Nucl. Fusion 52 (2012) 013013 (6pp)

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Tokamak plasma response to droplet spraying from melted plasma-facing components
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2012 Nucl. Fusion 52 013013
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IOP P UBLISHING and I NTERNATIONAL A TOMIC E NERGY A GENCY N UCLEAR F USION Nucl.Fusion52(2012)013013(6pp)doi:10.1088/0029-5515/52/1/013013
Tokamak plasma response to droplet spraying from melted plasma-facing components
M.Z.Tokar1,J.W.Coenen1,V.Philipps1,Y.Ueda2and
the TEXTOR Team1
1Institute for Energy and Climate Research-Plasma Physics,Forschungszentrum J¨u lich
GmbH,Association FZJ-Euratom,52425,J¨u lich,Germany
2Graduate School of Engineering,Osaka University,Osaka565-0871,Japan
Received28July2011,accepted for publication22November2011
Published23December2011
Online at /NF/52/013013
Abstract
High-Z materials such as tungsten are currently the potentially best candidates for plasma-facing components(PFCs) in future fusion devices.However,the threat of melting under uncontrolled conditions and the associated material redistribution and loss can place strict limits on the lifetime of PFCs and plasma operation conditions.In particular, material losses in the form offine sprayed droplets can provide a very intensive source of impurities in the plasma core.In this paper,the plasma response to radiation losses from impurity particles produced by droplet evaporation is modelled for the conditions found in the tokamak TEXTOR.The interplay between tungsten spraying and plasma behaviour,resulting in the reduction of power transferred to the limiter and diminution of droplet production,is taken into account.Calculations predict,in agreement with experimental observations,that this evolution results in a new steady state with significantly reduced central temperature and peaked impurity radiation profile.The efficiency of melt conversion into droplets,estimated by comparing experimental and computed plasma temperatures,is in reasonable agreement with the predictions from models for droplet generation.
1.Introduction
In future fusion reactors such as ITER and DEMO tungsten will be widely used for the fabrication of plasma-facing components(PFCs)[1].Such PFCs are tested currently on tokamaks TEXTOR[2]and ASDEX[3]and have been installed in JET as a part of the ITER-like wall[4].It has been observed in these experiments that under high heat loads,in particular,during transient events like edge-localized modes (ELMs),some surface layers of a mm width can be melted and droplets of boiled metal are sprayed around[5–7].The evaporation of droplets inside the plasma leads to a dramatic, by several orders of magnitude,if compared with physical sputtering of PFC,increase of the source of high-Z impurity ions.Enhanced radiation losses on W ions decrease the power transported from the plasma to the PFC and this reduces the droplet production rate.Finally,a new quasi-stationary state can be achieved where the central temperature is reduced and a significant fraction of heating power is transported from the confined plasma volume with the radiation of high-Z impurity ions,as has been observed in TEXTOR.The main aim of this study is a numerical modelling of modifications in plasma parameters caused by spraying of metal droplets from the W plates mounted,thermally isolated,on a limiter in TEXTOR.On the one hand,this gives a possibility to evaluate characteristics of particle and energy transport under experimental conditions in question.On the other hand, since the radiation losses are dependent on the intensity of impurity sources,models proposed to describe the mechanism of droplet generation can be validated.The latter is of principle importance by forecasting the behavior of tungsten PFCs under reactor conditions.
The rest of the paper is organized as follows.Equations used to model the penetration and evaporation of droplets sprayed from the limiter,the sources and transport of impurity ions,recycling neutrals of the working gas and main plasma components,the plasma heat balance,boundary conditions and numerical approaches to solve these equations are presented and discussed in the next section.In section3,the results of calculations for the conditions of discharges in TEXTOR with W limiter plates are compared with the data of experimental observations.The deduced values for the efficiency of melt conversion into droplets are compared with predictions from models for droplet formation.Finally,conclusions are drawn.
2.Basic equations
2.1.Metal droplets and impurity ions
At high enough heat loads onto a PFC of tungsten its surface can be melted and small metal droplets are sprayed into
0029-5515/12/013013+06$33.001©2012IAEA,Vienna Printed in the UK&the USA
the plasma [5–7].These are localized in a cloud with dimensions of 1cm.For plasma temperatures above several electronvolts the heat conduction along the magnetic field is large enough to maintain it isothermic on such distances.Thus,in the cross-section of the cloud by a magnetic surface all droplets interact with the plasma of nearly the same density n and temperature T .Moreover,although the energy loss to droplets is localized both toroidally and poloidally,the plasma parameters inside the cloud are close to those averaged on the surface far from the droplet cloud.This can be demonstrated by balancing this loss with heat supplied by conduction along the magnetic field from distant regions on the same surface.The corresponding heat influx may be estimated by applying the results of [8]and one finds that the balance is maintained by a temperature variation on the surface noticeably below then 1%.Owing to the pressure equilibration the plasma density is also well flattened.
