Thermonuclear inverse magnetic pumping power cycle for stellarator reactors

PPPL-2249 · UC20-D,G

Thermonuclear Inverse Magnetic Pumping Power Cycle for Stellarator Reactors

D. D.-M. Ho and R. M. Kulsrud, Plasma Physics Laboratory, Princeton University, P.O. Box 451, Princeton, New Jersey 08544. September 1985. Prepared for the U.S. Department of Energy under Contract DE-AC02-76-CHO-3073. Ho was at the time of publication at Lawrence Livermore National Laboratory, University of California, Livermore, California 94550.

Abstract

A novel power cycle for direct conversion of alpha-particle energy into electricity is proposed for an ignited plasma in a stellarator reactor. The plasma column is alternately compressed and expanded in minor radius by periodic variation of the toroidal magnetic field strength. As a result of the way a stellarator is expected to work, the plasma pressure during expansion is greater than the corresponding pressure during the compression. Therefore, negative work is done on the plasma during a complete cycle. This work manifests itself as a back-voltage in the toroidal field coils, and direct electrical energy is obtained from this voltage. For a typical reactor, the average power obtained from this cycle (with a minor radius compression factor on the order of 50%) can be as much as 50% of the electrical power obtained from the thermonuclear neutrons without compressing the plasma. Thus, if it is feasible to vary the toroidal field strength, the power cycle provides an alternative scheme of energy conversion for a deuterium-tritium fueled reactor. The cycle may become an important method of energy conversion for advanced neutron-lean fueled reactors. By operating two or more reactors in tandem, the cycle can be made self-sustaining.

1. Introduction

A novel power cycle for direct conversion of alpha-particle energy to electricity is proposed for stellarator reactors. This power cycle provides an alternative scheme of energy conversion for deuterium-tritium (D-T) fueled reactors, and could become an important method of energy conversion for advanced neutron-lean fueled (e.g., deuterium-helium-3) reactors. The direct energy conversion is achieved by alternating compression and expansion of the plasma minor radius. The cycle is composed of three stages as shown in Fig. 1.

At stage 0, ignited plasma is thermally stable and in thermal balance. The plasma beta is slightly above the critical beta. (Beta is the plasma pressure divided by the internal magnetic field pressure, i.e., the ratio of plasma kinetic pressure to the internal magnetic field pressure, and the critical beta, the maximum attainable beta, is set by the onset condition for MHD ballooning instabilities.) During the compression phase (from stage 0 to stage 1), plasma is compressed adiabatically by increasing the external toroidal magnetic field strength, and the plasma beta decreases. At stage 1, the end of the compression phase, beta is less than the critical beta and the plasma is no longer in thermal balance since the thermonuclear alpha-particle heating power exceeds the rate of energy loss (due to neoclassical transport, turbulent convection, and bremsstrahlung radiation). During the heating phase (from stage 1 to stage 2), as the plasma temperature and beta are driven up by the excess heating power, the plasma volume is kept constant by further increasing the external field strength. As beta approaches the critical beta, the rate of energy loss increases until it balances the alpha-particle heating power when beta reaches the critical value. The plasma is again in thermal balance and is thermally stable. This is stage 2 of the cycle.

To complete the cycle, the external field is then reduced so that the plasma expands back to its original radius during the expansion phase (from stage 2 to stage 0). When the plasma expands, the plasma beta tends to increase. However, beta is already at the critical beta at stage 2, hence ballooning instabilities force beta to stay at the critical value through turbulent convection during the entire expansion phase. As a result, the plasma pressure during the expansion is higher than the corresponding pressure during the compression. Therefore, negative work is done on the plasma during a complete cycle. Note that if there is no beta limit, then more work can be obtained during a complete cycle. This work manifests itself as a back-voltage in the toroidal field coils, and direct electrical energy is obtained from this voltage. By operating two or more reactors in tandem, the cycle can be made self-sustaining.

As an alternative cycle, net work can also be done on the external system by letting the plasma expand at constant pressure between stages 1 and 2, rather than keeping the plasma at constant volume. The reason that these two methods (heating phase at constant plasma volume and at constant plasma pressure) are chosen for discussion in this paper is their simplicity for analytical study. There are many other possible methods that can also result in net work done, e.g., the plasma minor radius varies sinusoidally during the cycle.

