R414 Philips Res. Repts 16, 66-84, 1961

THE BEHAVIOUR OFTHE MERCURY HIGH-PRESSUREARC UNDER MECHANICAL VIBRATIONS

by L. SCHMIEDER 621.327.534.2:534.13

SummaryIf a mercury high-pressure discharge is subjected to sinusoidal mechani-cal vibration perpendicular to its longitudinal axis, a definite increaseofthe are voltage is observed. The are will be quenched ifboth frequencyand velocityamplitude of the vibration exceed 'critical values. Thisphenomenon can be explained by assuming that the mechanical vibra-tions give rise to forced gas flow which, through convection, causes acertain additional loss of energy from within the column of the are,After the sudden application of a constant acceleration it takes sometime for the corresponding Poisseuille flow pattern to establish itself.This transient time explains the existence of a critical frequency ofmechanical vibration. Above this frequency the gas, through itsinertia, will behave like a frictionless fluid and the convection lossesbecome proportional to the velo city amplitude of the mechanicalvibration. The "rnechanical stability" of the are is defined as the velocityamplitude necessary for quenching it at frequencies above the criticalvalue. Theoretically, this quantity should be proportional to the tubediameter and inversely proportional to the mass of the gas per unit oftube length. The implications of the theory are qualitatively, and to areasonable degree also quantitatively, explained by experimental results.'

RésuméLorsqu'une décharge dans la vapeur de mercure à haute pression estsoumise à une vibration mécanique perpendiculaire à son axe longitu-dinal, it se produit un accroissement de la tension d'arc. L'arc estinterrompu dès que la fréquence et l'amplitude de la vitesse de vibrationdépassent des valeurs critiques. Ce phénomène peut s'expliquer ensupposant que les vibrations mécaniques provoquent une circulationforcée du gaz qui, par convection, entraîne une certaine perteadditionnelle d'énergie dans la colonne de l'arc. Après la brusqueapplication d'une accélération constante it faut un certain temps pourque s'établisse la forme de débit de Poisseuille correspondante. Ce tempsde transition explique l'existence d'une fréquence critique de la vibrationmécanique. Au-delà de cette fréquence, par suite de son inertie, le gazse comporte comme un fluide sans friction et les pertes par convectionsont proportionnelles à l'amplitude de la vitesse de la vibration mécani-que. La stabilité mécanique d'un arc est définie comme l'amplitude devitesse nécessaire à l'étouffernent de cet are au-delà de la valeur critique.Théoriquement, cette grandeur serait proportionnelle au diamètre dutube et à l'inverse de la masse du gaz par unité de longueur du tube.Les déductions de la théorie sont expliquées qualitativement, et dansune certaine mesure quantitativement, par les résultats expérimentaux.

ZusammenfassungWird eine Hochdruck-Quecksilberdampflampe einer sinusförmigenmechanischen Schwingung senkrecht zu ihrer Längsachse ausgesetzt,so beobachtet man' deutlich ein Ansteigen der Bogenspannung. Der'Bogen verlöscht, wenn die Frequenz und die Geschwindigkeitsamplitudeder Vibration kritische Werte überschreiten. Diese Erscheinung kanndurch die Annahme erklärt werden, daB die mechanischen Schwingun-

67------------------------------------.--------------------MERCURY HIGH-PRESSURE ARC UNDER MECHANICAL VIBRATIONS

gen einen erzwungenen GasfluB bewirken, der durch Konvektion einengewissen zusätzlichen Energieentzug aus dem Inneren der EntIadungs-säule verursacht. Nach plötzlichem Anlegen einer (konstanten) Be-schleunigung dauert es eine gewisse Zeit, bevor sich das entsprechendePoiseuille-Strömungsmuster einstellt. Diese Übergangszeit erklärt dieExistenz einer kritisch en Frequenz der mechanischen Schwingung.Oberhalb dieser Frequenz verhält sich das Gas infolge seiner Trägheitwie eine reibungsfreie Flüssigkeit, und die Konvektionsverluste werdendann der Geschwindigkeitsamplitude der mechanischen Schwingungproportional. Die "mechanische StabiIität" des Bogens ist definiertdurch diejenige Geschwindigkeitsamplitude, die zur Löschung beiFrequenzen oberhalb des kritischen Wertes erforderlich ist. Theoretischmüûte diese Gröûe dem Röhrendurchmesser proportional und derMasse des Gases pro Röhren-Längeneinheit umgekehrt proportionalsein. Die Konsequenzen der Theorie werden qualitativ und in hin-reichendem Malle auch quantitativ durch Versuchsergebnisse bestätigt.

1. Introduction

Mercury high-pressure lamps, which are fixed to masts of modern design(particularly aluminium), can sometimes be extinguished by strong impactsagainst the foot of the mast. After a few minutes of cooling the lamp ignitesagain. Examination of the masts showed that the energy transferred by theimpact to the foot of the mast travels as a surface wave to the tip of the mastwhere it is reflected and gives rise, to heavy impacts which lead to powerfulvibrations inside the lamp. In the further course of this work the are wastherefore subjected to sinusoidal vibrations of given frequency and amplitude.

2. Experimental results

The discharge tubes of norm al 125-W lamps (diameter 10mm, length 40 mm)were mounted on a flat steel spring and were caused to vibrate perpendicularto the longitudinal tube axis by means of an electrodynamic exciter (fig. I).To obtain an easy check on the behaviour of current and voltage, direct currentwas used (for experiments with alternating current the reader is referred to theend of this report). The are voltage was about 110V, the mains voltage 330 V,the current strength 1·15 A. ,The, vibration system was started up at theresonance frequency, which was adjustable by altering the length of the spring.To each frequency 'jI of the mechanical vibration there corresponds a certain

Fig. I. Experimental set-up: I clamp, 2 flat spring, 3 exciter, 4 insulating plates, 5 currentsupply, 6 discharge tube.

68 L. SCHMIEDER

I Loci with slow tóke..,upI of the vibration _.

~Loci with rapid take-tlfJ .of the vibration

Fig. 2. Extinction of the are dependent on the frequency and amplitude of the mechanicalvibration. El = 110 V (are voltage), Uo = 330 V (mains voltage), d = 9·5 mm (diameterof the tube), I = 36 mm (length of the are), m = 0·5 mg/rnm (amount of gas per mm).

amplitude of the deflection y at which the lamp is extinguished. In fig. 2 themeasured results are plotted, on a double logarithmic scale, so that the func-tions:. amplitude-of the deflectiony = constant,amplitude of the velocityo= 27TV Y = constant,amplitude of the acceleration á = (27T'V)2 Y = constant,

will appear as straight lines. It is seen from fig. 2 that in the frequency range'V > 16 cis the are always extinguishes at substantially the same velocity.amplitude u . 3·14m/sec. A possible reason for the irregularities, occurringwith rapid take-up of the vibration in the region between.40 and 16p· cf.s, ismèntioned at the. end of this report. . .. "'-

MERCURY HIGH-PRESSURE ARC UNDER MECHANICAL VIBRATIONS 69

Fig. 3. Horizontal are with a small vibrationamplitude.

In the following we shall define the "mechanical stability" as the velocityamplitude Vext at which the are will be quenched (if the frequency exceeds thecritical value).

Observations whith a stroboscope disc show that at smal! amplitudes theare first starts to lurch (fig. 3), while at larger amplitudes it moves in oppositionto the mechanical vibration (fig. 4a). Furthermore, at the oscillating, D.e.-operated are an A. C. voltage of twice the freq uency of the mechanical vibrationoccurs, corresponding to the movement of the are in opposite direction to themechanical vibration (fig. 4b). Just before quenching, strong lurching occurs.Then the A.c. voltage varies in every period. If the are is mounted with itsaxis horizontal and is vibrated in a veritcal direction, lurching starts near theelectrodes. When the are axis is vertical (horizontal direction of the vibration),lurching starts in the lower part of the are (fig. 6).

3. The increase of the arc voltage

Apart from the occurrence of an A.C. voltage there is an increase of the

b Fig. 4a. Stationary state of vibration with amoderate vibration amplitude at the are,Fig. 4b. Oscillogram of the A.C. voltage withmoderate vibration amplitude. The upper lineshows the voltage at the exciter.

a

70 L. SCHMIEDER

Fig. 5. Oscillogram of the A.C.voltage at the are near its extinc-tion point with a large vibrationamplitude.

Fig. 6. Oscillating are in verticalposition.

average are voltage, which can also be deduced theoretically. We shall firstconsider a stationary arc. It is well known that at high pressure the numberof elastic impacts far exceeds the number of the inelastic ones. As a result,the electrons heat up the gas and a temperature equilibrium is set up betweenthe electrons, the ions and the atoms. Near the envelope the temperature fallsas a result of heat conduction, towards the centre it rises up to about 6000 "Kso that light quanta are produced by thermal excitation. In accordance withthe law of energy, the quantity of electrical energy supplied must be equal tothe radiant energy increased by the amount of energy lost as a result of heatconduction. The radiant energy is determined by the Boltzmann equation,whereas the density of the charge carriers (and hence, after introducing themobility of the electrons, the current density) is given by the Saha equation *)

Therefore the radiation and current densities fall exponentially with fallingtemperature, thus justifying the introduetion of the columnar model: Withinthe are column the temperature Tc is constant (only radiation). Outside theare the temperature drops towards a value Ta at the envelope. This ideal modelis characterized by the following quantities (see also list of symbols): P = con-sumed electrical power; S = radiated power; Pcond = power lost by heat con-duction (all these taken per unit of are length). Further p = pressure; Tc =absolute temperature of the column; Ta = ditto at the envelope (interrelatedby a. = Ta/Tc ~ 0,15); E = electrical field strength; f3 = (radius of column)!(radius of envelope).

The pressure p being constant over a cross-section of the tube, we obtain

*) The deduction of the Saha equation is given, for example, by W. Weizei and R. Rompe,Theory of electric arcs and sparks, Joharm Ambrosius Barth, Leipzig, 1949.

MERCURY HIGH-PRESSURl> ARC UNDER MECHANICAL VIBRATIONS 71

the mass m of the gas per unit of tube length by introducing the coordinatex = (distance from centre ~f the tube)j(radius of envelope) as follows:

1 1 .

(x dx 7TPcd~pz 7Tpcd2m::= Clpd2/T = -2- - xdx = -2-f(f3,{J),

. uz. pc . .. 0 0

(1)

where Cl is a constant; see also list of symbols.For the columnar model the function f(f3,{}) is given by

f32 .I-f3[ . (l-f30)ln{)]f(f3,O) = - - -- (1 - (3) + ,. 2 I-{} l-{} (2)

which function is represented graphically in fig. lOa. .The introduetion of the mobility of the electrons (b~OCVTjp) and of eq. (1)

in the Saha equation yields the current density in the column, from which,with E, and the cross-section of the column, the power consumption P isobtained 1):

(3)

where C2 is a constant; for f(f3;{}) see eq. (2); see further list of constants.The introduetion of eq. (1) in the Boltzmann equation, and multiplication

with the cross-section of the column yields the radiated power S:I

• f32S= C3m ----:ct) exp (-eUajkTc),f(f3,v (4)

where C3 is a constant; see further list of symbols.According to Elenbaas 1), eqs (3) and (4) may be introduced into the energy

equation

P = S+ Pcond, (5)

whereupon, after eliminating the exponential functions, one obtains

(6)

'where C4 is a constant.The equation ofthe heat conduction Pcond = 27TÀ(oTzjox), where the thermal

conductivity À = 8·38.1O-5T3/4(Wjm,OK) 1), yields by integration .

Tc7/4(1-.f}7/4) . .;Pcond = 30,07.10-5 (Wjm for Tc ID K). (7)

In (Ijf3) .

Finally, for the electric circuit, the following equation is v~lid

72 L. SCHMIEDER

PRs = (Uo-IE)E; (8)

see list of symbols.The properties of the columnar model are apparently fully defined by Tc

_ and {3. In actual fact, according to eqs (4) and (7), Sand Pcond only depend onthese two quantities, and, according to eqs (6)'a~d (5); also on Pand E. .Equa-tion (8) correlates those pairs of Uo and R, which belong to one and the sameare as switching elements. The above six equations do not determine the sevencharacteristic quantities unambiguously, but lay down pairs 'of magnitudes {3and Tc. This is, of course, due to the fact that the columnar model containsquantities that have been neglected. An additional condition would be neededto select from all the possible columnar models the one corresponding mostclosely to reality. Within the framework of this article, however, it is sufficientto examine the two limiting cases:

S{3 = 0, the are reacts only by lowering the temperature *),ST . 0, the arc reacts only by reducing the radius of the column.

