Short-Circuit Current Interruption in a Low-Voltage Fuse with Ablating Walls. By S. Ramakrishnan and w.m.c. van den Heuvel

Transcription

1 Shrt-Circuit Current Interruptin in a Lw-Vltage Fuse with Ablating Walls By S. Ramakrishnan and w.m.c. van den Heuvel EUT Reprt 85-E-151 ISBN X ISSN August 1985

2 Eindhven University f Technlgy Research Reprts EINDHOVEN UNIVERSITY OF TECHNOLOGY Department f Electrical Engineering Eindhven The Netherlands SHORT-CIRCUIT CURRENT INTERRUPTION IN A LOW-VOLTAGE FUSE WITH ABLATING WALLS by S. Ramakrishnan and W.M.C. van den Heuvel EUT Reprt 85-E-151 ISBN X ISSN Cden: TEUEDE Eindhven August 1985

3 CIP-GEGEVENS KONINKLIJKE BIBLIOTHEEK, DEN HAAG Ramakrishnan, s. Shrt-circuit interruptin in a lw-vltage fuse with ablating walls / by S. Ramakrishnan and W.M.C. van den Heuvel. - Eindhven: University f Technlgy. - Fig. - (Eindhven University f Technlgy research reprts / Department f Electrical Engineering, ISSN , 85-E-151) Met lit. pg., reg. ISBN X SISO UDC UGI650 Trefw.: smeltveiligheden; laagspanning.

4 -iii- Abstract This reprt describes a cmputer simulatin study f the prcess f shrt-circuit current interruptin in lw-vltage fuses which have n sand filling. The current interruptin prcess in such fuses is aided by the ablatin f the wall material f the fuse which helps t cl the arc clumn inside the fuse. An algrithm has been develped t slve numerically the time-dependent energy balance equatin fr the arc clumn taking int accunt the ablatin f the wall material and the cnsequent pressure rise inside the fuse. The numerical algrithm is linked with the equatins describing a test circuit t prvide nearly 1500 A f prspective shrt-circuit current frm a 250 V, 50 Hz surce. This study suggests that the current interruptin prcess is dictated by the temperature distributin in the arc clumn immediately after the explsin f the fuse wire. The investigatin reveals that the mst likely reasn fr the successful peratin f the fuse under a set f test cnditins is that mst f the jule heating in the arc clumn immediately after the explsin f the fuse wire ccurs in the uter regins f the arc clumn clse t the wall f the tube. Results f temperature distributin in the arc clumn, arc vltage and current as a functin f time are presented fr tests with step arc currents and als fr shrt-circuit tests and cmpared with experimental results. Ramakrishnan, s. and W.M.C. van den Heuvel SHORT-CIRCUIT CURRENT INTERRUPTION IN A LOW-VOLTAGE FUSE WITH ABLATING WALLS. Department f Electrical Engineering, Eindhven University f Technlgy (Netherlands), EUT Reprt 85-E-151 Addresses f the authrs: S. Ramakrishnan, Schl f Electrical Engineering, University f Sydney, NSW 2006, Australia. W.M.C. van den Heuvel, Department f Electrical Engineering, Eindhven University f Technlgy, P.O. Bx 513, 5600 MB Eindhven, The Netherlands.

5 -iv- Cntents 1. Intrductin Arc mde Energy-Balance Equatin Calculatin f Pressure Rise in the Tube Rate f Mass Ablatin Pressure Rise when Carrier and Wall Gases have equal Mass Densities Pressure Rise when Carrier and Wall Gases have different Mass Densities Cnvective Cling Jule Heating Radiatin Lsses Material Functins Numerical Methd Experiments Experiments with Current Steps Shrt-circuit Current Interruptin Studies Results and Discussin Initial Temperature Prfile Arc Behaviur fr Steady Arc Currents Fuse Behaviur under Shrt-circuit Tests Discussin Summary and Cnclusin Re ferences.... page

6 -v- List f symbls c p c pc C pw E g h w i L m m c m w p r w R t T T w u u a x K p heat capacity (J/kg) heat capacity fr carrier gas (J/kg) heat capacity fr wall gas (J/kg) axial electric field in the arc (V/m) arc cnductance (S) energy per unit mass f wall material t ablate and raise t wall temperature (J/kg) arc current (A) length f the tube f the fuse (m) inductance (H) mass per unit length (kg/m) rate f mass ablating frm the wall per unit length (kg/sm) mass f a carrier gas per unit length (kg/m) mass f wall gas per unit length (kg/m) pressure (bar) reference pressure = 1 bar (bar) energy per unit time and length f arc (w/m) transparent radiatin lss per unit length (W/m) radial crdinate (m) inner radius f tube (m) electrical resistance (0) time (s) plasma temperature (K) wall temperature (K) net radiatin emissin (w/m3) arc vltage (V) surce vltage (V) 3 transparent radiatin emissin value (W/m ) velcity f the mass (m/s) rati number fr wall gass, depending upn the relative masses in a mixture f tw gasses clsing angle thermal cnductivity (W/m K) 3 plasma density (kg/m ) 3 plasma density at p = 1 bar (kg/m ) mass density f carrier gas (kg/m 3 ) 3 mass density f wall gas (kg/m ) electrical cnductivity (S/m)

