Basic geometrical parameters of fuel blocks in the HTR.
\\n\\n
Released this past November, the list is based on data collected from the Web of Science and highlights some of the world’s most influential scientific minds by naming the researchers whose publications over the previous decade have included a high number of Highly Cited Papers placing them among the top 1% most-cited.
\\n\\nWe wish to congratulate all of the researchers named and especially our authors on this amazing accomplishment! We are happy and proud to share in their success!
\\n"}]',published:!0,mainMedia:null},components:[{type:"htmlEditorComponent",content:'IntechOpen is proud to announce that 179 of our authors have made the Clarivate™ Highly Cited Researchers List for 2020, ranking them among the top 1% most-cited.
\n\nThroughout the years, the list has named a total of 252 IntechOpen authors as Highly Cited. Of those researchers, 69 have been featured on the list multiple times.
\n\n\n\nReleased this past November, the list is based on data collected from the Web of Science and highlights some of the world’s most influential scientific minds by naming the researchers whose publications over the previous decade have included a high number of Highly Cited Papers placing them among the top 1% most-cited.
\n\nWe wish to congratulate all of the researchers named and especially our authors on this amazing accomplishment! We are happy and proud to share in their success!
\n'}],latestNews:[{slug:"stanford-university-identifies-top-2-scientists-over-1-000-are-intechopen-authors-and-editors-20210122",title:"Stanford University Identifies Top 2% Scientists, Over 1,000 are IntechOpen Authors and Editors"},{slug:"intechopen-authors-included-in-the-highly-cited-researchers-list-for-2020-20210121",title:"IntechOpen Authors Included in the Highly Cited Researchers List for 2020"},{slug:"intechopen-maintains-position-as-the-world-s-largest-oa-book-publisher-20201218",title:"IntechOpen Maintains Position as the World’s Largest OA Book Publisher"},{slug:"all-intechopen-books-available-on-perlego-20201215",title:"All IntechOpen Books Available on Perlego"},{slug:"oiv-awards-recognizes-intechopen-s-editors-20201127",title:"OIV Awards Recognizes IntechOpen's Editors"},{slug:"intechopen-joins-crossref-s-initiative-for-open-abstracts-i4oa-to-boost-the-discovery-of-research-20201005",title:"IntechOpen joins Crossref's Initiative for Open Abstracts (I4OA) to Boost the Discovery of Research"},{slug:"intechopen-hits-milestone-5-000-open-access-books-published-20200908",title:"IntechOpen hits milestone: 5,000 Open Access books published!"},{slug:"intechopen-books-hosted-on-the-mathworks-book-program-20200819",title:"IntechOpen Books Hosted on the MathWorks Book Program"}]},book:{item:{type:"book",id:"3102",leadTitle:null,fullTitle:"Advances in Hurricane Research - Modelling, Meteorology, Preparedness and Impacts",title:"Advances in Hurricane Research",subtitle:"Modelling, Meteorology, Preparedness and Impacts",reviewType:"peer-reviewed",abstract:"This book provides a wealth of new information, ideas and analysis on some of the key unknowns in hurricane research. 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We aimed to compile information from diverse sources into a single volume and to give some real-life examples, extending the appreciation of organic fertilizers that may stimulate new research ideas and trends in relevant fields. The contributions in this field of research are gratefully acknowledged. The publication of this book is of great importance for those researchers, scientists, engineers, teachers, graduate students, agricultural agronomists, farmers and crop producers who can use these different investigations to understand the advantages of using organic fertilizers.",isbn:"978-1-78985-148-9",printIsbn:"978-1-78985-147-2",pdfIsbn:"978-1-78985-089-5",doi:"10.5772/intechopen.77847",price:119,priceEur:129,priceUsd:155,slug:"organic-fertilizers-history-production-and-applications",numberOfPages:144,isOpenForSubmission:!1,hash:"c139030955f06869c7b1c5f1ef9a492b",bookSignature:"Marcelo Larramendy and Sonia Soloneski",publishedDate:"November 20th 2019",coverURL:"https://cdn.intechopen.com/books/images_new/8017.jpg",keywords:null,numberOfDownloads:3120,numberOfWosCitations:0,numberOfCrossrefCitations:2,numberOfDimensionsCitations:9,numberOfTotalCitations:11,isAvailableForWebshopOrdering:!0,dateEndFirstStepPublish:"January 10th 2019",dateEndSecondStepPublish:"January 31st 2019",dateEndThirdStepPublish:"April 1st 2019",dateEndFourthStepPublish:"June 20th 2019",dateEndFifthStepPublish:"August 19th 2019",remainingDaysToSecondStep:"2 years",secondStepPassed:!0,currentStepOfPublishingProcess:5,editedByType:"Edited by",kuFlag:!1,biosketch:null,coeditorOneBiosketch:null,coeditorTwoBiosketch:null,coeditorThreeBiosketch:null,coeditorFourBiosketch:null,coeditorFiveBiosketch:null,editors:[{id:"14764",title:"Dr.",name:"Marcelo L.",middleName:null,surname:"Larramendy",slug:"marcelo-l.-larramendy",fullName:"Marcelo L. Larramendy",profilePictureURL:"https://mts.intechopen.com/storage/users/14764/images/system/14764.jpeg",biography:"Marcelo L. Larramendy, Ph.D., serves as a Professor of Molecular Cell Biology at the School of Natural Sciences and Museum (National University of La Plata, Argentina). He was appointed as Senior Researcher of the National Scientific and Technological Research Council of Argentina. He is a former member of the Executive Committee of the Latin American Association of Environmental Mutagenesis, Teratogenesis, and Carcinogenesis. He is the author of more than 450 contributions, including scientific publications, research communications, and conferences worldwide. He is the recipient of several national and international awards. Prof. Larramendy is a regular lecturer at the international A. Hollaender courses organized by the IAEMS and a former guest scientist at NIH (USA) and the University of Helsinki, (Finland). He is an expert in genetic toxicology and is, or has been, a referee for more than 20 international scientific journals. He was a member of the International Panel of Experts at the International Agency for Research on Cancer (IARC, WHO, Lyon, France) in 2015 for the evaluation of DDT, 2,4-D, and Lindane. Presently, Prof. Dr. Larramendy is Head of the Laboratory of Molecular Cytogenetics and Genotoxicology at the UNLP.",institutionString:"National University of La Plata",position:null,outsideEditionCount:0,totalCites:0,totalAuthoredChapters:"2",totalChapterViews:"0",totalEditedBooks:"17",institution:{name:"National University of La Plata",institutionURL:null,country:{name:"Argentina"}}}],coeditorOne:{id:"14863",title:"Dr.",name:"Sonia",middleName:null,surname:"Soloneski",slug:"sonia-soloneski",fullName:"Sonia Soloneski",profilePictureURL:"https://mts.intechopen.com/storage/users/14863/images/system/14863.jpeg",biography:"Sonia Soloneski has a Ph.D. in Natural Sciences and is an Assistant Professor of Molecular Cell Biology at the School of Natural Sciences and Museum of La Plata, National University of La Plata, Argentina. She is a member of the National Scientific and Technological Research Council (CONICET) of Argentina in the genetic toxicology field, the Latin American Association of Environmental Mutagenesis, Teratogenesis, and Carcinogenesis (ALAMCTA), the Argentinean Society of Toxicology (ATA), the Argentinean Society of Genetics (SAG), the Argentinean Society of Biology (SAB), and the Society of Environmental Toxicology and Chemistry (SETAC). She has authored more than 380 contributions in the field, including scientific publications in peer-reviewed journals and research communications. She has served as a review member for more than 30 scientific international journals. She has been a plenary speaker in scientific conferences and a member of scientific committees. She is a specialist in issues related to genetic toxicology, mutagenesis, and ecotoxicology.",institutionString:"National University of La Plata",position:null,outsideEditionCount:0,totalCites:0,totalAuthoredChapters:"2",totalChapterViews:"0",totalEditedBooks:"5",institution:{name:"National University of La Plata",institutionURL:null,country:{name:"Argentina"}}},coeditorTwo:null,coeditorThree:null,coeditorFour:null,coeditorFive:null,topics:[{id:"677",title:"Soil Fertility",slug:"soil-fertility"}],chapters:[{id:"68355",title:"Opening History: Gaining Perspectives",slug:"opening-history-gaining-perspectives",totalDownloads:332,totalCrossrefCites:0,authors:[null]},{id:"67534",title:"The State of the Soil Organic Matter and Nutrients in the Long-Term Field Experiments with Application of Organic and Mineral Fertilizers in Different Soil-Climate Conditions in the View of Expecting Climate Change",slug:"the-state-of-the-soil-organic-matter-and-nutrients-in-the-long-term-field-experiments-with-applicati",totalDownloads:439,totalCrossrefCites:1,authors:[null]},{id:"68604",title:"Composting",slug:"composting",totalDownloads:445,totalCrossrefCites:0,authors:[null]},{id:"68090",title:"Plant Growth Biostimulants from By-Products of Anaerobic Digestion of Organic Substances",slug:"plant-growth-biostimulants-from-by-products-of-anaerobic-digestion-of-organic-substances",totalDownloads:377,totalCrossrefCites:1,authors:[null]},{id:"68380",title:"Compost Tea Quality and Fertility",slug:"compost-tea-quality-and-fertility",totalDownloads:613,totalCrossrefCites:0,authors:[null]},{id:"67957",title:"Efficacy of Different Substrates on Vermicompost Production: A Biochemical Analysis",slug:"efficacy-of-different-substrates-on-vermicompost-production-a-biochemical-analysis",totalDownloads:433,totalCrossrefCites:0,authors:[null]},{id:"67917",title:"Organic Fertilizer Production and Application in Vietnam",slug:"organic-fertilizer-production-and-application-in-vietnam",totalDownloads:484,totalCrossrefCites:0,authors:[null]}],productType:{id:"1",title:"Edited Volume",chapterContentType:"chapter",authoredCaption:"Edited by"},personalPublishingAssistant:{id:"287827",firstName:"Gordan",lastName:"Tot",middleName:null,title:"Mr.",imageUrl:"https://mts.intechopen.com/storage/users/287827/images/8493_n.png",email:"gordan@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:"923",title:"Herbicides",subtitle:"Theory and Applications",isOpenForSubmission:!1,hash:"54a8eb808c05a5fe01c676e7047d4576",slug:"herbicides-theory-and-applications",bookSignature:"Sonia Soloneski and Marcelo L. Larramendy",coverURL:"https://cdn.intechopen.com/books/images_new/923.jpg",editedByType:"Edited by",editors:[{id:"14764",title:"Dr.",name:"Marcelo L.",surname:"Larramendy",slug:"marcelo-l.