P/2506 Denmark

Energy Balance in a Thermonuclear Reacting Plasmacontaining Deuterium, Tritium and Reaction Productsunder Isothermal Pulsed or Steady-State Conditions

By T. Hesselberg Jensen, O. Kofoed-Hansen and С F. Wandel*

The energy balance in a thermonuclear reactingplasma containing deuterium and tritium has beendiscussed in several places.1"4 By comparing the powerescaping as electromagnetic radiation with the powertransferred to charged reaction products in the nuclearreactions, a lower limit on the plasma temperaturenecessary to maintain a self-sustaining reaction isobtained.1 The limiting temperatures are 350 million°K in the case of pure deuterium and 50 million °Kin the case of deuterium and tritium in equal amounts.

Especially when only small fuel burnups are con-sidered, it is important to take into account, in theenergy balance, the amount of energy spent in heatingthe fuel to the reaction temperature. This point hasbeen investigated by Lawson2 where a pulsed systemhas been considered in which the fuel is instantan-eously heated to the reaction temperature and thenallowed to react in a definite time interval after whichthe plasma is again cooled to essentially zero tempera-ture. If one can make the apparently realistic assump-tion that the heat content of the plasma, like theradiative power, can be transformed into plasma heat(in a new pulse) only with an efficiency considerablyless than unity, then it can be concluded that not onlymust the temperature exceed a certain limit, but alsothe reaction must be sustained long enough for adefinite fraction of the fuel to be burnt.

It is clear that as soon as a considerable fraction ofthe fuel is burnt the reaction rates will decrease and,because of the higher nuclear charge of the reactionproducts, the radiative power will increase. Both ofthese effects will tend to reduce the power economy ofthe system. The first of these effects has been treatedby Lacombe et al.,3 who have considered a pulsedsystem and investigated the time dependence of thepower densities due to primary and secondary nuclearprocesses.

In Ref. 4, we have tried to incorporate all of theabove-mentioned effects by treating the problem of asteady-state thermonuclear reaction involving deuter-ium, tritium and their reaction products. To keep thereacting plasma in a steady state it is in general

* Danish Atomic Energy Commission Research Laboratory,Riso.

431

necessary to exchange matter and energy continuouslywith the surroundings. The rates of these exchangesare completely determined by only three independentparameters. These parameters are the temperature ofthe plasma, the tritium enrichment in the fuel and theproduct of the deuterium density and the plasmarenewal time. This renewal time corresponds roughlyto the pulse time in a pulsed thermonuclear reaction.Instead of the density-time product, a burnup para-meter indicating the fraction of the deuterium suppliedthat is actually burned in the reaction can be used.

It is conceivable that a thermonuclear reactor couldbe constructed so that the charged reaction productswould be retained long enough to reach the averageparticle energy, while it is less likely that the neutronsand the electromagnetic radiation can be preventedfrom escaping from the reaction region. Thus, anadequate measure of the ability of the thermonuclearreaction to become self-sustaining (in the sense thatthe amount of energy transferred to the chargedreaction products is sufficient to supply the energy re-quired for the electromagnetic radiation and to heat thefresh fuel to the reaction temperature) is given bythe reinjection fraction, e, defined as the ratio betweenthe power that must be injected into the plasma tokeep the system in a steady state and the total poweremitted from the plasma in the form of kinetic energyof neutrons, heat in extracted plasma and electromag-netic radiation. The reaction will obviously be self-sustaining when e is equal to zero, while positive valuesof e correspond to situations where energy mustartificially be supplied to the reaction region in orderto keep the plasma in a steady state.

When the conditions of a self-sustaining thermo-nuclear reaction have been reached it may be neces-sary to extract more energy than is emitted as kineticenergy of neutrons, heat in extracted plasma andelectromagnetic radiation in order to maintain thedesired reaction temperature. It is possible to extendthe definition of e to this case also. In this way, edescribes the criticality condition for the thermo-nuclear reaction: e > 0 meaning subcritical conditionsand e < 0 supercritical conditions.

A survey of the dependence of e on the three para-meters has been made in the case of the steady state

432 SESSION A-10 P/2506 T. HESSELBERG JENSEN etal.

and it is shown that optimum values of all threeparameters can be found. An investigation of theisothermal non-steady state shows great similaritywith the behaviour in the steady state, as has beendemonstrated by a few examples. The survey ofsteady-state systems can thus be said to cover thecase of pulsed isothermal systems, when they arecompared in terms of the deuterium burnup or interms of the density-time product.

