Advances of the HGPT-BU methodology for Burn-up analysis

Index

page Summary / Sommario ……….. 3 1 Introduction ……….………. 3 2 General Theory ………...………. 4

3 Control at End of Cycle ……….……… 8

3.1. Simple Example ……….………..………... 8

3.2. Two Nuclides ……… 10

3.3. Two Nuclides and Continuous Energy ……… 12

3.4. Two Nuclides, Continuous Energy and Z Zones .………... 16

3.5. M Nuclides and Z Zones ……….………. 20

3.6. Numerical Example ……….……….………... 21

Appendix. Control Modality ………... 23

1. Introduction

The methodology for the perturbative calculation of functionals of the density of neutrons and of the nuclides evolving during the evolution (burn-up) has been developed on the basis of the Generalized Perturbation Theory founded on the Heuristic Generalized Perturbation Theory (HGPT) [1, 2].

The present work has as its main objective the development of a general calculation procedure in view of its application to the perturbative and sensitivity analysis of physical projects relevant to new generation reactors, at critical or sub-critical conditions. It represents the continuation of previous works on the same topic [3, 4]. In these latter works various validation tests were made making use of the neutronic calculation code Eranos [5].

The cases studied with this methodology concerned in particular the quantities:  accumulation of a nuclide at the end of the burn-up cycle in an evolution zone  ratio of functionals linear with the neutron density

In the present work, we focused specifically on the functional 'Control parameter at end of cycle'. A general modeling of the neutron / nuclides field has been adopted considering systems from simplest to complex ones. This modeling may however be easily extended to any other functional.

2. General Theory

In order to derive the equation governing the variables describing the system, a number of assumptions has been made. Specifically:

 The densities of the nuclides that make up the fuel refer to average values relevant to the macro zones in which the core has been subdivided. They are represented with the vector cz

[z = 1, 2, ..., Z (number of sub-zones)]

 The neutron density (n), dependent on phase space and time, is given in a continuous energy distribution

 The diffusion approximation has been considered (which is widely used for burn-up calculations)

At the densities n(r,t) and cz(r), defined in the interval (to, tF), an intensive control variable is

associated, (t), such as to maintain constant the total power assigned W(t). The variable (t) may represent, for example, the total degree of insertion of the control rods in the core (not their relative movement, which may be generally described by parameters), or the average density of a neutron poison in the refrigerant. Generally, a fictitious control parameter is normally adopted, i.e., a coefficient that multiplies the fission source. The application of the methodology to different methods of control is however possible, this implying a different normalization to be imposed on a functions associated with the neutron density [1]. In a sub-critical system (ADS), (t) may represent the intensity of the source (via adjustment of the accelerator current intensity). The non-linear equations to which the variables n, c and  must satisfy may therefore be written, in the most general case, as:

                             .) ( ) p ( ) ( ) p | , c ( ) p |, , c , ( ) ( ) ( ) p | , c ( ) p |, , c , ( ) ( ) ( sottocrit reattore 0 s n t t n t n n m critico reattore 0 n t t n t n n m n o o n o o n B B (1) m_{( )}( ,c|p) c ( ,c|p)c(  )c s (p)0     _{o o c} c n t t t n E (2) m()(n,c|p )  _E f sy s T n dE  

_

 c Γσ = 0 , (3)

where <>sys indicates integration over the whole volume of the multiplying zone, B represents the

operator in diffusion or transport approximation (dependent on c and, generally, from ), the vector p the system parameters. sn and sc are source terms, while

M f 2 f 1 f f :: , , , σ      M 2 1 ... 0 0 :: :: :: :: 0 ... 0 0 ... 0     Γ (4)

where _{f ,m} is the microscopic fission cross section of the isotope m in group g while m represents

the amount of fission energy of the m-th element. The quantities m, , W and _{f ,m} are generically

represented as system parameters (pj). The terms in which the Dirac deltas appear are initial

conditions.

If, for simplicity, it is desired to replace the constant power condition with the constant fission rate condition, it will be sufficient to set m = 1.

In the following we will take into consideration the general case of both critical and sub-critical multiplying systems.

The source term sc in the second member of equation (2) is generally given by a sum of delta

functions defined at specific times to account for fuel loads and shuffling operations.

In the equation (1) for the sub-critical case, control over the source was chosen. If the control were in operator B, the methodology would be similar to that for the critical reactor.

