User:Mkovari/Sandbox

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 or all of this power comes from the fraction of the fusion power contained in charged products which remain in the plasma. For DT fusion, these are alpha particles. The rest comes from external sources required for heating, current drive and profile control. The plasma loses power to the walls of the chamber by radiation and loss of energetic particles.

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" and shield. Due to various exothermic and endothermic nuclear reactions, additional energy is released in the blanket and shield, but that will be neglected in the equations below. In a fusion power plant the heat deposited in the blanket and shield would heat a working medium such as helium gas or liquid lithium, which would be used to produce electricity or hydrogen fuel.

Some of the electrical power would be recirculated to run the reactor systems. Power is needed for lighting, pumping, producing magnetic fields, etc., but most is required for plasma heating and current drive.

If we define

fch = the fraction of the fusion power contained in charged products which remain in the plasma;

Pfus = the fusion power;

Pheat = power into the plasma from external sources;

ηelec = gross electricity production as a fraction of the power deposited in the blanket and shield;

Pelec = gross electrical power produced;

frecirc = fraction of the gross electrical power required to run the reactor.

ηheat = the efficiency with which electrical power is converted to the form of power needed to heat the plasma.

It is easy to show that

Pelec = ηelec(1-fch)Pfus, and

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 Q is then given by,

$$ Q \equiv \frac{P_{fus}}{P_{heat}} = \frac{1}{\eta_{heat} \cdot f_{recirc}\cdot \eta_{elec}\cdot  (1-f_{ch})} $$

For the D-T reaction, fch = 0.2. Efficiency values depend on design details but may be in the range of ηheat = 0.7 and ηelec = 0.4 (See for example 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 frecirc = 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.

A lower limit to the rate of energy loss can be derived from Bremsstrahlung radiation, which can be reduced but not eliminated. 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, for an optical thin (transparent) plasma, such as is used in magnetic confinement fusion. This ignition temperature is about 4 keV for the D-T reaction and about 35 keV for the D-D reaction.

Ignition
For magnetic confinement fusion, ignition has a precise definition. Ignition occurs when the power deposited in the plasma by the fusion reaction is equal to the rate at which the plasma loses energy to its surroundings. (Energy to the surroundings from neutrons is not included in the rate at which the plasma loses energy.)

The goal of ignition, a plasma which heats itself by fusion energy without any external input, corresponds to infinite Q. The Lawson criterion for magnetic confinement fusion specifies the requirements for ignition. 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. For Deuterium-Tritium fusion only 20% of the fusion energy yield is retained to heat the target. Therefore above Q = 5 the fusion heating power is greater than the external heating power.

Inertial confinement fusion always requires energy input to compress and heat the target, so ignition in the above sense can never be achieved. The term is nonetheless used - some candidate definitions are given below.

1. Ignition is obtained when the fusion target produces more energy than the laser energy required to heat it. . This corresponds to Q>1 (averaging Q over the duration of the reaction).

2. The authors of the JASON report on National Ignition Facility ignition state 'Our working definition of “ignition and propagating burn” (often written simply as “achieving ignition” in this report) is a fusion energy yield at least equal to the laser energy absorbed in the target.'. This definition makes clear that only laser energy actually absorbed by the target is included. At ignition, using this definition, only 20% of the heating power comes from fusion, bringing the term "propogating burn" into question. (Of the laser energy absorbed in the target, 10 - 20% is expected to be absorbed by the fuel capsule when indirect drive is used at NIF.)

The term is also used for thermonuclear bombs, to describe the liberation of a large amount of energy by fusion through the use of any combination of fission and fusion heating.