Home Search Collections Journals About Contact us My IOPscience Problems of a thermonuclear reactor with a rotating plasma This article has been downloaded from IOPscience. Please scroll down to see the full text article. 1980 Nucl. Fusion 20 579 (http://iopscience.iop.org/0029-5515/20/5/007) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 141.217.20.120 The article was downloaded on 15/03/2012 at 14:52 Please note that terms and conditions apply. PROBLEMS OF A THERMONUCLEAR REACTOR WITH A ROTATING PLASMA A.A. BEKHTENEV, V.I. VOLOSOV, V.E. PAL'CHIKOV, M.S. PEKKER, Yu.N. YUDIN Institute of Nuclear Physics, Novosibirsk, Union of Soviet Socialist Republics ABSTRACT. The authors consider the physical problems involved in the design of a thermonuclear reactor with a rotating plasma. Detailed consideration is given to a version of the reactor in which the plasma is stabilized mainly by radial variation of the rate of rotation of the plasma (electric shear). Such aspects as the heating, longitudinal confinement, stability and equilibrium of the plasma as well as the problem of impurities are considered and a calculation of the reactor's efficiency is made. The authors discuss the engineering problems of the creation of a high-intensity radial electric field in the plasma and describe a modification of this type of reactor — a system without magnetic mirrors (a 'centrifugal trap'). One possible way of solving the problem of controlled 1. DESCRIPTION OF THE REACTOR thermonuclear fusion is to use modified open magnetic traps, i.e. systems in which there is better confinement A schematic diagram of the rotating plasma reactor of the plasma along the magnetic field than in the is shown in Fig. 1. It consists of an open magnetic trap classical mirror trap [ 1 ]. Over the last few years, several with an axi-symmetric field, the mirrors of which contain systems have been advanced for solving this problem, the co-axial electrodes in contact with the plasma. Each for example, the ambipolar trap [2], the gas-kinetic electrode is fed a high voltage which shapes the required trap [3], the reversed magnetic field trap [4] etc. profile of the radial electric field E(r) in the plasma.1 Another such system is the rotating plasma trap, the As will be demonstrated below, the working main features of which as a possible thermonuclear temperature of the deuterium and tritium ions should device were discussed in a number of papers [5—7]. be of the order of 30—100 keV, the energy of A large number of experiments have been carried out rotation Wgo being of the order of 150-500 keV in traps of this kind in the 1960s at relatively low ion = mic2E2/2H2). The size of the magnetic field energy (10—100 eV), as described, for example, in the is determined by the plasma density n, the permissible review paper of Lehnert [6]. However, comprehensive values of (3, and the ratio a/r , where a is the width of 0 studies on the possibility of using rotating plasma traps the plasma layer and r is the plasma radius in the 0 as thermonuclear reactors present us with a number of centre of the trap. For n a 3 X 1013 cm"3, 0 = 0.25, problems that have not been studied before in detail. a/r = 0.1—0.3, the magnetic field intensity in the 0 Below we consider some of the problems arising in centre of the trap is of the order of 15—25 kG. connection with one of the possible designs for this Accordingly, the electric field intensity should be reactor. The characteristic feature of this design is ~ 100 keV • cm"1. Since the radial dimension of the stabilization of the plasma based on the establishment plasma has to be greater than the ion Larmor radius of an appropriate radial electric field profile by means (a/Pi ^10), for the given plasma parameters a ranges of a system of ring electrodes in contact with the plasma. between 15 and 50 cm; the full potential applied to This paper discusses the longitudinal confinement, the plasma is, accordingly, of the order of a few stability and equilibrium of the plasma as well as the megavolts (1.5—5 MV). The radius of the trap and its efficiency of the reactor and the possibility of improving length L are of the order of a few metres. Here we give it by recovering the energy of escaping ions and alpha- typical parameters of the reactor but they could particles; it also touches on a number of technical problems involved in constructing a device of this kind. Possible modifications of the magnetic system of the reactor are 1 The possibility of using the ring electrodes for shaping discussed, reactor operation in the steady state being the an