The mathematical model of non-stationary dividing processes of uranium enrichment in industrial centrifugal cascades which can be used in a computer simulator to prepare experts in dividing production and application as an expert system in the automated control system of technological circuit has been developed and realized

Filimonov, S. V.

Goldobin, D. N.

Orlov, А. А.

Skorynin, G. М.

Electronic archive of Tomsk Polytechnic University

Technical sciences71IntroductionIt is known that generally nonstationary processes individing processes are described by nonlinear equationof the second order in partial derivatives (with respect totime and the number of step) of parabolic type with variable coefficient, the analytical solutions of which areconstructed only for a limited number of problems [1].Much more perspective methods of researching nonstationary processes are those of numerical solution of equation describing these processes. Many works are devotedto the questions of numerical research of nonstationaryprocesses in dividing cascades, e. g. [2]. However, mathematical methods investigated in these works either have alimited practical application or take much time to solvethem numerically or do not take into consideration thefeatures of process flowsheet construction of centrifugalproduction. In this case the simplest symmetrical counterflow rectangularsectional cascade [3] in which thesections of the steps operate in the same hydraulic conditions. Meanwhile, the effect of centrifuge enrichment issignificantly higher (about 100 times) than that of diffusive devices and the length of centrifugal cascades is essentially less. As a rule, steps operate in dissimilar hydraulicmodes. Under these conditions the contribution of eachstep to the cascade operation increases, and, therefore,investigating the nonstationary processes it is importantto know the state of every step.In the work the mathematical model describing nearly all types of nonstationary processes both in a separate step and in cascades of optional construction differing from traditional symmetrical counterflow cascadeshas been proposed. Besides, all the multicascadetechnological circuit presenting a combination of cascades with their connecting lines of multicascade communications (MCC).The model is an addition to the developed earliermathematical model of nonstationary hydraulic processes in multicascade technological circuit (TC) ofcentrifugal plant [4, 5].1. Description of mathematical model of nonstationary dividing processes Multicascade TC of dividing plant consisting ofKcascades (fig. 1) is considered. Each cascade m(1≤m≤K) consists of Nm steps with numbering fromdump to selection.Рис.1. Многокаскадная ТСThe circuit of step connection in cascade can be optional, e.g. the socalled parallelseries (fig. 2).The step of number n consists of Snm parallel connected sections; section inm (1≤inm≤Snm) consists of Jinm parallel connected gas centrifuges. Gas centrifuges in thesection are considered to be identical; the sections canbe various (different number of GC, different geometrical sizes of communication, etc.). The design factor К(of order), Nm (of order 4...25 are defined by the nomenclature of the production, change of bundling andtechnological characteristics of GC.Fig. 2. Parallelseries circuit of step connection in cascadeIn the work involved a onevolume model of dividing step. Each step of cascades of multicascadetechnological circuit of centrifugal production is considered to be filled with binary mixture with the sameconcentration of isotope 235U equal to its concentrationin the supply collector of this step. Gas content of divin-1 n??1nG?nGn+1 ??1nG?nG??1nG??2nG??? 3,2 nnGm1K?1T?1T?mT?mT?KT?KT0TUDC 661.87:519SIMULATION OF NONSTATIONARY PROCESSES IN INDUSTRIAL CENTRIFUGAL CASCADES OF URANIUM ENRICHMENT S.V. Filimonov, G.М. Skorynin, А.А. Orlov*, D.N. GoldobinMC «Electrochemical plant», Zelenogorsk town*Tomsk polytechnic universityEmail: orlov@phtd.tpu.edu.ruThe mathematical model of nonstationary dividing processes of uranium enrichment in industrial centrifugal cascades which can be used in a computer simulator to prepare experts in dividing production and application as an expert system in the automated control system of technological circuit has been developed and realized.Bulletin of the Томsк Pоlyтеchnic University. 