In quasi-static problems, such as those involving burn-up studies, the derivative term t n  

is negligible. However, its notation is maintained to allow, as we will see later, for the determination of the correct operator that governs the importance function.

A general form of a (linear) response Q may be written in the form











           F o t t sy s T c sy s E dEhnn h dt Q h c (5)

with h_n,h_c,h_ρ quantities given.

From the linearization procedure, and recalling the coordinate complementation rule [1], we may obtain the linear equations that govern the derivative functions and the importance functions. The system of equations of the derivative functions is:

0 m ~ c (.) σ σ c (.) σ c ' ) ( ) ( ) ( / / / '                _              _       







m m p n 0 n dE (.) dE 0 t n dE t c n j j j j sy s E T f sy s E f T c sy s f E T n Γ Γ E B   (6)

whereΩ_n e Ω are the coupling operators_c c ) (   nB e n  (Ec) , respectively1

By reversing the operators, the importance functions result governed by the equations

0 h ~ c (.) ' c σ . (.) σ . c σ * * * , ' * * *              _           _         







h h n 0 0 dE n n dE t dE t c n E V E T f sy s E T f T i n sy s T f E c Γ E Γ B   (7)

Where Ω_n* and Ω_c* are the adjoints of operators Ω_n e Ω , respectively. _c Setting

sy s * * _~ _

 , (8)

the equation corresponding to the first row, concerning the importance of neutrons, is:

0 h n t n T c            _   * * * * *  c c S Γ B  (9)

while the equation corresponding to the second row, concerning the importance of the nuclides, results:          T _{n c} t c n n h c* _{* * * *} S E  . (10)

The equation corresponding to the third row, relative to the importance associated with the power control, results: 0 h n dE n sy s E V E T f       



' _, * ' c σ (11)

In case it is h _ 0 this equation indicates that the neutron importance function in critical systems is orthogonal in the space phase to the neutron density distribution, while in subcritical ones it is orthogonal to the neutron source distribution.

For critical systems it can be said that the neutron importance function is void of the fundamental mode.

Multiplying the equation relative to the first line of (7) by n on the left, and integrating, it is easily obtained: sy s T f E sy s c E dE n dE      





  * * * * c σ c Γ  (12)

The general expression of the variation Q resulting from a perturbation of the system parameters may be written in the form:

                      







j j c T j n T E t t J 1 j j p m p p m n dE dt p Q F o * * * m c . (13) 1_{The terms} c ) (   nB and n  (Ec)

represent Frechet derivatives [6] (applicable to expressions in which there appear linear operators).

3. Control at end of cycle

In the following we shall consider the application of the HGPT-BU methodology in relation to the functional ‘Control parameter at end of cycle’. For its full comprehension, we shall start from very simple cases up to most general ones. There is no difficulty in transferring the methodology to any other functional defined in the non-linear neutron/nuclide field.

3.1. Simple example

Consider a homogeneous bare system (slab, sphere, or infinite cylinder) with one fissile nuclide of density c, volume V and monoenergetic neutron density n. During the nuclide evolution the global fission rate density (W) is maintained constant by an intensive control function (), fictitiously assumed as a coefficient of the neutron fission source.

One energy group cross-sections are considered.

The functional to be calculated is the control function at end of cycle, i.e. (tF).

The equations relevant to functions n, c and result, setting for the fissile nuclide c=a +f and

defining with c the macroscopic capture cross section associated with all other elements

                             0 cn W 0 c ) t t ( c n dt dc 0 n cn ) ( n D dt dn f o , 1 o c c c f 2 (14)

Replacing in the first equation the leakage term with –DB2n and the densities c and n with their averages c and n , we may write:

                           0 n c V W 0 c ) t t ( c n dt c d 0 n n c ) ( n DB dt n d f o , 1 o c c c f 2 (15) with solutions: c DB c , ) t t ( V W c c , c V W n f c 2 c o f c o f               (16)

) t ( c DB c d ) t ( d F 2 f c 2 o F       (17)

For this elementary case the equations governing the importance functions result2:

0 1 0 0 ) t t ( ~ c n 0 0 (.) n c . n V . ) n dt d ( V (.) n ) ( . c V . c c ) ( D dt d F * * * f f c c f f c c c f 2                                                  _{     } (18)

The equation relevant to the third row gives:

* F * F * i i i fcn n (tt )0  n (tt )   (19)

which shows how at the final time tF the neutron importance n*_i is a singular function formed by a

normalized standard neutron adjoint  multiplied by the Dirac’s function. We may then write: *           F F * F * t t for 0 t t for ) t ( ) t t ( n (20)

From the equation corresponding to the first row we may easily verify that, for t<tF, it is

0 ) c

(_c *_i _f*  . Substituting the expression of n* in the equation relevant to c*, allows to obtain the (constant) solution

) t ( c DB ) t ( c c F 2 f c 2 F f c f *            (21)

which corresponds to the analytical solution obtained directly, Eq.(17).