-larramendy",fullName:"Marcelo L. 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Larramendy",coverURL:"https://cdn.intechopen.com/books/images_new/4616.jpg",editedByType:"Edited by",editors:[{id:"14764",title:"Dr.",name:"Marcelo L.",surname:"Larramendy",slug:"marcelo-l.-larramendy",fullName:"Marcelo L. Larramendy"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"5358",title:"Soil Contamination",subtitle:"Current Consequences and Further Solutions",isOpenForSubmission:!1,hash:"e4d136df9f1658ae17f3ba7b3c992460",slug:"soil-contamination-current-consequences-and-further-solutions",bookSignature:"Marcelo L. Larramendy and Sonia Soloneski",coverURL:"https://cdn.intechopen.com/books/images_new/5358.jpg",editedByType:"Edited by",editors:[{id:"14764",title:"Dr.",name:"Marcelo L.",surname:"Larramendy",slug:"marcelo-l.-larramendy",fullName:"Marcelo L. Larramendy"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}}]},chapter:{item:{type:"chapter",id:"55229",title:"Analysis of the Spatial Separation Effects of Thorium/Uranium Fuels in Block‐Type HTRs",doi:"10.5772/intechopen.68671",slug:"analysis-of-the-spatial-separation-effects-of-thorium-uranium-fuels-in-block-type-htrs",body:'During the mid‐1950s to the mid‐1970s, some different types of Th/U MOX fuels, for example (Th,U)O2 or (Th,U)C2, were tested and used as fuels in high‐temperature gas‐cooled reactors (HTRs), like AVR and THTR in Germany [1] and Fort St. Vrain in the USA [2]. Three main reasons inspired them to demonstrate Th‐U fuel cycle at that time. First, compared with kwon uranium, thorium is three times more abundant in nature. Second, Th‐232 has an attractive potential for breeding to fissile U‐233 efficiently in thermal neutron spectrum. U‐233 is also considered as the best compared with other two common fissile isotopes, U‐235 and Pu‐239, in epithermal or thermal spectrum from neutronic point of view because the number of fission neutrons per neutron absorbed is 10–20% higher than that of U‐235 and Pu‐239. Finally, uranium resources were believed to be insufficient to support the development of nuclear industries on a large scale at the early period of nuclear energy development.
Besides HTRs, thorium has been an interesting nuclear fuel for various reactor applications [3], such as molten‐salt reactors (MSRs) and water‐cooled reactors.Especially, combined with molten‐salt fuel in MSRs, thorium was used as a necessary composition of the “standard” fuel aiming to converse and even breed Th‐232 to U‐233 from the 1970s to mid‐1980s. In past 20 years, the potential of thorium has been extended to radioactive waste management and plutonium incineration [4]. In this application, Th‐232 is considered as a better fertile isotope than U‐238 because of a larger net destruction of plutonium, when weapon‐grade or reactor‐grade plutonium is burned in various nuclear reactors. Furthermore, thorium‐fueled reactors generate less long‐lived radioactive wastes than uranium‐fueled ones.
Combined with the commercial purpose and potential application scale, light‐water reactors (LWRs) are a natural choice from the reactor point of view in recent years because of a large amount of LWRs all over the world. The seed‐and‐blanket (S&B) concept has been reexamined for LWR application by the MIT group [5, 6]. The concepts originate from S&B configuration and were developed for the advanced water breeder application (AWBA) program and tested in the light water breeder reactor (LWBR) at Shippingport from 1977 to 1982 [7]. Further work on thorium‐fueled LWRs has been pursued by Radkowsky and Galperin [8]. The MIT group proposed the micro‐heterogeneous fuel assemblies and the whole assembly seed‐and‐blanket (WASB) concept. Moreover, compared with the past research, the recent MIT work is all based on low‐enriched uranium for proliferation resistance.
Because of the burnup limit of LWRs (usually 50 GWd/tHM), some research showed that the potential of thorium is limited in LWRs, and HTRs are a better choice for thorium‐based fuel for higher burnup and harder neutron spectrum. Recently, the concept of S&B fuel blockshas been introducedto the U‐battery [9], a small long‐life HTR, and a commercial‐level GT‐MHR [10], as well as advanced high‐temperature reactors (AHTRs) [11] with low‐pressure liquid‐salt (Flibe) as coolant, which enables the design of a high‐power (e.g., 2400–4000 MWth), high‐temperature (850–950°C) reactor with fully passive safety capability and the economic production of electricity orhydrogen.
The past studies have shown the distribution of thorium and uranium fuels in space is a very important factor for the performance of thorium‐fueled reactors and have proposed some interesting concepts of S&B and WASB. Actually, the different concepts represent different separation levels of thorium/uranium fuels in space. The chapter tries to systematically analyze the spatial separation effects of thorium/uranium fuels on the nuclear performance of block‐type HTRs and to quantitatively evaluate the difference by fuel cycle cost. The second section will describe the reactor, fuel block, and the calculation models, including the transport calculation model with burnup and fuel cycle cost model. The third section will present the calculation results of four different spatial separation levels and discuss the difference among them. The final section will conclude the chapter.
Modular block‐type HTRs [12] are one kind of inherently safe reactors, which passively remove decay heat by natural convection, conduction, and radiation from the reactor core and keep the fuel intact. As mentioned in Section 1, the block‐type HTRs are a better choice for thorium‐based fuel for higher burnup and harder neutron spectrum. Moreover, an advantage of block‐type HTRs over pebble‐bed HTRs is that the former provides multi‐level and fixed spatial distribution of thorium/uranium fuels in the reactor cores.The reactor core of the block‐type HTRs investigated, as shown in Figure 1(a), is comprised of an annular fuel zone, inner reflectors, and outer reflectors. The annular fuel zone is comprised of 1980 fuel blocks, as shown in Figure 1(b). The fuel blocks are stacked firmly against each other in columns that form an annulus between an inner and an outer reflector, both of which consist of rings of unfueled graphite blocks. The annular core configuration ensures higher thermal power and inherent safety under most accidents and transient conditions. Each fuel block includes 210 fuel channels, 108 coolant channels, and 6 fixed burnable poison channels which are filled with graphite in the flowing calculations.The cylindrical fuel compacts are stacked inside channels drilled into hexagonal graphite blocks. Other geometrical parameters ofthe fuel blocks are listed in Table 1.
Thorium-fueled HTR and fuel block, (a) Reactor core and (b) Fuel block. Does the query aim to use second style for the book?
Parameters | Width across flat [cm] | Height of block [cm] | Diameter/number of fuel channels [cm/−] | Diameter/number of coolant channels [cm/−] | Diameter of kernels [μm] | Thickness of TRISO layers [μm] | Density of TRISO layers [g/cm3] |
---|---|---|---|---|---|---|---|
Values | 36 | 79.3 | 1.27/210 | 1.588/102, 1.27/6 | 500 | 100/35/35/40 | 1.05/1.9/3.2/1.9 |
Basic geometrical parameters of fuel blocks in the HTR.
The most prominent feature of the block‐type HTRs is use of tristructural‐isotropic (TRISO) particles, which contain five layers from inside to outside: fuel kernel, porous carbon buffer, inner pyrolytic carbon (IPyC), silicon carbide (SiC), and outer pyrolytic carbon (OPyC). The porous carbon buffer layer is provided to protect the dense PyC from fission recoil damage and to provide void volume to limit the fission gas pressure. The dense PyC has good irradiation stability and capability to remain intact to perform fission production retention under severe exposure conditions. Sandwiched between the two PyC layers, the SiC layer provides metallic fission production retention and mechanical strength. TRISO fuel particles are blended and bonded together with a graphite matrix to form fuel compacts. Figure 2 illustrates how the TRISO particles are packaged into the annular core.
