heat transfer research paper

Lienhard Research Group

Seeking energy efficient solutions for clean water supplies, recent papers in heat transfer, heat diffusion during thin-film composite membrane formation.

Schematic illustration of heat transfer in during interfacial polymerization for the fabrication of thin-film composite membrane selective layers

Thin-film composite (TFC) membranes, the backbone of modern reverse osmosis and nanofiltration, combine the high separation performance of a thin selective layer with the robust mechanical support. Previous studies have shown that heat released during interfacial polymerization (IP) can have a significant impact on the physical and chemical structure of the selective layer. In this study, we develop a multilayer transient heat conduction model to analyze how the thermal properties of the materials used in TFC fabrication impact interfacial temperature, focusing on support-free (SFIP), conventional (CIP), and interlayer-modulated IP (IMIP). Using a combination of analytic solutions and computational models, we demonstrate that the thermal effusivities of fluid and material layers can have a significant effect on the temporal evolution of interfacial temperature during IP. In CIP, we show that the presence of a polymeric support adjacent to the reaction interface yields a 20% to 60% increase in interfacial temperature rise, lasting for ∼ 0.1 s. Furthermore, we demonstrate that inorganic or metallic interlayers, which have high thermal effusivities, can lead to short-lived order-of-magnitude reductions in interfacial temperature rise. Finally, we provide analytical approximations for transient heat conduction through multilayered systems, enabling rapid evaluation of the thermal impact of novel membrane support and interlayer materials and structures on interfacial temperature during TFC fabrication. Quantifying how the thermal properties of solvents, support layers, and interlayers affect interfacial temperature during IP is critical for the rational design of new TFC membranes.

A. Deshmukh, J.H. Lienhard, and M. Elimelech, “Heat Diffusion During Thin-Film Composite Membrane Formation,” J. Membrane Science , 696: 122493, March 2024. Editor’s Choice Article for March 2024. There are still fun things to do with classical heat conduction!

Heat transfer in flat-plate boundary layers: a correlation for laminar, transitional, and turbulent flow

Proposed correlation, Eq. (9), compared to constant heat flux data of Blair for three levels of free stream turbulence.

J.H. Lienhard V, “Heat transfer in flat-plate boundary layers: a correlation for laminar, transitional, and turbulent flow,” J. Heat Transfer , online 31 March 2020, 142 (6):061805, June 2020. ( doi: Open access ) ( presentation ) ( one-page summary ) (DSpace)

Accurate linearization of non-gray radiation heat exchange

The internal emissivity based on mean temperature is in good agreement with the exact heat exchange for even extremely non-gray surfaces.

J.H. Lienhard V, “Linearization of Non-gray Radiation Exchange: The Internal Fractional Function Reconsidered,” J. Heat Transfer , online 3 Dec. 2018, 141 (5):052701, May 2019. ( OPEN ACCESS ) (preprint) (presentation)

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Published  18  issues per year

ISSN Print: 1064-2285

ISSN Online: 2162-6561

AUTHOR INSTRUCTIONS HEAT TRANSFER RESEARCH

DESCRIPTION OF PAPERS CONSIDERED FOR PUBLICATION: Heat Transfer Research (HTR) presents archived theoretical, applied, and experimental papers selected globally. Selected papers from technical conference proceedings and academic laboratory reports are also published. Papers are selected and reviewed by a group of expert associate editors, guided by a distinguished advisory board, and represent the best of current work in the field. HTR is published under an exclusive license to Begell House, Inc., in full compliance with the International Copyright Convention. Subjects covered in HTR encompass the entire field of heat transfer and relevant areas of fluid dynamics, including conduction, convection and radiation, phase change phenomena including boiling and solidification, heat exchange design and testing, heat transfer in nuclear reactors, mass transfer, geothermal heat recovery, multi-scale heat transfer, heat and mass transfer in alternative energy systems, and thermophysical properties of materials.

GENERAL INSTRUCTIONS: Articles for Heat Transfer Research must be submitted through the Begell House Submission Site. If you are already a registered user begin at: http://submission.begellhouse.com/usr/login.html?prod_code=journals , enter your username and password, and click on the Login button. If you are not a registered user follow the instructions below.

SUBMISSION INSTRUCTIONS: All users of the Begell House submission site must be registered. If you are a first-time users you can begin at http://submission.begellhouse.com/usr/login.html?prod_code=journals , click the New User icon located at the top of the page, on the next screen complete the registration form and then click the Submit button located at the bottom left side of the form. Your form will be reviewed and an email confirmation will be sent to you within 24 hours (not including weekends and holidays). Please note that due to the high security that some affiliations have on their email programs, this email may go into your junk/spam folder; please check. Keep your username and password available for further use on the submission site. Logon information is case sensitive; always be sure to enter the appropriate upper and lower-case characters. Never register a second time on the submission site. Having more than one account on the submission site will create difficulties in working efficiently and being able to access all articles assigned to you. If you forget your password go to http://submission.begellhouse.com/usr/login.html?prod_code=journals , click Forgot Password , and enter the email address that you used when you registered. Within seconds you will receive an email providing you with your username and temporary password (again remember this email may go into your junk/spam folder). Once you receive the email containing your logon information, return to http://submission.begellhouse.com/usr/login.html?prod_code=journals , enter your username and temporary password, and then click Login . On the next screen you will be asked to provide a password of your choice. Enter the password and click Login . You will now be logged onto the submission site. If the email that you used when you originally registered is no longer available to you please contact [email protected] for assistance.

