Relation between reactor noise analysis and “not-linear” characteristics and comparative study on interpretation of analysed reactor noise

Progress in Nuclear Energy. 1982, Vol. 9, pp. 267-276 Printed in Great Britain. All rights reserved.

0079-6530/82/03267-1055.00/0 Copyright © 1982 Pergamon Press Ltd

R E L A T I O N BETWEEN R E A C T O R NOISE A N A L Y S I S AND "NOT-LINEAR" CHARACTERISTICS AND C O M P A R A T I V E S T U D Y ON I N T E R P R E T A T I O N OF A N A L Y S E D R E A C T O R NOISE H. KATAOKA 15-11, 1-Chrome, Minami (South), Asagaya, Suginami-ku, Tokyo, Japan

ABSTRACT Comparative studies were made on the interpretation of the analysed reactor-noise of MSRE, KSTR, and BWR: both the strange increments of low-frequency components of MSRE and discrepancies of the measured noise of BWR between by AR-method and by autocorrelation method. The Shot-nolse analogy was applied to the interpretation of the nolse-spectra figures of KSTR.

KEYWORDS Not-linear characteristics, benchmark tests, Molten Salt Reactor experiment, MSR, Kemma suspension Test Reactor, KSTR, Autoregressive method, Autocorrelation method, Shot-noise analogy.

INTRODUCTION Neutron noise analysis methods have been developed widely for the at-power reactor diagnosis, etc. but most of them still remain just as diagnosing disturbances outside the nuclear kinetics of the reactor, such as changes of neutron-path conditions from neutron-generatlon points to detec tors .i0) ~u~nermore, =.... targets of these analysis are majorly only in rather higher frequency range in neutron noise spectra (Shinohara, 1981), (SMORN-2, 1977) Because the theoretical analysis and experimental facts are well agreeable with each other for zero-power reactors neutron-noise analysis techniques have been able to be carried out by means of reactor kinetics. For at-power reactors, however, theoretical deduction and experimental reduction were not yet well agreeable, specially in very low frequency range, it was desperately difficult to apply the same kind of principle to them (Kataoka, 1975) The author and his collaborators already pointed out that the combination of "not-linear" characteristics (Akcasu, 1958), (Kataoka, etc. 1958), (Sandmeier, 1959), (Serdular, 1975) and output-to-reactivity feedback effects have important relations with the above-shown discrepancies (Kubo, Kataoka, 1966), (Kataoka, 1968), (Kataoka, Kubo, 1968), (Bridge, 1975) and some works to make this theory more concrete are presented in this paper. Concurrently the works expressed by this paper play the following role. Comparative studies of different analysers, like benchmark tests, are very important for establishing the analysis techniques, not only at the stage of analysing the fundamental data but also at the stage of fine interpreting the primitively analysed data. It is delightful and hopeful that the former side study is carried out successfully during SMORN-3 and the author practices from the latter viewpoint. He tries to interpret characteristics of the neutron noise of MSRE, KSTR, and BWR, based on the different standpoints from each original analysers.

COMPARATIVE STUDY ON MSRE NOISE Applying the above-mentloned theory, strange increments in low frequency range of at-power reactor noise can be interpreted as reasonable ones for Water Boiler Type JRR-I and two Tank