7 -1-1. Intrductin Prtectin f electrical equipment frm damage resulting frm electrical faults such as verlad and shrt circuit is ne f the majr factrs t be incrprated in the design f any electrical equipment r system. Electric fuses are widely used as prtectin devices and ffer advantages such as lw cst, simple design and cnstructinal feature, and current limiting capability. This reprt applies t a class f cartridge fuses, called "miniature fuses", which are nrmally used t prtect a single apparatus r instrument r a part f it. These fuses are used at a nminal vltage f 250 V and have typical uter dimensins f 5 mm diameter x 20 rm length. The present limit n the shrt-circuit current is 1.5 ka which is likely t increase in future. Present dimensins, interruptin ratings and test recmmendatins are cvered by lee-publicatin 127 [19]. These small fuses cnsist f a thin metallic wire f tinned cpper, silver r nickel stretched inside a tube made f glass, ceramic r suitable plastic. End cnnectins t the fuse wire inside the tube are prvided by means f tw metallic end caps, which make the fuse a ttally-enclsed prtective device. When the interrupting capacity f these fuses is as high as 1.5 ka, the tube is als filled with finegrain sand t absrb the arc energy liberated during the current interruptin prcess and als t absrb the mechanical energy generated as a result f high-pressure develpment during the vaprizatin and arcing f the fuse wire. The requirement f sand-filling inside the ~ube intrduces prblems in their manufacture thus increasing the manufacturing cst. One way t avid the necessity f sand filling inside the fuse is t make the tube f the fuse ut f a suitable plastic material with smewhat reduced inner diameter [1]. The plymer used t make the tube shuld then meet the fllwing tw imprtant requirements: (i) it shuld have sufficient mechanical strength t withstand the high pressure generated within the tube; and (ii) the vapur ablated frm the inner wall f the tube as a result f arcing inside the tube shuld have gd fl arc quenching" prperties, cmparable with thse f sand-filling.

8 -2- As fuses f this type perate under cnditins when gas is liberated frm the tube wall as a cnsequence f ablatin, these fuses can be called liablatin-dminated" fuses. Preliminary experiments described in this reprt indicate that it is feasible t cnstruct such ablatin-dminated fuses up t a nminal current rating f 3 A with a shr~-circuit rating f 1.5 ka. In rder t btain quantitative guide lines fr the design f ablatin-dminated fuses ver a range f currents, it is essential t understand the mechanism f current interruptin in the fuse and t be able t predict the pressure rise inside the fuse. Unfrtunately, there appears t be little published literature n the behaviur f ablatin-dminated fuses. Althugh there exist a few publicatins [2,3,4] n the behaviur f ablatin-dminated arcs, these are nt directly relevant in the cntext f fuse behaviur because these publicatins refer t arcing at very high current densities in tubes with pen ends thrugh which the plasma escapes t reduce the pressure rise inside the tube. This reprt attemps t develp a quantitative understanding f the fuse behaviur n the basis f a theretical mdel derived frm physical principles. OWing t the nn-linear nature f this timedependent prblem, a numerical slutin f the differential equatins describing the arc behaviur is cnsidered. The prblem cnsidered in this study deals nly with the arcing phase f the current interruptin prcess; the prearcing phase cnsisting f the melting f the fusing wire and its subsequent explsin is nt cnsidered. The study reveals that the prearcing phase which acts as the initial cnditin fr the arcing phase has a significant bearing n the interruptin prcess. 2. Arc mdel In a fuse f the type shwn in Figure 1, during a fault, the fuse wire initially melts, expldes if the current is high and then an arc clumn is established within the tube acrss the end caps. The flw f current thrugh the plasma results in jule heating within the arc and the heat is transprted by thermal cnductin and als by radiatin, if the plasma temperature is sufficiently high, t the inner wall f the

9 -3- ablating ~ wall arc clumn radius, r en~ cap fuse wire cap Figure 1 Fuse cnsidered in this study. tube. The wall material f the tube ablates and gets entrained int the arc clumn. As the fuse is a ttally enclsed tube, the additin f wall material increases the pressure inside the tube. The entrainment f wall vapur results in a thermal cnvectin and the cnvective flw is mainly in the radial directin twards the axis because the tube is cylindrically symmetric. This thermal cnvectin may be viewed as the energy required t elevate the wall vapur t the arc temperature frm its value adjacent t the wall. Thus, ablative cling f the arc clumn results. The prcess f arcing in a ttally enclsed tube is inherently nnstatinary in character. Even if the arc current is steady, cntinued ablatin f the wall results in ever increasing pressure inside the tube until, perhaps, a mechanical failure ccurs. On the theretical level, n steady-state slutin fr the prblem exists. A detailed mdelling f this type f arc shuld cnsider all three cnservatin equatins, viz. mass, mmentum and energy. As this is extremely difficult, in this study it is assumed that the mass liberated frm the wall has negligible inertia and hence distributes itself within the arc clumn instantaneusly. This assumptin is equivalent t assuming that the pressure in the radial directin at any instant f time is unifrm. This simplifying assumptin allws ne t discard the mmentum equatin. Further, as the tube is cylindrically symmetric, it is assumed that axial variatins in plasma prperties are negligible. Thus we seek slutins f plasma prperties in the radial directin nly.

10 -4- It is t be nted that at the instant the fuse wire expldes, the plasma is made up entirely f the carrier gas, which is a mixture f metal vapur and air. As time prgresses, the additin f wall gas changes the cmpsitin f the plasma and the rati f the mass f carrier gas t that f wall gas is a time-dependent functin. Calculatin f thermdynamic and transprt prperties f mixtures f gases at different temperatures and pressures is by n means simple. Hence, in this study, nly variatins in density and heat capacity as a functin f ga5- mixture rati are cnsidered at an apprximate level. These tw material prperties cntribute cnsiderably twards thermal cnvectin and the pressure rise inside the tube Energy-Balance Equatin Assuming that the plasma temperature T varies nly alng the radial crdinate r because f cylindrical symmetry, the energy balance equatin [5] fr the arc clumn can be written as at at 2 pc -;;-+Pc v-;;- =<JE + P t P r 1 r a (r K r at) ar - u (1) thermal thermal Jule strage cnvectin heating thermal cnductin radiatin where E is the axial electric field in the arc clumn, v the velcity f the plasma in the radial directin induced by the ablatin prcess as well as temperature changes and t the time. The material functins f the plasma are: P the plasma density in kg/m 3, c the heat capacity p in J/kg, <J the electrical cnductivity in S/m, K the thermal cnductivity in W/m K and u the net radiatin emissin in w/m3. The material functins are dependent upn temperature and pressure. The energy balance equatin (1) can be interpreted in simple terms as fllws: A fractin f the Jule heating as a result f electrical pwer input int the plasma is transprted by thermal cnductin; anther fractin is transprted as radiatin; and yet anther functin is expended in heating the ablated wall material t plasma temperature. The remaining is used up in raising the plasma temperature, as expressed in the thermal strage term in the equatin.

