The model of nonstationary hydraulic and dividing processes in rectangular symmetrical counterstream centrifugal cascades is considered. The calculation technique of centrifugal cascade parameters of transition processes has been developed. The results of numerical computation are presented

Butov, V. G.

Goldobin, D. N.

Orlov, А. А.

Timchenko, S. N.

Vlasov, V. А.

Electronic archive of Tomsk Polytechnic University

IntroductionEnriching uranium isotopes in dividing centrifugalcascade is a complex and expensive process, therefore,the questions of optimizing process flowsheet and conditions of device operation are of great significance forincreasing production efficiency. One of the ways of solving mentioned problems is design of appropriatemathematical models of the processes at isotope separation in centrifugal cascades.Here, the important role is played by modellingnonstationary hydraulic and dividing processes. Nonstationary processes in dividing centrifugal cascades areexplained by various reasons. For example, it could bechange in frequency of supply current, switching to thenew technological mode, switching off a part of equipment at both normal operation and irregularities in circuit operation connected with closing external cascadeflows, emergency switching off the part of equipment,power cut and etc.Simulation of nonstationary processes is also crucial for realisation of corresponding situations in real time scale in training simulator modelling operators’working places of separation centrifugal production.1.1. Model of counterstream symmetrical cascadeLet us consider rectangularstep centrifugal cascadefor separation of twocomponent isotope mixture. Letsupply flow F be fed to some intermediate point of cascade, but from the ends of cascade the flows Р (selection) and W (dump) be selected. Such a cascade is schematically presented in Fig. 1.The magnitudes F, Cf, W, Cw, P, Cp are the externalcascade parameters, but 5n of the magnitudes Ni, Gi+,Ci+, Gi–, Ci–(i=⎯1,n) shown in the Scheme are the internalones. Those parameters are obtained from the externalones in terms of their definitions and known characteristics of steps. The magnitudes Р, F and W are the timefunctions and connected with the balance equation:The model of cascade necessary for calculation ofnonstationary isotope separation processes includesthe equation set of nonstationary hydraulics and nonstationary transport in cascade. Solution of the obtainedcompatible system allows for determination of time forcascade changing over the stationary conditions, flows,selection and dump concentrations of steps and the cascade in general.The equation set describing nonstationary hydraulic processes is built in the following way [1]:Gas content of cascade consisting of n steps is broken into n volumes of gas.Gas content in each step is defined by gas mass rotating in centrifuge rotors(1)where M0 is the gas content of rotor gas centrifuge (GC)with the volume V=πraH, the radius ra and the height Hare defined for ideal gas with molar mass M, equilibriumdensity ρ, pressure p, at temperature T0, rotating aroundthe axis z as a single whole with the angular velocityΩ=Vm/ra; Vw is the rotor peripheral velocity. Gas contentMi(t) of ith section is calculated as the product of centrifuge quantity included in it by M02 2001 , ,2AaM reM V AA RTρ− Ω−= =,.F P WF P WF C P C W C= +⋅ = ⋅ + ⋅Bulletin of the Томsк Pоlytеchnic University. 2007. V. 310. № 296UDC 661.87.519SIMULATION OF NONSTATIONARY PROCESSES IN CENTRIFUGAL CASCADESV.А. Vlasov, V.G. Butov, D.N. Goldobin, А.