Joule heating and diameter of the cathode spot in a vacuum arc

Citation for published version (APA):

Daalder, J. E. (1973). Joule heating and diameter of the cathode spot in a vacuum arc. (EUT report. E, Fac. of Electrical Engineering; Vol. 73-E-33). Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1973

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JOULE HEATING AND DIAMETER OF THE CATHODE SPOT IN A VACUUM ARC.

by

VAKGROEP HOGE SPANNING EN HOGE STROMEN GROUP HIGH VOLTAGE AND HIGH CURRENTS

Joule heating and diameter of the cathode spot in a vacuum arc.

J • E. Daalder

january 1973

TH-Report 73 - E - 33 ISBN

90

6144 033 5

Page 15 under "COPPER" rb to read r a ra to read r_b·

In a previous paper [1] measurements regarding the influence of the cathode surface structure on the movement of a metal vapour discharge in vacuum were reported. The diameter of the cathode spot as a function of current has been studied.

This paper outlines some considerations in order to elucidate the experimental results.

Descriptions of the movement of the discharge and a theoretical calculation of the diameter of a single metal vapour discharge as a function of current is given.

It is shown experimentally [I] that the movement of·a vacuum discharge over the rough surface of a cathode is discontinuous,

the traces consist of a number of seperate craters. The distance between these craters is generally larger than their diameter. By photographing the splitting process of a discharge in vacuum it could be proved [2] that for current values up to about 100 A. one single discharge is most probable in case of a copper cathode. At current levels above 100 A the probability of division of the single discharge into two separate discharges strongly increases.

Assuming a single discharge for current values up to 100 A the movement of this discharge over the cathode surface can be explained as

follows.

The presence of distinct craters shows the discharge is fixed on one "spot" during a certain time. Energetical processes provide for the evaporation of the cathode material; a crater is formed.

The crater dimensions increase as a function of time. The positive space-charge due to ionization of the metal vapour just in front

of the cathode produces a strong electric field at the cathode surface. Under the influence of field emission or T-F emission [3] a new

discharge can be generated parallel to the existing discharge.

[It is even conceivable that for a short time several new discharges develop].The emission process is favoured by local protrusions on

the cathode surface, due to fieldenhancement.

The experiments with roughened cathode surfaces [I] show the craters being positioned on sharp ridges. In case the crater surface is highly polished the new discharge preferably starts on the rim of the crater previously formed. Separate craters are no longer distinguishable. This explains also the slight mobility of the discharge on a highly polished smooth surface.

If a new discharge develops, the current in the discharge already existing will begin to decrease. This leads to a smaller energy

supply for evaporation of material, which induces a further decreasing of the current. The process is cumulative, and leads to a new discharge. Consequently one may expect that starting from the momenta new

discharge develops the vapour production in the existing discharge will be very small until disappearance. This means that the dimensions of the crater, as found on the cathode surface are nearly identical with the dimensions of the crater formed by the total current just

prior to the moment the current starts to diminish.

As the crater diameter increases during the time the discharge is fixed above the crater, the current density calculated by the quotient of the total current and the area of the crater implies a minimum current density. This minimum current density value occurs on the moment preceding the disappearance of the discharge.

Generally speaking two mechanisms are known leading to evaporation of cathode material.

I. Energy supply from the discharge to the cathode.

By positive ion bombardment the evaporation energy is transmitted to the cathode spot [4,5,6, 7, 8].

2. Energy supply from the cathode.

The Joule heating is responsible for the evaporation of material [9, 10].

Usually it is assumed that the first mechanism is most likely to occur. The implication is that a considerable fraction of the metal vapour returns in ionized form to the cathode spot. The high vapour pressure and the fairly large particle density just in front of the crater is a limitation to this process and it seems more likely that if ionized vapour returns to the cathode it will condense on the area outside the cathode spot where the density is lower, leading to

The significance of Joule heating in the energy balance used is considered to be small

[a,

9] or is neglected [4,6].

