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Topic Editors

Laboratory on Convective Heat and Mass Transfer and Department of Theoretical Mechanics, Tomsk State University, 36 Lenin Ave., 634050 Tomsk, Russia
Laboratoire de Thermique Interfaces Environnement (LTIE), Université de Paris, Paris, France

Heat Transfer Enhancement and Applications

Abstract submission deadline
closed (31 August 2023)
Manuscript submission deadline
closed (30 October 2023)
Viewed by
24771

Topic Information

Dear Colleagues,

Heat transfer enhancement is a crucial challenge for engineers and scientists from different technical fields. Such improvement can be achieved using various aspects, including heat transfer fluids, special heat transfer surfaces, and additional engineering devices. Analysis of such elements can be considered using experimental methods or computational procedures. The first approach allows studying the considered phenomenon in physical life, while the second allows developing the mathematical model and studying the phenomenon in a wide range of governing parameters. Obtained outcomes can be used not only in mechanical engineering and power engineering, but also in building engineering, chemical engineering, medicine, and fundamental and practical science.

This article collection aims to bring together research on heat transfer enhancement and its applications in different fields of our life. Original research, as well as review articles and short communications, with a particular focus on (but not limited to) heat transfer enhancement and energy transport applications, are welcomed.

Prof. Dr. Mikhail Sheremet
Prof. Dr. Abderrahmane Baïri
Topic Editors

Keywords

  • convection
  • conduction
  • radiation
  • heat transfer enhancement
  • mass transfer
  • porous media
  • phase change material
  • nanofluids
  • extended heat transfer surfaces
  • fins
  • ribs
  • targeted drug delivery
  • heat exchangers
  • solar collectors

Participating Journals

Journal Name Impact Factor CiteScore Launched Year First Decision (median) APC
Coatings
coatings
2.9 5.0 2011 14.5 Days CHF 2600
Energies
energies
3.0 6.2 2008 16.8 Days CHF 2600
Entropy
entropy
2.1 4.9 1999 22.3 Days CHF 2600
Foundations
foundations
- - 2021 32.9 Days CHF 1000
Nanomaterials
nanomaterials
4.4 8.5 2010 14.1 Days CHF 2900
Symmetry
symmetry
2.2 5.4 2009 17.3 Days CHF 2400
Thermo
thermo
- 2.1 2021 22.8 Days CHF 1000

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Published Papers (8 papers)

