Research Laboratory for Nuclear Reactors, Tokyo Institute of Technology

Transcription

1 th IEEE/NPSS Symposium on Fusion Engineering (SOFE) October 14-17, 3, San Diego, CA USA Plasma Production in a Small High Field Force-alanced Coil Tokamak ased on Virial Theorem H.Tsutsui, T.Ito, H.Ajikawa, T.Enokida, K.Hayakawa, S.Nomura, S.Tsuji-Iio, R.Shimada Research Laboratory for Nuclear Reactors, Tokyo Institute of Technology

2 Introduction We studied the tokamak with the Force-alanced Coils which are hybrid helical coils of OH and TF coils and reduced the electromagnetic force. The virial theorem, which is derived only form the equilibrium, shows that the tension is required to hold the magnetic energy. The virial theorem in magnet systems is derived by the replacement of plasma pressure to stress. In this work, we extend the FC by the virial theorem, and obtain the minimal stress condition. The new small tokamak based on the virial theorem is constructed and plasma production is started.

3 Principle of Force-alanced Coil Centering Force by Poloidal Current Force-alanced Coil Centering force is much reduced, but stress distribution is not investigated. Hoop Force by Toroidal Current

4 ackground TODOROKI-I Parameter Toroidal Field Plasma Current Time of Discharge Value 1T 1kA 4ms The error field by FC made the control of plasma difficult. The force of toroidal direction was reduced in FC. Is it held in stress? Reduction Error Field Estimation of Stress Application of Virial Theorem

5 Virial Theorem j + S = = j: current density : magnetic field S: stress = µ j tensor Equilibrium Eq. Tr 3 i= 1 T ( T + S) ( T + S) = µ T: Maxwell stress tensor dv σ : principal stress i = I σ idv = dv U M > µ ~ V σ U σ 3 Positive stress (tension) is required to hold the field. Uniform tension is favorable. ~ 1 σ ~ ~ Theoretical limit is determined. 1 = σ = σ 3 = 3 i= 1 Ω M ~ σ i σ σ dv V Ω = 1

6 Application to Helical Coil ~ σ θ ~ σ φ ~ σ θ N = = + I I θ φ N A N ~ σ = 1 : Pitch of Coil A: Aspect Ratio φ + N A A A log8a A log8a A + N log8a A ~ <σ> 1-1 N ~ σ θ A= A=5 ~ σ φ ~ σ ~ θ σ = = φ 1 A=1 is optimal in energy. Virial-Limit Condition

7 Shape of Coils Relations of pitch number and aspect ratio of Virial- Limit Coil (VLC) etc. 3 <σ φ >= N=9 FC N 1 σ φ = A σ θ = σ θ =σ φ <σ φ >= <σ θ > N=6 VLC <σ θ >= N=4 Aspect Ratio A =4

8 Comparison of Toroidal Field <σ> ^max 1 ˆ σ σ θ = σ φ = V Ω U TF σ TFC σ θ =σ φ φ TFC 1 φ TFC σ θ =σ φ σ φ = = ˆ σ σ θ = A In the case of low aspect ratio, 1.5 times stronger magnetic field is created compared with traditional TF coil. A

9 Toroidal Effect Distribution of stress in the toroidal shell with circular cross section is derived analytically by use of magnetic pressure. z r R a ρ θ φ o µ φ θ p r R r r p r a r u ' )d ' ( ' ) ( ) ( ) ( R r u R r arp T r R r u T = = φ θ Equilibrium of Magnetic Pressure and Stress

10 p^ p σ θ σ φ Distribution of Stress A=1 θ <σ φ >= (large aspect ratio) <σ φ >=<σ θ > TFC 1 3 pˆ 1-1 V U T M σ^ φ θ µ When A=1, distribution of stress is flat. There is no advantage of helical winding. p^ TFC <σ φ >= <σ φ >=<σ θ > σ φ p A=1 θ σ θ σ^

11 ^ p 4 TFC <σ φ >= σ φ (π)=σ θ (π) <σ φ >=<σ θ > σ θ p σ φ Distribution of Stress (low aspect ratio) A= θ 4 N=3 ^ - σ When A<1, distribution of stress is important. Assumption of large aspect ratio is not held. Optimal distribution is achieved to minimize the stress at θ=π.

12 Uniaxial Stress Model Equilibrium of Electromagnetic Force and Stress dt ds df ds u F R T + R u + = t u + f u = T: tension, F: sharing force R: curvatur radius, f : electromagnetic force s: coordinate by length of coil T(s+ds) F u (s+ds) s F u (s) T(s)

13 Tension and ending Moment 3 T (kn) 1 N= N=3 N=4 VLC M u [k Nm] 4 VLC N= N=3 N=4 - π/ θ (rad) π π/ θ [rad] π The tension of coil with pitch=3 is reduced and its distribution is flat. In the fat cable, the bending stress (proportional to bending moment) is important. The distribution of bending moment in the coil with pitch 3 is flat.

