advance the development of more efficient and durable lubrication solutions for automotive engines, contributing to improved reliability, fuel efficiency, and environmental sustainability. In the present study, the heat and mass transformation of a Casson hybrid nanofluid (MoS2 + ZnO) based on engine oil over a stretched wall with chemical reaction and thermo-diffusion effect is analyzed. The governing nonlinear partial differential equations are simplified as ordinary differential equations (ODEs) by utilizing the relevant similarity variables. The MATLAB Bvp4c technique is used to solve the obtained linear ODE equations. The results are presented through graphs and tables for various parameters, namely, M, Q, β, Pr, Ec, Sc, Sr, Kp, Kr, and ϕ2* (hybrid nanolubricant parameters) and various state variables. A comparative survey of all the graphs is presented for the nanofluid (MoS2/engine oil) and the hybrid nanofluid (MoS2 + ZnO/engine oil). The results reveal that the velocity profile diminished against the values of M, Kp, and β, and the temperature profile rises with Ec and Q, whereas Pr decreases. The concentration profile is incremented (decremented) with the value of Sr (Sc and Kr). A comparison of the nanofluid and hybrid nanofluid suggests that the velocity f′ (η) becomes slower with the augmentation of ϕ2* whereas the temperature increases when ϕ2* = 0.6 become slower. Full article
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<p>Schematic diagram of the problem.</p>
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<p>Velocity <span class="html-italic">f</span>′ (<span class="html-italic">η</span>) fluctuation with <span class="html-italic">M</span>.</p>
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<p>Velocity <span class="html-italic">f</span>′ (<span class="html-italic">η</span>) fluctuation with <span class="html-italic">Kp</span>.</p>
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<p>Velocity <span class="html-italic">f</span>′ (<span class="html-italic">η</span>) fluctuation with <span class="html-italic">β.</span></p>
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<p><span class="html-italic">θ</span>(<span class="html-italic">η</span>) fluctuation with <span class="html-italic">Pr.</span></p>
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<p><span class="html-italic">θ</span>(<span class="html-italic">η</span>) fluctuation with <span class="html-italic">Q.</span></p>
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<p><span class="html-italic">θ</span>(<span class="html-italic">η</span>) fluctuation with <span class="html-italic">Ec.</span></p>
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<p>Fluctuation with <span class="html-italic">Sc</span>.</p>
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<p>Fluctuation with <span class="html-italic">Sr.</span></p>
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<p>Fluctuation with <span class="html-italic">Kr.</span></p>
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<p>Velocity <span class="html-italic">f</span>′ (<span class="html-italic">η</span>) fluctuation with <math display="inline"><semantics> <mrow> <msub> <mi>ϕ</mi> <mn>2</mn> </msub> <mo>*</mo> </mrow> </semantics></math></p>
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<p><span class="html-italic">θ</span>(<span class="html-italic">η</span>) fluctuation with <math display="inline"><semantics> <mrow> <msub> <mi>ϕ</mi> <mn>2</mn> </msub> <mo>*</mo> </mrow> </semantics></math></p>
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26 pages, 5186 KiB  
Article
Thermal Transportation in Heat Generating and Chemically Reacting MHD Maxwell Hybrid Nanofluid Flow Past Inclined Stretching Porous Sheet in Porous Medium with Solar Radiation Effects
by Mdi Begum Jeelani, Amir Abbas and Nouf Abdulrahman Alqahtani
Processes 2024, 12(6), 1196; https://doi.org/10.3390/pr12061196 - 11 Jun 2024
Viewed by 813
Abstract
The emerging concept of hybrid nanofluids has grabbed the attention of researchers and scientists due to improved thermal performance because of their remarkable thermal conductivities. These fluids have enormous applications in engineering and industrial sectors. Therefore, the present research study examines thermal and [...] Read more.
The emerging concept of hybrid nanofluids has grabbed the attention of researchers and scientists due to improved thermal performance because of their remarkable thermal conductivities. These fluids have enormous applications in engineering and industrial sectors. Therefore, the present research study examines thermal and mass transportation in hybrid nanofluid past an inclined linearly stretching sheet using the Maxwell fluid model. In the current problem, the hybrid nanofluid is engineered by suspending a mixture of aluminum oxide Al2O3  and copper Cu nanoparticles in ethylene glycol. The fluid flow is generated due to the linear stretching of the sheet and the sheet is kept inclined at the angle ζ=π/6 embedded in porous medium. The current proposed model also includes the Lorentz force, solar radiation, heat generation, linear chemical reactions, and permeability of the plate effects. Here, in the current simulation, the cylindrical shape of the nanoparticles is considered, as this shape has proven to be excellent for the thermal performance of the nanomaterials. The governing equations transformed into ordinary differential equations are solved using MATLAB bvp4c solver. The velocity field declines with increasing magnetic field parameter, Maxwell fluid parameter, volume fractions of nanoparticles, and porosity parameter but increases with growing suction parameter. The temperature drops with increasing magnetic field force and suction parameter values but increases with increasing radiation parameter and volume fraction values. The concentration profile increases with increasing magnetic field parameters, porosity parameters, and volume fractions but reduces with increasing chemical reaction parameters and suction parameters. It has been noted that the purpose of the inclusion of thermal radiation is to augment the temperature that is serving the purpose in the current work. The addition of Lorentz force slows down the speed of the fluid and raises the boundary layer thickness, which is visible in the current study. It has been concluded that, when heat generation parameters increase, the temperature field increases correspondingly for both nanofluids and hybrid nanofluids. The increase in the volume fraction of the nanoparticles is used to enhance the thermal performance of the hybrid nanofluid, which is evident in the current results. The current results are validated by comparing them with published ones. Full article