11 -5- In rder t slve equatin (1) tw bundary cnditins fr temperature are required. One f the tw bundary cnditins is at/ar (at r = 0) = 0 because the heat flux at the axis f a cylindrically symmetric arc shuld be zer. The secnd bundary cnditin is determined by the ablatin prcess at the wall. As the wall is cntinuusly ablating, the temperature at the wall shuld be equal t the vaprizing temperature f the wall material. It has been shwn by Kvitya and Lwke [3] that this temperature is apprximately 3000 K fr perspex and alumina. We, therefre, take the secnd bundary cnditin t be at r = r I T = T = 3000 K, where r and T are the radius f the tube w w w w and the temperature f plasma at the wall respectively Calculatin f Pressure Rise in the Tube. As the mass f gas inside an enclsed tube with nn-ablating wall remains cnstant, if the temperature f the plasma in the tube increases due t arcing, then the mass density f the plasma falls and hence the pressure increases. Similarly, if the temperature f plasma increases, the pressure decreases. If the tube wall ablates, then extra mass is added t the mass f gas already present in the tube and hence the pressure increases. The prcedure t calculate the pressure inside the tube at any instant f time shuld therefre cnsider pressure changes due t mass additin as a result f wall ablatin as well as thse due t temperature changes. The pressure changes due t bth these factrs can be estimated frm the mass cnservatin equatin, which is given by ap 1 a "t + (rpv) a r ar (2) In rder t use this equatin t estimate the pressure rise, we need t knw the dependence f density n pressure. This again intrduces difficulties because we need t knw the dependence f gas density as a functin f temperature, pressure and als the gas cmpsitin, which changes with time. Fr simplicity we assume that the density f the plasma is prprtinal t pressure. That is, if p (T) is the

12 -6- functinal dependence f plasma density n temperature at the reference pressure f P I which is taken as 1 atmsphere, then the density functin is given by p (p,t) p (T) (3) The abve equatin still des nt take int accunt the cmpsitin f the gas mixture inside the tube. This aspect will be discussed in subsectin f this reprt. In rder t illustrate the prcedure fr the calculatin f pressure rise, a simple case in which the carrier gas inside the tube has the same prperties as the wall gas ablated frm the wall is initially cnsidered. This illustrative prcedure is discussed in subsectin It is stressed that the arc mdel des nt use this illustrative prcedure; the mdel cnsiders the case when the carrier gas has different prperties t thse f the wall gas and this case is discussed in subsectin Rate f Mass Ablatin The rate ~ which the wall material is ablated and entrained int the arc clumn is determined by the rate at which energy is received by the wall frm the arc clumn and the energy required fr the wall material t vapurise. If q is the rate at which unit length f arc receives energy in W/m, hw the energy required by unit mass f wall material t ablate and raise t the wall temperature in J/kg and m the rate at which mass is liberated frm unit length f the wall in kg/s m, then q (4) The wall receives energy frm the arc clumn by means f thermal cnductin and transparent radiatin. Hence, q _ 2 ltr K at I r r == r w r w + I U 2lT r dr and can be estimated using the temperature prfile in the arc clumn. (5)

13 -7- The value f hw required t ablate the wall material and raise it t the wall temperature is nt knwn accurately. Niemeyer [2J has shwn fr bth plymer and ceramic materials, the value f hw lies in the range f 3 x 10 6 t 10 7 J/kg fr vapur temperatures in the range f 1000 t 5000 K. Kvitya and Lwke [3] used a value f 6.5 x 10 6 J/kg fr their studies n ablatin dminated arcs. In this study, the 6 value f h was taken t be 6.5 x 10 J/kg. w Pressure Rise when Carrier and Wall Gases have equal Mass Densities. Multiplying the mass cnservatin equatin (2) by 27f r dr and integrating frm r = 0 t r = r w' we get = - 2 7frc,. (6) The right-hand side f the abve equatin (6) is equal t the rate at which mass is crssing the bundary at r : r at any instant f time w and hence shuld be equal t the rate f mass liberatin frm the wall which is given by equatin (4). Thus, we have frw ap a;: 27frdr (7) As we have assumed in this case that the density functins fr the carrier and wall gases are ne and the same, we can estimate the pressure rise in the tube frm equatin (7) in a rather simple way_ Using the assumptin that density is prprtinal t pressure given by equatin (3), we get frm equatin (7) ~ dt r f w p (T) 27f r dr r f w ap (T) at p ~ 27f rdr r w f p (T) 27f r dr (8) where p is in bars and the reference pressure p 1.

14 -8- The abve equatin (8) gives the rate at which pressure changes within the tube and can be integrated t btain the pressure at any instant f time t. The first term n the right-hand side f equatin (8) gives the pressure rise due t mass additin, while the secnd term gives the pressure change due t temperature changes in the plasma within the tube Pressure Rise when Carrier and Wall Gases have different Mass Densities. The carrier gas in an enclsed fuse will be a mixture f cpper vapur (r ther metallic vapur),and air. It is very unlikely that the mass density f the wall gas will be the same as that f the carrier gas. As it is difficult t evaluate mass densities f cmplex gas mixtures, we use an apprximate averaging prcedure t calculate the density f the cmpsite plasma cnsisting f carrier and wall gases. We define the density functin P (T) f the cmpsite plasma at a reference pressure P f 1 bar as fllws: P (T) P (T) + xp c w 1 + x (T) (9) where P tt) and P (T) are the density functins at the reference pressure c w f the carrier gas and wall gas respectively and x is a rati which depends upn the relative masses f the tw gases. It can be seen that when x = 0 r when there is n wall gas, the density functin f the cmpsite plasma is equal t the density functin f the carrier gas. If the mass f wall gas is large in cmparisn with that f the carrier gas, then x is large and P (T) ~ P (T). Hence the fractinal densities f the carrier w and wall gases are respectively p (T) / (1+x) and xp (T) / (1+x). c W The value f rati x at any instant f time can be evaluated by frcing mass cnservatin fr the tw species individually. That is, if m and c m are,respectively,. the masses f carrier and wall gases per unit length w f the plasma clumn at time t, then we require: carrier gas m c P (T) c 1+x 211 r dr (10)