Four levels of spatial separation of thorium/uranium in block‐type HTRs.
When using thorium in HTRs, two important factors influence nuclear performance. One is the thorium content (the mass ratio of Th‐232 to all heavy metal isotopes), meaning the mixed proportion of thorium and uranium. The other is the spatial separation level of thorium and uranium, meaning how thorium and uranium are to be mixed. According to the level of spatial separation, four levels are in a block‐type HTR: (1) No separation (Th/U MOX level): with Th/U MOX fuel, the thorium and uranium are mixed in each fuel kernel as a form of (Th,U)O2, as shown in Figure 2(a). Thorium and uranium do not separate in the macrolevel, (2) TRISO‐level separation (SBT level): UO2 and ThO2 are made into different TRISO fuel particles (UO2 TRISO and ThO2 TRISO) separately, but the two kinds of TRISO fuel particles are mixed into the same fuel compact, as shown in Figure 2(b), (3) Channel‐level separation (SBU level): each fuel channel has only one kind of fuel compacts (UO2 fuel compacts or ThO2 fuel compacts), but a fuel block has both UO2 fuel channels and ThO2 fuel channels, as shown in Figure 2(c), and (4) Block‐level separation (WASB level): each fuel block only has a kind of fuel (UO2 or ThO2), but the core has both UO2 fuel blocks and ThO2 fuel blocks, as shown in Figure 2(d).
To only analyze the influence of spatial separation, the other parameters are kept the same including the fuel block geometry, as listed in Table 1, and the fuel shuffling scheme, as shown in Figure 3. A detailed full‐core 3D transport calculation in one step will require significant memory and a central processing unit (CPU) time as utilities in producing calculations or even in laboratories for design purposes. Therefore, a two‐step calculation scheme is typical: (1) a detailed calculation at the assembly level with reflective boundary conditions, which gives homogenized cross‐sections for the assemblies, condensed to a certain number of groups (lattice calculation step) and (2) a second calculation at the core level with homogenized properties in each assembly and usually small number of groups (full‐core calculation step).
Reactor core calculation model for 1/6th core.
In this chapter, the traditional two‐step calculation scheme is constructed based on the DRAGON V4 code system [13], as shown in Figure 4. An attractive feature of the DRAGON code is its ability to treat particle fuel in a graphite matrix in a full‐assembly calculation. It provides the possibility to define a stochastic mixture of spherical micro‐structures that can be distributed inside composite mixtures of the current macro‐geometry using the Hebert double‐heterogeneity model [14] or Sanchez‐Pogroming double‐heterogeneity model [15].
A schematic diagram of the two‐step calculation scheme.
In step (a) in Figure 4, a 2D fuel block is modeled including the structure of the TRISO particle. The method of characteristics (MOC) and 295‐group cross‐section library (
An evaluation criterion must be in the comparison of different spatial separationlevels. Usually, the fuel cycle cost is one of the criteria. To make a fair comparison, the adoption of a standardized methodology for fuel cycle cost calculations is prerequisite. Therefore, the levelized lifetime cost methodology is used in this study. The levelized lifetime cost methodology was developed by the Organization for Economic Cooperation and Development (OECD) [16], which uses an internationally accepted investment appraisal methodology to analyze the fuel cycle cost.
The cash flow for fuel cycle material and services commences before the reactor starts to generate electricity and continues well after the reactor ceases operation. The exact timing of payments for natural uranium purchase, conversion, enrichment, fabrication, spent fuel storage, and disposal depends on the associated lead and lag times for the fuel cycle components. To calculate the overall fuel cycle cost, the magnitude of each component cost and the appropriate time that it occurs must be identified. Fuel quantities are obtained from reactor neutronic calculations as described in Section 2.2.1. These quantities of materials and services are adjusted to allow for process losses in the various component stages of the nuclear fuel cycle and then multiplied by the unit costs to obtain the component costs. Finally, the total fuel cycle cost is gotten by discounting these component costs to their present values in a specified base year (usually the commission date of reactor), as shown in Figure 5. The levelized lifetime cost methodology provides costs per unit of electricity generated which are the ratios of lifetime expenses to total expected output, expressed in terms of the present value equivalent. This method derives economic merits by comparing their respective average levelized lifetime costs.
Time flow of nuclear fuel cycle cost with direct disposal option.
The reactor data and fuel cycle cost data for the HTR are shown in Tables 2 and 3, respectively. Table 2 gives the information about the reactor including the power, lifetime, cycle length, and back‐end options. Table 3 gives the unit price and lead or lag time for natural uranium purchase, thorium purchase, conversion, and enrichment. The price of natural uranium, conversion, and enrichment is the average price of spot market from 2010 to 2015 (
Items | Values |
---|---|
Thermal output | 600 MWth |
Electric output | 240 MWe |
Plant lifetime | 60 years |
Refueling cycle length | 2 years |
Load factor | 90% |
Back‐end options | Direct disposal |
Discount rate | 8% |
Reactor operation data for the HTR.
Components | Prices | Lead or lag times |
---|---|---|
Natural uranium purchase | 120.9 $/kg U | 24 months |
Thorium purchase | 120.9 $/kg Th | 24 months |
Conversion | 8.75 $/kg U | 18 months |
Enrichment | 126.5 $/SWU | 12 months |
Fabrication | 777 $/kg HM | 6 months |
Storage | 230 $/kg HM | 5 years |
Disposal | 610 $/kg HM | 40 years |
Fuel cycle cost data for the HTR.
For real HTRs with multi‐batch refueling and multi‐block, both the spatial separation levels defined in Section 2 and refueling patterns influence the performance of fuel in the reactor core, so the latter is excluded by ideally adopting one‐batch fixed‐patternrefueling mode in order to focus on the spatial separation effect of the thorium/uranium fuel. Based on the one‐batch fixed‐pattern refueling mode, as shown in Figure 3, thorium and uranium in the different spatial separation levels with the same mass are loaded into the geometrically same reactor core and are discharged when the keff reaches the same specified value. The core performance difference is due to the spatial separation levels.
In order to compare four different spatial separation levels in the core scale, the configurations or patterns of the thorium/uranium fuels in the channel‐level separation and block‐level separation are simply analyzedin this section because the former meets different configurationsor patterns in block scale and the latter does in core scale, as shown in Figure 2, and the configurations influence the performance of the reactor core. For the other two separation levels, that is no separation level and TRISO‐level separation, the same the fuel blocks exist in the whole reactor core.
For the channel‐level separation, the spatial configurations of thorium/uranium compacts in the fuel block are different even for the same thorium content because uranium and thorium compacts are located in different channels, that is, there are 2210 configurations theoretically. It is almost impossible to investigate all of them, but it is not necessary to do it because the performance is possibly similar for some similar configurations, and the comparison in the reactor core scale is of the main concern. Five typical configurations (SBU 1#–SBU 5#) are chosen and investigated for 46% of thorium content, as shown in Figure 6. The uranium fuel compacts are located in the central, middle, and outer regions of the fuel block, respectively, for the SBU 1#–SBU 3#, and are evenly located in the fuel block for the SBU 4# and SBU 5#. The uranium compacts are relatively concentrated in the former group of configurations and are relatively dispersed in the latter group of configurations. For other thorium content, the spatial separation of uranium/thorium compacts are located in the similar mode.
Five typical configurations of uranium/thorium fuels for 46% thorium content (red: uranium compacts, blue: Thorium compacts).
Figures 7 and 8 show the keff of the reactor core for 46 and 91% thorium contents, which are calculated based on the model described in Section 2.2. When the thorium content increases, the number of thorium compacts will increase and the number of uranium compacts will decrease. In order to maintain the same mass of U‐235 in the fuel block or in the reactor core, the enrichment of U‐235 has to be increased. For different thorium contents, when the enrichment of U‐235 is less than 20%, the five different spatial configurations of thorium/uranium fuels achieve the similar keff. However, when the enrichment is higher than 20%, for example 80%, the keff of the five configurations is different obviously. Moreover, the keff of the SBU 4# and SBU 5# is always higher than the other three configurations, and the influence of the spatial configurations becomes important. In the channel‐level separation, the dispersed uranium compacts are advantageous to transmute Th‐232 to U‐233 when the thorium content and, thus, the enrichment of U‐235 are high. Based on the calculations and analysis, the suggested spatial configurations of thorium/uranium compacts in the fuel block are shown in Figure 9 for the channel‐level separation. The uranium compacts are concentrated in the central region of the fuel block for low thorium content and are dispersed for high thorium content.
keff of five configurations for 46% thorium content.
keff of five configurations for 91% thorium content.
Configurations of thorium/uranium compacts for channel‐level separation (red: uranium compacts, blue: thorium compacts).
