Biuletyn Instytutu Spawalnictwa No. 01/2012

volume of plasma. The adopted assumption of
the axial-cylindrical shape of the column and
its stretching by length dl corresponds to an
increase in thermal power
dQ
dt
dQa
dl dl
= ql
(38) (38)
dl dt dt
a
=
where q l
– arc energy linear density. Hence
in simplifying conditions, the thermal power
necessary to generate the additional volume
of plasma is approximately proportional to the
length of the increment rate. The phenomenon
is accompanied by relaxation times resulting
from gas thermal inertia and additional cooling
of the column. Modified equations (2) and (3),
with a variable value of the Cassie-Berger
Representation of disturbance of arc
column length in modified Habedank
and TWV hybrid models
The series connection of nonlinear conductances
(16) corresponding to the Cassie-Berger
and Mayr-Kulakov models makes it possible
to obtain the modified Habedank model.
The conductance form is expressed by the following
formulas:
1
g
C
dg
dt
C
⎡
⎤
⎢
2
2
1
⎢
u ⎛ g ⎞
= −
⎢
⎜
⎟ 1
θ 2 1 ⎛ dl ⎞
⎢
() + ⎜ ⎟
⎝ ⎠
⎥ ⎥⎥⎥ C
gC
uC
l pv
⎣ gC
⎝ dt ⎠ ⎦
1 dg
M
i g dl
(44)
g dt
= 1 ⎡
⎤ 1
⎢
−1⎥
−
θ ⎛ dl ⎞
M
Ms ⎣ g
M
⋅l
⋅ EMstat
( i)
g
M ⎦ l dt
voltage, U
C
() t = U
C ⎜l,
⎟ give the conductance
⎝ dt ⎠
If one takes into consideration the series
form [13, 14]
connection of resistances (17), the form of the
⎛
⎞
⎜
model will be
2
1 dg
= 1 ⎜ ukol
−
⎜
(39)
⎛ ⎞
1
() ⎟ ⎟⎟⎟ (39)
g dt θ 2 1 dl
⎡
⎤
C
⎜ u l + p ⎜ ⎟
⎢
2
2
C
v
⎝ g ⎝ dt
1 drC
1
u
⎠ ⎠
⎢
⎛ rC
⎞
= 1−
(45)
The resistance form is
() ⎜ ⎟
r ⎢ 2 ⎛ dl
C
dt θC
⎞ ⎝ r
⎢
u
⎠
⎥ ⎥⎥⎥
C
l + rC
⋅ pv
⎜ ⎟
⎣
⎝ dt ⎠ ⎦
⎛
⎞
⎜
2
1 dr 1 ⎜ u
1 dr
kol
M
ir r dl
= 1−
(40)
(46)
() ⎟ ⎟⎟⎟ (40)
r dt θ ⎜
⎛ dl
C 2 ⎞
r dt
= 1 ⎡
⎤
M M
1
⎢1
−
⎥ +
θ M
Ms
⎜ u l + rp
⎣ l ⋅ EMstat
() i r ⎦ l dt
C
v⎜
⎟
⎝
⎝ dt ⎠ ⎠
where E Mstat
(i) – virtual characteristic of
where p v
(dl/dt) - power necessary to generate electric field intensity.
additional volume of plasma.
The hybrid model of the arc column, taking
Kulakov proposed a modification of model into consideration its length changes, associates,
by means of an appropriate tapering func-
(14) taking into consideration the modification
of the column length. Model I of the order in tion Ɛ(i), models (39) and (41) in the manner
the conductance form is [15]
(34). Thus, its form is as follows:
1 dg 1 ⎡ i ⎤ 1 dl
⎢ −1⎥
−
g dt
= θ g l E () i l dt (41)
Ms ⎣ ⋅ ⋅
(41)
1 dg
stat ⎦
g dt
=
⎧
⎫
where E stat
(i) – static characteristic of elec-
⎪
2
1
⎪
⎨[ − ukol
i
1 ε () i ]
+ ε () i
− ⎬
tric field intensity. The resistance form of the θ ⎪
⎛ ⎞ ⋅ ⋅
()
( )
1
2 1 dl g l Estat
iM
u
⎪
C
l + pv⎜
⎟
⎪⎩
g ⎝ dt ⎠
⎪⎭
model is described by the following formula:
dl
1 dr 1 ⎡ ir ⎤ 1 dl
−ε () i 1
= ⎢1
− ⎥ + (42)
(42)
l dt
(47)
r dt θ
Ms ⎣ l ⋅ Estat
() i ⎦ l dt
(43)
20 BIULETYN INSTYTUTU SPAWALNICTWA
NR 01/2012