15 -9- r w XP (T) w wall gas mw ~J l+x Cmpsite gas m m + mw = r 0 p c 0 0 2n r dr (11 ) p (T) 2" r dr 0 Frm equatins (lo) and (11) we can btain an expressin fr x which is (12) x m C r J w P w (T) 2" r dr (13) Knwing the temperature prfile in the plasma at time t and als the masses f carrier and wall gases at the same instant, the value f x can be calculated. The pressure within the tube at any instant f time, can be estimated frm the ttal mass cnservatin fr the cmpsite plasma given by equatin (12). We get fr pressure the expressin given belw: p = ( 14) rw P (T) + f { c 1 + 2" r dr ~~?nvective Cling The entrainment f the ablated wall material int the arc clumn results in a radial cnvectin directed frm the wall t the axis f the tube. This cnvectin results in a cling f the uter regins f the arc clumn and heating the inner regins. Or, it can be viewed as the energy absrbed by the wall gas in being raised frm the value f the temperature at the wall t the plasma temperature. A detailed study f this cnvectin requires the inclusin f the mmentum cnservatin equatin. Fr simplicity, we have assumed that the distributin pressure is unifrm acrss the tube and ignred the mechanical inertia f the fluid. This assumptin frmed the basis f ur pressure calculatin which used nly the mass cnservatin equatin.

16 -10- Using the mass cnservatin equatin (2), we can estimate the value f pv at different radial psitins f the plasma inside the tube. If the carrier and wall gases have equal mass densities, then the mass cnservatin (2) can be directly integrated frm r = 0 t r = r', the required radial psitin, t give the integral given belw: (pv) I r = r' 1 r' (15) The abve expressin is nt readily usable when the carrier and wall gases have unequal mass densities because the value f x als changes with time. Hence, we use the fllwing integral which deals with mass changes up t a given radius: (pv) I = r = rl 1 r' ( ""t{t rdr} (16) Fr r I = r w ' the abve integral gives: = as required by equatin (7) Jule Heating The lcal jule heating f the plasma is cre 2 as shwn in equatin (1) and is determined by the electrical pwer input int the plasma. The electrical pwer absrbed by the arc clumn is derived frm an electrical netwrk external t the arc clumn. The external circuit may be cnfigured t either impse a vltage acrss the arc clumn r drive a current thrugh it. In ther wrds, if the vltage acrss the arc clumn, u, is specified a by the circuit, we shuld be able t slve the energy balance equatin and slve fr the current thrugh the arc clumn. Or, if the current, i, thrugh the arc clumn is specified by the external circuit, then we shuld be able t estimate the vltage acrss the arc clumn. In this study, as the external test circuit is inductive in nature, it is easier

17 -11- t cnsider the current as the state variable fr the circuit and hence we can take the current i thrugh the arc clumn t be determined by the test circuit. Fr a specified value f arc current we can calculate the axial electric field E in the arc clumn using Ohm's law, which gives: E = i (17) a 2rr r dr As the arc clumn is assumed t be cylindrically symmetric, the electric field is unifrm axially and the arc vltage u can be calculated frm a u = E ~ a (18) where ~ is the length Of the tube f the fuse Radiatin Lsses As the investigatin discussed in this reprt was based n an heuristic apprach t determine the heat-lss mechanisms in the arc clumn during the current interruptin prcess in a fuse, the radiatin lss term was included in the energy balance equatin. Hwever, the calculatins shwed that if the arc temperature was as high as K fr the radiatin lsses t be dminant then the fuse wuld nt interrupt the current at all. This sectin, therefre, has been included in this reprt nly fr the sake f cmpleteness f the prblem under cnsideratin. A detailed treatment f radiatin lsses within an arc clumn is very cmplicated and requires integratin f the spectral radiative intensity ver space and wave length [6, 7, 8 J. Such treatments have been undertaken in simple plasmas cntaining nly Nitrgen, Argn r SF 6 In the case f ablatin dminated arcs, the prblem is even much mre cmplicated wing t the presence f metal vapur and wall material in the plasma. In a treatment f arcs in nzzle flws, Tuma and Lwke [9J use a simple prcedure by using net emissin f radiatin u at the arc axis. Since the value f u can be negative in the uter regins f the arc clumn wing t strng self-absrptin f radiatin, they used fr transparent radiatin a value u which was taken t be 0.1 U fr nitrgen plasma. It has been t shwn that fr air plasma cntaining cpper vapur r ther metallic vapur, the value f ut/u can be as high as 0.3 [10,11]. The mdel used by Tuma and Lwke [9] was an integral mdel with the temperature prfile in the radial directin assumed t be flat. Cnsequently, it was simple

18 -12- t intrduce the apprximatin that the transparent radiatin lsses are nly a small fractin f the radiatin lsses inside the arc clumn. In a tw-dimensinal treatment f free-burning arcs [12] and arcs in frced cnvectin, the values f u were arbitrarily made negative upt a certain radius and zer beynd s as t make the transparent radiatin lsses zer. In this study, this treatment has been refined further s that the functin f radiatin escaping the arc clumn as transparent radiatin can be specified as an input parameter. The values f net emissin cefficients u fr temperatures greater than K are psi ti ve and have been published fr ni trgen plasma [13]. Hwever, strng self-absrptin f radiatin takes place at a radius where the temperature in the plasma falls belw K. In this regin, the value f u is usually negative. This study chses the negative value f u in such a way that the fllwing cnditin is satisfied: = f, a specified fractin (19) Ju 2n rdr In the abve expressin, q d represents the transparent radiatin lsses ra escaping the arc clumn and is given by: r f wu 2n r dr (20) Fr the plasma inside the fuse, cntaining a cnsiderable amunt f metal vapur, the value f f was chsen t be 0.3 [101. The values f u were taken t be equal t thse f nitrgen frm the measurements f Ernst, Kpainsky and Maecker [131. The value f u f Ernst et a1. crrespnds t a pressure f 1 bar. Extraplatin f these values t higher pressures has been made in this investigatin by assuming that u is prprtinal t pressure alng the lines suggested by Kvitya and Lwke [3J.

