Results of studies of the equilibrium field coil system for a variety of tokamak fusion reactor configurations are presented. These results include the determination of the EF coil currents, sizes, Ampere-turns, and stored energies for various reactor sizes, toroidal fields, plasma shapes, and plasma parameters. Problems are found with highly D-shaped plasmas and with high toroidal fields for smaller reactors. A simple expression, which adequately matches the wide range of cases considered, is also given for the stored energy

Ehst, D. A.

Evans, K., Jr.

Messerschmidt, P.

UNT (University of North Texas) Digital Library

EQUILIBRIUM FIELD COIL CONSIDERATIONS FOR TOKAMAK REACTORSK. Evans, Jr., D. A. Ehst, and P. MesserschmidtNOTICE •Thu rrport waa prepared u in account or workiponlofed i y the United States Government. Neither theUnited Sutei noi the United Statei Department ofEnergy, nor any of their employee!, nor any or their'"ntraclon, lubconlractnrf, or their employed, make:*ay warranty, expreii tit implied, or aiaumet any legalliability or relponiibilily for the accuracy, campleleneuof uiefulneu of any Infarmal.'on, apparalui, product orproccia (liKlofeed, or repreienu that it i ule would nolInfringe privately owned rlitita.Prepared for3rd ANS Topical Meeting onThe Technology of Controlled Nuclear FusionMay 9-11 , 1978Santa Fe, NMUofC-AUA USDOE<*TB1S DOCCMBNT 0 UMJMTJ®ARGONNE NATIONAL LABORATORY, ARGONNE, ILLINOISOperated under Contract W-31-109-Eng-38 for theU. S. DEPARTMENT OF ENERGYThe facilities of Argonnu National Laboratory art; owned by the United States Govern-ment. Under the terms of a contract (W - i\ - 1 07- Eng- iH) between the U. S. Department of En-ergy, Arfionnc Universities AH sociation and I he University of Chicago, the University employsthe staff and operates the Laboratory in accordance with policies arid pro^ *ams formulated, ap-proved and reviewed by the Association.MKMI'.KKSOK AIU1ONNK UNI V KKSI'l IKS ASSOCIATIONThe Univers i ty of ArizonaCarneg ie -Mel lon Univers i tyK a n s a s State Univers i tyThe Univers i ty of KansasCase Western R e s e r v e Univers i ty Loyola Univers i tyThe Univers i ty of ChicagoUniversi ty of Cincinnat iIllinois Ins t i tu te of TechnologyUnivers i ty of I l l inoisIndiana UniversityIowa State UniversityThe University of IowaMarc j i i c t t e U n i v e r s i t yM i c h i g a n S ta t e U n i v e r s i t yT h e U n i v e r s i t y of Mic higaiU n i v e r s i t y of Minm-sol i iU n i v e r s i t y of M i s s o u r iN o r t h w e s t e r n U n i v e r s i t yU n i v e r s i t y of N o t r e Harm*T h e Ohio S ta t e U n i v e r s i t yOhio UniversityThe Pennsylvania Slate UniversityPurdue UniversitySainl Lnuis UniversitySoutht rn Illinois Universi tyThe Univers i ty of T e x a s at AustinWashington Univers i tyWayne State Un ive r s i tyThe Univers i ty of Wiscons in•NOTICE-This report was prepared as an account of work sponsoredby the United Slates Government. Neither the United Stalesnor the United States Department of Energy, nor any of theiremployees, nor any of their contractors, subcontractors, ortheir employees, makes any warranty, express or implied,or assumes any legal liability or responsibility for the ac-curacy, completeness or usefulness of any information, ap-paratus, product or process disclosed, or represents that itsuse would not infringe privately-owned rights. Mention ofcommercial products, their manufacturers, or their suppli-ers in this publication does not imply or connote approval ordisapproval of *b" product by Argonne National Laboratoryor the U. S. Department of Energy.EQUILIBRIUM FIELD COIL CONSIDERATIONS FOR TOKAMAK REACTORSK. E v a n s , J r . , D. A. Ehsc , and P . MesserschmidtAPPLIED PHYSICS DIVISION, ARGONNE NATIONAL LABORATORYARGONNE, ILLINOIS 60439Results of studies of the equilibrium field co i l system for a variety oftokamak fusion reactor configurations are presented. These results include thedetermination of the EF coi l currents, siaes, Ampere-turns, and stored energiesfor various realtor s izes , toroidal fields, plasma shapes, and plasma parameters.Problems are found with highly D-shaped plasmas and with high toroidal fields forsmaller reactors, A simple expression, which adequately matches the wide rangeof cases considered, i s also given for the stored energy.MODEL EQUILIBRIUM COIL SYSTEMIn order to have a model equilibrium field(EF) coil system suitable for comparing a largenumber of reactor systems with different parame-ters, the following, somewhat simplified systemwas chosen: The model consists of N coil pairs(one coil up and one coil down) equally spacedaround the toroidal field (TF) coils and a dis-tance, A, from them. The TF coils are assumed tobe constant tension, and their shape is calculated(1 2)by the approximate method of Moses and Young. 