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Advances in Applied Thermodynamics II

A special issue of Entropy (ISSN 1099-4300). This special issue belongs to the section "Thermodynamics".

Deadline for manuscript submissions: closed (30 November 2016) | Viewed by 105812

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NewRail - Newcastle Centre for Railway Research, Newcastle University, Newcastle upon Tyne NE17RU, UK
Interests: thermal power systems; refrigeration; combined cycles; internal combustion engines; finite time thermodynamics
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

You are invited to submit papers to the Special Issue, “Advances in Thermodynamics II”, focusing on the application of the Second Law of Thermodynamics to processes in several fields of study.

The concept of entropy originated in the period when thermodynamics was concerned with the conditions under which heat can be converted to work. It was formalized and named (from the Greek εντροπία, transformation) by Rudolf Clausius from considerations of reversible processes. Usually, today, an irreversible transformation is identified by the Clausius Inequality. In his later work, Clausius included irreversible process to derive the Second Law of Thermodynamics as an equality, and included a term to account for entropy generation by dissipative processes. A more generalized formulation of the entropy concept, developed by Boltzmann, is associated with disorder or the destruction of the coherence of an initial state. This has been widely adopted in many diverse fields of study including chemistry, biology, cosmology and information science. An indication of the importance of the Second Law of Thermodynamics can be gauged by the following statement made by Sir Arthur Eddington, "If someone points out to you that your pet theory of the universe is in disagreement with Maxwell's equations—then so much the worse for Maxwell's equations. If it is found to be contradicted by observation—well, these experimentalists do bungle things sometimes. But if your theory is found to be against the Second Law of Thermodynamics I can offer you no hope". The Second Law played a key role in the development of Classical Thermodynamics in the 20th century with entropy revealing some essential characteristics of the behavior of matter and energy. In moving away from equilibrium states and adopting mathematical techniques from other branches of science, the analysis of Carnot has been extended to include thermodynamic systems with fixed rates or durations and constraints on heat or mass transfer surfaces. This exciting development has established the conditions appropriate to time or rate constrained processes and the conditions for optimal configurations of heat and mass exchange processes. It is clear that such techniques will play an important part in many fields of activity that are important today. Papers will be welcome from a wide range of disciplines that are based upon the application of the Second Law of Thermodynamics.

Prof. Dr. Brian Agnew
Guest Editor

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Keywords

  • Second Law of Thermodynamics
  • Entropy generation minimization
  • Optimization

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Related Special Issue

Published Papers (14 papers)

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1095 KiB  
Article
A LiBr-H2O Absorption Refrigerator Incorporating a Thermally Activated Solution Pumping Mechanism
by Ian W. Eames
Entropy 2017, 19(3), 90; https://doi.org/10.3390/e19030090 - 26 Feb 2017
Cited by 5 | Viewed by 7743
Abstract
This paper provides an illustrated description of a proposed LiBr-H2O vapour absorption refrigerator which uses a thermally activated solution pumping mechanism that combines controlled variations in generator vapour pressure with changes it produces in static-head pressure difference to circulate the absorbent [...] Read more.
This paper provides an illustrated description of a proposed LiBr-H2O vapour absorption refrigerator which uses a thermally activated solution pumping mechanism that combines controlled variations in generator vapour pressure with changes it produces in static-head pressure difference to circulate the absorbent solution between the generator and absorber vessels. The proposed system is different and potentially more efficient than a bubble pump system previously proposed and avoids the need for an electrically powered circulation pump found in most conventional LiBr absorption refrigerators. The paper goes on to provide a sample set of calculations that show that the coefficient of performance values of the proposed cycle are similar to those found for conventional cycles. The theoretical results compare favourably with some preliminary experimental results, which are also presented for the first time in this paper. The paper ends by proposing an outline design for an innovative steam valve, which is a key component needed to control the solution pumping mechanism. Full article
(This article belongs to the Special Issue Advances in Applied Thermodynamics II)
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Figure 1

Figure 1
<p>A conventional single-effect absorption cycle.</p>
Full article ">Figure 2
<p>Schematic view of the novel cycle using a valve (V) to control the flow of solution.</p>
Full article ">Figure 3
<p>Some experimental results taken by Stephens and Eames [<a href="#B17-entropy-19-00090" class="html-bibr">17</a>].</p>
Full article ">Figure 4
<p>Showing a schematic view of an automatic steam valve in the generator of a VAR cycle due to Paurine et al. [<a href="#B19-entropy-19-00090" class="html-bibr">19</a>].</p>
Full article ">Figure 5
<p>Novel automatic solution pumping system controlled by a float valve.</p>
Full article ">Figure 6
<p>Operation of the float valve controlling the flow of solution between the absorber and generator. (<b>a</b>) Value part-open; (<b>b</b>) Value close; (<b>c</b>) Value about to open; (<b>d</b>) Value wide-open.</p>
Full article ">
12388 KiB  
Article
Response Surface Methodology Control Rod Position Optimization of a Pressurized Water Reactor Core Considering Both High Safety and Low Energy Dissipation
by Yi-Ning Zhang, Hao-Chun Zhang, Hai-Yan Yu and Chao Ma
Entropy 2017, 19(2), 63; https://doi.org/10.3390/e19020063 - 10 Feb 2017
Cited by 5 | Viewed by 5530
Abstract
Response Surface Methodology (RSM) is introduced to optimize the control rod positions in a pressurized water reactor (PWR) core. The widely used 3D-IAEA benchmark problem is selected as the typical PWR core and the neutron flux field is solved. Besides, some additional thermal [...] Read more.
Response Surface Methodology (RSM) is introduced to optimize the control rod positions in a pressurized water reactor (PWR) core. The widely used 3D-IAEA benchmark problem is selected as the typical PWR core and the neutron flux field is solved. Besides, some additional thermal parameters are assumed to obtain the temperature distribution. Then the total and local entropy production is calculated to evaluate the energy dissipation. Using RSM, three directions of optimization are taken, which aim to determine the minimum of power peak factor Pmax, peak temperature Tmax and total entropy production Stot. These parameters reflect the safety and energy dissipation in the core. Finally, an optimization scheme was obtained, which reduced Pmax, Tmax and Stot by 23%, 8.7% and 16%, respectively. The optimization results are satisfactory. Full article
(This article belongs to the Special Issue Advances in Applied Thermodynamics II)
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Graphical abstract

Graphical abstract
Full article ">Figure 1
<p>Horizontal cross section of the 3D-IAEA problem.</p>
Full article ">Figure 2
<p>Vertical cross section of the 3D-IAEA problem.</p>
Full article ">Figure 3
<p>Fast neutron flux at the diagonal line at the level of <span class="html-italic">z</span> = 195 cm.</p>
Full article ">Figure 4
<p>Thermal neutron flux at the diagonal line at the level of <span class="html-italic">z</span> = 195 cm.</p>
Full article ">Figure 5
<p>Local power distribution of standard problem (<b>left</b>: vertical cross section cloud picture at <span class="html-italic">y</span> = 0; <b>right-top</b>: horizontal cross section cloud picture at <span class="html-italic">z</span> = 315 cm; <b>right-bottom</b>: horizontal cross section cloud picture at <span class="html-italic">z</span> = 195 cm. <span class="html-italic">P</span><sub>avg</sub> represents the average local power of standard problem).</p>
Full article ">Figure 6
<p>Temperature distribution of standard problem (<b>left</b>: vertical cross section cloud picture at <span class="html-italic">y</span> = 0; <b>right-top</b>: horizontal cross section cloud picture at <span class="html-italic">z</span> = 315 cm; <b>right-bottom</b>: horizontal cross section cloud picture at <span class="html-italic">z</span> = 195 cm).</p>
Full article ">Figure 7
<p>Local entropy production distribution of standard problem (<b>left</b>: vertical cross section cloud picture at <span class="html-italic">y</span> = 0; <b>right-top</b>: horizontal cross section cloud picture at <span class="html-italic">z</span> = 315 cm; <b>right-bottom</b>: horizontal cross section cloud picture at <span class="html-italic">z</span> = 195 cm. <span class="html-italic">S</span><sub>0</sub> represents the average local entropy production of standard problem).</p>
Full article ">Figure 8
<p>Flow diagram of optimization procedure.</p>
Full article ">Figure 9
<p>Local power distribution of optimization scheme (<b>left</b>: vertical cross section cloud picture at <span class="html-italic">y</span> = 0; <b>right-top</b>: horizontal cross section cloud picture at <span class="html-italic">z</span> = 315 cm; <b>right-bottom</b>: horizontal cross section cloud picture at <span class="html-italic">z</span> = 195 cm. <span class="html-italic">P</span><sub>avg</sub> represents the average local power of standard problem).</p>
Full article ">Figure 10
<p>Temperature distribution of optimization scheme (<b>left</b>: vertical cross section cloud picture at <span class="html-italic">y</span> = 0; <b>right-top</b>: horizontal cross section cloud picture at <span class="html-italic">z</span> = 315 cm; <b>right-bottom</b>: horizontal cross section cloud picture at <span class="html-italic">z</span> = 195 cm).</p>
Full article ">Figure 11
<p>Local entropy production distribution of optimization scheme (<b>left</b>: vertical cross section cloud picture at <span class="html-italic">y</span> = 0; <b>right-top</b>: horizontal cross section cloud picture at <span class="html-italic">z</span> = 315 cm; <b>right-bottom</b>: horizontal cross section cloud picture at <span class="html-italic">z</span> = 195 cm. <span class="html-italic">S</span><sub>0</sub> represents the average local entropy production of standard problem).</p>
Full article ">Figure 12
<p>Local power, temperature and local entropy production of the standard problem and optimization scheme at the diagonal line on the midplane (the level of <span class="html-italic">z</span> = 195 cm).</p>
Full article ">
335 KiB  
Article
Scaling Relations of Lognormal Type Growth Process with an Extremal Principle of Entropy
by Zi-Niu Wu, Juan Li and Chen-Yuan Bai
Entropy 2017, 19(2), 56; https://doi.org/10.3390/e19020056 - 27 Jan 2017
Cited by 5 | Viewed by 7283
Abstract
The scale, inflexion point and maximum point are important scaling parameters for studying growth phenomena with a size following the lognormal function. The width of the size function and its entropy depend on the scale parameter (or the standard deviation) and measure the [...] Read more.
The scale, inflexion point and maximum point are important scaling parameters for studying growth phenomena with a size following the lognormal function. The width of the size function and its entropy depend on the scale parameter (or the standard deviation) and measure the relative importance of production and dissipation involved in the growth process. The Shannon entropy increases monotonically with the scale parameter, but the slope has a minimum at p 6/6. This value has been used previously to study spreading of spray and epidemical cases. In this paper, this approach of minimizing this entropy slope is discussed in a broader sense and applied to obtain the relationship between the inflexion point and maximum point. It is shown that this relationship is determined by the base of natural logarithm e ' 2.718 and exhibits some geometrical similarity to the minimal surface energy principle. The known data from a number of problems, including the swirling rate of the bathtub vortex, more data of droplet splashing, population growth, distribution of strokes in Chinese language characters and velocity profile of a turbulent jet, are used to assess to what extent the approach of minimizing the entropy slope can be regarded as useful. Full article
(This article belongs to the Special Issue Advances in Applied Thermodynamics II)
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Figure 1