19 Material Functins The plasma inside the tube f the fuse cnsists f a mixture f carrier gas and wall gas. N attempt has been made in this study t calculate material functins f the cmpsite plasma as a functin f temperature, pressure and cmpsitin. Fr simplicity, we assume that nly the mass density f the plasma is prprtinal t the pressure inside the tube. ther material functins, viz. heat capacity, thermal cnductivity and electrical cnductivity are assumed t be pressure independent. The carrier gas is made up f metal vapur frm the fuse wire and air. As material data fr such mixtures is rare, we have taken the fuse wire t be made frm cpper. Fr a 100 micrns diameter wire in a 3.2 mm diameter tube the mass rati f cpper t air is nearly 8. The values f density, heat capacity, thermal cnductivity and electrical cnductivity fr apprximately this mass rati at a pressure f lbar have been taken frm the publicatin f Shayler and Fang [14J. It is assumed that the transprt prperties, viz. thermal and electrical cnductivities fr the cmpsite plasma, are the same as thse fr the carrier gas as a first-rder apprximatin. It can be seen frm the energy balance equatin (1) that the mst dminant effect f ablatin results frm the thermal cnvectin term which is determined by the density and heat capacity f the cmpsite plasma. Hence, the density f the cmpsite plasma is estimated alng the lines discussed in sub-sectin frm the density functins f the carrier and wall gases. The heat capacity f the cmpsite plasma is estimated by summing ver the tw cmpnent gases [Shayler and Fang, 14l. the prduct f heat capacity and the mass fractin f the tw cmpnents. That is, the heat capacity C f the cmpsite plasma is given by p c p P I (1+x) c C P pc + c pw (21 ) where C and C are the heat capacities f carrier and wall gases pc pw respectively. The values f mass density and heat capacity fr wall gases f Perspex and Tefln have been taken frm the publicatin f Kvitya [15J.

20 2.7. Numerical Methd -14- An explicit scheme using finite-differences has been used t slve the energy-balance equatin (1). The regin frm r = 0 t r = r is divided w int (N-l) equal intervals, each interval having a width f ~r, t give N grid pints at which the temperature as a functin f time is evaluated. Discretizing the slutin in time by chsing an apprpriate small time step 6t, we seek the value f temperature at all grid pints '+1 ' i = 1 t N at time t J = t J + l>t knwing the temperature distributin at t = tj. That is, we attempt t estimate frm the energy-balance equatin changes in temperature, l>t~ at time t = t j fr all grid pints i = 1 t N. '+1 1 Then, at t = t J we get the temperature at all grid pints frm (22) The value f T j at all grid pints can be estimated frm the energyi balance equatin (1) by expressing the equatin in a finite-difference frm. The finite-difference algrithm used is based n the fllwing frmulae: 2 r j K j T,j - l>t l>t? [a?(e j 1 1 ) i i +- 1 pj j 1 r, l>r c 1 1 Pi r j _ i 1 K j T,j i-i i-i + (23) fr all i = 2 t (N-l) where T' j i T' j 1 is given by = j Ti T - T? 1 1 l>r fr all i 2 t (N-l) The bundary cnditins require that T j = T j because at I = 0 and T j 1 2 ar r = 0 T, the wall temperature. N w rhe abve frmulae (23) and (24) are used tgether with suitable integratin prcedures (a) t slve fr the pressure within the tube (equatin 14), (b) t determine the cnvectin term (equatin 16), (c) t estimate jule heating (equatin 17), (d) t accunt fr self-absrptin f radiatin t evaluate transparent radiatin lsses and (el t link with the external circuit. The material prperties at all grid pints are interplated frm reference tables.

21 -15- Fr calculatins, twenty-ne radial grid pints r twenty intervals were chsen. The accuracy f the slutin was checked by dubling the number f intervals. The value f time step ~t required fr the numeric&l algrithm was chsen t satisfy the vn Neumann criterin [16] s that stable slutins culd be btained. The numerical algrithm described by the frmulae (equatins 23 artd 24) is basically an integratin prcedure in time in which we march frward in time t seek slutins f temperature distributin as a functin f time, starting frm a set f initial cnditins. The set f initial endi tins crrespnds t a time t = t when the fuse wire expldes t e establish an arc clumn inside the fuse. We need t specify the pressure p, the mass f carrier gas me' the current i and the temperature distributin at t = t The mass f carrier gas was taken t be equal e t the sum f the masses f the cpper fuse wire and the surrunding air wi thin the fuse. The value f pressure at t = t was estimated frm the e mass f carrier gas and the temperature distributin at t = t. The e required cnditins n current and temperature distributin are discussed in sectin Experiments Preliminary experiments aimed at estimating the influence f wallablatin prcess n arc behaviur in ttally-enclsed tubes and n current-interruptin prcess in fuses were cnducted. These experiments als prvided a basis fr the chice f suitable initial cnditins fr the theretical mdel and its refinement during the theretical develpment. The prcess f arcing in fuses during current interruptin is inherently nn-statinary, nt nly because the arc temperature is nt steady but als because f the wall-ablatin prcess which makes the pressure in the fuse t rise even if the temperature is cnstant. In rder t gain an imprved understanding f the phenmenn arcing in fuses, the fllwing tw types f experiments were cnducted: (i) Fuses made frm different wall materials were tested in a 50 Hz pwer circuit fr shrt-circuit duty; these experiments retain the tw nn-statinary prcesses arising frm unsteady current as well as the wall ablatin. (ii) The respnse f vltage arcs in ttally-enclsed tubes made frm different materials t a step change in arc current was determined; these experiments characterize the nn-statinary feature arising frm wall-ablatin alne.

