DYNAMICAL EQUILIBRIUM AND STABILITY OF A D-T FUELED FUSION REACTOR. Anthony Michael Carver. Thesis submitted to the Graduate Faculty of the

Transcription

1 DYNAMICAL EQUILIBRIUM AND STABILITY OF A D-T FUELED FUSION REACTOR by Anthony Michael Carver Thesis submitted to the Graduate Faculty of the Virginia Polytechnic Institute and State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE in Nuclear Science and Engineering APPROVED: R. J. 'o~dga, Cha6\man H. A. Kur. i~s1t A. K. Furr October, 1975 Blacksburg, Virginia

2 ACKNOWLEDGEMENTS The author wishes to thank Dr. Ronald J. Onega for his suggestion of the problem. The author was also provided with invaluable assistance throughout the project by Dr. Onega and wishes to acknowledge his expertise and patience. ii

3 TABLE OF CONTENTS ACKNOWLEDGEMENTS ii LIST OF TABLES iv LIST OF FIGURES. v I. INTRODUCTION.. 1 A. Purpose and Scope. 1 B. Review of Literature 2 II. OVERVIEW OF NUCLEAR FUSION 4 A. The Fusion Process 4 B. Plasma Confinement and Reactor Description 5 III. MATHEMATICAL MODEL. 1 IV. A. Introduction and Assumptions B. Particle and Energy Balance Equations. EQUILIBRIUM STATE AND STABILITY ANALYSIS A. Equilibrium States B. Stability Analysis V. RESULTS AND CONCLUSIONS 28 A. B. Equilibrium and Stability Results. Conclusions VI. VII. BIBLIOGRAPHY VITA iii

4 LIST OF TABLES I. D-T Fusion Maxwell Averaged Cross-Sections II. Equilibrium States III. Initial and Final Values of x and y in Figures iv

5 LIST OF FIGURES Page 1. Cross-Sectional View of a D-T Fusion Reactor The Product of Ion Density and Confinement Time Vs. Fractional Burn-Up The Product of Ion Temperature and Confinement Time Vs. Fractional Burn-Up % Perturbation in x (Confinement Time 1.52 sec) % Perturbation in y (Confinement Time = 1.52 sec) % Perturbation in X (Confinement Time = 2.1 sec) % Perturbation in X (Confinement Time = 2.9 sec) % Perturbation in X (Confinement Time =.784 sec) v

6 I. INTRODUCTION A. Purpose and Scope The stability of a non-steady state plasma in a Controlled Thermonuclear Reactor (CTR) operating on the deuterium-tritium (D-T) fuel cycle is investigated in this paper. ' The reactor is assumed to be a practical working device. The first CTR to prove scientifically feasible will probably operate on the D-T cycle. The question of stability is important and needs to be examined especially at relatively low temperatures (5 kev to 3 kev) where the first generation CTRs will probably operate. Here, temperature is used synonymously with energy, and 1 ev = ll,6 K. The D-T reaction produces more energy per collision than the D-D (deuterium-deuterium) reaction at significant temperatures, less than 1 kev, and is also much more likely to undergo a reaction. The object of this study is to determine the stable equilibrium states in the operating range and the fractional fuel burn-ups corresponding to these equilibrium states. The scope of this paper includes developing a mathematical model of the energy balance of the fusion plasma. This set of equations forms a closed system consisting of both particle and energy balance equations. These are non-linear differential equations whose dependent variables can be functions of space and time, but in this model the 1

7 2 dependent variables are functions of time only. The non-linear differential equations in this study are solved by numerical techniques. Equilibrium conditions for the plasma are obtained by choosing certain operating parameters, the injected ion density per unit time and the confinement time scaling, and solving the steady state balance equations. Finally, dynamical simulation is used when perturbations are applied to the equilibrium points to investigate non-steady state stability. B. Review of Literature There have been numerous studies involving the energy balance and reaction rate equations in a fusion plasma. For example, in the late 1 19SO's, Kofoed-Hansen, et. al., investigated the reaction rate or particle density equations and their solutions analytically. Only the steady state problem was treated here. Rose 2 modeled the fusion plasma with and without helium buildup using confinement time scaling different from the Bohm scaled confinement time in this paper. Some of the more recent works include the energy balance and stability studies by Powell 3 4 and Hahn and Stacey Powell and Hahn investigated reactors burning pure deuterium with constant confinement time but varied plasma volume. Stacey scaled confinement time according to Bohm and neo-classical methods for closed and open systems. His model incorporated helium buildup and electrons. The model in this study employs a divertor to remove helium buildup. Stability of the energy balance and the control problem of the thermonuclear reactor were also investigated by Mills 5 ' 6

8 3 7 and Powell Mills modeled the plasma by including alpha particle buildup and temperature difference between ions and electrons. Powell followed his previous work and postulated various methods of control.

