UK Atomic Energy Authority (2017)

• Tritium Breeding Blankets
• Heat transfer simulations
• Parametric optimisation

CEA French Atomic Agency (2018)

• Started the development of FESTIM
• Early work on divertor modelling

PhD CEA - LSPM (2019 - Oct 2022)

• Estimation of the tritium inventory in the ITER divertor
• Influence of helium exposure

Helium: why should we care?

Why should we care?

Why should we care?

Bubbles

Tungsten fuzz

Thermo-mechanical properties

Tritium production

Hydrogen transport

How do we model this?

Let's consider the following reactions

Helium clustering (or emission)

Trap mutation

He$_1$

He$_2$

He$_4$

He$_3$

V$_1$He$_7$

V$_1$He$_8$

V$_1$He$_9$

V$_1$He$_{10}$

Our clustering scheme

Trap mutation

or

self-trapping

For each reaction

3-species model:

Rate constants:

N-species model:

Diffusion coefficients

Capture radii

Diffusion

Production

Reaction

Binding energy

Issue #1: Too many equations

$\frac{\partial c_1}{\partial t} = \nabla \cdot (D_1 \nabla c_1) + P_1 - 2 k^+_{1, 1} c_1^2 - \sum\limits_{i=2}^\infty k_{1,i}^+ c_1 c_i$

$$\vdots$$

$\frac{\partial c_i}{\partial t} = \nabla \cdot (D_i \nabla c_i) - k^+_{1, i} c_1 c_i + k_{1,i-1}^+ c_1 c_{i-1}$

$$\vdots$$

$\frac{\partial c_{N}}{\partial t} =$ $- k^+_{1, N} c_1 c_{N}$ $+ k_{1,N-1}^+ c_1 c_{N-1}$

$\frac{\partial c_{N+1}}{\partial t} =$ $- k^+_{1, N+1} c_1 c_{N+1}$ $+ k_{1,N}^+ c_1 c_{N}$

$\frac{\partial c_{N+2}}{\partial t} =$ $- k^+_{1, N+2} c_1 c_{N+2}$ $+ k_{1,N+1}^+ c_1 c_{N+1}$

$$\vdots$$

💪

$\frac{\partial c_1}{\partial t} = \nabla \cdot (D_1 \nabla c_1) + P_1 - 2 k^+_{1, 1} c_1^2 - \sum\limits_{i=2}^\infty k_{1,i}^+ c_1 c_i$

$$\vdots$$

$\frac{\partial c_i}{\partial t} = \nabla \cdot (D_i \nabla c_i) - k^+_{1, i} c_1 c_i + k_{1,i-1}^+ c_1 c_{i-1}$

$$\vdots$$

$\frac{\partial c_{N+1}}{\partial t} =$ $- k^+_{1, N+1} c_1 c_{N+1}$ $+ k_{1,N}^+ c_1 c_{N}$

$\frac{\partial c_{N+2}}{\partial t} =$ $- k^+_{1, N+2} c_1 c_{N+2}$ $+ k_{1,N+1}^+ c_1 c_{N+1}$

$\frac{\partial c_{N+3}}{\partial t} =$ $- k^+_{1, N+3} c_1 c_{N+3}$ $+ k_{1,N+2}^+ c_1 c_{N+2}$

$$\vdots$$

💪

Issue #1: Too many equations

$\frac{\partial c_1}{\partial t} = \nabla \cdot (D_1 \nabla c_1) + P_1 - 2 k^+_{1, 1} c_1^2 - \sum\limits_{i=2}^\infty k_{1,i}^+ c_1 c_i$

$$\vdots$$

$\frac{\partial c_i}{\partial t} = \nabla \cdot (D_i \nabla c_i) - k^+_{1, i} c_1 c_i + k_{1,i-1}^+ c_1 c_{i-1}$

$$\vdots$$

$\sum\limits_{i=N+1}^{\infty} \frac{\partial c_i}{\partial t} = k_{1,N}^+ c_1 c_{N}$

💪

Issue #1: Too many equations

$\frac{\partial c_1}{\partial t} = \nabla \cdot (D_1 \nabla c_1) + P_1 - 2 k^+_{1, 1} c_1^2 - \sum\limits_{i=2}^\infty k_{1,i}^+ c_1 c_i$

$$\vdots$$

$\frac{\partial c_i}{\partial t} = \nabla \cdot (D_i \nabla c_i) - k^+_{1, i} c_1 c_i + k_{1,i-1}^+ c_1 c_{i-1}$

