subcriticality level inferring in the ads systems: spatial corrective factors for area method

Forschungszentrum Karlsruhein der Helmholtz-Gemeinschaft

Subcriticality level inferring in the

ADS systems:

spatial corrective factors for Area

MethodF. Gabrielli

Forschungszentrum Karlsruhe, Germany

Institut für Kern- und Energietechnik (FZK/IKET)

Second IP-EUROTRANS Internal Training Course

June 7 – 10, 2006

Santiago de Compostela, Spain 1


Layout of the presentation

• Principle of Reactivity Measurements

• MUSE-4 Experiment

• PNS Area Method: a static approach

Analysis of the Experimental results: Area method analysis

• PNS α-fitting method: and p evaluation

Analysis of the Experimental results: Slope analysis by α-fitting method

• Conclusions

2


Principle of Reactivity Measurements

If point kinetics assumptions fail, correction factors are needed.

MUSE-4 experiment supplied a lot of information about this subject

Reactivity does not depend on the detector position, detector type, …

Some quantities, i.e. the mean neutron generation time Λ which is used in the slope method, do not depend on the subcritical level.

Several static/kinetics methods are available to infer the reactivity level of a subcritical system.

All these methods are based on the point kinetics assumption, then assuming that:

3


In this case, corrective spatial factors, evaluated by means of calculations,

should be applied to the experimental results analyzed by means of one of the

point kinetics based methods, in order to infer the actual subcriticality level of

the system.

Depending on the used method, corrective factors may have a different

amplitude. Thus, from a theoretical point of view, the reliability of a method for

inferring the reactivity will be given by the magnitude of the corrective factors

to be associated.

Depending on the subcriticality level and on the presence of spatial effects, the

subcriticality level of the system may not be inferred by the detectors

responses in different positions on the basis of a pure point kinetics approach.

Principle of Reactivity Measurements

4


MUSE-4 experiment: layout

MUSE (MUltiplication avec Source Externe) program was a series of zero-power experiments

carried out at the Cadarache MASURCA facility since 1995 to study the neutronics of ADS .

The main goal was investigating several subcritical configurations (keff is included in the

interval 0.95-1) driven by an external source at the reactor center by (d,d) and (d,t) reactions, the incident deuterons being provided by the GENEPI deuteron pulsed accelerator.

5


MUSE-4 experiment: layout and objectives

In particular, the MUSE-4 experimental phase aimed to analyze the system response to neutron pulses provided by GENEPI accelerator (with frequencies from 50 Hz to 4.5 kHz, and less than 1 μs wide), in order to investigate by means of several techniques the possibility to infer the subcritical level of a source driven system, in view of the extrapolation of these methods to an European Transmutation Demonstrator (ETD).

6


α-fitting method

Area method

Experimental techniques analyzed

7


is based on the

following relationship relative to the

areas subtended by the system

responses to a neutron pulse:

PNS Area Method

Concerning the method (which does not invoke the estimate of Λ), it is not possible

"a priori" to evaluate the order of magnitude of correction factors even if the system

response appears to be different from a point kinetics behaviour.

This aspect is strictly connected with the integral nature of the PNS area methods

d

p

eff I

I

areaneutrondelayed

areaneutronprompt

Because of spatial effects, reactivity is function of detector position. These spatial effects can be taken into account by solving inhomogeneous transport time-independent problems.

8


PNS Area Method: a static approach*

[*] S. Glasstone, G. I. Bell, ‘Nuclear Reactor Theory’, Van Nostrand Reinhold Company, 1970

Neutron source is represented by Q(r,,E,t)=Q(r,,E)δ+(t) and

a signal due to prompt neutrons alone is considered

The prompt flux p(r,,E,t) satisfies the transport equation

pp p p p

Φ1Φ σΦ SΦ (1-β)FΦ Q(r,Ω,E) (t)

v t p

With the usual free-surface boundary conditions and the initial condition p(r,,E,t)=0

Defining the prompt neutron flux Φp(r,Ω,E)=∫Φp(r,Ω,E,t)dt and after integrating over the time…

Where the initial condition was used and the fact that lim (t) Φp=0 because the reactor is subcritical

Therefore, the time integrated prompt-neutron flux satisfies the ordinary time-indipendent transport equation