For the block‐level separation, the spatial configurations of thorium/uranium fuel blocks in the reactor coremeet the same problem as the channel‐level separation. The uranium and thorium fuel blocks located in different positions in the reactor core lead to different configurations, that is, there are 2198 configurationsof the fuel blocks theoretically. Based on the same analysis method as presented in Section 3.1.1, every five typical configurations (WASB 1#–WASB 5#) were chosen and investigated for different thorium content. The results show that the dispersed thorium fuel blocks are advantageous to transmute Th‐232 to U‐233 in the block‐level separation. Based on the results, the suggested spatial configurations of thorium/uranium fuel blocks in the reactor core are shown in Figure 10 for the block‐level separation, which is chosen to be compared with other three separation levels.
Configurations of thorium/uranium fuel blocks for block‐level separation (red: uranium fuel blocks, blue: thorium fuel blocks).
As mentioned in Section 3.1, the highly‐enriched U‐235 possibly has to be used when the thorium content increases. Compared with thorium/uranium‐fueled reactor, the uranium‐fueled reactor only contains one fissile isotope, that is, U‐235, and one fertile isotope, that is, U‐238, in the fuel. The mass fraction of fissile isotopes, that is, U‐235, is the so‐called enrichment of U‐235, as defined in Eq. (1).,
When another fertile isotope, that is, Th‐232, appears in the reactor fuel, the mass fraction of fissile isotopes, as defined in Eq. (2)
is different from the traditional enrichment of U‐235, which is called effective enrichment of U‐235[19]. If the concept of the effective enrichment of U‐235 is adopted, the nominal mass fraction of fissile isotopes is far less than 20% for the thorium‐fueled HTRs as shown in Figure 11, which is the limit enrichment of U‐235 for low‐enriched fuel. Since the enrichment of U‐235 is no longer equal to the effective enrichment of U‐235 in a thorium‐loaded reactor, and the physical meaning of the latter is clearer, the latter is analyzed instead of the former.
Initial effective enrichment of U‐235 as a function of thorium content for different separation levels.
As shown in Figure 11, the required initial effective enrichment of U‐235 obviously decreases with the increase of separation level from Th/U MOX to SBU and WASB for the same 2‐year refueling cycle length, when the thorium content changes from 10 to 80%. When the thorium content is less than 10% or larger than 80%, the influence of spatial separation level on the initial εeff is weak because the fuel is nearly uranium or thorium and thus there is no obvious difference. Moreover, the initial εeff of SBU is nearly the same as that of theWASB when the thorium content is in the range of 20–70%. For the Th/U MOX fuel, the initial εeff increases when the thorium content increases from 0 to 30% and decreases with the further increase of thorium. For the SBT, SBU, and WASB, the initial εeff decreases with the increase of the thorium content.
Although the total initial inventory of heavy metal slightly decreases because of the density difference between ThO2 (9.4 g/cm3) and UO2 (10.4 g/cm3), as shown in Figure 12, the required initial inventory of U‐235,as shown in Figure 13, nearly has the same trend as the initial effective enrichment of U‐235, according to Eq. (2). Furthermore, the initial inventory of the enriched uranium also decreases with the increase of the thorium content, as shown in Figure 12, because more and more U‐238 is replaced by Th‐232 in the reactor core. The change of the initial inventory of U‐235 or the initial effective enrichment of U‐235 is a result of the difference of nuclear performance (e.g., initial effective multiplication factor and average conversion ratio) caused by the spatial separation levels, as furtherly discussed in Section 3.3.
Initial inventory of heavy metal as a function of thorium content for different separation levels.
Initial inventory of U‐235 as a function of thorium content for different separation levels.
Figure 14 presents the initial effective multiplication factors (keff) and average conversion ratios (ACRs) of four spatial separation levels as a function of thorium content. For each spatial separation level, the keff nearly decreases with the increase of thorium content. For the same thorium content, the initial keff usually increases when the spatial separation level increases from Th/U MOX to SBU. When the thorium content is larger than 80%, the difference among them becomes small. The more interesting rule is that the trend of ACR is always opposite to the initial keff, that is, when the ACR is higher, initial keff is smaller, even for the WASB level.
Initial keff and average conversion ratio of different spatial separation levels.
Figure 15 presents the keff as a function of operation time for two different ACRs in one refueling cycle. In these calculations, the refueling cycle length is 657 effective full power days (EFPDs), which means the coordinate of end of cycle (EOC) is fixed, as the point B (657,1.005) in Figure 15. On the other hand, the keff is a nearly linear function of burnup for the thorium‐fueled HTR as shown in Figure 15. Therefore, if the average conversion ratio of the refueling cycle is smaller, the reactivity drop is larger and thus the initial keff must be higher to guarantee a critical reactor at EOC.
keff as a function of EFPD for different average conversion ratios.
If Figure 14 is compared with Figure 11 or Figure 13, it is interesting to find that when the spatial separation level of thorium/uranium fuels changes from SBU to Th/U MOX, the required initial inventory of U‐235 is the most for the Th/U MOX in order to achieve the same operation time, but the initial keff is smallest and the main benefit is the most amount of Th‐232 transmuted into U‐233. Once the spatial separation of thorium/uranium fuels changes from SBU to WASB, the required initial inventory of U‐235 decreases. Moreover, the ACR increases with increase of thorium content, the initial keff decreases, and thus the reactivity drop or swingdecreases in the refueling length.
In order to analyze the influence of spatial separation levels and thorium content on the initial keff and thus the initial inventory of U‐235, four control groups of typical reactor cores including spatial separation levels and thorium content are calculated and discussed. The group 1 is to compare the Th/U MOX with SBU with 50% thorium content, in order to explain the reason of the decrease of the initial inventory of U‐235. The group 2 is to compare the SBU and WASB with 50% thorium content, in order to explain the difference between them. The groups 3 and 4 are to discuss the Th/U MOX with 0, 30, and 80% thorium content, respectively, in order to explain the influence of the thorium content.
Table 4 presents the five factors and initial keff of the reactor cores involved in the four control groups, which are calculated according to the validated method [10]. The five‐factor formula [20] can be written as
Reactor core | ηf | ε | p | PNL | keff |
---|---|---|---|---|---|
Th/U MOX‐0%Th | 1.7257 | 1.0804 | 0.6994 | 0.9550 | 1.24531 |
Th/U MOX‐30%Th | 1.6891 | 1.0861 | 0.6852 | 0.9551 | 1.20058 |
Th/U MOX‐80%Th | 1.5540 | 1.0701 | 0.7379 | 0.9570 | 1.17436 |
Th/U MOX‐50%Th | 1.6491 | 1.0830 | 0.6935 | 0.9556 | 1.18358 |
SBU‐50%Th | 1.5690 | 1.0687 | 0.7508 | 0.9567 | 1.20442 |
WASB‐50%Th | 1.5182 | 1.0734 | 0.7567 | 0.9615 | 1.18567 |
Five factors and initial keff of different reactor cores involved in four control groups.
where η, f, p, ε, PNL are, respectively, the reproduction factor, fuel utilization factor, resonance escape probability, fast fission factor, and nonleakage probability. However, in the core physics calculation described in Section 2.2.1, the homogenization of fuel block will mix the fuel, moderator, and coolant into a mixture, making the η and f inseparable. Thus, the production of η and f is regarded as the reproduction factor of the core. More information about the five‐factor formula can be found in Ref. [10].
To quantitatively describe the contribution of each factor to the variation of the initial effective multiplication factor, its contribution [21] is defined in terms of the components of five‐factor formula
where
The subscripts 1 and 2 represent two different cases, respectively. Table 5 presents the contribution of each factor for the four control groups.
Group number | Group | Δηf [pcm] | Δε [pcm] | Δp [pcm] | ΔPNL [pcm] | Δkeff [pcm] |
---|---|---|---|---|---|---|
1 | MOX‐50%Th | −4167 | −1112 | +6649 | +96 | +1460 |
SBU‐50%Th | (−285%) | (−76%) | (+455%) | (+6%) | (+100%) | |
2 | SBU‐50%Th | −2754 | +367 | +655 | +419 | −1313 |
WASB‐50%Th | (−210%) | (+28%) | (+50%) | (+32%) | (−100%) | |
3 | MOX‐0%Th | −1753 | +430 | −1677 | +9 | −2992 |
MOX‐30%Th | (−59%) | (+14%) | (−56%) | (+1%) | (−100%) | |
4 | MOX‐30%Th | −7017 | −1250 | +6236 | +171 | −1857 |
MOX‐80%Th | (−378%) | (−67%) | (+336%) | (+9%) | (−100%) |
Contribution of each factor to variation of initial keff.
