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Subcritical reactivity measurement by a source-jerk method
ReactorScienceand Technology(Journalof NuclearEnergyParts A/B) 1962, Vol. 16, pp. 455 to 463. PergamonPressLtd. Printedin
SUBCRITICAL
REACTIVITY SOURCE-JERK T. B.
MEASUREMENT METHOD*
Northern
Ireland
BY A
RYVES
National Physical Laboratory,
Teddington,
Middx.
M. C. SCOTT Dept. of Applied Physics, Harvard University, U.S.A. (Received 3 July 1962) Abstract-The theory of the source-jerk technique for the measurement of subcritical reactivity has been refined to take account of harmonics present in the neutron flux distribution. The method has been applied to the measurement of the subcritical reactivity in a large graphite-moderated uranium-fuelled reactor. The AK subcritical reactivity was found to be I-2.63 + 0.211 per cent 7 This value is about 10 per cent higher than estimates based on the subcritical approach to critical, and on the measured interaction of shut-off rods.
1. INTRODUCTION
source-jerk technique is a method on measuring the subcritical reactivity of a reactor. Accounts of the method have been given by JANKOWSKI et al. (1957) and SCHMID (1957). The present paper describes an extension of the theory to include the harmonics which are present in the flux distribution in a specified reactor geometry, and the application of the method to measure the subcritical reactivity of the reactor THE
BEPO. 2. PRINCIPLE
OF
THE
(ker - 1) k is a constant depending on the de;iyed and Am,, neutron data. (A,,,, is approximately 0.09 set in BEPO.) keff is the effective multiplication constant in the reactor. Equation (1) may be rearranged in the form where R,,,,
is the subcritical reactivity
A R meaS= - $= MO) mea3
P(~)l
(2)
METHOD
The steady neutron flux distribution in a subcritical reactor is suddenly altered at ‘zero’ time by the removal (or insertion) of a large neutron source. A neutron detector measures the change in flux until a steady state is again reached. The variation of the neutron density p with the time t is shown in Fig. l(a) where the vertical axis represents the integrated neutron density
‘P dt, the source having been s0 suddenly removed from the reactor at t = 0. The asymptotic slopes of the curve are equal to the steady neutron densities before and after moving the source, designated p(O) and p(co) respectively. The asymptote DB cuts the time axis at A and OA is the negative time intercept T. SCHMID (1957) has shown that :
(1) *The work described in this paper was done at A.E.R.E., Harwell, Berks. 455
as illustrated in Fig. I(b), C,,,,
being the intercept
OB in Figs. l(a) and l(b) i.e. mix - ~(~11dt. ~(0)s0 p(m) is the contribution to the steady neutron density from the moved neutron source alone; the presence of other fixed sources in the reactor has no influence on the value of R,,,, (apart, of course, from affecting the counting statistics). The limitation on equations (1) and (2) is that no allowance has been made for the departure of the spatial flux distribution in the subcritical state from the flux distribution attaining when the reactor is critical. The subcritical reactivity which is required is defined by RI = %,%S *f
(3)
where R, is the reactivity associated with the first harmonic of the neutron density distribution and f is a numerical factor which is calculated in the following section. This factor, f, depends on the geometry of the reactor and the relative positions of neutron source and neutron detector.
T B. RYVESand M.
456
c. SCOTT
(0)
FIG. I.-Variation
ib)
of integrated
neutron
2.1. The derivation off
It is assumed that in equation (2.1.2)
It is necessary to express R, in terms of R,,,,. This is done by assuming the simple model of a bare homogeneous reactor. Using age diffusion theory, the diffusion equations are
ax
[$nici+s]a(s, (2.1.1)
-=y(+ ae
ap
7
and
at
=
dCi = dt
-
LWp
-
p + TpPx@=o)
(2.1.2)
_A.C.
+
kh
(2.1.3)
%z
density with time in a source-jerk experiment.
PT
where 2 is the slowing down density of fast neutrons, p the thermal neutron density, 7 the thermal neutron lifetime, 0 the neutron age, p the resonance escape probability, k the infinite multiplication constant, t the time, L the thermal diffusion length and s the source term. The i-th group of delayed neutrons have a concentration of neutron-emitting precursors Ci, decay constant li and the fraction emitted per fission is pi. Pi is the fast non-leakage probability for the i-th group of delayed neutrons, and P, is the fast non-leakage probability for source neutrons. In the notation used the ‘prompt’ neutrons are associated with C,,, ,$,, A,,and P,, to produce homoThus 3 pi = 1 and A,,+ co. i=o Equation (2.1.1) contains the source term. These equations refer to the mns mode of the flux distribution, but in much of what follows these suffixes will be omitted to simplify the equations. [m, n and s are the integers which specify the eigenfunctions describing the mode.]
XPce=l))= 2 nic$pi + sps i
p. = e-B% 3
and that
(2.1.4) (2.1.5)
where B2 is the Buckling and 0, the age to thermal of neutrons of thej-th type. The operator V2 is replaced by -B2. In the ideal source-jerk experiment the thermal neutron density has a constant value p(O) before the neutron source is removed at t = 0. The neutron density then falls to zero as t + co because there is no background neutron source present in the reactor. Under these conditions equations (2.1.2) and (2.1.3) are integrated from t = 0 to t = co, and also using (2.1.4) and (2.1.5), it is readily shown after some rearrangement that R = -p(O) A/C.
(2.1.6)
This equation refers to the mns mode of the flux distribution, and R is the subcritical reactivity, A 03 is a constant, and C is the integral p dt. If k,, is s0 the effective multiplication constant, (2.1.7)
R = (k,,, - 1)/k,, and
kC &Pi k,, = 1 f L2B2
(2.1.8)
geneous equations.
T/k + z: PiPilJ, and
A=
(2.1.9) pk
*
The quantities k,, and A depend on the energies of the delayed neutrons, especially in the higher modes of the flux distribution.
Subcritical reactivity measurement
Solution of the steady-state age diffusion equations yields the value of p(O). TPSPS P(O) = 1 + LZB2 - k~@,P, i
(2.1.10)
Having derived the reactivity R in each mode of the flux distribution, it is now necessary to relate these quantities to the measured reactivity R,,,, in equations (1) and (2). Th e measured neutron density is the sum of the contributions from each flux mode of the reactor, so that (2.1.11)
by a source-jerk method
457
where =l,m=O &WI e?n=2,m>O m = 0, 1, 2, 3, . . . n= 1,2, 3, . . . integers S= 1,2,3,... i and (nbms) is a root of JJ+,,) = 0. It is easily shown that (2.2.2) Thus for a cylindrical reactor model the equation (2.1.15) becomes (2.2.3)
(2.1.12) The measured neutron density p$&, is equal to [p(O) - p(co)] of equation (2). In order to simplify the notation, p(O) will henceforth be written as p; the suffix 1 will refer to the first harmonic of the flux distribution, suffixes 2, 3, . . . . referring to the higher modes. Equation (2) may be rewritten as Pl 1 + P2IP2+ P3/P1 + * * * R meas= - A lneasc, i 1 + C& + C& + . . .1 (2.1.13) Combining
equations l
R, = R,,,,
(2.1.6) and (2.1.13) gives +
P2A2Rl PA&
.-$
+
P3A3R1 -+....
