9 Design of ITER (International Tokamak Experimental Reactor)
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15.9 Design of ITER (International Tokamak Experimental Reactor)
371
and the plasma current is
Ip (MA) =
5K 2 aBt
AqI
where K 2 = (1 + κ2s )/2 (Ip in MA, Bt in T and a in m).
The safety factor q95 is approximately given by (refer to (15.11))
q95 ≈ qI fδ fA
fδ =
1+
κ2s (1
+ 2δ − 1.2δ 3 )
1.17 − 0.65/A
fA =
.
2
1 + κs
(1 − 1/A2 )2
2
(15.39)
The volume average electron density n20 in unit of 1020 m−3 is
n20 = NG nG , nG ≡
Ip (MA)
πa2
(15.40)
where nG is Greenwald density(1020 m−3 ) and NG is Greenwald parameter. The beta
ratio of thermal plasma is expressed by
0.0403
( n20 Te keV + (fDT + fHe + fz ) n20 Ti keV )
Bt2
n20 Ti keV
= 0.0403(γT + fDT + fHe + fz )
.
(15.41)
Bt2
βth ≡
p
Bt2 /2μ0
=
Assuming that the spatial distributions of Te and Ti are the same, we have
γT ≡ nTe / nTi ≈ Te / Ti .
Scaling law of beta is
βth = fth βtotal , βtotal = 0.01βN
Ip (MA)
,
aBt
where βN is normalized beta. βtotal is the sum of βth (thermal plasma) and βfast (fast
(α) particle component) and fth = βth /βtotal . The notations fDT , fHe , and fI are the
ratios of fuel DT, He, and impurity densities to electron density respectively and the
unit of T is keV. X means volume average of X. Thermal energy of plasma Wth is
Wth (MJ) =
B2
3
βth t V = 0.5968βth Bt2 V
2 2μ0
(15.42)
where Wth is in unit of MJ and plasma volume V is in unit of m−3 . Plasma shape
with elongation ratio κs and triangularity δ is given by
372
15 Tokamak
R = R0 + a cos(θ + δ sin θ)
z = aκs sin θ
Plasma volume V is given by
V ≈ 2π 2 a2 Rκs fshape
where fshape is a correction factor due to triangularity
fshape = 1 −
δ
a
−
δ.
8 4R
In divertor configuration, V = 2π 2 a2 Rκs or S = πa2 κs are used to define the elongation ratio κs , S being the area of cross-section.
We utilize the thermal energy confinement scaling of IPB98y2 [38]
0.41 1.97 1.39 0.78 −0.69
a A κs Ph
τE = 0.0562 × 100.41 Hy2 I 0.93 Bt0.15 M 0.19 n20
2 1.34 0.41 0.05 1.49 2.49 −0.69
= 0.781Hy2 qI−1.34 M 0.19 κ0.78
NG A Bt a Ph
s (K )
(15.43)
where 0.0562 × 100.41 = 1.444, M(= 2.5) is average ion mass unit and Ph is the
heating power to compensate the loss power in MW by transport and is equal to
necessary absorbed heating power subtracted by radiation loss power Prad . The total
fusion output power Pfus is
Pfus =
Qfus 2
nDT σv
4
v
V
where Qfus = 17.58 MeV. σv v is a function of T . Since the fusion rate σv near
T = 10 keV is approximated by
σv
v
2
≈ 1.1 × 10−24 TkeV
(m3 /s)
the following Θ is introduced:
Θ( TkeV ) ≡
2
Then nDT
σv
v
2
nDT
σv
2
2
nDT
σv v
σv v
nDT
=
2
2
−24
−24
1.1 × 10
nDT TkeV
1.1 × 10
nDT TkeV
2
nDT T 2
.
2
nDT
T2
is expressed by
v
= 1.1 × 10−24 nDT TkeV 2 Θfprof ,
fprof =
2
T2
nDT
.
nDT T 2
15.9 Design of ITER (International Tokamak Experimental Reactor)
373
Θ is a function of average ion temperature TkeV in keV and is depend on the
spatial profiles of density and temperature. In the cases of spatial profiles of n(ρ) =
(1 + αn ) n (1 − ρ2 )αn , T (ρ) = (1 + αT ) T (1 − ρ2 )αT , (ρ2 = x 2 /a2 + y2 /(κs a)2 ),
the dependence of Θ on T is
Θ( Ti ) =
1 + 2αn + 2αT
1.1 × 10−24 (1 + αT )2 Ti
A fitting equation of T for σv
σv
v
=
v
1
2
(1 − ρ2 )2αn σv v 2ρdρ.