Hence, it can be determined by any of standard multigroup methods

E)Ω,Q(r,Φ~

β)F(1χΦ~

SΦ~

σΦ~

Ω ppppp

~

0

∞

9


Prompt Neutron Area = ∫ D(r,t)dt=∫∫∫σd(r,E)ΦpdVdΩdE∞ ~

0

PNS Area Method: a static approach*

0

pp t)dtE,Ω,(r,ΦE)Ω,(r,Φ~

E)Ω,Q(r,Φ~

β)F(1χΦ~

SΦ~

σΦ~

Ω ppppp

The time integrated prompt-neutron flux satisfies

the ordinary time-independent transport equation

The total time-integrated flux Φ(r,Ω,E) satisfies the same equation with χp(1-β) replaced by χ

Delayed Neutron Area =

-ρ($)=Prompt Neutron Area

Delayed Neutron Area

~

∫∫∫ σd(Φ - Φp)dVdΩdE~~

[*] S. Glasstone, G. I. Bell, ‘Nuclear Reactor Theory’, Van Nostrand Reinhold Company, 1970

10


ERANOS (European Reactor ANalysis Optimized System) calculation description

• A XY model of the configurations was assessed

• The reference reactivity level was tuned via buckling

• JEF2.2 neutron data library was used in ECCO (European Cell Code) cell code

• 33 energy groups transport calculations were performed by means of BISTRO

core calculation module

11


MUSE-4 SC0 1108 Fuel Cells Configuration – DT Source

The configuration with 3 SR up, SR 1 down and PR down was analyzed

Reference Reactivity:

-12.53 $

(Evaluations based on MSA*/MSM+

measurements in a previous configuration)

Experimental data from

E. González-Romero et al., "Pulsed Neutron Source

measurements of kinetic parameters in the source-driven fast

subcritical core MASURCA", Proc. of the "International

Workshop on P&T and ADS Development", SCK-CEN, Mol,

Belgium, October 6-8, 2003.

F. Mellier, ‘The MUSE Experiment for the subcritical

neutronics validation’, 5th European Framework Program

MUSE-4 Deliverable 6, CEA, June 2005.

12

*Modified Source Approximation

+Modified Source Multiplication


Sc0 results

Reactivity ρ($) Dispersion

Detector Experimental [*] Calculated Experimental Calculated (E-C)/C (%)

I -14.3 -13.1 0.8762 0.9561 +7.5

L -12.9 -13.0 0.9713 0.9658 -0.6

F -11.9 -11.8 1.0529 1.0603 +0.7

M -12.7 -12.8 0.9866 0.9783 -0.8

G -13.0 -12.4 0.9638 1.0121 +5.0

N -12.1 -11.8 1.0355 1.0587 +2.2

H -12.6 -12.1 0.9944 1.0369 +4.3

A -12.7 -12.4 0.9866 1.0140 +2.8

B -13.0 -12.8 0.9638 0.9824 +1.9

MUSE-4 SC0 1108 cells configuration, D-T Source, 3 SR up SR1 down PR downDispersion means the ratio ρ(MSM)/ ρ(AREA)exp or calc.

[*] E. Gonzáles-Romero (ADOPT ’03)

13

Mean/St.Dev: -12.6 ± 0.4



Reference Reactivity

(Rod Drop + MSM):

-8.7 ± 0.5 $

14


SC2 results



I -8.6 -8.6 1.012 1.012 0.0

L -8.8 -8.9 0.989 0.978 1.1

F -8.9 -9.0 0.978 0.967 1.1

C -8.7 -8.8 1.000 0.989 1.1

G -9.0 -8.8 0.967 0.989 -2.2

D -8.9 -8.7 0.978 1.000 -2.2

H -8.9 -8.7 0.978 1.000 -2.2

A -8.9 -8.8 0.978 0.989 -1.1

B -9.0 -8.8 0.967 0.989 -2.2

MUSE-4 SC2 1106 cells configuration, D-T Source

Dispersion means the ratio ρ(Reference)/ ρ(AREA)exp or calc.

[*] E. Gonzáles-Romero, ADOPT ‘03

15

Mean/St.Dev: -8.86 ± 0.16



Reference Reactivity

(Rod Drop + MSM):

-13.6 ± 0.8 $

16


SC3 results

MUSE-4 SC3 972 cells configuration, D-T Source

Dispersion means the ratio ρ(Reference)/ ρ(AREA)exp or calc.

