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Detection of subcriticality changes by Simmons-King and Sjöstrand methods
Annals of Nuclear Energy 138 (2020) 107209
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Annals of Nuclear Energy journal homepage: www.elsevier.com/locate/anucene
Detection of subcriticality changes by Simmons-King and Sjöstrand methods Yasunori Kitamura ⇑, Tsuyoshi Misawa Institute for Integrated Radiation and Nuclear Science, Kyoto University, Asashiro-Nishi, Kumatori-cho, Sennan-gun, Osaka 590-0494, Japan
a r t i c l e
i n f o
Article history: Received 18 June 2019 Received in revised form 8 October 2019 Accepted 11 November 2019
Keywords: Subcriticality change Time-response Simmons-King method Sjöstrand (or Area-ratio) method Prompt neutron decay constant Pulsed neutron source
a b s t r a c t The Simmons-King and the Sjöstrand (or the area-ratio) methods have been widely applied in measurement of the subcriticality of reactor systems driven by the pulsed neutron source and operated in the stationary state. In the present study, a theory-based investigation is conducted to examine the timeresponse of these two methods after perturbations in various parameters. As a result, the prompt neutron decay constant determined by the Simmons-King method shows a good trackability to the subcriticality. In the Sjöstrand method, calibration-free determination of the subcriticality is achieved, although it needs a long delay due to an asymptotic behaviour. Furthermore, it is found that the latter method can quickly detect the perturbation in neutron yield of the pulsed neutron source. Therefore, it is expected that a strong on-line monitoring tool for the subcritical reactor system driven by the pulsed neutron source can be developed by combining these two methods. Ó 2019 Elsevier Ltd. All rights reserved.
1. Introduction The subcriticality (i.e., the negative reactivity) is often discussed in the field of reactor physics, since it is one of the most fundamental quantities for the nuclear criticality safety of subcritical reactor systems. The Simmons-King and the Sjöstrand (or the area-ratio) methods, which are collectively referred to as the pulsed neutron source ones, have been widely applied in measurement of the subcriticality (Simmons and King, 1958; Sjöstrand, 1956). In these methods, pulsed neutrons generated by periodic bursts of the pulsed neutron source are injected into the subcritical reactor system. The neutron counting rate as a function of time after injection of pulsed neutrons is observed under the condition that a stationary state is maintained by periodic bursts of the pulsed neutron source. Owing to a great amount of efforts devoted by many researchers until the 1980s, the pulsed neutron source methods have been regarded as the most reliable and established ones for measuring the subcriticality (Krieger and Zweifel, 1959; Fultz, 1959; Beckurts, 1961; Gozani, 1962; Garelis and Russell, 1963; Aizawa and Yamamuro, 1967; Kaneko and Sumita, 1967; Sumita and Kaneko, 1967; Aizawa, 1969; Amano, 1969; Kosály and Valkó, 1971; Kosály and Fischer, 1972; Dragt, 1973; Difilippo
⇑ Corresponding author. E-mail address: [email protected] (Y. Kitamura). https://doi.org/10.1016/j.anucene.2019.107209 0306-4549/Ó 2019 Elsevier Ltd. All rights reserved.
et al., 1973; Kosály et al., 1974; Kosály and Valkó, 1975; Akino et al., 1980). Recently, a renewed necessity of the pulsed neutron source methods has been presented. During the last two decades, hence, the studies related to the pulsed neutron source methods have been activated again (Soule et al., 2004; Jammes et al., 2005; Persson et al., 2005; Kulik and Lee, 2006; Billebaud et al., 2007; Lebrat et al., 2008; Persson et al., 2008; Pyeon et al., 2008, Pyeon et al., 2009; Berglöf et al., 2010; Bécares et al., 2013, Bécares et al., 2013; Uyttenhove et al., 2014; Kitamura and Fukushima, 2014; Kitamura and Fukushima, 2015; Pyeon et al., 2014; Pyeon et al., 2015; Yamanaka et al., 2016; Pyeon et al., 2017; Iwamoto et al., 2017). Such a revival of the pulsed neutron source methods is mainly due to the accelerator-driven system (ADS). One application of the pulsed neutron source methods newly presented is development of the on-line subcriticality monitor that is desired to be equipped with the ADS to ensure and enhance the nuclear criticality safety (Iwamoto et al., 2017). It is expected that the pulsed neutron source methods will play an important role in such an application range. Among two methods of the pulsed neutron source ones, the Sjöstrand one has especially been attracted owing to the ability of calibration-free determination of the subcriticality (Sjöstrand, 1956). In order to realize the on-line subcriticality monitor, it is essential to confirm the time-response of the pulsed neutron source methods after perturbations in various parameters of the
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Nomenclature Parameters Before Perturbations kc probability that one neutron undergoes capture reaction per unit time. kd probability that one neutron undergoes detection reaction per unit time. kf probability that one neutron undergoes fission reaction per unit time. ka ¼ kc þ kd þ kf probability that one neutron undergoes one of absorption reactions per unit time. K ¼ kf 1hmi neutron generation time. a q ¼ kf khmf hik reactivity. mi bq a¼ K prompt neutron decay constant.
pS ðqÞ
probability that q neutrons are injected in one neutron burst. P hni ¼ þ1 q¼0 qpS ðqÞ average number of neutrons injected per neutron burst. Gn!n ðt Þ population of progeny neutron at time t originating from one ancestor neutron at time 0. Gn!c ðt Þ population of progeny delayed neutron precursor at time t originating from one ancestor neutron at time 0. Gc!n ðt Þ population of progeny neutron at time t originating from one ancestor delayed neutron precursor at time 0. Gc!c ðt Þ population of progeny delayed neutron precursor at time t originating from one ancestor delayed neutron precursor at time 0. ap larger time constant included in general solution of one-point reactor kinetic equation with one delayed neutron group. ad smaller time constant included in general solution of one-point reactor kinetic equation with one delayed neutron group. Parameters
after perturbations probability that one neutron undergoes capture reaction per unit time. kd probability that one neutron undergoes detection reaction per unit time. kf probability that one neutron undergoes fission reac tion per unit time. ka ¼ kc þ kd þ kf probability that one neutron undergoes one of absorption reactions per unit time. K ¼ 1 neutron generation time. kc
q ¼ kf hmika reactivity.
kf hmi
a ¼ bq
K
prompt neutron decay constant.
probability that q neutrons are injected in one neutron pS ðqÞ DE P burst. n ¼ þ1 q¼0 qpS ðqÞ average number of neutrons injected per neutron burst. Gn!n ðt Þ population of progeny neutron at time t originating from one ancestor neutron at time 0. Gn!c ðt Þ population of progeny delayed neutron precursor at time t originating from one ancestor neutron at time 0. Gc!n ðt Þ population of progeny neutron at time t originating from one ancestor delayed neutron precursor at time 0. Gc!c ðt Þ population of progeny delayed neutron precursor at time t originating from one ancestor delayed neutron precursor at time 0. ap larger time constant included in general solution of one-point reactor kinetic equation with one delayed neutron group. ad smaller time constant included in general solution of one-point reactor kinetic equation with one delayed neutron group. Other parameters k disintegration time constant of delayed neutron precursor. probability that n prompt neutrons and c delayed neupf ðn; cÞ tron precursors are simultaneously born in one fission reaction. mp average number of prompt neutrons per fission reaction. hmd i average number of delayed neutrons per fission reaction. average number of total neutrons per fission reaction. hmi b delayed neutron fraction. s periodicity of neutron burst. h time of first neutron burst after perturbations. Nðt Þ population of neutron at time t. C ðt Þ population of delayed neutron precursor at time t. Pðt Þ neutron counting rate at time t.
kf hmi
subcritical reactor system driven by the pulsed neutron source. However, as mentioned above, the pulsed neutron source methods are usually applied to the subcritical reactor system operated in the stationary state. Therefore, no theoretical study that focuses on the time-response after perturbations has not been carried out yet, as far as the authors know. In the present study, hence, a theory-based investigation for examining the time-response of the pulsed neutron source methods after perturbations is conducted as a part of the study for realizing the on-line subcriticality monitor for the ADS. In the following section, the notations employed in the present study are introduced. The theory of the pulsed neutron source methods that explicitly considers perturbations in various parameters of the subcritical reactor system driven by the pulsed neutron source is developed in Section 3. By using the theory thus developed, the time-response of the Simmons-King and the Sjöstrand methods after perturbations is discussed in Section 4. Finally, the conclusion is summarized in Section 5.
2. Notations 2.1. Parameters constant in time In the present study, a zero-power subcritical reactor system that consists of a multiplying medium, a pulsed neutron source, and a neutron detector is supposed. Delayed neutrons are rigorously taken into consideration, while they are assumed to be emitted from their precursors having a single disintegration time constant k, for simple discussions. Furthermore, formulation is thoroughly performed within a mono-energy one-point reactor model. The probability that n prompt neutrons and c delayed neutron precursors are simultaneously born in one fission reaction is denoted by pf ðn; cÞ with þ1 X þ1 X pf ðn; cÞ ¼ 1: n¼0 c¼0
ð1Þ
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By using this probability, the average numbers of prompt, delayed, and total neutrons per fission reaction are defined as
mp ¼
þ1 X þ1 X
npf ðn; cÞ;
ð2Þ
cpf ðn; cÞ;
ð3Þ
n¼0 c¼0
hmd i ¼
þ1 X þ1 X n¼0 c¼0
hmi ¼
þ1 X þ1 X
ðn þ cÞpf ðn; cÞ ¼
mp þ hmd i:
ð4Þ
n¼0 c¼0
With the quantities defined above, the delayed neutron fraction b of the subcritical reactor system is naturally introduced as follows:
b¼
hmd i : hmi
ð5Þ
In the present study, it is supposed that various parameters of the subcritical reactor system may be perturbed at time 0 to discuss the time-response of the Simmons-King and the Sjöstrand methods after the perturbations. We note here that the parameters with tilde are thoroughly used as those after the perturbations. By using the unit step function U ðt Þ, the probabilities with respect to the respective neutron interactions in the subcritical reactor system (i.e., capture, detection, and fission reactions) are written as
kx ðtÞ ¼ kx f1 U ðt Þg þ kx U ðt Þ;
1 ¼ K f1 U ðtÞg þ K U ðtÞ; kf ðt Þ hmi
ð8Þ
qðtÞ ¼
kf ðtÞ hmi ka ðt Þ ¼ q f1 U ðt Þg þ q U ðtÞ; kf ðt Þ hmi
ð9Þ
aðtÞ ¼
b qðtÞ ¼ a f1 U ðt Þg þ a U ðtÞ: Kð t Þ
x ¼ c; d; f;
ka ðtÞ ¼ kc ðtÞ þ kd ðt Þ þ kf ðt Þ ¼ ka f1 U ðtÞg þ ka U ðtÞ;
ð6Þ ð7Þ
where kc ðt Þ is the probability per unit time that one neutron undergoes a capture reaction in the multiplying medium at time t; kd ðtÞ the probability per unit time that one neutron undergoes a detection reaction in the neutron detector at time t; kf ðt Þ the probability per unit time that one neutron undergoes a fission reaction in the multiplying medium at time t, and ka ðtÞ the probability per unit time that one neutron undergoes one of the absorption reactions in the subcritical reactor system at time t. By using the quantities defined above, the neutron generation time Kðt Þ, the reactivity qðt Þ, and the prompt neutron decay constant aðt Þ of the subcritical reactor system are introduced as follows:
ð10Þ
The subcritical reactor system is supposed to be driven by periodic bursts of the pulsed neutron source, as illustrated in Fig. 1. When one denotes its periodicity by s and the time of the first neutron burst after the perturbations by h, one expresses the time of each neutron burst as follows:
tj ¼ js þ h;
j ¼ 1; . . . ; 1; 0; þ1; . . . ; þ1;
ð11Þ
with the following condition:
0 6 h < s:
2.2. Parameters vary in time
Kðt Þ ¼
ð12Þ
The probability that q neutrons are injected in one neutron burst is denoted by
pS ðq; t Þ ¼ pS ðqÞ f1 U ðt Þg þ pS ðqÞ U ðtÞ;
ð13Þ
with þ1 X pS ðqÞ ¼ 1;
ð14Þ
q¼0 þ1 X pS ðqÞ ¼ 1:
ð15Þ
q¼0
Using this probability, the average number of pulsed neutrons per neutron burst is defined as
hnðt Þi ¼
þ1 X
DE qpS ðq; t Þ ¼ hni f1 U ðt Þg þ n U ðt Þ:
ð16Þ
q¼0
3. Theory 3.1. Outline In this section, by using a formulation technique employed in our previous study (Kitamura and Misawa, 2017), the population of neutron N ðt Þ, that of delayed neutron precursor C ðt Þ, and the
Fig. 1. Timing diagram of neutron bursts and perturbations.
