HGPT based sensitivity time-dependent methods for the analysis of subcritical systems

Annals of Nuclear Energy 28 (2001) 1193±1217 www.elsevier.com/locate/anucene

HGPT based sensitivity time-dependent methods for the analysis of subcritical systems A. Gandini * University of Rome, Nuclear Engineering Department (DINCE), 131088 Corso Vittorio Emanuele II 224, 00186 Rome, Italy Received 2 October 2000; accepted 25 October 2000

Abstract In recent years an increasing interest is observed with respect to subcritical, accelerator driven systems (ADS). Considering the attention being given to these systems for their supposed ability to play a major role as actinides incinerators, as well as power production plants, the application of the heuristically-based generalized perturbation theory (HGPT) methodology for the cycle life analysis of these systems is reviewed and commented. It is discussed in particular the role of the importance function associated with the power control, and the de®nition of the concept of ``generalized reactivity'', merging into the standard concept of reactivity with the system approaching criticality. Basing on these results, a formulation is also described of a point kinetic equation, with physically signi®cant coecients, similar to those presented by Usachev in 1955 (Usachev, L.N., 1955. Atomnay a Energiya, 15, 4726), using the standard adjoint ¯ux as weighting function. # 2001 Elsevier Science Ltd. All rights reserved.

1. Introduction Since the beginning of nuclear reactor physics studies, perturbation theory has played an important role. As well known, it was ®rst proposed in 1945 by Wigner (1945) to study fundamental quantities such as the reactivity worths of di€erent materials in the reactor core. It is also well known that this ®rst formulation, today widely used by reactor analysts, makes a consistent use of the adjoint ¯ux concept. * Present address: ENEA-Centro Richerche Energetiche Casaccia ERG/SEIC, 00060 S. Maria di Galeria, Rome, Italy. Fax: +39-06-3048-4203. E-mail address: [email protected] 0306-4549/01/$ - see front matter # 2001 Elsevier Science Ltd. All rights reserved. PII: S0306-4549(00)00117-1

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The advantage of using perturbation theory lies in the fact that instead of making a new, often lengthy direct calculation of the eigenvalue (and then of the real ¯ux) for every perturbed system con®guration, a simple integration operation is required in terms of unperturbed quantities. It is interesting that as early as 1948 Soodak associated to the adjoint ¯ux the concept of importance, viewing it as proportional to the contribution of a neutron, inserted in a given point of a critical system, to the asymptotic power (Soodak, 1948). Along with the introduction of the concept of importance and, parallel to it, along with the development of calculation methods and machines, from the early 600 a ¯ourishing of perturbation methods, at ®rst in the linear domain and then in the nonlinear one, have been proposed for analysis of reactor core physics, shielding, thermohydraulics, as well as other ®elds. The perturbation formulations proposed by various authors may be subdivided into three main categories, according to the approach followed in their derivation: 1. The heuristic approach, making exclusive use of importance conservation concepts, adopted ®rst by Usachev (1963) and then extensively developed by Gandini (1967, 1969, 1976, 1981, 1983, 1987a,b). It will be referred to, in the following, as heuristic generalized perturbation theory (HGPT) method. 2. The variational approach adopted, in particular, by Lewins (1965), Pomraning (1967), Stacey (1976),Harris and Becker (1976) and Williams (1979). 3. The di€erential method, proposed by Oblow (1976) and extensively developed by Cacuci (1980), based on a formal di€erentiation of the response considered. Each of the above methods has its own merit, although all of them can be shown equivalent to each other (Greenspan, 1975). In this review we shall discuss the potential applications of the HGPT methodology to the analysis of subcritical systems A ®rst indication of the potential use of the HGPT methodology with respect to neutron kinetic analysis of critical and noncritical systems (with an external source) and to the possibility of analyzing integral experiments in reactor facilities at subcritical conditions was suggested in 1968 (Gandini). In particular, the neutron and precursor importances associated with a given response was considered. In subsequent articles (Gandini, 1976, 1981), the use of HGPT methods for time-dependent problems was again discussed. In particular, the composite neutron, precursor and multi-channel temperature ®eld, generally in presence of external neutron and enthalpy sources, was suggested for application of the HGPT methodology in dynamic studies. Considering the increasing attention being given to the subcritical, accelerator driven systems (ADS) for their supposed ability to play a major role as actinides incinerators, as well as power production plants, the application of the HGPT methodology for the cycle life analysis of these systems (Gandini, 1997) was proposed in 1997 basing on a previous procedure (Gandini 1987a,b; 1988) developed for critical ones. In the present paper, we shall shortly review and comment these works. In particular the role is discussed of the importance function associated with

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the power control, and the de®nition of the concept of ``generalized reactivity'', merging into the standard concept of reactivity with the system approaching criticality. Basing on these results, a formulation is ®nally described of a point kinetic equation, with physically signi®cant coecients, similar to that presented by Usachev (1955) using the standard adjoint ¯ux as weighting function and basing on a previous work by Hurwitz (1949). 2. The HGPT method In the HGPT method the importance function is uniquely de®ned in relation to a given system response, for example, a neutron dose, the quantity of plutonium in the core at end of cycle, the temperature of the outlet coolant. The HGPT method was ®rst derived in relation to the linear neutron density ®eld. Then it was extended to other linear ones. For all these ®elds the equation governing the importance function was obtained directly by imposing that on average the contribution to the chosen response from a particle (a neutron, or a nuclide, or an energy carrier) introduced at a given time in a given phase space point of the system is conserved through time (importance conservation principle). Obviously such importance will result generally dependent on the time, position, and, when the case, energy and direction, of the inserted particle. Consider a linear particle ®eld density represented by vector f governed by equation m…fjp† ˆ 0;