Since the actual rate of transfer of kinetic energybetween the different kinds of ions and the electronsin the plasma is comparable with the rate of energyproduction through nuclear reactions and with therate of energy exchange with the surroundings, itmust be expected that corrections must be made tothe results derived by assuming the energy exchangebetween particles to take place instantaneously.Calculations in the steady-state case, with theseeffects taken into account, show that the most impor-tant effect is a lowering of the electron temperaturewith respect to the ion temperature, in the cases wherethe reaction is nearly self-sustaining. This effect

increases somewhat the range of parameter valuesfor which a self-sustaining reaction is possible.

BASIC EQUATIONS

The reactions to be considered are listed in Table 1.In this table we have also indicated the notation to beused in the following. For the reaction rate para-meters <ot>> we have used Roman numerals accordingto the numbering of the reactions, and for the averageenergies of the particles participating in the reactionswe have, for example, used E41 for the triton inReaction 1. It is then obvious that for a reactionbetween particles of kinds i and k giving particles ofkinds r and s,

= Er+Es. (1)

Further, we use щ for the number of particles, percm3, of type ¿ = D, T, p, 3, 4, e, n for deuterons,tritons, protons, He3, He4, electrons and neutronsrespectively. The Q values5» 6 for the reactions arealso given in Table 1.

Table 1. Reactions of Interest, Q-values, Reaction Rates and Energy Partitions

Reaction Q -value, Mev Reaction rates Average energy of particles involved in reactions

1.2.

4.5.6.7.

D + D->T + pX) + D —5- He3 + nX -f D —>• He4 + nHe3 + D —> He4 + pT + T —> He4 + 2nT + He 3 - » H e 4 + DT + He3 ->• He4 + P + n

4.043.27

17.5818.3411.3214.3112.08

ET1

nTnsVII _ _ _ _ _

Reaction 5 of Table 1 needs only to be consideredfor fuel that is extremely rich in tritium. Reactions бand 7 need only to be considered for very high tem-peratures and for fuel that is very rich in both tritiumand He3. These reactions can, therefore, be neglectedfor our present purpose. At low temperatures (i.e.T < 10 kev) even Reaction 4 may be neglected.

For the calculation of the quantities mentioned inTable 1 we have assumed a Maxwellian velocity dis-tribution of each kind of particles with the tempera-ture Ti of the ¿th kind of particle. The most importantof these quantities, the reaction rate parameters, arefunctions only of a proper average temperature, T',for the temperatures of the reacting particles given by

where mt and nij are the masses of the reacting particlesand Ti and Tj are their temperatures. Furthermore,Mij is the reduced mass, and цц defined by (2) will beused later on and termed the temperature reducedmass. In Table 2 the reaction rate parameters aregiven as functions of T'. In the calculation 4> 7 of thereaction rate parameters, and the average energy ofparticles participating in reactions, we have usedcross sections compiled by Jarmie and Seagrave8 andby Bame and Perry.9

Particle Densities

First, we shall give a general set of equationscovering the kinetics of the ion densities in the plasma.At time t = 0 one cm3 of plasma contains щ ions oftype i and, if we impose the condition of charge

Table 2.

Temperature T' {kev)

I-IO1? (стз/sec)I I . 101? (стз/sec)

IILIO1? (стз/sec)IV-101? (стз/sec)

Reaction Rate

5

0.008620.008981.300.000727

Parameters

10

0.05750.0613

11.10.0223

as Functions of Temperature

25

0.3510.400

55.80.776

50

0.9811.17

85.45.68

100

2.262.76

84.517.85

150

3.424.20

70.424.6

ENERGY BALANCE IN A PLASMA 433

Table 3. Reaction Kinetic Equations

dnv¡dt =

(3)(4)(5)(6)(7)

neutrality, ne = TiitiiZi. All summations over г meansummations over ions only. In order to enable us todiscuss the steady state situation we may assumefresh fuel to be added at a rate of i>A deuteriumatoms, and nvrjA tritium atoms per cm3 sec. Wemay also assume that it is possible to extract arepresentative mixture of the gas at a rate of щЪ forthe ions and 8S<w<Z< for the electrons, again per cm3

sec. Under these conditions, the reaction kineticequations for the ion densities are as shown in Table 3.

