Fusion energy gain factor

The fusion energy gain factor, usually expressed with the symbol Q, is the ratio of fusion power produced in a nuclear fusion reactor to the power required to maintain the plasma in steady state.

In a fusion power reactor a plasma must be maintained at a high temperature in order that nuclear fusion can occur. Some of this power comes from the fraction of the fusion power contained in charged products which remain in the plasma. This power may be designated "f"_ch"P"_fus. The rest, designated "P"_heat comes from external sources required for heating, some of which may also serve additional purposes like current drive and profile control. This power is lost through various processes to the walls of the plasma chamber. In most reactor designs, various constraints result in this heat leaving the reactor chamber at a relatively low temperature, so that little or none of it can be recovered as electrical power. In these reactors, electrical power is produced from the fraction of the fusion power contained in neutrons, (1-"f"_ch)"P"_fus. The neutrons are not held back by the magnetic fields (in magnetic confinement fusion) or the dense plasma (in inertial confinement fusion) but are absorbed in a surrounding "blanket". Due to various exothermic and endothermic reactions, the blanket may have a power gain factor a few percent higher or lower than 100%, but that will be neglected here. The neutron power heats a working medium such as helium gas or liquid lithium to a high temperature, and the working medium is used to produce electricity at some efficiency η_elec, so that "P"_elec = η_elec(1-"f"_ch)"P"_fus. A fraction "f"_recirc of the electrical power is "recirculated" to run the reactor systems. Power is needed for lighting, pumping, producing magnetic fields, etc., but most is required for plasma heating so we can write "P"_heat = η_heat"P"_elec, where η_heat is substantially the efficiency with which electrical power is converted to the form of power needed to heat the plasma.

The heating power can thus be related to the fusion power by the following equation:

$P\_\{heat\}\; =\; eta\_\{heat\}\; cdot\; f\_\{recirc\}cdot\; eta\_\{elec\}cdot\; (1-f\_\{ch\})cdot\; P\_\{fus\}$

The fusion energy gain factor is then defined as:

$Q\; equiv\; frac\{P\_\{fus\{P\_\{heat\; =\; frac\{1\}\{eta\_\{heat\}\; cdot\; f\_\{recirc\}cdot\; eta\_\{elec\}cdot\; (1-f\_\{ch\})\}$

For the D-T reaction, "f"_ch = 0.2. Efficiency values depend on design details but may be in the range of η_heat = 0.7 and η_elec = 0.4 (See for example [http://www.efda.org/downloads_divers/ppcs.pdf "A CONCEPTUAL STUDY OF COMMERCIAL FUSION POWER PLANTS"] ). The purpose of a fusion reactor is to sell power, not to recirculate it, so a practical reactor must have "f"_recirc = 0.2 approximately. Lower would be better but will be hard to achieve. Using these values we find for a practical reactor "Q" = 22. Of course, "Q" = 15 might be enough and "Q" = 30 might be achievable, but this simple calculation shows the magnitude of fusion energy gain required.

The goal of ignition, a plasma which heats itself by fusion energy without any external input, corresponds to infinite "Q". Note that ignition is not a necessary condition for a practical reactor. On the other hand, achieving "Q" = 20 requires quality of confinement almost as good as that required to achieve ignition, so the Lawson criterion is still a useful figure of merit. The condition of "Q" = 1 is referred to as breakeven. It is somewhat arbitrary, but it does mean that a significant fraction (20%) of the heating power comes from fusion, so that fusion heating can be studied. Above "Q" = 5 the fusion heating power is greater than the external heating power.

The one channel of energy loss that is independent of the confinement scheme and practically impossible to avoid is Bremsstrahlung radiation. Like the fusion power density, the Bremsstrahlung power density depends on the square of the plasma density, but it does not increase as rapidly with temperature. By equating the two power densities, one can determine the lowest temperature for which the fusion power can overcome the Bremsstrahlung power. This ignition temperature is about 4 keV for the D-T reaction and about 35 keV for the D-D reaction.