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Y. Kitamura, T. Misawa / Annals of Nuclear Energy 138 (2020) 107209
neutron counting rate P ðt Þ at time t are derived. As illustrated in Fig. 2, one derives N ðtÞ as follows:
N ðt Þ ¼
j X
Nk ðtÞ;
tj 6 t < t jþ1 ;
ð17Þ
k¼1
where N k ðtÞ is the population of progeny neutron at time t originating from the neutron burst at time tk (6 t). One similarly derives C ðt Þ as
C ðt Þ ¼
j X
C k ðt Þ;
t j 6 t < tjþ1 ;
ð18Þ
k¼1
where C k ðt Þ is the population of progeny delayed neutron precursor at time t originating from the neutron burst at time t k (6 t). On the other hand, P ðt Þ is simply written as
PðtÞ ¼ kd ðtÞ NðtÞ:
ð19Þ
In the following two subsections, N ðtÞ; C ðt Þ, and P ðt Þ before and after the perturbations are separately derived.
In the present subsection, N ðtÞ; C ðtÞ, and P ðt Þ with respect to t < 0, i.e., before the perturbations, are derived. In order to describe the temporal evolution of population of neutron or delayed neutron precursor in the subcritical reactor system before the perturbations, the function Gy!z ðtÞ, which is derived for the multiplying medium where injection of pulsed neutrons, the neutron and precursor populations are absent, is introduced. With this function, one calculates the population of progeny particle z (¼ n; c) at time t originating from one ancestor particle y (¼ n; c) at time 0. We note here that ‘‘n” and ‘‘c” correspond to the neutron and the delayed neutron precursor, respectively. The function Gy!z ðt Þ is derived from the one-point reactor kinetic equation with one delayed neutron group for the unperturbed subcritical reactor system, which is written down as follows (Duderstadt and Hamilton, 1976):
dC ðtÞ b ¼ kC ðt Þ þ Nðt Þ; dt K
dGn!n ðtÞ ¼ aGn!n ðt Þ þ kGn!c ðt Þ þ dðt Þ; dt
ð22Þ
dGn!c ðtÞ b ¼ kGn!c ðt Þ þ Gn!n ðt Þ: dt K
ð23Þ
On the other hand, by replacing Nðt Þ with Gc!n ðtÞ and C ðt Þ with Gc!c ðt Þ then adding the Dirac’s delta dðt Þ to the right-hand side of Eq. (21), one obtains
dGc!n ðtÞ ¼ aGc!n ðt Þ þ kGc!c ðt Þ; dt
ð24Þ
dGc!c ðtÞ b ¼ kGc!c ðtÞ þ Gc!n ðtÞ þ dðtÞ: dt K
ð25Þ
Since the initial condition of Gy!z ðt Þ is read as
Gy!z ð0Þ ¼ 0;
3.2. Before perturbations
dNðt Þ ¼ aNðt Þ þ kC ðt Þ; dt
where Nðt Þ and C ðtÞ are the populations of neutron and delayed neutron precursor at time t, respectively. By replacing N ðt Þ with Gn!n ðtÞ and C ðtÞ with Gn!c ðtÞ then adding the Dirac’s delta dðtÞ to the right-hand side of Eq. (20), one obtains
z ¼ n; c;
ð26Þ
their Laplace transforms are calculated as
8 sþk ; > H ðsÞ > > > > b=K < ; H ðsÞ L Gy!z ðt Þ ¼ k > > H ðsÞ ; > > > : sþa ; H ðsÞ
y ¼ n;
z¼n
y ¼ n;
z¼c
y ¼ c;
z¼n
y ¼ c;
z¼c
;
ð27Þ
with
HðsÞ ¼ s2 þ ða þ kÞs and
ap ¼
ð20Þ
ad ¼ ð21Þ
y ¼ n; c;
ða þ kÞK þ
kq Y ¼ ðs þ ai Þ; K i¼p;d
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ða þ kÞ2 K2 þ 4kKq 2K
ða þ kÞK
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ða þ kÞ2 K2 þ 4kKq 2K
ð28Þ
;
ð29Þ
:
ð30Þ
One can re-writes Eq. (27) as
Fig. 2. Schematic illustration for deriving neutron population N ðt Þ and delayed neutron precursor population C ðt Þ.
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Y. Kitamura, T. Misawa / Annals of Nuclear Energy 138 (2020) 107209
X Xy!z;i L Gy!z ðt Þ ¼ ; s þ ai i¼p;d with
Xy!z;i ¼
8 ka i ; > > > ai ai > > > b=K > < a ai ;
y ¼ n; c;
z ¼ n; c;
y ¼ n;
z¼n
y ¼ n;
z¼c
y ¼ c;
z¼n
y ¼ c;
z¼c
i
k > > ai ai ; > > > > a k > : i ;
ai ai
ð31Þ
In the present subsection, N ðtÞ; C ðt Þ, and P ðt Þ with respect to 0 6 t, i.e., after the perturbations, are derived. In order to describe the temporal evolution of population of neutron or delayed neutron precursor in the subcritical reactor sys
;
i ¼ p; d;
i – i:
ð32Þ
Gy!z ðtÞ ¼
Xy!z;i eai t ;
y ¼ n; c;
z ¼ n; c:
ð33Þ
i¼p;d
When q neutrons are injected in the neutron burst at time t k (< 0), the populations of progeny neutron and delayed neutron precursor at time t (< 0) originating from these q neutrons are written as follows:
qGn!n ðt tk Þ;
t k 6 t < 0;
ð34Þ
qGn!c ðt tk Þ;
t k 6 t < 0:
ð35Þ
Since q can take various integer values ranging from 0 to þ1 according to the probability pS ðqÞ, by multiplying Eqs. (34) and (35) by pS ðqÞ and then summing over q, one obtains N k ðtÞ and C k ðt Þ with respect to t k 6 t < 0, which are the expected neutron and delayed neutron precursor populations at time t generated by the kth neutron burst, as
Nk ðt Þ ¼
þ1 X
qpS ðqÞGn!n ðt t k Þ ¼ hniGn!n ðt t k Þ;
dNðtÞ ¼ a Nðt Þ þ kC ðt Þ; dt
ð42Þ
dC ðt Þ b ¼ kC ðt Þ þ Nðt Þ: dt K
ð43Þ
As seen in Section 3.2, one derives Gy!z ðt Þ with respect to y ¼ n; c and z ¼ n; c as follows:
Gy!z ðt Þ ¼
qpS ðqÞGn!c ðt t k Þ ¼ hniGn!c ðt tk Þ;
X
Xy!z;i ea i t ;
y ¼ n; c;
z ¼ n; c;
with
Xy!z;i ¼
t k 6 t < 0:
8 kai > ; > > > ai ai > > > > > b= K > < ; ai ai
k > > ; > a a > > > i i > > a k > > :i ;
ai ai
q¼0
y ¼ n;
z¼n
y ¼ n;
z¼c
y ¼ c;
z¼n
y ¼ c;
z¼c
;
i ¼ p; d;
j P
N ðt Þ ¼ hni
ap ¼
Gn!n ðt tk Þ
P
i¼p;d
ð38Þ
Xn!n;i Dai s eai ðttj Þ ;
tj 6 t < t jþ1 1 dj;1 ; C ðtÞ ¼ hni
j P
j ¼ 1; . . . ; 2; 1;
P
i¼p;d
Gn!c ðt tk Þ ð39Þ
Xn!c;i Dai s eai ðttj Þ ;
t j 6 t < tjþ1 1 dj;1 ;
j ¼ 1; . . . ; 2; 1;
where
Dc ¼
1 : 1 e c
ð40Þ
Therefore, from Eqs. (19) and (38), one gets PðtÞ before the perturbations as
PðtÞ ¼ kd hni
P i¼p;d
ai ðtt j Þ
Xn!n;i Dai s e
j ¼ 1; . . . ; 2; 1:
2K
;
t j 6 t < tjþ1
1 dj;1 ;
ad ¼
a þk K
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 a þk K2 þ 4k K q
2K
;
ð46Þ
:
ð47Þ
When q neutrons are injected in the neutron burst at time tk (< 0), i.e., before the perturbations, the populations of progeny neutron and delayed neutron precursor at time 0 originating from these q neutrons are written as follows:
k¼1
¼ hni
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 a þk K þ a þk K2 þ 4k K q
k¼1
¼ hni
i – i;
ð45Þ
ð37Þ By substituting Eqs. (36) and (37) into Eqs. (17) and (18), respectively, the concrete expressions of Nðt Þ and C ðt Þ before the perturbations are obtained as
ð44Þ
i¼p;d
ð36Þ þ1 X
The function Gy!z ðtÞ is derived from the one-point reactor kinetic equation with one delayed neutron group for the perturbed subcritical reactor system, which is written down as follows (Duderstadt and Hamilton, 1976):
tk 6 t < 0;
q¼0
C k ðt Þ ¼
tem after the perturbations, the function Gy!z ðt Þ, which is derived for the multiplying medium where injection of pulsed neutrons, the neutron and precursor populations are absent, is introduced. With this function, one calculates the population of progeny particle z (¼ n; c) at time t originating from one ancestor particle y (¼ n; c) at time 0.