2_{Here we cannot replace the leakage term with -DB}2 _{since no space behaviour of the neutron importance has} been yet established.

The constancy over time of function c may be easily interpreted. In fact, due to the constraint over * the power (in our case, the global fission rate), a fuel nuclide inserted at any point during the time interval considered would be found at the end of cycle, with corresponding consequent equal reduction of .

3.2. Two nuclides

Consider the same case above with two nuclides, the first one transmuting into the second one. The system of equations becomes:

                                                0 n c c W 0 c t t c n c n dt dc 0 c t t c n dt dc 0 n n c c n D dt dn 2 2 f 1 1 f o 2 o 1 1 a 2 2 c 2 o 1 o 1 1 c 1 c 2 2 c 2 f 1 1 c 1 f 2 ) ( ) ( ) ( ] ) ( ) [( , , , , , , , , , , , (22)

For this elementary case the equations governing the importance functions result3

0 1 0 0 0 t t c c n 0 0 0 n c c n V n dt d 0 V n n c n V n dt d V n c c V c V c c c D dt d F 2 1 2 2 f 1 1 f 2 f 2 c 2 c 2 f 1 f 1 1 a 1 c 1 c 1 f 2 2 f 1 1 f 2 c 2 1 c 1 c 2 2 c 2 f 1 1 c 1 f 2                                                                                               ) ( ~ (.) ) ( . . ) ( (.) ) ( . . ) ( (.) ) ( . ) ( . . ] ) ( ) [( * * * * , , , , , , , , , , , , , , , , , , , (23)

3_{Here we cannot replace the leakage term with -DB}2 _{since no space behaviour of the neutron importance has} been yet established.

The equation relevant to the fourth row gives:

* F * I F * 2 2 , f 1 1 , f c c )n (t t ) 0 n (t t ) ( n           (24)

which shows how at the final time tF the neutron importance n*_i is a singular function formed by a

normalized standard neutron adjoint flux  (with negative sign) multiplied by the Dirac’s function. * We may then write:

          F F * F F * t t for 0 t t for ) t t ( n (25)

The equations relevant to second and third rows then result:

2 2 f 1 1 f 1 c 1 f 2 f 2 2 c F F 2 2 f F 1 1 f 2 c 2 f 2 1 f 1 1 c 2 1 1 a F F 2 2 f F 1 1 f 1 c 1 f 1 c c V 1 0 n c n t t c c V 1 dt c d 0 n c n c c n t t c c V 1 dt c d , , , , * , * , , , , , , , * * , * , * , , , , , , , * ) ( ) ( ) ( ) (                                               (26)

As well known, the coefficients of the delta functions correspond to the ‘final’ conditions for functions c₁* and c*₂. So we have, at tF,:

                       F 2 2 f F 1 1 f 2 c 2 f 2 2 2 f 1 1 f 1 c 1 f 1 c c V 1 c c c V 1 c , , , , , , * , , , , * ) ( (27)

From the equation relevant to the first row, for t<tF, we obtain:

2 2 , f 1 1 , f * 2 1 , a 1 2 , c 2 * 1 1 , c 1 * c c c ) c c ( c c           (28)

So, for t<tF, me may define the governing equations:

                     0 n c n dt c d 0 n c c n c n dt c d 2 f 2 2 c 2 1 f 2 1 1 a 1 1 c 1 * , * , * * , * , * , * (29)

3.3. Two nuclides and continuous energy

Consider the same case above with two nuclides and a continuous energy schematization for the neutron density and cross-sections. Rewriting the equations in terms of a group-wise notation will be straightforward. The governing equations will result, with t indicating the total microscopic

cross-section:                                                                                                  







