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A simple space dependent theory of the neutron noise in a boiling water reactor
Annals of Nuclear Energy, Vol. 2, pp. 315 to 321. Pergamon Press 1975. Printed in Northern Ireland
A SIMPLE SPACE D E P E N D E N T THEORY OF THE N E U T R O N NOISE IN A BOILING WATER REACTOR G. KOS.~,LY, L. MAR6TI and L. MESK6 Hungarian Academy of Sciences, Central Research Institute for Physics 1525, Budapest 114. P.O.B. 49, Hungary On the other hand the recent experimental In a recent paper [1], D. Wach and one of the results of Wach [1], [5] indicate clearly the influence present authors [G. K.] have developed a space of noise-sources affecting the core as an entity. The dependent theoretical model for the description of influence of such global driving sources is demonstrated also by the successful applications of the the neutron noise in a large BWR core. In contradiction to the models used in zero-power point reactor model to the description of the noise theory the model is rather heuristic, that is it neutron noise induced by bubbling [6, 7]. In view of the above assumption we consider the relies on physical intuition and its justification can be fluctuation of the detector current at the axial given only by using the model successfully in the position z [i(t, z)] as composed of two parts: a interpretation of noise measurements. In Chapter 1 of the present paper we put forward local one (it(t, z)) responding to local disturbances an improved version of the model and apply it to and a global one [igz(t, z)] representing the respond the interpretation of correlation measurements per- to perturbations affecting the reactor as a whole. formed by incore neutron detectors. The derivations We write therefore that of this chapter follow quite closely the ones given in i(t, z) = iz(t, z) + i~(t, z) (1.1) Chapter 3 of Reg. [1]. The alterations made in the It is one of the basic assumptions of the present derivation will be pointed out at the end of Chapter 1. work that the driving source of the local part of the In Chapter 2 the auto-spectral-density and the noise-field is the fluctuation of the steam-void mean-squared-noise-amplitude of the fluctuations of content at the axial position in question, that is the current of an incore neutron detector are calculated. At the end of this chapter some virtues and h(t, z) = ~oH~(t -- t') ~Vs~(t' , z) dt'. (1.2) weak points of the model are discussed. In Chapter 3, some experimental results (norrealized root mean square of the detector current) In equation (1.2) dVa(t, z) represents the fluctuation of W. Seifritz [2] are analysed using the results of the of the steam-void content at the axial position z. The transfer function H,(t) relates this noise source previous chapter. to detector current. For the description of the global part of the 1. T H E T H E O R E T I C A L MODEL AND ITS noise-field we adopt the point-reactor model, that is APPLICATION TO THE INTERPRETATION OF CORRELATION MEASUREMENTS the driving source of the global part is identified In the considerations a simple, axial dependent with the fluctuations of the reactivity. According to the point-reactor model the space model of the core is used. As it was done in Reg. [1] we start with the physical assumption that the dependence of the neutron noise is dictated by the fluctuations of the neutron density at a given space critical flux, that is the noise is space-independent point inside the core can be composed of two parts: in regions where the flux can be considered as flat. a local part driven by local disturbances, and a For the sake of simplicity such regions have been global one representing a more or less coupled considered in Reg. [1] and we restrict the treatment to such regions in the present paper as well. behaviour of the neutron field. We indicate the space-independence of the global That a considerable part of the neutron noise in a large BWR is driven by local disturbances was part of the noise field by the notation INTRODUCTION
demonstrated by several workers [2-5] who were able to measure steam-void velocity by correlating the signals of incore neutron detectors.
igz(t, z) =-- ig~(t).
(1.3)
To investigate the behaviour of the stochastic
315
G. KoS~,LY, L. MAR6TI and L. MESK6
316
variable 6 Vs,(t, z) we introduce the functions B(t, z, v) = Bo(z, v) + b(t, z, v)
(1.4a)
J(t, z, v) = Jo(z, v) + j ( t , z, v)
(1.4b)
B(t, z, v) and F(t, z, v) represent the number and the current of the steam-bubbles with volume v, at the axial position z, at time t, respectively. Obviously J(t, z, v) = V(z)B(t, z, v)
Using now equations (1.1), (1.2), (1.3) and (1.9) the functions CPSD~vz~(co ) and APSD~li(co) can be readily evaluated and using equation (1.10) the transfer function can be given. One obtains the following result*: 5
gld~o) =
(1.5)
where V(z) is the average velocity of the steambubbles at the level z. (Fluctuations of this velocity are not considered in the present treatment.) In equations (1.4a) and (1.4b) the notations Bo(Z, v) = (B(t, z, v))
(l.6a)
Jo(z, v) = (J(t, z, v)
(l.6b)
have been used. If one considers now two axial positions z 1 and z~ (z2 > Zx) and neglects bubble collapsing between these two positions one finds that*
j(t, zz, v) = j ( t - TIe, Z1, V) + ('" ")
(1.7a)
Kle-iC°rl2+ 1
V2
I71 = V(21)
Ve = V ( z , ) .
K1 + 1
(1.11)
K 1 = K ( ( o , Zx)
Here the notation K ( w , z)
APSDzh(co) APSDig~(o~)
(1.12)
was used. The quantity K(~o, z) being equal to the ratio of the APSD of the stochastic variable i~(t, z) and the APSD of the stochastic variable ig;(t) measures the importance of the local part of the detector current as compared to its global part. If for a given frequency Kx >)1 (significant dominance of the local effect), then equation (1.11) goes over into
that is His(to ) = V1 e_i,o,t, Ve
V(z2) b(t, z e, v) = V(Za) b(t - -q~, zl, v) + (" • ") (1.7b) In the above equations Tie stands for the mean transit time of the bubbles between z x and z e. The unspecified second terms on the RHS of the equations account for the bubbles being born between the two space points considered. Using the obvious relation 6Vst(t, z) =
f;
vb(t, z, v) dv
(1.8)
together with equation (1.7b) one obtains that 6Vs,(t, zz) = V ( z 1 ) 6Vs~(t -- "q~, zl) + ('" "). V(ze)
(1.9)
that is the mean transit time of the bubbles between Zx and z e can be measured by determining the phase of the transfer function [2-5]. On the other hand ifK 1 ~ 1 (significant dominance of the global effect) one obtains that Hl~(o~) = 1
_
i
co
CPSD~I'Z"( ) = I H x d o 0 l APSDz~/(~o)
g ~°12(c°1.
