A more comprehensive d-t isothermal bum analysis needs to include not only the destmction of deuterium and tritium by fusion reactions but also the injection and leakage rates of ions into and from the unit volume of interest. That is, the rate density equation, Eq.(7.3), referring to a fixed point in space is now given by the following:

The ion injection rate density, which is a reactor operations function, will be represented by the fueling rate density

where here, and in the following, we omit the + in front of the j. Ion leakage from a unit volume may be taken to be proportional to the ion density so that

where Xj is a characteristic time parameter representing the local mean residence time of the ions. The determination of this time quantity in terms of various reactor parameters is a continuing task of fusion research but for present purposes here, Xj is taken as a known parameter. Recalling our previous analysis leading to a global particle confinement time and noting that

L=E±=J_ r

x* Nj Nj{Tj(r, t)

suggests thereby that the global particle confinement time is the density-weighted volume average of the local mean residence time. Both these residence times
generally vary with time.

The more complete fuel balance equations for d-t bum are therefore given by the coupled set of equations dNd

dt

dN,

dt

or combining, as was done for Eq.(7.5), d

dt

(7.21)

(7.22)

(7.23)

where the subscript і refers to all fuel ions. This then leads to the dynamical equation for the fuel ion density at a fixed position in space and at some time t, for the three processes here identified, as

dNi

dt Т/ 2

This is a nonlinear first order differential equation, frequently called Ricatti’s equation, and like many nonlinear equations in general, possesses some subtle features. However, by combining several analytical, geometrical and physical notions, we can extract some useful qualitative information about the time variation of the fuel ion density N;(t) at a fixed position in space and hence about the power density Pdt(t), Eq.(7.2b).

For algebraic simplicity, we write Eq.(7.24) as

—— = ao + ai N + a.2 N2 (7.25)

dt

with ao = Fj, ai = -1/ti and a2= -<Ov>dt/2. For the special case when these parameters are constant, then the following information about the slope of an N vs. t plot at an arbitrary time can be specified:

ao > 0 for sufficiently small N

slope (=ao + aiN + a2N )=<

[~ a2 N <0 for sufficiently large N. ^ ^

Further, for the special case of as constant, any fixed points of Eq.(7.25)-to be represented by №-are the solutions of dN/dt = 0; that is

ao + ai № + a2( № f = 0

The above information about the slope of N(t) for small and large N’s, and the recognition that N(t) —» № for t —> °o-and the assumption that ao, aj and a2 can be adequately represented as suitable constants in time-allows us to sketch the solution of Nj(t) of Eq.(7.24) as depicted in Fig.7.3.