0
is already described in (1.5) of Chap. 1 as follows;
3.7 × 10−18 −2/3
−1/3
Ti
exp(−20Ti
) m3 /s,
h(Ti )
h(Ti ) =
5.45
Ti
+
.
37 3 + Ti (1 + (Ti /37.5)2.8 )
The dependences of Θ on average ion temperature T are shown in Fig. 15.23 in
the case of typical cases [51]. In the case of the flat density profile and parabolis
temperature profile (αn = 0, αT = 1), the profile parameter fprof is fprof = (αn +
αT + 1)2 /(2αn + 2αT + 1) = 4/3. In the case of flat density profile and peaked
temperature profile (αn = 0, αT = 2), the profile parameter is fprof = 9/5
Fusion output power Pfus is
2
fDT
fprof Θ( Ti )βth2 Bt4 V
(γT + fDT + fHe + fI )2
= 1.19fdil fprof Θ( Ti )βth2 Bt4 (2π 2 κs Aa3 )(MW)
Pfus = 4.77
= 2.35 × 10−3 fdil fprof Θ( Ti )κs fth2 βN2 Ip2 Bt2 Aa,
Fig. 15.23 Θ is function of
average temperature
T (keV) in cases with
profile parameters
(αT = 1.0, αn = 0.0),
(αT = 2.0, αn = 0.0),
(αT = 1.0, αn = 0.5) and
(αT = 2.0, αn = 0.3)
(15.44)
374
15 Tokamak
where Ip = (5K 2 /qI )(aBt /A). The dilution parameter of DT fuel due to He ions and
impurities fdil is
fdil ≡ 2
=
fDT
(γT + fDT + fHe + fz )
2
γT + 1
2
2
(1 − 2fHe − zfz )
1 − [fHe + (z − 1)fz ]/(γT + 1)
2
.
The output fusion power Pα due to α particle only is
Pα =
Pfus
.
5
since Pn : Pα = 4 : 1. When absorbed external hearing power is denoted by Pext and
the heating efficiency of α heating is fα , the total heating power is fα Pα + Pext . When
the fraction of radiation loss to total heating power is fR , the heating power Ph to
compensates transport loss power is
Ph = (1 − frad )(fα Pα + Pext ).
When Q value is defined by the ratio of total fusion power Pfus to external heating
power Pext
Pfus
Q=
,
Pext
Ph is
Ph = (1 − frad ) fα +
5
Pα .
Q
Therefore the burning condition is reduced to
5
Wth
Pα .
= (1 − frad ) fα +
τE
Q
(15.45)
From (15.42) and (15.45), we have
βth Bt2 τE =
1
2.503
.
(1 − frad )fdil (fprof Θ) (fα + 5/Q)
The term βth Bt2 τE is proportional to the fusion triple product n20 TkeV τE . From
(15.42) of Wth , (15.43) of τE , (15.44) of Pfus , the burning condition (15.45) is reduced
to, [52]
15.9 Design of ITER (International Tokamak Experimental Reactor)
Bt0.73 a0.42
A0.26
fα +
5
Q
0.31
= 2.99
×
375
0.31
1
(1 − frad )fdil (fprof Θ)
qI0.96 (fth βN )0.38
,
Hy2 M 0.19 NG0.41 K 1.92 κ0.09
s
(15.46)
that is,
1
qI0.96 (fth βN )0.38
fα
6.83
+
=
Q
5
(1 − frad )fdil (fprof Θ) Hy2 M 0.19 NG0.41 (K 2 )0.96 κ0.09
s
A0.26
a0.42 B0.73
3.226
.
When Kaye–Goldston scaling (L mode) described in Sect. 15.6 is used instead of
IPB98y2 scaling, the burning condition becomes
Ip A1.25
a0.12
fα +
5
Q
0.5
=
146.9
HKG
1
(1 − frad )fdil (fprof Θ)
0.5
,
where HKG is the improved parameter of the energy confinement of Kaye–Goldston
scaling of L mode (HKG = 1). Necessary value of HKG for ITER is 2.57. It is interesting to note that the burning condition mainly depends AIp in this case.
When the parameters a, Bt , A are specified in the case of inductive operation
scenario of ITER, Q value and the other parameters can be evaluated. Specified
parameters are listed in Table 15.4a and the results of evaluated parameters in the
case of inductive operation scenario are shown in Table 15.4b. The results of this
simple analysis is relatively consistent with ITER design parameters of inductive
operation, which is given in the lefthand side column of Table 15.5 [53].
