Lý thuyết làm giàu uranium theo cơ chế ly tâm

VIET NAM NATIONAL UNIVERSITY – HO CHI MINH CITY UNIVERSITY OF SCIENCE FACULTY OF PHYSICS – ENGINEERING PHYSICS DEPARTMENT OF NUCLEAR PHYSICS ***** UNDERGRADUATE THESIS THE THEORY OF URANIUM ENRICHMENT BY THE GAS CENTRIFUGE Student : NGUYEN THANH HUNG Supervisor : Prof CHAU VAN TAO Reviewer : M.Sc TRINH HOA LANG HO CHI MINH CITY – 2012 Acknowledgement I am extremely grateful to my supervisor, Prof Chau Van Tao His supervision and support really helped me during the process of my thesis Most of the materials for my work are found with his assistance My grateful thanks also go to M.Sc Trinh Hoa Lang for his help in review various chapters in my thesis I would like express gratitude to B.S Nguyen Hoang Anh My thesis is based on her research on the methods of uranium enrichment I appreciate all the contributions of the researchers in Department of Nuclear Physics who patient in helping me complete the thesis A special thank also to my friends who study with me at University of Science for their wise ideas throughout the process of my work Finally, I would like to thank my family, especially to my mother for giving me a great love and care Content Page Content Symbols List of figures Preface Chapter 1: Introduction to the gas centrifuge 1.1 Principle 1.2 History 1.3 Description of Centrifuge Chapter 2: Gas dynamics 2.1 Perturbations from the equilibrium 2.2 Equations of motion of the gas flow in cylindrical coordinates 11 2.3 Onsager’s pancake equation 17 2.3.1 Onsager’s equation for flow near the rotor wall 17 2.3.2 Boundary conditions at the rotor wall 18 2.3.3 Boundary conditions at the top and bottom end cap 20 Chapter 3: Solution to Onsager’s equation 22 3.1 Introduction to finite element method 22 3.2 Finite element method for the Onsager’s equation 23 3.2.1 Weak formulation 23 3.2.2 The finite element algorithm 25 3.3 Example and program 30 Chapter 4: Separation theory 38 4.1 The diffusion-convection equation 38 4.2 The Onsager-Cohen method 39 Conclusion and proposal 43 References 45 Appendix: Code of program 47 Symbols a : radius of the rotor dm : an infinitesimal mass of the gas dp : a differential of pressure of the gas dr : a differential of radius of the centrifuge E : total energy of the gas : centripetal force k : thermal conductivity of the gas m : mass of a given amount of the gas M : molecular mass of the gas p : pressure of the gas Pr : Prandtl number, Q : heat added to the gas r : radial position of the interested point Re : Reynolds number, T : temperature of the gas T0 : temperature of the gas in equilibrium conditions V : gas velocity Vr : component of gas velocity in the radial direction Vz : component of gas velocity in the axial direction Vθ : component of gas velocity in the azimuthal direction W : work done on the gas z0 : height of the rotor : angular velocity of the gas ρ : density of the gas : volume of the gas µ : viscosity of the gas ρ : density of the gas at the cyclinder wall γ : specific heat ratio List of figures Page Fig.1.1 Internal circulation in a gas centrifuge Fig.1.2 Schematic of Groth ZG5 centrifuge Fig.2.1 Cylindrical coordinates applied to the rotor of the gas centrifuge Fig.2.2 An infinitesimal volume of the gas and the pressures exerted on it 10 Fig.2.3 The domain of interest in Onsager’s equation 18 Fig.3.1 Partition of the intervals Fig.3.2 Stretched grid (the rotor wall is on the left) 28 and 27 Fig.3.3 The temperature of the rotor wall and end caps for thermal drive of a centrifuge .30 Fig.3.4 The general steps of the computer program 32 Fig.3.5 The algorithm for determining the interval [q1,q2] in the case of η-integral 35 Preface In the context of energy crisis in the world, the nuclear power attracts the interest of many nations Nowadays, the nuclear power plants provide 13-14% of the world electricity and this percentage grows relentlessly [14] However, uranium, the fuel for nuclear power plants, is unavailable in the coarse form In nature, uranium is found as U-238 (99.27%), U-235 (0.72%), and a very small amount of U-234 (0.01%) [15] In a nuclear fission, U-235 plays an important role because of its high cross section in thermal neutron absorption In order to process uranium ore for nuclear fuel, the percentage of U-235 needs increasing, this is called uranium enrichment In industry, the gas centrifuge method is most used to enrich U-235 due to its small consuming power compared with other methods In this method, the U238 component is separated from U-235 by the difference in centripetal force exerted on them Therefore, conducting the theoretical analysis of the gas centrifuge is necessary in nuclear fuel research Such