Links to other pages for this example : - Overview - Set-up

One Dimensional Analytical Solution

This example aims to demonstrate a simple set-up to get started with.

Considering a 1-dimensional domain with a single, homogeneous trap type, a known, constant concentration, $var element = document.getElementById("moose-equation-f76a15ea-f3c8-4391-b9c9-4355e966b1cd");katex.render("c_0", element, {displayMode:false,throwOnError:false});$ , on one boundary and an initial concentration of $var element = document.getElementById("moose-equation-5b2ed04e-6c9c-4bf9-a90e-ca8cb5792181");katex.render("0", element, {displayMode:false,throwOnError:false});$ on the other boundary. From Penalva et al. (2012) following Carslaw and Jaeger (1959), the particle flux through second boundary as a function of time can be solved analytically as shown in Equation Eq. (1) under the condition that $var element = document.getElementById("moose-equation-8eab02d8-b3fe-433b-8605-9f1a2e7049ce");katex.render("\\zeta >> c_m/n_1", element, {displayMode:false,throwOnError:false});$ .

$(1)var element = document.getElementById("moose-equation-60171631-2a78-456a-a1f3-51196f4d302f");katex.render(" \\varphi(t) = \\frac{c_0D}{l} \\left[ 1 + 2 \\sum_{m=1}^{\\infty}(-1)^m\\exp\\left(-m^2\\frac{\\pi^2D_{\\text{eff}}t}{l^2}\\right)\\right]", element, {displayMode:true,throwOnError:false});$

Where $var element = document.getElementById("moose-equation-d1049803-e090-44ee-88ef-6a78f3746987");katex.render("D_{\\text{eff}}", element, {displayMode:false,throwOnError:false});$ is:

$(2)var element = document.getElementById("moose-equation-8c4e2509-b453-4236-9dd0-98f68b6bdc20");katex.render(" D_{\\text{eff}} = \\frac{D}{1 +\\zeta}", element, {displayMode:true,throwOnError:false});$

And $var element = document.getElementById("moose-equation-d1701519-e2e3-4ab4-aad8-337e4ca168b9");katex.render("\\zeta", element, {displayMode:false,throwOnError:false});$ is given by:

$(3)var element = document.getElementById("moose-equation-8a9759cd-8a77-4439-be27-6a973d4e927f");katex.render(" \\zeta = \\frac{\\lambda^2n_{\\text{sol}}\\nu_0}{D_0n_1}\\exp \\left(\\frac{E_{\\text{diff}}- E_1}{k_BT} \\right) + \\frac{c_m}{n_1}", element, {displayMode:true,throwOnError:false});$

This can be simulated in Achlys and compared to the analytical solution. Example input parameters are shown in Table 1 and simulation set-up will be explained in more detail below.

Example input parameters for simple 1-dimensional effective diffusivity simulation taken from Delaporte-Mathurin et al. (2019).

