JAERI-Tech 98-059 PRELIMINARY THERMO-MECHANICAL ANALYSIS OF ITER BREEDING BLANKET January 1 9 9 9 Shigeto KIKUCHI, Toshimasa KURODA and Mikio ENOEDA Japan Atomic Energy Research Institute
Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
JAERI-Tech98-059
PRELIMINARY THERMO-MECHANICAL ANALYSIS OFITER BREEDING BLANKET
January 1999
Shigeto KIKUCHI, Toshimasa KURODA
and Mikio ENOEDA
Japan Atomic Energy Research Institute
(¥319-1195for, fc-^L«L</c'$v^0 tits, -
(T319-1195
This report is issued irregularly.Inquiries about availability of the reports should be addressed to Research
Information Division, Department of Intellectual Resources, Japan Atomic EnergyResearch Institute, Tokai-mura, Naka-gun, Ibaraki-ken ¥319—1195, Japan.
©Japan Atomic Energy Research Institute, 1999
JAERI-Tech 98-059
Preliminary Thermo-mechanical Analysis of ITER Breeding Blanket
Shigeto KIKUCHI, Toshimasa KURODA and Mikio ENOEDA
Department of Fusion Engineering Research
Naka Fusion Research Establishment
Japan Atomic Energy Research Institute
Naka-machi, Naka-gun, Ibaraki-ken
(Received December 9 ,1998)
Thermo-mechanical analysis has been conducted on ITER breeding blanket
taking into account thermo-mechanical characteristics peculiar to pebble beds. The
features of the analysis are to adopt an elasto-plastic constitutive model for pebble beds
and to take into account spatially varying thermal conductivity and heat transfer
coefficient, especially in the Be pebble bed, depending on the stress.
ABAQUS code and COUPLED TEMPERATURE-DISPLACEMENT procedure
of the code are selected so that thermal conductivity is automatically calculated in each
calculation point depending on the stress. The modified DRUCKER-PRAGER/Cap
plasticity model for granular materials of the code is selected so as to deal with such
mechanical features of pebble bed as shear failure flow and hydrostatic plastic
compression, and capability of the model is studied. The thermal property-stress
correlation used in the analysis is obtained based on the experimental results at FZK and
the results of additional thermo-mechanical analysis performed here. The thermo-
mechanical analysis of an ITER breeding blanket module has been performed for four
conditions : case A ; nominal case with spatial distribution of thermal conductivity and
heat transfer coefficient in Be pebble bed depending on the stress, case B ; constant
thermal conductivity, case C ; thermal conductivity = -20% of nominal case, and case D ;
thermal conductivity = +20% of nominal case. In the nominal case the temperature of
breending material (Li2ZrOs) ranges from 317 °C to 5 5 4 1 and the maximum
temperature of Be pebble bed is 4461 . It is concluded that the temperature distribution
is within the current design limits.
Though the analyses performed here are preliminary, the results exhibit well
the qualitative features of the pebble bed mechanical behaviors observed in experiments.
JAERI-Tech 98-059
For more detail quantitative estimates of the blanket performance, further investigation
on mechanical properties of pebble beds by experiment, including pebble-wall friction and
behaviors of pebbles subjected to tensile stresses and the improvement of the analysis
model and the calculation code are required.
Keywords : ITER, Breeding Blanket, Pebble Bed, Drucker-Prager, Thermal Analysis,
Stress Analysis
JAERI-Tech 98-059
B
mm mx-mm
(1998 ^L 12 ^ 9
tc#> I" Coupled Temperature —Displacement ProcedureJ
TDrucker-Prager/Cap plast ic i ty] *y!/l>&W:RlLt2 «t -5
A ^ -
— *<D-20%<Dfvf5^
( L i 2 Z r O 3 ) » 317°C *»?> 554°C fe *) , Be <D
"C,
: T 311-0193 801-1
This is a blank page.
JAERI-Tech 98-059
Contents
1. Introduction 1
2. Mechanical Analysis Method 4
2.1 Selection of Analysis Model 4
2.2 Trial Mechanical Analysis 4
2.2.1 Analysis Condition 4
2.2.2 Result of Analysis 7
3. Thermal Property-stress Correlation 20
3.1 FZK Experiment [5] 20
3.2 Analysis of the Heat Transfer Experiment 21
3.2.1 Analysis Condition 21
3.2.2 Results of Analysis 23
3.3 Thermal Property Correlation 23
4. Thermo-mechanical Analysis Method 40
4.1 Analysis Method 40
4.2 Verification of Thermo-mechanical Analysis 40
5. Analysis of Breeding Blanket 46
5.1 Analysis Condition 46
5.2 Results of Analysis 49
6. Summary 72
Acknowledgment 73
Reference 73
Appendix A 74
Appendix B 75
JAERI-Tech 98-059
l. liCfel: l
2. mmMvtJjm 42.1 fiW#&cDai£ 4
2.2 &H-& 4
2.2.1 A¥#f&# 42.2.2 A ? ^ * 7
3. fRl^tttlS^tOffiii 20
3. 1 FZK(DM7£ffi$L 20
3.2 mMvtnmvMW 213.2.1 MtixZkW 21
3.2.2 tetirt&R 23
3.3 ii4#ttffiilS: 23
4. m-mmM$rJT& 40
4.1 MWJffe 40
4.2 m-mmteVrttotfim 405. igfe-fyytry h(DMVt 46
5.1 MVfJlfe 46
5.2 MVtffi^ 49
6. £ b#> 72
73
73
A 74
B 75
VI
JAERI-Tech 98-059
1. Introduction
ITER (International Thermonuclear Experimental Reactor) is required to generatesome amount of tritium basically comparable to its consumption in the EnhancedPerformance Phase. Tritium production is accomplished by breeding blanket,which is composed of tritium breeding material and neutron multiplier. Thebreeding blanket of ITER [1,2] is designed to use currently Li2Zr03 pebblescontained in the breeder rods as tritium breeding material and Be pebbles filledaround the breeder rods in the basic cell as neutron multiplier as shown in Fig. 1.1.In the breeding blanket design, one of critical issues is caused by tight limitationson the breeder and Be temperatures for tritium recover from the breeder, materialsintegrity and safety aspect in case of accident. Therefore precise thermo-mechanical analysis is required. However, the analysis is very difficult becausepebble bed shows such complex thermal and mechanical features as:
©Pebble bed effective thermal conductivity and near wall thermal conductancespatially vary depending on its compressive stresses.
©Pebble bed shows such characteristics of granular materials as shear failureflows caused by shear stress and plastic consolidations caused by hydrostaticcompression [3].
©Thermal conductivity determining temperature distribution depends oncompressive stress and in turn differential thermal expansion determiningcompressive stress depends on temperature distribution.
So far very simple analysis model has been studied such that pebble bed wasmodeled as a continuum with only elastic property. Therefore pebble bed effectiveelastic constant and Poisson's ratio were mainly measured concerning mechanicalproperty of pebble bed, e.g. at UCLA [4]. As for the thermal property, effectivethermal conductivity and near wall conductance of Be binary pebble bed wasobtained in FZK as a function of AL/L, which is a measure for representingcompressive strain of pebble bed [5]. Based on this correlation, an effectivethermal conductivity averaged over a specified region in the breeding blanket canbe estimated.
The objective of present report is to investigate thermo-mechanical analysismethods and models so as to take into account those thermal and mechanicalcharacteristics of pebble beds, and to evaluate preliminarily the performance of theITER breeding blanket.
In Chapter 2 the mechanical analysis method used here and a trial analysis aredescribed. In Chapter 3 the correlation between thermal property and
-i
JAERI-Tech 98-059
compressive stress of Be binary pebble bed is evaluated with the experimentalresults at FZK. The thermo-mechanical analysis procedure of breeding blanket(pebble bed) used here and a trial analysis on the experiment by this procedure aredescribed in Chapter 4. Analysis of ITER breeding blanket is given in Chapter 5.
- 2 -
Shield Plate
Breeder Rods
Cooling Plate
First Wall
Detail of a basic cell (Be pebble bed removed for clarity)
tn
00
oen
Fig. 1.1 ITER Breeding Blanket [1]
JAERI-Tech 98-059
2. Mechanical Analysis Method2.1 Selection of Analysis Model
General purpose thermo-mechanical analysis codes such as NASTRAN andABAQUS are able to analyze pebble beds, which are generally modeled ascontinuum of plastic property in the analysis. The modified DRUCKER-PRAGER/Cap plasticity model (cap model) of ABAQUS code is selected herebecause this model can treat hydrostatic plastic compression most properly asmentioned below. The constitutive equation of the cap model can handle the twoyield surfaces expressing the features of pebble beds as shown in Fig. 2.1: 1) Shearfailure surface providing shearing flow, 2) "Cap" bounding the yield surface inhydrostatic compression, thus providing an plastic hardening mechanism torepresent plastic compaction. In the region bounded by the two yield surfaces,pebble beds show elastic behavior. If the stress condition changed to reach one ofthe surfaces, shear failure or plastic compaction occurs according to the surface.The cap position is generally enlarged by cap hardening effect when plasticcompression occurs. Fig. 2.2 shows flow potential of this model, defining its plasticflow. Associated flow in the cap region and non-associated flow in the shear failureregion are used in the model. Detailed explanation is given in the theory manualof ABAQUS code [6].
2.2 Trial Mechanical Analysis
Trial mechanical analysis using the cap model has been performed for the uniaxialcompressive experiments conducted by UCLA [4]. In the experiments, anapparatus shown in Fig. 2.3 was used, axial compressive force was loaded to singlesize pebble beds of Al and Li2Zr03, and the correlation between the axialcompressive stress and the axial compressive strain was obtained as shown in Figs.2.4 and 2.5. In Fig. 2.4 the axial compressive strain remaining after the 1st
unloading process represents hydrostatic plastic compaction which is not observedin case of metal. The hysterisis behavior is observed in the stress - strain plane asshown in Figs. 2.4 and 2.5.
