Parametric calculations of the first wall thermal response during plasma disruptions for the net

285

Journal of Nuclear Materials 155-157 (1988) 285-289 North-Holland, Amsterdam

PARAMETRIC CALCULATIONS OF THE FIRST WALL PLASMA DISRUPTIONS FOR THE NET E. FRANCONI Associazione

THERMAL

RESPONSE

DURING

and H. KROEGLER

EURA TOM-ENEA

sulla Fwone,

CRE Frascati. C.P. 65-00044-FrascatI,

Rome, Irn!v

The expected high energy fluxes on the first wall during plasma disruptions make it necessary to evaluate the thermal response of the first wall material. A model to predict material evaporation and melting, even considering the vapor shield effect, was developed assuming that the plasma ions and electrons are slowed down by the vapor. The model takes into account both the changes, solid-liquid and the liquid-vapor of the wall material. The dynamics of the plasma collapse is treated either as a function of time or as a constant during the disruptive event. The time evolution of the shielding effect is given as a function of the thermal energy released by the plasma. Specific results obtained by using the numerical values as working hypotheses for the NET project are given for AISI 316SS and graphite impregnated with Sic.

1. Introduction

variation is obtained method [4]:

The concept of wall protection from hard disruptions was first proposed by Sestero [l] and called oirtual limiter which is a dense cold high-Z gas blanket intercepting the particle flux escaping from the plasma and radiating isotropically away from the absorbed energy. As, to date, the tokamak discharges are never free from MHD instabilities which can cause minor and major disruptions accompanied by a partial or total energy loss and plasma confinement, and as neither the theory nor the experimental observations allow a prediction of the disruption phenomena, it is only possible to make a parametric study of the thermic load on the first wall due to the disruptions. The NET disruption scenario [2] assumes as a working hypothesis a fast energy quench phase with a duration of 2 ms and a slow current decay phase with a duration of 20 ms. In both cases a nonuniform energy deposition is assumed resulting in a maximum heat load up to 1000 J/cm’. The important aspects to characterize a plasma disruption are the duration T and the time shape of the heat pulse. A simple analysis assumes a constant heat flux; a more realistic model proposed by Onega et al. [3] is based on an expansion of the plasma with constant velocity u = u/r and a parabolic shape of the plasma temperature and density profile. The heat pulse is given

by the

Laplace

t) =2q,/K[(Dt/m)1’2

AT(x,

-x/2

transformation

exp(-x/4Dt)

erfcx/&]

(2)

Eq. (2) does not consider the internal heat generation rate due to eddy currents. As the proposed model includes two phase changes and temperature-depending material properties, an analytic solution is excluded. 2. The vapor shield concept Considering thesolid wall as a container of vapor molecules, the number of particles leaving the surface under a solid angle 2m about a point can be expressed by the evaporation flux J,” = (I/4)an(u),

(3)

where a is an evaporation coefficient, n is the vapor density and (u) is the mean value of the molecular speed assuming a Maxwell velocity distribution. During the evaporation process the evaporating molecules are partly backscattered by the molecule evaporated earlier

1.0 -

a, 9,

by

qO(t) = 3kTon,2nlou(l

- (1 - t/r)2)2(1

- t/T),

(1)

where T, and “9 are the temperature and the density in the plasma center, I is the length of the plasma column (axis) and a is the minor plasma radius. The first wall invested by a major disruption suffers a surface heat deposition from electrons and ions and a volumetric joule heating by eddy currents. The heat conduction problem is governed by the well-known Fourier equation which can be reduced for a one-dimensional geometry and resolved with the given boundary conditions. The resulting surface temperature 0022-3115/88/%03.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)

B.V.

05 :: 22 -12 =rb z#z=9 ZI

0000

0 I

0 0

1

10

5

tlma(ms) @

Fig. 1.

J/cm*

@

800J/cmz

@

1200 J/cm2

@

600J/cm*

@

1000 J/cm2

1400

Normalized heat flux qs deposited on the surface in function of time. Peaking factor 3, material AISI 316 SS.

_

286

E. Francow, H. Kroegler / Calculations of the first wall response

forming the vapor layer in front of the wall. The result is given by the net evaporation flux in the following expression J

NET

= 0

.lP,,,/~~

)

cm2/g] vapor:

of a particle

pdE/dx=

passing

a layer

dx[g/cm’]

of

(2se2/E,)(mP/mE)NA(Z/A)ln(E,~/I).