• 1. PREPARATION OF ELECTRONIC ARTICLE FILES: Write in clear, concise English. The author is responsible for all aspects of article preparation. Extensive changes to the article will not be undertaken by the Editor or during the production process. Each manuscript must be accompanied by a statement that it has not been published elsewhere and that it has not been simultaneously submitted for publication elsewhere.
• 2. ORIGINAL ARTICLES SUBMITTED for possible publication in Heat Transfer Research should be submitted in a single PDF file that includes all text, figures, and tables in single-column. Please use 12-point type and Times New Roman typeface for all text and figure captions. Please ensure that high quality figures and tables are provided and that any text that is an actual part of the figure be no less than 9-point type. Please remember that although your original submitted manuscript is required to be in PDF format for reviewing purposes, your accepted article must be provided to the publisher using your program files (see #4 below).
• a. Generative AI Declaration: Authors should only use generative artificial intelligence (AI) and AI-assisted technologies to improve readability and language in the writing process. AI can produce legitimate-seeming text that may be factually incorrect and problematic in other ways. Authors must disclose in their manuscript the use of AI and AI-assisted technologies in the writing process by adding a statement at the end of the main manuscript, prior to the references section. The statement will appear in the published article. Authors are ultimately responsible and accountable for the work, regardless of the tools used. The statement should read: During the preparation of this work the author(s) used [NAME TOOL / SERVICE] in order to [REASON]. After using this tool/service, the author(s) reviewed and edited the content as needed and take(s) full responsibility for the content of the publication. This declaration is only needed if AI was used in the writing process. It is not needed for the use of basic tools for checking grammar, spelling, references, etc. It is also not needed for the use of AI tools to analyze data as part of the research process. If there is nothing to disclose, there is no need to add a statement.
• 4. PERMISSIONS: Authors are responsible for obtaining permission to reproduce copyrighted material from other sources and are required to sign an agreement for the transfer of copyright to the publisher. Please note that if a figure or table was previously published the copyright for that figure is usually owned by the publisher not the author. ALL PERMISSIONS MUST BE UPLOADED WITH THE FINAL DRAFT OF YOUR ARTICLE BEFORE YOUR ARTICLE CAN BE PROCESSED .
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Heat Transfer Research

Subject Area and Category

• Fluid Flow and Transfer Processes
• Mechanical Engineering
• Condensed Matter Physics

Begell House Inc.

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1992-1993, 1995-2023

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Fluid Flow and Transfer Processes

Mechanical Engineering

Condensed Matter Physics

The Journal of Heat and Mass Transfer Research is an international, bi-annual publication (both in print and online) established in 2013 by Semnan University Press . The journal disseminates cutting-edge and innovative research findings in the domain of heat and mass transfer. This encompasses various subfields including conduction, convection, radiation, phase change phenomena, heat exchanger design and testing, nuclear reactors, geothermal heat recovery, and alternative energy systems. Additionally, it covers emerging technologies such as Micro-Electro-Mechanical Systems (MEMS), micro-channels, fuel cells, bio- and nano-technology, transport in porous media, ice formation and melting, reactor design, microreactors, multiphase reactors, multiscale modeling, and gas-solid reactions, as well as heat transfer in machinery and welding operations.

The Journal encourages contributions that facilitate the exchange of ideas and foster scholarly dialogue between practitioners, educators, and researchers in Mechanical and Chemical Engineering globally. The journal’s scope includes peer-reviewed original research articles, technical reports, review papers, short communications, and editorial notes. It welcomes high-quality research papers and reviews on any aspect of heat and mass transfer, whether theoretical, numerical, or experimental. The journal is published in English, with accepted papers available online immediately and subsequently in print.

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Heat transfer enhancement in a 3d-printed compact heat exchanger.

1. Introduction

1.1. additive manufacturing, 1.2. heat exchange in complex structures, 2. the experimental facility, 3. experimental data, 4. conclusions.

