Propagation of nuclear data uncertainties in fuel cycle calculations using MONTE-CARLO Technique

PROPAGATION OF NUCLEAR DATA

UNCERTAINTIES IN FUEL CYCLE

CALCULATIONS

CALCULATIONS

USING MONTE-CARLO TECHNIQUE

C.J. Díez

(1)

_{, O. Cabellos}

(1)

_{, J.S. Martínez}

(1) (1) _{Universidad Politécnica de Madrid (UPM)Universidad Politécnica de Madrid (UPM)}

International Conference on Mathematics and Computational Methods Applied to Nuclear Science and Engineering

(M&C 2011) (M&C 2011)

Abstract

The

uncertainty propagation

in fuel cycle calculations

due to Nuclear Data

(ND) is a

important issue for :

important issue for :

• Present fuel cycles (e.g. high burnup fuel programme)

• New fuel cycles designs (e.g. fast breeder reactors and ADS)

Different error propagation techniques

can be used:

• Sensitivity analysis

• Response Surface Method

Response Surface Method

•

Monte Carlo technique

Then, in this paper, it is assessed the

,

p p ,

impact of ND uncertainties on the decay heat

p

y

and radiotoxicity in two applications:

• Fission Pulse Decay Heat calculation (

y

(

FPDH

)

)

• Conceptual design of European Facility for Industrial Transmutation (

EFIT

)

The complete set of

uncertainty data

for

cross sections

(EAF2007/UN),

decay data

OUTLINE

PART I:

Methodology to propagate ND uncertainties

using Monte Carlo technique

PART II:

Application of Monte Carlo technique

pp

q

A. Fission Pulse Decay Heat calculation

B. EFIT fuel cycle calculation

Methodology to propagate ND uncertainties

PART I

Goal: “To analyse how ND uncertainties are transmitted

to response functions”

[ ]

N

[ ]

N

[

]

N

A

N

dN

λ

eff

Φ

₍

₎

eff

Φ

(

λ

)

N

N

[ ]

N

[ ]

N

[

]

N

A

N

dt

eff fiss

eff

⋅

Φ

+

⋅

Φ

=

⋅

+

=

λ

σ

(

γσ

)

N

_i

=

N

_i

(

λ

,

σ

,

γ

)

1) Sensitivity / Uncertainty Analysis (S/U)

First order Taylor series (

linear approximation

)

2)

Monte Carlo Uncertainty Analysis (MC)

T

h

l b l ff

f ll

l

d

i i

To treat the global effect of all nuclear data uncertainties

Methodology to propagate ND uncertainties

PART I

Monte Carlo technique

Individual / All together sampling

(

λ

,

σ

,

γ

)

PDFs

_N

_{l di t ib ti}

_{L N}

_{l di t ib ti}

(

)

) , 0 ( / log 10 1 M N → ⎟ ⎟ ⎟ ⎞ ⎜ ⎜ ⎜ ⎛

σ

σ

M

PDFs

)

,

0

(

)]

(

,

[

_{j0 j j j}

j

→

N

σ

DST

σ

⇒

ε

→

N

Δ

σ

0

Maybe

I

↑↑

Δ

⇒

<

⇒

f

σ

Normal distribution

LogNormal distribution

0

Always

σ

_i

>

(

_m / _m0

)

⎟_⎠

⎜

⎝

σ

σ

S

li

l

lib

i

ACA

l

0

Maybe

I

↑↑

Δ

⇒

<

⇒

f

_j

σ

_j

Samplig

γ

λ

Nuclear Data libraries

Collapsed

λ

σ

γ

λ

₁

,

₁

,

₁

LogNormal

distribution

N

N

₁

ACAB

Results

Mean Values

Uncertainties

(Standard Desv)

σ

γ

λ

,

,

n n

n

γ

σ

λ

σ

γ

λ

,

,

...

,

,

_{2 2}

2 n

N

N

...