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







             0 n dE V c n dE V c W 0 c ) t t ( V n dE V c V n dE V c V c dt d 0 c ) t t ( V n dE V c V c dt d 0 n ) ' E ( n ' dE n ' dE V c n ' dE V c ) ' E ( n ' dE V c ) ' E ( n ' dE V c n V c V c n D dt dn 2 , f E 2 1 , f E 1 o , 3 o 1 , a E 2 2 , c E 2 2 o , 1 o 1 , c E 1 1 t E ' E s ' E 2 , f ' E 2 1 , f ' E 1 E ' E 2 , s ' E 2 E ' E 1 , s ' E 1 2 2 , t 1 1 , t 2 (30)

Recalling the coordinates complementation rule, according to which the control  is replaced by

E , V ~  

 , the quantities c₁ and c₂ by ~ c_{1 E} and ~ c_{2 E}, respectively,. and setting n in place _x

of V n _V,E x    , with V n n  V,E , E , V E , V x x n n        , (31)









0 m m m m p c c n 0 n n c c dE 0 V n dt d V n 0 V n dt d c V c V c V n dE c n dE c V n E n dE E n dE V n E n dE E n dE E n dE c c dE c dE c dE c dE c D dt d 2 1 1 c c n j p 2 p 1 p p V E 2 f V E 1 f V 2 f 2 1 f 1 E E V 2 c E V 1 a E V 1 c 2 E V 2 c 1 E V 1 a 1 E V 1 c E V 2 f E 2 1 f E 1 E V 2 t E E 2 s E 2 f E E V 1 t E E 1 s E 1 f E t E E s E 2 2 t 1 1 t E E 2 s E 2 E E 1 s E 1 2 f E 2 1 f E 1 2                       _{  }           _{  }                                                                                                                                                                                                      

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) ( ) ( ) ( ) ( / / / / , , , , , , , , , , , , , , , , , , ' , ' , , ' , ' , ' , , ' , ' , ' ' ' , , ' , ' ' , ' , ' , ' ~ ~ ~ . . (.) . . . (.) (.) (.) . ' ' . ) ' ( ' ) ' ( ' . ) ' ( ' ) ' ( ' ) ' ( ' ~ (.) ' (.) ' (.) ' (.) ' (32)













0 1 0 0 0 ) t t ( ~ c ~ c ~ n 0 0 0 (.) ' dE n ) c c ( . n . n V . n dt d 0 V . n V . n dt d V (.) n V (.) n ' dE (.) ' dE n V (.) n V (.) n ' dE (.) ' dE n . ) c c ( V . ) c c ( V . c ) ' E ( n ' dE c c (.) ' dE c (.) ' dE c (.) ' dE c (.) ' dE c D dt d F * * 2 * 1 * E , V ' E 2 2 , f 1 1 , f E , V 2 , f E , V 1 , f E , V 2 , c E , V 1 , a E , V 1 , c E , V 2 , t ' E E , V ' E E 2 , s ' E 2 , f E , V 1 , t ' E E , V ' E E 1 , s ' E 1 , f E , V 2 , f 2 1 , f 1 E , V 2 2 , c 1 1 , a E , V 1 , c 1 t ' E E s ' E 2 2 , t 1 1 , t ' E E 2 , s ' E 2 ' E E 1 , s ' E 1 ' E 2 , f 2 ' E 1 , s 1 2                      _{  }               _{ }            _{ }                                                                                                                     





















               (33)

The equation corresponding to the third row results:





' ( ) , * ' , , F E V E 2 2 f 1 1 s c c dE n t t n 



    (34)

which shows how at the final time tF the neutron importance vector n* is a singular function formed

by the normalized standard neutron adjoint flux (with negative sign) multiplied by the Dirac’s function, i.e.:

* F * ) t t ( n    (35)

For t<tF the equation corresponding to the first row, considering that n* is identically equal to zero,

reduces to: 0 ) c c ( c ) c c ( c c c₁_{c,1 1}* ₂ _{a,1 1}_{c,2 2 2}*  ₁_f,1 ₂_f,2 *     (36)

Multiplying by n and integrating over energy, we may then write:





E , V 2 , f 2 1 , f 1 E , V * 2 1 1 , a 2 2 , c * 1 1 , c 1 * ) c c ( n c ) c c ( c c n             (37)

The equations corresponding to the second row, recalling the solution (35) for n* result:

                                                           _{   }            





    0 n c n t t V n V n dE E V n dt c d 0 n c n c n t t V n V n dE V n dt c d 2 f 2 2 c F E E V 2 t E V E E 2 s E V 1 f 2 1 f 2 1 a 1 1 c F E E V 1 t E V E E 1 s E V 1 f 1 * , * , ' , * , , * ' , , * , * * , * , * , ' , * , , * ' , , * , * ) ( ' ) ( ' (38)

with * given by equation (37). As well known, the coefficients of the delta functions correspond to the ‘final’ conditions for functions *

1 c and c*₂. So we have, at tF:                                         





    ' , * , , * ' , , * , * ' , * , , * ' , , * , * ' ' E E V 2 t E V E E 2 s E V 2 f 2 E E V 1 t E V E E 1 s E V 1 f 1 V n V n dE V n c V n V n dE V n c (39)

Since * is the standard adjoint flux, the values of c₁*(t_F)and c₂*(t_F) may be obtained following a procedure similar to that used for calculating a standard reactivity worth (keff/keff), where the

‘perturbation’ in this case would correspond to a single atom of nuclide c1, or c2 in the whole

volume considered4.

For t<tF the governing equations would result:

                    0 n c n dt c d 0 n c n c n dt c d * 2 , f * 2 2 , c * 2 * 1 , f * 2 1 , a * 1 1 , c * 1 (40)

3.4. Two nuclides, continuous energy and Z zones

Consider a case with 2 nuclides, continuous energy and the core subdivided into a number (Z) of active zones into which the densities of the fuel nuclides will be assumed space-averaged. The system of the equations governing the derivative functions in volume z results:

4 _{In practical cases the reactivity worth considered would be that corresponding to the real nuclide density} divided by the density itself.









0 m m m m p ~ c ~ c ~ n 0 . n . n (.) ) c c ( dE 0 V (.) n dt d V . n 0 E V (.) n dt d c V (.) c V (.) c V (.) . n ' dE c n ' dE c V . n ) ' E ( n ' dE ) ' E ( n ' dE V n n ) ' E ( n ' dE ) ' E ( n ' dE ) ' E ( n ' dE c c (.) ' dE c (.) ' dE c (.) ' dE c (.) ' dE c D dt d ) ( ) c ( ) c ( ) n ( j p / 2 p / 1 p / p / Z 1 ' z V E 2 , f Z 1 ' z V E 1 , f Z 1 ' z V ' z E 1 f,1 2 f,2 z E , V z 2 , c z E V 1 , a z E , V z 1 , c 2 z E , V 2 , c 1 z E , V 1 , a 1 z E , V 1 , c E , V 2 , f ' E 2 1 , f ' E 1 z E , V 2 , t E ' E 2 , s ' E 2 , f ' E z E , V 1 , a 1 , t E ' E 1 , s ' E 1 , f ' E t E ' E s ' E 2 2 , t 1 1 , t E ' E 2 , s ' E 2 E ' E 1 , s ' E 1 2 , f ' E 2 1 , s ' E 1 2 2 1 1 ' z ' z z z z z z z z z z z                           _{  }             _{  }                                                                                                                                                                                  





 























(41)













0 1 0 0 0 ) t t ( ~ c ~ c ~ n 0 0 0 (.) ' dE ) c c ( n . n . n V (.) n dt d 0 V . n V (.) n dt d V (.) n V (.) n ' dE V (.) ' dE n E V (.) n V (.) n ' dE V (.) ' dE n . c c V (.) ) c c ( V (.) c ) ' E ( n ' dE c c (.) ' dE c (.) ' dE c (.) ' dE ) c c ( D dt d F * * 2 * 1 * E , V ' E 2 2 , f 1 1 , f E . V 2 , f E . V 1 , f z E , V z 2 , c z E V 1 , a z E , V z 1 , c z E , V 2 , t ' E E , V ' E E 2 , s z E , V ' E 2 , f z E , V 1 , t ' E z E , V ' E E 1 , s z E , V ' E 1 , f E , V 2 , f 2 1 , f 1 z E , V z 2 2 , c 1 1 , a z E , V z 1 , c 1 t ' E E s ' E 2 2 , t 1 1 , t ' E E 2 , s ' E 2 ' E E 1 , s ' E 1 ' E 2 , f 2 1 , s 1 2 z z z z z z z z z z                         _{ }             _{ }                                                                                                                                                           



















              (42)



( c c )



dE' n (t t ) n _F E , V ' E * 2 2 , f 1 1 , f     

_

 (43)

which again shows how at the final time tF the neutron importance vector n* is a singular function

formed by the normalized standard neutron adjoint flux (with negative sign) multiplied by the Dirac’s function, i.e.,

* F * ) t t ( n    (44) .

