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The source-multiplication method for absolute reactivity determination
Journal
ofNuclear Energy, Vol.27,
pp. 129 to 138.
Persamon Press 1973.Printed
in
Northern Ireland
THE SOURCE-MULTIPLICATION METHOD FOR ABSOLUTE REACTIVITY DETERMINATION EHUD GREENSPAN Atomic EnergyCommission,NuclearResearchCentre-Negev, P.O.B. 9001, Beer-Sheva, Israel (Received 14 March 1972, and in revisedform 7 August 1972) Abstract-A new method for reactivity determination is presented. This method involves two measurements of the reactor response to a neutron source: (a) when subcritical, the sourcemultiplication, M, is measured and (b) when critical, the linear power rise, R, is measured. The reactivity is then determined from the relation #o//j= J _l M m+;y]’ [ or from a good approximation to it p/p w TJR/Ml. This method is compared with the asymptoticperiod method; expected accuracy is discussed, and possible measurement procedures are described. The feasibility of combining both methods for the determination of $/A is discussed. 1. INTRODUCTION THE SOURCE-MULTIPLICATION (SM)
method is a well known technique for the determination of relative changes in reactivity due to changes in the subcritical reactor. In criticality approach experiments (KEEPIN, 1968; PAXTON and KEEPIN, 1964) the SM method provides a convenient guide for the safe approach to criticality during processes such as fuel loading, moderator level rise, or control rod withdrawal. Various versions of the source multiplication method were recently employed for the measurement of absolute subcritical reactivities. These include, the standard source multiplication and the constant rod-drop techniques (LEHTO, DAUGHTRY, POND and CARPENTER, 1971), the modified neutron source multiplication method (BUHL, ACKERMANN, KRYTER and ROBINSON, 1972), and the asymmetric source method (WALTER and HENRY, 1968). In all of these methods, however, a calibration against results from either another experimental method or from calculations is required. In the present work it is shown that the SA4 method can be extended to measure absolute reactivities without the need for intercalibration against other measurements or calculations. This is done by supplementing the regular SM measurement with the measurement of the rate of the linear power (or detector response) rise in the critical reactor subjected to the external neutron source. The resulting method is designated the Source Multiplication and Rate of linear power rise (SMR) method. The theoretical formulation of the SMR method, experimental procedures and the accuracy expected from this method are discussed. The SMR method is then compared with the Asymptotic Period (AP) method (KEEPIN, 1968), with which it shares several features. The feasibility of combining the SMR and the AP methods to form a new source-period method for the experimental determination of j/A is finally investigated. 2. THEORETICAL
FORMULATION
Consider a subcritical reactor subjected to an external neutron source emitting S neutrons per second of the normalized distribution sO(r, u): dus,(r, u) = 1. 1
129
(1)
EHUD GREENSPAN
130
After the disappearance of transient phenomena, the neutron density distribution in this reactor, n(r, v), will approach a constant level given by (GREENSPAN and CADY, 1970):
s
(so,4.‘)
n(r, u) = i A n,(r, 4, j=o (-a,) (nj, +j+>
(2)
where +(r, v) is thejth mode of the solution of the eigenvalue equation for the neutron density: Hni(r, II) = a,n,(r, v)
j=o,1,2,...
(3)
H is the Boltzman operator, +,+ is the jth mode eigenfunction of the operator Hf, which is adjoint to the Boltzman operator: H+$,+(r, V) = c&+(r,
u)
j = 0, 1,2, . . .
(4)
aj is the eigenvalue of equations (3) and (4) and satisfies the relation 0 > a0 > aI > a2. . . and the inner product, (f, g) stands for an integration over the phase space:
(f, s> =
dr duf(r, Mr, 0). ss The neutron density eigenfunctions are normalized so that : dr dvn,(r, u) = 1. ss
(5)
(6)
As the reactor approaches criticality, the fundamental eigenvalue, or,, approaches zero and the first term in the sum of equation (2) becomes the dominant one. Hence :
s
(so,$0”)
n(r, u) = QO-‘~ ( -ao) (no, I#~+)no(r’ ‘)*
(7)
In equation (7) +,,+is the importance function and (s,,, &,+)/(n,, 4,‘) is the effectiveness of the source neutron relative to the effectiveness of the average neutron in the assembly. The reading of the detector located at (r,J in the subcritical reactor, Ds(rd), will be R(l;l) = r(rdWs = M,
(8)
where N
~
o
-- s (so9$0”) C-0)
(no,
do+>
is the amplitude of the neutron density in the subcritical reactor due to the external source S, and l;l(r& is the detector efficiency, defined as the response of the detector located at rd to n,,(rd, u). Equations (7) and (8) are accurate as long as the reactor is close enough to criticality,thatis,whena,,-Isa,-l,j= 1,2,. . . . As the reactor becomes more subcritical, the relative contribution of the higher modes increase. However, by locating the detector close to a node of a higher mode distribution, its contribution to the detector reading can still be kept low, even at a large subcritical state. For small perturbations a,, is related to the negative reactivity of the subcritical
The source-multiplication method for absolute reactivity determination
131
reactor, p, by the relation (GREENSPAN and CADY, 1970):
(9)
a0 = (-P>/A
where A is the prompt neutron generation time. Hence, from equations (7), (8) and (9)
(10) Equation (10) forms the basis for the “l/M” technique commonly used during criticality approach (KEEPIN, 1968; PAXTONand KEEPIN, 1964). Knowledge of detector efficiency, source strength and effectiveness are not needed for these measurements, that are relative ones. One can also eliminate the detector and source dependence by calibrating the source multiplication measurements against reactivity values measured by other techniques (LEHTO,DAUGHTRY,PONDand CARPENTER,1971; BUHL, ACKERMANN,KRYTERand ROBINSON,1972). In the SMR method presented here the dependence of reactivity on the detector efficiency and effective source strength is eliminated by supplementing the SM measurement with the measurement of the response of the critical reactor (p = 0) to the source. Thus consider a critical reactor with an initial neutron density distribution n(r, v; 0) = Nn,(r, u) into which the external neutron source Ss,(r, v) is inserted at time t = 0. After initial transients die out, the neutron density will increase linearly with time according to the relation (GREENSPANand CADY, 1970), n(r, v; t) = n(r, v; 0) +
mo,~o+>/(~o, $0’) no(r, v>t +
AiAi 1
,+B,tg
[
j=o 5 a&-,
v)
(11)
where j, is the effective fraction of the ith delayed neutron group, j = 1 pi, li is the decay constant of the ith delayed neutron group precursors and the coekients defined as follows (GREENSPANand CADY, 1970):
a, are
(12) and a, =
Sbo,A”> a&,
$j+> ’
j 2 1.