Figure 1
<p>The lognormal function and its derivative. The inflexion point (IP) is at <math display="inline"> <semantics> <mrow> <mi>t</mi> <mo>=</mo> <msub> <mi>t</mi> <mi>L</mi> </msub> </mrow> </semantics> </math> and the maximum point (MP) is at <math display="inline"> <semantics> <mrow> <mi>t</mi> <mo>=</mo> <msub> <mi>t</mi> <mi>D</mi> </msub> </mrow> </semantics> </math>.</p>
Full article ">Figure 2
<p>The curves of <math display="inline"> <semantics> <mrow> <mi>S</mi> <mo>(</mo> <mi>σ</mi> <mo>)</mo> </mrow> </semantics> </math> (with <math display="inline"> <semantics> <mrow> <msub> <mi>t</mi> <mi>D</mi> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics> </math>) and <math display="inline"> <semantics> <mrow> <msup> <mi>S</mi> <mo>′</mo> </msup> <mfenced open="(" close=")"> <mi>σ</mi> </mfenced> </mrow> </semantics> </math>.</p>
Full article ">Figure 3
<p>The curves of <math display="inline"> <semantics> <mrow> <mi>ϕ</mi> <mfenced open="(" close=")"> <mi>σ</mi> </mfenced> </mrow> </semantics> </math> and <math display="inline"> <semantics> <mrow> <mi>ψ</mi> <mfenced open="(" close=")"> <mi>σ</mi> </mfenced> </mrow> </semantics> </math>.</p>
Full article ">Figure 4
<p>Angular displacement of the floating disk for a bathtub vortex with reserval, original data from Figure 2 of Sibulkin [<a href="#B17-entropy-19-00056" class="html-bibr">17</a>].</p>
Full article ">Figure 5
<p>Rotation speed of the floating disk for a bathtub vortex with reserval, original data from Figure 2 of Sibulkin [<a href="#B17-entropy-19-00056" class="html-bibr">17</a>].</p>
Full article ">Figure 6
<p>Droplet size distribution for one set of data of Stow and Stainer [<a href="#B5-entropy-19-00056" class="html-bibr">5</a>].</p>
Full article ">Figure 7
<p>Population growth in the world.</p>
Full article ">Figure 8
<p>Traditional Chinese words and total strokes.</p>
Full article ">Figure 9
<p>Velocity profile in a turbulent jet.</p>
Full article ">Figure 10
<p>Velocity gradient profile in a turbulent jet.</p>
Full article ">
29008 KiB  
Article
Similarity Theory Based Radial Turbine Performance and Loss Mechanism Comparison between R245fa and Air for Heavy-Duty Diesel Engine Organic Rankine Cycles
by Lei Zhang, Weilin Zhuge, Yangjun Zhang and Tao Chen
Entropy 2017, 19(1), 25; https://doi.org/10.3390/e19010025 - 14 Jan 2017
Cited by 12 | Viewed by 7895
Abstract
Organic Rankine Cycles using radial turbines as expanders are considered as one of the most efficient technologies to convert heavy-duty diesel engine waste heat into useful work. Turbine similarity design based on the existing air turbine profiles is time saving. Due to totally [...] Read more.
Organic Rankine Cycles using radial turbines as expanders are considered as one of the most efficient technologies to convert heavy-duty diesel engine waste heat into useful work. Turbine similarity design based on the existing air turbine profiles is time saving. Due to totally different thermodynamic properties between organic fluids and air, its influence on turbine performance and loss mechanisms need to be analyzed. This paper numerically simulated a radial turbine under similar conditions between R245fa and air, and compared the differences of the turbine performance and loss mechanisms. Larger specific heat ratio of air leads to air turbine operating at higher pressure ratios. As R245fa gas constant is only about one-fifth of air gas constant, reduced rotating speeds of R245fa turbine are only 0.4-fold of those of air turbine, and reduced mass flow rates are about twice of those of air turbine. When using R245fa as working fluid, the nozzle shock wave losses decrease but rotor suction surface separation vortex losses increase, and eventually leads that isentropic efficiencies of R245fa turbine in the commonly used velocity ratio range from 0.5 to 0.9 are 3%–4% lower than those of air turbine. Full article
(This article belongs to the Special Issue Advances in Applied Thermodynamics II)
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Figure 1

