Analysis of neutron-temperature oscilations in neutron multiplying systems with delayed neutrons

External sources with a given frequency excite neutron and temperature oscillations. Analytical treatment and analysis of oscillations were performed under conditions of both the weak coupling and strong coupling between the oscillation branches.The neutron oscillation branch describes the wave prop...

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Datum:	2011
Hauptverfasser:	Vodyanitskii, A.A., Rudakov, V.A.
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Sprache:	English
Veröffentlicht:	Національний науковий центр «Харківський фізико-технічний інститут» НАН України 2011
Schriftenreihe:	Вопросы атомной науки и техники
Schlagworte:	Ядерно-физические методы и обработка данных
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spelling	nasplib_isofts_kiev_ua-123456789-1114752025-02-09T15:47:35Z Analysis of neutron-temperature oscilations in neutron multiplying systems with delayed neutrons Аналiз нейтронно-температурних коливань у розмножувальних системах з запiзнiлими нейтронами Анализ нейтронно-температурних колебаний в размножающих системах с запаздывающими нейтронами Vodyanitskii, A.A. Rudakov, V.A. Ядерно-физические методы и обработка данных External sources with a given frequency excite neutron and temperature oscillations. Analytical treatment and analysis of oscillations were performed under conditions of both the weak coupling and strong coupling between the oscillation branches.The neutron oscillation branch describes the wave propagation along the coolant flow with amplitude rising in the range of high flow velocity and with amplitude decreasing in the range of low velocity and high frequencies, as well as, the oscillation propagation in the opposite direction. The temperature oscillations undergo a weak decrease of amplitudes. The strongly coupled neutron-temperature oscillations are excited, as the microscopic fission cross-sections are dependent on the "effective" temperature of thermal neutrons.The influence of delayed neutrons is studied. Comparison between the analytical treatment data and the analysis of experimental measurements confirms these conclusions. Зовнішні джерела з заданою частотою в нейтронній розмножувальній системі збуджують нейтронні і температурні коливання. Проведено аналітичний розгляд і аналіз коливань в умовах як слабкjого, так і сильного зв'язків між гілками. Нейтронна гілка коливань описує їх розповсюдження уздовж течії теплоносія з наростанням амплітуди в області великих швидкостей його руху, а також, з затуханням амплітуди - в протилежному напрямку. Конвективна гілка температурних коливань зазнає слабкого затухання амплітуди. При залежності мікроскопічних перетинів поділу ядра від температури теплових нейтронів збуджуються сильно зв'язані нейтронно-температурні коливання. Враховано вплив запізнілих нейтронів. Аналітичний розгляд і аналіз експериментальних вимірювань підтверджують ці висновки. Внешние источники с заданной частотой в нейтронной размножающей системе возбуждают нейтронные и температурные колебания. Проведены аналитическое рассмотрение и анализ колебаний в условиях как слабой, так и сильной связей между ветвями. Нейтронная ветвь колебаний описывает их распространение вдоль течения теплоносителя с нарастанием их амплитуды в области больших скоростей его движения, а также с затуханием амплитуды - в противоположном направлении. Конвективная ветвь колебаний испытывает слабое затухание амплитуды. При зависимости микроскопических сечений деления ядер от температуры тепловых нейтронов возбуждаются сильно связанные нейтронно-температурные колебания. Учтено влияние запаздывающих нейтронов. Сравнение аналитического рассмотрения с анализом экспериментальных измерений подтверждает эти выводы. 2011 Article Analysis of neutron-temperature oscilations in neutron multiplying systems with delayed neutrons / A.A. Vodyanitskii, V.A. Rudakov// Вопросы атомной науки и техники. — 2011. — № 5. — С. 30-38. — Бібліогр.: 22 назв. — англ. 1562-6016 PACS: 24.60.-k https://nasplib.isofts.kiev.ua/handle/123456789/111475 en Вопросы атомной науки и техники application/pdf Національний науковий центр «Харківський фізико-технічний інститут» НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
topic	Ядерно-физические методы и обработка данных Ядерно-физические методы и обработка данных
spellingShingle	Ядерно-физические методы и обработка данных Ядерно-физические методы и обработка данных Vodyanitskii, A.A. Rudakov, V.A. Analysis of neutron-temperature oscilations in neutron multiplying systems with delayed neutrons Вопросы атомной науки и техники
description	External sources with a given frequency excite neutron and temperature oscillations. Analytical treatment and analysis of oscillations were performed under conditions of both the weak coupling and strong coupling between the oscillation branches.The neutron oscillation branch describes the wave propagation along the coolant flow with amplitude rising in the range of high flow velocity and with amplitude decreasing in the range of low velocity and high frequencies, as well as, the oscillation propagation in the opposite direction. The temperature oscillations undergo a weak decrease of amplitudes. The strongly coupled neutron-temperature oscillations are excited, as the microscopic fission cross-sections are dependent on the "effective" temperature of thermal neutrons.The influence of delayed neutrons is studied. Comparison between the analytical treatment data and the analysis of experimental measurements confirms these conclusions.
format	Article
author	Vodyanitskii, A.A. Rudakov, V.A.
author_facet	Vodyanitskii, A.A. Rudakov, V.A.
author_sort	Vodyanitskii, A.A.
title	Analysis of neutron-temperature oscilations in neutron multiplying systems with delayed neutrons
title_short	Analysis of neutron-temperature oscilations in neutron multiplying systems with delayed neutrons
title_full	Analysis of neutron-temperature oscilations in neutron multiplying systems with delayed neutrons
title_fullStr	Analysis of neutron-temperature oscilations in neutron multiplying systems with delayed neutrons
title_full_unstemmed	Analysis of neutron-temperature oscilations in neutron multiplying systems with delayed neutrons
title_sort	analysis of neutron-temperature oscilations in neutron multiplying systems with delayed neutrons
publisher	Національний науковий центр «Харківський фізико-технічний інститут» НАН України
publishDate	2011
topic_facet	Ядерно-физические методы и обработка данных
url	https://nasplib.isofts.kiev.ua/handle/123456789/111475
citation_txt	Analysis of neutron-temperature oscilations in neutron multiplying systems with delayed neutrons / A.A. Vodyanitskii, V.A. Rudakov// Вопросы атомной науки и техники. — 2011. — № 5. — С. 30-38. — Бібліогр.: 22 назв. — англ.