(This article belongs to the Section Energy Systems)
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<p>Flow structure.</p>
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<p>Evaluation of impact of <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>λ</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> </mrow> </semantics></math> on <math display="inline"><semantics> <mrow> <mi>f</mi> <mo>′</mo> </mrow> </semantics></math> when <math display="inline"><semantics> <mrow> <mi>P</mi> <mi>r</mi> <mo>=</mo> <mn>204</mn> <mo>,</mo> <mi>M</mi> <mo>=</mo> <mn>5.2</mn> <mo>,</mo> <msub> <mrow> <mo> </mo> <mi>λ</mi> </mrow> <mrow> <mi>T</mi> </mrow> </msub> <mo>=</mo> <mn>0.2</mn> <mo>,</mo> <msub> <mrow> <mo> </mo> <mi>λ</mi> </mrow> <mrow> <mi>C</mi> </mrow> </msub> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <msub> <mrow> <mo> </mo> <mi>K</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <mi>R</mi> <mi>d</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <mi>Q</mi> <mo>=</mo> <mn>0.2</mn> <mo>,</mo> <mi>S</mi> <mo>=</mo> <mn>0.5</mn> <mo>,</mo> <mi>S</mi> <mi>c</mi> <mo>=</mo> <mn>5.0</mn> <mo>,</mo> <mi>R</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <msub> <mrow> <mo> </mo> <mi>ϕ</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> <msub> <mrow> <mo> </mo> <mi>ϕ</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> <mi>n</mi> <mo>=</mo> <mn>6.0</mn> </mrow> </semantics></math> at <math display="inline"><semantics> <mrow> <mi>ζ</mi> <mo>=</mo> <mi>π</mi> <mo>/</mo> <mn>6</mn> </mrow> </semantics></math>.</p>
Full article ">Figure 3
<p>Evaluation of impact of <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>K</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> </mrow> </semantics></math> on <math display="inline"><semantics> <mrow> <mi>f</mi> <mo>′</mo> </mrow> </semantics></math> when <math display="inline"><semantics> <mrow> <mi>P</mi> <mi>r</mi> <mo>=</mo> <mn>204</mn> <mo>,</mo> <mi>M</mi> <mo>=</mo> <mn>2.0</mn> <mo>,</mo> <msub> <mrow> <mo> </mo> <mi>λ</mi> </mrow> <mrow> <mi>T</mi> </mrow> </msub> <mo>=</mo> <mn>0.2</mn> <mo>,</mo> <msub> <mrow> <mo> </mo> <mi>λ</mi> </mrow> <mrow> <mi>C</mi> </mrow> </msub> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <msub> <mrow> <mo> </mo> <mi>λ</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <mi>R</mi> <mi>d</mi> <mo>=</mo> <mn>0.2</mn> <mo>,</mo> <mi>Q</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <mi>S</mi> <mo>=</mo> <mn>0.5</mn> <mo>,</mo> <mi>S</mi> <mi>c</mi> <mo>=</mo> <mn>2.0</mn> <mo>,</mo> <mi>R</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <msub> <mrow> <mi>ϕ</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> <msub> <mrow> <mo> </mo> <mi>ϕ</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> <mi>n</mi> <mo>=</mo> <mn>6.0</mn> </mrow> </semantics></math> at <math display="inline"><semantics> <mrow> <mi>ζ</mi> <mo>=</mo> <mi>π</mi> <mo>/</mo> <mn>6</mn> </mrow> </semantics></math>.</p>
Full article ">Figure 4
<p>Evaluation of impact of <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>K</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> </mrow> </semantics></math> on <math display="inline"><semantics> <mrow> <mi>ϕ</mi> </mrow> </semantics></math> when <math display="inline"><semantics> <mrow> <mi>P</mi> <mi>r</mi> <mo>=</mo> <mn>204</mn> <mo>,</mo> <mi>M</mi> <mo>=</mo> <mn>2.0</mn> <mo>,</mo> <msub> <mrow> <mo> </mo> <mi>λ</mi> </mrow> <mrow> <mi>T</mi> </mrow> </msub> <mo>=</mo> <mn>0.2</mn> <mo>,</mo> <msub> <mrow> <mo> </mo> <mi>λ</mi> </mrow> <mrow> <mi>C</mi> </mrow> </msub> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <msub> <mrow> <mo> </mo> <mi>λ</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <mi>R</mi> <mi>d</mi> <mo>=</mo> <mn>0.2</mn> <mo>,</mo> <mi>Q</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <mi>S</mi> <mo>=</mo> <mn>0.5</mn> <mo>,</mo> <mi>S</mi> <mi>c</mi> <mo>=</mo> <mn>2.0</mn> <mo>,</mo> <mi>R</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <msub> <mrow> <mi>ϕ</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> <msub> <mrow> <mo> </mo> <mi>ϕ</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> <mi>n</mi> <mo>=</mo> <mn>6.0</mn> </mrow> </semantics></math> at <math display="inline"><semantics> <mrow> <mi>ζ</mi> <mo>=</mo> <mi>π</mi> <mo>/</mo> <mn>6</mn> </mrow> </semantics></math>.</p>
Full article ">Figure 5
<p>Evaluation of impact of <math display="inline"><semantics> <mrow> <mi>M</mi> </mrow> </semantics></math> on <math display="inline"><semantics> <mrow> <mi>f</mi> <mo>′</mo> </mrow> </semantics></math> when <math display="inline"><semantics> <mrow> <mi>P</mi> <mi>r</mi> <mo>=</mo> <mn>204</mn> <mo>,</mo> <msub> <mrow> <mo> </mo> <mi>λ</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>0.6</mn> <mo>,</mo> <msub> <mrow> <mo> </mo> <mi>λ</mi> </mrow> <mrow> <mi>T</mi> </mrow> </msub> <mo>=</mo> <mn>0.2</mn> <mo>,</mo> <msub> <mrow> <mo> </mo> <mi>λ</mi> </mrow> <mrow> <mi>C</mi> </mrow> </msub> <mo>=</mo> <mn>0.2</mn> <mo>,</mo> <msub> <mrow> <mo> </mo> <mi>K</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <mi>R</mi> <mi>d</mi> <mo>=</mo> <mn>0.3</mn> <mo>,</mo> <mi>Q</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <mi>S</mi> <mo>=</mo> <mn>0.5</mn> <mo>,</mo> <mi>S</mi> <mi>c</mi> <mo>=</mo> <mn>5.0</mn> <mo>,</mo> <mi>R</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <msub> <mrow> <mo> </mo> <mi>ϕ</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> <msub> <mrow> <mo> </mo> <mi>ϕ</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> <mi>n</mi> <mo>=</mo> <mn>6.0</mn> </mrow> </semantics></math> at <math display="inline"><semantics> <mrow> <mi>ζ</mi> <mo>=</mo> <mi>π</mi> <mo>/</mo> <mn>6</mn> </mrow> </semantics></math>.</p>
