Mean error and standard deviation between numeric and analytic data [34].
\r\n\tIn the book the theory and practice of microwave heating are discussed. The intended scope covers the results of recent research related to the generation, transmission and reception of microwave energy, its application in the field of organic and inorganic chemistry, physics of plasma processes, industrial microwave drying and sintering, as well as in medicine for therapeutic effects on internal organs and tissues of the human body and microbiology. Both theoretical and experimental studies are anticipated.
\r\n\r\n\tThe book aims to be of interest not only for specialists in the field of theory and practice of microwave heating but also for readers of non-specialists in the field of microwave technology and those who want to study in general terms the problem of interaction of the electromagnetic field with objects of living and nonliving nature.
",isbn:"978-1-83968-227-8",printIsbn:"978-1-83968-226-1",pdfIsbn:"978-1-83968-228-5",doi:null,price:0,priceEur:0,priceUsd:0,slug:null,numberOfPages:0,isOpenForSubmission:!1,hash:"8f6a41e4f5ce0e9c48628516d7c92050",bookSignature:"Prof. Gennadiy Churyumov",publishedDate:null,coverURL:"https://cdn.intechopen.com/books/images_new/10089.jpg",keywords:"Electromagnetic Wave, Microwave Energy Application, Electromagnetic Energy Generation, Intelligent Microwave Heating, Microwave Organic Chemistry, Microwave Reactor, Microwave Discharge, Microwave Plasma, Microwave Drying System, Tissue Microwave Heating, Measurement Automation, Industrial Microwave Process",numberOfDownloads:224,numberOfWosCitations:0,numberOfCrossrefCitations:0,numberOfDimensionsCitations:0,numberOfTotalCitations:0,isAvailableForWebshopOrdering:!0,dateEndFirstStepPublish:"July 3rd 2020",dateEndSecondStepPublish:"July 24th 2020",dateEndThirdStepPublish:"September 22nd 2020",dateEndFourthStepPublish:"December 11th 2020",dateEndFifthStepPublish:"February 9th 2021",remainingDaysToSecondStep:"7 months",secondStepPassed:!0,currentStepOfPublishingProcess:5,editedByType:null,kuFlag:!1,biosketch:"Prof. Gennadiy I. Churyumov is a professor at two universities: Kharkiv National University of Radio Electronics, and Harbin Institute of Technology and a senior IEEE member.",coeditorOneBiosketch:null,coeditorTwoBiosketch:null,coeditorThreeBiosketch:null,coeditorFourBiosketch:null,coeditorFiveBiosketch:null,editors:[{id:"216155",title:"Prof.",name:"Gennadiy",middleName:null,surname:"Churyumov",slug:"gennadiy-churyumov",fullName:"Gennadiy Churyumov",profilePictureURL:"https://mts.intechopen.com/storage/users/216155/images/system/216155.jfif",biography:"Gennadiy I. Churyumov (M’96–SM’00) received the Dipl.-Ing. degree in Electronics Engineering and his Ph.D. degree from the Kharkiv Institute of Radio Electronics, Kharkiv, Ukraine, in 1974 and 1981, respectively, as well as the D.Sc. degree from the Institute of Radio Physics and Electronics, National Academy of Sciences of Ukraine, Kharkiv, Ukraine, in 1997. \n\nHe is a professor at two universities: Kharkiv National University of Radio Electronics, and Harbin Institute of Technology. \n\nHe is currently the Head of a Microwave & Optoelectronics Lab at the Department of Electronics Engineering at the Kharkiv National University of Radio Electronics. \n\nHis general research interests lie in the area of 2-D and 3-D computer modeling of electron-wave processes in vacuum tubes (magnetrons and TWTs), simulation techniques of electromagnetic problems and nonlinear phenomena, as well as high-power microwaves, including electromagnetic compatibility and survivability. \n\nHis current activity concentrates on the practical aspects of the application of microwave technologies.",institutionString:"Kharkiv National University of Radio Electronics (NURE)",position:null,outsideEditionCount:0,totalCites:0,totalAuthoredChapters:"2",totalChapterViews:"0",totalEditedBooks:"0",institution:null}],coeditorOne:null,coeditorTwo:null,coeditorThree:null,coeditorFour:null,coeditorFive:null,topics:[{id:"24",title:"Technology",slug:"technology"}],chapters:[{id:"74623",title:"Influence of the Microwaves on the Sol-Gel Syntheses and on the Properties of the Resulting Oxide Nanostructures",slug:"influence-of-the-microwaves-on-the-sol-gel-syntheses-and-on-the-properties-of-the-resulting-oxide-na",totalDownloads:94,totalCrossrefCites:0,authors:[null]},{id:"75284",title:"Microwave-Assisted Extraction of Bioactive Compounds (Review)",slug:"microwave-assisted-extraction-of-bioactive-compounds-review",totalDownloads:12,totalCrossrefCites:0,authors:[null]},{id:"75087",title:"Experimental Investigation on the Effect of Microwave Heating on Rock Cracking and Their Mechanical Properties",slug:"experimental-investigation-on-the-effect-of-microwave-heating-on-rock-cracking-and-their-mechanical-",totalDownloads:28,totalCrossrefCites:0,authors:[null]},{id:"74338",title:"Microwave Synthesized Functional Dyes",slug:"microwave-synthesized-functional-dyes",totalDownloads:21,totalCrossrefCites:0,authors:[null]},{id:"74744",title:"Doping of Semiconductors at Nanoscale with Microwave Heating (Overview)",slug:"doping-of-semiconductors-at-nanoscale-with-microwave-heating-overview",totalDownloads:45,totalCrossrefCites:0,authors:[null]},{id:"74664",title:"Microwave-Assisted Solid Extraction from Natural Matrices",slug:"microwave-assisted-solid-extraction-from-natural-matrices",totalDownloads:25,totalCrossrefCites:0,authors:[null]}],productType:{id:"1",title:"Edited Volume",chapterContentType:"chapter",authoredCaption:"Edited by"},personalPublishingAssistant:{id:"252211",firstName:"Sara",lastName:"Debeuc",middleName:null,title:"Ms.",imageUrl:"https://mts.intechopen.com/storage/users/252211/images/7239_n.png",email:"sara.d@intechopen.com",biography:"As an Author Service Manager my responsibilities include monitoring and facilitating all publishing activities for authors and editors. From chapter submission and review, to approval and revision, copyediting and design, until final publication, I work closely with authors and editors to ensure a simple and easy publishing process. I maintain constant and effective communication with authors, editors and reviewers, which allows for a level of personal support that enables contributors to fully commit and concentrate on the chapters they are writing, editing, or reviewing. I assist authors in the preparation of their full chapter submissions and track important deadlines and ensure they are met. I help to coordinate internal processes such as linguistic review, and monitor the technical aspects of the process. As an ASM I am also involved in the acquisition of editors. Whether that be identifying an exceptional author and proposing an editorship collaboration, or contacting researchers who would like the opportunity to work with IntechOpen, I establish and help manage author and editor acquisition and contact."