Figure 1
<p>Demonstration of a regenerative organic Rankine cycle for heavy-duty diesel engine exhaust gas heat recovery: (<b>a</b>) configuration of the regenerative organic Rankine cycle; and (<b>b</b>) T-s diagram of the regenerative organic Rankine cycle.</p>
Full article ">Figure 2
<p>Current and future fields of application of ORC versus steam power systems in terms of average temperature of the energy source and power capacity [<a href="#B8-entropy-19-00025" class="html-bibr">8</a>].</p>
Full article ">Figure 3
<p>CFD domain including volute, full passages of nozzle ring, rotor wheel and exhaust pipe.</p>
Full article ">Figure 4
<p>Absolute Mach number in the 50% span of stationary domain: (<b>a</b>) absolute Mach number value of k-ω SST model; (<b>b</b>) difference between S-A and k-ω SST model; (<b>c</b>) difference between low Re k-ε and k-ω SST model; and (<b>d</b>) difference between EARSM and k-ω SST model.</p>
Full article ">Figure 4 Cont.
<p>Absolute Mach number in the 50% span of stationary domain: (<b>a</b>) absolute Mach number value of k-ω SST model; (<b>b</b>) difference between S-A and k-ω SST model; (<b>c</b>) difference between low Re k-ε and k-ω SST model; and (<b>d</b>) difference between EARSM and k-ω SST model.</p>
Full article ">Figure 5
<p>Relative Mach number in the 50% span of rotating domain: (<b>a</b>) relative Mach number value of k-ω SST model; (<b>b</b>) difference between S-A and k-ω SST model; (<b>c</b>) difference between low Re k-ε and k-ω SST model; and (<b>d</b>) difference between EARSM and k-ω SST model.</p>
Full article ">Figure 6
<p>Kinematic and dynamic similarity verification at the nominal and off-design operating conditions: (<b>a</b>) flow angle differences between air and R245fa at the nominal condition; (<b>b</b>) flow angle differences between air and R245fa at the off-design condition; (<b>c</b>) mach number differences between air and R245fa at the nominal condition; (<b>d</b>) mach number differences between air and R245fa at the off-design condition; (<b>e</b>) reynolds number differences between air and R245fa at the nominal condition; and (<b>f</b>) reynolds number differences between air and R245fa at the off-design condition.</p>
Full article ">Figure 6 Cont.
<p>Kinematic and dynamic similarity verification at the nominal and off-design operating conditions: (<b>a</b>) flow angle differences between air and R245fa at the nominal condition; (<b>b</b>) flow angle differences between air and R245fa at the off-design condition; (<b>c</b>) mach number differences between air and R245fa at the nominal condition; (<b>d</b>) mach number differences between air and R245fa at the off-design condition; (<b>e</b>) reynolds number differences between air and R245fa at the nominal condition; and (<b>f</b>) reynolds number differences between air and R245fa at the off-design condition.</p>
Full article ">Figure 7
<p>Total-to-static isentropic efficiency difference in percentage terms compared with the relative flow angle average difference.</p>
Full article ">Figure 8
<p>Turbine performance map comparison between R245fa (in red color) and air (in blue color): (<b>a</b>) reduced mass flow rates versus total-to-static pressure ratios at five reduced rotating speeds; and (<b>b</b>) total-to-static isentropic efficiencies versus velocity ratios.</p>
Full article ">Figure 9
<p>Entropy generation rate per unit volume in the nominal operating condition: (<b>a</b>) 10% span of blade-to-blade surface flow field of 245fa case; (<b>b</b>) 10% span of blade-to-blade surface flow field of air case; (<b>c</b>) 50% span of blade-to-blade surface flow field of 245fa case; (<b>d</b>) 50% span of blade-to-blade surface flow field of air case; (<b>e</b>) 90% span of blade-to-blade surface flow field of 245fa case; and (<b>f</b>) 90% span of blade-to-blade surface flow field of air case.</p>
Full article ">Figure 10
<p>Entropy generation rate per unit volume in the off-design operating condition: (<b>a</b>) 10% span of blade-to-blade surface flow field of 245fa case; (<b>b</b>) 10% span of blade-to-blade surface flow field of air case; (<b>c</b>) 50% span of blade-to-blade surface flow field of 245fa case; (<b>d</b>) 50% span of blade-to-blade surface flow field of air case; (<b>e</b>) 90% span of blade-to-blade surface flow field of 245fa case; and (<b>f</b>) 90% span of blade-to-blade surface flow field of air case.</p>
Full article ">Figure 11
<p>Absolute Mach number distribution in the nozzle ring at off-design operating condition: (<b>a</b>) R245fa case; and (<b>b</b>) air case.</p>
Full article ">Figure 12
<p>Normalized pressure comparison along the meridional length both on blade pressure and suction surfaces between R245fa (in red color) and air (in blue color).</p>
Full article ">Figure 13
<p>Velocity streamline and relative Mach number distribution in the rotor wheel at off-design operating condition: (<b>a</b>) R245fa case; and (<b>b</b>) air case.</p>
Full article ">
1977 KiB  
Article
Entropy Generation in Magnetohydrodynamic Mixed Convection Flow over an Inclined Stretching Sheet
by Muhammad Idrees Afridi, Muhammad Qasim, Ilyas Khan, Sharidan Shafie and Ali Saleh Alshomrani
Entropy 2017, 19(1), 10; https://doi.org/10.3390/e19010010 - 28 Dec 2016
Cited by 43 | Viewed by 7329
Abstract
This research focuses on entropy generation rate per unit volume in magneto-hydrodynamic (MHD) mixed convection boundary layer flow of a viscous fluid over an inclined stretching sheet. Analysis has been performed in the presence of viscous dissipation and non-isothermal boundary conditions. The governing [...] Read more.
This research focuses on entropy generation rate per unit volume in magneto-hydrodynamic (MHD) mixed convection boundary layer flow of a viscous fluid over an inclined stretching sheet. Analysis has been performed in the presence of viscous dissipation and non-isothermal boundary conditions. The governing boundary layer equations are transformed into ordinary differential equations by an appropriate similarity transformation. The transformed coupled nonlinear ordinary differential equations are then solved numerically by a shooting technique along with the Runge-Kutta method. Expressions for entropy generation (Ns) and Bejan number (Be) in the form of dimensionless variables are also obtained. Impact of various physical parameters on the quantities of interest is seen. Full article
(This article belongs to the Special Issue Advances in Applied Thermodynamics II)
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Figure 1

Figure 1
<p>Physical flow model and coordinatate system.</p>
Full article ">Figure 2
<p>(<b>a</b>) Variation of f’(<span class="html-italic">η</span>) with <span class="html-italic">M</span>; (<b>b</b>) variation of <span class="html-italic">θ</span>(<span class="html-italic">η</span>) with <span class="html-italic">M</span>; (<b>c</b>) variation of <span class="html-italic">Ns</span> with <span class="html-italic">M</span>; (<b>d</b>) variation of <span class="html-italic">Be</span> with <span class="html-italic">M</span>.</p>
Full article ">Figure 2 Cont.
<p>(<b>a</b>) Variation of f’(<span class="html-italic">η</span>) with <span class="html-italic">M</span>; (<b>b</b>) variation of <span class="html-italic">θ</span>(<span class="html-italic">η</span>) with <span class="html-italic">M</span>; (<b>c</b>) variation of <span class="html-italic">Ns</span> with <span class="html-italic">M</span>; (<b>d</b>) variation of <span class="html-italic">Be</span> with <span class="html-italic">M</span>.</p>
Full article ">Figure 3
<p>(<b>a</b>) Variation of <span class="html-italic">θ</span>(<span class="html-italic">η</span>) with Pr; (<b>b</b>) Variation of <span class="html-italic">Ns</span> with Pr; (<b>c</b>) Variation of <span class="html-italic">Be</span> with Pr.</p>
Full article ">Figure 3 Cont.
<p>(<b>a</b>) Variation of <span class="html-italic">θ</span>(<span class="html-italic">η</span>) with Pr; (<b>b</b>) Variation of <span class="html-italic">Ns</span> with Pr; (<b>c</b>) Variation of <span class="html-italic">Be</span> with Pr.</p>
Full article ">Figure 4
<p>(<b>a</b>) Variation of <span class="html-italic">f’</span>(<span class="html-italic">η</span>) with λ; (<b>b</b>) Variation of <span class="html-italic">θ</span>(<span class="html-italic">η</span>) with λ; (<b>c</b>) Variation of <span class="html-italic">Ns</span> with λ; (<b>d</b>) Variation of <span class="html-italic">Be</span> with λ.</p>
Full article ">Figure 4 Cont.
<p>(<b>a</b>) Variation of <span class="html-italic">f’</span>(<span class="html-italic">η</span>) with λ; (<b>b</b>) Variation of <span class="html-italic">θ</span>(<span class="html-italic">η</span>) with λ; (<b>c</b>) Variation of <span class="html-italic">Ns</span> with λ; (<b>d</b>) Variation of <span class="html-italic">Be</span> with λ.</p>
Full article ">Figure 5
<p>(<b>a</b>) Variation of <span class="html-italic">Ns</span> with <span class="html-italic">Ec</span>; (<b>b</b>) Variation of <span class="html-italic">Be</span> with <span class="html-italic">Ec</span>.</p>
Full article ">Figure 6
<p>(<b>a</b>) Variation of <span class="html-italic">Ns</span> with Ω; (<b>b</b>) Variation of <span class="html-italic">Be</span> with Ω.</p>
Full article ">
2182 KiB  
Article
Numerical Study of Entropy Generation in Mixed MHD Convection in a Square Lid-Driven Cavity Filled with Darcy–Brinkman–Forchheimer Porous Medium
by Rahma Bouabda, Mounir Bouabid, Ammar Ben Brahim and Mourad Magherbi
Entropy 2016, 18(12), 436; https://doi.org/10.3390/e18120436 - 6 Dec 2016
Cited by 6 | Viewed by 5223
Abstract
This investigation deals with the numerical simulation of entropy generation at mixed convection flow in a lid-driven saturated porous cavity submitted to a magnetic field. The magnetic field is applied in the direction that is normal to the cavity cross section. The governing [...] Read more.
This investigation deals with the numerical simulation of entropy generation at mixed convection flow in a lid-driven saturated porous cavity submitted to a magnetic field. The magnetic field is applied in the direction that is normal to the cavity cross section. The governing equations, written in the Darcy–Brinkman–Forchheimer formulation, are solved using a numerical code based on the Control Volume Finite Element Method. The flow structure and heat transfer are presented in the form of streamlines, isotherms and average Nusselt number. The entropy generation was studied for various values of Darcy number (10−3 ≤ Da ≤ 1) and for a range of Hartmann number (0 ≤ Ha ≤ 102). It was found that entropy generation is affected by the variations of the considered dimensionless physical parameters. Moreover, the form drag related to the Forchheimer effect remains significant until a critical Hartmann number value. Full article
(This article belongs to the Special Issue Advances in Applied Thermodynamics II)
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Figure 1