22 Experiments with Current Steps Experiments were cnducted with tubes 50 rm in length made f perspex, tefln, p.v.c. and glass in a pwer surce circuit develpment by Daalder [17]. The surce circuit was mdified t prduce flat current pulses f current level in the range f 30 A t 100 A lasting fr nearly 5 ms. Details f the experimental set-up and measurements are discussed elsewhere [IS). A typical experimental sht cnsisted f lading the tube with a cpper wire f 100 ~m diameter and hlding the tube between tw fixed blcks f brass which served as electrical cnnectins t the wire inside the tube and als as seals t prevent plasma escaping the tube. The current thrugh the wire was then initiated. Tests were cnducted with tubes f different inside diameters in the range f 2 mm t 10 mm. The variatin f the arc vltage with time was recrded using an scillscpe. It shwed the fllwing features (Figures 2 and 3) : ~~} trace l~r} trace u c = 3 kv, arc vltage 500 V/divisin arc current 75 A/divisin perspex tube 4 rm ~ a 1 a Figure 2 Typical Recrds f Vltage and Current btained in the current-step study.

23 -17- '" " Regin Regin time t Figure 3 Typical time-variatin f arc vltage in an enclsed tube after the explsin f the fuse wire. (i) The melting and subsequent vaprizatin f the fuse was characterized by a steep rise in the vltage acrss the tube (ii) fr a certain duratin ( ms) (Regin 1 in figure 3) after the explsin f the fuse wire, the vltage exhibited randm variatins with its mean value nearly cnstant r falling with time~ and (iii) after this duratin f smewhat randm behaviur, the recrded vltage was smth and decreased with time (regin 2 in figure 3); the rate f fall f vltage was fund t decrease steadily until the vltage reached nearly a cnstant value after 1 r 3 ms. It was inferred frm the experiments that regin 1 in figure 3 crrespnding t the fluctuating vltage trace crrespnds t the perid f establishment f an arc filling the tube frm the cnditins immediately after the explsin f the wire. Experiments with tubes f different materials and inside diameters shwed that (i) the vltage recrded fr the perspex tube was the highest; the ther materials in the rder f decreasing magnitude f vltage are: p.v.c., tefln and glass.

24 -18- (ii) the magnitude f vltage decreased if the tube diameter was increased. These experiments shw that the establishment f a well-defined arc clumn within a tube requires a certain duratin even when the impsed current is steady and that the ablatin f wall material has a significant influence n the arc vltage. Further experimentatin t investigate the prcess f arc establishment in the tube and t measure the pressure inside the tube is essential t gain a better understanding f the behaviur f arcs in enclsed tubes and fuses Shrt-circuit Current Interruptin Studies Experiments were als cnducted using fuses f standard uter dimensins in a test circuit built n lee recmmendatins [19] t deliver a prspective shrt-circuit current f nearly 1500 A (Figure 4). Althugh switch L = 0.3 mh u g surce clsing angle ~ with respect t surce vltage R fuse cnductance g i + 247V 50 Hz R shunt fuse vltage 11,56 mq fuse current Figure 4 Test circuit used fr current-interruptin study. the recmmendatin f the lee n the clsing angle fr initiatin f the circuit current relative t the surce vltage is in t h e range f 25 0 t 0 35, the clsing angle was varied frm 10 t 45 t investigate the severity f interruptin duty. The current thrugh the fuse and the vltage acrss it during shrt-circuit current interruptin were recrded using a cmputer cntrlled digital recrding system.

25 -19- Initial experiments cnducted using fuses made f a certain plymer material shwed that a fuse with an inside tube diameter f 3.2 rm and a cpper wire f 100 micrns in diameter, interrupted the current successfully fr a clsing angle f When the diameter f the fuse wire was increased t 200 micrns, the fuse failed t interrupt the current. Inspectin f the fuse after the test revealed that heavy arcing inside the fuse tgether with the high pressure had created large hles n the end caps thrugh which the plasma inside the tube had escaped. Further experiments were cnducted with fuses having a cpper wire f diameter f 100 micrns. Fuses made frm perspex, tefln, p.v.c. and ther materials were tested. It was fund that at a clsing angle f 29 0 the interruptin appeared t be critical. Fr example, the fuse made f p.v.c. develped a small hle n ne f the end caps. Other fuses shwed a small dimple n their end caps which might have resulted in the develpment f a hle. When the clsing angle was increased t 45, all the fuses failed. The failure was nt evident frm the current and vltage traces recrded during the test, but was due t the failure f the end caps. But, the fuse made f tefln was fund t explde during the test. The tests appeared t shw that when the fuses cleared the fault current, the current recrds were nt significantly different fr different tube materials. When a failure ccurred it was mainly due t a mechanical failure resulting frm arcing at the end caps and high pressure inside the tube. It was als fund frm tests with different clsing angles that if the cut-ff current fr a particular clsing angle was high due t high initial rate f rise f current, then the fuse failed t clear the current. This result shws that the cut-ff current r the current at which the fuse wire melts, vaprizes and expldes has a significant bearing n the subsequent arcing prcess during current interruptin; the higher the cut-ff current the mre severe is the shrt-circuit duty. 4. Results and Discussin The arc mdel presented in chapter 2 was used t calculate arc prperties and t predict interruptin behaviur and the calculatins were related t the experimental results discussed in sectin 3.