Full article ">Figure 6
<p>Evaluation of impact of <math display="inline"><semantics> <mrow> <mi>M</mi> </mrow> </semantics></math> on <math display="inline"><semantics> <mrow> <mi>θ</mi> </mrow> </semantics></math> when <math display="inline"><semantics> <mrow> <mi>P</mi> <mi>r</mi> <mo>=</mo> <mn>204</mn> <mo>,</mo> <msub> <mrow> <mo> </mo> <mi>λ</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>0.6</mn> <mo>,</mo> <msub> <mrow> <mo> </mo> <mi>λ</mi> </mrow> <mrow> <mi>T</mi> </mrow> </msub> <mo>=</mo> <mn>0.2</mn> <mo>,</mo> <msub> <mrow> <mi>λ</mi> </mrow> <mrow> <mi>C</mi> </mrow> </msub> <mo>=</mo> <mn>0.2</mn> <mo>,</mo> <msub> <mrow> <mo> </mo> <mi>K</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <mi>R</mi> <mi>d</mi> <mo>=</mo> <mn>0.3</mn> <mo>,</mo> <mi>Q</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <mi>S</mi> <mo>=</mo> <mn>0.5</mn> <mo>,</mo> <mi>S</mi> <mi>c</mi> <mo>=</mo> <mn>5.0</mn> <mo>,</mo> <mi>R</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <msub> <mrow> <mo> </mo> <mi>ϕ</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> <msub> <mrow> <mi>ϕ</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> <mi>n</mi> <mo>=</mo> <mn>6.0</mn> </mrow> </semantics></math> at <math display="inline"><semantics> <mrow> <mi>ζ</mi> <mo>=</mo> <mi>π</mi> <mo>/</mo> <mn>6</mn> </mrow> </semantics></math>.</p>
Full article ">Figure 7
<p>Evaluation of impact of <math display="inline"><semantics> <mrow> <mi>M</mi> </mrow> </semantics></math> on <math display="inline"><semantics> <mrow> <mi>ϕ</mi> </mrow> </semantics></math> when <math display="inline"><semantics> <mrow> <mi>P</mi> <mi>r</mi> <mo>=</mo> <mn>204</mn> <mo>,</mo> <msub> <mrow> <mo> </mo> <mi>λ</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>0.6</mn> <mo>,</mo> <msub> <mrow> <mo> </mo> <mi>λ</mi> </mrow> <mrow> <mi>T</mi> </mrow> </msub> <mo>=</mo> <mn>0.2</mn> <mo>,</mo> <msub> <mrow> <mo> </mo> <mi>λ</mi> </mrow> <mrow> <mi>C</mi> </mrow> </msub> <mo>=</mo> <mn>0.2</mn> <mo>,</mo> <msub> <mrow> <mo> </mo> <mi>K</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <mi>R</mi> <mi>d</mi> <mo>=</mo> <mn>0.3</mn> <mo>,</mo> <mi>Q</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <mi>S</mi> <mo>=</mo> <mn>0.5</mn> <mo>,</mo> <mi>S</mi> <mi>c</mi> <mo>=</mo> <mn>5.0</mn> <mo>,</mo> <mi>R</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <msub> <mrow> <mo> </mo> <mi>ϕ</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> <msub> <mrow> <mo> </mo> <mi>ϕ</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> <mi>n</mi> <mo>=</mo> <mn>6.0</mn> </mrow> </semantics></math> at <math display="inline"><semantics> <mrow> <mi>ζ</mi> <mo>=</mo> <mi>π</mi> <mo>/</mo> <mn>6</mn> </mrow> </semantics></math>.</p>
Full article ">Figure 8
<p>Evaluation of impact of <math display="inline"><semantics> <mrow> <mi>S</mi> </mrow> </semantics></math> on <math display="inline"><semantics> <mrow> <mi>f</mi> <mo>′</mo> </mrow> </semantics></math> when <math display="inline"><semantics> <mrow> <mi>P</mi> <mi>r</mi> <mo>=</mo> <mn>204</mn> <mo>,</mo> <mi>M</mi> <mo>=</mo> <mn>4.0</mn> <mo>,</mo> <msub> <mrow> <mo> </mo> <mi>λ</mi> </mrow> <mrow> <mi>T</mi> </mrow> </msub> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <msub> <mrow> <mi>δ</mi> <mo>=</mo> <mn>0.3</mn> <mo>,</mo> <mi>λ</mi> </mrow> <mrow> <mi>C</mi> </mrow> </msub> <mo>=</mo> <mn>0.3</mn> <mo>,</mo> <msub> <mrow> <mo> </mo> <mi>λ</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>1.0</mn> <mo>,</mo> <msub> <mrow> <mo> </mo> <mi>K</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <mi>R</mi> <mi>d</mi> <mo>=</mo> <mn>10.0</mn> <mo>,</mo> <mi>S</mi> <mi>c</mi> <mo>=</mo> <mn>2.0</mn> <mo>,</mo> <mi>R</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <msub> <mrow> <mo> </mo> <mi>ϕ</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> <msub> <mrow> <mo> </mo> <mi>ϕ</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> <mi>n</mi> <mo>=</mo> <mn>6.0</mn> </mrow> </semantics></math> at <math display="inline"><semantics> <mrow> <mi>ζ</mi> <mo>=</mo> <mi>π</mi> <mo>/</mo> <mn>6</mn> </mrow> </semantics></math>.</p>
Full article ">Figure 9
<p>Evaluation of impact of <math display="inline"><semantics> <mrow> <mi>S</mi> </mrow> </semantics></math> on <math display="inline"><semantics> <mrow> <mi>θ</mi> </mrow> </semantics></math> when <math display="inline"><semantics> <mrow> <mi>P</mi> <mi>r</mi> <mo>=</mo> <mn>204</mn> <mo>,</mo> <mi>M</mi> <mo>=</mo> <mn>4.0</mn> <mo>,</mo> <msub> <mrow> <mo> </mo> <mi>λ</mi> </mrow> <mrow> <mi>T</mi> </mrow> </msub> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <msub> <mrow> <mi>Q</mi> <mo>=</mo> <mn>0.3</mn> <mo>,</mo> <mi>λ</mi> </mrow> <mrow> <mi>C</mi> </mrow> </msub> <mo>=</mo> <mn>0.3</mn> <mo>,</mo> <msub> <mrow> <mo> </mo> <mi>λ</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>1.0</mn> <mo>,</mo> <msub> <mrow> <mo> </mo> <mi>K</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <mi>R</mi> <mi>d</mi> <mo>=</mo> <mn>10.0</mn> <mo>,</mo> <mi>S</mi> <mi>c</mi> <mo>=</mo> <mn>2.0</mn> <mo>,</mo> <mi>R</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <msub> <mrow> <mo> </mo> <mi>ϕ</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> <msub> <mrow> <mo> </mo> <mi>ϕ</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> <mi>n</mi> <mo>=</mo> <mn>6.0</mn> </mrow> </semantics></math> at <math display="inline"><semantics> <mrow> <mi>ζ</mi> <mo>=</mo> <mi>π</mi> <mo>/</mo> <mn>6</mn> </mrow> </semantics></math>.</p>