Furthermore, for the number of neutrons emittedper cm3 sec we have

(8)

and for the electrons we have

dne/dt = - SHiftiZi + wD(l + rj) Д (9)

With ne = TiifiiZi this last equation is alreadycontained in Eqs. (3)-(7), but it gives a convenient andoften used relation between S and A.

Energy Content

Next, we shall set up the kinetic equations describ-ing the energy content in each type of chargedparticles. In order to do this, we assume that thefresh fuel is added at zero temperature and as neutralatoms, and that the ionization energy is negligible.The ions and electrons extracted from the plasma areassumed to have the same average energy as thecorresponding type of particles in the plasma.Furthermore, we shall assume that the chargedreaction products are retained in the plasma while theneutrons and the bremsstrahlung escape from thereaction region.

The charged particles are assumed only to exchangeenergy by Coulomb encounters. A formula coveringthis energy exchange is given by the expression10

A

(10)

giving the average energy transfer per cm3 sec tothe particles of type i with density щ from particlesof type j of density щ.

The term In Л is given by

2 In Л = 2(sin -2(cos A) Ci{A)

whereA =

В =

(П)

(12)

(13)

and finally the Debye shielding distance, D, is given by

D = (kTel47rnee2)i (14)

In general, it will be necessary to inject or extractenergy from the plasma in order to initiate and controlthe thermonuclear reaction. The plasma will gainenergy from nuclear heating and energy will be lostas kinetic energy of neutrons, heat in extracted gasand radiated energy. Let us specify the injected orextracted power as рщ, the sign being positive forinjection and negative for extraction. Furthermore,let us assume that this power density is distributedwith a fraction, «i, to each type of charged particles.

Table 4. Energy Density Equations

ctt - Ге)/те + ( Г т

-Ге)/ре + (Г р

= <*3pin-n3{ikTz8 +Z,inik[Zi{T3-Te)f3e +{T3

(16)

(17)

(18)

(19)

(20)

434 SESSION A-10 P/2506 T. HESSELBERG JENSEN etal.

With these assumptions, we may write down theequations for the energy densities for the various typesof particles as shown in Table 4.

The a's fulfil the condition

2г-а$ + а е = 1 (21)

and the radiation power density pv is given byThompson11:

pr = о.

with kTe in kev.

(22)

Energy BalanceIn order to discuss the energy balance of the

system we shall add up the Eqs. (15)-(20), taking intoaccount Eqs. (1) and (21), and obtaining

pin = (23)

where pr is defined by Eq. (22); pn is the neutronkinetic power density,

pn = ЪПЪЩЕЦП + ПВПТШЕПШ ; (24)

pg is the power emitted as heat in extracted gas,

pë = ÇZàkT&i+tkTe Z<mZ,)8 ; (25)

and ^nuci is the nuclear power density,

. (26)

Finally, UQ stands for the total heat energy presentin the gas at any given time t,

UG = (27)

Power InjectionEquation (23) determines the value of the injected

power, pm, at any time t if the temperature conditionsare defined. The magnitude pin can assume negative orpositive values as a function of time, according towhether the reaction is momentarily self-sustainingor not.

In order to analyze the over-all power balance in aperiod of time t = 0 to t = f we define a powerreinjection fraction, e, through the equation

(\Pin\ ~pin)dt] (28)

The left-hand side of this equation is the totalenergy that must be injected into the plasma in orderto initiate and control the thermonuclear reaction,while the brackets on the right show the total energyemitted by the plasma to the surroundings duringthe same time interval.

When a pulsed system is considered, pin includesthe power shot necessary to achieve the high plasmatemperature and also the power recovered from theplasma at the end of the pulse. These contributionswill appear as positive and negative delta-functionsin time at t = 0 and t = f respectively.

Solving Eq. (28) for e and using Eq. (23), whileassuming UQ to be equal at t = 0 and t', one gets

(29)l +pin)df

The importance of the power reinjection fraction,e, can most readily be seen for the steady state. Inthis case, we may simply substitute the integrandsfor the integrals in Eq. (29) since they are constant intime. It is immediately seen that e can only assumevalues in the interval 0 < e < 1. e = 0 implies aself-sustaining reaction since in this case pin < 0.Since the e defined by (29) does not reflect to whatextent pin difíers from zero when the reaction is self-sustaining, it is convenient to modify (29) by intro-ducing an € defined by

1-е' = (29')

For positive values of e' the two parameters will beidentical in the steady state while for negative valuesof € one will have e = 0.