This allows us to proceed from the local density of droplets,being a function of all spatial coordinates,to their total number d in a cloud slice of a unit length in the radial direction r perpendicular to magnetic surfaces.This number depends on time t and r only and,since the droplets do not disappear before they are completely evaporated,obeys the conservation law:
d d d /d t =0(1)Her
e d d (···)/d t =∂(···)/∂t +∂(U d ···)/∂r ,where U d is
the average velocity of droplets in the radial direction r .Due to their small dimensions,of several µm ,and extremely high thermal conduction,the droplet temperature is constant throughout its volume and one can neglect a rocket force due to an inhomogeneity in the evaporation rate,see [7].Other forces discussed in [7]do not affect the droplet radial motion and we consider U d as constant during the whole life time of droplets.
The variation with t and r of the droplet radius ρd and temperature T d averaged over the cloud cross-sections can be computed from the particle and heat balances of the droplet material.These are used in an integral form governing the total number N d and thermal energy W d =3/2T d N d of atoms in slices of the droplet cloud by magnetic surfaces,independent of toroidal and poloidal coordinates:
d d N d /d t =−J d (2)d d W d /d t =Q d −L d .
(3)
Here J d = d S d j d is the outflow of atoms evaporated from
droplets with the averaged surface area S d =4πρ2
d ;th
e outflow density is given by the Clausius–Clapeyron law [9]:
j d =j d ,m T m
T d exp
E v T m −
E v T d (4)where j d ,m ≈4.83×1022m −2s −1is the j d -value at the melting
temperature T m and E v is the evaporation energy;the first term on the right-hand side (rhs)of heat balance equation,Q d = d S d γnc s T [10],is the heat flux convected from the plasma to the droplet surface,γ≈7.5the heat transmission coefficient and c s =√
2T /m i the sound speed of plasma ions with the mass m i ;the second term takes into account
the droplet cooling by atom evaporation and surface radiation,L d = d S d [j d (3T d /2+E v )+εW σT 4d ],with the greyness coefficient εW and Stefan–Boltzmann-constant σ;for tungsten T m =0.32eV ,E v =8.6eV ,and εW =0.347.
We take into account that N d = d V d n d ,with V d =4πρ3
d /3being th
e averaged droplet volume and n d the number density,not a mass one,o
f W atoms,measured in (m −3).This allows us to reduce equations (1)–(3)to explicit equations for ρd (t,r)and T d (t,r):
d d ρd /d t =−j d /n d
(5)
and
d d T d /d t =
γnc s T −j d E v −εW σT 4d /(n d ρd ).
(6)As boundary conditions to these equations we use the experimental values of ρd and T d at the last closed magnetic surface (LCMS),r =a ,distinguished by the limiter top,see [5,6];the boundary value of d is prescribed by the droplet influx d from the limiter, d (t,a)= d (t)/U d .Generally, d increases with the heat flux transferred to the melted limiter area S m ,Q m =S m sin ψγn(a)c s (a)T (a),with ψbeing the pitch-angle of the magnetic field with respect to the limiter surface.Existing models for the droplet generation from melt relate this phenomenon to metal boiling and hydrodynamic instabilities provoked by the growth and destruction of bubbles with saturated vapour [11,12].The maximum rate for generation of atoms filling such vapour bubbles is of Q m /E v ;however,only a small fraction α 1of the bubble content can be converted into droplets [12].Thus,we adopt d =αQ m /
V 0d n d E v (7)where V 0d is the averaged initial volume of droplets at the
moment when they are ejected into the confined plasma.In our calculations the conversion factor αis assumed as a free parameter.By comparing the experimental and computational results for the change of central plasma temperature caused by droplet spraying,the range of the most probable αvalues will be determined and contrasted with the droplet formation models [11,12].
Evaporated atoms move in the plasma chaotically with a thermal velocity v th and one can estimate their penetration
depth as λi ≈v th /(k W ion
n)with k W ion being the ionization rate coefficient for tungsten atoms.The droplet temperature does not exceed significantly the boiling one and for heavy metals such as W we get λi 1mm under the plasma conditions in question.This is by an order of magnitude less than a penetration depth of droplets of 1–2cm.Thus the outflow of atoms evaporated from droplets,J d ,provides directly the source density of impurity ions integrated over the magnetic surface,S I .