'The plasma boundary is given by the formR = Ro + a cos(8 + d sin 6)Z - Ka sin 6 .ic is the plasma elongation, d is related to theplasma D-shapedness, RQ is the major radius, and ais the minor radius of the plasma. A scrape-offregion, A = 0.2 m, is allowed around the plasma.The blanket thickness on the inside is assumed tobe ABe • 1.0 m, and on the outside is assumed toI BSbe A°g - 1.5 m. An extra distance, A^ « 0.5 m, isallowed on the outside for access. The TF coilmagnet thickness, A , depends on the major radiusand the toroidal field, 3 , at the coils..TABLE I. Coil ThicknessTable I gives the values used in this study. (2)* _ _I Work supported byTJ.~S."~Department oF Energy."; Graduate Student, Northwestern University: Evanaton, Illinois..^S»max' T6810 "70.3790.4180.45590.5020.5610.622111.0521.2501.459131.4701.7562.053These parameters are sufficient to determinethe locations of the TF coils and hence in thismodel the locations of the EF coils. A typicalexample is shown in Fig. 1. It should be notedthat the model allows coils in the central areainboard of the TF coils. This may not be possible' in practice, and configurations which require a: substantial amount of EF current in this areashould be looked at with suspicion.With the positions of the EF coils fixed, the1 current in them is determined by a simultaneous, least-squares fit to the desired external field'• inside the plasma and a minimization of the stored; energy in the EF system. This procedure is des-: cribed in detail in Refs. 3-4. The coil size is< adjusted by iteration to maintain a current den-sity of JEp » 17.6 MA/mz, so that the coil size is', proportional to the current carried. :~ The "desired external field is calculated numer-I ically by fixed boundary MHO calculations for givenJ_plasma_parameters_suchas_the_cross-sectional. shapeLaEquilibrium-Field Colls0FIGURE 1. Equilibrium field coil model for a 6-m major radius reactor with 9 Ttoroidal coils for a somewhat elongated D-shaped plasma with toroidal beta of 8%,plasma current of 9.8 MA, and aspect ratio of 3.5. (N = 10 and A = 1 m.)(determined by K,d); plasma b*ta, B ; safety fac-tor, q; and plasma pressure profile. All of theequilibria in this paper have q(0) = 1, q(a) = 3,and a relatively broad pressure profile. The basecase is (K - 1,65, d = 0.25, A = 3.5, S = 0.08,TFCRo = 6 m, B = 9 T), similar to that of themax ...Argonne 1977 EPR design. 'The external field that must be produced bythe EF coils is the total field from the MHD equi-librium less that part of the field that is pro-duced by the plasma current itself. Figure 2shows examples of these fields for various plasmacross sections.; By varying the relative importance of the ex-ternal field and the stored energy in the least-squaras minimization, one can get a range of con-figurations, trading off accuracy in matching theiequilibrium against stored energy. The EF systemparameters given in this paper correspond to aO.'.'X match to the desired external field. Experi-ence has shown that if the EF coil configuration0.33 m.the extreme cases.TFCBary MHD equilibrium to an average error of 0.4%,the corresponding free boundary MHD equlli-librium (where the EF coils are specified insteadof the boundary shape) will adequately match theparameters and boundary of the fixed boundaryequilibrium.The EF coil radius is restricted to a__ <This restriction only appears in a few ofOnly one reactor configuration» 6 m, ma = 13 T) calculated for this paperhad no solution with this restriction.A final caveat is that no attempt was made todecouple the EF and ohtnic heating coil systems,even though this Is most likely desirable in prac-tice. Decoupling would increase the EF systemstored energy, Ampere-turns, coil volume, etc. Theincrease is not too great, however, if there is areasonably large number of coils.NUMBER OF COILSFigure 3 shows the relation between the storedenergy and the number of colls. With too few coilsit is difficult to match the equilibrium. For moreeproduces the external field from the fixed bound-_ than about 10 coil pairs little improvement isgained and the access for vacuum ports, neutralbeams, and other penetrations is hampered. It• n••', H U M FO'.lQ - ?" v. if/'?Mi.)OT AT fl'j"-K»I.O K-\.l = l.65dsQ.OO0.25d=O.5O(1r SS.