,~,.E

9 - J

26"1

268

H. K A T A O K A

Swimming Pool Types (KI/R, etc), as already presented, (Kataoka, 1977) as well as MSRE, as presenting in the followings. MSRE at-power reactor-noise, measured by Fry, Robinson, Kryter, etc. (Fry, etc. 1966), (Robinson, etc. 1967) also show big increments of very low frequency components and these caused have been kept as unclear as other reactors. Since the investigation on the noise of other reactors was explained in previous papers, the author discusses MSRE noise. MSR is most hopeful type of the reactor for the thermal breeding potential, although its development has stopped in U.S.A. and INFCE (International Nuclear Fuel cycle Evaluation did not regard it as a very important reactor. (INFCE, 1980) Many Japanese nuclear scientists such as K. Furukawa, Y. Katoh, N. Nakamura, T. Satoh, S. Saitoh, E. Takeda, E. Nishiborl, S. Hokki, H. Hokubu etc. have made best efforts to research and develop this reactor (Furukawa, Kataoka, etc. 1977), (Kataoka, etc. 1980) and now its practical development is hopeful in cooperation with American nuclear scientists such as A.M. Weinberg, (Meinberg, 1979), (Furukawa, 1979) D.R. deBoisblanc, (ANST, 1980) J.R. Engel (Engel, Prince), etc. In the former period of excited development of MSR by ORNL, the reactor noise of MSRE and other homogeous type reactor ( Hirota ) were measured, analysed and interpreted. According to those reports, low frequency components of MSRE at-power reactor noise show unexpectedly large increments and moreover those data show that the figures of noise spectra have tendencies to increase more for the lower frequency direction. The cause of these increments were not clearly explained, hut their figure look like similar with those in many other types of at-power reactors (Griffir, Randall, 1958), (Shibata, Utsuro, 1967), (Yamada, Kazl, 1966) and these increments seem to occur colncidentally with the occurrence of output-to-reactivity feedback effects. Therefore the author points out the problem increments may be caused by the combination of "not-linear" kinetics and output-to-reactivity feedback effects in reactor kinetics, and practi es numerical analysis based on this principle. When we assume that the most effective frequencies (Kataoka, 1977) of MSRE noise for "notlinear" transfer in the following meaning are around 0.3 x i0 -b . The result of the abovementioned calculation fits well the figures of frequency-power spectra of MSRE reactor noise analysed by ORNL and the meaning of the significant figures of these noise spectra can get more understandable for us. The "not-linear" transfer characteristics from two different original components is shown as in the equation (i) (Kataoka, 1975)

nn n

= ((

+

l 81 82

+ (

~2

1 + i )(_h_+ 81 s1.~----q 81 s1+-----q 81 k2 (_x_L_+ 8~ ~

X 1 6k I 81 Si

) (

X~ 6k~ Sz S2

2

)+

~

)

} 8kl • 6k2

8z

) .....

(i)

Here each symbols show each conventional meanings and the suffixes i and 2 to the parameters show each parameters in the change rate 1 and 2 respectively. Then we put fl is smaller than f2, and 1 2 and 1 + 2 in the above suffixes for the difference and the s,--mation of the change rate 1 and 2. Modifying the equation (i), we get the equation (2), without feedback

an n

. {(

i 1 kI X2 ) (-r--+-r--)} 81.-------~ 81 + Sl S~+--------~

81 + (¢(si)

-~7 ) ~kl} {G(sz) ~kz} .....

(2)

~kl • ~k2

Reactor noise analysis and "not-linear" characteristics

269

Here G shows the transfer function from the reactivity input to the power output. When the change rate 1 is very low frequency component, the last term is most effective in the right side of the equation (2). When we assume the "not-linear" transfer effects are predominant for establishing some frequency component, this problem frequency component must be composed with the combination of many components each of those which are resultant components from comple of other frequency components. Among these original frequency couples, a couple composed with the well-balanced very low frequency component ~Fm and its partner frequency component is most effective for the above-mentioned resultant frequency component. This expression "well-balanced" means that the product of the frequency-component amplitude and the linear transfer function at that frequency is largest. This most effective frequency component and its neighbored frequency components are called the most effective frequencies for the "not-llnear" transfer. The above-shown neighbored frequency components involve frequency components which effect can be regarded as about the same as previously-described most effective frequency component. Then we can deduce the simplified equation (3) from the original equation (2), with feedback An(f) n & {G(Fm)

• ~k(Fm)}{C(fz)

" ~k(f2)}

& IG(Fm)

• 6k(Fm)}{G(fe)

" ~k(f2)

& {u

~n(Fm)} (=

~) " (G(f2)

An(f2%} G(Fm) 1

. ( An(Fm_____D } . {An~f~) n

} . G(Fm)

G'(f2) G(f2 )

•

'

• / 2Fm /

2Fm

.....

