Inverse kinetics for subcritical systems with external neutron source

Annals of Nuclear Energy 108 (2017) 343–350

Contents lists available at ScienceDirect

Annals of Nuclear Energy journal homepage: www.elsevier.com/locate/anucene

Inverse kinetics for subcritical systems with external neutron source Wemerson de Carvalho Gonçalves ⇑, Aquilino Senra Martinez, Fernando Carvalho da Silva Nuclear Engineering Department, COPPE/UFRJ, Av. Horácio Macedo, 2030, Bloco G – Sala 206 – Centro de Tecnologia, Cidade Universitária, Ilha do Fundão, 21941-914 Rio de Janeiro, RJ, Brazil

a r t i c l e

i n f o

Article history: Received 14 March 2017 Received in revised form 27 April 2017 Accepted 5 May 2017

Keywords: Reactor point kinetics Kinetic parameters Source-driven systems Inverse kinetics

a b s t r a c t Nuclear reactor reactivity is one of the most important properties since it is directly related to the reactor control during the power operation. This reactivity is influenced by the neutron behavior in the reactor core. The time-dependent neutrons behavior in response to any change in material composition is important for the reactor operation safety. Transient changes may occur during the reactor startup or shutdown and due to accidental disturbances of the reactor operation. Therefore, it is very important to predict the time-dependent neutron behavior population induced by changes in neutron multiplication. Reactivity determination in subcritical systems driven by an external neutron source can be obtained through the solution of the inverse kinetics equation for subcritical nuclear reactors. The main purpose of this paper is to find the solution of the inverse kinetics equation the main purpose of this paper is to device the inverse kinetics equations for subcritical systems based in a previous paper published by the authors (Gonçalves et al., 2015) and by (Gandini and Salvatores, 2002; Dulla et al., 2006). The solutions of those equations were also obtained. Formulations presented in this paper were tested for seven different values of keff with external neutrons source constant in time and for a powers ratio varying exponentially over time. Ó 2017 Elsevier Ltd. All rights reserved.

1. Introduction Nuclear reactor transient situations can be predicted only by the neutron flux modification and as a result, it is possible to make a sufficiently accurate prediction about the consequences of the transients. It is sufficient to relate the magnitude of the timedependent neutron flux with the neutron population in the nuclear reactor core. The point kinetics equations relate these functions and thus allow the study of the transients that may occur in a nuclear reactor, and its obtaining occurs from an approximations sequence from the neutron transport theory. The point kinetics equations may be performed directly from the neutron transport equation to the neutron diffusion equation, or by means of a heuristic procedure according to Duderstadt and Hamilton (1976) and Bell and Glasstone (1970). Reactivity can be predicted through the reactor inverse kinetic model. This model results from the space dependence separation, assuming a time-independent flux, separated from a timedependent amplitude function (Lamarsh and Baratta, 2001).

⇑ Corresponding author. E-mail addresses: [email protected] (W. de Carvalho Gonçalves), aquilino@ lmp.ufrj.br (A.S. Martinez), [email protected] (F.C. da Silva). http://dx.doi.org/10.1016/j.anucene.2017.05.004 0306-4549/Ó 2017 Elsevier Ltd. All rights reserved.

In practice, the use of point kinetics equations takes place in the so-called inverse kinetics where the reactivity is obtained from the nuclear power history (Duderstadt and Hamilton, 1976). There are only a few problems that it is possible to obtain an accurate analytical solution for the neutrons density given a specific reactivity. Indeed, it is often more appropriate to inverse the reactivity calculation problem to determine past neutron density behavior expressed from a direct nuclear power relationship. This procedure is more closely related to the nuclear reactor control methodology according to Bell and Glasstone (1970). When the neutron production rate via fission reactions is exactly balanced with neutrons leakage and absorption loss, the reactor operates at a constant power level. Any deviation from this balance condition will result in a time-dependent neutron population, consequently, the reactor power will also be time-dependent (Bell and Glasstone, 1970; Duderstadt and Hamilton, 1976; Caro, 1976). The formalism for reactivity will be developed from the set of point kinetic equations obtained by Gonçalves et al. (2015) in Section 2. The Sections 3 and 4 will present the reactivity obtained from the sets of point kinetic equations proposed by Gandini and Salvatores (2002), and Dulla et al. (2006). In order to analyze the inverse kinetic formulation developed in the Sections 2,3 and 4, the tests results as well as the analysis for seven different

