Electrical behaviour of fuse elements

DOI:

10.6100/IR43027

Document status and date: Published: 01/01/1969 Document Version:

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ELECTRICAL

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BEHAVIOUR

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OF FUSE

ELEMENTS

Proefschrift

ter verkrijging van de graad van doctor in de technische wetenschappen aan de Technische Hogeschool te Eindhoven, op gezag van de Rector Magnificus, prof.dr.ir. A.A.Th.M. van Trier, hoogleraar in de afdeling der Elektro_techniek, voor een

commissie uit de Senaat te verdedigen op dinsdag 6 mei, 1969, des namiddags te 4 uur

door

Leendert Vermij

Dit proefpchrift is goedgekeurd door de promotor Prof.Dr. D.Th.J. ter Horst

The investigations reported in this thesis were Cqrried out ~n

the High Voltage - High Current Labaratory of the T~ahnological

University Eindhoven, the Netherlands, under the greatly appreciated guidance and council of its head Prof.Dr. D.Th.J. ter Horst.

The author is indebted to Mr. G.M.V. van den Bosch of

this

laboratory, for his skilful assistance during the experiments, to Miss I.P.M. van Deutekom and Mr. C. Stoeller for their linguistic advice, to Mr. G. Abell wbo advised the lay-out of this thesis and to Miss H.C.G. Smalenaars for her valuable

CONTENTS 4

Symbols and Notation

CHAPTER I : Introduetion

I. I Fuses as protecting devices ~n electrical

circuits

I.2 Review of previous work

I.3 Survey of the investigation

CHAPTER 2 : The pre-arcing period

2. I The time duration of the melting process and

the associated increase of fuse resistance

2.2 The retardation of the evaporation

CHAPTER 3 : 3.1

3.2 3.3

3.4

2.2.1 The cammencement of the evaparatien

process

2.2.2 The instant at which the wire has just been completely evaporated

2.2.3 The superheating of liquid roetal The specific resistance of silver vapour Introduetion

The partiele densities in ionised silver vapour in the case of local thermadynamie equilibrium The computation of the specific resistance of silver vapour from the computed partiele densities

3.3.1 Introduetion

3.3.2 The callision cross-section Q of neutral silver atoms for electrans

3.3.3 The callision cross-section Qe+ of single ionised ions for electrans

3.3.4 The specific resistance

The validity of the computed specific

resistance in the case of a fusing silver wire

CHAPTER 4 : Fuse elements surrounded by a~r

4. 1 Introduetion

4.2 The behaviour of a fuse element surrounded by

9 1 1 I8 20 27 28 3I 32 34 34 40 40 4I 42 43 45 48 air. Phenomenological 48

4.3 The energy balans equation 55

4.4 Experimental proof of the energy balance

equation 58

4.5 Current interruption by a fusing silver wire~

surrounded by a~r 65 4.6 Conclusions 67 6 9 20 34 48

CHAPTER 5 : The initial fuse voltage Ef 69 5.1 Introduetion 69 5.2 The initial fuse voltage Ef as a function of

the energy input until the evaporation process starts 70 5.3 Multiple arcing and initial fuse voltage Ef 77

5. 3. 1 The deformation of a cylindrically shaped, liquid column by sUrface tension 78 5.3.2 The relation between the initial fuse

voltage Ef, fusing current I_{1 ,} diameter d0 and length ~ of a cylindrically

shaped fuse element 80 5.4 The duratio~ of the pre-arcing time 83 5.5 Summary and conclusions 84 CHAPTER 6 : Fuse elements surrounded by a solid matter

6. I Introduetion

6.2 Fuse elements stretched in air versus those embedded in fine-grained quartz-sand

6.3 The energy balance

6.4 Fuse elements enclosed in a small hole 6.5 Comparison with fuse elements embedded in

fine-grained quartz-sand

CHAPTER 7 : The voltage across and the current through a

86 86 86 89 92 95 fuse in a circuit 97 7.1 A fusing wire in a circuit with selfinductance

and capacitance 97 7.2 A fusing wire in an LC-circuit 103 7.3 The infl.uence of the rise-time of Rf on the

current and the fuse voltage 106 7. 3:·1 Interaction between rise-time and

time constant of the circuit 7.3.2 Some experiments of Baxter CHAPTER 8 : Summary and Conclusions

Samenvatting ~ist of References 106 1 1 1 114 116 120

SYMBOLS AND NOTATION 6

In this thesis the MKS unit system 1s employed unless otherwise stated.

Literature quoted is indicated by a figure placed between

brackets [ ]. This figure refers to the correspo~ding number in the List of References. Numbers of equations are placed between parentheses ( ).

The most customary symbols are listed below Al, A CM Cp,

c

cv D dl, d do + Ei , E· ₁ Ef Ef"~ ef E.!f* e G Hr h I I co Il Iz 11, 12' J Kl' K k L L ,~~,n, _\), m* me, mz, N nz nz+l 0 p Po Pm P ++

,

E· ₁ i mz+l + ++ ne n n n+ n cross section Meyer's constant capacity

speeific heat per unit mass at constant volume

Debye's radius diameter

diameter of the eylindrical fuse element in solid or liquid state

redueed ionisation energy initial fuse voltage

initial fuse voltage per unit wire length fuse voltage

fuse voltage per unit wire length unit charge thermadynamie potential heat flux Planck's constant current cut-off current fusing eurrent

ionisation energy of ions with charge z momentary values of the current

eurrent density constant Boltzmann's constant selfinduetance Coulomb logarithm length

mass per unit wire length mass of one partiele number of particles partiele densities surface

eleetrieal power input pressure

Pmax Po p Q Qe+ Q R R ·Re Rf R{'~ rf rf* r ro rab T To Tm Tmo Tv Tvo t to tl t2 ttm tie L'ltv

=

t)e- ttm IJ. t)v - t)e

u

U co Vv _VR. V V Ws

wl

w2

w

z·+

_l.

z.++

l.