}},relatedBooks:[{type:"book",id:"6826",title:"The Use of Technology in Sport",subtitle:"Emerging Challenges",isOpenForSubmission:!1,hash:"f17a3f9401ebfd1c9957c1b8f21c245b",slug:"the-use-of-technology-in-sport-emerging-challenges",bookSignature:"Daniel Almeida Marinho and Henrique Pereira Neiva",coverURL:"https://cdn.intechopen.com/books/images_new/6826.jpg",editedByType:"Edited by",editors:[{id:"177359",title:"Dr.",name:"Daniel Almeida",surname:"Marinho",slug:"daniel-almeida-marinho",fullName:"Daniel Almeida Marinho"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"8494",title:"Gyroscopes",subtitle:"Principles and 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Theophile"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"3092",title:"Anopheles mosquitoes",subtitle:"New insights into malaria vectors",isOpenForSubmission:!1,hash:"c9e622485316d5e296288bf24d2b0d64",slug:"anopheles-mosquitoes-new-insights-into-malaria-vectors",bookSignature:"Sylvie Manguin",coverURL:"https://cdn.intechopen.com/books/images_new/3092.jpg",editedByType:"Edited by",editors:[{id:"50017",title:"Prof.",name:"Sylvie",surname:"Manguin",slug:"sylvie-manguin",fullName:"Sylvie Manguin"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"3161",title:"Frontiers in Guided Wave Optics and Optoelectronics",subtitle:null,isOpenForSubmission:!1,hash:"deb44e9c99f82bbce1083abea743146c",slug:"frontiers-in-guided-wave-optics-and-optoelectronics",bookSignature:"Bishnu Pal",coverURL:"https://cdn.intechopen.com/books/images_new/3161.jpg",editedByType:"Edited by",editors:[{id:"4782",title:"Prof.",name:"Bishnu",surname:"Pal",slug:"bishnu-pal",fullName:"Bishnu Pal"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}}]},chapter:{item:{type:"chapter",id:"67835",title:"Single Axis Singularity Mapping for Mixed Skew Angle, Non-Redundant, Single Gimbaled CMG Systems",doi:"10.5772/intechopen.87179",slug:"single-axis-singularity-mapping-for-mixed-skew-angle-non-redundant-single-gimbaled-cmg-systems",body:'\nMechanical control has developed over centuries [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], expanding original theorems such as Chasle’s theorems of motion Phoronomics [23]. With increasing strike capability, advancements in spacecraft technology, and rising political tensions all over the globe, mechanical control has resurfaced as an important research front in order to further current technologies. Opposed nations frequently use satellites on orbit to gather critical intelligence on those around them, a mission that requires precise pointing and an extensive and expansive understanding of the mechanical control envelope provided by the spacecraft’s attitude control system. Recent research has been conducted in order to increase the maneuverability of spacecraft with control moment gyroscopes [24, 25, 26, 27, 28, 29, 30, 31, 32]. This research takes information and lessons learned from these previous research efforts and builds upon them.
\nDepending on a spacecraft’s mission, it will likely execute a particular kind of attitude maneuver many times during its life span. Characteristic attitude maneuvers should be considered when designing an attitude control system. The type and number of attitude control devices as well as their position within the spacecraft are design choices driven by the physical demands of the attitude maneuvers. These maneuvers should be considered in order to design an attitude control system that ensures the most angular momentum can be generated around that favored axis while also providing maneuverability in other directions.
\nConstant-speed, single-gimbaled control moment gyros (CMGs) are common spacecraft attitude control devices that, like reaction wheels, are momentum exchange devices that operate on the law of conservation of momentum in an undisturbed system. Unlike reaction wheels, CMGs do not change their rotational velocity to alter the spacecraft’s attitude but, rather, change their direction. Although this ability allows CMGs to uniquely control spacecraft attitude, it also poses challenges: CMGs can only provide torque in a plane orthogonal to their gimbal axis. When a desired torque orthogonal to this plane is commanded, the CMG encounters a mathematical singularity and attitude control is lost.
\nThe locations of these singularities can be plotted 3-dimensionally in order to gain an understanding of the singularity free angular momentum available to command. These singularity maps change based upon the CMG’s skew angle within the spacecraft and can be optimized to maximize the singularity free, angular momentum space about a particular axis.
\nIt is necessary to understand how CMGs are commanded and how they physically affect the spacecraft in order to understand how a mathematical singularity causes a spacecraft to lose control. Like any actuator system, a command is entered and a trajectory is generated to reach the commanded position from the initial position; applied to a CMG, a specific rotation is the command and Eq. (1) through Eq. (3) are the equations used to generate the attitude maneuver trajectory [33].
\nwhere θ is the gimbal angle, ω is the gimbal rotational velocity, and \n
The feedforward uses Eq. (5), an adapted version of Eq. (4), to calculate this torque command. Eq. (5) is the nonlinear feedforward control equation based off of the Newton-Euler equation written in the body frame. Since Eq. (5) directly describes the physics of the system, it is the best feedforward control to use.
\nwhere \n
At this point in the system topology, the torque command is converted to a voltage or current and sent directly to the actuators. The actuators move and torque is exerted on the spacecraft as described by Eq. (6).
\nwhere \n
For analysis purposes, a simplified, non-redundant, single gimbaled CMG system will be used. This system will consist of three CMG’s as pictured in Figure 1. To note, the CMG skew angle is defined as the angle between a vertical line parallel to the Z axis at each CMG location and the Z axis; in other words, the gimbal axis would be pointing out from the spacecraft in the x-y plane when β = 0° or would be pointing straight up when β = 90°. In Figure 1, β is annotated at its equivalent angle. Also, each angular momentum vector is drawn at its initial position, θ = 0°.
\n¾ CMG system [34].
Figure 1 provides a visual aid in generating a set of three equations that resolve the angular momentum of each CMG into the x, y, and z axes. These equations are described in Eqs. (7), (8), and (9).
\nwhere h is angular momentum about a particular axis, β is the skew angle of each CMG, θ is the angle the momentum vector has rotated about the CMG gimbal axis, and H is the maximum angular momentum a single CMG can produce.
\nThe desired torque given from the system described in Eqs. (7), (8), and (9) can be written as Eq. (10), where the desired torque is equal to the partial derivative of angular momentum with respect to the gimbal angle multiplied by the time derivative of the gimbal angle.
\nThe partial derivative of angular momentum with respect to the gimbal angle is found by taking the spatial gradient of Eqs. (7), (8), and (9) which produces a Jacobian matrix, the A matrix. The A matrix describes the components of torque provided by each CMG in each axis; this is represented in Eq. (11).