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26 pages, 6510 KiB  
Review
Design and Optimization of Heat Sinks for the Liquid Cooling of Electronics with Multiple Heat Sources: A Literature Review
by Yijun Li, Stéphane Roux, Cathy Castelain, Yilin Fan and Lingai Luo
Energies 2023, 16(22), 7468; https://doi.org/10.3390/en16227468 - 7 Nov 2023
Cited by 18 | Viewed by 5255
Abstract
This paper presents a detailed literature review on the thermal management issue faced by electronic devices, particularly concerning uneven heating and overheating problems. Special focus is given to the design and structural optimization of heat sinks for efficient single-phase liquid cooling. Firstly, the [...] Read more.
This paper presents a detailed literature review on the thermal management issue faced by electronic devices, particularly concerning uneven heating and overheating problems. Special focus is given to the design and structural optimization of heat sinks for efficient single-phase liquid cooling. Firstly, the paper highlights the common presence and detrimental consequences of electronics overheating resulting from multiple heat sources, supported by various illustrative examples. Subsequently, the emphasis is placed on single-phase liquid cooling as one of the effective thermal management technologies for power electronics, as well as on the enhancement of heat transfer in micro/mini channel heat sinks. Various studies on the design and structural optimization of heat sinks are then analyzed and categorized into five main areas: (1) optimization of channel cross-section shape, (2) optimization of channel flow passage, (3) flow distribution optimization for parallel straight channel heat sinks, (4) optimization of pin-fin shape and arrangement, and (5) topology optimization of global flow configuration. After presenting a broad and complete overview of the state of the art, the paper concludes with a critical analysis of the methods and results from the literature and highlights the research perspectives and challenges in the field. It is shown that the issue of uneven and overheating caused by multiple heat sources, which is commonly observed in modern electronics, has received less attention in the literature compared to uniform or single-peak heating. While several design and structural optimization techniques have been implemented to enhance the cooling performance of heat sinks, topology optimization has experienced significant advancements in recent years and appears to be the most promising technology due to its highest degree of freedom to treat the uneven heating problem. This paper can serve as an essential reference contributing to the development of liquid-cooling heat sinks for efficient thermal management of electronics. Full article
(This article belongs to the Topic Heat Transfer Enhancement and Applications)
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Figure 1
<p>Lithium-ion battery package with array arrangement of battery cells. (<b>a</b>) cylindrical type [<a href="#B9-energies-16-07468" class="html-bibr">9</a>]; (<b>b</b>) prismatic type [<a href="#B5-energies-16-07468" class="html-bibr">5</a>]; camera-measured temperature contours of the single battery cell (<b>c</b>) at the initial stage of discharging and (<b>d</b>) at the end of discharging, under a constant current [<a href="#B10-energies-16-07468" class="html-bibr">10</a>]. Reprinted/adapted with permission from Refs. [<a href="#B5-energies-16-07468" class="html-bibr">5</a>,<a href="#B9-energies-16-07468" class="html-bibr">9</a>,<a href="#B10-energies-16-07468" class="html-bibr">10</a>], October 2023, Elsevier.</p>
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<p>LED package. (<b>a</b>,<b>b</b>) array arrangement of LED chips and layer structure [<a href="#B15-energies-16-07468" class="html-bibr">15</a>]; (<b>c</b>) temperature distribution on the surface of a microchannel heat sink [<a href="#B16-energies-16-07468" class="html-bibr">16</a>]. Reprinted/adapted with permission from Refs. [<a href="#B15-energies-16-07468" class="html-bibr">15</a>,<a href="#B16-energies-16-07468" class="html-bibr">16</a>], October 2023, Elsevier.</p>
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<p>Thermal management for IGBT module. (<b>a</b>) a real opened IGBT module [<a href="#B19-energies-16-07468" class="html-bibr">19</a>]; (<b>b</b>) simulated temperature distribution of IGBT module [<a href="#B20-energies-16-07468" class="html-bibr">20</a>]; (<b>c</b>) air-cooling of IGBT module using a heat sink [<a href="#B21-energies-16-07468" class="html-bibr">21</a>].</p>
Full article ">Figure 4
<p>Thermal management issue for CPU chips. (<b>a</b>) an SCM with a heat sink [<a href="#B24-energies-16-07468" class="html-bibr">24</a>]; (<b>b</b>) multiple-peak heat generation [<a href="#B25-energies-16-07468" class="html-bibr">25</a>]. Reprinted/adapted with permission from Refs. [<a href="#B24-energies-16-07468" class="html-bibr">24</a>,<a href="#B25-energies-16-07468" class="html-bibr">25</a>], October 2023, Elsevier.</p>
Full article ">Figure 5
<p>High concentrator photovoltaics (HCPV) system cooled by a liquid multi-channel heat sink [<a href="#B34-energies-16-07468" class="html-bibr">34</a>]. (<b>a</b>) schematic diagram and working principle of HCPV; (<b>b</b>) power map of an array of CPV cell modules. Reprinted/adapted with permission from Ref. [<a href="#B34-energies-16-07468" class="html-bibr">34</a>], October 2023, Elsevier.</p>
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<p>Different categories and variants of heat sink for liquid cooling. (<b>a</b>) parallel straight channel [<a href="#B37-energies-16-07468" class="html-bibr">37</a>]; (<b>b</b>) parallel wavy channel [<a href="#B37-energies-16-07468" class="html-bibr">37</a>]; (<b>c</b>) pin-fin structure [<a href="#B41-energies-16-07468" class="html-bibr">41</a>]; (<b>d</b>) straight channel with cavities [<a href="#B38-energies-16-07468" class="html-bibr">38</a>]; (<b>e</b>) straight channel with ribs [<a href="#B39-energies-16-07468" class="html-bibr">39</a>]; (<b>f</b>) Complex (hybrid) structure [<a href="#B42-energies-16-07468" class="html-bibr">42</a>,<a href="#B43-energies-16-07468" class="html-bibr">43</a>]. Reprinted/adapted with permission from Refs. [<a href="#B37-energies-16-07468" class="html-bibr">37</a>,<a href="#B38-energies-16-07468" class="html-bibr">38</a>,<a href="#B39-energies-16-07468" class="html-bibr">39</a>,<a href="#B41-energies-16-07468" class="html-bibr">41</a>,<a href="#B42-energies-16-07468" class="html-bibr">42</a>,<a href="#B43-energies-16-07468" class="html-bibr">43</a>], October 2023, Elsevier.</p>
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<p>Local flow disturbance generated by different channel structuration for heat transfer enhancement. (<b>a</b>) zigzag and wary channel [<a href="#B44-energies-16-07468" class="html-bibr">44</a>]; (<b>b</b>) cavity channel [<a href="#B45-energies-16-07468" class="html-bibr">45</a>]; (<b>c</b>) ribs on the bottom walls [<a href="#B46-energies-16-07468" class="html-bibr">46</a>]; (<b>d</b>) ribs on the side walls [<a href="#B47-energies-16-07468" class="html-bibr">47</a>]. Reprinted/adapted with permission from Refs. [<a href="#B44-energies-16-07468" class="html-bibr">44</a>,<a href="#B46-energies-16-07468" class="html-bibr">46</a>], October 2023, Elsevier.</p>