Ti in Table 15.5 is calculated by
Ti =
nTi
n
Ti n
nTi 1
nT
(2)
=
,
, fprof
≡
(2)
nTi
n fprof
n T
(2)
where n and nT is given by (15.40) and (15.41) and fprof
≡ nT / n T ≈ (1 +
αn )(1 + αT )/(1 + αn + αT ).
There are constrains on plasma parameters by the engineering viewpoint of fusion
reactors.
When the distance of plasma separatrix and the conductor of toroidal field coil
is Δ and the maximum field of toroidal field coil is Bmax (see Fig. 15.24), there is a
constraint of
Δ 1
R−a−Δ
Bt
=1− 1+
=
Bmax
R
a A
By specification of Δ and Bmax , Bt is a function of a. Δ is the sum of thickness of
structural material of superconducting winding pack cM , of the blanket b and the
distance between the first wall and the separatrix of plasma dsep (Δ = cM + b +
376
15 Tokamak
Table 15.4 (a) Specified design parameters of inductive operation scenario. (b) Reduced parameters
of inductive operation scenario
(a)
a
Bt
A
qI
κs
NG
βN
fth
Hy2
fprof Θ frad
fα
2.0
5.3
3.1
2.22
1.7
0.85
1.8
0.95
1.063
1.35
0.27
0.95
(b)
Q
R
Ip
τE
n20
Ti
Te
Wth
Pfus
Pext
Prad
βtotal
9.8
6.2
15.0
3.75
1.01
8.01
8.81
338
424
42
33.2
0.025
fDT = 0.82, fHe = 0.04, fBe = 0.02. γT = 1.1αn = 0.1, αT = 1.0 are assumed
qI = 2.22 is specified to be Ip = 15.0. fprof Θ = 1.35 and Hy2 = 1.063 are specified in order to be
Q ≈ 10 (refer (15.46))
Pn , Pα , Pext , Prad are in the unit of MW, Wth is in the unit of MJ and Ip is in the unit of MA
Table 15.5 Parameters of ITER outline design [53–55]
Inductive operation
Ip (MA)
Bt (T)
R/a (m)
A
κs 95 /δ95
ne (1020 m−3 )
NG
Ti / Te
Wthermal (MJ)
τEtr (s)
Hy2
Pfus (MW)
Pext (MW)
Prad (MW)
Zeff
βt (%)
βp
βN
q95
qI
li
Q
fR
fDT /fHe (%)
fBe /fAr (%)
15
5.3
6.2/2.0
3.1
1.7/0.33
1.01
0.85
8.0/8.8
325
3.7
1.0
410
41
48
1.65
2.5
0.67
1.77
3.0
2.22
0.86
10
0.39
82/4.1
2/0.12
Non-inductive
operation
Ip
Bt
R/a
A
κs 95 /δ95
ne (0)
NG /nG
Te (0)/Ti (0)
9 (MA)
(5.17) (T)
(6.35/1.84) (m)
(3.45)
(1.84/0.41)
0.6
∼0.62/0.85
37/34
Hy2
1.5∼1.7
PNB (MW)
PEC (MW)
34
20
βt,th (%)
βp,th
βN,th
q95
∼1.9
∼1.2
∼2
7
Q
Ibs
Icd
5
4.5 (MA)
4.5 (MA)
15.9 Design of ITER (International Tokamak Experimental Reactor)
377
Fig. 15.24 Geometry of
plasma, toroidal field coil
and central solenoid of
current transformer
in tokamak
dsep ). The thickness of blanket b consists of vacuum chamber, neutron shield, tritium
breeding Li blanket and first wall. The optimistic value of b is [56]
b ≈ 1.2 m
and
cM = R0 {1 − εB − [(1 − εB )2 − αM ]1/2 }
εB =
(a + b)
B02
, αM =
R0
μ0 σmax
2εB
1 + εB
1
+ ln
1 + εB
2
1 − εB
where σmax is the maximum allowable stress of structural material of the toroidal
coil.
There is maximum allowable neutron power flux qN passing through the first wall
(qN < 4 MW/m2 ) [56]. Then qN is the total neutron power Pn divided by the total
area S of the first wall, where Pn = (4/5)Pfus and S = (2π)2 R0 a[(1 + κ2s )/2]1/2 .
