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Experimental Results and Validation of Thermo-Mechanical Models Used for the PREMUX Test Campaign, as Part of the Roadmap towards an Out-

of-Pile Testing of a Full Scale HCPB Breeder Unit Mock-up

F. Hernándeza, Y. Ganb, M. Kamlahc, M. Kolbc

aKarlsruhe Institute of Technology (KIT), Institute of Neutron Physics and Reactor Technology (INR), Germany. bThe University of Sidney, School of Civil Engineering, Australia. cKarlsruhe Institute of Technology (KIT), Institute for Applied Materials (IAM), Germany.

E-mail contact of main author: francisco.hernandez@kit.edu

Abstract. The Breeder Units are the key components of the Helium Cooled Test Blanket Module (HCPB TBM) in ITER, aimed at producing tritium and extracting high grade heat. The present strategy at the Karlsruhe Institute of Technology in Germany is to test the thermo-mechanical performance of a full-scale mock-up in an out-of pile experimental campaign. In order to achieve this milestone, several key research gaps are being addressed first in a roadmap comprising three pillars: (1) the design, development and manufacturing of a feasibility HCPB BU mock-up, (2) the development and construction of relevant testing tools and (3) the development of thermal and thermo-mechanical models. While the research gaps in the 1st pillar have been already bridged during the past years, this paper reports about the progress achieved in the 2nd and 3rd pillars of this roadmap. In the 2nd pillar, a relevant heating system has been developed and tested in the so-called “pre-test mock-up experiment” (PREMUX): the results of this campaign are reported here. In the 3rd pillar, two models have been developed. The first is a probabilistic thermal model based on response surface meta-modelling, which takes into account the inherent uncertainties of the model’s input parameters. This model has been applied to PREMUX and the results and validation procedure are reported here. The second is a simplified thermo-mechanical model of the pebble beds in the HCPB BU, which implements nonlinear elasticity, Drucker-Prager Cap plasticity, a non-associative flow rule and an isotropic hardening law. The results on the identification of the material parameters for both functional materials, as well as a preliminary validation of this model with available uniaxial compression tests and its application to PREMUX are reported as well here.

1. Introduction

The Helium Cooled Pebble Bed (HCPB) Test Blanket Module (TBM) (FIG.1 left) is one of the two European test blanket concepts in ITER. At the core of the HCPB TBM, 16 Breeder Units (BU, FIG.1 right) are assembled. These key functional components are the responsible of the tritium breeding and part of the heat extraction in the HCPB blankets of a future fusion reactor. The tritium breeding is performed inside each BU by a Li4SiO4 pebble bed. A second pebble bed of beryllium pebbles acts as neutron multiplier. The HCPB TBM structural material is EUROFER-97 steel (9Cr-1.1W-0.4Mn-Ta-V). An extensive program has been performed during the past years with the goal of obtaining a reference design for the HCPB TBM box [1] and the HCPB BU for ITER [2], taking into account DEMO relevancy aspects as well. One key milestone of the development of the HCPB TBM is the qualification of the thermo-mechanical performance of a BU mock-up in an out-of-pile experimental campaign at the Karlsruhe Institute of Technology (KIT). The test is planned to be performed in a dedicated high temperature helium loop (KATHELO), currently under construction at KIT.

Before this test can be carried out, several research gaps have been identified. First, the search for a heating system able to mimic the nuclear heat generation in the BU, which must not only

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be able to deploy a high power density in some areas of the Li4SiO4 pebble bed [3], but also must minimize the intrusion in the Li4SiO4 and Be pebble beds.

FIG. 1. HCPB TBM (left) and detail of a HCPB Breeder Unit (right)

Second and in view of the development of the breeding blankets for the future fusion demonstration reactor (DEMO), the overall thermal performance of the BU and the thermo-mechanics of the pebble beds must be assessed with reliable finite element (FE) models. On the one side, with a thermal model capable of take into account the inherent uncertainties of the breeder blanket parameters such as loads and boundary conditions. On the other side, the complex fully-coupled thermo-mechanical phenomena taking place in the pebble beds must be as well assessed during the development phase of these breeding blankets.

