An analysis of the exhaust of a fusion reactor to a divertor target by collisionless processes

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AN ANALYSIS OF THE EXHAUST OF A FUSION REACTORTO A DIVERTOR TARGET BY COLLISIONLESS PROCESSES

PJ. HARBOUR, M.F.A. HARRISONCulham Laboratory, Abingdon, Oxon.,(Euratom/UKAEA Fusion Association)United Kingdom

ABSTRACT. An analysis is presented of the collisionless transport of particles and energy from a tokamakreactor plasma to a divertor target. Account is taken of the cooling effects of both unburnt fuel from thereactor and of secondary electrons emitted from the target. Even so, the temperature of the exhaust plasmais high and the potential of the target sheath exceeds the thresholds for unipolar arcing unless the burn-upfraction is impracticably low, i.e. fB< 10~3. It would thus appear that the exhaust of a reactor by purelycollisionless processes will pose formidable problems, and it is desirable to introduce additional mechanismsfor cooling the electrons in the exhaust plasma.

1. INTRODUCTION

A tokamak reactor, fuelled with a deuterium-tritium mixture and operating in a steady state, mustcontinuously exhaust both thermonuclear a-particlesand unburnt fuel. It must also be capable of accom-modating the power associated with the formation ofthe 3.52-MeV a-particles. These a-particles slow downand thermalize with the plasma in a few seconds [1,2],a time comparable to the particle confinement timein a reactor, and so for the purposes of the presentpaper they are assumed to be in thermal equilibriumwith the plasma. However, a small percentage mighthit the first wall or enter the exhaust of the reactorbefore they have lost much energy [3]. The a-particlepower, some 800 MW for a 5000-MW (thermal) reactor,is used to heat incoming fuel to the plasma tempera-ture of, say, 10—15 keV, but in a steady state it musteither be radiated or conducted to the first wall orelse removed from the torus by particle transport intoa divertor. The choice has profound implications forthe design of a reactor [4, 5]: if all 800 MW go tothe first wall, the area and therefore the cost of thefirst wall and blanket must be increased or else thewall temperature will be higher and its life-time willbe decreased; the alternative strategy, to minimizepower losses to the first wall and to remove as muchof the a-particle power as possible into a divertor, willbe considered in this paper and for simplicity it isassumed that all of this power enters the divertor.This mode of operation is known as the unload mode.

Particle transport into the divertor could be underconditions of high plasma density with the objective

of exhausting the a-particle power by conduction andradiation within the divertor. However, such transportimplies a dense scrape-off plasma so that charge-exchange and impurity radiation losses to the first wallmay be unavoidable. An alternative, appropriate to apure D—T plasma, is to operate in a lower-densityregime so that, apart from the inevitable but smallbremsstrahlung losses, the energy is exhausted by non-radiative transport of particles into the divertor. Underthese conditions, the energy must be removed from thedivertor plasma before the particles can be exhausted,and one approach that has been studied in some detail(i.e. UWMAK III [6]) is to allow the plasma to impactupon a target. Practical considerations of the materialproperties of the target limit its power loading, andso the magnetic fields in the divertor must allow theplasma to expand and cover a large area at the target.Consequently, the plasma density is low and, for thepresent purposes, the exhaust plasma can be consideredas collisionless within the divertor although it mustoriginate in a much more dense region close to theseparatrix where particle collisions can take place. Itis the objective of this paper to consider some of theconsequences of such collisionless exhaust of particlesand energy from a tokamak reactor.