2.2.1 Analysis Condition
1) Analysis caseThe uniaxial compressive experiment with Al single size pebble bed conducted byUCLA
2) Analysis model
- 4 -
JAERI-Tech 98-059
2D cylindrical analysis model shown in Fig. 2.6. (Only the pebble bed is modeled,
thus the friction between the pebbles and the container wall is assumed perfectly
smooth.)
3) Analysis step
S tep l : 1st loading ; u=0.635mm (£ a=0.01)
Step2 : 1st unloading
Step3 : 2nd loading ; u=1.27mm (£ a=0.02)
In the loading steps, axial displacements (u) are loaded so tha t the expected axial
s t ra ins ( £ J are obtained. In the unloading step, the axial displacement loads was
fully removed.
4) Analysis code / option
ABAQUS5.7 / modified DRUCKER-PRAGER/Cap plasticity model
5) Mechanical da ta
Mechanical da ta used for the cap model analysis are listed in Table 2.1. The
elastic constant and cap hardening data are approximately es t imated with the
UCLA experimental da ta except for shear failure da ta as follows.
a) Elastic constant and cap hardening data
Young's modulus and cap hardening data are obtained by the assumption that , in
Fig. 2.4, the 1st loading step represents cap hardening process and the 1st unloading
step represents elastic process as redrawn in Fig. 2.7
Young's modulus
Young's modulus is calculated as described below:
E = 7 ,{aa, - 2var) = (l - 2vk0){ °aX > = 2AGPa (2.1)
E : Young's modulus
£ al, £ a2 : axial s t ra in (0.0377, 0.0296 ; fixed with Fig. 2.7)
<7al : axial s tress (23.5MPa ; fixed with Fig. 2.7)
v : Poisson's ratio ( 0.25 ; by UCLA [4])
k0 : =OjaA (0 .339 ; by UCLA [4])
Poisson's ratio
The Poisson's ratio evaluated by UCLA [4] is used.
- 5 -
JAERI-Tech 98-059
V =0.25
b) Cap hardening dataCap position is determined by a set of hydrostatic pressure and plastic volumestrain in the process of a hydrostatic plastic compression. For the cap position ofpoint A in Fig. 2.7,
p \ ( « r ) ( 0 ) a l (2.2)
£vOlP1 = £a2 = 0.0296
(Volume strain (£ vol) equals to axial strain (£ a) in a uni-axial case . The plasticstrain of point A is assumed to be the remaining strain after unloading (£ a2 at pointB).) The minimum cap position also has to be given, which is defined as thepressure at which hydrostatic plastic compression begins. It seems very low andcan not be clearly seen from Fig. 2.4. However since very low cap hardeningpressure makes convergence of the analysis deadly worse, the minimum capposition of 1 MPa (at £ vol
pl = 0) is assumed here. The inclination of Caphardening line defined as cap hardening pressure divided by plastic volume strainis 440 MPa (=13.2/0.0296) and is nearly 1/5 of Young's modulus (2.4 GPa).
c) Shear failure dataAmong the data related to shear failure summarized in Table 2.1, friction angle andcohesion are especially important.
Friction angleIt is assumed that the friction angle of Mohr-Coulomb model is 20° in the case of Beand Li2Zr03 binary pebble beds. Then, the friction angle of DRUCKER-PRAGER/Cap model is determined according to the analysis model or type ofelements as follows [6]:
- 2-D cylindrical model:
0 : friction angle of Mohr-Coulomb ( 20° : assumed at present)/? : friction angle of DRUCKER-PRAGER/Cap model
= 37.6° ( calculated with above equation)- 2-D X-Y model with plane strain condition:
/?=30.6° (for 0=20° [6])
CohesionThe cohesion is temporarily set to be 1/2 of the minimum cap pressure for Be pebble
- 6 -
JAERI-Tech 98-059
bed because of the convergence in computation.
2.2.2 Result of analysis
Analysis results for the uni-axial compressive experiment are shown in Figs. 2.8 -2.12. The mechanical behavior of the pebble bed is divided into the following sixprocesses as shown in these figures.
(1) Elastic compression (1st loading)(2) Hydrostatic plastic compression (1st loading)(3) Elastic expansion (1st unloading)(4) Shear failure (1st unloading)(5) Elastic compression (2nd loading)(6) Hydrostatic plastic compression (2nd loading)
The features of each process are described below.(1) Elastic compression (1st loading)
The axial compressive stress increases according to the Young's modulus (Fig.2.8 and Fig..2.10) to reach the minimum cap surface (Fig. 2.9).
(2) Hydrostatic plastic compression (1st loading)The inclination of hydrostatic plastic compressive process is lower than that ofthe elastic compressive process (Fig. 2.8). The axial plastic strain as well asthe radial plastic strain is caused by the hydrostatic compression (Fig. 2.11).
(3) Elastic expansion (1st unloading)The axial compressive stress decreases according to the Young's modulus (Fig.2.8) and becomes smaller than the radial stress in the unloading process (Fig.2.12). The latter behavior is observed in the UCLA experiment [4]. Theshear stress reaches the shear failure surface (Fig. 2.9). The dotted line inFig. 2.9 is drawn in order to show a presumed pass to supplement the lack ofanalysis points.
(4) Shear failure flow (1st unloading)The shear failure flow occurs (Fig. 2.9). The shear failure flow causes thehystrerisis behavior in the strain-stress plane as shown in Fig. 2.8. Thehysiterisis behavior is seen in the experimental result at UCLA (Fig. 2.4 and2.5).
(5) Elastic compression (2nd loading)
- 7 -
JAERI-Tech 98-059
The elastic range is enlarged by the cap hardening effect that is caused by the1st loading process.
(6) Hydrostatic plastic compression (2nd loading)The same behavior as the 1st loading is observed.
Consequently the results using the cap model qualitatively represent well thepebble bed mechanical behavior observed in the experiment, i.e. hydrostatic plasticcompression and hysteresis stress-strain curve due to shear failure. It can beconcluded from this trial analysis that the cap model is one of the promisingmethods to be used for the analysis of the breeding blanket. It should be notedthat since the mechanical behavior of the pebble bed is ruled by the inclination ofcap hardening line after the minimum cap position is reached, inclination of caphardening line and minimum cap position are very important on understanding andanalyzing the pebble bed behavior as well as effective Young's modulus andPoisson's ratio.
- 8 -
JAERI-Tech 98-059
Table 2.1 Mechanical data for analysis of uni-axial compressive experiment
a) Elastic constantYoung's modulus, EPoisson's ratio, V
2.4 GPa0.25
b) Shear failure datacohesion, dfriction angle, J3parameter for cap center shift, Rinitial plastic volume strain, £ voi
P1(0)parameter for transition surface, ayield stress ratio (tension/compression), K
0.5 MPa37.6°0.5*0 **0.*1.*
c) Cap hardening dataNo.12
p(MPa)1.
13.2
e ,P1
*•* vol0.0.0296
Commentminimum cap positionPoint A in Fig. 2.7
*: Typical values are temporarily assumed based on ABAQUS/Standard user's
manual [6].
**: No initial plastic volume strain is assumed.
- 9 -
JAERI-Tech 98-059
t : measure for maximum shear force= MISES stress
p : pressure = -1/3 ( <7n+ Cf22+ G 33)
Cap,Fe
Fig. 2.1 Modified Drucker-Prager/Cap model: yield surfaces in the p-t plane [6]
Similarellipses
Fig. 2.2 Modified Drucker-Prager/Cap model: flow potential in the p-t plane [6]
- 10 -
JAERI-Tech 98-059
INSTRONHydraulics Press Facility
ackedBed
Fig. 2.3 Uniaxial compression test apparatus (UCLA [4])
- 11 -
JAERI-Tech 98-059
5 10Axial Compressive Strain (%
IS
Fig. 2.4 Stress vs. Strain During Cyclic Loading and Unloading Tests (UCLA [4])(Aluminum Packed Bed)
0.2 0.4 O.S B-S 1-0Axial Compressive Strain (%)
Fig. 2.5 Stress vs. Strain During Cyclic Loading and Unloading Tests (UCLA [4])(Li2ZrO3 Packed Bed)
- 12 -
(2D cylindrical)
Forced displacement (u)
IIIPacked Bed
H
R
e
H: 63.5 mm*
R: 12.7 mm*
Boundary condition: no friction
Analvsis
Step
Step
Step
1: 1st
2: 1st
3:2nd
Step
loading u=0.635mm( £
unloading
loading u=1.27mm( e a
a=().01)
=0.02)
CD
CDO
00
Oen
(* assumed based on the figure in ref.[7])
Fig. 2.6 Analysis model of uniaxial compression test
JAERI-Tech 98-059
COQ_
WV)
ao
O
Is"x<
30
25
20
15
10
0
(ea1=0.0377, cra1=23.5MPa)(ea2=0.0296, aa2=0MPa)
0
Hydrostatic plasticcompression
3(ea2,aa2
A(e a1,cra1)
Elasticprocess
0.05
Fig.