2

Methodology to propagate ND uncertainties

PART I

Uncertainty data

Î

Cross section from activation-oriented nuclear data libraries

EAF2007-UN

E

_i+1

(eV)

W -180N,G 748033102

7.41800E+4 1.7840E+02 0 0 0 1748033102 0.0000E+00 0.0000E+00 0 102 0 1748033102 0.0000E+00 0.0000E+00 0 1 10 5748033102

i+1

(

)

e.g.:

180 0.0000E+00 0.0000E+00 0 1 10 5748033102

1.0000E-05 1.0000E+00 5.0000E+00 1.8404E-01 1.0140E+02 2.5000E-01748033102 2.0000E+07 2.5000E-01 6.0000E+07 0.0000E+00 748033102

E (eV)

Δ

2I=1 EAF

(relative error

_Δ

)~

_Δ

=

_Δ

/3

W

180

_(n,

γ

₎

.

E

_i

(eV)

Δ

I=1,EAF

(relative error,

_Δ

)~

_Δ

_I=1,EXP

=

_Δ

_I=1,EAF

/3

Î

Fission yield from evaluated nuclear data library

JEFF 3.1.1

γ

Th232→H3,400KeV

+

1

σ

γ

9.023200+4 2.300450+2 2 0 0 03486 8454 1 4.0000E+05 0.0000E+00 1 0 3664 9163486 8454 2 1.0010E+03 0.0000E+00 1.6073E-05 5.5423E-06 1.0020E+03 0.0000E+003486 8454 3 4.9121E-06 1.6564E-06 1.0030E+03 0.0000E+00 7.0081E-05 2.2139E-053486 8454 4

Th232

400 KeV

Methodology to propagate ND uncertainties

PART I

Processing and collapsing of nuclear data

Collapsing method:

C

ti

C

ti

f

ti

t

-Cross section: Conservation of reaction rate

T eff

i j E j i

i

j

E

E

dE

Rate

_→

=

∫

σ

_→

(

)

⋅

φ

(

)

⋅

=

σ

_→

⋅

φ

.

-Uncertainties: Using

Sandwich rule

(

Propagation of Momentum, first order)

ω

ω

T

V

=

Δ

2

H. Hiruta

et al.

, “

Few Group Collapsing of Covariance Matrix Data Based on a

ω

ω

V

=

Δ

H. Hiruta

et al.

, “

Few Group Collapsing of Covariance Matrix Data Based on a

p

p

g

Conservation Principle

”, Nuclear Data Sheets, vol. 109, 2801-2806, (Dec 2008)

Collapsing without losing information

p

p

g

Conservation Principle

”, Nuclear Data Sheets, vol. 109, 2801-2806, (Dec 2008)

Methodology to propagate ND uncertainties

PART I

Processing and collapsing of nuclear data

Given

V

the G-by-G variance matrix of the relative cross sections vector, the variance

Î

Cross section

y

,

Δ

2

_{of the relative spectrum-averaged cross section is:}

ω

ω

V

T

=

Δ

2 T G G

]

[

φ

1

σ

1

φ

σ

ω

=

L

φ

=

φ

1

+

φ

2

+

L

+

φ

G

with

.

eff

eff

,

,

]

[

σ

φ

σ

φ

ω

=

L

with

G G G eff

φ

φ

φ

σ

φ

σ

φ

σ

φ

σ

+

+

+

+

+

+

=

L

L

2 1 2 2 1 1

Î

Fission yield

Given

G

the M-by-M variance matrix of the relative fission yield vector, the variance

Δ

2

of the relative spectrum-averaged fission yield is:

Δ

2

=

ω

T

_G

ω

Î

Fission yield

G j fiss G j fiss G j fiss G i j G j fiss i j eff i j

φ

σ

φ

σ

φ

γ

σ

φ

σ

γ

γ

_, 1 , 1 , , 1 , 1 , 1 ,

...

...

