SOFE Short Course

Tritium transport in materials

June 21-22

Remi Delaporte-Mathurin

Agenda

Day 1 – Saturday, 21 June 2025

08:15 – 09:15 General Introduction, tritium in fusion – Remi Delaporte-Mathurin

09:15 – 09:30 Coffee Break

09:30 – 10:30 Modelling hydrogen transport: reactor scale and fuel cycle – Samuele Meschini

10:30 – 11:30 Modelling hydrogen transport: component to reactor scale – Remi Delaporte-Mathurin

11:30 – 12:30 Modelling hydrogen transport: atomistic scale – Stephen Lam

12:30 – 13:30 Lunch

13:30 – 15:30 Experimental techniques and material characterization – Thomas Fuerst, Hans Gietl

15:30 – 15:45 Coffee break

15:45 – 17:00 Q&A

Day 2 – Sunday, 22 June 2025

08:30 – 16:00 FESTIM workshop – Remi Delaporte-Mathurin, James Dark

Round of introductions

Introuction - Dr. Remi Delaporte-Mathurin

2022-today: Researcher at the Plasma Science and Fusion Center, MIT
2018-2019: Ph.D. French Atomic Energy Authority (CEA Cadarache) and University Sorbonne Paris Nord
2017: CEA Cadarache, IRFM, apprentice
2016: UK Atomic Energy Authority
2012-2018: M.Sc. Thermal Engineering and Energy Sciences

Joint European Torus (JET), Culham, UK

Introuction - Dr. Remi Delaporte-Mathurin

LIBRA tritium breeding experiment

Tritium contamination in a heat exchanger with FESTIM

Hydrogen gas-driven permeation experiment

General Introduction:

tritium in fusion

ITER

Plasma: mixture of Hydrogen (D-T) and Helium

Particle bombardment

Divertor

Why should we care?

T is rare

T is expensive

€£$

Material embrittlement

T is radioactive

☢

What's Tritium?

Protium

Deuterium

Tritium

Molar mass: 6.032 g/mol

What's Tritium?

Tritium

☢

Fuel self-sufficiency

Half-life: 12 years

\mathrm{T} \rightarrow \mathrm{He} + \mathrm{e}^-

☢

Consumption of a 1 GWth fusion reactor (1 year)
50 kg

Cost: $30,000 per gram

The breeding blanket

\mathrm{n} + ^6\mathrm{Li} \rightarrow \mathrm{T} + \mathrm{He} + 4.8 \ \mathrm{MeV}

\mathrm{n} + ^7\mathrm{Li} \rightarrow \mathrm{T} + \mathrm{He} + \mathrm{n} - 2.5 \ \mathrm{MeV}

\mathrm{TBR} = \mathrm{\frac{tritium \ produced}{tritium \ consumed}} > 1

\mathrm{D} + \mathrm{T} \rightarrow \mathrm{He} + \mathrm{n}

Lithium is used to breed tritium

→ Li6 enrichment is an option

DT fusion neutrons

\mathrm{D} + \mathrm{T} \rightarrow \mathrm{He} + \mathrm{n}

Magnet

Breeding blanket

\mathrm{n} + \mathrm{Li} \rightarrow \mathrm{T} + \mathrm{He}

The breeding blanket

Plasma

The breeding blanket is only one component of the fuel cycle

Samuele Meschini et al 2023 Nucl. Fusion 63 126005

Let's play with a simple fuel cycle model

Burns tritium

Breeds tritium (TBR)

Tritium Extraction System

Breeding Blanket

Plasma

Storage

neutrons

TBR

(constrained by technology)

Doubling time

(driven by economics)

Startup inventory

(constrained by safety)

Source: remdelaportemathurin.github.io/interactive-plots/fuel_cycle/

Startup inventory

Tritium Extraction System

Breeding Blanket

Plasma

Storage

Safety issues

Tritium is a health hazard

☢

No external hazard: electron stopped by skin
Once ingested, can cause damage to internal tissues
Forms: HT, HTO, methane, titrides...
Biological half-life of HTO: 10 days
Biological half-life of OBT (Organically Bound Tritium): 40 days

Augustin Janssens. ‘Emerging Issues on Tritium and Low Energy Beta Emitters”’. en. In: (Nov. 2007), p. 100

Country	Water limit (Bq/L)
EU	100
USA	740
UK	100
Canada	7,000
Finland	30,000
Australia	76,103
Russia	7,700
WHO	10,000

Source: Standards and Guidelines for Tritium in Drinking Water (INFO-0766)

standards.doe.gov/standards-documents/1100/1129-AStd-2015

Tritium US DOE Handbook

413 pages!

