Page 1
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
Page 2
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
Page 3
The power balance scheme
dVP
dVPQ
ext
fus
Plasma amplification factor
Local balance
lossessourcest
nj
j
Γ
eiextnfusiiiBi PPPPTknt
)(2
3J
sbi
einfuseeeBe PPPPPTknt
)(
23
J
Page 4
Electron – ion bremsstrahlung
i
eiie
i
ei
ieei dpppfpnnpddf
d
dnnP
0
23 4)()()( vv p
max
0
)( dd
dp
eiei
22222max )( cmcmcp ee
)/1()(4
)/exp()(
23
Kcmpf
e
2)/(1/1 cv)/( 2cmTk eeB
mec2 = 511 keV
)1(408.0exp)2ln4()2ln(43
203121
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
3121 )2ln(44
ibeiER 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
1
320
31
2
22
)1(408.0exp)2ln4()2ln(4)/1(K
ZnCP effebei
eiPd 22 )1)(/exp(
222
332
effebeiNR ZnCP Eeffeb
eiER CZnCP
2322 )2ln(12 CE = 0.5772...
i
iii
iieff nZnZZ 22 Pei – correction to the Born approximation
2
)2(ln2
3
21
eff
effeiNR
ei
BZ
Z
P
P– 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
Page 5
Electron – electron bremsstrahlung
dpdpdffud
dnP
ee
eee
23
13
21212 )()(),(
2
1pppp
2/322/14 ebF
eeNR nCCP 1/ 2
23
effeiNR
eeNR ZPP
CF = (5/9)(44–32) 8
EebeeER CncCP
4523 )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/(Zeff
2Cbne2)
Pee/(Cbne2)
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/(Zeff
2Cbne2)
Pee/(Cbne2)
Te, keV
1
1
2
3
3
numerical - - - - - numerical + Born corr. – – – – non relativistic 1 – fits 2 – McNally 3 – Dawson
Page 6
Synchrotron radiation losses
ees
e
eB
eBeeees nC
cm
Tk
Tkn
BncmrP
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
3
4
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
2
2
R
a
)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
,5.2
05.2
,,
a
eeffe rdrrna
n0
, )(2
a
eeffe drrTa
T0
, )(1
kaaeff
Page 7
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
Page 8
Some estimations
)(4)(
33c
sqf
vvv
High-energy approximation:2131
2 2
4
3
e
eB
i i
ii
e
e
e
ic m
Tk
m
nZ
n
mv
ee
eB
es
neZ
Tk
m
m42
2320
2
2
12
Optimal parameters: T 10 keV, Einj 100 keV, Pn Pinj ~ 4 MW/m3
keVkeV keV
MW/m3
m3/s
Page 9
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(
2
12/vvv
b
bbb
bb
baC u
uuu
uD )erf(
1)erf(
12
4
12/v
b
bb
bbb
aba
C uu
uum
mA )erf()erf(
1/2v
v2/
2
0
2
/4 a
babbaba
m
neZZ
)2/(2
bBbb Tkmu v
b
bbbNv EEvnA )/()(
2
1v
)(v a
af
L)(||
|| v a
af
L
0
sin),(2
1)( dfF aa vv
aa qds
0
sin)(2
1
Boundary conditions:
0),( 0 aaf vv
0),( vaf In the loss region
0),0(
vvaf
0)0,(
vaf0),(
vaf
Quasi isotropic velocity distribution function:
Page 10
Numerical scheme
220
420
04 a
a
m
eZn
)(4 00
a
a
a
qf
v
0
30
0 atv
0f
fy a
0
2 )(~
NC AA
a vvv
a
CDb
00
2~
v
v vv
0
0~
C
a Dc
v
Scales and dimensionless variables:
az
0vv
0
~t
tt
0
)(~t
v
Dimensionless equation (symbols “~” are not shown):
y
zyc
z
ybay
zt
yz 22 sin
sin
)(),1( szy
Page 11
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
yybyyb
h
yaya
h
yyz jkjkkjkjkkjkkjkkjkjk
k
jk
jkk
j
jkjkjjkjkjk
yz
h
yyyyc
,
,222
1,,,1,1
sin
)(sin)(sin
02
0,2,
h
yy kk 02
1,1,
h
yy 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
Page 12
21h
bA kk
j 1,...,2,1 Jj21
1
1 h
b
h
aB kkk
j 1,...,2,1 Jj
h
zz
h
c
h
bb
h
aC k
jk
k
j
jkkkkkjj
2
,
21
22
21
1
1, 1
sin
sin
1,...,2,1 Jj
1sin
sin
1
222
2,1h
cC kk
22
1,h
cC kk
jj 1,...,3,2 Jjj
jkkjj
h
cC
sin
sin 122
1, 2,...,2,1 Jj
1sin
sin
122
2,1J
JkkJJ
h
cC
h
yzD jk
kkj
,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
Page 13
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
Page 14
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
Page 15
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