Zi

z impact parameters power consumption cellision cross-sectien gas constant resistance

critical resistance of an LC-circuit initial fuse resistance

initial fuse resistance per unit wire length

fuse resistance

fuse resistance per unit wire length radius

radius of a cylindrical fuse element latent heat of transition from state a to state b

temperature (absolute) room temperature (absolute) melting temperature (absolute) melting temperature at atmospheric pressure (absolute)

evaporation temperature (absolute) evaporation temperature at atmospheric pressure (absolute)

time

instant at which the current starts to flow

instant of fusing

instant of current interruption

instant at which melting has just bee~ completed

instant at which the evaporation temperature is reached

instant at which evaporation has just been completed

time duration of the liquid phase

time duration of the evaporation process (rise time)

souree voltage

charging voltage of a capacitor volume

velocity energy

sum over internal states charge number

SYMBOLS AND NOTATION 8 y Eo n

e

el e

À Àz \Jo )J 7T p Po Pmo CJ Tf

TL

T T <Pe <Pz ~:z;+ 1 ct> <P w Wo constauts constant

temperature coefficient of the specific resistance

specific mass

dielectric constant of vacuum heat coefficient angle time constauts constant heat conductivity vacuum permeability chemical potential 3' 14 specific resistance

value of p at room temperature T0

value of p at melting temperature Tmo surface energy

time constauts of a circuit relaxation time

electrastatic potential angle

radial frequency

CHAPTER 1 INTRODUCTION

1.1 Fuses as protecting devicesin electrical circuits

Generally, a fuse is a current interrupting device incorporated in an electrical circuit to interrupt the electrical current in that circuit when this current attains - by whatever cause - a prohibitively high value. A fuse is capable to perform only once. In principle a fuse consists of a fuse element mounted in a cartridge. The construction of this cartridge is that of a removable electrical interconnection which fits in a fuse base incorporated in the circuit.

The fuse element is a conductor which has the shape of a round, flat or profile roetal strip, the roetal usually being silver. The element is mostly placed in an environment of air or purified fine-grained quartz-sand. When the electrical current in the circuit - and thus in the fuse - exceeds a certain value the fuse element will melt and evaporate as a result of the heat developed by Joule's effect. Subsequently, the electrical dis-charge which then occurs in the fuse must he extinguished and interruption of the current ensues.

Thus, a fuse is fundamentally a very simply constructed proteetion device.

Research has been applied to, and experiences have been gained with fuses during many decennia.

As early as 1884 Preece [l] studied the heating, effected by electrical current, of fuse elements. In this conneetion Preece refers to "wire protectors" which "act as a safety valve". Since then a gradual development of the fuse has taken place. Nowadays we have the use of a large choice of fuses for a varieP.f of applications. We are acquainted with, for example, the

miniature fuses for rated currents of some tenths up to several tens of amperes, primarily used for the proteetion of apparatus. The maximum current which these lew-voltage miniature fuses can interrupt is limited to a range from several hundreds to

approximately 1500 amperes. Next, we know the diazed cartridges for dornestic use, and the current limiting fuses for proteetion of lew-voltage circuits. The latter are designed for rated currents up to several hundreds of amperes; they can interrupt a short circuit current of some tens of kilo-amperes. Further-more there are the high-voltage cartridges which are

manufactured for rated voltages of up to + 30 kV and rated currents of up to + 200 amps. The maximum-current these

high-GRAPTER

10

voltage cartridges can interrupt also reaches up to several tens of kilo-amps.

The so-called currcnt limiting fnse is, especially in Europe, used most. In this fuse the element is fitted in a closed cartridge with dry purified fine-grained quartz-sand. Under short-circuit conditions the current through this type of fuse reaches an upper value (cut-off-current), which can be consider-ably lower than the maximum value of the prospective current. Fig. 1.1 illustrates the pattern of the voltage across and the current through a current limiting fuse which interrupts a short-circuit current. Ico is the cut-off-current, ef the voltage across the fuse. The latter is very small during pre--arcing time t0 < t < t1; during arcing time t1 < t < t 2 it may

reach a value higher than the nominal voltage of the circuit. After t = t2 the souree voltage comes across the f'..lse (recovery

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voltage).

The above terminology is in accordance with IEC

publication no. 66 [33] for fuses for voltages nat exceeding 1000 volts for AC and DC. In this IEC

publication the voltage ef for t 1 < t < _{t 2 is being}

referred to as arc-voltage; we, however, shall use the

term "fuse voltage", as it is doubtful whether arc--voltage indeed occurs-during the rapid increase of ef around the moment t1. Furthermore the current across the fuse during the pre-arcing period will be considered in this thesis. The term arc-voltage

there-Fig. 1. 1

Typical current and voltage traces of a current limiting fuse.

fore.does not fully cover the notion "voltage across the fuse", or fuse voltage, as referred to in this theses. Apart from the current limiting fuse other types exist, With respect to these, as well as for a more detailed description of the various makes and the behaviour of cu~rent limiting fuses, we refer to

literature (see e.g. [2,3,4]),

The lack of a clear idea about the exact .operation of fuses is remarkable, considering the extensive experience gained. The high fuse voltage occurring at time t = t₁ is not satisfactorily accounted for. The salution to this problem will appear to form an essential element in ·a sound clarification of the behaviour of fuses during their performance. The choice of fine-grained quartz-sand as a filler of cartridges too is a matter of experience rather than insight. Moreover, the processes occur-ring duoccur-ring arcing time - and which therefore seem to relate to

the interruption process - have been all but explained.

Due to this lack of insight the high voltage fuses particularly present a number of teehuical problems restricting the

application area, problems which up to now have only partly been solved. For instance: in a number of cases problems have been encountered with a high voltage fuse interrupting a current only a small multiple of the nominal current (overlaad condition). Moreover, there appears to exist an upper limit to the voltage

at which fuses can be successfully applied.

This thesis attempts to contribute to the clarification of the operation of fuses. In studying the phenomena relevant to fuses this contribution will be restricted to such aspects as are basic to the performance of the fuse element, Commercial fuses will not be considered. Due attention will he paid to the effect

the environment of the fuse element has on the interruption process. The investigation focusses on fuse elements in their simplest form, i.e. the cylindrical silver wires. The influence of the geometry of the fuse element on the characteristics of the fuse will be hardly or not at all considered in the

investigation. The above limitations of the area of investi-gation probably do not inflict any essential restrictions in arriving at a. satisfactory clarification of the behaviour of fuses.

1.2 Review of previous work

In the course of time the electrical current induced heating of fuse elements has been investigated rather thoroughly.

CHAPTER 12

As early as 1884, Preece [1] deducted arelation between the minimum current Imf (this is the current as a result of which

the element would melt after an infinite duration) and the diameter d of a round fusing element. He found a value C'd3/2 for lmf reasonably concurrent with the experiment. In literature of a later date (such as [3]) the constant C' is referred to as Preece's co-efficient. In 1906 Meyer [5], starting from the assumption that the fuse element is heated adiabatically,

calculated the duration of the pre-arcing period. He arrived at

t

the well-known equation

J

1 i2dt

=

~, in which CM is a constant

0

value determined by the material constauts of the element. The validity of "Meyer's equation" has amongst others been tested by van Liempt and de Vriend [6]. They found that Meyer's equation is only valid if the current causing the element to melt amounts to at least 20 times the minimum fusing current.