The tritium content needs to be limited

Ferrero, et al (2022) A Preliminary CFD and Tritium Transport Analysis for ARC Blanket, Fusion Science and Technology, 78:8, 617-630

1. Keep inventory at a minimum

Tritium limit in the ITER vacuum vessel: 1 kg

Delaporte-Mathurin et al. 2021. “Fuel Retention in WEST and ITER Divertors Based on FESTIM Monoblock Simulations.” Nuclear Fusion 61 (12): 126001.

1. Keep inventory at a minimum

2. Reduce inventory

Heating components help releasing their tritium content (cf. Basics of H transport)

Rémi Delaporte-Mathurin et al 2024 Nucl. Fusion 64 026003

The tritium content needs to be limited

1. Keep inventory at a minimum

2. Reduce inventory

3. Avoid contamination of coolants

Metal

Tritiated environment

"Clean" environment

Permeation

The tritium content needs to be limited

1. Keep inventory at a minimum

2. Reduce inventory

3. Avoid contamination of coolants

Metal

Tritiated environment

"Clean" environment

Permeation barrier

Permeation

The tritium content needs to be limited

1. Keep inventory at a minimum

2. Reduce inventory

3. Avoid contamination of coolants

Ceramics are promising candidates:

oxides
carbides
nitrides

Permeation barriers are caracterised by their PRF (Permeation Reduction Factor)

\mathrm{PRF} = \frac{\mathrm{uncoated \ flux}}{\mathrm{coated \ flux}}

Target for breeding blankets PRF ≈ 100-1000

Luo et al Surface and Coatings Technology 2020

The tritium content needs to be limited

Component embrittlement

Agglomeration of hydrogen can lead to blistering

Accumulation of H at defects:
- Voids
- Reaction with impurities (formation of methane)
- grain boundaries
Creation of a filled cavity
Growth of the cavity without bursting

Kuznetsov, Alexey S. et al. “Hydrogen-induced blistering of Mo/Si multilayers: Uptake and distribution.” Thin Solid Films 545 (2013): 571-579.

Cavities can lead to Hydrogen Induced Cracking

Review of HIC by Sofronis

https://doi.org/10.1016/j.jngse.2022.104547

Take aways

H transport

Safety

Fusion Economy

Materials

Minimising inventories
Limiting permeation
Protecting personel

Fuel cycle
Tritium breeding
Losses

Embrittlement
H induced cracks

DFT

Multi-scale hydrogen transport

Y. Ferro et al 2023 Nucl. Fusion 63 036017

Length scale

Time scale

Length scale

Time scale

DFT

potentials

Grigorev et al 2018 JNM, 508 451-458

Multi-scale hydrogen transport

Component scale modelling

Length scale

Time scale

DFT

D, S, other coeffs.

Multi-scale hydrogen transport

Length scale

Time scale

DFT

Component scale modelling

Fuel cycle modelling

Residency times, fluxes, ...

Multi-scale hydrogen transport

Length scale

Time scale

DFT

Component scale modelling

Fuel cycle modelling

Abstraction

Multi-scale hydrogen transport

Question time!

Tritium transport theory

Hydrogen transport in metals

Diffusion

Even more particles

continuity approximation

Single particle

Random walk

Many particles

Diffusion

\varphi = - D \ \nabla c

$ \varphi $: diffusion flux

$ D $: diffusion coefficient

$ c $: mobile hydrogen concentration

Fick's 1st law of diffusion

Diffusion

\varphi = - D \ \nabla c

$ \varphi$: diffusion flux

$ D $: diffusion coefficient

$ c $: mobile hydrogen concentration

$ S$: source term

Fick's 1st law of diffusion

\frac{\partial c}{\partial t} = - \nabla \cdot \varphi + S

Fick's 2nd law of diffusion

Diffusion

\varphi = - D \ \left( \nabla c + \frac{Q\ c}{R \ T^2} \ \nabla{T} - \frac{c \ V_H}{R \ T} \nabla \sigma \right)

$ \varphi$: diffusion flux

$ D $: diffusion coefficient

$ c $: mobile hydrogen concentration

$ S$: source term

Soret effect (or thermophoresis)

Stress assisted diffusion

Particle implantation

Ziegler et al. 2010. Nuclear Instruments and Methods in Physics Research Section B: Beam Interactions with Materials and Atoms, 268 (11): 1818–23. https://doi.org/10.1016/j.nimb.2010.02.091.

