thermo-mechanical modelling of ceramic breeder pebble beds ... - gan y - ther… · cbbi-14,...

Forschungszentrum Karlsruhein der Helmholtz-Gemeinschaft

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Gan and Kamlah, IMF II, Forschungszentrum Karlsruhe. CBBI-14, Petten, September 2006 1

Thermo-Mechanical Modelling of Ceramic Breeder Pebble Beds

and its Application in HELICA Mock-up

Yixiang Gan, Marc Kamlah

Institut für Materialforschung II, Forschungszentrum Karlsruhe


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Outline

Material model• Constitutive modelling• Identification of material parameters• Implementation and validation

Application in HELICA mock-up

Pebble-wall interaction• Thermal contact conductance• Friction


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Overview and cross-section of Li4

SiO4

(Reimann et al. 2006)

Material Modelling

Bulk material: Lithium-based ceramics (i.e. Li4

SiO4

)Microstructure: Spherical particlesSize:

d ~ 0.2mm-0.4mmProduction:

melt-spraying (Schott AG, Mainz)


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• Basic experimental results: Oedometric test

J. Reimann, L. Boccaccini, M. Enoeda, A. Y. Ying, Fusion Engineering and Design 61-62 (2002) 319-33

•

Pebble size (mm) is small compared to bed dimensions.•

Thermomechanical behaviour of ceramic breeder pebble beds can be described by the continuum-

mechanical approaches.


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• Nonlinear Elastic Law

• Drucker-Prager-Cap Theory (yielding and hardening)

• Creep Law based on the Drucker-Prager-Cap Model

• Volumetric Strain Dependent Conductivity

• Thermo-Mechanical Interaction (thermal expansion)

• Material Model for Pebble Beds


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Nonlinear Elastic Law (Coube, IWM Freiburg, 1998)

02/22 ])

21(3

31[ EpqAE s

e +−++

= νν

q

pContour Plot of Young’s Modulus at q-p

space

where,

ijij

ii

ssq

p

2331

=

−= σ is the equivalent pressure stress

is the Mises

pressure stress

Implemented in ABAQUS with USDFLD (user defined field).


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Drucker-Prager-Cap yield surface

pap( )βtanapdR +

bp

d

β

( )βα tanapd +

q

βtanapd +

shear failure, Fs

transition surface, Ft

cap, Fc

Drucker-Prager-Cap Theory

)tan1(,

)tan()()(

0tan22

β

β

β

RRdppwhere

pdRRqppF

dpqF

ba

aac

s

+−

=

+−+−=

=−−=


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Creep Mechanisms (strain-hardening)

Consolidation Creep:

If :No volumetric creep

p

q

ac ppp −=In ABAQUS:

Modification: ppc =for directly using the experimental data

]1)1[()/exp(1

1 1

0

10 −

Δ+−

+=Δ ++ mmn

c tttpTBA

mε

0111

011

])())/exp(1

1[( crmm

crmn

c tpTBAm

εεε −+Δ−+

=Δ +++(Buehler, 2002)

Implemented in ABAQUS with CREEP (user defined creep law).


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• From Experiments to Modelling

Nonlinear Elastic LawAe

, s, E0

, v

Drucker-Prager Cap ModelR, β, pb

, d

Cap CreepA, B, m, n

Oedometric test


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2σ

1σq

p

Deformation Mechanism in Oedometric Testing

Plasticity and Hardening (1)

⎪⎩

⎪⎨

⎧

=∂∂=

= 0|

0

311

σσσc

c

GF

2...),,( σβ Rdfpb = (*)

Yielding Criterion

Plastic Flow Law

pldε

pldε


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Reimann‘s Fit:LL CCLLL TCCC 54 1

1*

321 ])([ −+= εσUU CCUUU TCCC 54 1

1*

321 ])([ −+= εσ

plel εεε +=*

elεε =*

( , , )L Upl i ig C Cε σ= (**)

Eqn.(*) and Eqn.(**) plUi

Lib CCTRdp εβ ~...),,,,,(~

Implemented in ABAQUS with USDFLD (user defined field).

Plasticity and Hardening (2)


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plUi

Lib CCTRdp εβ ~...),,,,,(~

•

Using the experimental data by the method of trail and fail, with every set of material parameters.

The Advantages of the Improvement:

PresentBefore Modification

•

Introduce the temperature to the yielding and hardening laws.

