PLASMA KINETICS MODELS FOR FUSION

SYSTEMS BASED ON THE AXIALLY-SYMMETRIC

MIRROR DEVICES

A.Yu. Chirkov1), S.V. Ryzhkov1), P.A. Bagryansky2), A.V. Anikeev2)

1) Bauman Moscow State Technical University, Moscow, Russia

2) Budker Institute of Nuclear Physics, Novosibirsk, Russia

Injection of energetic neutrals

Neutron generator concept:T ~ 10..20 keV, n ~ 1019 m–3, a ~ 1 m, L ~ 10 m, B ~ 1..2 T in center solenoid, ~ 20 T in mirrors,

fast particle energy ~ 100..250 keV, Pn Pinj

Simple mirror geometry with long central solenoid

The power balance scheme

dVP

dVPQ

ext

fus

Plasma amplification factor

Local balance

lossessourcest

nj

j

Γ

eiextnfusiiiBi PPPPTknt

)(

23 J

sbi

einfuseeeBe PPPPPTknt

)(

23 J

Electron – ion bremsstrahlung

i

eiie

i

ei

ieei dpppfpnnpddf

ddnnP

0

23 4)()()( vv p

max

0

)( dd

dpei

ei22222

max )( cmcmcp ee

)/1()(4)/exp()(

23

Kcmpf

e

2)/(1/1 cv)/( 2cmTk eeB

mec2 = 511 keV

)1(408.0exp)2ln4()2ln(4 320

3121

ibei ZcC

32 cmrC eeb

1 2 3 4 5 6 7 8 9 10 0

10

20

30

40

50

60

70

80

90

100 eic/(CbZeff

2)

- - - - - numerical ––––– fit – – – – extreme relativistic

213

16ib

eiNR ZcC 3

121 )2ln(44 ib

eiER ZcC

Electron energy losses during slowing down on ions

1 10 100 103 104 1051

1.1

1.2

1.3

1.4

1.5

1.6

1

2 3

Te, eV

g––––– fit– - – - Elwert- - - - - Gould

Gaunt factors for low temperatures. Approximations of B: 1 – formula corresponds g 1 at Te 0; 2 – g gElwert at Te 0; 3 – by Gould

13

2031

2

22)1(408.0exp)2ln4()2ln(4

)/1(KZnC

P effebei

eiPd 22 )1)(/exp(

2223

32effeb

eiNR ZnCP Eeffeb

eiER CZnCP 2

322 )2ln(12 CE = 0.5772...

i

iii

iieff nZnZZ 22 Pei – correction to the Born approximation

2

)2(ln 2

3

21

eff

effeiNR

ei

BZ

Z

PP

– for Te ~ 1 keV [Gould]

Integral Gaunt factor: eieiNR PPg Kramers/

Approximation taking into account Gaunt factor for low temperatures:

)505exp(49.0)/008.0exp(139.02

3

eff

effB

Z

Z

Radiation losses

Electron – electron bremsstrahlung

dpdpdffud

dnPee

eee

23

13

21212 )()(),(

21 pppp

2/322/14 ebF

eeNR nCCP 1/ 2

23

effeiNR

eeNR ZPP

CF = (5/9)(44–32) 8

EebeeER CncCP 4

523 )2ln(24

CE = 0.5772...

Approximations of numerical results

222

332

effebei ZnCP )(07.2)4.4exp(32.068.0 B

2/322/14 ebF

ee nCCP )1.77.248.336.226.664.01( 65432

2

1

1 0 1 0 0 1 0 0 0 0.1

1

10

100 Pei/(Zeff2Cbne

2) Pee/(Cbne

2)

Te, keV

1 (ei)

2 (ee)

non rel.

ext. rel.

0 2 0 4 0 6 0 8 0 1 0 0 0

1

2

3

4

5 Pei/(Zeff2Cbne

2) Pee/(Cbne

2)

Te, keV

1

1

2

3

3

numerical - - - - - numerical + Born corr. – – – – non relativistic 1 – fits 2 – McNally 3 – Dawson

Synchrotron radiation losses

ees

e

eB

eBeeees nC

cmTk

TknBncmrP

2

22

20

2232

0 5.3323

32

Emission in unity volume of the plasma:

10 100 1000

Ps/Ps0

10–3

Te, keV

10–2

0.1

1

10–––– Trubnikov– – – Trubnikov + relativistic corr.- - - - Tamor, Te < 100 keV– - – - Tamor, Te = 100–1000 keV– - - – Kukushkin, et al.

