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Thermomechanical behavior of the first wall subjected to plasma disruption
Fusion Engineering and Design 5 (1987) 141-154 North-Holland, Amsterdam
141
THERMOMECHANICAL BEHAVIOR OF THE FIRST WALL SUBJECTED TO PLASMA DISRUPTION H. H A S H I Z U M E a n d K. M I Y A
Nuclear Engineering Research Laboratory, University of Tokyo, Japan M. S E K I
Japan Atomic Energy Research Institute, Tokaimura, Ibaraki-prefecture, Japan K. I O K I
Mitsubishi Heavy Industries. Tokyo, Japan
The first wall and plasma interactivecomponents of a fusion power reactor are subjected to heavy irradiation of high energy neutrons and high heat flux during normal operation. An extremely high heat flux is deposited in the components during a major plasma disruption categorized as abnormal operation. As a consequenceof the event the components melt and solidify • resulting in deterioration of the material, high residual stress, metallurgicalchange and initiation of small cracks. Quantitative evaluation of the consequence of plasma disruption is required to maintain the structural integrity of the components and predict their lifetimes. In the present study the whole process of melting, evaporation and resolidification is analysed using a newly developed computer code based on FEM. In addition, the elastoplastic thermal stress in the heated region during the event is described including residual stress. An experiment was carried out to verify the validity of the code.
1. Introduction The type of damage introduced in the first wall of a fusion power reactor can be categorized into three types: (1) radiation damage due to high energy neutrons, (2) electromagnetic-thermomechanical damage due to high thermal and electromagnetic stresses, and (3) metallurgical damage due to melting and resolidification caused by deposition of a large amount of heat. Radiation damage caused by heavy irradiation of neutrons has been investigated with limited test facilities, e.g. fission reactors and accelerators. There are several fission reactors in the world that are available for studies on damage, but differences in the spectrum and energy of the neutrons produced by fusion reactions make it difficult for such studies to simulate the actual damage. However, recent studies have shown remarkable progress revealing better possibilities for candidate materials to be used as first walls in fusion power reactors. The magnetic stress induced in the first wall is dynamic in its nature because it is caused only when an
eddy current is induced due to the change of the external magnetic field. A potential source of change is the plasma disruption. Thermal stress is caused by the temperature gradient of the the wall and becomes higher almost proportionally to the thickness of the wall. A thicker wall to resist the mechanical stress of the coolant pressure and protect against loss of thickness due to physical sputtering is preferable. Thus there a serious trade-off must be made between thermal and mechanical stress and physical sputtering. The damage accumulated by the stresses in the form of fatigue and creep result in a shorter lifetime of the first wall. The third type of damage is directly related to major plasma disruptions. It is conjectured that melting of the first wall is a natural consequence of deposition of a large amount of energy preserved in the plasmas. The melting followed by resolidification is a very complicated phenomenon that cannot be studied completely until the plasma disruption of an actual reactor is available. However, at present, it is reasonable to assume a thermomechanical model for the plasma disruption and study experimentally and theoretically the
0 9 2 0 - 3 7 9 6 / 8 7 / $ 0 3 . 5 0 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing Division)
142
H. Hashizume et al. / Thermomechanical behavior of the first wall
consequences of this, searching for a method of protecting the first wall against the disruption. The thermomechanical behavior of the first wall, as a consequence of the plasma disruption, was studied first in ref. [1-3]. Merrill et al. [1] discussed melting based on the solution of the heat conducting equation for the first wall taking into account the vaporization mass flux. Hassanein et al, [2] defined the sequence of the process ~s rapid heating, melting, intense evaporation, resolidification and cool-down, and calculated a detailed time history of the temperature distribution by solving a two-moving-boundary problem and the effect of vapor shielding on the melting. When the melting is caused by the disruption, the melt layer experiences an electromagnetic force and may be removed by a Rayleigh-Taylor instability. This was also discussed by Hassanein [3]. In ref. [4] surface melting and evaporation based on a comprehensive model, which includes effects of vapor shielding, pulse shape and pulse duration, are studied. On the other hand, experiments have recently been carried out to investigate the consequence of simulated plasma disruption [5-9]. In ref. [5] a test program using a large area surface heating facility at ANL is introduced. Experimental results with the facility are presented in ref. [6] where beam characteristics and the melting situation are demonstrated. Another simulation test with an electron beam was carried out at SNL [7] where time histories of the surface temperature and the melt layer cross section were investigated experimentally. The effect of repetition of the beam pulse is also discussed. In ref. [81 the thermal behavior and thermal shock resistance of a graphite limiter when heated by a 100 kW electron beam facility are reported. In ref. [9] the measured results of temperature and potential of the limiter in DOUBLET-III during disruption and neutral beam injection are reported. In ref. [10] pressures, stresses and strains induced during a plasma disruption in the first wall having the shape of a toroidal shell are discussed. The induced stress in the wall may be large but will not cause failure in the bulk structure. This force will have to be incorporated together with the behavior of the melted region. It can be concluded from the above mentioned references that a systematic comparison of experiment and numerical an.alysis has never been made: this will be done in the present paper. The model used in the present study consists of three processes, i.e. melting, evaporation and resolidification. The melting can be handled with consideration of the temperature rise to the melting point and latent heat needed for a further temperature rise. However, the
analytical treatment of this phenomenon required for a numerical formulation is rather difficult to do with mathematical accuracy. Application of the Marcal iteration for elasto-plastic stress analysis [11] is made here to deal with the melting. A small artificial temperature rise of the melted metal is assumed while it is actually kept constant until more latent heat is provided. The formulation is described in terms of an incremental representation based on the finite element method. The structural rigidity of the melted element is very smally reset as the melting progresses. Once the temperature distribution is determined, the elastoplastic stress state is calculated using the Marcal method and taking into account the strong dependence of the mechanical properties on the temperature.
2. Thermomechanical analysis of melting and resolidification The starting point of the present study is to solve the following equation of heat conduction,
pci'= k v r + Q,
(1)
where p = density, c = specific heat per unit mass, k = thermal conductivity, Q = heat generation rate, T = temperature, "= ~/Ot = derivative in time. The conventional scheme of the finite element formulation of eq. (1) is applied to derive the following system equation (see [12] for details), [ K ] { T } + [C]{/~} = { F ) ,
(2)
where
[c] = fo4
[K]
= f/~
'lqw] do,
[[NITx[N],.,, + [ N ]T.,,tN].,,
+[N]Tz[N],z ] +
do
f,,,[N]T[N] dS + f%[N]T[N] dS,
{F] = fo. tNydo- fqINydS+ f-,roINl'dS + fa,r,[Nl dS, where q = heat flux, a t = heat transfer coefficient, To -temperature of coolant, Tv = temperature of radiated
143
H. Hashizume et al. / Thermomechanical behavior of the first wall
£
/
Tm
= o
P : density c : specific heat
E
1 : latent heat
~"
&T: temperature change
/
/
,~
mixture
r
'--Latent heat
Slope
! liquid I
I = p.~==
.=-
~
Heat input
Heat input
(a) actual phase diagram
(b) artificial phase diagram
Fig. 1. Adoption of artificial phase diagram.
source and a~ is given by
(3)
a~ = 5 o F o ( T + T,)(T2 + T2),
where c = emissivity, o = Stefan-Boltzmann constant, F0 = shape constant depending on temperature (T + T~). [N] is a shape function and an eight-noded isoparametric element is used in the code.
o
Tm_ = T m - - ~ ¢,/ i
L
:
Iolenl
heal
,, i
1"!
. . . . . . ~!
i a ...... l" .... X
In order to apply eq. (2) to the melting phenomenon, an artificial technique is adopted. Although the temperature of the melting zone is kept constant at Tin(= melting temperature), a very small increment of temperature, a T , is introduced into the zone. This is similar to the assumption that the melted portion can be treated like a solid possessing a pseudo-specific heat capacity defined b y A T / L (L is latent heat). If AT is very small, accurate results may be expected. The relation between temperature and heat input in fig. l(a) is thus approximated by a polygonal line in fig. l(b). The Marcal iteration method used conventionally in structural analysis [11] can be applied to the present model as shown in fig. 2. Let the temperature at some point and at some time step be To as shown in the figure. To proceed to the next step from (Q0, To), the Marcal iteration is applied. Tt in the figure is a solution of the first iteration and is on the line with an original slope of a (= l/pc). If Tt is beyond the lower melting point Tin= ( = T=-AT/2), the following iteration is repeated until there is sufficient convergence.
,
(1) 1st step
I I,X 7
t-I/ ;/; ,
ill
I
I
1
!
!
i
,
',
I
Oo Om-Ol 0z
To estimate the degree of overshooting the temperature, two quantities a 1 a n d / ~ are defined as, al
I
0m÷ Heal inpu!
Fig. 2. Application of Marcal iteration technique.
Tm_-T 0 T I _ TO '
~1=1--0~
1
(4)
and using these variables a new reciprocal of pseudospecific heat, a], is defined as,
a] = a l a + ~ l b ,
(5)
144
H. Hashizume et al. / Thermomechanicalbehavior of the first wall and must be considered in the calculation of the melted zone size. The evaporation flux of atoms is given by ref. [13]
where af l/pc,
b=AT/L.
It can be verified from fig. 2 that
Q m - Qo ~1 = QI - Qo"
J(t)=p/2~Vt~m--~,
(6) J(t)
P
tAtv,
(14)
[tv-t~ ] 0.8 + 0.2 exp[ 1--6~,~ ] / ,
t>tv,
(15)
By using the n~w value al, a new temperature can be "calculated as T2, as shown in fig. 2. (2) 2rid step The heat input corresponding to T2 is given using the slope a I by Q2 = Qo + (7"2 - To)/al.
(7)
The temperature T2* corresponding to Q2 and located on the line with slope a is given by r ? = ( T 2 - To)a/a ~ + To.
(8)
Again, two quantifies of a and fl are calculated, similar to those in eq. (4): T=_-T o a2 = :72. _ To,
f12 = 1 - a 2.
(9)
Then, a new reciprocal of pseudo-specific heat a2 is given by a 2 = a2a + f12b.