electric field in a rotating plasma was noted for the first time main consideration. in Ref. [8] (see also Section 7). NCULEAR FUSION, Vol.20, No.S (1980) 579 BEKHTENEV et al. In such a system the particles are acted on by a centrifugal inertial force m £2| r, for which reason the particles escaping from the trap have to overcome the centrifugal potential barrier m £2 (ro — rp/2 (the E subscript k relates to the magnetic mirrors). Here the loss cone for ions in the velocity space is transformed into a two-sheet hyperboloid (Fig. 2): v| = vj (R-l) (r --rr22)) - *p - (4) o or, at Hr2 = const _2 FIG.l. Diagram of a rotating plasma trap: (I) ring electrodes; (5) (2) inner liner; (3) outer liner; (4) magnetic field coils; the m broken line shows the course of the magnetic field lines. where R is the mirror ratio (R = Hk/H = ro/r2,), V 0 Eu is the drift velocity in the centre of the trap, and

W|| = 0), their distribution function is close to the 6-function (2) where F = eE + (3) The last term in Eq. (3) is normally small, hence, allowing for the fact that the potential difference in the plasma between two fairly close magnetic surfaces is for all practical purposes constant along the magnetic field lines, we find on the magnetic surface i2 = — = const (2') LE Hr i.e. the so called 'isorotation law' [6]. The condition of confining the particles in this type FIG.2. Plasma confinement boundary in the phase space: of trap can be derived conveniently by a consideration (I) conventional mirror trap (T i-e. a current equal o Coulomb collisions, the ions are thermalized, and their to j || flows in the quasi-neutral plasma to the ring distribution function spreads in velocity space. This electrodes (here j|| is the current density, j (r) is the o approach to injection in a rotating plasma trap is more electron current density along the force lines [12]). or less equivalent to fast-neutral-atom injection in The radial current corresponding to this current, from conventional open traps, which was well studied in the condition div j = 0, heats the plasma up. Thus, in experiments [10, 11]. this case the electric current heating the plasma is Even if the region of ionization of neutral atoms entirely determined by the electron-ion flows from the is spread along z, i.e. ionization also occurs near the plasma along the magnetic field due to the Coulomb mirrors, the energy of the generated ions depends processes and is equal to their difference. Note that only slightly on the position of the point of ionization. this current is a/pgi times as low as j . 0 Let us use E and H to designate the fields in the For this injection to occur, the energy of the neutral o o centre of the trap and R = H(z)/H the ratio between atoms has to be sufficient for their path in the plasma z 0 the magnetic field at the point of generation of the to be of the order of either the transverse dimension ion and H , in which case the longitudinal and of the trap or the longitudinal dimension of the mirror. o transverse energies of the ion are For the reactor parameters considered above, these conditions are satisfied at velocities = 107 cms"1 Wi = W /R ; W|| = W d-l/R ) (10—100 eV). The neutral atom energy is much less EO Z E0 z than WEO- i.e. in velocity space the ions are generated on a sphere of radius Vgo • Allowance for the longitudinal electric field leads to a deformation of this sphere by the 2. LONGITUDINAL CONFINEMENT quantity ~ &P /WE0 = T /Ti. O e Let us analyse what macroscopic processes are Before the results of our accurate calculations will responsible for plasma heating, i.e. what is the source be presented, let us show, by a simple and crude estimate, of plasma energy. In case the radial currents in a that the longitudinal plasma confinement time in a rotating plasma trap are much higher than the longi- rotating plasma trap may be much longer than that valid tudinal ones and the plasma density and transverse in a classical mirror trap. Coulomb processes only are conductivity near the side walls are high enough, the taken into account in this estimate, the radial plasma plasma is heated up by radial plasma currents between structure is regarded as homogeneous. We shall compare the outside and inside walls of the installations. In the mean ion energy with the potential barrier for the another case realized in the reactor under consideration ions Uj = WEO (1—1/R) — e»^, assuming that the dimen- 0 the longitudinal currents for the ring electrodes are sion of the mirror is much smaller than the length of very significant. In this case, both density and the trap. Under