11211s
and the required values of S,,,, (2.2.1) and (2.2.2).
BiL,, are given by
2.3. One-Group approximation The expression for f becomes somewhat simpler if all the neutrons are assumed to have thermal energies. L2 may be replaced by M2, the migration area, and the Pi become unity. It can be shown that in this case the expression for I/foscillates infinitely on the reactor boundary. Consider a central neutron source (r,, = 0, z,, = j/2). The only value of m is zero since the angular dependence vanishes. In this case =$ P?W -
PIA,&
(2.1.14) Comparing equations result is found that
(2.1.14) and (3) the desired (2.3.1)
(2.1.15)
evaluated on the reactor boundary at r = a, z = l/2. The summation over n can be eliminated by making use of the result (found from Fourier Analysis) (2.3.2)
2.2. Application to cylindrical geometry f is a function of the reactor geometry. For a cylinder of extrapolated radius a and length I, using cylindrical co-ordinates, with the point neutron source at (r,c&,) a standard result (e.g. see MORSEand FESHBACH,1953) is E,
S(r,+,z) = const. x
mm
~0s 44
by identifying p with a function containing n$,, in equation (2.3.1). For large values of s, n/IOsincreases also, so that (2.3.1) becomes for large values of s
- $o)J, (%)J*(*) J~+k$ms)
(2.3.3)
sin [y)
sin (F)
(22 1)
. .
T. B. RYVESand M. C. SCOTT
458
which oscillates infinitely. The conclusion is that near the boundary of the reactor the expression for f becomes only slowly convergent, finally failing to converge on the boundary itself. The theory will also break down within several migration lengths of the source, where direct source neutrons will be detected.
Substituting (2.4.5) into (2.4.4) gives x)
To conclude this account of the theory of the source-jerk method, it is worthwhile to consider the simple case of a bare uniform sphere with a central neutron source, using the one-group approximation. f can here be expressed analytically, and gives a good physical picture of the expected behaviour off with detector position for various subcritical reactivities of the system. Moreover the result agrees with the expression found by CARTER(1960) by direct solution of the reactor kinetic equations and serves as a valuable check on the method. Let r be the radial co-ordinate of position and a the extrapolated radius of the sphere. In the n-th mode of the flux distribution
const* x
S, = const. x
and
k
sn _
1
(2.4.1)
&pB;
_
(2.4.6)
b6
2.4. Application to spherical geometry
Pn =
1
n2 sin (B,r) B,r ’
(2.4.2)
k-l Putting B, = %n, a = :r and /I” = M2B2 it is easy to
1.4 I.2 I I.0
0
f 0.0 05 0.4 0.2
0
V
I
1
O-6
0.8
I
0.2
0.4
I I.0
k/Q)
FIG.2.-Variation offwith (r/u) for a sphere withcentral neutron source for various subcriticalreactivitiesR,. where A, has been put equal to A,,,,. In Fig. 2 the r variation off with - is shown for various subi)a critical reactivities of the system. It is seen that the ‘nodal’ position, when f = 1, is dependent on the subcritical reactivity, and that f increases slowly to a
1
express 2 in terms of CI,p and n. The reactivity R, is Pl
k defined as kG , and keEn = Hence k effn 1 + M2B211’ substituting in equation (2.1.15)
.
Writing X, = 3
, equation
a
P
(2.4.3)
(2.4.3) becomes TI
(2.4.4)
By a Fourier Series expansion it can be shown that 0 < /‘3< 1.
(2.4.5)
maximum
value
of $$.(l
-p 7~cot p7r) as r
approaches a. The limiting value off
which occurs
when p = 0 lies on the parabola f = Gk i
2- ! . a1
3. EXPERIMENTAL APPARATUS Measurements were made in BEPO,a large graphitemoderated uranium-fuelled heterogeneous reactor which normally operates at 6 MW. The cylindrical core (length 20 ft, dia. 20 ft) lay horizontally in a graphite cube (side 26 ft) as reflector. A full description of Bmo has been given by FENNING(1954). There were many experimental channels and experimental rigs traversing the reactor core, and the twelve boron carbide shut-off rods were fully inserted in the reactor during the experiment. The background neutron source strength in BEPO was about IO7 n set-I, a factor of IO higher than the estimate of the neutron production from spontaneous fission in the 238U and 240Pupresent in the fuel. To give the requisite good counting statistics a large
Subcritical reactivity measurement by a source-jerk method
antimony-beryllium photoneutron source was designed (CROCKER and HENRY, 1956). A solid cylinder of antimony (length 12 in., dia. 1.25 in.) of radioactive strength 60 c could slide in a fixed hollow beryllium cylinder (length 12 in., o.d. 3.25 in., i.d. 1.5 in.). The maximum neutron output was about 4 x 1Osn see-I, and when the antimony was withdrawn 18 in. through a 6 in. lead collar, acting as a y-ray shield, the neutron output from the source fell to less than the background source in the reactor. The neutron source was operated mechanically, the antimony cylinder being connected by a 20 ft long rod to a piston situated outside the reactor driven by compressed air. When a valve was operated manually the antimony travelled a total distance of 18 in., the first 12 in. of travel taking about 0.13 sec. Details of the mechanical apparatus are given in Fig. 3. To give good lubrication the steel piston ‘connecting rod’ ran through a PTFE bush set in the steel cylinder head. The motion of the source was measured electrically using micro-switches operated by the movement of the connecting rod, to give the velocity of the source as a function of travel. The neutron source strength could be adjusted by moving the whole piston assembly away from the reactor face, thus altering the initial insertion of the antimony cylinder in the beryllium. The thermal neutron flux was measured with type 12EB40 BF, proportional counters encased in 1 in. thick lead shields to reduce the y-background from the reactor. Fission chambers could not be used because of their low neutron sensitivity, resulting in poor counting statistics. The pulses from each BF, counter channel, after amplification and discrimination, were fed into pairs of scalers used alternately. Theswitching of each pair of scalers was performed by a modified type 1009 scaler, counting the 50 counts/set mains frequency, which operated several d.c. coupled valve relay circuits every 40 sec. The counts produced in each 40-set interval could thus be recorded manually.