(10)
In this case, a 2 is also given by a2 =
Q m - - Qo Q 2 - Qo "
where % is the collision time of atoms, t v is the preheat time, m is the atomic mass, p is the vapor pressure and r is the Boltzmann constant. From the evaporation flux, the heat of vaporization can be evaluated by multiplying the atomic volume and the vaporization heat per unit mass. This quantity is treated as a negative heat flux. The dependence of the mechanical properties on the temperature has to be considered especially in the stress analysis of the problem where the temperature change is large enough to result in melting and resolidification of metal. For example, Young's modulus of stainless steel at 800°C is about half that at room temperature. Another important factor to be taken into consideration is the derivative of elastic constants with temperature. Stress increment is given as follows for elastic and plastic stress states [14], { Ao } = [ D e ] ( A ¢ t - A¢ a} - [ D e l ~ (for elastic),
(11)
{Ao} =[DP](A¢t-A¢ °)-[D
a.
*]-' {°}AT p] O [ D ~}T (for plastic), (! 7 )
(12)
Then, a. = a~a + fl, b.
(16)
aov 2 A T + ° v [ D C ] { ° ' } aT 3 S O
(3) Nth step It can easily be verified that a. is given by
Q = - - Qo Q n - Qo "
( o )AT
(13)
The above iteration is repeated until enough convergence of temperature is achieved. It is possible to analyrically prove that with this procedure convergence is assured for simplified cases of uniform heat generation and uniform temperature change; this is not shown here. On the other hand, some deposited heat is consumed for vaporization of the first wall material. The heat of vaporization is very intense compared to the latent heat
where { a o } = vector of stress increment, ( a d } = vector of total strain increment, {aE °} = vector of thermal strain increment, (o} = vector of elastoplastic stress, {o'} = vector of deviated stress, [D e] elastic stiffness matrix, [Dp] = elasto-plastic stiffness matrix, Oy yield stress, /iT = temperature increment, So = -~H' ~2+ {o,}T[D°](o,}, H t strain hardening rate, = equivalent stress, In fig. 3 the evaluation technique of the stiffness =
145
H. Hashizume et aL / Thermomechanical behavior of the first wall
¢. . . . . . . . . . . . . . .
O'y : y iel d _ _ s . . t y ~ 1 . ~
b~ b
J'
I
61
(a} case 1 : Ep=O
//
f
Strain, E
i
~,! ;i
b
:/
:YY
o'i : yield stress
__
Ik,
I
~/ - - '
,i
" - - ep ~!~ ~, " Strain, E (b) case 2 :Ep~O. , O'a-< O'b
----8p
%/ k/El ~
f
Il tT~=tT, liinltindirlnl,. O'i -~ ...........
I
~!- 8 , ~
L
Strain,8
(el case 3 :Sp~O , ~ > 0 " b
Fig. 3. Account of temperature change and application of Marcal iteration.
matrix when the temperature change cannot be neglected is shown. Case 1 in the figure corresponds to elastic deformation. When the temperature changes from Tn to T2 at a certain point, the Young's modulus changes from E n to E2, respectively. No change of mechanical boundary conditions is assumed and the strain q at the point is kept constant during the temp e r a t ~ e change, causing a change of stress from o~ to o2*. The difference of the stress, (02* - 01), corresponds to the second term of eq. (16). The Marcal iteration starts to be applied from the stress-strain state (02", q). The dependence of yield stress on temperature was considered in the stress analysis. A certain complication takes place when the stress is in a plastic state. In fig. 3(b) two curves are shown whose temperatures are T1 and T2. The stress oa at a point P is determined as the intersecting point of two lines, i.e. c ffi e= and o = E 2 ( e e - %) in fig. 3(b). It can be interpreted as the stress state that is attained when the temperature varies from 7"1 to T2 with the same boundary condition. The two stress states, { o 1} and (oa }, are given by (on) = [D~](%),
(18)
{o,) = [D~](%},
(19)
where [D~] and [D~] are elastic stiffness matrices at the respective temperatures of T1 and T 2. From eqs. (18) and (19), we have, {,,.)
(20)
=
which suggests the following simple relation with the notation of fig. 3(b),
test piece (SS316)
heating
l I
lJ
\ °)°
[-
n"
I
et~
I
, [mm]
E2 o, = E
The Marcal iteration starts from point P if oa is between 01 and O b. I n c a s e 2, 02* is o a g i v e n i n fig. 3(a). Ob is calculated from the relation o b = f ( T 2 , e e + ep) where f represents the stress-strain relation. When o a is larger than % , which corresponds to case 3 in fig. 3(c), 02* is set o b for loading and 02* is set oa for unloading. The material properties of stainless steel, molyb-
o,.
(21)
Fig. 4. Set-up view of test specimen.
H. Hashizume et al. / Thermomechanical behavior of the first wall
146
z
t Heal Flux q"
*
z [mm]
~
.
8
j
~
m
r
t
Heat Flux q" /I
8.0 i I 7.0 6.0 5.0
/i
//
4.0 3.0 2.0 1.0
1
2.0
5.0
IO.O
15.0
20.0
25.0
30.0
r[mm]
Fig. 5. Mesh division of disk subjected to incident beam.
denum and TiC used in the present analysis are given in the Appendix as a function of the absolute temperature T.
3. Experiment The experimental results should be compared with the numerical analysis to see the degree of agreement for the verification of the computer code. Experiments were therefore carried out as stated in the following.
diameter and heating period of 1.7 s. The time of 1.7 s is much longer than the order of a plasma disruption time, which is several tens of milliseconds. A mesh division of an 8 mm thick disk is shown in fig. 5 where an eight noded isoparametric element with nine integral points is adopted. The surfaces of the three elements near the center were heated for 1.7 s by a flow of Ar plasma. The temperatures at nine integral points in each element were calculated to determine the boundary of the melted region.
3.1. Apparatus and test condition The experimental apparatus shown in fig. 4 was provided for the melting test where temperature and strain were measured. A stainless steel circular plate (100 O x 8 mm thick) was clamped with six bolts. An arc torch of a plasma welder was set on the center of the plate and" an Ar plasmas hit the central surface to melt some part of the heated region. The beam power utilized in the experiment was 15 A × 30 V with a 3 mm beam
~a
Io
I
p
Q_
0.5
10
1,5
r [mm]
Shape of incidenl beam
0
110
1.7
lime [sec]
rectoncjulor shape
Fig. 6. Condition of incident b ~ in experl.ment.
147
H. Hashizume et aL / Thermomechanical behaoior of the first wall
In fig. 6 the beam profile and the time history of heat input are shown. The diameter of the circular beam is 3 mm and a rectangular shape is assumed for the history terminating at 1.7 s. The total energy provided from the plasma welder is 756 J in 1.7 s. Not all the energy is deposited in the stainless steel because some of the energy reflects from the surface and does not contribute to its temperature rise. The absorption efficiency is here taken as 0.45 [15]. Thus the energy of 340 J is deposited on an area of 0.07 cm2 in 1.7 s.
,°°I :.P'~'
'
300
¢:3mm
l[i~iT~/A
E =~i
~
o~ 2 0 0
m
experimen,l ol result at A ol B
I00
0
v ''W~ ~ i
t
i
i
i
i
1
3
4
5
6
7
2
Time
[see)
Fig. 7. Temperature change at points of plate subjected to heat flux.
04;
,:o 1.5~m]r
heol flux
400
0,5
liO
1.5 'r Cmm]
~..:-.....;.: oolllll
o4; o'5 ,1o
II
,
r
[mm]
_.04' 0
--2 7.-- ~ /
i
0 t'~ ~
'
0.5
1.0
1.5 r [ram]
.
100
~\ t ~.~
i
#i
t
2
/#3
4
~.'/
5
,
I
6
?
•
Time [sec]
-200
l = 1.8 (sec)
l : 1.2 (sec)
l O.o.~lclll 1 ,1 1
0.0
-:500
~ra302I"
-400
o
'5
-..-e-.-
¢'=3'0mm]H lil/I
~, 1oo
-
1.5
t
200
tm .2~
['m~]o 2
experimentol t e s u l l eloslic analysis (lemperature dependence of moteriol properties is not considered) elosloploslic onolysis (lemperolure dependence of moleriol properties
300
1=1.7 (see)
!°.°:i~!~l t I 1 1
0.4
- --=--
I . o41 I L, li 1 I l
t=0.8 (see)
1°21
In fig. 7 experimental and numerical results of the temperature change are compared. The temperature was measured at two points, point "A" on the top surface and point B on the bottom surface as indicated in the figure. Since A is very close to the heated region, the temperature at this point immediately follows the change of the beam power. The effect of evaporation is not taken into account in the numerical analysis of the data. This is justified by the fact that the experiment was carried out under atmospheric environment and thus the evaporation of metal is significantly suppressed. The plasma disruption, however, takes place in vacuum and the evaporation of first wall material is not suppressed.
I=1.5 (sec)
I=0.4 (see)
..... I Ill
3. 2. Experimental results
1.0
[mm]
1.5"r [mrrd
Fig. 8. Change of melting zone size (numerical result).
-500 Fig. 9. Thermal strain change at points of plate subjected to heat flux.
H. Hashizume et aL / Thermomechanical behavior of the first wall
148
Therefore the effect of evaporation is included in analyses of the plasma disruptions as shown later. In fig. 8 the progress of the melting zone with time is shown. The boundary between liquid and solid is not smooth since the size of an element is not small enough compared to that of the zone. The size is maximum at t = 1.7 s and resolidification of it is completed in a very short time: 0.1 s. The maximum depth is about 180 pro. In fig. 9 changes of radial strains at a rear point C 5.5 mm apart from the center line are shown. The measured strain is represented by a solid line. The results of the elastic and elastoplastic analyses are shown by triangles and circles, respectively. There is no remarkable difference between these. The agreement between the experiment and the analysis is very good up to 1.5 s. The large difference after 2 s could probably be attributed to a difference in the boundary condition, the neglection of the strain rate dependence of yield stress and the discrepancy between measured and calculated points. But the true reason for the difference is not clear at present. A similar comparison of tangential strain is shown in fig. 10. The strain is compressive for some time after the beginning of the heat input and becomes tensile roughly in 2 s. Peaks of the tangential strain are observed between 3 and 4 s. It should be noted that the resolidifi-
600
,,,(""- - ' - ' - ~
500
/
/~"'""*'. ""
4OO3oo
/
3 R
I00 0
'
2
-tOO I "" t~ / [200 _3001_
\~..