26 -20- As mentined previusly in sectin 2.7, the calculatin prcedure relies upn specifying apprximate initial cnditins fr the prblem. The imprtant initial cnditins are the current at which the fuse wire expldes and the temperature distributin within the arc clumn immediately after the explsin f the fuse wire. In the case f the shrt-circuit test circuit using a 50 Hz pwer surce, the current at which the fuse wire expldes is nt merely determined by the fuse wire but is related t circuit parameters as well. Further the dynamic interactin between the circuit and the arc inside the fuse makes the prblem cmplicated. Hence, initial calculatins were made fr the experiment using current steps. In this case, as the current remains nearly cnstant, the interactin f the arc with the circuit can be ignred and the current becmes a specified parameter fr calculatins. The chice f an apprpriate distributin fr temperature at the instant the fuse wire expldes, was made by cmparing the calculatins fr different initial distributins with experimental results f arc vltage Initial Temperature Prfile. N measured temperatured prfile fr the plasma inside the tube during the time immediately after the explsin f a fuse wire in an enclsed tube has been reprted in the published literature. Hence, an heuristic apprach was fllwed by chsing three different temperature prfiles. All these prfiles were chsen t satisfy the required bundary cnditins at radii 0 and rw. The three chsen prfiles are shwn in Figure 5 (t. 0) and can be seen t have temperatures larger than 3000 K fr cmputatinal cnvenience which depends upn the availability f material prperties f the plasma. Hwever, the energy required t raise the plasma within the tube t the temperatures shwn in the figure fr t = 0 is less than 0.5 J which means at a current f 100 A and a vltage f 1000 V immediately after the explsin f the fuse wire, it wuld take nly 5 ~s t heat up the plasma t the initial temperature distributin. Experiments n fuses <subsectin ) shw that typical interruptin times are 300 ~s r mre and the ttal energy absrbed during the interruptin prcess is 10 J r mre. As the values f time and energy

27 -21- required t heat the plasma t the assumed initial temperature are small, the errrs invlved in the cmputatinal prcedure are likely t be small. The three assumed initial temperature distributins are given belw: (1) Elevated cre, Fig. 5(a): The arc is assumed t have a thin cre near the axis with the cre temperature at 12000K. Calculatins shw that if the cre temperature is assumed t be lwer, then wing t Jule heating the temperature wuld rise rapidly t high temperatures. (ii) Unifrm, cre, Fig. 5(b): The temperature prfile is assumed t be unifrm radially with a value f 4000 K. (iii) Elevated Wall, Fig. 5(c): The temperature is assumed t be unifrm at 3000 K everywhere except near the wall where the temperature is raised t 4500 K. This elevated temperature gives larger electrical cnductivity near the wall and can als be viewed as an enhancement f electrical cnductivity near the wallwing t the presence f cpper r metal vapur near the wall rather than an elevated temperature. In the case f the assumptin f elevated cre (Figure 5(a», the temperature at the axis f the arc increases initially and als the prfile bradens t accmdate the impsed steady current which prduces

28 -22- ls000r , r , , ms ms ms (a) Case (i) Elevated cre 5000 t = 0 /" I I I g- ~ f< (]) -..., " "'" QJ!l' QJ f< 0.5 ms 0.3 ms ~ 5000 t- 0.1 ms "'- 0 I t 0 I (b) ease (ii) ~ Unifrm I ms ( c) 500 Case (iii) Elevated Wall Radius, r (em) Figure 5 Calculated Radial Distributin f Temperature in a 4. mm Diameter x S. mm Lng Perspex Tube fr a Step Current f 7SA.

29 -23- intense jule heating within the cre. The temperature at the axis then begins t drp, but the prfile cntinues t braden. The bradening f the temperature prfile results in an increase in the cnductance f the arc clumn and hence the vltage acrss the arc drps as shwn in Figure 6 (a) i. The rate at which the bradening f the temperature prfile ccurs drps as time prgresses because the ablated mass frm the wall nt nly cls the uter bundary but als increases the mass and hence the thermal inertia f the arc clumn. The cmparisn f the calculated time variatin f vltage with the experimental results fr a 75 A arc clumn in a 4 mm diameter x 50 mm lng perspex tube shws that the value f vltage is smaller than the measured value. In this case f assumptin f elevated cre, the ablatin f the wall material is mainly due t the transparent radiatin frm the arc clumn which is small as shwn by Figure 6(b). Cnsequently the arc cling as well as the pressure rise are small. The vltage predicted by this initial distributin is smaller than the measured ne because f an under-estimatin f the ablatin prcess. It is therefre cncluded that this temperature prfile is a very unlikely cnsequence f the explsin f the fuse wire. Calculatin f shrt-circuit behaviur f fuses using this temperature prfile als predictsthatthe fuse wuld fail t interrupt the current crrespnding t test cnditins f succesful experimental interruptin. The assumptin f unifrm temperature distributin within the plasma initially prduces an imprvement in the predicted vltage (Figure 6\a)ii), but again appears t under-estimate the ablatin prcess (Figure 6(b)ii, As ablative cling f the plasma is effective nly in the uter regins f the plasma, the temperature in the inner regins increases thereby increasing the cnductance f the arc. The predicted vltage, therefre, is smaller than the measured value. Fr wall ablatin t prvide arc cling as effective as shwn by the recrded vltage fr an impsed steady current, it is very likely that mst f the jule heating f the plasma immediately after the explsin f the fuse wire ccurs in the uter regin f the arc clumn. This mde f heating the plasma may ccur if cnsiderable amunt f cpper vapur is present in the uter regins near the wall f the cnfining tubes. If the jule heating is present nly in the uter regins, then the temperature in uter regins rises while the inner regin remains cler until heat diffuses t the inner regin as shwn in Figure 5(c).