In the model described here an evaluation is made of the influence of Joule heating on the evaporation process. By considering the moment the discharge extinguishes [which is a boundary condition in the discharge process] it is possible to determine the diameter of the crater if

evaporation solely is due to resistance heating. In this model use is made of the linear dependency of the specific resistance on temperature. The discontinuity of the specific resistance and the thermal conductivity at the melting temperature is taken .. into account. The error introduced by taking the specific resistance as a constant [5, 9] is clearly shown having in mind that in case of copper this quantity varies by a

factor larger than 17 in the range from room temperature to the normal boiling point.

When besides resistance heating other sources of energy supply participate in the energy balance of evaporation the value of the crater diameter calculated according the model described has to be appreciated as a minimum value. Increase of energy flow stimulates evaporation and thus an enlargement of the crater.

Using a model by which evaporation solely is due to Joule heating the following assumptions are made:

1. No energy supply from the discharge to the crater in the cathode. 2. Only the instant of time the total current starts to decrease

and the discharge extinguishes, is considered.

3. The crater formed is a hemisphere; the current is flowing radially into the homogeneous cathode material [13] •

solid zone.

- .

- - ' --

-_.

- - - ' .

- -

-

- _ .

'

-fig: 1.

rb denotes the radius of the crater.

Boiling temperature is reached here [which is not necessarily equal to the boiling temperature at n.p.t.J

2r

b corresponds with the crater diameter which has been measured experimentally [IJ.

ra denotes the value of the radius where the metal has attained melting temperature. (The melting temperature depends only slightly on pressure and therefore the melting temperature at atmospheric pressure is used here).

The evaporation process can be described by an energy balance inclu~ing three terms,

I. Heat production due to resistance heating. 2. Heat loss due to evaporation.

3. Heat loss due to conduction into the cathode.

The internal heat has been neglected. At the onset of evaporation i.e. at the first stage of the discharge the temperature will rise quickly as a function of time. We assume that at the end of the process the temperature reaches. a more or less constant value.

When the discharge process is in its last phase we already concluded that the evaporation rate is small. We therefore assume the term which represents the heat loss by evaporation as being zero and as boundary

aT

In other words: the discharge extinguishes because the heat produced is mainly balanced by heat-conduction.

The energy balance consisting of the two other terms is then given by

[ 211r2 K(T) dT

J

= 0 .--- (I)

. dr

where:

I the current carried by the discharge [A)

r the distance from P. (fig. I ) [m)

K(T) the heat conductivity coefficient

at r [~)

mOK

peT) the specific resistance at r [riM)

T the temperature at r [oK)

In case of pure metals the Wiedemann-Franz-Lorentz equation applies

pK

=

LT

---(2)

The relation holds for both solid and liquid metals like copper, silver and aluminum [II, 141. Here L is Lorentz' constant. This constant varies only to a small degree between both the solid and liquid states and with the type of metal.

Its order of magnitude is:

L = 2,4.10 -8 [volt)2

oK for the solid state and

-8 [Volt)2 L

=

3,0.10

OK for the liquid state [ 131

Two zones can be discerned for which the energy balance can be applied.

4.1. The solid state zone. This area runs from r = r _a where the melting temperature T is obtained to r + ~ where ambient

A

temperature T is reached.

~

4.2. The liquid state zone, beginning at r

=

ra to r

=

rb

where we assume the evaporation temperature Tk is reached.

In both zones the coefficient of heat-conduction K varies little with temperature (as far as is known for liquid metals);

however the specific resistance p depends nearly linear on temperature

[II, 12, 14, 151. At the melting temperature a discontinuity

occurs in p and K. The value of the jump in the specific resistance

is known for a number of metals [12, 141 and is about a factor two for closed packed metals (e.g. Cu, Zn, Cd). The same applies for K

[151. The values of K (experimentally unknown) can be calculated by using (2).

If K is taken as a constant in eq. (I), and eq. (2) is used, (I) can be written with

as 4

r dT +

dr with the solution

T = Acos ~ + Bsin ~

r r

2

'" T = 0 ---(3)

---(4)

Applying this result on both the solid and the liquid metal zone we find:

For a solid metal

For a liquid metal: T2 - C₃cos "'I - + r "'2 - + r (r a " r < "')---(5)

where: I "2 = 27TK2

~

4.3. Boundary conditions. For r = r _a For r -.. 'V For

I ) Continuity of the temperature : _TI

2) Continuity of the heat transport :

3) Room temperature TI

=

T~ 4) Negligible heat transport aT

2

by the metal vapour (-a:;-)r

b := T CI T 2 s aTI (K2 aT 2 (K I Tr)ra = Tr)ra = 0

With these boundary conditions and the

relation~ ~

I equations

(5) (6) enable us to calculate the values of ra

an~

.rb as a function of current and boiling temperature.