2. A MODEL COLLISIONLESS EXHAUST

For the purposes of this paper it is adequate toconsider a reactor exhaust in a highly idealized manner.The divertor channel is assumed to be of uniform areawith a magnetic field of uniform intensity that lies

NUCLEAR FUSION, Vol.19, No.6 (1979) 695

HARBOUR and HARRISON

region of particle collisions

separatrix

collisionless exhaust plasma

expanding magneticfield of diver tor

ions , D+, T+ He4"1" L

thermal electrons fromseparatrix region

thermal electronsreflected by sheath

secondary electrons

sheath

sheathinter-

actionregion

targets

ion pairsfrom

reactorplasma

thermalizedsecondaryelectrons

emittedsecondaryelectrons

FIG.l. Idealized exhaust by collisionless flow to a diver tor target.

normal to the target. Thus the effects of the field areneglected. When the exhaust plasma impacts upon thetarget an electron-repelling plasma sheath must beformed in order that the loss rate of electrons andions can be so balanced that the global loss of chargefrom the tokamak is zero. For a hot, reactor plasmaboth the electrons and ions that strike the target havesufficient energy to release appreciable numbers ofsecondary electrons. These electrons are acceleratedinto the exhaust plasma by the action of the sheathpotential. They can travel, together with electronsreflected by the sheath, back through the collisionlessexhaust plasma and enter the dense plasma near tothe separatrix where they thermalize before returningto the divertor region. It is, moreover, assumed thatthere are no electric fields in the exhaust plasmaexcept within the sheath, whose potential differenceis U. This concept is illustrated in Fig. 1.

The power input to the divertor is governed by theo-particle power and so the temperature of the collision-less exhaust plasma depends upon how many particlescan share in the energy (Ea= 3.52 MeV) of eacha-particle. Sharing can take place with the secondaryelectrons and also with any unburnt D-T fuel whoseburn-up fraction, ffi, can be defined as

and sinks of energy, the electron temperature of theexhaust plasma can be expressed as

+ n̂eutrons(1)

where ^ and $ are, respectively, the fluxes of particlesentering and leaving the reactor and the subscripts areself-explanatory. Neglecting relatively small sources

k T e ~ E a f B / 2 5 t (2)

where 5t is defined as an energy transport coefficient,i.e. 5tkTe units of energy are transported by each ionand its paired electron together with the associatedsecondary electrons.

The transport coefficient 5t can be determined bymeans of a one-dimensional analysis of a plasmasheath [7] which allows for the acceleration ofemitted secondary electrons and the return of theseelectrons to the sheath after thermalization to thetemperature Te. The effect of secondary electronemission on transport through a sheath was firstdescribed by Hobbs and Wesson [8], but the analysis ofRef.[7] is more exact and it applies over a wider rangeof conditions.

3. TRANSPORT PROPERTIES

Each particle striking the target is assumed to belost from the system and so an exhaust flow in thepositive x-direction can be characterized by thedistribution of ion and electron velocities shown inFigs 2a and b, respectively. These distributions in thex-direction are truncated Maxwellians; that for ions isallowed to have a superimposed drift velocity UJXwhereas the emitted secondary electrons move towards

696 NUCLEAR FUSION, Vol.19, No.6 (1979)

FUSION REACTOR EXHAUST

a) ions

nj(vix ) = A, exp - -

0 ui.

Secondaryelectrons

FIG.2. Ion and electron velocity distribution functions [ 7]for exhaust flow along a diver tor channel aligned in thex-direction.

the separatrix with a velocity (-2 eV/me)* corres-ponding to acceleration by a potential difference V < U.

The energy transported through the exhaust plasmaby ions is related, for mathematical convenience, tothe electron temperature: an ion energy transportcoefficient, 5i, is defined such that the average energytransported by an ion is 5ikTe. In addition, each ion isaccelerated within the sheath and so delivers (6i+0w)kTe

units of energy to the target; here 0W= eU/kTe is thenon-dimensional sheath potential. The relationshipbetween 8[ and the ion temperature is given inRef.[7] as

(3)

where the ion speed ratio si is defined as

(4)