As shown in sec. 4 below, the mechanical vibrations give rise to gas currents,which cause a forced additional loss of heat SPconv, and this has to be takeninto account ineqs (4) and (5) by replacing Pcond by Pcond + SPconv**). Afterintroducing the proportions w = Pcond/P, and g = apconv/Pcond, differentia-ting eqs (3), (5) and (8) and eliminating the quantities se, st and a{3, the sameresult is obtained for the two speciallimiting cases to a good degree of approxi-mation (see appendix I):

ss wg gE ~ 3(I-w) R:; (;'

where either a{3= ° or aTc = 0.Similar considerations show that the quantities {3, Tc, Pcond and P will de-

crease, although the rela,tive variations are considerably smaller than in E.The increase of the field strength, according to eq. (9), is independent of

the series resistance chosen, but it does depend on the are consumption, inthe sense that aE decreases if the are consumption is increased (g then becomessmaller). As aE represents the necessary reaction of the arc to the mechanicalvibration, any increase of P ought to make the are more stable.It follows also from (8) and (9) that

(9)

ap SE wg g- = - (I-Ra/Rs) R:; -- (I-Ra/Rs) R:; -(I-Ra/Rs), (10)PE. 3(I-w) 6

where Ra = total re~istance of the are. .

. *) By reducing Tand (3, the amount of radiant energy falls.**) Even in the stationary state the gas currents, due to gravitational acceleration, cause

only slight loss of heat in the lamps, as used in practice.

MERCURY HIGH-PRESSURE ARC UNDER MECHANICAL VmRATIONS 73

Ia \,by Ia \,by

++'<I-

"(£.1) x

"(£.l)

+_i

5 lI>1t

x +

x

Cb xil-"" 0

o +xo

at+

o~!

rJ+ +

"to 'o

>0<)(x

Q

08",,0

15 2mfsec

.£ 04225

Fig. 7a. Increase of the are voltage B(EI) as a function of the velocity amplitude-u of the'. mechanical vibration at 17 els.1= 1·15A, EI = 128 V, Uo = 320 V (0), 256 V (+), 192 V (x).Fig. 7b. The same at 29 els.Fig. 7c. The same at 65 els.

The variation in the overall consumption of the are (and hence obviouslyalso of the current. ~trength) is therefore dependent on the variation of E,and also on the series resistance selected. The are is most stable with R, in-finitely high. Then the increase in the consumption is equal to the increase in .the voltage, the current strength remaining constant.

The implications ofthe theory were tested experimentally. In figs 7a, band cthe increase id the voltage is plotted as a function of the velocityamplitude v.Quite clearly the, increase in the voltage was, in fact, independent of the seriesresistance, but it did vary with the frequency (for further interpretations ofthe measured results, see sec. 4). The measurements represented in figs 8a and bfurther confirm that the are becomes more stable by increasing P as well byincreasing n;

'"Vextm/sec

4

0 )l=T!c/s000 o )'=17C/s0..0 .2 0

0• 0 • )I=50c/s

ppwim '5 R/RaQ 12 4226

.2

Fig. 8a. Increase of the mechanical stability ;;;'"t as a result of increasing the specific areconsumption P.

Fig. 8b. Increase of the mechanical stability ;ext as a result of increasing the series resistance.

74 L. SCHMIEDER

yy

g

a

4227

Fig. 9a. Course of the flow of gas in the discharge tube.Fig. 9b. Gas flow with an enlarged cross-section of the tube.

4. The gas flow

According to the barometric formula, the dependence of the gas density pupon the "height" y in the direction of acceleration is as

ev>: pc Sp-- = - = 1- exp (- May/RTc) = 0.6.10-6 for a = 10 m/sec2, (11)pc pc '

in which M = 0·201 kg/mole (molecular weight ofmercury); R = gas constant= 8·3 joule/mole.Plc ; T = 4000 "K. Hence, for accelerations up to4000 m/sec2, as used experimentally, the gas may be considered as incom- •pressible. -