0.01 0.02 0.03 0.04
Axial Compressive Strain
2.7 Relation between the axial compressive strain andthe axial compressive stress for Al single size bed
(1st loading and 1st unloading in Fig. 2.4)
- 14 -
O.E+00
-1.E+06
-2.E+06
^ -3.E+06
^ -4.E+06(0w£ -5.E+06
C/D"To -6.E+06"x
** -7.E+06
-8.E+06
-9.E+06-1.E+07
-0.025
—•— 1 st Loading & Unloading
—•—2nd Loading
k
ml'(2)
-0.020 -0.015 -0.010
Axial Strain-0.005 0.000
>enI
00
oenO
Fig. 2.8 Analyzed results of the uni-axial compressive experiment(Axial stress vs. axial strain)
3.0E+06
1 st Loading & Unloading
Shear Failure Surface
Cap Surface(Minimum)
Cap Surface(Hardened)
0.0E+00-1.0E+06 0.0E+00 1.0E+06 2.0E+06 3.0E+06
Pressure (Pa)4.0E+06 5.0E+06
TO50
CDO
00I
oento
Fi. 2.9 Analyzed results of the uni-axial compressive experiment(Von Mises stress vs. hydrostatic pressure)
cto
C/)o
"+JCOCO
LLJ _
Radial Elastic Strain
Axial Elastic Strain
mSO
I
CDO
oo©
-1.2E-02 -1.0E-02 -8.0E-03 -6.0E-03 -4.0E-03 -2.0E-03 0.0E+00
Axial Strain (elastic+plastic)
Fig. 2.10 Analyzed results of the uni-axial compressive experiment(Elastic strain vs. total axial strain)
00
0.001
0.000
-0.001
-0.002
c'jo -0.003•»-»
COo -0.004
-2 -0.005Q_
-0.006
-0.007
-0.008
-0.009
—•— Radial Plastic Strain
—•— Axial Plastic Strain(2)
-0.012 -0.010 -0.008 -0.006 -0.004
Axial Strain (elastic+plastic)
-0.002 0.000
o
tooo
Ioen
Fig. 2.11 Analyzed results of the uni-axial compressive experiment(Plastic strain vs. total axial strain)
0.0E+00
-5.0E+05
-1.0E+06
-1.5E+06
CO
-2.0E+06
-2.5E+06
-3.0E+06
(1)
-1 st Loading & Unloading
• Radial stress = Axial stress
Radial stress = Axial stress
-6.0E+06 -5.0E+06 -4.0E+06 -3.0E+06 -2.0E+06 -1.0E+06 0.0E+00
Axial Stress (Pa)
>i—i
i
<DOsr<£>00o
Fig. 2.12 Analyzed results of the uni-axial compressive experiment(Radial stress vs. axial stress)
JAERI-Tech 98-059
3. Thermal Property-Stress Correlation
It was reported that effective thermal conductivity and wall heat transfercoefficients of pebble beds increased by compressive stress/force of the pebble bed [5].Correlation between those thermal properties and compressive stress are derivedfor Be binary pebble bed here through analysis of the stress distribution in thepebble bed of the heat transfer experiment performed by FZK [5]. Then, theobtained correlation on the thermal conductivity is compared to the experimentaldata measured for Al single sized pebble bed by UCLA [7].
3.1 FZK Experiment [5]
Effective thermal conductivity of Be binary pebble bed was measured from thetemperature gradient across a pebble bed contained in an annular cylinder.Typical experimental result of temperature profile is shown in Fig. 3.1. The radialcompressive stress was generated in the annular cylinder by a differential thermalexpansion among the inner and outer tubes and the pebble bed. Then themeasured thermal property was correlated to the compressive strain of pebble beddefined as AL/L by FZK. The relation for Be binary pebble bed is reported asfollows:
Packed pebbles : Be binary pebble (2mm 0(64.5%)+O.2mm 0(16.3%);total packing factor 80.8%)
X[W / mK] = (l 3145+ 1.00652 xlO-4Tm)i 1 + 7.259—[%]) (3.1)
h[W/cm2K] = 6.138xlO"2 • / . e00O35i32Trw (3.2)
with / = 4.023 + 54.63— for — [%] ;> 0.015L L L J
and / = 1 for —[%]< 0.015
Tm or Tw 130-600°CAL/L 0-0.1%
A = effective thermal conductivity of the bed [W/mK]
L = thickness of the bed in the direction of the heat flow (=R2-R1 [cm])Rl = outer radius of the inner heating tube [cm]R2 = inner radius of the outer containing tube [cm]T = Tm = average temperature of the bed [°C]
Tw = temperature of the outer surface of the inner tube
- 20 -
JAERI-Tech 98-059
h = heat transfer coefficient between bed and containing walls at the outersurface of the inner tube [W/cm2 °C]
a Be = thermal expansion coefficient of beryllium at Tm [K1]aSt = thermal expansion coefficient of the containing walls of stainless steel [K1]A L/L = percent difference between the thermal expansion of the bed and the
confinement walls referred to the thickness of the bed
• IOOXL (rm -ro)+ a » f - " » ' * ' T 0 - a » ' * ^ - a ^ T * 1 --(3.3)
[ K K KH J
3.2 Analysis of the Heat Transfer Experiment
Stress distribution of the heat transfer experiment system of FZK is analyzed sothat the measured thermal property is correlated to its compressive stress.
3.2.1 Analysis Condition
1) Analysis caseThe heat transfer experiment with water cooling on the outside surface forsubjecting compressive stresses to the pebble bed. Two cases were analyzed interms of minimum cap position (see 5), b)).
2) Analysis model• 2D cylindrical model shown in Fig. 3.2.
3) Analysis code / model• ABAQUS5.7• modified DRUCKER-PRAGER/Cap plasticity model
4) Thermal propertiesThe effective thermal conductivity (A) and heat transfer coefficient (h) of Be binarypebble bed are referred from the data experimentally evaluated by FZK [5]. Theheat transfer coefficients are taken into account by incorporating a modifiedthermal conductivity for near wall element as described below.
1 4 (3.4)AM Axxh A
AM : modified A for element to take into account
heat transfer coefficient
- 21 -
JAERI-Tech 98-059
A. ,h : effective thermal conductivity and heat transfer
coefficient of Be binary pebble bedAx: width of near wall element (Fig. 3.2)
Thermal properties used here are as follows:- A for pebble bed bulk region : 12.07 W/mK- AM for near wall element:
a) at inner tube wall: 3.38 W/mK( element width=0.5mm; A=12.07W/mK; h=0.94 W/cm2K)
b) at outer tube wall: 0.32 W/mK( element width:0.5mm; A=12.07W/mK; h=0.066 W/cm2KSince the h for the outer tube wall has not been measured by FZK,the h is evaluated with the correlation (Eq. (3.2)) using thecondition T=20°C and AL/L=0.)
- Thermal expansion coefficient of the pebble bed : the same values as basesolid materials
5) Mechanical propertiesMechanical properties for the cap model analysis are summarized in Table 3.1.
a) Young's modulusSince no experimental data is available for binary pebble beds at present, Young'smodulus for Be single size pebble bed is temporarily assumed. The Young'smodulus is estimated as 1.45 GPa by the analytic model of K.Walton [8] for uniaxialcompression of perfectly smooth spheres as shown in Table 3.2 (Appendix A).
b) Cap hardening dataCap hardening data are composed of a minimum cap position and at least one caphardening pressure corresponding to a plastic volume strain. The minimum capposition is parametrically assumed (1 MPa and 0.1 MPa). The other cap hardeningpressure is determined as shown in Table 3.1 based on the inclination of caphardening line calculated as 1/5 of Young's modulus by the assumption that theinclination for Be binary pebble bed is similar to that for Al single size pebble bed.Namely,
Z U50MPa (3.5)
£ volpl is assumed to be 0.1 which is never reached in the analysis.
c) Shear failure dataThe shear failure data are the same as those used in section 2.2.1 except for the
- 22 -
JAERI-Tech 98-059
cohesion data which are assumed to be half of minimum cap position (0.5 MPa, 0.05.MPa) as described in section 2.2.1.
3.2.2 Results of Analysis
Analyzed temperature distribution with uniform effective thermal conductivityindependent on the stress is shown in Fig. 3.3. Large temperature gap at the outerwall does not agree with the measured temperature profile shown in Fig. 3.1.Later in Chapter 4, it will be shown that this discrepancy is removed with the modeltaking into account thermal conductivity and heat transfer coefficient depending oncompressive stress.
Analyzed stress distributions in the pebble bed region are shown in Fig. 3.4 and Fig.3.5 for minimum cap positions of 1 MPa and 0.1 MPa, respectively. In both figures,the maximum radial compressive stress in the inner region is about 2.5 timeshigher than the minimum one in the outer region. Therefore it is found that thedifference of the minimum cap position causes only small change in the shape ofradial stress profile. On the other hand, rather large difference in absolute valuesof the radial stresses is given by the difference. For example, the maximumcompressive stress near the inner tube in the former case (Fig.3.4) is nearly 2.5 MPawhile it is about 1.0 MPa in the latter case (Fig.3.5).
3.3 Thermal Property Correlation
FZK has experimentally evaluated the relation between AL/L and the thermalproperties as described above. The AL/L is related to compressive stress herewith the analyzed stress distribution of the experiment system.
1) Effective thermal conductivityIt could be assumed that the A L/L is in proportion to the radial averagedcompressive stress in the experimental system because the A L/L is a sort ofcompressive strain defined as the difference between contraction of the containerdue to differential thermal expansion of inner and outer tubes and thermalexpansion of pebble bed. The local effective thermal conductivity (A(r)) of theexperimental system could be also expressed as a 1st order function of a radialcompressive stress ( <7 r(r)) (Eq. (3.6)) because the average effective thermalconductivity (Aavg) is expressed as a 1st order function of AL/L (Eq. (3.7) derivedfromEq. (3.1)).
- 23 -
JAERI-Tech 98-059
A ( r ) = A 0 + A x a r ( r ) (3.6)
(3.7)
A 0, A o': thermal conductivity without stress
A, A : constants
The average thermal conductivity of the experiment system is obtained byintegration of Eq. (3.6) with the cr(r) and weighting function w(r) of the system.