• The test results confirmed that the HTC value has a slight dependence on the heat flux density during refrigerant condensation. The heat exchange coefficient gradually increases with the heat flux density. However, a four-fold increase in q causes only a 10% increase in the value of the α on the cooled surface.
• The thermal power strongly depends on the type of coolant. The highest values were observed during cooling with a 10% mPCM slurry. The lowest heat flux values were noted for water cooling. It was also found that the heat flux increases during the PCM phase transition. The influence of mass share on the value of heat flux was also observed. The higher the mass fraction of mPCM in the mixture, the greater the heat flux.
• The OHTC also depends on the type of coolant. The lowest values were observed during water cooling. The highest overall heat exchange coefficient values were noted during the 10% mPCM slurry cooling. It was also found that the overall heat exchange coefficient increased during the PCM phase transition. For the 10% mPCM, there was an over 11% increase in the overall heat exchange coefficient. The influence of mass share on the value of heat flux was also observed. The higher the mass fraction of mPCM in the mixture, the greater the heat flux.
• The refrigerant’s HTC values depend on the thickness of the condensate film. The HTC values decrease significantly between 0.00003 and 0.001 m of the thickness of the condensate film.
• The value of the HTC is proportional to the increase in the velocity of the condensate, which results from the laminar nature of the condensate flow. An increase in the condensate velocity reduces the thickness of the condensate layer, which reduces the value of thermal resistance and increases the value of the HTC.
• The authors demonstrated the possibility of using Equation (9) to determine the value of the HTC during condensation of the refrigerant on the surface of a smooth pipe bundle inside a compact heat exchanger made by 3D printing. This equation can be used in compact heat exchanger projects.
• The research results indicate the need to conduct further experimental research on the heat exchange enhancement regarding the impact of process parameters such as Δt log , or the mass fraction of the mPCM in the cooling liquid, and the internal geometry of 3D-printed mini heat exchangers.
• The future direction of the experimental research is to determine the effect of the state of matter of the phase change material mixture on the flow resistance in compact heat exchangers, and the impact of deposits of cooling mixtures on the operating parameters of heat exchangers.

Author Contributions

Data availability statement, conflicts of interest, nomenclature.

• Tang, W.; Zou, C.; Zhou, H.; Zhang, L.; Zeng, Y.; Sun, L.; Zhao, Y.; Yan, M.; Fu, J.; Hu, J.; et al. A novel convective heat transfer enhancement method based on precise control of Gyroid-type TPMS lattice structure. Appl. Therm. Eng. 2023 , 230 , 120797. [ Google Scholar ] [ CrossRef ]
• Reynolds, B.W.; Fee, C.J.; Morison, K.R.; Holland, D.J. Characterisation of Heat Transfer within 3D Printed TPMS Heat Exchangers. Int. J. Heat Mass Transf. 2023 , 212 , 124264. [ Google Scholar ] [ CrossRef ]
• Wu, Y.; Zhi, C.; Wang, Z.; Chen, Y.; Wang, C.; Chen, Q.; Tan, G.; Ming, T. Enhanced thermal and mechanical performance of 3D architected micro-channel heat exchangers. Heliyon 2023 , 9 , e13902. [ Google Scholar ] [ CrossRef ]
• Haertel, J.H.K.; Nellis, G.F. A fully developed flow thermofluid model for topology optimization of 3D-printed air-cooled heat exchangers. Appl. Therm. Eng. 2017 , 119 , 3–24. [ Google Scholar ] [ CrossRef ]
• Ahmadi, B.; Cesarano, J.; Nawaz, K.; Ninos, N.; Bigham, S. A high-performance lung-inspired ceramic 3D-printed heat exchanger for high-temperature energy-efficient systems. Appl. Therm. Eng. 2023 , 219 , 119378. [ Google Scholar ] [ CrossRef ]
• Puttur, U.; Ahmadi, M.; Ahmadi, B.; Bigham, S. A novel lung-inspired 3D-printed desiccant-coated heat exchanger for high-performance humidity management in buildings. Energy Convers. Manag. 2022 , 252 , 115074. [ Google Scholar ] [ CrossRef ]
• Kus, K.; Wójcik, M.; Malecha, Z.; Rogala, Z. Numerical and experimental investigation of the gyroid heat exchanger. Int. J. Heat Mass Transf. 2024 , 231 , 125882, ISSN 0017-9310. [ Google Scholar ] [ CrossRef ]
• Pracht, S.; Will, J.; Klöppel, S.; Funke, T.; Quack, H.; Haberstroh, C. Experimental and numerical study of a 3D-printed aluminium cryogenic heat exchanger for compact Brayton refrigerators. Cryogenics 2022 , 123 , 103418. [ Google Scholar ] [ CrossRef ]
• Dixit, T.; Al-Hajri, E.; Paul, M.C.; Nithiarasu, P.; Kumar, S. High performance, microarchitected, compact heat exchanger enabled by 3D printing. Appl. Therm. Eng. 2022 , 210 , 118339. [ Google Scholar ] [ CrossRef ]
• Shin, D.; Jeon, J.; Kim, T.; Park, J.H.; Kim, S.J. Development of the thermal performance model using temperature gradient analysis for optimized design of steam surface condenser. Int. J. Heat Mass Transf. 2020 , 163 , 120411. [ Google Scholar ] [ CrossRef ]
• Li, Z.; Xing, W.; Sun, J.; Feng, X. Multiscale structural characteristics and Heat–Moisture properties of 3D printed building Walls: A review. Constr. Build. Mater. 2023 , 365 , 130102. [ Google Scholar ] [ CrossRef ]
• Ahmadi, B.; Bigham, S. Performance Evaluation of hi-k Lung-inspired 3D-printed Polymer Heat Exchangers. Appl. Therm. Eng. 2022 , 204 , 117993. [ Google Scholar ] [ CrossRef ]
• Wang, C.S.; Wang, E.S.; Huang, Y.J.; Liou, T.M. PIV and IRT measurements of hydrothermal performance in a U-shaped heat exchanger with 3D printed detached curved ribs. Int. J. Heat Mass Transf. 2023 , 201 , 123562. [ Google Scholar ] [ CrossRef ]
• Khan, H.H.; Aneesh, A.M.; Sharma, A.; Srivastava, A.; Chaudhuri, P. Thermal-hydraulic characteristics and performance of 3D wavy channel based printed circuit heat exchanger. Appl. Therm. Eng. 2015 , 87 , 3–528. [ Google Scholar ] [ CrossRef ]
• Aneesh, A.M.; Sharma, A.; Srivastava, A.; Vyas, K.N.; Chaudhuri, P. Thermal-hydraulic characteristics and performance of 3D straight channel based printed circuit heat exchanger. Appl. Therm. Eng. 2016 , 98 , 3–482. [ Google Scholar ] [ CrossRef ]
• Lowrey, S.; Hughes, C.; Sun, Z. Thermal-hydraulic performance investigation of an aluminium plate heat exchanger and a 3D-printed polymer plate heat exchanger. Appl. Therm. Eng. 2021 , 194 , 117060. [ Google Scholar ] [ CrossRef ]
• Bayaniahangar, R.; Okoh, I.; Nawaz, K.; Cesarano, J.; Bigham, S. Toward extreme high-temperature supercritical CO2 power cycles: Leakage characterization of ceramic 3D-printed heat exchangers. Addit. Manuf. 2022 , 54 , 102783. [ Google Scholar ] [ CrossRef ]
• Zhao, B.; Zhang, J.; Lian, W. Numerical Modeling of Heat Exchanger Filled with Octahedral Lattice Frame Porous Material. Aerospace 2022 , 9 , 238. [ Google Scholar ] [ CrossRef ]
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Kruzel, M.; Bohdal, T.; Dutkowski, K. Heat Transfer Enhancement in a 3D-Printed Compact Heat Exchanger. Energies 2024 , 17 , 4754. https://doi.org/10.3390/en17184754