If the external magnetic field is considered to be a piston, then the external system absorbs work done by the pumping action of the plasma on the piston. This action is referred to as magnetic pumping in this paper. It is inverse magnetic pumping over a full cycle because work is done on the external system by the plasma. Note that this concept of magnetic pumping differs from the traditional definition which refers to gyro-relaxation, as in Refs. 2 to 6.

This paper is organized as follows: In Sec. 2, the physics of the cycle is analyzed. In Sec. 3, the work done and performance of the cycle are calculated, and an analogy to a conventional internal combustion engine is made. In Sec. 4, the mode of operation and a sample calculation are presented. The conclusion is given in Sec. 5. Appendix A gives an alternative method (via the voltage-current relation) for calculating the electrical work extracted. In Appendix B, the thermonuclear thermal stability is discussed. Finally, a detailed analysis of the temporal evolution of the plasma temperature after compression is presented in Appendix C.

2. Physics of the Cycle

The detailed physical processes at, and between, various stages of the cycle are analyzed in this section. To simplify the analysis, the plasma column is modeled by a straight cylinder. The plasma density, temperature, and the magnetic field are assumed to be uniform across the minor radius except for a sharp discontinuity across the boundary between the plasma and the confining vacuum magnetic field. Throughout this analysis, the plasma is assumed to behave like an ideal MHD fluid.

Before the compression, the ignited plasma is in thermal balance, at a stable thermal equilibrium (see Appendix B). The plasma beta is slightly above the critical beta. Thus, the plasma pressure gradient exceeds the critical pressure gradient by a small amount. Turbulent convective cells are driven by this excess pressure gradient and carry part of the outward particle and energy fluxes; the rest of the fluxes are carried by neoclassical transport. This is stage 0 in the power cycle. The plasma and magnetic field parameters are the plasma beta, the plasma pressure, the plasma minor radius, the internal magnetic field and the external magnetic field, each carrying the subscript 0 to refer to the physical quantities at stage 0.

During the compression phase from stage 0 to stage 1, the plasma is compressed adiabatically as the external toroidal magnetic field strength is increased. Adiabatic compression means that the compression time is shorter than the burn time (the time for the thermonuclear alpha-particles to increase the plasma temperature by an amount comparable to itself). It is important for the compression to be adiabatic because otherwise, the alpha-particles could heat up the plasma and more work would be required during the compression phase. On the other hand, as explained later in this section, the compression phase must be carried out slowly compared with the ninety-degree deflection time for ions so that the compression is three-dimensional.

The plasma beta decreases during the compression because the internal magnetic field pressure increases faster than the plasma kinetic pressure. This can be shown explicitly if the plasma beta is expressed in terms of the pressure and internal field at stage 0. Using the adiabatic compression law (pressure times volume to the power gamma is constant, with gamma equal to five-thirds), the definition of beta, and the frozen flux condition, the plasma beta can be expressed as equation 1: the beta at time t equals the beta at stage 0 divided by the compression ratio at time t raised to the two-thirds power. Here the compression ratio at time t is the minor radius at stage 0 divided by the minor radius at time t. During the compression, the compression ratio increases from unity to its final value, and thus beta decreases. If the critical beta is roughly constant, then the plasma beta is below the critical beta after compression. This is stage 1 in the power cycle. The plasma and magnetic field parameters at stage 1 are: beta lower than the critical beta, pressure greater than at stage 0, minor radius smaller than at stage 0, internal magnetic field greater than at stage 0, and external magnetic field greater than at stage 0. The subscript 1 refers to the physical quantities at stage 1.

After the adiabatic compression, the increase in alpha-particle heating power per unit volume is larger than the increase in neoclassical energy diffusion and bremsstrahlung radiation losses per unit volume. Also, turbulent convective transport can be assumed to be absent after compression since beta is less than the critical beta. This imbalance between the heating power and the power loss causes the plasma temperature to increase while the plasma volume is kept constant by further increasing the external magnetic field strength. The increase in temperature can be understood in more detail from the volume-averaged energy balance equation, equation 2: three times the density times the rate of change of temperature equals the alpha-particle heating power minus the sum of the neoclassical energy loss rate, the convective energy loss rate and the radiation loss rate.