The function L(si) varies smoothly from L(0) = 0 toL(oo) = s?} and the sheath analysis is valid for all valuesof Sj and 5i. The speed ratio Si in a reactor will bestrongly influenced by magnetic mirroring, electricfields and collisions of particles, and, therefore, variousvalues of sj and 5j should be considered. However, itis stressed that although the velocity distributionfunctions for ions and electrons may be affected bythe magnetic field, and here we consider any value of

si from zero to infinity, the effect of a mirror loss-cone distribution function has been ignored. This isdiscussed in Section 6. The range of 8[ is restricted bythe Bohm sheath criterion [9], which requires that6i>0.5 in the case Si-+°° and when the criterion ismodified by secondary electron emission [4] thislower limit is only raised to 5j £ 0.53. In the collision-less exhaust of a reactor, 6j is likely to be larger thanthese limits. Firstly, Tj can decrease only if the randomion motion becomes directed along the magnetic fieldwhereupon there is a corresponding increase in L(sj).By analogy with supersonic expansion in a Lavalnozzle, 5[ may be as high as 2.5. Secondly, in thecollision-dominated region near the separatrix (seeFig.l), Te may be less than Tj because of the coolingeffects both of electrons reflected at the target sheathand of secondary electrons emitted by the target.Thus values of 5i greater than 2.5 should be considered.

4. CONSEQUENCES OFSECONDARY-ELECTRON EMISSION

Impact of both electrons and ions on the targetcauses the emission of secondary electrons, and it isconvenient to define an effective coefficient ofsecondary emission, i.e.

rs = 1-7o<rs<[rsL

Here, 7+ and 7" are, respectively, the emission coeffi-cients for ions and electrons incident upon the target,and thus Fs depends on the material of the target andthe energy of the incident particles [10]. The ionenergy is (5j+ 0w)kTe, and the electrons that strikethe target have the same distribution of velocities asthose entering the sheath, although their flux isattenuated by the repelling action of the sheath. How-ever, an upper limit [ r s ] m a x = [Fs]s a t is imposed byspace-charge saturation of electron emission in thesheath. The value of [Fs]satcan be determined bysolving the Poisson equation, and this limiting emissionis insufficient to cause collisionality in the exhaustplasma. Details of this analysis are given in Ref.[7]and the values of [Fs]sat so determined are shown inFig.3a as a function of 5j for a D-T plasma (mi= 2.5).The corresponding Bohm criterion for the existenceof a stable sheath is considered in Ref.[7], and itsassociated values of [FS]B are also plotted in Fig.3awhere they show the lower allowable limit in ion

NUCLEAR FUSION, Vol.19, N0.6 (1979) 697

HARBOUR and HARRISON

b)

Ion energy transport coefficient, 6j

FIG.3. Bounding values of parameters within which a stable sheath may exist:(a) coefficient, Fs, shown as a function o/Sj.

Values of Fs are allowed from zero to |Ts]sat but there is a lower limit to 8 [governed by theBohm criterion;

(b) corresponding limits to the sheath potential shown versus b\;(c) corresponding limits to the total energy transfer coefficient, 5 t , shown versus 5j.The parameters are evaluated for a D-T plasma (mx = 2.5 amu) with s-^ °° (monoenergetic incident ions).The shaded.regions indicate the domain in which a stable sheath may exist.

energy. Thus a stable sheath can exist in the para-metric regime bounded by [rs]s at , [FS]B and thecondition of zero emission of secondary electrons(shown in Fig. 3 a by Fs = 0). For mathematicalsimplicity the data here are presented for a mono-energetic flow of ions (si= °°) and for no net loss ofcharge from the exhaust. In these conditions, themaximum saturated emission corresponds to Fs«10at the Bohm limit of 6i ̂ 0.53.

The emission of secondary electrons causes thesheath potential to fall and conditions for chargeneutrality and current continuity in the exhaust plasma[7] lead to the following relationship between 0W

and Fs:

/2TTk F(si)

= (1 + r s ) V ? ( l +erfV0w) exp 0w+Fs0w (5)

The function F(si) is described in Ref.[7] and it isslowly varying in Si from F(0) = 1 to F(°°) = (2ir)~».