Then we have
1
R0 a (m2 ) =
4π 2
1+κ2s
2
0.8Pfus
2
> 5.04
1/2
qN
1 + κ2s
1/2
Pfus (GW).
In the case of κs = 1.7 and Pfus = 2.57 GW, there is constrains of R0 a > 9.8 m2 .
The ratio ξ of the flux swing ΔΦ of ohmic heating coil and the flux of plasma
ring Lp Ip is given by
ξ≡
2
)
ΔΦ
5Bmx ((ROH + dOH )2 + 0.5dOH
=
,
1/2
Lp Ip
(ln(8A/κs ) + li − 2)RIp
378
15 Tokamak
where ROH = R − (a + Δ + dTF + ds + dOH ), dTF and dOH being the thickness of
TF and OH coil conductors and ds being the separation of TF and OH coil conductors (refer Fig. 15.24). The average current densities jTF , jOH of TF and OH coil
conductors in MA/m2 = A/mm2 are
jTF =
1
2.5 Bmx
π dTF 1 − 0.5dTF /(R − a − Δ)
jOH =
2.5 Bmx
.
π dOH
Parameters of ITER outline design is listed in Table 15.5. q95 is the safety factor
in 95% flux surface. The maximum field of toroidal field coils is Bmax = 11.8 T. The
number of toroidal field coils is 18. Single null divertor configuration. One turn loop
voltage is Vloop = 89 mV. Inductive pulse flat-top under Q = 10 condition is several
hundred seconds.
In the case of non-inductive operation, the optimized parameters are different
from the optimized parameters in the case of inductive operation. The bootstrap
current is given by Ibs = cb (a/R)1/2 βp Ip from (5.29). Since βt = 0.01βN Ip /(aBt ),
Bp /Bt = μ0 Ip /(2πKaBt ) = 0.2(Ip /KaBt ), Bp /Bt = aK/RqI , βp is reduced to
βp = 0.25K 2 βN (aBt /Ip ) = 0.05AβN qI ,
the bootstrap current is given by
Ibs /Ip = Cbs A0.5 βN qI Cbs = 0.05cb .
When the driven current and the necessary power of the driver are denoted by Icd
and Pcd respectively, the current drive efficiency ηcd is defined by the equation
Icd =
ηcd
Icd
Pcd , ηcd ≡
n R.
nR
Pcd
(15.47)
All the current drive efficiencies of lower hybrid wave (11.47), electron cyclotron
wave (11.53) and neutral beam (11.63) are proportional to the electron temperature
Te . Therefore (15.47) is reduced to
Icd =
(ηcd / Te ) n Te
(ηcd / Te ) n ( Te + (fDT + fHe + fz ) Ti )
Pcd =
Pcd ,
2
n R
n 2 R(1 + (fDT + fHe + fz ) Ti / Te )
⎛
⎞
ηcd19 / Te keV
⎠
Icd (MA) ≈ 2.48 × 10−2 ⎝ (2)
fprof [1 + (fDT + fHe + fz )/γT ]
fth βN Ip (MA)Bt
Aa2 n 220
Pcd (MW),
15.9 Design of ITER (International Tokamak Experimental Reactor)
379
where ηcd19 is in unit of 1019 (A/Wm2 ) and Te keV is the volume averaged electron
temperature in keV unit and
βN Bt
Icd
= Ccd 2 2 Pcd (MW),
Ip
Aa n 20
Ccd = 2.48 × 10−2
(ηcd19 / Te keV )fth
(2)
[1 + (fDT + fHe + fz )/γT ]fprof
.
(15.48)
We must keep Ibs + Icd = Ip in the steady state operation, and the necessary driving
power Pcd is
Pcd =
2
2
Icd
aRn20
aRn20
=
Ccd βN Bt Ip
Ccd βN Bt
1−
Ibs
Ip
=
(1 − Cbs A0.5 βN qI )aRn2
.
Ccd βN Bt
Since the fusion output power Pfus is given by (15.44) as follows,
Pfus = Cfus βN2 Ip2 Bt2 Aa, Cfus = 2.35 × 10−3 fdil (fprof Θ)κs fth2 ,
Qcd ≡ Pfus /Pcd is [52, 55]
2
aR
(1 − Cbs A0.5 βN qI )n20
(1 − Cbs A0.5 βN qI )NG2
1
=
=
.