Kruzel M, Bohdal T, Dutkowski K. Heat Transfer Enhancement in a 3D-Printed Compact Heat Exchanger. Energies . 2024; 17(18):4754. https://doi.org/10.3390/en17184754

Kruzel, Marcin, Tadeusz Bohdal, and Krzysztof Dutkowski. 2024. "Heat Transfer Enhancement in a 3D-Printed Compact Heat Exchanger" Energies 17, no. 18: 4754. https://doi.org/10.3390/en17184754

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Hybrid Nanoparticle-Enhanced Fluid Flow and Heat Transfer Behaviors in a Parabolic Cavity with a Heat Source

• Research Article-Chemical Engineering
• Open access
• Published: 23 September 2024

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• Rasul Mohebbi 1 &
• Yuan Ma 2  

Natural convection of nanofluids holds considerable importance in both scientific research and engineering applications due to their exceptional heat transfer capabilities, which occur spontaneously without the need for additional energy input. In this paper, the natural convection of nanofluid inside a parabolic cavity containing a hot obstacle is studied numerically. The shape of the hot obstacle is selected as either circular or elliptical. Additionally, the effects of the Rayleigh number, nanoparticle volume fraction, and the position of the heat source are investigated. The computational fluid dynamics model was computed using COMSOL Multiphysics. It is observed that the average Nusselt number tends to increase with both the Rayleigh number and the volume fraction of nanoparticles in the fluid. When the heat source moves from the bottom region to the top area, the heat transfer performance of the heat source increases. When Ra ≤ 10 5 , the cases with circular heat sources exhibit better heat transfer performance than those with elliptical heat sources. However, at Ra = 10 6 , the average Nusselt number of the elliptical heat source is higher than that of the circular one.

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1 Introduction

Improving heat transfer efficiency is crucial not only for increasing energy utilization but also for reducing environmental pollution [ 1 , 2 ]. The improvement in heat transfer efficiency of a system is limited by the thermal properties of the heat transfer fluid. Therefore, it is essential to explore heat transfer fluids with excellent thermal properties to enhance heat transfer efficiency. Nanofluid is a composite material formed by suspending nanoparticles in a base fluid. These nanoparticles can be made of various nanomaterials, such as metals, oxides, or carbon nanotubes. Nanofluid possesses many unique properties, such as enhanced thermal conductivity, improved rheological properties, and higher surface activity.