One hence gets
X
3.3. After perturbations
ð41Þ
qGn!n ðt k Þ;
tk < 0;
ð48Þ
qGn!c ðt k Þ;
tk < 0:
ð49Þ
Hence, the populations of progeny neutron and delayed neutron precursor at time t ( 0), i.e., after the perturbations, originating from q neutrons generated at time t k (< 0), i.e., before the perturbations, are respectively expressed as follows:
n o q Gn!n ðtk ÞGn!n ðt Þ þ Gn!c ðt k ÞGc!n ðtÞ ;
t k < 0 6 t;
ð50Þ
n o q Gn!n ðtk ÞGn!c ðt Þ þ Gn!c ðt k ÞGc!c ðtÞ ;
t k < 0 6 t:
ð51Þ
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Y. Kitamura, T. Misawa / Annals of Nuclear Energy 138 (2020) 107209
Since q can take integer values ranging from 0 to þ1 according to the probability pS ðqÞ, by multiplying Eqs. (50) and (51) by pS ðqÞ and then summing over q, one obtains N k ðt Þ and C k ðt Þ with respect to t k < 0 6 t as N k ðt Þ ¼
n o qpS ðqÞ Gn!n ðt k ÞGn!n ðt Þ þ Gn!c ðt k ÞGc!n ðt Þ
þ1 P
ð52Þ
q¼0
n o ¼ hni Gn!n ðtk ÞGn!n ðtÞ þ Gn!c ðtk ÞGc!n ðt Þ ; C k ðt Þ ¼
þ 1 P
t k < 0 6 t;
n o qpS ðqÞ Gn!n ðtk ÞGn!c ðt Þ þ Gn!c ðtk ÞGc!c ðtÞ
ð53Þ
q¼0
n o ¼ hni Gn!n ðt k ÞGn!c ðtÞ þ Gn!c ðt k ÞGc!c ðt Þ ;
t k < 0 6 t:
C 0 ¼ hni
1 X
qGn!n ðt t k Þ;
qGn!c ðt tk Þ;
0 6 tk 6 t;
ð54Þ
0 6 tk 6 t:
ð55Þ
Since q can take integer values ranging from 0 to þ1 according to
the probability pS ðqÞ, by multiplying Eqs. (54) and (55) by pS ðqÞ and then summing over q, one obtains N k ðt Þ and C k ðt Þ with respect to 0 6 tk 6 t as
Nk ðt Þ ¼
þ1 X
D E qpS ðqÞGn!n ðt t k Þ ¼ n Gn!n ðt t k Þ;
0 6 tk 6 t;
q¼0
ð56Þ C k ðt Þ ¼
þ1 X
DE qpS ðqÞGn!c ðt t k Þ ¼ n Gn!c ðt t k Þ;
0 6 t k 6 t:
q¼0
ð57Þ By substituting Eqs. (52), (53), (56) and (57) into Eqs. (17) and (18), respectively, the concrete expressions of N ðtÞ and C ðt Þ after the perturbations are obtained as o DE P j 1 n P Gn!n ðtk ÞGn!n ðtÞ þ Gn!c ðtk ÞGc!n ðtÞ þ n Gn!n ðt tk Þ
N ðtÞ ¼ hni
k¼1
DE P j ¼ N0 Gn!n ðtÞ þ C 0 Gc!n ðtÞ þ n Gn!n ðt tk Þ k¼0 P N 0 Xn!n;i þ C 0 Xc!n;i eai t ¼
k¼0
i¼p;d
DE P n o þ n Xn!n;i Da s 1 eai ðjþ1Þs eai ðttj Þ ;
i¼p;d
n o DE P j Gn!n ðtk ÞGn!c ðtÞ þ Gn!c ðtk ÞGc!c ðtÞ þ n Gn!c ðt t k Þ
k¼1
DE P j ¼ N0 Gn!c ðtÞ þ C 0 Gc!c ðtÞ þ n Gn!c ðt tk Þ k¼0 P N 0 Xn!c;i þ C 0 Xc!c;i eai t ¼
k¼0
i¼p;d
DE P n o þ n Xn!c;i Da s 1 eai ðjþ1Þs eai ðttj Þ ; i¼p;d
i
tj 1 dj;1 6 t < tjþ1 ;
j ¼ 1; 0; . . . ; þ1; ð59Þ
with
N0 ¼ hni
1 X k¼1
ð61Þ
One easily understands from Eqs. (58) and (59) that the physical meanings of N0 and C 0 are the populations of neutron and delayed neutron precursor at time 0, i.e.,
Nð0Þ ¼ N0 ;
ð62Þ
C ð0Þ ¼ C 0 :
ð63Þ
Finally, from Eqs. (19) and (58), one gets Pðt Þ after the perturbations as
Pðt Þ ¼ kd
P
N0 Xn!n;i þ C 0 Xc!n;i eai t
i¼p;d
n o DE P þkd n Xn!n;i Da s 1 eai ðjþ1Þs eai ðttj Þ ;
i¼p;d
i
tj 1 dj;1 6 t < tjþ1 ;
j ¼ 1; 0; . . . ; þ1: ð64Þ
4. Discussions In this section, by using the theoretical expressions of P ðt Þ given in Eqs. (41) and (64), the time-response of the Simmons-King and the Sjöstrand methods to the subcritical reactor system after the perturbations is discussed. The parameters before the perturbations are listed in Table 1. The prompt neutron decay constant a and the subcriticality in dollar unit qsub (¼ q=b) before the perturbations are 1947:1 s1 and 3:837 $, respectively. Because of the neutron life time in the 105 s order, one understands that the subcritical reactor system supposed is one of the light water systems. In the present study, we introduce two kinds of perturbations, i.e., those in hni and kc . The magnitude of the perturbation in hni is supposed to be from þ75% to 75%. On the other hand, that in kc is supposed to be from þ4% to 4%. We note here that the latter perturbation results in the changes in a from 3275:0 s1 to 619:3 s1 and those in qsub from 7:136 $ to 0:538 $, as listed in Table 2. Figs. 3 and 4 provide P ðt Þ normalized by hni before and after the perturbations at time 0 in cases that hni and kc are independently perturbed. In the Simmons-King and the Sjöstrand methods, by means of the least-square fitting procedure, P ðt Þ within t j 6 t < tjþ1 (j ¼ 1; . . . ; 1; 0; 1; . . . ; þ1) is fitted by the following equation:
j ¼ 1; 0; . . . ; þ1; ð58Þ
C ðtÞ ¼ hni
Xn!c;i Dai s eai ðshÞ :
i
tj 1 dj;1 6 t < tjþ1 ;
1 P
X i¼p;d
k¼1
On the other hand, in cases that q neutrons are injected in the neutron burst at time tk ( 0), i.e., after the perturbations, the populations of progeny neutron and delayed neutron precursor at time t (P tk ) originating from these q neutrons are written as follows:
Gn!c ðt k Þ ¼ hni
Gn!n ðt k Þ ¼ hni
X i¼p;d
Xn!n;i Dai s eai ðshÞ ;
ð60Þ
Table 1 Parameters of subcritical reactor system before perturbations. Parameter
:
Value
k [s1 ] mp [–] hmd i [–] b [–] kc [s1 ] kd [s1 ] kf [s1 ] K [s]
: :
0:077 2:4564375
: : : : : :
0:0185625 0:0075 33196:231 334:850 21685:749
q [%Dk=k] a [s1 ] ap [s1 ] ad [s1 ]
: : :
1:863 105 2:8778 1947:144 1947:160
: :
0:0611 arbitrary numbers
hni [–]
7
Y. Kitamura, T. Misawa / Annals of Nuclear Energy 138 (2020) 107209 Table 2 Key parameters of subcritical reactor system after perturbations.
Perturbation in kc
a [s1 ]
ap [s1 ]
ad [s1 ]
q [%Dk=k]
q =b [$]
þ4% þ1% 0% 1% 4%
3274:993 2279:106 1947:144 1615:182 619:295
3275:002 2279:120 1947:160 1615:201 619:345
0:0675 0:0634 0:0611 0:0578 0:0269
5:3518 3:4963 2:8778 2:2593 0:4038
7:13579 4:66179 3:83712 3:01246 0:53846
Fig. 3. Counting rates normalized by hni with and without perturbation in n at time 0.
Fig. 4. Counting rates normalized by hni with and without perturbation in kc at time 0.
PðtÞ / Aeaðttj Þ þ B;
tj 6 t < t jþ1 ;
j ¼ 1; . . . ; 1; 0; 1; . . . ; þ1;
ð65Þ
where A and B are the respective amplitude factors related to the prompt and the delayed neutrons. Among these three parameters thus determined, a is utilized for measuring the subcriticality in
8
Y. Kitamura, T. Misawa / Annals of Nuclear Energy 138 (2020) 107209
the Simmons-King method (Simmons and King, 1958). On the other hand, in the Sjöstrand method (Sjöstrand, 1956), qsub is calculated from the following equation:
Rs
dt Aeat
qsub ¼ 0R s 0
dt B
¼
A 1 eas : B as
ð66Þ
Figs. 5 and 6 provide the results of subcriticality measurement by the Simmons-King and the Sjöstrand methods in cases of various magnitudes of the perturbation in hni with no perturbation in kc . In Fig. 5, a determined by Eq. (65) with the least-square fitting procedure was plotted as a function of tj after the perturbations. On the other hand, qsub calculated from Eq. (66) was
plotted as a function of tj in Fig. 6. Similarly, Figs. 7 and 8 provide the results in cases of various magnitudes of the perturbation in hni with þ4% perturbation in kc . Figs. 9 and 10 provide the results in cases of various magnitudes of the perturbation in hni with þ1% perturbation in kc . Figs. 11 and 12 provide the results in cases of various magnitudes of the perturbation in hni with 1% perturbation in kc . Figs. 13 and 14 provide the results in cases of various magnitudes of the perturbation in hni with 4% perturbation in kc . In these cases, the subcriticality changes are introduced depending on the magnitude of the perturbation in kc . One easily understands that a and qsub should be constant when no perturbation in kc is inserted. As expected, one observes in Fig. 5 that a determined by the Simmons-King method is unchanged and
Fig. 5. Determination of a by Simmons-King method in case of no perturbation in kc .
Fig. 6. Determination of qsub by Sjöstrand method in case of no perturbation in kc .