For the group 1, when the thorium content is 50% and spatial separation level changes from Th/U MOX level to SBU level, the nonleakage probability increases by 96 pcm and contributes +6% to the initial keff, because the spatial separation level strengthens thorium absorption, and the microscopic absorption cross section of Th‐232 (7.4 barns) is three times of U‐238 (2.7 barns). Based on the same reason, more thermal neutrons are absorbed by Th‐232 but cannot induce fission reactions, which leads to the reproduction factor of core (ηf) to decrease by 4167 pcm and contributes −285% to the initial keff. The fast fission cross‐section of Th‐232 (0.01 barns) is smaller than that of U‐238 (0.04 barns), which causes the fast fission factor (ε) to decrease by 1112 pcm and contribute −76%. However, because the thorium fuel is lumped in the thorium compacts for the SBU‐50%Th and a smaller resonance integral (RI) of Th‐232 (85) compared to U‐238’s (275), the resonance escape probability (p) of the SBU‐50%Th increases by 6649 pcm and contributes +455%. As a result of all effects, especially the contribution of the increase of p, the initial keff of the SBU‐50%Th is 1460 pcm larger than that of the Th/U MOX‐50%Th. The spatial self‐shielding effect is strengthened by Th‐232 and spatial separation levels, and thus the increase of resonance escape probability leads to the decrease of 188 kg initial inventory of U‐235.
For the group 2, when the spatial separation increases from SBU level to WASB level, although the resonance escape probability further increases by 655 pcm, the reproduction factor decreases by 2754 pcm because of the further lumping of the thorium fuel. As a result, the initial keff decreases by 1313 pcm causing a smaller reactivity swing in the refueling period. Moreover, the initial inventory of U‐235 decreases 6 kg because of a larger ACR.
For the group 3, for a fixed spatial separation level, for example, Th/U MOX level, when the thorium content increases from 0 to 30%, more thermal neutrons generated by fission are absorbed by fertile Th‐232 because of three‐time microscopic absorption cross section of Th‐232, which leads to the decrease of the reproduction factor by 1753 pcm. Moreover, because of harder neutron spectrum, the resonance escape probability decreases by 1677 pcm. As a result, the initial keff decreases by 2992 pcm and the required initial inventory of U‐235 increases by 77.8 kg. If the thorium content further increases from 30 to 80%, as shown in the group 4, a large amount of U‐238 is replaced by Th‐232 in the reactor core. Because of a smaller resonance integral of Th‐232 (85) compared to U‐238’s (275), the resonance escape probability (p) increases by 6236 pcm and contributes +336% but the reproduction factor decreases by 7017 pcm and contributes −378%. As a result, the initial keff decreases by 1857 pcm. Moreover, the required initial inventory of U‐235 decreases by 206 kg because of a larger ACR.
Using the levelized lifetime cost methodology described in Section 2.2.2, the fuel cycle cost of the four types of reactor cores (Th/U MOX, SBT, SBU, and WASB) analyzed in Sections 3.2 and 3.3 as a function of thorium content is shown in Figure 16. Compared with Figures 11 or 13, the fuel cycle cost changes with the same trend as the effective enrichment of U‐235 or the initial inventory of U‐235 in the reactor cores. When the thorium content is constant, the fuel cycle cost decreases with the increase of the spatial separation level. However, the difference of the SBU level and WASB level is small. The fuel cycle cost decreases with the increase of the thorium content for the SBT, SBU, and WASB levels. However, it increases in the range of thorium content from 0 to 40% and decreases when the thorium content is larger than 40%.
Fuel cycle cost of different spatial separation levels.
The tight relationship between fuel cycle cost and initial inventory of U‐235 mainly results from the composition of the cost. Figure 17 presents the composition of fuel cycle cost for three different reactor cores, including natural uranium purchase, thorium purchase, uranium conversion, uranium enrichment, the fabrication of fuel blocks, and storage and the deposal of spent fuels. Because the required amount of natural uranium is 13 times of the inventory of heavy metal in the reactor core due to 0.7% U‐235 in natural uranium, both the natural uranium purchase and uranium enrichment are 70% of total fuel cycle cost. The thorium purchase is only 2.5% because the thorium need not be enriched, and the required amount of thorium is by far less than the amount of the natural uranium. The fabrication cost is the highest unit price (777 $/kg HM) and the fabrication involves all heavy metals, so it is about 20% of the total cost. Although the unit price of the disposal of spent fuel is also high (610 $/kg HM), the disposal cost is only about 0.5% because of so‐called time value.
Composition of fuel cycle cost for some typical reactor cores.
The fuel cycle cost is mainly determined by natural uranium purchase (35–40%), uranium enrichment (32–35%), and the fabrication of fuel blocks (18–21%). The total of three items is 85–96% fuel cost. The fabrication cost is the same for all reactor cores because the inventory of heavy metal in the reactor core is the same. The natural uranium purchase and uranium enrichment are directly related to the initial inventory of U‐235. The more is the inventory of U‐235, the more are the required natural uranium and the uranium enrichment. As a result, the fuel cycle cost is the same trend as the initial inventory of U‐235 or the initial effective enrichment.
In order to utilize thorium fuel in block‐type HTRs with the features of inherent passive safety, high burnup, and hard neutron spectrum, two key factors of thorium content and spatial separation levels are chosen to be investigated. The thorium content represents the thorium/uranium fuel composition and the spatial separation level represents the spatial distribution of thorium/uranium fuels. For every thorium content,the spatial distribution of thorium and uranium fuels are concluded into four spatial separation levels, that is, no separation level (Th/U MOX), TRISO level (SBT), channel level (SBU), and block level (WASB) for the thorium‐fueled block‐type HTRs.
The nuclear performance of the reactor core, that is, initial effective multiplication factor, average conversion ratio, and initial inventory of U‐235, under four spatial separation levels are obtained in the one‐batch fixed‐pattern refueling mode by the two‐step calculation scheme developed based on the DRAGON. For every thorium content, the initial inventory of U‐235 decreases with the increase of the spatial separation level from Th/U MOX to WASB, because spatial self‐shielding effect is strengthened by the lumped thorium and uranium. However, the SBU level is nearly the same as the WASB level. If the multiple fuel batches and realistic refueling patterns are considered, the difference of the SBU from the WASB could be large. On the other hand, the initial inventory of U‐235 decreases with the increase of the thorium content for the SBT, SBU, and WASB levels. However, for Th/U MOX level, it increases in the range of thorium content from 0 to 30% and decreases when it is larger than 30% because of the better performance of U‐233 than Pu‐239 in thermal reactors.
The performance difference of the four spatial separation levels is synthetically evaluated bythe levelized lifetime cost method.The fuel cycle cost of the Th/U MOX, SBT, SBU, and WASB changes with the same trend as the effective enrichment of U‐235 or the initial inventory of U‐235 in the reactor cores because the latter determines 70% of the total cost.
This work is supported by the National Natural Science Foundation of China (11405036).
China has nearly 100,000 reservoir dams, of which earth-rock dams account for more than 95% [1, 2]. Most of these reservoir dams were built in the 1950s and 1970s. Due to economic and technical conditions at that time, the problem of dangerous reservoirs in China was outstanding [3]. According to statistics [4], from 1954 to 2018, 3541 reservoir dams broke in China. The “75·8” flood occurred in Henan in 1975, which led to the collapse of 2 large reservoirs in Banqiao and Shimantan (Figures 1 and 2), 2 medium-sized reservoirs in Tiangang and Zhugou, and 58 small reservoirs, causing heavy casualties and property losses [5]. In the twenty-first century, with the improvement of the dam safety management level and the comprehensive development of the reservoir’s risk elimination and reinforcement, the number of dam breaks has been significantly reduced, but due to the frequent occurrence of extreme weather events, dams’ breaching still occur frequently. On July 19, 2018, the Zenglongchang Reservoir in Inner Mongolia and the Sheyuegou Reservoir in Xinjiang on August 1, 2018, successively dams’ breaching [4] (Figures 3 and 4).
\nFinal breach of Banqiao dam.
Final breach of Shimantan dam.
Final breach of Zenglongchang dam.
Final breach of Sheyuegou dam.
Therefore, it is necessary to establish a mathematical model and numerical calculation method that reasonably simulates the process of overtopping and seepage failure collapse, improves the prediction accuracy of the flood flow process of earth-rock dam collapse, and provides theoretical and technical support for the evaluation of the consequences of dam collapse and the preparation of emergency plans. This article will briefly introduce the research progresses on the mechanisms and numerical models of earth-rock dams’ breaching, especially the latest research results of the author’s research team in recent years, and make suggestions for future research.
\nThe mathematical model of earth-rock dams’ breaching is generally divided into three categories [6]: The first category is the parameter model. Most of these models are based on statistical analysis of dam-break case data, and empirical formulas are used to calculate and obtain dam-break-related parameters. Although most models cannot consider the erosion characteristics of damming materials, but the parameter model formula is simple and fast to calculate and is also often used for rapid evaluation of the consequences of dams’ breaching. The second category is a simplified mathematical model based on the mechanism of failure. It is generally assumed that the shape of the fractured breach (rectangular, inverted trapezoidal, triangular, etc.) remains unchanged during the dams’ breaching. The method based on the flow shear stress and the critical shear stress of the dam material or the erosion formula of the dam material is used to calculate the breach development process; the weir flow is used (overtopping dam failure) or pore flow (seepage failure dam breaching) formulas are used to calculate the breach flow. The stability analysis of the breach slope mostly uses the limit equilibrium method; generally, the numerical calculation method based on time step iteration is used to simulate the breach development process and the breach flow process. The advantage of this type of model is that it considers the failure mechanism of earth-rock dams, and the calculation speed is relatively fast, which is the most widely used in the numerical simulation of earth-rock dams dam breaching process. The third category is a detailed mathematical model based on the failure mechanism. In recent years, a series of researches on one-dimensional, average two-dimensional, and three-dimensional mathematical models based on the hydrodynamic dam material erosion equation have made significant progress, which can simulate the dams’ breaching process of earth-rock dams in more detail. In order to deal with the diffuse overtopping flow composed of discontinuous mixed flow states, shock wave capturing methods such as approximate Riemann solution method and total variation declining (TVD) method are generally used, and finite volume method, level set method, and smooth particle hydrodynamic method are used to solve the governing equation. This type of model is a fast-developing simulation method in recent years, but it can only be used for the simulation of the overtopping collapse process of homogeneous dams or landslide dams. It has not been used to simulate the process of seepage and failure of earth-rock dams and the simulation of the process of overtopping failure of other types of earth-rock dams [6].
\nIn 1977, Kirkpatrick [7] proposed the first empirical formula for predicting peak outflow Qp\n, and then scholars from various countries proposed a series of models. With the continuous enrichment of dam failure case investigation data and the deepening of research, the dam failure parameter model has gradually evolved from the single-parameter model to a multi-parameter model, and the output results have increased from the original peak outflow of the breach to the final average width of the breach and the duration of the dam and can consider the shape of the dam body, reservoir capacity, dam material characteristics, etc. The peak outflow rate of breach is very important for the evaluation of the consequences of dam breaching. Therefore, domestic and foreign scholars have studied more. The commonly used parameter model of peak outflow rate is shown in Table 1.
\nModel | \nCase number | \nExpression | \n
---|---|---|
Kirkpatrick (1977) [7] | \n19 | \n\nQp\n = 1.268(hw\n + 0.3)2.5\n | \n
Soil Conservation Service (1981) [8] | \n13 | \n\nQp\n = 16.6hw\n\n1.85\n | \n
Hagen (1982) [9] | \n6 | \n\nQp\n = 0.54(hdS)0.5\n | \n
Singh and Snorrason (1984) [10] | \n28 | \n\nQp\n = 13.4hd\n\n1.89 or Qp\n = 1.776S\n0.47\n | \n
MacDonald and Langridge-Monopolis (1984) [11] | \n23 | \n\nQp\n = 1.154(Vwhw\n)0.412\n | \n
Costa (1985) [12] | \n31 | \n\nQp\n = 0.981(hdS)0.42\n | \n
Evans (1986) [13] | \n29 | \n\nQp\n = 0.72Vw\n\n0.53\n | \n
USBR (1988) [14] | \n21 | \n\nQp\n = 19.1hw\n\n1.85\n | \n
Froehlich (1995) [15] | \n22 | \n\nQp\n = 0.607Vw\n\n0.295\nhw\n\n1.24\n | \n
Walder and O’Connor (1997) [16] | \n18 | \n\nQp\n = 0.031 g\n0.5\nVw\n\n0.47\nhw\n\n0.15\nhb\n\n0.94\n | \n
Xu and Zhang\n1\n (2009) [17] | \n75 | \n\nQp\n = 0.175 g\n0.5\nVw\n\n5/6(hd\n/hr\n)0.199(Vw\n\n1/3/hw\n)−1.274\neB\n\n4\n | \n
Pierce et al. (2010) [18] | \n87 | \n\nQp\n = 0.0176(Vh)0.606or Qp\n = 0.038 V\n0.475 h\n1.09\n | \n
Thornton et al. (2011) [19] | \n38 | \n\nQp\n = 0.1202 L\n1.7856 or Qp\n = 0.863 V\n0.335\nhd\n\n1.833\nWave\n\n−0.663or\nQp\n = 0.012 V\n0.493\nhd\n\n1.205\nL\n0.226\n | \n
Lorenzo and Macchione (2014) [20] | \n14 | \n\nQp\n = 0.321 g\n0.258(0.07Vw\n)0.485\nhb\n\n0.802(overtopping) \nQp\n = 0.347 g\n0.263(0.07Vw\n)0.474\nhb\n\n−2.151\nhw\n\n2.992(seepage failure) | \n
Hooshyaripor et al. (2014) [21] | \n93 | \n\nQp\n = 0.0212 V\n0.5429\nh\n0.8713 or Qp\n = 0.0454 V\n0.448\nh\n1.156\n | \n
Azimi et al. (2015) [22] | \n70 | \n\nQp\n = 0.0166(gV)0.5\nh\n | \n
Froehlich\n2\n (2016) [23] | \n41 | \n\nQp\n = 0.0175kMkH\n(gVwhwhb\n\n2/Wave\n)0.5\n | \n
Mei Shiang et al. (2018) [24] | \n154 | \n\nQp = Vwg\n0.5\nhw\n\n−0.5(Vw\n\n1/3/hw\n)−1.58(hw\n/hb\n)−0.76(hd\n/h\n0)0.10e−4.55(homogeneous dam) \nQp = Vwg\n0.5\nhw\n\n−0.5(Vw\n\n1/3/hw\n)−1.51(hw\n/hb\n)−1.09(hd\n/h\n0)−0.12e−3.61 (core-wall dam) | \n
Parameter model of peak outflow rate.
The expression of parameter B\n4 is B\n4 = b\n3 + b\n4 + b\n5, for core-wall dam, concrete face rockfill dam or homogeneous dam, b\n3 is taken as −0.503, 0.591, or − 0.649, respectively; for overtopping or seepage failure, b\n4 is taken as −0.705 or − 1.039, respectively; for dam materials with high, medium, or low erosion rate, b5\n is taken as −0.007, −0.375, or − 1.362, respectively.
For overtopping dam failure, kM\n = 1.85; for seepage failure dam failure, kM\n = 1; when hb\n ≤ 6.1 m,kH\n = 1; when hb\n > 6.1 m, kH\n = (hb\n/6.1)1/8.
\nQp\n is the peak outflow of the breach; hw\n is the water depth above the bottom of the breach when the dam breaks; hd\n is the height of the dam; S is the reservoir capacity; Vw\n is the reservoir capacity above the bottom of the breach when the dam breaks; g is the gravity acceleration; hb\n is the depth of the dam breaks; hr\n is the reference dam height, take 15 m; V is the reservoir capacity at dam breaching; h is the water level at dam breaching; L is the length of the dam; Wave\n is the average width of the dam; kM\n and kH\n are coefficients.
In 1988, the US Bureau of Reclamation (USBR) [14] proposed the first empirical formula for predicting the final average width of the breach Bave\n, and then scholars from various countries put forward a series of models. The commonly used parameter model of the final average width of the breach is shown in Table 2.
\nModel | \nCase number | \nExpression | \n
---|---|---|
USBR (1988) [14] | \n21 | \n\nBave\n = 3hw\n\n | \n
Von Thun and Gillette\n1\n (1990) [25] | \n57 | \n\nBave\n = 2.5 hw\n + Cb\n\n | \n
Froehlich\n2\n (1995) [26] | \n22 | \n\nBave\n = 0.1803 K\n0(Vw\n)0.32(hb\n)0.19\n | \n
Xu and Zhang\n3\n (2009) [17] | \n75 | \n\nBave\n = 0.787(hb\n)(hd\n/hr\n)0.133(Vw\n\n1/3/hw\n)0.652\neB\n\n3\n | \n
Froehlich\n4\n (2016) [23] | \n41 | \n\nBave\n = 0.27kM\n(Vw\n)1/3\n | \n
Mei Shiang et al. (2018) [24] | \n154 | \n\nBave = hb\n(Vw\n\n1/3/hw\n)0.84(hw\n/hb\n)2.30(hd\n/h\n0)0.06e−0.90(homogeneous dam) \nBave = hb\n(Vw\n\n1/3/hw\n)0.55(hw\n/hb\n)1.97(hd\n/h\n0)−0.07e−0.09(core-wall dam) | \n
Parameter model of the final average width of the breach.
When S < 1.2335 × 106 m3, Cb\n = 6.096; when 1.2335 × 106 m3 ≤ S < 6.1676 × 106 m3, Cb\n = 18.288; when 6.1676 × 106 m3 ≤ S < 1.2335 × 107 m3, Cb\n = 42.672; when S ≥ 1.2335 × 107 m3, Cb\n = 54.864.
For overtopping dam failure, K\n0 = 1.4; for seepage failure dam breaching, K\n0 = 1.0.
\nhr\n is the dam height, which is 15 m; the expression of parameter B\n3 is B\n3 = b\n3 + b\n4 + b\n5, for core-wall dam, concrete face rockfill dam or homogeneous dam, b\n3 takes −0.041, 0.026 or 0.226; for overtopping or seepage failure, b\n4 = 0.149 or − 0.389, respectively; for dams with high, medium, or low erosion rate, b5\n is 0.291, 0.14, or 0.391, respectively.