Implantation range

Implantation range & width and reflection coefficient can be computed with SRIM, SDTRIM...

Mutzke et al, SDTrimSP Version 6.00 2019

Particle implantation

S = (1-r) \ \Gamma_\mathrm{incident} \ f(x)

$\Gamma_\mathrm{incident} $: incident flux (particle/m2/s)

$ f(x) $: Gaussian distribution (/m)

$r $: reflection coefficient

Surface effects

H2 molecules

Metal lattice

Surface effects

\varphi_\mathrm{dissociation} = K_\mathrm{d} \ P

Dissociation coefficient (H/m2/s/Pa)

Partial pressure of H (Pa)

Adsorbed H

Metal lattice

Surface effects

\varphi_\mathrm{dissociation} = K_\mathrm{d} \ P

\varphi_\mathrm{recombination} = K_\mathrm{r} \ c^2

Metal lattice

Recombination coefficient (m4/s)

Concentration (H/m3)

Surface effects

\varphi_\mathrm{dissociation} = K_\mathrm{d} \ P

\varphi_\mathrm{recombination} = K_\mathrm{r} \ c^2

Metal lattice

\varphi_\mathrm{net} = \varphi_\mathrm{dissociation} - \varphi_\mathrm{recombination}

Waelbroeck model

Surface effects

\varphi_\mathrm{dissociation} = K_\mathrm{d} \ P

\varphi_\mathrm{recombination} = K_\mathrm{r} \ c^2

Metal lattice

At equilibrium:

\varphi_\mathrm{net} = 0

c = \sqrt{\frac{K_\mathrm{d}}{K_\mathrm{r}}} \ \sqrt{P}

\varphi_\mathrm{net} = \varphi_\mathrm{dissociation} - \varphi_\mathrm{recombination}

c = K_S \ \sqrt{P}

Sievert's law of solubility

Surface effects

\varphi_\mathrm{dissociation} = K_\mathrm{d} \ P

\varphi_\mathrm{recombination} = K_\mathrm{r} \ c^1

Non-metallic liquid

At equilibrium:

\varphi_\mathrm{net} = 0

c = \frac{K_\mathrm{d}}{K_\mathrm{r}} \ P

\varphi_\mathrm{net} = \varphi_\mathrm{dissociation} - \varphi_\mathrm{recombination}

c = K_H \ P

Henry's law of solubility

Interfaces

Material 1

Material 2

Interfaces

Partial pressure and flux are continuous

Material 1

Material 2

Interfaces

Material 1

Material 2

Case 1:

Metal-Metal

- D_1 \nabla c_1 = - D_2 \nabla c_2

P = \left( \frac{c}{K_S} \right) ^2

\left( \frac{c_1}{K_{S,1}} \right)^2 =\left( \frac{c_2}{K_{S,2}} \right)^2

Sievert's law

Interfaces

Material 1

Material 2

Case 2:

Non metal-non metal

- D_1 \nabla c_1 = - D_2 \nabla c_2

P = \frac{c}{K_H}

\frac{c_1}{K_{H,1}} = \frac{c_2}{K_{H,2}}

Henry's law

Interfaces

Material 1

Material 2

Case 3:

Metal-Non metal

- D_1 \nabla c_1 = - D_2 \nabla c_2

P = \left( \frac{c}{K_S} \right) ^2

\left( \frac{c_1}{K_{S,1}} \right)^2 = \frac{c_2}{K_{H,2}}

Sievert's law

P = \frac{c}{K_H}

Henry's law

Interfaces

Material 1

Material 2

Steady state concentration profile

different diffusivities $\rightarrow$ different concentration gradients
different solubilities $ \rightarrow $ concentration discontinuity

$x$

$c$

⚠️Very little experimental validation data for interfaces

Permeation barriers are low solubility, low duffisivity

Metal

Tritiated environment

"Clean" environment

Permeation

Permeation barrier

Permeation barriers are low solubility, low duffisivity

Pressure $P$

High gradient = high flux

Low gradient = low flux

Pressure $P$

Trapping

Trap = anything binding to H

vacancy
grain boundary
impurity
chemical reaction
...