\nGiven the A matrix’s definition, Eq. (10) can be written inversely to find the commanded gimbal rotation rates as Eq.(12) where the inverse of A is equal to the reciprocal of the determinant of A multiplied by its cofactor [35].
\nEq. (12) encounters a mathematical singularity when the determinant of A equals zero; within the control system, the computer will continually attempt to calculate one over zero and, in the process, send the absurdly large results as torque commands to the CMGs. The CMG actuators follow the randomly large commands and the spacecraft loses attitude control. Physically, this kind of singularity is hit when a particular combination of gimbal angles is reached and the CMG cannot produce torque in the desired direction. These combinations of gimbal angles are defined by the determinant of the A matrix. For the CMG system in Figure 1 when all skew angles could be different, the determinant of A is evaluated in Eq. (13).
\nThere are a multitude of cases when Eq. (13) is equal to zero, causing a singularity. Within each of these cases, at any chosen combination of skew angles, there are numerous different gimbal angle combinations resulting in a singularity; each of these gimbal and skew angle combinations produces a certain angular momentum in the x, y, and z directions as calculated by Eqs. (7), (8), and (9) respectively. For a particular skew angle combination, there is a gimbal angle combination such that a singularity is hit with the smallest achievable angular momentum: this becomes the maximum angular momentum the entire CMG system can reach before encountering a singularity at that particular skew angle combination set up.
\nAlthough this reduction in the commandable angular momentum has been applied to many spacecraft on orbit, it is extremely limiting. Figure 2 illustrates this reduction with the black sphere representing the singularity free maximum angular momentum space while the space enclosed with the blue surface represents all valid angular momentum commands. Furthermore, the outer blue surface defines the angular momentum saturation limit for its particular CMG setup. In Figure 2, the CMG set up includes three CMGs at equivalent skew angles of 56°.
\nRestricted angular momentum sphere within entire command space [34].
In an attempt to remove this limit, Sands created a mechanism with which to penetrate this smallest angular momentum and expand the commandable angular momentum to everything up until saturation [32, 36, 37]. This mechanism is called singularity penetration with unit delay (SPUD) [32] and pierces the inner singularity surfaces by sending the CMG actuators valid control commands while the system passes through a singularity. This mechanism is critical in order to reach the maximum angular momentum at a particular axis.
\nDefining the maximum angular momentum achievable without encountering a singularity for a CMG system over all possible skew angle combinations can be calculated via two methods: numerically or analytically. To numerically define this surface, the skew angle combinations are discretized and the associated minimum angular momentum is calculated numerically. To analytically define the same surface, each case that makes the determinant of A equal to zero is identified. The equation defining each case is then evaluated for its minimum angular momentum over all gimbal angle combinations for every skew angle. The minimum angular momentum data for all cases is then plotted on a single graph and the minimum angular momentum out of each case is taken as the maximum angular momentum achievable for that skew angle combination.
\nFor this research, numerically calculating the maximum angular momentum without reaching a singularity for each discretized skew angle was chosen over the analytical method because the numeric solution creates a conservative model. The conservative nature of the numeric solution was determined by comparing a numerically calculated and analytically determined maximum angular momentum plot when all skew angles were equivalent. To compare these methods, however, a discretization size for the numeric solution had to be chosen. Three numeric solutions were plotted with discretizations of 0.1, 1, and 2°. One degree was chosen because using a smaller discretization, such as 0.1°, introduced noise into the plots while using a larger discretization, such as 2°, missed critical data points leading to important singularity locations. The 1° discretization plotted a smooth singularity location line while not skipping any important values. The plots using 0.1 and 2° are pictured in Figure 3 while the 1° discretization is plotted in Figure 4 with the analytic solution derived and created in Sands’ dissertation [36].
\n0.1° discretization (left) versus 2° discretization (right) for numerically determined maximum angular momentum [34].
Numeric versus analytic determination of maximum angular momentum [34].
Table 1 describes the mean error and standard deviation between the numerically obtained and analytically obtained data in Figure 4.
\nData points | \nμ | \nσ | \n
---|---|---|
1–37 | \n0.0333 | \n0.0388 | \n
38–60 | \n0.0811 | \n0.0707 | \n
61–90 | \n5.51e-5 | \n1.15e-4 | \n
Total | \n0.0344 | \n0.0530 | \n
Mean error and standard deviation between numeric and analytic data [34].
The numeric results vary from the analytic angular momentum values for most skew angles from 1 to 55° as can be seen in Figure 4 and Table 1. After 55° however, both the numeric and analytic data is equivalent; Figure 4 shows they plot along the same line while Table 1 confirms the mean error and standard deviation between the values are both approximately zero. Although the numerically obtained results differ from the analytic values before 55°, the numeric results claim a lower possible angular momentum is possible before reaching a singularity. Using these data points would provide a buffer between where the singularities are expected to be versus where they actually are, protecting the attitude control system from hitting a singularity. Because this buffer is on the “safe” side, the maximum angular momentum without hitting a singularity for a CMG system with different skew angles was determined numerically.
\nFigure 4 plotted the maximum angular momentum in any direction for a non-redundant CMG system with equivalent skew angles. In order to design an attitude control system for a spacecraft with a characteristic maneuver, a similar figure can be produced plotting only the maximum angular momentum in that favored axis. This research aims to characterize skew angle combination effects on maximum angular momentum around the spacecraft’s z axis, in other words, mixed skew angle effects on yaw maneuverability. To analyze this relationship, the maximum achievable angular momentum about the z axis was calculated for different skew angle combinations using the numerical method used to produce Figure 4. When creating the plots in Figures 5 and 6, the actual angular momentum values were plotted instead of strictly their magnitude; as a result, the plots are negative.
\n3-D maximum angular momentum on the z axis for β1 = 1°, β2 = β3 = free [34].
Maximum angular momentum for β1 = 1°,15°, 30°, 45°, 60°, 75°, 90°, β2 = β3 = free [34].
In order to plot the maximum achievable angular momentum about the z axis for all skew angle combinations, a four dimensional plot would be needed. Since this is not achievable, skew angle one was held constant while skew angles two and three were varied from 0 to 90°. Three dimensional plots were created as can be seen in Figure 5. However, due to the difficulty of orienting each graph to show the angular momentum magnitude, a color bar was employed instead. This allowed the same data to plot in two dimensions as can be seen in Figure 6.
\nFigure 6 illustrates the same trend for all β1: maximum achievable angular momentum is smallest when both β2 and β3 are close to 0° and largest when both β2 and β3 are equal to 90°. Additionally, the magnitude of achievable angular momentum increases with β1. For small β1, such as 1°, the maximum angular momentum when β2 and β3 are close to 0° is 0|H| while for large β1, such as 90°, the maximum angular momentum when β2 and β3 are close to 0° is 1|H|. Table 2 lists the maximum angular momentum and associated skew angles for each plot in Figure 6.