For overtopping dam failure, kM\n = 1.3; for seepage failure dam breaching, kM\n = 1.0.
In 1984, MacDonald and Langridge-Monopolis [42] proposed the first empirical formula for predicting the duration of dam failure, and then scholars from various countries proposed a series of models. Commonly used dam-break duration parameter model is shown in Table 3.
\nModel | \nCase number | \nExpression | \n
---|---|---|
MacDonald and Langridge-Monopolis (1984) [11] | \n23 | \n\nTf\n = 0.0179(0.0261(Vwhw\n)0.769)0.364\n | \n
USBR (1988) [14] | \n21 | \n\nTf\n = 0.011Bave\n\n | \n
Froehlich (1995) [26] | \n22 | \n\nTf\n = 0.00254(Vw\n)0.53(hb\n)−0.9\n | \n
Xu and Zhang\n1\n (2009) [17] | \n75 | \n\nTf\n = 0.304Tr\n(hd\n/hr\n)0.707(Vw\n\n1/3/hw\n)1.228\neB\n\n5\n | \n
Froehlich (2016) [23] | \n41 | \n\nTf\n = 63.2(Vw\n/(ghb\n\n2))0.5\n | \n
Mei Shiang et al.\n2\n (2018) [24] | \n154 | \n\nTf = T\n0(Vw\n\n1/3/hw\n)0.56(hw\n/hb\n)−0.85(hd\n/h\n0)−0.32e−0.20(homogeneous dam) \nTf = T\n0(Vw\n\n1/3/hw\n)1.52(hw\n/hb\n)−11.36(hd\n/h\n0)−0.43e−1.57(core-wall dam) | \n
Dam-break duration parameter model.
Tr means the duration of the reference dam break, take 1 h; the expression of parameter B5 is B5 = b3 + b4 + b5, for core-wall dam, concrete face rockfill dam, or homogeneous dam, b3 takes −0.327, −0.674, or − 0.189; for overtopping or seepage failure, b4 = −0.579 or − 0.611, respectively; for dam materials with high, medium, or low erosion rate, b5 is −1.205, −0.564, or 0.579 respectively.
T0 means unit duration, take 1 h.
Due to the difficulties in obtaining the dam-break duration, the relatively low accuracy of the data, and the small number of samples, the dam-break duration model has a large deviation in the calculation of individual cases.
\nIn order to fully consider the dam type, dam breach mode, reservoir characteristics, and breach characteristics, the reservoir capacity (Vw\n) is above the bottom of the breach at the dam break, the water depth (hw\n) above the bottom of the dam at the dam break (hd\n), and the final depth of the rupture (hb\n). For other parameters, the method of statistical regression is used to obtain the results of the peak flow of the breach, the final average width of the breach, and the duration of the dam breach. From the above statistics, it can be seen that the parameter model can simulate the dam-break parameters simply and quickly, which is an efficient and rapid evaluation method, but the parameter model cannot provide the dam-break flood flow process line.
\nIn the 1960s, European and American scholars began to study a simplified mathematical model based on the mechanism of collapse based on hydraulics and sediment transport formulas. This model is also the most widely used mathematical model of earth-rock dams’ breaching. In 1965, from the US Bureau of Reclamation, Cristofano [27] established the first mathematical model of homogeneous dam overtopping failure. Afterward, scholars from various countries proposed a series of mathematical models for simulating earth-rock dam collapse [6, 28]. The most widely used is the NWS BREACH model developed by Fread from the National Weather Service [29]. In recent years, the Nanjing Hydraulic Research Institute and China Institute of Water Resources and Hydropower Research have conducted systematic research work on the mathematical model of earth-rock dams’ breaching, establishing NHRI-DB series and DB-IWHR series dam-break mathematical models, respectively. The commonly used simplified mathematical model of earth-rock dam breaching is shown in Table 4.
\nModel | \nShape of breach | \nThe flow of the breach | \nErosion formula | \nMechanical analysis | \nBreach mode | \nType of dam | \n
---|---|---|---|---|---|---|
Cristofano (1965) [27] | \nTrapezoid | \nWide crest weir formula | \nCristofano formula | \nBreach without lateral collapse | \nOvertopping | \nHomogeneous | \n
BRDAM (1981) [30] | \nParabolic | \nWide crest weir formula, vent flow formula | \nSchoklitsch formula | \nBreach without lateral collapse (overtopping), top collapse (seepage) | \nOvertopping or seepage | \nHomogeneous | \n
DAMBRK (1984) [31] | \nTrapezoid or rectangle | \nWide crest weir formula | \nEven flush | \nBreach without lateral collapse | \nOvertopping | \nHomogeneous | \n
BEED (1985) [32] | \nTrapezoid | \nWide crest weir formula | \nEinstein and Brown formula, Meyer-Peter-Mueller formula | \nCollapse laterally | \nOvertopping | \nHomogeneous | \n
NWS BREACH (1988) [29] | \nTrapezoid or rectangle | \nWide crest weir formula, vent flow formula | \nCorrection Meyer-Peter-Mueller formula | \nCollapse laterally (overtopping), top collapse (seepage) | \nOvertopping or seepage | \nHomogeneous, core wall | \n
HR BREACH (2002, 2009) [33, 34] | \nEffective stress method | \n1D stable non-uniform weir flow formula | \nSediment transport formula or erosion rate formula | \nSingle (two) side erosion, collapse laterally, stability analysis of core wall | \nOvertopping or seepage | \nHomogeneous earth dam, core wall | \n
FIREBIRD (2006) [35] | \nTrapezoid | \nUnsteady Saint-Venant equation | \nSediment transport formula or erosion rate formula | \nCollapse laterally | \nOvertopping | \nHomogeneous | \n
WinDAM/SIMBA (2005, 2006, 2010) [36, 37, 38] | \nRectangle | \nWide crest weir formula | \nErosion rate formula | \nBreach without lateral collapse | \nOvertopping | \nHomogeneous | \n
DLBreach (2013) [40] | \nTrapezoid | \nWide crest weir formula, vent flow formula | \nSediment transport formula or erosion rate formula | \nSingle (two) side erosion, collapse laterally, stability analysis of core wall, dam foundation erosion | \nOvertopping or seepage | \nHomogeneous, core wall | \n
Hong Kong University of Science and Technology model [39] | \nTrapezoid | \nWide crest weir formula, vent flow formula | \nErosion rate formula | \nCollapse laterally (overtopping), top collapse (seepage) | \nOvertopping or seepage | \nHomogeneous, landslide | \n
DB-IWHR series dam-break mathematical model of China Institute of Water Resources and Hydropower Research [41, 42, 43] | \nTrapezoid | \nWide crest weir formula | \nErosion rate formula | \nCollapse laterally | \nOvertopping | \nHomogeneous, core wall, landslide dam | \n
NHRI-DB series dam-break mathematical model of Nanjing Hydraulic Research Institute [44, 45, 46, 47, 48] | \nTrapezoid | \nWide crest weir formula, vent flow formula | \nSediment transport formula or erosion rate formula | \nShearing or dumping of the core wall, panel break, collapse laterally (overtopping), top collapse (seepage) | \nOvertopping or seepage | \nHomogeneous, core wall, face dam, landslide | \n
Simplified mathematical model of earth-rock dam breaching.
It can be seen from the above analysis that this type of model is mainly aimed at the two failure modes of earth-rock dam overtopping and seepage failure. By assuming the shape of the breach, different flow calculation formulas and erosion formulas are used to simulate the scouring of the dam material, and different simulation methods are used to analyze the vertical undercut and lateral expansion of the breach. Most of the models use iterative numerical calculation methods based on time steps to simulate the process of dam break and can output the parameters of dam break (such as the flow of the breach, the size of the breach, the water level of the reservoir, etc.) at each time step.
\nFor example, based on the overtopping breach mechanism of the clay-core wall dam, a mathematical model to simulate its breach process is proposed. The model is based on the shape of the dam body and the characteristics of the flood flow to determine the initial scoring position of the downstream slope during erosion. The flow formula of the wide crested weir is used to calculate the rupture flow. The mechanical equilibrium method is used to simulate the tipping and shear failure of the core wall; the model can also consider the erosion of the dam body on one side, the erosion on both sides, and the erosion of the dam foundation and the process of water and soil coupling during dam break.
\nBased on the mechanism revealed by the model test of the overtopping breach of the homogeneous cohesive earth dam, the author has established a mathematical model that can simulate its collapse process (Figure 5). The specific modules of the model are as follows.
\nSchematic diagram of the author’s model calculation process. (1) Breach formation, (2) Scarp formation, (3) Scarp widen, (4) Headcut scour, (5) Breach widen, (6) Breach fully formed.