In the case of smaller currents or, in other words, for higher values of the pre-arcing period, the heat transport to the environment plays a role. In this case Meyer's equation no langer holds.

Wintergerst [7] in his calculations of the melting time account-ed for the heat transport to the environment. He confirmaccount-ed his calculations through measurements. Carne [8] too stuclied this problem and thereby gave emphasis to the temperature

distribution along the wire. A non-uniform temperature distribution along a homogeneaus wire is caused by the heat transport along the axis at the ends of the wire. This is

especially an essential factor in the calculation of the melting time of short fuse elements (1 centimeter or shorter). Novotny [9] devoted particular attention to this phenomenon. His

calculations of the effect of the heat transport along the axis were later verified through experiments. Also in this laboratory measurements and calculations were carried out regarding the temperature distribution along round wires in an environment of air as wellas embedded in fine-grained quartz-sand [10].

Satisfactory agreement between calculation and experiment was demonstrated.

The calculation of melting times and temperature distribution along a wire becomes substantially more complicated where, instead of round wires, profiled ribbons such as are aften applied in fuses, are concerned. This problem has been investigated by Adamsou and Viseshakul [11], by means of an

analog simulation.

carried out during past years and which have yielded us a rather thorough insight in the parameters determining the temperature of the fuse element and the duration of the pre-arcing period. These investigations have satisfactorily solved the questions concerning the nominal current and also partially those with regard to the fuse characteristics, i.e. the relation between fusing current and melting time.

A considerably less clear view has been obtained with respect to the behaviour of fuses as from the moment that melting and

evaporation has occurred. In an elucidation by Boehne

[12]

it is shown that the fuse voltage for t ~ _{t 1 determines for an}

essential part the electrical behaviour of a fuse in the arcing period. On the basis of a braader survey which Boehnè publishad previously [13] he developed a theory about the behaviour of fuses, which theory we shall bring out briefly.

A circuit consisting of an inductance 1 and a fuse F (see fig.

1.22

is at t=t0 connected to an a.c.souree with voltage

U= Usinwt. The relevant equation is in this case di

U = 1 - + e

dt f

(1.2.1)

where ef is the fuse voltage and i the current through the circuit.

In accordance with fuse practice we assume that ef ·"" o during the time interval t1 < t < _{t 2• Further, after Boehne, we assume that}

ef is a constant during t1 < t < t2 (see fig. 1.3).

From

(1.2.1)

follows

~

=

l

J

Udt -

l

J

e dt

1 1 f

i= o applies at the instant t = t2. I t follows t2 t2

J

Udt =

J

efdt

(1.2.2)

to t l

This means that the surfaces .A+C and B+C (see fig. 1.3) are equal. Consequently surface A is equal to surface B. If solely the voltage ef would act in the circuit a current if would be the result given by

• 1 t

1f

=

L

J

efdt

(1.2.3)

t l

CHAPTER 1 14

u

i

-L . Fig. I. 2 I

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- - - -r-

- r

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+-P-+---

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r~:--~:

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Fig. 1. 3

The current flowing through a current limiting fuse and the voltage across it, in accord~ce

with Boehne's conception.

during t1 ~ t ~ _{t 2 • From the principle of superpos1t1on applied}

to fig. 1.2 it fellows that at each time instant the current i through the fuse is formed by the difference between the

prospective. current ip and the current if• At the instant t = t2 the currents if and ip in fig. 1.3 intersect. This means i= o

for t ~ _{t 2• If ef} > U the slope of if is always greater than the

slope of ip. In this case the current i decreases as a function

of timefort > t1• In other words, the current i reaches an

absolute maximum at t

=

t1.

It fellows that for current limitation in a circuit with

self-inductance only it is implicit that at any moment ef > U. Also,

with a given value of ef, the arcing period t2- t1, and

consequently the discharge energy, are known. The effect of t₀

(the instant of closing the circuit, or in other words, the fault starting angle) on the are energy developed in total can

be traeed in this model as well. These, and a number of other

factors, Boehne has made a profound study of. Other authors (see

e.g. [3,14]) make also use of this methad for studying the

phenomena occurring with current limiting fuses.

Boehne [12] states explicitly that he does not enter into the causes which lead to the comparitively high fuse-voltage ef; he merely observes that it is there. However, his contemplations make it clear that the voltage across the fuse is a parameter determining the electrical behaviour of the fuse to an important extent. When it can be established what causes the high value of ef at the time t =t 1 and which parameters determine the pattern

of ef as a function of time, important headway has been made

towards a satisfactory clarification of the eperation of a fuse element. In literature one will come across three main

explanations for the origination and the further course of the voltage ef.

a. Baxter's [2,15] explanation is that ef comes into existence

through the occurrence of so-called multiple-arcing. In the

molten state the fuse element, under the influence of surface

tension and magnetic pinch pressure, is deformed into a number of globules. Amongst others, Kleen [16] has stuclied this phenomenon extensively. Between these globules short discharges take place, the voltage of them is mainly brought about as a result of the anode- and cathode-drop. The sum of

anode- and cathode-drop per discharge amounts to approx. 20

volts [15]. The value of the fuse voltageef is now determin-ed by the number of short discharges connectdetermin-ed in series at a given moment. Thus, if sufficiently large, the total number

CHAPTER I6

ini ti al fuse voltage Ef; this is the fuse voltage at t = t 1 •

With a greater length t of the fuse element, n increases proportionally to t, and so does Ef. This is in compliance with the experiment (see e.g.

[2,3]).

During t > t1, globules will evaporate, thereby repeatedly

causing several short discharges to link into one discharge.

A

drop in the voltage ef ensues.

After Baxter alsoother researchers, such as Lerstrup

[I4],

the Turner' s [

I

7] . and Mikulecky [

I8, I 9]

have set f orth u pon this explanation.

b. Läpple (20] explains the occurrence of ef at t = t1 by

proceeding from the assumption that in the first instanee a current interruption is brought about in the fuse at t =t1. He bases this on the results communicated by 0. Mayr [21]. Mayr states that roetal vapour below the thermal ionisation

temperature is a good insulator. Silver evaporates at a temperature too low to expect thermo-ionisation. Now if a current i through the fuse F is interrupted at t =t1, the ever present capacity Cp parallel to the fuse will be charged up to a certain voltage; the voltage being primarily

determined by the energy accumulated in self-induction at t

=

t1. This energy equals ~LI

1

2, in which 1_{1 is the current}

at t

=

_{t 1.}

It would follow then, that the voltage tends towards a value Êf = I1

v-ip .