9
Y. Kitamura, T. Misawa / Annals of Nuclear Energy 138 (2020) 107209
Fig. 7. Determination of a by Simmons-King method in case of þ4% perturbation in kc at time 0.
Fig. 8. Determination of qsub by Sjöstrand method in case of þ4% perturbation in kc at time 0.
remains to be 1947:1 s1 . One further observes in Figs. 7, 9, 11 that a determined by this method quickly tracks the subcriticality changes even when the multiple perturbations in hni and kc are simultaneously inserted. In order to discuss this virtue of the Simmons-King method, let us re-write Eq. (64) as
P ðt Þ ¼
X
~ i ðttj Þ a
Hi;j e
;
8 9 2 3 > > < = DE hni X 6 7 ~ i ÞðshÞ ~ ðai0 a Hi;j ¼ kd n Xn!n;i Da~ i s 41 1 DE li;i0 e eai ðjþ1Þs 5; > > : ; n i0 ¼p;d
i ¼ p; d;
li;i0
tj 1 dj;1 6 t < tjþ1 ;
i¼p;d
0 1 Dai0 s Xn!c;i0 Xc!n;i A @ ; ¼ Xn!n;i0 1þ Da s Xn!n;i0 X i n!n;i
ð68Þ
i ¼ p; d;
0
i ¼ p; d:
ð67Þ
ð69Þ
where Hi;j is the amplitude factor of the prompt and the delayed components that is defined as
In the pulsed neutron source methods, the repetition period s is usually set to be enough (at least several times) larger than
j ¼ 1; 0; . . . ; þ1;
10
Y. Kitamura, T. Misawa / Annals of Nuclear Energy 138 (2020) 107209
Fig. 9. Determination of a by Simmons-King method in case of þ1% perturbation in kc at time 0.
Fig. 10. Determination of qsub by Sjöstrand method in case of þ1% perturbation in kc at time 0.
a1 and much smaller than a1 p d . Hence, Eq. (67) is approximated as
PðtÞ ’ Hp;j eap ðttj Þ þ Hd;j ; ~
tj 1 dj;1 6 t < tjþ1 ;
j ¼ 1; 0; . . . ; þ1:
ð70Þ
One immediately finds that this equation is analogous to Eq. (65).
The Simons-King method, hence, properly determines ap even just after the perturbations. On the other hand, in the Sjöstrand method, as shown in Figs. 6, 8, 10, 12 and 14, calibration-free determination of qsub is achieved as expected, although it needs a long delay due to an asymptotic behaviour. Furthermore, one finds that the prompt response of
qsub determined by this method shows a dependency on the magnitude of the perturbation in hni. It is considered that this latter feature is rather useful because this would enable quick detection of the perturbation in neutron yield of the pulsed neutron source in addition to calibration-free determination of qsub . In order to discuss these results of the Sjöstrand method, let us calculate the area-ratio ðARÞj as R tjþ1 ðARÞj ¼ R with
tj tjþ1 tj
dt Hp;j eap ðttj Þ
dt Hd;j ead ðttj Þ
Xn!n;p
¼
ap
q
T j ’ T j; b Xn!n;d
ad
ð71Þ
11
Y. Kitamura, T. Misawa / Annals of Nuclear Energy 138 (2020) 107209
Fig. 11. Determination of a by Simmons-King method in case of 1% perturbation in kc at time 0.
Fig. 12. Determination of qsub by Sjöstrand method in case of 1% perturbation in kc at time 0.
8 9 < = X ~ 1 1 hni lp;i0 eðai0 ap ÞðshÞ ea~p ðjþ1Þs : ; n 0 i ¼p;d 8 9 Tj¼ : < = X ~ d ÞðshÞ hni ðai0 a ~ d ðjþ1Þs a 1 1 ld;i0 e e : ; n 0
ð72Þ
i ¼p;d
Here, we have to note that k K b and k K q were supposed in deriving the final form of Eq. (71). The coloured lines plotted in Figs. 6, 8, 10, 12 were calculated from Eq. (71). DE One immediately finds that the ratio of hni to n is explicitly included in T j . Hence, qsub determined by the Sjöstrand method shows a dependency on the magnitude of the perturbation in neu-
tron yield of the pulsed neutron source. Owing to ap ad , one sees that the numerator of T j quickly converges to unity as j is getting larger, while the denominator slowly converges to unity. Hence, the long-term response appears in the Sjöstrand method and its
behaviour is dominated by the smaller time constant ad . Let us further consider the cases when the multiplying medium is not perturbed but just the neutron yield of the pulsed neutron source is perturbed. When one removes the tildes on various parameters except for n, one calculates the parameter li;j as
(
li;i0 ¼
1; 0;
0
i ¼i ; 0 i –i
i ¼ p; d;
0
i ¼ p; d:
The area-ratio ðARÞj is hence written as follows:
ð73Þ
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Y. Kitamura, T. Misawa / Annals of Nuclear Energy 138 (2020) 107209
Fig. 13. Determination of a by Simmons-King method in case of 4% perturbation in kc at time 0.
Fig. 14. Determination of qsub by Sjöstrand method in case of 4% perturbation in kc at time 0.
q
ðARÞj ’ T j ; b
ð74Þ
with
1 1 hni eap ðjþ1Þs n Tj¼ : h 1 1 ni ead ðjþ1Þs
ð75Þ
One sees that the prompt response of the Sjöstrand method, hence, shows a dependency on the magnitude of the perturbation in the neutron yield of the pulsed neutron source even when the multiplying medium is not perturbed.
5. Conclusion
n
When one substitute j ¼ 0, the prompt response of the area-ratio ðARÞj becomes
D E 1 1 hni eap s n n q q q : ðARÞ0 ¼ T 0 ¼ ’ b b b hni 1 1 hni ead s n
ð76Þ
In the present study, a theory of the pulsed neutron source methods was developed by explicitly considering perturbations in various parameters of the subcritical reactor system driven by the pulsed neutron source. By using the theory thus developed, the time-response of the Simmons-King and the Sjöstrand (or the area-ratio) methods after the perturbations was examined. As a result, it was confirmed that the prompt neutron decay constant determined by the Simmons-King method shows a good trackability
Y. Kitamura, T. Misawa / Annals of Nuclear Energy 138 (2020) 107209
to the subcriticality changes. In the Sjöstrand method, as expected, it was observed that calibration-free determination of the subcriticality is achieved, although it needs a long delay due to an asymptotic behaviour. Furthermore, it was found that the latter method can quickly detect the changes in neutron yield of the pulsed neutron source. Therefore, we conclude that a strong on-line monitoring tool for the subcritical reactor system driven by the pulsed neutron source can be developed by combining these two methods. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at https://doi.org/10.1016/j.anucene.2019. 107209. References Aizawa, O., Yamamuro, N., 1967. J. Nucl. Sci. Technol. 4, 623. Aizawa, O., 1969. J. Nucl. Sci. Technol. 6, 498. Akino, F., Yasuda, H., Kaneko, Y., 1980. J. Nucl. Sci. Technol. 17, 593. Amano, F., 1969. J. Nucl. Sci. Technol. 6, 689. Bécares, V., Villamarín, D., Fernández-Ordóñez, M., González-Romero, E.M., Berglöf, C., Bournos, V., Fokov, Y., Mazanik, S., Serafimovich, I., 2013. Ann. Nucl. Energy 53, 40. Bécares, V., Villamarín, D., Fernández-Ordóñez, M., González-Romero, E.M., Berglöf, C., Bournos, V., Fokov, Y., Mazanik, S., Serafimovich, I., 2013. Ann. Nucl. Energy 53, 331. Beckurts, K.H., 1961. Nucl. Instr. Meth. 11, 144. Berglöf, C., Fernández-Ordóñez, M., Villamarín, D., Bécares, V., González-Romero, E. M., Bournos, V., Serafimovich, I., Mazanik, S., Fokov, Y., 2010. Nucl. Sci. Eng. 166, 134. Billebaud, A., Brissot, R., Le Brun, C., Liatard, E., Vollaire, J., 2007. Prog. Nucl. Energy 49, 142. Difilippo, F.C., Pieroni, N.B., Viez, J.C., 1973. Nucl. Sci. Eng. 51, 262. Dragt, J.B., 1973. Nucl. Sci. Eng. 50, 216. Duderstadt, J.J., Hamilton, L.J., 1976. Nuclear Reactor Analysis. John Wiley & Sons Inc, New York.
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Fultz, S.C., 1959. Nucl. Sci. Eng. 6, 313. Garelis, E., Russell Jr., J.L., 1963. Nucl. Sci. Eng. 16, 263. Gozani, T., 1962. Nukleonik 4, 348. Iwamoto, H., Nishihara, K., Yagi, T., Pyeon, C.-H., 2017. J. Nucl. Sci. Technol. 54, 432. Jammes, C., Geslot, B., Rosa, R., Imel, G., Fougeras, P., 2005. Ann. Nucl. Energy 32, 1131. Kaneko, Y., Sumita, K., 1967. J. Nucl. Sci. Technol. 4, 400. Kitamura, Y., Fukushima, M., 2014. J. Nucl. Sci. Technol. 51, 752. Kitamura, Y., Fukushima, M., 2015. J. Nucl. Sci. Technol. 52, 801. Kitamura, Y., Misawa, T., 2017. Ann. Nucl. Energy 103, 470. Kosály, G., Valkó, J., 1971. J. Nucl. Energy 25, 297. Kosály, G., Fischer, A., 1972. J. Nucl. Energy 26, 17. Kosály, G., Valkó, J., Fischer, A., 1974. Atomkernenergie 23, 251. Kosály, G., Valkó, J., 1975. Ann. Nucl. Energy 2, 477. Krieger, T.J., Zweifel, P.F., 1959. Nucl. Sci. Eng. 5, 21. Kulik, V.V., Lee, J.C., 2006. Nucl. Sci. Eng. 153, 69. Lebrat, J.F., Aliberti, G., D’Angelo, A., Billebaud, A., Brissot, R., Brockmann, H., Carta, M., Destouches, C., Gabrielli, F., Gonzalez, E., Hogenbirk, A., Klein-Meulenkamp, R., Le Brun, C., Liatard, E., Mellier, F., Messaoudi, N., Peluso, V., Plaschy, M., Thomas, M., Villamarín, D., Vollaire, J., 2008. Nucl. Sci. Eng. 158, 49. Persson, C.-M., Seltborg, P., Åhlander, A., Gudowski, W., Stummer, T., Kiyavitskaya, H., Bournos, V., Fokov, Y., Serafimovich, I., Chigrinov, S., 2005. Nucl. Instr. Meth. A 554, 374. Persson, C.-M., Fokau, A., Serafimovich, I., Bournos, V., Fokov, Y., Routkovskaia, C., Kiyavitskaya, H., Gudowski, W., 2008. Ann. Nucl. Energy 35, 2357. Pyeon, C.H., Hervault, M., Misawa, T., Unesaki, H., Iwasaki, T., Shiroya, S., 2008. J. Nucl. Sci. Technol. 45, 1171. Pyeon, C.H., Misawa, T., Lim, J.-Y., Unesaki, H., Ishi, Y., Kuriyama, Y., Uesugi, T., Mori, Y., Inoue, M., Nakajima, K., Mishima, K., Shiroya, S., 2009. J. Nucl. Sci. Technol. 46, 1091. Pyeon, C.H., Yagi, T., Sukawa, K., Yamaguchi, Y., Misawa, T., 2014. Nucl. Sci. Eng. 177, 156. Pyeon, C.H., Yagi, T., Takahashi, Y., Sano, T., Ito, D., Saito, Y., Nakajima, K., 2015. Prog. Nucl. Energy 82, 22. Pyeon, C.H., Yamanaka, M., Endo, T., van Rooijen, W.F.G., Chiba, G., 2017. Ann. Nucl. Energy 105, 346. Simmons, B.E., King, J.S., 1958. Nucl. Sci. Eng. 3, 598. Sjöstrand, N.G., 1956. Ark. Fys. 11, 233–246. Soule, R., Assal, W., Chaussonnet, P., Destouches, C., Domergue, C., Jammes, C., Laurens, J.-M., Lebrat, J.-F., Mellier, F., Perret, G., Rimpault, G., Servière, H., Imel, G., Thomas, G.M., Villamarin, D., Gonzalez-Romero, E., Plaschy, M., Chawla, R., Kloosterman, J.L., Rugama, Y., Billebaud, A., Brissot, R., Heuer, D., Kerveno, M., Le Brun, C., Liatard, E., Loiseaux, J.-M., Méplan, O., Merle, E., Perdu, F., Vollaire, J., Baeten, P., 2004. Nucl. Sci. Eng. 148, 124. Sumita, K., Kaneko, Y., 1967. J. Nucl. Sci. Technol. 4, 491. Uyttenhove, W., Lathouwers, D., Kloosterman, J.-L., van der Hagen, T.H.J.J., Van den Eynde, G., 2014. Ann. Nucl. Energy 72, 286. Yamanaka, M., Pyeon, C.H., Yagi, T., Misawa, T., 2016. Nucl. Sci. Eng. 183, 96.