2006. V. 309. № 472ding step is considered to be equal to the sum of gascontents of all the volumes forming the step and takeninto consideration in calculation of TC nonstationaryhydraulics [4]. At the output from the step (in the collectors of selection and dump) of concentration determined by the dividing properties of the steps, correspondingly higher and lower the supply concentration.The main equation of nonstationary division process is that of the balance of the enriched componentsupplying mixture in step n of cascade m of multicascade ТС:(1)where K – a number of ТС cascades; S – the quantity oftechnological sections in step n of cascade m; Mnm – gascontent of step n of cascade m; C0nm – concentration ofsupplying mixture of step n of cascade m; T0n(t), K0n(t) –consumption and concentration of the external supplyflow; δRkn – a sign of connection of selection flow TRknwith concentration KRkn of cascade k with the step supply n (k≠m) of cascade m (k≠m); δWkn – a sign of connection of dump flow TWkn with concentration KWkn ofcascade k with the supply of the step n (k≠m) of cascade m (k≠m); δnTp – a sign of connection of dump flow GTpwith concentration CTp of step n of cascade m with stepsupply n of cascade m; δnPw – a sign of connection of selection flow GPw with concentration CPw of step w of cascade m with step supply n of cascade m; δOP – a sign ofselective step (0 or 1); GЛNm, TOPm – correspondingly selection flow of selective step Nm (m = 1...K) and cascadem; CЗакр,m – flow concentration of cascade spin m; L01inm– supply flow of section i of step n of cascade m.In general case the forming the equation magnitudes(1) of gas content steps, cascade supply flows (externalwith respect to TC and intercascade flows being the selection and dump flows of other cascades as well as interstep flows) are the time functions. These functionscan be obtained by solution of hydrodynamic equationsconsidered in the works [4, 5].The concentrations of intercascade supply flow KRknand are KWkn defined through the corresponding concentrations of selection and dump flow of cascades taking into account transport delay τzad, by which the time of passinghydraulic excitement along the appropriate MCC lines isunderstood. The time τzad is defined by the solution of continuity equations for a steady onedimensioned flow isolated from the environment [6] with respect to unknown velocity of gas flowing υi along iarea of MCC line.where G – calculated value of flow in the MCC line (thesame for all areas of MCC line) g/sec; Si – square of section of iarea of MCC line, sm2; – gasdensity in iarea of MCC line, g/sm3; μ – molar mass ofgas, g/mol; Pi – calculated value of gas pressure in iarea of MCC line, mmHg; Rг – universal gas constant,erg/(molK); Ti – gas temperature, К; υi – gas velocityin iarea of MCC line, sm/sec.The values of pressure in i area of MCC line are defined form the accepted turbulent character of gas flowing known from the solution of hydraulic problem of some pressure value (e.g. the pressure in supply collectorof selective step) and assigned values of geometrical sizes of these areas.Selective steps of cascades have some features connected with the fact that the part of the selection flowsthrough the governors of spin come back as a supply ofthese steps. Besides, spin flow is fed to the parts of cascades after passing the purifying cascade (PC). Hence, theconcentration of spin flow CЗакр,m changes with some transport delay with respect to concentration of selection flowof selective step. The time of transport delay is also definedin the problem of nonstationary hydraulics [4, 5].Equation (1) is set up for each step of cascades ofmulticascade TC. The obtained set of equation consistsof KNmdifferential equations with 3 KNm unknowns,which are the concentrations of the step supply flow, ofselection and dump (C0nm, CPnm, CTnm). To reduce the dimension of the system let us use the equation of conservation of light component and the relation defining total coefficient of step division χnm.(2)where θnm – coefficient of flow division ( ).(3)From equations (2) and (3) one can define CTnm:(4)By means of substitution of the expressions for CTnmand C0nm set of equations (1) amounts to the set of differential expressions relatively KNm of unknown CPnm as time functions. The solution of this set of equations at thegiven initial values of CPnm simulates nonstationary dividing processes in TC cascades.2. The algorithm of equation solution of nonstationary divisionHaving denoted the right part of the equation (1) after substitution in it the expressions (2) and (4) throughψnm(t) we obtain:(5)where fCt(CPnm) – the function of concentration of selection step n of cascade m determined by the expression (4).The set of equations (5) presents a Cauchy problemwith the given initial conditions. The Cauchy problem,[ ( (1 ) ( ))]( ),nm nm Pnm nm Ct Pnm nmd M C f Ctdtθ θ− − = Ψ.