Due to their excellent thermal properties, nanofluids are widely utilized in the field of heat transfer. Yadav et al. [ 3 ] performed experiments and numerical simulations to study the Fe 3 O 4 /water nanofluid heat transfer in a tube. Ma et al. [ 4 ] compared the heat transfer performances of Al 2 O 3 /water and TiO 2 /water nanofluids natural convection inside a U-shaped enclosure. The results indicate that for nanoparticles with identical diameters, Al 2 O 3 exhibits greater effectiveness than TiO 2 in enhancing heat transfer. Furthermore, additional types of nanoparticles, such as CuO[ 5 ], Cu[ 6 ], clove-treated graphene nanoplatelet [ 7 ], and ZnO [ 8 ], have also been investigated in this study.

Besides nanofluids composed of a single type of nanoparticle, those consisting of two or more types are referred to as hybrid nanofluids. Al 2 O 3 -MWCNT/water hybrid nanofluid has been used by Giwa et al. [ 9 ] to enhance the heat transfer performance in a differentially heated square enclosure. The integration of hybrid nanofluids has been noted to improve both the thermal and flow characteristics of the base fluid. As a result, the natural convection behavior of this novel category of nanofluids is improved. Goudarzi et al. [ 10 ] analyzed the effect of nanoparticle migration on the natural convection behavior of Ag-MgO/Water hybrid nanofluid. The findings of the study demonstrate that an increase in thermophoresis diffusion for both types of nanoparticles leads to an 11% increase in the Nusselt number. Reddy et al. [ 11 ] numerically studied the Ag-TiO 2 /water hybrid nanofluid natural convection in an inclined cavity. The Zn-TiO2/water hybrid nanofluid is used by Yasir et al. [ 12 ] to enhance the mixed convective flow. The results from their investigation suggest that the mixed convection parameter exerts a positive influence on rate of heat transport for the upper branch while exerting a negative influence on these factors for the lower branch.

In recent years, there has been increasing attention on natural convection inside wavy enclosures, attributed to the complexity and variability of the boundary. Chuhan et al. [ 13 ] investigated the influence of an inclined magnetic field on entropy generation within natural convective flow in a wavy enclosure filled with a non-Newtonian Casson fluid. They discovered that there is an inverse correlation between the Nusselt number and Lewis number, while the Sherwood number demonstrates a positive correlation with the Lewis number. Rashid et al.[ 14 ] conducted a numerical investigation on the mass and heat transfer characteristics of magnetohydrodynamic (MHD) Casson fluid flow within a wavy cavity containing a circular heat source. They observed that the heat and mass transfer rates decreased as the Hartmann number increased. Roy and Monira [ 15 ] conducted a study on the natural convection behavior of a reacting hybrid nanofluid within an open porous cavity featuring vertical wavy walls while considering the effects of an inclined magnetic field. They observed that regardless of the parameter values, the streamlines within the cavity formed two counter-rotating cells, with the highest intensities occurring near the open end. Geridonmez and Atilgan [ 16 ] utilized machine learning modeling to obtain the average Nusselt number, which resulted from numerical simulations of the natural convection of nanofluid in a wavy cavity. Their findings reveal that trilayer neural network modeling is a feasible alternative for obtaining immediate and expected outcomes, as opposed to repeatedly performing numerical simulations for the average Nusselt number at specific parameter settings. Alazzam et al. [ 17 ] performed a numerical investigation on the natural convection of NePCM confined within a wavy enclosure. Saha et al. [ 18 ] nanofluid natural convection inside a cavity with a top wavy wall and a heated single fin.

Conversely, natural convection within cavities containing internal heat sources has found widespread applications and is gaining increasing attention. Nabwey et al. [ 19 ] conducted a numerical investigation of natural convective flow in an inclined wavy porous cavity filled with a square obstacle. Kumar and Mahapatra [ 20 ] conducted an experimental and numerical analysis focusing on natural convection within a partially open enclosure containing a cylindrical obstacle. Franco et al. [ 21 ] explored the impacts of discrete conductive blocks on natural convection within an open cavity. Their findings revealed that as the solid is progressively diminished in size and augmented in quantity, it hampers the growth of the boundary layer along the hot wall, thereby causing a decrease in the heat transfer process.