9 II. OVERVIEW OF NUCLEAR FUSION A. The Fusion Process Nuclear fusion is often thought of as the solution to mans energy problems. Some difficulties have to be overcome, however, before practical power can be obtained from a fusion reactor. 8 The future fusion plant would include three essential components. The thermonuclear plasma in the reactor itself is the heat source. The heat generated must be removed and converted to electrical energy. This requires that a steam generator and turbine or some other means of heat removal and conversion must be used. There must also be a method of supplying the fuel to the fusion reactor. The hydrogen isotopes, deuterium and tritium, are the most practical fusion fuels. The most important reaction in a D-T cycle 3 reactor is D 1 + T 1 ~ He 2 (3.52 MeV) + n(l4.8 MeV). (1) In the thermonuclear plasma, a significant fusion energy release is achieved. The rate of this energy release can be calculated once the collision cross-section is known. The reaction rate increases rapidly as the ionic temperature increases. If a binary reaction is considered in a system containing n 1 nuclei/cm 3 of one reacting species and n 2 nuclei/cm 3 of the other, the thermonuclear reaction rate can be 9 expressed as 3 = n 1 n 2 <av> 12 interactions/cm -sec. 4

10 5 Here RR 12 is the reaction rate, o(cm 2 ) is the ~usion reaction crosssection and vis the relative velocity (cm/sec). Since the reaction cross-section is dependent on the velocity, the brackets indicate that the product <ov> 12 is averaged over a whole range of relative velocities such as a Maxwellian distribution of velocities. B. Plasma Confinement and Reactor Description The problem of confinement of the plasma is formidable and much research has been done in this area. The plasma must be confined for a time long enough to extract the heat produced. In a practical D-T 5 cycle CTR ionic temperatures would be of the order of 1 kev. such temperatures, much higher than the ionization potentials of most At of the elements, the plasma is fully ionized. Since the plasma is fully ionized, it is an excellent conductor and can be confined by magnetic fields. Physical and economic considerations in the individual CTR determine important plasma parameters. In general, the main requirement for a D-T cycle to achieve net energy output or "break even" is the Lawson Criterion. The Lawson Criterion states that the product of the plasma ion density n. and the confinement time T should be equal to or 1 exceed the approximate value 3 x 1 14 sec/cm 3 This is expressed as 1 Lawson's criterion also establishes the ion temperature to be approximately 5 kev corresponding with the above value of the product n.t. 1

11 6 A "Lawson-like" criterion for the plasma is derived from a plasma energy balance equation to achieve zero energy output. Lawson's criterion for a fusion plant would take into account the overall thermodynamic efficiency to achieve or exceed "break even" values for the product n.t l. and ionic temperature. The problems involved in achieving the ionic temperature necessary for net energy output are difficult and varied. The heating of the plasma from ~era energy to Lawson criterion temperatures is beyond the scope of this work. Plasma is lost continuously by the diffusion of particles across magnetic field lines. Coulombic collisions or microinstabilities can cause the particles to move across field lines. Microinstabilities are based on the velocity-space distribution of the particles and the microfields produced by particles. Collision processes correspond to the classical diffusion mechanism and microinstabilities correspond to the Bohm diffusion mechanism. 11 The classical diffusion coefficient, D!, perpendicular to the magnetic lines of force is related to the temperature and magnetic field B as 1 ~ 2 T B The classical confinement time 'l is inversly proportional to D 1 and. 11 l.s The relation for the Bohm diffusion coefficient DB is given by the proportionality

12 7 where Te is the electron temperature. The Bohm confinement time 'B is 1 'B a: DB It should be noted that the confinement time scalings discussed here could be modified to be more applicable to open-ended (mirror) systems. Diffusion across magnetic field lines is the dominant particle loss mechanism in closed machines. Scattering into the loss cone is the dominant loss mechanism in open machines. This scattering is parallel to field lines. The ratio of the plasma pressure to the magnetic pressure is often used to compare plasma confinements. This ratio is usually quite sma 11, a b out 5 to lo percent, f or toro id a 1 co nf 1gurat1ons For confinement, it is obvious that this ratio must be less than or equal to one. Figure 1 shows a cross-section of the Tokamak reactor, a fusion reactor of toroidal configuration. The plasma is surrounded by the vacuum wall through which the 14 MeV neutrons should pass from the plasma to the neutron moderating and tritium breeding blanket. The blanket should be composed at least partially of lithium in order to 2 breed tritium for a D-T cycle reactor. The vacuum wall and the blanket assembly structure are subject to material science problems because of heat, radiation, and neutron bombardment.