The result is:

I"S"

arccos T ~ arccos

---(7)

r

=

b I~ ---:T,;:---:c:---=T,-- ---(8) [ ~ K2 _s]

211KI arccos (I - - ) arccos

TK KI TK

The equations show that at a given boiling temperature TK the relation between the diameter of the crater rb and the current I is linear. The same applies to the diameter of the molten shell and the current.

rb ra The functions

-r

=

f{T K} and

:r

=

g {TK} are [Ts

=

1356 OK; LI

=

L2 • 2,23.10- 8 [v~rt]2 ; K K2 I

-

=

-2] and graphically shown in fig. 2. KI

calculated for copper

KI = 3,6.102

[~]

rb

It appears that the variable

-y

depends only slightly on temperature. The variation is less than 20% between temperatures of 20000K and 60000K. The value of this temperature attained by the crater wall is not known experimentally.

However a minimum value can be calculated, in a way proposed by Lee

[3].

The average pressure of the metal vapour produced has to be larger than the average magnetic pressure, due to the magnetic field of the discharge, in order to avoid pinch effect.

The average magnetic pressure Pm can be calculated by

where p in atmospheres (~ bars).

m

This relation is applicable only in the case the collision frequency of the electrons

I f we apply the

is large. compared to the rb

relation

-y

= f {T

K}, the

cyclotron frequency. magnetic pressure can be calculated as a function of temperature as is shown in fig. 3.

If one assumes that the temperature of the metal vapour is the same as the temperature of the crater wall, the vapour pressure of copper as a function of temperature is given by [17]:

1,6030 104

10 p

=

5,6602

-log atm TK TK in degree Kelvin.

The curves intersect at a temperature of about 3800 oK. This means that the temperature of the cathode spot should have been at least 3800 oK. For temperatures beyond 3800 oK we found a linear relation between the radius rb of the cathode spot and the discharge current, largely independent of temperature.

This relation is (T

K ~ 4000 oK)

rb -8

Experimentally [I] it has been shown that the most probable craterradius for a single discharge is linearly dependent on the current according to

-8

5,6.10 m/A

The agreement seems reasonable.

Froome [18, 19, 20] investigated the behaviour of the mercury discharge using Kerr-cell apparatus.

By means of very short exposure times (in the order of tenths

of microseconds) he determined the size of the luminous cathode spot. In the current range of 0 - 200 A. the spot was rectangular and

moved in·a direction perpendicular to its longest side (fig. 4).

-5

The width d amounted to 10 m, the length 1 was linearly

dependent on the current according -6

to 1

=

9.10 I, where I.in Amperes

and 1 in meters.

I.

f

direction of motion.

fig: 4.

.I

From his and other descriptions [21] it is clear that the spot in

fact consisted of a number of separate cathode spots oriented all along one line. These cathode spots have a circular structure

[211.

From this picture it follows that the width d of the rectangular agrees with the maximum diameter of the cathode spot of a single

-5

discharge: d

=

2r

b

=

10 m.

It is known

[21]

that the number of partial discharges on mercury is proportional to the

This means that 1 =

current.

-6 n.2r

b

=

9.10 I meter where partial discharges for a given current value.

It follows that the

i

1,1 A, having a

- = n

maximum current value -5 spot diameter of 10

of a single discharge m.

The Joule heating model applied to solid metals can be used to calculate the radius of the cathode spot in case of liquid metals. Starting point is the formula derived from the energy balance (6). This equation is now valid for the zone rb < r < ~ • For r + ~ there is room temperature. At r

=

rb the boiling temperature of mercury is

attained. We assume that this temperature is about the boiling temperature at atmospheric conditions: TK

=

630oK. The solution of (6) is

IlL

T", 2nK arccos TK

The relevant numerical values of mercury are:

L

=

2,8.10- 8 [~]2 oK K

=

8 _Watt/moK T = 630 oK K T

=

300 oK ~

Using these values we· find the relation rb

[ I 1 ]

[22]

-6

= 3,1.10 I meter.

