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Temperature distribution of fuse elements during the pre-arcing period
Electric Power Systems Research 61 (2002) 161– 167 www.elsevier.com/locate/epsr
Temperature distribution of fuse elements during the pre-arcing period C.S. Psomopoulos *, C.G. Karagiannopoulos Department of Electrical and Computer Engineering, Di6ision of Industrial Electric De6ices and Decision Systems, National Technical Uni6ersity of Athens, 9 Iroon Polytechniou Str., Zografou, Athens, GR-15773, Greece Received 13 March 2001; accepted 13 June 2001
Abstract This work models the function of medium and low voltage fuses so as to estimate the increase in temperature across the fuse elements during nominal current operation for different types, diameters, element lengths and currents. The power balance in a thin metallic conductor, which simulates the fuse element, was derived and the resulting differential equation solved analytically. The theoretical results were close to the experimental measurements that we made, and the experimental work of other authors. The mathematical model that was developed will be useful in the design of fuses, as the computation of the axial temperature distribution across the fuse elements for the different parameters mentioned above is relatively simple. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Fusible elements; Temperature distribution; Pre-arcing period
1. Introduction In electrical installations of the low and medium voltage distribution network, fuses have been in use for over 100 years and their behavior has been adequately studied. They generally operate either below their nominal current or momentarily under excess, or short-circuit, currents. During operation under nominal current or less, the Joule heating produced on the fuse element dissipates to the surrounding area of the element, and thermal equilibrium is attained after a period of time [1 – 4]. For operation under excessive or heavy fault currents, the design of fuses is based on the well-known fundamental principle that they must interrupt those currents in a very short time. The phenomena developed are briefly as follows. The rapid rise of the temperature due mainly to Joule (I 2R) heating, which is aided by the resulting resistance increase, continues until the melting point is reached. The latent heat of fusion is produced gradually by the current during the * Corresponding author. E-mail addresses: [email protected] (C.S. Psomopoulos), [email protected] (C.G. Karagiannopoulos).
melting time. During the short times here in question, gravitational forces do not play any role, and thus the now liquid metal will remain in place and be heated further until the material is completely vaporized [2,3,5]. When the material of the element vaporizes, an electric arc is struck between the remaining solid parts of the element. This dynamic process causes a rapid temperature increase, while the current decreases rapidly until its flow is interrupted. The fundamentals of fuse operation have been described extensively by P.G. Newbery, A. Wright, E. Jacks, R. Ru¨denberg, D.R. Barrow, T. Chicata et al. [1–7]. Present research activities are mainly focused on the arcing period, and a number of models have been developed to simulate the fundamental operation. This operation is well understood, in general, but all the physicochemical processes during the arcing period are not so completely known [8–10]. However, some models have been developed which can simulate fuse operation under fault current during the arcing period [1,6 –10]. This work attempts to model the operation of medium and low voltage fuses carrying nominal current, so as to estimate the temperature across the fuse, considering the geometric characteristics and the material of the element as well as the current through the
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C.S. Psomopoulos, C.G. Karagiannopoulos / Electric Power Systems Research 61 (2002) 161–167
fuse. The relevant papers in the field of the pre-arcing period of fuse elements have been published by A. Wright and P.G. Newbery [1,3] and by M.S. Agarwal et al. [11]. They examine strip form elements and cylindrical fusible elements, respectively. Regarding the processes adopted, they both use finite differential methods for the development of their models. This paper proposes the development of a new model, which is confirmed by laboratory observations and results. Moreover, it attempts a comparison with the results of the model proposed by M.S. Agarwal et al. and solved with the Crack–Nicolson implicit finite difference method. The M.S. Agarwal et al. model examines the arrangement of fuse elements of two different cross-sections, whereas the model proposed hereafter considers fuse elements of even diameter, as is the condition in most cases that fuses are applied. 2. The proposed fuse model Fuses consist of one or more metallic conductors, known as the fuse element, and usually have a cylindrical or flat strip form. The surrounding media is granular silica quartz (SiO2) in high breaking capacity (HBC) fuses, and boric acid in expansion fuses. The element and the surroundings are housed in a body of insulating material (ceramic, fibre, melamine, etc.). The fuse element’s edges are usually soldered to, or in electric contact with, the fuse end caps; sometimes the connection is by a spring. In this work, fuses of low and medium voltage were simulated by the following simplified model. The element is considered to be a thin metallic wire or strip and the surrounding medium a liquid. Two conductors with much greater cross-section (compared with the fuse element’s cross-section) are in contact with the edges of the thin metallic wire (Fig. 1). This simplification seems to simulate the real conditions, because after the fuse’s end tags and fuse-carrier’s contacts, the circuit continues either with the proper cross-section conductors that result from the current specifications, or with conducting bars [VDE0670]. In both cases, the metallic cross-section of the fuse’s end tags, the fusecarrier’s contacts and the conductors or bars, is much greater than the fuse element’s cross-section. The simplified model proposed is illustrated in Fig. 1, where, for the purposes of symmetry, the center of the fuse element corresponds to x = 0.
