Nuclear Engineering

Fusion Reactions in the Plasma and Reactor Design Aspects of Fusion Reactor

Now from the previous lectures it is clear that to realize fusion reaction as an energy producing source, plasma state of matter is inevitable and hence it is important to know the engineering aspects related to it. First, let us see fusion reactions in the plasma.

There are four feasible hydrogen reactions, all of which probably take place in a hydrogen plasma. These are:

Considering all the above reactions, we could imagine the overall conversion of six deuterons as follows:

This is equivalent to the production of about 10⁵ kW h per gram of deuterium. In the above reactions, when there are only two product particles, the lighter particles carry away the majority of the energy, so that in the first reaction the neutron takes with it three quarters of the reaction energy, i.e. about 2.4 MeV, and so could be detected as a fast neutron. Considering reaction (m3.75) neutron take 14.0 MeV of the Q-value and can be used to replace the used tritium from the lithium breeder reactions

and

where R ^~10^-14 m.

These equations are essential for the design of fusion power reactors.

Now I am going to discuss the design aspects of a fusion reactor.

Design criteria 1: Lawson Criterion

The fusion reactions in the plasma primly depend upon two parameters: confinement time τ of the plasma, and the number density of nuclei. To have an approximate idea, let us consider a cylindrical plasma of length vτ and cross-section α, where v = r.m.s. velocity and α = fusion reaction cross-section. It may be assumed that an average particle traverses the volume αvτ in a time τ, when any other particle within this volume can cause fusion reaction with the test particle. If the number density of particles participating in the fusion reactions or collisions is n, the average number of such particles in the above volume is nαvτ. Therefore, for high probability of fusion, we must have

Now v = (3kT/m)^1/2, where m is the nuclear mass. If T is assumed to be 300 keV, then v

6.5 x 10⁶ m/s (for the D-D reaction). For this reaction and above parameters, the reaction cross-section α

10^-29 m². Therefore, the condition for the high probability of fusion can be written as

; if n is in m^-3 and τ is in seconds, then

This condition was first given by J. D. Lawson, hence it is known as Lawson criterion. For the D-T reaction, the minimum value of nτ obtained is 10²⁰ m^-3s.^2,5

Design criteria 2: Heating of Plasma

There are several physical processes by which plasma can be heated. First heating process is known as ohmic heating. Since plasma is a conductor, it is possible to heat it by passing a current through it, which is nothing but the Joule’s heating. A high temperature can be achieved by passing a large current through it. If J is the current density passing through the plasma and ρ is the resistivity, the rate of heat dissipation per unit volume is ρJ². If the number density of plasma is n, the energy required to raise the temperature upto T is nkT, where k is the Boltzmann constant. For n = 10¹⁶particles per cc and T = 10⁵ K, we have,

nkT = 10¹⁶ x 1.37 x 10^-16 x 10⁵ ergs = 1.37 x 10⁵ ergs.

For a liter of such an assembly, the energy required is 1.37 x 10⁸ ergs = 13.7 joules.⁵

The second method is by magnetic compression. This method is identical with the compression of a gas by adiabatic method, where the rise of temperature of a gas is achieved by increasing the pressure of the gas, and the principle provides a method of heating the ions directly.

A very efficient way heating the plasma is ion cyclotron heating, which is based on the same principle of acceleration of charged particles in a cyclotron. If the plasma is confined by a static magnetic field of strength H, the ions will gyrate about the lines of force with a frequency

where, M is the mass of the ion. If the plasma consists of deuterium ions, the ion cyclotron frequency is ~ 8 x 10² H cycles/sec where H is in gauss and generally for the field used for the confinement of the plasma the frequency laid in the radiofrequency region. If now a radiofrequency field of the same frequency, as ion cyclotron frequency, is superimposed on the plasma by passing a radiofrequency current through a coil concentric with that producing the magnetic field, then the oscillating field will change its direction as the ion completes each half cycle of its motion, and will thus continuously accelerate the ions, which will eventually increase the kinetic energy of the ions and hence will lead to a very high temperature.