(4)

(5)

obtained by a very crude evaporation model, but sufficient accurate to treat our problem. There are more elaborated solutions, such as the one proposed by Anisimov in his paper [.5] which features correct hydrodynamic boundary conditions on the metal surface and allows the solution of the solid-phase thermal conductivity problem to be related to the gas dynamic problem of vapor motion.

where: mp is the mass of the incident particle: mE is the electron mass; e is the electron charge; nE is the electron density of the vapor; v is the velocity of the incident particle; I is the mean value of the ionization potential of the vapor atoms; Z is the atomic number of the vapor atoms; A is the atomic weight of the E, is the thermal vapor; NA is Avogadro’s number; energy of the plasma particle. The total energy lost by a particle in the vapor results as a function of time

2.1. The plasma vapor interaction

W,(At)

= gJr2JNET

dr.

*I Assuming that all electric and magnetic field related effects are negligible, the plasma-vapor interaction process can be reduced to a problem of energy loss of charged particles in matter. The stopping power of a charged particle is given by the well-known Bethe formula (61 as the energy loss [eV

Based on the hypothesis that the vapor cloud can be considered as a blackbody, the whole assorbed energy is released by radiation. The energy flux coming from the plasma is q,, = EqN/A,r, where

N is the total number

(7) of particles

involved

and

r=2ms

qO=cOstanf 100 J/cm2

200 J/cm* 400 J /cm2 600 J /cm2

200 J/cm* 400 600 600 1000

600 J/cm2 1000 J/cm2 1200 J/cm* 1400 J/cm2

J/ cm2 J I cm* J / cm2 J / cm*

1600 J / cm2 I 0

I 0.5

I1

1 1.0

I 1.5

I1

I

2.0

2.5

time Ims) Fig. 2. Surface temperature in function of time for different energy densities, heat flux constant, surface material AISI 316 SS.

Fig. 3. Surface temperature in function of time for different energy densities, heat flux variable eq. (1) material AK1 316 SS.

281

E. Franconi, H. Kroegler / Calculations of the first wall response A,

is the first wall area hidden by the flux. The energy

flux stopped

in the vapor is

qv = WvN/A,r, and

the part

becomes

(8) of the

flux

just the difference

Assuming proximation

through

the vapor

q0 - qv.

now a neutral is to replace

passing D-T

plasma,

the different

surface is crucial for the efficiency of the virtual limiter concept. If we suppose a uniform deposition over all the first wall, the vapor layer cannot redistribute the heat flow and no attenuation will occur. The normalized eq.(9) is plotted in fig. 1 for different values of incident energy densities q0 and as a function of time.

a useful ap-

ionic species with

average ion which has an weight of 2.5 times the proton mass. The total heat flux qs impinging on the first wall is the sum of the attenuated electron flux, the ion heat flux, and the irradiated heat flux by the vapor layer due to the energy deposited in the vapor by the electrons and ions which gives finally

3. Computational tools

a single

4s = ‘W,[2

+ (Ad&t

-

1)(Wv,+

(9)

~V,VJ%l~

is the total area of the first wall, WV, is the energy lost by the electrons and WV, is the energy lost by the ions in the vapor layer. For NET we could expect and energy deposition on the top, the bottom, and the inboard region of the first wall with a peaking factor of about 3 which means AH/A,,,, = 0.33. The fact that the main part of the thermal energy will be deposited on a limited part of the first wall where A,,,

One-dimensional geometry. To calculate the temperature history of the first wall during the disruption, a finite difference equation substitutes the general Fourier heat transfer equation. For this purpose the HEATING6 code [7] is used. In this code the thermal properties of the materials may be temperature-dependent and may undergo a change of phase such as solid-liquid. The expected phase change liquid-vapor is included by adding a subroutine. The vapor is treated in the same way, based on eq. (9). The initial temperature distribution is obtained by a steady-state calculation with a total front surface heat flux during the plasma bum (before the disruption) of 13.6. lo4 W/m3 and a heat transport coefficient of 2.8 W/cm2 by water cooling (T = 200 o C) on the rear surface of the lOmm-thick wall. The internal

400 J / cm* 600 J / cm* 800 J / cm2 IO00 J /cm* 1200 J /cm*

1400 J /cm* 1600 J

qo=600 J/ ,m2

/cm*

5

0

5

10

15

20 tlmc

25 (ms)

Fig. 4. Surface temperature in function of time for different energy densities, heat flux variable eq. (1) material graphite/ Sic-composite.

