TRANSPORT CALCULATIONS FOR REACTOR …multiscale/Multiscale_Simone...Transport calculations for Reactor Physics. The challenge of Generation IV DM2S/SERMA/LENR 2 What is a fourth generation

Transport calculations for Reactor Physics. The challenge of Generation IVDM2S/SERMA/LENR 1

TRANSPORT CALCULATIONS FOR REACTOR PHYSICS. THE CHALLENGE OF GENERATION IV

BY

SIMONE SANTANDREA & RICHARD SANCHEZ


What is a fourth generation Reactor?

• Reactors actually on operation are of 2nd generation.

• Third generation reactors are advanced concepts of 2nd

generation ones. Some of them (EPR) are now underconstruction

• Fourth generation reactors are completely innovativeconcepts that should be available to the market in 20 or more years.


Classical Homogenization approach

Typical PWR Reactor Geometry

zircaloy clad

fuel rod

moderator

reactor core

assembly

cell


Classical homogeneization theory

If the reactor core has a size ∼O(1) and we have to solve mono-energetic the transport problem:

and Xsection are ε-periodic functions, one can write the solution as:

where the dependence on x is weak at ε scale, while the dependence on y is fast variating and ε-periodical. One then gets:

where is the solution of the infinite lattice transport problem, while u:

where D and σ are ψ∞ dependent

( ) ( ) ( ) ( )fx d f x x d x x on D

on D(4 ) (4 )

, ' ( , ', ) , ' ' ( , ', ) , ' ,

0π π

ψ ψ λ ψ

ψ −

Ω∇ + Σ Ω = Ω Ω Ω Ω + Ω Σ Ω Ω Ω

= ∂

∫ ∫

x x xx x x x0 1 2

0 1 2

( , ) ( , , ) ( , , ) ² ( , , ) ..

² ..

ψ ψ εψ ε ψε ε ε

λ λ ελ ε λ

Ω Ω + Ω + Ω +

= + + +

∼

x y u x y0

0 1

( , , ) ( ) ( , )

, 0

ψ ψ

λ λ λ∞

∞

Ω = Ω

= =

i i x y( , , )ψ ψ= Ω

y( , )ψ∞ Ω

fD u u on D

u on D

,

0 ,

σ−∇ ∇ =

= ∂


Non periodic reactorsProject Calculations : Project Calculations :

the RJH reactorthe RJH reactor

Model

UMoAl235U 19.75%thickness 0.61 mm

Clad235U 19.75%thickness 0.31 mm

stiffener

Water gap Thickness 0.184

Aluminum filler orControl rod orTest device


Limits of the classical homogeneization: the EPR case

Third generation Pressurized Water reactor

reactor layout: assembliescompositions1/8 EPR computational mesh

9 8 7 6 5 4 3 2 1

Cquarts :

N C

D N D

E N N E

F N N F

G N N N G

H N N N H

I N N N N I

9 8 7 6 5 4 3 2 1

1er cycle 4e cycle

2e cycle 5e cycle N Grappes de contrôle

3e cycle

Assemblages gadoliniés

a b

c d


Discrepancies on fission ratesComparisons done with a reference Montecarlo Calculation

Transport calculation: error on ¼ assemblies

Diffusion calculation: error on assemblies

9 8 7 6 5 4 3 2 1

C -3,7% C

D +3,1% +1,6% -2,4% D

E +1,1% +1,8% +1,7% -2,2% E

F +1,2% +1,2% +2,3% +1,9% +0,8% -5,5% F

G +0,2% -0,2% +1,3% +0,9% +2,2% +1,6% -2,5% G

H -1,2% -0,6% -0,6% +0,1% +0,6% +1,5% +2,1% -3,2% H

I -1,3% -1,7% -1,4% -0,1% +0,4% +1,3% +0,7% +1,2% -2,2% I

9 8 7 6 5 4 3 2 1

Even with a not fully converged transport calculation, the interface reactor-reflector errors are strongly reduced