On account of the lower gas density pc of the are as ~ompared to its sur-roundings, a downward acceleration will force the are upwards and this isconnected with a flow of gas (see fig. 9a). The Navier-Stokes equations de-scribing this problem become very complicated, because the viscosity 7J varieswith the temperature. However, a straight portion might be imagined insertedbetween the upper and lower semi-circular halves, within which the streaminglines must have a parallel course (fig. 9b). Within a narrow strip on either sideof the x-axis the differential equation, giving the gas-flow velocity V in they-direction as a function of x, is as follows:

4 () ( ()V) ()P .- - 7J - = - + pza = (npc + pz)a,d2 öx öx ()y

in which n is a dimensionless constant; see further list of symbols., The viscosity constant 7J, according to Elenbaas 1), may be written as

7J = 5,5.10-7 T3/4 (kg/m.sec for T in OK) ..

With the boundary conditions

(12)

(13)

MERCURY IDGH-PRESSURE ARC UNDER MECHANICAL VmRATIONS 75

()V- = 0 (x = 0) and V = 0 (x = 1),öx . \

and the requirement that the amounts of gas transported upwards and down-wards must be equal

1

Jpa; Vdx = 0,

o

eq_ (12) may be integrated in two stages, separately over the ranges:

o ~ x ~ {3 and {3 ~ x ~ 1.

For the flow velocity Vo at the centre (x=O) the solution takes the following~~: -

Vo = a pcd2 cf>({3,#)= 2ma cfo({3,#) ,'1Jc 7T'1Jc f({3,#)

(14)

where the index c stands for conditions within the are (see also list of symbols).The new function cfo({3,#) is represented in :fig. lOa; the ratio cfo({3,#)/f({3,#)

in :fig. lOb.

f((J,

~'mech

(.1=0

/

g 4228 .

Fig. lOa. Graphic representation of the functions f«(3;{}) and </>«(3,{}) •. Fig. lOb.Graphic representation ofthefunctions VO/Vmech (fullydrawn lines) and </>«(3,{})/f«(3,{})

. (dotted lines). .

76 L ..SCHMIEDER ,. ., "

The flow velocity, as expected in the case of viscous flow, is proportional tothe acceleration a. The quantity pcd2/7}c has the dimension of time and signifies,as we know from acoustics, the time constant with which the gas goes overfrom the state of rest into the stationary Poiseuille flow 2) .. It would.be useful to call the reciprocal value'

(15)

the "Reynolds frequency", while the velocity deduced from the Reynoldsnumber Re

(16)

should be called the "Reynolds velocity".An estimate of 'JIRe is obtained from the following example: With the values

of (3 = 0,35, -& = 0·15 (see appendix I) it follows from fig. lOa that f((3,-&) =1·20. It follows further from eq. (1) that

pc = Zm]» d2 f((3,f)),

whence, with m = 5.10-4 kg/m, d = 1O-2m, it follows that pc = 2·50 kg/m''.From T ~ 5700 OK (see appendix I) we obtain for the viscosity:7}c = 3.60.10-4 kg/m.sec, If, in eq. (15) the radius ofthe column r = 1·75mmis substituted for r, it follows that 'JIRe = 47 clsec. The measured value (seefig. 2) was 'JIRe = 16 cisec. This experimentally measured Reynolds frequencydoes, however, vary between 10 and 20 c/sec (see fig. lIb) according to theexperimental arrangement. Because r was chosen rather at random, a coinci-dence can only be expected in the order of magnitude.At stationary values of a, or frequencies low with respect to 'JIRe, eq. (14)

yields, with a = 400 m/sec2 (see fig. 2) and cp((3,-&)= 0·023 (see fig. lOa),a value of the velocity at the centre Vo = 6·3 m/sec.At frequencies above 'JIRe only the forces of inertia are effective. In this case

the differential equation for the problem given in fig. 9b becomes

b Pv (t) .- V(x,t) = - _.--a (t),b( p (x)

(17)

where all quantities are considered as functions of the arguments mentionedin parentheses. The boundary conditions are

bVV(x,O)= 0; _ = 0 (x = 0)

bx

as well as the requirement1

r fp(X) V(x,t)dx = O.

o

MERCURY HIGH-PRESSURE ARC UNDER MECHANICAL VIBRATIONS 77

Fig. lla. Reynolds frequency as a function of m/d.Fig. lIb. Mechanical stability as a function of m/d.