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<p>Cross-section optimization of straight channel heat sinks. (<b>a</b>) single channel cross-section geometrical parameters optimization [<a href="#B59-energies-16-07468" class="html-bibr">59</a>]; (<b>b</b>) single channel cross-section shape optimization [<a href="#B67-energies-16-07468" class="html-bibr">67</a>]; (<b>c</b>) entire cross-section topology optimization [<a href="#B68-energies-16-07468" class="html-bibr">68</a>]. Reprinted/adapted with permission from Refs. [<a href="#B67-energies-16-07468" class="html-bibr">67</a>,<a href="#B68-energies-16-07468" class="html-bibr">68</a>], October 2023, Elsevier.</p>
Full article ">Figure 9
<p>Different types of heat sink channel flow passage optimization. (<b>a</b>) backward right-angled trapezoidal grooves in channel sidewalls [<a href="#B79-energies-16-07468" class="html-bibr">79</a>]; (<b>b</b>) channels added with fins [<a href="#B86-energies-16-07468" class="html-bibr">86</a>]; (<b>c</b>) channels with cross-rib [<a href="#B81-energies-16-07468" class="html-bibr">81</a>]; (<b>d</b>) channels with deflectors [<a href="#B89-energies-16-07468" class="html-bibr">89</a>]; (<b>e</b>) Channels with various cavities [<a href="#B88-energies-16-07468" class="html-bibr">88</a>]; (<b>f</b>) free-shape twisted channel [<a href="#B90-energies-16-07468" class="html-bibr">90</a>]. Reprinted/adapted with permission from Refs. [<a href="#B79-energies-16-07468" class="html-bibr">79</a>,<a href="#B88-energies-16-07468" class="html-bibr">88</a>,<a href="#B89-energies-16-07468" class="html-bibr">89</a>,<a href="#B90-energies-16-07468" class="html-bibr">90</a>], October 2023, Elsevier.</p>
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<p>Different arrangements of global inlet-outlet position for parallel straight channel heat sink [<a href="#B99-energies-16-07468" class="html-bibr">99</a>]. Reprinted/adapted with permission from Ref. [<a href="#B99-energies-16-07468" class="html-bibr">99</a>], October 2023, Elsevier.</p>
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<p>Design and structuration of the manifolds (headers) by (<b>a</b>) pin-fin structure [<a href="#B103-energies-16-07468" class="html-bibr">103</a>], (<b>b</b>) baffle [<a href="#B104-energies-16-07468" class="html-bibr">104</a>], and (<b>c</b>) baffle and mini-baffle [<a href="#B107-energies-16-07468" class="html-bibr">107</a>]. Reprinted/adapted with permission from Refs. [<a href="#B103-energies-16-07468" class="html-bibr">103</a>,<a href="#B104-energies-16-07468" class="html-bibr">104</a>,<a href="#B107-energies-16-07468" class="html-bibr">107</a>], October 2023, Elsevier.</p>
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<p>Structural optimization of pin-fin type heat sinks. (<b>a</b>) pin-fin size and spacing optimization [<a href="#B131-energies-16-07468" class="html-bibr">131</a>]; (<b>b</b>) single fin-shape optimization [<a href="#B132-energies-16-07468" class="html-bibr">132</a>]; (<b>c</b>) topology optimization [<a href="#B133-energies-16-07468" class="html-bibr">133</a>]. Reprinted/adapted with permission from Refs. [<a href="#B131-energies-16-07468" class="html-bibr">131</a>,<a href="#B132-energies-16-07468" class="html-bibr">132</a>], October 2023, Elsevier.</p>
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<p>TO of global channel configuration of heat sinks. (<b>a</b>) flow channel generation by density-based gradient adjoint optimizer [<a href="#B158-energies-16-07468" class="html-bibr">158</a>]; (<b>b</b>) level-set TO for thermal-fluid problem [<a href="#B159-energies-16-07468" class="html-bibr">159</a>]; (<b>c</b>) non-gradient GA-based TO approach [<a href="#B147-energies-16-07468" class="html-bibr">147</a>]; (<b>d</b>) non-gradient Bayesian method for TO of heat sink in natural convection [<a href="#B145-energies-16-07468" class="html-bibr">145</a>]. Reprinted/adapted with permission from Refs. [<a href="#B147-energies-16-07468" class="html-bibr">147</a>,<a href="#B158-energies-16-07468" class="html-bibr">158</a>], October 2023, Elsevier; from Ref. [<a href="#B159-energies-16-07468" class="html-bibr">159</a>], October 2023, John Wiley and Sons.</p>
Full article ">
19 pages, 7229 KiB  
Article
Assessment of Semi-Empirical Soot Modelling in Turbulent Buoyant Pool Fires from Various Fuels
by Lahna Acherar, Hui-Ying Wang, Bruno Coudour and Jean Pierre Garo
Thermo 2023, 3(3), 424-442; https://doi.org/10.3390/thermo3030026 - 7 Aug 2023
Cited by 2 | Viewed by 1368
Abstract
The objective of this work is to assess the accuracy and limitations of two different semi-empirical soot models: the Laminar Smoke Point (LSP) and soot-yield approach. A global soot formation model based on the LSP concept is embedded within FDS6.7. Quantitative comparisons were [...] Read more.
The objective of this work is to assess the accuracy and limitations of two different semi-empirical soot models: the Laminar Smoke Point (LSP) and soot-yield approach. A global soot formation model based on the LSP concept is embedded within FDS6.7. Quantitative comparisons were made from turbulent buoyant pool fires between several computational results and well-instrumented experimental databases on the soot volume fraction, mass loss rate, heat release rate and gas temperature. The LSP model in combination with soot oxidation and surface growth is validated for most of the methane, ethylene and heptane turbulent buoyant pool fires, covering a wide range of fuel likely to form soot. This paper aims to broaden the scope of the validation of the available semi-empirical soot modelling. For the porous methane and ethylene burner, the LSP model was found to provide a better description of the soot volume fraction. The overall visual soot distribution is also numerically reproduced with the soot-yield approach, but as expected, there are some differences between the prediction and the measurement regarding the magnitude of soot volume fraction. The computed radiant heat flux was compared with experimental data for heptane flame, showing that predictions using both the LSP and soot-yield models were found to be twice the value of experimental data, although the measured HRR (Heat Release Rate) is reliably reproduced in the numerical simulation. For the heptane buoyant pool fires, a sufficient accuracy of the numerical model is confirmed only in some of the locations as compared to the experimental results. It is demonstrated that neither the temperature nor the soot volume fraction can be reliably calculated in the necking flame flapping region when the pyrolysis rate of condensed fuel (heptane) is coupled with radiation/convection heat feedback. This implies that an accuracy of prediction on the turbulent buoyant pool fires depends on the studied fire scenario regardless of the semi-empirical soot models. Full article
(This article belongs to the Topic Heat Transfer Enhancement and Applications)
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Figure 1
<p>Computational domain and coordinate system for the numerical simulation.</p>
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<p>Computed fields of typical temperature (°C) and soot volume fraction above methane burner.</p>
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<p>Comparison between the predicted and measured [<a href="#B10-thermo-03-00026" class="html-bibr">10</a>] radial soot volume fraction profiles at 4 different heights for methane burner.</p>