The first term on the right-hand side of this equation is the heating power from the thermonuclear alpha-particles. The first term inside the bracket is the energy loss rate due to neoclassical diffusion, equation 3, which is the sum of the electron and ion energy contents divided by their respective energy confinement times. From Ref. 8 the electron energy confinement time is given by equation 4 (not reproduced; the expression did not survive the scan) and the ion energy confinement time is equal to the electron energy confinement time. Here, the minor radius and the major radius are in meters, the magnetic field is in units of 10 kG, electron or ion density is in units of 10 to the 14 per cubic centimetre, plasma temperature is in units of 10 keV, h is the depth of the helical magnetic well caused by external helical windings, and the normalized ambipolar electric field strength can be approximated by unity. The second term inside the bracket in Eq. 2 is the convective energy loss rate. The last term is the energy loss rate due to bremsstrahlung radiation and is equal to a constant times the density squared times the square root of the temperature. Note that density is not a function of time during the heating phase since the plasma volume is held fixed. Also note that before the compression, thermal balance means that the rate of change of temperature is zero.

The ratio of the value of each term on the right-hand side of Eq. 2 after the adiabatic compression to the value of the corresponding term before the compression will now be studied. Because of the increase in plasma density and temperature after the adiabatic compression, the rate of thermonuclear alpha-particle energy production is larger than that before the compression by the ratio given in equation 5, which is the square of the density ratio times the ratio of the Maxwellian reactivities at stage 1 and stage 0. Here, 3.5 MeV is the energy of an alpha-particle generated from D-T reactions. If the compression ratio is one over 0.6 and the temperature at stage 0 is 10 keV, then the temperature at stage 1 is 19.8 keV. Using the formula for the D-T reactivity given in Ref. 9, Eq. 5 has a value of 32.1.

To obtain the ratio of the neoclassical energy loss rate per unit volume after the compression to that before the compression, we let the neoclassical loss be three times the density times the temperature divided by the confinement time. Then, it can be shown, equation 6, that this ratio is the compression ratio raised to the fourth power, which has a value of 7.72 for a compression ratio of one over 0.6. The turbulent convective transport is assumed to vanish at the end of the compression phase since beta drops below the critical beta. Finally, the ratio of the bremsstrahlung radiation energy loss rate after compression to that before the compression, equation 7, is the compression ratio raised to the fourteen-thirds power, which has a value of 10.85 for a compression ratio of one over 0.6.

From Eqs. 5 to 7 we can conclude that alpha-particle heating will further increase the plasma temperature after the adiabatic compression since the relative increase in the alpha-particle heating power is larger than the relative increase in the energy loss rates. This phenomenon of plasma temperature rise after compression is of fundamental importance to the success of the power cycle, since the plasma must have a higher pressure during the expansion phase than during the compression. As the temperature rises, the plasma beta increases since the plasma volume is held fixed. As beta approaches the critical beta, the convective energy loss re-emerges and the convective energy loss rate gradually catches up with the increase in the alpha-particle heating power. After the compression, the plasma temperature asymptotically approaches the limit at which beta equals the critical beta in a characteristic time defined as the thermal relaxation time (see Fig. 8). (In Appendix C, the temporal evolution of the plasma temperature is treated in detail and the thermal relaxation time is estimated.) At stage 2 in the power cycle, the plasma is again thermally stable and in thermal balance at a stable equilibrium. The plasma and magnetic field parameters at stage 2 are: beta equal to the critical beta, pressure greater than at stage 1, and the minor radius, internal magnetic field and external magnetic field as listed in the report.

To complete the cycle, the external magnetic field is decreased so that the plasma expands back to its original minor radius. This is the expansion phase. When the plasma expands, the plasma beta tends to increase. (According to Eq. 1, beta increases as the compression ratio decreases.) However, beta is already at the critical beta at stage 2, hence ballooning instabilities force beta to stay at the critical value (recall that the critical beta is assumed to be a constant) through turbulent convection during the entire expansion phase. Consequently, the plasma pressure during the expansion phase can be expressed in terms of the external field and the critical beta, and upon applying the frozen flux condition this expression can be written as equation 8, in which the pressure varies as the inverse square of the plasma volume, i.e., exactly as though it had an adiabatic index gamma equal to 2.