13 MAGNET COIL VACUUM VJ ALL LITHIUM BLANKET THERMAL INSULATION VACUUM WALL COOLANf BIOLOGICAL SHIELD FIG. I CROSS- SECTIONAL VIEW OF A D-T FUS!ON REACTOR CXl

14 9 A steady state reactor will require a continuing supply of new fuel injected into the plasma, since fuel will be consumed in the burnup process, and since some fuel will escape the magnetic field containment. It seems essential for particles to be extracted in closed 12 systems by a divertor. This is due to economic considerations for the reactor and maintaining operating temperatures within the plasma. 12 For open-ended systems the majority of particles will be lost through open loss cones and pumped away. This vacuum pumping problem is separated from the containment region. For closed line systems, the principle is to modify the magnetic field at the outer surfaces of the plasma column, diverting the field lines out through apertures in the vacuum wall accomplishing extraction of particles. Stability of the plasma, as it is used in this study, is that due to dynamical considerations and not that of holding the plasma in place, i.e., the resistance to physical break-up.

15 III. MATHEMATICAL MODEL A. Introduction and Assumptions In this study, a number of simplifying assumptions are made to help obtain the model for the plasma energy balance and particle equations. This model is developed from phenomenological considerations. It may be possible to develop the plasma particle and energy balance equations from plasma kinetic equations such as the Vlasov equation. A toroidal, closed magnetically confined reactor operating on a D-T cycle shall be considered in order for some of the following assumptions to be more valid. 1. A point kinetics calculational model is assumed. 2. A 5% deuterium - 5% tritium fuel mixture is burned in the CTR. In this case, the only reaction of any importance is the T(D,n) 4 He reaction. 3. The electrons and ions have Maxwellian distributions. Since the particles in a plasma are charged, many of the properties are the result of long-range Coulomb interactions. Since the range of interaction is long, many particles interact simultaneously. If this group of particles is in equilibrium, ideally this gas relaxes to a Maxwellian distribution of velocities. 4. The ions and electrons are at common temperatures. Actually, the electron and ion temperatures may differ in the plasma, 5 but this difference is ignored here. 1

16 11 5. The confinement time is scaled according to the Bo1un diffusion mechanism. It is not known exactly how the confinement time will vary with temperature or other reactor or plasma 11 parameters. 6. The confinement time is the same for all ions and electrons. 7. Fuel is injected in the form of small droplets of liquid deuterium-tritium, zero energy droplets, at a constant rate. The cold particles are assumed to instantaneously equilibrate with the ions because of alpha particle heat. 8. A divertor is employed in the CTR to remove alpha particles, ions, and impurities. It is assumed that less than 1% 13 of the plasma density is alpha particles and impurities so that these may be neglected in the model. This also means that the number of ions is equal to the number of electrons in this study. 9. Cyclotron and black-body radiation losses will be neglected. Cyclotron radiation becomes significant only at high operating 8 temperatures. Since cyclotron radiation is expected to mainly be in the infrared and conventional microwave regions 9 of the spectrum, it is probable that the cyclotron radiation will be partially absorbed in the plasma. Black-body radiation11 occurs only at low frequency ranges and is insignificant. 1. The magnetic field strength is constant. 11. The volume of the plasma is assumed constant.

17 12 noted. The system of units in this work are cgs units unless otherwise B. Particle and Energy Balance Equations Using these assumptions, the particle equations can be written as (2) and (3) SD and ST are the deuterium and tritium source terms respectively; ~ and r are the number densities, and T is the confinement time which is the same for deuterium and tritium. <ov>dt is the product of the fusion reaction cross-section and the relative velocity averaged over a Maxwellian distribution. The reaction parameter <ov>dt is a function of ion temperature, 14 and the values given by Greene are used in this study. These values are presented in Table I. The assumption of a Maxwellian-averaged <av> is only approximate, especially for the ions. The reaction rates depending on the precise ion distributions will eventually have to be worked out for individual cases. 8 When the number of ions is equivalent to the number of electrons, energy conservation in the plasma requires d % (~ + r) k(ti + Te) VP = dt