Kriechbaum

[22]

shows that if in this relation we were to substitute values derived from practice for 1_{1 , L}

and CP, the voltage Êf would attain an improbably high value.

A di-electricAbreakdown through the roetal vapour occurs at a voltage Ef < Ef•

Kriechbaumleaves open the possibility that the wire

evaporates only at a temperature at which the roetal vapour can be in the ionised state and is therefore conductive. He states, however, that as a result of adiabatic expansion the temperature will drop and within microsecouds makes the roetal vapour reach the state of a good insulator. During the

increase of the voltage Êf the vapour will expand further; the subsequent drop in partiele density will cause the break--down voltage of th~ roetal vapour to decrease. So, with a slower increase of Ef, i.e. with a lower natural frequency of the circuit w0

=

(LCP)-! , Ef will diminish. Kriechbaum shows

experiments which are in coneerdance with the above.

c. Lohausen

[23]

already explained in

I934

the appearance of Ef, proceeding from the assumption that the evaporation process

of the fusing element involves a sharp r1se of the resistance of the fuseo In his further works (see e.g. [24]) Lohausen elucidates this point of view. Also Mocsáry [25,26] in his investigations proceeds from this explanation. In Lohausen's explanation the sharp rise of resistance is caused by the fact that the fulgurite - which is formed around a fuse element brought to evaporation in an environment of fine--grained quartz-sand - possesses a high finite resistance at the pertinent temperatures. The metal loses its conductivity when it has evaporated, but the conduction is then taken over by the fulguriteo The resistance of the fulgurite is subject to temperature. The interruption of the current is attained when the balance of the energy transport to the environment of the fulgurite on the one hand, and the energy production within the fulgurite on the other hand, is such that the resistance during the arcing period increases and eventually tends towards an infinite value.

The above summarized explanations for the appearance and the pattern of ef are contained in literature dealing with fuses. Still another explanation for the occurrence of ef has evolved from investigations on exploding wires. Investigations by David

[27], Keilhacker [28], Lebedev [29] and others have shown the probability that the fuse element, after reaching the evaporation temperature under atmospheric conditions, can still be highly superheated in the liquid state. Evaporation takes place at a much higher temperature, at which thermal ionisation can also be

expected. A high pressure metal plasma with a comparatively high specific resistance for the positive column is then formed, Of thi-s metal plasma the total resistance causes the fuse

voltageef fort> t 1• Mikulecky [19] reports this opinion, yet he gives the impression that multiple arcing could be the cause of ef in fuses. To which extent this explanation, derived from exploding wires research, is applicable to fuses, has not been the subject of his investigations.

As appears from literature the size and the course of ef for

t > t1 is influenced by the environment, by the material and by

the geometry of the fuse element. Several researchers (such as Läpple [20] and Baxter [2]) have investigated whether the filler of the cartridge should be inert or gas evolving under influence of the temperature. Also the effect of a gas evolving spider, around which the fusing element has been wound in a spiral shape, has been investigated a.o" by Mikulecky [18]. The latter believes to abserve a favourable effect; Baxter, however, is quite

CHAPTER 18

medium. Baxter [2] does demonstrate experimentally that the size of the filler grains has an apparent effect on ef.

Several investigatians have extensively probed into the influence of the geometry of the fuse element; we suffice with a

reference to some of the numerous publications in this field (e.g. [18,19,24,26,30,31,32]), from which it appears that the geometry of the fuse element controls for an important part the behaviour of ef for t > t1.

The above bibliography-concerns mainly that literature in which in one way or another reference is made to the fundamental operatien of fuses. Many of the publications mentioned discuss the influence of the circuit parameters on the behaviour of fuses (e.g. see [2,4,12,22,25,30]). The available literature conveys to us that it is of doubtful consequence to discuss the operatien and the behaviour of fuses without including in our considerations the circuit of which the fuse farms a part. A manifest interaction always exists between the behaviour of the fuse and the parameters of the circuit in which it is

incorporated.

1.3 Survey of the investigation

We argued that the answer to the question, why a fuse is capable to interrupt a current, must be found in clarifying the

behaviour of the fuse voltage ef during the interruption process. To achieve this we will try in the following chapters to find the answers to the questions:

1. What causes the voltage Ef at t = _{t 1 (see fig. 1.3) and which}

parameters determine its magnitude?

2. What controls the pattern of the voltage ef for t1 < t < t₂? 3. Why is, that in a well-functioning fuse, the current is

def ini tely interrupted at the moment t

=

t₂?

Posing the questions 1 and 2 implies that present knowledge and opinion~, as summarized under par. 1.2, cannot render a

satisfactory answer to these questions. Indeed, the simultanious existence of several mutually excluding interpretations about the occurrence of Ef at t = t1 (see par. 1 .2) demonstrates that the real cause of this phenomenon has as yet nat been fully understood.

To explain the operatien and the behaviour of fuses we shall praeeed from a model of beunding resistance. This means that the behaviour of fuses can be characterized by a sharp increase of resistance during the evaparatien process.

In the first place we shall endeavour to show in chapter 2 the plausibility of the assumption that the fusing element

evaporates at temperatures much higher than the evaparatien temperature under atmospheric conditions. This retardation of the evaporation does not only occur with exploding wires, as shown by David [27], Keilhacker [28], Lebedev [29] and ethers, but also with fuse elements; i.e. in current densities

significantly lower than are common in the case of exploding wires.

Next, in chapter 3, we shall prove through calculations that thermal ionisation can be expected at the actual evaparatien temperature. In this chapter the specific resistance of silver vapeur will be approximated as a function of pressure and temperature.

Now the rapid increase of resistance during the evaparatien process evolves from a very rapid transition from metallic conduction, through the still wholly or partly molten fuse element, into a gaseaus conduction.

In chapter 4 we shall establish the conclusions to the above considerations with respect to wires in an environment of air. An energy balance equation will be composed and tried on

experiments. It will be shown that the theoretical as well as the experimental results of this chapter strongly support the model of beunding resistance. Proceeding from the energy balance we shall furthermore establish imperative conditions for

attaining current interruption. In chapter 5 the initial fuse voltage Ef will be subject of further investigation.