344

W. de Carvalho Gonçalves et al. / Annals of Nuclear Energy 108 (2017) 343–350

Table 1 Kinetic parameters by Gonçalves et al. (2015). Kinetic parameters

0.930

0.940

0.950

0.960

0.970

0.980

0.990

KG CG qG bG1 bG2 bG3

1.64812E03 7.52316E02 7.52316E02 2.50047E04

1.62131E03 6.38221E02 6.38221E02 2.49830E04

1.59587E03 5.25452E02 5.25452E02 2.49692E04

1.57239E03 4.16567E02 4.16567E02 2.49631E04

1.55020E03 3.08541E02 3.08541E02 2.49643E04

1.52969E03 2.03282E02 2.03282E 02 2.49725E04

1.51078E03 1.00475E02 1.00475E02 2.49872E04

1.40101E03

1.39985E03

1.39911E03

1.39878E03

1.39884E03

1.39927E003

1.40005E03

1.23789E03

1.23676E03

1.23604E03

1.23572E03

1.23578E03

1.23620E003

1.23697E03

bG4

2.67391E03

2.67191E03

2.67064E03

2.67009E03

2.67019E03

2.67094E003

2.67228E03 8.40963E04

bG5

8.41506E04

8.40834E04

8.40406E04

8.40219E04

8.40256E04

8.40507E004

bG6

1.70891E04

1.70758E04

1.70673E04

1.70636E04

1.70643E04

1.70693E004

1.70784E04

bG

6.57526E03

6.56994E03

6.56656E03

6.56507E03

6.56535E03

6.56733E003

6.57091E03

Table 2 Kinetic parameters by Gandini and Salvatores (2002). Kinetic parameters

0.930

0.940

0.950

0.960

0.970

0.980

0.990

KGS CGS bGS 1

1.62603E03 6.96697E02 1.00960E+00 2.32276E04

1.60399E003 5.96893E02 1.00917E+00 2.32262E04

1.58284E03 4.96521E02 1.00896E+00 2.32255E04

1.56306E03 3.97795E02 1.00895E+00 2.32255E04

1.54407E03 2.97903E02 1.00914E+00 2.32261E04

1.52618E03 1.98512E02 1.00951E+00 2.32273E04

1.50930E03 9.92636E03 1.01005E+00 2.32291E04 1.35852E03

a bGS 2

1.35850E03

1.35848E03

1.35847E03

1.35847E03

1.35848E03

1.35850E03

bGS 3

1.09748E03

1.09737E03

1.09732E03

1.09732E03

1.09737E03

1.09746E03

1.09760E03

bGS 4

2.80567E03

2.80580E03

2.80587E03

2.80587E03

2.80581E03

2.80569E03

2.80552E03

bGS 5

8.38913E04

8.38917E04

8.38920E04

8.38920E04

8.38918E04

8.38914E04

8.38909E04

bGS 6

1.73161E04

1.73165E04

1.73166E04

1.73166E04

1.73165E04

1.73162E04

1.73158E04

bGS

6.50600E03

6.50600E03

6.50600E03

6.50600E03

6.50600E03

6.50600E03

6.50600E03

Table 3 Kinetic parameters by Dulla et al. (2006). Kinetic parameters

0.930

0.940

0.950

0.960

0.970

0.980

0.990

KD

1.51958E03 6.51252E02

1.51317E03 5.63223E02

1.50758E03 4.73001E02

1.50295E03 3.82555E02

1.49917E03 2.89273E02

1.49632E03 1.94641E02

1.49439E03 9.82878E03

bD 1

6.51252E02 2.33428E04

5.63223E02 2.35531E04

4.73001E02 2.37741E04

3.82555E02 2.40006E04

2.89273E02 2.42393E04

1.94641E02 2.44863E04

9.82878E03 2.47424E04

qD;0 qD bD 2

1.30779E03