As a result, the plasma pressure during the expansion is higher than the corresponding pressure during the compression. Therefore, negative work is done on the plasma during a complete cycle. This work manifests itself as a mean back-voltage in the toroidal field coils, and direct electrical energy is obtained from this voltage. Using the pressure-volume relation, the amount of work done on the coils and the thermal efficiency of the cycle are calculated in Sec. 3. This work is also calculated by using the voltage-current relation in Appendix A, and the result agrees with that obtained from using the pressure-volume relation. It is now clear that if the compression phase had been carried out faster than the ninety-degree deflection time for ions, then the compression would be two-dimensional (gamma equal to 2) and the pressure variation during the compression would follow Eq. 8 instead of the five-thirds adiabatic law. Consequently, no net work would be done on the external system since the plasma pressure during the compression phase would equal the corresponding pressure during expansion.

Finally, the cycle described here satisfies the Kelvin-Planck statement of the second law of thermodynamics by losing heat (obtained from thermonuclear alpha-particles) to the outside through turbulent convection.

3. Cycle Energetics and Performance

During the heating phase, between stages 1 and 2, the plasma can either be held at constant volume or be allowed to expand at constant pressure while receiving energy from the thermonuclear alpha-particles. In a complete cycle, both methods result in net work done on the external system. The work done and thermal efficiency for each of these methods are calculated in this section.

3.1. The thermonuclear inverse magnetic pumping power cycle with heating phase at constant plasma volume

To calculate the amount of work delivered during this cycle, it is only necessary to consider the work performed by the plasma during the compression and expansion phases. No work is performed during the heating phase because the plasma volume is constant.

The work done on the plasma during the adiabatic compression is given by equation 10 (not reproduced; the expression did not survive the scan). Invoking the relation that the pressure at stage 0 is the critical beta times the internal magnetic field pressure, together with the pressure balance equation stating that the plasma pressure plus the internal magnetic field pressure equals the external magnetic field pressure, the pressure at stage 0 can be expressed as equation 9: the critical beta divided by one plus the critical beta, multiplied by the external magnetic field pressure at stage 0. Using this expression, the compression work can be rewritten in terms of the machine parameters, where the length of the system is twice pi times the major radius. The work done by the plasma during the expansion phase is calculated by using Eq. 9, giving equation 11 (not reproduced), and therefore the amount of net work done on the coils during a complete cycle is given by equation 12 (not reproduced).

This expression shows that either a higher compression ratio or a larger critical beta will give a larger amount of delivered work. The net work done normalized by the machine dependent parameters is given by equation 13: the compression ratio squared, minus three halves of the compression ratio raised to the four-thirds power, plus one half, which is plotted versus the compression ratio in Fig. 2. Although it is desirable to operate the cycle at a high compression ratio, the attainable compression ratio may depend on the highest achievable toroidal magnetic field. This field is limited by the allowable static and dynamic loading on the machine structure. The amount of work delivered by the cycle can be represented by the area of the triangle 0-1-2 in the pressure-volume diagram shown in Fig. 3.

The performance of this cycle is characterized by the thermal efficiency, which is defined as the net work divided by the heat obtained by the cycle (engine) from a heat source (thermonuclear alpha-particles). To calculate the heat received by the plasma during the heating phase, use the first law of thermodynamics, equation 14: the heat supplied equals the change in plasma internal energy between stages 1 and 2 plus the work done between stages 1 and 2. Since the plasma volume is held fixed during the heating phase, the heat supplied equals the change in internal energy. From the relation between the pressure at stage 2, the critical beta and the internal field, the frozen flux condition, and the adiabatic compression law, it can be shown that the heat supplied is given by equation 15 (not reproduced), and using Eqs. 12 and 15 the thermal efficiency can be expressed as equation 16 (not reproduced).

The important thing to note is that the efficiency of this cycle is a function only of the compression ratio and increases with it. The thermal efficiency is plotted versus the compression ratio in Fig. 4. As the compression ratio approaches infinity, the thermal efficiency approaches a value of two-thirds.

The thermal efficiency can be visualized by looking at the temperature-entropy diagram for the cycle, Fig. 5. The thermal efficiency is the ratio of the area of the triangle 0-1-2 to the area beneath the path 1-2.