Figure 1
<p>Schematic view of the physical model.</p>
Full article ">Figure 2
<p>(<b>a</b>) Streamlines; (<b>b</b>) Isotherms; (<b>c</b>) Isentropic lines (Re = 10, Ra = 10<sup>5</sup>, Da = 10<sup>−2</sup>).</p>
Full article ">Figure 2 Cont.
<p>(<b>a</b>) Streamlines; (<b>b</b>) Isotherms; (<b>c</b>) Isentropic lines (Re = 10, Ra = 10<sup>5</sup>, Da = 10<sup>−2</sup>).</p>
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<p>Average Nusselt number versus Hartmann number for different Darcy numbers (Ra = 10<sup>5</sup>, Re = 10).</p>
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<p>Entropy generation rate versus Hartmann number for different Darcy numbers (Ra = 10<sup>5</sup>, Re = 10).</p>
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<p>Average Nusselt number versus Hartmann number for different Reynolds numbers (Ra = 10<sup>5</sup>, Da = 10<sup>−2</sup>).</p>
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<p>Entropy generation rate versus Hartmann number for different Reynolds numbers (Ra = 10<sup>5</sup>, Da = 10<sup>−2</sup>).</p>
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<p>Entropy generation rate versus Hartmann number for Ra = 10<sup>4</sup> and Ra = 10<sup>5</sup> (Re = 10, Da = 10<sup>−2</sup>).</p>
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<p>Entropy generation rate versus Hartmann number for three Forchheimer parameter values of 0.25, 0.4 and 0.87 (Ra = 10<sup>5</sup>, Da = 10<sup>−2</sup>).</p>
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1201 KiB  
Article
Energy Efficiency Improvement in a Modified Ethanol Process from Acetic Acid
by Young Han Kim
Entropy 2016, 18(12), 422; https://doi.org/10.3390/e18120422 - 24 Nov 2016
Cited by 2 | Viewed by 8652
Abstract
For the high utilization of abundant lignocellulose, which is difficult to directly convert into ethanol, an energy-efficient ethanol production process using acetic acid was examined, and its energy saving performance, economics, and thermodynamic efficiency were compared with the conventional process. The raw ethanol [...] Read more.
For the high utilization of abundant lignocellulose, which is difficult to directly convert into ethanol, an energy-efficient ethanol production process using acetic acid was examined, and its energy saving performance, economics, and thermodynamic efficiency were compared with the conventional process. The raw ethanol synthesized from acetic acid and hydrogen was fed to the proposed ethanol concentration process. The proposed process utilized an extended divided wall column (DWC), for which the performance was investigated with the HYSYS simulation. The performance improvement of the proposed process includes a 27% saving in heating duty and a 41% reduction in cooling duty. The economics shows a 16% saving in investment cost and a 24% decrease in utility cost over the conventional ethanol concentration process. The exergy analysis shows a 9.6% improvement in thermodynamic efficiency for the proposed process. Full article
(This article belongs to the Special Issue Advances in Applied Thermodynamics II)
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Figure 1
<p>Schematic diagram of distillation columns in the conventional process for the ethanol concentration. The numbers in the columns indicate tray numbers, counted from the top.</p>
Full article ">Figure 2
<p>Ternary plots of distillation lines: (<b>a</b>) water (100 °C)–ethanol (78 °C)–acetaldehyde (21 °C) system; (<b>b</b>) water (100 °C)–ethanol (78 °C)–ethyl acetate (77 °C) system.</p>
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<p>Schematic diagram of the extended divided wall column for the ethanol concentration. The numbers in the column indicate the tray number from the top.</p>
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<p>Enthalpy–Carnot factor diagrams: (<b>a</b>) first column in the conventional system; (<b>b</b>) main column in the extended DWC.</p>
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234 KiB  
Article
On Thermodynamics Problems in the Single-Phase-Lagging Heat Conduction Model
by Shu-Nan Li and Bing-Yang Cao
Entropy 2016, 18(11), 391; https://doi.org/10.3390/e18110391 - 9 Nov 2016
Cited by 4 | Viewed by 4443
Abstract
Thermodynamics problems for the single-phase-lagging (SPL) model have not been much studied. In this paper, the violation of the second law of thermodynamics by the SPL model is studied from two perspectives, which are the negative entropy production rate and breaking equilibrium spontaneously. [...] Read more.
Thermodynamics problems for the single-phase-lagging (SPL) model have not been much studied. In this paper, the violation of the second law of thermodynamics by the SPL model is studied from two perspectives, which are the negative entropy production rate and breaking equilibrium spontaneously. The methods for the SPL model to avoid the negative entropy production rate are proposed, which are extended irreversible thermodynamics and the thermal relaxation time. Modifying the entropy production rate positive or zero is not enough to avoid the violation of the second law of thermodynamics for the SPL model, because the SPL model could cause breaking equilibrium spontaneously in some special circumstances. As comparison, it is shown that Fourier’s law and the CV model cannot break equilibrium spontaneously by analyzing mathematical energy integral. Full article
(This article belongs to the Special Issue Advances in Applied Thermodynamics II)
4063 KiB  
Article
Analysis of Entropy Generation in Mixed Convective Peristaltic Flow of Nanofluid
by Tasawar Hayat, Sadaf Nawaz, Ahmed Alsaedi and Maimona Rafiq
Entropy 2016, 18(10), 355; https://doi.org/10.3390/e18100355 - 30 Sep 2016
Cited by 18 | Viewed by 6291
Abstract
This article examines entropy generation in the peristaltic transport of nanofluid in a channel with flexible walls. Single walled carbon nanotubes (SWCNT) and multiple walled carbon nanotubes (MWCNT) with water as base fluid are utilized in this study. Mixed convection is also considered [...] Read more.
This article examines entropy generation in the peristaltic transport of nanofluid in a channel with flexible walls. Single walled carbon nanotubes (SWCNT) and multiple walled carbon nanotubes (MWCNT) with water as base fluid are utilized in this study. Mixed convection is also considered in the present analysis. Viscous dissipation effect is present. Moreover, slip conditions are encountered for both velocity and temperature at the boundaries. Analysis is prepared in the presence of long wavelength and small Reynolds number assumptions. Two phase model for nanofluids are employed. Nonlinear system of equations for small Grashof number is solved. Velocity and temperature are examined for different parameters via graphs. Streamlines are also constructed to analyze the trapping. Results show that axial velocity and temperature of the nanofluid decrease when we enhance the nanoparticle volume fraction. Moreover, the wall elastance parameter shows increase in axial velocity and temperature, whereas decrease in both quantities is noticed for damping coefficient. Decrease is notified in Entropy generation and Bejan number for increasing values of nanoparticle volume fraction. Full article
(This article belongs to the Special Issue Advances in Applied Thermodynamics II)
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Figure 1
<p>Flow Geometry.</p>
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<p><math display="inline"> <semantics> <mi mathvariant="sans-serif">φ</mi> </semantics> </math> versus <math display="inline"> <semantics> <mi>u</mi> </semantics> </math> when <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.02</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi>x</mi> <mo>=</mo> <mn>0.2</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi mathvariant="sans-serif">ε</mi> <mo>=</mo> <mn>0.2</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi>B</mi> <mi>r</mi> <mo>=</mo> <mn>3.0</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi>G</mi> <mi>r</mi> <mo>=</mo> <mn>3.0</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi mathvariant="sans-serif">β</mi> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi mathvariant="sans-serif">γ</mi> <mo>=</mo> <mn>0.01</mn> <mo>.</mo> </mrow> </semantics> </math></p>
Full article ">Figure 3
<p><math display="inline"> <semantics> <mi mathvariant="sans-serif">β</mi> </semantics> </math> versus <math display="inline"> <semantics> <mi>u</mi> </semantics> </math> when <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.02</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi>x</mi> <mo>=</mo> <mn>0.2</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi mathvariant="sans-serif">ε</mi> <mo>=</mo> <mn>0.2</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi>B</mi> <mi>r</mi> <mo>=</mo> <mn>3.0</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi>G</mi> <mi>r</mi> <mo>=</mo> <mn>3.0</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi mathvariant="sans-serif">φ</mi> <mo>=</mo> <mn>0.15</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi mathvariant="sans-serif">γ</mi> <mo>=</mo> <mn>0.01</mn> <mo>.</mo> </mrow> </semantics> </math></p>
Full article ">Figure 4
<p><math display="inline"> <semantics> <mrow> <mi>G</mi> <mi>r</mi> </mrow> </semantics> </math> versus <math display="inline"> <semantics> <mi>u</mi> </semantics> </math> when <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.02</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi>x</mi> <mo>=</mo> <mn>0.2</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi mathvariant="sans-serif">ε</mi> <mo>=</mo> <mn>0.2</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi>B</mi> <mi>r</mi> <mo>=</mo> <mn>3.0</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi mathvariant="sans-serif">φ</mi> <mo>=</mo> <mn>0.15</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi mathvariant="sans-serif">β</mi> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi mathvariant="sans-serif">γ</mi> <mo>=</mo> <mn>0.01</mn> <mo>.</mo> </mrow> </semantics> </math></p>
Full article ">Figure 5
<p><math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>1</mn> </msub> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>2</mn> </msub> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>3</mn> </msub> </mrow> </semantics> </math> versus <math display="inline"> <semantics> <mi>u</mi> </semantics> </math> when <math display="inline"> <semantics> <mrow> <mi mathvariant="sans-serif">φ</mi> <mo>=</mo> <mn>0.15</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi>x</mi> <mo>=</mo> <mn>0.2</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi mathvariant="sans-serif">ε</mi> <mo>=</mo> <mn>0.2</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi>B</mi> <mi>r</mi> <mo>=</mo> <mn>3.0</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi>G</mi> <mi>r</mi> <mo>=</mo> <mn>3.0</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi mathvariant="sans-serif">β</mi> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi mathvariant="sans-serif">γ</mi> <mo>=</mo> <mn>0.01</mn> <mo>.</mo> </mrow> </semantics> </math></p>
Full article ">Figure 6
<p><math display="inline"> <semantics> <mi mathvariant="sans-serif">φ</mi> </semantics> </math> versus <math display="inline"> <semantics> <mi mathvariant="sans-serif">θ</mi> </semantics> </math> when <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.02</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi>x</mi> <mo>=</mo> <mn>0.2</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi mathvariant="sans-serif">ε</mi> <mo>=</mo> <mn>0.2</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi>B</mi> <mi>r</mi> <mo>=</mo> <mn>3.0</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi>G</mi> <mi>r</mi> <mo>=</mo> <mn>3.0</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi mathvariant="sans-serif">β</mi> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi mathvariant="sans-serif">γ</mi> <mo>=</mo> <mn>0.01</mn> <mo>.</mo> </mrow> </semantics> </math></p>
Full article ">Figure 7
<p><math display="inline"> <semantics> <mi mathvariant="sans-serif">γ</mi> </semantics> </math> versus <math display="inline"> <semantics> <mi mathvariant="sans-serif">θ</mi> </semantics> </math> when <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.02</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi>x</mi> <mo>=</mo> <mn>0.2</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi mathvariant="sans-serif">ε</mi> <mo>=</mo> <mn>0.2</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi>B</mi> <mi>r</mi> <mo>=</mo> <mn>3.0</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi>G</mi> <mi>r</mi> <mo>=</mo> <mn>3.0</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi mathvariant="sans-serif">β</mi> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi mathvariant="sans-serif">φ</mi> <mo>=</mo> <mn>0.15</mn> <mo>.</mo> </mrow> </semantics> </math></p>
Full article ">Figure 8
<p><math display="inline"> <semantics> <mrow> <mi>G</mi> <mi>r</mi> </mrow> </semantics> </math> versus <math display="inline"> <semantics> <mi mathvariant="sans-serif">θ</mi> </semantics> </math> when <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.02</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi>x</mi> <mo>=</mo> <mn>0.2</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi mathvariant="sans-serif">ε</mi> <mo>=</mo> <mn>0.2</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi>B</mi> <mi>r</mi> <mo>=</mo> <mn>3.0</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi mathvariant="sans-serif">φ</mi> <mo>=</mo> <mn>0.15</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi mathvariant="sans-serif">β</mi> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi mathvariant="sans-serif">γ</mi> <mo>=</mo> <mn>0.01</mn> <mo>.</mo> </mrow> </semantics> </math></p>
Full article ">Figure 9
<p><math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>1</mn> </msub> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>2</mn> </msub> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>3</mn> </msub> </mrow> </semantics> </math> versus <math display="inline"> <semantics> <mi mathvariant="sans-serif">θ</mi> </semantics> </math> when <math display="inline"> <semantics> <mrow> <mi mathvariant="sans-serif">φ</mi> <mo>=</mo> <mn>0.15</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi>x</mi> <mo>=</mo> <mn>0.2</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi mathvariant="sans-serif">ε</mi> <mo>=</mo> <mn>0.2</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi>B</mi> <mi>r</mi> <mo>=</mo> <mn>3.0</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi>G</mi> <mi>r</mi> <mo>=</mo> <mn>3.0</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi mathvariant="sans-serif">β</mi> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi mathvariant="sans-serif">γ</mi> <mo>=</mo> <mn>0.01</mn> <mo>.</mo> </mrow> </semantics> </math></p>
Full article ">Figure 10
<p><math display="inline"> <semantics> <mi mathvariant="sans-serif">φ</mi> </semantics> </math> versus <math display="inline"> <semantics> <mrow> <mi>N</mi> <mi>s</mi> </mrow> </semantics> </math> when <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.02</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi>x</mi> <mo>=</mo> <mn>0.2</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi mathvariant="sans-serif">ε</mi> <mo>=</mo> <mn>0.2</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi>B</mi> <mi>r</mi> <msup> <mi mathvariant="normal">Λ</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mn>1.0</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi>G</mi> <mi>r</mi> <mo>=</mo> <mn>3.0</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi mathvariant="sans-serif">β</mi> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi mathvariant="sans-serif">γ</mi> <mo>=</mo> <mn>0.01</mn> <mo>.</mo> </mrow> </semantics> </math></p>