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Annals of Nuclear Energy journal homepage: www.elsevier.com/locate/anucene
Detection of subcriticality changes by Simmons-King and Sjöstrand methods Yasunori Kitamura ⇑, Tsuyoshi Misawa Institute for Integrated Radiation and Nuclear Science, Kyoto University, Asashiro-Nishi, Kumatori-cho, Sennan-gun, Osaka 590-0494, Japan
a r t i c l e
i n f o
Article history: Received 18 June 2019 Received in revised form 8 October 2019 Accepted 11 November 2019
Keywords: Subcriticality change Time-response Simmons-King method Sjöstrand (or Area-ratio) method Prompt neutron decay constant Pulsed neutron source
a b s t r a c t The Simmons-King and the Sjöstrand (or the area-ratio) methods have been widely applied in measurement of the subcriticality of reactor systems driven by the pulsed neutron source and operated in the stationary state. In the present study, a theory-based investigation is conducted to examine the timeresponse of these two methods after perturbations in various parameters. As a result, the prompt neutron decay constant determined by the Simmons-King method shows a good trackability to the subcriticality. In the Sjöstrand method, calibration-free determination of the subcriticality is achieved, although it needs a long delay due to an asymptotic behaviour. Furthermore, it is found that the latter method can quickly detect the perturbation in neutron yield of the pulsed neutron source. Therefore, it is expected that a strong on-line monitoring tool for the subcritical reactor system driven by the pulsed neutron source can be developed by combining these two methods. Ó 2019 Elsevier Ltd. All rights reserved.
1. Introduction The subcriticality (i.e., the negative reactivity) is often discussed in the field of reactor physics, since it is one of the most fundamental quantities for the nuclear criticality safety of subcritical reactor systems. The Simmons-King and the Sjöstrand (or the area-ratio) methods, which are collectively referred to as the pulsed neutron source ones, have been widely applied in measurement of the subcriticality (Simmons and King, 1958; Sjöstrand, 1956). In these methods, pulsed neutrons generated by periodic bursts of the pulsed neutron source are injected into the subcritical reactor system. The neutron counting rate as a function of time after injection of pulsed neutrons is observed under the condition that a stationary state is maintained by periodic bursts of the pulsed neutron source. Owing to a great amount of efforts devoted by many researchers until the 1980s, the pulsed neutron source methods have been regarded as the most reliable and established ones for measuring the subcriticality (Krieger and Zweifel, 1959; Fultz, 1959; Beckurts, 1961; Gozani, 1962; Garelis and Russell, 1963; Aizawa and Yamamuro, 1967; Kaneko and Sumita, 1967; Sumita and Kaneko, 1967; Aizawa, 1969; Amano, 1969; Kosály and Valkó, 1971; Kosály and Fischer, 1972; Dragt, 1973; Difilippo
⇑ Corresponding author. E-mail address: [email protected] (Y. Kitamura). https://doi.org/10.1016/j.anucene.2019.107209 0306-4549/Ó 2019 Elsevier Ltd. All rights reserved.
et al., 1973; Kosály et al., 1974; Kosály and Valkó, 1975; Akino et al., 1980). Recently, a renewed necessity of the pulsed neutron source methods has been presented. During the last two decades, hence, the studies related to the pulsed neutron source methods have been activated again (Soule et al., 2004; Jammes et al., 2005; Persson et al., 2005; Kulik and Lee, 2006; Billebaud et al., 2007; Lebrat et al., 2008; Persson et al., 2008; Pyeon et al., 2008, Pyeon et al., 2009; Berglöf et al., 2010; Bécares et al., 2013, Bécares et al., 2013; Uyttenhove et al., 2014; Kitamura and Fukushima, 2014; Kitamura and Fukushima, 2015; Pyeon et al., 2014; Pyeon et al., 2015; Yamanaka et al., 2016; Pyeon et al., 2017; Iwamoto et al., 2017). Such a revival of the pulsed neutron source methods is mainly due to the accelerator-driven system (ADS). One application of the pulsed neutron source methods newly presented is development of the on-line subcriticality monitor that is desired to be equipped with the ADS to ensure and enhance the nuclear criticality safety (Iwamoto et al., 2017). It is expected that the pulsed neutron source methods will play an important role in such an application range. Among two methods of the pulsed neutron source ones, the Sjöstrand one has especially been attracted owing to the ability of calibration-free determination of the subcriticality (Sjöstrand, 1956). In order to realize the on-line subcriticality monitor, it is essential to confirm the time-response of the pulsed neutron source methods after perturbations in various parameters of the
2
Y. Kitamura, T. Misawa / Annals of Nuclear Energy 138 (2020) 107209
Nomenclature Parameters Before Perturbations kc probability that one neutron undergoes capture reaction per unit time. kd probability that one neutron undergoes detection reaction per unit time. kf probability that one neutron undergoes fission reaction per unit time. ka ¼ kc þ kd þ kf probability that one neutron undergoes one of absorption reactions per unit time. K ¼ kf 1hmi neutron generation time. a q ¼ kf khmf hik reactivity. mi bq a¼ K prompt neutron decay constant.
pS ðqÞ
probability that q neutrons are injected in one neutron burst. P hni ¼ þ1 q¼0 qpS ðqÞ average number of neutrons injected per neutron burst. Gn!n ðt Þ population of progeny neutron at time t originating from one ancestor neutron at time 0. Gn!c ðt Þ population of progeny delayed neutron precursor at time t originating from one ancestor neutron at time 0. Gc!n ðt Þ population of progeny neutron at time t originating from one ancestor delayed neutron precursor at time 0. Gc!c ðt Þ population of progeny delayed neutron precursor at time t originating from one ancestor delayed neutron precursor at time 0. ap larger time constant included in general solution of one-point reactor kinetic equation with one delayed neutron group. ad smaller time constant included in general solution of one-point reactor kinetic equation with one delayed neutron group. Parameters
after perturbations probability that one neutron undergoes capture reaction per unit time. kd probability that one neutron undergoes detection reaction per unit time. kf probability that one neutron undergoes fission reac tion per unit time. ka ¼ kc þ kd þ kf probability that one neutron undergoes one of absorption reactions per unit time. K ¼ 1 neutron generation time. kc
q ¼ kf hmika reactivity.
kf hmi
a ¼ bq
K
prompt neutron decay constant.
probability that q neutrons are injected in one neutron pS ðqÞ DE P burst. n ¼ þ1 q¼0 qpS ðqÞ average number of neutrons injected per neutron burst. Gn!n ðt Þ population of progeny neutron at time t originating from one ancestor neutron at time 0. Gn!c ðt Þ population of progeny delayed neutron precursor at time t originating from one ancestor neutron at time 0. Gc!n ðt Þ population of progeny neutron at time t originating from one ancestor delayed neutron precursor at time 0. Gc!c ðt Þ population of progeny delayed neutron precursor at time t originating from one ancestor delayed neutron precursor at time 0. ap larger time constant included in general solution of one-point reactor kinetic equation with one delayed neutron group. ad smaller time constant included in general solution of one-point reactor kinetic equation with one delayed neutron group. Other parameters k disintegration time constant of delayed neutron precursor. probability that n prompt neutrons and c delayed neupf ðn; cÞ tron precursors are simultaneously born in one fission reaction. mp average number of prompt neutrons per fission reaction. hmd i average number of delayed neutrons per fission reaction. average number of total neutrons per fission reaction. hmi b delayed neutron fraction. s periodicity of neutron burst. h time of first neutron burst after perturbations. Nðt Þ population of neutron at time t. C ðt Þ population of delayed neutron precursor at time t. Pðt Þ neutron counting rate at time t.
kf hmi
subcritical reactor system driven by the pulsed neutron source. However, as mentioned above, the pulsed neutron source methods are usually applied to the subcritical reactor system operated in the stationary state. Therefore, no theoretical study that focuses on the time-response after perturbations has not been carried out yet, as far as the authors know. In the present study, hence, a theory-based investigation for examining the time-response of the pulsed neutron source methods after perturbations is conducted as a part of the study for realizing the on-line subcriticality monitor for the ADS. In the following section, the notations employed in the present study are introduced. The theory of the pulsed neutron source methods that explicitly considers perturbations in various parameters of the subcritical reactor system driven by the pulsed neutron source is developed in Section 3. By using the theory thus developed, the time-response of the Simmons-King and the Sjöstrand methods after perturbations is discussed in Section 4. Finally, the conclusion is summarized in Section 5.