Fig. 1. Schematic illustration of the model. (1) Conductor, (2) fuse element.
From the above, it can be assumed that the fuse end caps and tags, because of their great volume and, consequently, their external surface area (compared with the fuse element), operate as sinks for the heat that comes from, and through, the fuse element. In practice, for nominal operating conditions, it is true that the temperature in the fuse’s end tags, fuse-carrier’s contacts and conductors or bars, is the same as the temperature that would have existed, if the element was not interposed. This is particularly true in medium voltage installations where the cross-section of the bars, which result from calculations under short-circuit conditions, is far greater than the fuse element’s cross section. Therefore, the temperature rise above the environment temperature is only a few degrees under nominal operating conditions. Of course, that implies the stationary contacts between the fuse and the fuse-carrier operate within the normal standards [VDE0670]. The heat balance condition in the elementary part dx is mathematically expressed [12,13] by: Pc + Pa = Pl + Pr
(1)
where Pl is the thermal power deriving from and through the element and remaining in the elementary part dx; Pr the thermal power due to Joule shelf-heating of the elementary part dx; Pc the thermal power, due to heat capacity, remaining in the elementary part dx; and Pa the thermal power dissipated to the surrounding area by the external surface of the elementary part dx. The radiation is neglected, as in nominal current operation the temperature differences are relatively low [11]. For Pl, Pr, Pc and Pa the following equations [12,13] are valid: Pc = FCm
dDT dx dt
Pa = sSdxDT Pl = uF
d DT dx dx 2
(2) (3)
2
dx Pr = I 2z(1+ hDT) F
(4) (5)
Eqs. (1)–(5) lead to: FCm
dDT d2DT I 2z + sSDT =uF + (1+ hDT) dt dx F
d2DT 1 dDT + m 2DT − + c= 0 2 dx A dt
(6) (7)
Eq. (7) can be employed in order to calculate D? as a function of the distance x, and the operating time t, of the system (see Fig. 1). The limit conditions are formulated as follows: (a) at the instance t= 0, there is no current flow, so:
C.S. Psomopoulos, C.G. Karagiannopoulos / Electric Power Systems Research 61 (2002) 161–167
DT(x, t) t = 0 = 0
(8)
(b) for x= 9 l, the temperature change is zero, independent of time, as was mentioned, and, therefore:
)
dDT(x, t) =0 dt x 9l
(9)
(c) due to the symmetry of the model it can be easily assumed that in the element’s middle, x =0, temperature reaches the maximum value. Hence the limit condition at x = 0 is:
)
dDT(x, t) =0 dx x=0
(10)
(d) for t= the temperature across the element in nominal (or lower) current operation is stable, and, therefore: DT(x, t) t = =const
(11)
To solve differential Eq. (7) the Laplace transform is used, that is: (12) (13)
or
m2−
163
p cA D= A p(p− m 2A)
(20)
From relations Eqs. (15), (17) and (20) it follows that: 2
[(x, p)= b1e − x m
− p/A
2
+ b2e x m
− p/A
+
cA p(p−m 2A) (21)
where i1 and i2 are determined by the use of the limit conditions. Thus, relation Eq. (21) becomes: [(x, p) =
cA cA cos h[(x/ A) p −m 2A] − p(p− m 2A) p(p− m 2A) cos h[(l/ A) p −m 2A] (22)
To calculate DT(x, t) from [(x, p), the inverse Laplace transform is used, and finally, at each position x of the fuse element, DT can be given by two different equations (the solution has two forms): First form of DT: DT(x, t) =
16cl 2 (− 1)n + 1 (2n − 1)^x % cos ^ n = 1 2n−1 2l
n
2
e[m A − (2n − 1) ^ A/4l ]t − 1 [4l 2m 2 − (2n − 1)2^ 2] 2
2
2
(23)