(13)
The reading of the detector at rd, D(rd; t), will increase linearly with time (after initial transients die out) with the rate R: (14)
132
Enun GREENSPAN
By combining equations (10) and (14), the detector and source dependent parameters can be eliminated to give
Or in units of dollars, the reactivity is:
(16) Equation (16) provides a relation between the reactivity and two measurable parameters: R (Rate of linear power rise) and M (Multiplication of the subcritical reactor) with the reactor kinetic parameters serving as a scale factor. In many cases, (17) and equation (16) is simplified to be
When the relative effective yield of the delayed neutron group (/$/#j), is approximated by the relative physical yield x1 = pJj3: *
and the connecting chain between the reactivity and the measured parameters, R and M, are the fissile isotopes delayed neutron parameters. The physical meaning of J$ (a,/&) is the m ean life, Td, of the delayed neutrons. These mean-lives can be measured directly [Prompt-Burst experiments (KEEPIN, 1965)]. When photoneutrons are present (in particular in Be and D,O moderated reactors) their contribution to the mean life of the delayed neutrons should be included. For future reference, the effective mean life of the delayed neutrons will be defined as
3. MEASUREMENT
TECHNIQUES
The determination of (p/B) by the SMR method involves the measurement of two quantities : (a) The rate of the linear rise of the detector response, R, due to the insertion of the source into the critical reactor, and (b) The source multiplication, M, when the reactor is subcritical by the reactivity amount to be measured. * This CC‘ has no connection to the CC’employed S earlier.
The source-multiplicationmethod for absolute reactivity determination
133
An example for a possible measurement sequence of R and M, aimed at the absolute determination of the reactivity of a small sample is: -The reactor is brought to a critical state with the desired control-rod configuration. -The sample, whose reactivity is to be measured, is then inserted into the core. The reading of a fixed detector (say in the reflector) is recorded after the flux level stabilizes. This reading, Dn, is the contribution of background and inherent sources in the reactor. -An external neutron source (such as a californium-252 spontaneous fission source or the reactor start-up source) is inserted into the reactor. The contribution of the source to the detector reading, D,,is recorded after the reactor flux level stabilizes. -The sample is withdrawn from the core while the external source is left. The timedependent reading of the detector, D(t), is recorded and the reading-rate increase, R, is deduced from it after it becomes constant with time. The separation of the effects of the external neutron source from the background neutron sources is not really required. Any superposition of such sources is permissible, as long as the source strength and distribution remain constant throughout the experiment. When the sample has a positive reactivity, the measurement sequence should be slightly modified : -The -The -The
reactor is brought to critical with the sample in it. external neutron source is inserted and R is determined. sample is withdrawn, and the subcritical multiplication, M, is measured.
The procedure can be simplified even more if only relative reactivity values are desired. The reactivity p, is related to a reference reactivity, p,,, by the relation:
(21)
Hence, no linear power rise measurements are required and the SMR method reduces to the Source Multiplication method. 4. ACCURACY OF THE METHOD The SMR method is essentially composed of two known experimental techniques: The Linear Power Rise (LPR)measurement and the Source Multiplication (S&f) measurement. Based on experience gained in applying the LPR and S&f measurements, the following observations can be made: (a) LPR experiments have been performed in recent years in fast and in thermal Zero Power Reactors. KARAM(1969) measured the slope, R, of the LPR in the fast critical assemblies ZPR-6 and ZPR-9. Employing a 1 lug C,-252 spontaneous fission neutron source (yielding about 3 x 10”n/set) he could determine the slope, R, to an accuracy of 1 per cent. In similar measurements at the Cornell University ZPR (H,O moderated, slightly enriched UO, fueled), employing a O-1,ug C,-252 source, HAGA (1967) also determined R to an accuracy of about 1 per cent.