Thermoplasticity

•

Directly using the empirical curve fittings, Reimann fit for experiments. Flexibilities in material parameters.

•

The yielding surface and hardening law based on Drucker-

Prager Cap theory, in stresses space.


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Validation

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.50

1

2

3

4

5

6

Com

pres

sive

Stre

ss (M

Pa)

Strain (%)

FEM at 500C Exp. at 500C FEM at 5500C Exp. at 5500C FEM at 7500C Exp. at 7500C FEM at 7500C Exp. at 7500C

HELICA Material Calibration: Lithium Orthosilicate

0 2000 4000 60000

1

2

3

4

5

Cre

ep S

train

(%)

Time (min)

T(0C) Stress(MPa) F.E.M. Exp. 840 4.1 850 2.1 765 2.1 700 2.1

Oedometric compression:T = 50~850 oC

Creep experiments

Experimental data for Li4SiO4 used in HELICA mock-up, from Reimann et al. 2006


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Mechanical interaction:Normal contact force, averaged as surface pressureFriction, μ=0.02/0.05

Thermal interaction:Thermal contact conductance (TCC) model

Contact area

Pebble

Pebble-Wall Interaction

Multiple spot contact conductance, hot-spots distributed randomly along the contact plane.(N contact pairs in the total area S; radius of pebble r)

QhT A

=Δ

2

SsNr

=


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b (>>a)

a

1 1 1, ,k E ν

2 2 2, ,k E ν

Solid spot contact conductance Hertzian

solution for contact problem

s gh h h= +

2

22 /( )sakh Nak S

sr= =

1 2

1 2

2k kkk k

=+

(Madhusudana, 1995,Chan and Tien, 1973)

1/31 23 ( )[ ]4

P K K ra π +=

1 ;2 2(1 )

i ii i

i i

EK GGν

π ν−

= =+

2P psr=

1/3 1/31 22 /3

2 [3 ( ) / 4] gkh K K p h

s rπ= + +


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T91 Steel(Fokkens, 2005)

Inconel 718(Fokkens, 2005)

Li4SiO4 (bulk)(Futamura, 2005)

Poisson Ratio 0.3 0.298 0.24

Young’s modulus 206GPa 207GPa ~95GPa

Thermal conductivity 25.9W/moC 11.2 W/moC ~2 W/moC

radius ∞ ∞0.1125 mm(Reimann

et al., 2006)

s Pebble-

T91 steel

Pebble-

Inconel 718

3 700.482 640.012

4 578.235 528.318

5 498.308 455.291

6 441.276 403.182

7 398.180 363.806

1/3gh d p h= +

Material properties at room temperature

The values of d in Pebble packing topology:(111)-surface of FCC: 3.464(001)-surface of SC: 4(001)-surface of FCC: 4(001)-surface of BCC: 16/3

• Dependent on both local temperature and pressure (initial pressure).•

First simulation, pressure dependent TCC model, using RT parameters. (Aquaro, 2006)


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Prediction of present model and compared with experiments:Abou-Sena, Ying and Abdou

(2003) Fusion Science and Technology

Validation of TCC model

Lack of experimental data of ceramic breeder pebble-wall interaction.

Using other materials’ data sets (Beryllium-SiC), in the same framework of TCC model for validation purpose.

Validated with existing beryllium pebble-wall experiments, well-consist with experimental data while concerning the “initial pressure” (i.e. gravity…).

0.0 0.5 1.0 1.5 2.0 2.5500

1000

1500

2000

2500

3000

3500

4000

4500

Inte

rface

Con

duct

ance

Coe

ffici

ent

(W/m

2 K)

External Applied Load (MPa)

TCC model Experiments

(Abou-Sena et al. 2003)

1/33000( 0.1)Be SiCh p− = +


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• Materials:T91 steel; AISI 316L; Inconel

718; Kathal; Gasket;Li4

SiO4 (Fokkens 2005; Reimann et al. 2006)• FE model:Extracted from IGES file from ENEA BrasimoneModel symmetry2D generalized plane strain element: 3664• Pebble-wall interactionThermal contact conductance modelFriction (0.02/0.05)

HELICA Mock-up


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Results

• Temperature FieldProfiles at different heating steps

• FEM results vs. ExperimentsComparison between FEM model and thermal couples, LVDTs

• Stress-strain fieldhydrostatic pressure, von-Mises

stress, volumetric inelastic strains• Thermal conductivity profiles


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Temperature profiles

Heat flux step-428 kW/m2



CPU time: 4 hours for whole analysis. Convenient for studying the variations of model parameters,

i.e. surface interactions, and physical meaning.