Modern methods for full scale reactors problems: MOC

Angular approximation: SN

Flat flux approximation over homogeneous regions

Robust and positive method

Arbitrary collision anisotropy

i ii

( , ) ( ) ( )ψ ψ θΩ Ω∑r r∼

iterative solution ( 1) ( )

( 1) ( )0

( ) n n

n noutin

qψ

ψ βψ ψ

+

+

⎫Ω ⋅ ∇ + Σ = ⎪⎪⎪⎬⎪= + ⎪⎪⎭

( ) ( )

:

n n

out in

q H S

albedo operator

ψ

β ψ ψ

= +

→


MOC 2

The numerical implementation is based on trajectories

Balance and transmission equation over a trajectory

R(t,Ω)

ψi(Ω)ψ+,i(t,Ω)

ψ-,i(t,Ω)W⊥(t,Ω)

trajectory (t,Ω) cell i

nn nii i

nR Rqi( 1)( 1) ( 1)

, ,( )ψ ψ ψ ++ +

+ −− + =Σ

n ni i

nT E qi( 1) ( 1), ,

( )ψ ψ+ ++ −= × + ×


Synthetic acceleration approach

Free iterations :

Problem for the error

The transport operator is substituted with a low-order one :

Correct the values of free iterations :

Low-order operator : DPN. It is based on

• Balance equation over a computational region.

• Approximated transmission equation based on surface angular

fluxes.

1( n ) ( n )B H Sψ ψ −= +

( ) ( )1(n) ( ) (n) (n) (n) (n )B H Hε ψ ψ ε ψ ψ∞ −= − → − = −

( ) ( )n n ( n )acc libresψ ψ δψ= +


DPN Synthetic Acceleration Operator

J+,αi(t,Ω)

J-,βi(t,Ω)

Surface β

Surface α

cell i

Tiαβ

( )

( ) ,

( , )

( , ) ( )

s exti i iV

Si

q A q q q

A ααα

φ

ψ ψ θ± ±∈∂

Ω Ω ⋅ = Σ +

Ω ⋅Ω ∑

r

r r

∼

∼

( )

, , ,

,,

,

1 .

i ii

i i ii i

A J T V E q

A SA S BqV

ρυ υ ρ ρυ υα αα α αβ β

β

α α ααα

ψ ψ

ψ ψ

+ + −∈∂

−+∈∂

= = +

− + Σ Φ =

∑

∑

balance & propagation equations :

( ) ( ) ( )(2 )

, A dSd A A S parity of Aα α π×= ⋅ ⋅ =∫∫ nΩ Ω Ω Ω Ω


DPN 2

transmission and escape probabilities (symmetry and conservation preserved by

numerical scheme)

numerical evaluation (coherence with transport)

( ) ( ) ( , ) ( )

1( )

iR

i i i

T T dS d A A e symmetry

E A T conservationV

ρυ ρυ ρ υαβ βα α β α

ρυ ρυ ρυα α αβ

α

−Σ→

∈∂

= = Ω ⋅

⎛ ⎞⎟⎜= − ⎟⎜ ⎟⎟⎜Σ ⎝ ⎠

∫ ∫

∑

rn ΩΩ Ω Ω

( ) ( ) ( , )

( , )w A A w ( , ) iR t

tT t eρυ ρ υαβ

β α

−Σ⊥

∈ →∑ ∑ Ω

ΩΩ Ω

Ω Ω Ω∼


1D Fourier analisys

water waterfuelBenchmark 1D case :

1 1 1

• ρDP0=0.17

• ρFree= 0.96

• ρDP1=0.05

Spectral radius in function of cells width

0

0,05

0,1

0,15

0,2

δ=1. δ=0.5 δ=0.2 δ=0.1 δ=0.05 δ=0.02 δ=0.01

cells width

Spec

tral

radi

us

DP0DP1

Spectral radius for different integration steps.