134
EHUDGREENSPAN
(b) Accuracy limitations of the SM method, especially in a far-subcritical reactor, are due to: -the contribution of higher modes to the detector reading or, in other words, the errors caused by approximating equation (2) by equation (7). Higher mode contamination can be controlled, to some extent, by locating the neutron source so that it will not strongly excite the dominant higher modes and also by locating the detector at a node of the dominant higher modes. -The substitution of the effective detector efficiency and source in the critical reactor for the corresponding quantities in the subcritical reactor. -The deviation from the unperturbed fundamental mode distribution caused by sample perturbation. When the reactor is only a slightly subcritical, higher modes contamination and perturbation are not expected to impair the accuracy of the SM method. Results of relative SM measurements at the Cornell University ZPR, using negative reactivities of the order of 10 cents, show (GREENSPAN, 1966) very good agreement with the relation of equation (10). The accuracy in these measurements was determined by the reproducibility of the experimental set-up, including reactor criticality, sample location, and temperature drifts. Reproducibility problems are, however, common also to other methods, such as the AP method. In recent measurements in fast reactors, source multiplication methods are reported (LEHTO et al. 1971; BUHL et al. 1972) to give reasonable results down to several dollars subcriticality. In cases where the effects of higher modes contamination and perturbations are non-negligible, the SMR method can still be applied provided the measurements will be corrected to take account of the deviations from the conditions assumed for the derivation of relation (16). Correction factors can be calculated in a manner similar to correction factors calculated for other reactivity measurement methods. For example, in the asymmetric source method (WALTER and HENRY, 1968), in the modified neutrons source multiplication (BUHL et al. 1972) as well as in the kinetic techniques based on the measurement of prompt neutron multiplication or of prompt die-away time constants (BUHL et al. 1972). Another question of concern in performing the SM measurement is the time it takes for the neutron flux level to reach its asymptotic value. This transient time depends strongly on the initial conditions for the SMmeasurements and on the degree of subcriticality. The further the initial flux level is from the asymptotic one (either above or below) and the closer the reactor is to criticality, the longer lasts the transient period. Hence, by proper design of the experiment and the selection of initial flux level it is possible to reduce the transient time. The intensity of the neutron source should be high enough to give signals easily separable from the background. Neutron sources employed for the reactor start-up in low power reactors will usually give signal large enough for the measurement of R and M with good statistics. In reactors having large inherent neutron sources, these sources may be employed directly for the measurement. 5. COMPARISON
WITH THE ASYMPTOTIC
PERIOD
METHOD
The AP is perhaps the method mostly used for the determination of small reactivity values. In this method one measures the asymptotic period, T, resulting from the addition of (usually positive) reactivity to a just critical reactor. The reactivity is
The source-multiplication method for absolute reactivity determination
135
then related to the measured period through the Inhour equation:
(22) The SMR method is also valid for small reactivity determination, and can use standard reactor equipment for its application. Hence the SMR method shares several essential features with the AP method. The following are features the SMR and the AP methods have in common: -No special equipment is needed for the experiments. Standard neutron measurement channels, and for the SMR method also start-up neutron source, will usually suffice. -The critical state of the reactor has to be established and maintained. -The rate of change of the reactor power level is measured. Expected advantages of the AP method are: -Only one measured parameter (T) is required as compared to two parameters (R and M) needed for the SMR method. -It is easy to take care of the effect of background neutron sources, and of statistical fluctuations in the neutron density by carrying the measurements at high enough power levels. -Transient times will, in general, be shorter with the AP method. But when the initial amplitude of the neutron density is properly selected the SMR method may have shorter transient times. Possible advantages of the SMR method over the AP method include: -The rate of the LPR, R, does not depend on the reactivity value to be measured. It can be chosen by selecting the strength and location of the neutron source. The asymptotic period, on the other hand, strongly depends on the reactivity. -The SMR method enables to measure a wider range of reactivities. The lowest value of reactivities possible to measure with the AP and the SMR methods is practically limited by the large times it takes for the transients to decay. The upper limit of reactivity possible to measure with the AP method (in a single step) is set by safety considerations (too short a period) to about 50 cents. No safety considerations limit the range of applicability of the SMR method. Very large negative reactivities can be measured, though with limited accuracy, or else, corrected with calculations. -The error in the calculated reactivity caused by an error in the value of @/A) is, in the AP method, at least as large as in the SMR method. This can be seen from the relative contribution of the (/?/A) term to (p/p): (23) The equality is obtained with T > l/&. This implies that uncertainties in the value of the prompt neutron generation time, A, or the kinetic parameter @/A), will be reflected in a small error in the reactivity when the SMR method is employed. To demonstrate the magnitude of the expected effects, the relative error in the p/i? caused by a total neglect of the @/A) term in the SMR method [equation (I@]
136
EHUD
GREENSPAN
has been evaluated. Figure I shows this relative error for the three fissile nuclides as a function of @/A). The results were computed from the expression l/V + (B/A) z with the values of z
(a,/&)
(4Ul
from KEEPIN (;968).