Full article ">Figure 11
<p><math display="inline"> <semantics> <mrow> <mi>G</mi> <mi>r</mi> </mrow> </semantics> </math> versus <math display="inline"> <semantics> <mrow> <mi>N</mi> <mi>s</mi> </mrow> </semantics> </math> when <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.02</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi>x</mi> <mo>=</mo> <mn>0.2</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi mathvariant="sans-serif">ε</mi> <mo>=</mo> <mn>0.2</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi>B</mi> <mi>r</mi> <msup> <mi mathvariant="normal">Λ</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mn>1.0</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi mathvariant="sans-serif">φ</mi> <mo>=</mo> <mn>0.15</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi mathvariant="sans-serif">β</mi> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi mathvariant="sans-serif">γ</mi> <mo>=</mo> <mn>0.01</mn> <mo>.</mo> </mrow> </semantics> </math></p>
Full article ">Figure 12
<p><math display="inline"> <semantics> <mrow> <mi>B</mi> <mi>r</mi> <msup> <mi mathvariant="normal">Λ</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </semantics> </math> versus <math display="inline"> <semantics> <mrow> <mi>N</mi> <mi>s</mi> </mrow> </semantics> </math> when <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.02</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi>x</mi> <mo>=</mo> <mn>0.2</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi mathvariant="sans-serif">ε</mi> <mo>=</mo> <mn>0.2</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi mathvariant="sans-serif">φ</mi> <mo>=</mo> <mn>0.15</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi>G</mi> <mi>r</mi> <mo>=</mo> <mn>3.0</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi mathvariant="sans-serif">β</mi> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi mathvariant="sans-serif">γ</mi> <mo>=</mo> <mn>0.01</mn> <mo>.</mo> </mrow> </semantics> </math></p>
Full article ">Figure 13
<p><math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>1</mn> </msub> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>2</mn> </msub> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>3</mn> </msub> </mrow> </semantics> </math> versus <math display="inline"> <semantics> <mrow> <mi>N</mi> <mi>s</mi> </mrow> </semantics> </math> when <math display="inline"> <semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi>x</mi> <mo>=</mo> <mn>0.2</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi mathvariant="sans-serif">ε</mi> <mo>=</mo> <mn>0.2</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi mathvariant="sans-serif">φ</mi> <mo>=</mo> <mn>0.15</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi>B</mi> <mi>r</mi> <msup> <mi mathvariant="normal">Λ</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi>G</mi> <mi>r</mi> <mo>=</mo> <mn>3.0</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi mathvariant="sans-serif">β</mi> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi mathvariant="sans-serif">γ</mi> <mo>=</mo> <mn>0.01</mn> <mo>.</mo> </mrow> </semantics> </math></p>
Full article ">Figure 14
<p><math display="inline"> <semantics> <mi mathvariant="sans-serif">φ</mi> </semantics> </math> versus <math display="inline"> <semantics> <mrow> <mi>B</mi> <mi>e</mi> </mrow> </semantics> </math> when <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.02</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi>x</mi> <mo>=</mo> <mn>0.2</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi mathvariant="sans-serif">ε</mi> <mo>=</mo> <mn>0.2</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi>B</mi> <mi>r</mi> <msup> <mi mathvariant="normal">Λ</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mn>1.0</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi>G</mi> <mi>r</mi> <mo>=</mo> <mn>3.0</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi mathvariant="sans-serif">β</mi> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi mathvariant="sans-serif">γ</mi> <mo>=</mo> <mn>0.01</mn> <mo>.</mo> </mrow> </semantics> </math></p>
Full article ">Figure 15
<p><math display="inline"> <semantics> <mrow> <mi>G</mi> <mi>r</mi> </mrow> </semantics> </math> versus <math display="inline"> <semantics> <mrow> <mi>B</mi> <mi>e</mi> </mrow> </semantics> </math> when <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.02</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi>x</mi> <mo>=</mo> <mn>0.2</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi mathvariant="sans-serif">ε</mi> <mo>=</mo> <mn>0.2</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi>B</mi> <mi>r</mi> <msup> <mi mathvariant="normal">Λ</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mn>1.0</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi mathvariant="sans-serif">φ</mi> <mo>=</mo> <mn>0.15</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi mathvariant="sans-serif">β</mi> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi mathvariant="sans-serif">γ</mi> <mo>=</mo> <mn>0.01</mn> <mo>.</mo> </mrow> </semantics> </math></p>
Full article ">Figure 16
<p><math display="inline"> <semantics> <mrow> <mi>B</mi> <mi>r</mi> <msup> <mi mathvariant="normal">Λ</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </semantics> </math> versus <math display="inline"> <semantics> <mrow> <mi>B</mi> <mi>e</mi> </mrow> </semantics> </math> when <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.02</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi>x</mi> <mo>=</mo> <mn>0.2</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi mathvariant="sans-serif">ε</mi> <mo>=</mo> <mn>0.2</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi>B</mi> <mi>r</mi> <msup> <mi mathvariant="normal">Λ</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mn>1.0</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi mathvariant="sans-serif">φ</mi> <mo>=</mo> <mn>0.15</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi>G</mi> <mi>r</mi> <mo>=</mo> <mn>3.0</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi mathvariant="sans-serif">β</mi> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi mathvariant="sans-serif">γ</mi> <mo>=</mo> <mn>0.01</mn> <mo>.</mo> </mrow> </semantics> </math></p>
Full article ">Figure 17
<p><math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>1</mn> </msub> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>2</mn> </msub> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>3</mn> </msub> </mrow> </semantics> </math> versus <math display="inline"> <semantics> <mrow> <mi>B</mi> <mi>e</mi> </mrow> </semantics> </math> when <math display="inline"> <semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi>x</mi> <mo>=</mo> <mn>0.2</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi mathvariant="sans-serif">ε</mi> <mo>=</mo> <mn>0.2</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi>B</mi> <mi>r</mi> <msup> <mi mathvariant="normal">Λ</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mn>1.0</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi mathvariant="sans-serif">φ</mi> <mo>=</mo> <mn>0.15</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi>G</mi> <mi>r</mi> <mo>=</mo> <mn>3.0</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi mathvariant="sans-serif">β</mi> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi mathvariant="sans-serif">γ</mi> <mo>=</mo> <mn>0.01</mn> <mo>.</mo> </mrow> </semantics> </math></p>
Full article ">Figure 18
<p><math display="inline"> <semantics> <mi mathvariant="sans-serif">ψ</mi> </semantics> </math> versus <math display="inline"> <semantics> <mi mathvariant="sans-serif">β</mi> </semantics> </math> for SWCNT when <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.02</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi>B</mi> <mi>r</mi> <mo>=</mo> <mn>3.0</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi>G</mi> <mi>r</mi> <mo>=</mo> <mn>3.0</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi mathvariant="sans-serif">ε</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi mathvariant="sans-serif">γ</mi> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi mathvariant="sans-serif">φ</mi> <mo>=</mo> <mn>0.2</mn> </mrow> </semantics> </math>: (<b>a</b>) <math display="inline"> <semantics> <mrow> <mi mathvariant="sans-serif">β</mi> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics> </math>; and (<b>b</b>) <math display="inline"> <semantics> <mrow> <mi mathvariant="sans-serif">β</mi> <mo>=</mo> <mn>0.03</mn> <mo>.</mo> </mrow> </semantics> </math></p>
Full article ">Figure 19
<p><math display="inline"> <semantics> <mi mathvariant="sans-serif">ψ</mi> </semantics> </math> versus <math display="inline"> <semantics> <mi mathvariant="sans-serif">β</mi> </semantics> </math> for MWCNT when <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.02</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi>B</mi> <mi>r</mi> <mo>=</mo> <mn>3.0</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi>G</mi> <mi>r</mi> <mo>=</mo> <mn>3.0</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi mathvariant="sans-serif">ε</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi mathvariant="sans-serif">γ</mi> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi mathvariant="sans-serif">φ</mi> <mo>=</mo> <mn>0.2</mn> </mrow> </semantics> </math>: (<b>a</b>) <math display="inline"> <semantics> <mrow> <mi mathvariant="sans-serif">β</mi> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics> </math>; and (<b>b</b>) <math display="inline"> <semantics> <mrow> <mi mathvariant="sans-serif">β</mi> <mo>=</mo> <mn>0.03</mn> <mo>.</mo> </mrow> </semantics> </math></p>
Full article ">Figure 20
<p><math display="inline"> <semantics> <mi mathvariant="sans-serif">ψ</mi> </semantics> </math> versus <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>1</mn> </msub> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>2</mn> </msub> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>3</mn> </msub> </mrow> </semantics> </math> for SWCNT when <math display="inline"> <semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi>B</mi> <mi>r</mi> <mo>=</mo> <mn>3.0</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi>G</mi> <mi>r</mi> <mo>=</mo> <mn>3.0</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi mathvariant="sans-serif">ε</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi mathvariant="sans-serif">β</mi> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi mathvariant="sans-serif">γ</mi> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi mathvariant="sans-serif">φ</mi> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics> </math>: (<b>a</b>) <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.03</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics> </math>; (<b>b</b>) <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.06</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.03</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics> </math>; (<b>c</b>) <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.07</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics> </math>; and (<b>d</b>) <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.03</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>0.02</mn> <mo>.</mo> </mrow> </semantics> </math></p>
Full article ">Figure 21
<p><math display="inline"> <semantics> <mrow> <mi mathvariant="sans-serif">ψ</mi> </mrow> </semantics> </math> versus <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>1</mn> </msub> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>2</mn> </msub> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>3</mn> </msub> </mrow> </semantics> </math> for MWCNT when <math display="inline"> <semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi>B</mi> <mi>r</mi> <mo>=</mo> <mn>3.0</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi>G</mi> <mi>r</mi> <mo>=</mo> <mn>3.0</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi mathvariant="sans-serif">ε</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi mathvariant="sans-serif">β</mi> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi mathvariant="sans-serif">γ</mi> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <mi mathvariant="sans-serif">φ</mi> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics> </math>: (<b>a</b>) <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.03</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics> </math>; (<b>b</b>) <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.06</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.03</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics> </math>; (<b>c</b>) <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.07</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics> </math>; and (<b>d</b>) <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.03</mn> <mo>,</mo> </mrow> </semantics> </math> <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>0.02</mn> <mo>.</mo> </mrow> </semantics> </math></p>
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4958 KiB  
Article
Heat Transfer and Entropy Generation of Non-Newtonian Laminar Flow in Microchannels with Four Flow Control Structures
by Ke Yang, Di Zhang, Yonghui Xie and Gongnan Xie
Entropy 2016, 18(8), 302; https://doi.org/10.3390/e18080302 - 12 Aug 2016
Cited by 7 | Viewed by 5495
Abstract
Flow characteristics and heat transfer performances of carboxymethyl cellulose (CMC) aqueous solutions in the microchannels with flow control structures were investigated in this study. The researches were carried out with various flow rates and concentrations of the CMC aqueous solutions. The results reveal [...] Read more.
Flow characteristics and heat transfer performances of carboxymethyl cellulose (CMC) aqueous solutions in the microchannels with flow control structures were investigated in this study. The researches were carried out with various flow rates and concentrations of the CMC aqueous solutions. The results reveal that the pin-finned microchannel has the most uniform temperature distribution on the structured walls, and the average temperature on the structured wall reaches the minimum value in cylinder-ribbed microchannels at the same flow rate and CMC concentration. Moreover, the protruded microchannel obtains the minimum relative Fanning friction factor f/f0, while, the maximum f/f0 is observed in the cylinder-ribbed microchannel. Furthermore, the minimum f/f0 is reached at the cases with CMC2000, and also, the relative Nusselt number Nu/Nu0 of CMC2000 cases is larger than that of other cases in the four structured microchannels. Therefore, 2000 ppm is the recommended concentration of CMC aqueous solutions in all the cases with different flow rates and flow control structures. Pin-finned microchannels are preferred in low flow rate cases, while, V-grooved microchannels have the minimum relative entropy generation S’/S0 and best thermal performance TP at CMC2000 in high flow rates. Full article
(This article belongs to the Special Issue Advances in Applied Thermodynamics II)
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Graphical abstract