18 13 Table I. D-T Fusion Maxwell Averaged Cross-Sections Ion Temperature <rjv> DT Ion Temperature <rjv> DT (kev) 3 (cm I sec) (kev) 3 (cm I sec) x x x x x x x x x x lo x x lo x x lo x l x x x lo x l x x x x x x x lo x x x x x x x x x x x x x l x x x 1-17

19 14 Table I. (Continued) Ion Temperature <(JV> DT Ion Temperature <(JV> DT (kev) 3 (cm /sec) (kev) 3 (cm /sec) x x x x x x x io x x x x x x x x x x x x x o 5.39 x x x x x x x x x x x x x x x x x x x x x x 1-16

20 15 Table I. (Continued) Ion Temperature (kev) <av> DT 3 (cm /sec) Ion Temperature (kev) <av> DT 3 (cm /sec) x x io x io x io x io x x x x x l x x 1-16

21 16 where k is Boltzmann's constant. Recalling that the ionic temperature Ti is equal to the electron temperature Te and the volume of the plasma V is constant, the energy balance becomes p d 3(1)) + r) kt dt (4) T is the common temperature for ions and electrons. Pf is the fusion power density; Pb is the bremsstrahlung radiation power density; and P c is the charged-particle power density. The fusion power density is P = <crv> IL n Q f DT il T a' 4 where Qa is the energy liberated by the 2 He particle in equation (1). It is assumed the neutron escapes the plasma and is absorbed in the blanket. The bremsstrahlung power density is approximately 9 ~ 3 h C is 3.34 x w ere 1 l- 15 kev sec - cm kev. The amount of energy that an ion or electron possesses is 1. kt 2 The charged particle power density is p = c

22 17 because of electron and ion escape from the system. The temperature T is used $ynonymously with the energy kt. Let = kt in equation (4) to give 3(~ + ~) (5) Recognizing that ~' nt, and 9 are all functions of time, equation (5) becomes d(~ + ~) d dt + dt (~ + ~) = <av> IL 1L Q DT lj T CL 3 Adding equations (2) and (3) and substituting gives 2<av> IL n_ IL n_ <av> Q + DT lj T + lj T DT CL ~ + ~ 3(~ + ~) (6)

23 18 Equations (2) and (3) require and SD = ST. Let n = ~ = ~ and in equations (2) and (3) to give s = s = ST D dn 2 dt = S - <ov>dt n n - - (7) After making the above substitutions equation (6) becomes 2 c n ~ Q <ov> n 1 + _a D_T d_e = s_e + < > dt n 3 ov DT n 6 (8) The model of the plasma is now described by equations (7) and (8) forming the closed system of particle and energy balance equations if initial conditions are specified. Since this study deals with the relatively low temperature range of operation in a CTR, it is important to know the ignition tempera-

24 19 ture. Below the ignition temperature no power is generated and the plasma is lost. The ignition temperature for this system can be found by equating the energy balance equation (4) to zero indicating steady state, thus Solving for the product nt gives (9) The denominator of the right-hand side term of equation (9) is equated to zero and solved for. 1 nition temperature. This value of is by definition the ig- Fractional burn-up is now introduced and defined as the fraction of injected ions that undergo fusion. The fractional fuel burn-up can be thought of as a measure of efficiency of fuel utilization. From equation (7) it is seen that <crv> DT 1, <crv> + ~ DT UT (1) where fb is the fractional burn-up at steady state. It was mentioned previously that the confinement time is not known explicitly as a dependent function. The confinement time will

25 2 depend on the operating temperature, magnetic field, and characteristic size and will eventually have to be calculated for each individual reactor. In this study, the confinement time scaling according to the Bohm diffusion mechanism is chosen because of experimental data 11 indicating that the plasma may diffuse in this way. The confinement time T is given by (11) where c is a constant which takes into account the reactor parameters 2 (characteristic size, magnetic field strength) and other constants involved in Bohm diffusion. This constant c is varied in this work to 2 discover different equilibrium values of n and e. It is convenient to introduce definitions of x and y to make equations (7) and (8) less cumbersome numerically. Incorporate equation (11) into equation (7) and let n x =- no and in equations (7) and (8) to give dx -= dt (12)

26 21 and ~- dt - x (13) where and b3 = 6---eo In the preceding equations n and 9 are the chosen values of particle density and temperature respectively. n and 9 are chosen within typical operating ranges of a CTR in order to make x and y simpler numerically.