Trapping

Potential energy

Distance

Diffusion barrier

Energy barrier = activation energy

Trap binding energy

Trapping energy

Common assumption:

$ E_k = E_D $

Trapping

\mathrm{H} + [\ \ \ ]_\mathrm{trap} \ \substack{p \\[-1em] \longleftarrow\\[-1em] \longrightarrow \\[-1em] k} \ [\mathrm{H}]_\mathrm{trap}

\frac{\partial c_\mathrm{t}}{\partial t} = -\frac{\partial c_\mathrm{m}}{\partial t} = k \ c_\mathrm{m} \ n_\mathrm{free \ trap} - p c_\mathrm{t}

c_\mathrm{t}

Since $n_\mathrm{trap} = n_\mathrm{free \ trap} + c_\mathrm{t} $

c_\mathrm{m}

n_\mathrm{free \ trap}

Trapping

\mathrm{H} + [\ \ \ ]_\mathrm{trap} \ \substack{p \\[-1em] \longleftarrow\\[-1em] \longrightarrow \\[-1em] k} \ [\mathrm{H}]_\mathrm{trap}

\frac{\partial c_\mathrm{t}}{\partial t} = -\frac{\partial c_\mathrm{m}}{\partial t} = k \ c_\mathrm{m} \ (n_\mathrm{trap,0} - c_\mathrm{t}) - p c_\mathrm{t}

c_\mathrm{m}

n_\mathrm{free \ trap}

c_\mathrm{t}

Total concentration of traps

Trapping

\frac{\partial c_\mathrm{t}}{\partial t} = -\frac{\partial c_\mathrm{m}}{\partial t} = k \ c_\mathrm{m} \ (n_\mathrm{trap,0} - c_\mathrm{t}) - p c_\mathrm{t}

With diffusion

and

1 trap

\frac{\partial c_\mathrm{m}}{\partial t} = \nabla \cdot \left(D\nabla c_\mathrm{m} \right) + S - \frac{\partial c_\mathrm{t}}{\partial t}

\frac{\partial c_\mathrm{t}}{\partial t} = k \ c_\mathrm{m} \ (n_\mathrm{trap,0} - c_\mathrm{t}) - p c_\mathrm{t}

Trapping

N traps

\frac{\partial c_\mathrm{m}}{\partial t} = \nabla \cdot \left(D\nabla c_\mathrm{m} \right) + S - \sum \frac{\partial c_\mathrm{t,i}}{\partial t}

\frac{\partial c_\mathrm{t, 1}}{\partial t} = k_1 \ c_\mathrm{m} \ (n_\mathrm{trap,1} - c_\mathrm{t,1}) - p_1 c_\mathrm{t,1}

\frac{\partial c_\mathrm{t, 2}}{\partial t} = k_2 \ c_\mathrm{m} \ (n_\mathrm{trap,2} - c_\mathrm{t,2}) - p_2 c_\mathrm{t,2}

\frac{\partial c_\mathrm{t, 3}}{\partial t} = k_3 \ c_\mathrm{m} \ (n_\mathrm{trap,3} - c_\mathrm{t,3}) - p_3 c_\mathrm{t,3}

McNabb & Foster model

Trapping

Other models assume traps can hold more than one H

\mathrm{H} + [\ \ \ ]_\mathrm{trap} \ \substack{p_1 \\[-1em] \longleftarrow\\[-1em] \longrightarrow \\[-1em] k_1} \ [\mathrm{H}_1]_\mathrm{trap}

\mathrm{H} + [\mathrm{H}_1]_\mathrm{trap} \ \substack{p_2 \\[-1em] \longleftarrow\\[-1em] \longrightarrow \\[-1em] k_2} \ [\mathrm{H}_2]_\mathrm{trap}

