Los supuestos de la utilización de un instrumento intraneus por parte del

The difference between the power supplied to and the power dissipated from a pair of electrodes is used in raising their temperatures and in melting and vaporising them.

The power available for this purpose at an anode was given by Cobine and Burger [24] as:

power= (Vaf+ Vwf + VT)× i (3.2)

Since the ratio of electronic to ionic currents at a cathode is unknown it is not possible readily to determine the power supplied to it. X-ray photographs of elements which have passed high currents invariably show that associated pairs of anodes and cathodes have burned back almost equally and therefore it is reasonable to assume that the power supplied to a cathode is the same as that given to an anode.

An important factor, which affects the burnback, is that only a fraction of the element material which initially occupies the space in which a gap is ultimately formed is vaporised during the arcing process, the remainder flowing away in liquid form. This fact was revealed by Wright and Beaumont when elements which had cleared large currents were examined microscopically. The elements together with the fused quartz filler, known as fulgurite, were sectioned longitudinally and in a plane orthogonal to that of the element strip, for examination. An example is shown in Figure 3.4 and the metal which flowed out in liquid form can be seen clearly. Further tests showed that the metal which had vaporised was deposited finely in the quartz filling material. This behaviour is presumably caused by the pressure distribution during arcing.

In calculating the arcing performance by taking account of energy requirements and interchanges, it is clearly necessary to determine the proportion of the element which is vaporised as this requires more energy per unit mass than the material which runs away in liquid form. To assess the proportions, Wright and Beaumont used a number of fuselinks containing silver elements and quartz filler to clear large currents.

They were subsequently X-rayed to determine the final lengths of the arcs. The time integrals of the currents during the arcing periods were determined from oscillograph records of the currents and the energy available for melting and vaporising was taken to be:

E_MV = 2(Vaf+ Vwf + VT)

 t_a 0

i dt (3.3)

in which tais the duration of the arcing period.

This was equated to the energy needed to remove the electrode material, i.e.

E_MV = mv(latent heat of vaporisation)+ mt(latent heat of fusion)

+ mt(melting temperature− 200) × specific heat (3.4)

d

a

b

c

Figure 3.4 Microscopic picture of fulgurite section Where

a electrode

b electrode material which has flowed away in liquid form c grains of filler

d arc cavity which has been filled with epoxy adhesive to enable the section to be made through the fulgurite

in which mt is the total mass of material removed from the element, a quantity determinable from the X-ray photographs, and mvis the mass of the vaporised metal.

It will be noted that this latter expression neglects the energy required to raise the temperature of the element material from the melting to boiling temperature, a quantity which is very small compared with the latent heat of vaporisation. It will also be seen that the bulk temperature of the electrode material is assumed to be raised to 200^◦C during the pre-arcing period.

In all the tests which were done it was found that eqns. 3.3 and 3.4 were satisfied when m_v 0·40 mt, i.e. the mass of material vaporised was only about 40 per cent of the total mass which was melted. For this condition, it is found that 20 per cent of the total energy supplied is used to provide the latent heat of fusion L_fand consequently the following expression may be used to determine the lengthening of the gap between a pair of electrodes, that is the burnback in an interval of time δt:

l₂− l1= mass of electrode removed in δt A_e× density of electrode material

= 2(V_af + Vwf+ VT)× {(i1+ i2)/2}δt × (0.2/Lf)

A_e× density of electrode material (3.5) where l₁and l₂and i₁and i₂are the gap lengths and currents at the beginning and end of the interval, respectively, and A_eis the cross-sectional area of the electrode at the arc roots over the time being considered.

By continuously calculating the burnback for each time interval the length of the positive column at any instant in the arcing period can readily be determined.

3.3.2 Cross-sectional area of a positive column

When a positive column is established in a fuse it receives power from the electrical system equal to the product of the voltage drop along the column and the total current in it. In any interval of time δt the energy given to the column may be expressed as:

energy input (E_a)= ¹₂(v_a1i₁+ va2i₂)δt (3.6) in which v_a1 and v_a2are the column voltages and i₁ and i₂are the currents at the beginning and end of the interval, respectively.