( 1)PnmTnmnm Pnm nmCCCχ χ= − −(1 ).(1 )Pnm TnmnmTnm PnmC CC Cχ −= −0PnmnmnmGGθ =0 ( 1) ,nm nm Pnm nm TnmC C Cθ θ= + −1333iiг iPR Tμρ =,i i iG S ρ υ=00 0111 1 1, 0 01 01( )( ) ( )( ( )) ,Knm nmn n Rkn Rkn RknkK N nWkn Wkn Wkn nTp Tp Tp nPw Pw Pwk p n wSOP Закр m ЛNm Pm inm nmid M CT t K t T KdtT K G C G CC G T t L Cδδ δ δδ=−= = + === + ++ + + ++ − −∑∑ ∑ ∑∑with few exceptions, does not have analytical methodsof solution. In the numerical method the values of thesoughtfor functions in discrete moment of time are calculated instead of seeking continuous functions of time,the set of equations being replaced by difference equations in this or that way. In the algorithm applied, similar to hydraulic problem [4, 5], Euler implicit modelwith recomputation [7], which possesses the third order of accuracy on the step and the second interval is used. According to it we have:(6)Having written down the equations (6) for all thesteps of TC cascade, we obtain the set of algebraic nonlinear equations relatively CPnm(K+1). Nonlinearity is explained by the presence of terms containing the expressionfCt(K+1)CPnm(K+1). At each time step the linearization of its nonlinear terms is performed. The algorithm proposed itconsists in substitution of the fuction fCt(K+1)CPnm(K+1) the equation of tangent in point .(7)Coefficients aTnm(K) and bTnm(K) are calculated from thecondition of equality of linear (7) and nonlinear (4) expressions as well as their derivatives in some point C0Pnm.(8)(9)From the equation (8, 9) we obtain:After linearization we get the set of linear equationwith respect to unknowns CPnm(K+1), which are approximationto the magnitudes CPnm(t+Δt). The set is solved by Gaussmethod of exception [7, 8]. As initial conditions the initial magnitudes CPnm(0) =CPnm(t=0) are taken as calculateddata of initial stationary mode of the TC investigated.The values of coefficients of total division χnm ofsteps are determined by dividing properties of the steps.Dividing properties of a separate centrifuge (GC) aredescribed by empirical dependence defined as a result ofmathematical treatment of experimental data obtainedin testing the equipment of GC of the given type.(10)where g, θ – supply consumption and coefficient offlow division of separate GC.Considering magnitude χ invariant, independent onconcentration of supply flows of the steps one can connect the magnitudes CTnm and CPnm by the formula (4) atoperation of steps in concentration region differingfrom concentration of natural material.Dividing characteristic χnm(g,θ) is defined at GCoperation in stationary conditions when there is a substance balance and light component. Application of thischaracteristic in nonstationary process is explained bythe fact that the processes of attaining equilibrium in aseparate GC proceed significantly faster than they do inTC cascades. The centrifuge is considered as one operating in stationary conditions which is slowly changing.According to the model described in BorlandDelphi 5.0 medium bundled software is realized and thecalculations for the following cases of nonstationaryTC excitements: change of selection and supply flow;transfer of supply delivery points; closing steps and sections of cascades; change of frequency of GC supplyingcurrent; change of equipment loading; change of external supply flow concentration. As a dividing one the approximate model characteristic differing from the realone, but showing the main tendencies of real processproceeding is used.As an example, present the results of numerical calculation of nonstationary processes of division appearing atclosing separate steps of a ninestep cascade (fig. 3).Fig. 3. Time dependence of concentration deflection of selection cascade with respect to reference quantity in standard units (1 s.u.=0,001 %). Closing steps: 1) calculation;2) real – selective and 3) calculation; 4) real – predumpComparative analysis of the pairs of curves 1, 2 and3, 4 permits to conclude that the developed modeldescribes the behaviour of the real object adequately.The increase of enrichment of cascade selectionwith time (1, 2) is explained by the fact that from themoment of step closing the cascade passes to the hydraulic mode with high loading on the remaining steps atwhich the magnitude χ has great value. The broken character curves 2, 4 is explained by the technique of concentration measurement of 235U based on recording ofγradiation of working gas mixture, intensivity of whichhas a probabilistic nature.From the figure it is also seen that the influence ofselective step on the final product is significantly higherthan that of dump steps.The calculations performed show that properties ofquick action of the model permit to carry out calculations1 234-300-250-200-150-100-50050 0: 0: 0  0: 3:20  0: 6:40  0:10: 0  0:13:20  0:16:40  0:20: 0  0:23:20  0:26:40 0:30: t, ???.???????????????????????????????????,?.?.