I
I
3
4
5
6
7
Time (see] -experimentolresull -'-,~--- elastic analysis(temperature dependenceof material properties is no! considered) ---.e-.- elastoploslicanalysis(temperature dependenceof material properties is considered)
Fig. 10. Thermal strain change at points of plate subjected to heat flux.
q1 500
/
I
/
]
0L
' ~ I 7lsec)
400 ~
~a
300
--
b~
ioo
\
~
~, t=70.
\ b\
t=00$ [sec] t:03 [sec] t = I 7 [sec]
-----o-- t : 2 0 [sec] ----o-- t : 7 0 [sec)
\
^
~
o
elapsed time " --~
, q 01mm f Ill ,
suaoce
_\_
m~lted ~l= 2,0
i "k2 ~ 6__ 8 I \\ ~ t=005 -100[ V ~ . / / ~ _2ooI_
-3001~ /
~o La~a.on ~C~mJ
s-'~
'
/,~oy./
~
~ ~ t
lit=03
Fig. 11. Change of thermal stress distribution of stainless steel.
cation of the melted zone is completed in at least 0.1 s after completion of preheating. The temperature changes continue until 10 s. In fig. 11 distributions of radial stress at various times are shown. The stress is taken on the line A - B 0.1 mm below the top surface. A location near point A does not melt at 0.05 s, while heating continues, but melts in 0.3 s resulting in zero stress. The stress distributions are almost the same for the elapsed times of 0.3 and 1.7 s. However, when resolidification is complete and the temperature decreases, the stress in the melted region becomes tensile due to contraction of the portion. The stress distribution at 7.0 s may be considered as a residual stress distribution. Thus the maximum tensile stress of 42 kg/mm 2 remains. This high residual stress could cause microcracks around the melted region. In many cases hair cracks were observed in a metallurgically polished specimen. Initiation of hair cracks during the process of resolidification is very hazardous for the structural integrity of the first wall subjected to a large plasma disruption. It is crucial to find some engineering solution to prevent cracking but this seems very difficult.
4. Numerical analysis of plasma disruption Two kinds of energy, magnetic and thermal, are deposited in the first wall and the plasma interactive components on plasma disruption. It is important to
tl. Hashizume et aL / Thermomechanical behavior of the first wall
know the amount of energy deposited in the wall as well as the disruption time. According to recent experimental results the disruption time becomes longer as the size of the plasma grows. A longer disruption time can reduce the design requirements because the amount of eddy current is reduced and the melting zone size is restricted. Therefore, the amount of deposited heat and the length of the disruption time are key parameters which have a strong effect on the melting phenomenon. In the following calculation disruption times of 10 to 100 ms are considered. The deposited energy is varied from 200 to 1000 J / c m 2. In fig. 12 the dependence of the evaporation thickness on the deposited energy is shown. It is evident from the results that evaporation thickness increases with the deposited energy and decrease of disruption time. When the disruption time is longer, the temperature rise can be suppressed because of heat diffusion. The evaporation thickness is 70 #m for a disruption time of 10 ms and 12 #m for 100 ms. A difference of 70 /~m and 10/~m is influential in the design specification of the first wall from the viewpoint of reducing its thickness. For example, the assumption that the disruption of 10 ms and 800 J / c m 2 takes place 100 times during the lifetime of the first wall will mean erosion of the surface of the wall by 7 mm in the case of stainless steel. The thickness of the first wall of a fusion power reactor has to be less than 10 ram, for else a very large thermal stress will be induced resulting in a shorter lifetime of the wall due to fatigue and creep damages. The evaporation thickness of 1 #m per disruption could be allowed if the number is 1000 or less. Therefore, it is very important to obtain a data base on the evaporation
2000
disruption time ~, I 0 [msec] I o 20 presenl result o 50 ] v IO0 I0 l ref from [4,13] ..... 20
149
thickness as function of the disruption time and the deposited energy. In fig. 12 numerical results from Hassanein [4,13] are shown to compare with the present results. SoLid and chain lines show the evaporation thickness for a disruption time of 10 ms and 20 ms, respectively. The agreement is quite excellent as shown in the figure. In fig. 13 the melting thickness is shown as a function of the deposited energy with as parameter the disruption time. Results obtained by Hassanein [4,13] are shown with solid and dotted curves for comparison. In this case, the agreement between both results is not good. This may be due to neglect of the receding surface caused by evaporation in the present analysis. As can be seen in fig. 12, the thermal energy consumed in evaporation becomes greater as the disruption time becomes shorter. This is a primary reason for the fact that the melting thickness for the disruption of 10 ms is the lowest. However, the disruption time of 100 ms is smaller than that of 50 ms. Therefore, in order to understand this tendency, the ratio of heat consumed in evaporation and diffused toward a low temperature region should be taken into account. The heat consumed for evaporation is a decreasing function of the disruption time while the heat diffused away from the surface is an increasing function of it. The total energy deposited is partitioned into two types and a maximum value of the melting depth is expected to occur. In fig. 14 the melting depth and the evaporated thickness as function of the disruption time are shown. It is quite clear that the evaporated thickness is linearly decreased in a semi-log plot while the melting depth shows a peak around 50 ms. The comparatively shorter disruption
time
I0 [msec]
o a v
500
20 50 I00
" • •
tO
•
- -
400
- - - 20
"]
t present
•
r6|. from ~.131
J" ...o-
~. 3oo
I00,0 =
50.0
f ~
.j/
i,~
IO0
200
.~" / 0
~.0
~---~.~
100
I
400
F
I
,
I
j
600 800 I000 Deposited energy [J/cm z]
Fig. 12. Relation between evaporated thickness and deposited energy on major plasma disruption.
I
400
i
I
600
Oeposiled
i
I
800 energy
i
I
i
I000 [ J / c m z]
Fig: 13. Relation between melting thickness and deposited energy on major plasma disruption.
150
H. Hashizume et aL
/ Thermomechanical behaoior of the first wall 1:4 (msec)
70
350 Xx
300
6O
xxx
Z
[#) 200 x
250
50 =t \\\
20O
40
X\
t
0'5
4006
~,
100
I=8 (msec)
!
~x
I0
2
r0
~
'
J
30 40 50
Disruplion lime
~
100
05
I0
15 r
~
l0
0
o
o 0.5
[mini 1:20 (reset)
ol I 1 I ly,×,~.,~ I 1 1
I. ~..../,,:,,/ • . :.. . ".'." V,/ ,, I:. .....;;y:.<.-/-~ t,,,ool> ...~;~.:: :~.S-~: ;If
t,,, o o ~
\x
4000
[Fnrn]
I
50
0
1.5 r
Ii0
30 xx
{msec)
t ~ 2oo "::;.%:ii.- ;:/ :::~
az:
~ 150
1 = 16
01!.!I I I I !
o1111111
~
1.0
I.SEr]mm 4000
;
I=12 (msec)
0
0.5
"::;1.0.....1.5 -
,
r
Cram]
I=24 (msec)
(msec) z
Fig. 14. Effect of disruption time on melting depth and evaporated thickness.
,
~"~ ~oo ;'!~< 400
c, ~oo
0.5
1.0
1.5 r
.
.
[mm]
disruption is more severe since the loss of material due to evaporation and the melting depth are large. In comparison the disruption time of 10 ms, that of 100 ms is milder with regard to the evaporation thickness. In fig. 15 the progress of the melting zone for a heat input of 800 J / c m 2 lasting for 20 ms is shown. There are 3 x 2 = 6 elements in a region of 1.5 mm X 0.4 mm in the figure. However, though mesh division is coarse, nine evaluating points for numerical integration are included in each element. Thus it is possible to determine whether a smaller region in the element melts or not. It goes without saying that the melting develops first in the radial direction along the surface and then in the direction of plate thickness. As assumed in the calculation heat deposition ceases at the time of 20 ms of course a large m o u n t of energy is conserved in the melted region at that time. The energy diffuses to the neighboring region through liquid-solid interface indicaring the possibility that further development of the melted region takes place. A precise examination of the numerical results does not show a visible development at all, indicating that the energy stored in the melted region is not enough to exceed the latent heat of the neigfiboring element. The maximum melting depth of 250/xm is achieved at t = 20 ms. In fig. 16 the relation between melting depth and deposited energy with the disruption time as parameter is shown. In the disruption time range of 4 to 20 ms, the melting depth is almost independent of the disruption time but strongly depends on the deposited energy. This
1:5 'r
[mini
SUS, 0=800 [Jlcm 2] Td =20 [msec]
Fig. 15. Change of melted zone due to plasma disruption of Q = 800 J / c m 2 and Td = 20 ms.
is observed clearly as the threshold energy for melting, which is about 200 J / c m 2 for stainless steel. This is very important information for the thermomechanical design
30O ff
o
Td (msec) 4
200 --x--
B
E o
~ 100 6 I f
00 --o~,~
~
500
1000
0ep0siled energy (d/cm 2 I Fig. 16. Relation between melted depth and deposited energy with as parameter disruption time.