Full article ">Figure 10
<p>Evaluation of impact of <math display="inline"><semantics> <mrow> <mi>Q</mi> </mrow> </semantics></math> on <math display="inline"><semantics> <mrow> <mi>ϕ</mi> </mrow> </semantics></math> when <math display="inline"><semantics> <mrow> <mi>P</mi> <mi>r</mi> <mo>=</mo> <mn>204</mn> <mo>,</mo> <mi>M</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <msub> <mrow> <mi>λ</mi> </mrow> <mrow> <mi>T</mi> </mrow> </msub> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <msub> <mrow> <mi>λ</mi> </mrow> <mrow> <mi>C</mi> </mrow> </msub> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <msub> <mrow> <mo> </mo> <mi>λ</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <msub> <mrow> <mo> </mo> <mi>K</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>1.1</mn> <mo>,</mo> <mi>R</mi> <mi>d</mi> <mo>=</mo> <mn>15.0</mn> <mo>,</mo> <mi>S</mi> <mi>c</mi> <mo>=</mo> <mn>2.1</mn> <mo>,</mo> <mi>R</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <mi>S</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <msub> <mrow> <mo> </mo> <mi>ϕ</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> <msub> <mrow> <mo> </mo> <mi>ϕ</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> <mi>n</mi> <mo>=</mo> <mn>6.0</mn> </mrow> </semantics></math> at <math display="inline"><semantics> <mrow> <mi>ζ</mi> <mo>=</mo> <mi>π</mi> <mo>/</mo> <mn>6</mn> </mrow> </semantics></math>.</p>
Full article ">Figure 11
<p>Evaluation of impact of <math display="inline"><semantics> <mrow> <mi>R</mi> <mi>d</mi> </mrow> </semantics></math> on <math display="inline"><semantics> <mrow> <mi>θ</mi> </mrow> </semantics></math> when <math display="inline"><semantics> <mrow> <mi>P</mi> <mi>r</mi> <mo>=</mo> <mn>204</mn> <mo>,</mo> <mi>M</mi> <mo>=</mo> <mn>3.0</mn> <mo>,</mo> <msub> <mrow> <mi>λ</mi> </mrow> <mrow> <mi>T</mi> </mrow> </msub> <mo>=</mo> <mn>0.2</mn> <mo>,</mo> <msub> <mrow> <mi>Q</mi> <mo>=</mo> <mn>0.3</mn> <mo>,</mo> <mi>λ</mi> </mrow> <mrow> <mi>C</mi> </mrow> </msub> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <msub> <mrow> <mo> </mo> <mi>λ</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>1.0</mn> <mo>,</mo> <mi>R</mi> <mi>d</mi> <mo>=</mo> <mn>0.2</mn> <mo>,</mo> <mi>S</mi> <mo>=</mo> <mn>0.5</mn> <mo>,</mo> <mi>S</mi> <mi>c</mi> <mo>=</mo> <mn>2.0</mn> <mo>,</mo> <mi>R</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <msub> <mrow> <mo> </mo> <mi>ϕ</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> <msub> <mrow> <mo> </mo> <mi>ϕ</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> <mi>n</mi> <mo>=</mo> <mn>6.0</mn> </mrow> </semantics></math> at <math display="inline"><semantics> <mrow> <mi>ζ</mi> <mo>=</mo> <mi>π</mi> <mo>/</mo> <mn>6</mn> </mrow> </semantics></math>.</p>
Full article ">Figure 12
<p>Evaluation of impact of <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>ϕ</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mrow> <mi>ϕ</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> </mrow> </semantics></math> on <math display="inline"><semantics> <mrow> <mi>f</mi> <mo>′</mo> </mrow> </semantics></math> when <math display="inline"><semantics> <mrow> <mi>P</mi> <mi>r</mi> <mo>=</mo> <mn>204</mn> <mo>,</mo> <mi>M</mi> <mo>=</mo> <mn>5.0</mn> <mo>,</mo> <msub> <mrow> <mo> </mo> <mi>λ</mi> </mrow> <mrow> <mi>T</mi> </mrow> </msub> <mo>=</mo> <mn>0.3</mn> <mo>,</mo> <msub> <mrow> <mi>Q</mi> <mo>=</mo> <mn>0.5</mn> <mo>,</mo> <mi>λ</mi> </mrow> <mrow> <mi>C</mi> </mrow> </msub> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <msub> <mrow> <mi>λ</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <msub> <mrow> <mo> </mo> <mi>K</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <mi>R</mi> <mi>d</mi> <mo>=</mo> <mn>13.0</mn> <mo>,</mo> <mi>S</mi> <mi>c</mi> <mo>=</mo> <mn>2.0</mn> <mo>,</mo> <mi>R</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <mi>S</mi> <mo>=</mo> <mn>0.5</mn> <mo>,</mo> <mi>n</mi> <mo>=</mo> <mn>6.0</mn> </mrow> </semantics></math> at <math display="inline"><semantics> <mrow> <mi>ζ</mi> <mo>=</mo> <mi>π</mi> <mo>/</mo> <mn>6</mn> </mrow> </semantics></math>.</p>
Full article ">Figure 13
<p>Evaluation of impact of <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>ϕ</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mrow> <mi>ϕ</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> </mrow> </semantics></math> on <math display="inline"><semantics> <mrow> <mi>θ</mi> </mrow> </semantics></math> when <math display="inline"><semantics> <mrow> <mi>P</mi> <mi>r</mi> <mo>=</mo> <mn>204</mn> <mo>,</mo> <mi>M</mi> <mo>=</mo> <mn>5.0</mn> <mo>,</mo> <msub> <mrow> <mi>λ</mi> </mrow> <mrow> <mi>T</mi> </mrow> </msub> <mo>=</mo> <mn>0.3</mn> <mo>,</mo> <msub> <mrow> <mi>Q</mi> <mo>=</mo> <mn>0.5</mn> <mo>,</mo> <mi>λ</mi> </mrow> <mrow> <mi>C</mi> </mrow> </msub> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <msub> <mrow> <mo> </mo> <mi>λ</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <msub> <mrow> <mo> </mo> <mi>K</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <mi>R</mi> <mi>d</mi> <mo>=</mo> <mn>13.0</mn> <mo>,</mo> <mi>S</mi> <mi>c</mi> <mo>=</mo> <mn>2.0</mn> <mo>,</mo> <mi>R</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <mi>S</mi> <mo>=</mo> <mn>0.5</mn> <mo>,</mo> <mi>n</mi> <mo>=</mo> <mn>6.0</mn> </mrow> </semantics></math> at <math display="inline"><semantics> <mrow> <mi>ζ</mi> <mo>=</mo> <mi>π</mi> <mo>/</mo> <mn>6</mn> </mrow> </semantics></math>.</p>
Full article ">Figure 14