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<p>Computed fields of the typical temperature (°C) above ethylene burner.</p>
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<p>Computed fields of soot volume fraction above ethylene burner.</p>
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<p>Comparison between the predicted and measured [<a href="#B10-thermo-03-00026" class="html-bibr">10</a>] radial profiles of soot volume fraction at 4 different heights of 1D, 2D, 3D and 4D above the burner, where D is the diameter of the burner.</p>
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<p>Computed pyrolysis rate and heat release rate for heptane pool fire.</p>
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<p>Computed time-averaged temperature fields with different soot models for heptane pool fire.</p>
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<p>Computed fields of soot volume fraction above heptane pool fire of 30 cm for the different soot models.</p>
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<p>Comparison between the predicted and measured radial soot volume fraction profiles and its RMS component at 4 different heights.</p>
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<p>Comparison between prediction and experiment for the radial distribution of mean temperature and its RMS component at 4 different heights.</p>
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<p>Comparison between the computed and measured MLR and HRR.</p>
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<p>Predicted temperature (°C) fields above heptane pool fire of 23 cm for the different soot models.</p>
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<p>Experimental temperature (°C) field above heptane pool fire of 23 cm.</p>
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<p>Comparison between the numerical and experimental profiles of gas temperature along the axis of the heptane pool fire at five radial positions.</p>
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<p>Predicted fields of soot volume fraction above heptane pool fire of 23 cm by using different soot models.</p>
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<p>Computed and measured radiant heat flux over the heptane surface.</p>
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14 pages, 8433 KiB  
Article
Stability and Photothermal Properties of Fe3O4-H2O Magnetic Nanofluids
by Chengya Zhang, Lei Gao, Xiaofeng Zhou and Xiaohu Wu
Nanomaterials 2023, 13(13), 1962; https://doi.org/10.3390/nano13131962 - 28 Jun 2023
Cited by 8 | Viewed by 1700
Abstract
Solar collectors are more efficient and commercial devices for collecting solar energy, compared to other solar energy utilizations. To improve the efficiency of solar collectors, it is important to prepare a liquid heat-collecting medium, which is stable and has high photothermal properties. Therefore, [...] Read more.
Solar collectors are more efficient and commercial devices for collecting solar energy, compared to other solar energy utilizations. To improve the efficiency of solar collectors, it is important to prepare a liquid heat-collecting medium, which is stable and has high photothermal properties. Therefore, in this work, we develop a droplet–droplet mixing technique to prepare Fe3O4-H2O magnetic nanofluid. The results show that magnetic nanofluids prepared using the droplet–droplet mixing technique have more stable performance and a better encapsulation of dispersants than those prepared via traditional liquid–liquid mixing. Then, the thermal conductivity and photothermal properties of Fe3O4-H2O magnetic nanofluids are investigated experimentally and theoretically. The thermal conductivity and temperature of the magnetic nanofluid with Fe3O4 nanoparticles of a 1.0% volume fraction can reach the maximum value of 0.95 W/m∙K and 73.9 °C when the magnetic field strength is equal to the saturation magnetic field of 800 Gs. These findings provide insights into the potential applications of Fe3O4-H2O magnetic nanofluids in direct absorption solar collectors, heat exchangers, automobile radiators, etc. Full article
(This article belongs to the Topic Heat Transfer Enhancement and Applications)
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Figure 1
<p>Comparison of the mixing technique in the second step, (<b>a</b>) the traditional liquid–liquid mixing technique and (<b>b</b>) the droplet–droplet mixing technique.</p>
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<p>Characterization diagram of Fe<sub>3</sub>O<sub>4</sub> nanoparticles. (<b>a</b>) XRD pattern; (<b>b</b>) SEM pattern; and (<b>c</b>) size distributions of Fe<sub>3</sub>O<sub>4</sub> nanoparticles with different sizes.</p>
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<p>Stability of magnetic nanofluids. (<b>a</b>) The magnetic nanofluids with different volume fractions and (<b>b</b>) zeta potential versus time for magnetic nanofluids.</p>
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<p>Effects of the magnetic field on the distribution of magnetic particles (<b>a</b>) without the magnetic field and (<b>b</b>) with the magnetic field; <span class="html-italic">l</span> is the length of the chain-like aggregate, and <span class="html-italic">w</span> refers to the width of the chain-like aggregate.</p>
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<p>(<b>a</b>) The schematic diagram for measuring the photothermal property and (<b>b</b>) the cross-section of the test tube. One probe is located at 7 mm, and another probe is located at 22 mm.</p>
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<p>Influence of the magnetic field on the effective thermal conductivity of the Fe<sub>3</sub>O<sub>4</sub>-H<sub>2</sub>O magnetic nanofluids for various fractions.</p>
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<p>Theoretical results for the temperature of the magnetic nanofluids as a function of the applied magnetic field for (<b>a</b>) <span class="html-italic">f</span> = 0.2%, (<b>b</b>) <span class="html-italic">f</span> = 0.5%, and (<b>c</b>) <span class="html-italic">f</span> = 1.0%.</p>
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<p>Temperature of Fe<sub>3</sub>O<sub>4</sub> magnetic nanofluids with different volume fractions: (<b>a</b>) 0.2%, (<b>b</b>) 0.5%, and (<b>c</b>) 1.0% for <span class="html-italic">H</span> = 0 Gs, and (<b>d</b>) 0.2%, (<b>e</b>) 0.5%, and (<b>f</b>) 1.0% for <span class="html-italic">H</span> = 700 Gs.</p>
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<p>Experimental results on the temperature of magnetic nanofluids at <span class="html-italic">h</span> = 7 mm under different magnetic fields for (<b>a</b>) <span class="html-italic">f</span> = 0.2% and (<b>b</b>) <span class="html-italic">f</span> = 0.5%.</p>
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<p>The photothermal conversion efficiency of Fe<sub>3</sub>O<sub>4</sub>-H<sub>2</sub>O magnetic nanofluids with different volume fractions.</p>
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15 pages, 6147 KiB  
Article
Experimental and Numerical Investigation of the Heat Transfer of Honeycomb-Structured Tubes
by Eileen Trampe, Dominik Büschgens and Herbert Pfeifer
Thermo 2023, 3(2), 331-345; https://doi.org/10.3390/thermo3020021 - 20 Jun 2023
Viewed by 1680
Abstract