(1.14)
which is the well known result of point-reactor theory. From equation (1.12) it is easy to derive the gain and the phase of the transfer function. One obtains that
Let us define now the transfer function Hie as the ratio of the CPSD of the two incore detectors placed at the positions z 1 and zz respectively, and the APSD of the signals of the incore detector placed at zx. Hie(co)
(1.13)
(/(1 + 1)~ _2V_A K1 V~(K1 + 1) ~[1 - c ° s ( ° ~ l e ) ]
(1.15)
(1.10)
* To derive equations (1.7a) and (1.7b) the expansion of the volume of the bubbles while propagating along the core is neglected as well. In view of the small differences between the values of temperature, and pressure at the top and at the bottom of the core in a BWR this assumption seems to be justified.
~0xe
c° / arctg /~_~2 1s'-in "(~''''12) V1 _ _ + cos (co~lz)| K, )
(1.16)
* Some details of the derivation not specified here can be found in Ref. [1].
A simple space dependent theory of the neutron noise in a boiling water reactor which equations go over into the corresponding results of Ref. [1], if V(zl) -- V(z2) is assumed. The differences between the above derivation and the one given in Ref. [1] are twofold. Firstly, in Ref. [1] the fluctuation of the number of bubbles rather than the fluctuation of the steam-void content was considered as the driving source of the local component of the noise field. As the steam-void content fluctuates because of the fluctuation of both the number and the volume of the bubbles we feel that the present choice is the one which is more conceivable physically. Secondly, in Ref. [1] it was assumed tacitly that the velocity of the steam bubbles is constant along the axis of the reactor, an assumption which is certainly not valid. As regards the consequence of the above alterations it can be seen that it is only the change of the velocity of the bubbles along the core which affects the expression of the transfer function. The considerations of Ref. [1 ] have been prompted by the fact that the gains and the phases of the transfer functions measured by Wach at the Lingen BWR, [1 ], [5] showed a very characteristic structure which could be understood neither on the basis of fluctuations driven by local disturbances, nor on the basis of point reactor theory, but was easily interpreted by the combination of the two (for details see Ref. [1 ]. It is important to emphasize that the appearance of the factor V1//12 :~ 1 in equations (1.15) and (1.16) influences neither the locations of the dips and the bumps of the gain, nor the general behaviour of the phase, that is the alterations introduced in the present work do not spoil the striking agreement between theory and experiment which was pointed out in Ref. [1 ].
In equation (2.1) APSDzS~(to) is the auto-spectraldensity of the fluctuation of the steam-void content at the axial position. In order to analyse the function APSDzSt(~o) we use equations (1.8) and (1.5) leading to 1
~o
6Vst(t, z) ~- ~ f vj(t, z, v) dr. V(z) Jo
1
£.oo
APSD~St(to) = VZ(z)J °
/-co
do'J0 dv"v'v"
× CPSD~.~,;z.¢(co). (2.3) Here CPSD~.v,z.~, is the cross-spectral-density of the stochastic variablesj(t, z, v', v-1) andj(t, z, v"). In the Appendix we show using rather plausible arguments that CPSD~.~:~,f(~o) = Jo(z, v') 6(v' -- v")
(2.4)
that is the above cross-spectral-density can be expressed readily by the mean value of the current of the bubbles. Using equations (2.3) and (2.4) one gets that
vZJo(Z, v) dv
APSD~ ~t(~o) = ~
V2(z) fO°°
= V (1z )
ff
v2Bo(z, v) dr.
(2.5)
Introducing now the second moment of the volume of the bubbles at the axial position z by the definition
2
f f v 2 B o ( z , v) dv
(v)~ = -
(2.6)
fo
°°Bo(z, v) dv
one may write equation (2.5) as A P S D zSt(~o) = (v2)z Bo(Z)
V(z)
1.3)*.
(2.7)
where
APSDzi(co) ~ [H~(~o)]2 APSDzS~(~) + APSD%(co).
(2.2)
From equation (2.2) one obtains by standard techniques that
2. C A L C U L A T I O N OF T H E APSD AND T H E N R M S OF T H E F L U C T U A T I O N OF T H E DETECTOR CURRENT
The model used in the previous chapter is represented mathematically by equations (1.1), (1.2), (1.3) and (1.9). To derive the APSD of the stochastic variable i(t, z) we use equations (1.1-
317
(2.1)
* To derive equation (2.1) it is assumed that the stochastic variables tSVs~(t,z) and i~(t) are independent. ig~(t) is obviously proportional to the fluctuation of the reactivity, that is its fluctuation is influenced by the entire steam content of the reactor, which in turn, is rather weakly correlated to the steam content at a particular point of the system.
Bo(z) =
~0°°Bo(z, v) dv
(2.8)
is the average number of bubbles at the height z. Assuming that (v2)z ~ (v)z2 (2.9) and expressing the equilibrium value of the volume of the steam per unit volume of the steam-liquid
G. KOSALY, L. MAR6TI and K. MESK6
318
mixture (local steam-void fraction) as
~(z) = (v>~Bo(z)
(2.10)
A P S D s~(co) = (v)z ~(z) V(z)
(2.11)
the result
is obtained. It is a very characteristic consequence of equation (2.4) that the auto-spectral-density of the steam-void fluctuation does not depend on the frequency (el. equations 2.5, 2.7, 2.11). On the other hand it does depend on the axial position via the axial dependence of the average volume of the bubbles ((v)z), steam velocity (V(z)) and average steam-void fraction (~(z)). According to thermohydraulic predictions (see Ref. [8]) (v)z depends but weakly on the axial position, that is the auto-spectral-density function of the fluctuations of the detector current can be given by the expression: APSD~(os) =
IH,(o01u ~ ( z )
Z(z) + APSDtg~(oJ)
(2.12)
In order to characterize the amplitude of the neutron noise the so-called NRMS-value (Normalised Root Mean Square) is often introduced [2, 3]. By definition NRMS(z)
x/(i2(t, z))
Io( z )
(2.13)
where (is(t, z)) and Io(z) are the second and the moment of the detector output current respectively. As
first
(i2(t, z)) =
APSDz/(¢o) dco
(2.14)
one obtains using equation (2.12) that* ~(z) [NRMS(z)] 2 = m ~ + C V(z)
(2.16)
where C =
APSD%(o) do; TM
m
I0 ~J° IH~(co)l2 do~.