In order to bridge these research gaps, a three-pillar roadmap is being performed at KIT with the goal of (1) obtain a reference design of the HCPB BU, (2) developing and qualifying testing tools and (3), developing of FE models for the assessment of the thermal and thermo-mechanical performance of the HCPB BU. While the 1st pillar has been developed in the past [2], the progress and results of the 2nd and 3rd pillars are reported here.

2. Pre-test Mock-Up eXperiment (PREMUX) of a relevant HCPB BU section

Studies on different alternatives for reproducing the nuclear heating in an out-of-pile experiment has found that a 3-dimensional matrix of bent heater cables (∅1mm) and regularly spaced in the BU in a hexagonal pattern (FIG.2 top-left) can meet the requirements of low intrusion and high flexibility needed for the out-of-pile testing of a HCPB BU mock-up [4].

The performance of such heater concept has been tested in PREMUX by reproducing the most thermally loaded region of the Li4SiO4 pebble bed in the BU (FIG.2 top-left). The test section consists in a test box (FIG.2 top-right) of P92 steel (9Cr-0.5Mo-1.75W-V-Nb). The box is actively cooled by air flowing at 3 g/s, 2 bar and ≈20 °C through each of its cooling channels. The heater system is composed by 3 heaters (FIG.2. top-right): 2 based on a heater cable of 2.15 m length and ∅1 mm, bent 11 times to obtain the hexagonal pattern and 1 central heater consisting of a straight cable of ∅1 mm and 175 mm long. The heater system has a dense layout of thermocouples (FIG.2. bottom), whose positions have been determined by 3D laser scanning before its assembly in PREMUX. The Li4SiO4 pebble bed is kept under a static He atmosphere (4 bar), which simulates the purge gas in the BU during operation. Further details about the goals, the design and construction of PREMUX are described in [4].

In this test, steady state test runs and transient thermal cycles reproducing the ITER power pulses have been performed. As additional goals, the test has aimed as well at obtaining the effective thermal conductivity of the Li4SiO4 pebble bed at room temperature and at testing

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the robustness of the system against strong variations in the coolant mass flow rate. The steady state runs of PREMUX have been used as a benchmark data for the validation of the thermal and thermo-mechanical models presented in the sections 3 and 4 of this paper.

FIG. 2. Top-left: BU mock-up thermal analysis with a detail (lower right corner) of the heating blocks and superimposed picture of the PREMUX test box. Top-right: 3D section cut of the PREMUX test

box. Bottom: layout of the thermocouple positions in the Li4SiO4 pebble bed test section.

2.1. Test campaigns and experimental results

Steady state runs: A total of 5 power levels have been tested in PREMUX in steady state regime, namely “very high” (“VHI”, heating power of ���� = 1400 ), “high” (“HI”, ��� = 1100 ), “medium” (“MED”, ��� = 900 ), “low” (“LO”, ��� = 600 ) and “very low” (“VLO”, ���� = 300 ). For each test, a power ramp-up of 120 s to the corresponding power level is applied. Nominal values of the coolant (3 g/s and 2 bar) and for the He purge gas (4 bar) are specified and the experiments have been repeated randomly 6 times for each power level. The temperature measurements are shown in FIG.4, along with the error bars showing a 95% confidence interval of the measurement. These runs have been oriented to be used as benchmark data for the validation of the modeling tools.

Transient runs: The transient run has served to proof the concept of the heater system for its use in a future BU test. FIG.3 shows 3 successful power cycles reproducing 3 relevant ITER power pulses have been performed, consisting of a power ramp-up of 30 s, a flat top of 430 s at VHI and a ramp-down of 60 s to 0 W, followed by 510 s of rest until the next pulse.

3. Development of a probabilistic thermal model

3.1. Computational Design of Experiments, response surface and stochastic sampling

A probabilistic thermal model has been developed using the ANSYS Workbench 14.5 code. This model takes into consideration not only the nominal values of the thermal loads and boundary conditions on the HCPB BU, but also their associated uncertainties. The model is built in 5 steps. In a 1st step, a deterministic 3D model of PREMUX consisting of a slice of

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the test box containing the test section under study is performed taking into account the nominal values of the model’s input parameters (power levels, material properties and boundary conditions). In a 2nd step, a probability density distribution is associated to each model input that is considered to be stochastic. By means of ANSYS DesignXplorer in a 3rd step, a computational Design of Experiments (DoE) is performed by using an optimal-filling Design. In the 4th step and once the DoE is finished, a meta-model based on Gaussian process substitutes the FE model by a response surface. In the last 5th step, stochastic sampling by weighed Latin Hypercube Sampling method is used to determine the expected values and the uncertainty propagation of the temperature distribution of the PREMUX FE model.

FIG. 3. Experimental results of a transient run: plot of the temperature evolution during 3 power

cycles reproducing 3 ITER power pulses in the central thermocouples of the test section.

3.2. Results and validation using PREMUX as benchmark

FIG.4 depicts the results of the probabilistic analysis. The central pebble bed temperatures at the 5 steady state power levels from VLO to VHI are plotted in FIG.4 (a) to (e) respectively (see FIG.2. bottom to refer to the thermocouple nomenclature). FIG.4 (f) shows the statistical validation plot proposed for the thermal model. It is done by plotting the measured � ��� vs. predicted � ��� temperatures together with the regression line � ��� = � � ���. The slope � is 1.001 with a 95% confidence interval (0.993, 1.009). Therefore, it is concluded that the probabilistic model agrees with the experimental results with a significance level � 5%. The adjusted goodness of fit of the validation regression line is 2 -adj= 99.8 % , with a RMSE = 20.65 °). In order to generalize and evaluate the model’s accuracy, a RMSE% has been defined relative to the measured mean temperature of the Li4SiO4 pebble bed in the range of the 5 power levels performed in the experimental campaign (371.5 °)). This yields to a RMSE% = 5.56 %. As the slope of the regression line is 1.001, it indicates that the model marginally overestimates the temperature measurements an average of 0.1%, thus making the model slightly conservative, which is also an advantageous property of the model.

4. Development of a simplified thermo-mechanical model for the pebble beds

ANSYS has been the favored by the ITER organization and its parties for the development of the reactor’s components and therefore for the development of the HCPB TBM, including the BU. An essential part of the development of future breeding blankets will be to understand and to assess the nonlinear, fully coupled thermo-mechanical phenomena occurring in the pebble beds. In the last years, important efforts have been performed in this direction, but

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practically all of them developed with the ABAQUS code. Here, a simplified fully coupled thermo-mechanical FEM developed in ANSYS is presented.

FIG. 4. Probabilistic thermal model results vs PREMUX experimental results. Plots (a) to (e):

comparison from VLO to VHI respectively. Plot (f): Statistical validation of the model. The uncertainty bars represent a 95% confidence interval of the measurements.

4.1. Nonlinear elasticity and stress-strain dependent material properties in ANSYS

Relatively complex and specialized user defined material subroutines are normally developed to describe the material properties of the pebble beds for fusion reactors. Here, a methodology to use exclusively the verified built-in subroutines of ANSYS has been developed. However, these built-in subroutines have been found to have 2 severe limitations for the pebble beds thermo-mechanical modeling: (1) the absence of an element type that can combine nonlinear elasticity with Drucker-Prager Cap plasticity, and (2) the inability to work with stress-strain dependent material properties, which is fundamental to describe the thermal conductivity + of the pebble beds, as they are function of the temperature � and the inelastic volumetric strains ,-./01 .

In order to overcome these shortcomings, a workaround solution consisting of a multi-linear elastic-plastic algorithm (MEPLAS) during the solution process (FIG.5 left) and a treatment of the finite element domain as a subdomain of a finite meta-elements domain (FIG.5 right) has been applied. The MEPLAS algorithm assumes linear material properties (Young modulus

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2��3 and thermal conductivity +��3) during each load step 450 of the the iterative process. At the end of 450, the stress-dependent 2��3+1and +��3+1 are recalculated for 450+1 with the new stress state of the pebble bed, reproducing in this way the real stress-strain curve.

