Subcriticality level inferring in the ADS systems: spatial corrective factors for Area Method
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Transcript of 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
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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: and p evaluation
Analysis of the Experimental results: Slope analysis by α-fitting method
• Conclusions
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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:
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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
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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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α-fitting method
Area method
Experimental techniques analyzed
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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.
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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
∞
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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
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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)
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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*Modified Source Approximation
+Modified Source Multiplication
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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)
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Mean/St.Dev: -12.6 ± 0.4
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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
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
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
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Mean/St.Dev: -8.86 ± 0.16
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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
MUSE-4 SC3 972 cells configuration, D-T Source
Dispersion means the ratio ρ(Reference)/ ρ(AREA)exp or calc.
[*] From Y. Rugama
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
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Mean/St.Dev: -13.7 ± 0.5
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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:
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
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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
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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:
K
effKp, Λ
βρα
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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:
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
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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)
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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
.
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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.
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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.
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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.
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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
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
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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%
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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
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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)
φk eigenfunctions (k calculation)
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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)
φk eigenfunctions (k 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