+

+

+

+

=

T eff fi fiss G G eff fi fiss

]

,

,

[

1 1, ,

σ

σ

φ

φ

σ

σ

φ

φ

ω

=

L

of the relative spectrum-averaged fission yield is:

_Δ

₌

ω

_G

ω

with

where

G G

φ

φ

1 1 fiss

fiss

φ

σ

Methodology to propagate ND uncertainties

PART I

Advantages & Disadvantages of Monte Carlo Technique

Î

Advantages

ƒ

Collapsing to one energy group

→

Reduce amount of variables

to sample

ƒ

No sensitivity coefficients should be calculated

ƒ

No approximation on equations

→

Take into account non-linear effects

Î

Disadvantages

ƒ

How to check if the phase space is well sampled ?

ƒ

Which PDFs should be taken ?

APPLICATIONS

PART II

APPLICATIONS:

A. Fission Pulse Decay Heat calculation

APPLICATIONS

PART II

A. Fission Pulse Decay Heat calculation

PART II

Description of the problem

Description of the problem

•

Decay heat

of a

single thermal fission event in Pu239

I t

l Fi

i

P

d

t

(FP )

• Isotopes:

only Fission Products

(FPs)

• Only

Fission yield (FY) and Decay data

(Energy/Decay constant)

uncertainties are propagated

FPs

97

S

DH

=

λ

N

E

Fission event Pu239

38 97

Sr

104

X Pu239→

γ

_L

DH

x

=

λ

x

⋅

N

x

⋅

E

x

−

β

104

−

β

41 104

Nb

m

β β

λ

E

DH

42 104

Mo

β β

λ

E

DH

L

A. Fission Pulse Decay Heat calculation

PART II

Calculations

10.00

12.00

•

Histories launched/case: 300

R l ti

f ll

d

6.00 8.00

tim

e s

tep

s (%

)

• Relative error followed

Case studied

2.00

4.00

E

rro

r i

n

•

Total decay heat

• Beta decay heat

Only known uncertainties // All with uncertainties

0.00

0 50 100 150 200 250 300 Number of histories

Beta decay heat

• Gamma decay heat

y

Decay Mode

Uncertainty

For unkown uncertainties

Compared with:

Alfa

10%

Beta

15%

G

15%

Compared with:

- JEFF report 20

A. Fission Pulse Decay Heat calculation

PART II

Total decay heat

C/E Mean Value JEFF 3 1 1

reference value C/E JEFF3 1 1

Tobias 1989

1.15

1.2

C/E Mean Value JEFF 3.1.1

reference value C/E JEFF3.1.1

Tobias 1989

1

1.05

1.1

C/

E

0.85

0.9

0.95

1.00E+00

1.00E+01

1.00E+02

1.00E+03

1.00E+04

1.00E+05

APPLICATIONS

PART II

B. EFIT fuel cycle calculation

PART II

Reference system

Coolant Pure Lead

One of the preliminary conceptual

designs of the

E

uropean

F

acility for

I

ndustrial

T

ransmutation (

EFIT

)

Thermal Power 400 MWth

Fuel (Pu, Am)O₂+ MgO

Initial mass of actinides 2.074 tonnes

(

)

Constant neutron environment:

1,E-03

Initial

Constant neutron environment:

- neutron flux: 3.12 x 10

15

_n/cm

2

_s

- average energy <E> = 0.37 MeV

C l

l ti

f

di

h

b

1,E-05 1,E-04

ron Flux

400 days

Calculations for discharge burn-up:

- 150 GWd/tHM (778 irradiation days)

- 500 GWd/tHM (3225 irradiation days)

1,E-07 1,E-06

Nor

m

alized Neut

Initial total flux intensity = 2.84E+15 n cm-2_s-1

2 1

1,E-09 1,E-08

1,E-06 1,E-05 1,E-04 1,E-03 1,E-02 1,E-01 1,E+00 1,E+01 1,E+02

400 days total flux intensity = 3.12E+15 n cm-2_s-1

B. EFIT fuel cycle calculation

PART II

Calculations

6 00

7.00

step

•

Histories launched:

1000/DH case

300/RTX

3.00 4.00 5.00 6.00

r p

er

ea

ch

t

im

e

s

(%

)

300/RTX case

Case studied

0.00 1.00 2.00

1 10 100 1000

re

la

ti

v

e err

o

r

Number of histories

Case studied

1. Decay heat

Number of histories

_~300

2. Radiotoxicity

a.Inhalation dose

All uncertainties are propagated:

- Individually

σ

_,

γ

_,

λ

B. EFIT fuel cycle calculation

PART II

Decay heat for

150 GWd/tHM

∑

∑

N

var(

y

_i

)

>>

N

cov(

y

_i

,

y

_j

)

∑

=

_i

total

DH

DH

Main contributors analysis

≠ =

= i j i j

i 1 , 1;

∑

∑

= =

⋅

=

⋅

=

N i i i N i i i y x

x

y

y

error

x

y

y

x

i 1 2 2 2 1 2 2 2 2 2 2

)

(

σ

σ

∑

=isotope i

∑

∑

≠ = =

+

=

N j i j i j i N i

i

y

y

y

y

; 1 , 1

)

,

cov(

)

var(

)

var(

⇒

1.40

1.60 ∑var(i)/var(∑) ∑cov(i,j)/var(∑)

i

i 1

y

i 1

0.60 0.80 1.00 1.20 0.00 0.20 0.40

B. EFIT fuel cycle calculation

PART II

Decay heat for

150 GWd/tHM

Cm242 Cm244 Pu238

Main contributors

TOTAL CM242 CM244 PU238 AM241 PU240 PU239 PO214 PO213

Am241

Pu240

Pu239

5.00 6.00 7.00

2.00 3.00 4.00

erro

r (

%

)

0.00 1.00

1.0E-03 1.0E-02 1.0E-01 1.0E+00 1.0E+01 1.0E+02 1.0E+03 1.0E+04 1.0E+05 1.0E+06

B. EFIT fuel cycle calculation

PART II

Radiotoxicity for

150 GWd/tHM

Error XS+FY+DECAY Error XS Error FY Error DECAY

2 50 3.00 3.50 4.00 4.50 ror ( % )

Error XS+FY+DECAY Error XS Error FY Error DECAY

0.50 1.00 1.50 2.00 2.50 In g es ti o n er r

Ingestion

0.00

1.0E-03 1.0E-02 1.0E-01 1.0E+00 1.0E+01 1.0E+02 1.0E+03 1.0E+04 1.0E+05 1.0E+06

Cooling time (years)

4.50

TOTAL XE133 CM244 PU238 AM241 RN222

Xe133

Cm244

Pu238

Am241

Rn222

2 00 2.50 3.00 3.50 4.00 n er ro r (% )

Due to FY / XS error

0.00 0.50 1.00 1.50 2.00

1 0E 03 1 0E 02 1 0E 01 1 0E+00 1 0E+01 1 0E+02 1 0E+03 1 0E+04 1 0E+05 1 0E+06

In

g

est

io

CONCLUSIONS

Monte Carlo technique for ND uncertainty propagation in activation calculations

Pre-proccesing of nuclear data is needed:

- Identifying uncertainties

- Collapsing of nuclear data

Implemented on ACAB code

Implemented on ACAB code

Monte Carlo technique

VS

deterministic calculations / experimental data

¾

A good agreement is found between both

A method to identify

main contributors to error

is developed based on MC results

A method to identify

main contributors to error

is developed based on MC results

ACKNOWLEGMENTS

This work is partially supported by:

p

y

pp

y

- FP7-EURATOM-FISSION-2009

Project ANDES/249671

-

Ministerio de Educación y Ciencia, SPAIN

(Spanish Science and Innovation Ministry)

( p

y)

THANK YOU

THANK YOU