FIG. 5. Left: approximation of the stress-strain curve by multi-linear segments. Right: reorganization

of the finite element domain in meta-elements.

The nonlinear elastic modulus is calculated according to the work of Coube as shown by Gan in [5] as 2 = 20 + :;[32 (1 − 2?)@2 + 1+A3 B2]D2 . Here, 20 is a residual value (20 → 0), ? = 0.25 is the Poisson ratio of the pebble bed, @ and B the hydrostatic pressure and von Mises stresses respectively and G and :; are material parameters to be determined experimentally. According to Gan [5], G = )5I . Though, a slightly modified expression derived from Gan [5] is proposed for :; here:

:; = 2)1I()2I+)3I� K4M) [(1 + ?)(1 − 2?)2(1 − ?) ]1−12 . (1)

)1I , )2I , )3I , )4I and )5I are the coefficients of the “Reimann’s fits” [6]. Due to the dependence of :; with � , the coupled thermal-mechanical behavior of the pebble bed during the elastic regime can be modeled. The strain dependent thermal conductivity of the Li 4SiO4 and Be pebble beds are implemented using the relationships reported in [6].

4.2. Drucker-Prager Cap plasticity, plastic flow potential and isotropic hardening law

In order to model the onset of plasticity in pebble beds for fusion blankets, the Drucker-Prager Cap yield surface R SK��� is available in ANSYS. This yield surface is expressed in terms of the 3 invariants T1, U2, U3 of the Cauchy stress tensor and consists of a product of 3 surfaces, namely the shear envelope RV, the compaction cap RV and the expansion cap RW [7]:

R SK��� ≡ Γ2(Z�)U2 − RV2(T1)R[(T1,\0)RW(T1) = Γ2(Z�)U2 − (]0 − :^(_` �1) − ab T1)2 ×

× [1 − e(\0 − T1)( T1 − \0 [b RV(\0))2] × [1 − e(T1)( T1 Wb RV(0))

2] = 0, (2)

where e is the Heaviside function, \0 = i. + [b RV(\0) is a “flag” point controlled by the evolution of the “marker” point i. along the −T1 axis and

Γ(Z�) = 12(1 + sin 3Z� + 1n (1 − sin 3Z�)) ,with Z�(U2, U3) = −12 sin−1 (3√3U32U232 ) . (3)

The constants n, ]0, :, Zb , ab , [b , Wb are material parameters to be identified.

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The plastic flow potential function t���, whose derivatives defines the direction of the inelastic strain increments, is constructed with eqn. (3) and substituting the parameters Zb , ab , [b , Wb by Zu , au , [u , Wu [7], which must be also identified separately.

Once the onset of plasticity has reached, the pebble bed experiences inelastic strains and isotropic hardening, enlarging the shape of the R SK���. For the pebble beds of breeding blankets, hardening is isotropic and considered only in the compaction cap surface [5]. The enlargement of the compaction cap is controlled in ANSYS by the “marker” i. as a function of the inelastic volumetric strains ,-./01 (i0) [7]:

,-./01 (i0) = 1,w3 {^[ 1,{3(}0−}0,3)− 2,{3(}0−}0,3)2] − 1} , (4)

where i0,0 is an initial value of i0 and 1,w3 ,�1,w3 and �2,w3 are also material parameters to be identified. The temperature dependency of ,-./01 (i0) can be modelled by obtaining sets of 1,w3 ,�1,w3 and �2,w3 for a temperature �0 : ANSYS then linearly interpolates between these parameters sets to obtain ,-./01 (i0) for other temperatures. This is necessary for the Li4SiO4 pebble beds, as the loading path for this pebble bed in the “Reimann fits” [6] has a temperature dependency; this dependency is not present in the Be pebble bed.

4.3. Identification of material parameters

A total of 4 material parameters for the nonlinear elastic expressions, 11 for the plasticity formulations and the sets of 3 temperature-dependent parameters for the hardening law must be determined for each pebble bed type so as to represent their thermo-mechanical behavior.