For optimizing the parameters, e' is more convenientthan e since it has a smooth variation in changingfrom positive to negative values, thus permitting aminimum point to be determined instead of aminimum region.

In the pulsed isothermal case, e is the most con-venient for comparison with the steady state. Al-though e will always be larger than zero in this case, amomentarily self-sustaining reaction will be achievedin a period of time in which pin < 0.

The calculation of e is our ultimate aim andEqs. (3)-(7) and (15)-(20) are our basic coupledequations. To solve this set of equations in all detailcan only be attempted by means of electronic com-puters. This falls outside our present program and wehave only treated problems to which analyticalsolutions may be found; namely, the steady-stateproblem and the isothermal kinetics, neglecting alltemperature difference T\ — Ta which means assump-tion of infinite energy transmission parameters. Asurvey of this work is presented in the followingsections.

RESULTS

Steady-state SystemsIn this section we shall assume a steady-state

operation in time. This means that Eqs. (3-7) and(15-20) are equated to zero. We shall also make thesimplifying assumption that all temperatures areequal to a common temperature T. Mathematically,this means assuming al l / i k-> oo. Thus, Eqs. (15-20)are of no consequences except for their sum leading tothe definition of e by Eq. (29').

We then solve Eqs. (3-7) and use the results for thecomputation of e'. In Ref. 4 we give the explicitexpressions for the solutions of Eqs. (3-7) in this case.

The solutions depend on three parameters only: thetemperature T, the tritium enrichment 77 and the

ENERGY BALANCE IN A PLASMA 435

magnitude n-^jh which is a product of density andtime since 1/8 is the mean renewal time for the plasma.Instead of the third parameter we may also use thedeuterium burnup, p, defined by

p = (А-8)/Д. (30)

The parameter p is related, in a one to one corre-spondance, to njy/8 through Eqs. (3-7).

Some of the results of the calculations are illu-strated in Figs. 1 and 2. In Fig. 1 we shall first discussthe case for 77 = 0. For this case, we have shown thee'(j8) curves for the cases T = 25,50, 100 and 150 kev.In each of these cases, an optimum in p is found and ifa curve is drawn through these optima an optimum inT results. The total optimum in p is further illustratedby the fat curve which is the envelope of all the e'(jS)curves. It is easy to understand the appearance ofoptimum values for T and p from physical arguments.If the fuel is pumped through the reactor at a fastrate, relative to the reaction rates, the burnup ratiowill be low and the heating and cooling of the fuel willdominate the power balance, with the result that€ ->- 1. On the other hand, if the fuel is permitted tostay for a long time in the reactor the burnup will behigh, but there will also be a high buildup of passivereaction products leading to severe radiation loss:the power balance will then be dominated by theradiation losses and the heating necessary to com-pensate for these losses, again with the result thate —> 1. Thus, a minimum in c' must be expected forintermediate burnup ratios. Similarly, for the optimumtemperature. For low temperatures, radiation lossesdominate over the nuclear heating as demonstratedby Post.1 On the other hand, for very high tempera-tures, the reaction rate parameters (ov) level off as afunction of temperature while at the same time boththe radiation loss from electron-electron collisions,increasing as Г3^2, and the fuel heating, increasing asT, become more important and finally dominate forthe highest temperatures. Again, in the two extremes,e' -> 1 and at intermediate temperatures an optimummust exist.

We have already mentioned that an envelope of the€f(P) curves for different temperatures can be found.In the remaining part of Fig. 1 we show these enve-lopes for different values of 77. Again an optimum forthis parameter is found, as illustrated by the trend inthe entire valley created by these envelopes as func-tions of 77 and p. The slopes of the valley are rathersteep for small values of 77 and because of this and inorder not to have too many intersecting lines in thefigure we have chosen to plot the envelopes as afunction of 77e rather than 77. Also, in the case of 77, asimple physical interpretation of the optimum can begiven. That tritium enrichment improves e is a resultof the fact that III > I+11. However, if pure tritiumis used, Reaction 3 disappears again and only Reaction5 contributes to the power balance and again / / / > V.Thus, an optimum is found for such values of nn andпт that the condition n^W ^ п-цп^Ш and at thesame time n^n-^III ^ пц2(1 + 11). However, the

16

n-oo

Figure 1. The power reinjection fraction, c', as a function of thedeuterium burnup, ft and tritium enrichment, 17, optimizedwith respect to the temperature, 7. For 17 = 0 the individual ¿

curves for different temperatures are also given

optimum is not very pronounced, since the valley isvery flat as a function of 77. In the optimum the D-Treaction dominates, and it is thus obvious that theoptimum condition is reached when пъ = пт, whichmeans 77 = 1.