…2:1†

pj …j ˆ 1; 2; . . .† representing system parameters, and a response Q of the type … tF Qˆ < h‡T f > dt  h‡ ; f ; …2:2† to

where h‡ is an assigned vector function and where   indicate integration over the phase space and time. Weighting all the particles inserted into the system from an external source h with their corresponding importance (f), the sum of their contribution will obviously give the response itself. This allows to write an important reciprocity relationship (Gandini, 1987a,b):  f ; h ˆ Q ˆ h‡ ; f ;

…2:3†

Along with the HGPT methodology, the importance function f obeys the equation H f ‡ h‡ ˆ 0;

…2:4†

where H is the operator obtained by reversing the Jacobian operator H (  @m=@f), this implying transposing matrix elements, changing sign of odd derivatives, inverting

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the order of the operators. The sensitivity sj with respect to a generic system parameter pj results sj 

dQ @h‡ @m ˆ ; f  ‡  f ; : dpj @pj @pj

…2:5†

The HGPT method was extended to any linear (or linearized) ®eld governed by operators for which the rules for their reversal were known. In Appendix A the derivation of the governing equations relevant to the neutron and nuclide densities in a critical nuclear reactor system is illustrated. 3. Source driven systems The methodology described in Appendix A for long term nuclide/neutron core cycle evolution analysis may be very well applied to source driven, subcritical systems. One of the advantages often claimed for the subcritical source driven power systems is associated to the fact that the power level may be basically controlled by the source strength (via the regulation of the accelerator current). So, no control, or regulating elements would be necessary, if a sucient breeding is available (and/or an appropriate core burnable poison distribution is provided at the beginning of cycle) in the core for compensating the reactivity loss during burnup. To represent this, we shall rewrite Eqs. (A1), (A2) and (A3), relevant to the neutron density n, the nuclide density c and the control function , in the form m…n† …n; c;jp† ˆ

@n ‡ Bn ‡ sn ˆ 0 @t @c ‡ Ec ‡ sc ˆ 0 @t

m…c† …n; cjp† ˆ

m…† …n; cjp† ˆ< c;Sn > -W ˆ 0

…3:1† …3:2† …3:3†

where B and E depend on fuel and neutron densities c and n, respectively. Since we generally consider systems at quasi-static, i.e. stationary conditions, the time derivative at second member of Eq. (3.1) may be neglected in the course of the integration process. Any response, functional of variables n, c, and , could be considered for analysis. We think instructive to limit here consideration to the response de®ned by the expression … tF Q ˆ …tF † ‡ …t tF †…t†dt …3:4† to

which corresponds to the relative source strength required at tF to assure the power level imposed. We may assume that, at unperturbed conditions, …t† ˆ 1 in the

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interval (to ; tF ). If some system parameter (for instance, the initial enrichment, or some other material density) is altered, as in an optimization search analysis, it may be of interest to evaluate the corresponding change of  at the end of cycle, to make sure that given upper limit speci®cations of the source strength are non exceeded. Along with the HGPT methodology, the equations for the corresponding importance functions result -

@n ˆ B n ‡ c c ‡ ST c @t

…3:5†

-

@c ˆ E n ‡ n n ‡ Sn @t

…3:6†

< n ; sn > ‡…t

tF † ˆ 0

…3:7†

c and n being coupling operators de®ned with Eq. (A8). Eq. (3.7) corresponds to an orthonormal condition for n. In order to determine the `®nal' value n …tF † required for starting the integration of Eq. (3.5), in consideration of the nature of the above governing equations, we shall ®rst write n and  in the form1 n …r; t† ˆ nF …t  …t† ˆ F …t

tF † ‡ n~  …r; t† tF † ‡ ~  …t†

with n~  …r; t†and ~  …r† being ®nite functions, vanishing at tF . Replacing into Eq. (3.5), integrating in the interval (tF making " ! 0, we obtain the equation B nF ‡ ST c…tF †F ˆ 0

…3:8† …3:9† "; tF ‡ "), and then

…3:10†

Let us now de®ne n F as obeying equation B n F ‡ ST c…tF † ˆ 0

…3:11†

1 The diverging of n …r; t† at tF may be explained on physical grounds recalling the meaning of importance (in this case, the contribution to the given response by a neutron with the same space/time coordinates) and considering that the response here is …tF † i.e. the control assumed to maintain the power at a pre®xed level. A neutron introduced at tF into the system would in fact produce a (delta-like) impulse of control  to balance its e€ect on the power level. Then, the importance associated to such neutron would be characterized by a similar delta-like behaviour. A quite similar reasoning applies in relation to the diverging of importance  …t† at tF , considering that it physical meaning corresponds to the contribution to the response [de®ned as …tF †] due to a unit energy insertion at tF or, which is the same, to an overall power pulse …t tF †.

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We note that n F corresponds to the importance relevant to functional < c…tF †; Sn…tF † >, i.e. to the system power W. From the source reciprocity relationship (Section 2), we may write < c…tF †; Sn…tF † >ˆ< n F ; sn >ˆ W:

…3:12†

From constraint, Eq. (3.7), we easily obtain F ˆ

1 ˆ < n F ; sn >

1 W

…3:13†

and then nF ˆ n F F ˆ

n F : W

…3:14†

From this `®nal' value, a recurrent calculation scheme may be de®ned starting from tF and proceeding backward. Along the HGPT methodology, the sensitivity coecient relevant to the k'th parameter pk is found as … tF @…tF † @  @  ˆ F ‰< n F …Bn ‡ sn † > ‡ …< c; Sn > W†ŠtF ‡ @pk @pk @pk to  @  @E  @ ‰< n~ ; …Bn ‡ sn †‡ < c ; c > ‡~ …< c; Sn > W†Šdt @pk @pk @pk