Integration of eq. (17) with respect to x yields1 1

~fp(X)V(X,t) dx = - ~aCt) p (x) +py (t)} dx = O.~t )1 .o 0

Obviously the right-hand side only disappears for random values of t if1

Pv (t) f ~ (1-,8) lnif)~-- = const. =- p(x)dx = + p» ,8- =- pa(t). 1~ if

oThis is substituted in eq. (17), which is then integrated with respect to t. Itfollows that

t

V(x,t) = - (1 - P) (a( T)dT.p(x) .

o

In the case of a periodic acceleration we find on the right-hand side the velocityamplitude of the mechanical vibration. The expression

78 L. SCHMIEDER

Vo/Vmech = p - 1 = - (1-,8)~ 1 + ln-& ~. Pc ? 1--& ~

is shown graphically in fig. lOb. With ,8= 0,35; -&= 0,15, VO/Vmech= 0·68.From Vmech= 3·14 m/sec (see fig. 2) it follows thatFn = 2·12 m/sec, which

is of the same order of magnitude as the value (6,3 m/sec) calculated earlierin this section under the supposition of a continuing stationary acceleration.An upper limit for the amount of heat carried away by the flow of gas isobtained assuming that the gas at the envelope is cooled down to Ta = -&Tc:

(18)

2 ,8d (1--&)SPconv= - Vo pc Tc cp.

7T 2(19)

Introducing the above numerical values, and with Cp = 2·5 (R/M), one obtains

p = Pc TcR/M = 5.89.105 N/m2,

SP = Vo 5,8 dp (1-#) / 2 7T = 2900 W/m. (20)

A different estimate is given by eq. (9), when inserting the value SE/E = 0·3(this increase of voltage was roughly measured at a moment of extinction),and x = t, Pcond . 103 W/m. Hence we find SPconv . 1800 W/m (the totalconsumption with the are stationary amounts to 3600 W /m). As' the voltageincrease is proportional to SPconv, the variation 'of the function SE/ E =j( Vmech)can be predicted .. With very small amplitudes, the temperature drop is pro-portional to the displacement, but so is the flow velocity of the gas, so that theamount of heat carried away SPconvwill rise with the square ofvmech. Whenthe gas reaches the envelope (of which the heat-conducting power is about10 times that of the gas), the temperature of the cooled gas no longer dropsand the amou~t of heat carried away will rise in direct proportion to the velo-city amplitude, If one 'works in the frequency. range below the Reynoldsfrequency, the Poiseuille velocity will finally be reached, in which case theflow velocity, and hence the amount' ~f heat carried a~ay, will increase con-siderably more slowly than before.' .

These predictions of the theory are confirmed by the measurements repro-duced in figs 7a, band c. The Poiseuille velocity is attained in fig. 7a, while inthe measurements shown in figs 7band c the are is extinguished even before

.' .reaching the point of inflection;

If; in eq. (19), Vo is replaced by Vmechin accordance with eq. (18), pcd2 bym according to eq. (1), and if it be furth~r assumed that Pconv (max) is prop or-

• _. • f

tional to P, one obtains the relationship' . .

(Pd) ( ln-& )Vmechcc -;; j({3,1J)/{3 (1- (3)(1- -&) 1 + 1 _ -& • (21)

MERCURY HIGH-PRESSURE ARC UNDER MECHANICAL VIBRATIONS 79

The mechanical stability is therefore improved by increasing the ratio of djmor by increasing the consumption P. The second form alters so little that onlya small effect on the mechanical stability of the are is to be expected.

An experimental investigation of the increase of the mechanical stabilityVext with increasing P was undertaken (see fig. 8a). The curve, admittedly,does not show a strictly linear variation. Furthermore, measurements werecarried out with discharge tubes of varying m/d. Here the linear dependenceof Vext is fairly well exhibited (fig. lIb). According to eq. (15), the Reynoldsfrequency should be inversely proportional to m/d. This prediction, accordingto our measurements, is not borne out so well in reality (fig. lla).

If lamps of various diameters are compared at equal pressure, it followsthat p o: mld» = constant; thus mld = con st. d. Therefore the are in a lampwith a smaller diameter is mechanically more stable.

5. Behaviour of the are under impact

To investigate the behaviour under a single, powerful impact, the dischargetubes were mounted with their axis vertical on an anvil which, by means of

o la 19

33 57 100

200 330 500

Fig. 12. Behaviour of the are after impact. Exposure technique: 3000 frames per second.The numbers below the photographs indicate the picture number.

80 L. SCHMIEDER

an impact from a falling hammer, was brought to a velocity of 4·16 m/secwithin half a millisecond. After the anvil, with the tube, had passed througha distance of about O·3 m it was braked by means of a spring. This processwas filmed with a camera giving 3000 pictures per second. In fig. 12, ninedifferent stages of the phenomenon are shown for a D.C.-operated are.Picture 0 showsthe are immediately before the impact. Inpicture 10 (on the

right the after-glow can still be seen of the flash lamp which was ignited whenthe hammer hit the anvil) the are has been displaced 2·5 mm to the right, inthe lower half of the discharge. This is obviously due to the fact that the spacebelow the electrode contains cold gas, which causes an additional gas flow.In picture 19, the displacement of the are reaches its maximum.

After 11 msec (picture 33) the displacement of the are has again returnedto zero, but afterwards it becomes displaced to the left (picture 57). This factis surprising, because at this time there is no effective acceleration, as may beseen from 13, in which the acceleration of the tube is plotted against time or,respectively, the picture number. An explanation for the rebound can be soughtin the shape of the curve which shows the displacement of the are in dependenceon, time (figs 13 and 15). It is seen that the are rebounds with a high velocityas soon as it touches the wall (4 mm displacement). After 33 msec (picture 100)

9000mjsec2 5300m/sec2

. r t300

o

5 mm2.5

r-----------~--~O-2.5

-5

200

100

-100

-200

500Picture number

~-5800m/sec2

4230

Fig. 13. Variation of the external acceleration and of the displacement of the are after impactas ,a function of time or, respectively. of the picture number, Iron/iron impact.

MERCURY HIGH-PRESSURE ARC UNDER l'4ECHANICAL vmRATIONS 81

m/sec2300

200

»r=:A~; disp7acement700

300~~ __ ~ __ L_~ __ ~_'4231 0

Fig. 14. As fig. 13, iron/rubber impact.

the are has again swung back, after 67 msec the braking begins to make itselffelt, and 90 msec after braking the are comes to rest. Thé test was repeated,and this time a rubber cushion was placed between hammer and anvil (fig. 14).The impact transmitted was somewhat greater (the initial velocity of the anvilamounted to 5·7 m/sec), the maximum actual acceleration was considerablysmaller (400 m/sec2) than in the first test. The rebound of the are was slower;apparently contact with the wall was less intense.From fig. 15 it is seen that in both tests the are approached -the wall with

constant velocity as soon as the effect of the acceleration had ceased.

4232

iron/rubber

Picture numberFig. IS. Are displacement as a function of ti,me.

82 L. SCHMIEDER

The are behaves rather like a rubber ball jumping to and fro between thebottom and the lid of a box. In such a system, just as in an ordinary oscillatingsystem, resonance phenomena may occur. Whether the decrease of mechanicalstability in the frequency range between 40 and 120 c/sec' (see fig. 2) should beconsidered as a resonance phenomenon has not been finally proved, as this _.,phenomenon occurs only with rapid setting off, when non-linear distortionsmay occur during the oscillating.

Oscillation tests in which the are was supplied with alternating current didnot bring to light any new principles. For the same mains voltage, the stabilityunder mechanical vibrations was about half as great as with direct current.A 50-% improvement in stability is obtained when a capacitor is connected inseries to the series choke, but this cannot be considered in further detail here.

Acknowledgement

My thanks are due to Dr W. Elenbaas, Dr E. W. van Reuven and Dr D.Vermeulen for the formulation of the problem, supplying the apparatus forthe experiments and the' helpful and inspiring discussions.

Eindhoven, September 1960

REFERENCES

1) W. Elenbaas, The high pressure mercury vapour discharge, North-Holland Pub), Comp.,Amsterdam, 1951.

2) C. Zwikker and C. W. Kosten, Sound absorbing materials, Elsevier Pub), Comp., Inc.,Amsterdam, 1951.