The surface averaged density of impurity ions in all charged states,n I ,is governed by the continuity equation:
∂n I ∂t +1r ∂∂r (r I )=S I
4π2rR
(8)
where R is the major radius of the magnetic surface and I is the surface-averaged density of the radial impurity ion flux.Henceforth it is assumed that the latter has both
2
diffusive and convective contributions, I=−D I∂n I/∂r+ V I n I,characterized by the diffusivity D I and pinch-velocity V I,respectively.At the LCMS the e-folding length for the density decay,δI,is prescribed:∂n I/∂r=−n I/δI.The latter is determined by the transport phenomena in the scrape-off layer(SOL)being out of scope of this paper.
2.2.Plasma parameters
On the one hand,the plasma parameters control the heatflux to droplets and intensity of their evaporation.On the other hand,the radiation of impurity ions produced from droplets affects the plasma state and reduces the heat outflow from the plasma volume to the limiter surface.This in turn results in the reduction of melt production.Thus,the behavior of droplets and plasma has to be described self-consistently.In this study we presume that the main plasma components,electrons and deuterons,have the identical density n and temperature T. This is a satisfactory approximation under the conditions of experiments with W limiter plates in TEXTOR[5,6]:on the one hand,the measured concentration of W ions does not exceed10−4and the plasma dilution by tungsten is well below 1%;on the other hand,electron–ion Coulomb collisions are often enough to maintain thermal equilibrium.Because of the homogeneity of plasma parameters on magnetic surfaces, see the previous section,one can average normally three-dimensional plasma transport equations over the toroidal and poloidal angles.This provides equations which describe the time evolution of the radial profiles of surface-averaged parameter values.In particular,the plasma temperature radial profile is governed by the following heat balance equation:
∂∂t (3nT)+
1
r
∂
∂r
(rq)=Q heat−Q rad−
Q d
4πrR
(9)
where q=−κ∂T/∂r+3 T is the density of the radial heat
flux with the heat conductionκand particleflux density ;
in the discharges in question,with the main contribution to
heating from the neutral beam injection,the radial profile of the
heating power density is well approximated by a Gaussian one,
Q heat(r)=Q heat(0)×exp(−r2/r2∗),with the maximum value
Q heat(0)=P heat/(2π2Rr2∗)/[1−exp(−a2/r2∗)]defined by the
heating power P heat and e-folding radius r∗;Q rad=nn I L I is the density of radiation energy losses with the cooling rate
L I(T)computed in corona approximation[13].This is a good
guess for heavy impurities in the plasma core where impacts
of transport on the charge distribution of impurity ions are
normally weak[14].Recent calculations with plane-wave
functions in a Born approximation for diverse levels in each
charge state[15]provide a cooling rate deviating less than by
20%from that found in[13].The last term in equation(9)is
the energy absorbed by droplets.
To solve equation(9)one has to know the density of
plasma particleflux involved in the convection contribution
to the heatflux density q.This contribution is particularly
important in TEXTOR,being a limiter machine,where
recycling neutrals of the working gas penetrate through the
SOL deeply into the confined plasma volume.As in the case
of impurity ions, is considered as compounded of diffusive
and convective contributions, =−D∂n/∂r+V n;the plasma
density n is governed by the continuity equation:
∂n
∂t
+
1
r
∂
∂r
(r )=k ion nn a.(10)
The source term in the rhs is due to ionization by electrons of recycling neutrals of the working gas,with the rate coefficient k ion being a function of electron temperature and density[16]. To integrate equation(10)we need the radial profile of the neutral density n a contained in the source term.This is determined by the equation:
∂n a
∂t
+
1
r
∂
∂r
(rj a)=−k ion nn a(11)
where the neutralflux density j a is computed in a diffusion approximation[17]
j a=−
T a
m a(k i+k cx)n
∂n a
∂r
.(12)
This approximation takes into account that the rate coefficient for the charge-exchange of neutrals with background ions, k cx,is noticeably larger than k ion.The neutral temperature T a involved in j a is calculated from heat balance equation:
3
2
n a
∂T a
∂t
+
3
2
j a
∂T a
∂r
=3
2
(T−T a)k cx nn a(13)
taking into account that owing to charge-exchange collisions cold primary neutrals,entering the plasma volume,acquire the energy of ions.