\n|H| | \nβ1 (°) | \nβ2 (°) | \nβ3 (°) | \n
---|---|---|---|
2.017 | \n1 | \n90 | \n90 | \n
2.259 | \n15 | \n90 | \n90 | \n
2.5 | \n30 | \n90 | \n90 | \n
2.707 | \n45 | \n90 | \n90 | \n
2.866 | \n60 | \n90 | \n90 | \n
2.966 | \n75 | \n90 | \n90 | \n
3 | \n90 | \n90 | \n90 | \n
Maximum yaw maneuverability skew angle combinations [34].
Plotting the singularity maps for the skew angle combinations listed in Table 2 visualizes the commandable angular momentum on the z axis. These mixed skew angle combinations produce the singularity maps pictured in Figure 7.
\nSingularity maps [34].
Within Figure 7, the highlighted blue surface in each plot contains the singularity defining the maximum achievable angular momentum about the z axis. For skew angle combinations with β1 lower than 45° and β2 and β3 equal to 90°, the saturation limit on the z axis is defined by one of the inner singularity surfaces. For β1 larger than 45° and β2 and β3 equal to 90°, the saturation limit is defined by the outer singularity surface. As long as β1 is larger than 0°, there are no singularities exactly on the z axis before the saturation limit because there are at least two CMG’s capable of exerting maximum angular momentum in the z direction. Since angular momentum can be commanded in that direction regardless of the orientation of the third CMG, there is no singularity until the saturation limit.
\nDrawing from the key points of this research, it is clear that different skew angles create drastically different singularity plots. These singularity plots map out the unattainable torque commands for a particular CMG system, ultimately defining the attitude envelope a spacecraft can achieve within a defined amount of time. As a result of this important relationship, CMG skew angles should be carefully chosen when designing a spacecraft attitude control system.
\nWhen designing a non-redundant CMG attitude control system for a spacecraft that needs to maximize its yaw maneuverability, a CMG system with all skew angles equal to 90° would maximize the commandable angular momentum about the z axis as Figure 6, Table 2, and Figure 7 all show. The next greatest combination would be to set two of the skew angles equal to 90° and the third skew angle equal to something greater than zero in order to avoid a singularity at the origin.
\nThe outward convex corrugated tube heat exchanger (CTHE) is a novel kind of shell and tube heat exchangers, which can be applied in many applications. Designing this kind of heat exchangers is considerable flexibility because the geometrical structure can be varied easily by altering the tube diameter, length, and arrangement [1, 2]. The exchanger can be designed for suffering high pressure condition. The exchangers are applied primarily for single phase and phase change heat transfer application. They could also be used for heat transfer applications with high operating temperature and/or pressure.
\nFigure 1 shows a bundle of outward convex corrugated tubes (CT) fabricated in the tubesheets, which is the most significant components in the CTHE. Two kinds of fluids flow inside and outside of CT, respectively. Except the tube bundles, the major components of this exchanger also include shell, front-end head, and rear-end head.
\nThe schematic of out outward convex corrugated tube heat exchangers.
The exchangers could be widely used in industry for the following reasons. (a) Wide capacity and operating conditions, such as from high vacuum to ultrahigh pressure (over 100 MPa) and from cryogenics to high temperatures (about 1100°C). (b) Special operating conditions: vibration, heavy fouling, highly viscous fluids, erosion, corrosion, toxicity, radioactivity, multicomponent mixtures, and so on. (c) The most versatile exchangers, made from a variety of metal and nonmetal materials (such as graphite, glass, and Teflon) and range in size from small to supergiant surface area. (d) Extensively applications: petroleum-refining and chemical industries; as steam generators, condensers, boiler feedwater heaters, and oil cooler in power plants; as condensers and evaporators in some air-conditioning and refrigeration applications; in waste heat recovery applications; and in environmental control [3, 4, 5].
\nThe main difference between the CTHE and traditional heat exchangers is the adopted tube type. Traditionally, the inward intermittent or continuous type corrugated tubes are employed, as an example for both helically corrugated and transverse corrugated tubes, owing to their ease of realization. However, in engineering devices, it is necessary to adopt CT, which could be conveniently and periodically inspected with complete accessibility [6].
\nA schematic view of the CT configuration currently investigated is shown in Figure 2. The structure parameters of the outward corrugated tube include inner diameter (D), tube length (L), corrugation height (H), corrugation pitch (p), corrugation crest radius (R), and corrugation trough radius (r).
\nThe real and schematic view of the outward convex corrugated tube.
The design and improvement of the CT are considered a significant aspect of researches in terms of heat and mass transfer. Almost all of the heat transfer augmentation techniques have been introduced to improve the overall thermo-hydraulic performance. Thus, these techniques achieved reductions in the size and cost of heat exchangers.
\nManufacture consideration could be divided into manufacturing equipment, processing, and other qualitative criteria. The equipment considerations determine which design could be selected, which include existing and new tooling, availability and limitations of equipment, offline production, and investment funding. Processing considerations make sure how individual parts and components of a heat exchanger are manufactured and assembled, which including manufacture of individual parts, stacking of a heat exchanger core and eventual brazing, mounting of pipes, washing/cleaning of the exchanger, and leak testing in the system. When a heat exchanger is designed, the manufacturing equipment and the complete processing considerations must be evaluated previously, particularly for an extended surface heat exchanger [11, 12].
\nIn the novel tube and shell heat exchanger, the structure of the outward convex corrugated tube is special, composed of alternating corrugated segment and straight pipe section. The main difference from traditional heat exchanger is the adopted structure, so the manufacture processing for the novel tube type is highlighted in this section. The working conditions of the heat exchanger are mainly for high temperature and pressure operation condition. To ensure the safe operation of heat exchanger, a thick-walled stainless steel tube with strong pressure resistance is selected as the base tube. For example, the mechanical properties of stainless steel tube material are as follows: yield strength is 390 MPa, material hardening index is 0.148, material strength coefficient is 764 MPa, material anisotropy coefficient is 0.83, material modulus of elasticity is 207GPa, and Poisson ratio is 0.28.
\nThe outward convex corrugated tube is manufactured according to high pressure hydraulic bulking based on the smooth stainless steel tube. The hydraulic bulking equipment is 10,000 KN. As shown in Figure 3(a), the equipment is assembled with 400 MPa internal high-pressure forming system, which is mainly composed of the supercharger, two horizontal push cylinder hydraulic servo system, and computer control system. The manufacturing process needs to be supplemented with the corresponding mold, installed on the hydraulic bulking equipment. The mold consists of three parts, which includes upper module as shown in Figure 3(b), lower module as shown in Figure 3(c) and sealing punch. The inner mosaic block with corrugation shape is inserted in the mold as shown in Figure 3(d). High pressure liquid (water or oil) is provided inside the smooth stainless steel tube and finally hydroforms the outward convex corrugated tube.
\nThe hydraulic bulging machine and mold. Based on the modified order as the above sticky. (a) Hydraulic bulking equipment; (b) upper module; (c) lower module; (d) inner mosaic block.