To the author's knowledge, no prior research exists on the natural convection of nanofluids within a parabolic-shaped cavity containing a circular heat source. Nonetheless, exploring this phenomenon holds considerable significance owing to the distinctive surface boundary of the parabolic cavity [ 22 , 23 ]. Moreover, the coupling effect between internal heat sources and nanofluids is expected to significantly influence both flow patterns and heat transfer processes. This study investigates the natural convection of nanofluids within a parabolic-shaped cavity featuring a circular heat source. We examine the impacts of Rayleigh number (Ra), volume fraction of nanoparticles, positions, and shapes of obstacles on flow characteristics and heat transfer rates.

2 Physical and Mathematical Depiction

Figure  1 shows the schematic diagram of the present work. As seen in this figure, a heat source is placed inside a parabolic-shaped enclosure. It is worth noting that, in the current study, the heat source has two shapes: circular and elliptical.

Schematic diagram of nanofluid natural convection inside a parabolic-shaped enclosure with an obstacle

The MWCNT-Fe 3 O 4 /water hybrid nanofluid is used in the present work to enhance the heat transfer performance. The parabolic shape of the cavity is determined by the following equation:

where H represents the height of the cavity, and L represents the width of the cavity.

As illustrated in the figure, the heat source is located at the position along the central axis of the cavity, at a distance (a) from the upper wall. The top wall surface of the cavity serves as the cold wall, with a fixed temperature of T c . The heat source surface is also set at a fixed temperature, T h . The remaining walls, which are the curved surfaces of the cavity, serve as adiabatic walls.

3 Numerical Method and Strategy

When the fluid is subjected to non-uniform temperature and gravity fields, it often exhibits natural convective transport phenomena observed in various real-world scenarios. This natural phenomenon drives the circulation of fluids and the transfer of energy within a system. Specifically, the Boussinesq approximation can govern fluid and heat transport in the system by utilizing continuity, momentum, and energy equations. These equations describe the relationships between various parameters such as the velocity vector ( u ), density of the nanofluid ( \({\rho }_{nf}\) ), time ( t ), pressure ( p ), dynamic viscosity of nanofluid ( \({\mu }_{nf}\) ), thermal conductivity of nanofluid ( \({k}_{nf}\) ), specific heat capacity at constant pressure ( C p ), and the local ( T ) and reference temperatures ( T r ). By analyzing and understanding these equations, we can gain valuable insights into the dynamics of the natural convective transport phenomenon and how it affects fluid and energy transfer in different systems. The continuity, momentum, and energy equations can be obtained as follows:

The dimensional analysis of the governing equations mentioned earlier reveals that the problem is primarily characterized by two dimensionless parameters: the Rayleigh number and the Prandtl number. The former is determined by the thermal buoyancy force ( β ), the gravitational acceleration ( g ), the height of the cavity ( H ), and the temperature difference between the hot and cold walls ( T h and T c ), divided by the product of kinematic viscosity ( v ), and thermal diffusivity ( α ), hence expressed as:

Likewise, the latter is measured by the ratio of kinematic viscosity, v, to thermal diffusivity, α, as indicated:

For simulating nanofluids, a single-phase nanofluid model is employed. Specific details can be referred to in previous literature 24 . Table 1 displays the thermal properties of MWCNT-Fe 3 O 4 /water hybrid nanofluid at different volume fractions.

To quantitatively evaluate the heat transfer characteristics of the system and assess the impact of different parameters on these characteristics, we introduce the Nusselt number. The Nusselt number allows for the determination of the convective heat transfer rate by comparing the conductive and convective heat transfer at a fluid–solid interface. The local Nusselt number ( Nu loc ) which is defined as

where n is the coordinate normal to the wall. In some cases, it is necessary to determine an average Nusselt number over a given time or spatial domain. The average Nusselt number, also called Nu ave , is obtained by averaging Nu loc over both time and space. Its value provides insight into the overall heat transfer characteristics of the system being studied. Importantly, it should be noted that for steady-state flow (when conditions within the system are unchanging over time), the average Nusselt number is solely determined by the spatial average of Nu loc . This is because the time-averaged Nusselt number is equal to the steady-state Nusselt number.

4 Validation and Grid Independence

Figure  2 shows the isothermal lines and streamlines of a benchmark problem of natural convection inside a square cavity at different Rayleigh numbers. The present results can be compared with the works by Davis 25 . By comparison, it can be observed that both the isothermal lines and streamlines are consistent with the findings from previous literature. To quantitatively assess the accuracy of the obtained results, we performed calculations for natural convection inside a square cavity with a circular heat source. The average Nusselt number over the surface of the circular heat source was then calculated. Table 2 shows the comparisons of the average Nusselt numbers.

The isothermal lines and streamlines by the present work [ 25 ]

To select a suitable grid while ensuring computational accuracy and precision, a grid-independent study was conducted. The case of a circular heat source located above the cavity with φ = 0.003 and Ra = 10 6 was simulated using various grids. Table 3 illustrates the average Nusselt number obtained on the heated surface for different mesh numbers. From the figure, it can be observed that the difference in the average Nusselt numbers calculated for grid resolutions of 16,785 and 21,461 is 0.013. Therefore, to account for both the accuracy of the computational results and computational resources, the grid resolution of 16,785 is adopted.