•avg fw(r)dr(3.8)
By comparing Eq. (3.8) and Eq. (3.7), the next correlations are obtained because Eq.(3.8) and Eq. (3.7) must coincide in any experimental condition.
(3.9)
A
Ax
a(3.10)
r-avg
By substituting Eq. (3.9) and Eq. (3.10) into Eq. (3.6),
A(r)=A0avg
axa(r)= Ao +Ax
r-avg a r-avg
(3.11)
Comparison of Eq. (3.11) and Eq. (3.7) shows that local effective thermalconductivity can be obtained by Eq. (3.1) with local AL/L defined by the nextequation.
M
(3.12)
Selection of the weighting function and range of integration in Eq. (3.8) fullydepends on how the effective thermal conductivity is determined with the measured
- 24 -
JAERI-Tech 98-059
temperature distributions. In this preliminary analysis, most general weightingfunction (w(r)=r) is selected for simple volumetric average.
far(r)rdr(*,—) = J
r j (3-13)
Jrdr
Calculated results are:OCase for minimum cap position = IMPa
AL, > 0.0927 / O 1 i l .— ( C T ) = xo (3.14)L V ' 1.37
(<7 ^ 2.45 MPa (maximum stress in the analysis))
OCase for minimum cap position = O.lMPaAL, x 0.0927L V ; 0.54
((7 S 0.969 MPa (maximum stress in the analysis))
Using the above equations and Eq. (3.1), the correlation between the effectivethermal conductivity and compressive stress is obtained.
OCase for minimum cap position = IMPa
X\WImK] - (7.3145 + 1.00652xl0-4rm)(l + 0.491xa[M/Jfl]) (3.16)
(a ^ 2.45 MPa)
OCase for minimum cap position = O.lMPa
X[W ImK] = (7.3145 +1.00652 xlO~4Jm)(l +1.25 xa[MPa]) (3.17)
(O ^ 0.969 MPa)
With the above equations effective thermal conductivity is calculated as shown inFig. 3.6. Though these relation is strongly affected by the data used for the stressanalysis (a minimum cap position), the calculated thermal conductivity has smalldifference through employment of the consistent analytic data, as described in thelater Chapter 4 (Fig. 4.3). For with a bigger minimum cap position, rate ofincrease in thermal conductivity by compressive stress get lower as shown in Fig.3.6. With the same minimum cap position, compressive stress is calculated to behigher. Then the lower increase rate in thermal conductivity and higher compressstress is expected to cancel out in calculation of thermal conductivity.The same correlation is measured for Al single pebble bed at UCLA [7]. Forreference the above correlation for Be binary pebble bed is compared with the
- 25 -
JAERI-Tech 98-059
experimental results as shown in Fig. 3.7 and Fig. 3.8 (normalized at zero MPa).Fig. 3.8 shows that the correlation obtained above for binary pebble bed does notdiffer so much from the measured correlation for single size pebble bed. Howeverquantitative investigation can not be conducted from these figures because of thedifference in packed mode (binary, single size) and pebble material (Be, Al).Some supplements are given here as for the weighting function. The integration ofEq. (3.8) should intuitively be done using d(ln(r)) because the measuredtemperature distribution may be fitted with the logarithmic axis as shown in Fig.3.1. Then the weighting function is selected as " 1/r" on the contrary to the previous"r" as described below.
( )dr r
With the weighting function (1/r), cravg may be calculated to be larger than previousvalue because the stress in inner region is lager than that in outer region. Furtherstudy is needed for the weighting function.
2) Heat transfer coefficientThe relation between compressive stress andAL/L for heat transfer coefficient can
be obtained as follows:
(AL/L)avg: AL/L defined in Eq. (3.2)(<7r)in : analyzed inner wall compressive stress
(Heat transfer coefficient is measured for only inner wall.)
From the analysis, next correlation is obtained.OCase for minimum cap position = IMPa
( o . ^ 2.45 MPa) (3.20)L 2.45
OCase for minimum cap position = O.lMPa
AL 0.0927L 0.969
xo (a S 0.969MPa) (3.21)
Using the above equations, value of fin Eq. (3.2) is calculated as described belowand the correlation between the heat transfer coefficient and compressive stressnear wall is obtained with this value.
OCase for minimum cap position = IMPa
- 26 -
JAERI-Tech 98-059
with / - 4.023 + 2.067 x o for a 2 0A0[MPa]
and / = 1 for o < 0AC{MPa]
(a ^ 2.45 MPa)
OCase for minimum cap position = IMPawith / = 4.023 + 5.226x o for ex ;> 0.16[MPa]and / = 1 for a < 0.16[MPa]
( a ^ 0.969 MPa)
With the above correlation, the modified thermal conductivity for heat transfer (Eq.(3.4)) is calculated as shown in Fig. 3.9. Though there is a radical change in thethermal conductivity at cr = ~ 0.4MPa, due to the definition by Eq. (3.2) of nocompressive effect for AL/L less than 0.015%, a linear change shown by a dottedline in the figure is assumed for convergence in computation. Thermalconductivity at near wall element is shown as a function of temperature in Fig. 3.10and Fig. 3.11 for two minimum cap positions, respectively.
- 27 -
JAERI-Tech 98-059
Table 3.1 Mechanical data for analysis of heat transfer experiment
d) Elastic constantYoung's modulus, EPoisson's ratio, v
1.45* GPa0.25
e) Shear failure datacohesion, dfriction angle, /3parameter for cap center shift, Rinitial plastic volume strain, £ voi
PI(0)parameter for transition surface, ayield stress ratio (tension/compression), K
0.5 / 0.05 MPa37.6°0.5**Q * * *
0.**1 * *
f) Cap hardening dataNo.12
p(MPa)1./0.1
29.
fc vol
0.0.1
Commentminimum cap position
*: Estimated by an analytic model of K. Walton [8] (Appendix A)
*: Typical values are temporary assumed based on ABAQUS/Standard user's
manual [6].
*: No initial plastic volume strain is assumed.
Table 3. 2 Young's modulus of Be pebble bed
Itemporosity, $temperature, TYoung's modulus for Be bulk, EPoisson's ratio, vcontacts per sphere, 7?axial strain, £Young's modulus for Be pebble bed, Einclination of cap hardening line E(cap)
FZK experiment0.192 [5]50 °C
295 GPa0.076.4 [9]0.001*1.45 GPa**0.29 GPa***
Breeding blanket0.192 [5]300 °C281GPa0.076.4 [9]0.001*1.38 GPa**0.28 GPa***
* : Uniaxial strain is assumed
**: Estimated with the analytic model developed by K. Walton [8] (Appendix A).
Temporarily assumed as single size pebble bed due to the lack of data for binary
pebble bed.
*** : Assumed as 1/5 of pebble bed Young's modulus
- 28 -
JAERI-Tech 98-059
Qo : 9.87 [W/cm2]T : 50.8 [°C]P : 1.0 [bar]A : 12.07 [w/mK]h : 0.94 r\V/cm3l
( heat flux at the inner wall)(average temperature of the(helium pressure )(thermal conductivity of the( heat transfer coefficient)
bed)
bed)
sg
I
0.1 <U (UO91.0
R/R2
Fig. 3.1 Radial temperature distribution in the bed [5]
- 29 -
JAERI-Tech 98-059
A) Experiment system (Cross-section of test equipment)
(The thickness of inner and outer tubesis assumed to be 2 mm each.)
AL/L=0.0927(evaluated by Eq. 3.3)
aBe = 1.4 X10"5
ass- 1.6X105
B) Analysis modelO2D cylindrical
Inner tube (ss)
R2:51mmTw2:19.2°C
Tm=50.8°C Rl: 8mmTwl:176.4°C
i Be Debbie bed
T775'^?I7^l
2mm
/
Outer tube (sBoundary condition for thermal analysis :
inner surface of inner tube (SS) : 180°Couter surface of outer tube (SS) : 18°C
(This condition is estimated with experimental temperaturedistribution in Fig. 3.1)
Fig. 3.2 Analysis model of FZK experiment system
- 30 -
IQ.
O
200
180
160
140
120
100
80
60
40
20
00.001
Thremal conductiovityA Be = 10.07 W/mKA Min = 3.3 W/mKA Mout = 0.33 W/mK
Boundary ConditionTin = 185 °CTout= 15°C
Inner tube
!Outer tube
N^ j
^^ . !
- 1i
!
1
I
(V
osr<£>ooo
0.010
Radial distance (m)
0.100
Fig. 3.3 Analyzed temperature distribution of the heat transfer experiment( Constant thermal conductivity)
to
0.0E+00
-5.0E+05
-1.0E+06Q_
Jg -1.5E+06
C/D-2.0E+06
-2.5E+06
-3.0E+060.00
>
i
O3-
O00
oento
0.01 0.02 0.03
Radius (m)
0.04 0.05 0.06
Fig. 3.4 Analyzed stress distribution in the heat transfer experiment(Minimum cap position = 1 MPa)
O.OE+00
-2.0E+05
_ -4.0E+05(0
Q.