* One has to remember that the critical flux was assumed to be fiat throughout the present paper, that is t0(z) = to.
Equation (2.16) is a mathematical form of an earlier observation of Stegemann, Gebureck, Mikulski and Seifritz [3] who found by the inspection of their experimental results that the noise amplitude gives a measure of the steam-void fraction and boiling intensity in the respective core region. Equations (2.12) and (2.16) demonstrates both the virtues and the weak points of the present treatment. One of the virtues of the model is that in spite of being a space dependent model it leads to simple, clear results which are rather conceivable by physical intuition. The main shortcoming of the model is its phenomenological character. This is why in the results one always has unknown quantities as K(oJ, z) in equation (1.11), IH~(,o) I~and APSD~g~(o0 in equation (2.12) and the constans m and C in equation (2.16). This shortcoming of the model cannot be removed without the derivation of the model from transport or diffusion theory and necessarily restricts the usefulness of the model both in theoretical work and in the interpretation of experiments. Considering the virtues and the shortcoming one has to conclude that it is certainly the comparison to experimental results which has to decide whether the model used in this work can be considered as a valid "working description" of the noise in a BWR. We feel that the considerations of Ref. [1] indicate that the model is applicable to the interpretation of correlation measurements performed by incore neutron detectors. To seek further corroboration of the model, in the next chapter we use equation (2.16) to the interpretation of the experimental results of Seifritz [2]. 3. ANALYSIS OF THE MEASURED VALUES OF THE NRMS OF THE FLUCTUATION OF THE DETECTOR OUTPUT VIA EQUATION (2.16) Figure 1 is taken from Ref. [2] and shows the axial distribution of the mean detector current (critical flux) as measured by Seifritz at the Lingen Nuclear Power Station (KWL). The figure shows that in the almost central radial position where the measurements have been performed the flux is approximately constant between the measuring positions No. 9 and No. 2. Figure 2 is taken also from Ref. [2] and shows the measured values of the mean square noise amplitude as the function of the axial detector position.* * In Ref. [2] measured values of the quantity NRMS are given, while on our Fig. 2 the values of its square are indicated.
A simple space dependent theory of the neutron noise in a boiling water reactor o 03
x
P
~
OI
~0
0
Lower edge of c o r e
1(30
150
200
Axiol coordinofe,
250
z~cm
30o
Upper edge of core
Fig. I. Mean detector net current at 11 axial positions (from the paper of W. Seifritz [23].
319
On Figs. 3-5 the results of the calculation can be seen. It is to be noted that the calculated curve of ~(z) agrees quite closely with the curve published in Ref. [2]. On Fig. 6 measured values of [NRMS(z)] 2 (taken from Fig. 2) versus calculated values of o~(z)/V(z) (taken from Fig. 5) are shown. The numbers labelling the points indicate the respective measuring positions. Figure 6 shows that if the measuring positions falling outside the flat-flux region (No. 11, No. 10 and No. 1) are excluded, and the experimental point No. 5 is considered to be a stray away point, the rest of the points follow quite closely a straight line as they should according to equation (2.16).
08 e4 N v t.O
cr Z
060!
0 50 07
Filled curve
-o-. 040
2
o>
030
~
0.20
m
t~ o _1
0 I0
m
E i
0.6
.o Z i 03 13" U) i
50
05
I
I
l
I00
t50
200
I
I
250
500
ring positions Z,
(I t) 0
,50
1(30
150 Z,
Fig. 2. Measured
200
250
300
cm
Fig. 3. Axial distribution of the local steam-void fraction.
Cm
values of the mean-square-noise
amplitude. (The experimental points are from the paper of W. Seifritz [231.
5 x
In order to use equation (2.16) the quantity o:(z)/V(z) was computed using the program VOID-1 [9]. VOID-1 calculates the quality of the coolant (x(z)) by heat balance considerations using the formalism of Zuber [10], which applies to the case of non-equilibrium two-phase flow. To calculate ~(z) and V(z) the prescriptions of Rouhani [11] are applied. For the calculation the following input parameters have been used (all parameters refer to a single rod
>
.g
>~
5
s
bundel) [12]:
Coolant flow area = 6.33 x 10-a m 2 Heat transfer area ----4.75 m S Average heat flux = 32.8 W/era 2 Inlet temperature = 280°C Inlet velocity -- 2.21 m/sec.
*~ rio
<
I
I
1
I
50
I00
150
200 Z,
I
I
250
500
cm
Fig. 4. Axial dependence of the average velocity of steam bubbles.
G. KosJ~LY, L. MAR6TI and L. MESK6
320
that the radial heat flux must have been approximately flat in rather big parts of the core, that is the actual average heat flux at the central position where the measurement was performed might differ not too strongly from the average heat flux of all rod bundles. Certainly the real job would be to put an end to the above inconsistency by repeating the calculations using the average heat flux at the actual radial position. Unfortunately this value is dependent on control rod position, burnup etc. at the date of the measurement, and these parameters are not known by us.
m
8
O
6
N 2
T
50
DO
150
200
Z,
I
250
~00
CONCLUSIONS
cm
Fig. 5. Axial dependence of the ratio of the steam-void fraction and steam bubble velocity. o-? -
o(i)
~
.(51
0.6 ~:
(-8 ) / I
.z.