In order to test the heat transfer and resistance performance of the corrugated tube heat exchangers, experimental study on the corrugated tube heat exchanger must be performed. We adopted steady-state techniques to establish the relationship between Nu and Re. Different data acquisition and reduction methods are used, depending on whether the test fluid is primarily a gas (air) or a liquid. A gas to gas heat exchange will be conducted in our experimental test.
\nThe schematic of the experimental apparatus for outward corrugated tube is depicted in Figure 4. The system comprises a screw air compressor (the highest discharging pressure is 1.3 MPa, and the air displacement is fixed at 1.81 m3/min), two pressure-regulating valves (0.3 MPa on the hot circuit and 0.9 MPa on the cold circuit), a heater (the temperature range is 50–500°C), a test section (operating with two groups of switching valves), a measuring system (two critical Venturi flowmeters, two pressure transducers, and two temperature transducers), a data acquisition system (DAS), and a pipe system (304 stainless steel tube).
\nSystem drawing of test bed. 1. Screw air compressor, 2. Pressure-regulating valve in hot circuit, 3. Pressure-regulating valve in cold circuit, 4. Critical Venturi flowmeter in hot circuit, 5. Critical flow meters in cold circuit, 6. Air heater, 7. Switching valves in hot circuit, 8. Switching valves in cold circuit, 9. Test section, 10. Data acquisition system, 11. Muffler.
The experimental medium was air, which was compressed by the helical-lobe compressor to a pressure of 1.25 MPa. The system is made of stainless steel devices and consists of the hot circuit and cold circuit. The pressure-regulating valves adjust the air pressure to 0.3 MPa on the hot circuit and 0.9 MPa on the cold circuit with an accuracy of ±2%. The critical Venturi flowmeters control the mass flow rate in the hot and cold circuits. The air in the hot circuit is heated by the heater exchanger and then flows into the tube side of the test section, whereas the air in the cold circuit directly flows into the shell side. The section has a detachable structure, which enables convenient changes in various tube components. Moreover, the valve group in the vicinity of the test section makes the air flow into the tube, through either inlet of the tube side or the shell side, thus creating a uniform-current flow and a counter-current flow for each respective flow direction. Finally, the hot air and the cold air complete the heat exchange in the annular tubes of the test section, and then noise of them will be reduced through the muffler.
\nIn the measuring system, the mass flow rates can be measured with two critical Venturi flowmeters on both circuits, with an accuracy of ±0.2%. The flow meter in the hot circuit was installed before the air heater because hot air may damage the flow meter or reduce the measurement accuracy (precision). After the heater, a temperature transducer was installed to monitor the air temperature. The DAS obtained the flow rate signal, which was transferred to a programmable logic controller (PLC) in the industrial personal computer (IPC), and the accuracy of the transformation module was ±0.05%. The pressure and temperature transducers were installed at the inlet and outlet of the section to measure the pressure and temperature of the air on both sides. All thermocouples were calibrated with an accuracy of ±0.1% of the test data. The pressure drop of the test section was measured with pressure transducers, which have an accuracy of ±0.2% and a measuring range of 0–5 kPa. The values were collected and displayed on the IPC and were automatically recorded.
\nThe uncertainty is estimated with the method suggested by Kline and Moffat. As mentioned above, the measurement uncertainties of tube length and tube diameters are about 0.05 and 0.1%, respectively. In addition, the measurement accuracy of temperature is 0.14%, the measurement error of the differential pressure meter is 2.06%, and the critical Venturi flowmeter has a precision of 3.11%. According to the uncertainty propagation equation, the uncertainties in the values of experimental parameters like the Reynolds number, Nusselt number, and friction factor are 3.89, 4.41, and 4.87%, respectively.
\nThe main purpose of our experimental study is to construct the relationship among the heat transfer rate q, heat transfer surface area A, heat capacity rate c of each fluid, overall heat transfer coefficient U, and fluid terminal temperatures [10]. To conduct the heat transfer analysis of an exchanger, the basic relationships that are applied for this purpose are the energy balance based on the first law of thermodynamics, as outlined in Eq. (1).
\nwhere \n
As shown in Figure 5, a two-fluid counterflow exchanger is considered to present variables relating to its thermal performance. Although flow arrangement may be different for different exchangers, the basic concept of modeling remains the same. The following analysis is intended to introduce important variables for heat exchanger.
\nThe energy balance for the hot and cold fluids of a two-fluid heat exchanger.
If the fluids do not undergo a phase change and have constant specific heats with di = cp · dT, heat transfer rate released from the hot fluid (Qh) and absorbed by the cold air (Qc) can be expressed as
\nand
\nThe subscripts h and c refer to the hot and cold fluids, and the numbers 1 and 2 designate the fluid inlet and outlet conditions, respectively.
\nThus, the average value of the heat transfer rate is calculated as
\nEq. (5) reflects a convection-conduction heat transfer phenomenon in a two-fluid heat exchanger. The temperature difference between the hot and cold fluids (ΔT = Th−Tc) constantly changes along with heat exchanger. Therefore, in order to conveniently analyze the heat transfer performance of heat exchanger, it is important to establish an appropriate mean value of the temperature difference between the hot and cold fluids such that the total heat transfer rate Q between the fluids can be determined from
\nThe heat transfer rate Q is proportional to the heat transfer area A, the average overall heat transfer coefficient based on the area U, and mean temperature difference \n
\n\n
In the experiments, the tube-wall temperature was not measured directly. The heat transfer coefficient of the tube side (hi) is determined from:
\nwhere ri and ro are the inner radius and outer radius of the test tube, respectively. Ai and Ao are the inner and outer surface area of the tube, respectively. k is the thermal conductivity of tube material, L is the length of the heat exchange tube, and hi and ho are the heat transfer coefficients for inside and outside flows, respectively.
\nThe Nusselt number can be calculated as
\nwhere D is the characteristic diameter; the thermal conductivity k is calculated from the fluid properties at the local mean bulk fluid temperature.
\nThe Reynolds number is based on the average flow rate of the test section.
\nwhere μ is the dynamic viscosity of the working fluid, and u is the mean velocity.
\nThe friction factor (f) can be written as
\nwhere Δp is the pressure drop in the test section.
\nThe performance evaluation criterion (PEC) is a dimensionless ratio, which is used for the evaluation of the overall performance of the enhanced tube and defined as follows:
\nWhen PEC > 1, it indicates that the enhanced tube has an advantage over the smooth tube; otherwise, the corrugated heat transfer component compares unfavorably with the smooth tube.
\nFor the engineering applications and to design exchangers, the prediction of heat and mass transfer performance is important. We presented experimental data on the Nusselt numbers for turbulent regimes. In our experimental study, the hot fluid is at the tube side, and the cold fluid is at the shell side.