During the earlier part of the arcing period some of the energy input is retained within the column and is responsible for its increases in dimensions and temperature, while in the later part of the period the column reduces in cross-sectional area and the energy within it decreases. The changes in the energy present in a column during a short period may, however, be shown to be very small compared with the total energy input and so also may the energy given up to the electrodes. It may therefore be assumed that the power given to a column at any instant is dissipated to the surrounding filler which will consequently melt back progressively. About 2100 J are required to produce a gram of molten quartz and thus about E_a/2100 g of molten quartz [25] must be produced in the time interval δt.

Fuselinks are vibrated after the filling material is put into them and this should achieve random packing, in which state about 60 per cent of each unit of volume within the body should be occupied by quartz, the remainder being taken up by air.

On melting, quartz increases in volume by about 7 per cent but the grains fuse together and as a result the volume of the liquid quartz is only about 64 per cent (i.e. 1·07×60) of that originally occupied by the quartz–air mixture. A separate air space or cavity is formed into which the column can expand. In addition, a column expands into the space previously occupied by the part of the element which has melted. The total increase in the volume of a column in an interval δt may therefore be taken to be

vol_a2− vola1=

in which l₁ and l₂are the column lengths at the beginning and end of the interval, respectively, and A_e is the element area at the arc roots. It should be noted that the above expression uses a value of 1·6 × 10⁶g/m² for the density of dry, randomly packed sand.

Examinations of fuses which have cleared large currents have confirmed that there is always a cavity in the fused filler, surrounding each of the positions at which there had been a restriction in the fuse element. Although the cavities do not have a

constant cross-sectional area it may be thought reasonable for mathematical purposes to neglect this effect and to take the column area at the end of the interval as being

A_a2=vol_a2

l₂ (3.8)

3.3.3 Electrical conductivity of a positive column

When a positive column is established, material evaporated from the electrodes often enters it in the form of jets. Wright and Beaumont conducted a number of tests on fuselinks, including the taking of high-speed films of the arcs, and it was established that jets were present. The jets from opposite electrodes collide and create turbulence which in turn causes the temperatures and electron densities within each column to be relatively uniform.

In any short interval of time δt, a number of atoms of the vaporised electrode material are accelerated in jets into the column and a fraction X of them become ionised. During the same interval an almost equal number of atoms and ions must be scattered out of the column, as otherwise untenable pressures would be set up and the conductivity would be much above the values it can be shown to have by analysing voltage and current traces obtained during the testing of fuselinks. For the purposes of modelling, it may be possible to assume that the following equations apply for any time interval:

N_ai= Na (3.9)

and

N_e= XNa (3.10)

in which N_ai and N_eare the numbers of atoms and ions and electrons, respectively, scattered out of the arc in δt seconds and N_ais the number of atoms evaporated from the electrodes in the same period.

It was stated in Section 3.3.1.5 that the total mass of electrode material which is melted in a time interval is given by

mass melted = 2(Vaf+ Vwf+ VT)

Of this amount 40 per cent is vaporised, and therefore knowing the number of atoms per gram (N_g) of the element material (for silver N_g= 5.66 × 10²¹) it is clear that the number of atoms injected into the jets in an interval δt is

N_a= Ngm_v

A positive column loses energy in the following main ways:

(a) The kinetic energy of the atoms and ions (N_ai) which are scattered out of the column is given to the surrounding filler. The amount given up in an interval δt is

KE_ai= 1·5Naik_B(θ_a1+ θa2)

2 (3.11)

where k_B is Boltzmann’s constant which is equal to 1·38 × 10⁻²³J/K, and θ_a1and θ_a2are the column temperatures at the beginning and end of the interval, respectively.