Tube bundle recuperators are generally designed to operate with smooth tubes. Structured tubes can be used to increase the efficiency of recuperators. Compared to smooth tubes, the surface for heat transfer is increased and thus heat transfer is enhanced. This effect is accompanied [...] Read more.
Tube bundle recuperators are generally designed to operate with smooth tubes. Structured tubes can be used to increase the efficiency of recuperators. Compared to smooth tubes, the surface for heat transfer is increased and thus heat transfer is enhanced. This effect is accompanied by an increased pressure loss, which must be kept as low as possible. Four tube geometries with different honeycomb structures are examined. The results are compared with the performance of a smooth tube. The investigations were carried out both numerically and experimentally at different off-gas and combustion air velocities. The experimental results show that the highest heat transfer is achieved with the concave 6 mm structured tube. The greatest pressure loss also occurs here. The validation of the numerical model has shown issues in resolving the turbulence. Full article
(This article belongs to the Topic Heat Transfer Enhancement and Applications)
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Figure 1
<p>(<b>a</b>) Schematic operating principle of a tube bundle recuperator [<a href="#B2-thermo-03-00021" class="html-bibr">2</a>] and its (<b>b</b>) gas exchange.</p>
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<p>Examined test tubes (<b>a</b>) convex 3 mm (<b>b</b>) convex 6 mm (<b>c</b>) concave 3 mm and (<b>d</b>) concave 6 mm with a tube length of <span class="html-italic">l<sub>tube</sub></span> = 1000 mm.</p>
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<p>Flow channel for investigating the heat transfer of various tubes.</p>
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<p>Cross-section of the test bench with test tube and measurement technology.</p>
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<p>Uncertainty of (<b>a</b>) the heat flux <math display="inline"><semantics> <mrow> <mover> <mi>Q</mi> <mo>˙</mo> </mover> </mrow> </semantics></math> and (<b>b</b>) the heat transfer coefficient <span class="html-italic">α</span>.</p>
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<p>Thermographic images of the convex 6 mm structured tube at three test points.</p>
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<p>Tube wall temperatures measured with the IR camera at three test points at (<b>a</b>) <span class="html-italic">T<sub>off-gas</sub></span> = 100 °C and (<b>b</b>) <span class="html-italic">T<sub>off-gas</sub></span> = 200 °C.</p>
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<p>Correlation of (<b>a</b>) the heat flux <math display="inline"><semantics> <mrow> <mover> <mi>Q</mi> <mo>˙</mo> </mover> </mrow> </semantics></math> and (<b>b</b>) the internal heat transfer coefficient <span class="html-italic">α<sub>internal</sub></span> to the combustion air velocity <span class="html-italic">v<sub>air</sub>.</span></p>
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<p>Influence of the combustion air velocity v<sub>air</sub> on the pressure loss ∆<span class="html-italic">p</span> inside the tube.</p>
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<p>Main-effect diagram of the transferred heat flux <math display="inline"><semantics> <mrow> <mover> <mi>Q</mi> <mo>Ë™</mo> </mover> </mrow> </semantics></math>.</p>
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<p>Main-effect diagram of the internal heat transfer coefficient <span class="html-italic">α</span><sub>internal</sub>.</p>
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<p>Main-effect diagram of the external heat transfer coefficient <span class="html-italic">α</span><sub>external</sub>.</p>
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<p>Main-effect diagram of pressure loss ∆<span class="html-italic">p</span> inside the tube.</p>
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<p>Comparison of the experimental and numerical increases of the test tubes compared to the smooth tube in terms of (<b>a</b>) pressure loss ∆<span class="html-italic">p</span>, (<b>b</b>) heat transfer coefficient <span class="html-italic">α</span><sub>internal</sub>, and (<b>c</b>) heat flux <math display="inline"><semantics> <mrow> <mover> <mi>Q</mi> <mo>˙</mo> </mover> </mrow> </semantics></math>.</p>
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17 pages, 1305 KiB  
Article
Conjugate Heat Transfer Analysis for Cooling of a Conductive Panel by Combined Utilization of Nanoimpinging Jets and Double Rotating Cylinders
by Lioua Kolsi, Fatih Selimefendigil, Hatem Gasmi and Badr M. Alshammari
Nanomaterials 2023, 13(3), 500; https://doi.org/10.3390/nano13030500 - 26 Jan 2023
Cited by 5 | Viewed by 1666
Abstract
In this work, double rotating active cylinders and slot nanojet impingement are considered for the cooling system of a conductive panel. Colder surface temperatures of the cylinders are used, while different rotational speeds are assigned for each of the cylinders. The impacts of [...] Read more.
In this work, double rotating active cylinders and slot nanojet impingement are considered for the cooling system of a conductive panel. Colder surface temperatures of the cylinders are used, while different rotational speeds are assigned for each of the cylinders. The impacts of cylinder rotational speeds, size and distance between them on the cooling performance are evaluated. The rotational effects and size of the cylinders are found to be very effective on the overall thermal performance. At the highest rotational speeds of the cylinders, the average Nusselt number (Nu) rises by about 30.8%, while the panel temperature drops by about 5.84 °C. When increasing the cylinder sizes, temperature drops become 7 °C, while they are only 1.75 °C when varying the distance between the cylinders. Subcooling and nanofluid utilization contributes positively to the cooling performance, while 1.25 °C and 10 °C temperature drops are found by varying the subcooled temperature and solid volume fraction. An artificial neural network is used for the estimation of maximum and average panel temperatures when double cylinder parameters are used as the input. Full article
(This article belongs to the Topic Heat Transfer Enhancement and Applications)
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Figure 1
<p>Conductive panel cooling system with slot jet impinging and active RCCs (not to scale).</p>
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<p>GIT results for average PST at two different rotational speeds of the first cylinder (<b>a</b>) and mesh distribution near the RCCs and panel (<b>b</b>).</p>
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<p>Code validation studies: (<b>a</b>) Local Nu distribution comparisons with Ref. [<a href="#B49-nanomaterials-13-00500" class="html-bibr">49</a>] at Re = 250 for confined slot jet impingement system and (<b>b</b>) average Nu comparisons with Ref. [<a href="#B50-nanomaterials-13-00500" class="html-bibr">50</a>] in a differentially heated cavity under the effects of RCC (Case 1 − <math display="inline"><semantics> <mrow> <mi>R</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <mi mathvariant="normal">Ω</mi> <mo>=</mo> <mn>500</mn> </mrow> </semantics></math> and Case 2 − <math display="inline"><semantics> <mrow> <mi>R</mi> <mo>=</mo> <mn>0.2</mn> <mo>,</mo> <mi mathvariant="normal">Ω</mi> <mo>=</mo> <mn>1000</mn> </mrow> </semantics></math>).</p>