calculations can be used to guide experiments and provide the understanding of how the flow affects isotope separation For all these reasons, the research on the theory of uranium enrichment by the gas centrifuge is undertaken There are two points considered in this theory: the gas dynamics of the fluid circulation in the boundary layer on the rotor wall as well as on the top and bottom end caps of the gas centrifuge including the Onsager’s equation and the separation theory including the diffusion-convection equation This thesis is organized into the following four chapters:  Chapter 1: Introduction to the gas centrifuge  Chapter 2: Gas dynamics  Chapter 3: Solution to Onsager’s equation  Chapter 4: Separation theory An overview of uranium enrichment by the gas centrifuge is shown in the first chapter It begins with the presentation of the principle of this method Through the research stages, the gas centrifuge design has a lot of changes However, all the machines are based on the same principle which is presented in this text The next part in this chapter mentions the history of gas centrifuge development This method, in the beginning, was complex and contained a great deal of weakness Under the influence of growing market for enriched uranium, many scientists and engineers had improved the machine and increased its efficiency in isotope separation The development of gas centrifuge has provided a significant amount of enriched uranium in low and high grade The last part of this chapter describes Groth’s ZG5 machine, a type of gas centrifuge A significant advantage of this machine is it controls the internal circulation by convective heating and cooling, this permitted attainment of 75 percent of the maximum theoretical separative capacity [7] The gas initially rotates as a rigid body but the flow field becomes complex when circulation currents are generated The analysis of gas dynamic in the centrifuge is presented in the second chapter There are six equations describing the flow which resulted from Onsager’s analysis [3] These equations were used to evolve Onsager’s equation The requisite boundary conditions to solve the equation are also considered In the third chapter, the finite element method proposed by Max D Gunzburger to find an approximate solution to Onsager’s equation is presented The computer program is written to implement the algorithm described in the method In the last chapter, we mention the separation theory The study of this theory is a crucial step in gas centrifuge design Its object is to determine the product and waste concentrations [3] The diffusion-convection equation, the essence of separation analysis, must be solved numerically Seeking an approximate solution to this equation is the goal of our research in the future Chapter INTRODUCTION TO THE GAS CENTRIFUGE 1.1 Principle [3], [7], [12] The gas centrifuge consists of a long, thin, vertical cylinder made of material with high strength-to-density ratio This device rotates about its axis with high peripheral speed Inside the cylinder, the gas molecules are accelerated by the centripetal force Due to the difference in mass, the heavier molecules move towards the wall while the lighter ones remain close to the center If the countercurrent flow is induced, the lighter molecules are carried to the top while the heavier ones settle at the bottom, from which they can be continuously withdrawn There are four methods to induce the countercurrent flow:  Wall thermal drive: convection currents are set up by establishing temperature gradient along the wall  End cap thermal drive: results from removing heat from the top end of the rotor and adding heat through the bottom end cap  Feed drive: the flow is induced by pumps external to the machine which inject the feed stream at or near the middle of the rotor  Scoop drive: the flow is developed by the interaction of the internal scoops and baffles with the spinning gas 1.2 History [2], [7] The concept of separating isotopes in centrifugal field was first proposed by Lindemann and Aston in 1919 The first successful use of this method was by Dr Jesse Beams in 1935 at the University of Virginia He demonstrated experimentally the separation of chlorine isotopes employing an ultra high speed centrifuge In 1939, Urey suggested use of countercurrent flow to multiply single-stage enrichment by heating the bottom of the rotor and cooling the top In 1940, an alternative thermal means of generating the internal circulation was proposed by Bramley, Brewer, Martin and Kuhn The German engineer, G Zippe, invested a simple method of inducing the countercurrent flow by internal scoops and baffles during the World War II After