27 IV. EQUILIBRIUM STATE AND STABILITY ANALYSIS A. Equilibrium States The state of a fusion reactor is now described at any time t by the set of state variables x(t) and y(t). These variables are taken to change in time in accordance with equations (12) and (13) which can be expressed as and _<_!y ( ) dt - f2 x,y ' y(o) = c2; where c and c are values at t 1 2 = of x and y respectively. Here, f (x,y) and f (x,y) indicate that the right-hand sides of equations 1 2 (12) and (13) can be expressed as functions of the state variables x and y. An equilibrium state is a set of values (a,a ) which possesses 1 2 the property that where is the vector 22

28 23 These values furnish a point solution of equations (12) and (13). Without the intervention of an internal or external force the system will remain in the state specified by x = a 1 and y = a 2, if it starts in this state. In this study several equilibrium operating points are considered. The equilibrium points for equations (12) and (13) are found using the Newton-Raphson iterative method 15 on a computer. Various operating regimes of CTRs are simulated by varying the constant c in equation 2 (12) to account for various reactor parameters mentioned previously. This produces different equilibrium values of particle density, ion temperature, confinement time, and fractional burn-up. Values of c 2 vary from 5. to 15. in increments of.25 in this work. The iterative technique uses a subroutine to reproduce the temperature dependent values of <av>dt shown previously in Table I. The arbitrary values of n and are chosen to be 5 x 1 14 atoms/cm 3 and 1 kev respectively. The source density S is fixed at an arbitrary 4.46 x 1 14 atoms/cm 3 - sec for all regimes. Only the relevant values of c and S are chosen and fixed in this model of the plasma. 2 In order to use the Newton-Raphson method for equilibrium point solutions to equations (12) and (13) it is necessary to take partial. d<av>dt derivatives with respect to x and y. This poses no problem if y is evaluated numerically. This is accomplished by using a Lagrangian

29 24 interpolating polynomial that fits a parabola to three values of <av> DT The partial derivative of the polynomial can be taken without difficulty. a <av> DT A subroutine in the iterative program produces f a y for each y. In order to do this <av>dt must be transformed from a function of El to a function of y. This is easily accomplished remembering that El y =- eo Let R(e) = <av>dt (El) and R(y) = <av> DT (y); the variables in parentheses indicating the dependency of the function R or <av>dt. To transform variables, the equation R(9) del = R(y) dy must be true. Therefore, R(y) = R(El) :~~~ where

30 25 Finally, a <crv> (y) ----"'-DT~- = dr(y) a y dy dr(g) 2 8 de If y is less than.1 (corresponding to a temperature of 1 kev) at any point in the iterative process, that particular set of equilibrium points is not calculated. The first reaction cross-section corresponds to 1 kev in Table I and this is well below ignition temperature. The point solutions of equations (12) and (13) are now straightforward. B. Stability Analysis Now that the equilibrium points are established the stability of the system can be analyzed. For small departures from equilibrium equations (12) and (13) can be linearized and investigated mathematically for stability. The procedure is tedious and involves some complicated algebra. Computer simulation, using the non-linear equations (12) and (13), seems more informative and easier. A fourth order R unge-k utta a 1 gorit. hm 15 is. use d to so 1 ve t h e equations.. An initial positive perturbation of +1% in x and y is introduced to disturb the equilibrium condition, then the values of x and y are calculated as functions of time. A few equilibrium operating points are considered. It becomes necessary to define some terms concerning stability as they are used in this study. The evolution of particle densities and temperatures following initial perturbation can be characterized by one of the following terms: stable in the sense of Liapunov; Lagrange

31 26 16 stable; or unstable. The dynamical system.... 'x, t) n is considered where i = 1,, n. Given initial equilibrium values at time t, x. (t ) = x. 1 1 perturbations are applied and the behavior of the system for t > t investigated. is It is convenient to adopt a definition of distance in the state space. Regarding the x. as components of a vector let the displacement 1 from the equilibrium state be measured by the Euclidean norm 16-2 k x(t) = {E.[x.(t)] } A system is stable in the sense of Liapunov if, for any given positive number, a positive number o can be found such that implies x(t ) < a x(t) < for all t > t In general, o is a function of. In some cases, the demand for stability in the sense of Liapunov is an unnecessarily great restriction on the design and operation of a sys.tern. Some systems that are not stable in the sense of Liapunov can operate satisfactorily as long as the physical variables remain within

32 27 limits fixed by design considerations, material properties, or other constraints. This is the idea behind Lagrange stability. The system is Lagrange stable if the norm x(t) < m, t (O, ) where mis a positive number; i.e., x(t) has a finite upper bound for all t following an initial disturbance. When all solutions are bounded, the system possesses Lagrange stability. The system is unstable if it is not stable in the sense of Liapunov or Lagrange stable.