Second form of DT: Based on the limit condition (a), it ensues that:
DT(x, t)=
c m2At (e − 1) m2
Based on relations 12 and 13, differential Eq. (7) becomes: d2[(x, p) p c + m 2 − [(x, p) + = 0 2 dx A p
cos
(15)
The solution of this differential equation has two terms: (a) the solution of the homogeneous [1(x, p):
d [1(x, p) p = − m 2 − [1(x, p) 2 dx A 2
(16)
2 − p/A
2 − p/A
+b2 ex m
(17)
(2n − 1)^x 2l
2
e[m A − (2n − 1) ^ A/4l ]t − 1 4m 2l 2 − (2n − 1)2^ 2
+
that is: [1(x, p)=b1 e − x m
4c^ % (− 1)n + 2(2n − 1) m 2 n=1
+
2
e[m
2
2
2A − (2n − 1)2^2A/4l2]t
(2n − 1)2^ 2
n
2At
− em
(24)
Comparing the two forms it is obvious that the first form is the simpler, as the resulting mathematical series is simpler. Hence, the first form will be used in the following application.
(b) and a partial solution [2(x, p). 3. Results and discussion
If it is assumed that: [2(x, p)=D
(18)
substituting this in Eq. (15) yields:
m2−
p c D + =0 A p
(19)
By the use of appropriate software, which was created for the analytical solution of Eq. (23), calculations of the temperature rise along the length of the fuse element were made, and characteristics DT = f(x) and DT = f(t) were drawn, for different parameters of the
164
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Fig. 2. (a) Temperature distribution lengthwise, and (b) temperature rise to the mid-point, as a function of time, of a thin copper wire simulating a fuse element with current value as parameter. Experimental characteristic DT= f(t) for a current value 40 A is also illustrated with the dashed and dotted line. Geometric characteristics of the copper wire: cross-section 0.55 mm2, and length 0.06 m.
model (current values and element’s geometric characteristics). Fig. 2a and b, Fig. 3a and b show results for copper elements with length 6 cm and cross-section 0.55 and 0.7 mm2, respectively, for different current values. The lengths and the cross-sections of the elements were selected to correspond to low and medium voltage fuses, which are employed in the distribution network. In order to evaluate the relation given in Eq. (23), measurements were carried out on thin wires with the above geometric characteristics, simulating fuse elements of practical applications. Fig. 4 shows the sim-
plified schematic diagram of the experimental set-up. A stabilized current supply provided undistorted 50 Hz sinusoidal current waveforms, even for short-circuit operating conditions. The sensors placed along the thin wires, which simulated the fuse element, were Pt-1000 resistances as well as Fe– CuNi thermocouples. Temperature measurements could be taken simultaneously using a digital recorder, thereby allowing temperature monitoring during the experiment. Potential measurements were performed at the ends of the element, implementing a digital voltage tracing system. As a first
C.S. Psomopoulos, C.G. Karagiannopoulos / Electric Power Systems Research 61 (2002) 161–167
step, indicating measurements were performed and experimental characteristics DT =f(t) (for constant current value) were drawn. Those characteristics are presented in Fig. 2b and Eq. (3)b. Fig. 5a shows theoretical results, DT = f(x), for copper elements with length 20 cm and cross-section 0.7 mm2, with the current value as the variable parameter. Fig. 5b shows measured and calculated steady-state temperatures at the mid-point of the fuse wire as functions of the current, DT =f(I), for copper elements with the above geometric characteristics. In the same figure, the experimental results of M.S. Agarwal et al. [11] are presented.
