analysis of thermomechanical interactions and properties of ceramic breeder blankets

Fusion Engineering and Design 39–40 (1998) 651–657

Analysis of thermomechanical interactions and properties ofceramic breeder blankets

Alice Ying *, Mohamed AbdouMechanical and Aerospace Engineering Department, UCLA, Los Angeles CA 90095-1597, USA

Abstract

Analysis of the thermomechanical interactions of fusion blanket materials and physical elements, particularly thoseinvolving particle bed design concepts, have been performed. The results show that the stress caused by the highthermal expansion of the blanket particle bed material is not excessive enough to be of concern for the structuralmaterial. However, its magnitude could lead to ceramic particle breakage if the bed behaves like a solid material.Furthermore, the calculations show that the clad deflection caused by this stress could lead to a possible separationbetween the bed and clad, which would add an unfavorable thermal resistance to the region and increase localtemperature. A comparison between the estimated stress extrapolated from the experimental data and the analyticalresults shows that the thermomechanical performance of the pebble bed approaches that of solid materials. © 1998Elsevier Science S.A. All rights reserved.

1. Introduction

High volumetric heat generation rates and lowthermal conductivity cause large temperature gra-dients in the breeder elements. These temperaturedifferences produce expansion of materials and ifno restraints, both internal and external, areplaced on materials, they expand in a stress-freemanner. However, interconnecting componentsproduce external restraints on expansion and ther-mal stresses result in blanket elements. Thus, po-tential failure due to thermal stresses placesconstrains on the blanket performance. One mightargue that ceramic breeders have no structuralrole; however, their thermomechanical behavior isimportant from a cracking-resistance perspective.

Breeder cracking is undesirable since it alters heattransfer characteristics and may cause tritiumpurge channel plugging due to transport and relo-cation of breeder fragments by the purge gas.Consequently, it is very important to be able toaccurately predict the state of the stress in theblanket element to insure that a failure does notoccur.

Substantial experimental and modeling effortshave been carried out to characterize the mostimportant properties that directly affect the blan-ket thermal performances. These properties in-clude effective thermal conductivity, pebble bedwall conductance, and interface conductance be-tween sintered Be and stainless steel cladding.Good agreement has been obtained for the effec-tive thermal conductivity of single-size pebblebeds. The most uncertainty exists in the experi-* Corresponding author.

0920-3796/98/$19.00 © 1998 Elsevier Science S.A. All rights reserved.

PII S0920-3796(98)00198-7

A. Ying, M. Abdou / Fusion Engineering and Design 39–40 (1998) 651–657652

mental database and model prediction capabilityof the pebble bed wall conductance [1] which isgenerally influenced by less quantifiable parame-ters such as packing geometry, contact force andporosity distributions. It is further affected by thebed/clad interaction. For example, low value ofinterface conductance obtained experimentallymight be the result of the bed/clad separation dueto thin clad bowing during operation [2]. On theother hand, it is known that the induced thermalstresses alter particle contact characteristics andincrease the amount of contact area which couldlead to enhancement of the packed bed effectivethermal conductivity. With this in mind, this pa-per focuses on the analysis of the thermomechani-cal interaction of fusion blanket elementsparticularly those involving particle bed designconcepts. Experimental data related to the ther-momechanical interaction are first examined fol-lowed by the thermomechanical analysis of theblanket unit cell. The stress exerted on the clad-ding wall due to bed thermal expansion is calcu-lated utilizing experimental particle bed effectivemodulus data. The force transmitted at the con-tact point is discussed and compared with avail-able experimental data.

2. Analysis of experimental data related tothermomechanical interactions

In this section, previous experimental studies onthe thermomechanical interactions related to theceramic breeder blankets are discussed. The heattransfer data from Dalle Donne et al. [3] hasshown that for the same average bed temperature,a larger temperature gradient across the blanketprovides a higher effective thermal conductivity.These experiments were carried out for aluminumparticle beds (to simulate beryllium packed beds)at different temperature gradients (DT). The heattransfer enhancement is caused by the differencein the thermal expansion coeffcient of the alu-minum and cladding material. Aluminum has ahigher thermal expansion coefficient and conse-quently the aluminum particles are pressed to-gether more strongly and the contact area

between the pebbles increases. To reflect this phe-nomenon the data of the bed effective thermalconductivity (k) has been correlated as [3]

k=C0[T(,C)]0.1721[DT(oC)]0.1173 (1)

where C0 is a constant. The increase in the effec-tive thermal conductivity caused by the tempera-ture difference can be related to the increase in thecontact area of the particulate. In Fig. 1, thecalculated average increase in the fractional con-tact area (based on the analysis of Wakao et al.[4]) and the corresponding normal constraineddisplacement at the contact point causing thiscontact area are plotted as a function of tempera-ture difference. These calculations show that aslight increase in the contact area of 0.024% at theparticle contact point can lead to an increase ofthe aluminum bed effective thermal conductivityby a factor of 2.