2. Notations 2.1. Parameters constant in time In the present study, a zero-power subcritical reactor system that consists of a multiplying medium, a pulsed neutron source, and a neutron detector is supposed. Delayed neutrons are rigorously taken into consideration, while they are assumed to be emitted from their precursors having a single disintegration time constant k, for simple discussions. Furthermore, formulation is thoroughly performed within a mono-energy one-point reactor model. The probability that n prompt neutrons and c delayed neutron precursors are simultaneously born in one fission reaction is denoted by pf ðn; cÞ with þ1 X þ1 X pf ðn; cÞ ¼ 1: n¼0 c¼0
ð1Þ
3
Y. Kitamura, T. Misawa / Annals of Nuclear Energy 138 (2020) 107209
By using this probability, the average numbers of prompt, delayed, and total neutrons per fission reaction are defined as
mp ¼
þ1 X þ1 X
npf ðn; cÞ;
ð2Þ
cpf ðn; cÞ;
ð3Þ
n¼0 c¼0
hmd i ¼
þ1 X þ1 X n¼0 c¼0
hmi ¼
þ1 X þ1 X
ðn þ cÞpf ðn; cÞ ¼
mp þ hmd i:
ð4Þ
n¼0 c¼0
With the quantities defined above, the delayed neutron fraction b of the subcritical reactor system is naturally introduced as follows:
b¼
hmd i : hmi
ð5Þ
In the present study, it is supposed that various parameters of the subcritical reactor system may be perturbed at time 0 to discuss the time-response of the Simmons-King and the Sjöstrand methods after the perturbations. We note here that the parameters with tilde are thoroughly used as those after the perturbations. By using the unit step function U ðt Þ, the probabilities with respect to the respective neutron interactions in the subcritical reactor system (i.e., capture, detection, and fission reactions) are written as
kx ðtÞ ¼ kx f1 U ðt Þg þ kx U ðt Þ;
1 ¼ K f1 U ðtÞg þ K U ðtÞ; kf ðt Þ hmi
ð8Þ
qðtÞ ¼
kf ðtÞ hmi ka ðt Þ ¼ q f1 U ðt Þg þ q U ðtÞ; kf ðt Þ hmi
ð9Þ
aðtÞ ¼
b qðtÞ ¼ a f1 U ðt Þg þ a U ðtÞ: Kð t Þ
x ¼ c; d; f;
ka ðtÞ ¼ kc ðtÞ þ kd ðt Þ þ kf ðt Þ ¼ ka f1 U ðtÞg þ ka U ðtÞ;
ð6Þ ð7Þ
where kc ðt Þ is the probability per unit time that one neutron undergoes a capture reaction in the multiplying medium at time t; kd ðtÞ the probability per unit time that one neutron undergoes a detection reaction in the neutron detector at time t; kf ðt Þ the probability per unit time that one neutron undergoes a fission reaction in the multiplying medium at time t, and ka ðtÞ the probability per unit time that one neutron undergoes one of the absorption reactions in the subcritical reactor system at time t. By using the quantities defined above, the neutron generation time Kðt Þ, the reactivity qðt Þ, and the prompt neutron decay constant aðt Þ of the subcritical reactor system are introduced as follows:
ð10Þ
The subcritical reactor system is supposed to be driven by periodic bursts of the pulsed neutron source, as illustrated in Fig. 1. When one denotes its periodicity by s and the time of the first neutron burst after the perturbations by h, one expresses the time of each neutron burst as follows:
tj ¼ js þ h;
j ¼ 1; . . . ; 1; 0; þ1; . . . ; þ1;
ð11Þ
with the following condition:
0 6 h < s:
2.2. Parameters vary in time
Kðt Þ ¼
ð12Þ
The probability that q neutrons are injected in one neutron burst is denoted by
pS ðq; t Þ ¼ pS ðqÞ f1 U ðt Þg þ pS ðqÞ U ðtÞ;
ð13Þ
with þ1 X pS ðqÞ ¼ 1;
ð14Þ
q¼0 þ1 X pS ðqÞ ¼ 1:
ð15Þ
q¼0
Using this probability, the average number of pulsed neutrons per neutron burst is defined as
hnðt Þi ¼
þ1 X
DE qpS ðq; t Þ ¼ hni f1 U ðt Þg þ n U ðt Þ:
ð16Þ
q¼0
3. Theory 3.1. Outline In this section, by using a formulation technique employed in our previous study (Kitamura and Misawa, 2017), the population of neutron N ðt Þ, that of delayed neutron precursor C ðt Þ, and the
Fig. 1. Timing diagram of neutron bursts and perturbations.
4
Y. Kitamura, T. Misawa / Annals of Nuclear Energy 138 (2020) 107209
neutron counting rate P ðt Þ at time t are derived. As illustrated in Fig. 2, one derives N ðtÞ as follows:
N ðt Þ ¼
j X
Nk ðtÞ;
tj 6 t < t jþ1 ;
ð17Þ
k¼1
where N k ðtÞ is the population of progeny neutron at time t originating from the neutron burst at time tk (6 t). One similarly derives C ðt Þ as
C ðt Þ ¼
j X
C k ðt Þ;
t j 6 t < tjþ1 ;
ð18Þ
k¼1
where C k ðt Þ is the population of progeny delayed neutron precursor at time t originating from the neutron burst at time t k (6 t). On the other hand, P ðt Þ is simply written as
PðtÞ ¼ kd ðtÞ NðtÞ:
ð19Þ
In the following two subsections, N ðtÞ; C ðt Þ, and P ðt Þ before and after the perturbations are separately derived.
In the present subsection, N ðtÞ; C ðtÞ, and P ðt Þ with respect to t < 0, i.e., before the perturbations, are derived. In order to describe the temporal evolution of population of neutron or delayed neutron precursor in the subcritical reactor system before the perturbations, the function Gy!z ðtÞ, which is derived for the multiplying medium where injection of pulsed neutrons, the neutron and precursor populations are absent, is introduced. With this function, one calculates the population of progeny particle z (¼ n; c) at time t originating from one ancestor particle y (¼ n; c) at time 0. We note here that ‘‘n” and ‘‘c” correspond to the neutron and the delayed neutron precursor, respectively. The function Gy!z ðt Þ is derived from the one-point reactor kinetic equation with one delayed neutron group for the unperturbed subcritical reactor system, which is written down as follows (Duderstadt and Hamilton, 1976):
dC ðtÞ b ¼ kC ðt Þ þ Nðt Þ; dt K
dGn!n ðtÞ ¼ aGn!n ðt Þ þ kGn!c ðt Þ þ dðt Þ; dt
ð22Þ
dGn!c ðtÞ b ¼ kGn!c ðt Þ þ Gn!n ðt Þ: dt K
ð23Þ
On the other hand, by replacing Nðt Þ with Gc!n ðtÞ and C ðt Þ with Gc!c ðt Þ then adding the Dirac’s delta dðt Þ to the right-hand side of Eq. (21), one obtains
dGc!n ðtÞ ¼ aGc!n ðt Þ þ kGc!c ðt Þ; dt
ð24Þ
dGc!c ðtÞ b ¼ kGc!c ðtÞ þ Gc!n ðtÞ þ dðtÞ: dt K
ð25Þ
Since the initial condition of Gy!z ðt Þ is read as
Gy!z ð0Þ ¼ 0;
3.2. Before perturbations
dNðt Þ ¼ aNðt Þ þ kC ðt Þ; dt
where Nðt Þ and C ðtÞ are the populations of neutron and delayed neutron precursor at time t, respectively. By replacing N ðt Þ with Gn!n ðtÞ and C ðtÞ with Gn!c ðtÞ then adding the Dirac’s delta dðtÞ to the right-hand side of Eq. (20), one obtains
z ¼ n; c;
ð26Þ
their Laplace transforms are calculated as
8 sþk ; > H ðsÞ > > > > b=K < ; H ðsÞ L Gy!z ðt Þ ¼ k > > H ðsÞ ; > > > : sþa ; H ðsÞ
y ¼ n;
z¼n
y ¼ n;
z¼c
y ¼ c;
z¼n
y ¼ c;
z¼c
;
ð27Þ
with
HðsÞ ¼ s2 þ ða þ kÞs and
ap ¼
ð20Þ
ad ¼ ð21Þ
y ¼ n; c;
ða þ kÞK þ
kq Y ¼ ðs þ ai Þ; K i¼p;d
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ða þ kÞ2 K2 þ 4kKq 2K
ða þ kÞK
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ða þ kÞ2 K2 þ 4kKq 2K
ð28Þ
;
ð29Þ
:
ð30Þ
One can re-writes Eq. (27) as
Fig. 2. Schematic illustration for deriving neutron population N ðt Þ and delayed neutron precursor population C ðt Þ.