…3:15†

with F given by Eq. (3.13). The ®rst term at right side accounts for e€ects on …tF † due to parameter changes at tF , in particular, if pk  W, it gives the (trivial) result @…tF † 1 W. The second, integral term accounts for analogous e€ects on …tF † pro@W ˆ duced by parameter changes at times t < tF . Rather than on the source term, a control on the neutron absorption in the multiplying region could be of interest. In this case, the (intensive) control variable  would represent the average penetration of the control elements, or the average density of the soluble boron in the coolant, and then would enter into the (transport, or di€usion) operator B. The orthonormal condition for the neutron importance n would now be, rather than Eq. (3.7), < n ;

@B n > ‡…t @

tF † ˆ 0:

…3:16†

In this case, the sensitivity coecient with respect to a given parameter pk would always be given by Eq. (3.15), with n F obeying Eq. (3.11), but with

A. Gandini / Annals of Nuclear Energy 28 (2001) 1193±1217

F ˆ

1 : @B < n F ; n > @

1199

…3:17†

In general, a control strategy, by which an automatic resetting of the imposed overall power is actuated, might imply a control intervention on both the neutron source strength and the absorbing elements within the multiplying region. In this case,  (which remains a unique, intensive control variable) would a€ect both operator B and the neutron source [in this latter case, via an appropriate  and parameter dependent coecient …jp†, assumed unity at unperturbed conditions]. The distribution between these two control mechanisms could be described by appropriate parameters (subject to perturbation analysis). The sensitivity coecient, in this case, with respect to a given parameter pk would always be given by Eq. (3.15), with n F obeying Eq. (3.11), but with F ˆ

1 : @B @   < n F ; … n ‡ sn † > @ @

…3:18†

3.1. Stationary case To study a given subcritical system at stationary conditions (which may be interpreted at the beginning of its cycle life), we may consider the same system above in which the neutron source and the nuclide density are assumed time-independent during an arbitrary time interval (to ; tB ). We assume that at to the neutron density (no ), as well as the control (o ) have already reached stationary conditions. So, also these two quantities are time-independent in the same time interval. Their governing equations can then be written, in case the power level is controlled by the source strength, Bno ‡ o sn;o ˆ 0

…3:19†

< co ; Sno >

…3:20†

W ˆ 0:

Also here we shall assume that at unperturbed conditions o ˆ 1. The same equations derived previously are applicable to this case, with the advertence of replacing tF with tB and setting the coupling operators c and n appearing in Eqs. (3.5) and (3.6) equal to zero. The sensitivity coecient of the response …tB †‰ˆ …t† ˆ o , i.e. constant in the whole interval …to ; tB †Š relevant to the jth parameter pk can then be obtained. Since in this case c , as well as n~ …r; t† and ~  …r; t† vanish, recalling Eq. (3.15), we obtain @o @ @ ˆ o ‰< no ; …Bno ‡ sn;o † > ‡ …< co ; Sno > @pk @pk @pk

Wo †Š

…3:21†

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A. Gandini / Annals of Nuclear Energy 28 (2001) 1193±1217

where o ˆ

1 Wo

…3:22†

and no obeys equation B no ‡ ST co ˆ 0

…3:23†

If, rather than via the source strength, the power level reset control is assumed to be regulated via neutron absorption, so that the control o would enter into operator B, the sensitivity coecient would be given always by Eq. (3.21), but with o ˆ

1 : @B  < no ; n > @

…3:24†

We might as well consider a (®ctitious) control mechanism a€ecting the ®ssion source, rather than the neutron absorption, i.e. we might choose as control a coecient multiplying the ®ssion matrix (F) and, therefore, entering into the Boltzmann, or di€usion, operator B…ˆ A ‡ o F†. The sensitivity coecient would be given again by Eq. (3.21), but with o ˆ

1 : < no ; Fno >

…3:25†

3.2. Reactivity of subcritical systems For resetting the power level, we have considered above di€erent control mechanisms to which the following types of equations governing the neutron density may be associated: B…p†no ‡ o sn;o …p† ˆ 0

…source control†

…3:26†

B…o jp†no ‡ sn;o …p† ˆ 0

…neutron absorption; or fission control†

…3:27†

B…o jp†no ‡ …o jp†sno …p† ˆ 0

2

…mixed control†2

…3:28†

A mixed control strategy may be considered also using Eqs. (3.26), or (3.27). Adopting, for instance, Eq. (3.26), relevant to the neutron source control, part of the power level would be taken care of parametrically (e.g. by properly changing the control rod position, or the soluble boron density). The remaining reset would be taken care of intrinsically, by the -control chosen.

A. Gandini / Annals of Nuclear Energy 28 (2001) 1193±1217

1201

where the control and parameter dependence is indicated. Coecient  is given and re¯ects the mixed strategy chosen. Eqs. (3.26), (3.27) and (3.28) may be generally represented by equation m…n;o† …no ; o jp† ˆ 0:

…3:29†

The sensitivity expression (3.21) may be generalized so that do ˆ dpj

< no ;

@m…n;o† @ >‡ …< co ; Sno > @pj @pj @m…n;o† > < no ; @o

Wo † ;

…3:30†

with no obeying Eq. (3.23). A corresponding perturbation expression may now be obtained. Assuming that the power Wo appearing in Eq. (3.30) is not subject to perturbation, we may write: o ˆ

< no ; m…n;o† > ‡ < no ; …ST co † > ; @m…n;o† < no ; > @o

where m…n;o† ˆ

P

j pj

@m…n;o† @pj

and …ST co † ˆ

P

j pj

…3:31†

@…ST co † @pj .