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<p>FP variations for different (<math display="inline"><semantics> <mrow> <msub> <mi>Rew</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>Rew</mi> <mn>2</mn> </msub> </mrow> </semantics></math>) combinations at (Rc = 0.1H, Sc = 6Rc, <math display="inline"><semantics> <mrow> <mo>Δ</mo> <mi>T</mi> <mspace width="3.33333pt"/> <mo>=</mo> <mspace width="3.33333pt"/> <mn>10</mn> </mrow> </semantics></math>, SVF = 3%).</p>
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<p>Average Nu (<b>a</b>) and average PST (<b>b</b>) for different rotational speeds of the cylinders (Rc = 0.1H, Sc = 6Rc, <math display="inline"><semantics> <mrow> <mo>Δ</mo> <mi>T</mi> <mspace width="3.33333pt"/> <mo>=</mo> <mspace width="3.33333pt"/> <mn>10</mn> </mrow> </semantics></math>, SVF = 3%).</p>
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<p>Impacts of RCC size (Sc = 6Rc, (<b>a</b>–<b>c</b>)) distance between the cylinders (Rc = 0.1H, (<b>d</b>–<b>f</b>)) on the FP distributions (<math display="inline"><semantics> <mrow> <mrow> <mo stretchy="false">(</mo> <msub> <mi>Rew</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>Rew</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> <mspace width="3.33333pt"/> <mo>=</mo> <mspace width="3.33333pt"/> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>20</mn> <mo>,</mo> <mo>−</mo> <mn>20</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>Δ</mo> <mi>T</mi> <mspace width="3.33333pt"/> <mo>=</mo> <mspace width="3.33333pt"/> <mn>10</mn> </mrow> </semantics></math>, SVF = 3%.)</p>
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<p>Effects of RCC size (Sc = 6Rc, (<b>a</b>)) and distance between the cylinders (Rc = 0.1H, (<b>b</b>)) on the average Nu variations (<math display="inline"><semantics> <mrow> <mrow> <mo stretchy="false">(</mo> <msub> <mi>Rew</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>Rew</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> <mspace width="3.33333pt"/> <mo>=</mo> <mspace width="3.33333pt"/> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>20</mn> <mo>,</mo> <mo>−</mo> <mn>20</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>Δ</mo> <mi>T</mi> <mspace width="3.33333pt"/> <mo>=</mo> <mspace width="3.33333pt"/> <mn>10</mn> </mrow> </semantics></math>, SVF = 3%).</p>
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<p>Average PST variations for different RCC sizes (Sc = 6Rc, (<b>a</b>)) and for various distances between the cylinders (Rc = 0.1H, (<b>b</b>)) (<math display="inline"><semantics> <mrow> <mrow> <mo stretchy="false">(</mo> <msub> <mi>Rew</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>Rew</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> <mspace width="3.33333pt"/> <mo>=</mo> <mspace width="3.33333pt"/> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>20</mn> <mo>,</mo> <mo>−</mo> <mn>20</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>Δ</mo> <mi>T</mi> <mspace width="3.33333pt"/> <mo>=</mo> <mspace width="3.33333pt"/> <mn>10</mn> </mrow> </semantics></math>, SVF = 3%).</p>
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<p>Impacts of subcooled temperature ((<b>a</b>), SVF = 3%) and SVF of NPs (<math display="inline"><semantics> <mrow> <mo>Δ</mo> <mi>T</mi> <mspace width="3.33333pt"/> <mo>=</mo> <mspace width="3.33333pt"/> <mn>10</mn> </mrow> </semantics></math>, (<b>b</b>)) on the variation of average Nu (<math display="inline"><semantics> <mrow> <mrow> <mo stretchy="false">(</mo> <msub> <mi>Rew</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>Rew</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> <mspace width="3.33333pt"/> <mo>=</mo> <mspace width="3.33333pt"/> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>20</mn> <mo>,</mo> <mo>−</mo> <mn>20</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics></math>, Rc = 0.1H, Sc = 6Rc).</p>
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<p>Average panel temperature for varying subcooled temperatures of RCCs ((<b>a</b>), SVF = 3%) and SVF of NPs (<math display="inline"><semantics> <mrow> <mo>Δ</mo> <mi>T</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math>, (<b>b</b>)) (<math display="inline"><semantics> <mrow> <mrow> <mo stretchy="false">(</mo> <msub> <mi>Rew</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>Rew</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> <mspace width="3.33333pt"/> <mo>=</mo> <mspace width="3.33333pt"/> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>20</mn> <mo>,</mo> <mo>−</mo> <mn>20</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics></math>, Rc = 0.1H, Sc = 6Rc).</p>
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<p>Schematic of the ANN with inputs–outputs (<b>a</b>) and network architecture with different layers (<b>b</b>).</p>
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<p>Variation of MSE for different epochs considering different data sets.</p>
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<p>Estimation of maximum (<b>a</b>) and average (<b>b</b>) panel temperatures by using ANN.</p>
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23 pages, 13489 KiB  
Article
Combined Energetic and Exergetic Performance Analysis of Air Bubbles Injection into a Plate Heat Exchanger: An Experimental Study
by Zakaria M. Marouf and Mahmoud A. Fouad
Energies 2023, 16(3), 1164; https://doi.org/10.3390/en16031164 - 20 Jan 2023
Cited by 4 | Viewed by 2470
Abstract
This paper aims to give a comprehensive energetic-exergetic performance analysis on the impacts of injecting-submillimeter of air bubbles into both sides of cold and hot water streams before the entrance port of a corrugated plate heat exchanger (C-PHE) having ten plates within counterflow [...] Read more.
This paper aims to give a comprehensive energetic-exergetic performance analysis on the impacts of injecting-submillimeter of air bubbles into both sides of cold and hot water streams before the entrance port of a corrugated plate heat exchanger (C-PHE) having ten plates within counterflow configuration. Hence, optimize the energy and exergy effectiveness at different operating conditions for counter and parallel fluid flow configurations. Hot streams were studied in seven flow rates ranging from 280 L/h to 880 L/h with a regular step of 100 L/h, and constant hot water temperature and cold-water stream of 50 °C and 290 L/h, respectively. Hence, the air was discharged with four flow rates ranging between 150 and 840 L/h. The obtained results showed the vital role of the ABI technique in enhancing the NTU and effectiveness by 59% and 18.6%, respectively, for CWS. The entropy generation was reduced to 0.038 W/K and the augmentation entropy generation number to 0.087 at the low airflow rate for CWS, which are the main parameters for evaluating the EGM. These two parameters increase the Witte-Shamsundar-efficiency to a maximum value of 98.6% at the same operating conditions. Moreover, the exergy effectiveness was enhanced to a maximum value of 80.9% at a high ABI flow rate and low volumetric rate of the hot stream at CWS. The thermo-economic assessment has been carried out, which revelers the positive effects of ABI on the combined energetic and exergetic performance on both sides, i.e., hot and cold sides. Full article
(This article belongs to the Topic Heat Transfer Enhancement and Applications)
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Figure 1