High Temperature Gas Reactor: the double heterogeneity problem

1/12th Core

7 m

Fuel particle

1mm

1cm

Compact

36 cm

Fuel element

GTGT--MHR : Heterogeneous 2D MHR : Heterogeneous 2D MOC MOC

Depletion CalculationDepletion Calculation


Stochastic media• Statistical set:

• Multimaterial renewal statistics:

• Homogeneous line statistics• Ensemble average:

, ( )pω ωΩ =

UO2

Gd grain

Realization with density of probability p( )

[ ]0 ,lim ( ) 0y x x y

p dαβγ

ω ω− → Ω=∫

x y

( ) ( ) ( ) ( ) ( )x p d x p xω α αα

ψ ω ωψ ψΩ

= =∑∫ r

( , , )x E= r Ω


Ensemble averaging• Material flux:

• equation

• Material averaging operator

( ) ( )( ) ( ) ( )/ ( )x p d x p d

α αα ωψ ω ωψ ω ω

Ω Ω= ∫ ∫r r

( ) ,H Sω ω ω ω ωψ ψ ω⋅ + Σ = + ∈ ΩΩ ∇∂∫

( )[ ( ) ]( ) ( ) ( )M x p d x

αα ω ωψ ω ωψ

Ω= ∫ r

r

( ) ?Mα Ω ⋅ ∇r

0

( )

1lim [ ( ) ( ,..) ( ) ( ,..)]

M

M M

α

α αε

ψ

ε ψ ε ψε+→

⋅ ∇ =

+ + −

r

r r r r

Ω

Ω Ω

Ensemble averaging


r r+(α)=any material different from α

( )

( )

( ) ( , ) ( , )

( ) ( , ) ( , )

α αα α α

α αα α α

ε ε

ε ε ε

Ω = Ω + ∪ Ω +

Ω + = Ω + ∪ Ω +

r r r r r

r r r r r

Ω Ω

Ω Ω Ω

( )

( )

( ) ( ,..)

[ ( , ) ( , )] ( ,..)

( ) ( ,..)

[ ( , ) ( , )] ( ,..)

M

M M

M

M M

α

αα α α

α

αα α α

ε ψ ε

ε ε ψ ε

ψ

ε ε ψ

+ + =

+ + + +

=

+ + +

r r

r r r r r

r r

r r r r r

Ω Ω

Ω Ω Ω

Ω Ω

Ensemble averaging


( ) ( )( ) ( ) ( , ) ( , )M M M Mα α α α α α⋅ ∇ ≡ ⋅ ∇ + ∂ − ∂r r r rΩ Ω Ω Ω

( )0( ) ( , )

1[ ( , ) ]( ) lim ( ) ( )

( , ) ( )

M x p d x

p x

α αω ωεα α

αβ αββ α

ψ ω ωψε

ψ

+→ Ω

≠

∂ = =

=

∫

∑

rr

r

ΩΩ

Ω

( ) [ ( , ) ]( )/[ ( , )1]( )x M x M xωαβ αβ αβψ ψ= ∂ ∂r rΩ Ω

hence

with

Interface flux:

Ensemble averaging


• First balance equation

• Markovian closure:

• Markovian model

( ) ( )

( )

p p H S

p p

α α α α α α α

βα βα αβ αββ α

ψ ψ

ψ ψ≠

⋅ + Σ = +

+ −∑

Ω ∇

( ) ( )

Markovian statisticsx x

Collisionless transportααβψ ψ+ ⇒ =

( ) ( )

1( ( ) )

( )L

p p H S

t

α α α α α α α

αβα βα β α

ψ ψ

ψ ψλ ≠

⋅ + Σ = +

+ −∑

Ω ∇

ΩΩ


Renewal approach 1transition point = position at the interface between two materials Renewal statistics ‘the conditional probability for the material distribution to the

right of a transition point is independent of the material distribution to the left’: the probability density depends only on what there is at the right of point y

Renewal equations for material fluxes:

Interior probabilities for length l of material αprobability for length

dens. of prob. for length

( )0

( )

[ ( ) ( ) ( ) ]

( )