The bounds in 0W corresponding to those in Fs havebeen derived from Eq.(5) and are shown as functionsof 5i in Fig.3b. The emission of secondary electronssubstantially decreases the sheath potential but, evenwith saturated emission, the potential is still 0.87 kTe/eat the Bohm limit and only falls to 0.5 kTe/e when5j = 2.5. The energy transport coefficient 5t can nowbe determined from

(6)

where the coefficient 2FS is associated with thesecondary electrons and [(5j + 0w) + 2] with the ionpairs exhausted from the reactor plasma. It is evidentthat most of the energy is transported by electrons,and so 51 is not particularly sensitive to 5i and si. Thebounds in 5t that correspond to those in Fs are plottedin Fig.3c as functions of 8[. The maximum transportcoefficient, 5 t^ 23, occurs at the Bohm limit and it isabout 4 times greater than the value without secondaryelectron emission. The effect of secondary electronemission becomes significantly less at higher values of 5j.

698 NUCLEAR FUSION, Vol.19, N0.6 (1979)

FUSION REACTOR EXHAUST

10"3 10 "2

Burn-up f ract ion , fB

10'3 10"2 10"1

Burn-up fraction, fB

FIG.4. Electron temperature (a) and potential (b) of theexhaust plasma plotted versus burn-up fraction, fy. Twoconditions are illustrated: no secondary electron emission(8t = 5.8, 5j= 0.5, 0W= 3.3) and saturated electron emission(8t = 23, 5i= 0.53 and 0W = 0.865).

5. REACTOR EXHAUST CONDITIONS

The exhaust temperature, irrespective of reactorpower, can be determined to a close approximation bysubstitution of values 5t in Eq.(2). The temperature isshown in Fig.4a plotted as a function of burn-upfraction for values of 6t corresponding to no secondaryemission (i.e. 5t « 5.8) and to saturated emission(5 t « 23). Energy is transported in a collisionlessexhaust by particle convection without radiation andso the temperature can only be reduced by increasingthe number of particles in the exhaust. This can beachieved either by reducing the burn-up fraction orby increasing the number of secondary electrons up tothe limit of space-charge saturation. Even for saturatedemission the exhaust temperatures are high, about4 keV for fB = 0.05. The temperature of the exhaustplasma cannot exceed the temperature at the separa-trix which itself must be lower than the averagetemperature ( « 15 keV) of a reactor plasma. Thus, itis evident that the burn-up fraction for a reactor witha collisionless exhaust must not exceed a few per centand indeed ffi % 0.05 could well represent an upperlimit.

Corresponding values of the sheath potential(U= 0wkTe/e) can also be determined by using Eq.(5)and are shown in Fig.4b. These are also high, forexample about 3.5 kV for fe = 0.05 and for saturatedemission. Even under the most favourable conditionsof saturated secondary electron emission and low

burn-up, the sheath potentials exceed the thresholdsfor unipolar arcing [10—12] and also for sputtering,which has been discussed by Conn [5].

The power density on the target due to convectivetransport of particles can be expressed as

P =frn / fij

F(Si) \7rmi(kTe)

3/2(7)

where n is the density of the exhaust plasma. Practicallevels of power density lie in the range 1 to 10 MW • m"2

and the appropriate values of n and Te for no emissionof secondary electrons [i.e. 5 t « 5.8 substituted intoEq.(7)] are shown in Fig.5 by the curves (D,A) and(C,B), respectively. These curves are intercepted bythe temperature levels (D,C) and (A,B) that correspond,respectively, to burn-up fractions of 0.01 and 0.1.Thus, the domain (A,B,C,D) defines some practicaloperating limits for a collisionless exhaust in whichthere is no emission of secondary electrons from thedivertor target. A similar domain (A', B', C', D') canbe determined for a reactor exhaust when emissionfrom the target is saturated (i.e. 6 t« 23). In both ofthese domains the density and temperature are suchthat the plasma is well within the collisionless regimewhich occurs if the transit time through the charac-teristic length, L, of the divertor is less than theelectron-electron collision time, so that the regime isdefined by

n(nf3) <1.11 X1017

L(m)ln ATl (eV)

10'

Exhaust plasma density, n/(m3)

FIG. 5. Parameters of a collisionless exhaust plasma. Theregion ABCD corresponds to no secondary electron emissionand the shaded region A'B'C'D' to saturated electron emission.The plasma conditions are described in the text.