steady-state conditions we make the conductivity of the plasma near the side walls are low energy received by the ion and electron from the or equal to zero (n(rj) = n (r2) = 0). In the stationary electric field equal to the mean energy removed by dense rotating plasma in this trap (at a/pj > 1), the them from the trap (in the rotating co-ordinate system) electrons and ions generated in the trap escape from the plasma along the force lines during the ionization w (i- Q = W E0 ie EO time. Because of the radial density gradient, the magnitudes of the electron and ion longitudinal flows (6) are not equal since the rates of generation of electrons and ions at point r are not equal either. The rate of where W-* and W* are the mean transverse energies of ion generation at r is determined by the rate of the ions and electrons when escaping from the trap, ionization and charge exchange at point r + pgi and Qi is the energy transferred from the ions to the e (PEi = miVEOc/eHo)> because the ion, after ionization electrons. (or charge exchange), shifts, on an average, by a Larmor Summing these expressions, we obtain radius in radial direction; the rate of electron generation at point r is also determined by the ionization at W /R = Wf (7) E0 point r — PEe- Accordingly, in the stationary regime, the longitudinal currents of the outgoing ions and Use is made here of an approximate relation linking the electrons differ by a small quantity determined by this energy carried off by the particle through the potential 581 NUCLEAR FUSION, Vol.20, No.S (1980) BEKHTENEV et al. barrier from the magnetic trap and its temperature, on where the assumption that the barrier is much higher than T [13]. The electric potential between the trap centre x = V/VEO ; X = 0 at x < and the mirror can be found from the condition of e equality between the flow of escaping electrons and ions [14]: -e 3, as compared with nr for the conventional (10) 2x3 3x magnetic mirror trap. Figure 3b shows the dependence 582 NUCLEAR FUSION, Vol.20, No.5 (1980) ROTATING-PLASMA REACTOR of Tj T on R. These plots are shown in dimensionless ; e form since to with an accuracy of up to 5% over the 10-200-keV injection energy range. Normally, the efficiency of a thermonuclear reactor is defined by the quantity Q, which is the ratio of the energy released in the plasma in the thermonuclear reaction to the energy introduced into the plasma. Figure 4 shows Q as a function of R; the equation for determining Q, i.e. Q = TIT. WR/8WE0 was r derived with allowance for the features of a trap with a rotating plasma. Here a is the thermonuclear reaction cross-section, and W is the energy released r as a result of one reaction event. It is normally considered that W = 22.4 MeV, with allowance for r the energy released in the lithium blanket [17]. In this trap the energy expended on generating an ion- electron pair is not equal to 2WEO> but to 2WEO/R, since when the ion and electron escape from the trap along the magnetic field some of their energy returns to the plasma. 3. EFFICIENCY OF THE REACTOR The above-derived values of Q and nr for a rotating plasma trap show the advantages of this system over the classical open trap, but these calculations are valid only for 'ideal' traps. To find the actual reactor parameters we have to take into account a number of important physical processes affecting its operation and the Q factor values attainable, such as ionization and charge-exchange processes, the possibility of partial recovery of the ion and a-particle energy at the 0.2 end electrodes, and the possibility of recycling the nuclear fuel to the trap ('re-injection'). Accordingly, the corrections to the Fokker-Planck equations caused by these processes have to be taken into account. It is o.l =1 1 clear that they only lead to a change in the source function in Eq. (10). The character of the corrections associated with the processes mentioned above as well as the results of the calculations are given below. Effects associated with the radial structure of the plasma FIG.3. Dependence of nr and T/WEO on R for a deuterium in a rotating plasma trap have been omitted here on the plasma. Injection energy W = 20 keV; source power E0 assumption that the transverse dimension of the plasma S = 10!Sparticles • cm'3; (I) conventional mirror trap [ 17]; is fairly large (see also Section 6). (II) rotating plasma trap. 583 NUCLEAR FUSION, Vol.20, No.S (1980) BEKHTENEV et al. Q where OQ[ is the charge-exchange cross-section, and (7j, a are the cross-sections for ionization by the ions e and the electrons, respectively; v is the velocity of charged particles in the laboratory frame; here, the (8 velocity of the neutral particles is much less than v. However, even if condition (12) is known to be satisfied, we have to take into account the fact that 16 the charge exchange leads to additional volume energy losses, which reduces the efficiency of the reactor. Ionization leads to generation of particles with an energy of W|| = 0, Wi = m Vg /2, while charge exchange Q creates ions with an energy W|| = 0; Wi = WEO> instead of ions with an energy of the order of Tj if either process is analysed in the rotating system of co-ordinates. 