In Fig. 2 we give a further illustration of theoptimization. Here, we have given contour curves fore' in the (»D/8, T)-plane for two values of 77, namely77 = 0 and 7] = 1. Since we know from Fig. 1 thatthe optimum 77 is close to 1 we see from Fig. 2 that theoptimum temperature lies near 25 kev and theoptimum in п-о/Ь near 1015 sec/cm3 corresponding, forthat case, to a burnup of ~20%. From Fig. 2 wealso see that in spite of the fact that the reaction rateparameter III for any given temperature is muchlarger than / + / / , only a factor of 10 in the optimum»D/8 is obtained with equal amounts of deuterium andtritium.

The region in 77, T and p, in which <•' < 0, is notgiven with any reasonable precision by the ratherunrealistic steady-state calculations with the sametemperature for all particles. If we use the extendedset of equations, with individual ion and electron

- ^f 553

SBA ъЬ5§

:—'-

02ai

10 20 50

kT

100 200

key

Figure 2. Contour curves for the power reinjection fraction, c',in the (HD/S, T)-pláne where no is the deuterium ion density,S is the inverse plasma renewal time and Tthe temperature. Twosets of contours are given, one for the tritium enrichment 17 = 0

and another for 17 = 1

436 SESSION A-10 P/2506 T. HESSELBERG JENSEN et al.

temperatures, e' will be diminished; also, if we turn topulsed operation, the region of e ~ 0 is shifted towardshigher burnups.

In Refs. 4 and 12 we have given a more detaileddescription of the steady-state calculations.

Effects of Finite Energy Transmission Rates for ChargedParticles

The aim of this section is to investigate the influenceon the reinjection fraction, e', of leaving out theassumption of a common temperature for all particlesin the plasma. However, a Maxwellian velocity dis-tribution of each kind of particles is still assumed.

We now consider the full set of equations, (3)-(9)and (15)-(21), where expressions (3)-(7) and (15)-(20)are equated to zero. This set of equations is rathercomplicated and will only be solved approximately.

In order to describe a steady state in this case wemay again use the parameters rj, пв/S, and a character-istic temperature. As it is seen, however, we must alsoascribe definite values to the as, indicating how theinjected power ft m is distributed between the differentkinds of charged particles. Thus, it is seen that thisproblem contains more parameters than the previousone.

It is possible to avoid a specific choice of the a's byconsidering only the special case ftin = 0, or e = 0.

If we compare two systems with the same averageion temperatures, one in which all particles areassumed to have the same temperature and anotherwhere this assumption is not made, then it is foundthat € differs in the two cases mainly because of thefollowing two effects. First, the different ion tempera-tures modify the reaction rate parameters, therebyshifting the relative density of the different kinds ofions and thus changing the value of e'. Secondly, theelectron temperature is lowered, as discussed by Post,1

which modifies >rad and ftg and, consequently, e'.It is found that the second effect is the stronger.

We therefore assume, as a first approximation, thatall the ions have the same temperature, which wenow choose as our temperature parameter. Eqs.(3)-(8) can then be solved separately as in the pre-vious section, and Eqs. (15)—(19) can be neglected,since Te is determined by Eq. (20) alone. In this

«p

uf

I*

6'= 0.)

qsO

E9

kT10 20 50 100 200

key

Figure 3. Contour curves for the power reinjection fraction, c',in the (nD/S, 7)-plane. Curves A include the effects of finiteenergy transmission rates. Curves В are taken from Fig. 2

for comparison

equation it is convenient to put ae = 1; e is thencalculated and only the cases where e' = 0 are con-sidered since, for these particular cases, the resultsare independent of the choice ae = 1. These resultsare, however, slightly dependent on the density ofparticles and the temperature through the energytransmission rates between ions and electrons. In thecalculations we have set In Л = 14, which, for theactual temperatures, roughly corresponds to particledensities between 1014 and 1018 cm"3. For furtherdetails see Ref. 10.