459
neously in different horizontal channels traversing the reactor, at right angles to the axis of the horizontal cylindrical core. In all, measurements were taken in six widely separated channels (E1/6,E1/9, El/IO, E1/12, E2/2 and E2/8), keeping the neutron source position fixed, about 4 ft below the geometrical centre of the reactor (Fig. 4). As each channel was scanned, by moving the neutron detectors on aluminiumpush-rods, the strength of the neutron source was adjusted to give an initial count rate near 1.5 x IO4 counts/set. Counter readings were taken for about 5 min before the source was fired, and for 25 min afterwards. The results were used in equation (1) to find R,,,, The actual data were processed with a computer programme written by ARNOLDand FREEMANTLE (1959) which was designed to reveal any serious drifts in counter sensitivities, and which calculated the statistical errors. In several cases experimental runs were spoiled by large bursts of spurious counts, or by faults in the mechanical registers of several scalers but these were easily detected in the analysis, both by the computer programme print-out and by taking first differences of the integrated counts in successive time intervals. The relay change-overs in the scaler switching unit produced a few random spurious counts but these were shown to be insignificant at the high count rates used; they merely contributed to the effective neutron background of the reactor. The time of travel of the piston operating the neutron source was measured for each run. There were several small corrections to be applied to the computed values of R,,,, which are described below. 5. EXPERIMENTAL CORRECTIONS AND ERRORS Four main corrections and errors were considered, the other corrections being small. 5.1. Neutron counter dead-time correction Two methods were used to measure the effective dead-times of the counting channels. (i) A235U fission chamber was placed with a BF,
4. EXPERIMENTAL PROCEDURE Three BF, neutron detectors were used simultamicroswitches
Longitudinal
\
Microswitches
Base - bolted
to pile
.
face
FIG. 3.-General
layout of apparatus.
pile
T. B. RYVES and M. C. SCOTT
460 Vert
calculated for each detector position by detailed statistical weighting, the calculations being based on a set of reactivity measurements made in the GLEEP reactor train, e.g. see FENNING (1954). The total AK reactivity absorption, between 0 and 0.08 per cent K
L
(with an error of 30 per cent) was subtracted from each result to give a constant subcritical state of the reactor. 5.5 Other errors
i Note:
FIG. 4.-Side
Not
to scale
view of BEPO showing channel positions.
counter in two positions in the reflector of the reactor. The count rates of each pair of counters were compared as the reactor diverged with a slow period. The fission chamber counter channels were assumed to have a 2 pusecdead-time, (which is not very important since their counting rates were much lower than those of the associated BF, counters) and the BFB counter channels were found to have dead-times of 2.7 i 0.6 and 3.1 i 0.6 ,usec. (ii) The measured reactivities were compared for three source-jerk experiments done with fixed detector positions and varying initial neutron source strength. The dead-time of the BF, channels was 3 f 1 psec. Since (ii) applied directly to the experimental conditions, with a high y-flux and rapid rate of change of neutron flux when the source was moved, the adopted dead-time was 3 f 1 psec. As a result the measured reactivity was increased by about 5% in most cases, with an error of 1.7 per cent. 5.2. Motion of the source The correction was to approximate the ramp change in source strength to an equivalent step change. Knowing the motion and time of travel of the source for each experiment, this was calculated for each case, and the measured reactivity increased by 2 to 3 per cent. 5.3. Statistical counting errors These were computed, as stated in Section 4, and were always less than &l per cent. 5.4. The reactivity absorption of the neutron detectors As the counters were moved, the change in their effective neutron capture cross sections varied the subcritical reactivity of the reactor. The effect was
The 135Xe decay in the reactors was negligible since the reactor had been shut down for two weeks. Reactivity changes due to changes in air coolant pressure and temperature were neglected < & 0.02 C AK The errors in timing and the radioper cent K 1 * active decay of the antimony source were negligible. The few spurious counts produced by the switching relays were shown to be unimportant. The line neutron source was approximated to a fixed point source; this assumption simplified the analysis and was thought to be quite accurate at distances greater than about three migration lengths (80 cm) from the source. During the motion of the source, the addition of reactivity to the reactor was about 0.01 per cent AK and was neglected in the analysis. As the source K’ strength was altered by changing the relative initial positions of the antimony and beryllium, the effective source centre moved over a range of several inches; this effect was thought to be small in such a large reactor. During the whole set of measurements the reactor was not taken critical, so that the neutron flux remained very small. This was done to ensure that the fission product high-energy y-activity remained more or less steady, and did not produce a rapidly decaying source of photoneutrons in the beryllium cylinder. Changes in BF, counter neutron detection efficiencies with position, due to alterations in the y-field around th e counters, did not affect the measurements. 5.6. Estimate of the experimental errors The combined errors of 5-l to 5.4 are 13 per cent, which is less than the uncertainty in the theory used to find the factor f. 6. EXPERIMENTAL RESULTS The experimental results are given in Table 1. The values of R,,,, have been corrected according to Section 5. The positions of the measuring channels
Subcritical reactivity measurement
are shown in Fig. 4. The computed values off and the values of RI calculated using equation (3) are also included in Table 1. To findf, the delayed neutron data of KEEPIN et al. (1957) . , were used, corrected for fast fission in !WJ. Hence i /Ii = 0.00706 and $’ pi/,$ = i=1
i=l
0.0881 sec. Average vai&of R, for each channel are given in Table 2. These averages werefoundgraphically (e.g. see Figs. 5 and 6) and the assigned errors are based on the spread of maximum and mimimum values of RI. The average value of RI was -2*63& 0.21 per cent F,
found by giving each of the channels
an equal weight, 7. DISCUSSION
OF THE
RESULTS
The experimental reactivities R,,,, decreased from a TABLE I.-VALUES
- hneas (%I
343
1.94 1.95 2.02 3.98 2.14 2.16 2.18 2.18 2.28 2.28 2.50 2.75 2.72 2.98 2.97 3.05 2.78 2.45 2.41 2.07 2.04
282 228
190 152 114 76 38 -38 -114 -190
_______
d (cm) 343 267 228 190 114 38 -38 -114
1.56 1.56 1.52 1.51 1.52 1.53 1.52 1.51
f
AND R,
El/10
El/12
f 1.37 1.34 1.29
1.22 1.12 1.04 0.97 0.92 0.92 1.04 1.22
-R,
(%)
2.66 2.67 2.70 2.65 2.76 2.79 2.81 2.81 2.78 2.78 2.80 2.86 2.83 2.89 2.73 2.80 2.56 2.55 2.51 2.53 2.49
d (cm)
--R,,,, (%I
343 335
2.19 2.26 2.19 2.20 2.26 2.28 2.28 2.30 2.31 2.32 2.50 2.57 2.56 2.79 2.89 2.85 2.71 3.04 3.19
328 321 297 267
206 190 175 160 152 137 107 99
____I___
E 2/Z -Rlil,,, (%I
maximum value near the centre of a channel traversing the core, to a steady value in the reflector region. This behaviour is predicted by the expression for f derived for a bare sphere with central neutron source (Section 2.5 and Fig. 2). It was found that the results were asymmetrical (e.g. Channel E l/9 in Fig. 6) along channels through the core of the reactor, probably due to the known irregular loading of the core with heavy neutron absorbers. A calculation offwas made along one channel assuming that the pile centre was displaced horizontally and radially 30 cm relative to neutron source and detector positions (El/9 in Fig. 7). The average reactivity R, was almost unchanged, although the shape of the curve for R, altered considerably. A good constant value of R, would probably have been obtained for a radial displacement of 15 cm from the pile centre. The higher values of RI found in channels
OP R,,,,,
E 119
d (cm)
3.56 3.26
d
f 1.28 1.27 1.27 1.26 1.24 1.22
1.13 1.10 1.06 1.02 1.00 0.97 0.90 0.88
(cm) 2.80 2.87 2.78 2.80 2.84 2.87 2.82 2.80 2.82 2.83 2.83 2.90 2.82 2.96 2.95 2.91 2.71 2.95 2.87
350 328 282 252
183 145 130 114 99
-76 -160
.f 1.60 160 1.60 160 1.60 1.60 I.60 1.60
(%I 2.50 2.50 2.43 2.42 2.43 2.45 2.43 2.42
d (cm)
--RIM,, (%I
343
1.76 1.75 1.76 1.91 2.06 2.08 2.13 2*20
267 190 114 84 8
R mem (%I 1.86 2.24 2.24 2.21 2.25 2.66 (a) 2.54 2.71 2.81 2.98 3.13 3.29 3.39 3.45 3.46 3.22 2.58 b 2.31 I2.36 2.17 2.11
1.13 2.87
-236
-4
--ma, (%I 1.68 1.81 1.80 1.97 1.92 2.01 1.87 b 1.96 I1.93
El/6
_-RI
461
by a source-jerk method
--R, f 1.28 1.27 1.23 1.20
1.08 0.98 0.95 0.88
0.83 1.02
1.18
(%I 2.38 284 2.76 2.72 2.70 3.19 3.05 2.93 3.04 2.92 2.97 3.12 2.98 3.04 3.04 2.67 2.63 2.36 2.41 2.56 2.49
E Z/S
.f
(73
d (cm)
1.46
2.57 2.56 2.54 2.60 2.55 2.54 2.60 2.64
343 267 190 114 38 -38 -114 -190
144 1.36 1.24 1.22 1.20
_-RI
.f
(73
1.48 1.45 1.39 1.31 1.27 1.27 1.31 1.39
2.48 2.62 2.50 2.58 2.44 2.55 2.45 2.72 2.68
Note: d is distance of effective centre of neutron counter from channel centre in cm. R meaBis corrected for (1) dead-time (2) source motion (3) counter neutron absorption. Reactivities are in % AK/K. (a) result is ignored due to a large burst of spurious counts (b) 1st of pair is uncorrected for small spurious counts; 2nd is corrected.