~Fitted
curve
0.5 (I0) ]
0
I
r
~
~
l
r
~
~
5
~
[
I0
a(z] xlO-4 (crnls]-t V(z)
Fig. 6. Measured values of the mean-square-noise amplitude versus computed values of ~(z)/v(z). Using the points from No 9 to No. 2 (No. 5 excluded) a straight line was fitted determining by this the optimal values of the constants m and C. The solid curve on Fig. 2 was drawn using at the RHS of equation (2.16) these optimal values together with calculated values of ~(z)[ V(z). We feel that the results shown on Figs. 1 and 6 provide a rather convincing indication of the validity of the model used in the present work. On the other hand to remain fair we have to point out an inconsistent step in our derivation. Among the input parameters used in the thermohydraulic calculations, as average heat flux, the average heat flux of all bundles (axial and radial average) was used instead of the average heat flux at the particular radial position (axial average only) in question. In view of the well behaving results shown on Figs. 1 and 6 one may speculate hopefully
In the present work a simple heuristic model of the neutron noise in a large BWR core is given. Mathematically the model is represented by equations (1.1), (1.2), (1.3), (1.9) and (2.12). Throughout the paper the critical flux was assumed to be constant. This assumption is not an essential part of the model and should be removed in further work. As it was explained at the end of Section 2. an important task would be to derive the present model from more fundamental knowledge. By such a derivation the unknown quantities listed at the end to Section 2, could be determined. As regards the comparison of the results of the model to experimental results we feel, that up to now such comparisons were rather successful indicating the validity of the model. It is to be noted though, that all the experimental results used to corroborate the model refer to the Lingen BWR. Besides a more profound comparison of theoretical and experimental results at this reactor (see the remark at the end of Section 3.) the use of experimental data referring to other reactors would be needed. As a closing remark we would like to emphasize that if the validity of equation (2.16) could be fully established that could open the way to the determination of the average steam-void fraction by incore neutron noise measurements.
Acknowledgements--Authors are indebted to Messrs. Z. Szatmfiry and J. Valk6 for some clarifying discussions. Some useful information given by Mr. D. Ceelen from Institut fiir Kerntechnik, Hannover, is also greatly acknowledged. REFERENCES 1. D. Wach and G. Kos~ily, Investigation of the joint effect of local and global driving sources in incore neutron noise measurements. Atomkernergie 23,244 (1975).
A simple space dependent theory of the neutron noise in a boiling water reactor 2. W. Seifritz, Zur Analyse der Reaktorrauschens in Siedewasser-reaktoren. Habilitationschrift, Technische Universitat Hannover 1972; W. Seifritz, An analysis of the space dependent neutron flux density fluctuations at the lingen boiling water reactor (KWL) by methods of stochastic processes. Atomkernenergie 19, 271 (1972). 3. D. Stegemann, P. Gebureck, A. T. Mikuski and W. Seifritz, Operating characteristics of a boiling water reactor deduced from incore measurements. Paper presented at the Symposium on Power Plant Dynamics, Control and Testing, Knoxville, Tenn. Oct. 8-10 (1973). 4. W. Seifritz and F. Cioli: On-load monitoring of local steam velocity in BWR cores by neutron noise analysis Trans. Am. Nucl. Soc. 17, 451 (1973). 5. D. Wach, Ermittlung lokaler Dampfblasengeschwindigkeit aus Rauschsignalen yon Incore-Ionisationskammern Atomwirtschaft (1973). 6. T. Nomura, Noise analysis of a boiling water reactor. Proc. o f the Japan-United States Seminar on Reactor Noise Analysis, Tokio and Kyoto (1968). 7. A. I. Mogilner, Noise induced by heterogeneities of the coolant (in Russian) Atomnaya Energiya 30, 510 (1971). 8. O. Sandervag, Thermal non-equilibrium and bubble size distributions in an upward steam water flow. Kjeller Report KR-144 (1971). 9. L. Szabados and Zs. T6chy, VOID-l, Computer program for the determination of steam content in the channels of a nuclear reactor (in Hungarian). Central Research Institute for Physics, Budapest, Rep. No. KFKI-73-28. 10. N. Zuber et al. Steady State and Transient Void Fraction in Twophase Flow Systems. Volume I. EURAEC-GEAP 5417 (1967). 11. Z. Rouhani, Modified correlations for void and two-phase pressure drop. Report AE-RTV-841, AB Atomenergi, Sweden (1969). 12. Kernkraftwerk Lingen. Atomwirtschaft-Atomtechnik 13, 138 (1968). APPENDIX Derivation of equation 2.4 We define S(t~ z~ v) as the source of the bubbles, that is it gives the number of bubbles "emitted" in unit time with the volume v and at the anxial position z. The source is composed of an equilibrium part and a fluctuating part.
So(z, v) = iS(t, z, v)).
(A.1)
Obviously (t, z, v) = I~ s(t -- ~(z o, z), Zo, v) dz o do
(A.2)
where ~-(z0,z) is the mean transit time of the bubbles between z 0 and z. The position z = 0 is at the bottom of the core. 18
From equation (A.2) the relation J CPSD~j;~.~.(~o)
= f dzl f S dz, exp (io~[-r(z', z) -- "r(z", z)]) • CPSD~,.,,;,-.,-(to)
(A.3)
can be obtained by usual techniques. Considering now the stochastic variable s(t, z, v) dt dz dv to be a Poisson-process and assuming furthermore that (s(t~ z', v') s(t, z", v")) = O, if z'@- z" and/or v" @ v" (A.4) one obtains that CPSD~,.,,;~..~.(co) = So(z', v') 6(z"
-
-
z") 6(v" -- v") (A.5)
Substituting equation (A.5) into equation (A.3) the result CPSD~.~,;~.~-(og) = d(v' -- v")
S0(z', v') dz' =
= 6(v" -- v") Jo(z, v')
(A.6)
is obtained. Note 1 In Section 1 of the paper, to derive equation (1.9) the concept of individual bubbles was used. The use of this concept can be avoided by introducing the concept of the fluctuation of the current of the volume of steam by the definition: j(t, z) = V(z) 6V,(t, z). On the other hand the relation j(t, zz) = j ( t -- ~1~, zO + ( . . . ) is a simple consequence of the equation of continuity. From the above two equations, (1.9) follows simply• The situation is rather different in Section 2 and in the Appendix. Neither equation (2.4) nor (A.5) could be derived without assuming that the stochastic variables in question form a Poisson-impulse process. This assumption obviously cannot be even formulated in "continuum physics", that is the use of the bubble concept seems to be necessary. Note 2 A refinement of the derivation of equation (2.12) and (2.16) starts from equation (2.7). If instead of equations (2.9-11) one simply writes that iv~L V(z) Bo(z)
S(t, z, v) = So(z, v) + s(t, z, v)
321
ivY), ~(z) iv), V(z)
then the rather unrealistic assumption given in equation (2.9) can be avoided. On the other hand according to Ref. [8] not only the quantity iv), but the distribution of bubble volume depends but weakly on the axial position, that is equation (2.12) can be replaced by the relation iv~) ~(z) APSD~'(w) = IHz(~o)[~iv) V(z) + APSD'o@o). From the above equation, equation (2.16) follows readily.