In chapter 6 we shall investigate the behaviour of a wire embedded in fine-grained quartz-sand and the characteristic differences between this behaviour and that of a wire stretched in air. For this case too we shall establish an energy balance and campare this with the experiment.

In chapter 7 a study will be made of the effect of the model of beunding resistance on the pattern of currents and voltages in a circuit.

In several cases the results of the chapters 5 and 7 are directly comparable with data from literature.

CHAPTER 2

20

CHAPTER 2

THE PRE-ARCING PERIOD

2.1 The time duration of the melting process and the associated increase of fuse resistance

As already stated in paragraph 1.2, the research in the field of exploding wires has revealed that the evaparatien process takes place at temperatures well above the evaporation temperature at atmospheric pressure. The authors mentioned befare [27,28,29],

have based this conclusion firstly upon a careful investigation of the energy input to a fusing wire until the instant of

evaparatien and secondly upon the variatien of resistance during the pre-arcing period. With a similar experimental methad we shall examine whether the above conclusion is also justified in the case of much smaller current densities as they appear with fuses.

The heating of each element of a cylindrical wire, carrying an electric current, is given by

2 dT

J p

0 (1 +ST)= cvy dt (2.1.1)

where J current density (A.m~2)

p₀ specific resistance at room temperature T₀ (Qm) S temperature coefficient of the specific resistance

(OK-1)

T temperature (OK)

cv specific heat at constant volume per unit ma ss (J. OK'""l.kg-1)

y specific mass (kg. m- 3) t time (sec)

Integration of eq. (2.1.1) yields tlm _{cvy l+ST}

J

J2dt = ln ma

SPO 1 + ST

0 0

In this equation t ₁m is the instant at which the melting temperature Tmo is reached.

(2. I. 2)

If we substitute numerical valuesfor cv, y, S, Po and Tmo quoted from physical handbooks, we obtain Meyer's equation (see

paragraph 1.2). If we know the current density Jas a function of time, we are able to compute the melting time t]m from eq.

In the same way as indicated above, for the interval of time lltv = t 1 e-t 1v during which the liquid me tal is heated from melting temperature to evaporation temperature Tvo (see fig.2.2) can be derived

cvy 1+8 T

ln m vo

e

_{m mo} P 1

+s

_{m mo} T (2.1.3)

where Pmo is the specific resistance of the liquid metal at the

melting temperature Tmo and Sm the temperature cöefficient of

the specific resistance Pm of the liquid metal. Introducing numerical values the right hand side of eq. (2. 1.3) can also be computed.

Further the latent melting heat supplied to each volume element of the fuse wire during the melting process is known and there-fore the value of Meyer's integral can be computed for each roetal between the boundaries 0 and t 1e, at least in first approximation. Gibson [3] carries out these computations.

Let us consider the pattern of the fuse voltage ef as a function of time during the pre-arcing period. The oscillograms of fig. 2.1 show typical casesfora silver wire which is stretched in air (fig. 2.Ia) and fora silver wire which is embedded in fine--grained quartz-sand (fig. 2.Ib) resp. To obtain the oscillogram of fig. 2. 1b the vertical amplifier of the oscilloscope was adjusted so as to give a maximum deflection of ef far beyond the

reach of the screen of the oscilloscope.

The fuse voltage as a function of time is shown in diagram in fig. 2. 2.

It can be shown that at t= t1m (see fig. 2.2) the wire has just completely been melted. At this moment of time the temperature Tm may be somewhat higher than the melting temperature Tmo at atmospheric pressure [28,34]. In first approximation we assume

Tm=Tmo• As will become evident this is no essential neglect. We

assume that the evaporation process of the wire starts at t

=

tle'

whereas the heating from melting temperature Tmo to evaporation temperature Tv takes place during the interval of time

lltv =t1e- t1m•

From a large number of oscillographic records, as shown in fig.

2. l, bath the time interval lltv and the voltages efm and efe

(see fig. 2.2) have been determined.

All oscillograms were taken with a type 555 Tektronix oscillo-scope and have been obtained by fusing cylindrical shaped silver wires 0.2; 0.3 or 0.4 mm in diameter and 7 cm up to 25 cm in

CHAPTER 2 22

length. The wires were embedded in fine-grained quartz-sand

(grain size < 0.3 mm diameter) or stretched in air and were part

of an LC-circuit with a natural frequency of 50 Hz or 500 Hz. The magnitude of the current could be adjusted with the aid of

the charging voltage of the capacitor (maximum 15 kV). The

current has been measured with a coaxial shunt as described by Park [35] whereas the voltage has been measured with a mixed

(capacitive~resistive) voltage divider. Both coaxial shunt and

voltage divider have a flat frequency response up to appr.

7

MHz

as measured in the experimental set-up.

The time interval ~tv is usually so short that the current may

he considered to be a constant onè and equal to 1_{1 ,} as can be

seen from fig. 2.1 (I₁ is the current at the instant of the wire

evaporating). Then follows in first approximation

_

...

,

I

f

I~~

++-+ 1 +++J_. 11 I ; I I I I ~ I I I -l ! ' I _til~ I l i l ! ... ~ ... ···~· . . .. t . . . ... ...

i-- .

·

:

·-ti tO • ,, l.. .. _ ':' - . ., .

-

,

. l~ •

~i

I -Fig. 2. I

Oscillographic records of the current and the voltage during the fusing process of a fusing silver wire 0.2 mm diameter.

a. wire stretched in air b. wire embedded in

fine--grained quartz-sand.

Fig. 2. 2

The fuse voltage as a function of time.

yc I+S T

2 v m vo

Jl (tle- tlm) = S ln I+S T

mpmo m mo

(2.1.4) where J1 is the current density corresponding with the fusing current I 1. With numerical values taken from Landolt-Bornstein [36] , viz: y =I 0, 5 0 I o3kg.m"'"'3; Cv"' 320JJ<gl; oc-1, Sm= 5, 8. I0-4

.<t

-1

at T ~ 1200°C; Pm₀ =17,2.I0-8

nm,

Tv₀ =2075°C and Tm₀ =96ooc we can compute the value of tie- tlm for every value of J_{1 •} Th.ese computed values of t 1 e-t lm can be compared with the values of

~tv obtained from experiments.

In general the term J12Ctte- t1m) and consequently at a given current I1 also the factor (tie- t1m) yields a constant K,

irrespective of the values of cv, y and Sm or the dependency of these factors on the temperature T. So if the values of cv, y, Po and Sm are properly selected, the experiment should yield

~t

V

(~t is shown in fig. 2.2)

V

~t

From the experiments it became evident that the quotient v tte-tlm has not a constant value, but increases as a function of J₁2 , as shown in fig. 2.3.