( )

x in

x x y in

x e

e x y q y x y dy

x R

R g y

α

α

α α

α α α α

ψ ψ

ψ

−Σ

−Σ −+ − + −

=

∫

( )R lα =( )g lα =

l≥

l=


Renewal approach 2

• statistical average of the flux entering material α at x

• conditional transition probability

• Interface flux

• Renewal equations for interface fluxes:

( )in xαψ = ( ) ( )in outtx xα βα ββ α

ψ ψ≠

= ∑

( )out xαψ

( )0

( )

[ ( ) ( ) ( ) ]

( )

( )

out x in

x x y in

x e

e x y q y x y dy

x R

R g y

α

α

α α

α α α α

ψ ψ

ψ

−Σ

−Σ −+ − + −

=

∫

tβα =


Renewal approach 3• Markovian matrix (m) + spherical grains (g)

• grains do not collapse

• renewal, homogeneous, line statistics

• Isotropic grain chord distribution

• The statistics are completely defined by:• pg = volumetric proportion of grain of type g• Rg = radius of grain of type g

/1( ) mlm

mef l λ

λ−=

1mgt =

( ) ( ), ( ), gmgm

m g

pp

f l R l g l tα α αλ

λ=→

24,32

( )g g gg

l RR

f l λ ==


Renewal approach 4Interior analytical solution adapted to the MOC

Interfaces fluxes:

with

mean fluxes:

( ) (1 ) , 0,

( ) ( ) , 2

out x in x asm

out x in x as outg gg g g g

x e e x

x T e T T e x R

ψ ψ ψ

ψ ψ ψ ψ

−Σ −Σ

−Σ −Σ

= + − ≥

= + − + ≥

1 1( ),as out outm gm g g g g g

m gq t q Eψ ψ ψ λ

λ= + =

Σ ∑

1 1 ˆ(1 ), (1 )m gm g m gm gm mg g

t T t Tλ λ

Σ = Σ + − Σ = Σ + −∑ ∑

( )[ ]0

0

/ / ,

( ) as outg gg gg

q l

E E E

ψ ψ ψ

ψ ψ ψ ψ

− += + − Σ

= + − +


High Temperature Gaz Reactor

Comparisons of keffective et of the ratio Production/Absorption inside a representative fuel element

keffective

(variations in pcm) P/A inside the annular zone

(variations in pcm) TRIPOLI4

Particles described with a regular hexagonal lattice

1,4316 ± 45 pcm

1,5753

TRIPOLI4 Particles distributed stochastically inside

the fuel element

1,4312 ± 32 pcm (− 30 )

1,5760 (+ 40)

APOLLO2

Transport 99 gr X sections P0 corrected ∆r = 5.0*10-2 / ∆N = 12

1,43037 (− 80 )

1,57666 (+ 86)

Reactor modelisation in APOLLO2:

- 40000 computational points for the flux calculation

- 35000 s for 1 calculation (99 gr – xsections P0 corrected)


Another scale level: Pebble Bed Reactors (PBR)

PBRs fuel is constituted by pebbles of different types that are stochastically piled up over a cylindrical core. An internal and external circular reflectors delimit the internal and external boundaries for the power region. The cooling is granted by the flow of an Helium gas that pass trough the inter-pebbles space

Horizontal section Vertical section

free (interface) surfaces

grap

hite

albe

doco

nditi

on

externalgraphitereflectorinternal

graphitereflector

Fuel composition and position are therefore only approximativelyknown.


PBRsSeveral channels exist that drive pebbles toward the lower part of the core where they are progressively discarded. During their path toward the reactor bottom pebbles change their isotope concentration, due to a different burn up.


Numerical techniques for PBRs calculationso We try to make a diffusion calculation using an engineering equivalence

theory. This technique consists in two steps.• First step: compute an infinite lattice transport solution, used to compute

some reference reaction rates.• Second step: force in the diffusion calculation the conservation of

reaction rates of the infinite transport lattice case over macro groups and layers.

Note that so that the problem is non linear!

Generally some supplementary constraints are imposed to the equivalence, such as the conservation of the integrated reference flux.