(3')

.....

(3)

.....

(4)

/ 2Fm

• = • / 2Fm

Here G(f) and G'(f) express the transfer functions without feedback and with feedback respectively. f = f2 ± Fm & f= .....

(5)

Then we can find that the relation shown by (3) is free from fl and duduce the approximate relation shown by the equation (6) for the steady state or the quasi-steady state,

{

~n(Fm)

)

• C(Fm)

• =

• / 2Fm

~ i

.....

(6)

n

The fine figures of frequency power spectra of the reactor noise are decided by the additional effect of the minor factors to the effect of the above-descrlbed principal factor. Although Fm stands in very low frequency side and can may estimate Fm to satisfy the equation (6).

not be easily measured directly, we

Resultantly we get around 0 . 0 0 0 5 ~ 0 . 0 0 1 for Water Boiler Type JRR-1, around 10 -4 for Tank Swimming Pool Types KUR and one more in each problem at-power operation conditions, as already presented• In this way, when we put around 0.3 x 10 -4 for Fm, the equation (6) is about satisfied for MSRE at 5 MW operatlon condition• As the result of these studies, the reliability on the reactor noise analysis by the classical principles can be supported for MSR.

COMPARATIVE STUDY ON KSTR N O I S E

The Light or Heavy Water Suspension Reactor has also the thermal breeding potential by use of Th fuel and Netherland etc. have made efforts to develop it. Veer presented the analysis of KSTR 0KEI4~ Suspension Test Reactor) reactor noise at SMORN-2. (Hoekstra, Veer, 1977)

270

H. KATAOKA

In this paper, KSTR noise was handled based on the viewpoint following the current tendenty common in the at-power reactor noise field. Recently it has been believed that the reactor output or neutron-flux level is proportional to the powers of frequency components in noise spectra of the O-power reactor but proportional to the amplitudes of frequency components of the at-power reactor. In the latter principle, the at-power reactor nnise is handled as the product of the reactivity input and purely linearized transfer function of reactor kinetics. Surely this fundamental priciple of handling at-power reactor noise is available to the neutron noise caused by disturbances outside or independent of nuclear kinetics, but the foundamental handling of the at-power reactor noise source closely connected or compounded with nuclear kinetics is possible to follow the classical principle to handle the reactor noise. In the original paper, the comparison among the data at various power levels was not yet written clearly, and the discrepancies among the noise spectra at different conditions seem to be interpreted not so easily. Therefore the author investigates and trys to interpret KSTR noise from the new viewpoint. In the original paper, the data were arranged in the graph following the noise-proportlonal-to-power-level principle. When the data are arranged following the noise-proportional-to-square-root-of power-level principle, the comparison among noise spectra at different power levels seem more interpretable. When the normalization reference of noise spectral density is changed from the power level into its square in the graph of noise spectra of KSTR, the values at different power levels on the ordinate agree in each other rather well in higher frequency side and little differ from each other in lower frequency side. Both this agreement and little difference seem to be more understandable than in the comparison in the original paper's graph. Usually the essential characteristic of the prevlously-descrlbed phenomena following the fluctuatlon-proportional-to-square-root-of-level condition is based on the principle which is similar to the Shot-noise theory. In the case of KSTR, if we assume that the suspensions flow in the formation of element groups which flow randomly, we can regard this reactor as Shot noise-analogy model or well-known Cohn's type noise source.(Moore, 1958), (Cohn, 1959), (Cohn, 1960). Replacing one electron in Shot -nolse-theory basic equation or one fission neutron in Cohn's fundamental reactor-nolse equation by the above-mentloned element group of suspensions, we get the following basic equation (Gotoh , 1975) for KSTR noise, - = 2( n__.n__) . q . G(f)2 ..... N2 ~*

(7)