Characteristical
values of @/A)
fall in the range Af l-100 set-l for natural uranium graphite or heavy-water moderated reactors and of 100-200 set-l for typical slightly enriched light-water
0.01
Fxo. 1 -Relative
error in (p/n
due to the neglectof the @/A)term in the SMR method.
moderated reactors. It is even larger for fast reactors. With plutonium fuel, the value of (j/A) will be about 6 of its value in the corresponding reactor with 235Uas the fuel. Referring to Fig. 1, it is seen that the error in p/j caused by the neglect of the ,!?/A term will be less than O-1 per cent for light-water thermal reactors, and of the order of 1 per cent for the natural uranium thermal reactors. Figure 2 gives the ratio of the error introduced in the AP method by the neglect of j/A, relative to the error in the SMR method. The results were calculated from equation (23) for two extreme values of (j/A) with 235U as fuel. It is seen that for periods of about 30 set, the error in the AP method is approximately twice as large as with the SMR method. The relative error increases as the period shortens and the value of (B/A) becomes larger. -Only one delayed neutron parameter, except for the value of /?/A, is required for the SMR method. This parameter is the effective delayed neutron mean life, Td. The Inhour equation in use in the AP method, requires the fractional yield and decay constants of all delayed neutron groups. The accuracy with which these group parameters are known is inferior to that of the mean-life. -There is a linear relationship between the reactivity and the measured parameters
The source-multiplication
method for absolute reactivity determination
i
lb0
Period,
FIG. 2.-Error
137
set
in (p/B) due to the neglect of the (p/A) term in the AP method relative to the SMR method. Fissile fuel is ‘W.
(R and ikf) in the Sh4R method. The SA4R equation (16) is therefore simpler to use than the Inhour equation. The SMR method and the AP method can, in principle, be combined for the determination of the kinetic parameter b/A. Eliminating the reactivity from equations (16) and (22) one obtains the following expression for @/A), the ratio of the effective delayed neutron yield to the prompt neutron generation time:
Hence, by measuring the asymptotic period corresponding to the reactivity employed for the subcritical multiplication measurement (M) in addition to the measurement of R and M by the SMR method, one could, in principle, calculate (j/A) from equation (24). The accuracy to which (j/A) can be determined by this method, to be called the Source-Period method, is however, very limited. This is due to the fact that the denominator of equation (24) consists of a difference between two terms of very similar value. Hence, an error in the value of R/M is amplified in the calculated value of (p/b). This amplification becomes larger with larger periods and larger values of @/A). Even for extreme conditions of periods as short as 10 set, and (/?/A) as small as 1 set-l, the error amplification is of the order of 20!. 6. SUMMARY
Based on theoretical formulation, numerical investigation, and analysis of experimental results we conclude that the SMR can be a useful method for determining
138
EHUD GREENSPAN
absolute reactivity values (in units of b). Heigher mode contamination and perturbation and make it valid for measuring small reactivities. Larger reactivities can also be measured, but calculated correction factors to the SMR measurements may be required for getting satisfactory accuracy. The SMR method is expected to have similar accuracy as the AP method in the reactivity range common to both methods. The SMR method shares two other basic features with the AP method: (1) Simplicity of application, requiring only standard reactor equipment and (2) Criticality has to be established during the measurement procedure. Hence, from the point of view of accuracy range and ease of applicability, the SMR method can be classified in a category with the AP method. Unlike the AP method, no safety considerations limit the range of reactivities to be measured at one single step with the SMR method. The equation connecting reactivity with the measured parameters in the SMR method is easier to apply than the Inhour equation of the AP method. Except for /?/A, only the average effective delayed neutron mean life, Td, is needed for the SMR equation. Moreover, the calculated reactivity in the SMR method is less sensitive to inaccuracies (or even omission) of the @/A term. Its complete neglect introduces an error of the order of 1 per cent for natural uranium reactors, and less than 0.1 per cent for light-water reactors. A good approximation for the SMR equation is thus: (PI/% = WW). The major sources of inaccuracy in the SMR methods are expected to be effects of transients, establishment and maintenance of criticality, and for large negative reactivities also higher modes contamination, and distortion of fundamental mode distributions. As is the case with other reactivity measurement methods, the effects of these sources of inaccuracy can, in general, be reduced by careful design of the experiments. They can also be reduced by applying calculated correction factors. An experimental investigation of all these effects is due before final conclusions about the accuracy of the SMR method can be reached. The SM and the AP methods can be formally combined to measure the kinetic parameter @/A). This Source-Period method is very inaccurate because the error in the measured ratio, RIM, is highly amplified in this method. REFERENCES BUHL A. R., ACKERMANN. J., KR~R R. C. and ROBINSONJ. C. (1972) Trans. ANS 15 (1) 423. BUHL A. R., ROBINSONJ. C. and ACKERMANN. J. (1972) Trans. ANS 15 (1) 493. GREENSPANE. (1966) Cornell University Report No. CURL-15 GREENSPANE. and CADY K. B. (1970) J. nucf. Energy, 24,529. HAGA H. (1967) Cornell University Report No. CURL-22. KARAM R. A. (1969) Nucl. Sci. Eng. 37 (2) 192. KEEPIN G. R. (1965) Physics of Nuclear Kinetics, Addison-Wesley. LEHTO W. K., DAUGHTRYJ. W., POND R. P. and CARPENTERS. G. (1971) Trans. ANS 14 (1) 42. PAXTONH. C. and KEEPIN G. R. (1964) The Technology of Nuclear Reactor Safety, Vol. I, Criticality, Chap. 5, Edited by THOMPSONT. J. and BECKERLY J. G., MIT Press. WALTER J. F. and HENRY A. F. (1968) Nucl. Sci. Erg. 32,332. WALTZ W. R. and WALTER B. F. (1971) Nucl. Tech. 10 (2) 160.
ofNuclear Energy, Vol.27,
pp. 129 to 138.