series	Вопросы атомной науки и техники
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fulltext	NUCLEAR-PHYSICAL METHODS AND PROCESSING OF DATA ANALYSIS OF NEUTRON-TEMPERATURE OSCILATIONS IN NEUTRON MULTIPLYING SYSTEMS WITH DELAYED NEUTRONS A.A. Vodyanitskĭı∗, V.A. Rudakov National Science Center ”Kharkov Institute of Physics and Technology”, 61108, Kharkov, Ukraine (Received July 21, 2011) External sources with a given frequency excite neutron and temperature oscillations. Analytical treatment and analysis of oscillations were performed under conditions of both the weak coupling and strong coupling between the oscillation branches.The neutron oscillation branch describes the wave propagation along the coolant flow with amplitude rising in the range of high flow velocity and with amplitude decreasing in the range of low velocity and high frequencies, as well as, the oscillation propagation in the opposite direction. The temperature oscillations undergo a weak decrease of amplitudes. The strongly coupled neutron-temperature oscillations are excited, as the microscopic fission cross-sections are dependent on the ”effective” temperature of thermal neutrons.The influence of delayed neutrons is studied. Comparison between the analytical treatment data and the analysis of experimental measurements confirms these conclusions. PACS: 24.60.-k 1. INTRODUCTION The in-reactor wave processes can possess features characteristic for phenomena of spatial damping and amplification. Therefore, to develop in-reactor con- trol systems it is necessary to investigate physical processes, taking place in the neutron-multiplying system, and their wave properties characteristic for fission reactors [1] - [6]. The goal of the paper is to analyze the propa- gation and spatial damping (amplification) of neu- tron field and cool-ant oscillations excited by exter- nal sources. Oscillations in the coolant-containing neutron-multiplying systems (cooled reactors) may be described by the set of nonequilibrium statisti- cal physics equations. The paper presents the analy- sis of wave processes of neutron-temperature oscil- lation transport in the cool-ant-containing neutron- multiplying system. The coolant moving through the core makes a con- vective heat and neutron transport and causes some non-equilibrium that can lead to the oscillation am- plification in the dissipative media. This effect is not evident for our system because the neutron diffusion and capture by nuclei act as stabilizing factors. Be- sides, the choice of a steady state being stable in time does not guarantee the spatial damping of oscillations unlike the linear and nonlinear theory of waves in weakly nonequilibrium media. The previous papers have considered a problem of excitation of neutron field oscillations by the acoustic wave [7]. In addition, there investigated was a prob- lem of neutron field modulation by the external lo- calized source with a given frequency through the acoustic, neutron and temperature channels [8]. The paper [20] briefly considers the conditions for propa- gation of neutron-temperature oscillations in the mul- tiplying systems without taking into account the de- layed neutrons and temperature dependence of dif- fusion coefficients. In the present paper, within the framework of the model applied, we comprehensively studied and analyzed the conditions for propagation, damping and amplification of oscillations. The ki- netics of centers, emitting the delayed neutrons and the dependence of linearized diffusion terms on the temperature perturbations, are taken into account. To solve the engineering tasks one uses an incon- sistent approximation in definition and solving the problem. The influence of heat release in the fission reactions on the neutron diffusion is related with the neutron density via some transfer function introduced in the design formula as an unknown and should be determined from the experimental analysis of quan- tities being measured. To the equation of coolant heat balance one adds the equation of a fuel heat balance with an unknown heat-transfer coefficient. Thus, there is no self-consistent description in the en- gineering tasks. Probably, such an approximation is quite acceptable in the engineering interpretation of the models of noise diagnostics including some data from the experiments of [3] - [5] (more complete list of references see in [9]). Note that the first works on the neutron physics of multiplying systems were per- ∗Corresponding author E-mail address: vodyanitskii@kipt.kharkov.ua 30 PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY, 2011, N5. Series: Nuclear Physics Investigations (56), p.30-38. formed in the understandable language of theoretical physics [10]-[15]. 2. SELF-CONSISTENT SYSTEM OF EQUATIONS The influence of hydrodynamic parameter variations on the neutron density oscillations in the multiplying system with coolant and delayed neutrons requires the description by the methods of non-equilibrium statistical physics. To analyze this influence let us consider the set of equations for diffusion of thermal neutrons with their multiplication and capture and the set of hydrodynamic equations for the coolant being a moderator of fast neutrons [15]. ∂N ∂t +∇(~vN)=∇(D∇N)+Da4N+νc(Kβ−1)N+S, (1) ρcp(∂T/∂t + ~v · ∇T ) = ∇(κ∇T ) + +Q(~r, t; N) + qT , (2) ρ(∂~v/∂t + (~v · ∇)~v) = −∇P + η4~v, (3) ∂ρ/∂t + div(ρ~v), P = P (ρ, T ). (4) In diffusion equation (1) the term ∇(~vN) = ~v∇N + N∇~v, obtained in [16], represents the con- servation of neutron density during the neutron con- vection with a velocity of the coolant taking into ac- count its compressibility. Then, Kβ = K(1 − β), where K = νϕθ is the neutron multiplication fac- tor including three factors from which ν is the num- ber of generated neutrons in the one fission act, ψ is the probability to avoid the neutron capture in the process of its slowing-down. The