According to Hertz’s theory [5], it is conceiv-able to estimate the normal force (P) exerted onthe contact point to respond to this normal con-strained displacement:

P=M�3d

R�1.5

(2)

Fig. 1. Fractional contact area and normalized constraineddisplacement as a function of temperature difference.

A. Ying, M. Abdou / Fusion Engineering and Design 39–40 (1998) 651–657 653

Fig. 2. Comparison of mechanically applied forces and ther-mally induced forces for achieving the same magnitudes ofthermal conductivity enhancement.

3. Thermomechanical interaction analysis model

In an attempt to explain how the heat transferand contact area increase and to estimate thestress of the state at the containment wall as aresult of the differential thermal expansion, athermomechanical model is developed for a typi-cal blanket unit cell involving a packed bedregime with its associated containment clad. Themodel is derived from Euler-Bernoulli’s equation,which was also used in a previous thermomechan-ical analysis [7]. When compared to the previousmodel, the present analysis has the following dis-tinguished features: (1) it utilizes an experimentalvalue of the packed bed effective modulus forstress [8] calculations; (2) since there is, as of yet,no adequate model for describing particle bedmacroscopic mechanical behavior, analyses areperformed for both solid- and fluid-like behaviors,and (3) the model allows no stress to exert on thecontainment wall where the bed expands less thanthe wall deformation. Furthermore, the modeldecouples the microscopic effect from the macro-scale behavior considering that no adequate mi-crophysical properties and constitutive equationsare available for describing the complex phenom-ena involved at the microscopic level. The inter-particle mechanical quantities at microscopic levelare analyzed once the stress is calculated.

For the unit cell as shown in Fig. 3, the totalunrestrained deformation of the particle bed inthe y direction due to a uniform temperature riseDT given by:

Dy=ap DTW (4)

If the particle bed is assumed to behave like asolid continuous material, then the bed-clad con-tact stress (s) would be equal to

where d and R are the normal displacement andparticle radius, respectively, and M is given interms of the shear modulus G and Poisson’s ratiov of the material of the spheres by:

M=8

93

GR2

(1−6)(3)

The calculated P is plotted as a function ofthermal conductivity enhancement in Fig. 2 andcompared with the experimental data of Tehra-niam et al. [6] in which the experiments werecarried out to study the effect of the appliedmechanical load on heat transfer enhancement.The comparison shows a reasonable agreementbetween the applied mechanical loads and thethermally induced stresses that obtain the samemagnitude of bed heat transfer enhancement.Since this normal force originates from the con-strained bed thermal expansion which is causedby the temperature rise of the bed, the relation-ship between the macro-scale stress exerted on theclad and the aforementioned force is addressed inthe later section.

Fig. 3. Schematic view of thermomechanical interaction ana-lytical model (clamped boundaries at both ends).


s=Ep

Dy−dxW

(5)

where Ep is the particle bed effective Young’smodulus, W is the bed half-width, and d(x) is theamount of clad deflection. If DyBd(x), thismeans that no contact is established at the struc-tural/bed interface and accordingly the aforemen-tioned stress is set to 0. By assuming that thedeflection of the structural clad is governed by theEuler-Bernau equations, which states:

EsId4d(x)

dx4 =sD (6)

where I is given as

I=h3D12

(7)

and Es is the Young’s modulus of the containingstructural wall, and D and h are the bed heightand containment clad thickness. Substituting Eqs.(4) and (5) into Eq. (6) we have

d4d(x)dx4 =

EpDEsI

apDTW−EpDEsIW

d(x) if

d(x)5apDTW (8)

and

d4d(x)dx4 =0 if d(x)\apDTW (9)

with the boundary conditions of

d�

−l2�

=d� l

2�

=0 (10)

d(x0−)=d(x0

+)=apDTW (11)

and

d %�

−l2�

=d %� l

2�

=0 (12)

d %(x0−)=d(x0

+) (13)

here x0 is the location where the amount of claddeflection is equal to that of the bed thermalexpansion and l is the bed or clad length.