А. Orlov, S.N. TimchenkoTomsk Polytechnic UniversityEmail: richi@inet.tsu.ruThe model of nonstationary hydraulic and dividing processes in rectangular symmetrical counterstream centrifugal cascades is considered. The calculation technique of centrifugal cascade parameters of transition processes has been developed. The results of numericalcomputation are presented.Fig. 1. Counterstream symmetrical cascade: i is the number of step; N is the quantity of steps; W, Cw is the flow and concentration ofcascade dump; P, Cp is the flow and concentration of cascade selection; F, Cfi is the flow and concentration of external supplyfed to ith step; Gi+, Ci+ is the flow and concentration of ith step selection; Gi–, Ci– is the flow and concentration of ith stepdump; Gi, Ci is the flow and concentration of ith step supply( ɪ) ( )P,CPW, CWGɡɚɤɪ1 i NN–1F, Cf1G 2GiG NG1G 2G3G1iGiG1iG1NG2NG1NGRewrite this relationship taking into account theequ. (2), replacing ρ=Mp/RT0:(2)In general case the flows entering the ith volume arethe selection flow of previous steps G+i–1; the dump flowof the next steps G –i+1; external supply flow of the givenstep F, being the specified time function t. Besides, incase of closing the neighbouring steps, in ith volumethe selection G+i–2 and dump flows come G –i+2 from thenearest operating step.From the volume of step the selection G+i and dumpflows come G –i out.1.2. Basic equations of nonstationary hydraulicsThese are the equations of substance balance in theset off volumes. The change in gas content in the set offvolume is determined by the difference of entering andcoming out flows.For ith step one can write down the following equations of nonstationary hydraulics:(3)where: δF, δ +i–1, δ +i–2, δ –i+1, δ –i+2 are the indicators (0 or 1) offlow in section, t is the time. Therefore, in general casegas content of the gas centrifuge depends on the frequency of its rotation:The initial condition for the equation (3) is obtainedfor Ω(0)=Ω0, A(0)=A0 and reduces the equ. (2) to the view(4)Where pi(0) is the initial pressure in ith step.1.3. Equations of flowsThe supply flow of separate GC comes through critical cell. The expression for GC supply flow of the ithstep can be written down in the form:where: Ni is the number of GC in the ith step; Ki is thedischarge coefficient; Pi is the supply pressure at GCoutput.Following from laminar character of gas flow in thesorter of GC sinking fraction for the GC dump flow ofthe ith section G –i we have [3]:where ζ –i is the coefficient of hydraulic resistance.The GC selection flow of the ith step G+i we defineby means of formalized universal hydraulic characteristic of GC in the form:where b1i–b6i are the coefficients of hydraulic characteristics in general case depending on the GC type, rotortemperature, concentrations of light impurities, rotation frequency Ω, etc.1.4. Basic equations of nonstationary transferThe transfer of light isotope from dump to selectionin enriching and regeneration parts of cascade are respectively written down in the following form [2, 4, 5](5)(6)where: ε is the coefficient of complete enrichment.Here, because of the smooth change of enrichmentin cascade the number of step can be considered a variable magnitude, then the concentration c=c(n,t),G=G(n,t), τ=τ(n,t) and τ'=τ'(n,t).After substitution of τl and τl' from (5) and (6) intothe equation of transfer process in centrifuge cascadefor enriching part we obtain(7)for regeneration part(8)To solve the equations (7) and (8) one should set theinitial condition and the boundary conditions at selection and dump ends of the cascade. It is assumed thatconcentration distribution at t=0 corresponds to somestationary state of the cascade(9)The boundary condition at selection end of cascade(n=N) at unselection mode of operation is determinedin such a wayHence, we obtain(10)2 (1 ).n Ndc c cdnε== −( , ) 0; (1 ) 0.2l n Nc GN t c G c cnτ τ τ ε=∂⎡ ⎤= = + − − =⎢ ⎥∂⎣ ⎦0( ,0) ( ).