Arbeiter, Frederik

Dagan, Ron

Nagatou, Kaori

Pasler, Volker

Schulz, Marvin R.

von der Weth, Axel

English

Weth, Axel von der

KITopen

Analytical solution of a gas release problemconsidering permeation with time dependentboundary conditionsMarvin R. Schulz | September 26, 2019Joint Work With: Kaori Nagatou, Axel von der WethFrederik Arbeiter, Ron Dagan, Volker PaslerKIT – Die Forschungsuniversität in der Helmholtz-Gemeinschaft www.kit.eduMotivationInfluence of radiation damage on Diffusion and Sieverts’ constantSetup-Material?Heating heals defects!How to recover D and ksfrom data?Experiments like this are relevant for:Reactor and Fusion-technology,Green energy: Storage and transport of H2,repository exploration, . . .Marvin R. Schulz – Analytical Solution of a gas release problem September 26, 2019 1/16OutlineMotivationExperiment, SetupDiffusion-Equation with time dependent boundary-conditionsExplicit Solution for each time interval [Sample]Partial pressure of Hydrogen during Release-IntervalMarvin R. Schulz – Analytical Solution of a gas release problem September 26, 2019 2/16Experimental SetupCylinder geometricisovolumetric setupLoading with Hydrogen atdefined pressure pL (andTemperature T )Evacuate the System→ p(t) = pL exp[−λt]Recover Sieverts’ andDiffusion ConstantR0 = 3.0 10−3 m, L0 = 3.0 10−2 mR1 = 0.01 m, R2 = 0.02 m, L1 = 0.04 mMarvin R. Schulz – Analytical Solution of a gas release problem September 26, 2019 3/16Boundary ConditionsNote: (Sieverts’ Law)On the surfaces the Concentration of(Mono-atomic)-Hydrogen is given byc ≡ k√p(t) ∀t ≥ 0p(t) =pL, t ∈ [t0 = 0, t1]pL exp[−λt], t ∈ [t1, t2]unclear, t ∈ [t2, t3]Note: (Diffusion-Equation)∂tc(n) − Dj∆c(n) = 0, j ∈ {s, c} t ∈ [tn, tn+1]c(n)(r , z, t) = kj√p(t) for{r ∈ {R0,R1}z ∈ {±L0,±L1}c(n)(R2, z, t) ≡ 0c(n)(r , z, tn) = c(n−1)(r , z, tn−1), c(−1) ≡ 0Marvin R. Schulz – Analytical Solution of a gas release problem September 26, 2019 4/16Solving These EquationsZero-Boundary condition [Sample]gS(r , z, t) := c(r , z, t)− ks√p(t), r ∈ [0,R0], z ∈ [−L0, L0]⇒ ∂tgS − Ds∆gS = ks∂t√pZero-Boundary condition [Containment, neglecting z]gC(r , t) := c(r , t)− τ(r)kc√p(t), τ(r) =log[r ]− log[R2]log[R1]− log[R2], r ∈ [R1,R2]⇒ ∂tgC − Dc∆gC = τks∂t√pHom. PDE →Separation of variablesODEs →SolvableFundamental Solutions→Variation of ConstantsFull SolutionMarvin R. Schulz – Analytical Solution of a gas release problem September 26, 2019 5/16Sample: Analytical Solutiont ∈ [0, t1], p(t) ≡ 3 kPa, ks = 1.829 10−3 mol√Pam3 , Ds = 7.879 10−9 m2sSpecimen Concentration Profile for z = 0. Relative error of approximation.gS(r , z, t) =∑n,mbn,m exp[−γ2m,nt]cos((2n + 1)π2L0z)J0(xmrR0),γ2m,n = Ds(x2mR20+(2n + 1)2π24L20), bm,n = 8(−1)n+1ks√pLπ(2n + 1)xmJ1(xm)Marvin R. Schulz – Analytical Solution of a gas release problem September 26, 2019 6/16Compare: Numerical (Fortran),Analytical (Python), OpenFOAMDifferent approaches show similar results!AnalyticalSolutionPython +FortranNumericalSolutionFortranNumericalSimulationOpenFOAMMarvin R. Schulz – Analytical Solution of a gas release problem September 26, 2019 7/16Compare: Numerical (Fortran),Analytical (Python), OpenFOAMDifferent approaches show similar results!AnalyticalSolutionPython +FortranNumericalSolutionFortranNumericalSimulationOpeanFOAMMarvin R. Schulz – Analytical Solution of a gas release problem September 26, 2019 8/16Containment: Analytical Solutiont ∈ [0, t1], p(t) ≡ 3 kPa, kc = 5.941 10−4 mol√Pam3 , Dc = 8.257 10−10 m2sContainmentConcentration Profile for z = 0. Stored amount of Hydrogen.gc(r , z, t) =∞∑n=1An exp[−λ2nDR21t](−Y0(λn)J0(λn)J0(λnrR1) + Y0(λnrR1))det(J0(λn) Y0(λn)J0(λn