The purpose of a fusion energy device is to initiate and sustain nuclear fusion reactions under acceptable operational conditions. An understanding of the reaction rates and the time dependence of bum processes is therefore of importance and will now be considered.

7.1 Elementary D-T Burn

Consider a deuterium-tritium ion population at thermonuclear temperatures in a suitable unit volume of interest. Regardless of what approach is used to heat and confine the ions, a quantity of energy Qdt is released whenever the event

d + t —^ n + ex (7.1)

occurs. As discussed in Sec. 2.5, these reactions proceed at the rate

Rdt(r>t) = JJNd(r>vdd)Nt(r>vt.^CTarflvd • v(|)|vd — td3 vd d3 v,

= N*d N*, JJ /d (r, vd, t) f t (r, v,, t)Udt ([vd — v,|)[Vd — v«I d3 vd d3 v,

,t)ft(r,t, t)(jdt(

Vd — vt|)|vd — Ytd3vd d3v,

\fd(T’yd>t)ft(r, vt, t)d3vdd3v, = Nd(r. t)N,(r, t)«JV>dl(r, t),

(7.2a)

in a unit volume where use has been made of Eqs.(6.40) and (6.41). Note that <av>dt appears correctly now as a function of position and time, arising from the averaging process over the particle distribution functions which explicitly depend upon these variables. Accordingly, the rate of energy-density release-that is the fusion power density-is given by

Pdi(r, t)= Rdt(r, t)Qdt = Nd(Г, t)N,(r, t)<CFv>dt(r, t)Qdt. (7.2b)

Recall that Qdt = 17.6 MeV and <CTv>dt is, of course, primarily a function of the ion temperature and is graphically displayed in Fig.2.5 with a tabulation listed in Appendix C.

Evidently then, the fusion power in a unit volume will change if the ion temperature changes with time or if the fusile ion densities vary with time. These variations are governed by the occurring reactions, by the injection rates, and by the leakage rates, all of which add to or subtract from the particle and energy

densities. Adopting our previous rate equation analysis of the form of Eq.(6.4) to that of particle density and energy density variations at a fixed point in space, i. e., г Ф r(t)-or for the case of uniform ion distributions-so that total derivatives can be used, we choose as an appropriate description for a particular ion density Nj(r, t) the general rate equation

^i={F+j-F-j)+{R+j — R. j). (7.3)

Here, as well as in the following, we omit the space-dependent notation in view of the fact that we are only considering the time dependence of the particle density. Further, F+J and F. j are the ion injection rate and leakage rate densities, with F. j equivalent to the term Vxjj(r, t) in the Continuity Equation, Eq.(6.10b), and R+j is the reaction rate density which adds to Nj(t) while R. j is the reaction rate density which removes particles from the population Nj(t) in the fixed unit volume.

For the special case of zero ion leakage and no ion injection, or if these two processes are exactly equal and hence cancel each other in Eq.(7.3), and for R+j=0, the instantaneous time dependence of the deuterium and tritium ions’ densities is therefore specified by the rate equations

= — Rdt(t) = — Nd(?) N, (t) < ov >dt (7.4a)

dt

and similarly

^±=-Nd(t)N,(t)<ow>d, ■ (7.4b)

dt

These equations can be combined in the form

4~(Nd + N,) = -2Nd(t)N,(t)«n>d, • (7-5)

dtv ‘

Suppose that a 50:50% ratio between the deuterium and tritium ions exists so

that

Ni (t) = Nd (t) + N,(t) = j N; (?) + j N; (?) (7.6)

is the total fuel ion density at an arbitrary time during the bum; this gives for Eq.(7.5) therefore

ёЫ±=.5а1>* iVf(?). (7.7)

dt 2

This differential equation is valid in the unit volume of interest and for a bum time under conditions of F+i — F., = 0. Bum times can be very short, of the order of 10‘8 s for inertially confined fusion, or of the order of seconds-and envisaged to be much longer-for magnetically confined systems.

The associated instantaneous fusion power density generated in this bum is obtained by multiplying the d-t reaction rate density with the energy release Qdt per fusion event and thus

Pd, U) = Nd (*) N, (t) < OV >dt Qdt = <aV>* Qd> Nf (t)

A determination of the time dependence of the bum requires a knowledge of how the fuel density Nj(t) at a fixed position in space varies during a bum time Хь — This variation is described by Eq.(7.7) for the case F+i — Fd = 0 and R+1 = 0, and represents a differential equation which is separable to give

jj<av>dl dt 0

for 0 < t < Ть, with Nj(0) = Nj?0 as an initial condition. The left part can be integrated so that

——— + — = -±i<crv>d[ dt (7.10a)

Ni(t) Ni,0 2j0

or

N,(t) =————— ^————— . (7.10b)

— + i<av>dl dt

Ni, o 0

If the ion temperature remains sufficiently constant during the bum-implying that, for example, the alpha particle reaction products transfer some of their kinetic energy to the plasma and so compensate for energy loss by bremsstrahlung radiation, or that external heating is supplied or that excess energy is removed-then <Gv>d, is also a constant and the resultant isothermal fuel ion density variation with time is simply

Ni(t) = —————————— (7.11)

—— + L2«n>dlt

Ni,0

Thus, the important consequence is that the bumup fraction is determined by the product of three factors: NU), <Gv>dt, and Хь — Figure 7.1 illustrates this variation of the bum fraction with the ion temperature T for several bum times.

The fusion power released in a fixed unit volume, Eq.(7.8), is now given by

<ov>* Qd,

Г “|2

and evidently tends to decrease more rapidly with time than the fuel ion density, Fig.7.2.

Fig. 7.1: Fusion fuel burn fraction as a function of ion temperature and burn time for d-t fusion. The initial ion density inventory has been taken as Ni o = 1020 m"3.

Fig. 7.2: Dominant variation of the power and fuel ion density with time.

Finally, the total energy released in a unit volume at a fixed r during the isothermal fusion bum is given by

E*~

In general, with the ion temperature varying during the bum time, the more exact non-isothermal expression can be shown to be