S -6.0E+05
W -8.0E+05
-1.0E+06
-1.2E+06
0.00
O
00
oenCO
0.01 0.02 0.03
Radius (m)
0.04 0.05 0.06
Fig.3.5 Analyzed stress distribution in the heat transfer experiment(Minimum cap position = 0.1 MPa)
18
LU
MC: minimum cap positionT : Temperature of pebble bed
0.0
, T=O°CMC=1MPa, T=600°CMC=0.1MPa, T=O°CMC=0.1MPa, T=600°C
i
CDO
to00
Iov\to
1.0 2.0
Compressive Stress (MPa)
3.0
Fig. 3.6 Evaluated relation between effective thermal condcutivity andcompressive stress (Be binary pebble bed)
toen
o"Ocoo
E
O
LU
18
16
14
12
10
8
6
0
MC : Minimum Cap positionT : Temperature of pebble bed
//
• I
/ u^00^ Calculation : Be binary pebble, MC=1 MPa,/ ^ T=20°C
£^ — Calculation : Be bainary pebble,MC=0.1MPa, T=20°C i
i
- • - Experiment: single size A.l pebble [~
T ^ I
I
o3 "
<£>00
Ocna
0.0 0.5 1.0 1.5 2.0 2.5
Compressive Stress (MPa)
3.0
Fig. 3.7 Comparison of relations between effective thermal condcutivity andcompressive stress
to(Si
<D
<D
0.5
MC : Minimum Cap positionT : Temperature of pebble bed
•Calculation : Be binary pebble, MC=1MPa, T=20°C
Calculation : Be binary pebble, MC=0.1MPa, T=20°C
-Experiment: single size Al pebble
§ o.ow 0.0
mi—i
i- 9CDO3 "
«300
Ocn<£>
0.5 1.0 1.5 2.0
Stress (MPa)
2.5 3.0
Fig. 3.8 Comparison of relations between normalized effective thermal condcutivityand compressive stress
8.0
^ 7>0
^ 6.0
£5.0
H 4.0
§ 3.0(0
E03 2.0
1.0
0.0
—•—400°C
i
•changed line ; ^^**~^
1
r
r
i
. _—_—
modified line in order to avoiddivergence probrems in thethermo-mechanical analysis
1
0.0
m
I
O
ooooC7I
0.5 1.0 1.5 2.0
Compressive stress (MPa)2.5 3.0
Fig. 3.9 Modification in relation between thermal conductivity forheat transfer and compressive stress (Minimum cap position=1 MPa)
to00
>>
12.0
10.0
8.0
0.0
-B-O°C-*-100°C--X~200oC
tooo
o
0.5 1.0 1.5 2.0
Compressive stress (MPa)
2.5 3.0
Fig. 3.10 Relation between modified thermal conductivity forheat transfer and compressive stress
(Minimum cap positional MPa)
Ito
12.0
10.0E
8.0
§ 6.0•o
oo•s 4.0
0>
2.0
0.0
-B-0°C j
!-#-100°C :
-»-600oC
m u
_—-4-—•—-"" ~~
0.0 0.2 0.4 0.6 0.8Compressive stress (MPa)
1.0 1.2
I
CO00
Otnto
Fig. 3.11 Relation between modified thermal conductivity forheat transfer and compressive stress(Minimum cap positioned MPa)
JAERI-Tech 98-059
4.Thermo-mechanical Analysis Method4.1 Analysis Method
The coupled temperature-displacement analysis procedure of ABAQUS code isselected as shown in Fig. 4.1 in order to take into account the dependencies ofthermal conductivity and heat transfer coefficient on compressive stress. Thermalconductivity is automatically calculated based on the iterated stress (<7) at everycalculation point.
4.2 Verification of Thermo-mechanical Analysis
The heat transfer experiment [5] is analyzed with the coupled temperature-displacement analysis procedure using correlation obtained in 3.3 and the samemechanical data as in 3.3 (Table 3.1). Analyzed thermal conductivity distributionand temperature distribution are shown in Fig. 4.2 and Fig. 4.3, respectively. Incomparison with the previous thermal analysis results (Fig. 3.3), the temperaturejump at the outer wall surface is reduced to the same magnitude as theexperimental results in Fig. 3.1. Because of the reduction of the temperature jump,the averaged temperature in the pebble bed becomes lower than the case withoutthe effect of compressive stress on effective thermal conductivity. The temperaturedistribution in the pebble bed region shows slight convex curvature since thethermal conductivity of outer region is lower than that of more compressed innerregion. The convex curvature can be also seen for the measured temperaturedistribution plotted in Fig. 3.1, which may demonstrate the compressive effect onthe thermal conductivity. The analyzed stress distribution in the pebble bed isshown in Fig. 4.4 and Fig. 4.5. The analyzed stress is lower than the case withoutcompressive effect because of its lower temperature.
- 40 -
Correlation based on experiment Breeding blanket analysis
Thermal analysis
1 Stress analysis II
A vs. T, AL/Lh v s . T,AL/L
(1997 FZK correlation)
AL/L vs. bulk averaged aAL/L vs. near waller
r
A vs. %OAM*vs.T,C7
Coupled temperature-displacement analysis(Analysis procedure of ABAQUS code)
I Thermal analysis | | — • T —•j) Stress analysis | |
FV
SUBROUTINE USDFLDF V l = o - l l , FV2=CT22,FV3=CT33
CD33
I
O
o00o
*: modified for heattransfer coefficient
1)2)
aAare
is defined as filedvs. T,FV(=cr)&Ainput
variables (FV).Mvs.T,FV(=cr)
3) FV is set to be O in USDFLD at amater ia l point
4) Thermal conductivity is calculatedwith redefined FV
Fig. 4.1 Flow of thermo-mechanical analysis of breeding blanket
(S3
16
14
« 10oc°o
15
8
>
m o
MC : Minimum cap position
•MC=1MPa•MC=0.1MPa
m3
O
00I
O
to
0.00 0.01 0.02 0.03 0.04
Radial (m)0.05 0.06
Fig. 4.2 Evaluated distribution of effective thermal conductivityin the heat transfer experiment system
MC: Minimum cap position
p
I93Q.Ea)
200
180
160
140
120
100
80
60
40
20
0.001
...— MC=1MP
---MC=0.1M
aP a ^
Inner tube-
Outer tube ^
\X
\
>CD
o
oen
0.010
Radial (m)
0.100
Fig. 4.3 Analyzed temperature distribution with coupledtemperature-displacement procedure
O.OE+00
-2.5E+06
0.00
>m70
CDO
00
O
to
0.01 0.02 0.03 0.04 0.05 0.06
Radial distance (m)
Fig. 4.4 Analyzed stress distribution with coupledtemperature-displacement procedure(Minimum cap position = 1Mpa)
O.OE+00
-1 .OE+05
-2.0E+05
_ -3.0E+05" T O
w -4.0E+05</}
£ -5.0E+05
W -6.0E+05
-7.0E+05
-8.0E+05
-9.0E+05
0.00
rnsoI
no
00
oC71
0.01 0.02 0.03 0.04 0.05 0.06
Radial distance (m)
Fig. 4.5 Analyzed stress distribution with coupledtemperature-displacement procedure
(Minimum cap position = 0.1 Mpa)
JAERI-Tech 98-059
5. Analysis of Breeding Blanket
Thermo-Mechanical analysis is conducted on ITER breeding blanket by the methodand data discussed above.
5.1 Analysis Condition
1) Analysis case• Unit cell of ITER #19 (outboard mid-plane) breeding blanket (Fig. 5.1)
Case A: nominal case with effective thermal conductivity of Be pebble bed
dependent on the stressCase B: constant thermal conductivity in the Be pebble bed bulk region
(A = 13.43 W/mK) (referred from [1])Case C: A = -20% of nominal case in the Be pebble bed bulk regionCase D: A = +20% of nominal case in the Be pebble bed bulk region
2) Analysis model• Unit cell of ITER #19 (outboard mid-plane) breeding blanket• 2D X-Y model with plane strain elements as shown in Fig. 5.2. (Generalized
plane strain condition is desirable but unavailable for the CoupledTemperature-Displacement analysis at present.)
3) Analysis code / model• ABAQUS5.7 / modified DRUCKER-PRAGER/Cap plasticity model
(Coupled Temperature-Displacement analysis procedure was for case A, C and D)
4)a)
Thermal dataThermal loading [1]
- Heat flux- Volumetric heating
first wallBe
Li2Zr03
SS316LN
0.5 MW/m2
6.63*EXP(-6.41*X) MW/m3
Table 5.3
9.52*EXP(-5.085*X) MW/m3
(X: distance from first wall)b) Heat transfer coefficient (h), coolant temperature (T) [1]
- First wall channel h=27400 W/m2 K, T=145°C- Cooling plate channel h=13000 W/m2 K, T=170°C- Cooling plate header h=22000 W/m2 K, T=190°C
c) Pebble bed effective thermal conductivity
- 46 -
JAERI-Tech 98-059
Be pebble bedCase A:
Among two thermal conductivity correlations described in 3.2, the correlation forthe minimum cap position of 0.1 MPa (Eq. (3.17)) failed to converge in the analysis.Therefore one for the minimum cap position of IMPa (Eq. (3.16)) is used here. Atsmaller compressive stresses than 0 MPa, the effective thermal conductivity is setas 7.31 W/raK corresponding to a value at 0 MPa. At lager ones than 2.45 MPa,the effective thermal conductivity is set as 16.12 W/raK corresponding to a value at2.45 MPa which is the maximum stress of the analyzed heat transfer experimentalsystem. Between the two compressive stresses, thermal conductivity is estimatedby Eq. (3.16).Case B:
13.43 W/mKCase C:
-20% of case ACase D:
+20% of case A
Effective thermal conductivity of Li2Zr03 pebble bed is estimated with SZB analyticmodel [10] as shown in Fig. 5.3. The effect of compressive stress on the effectivethermal conductivity is not taken into account because it is reported that thecompressive effect has not been significantly observed for Li2Zr03 pebble bed in theexperiment [7].
Calculation condition:Li2Zr03 pebble bed 2 0 (65%)+0.2 <j> (15%)SZB model Contact area = 0.