151
H. Hashizume et aL / Thermomechanical behavior of the first wall
of the first wall, from which it is probably possible to allow heat deposition below the threshold value many times if fatigue damage due to cyclic thermal stress is neglected. Other important information seen in the figure is that the melting depth does not increase significantly for energy deposition over 500 J / c m 2. As stated previously, this is due primarly to the energy consumption in the evaporation and this is remarkable for the shorter plasma disruption. This situation may be different for materials such as graphite that are easy to evaporate or sublimate. In fig. 17 distributions of radial stress with elapsed time as parameter are shown. The stress on the line A - B 0.1 mm underneath the surface is plotted. Similar to the distribution in fig. 11, the stress state is compressive in the whole region at the beginning of heat deposition. The stress state reverses to a tensile one in the melted region at a time of at least 80 ms. The maximum residual stress is about 250 MPa in this case and it is smaller compared to the maximum value in fig. 11. In general, the residual stress is higher in a larger melted region larger because the amount of contraction is proportit)nal to the size. Compared with the case of fig. 17 the deposited energy is very large in the melting experiment. This is a primary reason for the difference of the maximum residual stress. Although it is not verified systematically, it is possible to consider that cracking could take place in the case of a large amount of heat input with high residual stress (larger than 400 MPa),
but not in the case of residual stress less than 300 MP& Information on the possibility of cracking is very important for the structural integrity of the first wall and in the near future systematic experiments with regard to the criterion of cracking are necessary. There are many cases where coated materials are used for the first wall. A typical example of this is TiC coated molybdenum. This has been used for the first wall of JT-60. Thus it is important to know the thermomechanical behavior of the TiC coated Mo when i t i s subjected to plasma disruption. In fig. 18 the surface temperature changes of TiC, Mo and TiC coated Mo are shown. Deposited heat and disruption time are 200 J / c m 2 and 1 ms, respectively. The temperature of TiC coated Mo is between those of TiC and Mo. The maximum temperature is about 4200 ° C for all the materials, where these melt on their surfaces. The temperature starts to drop rapidly at the time of 1.0 ms when the heat input ceases. In fig. 19 the development of the melting region for Mo is shown. The maximum depth is 70 # m that is achieved at t = 1.0 ms. Fig. 20 shows the development of the melting region for TiC. A maximum depth of 70 # m is achieved; this is the same as for Mo.
5000]
4000
q
, 300 -
.'--"' /,~/~
.,oo=:m:c,
'm°
i 20 [msec] --.o..- 80 [mse¢]
~
200
~
T.m.
(TIC)
3000 . . . . . . . . . . . . , ,',~
(Moli . . . . . . . . . . . . . . . . . . . . .
"---
lO0
i
.
.
.
.
.
,,o.
-I00
• TiC ,T.m.:3067"C,
~' 20001 y / ~ I/I
. . . .
TiC+Mo
m -200
'°°° rl/
,
,
,
,
,
0.2
0.4
0.6
0.8
1.0
,~
-300
Q=800J/cm2 , Td: 20 msec
Fig. 17. Change of thermal stress distribution in SUS first wall due to plasma disruption.
O0
Elopsed lime [msec]
Fig. 18. Change of surface temperature.
1.2
H. Hashizume et aL / Thermomechanical behavior of the first wall
152 I=0.4 (reset)
I=0.4
I=0.8 (reset)
(msec)
o¢,,,,,, I
oIIJllll
TiC No
1 ~]
ol
I=0.8 lmsec)
fill
L:;';~';'//"/~~ J ,
.... , ,,..:,/. ,:,,×
[~150 [
1001 000
0~.5
JO
1.5'r [ram}
'00;
0'3
1=0.6 (msec)
I=1.0
ollllllll [/:250
1
IO0 •
0
1'.0
1.5"r [mm}
~
~
1.0
.
1.5 r
Cram]
I
1.0
.
1.5 r
Cmm]
0
0.5
1.5 r
L0
tram]
t : 1.0(msec)
[/1150I
000~
0.5
.0
1.5 r
,00;
0'.5
L'O
1.5"r
[mrn]
Cmm] e
+
e
100 ~ 0
0.5
1.5 r
1.0
[mm]
•
TiClrn.p.306? C) Mo(m.p.2622 C), O:200[J/cm2], Td=ICmsec3
O:200[Jlcm 2], Td=l[msec]
Mo,
i 0.5
I=0.6 (reset)
(msec)
[/050
0.5
0
Fig. 19. Change of melted zone due to plasma disruption.
Fig. 21. Change of melted zone due to plasma disruption.
In fig. 21 the development of the melting region of TiC coated Mo is shown. The melting behavior of TiC is different from that of Mo. A portion near the surface is melted for TiC and Mo at t = 0.4 ms. However, TiC coated Mo is not melted at this time. This may be due to a difference in the m o u n t of heat conducted through the interface between both materials. It is also interesting to observe an unmelted region in TiC where a larger region of Mo is melted. This is due to the difference in t h e malting temperature of both materials. From this
fact it is possible to predict that radical internal evaporation could take place and blow off the surface material if much more energy is deposited in a short time and in a localized way. The blow-off phenomenon may be related not only to the internal vaporization to result in high pressure but also to thermal shock. The phenomenon seen in graphite is primarily due to brittle cracking caused by the thermal shock because the graphite is not melted but sublimates or evaporates. However, in the case of TiC coated Mo, there is a possibility of blow off due to a sudden increase of the internal pressure. This kind of blow-off phenomenon has been observed in JET [16]. Not only plasma disruption but also arcing between the plasma and the first wall has a possibility of causing the blow-off of material. Study of this phenomenon has not been undertaken up to now but it is crucial for the engineering feasibility of a fusion reactor. Therefore we strongly urge that such an experiment or theoretical study be carried out to reveal the phenomenon quantitatively and establish a method of protection. Numerical results, some of Which are compared to experimental results to verify the validity Of the computer code, demonstrate that residual stress achieved after the resolidificafion reaches the yield stress level of the first wall material and could cause micro-crackings near the boundary of the resolidified region. Another important result is that the threshold energy for melting seems to be about 200 J / c m 2 for stainless steel. The plasma disruption, thus, can be allowed to take place if its heat deposition is below the threshold value.
I=0.4 (msec)
LL.L,I
[/~1~ i00 ~ 0
I= 0 . 8
1
r
0.5
f 1.0
I00 ~ 0
1,5 r [mm]
I=0.6 (msec)
,
0.5
~
1.0
~
,
1.5 r [ram]
l= 1.0 {msec )
ollLlll|
1
"27 " × " / z
"~
t~ 5o"';"×" " ~'×
[la]50I I00 ~ 0
(msec)
0i I I I l J I 1
"
~
0.5
1.0
,4,
,0%
o'.5
Cmm] TiC,
0:200
[dtcmZ],
,Jo
,.5, Cram]
Td:l[msec]
Fig. 20. Change of melted zone due to plasma disruption.
H. Hashizume et al. / Thermomechanical behavior of the first wall
153
5. C o n c l u s i o n s
Heat of vaporization L~ [J/kg]
Conclusions obtained from the present study are summarized as follows: (1) The computer code was made based on F E M which enables us to perform analyses of malting, evaporation, solidification, temperature distribution and elastoplastic stress-strain. Its validity was verified through the experiment. (2) The amount of evaporated mass increases with decrease of the disruption time. (3) There exists a threshold energy for initiation of melting, which is about 200 J / e r a 2 for stainless steel. (4) Thermal stress taking place during the resolidification process is large and may cause small cracks in the melted region. (5) Further study is required to reveal the blow-off phenomenon seen in the existing facilities.
L v = 7.46 x 106. Young's modulus E [Pa] E = 2.23 x 10 n - 8.09 x 107T. Poisson's ratio v vffi 0.3.
Yield stress o v [Pa] o v = 4.82 x 10 s - 3.00 X 105T. Strain hardening rate H ' H ' = 5.0 x 109 - d H ' / d T x
where d H ' / d T = d E / d T x H ' / E .
Appendix
Molybdenum
Stainless steel
Heat capacity c [ J / k g K]
Heat capacity c [ J / k g K] c s = 3.04 × 102 + 6.70 x 1 0 - 1 T - 4.78 × 1 0 - 4 T 2 + 1.27 x 1 0 - 7 T 3,
T,
cs = 4.15 × 102 - 1.68 x 1 0 - 1 T + 7.42 × 1 0 - S T 2, ct = 4.23 X 10 2. Density p [ k g / m 2]
c I = 7.12 x 102.
Ps.t = 1.03 x 104 - 2.4 × 10-1T.
Density p [ k g / m 3]
Thermal diffusivity a [n~/s]
Ps = 8.03 x 103 - 4.21 x 1 0 - 1 T - 3.89 x 1 0 - 5 T 2,
a s = 5.00 x 10 -5 - 1.10 × 10-ST,
pt = 7.43 x 103 + 2.93 × 1 0 - 2 T - 1.80 x 1 0 - 5 T 2.
a t = 1.75 X 10 -5 - 3.00 X 10-1°T.
Thermal diffusivity a [m2/s]
Melting point Tm [K]
a s = 3.87 × 10 -6 + 4.90 × 1 0 - 9 T - 4.17 × 10-13T2,
Tm -- 2895.
a t = 5.80 × 10 -6.
Heat of fusion L [J/kg]
Melting point Tm [K]
L -- 2.90 × 105.
Tm = 1700. Heat of fusion L [J/kg]
Vapor pressure Ps [Pa] Ps = exp(26.1 - 7.19 × 1 0 4 / T ) .
L -- 2.70 × 105. heat of vaporization L v (J/kg] Vapor pressure Ps [Pal Ps = exp(25.6 - 4.35 x 1 0 4 / T ) .
L v = 6.18 × 106.
154
H. Hashizume et al. / Thermomechanical behavior of the first wall
T/C
Heat capacity c [ J / k g K] c s = 8.02 × 102 + 6.20 × 10-2T, ct-- 1.12 × 103. • Density p [ k g / m 3] Ps./-- 4.60 x 103. Thermal diffusivity a [m2/s] a s -- 8.18 x 10 -6 + 6.83 x 10-ST, a I = 1.05 × 10 -5. Melting point Tm [K] Tm = 3340. Heat of fusion L [J/kg] L = 3.88 × 105. Vapor pressure Ps [Pa] Ps = exp(2.91 - 7.74 x 1 0 4 / T ) . Heat of vaporization L v [J/kg] L v = 7.92 × 106.