T. B. RYWS and M. C. Scorr
462 TABLE&--AVERAGEEXPERIMENTALRESULTS FOREACHCHANNEL
Channel
RI (%) -268
+ -2.76 + -2.45 f -2.56 + -2.57 f
El/9 El/10 E l/12 1 E 212 E 218 E l/6 Average
-2.63
0.15 0.19 0.18 0.16 0.03 0.08 0.06 @03
i 0.21
El/l0 and E/12 might be due to the proximity of the source with some direct source neutrons being detected. Measurements taken near the shut-off rods in regions of large flux depression did not show any noticeable fmctuations from other measurements. The simple model of BEnoassumed gave an adequate overall description of the observed experimental results R,,,,, which vary from -1.5 per cent to -3.7
per cent %(inferred).
In principle a reflector
could have been included in the reactor model, but the
detailed analysis would be much more complicated. The assumption that the reactor was homogeneous, the extrapolated boundaries remaining fixed, is certainly not true in reality. In the calculations of f,k was altered empirically, until R, found from equation (3) produced the nearest constant value along each channel. Sincef depends on R, equation (3) is really a transcendental equation in R, and may be solved by iteration, but in practice the shape off is not very sensitive to the value of R, selected(which depends on the value of k selected). It was found that the values off obtained by the age-diffusion and one-group calculations to give the ‘best’ fit to the experimental data did not vary significantly, but the values of R, equivalent to the k’s used were -2.63 per cent and AK -2.40 per cent K respectively. Thus the value of RI used in the age-diffusion calculation off agreed exactly with the final average value of R, from the experimental measurements. The absolute error of the measurement is uncertain. The quoted error of f8 per cent is based on the spread of the results. The simple graphical method
1 I.6
f
1.4 f I.2 I.0
2.8; 26 - __-----_________________________---__-----------------~ d
2.4
c;'S--------
‘T.
"
c R me’2
1,6I.40
11 20
40
I 60
I 80
II 100
120
Distance FIG.
11 140 from
160
11 180 200
channel
centre.
' 220
11 240 260
‘I” 280
300
320 340
Cm
vs. position (E l/6).
5.-Reactivity
6 4 2 0 f 1.8 1.6 2O8642Oa6.4 I
0
-150
I
1
I
-100 -50 Distance FIG.
I
0 50 from channel
6.-Reactivity
I
1
100 centre,
150
I
200 cm
I
I
I
250
300
3
vs. position (E l/9).
360
Subcritical reactivity measurement
by a source-jerk method
463
I
i
(I.6
$$;le;
-200
I -150
I -100
I -50 Dlslonce
FIG. 7.-Reactivity
I 0
, 50 from
, 100 channel
I 150 centre,
vs. position-off-centre
used to average R, along each channel, followed by a simple average taken over all the channels is not very satisfactory. By combining detailed statistical weighting with a graphical method to smooth the results along each channel a more satisfactory average result might be obtained. It is important to make sufficient measurements to cover the whole reactor. Reactivity estimates based on a subcritical approach to critical and on the measured interaction of shut-off rods are about 10 per cent lower, but do not apply to the same condition of the reactor. A calculation based on the method of Nordheim and Scalettar (GLASSTONE and EDLUND, 1952) is about 20 per cent lower. However none of these estimates is very reliable. 8. CONCLUSION The source-jerk method of subcritical reactivity measurement appears to have certain advantages over other methods, especially on very large power reactors with extremely limited experimental time available. Itreveals asymmetries inthe effective fuelloading of the reactor, such as might be introduced by the presence of neutron absorbers or by the effects of temperature gradients on fuel burn-up. The method gives individual results reproducible within the experimental error of about +3 per cent, and could be used to measure long-term changes of reactivity, due to fuel burn-up
I 200
I 250
I 300
I 350
Cm case for E l/9.
for example. An assessment of the absolute accuracy of the method is needed over the range of subcritical reactivities likely to be met in practice. Acknowledgments-The authors wish to thank DR. V. S. CROCKER for his encouragement during the experiments.
Valuable assistance was received from DR. R. G. FREEMANTLE, MR. M. J. ARNOLD, DR. C. CARTERand MR. C. J. ENGLAND. MR. R. PANTER and the staff of BEPO helped to install the equipment. The authors wish to thank the Director of AERE for permission to publish this paper. REFERENCES ARNOLD M. J. and FREEMANTLER. G. (1959) Private communication. CARTERC. (1960) Private communication. CROCKERV. S. and HENRY K. J. (1956) The Design ofa Large Antimony Beryllium Neutron Source for Use with Exponential Assemblies. AERE RP/M72. FENNING F. W. (1954)Progress in Nuclear Energy-Reactors,
Pergamon Press, London. GLASSTONES. and EDLUND M. C. (1952) The Elements of Nuclear Reactor Theory, Section 11, pp. 34-35, Van Nostrand Inc. JANKOWSKIF. J., KLEIN D. and MILLERT. M. (1957) Nucl. Sci. Engng. 2,288. KEEPIN G. R., WIMETTT. F. and ZEIGLERR. K. (1957) Phys. Rev. 107, 1044. MORSE P. M. and FESHBACHH. (1953) Methods of Theoretical Physics, McGraw-Hill. SCHMID P. (1957). Absolute Reactivity Measurements from Transient Behaviour of a Subcritical Nuclear Reactor R.A.G.