third fac- tor, the coefficient of thermal neutron utilization θ = νf/(νf + νa) , is equal to the relation of the inverse time (”frequency”) of neutron capture by fis- sion nuclei νf = σfNF v to the sum of frequencies of neutron absorption by fission nuclei and in other processes νa = Σiσ i aN i av. Here i is the index of nu- clear kind; σf is the nuclear fission reaction cross- section; Nf is the nuclear concentration and v is the average velocity of thermal neutrons. The effective coefficient of neutron multiplication, Kβ , includes a factor where β is the sum of probabilities for forma- tion of all the sources of delayed neutron emission source. The coefficient of diffusion D = (1/3)vl and the diffusion-type transport coefficient Da = Kτνc, related with the age of slowing-down neutrons, in the linear approximation form an effective coefficient of diffusion equal to Def = D + Kτνc. As is shown by the authors of [14], the age parameter is equal to τ = l2ln(E0/E), where l2 = 1/(3ξsΣsΣtr) or to 1/l = ΣαNασα s in terms of [12] for the scatter- ing by different-type nuclei with concentrations Nα and scattering cross-sections σα s . In these formulae ξs ≈ 1, and Σtr is the macroscopic transport cross- section Σtr ≈ Σs, approximately equal to the scatter- ing cross-section Σs = Nsσs, where σs is the micro- scopic scattering cross-section, Ns is the density of scattering nuclei. In diffusion equation (1) S is the density of thermal neutron sources, either external or delayer neutrons, for which S = ΣiνiNi, where Ni is the concentration of i-type fission products radi- ating delayed neutrons and 1/τi = νi is the inverse ”keeping” or delay time of neutron radiation. In the last case the set of equations under consideration is complemented by the following equation ∂Ni/∂t = βi fN −Ni/τi, (5) where βi f = νβiνf and νβi is ”the multiplication of probability of the i-type radiation source formation in the process of fission on the probability of neu- tron emission by these fission product” [13] and νf is denoted above. The hydrodynamic equations for the coolant are with standard notations. Note that Q = Q(~r, t) is the thermal-power density per time unit of fission nuclear reactions. For the heat release we introduce EN = Q/(ρcp), where N denotes the neutron den- sity. The last relation in the line (4) is the coolant state equation. 3. MODEL CHOICE Propagation of neutron-temperature oscillations occurs when the multiplying medium is in the station- ary or quasi-stationary state. The simplest model of the stationary state is a one-dimensional model with a coolant having the constant density ρ0 and veloc- ity U , neutron density N0, temperature T0 and con- centration of centers radiating delayed neutrons Ni0. These quantities satisfy the equations (here we ne- glect the inhomogeneity in the velocity and pressure supposing their changes being insignificant) U dN0 dz = d dz ( D dN0 dz ) +Da d2N0 dz2 +νc(Kβ−1)N0 +S, U dT0 dz = EN0 + χ d2T0 dz2 , βi fN0 = Ni0 τi . The choice of the model is accompanied by the inves- tigation of the stationary state stability. (Mathemat- ical aspects of the theory on the existence, uniqueness and stability of stationary states in nuclear reactors, as well as, in a slab model are under consideration in the monograph [17]). Let us formulate the set of equations for perturba- tions of stationary values of system parameters. As- sume that N = N0 +n(z, t), Ni = N0i +ni(z, t), T = T0 + T1(z, t), ρ = ρ0 + ρ1(z, t) and ~v = (0, 0, U + U1). We will take the linear terms in each of equations (1) - (5) considering the dependence of coefficients from the system parameters. The linear dependence of the capture frequency on the medium density is natural if the relative concentrations of nuclei are not varying, i.e. when ρj = Njmj/ρ = const, where Nj and mj , are the density and mass of j-type nuclei. Under these conditions all the macroscopic cross-sections of capture and scattering and inverse times of neutron capture are directly proportional to the medium den- sity Σj = σjNj = rΣj0, 1/Tc = νc = νcor, where r = ρ/ρ0. Here and below the index 0 denotes the values of quantities in the stationary state. 31 The terms, describing the multiplication of neu- trons and their capture in equation (1) gain the de- pendence on the medium density and on the ”effec- tive” neutron temperature Tn in the form νc(Kβ − 1) = rϑδ nνc0(Kβ0), where ϑn = Tn/T0. The contribution of temperature perturbations to the multiplication and capture of neutrons in the same equation (1) gives the term T1d ( νc(Kβ−1) ) / dT = T1 ( νc(Kβ−1)+νa ) δ/T0, (6) where νc = νa + νf is the frequency of all the cap- ture types. The frequency of thermal neutron absorp- tion by the nuclei without fission νa, in view of the dependence of absorption cross-sections on the aver- age thermal velocity in the form of σα ∼ 1/v, does not depend on the energy. In formula (6) assumed was a dependence of the frequency of capture by fis- sion nuclei on the ”effective” neutron temperature in the form of νf ∼ T δ. There used was equation (38) from Appendix 1 and the value of the deriva- tive (∂νf/∂Tf ) ·∂Tn/∂T = νf (Tf )δ/T , containing, as a multiplier, the inverse medium temperature 1/T . (Details on the physical situation concerning the ”ef- fective” temperature in nuclear reactors see in Ap- pendix 1). Here we give the results obtained in Appendix 2 on the temperature perturbation contribution into the diffusion transport of neutrons. The depen- dence of diffusion coefficients on density variations ρ1 = ρ− ρ0 by the equation of motion (9) we express in terms of temperature perturbations neglecting a medium local acceleration. Then the linearized terms of diffusion, being dependent on the temperature per- turbations in the one-dimensional approximation, are added giving the sum L̂T { ∂ ∂z ( D ∂N ∂z ) + Da ∂2N ∂z2 } = = D′ G,T ( N ′′ 0 T1 + N ′ 0 ∂T1 ∂z ) + D′ Ga,T N ′′ 0 T1 + Rext, (7) where: Rext = {D0∂ (P extN ′ 0) /∂z + N ′′ 0 Da0P ext} /(ρs2), N ′ 0 = dN0/dz, N ′′ 0 = d2N0/dz2, D′ G,T = D0G/T0, D′ Ga,T = Da0Ga/T0. Parameters G and Ga are given in Appendix 2 after formulae (39) and (40) and con- tain not only the temperature perturbation contribu- tion but also the medium density perturbation con- tribution. In the expression given the dependence of stationary values of T0, D0 and G on the ”nonuni- form” coordinate z is neglected. The neglect condi- tion is fulfilled for perturbations having the scales of characteristic lengths significantly less than the char- acteristic dimensions of uniformities in the stationary state. The linearized equations of coolant motion and continuity ρ0∂U1/∂t=−s2∂ρ1/∂z−( P ′T ) ρ∂T1/∂z−∂P ext/∂z, (8) ∂ρ1/∂t + ∂(ρ1U + ρ0U1)/∂z = 0, (9) require explications. As is shown in [9], the pertur- bations in coolant local acceleration and coolant den- sity inertia cause the excitation of acoustic waves. However, the sound branch of oscillations is sepa- rated from the neutron-temperature oscillations and the above-mentioned time derivatives are neglected. Equation (8) includes the force action of the external pressure gradient on the coolant. The perturbations in the medium density ρ1 and in the coolant velocity will be removed from the set of perturbations in the linearized system of equa- tions by equations (8) and (9) by neglecting the local medium acceleration and the term in the continuity equation ∂ρ/∂t. At the same time, the perturbations of neutron density, temperature and concentration of sources radiating delayed neutrons satisfy the set of equations −iω̂Dn(z, t)−ĤT1− ∑ i Niνi = R̂P ext / E ≡ b1, (10) En(z, t)−i ( ω̂χ+iΓ ) T1 = EN0P ext / ρs2−qT ≡ b2, (11) ∂ni/∂t = βi fn− νini. (12) Here Rext is given in the line after formula (7) and the operator quantities and some parameters are equal to the following −iω̂D = ∂/∂t + U∂/∂z −Def∂ 2/∂z2 − νc(Kβ − 1), −iω̂χ = ∂/∂t + U∂/∂z + E′ T N0, (13) ĤE =−ΓΩ̂ +ED′ G,T ( N ′′ 0 +N ′ 0∂/∂z ) +EN0(λ+νa)δ/T0, (14) Γ = γ ( P ′T ) ρ EN0 ρs2 , λ = νc(Kβ − 1), Ω̂ = λ + ikU. (15) The set of equations (10)-(12) describes both the mul- tiplying medium stationary state stability and the propagation of coupled neutron-temperature oscilla- tions excited by the external sources of force P ext and thermal action qT . For the conditions of oscillation propagation an asymptotic solution is found in the form of quasi- classical exponents or exponents with expanded in- dexes exp ( i ∫ z0 k(z′)dz′ ) ≈ exp(ik(z1)z0), where z1 = εz0 is a ”slow” coordinate for ε ¿ 1, and z0 = z is the ”fast” one [18]. The coupling of neutron and temperature perturbations is performed with the help of two factors. The first takes place in the thermal balance equation in the form of the heat releasing from the nuclear fission reactions proportional to the thermal neutron density. The second factor intro- duces the temperature perturbation into the neutron diffusion equation and is proportional to the sum of four terms: H =−Γ Ω E +D′ G,T ( N ′′ 0 + ikN ′ 0 ) +D′ Ga,T N ′′ 0 +N0(λ+νa) δ T0 , (16) where the notations correspond to these used in for- mulae (13)-(15). The first term in (16) on the right part expresses the medium density perturbation con- tribution into the macroscopic capture cross-section 32 Σc = ∑ i N i cσ i c ∼ ρ. As is noted above, the medium density-and-temperature perturbations are interre- lated by equation (8) with a left part being equal to zero. The second and the third terms in (16) de- scribe the contribution of dependence of diffusion co- efficients on the medium density and medium temper- ature given in Appendix 2 (see the text with formulae (39) and (40)). The last term in (16) is obtained by taking into account the microscopic nuclear fission cross-sections as a function of the thermal neutron ”effective” temperature. 4. CHARACTERISTIC EQUATION FOR NEUTRON MULTIPLYING MEDIA WITH DELAYED NEUTRONS The propagation of waves with a given frequency oscillations ω from the external sources is determined by the values of complex wave numbers k from the characteristic (or dispersion) equation obtained as a solvability condition for a homogeneous system of equations corresponding to equations (10)-(12) in the k-representation. The characteristic equation has the form ( ωD−i ∑ j νjβ j f / (νj−iω) )( ωχ+iΓ ) = a1+a2+A, (17) where the quantities ωD = ω − kU + ik2Def − i(Kβ − 1)νc, (18) ωχ = ω − kU + ik2χ− iE′ T N0, are the same as in formulae (13)-(15) with the sub- stitution ∂/∂t⇒−iω, ∂/∂z⇒ ik. In the right part of (17), the sum is presented by the coupling factors a1 =Γ(λ + ikU), a2 =−D′ G,T (N ′′ 0 + ikN ′ 0), (19) A=−EN0(λ+νa)δ / T0, δ < 0. (20) In the left part of equation (17) in the parenthesis the term with an index of summation is a contribution to the propagation of neutron oscillations of delayed neutrons from all the sources of their radiation. Equation (17) without delayed neutron contribu- tion is obtained in [8]. We will start the analysis of the solutions from the study of the effect equation of delayed neutrons. For low oscillation frequencies ω, which are much less than the inverse time of de- layed neutron retention time ω ¿ 1/τi = νi, the sum∑ i νiβ i f / (νi − iω) ≈ ∑ i βi f = Kνcβ is well simpli- fied (here we used the notations K = νθ, θ = νf / νc, β = ∑ i βi and βi f = νβiνf ). For high frequencies ω À 1/τi = νi, as compared to the inverse times 1/τi, both the reactive contribu- tion and the dissipative contribution are insignificant due to the smallness of νi/ω ¿ 1: Σiνiβ i f / (νi−ω) ≈ Kνc ( iΣiβiνi / ω+Σiβi(νi/ω)2 ) . If the frequency ω is close to one of the frequen- cies νi then the influence of delayed neutrons may be significant. The frequency band f = ω/2π in this case ranges to the tenth of hertz for the times of fis- sion reaction half-life equal to 2 s and 0.45 s [11]. The contribution of a j-type emitter into the quantity