Persamon Press 1973.Printed
in
Northern Ireland
THE SOURCE-MULTIPLICATION METHOD FOR ABSOLUTE REACTIVITY DETERMINATION EHUD GREENSPAN Atomic EnergyCommission,NuclearResearchCentre-Negev, P.O.B. 9001, Beer-Sheva, Israel (Received 14 March 1972, and in revisedform 7 August 1972) Abstract-A new method for reactivity determination is presented. This method involves two measurements of the reactor response to a neutron source: (a) when subcritical, the sourcemultiplication, M, is measured and (b) when critical, the linear power rise, R, is measured. The reactivity is then determined from the relation #o//j= J _l M m+;y]’ [ or from a good approximation to it p/p w TJR/Ml. This method is compared with the asymptoticperiod method; expected accuracy is discussed, and possible measurement procedures are described. The feasibility of combining both methods for the determination of $/A is discussed. 1. INTRODUCTION THE SOURCE-MULTIPLICATION (SM)
method is a well known technique for the determination of relative changes in reactivity due to changes in the subcritical reactor. In criticality approach experiments (KEEPIN, 1968; PAXTON and KEEPIN, 1964) the SM method provides a convenient guide for the safe approach to criticality during processes such as fuel loading, moderator level rise, or control rod withdrawal. Various versions of the source multiplication method were recently employed for the measurement of absolute subcritical reactivities. These include, the standard source multiplication and the constant rod-drop techniques (LEHTO, DAUGHTRY, POND and CARPENTER, 1971), the modified neutron source multiplication method (BUHL, ACKERMANN, KRYTER and ROBINSON, 1972), and the asymmetric source method (WALTER and HENRY, 1968). In all of these methods, however, a calibration against results from either another experimental method or from calculations is required. In the present work it is shown that the SA4 method can be extended to measure absolute reactivities without the need for intercalibration against other measurements or calculations. This is done by supplementing the regular SM measurement with the measurement of the rate of the linear power (or detector response) rise in the critical reactor subjected to the external neutron source. The resulting method is designated the Source Multiplication and Rate of linear power rise (SMR) method. The theoretical formulation of the SMR method, experimental procedures and the accuracy expected from this method are discussed. The SMR method is then compared with the Asymptotic Period (AP) method (KEEPIN, 1968), with which it shares several features. The feasibility of combining the SMR and the AP methods to form a new source-period method for the experimental determination of j/A is finally investigated. 2. THEORETICAL
FORMULATION
Consider a subcritical reactor subjected to an external neutron source emitting S neutrons per second of the normalized distribution sO(r, u): dus,(r, u) = 1. 1
129
(1)
EHUD GREENSPAN
130
After the disappearance of transient phenomena, the neutron density distribution in this reactor, n(r, v), will approach a constant level given by (GREENSPAN and CADY, 1970):
s
(so,4.‘)
n(r, u) = i A n,(r, 4, j=o (-a,) (nj, +j+>
(2)
where +(r, v) is thejth mode of the solution of the eigenvalue equation for the neutron density: Hni(r, II) = a,n,(r, v)
j=o,1,2,...
(3)
H is the Boltzman operator, +,+ is the jth mode eigenfunction of the operator Hf, which is adjoint to the Boltzman operator: H+$,+(r, V) = c&+(r,
u)
j = 0, 1,2, . . .
(4)
aj is the eigenvalue of equations (3) and (4) and satisfies the relation 0 > a0 > aI > a2. . . and the inner product, (f, g) stands for an integration over the phase space:
(f, s> =
dr duf(r, Mr, 0). ss The neutron density eigenfunctions are normalized so that : dr dvn,(r, u) = 1. ss
(5)
(6)
As the reactor approaches criticality, the fundamental eigenvalue, or,, approaches zero and the first term in the sum of equation (2) becomes the dominant one. Hence :
s
(so,$0”)
n(r, u) = QO-‘~ ( -ao) (no, I#~+)no(r’ ‘)*
(7)
In equation (7) +,,+is the importance function and (s,,, &,+)/(n,, 4,‘) is the effectiveness of the source neutron relative to the effectiveness of the average neutron in the assembly. The reading of the detector located at (r,J in the subcritical reactor, Ds(rd), will be R(l;l) = r(rdWs = M,
(8)
where N
~
o
-- s (so9$0”) C-0)
(no,
do+>
is the amplitude of the neutron density in the subcritical reactor due to the external source S, and l;l(r& is the detector efficiency, defined as the response of the detector located at rd to n,,(rd, u). Equations (7) and (8) are accurate as long as the reactor is close enough to criticality,thatis,whena,,-Isa,-l,j= 1,2,. . . . As the reactor becomes more subcritical, the relative contribution of the higher modes increase. However, by locating the detector close to a node of a higher mode distribution, its contribution to the detector reading can still be kept low, even at a large subcritical state. For small perturbations a,, is related to the negative reactivity of the subcritical
The source-multiplication method for absolute reactivity determination
131
reactor, p, by the relation (GREENSPAN and CADY, 1970):
(9)
a0 = (-P>/A
where A is the prompt neutron generation time. Hence, from equations (7), (8) and (9)
(10) Equation (10) forms the basis for the “l/M” technique commonly used during criticality approach (KEEPIN, 1968; PAXTONand KEEPIN, 1964). Knowledge of detector efficiency, source strength and effectiveness are not needed for these measurements, that are relative ones. One can also eliminate the detector and source dependence by calibrating the source multiplication measurements against reactivity values measured by other techniques (LEHTO,DAUGHTRY,PONDand CARPENTER,1971; BUHL, ACKERMANN,KRYTERand ROBINSON,1972). In the SMR method presented here the dependence of reactivity on the detector efficiency and effective source strength is eliminated by supplementing the SM measurement with the measurement of the response of the critical reactor (p = 0) to the source. Thus consider a critical reactor with an initial neutron density distribution n(r, v; 0) = Nn,(r, u) into which the external neutron source Ss,(r, v) is inserted at time t = 0. After initial transients die out, the neutron density will increase linearly with time according to the relation (GREENSPANand CADY, 1970), n(r, v; t) = n(r, v; 0) +
mo,~o+>/(~o, $0’) no(r, v>t +
AiAi 1
,+B,tg
[
j=o 5 a&-,
v)
(11)
where j, is the effective fraction of the ith delayed neutron group, j = 1 pi, li is the decay constant of the ith delayed neutron group precursors and the coekients defined as follows (GREENSPANand CADY, 1970):
a, are
(12) and a, =
Sbo,A”> a&,
$j+> ’
j 2 1.