$$\vdots$$

$\frac{\partial c_b}{\partial t} = k_{1,N}^+ c_1 c_{N}$

$c_b = \sum\limits_{i=N+1}^{\infty} c_i$ : bubble concentration

💪

Issue #1: Too many equations

$\frac{\partial c_1}{\partial t} = \nabla \cdot (D_1 \nabla c_1) + P_1 - 2 k^+_{1, 1} c_1^2 - \sum\limits_{i=2}^{N}k_{1,i}^+ c_1 c_i - \sum\limits_{i=N+1}^\infty k_{i, 1}^+c_i c_1$

$$\vdots$$

$\frac{\partial c_i}{\partial t} = \nabla \cdot (D_i \nabla c_i) - k^+_{1, i} c_1 c_i + k_{1,i-1}^+ c_1 c_{i-1}$

$$\vdots$$

$\frac{\partial c_b}{\partial t} = k_{1,N}^+ c_1 c_{N}$

$c_b = \sum\limits_{i=N+1}^{\infty} c_i$ : bubble concentration

💪

Issue #1: Too many equations

$\frac{\partial c_1}{\partial t} = \nabla \cdot (D_1 \nabla c_1) + P_1 - 2 k^+_{1, 1} c_1^2 - \sum\limits_{i=2}^{N}k_{1,i}^+ c_1 c_i - \langle k_b^+\rangle c_1 c_b$

$$\vdots$$

$\frac{\partial c_i}{\partial t} = \nabla \cdot (D_i \nabla c_i) - k^+_{1, i} c_1 c_i + k_{1,i-1}^+ c_1 c_{i-1}$

$$\vdots$$

$\frac{\partial c_b}{\partial t} = k_{1,N}^+ c_1 c_{N}$

$c_b = \sum\limits_{i=N+1}^{\infty} c_i$ : bubble concentration

$\langle k_b^+ \rangle = \left( \sum\limits_{i=N+1}^{\infty} k_ {i, 1}^+ c_i \right) / c_b$ : average clustering rate in bubbles

💪

Issue #1: Too many equations

$c_b = \sum\limits_{i=N+1}^{\infty} c_i$ : bubble concentration

$\langle k_b^+ \rangle = \left( \sum\limits_{i=N+1}^{\infty} k_ {i, 1}^+ c_i \right) / c_b$

$= \left(\sum\limits_{i=N+1}^{\infty} 4 \pi D_1 (r_1 + r_i) c_i\right) / c_b$

💪

average clustering rate in bubbles

Issue #2: what's the value of $\langle k_b^+ \rangle$?

Issue #2: what's the value of $\langle k_b^+ \rangle$?

$c_b = \sum\limits_{i=N+1}^{\infty} c_i$ : bubble concentration

$\langle r_b \rangle = \left( \sum\limits_{i=N+1}^{\infty} r_ i c_i \right) / c_b$ : average bubble radius

$\langle k_b^+ \rangle$$= \left( \sum\limits_{i=N+1}^{\infty} k_ {i, 1}^+ c_i \right) / c_b$

$= \left(\sum\limits_{i=N+1}^{\infty} 4 \pi D_1 (r_1 + r_i) c_i\right) / c_b$

$=4 \pi D_1 (r_1 + \langle r_b \rangle)$

💪

average clustering rate in bubbles

$r_i = r_{\mathrm{He}_0\mathrm{V}_1} + \left( \frac{3}{4\pi} \frac{a_0^3}{2} n_{\mathrm{V},i} \right)^{1/3} - \left( \frac{3}{4\pi} \frac{a_0^3}{2} \right)^{1/3}$

💪

$c_b = \sum\limits_{i=N+1}^{\infty} c_i$ : bubble concentration

$\langle k_b^+ \rangle = 4 \pi D_1 (r_1 + \langle r_b \rangle)$ : average clustering rate in bubbles

$\langle r_b \rangle = \left( \sum\limits_{i=N+1}^{\infty} r_ i c_i \right) / c_b$ : average bubble radius

Issue #3: what's the value of $\langle r_b \rangle$?

$r_i = r_{\mathrm{He}_0\mathrm{V}_1} + \left( \frac{3}{4\pi} \frac{a_0^3}{2} \frac{i}{4} \right)^{1/3} - \left( \frac{3}{4\pi} \frac{a_0^3}{2} \right)^{1/3}$

$n_{\mathrm{V},i} = i/4$ : 4 He per vacancy

💪

$c_b = \sum\limits_{i=N+1}^{\infty} c_i$ : bubble concentration

$\langle k_b^+ \rangle = 4 \pi D_1 (r_1 + \langle r_b \rangle)$ : average clustering rate in bubbles

$\langle r_b \rangle = \left( \sum\limits_{i=N+1}^{\infty} r_ i c_i \right) / c_b$ : average bubble radius

Issue #3: what's the value of $\langle r_b \rangle$?