References [1] B.J. Merrill and T.L. Jones, J. Nucl. Mater. 111&112 (1982). [2] A.M. Hassanein, G.L. Kulcinski and W.G. Wolfer, J. Nucl. Mater. 111&112 (1982). [3] W.G. Wolfer and A.M. Hassanein, J. Nucl. Mater. 111& 112 (1982). [4] A.M. Hassanein, G.U Kulcinski and W.G. Wolfer, Nuc. Engrg. Des. Fusion 1 (1984). [5] H.D. Michael, J. Lempert, J. Chi and R.P. Rose; Nuc. Tech./Fusion, Vol. 4 (1983) [6] J.IL Easoz and R. Bajaj, Proc. 10th Symp. of Fusion Engineering (1983). [7] S.R. Picraux, J.A. Knapp and M.J. Davis, J. Nucl. Mater. 120 (1984). [8] T. Uchikawa et al., 6th Topical Meeting on the Technology of Fusion Energy (1985). [9] T. Hino et al., J. Nuci. Mater. 121 (1984). [10] M.S. Tillack, M.S. KazJmi and L.M. Lidsky, Fusion Teehnol. 8 (1985). [11] P.V. Marcal and l.P. King, Int. J. Mech. Sci. 9 (1967). [12] O.C. Zienkiewicz, Finite Element Method in Engineering Science (McGraw-Hill, New York, 1961). [13] H. Nakamura, T. Hiraoka, A.H. Hassanein, G.L. Kulcinski and W.G. Wolfer, First wall erosion during a plasma disruption in tokamak, JAERI-M, 83-058 (1983). [14] Y. Uoda and T. Yamakawa, Trans. Japan Welding Society, Vol. 2 (1971). [15] S.L. Engel, Laser Focus 44 (1976). [16] W.M. Lomer, Proc. First Int. Conf. Fusion Reactor Materials, Dec. (1984).
141
THERMOMECHANICAL BEHAVIOR OF THE FIRST WALL SUBJECTED TO PLASMA DISRUPTION H. H A S H I Z U M E a n d K. M I Y A
Nuclear Engineering Research Laboratory, University of Tokyo, Japan M. S E K I
Japan Atomic Energy Research Institute, Tokaimura, Ibaraki-prefecture, Japan K. I O K I
Mitsubishi Heavy Industries. Tokyo, Japan
The first wall and plasma interactivecomponents of a fusion power reactor are subjected to heavy irradiation of high energy neutrons and high heat flux during normal operation. An extremely high heat flux is deposited in the components during a major plasma disruption categorized as abnormal operation. As a consequenceof the event the components melt and solidify • resulting in deterioration of the material, high residual stress, metallurgicalchange and initiation of small cracks. Quantitative evaluation of the consequence of plasma disruption is required to maintain the structural integrity of the components and predict their lifetimes. In the present study the whole process of melting, evaporation and resolidification is analysed using a newly developed computer code based on FEM. In addition, the elastoplastic thermal stress in the heated region during the event is described including residual stress. An experiment was carried out to verify the validity of the code.
1. Introduction The type of damage introduced in the first wall of a fusion power reactor can be categorized into three types: (1) radiation damage due to high energy neutrons, (2) electromagnetic-thermomechanical damage due to high thermal and electromagnetic stresses, and (3) metallurgical damage due to melting and resolidification caused by deposition of a large amount of heat. Radiation damage caused by heavy irradiation of neutrons has been investigated with limited test facilities, e.g. fission reactors and accelerators. There are several fission reactors in the world that are available for studies on damage, but differences in the spectrum and energy of the neutrons produced by fusion reactions make it difficult for such studies to simulate the actual damage. However, recent studies have shown remarkable progress revealing better possibilities for candidate materials to be used as first walls in fusion power reactors. The magnetic stress induced in the first wall is dynamic in its nature because it is caused only when an
eddy current is induced due to the change of the external magnetic field. A potential source of change is the plasma disruption. Thermal stress is caused by the temperature gradient of the the wall and becomes higher almost proportionally to the thickness of the wall. A thicker wall to resist the mechanical stress of the coolant pressure and protect against loss of thickness due to physical sputtering is preferable. Thus there a serious trade-off must be made between thermal and mechanical stress and physical sputtering. The damage accumulated by the stresses in the form of fatigue and creep result in a shorter lifetime of the first wall. The third type of damage is directly related to major plasma disruptions. It is conjectured that melting of the first wall is a natural consequence of deposition of a large amount of energy preserved in the plasmas. The melting followed by resolidification is a very complicated phenomenon that cannot be studied completely until the plasma disruption of an actual reactor is available. However, at present, it is reasonable to assume a thermomechanical model for the plasma disruption and study experimentally and theoretically the
0 9 2 0 - 3 7 9 6 / 8 7 / $ 0 3 . 5 0 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing Division)
142
H. Hashizume et al. / Thermomechanical behavior of the first wall
consequences of this, searching for a method of protecting the first wall against the disruption. The thermomechanical behavior of the first wall, as a consequence of the plasma disruption, was studied first in ref. [1-3]. Merrill et al. [1] discussed melting based on the solution of the heat conducting equation for the first wall taking into account the vaporization mass flux. Hassanein et al, [2] defined the sequence of the process ~s rapid heating, melting, intense evaporation, resolidification and cool-down, and calculated a detailed time history of the temperature distribution by solving a two-moving-boundary problem and the effect of vapor shielding on the melting. When the melting is caused by the disruption, the melt layer experiences an electromagnetic force and may be removed by a Rayleigh-Taylor instability. This was also discussed by Hassanein [3]. In ref. [4] surface melting and evaporation based on a comprehensive model, which includes effects of vapor shielding, pulse shape and pulse duration, are studied. On the other hand, experiments have recently been carried out to investigate the consequence of simulated plasma disruption [5-9]. In ref. [5] a test program using a large area surface heating facility at ANL is introduced. Experimental results with the facility are presented in ref. [6] where beam characteristics and the melting situation are demonstrated. Another simulation test with an electron beam was carried out at SNL [7] where time histories of the surface temperature and the melt layer cross section were investigated experimentally. The effect of repetition of the beam pulse is also discussed. In ref. [81 the thermal behavior and thermal shock resistance of a graphite limiter when heated by a 100 kW electron beam facility are reported. In ref. [9] the measured results of temperature and potential of the limiter in DOUBLET-III during disruption and neutral beam injection are reported. In ref. [10] pressures, stresses and strains induced during a plasma disruption in the first wall having the shape of a toroidal shell are discussed. The induced stress in the wall may be large but will not cause failure in the bulk structure. This force will have to be incorporated together with the behavior of the melted region. It can be concluded from the above mentioned references that a systematic comparison of experiment and numerical an.alysis has never been made: this will be done in the present paper. The model used in the present study consists of three processes, i.e. melting, evaporation and resolidification. The melting can be handled with consideration of the temperature rise to the melting point and latent heat needed for a further temperature rise. However, the
analytical treatment of this phenomenon required for a numerical formulation is rather difficult to do with mathematical accuracy. Application of the Marcal iteration for elasto-plastic stress analysis [11] is made here to deal with the melting. A small artificial temperature rise of the melted metal is assumed while it is actually kept constant until more latent heat is provided. The formulation is described in terms of an incremental representation based on the finite element method. The structural rigidity of the melted element is very smally reset as the melting progresses. Once the temperature distribution is determined, the elastoplastic stress state is calculated using the Marcal method and taking into account the strong dependence of the mechanical properties on the temperature.
2. Thermomechanical analysis of melting and resolidification The starting point of the present study is to solve the following equation of heat conduction,
pci'= k v r + Q,
(1)
where p = density, c = specific heat per unit mass, k = thermal conductivity, Q = heat generation rate, T = temperature, "= ~/Ot = derivative in time. The conventional scheme of the finite element formulation of eq. (1) is applied to derive the following system equation (see [12] for details), [ K ] { T } + [C]{/~} = { F ) ,
(2)
where
[c] = fo4
[K]
= f/~
'lqw] do,
[[NITx[N],.,, + [ N ]T.,,tN].,,
+[N]Tz[N],z ] +
do
f,,,[N]T[N] dS + f%[N]T[N] dS,
{F] = fo. tNydo- fqINydS+ f-,roINl'dS + fa,r,[Nl dS, where q = heat flux, a t = heat transfer coefficient, To -temperature of coolant, Tv = temperature of radiated
143
H. Hashizume et al. / Thermomechanical behavior of the first wall
£
/
Tm
= o
P : density c : specific heat
E
1 : latent heat
~"
&T: temperature change
/
/
,~
mixture
r
'--Latent heat
Slope
! liquid I
I = p.~==
.=-
~
Heat input
Heat input
(a) actual phase diagram
(b) artificial phase diagram
Fig. 1. Adoption of artificial phase diagram.
source and a~ is given by
(3)
a~ = 5 o F o ( T + T,)(T2 + T2),
where c = emissivity, o = Stefan-Boltzmann constant, F0 = shape constant depending on temperature (T + T~). [N] is a shape function and an eight-noded isoparametric element is used in the code.
o
Tm_ = T m - - ~ ¢,/ i
L
:
Iolenl
heal
,, i
1"!
. . . . . . ~!
i a ...... l" .... X
In order to apply eq. (2) to the melting phenomenon, an artificial technique is adopted. Although the temperature of the melting zone is kept constant at Tin(= melting temperature), a very small increment of temperature, a T , is introduced into the zone. This is similar to the assumption that the melted portion can be treated like a solid possessing a pseudo-specific heat capacity defined b y A T / L (L is latent heat). If AT is very small, accurate results may be expected. The relation between temperature and heat input in fig. l(a) is thus approximated by a polygonal line in fig. l(b). The Marcal iteration method used conventionally in structural analysis [11] can be applied to the present model as shown in fig. 2. Let the temperature at some point and at some time step be To as shown in the figure. To proceed to the next step from (Q0, To), the Marcal iteration is applied. Tt in the figure is a solution of the first iteration and is on the line with an original slope of a (= l/pc). If Tt is beyond the lower melting point Tin= ( = T=-AT/2), the following iteration is repeated until there is sufficient convergence.
,
(1) 1st step
I I,X 7
t-I/ ;/; ,
ill
I
I
1
!
!
i
,
',
I
Oo Om-Ol 0z
To estimate the degree of overshooting the temperature, two quantities a 1 a n d / ~ are defined as, al
I
0m÷ Heal inpu!
Fig. 2. Application of Marcal iteration technique.
Tm_-T 0 T I _ TO '
~1=1--0~
1
(4)
and using these variables a new reciprocal of pseudospecific heat, a], is defined as,
a] = a l a + ~ l b ,
(5)
144
H. Hashizume et al. / Thermomechanicalbehavior of the first wall and must be considered in the calculation of the melted zone size. The evaporation flux of atoms is given by ref. [13]
where af l/pc,
b=AT/L.