Accommodation factor=0.4
d) Modified thermal conductivity for near wall element to take heat transfercoefficient at wall into accountBe pebble bedThe modified thermal conductivity described in section 3.3, 2) is used in case A, Cand D. The used thermal conductivity is shown in Fig. 3.10. In case B, heattransfer coefficient is not taken into account.Li2ZrO3 pebble bedThe heat transfer coefficient between Li2Zr03 pebble bed and the tube wall was nottaken into account at present.
- 47 -
JAERI-Tech 98-059
e) Thermal expansion coefficient of pebble bedThe same values as base materials (Be, Li2Zr03) as shown in Table 5.1 are used.
5) Mechanical data
Mechanical data for the cap model analysis are summarized in Table 5.2.
a) Young's modulusBe pebble bedThe Young's modulus for Be single size pebble bed is tentatively applied, which isestimated as 1.38 GPa by the analytic model of K.Walton in the same way asdescribed in the section 3.2.1, 5). This value is slightly less than that used in theanalysis of the heat transfer experimental system because of higher temperature ofthe breeding blanket as shown in Table 3.2.
Li2ZrO3The Young's modulus used here is 0.5 GPa which is experimentally obtained forLi2Zr03 single size pebble bed at axial compressive strain = nearly 0.3% [4].
b) Cap hardening dataBe pebble bedMinimum cap position is assumed to be 1 MPa. Another cap hardening pressure isdetermined as shown in Table 3.2 based on the inclination of cap hardening lineassumed as 1/5 of Young's modulus.
Li2ZrO3
Minimum cap position is assumed to be 1 MPa. Another cap hardening pressure isdetermined with the inclination of cap hardening line calculated using Fig. 5.4 inwhich 1st loading and 1st unloading lines are drawn based on Fig. 2.5.
(5.1)
p : hydrostat ic pressure
£ a l ) £ a2 : axial s t ra in (0.0105, 0.006 ; Fig. 5.4)
<7al : axial s t ress (3.8MPa ; Fig. 5.4)
v : Poisson's rat io ( 0.25 [4])
k0 : =<7jo'a (0.339; temporari ly assumed to be the same
as tha t for Al pebbles because of no available da ta for
Li 2 Zr0 3 a t present)
£VoiP1 = £ a 2 = 0.006
Since the inclination of Cap hardening line is calculated as 350 MPa(=2. lMPa/0.006), the hydrostatic pressure is 350 MPa at £ vol
pl = 1.
- 48 -
JAERI-Tech 98-059
c) Shear failure dataThe shear failure data are the same as those used in section 3.1 (Table 2.1) exceptfor the friction angle. The friction angle used here is 30.6° for 2-D X-Y model withplane strain condition.
5.2 Results of Analysis
1) Case A (nominal case)The analyzed temperature distribution is shown in Fig. 5.5. The maximum andminimum temperatures of breeding material (Li2Zr03) are summarized for eachbreeder tube in Table 5.3. The maximum temperatures for other materials are alsosummarized in Table 5.4. In the design of the breeding blanket, temperaturelimits are tentatively considered to be within 300-350°C to 800°C for Li2Zr03 andunder 500°C for Be [1].
The temperature of the breeding material ranges from 317°C (No. 8 breeder tube) to554°C (No. 6 breeder tube). Since the heat transfer coefficient between thebreeder pebbles and the tube surface are not taken into account in this analysis, thetemperature would become slightly higher with this effect, but would be still withinthe limits. The maximum temperature of Be pebble region is 446°C which is belowthe present limit (500°C).
The stress distributions in the X and Y directions are shown in Figs. 5.6 and 5.7,and the minimum and maximum stresses are summarized in Table 5.5. Tensilestress is found in the entire breeding region because the thermal expansioncoefficient of Li2Zr03 is less than that of tube material (SS). The stress in the Bepebble region spatially varies from -2.5MPa to 0.47MPa as shown in Table 5.5.Strong compressive force is generated in the higher temperature region, socompressive stresses in the region near the first wall and far from the cooling panelare higher than the other region.
Stresses in SS (first wall, tube, rib and back wall) are extremely high because of theplane-strain condition. However, the plane-strain condition is impractically severe,and it is desirable that the code would be improved to apply generalized plane-strain condition.
2) Case B (case for constant thermal conductivity)Analyzed temperature and stress distributions are shown in Figs. 5.8-5.10. Theseare similar to the results of the nominal case (case A) as also seen from Table 5.3-5.5.
- 49 -
JAERI-Tech 98-059
Slight difference from the nominal case, the temperatures of the 8th breeder tubeand the shielding plate of case A are higher than those of case B. case, is found.This feature is appreciated by the compressive effect. In case A the compressivestresses in the Be pebble bed region located between 8th (last) breeder tube andshielding plate are lower than the other region as shown in Figs. 5.11 and 5.12where stress distributions are drawn focusing on Be pebble bed. The lowercompressive stresses cause the thermal conductivity of the region lower and thenthe temperature higher.
Consequently it might be concluded that the spatially constant thermalconductivity estimated by FZK (Eq. 3.1)) is fairly good as a whole for ITER typebreeding blanket. Naturally detailed treatment of tmermo-mechanical propertyincluding spatially varying thermal conductivity depending on the stress is requiredfor accurate analysis, particularly in the case of time dependent thermo-mechanicalanalysis
3) Case C, D (case for conductivity of ±20%)Analyzed temperature and stress distributions are shown for case C and D in Figs.
5.13-5.18. The temperatures of the breeding material (Li2Zr03 pebble bed) for caseC (A Be=-20%) are about 20-30°C higher than those for case A (nominal) as shown inTable 5.3. The maximum temperature of the multiplier (Be pebble bed) for case Care 45°C higher than that for case A. The temperatures for case D (ABe=+20%) areabout 12-16°C lower in the Li2Zr03 pebble bed and 31°C lower in the Be pebble bed.
Current design of ITER breeding blanket is evaluated to permit ± 20% change inthermal conductivity of Be pebble bed.
For more detail quantitative estimates of the breeding blanket, further studies arerequired as for:
• elaborate investigation of thermal and mechanical properties of binary pebblebed, including pebble-wall friction and behaviors of pebbles subjected totensile stresses
• establishment of analysis methods and constitutive equation to describe thesepebbles behavior based on plastic theory
• incorporation of the analysis method and the constitutive equation intoavailable thermo-mechanical analysis code
- 50 -
JAERI-Tech 98-059
Table 5.1 Thermal expansion coefficient(1/K)
T(°C)
2050100150200250300350400450500550600
SS[11]
1.59E-051.61E-051.64E-051.67E-051.70E-051.72E-051.75E-051.77E-051.79E-051.81E-051.83E-051.85E-051.87E-05
Be[11]
1.13E-051.19E-051.29E-051.38E-051.47E-051.55E-051.63E-051.70E-051.77E-051.83E-051.88E-051.94E-051.99E-05
Li2ZrO3[12]
9.92E-069.93E-069.93E-069.94E-069.94E-069.95E-069.95E-069.96E-069.97E-069.97E-069.98E-069.98E-069.99E-06
- 51
JAERI-Tech 98-059
Table 5.2 Mechanical data for analysis of breeding blanket
g) Elastic constant
Young's modulus, EPoisson's ratio, v
Be1.38 GPa0.25
Li2ZrO30.5 GPa0.25
h) Shear failure data
cohesion, dfriction angle, J3parameter for cap center shift, Rinitial plastic volume strain, £ voi
P1(0)parameter for transition surface, ayield stress ratio (tension/compression), K
Be0.5 MPa30.6°0.5*0 **0.*1.*
Li2ZrO30.5 MPa30.6°0.5*0.**0.*1.*
i) Cap hardening data
(Three sets of hydrostatic compression pressure (p) and plastic volume strain £
(volpl) are used, The third set is added because plastic volume strain may exceed
that of the second set (0.1))
No.123
Bep(MPa)
1.27.6
276.
F P1£ vol
0.0.11.
Li2ZrO3p(MPa)
1.35.
350.
F P1£ vol
0.0.11.
Commentminimum cap position
*: Typical values are temporarily assumed based on ABAQUS/Standard user's
manual [6].
**: No initial plastic volume strain is assumed.
- 52 -
Table 5.3 Analyzed temperature of bleeding material (Li2Zr03)
enCO
TubeNo.