Here q is generated neutrons from one element grouped-flow suspensions and other symbols have each conventional meanings. In the case of this calculation, it can be estimated that the fuel material in one element group is around 0.5 x 10-7 of the total fuel material in the core. This ratio seems to be not unreasonable. The above works on MSR and KSTR make us understand the case of at-power reactor noise from the deeper aspects and verify the significance of the use of the basic noise theory, such as Cohn's Shot noise analogy as well as the author's "not-linear" deduction. In these homogeneous and semi-homogeneous types of reactors, mechanical disturbances and faults occur much less often than in conventional solld-fuel types of reactors. Therefore the neutron. noise analysis techniques by the classical p~inciple will be much more effective for MSR and STR than for conventional solid-fuel types of reactors. Moreover it is possible that the above classical principle will be more effective than the previously-mentioned current principle with the fluctuation-proportional-to-power-level noise model for these homogeneous or semihomogeneous type of reactors. Concurrently the characteristics of MSR and Suspension Reactors, which were not well-known yet, can be understood more clearly as the results of the above works and it is convinced that these works push developments of these useful types of power-generating and fuel-breeding reactors.

Reactor noise analysis and "not-linear" characteristics

271

COMPARATIVE STUDY ON B N R N O I S E , SPECIALLY FOR AUTOCORRELATION METHOD AR method and autocorrelation method were compared in the analysis of BWR noise by Fukunishi (Fukunlshl, 1976), (Fukunish, 1977), (Fukunishi, 1976) and large discrepany occured on low frequency side. This research must be respected as very valuable one, that is, a kind of benchmark test for different ways by the same researcher. In Japan, the noise analysis of the same reactor bv th~ different researchers were practiced rerely such as for JRR=I and Swimming Pool Types. The main targets of each analysis, however, were a little different from each other and so the purpose ot the comparatlve studies were not so straightly satisfied by these competitive works. In Fukunishi's work, the result of computation by A R method is larger than that by the autocorrelation method, that is to say, the former's values were several times of the latter's in low frequency range. The cause of this discrepany was also not made clear. Here the author trys to get deeper insight for the autocorrelatlon method and will extend his research to other methods including AR method later. The autocorrelation method has the unique characteristic, because it squares the data at first and then analize into frequency components, but other methods such as A R method direct Fourier transform method, usual bandpass filter method, etc. analize the data into frequency components at first and then square them. The problem points may be classified into the following cathegories. (i)

The occurence of experimental error or analytical error, accidentally or inevitably.

Becuase the original paper's author is very reliable, this paper's author believes we need not mind this problem here. (2)

The effect of smoothing function.

Usually in the autocorrelatlon method, the smoothing function is multiplied in the analytical process. This effect make the resultant data somehow lower but this influence must be impressed not only in ~ow frequency side but about equally over all frequency range. (3)

Phase Coherency of noise spectra.

It is conventionally believed that the phases of different frequency components are independent of each other in the reactor noise but this belief can not hold good under the influence of specific characteristics in reactor kinetics such as "not-llnear" characteristics, as already mentioned in SMORN-2. (4)

Line spectra formation of reactor noise.

It is believed now that frequency spectra of reactor noise distribute continuously and Cohn's and Moore's theory suggests that it is true, but the truth is not yet clearly disclosed for us and these spectra may be possible to be the group of llne spectra. (5)

The frequency band width cover by a nominal problem frequency

It is impossible for us to practice strictly continuous frequency anaysis of the data. Furthermore when the frequency spectra are really continuous, the spectral density of the strictly point frequency component must be zero. Then usually and inevitably we get some grouped frequency components distributing over some width around the problem frequency component as nominally the problem frequency component. The width and effect of these grouped frequency components are naturally different from each other between the autocorrelation method and other ways. In the autoenrrelation method such grouping process occurrs twice, i.e. at the first autocorrelation process and the second Fourier transform the above-mentloned grouping problem becomes more complicated. When the widths of these frequency group are larger, total power of grouped frequency components may become larger but concurrently mutual cancellation may occur depending on these frequency

272

H. K A T A O K A

component's phases. These effects possibly influence the analysis incidentally like the orlginal paper's result. However, since the result of this effect may be flexible depending on each freouencv eomDonentsohase, the effect sometimes may Rive the influence like the above result but sometimes may give the influence on the opposite direction of the above result. Therefore, if the result shown in the original paper is gained constantly, this problem's effect is doubtful to be the only one cause of the above-described discrepancy. (6)

Combination of "not-linear"

tranfer characteristics and autocorrelating process.