Bericht No. 1.
SUBCRITICAL
REACTIVITY SOURCE-JERK T. B.
MEASUREMENT METHOD*
Northern
Ireland
BY A
RYVES
National Physical Laboratory,
Teddington,
Middx.
M. C. SCOTT Dept. of Applied Physics, Harvard University, U.S.A. (Received 3 July 1962) Abstract-The theory of the source-jerk technique for the measurement of subcritical reactivity has been refined to take account of harmonics present in the neutron flux distribution. The method has been applied to the measurement of the subcritical reactivity in a large graphite-moderated uranium-fuelled reactor. The AK subcritical reactivity was found to be I-2.63 + 0.211 per cent 7 This value is about 10 per cent higher than estimates based on the subcritical approach to critical, and on the measured interaction of shut-off rods.
1. INTRODUCTION
source-jerk technique is a method on measuring the subcritical reactivity of a reactor. Accounts of the method have been given by JANKOWSKI et al. (1957) and SCHMID (1957). The present paper describes an extension of the theory to include the harmonics which are present in the flux distribution in a specified reactor geometry, and the application of the method to measure the subcritical reactivity of the reactor THE
BEPO. 2. PRINCIPLE
OF
THE
(ker - 1) k is a constant depending on the de;iyed and Am,, neutron data. (A,,,, is approximately 0.09 set in BEPO.) keff is the effective multiplication constant in the reactor. Equation (1) may be rearranged in the form where R,,,,
is the subcritical reactivity
A R meaS= - $= MO) mea3
P(~)l
(2)
METHOD
The steady neutron flux distribution in a subcritical reactor is suddenly altered at ‘zero’ time by the removal (or insertion) of a large neutron source. A neutron detector measures the change in flux until a steady state is again reached. The variation of the neutron density p with the time t is shown in Fig. l(a) where the vertical axis represents the integrated neutron density
‘P dt, the source having been s0 suddenly removed from the reactor at t = 0. The asymptotic slopes of the curve are equal to the steady neutron densities before and after moving the source, designated p(O) and p(co) respectively. The asymptote DB cuts the time axis at A and OA is the negative time intercept T. SCHMID (1957) has shown that :
(1) *The work described in this paper was done at A.E.R.E., Harwell, Berks. 455
as illustrated in Fig. I(b), C,,,,
being the intercept
OB in Figs. l(a) and l(b) i.e. mix - ~(~11dt. ~(0)s0 p(m) is the contribution to the steady neutron density from the moved neutron source alone; the presence of other fixed sources in the reactor has no influence on the value of R,,,, (apart, of course, from affecting the counting statistics). The limitation on equations (1) and (2) is that no allowance has been made for the departure of the spatial flux distribution in the subcritical state from the flux distribution attaining when the reactor is critical. The subcritical reactivity which is required is defined by RI = %,%S *f
(3)
where R, is the reactivity associated with the first harmonic of the neutron density distribution and f is a numerical factor which is calculated in the following section. This factor, f, depends on the geometry of the reactor and the relative positions of neutron source and neutron detector.
T B. RYVESand M.
456
c. SCOTT
(0)
FIG. I.-Variation
ib)
of integrated
neutron
2.1. The derivation off
It is assumed that in equation (2.1.2)
It is necessary to express R, in terms of R,,,,. This is done by assuming the simple model of a bare homogeneous reactor. Using age diffusion theory, the diffusion equations are
ax
[$nici+s]a(s, (2.1.1)
-=y(+ ae
ap
7
and
at
=
dCi = dt
-
LWp
-
p + TpPx@=o)
(2.1.2)
_A.C.
+
kh
(2.1.3)
%z
density with time in a source-jerk experiment.
PT
where 2 is the slowing down density of fast neutrons, p the thermal neutron density, 7 the thermal neutron lifetime, 0 the neutron age, p the resonance escape probability, k the infinite multiplication constant, t the time, L the thermal diffusion length and s the source term. The i-th group of delayed neutrons have a concentration of neutron-emitting precursors Ci, decay constant li and the fraction emitted per fission is pi. Pi is the fast non-leakage probability for the i-th group of delayed neutrons, and P, is the fast non-leakage probability for source neutrons. In the notation used the ‘prompt’ neutrons are associated with C,,, ,$,, A,,and P,, to produce homoThus 3 pi = 1 and A,,+ co. i=o Equation (2.1.1) contains the source term. These equations refer to the mns mode of the flux distribution, but in much of what follows these suffixes will be omitted to simplify the equations. [m, n and s are the integers which specify the eigenfunctions describing the mode.]
XPce=l))= 2 nic$pi + sps i
p. = e-B% 3
and that
(2.1.4) (2.1.5)
where B2 is the Buckling and 0, the age to thermal of neutrons of thej-th type. The operator V2 is replaced by -B2. In the ideal source-jerk experiment the thermal neutron density has a constant value p(O) before the neutron source is removed at t = 0. The neutron density then falls to zero as t + co because there is no background neutron source present in the reactor. Under these conditions equations (2.1.2) and (2.1.3) are integrated from t = 0 to t = co, and also using (2.1.4) and (2.1.5), it is readily shown after some rearrangement that R = -p(O) A/C.
(2.1.6)
This equation refers to the mns mode of the flux distribution, and R is the subcritical reactivity, A 03 is a constant, and C is the integral p dt. If k,, is s0 the effective multiplication constant, (2.1.7)
R = (k,,, - 1)/k,, and
kC &Pi k,, = 1 f L2B2
(2.1.8)
geneous equations.
T/k + z: PiPilJ, and
A=
(2.1.9) pk
*
The quantities k,, and A depend on the energies of the delayed neutrons, especially in the higher modes of the flux distribution.
Subcritical reactivity measurement
Solution of the steady-state age diffusion equations yields the value of p(O). TPSPS P(O) = 1 + LZB2 - k~@,P, i
(2.1.10)
Having derived the reactivity R in each mode of the flux distribution, it is now necessary to relate these quantities to the measured reactivity R,,,, in equations (1) and (2). Th e measured neutron density is the sum of the contributions from each flux mode of the reactor, so that (2.1.11)
by a source-jerk method
457
where =l,m=O &WI e?n=2,m>O m = 0, 1, 2, 3, . . . n= 1,2, 3, . . . integers S= 1,2,3,... i and (nbms) is a root of JJ+,,) = 0. It is easily shown that (2.2.2) Thus for a cylindrical reactor model the equation (2.1.15) becomes (2.2.3)
(2.1.12) The measured neutron density p$&, is equal to [p(O) - p(co)] of equation (2). In order to simplify the notation, p(O) will henceforth be written as p; the suffix 1 will refer to the first harmonic of the flux distribution, suffixes 2, 3, . . . . referring to the higher modes. Equation (2) may be rewritten as Pl 1 + P2IP2+ P3/P1 + * * * R meas= - A lneasc, i 1 + C& + C& + . . .1 (2.1.13) Combining
equations l
R, = R,,,,
(2.1.6) and (2.1.13) gives +
P2A2Rl PA&
.-$
+
P3A3R1 -+....