The value of the radius of the spot having a discharge current -6

of 1,1 A amounts to 3,4.10 m.

The radius determined from Froomes measurements is 5.10- 6 meter. The discrepancy is small, taking into account that the evaporating temperature is not well established and Froome already mentioned that the width d of the line (10-5 m) probably is smaller [[20] page 812].

Using a model based on metal evaporation being mainly due to

Joule heating a linear relation between the diameter of the crater formed by a single discharge and the passing current was established. This relation is nearly independent of the temperature at the crater wall.

The results are in accordance with measurements on copper and mercury.

These results indicate that Joule heating is important for the evaporation process in a metal vapour arc.

The author wishes to record the stimulating discussions with Prof.Dr. D.Th.J. ter Horst, head of the laboratory and Dr.lr. W.M.C. van den Heuvel, staff member.

[ I] Daalder, J.E. and Vos, C.W.M.

[ 2] Djakov, B.E., Holmes, R. [ 3] Lee, T .H., Greenwood, A. [ 4] Lee, T.H. [ 5] Kozlov, N.P., Khvesyuk, V. 1. [ 6] Holm, R. [ 7] Kulyapin, VIM. [ 8] Rondeel, W.G.J. [ 9] Rich, J .A. [101 l1'm, V.E., Lebedev, s.v. [ I I] Jacob, M [12] Gubanov, A.I. [13] Holm, R. [14] Landolt - Bornstein [15] Goldsmith, A, Waterman, Th., Hirchhorn, H.J.

Distributionfunctions of the spot diameter for single and multi cathode discharges in vacuum. TH-Report 73 - E - 032.

ISBN 90 6144 032 7.

J. Phys. D. Appl. Phys. 4 (197 I)

J. Appl. Phys. 32 5 (196 I) 916. J. Appl. Phys. 30 2 (1959) 166.

504.

Sov. Phys. Techn. Phys. 16 I (197 I ) 97. J. Appl. Phys. 20 (1949) 715.

Sov. Phys. Techn. Phys. ~ 2 (1971) 287. Electrode erosion and energy balance of a metal vapour arc.

Norwegian Institute of Technology Trondheim Norway (1971).

J. Appl. Phys. 32 ~ (1961) 1023.

Sov. Phys. Techn. Phys. 16 2 (1971) 287. Heat transfer.

Wiley (New York) 1957. I p.112; II p.565. Quantum electron theory of amorphous

conductors.

Consultants bureau N.Y. (1965) p. 2, 6. Electrical Contacts Handbook.

Springer (1958) 429.

Zahlenwerte und Functionen. Springer, Berlin (1959) 24, 97.

Handbook of thermophysical properties of solid materials.

[16] Gundlach, H.C.W. [17] Nesmeyanov, A.N. [ 18] Froome, K.D. [19] Froome, K.D. [20] Froome, K.D. [21] Kesaer, I.G. th

Proc. V Int. symp. Discharges and Electrical Insulation in vacuum. Poznan Poland (1972).

Vapor pressure of the chemical elements. Elsevier publ. compo (1963) 151

Proc. Phys. Soc. London 63 ~ (1950) 377. Proc. Phys. Soc. London 60 (1948) 424. Proc. Phys. Soc. London ~ (1949) 805. Cathode processes in· the mercury arc. Consultants bureau N.Y. (1964).

[22] Handbook of chemistry and physics. Chemical rubber publishing Co. (1962).

rb radius of the melted zone.

ra,~ (m/A) _{ra radius of the evaporated zone.} I 1 r.

,,/

V'

-1 20

V

V'

,,/

V

,,/

...

l /

...

. /

V

V

10

~

~.

o

1000 2000 3000 4000

melting temperature _{boiling tem~erature} (latm)

Fig:2 / '

V

"'"

V

V

V'

rb - I 5000

L.

7

Tk (OK)

•

L

6000 I V> I

10 2 iii. 10 I 2000 A 8: vapour pressure. J

/

/

'I

/

V

I

V

J

I

I

I

If

/

/

V

3000 4000 fig:3.

/

/

/

1/

II

V.

I

I

II

T (OK)

•

5000