Up to this point, the discussions have been restricted to the case in which there is a limiting beta. However, the cycle will become more efficient and more work can be obtained if the critical beta is very high or if there is actually no beta limit (this situation may occur in Heliac, Ref. 11). For this case the work required to compress the plasma is given by Eq. 10. The plasma pressure at stage 2 may now reach a higher value than in the case where there is a beta limit. (The temperature at stage 2 can be obtained by the graphical method discussed in Appendix B. Note that turbulent convection is absent here since there is no beta limit.) During the expansion, the plasma pressure variation follows the five-thirds adiabatic law since beta will no longer be prevented from rising. Hence the expansion work is given by equation 17 (not reproduced), where the subscript zero-prime denotes conditions at stage 0-prime, the moment when the expansion phase is completed and the plasma volume returns to its original value at stage 0. At stage 0-prime the pressure is greater than at stage 0 and the power losses are greater than the alpha-particle heating power. Thus, the plasma temperature will decrease. While the temperature is decreasing, the external magnetic field is reduced in order to keep the plasma at constant volume, and heat diffuses to the outside by neoclassical transport. The cycle is completed when the plasma temperature and external magnetic field return to their original values at stage 0 (point C in Fig. 7).

The net work is given by an expression in which the temperature at stage 0-prime can be related to the temperature at stage 1 using the adiabatic law. The thermal efficiency, equation 18, is one minus one divided by the compression ratio raised to the four-thirds power, which is the Otto cycle efficiency. Note that this cycle is identical to the Otto cycle.

3.2. The thermonuclear inverse magnetic pumping power cycle with heating phase at constant plasma pressure

In this power cycle, the compressed plasma is allowed to expand at constant pressure until the plasma beta reaches the critical beta. At this point (stage 2), the plasma radius is between the minor radius before the compression and that after the compression. Next, the plasma column is expanded further with beta staying at the critical value until the minor radius reaches the precompression value. The cycle is now complete.

To calculate the amount of work delivered during this cycle, note that the work required to compress the plasma is the same as that of the previous cycle (see Eq. 10). To calculate the work done by the plasma on the confining field it is necessary to know the plasma minor radius at the end of the constant pressure heating phase. Starting with the relation between the pressure at stage 1, the critical beta and the internal field at stage 2, using the frozen flux condition, and noting that the pressure at stage 2 equals the pressure at stage 1, it can be shown, equation 19, that the minor radius at stage 2 is the minor radius at stage 1 multiplied by the compression ratio raised to the one-sixth power.

The work done by the plasma on the coils during the constant pressure heating phase is given by an integral of pressure over volume between stages 1 and 2, and using Eq. 9 the work done by the plasma during the final constant-beta expansion phase is given by a further integral. Therefore the amount of net work done on the coils during a complete cycle is given by equation 20, in which the normalized work is again a function of the compression ratio alone and is plotted versus the compression ratio in Fig. 2. (The exponents in the normalized work expression did not survive the scan and it is not reproduced.) Again, the amount of work delivered by this cycle is represented by the area of the triangle 0-1-3 in the pressure-volume diagram, Fig. 3.

The thermal efficiency, which again depends only on the compression ratio, is given by equation 21 (not reproduced). The thermal efficiency is plotted versus the compression ratio in Fig. 4. As the compression ratio approaches infinity, the thermal efficiency approaches a value of four-fifths. Now, the thermal efficiency is the ratio of the area of triangle 0-1-3 to the area beneath the path 1-3 in Fig. 5. Thus, for the same compression ratio, this cycle delivers less work but at a higher thermal efficiency than the previous cycle. From the standpoint of reactor economics, it is probably the net work done, rather than the thermal efficiency, that is important. Hence, the cycle described in Sec. 3.1 appears to be an appropriate choice.

4. Mode of Operation and Sample Calculation of Output Power

4.1. Mode of operation

To make the power cycle self-sustaining, part of the work done by the plasma on the coils during the expansion phase must be stored in order to supply power for the next compression phase. However, the amount of energy to be stored is large compared with the net work done by the plasma during the cycle. Thus, even if the ratio of the energy lost (during the transfer to compensate for the loss in the storage system) to the total energy transferred from the reactor to the storage system (this ratio is defined as the recirculation inefficiency) is only a few percent, the total energy lost in recirculation may still exceed the work done during the cycle. Consequently, the power cycle may require a lower recirculation inefficiency than can be provided by conventional energy storage systems, e.g., capacitive, inductive, and inertial (motor-generator-flywheel). To obviate this, we propose to operate two (or more) stellarator reactors in tandem. The major electrical energy loss in this system is the resistive dissipation in the coils, but this loss is small compared with the net work performed by the cycle as is shown in the second part of this section.