The parameters for the nonlinear elasticity can be directly implemented from the work of Gan [5]. The parameters for the yield surface in ANSYS are identified by transforming the analog formulations of ABAQUS, which are expressed in terms of @ and B, to the invariant space, comparing the resulting expressions and obtaining equations relating the material parameters of the ANSYS and ABAQUS. Then, the values of the material parameters in ANSYS are calculated by using the material identification done by Gan in ABAQUS [5].

No analytic expression has been found that relates the parameters of the flow potential between both codes. Therefore, the flow potential parameters for ANSYS have been derived by estimating the dilatancy angle of the pebble beds, which for a typical friction coefficient of 0.2 has been found to be 38.3°.

The equivalent expression for the hardening law in ABAQUS of Gan [5] has been converted into invariant formulation. By means of curve fitting, sets of 1,w3 ,�1,w3 and �2,w3 have been found for 50 °C, 550 °C, 750 °C and 850 °C in the case of the Li4SiO4 pebble bed. Only 1 set is needed for the case of the Be pebble bed, as there is no temperature dependence here.

4.4. Preliminary validation using uniaxial compression tests

As a first validation benchmark, the database of temperature dependent uniaxial compression tests (UCT) of Reimann et al. for Li4SiO4 and Be pebble beds [6] has been used. For the validation, a 2D axisymmetric model of the cylindrical set-up used by Reimann et al. for these tests have been built in ANSYS Workbench 14.5. The tests have been reproduced for 50 °C, 550 °C, 750 °C and 850 °C with an axial load from 0 MPa to 6 MPa. The results of the benchmark are shown in FIG.6 and reflect a good agreement of the thermo-mechanical model with the experimental results for both types of pebble beds.

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FIG. 6. Preliminary validation of the ANSYS thermo-mechanical model of the pebble beds with UCT

data of Reimann et al. [6]. Left: Li4SiO4 pebble bed. Right: beryllium pebble bed

5. Summary and conclusions

This paper has presented the progress done in the development of testing and modeling tools as a preparation for a future out-of-pile test of a HCPB BU mock-up. The experimental tests performed in PREMUX have proved the principle of a new heater system concept, based on a 3D matrix of cable heaters, to reproduce the nuclear heating in the BU in an out-of-pile test. Moreover, two models have been developed and preliminary validated. The first is a probabilistic thermal model of PREMUX, which allows assessing the thermal performance of a system in a stochastic manner and not only in terms of nominal values, so as to evaluate the uncertainties in the model outputs. The second is a simplified thermo-mechanical model of the pebble beds in the BU, developed in ANSYS Workbench using the well-verified built-in subroutines of the code. The model implements temperature and stress-dependent nonlinear elasticity, Drucker-Prager Cap yield plasticity with non-associative flow rule and temperature dependent isotropic hardening and a first validation exercise shows a good agreement with the available UCT as function of the temperature.

References

[1] BOCCACCINI, L.V., et al., “Present status of the conceptual design of the EU test blanket systems”, Fusion Engineering and Design 86 (2011) 478-483.

[2] HERNANDEZ, F., et al., “Thermo-mechanical analyses and assessment with respect to the design codes and standards of the HCPB-TBM Breeder Unit”, Fusion Engineering and Design 87 (2012) 1111-1117.

[3] PERESLAVTSEV, P., “Neutronic analysis of the HCPB TBM in ITER utilizing an advanced integral approach”, Fusion Engineering and Design 85 (2010) 1653-1658.

[4] HERNANDEZ, F., et al., “Construction of PREMUX and preliminary experimental results, as preparation for the HCPB breeder unit mock-up testing”, Fusion Engineering and Design 89 (2014) 1257-1262.

[5] GAN, Y., “Thermo-mechanics of pebble beds in fusion blankets”, PhD Thesis, Karlsruhe Institute of Technology (2014). [6]

[6] REIMANN, J., et al., “Final report on new compilation of the material data base and the material assessment report”, Final Report TW5-TTBB-006, Institut für Kern- und Energietechnik, Karlsruhe Institute of Technology (2005).

[7] ANSYS, “ANSYS Theory Reference”, Release 14.5 (2013).