\mathrm{H} + [\mathrm{H}_2]_\mathrm{trap} \ \substack{p_3 \\[-1em] \longleftarrow\\[-1em] \longrightarrow \\[-1em] k_3} \ [\mathrm{H}_3]_\mathrm{trap}

\frac{\partial c_\mathrm{m}}{\partial t} = \nabla \cdot \left(D\nabla c_\mathrm{m} \right) + S - \sum{k_{i} \ c_\mathrm{m} \ c_{\mathrm{t},i-1} - p_i c_{\mathrm{t},i}}

\frac{\partial c_{\mathrm{t},i}}{\partial t} = k_{i} \ c_\mathrm{m} \ c_{\mathrm{t},i-1} - p_i c_{\mathrm{t},i} - k_{i+1} \ c_\mathrm{m} \ c_{\mathrm{t},i} + p_{i+1} c_{\mathrm{t},i+1}

Many of these processes are thermally activated

Recombination

Dissociation

Absorption

Trapping

Detrapping

Diffusion

Arrhenius law

X = X_0 \ \exp{\left(\frac{-E_X}{k_B \ T}\right)}

Pre-exponential factor

Activation energy (eV/H)

Temperature (K)

Boltzmann constant (eV/H/K)

k_B = 8.617\times 10^{-5} \ \mathrm{eV/H/K}

Arrhenius law

X = X_0 \ \exp{\left(\frac{-E_X}{R \ T}\right)}

Pre-exponential factor

Activation energy (J/mol)

Temperature (K)

Gas constant (J/mol/K)

k_B = 8.617\times 10^{-5} \ \mathrm{eV/H/K}

R = 8.314 \ \mathrm{J/mol/K}

\frac{E_D [eV/H]}{k_B} = \frac{E_D [J/mol]}{R}

Conversion:

Arrhenius law

X = X_0 \ \exp{\left(\frac{-E_X}{R \ T}\right)}

$ 1/T $ (1/K)

E_X > 0

E_X < 0

Arrhenius law

X = X_0 \ \exp{\left(\frac{-E_X}{R \ T}\right)}

\log{(X)} = \log{(X_0)} \ - \frac{E_X}{k_B \ T}

\log{(X)} = \log{(X_0)} \ - \frac{E_X}{k_B} \ \frac{1}{T}

Intercept

+ Slope

Y =

Arrhenius law

Arrhenius parameters:

Diffusivity
Solubility
Permeability
Recombination coeff.
Dissociation coeff.
Trapping rate
Detrapping rate

Check out the HTM project!

github.com/RemDelaporteMathurin/h-transport-materials

Component scale modelling

\frac{\partial c_\mathrm{m}}{\partial t} = \nabla \cdot \varphi_\mathrm{diffusion} + S - \sum \frac{\partial c_\mathrm{t,i}}{\partial t}

\frac{\partial c_\mathrm{t, i}}{\partial t} = k_i \ c_\mathrm{m} \ (n_\mathrm{trap,i} - c_\mathrm{t}) - p_i c_\mathrm{t}

McNabb & Foster model

Challenges

Number of degrees of freedom
Interface discontinuities

Governing equations

- D_1 \nabla c_1 = - D_2 \nabla c_2

\left( \frac{c_1}{K_{1}} \right)^2 =\left( \frac{c_2}{K_{2}} \right)^{\{1,2\} }

Let's solve the system analytically

\frac{\partial c_\mathrm{m}}{\partial t} = \nabla \cdot \left(D\nabla c_\mathrm{m} \right) + S - \sum \frac{\partial c_\mathrm{t,i}}{\partial t}

\frac{\partial c_\mathrm{t, i}}{\partial t} = k_i \ c_\mathrm{m} \ (n_\mathrm{trap,i} - c_\mathrm{t,i}) - p_i c_\mathrm{t,i}

Let's solve the system analytically

Simplification #1: 1D domain $L$

\frac{\partial c_\mathrm{m}}{\partial t} = \frac{\partial}{\partial x} \left(D \frac{\partial}{\partial x} c_\mathrm{m} \right) + S - \sum \frac{\partial c_\mathrm{t,i}}{\partial t}

\frac{\partial c_\mathrm{t, i}}{\partial t} = k_i \ c_\mathrm{m} \ (n_\mathrm{trap,i} - c_\mathrm{t,i}) - p_i c_\mathrm{t,i}