A SIMPLE SPACE D E P E N D E N T THEORY OF THE N E U T R O N NOISE IN A BOILING WATER REACTOR G. KOS.~,LY, L. MAR6TI and L. MESK6 Hungarian Academy of Sciences, Central Research Institute for Physics 1525, Budapest 114. P.O.B. 49, Hungary On the other hand the recent experimental In a recent paper [1], D. Wach and one of the results of Wach [1], [5] indicate clearly the influence present authors [G. K.] have developed a space of noise-sources affecting the core as an entity. The dependent theoretical model for the description of influence of such global driving sources is demonstrated also by the successful applications of the the neutron noise in a large BWR core. In contradiction to the models used in zero-power point reactor model to the description of the noise theory the model is rather heuristic, that is it neutron noise induced by bubbling [6, 7]. In view of the above assumption we consider the relies on physical intuition and its justification can be fluctuation of the detector current at the axial given only by using the model successfully in the position z [i(t, z)] as composed of two parts: a interpretation of noise measurements. In Chapter 1 of the present paper we put forward local one (it(t, z)) responding to local disturbances an improved version of the model and apply it to and a global one [igz(t, z)] representing the respond the interpretation of correlation measurements per- to perturbations affecting the reactor as a whole. formed by incore neutron detectors. The derivations We write therefore that of this chapter follow quite closely the ones given in i(t, z) = iz(t, z) + i~(t, z) (1.1) Chapter 3 of Reg. [1]. The alterations made in the It is one of the basic assumptions of the present derivation will be pointed out at the end of Chapter 1. work that the driving source of the local part of the In Chapter 2 the auto-spectral-density and the noise-field is the fluctuation of the steam-void mean-squared-noise-amplitude of the fluctuations of content at the axial position in question, that is the current of an incore neutron detector are calculated. At the end of this chapter some virtues and h(t, z) = ~oH~(t -- t') ~Vs~(t' , z) dt'. (1.2) weak points of the model are discussed. In Chapter 3, some experimental results (norrealized root mean square of the detector current) In equation (1.2) dVa(t, z) represents the fluctuation of W. Seifritz [2] are analysed using the results of the of the steam-void content at the axial position z. The transfer function H,(t) relates this noise source previous chapter. to detector current. For the description of the global part of the 1. T H E T H E O R E T I C A L MODEL AND ITS noise-field we adopt the point-reactor model, that is APPLICATION TO THE INTERPRETATION OF CORRELATION MEASUREMENTS the driving source of the global part is identified In the considerations a simple, axial dependent with the fluctuations of the reactivity. According to the point-reactor model the space model of the core is used. As it was done in Reg. [1] we start with the physical assumption that the dependence of the neutron noise is dictated by the fluctuations of the neutron density at a given space critical flux, that is the noise is space-independent point inside the core can be composed of two parts: in regions where the flux can be considered as flat. a local part driven by local disturbances, and a For the sake of simplicity such regions have been global one representing a more or less coupled considered in Reg. [1] and we restrict the treatment to such regions in the present paper as well. behaviour of the neutron field. We indicate the space-independence of the global That a considerable part of the neutron noise in a large BWR is driven by local disturbances was part of the noise field by the notation INTRODUCTION
demonstrated by several workers [2-5] who were able to measure steam-void velocity by correlating the signals of incore neutron detectors.
igz(t, z) =-- ig~(t).
(1.3)
To investigate the behaviour of the stochastic
315
G. KoS~,LY, L. MAR6TI and L. MESK6
316
variable 6 Vs,(t, z) we introduce the functions B(t, z, v) = Bo(z, v) + b(t, z, v)
(1.4a)
J(t, z, v) = Jo(z, v) + j ( t , z, v)
(1.4b)
B(t, z, v) and F(t, z, v) represent the number and the current of the steam-bubbles with volume v, at the axial position z, at time t, respectively. Obviously J(t, z, v) = V(z)B(t, z, v)
Using now equations (1.1), (1.2), (1.3) and (1.9) the functions CPSD~vz~(co ) and APSD~li(co) can be readily evaluated and using equation (1.10) the transfer function can be given. One obtains the following result*: 5
gld~o) =
(1.5)
where V(z) is the average velocity of the steambubbles at the level z. (Fluctuations of this velocity are not considered in the present treatment.) In equations (1.4a) and (1.4b) the notations Bo(Z, v) = (B(t, z, v))
(l.6a)
Jo(z, v) = (J(t, z, v)
(l.6b)
have been used. If one considers now two axial positions z 1 and z~ (z2 > Zx) and neglects bubble collapsing between these two positions one finds that*
j(t, zz, v) = j ( t - TIe, Z1, V) + ('" ")
(1.7a)
Kle-iC°rl2+ 1
V2
I71 = V(21)
Ve = V ( z , ) .
K1 + 1
(1.11)
K 1 = K ( ( o , Zx)
Here the notation K ( w , z)
APSDzh(co) APSDig~(o~)
(1.12)
was used. The quantity K(~o, z) being equal to the ratio of the APSD of the stochastic variable i~(t, z) and the APSD of the stochastic variable ig;(t) measures the importance of the local part of the detector current as compared to its global part. If for a given frequency Kx >)1 (significant dominance of the local effect), then equation (1.11) goes over into
that is His(to ) = V1 e_i,o,t, Ve
V(z2) b(t, z e, v) = V(Za) b(t - -q~, zl, v) + (" • ") (1.7b) In the above equations Tie stands for the mean transit time of the bubbles between z x and z e. The unspecified second terms on the RHS of the equations account for the bubbles being born between the two space points considered. Using the obvious relation 6Vst(t, z) =
f;
vb(t, z, v) dv
(1.8)
together with equation (1.7b) one obtains that 6Vs,(t, zz) = V ( z 1 ) 6Vs~(t -- "q~, zl) + ('" "). V(ze)
(1.9)
that is the mean transit time of the bubbles between Zx and z e can be measured by determining the phase of the transfer function [2-5]. On the other hand ifK 1 ~ 1 (significant dominance of the global effect) one obtains that Hl~(o~) = 1
_
i
co
CPSD~I'Z"( ) = I H x d o 0 l APSDz~/(~o)
g ~°12(c°1.