It can be verified from fig. 2 that
Q m - Qo ~1 = QI - Qo"
J(t)=p/2~Vt~m--~,
(6) J(t)
P
tAtv,
(14)
[tv-t~ ] 0.8 + 0.2 exp[ 1--6~,~ ] / ,
t>tv,
(15)
By using the n~w value al, a new temperature can be "calculated as T2, as shown in fig. 2. (2) 2rid step The heat input corresponding to T2 is given using the slope a I by Q2 = Qo + (7"2 - To)/al.
(7)
The temperature T2* corresponding to Q2 and located on the line with slope a is given by r ? = ( T 2 - To)a/a ~ + To.
(8)
Again, two quantifies of a and fl are calculated, similar to those in eq. (4): T=_-T o a2 = :72. _ To,
f12 = 1 - a 2.
(9)
Then, a new reciprocal of pseudo-specific heat a2 is given by a 2 = a2a + f12b.
(10)
In this case, a 2 is also given by a2 =
Q m - - Qo Q 2 - Qo "
where % is the collision time of atoms, t v is the preheat time, m is the atomic mass, p is the vapor pressure and r is the Boltzmann constant. From the evaporation flux, the heat of vaporization can be evaluated by multiplying the atomic volume and the vaporization heat per unit mass. This quantity is treated as a negative heat flux. The dependence of the mechanical properties on the temperature has to be considered especially in the stress analysis of the problem where the temperature change is large enough to result in melting and resolidification of metal. For example, Young's modulus of stainless steel at 800°C is about half that at room temperature. Another important factor to be taken into consideration is the derivative of elastic constants with temperature. Stress increment is given as follows for elastic and plastic stress states [14], { Ao } = [ D e ] ( A ¢ t - A¢ a} - [ D e l ~ (for elastic),
(11)
{Ao} =[DP](A¢t-A¢ °)-[D
a.
*]-' {°}AT p] O [ D ~}T (for plastic), (! 7 )
(12)
Then, a. = a~a + fl, b.
(16)
aov 2 A T + ° v [ D C ] { ° ' } aT 3 S O
(3) Nth step It can easily be verified that a. is given by
Q = - - Qo Q n - Qo "
( o )AT
(13)
The above iteration is repeated until enough convergence of temperature is achieved. It is possible to analyrically prove that with this procedure convergence is assured for simplified cases of uniform heat generation and uniform temperature change; this is not shown here. On the other hand, some deposited heat is consumed for vaporization of the first wall material. The heat of vaporization is very intense compared to the latent heat
where { a o } = vector of stress increment, ( a d } = vector of total strain increment, {aE °} = vector of thermal strain increment, (o} = vector of elastoplastic stress, {o'} = vector of deviated stress, [D e] elastic stiffness matrix, [Dp] = elasto-plastic stiffness matrix, Oy yield stress, /iT = temperature increment, So = -~H' ~2+ {o,}T[D°](o,}, H t strain hardening rate, = equivalent stress, In fig. 3 the evaluation technique of the stiffness =
145
H. Hashizume et aL / Thermomechanical behavior of the first wall
¢. . . . . . . . . . . . . . .
O'y : y iel d _ _ s . . t y ~ 1 . ~
b~ b
J'
I
61
(a} case 1 : Ep=O
//
f
Strain, E
i
~,! ;i
b
:/
:YY
o'i : yield stress
__
Ik,
I
~/ - - '
,i
" - - ep ~!~ ~, " Strain, E (b) case 2 :Ep~O. , O'a-< O'b
----8p
%/ k/El ~
f
Il tT~=tT, liinltindirlnl,. O'i -~ ...........
I
~!- 8 , ~
L
Strain,8
(el case 3 :Sp~O , ~ > 0 " b
Fig. 3. Account of temperature change and application of Marcal iteration.
matrix when the temperature change cannot be neglected is shown. Case 1 in the figure corresponds to elastic deformation. When the temperature changes from Tn to T2 at a certain point, the Young's modulus changes from E n to E2, respectively. No change of mechanical boundary conditions is assumed and the strain q at the point is kept constant during the temp e r a t ~ e change, causing a change of stress from o~ to o2*. The difference of the stress, (02* - 01), corresponds to the second term of eq. (16). The Marcal iteration starts to be applied from the stress-strain state (02", q). The dependence of yield stress on temperature was considered in the stress analysis. A certain complication takes place when the stress is in a plastic state. In fig. 3(b) two curves are shown whose temperatures are T1 and T2. The stress oa at a point P is determined as the intersecting point of two lines, i.e. c ffi e= and o = E 2 ( e e - %) in fig. 3(b). It can be interpreted as the stress state that is attained when the temperature varies from 7"1 to T2 with the same boundary condition. The two stress states, { o 1} and (oa }, are given by (on) = [D~](%),
(18)
{o,) = [D~](%},
(19)
where [D~] and [D~] are elastic stiffness matrices at the respective temperatures of T1 and T 2. From eqs. (18) and (19), we have, {,,.)
(20)
=
which suggests the following simple relation with the notation of fig. 3(b),
test piece (SS316)
heating
l I
lJ
\ °)°
[-
n"
I
et~
I
, [mm]
E2 o, = E
The Marcal iteration starts from point P if oa is between 01 and O b. I n c a s e 2, 02* is o a g i v e n i n fig. 3(a). Ob is calculated from the relation o b = f ( T 2 , e e + ep) where f represents the stress-strain relation. When o a is larger than % , which corresponds to case 3 in fig. 3(c), 02* is set o b for loading and 02* is set oa for unloading. The material properties of stainless steel, molyb-
o,.
(21)
Fig. 4. Set-up view of test specimen.
H. Hashizume et al. / Thermomechanical behavior of the first wall
146
z
t Heal Flux q"
*
z [mm]
~
.
8
j
~
m
r
t
Heat Flux q" /I
8.0 i I 7.0 6.0 5.0
/i
//
4.0 3.0 2.0 1.0
1
2.0
5.0
IO.O
15.0
20.0
25.0
30.0
r[mm]
Fig. 5. Mesh division of disk subjected to incident beam.
denum and TiC used in the present analysis are given in the Appendix as a function of the absolute temperature T.
3. Experiment The experimental results should be compared with the numerical analysis to see the degree of agreement for the verification of the computer code. Experiments were therefore carried out as stated in the following.
diameter and heating period of 1.7 s. The time of 1.7 s is much longer than the order of a plasma disruption time, which is several tens of milliseconds. A mesh division of an 8 mm thick disk is shown in fig. 5 where an eight noded isoparametric element with nine integral points is adopted. The surfaces of the three elements near the center were heated for 1.7 s by a flow of Ar plasma. The temperatures at nine integral points in each element were calculated to determine the boundary of the melted region.
3.1. Apparatus and test condition The experimental apparatus shown in fig. 4 was provided for the melting test where temperature and strain were measured. A stainless steel circular plate (100 O x 8 mm thick) was clamped with six bolts. An arc torch of a plasma welder was set on the center of the plate and" an Ar plasmas hit the central surface to melt some part of the heated region. The beam power utilized in the experiment was 15 A × 30 V with a 3 mm beam
~a
Io
I
p
Q_
0.5
10
1,5
r [mm]
Shape of incidenl beam
0
110
1.7
lime [sec]
rectoncjulor shape
Fig. 6. Condition of incident b ~ in experl.ment.
147
H. Hashizume et aL / Thermomechanical behaoior of the first wall
In fig. 6 the beam profile and the time history of heat input are shown. The diameter of the circular beam is 3 mm and a rectangular shape is assumed for the history terminating at 1.7 s. The total energy provided from the plasma welder is 756 J in 1.7 s. Not all the energy is deposited in the stainless steel because some of the energy reflects from the surface and does not contribute to its temperature rise. The absorption efficiency is here taken as 0.45 [15]. Thus the energy of 340 J is deposited on an area of 0.07 cm2 in 1.7 s.
,°°I :.P'~'
'
300
¢:3mm
l[i~iT~/A
E =~i
~
o~ 2 0 0
m
experimen,l ol result at A ol B
I00
0
v ''W~ ~ i
t
i
i
i
i
1
3
4
5
6
7
2
Time
[see)
Fig. 7. Temperature change at points of plate subjected to heat flux.
04;
,:o 1.5~m]r
heol flux
400
0,5
liO
1.5 'r Cmm]
~..:-.....;.: oolllll
o4; o'5 ,1o
II
,
r
[mm]
_.04' 0
--2 7.-- ~ /
i
0 t'~ ~
'
0.5
1.0
1.5 r [ram]
.
100
~\ t ~.~
i
#i
t
2
/#3
4
~.'/
5
,
I
6
?
•
Time [sec]
-200
l = 1.8 (sec)
l : 1.2 (sec)
l O.o.~lclll 1 ,1 1
0.0
-:500
~ra302I"
-400
o
'5
-..-e-.-
¢'=3'0mm]H lil/I
~, 1oo
-
1.5
t
200
tm .2~
['m~]o 2
experimentol t e s u l l eloslic analysis (lemperature dependence of moteriol properties is not considered) elosloploslic onolysis (lemperolure dependence of moleriol properties
300
1=1.7 (see)
!°.°:i~!~l t I 1 1
0.4
- --=--
I . o41 I L, li 1 I l
t=0.8 (see)
1°21
In fig. 7 experimental and numerical results of the temperature change are compared. The temperature was measured at two points, point "A" on the top surface and point B on the bottom surface as indicated in the figure. Since A is very close to the heated region, the temperature at this point immediately follows the change of the beam power. The effect of evaporation is not taken into account in the numerical analysis of the data. This is justified by the fact that the experiment was carried out under atmospheric environment and thus the evaporation of metal is significantly suppressed. The plasma disruption, however, takes place in vacuum and the evaporation of first wall material is not suppressed.
I=1.5 (sec)
I=0.4 (see)
..... I Ill
3. 2. Experimental results
1.0
[mm]
1.5"r [mrrd
Fig. 8. Change of melting zone size (numerical result).
-500 Fig. 9. Thermal strain change at points of plate subjected to heat flux.