ωD in equation (17) is estimated as (1 + i) βjKνc/(2ω) and can be significant for the case of a great proba- bility βj of formation in the process of j-type emitter fission. The neutron multiplication factor includes only prompt neutrons and their part from the to- tal number of thermal neutrons is decreased by the sum of probabilities for all-type emitter formation β = ∑ j βj (a part of thermal neutrons arrives from the delayed ones). As is seen from the above analysis, the contri- bution of delayed neutrons includes the reactive and dissipative parts. In the case of general statement, it can be combined in the dispersion equation with the terms of the same type. Introduce the notations ωβ = ω + ωKνc ∑ i νiβi /( ω2 + ν2 i ) , (21) Kβω = Kβ + Kβω, βω = ∑ i ν2 i βi /( ω2 + ν2 i ) , (22) with which the investigation of limiting cases is eas- ier. The dispersion function of proper oscillations together with delayed neutrons, taking into account the introduced notations, is written in the following form DN (ω, k) = ωD − i ∑ i νiβ i f / (νi − iω) = = ωβ − kU + ik2Def − iνc(Kβω − 1). (23) 5. PROPAGATION OF OSCILLATIONS IN NEUTRON MULTIPLYING MEDIUM WITH DELAYED NEUTRONS Let us present the results of dispersion equation solution and their analysis. Of importance are not only the conditions for perturbation theory applica- tion in the range of values of the multiplying medium but also its physical characteristics, e.g. dependence on the temperature of medium of the nuclear fission microscopic cross-sections and on the frequencies of capture by these thermal neutron nuclei. It is possi- ble to estimate the influence of these processes in the noise diagnostics of the multiplying system by study- ing the conditions of neutron-temperature oscillation propagation. In the disperse equation with taking into account the introduced notations DN (ω, k) ( ωχ + iΓ ) = a1 + a2 + A, (24) the sum of partial coupling coefficients of neutron- and-temperature perturbations is given in formula (17) with notations on the line of (18), DN is in for- mulae (23). The solution of this equation we search by the perturbation theory in the approximation of weak coupling between neutron and temperature os- cillations. The wave number of neutron oscillations is found in the form k = kN +δkN , where \|kN \| À \|δkN \|. 33 In the zeroth approximation k = k1,2 are the solutions of quadratic equation ωβ − kU + ik2Def − (Kβ − 1)νc = 0. (25) The real and imaginary parts of the complex roots of these equations are obtained in [8] k1,2 = ( 1 /√ Def )[−iU /( 2 √ Def )± ± √ −U2/(4Def) + iωβ + νc(Kβ − 1) ] , (26) are written as Rek1,2 =± 1√ 2 [√ ωβ+ √ ω2 β +B2 + √√ ω2 β +B2−ωβ ] , (27) Imk1,2 = 1√ Def { −ib± ±1 2 [√ ωβ + √ ω2 β + B2 − √√ ω2 β + B2 − ωβ ]} . (28) Here the notations b =U /( 2 √ Def ) and B = νc ( Kβ − 1 )− b2 are introduced. When an above-critical state can be realized and when B > 0 (B À ωβ or B ¿ ωβ) it is easy to find approximated expressions using the perturbation theory. In particular, in the case of low values of the squared frequency ω2 β ¿ B2 the real and imaginary parts of wave numbers take the values Rek1,2 = ± √ B/Def ( 1 + ω2 / (8B2) ) , (29) Imk1,2 = ( 1 / √ Def )(−b + ω / (2 √ B) ) , where the notations of formulae (27) and (28) are taken. The frequency dependence of the real part of the wave number about its low values looks like a branch of a parabola with vertical axis and the imag- inary part of the wave number is a linear function. As the frequency increases with its high values ω2 β À B2, the approximated values of the real and imaginary parts of wave numbers are equal Rek1,2 = ± √ ωβ/Def ( 1 + B / (2ωβ ) , Imk1,2 = ( 1 / √ Def )(−b±√ 2ωβ ( 1−B / (2ωβ )) . (30) Besides, the fundamental approximation of mod- ules Rek1,2 expresses their growth depending on the frequency by the parabolic law with a horizontal parabola axis. As the frequency Imk1 increases and becomes positive, and Imk2 decreases and remains negative. The expressions for Rek1,2 and Imk1,2 determine the complex wave numbers of proper neutron oscilla- tions including the diffusion kinetics of thermal neu- trons with capture and multiplication by fission nu- clei, as well as, the processes of delayed neutron radi- ation. The waves are propagating in opposite direc- tions. The wave, propagating in the direction oppo- site to the direction of the coolant flow rate, under- goes the spatial damping at all the values of parame- ters being in formula (28). The neutron wave, moving along the flow having a low rate and a high oscillation frequency, undergoes the amplitude decreasing too. When the flow rate increases or the frequency de- creases , this wave changes the propagation character from the weakening to its amplification. Indeed, the condition of amplification Imk1 < 0 for the wave with Rek1 > 0 leads to the inequality ω2 β < 4b2 ( B + b2 ) or in the reference variables ω2 β < νc ( Kβ − 1 )( U2/Def ) . (31) If the frequency values exceed the inverse decay time of all the fission products being the sources of delayed neutron radiation ω À 1/τi, condition (31) simplifies and takes the form ω2 < νc ( K(1− β)− 1 )( U2/Def ) , β = ∑ i βi, where is the sum of probabilities for formation of these sources. To find additives to the wave number of neutron oscillations (being small because of a weak coupling between the neutron field oscillations and coolant hydrodynamic oscillations), in formula (20) used is the expansion DN (ω, k) = DN (ω, kN ) + δDN , where DN (ω, kN ) = 0 and δDN = (−U + 2ikNDef)δkN , ωχ(kN ) = ω − kNU + iξ (last term is small, ξ = Γ+ k2 N − E′ T N0 > 0 and \|E′ T N0/U \| ¿ \|kN \|, where E′ T N0 = ET N0δ/T ). Substituting this expansion into equation (24) we find the sought correction to the wave number δkN = ( a(kN ) )/[ ωχ(kN ) ( U−2ikNDef )] , (32) containing the dependence on the delayed neutrons in the quantity kN , one of the complex roots in formula (26). In