0.2 0.4 0.6 0.80 10

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2Trrel

0

12

34

1 – Te = Ti = 30 keV 2 – 50 keV3 – 70 keV4 – 90 keV

a = 2 m, Rw = 0.7, Bext = 7 T

Output factors at a = 2 m, Rw = 0.7, Bext = 7 T, 0 = 0.1 (upper curves) and 0 = 0.5 (down)

Output factor vs 0 at a = 2 m, Rw = 0.7, Bext = 7 T, Te = Ti = 30 keV (1), 50 (2), 70 (3), and 90 keV (4)

VRaBTnP wexteetots 11)1(414.0 2/125.1

05.25.2

Losses from plasma volume (Trubnikov):

Output factor: 1160 2/12/3

Tr wR – Trubnikov

])511()1/(3201/[ 2/3relTam

wR

/11

5.15.21rel – relativistic correction [Tamor]

)/(2cep ca

22

Ra

)511/(1039.0 Generalized Trubnikov’s formula for non-uniform plasma [Kukushkin et al., 2008]:

VT

a

RBTnP effe

eff

weffeeffetots

5115.21

1)(414.0 ,

5.20

5.2,,

a

eeffe rdrrna

n0

, )(2a

eeffe drrTa

T0

, )(1kaaeff

Proton slow-down rate (a) and cross section (b) for interaction with electrons (- - - - -), deuterium ions (–––––) and helium-3 ions (– - – - –):

1, 2 – Coulomb collisions, 3 – nuclear elastic scattering

D–T reaction and slow-down cross sections ratio for tritium ions in the deuterium plasma with Ti = Te = T

Fast particle kinetics

b

Some estimations

)(4)( 33

c

sqfvv

v

High-energy approximation:2131

2 24

3

e

eB

i i

ii

e

e

e

ic m

Tkm

nZnmv

ee

eB

es

neZTk

mm

42

2320

2

212

Optimal parameters: T 10 keV, Einj 100 keV, Pn Pinj ~ 4 MW/m3

keVkeV keV

MW/m3m3/s

The Fokker – Planck equation

aaa

aaCa

NCaCa Lsf

DfAAf

Dt

f

)(4

)(sin

sin

1)(102

02

22

vvvvv

vvv vvvv

b

bbbb

bba

C uuuu

uD )erf(1)erf(

21

2/vvv

bb

bbb

bba

C uuu

uu

D )erf(1)erf(1241

2/v

b

bb

bbb

aba

C uu

uumm

A )erf()erf(1/2v

v2/

2

0

2

/4 a

babbaba

m

neZZ

)2/(2

bBbb Tkmu v

b

bbbNv EEvnA )/()(

21 v

)(v a

af

L)(||

|| v a

af

L

0

sin),(21)( dfF aa vv

aa qds

0

sin)(21

Boundary conditions:

0),( 0 aaf vv

0),( vaf In the loss region

0),0( vvaf

0)0,( vaf 0),(

vaf

Quasi isotropic velocity distribution function:

Numerical scheme

220

420

04 a

a

meZn

)(4 00

a

aaq

fv

0

30

0 atv

0ffy a

0

2 )(~

NC AA

a vvva

CDb

00

2~v

v vv

0

0~

C

a Dc

v

Scales and dimensionless variables:

az

0vv

0

~ttt

0

)(~tv

Dimensionless equation (symbols “~” are not shown):

yzyc

zybay

ztyz 22 sin

sin

)(),1( szy

Numerical scheme

Kkkhzk ,...,2,1,0,1 Jjjhj ,...,2,1,0,2 ,...2,1,0, nnhtnGreed:

Finite difference equations:

21

,1,,,11

1

,,11,,2 )()(

h

yybyybh

yayah

yyz jkjkkjkjkkjkkjkkjkjk

k

jk

jkk

j

jkjkjjkjkjk

yz

h

yyyyc

,

,222

1,,,1,1

sin

)(sin)(sin

02

0,2,

hyy kk 0

2

1,1,

hyy JkJk

Matrix form:

1,...,2,1,11 KkYYY kkkkkkk DBCA

01 YY 1,...,2,1

)(

JjjaK sY

1,...,2,1,

Jjjkk yY

1,...,2,1,

Jjjkk DD

)1()1(,

JJjkk AA

)1()1(,

JJjkk BB )1()1(

1,,1,...................