Let's solve the system analytically

Simplification #1: 1D domain $L$

Simplification #2: 1 material

\frac{\partial c_\mathrm{m}}{\partial t} = \frac{\partial}{\partial x} \left(D \frac{\partial}{\partial x} c_\mathrm{m} \right) + S - \sum \frac{\partial c_\mathrm{t,i}}{\partial t}

\frac{\partial c_\mathrm{t, i}}{\partial t} = k_i \ c_\mathrm{m} \ (n_\mathrm{trap,i} - c_\mathrm{t,i}) - p_i c_\mathrm{t,i}

Let's solve the system analytically

D, S, k, p, n = \mathrm{constants}

Simplification #1: 1D domain $L=1$

Simplification #2: 1 material

Simplification #3:

\frac{\partial c_\mathrm{m}}{\partial t} = D \frac{\partial^2 c_\mathrm{m}}{\partial x^2} + S - \sum \frac{\partial c_\mathrm{t,i}}{\partial t}

\frac{\partial c_\mathrm{t, i}}{\partial t} = k_\mathrm{i} \ c_\mathrm{m} \ (n_\mathrm{trap,i} - c_\mathrm{t,i}) - p_\mathrm{i} c_\mathrm{t,i}

Let's solve the system analytically

D, S, k, p, n = \mathrm{constants}

Simplification #1: 1D domain $L$

Simplification #2: 1 material

Simplification #3:

Simplification #4: Steady state

0 = k_\mathrm{i} \ c_\mathrm{m} \ (n_\mathrm{trap,i} - c_\mathrm{t,i}) - p_\mathrm{i} c_\mathrm{t,i}

0 = D \frac{\partial^2 c_\mathrm{m}}{\partial x^2} + S

💡Tip

At steady state, the mobile concentration is independent of trapping

Let's solve the system analytically

0 = k_\mathrm{i} \ c_\mathrm{m} \ (n_\mathrm{trap,i} - c_\mathrm{t,i}) - p_\mathrm{i} c_\mathrm{t,i}

0 = D \frac{\partial^2 c_\mathrm{m}}{\partial x^2} + S

c_\mathrm{t,i} = \frac{n_\mathrm{trap,i}}{1+ \frac{p_\mathrm{i}}{k_\mathrm{i} \ c_\mathrm{m}}}

R.A Oriani, Acta Metallurgica, 1970

Oriani's model

Let's solve the system analytically

0 = D \frac{\partial^2 c_\mathrm{m}}{\partial x^2} + S

c_\mathrm{t,i} = \frac{n_\mathrm{trap,i}}{1+ \frac{p_\mathrm{i}}{k_\mathrm{i} \ c_\mathrm{m}}}

\frac{\partial^2 c_\mathrm{m}}{\partial x^2} = \frac{-S}{D}

\frac{\partial c_\mathrm{m}}{\partial x} = \frac{-S}{D} x + K_1

c_\mathrm{m} = \frac{-S}{2\ D} x^2 + K_1 \ x + K_2

2 unknowns =

2 equations =

2 boundary conditions

Let's solve the system analytically

c_\mathrm{t,i} = \frac{n_\mathrm{trap,i}}{1+ \frac{p_\mathrm{i}}{k_\mathrm{i} \ c_\mathrm{m}}}

c_\mathrm{m} = \frac{-S}{2\ D} x^2 + K_1 \ x + K_2

c_\mathrm{m}(x=0) = 1 \\ c_\mathrm{m}(x=1) = 0

Let's solve the system analytically

c_\mathrm{t,i} = \frac{n_\mathrm{trap,i}}{1+ \frac{p_\mathrm{i}}{k_\mathrm{i} \ c_\mathrm{m}}}

c_\mathrm{m} = \frac{-S}{2\ D} x^2 + K_1 \ x + K_2

K_2 = 1 \\ \frac{-S}{2\ D} + K_1 + K_2 = 0

K_2 = 1 \\ K_1= \frac{-S}{2\ D} - 1

Let's solve the system analytically

c_\mathrm{t,i} = \frac{n_\mathrm{trap,i}}{1+ \frac{p_\mathrm{i}}{k_\mathrm{i} \ c_\mathrm{m}}}

c_\mathrm{m} = \frac{-S}{2\ D} x^2 - (\frac{S}{2\ D} + 1) \ x + 1

Steady state
1 material
Homogeneous temperature
1D

We can solve it numerically

Component modelling

Experimental analysis

3 main numerical methods

Finite Difference Method (FDM)