(1.14)
which is the well known result of point-reactor theory. From equation (1.12) it is easy to derive the gain and the phase of the transfer function. One obtains that
Let us define now the transfer function Hie as the ratio of the CPSD of the two incore detectors placed at the positions z 1 and zz respectively, and the APSD of the signals of the incore detector placed at zx. Hie(co)
(1.13)
(/(1 + 1)~ _2V_A K1 V~(K1 + 1) ~[1 - c ° s ( ° ~ l e ) ]
(1.15)
(1.10)
* To derive equations (1.7a) and (1.7b) the expansion of the volume of the bubbles while propagating along the core is neglected as well. In view of the small differences between the values of temperature, and pressure at the top and at the bottom of the core in a BWR this assumption seems to be justified.
~0xe
c° / arctg /~_~2 1s'-in "(~''''12) V1 _ _ + cos (co~lz)| K, )
(1.16)
* Some details of the derivation not specified here can be found in Ref. [1].
A simple space dependent theory of the neutron noise in a boiling water reactor which equations go over into the corresponding results of Ref. [1], if V(zl) -- V(z2) is assumed. The differences between the above derivation and the one given in Ref. [1] are twofold. Firstly, in Ref. [1] the fluctuation of the number of bubbles rather than the fluctuation of the steam-void content was considered as the driving source of the local component of the noise field. As the steam-void content fluctuates because of the fluctuation of both the number and the volume of the bubbles we feel that the present choice is the one which is more conceivable physically. Secondly, in Ref. [1] it was assumed tacitly that the velocity of the steam bubbles is constant along the axis of the reactor, an assumption which is certainly not valid. As regards the consequence of the above alterations it can be seen that it is only the change of the velocity of the bubbles along the core which affects the expression of the transfer function. The considerations of Ref. [1 ] have been prompted by the fact that the gains and the phases of the transfer functions measured by Wach at the Lingen BWR, [1 ], [5] showed a very characteristic structure which could be understood neither on the basis of fluctuations driven by local disturbances, nor on the basis of point reactor theory, but was easily interpreted by the combination of the two (for details see Ref. [1 ]. It is important to emphasize that the appearance of the factor V1//12 :~ 1 in equations (1.15) and (1.16) influences neither the locations of the dips and the bumps of the gain, nor the general behaviour of the phase, that is the alterations introduced in the present work do not spoil the striking agreement between theory and experiment which was pointed out in Ref. [1 ].
In equation (2.1) APSDzS~(to) is the auto-spectraldensity of the fluctuation of the steam-void content at the axial position. In order to analyse the function APSDzSt(~o) we use equations (1.8) and (1.5) leading to 1
~o
6Vst(t, z) ~- ~ f vj(t, z, v) dr. V(z) Jo
1
£.oo
APSD~St(to) = VZ(z)J °
/-co
do'J0 dv"v'v"
× CPSD~.~,;z.¢(co). (2.3) Here CPSD~.v,z.~, is the cross-spectral-density of the stochastic variablesj(t, z, v', v-1) andj(t, z, v"). In the Appendix we show using rather plausible arguments that CPSD~.~:~,f(~o) = Jo(z, v') 6(v' -- v")
(2.4)
that is the above cross-spectral-density can be expressed readily by the mean value of the current of the bubbles. Using equations (2.3) and (2.4) one gets that
vZJo(Z, v) dv
APSD~ ~t(~o) = ~
V2(z) fO°°
= V (1z )
ff
v2Bo(z, v) dr.
(2.5)
Introducing now the second moment of the volume of the bubbles at the axial position z by the definition
2
f f v 2 B o ( z , v) dv
(v)~ = -
(2.6)
fo
°°Bo(z, v) dv
one may write equation (2.5) as A P S D zSt(~o) = (v2)z Bo(Z)
V(z)
1.3)*.
(2.7)
where
APSDzi(co) ~ [H~(~o)]2 APSDzS~(~) + APSD%(co).
(2.2)
From equation (2.2) one obtains by standard techniques that
2. C A L C U L A T I O N OF T H E APSD AND T H E N R M S OF T H E F L U C T U A T I O N OF T H E DETECTOR CURRENT
The model used in the previous chapter is represented mathematically by equations (1.1), (1.2), (1.3) and (1.9). To derive the APSD of the stochastic variable i(t, z) we use equations (1.1-
317
(2.1)
* To derive equation (2.1) it is assumed that the stochastic variables tSVs~(t,z) and i~(t) are independent. ig~(t) is obviously proportional to the fluctuation of the reactivity, that is its fluctuation is influenced by the entire steam content of the reactor, which in turn, is rather weakly correlated to the steam content at a particular point of the system.
Bo(z) =
~0°°Bo(z, v) dv
(2.8)
is the average number of bubbles at the height z. Assuming that (v2)z ~ (v)z2 (2.9) and expressing the equilibrium value of the volume of the steam per unit volume of the steam-liquid
G. KOSALY, L. MAR6TI and K. MESK6
318
mixture (local steam-void fraction) as
~(z) = (v>~Bo(z)
(2.10)
A P S D s~(co) = (v)z ~(z) V(z)
(2.11)
the result
is obtained. It is a very characteristic consequence of equation (2.4) that the auto-spectral-density of the steam-void fluctuation does not depend on the frequency (el. equations 2.5, 2.7, 2.11). On the other hand it does depend on the axial position via the axial dependence of the average volume of the bubbles ((v)z), steam velocity (V(z)) and average steam-void fraction (~(z)). According to thermohydraulic predictions (see Ref. [8]) (v)z depends but weakly on the axial position, that is the auto-spectral-density function of the fluctuations of the detector current can be given by the expression: APSD~(os) =
IH,(o01u
Z(z) + APSDtg~(oJ)
(2.12)
In order to characterize the amplitude of the neutron noise the so-called NRMS-value (Normalised Root Mean Square) is often introduced [2, 3]. By definition NRMS(z)
x/(i2(t, z))
Io( z )
(2.13)
where (is(t, z)) and Io(z) are the second and the moment of the detector output current respectively. As
first
(i2(t, z)) =
APSDz/(¢o) dco
(2.14)
one obtains using equation (2.12) that* ~(z) [NRMS(z)] 2 = m ~ + C V(z)
(2.16)
where C =
APSD%(o) do; TM
m
I0 ~J° IH~(co)l2 do~.