The combination of the "not-linear" transfer and the autocorrelation process produces a couple of factors which cansel each other for some frequency component. Here we put two frequency components

Anl = An(fl) = An0(fl) sin (wlt + a)

. . . . .

(7)

An 2 - An(f~) = An0(f2) sin (w2t + 8)

. . . . .

(8)

.....

(9)

fl > f2

Suppose "not-linear" transfre effects are predominant :in the formation of n(f I) and n(f2), then we get the next relations 6ki " An(fl)

..... (I0)

Ak2 ~ An2 • a

..... (ii)

Then, in the same way as the equation (3)', we can get the equation (12).

An(fl + f2) n

Anl = G(fl) " { - } " G(f2) n

" (

~nz n )

G'(fl + f2) .....

(12)

G(fl + f2) Following the similar deduction to (3), (4), (5), we get the following factor (13) as the principal part of the right side in the equation (12).

l

{J

r

sin (wlt + e) dt} { J

sin (w2t + 8) dr}

G' (fl + f2) ..... (13) G(fl + f2) When we regard that the sunm~ation of fl and f2 is good large, the phase factor in {G'(fl + f2)/G(fl + f2)} may be neglected approximately. Then we get the following factor (14) as the principal part of (13). cos {wl + w2)t + (= + B)}

..... (14)

Reactor noise analysis and "not-linear" characteristics Then principal part of the autocorrelation of the components and (16). __

i

W2

cos

[

(w2t - w 1 ~ + B ) ]

= i___ c o s

273

(7) and (14) is expressed as (15)

t = T - r t 0

..... (15)

[cos (w2(T - T ) ) - I]

(s - w z z )

%;2 _ i___ sin (81 - w1~) w2

[ sin ( w z ( T - T))]

..... (16)

In the same way as this process from 8) and (14), w e get the principal part of the autocorrelation, (17)

i__ [ COS (wit - w2z + a)]~ = T - z wt 0 = l__cos

wl

(~ - wzT)

- I___ sin (S - wz~) wl

..... (17)

[cos (w1(Z - ~ ) ) - i]

[sin {wt(T - z)}]

H e r e T shows the total interval of the aquired tion interval.

..... (18)

data and T shows the correla-

M u l t i p l i e d by cos wit and cos w21, we get (19) and (20) from (16) and (18), respectively.

!(cosS+cos

(B - 2 w i T ) }

[cos

(w2(T-

T)) -

t]

W2

_!

(slnB - sin (S - 2WIT)}

[sin (w2(T - T))]

..... (19)

w2

!(cosS

+cos

(S - 2w2T))

[cos (w1(T - ~)) - l)

W2

_

1 (sins -sin (S - 2w2r)} wx

[sin (w1(T - ~)}]

..... (20)

On the other hand, w h e n f, and f2 are not far from each other, (fl - f2 ) may be regarded as very small and in this condition the first term may be predominant in the right side of the equation (2). Then w e get the following factor as the principal factor of in the similar way to the p r e v i o u s l y - d e s c r l b e d process for cos ((wl - w2)C + (a - S)}

..... (21)

The principal parts of the autocorrelations of (21) w i t h and (24) respectively.

-

I [cos w--~

(w2t

- ----I c o s

+w,T

+

S)]

(B + w 1 ~ )

t = T t = 0

[cos

(7) and (8) may be expressed by (22)

- T

(w2(r

- T))

-

.....

(22)

.....

(23)

.....