Another heating process is by neutral beam injection to plasma. Neutral beam injection consists in shooting high energy particles into the plasma. Charged deuterium particles are accelerated to the required energy level and these accelerated ions then pass through an “ion beam neutralizer” where their electrical charge is removed. The high velocity neutral particles can then be injected into the heart of the plasma where, by way of rapid collision, they transfer their energy to the plasma particles.

Design criteria 3: Confinement of Plasma

The third and the most important design criteria of the thermonuclear power generation is the confinement of plasma. The simplest way to confine plasma is to apply a static electric field. An electrostatic cage can be made by means of a set of charged electrodes, where the pressure produced by electrostatic field must be greater than the thermodynamic pressure created on plasma. This is not an efficient way as plasma loss from the cage cannot be controlled.

The second approach is to confine plasma in a ‘pinch’ device. In this device an extremely high current (million amperes) is sent down the plasma tube. This produces a circular magnetic field that reacts with the plasma to ‘pinch’ it down to a thin filament. This does not ensure loss of plasma through the open ends of the tube as shown in Fig. m3.7.

FIGURE m3.7 Schematic of pinch effect in hot plasma.

In order to prevent loss of plasma through the end points a magnetic mirror arrangement can be designed. In this arrangement, the magnetic field is applied in such a manner that the charged particle is trapped in the central part due to reflection from the end. The magnetic field strength is increased at the end, while it is less at the centre. Such a system is known as magnetic mirror, and under suitable conditions, particles moving from the region of weaker field strength to stronger region will be reflected back and confined2. The lines of force in a magnetic mirror are sketched in Fig. m3.8 as below:

FIGURE m3.8 Magnetic field in magnetic mirror arrangement

In this arrangement ions within certain angular range can only be trapped by reflection.

The shortcomings of both the above confinement scheme was solved in a torus geometry popularly known as Tokamak. Tokamak is a machine where a combination of toroidal and poloidal magnetic field is applied for confining plasma. It is one of the several types of magnetic confinement devices and the leading candidate for producing fusion energy. It was invented in the 1950s by Igor Yevgenyevich Tamm and Andrei Sakharov. It is the most promising of all the devices that have been suggested for the confinement of plasma. The tokamak is characterized by azimuthal symmetry and the use of the plasma current to generate the helical component of the magnetic field necessary for stable equilibrium. In a tokamak, achievable plasma densities (i.e. the number of particles per unit volume) are around 10²⁰ per cubic meter (m^-3). Plasma confinement in a tokamak is based on the property that charged particles make a helical trajectory around the magnetic field lines [Fig. m3.9].

FIGURE m3.9 Toroidal magnetic field in a tokamak.

The radius of gyration, called Larmor radius depends on the intensity of the magnetic field, the mass and charge of the particle, and its energy. The stronger the magnetic field, the smaller the Larmor radius. Moreover, the electrons, much lighter than the ions, have a smaller Larmor radius of the same energy. Very energetic particles have much larger Larmor radius than low energy particles, and are therefore more difficult to confine. The Larmor radius may typically vary from several millimeters for not very energetic particles with an intense magnetic field, to tens of centimeters for highly energetic particles. Due to gradient and curvature of magnetic field, positive and negative particles move in opposite directions, resulting in the separation of charges. The plasma is thus driven out due to the electric field developed in this way. In order to prevent this, a relatively weak poloidal magnetic field is created by sending a current along the toroidal direction (Fig. m3.10). This field provides a ‘pinch’ to the plasma and confine it well within the torus.

FIGURE m3.10 Poloidal magnetic field configuration

If the ratio of the toroidal and poloidal magnetic fields is greater than a critical value, the instability of the plasma is minimized. The critical value depends on the shape and volume of the torus. For an ideal case, the theory of tokamak defines a quantity called merit factor (m), which is^x

where r is the minor radius of the torus and R is the major radius. Maximum plasma stability is obtained when q ≥ 3, and this implies that for a workable size of torus with r:R about 1:3 the toroidal field B_t must be much greater than poloidal field B_p. This is the basis of all tokamak configuration^x.

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Fusion Reactions in the Plasma and Reactor Design Aspects of Fusion Reactor

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