14 Stress Analysis of VLC Why FEM analysis? 3D analysis is required because the virial-limit condition is obtained from the D shell model. Conditions in FEM Analysis Current layer coincide with magnetic surface. VLC with N=3 FC with N=4 HC with N=3 Models in Analysis HC with N=4

15 Stress Analysis of VLC Distributions of von Mises Stress Max Max Max Max VLC(N=3) has no stress concentration and minimum stress compared with those of other coils. VLC realize both nearly uniform distribution of

16 Outline of Tokamak utline of Design asy winding of coil integer winding pitch arge Plasma Volume low aspect ratio Coil Support Vacuum Vessel oil orbit between their supports is nearly straight. Parameters of Tokamak VLC (N=3) Todoroki-II Materials

17 Vertical Field ow to design Vertical Field Coil (VFC) Controllability minimization of mutual inductances to VLC. Positional Instability n-index: n (stable condition: <n<1.5) -index Positions of VFC on the cross sectio

18 Effects of Plasma Current Electromagnetic force ( N/m ) VLC VLC & PLASMA VLC & PFC VLC & PFC & PLASMA Normal stress in tangent direction ( MPa ) Poloidal angle ( deg. ) FEM analysis VLC VLC & PLASMA VLC & PFC VLC & PFC & PLASMA Poloidal angle ( deg. ) In an actual system, electromagnetic force is modified by magnetic field produced by plasma and PF coil currents. Their influences are small enough to keep the advantage of VLC.

19 Stress with Supporting oard Cables are supported by boards, but freely movable to the tangential direction t. Max: 14MPa<144MPa Normal stress in tangent direction ( MPa ) nearly flat distribution of tension allowable tension is about 6 MPa VLC VLC & PLASMA VLC & PFC VLC & PFC & PLASMA Poloidal angle ( deg. ) Maximum stress in coils and boards is much smaller than their allowable stress. Sharing Stress in a oard

20 Power Supply and Its Operation ower Supply Discharge period: about 1 msec peration In the initial stage, there is no toroidal field because VLC is a hybrid coil of CS and TF coil φ Ignitron VLC current I pole (ka) (T) Fist Ignitron Capacitors Todoroki-II 1 5 V loop (V) First: Toroidal Field Second: Loop Voltage Plasma current(ka)

21 reakdown Condition ormalized Poloidal Field P T = R + Z T P/T Experiment Case 1 Case Case 1: Ideal VLC Case : Designed VLC Case 3: Designed VLC+VFC Case 1: Ideal Coil Orbit Case : Designed Coil Orbit equired Loop Voltage in reakdown Next equation is obtained from Townsend Avalanche Model. V min.1 ea = πr log( Ne N) A R major radius A 1, A constants of gas species X p limiter radius n e /n multiplication factor of electron p T R (mm) ( ) Additional vertical field is required for breakdown. X P Loop voltage in case 1

22 Plasma reakdown Maximum Electric Field in Todoroki-II Eφ ( V/m) p/t =.1.1 Large Tokamak P (torr).1 reakdown region with pre-ionization H Expected breakdown region Next equation is obtained from Townsend Avalanche Model. α 1 ( E) α = Apexp p = L =. 5a eff A, : constants of gas species p : gas pressure E : toroidal electric field a eff : limiter radius T P Error fields by eddy current require pre-ionization.

23 Eddy current analysis A = µ 4 π J ds ρ J= V V: current vector potential s : distance between source and observed points. Vacuum vessel in FEM 1 Vacuum vessel material sus34 width.81-3 m resistivity m major radius.3m minor radius.8 m A vacuum vessel is constructed by 3 insulated blocks, and has a periodicity of 1 degree. Ports

24 Eddy current and error field A result by FEM (EDDYCAL) Normalized poloidal field P T = R + Z T.9 Vacuum vessel Without eddy current With eddy current Error fields by eddy current prevent plasma breakdown.

25 Plamsma position control Vertical field for equilibrium z (gauss) z µ I p 8R lin 3 () r = ln + + β p 4πR a - -4 I p plasma current l in plasma internal inductance p poloidal beta Required field Applied field Time (ms) Plasma current (A) Plasma position δr (mm) W/O VF V.5 3 (ms) W/O VF V.5 3 Time(ms) Limiters Maximum current is enhanced to about 1.6 times.

26 Peak plasma current q a =3 Plasma current and safety factor 1 Peak plasma current (ka) With VF Without VF Parallel connection Series connection Same toroidal field Limited by toroidal field Limited by loop voltage V φ ( toroidal field) (V) 5 3 q Safety factor on plasma surface a = at R P R Major radius Plasma Current ka 4 q a Time πa µ R (msec) I T P 3

27 Cauchy Condition Surface (CCS) method CCS method is based on an exact integral equation. σ φ( x) + [ G( x, y) gradφ φ( y) gradg] ds/ r = µ j Ω P y Ω P c( y) G( x, y) dv( y) / r y Magnetic field by plasma current is replaced by surface integrals of and φ on CCS. and φ on CCS are determined from sensor signals. Accuracy of calculation increases with the number of sensor signals. Axisymmetry is assumed. Plasma current is also evaluated from magnetic data.

28 Visualization by CCS Without vertical field With vertical field [ka] Ip Hoop direction Rogowski Coil CCS y vertical field [ka] Ip Rogowski Coil CCS

29 Summary I The relation of toroidal field and stress is obtained by virial theorem, which shows that the optimal stress configuration is uniform tensile stress. When A=, VLC makes 1.4 times stronger magnetic field than TF coil. VLC winding generates small error fields, and makes room for blanket and other parts in conventional tokamak reactors. Nearly uniform stress distribution with VLC configuration is obtained from both uniaxial model and FEM analysis. A small VLC tokamak Todoroki-II was constructed and its experiments started.

30 Summary II Peak plasma current with an additional vertical field was increased to 1.6 times larger than that of nonvertical field discharge. Error field is estimated from eddy current by FEM. It was shown that plasma current was limited by loop voltage or toroidal field strength. Plasma current and surface were evaluated by CCS method, and validity of CCS method for a small pulsed tokamak was verified. In order to increase discharge time, current control with arbitrary wave form is required.