For t<tF the equation corresponding to the first row, considering that n* is identically equal to zero,

reduces to: 0 ) c c ( c ) c c ( c c c₁_{c,1 1}* ₂ _{a,1 1}_{c,2 2 2}*  ₁_f,1 ₂_f,2 *     (45)

Multiplying by n and integrating over energy, we may then write:









E , V 2 , f 2 1 , f 1 E , V * 2 1 1 , a 2 2 , c * 1 1 , c 1 * c c c ) c c ( c c n             (46)

The equations corresponding to the second row, recalling the solution (43) for n* result within each zone z:                               _{  }                               _{  }            









      0 c n t t V n V n dE V dE n dt c d 0 c n c n t t V n V n dE V dE n dt c d 2 f 2 2 c F E z E V 2 t z E V E E 2 s z E V E 2 f 2 1 f 2 1 a 1 1 c F E z E V 1 t z E V E E 1 s z E V E 1 f 1 z z z z z z * , * , ' , * , , * ' , , ' * , * * , * , * , ' , * , , * ' , , ' * , * ) ( ' ' ) ( ' ' (47)

with * given by equation (46). As well known, the coefficients of the delta functions correspond to the ‘final’ conditions for functions *

1

                                          









      ' , * , , * ' , , * ' , * ' , * , , * ' , , ' * , * ' ' ' ' E z E V 2 t z E V E E 2 s z E V E 2 f 2 E z E V 1 t z E V E E 1 s z E V E 1 f 1 V n V n dE V dE n c V n V n dE V dE n c z z z z z z (48)

As in previous case, the values of c₁*(t_F)and c*₂(t_F) may be obtained following a procedure similar to that used for calculating a standard reactivity worth.

For t<tF the governing equations would result:

                    0 n c n dt c d 0 n c n c n dt c d * 2 , f * 2 2 , c * 2 * 1 , f * 2 1 , a * 1 1 , c * 1 (49)

3.5. M nuclides and Z zones

Consider the same case of previous section with M nuclides. The resulting equations may be easily obtained.

For *, defining with cm (m=1, 2,…, M) a nuclide inserted at the beginning of cycle (denoted as

‘primitive’) and with t

m

c (t=1,…,Tm) one produced from cm after t transmutations, we may write:

E , V m , f m T 1 m t m , f m M 1 m E , V * m m , a m T 1 m t * m m , c m M 1 m * t t m t ) 1 t ( ) 1 t ( m c c c c c c n                    









          (50)

0 terms other t t V n V n dE V dE n dt c d F E z E V x t z E V E E x s z E V E x f x z z z             _{  }            



_

   ) ( ' ' ' , * , , * ' , , ' * , * (51)

As well known, the coefficients of the delta functions correspond to the ‘final’ conditions for functions c₁* and c*₂. So we have, at tF, and at each zone z:





     _     _        ' , * , , * ' , , * ' , * ' ' E z E V x t z E V E E x s z E V E x f x V n V n dE V dE n c z z z (52)

As in previous cases, the value of c_x*(t_F) may be obtained following a procedure similar to that used for calculating a standard reactivity worth.

At t<tF, in relation to each primitive nuclide cm, we would have:

                                   0 n c n dt c d ... ... ... ... ... ... 0 n c n c n dt c d 0 n c n c n dt c d * m , f * m m , c * m * m , f * m m , a * m m , c * m * m , f * m m , a * m m , c * m m T m T m T m T 1 2 1 1 1 1 1 (53) 3.6. Numerical example

We shall consider now a simple example to illustrate the methodology, as that described above at paragraph 3.1. In this case the sensitivity coefficient results:

) t ( c DB c d ) t ( d F 2 f c 2 o F       (54)

Assuming a core representative of a lead cooled fast reactor fuelled with Pu-239 and the values given in Table 1, the sensitivity coefficient

o F c d t d( )

calculation coincide and result equal to -2.29E-20. In case the Pu239 density in the core is increased by 1% at beginning of cycle, this would correspond to a change of the control parameter  by -0.85%.