165
As can be seen in Fig. 2b Fig. 3b Fig. 5b, the theoretical results are in good agreement with the experimental measurements that were carried out, as well as with the experimental results of other authors. Small differences can be observed, which probably exist because of the simplifications, which were made during the model’s development. These simplifications will be discussed further. The thermal dissipation capability is a linear function of the heat transfer coefficient (s) and of the surface that dissipates the heat [12,13]. The insulating materials that support the conductors (insulators, etc.) do not have the capability to conduct, and hence dissipate,
Fig. 3. (a) Temperature distribution lengthwise; and (b) temperature rise to the mid-point, as a function of time, of a thin copper wire simulating a fuse element with current value as parameter. Experimental characteristic DT= f(t) for a current value 50 A is also illustrated with the dashed and dotted line. Geometric characteristics of the copper wire: cross-section 0.7 mm2 and length 0.06 m.
166
C.S. Psomopoulos, C.G. Karagiannopoulos / Electric Power Systems Research 61 (2002) 161–167
Fig. 4. Simplified schematic diagram of the experimental arrangement. 1: current stabilizer; 2: thin copper wire; 3: digital temperature recorder; 4: digital voltage tracing system.
heat through their surface. Therefore, it can be reasonably assumed that the heat produced on the fuse element dissipates to the environment only through the fuse end tags, fuse caps and conductors. In addition, the clamping of the conductors at the terminals of the fuse-carriers must meet the specification requirements concerning the suitability of the cross-section, and the size of the bindings and the coupling bolts. Moreover, when the clamping of the two parts (terminal/conductor) takes place after appropriate preparatory work [14], contact heating losses do not appear in practice. As regards the stationary contacts between fuses and fuse-carriers, the contact resistance’s value is of the order of 10 − 4 –10 – 5 V [14]. These heat sources [12,13] in the suggested model have been neglected. This seems to be appropriate and realistic because the produced
Fig. 5. (a) Temperature distribution lengthwise with parameter the current value; and temperature rise to the mid-point, as a function of the current value of a thin copper wire simulating a fuse element. Experimental characteristics DT=f(I) are also illustrated. Geometric characteristics of the copper wire: cross-section 0.7 mm2, and length 0.20 m.
C.S. Psomopoulos, C.G. Karagiannopoulos / Electric Power Systems Research 61 (2002) 161–167
heat in these contacts, in nominal current operation, is negligible compared with the heat produced in the thin fuse element (resistance of the order of 10 − 2 V). Furthermore, the main reason these sources are neglected is that the external surface of the conductors, the terminals of the fuse-carriers and the fuse end tags are much greater (by a factor of at least 1000) than the fuse element’s surface, resulting in a high heat dissipation rate. Besides that, the thermal capacity of the above, because of their great volume compared with the fuse element, stabilizes the temperature in continuous nominal operation. Thus, the assumption that in the fuse element’s edges there are heat sinks with good temperature stability seems to be reasonable. Fig. 2b Fig. 3b shows that in a short time period (a few seconds) the temperature on the element is stabilized, and it can also be observed that the highest temperature is reached in the center of the fuse element, as was expected. Besides, as can be seen by comparing Fig. 3a Fig. 5a, the length of the element seriously affects the maximum temperature of the element (for the same current values). Of course, the parameter that determines the fuse element’s length must be the nominal voltage at which the fuse operates. Arcing between the end tags of the fuse for a long time is the most undesirable event when a short-circuit is being interrupted. According to the above, the proposed model can estimate with satisfactory accuracy, the temperature distribution along the fuse element for different current values during nominal operation. The parameters needed for this calculation are: the fuse element’s geometric characteristics (length and cross-section), the element’s material and the surrounding agent. To our opinion, this model could be useful in the design of fuses, as many experimental tests concerning the estimation of the nominal current may be omitted.
4. Conclusions The analytic expression for the temperature of the fuse elements, under nominal current operation, seems to be the main advantage of this theoretical model compared with previous models. This mathematical expression give results that are in good agreement with the experimental measurements carried out, as well as with the work of others. The developed mathematical model should be useful in the design of fuses.
