94
E. L. Stupitsky et al.
3
2
n e
dT e
dt
+ n e T e div
− →
U = −
J +
3
2
T e
· n e j ei ·
n 0 K − n
2
e
− E 01 n e (n 0 j 01 − n 1 j 10 )
+ T e
3
2
− F
n
2
e j
ν
ei − S
ν
e + Q e0 + Q ei + Q g
(7.20)
3
2
n
dT
dt
+ nT div
− →
U = Q 0e + Q ie
(7.21)
From the equation of continuity
∂n
∂t
+ ndivU = 0 for the term describing the
adiabatic expansion, we obtain Eq. 7.22 where V = 2π Rπa
2 is the torus volume.
divU =
1
V
dV
dt
(7.22)
All designations are generally accepted. Expressions for the rate constants and
energy exchange are given in [6]. For Joule heating of electrons, the approximate
expression can be obtained Q g = I
2
/σ π
2 a
4 k.
To solve the most complex equations for T e (t) and T (t), the splitting method for
physical processes was used: y e = T e /T g , y i = T /T g , where T g is determined
by adiabatic expansion T g V
2/3
= const. At that for y e and y i , ordinary differential
equations are obtained for [5]. To get an idea of the interaction of the geometric
characteristics with the current and the magnetic field, the problem was first solved
under the assumption: T e = T i = T, y e = y i = 1, α = α 0 , i.e., practically in the
adiabatic approximation.
Since the initial determining parameters are current I 0 and temperature T 0 , then
the quantity β(t = 0) =
P 0
B
2
0 /8π
= 0.0204
T 0
I
2
0
eV
MA
depends on them.
7.4 Numerical Results and Their Analysis
Calculations performed in the adiabatic approximation demonstrate that the behavior
of a(t), U a (t), T (t), and n(t) significantly depends on the relation between the
thermal pressure inside the torus and magnetic pressure created by the current inside
the torus, i.e., on β(t = 0). The calculations were performed for different T 0 and
I 0 to implement different values of β(t = 0) and, at the same time, to keep them
within possible experimental values. Figure 7.2 shows some values of β(t = 0).
One should emphasize that main initial parameters determining TPB behavior after
leaving the generator—T 0 and I 0 —are not related to each other directly because T 0
is determined by the discharge power and I 0 is determined primarily by the presence
of the radial component of the magnetic field created in the generator in some way.
Thus, the effect of T 0 and I 0 on the change in TPB characteristics after leaving the
generator was first studied.
Various calculation options were obtained with varying values β. The results
are presented in Fig. 7.2. The behavior of the main parameters at an early stage
E. L. Stupitsky et al.
3
2
n e
dT e
dt
+ n e T e div
− →
U = −
J +
3
2
T e
· n e j ei ·
n 0 K − n
2
e
− E 01 n e (n 0 j 01 − n 1 j 10 )
+ T e
3
2
− F
n
2
e j
ν
ei − S
ν
e + Q e0 + Q ei + Q g
(7.20)
3
2
n
dT
dt
+ nT div
− →
U = Q 0e + Q ie
(7.21)
From the equation of continuity
∂n
∂t
+ ndivU = 0 for the term describing the
adiabatic expansion, we obtain Eq. 7.22 where V = 2π Rπa
2 is the torus volume.
divU =
1
V
dV
dt
(7.22)
All designations are generally accepted. Expressions for the rate constants and
energy exchange are given in [6]. For Joule heating of electrons, the approximate
expression can be obtained Q g = I
2
/σ π
2 a
4 k.
To solve the most complex equations for T e (t) and T (t), the splitting method for
physical processes was used: y e = T e /T g , y i = T /T g , where T g is determined
by adiabatic expansion T g V
2/3
= const. At that for y e and y i , ordinary differential
equations are obtained for [5]. To get an idea of the interaction of the geometric
characteristics with the current and the magnetic field, the problem was first solved
under the assumption: T e = T i = T, y e = y i = 1, α = α 0 , i.e., practically in the
adiabatic approximation.
Since the initial determining parameters are current I 0 and temperature T 0 , then
the quantity β(t = 0) =
P 0
B
2
0 /8π
= 0.0204
T 0
I
2
0
eV
MA
depends on them.
7.4 Numerical Results and Their Analysis
Calculations performed in the adiabatic approximation demonstrate that the behavior
of a(t), U a (t), T (t), and n(t) significantly depends on the relation between the
thermal pressure inside the torus and magnetic pressure created by the current inside
the torus, i.e., on β(t = 0). The calculations were performed for different T 0 and
I 0 to implement different values of β(t = 0) and, at the same time, to keep them
within possible experimental values. Figure 7.2 shows some values of β(t = 0).
One should emphasize that main initial parameters determining TPB behavior after
leaving the generator—T 0 and I 0 —are not related to each other directly because T 0
is determined by the discharge power and I 0 is determined primarily by the presence
of the radial component of the magnetic field created in the generator in some way.
Thus, the effect of T 0 and I 0 on the change in TPB characteristics after leaving the
generator was first studied.
Various calculation options were obtained with varying values β. The results
are presented in Fig. 7.2. The behavior of the main parameters at an early stage