5
Y. Kitamura, T. Misawa / Annals of Nuclear Energy 138 (2020) 107209
X Xy!z;i L Gy!z ðt Þ ¼ ; s þ ai i¼p;d with
Xy!z;i ¼
8 ka i ; > > > ai ai > > > b=K > < a ai ;
y ¼ n; c;
z ¼ n; c;
y ¼ n;
z¼n
y ¼ n;
z¼c
y ¼ c;
z¼n
y ¼ c;
z¼c
i
k > > ai ai ; > > > > a k > : i ;
ai ai
ð31Þ
In the present subsection, N ðtÞ; C ðt Þ, and P ðt Þ with respect to 0 6 t, i.e., after the perturbations, are derived. In order to describe the temporal evolution of population of neutron or delayed neutron precursor in the subcritical reactor sys
;
i ¼ p; d;
i – i:
ð32Þ
Gy!z ðtÞ ¼
Xy!z;i eai t ;
y ¼ n; c;
z ¼ n; c:
ð33Þ
i¼p;d
When q neutrons are injected in the neutron burst at time t k (< 0), the populations of progeny neutron and delayed neutron precursor at time t (< 0) originating from these q neutrons are written as follows:
qGn!n ðt tk Þ;
t k 6 t < 0;
ð34Þ
qGn!c ðt tk Þ;
t k 6 t < 0:
ð35Þ
Since q can take various integer values ranging from 0 to þ1 according to the probability pS ðqÞ, by multiplying Eqs. (34) and (35) by pS ðqÞ and then summing over q, one obtains N k ðtÞ and C k ðt Þ with respect to t k 6 t < 0, which are the expected neutron and delayed neutron precursor populations at time t generated by the kth neutron burst, as
Nk ðt Þ ¼
þ1 X
qpS ðqÞGn!n ðt t k Þ ¼ hniGn!n ðt t k Þ;
dNðtÞ ¼ a Nðt Þ þ kC ðt Þ; dt
ð42Þ
dC ðt Þ b ¼ kC ðt Þ þ Nðt Þ: dt K
ð43Þ
As seen in Section 3.2, one derives Gy!z ðt Þ with respect to y ¼ n; c and z ¼ n; c as follows:
Gy!z ðt Þ ¼
qpS ðqÞGn!c ðt t k Þ ¼ hniGn!c ðt tk Þ;
X
Xy!z;i ea i t ;
y ¼ n; c;
z ¼ n; c;
with
Xy!z;i ¼
t k 6 t < 0:
8 kai > ; > > > ai ai > > > > > b= K > < ; ai ai
k > > ; > a a > > > i i > > a k > > :i ;
ai ai
q¼0
y ¼ n;
z¼n
y ¼ n;
z¼c
y ¼ c;
z¼n
y ¼ c;
z¼c
;
i ¼ p; d;
j P
N ðt Þ ¼ hni
ap ¼
Gn!n ðt tk Þ
P
i¼p;d
ð38Þ
Xn!n;i Dai s eai ðttj Þ ;
tj 6 t < t jþ1 1 dj;1 ; C ðtÞ ¼ hni
j P
j ¼ 1; . . . ; 2; 1;
P
i¼p;d
Gn!c ðt tk Þ ð39Þ
Xn!c;i Dai s eai ðttj Þ ;
t j 6 t < tjþ1 1 dj;1 ;
j ¼ 1; . . . ; 2; 1;
where
Dc ¼
1 : 1 e c
ð40Þ
Therefore, from Eqs. (19) and (38), one gets PðtÞ before the perturbations as
PðtÞ ¼ kd hni
P i¼p;d
ai ðtt j Þ
Xn!n;i Dai s e
j ¼ 1; . . . ; 2; 1:
2K
;
t j 6 t < tjþ1
1 dj;1 ;
ad ¼
a þk K
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 a þk K2 þ 4k K q
2K
;
ð46Þ
:
ð47Þ
When q neutrons are injected in the neutron burst at time tk (< 0), i.e., before the perturbations, the populations of progeny neutron and delayed neutron precursor at time 0 originating from these q neutrons are written as follows:
k¼1
¼ hni
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 a þk K þ a þk K2 þ 4k K q
k¼1
¼ hni
i – i;
ð45Þ
ð37Þ By substituting Eqs. (36) and (37) into Eqs. (17) and (18), respectively, the concrete expressions of Nðt Þ and C ðt Þ before the perturbations are obtained as
ð44Þ
i¼p;d
ð36Þ þ1 X
The function Gy!z ðtÞ is derived from the one-point reactor kinetic equation with one delayed neutron group for the perturbed subcritical reactor system, which is written down as follows (Duderstadt and Hamilton, 1976):
tk 6 t < 0;
q¼0
C k ðt Þ ¼
tem after the perturbations, the function Gy!z ðt Þ, which is derived for the multiplying medium where injection of pulsed neutrons, the neutron and precursor populations are absent, is introduced. With this function, one calculates the population of progeny particle z (¼ n; c) at time t originating from one ancestor particle y (¼ n; c) at time 0.
One hence gets
X
3.3. After perturbations
ð41Þ
qGn!n ðt k Þ;
tk < 0;
ð48Þ
qGn!c ðt k Þ;
tk < 0:
ð49Þ
Hence, the populations of progeny neutron and delayed neutron precursor at time t ( 0), i.e., after the perturbations, originating from q neutrons generated at time t k (< 0), i.e., before the perturbations, are respectively expressed as follows:
n o q Gn!n ðtk ÞGn!n ðt Þ þ Gn!c ðt k ÞGc!n ðtÞ ;
t k < 0 6 t;
ð50Þ
n o q Gn!n ðtk ÞGn!c ðt Þ þ Gn!c ðt k ÞGc!c ðtÞ ;
t k < 0 6 t:
ð51Þ
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Y. Kitamura, T. Misawa / Annals of Nuclear Energy 138 (2020) 107209
Since q can take integer values ranging from 0 to þ1 according to the probability pS ðqÞ, by multiplying Eqs. (50) and (51) by pS ðqÞ and then summing over q, one obtains N k ðt Þ and C k ðt Þ with respect to t k < 0 6 t as N k ðt Þ ¼
n o qpS ðqÞ Gn!n ðt k ÞGn!n ðt Þ þ Gn!c ðt k ÞGc!n ðt Þ
þ1 P
ð52Þ
q¼0
n o ¼ hni Gn!n ðtk ÞGn!n ðtÞ þ Gn!c ðtk ÞGc!n ðt Þ ; C k ðt Þ ¼
þ 1 P
t k < 0 6 t;
n o qpS ðqÞ Gn!n ðtk ÞGn!c ðt Þ þ Gn!c ðtk ÞGc!c ðtÞ
ð53Þ
q¼0
n o ¼ hni Gn!n ðt k ÞGn!c ðtÞ þ Gn!c ðt k ÞGc!c ðt Þ ;
t k < 0 6 t:
C 0 ¼ hni
1 X
qGn!n ðt t k Þ;
qGn!c ðt tk Þ;
0 6 tk 6 t;
ð54Þ
0 6 tk 6 t:
ð55Þ
Since q can take integer values ranging from 0 to þ1 according to
the probability pS ðqÞ, by multiplying Eqs. (54) and (55) by pS ðqÞ and then summing over q, one obtains N k ðt Þ and C k ðt Þ with respect to 0 6 tk 6 t as
Nk ðt Þ ¼
þ1 X
D E qpS ðqÞGn!n ðt t k Þ ¼ n Gn!n ðt t k Þ;
0 6 tk 6 t;
q¼0
ð56Þ C k ðt Þ ¼
þ1 X
DE qpS ðqÞGn!c ðt t k Þ ¼ n Gn!c ðt t k Þ;
0 6 t k 6 t:
q¼0
ð57Þ By substituting Eqs. (52), (53), (56) and (57) into Eqs. (17) and (18), respectively, the concrete expressions of N ðtÞ and C ðt Þ after the perturbations are obtained as o DE P j 1 n P Gn!n ðtk ÞGn!n ðtÞ þ Gn!c ðtk ÞGc!n ðtÞ þ n Gn!n ðt tk Þ
N ðtÞ ¼ hni
k¼1
DE P j ¼ N0 Gn!n ðtÞ þ C 0 Gc!n ðtÞ þ n Gn!n ðt tk Þ k¼0 P N 0 Xn!n;i þ C 0 Xc!n;i eai t ¼
k¼0
i¼p;d
DE P n o þ n Xn!n;i Da s 1 eai ðjþ1Þs eai ðttj Þ ;
i¼p;d
n o DE P j Gn!n ðtk ÞGn!c ðtÞ þ Gn!c ðtk ÞGc!c ðtÞ þ n Gn!c ðt t k Þ
k¼1
DE P j ¼ N0 Gn!c ðtÞ þ C 0 Gc!c ðtÞ þ n Gn!c ðt tk Þ k¼0 P N 0 Xn!c;i þ C 0 Xc!c;i eai t ¼
k¼0
i¼p;d
DE P n o þ n Xn!c;i Da s 1 eai ðjþ1Þs eai ðttj Þ ; i¼p;d
i
tj 1 dj;1 6 t < tjþ1 ;
j ¼ 1; 0; . . . ; þ1; ð59Þ
with
N0 ¼ hni
1 X k¼1
ð61Þ
One easily understands from Eqs. (58) and (59) that the physical meanings of N0 and C 0 are the populations of neutron and delayed neutron precursor at time 0, i.e.,
Nð0Þ ¼ N0 ;
ð62Þ
C ð0Þ ¼ C 0 :
ð63Þ
Finally, from Eqs. (19) and (58), one gets Pðt Þ after the perturbations as
Pðt Þ ¼ kd
P
N0 Xn!n;i þ C 0 Xc!n;i eai t
i¼p;d
n o DE P þkd n Xn!n;i Da s 1 eai ðjþ1Þs eai ðttj Þ ;
i¼p;d
i
tj 1 dj;1 6 t < tjþ1 ;
j ¼ 1; 0; . . . ; þ1: ð64Þ
4. Discussions In this section, by using the theoretical expressions of P ðt Þ given in Eqs. (41) and (64), the time-response of the Simmons-King and the Sjöstrand methods to the subcritical reactor system after the perturbations is discussed. The parameters before the perturbations are listed in Table 1. The prompt neutron decay constant a and the subcriticality in dollar unit qsub (¼ q=b) before the perturbations are 1947:1 s1 and 3:837 $, respectively. Because of the neutron life time in the 105 s order, one understands that the subcritical reactor system supposed is one of the light water systems. In the present study, we introduce two kinds of perturbations, i.e., those in hni and kc . The magnitude of the perturbation in hni is supposed to be from þ75% to 75%. On the other hand, that in kc is supposed to be from þ4% to 4%. We note here that the latter perturbation results in the changes in a from 3275:0 s1 to 619:3 s1 and those in qsub from 7:136 $ to 0:538 $, as listed in Table 2. Figs. 3 and 4 provide P ðt Þ normalized by hni before and after the perturbations at time 0 in cases that hni and kc are independently perturbed. In the Simmons-King and the Sjöstrand methods, by means of the least-square fitting procedure, P ðt Þ within t j 6 t < tjþ1 (j ¼ 1; . . . ; 1; 0; 1; . . . ; þ1) is fitted by the following equation:
j ¼ 1; 0; . . . ; þ1; ð58Þ
C ðtÞ ¼ hni
Xn!c;i Dai s eai ðshÞ :
i
tj 1 dj;1 6 t < tjþ1 ;
1 P
X i¼p;d
k¼1
On the other hand, in cases that q neutrons are injected in the neutron burst at time tk ( 0), i.e., after the perturbations, the populations of progeny neutron and delayed neutron precursor at time t (P tk ) originating from these q neutrons are written as follows:
Gn!c ðt k Þ ¼ hni
Gn!n ðt k Þ ¼ hni
X i¼p;d
Xn!n;i Dai s eai ðshÞ ;
ð60Þ
Table 1 Parameters of subcritical reactor system before perturbations. Parameter
:
Value
k [s1 ] mp [–] hmd i [–] b [–] kc [s1 ] kd [s1 ] kf [s1 ] K [s]
: :
0:077 2:4564375
: : : : : :
0:0185625 0:0075 33196:231 334:850 21685:749
q [%Dk=k] a [s1 ] ap [s1 ] ad [s1 ]
: : :
1:863 105 2:8778 1947:144 1947:160
: :
0:0611 arbitrary numbers
hni [–]
7
Y. Kitamura, T. Misawa / Annals of Nuclear Energy 138 (2020) 107209 Table 2 Key parameters of subcritical reactor system after perturbations.