5 Numerical Results and Discussions

In this section, we will investigate the effects of different Rayleigh numbers, volume fractions of nanofluids, positions of the heat source, and shapes of the heat source on flow dynamics, temperature fields, and heat transfer characteristics.

Figure  3 depicts the comparison of streamlines between the pure water (φ = 0) and nanofluid (φ = 0.03) for the case when the circular heat source located on the bottom of the cavity for low (Ra = 10 3 ) and high Rayleigh numbers (Ra = 10 6 ). Firstly, due to the symmetry of the geometry and the steady-state nature of the flow, the flow structure within the cavity is also symmetrical. At Ra = 10 3 , the shape of the vortex is approximately elliptical, and the streamlines around the vortex are nearly identical, with only variations in size. When the Ra increases to 10 6 , the shape of the streamlines in the inner and outer layers undergoes a change. The streamlines in the outer layer become closer to the shape of the wall boundary. This is due to the increase in convection intensity caused by the increase in the Rayleigh number. The subsequent analyses will involve a quantitative assessment of the impact of the nanofluid.

Comparison of streamlines for the case when the circular heat source is located on the bottom of the cavity for different Ra

Figure  4 illustrates the isothermal lines for the case when the circular heat sources are located on the bottom for different Ra at φ = 0 and φ = 0.003. It should be noted that the color bar used in the temperature contour maps remains consistent. Therefore, the color bar will not be displayed in the subsequent sections. It can be observed that at Ra = 10 3 , the main heat transfer mode is conducted through heat conduction due to the weak convective intensity and low flow velocity within the cavity compared to Ra = 10 6 . Therefore, the isotherms are relatively flat and evenly spaced. When the Ra increases to 10 6 , thermal plumes form above the heat source and directly impinge on the top wall of the cavity, leading to the formation of two vortices on the left and right sides. It is worth noting that at high Rayleigh numbers, due to the generation of thermal plumes and the increase in convective intensity, the isotherms are no longer evenly distributed.

Isothermal lines for the case when the circular heat source located on the bottom of the cavity for different Ra at φ = 0 and φ = 0.003

To quantitatively evaluate the heat transfer performance of the system, Fig.  5 presents the Nusselt number for the case when the circular heat source is located at the bottom for different values of φ and Ra. It can be observed that the average Nu increases with increasing φ and Ra. It is noteworthy that as Ra increases, the improvement in heat transfer characteristics achieved by using nanofluids becomes more pronounced compared to pure fluids.

The Nusselt number for the case when the circular heat source is located on the bottom for different φ and Ra

On another hand, in addition to the Rayleigh number and the nanoparticle volume fraction φ, this paper also investigates the impact of the vertical position of the heat source. Figure  6 shows the streamlines and isothermal lines for the case when the circular heat source is located on the top of the cavity for different Ra at φ = 0. It can be observed that at Ra = 10 3 and 10 4 , the streamlines and isothermal lines exhibit similar patterns. The distribution of the isothermal lines appears relatively uniform, without the presence of thermal plumes. However, as the Ra increases to 10 5 and 10 6 , the isothermal lines become noticeably curved, indicating the presence of significant thermal plumes. The streamlines also indicate an evident increase in convection intensity. Based on the analysis above, it can be concluded that when the Ra is below 10 4 , the dominant heat transfer mechanism is conduction.

Streamlines and isothermal lines for the case when the circular heat source is located on the top of the cavity for different Ra at φ = 0

On the other hand, when Ra exceeds 10 5 , the primary heat transfer mechanism is thermal convection. Therefore, the critical Ra at which the transition between these two heat transfer mechanisms occurs lies between 10 4 and 10 5 . Additionally, according to Fig.  3 , it can be observed that when the heat source is located below the cavity, even at a high Ra number (10 6 ), only two vortices form within the cavity. However, when the heat source is positioned at the top (Fig.  6 ), in close proximity to the cold top wall, two smaller vortices form between the cold and heat sources. This corresponds to the presence of two thermal plumes at the top of the heat source.

Figure  7 shows the average Nusselt numbers at different φ and Ra values when the heat source is located at the top. By comparing Fig.  5 and Fig.  7 , it can be observed that when the heat source moves from the bottom to the top, the average Nusselt number increases at any Ra value. This is because when the heat source is located at the top, it is closer to the cold source, leading to a significant improvement in heat transfer characteristics.

The Nusselt number for the case when the circular heat source is located on the top for different φ and Ra

Figure  8 presents the streamlines and isothermal lines for the scenario where the circular heat source is positioned in the middle of the cavity, with Ra = 10 5 and φ = 0.001. From the streamlines and isothermal lines, it can be observed that thermal plumes have emerged, and the main mechanism of heat transfer is convection. This is consistent with the previous analysis, which indicates that the change in the heat transfer mechanism is mainly affected by the Rayleigh number, with little influence from the heat source position.