To justify the assumption of equating all iontemperatures, we have, in some characteristic cases,solved Eqs. (15)-(19) for the ion temperatures in afirst order approximation. To solve this set of equationswe used the previously found electron temperature.To calculate the energy transmission rate para-meters, the mean energy of reacting particles and thereaction rate parameters, it is also necessary to assumea temperature for the ions. This temperature waschosen as the average ion temperature. Using thevalues of the ion temperatures found in this way onegets a first order correction to e', a correction foundto be insignificant. Also, one can, from Eq. 20, find acorrection to the previously found electron tempera-ture, but this correction was also found insignificant.For further details, see Ref. 7.

The results are given in Fig. 3. Here, contourcurves corresponding to e' = 0 are given in the(пт>/8, T)-plane for rj = 0 and 1. Curves A correspondto the assumption of different temperatures for theions and electrons; and В for comparison, to the casewhere the same temperature is assumed for allparticles. As e' did not reach zero for 77 = 0 in thelatter case, the contour curve for e' = 0.1 is showninstead.

First, it is seen that modifications are not verysignificant. For the parameter пт>/8 the interval withe < 0 is somewhat broadened, because both ftg andftiaa> which dominate the energy balance at large andsmall values of n^/S, are lowered by the lower electrontemperature. This effect is somewhat more pro-nounced at the higher temperatures, mainly becausethe relative difference between the ion and electrontemperatures is larger at higher temperatures sincethe energy transmission rate parameters between ionsand electrons are proportional to Te~

3f2.

The Pulsed Isothermal SystemSince some of the features of the steady-state

system seem rather unrealistic from the point of viewof present-day ideas and lines of development, wethought it appropriate to investigate a system moreclosely connected with the physical situation en-countered in a pulsed gas discharge.

In the model chosen, a mixture of tritium anddeuterium in the ratio rj to 1 is instantaneouslyheated, at time t = 0, to a uniform temperature, Г,and then kept at this temperature until the plasma isinstantaneously cooled down to essentially zerotemperature at t — V. The function ftm in Eqs. (15)-(20)

ENERGY BALANCE IN A PLASMA 437

has thus the character of a delta-function at t = 0 andi = V while it is adjusted in between so as to keep thetemperature of the mixture of fuel and reaction pro-ducts constant. Since we do not, in this case, extractor inject matter we have Д = 8 = 0 in Eqs. (3)-(7)and (15)-(20). Accordingly, we wiD not have a steadystate and àn%\it will, in general, be different from zero.It is further assumed, for the sake of simplicity, thatthere is a common temperature for all kinds of par-ticles in the plasma. These assumptions considerablysimplify Eqs. (3)-(9) and (15)-(20). In Ref. 13 it isshown how Eqs. (3)-(9) can be solved in this case.Only the main results will be presented here.

Three linearly independent integrals of the type^ч^гЩ + ХцПп = Nj can be obtained where the A'sare constants satisfying the conditions

solutions of the form

A4 = — A© + AT+A3

Ap = 2AD — AT

An = 2AD —A3,

(31)

where AD, AT and A3 can be chosen arbitrarily, whilethe Nj's are arbitrary constants to be determined fromthe initial conditions

nj> = Щ, пт = пщ, % = пъ = щ = пп = 0. (32)

The integrals can, for example, be chosen to be

(a) particle conservation:

(33)

It is not,possible to express the Щ'Б in terms ofreal time in any simple form, but if the parametertime, г, defined by

T = JO*'(WD/WO)^ (34)

is introduced, three further integrals are obtained, ofthe type

(b) charge conservation:

"ZimZi = no{l+71)

(c) "tritium conservation":

= пщ

>T. (35)

where y% is any one of the three roots in the equation

y3 + ( / + / / + / / / +JF)y2

+f(J+77).///./F = 0 2 (36)

while Ci is an integration constant determined by the

initial conditions (32),

As may be seen from Eq. (34), the parameter time, r,is approximately equal to t' for small deuteriumburnups.

