Page 108 - Fundamentals of Magnetic Thermonuclear Reactor Design

δk* 92 Fundamentals of Magnetic Thermonuclear Reactor Design

th
*
z ). Ripple δ corresponding to the k field harmonic is computed according
j
k
δk*=ak2+bk2/0.5a 0 to δ = (a k 2 + b k 2 ) / 0.5 . To evaluate ripple-induced particle losses in ITER,
*
a
k
0
*
δ18* mode n = 1 is assumed to be correspondent to ripple modeδ . In the general
18
*
*
δ36*
δn=δ18n* case, δ = δ 18 n , which means that the second mode (n = 2) corresponds to δ ,
n
36
and so on.
The spectrum and amplitude of higher harmonics may be adjusted by vary-
ing the inserts geometry and positions.
δTFCi The ripple of a toroidal field δ(i) due to TFCs (δ TFC () i ) and ferromagnetic
δ ( ) j occurring in characteristic points ∈1,8 is derived from
∑j=18CjδFei,j∈1,8 inserts ∑ 8 = j 1 C jFe i , i
i
the superposition of their contributions:
8
i
C
ij,,
δi=δTFC+∑j=18CjδFei,j, i∈1,8, δ () = δ TFC + ∑ j δ Fe ( ) i ∈1,8, (4.26)
= j 1
where
() 1
i
i
δTFCi=BTFC2i−BTFC1i/2Bi, i∈1,8 δ TFC () =  B () 2 () −i B TFC ()  /2 B () i , i ∈1,8 (4.27)

 TFC
δ ( ) =  B () 2 ( ) −i j, B () 1 ( )  / 2 Bi i ∈1,8, j ∈1,8. (4.28)
ij
,
ij
,
(),
δFei,j=BFe2i,j−BFe1i,j/2Bi, i∈ Fe  Fe Fe 
1,8, j∈1,8.
In Eqs. (4.26)–(4.28), index ‘2’ refers to a toroidal field beneath a coil in
the coil rotational symmetry plane, and index ‘1’ to a field between two neigh-
() 1
Bi=0,5BTFC2i−BTFC1i bouring coils; () =Bi 0,5 ( B () 2 () −i B TFC () i ) is an average field on a circle
TFC
with coordinates (r , z ), and C is the filling factor showing the percentage
j
i
j
th
of the j subregion A – E filled with a ferromagnetic material. Eq. (4.28) is
1
2
consistent with the ferromagnetic insets contribution to the field ripple with
Cj=1, j∈1,8. C j = 1, j ∈1,8.
Cjopt=1, j∈1,8 Optimised filling factors (C opt = 1, j ∈1,8) must meet the following require-
j
ments:
Cjopt≤Cmax=0,6, j∈1,8 C opt ≤ C max = 0,6, j ∈1,8 (4.29)
j
8
i
i
,
C
j
δi=δTFCi+∑j=18CjoptδFei,j>0 δ () = δ TFC () +i ∑ j opt δ ( ) > 0, i ∈1,8, (4.30)
Fe
, i∈1,8, = j 1
which define the admissible set of solutions for C . j
The first requirement is due to the design constraints. The second is applied
to avoid ripple overcompensation with the inserts and its increase during the
tokamak operation with hydrogen plasma at half the nominal TFC current. In
that case, we would have a double increase in δ , while δ TFC is the same at any
Fe
operation mode.
Cjopt C opt under these constraints should be determined as a solution to the fol-
j
lowing optimisation problem: