Comparing the Diwekar [11] and Narváez-García et al. [10] shortcut method using variable reflux policy.
\r\n\tThe study of populations and plant communities in their different aspects; ecological, structural, functional and dynamic, it is essential to establish a posteriori models of forest and agricultural management.
\r\n\r\n\tFor this, the methodological approaches on the type of sampling are considered essential, since there are differences between the purely ecological and the phytosociological methods, despite the fact that both pursue the same objective.
\r\n\tAlthough the ecological method for the knowledge of the vegetation is widely extended, the phytosociological one is no less so, since in the European Union it has been developed as a consequence of policies on sustainability, through which regulations have been issued, such as the habitats directive.
\r\n\tOn the other hand, research on plant dynamics and knowledge of the landscape in an integral way, have multiplied in the last 30 years, which has favored a deep knowledge of the floristic and phytocenotic wealth, which is fundamental for agricultural management, livestock and forestry.
",isbn:"978-1-83969-386-1",printIsbn:"978-1-83969-385-4",pdfIsbn:"978-1-83969-387-8",doi:null,price:0,priceEur:0,priceUsd:0,slug:null,numberOfPages:0,isOpenForSubmission:!0,hash:"0abf2a59ee63fc1ba4fb64d77c9b1be7",bookSignature:"Dr. Eusebio Cano Carmona, Dr. Ricardo Quinto Canas, Dr. Ana Cano Ortiz and Dr. Carmelo Maria Musarella",publishedDate:null,coverURL:"https://cdn.intechopen.com/books/images_new/9662.jpg",keywords:"Climatic Factors, Bioclimate, Thermotype, Flora, Conservation, Phytocenosis, Plant Dynamics, Landscape, Cartography, Vegetation Series, Crops, Reforestation",numberOfDownloads:null,numberOfWosCitations:0,numberOfCrossrefCitations:null,numberOfDimensionsCitations:null,numberOfTotalCitations:null,isAvailableForWebshopOrdering:!0,dateEndFirstStepPublish:"November 23rd 2020",dateEndSecondStepPublish:"January 25th 2021",dateEndThirdStepPublish:"March 26th 2021",dateEndFourthStepPublish:"June 14th 2021",dateEndFifthStepPublish:"August 13th 2021",remainingDaysToSecondStep:"a month",secondStepPassed:!0,currentStepOfPublishingProcess:3,editedByType:null,kuFlag:!1,biosketch:"Dr. Cano Carmona and colleagues have directed 12 doctoral theses and more than 200 publications among articles, books, and book chapters. He has participated in national and international congresses with about 250 papers. He has held a number of different academic positions, including Dean of the Faculty of Experimental Sciences at the University of Jaen, Spain, and founder and director of the International Seminar on Management and Conservation of Biodiversity.",coeditorOneBiosketch:"Ricardo Jorge Quinto Canas is currently an Invited Assistant Professor in the Faculty of Sciences and Technology at the University of Algarve – Portugal, and a member of the Centre of Marine Sciences (CCMAR), University of Algarve. His current research projects focus on Botany, Vegetation Science (Geobotany), Biogeography, Plant Ecology, and Biology Conservation, aiming to support Nature Conservation.",coeditorTwoBiosketch:"Ana Cano Ortiz's fundamental line of research is related to botanical bioindicators. She has worked in Spain, Italy, Portugal, and Central America. It presents more than one hundred works published in various national and international journals, as well as books and book chapters; and has presented a hundred papers to national and international congresses.",coeditorThreeBiosketch:"Carmelo Maria Musarella is a biologist, specialized in Plant Biology. He is a member of the permanent scientific committee of the International Seminar on “Biodiversity Conservation and Management” guested by several European universities. He has participated in several international and national congresses, seminars, and workshops and presented oral communications and posters.",coeditorFourBiosketch:null,coeditorFiveBiosketch:null,editors:[{id:"87846",title:"Dr.",name:"Eusebio",middleName:null,surname:"Cano Carmona",slug:"eusebio-cano-carmona",fullName:"Eusebio Cano Carmona",profilePictureURL:"https://mts.intechopen.com/storage/users/87846/images/system/87846.png",biography:"Eusebio Cano Carmona obtained a PhD in Sciences from the\nUniversity of Granada, Spain. He is Professor of Botany at the\nUniversity of Jaén, Spain. His focus is flora and vegetation and he\nhas conducted research in Spain, Italy, Portugal, Palestine, the\nCaribbean islands and Mexico. As a result of these investigations,\nDr. Cano Carmona and colleagues have directed 12 doctoral theses\nand more than 200 publications among articles, books and book\nchapters. He has participated in national and international congresses with about\n250 papers/communications. He has held a number of different academic positions,\nincluding Dean of the Faculty of Experimental Sciences at the University of Jaen,\nSpain and founder and director of the International Seminar on Management and\nConservation of Biodiversity, a position he has held for 13 years. He is also a member of the Spanish, Portuguese and Italian societies of Geobotany.",institutionString:"University of Jaén",position:null,outsideEditionCount:0,totalCites:0,totalAuthoredChapters:"5",totalChapterViews:"0",totalEditedBooks:"1",institution:{name:"University of Jaén",institutionURL:null,country:{name:"Spain"}}}],coeditorOne:{id:"216982",title:"Dr.",name:"Ricardo Quinto",middleName:null,surname:"Canas",slug:"ricardo-quinto-canas",fullName:"Ricardo Quinto Canas",profilePictureURL:"https://mts.intechopen.com/storage/users/216982/images/system/216982.JPG",biography:"Ricardo Quinto Canas, Phd in Analysis and Management of Ecosystems, is currently an Invited Assistant Professor in the Faculty\nof Sciences and Technology at the University of Algarve, Portugal, and member of the Centre of Marine Sciences (CCMAR),\nUniversity of Algarve. He is also the Head of Division of Environmental Impact Assessment - Algarve Regional Coordination\nand Development Commission (CCDR - Algarve). His current\nresearch projects focus on Botany, Vegetation Science (Geobotany), Biogeography,\nPlant Ecology and Biology Conservation, aiming to support Nature Conservation.\nDr. Quinto Canas has co-authored many cited journal publication, conference articles and book chapters in above-mentioned topics.",institutionString:"University of Algarve",position:null,outsideEditionCount:0,totalCites:0,totalAuthoredChapters:"3",totalChapterViews:"0",totalEditedBooks:"0",institution:null},coeditorTwo:{id:"203697",title:"Dr.",name:"Ana",middleName:null,surname:"Cano Ortiz",slug:"ana-cano-ortiz",fullName:"Ana Cano Ortiz",profilePictureURL:"https://mts.intechopen.com/storage/users/203697/images/system/203697.png",biography:"Ana Cano Ortiz holds a PhD in Botany from the University of\nJaén, Spain. She has worked in private enterprise, in university\nand in secondary education. She is co-director of four doctoral\ntheses. Her research focus is related to botanical bioindicators.\nDr. Ortiz has worked in Spain, Italy, Portugal and Central America. She has published more than 100 works in various national\nand international journals, as well as books and book chapters.\nShe has also presented a great number of papers/communications to national and\ninternational congresses.",institutionString:"University of Jaén",position:null,outsideEditionCount:0,totalCites:0,totalAuthoredChapters:"6",totalChapterViews:"0",totalEditedBooks:"0",institution:{name:"University of Jaén",institutionURL:null,country:{name:"Spain"}}},coeditorThree:{id:"276295",title:"Dr.",name:"Carmelo Maria",middleName:null,surname:"Musarella",slug:"carmelo-maria-musarella",fullName:"Carmelo Maria Musarella",profilePictureURL:"https://mts.intechopen.com/storage/users/276295/images/system/276295.jpg",biography:"Carmelo Maria Musarella, PhD (Reggio Calabria, Italy –\n23/01/1975) is a biologist, specializing in plant biology. He\nstudied and worked in several European Universities: Messina,\nCatania, Reggio Calabria, Rome (Italy), Valencia, Jaén, Almeria\n(Spain), and Evora (Portugal). He was the Adjunct Professor\nof Plant Biology at the “Mediterranea” University of Reggio\nCalabria (Italy). His research topics are: floristic, vegetation,\nhabitat, biogeography, taxonomy, ethnobotany, endemisms, alien species, and\nbiodiversity conservation. He has authored many research articles published in\nindexed journals and books. He has been the guest editor for Plant Biosystems and a\nreferee for this same journal and others. He is a member of the permanent scientific\ncommittee of International Seminar on “Biodiversity Conservation and Management”, which includes several European universities. He has participated in several\ninternational and national congresses, seminars, workshops, and presentations of\noral communications and posters.",institutionString:'"Mediterranea" University of Reggio Calabria',position:null,outsideEditionCount:0,totalCites:0,totalAuthoredChapters:"6",totalChapterViews:"0",totalEditedBooks:"1",institution:null},coeditorFour:null,coeditorFive:null,topics:[{id:"5",title:"Agricultural and Biological Sciences",slug:"agricultural-and-biological-sciences"}],chapters:null,productType:{id:"1",title:"Edited Volume",chapterContentType:"chapter",authoredCaption:"Edited by"},personalPublishingAssistant:{id:"247865",firstName:"Jasna",lastName:"Bozic",middleName:null,title:"Ms.",imageUrl:"https://mts.intechopen.com/storage/users/247865/images/7225_n.jpg",email:"jasna.b@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:"6893",title:"Endemic Species",subtitle:null,isOpenForSubmission:!1,hash:"3290be83fff5bc015f5bd3d78ae9c6c7",slug:"endemic-species",bookSignature:"Eusebio Cano Carmona, Carmelo Maria Musarella and Ana Cano Ortiz",coverURL:"https://cdn.intechopen.com/books/images_new/6893.jpg",editedByType:"Edited by",editors:[{id:"87846",title:"Dr.",name:"Eusebio",surname:"Cano Carmona",slug:"eusebio-cano-carmona",fullName:"Eusebio Cano Carmona"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"6418",title:"Hyperspectral Imaging in Agriculture, Food and Environment",subtitle:null,isOpenForSubmission:!1,hash:"9005c36534a5dc065577a011aea13d4d",slug:"hyperspectral-imaging-in-agriculture-food-and-environment",bookSignature:"Alejandro Isabel Luna Maldonado, Humberto Rodríguez Fuentes and Juan Antonio Vidales Contreras",coverURL:"https://cdn.intechopen.com/books/images_new/6418.jpg",editedByType:"Edited by",editors:[{id:"105774",title:"Prof.",name:"Alejandro Isabel",surname:"Luna Maldonado",slug:"alejandro-isabel-luna-maldonado",fullName:"Alejandro Isabel Luna Maldonado"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"1591",title:"Infrared Spectroscopy",subtitle:"Materials Science, Engineering and Technology",isOpenForSubmission:!1,hash:"99b4b7b71a8caeb693ed762b40b017f4",slug:"infrared-spectroscopy-materials-science-engineering-and-technology",bookSignature:"Theophile Theophanides",coverURL:"https://cdn.intechopen.com/books/images_new/1591.jpg",editedByType:"Edited by",editors:[{id:"37194",title:"Dr.",name:"Theophanides",surname:"Theophile",slug:"theophanides-theophile",fullName:"Theophanides 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"}},{type:"book",id:"72",title:"Ionic Liquids",subtitle:"Theory, Properties, New Approaches",isOpenForSubmission:!1,hash:"d94ffa3cfa10505e3b1d676d46fcd3f5",slug:"ionic-liquids-theory-properties-new-approaches",bookSignature:"Alexander Kokorin",coverURL:"https://cdn.intechopen.com/books/images_new/72.jpg",editedByType:"Edited by",editors:[{id:"19816",title:"Prof.",name:"Alexander",surname:"Kokorin",slug:"alexander-kokorin",fullName:"Alexander Kokorin"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"1373",title:"Ionic Liquids",subtitle:"Applications and Perspectives",isOpenForSubmission:!1,hash:"5e9ae5ae9167cde4b344e499a792c41c",slug:"ionic-liquids-applications-and-perspectives",bookSignature:"Alexander Kokorin",coverURL:"https://cdn.intechopen.com/books/images_new/1373.jpg",editedByType:"Edited by",editors:[{id:"19816",title:"Prof.",name:"Alexander",surname:"Kokorin",slug:"alexander-kokorin",fullName:"Alexander Kokorin"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"57",title:"Physics and Applications of Graphene",subtitle:"Experiments",isOpenForSubmission:!1,hash:"0e6622a71cf4f02f45bfdd5691e1189a",slug:"physics-and-applications-of-graphene-experiments",bookSignature:"Sergey Mikhailov",coverURL:"https://cdn.intechopen.com/books/images_new/57.jpg",editedByType:"Edited by",editors:[{id:"16042",title:"Dr.",name:"Sergey",surname:"Mikhailov",slug:"sergey-mikhailov",fullName:"Sergey Mikhailov"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"371",title:"Abiotic Stress in Plants",subtitle:"Mechanisms and Adaptations",isOpenForSubmission:!1,hash:"588466f487e307619849d72389178a74",slug:"abiotic-stress-in-plants-mechanisms-and-adaptations",bookSignature:"Arun Shanker and B. Venkateswarlu",coverURL:"https://cdn.intechopen.com/books/images_new/371.jpg",editedByType:"Edited by",editors:[{id:"58592",title:"Dr.",name:"Arun",surname:"Shanker",slug:"arun-shanker",fullName:"Arun Shanker"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"878",title:"Phytochemicals",subtitle:"A Global Perspective of Their Role in Nutrition and Health",isOpenForSubmission:!1,hash:"ec77671f63975ef2d16192897deb6835",slug:"phytochemicals-a-global-perspective-of-their-role-in-nutrition-and-health",bookSignature:"Venketeshwer Rao",coverURL:"https://cdn.intechopen.com/books/images_new/878.jpg",editedByType:"Edited by",editors:[{id:"82663",title:"Dr.",name:"Venketeshwer",surname:"Rao",slug:"venketeshwer-rao",fullName:"Venketeshwer Rao"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}}]},chapter:{item:{type:"chapter",id:"54045",title:"Short-Cut Methods for Multicomponent Batch Distillation",doi:"10.5772/66830",slug:"short-cut-methods-for-multicomponent-batch-distillation",body:'Batch distillation is a process widely used for separation of small quantities of chemical compound of the one mixture as the recovery of small quantities of hazardous materials in waste streams, recovery of solvents, as well as, for pharmaceutical and biotechnological products with high added value, among others. Therefore, the development of mathematical models for the prediction of a process has a high interest in recent times [1–4].
Batch distillation is a flexible process because one equipment can obtain the separation of all the components of the mixture, while the continuous process [5] requires a number of columns distillation equal to the number of components minus one (n − 1). Another advantage of batch distillation process is the use of the same equipment for the mixture separation with different compositions or different mixtures [6].
On the other hand, the disadvantage of the batch distillation with respect to continuous distillation is that only small amounts of products can be obtained of the mixture. Another disadvantage is the production of waste unwanted for each cuts, however, these residual cuts can be separated into the same column [7].
A batch distillation column can be operated using any of the following policies [1]:
Constant reflux.
Variable reflux.
Optimal reflux.
Reflux profile.
The process behavior can be predicted by developing mathematical models based on mass and energy balances. The mathematical models obtained can be classified as [1, 8, 9]:
Simplified (shortcut method).
Semirigorous.
Rigorous.
Order reduction.
Currently the rigorous models have an area of great interest and these require especially the use of computers with high accuracy and processing capabilities; however, simplified methods can be applied with the use of the equipment such as tablets, smart phones, and/or laptops with smaller capacity of data processing, which makes possible the search for predicting the behavior of the process[2, 10]. In addition, the use of this kind of methods is a tool for obtaining initial data for the mathematical optimization.
Unlike the rigorous methods that considered the dynamics of the complete column, the shortcut methods are mathematical models that predict the behavior of the process considering the least amount of equations, usually making an overall material balance and partial balances considering a component any “i.” The main limiting factor of these shortcut methods is to find a functional relationship between the concentrations of the dome and the bottom.
The shortcut methods are justified because these require a minor calculus time and memory requirements, as well as, an acceptable accuracy in the results obtained with respect to the rigorous method. These are an appropriate tool to obtain initial values for the mathematical optimization of some process, when the complexity of the methods required data very close to the solution.
The shortcut method is also used for the columns design and obtaining of limit conditions as minimum reflux ratio, Rmin and minimum number of stage Nmin. On the other hand, the shortcut methods are very simple to apply and to program, therefore, are useful in the teaching-learning process.
The two most important shortcut methods reported in the literature made use of the Fenske-Underwood-Gilliland (FUG) method developed for continuous distillation, but considering that the feed changes at every instant; that is, the bottom product in the current time is the feed for the next time (step).
The first of the shortcut methods was developed by Diwekar [11] and reported in the literature by Diwekar and Madhavan [12]. This method was developed considering the policies of constant and variable reflux. This method used the Hengstebeck-Geddes equation. This method also performs the comparison between the values of the minimum reflux ratio of Underwood and minimum reflux ratio of Gilliland, which increases the computational time because it uses an additional iterative process.
The second method was reported by Sundaram and Evans [13] and only considered the constant reflux policy and the Fenske Equation. The model obtains a solution in two parts; an outer loop that solves material balances and internal one that solves the functional relationship between the compositions of the dome and the bottom using the FUG method. The mathematical model developed initially considered:
Constant relative volatilities.
Constant molar flow.
Negligible vapor and liquid accumulation in trays and the condenser.
Based on the work of Sundaram and Evans [13], Narváez-García et al. [10] developed a mathematical model for batch distillation process using a variable reflux policy.
The present studies show the most important shortcut methods used to predict the behavior of the batch distillation process.
For the use of the Underwood equations, this work considered separations Class I and Class II. In according to Shiras et al. [14] defined Class I and Class II as follows:
Class I: “Separations such that, with infinite plates, all components of the feed are present in both the top product and bottom product.”
Class II: “Separations such that, with infinite plates, some of the components are completely in the top product or completely in the bottom product.”
Similarly, an important concept in model developments is the key component light (lk) and heavy key component (hk) defined as follows:
Light key component (lk): Light component that is present in the residue in important amounts.
Heavy key component (hk): Heavy component that is present in the distillate in important amounts.
When a fraction of the product obtained is fed back into the process and this can be done on four operations of the process: (1) constant reflux, (2) variable reflux, (3) optimum reflux, and (4) profile of reflux. In either case, the reflux ratio (R) is defined as
where L is the reflux in the dome and D the product flow.
For some type of reflux used, it should be considered if there is accumulation of liquid and vapor in each of the trays as well as in the reflux tank. Another aspect that should be considered is where the initial feed is introduced because when it is performed from the reflux tank, the accumulation in each of the stages is equal to feed initial concentration, if conversely, the feed is introduced in the reboiler and column is operated without reflux, the concentration of each of the trays is equal to the concentration of the vapor phase and will be in equilibrium with the feed [15].
Constant reflux: In constant reflux policy, the product concentration varies with time because new feed input does not exist, so that the initial mole fractions of the more volatile are depleted and the molar fraction of the final distillate is an average.
Variable reflux: The batch distillation process with a variable reflux policy is used when it is desired to obtain a constant product concentration. In others words, reflux ratio is modified such that at each instant the same concentration of distillate is obtained.
Optimal reflux: For optimum reflux policy, the process used an objective function directly related to a control variable, which usually is the reflux ratio. This function is solved by applying mathematical methods such as dynamic programming, variation calculation, pontryagin maximum principle or nonlinear programming (NLP), among others. In general, the process is considered as an optimal control problem and the most common cases studied in the literature are [16]: (a) maximum distillate problem, (b) minimum time problem, (c) maximum profit problem, (d) minimum energy problem, and (e) maximum thermodynamic efficiency problem.
Reflux profile: For this case, a combination of constant reflux and variable reflux are used for obtaining a given concentration of the desired product in a time given. This operation policy is a derivation of the optimization process.
In this work is considered a batch distillation column with the following characteristics:
Adiabatic column.
Theoretical trays.
Partial reboiler.
Total condenser.
Constant pressure.
Constant relative volatility.
Negligible accumulation (holdup) of liquid and vapor.
Constant molar flows through the column.
The mathematical model of the column is obtained by performing a total mass balance and partial mass balances to component “i.” The Fenske-Underwood-Gilliland method is used to find the functional relationship between the compositions of the bottom and the dome of the column.
Although it presents the development of the model, considering the policies of operation of constant and variable reflux, these are presented in only four cases of study for the shortcut method to reflux variable considering the contribution to the state of the art of the authors.
In each case, it is considered that the mixture is fed to the boiling temperature. The error tolerance is 10−4, the integration step is Δt = 10−1 h and the time of production is required to deplete the lighter component. It has been considered a feed of 200 kmol and a vapor flow of 110 kmol h−1. For these cases, it is considered that the relative volatility is constant throughout the process.
The value of the vapor flow was established so that it allows to deplete the most volatile component in a small operation time. For the ternary and quaternary mixtures only first cut is considered. For validation, the results of both methods, Diwekar [11] and Narváez-García et al. [10], are compared with the results using the rigorous method presented by Domenech and Enjalbert [17]. This model is used because it is considered as a low holdup. To solve each one of the cases was made a program in Fortran language.
The complete mathematical model of a batch distillation column considering the dynamics of the process consists of a system of differential equations and algebra (DAEs) added by equations that allow the calculation of the thermodynamic properties and hydraulic conditions of the column. The solution of the system can be very complex depending on the state equations used to predict the behavior of the gas phase (Soave, Redlich-Kwong Peng-Robinson, etc.) or the solution of the models used to predict the liquid phase behavior (Wilson, NRTL, UNIQUAC, UNIFAC, etc.).
According to Diwekar [9], the number of equations in a rigorous mathematical model of a batch distillation, which considers a mixture of components nc and N trays is equal to (N + 2) (2nc + 1), where N + 2 considers the reboiler and condenser-tank reflux. Further, the number of equations consider the total restrictions in each of the stages (
The solution of this equation system is complex and requires intensive use of computers with adequate processing capacity, which affects costs in the area of process simulation. Therefore, it is needed to consider some simplifications to the mathematical model to reduce the data processing time.
Reductions to the batch distillation mathematical model are possible if it is considered that the process is continuous, with a feed that changes in every moment as shown in Figure 1 [1, 10], which allows to use equations Fenske [18], Underwood [19], and Gilliland [20] of continuous distillation (FUG method).
Scheme of a batch distillation column for the shortcut method.
Gilliland correlation can be replaced by the correlation Eduljee [21] because the mathematical expression is simpler for numerical works. The shortcut method considers:
Constant molar flow along the column.
Constant relative volatilities throughout the process.
Negligible fluid and vapor accumulation within the column.
Constant molar flow is based on the assumption that the enthalpy of vaporization is the same for all components, which is correct if the mixture consists of very similar compounds.
The simplification is more restrictive from shortcut method to consider constant relative volatilities throughout the process. This consideration significantly reduces the number of calculations in the model, especially because the iterative processes of liquid-vapor equilibrium does not apply. When the relative volatility cannot be considered as a constant amount along the time or the column, polynomial expressions or Winn [22] equation can be used to estimate the changes; therefore, Diwekar [11] suggested that the relative volatilities can be used to calculate in every moment of the process using an average between the values of the bottom and the dome.
Finally, the vapor accumulation in a distillation column can be neglected because it is much less than the cumulative amount of liquid, and the accumulation of fluid in the column can be neglected, considering that this accumulation is less than the liquid accumulated in the reboiler. Under these circumstances, the two most important shortcut methods for batch distillation in the literature were reported by Diwekar [11], Sundaram and Evans [13], and Narvaez-García et al. [17].
The first shortcut method for batch distillation presented here was developed by Diwekar [11]. This method considers a global balance in the column and their respective partial balances (component “i”); each of the equations used in the method are presented below:
Global balance:
where D is the distillate obtained by a mass balance in the dome of the column:
where R is the reflux ratio:
Partial balance with respect to a component (i):
and substituting in Eq. (2) is obtained:
substituting in Eq. (3) is obtained:
A material balance is also performed for the reference component (k) and the ratio between the component (i) and the component (k).
Eqs. (2) and (14) can be discretized if considered very small changes, then considering Eq. (2), is obtained:
Using Eq. (3) is obtained:
Considering Eq. (14) is obtained:
The functional relationship between the concentrations of the dome and the bottom for this method is obtained using the equation Hengestebeck-Geddes:
being α the relative volatility and C1 a constant equal to the minimum number of trays of Fenske equation. It is also necessary to use the equations of Underwood and Gilliland or Eduljee:
The Eduljee correlation is
being, in both cases:
In the case of a constant reflux policy, Diwekar [11] uses a function that relates the minimum reflux ratio obtained with Underwood equation (RminU) and the minimum reflux ratio obtained with Gilliland Equation (RminG) can be expressed as
In the variable reflux case, Diwekar [11] uses a function obtained from the equation Hengestebeck-Geddes (Eq. (21)), considering the sum of all components, and applying the sum in both members of this equation is obtained:
Therefore, the function is
In this method, it is necessary to consider a reference component, hence, using the equation Hengstebeck-Geddes (Eq. (21)) and separating the concentration of the reference component (k):
With the developed mathematical models, it is necessary to consider finding the solution; therefore, the solution algorithm is shown below.
The shortcut method developed by Diwekar [11] is setup by the system of Eqs. (1)–(32). To solve this system of equations, it is necessary to follow the next steps:
Steady-state concentrations are calculated considering that C1 = N.
To propose an initial value of C1, which will be adjusted by an iterative process.
To calculate the reference component concentration in the dome (Eq. (32)).
Other concentrations are calculated using Equation Hengestebeck-Geddes (Eq. (21)).
To solve Gilliland equation.
Verify that the obtained value by Underwood equations is the same to that obtained by Equation Gilliland.
To use Eq. (31).
If this is not true it is necessary to change the value of C1 with some iterative process as the Newton-Raphson method.
The process is repeated from step 2 to achieve convergence.
If this is true go to step 8.
Increase the time (∆t).
Calculate new concentrations (Eq. (20)) and the remaining amount in the reboiler (Eq. (18)).
The process is repeated until the desired time production.
Steady-state concentrations are calculated considering that C1 = N.
To propose an initial value of C1, which will be adjusted by an iterative process.
Verify that Eq. (31) is zero.
If this is not true it is necessary to change the value of C1 with some iterative process as the Newton-Raphson method until converge.
Other concentrations are calculated using Equation Hengestebeck-Geddes (Eq. (21)).
To solve Gilliland equation.
Calculate the value of the reflux ratio R with Eq. (26).
Increase the time (∆t).
Calculate new concentrations (Eq. (20)) and the remaining amount in the reboiler (Eq. (18)).
The process is repeated until the desired time production.
In the method of Sundaram and Evans [13], the total material balance (Eq. (2)) and partial (Eq. (4)) are similar to the method of Diwekar [11], and Eqs. (18) and (20) are the same; however, Eq. (20) may be a function of the remaining liquid in the bottom; therefore, Eq. (6) calculates the change in the mole fractions in the bottom, then:
Considering very small changes in the above equation; therefore, it is obtained:
Using the reference component (k) in the partial balance, instead of component (i), it is obtained:
and considering very small changes in Eq. (24) can be obtained:
Substituting Eq. (37) into Eq. (35), Eq. (20) is obtained. Eq. (28) is easily solved; however, Eq. (30) and Eq. (35) are much more complex because they require the functional relationship between the concentrations of the bottom and the dome.
The functional relationship between the concentrations of the dome and the bottom is calculated using the Fenske equation considering the minimum number of separation stages (Nmin) with the mole fractions of the dome (xD) and bottom (xB):
The Underwood equation for Class I mixtures and the Gilliland or Eduljee equations are used. In the original work of Sundaram and Evans [13] the equations for mixtures Class II were not considered.
When the mixture is classified as a multicomponent system Class I, the Underwood equation that relates the Rmin it can be expressed as [22]:
In this method, it is also necessary to consider the composition of a component (k) of reference; then, using the Fenske equation, the composition of the reference component is isolated. Therefore, it is necessary to considerer the sum of all components. From Eq. (35), the following is obtained:
With all the mathematical equations of the shortcut method, the next step is to provide an appropriate methodology for the solution.
The mathematical model of Sundaram and Evans is formed by the system of Eqs. (35), (37)–(42). To solve this system, follow the next steps:
This proposal is based on the concepts of Sundaram and Evans [13]. It is initiated by calculating the reflux ratio required to obtain the desired product; therefore, using Eq. (26) and solving it, the following is obtained:
Eq. (43) requires the calculation of Rmin, which you can get from one of the equations of Underwood (Eq. (23)) as shown below:
Eq. (23) requires the calculation of θ, so Eq. (22) is used:
Also, it is necessary to obtain the concentrations in the dome, therefore using the Fenske Equation (38) for the component (i):
Applying a sum process considering all the components of the mixture and calculating the composition of the reference component (k) and is obtained:
Eqs. (40) and (42) require the calculation of the Nmin. Eq. (23) also requires the calculation of X; therefore, using Eqs. (25) and (27), the following is obtained;
Eq. (48) requires the Nmin value, therefore, it is necessary to obtain Nmin for the shortcut method. In this sense the Fenske equation allows to calculate the minimum number of trays when the light key component (lk) and the heavy key component (hk = k) are considered, then:
where:
To change Eq. (49) in function of known values,
Substituting Eq. (40) into Eq. (51), the following is obtained:
Substituting Eq. (50) in Eq. (52), the following is obtained:
and, Eq. (49) becomes:
Eq. (55) requires an iterative process for the solution of Nmin; therefore, using the Newton-Raphson method, the following is obtained:
With the Nmin value, the reflux ratio and other values relating to the variable Nmin can be calculated. The proposed solution to the developed method is addressed in the following section.
It is notable that both the model of Diwekar [11] and this model started from the same material balances (global and partial), and in other words, both works are developed following the same method; however, the functional relationship between the concentrations of the dome and bottom is different equations. Table 1 presents a comparison between the equations of the two models.
Diwekar [11] | Narváez-García et al. [10] |
---|---|
Not considered | |
In fact the equations of Underwood and Gilliland are the same in each model, and the difference is the way of how the values of the Nmin are obtained. Narváez-García et al. [10] used the Fenske equation, while Diwekar [11] used the equation of Hengestebeck-Geddes.
While calculation times are similar in both models, the Narváez-García et al. model has an advantage over the Diwekar model when the separation of mixtures Class I is performed due to the use of a simplified Underwood equation (Eq. (39)). This does not happen with the model of Diwekar because the original equations of Underwood are always considered.
The mathematical model by Narváez-García et al. [10] is conformed for the system of Eqs. (22) and (23) or (39), (35), (37), (43), (48), (49), (54), (55), (57), (58), and (59). The main objective of this system of equations is to calculate the value of the reflux ratio and for this Eq. (40) is used.
Eq. (43) requires the value of X and Rmin. The value of X is related to the minimum number of trays (Nmin) through Eqs. (48) and (49); therefore, first Nmin is calculated, starting with an assumed value and is iterated until it converges to the correct value of Nmin.
The Newton-Raphson iterative method used Eqs. (57)–(59). These equations are only function of the dome and bottom concentrations of as well as of the relative volatilities.
The value obtained of X allows to find the value of Rmin, which is solve using the Underwood equation (39). However, to get the value of Rmin before it is necessary to calculate the mole fractions of the dome (xD) using Eqs. (40) and (54). With the values of X and Rmin will be calculated the reflux ratio R (Eq. (43)) and now it is possible to calculate the amount remaining in the reboiler using Eq. (37) and the bottom concentration using Eq. (35).
The mathematical models of the shortcut method presented in this work have been solved considering various mixtures: binary, ternary, and quaternary. Being the variable reflux policy more complicated than the constant reflux policy, only are presented cases considering the variable reflux policy. The input conditions to the process are shown in Table 2.
Case | Feed molar fraction | Relatives volatilities (α) | N+ | k | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|
1 | 2 | 3 | 4 | 1 | 2 | 3 | 4 | ||||
1 | 0.40 | 0.20 | 0.30 | 0.10 | 1.67 | 1.25 | 1.00 | 0.83 | 5 | 3 | 0.70 |
2 | 0.33 | 0.33 | 0.34 | – | 1.33 | 1.00 | 0.67 | – | 10 | 2 | 0.80 |
3 | 0.50 | 0.50 | – | – | 2.40 | 1.00 | – | – | 9 | 2 | 0.95 |
Input conditions for cases of study.
N+ = Number of trays, k= Reference, Component = 1, 2, 3, 4.
The results of the cases of the study are shown below. Considering that, the mole fraction of the desired component is a constant amount, the profiles of the reflux ratios, the remaining amounts in the bottom and its concentrations are obtained. To validate the results of the reflux rate obtained by the shortcut methods, a comparison between the profile of the reflux ratio obtained and the profile obtained with a rigorous method was performed.
Figures 2, 4, and 5 show the results obtained with the short methods of Diwekar [11] and Narváez-García et al. [10]. Figure 3 shows the result comparison between the shortcut method and the rigorous method.
Reflux ratio profiles obtained with the shortcut methods.
Comparison of profiles of reflux ratio using the shortcut method and a rigorous method.
Remaining amount in the reboiler obtained by shortcut methods.
Concentration profiles in the reboiler obtained by shortcut methods.
The comparison of the results between the two shortcut methods (Figures 2, 4, and 5) allows to establish that there are no significant differences. The maximum deviation for reflux was 1.5%, the amount remaining in the reboiler was 0.55 %, and concentrations in the bottom was 2%.
As for the comparison between the shortcut method and the rigorous method (Figure 3), the deviations are within an acceptable range of 9.7% maximum considering the reflux ratio is calculated.
Figures 6, 8, and 9 show the results obtained with the short methods of Diwekar [11] and Narváez-García et al. [10]. Figure 3 shows the results comparison between the shortcut method and the rigorous method.
Reflux ratio profiles obtained by shortcut methods.
Comparison of profiles of reflux ratio obtained using the shortcut method and a rigorous method.
Remaining amount in the reboiler obtained by shortcut methods.
Concentration profiles in the reboiler obtained by shortcut methods.
The results between both short methods (Figures 6, 8, and 9) allow to establish that there are no significant differences. The maximum deviation for calculated reflux ratio was 2.2%, for the amount remaining in the reboiler was 0.29%, and the deviation in the bottom concentrations was 0.67%. As for the comparison between the shortcut method and the rigorous method (Figure 7), the deviations are within an acceptable range of 9.7% maximum considering the reflux ratio is calculated.
Figures 10, 12, and 13 show the results obtained with the short methods of Diwekar [11] and Narváez-García et al. [10]. Figure 11 shows the results comparison between the shortcut method and the rigorous method.
Reflux ratio profiles obtained by shortcut methods.
Comparison of profiles of reflux ratio obtained using the shortcut method and a rigorous method.
Remaining amount in the reboiler obtained by shortcut methods.
Concentration profiles in the reboiler obtained by shortcut methods.
The results between both short methods (Figures 10, 12, and 13) allow to establish that there are no significant differences. The maximum deviation for calculated reflux ratio was 2.7%, the amount remaining in the reboiler was 0.45%, and the deviation in the bottom concentrations was 0.63%. As for the comparison between the shortcut method and the rigorous method (Figure 11), the deviations are within an acceptable range of 3.8% maximum considering the reflux ratio is calculated.
In general, as shown in each of the figures, the maximum deviation found between the two shortcut methods considering a policy of variable reflux is less than 3% and, in this sense, the use of either of the two depends on the ease of application of the method.
In this case, the method developed by Narváez-García et al. [10] is better because it is adjusted to the original equations of the FUG method.
Similarly, to validate the shortcut methods considering a variable reflux policy, we had presented a comparison between the profiles of the reflux ratio which shows that you can have up to 9.7% difference between the results of the shortcut method and the rigorous method, of course, this difference is due to the simplifications of the short method, however, the difference falls within an acceptable range and this validated the shortcut methods presented. The maximum difference found between the concentrations of the bottom was less than 2%.
In all cases, the behavior of the profiles is adequate for the batch distillation process; in other words, greater process time is necessary for a greater reflux and the more volatile component is depleted.
In this chapter, we have presented the shortcut methods developed by Diwekar [11], Sund-aram and Evans [13], and Narváez-García et al. [10]. Considering the complexity of the solution only, the shortcut method with a variable reflux policy is solved. The results were validated using a rigorous method. It is considered that the results of the shortcut methods are very close with respect to the rigorous method results.
The era of nuclear energy in the world started in 1954 by putting into service the first nuclear power plant (NPP)—the Obninsk NPP with a channel-type reactor and power of 5 MW. Since then, leading countries of the world (the USSR-Russia, the USA, Great Britain, France, etc.) have come up with a whole spectrum of a new type of power supply—nuclear-powered.
\nBy 2019, in the Russian Federation, 10 NPPs with 35 power-generating units with a total power of 29 GW are operational. In model of the NPP of Russia, there are 20 pressurized water reactors, including water-moderated power reactors (12 units of VVER-1000 type, 1 unit of VVER-1100 type, 2 units of VVER-1200 type, 5 units of VVER-400 type, and 1 unit of VVER-417 type). There are also 13 units of channel boiling water reactors of a high power of RBMK type (channel-type graphite-moderated power reactor—GMPR)—(10 units of RBMK-1000 type and 3 units of type EPG-6 type with power of 12 MW) and 2 units of fast-neutron reactor (FNPR) of BN type (BN-600 type and BN-800 type).
\nIn 56 states of the world, more than 430 nuclear reactors with a total power 370 GW is now operated. The NPPs in the world produce about 11% of the consumable electric power. Leaders in this production are France (80%), South Korea (32%), and Ukraine (30%). In Russia, this share amounts to 16%. In the long term of 20–25 years, probably accretion of this share will be about 25%.
\nOn changeover to reactors of power plants of first generations of 1960–1970 reactors of new third and fourth breeds come. And if the first reactors were considered as “nuclear boilers” and designed on norms of boiler fabrication for thermal power, up-to-date reactors develop on these details both on scientifically well-founded norms and on methods of national (Russia, the USA, Great Britain, France, and Germany) and international levels (IAEA).
\nFrom stands of classes of hazards detection for technosphere objects, nuclear reactors undoubtedly fall into critically (CRO) and strategically (SRO) relevant objects. These are facts that demand the profound combined analysis and a justification of all design and service solutions for all stages of their life cycle.
\nIn the proximal (till 2020), midrange (till 2030), and kept away (till 2050) prospects, the evolution of nuclear energetics will be carried out on the basis of operating, built, and designed nuclear power plants. Basis of the fundamental and application analysis of strength, life time, reliability, and safety of operation of NPP elements with reactors of VVER, RBMK, and BN types (Figure 1) in regular both emergency situations are the equations and criteria linear and nonlinear mechanics of deformation and fracture [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. They contain in home and foreign strength standards and are used as at design, so at manufacture and operation of working in extremely conditions, a high-loaded power-generating plants with use physical and mathematical modeling [1, 12, 13, 14, 15, 16, 17].
\nThe Russian reactors of VVER (water-moderated power reactor) (a), RBMK (channel graphite-moderated power reactor) (b) and BN (fast-neutron reactor) (c) types.
Results of traditional researches and a standardization of strength and life time of NPP in the determined statement in Russia and abroad are both initial scientific baseline of normative documents on design and actual baseline of making of perspective methods of a reliability estimate, survivability, initiation, and evolution of accidents and disasters by risk criteria, and also of makings of new principles, technologies, and engineering complexes ensuring safety service of NPP. These are conditions that are scientifically grounded to prevent initiation of the emergency and disastrous situations and also to minimize probable losses at their initiation at all stages of life cycle. Such situations within the limits of usual normative approaches and methods, as a rule, remained the least investigated from the scientific and application points of view owing to the complication, small predictability, and recurrence. At the same time, survivability of power-generating units in emergency situations and risk analysis of probable aftereffects should become weighable arguments in favor of building of nuclear units with a life expectancy from 60 to 100 years.
\nThe analysis of sources, the reasons, and aftereffects of the heavy disasters occurring during installations of nuclear energetics display both their likeness and essential difference. Accidents known to the world on NPP with radioactivity ejection in a circumambient manner in the USA (the NPP “Three Mile Island (TMI)”—Figure 2), in the USSR (the Chernobyl NPP (CNPP)—Figure 3), and in Japan (the NPP “Fukushima-1—Figure 4) were the heaviest [3, 6, 8, 11, 18].
\nThe “Three Mile Island” NPP (TMI).
The accident on the Chernobyl NPP (CNPP).
The accident on the “Fukushima-1” NPP.
A common after effect of NPP accidents and disasters was that direct and indirect economical losses from them reached tens and hundreds of billions of USD. For their forestalling and preventing in the subsequent, the principal changes were made to designer, technological, and service solutions. Heavy emergency situations for NPP service arose earlier at the time of damage to their equipment, such as runners, steam plants, main coolant pumps, heat exchanger pipes, gate valves, and legs of reactor internals [11, 17].
\nThe abovementioned NPP heavy accidents and disasters originated from unapproved impacts of human controllers, non-observance of technological discipline at emergency situation (TMI, CNPP), heavy-lift seismic loads, and a tsunami (Fukushima-1). Regular systems of the automatize guard of the NPP have been unreasonably disconnected (CNPP) or could not work in an emergency situation (TMI, Fukushima-1). Heavy emergency situations on turbine runners, steam plants, gate valves, and legs arose due to the lack of suitable technical diagnostics of these situations [11, 19], when faults in the form of cracks because of technological or operational fault attained of the limiting, intolerable sizes (102 to 1.5 × 103 mm), affecting 50–70% of carrying cross-section and creating sharp magnification of runner chattering. Thus, the analysis of such situations was not envisioned by normative calculations.
\nFor installations of a nuclear energy in our country and abroad in the second half of twentieth century, the whole complex of fundamental and application developments [1, 2, 3, 4, 5, 6, 7, 11, 12, 13, 14, 20, 21, 22, 23] on the creation of normative strength calculations of the equipment and pipelines for nuclear power plants has been executed. Thus in our country special meaning had the solution of policy-making bodies that the scientific adviser of research developments on a justification of norms had been defined the Academy of Sciences of the USSR (The A.A. Blagonravov Institute for Machine Sciences—the IMASH), and the head development engineer of norms—the Ministry of medium machine building of the USSR (The N.A. Dollezhal Research and Development Institute of Power Engineering—NIKIET).
\nThe same organizations making all prototype models of reactors for the NPP established the total statement about the strength before starting a reactor in service. Such norms developed both in the USSR [1, 12] and in the USA [14] subsequently were developed according to international standards set by the International Atomic Energy Agency—IAEA [13]. Compared to home norms of an NPP design [1, 12], basic sections on calculations, monitoring, probability safety assessment, and a justification of life time extension have been included.
\nLong-term experience of home nuclear branch organizations and the academic institutes has allowed to form (Figure 5) the schematic diagram of the combined solution of tasks in view:
The determined and statistical researches of deformation and fracture processes of laboratory specimens (with groups from 3–10 to 100–200 specimens of one steel)
Model tests of the metallic specimens imitating most important parts (for example, studs of threaded connections with a diameter from 24 to 110 mm) and also nonmetallic specimens of studs with a diameter from 60 to 210 mm
Tests of the modeling reactor vessels fabricated of nonmetallic materials in scale 1:10 and from metallic materials in scale 1:5
Full-scale prestarting and starting tests of reactor prototype models of VVER, RBMK, and BN types
The structure of the main task solution at making and service of the NPP equipment.
In considered norms, there are two cores sections: calculation of principal dimensions predominantly by criteria of a static strength and the verification calculations on a different combination of limiting states at low-cycle and high-cycle, long-term, vibration, seismic loads with initiation of static, cyclic, brittle, corrosion fracture, and also cyclic forming and radiation damage.
\nIn the capacity of the most responsible and dangerous NPP components, nuclear reactor vessels, pipelines, pumps, steam generators, reactors, and machine halls have been accepted (Figure 6).
\nThe flow chart of strain-gauging of power equipment: 1—steam generator, 2—reactor, 3—pressure compensator, 4—bubble tank, 5—a main coolant pump, 6—commutators, and 7—registering apparatuses.
In an NPP with water-moderated power reactors (VVER) in the capacity of the major critical parts, it is possible to consider also the basic attachment fittings of reactor covers such as studs. Thus, the computational-experimental analysis of stress-strain states, strengths, and life times of a connection joint of reactor covers is conducted by improved methods in more detail (Figure 7).
\nThe scheme of strain-gauging of a threaded connection of an attachment fitting of a cover.
For reactor installations of home production, such analysis was fulfilled [2, 3, 4, 11, 15, 16] jointly by the academic institutes, head branch research, and designer organizations on all prototype models of reactors in our country and abroad (Bulgaria, Finland, Hungary, Czech, and China) with application of the foremost methods: model researches of covers, studs, pressing rings on models from stress-optical and metallic materials, full-scale researches on reactors at preoperational tests on all regimes (including emergency), and also at an initial stage (till 1–3 years) of service.
\nIn particular, the fifth unit of the Kozloduy NPP (Bulgaria) has been developed and implemented [15] after a most complicated program of full-scale researches by methods of a strain measurement, a thermometry, a vibrometry for all components of a primary loop with 1000 measuring points of local stresses, pressure pulsations, and temperatures (Figure 8).
\nZones and points of placing of measuring gauges on the NPP equipment.
Modeling and full-scale researches have allowed to define detailed stress distributions on threads (Figure 9) and in a cover (Figure 10). These facts have given the chance to obtain real history of service impacts and nominal and local stresses on all parts of a reactor main joint.
\nA stress loading of a stud attachment fitting of a reactor cover.
Stress distribution diagrams in a cover, flanges, and studs.
Computational and special experimental test bench researches of a dynamic stress loading and cyclical damages from seismic loads had a particular actuality.
\nOn metallic modeling studs with a diameter from М12 to М110, data about life time on the basis of 104–105 cycles have been obtained. These data have allowed to justify improved margins on strength and life time of analyzed studs.
\nThe principal great value in results these researches had that facts that the maximum accumulated damages (to 70%) arose in regimes multiple tightening and seal failure of caps (Figure 11). This fact has demanded work on special activities to decrease the indicated damages [15, 16].
\nThe diagram of stresses change in studs at sealing of the main joint of the VVER-1000 reactor.
Formation of development trends at the standardization instituting serviceability and safety of a nuclear (power-generating equipment went in a direction of specification and complicating of applied methods and criteria [1, 2, 3, 11, 20, 21, 22, 23]. Thus, accidents and disasters (the TMI in the USA, the CNPP in the USSR, and the Fukushima-1 in Japan) added additional information baseline for such development.
\nTo the traditional solution of a problem of service safety [2, 6, 7, 8, 9, 10, 20, 21, 22, 23, 24, 25], three groups of approaches had a direct ratio:
From the position of strengths (in its multicriteria expression)
From the position of life time (in time and cyclic statement)
From the position of inadmissibility of large plastic strains
Traditional methods of strength justification were founded on a complex of determined characteristics of mechanical properties of materials and fracture criteria (yield point—σ\ny\n, ultimate strength—σ\nu\n, fatigue limit—σ−1, and long-term strength—σ\nlt\n). On the basis of these parameters of strength and fracture (present in standard and technical specifications for reactor structural materials), the status of safety and life time margins (n\nσ, nN\n, nτ\n) has been generated. These margins are included in the reference, educational, and standard literature [1, 2, 12, 20, 21, 22, 23, 24, 25, 26]. Today, a common system of criteria and strength margins guaranteeing a fracture of nonadmission for equipment components at observance of the given service conditions is developed.
\nMathematical modeling at the determined normative requirements to strength and life time came down to two approaches:
To modeling parts of rods, plates, and thin shell types on the basis of analytical solutions of the theory of a strength of materials and theory of elasticity
To modeling real objects on the basis of numerical solutions by finite-element method, finite difference method, and integral equations method
Research of seismic impacts was the most complicated at computational and experimental modeling:
By finite-element method (FEM) for all parts of the first circuit (Figure 12)
By methods of physical modeling of a reactor with reactor internals (Figure 13)
Loads and stresses in a connecting pipes zone of a reactor vessel at seismic impacts for YOZ plain—the computational scheme (a), response stresses (MPa) on outside (b) and interior (c) surfaces; for XOZ plain—the calculation scheme (d) and stresses (e) on an interior surface.
A research of a dynamic state of a reactor simulator at seismic excitation.
It has thus appeared that most high stresses and damages from seismic loads occur at the zone of attaching of pipelines to a reactor vessel.
\nOn the basis of such modeling, nominal σ\nn\n and maximum local σmax stresses in concentration zones were defined. However, in these traditional approaches, normative materials often did not contain the direct data quantitatively instituting strength and life time of considered objects taking into account a statistical property of parameters σ\ny\n, σ\nu\n, σ−1, and σ\nlt\n. Occurring actually dissipation of parameters for strength calculation and life time of a NPP environment is caused by instability of manufacturing procedures at production of structural materials and NPP bearing parts (reactor vessels, pipelines, pumps, and heat exchangers). In the last decades, this deficiency has been eliminated, and the sphere of the traditional analysis of serviceability of the NPP equipment includes the theory and criteria of life time and reliability [2, 20, 21, 22, 23, 24, 25, 26, 27].
\nIn addition to normative calculations of reactors on [1] at the complicated regimes (Figure 14) of an assembly, test and service loading (assembly, a tightening of studs, a hydroshaping testing, launch, capacity change, emergency operations, and shut-down) for events of occurrence of high levels of stresses improved strength, and life time calculations were carried out on the equations type
\nThe diagram of change of service loading parameters.
where ea\n is the amplitude of strain at a design regime; N is the life time at a crack initiation stage, in cycles; σ\nb\n is the ultimate strength of a material (400 ≤ σ\nb\n ≤ 950 MPa); ψ\nc\n is the reduction of area in a neck of a specimen at single-pass rupture (0.3 ≤ ψ\nc\n ≤ 0.7); re\n, r\nσ are the cycle asymmetry parameters on strains and stresses, accordingly; and me\n, m\nσ are the characteristics of a real material (0.5 ≤ me\n ≤ 0.6), (0.08 ≤ m\nσ ≤ 0.12). Values of parameters in Eq. (1)\nea, ψ\nc, re\n, r\nσ\n, me\n, and m\nσ are relative and dimensionless.
\nCalculation on Eq. (1) with the use of deformation criteria can be brought together to calculate by force criteria (оn stresses) to accept \n
\nEquation (1) is true for a wide band of life times (100 ≤ N ≤ 1012). Permissible regimes of a stress loading are established in Eq. (1) with introduction of two margins \n
For the complicated regimes of a two-frequency loading (low-frequency with frequency fl\n = f\n0 (hertz) and amplitude of stress \n
where χ and η are dimensionless characteristics of a material and parameters of a two-frequency regime.
\nThe same approach is used to calculate life time taking into account the presence of contact (wear resistance) and seismic impacts.
\nThe presence of initial or service defects of cracks type with depth l is reflected in calculations of survivability on the basis of the equations of linear and a nonlinear fracture mechanics by change of stresses KI\n (MPa⋅m1/2) and strains KIe\n intensity factors [2, 20, 29]. For one-time brittle or a ductile fracture,
\nwhere KIc\n and KIec\n are the critical (fracture) stresses and strains intensity factors, accordingly; \n
Reliability of equipment PQR\n(τ) along with the account of the probabilistic approach to estimations of mechanical properties of a structural material is defined also (Figure 15) on probabilistic characteristics of service stress loading Q(τ) and life time RNτ\n(τ) on the basis of distribution functions f of service impacts Qs\n(τ) and of ultimate loads Qc\n(τ) for the given life times Nc\n, τc\n. Thus, usually “trees of events” and “trees of failures” on experience of previous service of analogous technosphere objects are used. In such statement, the risk can be defined as
\nThe scheme of determination of reliability, failures, accidents, and disaster probability \n\n\nР\nQR\n\n\nτ\n\n\n.
More oriented on the quantitative solution of a safety problem for complicated NPP installations, capable to cause severe accidents and disasters, are new methods and criteria of the following groups [2, 6, 7, 8, 11, 18, 19, 20, 21, 24, 25, 26, 29, 30, 31, 32, 33]:
Survivability (ability and steadiness of operation at occurrence of damages at different stages of accidents and disaster evolution)
Safety (taking into account the risk criteria and characteristics of accidents and disasters)
Risk (in probability-economic statement)
From the above-stated, the up-to-date justification of strength, life time, reliability, survivability, safety, and risks (Figure 16) should be based on results of corresponding calculations and tests with observance of the special and new requirements established by corresponding normative-legal documents.
\nA structure and evolution of standardization methods on the determined justification of strength and life time with the use of physical modeling of materials behavior at a static, cyclic, and long-term loading.
For long-term operated high-risk installations of a nuclear energetic to which the NPPs with reactors of the VVER concern, the BN and the RBMK types’ rate, initial parameters of strength, life time, risk, and safety were defined in an explicit and implicit kinds on stages of their design and commissioning on acting then norms and rules which place at the different displayed in Figure 16 footsteps (on time and analysis level).
\nThereupon, during estimations of their state, two scientific and application approaches are possible:
To realize stage by stage an estimation of the initial, exhausted, and remaining life time
To estimate current life time, as initial for the given level of the service damage that has been accumulated in the previous operating period
At the present time, the first approach was found to be the largest application. However, subsequently, the second approach appears to be deciding owing to its higher precision at estimations of the remaining strength, life time, and safety.
\nOn the basis of the normative documents developed and accepted to present safety of power engineering as a whole, and NPPs in particular, the level of individual risks and risks of a possibility of accidents and disaster initiation should be estimated. In the process of perfecting NPPs and their nuclear reactors, these risks were reduced and will be reduced from 10−4 to 10−8 1/year and less. For example, the reactor of natural safety with plumbeous heat-transfer agent will have a probability of fracture considerably below 10−8 1/year [8, 11]. Individual risks of nonnuclear power engineering lay within the limits 10−4–10−7 1/year (Table 1).
\nComparative data about a radiation-ecological risk for different directions of the electric power manufacture.
The great importance for the analysis, support, and improvement of safety of the considered equipment within the limits of dominating and active concepts, strategies, norms, orders, and margins has the level of a scientific-practical justification of the predictable and acceptable risks characterizing generally regular and limiting states of these installations.
\nFor all spectrum of technosphere installation types of emergency and catastrophic situations, the level of their protectability and types of accompanying risks at transition from standard conditions operation in regular states to emergency and catastrophic at service can be described (Table 2) as:
Regular situations—occurring at installations operation in the breaking points established by norms and rules; risks for them controlled; and protectability from them increased
Regime emergency situations—occurring at a shift from service standard conditions at regular operation of potentially dangerous installations; aftereffects from them predicted, risks for them controlled; and protectability from them sufficient
Design emergency situations—arise at a runout of installation out of breaking points of regular regimes with predicted and acceptable aftereffects; risks for them analyzed; and protectability from them partial
Out-of-design emergency situations—arise at nonreversible damages of important parts of installation with high losses and human sacrifices and with necessity of carrying out a recovery work; risks for them heightened; and the level of protectability from them insufficient
Hypothetical emergency situations—can arise at the not forecast in advance scenarios of evolution with the greatest possible losses and sacrifices; are characterized by high risks; protectability from them low; and restoration of installations is impossible
Types extreme (emergency and catastrophic) situations and level of protectability from them of high-risk installations.