30 -24- Initial Cnditins fr Temperature (i) Elevated cre (ii) Unifrm (iii) Elevated Wall Experimental Results

31 -25- This figure crrespnds t the case f elevated-wall temperature. The vltage estimated using this initial temperature distributin cmpares favurably with the experimental result fr the regin 2 f Figure 3. Regin 1 in Figure 3 which crrespnds t the establishment f a welldefined arc clumn, hwever, is nt explained by this assumptin n initial temperature distributin. Preliminary investigatins [18] using framing phtgraphy at frames/secnd f the initial phase f arcing in an enclsed fuse shw that the arc resides near the wall f the tube in the frm f a cre initially and after a certain perid an arc clumn which fills the tube is established. This preliminary experimental test result tends t supprt the idea that the jule heating ccurs mainly in the uter regins f the tube initially. In reality, the jule heating is mre lcalised than what has been assumed because the initial distributin with elevated wall temperature cnsiders heating all arund the inner wall. A treatment f this lcalised cre near the inner wall f the tube immediately after the explsin f the fuse wire is cmplicated and requires slutin f the energy-balance equatin in tw dimensins, viz. radial and azimuthal crdinates. As a first-rder apprximatin t this cmplicated prblem, we use the initial temperatue distributin t be ne f elevated-wall type fr the predictin f fuse behaviur Arc Behaviur fr Steady Arc Currents Calculated results using the mdel fr a 75 A arc in a 4 mm diameter x 50 mm lng tube are shwn in Figure 7(a) and (b). The results f time variatin f vltage after the explsin f the fuse wire fr tubes made f perspex, tefln and nn-ablating material shw that perspex has the highest vltage fllwed by tefln. A tube whse wall des nt ablate results in the smalles vltage. This result is cnsistent with experimental results. Figure 7(b) shws the calculated variatin f pressure inside the tube with time. The pressure inside a perspex tube is higher than that in a tefln tube because perspex vapur has a lwer density and additin f lwer density wall gas t the carrier gas results in a reductin f the carrier gas density at the reference pressure f 1 bar. The pressure in a tube f nn-ablating material als increases because f

32 : "'" <D...,.-l '" ~ u <t; '" '0> 1000, mm diameter x 5111 ~ /' mm ln2 tubes 75A current step..,,.' ~ (a) - ~ :-~ ~~ tefln -::::::-- ~ --==---- n ablatin time, t (ms) (after fuse wire explsin) ~~--~--~--~--~~--~--~--~~ Ul '" :9 '" <D Ul "'" Ul <D '" 150,0 100,0 50,0 --,,/" ""'-;efln -- n ablatin (b) 0.5 time, t (ms) 1.0 (after fuse wire explsin) Figure 7 Calculated dependence f time-variatin f arc vltage and pressure n wall material fr an impsed current step f 75A.

33 -27- the rise in temperature inside the tube. It is likely that the calculatins verestimate the pressure inside the tube because we have assumed that the whle f the circumference f the arc gets heated and prduces ablatin whereas in practice the heated zne might be mre lcalised near the wall f the tube Fuse Behaviur under Shrt-'Circui t Tests In rder t predict the behaviur f fuses under shrt-circuit test cnditins, the arc mdel described in sectin 2 has been linked with the equatins describing the test circuit shwn in Figure 4. The necessary circuit equatins are given belw: i = il + ir ( 25) L d' ~L u - irl - i R -u (26) dt g shunt a and d' ~L i L = ir R2 u = dt a g (27) where the surce vltage U g is given by u = u sin (wt+l/j) g m (28) ~being the clsing angle and g the arc cnductance. Befre clsing the switch in the test circuit, the circuit currents are zer which are taken as the ini tial cnditins. The inclusin f the test circuit which prduces 50 Hz current with superimpsed transients intrduces the prblem f finding the current and the instant at which the fuse wire expldes. The circuit equatins (25) - (28) were initially slved numerically withut the inclusin f the arc mdel until melting f the fuse wire ccurs. The initial value f cndunctance g f the fuse wire was estimated frm the dimensins f fuse wire. The value f g was updated at each time step f integratin f equatin (26) by calculating apprximately the temperature f the fuse wire and using the temperature cefficient f the resistance f the fuse wire. The instant at which the i 2 t value reached a value crrespnding t the i 2 t value fr melting estimated using the Mayr's cnstant was taken t give the instant f melting f the fuse wire. The circuit current at the instant the fuse wire melts, gives the cut-ff current i. c

34 -28- The value f the cut-ff current and the instant f mel':in9 we"re used as the initial cnditins fr the arc mdel. Frm this instant f melting f the fuse wire, the circuit equatins were slved in cnjunctin with the arc mdel. The temperature distributin in the arc clumn at the instant f melting f the fuse wire was taken t be f the elevated wall type. At each time step f numerical calculatin, the value f current given by the circuit equatins was used in the arc mdel t update the temperature distributin in the arc clumn, the cnductance f the fuse and the vltage u acrss the fuse. Using the updated value f a arc cnductance the arc current i was updated using the circuit equatins. This prcess was cntinued t estimate fuse current, fuse vltage, pressure inside the tube and the temperature distributin in the arc clumn, all as a functin f time. There was, hwever, a difficulty in the determinatin f the exact initial cnditin fr the current at which the fuse wire explded. The cut-ff current estimated frm i 2 t values was fund t be higher than the value btained experimentally. This discrepancy was due t an versimplificatin in this study f the melting prcess in the fuse wire. Calculated results (Figure 8) f current and vltage f the fuse as a functin f time are cmpared with the experimental results fr a fuse made f perspex tube f inner diameter 3 mm and having a cpper fuse wire f 100 micrn in diameter. Case (1) the cut-ff current is estimated frm crrespnds t the case where 2 it-values. As the cut-ff current in case (i) is higher than the value btained in the experiment, it is fund that intense arc heating ccurs immediately after the explsin f the fuse. The arc current drps, but nt significantly enugh. Calculatins shw that at the end f 0.6 ms after the initiatin f the current thrugh the fuse, the current is still nearly 60 A and the pressure is nearly 190 bars. It is very likely that a fuse under these cnditins wuld fail mechanically, thus failing t interrupt the current. In case (ii) we have taken the value f the cut-ff current t be 185 A as btained in the experiment. In this case, the fuse appears t interrupt the current. Results were als btained fr current interruptin by a perspex fuse with a cpper wire f 100 )lm in diameter at a clsing angle f 10. In this case, as the initial rate f rise f current thrugh the fuse wire is lwer than the value at a clsing angle f 29, the cut-ff