the war, he moved to U.S and continued his work Feed Rotor Enriching section Casing Enriched stream Feed injection Post Stripping section Depleted stream Fig.1.1 Internal circulation in a gas centrifuge with Beams at the University of Virginia from 1958 to 1960 Groth and his coworkers, in Germany, developed and built a series of progressively larger gas centrifuge in 1950s In the 1960s, the United States, Great Britain, the Netherlands and West Germany agreed to set restriction on description of the technology due to the competition for large-scale uranium enrichment Several improved production sized machines were tested between 1970 and 1976 There were thousands of gas centrifuges tested individually and in cascades to evaluate reliability, operability and performance over a wide range of conditions Up to present, the gas centrifuge has been being developed to reach higher capacity for isotopes separation 1.3 Description of Centrifuge [7] Groth’s ZG5 machine is chosen to describe Figure 1.2 is a schematic drawing of this type of centrifuge Feed Outlet Thermocouple Electromagnet Scoop Baffle Case Rotor Stationary tube Baffle Scoop Thermocouple Cooling coil Outlet Fig.1.2 Schematic of Groth ZG5 centrifuge [6] 34 Step 2: Invoke the procedure which is used to save the domain [q1, q2] into an array consisting of elements This procedure gives two outputs: mangj for ( ) and mangn for ( ) Step 3: Take the products of elements of mangj and ones of mangn The output of this step is an array denoted by mangjn Step 4: For each element of mangjn, we do: Substitute ( ) ( ) into mangjn(i) where i denotes the position of the element, a, b are the limits of the interval in η direction corresponding to the element Set Syi as an array consisting of elements By the use of equation (3.32), each element of Syi is determined as follow: () Step 5: Output ∫ ( ) ( ) [ ( ∑ √ * ( √ *] ( ) From the equation (3.31) and the definitions of ( ), ( ), ( ) and ( ), Alk=0 if |j-n| or |i-m| This means that A is a banded matrix In addition, Alk=Akl or A is a symmetric matrix Both of these characteristics are used to simplify the programming With regard to the approximation that ( ) ( ) (see page 31) and equation (3.26), equation (3.23) becomes: ∫ The algorithm to determine ∫ ( ) ( ) ( ) ( ) 3.33 is fairly similar to the one discussed above The matrix c is then figured out by: 3.34 where A-1 is the inverse matrix of A 35 Start Input j,n No q1= ηn-2 q2=ηj+2 Yes q1= ηj-2 q2=ηn+2 Yes Output S=0 or No Yes q1ξM q2=ξM No Output S=[q1,q2] End Fig.3.5 The algorithm for determining the interval [q1,q2] in the case of η-integral 36 The result of the calculations is presented with M=N=2, ξT=8 In this case, A , d and c is the column vectors of R5 The computer program’s is a matrix of output is: 3.35 [ ] to the Onsager’s equation is given by Therefore, the approximate solution equation (3.19a): ( )∑ ( ) 3.36 By the use of equation (2.48), the axial momentum ρeqw is determined as follow: , ( )- , ( ) ( ) ( ) ( ) ( )- is a piecewise continuous function By the use of equations (3.14) and (3.15), the function is defined by: for for { (3.37) The stream function is defined in (2.44) and in the case we are considering, it is given by: , , ( )- , ( ) ξ ξ ( ) ( ) ξ ξ ( ) for for ( )ξ ξ 3.38 37 By the use of equations (2.47), the radial momentum ρequ is defined as follow: { for for 3.39 The roles of radial and axial momenta will be shown in the next chapter when the separation theory is considered 38 Chapter SEPARATION THEORY 4.1 The diffusion-convection equation [3], [7] In the previous chapters, only the mono constituent gas is considered Now, we pay attention to a binary mixture of gases consisting of 235UF6 and 238UF6 In this case, equation (2.9) comes into: * ( ( ( ) )+ * ) ( )+ 4.2 where x is the mole fraction of light component; M1, M2 are molecular weight of light and heavy component of gas respectively Dividing (4.1) by (4.2) results in: ( ) ( * ) ( )+ 4.3 where Taking derivative of equation (4.3) yields: ( / ) The equation (4.4) is considered in the equilibrium conditions The composition gradient is given by: ( ( * 4.5 ⃗ , has two components: The mass velocity of gas, direction and ) in the axial The angular mass velocity in the radial is equal to zero The fact that is a solenoidal vector field in the gas centrifuge gives: ⃗ where ( ) 4.6 39 ρ* ( (4.7) * + 4.8 Substitute (4.7), (4.8) and J𝜃=0 into (4.6) and make use of the approximation : ( * ) ( ) + 4.9 This is the diffusion-convection equation which determines the product and waste concentration for specified feed concentration In this equation, the density-self diffusion coefficient product, , is a constant at a