The principle of magnetic induction is based on the phenomenon that a time varying magnetic field will induce a current flow in an electrical conductor. Faraday’s law provides the important relation in the form

V = — n^-¥n (16.16)

dt

where n is the number of identical turns, V is the induced voltage across the conductor ends, and 4/m(t) is the spatially integrated magnetic flux crossing the coil area A

4>Jt) = m-dk. (16.17)

A

Here, B(t) is the time varying magnetic field and dA is the normal differential area. Evidently, a pulsed fusion reactor is necessary.

A schematic of how the magnetic field in a plasma could be coupled to an external circuit is suggested in Fig.16.4. The fundamental energy transformation mechanism is that any expansion of the confined plasma, working against the magnetic field, appears as an induced voltage across the coils, Eq.(16.16).

L———— 1

Fusion Energy

Plasma

чжшии

Г-ШШШЬ

—w————————

Load

Fig. 16.4: Depiction of magnetic coupling by compression-expansion of a long cylindrical
plasma. Part (a) represents the plasma-coil configuration while part (b) suggests its

equivalent circuit.

Two versions of conceivable systems are envisaged. In one, the plasma is first compressed magnetically to ignition with the expansion of the plasma against the magnetic field providing for energy transformation by induction. A
related approach is to move the fusion plasma through a linear bum chamber and plasma energy could be removed by having the plasmoid exit through an expanding section of magnetic field.

For either case, the analysis of electromagnetic coupling follows from the equivalent circuit of Fig. 16.4. The power induced in the secondary magnet coils of n turns is given by

p* =vi = — nl^^-i2 Rc (16.18)

dt

where dT’m/dt is the total rate of change in magnetic flux caused by the expanding plasma, I is the induced current, and I2RC represents Joule-heat losses in the coil. Neglecting Joule-losses, the energy transferred during an expansion

with the plasma radius varying from r2 to r3 in Fig. 16.4 follows by integration

з

£* = — nJ/J4V (16.19)

2

Consider now an isobaric expansion in which a constant magnet current I is required to maintain a constant field В and in which the magnetic field associated with the coils is

B = 4m-L (16.20)

with L as the plasma length. The magnetic flux in the region between the plasma and coil is

Vmj = n(R2-r2pj)B, (16.21)

where the subscript j signifies the radius position so that the corresponding

plasma volume is given by

Vpi = 7Cr2L. (16.22)

Combining these relations we find that the energy transferred to the secondary coil to be

E* =-п1(‘¥т3-Ч>т1) = ^{уp3-VРг) ■ (16.23)

To understand the origin of the transferred energy E*, we consider the various individual components of the expansion work. To simplify the analysis, we take p-the ratio of plasma to magnetic field pressures-as a constant. The expansion work qp+ done by the plasma pressure pp is then

3 2

qP = J PpdV = 0^-(y,3 — Vp2) (16.24)

where we have used Eqs. (9.10) and (9.11) to substitute for the plasma pressure. Flowever, the trapped field Bp in the plasma simultaneously expands and performs work against the confining field. This work is

3 2

4′ = }-&dv=1(,-^Tf(v^-v^ (16-25)

2 2^o 2Я>

where we have introduced the relation Bp2 = (1 — (3)B2 reflecting the condition that the confining field В has to balance the total of kinetic pressure, pp, and the trapped-magnetic-field energy density, Bp2/(2fi0), in the plasma. In addition, as the plasma and trapped field expand, it displaces the confining field originally contained in the volume between r3 and r2. The corresponding energy Ed must appear in the secondary coil along with pdV-work; this term is given by

Ed = y—{vP3-VP2)- (16.26)

The total energy transferred to the coil is then the sum of these three components:

E* = qP+ + ql+Ed (16.27)

which, upon substitution of Eqs.(16.24) through (16.26), agrees with Eq.(16.23).

Similar expressions can be derived for all the components in a cycle; expansion steps transfer energy to the coil, while compression steps require an input energy. Note, however, that the net output will be determined by the plasma-work terms Eq.( 16.24) alone because the energy associated with the trapped and displaced magnetic fields must cancel out in a complete cycle. That is, in order to return to the starting point, the displaced field energy transferred to the coils during expansion steps must be replaced during compression steps. For this reason, we will compute work terms based on the plasma term alone. It must be remembered, however, that the actual energy transferred during any given stage will exceed the plasma term; this becomes important in the actual sizing of coils and the evaluation of joule losses.