At the plasma axis,r=0,the densities and temperatures of charged particles and neutrals have zero derivatives.At the LCMS we prescribe the e-folding lengths for T and n,δT and δn,respectively,and the magnitudes of n a and T a.
2.3.Numerical approach
After discretization of the left-hand side(lhs)of equations(5) and(6)usingfinite differences,e.g.,
d dρ
d t
(t,r)≈
ρd(t,r)−ρd(t−τ,r)
τ
+U d
ρd(t,r+h)−ρd(t,r)
h
whereτand h are time and radial grid steps,respectively,one can get recurrent relations which allow us tofindρd(t,r)and T d(t,r)forfixed initial and boundary conditions.Stability of such a procedure depends,however,crucially on the approximations for the rhs.In the case of equation(6),a fully explicit scheme with the rhs computed for parameters at the previous time moment t−τbecomes unstable for extremely smallτ,which practically does not allow us to model the whole plasma evolution caused by droplet spraying.The reason for this behaviour is the extremely non-linear temperature dependence of the rhs in equation(6).These deficits of explicit approach can be eliminated to a certain extent using higher order schemes by discretizing partial derivatives withfinite differences,e.g.back differentiation formulae,see[18].Here we apply,however,a fully implicit approach with the rhs evaluated at T d(t,r).In this case,the discretized equation(6) results in a non-linear algebraic equation for T d(t,r).It is possible to prove that this equation has only one physically 3
01
2
Figure 1.Experimental radial profiles of the electron temperature in TEXTOR discharges 112060and 112062:before the initiation of W droplet spraying from a melted limiter (solid and dashed curves,respectively)and during the spraying stage (dotted and dashed–dotted curves,correspondingly).
reasonable positive solution and to estimate the upper limit for this.Finally,this equation is solved numerically by bisection or the Newton–Raphson method [19].This approach is somewhat similar to that used in [20]for integration of diffusion equations with a diffusivity dependent on gradient.
Diffusion-like equations (9)–(11)have been integrated by the method outlined in [21,22].The variable change proposed there is of principal importance to get the solution converged for every time moment and for describing the evolution with a sufficient accuracy for reasonably large time steps.This is not a trivial result for physical conditions under consideration fraught with instabilities,e.g.due to impurity radiation,see, e.g.,[23].The transport coefficients and boundary conditions have been chosen to reproduce reasonably the absolute magnitudes and shapes of the plasma parameters in a stationary state without droplets.Thus for the plasma density profile and the measured intensity of D α-signal satisfactory agreement has been achieved with the following choice:
D =A 0+A 1·r 2/a 2,V =−0.3rD/a 2
where A 0=0.2m 2s −1,A 1=0.8m 2s −1and n a (a)=2×1016m −3,δn =0.015m;the latter is in good agreement with the density measurements in the SOL [24].The same transport characteristics are adopted for impurity description.For plasma heat conduction κwe use a similar parabolic profile as above for the particle diffusivity but with the coefficients A 0=A 1=2×1019m −1s −1;δT =0.025m.
3.Results of calculations
Figure 1displays the experimentally measured profiles of the electron temperature before and after the beginning of droplet
spraying in two similar shots in TEXTOR (R =1.75m,a =0.46m)where tiles of tungsten were mounted on test limiters.In these discharges,the total heating power P heat ,including ohmic dissipation and neutral beam injection,was 1.5MW and the line-averaged density 3.5×1019m −3.The heat load onto the limiter perpendicular to its surface was up to 30MW m −2and a molten surface of 1cm 2with a typical melt-layer thickness of 1–1.5mm was produced.The melt existed during time intervals up to 0.5s and moved across the magnetic field with a velocity of 0.2–0.5m s −1[5,6].The behaviour of droplets, e.g.their penetration depth in the confined volume,and modifications in plasma parameters observed in the experiments in question are typical for all discharges with high enough heat loads onto tungsten limiter tiles.Although such tiles do not belong to a standard TEXTOR setup,droplet spraying and cooling of the plasma core have been observed in more than 15shots to date.
Figure 2shows the time evolution of the radial profiles for n and T ,density of W impurity ions and their radiation losses computed under the assumption that droplet spraying is initiated at t =2s;droplets have initial radius ρ0≡ρ(t,a)=4µm,corresponding to the maximum of the experimentally found droplet distribution over ρ0,and move with the velocity U d =1m s −1[5,6];the efficiency of melt conversion into droplets,α=10−3,was adopted.One can see that after the transition stage with a duration of a hundred ms,a steady state with significantly reduced central temperature and peaked radiation sets in the plasma.In this state the rate of droplet generation is reduced compared with its initial level because the heat flow to the limiter is diminished by the impurity radiation from the plasma core.