(13)
The reading of the detector at rd, D(rd; t), will increase linearly with time (after initial transients die out) with the rate R: (14)
132
Enun GREENSPAN
By combining equations (10) and (14), the detector and source dependent parameters can be eliminated to give
Or in units of dollars, the reactivity is:
(16) Equation (16) provides a relation between the reactivity and two measurable parameters: R (Rate of linear power rise) and M (Multiplication of the subcritical reactor) with the reactor kinetic parameters serving as a scale factor. In many cases, (17) and equation (16) is simplified to be
When the relative effective yield of the delayed neutron group (/$/#j), is approximated by the relative physical yield x1 = pJj3: *
and the connecting chain between the reactivity and the measured parameters, R and M, are the fissile isotopes delayed neutron parameters. The physical meaning of J$ (a,/&) is the m ean life, Td, of the delayed neutrons. These mean-lives can be measured directly [Prompt-Burst experiments (KEEPIN, 1965)]. When photoneutrons are present (in particular in Be and D,O moderated reactors) their contribution to the mean life of the delayed neutrons should be included. For future reference, the effective mean life of the delayed neutrons will be defined as
3. MEASUREMENT
TECHNIQUES
The determination of (p/B) by the SMR method involves the measurement of two quantities : (a) The rate of the linear rise of the detector response, R, due to the insertion of the source into the critical reactor, and (b) The source multiplication, M, when the reactor is subcritical by the reactivity amount to be measured. * This CC‘ has no connection to the CC’employed S earlier.
The source-multiplicationmethod for absolute reactivity determination
133
An example for a possible measurement sequence of R and M, aimed at the absolute determination of the reactivity of a small sample is: -The reactor is brought to a critical state with the desired control-rod configuration. -The sample, whose reactivity is to be measured, is then inserted into the core. The reading of a fixed detector (say in the reflector) is recorded after the flux level stabilizes. This reading, Dn, is the contribution of background and inherent sources in the reactor. -An external neutron source (such as a californium-252 spontaneous fission source or the reactor start-up source) is inserted into the reactor. The contribution of the source to the detector reading, D,,is recorded after the reactor flux level stabilizes. -The sample is withdrawn from the core while the external source is left. The timedependent reading of the detector, D(t), is recorded and the reading-rate increase, R, is deduced from it after it becomes constant with time. The separation of the effects of the external neutron source from the background neutron sources is not really required. Any superposition of such sources is permissible, as long as the source strength and distribution remain constant throughout the experiment. When the sample has a positive reactivity, the measurement sequence should be slightly modified : -The -The -The
reactor is brought to critical with the sample in it. external neutron source is inserted and R is determined. sample is withdrawn, and the subcritical multiplication, M, is measured.
The procedure can be simplified even more if only relative reactivity values are desired. The reactivity p, is related to a reference reactivity, p,,, by the relation:
(21)
Hence, no linear power rise measurements are required and the SMR method reduces to the Source Multiplication method. 4. ACCURACY OF THE METHOD The SMR method is essentially composed of two known experimental techniques: The Linear Power Rise (LPR)measurement and the Source Multiplication (S&f) measurement. Based on experience gained in applying the LPR and S&f measurements, the following observations can be made: (a) LPR experiments have been performed in recent years in fast and in thermal Zero Power Reactors. KARAM(1969) measured the slope, R, of the LPR in the fast critical assemblies ZPR-6 and ZPR-9. Employing a 1 lug C,-252 spontaneous fission neutron source (yielding about 3 x 10”n/set) he could determine the slope, R, to an accuracy of 1 per cent. In similar measurements at the Cornell University ZPR (H,O moderated, slightly enriched UO, fueled), employing a O-1,ug C,-252 source, HAGA (1967) also determined R to an accuracy of about 1 per cent.