This model is based on the shape of the dam body and the characteristics of the flow at the top of the crater to determine the formation position of the “dark ridge.” The traceable erosion formula that can consider the physical and mechanical characteristics of the dam material is used to simulate the movement of the “dark ridge.” The collapse of the dam body: choose a reasonable erosion formula of the dam material to simulate the development of the dam crest and the downstream slope failure, and use the limit equilibrium method to simulate the failure of the collapse slope. The model considers incomplete dam failure and erosion of the dam foundation, as well as erosion on one side and both sides of the dam body.
\nThe flow chart of the model calculation process of the collapse process of the homogeneous earth dam is shown in Figure 6.
\nCalculation flow chart of the process of overburden collapse of the homogeneous earth dam.
There are two major highlights of the NHRI-DB concrete-face dam-break mathematical model [47]: the adoption of total-load nonequilibrium transport equation (Eq. (1)) [72] to simulate the erosion process of sand gravels with a wide range of gradation and the establishment of an analogy to simulate the failure process of each concrete-face slab under various loads during the dam breaching process.
\nwhere t = time; x = longitudinal coordinate; A = cross-sectional flow area in the breach channel; Ct\n = actual total-load sediment concentration; Ct\n = sediment concentration at the equilibrium state; and L\ns = adaptation length characterizing the adjustment of sediment from a nonequilibrium state to equilibrium state.
\nIn the NHRI-DB core dam-break mathematical model [45], a hydraulic method was used to predict the initial scour position for high dam. A time averaged erosion equation was adopted to simulate the backward erosion of dam’s shoulder. The broad-crested weir equation (Eqs. (2) and (3)) [73, 74] was adopted to calculate the breach flow discharge. Furthermore, the sliding or overturning failure was adopted as the key mechanism for the core, which was judged based upon numerical analysis. The calculated results show that the proposed model gives reasonable peak outflow, final breach width, and failure time.
\nwhere Bb\n is the bottom width of the breach (m), H represents the difference “zs − zb\n” (m), in which zb\n is the elevation of the breach bottom (m), m is the side slope (horizontal/vertical) of the breach, c1\n and c2\n are the discharge coefficients with values of 1.7 m0.5/s and 1.3 m0.5/s [73], and ksm is the submergence correction factor for tailwater effects on weir outflow.
\nwhere zt\n is the tailwater level (m).
\nThe advantage of this type of model is that it can consider the failure mechanism of the earth-rock dam and can use a short calculation time to complete the simulation of the dam-break process; however, most models cannot really consider the water-soil coupling effect during the dam-break process.
\nIn order to fully describe the water-soil coupling effect in the process of dams’ breaching, in recent years, with the improvement of computer performance and the development of sediment science and computational fluid dynamics, a series of nonequilibrium dam material transport theory has emerged based on shallow water hypothetical detailed mathematical model for dam failure [49]. The commonly used detailed mathematical model of earth-rock dams’ breaching is shown in Table 5.
\nModel | \nDetermination method of breach shape | \nFlow of breach | \nDam material erosion | \nMechanical analysis | \nCalculation method | \n
---|---|---|---|---|---|
Wang and Bowles (2006) [50] | \nScour without sediment motion | \nShallow water equations | \nErosion rate formula | \nThree-dimensional collapse laterally | \nFinite different method | \n
Faeh (2007) [51] | \nTwo-dimensional Exner equations | \nShallow water equations | \nTraction load and suspended load formula | \nCollapse laterally | \nFinite volume method | \n
Wu et al. (2007, 2012) [52, 53] | \nOne- and two-dimensional nonequilibrium total sand transport equations | \nGeneral shallow water equations | \nTotal sand transport formula | \nCollapse laterally | \nFinite volume method | \n
Swartenbroekx et al. (2010) [54] | \nTwo-dimensional Exner equations | \nShallow water equations | \nTraction load formula | \nCollapse laterally | \nFinite volume method | \n
Li et al. (2011) [55] | \nTwo-dimensional nonequilibrium sediment transport equations (suspended load) | \nShallow water equations | \nEmpirical formulas of sediment carrying rate and Sedimentation rate | \nBreach without lateral collapse | \nFinite volume method | \n
Cao et al. (2011) [56] | \nTwo-dimensional nonequilibrium total sediment transport equations | \nGeneral shallow water equations | \nTraction load formula | \nCollapse laterally | \nFinite volume method | \n
Rosatti and Begnudelli (2013) [57, 58] | \nTwo-dimensional mass conservation and energy conservation equations (solid phases) | \nShallow water equations (liquid) | \nFloe concentration formula | \nBreach without lateral collapse | \nFinite volume method | \n
Juez et al. (2013, 2014) [59, 60] | \nOne- and two-dimensional Exner equations | \nSaint-Venant equations and shallow water equations | \n10 different erosion formulas | \nBreach without lateral collapse | \nFinite volume method | \n
Swartenbroekx et al. (2013) [61] | \nTwo-dimensional mass conservation and energy conservation equations (traction load) | \nShallow water equations (clean water) | \nErosion rate formula | \nBreach without lateral collapse | \nFinite volume method | \n
Guan et al. (2014) [62] | \nTwo-dimensional nonequilibrium sediment transport equations (traction load) | \nShallow water equations (pure water) | \nTraction load formula | \nCollapse laterally | \nFinite volume method | \n
Kesserwani et al. (2014) [63] | \nTwo-dimensional nonequilibrium sediment transport equations (suspended load) | \nShallow water equations | \nEmpirical formulas of sediment carrying rate and sedimentation rate | \nBreach without lateral collapse | \nIntermittent Galerkin method | \n
Razavitoosi et al. (2014) [64] | \nN-S equations (solid phases, non-Newtonian fluid) | \nN-S equations (liquid, non-Newtonian fluid) | \n/ | \nBreach without lateral collapse | \nSmoothed particle hydrodynamics method | \n
Marsooli and Wu (2015) [65] | \nThree-dimensional nonequilibrium sediment transport equations | \nN-S equations | \nTraction load and suspended load formula | \nBreach without lateral collapse | \nFinite volume method and volume of fluid | \n
Abderrezza et al. (2016) [66] | \nTwo-dimensional Exner equations | \nShallow water equations | \nTraction load formula | \nCollapse laterally | \nFinite volume method | \n
Cantero-Chinchilla et al. (2016) [67] | \nOne-dimensional nonequilibrium sediment transport equations | \nSaint-Venant equations, vertical momentum equation | \nTraction load and suspended load formula | \nBreach without lateral collapse | \nFinite volume method | \n
Cristo et al. (2016, 2018) [68, 69] | \nTwo-dimensional mass conservation and energy conservation equations (solid phase) | \nShallow water equations (liquid) | \nTraction load formula | \nBed collapse algorithm | \nFinite volume method | \n
YAN Zhikun et al. (2019) [70] | \nTwo-dimensional nonequilibrium sediment transport equations | \nGeneral shallow water equations | \nTotal sand transport formula considering bed slope | \nBed collapse algorithm | \nFinite volume method | \n
Detailed mathematical model of earth-rock dams’ breaching.
It can be seen from the above statistics that this type of model is mainly based on the continuity equations of water flow (Eq. (4)), momentum equations (Eq. (5)), and energy equations (Eq. (6)), coupled with the sediment movement equation, and the finite volume method and other numerical simulation methods are used to discretely solve the governing equations
\nIn Yan Zhikun’s model [70], based on the continuity equations of water flow, momentum equations, and nonequilibrium sediment transport equations, a planar two-dimensional mathematical model of dam rupture along the depth average is proposed. The sand capacity and the collapse mechanism of the two-dimensional slope during the dam-break process. The fully coupled method is used to convert the hydrodynamic equation and the nonequilibrium sediment transport equation into a shallow water equation with source terms and is based on the finite volume method under a rectangular grid. Discrete processing, using conservative, non-negative water depth numerical reconstruction format to make the model have second-order accuracy in the space–time direction, using HLLC [71] approximate Riemann solver to calculate grid boundary flux, SGM (Surface Gradient Method) format to calculate water surface gradient source terms, semi-implicit format. For the bottom bed friction term, the explicit gradient calculation of the source term of the concentration gradient is used to numerically solve the control equation.
\nSuch models can achieve detailed simulation of the dam-break process, but the calculation speed is slow, and it can only be used for the numerical simulation of the overtopping dams’ breaching. However, this method can fully consider the coupling effect of water-soil coupling in the process of dam failure and can simulate complex boundary conditions, which is the development direction of numerical simulation of earth-rock dams’ breaching.
\nEarth-rock dams’ breaching mechanism and dam-break process simulation are the foundation of dam-break disaster assessment and emergency response. They involve fluid mechanics, sediment kinematics, soil mechanics, and other disciplines. They are complex water-soil coupling problems. After decades of research and exploration, various mathematical models of dams’ breaching have been developed and made a series of innovative achievements, which provide theoretical support for improving the accuracy of flood disaster prediction of earth-rock dams. It is suggested that in the future, research efforts should be intensified on the mathematical model of the detailed simulation of the earth-rock dam breaching process, focusing on the application of visualization technology in the simulation of the dam-break process and accelerating the development of a universal and friendly simulation of the earth-rock dams’ breaching and the visual calculation of the disaster-causing process software.
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