As said previously, o corresponds to the control change necessary to reestablish the power level existing before the
perturbation m…n;o† . We may as well say that the perturbation m…n;o† [and  ST co ] would produce a power level change equivalent to that produced by a control change K given by the equation K ˆ

< no ; m…n;o† > ‡ < no ; …ST co † > : @m…n;o† < no ; > @o

…3:32†

In the case of the (®ctitious) control on the neutron ®ssion, setting l in place of  to distinguish this peculiar case, we may explicitly write Kl ˆ

< no ; Bno > < no ; sn;o > < no ; …ST co † > ‡ ‡ : < no ; Fno > < no ; Fno > < no ; Fno >

…3:33†

The ®rst term at the right side closely resembles the reactivity expression for critical systems.3 So, we shall call a quantity Kl as given by expression (3.33) a `generalized 3 The ®rst term at right hand side of Eq. (3.33) can be demonstrated to formally approach the standard reactivity expression as the (reference) system considered gets close to criticality conditions (Gandini, 1997).

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A. Gandini / Annals of Nuclear Energy 28 (2001) 1193±1217

reactivity'. The second term may be de®ned the ``source reactivity'', whereas the last one a ``direct e€ect''. To account for a generic -mode control mechanism, we shall extend this de®nition to K , similarly de®ned by Eq. (3.32), i.e. K ˆ

< no ; Bno > < no ; sn;o > < no ; …ST co † > ‡ ‡ : @m…n;o† @m…n;o† @m…n;o† < no ; > < no ; > < no ; > @o @o @o

…3:34†

and call it generalized -mode reactivity. 3.3. Point kinetics Let us now consider equations governing the neutron ¯ux  (  Vn) and precursor mi …i ˆ 1; 2; . . . ; I† in a multigroup (G groups) neutron energy scheme: V

1

d ˆ A ‡ …1 dt

dmi ˆ i Tf  dt

†P SG f  ‡ D u

I X li m i ‡ s n

…3:35†

iˆ1

li mi

…3:36†

where A is the transport, capture and scattering matrix operator, V the diagonal neutron velocity matrix, u is a unit (G component) vector and ::: ::: :::

f;G ::: f;G

::: ::: :::

D;1 ::: ;  ˆ diag l1 D;G …GxI†

f;1 ::: SX ˆ f f;1

; Tf ˆ f;1


f;G ; z ˆ diag z;1

::: z;G

…X rows†

Setting D;1 XD ˆ :::  D;G

:::

lI ; B ˆ diag 1

::: I

Eqs. (3.35) and (3.36) may be written V

1

d ˆ A ‡ …1 dt

dm ˆ BSIf  dt or, in matrix form,

m

†P SG f  ‡ XD Lm ‡ sn

…3:37† …3:38†

A. Gandini / Annals of Nuclear Energy 28 (2001) 1193±1217

d V 1  A ‡ …1 †P SG ‡ XD   sn f ‡ ˆ BSIf  m 0 dt m

1203

…3:39†

At unperturbed, steady state conditions Eq. (3.39) reduces to: A ‡ …1 † SG X  sno o P f;o D o ˆ0 ‡ BSIf;o  mo 0

…3:40†

or Ao o ‡ ‰P …1

† ‡ D ŠSf;o o ‡ sn ˆ 0

…3:41†

Consider the neutron importance ns;o associated to the source power control, as de®ned by Eq. (3.14), and the corresponding precursor density importance ms;o (Gandini, 1976). These importances are governed by the equation  P Ao ‡ …1 †SG;T f;o T XD

n SI;T B  f;o f;o ˆ0 s;o  ‡ Wo m  s;o 0

…3:42†

being the number of energy units per ®ssion and Wo the system power at stationary, unperturbed conditions. We may also write: Ao ns;o ‡ STf;o ‰…1

†P ‡ D Šns;o ‡

f;o ˆ 0 Wo

…3:43†

Function ci results, by de®nition of importance: ms;i;o  ms;o ˆ uT D ns;o

…3:44†

Rewrite Eq. (3.39) in the form (writing Sf rather than SG f ): V

1

d ˆ …Ao ‡ A† ‡ …1 dt

dmi ˆ i Tf  dt

li mi

†P …Sf;o ‡ Sf † ‡ D u

I X li m i ‡ s n

…3:45†

iˆ1

…3:46†

Multiplying Eqs. (3.45) and (3.46) on the left by n Ts;o ,and, ms;o , respectively, space-integrating and recalling expression (3.44), we obtain

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d < ns;o ; V 1  > ˆ < ns;o ; ‰…Ao ‡ A† ‡ ‰…1 †P …Sf;o ‡ Sf †Š > dt I X ‡ li < mo mi > ‡ < ns;o ; …sn;o ‡ sn † >

…3:47†

iˆ1

d ˆ i < ms;o Tf  > dt

li < ms;o ci >

…3:48†

Recalling Eq. (3.43) governing the importance function ns;o and the importance reciprocity relationship

< fo; ; o >ˆ< n s;o ; sn;o > …ˆ 1†; Wo

…3:49†

adding and subtracting the term < ns;o ; D Sf  > at the left side of Eq. (3.47), after some manipulations this transforms into  d < ns;o ; V 1  > ˆ < ns;o ; A ‡ ‰…1 †P ‡ D ŠSf  > ‡ dt M X < ns;o ; sn > ‡ li < ms;o mi;o > < n s;o ; D Sf  > iˆ1

‡1

W

‡ < f ;  > : Wo Wo

…3:50†

Let us de®ne the source term < ns;o ; Sf  > …1

† < ns;o ; P Sf  > ‡ < ns;o ; D Sf  >

…3:51†

and assume that " # d < ns;o ; V 1  > d < ns;o ; V 1  > < ns;o ; Sf  > < f ;  > ˆ dt < ns;o ; Sf  > < f ;  > dt " # d < ns;o ; V 1 o > < ns;o ; Sf o >  < f ;  > dt < ns;o ; Sf o > < f ; o >