$c_b = \sum\limits_{i=N+1}^{\infty} c_i$ : bubble concentration

$\langle k_b^+ \rangle = 4 \pi D_1 (r_1 + \langle r_b \rangle)$ : average clustering rate in bubbles

$\langle i_b \rangle = \left( \sum\limits_{i=N+1}^{\infty} i c_i \right) / c_b$ : average He content in bubbles

$\langle i_b \rangle = \left( \sum\limits_{i=N+1}^{\infty} i c_i \right) / c_b$

💪

Issue #4: what's the value of $\langle i_b \rangle$?

$\langle i_b \rangle = \left( \sum\limits_{i=N+1}^{\infty} i c_i \right) / c_b$

$\langle i_b \rangle c_b=  \sum\limits_{i=N+1}^{\infty} i c_i$

💪

$c_b = \sum\limits_{i=N+1}^{\infty} c_i$ : bubble concentration

$\langle k_b^+ \rangle = 4 \pi D_1 (r_1 + \langle r_b \rangle)$ : average clustering rate in bubbles

$\langle i_b \rangle = \left( \sum\limits_{i=N+1}^{\infty} i c_i \right) / c_b$ : average He content in bubbles

Issue #4: what's the value of $\langle i_b \rangle$?

$\langle i_b \rangle = \left( \sum\limits_{i=N+1}^{\infty} i c_i \right) / c_b$

$\langle i_b \rangle c_b=  \sum\limits_{i=N+1}^{\infty} i c_i$

$\frac{\partial \langle i_b \rangle c_b}{\partial t} = \sum\limits_{i=N+1}^{\infty} i \frac{\partial c_i}{\partial t}$

💪

$c_b = \sum\limits_{i=N+1}^{\infty} c_i$ : bubble concentration

$\langle k_b^+ \rangle = 4 \pi D_1 (r_1 + \langle r_b \rangle)$ : average clustering rate in bubbles

$\langle i_b \rangle = \left( \sum\limits_{i=N+1}^{\infty} i c_i \right) / c_b$ : average He content in bubbles

Issue #4: what's the value of $\langle i_b \rangle$?

$\langle i_b \rangle = \left( \sum\limits_{i=N+1}^{\infty} i c_i \right) / c_b$

$\langle i_b \rangle c_b=  \sum\limits_{i=N+1}^{\infty} i c_i$

$\frac{\partial \langle i_b \rangle c_b}{\partial t} = \sum\limits_{i=N+1}^{\infty} i \frac{\partial c_i}{\partial t}$

💪

$\frac{\partial \langle i_b \rangle c_b}{\partial t} = (N+1) k_{1, N}^+ c_1 c_N + \langle k_b^+ \rangle c_1 c_b$

↓ trust me

$c_b = \sum\limits_{i=N+1}^{\infty} c_i$ : bubble concentration

$\langle k_b^+ \rangle = 4 \pi D_1 (r_1 + \langle r_b \rangle)$ : average clustering rate in bubbles

$\langle i_b \rangle = \left( \sum\limits_{i=N+1}^{\infty} i c_i \right) / c_b$ : average He content in bubbles

Issue #4: what's the value of $\langle i_b \rangle$?

$\frac{\partial c_1}{\partial t} = \nabla \cdot (D_1 \nabla c_1) + P_1 - 2 k^+_{1, 1} c_1^2 - \sum\limits_{i=2}^{N-1}k_{1,i}^+ c_1 c_i - \langle k_b^+\rangle c_1 c_b$

$$\vdots$$

$\frac{\partial c_i}{\partial t} = \nabla \cdot (D_i \nabla c_i) - k^+_{1, i} c_1 c_i + k_{1,i-1}^+ c_1 c_{i-1}$

$$\vdots$$

$\frac{\partial c_b}{\partial t} = k_{1,N}^+ c_1 c_{N}$

💪

$\frac{\partial \langle i_b \rangle c_b}{\partial t} = (N+1) k_{1, N}^+ c_1 c_N + \langle k_b^+ \rangle c_1 c_b$

$c_b = \sum\limits_{i=N+1}^{\infty} c_i$ : bubble concentration

$\langle k_b^+ \rangle = 4 \pi D_1 (r_1 + \langle r_b \rangle)$ : average clustering rate in bubbles

$\langle i_b \rangle = \left( \sum\limits_{i=N+1}^{\infty} i c_i \right) / c_b$ : average He content in bubbles

Issue #4: what's the value of $\langle i_b \rangle$?