Finite Element Method (FEM)

Finite Volume Method (FVM)

Let's not bother

\frac{\partial c }{\partial x} \approx \frac{c(x=x_{i+1}) - c(x=x_{i})}{x_{i+1} - x_{i}}

There are a handful of H transport codes

TMAP7

TMAP8

MHIMS

FESTIM

COMSOL

Abaqus

There are a handful of H transport codes

TMAP7

TMAP8

MHIMS

FESTIM

COMSOL

Abaqus

1D/2D/3D

TESSIM

There are a handful of H transport codes

TMAP7

MHIMS

FDM

FEM

TESSIM

TMAP8

FESTIM

COMSOL

Abaqus

There are a handful of H transport codes

TMAP7

TMAP8

MHIMS

FESTIM

COMSOL

Abaqus

Proprietary

Open-source

Closed-source

TESSIM

Ferrero, et al (2022) A Preliminary CFD and Tritium Transport Analysis for ARC Blanket, Fusion Science and Technology, 78:8, 617-630

COMSOL

ARC's breeding blanket
Source of tritium from neutronics calculations
Heat transfer + Fluid dynamics + H transport
Steady state
Computation time 8h

TMAP7 and TESSIM

R. Arredondo, et al (2021) Preliminary estimates of tritium permeation and retention in the first wall of DEMO due to ion bombardment, NME

W on Eurofer
1D
Agreement between TMAP and TESSIM

How to handle large models

x 10,000

💡Build a surrogate model!

What is the inventory of the whole divertor?

Large problem

Delaporte-Mathurin, PhD thesis, 2022

A monoblock surrogate model

t = 10^7 \, \mathrm{s}

\mathrm{inventory} = \int c \, dV

T_\mathrm{surface} = 700 \, \mathrm{K}

c_\mathrm{m} = 10^{21} \, \mathrm{H\,m^{-3}}

c_\mathrm{m} = 10^{20} \, \mathrm{H\,m^{-3}}

T_\mathrm{surface} = 1000 \, \mathrm{K}

Delaporte-Mathurin, PhD thesis, 2022

A monoblock surrogate model

Delaporte-Mathurin, PhD thesis, 2022

A monoblock surrogate model

Delaporte-Mathurin, PhD thesis, 2022

A monoblock surrogate model

\mathrm{inventory} = \int c \, dV

t = 10^7 \, \mathrm{s}

Gaussian Process Regression (GPR)

Chris Bowman. ‘C-Bowman/Inference-Tools: 0.5.3 Release’. Zenodo, 20 April 2020.

Tritium contamination in a heat exchanger

$ c = K_H \ P_\mathrm{up} $

$ c = 0 $

Permeation through the crucible wall

FLiBe

2D permeation through molten salts

\mathrm{flux} = 2 \pi \ \int_0^R r \ D \nabla c \ \cdot \mathbf{n} \ dr

HYPERION permeation rig

Tritium Breeding Blanket

DEMO WCLL
Complex 3D geometry
Coupled to fluid dynamics
Tritium generation in the LiPb volume (computed from neutronics)

James Dark et al 2021 Nucl. Fusion 61 116076

LIBRA produces validation data for multiphysics models

Tritium production map from neutronics (OpenMC)

Tritium concentration field (FESTIM)

Max conc. 1.4E13 T/m3

[T/n/cm3]

Natural convection neglected

ARC Breeding Blanket

ARC blanket geometry

T source

Heat source

Velocity

Temperature

Turbulent viscosity

Tritium concentration

Tritium extraction system

Dunnell and Delaporte-Mathurin, MIT NSE Beyond Oppenheimer (2024) 10.13140/RG.2.2.13923.16164

Permeation Against Vacuum
Complex 3D geometry
Coupled with fluid dynamics
Tritium extraction from permeable membranes

HISP project: coupling FESTIM to plasma codes

RISP pulse

ITER FW divided in 60 bins

Data from DINA

Goal: find the best strategy for minimising ITER T inventory

HISP project: coupling FESTIM to plasma codes

10 DT FP pulses

ICWC + RISP

GDC

Simulation time:
~ 60 s per bin (~ hour full reactor)

Question time!