11211s
and the required values of S,,,, (2.2.1) and (2.2.2).
BiL,, are given by
2.3. One-Group approximation The expression for f becomes somewhat simpler if all the neutrons are assumed to have thermal energies. L2 may be replaced by M2, the migration area, and the Pi become unity. It can be shown that in this case the expression for I/foscillates infinitely on the reactor boundary. Consider a central neutron source (r,, = 0, z,, = j/2). The only value of m is zero since the angular dependence vanishes. In this case =$ P?W -
PIA,&
(2.1.14) Comparing equations result is found that
(2.1.14) and (3) the desired (2.3.1)
(2.1.15)
evaluated on the reactor boundary at r = a, z = l/2. The summation over n can be eliminated by making use of the result (found from Fourier Analysis) (2.3.2)
2.2. Application to cylindrical geometry f is a function of the reactor geometry. For a cylinder of extrapolated radius a and length I, using cylindrical co-ordinates, with the point neutron source at (r,c&,) a standard result (e.g. see MORSEand FESHBACH,1953) is E,
S(r,+,z) = const. x
mm
~0s 44
by identifying p with a function containing n$,, in equation (2.3.1). For large values of s, n/IOsincreases also, so that (2.3.1) becomes for large values of s
- $o)J, (%)J*(*) J~+k$ms)
(2.3.3)
sin [y)
sin (F)
(22 1)
. .
T. B. RYVESand M. C. SCOTT
458
which oscillates infinitely. The conclusion is that near the boundary of the reactor the expression for f becomes only slowly convergent, finally failing to converge on the boundary itself. The theory will also break down within several migration lengths of the source, where direct source neutrons will be detected.
Substituting (2.4.5) into (2.4.4) gives x)
To conclude this account of the theory of the source-jerk method, it is worthwhile to consider the simple case of a bare uniform sphere with a central neutron source, using the one-group approximation. f can here be expressed analytically, and gives a good physical picture of the expected behaviour off with detector position for various subcritical reactivities of the system. Moreover the result agrees with the expression found by CARTER(1960) by direct solution of the reactor kinetic equations and serves as a valuable check on the method. Let r be the radial co-ordinate of position and a the extrapolated radius of the sphere. In the n-th mode of the flux distribution
const* x
S, = const. x
and
k
sn _
1
(2.4.1)
&pB;
_
(2.4.6)
b6
2.4. Application to spherical geometry
Pn =
1
n2 sin (B,r) B,r ’
(2.4.2)
k-l Putting B, = %n, a = :r and /I” = M2B2 it is easy to
1.4 I.2 I I.0
0
f 0.0 05 0.4 0.2
0
V
I
1
O-6
0.8
I
0.2
0.4
I I.0
k/Q)
FIG.2.-Variation offwith (r/u) for a sphere withcentral neutron source for various subcriticalreactivitiesR,. where A, has been put equal to A,,,,. In Fig. 2 the r variation off with - is shown for various subi)a critical reactivities of the system. It is seen that the ‘nodal’ position, when f = 1, is dependent on the subcritical reactivity, and that f increases slowly to a
1
express 2 in terms of CI,p and n. The reactivity R, is Pl
k defined as kG , and keEn = Hence k effn 1 + M2B211’ substituting in equation (2.1.15)
.
Writing X, = 3
, equation
a
P
(2.4.3)
(2.4.3) becomes TI
(2.4.4)
By a Fourier Series expansion it can be shown that 0 < /‘3< 1.
(2.4.5)
maximum
value
of $$.(l
-p 7~cot p7r) as r
approaches a. The limiting value off
which occurs
when p = 0 lies on the parabola f = Gk i
2- ! . a1
3. EXPERIMENTAL APPARATUS Measurements were made in BEPO,a large graphitemoderated uranium-fuelled heterogeneous reactor which normally operates at 6 MW. The cylindrical core (length 20 ft, dia. 20 ft) lay horizontally in a graphite cube (side 26 ft) as reflector. A full description of Bmo has been given by FENNING(1954). There were many experimental channels and experimental rigs traversing the reactor core, and the twelve boron carbide shut-off rods were fully inserted in the reactor during the experiment. The background neutron source strength in BEPO was about IO7 n set-I, a factor of IO higher than the estimate of the neutron production from spontaneous fission in the 238U and 240Pupresent in the fuel. To give the requisite good counting statistics a large
Subcritical reactivity measurement by a source-jerk method
antimony-beryllium photoneutron source was designed (CROCKER and HENRY, 1956). A solid cylinder of antimony (length 12 in., dia. 1.25 in.) of radioactive strength 60 c could slide in a fixed hollow beryllium cylinder (length 12 in., o.d. 3.25 in., i.d. 1.5 in.). The maximum neutron output was about 4 x 1Osn see-I, and when the antimony was withdrawn 18 in. through a 6 in. lead collar, acting as a y-ray shield, the neutron output from the source fell to less than the background source in the reactor. The neutron source was operated mechanically, the antimony cylinder being connected by a 20 ft long rod to a piston situated outside the reactor driven by compressed air. When a valve was operated manually the antimony travelled a total distance of 18 in., the first 12 in. of travel taking about 0.13 sec. Details of the mechanical apparatus are given in Fig. 3. To give good lubrication the steel piston ‘connecting rod’ ran through a PTFE bush set in the steel cylinder head. The motion of the source was measured electrically using micro-switches operated by the movement of the connecting rod, to give the velocity of the source as a function of travel. The neutron source strength could be adjusted by moving the whole piston assembly away from the reactor face, thus altering the initial insertion of the antimony cylinder in the beryllium. The thermal neutron flux was measured with type 12EB40 BF, proportional counters encased in 1 in. thick lead shields to reduce the y-background from the reactor. Fission chambers could not be used because of their low neutron sensitivity, resulting in poor counting statistics. The pulses from each BF, counter channel, after amplification and discrimination, were fed into pairs of scalers used alternately. Theswitching of each pair of scalers was performed by a modified type 1009 scaler, counting the 50 counts/set mains frequency, which operated several d.c. coupled valve relay circuits every 40 sec. The counts produced in each 40-set interval could thus be recorded manually.