\o>oUiUiaUJocoi -tnu.uiFIGURE 2. Required equilibrium fields for several plasma cross section's. The equilibriaall have Bt = 0.08, q(0) • 1, q(a) = 3, and A = 3.5. (The cases with field lines concaveto the right are unstable to vertical displacements.)should be noted that Fig. 3 is for a relativelydifficult (K = 1.65, d = 0.25) plasma shape.Fewer coils may be necessary for a more nearly cir-cular shape. In addition, if some attempt is madeto optimize the coil locations for a particular: device, fewer coils may be necessary. On the- other hand, in order to maintain the flexibilityto handle, for example, equilibria with differentparameters, it is desirable to Include a reasona-ble number of coils. In the rest of this studyi the number of coils was held constant at N = 10. coil pairs. :0 5 10 15 20NUMBER OF EF COIL PAIRSFIGURE 3. Stored energy as a function of thenumber of .coils. Base case equilibrium. - -•DISTANCE FROM TF COIL ;The stored energy and other EF system parame-ters axe very sensitive to the distance, A, of the-EF coils from the TF coils. The relationship be-tween the stored energy and this distance is shownin Fig. 4. The EF coils must be sufficiently far, ooUl2OI—(AUlVO0-00-0 0*5 1*0 1-5 2'0DISTANCE FROM TF COIL, mFIGURE 4. Stored energy as a function of the dis-tance from the TF coils. Base case equilibrium.from the TF coils that the heating due to currentsinduced in the TF coils by the EF coils is not ex-cessive. In addition, the stray field from the EFcoils reduces the useable toroidal field in the TFcoils, so that the EF coils must also be far enoughaway to keep the stray field low.Only EF coils outside the TF colls are con-sidered in this study for the following reasons:(1) Superconducting coils inside the TF coilswould be hard to wind and normal conducting coilswould most likely have too great a power loss;(2) the space inside the TF coils is well occupiedby other systems such as the blanket, vacuum ducts,etc.; (3) inside coils would interfere with accessand remote maintenance.j The remainder of this study assumes the EFcoils are fl - 1.0 m from and outside the TF coils.Further study of considerations beyond the scopeof this paper would be necessary to determine amore accurate dependence of & on the reactorparameters such as size and toroidal field.ASPECT RATIO AMD 8tThe EF system is not especially sensitive toaspect ratio,, A •» Ro/a, nor 8fc as shown in Fins.|5-6. The stored energy varies approximately asJA"1. Both the lower aspect ratios (with Ra fixed)and higher betas have higher currents, which-accounts for the increase in stored energy.OOuiQUJonoI—V)UJ10-0-02-5 3-0ASPECT RATIO4-0FIGURE 5. Stored energy as a function of aspectratio. (RQ is fixed.) Base case equilibrium.OoUlQUl0-050 0-075PLASMA BETA0-100FIGURE 6. Stored energy as a function k»£ plasmabeta. Base case equilibrium. :PLASMA SHAPEThe EF system is quite sensitive to the plasmacross section as shown in Fig. 7. In particular,the more highly D-shaped plasmas are hard to pro-duce: the natural configuration seems:to be anellipse. In particular, as can be seek in Fig. 8,the more highly 0-shaped plasmas have 1 largeamount of EF current in the central region inboardof the TF coils. It has been found previouslythat if EF coils are not allowed in this region,(Athen these higher values of d cannot be produced.-The difficulty in-producing the more W-shaped•' ! . d . O ' i ' O I . I O - • ? • • *ooLJ2OSO1-0O'O0-00 0*25 0-50PLASMA D-SHAPEDNESSFIGURE 7. Stored energy as a function of elonga-tion and D-shapedness. The solid line is for < =1.65; the dashed line is for < = 1.3; and tha dash-dot line is for K » 1.0. The reason for the cross-overs is that a given value of d represents moredistortion from the d = 0 case when the elongationis less, as can be seen in Figure 2.plasmas must be traded off against the fact thatsuch plasmas are predicted to have higher 8 tvalues and hence increased performance.It should be noted as well that the morehighly elongated, D-shaped