From the oscillographic records we can also determine the ~ncrease ~Rf = Rfv- Rfm of the resistance Rf during the time

interval ~tv, viz, from the differences between the fuse volt~

efe and efm and the current I1• These values of ~Rf can be compared with the calculated values according

Rfv = Rf _m [ 1 + S (T _{m vo} - T ) ] _mo (2.1.5) The resistance Rfm can be determined from handhook data, taking into account the sudden increase of the specific resistance at the transition from the solid to the liquid phase (for silver a factor 2. 1). From each oscillogram wedetermine the quotient R

~~i

.

The result is graphically plotted in fig. 2.4.

~ fu .

In principle it would be possible to determine the real

evaporation temperature Tv from experimental data as shown ~n

figures 2.3 and 2.4. However, during the liquid phase a deformation of the wire takes place, as will be discussed in more extensive detail in chapter 4 and 5. This deformation process causes a reduction of the time interval tte- ttm• This ~s due to the fact that during this deformation process the

CHAPTER 2 24 1.2

t

t1e- '1m At" 0.8 0.6 0.4 0.2 0 Fig. 2.3

"

0.2 0.4 0.6 0.8 1.2.1021

Dependency of the melting the current density

J.

period 6tv of a fusing silver w~re on x 0.2 mm dia (sand) v 0.2 mm dia (air) 0. 3 mm dia (s·and) 3

t

Rtv- Rtm 6Rt 2

_.

-.

.

--

--

--

.

0 0.4 Fig. 2. 4 +

0.4

mm dia (sand) o

0.4

mm dia (air)

-

~

~

~ 0 0 0

V.

0 ...,.. 0

T

•

_.--:

.;..

=---

~---0.6 0.8 0 0 0

-

~

~

, -u

•

·---~-

...

I

•

J2(A2fm4) 1.2.1021

The increase of resistan~e 6Rf during the melting period of a fJ.Ising sîlver wire, as a function of J2 •

curve 1 silver wires stretched in air

current density in the parts with the smalles diameter

increaseso This means that the actual duration of the melting period ~tv will beshorter than the time interval tie -tlm as determined from (2.1.4), providing all the other parameters remain unchanged. It seems therefore plausible that the quotient

~tv/(t)e- t1m) can be smaller than 1, as is shown in fig. 5.3. However, the increase of the quantity ~tv/ (tIe- t Jm) as a function of J₁2 cannot be explained by the above mentioned

deformation process. This can be shown as follows.

Eq. (2. lo3) is valid for each tiny part of the fuseelementin the liquid stateo The time interval ~tv has been determined from oscillographic records consiclering the first discontinuity of the voltage trace. It can be assumed that at this instant of time the first tiny part of the fuse element has just been evaporated (see chapter 4). So ~tv is comparable with the time interval t1e- t)m according to eqo (2.1.3),

In case the fuse element does nat deform during the liquid stage it follows [see eq. (2. 1.4)]

K

t - t

=

le lm J_{1 2}

where K is a constant.

The deformation process causes an increase of J₁2 with time in

the first evaporating part of the fuse element. If we assume that the increase of J₁ can be written as Jf2 =J₁2(1+o.t), it follows for the reduced time interval tie- t ₁m < t1e- tlm

t1'e-tlm _.!.._-

~

(t'2- t2 ) J12 2 Ie lm So i t f o 11 ows I tle-tlm tle-tlm

From this equation it seems very improbable that the time interval t;e- tlm increases as a function of J12 , whatever the

dependency on t~me of J₁2 may be. It may be concluded therefore that the increase of ~tv/Ct te- tJm) with J12 , as determined

experimentally, cannot be caused by the deformation process. Then it follows that the increase of ~tv/(t]e- t1m) as a function of J₁2, due tothefact that the numerical value of J₁2 (t]e-tlm)

can only be varied if the integration limits of the right hand side of eq. (2.1,3) are changed, Assuming that the lower limit Tmo remains unchanged it must follow that the upper limit

CHAPTER 2 26

increases to a value of the temperature Tv > Tvo•

A lso the increase of l:.Rf

I

(Rfv- Rfm) as a function of J 1 2 , as

shown in fig. 2.4, indicates an increase of Tv with increasing current density.

If no deformation of the liquid wire takes place, and if the entire fuse element would be heated until the actual evaporation temperature Tv, it can be easily shown that

T - T

V IDO

T -T

VO IDO

(2.1.6)

From fig. 2.4 it appears that l:.Rf/(Rfv- Rfm) increases

approximately withafactor 2.5. According to (2.1.6) this would mean that the actual evaporation temperature Tv amounts to appr.

3800°C.

Due to the deformation process, however, it is very unlikely that the wire reaches the temperature Tv over its entire length. It seems therefore probable that at the first part of the wire the evaporation process may start at a temperature Tv larger than 38oooc.

The fact that in nearly all the cases the value of l:.Rf/(Rfv-Rfm)

is larger than I (see fig. 2.4) indicates also that the

deformation process plays a part.

From the rate of increase of the quotient l:.tv/(tle -t1m) the actual temperature Tv can also be estimated. If we assume that

the wire melts at T=Tm_{0 ,} then from the measure::l time intervals

l:.tv the real evaporation temperature Tv can be determined

according to

yc + B T

Jr2t:.t V ln mv

V _8mPmo +B _{m mo} T (2.1.7)

Combination of (2, 1.4) and (2. 1. 7) leads to

1 + S T ln m v t:.t I+ B T V mmo = t1e-tlm 1 + B T ln m VO 1 + B T m mo (2.1.8)

Eq. (2.1.8) is graphically plotted in fig. 2.5 for the above

mentioned values of Sm, Tvo and Tmo· From this figure it ensues

that an increase of l:.tv/Ct1e- t1m) withafactor 4 corresponds with a temperature Tv= 90QQOC, This increase of l:.tv/(tle- ttm)

1~----_L ______ L_ ____ _L~==~

2 6 10 14 18.103

appears approximately from fig. 2.3.

Fig. 2.5

The time interval lltv versus the evaporation temperature Tv according to eq.

(2.1.6).

Based o~ the ~hove experiments and considerations it seems justifiable t-.. conclude that already at current densities occurring in fuse practice a considerable retardation of the evaporation and consequently a substantial superheating can be

expected. This superheating increases with the increasing of the fusing current I1.