ñ n c n=22( ) [ (1 ) ].2Mc G c G c c ct n nε τ∂ ∂ ∂ ′= − − −∂ ∂ ∂22( ) [ (1 ) ],2Mc G c G c c ct n nε τ∂ ∂ ∂= − − +∂ ∂ ∂( ) ( ),l lMc Mct n t nτ τ ′∂ ∂∂ ∂= − = −∂ ∂ ∂ ∂(1 ) ,2lc Gc G c cnτ τ ε ∂′ ′= − + − − ∂(1 ) ,2lc Gc G c cnτ τ ε ∂= + − − ∂221 2 3,4 5 ( ) 6i ii i ii ii iii i i i i iiG Gb b bN NG NGb P b P b PN+− − −⎛ ⎞⎛ ⎞⎜ ⎟+ + +⎜ ⎟⎜ ⎟⎝ ⎠= ⎜ ⎟⎜ ⎟+ + +⎜ ⎟⎝ ⎠2 2( ) ,i i w iiG P PNς− − −= −,i i i iG N K P= ⋅ ⋅00 01(0) (0),ii iAN M V eM pRT A−−=.dM dp dEE pdt dt dt= +1 1 2 21 1 2 2 ,iF i i i ii i i i i idM F G GdtG G G Gδ δ δδ δ+ + + +− − − −− − − − + −+ + + += + + ++ + − −01, .AiiN VM eM Ep ERT A−−= = ⋅0 , 1... .i iM N M i n= ⋅ =Mathematics and mechanics. Physics97At dump end of the cascade (n=1) at dumpfreemode of operation the boundary condition is determined in such a wayHence, we obtain(11)In the case of operation in unselectiondumpfreemode with reservoirs having the gas content MPT and MPЛat heavy and light end of the cascade respectively, theboundary conditions have the view:at    at    (12)At the joints of steps of different capacity the following continuity conditions are to be met(13)Signs «+» and «–» denote the meanings of parameters to the right and left of the kth joint of steps. Aftersubstitution of (5) and (6) into (12) we obtain(14)Here signs «+» and «–» of derivatives have the samemeaning, but c*=c(+nk,t)=c(–nk,t).If at «joint» of steps (n=nk) the supply flow is fed,then instead of the equations (17) the following equations are solvedAfter substitution of (10) and (11) into (19) we obtain(15)2.1. Numerical calculation of nonstationary processes in separation cascadeNumerical calculation of equation system (3) fordetermination of gas content and (7), (8) of nonstationary transfer is performed in the following way:The initial gas content in the steps according to (4)and initial concentration along the cascade in terms ofthe condition (9) is calculated. Then the minimal stepin time Δτ is calculated for the equations (3) and (7),(8). For the equations (7), (8) the step Δn of the derivative n is also introduced.For numerical solution of the equations (3) the Euler’s modified method of the second order of accuracy [6]in time is used. When solving the equations (7) and (8)the twolevel scheme of calculation providing the secondorder of accuracy in the variables t and n [6] is used.In the same manner as in the works [7] for calculation with the general second order of accuracy at approximation of conditions at the selection (10) and dump(11) ends of cascade, at the joint of steps of different capacity without (13) and with supply (14) the onesideddifference schemes are used.which are obtained from superposition of the Taylor series expansion of the function с at the points with the indexes k–1, k–2, k–3, k+1, k+2, k+3 respectively.2.2. Results of calculationsThe calculation of nonstationary separation process in symmetrical counterstream cascade consistingof 6 separating steps has been performed. As an initialcondition the initial concentration CF=0,00711 was set.It was considered that all steps of the cascade are filledwith the gas mixture at the initial moment of time at thespecified magnitude Mн. The external supply flowF=F(s3,t) was set as the time function and that of the point supply flow delivery into the cascade. The results ofcalculation are presented in Fig. 2–4.Fig. 2. Change of relative concentration in time at the selectionend of separating cascadeIn Fig. 2 the time distribution (GN, MN isthe flow and gas content of the last step at stationarymode of cascade operation) of relative concentration at the selection end of