Table 1

Pu239 density D B2 Fission rate Fuel cycle length

Appendix

Control modality

In the procedure illustrated in the present work, a power control mechanism (or global fission rate) corresponding to a fictitious parameter () coefficient of the fission source has been assumed for simplicity. In the mathematical formalism it was represented in the equation that governs the neutron density with the term:

n dE _f E T _σ c ' '  



 . (A.1)

As we have seen above, this criterion corresponds to the orthonormality relationship ) ( ' c σ , * ' , , * F E V E T 1 s E V E n n dE n t t B n dE        





  (A.2)

The control modality over the fission source may be considered adequate in the majority of cases of interest for studies on the evolution of the core in critical systems during the burn-up. In particular cases, however, different control methods may be of interest, for example, the control on the density of the soluble boron in a water-cooled reactor, or the regulation of a control rod. Consider the latter case. The control can be parametrized by a coordinate, say zc, which establishes the lower insertion

point in the core (assuming that the rod is adjusted from above). The displacement zc of the rod

may be represented with the corresponding replacement of material in section zc: boron in case of

insertion, refrigerant otherwise. In mathematical formalism, this mechanism can be represented in the equation that governs the neutron density with a term of the type:

-



H z z dEΣ n H z z dEΣ n



E Pb c E B c c 



_  



_  ( ) ( ) (A.3)

where c is a parameter equal to 1 within the control rod and zero otherwise, B and Pb are the

macroscopic capture cross sections of boron and lead, respectively, while (z-zc) is the Heaviside

function (equal to 1 for z ≥zc and zero otherwise) while H is its complement (equal to 1 for z <zc

and zero otherwise). The effect of a control rod displacement is almost exclusively due to the greater or lesser presence of boron in the core, while, due to the quantities involved, the effect due to the displacement of lead may be considered negligible. The term (A.1) can therefore be replaced by the simple term:

- H z z dEΣ n

E B

c

c 



_

To note that, assuming the z-coordinate oriented upwards, a greater insertion downwards would result in a decrease in the value of zc.

By assuming this power control mechanism, the normalization condition for the adjoint flux at tF in

this case becomes:

1 n Σ dE z z _V E F B c c      



 * ) ( (A.5)

Consider now the standard adjoint flux  and the normalization condition: *_ 1 n F dE _V E    



  * . (A.6)

It easy to verify that the relationship between  and *_  results: *_

* * ) (  _  _F 1 (A.7) with ) ( ) ( ) ( ) ( ) ( ) ( * * * * c E c B c c E c B c V E B c c V E B c c z n Σ z dE S z n Σ z dE S 1 n Σ dE z z n Σ dE z z 1









                           (A.8)

References

1- A. Gandini, "Generalized Perturbation Theory (GPT) Methods. A Heuristic Approach", in

Advances in Nuclear Science and Technology, Vol. 19, Plenum Publishing Corporation,

New York, 1987.

2- A. Gandini, "Sensitivity Analysis of Source Driven Subcritical Systems by the HGPT Methodology", Proc. Intern. IAEA Techn. Committee Meet., Madrid 17-19 Sept. 1997 (IAEA-TC-903.3, p. 377), and: Annals of Nuclear Energy, 24, 1241 (1997)

3- A. Gandini, M. Frullini, V. Peluso, “Implementazione del codice ERANOS per l’analisi perturbativa e di sensitività nel campo non lineare neutroni/nuclidi”,

http://www.enea.it/it/Ricerca_sviluppo/documenti/ricerca-di-sistema-elettrico/nucleare- iv-gen/2014/rds-par2014-191.pdf

4- A. Gandini, V. Peluso, “Metodologia HGPT-BU per l’analisi perturbativa nel campo neutroni/nuclidi. Implementazione e validazione del codice Eranos, Rapporto Tecnico ENEA ADP-LP2-143 (2017)

5- G. Rimpault et al., “The ERANOS code and data system for fast reactor neutronic analyses”, Proc. PHYSOR 2002, Seoul, Korea, October 7-10, 2002

6- L. A. Linsternik and V. J. Sobelev, “Elements of functional analysis”, Frederik Ungar Publishing Company, New York (1961)

7- G. Soodak, “The Science and Engineering of Nuclear Power”, United Nations, New York (1948)