Perturbation in kc
a [s1 ]
ap [s1 ]
ad [s1 ]
q [%Dk=k]
q =b [$]
þ4% þ1% 0% 1% 4%
3274:993 2279:106 1947:144 1615:182 619:295
3275:002 2279:120 1947:160 1615:201 619:345
0:0675 0:0634 0:0611 0:0578 0:0269
5:3518 3:4963 2:8778 2:2593 0:4038
7:13579 4:66179 3:83712 3:01246 0:53846
Fig. 3. Counting rates normalized by hni with and without perturbation in n at time 0.
Fig. 4. Counting rates normalized by hni with and without perturbation in kc at time 0.
PðtÞ / Aeaðttj Þ þ B;
tj 6 t < t jþ1 ;
j ¼ 1; . . . ; 1; 0; 1; . . . ; þ1;
ð65Þ
where A and B are the respective amplitude factors related to the prompt and the delayed neutrons. Among these three parameters thus determined, a is utilized for measuring the subcriticality in
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Y. Kitamura, T. Misawa / Annals of Nuclear Energy 138 (2020) 107209
the Simmons-King method (Simmons and King, 1958). On the other hand, in the Sjöstrand method (Sjöstrand, 1956), qsub is calculated from the following equation:
Rs
dt Aeat
qsub ¼ 0R s 0
dt B
¼
A 1 eas : B as
ð66Þ
Figs. 5 and 6 provide the results of subcriticality measurement by the Simmons-King and the Sjöstrand methods in cases of various magnitudes of the perturbation in hni with no perturbation in kc . In Fig. 5, a determined by Eq. (65) with the least-square fitting procedure was plotted as a function of tj after the perturbations. On the other hand, qsub calculated from Eq. (66) was
plotted as a function of tj in Fig. 6. Similarly, Figs. 7 and 8 provide the results in cases of various magnitudes of the perturbation in hni with þ4% perturbation in kc . Figs. 9 and 10 provide the results in cases of various magnitudes of the perturbation in hni with þ1% perturbation in kc . Figs. 11 and 12 provide the results in cases of various magnitudes of the perturbation in hni with 1% perturbation in kc . Figs. 13 and 14 provide the results in cases of various magnitudes of the perturbation in hni with 4% perturbation in kc . In these cases, the subcriticality changes are introduced depending on the magnitude of the perturbation in kc . One easily understands that a and qsub should be constant when no perturbation in kc is inserted. As expected, one observes in Fig. 5 that a determined by the Simmons-King method is unchanged and
Fig. 5. Determination of a by Simmons-King method in case of no perturbation in kc .
Fig. 6. Determination of qsub by Sjöstrand method in case of no perturbation in kc .
9
Y. Kitamura, T. Misawa / Annals of Nuclear Energy 138 (2020) 107209
Fig. 7. Determination of a by Simmons-King method in case of þ4% perturbation in kc at time 0.
Fig. 8. Determination of qsub by Sjöstrand method in case of þ4% perturbation in kc at time 0.
remains to be 1947:1 s1 . One further observes in Figs. 7, 9, 11 that a determined by this method quickly tracks the subcriticality changes even when the multiple perturbations in hni and kc are simultaneously inserted. In order to discuss this virtue of the Simmons-King method, let us re-write Eq. (64) as
P ðt Þ ¼
X
~ i ðttj Þ a
Hi;j e
;
8 9 2 3 > > < = DE hni X 6 7 ~ i ÞðshÞ ~ ðai0 a Hi;j ¼ kd n Xn!n;i Da~ i s 41 1 DE li;i0 e eai ðjþ1Þs 5; > > : ; n i0 ¼p;d
i ¼ p; d;
li;i0
tj 1 dj;1 6 t < tjþ1 ;
i¼p;d
0 1 Dai0 s Xn!c;i0 Xc!n;i A @ ; ¼ Xn!n;i0 1þ Da s Xn!n;i0 X i n!n;i
ð68Þ
i ¼ p; d;
0
i ¼ p; d:
ð67Þ
ð69Þ
where Hi;j is the amplitude factor of the prompt and the delayed components that is defined as
In the pulsed neutron source methods, the repetition period s is usually set to be enough (at least several times) larger than
j ¼ 1; 0; . . . ; þ1;
10
Y. Kitamura, T. Misawa / Annals of Nuclear Energy 138 (2020) 107209
Fig. 9. Determination of a by Simmons-King method in case of þ1% perturbation in kc at time 0.
Fig. 10. Determination of qsub by Sjöstrand method in case of þ1% perturbation in kc at time 0.
a1 and much smaller than a1 p d . Hence, Eq. (67) is approximated as
PðtÞ ’ Hp;j eap ðttj Þ þ Hd;j ; ~
tj 1 dj;1 6 t < tjþ1 ;
j ¼ 1; 0; . . . ; þ1:
ð70Þ
One immediately finds that this equation is analogous to Eq. (65).
The Simons-King method, hence, properly determines ap even just after the perturbations. On the other hand, in the Sjöstrand method, as shown in Figs. 6, 8, 10, 12 and 14, calibration-free determination of qsub is achieved as expected, although it needs a long delay due to an asymptotic behaviour. Furthermore, one finds that the prompt response of
qsub determined by this method shows a dependency on the magnitude of the perturbation in hni. It is considered that this latter feature is rather useful because this would enable quick detection of the perturbation in neutron yield of the pulsed neutron source in addition to calibration-free determination of qsub . In order to discuss these results of the Sjöstrand method, let us calculate the area-ratio ðARÞj as R tjþ1 ðARÞj ¼ R with
tj tjþ1 tj
dt Hp;j eap ðttj Þ
dt Hd;j ead ðttj Þ
Xn!n;p
¼
ap
q
T j ’ T j; b Xn!n;d
ad
ð71Þ
11
Y. Kitamura, T. Misawa / Annals of Nuclear Energy 138 (2020) 107209
Fig. 11. Determination of a by Simmons-King method in case of 1% perturbation in kc at time 0.
Fig. 12. Determination of qsub by Sjöstrand method in case of 1% perturbation in kc at time 0.
8 9 < = X ~ 1 1 hni lp;i0 eðai0 ap ÞðshÞ ea~p ðjþ1Þs : ; n 0 i ¼p;d 8 9 Tj¼ : < = X ~ d ÞðshÞ hni ðai0 a ~ d ðjþ1Þs a 1 1 ld;i0 e e : ; n 0
ð72Þ
i ¼p;d
Here, we have to note that k K b and k K q were supposed in deriving the final form of Eq. (71). The coloured lines plotted in Figs. 6, 8, 10, 12 were calculated from Eq. (71). DE One immediately finds that the ratio of hni to n is explicitly included in T j . Hence, qsub determined by the Sjöstrand method shows a dependency on the magnitude of the perturbation in neu-
tron yield of the pulsed neutron source. Owing to ap ad , one sees that the numerator of T j quickly converges to unity as j is getting larger, while the denominator slowly converges to unity. Hence, the long-term response appears in the Sjöstrand method and its
behaviour is dominated by the smaller time constant ad . Let us further consider the cases when the multiplying medium is not perturbed but just the neutron yield of the pulsed neutron source is perturbed. When one removes the tildes on various parameters except for n, one calculates the parameter li;j as
(
li;i0 ¼
1; 0;
0
i ¼i ; 0 i –i
i ¼ p; d;
0
i ¼ p; d:
The area-ratio ðARÞj is hence written as follows:
ð73Þ
12
Y. Kitamura, T. Misawa / Annals of Nuclear Energy 138 (2020) 107209
Fig. 13. Determination of a by Simmons-King method in case of 4% perturbation in kc at time 0.
Fig. 14. Determination of qsub by Sjöstrand method in case of 4% perturbation in kc at time 0.
q
ðARÞj ’ T j ; b
ð74Þ
with
1 1 hni eap ðjþ1Þs n Tj¼ : h 1 1 ni ead ðjþ1Þs
ð75Þ
One sees that the prompt response of the Sjöstrand method, hence, shows a dependency on the magnitude of the perturbation in the neutron yield of the pulsed neutron source even when the multiplying medium is not perturbed.
5. Conclusion
n
When one substitute j ¼ 0, the prompt response of the area-ratio ðARÞj becomes
D E 1 1 hni eap s n n q q q : ðARÞ0 ¼ T 0 ¼ ’ b b b hni 1 1 hni ead s n
ð76Þ
In the present study, a theory of the pulsed neutron source methods was developed by explicitly considering perturbations in various parameters of the subcritical reactor system driven by the pulsed neutron source. By using the theory thus developed, the time-response of the Simmons-King and the Sjöstrand (or the area-ratio) methods after the perturbations was examined. As a result, it was confirmed that the prompt neutron decay constant determined by the Simmons-King method shows a good trackability
Y. Kitamura, T. Misawa / Annals of Nuclear Energy 138 (2020) 107209
to the subcriticality changes. In the Sjöstrand method, as expected, it was observed that calibration-free determination of the subcriticality is achieved, although it needs a long delay due to an asymptotic behaviour. Furthermore, it was found that the latter method can quickly detect the changes in neutron yield of the pulsed neutron source. Therefore, we conclude that a strong on-line monitoring tool for the subcritical reactor system driven by the pulsed neutron source can be developed by combining these two methods. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at https://doi.org/10.1016/j.anucene.2019. 107209. References Aizawa, O., Yamamuro, N., 1967. J. Nucl. Sci. Technol. 4, 623. Aizawa, O., 1969. J. Nucl. Sci. Technol. 6, 498. Akino, F., Yasuda, H., Kaneko, Y., 1980. J. Nucl. Sci. Technol. 17, 593. Amano, F., 1969. J. Nucl. Sci. Technol. 6, 689. Bécares, V., Villamarín, D., Fernández-Ordóñez, M., González-Romero, E.M., Berglöf, C., Bournos, V., Fokov, Y., Mazanik, S., Serafimovich, I., 2013. Ann. Nucl. Energy 53, 40. Bécares, V., Villamarín, D., Fernández-Ordóñez, M., González-Romero, E.M., Berglöf, C., Bournos, V., Fokov, Y., Mazanik, S., Serafimovich, I., 2013. Ann. Nucl. Energy 53, 331. Beckurts, K.H., 1961. Nucl. Instr. Meth. 11, 144. Berglöf, C., Fernández-Ordóñez, M., Villamarín, D., Bécares, V., González-Romero, E. M., Bournos, V., Serafimovich, I., Mazanik, S., Fokov, Y., 2010. Nucl. Sci. Eng. 166, 134. Billebaud, A., Brissot, R., Le Brun, C., Liatard, E., Vollaire, J., 2007. Prog. Nucl. Energy 49, 142. Difilippo, F.C., Pieroni, N.B., Viez, J.C., 1973. Nucl. Sci. Eng. 51, 262. Dragt, J.B., 1973. Nucl. Sci. Eng. 50, 216. Duderstadt, J.J., Hamilton, L.J., 1976. Nuclear Reactor Analysis. John Wiley & Sons Inc, New York.
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