(24)

I]

W2

+ _ _ I sin (8 + wiT) w2 ___1 wI

[ sin (wz(T - T))]

[cos (wit +w2T + a ) ] t - T t ~0

274

H. KATAOKA

! ----cos

(8 + w 2 T )

wl

+ _ L sin (8 + w2r) wl

[cos ( w z ( T -

T)} -

t]

[sin {wl(T - r)}]

..... (25)

Here we may pnt G'(fl - fz) G (fl - f=)

=

i S(fz - f2)

..... (26)

since (fl - f2 ) may be regarded as very low. Multiplied by cos w, and cos w2, , we get (26) and (27) from (23) and (25) respectively in respectively in the same way as (19) and (20). I. - -w2 -tcosS

+ cos (S + 2wlz)} [cos (w2(T - z)}

I]

+ 1w2 (sinB + sin (8 + 2wzT )} [sin (w2(T - r)}]

.....

(27)

___i wl (toss + cos (S + 2w2r)} [cos {wz(T - r)} - i] + _~i wl {slnB + sin (8 + 2wzT)} [sin (w1(T - r))]

..... (28)

Comparing (19) and (20) with (27) and (28) respectively, we can find clearly that the major factors of (19) and (27) as well as (20) and (28) cancel each other. Because (woT) must be larger than (2~), all of cos (8 Z 2Wlor2r) and sin (8 + 2Wzor2T) are very changeable depending on and can not hold some deterministic value llke c0s8 or sin~. Moreover

T I "~ J 0

{COS (S ± 2W f o r 2 T) or sin (~ ± 2w I o r 2 ~ ) } d ~ l

must be smaller than (i/dw) of the integration interval, E. Thus, generally the effect of these terms may be neglected comparing with the effect of the term cos8 or sln~ In (19), (20), (27), (28). Changing the above multiplication factor from cos w~ (w = w, or w2) , a couple of two factors of the similar results to the above don't cancel each other but associate with each other. Therefore the above-shown cancellation occurs strongly only when conslne transform is used for Fourier transform at the second stage of the autocorrelatlon method and such cosine transform is used in the original paper. The formation of the above-descrlbed cancelling condition by the suitable combination of two ways can be made in the moderately low frequency range. Resultantly we may say that the discrepancy of the dlscrepany between the autocorrelatlon method and AR method does not seem so strange.

ACKNOWLEDGEMENT The author has been completely obliged to and would like to present may thanks to, for thelr constant help for long years and well-timed useful advlces: in general research, Dr. T. Mukalbo, Japan's AEC, Prof. Nishlno, Kohgaku-ln University, Prof. A. Seklguchl and Prof. Y.

Reactor noise analysis and "not-linear" characteristics

275

Togoh, University of Tokyo, Dr. J. Hirota and Dr. Y. Shinohara, JAERI, Prof. H. Kawal, Kinkl University, Dr. C. E. Cohn, ANL, Prof. A. Z. Akcasu, University of Michigan; about MSR research, Dr. K. Furukawa and Dr. Y. Katoh, JAERI, Dr. T. Satoh, Sumitomo Co~mmerce Co., Prof. N. Saltoh, Toho University, Dr. E. Nishlbori; Dr. D.R. deBolblanc, Ebasco Service Co. about STR research, Dr. J.H.C.v.d. Veer, N.V. KEMA, Dr. S. Hohkl, Sumltomo Atomic Indus. Co, about BWR research, Dr. K. Fukunlshi, Hitachi Ltd.