formula (32) the terms a(kN ) = a1+a2(kN )+A are determined in lines (19-20). Prop- agation of temperature oscillations in the basic ap- proximation occurs with a wave number of the con- vective transport kT = ω/U . By analogy with the above case, the correction is found from the equation of neutron oscillations written in the form k−ω / U = a / [UDN (ω, k)]+ iξ / U . Assuming that both terms in the right part are small and k = kT +δT , \|kT \| À \|δkT \| by the perturbation theory of we find δkT = iξ / U − a(kT ) / [ UDN (ω, kT ) ] , (33) where DN (ω, kT ) = ωβ −ω + ik2 T Def − iνc(Kβω − 1). The first term in (33) leads to the oscillation damping unlike the second term, which under conditions of su- percriticality νc(Kβ−1) > kT Def gives the correction decreasing the oscillation damping. Note here the value of the factor of coupling be- tween the neutron and the temperature oscillations. The dependence of the frequency of thermal neutron capture by the fission nuclei νf ∼ T δ n, even with low values of the index \|δ\|, leads to the strong coupling 34 between the temperature perturbations and the neu- tron density. In dispersion equation (24) the term A performs a strong coupling, and is proportional not to the supercriticality coefficient νc(Kβ−1), but sim- ply νaK, to νc(Kβ − 1)δ/T that exceeds by several orders of magnitude. By virtue of a mentioned property, characteristic for nuclear-physical kinetics of thermal neutron cap- ture by the fission nuclei, it is interesting to carry out the analysis of two limiting situations with the strong coupling between the proper neutron wave os- cillations and the coolant oscillations. For this pur- pose in dispersion equation (24) we transpose into the left part the terms independent on the wave number k, (ω + iξ) ( ωβ − iνc(Kβ − 1) )− α = kϕ(k, ω), (34) where a = α + iγ; α = a1 + Rea2 + A is the real part of the sum of coupling factors and the polynomial is equal to the equation ϕ(k, ω) = U [ ωβ + iγ − iνc(Kβ − 1)− ω − iξ ] + (35) +ikDef(ω + iξ)− kU2 + ik2UDef . After the transformation, the dispersion equation takes the form ωωβ − α + 2ξλ + iωλ = kϕ(k, ω), (36) where the parameter λ = νc(Kβ − 1) is proportional to the supercriticality of the neutron multiplying sys- tem. Of a particular interest are the strongly coupled neutron-temperature oscillations with high values of the frequency and coupling factors. Therefore, when searching the solution of a dispersion equation one supposes that the oscillation frequency contribution is much higher than the contribution into the dis- persion equation of diffusion, convection and balance between the multiplication and capture of neutrons. It means that the wave numbers of oscillations (at least, one of the branches) will be low and at a given frequency of the external oscillation source undergoes weak spatial variations. Such a behavior is important for the oscillation recording in the core of the multi- plying medium. The left part of equation (36) is small at low val- ues of the parameters δα = ωωβ − α and ξλ. Then, assuming that \|k\| is low, in the right part it is possi- ble to restrict oneself by the term linear on k, namely: kϕ(0), where ϕ(0) = U [ ωβ+iγ− iνc(Kβ−1)−ω−iξ ] . Neglecting the low terms we have k1 = q1 + κ1 = ( δα + iξωβ − iωλ ) / ϕ(0). (37) Besides the roots determined there are another two roots with the same assumptions about the parame- ter smallness. These roots do not contain at all the smallness of the above-mentioned parameters and are close to zeros k2 and k3 of quadratic trinomial (35), ϕ(k2,3) = 0, with minor corrections. However, here we do not given them and restrict ourselves by the following interpretation of conditions of calculation validity. The equality to zero of the parameter δα = ωωβ − α = 0 with high values of its parameters, as well as the equality ξωβ − ωλ = 0 in root expres- sion (37) does not mean ”automatically” the amplifi- cation or nonpassage of oscillations with frequencies from the vicinity of their values ω1 and ω2, q(ω1) = 0 and κ(ω2) = 0. There required is an adequate prob- lem definition and choice of parameter values such as, for example, in guide [19] about amplification or nonpassage of waves in the stable and absolutely- or convectively unstable medium. For example, for a medium for the problem with initially boundary con- ditions it is a choice of ”immediately including” ex- ternal source in the form of g(x, t) ∼ δ(x)exp(−iω0t) at t > 0 and g(x, t) = 0 at t < 0 corresponding to the initially boundary problem with the point local- ization and given frequency ω0. Far from the source, the steady-state condition at t → ∞ with amplifi- cation or nonpassage of oscillations at \|x\| → ∞ is possible only in the problem ” on the convective un- stable (or stable) system. In the case of absolute in- stability the perturbation is increasing unrestrictedly in all the points of space, so that it is impossible at all to reach a steady-state condition” [19] p.332. Note that the low wave number values violate the condition of applicability of the quasi-classical approximation∣∣(d/dz)(1/k(z) ∣∣ = ∣∣(1/k(z)2(dk(z)/dz) ∣∣ ¿ 1, within the framework of which the treatment accepted in this paper is valid. Therefore, a special asymptotic expansion [21] of the desired solution and numerical computation are required. Besides, it is of interest to consider the case with high modulo wave numbers. We solve the problem on a spatial behavior of strongly coupled neutron and temperature oscillations. Their coupling occurs through the density and temperature oscillations of both the media and the neutron field. To be exact: the neutron density variation in the term being the released heat contribution causes temperature varia- tions in the medium. Under conditions of neglecting the inertia and weak convective transfer of a hydrody- namic medium pulse to these temperature variations ”fitted” are the medium density variations together with the density of fuel nuclei. In the neutron diffu- sion equation among the terms of neutron self-action we take into account only the diffusion of neutrons and neglect their multiplication and nuclear capture (the matter concerns the spatial behavior of excita- tions over the stationary state of the active multiply- ing medium not the stationary state itself). In the right part of equation (36) the largest wave-number cubic term remains. Neglecting the small terms we obtain from (34) the equation δα ≡ ωωβ − α = ik3UDef . When δα > 0 the solution of this equation is kj = ρ exp ( i2πj/3− iπ/6 ) , where j is the integer and ρ = 3 √ δα/(UDef) or k0 = ρ exp (−iπ/6 ) , k1 = iρ and k−1 = ρ exp(−i5π/6). 