1.31979E03

1.33218E03

1.34487E03

1.35822E03

1.37202E03

1.38633E03

bD 3

1.15554E03

1.16592E03

1.17684E03

1.18806E03

1.19989E03

1.21214E03

1.22485E03

bD 4

2.49671E03

2.51934E03

2.54303E03

2.56725E03

2.59269E03

2.61894E03

2.64609E03

bD 5

7.85669E04

7.92772E04

8.00220E04

8.07844E04

8.15861E04

8.24144E04

8.32721E04

bD 6

1.59556E04

1.61000E04

1.62513E04

1.64062E04

1.65689E04

1.67370E04

1.69110E04

bD

6.13889E03

6.19436E03

6.25253E03

6.31209E03

6.37474E003

6.43947E03

6.50652E03

subcritical systems, characterized by different effective multiplication factors (keff = 0.930; 0.940; 0.950; 0.960; 0.970; 0.980; 0.990) with external neutrons source constant in time and power ratio (TðtÞ ¼ e0:12353t þ 1) will be presented in Section 5. The paper conclusions will be presented in Section 6.

Gonçalves et al. (2015) presented a new point kinetic model for ADS subcritical nuclear reactors based on the concept of the Heuristic Generalized Perturbation Theory (HGPT) importance function. The proposed importance function is related to the system subcriticality, with the value of the neutrons external source, represented by the following equation:

Lþ0

 ~ G ðr; E;

W

^Þ ¼ X

F þ0

 ~ G ðr; E;

W

0

^Þ  X

qsub wsource

mðEÞRf ð~r; E; t0 Þ

vðE Þ ^  ~ 0 ^ 0 0 ^ 0  u ðr; E ; X ÞdE dX 4p 0

4p

Z

Z Z

1

0

ð1Þ

Z

1

wsource  V

4p

0

^ Þs0 ð~ ^ ÞdEdX ^ d3 r ^ 0 ð~ u r; E; X r; E; X

ð2Þ

The set of equations presented by Gonçalves et al. (2015) is:

KG

2. Inverse kinetics equation for ADS reactor from the set of point kinetics equations proposed by Gonçalves et al.

Z

^ Þ is the neutron source and where s0 ð~ r; E; X

6 X dT G ðtÞ ¼ fqG ðtÞ  bG gT G ðtÞ þ ki nGi ðtÞ þ CG T G ðtÞ þ qG ðtÞ dt i¼1

ð3Þ

and

d G n ðtÞ ¼ bGi T G ðtÞ  ki nGi ðtÞ; dt i where:

KG 

1 IF G

qG ðtÞ 

Z Z V

1 IF G

Z 4p

1 0

Z Z V

4p

^Þ WG ð~ r; E; X

Z 0

1

ð4Þ

1

v ðEÞ

^ ÞdEdX ^ d3 r u0 ð~r; E; X

^ ÞfdF  dLgu ð~ ^ ^ 3 WG ð~ r; E; X 0 r; E; XÞdEdXd r

ð5Þ

ð6Þ

345

W. de Carvalho Gonçalves et al. / Annals of Nuclear Energy 108 (2017) 343–350

Fig. 1. Reactivities variation assuming keff ¼ 0:930.

Fig. 2. Reactivities variation assuming keff ¼ 0:940.

bG ¼

6 X bGi

ð7Þ

CG 

i¼1

bGi

1  IF G

Z Z V

1 nGi ðtÞ  IF G

Z

4p

0

Z Z V

1

4p

 ~ G ðr; E;

W

^ ÞF i u ð~ ^ ^ 3 X 0 r; E; XÞdEdXd r

ð8Þ

1 q IF G sub

qG ðtÞ 

1 IF G

Z Z 4p

V

Z Z 4p

V

Z

1

0

Z 0

1

^ ÞF 0 u ð~ ^ ^ 3 ^ 0 ð~ u r; E; X 0 r; E; XÞdEdXd r

^ Þsð~ ^ ; tÞdEdX ^ d3 r WG ð~ r; E; X r; E; X

ð10Þ

ð11Þ

and

Z 0

1

 ~ G ðr; E;