12345678
TubeRadius(mm)444446.56.56.6
Power density(W/cm3)45.539.236.834.93220.215.510.9
Case A (Nominal)
Min.362372376363364348337317
Max.513527522500482554506448
CaseBABe=13.43W/mKMin.349372370355355331306277
Max.506527523499479544477404
CaseCABe=-20%Min.380404402386389372362340
Max.537561556532514587538478
CaseDABe=+20%Min.350363360348348333321302
Max.497504498478461532483427
i—ii
O
00
©
to
Table 5.4 Analyzed maximum temperature of Armor(Be), multiplier (Be), structure (SS)
Region
Armor (Be)Multiplier (Be)Structure (SS)
Case A (Nominal)
290446444
Case B: ABe=13.43 W/mK
290455449
Case C:ABe=-20%
290491486
Case D:ABe=+20%
290415532
Table 5.5 Analyzed stresses of breeding blanket
en
I
First wall(SS)
Multiplier(Be pebble)
Breeder(Li2ZrO3 pebble)
Structure(Tube, rib.back wall)
Cr-X
a-Ycr-ZMISEScr-Xor-Ycr-ZMISESo--Xa-Ycr-ZMISESa-Xa-Ya-ZMISES
Case A (Nominal)Min.-2.65E+074.68E+08
-1.22E+091.50E+09
-2.52E+06-2.07 E+06-3.33E+063.24E+051.11E+052.08E+05
-2.25E+053.21 E+05
-6.52E+07-3.08E+08-1.20E+094.51 E+08
Max.1.42E+075.93E+08
-1.13E+091.54E+094.70E+051.35E+051.79E+051.32E+064.21 E+054.11 E+058.34E+044.63E+057.08E+077.10E+08
-2.19E+081.19E+09
CaseB: A Be=13.43 W/mKMin.-2.48E+073.70E+08
-1.31E+091.51E+09
-2.68E+06-2.37E+06-3.43E+063.38E+051.11 E+051.98E+05
-2.49E+053.41 E+05
-6.48E+07-2.54E+08-1.28E+095.23E+08
Max.1.31E+074.88E+08
-1.22E+091.55E+094.90E+051 47E+051.89E+051.28E+063.83E+054.02E+054.25E+044.75E+059.78E+075.88E+08
-3.27E+081 26E+09
Case B: A Be="20%
Min.-2.80E+075.17E+08
-1.22E+091.54E+09
-2.99E+06-2.53E+06-3.83E+063.33E+051.25E+052.07E+05
-1.48E+053.10E+05
-6.83E+07-3.18E+08-1.31E+094.51 E+08
Max.1.50E+076.44E+08
-1.12E+091.58E+094.67E+051.23E+051.75E+051.42E+064.36E+054.27E+051.11 E+054.36E+057.75E+077.49E+08
-2.08E+081.29E+09
Case B: A Be=+20%
Min.-2.54E+074.31 E+08-1.22E+091.48E+09
-2.20E+06-1.74E+06-3.00E+063.28E+051.01 E+052.16E+05
-2.80E+053.22E+05
-6.28E+07-2.90E+08-1.13E+094.51 E+08
Max.1.36E+075.55E+08
-1.13E+O91.52E+094.71 E+051.33E+051.73E+051.25E+064.09E+053.98E+055.99E+044.86E+056.81 E+076.81 E+08
-2.27E+081.11E+09
i—3CDO
COI
oen
JAERI-Tech 98-059
-3.0 Plasma
r-4.0f3.0
— W.0\ZD CD CD CD
- —j f—5.0
cCLC
II;;
O OO O
8e binary bed
o oo
o o
O O
O OO O
Shielding Plate 30.0
246.5
24.5-
Fig. 5.1 Module #19 basic cell [1]
- 55 -
Ien
Be armor
Breeder tube No.l No.2 No.3 No.4 No.5 No.6 No.7 No.8
CDOHT
CO00oento
Fig. 5.2 Analysis model of ITER #19 breeding blanket
JAERI-Tech 98-059
1.6
2 ,.4
I 1-2•f 1.0
% 0.8•o
§ 0.6(0
0.4
0.2
0.0
—. —
Li2ZrO3 pebble2 0(65%)+O.20(15%)
SZB modelContact area = 0.Accommodation factor =0.4
-
0 200 400 600
T(°C)
800 1000
Fig. 5.3 Thermal conductivity of Li2ZrO3 binary pebble bed(Calculated with SZB model)
Q.
</> 4OJ
+JCO 3
<D
£ 9(1) /Q.
O 1
H
< o <
(ea1=0.0105,o-a1=3.8MPa)(ea2=0.006,aa2=0.MPa)
Hydrostatic plasticcompression
\
B(ea1,cra1)
^ — 4
; obtained usingFig. 2.4
A(e a1,aa1)
/7///
Plasticprocess
0.002 0.004 0.006 0.008 0.01
Axial Compressive Strain
0.012
Fig. 5.4 Relation between the axial compressive strain andthe axial compressive stress for Li2ZrO3 single size
pebble bed based on Fig. 2.5
- 57 -
JAERI-Tech 98-059
*
sc*5. fl™i
3
tf ftPH ©
« S^ o
- 58 -
JAERI-Tech 98-059
I I U-'-UHL
•'•'.^'fo'X >vt ><t
:': ' >* ; -::::> :::->• r . *•• & •*•
: • • • • x . , - , : • . . • : .
I; I'm::;:::^S.:;:::;x:;;;:::;:::
111* . • ' * • • : : -
••••Y- SS>: - ^
•11: • : •> :
|
ii: • :<• :f
:6ft:;s£<::;iSft:j:
wmm
Hi
CD
OS
CO
tf
HH b
4H ^
co - gCD gS-i OCDN
CDCO
O
CD
bbfa
- 59
1 * s ss
m
i
(tio
ooi
oen
Fig. 5.7 Analyzed result of ITER #19 bleeding blanket( Case A-nominal: a -Y distribution)
JAERI-Tech 98-059
* Jwhiv
ti
4. ii-C
CD
3CO
0
m
>,fa.
8 E
rH «it ^
0 rsj
1.(403 0^ co
cCO
00 w
bbe
- 61 -
JAERI-Tech 98-059
- . - r -f—'-<-*r J
CD
1! ^•H Vl
fl3 CO
C7i
o w^ CO
co TT1
CD II
r 2 ^ COco cc
»3 o
bhs
- 62 -
>tTiSO
CDO
Oento
Fig. 5.9 Analyzed result of ITER #19 bleeding blanket( Case B- X =13.43W/mK : a -X distribution)
- 3CDO
00I
Ocnto
Fig. 5.11 Analyzed result of ITER #19 bleeding blanket( Case A-nominal: a -X distribution in Be pebble bed)
JAERI-Tech 98-059
E- i - * - * - T —-f-f , -
< -1-y f
u^ r-r- — • — - *
^ t
V
• + s % S ~ *
CD
5U 0)cd a
bJD«
.9 .9CU O
CD - rH
cdto
O
IO
- 65 -
JAERI-Tech 98-059
CCD
bo
•
)leed
,—i
WEH
o
CO<D
o• i H
•5<D
IdCD(~>M
' 00 s -0
ii
CO
E
- 66 -
50h—<I
•HCDO
ocnCO
Fig. 5.14 Analyzed result of ITER #19 bleeding blanket( Case B- X =-20%: a -X distribution)
JAERI-Tech 98-059
CD
03
bo
leed
i
Wh-1
O-1-3
CO<D
0N
itio
n;r
ibi
- -CO
'B1
b
o1
1 1
1!
W
CO
Q
bb
- 68 -
JAERT-Tech 98-059
'-'% '
CD
-fri
b£• l-H
0CD
- f t
tfWH
i -.i - H
CO0
0N
r-HOS
CDi—1
idbe
CO
0
cd0ft
£0H
0 s -O
+i iII
•
PQ0COcd
O
- 69 -
JAERI-Tech 98-059
be
0
1
1
1
1
1
1
1
wH(—i
O-1-3
'SCOO>
b• •
N O
O<N_i_\\1 J
N
PQCO
c3
ti
- 70
JAERI-Tech 98-059
-_U-l-4-' - / f ,
afe«lc
£#&&*
(D
r—H
d
to
^wrt'jj
O O
IfCD r <
-d PQ
13 o00
$fe
- 71 -
JAERI-Tech 98-059
6. Summary
Thermo-mechanical analysis of #19 ITER breeding blanket module has beenconducted taking into account spatially varying thermal conductivity and heattransfer coefficient in Be pebble bed depending on the stresses due to thedifferential thermal expansion of the blanket.
The modified DRUCKER-PRAGER/Cap plasticity model of ABAQUS code is usedbecause it can deal with such mechanical features of pebble bed as shear failureflow and plastic consolidation. The capability of the model is studied and provedthrough analysis of the uni-axial compressive experiment.
The thermal conductivity - stress correlation and heat transfer coefficient - stresscorrelation for Be pebble beds are obtained based on the experimental results andadditional thermo-mechanical analysis.
The COUPLED TEMPERATURE-DISPLACEMENT procedure of ABAQUS codeis used so that thermal conductivity is automatically calculated in each calculationpoint depending on the stress.
Thermo-mechanical analysis of the ITER breeding blanket module has beenperformed for four conditions: case A; nominal case, case B; constant thermalconductivity (13.43 W/mK), case C; thermal conductivity = -20% of nominalcase ,and case D; thermal conductivity = +20% of nominal case.- In the nominal case the temperature of breeding material (Li2Zr03) ranges from
317°C to 554°C. The maximum temperature of Be pebble bed is 446°C.- The breeder temperatures in 8th tube and the shielding plate in case A are
higher than those for case B because of lower thermal conductivity caused bylower compressive stresses. It might be concluded that the spatially constantthermal conductivity estimated by FZK is fairly good as a whole for ITER typebreeding blanket.
- Current design of ITER breeding blanket is evaluated to permit ± 20% changein thermal conductivity of Be pebble bed.
From the analysis for the uni-axial compressive and the heat transfer experiment,it is confirmed that the analysis method and model taken here qualitativelyrepresent well the pebble bed behaviors observed in the experiment.Preliminarily analysis on the ITER breeding blanket shows the present design willsatisfy the currently specified material temperature limits. For more detailquantitative estimates of the blanket performance, further investigation onmechanical properties of pebble beds by experiment and the improvement of theanalysis model and the calculation code are required.
- 72 -
JAERI-Tech 98-059
Acknowledgment
The authors wish to acknowledge Drs. S. Matsuda, Y. Seki, T. Nagashima, T.Tsunematsu and M. Seki for their support. They are also grateful to Dr. Takatsuand Dr. Ohara for his continuous encouragement. This work has been performedin the framework of ITER Design Task. Then the authors are also grateful to Drs.Y. Gohar, K. Mohri and K. Ioki of the ITER Joint Central Team for valuableinformation on breeding blanket configuration and constructive discussions.
Reference
[I] Y. Gohar, Personal Communication, Breeding Blanket group of ITER JointCentral Team, 1997.
[2] M. Ferrari et al., ITER Reference Breeding Blanket Design, 20th Symp. FusionTechnol. Sept., 1998, Marseille, France.;
[3] J. Feda, Mechanics of Particle Materials The Principles, Elsevier ScientificPublishing Company, 1982.