[*] From Y. Rugama



I -12.9 -13.0 1.054 1.046 0.8

L -14.4 -13.8 0.944 0.986 -4.2

F -14.0 -14.0 0.971 0.971 0.0

C -13.7 -13.7 0.993 0.993 0.0

A -13.8 -13.6 0.986 1.000 -1.4

B -13.8 -13.6 0.986 1.000 -1.4

J -12.9 -12.9 1.054 1.054 0.0

K -12.9 -12.8 1.054 1.063 -0.8

17

Mean/St.Dev: -13.7 ± 0.5


Experimental results for α-fitting analysis

18


PNS α-fitting analysis in MUSE-4

Concerning the PNS α-fitting method (which invokes the evaluation of Λ), three

types of possible MUSE-4 responses to a short pulse may be obtained:

a) The system responses show the same 1/τ-slope in all the positions (core, reflector and shield), thus the system behaves as a point.

b) The system responses show a 1/τ-slope only in some positions, but not all the slopes are equal; the system does not show an ‘integral’ point kinetics behavior and a reactivity value position-depending will be evaluated. Thus, corrective factors have to be applied in order to take into account the reactivity spatial effects.

c) The system responses do not show any 1/τ-slopes; the system does not behave anywhere as a point and only experimental data fitting can try to solve the problem. As in the previous case, corrective factors have to be applied.

19


Corrective factors approach to the α-fitting analysis

When PNS α-fitting method is performed, we assumed that, at least in the prompt time domain, the flux behaves like:

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 20 40 60 80 100t (s)

(u.a.)if we are coherent with this hypothesis, we have to

perform the substitution of our factorised flux into:

Consequently in the prompt time domain, the (time-constant) shape of the flux obeys the eigenvalue relationship:

)t(t

)t(

v

1

),E,(e)t,,E,( tp rr

20


Corrective factors approach to the α-fitting analysis: flow chart

Directly evaluated by the α-eigenvalue equation

“Prompt version” of the inhour equation

(p>>i)

d

deff,p Λ

βρα

21


Corrective factors approach to the α-fitting analysis: flow chart

It is possible to follow the standard way to calculate αp starting from the k

eigenvalue equation:

K

effKp, Λ

βρα

22


Prompt α Calculation procedure performed by means of ERANOS

ERANOS core calculation transport spatial modules (BISTRO and TGV/VARIANT)

solve the k eigenvalue equation:

While, for our purpose, the following eigenvalue relationship has to be solved:

0)1(K

1

v fpinsp

t

K=1

g

pg,z,c

modg,z,c v

g,pz

modg,z )1(

…that means performing the following substitution if ERANOS is used

23


Prompt α Calculation procedure: MUSE-4 SC0 analysis

keff ρ βeff ΛK(ms) αp,k (s-1)

k calculation 0.95970 -0.04200 0.00335 0.4683 -96821

kd ρ βeff,d Λd(ms) αp (s-1)

α calculation 0.95843 -0.04337 0.00368 1.0069 -46730

Red data indicate eigenvalues directly evaluated by ERANOS (XY model)

+47% -48%

1108 Fuel Cells Configuration (3 SR up, SR 1 down and PR down) – DT Source

Reactivity values calculated by using φK and ψ eigenfunctions are similar

(compensation in the product α· Λ)

24


0.0E+00

5.0E-02

1.0E-01

1.5E-01

2.0E-01

2.5E-01

1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 1.E+08

Energy (eV)

Nor

mal

ized

Neu

tron

Spec

trum

a.u

.

25

Spectra in the shielding and in the reflector

0.0E+00

5.0E-02

1.0E-01

1.5E-01

2.0E-01

2.5E-01

1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 1.E+08

Energy (eV)

Nor

mal

ized

Neu

tron

Spe

ctru

m a

.u.

ψ eigenfunctions (α calculation)

φk eigenfunctions (k calculation)

Reflector Shielding

According to the theory, differences between ψ and φk eigenfunctions energy profiles at low energies are

mainly observed in the reflector and in the shielding regions: in fact, besides the different fission spectrum,

the main differences will be localized in the spatial and energetic regions where α/v is equal or greater than the

Σt term. Such happens at low energies and inside, or near, reflecting regions at low absorption, where the

profile of the ψ shapes functions spectra will be more marked than those of the φk functions, because of the

lower absorptions.


Comparison among the calculated results

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

0 25 50 75 100 125 150 175 200Time ( s)

Arb

itrar

y U

nit

y = exp(alp ha *t) (from alpha eigenva lue calculation)

K IN3D: Detector F (Core)

KIN 3D: Dete ctor N (Re flector )

KIN 3D : Detector A (Shield )

MCNP: De tector F (Core)

MCN P: Detector N (Re flector)

M CN P: Detec tor A (Shield)

y=exp(αpt)

Results seem to provide a coherent picture concerning the system location where α-fitting method (with refined Λ evaluation) could be applied, i.e. far from the source.