33 V. RESULTS AND CONCLUSIONS A. Equilibrium and Stability Results This model of the plasma is essentially reactor independent. As was mentioned previously, the only chosen values that would affect the particle density and temperature in this model are S and c 2. It is assumed that certain necessary confinement times proportional to c 2 can be achieved and that the value of the source density can be controlled in individual reactors. The ignition temperature for this system is found to be approximately 4.32 kev. Provided Lawson's Criterion for the system is met, it is possible to have a power producing reactor above this ignition temperature. Table II shows the equilibrium values of particle density and temperature, confinement time, the product of particle density and confinement time, the product of temperature and confinement time, and fractional burn-up. It should be noted that the product of temperature and confinement time is simply c 2 Figures 2 and 3 show the products nt and T plotted as a function of fractional burn-up, fb. Except in the low fb-low confinement time regime, the curve is monotonically increasing in both figures. In Figure 2, both n and T are increasing as the fractional burn-up increases. The fractional burn-up is expected to increase with increasing confinement time and it does. In Figure 3, the temperature is decreasing and confinement time is increasing with fractional burn-up. 28

34 Table II. Equilibrium States JI x y n e '( IlT e. (atoms/ cm 3 ) 3 (kev) (sec) (sec/cm ) (kev-sec) fb x x x x x x x x x x x x N \ x x x x x x x x x x x x x x x x x x

35 Table II. (Continued) II x y n e T n-r e. (atoms/cm 3 ) 3 (kev) (sec) (sec/cm ) (kev-sec) fb x x x x x x x x x x x x VJ x x x x x x x x x x x x x x x x x x

36 31 Ll'l,..._ ("") - e u ("") u a> en ~ o x N. r FIG. 2 lh: PRODUCT OF ICX'J DEHSilY MID CONFINE:rfNT TI~ VS. FRACTIONAL BURN-UP

37 32. C"'l r-1. N r-1... (.) Q) CJ) I ~ I-' <D. r-1 r-1. r-1. ' co., FIG. 3 TP.E PRODUCT OF Ia~ lejiperature AND CONFINEMJ\ff TI~ VS. FRACTIQ\JAL BURN-UP

38 33 The magnitude of the increase in confinement time is greater than the magnitude of the decrease in temperature causing T to increase with increasing fb. Figure 2 compares favorably with the general shape of 4 Stacey's curve of nt versus fb. The fluctuation in the low fb-low confinement time regime of both figures is caused by the relatively large change in n from 3.44 x 1 14 atoms/cm 3 to 4.6 x 1 14 atoms/cm 3 (see Table II). This magnitude of change in n causes fb to decrease. The change in n between regimes #3 and #4 in Table II is smaller and representative of the change in n in the regimes following #4. To clarify this phenomenon the change in fb which is a function of n and only is investigated, i.e., fb = fb(n,). The change in fb can be expressed as or (14) In this study the first term on the right-hand side is positive at all times and the second and third terms are negative. When the magnitude of the second and third terms is greater than the magnitude of the first, 6fb will be negative. This is the case for the lowest fb regimes of Table II cited previously. The right-hand side terms of equation (14) are given below: fb =<av> (1 - fb) is always positive (fb < 1); DT

39 34 o<crv> DT ae is negative in this case (see Table II); ~e is negative in this case; fb 2 ( 1 ) - <av> --2 ; DT n-r a. ae is always negative; 2 fb ( 1 ) - <av> -2- ; DT n -r and ~n is positive in this case. Equilibrium regimes #1, #9, #16, and #3 are chosen from Table II to be investigated for stability. Figure 4 shows the plot of x and y in (12) and (13) against time for a 1% perturbation in x of regime #9 of Table II. Figure 5 shows a 1% perturbation in y of regime #9. Figures 6, 7, and 8 plot x and y against time for regimes #16, #3, and #1 respectively with a 1% perturbation in x. In the iterative technique used to obtain data for Figures 4 through 8 a time step of.25 seconds is chosen and time goes from to 9. seconds in Figures 4 through 7. The time step interval was chosen to be less than the inverse of any coefficient of yin equation (13). The units of the -1 coefficients of y are sec The 9 second time span was chosen to be about four times a typical confinement time. In Figure 8, time goes from to about 4.5 seconds.