\nThe heat transfer and resistance performance of corrugated tube are compared to smooth tube, aiming to reflect the superior of the corrugated tube. Ratio of Nu in the corrugated tube to that in the smooth tube (Nuc/Nus) and ratio of f in the corrugated tube to that in the smooth tube (fc/fs) are adopted to indicate the enhancement degree of heat transfer and flow resistance performance.
\nFigure 6 shows the effect of Rec (Re of the cold fluid) on Nuc/Nus, fc/fs, and PEC, along with the changing Reh (Re of the hot fluid). The figure exhibits that with the increase of Re, Nuc/Nus, fc/fs, and PEC decline. The decreasing rate of Nuc/Nus and PEC is almost linear, but fc/fs is decelerated.
\nFlow and mass transfer performance. (a) Nuc/Nus; (b) f/fs; (c) PEC.
The first task to accomplish in a numerical simulation is the definition of the geometry followed by the mesh generation. The geometry of the design needs to be created from the initial design. Any modeling software can be used for modeling and shifted to other simulation software for analysis purpose.
\nFigure 7 shows a schematic view of the structural parameters for corrugated tube investigated in this chapter, which include inner diameter (D), tube length (L), corrugation height (H), corrugation pitch (P), corrugation crest radius (R), and corrugation trough radius (r). Since the investigated corrugated tubes are used in tube-shell type heat exchanger, the flow region inside of tube is named “tube side” and out of tube is named “shell side.”
\nStructure parameters of outward convex corrugated tube.
Mesh generation is the process of subdividing a region to be modeled into a set of small control volumes. In general, a control volume model is defined by a mesh network, which is made up of the geometric arrangement of control volumes and nodes. Nodes represent points at which features such as displacements are calculated. Control volumes are bounded by set of nodes and also defined by the number of mesh. One or more values of dependent flow variable (e.g. velocity, pressure, temperature, etc.) will be contained in each control volume. Usually, these represent some type of locally averaged values. Numerical algorithms representing approximation to the conservation law of mass, momentum, and energy are then used to compute these variables in each control volume.
\nMesh generation is often considered as the most important and most time consuming part of CFD simulation [13]. The quality of the mesh plays a direct role on the quality of the analysis, regardless of the flow solver used. In this work, a 3D non-uniform mesh system of hexahedral elements was established via the professional mesh generation software ICEM to accurately control the size and number of cells in the domain, as illustrated in Figure 8. The near-wall vicinity should be present drastic velocity and temperature gradients, so a high density of gradient elements was applied in this region. Nevertheless, the remaining domain was modeled with relatively sparse elements. The first layer of thickness should satisfy y+ ≈ 1.
\nSchematic diagram of meshing system for the simulated corrugated tube.
Mathematical model should be constructed to numerically describe flow and heat transfer of corrugated tube. The Navier-Stokes equations generally are adopted to describe the laminar and turbulent flows, which could be solved by various kinds of simulation model including DNS, LES, and RANS. The direct numerical simulation (DNS) can solve accurately the turbulent fluctuation, but these models require huge computing power, which is many orders of magnitude higher than other models. Reynolds-averaged Navier-Stokes (RANS) is a high efficient model that can be used to approximate turbulence by time-averaged turbulent fluctuation, but the accuracy of the models is much less than DNS. The accuracy and efficient of LES are between the DNS and RANS.
\nThe k-ε (k-epsilon) model is one of the most prominent RANS models, which has been implemented in most CFD codes and is considered the most common industry model. The stability and robustness of the models have a well-established regime of predictive capability, satisfying general purpose simulation by offering a comparative good accuracy. In our research work for outward convex corrugated tube, we use standard k-ε model for numerical simulation research.
\nThe governing equations in a RANS (Reynolds Averaged Navier-Stokes) manner are given below.
\nContinuity equation:
\nMomentum equation:
\nEnergy equation:
\nThe standard k-ε model is adopted here to close governing equations:
\nwhere μt is the turbulent or eddy viscosity, and \n
The next step in preprocessing is setting up the boundary conditions. Boundary conditions refer to the conditions that the solution of the equations should satisfy at the boundary of the moving fluid. Boundary condition will be different for each type of problem. In our research work, the initial and boundary conditions of the outward convex corrugated tube heat exchangers are shown as follows:
The inlet conditions at the shell side are as follows: velocity inlet U = Uin, Tin = 563.15 K, and the inlet turbulence specifications are a turbulence intensity of I = 5% and hydraulic diameter D = 20 mm.
The outlet conditions at the shell side are as follows: pressure outlet, Po = 7 MPa, and the outlet turbulence specifications are a turbulence intensity of I = 5% and a viscosity ratio μt/μlam = 5%.
The wall conditions are as follows: the outer wall temperature boundary condition is constant, Tw = 700 K, and the inner wall-coupled boundary condition was set as a no-slip boundary, u = v = w = 0, T = Tw, and q = qw.
The final step in preprocessing is setting up the numerical procedure, which includes solver, discretization, and convergence criterion. In our work, the governing equations are discretized by the finite volume method and solved by the steady-state implicit format. The SIMPLE algorithm is used to couple the velocity and pressure fields. The second-order upwind scheme is applied herein. The convergence criterion for energy is set to be 10−7 relative error and 10−4 relative error for other variables.
\nThe variable distribution exhibits the opposite similar tendency at the shell side compared with that at the tube side. In this chapter, we mainly analyze the distribution of velocity, temperature, and turbulence kinetic energy.
\nFigure 9(a) shows the velocity vector distribution in the tube side of outward convex corrugation tube. As shown from this figure, when fluid flow starts to cross the corrugation section bended from the straight segment, the flow boundary layer separates into two parts: one is the wall boundary layer developed at the near wall region; the other is shear layer associated with an inflection point of large velocity gradient developed away from the wall, which moves away from the surface at the separation point and forms a free shear layer. When the fluid flows through the upstream of the corrugation, the flow velocity decreases and the pressure increases due to the narrowing of the flow cross section. The fluid layer near the wall is gradually difficult to overcome the rising pressure due to the small amount of momentum, resulting in a reflow of the original flow direction. The recirculating zone between the separating streamline and the free boundary streamline is generated at the upstream of the corrugation.
\nVelocity vector distribution at tube side and at shell side. (a) tube side; (b) shell side.
Figure 9(b) indicates the velocity vector distribution in the shell side of outward convex corrugation tube. As shown in this figure, the upstream side boundary of the corrugation is influenced by the accelerating outer-flow, that is, a favorable gradient. As the boundary layer thickens, instabilities occur when the near-wall fluid begins to decelerate as shown in Figure 7. The flow separates at the downgrade of the corrugation crest, which is associated with an inflection point of the large velocity gradient developed away the wall.