Appendix I

By differentation of eqs (3), (4) and (8) one obtains, respectively,

SP ,SE (2 . 1 . 'Of«(3,1J)) e Ui, STcP = 2]i + tl + 2f«(3,{J) . '0(3 S(3 + 2kTc Tc

(3')

(in the differentiation with respect to Tc only the exponential function hasbeen considered),

SE 1 (1 7 'Of«(3,1J) 1/(3 W)']i = - '3 tl + 4f«(3,{)) '0(3 -ln(1/(3) 1 - w S(3+

(7w 1) STc 1- 3w SP . wç

+ 12(1 - w) - '8 -Tc + -6(-1--w)P + -3(-I-j-v) , (4')

and

MERCURY-HIGH-PRESSURE ARC UNDER MECHANICAL VffiRATIO·NS 83

SP = SE ( 1_ Ra) _ (8')P . E RB

In practice, generally, W R;j t (according to eq_ (5); the field strength is thenat its minimum), Furthermore, Ra/RB R:;; 1 (the power supplied by the sourceis then at its maximum).Insertion of (8') in (4') shows that the penultimate member on the right-

hand side of eq. (4') may be neglected. Elimination of SP/P from the abovethree equations gives

SE 1 (1 7 bf((:1,19) 1/(:1 W)E + 3 P + 4f((:1,{}) 'b(:1 -ln(I/(:1) 1- IV S(:1-

(7w STe

12(I-w) Tc1) wç"8 = 3(1- w)

and

SE ( Ra) (2 1 bf((:1,{}») e Ui, STeE 1 + RB + P + 2/((:1,19) b(:1 S(:1+ 2kTe Tc = O.

Hence, in case S(:1= 0

SE !Vç [ (7W 1) 1+ Ra/RB] -1

E = 3(1- w) 1 + 12(1-~-"8 eUd2kTe •

In case STe = 0:

SE wç-=----E 3(1- w)

1 + RaRB

1---3

!.+ 7 b/ ((:1,19)_ _!f!_ ___!!__(:1 4f((:1,19) b(:1 ln(I/(:1)1- IV

2 1 bf((:1,{})P + f((:1,19) b(:1

-1

Numerical example: for a discharge, mentioned by Elenbaas 1), the followingvalues were measured: d = 4·1 mm; P = 3500 W/m; Ta = 765 "K; m = 1.2g/m; E = 580 V/m; J?cond = 1000 W/m.The columnar model, having the same values of Sand Pcond, is defined by:

(:1= 0,35; Tc = 5700 "K; 3eU'!/4kTe = 15:9,and [lff((:1,{})] bf((:1,O')/b(:1= -0,75.If these values, together with the appropirate values of wand ç are inserted

in eq. (9) one obtains

SE IVÇ- = (1- 0·02)E 3(1- w) (if S(:1= 0),

SE !VçE ) ( 1+ 0·03)3(1 -:- IV

(if STe = 0).

84 L. SCHMIEDER

Appendix 11

List of used symbols

a (á)dEegIkImPPcondPCOll\'

PRRaReRsI'

STaTcTz

UoUtUa

VVmech (iJ)wXye;)fJ3 ..7]

-0'11

pcpzg

,/

acceleration (amplitude) of the' mechanical vibrationdiameter of the discharge tubeelectrical field strengthcharge of the electron = 1·6 . 10-19 coulombgravitational acceleration = 9·81 m/sec2electric currentBoltzmann's constant = 1.38.10-9 jouletKlength of the aremass of gas per unit of tube length.consumed electric power per unit of are lengthpower of dissipated heat per unit of are lengthextra power carried away by forced mechanical convectiongas pressure 'gas constant = 8·3 joule/mole.fX.electric resistance of the are = El/lReynolds' numberelectric resistance in series with discharge tuberadius of the discharge tube = tdradiation power per unit of are lengthabsolute temperature at the envelope of the discharge tubeabsolute temperature of the areabsolute temperature at location xmains voltageionization potential (= 10·4 electron volts for Hg)average excitation potential (= 7·81 electron volts (after Elen-baas 1»flow velocity of the gasvelocity (amplitude) of the mechanical vibrationratio Pcond/Plocation: (distance from centre)/(radius of tube)deflection (amplitude) of the mechanical vibrationratio (diameter of are column)/dfinite variation of ...viscosity constant of the gasratio Ta/Tc (:::::;0·15)frequency of the mechanical vibrationgas density within the are columngas density at location xratio: 3Pconv/Pcond