40 35 x l/1 I'- o>rri frt C'i 2....:t. a s. a 8. TI ME fsecj FIG. 4 PERTURBATION IN X C CONF. Tr NE l. 52 SEC. l

41 36 lf1 ~ Cl cd ui 8 T x,... 1/1 frt a>rri C'i 2...i:t. a s. o a. o TIME fsecj FIG. 5 PERTURBATION IN Y C CONF. TINE L 52 SEC. l

42 37 rrj cd 1/1 N ui,.: o>rri N ll1 2. "t. 6. a.a TI ME CSE CJ FIG. 6 PERTURBATION IN X C CONF. TINE SEC. l

43 38 a N a aj a>cd a _:a a T U1 a N 2. '4. 6. a.a TIME CSECJ FIG. 7 PERTURBATION IN X C CONF. TIME SEC. l

44 39 rri C'i x w cn> Ul. l.o 't. TIME CSECJ FIG. 8 PERTURBATION IN X C CONF. TINE. 784' SEC. l

45 4 Before conclusions are dr~wn from Figures 4 through 8 it is interesting to discuss the vario~s physical mechanisms which affect stability and indicate trends when equations (12) and (13) are analyzed. An initial positive perturbation in the ion density is considered first. This perturbation causes an increase in the ion leakage and fusion rates. The increased fusion and ion leakage rates in turn lessen the positive perturbation in ion density causing a slightly stabilizing effect. This initial increase in the ion density and resulting increasing fusion rate causes the ionic temperature to increase because of the increase in the alpha heating of the ions. Thus, a positive ion density perturbation induces a rise in temperature which in turn increases <ov>dt" An increase in <ov>dt causes an increase in ion consumption; therefore, a decrease in ion density occurs. For Bohm scaling of the confinement time, the leakage rate of ions increases with an increase in ionic temperature. Now, if an initial positive perturbation in the ionic temperature is considered this increases the fraction of the alpha particle energy, increases <ov>dt' and increases the loss of ion energy by fusion. This is accompanied by an increase in ion loss and effects occur as explained above. As long as the temperature is less than about 65 kev (see Table I), <ov>dt will increase with an increase in ionic temperature. This type of discussion gives some insight on effects, but it is not possible to determine stability or instability of a system by these conjectures. Based on terms defined earlier within this paper, Figures 4 through 7 indicate Lagrange stability for regimes #9, #16, and #3.

46 41 For regime #9 the perturbed x value is approximately 1.35 initially in Figure 4 and y has the equilibrium value.542. Figure 4 indicates that the values of x and y at the end of 9 seconds are.383 and 1.9 respectively, and the values are almost constant in time. This first value of y corresponds to a temperature of 19 kev for this system. The initial and final values of x and y for Figures 4 through 7 are shown in Table III where the subscripts i and f indicate initial and final values respectively. Figure 8 indicates that regime #1 is unstable. The values for Figure 8 are xi=.695 and yi =.797 with the final value of y going below.4 which corresponds to an ionic temperature below the ignition temperature for the system. The plasma is lost in this case. In fact, for the first seven equilibrium states (#1 through #7) in Table IL, the plasma is lost. Equilibrium state #8 is Lagrange stable and regimes #9, #16, and U3 have been mentioned previously as Lagrange stable. This would indicate that important values of nt and T are between regimes #7 and #8. B. Conclusions For this model of the plasma the following statements can be made: the system is Lagrange stable if 3.62 x 1 15 sec/cm 3 ~ nt ~ 9.17 x 1 14 sec/cm 3 and 13.5 kev-sec ~ T > 8. kev-sec; the system is unstable and the plasma is lost if nt ~ 8.23 x 1 14 sec/cm 3 and T < 7.75 kev-sec. The smaller the increment of C chosen the more 2 accurate one becomes in predicting the OT and T values for stability.