Table 1: Example innput parameters

parameter name	symbol	value	Unit
Density	$var element = document.getElementById("moose-equation-d9d473e3-c3a4-4f03-b1d0-e536f2c19922");katex.render("\\rho", element, {displayMode:false,throwOnError:false});$	$var element = document.getElementById("moose-equation-55ecf259-a954-4e74-af84-f5b86f00a374");katex.render("3.16 \\times 10^{22}", element, {displayMode:false,throwOnError:false});$	$var element = document.getElementById("moose-equation-90fffc23-89dc-4624-99f5-b8a82e3f74f2");katex.render("m^{-3}", element, {displayMode:false,throwOnError:false});$
Lattice sites	$var element = document.getElementById("moose-equation-366942e2-3eda-4741-84a2-e5b3b73b1782");katex.render("N_{\\text{sol}}", element, {displayMode:false,throwOnError:false});$	$var element = document.getElementById("moose-equation-53c8c793-4221-4586-bf1b-35edd02f6a4f");katex.render("2.0 \\rho", element, {displayMode:false,throwOnError:false});$	$var element = document.getElementById("moose-equation-d8aa1a7d-17f8-4f9f-8714-9ab207bc8280");katex.render("m^{-3}", element, {displayMode:false,throwOnError:false});$
Trap Density	$var element = document.getElementById("moose-equation-54951824-9c02-4a3f-8d6c-874b5fad952b");katex.render("n_1", element, {displayMode:false,throwOnError:false});$	$var element = document.getElementById("moose-equation-0f99714d-28a1-4bfe-90c3-dc9e23ce24a7");katex.render("1.00\\times 10^{-1}\\rho", element, {displayMode:false,throwOnError:false});$	$var element = document.getElementById("moose-equation-5d3085bd-265e-461f-80f9-056b6111c46d");katex.render("m^{-3}", element, {displayMode:false,throwOnError:false});$
Surface Concentration	$var element = document.getElementById("moose-equation-3d91c157-f896-441f-9ca6-ed367cb304c6");katex.render("c_0", element, {displayMode:false,throwOnError:false});$	$var element = document.getElementById("moose-equation-c17c8951-f85c-4147-89a1-b66c5ec6ee8a");katex.render("1.00\\times 10^{-4} \\rho", element, {displayMode:false,throwOnError:false});$	$var element = document.getElementById("moose-equation-b00ebc7c-90c8-47c2-9c33-7faeec389178");katex.render("m^{-3}", element, {displayMode:false,throwOnError:false});$
Diffusion rate pre-exponential factor	$var element = document.getElementById("moose-equation-bc9f2c03-c446-48d2-919e-65246b9b5dfa");katex.render("D_0", element, {displayMode:false,throwOnError:false});$	1.0	$var element = document.getElementById("moose-equation-d0a07912-d5ce-40c1-a23d-00157ca7903f");katex.render("m^{2}s^{-1}", element, {displayMode:false,throwOnError:false});$
Diffusion Energy	$var element = document.getElementById("moose-equation-e493a720-1d2e-46bc-b08d-96df3606fd57");katex.render("E_{\\text{diff}}", element, {displayMode:false,throwOnError:false});$	0.0	$var element = document.getElementById("moose-equation-949df6bc-4fe6-4369-8878-60fc8325a246");katex.render("eV", element, {displayMode:false,throwOnError:false});$
Lattice constant	$var element = document.getElementById("moose-equation-abb1ec06-4459-43fc-b236-a0fdb6d9e0a0");katex.render("\\lambda", element, {displayMode:false,throwOnError:false});$	$var element = document.getElementById("moose-equation-cbfa8f3f-e052-4f82-ac66-1c9a8dcc2a91");katex.render("3.16\\times 10^{-8}", element, {displayMode:false,throwOnError:false});$	$var element = document.getElementById("moose-equation-82a2f6e3-5fb2-4e73-9813-b1afa24422d8");katex.render("m", element, {displayMode:false,throwOnError:false});$
De-trapping pre-exponential factor	$var element = document.getElementById("moose-equation-3ef56883-5b63-4b9c-8e3f-d24dd9a4baaa");katex.render("\\nu_0", element, {displayMode:false,throwOnError:false});$	$var element = document.getElementById("moose-equation-05e4f7a8-a054-4c2e-9e41-3e0a3ae94053");katex.render("1.0\\times 10^{-13}", element, {displayMode:false,throwOnError:false});$	$var element = document.getElementById("moose-equation-30cbe2e2-1337-471a-9613-0e75383a5757");katex.render("s^{-1}", element, {displayMode:false,throwOnError:false});$
De-trapping Energy, trap 1	$var element = document.getElementById("moose-equation-76102257-566d-408b-9335-e98e4547f497");katex.render("E_{1}", element, {displayMode:false,throwOnError:false});$	$var element = document.getElementById("moose-equation-485a9c9f-83e6-4da8-9072-c7525d0cdc8a");katex.render("8.6\\times 10^{-3}", element, {displayMode:false,throwOnError:false});$	$var element = document.getElementById("moose-equation-6824b082-4506-48f8-9333-db1b085fe3a7");katex.render("eV", element, {displayMode:false,throwOnError:false});$
Temperature	$var element = document.getElementById("moose-equation-c62b78eb-41e3-4f23-9c27-1f23186dc1c7");katex.render("T_{\\text{imp}}", element, {displayMode:false,throwOnError:false});$	$var element = document.getElementById("moose-equation-fea9f0be-e055-49d4-b403-06dd4559309d");katex.render("1000", element, {displayMode:false,throwOnError:false});$	$var element = document.getElementById("moose-equation-a716ab7e-7a64-445b-a79b-788d1a655f20");katex.render("K", element, {displayMode:false,throwOnError:false});$
Simulation time	$var element = document.getElementById("moose-equation-a88b35ae-66ca-45ec-afe6-5fccf8e2cf2c");katex.render("t_{\\text{sim}}", element, {displayMode:false,throwOnError:false});$	$var element = document.getElementById("moose-equation-9e04f18e-63cb-4b8b-9248-cc862f63721e");katex.render("1.0\\times 10^{-8}", element, {displayMode:false,throwOnError:false});$	$var element = document.getElementById("moose-equation-9c5ebbd4-f032-4030-90c3-82692db2d710");katex.render("s", element, {displayMode:false,throwOnError:false});$