This proposed reactor system consists of two identical stellarator reactors A and B. The cycle with heating phase at constant volume will be used for discussions here. The operation scenario is as follows (see Fig. 6). When the plasma in reactor A is expanding, some of the resulting work is used to compress the plasma in reactor B. When the expansion phase in reactor A is completed, the plasma in that reactor returns to its pre-compression state (stage 0) while the plasma in reactor B completes the compression phase and is at stage 1. The plasma in reactor A is now maintained at stage 0 temporarily while the external magnetic field strength in reactor B is increasing so that the plasma volume in reactor B can be kept constant as the plasma temperature is increasing. During this period, the power used to increase the magnetic field strength in reactor B comes either from the electrical power generated by the thermonuclear neutrons in reactor A or from that in reactor B itself. When the plasma in reactor B reaches thermal balance (stage 2), it is allowed to expand. Some of the resulting work obtained from reactor B during its expansion phase is used to compress the plasma in reactor A. The cycle continues in this manner. The current in the coils of reactors A and B as a function of time is illustrated in Fig. 6. Note that this proposed coupled reactor system is analogous to a two-cylinder, internal combustion engine.

4.2. Sample calculation of output power

The averaged total electrical output power obtained from direct conversion is defined, equation 22, as the net work done during the cycle minus the cyclic time integral of the resistive power dissipation in the coils, all divided by the period of one complete cycle. In this section, only the power cycle with heating phase at constant plasma volume is considered. Also note that the neoclassical energy loss, which could in principle be converted into electricity, is not included in Eq. 22.

As an example, for a stellarator reactor with reactor parameters given in Table I, the thermal relaxation time required by the plasma to reach stage 2 after the adiabatic compression with a compression ratio of one over 0.6 is approximately 0.16 sec as indicated in Appendix C. If the compression and expansion phases are each carried out in 0.01 sec (note that the ninety-degree deflection time is of order 10 to the minus 3 sec for the plasma with parameters given in Table I), then the cycle period is about 0.18 sec. With this information and using Eq. 12, it can be shown that the net work divided by the period is 2.2 GW.

If the plasma radius remains constant, then at the plasma radius the thermonuclear neutron power is about 12.2 GW. The electrical power obtained from the neutrons at a conversion efficiency of one-third is about 4.1 GW. Thus, the net work divided by the period is about 50% of the electrical power from the neutrons. Note that the net work divided by the period for each reactor in the coupled stellarator system may be lower than the 2.2 GW since the period for the coupled system is longer (see Fig. 6). Also note that the actual values of the net work divided by the period and of the neutron power should be lower than the results obtained here since the radial profiles of density and temperature must be taken into account properly.

To calculate the resistive power dissipation in the coils, note equation 23: the dissipation is the current squared times the coil resistance, multiplied by a factor which is the room temperature minus the coil temperature, divided by the coil temperature. That factor is an approximation for the thermodynamic efficiency of the refrigeration cycle. Here the room temperature and the cryogenic temperature of the cooled coils are as stated. The current in the coils is given by the magnetic field divided by four pi and by the number of turns of the coil per meter, multiplied by the speed of light. The coil resistance is expressed in terms of the resistivity, the total length of the coil (which is two pi times the distance between the coil and the center of the plasma, times the length of the system, times the number of turns per meter) and the coil cross-sectional area. If b is the width of the coil (measured along the direction of the minor radius), then the cross-sectional area is b divided by the number of turns per meter.

Using the reactor parameters in Table I and setting the coil radius to 3 m, the coil width b to 0.5 m, the room temperature to 293 K, and the coil temperature to 77 K (the boiling point for liquid nitrogen), it can be shown that the total resistive power dissipation in a copper coil during a complete cycle is on the order of a few percent of the net work. Hence, the cyclic integral of the dissipated power can be neglected and the direct-conversion power is 2.2 GW.

It may be thought that more neutron power could be obtained by increasing the confining magnetic field strength and operating the plasma at a higher pressure without pulsing the minor radius. However, this mode of operation is undesirable since the neutron wall load would exceed the present day engineering limit which is below 10 MW per square metre.