10 Allowance for effects associated with the charge- exchange process introduces additional collision terms into the Fokker-Planck equation. Then, the equation for fj in dimensionless variables has the form: e (13) Here, account is taken of the fact that the intensity of WO ZOO 300 HOO (keV) the ion source is given by FIG.4. Dependence ofQ on the deuteron injection energy WED = mp- V£0/2 for an ideal DT reactor; at nD = nT; So = n n0 ) W = (3/2)W ; (I)R = 5; (II) R = 3. ET ED S is assumed to be independent of time. In these o expression, n is the ion density, n the density of 0 neutral atoms, and < > denotes an average over Charge exchange and ionization the distribution functions of the colliding particles. The additional terms in Eq. (13) are due to charge- exchange processes. The second term in the parenthesis The ionization and charge-exchange processes in a of Eq. (13) describes the generation of ions with an rotating plasma create radial plasma flow and, accordingly, energy of WEO- The third term on the right-hand side radial electric currents. The occurrence of the flow is of this equation describes the loss of ions with an due to the fact that after each ionization or charge- energy of the order of Ti. Note that allowance for exchange event in the rotating plasma trap, the ion is the charge-exchange process does not change the displaced along the electric field by a value of equation for the electron distribution function. For Q the order of the Larmor radius. The condition for to be defined, the fact should be taken into account which this flow does not affect the escape of ions from that charge exchange leads to volume energy losses the trap along the magnetic field, i.e. it need not be in addition to those occurring in 'ideal' traps. The taken into account when solving the Fokker-Planck energy flow removed from the plasma by the fast equations, reduces to the condition of smallness of the 'charge-exchange' atoms escaping from the trap is radial ji currents compared with the longitudinal j 0 given by (seeRef. [12]): (12) Pi = 1 = So WE0 0i (14) JO VE0 584 NUCLEAR FUSION, Vol.20, No.5 (1980) ROTATING-PLASMA REACTOR The additional energy flow heating up the plasma and, accordingly, carried off by the ions escaping along the magnetic field is equal to H m; iv> V| - - (15) Jiv) + ft The expression for P is derived after integration of 2 Eq. (13) over d3x with the factor x2 due to additional charge-exchange terms. The efficiency of the thermo- FIG.5. Diagram illustrating the motion of an ion near the end nuclear reactor, taking charge exchange into account, electrodes. is determined by the expression nr W X is the mean free path of the ion in the trap, wj « vj.. r Q = (16) Hence 2W E0 R 2irp* /L 4 "" A and for Recovery of charged-particle energy O ^ T "7*-=4; -=10"5; we have ^s 30° A A The ion and electron escaping from the trap along the magnetic field overcome the corresponding potential barriers and transfer a portion of their energy equal to The mean energy of the ion in its collision with the WEO (1-1/R) to the electric field. This effect was taken electrode is determined by the expression into account above in the calculation of Q. Let us consider one more mechanism for the recovery of the ion energy. Since the energy of an ion moving in a cycloid varies along its trajectory, we can arrange the geometry of the end electrodes such that the ions hit the electrode with minimal kinetic energy. For this mj VE0 2viVE0 cos v? (18) purpose the end surface of the electrodes is inclined towards a plane perpendicular to the z axis at the angle i// (Fig. 5) so that the ion collides with the most where we average with respect to r and the transverse 'positive' electrode (the top one in Fig. 5). ion energy in the mirror. To evaluate Wj we apply the The angle \p is found from the condition that over Pastukhov formulas [13]. In the case where 2p* > A the time taken to revolve around the Larmor orbit the we obtain displacement