Furthermore, the experiments show a remarkable quantitative difference in the behaviour of fuse elements stretched in air and fuse elements embedded in fine-grained quartz-sand. This difference indicates more or less that liquid fuse elements stretched in air deform to a greater. extent than the ones embedded in fine-grained quartz-sand. This conclusion is in concurrence with the experimental data described in chapter 5. 2.2 The retardation of the evaporation

In this patagraph we shall try to give a very rough and brief indication on some possible causes of the above mentioned super-heating.

In general the c,:va.poration process cammences at the circumference of the wire [34,38]. Due to the magnetic pressure evaporation starts at a temperature Tv above the evaporation temperature at atmospheric pressure [28]. If an outershellof the wire has been evaporated the vapour pressure which is present at that certain moment will cause the evaporation of the remaining

CHAPTER 2 28

hence the increase of Tv at a given instant of time depends on the expansion speed of the metal vapour. Therefore it may be probable that the surrounding of the wire, as far as this influences the expansion speed, effects the evaporation temperature during the evaporation process.

We shall try from a qualitative and rough consideration to find out the influence of the magnetic pressure and the vapour

pressure on the evaporation temperature Tv•

We start from Clapeyron's equation and assume that the volume of the evaporated roetal Vv is much larger than the volume of the liquid metal Vt• Then will apply

ar 12

T V

V V

(2.2.1) where r12 is the evaporation heat (kcal/kmol) and a the number of kmols of the evaporated metal.

Assume r1 2 to be a constant. Then we obtain from (2,2.1) with

pVv

=

aRTv

P2 r 12 (

l n - =

p1 R

where Tv] the evaporation temperature at pressure p = _{p 1}

Tv2 the evaporation temperature at pressure p

=

_{p 2}

R the universal gas constant

The pressure p could be the magnetic pressure, the vapour pressure, or both.

(2.2.2)

In the paragraphs (2.2.1) and (2.2.2) we shall examine two cases more closely.

~~~~l

__

!h~-~~~~~~~~~~!-~É_!h~-~~~E~E~!i~~-EE~~~~~

At the beginning of the evaporation process the additional pressure on the wire is mainly formed by the magnetic pressure. In this case Tvl according eq. (2.2.2) the evaporation

temperature equals Tvo

=

2075oc at atmospheric pressure

P1 :: p₀

=

I05N/m2 • Consiclering a cylindrically .shaped silver

wire the magnatie pressure

Pm

is given by

I2 -7

Pm

=

2Tir2 .10 N/m2 (2.2.3)

In this equation r is the radius of the wire and I the current flowing through it.

From Landolt-Bornstein [36] we take r1 2 = 6.104 kcal/krnol for silver at atmospheric pressure.

Substituting (2.2.3) in (2.2.2), introducing numerical values for rlz, Rand Tvo and replacing the current I by the current density J it fellows with pz=p₀ +pm

ln(l + !Tir2J2.lo-12) = 3.104( 20175

--f-)

V

(2.2.4) This equation is plotted graphically in fig. 2.6 for three

different values of the wire radius r. From this figure it can be seen that at current densities mentioned in figures 2.3 and 2.4 no evaparatien temperatures occur which are comparable with the estimation of paragraph 2.1. The superheating due to

magnetic pressure is limited, only at high values of I and hence of J a considerable superheating can occur. However, in this region remarkable deviations from Clapeyron's equation can be expected due to the dependency of r1 2 on temperature.

Keilhacker connects Pro and Tv in the case of capper wires starting from an equation of state given by Himpan [28]

(p + (V-b)(TV-c) )(V-d) = RT (2.2.5)

where a, b, c and d are constauts to be determined. Keilhacker states that evaparatien and hence rapid expansion of the wire can only occur if the vapeur pressure balances the magnetic pressure. If the magnetic pressure Pro is known Tv can be determined from the equation of state (2.2.5).

From this consideration Keilhacker clarifies indeed the ocur-rence of strongly increased values of Tv· However, he considers current densities of the order of 10 12 A/m2 , these are much higher than the ones used in our experiments (see paragraph 2.1). In order to examine whether the above conclusions regarding Tv can also be applied to the case of smaller current densities we have computed the equation of state (2.2.5) with the aid of a digital computer.

True to Keilhacker we computed the critical values of silver from similarity relations as were given by David [27]. We found: Tcrit"' 6150°C; Peri~"' 5600 atm; Vcrit"' 27 cm3/mol. This is the first point with wh1ch eq. (2.2.5) can be solved. As a secend point we take the liquid phase at boiling point at 1 atm.

(Tv₀ = 2075°C, V₀ = 10,5 cm3/mol).

The computations yield similar isotherms as given by Keilhacker. For each isotherm the value p =pv is computed at which the

CHAPTER2 30 r--··

r.~vec>

-32 3 2 26 24 2 2

---=::-...=.

.--2 0 . ₁₀19 Fig.

2.6

I./ ,."~ ~· io""

v·

"."..

_,.,.,.

" ,

~

_.""..

-

-/

/ /

_/

/ /

_./

,·

/ /

V_/

_,' .

'/

/

"""

~, ./';!"

_V

,

. / ' _

·- "!/

..

, r=02~""' .~

~m

/ .~

r/

i.oo' / ~ ~ ~

v·

/

V"

i

2-4 <Am)

_i

1022

The evaporation temperature Tv·of silver at the onset of the

evaporation process, according to eq. (2.2.4). r is the radius

of the cylincrical s iJ.~ er wire.

_-

_{- ...}

--

trvf'K)

_!

~V

17

4 400 ~ Fig. 2.7 Typical isotherm. _,/

'/

i-'~

/'fl'

...

·

-

~ " ,

".~

'/

""".

--

i

·-

₇

!

Jz=fJIII~ ~l'Alm " 2200

11

I

1-l

I

PyfllmJ I Fig.

2.8

The vapour pressure Pv as a function of the evaporation temperature Tv according to eq.

conditions the evaporation process starts at this pressure Pv• The result of the computations of pressures Pv is plotted in fig. 2.8, In this figure two values of the current densities are also mentioned causing a magnetic pressure on a 0.2 mm wire as indicated on the Pv-axis.

Fig. 2.8 also shows that only at very high current densities a considerable superheating may be expected. At current densities used in our experiments (par. 2.1) the superheating effect is

limited,

From the above said it can be concluded that the magnetic

pressure cannot be the sole cause of the observed superheating. Also from experiments to be described in subsequent chapters it will be seen that the actual evaporation temperature becomes much higher than can be explained from the magnetic pressure only.

~~f~

__

!t~_ig~!~g!_~!-~ti~t_!t~-~iE~-t~~-i~~!-~~~g-~~~El~!~l~

~Y~E2!~f~~

During the evaporation process the already developed vapour expands. Assume that at the instant at which the wire has just been completely evaporated the metal vapour occupies a volume Vz. Roughly at that instant of time p₂

v

₂ = aRTv₂• In this case Tv2 is the temperature at which the last part of the liquid metal has been evaporated. Tv2 is therefore determined by the pressure p at this instant of time. In other words, the relation between p₁ and Tv2 is given by Clapeyron's equation from which eq. (2.2.2) has been derived. So in eq. (2.2.2) the quantities Pl and Tv! now stand for the pressure and the temperature respectively at the beginning of the evaporation proces. From eq, (2.2.2) it fellows Tv2 >Tv! if P2 > Pl•

Consequently the expansion speed determines to a great extent whether the evaporation temperature increases during the

evaporation process. From eq. (2.2.2) and with the aid of some experimental data it is possible to estimate roughly the order of magnitude of P2 and Tv2·

For the ratio between the volumes of silver in the liquid state Vt and in the gaseous state Vv one can derive

V

V

Vt

where p ~s expressed in N/m2 •

(2.2.6)

Experimentsas described in chapter 4 reveal that if a 0.1 mm silver wire surrounded by air under atmospheric conditions ~s

CHAPTER 2 32

fused by a current of about 500 amps, the ratio Vv/V~ aroounts to approximately 300. This current causes a magnetic pressure of appr. 16 atm. So the pressure Pl aroounts to appr. 17 atm. From eq. (2.2.2) it can now be computed that the evaparatien process starts at a temperature Tvt "'24oooc.

I f we combine the equations (2~2.2) and (2.2.6) it can be

computed that in this case the temperature Tv2 amounts to appr. 2750°C. Sa during the evaparatien process of the wire no

considerable increase of temperature occurs.

Based on the foregoing considerations it can be expected that if free expansion is nat possible, as for instanee is the case if the wire is embedded in fine-grained quartz-sand, the

evaparatien temperature during the evaparatien process will

~ncrease more.