\nFigure 10(a) shows the temperature distribution in the tube side of outward convex corrugation tube. As shown in Figure 8, the wall velocity boundary layer becomes thicker at the upstream side of the corrugation accompany gradually, while the temperature boundary layer gets thicker along the flow direction, due to the eddy generating. Then it goes into thinner at the downstream side of the corrugation with the velocity boundary layer getting thinner, due to the scouring action of the fluid.
\nTemperature distribution at tube side and at shell side. (a) tube side; (b) shell side.
Figure 10(b) shows the temperature distribution in the tube side of outward convex corrugation tube. As shown in Figure 9, the wall velocity boundary layer becomes thicker at the downstream side of the corrugation accompany gradually, while the temperature boundary layer gets thicker along the flow direction, due to the eddy generating. Then, it goes into thinner at the upstream side of the corrugation with the velocity boundary layer getting thinner, due to the scouring action of the fluid. The thinnest temperature boundary layer occurs at the corrugation crest.
\nTurbulence kinetic energy (TKE) is one of the most important variables in boundary layer since it is a measure of the turbulence intensity, which is tightly related to the velocity profile. Figure 11(a) shows the turbulence kinetic energy distribution in the tube side of outward convex corrugation tube. As shown in this figure, the magnitude of the TKE gradient increases past upstream side section of corrugation with a noticed reduction after the flow reattaches as it enters downstream side section of corrugation. The location of the maximum turbulence kinetic energy extends over most of the corrugation, before descending when passing the downstream section of the wave trough.
\nTKE distribution at tube side and at shell side. (a) tube side; (b) shell side.
Figure 11(b) shows the turbulence kinetic energy distribution in the tube side of outward convex corrugation tube. As shown in this figure, the magnitude of the high TKE extends fairly constant past most of the corrugation with a noticed reduction after the flow reattaches. The location of the high TKE extends over most of corrugation at a height, which roughly equals to the maximum corrugation height, before subsiding toward the corrugation trough.
\nHeat transfer enhancement methods are classified into three classifications: active, passive, and compound. The active methods include electrostatic and magnetic fields, induced pulsation, mechanical aid, vibration, and jet impingement. These methods require external activating power to enhance the heat transfer [3, 4, 5, 6]. Passive methods modify the geometrical structure to expand the effective surface area to disturb the actual boundary layer. Compound methods combine the two heat transfer augmentation methods to increase heat transfer performance. In the above-mentioned methods, passive methods have attracted significant attention from researchers and engineers since they are user-friendly and affordable. Extensive research has been devoted to develop highly efficient heat transfer components to better understand the physical mechanisms and optimal parameters of passive heat transfer augmentation methods.
\nThe heat transfer enhancement mechanism in the corrugated tube is described as follows. The periodically corrugated structure on the tube wall arouses periodic alteration of velocity gradient, leading to adverse and favorable pressure gradient locally. The recurrent alternation of axial pressure gradient induces the secondary disturbance, and then the produced intensive eddy destroys the flow boundary layer. The eddy also increases the turbulence intensity of the flow. The disturbance caused by corrugated structures thus increases the heat transfer coefficient drastically.
\nFigure 12 shows the effect of Re on Nuc with various p/D and H/D. The Nuc tends to increase linearly with the increasing Re with a fixed structure of the corrugated tube. This behavior occurs because the increases of flow velocity break wall thermal boundary layer and could obtain higher convective heat transfer coefficient. Moreover, with the decreasing p/D and increasing H/D, the values of the Nuc increase.
\nEffect of Re on Nu with various p/D and H/D in the tube side. (a) various p/D (b) various H/D.
In order to compare the performance between corrugated tube and smooth tube, the ratio of Nu in the corrugated tube to that in the smooth tube (Nuc/Nus) is adopted to indicate the relative grow rate of heat transfer performance. Figure 13 shows the effect of Re on Nuc/Nus with various p/D and H/D, and the figure exhibits that with the increase of Re, Nuc/Nus declines deceleratedly. Moreover, the Nuc/Nus increases with the decreasing p/D and increasing H/D.
\nEffect of Re on Nuc/Nus with various p/D and H/D in the tube side. (a) various p/D (b) various H/D.
Figure 14 shows the effect of Re on Nuc with various p/D and H/D in the shell side. Compared with Figure 12, the changing tendency of Nuc along with Re, p/D, and H/D is consistent, but the Nuc in the shell side is obviously higher than in the tube side.
\nEffect of Re on Nu with various p/D and H/D in the shell side. (a) various p/D (b) various H/D.
Figure 15 shows the effect of Re on Nuc/Nus with various p/D and H/D in the shell side. It can be found when compared with Figure 13, the changing tendency of Nuc/Nus along with Re, p/D, and H/D is also consistent, but the Nuc/Nus in the shell side is obviously higher than in the tube side.
\nEffect of Re on Nu with various p/D and H/D in the shell side. (a) various p/D (b) various H/D.
Generally, heat transfer enhancement accompanies with a penalty of flow resistance when a heat transfer enhancement component (corrugated tube in this paper) is utilized in a heat exchanger compared to the smooth tube. Therefore, an assessment criterion needs to be constructed to evaluate the overall heat transfer performance for the investigated corrugated tube. The function of overall heat transfer performance is adopted as follows:
\nFigure 16 indicates the effect of Re on overall heat transfer performance (η) with various p/D and H/D in the tube side of outward convex corrugated tube. The figure displays that with the increase of Re, η declines deceleratedly. This is because the Nuc/Nus gradually decreases along with increasing Re. In addition, with the increase of p/D, η decreases when Re < 30,000, but increases when Re > 30,000. This can be explained from the fact that decreasing extent of Nuc/Nus is larger than that of fc/fs with increase in p/D when Re < 30,000, but lower when Re > 30,000. Moreover, the η decreases obviously with the increasing H/D, and the decreasing extent from H/D = 0.02 to H/D = 0.06 is more obvious than that from H/D = 0.06 to H/D = 0.10. This variation is quite intuitive because of the fact that increasing extent of Nuc/Nus is larger than that of fc/fs along with increasing H/D.
\nEffect of Re on η with various p/D and H/D in the tube side. (a) various p/D (b) various H/D.
It can be observed from Figure 17 that the changing trend of η with various p/D and H/D in the shell side is almost the same from the tube side. However, the values of η in the shell side are larger than in the tube side. Therefore, the overall heat transfer enhancement in the shell side is superior to the tube side.
\nEffect of Re on η with various p/D and H/D in the shell side. (a) various p/D (b) various H/D.
Response surface methodology (RSM) is composed of a series of statistical and mathematical method for analyzing empirical results, which can construct connection between effect factors and objective functions. The sensitivity of each effect factor and the interactions between two factors can also be analyzed to the objective functions. Recently, RSM has been extensively used to study on the optimal design of heat exchangers, which is able to efficiently and accurately provide the design consideration for heat exchangers [7, 8, 9].