<p>Evaluation of impact of <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>ϕ</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mrow> <mi>ϕ</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> </mrow> </semantics></math> on <math display="inline"><semantics> <mrow> <mi>ϕ</mi> </mrow> </semantics></math> when <math display="inline"><semantics> <mrow> <mi>P</mi> <mi>r</mi> <mo>=</mo> <mn>204</mn> <mo>,</mo> <mi>M</mi> <mo>=</mo> <mn>5.0</mn> <mo>,</mo> <msub> <mrow> <mo> </mo> <mi>λ</mi> </mrow> <mrow> <mi>T</mi> </mrow> </msub> <mo>=</mo> <mn>0.3</mn> <mo>,</mo> <msub> <mrow> <mi>Q</mi> <mo>=</mo> <mn>0.5</mn> <mo>,</mo> <mi>λ</mi> </mrow> <mrow> <mi>C</mi> </mrow> </msub> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <msub> <mrow> <mi>λ</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <msub> <mrow> <mo> </mo> <mi>K</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <mi>R</mi> <mi>d</mi> <mo>=</mo> <mn>13</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>6.0</mn> </mrow> </semantics></math> at <math display="inline"><semantics> <mrow> <mi>ζ</mi> <mo>=</mo> <mi>π</mi> <mo>/</mo> <mn>6</mn> </mrow> </semantics></math>.</p>
Full article ">Figure 15
<p>Evaluation of impact of <math display="inline"><semantics> <mrow> <mi>R</mi> </mrow> </semantics></math> on <math display="inline"><semantics> <mrow> <mi>ϕ</mi> </mrow> </semantics></math> when <math display="inline"><semantics> <mrow> <mi>P</mi> <mi>r</mi> <mo>=</mo> <mn>204</mn> <mo>,</mo> <mi>M</mi> <mo>=</mo> <mn>5.0</mn> <mo>,</mo> <msub> <mrow> <mo> </mo> <mi>λ</mi> </mrow> <mrow> <mi>T</mi> </mrow> </msub> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <msub> <mrow> <mi>δ</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <mi>λ</mi> </mrow> <mrow> <mi>C</mi> </mrow> </msub> <mo>=</mo> <mn>0.3</mn> <mo>,</mo> <msub> <mrow> <mi>λ</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <msub> <mrow> <mo> </mo> <mi>K</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <mi>R</mi> <mi>d</mi> <mo>=</mo> <mn>10.0</mn> <mo>,</mo> <mi>S</mi> <mi>c</mi> <mo>=</mo> <mn>1.2</mn> <mo>,</mo> <mi>S</mi> <mo>=</mo> <mn>0.4</mn> <mo>,</mo> <msub> <mrow> <mo> </mo> <mi>ϕ</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> <msub> <mrow> <mo> </mo> <mi>ϕ</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> <mi>n</mi> <mo>=</mo> <mn>6.0</mn> </mrow> </semantics></math> at <math display="inline"><semantics> <mrow> <mi>ζ</mi> <mo>=</mo> <mi>π</mi> <mo>/</mo> <mn>6</mn> </mrow> </semantics></math>.</p>
Full article ">Figure 16
<p>Evaluation of impact of <math display="inline"><semantics> <mrow> <mi>Q</mi> </mrow> </semantics></math> on <math display="inline"><semantics> <mrow> <mi>R</mi> <msup> <mrow> <mi>e</mi> </mrow> <mrow> <mn>1</mn> <mo>/</mo> <mn>2</mn> </mrow> </msup> <msub> <mrow> <mi>C</mi> </mrow> <mrow> <mi>f</mi> </mrow> </msub> </mrow> </semantics></math> when <math display="inline"><semantics> <mrow> <mi>P</mi> <mi>r</mi> <mo>=</mo> <mn>204</mn> <mo>,</mo> <mi>M</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <msub> <mrow> <mo> </mo> <mi>λ</mi> </mrow> <mrow> <mi>T</mi> </mrow> </msub> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <msub> <mrow> <mi>λ</mi> </mrow> <mrow> <mi>C</mi> </mrow> </msub> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <msub> <mrow> <mo> </mo> <mi>λ</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <msub> <mrow> <mo> </mo> <mi>K</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>1.1</mn> <mo>,</mo> <mi>R</mi> <mi>d</mi> <mo>=</mo> <mn>15.0</mn> <mo>,</mo> <mi>S</mi> <mi>c</mi> <mo>=</mo> <mn>2.1</mn> <mo>,</mo> <mi>R</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <mi>S</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <msub> <mrow> <mo> </mo> <mi>ϕ</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> <msub> <mrow> <mo> </mo> <mi>ϕ</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> <mi>n</mi> <mo>=</mo> <mn>6.0</mn> </mrow> </semantics></math> at <math display="inline"><semantics> <mrow> <mi>ζ</mi> <mo>=</mo> <mi>π</mi> <mo>/</mo> <mn>6</mn> </mrow> </semantics></math>.</p>
Full article ">Figure 17
<p>Evaluation of impact of <math display="inline"><semantics> <mrow> <mi>Q</mi> </mrow> </semantics></math> on <math display="inline"><semantics> <mrow> <mi>R</mi> <msup> <mrow> <mi>e</mi> </mrow> <mrow> <mo>−</mo> <mn>1</mn> <mo>/</mo> <mn>2</mn> </mrow> </msup> <mi>N</mi> <mi>u</mi> </mrow> </semantics></math> when <math display="inline"><semantics> <mrow> <mi>P</mi> <mi>r</mi> <mo>=</mo> <mn>204</mn> <mo>,</mo> <mi>M</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <msub> <mrow> <mo> </mo> <mi>λ</mi> </mrow> <mrow> <mi>T</mi> </mrow> </msub> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <msub> <mrow> <mi>λ</mi> </mrow> <mrow> <mi>C</mi> </mrow> </msub> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <msub> <mrow> <mi>λ</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <msub> <mrow> <mo> </mo> <mi>K</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>1.1</mn> <mo>,</mo> <mi>R</mi> <mi>d</mi> <mo>=</mo> <mn>15.0</mn> <mo>,</mo> <mi>S</mi> <mi>c</mi> <mo>=</mo> <mn>2.1</mn> <mo>,</mo> <mi>R</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <mi>S</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <msub> <mrow> <mo> </mo> <mi>ϕ</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> <msub> <mrow> <mo> </mo> <mi>ϕ</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> <mi>n</mi> <mo>=</mo> <mn>6.0</mn> </mrow> </semantics></math> at <math display="inline"><semantics> <mrow> <mi>ζ</mi> <mo>=</mo> <mi>π</mi> <mo>/</mo> <mn>6</mn> </mrow> </semantics></math>.</p>
Full article ">Figure 18