Figure 1
<p>Schematic diagram for the experimental setup components of ABI into CWS.</p>
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<p>Experimental setup components consist of (<b>1</b>) CWB, (<b>2</b>) HWB, (<b>3</b>) C-PHE, and (<b>4</b>) corrugated plate channel of PHE.</p>
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<p>Schematic diagram for the mixing chamber.</p>
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<p>Effectiveness vs. NTU for the original case of C-PHE with counter and parallel fluid flows arrangement.</p>
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<p>The relationship between pressure losses (<math display="inline"><semantics> <mrow> <mo mathvariant="sans-serif">Δ</mo> <mi>P</mi> </mrow> </semantics></math>) and counter and parallel hot water flow rates at the original case.</p>
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<p>(<b>a</b>) Effectiveness and (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>ε</mi> <mo>−</mo> <mi>R</mi> <mi>a</mi> <mi>t</mi> <mi>i</mi> <mi>o</mi> </mrow> </semantics></math> vs. hot water flow rates at different operating conditions for single-phase, and two-phase flow of ABI into CWS and HWS.</p>
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<p>Relationship between (<b>a</b>) NTU and (<b>b</b>) NTU Ratio vs. <math display="inline"><semantics> <mrow> <msub> <mi>Q</mi> <mrow> <mi>h</mi> <mo> </mo> </mrow> </msub> </mrow> </semantics></math>at different operating conditions for the original case (without ABI); and with ABI into CWS, HWS.</p>
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<p>Variations between <math display="inline"><semantics> <mrow> <msub> <mover accent="true"> <mi>S</mi> <mo>˙</mo> </mover> <mrow> <mi>g</mi> <mi>e</mi> <mi>n</mi> </mrow> </msub> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>Q</mi> <mrow> <mi>h</mi> <mo> </mo> </mrow> </msub> </mrow> </semantics></math>for original case and different ABI of: CWS; HWS.</p>
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<p>Relationship between <math display="inline"><semantics> <mrow> <msub> <mi>N</mi> <mi>s</mi> </msub> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>Q</mi> <mrow> <mi>h</mi> <mo> </mo> </mrow> </msub> </mrow> </semantics></math> at different operating conditions without/with ABI.</p>
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<p><math display="inline"><semantics> <mrow> <msub> <mi>N</mi> <mrow> <mi>s</mi> <mo>,</mo> <mi>a</mi> <mi>u</mi> <mi>g</mi> </mrow> </msub> </mrow> </semantics></math> vs. <math display="inline"><semantics> <mrow> <msub> <mi>Q</mi> <mrow> <mi>h</mi> <mo> </mo> </mrow> </msub> </mrow> </semantics></math>at different ABI into both CWS and HWS.</p>
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<p><math display="inline"><semantics> <mrow> <msub> <mrow> <mover> <mrow> <mi>E</mi> <mi>x</mi> </mrow> <mo>˙</mo> </mover> </mrow> <mrow> <mi>d</mi> <mi>e</mi> <mi>s</mi> </mrow> </msub> </mrow> </semantics></math> vs. <math display="inline"><semantics> <mrow> <msub> <mi>Q</mi> <mrow> <mi>h</mi> <mo> </mo> </mrow> </msub> </mrow> </semantics></math>for original cases and different ABI into CWS and HWS.</p>
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<p>Relationship between <math display="inline"><semantics> <mrow> <msub> <mi>ε</mi> <mrow> <mi>e</mi> <mi>x</mi> </mrow> </msub> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>Q</mi> <mrow> <mi>h</mi> <mo> </mo> </mrow> </msub> </mrow> </semantics></math>for original case and, different ABI into both CWS and HWS.</p>
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<p>The variations of <math display="inline"><semantics> <mrow> <msub> <mi>η</mi> <mrow> <mi>W</mi> <mo>−</mo> <mi>S</mi> </mrow> </msub> </mrow> </semantics></math> at different <math display="inline"><semantics> <mrow> <msub> <mi>Q</mi> <mrow> <mi>h</mi> <mo> </mo> </mrow> </msub> </mrow> </semantics></math>and <math display="inline"><semantics> <mrow> <msub> <mi>Q</mi> <mrow> <mi>a</mi> <mi>i</mi> <mi>r</mi> <mo> </mo> </mrow> </msub> </mrow> </semantics></math> at different operating conditions.</p>
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<p><math display="inline"><semantics> <mrow> <msub> <mi>η</mi> <mi>P</mi> </msub> </mrow> </semantics></math> vs. hot water flow rate at different operating conditions.</p>
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<p>Pareto-front of <math display="inline"><semantics> <mrow> <mi>ε</mi> <mrow> <mo> </mo> <mi>vs</mi> </mrow> <mo>.</mo> <mo> </mo> <msub> <mi>ε</mi> <mrow> <mi>e</mi> <mi>x</mi> </mrow> </msub> </mrow> </semantics></math> at different operating conditions.</p>
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18 pages, 1859 KiB  
Review
PHE (Plate Heat Exchanger) for Condensing Duties: Recent Advances and Future Prospects
by Petro Kapustenko, Jiří Jaromír Klemeš, Olga Arsenyeva and Leonid Tovazhnyanskyy
Energies 2023, 16(1), 524; https://doi.org/10.3390/en16010524 - 3 Jan 2023
Cited by 8 | Viewed by 4468
Abstract
Increasing energy usage efficiency requires enhanced heat energy recuperation between process streams in the industry and civic sector with waste heat utilization. The condensation of different vapours is the process encountered in many industrial applications. Increasing the heat recuperation in this process is [...] Read more.
Increasing energy usage efficiency requires enhanced heat energy recuperation between process streams in the industry and civic sector with waste heat utilization. The condensation of different vapours is the process encountered in many industrial applications. Increasing the heat recuperation in this process is possible with efficient heat transfer equipment, among which a Plate Heat Exchanger (PHE) is at the leading position. A number of research works have been conducted in recent years concerning construction development and heat transfer enhancement in conditions of limited pressure drop to increase PHE performance in condensation processes. The results of studies on heat transfer and pressure drop in the two-phase condensing flow inside channels of PHE with different geometries of corrugations are discussed. In many implementations, the total pressure drop allowable for gaseous streams in heat exchangers is relatively small. The structure of two-phase flow in PHE channels of complex geometry is very different than in tubes and flat wall channels. The relative differences in approaches to enhance PHE performance in condensation processes based on its modelling, optimisation and design are analyzed. The directions and prospects for future developments are formulated, and potential savings for the economy and the environmental footprint is presented. Full article
