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Subcriticality level inferring in the ADS systems:
spatial corrective factors for Area Method
F. 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
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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: L and ap evaluation
 Analysis of the Experimental results: Slope analysis by α-fitting method
• Conclusions
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Principle of Reactivity Measurements
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:
 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.
If point kinetics assumptions fail, correction factors are needed.
MUSE-4 experiment supplied a lot of information about this subject
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Principle of Reactivity Measurements
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.
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.
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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.
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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).
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Experimental techniques analyzed
α-fitting method
Area method
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PNS Area Method is
based
on
the
following relationship relative to the
areas
subtended
by
the
system
responses to a neutron pulse:
Ip

prompt neutron area


eff
delayed neutron area
Id
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
Because of spatial effects, reactivity is function of detector position. These spatial
effects can be taken into account by solving inhomogeneous transport timeindependent problems.
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PNS Area Method: a static approach*
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
1 Φ p
  Φ p  σΦ p  SΦ p   p (1-β)FΦ p  Q(r,Ω,E)  (t)
v t
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…
0
~
~
~
~
Ω  Φ p  σΦ p  SΦ p  χ p (1  β)FΦ p  Q(r, Ω, E)
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 timeindipendent transport equation
Hence, it can be determined by any of standard multigroup methods
[*] S. Glasstone, G. I. Bell, ‘Nuclear Reactor Theory’, Van Nostrand Reinhold Company, 1970
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PNS Area Method: a static approach*

~
Φ p (r, Ω, E)   Φ p (r, Ω, E, t)dt
~
~
~ 0
~
Ω  Φ p  σΦ p  SΦ p  χ p (1  β)FΦ p  Q(r, Ω, E)
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
Prompt Neutron Area =
∞
~
∫ D(r,t)dt=∫∫∫σd(r,E)ΦpdVdΩdE
0
-ρ($)=
Delayed Neutron Area =
~
Prompt Neutron Area
Delayed Neutron Area
~
∫∫∫ σd(Φ - Φp)dVdΩdE
[*] S. Glasstone, G. I. Bell, ‘Nuclear Reactor Theory’, Van Nostrand Reinhold Company, 1970
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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
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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)
*Modified Source Approximation
+Modified Source Multiplication
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.
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Sc0 results
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MUSE-4 SC0 1108 cells configuration, D-T Source, 3 SR up SR1 down PR down
Dispersion means the ratio ρ(MSM)/ ρ(AREA)exp or calc.
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
Mean/St.Dev: -12.6 ± 0.4
[*] E. Gonzáles-Romero (ADOPT ’03)
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MUSE-4 SC2 1106 Fuel Cells Configuration – DT Source
Reference Reactivity
(Rod Drop + MSM):
-8.7 ± 0.5 $
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SC2 results
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MUSE-4 SC2 1106 cells configuration, D-T Source
Dispersion means the ratio ρ(Reference)/ ρ(AREA)exp or calc.
Reactivity ρ($)
Dispersion
Detector
Experimental [*]
Calculated
Experimental
Calculated
(E-C)/C (%)
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
Mean/St.Dev: -8.86 ± 0.16
[*] E. Gonzáles-Romero, ADOPT ‘03
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MUSE-4 SC3 1104 Fuel Cells Configuration – DT Source
Reference Reactivity
(Rod Drop + MSM):
-13.6 ± 0.8 $
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SC3 results
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MUSE-4 SC3 972 cells configuration, D-T Source
Dispersion means the ratio ρ(Reference)/ ρ(AREA)exp or calc.
Reactivity ρ($)
Dispersion
Detector
Experimental [*]
Calculated
Experimental
Calculated
(E-C)/C (%)
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
Mean/St.Dev: -13.7 ± 0.5
[*] From Y. Rugama
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Experimental results for α-fitting analysis
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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.
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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:
(r , E, , t )  e
apt
 (r , E,  )
1.2
1.0
0.8
if we are coherent with this hypothesis, we have to
perform the substitution of our factorised flux into:
 (u.a.) 0.6
0.4
0.2
1 ( t )
 (t )
v t
0.0
0
20
40
60
80
100
t ( s)
Consequently in the prompt time domain, the (time-constant) shape of the flux obeys
the eigenvalue relationship:
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Corrective factors approach to the α-fitting analysis: flow chart
“Prompt version” of the
inhour equation (ap>>li)
αp 
ρ  β eff, d
Λd
Directly evaluated by the
α-eigenvalue equation
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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:
α p,K
ρ  β eff

ΛK
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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:
…that means performing the following substitution if ERANOS is used
ap 