35 It is possible to consider the definition of a bound- ary problem with oscillations being stationary in time at a given frequency. Then the position in the com- plex plane of a single root k0 for oscillations in the region where z > 0, on the right of their excitation source site, expresses the amplification of these os- cillations. Two other roots determine the damped oscillations in the infinite medium. Similar consider- ations are convenient if δα < 0. In the problems on the excitation of oscillations in finite systems, for con- struction of solutions with in-time stable behavior, one uses all three roots of the dispersion equations. The application of a method of variation of arbitrary constants using the exponents makes it possible to solve the problems with an arbitrary dependence on the spatial variable (see also in [19] § 65 on the finite system instability). 6. CONCLUSIONS The conditions of excitation and propagation of neutron-temperature oscillations in the multiplying system with delayed neutrons are studied. It is ob- tained and investigated the characteristic equation that is equation of the third degree in relation to the wave number of complex kind. The roots behavior of equation depending on external source frequency and other parameters of the system is studied, in particu- lar, values of coefficient of reproduction of neutrons, their ”effective” temperature, times of capture and of coolant velocity. The calculation analysis data show that in the case of a weak coupling between the neutron field and temperature the dependence of the wave number real part on the frequency, near its low values, looks like a branch of a parabola with vertical axis and the imaginary part of the wave number is a linear func- tion. As the frequency of external source oscillations increases, the values of the wave number real part, as well as, the neutron oscillation spatial damping are increasing too (at high frequencies it is as square root of ω ). The signal amplitudes are noticeably decreas- ing when the frequency increases. Numerical analysis shows [22], that for standard parameter values of WWER-1000 type reactors, the neutron wave frequencies can be within the range of values from fractions and units of Hz to several tens of Hz with taking into account the time of thermal neutron capture by fission nuclei as a function of their ”effective” temperature. If the frequency oscillations is close to one of the inverse times of delayed neutron emission their influ- ence may be significant. Found out the noticeable increase of real values of wave numbers and coeffi- cients of the spatial damping of neutron branch of oscillations at diminishing of coefficient of diffusion of neutrons. Under similar conditions, the neutron fields can experience the transport by the convective temperature mode undergoing a slight damping with a weak connection of neutron and temperature per- turbations. Dependence of time of capture of thermal-neutron from their ”effective” temperature by nuclei of divi- sion plays a key role at the analysis of conditions of propagation of neutron-temperature oscillations. As a result of its account the strongly coupled neutron- temperature oscillation are formed. The mentioned dependence is a problem of nu- clear kinetics of the neutron field evolution in time. Namely, it is necessary to formulate the meth- ods describing the different non-stationary physical processes during slowing-down of fast neutrons from the nuclear fission reactions to the thermal energies simultaneously with their scattering and nuclear cap- ture. The conditions for propagation of analyzed os- cillations revealed in experiment [22] allows one to solve the problem of the ”effective” temperature exis- tence and dependence of the time of thermal neutron capture by fission nuclei on this temperature. The authors are grateful to S.V.Peletminsky and Yu.V.Slusarenko for helpful discussions of the matter and investigation results. The subject of the investigation fulfilled is in- cluded in the Government Program according to Con- tract No X-2-204. APPENDIX 1 ”EFFECTIVE” TEMPERATURE OF THERMAL NEUTRON The question concerning the dependence of the fission nuclear capture frequency νf on the ”effective” neu- tron temperature Tn and its relation with the medium temperature T is very complicated. One knows that in the reactor core the neutron temperature is by 50-100 degree higher than the medium temperature. In the region of thermal neutron energies, a steady- state connection was established [13] from the condi- tion of a balance between the flow of neutrons slowing down in the scattering processes and the flow of ther- mal neutrons captured by nuclei. The foregoing is expressed by the equation Tn = ( 1 + ∆ · Σa / Σs ) T. (38) Here represented are the medium temperature T , the ”effective” neutron temperature Tn and the macro- scopic capture cross-sections ∑ a = σaNa and scat- tering cross-sections ∑ s = σsNs, where σa and σs are the corresponding microscopic cross-sections of nuclei with concentrations Na and Ns. The parameter 4 depends on the moderator type (water, graphite etc.) and equals to the tenth part of unity. Phase