…3:52†

If we de®ne then the quantities: P…t† ˆ

W…t† Wo

…relative power†

…3:53†

A. Gandini / Annals of Nuclear Energy 28 (2001) 1193±1217

`eff ˆ

< ns;o ; V 1 o > < n s;o ; Sf o >

gen ˆ

 < ns;o ; A ‡ …1

…effective prompt neutron lifetime†

ˆ

…3:55†

< ns;o ; sn > …source reactivity† < ns;o ; Sf;o o >

< ns;o ; D Sf;o o > < ns;o ; Sf;o o >

<

i ˆ

ns;o ; Sf;o 

…3:56†

…3:57†

1

ˆ

…3:54†



†P ‡ D ŠSf o > ‡ < f ; o > Wo < ns;o ; Sf o >

…generalized reactivity† source ˆ

1205

…3:58†

>

< ms;o mi > < ns;o ; Sf;o o >

…3:59†

Eqs. (3.50) and (3.48) may then be written in the form `eff

dP ˆ …gen dt

di ˆ i P dt

 †P ‡ 

li i

I X li i ‡ …1

P† ‡ source

…3:60†

iˆ1

…3:61†

with P ˆ Po ˆ 1 and i ˆ i =li at steady state conditions. The expression for gen was discussed in the previous section. It is interesting also to note that, with the system approaching criticality, quantity  vanish. Consequently, the last term at the right side of Eq. (3.60) also vanishes, whereas the space distribution of ns;o approaches the standard adjoint ¯ux o (Gandini, 1997). In this case, Eqs. (3.60) and (3.61) reduce to the homogeneous, standard form of the point kinetics equations. Searching solutions for functions P and i of the form e !t , we may arrive at the expression  ˆ `eff ! ‡ 

I X ! i ! ‡ li iˆ1

…3:62†

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A. Gandini / Annals of Nuclear Energy 28 (2001) 1193±1217

with ˆ

 < o ; A ‡ ‰…1 †P ‡ D ŠSf o > < o ; Sf o >

…3:63†

and with `eff and  given by Eqs (3.54) and (3.57) with ns;o replaced by o . The general solution will be then given by the superimposition of the solutions corresponding to the (M ‡ 1) roots !` . Eqs. (3.60) and (3.61) may be considered an extension of the point kinetic equation to subcritical systems. Solving Eq. (3.62), with gen given by Eq. (3.63) in place of , and with `eff and  given by Eqs. (3.54) and (3.56), shall give the (M ‡ 1) roots !i relevant the exponential solutions of the homogeneous equation associated with Eqs. (3.60) and (3.61). As well known, the general solution shall be given by the sum of the solution of the equivalent homogeneous equation and a particular one. Asymptotically, if after the perturbation the system is still subcritical, a new (relative) power level Pas will be reached, given by the expression Pas ˆ

 ‡ source ;  gen

…3:64†

which, as expected, increases with source and gen . Quantity  plays the role of a measure of the system subcriticality. To show this, consider ®rst the two subcriticality measures so far generally adopted Keff ˆ

< o ; Sf;o o > < o ; sn;o > ‡ < o ; Sf;o o >

…3:65†

< u; Sf;o o > < u; sn;o > ‡ < u; Sf;o o >

…3:66†

Ksource ˆ

with u a unit vector. Keff is associated with the fundamental mode of the neutron. It has relevance for safety studies implying accidents bringing the system to overcritical conditions. Ksource is a multiplication factor implying the actual ¯ux, in a source driven system generally formed by a superposition of eigenfunctions. It does not take into account the importance of ®ssion and source neutrons with respect to the power. So, taking this importances into account, and recalling that < no sn;o >ˆ 1, me may de®ne the multiplication coecient

Ksub ˆ

< ns;o ; Sf;o o > : 1‡ < ns;o ; Sf;o o >

…3:67†

A. Gandini / Annals of Nuclear Energy 28 (2001) 1193±1217

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Quantity  then may be written as ˆ

1

Ksub Ksub

;

…3:68†

and may be clearly taken as a consistent measure of the distance of the system from criticality. It was shown (Gandini, 1997) that for Ksub approaching unity, function ns;o diverges, its space shape approaching that of the standard adjoint ¯ux. Correspondingly, gen converges to the standard form of reactivity, Eq. (3.63) We have seen that the quantity gen plays a role analogous to that of the reactivity in the point kinetics equation for critical systems. We may also verify that this quantity, for the same parameter perturbation, gives a decreasing contribution to the power change with the system subcriticality increasing. This is due to the presence of the source-related term …1 P† at the right side of Eq. (3.60), where  increases with the subcriticality. As we have seen, the coecients appearing in Eq. (3.60) are all physically meaningful. The generalized reactivity, gen , in particular, may be determined by measurement. In fact, as shown in the previous section, it is given by the product of the source-mode generalized reactivity associated with the source control [cfr. Eq. (3.21)]  gen;s ˆ< ns;o ; A ‡ …1



†P ‡ D ŠSf o > ‡ < f ;  > Wo

…3:69†

by the quantity , given by expression (3.58). Since gen;s corresponds to the source strength change necessary to reset the power level  afterthe perturbation, it is clearly a measurable quantity. For what concerns   1 KKsubsub , this quantity doesn't seem

easily amenable to experimental evaluation. It seems easier to determine its counterpart, with ns;o replaced by standard adjoint function o , for instance, via fundamental mode period measurements.  could be then evaluated by multiplying its calculated value by a bias factor, i.e. 