Once a macro flux is obtained from the solution of the diffusion calculation in each layer, the pebble region fluxes are computed from the following factorization:

G G G g gh h pebbles pebbles pebbles

pebbles g GV Vτ ϕ

∈Σ Φ = = Σ∑ ∑

( )G G Gh h hΦ = Φ Σ

,g G greconstructed pebbles Diffusion pebblesφ ϕΦ = ×


Step 1: infinite lattice calculation

• For each layer we write

where X is the phase space associated to the layer. Supposing that eachpebble occupies every available position with the same probability, andsupposing isotropic and uniform entering flux, the problem for pebbles oftype “i” is given by:

where J- is the uniform entering current in pebbles. Problem (A) is 1D and can be treated with the CP formalism as

A relation still remain to be found between entering and exiting currents.

( )

( ) ( )

1 ,

,

H P x X

J E J E x X

ψ ψ ψλ

− −+

Ω∇+Σ = + ∈

= ∈ ∂

( )( ) ,

1 ,

/ ,

i i i i i i i

i i

H P x X

J E x X

ψ ψ ψλ

ψ π− −

Ω∇+Σ = + ∈

= ∈∂(A)

, , ,, , ,

, , , ,

i i i ii i i

i i i ii i i

s fiss

P V F I AJ

AJ E V F TAJ

F Q

α α ααβ β ββ

β β ββ

φ

φ

−

−+

Σ = +

= +

=Σ +

∑

∑(B)


Include the Helium contribution• To find this last relation we model the inter-pebbles Helium space. The total

current of neutrons exiting the spheres and entering other spheres can be written as:

• Supposing that the source in Helium is uniform and isotropic, as well as the pebble exiting flux one gets

where l is an arbitrary chord length. Making then an average and integrating over surface gives:

using the markovian distribution one gets

( )( )2

, i iiS

AJ E dS d n A N Aπ

ψ−

− −= Ω Ω⋅ =∑∫ ∫

( ) ( ) ( ')

[0, ]

, , , , ' ( ' , , )He Hel l lHe

l

r E e r E dl e q r l Eψ ψ−Σ −Σ −− +Ω = Ω + − Ω Ω∫

( ) ( )( )( ), , 1

4He Hel lHe

He

F Er E e E eψ ψ−Σ −Σ

− +Ω = + −Σ

( ) ( ) ( ) ( )

0

1 , ( )4

He He He Hel l l lHe

He

F EJ E e J E e with e dlp l e

∞−Σ −Σ −Σ −Σ

− += + − =Σ ∫

/( )

Hel

He

ep lλ

λ

−=

( ) ( )( )1 [ ]

1 4He He

He He

F EJ E J E

λλ− += ++ Σ

(C)


Final form of the infinite lattice solution• Add to (B) and (C) a balance equation for Helium

since

one has, for a suitable constant C :

that inserted back in (B) gives the final problem to be solved:

that is an eigenvalue problem for the infinite lattice problem.

( )He He He He HeV V F A J Jφ −+Σ = + −

( ),i i ii

i ii

N AJ EJ

N A+

+ =∑

∑

( ) ( )i i i i He He HeiJ E C N E V F E V Fα α α

α− = +∑ ∑

V PVFφΣ =


Conclusions

• Many challenges are proposed to researchers by innovative 4th generation reactors.

• Strong geometrical heterogeneities prevent traditional homogeneization approaches.

• The presence of stochastic media, both at microscopic and macroscopic level ask to manage with difficult equations.

• More work has to be done…


References

• For the MOCSantandrea & Sanchez, Annals of Nuclear Energy,

32,163-193, (2005) (others refs inside)• For the Double Heterogeneity problem for

exampleSanchez & Pomraning, Annals of Nuclear Energy,

18, 371-.., (1991)

Documents

TRANSPORT CALCULATIONS FOR REACTOR …multiscale/Multiscale_Simone...Transport calculations for Reactor Physics. The challenge of Generation IV DM2S/SERMA/LENR 2 What is a fourth generation