NUCLEAR FUSION, Vol.19, No.6 (1979) 699

HARBOUR and HARRISON

Here A is the usual ratio of the Debye shieldingdistance to the minimum impact parameter forelectron-proton collisions [13].

6. CONCLUSIONS

It is evident from Section 5 that the additionalcooling of the coUisionless exhaust plasma by secondaryelectrons allows reactor operation at higher burn-upfractions. Thus, the emission can be accompanied bya reduction in the throughput of fuel and a mitigationof the problems associated with fuel injection,divertor pumping, tritium inventory and tritiumrecovery and recirculation. However, the idealizedmodel presented here is likely to overemphasize theeffects of secondary electrons because the followingpractical considerations should be taken into account:

(a) The magnetic field will not be uniform because itmust be expanded to intersect a large area oftarget in order to reduce wall loading at the target.The magnetic field strength in the vicinity of thetarget could be tens of gauss. Thus, electronstravelling from the sheath towards the separatrixwill encounter an increasing field. Mirroring willoccur within the divertor and some of the electronswill be returned to the target, thereby reducingtheir probability for thermalization in the plasmanear the separatrix. Energy transport by electronswill thereby be reduced. However, transport in amagnetic mirror field may be affected by micro-instabilities. These have been assessed byMense et al. [14] for the plasma entering thedivertor, and it was concluded that the scrape-offplasma seems to be unstable to a wide variety ofmicroinstabilities driven by the free energy asso-ciated with a loss-cone distribution so the magneticmirror would not significantly impede the entry ofparticles into the divertor. However, for particlesreturning from the divertor to the scrape-offplasma the mirror ratio is orders of magnitudehigher and the density is orders of magnitude lowerthan their respective values in Ref.[14]. Conse-quently, the various assumptions made byMense et al. are inapplicable. For example,adopting the notation of Ref.[14], the scale length,n̂ ~ Pi \A m i / m e) exceeds the length for a

coUisionless divertor so infinite-medium theorydoes not apply. Moreover, although the ionplasma frequency exceeds the ion gyrofrequency,the latter is so low that the growth rate of all

standing-wave modes may be too low to beimportant. An assumption [14] concerning con-vective loss-cone instabilities is that cope < £2e butin the present analysis cop e~ 10 ^2e- The conclu-sion is that electrons returning from the divertorto the scrape-off layer will encounter a powerfulmagnetic mirror which will reduce energy transport,but the effect of this reduction is uncertain becauseof the unknown role of microinstabilities withinthe divertor itself.

(b) In many concepts of coUisionless divertors (seee.g. Refs [15-17]) the area ratio of target tofield is maximized by inclining the target to thefield and also by using non-planar targets. Deviationof the field from normal incidence influences theeffective yield of secondary electrons, and eitheran increase or a decrease is possible. In addition,when the electric field of the sheath is inclined tothe magnetic field, then the electrons that leavethe sheath have a substantial ratio of vei/veii andhence an enhanced probability of being reflectedby the magnetic mirror.

Several important plasma properties have not beenanalysed in this simple model. Firstly, it is necessaryto balance outward diffusion of particles towards theseparatrix of the tokamak to the convective exhaustflow along the field lines towards the divertor target.Secondly, instabilities in the electron velocity distribu-tion function may occur and, thirdly, the target andexhaust flow characteristics could be affected byspatial or temporal non-uniformities in plasma para-meters and by incomplete thermalization of a-particlesentering the divertor. The present model shows thatthe sheath potential will be in excess of the thresholdsfor unipolar arcing and, moreover, if the model wererefined to account for the restrictions to energytransfer by secondary electrons, then the predictedsheath potential would be even larger. Only if theburn-up fraction is reduced to an impracticable regionbelow 10~3 does the present model predict sheathpotentials below the unipolar arc threshold. Theoperational life-time of the target will be furtherreduced by sputtering due to impact of ions acceleratedthrough the high potential of the sheath [5]. Theconsequences of arcing are not treated in this paper,but arcs could be substantial sources of impurityelements and target erosion. The electrons emittedfrom arc spots would decrease plasma potential andthe enhanced radiation from impurities and neutralspecies would serve to cool the electrons and so furtherto decrease plasma potential. It is a matter of specu-lation whether the effect on plasma potential would