FESTIM hands-on workshop

User inputs

Material properties
Trap properties
Geometry
Boundary conditions
Initial conditions
...

Heat transfer model

Hydrogen transport model

McNabb & Foster
Multi-level trapping
Multi-isotopes
...

FESTIM

Outputs

H concentration fields $c(x,t)$
Temperature field $T(x,t)$

surface fluxes
inventories
average concentration
...

2019

Start of development

We need a new tool!

Problem: predict T retention and permeation

Multi-dimensional
Multi-material
non homogeneous temperature

CuCrZr

Particle and heat fluxes

Convection

14 mm

2022

License Apache 2.0

Available on Github

github.com/festim-dev/FESTIM

✅ More transparency

Reproducibility in papers
Source code available

✅ More collaborations

No need for complex collaboration agreements
External contributions

✅ More flexibility

Can be adapted to your needs
Better interoperability

2019

Open source

We need a numerical tool

Problem: predict T retention and permeation

Multi-dimensional
Multi-material
non homogeneous temperature

CuCrZr

Particle and heat fluxes

Convection

14 mm

2022

License Apache 2.0

Available on Github

github.com/festim-dev/FESTIM

✅ More transparency

Reproducibility in papers
Source code available

✅ More collaborations

No need for complex collaboration agreements
External contributions

✅ More flexibility

Can be adapted to your needs
Better interoperability

Oct 2023

v1.0 release

2019

v1.0 release

We need a numerical tool

Problem: predict T retention and permeation

Multi-dimensional
Multi-material
non homogeneous temperature

CuCrZr

Particle and heat fluxes

Convection

14 mm

2022

Oct 2023

April 2024

Non-profit organisation supporting open source software for science and research

More info at numfocus.org

2019

We need a numerical tool

Problem: predict T retention and permeation

Multi-dimensional
Multi-material
non homogeneous temperature

CuCrZr

Particle and heat fluxes

Convection

14 mm

2022

Oct 2023

April 2024

Non-profit organisation supporting open source software for science and research

More info at numfocus.org

Meet the development team

Jonathan Dufour

CEA, France

+ other contributors

Remi Delaporte-Mathurin

MIT, USA

@RemDelaporteMathurin

Samuele Meschini

PoliTo, Italy

@SamueleMeschini

Etienne Hodille

CEA, France

@ehodille

Gabriele Ferrero

PoliTo, Italy

@gabriele-ferrero

James Dark

MIT, USA

@jhdark

Vladimir Kulagin

MEPhI, Russia

@KulaginVladimir

FESTIM is open-source

Check it out on Github

github.com/festim-dev/FESTIM

✅ More transparency

✅ More collaborations

✅ More flexibility

The code is missing a feature?

Just add it!

Found a bug?

Report it and we'll fix it!

FESTIM is open-source

ossfe.org

Open-Source Software for Fusion Energy 2026 conference

Call for abstracts open!

FESTIM's development workflow

Anyone can create their own copy (fork) of the FESTIM repository and make changes
PRs are a place where FESTIM's maintainers review the proposed changes
~500 tests are automatically run
The PR is only merged when all the tests pass ✅

fork

pull request

FESTIM

Jane Doe

FESTIM

John Doe

FESTIM

festim-dev

pull request

FESTIM is user-friendly

Very easy to learn
Plenty of libraries (numpy, scipy, matplotlib, HTM...)
Inuitive interface

conda install -c conda-forge festim

One-line install!

import festim as F
import numpy as np

my_model = F.Simulation()

my_model.mesh = F.MeshFromVertices(
    vertices=np.linspace(0, 1e-6, num=1001)
)

my_model.materials = F.Material(id=1, D_0=1.9e-7, E_D=0.2)

my_model.T = 500  # K

my_model.boundary_conditions = [
    F.DirichletBC(
        surfaces=[1, 2],
        value=1e15,  # H/m3/s
        field=0
        )
]

my_model.settings = F.Settings(
    absolute_tolerance=1e10,
    relative_tolerance=1e-10,
    final_time=100  # s
    )


my_model.dt = F.Stepsize(0.1)  # s

my_model.initialise()

my_model.run()

festim.discourse.group