Graphical abstract
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<p>Schematic diagram of boundary conditions.</p>
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<p>Geometrical structures of flow field: (<b>a</b>) protrusion; (<b>b</b>) pin fin; (<b>c</b>) cylinder rib; (<b>d</b>) V-groove.</p>
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<p>Temperature distributions (unit: K) and limiting streamlines on the structured wall (Case: <span class="html-italic">Q</span> = 4 × 10<sup>−5</sup> kg·s<sup>−1</sup> with protruded wall): (<b>a</b>) CMC100; (<b>b</b>) CMC500; (<b>c</b>) CMC2000; (<b>d</b>) CMC4000.</p>
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<p>Dynamic viscosity distributions (unit: Pa·s) and streamlines in the stream-wise middle sections (Case: <span class="html-italic">Q</span> = 4 × 10<sup>−5</sup> kg·s<sup>−1</sup> with protruded wall): (<b>a</b>) CMC100; (<b>b</b>) CMC500; (<b>c</b>) CMC2000; (<b>d</b>) CMC4000.</p>
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<p>Dynamic viscosity distributions (unit: Pa·s) and streamlines in the perpendicular to stream-wise middle sections (Case: <span class="html-italic">Q</span> = 4 × 10<sup>−5</sup> kg·s<sup>−1</sup> with protruded wall): (<b>a</b>) CMC100; (<b>b</b>) CMC500; (<b>c</b>) CMC2000; (<b>d</b>) CMC4000.</p>
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<p>Temperature distributions (Unit: K) and limiting streamlines on structured wall (Case: <span class="html-italic">Q</span> = 2 × 10<sup>−5</sup> kg·s<sup>−1</sup>; CMC500): (<b>a</b>) protrusion; (<b>b</b>) pin-fin; (<b>c</b>) cylinder-rib; (<b>d</b>) V-groove.</p>
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<p>Velocity contours (unit: m·s<sup>−1</sup>) and streamlines in the perpendicular to stream-wise sections (Case: <span class="html-italic">Q</span> = 2 × 10<sup>−5</sup> kg·s<sup>−1</sup>; CMC500; protruded microchannel).</p>
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<p>Variations of <span class="html-italic">f</span>/<span class="html-italic">f</span><sub>0</sub> with flow rate and CMC concentration in protruded and cylinder-ribbed microchannels.</p>
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<p>Variations of <span class="html-italic">f</span>/<span class="html-italic">f</span><sub>0</sub> with flow rate and CMC concentration in pin-finned and V-grooved microchannels.</p>
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<p>Variations of <span class="html-italic">Nu</span>/<span class="html-italic">Nu</span><sub>0</sub> with flow rate and CMC concentration in protruded and cylinder-ribbed microchannels.</p>
Full article ">Figure 11
<p>Variations of <span class="html-italic">Nu</span>/<span class="html-italic">Nu</span><sub>0</sub> with flow rate and CMC concentration in pin-finned and V-grooved microchannels.</p>
Full article ">Figure 12
<p>Variations of <span class="html-italic">S’</span>/<span class="html-italic">S</span><sub>0</sub><span class="html-italic">’</span> with flow rate and CMC concentration in protruded and cylinder-ribbed microchannels.</p>
Full article ">Figure 13
<p>Variations of <span class="html-italic">S’</span>/<span class="html-italic">S</span><sub>0</sub><span class="html-italic">’</span> with flow rate and CMC concentration in pin-finned and V-grooved microchannels.</p>
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<p>Variations of <span class="html-italic">TP</span> with flow rate and CMC concentration in protruded and cylinder-ribbed microchannels.</p>
Full article ">Figure 15
<p>Variations of <span class="html-italic">TP</span> with flow rate and CMC concentration in pin-finned and V-grooved microchannels.</p>
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4767 KiB  
Article
A Critical Reassessment of the Hess–Murray Law
by Enrico Sciubba
Entropy 2016, 18(8), 283; https://doi.org/10.3390/e18080283 - 5 Aug 2016
Cited by 22 | Viewed by 9102
Abstract
The Hess–Murray law is a correlation between the radii of successive branchings in bi/trifurcated vessels in biological tissues. First proposed by the Swiss physiologist and Nobel laureate Walter Rudolf Hess in his 1914 doctoral thesis and published in 1917, the law was “rediscovered” [...] Read more.
The Hess–Murray law is a correlation between the radii of successive branchings in bi/trifurcated vessels in biological tissues. First proposed by the Swiss physiologist and Nobel laureate Walter Rudolf Hess in his 1914 doctoral thesis and published in 1917, the law was “rediscovered” by the American physiologist Cecil Dunmore Murray in 1926. The law is based on the assumption that blood or lymph circulation in living organisms is governed by a “work minimization” principle that—under a certain set of specified conditions—leads to an “optimal branching ratio” of r i + 1 r i = 1 2 3 = 0.7937 . This “cubic root of 2” correlation underwent extensive theoretical and experimental reassessment in the second half of the 20th century, and the results indicate that—under a well-defined series of conditions—the law is sufficiently accurate for the smallest vessels (r of the order of fractions of millimeter) but fails for the larger ones; moreover, it cannot be successfully extended to turbulent flows. Recent comparisons with numerical investigations of branched flows led to similar conclusions. More recently, the Hess–Murray law came back into the limelight when it was taken as a founding paradigm of the Constructal Law, a theory that employs physical intuition and mathematical reasoning to derive “optimal paths” for the transport of matter and energy between a source and a sink, regardless of the mode of transportation (continuous, like in convection and conduction, or discrete, like in the transportation of goods and people). This paper examines the foundation of the law and argues that both for natural flows and for engineering designs, a minimization of the irreversibility under physically sound boundary conditions leads to somewhat different results. It is also shown that, in the light of an exergy-based resource analysis, an amended version of the Hess–Murray law may still hold an important position in engineering and biological sciences. Full article
(This article belongs to the Special Issue Advances in Applied Thermodynamics II)
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Graphical abstract
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<p>Bifurcated vessels: geometry definition.</p>
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<p>Murray’s original drawing for the derivation of his branching angles rule [<a href="#B5-entropy-18-00283" class="html-bibr">5</a>] (in the notation of this paper, x = α<sub>1</sub>, y = α<sub>2</sub>).</p>
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<p>Minimum irreversibility ratio S/S<sub>0</sub> as a function of the aspect ratio <span class="html-italic">H</span>/<span class="html-italic">L</span> and of the splitting angle α<sub>1</sub> for different physical assumptions [<a href="#B29-entropy-18-00283" class="html-bibr">29</a>].</p>
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<p>Each bifurcation level introduces two additional “delivery points”.</p>
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<p>Schematic representation of the function of capillary networks in biological tissue. (<b>a</b>) functional scheme of a capillary network [<a href="#B32-entropy-18-00283" class="html-bibr">32</a>]; (<b>b</b>) its realistic representation (adapted from [<a href="#B16-entropy-18-00283" class="html-bibr">16</a>]).</p>
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<p>Microscopic images of natural capillary networks. (<b>a</b>) Capillary network in fat tissue [<a href="#B33-entropy-18-00283" class="html-bibr">33</a>]; (<b>b</b>) Hydrangea leaf [<a href="#B34-entropy-18-00283" class="html-bibr">34</a>]; (<b>c</b>) Capillary network in the human retina [<a href="#B35-entropy-18-00283" class="html-bibr">35</a>]; (<b>d</b>) Nettle leaves [<a href="#B36-entropy-18-00283" class="html-bibr">36</a>].</p>
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<p>The two regions “served” by the main and by the daughter branches.</p>
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<p>Optimal geometries for equal area drain.</p>
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3798 KiB  
Article
Entropy Generation through Non-Equilibrium Ordered Structures in Corner Flows with Sidewall Mass Injection
by LaVar King Isaacson
Entropy 2016, 18(8), 279; https://doi.org/10.3390/e18080279 - 28 Jul 2016
Cited by 2 | Viewed by 5822
Abstract
Additional entropy generation rates through non-equilibrium ordered structures are predicted for corner flows with sidewall mass injection. Well-defined non-equilibrium ordered structures are predicted at a normalized vertical station of approximately eighteen percent of the boundary-layer thickness. These structures are in addition to the [...] Read more.
Additional entropy generation rates through non-equilibrium ordered structures are predicted for corner flows with sidewall mass injection. Well-defined non-equilibrium ordered structures are predicted at a normalized vertical station of approximately eighteen percent of the boundary-layer thickness. These structures are in addition to the ordered structures previously reported at approximately thirty-eight percent of the boundary layer thickness. The computational procedure is used to determine the entropy generation rate for each spectral velocity component at each of several stream wise stations and for each of several injection velocity values. Application of the procedure to possible thermal system processes is discussed. These results indicate that cooling sidewall mass injection into a horizontal laminar boundary layer may actually increase the heat transfer to the horizontal surface. Full article
(This article belongs to the Special Issue Advances in Applied Thermodynamics II)
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Graphical abstract