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EHUDGREENSPAN
(b) Accuracy limitations of the SM method, especially in a far-subcritical reactor, are due to: -the contribution of higher modes to the detector reading or, in other words, the errors caused by approximating equation (2) by equation (7). Higher mode contamination can be controlled, to some extent, by locating the neutron source so that it will not strongly excite the dominant higher modes and also by locating the detector at a node of the dominant higher modes. -The substitution of the effective detector efficiency and source in the critical reactor for the corresponding quantities in the subcritical reactor. -The deviation from the unperturbed fundamental mode distribution caused by sample perturbation. When the reactor is only a slightly subcritical, higher modes contamination and perturbation are not expected to impair the accuracy of the SM method. Results of relative SM measurements at the Cornell University ZPR, using negative reactivities of the order of 10 cents, show (GREENSPAN, 1966) very good agreement with the relation of equation (10). The accuracy in these measurements was determined by the reproducibility of the experimental set-up, including reactor criticality, sample location, and temperature drifts. Reproducibility problems are, however, common also to other methods, such as the AP method. In recent measurements in fast reactors, source multiplication methods are reported (LEHTO et al. 1971; BUHL et al. 1972) to give reasonable results down to several dollars subcriticality. In cases where the effects of higher modes contamination and perturbations are non-negligible, the SMR method can still be applied provided the measurements will be corrected to take account of the deviations from the conditions assumed for the derivation of relation (16). Correction factors can be calculated in a manner similar to correction factors calculated for other reactivity measurement methods. For example, in the asymmetric source method (WALTER and HENRY, 1968), in the modified neutrons source multiplication (BUHL et al. 1972) as well as in the kinetic techniques based on the measurement of prompt neutron multiplication or of prompt die-away time constants (BUHL et al. 1972). Another question of concern in performing the SM measurement is the time it takes for the neutron flux level to reach its asymptotic value. This transient time depends strongly on the initial conditions for the SMmeasurements and on the degree of subcriticality. The further the initial flux level is from the asymptotic one (either above or below) and the closer the reactor is to criticality, the longer lasts the transient period. Hence, by proper design of the experiment and the selection of initial flux level it is possible to reduce the transient time. The intensity of the neutron source should be high enough to give signals easily separable from the background. Neutron sources employed for the reactor start-up in low power reactors will usually give signal large enough for the measurement of R and M with good statistics. In reactors having large inherent neutron sources, these sources may be employed directly for the measurement. 5. COMPARISON
WITH THE ASYMPTOTIC
PERIOD
METHOD
The AP is perhaps the method mostly used for the determination of small reactivity values. In this method one measures the asymptotic period, T, resulting from the addition of (usually positive) reactivity to a just critical reactor. The reactivity is
The source-multiplication method for absolute reactivity determination
135
then related to the measured period through the Inhour equation:
(22) The SMR method is also valid for small reactivity determination, and can use standard reactor equipment for its application. Hence the SMR method shares several essential features with the AP method. The following are features the SMR and the AP methods have in common: -No special equipment is needed for the experiments. Standard neutron measurement channels, and for the SMR method also start-up neutron source, will usually suffice. -The critical state of the reactor has to be established and maintained. -The rate of change of the reactor power level is measured. Expected advantages of the AP method are: -Only one measured parameter (T) is required as compared to two parameters (R and M) needed for the SMR method. -It is easy to take care of the effect of background neutron sources, and of statistical fluctuations in the neutron density by carrying the measurements at high enough power levels. -Transient times will, in general, be shorter with the AP method. But when the initial amplitude of the neutron density is properly selected the SMR method may have shorter transient times. Possible advantages of the SMR method over the AP method include: -The rate of the LPR, R, does not depend on the reactivity value to be measured. It can be chosen by selecting the strength and location of the neutron source. The asymptotic period, on the other hand, strongly depends on the reactivity. -The SMR method enables to measure a wider range of reactivities. The lowest value of reactivities possible to measure with the AP and the SMR methods is practically limited by the large times it takes for the transients to decay. The upper limit of reactivity possible to measure with the AP method (in a single step) is set by safety considerations (too short a period) to about 50 cents. No safety considerations limit the range of applicability of the SMR method. Very large negative reactivities can be measured, though with limited accuracy, or else, corrected with calculations. -The error in the calculated reactivity caused by an error in the value of @/A) is, in the AP method, at least as large as in the SMR method. This can be seen from the relative contribution of the (/?/A) term to (p/p): (23) The equality is obtained with T > l/&. This implies that uncertainties in the value of the prompt neutron generation time, A, or the kinetic parameter @/A), will be reflected in a small error in the reactivity when the SMR method is employed. To demonstrate the magnitude of the expected effects, the relative error in the p/i? caused by a total neglect of the @/A) term in the SMR method [equation (I@]
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EHUD
GREENSPAN
has been evaluated. Figure I shows this relative error for the three fissile nuclides as a function of @/A). The results were computed from the expression l/V + (B/A) z with the values of z
(a,/&)
(4Ul
from KEEPIN (;968).