REFERENCES Akcasu, A.Z: General solution of the reactor kinetic equation wlthou feedback, Nucl. Scl. Engng. 3(4), 456-467(1958) ANSJ: Recent development of MSBR techniques, note on the meeting wlth Dr. deBolsblanc, Renlsed and Enlarged Edition , Apr. 1981. (the same content in J. of ANSJ. V01.22, No.3, 1980) Bridge M.J.: A review of SMORN-I, Nucl. Energ. Vol.2. No.2-5. June, 1975. p.55-58. Cohn, C.E. : A simplified theory of pile noise. Nucl. Scl. Engng 7, 47 475(1960). Engel J.R., Prince B.E.: Zero-Power Experiments with 233V in the MSRE, ORNL-TM-3963 Fukunishi K.: Coherence analysis of BWR noise, JNST. Vol.14, No.5. May 1977. p.351-358 Fukunlshl K.: Dynamical analysis of BWR by multivariable autoregresslve midel, JNST. Vol.13, No.3, March 1976. p.139-140 Fry D.N., Roux D.P., Ricker C.W., Stephenson S.E., Hanauer S.H., Trlnko J.R.: Nuetronfluctuation measurements at Oak Ridge National Laboratory, Symp. at Galnesmille, Florida, Feb. 14-16, 1966. p.463-474. Furukawa K., Kataoka H., Satoh T., Takeda E., Saltoh S., etc.: Molten salt breeder reactors, Japan Atomic Energy Society Report, 1977. Furukawa K. etc.: Trans. into Japanese , Chuoh-Kohron, Winter, 1979 Gotoh Y.: Study of the power spectral density by a nonlinear responce to the stochastic input, Nucl. Energy. Vol.2. No.2-5. p.i19-125. Jun. 1975. Griffin C.W., Randall R.L.: At-power low-frequency reactor-power°spectrum measurements and comparison with oscillation measurements, Nucl. Sci. Engng. 3(4), 456-267, 1958. Hirota J.: Statistical analysis of small power oscillation in HRT, CF-60-I-I07, 1960. Hoekstra J., Veer J.H.C.v.d.: Noise measurements on the KEMA Suspension Test Reactor (KSTR) and the Dodeward Boiling Water Reactor, Nucl. Energ. Vol.l. No.2-4, p.583-503. 1977 (Proceeding of SMORN-2) INFCE WG8: INFCE/PC/2/8 (1980) Kataoka H., Ishikawa M., Tsuchida T.: Nonlinear instability of proper characteristics in nuclear kinetics of reactor, Trans. Joint Convention of Four Electric Institutes of Japan, Dec. 1958. Kataoka H.: Relation between nonlinear or "not-llnear" characteristics in nuclear kinetics and noise analysis of neutron flux, Ann. Nucl. Energ. Vo;.2 No.2-5. 407-413, Jun. 1975 (Proceedings of SMORN-I) Kataoka H.: Effect of nonlinear or "not-llnear" characteristics of reactor kinetics on power reactor noise analysis and proposal of appplication of some newly developing techniques to power reactor noise analysis, Nucl. Energ. Vol.l No.2-4. P.575-582. 1977. (Proceedings of SMORN-2) Kubo M., Kataoka H.: Identification of transfer function of reactor control system by noise analysis, Symp. at Galnesmille, Florida, Feb. 14-16, 1966, p.439-453. Kataoka H.: Relation between nonlinear or "not-llnear" characteristics in nuclear kinetics & noise analysis of neutron flux, Japan-Unlted States Seminar on Nuclear Reactor Noise Analysis, 2-7th, Sep. 1968. Kataoka H. Kuho M.: Experiment and analysis for strange increment 6f low frequency neutron noise at high power operation, Japan-Unlted States Seminar on Nuclear Reactor Noise Analysis. 2nd-7th SeD. 1968. Kataoka J., Furukawa K., Shlba K., Konno K., Hokkl S., Saitoh S., Koklbu H., Takahashi Y., etc.: "Th cycles, Present status and future aspect", Japan Atomic Energy Society Report, Oct. 1980. Kataoka H. and Kuho M., Identification of transfer function of reactor control system by noise analysis. Proc. Int. Symp. "Neutron Noise, Wave and Pulse Propagation." USAEC, 1967. p.439454 (Symp. at Ganesville, Florida, Feb. 14-16, 1966). Kebadze B.V., Adamovskl L.A.: Non-llnearity consideration when analysing reactor noise statistical characteristics, Nucl. Energy. Vol.2 No.2-5. p.337-34-. Jun. 1975 Moore M.N., The determination of reactor transfer function from measurement at steady operation. Nucl. Sct. Engng 3, 387-304(1958) Robinson J.C., Fry D.N.: Determination of the void fraction in the MSRE using small induced pressure perturbations, ORNL-TM-2318, Feb.6.1967.

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