H. Hashizume et aL / Thermomechanical behavior of the first wall
148
Therefore the effect of evaporation is included in analyses of the plasma disruptions as shown later. In fig. 8 the progress of the melting zone with time is shown. The boundary between liquid and solid is not smooth since the size of an element is not small enough compared to that of the zone. The size is maximum at t = 1.7 s and resolidification of it is completed in a very short time: 0.1 s. The maximum depth is about 180 pro. In fig. 9 changes of radial strains at a rear point C 5.5 mm apart from the center line are shown. The measured strain is represented by a solid line. The results of the elastic and elastoplastic analyses are shown by triangles and circles, respectively. There is no remarkable difference between these. The agreement between the experiment and the analysis is very good up to 1.5 s. The large difference after 2 s could probably be attributed to a difference in the boundary condition, the neglection of the strain rate dependence of yield stress and the discrepancy between measured and calculated points. But the true reason for the difference is not clear at present. A similar comparison of tangential strain is shown in fig. 10. The strain is compressive for some time after the beginning of the heat input and becomes tensile roughly in 2 s. Peaks of the tangential strain are observed between 3 and 4 s. It should be noted that the resolidifi-
600
,,,(""- - ' - ' - ~
500
/
/~"'""*'. ""
4OO3oo
/
3 R
I00 0
'
2
-tOO I "" t~ / [200 _3001_
\~..
I
I
3
4
5
6
7
Time (see] -experimentolresull -'-,~--- elastic analysis(temperature dependenceof material properties is no! considered) ---.e-.- elastoploslicanalysis(temperature dependenceof material properties is considered)
Fig. 10. Thermal strain change at points of plate subjected to heat flux.
q1 500
/
I
/
]
0L
' ~ I 7lsec)
400 ~
~a
300
--
b~
ioo
\
~
~, t=70.
\ b\
t=00$ [sec] t:03 [sec] t = I 7 [sec]
-----o-- t : 2 0 [sec] ----o-- t : 7 0 [sec)
\
^
~
o
elapsed time " --~
, q 01mm f Ill ,
suaoce
_\_
m~lted ~l= 2,0
i "k2 ~ 6__ 8 I \\ ~ t=005 -100[ V ~ . / / ~ _2ooI_
-3001~ /
~o La~a.on ~C~mJ
s-'~
'
/,~oy./
~
~ ~ t
lit=03
Fig. 11. Change of thermal stress distribution of stainless steel.
cation of the melted zone is completed in at least 0.1 s after completion of preheating. The temperature changes continue until 10 s. In fig. 11 distributions of radial stress at various times are shown. The stress is taken on the line A - B 0.1 mm below the top surface. A location near point A does not melt at 0.05 s, while heating continues, but melts in 0.3 s resulting in zero stress. The stress distributions are almost the same for the elapsed times of 0.3 and 1.7 s. However, when resolidification is complete and the temperature decreases, the stress in the melted region becomes tensile due to contraction of the portion. The stress distribution at 7.0 s may be considered as a residual stress distribution. Thus the maximum tensile stress of 42 kg/mm 2 remains. This high residual stress could cause microcracks around the melted region. In many cases hair cracks were observed in a metallurgically polished specimen. Initiation of hair cracks during the process of resolidification is very hazardous for the structural integrity of the first wall subjected to a large plasma disruption. It is crucial to find some engineering solution to prevent cracking but this seems very difficult.
4. Numerical analysis of plasma disruption Two kinds of energy, magnetic and thermal, are deposited in the first wall and the plasma interactive components on plasma disruption. It is important to
tl. Hashizume et aL / Thermomechanical behavior of the first wall
know the amount of energy deposited in the wall as well as the disruption time. According to recent experimental results the disruption time becomes longer as the size of the plasma grows. A longer disruption time can reduce the design requirements because the amount of eddy current is reduced and the melting zone size is restricted. Therefore, the amount of deposited heat and the length of the disruption time are key parameters which have a strong effect on the melting phenomenon. In the following calculation disruption times of 10 to 100 ms are considered. The deposited energy is varied from 200 to 1000 J / c m 2. In fig. 12 the dependence of the evaporation thickness on the deposited energy is shown. It is evident from the results that evaporation thickness increases with the deposited energy and decrease of disruption time. When the disruption time is longer, the temperature rise can be suppressed because of heat diffusion. The evaporation thickness is 70 #m for a disruption time of 10 ms and 12 #m for 100 ms. A difference of 70 /~m and 10/~m is influential in the design specification of the first wall from the viewpoint of reducing its thickness. For example, the assumption that the disruption of 10 ms and 800 J / c m 2 takes place 100 times during the lifetime of the first wall will mean erosion of the surface of the wall by 7 mm in the case of stainless steel. The thickness of the first wall of a fusion power reactor has to be less than 10 ram, for else a very large thermal stress will be induced resulting in a shorter lifetime of the wall due to fatigue and creep damages. The evaporation thickness of 1 #m per disruption could be allowed if the number is 1000 or less. Therefore, it is very important to obtain a data base on the evaporation
2000
disruption time ~, I 0 [msec] I o 20 presenl result o 50 ] v IO0 I0 l ref from [4,13] ..... 20
149
thickness as function of the disruption time and the deposited energy. In fig. 12 numerical results from Hassanein [4,13] are shown to compare with the present results. SoLid and chain lines show the evaporation thickness for a disruption time of 10 ms and 20 ms, respectively. The agreement is quite excellent as shown in the figure. In fig. 13 the melting thickness is shown as a function of the deposited energy with as parameter the disruption time. Results obtained by Hassanein [4,13] are shown with solid and dotted curves for comparison. In this case, the agreement between both results is not good. This may be due to neglect of the receding surface caused by evaporation in the present analysis. As can be seen in fig. 12, the thermal energy consumed in evaporation becomes greater as the disruption time becomes shorter. This is a primary reason for the fact that the melting thickness for the disruption of 10 ms is the lowest. However, the disruption time of 100 ms is smaller than that of 50 ms. Therefore, in order to understand this tendency, the ratio of heat consumed in evaporation and diffused toward a low temperature region should be taken into account. The heat consumed for evaporation is a decreasing function of the disruption time while the heat diffused away from the surface is an increasing function of it. The total energy deposited is partitioned into two types and a maximum value of the melting depth is expected to occur. In fig. 14 the melting depth and the evaporated thickness as function of the disruption time are shown. It is quite clear that the evaporated thickness is linearly decreased in a semi-log plot while the melting depth shows a peak around 50 ms. The comparatively shorter disruption
time
I0 [msec]
o a v
500
20 50 I00
" • •
tO
•
- -
400
- - - 20
"]
t present
•
r6|. from ~.131
J" ...o-
~. 3oo
I00,0 =
50.0
f ~
.j/
i,~
IO0
200
.~" / 0
~.0
~---~.~
100
I
400
F
I
,
I
j
600 800 I000 Deposited energy [J/cm z]
Fig. 12. Relation between evaporated thickness and deposited energy on major plasma disruption.
I
400
i
I
600
Oeposiled
i
I
800 energy
i
I
i
I000 [ J / c m z]
Fig: 13. Relation between melting thickness and deposited energy on major plasma disruption.
150
H. Hashizume et aL
/ Thermomechanical behaoior of the first wall 1:4 (msec)
70
350 Xx
300
6O
xxx
Z
[#) 200 x
250
50 =t \\\
20O
40
X\
t
0'5
4006
~,
100
I=8 (msec)
!
~x
I0
2
r0
~
'
J
30 40 50
Disruplion lime
~
100
05
I0
15 r
~
l0
0
o
o 0.5
[mini 1:20 (reset)
ol I 1 I ly,×,~.,~ I 1 1
I. ~..../,,:,,/ • . :.. . ".'." V,/ ,, I:. .....;;y:.<.-/-~ t,,,ool> ...~;~.:: :~.S-~: ;If
t,,, o o ~
\x
4000
[Fnrn]
I
50
0
1.5 r
Ii0
30 xx
{msec)
t ~ 2oo "::;.%:ii.- ;:/ :::~
az:
~ 150
1 = 16
01!.!I I I I !
o1111111
~
1.0
I.SEr]mm 4000
;
I=12 (msec)
0
0.5
"::;1.0.....1.5 -
,
r
Cram]
I=24 (msec)
(msec) z
Fig. 14. Effect of disruption time on melting depth and evaporated thickness.
,
~"~ ~oo ;'!~< 400
c, ~oo
0.5
1.0
1.5 r
.
.
[mm]
disruption is more severe since the loss of material due to evaporation and the melting depth are large. In comparison the disruption time of 10 ms, that of 100 ms is milder with regard to the evaporation thickness. In fig. 15 the progress of the melting zone for a heat input of 800 J / c m 2 lasting for 20 ms is shown. There are 3 x 2 = 6 elements in a region of 1.5 mm X 0.4 mm in the figure. However, though mesh division is coarse, nine evaluating points for numerical integration are included in each element. Thus it is possible to determine whether a smaller region in the element melts or not. It goes without saying that the melting develops first in the radial direction along the surface and then in the direction of plate thickness. As assumed in the calculation heat deposition ceases at the time of 20 ms of course a large m o u n t of energy is conserved in the melted region at that time. The energy diffuses to the neighboring region through liquid-solid interface indicaring the possibility that further development of the melted region takes place. A precise examination of the numerical results does not show a visible development at all, indicating that the energy stored in the melted region is not enough to exceed the latent heat of the neigfiboring element. The maximum melting depth of 250/xm is achieved at t = 20 ms. In fig. 16 the relation between melting depth and deposited energy with the disruption time as parameter is shown. In the disruption time range of 4 to 20 ms, the melting depth is almost independent of the disruption time but strongly depends on the deposited energy. This
1:5 'r
[mini
SUS, 0=800 [Jlcm 2] Td =20 [msec]
Fig. 15. Change of melted zone due to plasma disruption of Q = 800 J / c m 2 and Td = 20 ms.
is observed clearly as the threshold energy for melting, which is about 200 J / c m 2 for stainless steel. This is very important information for the thermomechanical design
30O ff
o
Td (msec) 4
200 --x--
B
E o
~ 100 6 I f
00 --o~,~
~
500
1000
0ep0siled energy (d/cm 2 I Fig. 16. Relation between melted depth and deposited energy with as parameter disruption time.