cascadeis presented where CDis the required concentration of base product. In Fig. 3 the distribution of relative concentration over thePDCC,PDCC2NNG tMετ =Relative concentrationTime3 2 13 2 1( 2 9 18 11 ) / 6 ,(2 9 18 11 ) / 6 ,k k k kk k k ki i i in n n ni i i in n n nc c c c c nnc c c c c nn−− − −+− − −∂ = − ⋅ + ⋅ − ⋅ + ⋅ Δ∂∂ = ⋅ − ⋅ + ⋅ − ⋅ Δ∂( , ) ( , )2 2[ ( , ) ( , )] (1 ) .k kk k f f fG n t G n tc cn nG n t G n t c c c Fε− +− +∂ ∂⋅ − ⋅ +∂ ∂+ + − − ⋅ − =( , ) ( , ) ,( , ) ( , ) ,( , ) ( , ).l lk kk k ff k kn t n t Fn t n t c Fc c n t c n tτ ττ τ+ − − =+ − − == + = −* *( , ) ( , )2 2[ ( , ) ( , )] (1 ) 0.k kk kG n t G n tc cn nG n t G n t c cε− +− +∂ ∂⋅ − ⋅ +∂ ∂+ + − − ⋅ − =( , ) ( , ), ( , ) ( , ),( , ) ( , ).l lk k k kk kn t n t n t n tc n t c n tτ τ τ τ+ = − + = −+ = −; 2 2 .ÒPMdc cn N cdn G tε ∂= − = − ∂1; 2 2 ,ËPMdc cn cdn G tε ∂= − = ∂12 (1 ).ndc c cdnε== −1(1, ) 0; (1 ) 0.2l nc Gt c G c cnτ τ τ ε=∂⎡ ⎤′= = − + − − =⎢ ⎥∂⎣ ⎦Bulletin of the Томsк Pоlytеchnic University. 2007. V. 310. № 298cascade changing into stationary operation mode is shown. Relative distribution of gas content overthe cascade steps when changing into stationary operation mode is shown in Fig. 4.Fig. 3. Distribution of relative concentration over the cascadewhen switched to stationary modeConclusionThe mathematical model for application of nonstationary separation processes in symmetrical counterstream cascade has been designed. The system of equationsdescribing given processes consists of the equations ofnonstationary hydraulics which are the equations of substance balance in the isolated volumes. The process oftransferring light isotope from dump to selection in theenriching and regeneration parts of cascade is describedby two equations in partial derivatives of the second order.At their simultaneous solution for the equation ofnonstationary hydraulics the Euler’s modified methodof the second accuracy order in time is used, in the equations of separation the twolevel scheme of calculationproviding for the second order of accuracy of variables tand n is used.The techniques has been developed and the resultsof simultaneous numerical simulation for nonstationary parameters of hydraulic and separation processes incounterstream symmetrical cascade have been obtained. They allow for simulation of different transfer situations in centrifugal separation cascades.Relative concentrationiNMMMathematics and mechanics. Physics99Fig. 4. Distribution of relative gas content over the cascadewhen switched to stationary modeɈɬɧɨɫɢɬɟɥɶɧɨɟɝɚɡɨɫɨɞɟɪɠɚɧɢɟɋɬɭɩɟɧɶStepRelative gas contentREFERENCES1. Filimonov S.V., Skorynin G.М., Goldobin D.N. Simulation of nonstationary hydraulic processes // Izvestiya Vuzov. Physics. – 2004. –V. 47. – № 11. – P. 173–177.2. Enrichment of uranium / Ed. by S. Willaney. Trans. from Eng. Ed.by I.К. Kikoin. – Moscow: Energoatomizdat, 1983.3. Idelchik I.Е. Reference book on hydraulic resistance. – Moscow:Gosenergoizdat, 1960. – 464 p.4. Cohen K. The Theory of Isotope Separation as Applied to the LargeScale Production of U235. – N.Y.: McGrawHill, 1951.5. Rabinovich G.D., Gurevich R.Ya., Bobrov G.I. Thermaldiffusion isolations of liquid mixtures. – Minsk: Nauka i Technika, 1971. – 207 p.6. Samarski А.А. Theory of difference schemes. – Moscow: Nauka,1977. – 656 p.7. Levin Е.V., Laguntsov N.I., Sulaberidze G.А. On one method of numerical study of nonstationary processes in multisectional installations for separation of multicomponent isotope mixtures // Inzhenerny Physicheski Zhurnal. – 1982. – V. 43. – № 3. – P. 456–462.Received on 20.12.2006

Simulation of non-stationary processes in centrifugal cascades

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