A fusion energy system will contain energy in various forms. The relative magnitude of each form and an understanding of how changes in each energy component can be affected need to be considered and assessed. We begin with an overall system energy balance and subsequently examine selected details of energy forms and transformations.

8.1 System Energy Balance

A fundamental requirement of any power system is that it be a net energy producer. That is, during some time interval X, we require the overall net energy

Enet = Е*ош ‘ E*n > 0 (8.1)

where Ein* is the total energy externally supplied to sustain fusion reactions and associated processes and Eout* is the total recovered energy. Note that the asterisk notation is again used to indicate magnitudes referring to the entire reaction volume as opposed to the respective density expressions. While energy can exist in various forms-thermal, kinetic, radiation, electrical, magnetic, etc .-common units need to be employed in energy balance equations; for that, energy in electrical form is particularly useful.

In Fig.8.1, we depict a power plant’s energy flow in a simple general form. The energy eventually delivered to the reaction chamber differs from the provided input energy, Ein, by a factor T|in which, obviously, is the efficiency of converting the different forms of energy involved and of coupling specific energy forms into the fusion plasma. Analogously, the energy released from the fusion domain, which may consist of thermal energy as well as electromagnetic and neutron radiation, will have to be converted-for external use-into electricity with some overall efficiency Т|0ш typical for the employed energy conversion cycles.

Л *-j| 0

Recall the connection between energy and power P0 = dE0 /dt. Indeed, since time variations are of dominant interest, particularly for pulsed systems, it is common in energy accounting to consider the time variations of power. Thus, the basic energy balance Eq.(8.1), may equally be written

We proceed now to identify individual power or energy components

Fig. 8.1: Depiction of energy components associated with a general fusion power plant.

It is useful to recognize that, unlike fission reactors which sustain a fission chain by particle feedback (neutrons) requiring only a critical mass, fusion reactors will need a critical plasma energy concentration. For that, energy has to be initially supplied to the plasma which, upon reaching the critical heat condition, will be capable of sustaining the fusion processes by self-heating due to fusion power deposition in the plasma. Note therefore, that the nuclear fusion chain is maintained by energy feedback.

The achievement of the mentioned critical energy concentration in the plasma is yet an ongoing task in fusion research. Depending on the fusion device and fuel cycle deployed, today’s fusion experiments are far from demonstrating energy self-sufficiency.

In general, the fusion plasma energy will be balanced by external (auxiliary) and internal (fusion power deposition) heating against the several losses due to radiation and other leakage. Considering the thermal energy content in the total plasma volume, that is we take here E = Eth*, we integrate its temporal variation

rate over a typical fusion bum time Tb to relate

tb, ч,

f Щ-dt = E*aux + Efu — El — Elad — f—dt (8.3)

о dt ‘ JoV

where Eaux* is the auxiliary heat supply, Efu represents the fusion energy production, En is the energy of fusion neutrons escaping from the plasma, and

Erad* denotes the radiation losses-all over time Tb; the last term of Eq.(8.3) accounts for energy leakage through plasma transport processes as suggested by Eq.(6.53). Inspection of Fig.8.1 suggests taking

E*aux ~ ‘Піп E*in (8.4)

with T)jn denoting the efficiency of input energy coupling to the plasma. Further, since the ratio

is a constant for the fusion cycle considered (see Table 7.1), we take

E*fu-E*n = fcE)u (8.5b)

with fc representing the charged particle fraction of fusion product energy (recall that fc, dt= 0.2 only). Commonly, the definition of the plasma fusion multiplication or, briefly called the plasma Q-value,

E fu

Пт Eu

Note that Qp is a measure for how efficiently an energy input to the plasma is converted into fusion energy. In the case of an energetically self-sufficient fusion plasma-later on we will call this state an ignited one-that is when the plasma meets its power demands by the fusion energy deposition alone and hence auxiliary heating is no longer required, the plasma Q-value approaches infinity as Ein* —> 0. The left hand side of Eq. (8.7) can be immediately integrated to give

‘■b

jdE*,h= Ел(ть)-Ел(0).

E*rad+ — dt 0 TE’

Obviously then, if the plasma is restored to the same energetic state after the bum, that is if Ел*(Ть) = Eth*(0), the fusion energy generated in this bum is found from Eq.(8.7) to balance the losses by

For the case of energy self-sufficiency, Qp —> «>, the fusion energy delivered to the plasma via the charged reaction products is seen to balance the total energy loss from the plasma. Considering a d-t plasma and collecting the relevant power densities previously derived, we use the explicit expressions given in Eqs.(2.21a),

(7.2b), (3.44) and (5.88), whereby we suggest Erad* to be due to bremsstrahlung

ld3rEth(r, t)_ EiM I т E(r>t) т E'(t)

with TE(r, t) representing the local energy confinement time.

As mentioned previously, the plasma state described by Eq.(8.10), that is when the internal fusion power deposition is capable of balancing the plasma power losses, is commonly referred to as ignition. An auxiliary energy supply is thus no longer needed.

For a fusion reactor operating in steady state the time integrals in Eq.(8.10) may be omitted thus leaving behind a global plasma power balance. Further, if ignition is assumed to occur everywhere throughout the plasma volume (e. g. in a homogeneous plasma), then we may disregard the volume integration as well and finally derive the local d-t fusion ignition condition

fed, Pd, ( Ni ,Ti)= Pbr( Ni, Ne>Te)+ Pc%( Ne, Te) + (8-12)

2 Те

where the thermal plasma energy density has been taken as

E, h,j = 1 NjkTj, j = i, e (8.13)

according to the average particle energy in a Maxwellian distribution, Eq.(2.19c), multiplied by the respective number density. Explicit expressions for the several power terms in Eq.(8.12) may be readily introduced via their definitions in previous sections and thus reveal the complex interrelation between the plasma density and its temperature as required for ignition.