(b) The kinetic energy of the electrons (N_e)scattered out of the column is also given to the filler. The amount given up in an interval δt is

KE_e= 1·5Nek_B(θ_a1+ θa2)

2 (3.12)

(c) The energy required to ionise N_eatoms is taken out of the column in a time δt and given up to the filler material when recombination occurs. The energy is given by

IE= NeE_J (3.13)

in which E_Jis the ionisation energy per atom, the value of which is 12.1×10⁻¹⁹J for silver.

(d) Energy is lost by radiation to the surroundings of the column. Because the column temperature and pressure are both high, the emissivity may reason-ably be assumed to be unity, i.e. black-body radiation. To further simplify the calculation, the temperature difference may be taken to be the temperature of the column because of the relatively low temperature of its surroundings. With these approximations the radiation energy loss may be expressed as:

RE= surface area of column × ks

θ_a1+ θa2

2

₄

(3.14) where k_s is the Stefan–Boltzmann constant which is equal to 5·67 × 10⁻⁸W/(m²K⁴).

For modelling purposes, Wright and Beaumont assumed the column to be cylindrical and its diameter was taken to be

d_a2=

4 πA_a2

0·5

in which A_a2is the cross-sectional area of the column at the end of the time interval.

The surface area of the column is then

surface area= πda2l₂= 2l2(π A_a2)^0·5 (3.15) Energy is also lost from a column in other ways including conduction to the filler and radiation to the electrodes, but the amounts may be shown to be relatively small and the thermal time constants are high.

The energy which enters a column is the time integral of the product of the voltage along it and the current flowing through it. For a short interval of time this may be expressed as

input energy=V_a1i₁+ Va2i₂

2 δt

The following energy-balance equation must apply:

V_a1i₁+ Va2i₂

2 δt= KEai+ KEe+ IE + RE Substitution in this equation from eqns. 3.9–3.15 gives

V_a1i₁+ Va2i₂ At any time, the temperatures of the atoms and electrons in a column will be the same because the mean free path is very short and they must experience many collisions before leaving the column. In these circumstances Saha’s equation [26], which relates electron density, ionisation fraction and temperature is applicable, i.e.

n_e2=1− X2 in which n_e2is the electron density at the end of the interval. The atoms evaporated in an interval δt flow in the vapour jets which may, for modelling purposes, be assumed to be of the cross-sectional areas of their associated electrodes and to have a constant velocity throughout the column.

The volume (V_a) occupied by the atoms is given by V_a= AaV_Jδt

in which V_Jis the velocity of the jets.

The atomic density of this material is therefore n_a= N_a

A_eV_Jδt

and the corresponding electron density is n_e= XNa= N_aX

A_eV_Jδt (3.18)

Wright and Beaumont further assumed that the ions and electrons reach terminal velocities in a column before being scattered out of it, and this enabled ionic and electronic conductivities to be assigned. Because the electrons move at much higher velocities than the ions, a further simplification of neglecting the ionic component of the current may be made.

At ionisation levels of more than 0·01 per cent which is certainly the situation in fuse arcs, Spitzer’s conductivity equation [27] given below is valid:

σ₂= 1·55 × 10⁻²θ_a2^3/2

log_e{(1·242 × 10⁴θ_a2^3/2)/n⁰_e2^·5} (3.19)

Eqns. 3.16–3.19 are non-linear and as they relate five unknown terms, i.e.

σ₂, n_e2, θ_a2, V_Jand X₂they cannot be solved.

Wright and Beaumont performed a range of tests on fuselinks, and the variations of the electrical conductivities were determined from the current and voltage traces.

These values were then substituted in eqns. 3.16–3.19 and the other four unknown quantities were calculated. It was found that the jet velocity VJvaried over a consid-erable range, which did, however, correspond well with values determined by other workers [27].

Because the variations in jet velocity did not have a great effect on the column conductivity, Wright and Beaumont used a value of vapour-jet velocity in the above equations, which minimised the errors between the computed and measured values of conductivity for the tests which had been done.