Solving for щ in Integrals (33) and (35), one gets

щ3

= 2 (38)

where the K's and L's are constants.In order to compare the pulsed system with the

steady state it is necessary to consider the over-allenergy balance in the time interval t = 0 to t = t'.This is reflected in the definition of e, in Eqs. (28) and(29) by the fact that time-integrals of the powerdensities are considered. Thus, it is perfectly possiblethat pin can be negative during a part of the pulsetime, corresponding to nuclear "ignition/' whereasthe pulsed system is never self-sustaining in the moreexacting over-all sense defined by e = 0 because ofthe necessity of injecting power initially.

All the integrations necessary to determine e canbe performed analytically except for the integration ofthe electromagnetic radiative power which must beevaluated numerically. As in the steady state, e is afunction of the temperature, T, and of the tritiumenrichment, 77. The third parameter can either bechosen as the burnup, j8, defined by

which can directly be compared with the burnup inthe steady-state calculations, or it can be chosen asпот, which must then be compared with the parameterпв/й in the steady state.

Computations of e as a function of the deuteriumburnup, j8, has been made in the two cases:

1. rj = 0 and kT = 100 kev; and

2. rj = 0.983 and kT = 5 kev.

In the last case it was permissible to neglect Reaction 4because of the relatively low temperature. The resultsare plotted in Fig. 4, together with the correspondingcurves for the steady-state case. The great similaritybetween the non-steady and the steady case isimmediately noticed. The largest differences are to be

1 8 2 ЯГ1 В 2 Ю*2 б 2

КГ1

kTs5kevг)* 0 3 8 3 — pulsedq.tOQO—steady

ю-2

Figure 4. The power reinjection fraction, e', as a function of thedeuterium burnup, j8, for the isothermal pulsed cases where

r¡ = 0, kT = 100 kev and 1? = 0.983, kT = 5 kevThe corresponding steady-state curves for 77 = 0, kT = 100 kev

and 7) = 1, kT = 5 kev are given for comparison

438 SESSION A-10 P/2506 T. HESSELBERG JENSEN etal.

found for the large burnups corresponding to longpulse times and slow renewal rates respectively. Thisis due to the fact that the pulsed system, because ofthe retarding eñect of the finite reaction times, isrelatively richer in the active primary reaction pro-ducts^ T and He3, while it is poorer in the passivesecondary product He4 compared with the equili-brium values of these isotopes in the steady-statesystem. Nevertheless, it is obvious that both thetrends and the actual values of e as a function of theparameters T, rj and j8 are quite well reproduced bythe steady-state calculations. In particular, it will betrue that optimum values for the three parameterswill exist also for the pulsed system and that theywill not be very much different from the steady-statevalues. For all practical purposes, at the present time,the analysis of the steady-state problem given abovecan be taken as valid also for the pulsed system whenthe comparison is made in terms of the deuteriumburnup or the equivalent parameters п^Ь and n^rrespectively.

DISCUSSION

To be able to treat the problem above it has beennecessary to make some approximations and assump-tions that limit the results. In this section we shalldiscuss some of these limitations.

In order to solve the reaction kinetic equations ithas been necessary to neglect Reactions 5 to 7.Fortunately, these reactions contribute very little tothe power economy in the cases treated here (seeRefs. 3, 4 and 12): over most of the parameter rangesconsidered they contribute less than 10~3 in с

Since not all of the power losses conceivable havebeen taken into account in the power balance calcula-tions, the results must be considered as optimistic.The kind of losses which have been neglected are, forinstance, energy dissipated in magnetic fields neces-sary for containment of the plasma, excessive radia-tion due to contamination of the plasma with heavyatoms, and losses due to particle collisions with thewalls in a finite reactor. Also, the optimum conditionscan only beràchieved in a restricted region in space andtime since temperature gradients and finite heatingtimes are likely to be features of a more realistic fusionreactor.

Although we have tried to keep the number ofspecific assumptions as low as possible, in order toconserve the generality of the results, it has still beennecessary to make some more or less arbitrary deci-sions. This concerns especially the fuel cycling. Thepower balance would obviously be improved if a non-representative mixture of the plasma could be ex-

tracted, containing mainly passive reaction productslike protons and He4 at a temperature lower than theaverage. Since this possibility did not seem verylikely to us we chose to assume a representativemixture at the average temperature to be extracted.The steady-state concept also implies that the fuel ishomogeneously and eontinuausly injected in theplasma. The unrealistic character of this assumptionis largely removed, we think, by the demonstratedsimilarity with the pulsed system. It has also beennecessary to make assumptions about the energydistribution of the particles involved. For obviousphysical reasons, the Maxwellian distribution waschosen; apart from the fact that considerable mathe-matical simplicity was thereby also achieved.