459
neously in different horizontal channels traversing the reactor, at right angles to the axis of the horizontal cylindrical core. In all, measurements were taken in six widely separated channels (E1/6,E1/9, El/IO, E1/12, E2/2 and E2/8), keeping the neutron source position fixed, about 4 ft below the geometrical centre of the reactor (Fig. 4). As each channel was scanned, by moving the neutron detectors on aluminiumpush-rods, the strength of the neutron source was adjusted to give an initial count rate near 1.5 x IO4 counts/set. Counter readings were taken for about 5 min before the source was fired, and for 25 min afterwards. The results were used in equation (1) to find R,,,, The actual data were processed with a computer programme written by ARNOLDand FREEMANTLE (1959) which was designed to reveal any serious drifts in counter sensitivities, and which calculated the statistical errors. In several cases experimental runs were spoiled by large bursts of spurious counts, or by faults in the mechanical registers of several scalers but these were easily detected in the analysis, both by the computer programme print-out and by taking first differences of the integrated counts in successive time intervals. The relay change-overs in the scaler switching unit produced a few random spurious counts but these were shown to be insignificant at the high count rates used; they merely contributed to the effective neutron background of the reactor. The time of travel of the piston operating the neutron source was measured for each run. There were several small corrections to be applied to the computed values of R,,,, which are described below. 5. EXPERIMENTAL CORRECTIONS AND ERRORS Four main corrections and errors were considered, the other corrections being small. 5.1. Neutron counter dead-time correction Two methods were used to measure the effective dead-times of the counting channels. (i) A235U fission chamber was placed with a BF,
4. EXPERIMENTAL PROCEDURE Three BF, neutron detectors were used simultamicroswitches
Longitudinal
\
Microswitches
Base - bolted
to pile
.
face
FIG. 3.-General
layout of apparatus.
pile
T. B. RYVES and M. C. SCOTT
460 Vert
calculated for each detector position by detailed statistical weighting, the calculations being based on a set of reactivity measurements made in the GLEEP reactor train, e.g. see FENNING (1954). The total AK reactivity absorption, between 0 and 0.08 per cent K
L
(with an error of 30 per cent) was subtracted from each result to give a constant subcritical state of the reactor. 5.5 Other errors
i Note:
FIG. 4.-Side
Not
to scale
view of BEPO showing channel positions.
counter in two positions in the reflector of the reactor. The count rates of each pair of counters were compared as the reactor diverged with a slow period. The fission chamber counter channels were assumed to have a 2 pusecdead-time, (which is not very important since their counting rates were much lower than those of the associated BF, counters) and the BFB counter channels were found to have dead-times of 2.7 i 0.6 and 3.1 i 0.6 ,usec. (ii) The measured reactivities were compared for three source-jerk experiments done with fixed detector positions and varying initial neutron source strength. The dead-time of the BF, channels was 3 f 1 psec. Since (ii) applied directly to the experimental conditions, with a high y-flux and rapid rate of change of neutron flux when the source was moved, the adopted dead-time was 3 f 1 psec. As a result the measured reactivity was increased by about 5% in most cases, with an error of 1.7 per cent. 5.2. Motion of the source The correction was to approximate the ramp change in source strength to an equivalent step change. Knowing the motion and time of travel of the source for each experiment, this was calculated for each case, and the measured reactivity increased by 2 to 3 per cent. 5.3. Statistical counting errors These were computed, as stated in Section 4, and were always less than &l per cent. 5.4. The reactivity absorption of the neutron detectors As the counters were moved, the change in their effective neutron capture cross sections varied the subcritical reactivity of the reactor. The effect was
The 135Xe decay in the reactors was negligible since the reactor had been shut down for two weeks. Reactivity changes due to changes in air coolant pressure and temperature were neglected < & 0.02 C AK The errors in timing and the radioper cent K 1 * active decay of the antimony source were negligible. The few spurious counts produced by the switching relays were shown to be unimportant. The line neutron source was approximated to a fixed point source; this assumption simplified the analysis and was thought to be quite accurate at distances greater than about three migration lengths (80 cm) from the source. During the motion of the source, the addition of reactivity to the reactor was about 0.01 per cent AK and was neglected in the analysis. As the source K’ strength was altered by changing the relative initial positions of the antimony and beryllium, the effective source centre moved over a range of several inches; this effect was thought to be small in such a large reactor. During the whole set of measurements the reactor was not taken critical, so that the neutron flux remained very small. This was done to ensure that the fission product high-energy y-activity remained more or less steady, and did not produce a rapidly decaying source of photoneutrons in the beryllium cylinder. Changes in BF, counter neutron detection efficiencies with position, due to alterations in the y-field around th e counters, did not affect the measurements. 5.6. Estimate of the experimental errors The combined errors of 5-l to 5.4 are 13 per cent, which is less than the uncertainty in the theory used to find the factor f. 6. EXPERIMENTAL RESULTS The experimental results are given in Table 1. The values of R,,,, have been corrected according to Section 5. The positions of the measuring channels
Subcritical reactivity measurement
are shown in Fig. 4. The computed values off and the values of RI calculated using equation (3) are also included in Table 1. To findf, the delayed neutron data of KEEPIN et al. (1957) . , were used, corrected for fast fission in !WJ. Hence i /Ii = 0.00706 and $’ pi/,$ = i=1
i=l
0.0881 sec. Average vai&of R, for each channel are given in Table 2. These averages werefoundgraphically (e.g. see Figs. 5 and 6) and the assigned errors are based on the spread of maximum and mimimum values of RI. The average value of RI was -2*63& 0.21 per cent F,
found by giving each of the channels
an equal weight, 7. DISCUSSION
OF THE
RESULTS
The experimental reactivities R,,,, decreased from a TABLE I.-VALUES
- hneas (%I
343
1.94 1.95 2.02 3.98 2.14 2.16 2.18 2.18 2.28 2.28 2.50 2.75 2.72 2.98 2.97 3.05 2.78 2.45 2.41 2.07 2.04
282 228
190 152 114 76 38 -38 -114 -190
_______
d (cm) 343 267 228 190 114 38 -38 -114
1.56 1.56 1.52 1.51 1.52 1.53 1.52 1.51
f
AND R,
El/10
El/12
f 1.37 1.34 1.29
1.22 1.12 1.04 0.97 0.92 0.92 1.04 1.22
-R,
(%)
2.66 2.67 2.70 2.65 2.76 2.79 2.81 2.81 2.78 2.78 2.80 2.86 2.83 2.89 2.73 2.80 2.56 2.55 2.51 2.53 2.49
d (cm)
--R,,,, (%I
343 335
2.19 2.26 2.19 2.20 2.26 2.28 2.28 2.30 2.31 2.32 2.50 2.57 2.56 2.79 2.89 2.85 2.71 3.04 3.19
328 321 297 267
206 190 175 160 152 137 107 99
____I___
E 2/Z -Rlil,,, (%I
maximum value near the centre of a channel traversing the core, to a steady value in the reflector region. This behaviour is predicted by the expression for f derived for a bare sphere with central neutron source (Section 2.5 and Fig. 2). It was found that the results were asymmetrical (e.g. Channel E l/9 in Fig. 6) along channels through the core of the reactor, probably due to the known irregular loading of the core with heavy neutron absorbers. A calculation offwas made along one channel assuming that the pile centre was displaced horizontally and radially 30 cm relative to neutron source and detector positions (El/9 in Fig. 7). The average reactivity R, was almost unchanged, although the shape of the curve for R, altered considerably. A good constant value of R, would probably have been obtained for a radial displacement of 15 cm from the pile centre. The higher values of RI found in channels
OP R,,,,,
E 119
d (cm)
3.56 3.26
d
f 1.28 1.27 1.27 1.26 1.24 1.22
1.13 1.10 1.06 1.02 1.00 0.97 0.90 0.88
(cm) 2.80 2.87 2.78 2.80 2.84 2.87 2.82 2.80 2.82 2.83 2.83 2.90 2.82 2.96 2.95 2.91 2.71 2.95 2.87
350 328 282 252
183 145 130 114 99
-76 -160
.f 1.60 160 1.60 160 1.60 1.60 I.60 1.60
(%I 2.50 2.50 2.43 2.42 2.43 2.45 2.43 2.42
d (cm)
--RIM,, (%I
343
1.76 1.75 1.76 1.91 2.06 2.08 2.13 2*20
267 190 114 84 8
R mem (%I 1.86 2.24 2.24 2.21 2.25 2.66 (a) 2.54 2.71 2.81 2.98 3.13 3.29 3.39 3.45 3.46 3.22 2.58 b 2.31 I2.36 2.17 2.11
1.13 2.87
-236
-4
--ma, (%I 1.68 1.81 1.80 1.97 1.92 2.01 1.87 b 1.96 I1.93
El/6
_-RI
461
by a source-jerk method
--R, f 1.28 1.27 1.23 1.20
1.08 0.98 0.95 0.88
0.83 1.02
1.18
(%I 2.38 284 2.76 2.72 2.70 3.19 3.05 2.93 3.04 2.92 2.97 3.12 2.98 3.04 3.04 2.67 2.63 2.36 2.41 2.56 2.49
E Z/S
.f
(73
d (cm)
1.46
2.57 2.56 2.54 2.60 2.55 2.54 2.60 2.64
343 267 190 114 38 -38 -114 -190
144 1.36 1.24 1.22 1.20
_-RI
.f
(73
1.48 1.45 1.39 1.31 1.27 1.27 1.31 1.39
2.48 2.62 2.50 2.58 2.44 2.55 2.45 2.72 2.68
Note: d is distance of effective centre of neutron counter from channel centre in cm. R meaBis corrected for (1) dead-time (2) source motion (3) counter neutron absorption. Reactivities are in % AK/K. (a) result is ignored due to a large burst of spurious counts (b) 1st of pair is uncorrected for small spurious counts; 2nd is corrected.