<p>Evaluation of impact of <math display="inline"><semantics> <mrow> <mi>Q</mi> </mrow> </semantics></math> on <math display="inline"><semantics> <mrow> <mi>R</mi> <msup> <mrow> <mi>e</mi> </mrow> <mrow> <mo>−</mo> <mn>1</mn> <mo>/</mo> <mn>2</mn> </mrow> </msup> <mi>S</mi> <mi>h</mi> </mrow> </semantics></math> when <math display="inline"><semantics> <mrow> <mi>P</mi> <mi>r</mi> <mo>=</mo> <mn>204</mn> <mo>,</mo> <mi>M</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <msub> <mrow> <mi>λ</mi> </mrow> <mrow> <mi>T</mi> </mrow> </msub> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <msub> <mrow> <mi>λ</mi> </mrow> <mrow> <mi>C</mi> </mrow> </msub> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <msub> <mrow> <mo> </mo> <mi>λ</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <msub> <mrow> <mo> </mo> <mi>K</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>1.1</mn> <mo>,</mo> <mi>R</mi> <mi>d</mi> <mo>=</mo> <mn>15.0</mn> <mo>,</mo> <mi>S</mi> <mi>c</mi> <mo>=</mo> <mn>2.1</mn> <mo>,</mo> <mi>R</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <mi>S</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <msub> <mrow> <mo> </mo> <mi>ϕ</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> <msub> <mrow> <mo> </mo> <mi>ϕ</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> <mi>n</mi> <mo>=</mo> <mn>6.0</mn> </mrow> </semantics></math> at <math display="inline"><semantics> <mrow> <mi>ζ</mi> <mo>=</mo> <mi>π</mi> <mo>/</mo> <mn>6</mn> </mrow> </semantics></math>.</p>
Full article ">Figure 19
<p>Evaluation of impact of <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>λ</mi> </mrow> <mrow> <mi>T</mi> </mrow> </msub> </mrow> </semantics></math> on <math display="inline"><semantics> <mrow> <mi>R</mi> <msup> <mrow> <mi>e</mi> </mrow> <mrow> <mn>1</mn> <mo>/</mo> <mn>2</mn> </mrow> </msup> <msub> <mrow> <mi>C</mi> </mrow> <mrow> <mi>f</mi> </mrow> </msub> </mrow> </semantics></math> when <math display="inline"><semantics> <mrow> <mi>P</mi> <mi>r</mi> <mo>=</mo> <mn>204</mn> <mo>,</mo> <mi>M</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <mi>Q</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <msub> <mrow> <mi>λ</mi> </mrow> <mrow> <mi>C</mi> </mrow> </msub> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <msub> <mrow> <mo> </mo> <mi>λ</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <msub> <mrow> <mi>K</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>1.1</mn> <mo>,</mo> <mi>R</mi> <mi>d</mi> <mo>=</mo> <mn>15.0</mn> <mo>,</mo> <mi>S</mi> <mi>c</mi> <mo>=</mo> <mn>2.1</mn> <mo>,</mo> <mi>R</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <mi>S</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <msub> <mrow> <mo> </mo> <mi>ϕ</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> <msub> <mrow> <mi>ϕ</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> <mi>n</mi> <mo>=</mo> <mn>6.0</mn> </mrow> </semantics></math> at <math display="inline"><semantics> <mrow> <mi>ζ</mi> <mo>=</mo> <mi>π</mi> <mo>/</mo> <mn>6</mn> </mrow> </semantics></math>.</p>
Full article ">Figure 20
<p>Evaluation of impact of <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>λ</mi> </mrow> <mrow> <mi>T</mi> </mrow> </msub> </mrow> </semantics></math> on <math display="inline"><semantics> <mrow> <mi>R</mi> <msup> <mrow> <mi>e</mi> </mrow> <mrow> <mo>−</mo> <mn>1</mn> <mo>/</mo> <mn>2</mn> </mrow> </msup> <mi>N</mi> <mi>u</mi> </mrow> </semantics></math> when <math display="inline"><semantics> <mrow> <mi>P</mi> <mi>r</mi> <mo>=</mo> <mn>204</mn> <mo>,</mo> <mi>M</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <mi>Q</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <msub> <mrow> <mi>λ</mi> </mrow> <mrow> <mi>C</mi> </mrow> </msub> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <msub> <mrow> <mo> </mo> <mi>λ</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <msub> <mrow> <mo> </mo> <mi>K</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>1.1</mn> <mo>,</mo> <mi>R</mi> <mi>d</mi> <mo>=</mo> <mn>15.0</mn> <mo>,</mo> <mi>S</mi> <mi>c</mi> <mo>=</mo> <mn>2.1</mn> <mo>,</mo> <mi>R</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <mi>S</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <msub> <mrow> <mi>ϕ</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> <msub> <mrow> <mi>ϕ</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> <mi>n</mi> <mo>=</mo> <mn>6.0</mn> </mrow> </semantics></math> at <math display="inline"><semantics> <mrow> <mi>ζ</mi> <mo>=</mo> <mi>π</mi> <mo>/</mo> <mn>6</mn> </mrow> </semantics></math>.</p>
Full article ">Figure 21
<p>Evaluation of impact of <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>λ</mi> </mrow> <mrow> <mi>T</mi> </mrow> </msub> </mrow> </semantics></math> on <math display="inline"><semantics> <mrow> <mi>R</mi> <msup> <mrow> <mi>e</mi> </mrow> <mrow> <mo>−</mo> <mn>1</mn> <mo>/</mo> <mn>2</mn> </mrow> </msup> <mi>S</mi> <mi>h</mi> </mrow> </semantics></math> when <math display="inline"><semantics> <mrow> <mi>P</mi> <mi>r</mi> <mo>=</mo> <mn>204</mn> <mo>,</mo> <mi>M</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <mi>Q</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <msub> <mrow> <mi>λ</mi> </mrow> <mrow> <mi>C</mi> </mrow> </msub> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <msub> <mrow> <mo> </mo> <mi>λ</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <msub> <mrow> <mo> </mo> <mi>K</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>1.1</mn> <mo>,</mo> <mi>R</mi> <mi>d</mi> <mo>=</mo> <mn>15.0</mn> <mo>,</mo> <mi>S</mi> <mi>c</mi> <mo>=</mo> <mn>2.1</mn> <mo>,</mo> <mi>R</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <mi>S</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <msub> <mrow> <mo> </mo> <mi>ϕ</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> <msub> <mrow> <mo> </mo> <mi>ϕ</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> <mi>n</mi> <mo>=</mo> <mn>6.0</mn> </mrow> </semantics></math> at <math display="inline"><semantics> <mrow> <mi>ζ</mi> <mo>=</mo> <mi>π</mi> <mo>/</mo> <mn>6</mn> </mrow> </semantics></math>.</p>
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34 pages, 15586 KiB  
Review
Constrained Volume Micro- and Nanoparticle Collection Methods in Microfluidic Systems
by Tanner N. Wells, Holger Schmidt and Aaron R. Hawkins
Micromachines 2024, 15(6), 699; https://doi.org/10.3390/mi15060699 - 25 May 2024
Viewed by 803
Abstract
Particle trapping and enrichment into confined volumes can be useful in particle processing and analysis. This review is an evaluation of the methods used to trap and enrich particles into constrained volumes in microfluidic and nanofluidic systems. These methods include physical, optical, electrical, [...] Read more.