In any case, point kinetics αp slope

seems to agree with exponential 1/τ-

slope only in the shield and for a

short time period.

26


MCNP Vs Experimental results

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

0 25 50 75 100 125 150 175 200Time (s )

Arb

itra

ry U

nit

t

MCN P: Detector F (Core )

M CN P: Detec tor N (Reflector)

M CNP: De tector A (Shield)

Experimental: De tector A (Shield )

Experimental: Detector F (Fuel )

Experimental: Detector N (Reflector)

Reflector and shield experimental slopes show a double exponential behavior which is not reproduced by MCNP calculations; on the contrary, it looks evident a good agreement for a short time period.

Experimental results show that for large subcriticalities, 1/τ-slopes are different for core, reflector and shield detectors positions. MCNP results well reproduce in the core the experimental responses.

27


Conclusions

1. For large subcriticalities, PNS area method seems to be more reliable respect

to a-fitting method, for what concerns the order of magnitude of the spatial

correction factors (about 5%).

2. Concerning the application to the ADS situation, because of the beam time

structure required for an ADS, it does not allow an on-line subcritical level

monitoring, but can be used as “calibration” technique with regards to some

selected positions in the system to be analyzed by alternative methods, like

Source Jerk/Prompt Jump (which can work also on-line).

3. Codes and data are able to predict the MUSE time-dependent behavior in the

core region. The presence of a second exponential behavior in the reflector

and shield regions is not evidenced either by the deterministic or by the MCNP

simulations.

28


THANK YOU FOR YOUR

KIND HOSPITALITY


Prompt α Calculation procedure: pre-analysis

Reflector NA/SS

MOX1

Radial Shielding

Axial Shielding

169.6

159

148.4

137.8

121.9

116.6

100.7

95.4

84.8

74.2

63.6

42.4

31.8

21.2

10.6

8.28 18.5 33.1 39.7 55.9 97.03

Lead

MOX3

Homogenized Beam Pipe

MUSE-4 Sub-Critical ERANOS RZ model: symmetry axis around the Genepi Beam Pipe axis

Z (cm)

R (cm)

Positions for neutron spectra analysis

Core

17 cm,92.8 cm

Reflector

57.5 cm, 92.8 cm

Shield

66.4 cm, 129.9 cm

a1


Prompt α Calculation procedure: pre analysis results

Red data indicate eigenvalues directly evaluated by ERANOS (RZ model)

keff ρ βeff ΛK(ms) αp,k (s-1)

k calculation 0.97124 -0.02961 0.00335 0.51634 -63834

kd ρ βeff,d Λd(ms) αp (s-1)

α calculation 0.97166 -0.02916 0.00369 0.81633 -40240

+37% -37%

a2


αp / αp,k Ratio at Different Reactivity Levels

0

0.2

0.4

0.6

0.8

1

1.2

0.96 0.97 0.98 0.99 1 1.01 1.02 1.03

αp / αp,k

keff

Far from criticality, the deviation is mainly due to the differences between the mean neutron generation times ΛK and Λd

evaluated using respectively φK and ψ

eigenfunctions.

αp/αp,k ratio deviates from

the unity depending on the subriticality level

a3


Spectra in the core

0.0E+00

5.0E-02

1.0E-01

1.5E-01

2.0E-01

2.5E-01

1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 1.E+08

Energy (eV)

Nor

mal

ized

Neu

tron

Spe

ctru

m a

.u.

Core

ψ eigenfunctions (α calculation)


a4


0.0E+00

5.0E-02

1.0E-01

1.5E-01

2.0E-01

2.5E-01

1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 1.E+08

Energy (eV)

Nor

mal

ized

Neu

tron

Spe

ctru

m a

.u.

Spectra in the shielding and core selected positionsψ eigenfunctions (α calculation)


0.0E+00

5.0E-02

1.0E-01

1.5E-01

2.0E-01

2.5E-01

1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 1.E+08

Energy (eV)

Nor

mal

ized

Neu

tron

Spec

trum

a.u

.

Reflector Shielding

According to the theory, differences between ψ and φk eigenfunctions energy profiles at low energies are

mainly observed in the reflector and in the shielding regions: in fact, besides the different fission spectrum,

the main differences will be localized in the spatial and energetic regions where α/v is equal or greater than the

Σt term. Such happens at low energies and inside, or near, reflecting regions at low absorption, where the

profile of the ψ shapes functions spectra will be more marked than those of the φk functions, because of the

lower absorptions. a5

subcriticality level inferring in the ads systems: spatial corrective factors for area method

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