Page 90 - Fundamentals of Magnetic Thermonuclear Reactor Design

74 Fundamentals of Magnetic Thermonuclear Reactor Design

some details, which, however, are essential for the EM processes under analy-
sis. Finally, computations for the design support should include, first, detailed
cross-check computations for all the structures and, second, the development
of predictive and process simulators for the reactor systems at the adjustment/
operation stages.
While these approaches remain methodologically unchanged, software
functionality may change with time due to the increase in the capabilities of
state-of-the-art computers.

4.2 STATIONARY AND QUASI-STATIONARY FIELDS
Field simulation is based on a system of Maxwell’s equations that express
the classical electrodynamics postulates [11–13]. The differential form of
these equations links the electrical and magnetic field intensity vectors (E
and H, respectively) and the electrical and magnetic field induction vectors
(D and B, respectively) between themselves and with the volume density of
free electric charges ρ and the volume density of current (conduction current)
/
t
j
)
∇⋅j+(∂ρ/∂t)=0 vector j. From these equations, the continuity equation ∇⋅ +∂( ρ ∂= 0
, can be derived, where ∇ is the Hamiltonian expressing the law of time-
dependent conservation of an electric quantity. The macroscopic variables
of this equation can be measured independently and, hence, have physical
meanings.
Complementing the four Maxwell equations are the material equations that
characterise the medium. Equations D = ε E, B = µ H, where ε and µ are the
a
a
a
a
E- and H-independent absolute dielectric permittivity and absolute magnetic
permeability of the medium, respectively, are valid for any homogeneous iso-
tropic conducting medium. For current density, the same approximation is given
by the differential form of Ohm’s law for a static medium as j = σE, where σ
is the specific electrical conductivity of the medium. For a linear anisotropic
medium, the ε , µ and σ parameters are described by tensors. Vector j is consid-
a
a
ered to mean the density of a conduction current produced by an electric field. In
many modelling applications, it is preferred to assume that not only j, but also
e
the extrinsic current j is found in a medium. The extrinsic current, generated
by some process (halo current as an example), is regarded as one of the sources
of an EM field. To take extrinsic currents into account, the j vector in the first
e
Maxwell equation is substituted for j + j .
Complemented by material equations, Ohm’s law and boundary and ini-
tial conditions, Maxwell’s system of equations allows macroscopic electrody-
namic problems to be solved unambiguously. These equations are simplified
for particular cases. There are two classes of problems typical for tokamak
modelling.
Stationary Problems: For these problems, we know how direct extrin-
sic currents are distributed. The initial system of equations is converted