The complex calculation-experimental analysis of the initial and remaining service life of an NPP is founded first of all on an estimation of service damages accumulation conditions at different service regimes taking into account corresponding state equations, and also on the study of conditions of transition in limiting states taking into account service kinetics of mechanical properties of materials, criteria of strength, crack resistance, and survivability.
\nGenerally termed procedures are implemented with the use of a complex criteria equations, computational equations, and design parameters applied to the analysis and definition of regular and limiting states of engineering objects. The complex criteria include the following equations:
\nFor an estimation of static and long-term strength,
\nwhere FQ\n is the functional characterizing dependence of stresses from actual force impacts Q; σ, е are the operating in time τ at temperature t stresses and strains; f\n1 is the functional dependence, which includes \n
Для оценки ресурса по параметрам числа N циклов и времени τ\n
\nwhere \n
For a crack resistance estimation,
\nwhere \n
For a survivability estimation,
\nwhere \n
For a risk and safety estimation,
\nwhere \n
Thus, the level of installation safety functionally (\n
The mentioned complex functional criteria in Eqs. (1)–(10) allow to implement the full sequence of installation calculation for the purpose of providing for its service safety, beginning from strength parameters and completing at protectability parameters with acceptable values of risk both on a design stage, and at concrete stages of service, including a decision made about life time extension.
\nAt an estimation of the remaining life time on resistance to cyclic fracture, levels of cyclical stresses, cycle asymmetry parameters, a stress concentration, cyclical properties of a material, service temperatures, special conditions of loading, and residual stresses and strains are subject to analysis. Under these data calculation processes and parameters of impacts, fracture stresses and life time are defined. On the basis of such definition are the functionals that resulted above in Eqs. (4)–(10), which include calculation dependences (state equations, curve of deformations and fractures, and strain and force criteria). In improved calculation zones of welded joints, a plastic deformation in the most loaded zones, variety of operating conditions and impacts, and dispersion of characteristics of mechanical properties [2, 10, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 34, 35, 36] are considered.
\nAs appears from Eqs. (1)–(10) the computational-experimental justification of static, long-term, and cyclic strength, life time, and risks included in comprehensive analysis of conditions of safety service of the NPP equipment at regular and unnominal situations, sampling of types of limiting states, calculation schemes and calculation cases, methods of the analysis of stress-strain states, methods of preliminary diagnostics of technical state, assignment of margins on strength and on life times, study of probabilities of limiting states reaching, an estimation of risks of accidents and disasters [2, 9, 10, 11, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36].
\nThe built-up calculation of curve (permissible amplitudes of stresses and life time at a cyclic loading, and also of the maximum stresses and time before fracture in the long term) is carried out for an estimation of initial and remaining life time on the basis of a schematization of history of loading, sampling of computational schemes, and computational cases. The calculation of initial and remaining life time is carried out in two alternatives: an approximate calculation and improved calculation.
\nThe concept of an estimation, a diagnosis, and a prediction of service life of the NPP is correlated with the sampling of state variables of the equipment on the level of wearing and life time exhaustion. To define the factors and parameters influencing on life time, it is necessary to attribute maximum deviations of wall width and errors in measurement, a staging of prediction of life time, results of resource and strength researches, levels of diagnosing of installations, and influence of engineering preliminary diagnostics efficiency on the level of a fracture risk.
\nOn the basis of summarizing of results of a life time design justification of reactors, it is possible to establish a dependence of life time on commissioning terms, for example, an NPP with VVER type reactor of all generations (Figure 17). To a twenty-first century kickoff in our country and abroad, the design life time (expected life) has increased to 40–60 years; by 2025, the design life time can increase to 100 years [1, 3, 7, 11, 24].
\nCharacteristics of initial design (full line) and the prolonged expected life (lives times) of the NPPs with type reactors VVER of the first–the fifth generations.
Thus, the key problems of design, manufacture, service, upgrading, and a leading-out from service of nuclear units of the following (the fourth and the fifth) generations with heightened characteristics of life time and safety are:
Transition to new principles of reactor core build-up, sharply reducing severe accident possibility with its melting
Use of joint guard from severe accidents by new organization of working master schedules both in regular and in the emergency situations promoting to decrease of negative and dangerous aftereffects of accident propagation
Introduction in practice of making and service of reactors with an in-depth analysis of risks of occurrence and propagation of the emergency and catastrophic situations, considering both probabilities of these situations and their aftereffects
Inclusion in the analysis of heightened life time, risks and safety of reactors of such base criteria as strength, life time, reliability, survivability, physical protectability, and economic justification
Orientation to escalating requirements to safety of the NPP formed by national and international laws, norms, and rules
Elimination of unreasonable conservatism in already accepted normative and technical documents and introduction in the safety analysis of new threats and risks (including risks of terrorism)
Statement as the corner-stone fundamental and applied researches of safety of nuclear reactors of problems of forming of unified methodical baseline on integrated study of external and interior impacts of a wide spectrum, responses to these impacts of critical important bearing elements of the NPP in linear and nonlinear fields of a deformation, damages, and fractures
Setting, justification, control, and monitoring of the major parameters of life time and safety of the NPP operation at regular and emergency situations for confinement of margins on strength, life time, and risks in safety breaking points
Problems of safety maintenance on the basis of the concept of risks generally should to be decided with the use of the determined, statistical, probability, and combined methods of fracture mechanics and mechanics of disasters. Probabilities PS\n of realization in an NPP of system threats can be presented with the use of functional FPS\n [2, 6, 8, 18, 24, 25, 26, 29, 32, 33]
\nwhere PN\n is the probability of occurrence of the unfavorable event, stipulated by the human factor; PT\n is the probability of such event stipulated by a state of an NPP components; and PО\n is the probability of its occurrence stipulated by an environmental exposure.
\nThe type of functional Eq. (11) remains the same and for probabilities of risks realization included in the analysis at design, making, and service of the NPP. The great importance thus has that facts that the role of the human factor in appraisal PS\n at change PN\n is defined not only human controllers and the personnel, their professional qualities and a physiological state, but the experts, making solutions on all level of the hierarchy by safety of the NPP.
\nProbabilities PT\n essentially depend on the level of protectability of the NPP from accidents and disasters. This protectability is defined by quality of their initial and current state, extent of degradation of installations at the given stage of service, and diagnosing and monitoring level. Such position indicates direct interacting of parameters PT\n and PN\n taking into account base parameters of reliability and quality of technosphere installations.
\nProbabilities PS\n, as it is known, depend on occurrence of dangerous natural processes (earthquakes, floods, hurricanes, tsunami, landslides, etc.) and also from a state of the NPP installations and, hence, from PT\n. Adoption unreasonable (from the point of view of risks) R (τ) solutions on arrangement of technosphere installations and zones of population residing does parameter PS\n dependent and from PN\n.
\nLosses US\n from realization of system threats generally can be recorded through the functional FUS\n\n
\nwhere UN\n is the losses caused to the population at interacting of primary and secondary knocking factors at realization of strategic system threats; UT\n is the losses caused to technosphere installations; and UO\n is the losses caused to an environment.
\nValues UN\n, UT\n, and UO\n can be measured both in natural units (for example, a death-roll of people, number of the blasted installations, and the square of injured territories) and in equivalents (for example, in economic, monetary parameters).
\nAs a whole, in Russia, taking into account social and economic transformations, global processes to power supply and experience and prospects of nuclear energetics development based characteristics of risks R of accidents and disasters of the natural-technogenic character, defined by their losses U (or severity) and probability P (or quantity), have rather complicated character of a time history τ with a common trend to increment (Figure 18).
\nThe time history of the relative risk levels and protectability.
Accepting that the relative risks \n
where the fact of accident and disaster occurrence will correspond to the condition
\nSuch conditions occurred at the moment of Chernobyl disaster (1986), last years the twentieth centuries at damages of collecting channels of steam generators PGV-1000 type, on boundary line of centuries at damages of welds to a weld zone of the principal circuital pipeline to the steam generator [4, 11].
\nIn Figure 18 the major role of improving of all service parameters of the NPP, and first of all life time and safety which promote decrease of probabilities of accidents and disasters occurrence P(τ) and accompanying them losses U(τ) is visible.
\nWhen for the equipment of the concrete NPP, the relative system risks \n
The surfaces dangerous and safe states on values of risks.
To reach the acceptable protectability of the NPP equipment, implementation of complex steps on the decrease of system risks \n
If on axes \n
For an NPP transfer in safe states with the use of risk criteria \n
or to reduce the relative risks of accidents and disasters \n
This result can be attained by the creation of monitoring systems for diagnostics and monitoring of risk parameters \n
The state, regional and object control, regulating and providing of safety \n
where nS\n is the safety factor on system risks; \n
Safety of the NPP by criteria of risks can be considered ensured if the inequality nS\n ≥ 1 is attained.