Fig. 5. Surface temperature considering the vapor shield (lower curve) and without vapor shield (upper curve): material AK1 316 SS, - - - graphite/Sic-composite.

E. Franconi, H. Kroegler / Calculatmzs of the firs! wall response

288

heat generation caused by eddy currents is not considered because it contributes only 1% to the total change of the wall temperature produced by the surface flux. The plasma is assumed to be neutral and composed of deuterons and tritons with a temperature of 10 keV. Cylindrical coordinates. To check the validity of the theoretical model a comparision between the melted zone produced in a material probe by an electron beam and the calculating melting zone is proposed. The beam and the probe are simulated by cylindrical coordinates; the thermal energy of the electrons of the beam is 30 keV. The profile of the energy density of the beam is assumed axial symmetric with a Gaussian distribution. Contrary to the semi-infinite model, where all evaporated molecules contribute to the shielding layer, in the cylindrical geometry only the vapor remaining in the beam diameter contributes to the attenuation of the electrons.

4. Results The surface temperature evolution less steel and graphite/Sic-composite

of AN1316 stain(Schunk & EBE

3000

Ts,,,,

SiC30) as a function of time for different energy densities and for different pulse shapes is shown in figs. 2. 3, 4. All plots comprise the vapor shielding. The thermal properties such as thermal conductivity, density, and specific heat of Sic30 materials are not well known so far, hence, it follows on the Sic30 results persist some indetermination. The initial temperature is 362 o C resulting from the steady-state calculation. Fig. 5 is an example to evidence the vapor shielding effect. The maximum surface temperature of graphite material is about 20% higher than that of AISI contrary to the higher thermal conductivity of Sic30 (AISI: K = 9.2 W/m K. Sic: K = 125 W/m K). This can be explained by the smaller shielding efficiency due to the lower Z and the higher latent heat of SiC30. Fig. 6 shows the evaporated and melted layer thickness for different disruption energy densities. The effect of vapor shielding is well evidenced. Fig. 7 shows a comparison between the experimental melting zone thickness and the results obtained from the numerical model assuming the same conditions as in the experimental work [8]. The cooling rate of the layers. determined by measuring the interdendritic distance of the second order dendrites. results as lo4 K/s in the molten zone and may reach 600

(“c)

2500

400

800

1600

1200 J/cm2

Fig. 6. Total evaporated and melted layer from AISI 316 SS and SIC first walls as a function of disruption energy densities: with vapor shield. without vapor shield, -

ENERGY

DENSITY

(J/cd)

Fig. 7. Comparison between the experimental melting layer thickness and the calculated one for different energy densities.

E. Franconi, H. Kroegler / Calculations of the first wall response BEAM 04

0

08

RADIUS

can give rise to high temperature phase transformation near the irradiated surface of the wall. Finally, in fig. 8 the result of the calculation is plotted in cylindrical coordinates evidencing the melting zone shape due to an electron beam with a Gaussian density profile.

(mm) 12

289

16

References

600

1

I

zone due to an electron beam for different energy densities, surface material AISI 316 SS.

Fig. 8. Shape of the melted

lo5 K/s in the shallow layer. The numerical simulation gives a cooling rate from 6.25 X lo4 to 1.6 x 10’ K/s which is well in agreement with the experimental values. It means that high cooling rates, starting from approximately 1300 o C up to melting point, can stabilize the S-ferrite phase down to room temperature, which

[l] A. Sestero, Nucl. Fusion 17 (1977) 115. [2] NET Team, NET Status Report (1985). [3] R.J. Onega, W.R. Becraft, C.A. Kukielka, Nucl. Sci. Eng. 75 (1980) 243. [4] H.W. Carslaw, J.C. Jaeger, Conduction of Heat in Solids (Clarendon, Oxford, 1959). [5] S.I. Anisimov, Sov. Phys. JETP 27 (1) (1968) 182. [6] H.A. Bethe and J. Ashkin, in: Experimental Nuclear Physics, Ed. E. Segre, Vol l., (Wiley, New York, 1953). [7] D.C. Elrod, G.E. Giles and W.D. Turner, HEATING6: A Multidimensional Heat Conduction Analysis with the Finite-Difference Formulation, NUREG/CR-0200, Vol. 2, Section f 10, ORNL/NUREG/CSD-2/V2. [8] F. Brossa, E. Franconi, P. Moretto and G. Rigon, in: Proc. 2nd Int. Conf. on Fusion Reactor Material, Chicago, 1986, J. Nucl. Mater. 141-143 (1986) 210.