47 42 Table III. Initial and Final Values of x and y in Figures 4-7 Figure

48 43 Figures 2 and 3 and Table II indicate that the fractional burn-up could vary from approximately.173 to.362, and the system would still remain Lagrange stable. This model was based on a number of simplifying assumptions; some may deserve comment. The Maxwellian distribution assumed to apply to the particles and the common temperature of ions and electrons are the most severe assumptions. The electron and ion temperature difference could have been accounted for in this model, possibly following Mills 5 ' 6 but was neglected. Bohm scaling of the confinement time was chosen for this model, but up to date and promising results of experiments by Mills 17 at Princeton indicates that the confinement time scales in the Synnnetric 3 Tokamak as T 2 corresponding to neo-classical or classical confinement e time scaling. Future studies on stability and control should be made 4 with this in mind. However, Stacey and others have indicated the stabilizing influence of Bohm scaling when compared with neo-classical or classical scaling. Energy-balance stabilization and control of fusion reactors are still problems for the future. Studies indicate that it should be possible to control a reactor by controlling the injected ion rate or source intensity or both. Controlling by means of the confinement time and controlling radiation power loss are two other possibilities.

49 VI. BIBLIOGRAPHY 1. Kofoed-Hansen,., Jensen, T. H., Sillesen, A. H. and Wandel, C. F., "Some Criteria for a Self-Sustaining Steady State Thermonuclear Reaction," Danish A.E.C., Research Establishment Riso Report No. 2, Rose, D. J. and Clark, M., Jr., Plasmas and Controlled Fusion, John Wiley and Sons Inc., New York-London, 1961, p. 31, p. 316, p Powell, C. and Hahn,. J., "Energy Balance Instabilities in Fusion Plasmas," Nuclear Fusion, 12, 1972, pp Stacey, W. M., Jr., "Operating Regimes of Controlled Thermonuclear Reactors and Stability Against Fundamental-Mode Excursions in Particle Densities and Temperatures," Nuclear Fusion, 13, 1973, pp Mills, R. G., "The Problem of Control of Thermonuclear Reactors," Engineering Problems of Fusion Research (Proceedings Symposium Los Alamos Scientific Laboratory, 1969), Los Alamos Report, LA 425, (Bl.1-Bl.5). 6. Mills, R. G., "Time Dependent Behavior of Fusion Reactors," B.N.E.S. Nuclear Fusion Reactor Conference at Culham Laboratory, September 1969, pp Powell, C., "Control of Energy Balance in a Fusion Plasma," Plasma Physics, Vol. 15, 1973, pp Rose, D. J., "Engineering Feasibility of Controlled Fusion" (A Review), Nuclear Fusion, 9, 1969, pp Glasstone, J. and Lovberg, R. H., Controlled Thermonuclear Reactions, D. Van Nostrand Inc., Princeton, New Jersey, 196, pp "A Short Course in Fusion Power," published by Princeton University, Kettani, M. A. and Hoyanx, M. F.' Plasma Engineering, John Wiley and Sons Inc., New York-Toronto, 1973, PP , P Carruthers, R., "Engineering Parameters of a Fusion Reactor," B.N.E.S., Nuclear Fusion Reactor Conference at Culham Laboratory, September 1969, pp

50 McCracken, G. M. and Erents, S. K., "Ion Burial in the Divertor of a Fusion Reactor," B.N.E.S. Nuclear Fusion Reactor Conference at Culham Laboratory, September 1969, p Greene, S. L., Jr., "Maxwell Averaged Cross-Sections for Some Thermonuclear Reactions on Light Isotopes," UCRL-7522, May Carnahan, B., Luther, H. A. and Wilkes, J.., Applied Numerical Methods, John Wiley and Sons Inc., New York, 1969, p. 319, p Hetrick, D. L., Dynamics of Nuclear Reactors, University of Chicago Press, 1971, p Mills, R. G., "Problems and Promises of Controlled Fusion Power," Mechanical Engineering, Vol. 97, September 1975, p. 21.

51 The vita has been removed from the scanned document

52 DYNAMICAL EQUILIBRIUM AND STABILITY OF A D-T FUELED FUSION REACTOR by Anthony Michael Carver (ABSTRACT) A point kinetics model of particle densities and temperatures within a CTR was formulated with certain assumptions. Common temperatures were assumed for ions and electrons. The number densities of each ionic species (deuterium and tritium) were equal. A divertor was employed in the reactor to remove ions, alpha particles, and impurities. Equilibrium point solutions and fractional burn-ups were obtained after simulating different operating regimes of a D-T fueled fusion reactor. This was done essentially by varying a temperature dependent confinement time. The source density remained constant throughout all calculations. The stability of some equilibrium points against perturbations in particle densities and temperatures was investigated by dynamic simulation. Some regimes were found to be unstable and others were found to be Lagrange stable.