given temperature 4.2 The Onsager-Cohen method [3], [7] Making use of the following assumptions which are proposed by Cohen to simplify equation (4.9): x(1-x) is treated as a constant is neglected is independent of r ρeqw is independent of z Multiplying (4.7) by r and integrating with respect to from to result in: ( In this equation, ( ) ) ∫ because v is so smaller than that it can be neglected The net flows (mass of the gas passes the section of the centrifuge in a unit of time) in the enriching section and the stripping section are respectively given by: ∫ a 40 ∫ b The minus appears since the depleted stream runs towards the bottom end cap (the direction of enriched stream is chosen as the positive) The net flow of light component is: ∫ ∫ ∫ where xP is the mole fraction of light component in enriched stream (see fig.1.1) Integrating the first term of the right hand side by parts yields: ∫ ∫ ∫ ∫ Take account of equation (4.11a) and substitute (4.10) into (4.13): ( ) ( ∫ ) ∫ ∫ (∫ ) ∫ Because 4 is independent of r, (4.14) can be rewritten in the form: ( ∫ ) ( ∫ ) ∫ (∫ ) Equation (4.15) is called the axial enrichment equation Instead of seeking the solution to the diffusion-convection equation, we can look for the simpler solution to equation (4.15) The aim of this work is determining the product and waste compositions: ( ∫ ∫ ) ( ( ) ( ) 6a ) where xW is the mole fraction of light component in depleted stream 6b 41 Cohen used the following notation to simplify the appearance of equation (4.15): ∫ ( ) ∫ , ( )- where F(r) is the flow function which is given by: ( ) ∫ By the use of this notation and with respect to the fact that variation of x with r has been neglected, (4.15) becomes: ( ) ( ) The similar equation can be derived for the stripping section by substituting –W for P in (4.17): ( ) ( ) Equations (4.17) and (4.18) cannot be solved analytically, so the following numerical scheme is considered to find the approximate solution to them: The parameters of the centrifuge are specified The gas dynamics analysis in chapter and the solution to Onsager’s equation in chapter provide the axial mass velocity A value of is guessed The value of xw is calculated from the value of xp by the relation: where xF and L are the mole fraction of light component in the feed stream and the net flow of feed stream respectively Equation (4.17) is integrated numerically from the position of feed injection z= zF to the top end cap z=z0 The integration of equation (4.18) is 42 conducted over the integral [0, zF] The result of the calculation, x, is substituted into equation (4.16a) to determine xP The result of step is compared with the guessed value of xP in step If they not agree with each other, the initial value is improved and the calculation repeated If the guessed in accord with the calculated value of xP, the separative power (kg U-235/year) can be determined This is an important quantity to define the capacity of the gas centrifuge 43 Conclusion and proposal Conclusion In this thesis, the gas dynamics and thermodynamic analyses of the countercurrent flow in a gas centrifuge are reported Relying on the equations of motion of the gas flow in cylindrical coordinates and the characteristics of the flow in a gas centrifuge, the set of equations describing the movement and the state of the flow is derived The steps to evolve the Onsager’s pancake equation and its boundary conditions are also summarized In order to find solution to the Onsager’s equation, we rely on the finite element method described in Gunzburger’s paper [8] to write the computer program by the use of Maple programming tool This program implements the algorithm to derive the approximate solution to the equation In our work, only the homogeneous form of Onsager’s equation is considered and the solution is presented in the case of linear gradient temperature along the wall The results of our program are presented in (3.35), (3.37), (3.38) and (3.39) However, as many materials of the gas centrifuge technique are not published, the gas parameters we used in our program are not completely valid We try to restrict the error as much as possible by the use of parameters of the gas fairly similar to UF6 (B, Pr, Re and γ in Table 3.1 are the parameters of SF6 employed in the program [6]) In addition, the Gauss–Legendre quadrature for points reduces significantly the program’s validity Moreover, because the procedures to calculate the variables ρw, and ρu are not completed, their values only presented with M=N=2 The small number of sub-intervals makes the definitions of these variables vulnerable In the last chapter, the concept of separation theory and the Onsager-Cohen method to solve