* One has to remember that the critical flux was assumed to be fiat throughout the present paper, that is t0(z) = to.
Equation (2.16) is a mathematical form of an earlier observation of Stegemann, Gebureck, Mikulski and Seifritz [3] who found by the inspection of their experimental results that the noise amplitude gives a measure of the steam-void fraction and boiling intensity in the respective core region. Equations (2.12) and (2.16) demonstrates both the virtues and the weak points of the present treatment. One of the virtues of the model is that in spite of being a space dependent model it leads to simple, clear results which are rather conceivable by physical intuition. The main shortcoming of the model is its phenomenological character. This is why in the results one always has unknown quantities as K(oJ, z) in equation (1.11), IH~(,o) I~and APSD~g~(o0 in equation (2.12) and the constans m and C in equation (2.16). This shortcoming of the model cannot be removed without the derivation of the model from transport or diffusion theory and necessarily restricts the usefulness of the model both in theoretical work and in the interpretation of experiments. Considering the virtues and the shortcoming one has to conclude that it is certainly the comparison to experimental results which has to decide whether the model used in this work can be considered as a valid "working description" of the noise in a BWR. We feel that the considerations of Ref. [1] indicate that the model is applicable to the interpretation of correlation measurements performed by incore neutron detectors. To seek further corroboration of the model, in the next chapter we use equation (2.16) to the interpretation of the experimental results of Seifritz [2]. 3. ANALYSIS OF THE MEASURED VALUES OF THE NRMS OF THE FLUCTUATION OF THE DETECTOR OUTPUT VIA EQUATION (2.16) Figure 1 is taken from Ref. [2] and shows the axial distribution of the mean detector current (critical flux) as measured by Seifritz at the Lingen Nuclear Power Station (KWL). The figure shows that in the almost central radial position where the measurements have been performed the flux is approximately constant between the measuring positions No. 9 and No. 2. Figure 2 is taken also from Ref. [2] and shows the measured values of the mean square noise amplitude as the function of the axial detector position.* * In Ref. [2] measured values of the quantity NRMS are given, while on our Fig. 2 the values of its square are indicated.
A simple space dependent theory of the neutron noise in a boiling water reactor o 03
x
P
~
OI
~0
0
Lower edge of c o r e
1(30
150
200
Axiol coordinofe,
250
z~cm
30o
Upper edge of core
Fig. I. Mean detector net current at 11 axial positions (from the paper of W. Seifritz [23].
319
On Figs. 3-5 the results of the calculation can be seen. It is to be noted that the calculated curve of ~(z) agrees quite closely with the curve published in Ref. [2]. On Fig. 6 measured values of [NRMS(z)] 2 (taken from Fig. 2) versus calculated values of o~(z)/V(z) (taken from Fig. 5) are shown. The numbers labelling the points indicate the respective measuring positions. Figure 6 shows that if the measuring positions falling outside the flat-flux region (No. 11, No. 10 and No. 1) are excluded, and the experimental point No. 5 is considered to be a stray away point, the rest of the points follow quite closely a straight line as they should according to equation (2.16).
08 e4 N v t.O
cr Z
060!
0 50 07
Filled curve
-o-. 040
2
o>
030
~
0.20
m
t~ o _1
0 I0
m
E i
0.6
.o Z i 03 13" U) i
50
05
I
I
l
I00
t50
200
I
I
250
500
ring positions Z,
(I t) 0
,50
1(30
150 Z,
Fig. 2. Measured
200
250
300
cm
Fig. 3. Axial distribution of the local steam-void fraction.
Cm
values of the mean-square-noise
amplitude. (The experimental points are from the paper of W. Seifritz [231.
5 x
In order to use equation (2.16) the quantity o:(z)/V(z) was computed using the program VOID-1 [9]. VOID-1 calculates the quality of the coolant (x(z)) by heat balance considerations using the formalism of Zuber [10], which applies to the case of non-equilibrium two-phase flow. To calculate ~(z) and V(z) the prescriptions of Rouhani [11] are applied. For the calculation the following input parameters have been used (all parameters refer to a single rod
>
.g
>~
5
s
bundel) [12]:
Coolant flow area = 6.33 x 10-a m 2 Heat transfer area ----4.75 m S Average heat flux = 32.8 W/era 2 Inlet temperature = 280°C Inlet velocity -- 2.21 m/sec.
*~ rio
<
I
I
1
I
50
I00
150
200 Z,
I
I
250
500
cm
Fig. 4. Axial dependence of the average velocity of steam bubbles.
G. KosJ~LY, L. MAR6TI and L. MESK6
320
that the radial heat flux must have been approximately flat in rather big parts of the core, that is the actual average heat flux at the central position where the measurement was performed might differ not too strongly from the average heat flux of all rod bundles. Certainly the real job would be to put an end to the above inconsistency by repeating the calculations using the average heat flux at the actual radial position. Unfortunately this value is dependent on control rod position, burnup etc. at the date of the measurement, and these parameters are not known by us.
m
8
O
6
N 2
T
50
DO
150
200
Z,
I
250
~00
CONCLUSIONS
cm
Fig. 5. Axial dependence of the ratio of the steam-void fraction and steam bubble velocity. o-? -
o(i)
~
.(51
0.6 ~:
(-8 ) / I
.z.