of the ion along z is less than d, i.e. ^jfi- 2 <> V ) ^ ^ (2 VS Vi E0 (17) (19) where p? is the ion Larmor radius in the mirror. The A more accurate evaluation taking into account the quantity v|| in the mirror (as in Ref. [18]) is given by dependence of Wj on 2p*/A takes the form II = (l-exp[-(A/2pf)2])) v (20) NUCLEAR FUSIDN, Vol.20, No.S (1980) • 585 BEKHTENEV et al. Transfer of a-particle energy to the plasma Alpha particles generated outside the loss cone remain inside the trap and transfer energy to the plasma by means of Coulomb collisions; a-particles generated inside the loss cone move to the electrodes. Like the ions escaping along z, they leave some of then- rotation energy in the plasma, i.e. (22) 0.25 where Wga = m V| /2. The energy WE<* is the a 0 portion of the energy of rotation of the centre of 90 gravity of the DT system that is distributed, after the thermonuclear reaction, between the a-particle and FIG.6. Function I(ty). the neutron. In the reactor with 'skew' electrodes the a-particles transfer some of their transverse energy to the electric field (5W ) and, correspondingly, to the plasma, in the a same way as the ions. When considering this effect it should be borne in mind that the longitudinal energy In Eqs (19) and (20), it was assumed that of the a-particles is comparable with their transverse = V£ /R. energy (as distinct from the case of the ions). 0 Electrons escaping from the trap are accelerated by This energy, after averaging over all a-particles the electric field in the inter-electrode gap and reach generated (isotropic distribution function;W = const) a the positive electrode (see Fig. 5) with an additional and over all possible angles of collisions between energy equal to W — eE A/2, from which we obtain the a-particles and electrodes, is given by e mean energy removed from the trap by the ion-electron pair W: (23) 4pf w = w w ^ i + e where W is the kinetic energy received by an a-particle a in the thermonuclear reaction. Figure 6 shows a plot of I(i//). When calculating 5W we did not take into a 277WEQ X(l-exp[-(A/2pf)2]) -P)f/ = R (21) account effects associated with the discreteness of the electrodes, since p > A. For a-particles the efficiency a with which the energy is transferred to the field is The contribution of the secondary electrons to the much lower than for ions since Wa > WE<*/R- AS energy balance in this system is fairly small. Indeed, is clear from this calculation, for Q > 50 (taking into since the electrons escaping from the plasma reach the account the heating of the plasma by the a-particles), positive electrode (see Figs 5 and 9), the secondary the reactor becomes self-sustaining, i.e. an emf sufficient electrons generated by them cannot go into the plasma to sustain the reaction is created at the end electrodes and, because of the retarding potential, return to the through the slowing-down of the a-particles in the same electrode. A small number of secondary electrons electric fields. generated on the end surface of the electrode (the thickness of the electrode is A') enter the plasma and are an additional source of electrons which differs from Re-injection the main source of electrons by the factor 5* A'/(A + A'); 6* is the secondary emission coefficient. This effect An ion escaping from the trap and recombining on has not been taken into account in the calculations the end electrodes may then, with a fairly high degree because it is assumed that A'/(A' + A) for the reactor of probability, return to the space occupied by the can be of the order of 0.1, and 5* < 1 at T = 5-20 keV. plasma in the form of a neutral atom, re-ionize and heat e 586 NUCLEAR FUSION, Vol.20, No.S (1980) ROTATING-PLASMA REACTOR up in the crossed fields. In this way, the D and T nuclei We can then evaluate the mean distance covered by escaping from the reactor are collected and returned to the atom before it is ionized: the plasma, i.e. there is a sort of 're-injection'. This process may substantially improve the actual burnout 1 2X (z )( /R-l) i 0 N coefficient (c^) of the fuel, provided that the losses = Z0 11 + In (26) 2(R- /R) N of deuterium and tritium nuclei through incomplete return to the plasma are less than the 'losses' due to where Xj(z ) is the mean free path of the neutral atom 0 occurrence of the fusion reaction: at the centre of the trap, which is determined by ionization and charge-exchange processes (here, Xj(z ) < z ). 0 0 (24) Applying Eq. (26), we find

Problems of a thermonuclear reactor with a rotating plasma

Similar Problems of a thermonuclear reactor with a rotating plasma