~~~~]

__

!b~-~~E~Eb~~~!~g-~~-~-±!s~!~-~~~~1

Spiller, Grass and Perschke [39] describe an experiment with potassium from which it became evident that superheating of the liquid roetal can occur befare the evaparatien process starts. Heating of the potassium column was acquired through means of an electric current flowing through it (Joule heating), Intheir experiments the rate of temperature rise dT/dt was of the order of 30 to 80°C per second. So the temperature increased far slow~ than in the case of a fusing wire. They found that the

evaparatien process began at a temperature Tv approximately 1.7-2 times the saturation temperature Tvo•

One could ask oneself whether such a superheating would also appear if a fusing element is evaporated by an electric current. As far as we know no experimental data are available with regard to this. However, as early as 1954 Chace [40] suggests the

possibiritythat with exploding wires the evaparatien temperature .

rises far above the normal boiling point due to the lack of evaparatien nuclei.

A rough consideration on the basis of van der Waals equation of state may give an impression about the possible superheating of liquid roetaL ·

In fig. 2,9 some reduced isobars are plotted which are computed from the reduced van der Waals equation of state

(

p

*

+

""""*2 (

3 ) 3V -

*

I )

=

8T

*

*

where p V =-.._P_ Pcrit

v*

=

V V . cr~t T*

= __

T_ T . cr~t (2.2.7)

2.5 ~

·---I

.,

fr*

I i 2 tS -1 ---

/

~

-"''

I('

-I

0

I

nr; 01 Fig. 2. 9 ~

...

I/~/

--- -

~

I

V

I

-

- -

· -

V

•

P=Y

j

V

_A.,

7

,__ __ ~

'

/

Vp~O.t

""'

~

...

p:QD1

•

b--"' ... --~OilDl

•

1__!.._

10

Isobars obeying to the reduced van der Waals equation of state. This figure shows that also at low outer pressure a superheating is possible up to a temperature of the same order of magnitude as the critical temperature. Furthermore a perceptible expansion seems possible during superheating. This would agree with the very nice streak records of exploding wires which were taken by Thomas and Hearst [41]. These streak records show a moderate expansion befare a rapid expansion combined with a strong light emission, occurs.

We would like to remark that the preceeding considerations intend only to indicate a possible cause of the observed retardation of the boiling point. In our apinion

it.

should require a rather extensive investigation to find out the cause of the superheating, which investigation is believed to be beyond the scope of this thesis.

Furthermore we would like to state that it is only significant to consider the liquid phase with the help of the equations of state (2.2.5) and (2.2.7). These equations of state can only teach us something about the beginning of the evaparatien

process. The validity of these equations is doubtful in case of evaporated roetal at higher temperatures, because partly or fully ionisation can be expected at these temperatures.

CHAPTER 3 34

CHAPTER 3

THE SPECIFIC RESTSTANCE OF SILVER VAPOUR 3. I Introduetion

In chapter 2 it is shown experimentally that during the

evaporation process the fuse element can be considerably super-heated, Therefore it does not seem unreasonable to assume that during and immediately after the evaporation process, thermal ionisation of the metal vapour can occur. If this is the case, no interruption of the current takes place at the moment of evaporation of the fuse element, but a rapid transition from metallic to gaseaus conduction. In principle this process can be accompanied by a large and rapid increase of the specific

electrical resistance p of the fuse element.

In this chapter we try to calculate the order of magnitude of the specific resistance of silver vapour as a function of temperature.

We firstly compute the partiele densities of neutral atoms, electrans and ions as a function of pressure p and temperature T

(par. 3.2), and secondly (par. 3.3) wedetermine pas a function of T at some values of the pressure p from the computed partiele densities.

All these computations are carried out assuming local

thermadynamie equilibrium of the silver vapour. In paragraph 3.4 we shall try to prove whether this assumption is reasonable in

the case of a fusing silver wire.

3.2 The partiele densities in ionised silver vapour 1n the case of local thermadynamie equilibrium

Silver vapour is a mono-atomie gas. In this case the thermadynamie potential G is given by the equation [43]

ph3 G = nkT ln ---A-~--- (3. 2. I) where n k h T p m Z· ₁ (2nm)3/2(kT)5/2zi

number of particles of one kind per unit volume (m-3) Boltzmann's constant= 1,38.10-23J°K-l

Planck's constant= 6,63.I0- 34J.sec temperature (°K)

pressure (N.m-2)

mass of one partiele (kg)

sum over internal states of the particle, referred to a ground level.