plasmas are unstable tovertical displacements; that is, the J x I force,which is perpendicular to the external field linesas shown in Fig. 2, has a component away from themidplane. Even if the overall position can bestabilized by feedback, locally this force maytend to distort the plasma.MAJOR RADIUSThe EF system difficulty increases more slowlywith size r.han the volume does. This is primarilybecause the plasma current increases slowly withsize for a given cross section and equilibrium.The stored energy, in particular, increases alittle faster than Rp as seen in Fig. 9. Thisscaling is favorable to large reactors.IC=I.Od * 0.00o od > 0.25d»0.50FIGURE 8. The EF system model as a function of elongation and D-shapedness.s i r ' • ' . • '. • . * • • " . : : ! : .>oou.16*012-0-6-0 8 0MAJOR RADIUS, m10-0FIGURE 9. Stored energy as a function of majorradius. The solid line is for BTFC = 7 T; thedashed line is for 9 T; the dash-Sot line is for11 T; and the dotted line is for 13 T.TOROIDAL FIELD, There is a strong dependence of the EF systemon the toroidal field, as shown in Eig. 10. Thestored energy, in particular, increases almost asUL__] • This is faster than the square dependenceone might expect, because the higher field magnetsare thicker, making the EF coil further from theplasma.Figures 11 and 12 show the EF system model forseveral values of toroidal field. It can be seenthat for the smaller reactor, fields higher thanabout 9 T probably have too nuch current in thecentral regioa to be practical. (Less current,however, would be required for a more nearlycircular plasma.) There seems to be no essentialproblem with high fields in the larger reactor,but the EF coils are quite large and carry a sub-stantial amount of current.FIT TO THE DATAThe stored energy for the range of casestreated in this paper can be fit within an errorof 20% by the following formula:u(GJ) - 1.3 * 10-uF(K,d)A-1 exp[0.53A(m)]where F(ic,d) is a form factor with value unity forthe base case (K = 1.65, d = 0.25). The dependenceof this form factor is quite model dependent. Thedata in this paper (see Fig. 7) are matched by:F(K,d) • 1.9 - 2.5 K + 1.4 K 2 + (11.7 - 20.0 K+7.0 K2)d + (17.1 + 1.4 K - 4.0 K2)d2 .It should be kept in mind that d values greaterthan 0.25 have a substantial amount of current inthe central region, and that a realistic systemwould probably have a stronger increase of F(ic,d)with d. It should also be recalled that small,high-field reactors are more difficult than thisformula would indicate.ooOSo16-012-0-40-0- 250200-OO7-0 9 0 110 130TOROIDAL FIELD AT COILS. T7-0 9-0 1M> 130TOROIDAL FIELD AT COILS, T50 -7-o 9-o no is-oTOROIDAL FIELD AT COILS. TFIGURE 10. Stored energy, Ampere-turns, and co i l volume as a function of to ro ida l f ie ld .The solid l i ne i s for R » 6 m, the dashed l ine i s for R «• 8 m, and the dash-dot l ine i sfor R - 10 m.LJTill'.! fi.""; r01.10 - y * o:/"R0=6mB T F C B T F C =maxFIGURE 11. The EF system model as a function of' toroidal field for a small reactor.No solution exists for the B T F C = 13 T case if the coils are constrained to be lessthan 0.33 m, and the case shown is obviously impractical.: U J ; . t <.-<'_\. -R a 10 moo/of 00 /Ofo>\ oBTFC=9T0oo<!o „0oO0/O o\ \ \oFIGURE 12. The EF system model as a function of toroidal field for a large reactor.REFERENCES1; R. W- MOSES, JR., and W. C. YOUNG, "Analytic1 Expressions for Magnetic Forces on Sectoredj Toroidal Coils," Proa. 6th Symp. Engineering[ Problems of Fusion Reaearah (1975), p. 917.2. L. R, TURNER and Ma A. ABDOU, "ComputationalModel for Superconducting Toroidal-Field Mag-nets for a Tokamak Reactor," Argonne Hationsl; laboratory, AHL/FPP/TM-88 (1977).3. A. M. M. TODD, "Calculation of Coil Currentsj to Produce a Giver Vacuum Poloidal Field,"; Princeton Plasma Physics Laboratory, PPPL-i TM-300 (1977).5 .J . BBDOKS, K. EVANS, J R . , H. STEVENS, andL. TURNER, "The Equ i l i b r ium F i e l d Coil Designf o r the Argonne EPR Des ign , " Proa. 7th Symp.Engineering Vvohlems of Fusion Research,Knoxville, Tenn. (1977).K. EVANS, JR., "High 8 t Equilibria in Tokamaks,"Argonne National Laboratory, ANL/FPP/TM-98(1977).

Equilibrium field coil considerations for tokamak reactors

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Equilibrium field coil considerations for tokamak reactors

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