W

^ Þ vi ðEÞ C G ð~ ^ d3 r X r; tÞdEdX 4p i

ð9Þ

IF G 

1 4p

Z Z V

Z 4p

1 0

^Þ WG ð~ r; E; X

Z

Z 4p

o ^ 0 ÞdE0 dX ^ 0 dEdX ^ d3 r:  u0 ð~ r;E0 ; X

1 0

vðEÞmðE0 ÞRf ð~r; E0 ; t0 Þ ð12Þ

346

W. de Carvalho Gonçalves et al. / Annals of Nuclear Energy 108 (2017) 343–350

Fig. 3. Reactivities variation assuming keff ¼ 0:950.

Fig. 4. Reactivities variation assuming keff ¼ 0:960.

To obtain the reactivity expression, first multiply both sides of Eq. (4) by eki t . This allows regrouping the resulting equation in order to obtain the following expression:

d G fn ðtÞeki t g ¼ bGi T G ðtÞeki t dt i

ð13Þ

By integrating Eq. (13) in the time 0 to t (Hildebrand, 1963), it follows that:

nGi ðtÞ

¼

nGi ð0Þeki t

Z þ

bGi

t

0

T G ðt Þe 0

ki

ðt 0 tÞ

dt

0

ð14Þ

Assuming that for t ¼ 0, the change in precursors concentration is zero, that is, dnGi ðtÞ=dt ¼ 0 of Eq. (4), it follows that:

nGi ð0Þ ¼

bGi T G0 ki

ð15Þ

Substituting Eq. (15) into Eq. (14), and rewriting the concentration of the precursors as follows:

nGi ðtÞ ¼

bGi T G0 ki t e þ bGi ki

Z 0

t

0

T G ðt0 Þeki ðt tÞ dt

0

ð16Þ

W. de Carvalho Gonçalves et al. / Annals of Nuclear Energy 108 (2017) 343–350

347

Fig. 5. Reactivities variation assuming keff ¼ 0:970.

Fig. 6. Reactivities variation assuming keff ¼ 0:980.

Returning to Eq. (3) and isolating the reactivity qG ðtÞ, we have:

qG ðtÞ ¼ KG

6 1 dT G ðtÞ q ðtÞ 1 X þ bG  CG  G  ki nG ðtÞ T G ðtÞ dt T G ðtÞ T G ðtÞ i¼1 i

ð17Þ

Substituting Eq. (16) into Eq. (17), obtains

1 dT G ðtÞ q ðtÞ þ bG  CG  G T G ðtÞ dt T G ðtÞ   Z t 6 1 X T G0 ki t 0 0  ki bGi e þþ T G ðt 0 Þeki ðt tÞ dt T G ðtÞ i¼1 ki 0

qG ðtÞ ¼ KG

Eq. (18) is the mathematical expression of the subcritical systems reactivity with external source obtained from the set of point kinetic equations proposed by Gonçalves et al. (2015). Assuming that TðtÞ ¼ ext þ a, where x is the positive root of the inhour equation and x is a constant determined using Eqs. (3) and (4), which it can be shown that a ¼ 1, that is, TðtÞ ¼ ext þ 1, the Eq. (18) is rewritten as follows:

qG ðtÞ xext þ KG x t T G ðtÞ e þ1   k t 6 X 1 2e i ðext  eki t Þ ð1  eki t Þ  xt ki bGi þ þ e þ 1 i¼1 ki ki x þ ki

qG ðtÞ ¼ bG  CG  ð18Þ

ð19Þ

348

W. de Carvalho Gonçalves et al. / Annals of Nuclear Energy 108 (2017) 343–350

Fig. 7. Reactivities variation assuming keff ¼ 0:990.