$\frac{\partial c_1}{\partial t} = \nabla \cdot (D_1 \nabla c_1) + P_1 - 2 k^+_{1, 1} c_1^2 - \sum\limits_{i=2}^{6}k_{1,i}^+ c_1 c_i - \langle k_b^+\rangle c_1 c_b$

$$\vdots$$

$\frac{\partial c_6}{\partial t} = \nabla \cdot (D_6 \nabla c_6) - k^+_{1, 6} c_1 c_6 + k_{1,5}^+ c_1 c_{5}$

$\frac{\partial c_b}{\partial t} = k_{1,6}^+ c_1 c_{6}$

$\frac{\partial \langle i_b \rangle c_b}{\partial t} = 7 k_{1, 6}^+ c_1 c_6 + \langle k_b^+ \rangle c_1 c_b$

Our clustering scheme

average clustering rate in bubbles

$\langle k_b^+ \rangle = 4 \pi D_1 (r_1 + \langle r_b \rangle)$

average bubble radius

$\langle r_b \rangle = r_{\mathrm{He}_0\mathrm{V}_1} + \left( \frac{3}{4\pi} \frac{a_0^3}{2} \frac{\langle i_b \rangle}{4} \right)^{1/3} - \left( \frac{3}{4\pi} \frac{a_0^3}{2} \right)^{1/3}$

Only 8 equations:

Let's check our implementation... 

Comparison with experiments

Mykola Ialovega's PhD research

Comparison with experiments

Mykola Ialovega's PhD research

Code verification & validation

✔️Results in agreement with more complex models → additional assumptions are valid

✔️The model is in qualitative agreement with experiments

Let's run a standard case

Half-slab case

• "Semi-infinite" (0.6 mm)
• Helium source
  • ​100 eV
  • Gaussian distribution
    • ​ $\mu =1.5 \mathrm{nm} \, \sigma = 1.5 \mathrm{nm}$
  • Flux $10^{22} \, \mathrm{m^{-2} \, s^{-1} }$
• Temperature 1000 K

$c_i = 0$

0.6 mm

Half-slab case

Can be compared to experiments!

What's the influence of exposure conditions on bubble growth?

Divertor exposure conditions

Surface temperature: $350 \mathrm{-} 2000 \mathrm{K}$

Helium flux: $10^{19} \mathrm{-} 10^{21} \mathrm{m^{-2} \, s^{-1}}$ 

Parametric study

Varying temperature and particle flux

Parametric study

$c_\mathrm{He_1 \, ideal} = \frac{\varphi_\mathrm{imp} R_p}{D_1(T)}$

Varying temperature and particle flux

Parametric study

1 h

$\int c_b \langle i_b \rangle dx$

He inventory in bubbles

Parametric study

1 h

$\int c_b \langle i_b \rangle dx$

He inventory in bubbles

Parametric study

1 h tert

$\bar{\langle i_b \rangle} = \frac{\int c_b \langle i_b \rangle dx}{\int c_b dx} = \frac{\mathrm{inventory}}{\mathrm{total \, bubbles}}$

Mean helium content in bubbles

Parametric study

1 h

$\bar{\langle i_b \rangle} = \frac{\int c_b \langle i_b \rangle dx}{\int c_b dx} = \frac{\mathrm{inventory}}{\mathrm{total \, bubbles}}$

Mean helium content in bubbles

Parametric study

 $\int c_b dx$

Total bubbles

1 h  efef

Parametric study

 $\int c_b dx$

Total bubbles

1 h  efef

Two regimes can be identified

Two regimes can be identified

$\langle i_b \rangle$ is low

$\langle k_b^+ \rangle$ is low

Two regimes can be identified

Nucleation

🡺Self trapping

🡺$c_b$ increases

$\langle i_b \rangle$ is low

$\langle k_b^+ \rangle$ is low

Two regimes can be identified

Nucleation

🡺Self trapping

🡺$c_b$ increases

Growth

🡺$\langle i_b \rangle$ increases

🡺$\langle k_b^+ \rangle$ increases

$\langle i_b \rangle$ is low

$\langle k_b^+ \rangle$ is low

When $c_b$ is large enough

Where to go from here?

Bursting needs to be added

WIP: Collaboration with University of Tennesse, Knoxville

Influence on hydrogen transport

Coupled to the H transport code FESTIM

Reproduced TDS of deuterium in W pre-damaged with helium

Traps induced by bubbles!

Indirect sources of helium

Neutronics monoblock simulations with OpenMC

Indirect sources of helium

Tritium decay simulations with FESTIM

Neutronics monoblock simulations with OpenMC

Indirect sources of helium

Tritium decay simulations with FESTIM

Neutronics monoblock simulations with OpenMC

Main conclusions

• The grouping technique is a powerful technique to lower the complexity of clustering schemes.

• Its implementation has been compared to published numerical results and experiments.

• Two bubble growth regimes were identified: nucleation and growth.

• The model needs to be improved to account for bubble bursting.

Thank you for your attention!

Any questions?