Particle trapping and enrichment into confined volumes can be useful in particle processing and analysis. This review is an evaluation of the methods used to trap and enrich particles into constrained volumes in microfluidic and nanofluidic systems. These methods include physical, optical, electrical, magnetic, acoustic, and some hybrid techniques, all capable of locally enhancing nano- and microparticle concentrations on a microscale. Some key qualitative and quantitative comparison points are also explored, illustrating the specific applicability and challenges of each method. A few applications of these types of particle trapping are also discussed, including enhancing biological and chemical sensors, particle washing techniques, and fluid medium exchange systems. Full article
(This article belongs to the Section C:Chemistry)
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Figure 1

Figure 1
<p>An example of physical trapping of particles into individual trapping sites: (<b>a</b>) the hydrodynamic method used to trap these particles is depicted; (<b>b</b>) a hydraulic resistance model is shown to describe the trapping effect. This mechanism was used to capture microbeads (red in (<b>a</b>) and yellow in (<b>b</b>)) with exosomes attached to their surfaces, as depicted in (<b>a</b>). Adapted with permission from [<a href="#B41-micromachines-15-00699" class="html-bibr">41</a>]. Copyright 2020 American Chemical Society.</p>
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<p>An example of dual-height particle trapping: (<b>a</b>) top-view depiction of a series of parallel channels, each with 2 sections of different heights; (<b>b</b>) cross-sectional depiction of one of these channels, showing the mechanism used for particle collection. Reprinted with permission from [<a href="#B43-micromachines-15-00699" class="html-bibr">43</a>]. Copyright 2009 Royal Society of Chemistry.</p>
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<p>Two related examples of dual-height particle trapping. (<b>a</b>) Three-dimensional view and (<b>b</b>) cross-section of membrane-covered physical trapping mechanism from Ref [<a href="#B12-micromachines-15-00699" class="html-bibr">12</a>]; (<b>c</b>) top view and (<b>d</b>) cross-section of V-shaped physical trapping mechanism from Ref [<a href="#B47-micromachines-15-00699" class="html-bibr">47</a>]. Reprinted from [<a href="#B12-micromachines-15-00699" class="html-bibr">12</a>,<a href="#B47-micromachines-15-00699" class="html-bibr">47</a>] with the permission of AIP Publishing.</p>
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<p>Flow-based trapping mechanism from [<a href="#B36-micromachines-15-00699" class="html-bibr">36</a>]. (<b>a</b>) Selective trapping of large particles is depicted; (<b>b</b>) a top view of the channel design; (<b>c</b>) a top view showing fluorescent particles trapped in vortices; (<b>d</b>) a depiction of the underlying selective trapping principle. Reprinted from [<a href="#B36-micromachines-15-00699" class="html-bibr">36</a>] with the permission of AIP Publishing.</p>
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<p>(<b>a</b>–<b>h</b>) Structured particle trapping by a photonic crystal cavity. The hollow triangles point to the trapped particles, and the insets with hollow circles show the trapped particle structure. Reprinted from [<a href="#B59-micromachines-15-00699" class="html-bibr">59</a>] with the permission of AIP Publishing.</p>
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<p>(<b>a</b>,<b>b</b>) Two counter-propagating waveguides attract particles using optical gradient forces from the evanescent field near the waveguide, then direct these particles to a collection point using optical radiation pressure. Reprinted with permission from [<a href="#B62-micromachines-15-00699" class="html-bibr">62</a>]. Copyright 2012 Royal Society of Chemistry.</p>
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<p>Optical particle trapping into a protrusion in an enclosed fluid channel using (<b>a</b>) optical radiation pressure and (<b>b</b>) a combination of optical radiation pressure and optical gradient force, in both cases provided by an integrated waveguide. Reprinted from [<a href="#B34-micromachines-15-00699" class="html-bibr">34</a>] under open access Creative Commons CC BY license.</p>
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<p>A dielectrophoresis-based particle trapping and manipulation device. (<b>a</b>) A top-down picture of the device, showing 4 upper electrodes and 4 lower ones; (<b>b</b>) particles trapped in suspension, out of contact with the device surfaces; (<b>c</b>,<b>d</b>) the shape of the trapped particle cluster can be manipulated by changing the electrical phase difference between the alternating current signals of the separate probes. The scale is equal for (<b>b</b>–<b>d</b>). Reprinted in part from [<a href="#B72-micromachines-15-00699" class="html-bibr">72</a>], with permission from Elsevier.</p>
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<p>Particle trapping by dielectrophoresis, enhanced by electro-osmotic flow. (<b>a</b>) A brightfield image of the electrodes; (<b>b</b>,<b>c</b>) a fluorescence image of particles collecting on the electrodes as time progresses. Particles collect on the electrode pads due to DEP, and vortices produced by ACEO (depicted by the circulating arrows in (<b>d</b>)) allow wide-scale circulation of particles to enhance the range of the trapping effect. Reprinted with permission from [<a href="#B85-micromachines-15-00699" class="html-bibr">85</a>]. Copyright IOP Publishing. All rights reserved.</p>
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<p>Alternating current electro-osmosis (ACEO) particle trapping using induced vortex flows. (<b>a</b>) A depiction of vortex flows generated by ACEO; (<b>b</b>,<b>c</b>) <span class="html-italic">E. coli</span> particles collected in a line on the center of electrodes. Adapted with permission from [<a href="#B96-micromachines-15-00699" class="html-bibr">96</a>]. Copyright 2005 American Chemical Society.</p>
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<p>Particle collection using the alternating current electrothermal effect (ACET). Vortices generated by ACET cause particles to migrate (<b>a</b>) to the bottom of a droplet or (<b>b</b>) to the top of a droplet. The droplet can then be divided to separate particle-carrying fluid from particle-free fluid. The magnified images indicated by the arrows in (<b>a</b>,<b>b</b>) show the distribution of particles during processing (red) and after particles have been concentrated (blue). Reprinted with permission from [<a href="#B103-micromachines-15-00699" class="html-bibr">103</a>]. Copyright 2021 Royal Society of Chemistry.</p>
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<p>Particle collection onto a heated section of an electrode. (<b>a</b>) A depiction of the forces acting on particles to cause them to collect; (<b>b</b>) a top-view image of particles collected in a monolayer above a heater. The formation of a monolayer of particles is due to electrohydrodynamic flows from an electrode and is further induced by localized heating of the electrode. Reprinted from [<a href="#B109-micromachines-15-00699" class="html-bibr">109</a>]. Copyright 2012, with permission from Elsevier.</p>
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<p>A method of trapping magnetic and diamagnetic polystyrene (PS) beads in a capillary using two external permanent magnets. Particles are suspended in a paramagnetic solution to allow for the trapping of diamagnetic particles. Reprinted with permission from [<a href="#B112-micromachines-15-00699" class="html-bibr">112</a>]. Copyright 2021 Royal Society of Chemistry.</p>
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<p>A method of both collecting and sensing magnetic particles using an integrated electromagnet. (<b>a</b>) Suspended particles are at first unaffected; (<b>b</b>–<b>d</b>) voltages are sequentially applied to the outer, middle, and inner ring electrodes to attract particles and move them to the device center. A magnetic sensor is placed at the center of the device, allowing particles to be evaluated. Reprinted from [<a href="#B119-micromachines-15-00699" class="html-bibr">119</a>], with the permission of AIP Publishing.</p>