Appendix A. Nomenclature I
current through the fuse element (RMS value)
167
sustained short-circuit current cross-section of the fuse element 2yR perimeter of the fuse element density of the fuse element specific heat of the fuse element coefficient of the heat transfer from the fuse element to the surrounding area u thermal conductivity of the fuse element z resistivity of the fuse element temperature coefficient of ? h t time since the beginning of the fuse element operation dx elementary part of the fuse element T temperature of the fuse element Th ambient temperature DT T−Ta A u/Cm thermal diffusivity of the fuse element m 2 −(sSF−I 2hz)/F 2u by definition c I 2z/F 2u by definition Id F S m C s
References [1] J.G. Leach, P.G. Newbery, A. Wright, Analysis of high-rupturing-capacity fuselinks prearcing phenomena by a finite difference method, Proc. IEE 120 (9) (1973) 987 – 993. [2] E. Jacks, High rupturing capacity fuses, Design and Applications for safety in electrical systems, E. & F. N. SPON Ltd, London, 1975. [3] A. Wright, P.G. Newbery, Electric fuses, Peter Peregrinus, London, 1982. [4] R. Wilkins, S. Wade, J.S. Floyd, A suite of interactive programs for fuse design and development, Proc. of Int. Conf. on Electric Fuses and their Applications, Trondheim, Norway, 1984, pp. 227 – 235. [5] R. Reinhold, Transient performance of electric power systems, MIT Press, 1970. [6] T. Chicata, Y. Ueda, Y. Murai, T. Miyamoto, Spectroscopic observations of arcs in current limiting fuse through sand, Proc. of Int. Conf. on Electric Fuses and their Applications, Liverpool, 1976, pp. 114 – 121. [7] V.N. Narancic, G. Fecteau, Arc energy and critical tests for HV current-limiting fuses, Ibid., 1975, pp. 236 – 251. [8] D.R. Barrow, A.F. Howe, A. Wrigth, Methods of determining fuse arc parameters, in: Int. Conf. on Electrical Contacts, Arcs Apparatus and their Applications, Xi’an Jiaotong University, Xi’an, China, 1989. [9] D.R. Barrow, A.F. Howe, N. Cook, The chemistry of electric fuse arcing, IEE Proc.-A 138 (1) (1991) 83 – 88. [10] L.A.V. Cheim, A.F. Howe, Spectroscopic observation of high breaking capacity fuse arcs, IEE Proc.-Sci. Meas. Technol. 141 (2) (1994) 123 – 128. [11] M.S. Agarwal, A.D. Stokest, P. Kovitas, Pre-arcing behaviour of open fuse wire, J. Phys. D: Appl. Phys. 20 (1987) 1237 – 1242. [12] Alan J. Chapman, Heat Transfer, 4th ed., Maxwell Macmillan International, New York, 1989. [13] J. Taine, J.-P. Petit, Heat Transfer, 1st ed., Prentice Hall International, UK, 1993. [14] R. Holm, Electric contacts theory and applications, 4th ed., Springer, Berlin, 1979.
Temperature distribution of fuse elements during the pre-arcing period C.S. Psomopoulos *, C.G. Karagiannopoulos Department of Electrical and Computer Engineering, Di6ision of Industrial Electric De6ices and Decision Systems, National Technical Uni6ersity of Athens, 9 Iroon Polytechniou Str., Zografou, Athens, GR-15773, Greece Received 13 March 2001; accepted 13 June 2001
Abstract This work models the function of medium and low voltage fuses so as to estimate the increase in temperature across the fuse elements during nominal current operation for different types, diameters, element lengths and currents. The power balance in a thin metallic conductor, which simulates the fuse element, was derived and the resulting differential equation solved analytically. The theoretical results were close to the experimental measurements that we made, and the experimental work of other authors. The mathematical model that was developed will be useful in the design of fuses, as the computation of the axial temperature distribution across the fuse elements for the different parameters mentioned above is relatively simple. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Fusible elements; Temperature distribution; Pre-arcing period
1. Introduction In electrical installations of the low and medium voltage distribution network, fuses have been in use for over 100 years and their behavior has been adequately studied. They generally operate either below their nominal current or momentarily under excess, or short-circuit, currents. During operation under nominal current or less, the Joule heating produced on the fuse element dissipates to the surrounding area of the element, and thermal equilibrium is attained after a period of time [1 – 4]. For operation under excessive or heavy fault currents, the design of fuses is based on the well-known fundamental principle that they must interrupt those currents in a very short time. The phenomena developed are briefly as follows. The rapid rise of the temperature due mainly to Joule (I 2R) heating, which is aided by the resulting resistance increase, continues until the melting point is reached. The latent heat of fusion is produced gradually by the current during the * Corresponding author. E-mail addresses: [email protected] (C.S. Psomopoulos), [email protected] (C.G. Karagiannopoulos).
