Physics of fusion power

Lecture 2: Lawson criterion / Approaches to fusion.

Key problem of fusion

…. Is the Coulomb barrier

Cross section

The cross section is the effective area connected with the likelihood of occurrence of a reaction

For snooker balls the cross section is r2 (with r the radius of the ball)

The cross section of various fusion reactions as a function of the energy. (Note logarithmic scale)

1 barn = 10-28 m2

Averaged reaction rate

One particle (B) colliding with many particles (A)

Number of reactions in t is

Both as well as v depend on the energy which is not the same for all particles. One builds the average

The cross section

Schematic picture of the number of reactions in a time interval t

Averaged reaction rate …..

The cross section must be averaged over the energies of the particles. Assume a Maxwellian

Averaged reaction rates for various fusion reactions as a function of the temperature (in keV)

Sizeable number of fusion reactions even at relatively low temperatures

Even for temperatures below the energy at which the cross section reaches its maximum, there is a sufficient amount of fusion reactions due to the number of particles in the tail of the Maxwell distribution Schematic picture of the calculation

of the averaged reaction rate (Integrand as a function of energy)

The Maxwellian (multiplied by the velocity)

The cross section

The product of distribution and cross section

Compare the two

Cross section as a function of energy

Averaged reaction rate as a function of Temperature

The averaged reaction rate does not fall off as strongly when going to lower energies

Current fusion reactor concepts

are designed to operate at around 10 keV (note this is still 100 million Kelvin, matter is fully ionized or in the plasma state)

Are based on a mixture of Deuterium and Tritium

Both decisions are related to the cross section

Averaged reaction rates for various fusion reactions as a function of the temperature (in keV)

Limitations due to the high temperature

10 keV is still 100 million Kelvin (matter is fully ionized, i.e. in the plasma state)

Some time scales can be estimated using the thermal velocity

This is 106 m/s for Deuterium and 6 107 m/s for the electrons

In a reactor of 10m size the particles would be lost in 10 s.

Two approaches to fusion

One is based on the rapid compression, and heating of a solid fuel pellet through the use of laser or particle beams. In this approach one tries to obtain a sufficient number of fusion reactions before the material flies apart, hence the name, inertial confinement fusion (ICF).

Week five ……

Guest lecturer and international celebrity Dr. D. Gericke will give an overview of inertial confinement fusion …..

Magnetic confinement ..

The Lorentz force connected with a magnetic field ensures that the charged particles cannot move over large distances across the magnetic field

They gyrate around the field lines with a typical radius

At 10 keV and 5 Tesla this radius of 4 mm for Deuterium and 0.07 mm for the electrons

Lawson criterion

Derives the condition under which efficient production of fusion energy is possible

Essentially it compares the generated fusion power with any additional power required

The reaction rate of one particle B due to many particles A is

In the case of more than one particle B one obtains

Fusion power

The total fusion power then is

Using quasi-neutrality

For a 50-50% mixture of Deuterium and Tritium

Fusion power

To proceed one needs to specify the average of the cross section. In the relevant temperature range 6-20 keV

The fusion power can then be expressed as

The power loss

The fusion power must be compared with the power loss from the plasma

For this we introduce the energy confinement time E

Where W is the stored energy

Ratio of fusion power to heating power

If the plasma is stationary

Compare this with the fusion power

One can derive the so called n-T-tau product

Break-even

The break-even condition is defined as the state in which the total fusion power is equal to the heating power

Note that this does not imply that all the heating power is generated by the fusion reactions

Ignition condition

Ignition is defined as the state in which the energy produced by the fusion reactions is sufficient to heat the plasma.

Only the He ions are confined (neutrons escape magnetic field and plasma) and therefore only 20% of the total fusion power is available for plasma heating

n-T-tau

Difference between inertial confinement and magnetic confinement: Inertial short tE but large density. Magnetic confinement the other way around

Magnetic confinement: Confinement time is around 3 seconds

Note that the electrons move over a distance of 200.000 km in this time

Energy Cycle in a Power Plant

External Power In

Neutrons (80% of fusion power)

Energy losses due to imperfect confinement

D

T

He (20% of power)

Steam Turbine

Electrical power out

3GW

1GW0.2GW

Waste heat

1.8GW

n-T-tau is a measure of progress

Over the years the n-T-tau product shows an exponential increase

Current experiments are close to break-even

The next step ITER is expected to operate well above break-even but still somewhat below ignition

Some landmarks in fusion energy Research

Initial experiments using charged grids to focus ion beams at point focus (30s).

Early MCF devices: mirrors and Z-pinches.

Tokamak invented in Russia in late 50s: T3 and T4

JET tokamak runs near break-even 1990s

Other MCF concepts like stellarators also in development.

Recently, massive improvements in laser technology have allowed ICF to come close to ignition: planned for this year.

Alternative fusion concepts

End of lecture 2

Force on the plasma

The force on an individual particle due to the electro-magnetic field (s is species index)

Assume a small volume such that

Then the force per unit of volume is

Force on the plasma

For the electric field

Define an average velocity

Then for the magnetic field

Force on the plasma

Averaged over all particles

Now sum over all species

The total force density therefore is

Quasi-neutrality For length scales larger than the Debye length the charge

separation is close to zero. One can use the approximation of quasi-neutrality

Note that this does not mean that there is no electric field in the plasma.

Under the quasi-neutrality approximation the Poisson equation can no longer be used directly to calculate the electric field

The charge densities are the dominant terms in the equation, and implicitly depend on E.

Divergence free current

Using the continuity of charge

Where J is the current density

One directly obtains that the current density must be divergence free

Also the displacement current must be neglected

From the Maxwell equation

Taking the divergence and using that the current is divergence free one obtains the unphysical result:

To avoid this, the displacement current must be neglected, so the Maxwell equation becomes

Quasi-neutrality

The charge density is assumed zero (but a finite electric field does exist)

One can not use the Poisson equation to calculate this electric field (since it would give a zero field)

Valid when length scales of the phenomena are larger than the Debye length

The current is divergence free The displacement current is neglected (this

assumption restricts us to low frequency waves: no light waves).

Force on the plasma

This force contains only the electro-magnetic part. For a fluid with a finite temperature one has to add the pressure force

Reformulating the Lorentz force

Using The force can be written as

Then using the vector identity

Force on the plasma

One obtains

Important parameter (also efficiency parameter) the plasma-beta

Magnetic field pressure Magnetic field tension