After the reactivity expression qG ðtÞ was obtained, the expressions for the reactivity calculation from the sets of point kinetic equations proposed by Gandini and Salvatores (2002), and Dulla et al. (2006) will be presented in Sections 3 and 4. Also, all tests will be presented in the Section 5. 3. Inverse kinetics equation for ADS reactor from the set of point kinetics equations proposed by Gandini and Salvatores The equation for the calculation of reactivity obtained from the set of point kinetics equations proposed by Gandini and Salvatores, 2002 is identical to the one obtained in the previous section. Therefore, it can be verified that:

qGS ðtÞ ¼ abGS þ

CGS ext  qsource

V

nR R o 1 ^ 0 ÞdE0 dX ^ 0 dEdX ^ d3 r mðE0 ÞRf ð~r; E0 ; tÞu0 ð~r; E0 ; X 4p 0 n o R R1 R1  0 ^0 3 0 0 0 ^0 ^ ^ WGS ð~ r; E; XÞvD ðEÞ 4p 0 mðE ÞRf ð~ r; E ; tÞu0 ð~ r; E ; X ÞdE dX dEdXd r 4p 0 R1

4p

V

0

^ Þb v ðEÞ WGS ð~ r; E; X i i

ð25Þ

1 4pIF GS

nGS i ðtÞ 

CD 

Z Z

Z

4p

V

1

0

^ Þv ðEÞC i ð~ ^ d3 r WGS ð~ r; E; X r; tÞdEdX i

ð26Þ

1 IF GS

ð27Þ

qsource ðtÞ 

Z Z

1 IF GS

Z

4p

V

1

0

^ Þdsð~ ^ ; tÞdEdX ^ d3 r WGS ð~ r; E; X r; E; X

ð28Þ

and

xext

þ KGS

R R bGS i  R R

ext þ 1 ext þ 1  k t  6 X i a 2e ðext  eki t Þ ð1  eki t Þ ki bGS þ þ  xt ki e þ 1 i¼1 i ki x þ ki

IF GS 

Z Z 1 4p V 4p Z Z 1 ^Þ  WGS ð~ r; E; X

4p

0

Z 0

1



^ d3 r ^ 0 ÞdE0 dX ^ 0 dEdX vðEÞmðE0 ÞRf ð~r; E0 ; t0 Þu0 ð~r; E0 ; X ð29Þ

ð20Þ The kinetic parameters that appear in Eq. (20) is defined as follows:

KGS 

1 IF GS

qGS ðtÞ 

Z Z V

1 IF GS

Z

4p

0

Z Z V

1

4p

^Þ WGS ð~ r; E; X

Z

1

0

1

v ðEÞ

^ ÞdEdX ^ d3 r u0 ð~r; E; X

ð21Þ

^ Þ½dF WGS ð~ r; E; X

^ ÞdEdX ^d r r; E; X  dLu0 ð~ 3

a

ð22Þ

The equation for the reactivity determination from the set of the point kinetics equations proposed by Dulla et al. (2006) is given as follows:

qD ðtÞ xext þ KD xt x t e þ1 e þ1  k t  6 i 1 X 2e ðext  eki t Þ ð1  eki t Þ  xt ki bDi þ þ e þ 1 i¼1 ki ki x þ ki

qD ðtÞ ¼ bD  q0;D 

Z Z

1 IF GS Z 

V 1

0

4P

(Z

^ Þv ð~ WGS ð~ r; ~ E; X D EÞ

4p

Z 0

1

)

mð~ E0 Þ ^ 0 Þd~ ^ 0 d~ ^ d3~ Rf ð~ r; ~ E0 ; tÞu0 ð~ r; ~ E0 ; X E0 dX EdX r 4p ð23Þ

bGS 

4. Inverse kinetics equation for ADS reactor from the set of point kinetics equations proposed by Dulla et al.

6 X bGS i i¼1

ð24Þ

where:

KD 

1 IF D

Z Z V

Z 4p

0

1

^Þ WD ð~ r; E; X

qD ðtÞ  q~ D ðtÞ þ qD;0 ðtÞ

1

v ðEÞ

^ ÞdEdX ^ d3 r u0 ð~r; E; X

ð30Þ

ð31Þ ð32Þ

W. de Carvalho Gonçalves et al. / Annals of Nuclear Energy 108 (2017) 343–350

1 IF D

q~ D ðtÞ 

Z Z

Z

4p

V

1

0

^ Þ½dF WD ð~ r; E; X

^ ÞdEdX ^d r r; E; X  dLu0 ð~ 3

qD;0  

Z Z

1 IF D

Z

4p

V

1

0

ð33Þ

^ Þs0 ð~ ^ ÞdEdX ^ d3 r WD ð~ r; E; X r; E; X

ð34Þ

6 X bDi ðtÞ

bD ðtÞ 

ð35Þ

i¼1

Z Z

bDi ðtÞ 

1 IF D

nDi ðtÞ 

1 4pIF D

qD ðtÞ 

1 IF D

Z

4p

V

Z Z

Z

4p

Z 4p

V

^ ÞF i u ð~ ^ ^ 3 WD ð~ r; E; X 0 r; E; XÞdEdXd r

0

Z Z V

1

1

^ Þv ðEÞC i ð~ ^ d3 r WD ð~ r; E; X r; tÞdEdX i

0 1

0

ð36Þ

ð37Þ

^ Þsð~ ^ ; tÞdEdX ^ d3 r WD ð~ r; E; X r; E; X

ð38Þ

The term IF is expressed as Z Z IF D 

V

Z

4P

 0

1

^ Þvð~ WD ð~ r; ~ E; X EÞ

Z Z

Z

þ V

4p

0

1

(Z 4p

Z 0

1

)

mð~ E0 Þ ^ 0 Þd~ ^ 0 d~ ^ d3~ r Rf ð~ r; ~ E0 ; tÞu0 ð~ r; ~ E0 ; X E0 dX EdX 4p

^ Þsð~ ^ ; tÞdEdX ^ d3 r: WD ð~ r; E; X r; E; X ð39Þ

349

However, the generalized source reactivity parameter C increase for the set of point kinetic equations proposed by Gonçalves et al. (2015) as the multiplication factor increases, and decreases for the set of point kinetics equations presented by Gandini and Salvatores (2002). After the kinetic parameters presented in the tables above were determined, tests were performed for one type of nuclear power variation TðtÞ ¼ e0:12353t þ 1 for each of the seven values of the effective multiplication factor. The results of these tests are shown in Figs. 1 to 7. As can be seen in Figs. 1 to 7, the reactivity obtained from the sets of point kinetics equations proposed by Gonçalves et al. (2015), Gandini and Salvatores (2002) and Dulla et al. (2006) initially increases to a certain value. After that, it slowly decreases and stabilizes. The same behavior is presented for the sets of point kinetics equations in ADS reactors proposed by Gonçalves et al. (2015), Gandini and Salvatores (2002) and Dulla et al. (2006), which the reactivity increases to a certain value and then qG ðtÞ; qGS ðtÞ and qD ðtÞ decrease slightly and stabilize. It was also verified that, for all cases, the reactivity qG ðtÞ obtained from the set of point kinetics equations proposed by Gonçalves et al. (2015) always presented a slightly higher value than those qGS ðtÞ and qD ðtÞ obtained from the sets of equations proposed by Gandini and Salvatores (2002) and Dulla et al. (2006). Also, the reactivity qGS ðtÞ always presented an intermediate value and qD ðtÞ always presented a value below the other two. Another important fact highlighted in Figs. 1 to 7 is that as we approach the criticality, the values of the reactivities qG ðtÞ; qGS ðtÞ andqD ðtÞ are approximate even more, whereas for keff ¼ 0:990 the curves are practically superimposed, as shown in Fig. 7.