\nThe interval of time Δτ for which risks \n
According to Eqs. (15) and (16), control and planning with the use of the criteria baseline grounded on risks come to following primal tasks:
\nTo the development of scientifically well-founded methods of the analysis of risks \n
To decision making about the level of allowable values [\n
To scientifically well-founded level of definition of necessary expenditures [\n
Thus, predicting, monitoring, and forestalling of accidents and disasters for an NPP (including by improving of all parameters of strength, life time and survivability) appear to be essentially more effective, than liquidating of aftereffects of catastrophic situations (type of the TMI, the CNPP, and the Fukushima-1). Values \n
As it was already mentioned, safety of nuclear energy installations S(τ), as well as all other complicated engineering systems, on the given interval of time τ is defined in Eq. (13) by two basic quantities: probability P(τ) of unfavorable event occurrence (an unfavorable situation) and probable loss U(τ) from this event. Values P(τ) and U(τ) are generally statistically uncertain, demanding for their quantitative assessment of great volumes of the information on the nature, behaviors, sources, and scenarios of unfavorable events both for each of considered installations and for the given set of installations (group, batch, and series) at occurrence and propagation of unfavorable events and also the information on aftereffects for installations, persons, and an environment at occurrence, propagation, and liquidation of unfavorable events.
\nIn nuclear energetics with reactors of all types and all generations (from the first to the fourth) prior to the beginning of the twenty-first century, at failure analysis, the basic attention was given to parameter P(τ) that defined reliability of safety operation of the NPP. Special meaning was added thus to the forestalling and prevention of the heaviest on the aftereffects of catastrophic situations with the peak damages—melting of the core and a radioactivity runout for breaking points of all guard barriers—casings of the fuel element, cartridge, reactor vessel, reactor hall, and containment. In this case, reactor vessel fracture is extremely dangerous. This event concerns the seventh group of limiting states.
\nSignificant aftereffects arise also at fracture of the basic elements of the first circuit of a reactor vessel and collecting channels of steam generators, pumps, volume compensators, bubbler tanks, and also housings and runners of turbines in the second circuit. These fractures amount the sixth group of the limiting states creating threats to the population, the NPP, and the environment.
\nIf while in service of the NPP because of occurrence of damages of parts of the first circuit has arisen a radioactivity outside breaking points of the NPP and there were thus threats of bombarding radiation for the population, then it is necessary to attribute these events to the fifth group of dangerous limiting states.
\nThe leakages caused by partial damages (faults of crack type or depressurizations of connectors) and creating threats for human controllers and the personnel in the NPP concern the fourth group of limiting states.
\nThe third group of limiting states should be bundled to the considerable damages of the above-termed parts of the first and the second circuit without a radioactivity runout for breaking points of an NPP, which are not demanding their mandatory substitution.
\nThe second group of limiting states concern occurrence in bearing structures of the NPP of partial damages without a radioactivity runout for breaking points of the first circuit, not demanding their substitution, but demanding carrying out of repair-and-renewal operations.
\nThe first group of limiting states is amounted by those of them which are bundled to damages and the faults that have fallen outside the limits admissible under inspection norms and calculation, but not demanding mandatory carrying out of repair-and-renewal operations and that can be admitted to prolongation of service before the next examination.
\nThese facts allow to execute summary classification by groups of limiting states for the NPP equipment (Table 3) from the most dangerous admissible (the seventh group of limiting states LS-7) to the least dangerous admissible (the first group of limiting states LS-1).
\nGroups of limiting states for the analysis of the NPP safety.
For the groups of limiting states indicated in Table 3 taking into account summarizing of great volume of normative and technical materials and results of the executed researches, it is possible to describe demanded (admissible) probabilities [P(τ)] occurrence of unfavorable events. To such probabilities there correspond their actual levels obtained from statistics of their occurrence while in service of NPPs of all generations. Each severe accident or disaster on an NPP, happening at the moment τc\n, was accompanied by comprehensive analysis of their reasons and sources, and also realization of considerable on volumes and expenditures of activities for safety improving. Eventually, at τs\n > τc\n, after such accidents or disasters, decrease of probabilities from P(τc\n) to P(τs\n) was observed.
\nFor values of probabilities P(τc\n) and P(τs\n) for all reactors operated in the world at \n
where Nd\n is the quantity of the reactors that have obtained damages at the given i-th type of limiting state under Table 3; Ntc\n is the total of reactors to the time τc\n of occurrence of the given i-th type of damage; Nts\n is the total of reactors to the time τs\n; τc\n is the mean time (years) of service of one reactor to the time of reaching of the given i-th type of limiting state; and τs\n is the mean time of the service of one reactor.
\nAs it was already mentioned, unfavorable events on an NPP (disasters, accidents, failures, and disruptions) are accompanied by corresponding losses U(τ) both at the moment of occurrence of these events τc\n and after them (τ ≥ τc\n). These losses are caused to the person (to human controllers, the personnel, and the population), to technosphere installations (to an NPP and other installations of its infrastructure), and also to the environment. Now while miss direct legal and normative documents by the quantitative definition of these losses. Some suggestions on this problem are stated below.
\nFor a tentative estimation of loss U(τ), it is possible to use the simplified statistical and expert information on such losses. Generally, values of losses are defined by two basic parameters:
Losses of human lives or health at occurrence and progressing of unfavorable situations
Economical losses (for example, in Rubles or USD) from a loss of life, from maiming to people, and from fractures and damages of technosphere installations and the environment
Direct loss U(τ) for the LS-7 limiting state interlinked immediately to fracture of the NPP or full termination of its service. Then, the datum of loss U(τ) can be accepted to the equal cost of the NPP. In this, the loss can and should include charges U(τ\n1) within 1–2 years on a primary elimination of the consequences of disaster or accident (realization of protective measures, evacuation of the population, and termination of infrastructure installation operation). These charges at (τ\n1 ≥ τ) several times (2–4) can exceed the initial loss U(τc\n). Decrease of secondary consequences of heavy disasters on an NPP (making of shelters, recultivation, medical examination and the help, and compensating payments) demands complementary essential annual expenditures U(τ\n2) for a long time \n
The time-history and schematization of losses U(τ).
With the reduction of the hazard level of accidents and disasters (at transition of limiting states from the LS-7 to the LS-1), value U (τc\n) and \n
From assemblage of tens methods for definition of risks parameters as the most simple is the statistical or determined-statistical method according to which it is possible to write
\nwhere τi\n is the time for which one the risk assessment is conducted and P(τi\n) and U(τi\n) are the probabilities and losses for time τi\n.
\nIf under τi\n is fathomed the time of unfavorable event occurrence of τc\n, then according to Eq. (19), it is possible to obtain
\nRisk R(τc\n) is possible to consider as risks of the implemented unfavorable events at τi\n = τc\n and to use them for prediction of events for times τi\n ≥ τc\n. One such prospective risk appears as the risk for the current phase of service τi\n = τs\n. In this case, on the basis of Eq. (19), it is possible to write
\nwhere τs\n is the time after unfavorable event (τs\n ≥ τc\n).
\nThis time can be situated in the interval τc\n ≤ τ\n1 ≤ τ\n2. Then, for one operated unit of the NPP, the common risk at reaching the given i-group of limiting state from the LS-7 to the LS-1 will constitute
\nIf at loss estimations to consider not only direct losses at occurrence of unfavorable event U(τ\nк) together with complementary losses U(τ\n1) and U(τ\n2), then it is possible to define common (integral) losses as
\nThese integral losses respond to the appropriate risks
\nOn the basis of results of an estimation considered above risk components, it is possible to build dependences between basic parameters of risk for the NPP—probabilities P(τ) occurrence of unfavorable situations and losses U(τ) from them (Figure 21).
\nParameters of risks for the NPP with reactors of VVER types.
The line had above and design points in the Figure 21 belong to probabilities P(τc\n) and to losses U(τc\n) for the moment of accident or disaster occurrence on the NPPs. The lower line made like overhead characterizes a negligible zone of risk parameters \n
From stated above follows that the major problems which have been not decided while to the full for a NPP there are problems of provision of their protectability and safety on the basis of new scientific fundamental and application researches on mechanics, hydrodynamics, economics, mathematical and physical modeling of dangerous processes resulting to heavy disasters, and also development of detailed methods of the analysis of risks for heavy disasters.
\nResults of the fulfilled scientific researches and developments in this direction, integrated [3, 4, 5, 6, 7, 8, 15, 16, 17] in the serial of monographic publications on strength, life time, and safety of power nuclear reactors, are initial scientific baseline for the applicable normative, designer, technological solutions on provision of protectability of the NPP equipment from heavy disasters on the basis of criteria of acceptable risks.
\nThe above-mentioned results of analytical and experimental researches can be considered in the capacity of a theoretical basis for the subsequent development of practical models of the computational analysis of risks for strategically relevant installations of a nuclear energetic on the basis of the complex Eqs. (1)–(24). Development of such models, and the most important—their filling up statistically reliable probability distribution of fractures on groups of limiting states (see Table 3) on the one hand, and economical computations of losses, with another, it is necessary to consider as the major task for a solution of a problem of safe development of power supply of human community.
\nAt up-to-date and subsequent stages of evolution of power engineering in Russia in the capacity of a basic recommended position, it is necessary to use the position about provision of an acceptable risk level of occurrence of accidents and disasters. In this connection, it is not obviously possible to ensure from social-economic and technological stands the declared principle of absolute safety with null risks (R(τ) = 0). Owing to it, the solution of the delivered problem is brought together to determination of scientifically well-founded admissibility of occurrence of the emergency situations with possible minimization of loss caused by them, with an estimation of the greatest possible, acceptable, and controlled risk both at probable occurrence of global and national accidents and disasters, and their realization at regional and local levels.
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