1
      t  
     ins    (1  ) p    f   0
v 
K

 cmod
,z ,g   c ,z ,g 
ap
vg
K=1
 mod
z , g  (1   z ) p, g
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Prompt α Calculation procedure: MUSE-4 SC0 analysis
1108 Fuel Cells Configuration (3 SR up, SR 1 down and PR down) – DT Source
Red data indicate eigenvalues directly evaluated by ERANOS (XY model)
k calculation
α calculation
keff
ρ
βeff
ΛK(ms)
αp,k (s-1)
0.95970
-0.04200
0.00335
0.4683
-96821
kd
ρ
βeff,d
Λd(ms)
αp (s-1)
0.95843
-0.04337
0.00368
1.0069
-46730
Reactivity values calculated by using
φK and ψ eigenfunctions are similar
(compensation in the product α· Λ)
+47%
-48%
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ψ eigenfunctions (α calculation)
φk eigenfunctions (k calculation)
Spectra in the shielding and in the reflector
2.5E-01
2.5E-01
Reflector
Shielding
2.0E-01
Normalized Neutron Spectrum a.u.
Normalized Neutron Spectrum a.u.
2.0E-01
1.5E-01
1.0E-01
1.5E-01
1.0E-01
5.0E-02
5.0E-02
0.0E+00
1.E-01
0.0E+00
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)
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)
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.
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Comparison among the calculated results
y=exp(α
y = exp(alp
pt) ha *t) (from alpha eig en va lue calcul ation)
1.E+0 0
K IN3D: Det ect or F (Co re)
K IN 3D: Dete ct o r N (Re flect or )
KIN 3D : Det ector A (S hi eld )
MC NP : De tect or F (Co re)
1.E-0 1
MC N P: Detect o r N (Re flector)
Arb itrary Un it
M CN P : Detec tor A (S hi eld)
1.E-0 2
1.E-0 3
Results
seempoint
to provide
a coherent
In
any case,
kinetics
αp slope
picture
concerning the system
1.E-0
seems
to4 agree with exponential 1/τlocation where α-fitting method
(with refined
evaluation)
could
slope
only inΛthe
shield and
forbea
applied, i.e. far from the source.
short1.E-0
time
period.
5
0
25
50
75
10 0
12 5
15 0
175
20 0
Ti me (? s)
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MCNP Vs Experimental results
MCN P: Detecto r F (C ore )
1.E +00
M CN P: Det ec tor N (Reflect o r)
M CN P: De tector A (Shield)
Exp erimen tal: De tector A (Shield )
1.E-01
Ex perim en tal: Det ector F (Fuel )
Arb it rary Unitt
Ex peri m ent al: Det ector N (Refl ect o r)
1.E-02
1.E-03
Experimental
Reflector andresults
shield show
experimental
that forslopes
large
subcriticalities,
show a double exponential
1/τ-slopes are
behavior
different
which
for
1.E-04
core,
is not reproduced
reflector and
by MCNP
shieldcalculations;
detectors
positions.
on the contrary,
MCNP results
it lookswell
evident
reproduce
a good
in
the
agreement
core the for
experimental
a short time
responses.
period.
1.E-05
0
25
50
75
1 00
1 25
1 50
1 75
2 00
Tim e ( ? s )
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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.
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Prompt α Calculation procedure: pre-analysis
169.6
159
Positions for neutron
spectra analysis
148.4
Lead
137.8
Shield
121.9
Reflector NA/SS
66.4 cm, 129.9 cm
116.6
100.7
MOX1
Reflector
95.4
57.5 cm, 92.8 cm
84.8
Radial Shielding
74.2
Core
63.6
17 cm,92.8 cm
Axial Shielding
Homogenized Beam Pipe
42.4
31.8
MOX3
Z (cm)
21.2
10.6
R (cm)
8.28
18.5
33.1
39.7
55.9
97.03
MUSE-4 Sub-Critical ERANOS RZ model: symmetry axis around the Genepi Beam Pipe axis
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Prompt α Calculation procedure: pre analysis results
Red data indicate eigenvalues directly evaluated by ERANOS (RZ model)
k calculation
α calculation
keff
ρ
βeff
ΛK(ms)
αp,k (s-1)
0.97124
-0.02961
0.00335
0.51634
-63834
kd
ρ
βeff,d
Λd(ms)
αp (s-1)
0.97166
-0.02916
0.00369
0.81633
-40240
+37%
-37%
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αp / αp,k Ratio at Different Reactivity Levels
1.2
1
0.8
αp / αp,k
αp/αp,k ratio deviates from
the unity depending on the
subriticality level
0.6
0.4
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.
0.2
0
0.96
0.97
0.98
0.99
1
1.01
1.02
1.03
keff
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Spectra in the core
2.5E-01
Normalized Neutron Spectrum a.u.
2.0E-01
Core
1.5E-01
ψ eigenfunctions (α calculation)
φk eigenfunctions (k calculation)
1.0E-01
5.0E-02
0.0E+00
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)
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ψ eigenfunctions (α calculation)
φk eigenfunctions (k calculation)
Spectra in the shielding and core selected positions
2.5E-01
2.5E-01
Reflector
Shielding
2.0E-01
Normalized Neutron Spectrum a.u.
Normalized Neutron Spectrum a.u.
2.0E-01
1.5E-01
1.0E-01
1.5E-01
1.0E-01
5.0E-02
5.0E-02
0.0E+00
1.E-01
0.0E+00
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)
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)
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