densi- ties (their energy distributions ) of slowing-down neu- trons are not Maxwellian ones. It has been found [13] that the value of ”effective” temperature, at which the neutron distribution is maximum, exceeds the medium temperature . The evolution processes of neutron slowing-down flow during 10−5 s and their capture time is less than 10−4 s (the lifetimes of thermal neutrons). As the periods of neutron slowing-down exceeds 2 · 10−2 s, 36 it can be assumed that, in the oscillation processes too, the ”effective temperature of neutrons is related with the medium temperature by the relationship of (38)-type established for the stationary conditions. APPENDIX 2 Linearization of diffusion terms Let us go on to the problem of linearization in the neutron diffu- sion equation. Classical diffusion of thermal neu- trons and their transport, represented by the term of a diffusion type and related with the age of neutrons being slowing-down, are written in equa- tion (1) in the form of two terms with the co- efficients D and Da. Regard the explicit depen- dence of the temperature and density in the diffu- sion constants. By changing them for dimension- less ones (relatively to the values in the stationary state) ϑ = T/Ta and these introduced above ϑn and r we obtain, D = (1/3)v/(σtrNtr) = ϑ1/2+µD0/r, Da = νcKτ = ϑδ nDa0/r. The values of derivatives of diffusion constant are equal to ∂D/∂ρ = −D/ρ, ∂D/∂T = −(µ + 1/2)D/T and ∂Da/∂ρ = −Da/ρ, ∂Da/∂T = δ − 1/ ln(E0/T )Da/T . Here µ ≈ 0.1 rep- resents the weak dependence of the scattering cross- section on the medium temperature σs ≈ σtr ∼ ϑ−µ. The above-mentioned formulae takes into account the dependence of the inverse time of thermal neutron capture by fission nuclei on the density and medium temperature T and the dependence on the ”effective” temperature Tn. The result of diffusion term L̂ lin- earization takes the form L̂∇(D∇N)=∇{ D0∇n + [ T1G/T0 + P ext /( ρs2 )] D0∇N0}. (39) Here introduced is the notation for the factor G = 1/2 + µ+ ( γcT0 / (%s2) ) (P ′T )ρ, where the first two terms show the dependence of the diffusion con- stant on the temperature, and the last term is ob- tained by changing its density perturbation for ρ1 = −( T1/s2 ) γc(P ′T )ρ− P ext / s2 via the perturbation of temperature T1 in accordance with equation (9) with- out an inertial term and with zero boundary condi- tions ρ1(~r0) = T1(~r0) = P ext(~r0) = 0. Linearization of the diffusion-type term, related with the slowing- down neutron age, leads to the expression (a zero in- dex denotes, as before, the stationary values of quan- tities) L̂Da4N)=D04n + ( Da0Ga T1 T0 + D0 ρs2 P ext ) 4N0, (40) where the factor Ga = δ + ( γcT0 / ρs2 ) (P ′T )ρ− 1/ ln(E0/T ) contains the second terms obtained from the parameter of neutron age τ = l2 ln(E0/T ). When in the fission reaction the energy of neutron generated is E0 ≈ 5 MeV and the temperature T ≈ 0.05 eV this term is equal to ln(E0/T ' 1/20. The result of diffusion term linearization in the one-dimensional approximation is given in formulae (39), (40) and (7) of the paper text. References 1. Yu.M. Semchenko, V.A. Milto, A.A. Pinegin, B.E. Shumsky. Analysis of neutral flux noises produced by the of coolant parameters fluctua- tions in the WWER core // Atomnaya ehnergiya. 2007, v.103, N5, p.283-286 (in Russian). 2. Yu.M. Semchenko, V.A. Milto, B.E. Shumsky. 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Rudakov. Neutron tem- perature oscillations in neutron multiplication systems and in blankets of thermonuclear reac- tors // Problems of Atomic Science and Tech- nology. Series ”Plasma Physics”, 2011, N1(17), p.194-196. 21. C.C. Lin. The theory of hydrodynamic stability. ”Cambridge at University Press”, 1955, 194p. 22. V.A.Rudakov, A.A. Vodyanitskĭı. Numerical cal- culus of neutron-temperature oscillations in the controlled nuclear systems // Abstract on 2nd Conference QEDSP2011, Kharkov, 2011. АНАЛИЗ НЕЙТРОННО-ТЕМПЕРАТУРНИХ КОЛЕБАНИЙ В РАЗМНОЖАЮЩИХ СИСТЕМАХ С ЗАПАЗДЫВАЮЩИМИ НЕЙТРОНАМИ А.А. Водяницкий, В.А. Рудаков Внешние источники с заданной частотой в нейтронной размножающей системе возбуждают нейтрон- ные и температурные колебания. Проведены аналитическое рассмотрение и анализ колебаний в усло- виях как слабой, так и сильной связей между ветвями. Нейтронная ветвь колебаний описывает их распространение вдоль течения теплоносителя с нарастанием их амплитуды в области больших скоро- стей его движения, а также с затуханием амплитуды – в противоположном направлении. Конвективная ветвь колебаний испытывает слабое затухание амплитуды. При зависимости микроскопических сече- ний деления ядер от температуры тепловых нейтронов возбуждаются сильно связанные нейтронно- температурные колебания. Учтено влияние запаздывающих нейтронов. Сравнение аналитического рас- смотрения с анализом экспериментальных измерений подтверждает эти выводы. АНАЛIЗ НЕЙТРОННО-ТЕМПЕРАТУРНИХ КОЛИВАНЬ У РОЗМНОЖУВАЛЬНИХ СИСТЕМАХ З ЗАПIЗНIЛИМИ НЕЙТРОНАМИ А.А. Водяницький, В.А. Рудаков Зовнiшнi джерела з заданою частотою в нейтроннiй розмножувальнiй системi збуджують нейтроннi i температурнi коливання. Проведено аналiтичний розгляд i аналiз коливань в умовах як слабкого, так i сильного зв’язкiв мiж гiлками. Нейтронна гiлка коливань описує їх розповсюдження уздовж течiї теплоносiя з наростанням амплiтуди в областi великих швидкостей його руху, а також, з зату- ханням амплiтуди – в протилежному напрямку. Конвективна гiлка температурних коливань зазнає слабкого затухання амплiтуди. При залежностi мiкроскопiчних перетинiв подiлу ядер вiд температури теплових нейтронiв збуджуються сильно зв’язанi нейтронно-температурнi коливання. Враховано вплив запiзнiлих нейтронiв. Аналiтичний розгляд i аналiз експериментальних вимiрювань пiдтверджують цi висновки. 38

Analysis of neutron-temperature oscilations in neutron multiplying systems with delayed neutrons

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