\nRSM constructs the relationship between objective functions and design variables using a series of statistical and mathematical methodology. The function expression of the relationship could be written as follows:
\nwhere G represents the objective functions and X1, X2, …, Xk stand for design variables, f represents an approximate function, and ε is the residual error between the real value and the approximate value. The approximate functions are described as a quadratic polynomial, aiming to reflect the nonlinear characteristic between objective functions and design variables. In this study, the quadratic polynomial function, including the linear, square, and interaction terms, can be expressed as follows:
\nwhere bI represents the linear effect of design variable XI, bI,I represents the quadratic effect of XI, and bI,J represents the linear-linear interactions between XI and XJ.
\nIn our present work, we adopted the flow chart of optimization procedure as shown in Figure 18. Three objective functions including heat transfer, pressure drop, and overall heat transfer performance in a heat exchanger tube are selected for optimization. In this simulation plan, a most popular design method called the design of experiment (DOE) and central composite design (CCD) is applied. As shown in Figure 19, points including factorial points and center points augmented by axial points are set in CCD. The numerical results for DOE runs are utilized in reflecting the behavior of responses with geometrical and flow parameters.
\nFlow chart of optimization procedure.
CCD model. (a) Two factors. (b) Three factors.
Nondominated sorting genetic algorithm II (NSGA-II) combined with the multi-objective optimization is adopted in this study. The advantages of NSGA-II are a uniformly distributed Pareto-optimal front, which can suitably detect Pareto-optimal front for multi-objective problems, decrease time consuming, and present solutions with a single run.
\nFigure 20 shows the NSGA-II flowchart. As specified in Figure 18, the RSM has been employed to determine the fitness functions in the optimization algorithm. As well, cross over and mutation contained in genetic operators are used in order to generate a new population. Finally, the optimization process is wrapped up with condition of repetitions number.
\nNSGA-II flowchart.
This algorithm uses two functions including nondominated sorting function and crowding distance function, respectively. This subprogram takes population members as input, ranks them, and puts them into fronts in proportion to their ranks. Crowding distance function has been designed to avoid the accumulation of population members in a limited distance. On the other hand, there are no blank intervals in the domain by using crowding distance function. The function is applied for comparison between members of a front that has equal ranks. Compared to the previous and the next member and also the first and the last member of the population, the normalization Euclidean distance of each solution of the front is for each reference point. Normalization is applied to avoid the problem that the objectives are in the different scale.
\nANOVA is one statistical analysis method used to evaluate the fitness of regression models, perform significance testing, and construct simplified regression models between design factors and objective functions. Tables 1 and 2 are ANOVA for Nuc and fc. According to the values of R2, the fitting degree of the RSM is estimated. The F value and P value indicate the influencing significances of the model terms, judging the significant degree of each model term for the global sensitivity analysis. If the model term is the most significant, the corresponding P value is minimum, and F value is maximum. Generally, the terms having a P value >0.05 are considered insignificant and are removed from the models.
\nAnalysis of variable (ANOVA) for Nuc.
Analysis of variable (ANOVA) for fc.
The regression response surface models are described in quadratic polynomial form. Coefficients in the models are determined based on a series of statistical and mathematical methods. The models evaluate the objective functions G including Nuc/Nus, fc/fs, and η, which are expressed as:
\nIn our optimum work, the regression response surface models for evaluating Nuc and fc are expressed as:
\nWe applied 2D response surface contour plots to describe the regression response surface model, in order to display the interaction influence of each pair of design variables on the required responses. From the 2D response surface contour plots, the regulation of objective functions with changing design variables can be clearly observed, distinguished by contour plot color. Figures 21 and 22 show the 2D surface plots of the combined effects for the standard deviation of Nuc and fc. It can be observed that the decrease of p/D, the increase of H/D, the decrease of r/D, and the increase of Re result in the augment of Nu. Moreover, it can be also seen that the decrease of p/D, the decrease of H/D, the increase of r/D, and the decrease of Re result in the weak of fc.
\nResponse surfaces contour plots of combined effects for Nuc.
Response surfaces contour plots of combined effects for fc.
By inspecting the numerical results of Nuc/Nus and fc/fs, it is found that these two responses are varied with the changes of the design parameters. There must exist design parameters corresponding to the optimal objective functions. The goal of optimization for a corrugated tube subjected the design constrains of structural limitation in this study is to find the optimal values of designing parameters to maximize Nuc/Nus and minimize fc/fs. In this study, the multi-objective optimization is executed by NSGA-II. The results for Pareto-optimal curve are shown in Figure 23, which clearly reveal the conflict between the two responses, Nuc/Nus and fc/fs. Any changed design parameter that increases Nuc/Nus leads to an increase of fc/fs. It is worth noting that the minimum values of fc/fs with Nuc/Nus for various points on Pareto optimal front. Therefore, the reported results are applicable for a problem with one objective function (fc/fs) and specific constraint (the value of selected or input Nuc/Nus). This means that the presented multi-objective optimization method provides a general optimal solution in simplified form, and one may obtain an optimum design (minimum of fc/fs and maximum of η) with a specified Nuc/Nus.
\nPareto-optimal curve.
A deep investigation of the heat and mass transfer was given in outward convex corrugated tube heat exchangers in this chapter. The detailed structure of the novel tube has been introduced, in which the heat and mass transfer mechanism is different with the traditional tube type. A specific manufacturing procedure by hydraulic bulking system has been presented for the novel tube type. The experimental setup and measuring system for the novel tube type have been depicted. From obtained experimental data, we found that with the increase of Re, Nuc/Nus, fc/fs, and PEC decline. The decreasing rate of Nuc/Nus and PEC is almost linear, but fc/fs is decelerated. The numerical study on the heat and mass transfer at outward convex corrugated tube heat exchangers has been displayed. The distribution of velocity, temperature, and turbulence kinetic energy has been analyzed. The recirculating zone between the separating streamline and the free boundary streamline is generated, which breaks the thermal boundary layer to enhance the heat transfer performance. Turbulence kinetic energy is improved at the recirculating zone. Heat and mass transfer enhancement of outward convex corrugated tube heat exchangers has been revealed. Both on the tube side and shell side, with the decreasing p/D and increasing H/D, the values of Nuc and Nuc/Nus increase. Moreover, with the increase of p/D, η decreases when Re < 30,000, but increases when Re > 30,000; the η decreases obviously with the increasing H/D. The multi-objective optimization is executed by RSM combined with NSGA-II. ANOVA is used to evaluate the fitness of regression models, perform significance testing, and construct simplified regression models between design factors and objective functions. 2D response surface contour plots are applied to describe the regression response surface model. Multi-objective optimization method provides a general optimal solution in simplified form, and one may obtain an optimum design (minimum of fc/fs and maximum of η) with a specified Nuc/Nus.
\nThe authors gratefully acknowledge the support by the National Natural Science Fund (Grant No. 51506034).
\nThe article has not been previously published, is not currently submitted for review to any other journal, and will not be submitted elsewhere before one decision is made.
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