Qcd
Cfus (βN Bt )2 Ip (MA)2 AaCcd βN Bt
π 2 Ccd Cfus (βN Bt a)3
(15.49)
The increase of A1/2 βN qI is favorable to increase the bootstrap current and the increase
of (βN Bt a)3 /NG2 is favorable to increase of Qcd . However the increase of qI ∝ 1/Ip
(decrease of plasma current Ip ) and the decrease of ne degrade confinement time and
needs larger confinement enhance factor Hy2 .
Non-inductive steady state operation reference scenario 4, type II in [55] is examined. In non-inductive operation scenario, the bootstrap current and driven current
are 4.5 MA and 4.5 MA, respectively. The parameters of R, a, Bt , κs 95 /δ95 in noninductive operation are referred from [54]. NG , βt,th , βp,th , βN,th in non-inductive
operation are estimated values of Greenwald parameter and the thermal component
of β’s from [55] respectively. These data are used in Table 15.6a. Reduced parameters in the case of non-inductive operation scenario are listed in Table 15.6b. These
results are consistent to the parameters of non-inductive operation scenario in the
righthand side column of Table 15.5 [52].
The specified bootstrap current is Ibs = 4.5 MA. This specification requires Cbs =
0.0374, that is cb = 0.748. In the full non-inductive current drive experiment in JT60U (a/R = 0.24, βp = 2.7, reverse shear configuration), the estimated value of cb
is 0.6 (refer to Sect. 15.8).
380
15 Tokamak
Table 15.6 (a) Specified parameters in the case of non-inductive operation scenario. (b) Reduced
parameters in the case of non-inductive operation scenario
(a)
a
Bt
A
qI
κs
NG
βN
fth
Hy2
fprof Θ frad
fα
1.84
5.17
3.45
3.35
1.84
(b)
Q
R
Ip
τE
n20
5.01
6.35
9.02
3.88
0.534
0.63
2.15
0.95
1.702
1.20
0.3
0.95
Ti
Te
Wth
Pfus
Pext
Prad
βtotal
12.0
13.0
241
228
45.5
26.6
0.020
fDT = 0.82, fHe = 0.04, fBe = 0.02 δ = 0.41. αn = 0.03, αT = 2.0, γT = 1.08
qI = 3.34 is specified to be Ip = 9.0MA. fprof Θ( Ti ) = 1.2 and Hy2 = 1.702 are specified to be
Q≈5
Pext = Pcd . T (0) = (1 + αT ) T ≈ 3 T . The value of the approximate equation q95 ≈ qI fδ fA is
4.69 and is different from the q95 ≈ 7 in Table 15.5
The specified driven current is Icd = 4.5 MA and the driving power is Pcd = Pext =
41.4MW . The necessary value of Ccd is Ccd = 0.329 × 10−2 and the necessary
current drive efficiency ηcd is given by (15.48) to be
ηcd19 = 0.133
(2)
[1 + (fDT + fHe + fz )/γT ]
fprof
fth
Te keV ≈ 0.259 Te keV .
The Q value and Qcd are quite different quantities with each other. Qcd does not
depend on Hy2 , while Q does not depend on Cbs and Ccd . However the sensitivities
of Q and Qcd on NG , qI , βN , . . . are different. It is necessary to keep Qcd = Q by
feedback control of sensitive parameters [52].
The conceptual design of tokamak reactors has been actively pursued according
to the development of tokamak experimental research. INTOR1 (International Tokamak Reactor) [57] and ITER (International Thermonuclear Experimental Reactor)
[53, 55] are representative of international activity in this field.
ITER aims achievement of extended burn in inductively driven plasmas with
Q ∼ 10 and aims at demonstrating steady state operation using non-inductive
drive with Q ∼ 5. Now ITER device is under the construction in Cadarache, France.
The experiment is scheduled to start in 2020. The cross section of outline design of
ITER and the bird-eye view of ITER are shown in Fig. 15.25 [53] and Fig. 15.26
[58] respectively.
1 The
working group of INTOR consisted of 4 parties namely Euratom, Japan, USA and USSR. A
note with the title of ‘Who’s job is it?’ was pinned on the wall of meeting room of INTOR in IAEA
building in Vienna. (This is a story about four peoples named everybody, somebody, anybody and
nobody. There was an important job to be done. Everybody was asked to do it. Everybody was sure
somebody would do it. Anybody could have done it, but nobody did it. Somebody got angry about
that, because it was everybody’s job. Everybody thought anybody could do it, but nobody realized
that everybody would’nt do it. It ends up that everybody blamed somebody when nobody did what
everybody could have done.)
15.9 Design of ITER (International Tokamak Experimental Reactor)
Fig. 15.25 The cross section of outline design of ITER
381