~Fitted
curve
0.5 (I0) ]
0
I
r
~
~
l
r
~
~
5
~
[
I0
a(z] xlO-4 (crnls]-t V(z)
Fig. 6. Measured values of the mean-square-noise amplitude versus computed values of ~(z)/v(z). Using the points from No 9 to No. 2 (No. 5 excluded) a straight line was fitted determining by this the optimal values of the constants m and C. The solid curve on Fig. 2 was drawn using at the RHS of equation (2.16) these optimal values together with calculated values of ~(z)[ V(z). We feel that the results shown on Figs. 1 and 6 provide a rather convincing indication of the validity of the model used in the present work. On the other hand to remain fair we have to point out an inconsistent step in our derivation. Among the input parameters used in the thermohydraulic calculations, as average heat flux, the average heat flux of all bundles (axial and radial average) was used instead of the average heat flux at the particular radial position (axial average only) in question. In view of the well behaving results shown on Figs. 1 and 6 one may speculate hopefully
In the present work a simple heuristic model of the neutron noise in a large BWR core is given. Mathematically the model is represented by equations (1.1), (1.2), (1.3), (1.9) and (2.12). Throughout the paper the critical flux was assumed to be constant. This assumption is not an essential part of the model and should be removed in further work. As it was explained at the end of Section 2. an important task would be to derive the present model from more fundamental knowledge. By such a derivation the unknown quantities listed at the end to Section 2, could be determined. As regards the comparison of the results of the model to experimental results we feel, that up to now such comparisons were rather successful indicating the validity of the model. It is to be noted though, that all the experimental results used to corroborate the model refer to the Lingen BWR. Besides a more profound comparison of theoretical and experimental results at this reactor (see the remark at the end of Section 3.) the use of experimental data referring to other reactors would be needed. As a closing remark we would like to emphasize that if the validity of equation (2.16) could be fully established that could open the way to the determination of the average steam-void fraction by incore neutron noise measurements.
Acknowledgements--Authors are indebted to Messrs. Z. Szatmfiry and J. Valk6 for some clarifying discussions. Some useful information given by Mr. D. Ceelen from Institut fiir Kerntechnik, Hannover, is also greatly acknowledged. REFERENCES 1. D. Wach and G. Kos~ily, Investigation of the joint effect of local and global driving sources in incore neutron noise measurements. Atomkernergie 23,244 (1975).
A simple space dependent theory of the neutron noise in a boiling water reactor 2. W. Seifritz, Zur Analyse der Reaktorrauschens in Siedewasser-reaktoren. Habilitationschrift, Technische Universitat Hannover 1972; W. Seifritz, An analysis of the space dependent neutron flux density fluctuations at the lingen boiling water reactor (KWL) by methods of stochastic processes. Atomkernenergie 19, 271 (1972). 3. D. Stegemann, P. Gebureck, A. T. Mikuski and W. Seifritz, Operating characteristics of a boiling water reactor deduced from incore measurements. Paper presented at the Symposium on Power Plant Dynamics, Control and Testing, Knoxville, Tenn. Oct. 8-10 (1973). 4. W. Seifritz and F. Cioli: On-load monitoring of local steam velocity in BWR cores by neutron noise analysis Trans. Am. Nucl. Soc. 17, 451 (1973). 5. D. Wach, Ermittlung lokaler Dampfblasengeschwindigkeit aus Rauschsignalen yon Incore-Ionisationskammern Atomwirtschaft (1973). 6. T. Nomura, Noise analysis of a boiling water reactor. Proc. o f the Japan-United States Seminar on Reactor Noise Analysis, Tokio and Kyoto (1968). 7. A. I. Mogilner, Noise induced by heterogeneities of the coolant (in Russian) Atomnaya Energiya 30, 510 (1971). 8. O. Sandervag, Thermal non-equilibrium and bubble size distributions in an upward steam water flow. Kjeller Report KR-144 (1971). 9. L. Szabados and Zs. T6chy, VOID-l, Computer program for the determination of steam content in the channels of a nuclear reactor (in Hungarian). Central Research Institute for Physics, Budapest, Rep. No. KFKI-73-28. 10. N. Zuber et al. Steady State and Transient Void Fraction in Twophase Flow Systems. Volume I. EURAEC-GEAP 5417 (1967). 11. Z. Rouhani, Modified correlations for void and two-phase pressure drop. Report AE-RTV-841, AB Atomenergi, Sweden (1969). 12. Kernkraftwerk Lingen. Atomwirtschaft-Atomtechnik 13, 138 (1968). APPENDIX Derivation of equation 2.4 We define S(t~ z~ v) as the source of the bubbles, that is it gives the number of bubbles "emitted" in unit time with the volume v and at the anxial position z. The source is composed of an equilibrium part and a fluctuating part.
So(z, v) = iS(t, z, v)).
(A.1)
Obviously (t, z, v) = I~ s(t -- ~(z o, z), Zo, v) dz o do
(A.2)
where ~-(z0,z) is the mean transit time of the bubbles between z 0 and z. The position z = 0 is at the bottom of the core. 18
From equation (A.2) the relation J CPSD~j;~.~.(~o)
= f dzl f S dz, exp (io~[-r(z', z) -- "r(z", z)]) • CPSD~,.,,;,-.,-(to)
(A.3)
can be obtained by usual techniques. Considering now the stochastic variable s(t, z, v) dt dz dv to be a Poisson-process and assuming furthermore that (s(t~ z', v') s(t, z", v")) = O, if z'@- z" and/or v" @ v" (A.4) one obtains that CPSD~,.,,;~..~.(co) = So(z', v') 6(z"
-
-
z") 6(v" -- v") (A.5)
Substituting equation (A.5) into equation (A.3) the result CPSD~.~,;~.~-(og) = d(v' -- v")
S0(z', v') dz' =
= 6(v" -- v") Jo(z, v')
(A.6)
is obtained. Note 1 In Section 1 of the paper, to derive equation (1.9) the concept of individual bubbles was used. The use of this concept can be avoided by introducing the concept of the fluctuation of the current of the volume of steam by the definition: j(t, z) = V(z) 6V,(t, z). On the other hand the relation j(t, zz) = j ( t -- ~1~, zO + ( . . . ) is a simple consequence of the equation of continuity. From the above two equations, (1.9) follows simply• The situation is rather different in Section 2 and in the Appendix. Neither equation (2.4) nor (A.5) could be derived without assuming that the stochastic variables in question form a Poisson-impulse process. This assumption obviously cannot be even formulated in "continuum physics", that is the use of the bubble concept seems to be necessary. Note 2 A refinement of the derivation of equation (2.12) and (2.16) starts from equation (2.7). If instead of equations (2.9-11) one simply writes that iv~L V(z) Bo(z)
S(t, z, v) = So(z, v) + s(t, z, v)
321
ivY), ~(z) iv), V(z)
then the rather unrealistic assumption given in equation (2.9) can be avoided. On the other hand according to Ref. [8] not only the quantity iv), but the distribution of bubble volume depends but weakly on the axial position, that is equation (2.12) can be replaced by the relation iv~) ~(z) APSD~'(w) = IHz(~o)[~iv) V(z) + APSD'o@o). From the above equation, equation (2.16) follows readily.