5. Conclusions

We have proposed a power cycle for direct conversion of alpha-particle energy into electricity for stellarator reactors. The direct energy conversion is achieved by alternating compression and expansion of the plasma minor radius. In analyzing each phase of the power cycle, there appear to be no major physical or engineering obstacles that would make the cycle impractical. For the stellarator reactor with parameters given in Table I, the averaged power obtained from the cycle with compression ratio 1/0.6 is about 50% of the electrical power obtained from the thermonuclear neutrons from the same reactor without compressing the plasma. Thus, the cycle may provide an alternative scheme for extracting energy from D-T fueled reactors. This scheme can be used either alone or as a supplement to the electrical power generated from thermonuclear neutrons. For advanced neutron-lean fueled reactors, this power cycle may become an important scheme for energy conversion. The possibility of using this cycle for tokamak reactors, however, has not been pursued in this paper because of the uncertainty in the behavior of anomalous transport in tokamaks.

Acknowledgments

It is a great pleasure to thank D. L. Jassby, J. L. Johnson, D. W. Kerst, R. G. Mills, S. Yoshikawa, D. A. Buchenauer, R. G. Hay, B. J. Micklich, and E. R. Salberta for illuminating discussions. This work is supported by United States DoE Contract No. DE-AC02-76-CHO-3073.

(Appendix A, calculation of the power cycle work done from the voltage-current relation; Appendix B, thermonuclear thermal stability; and Appendix C, temporal evolution of the plasma temperature after adiabatic compression, are omitted for length. The complete report is at the source.)

Table I. Stellarator reactor parameters

| Parameter | Value | | --- | --- | | Plasma density (per cubic centimetre) | 6 times 10 to the 14 | | Plasma temperature (keV) | 10 | | Toroidal magnetic field (kG) | 50 | | Plasma beta (%) | 19.3 | | Major radius (m) | 10 | | Minor radius (m) | 2 |

Figure captions

Fig. 1. Minor cross section of a stellarator reactor at various stages during the thermonuclear inverse magnetic pumping power cycle.

Fig. 2. Net work done normalized to machine dependent parameters versus compression ratio. (a) Power cycle with heating phase at constant plasma volume. (b) Power cycle with heating phase at constant plasma pressure.

Fig. 3. Pressure-volume schematic. Path 0 to 1 is the compression phase, paths 1 to 2 and 1 to 3 are, respectively, the heating phase at constant plasma volume and constant plasma pressure. Path 2 to 0 (3 to 0) is the expansion phase.

Fig. 4. Thermal efficiency versus compression ratio. (a) Power cycle with heating phase at constant plasma volume. (b) Power cycle with heating phase at constant plasma pressure.

Fig. 5. Temperature-entropy schematic. Path 0 to 1 is the compression phase, paths 1 to 2 and 1 to 3 are, respectively, the heating phase at constant plasma volume and constant plasma pressure. Path 2 to 0 (3 to 0) is the expansion phase.

Fig. 6. Schematic representation of currents in coils and electrical energy flow between reactors as a function of time for a coupled reactor system (for the power cycle with heating phase at constant plasma volume). The current in the coils when the reactor is at stage 0, and the period of one complete cycle, are marked.

Fig. 7. Comparison of heating power with power losses for the plasma in the stellarator with machine parameters given in Table I. The solid line represents alpha-particle heating power per unit volume, the dashed line represents power loss per unit volume due to neoclassical diffusion and bremsstrahlung radiation, and the broken line represents the total power loss per unit volume (plus turbulent convection). Point A is an unstable thermal equilibrium, points B and C are stable thermal equilibria.

Fig. 8. Temporal evolution of the plasma temperature after adiabatic compression for the reactor with parameters given in Table I. The solid curve is obtained using Eq. C8 and the dashed curve is obtained using the approximation for the alpha heating power given by Eq. C10.

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Companion sheets on this site: for the other route to taking electricity straight out of a burn without a steam cycle, see the aneutronic fusion reference documents DIRD Aneutronic Fusion Propulsion and DIRD Aneutronic Fusion Propulsion II, and the field-reversed configuration decadal study Clean, multi-purpose fusion power from small FRC reactors. For a proton-boron fuel that returns almost nothing but charged particles, see HB11 — Understanding Hydrogen-Boron Fusion as a New Clean Energy Source. For a device that also drives its plasma by pulsing the confining field, see the Plasma Compression Fusion Device. For the relaxation physics that governs what a magnetically confined plasma settles into, see The spheromak confinement device and Relaxation of Coaxial Nonneutral Magnetized Plasmas.