the diffusion-convection equation are presented This chapter clarifies the meanings of the quantities considered in the previous ones The axial enrichment equations in enriching and stripping sections play an important role in determining the separative unit, the key to the gas centrifuge design Nevertheless, 44 they require a numerical method to find approximate solutions, which needs the help of a computer program Developing the program to solve this problem is the aim of our work in the future Proposal The computer program we construct in our work has a lot of drawbacks To obtain a more exact solution for Onsager’s equation, the following tasks need undertaking: Obtain the exact parameters of the gas centrifuge, especially the ones of the gas UF6 Involve the feed injection and gas removing In other words, we consider Onsager’s equation in the general form with the right side different from zero To increase the accuracy of the calculation, the Gauss–Legendre quadrature for points should be used in the computer program This method reduces the error of computing the integrals Optimize the algorithm to reduce the time of calculation If the consuming time decreases, the program is able to figure Master potential out in the case of the interval’s width, h, approaches zero Implement the stretched grid as mentioned in 3.2.2 Complete the procedures to determine the variables ρu, ρw, Take account of the other types of driven mechanisms In a real gas centrifuge, the different types of mechanisms to generate the countercurrent flow are often employed together Therefore, the total circulation rate is a sum of the contributions due to driving mechanisms After obtaining the solution to Onsager’s equation, the computer program needs developing to determine the approximate solution to the axial enrichment equation This is a crucial step to expand the gas centrifuge analysis 45 REFERENCES English [1] Claes Johnson (1987), Numerical solution of partial differential equations by the finite element method, Cambridge University Press, New York [2] Dean A Waters (2003), “The American gas centrifuge, past, present, and future”, SPLG workshop [3] Donald R.Olander (1981), “The theory of uranium enrichment by the gas centrifuge”, Progress in Nuclear Energy, 8, pp 1-31 [4] Frank M.White (1991), Viscous fluid flow, 2nd Edition, McGraw-Hill, New York [5] H.G.Wood and J.B.Morton (1980), “Onsager’s pancake approximation for the fluid dynamics of a gas centrifuge”, J.Fluid Mech, 101(1), pp 1-31 [6] L.D Cloutman, R.A Gentry (1983), “Numerical simulation of the countercurrent flow in a gas centrifuge”, Los Alamos National Laboratory [7] Manson Benedict, Thomas H Pigford, Hans Wolfgang Levi (1981), Nuclear chemical engineering, 2nd Edition, McGraw-Hill, New York [8] Max D Gunzburger (1982), “A finite element method for the Onsager pancake equation”, Computer methods in applied mechanics and engineering, 31, pp 43-59 [9] M de Stadler, K Chand (2007), “A finite-different numerical method for Onsager’s pancake approximation for fluid flow in a gas centrifuge”, Lawrence Livermore National Laboratory [10] Nguyen Hoang Anh (2011), Methods of enriching U-235, Undergraduate thesis, University of Science, HCM city Vietnamese [11] Đặng Văn Liệt (2004), Giải tích số, NXB Đại học Quốc gia, TP Hồ Chí Minh Website [12] Gas centrifuge - http://www.coleparmer.com/TechLibraryArticle/30 [13] Linearization - http://www.scholarpedia.org/article/Siegel_disks/Linearization 46 [14] Nuclear power - http://www.world-nuclear news.org/newsarticle.aspx?id= 27665&terms=another+drop+ [15] Uranium isotopes - http://www.globalsecurity.org/wmd/intro/u-isotopes.htm 47 Appendix – Code of program The program consists of files, each of them is used to save the procedure programmed to solve particular problem When a procedure is used, the file containing it is loaded into computer’s memory The purpose of this way of programming is to save the memory and as a result increase speed of the program The files in the program are: Page laptrinh 49 tinhAlk 50 tinhcum1theox 51 tinhcum1theoy 52 tinhcum2theox 53 tinhcum2theoy 54 tinhcum3theox 55 tinhcum3theoy 56 tinhdl 60 tinhsjnTH1 59 tinhsjnTH1cum2 60 tinhsjnTH2 61 tinhsjnTH2cum2 62 tinhsjnTH3 63 tinhsjnTH3cum2 65 tinhsjnTH4 68 tinhsjnTH4cum2 71 tinhsjnTH5 74 tinhsjnTH5cum2 76 tinhsjnTH6 78 tinhsjnTH6cum2 80 tinhsn 82 tinhxichmaimTH1 85 48 tinhxichmaimTH1cum2 86 tinhxichmaimTH1cum3 87 tinhxichmaimTH2 88 tinhxichmaimTH2cum2 91 tinhxichmaimTH2cum3 93 tinhxichmaimTH3 95 tinhxichmaimTH3cum2 98 tinhxichmaimTH3cum3 100 tinhxichmaimTH4 102 tinhxichmaimTH4cum2 103 tinhxichmaimTH4cum3 104 tinhxichmaimTH5 105 tinhxichmaimTH5cum2 106 tinhxichmaimTH5cum3 107 thutuc (containing procedures: tinhs1, tinhijmn, tinhmn, xacdinhpts1, tinhmuyk, tinhnuyk) 108 [...]