1

exp ˆ cal  1

Keff Keff Keff

exp cal :

…3:70†

Keff Of course, a similar procedure could be also followed for determining via a bias exp factor exp . gen starting from the measurement of a standard reactivity value  In above expressions we have assumed, for simplicity of presentation, constant values for the delayed neutron fractions and j . In reality these quantities are generally dependent on energy and space, in correspondence to the space

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A. Gandini / Annals of Nuclear Energy 28 (2001) 1193±1217

distribution of the fuel elements composition and of the neutron energy spectrum. The correct values to be adopted in the above equations are discussed in Appendix B. 3.4. Illustrative example Let us consider the simple case of one-group, one precursor, in®nite system. In this case Eqs. (3.35) and (3.36) become 1 df ˆ v dt

c f ‡ …1

dm ˆ f f dt

†f f ‡ lm ‡ sn

lm

…3:71†

…3:72†

At unperturbed conditions it is: c;o fo ‡ f;o fo ‡ sn;o ˆ 0

…3:73†

with solutions fo ˆ

1 sn;o c;o 1 Ko

…3:74†

mo ˆ

Ko f fo ˆ sn;o l l 1 Ko

…3:75†

The importance function ns;o is governed by the equation co ns;o ‡ fo ns;o ‡

1

fo ˆ 0 Wo

…3:76†

with the solution ns;o ˆ

1

fo 1 Ko  Wo co fo Wo  1 Ko

…3:77†

As the (reference) system approaches criticality, and then sn;o , for the same power, goes to zero, the importance ns;o diverges. If, on the contrary, it become increasingly

A. Gandini / Annals of Nuclear Energy 28 (2001) 1193±1217

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subcritical, it correspondingly reduces, vanishing with f;o approaching zero. This is expected recalling the meaning of importance.4 Consider a perturbation altering the system parameters. The governing equations will result `eff

dP ˆ …gen dt

d ˆ P dt

†P ‡ l ‡

1

Ko …1 Ko

l:

P† ‡ source

…3:78†

…3:79†

If after the perturbation the system is still subcritical, the new power asymptotic level will be Pas ˆ

1 1

Ko ‡ Ko source : …Ko ‡ Ko gen †

…3:80†

As expected, the condition for remaining at subcriticality condition is that gen < :

1 Ko Ko

Assume now the values: `eff ˆ 10 3 ;

l ˆ 0:3;

ˆ 0:007:

A number of illustrative examples relevant to di€erent reactivity insertions as shown in Fig. 1±5 (showing P vs. s). 4

An importance function (Gandini, 1987a,b) is strictly associated with a response de®ned in a given space interval (at a limit, at a given time point). To exemplify, with the above one-group, in®nite medium, the importance f relevant to the power de®ned at an arbitrary time t0 would be governed by the equation: df ˆ dt

Sc;o f  ‡ Sf;o f  ‡

1

Sf;o …t Wo

t0 †

 Integrating from 1 and t‡ o , recalling that for a subcritical, dissipative system f vanishes for t ! and at t > t0 , and de®ning the integrated importance

no ˆ

… t0 1

f  …t†dt;

…a† 1

…b†

we easily obtain Eq. (3.75). It is also clear that for the system approaching criticality (since the introduction of a neutron at an asymptotic negative time increasingly a€ect the power value at t0 ) the value no given by Eq. (b) diverges.

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A. Gandini / Annals of Nuclear Energy 28 (2001) 1193±1217

Fig. 1. gen ˆ 0:005 (asymptotic value: P=1.11).

Fig. 2. gen ˆ

0:005 (asymptotic value P=0.91).

Fig. 3. gen ˆ 0:0526 (critical conditions).

A. Gandini / Annals of Nuclear Energy 28 (2001) 1193±1217

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Fig. 4. gen ˆ 0:07 (over pprompt critical).

Fig. 5. source ˆ

1 (source removal).

Appendix A. Cycle analysis To the neutron and fuel nuclide densities, represented by vectors n(r,t) and c(r,t), respectively, de®ned in the reactor cycle interval (to,tF), a speci®ed intensive control variable, …t†, is associated so that the assigned, overall power history W…t† is maintained. Vector n represents the space- and time-dependent neutron density in a multigroup energy form, whereas vector c the space- and time-dependent density of the various fuel nuclide species. The intensive, time-dependent, control variable …t† may represent, for instance, the overall control rod bank penetration into the core [not their relative movement, which is generally described by parameters pk …k ˆ 1; 2; . . .†], or the average neutron poison material density. The nonlinear governing equations can then be written formally as

m…n† …n; c;jp† ˆ

@n ‡ Bn ‡ sn ˆ 0 @t

…A1†

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A. Gandini / Annals of Nuclear Energy 28 (2001) 1193±1217

m…c† …n; cjp† ˆ

@c ‡ Ec ‡ sc ˆ 0 @t

m…† …n; cjp† ˆ< c; Sn >

Wˆ0

…A2† …A3†

where B is the neutron di€usion, or transport, matrix operator (depending on c and ), E the nuclide evolution matrix (depending on n), sn and sc are given source terms,5 while 1 1 f;1 ::: f;G …A4† S ˆ ::: ::: ::: V; J :::  J f;1

f;G

j

being the amount of energy per ®ssion, and f;g the microscopic gth group ®ssion cross-section of the jth heavy isotope. V is the diagonal neutron velocity matrix. j Quantities , V, W and f;g may be considered generally represented by (or function of) system parameters pk . Source terms sn and sc are also parameter dependent. In quasi-static problems, as those of interest here, the derivative @n @t is negligible. If we introduce the ®eld n …A5† f…r; t† ˆ c 

the system of Eqs. (A1), (A2) and (A3) may be represented in the compact symbolic form, Eq. (2.1), and the HGPT methodology described above applied. Consider a functional n…r; t† … ‡ ‡ ‡ Q ˆ< sn sc s to tF f > c…r; t† dt: …A6† …t† Q may represent, for instance, the amount of a given nuclide built up at time tF [in ‡ ‡ this case s‡ tF †], or the ¯uence at a n ˆ 0, s ˆ 0 and sc includes a delta function …t ‡ ‡ speci®c point r [in this case sc ˆ 0, s‡ r†.  ˆ 0 and sn includes a delta function …r The importance function T f …r; t† ˆ n …r; t† c …r; t†  …t†

…A7†

can then be de®ned, and results governed by Eq. (2.4), with H and h+ given by expressions: 5 sn is generally assumed zero during burnup, except a delta-like source at to to represent initial conditions (usually considered at steady-state), whereas sc is generally given by a sum of delta functions de®ned at to and at given times to account for fuel feed and shu‚ing operations.