Characteristical
values of @/A)
fall in the range Af l-100 set-l for natural uranium graphite or heavy-water moderated reactors and of 100-200 set-l for typical slightly enriched light-water
0.01
Fxo. 1 -Relative
error in (p/n
due to the neglectof the @/A)term in the SMR method.
moderated reactors. It is even larger for fast reactors. With plutonium fuel, the value of (j/A) will be about 6 of its value in the corresponding reactor with 235Uas the fuel. Referring to Fig. 1, it is seen that the error in p/j caused by the neglect of the ,!?/A term will be less than O-1 per cent for light-water thermal reactors, and of the order of 1 per cent for the natural uranium thermal reactors. Figure 2 gives the ratio of the error introduced in the AP method by the neglect of j/A, relative to the error in the SMR method. The results were calculated from equation (23) for two extreme values of (j/A) with 235U as fuel. It is seen that for periods of about 30 set, the error in the AP method is approximately twice as large as with the SMR method. The relative error increases as the period shortens and the value of (B/A) becomes larger. -Only one delayed neutron parameter, except for the value of /?/A, is required for the SMR method. This parameter is the effective delayed neutron mean life, Td. The Inhour equation in use in the AP method, requires the fractional yield and decay constants of all delayed neutron groups. The accuracy with which these group parameters are known is inferior to that of the mean-life. -There is a linear relationship between the reactivity and the measured parameters
The source-multiplication
method for absolute reactivity determination
i
lb0
Period,
FIG. 2.-Error
137
set
in (p/B) due to the neglect of the (p/A) term in the AP method relative to the SMR method. Fissile fuel is ‘W.
(R and ikf) in the Sh4R method. The SA4R equation (16) is therefore simpler to use than the Inhour equation. The SMR method and the AP method can, in principle, be combined for the determination of the kinetic parameter b/A. Eliminating the reactivity from equations (16) and (22) one obtains the following expression for @/A), the ratio of the effective delayed neutron yield to the prompt neutron generation time:
Hence, by measuring the asymptotic period corresponding to the reactivity employed for the subcritical multiplication measurement (M) in addition to the measurement of R and M by the SMR method, one could, in principle, calculate (j/A) from equation (24). The accuracy to which (j/A) can be determined by this method, to be called the Source-Period method, is however, very limited. This is due to the fact that the denominator of equation (24) consists of a difference between two terms of very similar value. Hence, an error in the value of R/M is amplified in the calculated value of (p/b). This amplification becomes larger with larger periods and larger values of @/A). Even for extreme conditions of periods as short as 10 set, and (/?/A) as small as 1 set-l, the error amplification is of the order of 20!. 6. SUMMARY
Based on theoretical formulation, numerical investigation, and analysis of experimental results we conclude that the SMR can be a useful method for determining
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EHUD GREENSPAN
absolute reactivity values (in units of b). Heigher mode contamination and perturbation and make it valid for measuring small reactivities. Larger reactivities can also be measured, but calculated correction factors to the SMR measurements may be required for getting satisfactory accuracy. The SMR method is expected to have similar accuracy as the AP method in the reactivity range common to both methods. The SMR method shares two other basic features with the AP method: (1) Simplicity of application, requiring only standard reactor equipment and (2) Criticality has to be established during the measurement procedure. Hence, from the point of view of accuracy range and ease of applicability, the SMR method can be classified in a category with the AP method. Unlike the AP method, no safety considerations limit the range of reactivities to be measured at one single step with the SMR method. The equation connecting reactivity with the measured parameters in the SMR method is easier to apply than the Inhour equation of the AP method. Except for /?/A, only the average effective delayed neutron mean life, Td, is needed for the SMR equation. Moreover, the calculated reactivity in the SMR method is less sensitive to inaccuracies (or even omission) of the @/A term. Its complete neglect introduces an error of the order of 1 per cent for natural uranium reactors, and less than 0.1 per cent for light-water reactors. A good approximation for the SMR equation is thus: (PI/% = WW). The major sources of inaccuracy in the SMR methods are expected to be effects of transients, establishment and maintenance of criticality, and for large negative reactivities also higher modes contamination, and distortion of fundamental mode distributions. As is the case with other reactivity measurement methods, the effects of these sources of inaccuracy can, in general, be reduced by careful design of the experiments. They can also be reduced by applying calculated correction factors. An experimental investigation of all these effects is due before final conclusions about the accuracy of the SMR method can be reached. The SM and the AP methods can be formally combined to measure the kinetic parameter @/A). This Source-Period method is very inaccurate because the error in the measured ratio, RIM, is highly amplified in this method. REFERENCES BUHL A. R., ACKERMANN. J., KR~R R. C. and ROBINSONJ. C. (1972) Trans. ANS 15 (1) 423. BUHL A. R., ROBINSONJ. C. and ACKERMANN. J. (1972) Trans. ANS 15 (1) 493. GREENSPANE. (1966) Cornell University Report No. CURL-15 GREENSPANE. and CADY K. B. (1970) J. nucf. Energy, 24,529. HAGA H. (1967) Cornell University Report No. CURL-22. KARAM R. A. (1969) Nucl. Sci. Eng. 37 (2) 192. KEEPIN G. R. (1965) Physics of Nuclear Kinetics, Addison-Wesley. LEHTO W. K., DAUGHTRYJ. W., POND R. P. and CARPENTERS. G. (1971) Trans. ANS 14 (1) 42. PAXTONH. C. and KEEPIN G. R. (1964) The Technology of Nuclear Reactor Safety, Vol. I, Criticality, Chap. 5, Edited by THOMPSONT. J. and BECKERLY J. G., MIT Press. WALTER J. F. and HENRY A. F. (1968) Nucl. Sci. Erg. 32,332. WALTZ W. R. and WALTER B. F. (1971) Nucl. Tech. 10 (2) 160.