Page 112 - Fundamentals of Magnetic Thermonuclear Reactor Design

96 Fundamentals of Magnetic Thermonuclear Reactor Design

It is clear that a detailed description of EM transients is only possible through
a solution of coupled electrodynamic and thermal problems taking into account
the system configuration.
Many tokamak conductive components present structures with walls that
are thin compared with other linear dimensions [32]. For ITER, they are the
vacuum vessels, thermal shield and cryostat. EM loads occurring in such struc-
tures often become the key design constraint.
In ITER, the following procedure is accepted for the analysis of EM tran-
sients. The R&D stage for the reactor involves axisymmetric DINA simulations
of plasma scenarios (computer codes other than DINA can also be used). Then
the results of solving a related EM problem are used as initial data for 3D com-
putations.
For a qualitative description of transient processes as per, for instance [28]
one can use existing analytical solutions to the problem of a field penetration
into a conductive medium. Such a solution was first found [28,33] in describing
a uniform thermal field diffusion in an infinitely long flat layer with a thick-
ness of d. In our case, boundary conditions take the form: B(0, t) = B (t), B(d,
1
t) = B (t), with the initial condition B(x,0) = f(x).
2
The x coordinate is directed normally from the medium boundary inwards
the layer. At zero boundary conditions, B = B = 0 and
1
2
/
fx=B sinπx/d fx 0 π ( x d )
() = B sin
0
2
( t,
/
2
0
0
0
Bx,t=B sinπx/dexp−π ρ t/µ d 2 Bx ) = B sin π ( xd )exp (− πρ t / µ d 2 )
0
0
0
As seen, the field decay occurs without any spatial re-distribution. The char-
2
τ =µ d /π ρ 0 acteristic decay time for the first harmonic is τ = µ d / 2 ρ . Higher harmon-
2
2
0
0
0
0
0
τ 0 ics tend to decay more rapidly. For a 60-mm-thick steel wallτ is 3 ms.
0
Similar analytical solutions for a problem involving field diffusion through
τ 0 a flat layer contain the same characteristic time τ and linear dimension d.
0
They are used in numerical analyses to assess spatial and temporal steps for FE
models.
Solving a diffusion problem for a field induced by external currents that run
parallel to the layer, one can obtain the asymptotic values of a tangential compo-
,
BF(−d,t) nent B ( −d t) behind the conducting layer with boundaries x = 0, x =−d. One
F
example is related to current filaments. If the depth of field penetration into the
layer is much smaller than its thickness and the characteristic linear dimensions,
the solution for a current filament loop lying in plane x = h can be reduced to the
principal term of the expansion in powers with respect to the minor parameter
ρ t/µ d 2 ρ t / µ d 2 . This solution is the more accurate the smaller t is. For an instant
0
0
0
0
current step from 0 to i , the tangential field [28]
0
2
h
i
B (−d t, ) = −∂8 b () ∂⋅ ⋅ t / µ d 2 ) 3 2 ⋅d / π ⋅ exp(− µ d /4 ρ ) t .
0/
2
BF−d,t=−8∂bF0/∂h⋅i ⋅ρ t/µ d 32 F F 0 ρ ( 0 0 0 0
0
0
0
2
⋅d/π⋅exp−µ d /4ρ t.
0
0