Graphical abstract
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<p>Schematic of the corner flow boundary layer configuration is shown [<a href="#B1-entropy-18-00279" class="html-bibr">1</a>].</p>
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<p>Shown is a three-dimensional deterministic trajectory of the spectral velocity components, <span class="html-italic">a</span><sub><span class="html-italic">x</span>4</sub>, <span class="html-italic">a</span><sub><span class="html-italic">y</span>4</sub>, and <span class="html-italic">a</span><sub><span class="html-italic">z</span>4</sub>, at <span class="html-italic">x</span> = 0.140 for η = 1.40 and <span class="html-italic">w<sub>e</sub></span> = 0.0800.</p>
Full article ">Figure 3
<p>The phase diagram of the span wise and normal spectral velocity components, <span class="html-italic">a</span><sub><span class="html-italic">z</span>4</sub>–<span class="html-italic">a</span><sub><span class="html-italic">y</span>4</sub>, is shown for <span class="html-italic">x</span> = 0.140, η = 1.40 and <span class="html-italic">w<sub>e</sub></span> = 0.0800.</p>
Full article ">Figure 4
<p>A three-dimensional representation of the deterministic trajectories of the spectral velocity components, <span class="html-italic">a</span><sub><span class="html-italic">x</span>4</sub>, <span class="html-italic">a</span><sub><span class="html-italic">y</span>4</sub>, and <span class="html-italic">a</span><sub><span class="html-italic">z</span>4</sub>, is shown for <span class="html-italic">x</span> = 0.140, η = 1.40 and <span class="html-italic">w<sub>e</sub></span> = 0.0925.</p>
Full article ">Figure 5
<p>The phase diagram of the span wise and normal spectral velocity components, <span class="html-italic">a</span><sub><span class="html-italic">z</span>4</sub>–<span class="html-italic">a</span><sub><span class="html-italic">y</span>4</sub>, is shown for <span class="html-italic">x</span> = 0.140, η = 1.40 and <span class="html-italic">w<sub>e</sub></span> = 0.0925.</p>
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<p>The power spectral density for the normal spectral velocity component is shown for <span class="html-italic">x</span> = 0.140, η = 1.40 and <span class="html-italic">w<sub>e</sub></span> = 0.0800.</p>
Full article ">Figure 7
<p>The power spectral density for the span wise spectral velocity component is shown for <span class="html-italic">x</span> = 0.140, η = 1.40 and <span class="html-italic">w<sub>e</sub></span> = 0.0800.</p>
Full article ">Figure 8
<p>The power spectral density for the normal spectral velocity component is shown for <span class="html-italic">x</span> = 0.140, η = 1.40 and <span class="html-italic">w<sub>e</sub></span> = 0.0925.</p>
Full article ">Figure 9
<p>The power spectral density for the span wise spectral velocity component is shown for <span class="html-italic">x</span> = 0.140, η = 1.40 and <span class="html-italic">w<sub>e</sub></span> = 0.0925.</p>
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<p>The empirical entropic index for the normal spectral velocity component is shown as a function of the empirical mode, <span class="html-italic">j</span>, for <span class="html-italic">x</span> = 0.140, η = 1.40 and <span class="html-italic">w<sub>e</sub></span> = 0.0800.</p>
Full article ">Figure 11
<p>The empirical entropic index for the normal spectral velocity component is shown as a function of the empirical mode, <span class="html-italic">j</span>, for <span class="html-italic">x</span> = 0.140, η = 1.40 and <span class="html-italic">w<sub>e</sub></span> = 0.0925.</p>
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<p>The empirical intermittency exponent for the normal spectral velocity component is shown as a function of the empirical mode, <span class="html-italic">j</span>, for <span class="html-italic">x</span> = 0.140, η = 1.40 and <span class="html-italic">w<sub>e</sub></span> = 0.0800.</p>
Full article ">Figure 13
<p>The empirical intermittency exponent for the normal spectral velocity component is shown as a function of the empirical mode, <span class="html-italic">j</span>, for <span class="html-italic">x</span> = 0.140, η = 1.40 and <span class="html-italic">w<sub>e</sub></span> = 0.0925.</p>
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<p>The entropy generation rates for the non-equilibrium ordered structures at a normalized vertical station η = 1.40 are shown as a function of the <span class="html-italic">x</span>-direction stations for several applied crosswind velocities.</p>
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<p>The entropy generation rates for the non-equilibrium ordered structures at a normalized vertical station η =3.00 are shown as a function of the <span class="html-italic">x</span>-direction stations for several applied crosswind velocities [<a href="#B1-entropy-18-00279" class="html-bibr">1</a>].</p>
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<p>The entropy generation rates for various crosswind velocities are shown at normalized distances of η = 1.40 and η = 3.00 at a distance of <span class="html-italic">x</span> = 0.140. Also shown is the distribution of the entropy generation rate at a distance of <span class="html-italic">x</span> = 0.140 for a turbulent boundary layer initiated at <span class="html-italic">x</span> = 0.02 on the horizontal surface.</p>
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Review

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1228 KiB  
Review
Nonequilibrium Thermodynamics of Ion Flux through Membrane Channels
by Chi-Pan Hsieh
Entropy 2017, 19(1), 40; https://doi.org/10.3390/e19010040 - 19 Jan 2017
Cited by 3 | Viewed by 6931
Abstract
Ion flux through membrane channels is passively driven by the electrochemical potential differences across the cell membrane. Nonequilibrium thermodynamics has been successful in explaining transport mechanisms, including the ion transport phenomenon. However, physiologists may not be familiar with biophysical concepts based on the [...] Read more.
Ion flux through membrane channels is passively driven by the electrochemical potential differences across the cell membrane. Nonequilibrium thermodynamics has been successful in explaining transport mechanisms, including the ion transport phenomenon. However, physiologists may not be familiar with biophysical concepts based on the view of entropy production. In this paper, I have reviewed the physical meanings and connections between nonequilibrium thermodynamics and the expressions commonly used in describing ion fluxes in membrane physiology. The fluctuation theorem can be applied to interpret the flux ratio in the small molecular systems. The multi-ion single-file feature of the ion channel facilitates the utilization of the natural tendency of electrochemical driving force to couple specific biophysical processes and biochemical reactions on the membrane. Full article
(This article belongs to the Special Issue Advances in Applied Thermodynamics II)
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Figure 1
<p>(<b>a</b>) The isolated two-compartment system, enclosed by rigid adiabatic walls, for describing entropy change in irreversible processes; (<b>b</b>) The two-compartment system separated by a cell membrane containing an ion channel in the voltage-clamp patch-clamp recording.</p>
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<p>The K<sup>+</sup> ion conduction model is shown with the structure of the KcsA channel using Rasmol. The K<sup>+</sup> channel switches rapidly between (S2, S4) and (S1, S3) configurations to transport K<sup>+</sup> ions. The ratio of the outward transport event to the inward transport event can be estimated from the fluctuation theorem.</p>
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<p>(<b>a</b>) The structure model of the inward rectifier K<sup>+</sup> (Kir) channel reveals an elongated multi-ion single-file cytoplasmic pore beyond the selectivity filter [<a href="#B32-entropy-19-00040" class="html-bibr">32</a>,<a href="#B33-entropy-19-00040" class="html-bibr">33</a>], facilitating the flux-coupled blockage. The strands presentation of the Kir2.1 channel is retrieved from the SWISS-MODEL Repository based on the X-ray crystallography template structure of a Kir2.2 channel (PDB: 3sph) [<a href="#B34-entropy-19-00040" class="html-bibr">34</a>]; (<b>b</b>) The current–voltage relationship of the Kir channel at symmetrical K<sup>+</sup> concentrations (<math display="inline"> <semantics> <mrow> <msub> <mi>V</mi> <mrow> <mi>e</mi> <mi>q</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mtext> </mtext> <mi>mV</mi> <mo stretchy="false">)</mo> </mrow> </semantics> </math> in the presence of the intracellular blocker. The strong inward rectification results from the “driving force”-dependent or flux-coupled block by the intracellular blocker [<a href="#B1-entropy-19-00040" class="html-bibr">1</a>,<a href="#B2-entropy-19-00040" class="html-bibr">2</a>].</p>
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13147 KiB  
Review
Generalized Thermodynamic Optimization for Iron and Steel Production Processes: Theoretical Exploration and Application Cases
by Lingen Chen, Huijun Feng and Zhihui Xie
Entropy 2016, 18(10), 353; https://doi.org/10.3390/e18100353 - 29 Sep 2016
Cited by 128 | Viewed by 17118
Abstract
Combining modern thermodynamics theory branches, including finite time thermodynamics or entropy generation minimization, constructal theory and entransy theory, with metallurgical process engineering, this paper provides a new exploration on generalized thermodynamic optimization theory for iron and steel production processes. The theoretical core is [...] Read more.
Combining modern thermodynamics theory branches, including finite time thermodynamics or entropy generation minimization, constructal theory and entransy theory, with metallurgical process engineering, this paper provides a new exploration on generalized thermodynamic optimization theory for iron and steel production processes. The theoretical core is to thermodynamically optimize performances of elemental packages, working procedure modules, functional subsystems, and whole process of iron and steel production processes with real finite-resource and/or finite-size constraints with various irreversibilities toward saving energy, decreasing consumption, reducing emission and increasing yield, and to achieve the comprehensive coordination among the material flow, energy flow and environment of the hierarchical process systems. A series of application cases of the theory are reviewed. It can provide a new angle of view for the iron and steel production processes from thermodynamics, and can also provide some guidelines for other process industries. Full article
(This article belongs to the Special Issue Advances in Applied Thermodynamics II)
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<p>Schematic diagram of iron and steel production process.</p>
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<p>Energy flow of coking procedure.</p>
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<p>Effect of final moisture content on the energy consumption per ton product.</p>
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<p>Energy value balance of the sintering process.</p>
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<p>Effect of coke on the minimum energy value of sinter.</p>
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<p>Two-dimensional unsteady model of continuous cooling process of sintered ore.</p>
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<p>Effect of heat transfer ratio on temperature of waste gas.</p>
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<p>Schematic diagram of sinter cooling process.</p>
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<p>Effect of the porosity on the optimization result.</p>
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<p>Flow and heat transfer model in vertical tank.</p>
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<p>Effects of layer height on the field synergy number.</p>
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<p>Heat transfer model of a blast furnace wall.</p>
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<p>Physical model of blast furnace iron-making elemental package.</p>
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<p>Physical model of blast furnace iron-making procedure.</p>
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<p>Optimal cost distribution for a blast furnace iron-making process.</p>
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<p>Physical model of blast furnace iron-making procedure.</p>
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<p>The schematic diagram of the open simple Brayton power plant model.</p>
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<p>Optimal cost distribution and useful energy distribution.</p>
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<p>Physical model for a converter steel-making process.</p>
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<p>Effect of Si content on the optimal results.</p>
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<p>Physical model for a converter steel-making procedure.</p>
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<p>Effect of steel slag basicity on the optimal results.</p>
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<p>Schematic diagram of thin slab continuous casting and rolling procedures.</p>
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<p>Effect of water flow distribution in the secondary cooling zone on the final rolling temperature and final cooling temperature.</p>
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<p>Schematic diagram of slab continuous casting process.</p>
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<p>Temperature distributions of the slab for initial and optimal schedules.</p>
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<p>Model of a reheating furnace wall with multi-layer insulation structures.</p>
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<p>Comparisons of the optimal results based on different optimization objectives.</p>
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<p>Schematic diagram of a strip laminar cooling process.</p>
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<p>Effect of cooling mode on the complex function.</p>
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<p>Flow chart of hot blast stove flue gas sensible heat recovery and utilization.</p>
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<p>Flow chart for the sintering waste heat utilization.</p>
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<p>Model of tubular plug flow reactor.</p>
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<p>Effect of input temperature on the reacting rate versus piling catalyst mass.</p>
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<p>System layout of an air Brayton cycle driven by waste heat of blast furnace slag.</p>
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<p>System layout of an open simple Brayton cycle driven by residual energy of converter gas.</p>
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<p>Disc-shaped model of solid-gas reactor.</p>
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<p>Schematic diagram of a one-stage air-cooling thermoelectric power generator device driven by waste water.</p>
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<p>Waste heat recovery net work in iron and steel factory.</p>
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<p>Input-output relationship of iron-making system.</p>
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<p>Diagram of energy flow for electric arc furnace steel-making process.</p>
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<p>Causal loop diagram of ISPP.</p>
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<p>Response characteristic of the global iron-flow network to returned iron-flow of rolling procedure.</p>
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<p>Dissection of energy consumption structure for ISPP.</p>
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<p>Schematic diagram of the ISPP for constructal optimization.</p>
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