A. Gandini / Annals of Nuclear Energy 28 (2001) 1193±1217

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@

c ST c … ‡ B † @t @ T  

n … ‡ E † Sn H ˆ @t   < n; @B …
† > 0 0 @

…A8†

‡ sn h‡ ˆ s‡ c s‡ 

…A9†

h i h i @…Bn†

c and n being operators adjoint of C ˆ @…Ec† ˆ and

n @n @c , respectively. The equation relevant to function  corresponds to a relationship between n and n, i.e. < n;

@B  n >ˆ s‡  @

…A10†

In case s‡  ˆ 0, Eq. (A10) corresponds to an orthogonality relationship. To solve the equations relevant to n and c di€erent resolution recurrent schemes may be considered, starting from the `®nal' time tF and proceeding backward, along with the same time discretisation adopted in the forward reference calculation. It can be shown (Gandini, 1987a,b) that, at quasi static conditions, the equations to be solved reduce to the types: B  n ‡ h‡ n ˆ 0

-

@c ˆ E T c  ‡ h‡ c @t

…A11†

…A12†

‡ where h‡ n and hc correspond to known source terms determined during the recurrent calculation procedure. Therefore, existing, well established codes can be used for their solution. dQ with respect to a given parameter pk may then be The sensitivity coecient dp k obtained from Eq. (2.5), with vector m made of components m…n† , m…c† and m…† de®ned in Eqs. (A1), (A2) and (A3), respectively. A general problem we are faced with is the following: how does the control criticality reset () strategy a€ect the sensitivity analysis results? To answer this question, let us consider Eq. (A11) governing n. We note that, given a particular solution npart , the general one may be written as

n ˆ npart ‡ a

…A13†

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A. Gandini / Annals of Nuclear Energy 28 (2001) 1193±1217

where a is an arbitrary coecient and  the conventional adjoint function obeying the homogeneous equation B  ˆ 0:

…A14†

Once a solution npart has been obtained, the solution desired can then be derived by proper ®ltering from the fundamental mode, i.e. it will be given by Eq. (A13), with coecient a determined by imposing condition (A10). Assuming s‡  ˆ 0, we shall have

n ˆ npart

@B n> @  :  @B < ; n> @

< npart ;

…A15†

The dependence of the importance function n on the control mode adopted is evident. When calculating the sensitivity coecient of a response Q with respect to a given parameter pk (or its change Q with respect to parameter alterations pk ), the ®ltering of the importance function as shown in Eq. (A15) corresponds to implicitly accounting for the -mode control reset of the criticality (in the following we shall refer to it simply as -mode reset). The above result may have important implications, in the sense that in many circumstances, prior to a sensitivity study, it may be necessary to consider the proper reactivity control mode to be adopted. On the other hand, within many existing codes used with the HGPT methodology, the ®ctitious ``l'' reset control is implicitly assumed, i.e. that related to the coecient (eigenvalue) l multiplying the ®ssion source term (Fn) in the Boltzmann (or di€usion) equation. In this circumstance expression (A15) for the importance n will result, recalling that in this case @B @l ˆ F n ˆ npart

< npart ; Fn >   : <  ; Fn >

…A16†

Using this l-mode ®ltering, rather than the correct r-mode one, may lead to erroneous results. Consider, for instance, the case of a sensitivity analysis with respect to core breeding, or conversion ratio, a quantity clearly dependent on the neutron energy spectrum. Assuming that the reactivity compensation corresponding to the change of a system parameter (for instance, the initial fuel enrichment) is e€ected, as it may very well be the case for a thermal reactor, by an alteration of the average (boron) poison concentration in the coolant, the correct choice of the control mode reset would clearly have the e€ect of hardening (if boron is added), or softening (if boron is subtracted) the neutron spectrum. Instead, if a l-mode reset would have been implicitly adopted (as is often done with existing codes), no signi®cant neutron

A. Gandini / Annals of Nuclear Energy 28 (2001) 1193±1217

1215

energy shift would have been taken into account, and, consequently, an erroneous sensitivity coecient would result. It is also true that in principle one could calculate separately the amount of control poison (referring to the above example) to reset the criticality and consider the overall parameter plus control change along with the l-mode methodology. But this would imply a reactivity reset calculation to be performed for each parameter considered. On the other hand, the correct fundamental r-mode ®ltering may be a quite straightforward procedure. In fact, it can be e€ected ``a posteriori'' adopting expression (A15) in which npart would correspond to a preliminary l-mode calculation with an existing code. Appendix B. Beta e€ective In the expressions derived in Section 3 we have assumed, for simplicity of presentation, constant values for the delayed neutron fractions and i In reality these quantities are generally dependent on energy and space, in correspondence to the space distribution of the fuel elements composition and of the neutron energy spectrum. To obtain the e€ective quantities to be inserted in the point kinetic equations, we should consider, in place of Eqs. (3.37) and (3.38), the following ones, indicating with cj …j ˆ 1; 2; . . . ; J† the density of the jth heavy isotope, V