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0 60 120 180 240 300 360 420 480

200

300

400

500

600

700

800FEM Experiments

TE211D (°C) TE306S (°C) TE222D (°C) TE351D (°C)

Tem

pera

ture

(o C)

Time (min)

TC 150mm

0 60 120 180 240 300 360 420 480

200

300

400

500

600

700

800

Tem

pera

ture

(o C)

Time (min)

FEM Experiments TE223D (°C) TE304S (°C) TE105D (°C) TE306D (°C)

TC 55mmThermal couplesCassette

Pebble

Heater

Pebble

Experimental data from A. Tincani, ENEA Brasimone

0 60 120 180 240 300 360 420 480

200

300

400

500

600

700

800FEM Experiments

TE231D (°C) TE305S (°C) TE212D (°C) TE305D (°C)

Tem

pera

ture

(o C)

Time (min)

TC 100mm

Surface heat flux is not uniform in experiments.

55mm100mm150mm


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1 2 3 4 5 660

70

80

90

100

110

120

130

60

70

80

90

100

110

120

130

Tem

pera

ture

Incr

emen

t (0 C

)

Step Number

Experiment FEM Position 55mm 100mm 155mm

55/100/150 mmPressure-dependent TCC modelIncrement of temperature at each step

•

Increment of temperature decreases due to the contact pressure increasing.

• The maximum decrement

(25% of maximum increment) in the experimental observation.•

The trends predicted by FEM compared well with the experiments.•

The initial TCC value is higher in experiments, due to the initial pressure.

; ;p h T↑ ↑ Δ ↓

0( ) 26.5i jMax T T CΔ −Δ =


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0 60 120 180 240 300 360 420 480-0.35

-0.30

-0.25

-0.20

-0.15

-0.10

-0.05

0.00

LVD

T (m

m)

Time (min)

LVDT 01-03 LVDT 04-06 Average value FEM

Displacement vs. LVDTs

0.000 0.005 0.010 0.015 0.020 0.025

400

450

500

550

600

650

700

750

800

Tem

pera

ture

(0 C)

Distance (m)

FEM Experiment 55mm 100mm 150mm

Heater Cassette

Temperature distribution at 42kW/m2

Temperature distribution along the transversal direction

55mm100mm150mm

Experimental data from A. Tincani, ENEA Brasimone

Y


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Hydrostatic Pressure (Pa) at heat flux step-6

von Mises

stress (Pa) at heat flux step-6

• Maximum hydrostatic pressure: 4.28 MPa (average <2 MPa)• Locally distributed at the end of pebble layers• Related to the pebble-wall friction

• Shear region


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0 60 120 180 240 300 360 420 4800.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

von

Mis

es S

tress

(MP

a)

Time (min)

55mm 100mm 150mm

0 60 120 180 240 300 360 420 4800.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

55mm 100mm 150mm

Hyd

rost

atic

Pre

ssur

e (M

Pa)

Time (min)

Stresses along symmetric axis vs. Time

• von Mises

and hydrostatic stresses • Decrease by creep effects• non-positive stresses after unloading.

55/100/150 mm


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Volumetric plastic strain

Volumetric creep strain


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Plastic strain magnitude

Creep strain magnitude


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Thermal Conductivity

• Thermal conductivity depends on both temperature and strain values.• Different values while loading and unloading, decreasing value after unloading.

0.83~1.08W/moC step-6

0.77~0.79 W/moC after unloading


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Conclusion

• Continuum approach for modelling of pebble bedNonlinear elastic law (Coube, 1998)Drucker-Prager-Cap theory (yielding, hardening and creep law)Volumetric strain dependent conductivity (Reimann et al. 2006)Thermo-mechanical interaction (thermal expansion)Identification of material parameters (Gan and Kamlah, 2006)

• Pebble-wall interactionPebble-wall frictionThermal contact conductance

• Simulation of HELICA mock-up2D generilized plane strain modelFully coupled thermal-mechanical analysis


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Dank u zeer!

Thank you!

thermo-mechanical modelling of ceramic breeder pebble beds ... - gan y - ther… · cbbi-14,...

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