two-region diffusion model for improved analysis of ads ...€¦ · !comparison with experimental...
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Varvara Glivici-Cotruţă | Institute of Safety Research | www.hzdr.de
Two-Region Diffusion Model for Improved Analysis of ADS Experiments
Varvara Glivici-Cotruţă, Bruno Merk
The 22nd International Conference on Transport Theory Portland, Oregon, September 12-16, 2011
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Motivation
Radiotoxicity and long-term disposal of irradiated nuclear fuel
Around 200,000 m2 of low- and intermediate-level radioactive waste and 10,000 m2 of high-level waste is generated in the
world every year
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Argoncirculation
Proton beam in evacuated tube
Proton accelerator (zyclotron)
Heatexchanger
Protons
Direction oflead
circulation
Argon
Liquid leadNeutrons
Reactor core (thorium oxide)
Breedingreaction
Lead target
Accelerator-Driven Systems
ü A sub-critical reactor ü A particle accelerator ü Spallation reaction ü A 1000 MeV beam creates 20-30 neutrons per proton ü A 20 mA accelerator can transmute annually the 129I generated by more than 10 reactors
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Accelerator-Driven Systems
On-line monitoring of the reactivity level in an accelerator-driven system is of major importance
for safe operation!
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Yalina-Booster Experiment
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Yalina-Booster Core -Fast zone: 36% and 90% enriched UO2 in lead -Valve zone: natural uranium and B4C pins -Thermal zone: 10% enriched UO2 in polyethylene -Graphite reflector -B4C control rods
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Results of the Experiments
Calle Berglöf, 2nd EROTRANS Progress Meeting, 2009, France
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Comparison of Analytical Solutions with Experiments
Normalized Time
EC1B EC2B EC3B Source
Coun
ts per Pulse [s
-‐1]
B. Merk, V. Glivici-Cotruţă, F.P. Weiß, XXI International Conference on Transport Theory, 2009, Italy
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Comparison with Experiments
EC1B EC2B EC3B Source
Cou
nts p
er P
ulse
[s-1
]
Normalized Time
Results for the fast zone: ü good reproduction of the
detector response
ü good agreement between diffusion and P1 transport
ü keff ≈ 0.6 for the fast zone only
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Green’s Function for the Diffusion Equation in a Composite Slab
0 ,0
),()(),(),(),(1112
12
11
1
><<−
−+=ΦΣ+∂Φ∂−
∂Φ∂
txa
txtxxtxD
ttx
v a τδξδ
0 , , ,
0 ,0 ,0),(),(),(1222
22
22
2
>
><<=ΦΣ+∂Φ∂−
∂Φ∂
τξba
tbxtxxtxD
ttx
v a
Initial conditions: bxx <<=Φ 0 ,0)0,(2
0 ,0)0,(1 <<−=Φ xax
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Green’s Function for the Diffusion Equation in a Composite Slab
Conditions at the interface:
0 ),,0(),0( 21 >Φ=Φ ttt
0 ,),(),(0
220
11 >
∂Φ∂⋅=
∂Φ∂⋅ == t
xtxD
xtxD xx
Boundary conditions: 0 ,0),(1 >=∂
Φ∂−= t
xtx
ax
0 ,0),(2 >=∂
Φ∂= t
xtx
bx
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0 -‐a b x
Region 1 Region 2 x=-‐ᵹ
Geometry
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Green’s Function for the Diffusion Equation in a Composite Slab
Notation: 1,2i , ,11 ==Σ
= ii
ia
iii
cDkDv
Substitution: 1,2i ,i ==Φ − tcki
iieu
0 ,0
),()(),(1),(
1
1
12
12 11
><<−
−+−=∂
∂−∂
∂
txa
txDe
ttxu
kxtxu tck
τδξδ
0 ,0 ,0),(1),( 2
22
22
><<=∂
∂−∂
∂ tbxttxu
kxtxu
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Green’s Function for the Diffusion Equation in a Composite Slab
0 ,0)0,(1 <<−= xaxuInitial conditions:
bxxu <<= 0 ,0)0,(2
Boundary conditions: 0 ,0),(111 >=
∂∂
−=− te
xtxu
axtck
0 ,0),(222 >=
∂∂
=− te
xtxu
bxtck
Conditions at the interface:
0 ,),0(),0( 221121 >= −− tetuetu tcktck
0 ,),(),(0
220
11
2211 >∂
∂⋅=∂
∂⋅ =−
=− te
xtxuDe
xtxuD x
tckx
tck
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Green’s Function for the Diffusion Equation in a Composite Slab
Laplace transform:
0 ),(1
)(
11
21
2 11
<<−+−=−∂∂ −−
xaxD
eGks
xG cks
ξδτ
bxtxGks
xtxG <<=−
∂∂ 0 ,0),(),(
22
22
2
Boundary conditions: 01 1
11
=∂∂
+ −= axxG
cks
01 2
22
=∂∂
+ =bxxG
cks
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Green’s Function for the Diffusion Equation in a Composite Slab
Conditions at the interface:
),0(1),0(12
221
11
sGcks
sGcks +
=+
02
2220
1
111
),(1),(1== ∂
∂⋅+
=∂
∂⋅+ xx x
sxGcks
DxsxG
cksD
Notation: 2
12
1
2 ,kk
ks =−= λβ
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Green’s Function for the Diffusion Equation in a Composite Slab
))cos(sincoscossincoscossinsin
)(sin)((1),(
12
12
)(
11
11
axabDabD
bDbD
xxHeD
sxG cks
++−+
+++−= −−
βλββλβλβξλββξλβλ
ξβξβ
τ
)sincoscossin()(cos)(cos),(
12
)(
11
222
11
abDabDbxae
cksckssxG
cks
βλββλβλβλβξβτ
+−−⋅
++−=
−−
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Green’s Function for the Diffusion Equation in a Composite Slab
Poles are:
)(')(by given are residues the where,
0
21
11
n
nnn sq
spks
ckss
β−=
−==
dsaxabDabD
bDbD
xxHeDi
txu cksi
i
))cos(sincoscossincoscossinsin
)(sin)((121),(
12
12
)(
11
11
++−+
+++−= −−∞+
∞−∫
βλββλβλβξλββξλβλ
ξβξβπ
τγ
γ
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abaDbDbDaDabbxae
ckscks
acbcDbcacDckbxcacckck
bDaDcDecktxu
nnnn
nncks
n
n
ck
n
βλβλλλβλβλβξβ
λλλξ
λ
τ
τ
sinsin)()(coscos)(cos)(cos
2)(
)cossincossin()(cos)(cos)(
)(2),(
2122
1
)(
11
22
11211111
112211
22
111
222
11
11
−−++−−×
×+−
+−
+−−+−
+
++
−=
−−
Green’s Function for the Diffusion Equation in a Composite Slab
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Summary and Conclusions
ü Full analytical solution for the P1 and diffusion equations with external source is developed with Green’s function method ü Comparison with experimental results from YALINA-Booster ü Evaluation of difference between time dependent P1 transport and diffusion ü Solution of a problem for the two regions diffusion equation with appropriate boundary conditions for YALINA experiments
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Further Steps
ü Development of something comparable with widely used integral reactor physical parameters like “corrected reactivity” ü Development for an analytical solution for the shut down of the source ü Evaluation of the mathematical model for the Guinevere experiment
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Thank you for your attention!