Streamlines and isothermal lines for the case when the circular heat source located in the middle of the cavity for Ra = 10. 5 and φ = 0.001

To further analyze the impact of the transition in heat transfer mechanisms on heat transfer efficiency, a plot of the variation in local Nusselt number along the surface of the heat source is depicted when the obstacle is in the middle of the cavity. Figure  9 illustrates the distribution of the local Nusselt number on the heated surface at φ = 0.001 for varying Rayleigh numbers. It can be clearly observed that the local Nusselt number remains relatively low and exhibits minimal variation when the Rayleigh number is equal to 10 3 and 10 4 . However, for Ra = 10 5 and 10 6 , the curve of Nu loc exhibits a consistent pattern, featuring a trough corresponding to the location of thermal plumes on the heated surface. This is due to different heat transfer mechanisms. As above-mentioned, at Ra = 10 3 , heat conduction is the primary heat transfer mechanism. As shown in Fig.  4 , the isotherms are relatively evenly distributed, with the density of isotherms near the heat source directly above significantly higher than on both sides, resulting in a higher Nu loc . However, when Ra = 10 6 , the convective intensity significantly increases, forming plumes above the heat source, creating two vortices on the left and right. The cold fluid will impact the sides of the heat source, leading to higher Nu loc on both sides. The occurrence of thermal plumes causes a significant increase in the Nu loc in the surrounding area, resulting in a substantial enhancement of the average Nusselt number.

The distribution of local Nusselt number on the heated surface at φ = 0.001 for different Ra

Figure  10 shows the Nusselt number for the case when the circular heat source is located in the middle for different φ and Ra. By comparing with Fig.  5 and Fig.  7 , it can be observed that when the heat source is positioned at the center of the cavity, the heat transfer characteristics also lie in the intermediate range. In other words, as the heat source moves upward from the bottom, there is an increase in the heat transfer characteristics.

The Nusselt number for the case when the circular heat source is located in the middle for different φ and Ra

The influence of the shape of the heat source was introduced in the following sections. Figure  11 shows the streamlines inside the cavity with circular and elliptic heat sources at φ = 0.003 and different Ra. As previously mentioned, when the heat source is circular, there is no need to elaborate further here; this section will focus solely on the impact of shape variations. When the shape of the heat source changes from circular to elliptical, the number of vortices remains unchanged, but the shape of the vortices differs due to variations in the boundary.

Streamlines inside the cavity with a circular and elliptical heat source at φ = 0.003 and different Ra

The similar changes can be observed in the temperature distribution. Figure  12 depicts the temperature distribution inside the cavity with circular and elliptical heat sources at φ = 0.003 and different Ra. When the heat source changes into an elliptical shape, at low Ra (10 3 ), the temperature distribution remains similar with only a change in shape. However, due to an increase in length in the horizontal direction, the distance at which thermal plumes occur increases at high Ra (10 6 ).

Temperature distribution inside the cavity with circular and elliptical heat sources at φ = 0.003 and different Ra

Figure  13 shows the average Nusselt number of the circular and elliptical heat sources at φ = 0.003 and different Ra. It can be observed that Nu ave increases with increasing Ra and φ, regardless of the circular or elliptical heat source. Moreover, at Ra ≤ 10 5 , the case with circular heat sources exhibits better heat transfer performance. However, at Ra = 10 6 , the average Nusselt number of the elliptical heat source is higher than that of the circular one.

The average Nusselt number of the circular and elliptical heat source at φ = 0.003 and different Ra

6 Conclusion

In this study, we investigate the natural convection of nanofluid inside a parabolic cavity with a heat source, which can be either circular or elliptical in shape. We analyze the impact of Ra, nanoparticle volume fraction, and heat source position on the convection process. Our observations reveal that the average Nusselt number increases with higher Rayleigh numbers and nanoparticle volume fractions. Additionally, relocating the heat source from the bottom region to the top area enhances its heat transfer performance. For Rayleigh numbers less than or equal to 10 5 , the circular heat source exhibits better heat transfer performance compared to the elliptical heat source. However, at a Rayleigh number of 10 6 , the average Nusselt number of the elliptical heat source surpasses that of the circular one.

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Rasul Mohebbi

Department of Civil and Environmental Engineering, The Hong Kong Polytechnic University, Hong Kong SAR, People’s Republic of China

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Mohebbi, R., Ma, Y. Hybrid Nanoparticle-Enhanced Fluid Flow and Heat Transfer Behaviors in a Parabolic Cavity with a Heat Source. Arab J Sci Eng (2024). https://doi.org/10.1007/s13369-024-09586-2

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Received : 19 March 2024

Accepted : 09 September 2024

Published : 23 September 2024

DOI : https://doi.org/10.1007/s13369-024-09586-2

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