151
H. Hashizume et aL / Thermomechanical behavior of the first wall
of the first wall, from which it is probably possible to allow heat deposition below the threshold value many times if fatigue damage due to cyclic thermal stress is neglected. Other important information seen in the figure is that the melting depth does not increase significantly for energy deposition over 500 J / c m 2. As stated previously, this is due primarly to the energy consumption in the evaporation and this is remarkable for the shorter plasma disruption. This situation may be different for materials such as graphite that are easy to evaporate or sublimate. In fig. 17 distributions of radial stress with elapsed time as parameter are shown. The stress on the line A - B 0.1 mm underneath the surface is plotted. Similar to the distribution in fig. 11, the stress state is compressive in the whole region at the beginning of heat deposition. The stress state reverses to a tensile one in the melted region at a time of at least 80 ms. The maximum residual stress is about 250 MPa in this case and it is smaller compared to the maximum value in fig. 11. In general, the residual stress is higher in a larger melted region larger because the amount of contraction is proportit)nal to the size. Compared with the case of fig. 17 the deposited energy is very large in the melting experiment. This is a primary reason for the difference of the maximum residual stress. Although it is not verified systematically, it is possible to consider that cracking could take place in the case of a large amount of heat input with high residual stress (larger than 400 MPa),
but not in the case of residual stress less than 300 MP& Information on the possibility of cracking is very important for the structural integrity of the first wall and in the near future systematic experiments with regard to the criterion of cracking are necessary. There are many cases where coated materials are used for the first wall. A typical example of this is TiC coated molybdenum. This has been used for the first wall of JT-60. Thus it is important to know the thermomechanical behavior of the TiC coated Mo when i t i s subjected to plasma disruption. In fig. 18 the surface temperature changes of TiC, Mo and TiC coated Mo are shown. Deposited heat and disruption time are 200 J / c m 2 and 1 ms, respectively. The temperature of TiC coated Mo is between those of TiC and Mo. The maximum temperature is about 4200 ° C for all the materials, where these melt on their surfaces. The temperature starts to drop rapidly at the time of 1.0 ms when the heat input ceases. In fig. 19 the development of the melting region for Mo is shown. The maximum depth is 70 # m that is achieved at t = 1.0 ms. Fig. 20 shows the development of the melting region for TiC. A maximum depth of 70 # m is achieved; this is the same as for Mo.
5000]
4000
q
, 300 -
.'--"' /,~/~
.,oo=:m:c,
'm°
i 20 [msec] --.o..- 80 [mse¢]
~
200
~
T.m.
(TIC)
3000 . . . . . . . . . . . . , ,',~
(Moli . . . . . . . . . . . . . . . . . . . . .
"---
lO0
i
.
.
.
.
.
,,o.
-I00
• TiC ,T.m.:3067"C,
~' 20001 y / ~ I/I
. . . .
TiC+Mo
m -200
'°°° rl/
,
,
,
,
,
0.2
0.4
0.6
0.8
1.0
,~
-300
Q=800J/cm2 , Td: 20 msec
Fig. 17. Change of thermal stress distribution in SUS first wall due to plasma disruption.
O0
Elopsed lime [msec]
Fig. 18. Change of surface temperature.
1.2
H. Hashizume et aL / Thermomechanical behavior of the first wall
152 I=0.4 (reset)
I=0.4
I=0.8 (reset)
(msec)
o¢,,,,,, I
oIIJllll
TiC No
1 ~]
ol
I=0.8 lmsec)
fill
L:;';~';'//"/~~ J ,
.... , ,,..:,/. ,:,,×
[~150 [
1001 000
0~.5
JO
1.5'r [ram}
'00;
0'3
1=0.6 (msec)
I=1.0
ollllllll [/:250
1
IO0 •
0
1'.0
1.5"r [mm}
~
~
1.0
.
1.5 r
Cram]
I
1.0
.
1.5 r
Cmm]
0
0.5
1.5 r
L0
tram]
t : 1.0(msec)
[/1150I
000~
0.5
.0
1.5 r
,00;
0'.5
L'O
1.5"r
[mrn]
Cmm] e
+
e
100 ~ 0
0.5
1.5 r
1.0
[mm]
•
TiClrn.p.306? C) Mo(m.p.2622 C), O:200[J/cm2], Td=ICmsec3
O:200[Jlcm 2], Td=l[msec]
Mo,
i 0.5
I=0.6 (reset)
(msec)
[/050
0.5
0
Fig. 19. Change of melted zone due to plasma disruption.
Fig. 21. Change of melted zone due to plasma disruption.
In fig. 21 the development of the melting region of TiC coated Mo is shown. The melting behavior of TiC is different from that of Mo. A portion near the surface is melted for TiC and Mo at t = 0.4 ms. However, TiC coated Mo is not melted at this time. This may be due to a difference in the m o u n t of heat conducted through the interface between both materials. It is also interesting to observe an unmelted region in TiC where a larger region of Mo is melted. This is due to the difference in t h e malting temperature of both materials. From this
fact it is possible to predict that radical internal evaporation could take place and blow off the surface material if much more energy is deposited in a short time and in a localized way. The blow-off phenomenon may be related not only to the internal vaporization to result in high pressure but also to thermal shock. The phenomenon seen in graphite is primarily due to brittle cracking caused by the thermal shock because the graphite is not melted but sublimates or evaporates. However, in the case of TiC coated Mo, there is a possibility of blow off due to a sudden increase of the internal pressure. This kind of blow-off phenomenon has been observed in JET [16]. Not only plasma disruption but also arcing between the plasma and the first wall has a possibility of causing the blow-off of material. Study of this phenomenon has not been undertaken up to now but it is crucial for the engineering feasibility of a fusion reactor. Therefore we strongly urge that such an experiment or theoretical study be carried out to reveal the phenomenon quantitatively and establish a method of protection. Numerical results, some of Which are compared to experimental results to verify the validity Of the computer code, demonstrate that residual stress achieved after the resolidificafion reaches the yield stress level of the first wall material and could cause micro-crackings near the boundary of the resolidified region. Another important result is that the threshold energy for melting seems to be about 200 J / c m 2 for stainless steel. The plasma disruption, thus, can be allowed to take place if its heat deposition is below the threshold value.
I=0.4 (msec)
LL.L,I
[/~1~ i00 ~ 0
I= 0 . 8
1
r
0.5
f 1.0
I00 ~ 0
1,5 r [mm]
I=0.6 (msec)
,
0.5
~
1.0
~
,
1.5 r [ram]
l= 1.0 {msec )
ollLlll|
1
"27 " × " / z
"~
t~ 5o"';"×" " ~'×
[la]50I I00 ~ 0
(msec)
0i I I I l J I 1
"
~
0.5
1.0
,4,
,0%
o'.5
Cmm] TiC,
0:200
[dtcmZ],
,Jo
,.5, Cram]
Td:l[msec]
Fig. 20. Change of melted zone due to plasma disruption.
H. Hashizume et al. / Thermomechanical behavior of the first wall
153
5. C o n c l u s i o n s
Heat of vaporization L~ [J/kg]
Conclusions obtained from the present study are summarized as follows: (1) The computer code was made based on F E M which enables us to perform analyses of malting, evaporation, solidification, temperature distribution and elastoplastic stress-strain. Its validity was verified through the experiment. (2) The amount of evaporated mass increases with decrease of the disruption time. (3) There exists a threshold energy for initiation of melting, which is about 200 J / e r a 2 for stainless steel. (4) Thermal stress taking place during the resolidification process is large and may cause small cracks in the melted region. (5) Further study is required to reveal the blow-off phenomenon seen in the existing facilities.
L v = 7.46 x 106. Young's modulus E [Pa] E = 2.23 x 10 n - 8.09 x 107T. Poisson's ratio v vffi 0.3.
Yield stress o v [Pa] o v = 4.82 x 10 s - 3.00 X 105T. Strain hardening rate H ' H ' = 5.0 x 109 - d H ' / d T x
where d H ' / d T = d E / d T x H ' / E .
Appendix
Molybdenum
Stainless steel
Heat capacity c [ J / k g K]
Heat capacity c [ J / k g K] c s = 3.04 × 102 + 6.70 x 1 0 - 1 T - 4.78 × 1 0 - 4 T 2 + 1.27 x 1 0 - 7 T 3,
T,
cs = 4.15 × 102 - 1.68 x 1 0 - 1 T + 7.42 × 1 0 - S T 2, ct = 4.23 X 10 2. Density p [ k g / m 2]
c I = 7.12 x 102.
Ps.t = 1.03 x 104 - 2.4 × 10-1T.
Density p [ k g / m 3]
Thermal diffusivity a [n~/s]
Ps = 8.03 x 103 - 4.21 x 1 0 - 1 T - 3.89 x 1 0 - 5 T 2,
a s = 5.00 x 10 -5 - 1.10 × 10-ST,
pt = 7.43 x 103 + 2.93 × 1 0 - 2 T - 1.80 x 1 0 - 5 T 2.
a t = 1.75 X 10 -5 - 3.00 X 10-1°T.
Thermal diffusivity a [m2/s]
Melting point Tm [K]
a s = 3.87 × 10 -6 + 4.90 × 1 0 - 9 T - 4.17 × 10-13T2,
Tm -- 2895.
a t = 5.80 × 10 -6.
Heat of fusion L [J/kg]
Melting point Tm [K]
L -- 2.90 × 105.
Tm = 1700. Heat of fusion L [J/kg]
Vapor pressure Ps [Pa] Ps = exp(26.1 - 7.19 × 1 0 4 / T ) .
L -- 2.70 × 105. heat of vaporization L v (J/kg] Vapor pressure Ps [Pal Ps = exp(25.6 - 4.35 x 1 0 4 / T ) .
L v = 6.18 × 106.
154
H. Hashizume et al. / Thermomechanical behavior of the first wall
T/C
Heat capacity c [ J / k g K] c s = 8.02 × 102 + 6.20 × 10-2T, ct-- 1.12 × 103. • Density p [ k g / m 3] Ps./-- 4.60 x 103. Thermal diffusivity a [m2/s] a s -- 8.18 x 10 -6 + 6.83 x 10-ST, a I = 1.05 × 10 -5. Melting point Tm [K] Tm = 3340. Heat of fusion L [J/kg] L = 3.88 × 105. Vapor pressure Ps [Pa] Ps = exp(2.91 - 7.74 x 1 0 4 / T ) . Heat of vaporization L v [J/kg] L v = 7.92 × 106.
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