It is important to see that figure 2demonstrates radial profiles of parameters averaged over the magnetic surfaces.In the case of the plasma temperature and density,figures 2(a )and (b ),their local values are very close everywhere to the averaged values because very fast transport processes on the surfaces lead to a nearly perfect flattening of T and n ,see remarks in the previous sections.The impurity density and radiation can be,however,strongly localized in toroidal and poloidal directions,especially at the initial stage of droplet spraying.This localization does not affect,nonetheless,the plasma parameter profiles.Indeed,if these are flat on magnetic surfaces one-dimensional transport equations (9)and (10),describing the radial profiles of T and n ,result directly from three-dimensional transport equations by averaging these in toroidal and poloidal directions.Equations (9)and (10)contain surface-averaged impurity characteristics only,independent of their variation on the surface.Thus,even if the time of impurity spreading over the magnetic surfaces is larger than the transition time to the new plasma state this state should sustain when the spreading process is finished.At the same time,some features of this process,e.g.the time evolution of the radiation loss localization,are of importance.They can and will be described in the framework of a non-stationary ‘shell’model developed recently [25,26].
In figure 3,the radial profiles of the surface integrated density of the impurity ion source,S I (r),is displayed in the final stationary state both for the nominal initial droplet radius 4µm and for two other ρ0of 2and 8µm,corresponding to the edges of the droplet distribution over ρ0.A characteristic
4
Figure 2.Calculated time evolution of the radial profiles of the plasma temperature (a ),density (b ),W ion density (c )and radiation loss density (d )for TEXTOR conditions with droplet spraying initiated at t =2s.
01
2
3
Figure 3.Radial profiles in the final stationary state for the density of the impurity ion source integrated over the magnetic surfaces and calculated with different initial values of the droplet radius:4µm (solid curve),2µm (dashed curve)and 8µm (dashed–dotted curve).
radial width of the source of 2cm is in good agreement with the observations and significantly,by an order of magnitude,exceeds a penetration depth of 1–2mm for evaporated W atoms in the plasma.
In figure 4we demonstrate the calculated central plasma temperature in steady states with droplet spraying versus the conversion efficiency factor αfor three magnitudes of ρ0mentioned above.The range of experimentally observed T (0)values,0.8–1.5keV ,corresponds to αvalues in the interval 5×10−4–5×10−3.This is in agreement with the modelling results in [11,12]predicting α∼10−3:(i)due to vapour-shielding phenomenon the incident heat flux onto the surface can be reduced by an order of magnitude and (ii)roughly 1%of the mass of vapour bubbles arising in boiling melt of tungsten is converted into droplets.
0.5
1.0
1.5
2.0
10
10
10
10
Figure 4.Central plasma temperature in steady states with droplet spraying versus the efficiency of melt conversion into droplets calculated for different initial values of the droplet radius:4µm (solid curve),2µm (dashed curve)and 8µm (dashed–dotted curve).
4.Conclusion
The plasma response on radiation from impurities produced by the evaporation of droplets sprayed from a melted tungsten limiter in tokamak TEXTOR has been numerically modelled.The transport characteristics for charged particles and heat across magnetic surfaces have been deduced by comparing the measured and calculated radial profiles of the plasma density and temperature under conditions without droplets and applied to study the situation with droplet spraying.Calculations show in agreement with observations that as a result of droplet spraying a new steady state with significantly reduced central temperature and peaked impurity radiation profile is achieved.The radial extension of the impurity ion source inside the
5
plasma due to droplet evaporation is of2cm and in agreement with experimental observations.This source extension significantly exceeds a typical penetration depth of evaporated atoms.The efficiency of melted material conversion into droplets is estimated by comparing the measured and computed values of the electron temperature in the plasma core.The found value of10−3agrees qualitatively with predictions of models for droplet formation.Unfortunately the model developed here cannot be directly applied to other experimental devices with PFC of tungsten,ASDEX Upgrate,JET and ITER,because of obvious distinctions from TEXTOR:divertor configuration,normally H-mode of confinement with edge transport barrier,etc.Nonetheless,the heat loads on the TEXTOR limiter are comparable to those expected in these machines on unshaped tiles,in regimes with loss of divertor detachment.Therefore,the efficiency of melt conversion into droplets assessed in this study can be used,to our mind, by interpreting experiments with tungsten PFC also under conditions more relevant for a fusion reactor.
©Euratom2012.
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