Post on 18-Jan-2018
description
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!