5. Numerical tests

6. Conclusions

This section is dedicated to the calculations and analyzes of the reactivity expressions obtained in the last three sections by the inverse kinetics method from the set of point kinetics equations for the ADS as proposed by Gonçalves et al. (2015), Gandini and Salvatores (2002) and Dulla et al. (2006). The kinetic parameters in Eqs. (19), (20) and (30) and defined in Eqs. (5)–(11), Eqs. (21)–(28) and Eqs. (31)–(38), respectively, were calculated for seven different subcritical systems, characterized by different effective multiplication factors ðkeff ¼ 0:930; 0:940; 0:950; 0:960; 0:970; 0:980; 0:990Þ with external neutrons source con^ Þ. These parameters are shown ^ ; tÞ ¼ s0 ð~ r; E; X stant in time sð~ r; E; X in Tables 1, 2 and 3, respectively. The results presented in table 1 shown that as the values come closer to the criticality, the delayed neutrons fraction b decrease from keff ¼ 0:930 to keff ¼ 0:960, and increase from keff ¼ 0:970 to keff ¼ 0:990 for the set of point kinetic equations proposed by Gonçalves et al. (2015). The set equations presented by Dulla et al. (2006), in table 2, the b increase since the system subcriticality decrease. In Table 3, the set equations proposed by Gandini and Salvatores (2002), the b remain constant. However, differently from the other set of point kinetic equations definitions, the one proposed by Gandini and Salvatores (2002) has a factor a:bGS , while

This paper presents formulations for the reactivity calculation of ADS reactors based on the sets of the point kinetics equations proposed by Gonçalves et al. (2015), Gandini and Salvatores (2002) and Dulla et al. (2006). The formulations presented were tested for seven different values of keff with external neutrons source constant in time and for a powers ratio varying exponentially over time. The tests demonstrate that the developed formulations present very satisfactory and consistent results, since the reactivity stabilized in all cases, as expected. It has also been shown that the three formulations for reactivity analyzed in this work approximate each other as the system approaches criticality, so that for keff ¼ 0:990 they reproduce practically the same result.

bGS remains constant, the parameter a decrease from keff ¼ 0:930 to keff ¼ 0:960, and increase from keff ¼ 0:970 to keff ¼ 0:990. It is important to note that bGS has different meaning of bG and bD . The product a:bGS has the same physical meaning of delay neutron fractions bG and bD . The parameter mean neutron lifetime K decrease from keff ¼ 0:930 to keff ¼ 0:990 for all set equations. This same behavior was observed with the neutron external source parameter q.

Acknowledgements This research was support by Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES-BRAZIL), Conselho Nacional de Desenvolvimento Científico and Tecnológico (CNPq-BRAZIL) and Instituto Nacional de Ciência and Tecnologia de Reatores Nucleares Inovadores (INCT-RNI-BRAZIL). References Bell, G.I., Glasstone, S., 1970. Nuclear Reactor Theory. Van Nostrand Reinhold, New York, N. Y.. Caro, R., 1976. Fisica De Reactores Nucleares. Seccion de Publicaciones de LA J.E.M, Madri. Duderstadt, J.J., Hamilton, L.J., 1976. Nuclear Reactor Analysis. John Wiley and Sons, New York. Dulla, S., Ravetto, P., Carta, M., D’angelo, A., 2006. Kinetic Parameters for Source Driven Systems’, PHYSOR-2006. ANS Topical Meeting on Reactor Physics Organized and hosted by the Canadian Nuclear Society. Vancouver, BC, Canada.

350

W. de Carvalho Gonçalves et al. / Annals of Nuclear Energy 108 (2017) 343–350

Gandini, A., Salvatores, M., 2002. The physics of subcritical multiplying systems. J. Nucl. Sci. Technol. 39 (6), 673–686. Gonçalves, W.C., Martinez, A.S., Silva, F.C., 2015. Point kinetics equations for subcritical systems based on the importance function associated to an external neutron source. Ann. Nucl. Energy 79, 1–8.

Hildebrand, F.B., 1963. Advanced Calculus for Applications. Prentice Hall Inc, Englewood Cliffs, New Jersey. Lamarsh, J.R., Baratta, A.J., 2001. Introduction to Nuclear Engineering. Prentice Hall Inc, Englewood Cliffs, New Jersey.