T. B. RYWS and M. C. Scorr
462 TABLE&--AVERAGEEXPERIMENTALRESULTS FOREACHCHANNEL
Channel
RI (%) -268
+ -2.76 + -2.45 f -2.56 + -2.57 f
El/9 El/10 E l/12 1 E 212 E 218 E l/6 Average
-2.63
0.15 0.19 0.18 0.16 0.03 0.08 0.06 @03
i 0.21
El/l0 and E/12 might be due to the proximity of the source with some direct source neutrons being detected. Measurements taken near the shut-off rods in regions of large flux depression did not show any noticeable fmctuations from other measurements. The simple model of BEnoassumed gave an adequate overall description of the observed experimental results R,,,,, which vary from -1.5 per cent to -3.7
per cent %(inferred).
In principle a reflector
could have been included in the reactor model, but the
detailed analysis would be much more complicated. The assumption that the reactor was homogeneous, the extrapolated boundaries remaining fixed, is certainly not true in reality. In the calculations of f,k was altered empirically, until R, found from equation (3) produced the nearest constant value along each channel. Sincef depends on R, equation (3) is really a transcendental equation in R, and may be solved by iteration, but in practice the shape off is not very sensitive to the value of R, selected(which depends on the value of k selected). It was found that the values off obtained by the age-diffusion and one-group calculations to give the ‘best’ fit to the experimental data did not vary significantly, but the values of R, equivalent to the k’s used were -2.63 per cent and AK -2.40 per cent K respectively. Thus the value of RI used in the age-diffusion calculation off agreed exactly with the final average value of R, from the experimental measurements. The absolute error of the measurement is uncertain. The quoted error of f8 per cent is based on the spread of the results. The simple graphical method
1 I.6
f
1.4 f I.2 I.0
2.8; 26 - __-----_________________________---__-----------------~ d
2.4
c;'S--------
‘T.
"
c R me’2
1,6I.40
11 20
40
I 60
I 80
II 100
120
Distance FIG.
11 140 from
160
11 180 200
channel
centre.
' 220
11 240 260
‘I” 280
300
320 340
Cm
vs. position (E l/6).
5.-Reactivity
6 4 2 0 f 1.8 1.6 2O8642Oa6.4 I
0
-150
I
1
I
-100 -50 Distance FIG.
I
0 50 from channel
6.-Reactivity
I
1
100 centre,
150
I
200 cm
I
I
I
250
300
3
vs. position (E l/9).
360
Subcritical reactivity measurement
by a source-jerk method
463
I
i
(I.6
$$;le;
-200
I -150
I -100
I -50 Dlslonce
FIG. 7.-Reactivity
I 0
, 50 from
, 100 channel
I 150 centre,
vs. position-off-centre
used to average R, along each channel, followed by a simple average taken over all the channels is not very satisfactory. By combining detailed statistical weighting with a graphical method to smooth the results along each channel a more satisfactory average result might be obtained. It is important to make sufficient measurements to cover the whole reactor. Reactivity estimates based on a subcritical approach to critical and on the measured interaction of shut-off rods are about 10 per cent lower, but do not apply to the same condition of the reactor. A calculation based on the method of Nordheim and Scalettar (GLASSTONE and EDLUND, 1952) is about 20 per cent lower. However none of these estimates is very reliable. 8. CONCLUSION The source-jerk method of subcritical reactivity measurement appears to have certain advantages over other methods, especially on very large power reactors with extremely limited experimental time available. Itreveals asymmetries inthe effective fuelloading of the reactor, such as might be introduced by the presence of neutron absorbers or by the effects of temperature gradients on fuel burn-up. The method gives individual results reproducible within the experimental error of about +3 per cent, and could be used to measure long-term changes of reactivity, due to fuel burn-up
I 200
I 250
I 300
I 350
Cm case for E l/9.
for example. An assessment of the absolute accuracy of the method is needed over the range of subcritical reactivities likely to be met in practice. Acknowledgments-The authors wish to thank DR. V. S. CROCKER for his encouragement during the experiments.
Valuable assistance was received from DR. R. G. FREEMANTLE, MR. M. J. ARNOLD, DR. C. CARTERand MR. C. J. ENGLAND. MR. R. PANTER and the staff of BEPO helped to install the equipment. The authors wish to thank the Director of AERE for permission to publish this paper. REFERENCES ARNOLD M. J. and FREEMANTLER. G. (1959) Private communication. CARTERC. (1960) Private communication. CROCKERV. S. and HENRY K. J. (1956) The Design ofa Large Antimony Beryllium Neutron Source for Use with Exponential Assemblies. AERE RP/M72. FENNING F. W. (1954)Progress in Nuclear Energy-Reactors,
Pergamon Press, London. GLASSTONES. and EDLUND M. C. (1952) The Elements of Nuclear Reactor Theory, Section 11, pp. 34-35, Van Nostrand Inc. JANKOWSKIF. J., KLEIN D. and MILLERT. M. (1957) Nucl. Sci. Engng. 2,288. KEEPIN G. R., WIMETTT. F. and ZEIGLERR. K. (1957) Phys. Rev. 107, 1044. MORSE P. M. and FESHBACHH. (1953) Methods of Theoretical Physics, McGraw-Hill. SCHMID P. (1957). Absolute Reactivity Measurements from Transient Behaviour of a Subcritical Nuclear Reactor R.A.G.
Bericht No. 1.