700 NUCLEAR FUSION, Vol.19, No.6 (1979)

FUSION REACTOR EXHAUST

be sufficient to achieve a steady state before the

backflow of impurities from the divertor would cause

the main reactor discharge to be extinguished because

of radiative cooling. Thus the most important conse-

quence of arcing is that the exhaust plasma is likely

to become collisional so that the present model of

idealized convective transport is no longer applicable.

Thus, it appears that the concept of a reactor

exhausted by purely collisionless processes presents

formidable practicable problems unless some means

can be found for cooling the electrons and so reducing

the sheath potential. It may be that some compact

form of direct energy recovery system might eventually

achieve this objective. Failing this, an alternative

would be to exhaust in the collision-dominated mode

whereby electrons lose energy by radiative collisions

in a high-density exhaust plasma.

[ i ]

[21

[3]

[4]

REFERENCES

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[5] CONN, R.W., J. Nucl. Mater. 76 and 77 (1978) 103.

[6] CONN, R.W., KULCINSKI, G.L., MAYNARD, C.W.,ARONSTEIN, R., BOOM, R.W., BOWLES, A.,CLEMMER, R.G., DAVIS, J., EMMERT, G.A.,CHOSE, S., GOHAR, Y., KESNER, J., KUO, S.,LARSEN, E., SCHARER, J., SVIATOSLAVSKY, I.,SZE, D.K., VOGELSANG, W.F., YANG, T.F.,YOUNG, W.D., in Plasma Physics and Controlled NuclearFusion Research (Proc. 6th Int. Conf. Berchtesgaden,1976) Vol.3, IAEA, Vienna (1977) 203.

[7] HARBOUR, P.J., The Effect of Secondary ElectronEmission on a Plasma Sheath, to be published, Phys.Fluids (available as Culham Laboratory PreprintCLM-P535).

[8] HOBBS, G.D., WESSON, J.A., Plasma Phys. 9 (1967) 85.[9] BOHM, D., Characteristics of Electrical Discharges in

Magnetic Fields (GUTHRIE, A., WAKERLING, R.K.,Eds), McGraw Hill (1949) 77.

[10] HARBOUR, P.J., HARRISON, M.F.A., J. Nucl. Mater.76 and 77 (1978) 513.

[11] ROBSON, A.E., THONEMANN, P.C., Proc. Phys. Soc. 73(1959)508.

[12] ECKER, G., in Plasma Wall Interaction (Proc. Int. Symp.Julich, 1976) Pergamon Press (1977) 245.

[13] SPITZER, L., Physics of Fully Ionised Gases, 2nd ed.,Interscience (1962).

[14] MENSE, A.T., EMMERT, G.A., CALLEN, J.D., Nucl.Fusion 15(1975)703.

[15] SANDERSON, A.D., STOTT, P.E., J. Nucl. Mater. 76and 77 (1978) 530.

[16] YANG, T.-F, LEE, A.Y., RUCK, G.W., WestinghouseCompact Poloidal Divertor Reference Design, WFPS-TME-042, August 1977.

[17] EMMERT, G.A., MENSE, A.T., DONHOWE, J.M., APoloidal Divertor for the UWMAK-I Tokamak Reactor,paper presented at 1st Top. Meeting on the Technologyof Controlled Nuclear Fusion (U.S.A.E.C.), San Diego(April 1974). Also available as University of WisconsinReport UWFDM-93.

(Manuscript received 10 August 1978

Final version received 6 March 1979)

NUCLEAR FUSION, Vol.19, No.6 (1979) 701