R1R2 ) Y0(λnR1R2))= 0,An = −k√pL∫ R2R1τ(r)un(r)rdr∫ R2R1un(r)2rdrMarvin R. Schulz – Analytical Solution of a gas release problem September 26, 2019 9/16Pumping IntervalFick’s Law→ c(r , z, t) = pL exp[−λt], for r ∈ {R0,R1}, z ∈ {L0, L1}Fixed Ratio: 97 % He and 3 % H2Remaining Partial Pressure:⇒ pH2(0.9s) = 1PaSpecimen ContainmentmsH(800 s+0.9 s)ks√PLVsample≈ 96% mcH(800 s+0.9 s)ks√PLVsample≈ 124%Marvin R. Schulz – Analytical Solution of a gas release problem September 26, 2019 10/16Release IntervalProblem: p(t) in gaseous phase unknown⇒ No Boundary ConditionsNote: (Initial Guess)√p(t) =√pf −√p01− exp[−βT ](1− exp[−βt]) +√p0 (1)[Sedano et al., J. Nucl. Mater. (1999)]∂tgS − Ds∆gS = ks ∂t√p︸ ︷︷ ︸Now Easy, ∂tgC − Dc∆gC = τkc∂t√pNote:β,T and pf free parameters,Sieverts’ Law still holds for the pressure above.Marvin R. Schulz – Analytical Solution of a gas release problem September 26, 2019 11/16Release Interval GraphicsSpecimenjs1(t) = −2π Ds∫ L−LR0∂r c(R0, z, t)dzjs2(t) = −4π Ds∫ R00∂zc(r , L0, t)rdrjc3 (t) = −2π Dc∫ L1−L1R1∂r c(R1, z, t)dzContainmentFlux of Hydrogen:Coat Containment jc3 ,(2x Caps Containment),Coat Specimen js1 ,Caps Specimen js2 .Marvin R. Schulz – Analytical Solution of a gas release problem September 26, 2019 12/16Pressure Increasem(t) =∑k∫ t0jk (τ)dτ, pH2(t) =(msH(t) + mcH(t)) RT2 (V0 − VS).Marvin R. Schulz – Analytical Solution of a gas release problem September 26, 2019 13/16ConclusionFortran FDM and analytic (Python) model show similar results,Measure containment effect,Recover D and k with B&B (fast evaluation -> analytic solution),Estimate: Duration of Experiment,copper containment may be usableOpen Problems:Guess function√p(t) does not exactly fit. (Fixed-point Iteration),Melting Point of copper ≈ 1.300 K.Marvin R. Schulz – Analytical Solution of a gas release problem September 26, 2019 14/16Acknowledgement:The current talk summarizes results of two running projects:Main goal of the first one is the determination of transport parameters ofhydrogen in structural metallic materials used for components in fusionpower stations: This work has been carried out within the framework of theEUROfusion Consortium, and has received funding from the Euratomresearch and training program 2014-2018 under grant agreement No.633053. The views and opinions expressed herein do not necessarilyreflect those of the European CommissionThe authors are also thanking for support and fundings by MathSEEregarding the second project: Neue Lösungen derKontinuitätsdiffernetialgleichung mit Phasengleichgewicht zurVerbesserung der Ergebnisse bei der Auswertung von Experimenten.Marvin R. Schulz – Analytical Solution of a gas release problem September 26, 2019 15/16BibliographySedano, Perujo, Wu, Intrinsic hydrogen transport constants in theCFC matrix and fibres derived from isovolumetric desportionexperiments, in J. Nucl. Mater. (1999)Parsons, Reichanadter, Vicksman, Segur, Explicit Solution forCylindrical Heat Conduction, in American Journal of UndergraduateResearch (2016)von der Weth, Nagatou, Arbeiter, Dagan, Klimenko, Schulz,Numerical analysis of an isovolumetric thermal desorptionexperiment, Defect and Diffusion Forum (2019)Marvin R. Schulz – Analytical Solution of a gas release problem September 26, 2019 16/16

Analytical solution of a gas release problem considering permeation with time dependent boundary conditions

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Analytical solution of a gas release problem considering permeation with time dependent boundary conditions

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