In the case where the particle bed behaves likethe fluid, the stress exerted on the structural claddue to the particle bed thermal expansion will be

equal to a constant. Hence, the Euler–Bernoulli’sequation becomes

EsId4d f(x)

dx4 =s fD (14)

with the boundary conditions of

d f�−12�

=d f�12�

=0

d f%�−12�

=d f%�12�

=0. (15)

The stress can further be related to the strainthrough Hooke’s law

DVV

=2�

apDT−s f

Ep

�(16)

where DV=DAD for a 2-D deflection and iscalculated as

DA=& 1/2

−1/2

d f(x) dx (17)

4. Results

The calculated clad deflection and stress exertedon the cladding wall based on different models ofbed behavior are plotted in Figs. 4 and 5 for theceramic packed bed, respectively, as a function ofnormalized clad position. In spite of the fact thatthe experimental data of the packed bed thermalexpansion obtained presently at UCLA reveals itsdependence on the state of the local stress, thedata is not yet comprehensive. Consequently,these calculations are performed using the thermalexpansion coefficient data for solid material. Thecalculations show that the stress caused by thedifference in the thermal expansions of the ce-ramic pebbles and the containment clad could beas high as 4.2 MPa at the corners and drop tozero where the clad deflects more than the bedexpands as calculated by the solid model. Accord-ing to the mechanical test results for Li2ZrO3

particles (as fabricated), they begin to break at thestress level of about 2.6 Mpa [8], this implies thata fraction of particles would be broken duringoperation according to the solid model calcula-tions. This consequence is also affected by the bed


Fig. 4. Calculated ceramic breeder bed clad stress vs normal-ized clad position using different models.

Fig. 5. Calculated ceramic breeder bed clad deflection vsnormalized clad position using different models.

s=P(1−f)n

4pR2 (18)

where f is the porosity of the packed bed (=0.375), n the average number of contacts per

length, the shorter the bed the larger will be theportion of the containment clad that would exertthe stress. The clad deflects more if the bed’sbehavior approaches that of a fluid. The largestdeflection of 175 mm takes place at the center ofthe clad and is independent of the clad length.The stress on the clad is significantly low based onthe fluid model and is about 0.1 MPa for a cladlength of 20 cm. The stress is even lower for alonger bed.

Because the effective modulus of the aluminumpacked bed is higher than that of the ceramicpebble bed, both the stress and deformation of thewall resulting from the aluminum packed bedthermal expansion are much higher as shown inFigs. 6 and 7, respectively. Again, the magnitudeof the stress is much lower with the fluid modelassumption. To resolve which model provides acloser solution to the experimental data, the stresswhich would give an average contact force of Pequal to 0.7697 N, corresponding to an averagebed temperature rise of 400°C as presented in theprevious section, is estimated using [9]:

Fig. 6. Calculated aluminum bed clad stress vs normalized cladposition using different models.


Fig. 7. Calculated aluminum bed clad deflection vs normalizedclad position using different models.

5. Conclusions

The thermomechanical analysis has shownthat the stress resulting from the high thermalexpansion of the blanket particle bed material isnot excessive enough to be of a concern for thestructural material. However, its magnitudecould lead to ceramic particle breakage if thebed behaves like the solid material. Further-more, the calculations show that the clad deflec-tion caused by this stress could lead to apossible bed/clad separation which would add anundesirable thermal resistance to the region andincrease local temperature. The calculationshows that the stress exerted on the clad ismuch smaller if a fluid model is assumed for thebed behavior. The comparison between the esti-mated stress extrapolated from the experimentaldata and the analytical results shows that thethermomechanical performance of the pebblebed approaches that of solid material. However,in order to accurately predict bed performanceand to understand how the induced/external me-chanical force is transmitted through the particlecontact points and alters the contact characteris-tics, it is necessary to better quantify bed behav-ior and particle bed mechanical properties. Moreexperimental work is needed to determine macrothermomechanical effective properties andmicromechanics parameters to better understandthe bed thermomechanical interaction. Theseparameters include the intra-particle friction co-efficient, skin friction and contact stiffness undervarious levels of constraints.

Acknowledgements

This work was performed under US Depart-ment of Energy Contract DE-F603 86ER52123.

References

[1] M.C. Billone, Update of beryllium properties and perfor-mance, Intra-laboratory Memo, ANL, November, 1995.

particle (=8) and R particle radius (=1 mm).The calculated result of 1.24 MPa as shown inFig. 6 is close to the average value from thesolid model and is much higher than the resultcomputed from the fluid model. Although thecomparison appears promising for the solidmodel, one has to realize that the mechanicalcharacteristics at the microscopic level, such ascontact points, is much more complicated. It isknown that the displacement of a contact pointrelative to the center of its sphere includes notonly the normal displacement but tangential dis-placement which consists of two terms describ-ing translational displacement between particlecenters and rotational. For a dense system wherethe rotational displacement can be neglected, theforce exerted by the sphere at a contact wouldthen have parallel components other than nor-mal components considered in the calculations.The analysis is further complicated by the factthat the packed bed effective modulus is ex-pected to increase as the average bed tempera-ture increases. Thus, in order to accuratelypredict bed thermomechanical performance, theeffective macromechanics parameter, such as thebed effective modulus, and the micromechanicsparameter, such as inter-particle friction coeffi-cient, are needed.


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analysis of thermomechanical interactions and properties of ceramic breeder blankets

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