CONCLUSIONS

Finally, we state a few conclusions which can bedrawn from the material presented.

{a) The similarity between results for the steadystate and the isothermal pulsed case indicate thegenerality of the results derived in the former case.Comparisons can be made in terms of the parametersT, r¡ and j8 indicating the temperature, the tritiumenrichment and the deuterium burnup respectively.The burnup parameter can be replaced by the para-meter щт = ¡Qvnjydt which, in the steady state,reduces to njyjb.

(b) The effects of finite energy transfer rates be-tween the charged particles slightly increase the para-meter range for which self-sustaining thermonuclearreactions can be achieved, the extension being mostmarked in the high temperature region, above50 kev.

(c) An optimum value of the power-reinjectionfraction, e', can be obtained for the following values ofthe temperature, Г, the tritium enrichment, 77, andthe deuterium burnup, p; viz., kT = 25 kev, rj = 1 andj8 = 0.3. A self-sustaining reaction can be achievedat a temperature kT « 5 kev in the case of a one-to-one deuterium-tritium mixture, with a correspondingvalue of ¡tf'nvdt « 1016 sec cm~3 and j8 = 0.1.

(d) Even a slight tritium enrichment considerablylowers the temperature necessary to achieve a self-sustaining reaction. (From about kT = 50 kev for77 = 0 to about kT = 9 kev for rj = 0.1 in the steady-state case.) In itself, this has not great consequencessince, at these temperatures, the reactor will mainlybe a tritium burner, but it opens up the possibility ofachieving the higher temperatures by nuclear heatingand thereby initiating the D-D reaction. The copiousneutron production could then be used to produce thetritium necessary for the nuclear ignition.

REFERENCES1. R. F. Post, Controlled Fusion Research—An Application

of the Physics of High Temperature Plasmas, Rev. Mod.Phys., 28, 338-62 (1956).

2. J. D. Lawson, Some Criteria for a Power ProducingThermonuclear Reactor, Proc. Phys. Soc, 70 B, 6-10 (1957).

3. E. Lacombe, D. Magnac-Valette and P. Ciier, EvolutionEnergétique d'un Plasma Thermonuclèaire de Deuterium,Compt. Rend., 246, 744-6 (1958).

4. T. Hesselberg Jensen, O. Kofoed-Hansen, A. H. Sillesenand C. F. Wandel, Some Criteria for a S elf-Sustaining

ENERGY BALANCE IN A PLASMA 439

Steady State Thermonuclear Reaction, Rise Report No. 2,1-20 (1958).

5. F. Ajzenberg and T. bauritsen, Energy Levels of LightNuclei. V, Rev. Mod. Phys., 27, 77-166 (1955).

6. D. N. Van Patter and W. Whaling, Nuclear DisintegrationEnergies, II, Rev. Mod. Phys., 29, 757-66 (1956).

7. T. Hesselberg Jensen and F. Heikel Vinther, On theEffects on a Steady State Thermonuclear Reaction of FiniteEnergy Transfer Rates Between the Charged Particles,Riso Report No. 5 (1958), in press.

8. N. Jarmie and J. D. Seagrave, Charged Particle CrossSections, LA-2014 (1956).

9. S. J. Bame Jr. and J. E. Perry Jr., T(d,n)He* Reaction,Phys. Rev., 107, 1616-20 (1957).

10. C. F. Wandel, Energy Exchange by Coulomb Encounters forMaxwellian Ion Mixtures in the Plasma State, Riso Re-port No. 4 (1958), in press.

11. W. B. Thompson, Radiation from a Plasma, AERET/M 73, 1-8 (1957).

12. T. Hesselberg Jensen, On the Optimization of the TritiumEnrichment with Respect to the Energy Balance in SteadyState Thermonuclear Reactions, Riso Report, No. 6 (1958),in press.

13. O. Kofoed-Hansen and C. F. Wandel, On the EnergyBalance and Reaction Kinetics for Isothermal Thermo-nuclear Reactions Involving Deuterium and Tritium, RisoReport No. 7 (1958), in press.