The expected results are shown in Figure Figure 1. Here the units have been converted from the raw result into SI units by multiplying by the atomic number density $var element = document.getElementById("moose-equation-8a95fcbe-15a3-4219-aa80-b6dd22a5fadf");katex.render("\\rho = 3.16\\times 10^{22}", element, {displayMode:false,throwOnError:false});$ . Note that the differences between the simulation and analytical results are due to the coarse mesh used here. Accuracy can be improved by refining the mesh further, however, the calculation time will also increase.

Figure 1: Comparison of Achlys with analytical case

References

H S Carslaw and J C Jaeger. Conduction of heat in solids. Clarendon Press, 1959.

@book{Carslaw1959,
    author = "Carslaw, H S and Jaeger, J C",
    title = "Conduction of heat in solids",
    date = "1959",
    year = "1959",
    OPTpublisher = "Clarendon Press",
    publisher = "Clarendon Press",
    OPTisbn = "0198533683"
}

Rémi Delaporte-Mathurin, Etienne A. Hodille, Jonathan Mougenot, Yann Charles, and Christian Grisolia. Finite element analysis of hydrogen retention in iter plasma facing components using festim. Nuclear Materials and Energy, 21:100709, 2019. URL: https://www.sciencedirect.com/science/article/pii/S2352179119300547, doi:https://doi.org/10.1016/j.nme.2019.100709.

@article{DELAPORTE2019,
    author = "Delaporte-Mathurin, Rémi and Hodille, Etienne A. and Mougenot, Jonathan and Charles, Yann and Grisolia, Christian",
    title = "Finite element analysis of hydrogen retention in ITER plasma facing components using FESTIM",
    journal = "Nuclear Materials and Energy",
    volume = "21",
    pages = "100709",
    year = "2019",
    issn = "2352-1791",
    doi = "https://doi.org/10.1016/j.nme.2019.100709",
    url = "https://www.sciencedirect.com/science/article/pii/S2352179119300547",
    keywords = "Fusion, Tritium, Finite elements, Plasma facing components, Hydrogen isotopes",
    abstract = "The behaviour of hydrogen isotopes in ITER monoblocks was studied using the code FESTIM (Finite Element Simulation of Tritium In Materials) which is introduced in this publication. FESTIM has been validated by reproducing experimental data and the Method of Manufactured Solutions was used for analytical verification. Following relevant plasma scenarios, both transient heat transfer and hydrogen isotopes (HIs) diffusion have been simulated in order to assess HIs retention in monoblocks. Relevant materials properties have been used. Each plasma cycle is composed of a current ramp up, a current plateau, a current ramp down and a resting phase before the following shot. 100 cycles are simulated. The total HIs inventory in the tokamak during resting phases reaches 1.8×10−3mg whereas during the implantation phases it keeps increasing as a power law of time. Particle flux on the cooling channel of the monoblock is also computed. The breakthrough time is estimated to be t=1×105s which corresponds to 24 cycles. Relevance of 2D modelling has been demonstrated by comparing the total HIs inventory obtained by 2D and 1D simulations. Using 1D simulations, a relative error is observed compared to 2D simulations which can reach -25\% during the resting phase. The error during implantation phases keeps increasing."
}

I. Penalva, G. Alberro, Fernando Legarda, G. Esteban, and B. Riccardi. Interaction of Copper Alloys with Hydrogen, pages. IntechOpen, 03 2012. doi:10.5772/34469.

@inbook{PENALVA2012,
    author = "Penalva, I. and Alberro, G. and Legarda, Fernando and Esteban, G. and Riccardi, B.",
    year = "2012",
    month = "03",
    pages = "",
    title = "Interaction of Copper Alloys with Hydrogen",
    isbn = "978-953-51-0160-4",
    doi = "10.5772/34469",
    publisher = "IntechOpen"
}

References

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One Dimensional Analytical Solution

References