( , ),nm nm gχ χ θ=( ) 2 ( )( )( ) ( ) ( ) 2( )( )( ) ( ) ( ) 2( ) ( 1),[ ( 1)].[ ( 1)]K Kk Pnm nmTnm K K Knm Pnm nmKk nmTnm K K Knm Pnm nmCaCbCχχ χχχ χ−= − − −= − −'20( ) .[ ( 1)]nmTnm Tnmnm Pnm nmC bCχχ χ= =− −000,( 1)PnmTnm Tnm Pnmnm Pnm nmCa b CCχ χ = +− −( 1) ( ) ( ) ( 1) .K k k kTnm Tnm Tnm PnmC a b C+ += +( 1) ( 1) ( 1) ( 1) ( 1) ( 1)( ) ( ) ( ) ( ) ( ) ( )( 1) ( )( (1 ) ( ))( (1 ) ( ))( ).2θ θθ θ+ + + + + ++− − −− − =Δ= Ψ + ΨK k k k k knm nm Pnm nm Ct Pnmk k k k k knm nm Pnm nm Ct Pnmk knm nmM C f CM C f CtTechnical sciences73Bulletin of the Томsк Pоlyтеchnic University. 2006. V. 309. № 474of multicascade technological circuit efficiently in thecondition of real time. Thus, for example, the time of calculation of fourhour time interval of nonstationary process for a fourcascade TC with total number of steps ofthe order of 50 and the time step equal to 1 sec. amountsabout 140 sec. In this connection the developed modelhas found application in ACSTC as an expert system. Besides, the developed model is used in computer simulatorto train specialists if dividing production [9, 10].ConclusionMathematical model of nonstationary dividing processes in the industrial centrifugal cascades taking intoaccount the construction distinguished features if thetechnological circuit involved in dividing production hasbeen proposed. It presents a set of differential equationsof the first order with specified initial conditions. The algorithm of equation solution describing nonstationaryprocesses in centrifugal cascades is developed.The model realized allows to obtain solutions in real time conditions at optional set of exciting actions andresearch the nonstationary processes occurring at thechange of external flow values, their disconnection; thechange of frequency and current interruption; the change of supply delivery point; failure and realignment ofmultistep governor; switching off the part of the equipment; failure in the operation of equipment of MCC lines; at different combination of these factors.The model is introduced and used in ACSTC of dividing production at the Electrochemical plant an expert system as well as in computer simulator to trainspecialists of dividing production.Literature1. Cohen K. The Theory of Isotope Separation as Applied to the Large– Scale Production of U235 // National Nuclear Energy Series. –1951. – V. 1B. – Div. 3. – 156 p.2. Laguntsov N.I., Levin J.V., Sulaberidze G.A. On the calculationwith computer of nonstationary processes in multistep cascades fordivision of bicomponent isotope mixtures // Engineering physicaljournal. – 1976. – № 3. – V. 31. – P. 507–513.3. Uranium enrichment / Ed. by S. Villani. Trans. from English, editedby I.K. Kikoin. – Moscow: Energoatomizdat, 1983. – 320 p.4. Filimonov S.V., Skorynin G.М., Goldobin D.N. Simulation of nonstationary processes in TC of ACS of industrial uranium enrichment// Proceedings of high schools. Physics. – 2004. – V. 47. – № 11. –P. 173–178.5. Filimonov S.V., Skorynin G.М., Orlov А.А., Goldobin D.N. Simulation of nonstationary hydraulic processes in the industrial centrifugal cascades of uranium enrichment // Proceedings of Tomsk polytechnic university. – 2006. – V. 309. – № 3. – P. 85–88.6. Deich М.J. Technical gasodynamics. – Moscow: Energia, 1974. –592 p.7. Pirumov U.G. Numerical methods. – Мoscow: Drofa, 2003. – 221p.8. Bakhvalov N.S. and others. Numerical methods. – Moscow: Nauka,1987. – 598 p.9. Vlasov V.А., Shubin А.N., Filimonov S.V., Orlov А.А., GoldobinD.N. Kolpakov G.А. and others. Simulator for operations staff ofproduction on uranium isotope division by centrifugal method //Proceedings of Tomsk polytechnic university. – 2004. – V. 307. –№ 1. – P. 176–181.10. Pat. 2250509 RF. IPC6 G09В 9/00. Simulator of control system operators of technological process of uranium isotope division by centrifugal method in standard and emergency conditions / V.А. Vlasov,А.А. Orlov, А.N. Shubin, S.V. Filimonov, G.А. Kolpakov. Published.20.04.2005, Bull. № 11.

Simulation of non-stationary processes in industrial centrifugal cascades of uranium enrichment

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