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<p>An external magnetic field-based device for selective localized cell capture. The device uses an external magnetic field that couples to ferromagnetic pillars, enhancing the local magnetic field gradient and attracting surface-modified magnetic nanoparticles. These particles in turn capture cancer cells on their modified surfaces. (<b>a</b>) A depiction of the trapping device; (<b>b</b>) a depiction of a single pillar capturing a cell; (<b>c</b>) a microscope image of a cell trapped on a pillar. Reprinted with permission from [<a href="#B122-micromachines-15-00699" class="html-bibr">122</a>]. Copyright 2013 WILEY-VCH Verlag GmbH &amp; Co. KGaA, Weinheim, Germany.</p>
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<p>Acoustic trapping and holding of cells during a 6 h cell culture. (<b>a</b>) A cross-sectional depiction of the device; (<b>b</b>) a depiction of the pressure distribution produced by the acoustic transducer; (<b>c</b>) an immobilized cell cluster is observed through the course of a 6 h culture experiment. Reprinted with permission from [<a href="#B10-micromachines-15-00699" class="html-bibr">10</a>]. Copyright 2007 American Chemical Society.</p>
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<p>Particle enrichment and medium exchange in a droplet through a surface acoustic wave (SAW). (<b>a</b>) A droplet with 6 µm particles is shown entering the processing region; (<b>b</b>,<b>c</b>) particles collect in a confined region of the droplet due to the SAW; (<b>d</b>) the droplet is cut, increasing particle concentration; (<b>e</b>) the new suspension medium is introduced; (<b>f</b>) the particles are moved to the new medium; (<b>g</b>) the new droplet is cut, again enriching the particles; (<b>h</b>,<b>i</b>) the process repeats. Reprinted with permission from [<a href="#B9-micromachines-15-00699" class="html-bibr">9</a>]. Copyright 2018 Royal Society of Chemistry.</p>
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<p>Particle filtration using an acoustic “membrane”. Large particles are selectively trapped by the surface acoustic wave, enriching them at the trapping location. (<b>a</b>) An expanded depiction of the acoustic trapping channel with larger (red) particles becoming trapped and smaller (blue) particles unaffected; (<b>b</b>) a magnified depiction of the trapping region showing the virtual membrane (green) trapping larger (red) particles but allowing smaller (blue) particles to pass. Reprinted with permission from [<a href="#B140-micromachines-15-00699" class="html-bibr">140</a>]. Copyright 2016 Royal Society of Chemistry.</p>
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<p>A hybrid particle trapping method that combines physical barrier-based trapping of larger 10 µm particles with acoustic trapping of much smaller nanoparticles. This allows the surface acoustic waves (SAWs) to capture much smaller particles than would otherwise be feasible. (<b>a</b>) A depiction of the particle trapping device showing the packed bed of larger (green) particles and acoustically trapped smaller (red) particles; (<b>b</b>) a depiction of the trapping mechanism showing physical trapping of larger (blue) particles and acoustic trapping and release of smaller (red) particles. Reprinted with permission from [<a href="#B150-micromachines-15-00699" class="html-bibr">150</a>]. Copyright 2019 Royal Society of Chemistry.</p>
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<p>An example setup for a particle processing application. Particles suspended in fluid 1 flow into the trapping volume and are collected. Fluid 2 could be either a new medium for the particles (in the case of washing) or another solution, perhaps containing a particle dye or labelling molecule. Excess or used fluid is removed through the outlet channel.</p>
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25 pages, 8756 KiB  
Article
Optimizing Nanofluid Hybrid Solar Collectors through Artificial Intelligence Models
by Safae Margoum, Bekkay Hajji, Stefano Aneli, Giuseppe Marco Tina and Antonio Gagliano
Energies 2024, 17(10), 2307; https://doi.org/10.3390/en17102307 - 10 May 2024
Cited by 1 | Viewed by 830
Abstract
This study systematically explores and compares the performance of various artificial-intelligence (AI)-based models to predict the electrical and thermal efficiency of photovoltaic–thermal systems (PVTs) cooled by nanofluids. Employing extreme gradient boosting (XGB), extra tree regression (ETR), and k-nearest-neighbor (KNN) regression models, their accuracy [...] Read more.
This study systematically explores and compares the performance of various artificial-intelligence (AI)-based models to predict the electrical and thermal efficiency of photovoltaic–thermal systems (PVTs) cooled by nanofluids. Employing extreme gradient boosting (XGB), extra tree regression (ETR), and k-nearest-neighbor (KNN) regression models, their accuracy is quantitatively evaluated, and their effectiveness measured. The results demonstrate that both XGB and ETR models consistently outperform KNN in accurately predicting both electrical and thermal efficiency. Specifically, the XGB model achieves remarkable correlation coefficient (R2) values of approximately 0.99999, signifying its superior predictive capabilities. Notably, the XGB model exhibits a slightly superior performance compared to ETR in estimating electrical efficiency. Furthermore, when predicting thermal efficiency, both XGB and ETR models demonstrate excellence, with the XGB model showing a slight edge based on R2 values. Validation against new data points reveals outstanding predictive performance, with the XGB model attaining R2 values of 0.99997 for electrical efficiency and 0.99995 for thermal efficiency. These quantitative findings underscore the accuracy and reliability of the XGB and ETR models in predicting the electrical and thermal efficiency of PVT systems when cooled by nanofluids. The study’s implications are significant for PVT system designers and industry professionals, as the incorporation of AI-based models offers improved accuracy, faster prediction times, and the ability to handle large datasets. The models presented in this study contribute to system optimization, performance evaluation, and decision-making in the field. Additionally, robust validation against new data enhances the credibility of these models, advancing the overall understanding and applicability of AI in PVT systems. Full article
(This article belongs to the Special Issue Advanced Solar Technologies and Thermal Energy Storage)
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<p>Python libraries used for data analysis and machine learning in photovoltaic–thermal system efficiency study.</p>
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<p>The geographic areas where the analyses were carried out.</p>
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<p>Electrical efficiency and thermal efficiency, respectively, of PVT with (2% of volume fraction) and in the case of laminar flow (0.0085 Kg/s of mass flow rate) in [<a href="#B55-energies-17-02307" class="html-bibr">55</a>].</p>
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<p>Design of integrated PVT system.</p>
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<p>Flowchart: performance estimation and prediction in photovoltaic–thermal (PVT) systems.</p>
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<p>Correlation coefficient in k-fold function for electrical (<b>left</b>) and thermal efficiency (<b>right</b>).</p>
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<p>An extract from the first five splits for training and testing for (<b>a</b>) electrical efficiency and (<b>b</b>) thermal efficiency.</p>
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<p>Electrical efficiency regression plot: experimental vs. estimated data from the three models.</p>
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<p>Thermal efficiency regression plot: experimental vs. estimated data from the three models.</p>
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<p>Relative deviations of electrical efficiency: experimental vs. predicted by the three models.</p>
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<p>Relative deviations of electrical efficiency: experimental vs. predicted by the three models.</p>
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<p>Experimental vs. predicted electrical efficiency for the three models.</p>
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<p>Experimental vs. predicted thermal efficiency values from the three models.</p>
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<p>Comparison of the predicted electrical (<b>left</b>) and thermal efficiency (<b>right</b>) with the experimental values (derived from [<a href="#B55-energies-17-02307" class="html-bibr">55</a>]).</p>
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