As illustrated in Fig. 16.5, a Carnot cycle consists of two isothermal and two adiabatic processes. The term "adiabatic" is used here in a dual sense: first, we assume no heat-transfer or radiation losses occur during the compression or expansion legs; second, since the Carnot cycle is by definition an equilibrium cycle, these stages use slow reversible processes where magnetic-field lines remain "frozen" in the plasma.

A stable plasma with an ideal beta~l confinement is assumed throughout. Furthermore, the number of plasma particles is taken to be constant, corresponding to negligible leakage and bumup; energy carried off by neutrons and radiation is omitted, and additional energy losses such as Joule-heating losses in the magnetic coils are also neglected.

The cycle begins at point 1, where a total of 2N plasma particles (Nj* = Ne* = N *) are assumed to be trapped in the magnetic field. The following steps then follow:

Stage 1 to 2: Slow-adiabatic compression to a maximum field value Brrax and temperature T^. A compression work input q. is required. In the present idealized cycle, any fusion input is neglected prior to

ignition at point 2.

Fusion stage. A constant temperature is maintained during the fusion bum (qf* = fusion-energy input to the plasma) by expanding the plasma. Thus, work q+ is performed against the confining field.

Slow-adiabatic expansion to Bmin, extracting work q+. It is assumed that the temperature and density are reduced so rapidly that fusion input can be neglected after point 3.

Cooling stage. Radiative cooling qr is achieved at constant temperature Tmin by simultaneously compressing the plasma (q. = work input).

During the fusion bum (stage 2 to 3), the plasma expansion work must just balance the fusion input so that isothermal conditions are maintained. To a good approximation the plasma pressure-temperature-volume relations can be taken from ideal gas laws with care taken to incorporate the trapped-field defined (3 parameter. This gives

* ‘ }

?/ = ?+ =J PpdVP

2

= 2N‘kTw In

However, since pressure balance requires that B2 «= V"1 during this stage, Eq.( 16.28) can be written as

During the slow-adiabatic compression (1 to 2) and expansion (3 to 4) stages, ideal gas laws apply so that

1 1

(16.31)

with у denoting the ratio of specific heats. However, because of the isothermal stages, T3 = T2 and T4 = Ti,

Bi _ B4 B2 Вз

Using this result in the relations for qf and qr, we obtain the expansion cycle efficiency

net work out

Bc = —

(16.33)

__ j 1 min

Tmax

As expected, this result corresponds to the classical Carnot efficiency. Indeed, since Tmin could be in the range of 10 eV while Тща* is on the order of 103 eV, a Carnot limit approaching 100% is indicated. While this is encouraging, a basic question remains: How close can an actual cycle approach this ideal limit? Additional losses enter by non-ideal effects which have been neglected here. Another problem is that, for various reasons, it may be impractical to follow a Carnot cycle; for example, internal combustion engines typically utilize an Otto cycle rather than a Carnot cycle. Here, the control necessary to achieve an isothermal fusion bum may be difficult. Indeed pulsed reactor conceptual design studies to date have assumed an isobaric bum.

Another important point to stress is that the expansion cycle efficiency only applies to that portion of the fusion energy that remains in the plasma. Thus neutron and radiation energy that is processed through a thermal cycle must be included in the overall efficiency using the fractional energy flows as weighting factors with each cycle efficiency. Advanced fusion fuels with a larger fraction of
energy retained in charged particles would more closely approach the electro­magnetic cycle efficiency, but even with p-nB, radiation losses from the plasma would probably involve significant energy flows and be processed through a blanket thermal cycle.

A practical consideration in designing devices to operate on expansion — compression cycles is the need for a large plasma chamber in order to accommodate a relatively large expansion ratio. Thus the blanket and magnetic coils in a magnetic confinement device must have large dimensions making them relatively costly. Application to inertial confinement fusion has the advantage that the chamber dimensions are large anyway, compared to the target, in order to accommodate the micro-explosion shock effects. All of these various considerations, fuel cycle, confinement approach, economic and complexity tradeoffs, must be factored into the decision to select a best energy conversion system for a given device.