.........................

JJ

kjj

kjj

kjjk CCCC

21h

bA kkj 1,...,2,1 Jj

21

1

1 hb

haB kkk

j 1,...,2,1 Jj

hzz

hc

hbb

haC k

jk

k

j

jkkkkkjj

2

,

21

22

21

1

1, 1

sinsin

1,...,2,1 Jj

1sinsin

1

222

2,1hcC kk

22

1,hcC kk

jj 1,...,3,2 Jjj

jkkjj

hcC

sinsin 1

22

1, 2,...,2,1 Jj

1sinsin

122

2,1J

JkkJJ

hcC

hy

zD jkk

kj

,2 1,...,2,1 Jj

Solution:

0,1,2,...,2,1,111 KKkYY kkkk

1,...,2,1,)( 11 Kkkkkkk BAC

1,...,2,1,)()( 11 Kkkkkkkkk DAAC

Examples of numerical calculations

Velocity distribution function of tritium ions and its contours at time moments after injection swich on t = 0.1s (а), 0.3s (b) и 10s (c). Deuterium density nD = 3.31019 м–3, energy of injected particles 250 keV, injection angle 455, injection power 2 MW/m3, Ti = Te = 20 keV, = 10

keV, slow-down time s = 4.5 s, transversal loss time = s

Role of particles in D–T fusion mirror systems

5 10 15 200

0.004

0.008

0.012

T, keV0

0.1

0.2

0.3

n /n0

p /p0

5 10 15 201.6

2.0

2.4

2.8

3.2

0.02

0.04

0.06

0.08

0.10

T, keV

/s

WL /W0

Relative pressure and density of alphas in D–T plasma (D:T = 1:1):

–––––– isotropic plasma (no loss cone)– – – – mirror plasma with loss cone

n0 = nD + nT = 2nD p0 = pD = pT

Energy losses (WL) due to the scattering into the loss cone and corresponding energy loss time () of

alphas in D–T mirror plasmaW0 is total initial energy of alphas (3.5 MeV/particle)

s is slowdown time

Parameters of mirror fusion systems: Neutron generator and reactors with D–T and D–3He fuels

Parameter Neutron generator regimes Tandem mirror reactors

Ver. # 1 Ver. # 2 Ver. # 3 Ver. # 4 D–T fuel D–3He fuel

Plasma radius a, m 1 1 1 1 1 1

Plasma length L, m 10 10 10 10 10 44

Magnetic field of the central solenoid B0, T 1.5 1.5 2 2 3.3 5.4

Magnetic field in plugs (mirrors) Bm, T 11 11 14 14 14.8 14.8

Averaged 0.5 0.5 0.5 0.5 0.2 0.7

Deuterium density nD, 1020 m–3 0.26 0.22 0.21 0.415 0.82 1.35

Ion temperature Ti, keV 10 11 22 22 15 65

Electron temperature Te, keV 10.5 8.5 18 19 15 65

Ion electrostatic barrier , keV 15 16.5 33 44 60 260

Injection power Pinj, MW 60 74 60 55 – –

ECRH power PRH, MW 18 0 0 0 – –

Neutron power Pn, MW 24 30 43 59 – –

Plasma amplification factor Qpl = Pfus/(Pinj + PRH) 0.38 0.5 0.9 1.34 10 10

Total neutron output N, 1018 neutrons/s 11 13 19 26.5 – –

Neutron energy flux out of plasma Jn, MW/m2 0.4 0.4 0.7 1 2 0.04

Heat flux out of plasma JH, MW/m2 1.2 1.8 1.8 2.0 2.4 0.94

Thank you!