... first order only [3], [5] It is regarded as a basis of the analysis of the countercurrent flow in gas centrifuge 17 2.3 Onsager’s pancake equation 2.3.1 Onsager’s equation for flow near the rotor wall [3] By the use of equation (2.11), only regions next to solid boundaries contain gas at significant density The gas kept close to the rotor wall by the strong centrifugal force flows primarily in axial... the first law of thermodynamics is conveniently written as a time rate of change [4]: 2.35 where E, Q and W are the total energy of the gas, the heat added to the gas and the work done on the gas respectively This equation can be linearized about the equilibrium solution, which results in: Hence, (2.35) becomes: 2.36 The gas pressure and density primarily change in radial direction, thus dW is work... specific heat ratio respectively The radial and axial mass flows are: 18 2.47 and 2.48 2.3.2 Radial boundary conditions [3], [5], [8] Onsager’s equation requires boundary conditions to be resolved There are 6 radial boundary conditions Three of them are apply at the rotor wall and the others prescribe the gas behavior at the inner edge of Stewartson layer As we discussed previously, due to the high speed... to the high speed of rotation the gas is confined to the region near the rotor wall Hence, the Onsager’s equation is only considered in the domain which is given by the set * +, where and ξT is chosen to simulate “the top of the atmosphere” at which the gas density becomes extremely small Top end cap η=ηT Rotor Rotor D axis wall η= ξ=A2 ξ=ξT ξ=0 Bottom end cap Fig.2.3 The domain of interest in Onsager’s... boundary conditions at the inner boundary of Stewartson layer 2.3.3 Boundary conditions at the bottom and top end cap [3], [5], [8] Ekman layer analysis provides axial boundary conditions for the Onsager’s equation In this layer, the gradients of velocity are primarily in z-direction, thus we 21 can neglect all radial derivatives in the set of conservation equations Imposing the Carrier-Maslen method for... was introduced by engineers in the late 5 ’s and early 6 ’s for the numerical solution of partial differential equations in structural engineering During the 6 ’s and 7 ’s, the method was developed by engineers and scientists and it had many applications in various areas of science and engineering At present, the finite element method is used extensively in structural engineering, material science, fluid... one Nevertheless, the variational problem cannot be solved exactly; this leads to a finite-dimensional variational problem This problem is equivalent to a system of linear or nonlinear equations On the whole, to find approximate solution to a given differential or integral equation by the finite element method, we have to go through basically the following: 1 Define variational form of the given problem... D Gunzburger We now review his method and analyze the algorithm which is used to solve our problem 3.2.1 Variational formulation To solve the Onsager’s equation, the given problem needs reformulating into a variational form (Galerkin form) Assuming is a solution to our problem, we rewrite (2.42) as follow: ( ( ,( where ) ) ) 3 /, ( ( ) ) { /} and Multiplying (3.1) by a smooth function 𝛟 and then... radial gradient of the temperature and the azimuthal velocity are assumed to vanish at the top of the atmosphere: ( * This results in the boundary condition: ( * 2.57 Because the variables are always only related to derivatives of the Master potential, therefore, without loss of generality, the following condition can be deduced: ( ) ( * 2.58 At the inner edge of the Stewartson layer, the axial velocity... pressure and shear stress component The pressure can be determined by equation (2.4): 2 9 The shear stress component in radial direction defined in cylindrical coordinates is [4]: θ ( θ (2.2 ) * where ( Relying on the fact that * , equation (2.18) becomes: ( or ( * ) 13 , Consider the function ( [ ) - ( )] 2.2 , f is linearized by expanding around the point t=0 [13]: ( ) ( ) ( )( ) where X represents other ... 50 tinhcum1theox 51 tinhcum1theoy 52 tinhcum2theox 53 tinhcum2theoy 54 tinhcum3theox 55 tinhcum3theoy ... electricity and this percentage grows relentlessly [14] However, uranium, the fuel for nuclear power plants, is unavailable in the coarse form In nature, uranium is found as U-238 (99.27%), U-235 (0.72%),... For all these reasons, the research on the theory of uranium enrichment by the gas centrifuge is undertaken There are two points considered in this theory: the gas dynamics of the fluid circulation