[4] Alice Y.Ying et al., Mechanical Behavior and Design Database of Packed Bedsfor Blanket Designs, ISFNT-4, Tokyo, Japan, April(1997)
[5] M. Dalle Donne et al., Measurement of the Thermal conductivity and HeatTransfer Coefficient of a Binary Bed of beryllium Pebbles, Proc, 3rd IEAInternational Work shop on Beryllium Technology for Fusion, Mito, Japan,Oct., 1997.
[6] ABAQUS THEORY MANUAL and ABAQUS/Standard User's Manual,Hibbitt, Karlsson & Sornsen, INC
[7] F. Tehranian and M. Abdou, Experimental Study of the Effect of ExternalPressure on Particle Bed Effective Thermal Properties, Fus. Technol.,27(1995)
[8] K. Walton, "The effective Elastic Modulus of a Random Packing of Sphere", J.Mech. Phis. Solids, Vol. 35, No. 2, pp. 213-226, 1987
[9] A. L. Endres, The Effect of Contact Generation on the Elastic Properties of aGranular Medium, Trans. ASME, 57(1990)330-336
[10] E. U. Schlunder and R. Bauer, Inter. Chem. Eng. 18(1978), 181
[II] ITER Material Properties Handbook, Oct. 1997.[12] M. C. Billone, et al., ITER Solid Breeder Blanket materials Database,
ANL/FPP/TM-263, Nov., 1993.[13] A. C. Paine, Elastic Properties of Granular Materials, Univ. of Bath, 1998.
- 73 -
JAERI-Tech 98-059
Appendix ACalculations of Young's modulus of pebble bed
An analytic method for estimating Young's modulus of a random packed single sizepebble bed is reported by K.Walton [8]. This method is used here despite of itsapplicability for only single size pebble bed. It is because a newly proposed methodimproved to deal with binary pebble beds [13] cannot be used here due to lack ofavailable data on binary pebble bed at present (e.g. number of contact pointsbetween pebbles). In the method of K.Walton, Young's modulus of pebble bed iscalculated as a proportional coefficient between stress and strain averaged overpebble bed. The stress is computed with a given strain by an elastic theory oncontact of pebbles. The relation between the strain and the stress is analyzed inincremental form since young's modulus generally depends on a strain in pebblebed.
Young's modulus is given by the next equation for uniaxial strain and no frictionbetween pebbles.
a = 32JT2B
~l'n (Young's modulus) = 3a
$ : 0.808 (packing factor)n : 6.4 (number of average contacts points per one pebble )e : 0.001 (strain )
„ W l 1 \4JV 1 fi A + n J
A: 2.14 X 1010 [Pa] ( Lame's constants )ju: 1.31 x 1011 [Pa] ( Lame's constants )
C,*, : 1.38 X 109 [Pa] (Young's modulus for pebble bed)
(Above variables were used to calculate Young's modulus of Be pebble bed in Table5.2)
74
Appendix B Thermal and mechanical property
SS316
Temperature
(°C)2050
100150200250300350400450500550600650700800
ThermalConductivity
(W/m-K)13.9414.3715.08
15.816.5217.2417.9518.6719.3920.1
20.8221.5422.2622.9723.6925.12
ThermalExpansion
(1/K)1.59E-051.61E-051.64E-051.67E-051.70E-051.72E-051.75E-051.77E-051.79E-051.81E-051.83E-051.85E-051.87E-051.89E-051.91 E-051.93E-05
Young'sModulus
(Pa)1.92E+111.90E+111.86E+111.82E+111.78E+111.74E+111.70E+111.66E+111.61 E+111.57E+111.53E+111.49E+111.45E+111.41 E+111.37E+111.29E+11
Poisson'sRatio
0.30.30.30.30.30.30.30.30.30.30.30.30.30.30.30.3
Be
Temperature
(°C)2050
100150200250300350400450500550600650700800
ThermalConductivity
(W/m-K)184.51176.95
165.3154.77145.29136.77129.14122.33116.26110.86106.05101.7597.8994.3991.17
85.3
ThermalExpansion
(1/K)1.13E-051.19E-051.29E-051.38E-051.47E-051.55E-051.63E-051.70E-051.77E-051.83E-051.88E-051.94E-051.99E-052.03E-052.07E-052.15E-05
Young'sModulus
(Pa)3.08E+113.06E+113.04E+113.03E+113.02E+113.00E+112.98E+112.94E+112.88E+112.79E+112.67E+112.51 E+112.32E+112.07E+111.76E+119.70E+10
Poisson'sRatio
0.0710.07
0.0690.0680.0670.0650.0640.0630.062
0.060.0590.0580.0570.0550.0540.052
i—t
O
tooo
oaito
(si) t
ft
%
ft¥ Si
* 1*
smN\»it
SJS
ftIS
ftft
> -
wr y
* y
7 'y
Xx 7
*
7
r
T
y ' T
A
Ty
IV
7
y
y
IB *j
m
kgs
A
K
mol
cd
rad
sr
« 3
ffi
Ifx ;
X
mif
ffi
•fe
)t
mft
J
>*
f) ,r-'M^-,
* ,
% ft
ay y i
y y' '
IV •> f
sft
t
j
it; ^i
m #t *> s £t
E, I3«/]
ffi fit7 •? y x
ffi IS1 9 y x
:> x fig
IS
1* &
•y'
7
7
• y
>>
-f' N
-b
IV
IV
9'•y
^ - t-x *
— D
IV
7 7
-
— / y
j -
X
y 'JVU > *} X
- >
— *< IV
•y
y
IV
IV
Vy
h
KA
X
• • ;
7
-
ISy
X
;u
-f
h
Hz
N
PaJ
W
C
V
F
ns
Wb
T
H
°Clm
lx
Bq
Gy
Sv
ffi© si ma
s '
m-kg/s
N/m
N-m
J/s
A-sW/A
C/V
V/A
A/V
V-sWb/m2
Wb/A
cd-sr
lm/m2
s"'
J/kg
J/kg
a.IJ
1
1
Kf, H
5}. fit
y
Id
min,•
1, Lt
eV
u
eV=1.60218xl0-
u=1.66054xl0^2'
h, d
' , "
9J
kg
/ < •
/< -
if
i y t- y
7
A
y
;u
-y
KA
12 tA
b
bar
Gal
CiR
radr e m
A=0.1 nm=10-'0m
b=100fm2=10^2'm2
bar=0.1MPa=10sPa
1 Ci=3.7xlO'°Bq
1 R=2.58xlO-(C/kg
1 rad = lcGy=10"2Gy
1 rem=l cSv=10"2Sv
1 0 "
10"
10'2
10"
106
103
102
10'
10 '
io-2
10"3
io-6
io-io-'2
io-' s
I O - "
t)l. Ml
ffiicj;
2. iJ4
X ? -If
T 7
+" *"^ a+ a-s ? (•
7" 7
T -y
-b y -f
5 'J
+ /t° 3
7 i A h
T t-
- 5 ii rap
E
P
T
G
Mk
h
da
d
c
m
ti
n
Pf
a
smitmj m 5 K, mmHTI£££O AI^IL. 1 eV
1 u Oft l i CODATA CO 1986Sf}t?t
->tzo
/ ••/ v, y — iv, 'N ? -7
3. barli,
r, barniJi
1)
f£
«/*
X
i
tt
ft
w
N( = lOsdyn)
I
9.80665
4.44822
kgf
0.101972
1
0.453592
lbf
0.224809
2.20462
1
If I Pa-s(N-s/m2)=10P(.t:rx')(g/(cm.s))
£ffi lmVs=l0
J( = lO'erg)
I
9.80665
3.6 xlO6
4.18605
1055.06
1.35582
1.60218x10-"
Bq
1
3.7 x 10'°
'St( x h-i> x)
kgf-m
0.101972
1
3.67098x10'
0.426858
107.586
0.138255
1.63377 x 10'20
Ci
2.70270x10-"
1
( c m ' / s )
kW- h
2.77778x10-'
2.72407 x 10-'
1
1.16279 x 10-'
2.93072 x 10-'
3.76616 x 10-'
4.45050 xlO-2S
»&
mm.
rfi MPa( = 10bar)
1
0.0980665
0.101325
1.33322 x 10-'
6.89476 x 10"'
caKs+ftffi)
0.238889
2.34270
8.59999 x 10
1
252.042
0.323890
3.82743 x 10
Gy
1
0.01
rad
100
1
5
kgf/cm2
10.1972
1
1.03323
1.35951 x 1 0 s
7.03070 x 10-2
Btu
9.47813x10-
9.29487 x 10-
3412.13
3.96759 x 10
1
1.28506 x 10-
1.51857x 10"
pa
m
«
3
3
3
a tm
9.86923
0.967841
1
1.31579 x 10'3
6.80460 x 10-2
ft • lbf
0.737562
7.23301
2.65522 x 10
3.08747
778.172
1
1.18171 x 10"
C/kg
1
2.58 x 1Q-'
19
eV
6.24150 x 10
6.12082 x 10
2.24694 x 10
2.61272x 10
6.58515 x 10
8.46233 x 10
1
R
3876
1
mmHg(Torr)
7.50062 x 101
735.559
760
1
51.7149
Ibf/in2(psi)
145.038
14.2233
14.6959
1.93368 x 10'2
1
1 cal = 4.18605 J ( l t i i t )
= 4.184 J (8Wt£)
9 = 4.1855 J (15 °C)
" =4.1868J(B^^a
" f±ff* lPS(1L$tl)
21 -75kgf -m/s
8 = 735.499 W
1
Sv
1
0.01
rem
100
1
PRELIMINARY THERMO-MECHANICAL ANALYSIS OF ITER BREEDING BLANKET
n
Related Documents