1

J X I X d ˆ A ‡ cj jP …U dt jˆ1 iˆ1

dmj ˆ Bj Sj  dt

Bi;j †Sj  ‡

J X jˆ1

cj XjD mj ‡ sn

…B1†

mj

…B2†

where we also accounted for a general dependence of the prompt and delayed ®ssion neutron spectrum on the heavy nuclide species undergoing ®ssion, and where, with evident notation, j 1 ::: 1 i;1 U ˆ ::: ::: ::: Bi;j ˆ ::: 1 ::: 1 j i;1 …GG† j  ::: 0 f;1 ::: : Sj ˆ ::: ::: j 0 ::: f;G

::: ::: :::

ji;G ::: ji;G …GG†

j 1;1 Bj ˆ ::: j I;1

::: ::: :::

j1;G ::: jI;G …IxG†

Eqs. (B1) and (B2) could be written in the form of Eqs. (3.37) and (3.38) if we replace i and by the expressions, assuming for the i'th delayed neutron fraction a general dependence on nuclide species and incident neutron energy,

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A. Gandini / Annals of Nuclear Energy 28 (2001) 1193±1217

ji;eff ˆ

G P

< ns;o ;

j jD u ji;g f;g fg >

gˆ1

< ns;o ;

G P

gˆ1

J P I P

< ns;o ;

jˆ1iˆ1

eff ˆ

cj jP Bi;j Sj  >

J P I P

< ns;o ;

…B3†

j jD uf;g fg >

jˆ1iˆ1

cj jP USj  >

:

…B4†

In place of Eqs. (3.60) and (3.61) we would then have: `eff

dP ˆ …gen dt

dij ˆ ji;eff P dt

 eff †P ‡ 

J X I X cj li ij ‡ …1

P† ‡ source

…B5†

jˆ1 iˆ1

li ij

…B6†

with ij ˆ

< ms;o mji > < ns;o ; Sf;o o >

ji;eff ˆ

< ns;o ; <

G P

…B7†

j jD u ji;g f;g fg >

gˆ1

ns;o ;

G P

j jD uf;g fg gˆ1

…B8† >

and < ns;o ; Sf  > …1 < ns;o ;

<

J X I X jˆ1 iˆ1

J P I P

cj jD USj  jˆ1iˆ1 ns;o ; Sf;o o >

< ns;o ; ˆ

eff † < ns;o ;

We may also de®ne i;eff as:

J X I X cj jP USj  > ‡ eff jˆ1 iˆ1

cj jD USj  >

…B9†

> :

…B10†

A. Gandini / Annals of Nuclear Energy 28 (2001) 1193±1217

< ns;o ; i;eff ˆ

J P G P jˆ1gˆ1

< ns;o ;

j cj jD u ji;g f;g fg >

J P G P

j cj jD uf;g fg >

< ns;o ; 

jˆ1gˆ1

J P jˆ1

< ns;o ;

1217

cj jD Bi;j Sj  >

J P

cj jD USj  >

:

…B11†

jˆ1

Eqs. (B5) and (B6) could then be written as `eff

dP ˆ …gen dt

di ˆ i;eff P dt

 eff †P ‡ 

I X li i ‡ …1

P† ‡ source

…B12†

iˆ1

li i :

…B13†

which correspond to Eqs. (3.60) and (3.61), with and i replaced with eff and i;eff . P To note that eff does not generally correspond with the sum Iiˆ1 i;eff , as may be inferred from expressions (B4) and (B11) above. Considering that, from the system safety point of view, eff is a crucial quantity (whereas 1;eff is related only with the delayed ®ssion neutron P release), expression (B4) should be generally used for it, rather than the sum Iiˆ1 i;eff . References Cacuci, D.G., Oblow, E.M., Marable, J.H., Weber, C.F., 1980. Nucl. Sci. Eng. 75, 88. Gandini, A., 1967. J. Nucl. Energy 21, 755. Gandini, A., 1969. Nucl. Sci. Eng. 35, 141. Gandini, A., 1976. Nucl. Sci. Eng. 59, 60. Gandini, A., 1981. Nucl. Sci. Eng. 77, 316. Gandini, A., 1983. Trans. Am. Nucl. Soc. 45, 325. Gandini, A., 1987a. Generalized perturbation theory methods. a heuristic approach. In: Lewins, J., Becker, M. (Eds.), Advances in Nuclear Sci. and Techn., Vol 19. Plenum Press, New York, pp. 205±. Gandini, A., 1987b. Ann. Nucl. Energy 14, 273. Gandini, A., 1988. Ann. Nucl. Energy 15, 327. Gandini, A., 1997. Ann. Nucl. Energy, 24, 1241 and A sensitivity approach to the analysis of source driven systems. Meeting on Feasibility and Motivation for Hybrid Concepts for Nuclear Energy Generation and Transmutation, Madrid, 17±19 September 1997. Greenspan, E.M., 1975. Nucl. Sci. Eng. 57, 250. Harris, D.R., Becker, M., 1976. Trans. Am. Soc. 23, 534. Hurwitz, H., 1949. Nucleonics 5 (1), 62. Lewins, J., 1965. Importance, the adjoint function. Pergamon Press, Oxford. Oblow, E.M., 1976. Nucl. Sci. Eng. 59, 187. Pomraning, G., 1967. J. Math. Phys. 8, 149. Soodak, H., 1948. The science and engineering of nuclear power. United Nations, New York. Stacey Jr., W.M., 1974. Variational methods in nuclear reactor physics. Acad. Press, New York. Usachev, L.N., 1955. 1st ICPUE-UN, Geneva, 5, 503. Usachev, L.N., 1963. Atomnaya Energiya 15, 472. Wigner, E.P., 1945. E€ect of small perturbations on pile period, Chicago Report CP-G-3048. Williams, M.L., 1979. Nucl. Sci. Eng. 70, 20.