(This article belongs to the Topic Heat Transfer Enhancement and Applications)
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Figure 1
<p>A typical corrugated plate of commercially produced GPHE: 1—ports for streams entrances and exits; 2, 5—zones for flow distribution; 3—gasket; 4—the major heat transfer field.</p>
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<p>The main field of a channel between two corrugated PHE plates (schematically): (<b>a</b>) corrugations crossing; (<b>b</b>,<b>d</b>) sine-shaped corrugations; (<b>c</b>,<b>e</b>) triangle-shaped corrugations.</p>
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<p>The number of experimental points data received in studies of PHEs with different angle of plates corrugation.</p>
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<p>The number of experimental points data received in studies of PHEs with different condensing vapours.</p>
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14 pages, 2249 KiB  
Article
Multi-Objective Constructal Design for Quadrilateral Heat Generation Body with Vein-Shaped High Thermal Conductivity Channel
by Hongwei Zhu, Lingen Chen, Yanlin Ge, Shuangshuang Shi and Huijun Feng
Entropy 2022, 24(10), 1403; https://doi.org/10.3390/e24101403 - 1 Oct 2022
Cited by 12 | Viewed by 1895
Abstract
Based on the quadrilateral heat generation body (HGB) proposed by previous literature, the multi-objective constructal design is performed. Firstly, the constructal design is performed by minimizing the complex function composed of the maximum temperature difference (MTD) and entropy generation rate (EGR), and the [...] Read more.
Based on the quadrilateral heat generation body (HGB) proposed by previous literature, the multi-objective constructal design is performed. Firstly, the constructal design is performed by minimizing the complex function composed of the maximum temperature difference (MTD) and entropy generation rate (EGR), and the influence of the weighting coefficient (a0) on the optimal constructal is studied. Secondly, the multi-objective optimization (MOO) with the MTD and EGR as optimization objectives is performed, and the Pareto frontier with an optimal set is obtained by using NSGA-II. The optimization results are selected from the Pareto frontier through LINMAP, TOPSIS, and Shannon Entropy decision methods, and the deviation indexes of different objectives and decision methods are compared. The research of the quadrilateral HGB shows that the optimal constructal can be gained by minimizing the complex function with the objectives of the MTD and the EGR, the complex function after the constructal design is reduced by up to 2% compared with its initial value, and the complex function of the two reflects the compromise between the maximum thermal resistance and the irreversible loss of heat transfer. The Pareto frontier includes the optimization results of different objectives, and when the weighting coefficient of a complex function changes, the optimization results obtained by minimizing the complex function will also be distributed in the Pareto frontier. The deviation index of the TOPSIS decision method is 0.127, which is the lowest one among the discussed decision methods. Full article
(This article belongs to the Topic Heat Transfer Enhancement and Applications)
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Figure 1
<p>Quadrilateral HGB with vein shaped HTCCs [<a href="#B50-entropy-24-01403" class="html-bibr">50</a>].</p>
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<p>Elemental unit based on any branch HTCC [<a href="#B50-entropy-24-01403" class="html-bibr">50</a>].</p>
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<p>Central HTCC [<a href="#B50-entropy-24-01403" class="html-bibr">50</a>].</p>
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<p>Relationships of <math display="inline"><semantics> <mrow> <mo>Δ</mo> <mi>T</mi> </mrow> </semantics></math> and <math display="inline"><semantics> <mi>σ</mi> </semantics></math> versus <math display="inline"><semantics> <mrow> <msub> <mi>H</mi> <mn>1</mn> </msub> <mo>/</mo> <msub> <mi>L</mi> <mn>1</mn> </msub> </mrow> </semantics></math> [<a href="#B50-entropy-24-01403" class="html-bibr">50</a>,<a href="#B56-entropy-24-01403" class="html-bibr">56</a>].</p>
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<p>Relationships of <math display="inline"><semantics> <mrow> <msub> <mi>F</mi> <mrow> <mi>ST</mi> </mrow> </msub> </mrow> </semantics></math> versus <math display="inline"><semantics> <mrow> <msub> <mi>H</mi> <mn>1</mn> </msub> <mo>/</mo> <msub> <mi>L</mi> <mn>1</mn> </msub> </mrow> </semantics></math> for <math display="inline"><semantics> <mrow> <msub> <mi>a</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>.</p>
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<p>Effects of <math display="inline"><semantics> <mrow> <msub> <mi>a</mi> <mn>0</mn> </msub> </mrow> </semantics></math> on <math display="inline"><semantics> <mrow> <msub> <mi>F</mi> <mrow> <mi>ST</mi> <mo>,</mo> <mi>min</mi> </mrow> </msub> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>H</mi> <mn>1</mn> </msub> <mo>/</mo> <msub> <mi>L</mi> <mn>1</mn> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>opt</mi> </mrow> </msub> </mrow> </semantics></math>.</p>
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<p>Flow chart of NSGA-II.</p>
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<p>Pareto frontier for multi-objective optimization of the quadrilateral HGB.</p>
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<p>Relationships of <math display="inline"><semantics> <mover accent="true"> <mi>σ</mi> <mo>˜</mo> </mover> </semantics></math> and <math display="inline"><semantics> <mrow> <mo>Δ</mo> <mover accent="true"> <mi>T</mi> <mo>˜</mo> </mover> </mrow> </semantics></math> versus <math display="inline"><semantics> <mrow> <msub> <mi>H</mi> <mn>1</mn> </msub> <mo>/</mo> <msub> <mi>L</mi> <mn>1</mn> </msub> </mrow> </semantics></math> in Pareto frontier of the quadrilateral HGB.</p>
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