melting time. During the short times here in question, gravitational forces do not play any role, and thus the now liquid metal will remain in place and be heated further until the material is completely vaporized [2,3,5]. When the material of the element vaporizes, an electric arc is struck between the remaining solid parts of the element. This dynamic process causes a rapid temperature increase, while the current decreases rapidly until its flow is interrupted. The fundamentals of fuse operation have been described extensively by P.G. Newbery, A. Wright, E. Jacks, R. Ru¨denberg, D.R. Barrow, T. Chicata et al. [1–7]. Present research activities are mainly focused on the arcing period, and a number of models have been developed to simulate the fundamental operation. This operation is well understood, in general, but all the physicochemical processes during the arcing period are not so completely known [8–10]. However, some models have been developed which can simulate fuse operation under fault current during the arcing period [1,6 –10]. This work attempts to model the operation of medium and low voltage fuses carrying nominal current, so as to estimate the temperature across the fuse, considering the geometric characteristics and the material of the element as well as the current through the
0378-7796/02/$ - see front matter © 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 7 7 9 6 ( 0 2 ) 0 0 0 1 4 - 7
162
C.S. Psomopoulos, C.G. Karagiannopoulos / Electric Power Systems Research 61 (2002) 161–167
fuse. The relevant papers in the field of the pre-arcing period of fuse elements have been published by A. Wright and P.G. Newbery [1,3] and by M.S. Agarwal et al. [11]. They examine strip form elements and cylindrical fusible elements, respectively. Regarding the processes adopted, they both use finite differential methods for the development of their models. This paper proposes the development of a new model, which is confirmed by laboratory observations and results. Moreover, it attempts a comparison with the results of the model proposed by M.S. Agarwal et al. and solved with the Crack–Nicolson implicit finite difference method. The M.S. Agarwal et al. model examines the arrangement of fuse elements of two different cross-sections, whereas the model proposed hereafter considers fuse elements of even diameter, as is the condition in most cases that fuses are applied. 2. The proposed fuse model Fuses consist of one or more metallic conductors, known as the fuse element, and usually have a cylindrical or flat strip form. The surrounding media is granular silica quartz (SiO2) in high breaking capacity (HBC) fuses, and boric acid in expansion fuses. The element and the surroundings are housed in a body of insulating material (ceramic, fibre, melamine, etc.). The fuse element’s edges are usually soldered to, or in electric contact with, the fuse end caps; sometimes the connection is by a spring. In this work, fuses of low and medium voltage were simulated by the following simplified model. The element is considered to be a thin metallic wire or strip and the surrounding medium a liquid. Two conductors with much greater cross-section (compared with the fuse element’s cross-section) are in contact with the edges of the thin metallic wire (Fig. 1). This simplification seems to simulate the real conditions, because after the fuse’s end tags and fuse-carrier’s contacts, the circuit continues either with the proper cross-section conductors that result from the current specifications, or with conducting bars [VDE0670]. In both cases, the metallic cross-section of the fuse’s end tags, the fusecarrier’s contacts and the conductors or bars, is much greater than the fuse element’s cross-section. The simplified model proposed is illustrated in Fig. 1, where, for the purposes of symmetry, the center of the fuse element corresponds to x = 0.
Fig. 1. Schematic illustration of the model. (1) Conductor, (2) fuse element.
From the above, it can be assumed that the fuse end caps and tags, because of their great volume and, consequently, their external surface area (compared with the fuse element), operate as sinks for the heat that comes from, and through, the fuse element. In practice, for nominal operating conditions, it is true that the temperature in the fuse’s end tags, fuse-carrier’s contacts and conductors or bars, is the same as the temperature that would have existed, if the element was not interposed. This is particularly true in medium voltage installations where the cross-section of the bars, which result from calculations under short-circuit conditions, is far greater than the fuse element’s cross section. Therefore, the temperature rise above the environment temperature is only a few degrees under nominal operating conditions. Of course, that implies the stationary contacts between the fuse and the fuse-carrier operate within the normal standards [VDE0670]. The heat balance condition in the elementary part dx is mathematically expressed [12,13] by: Pc + Pa = Pl + Pr
(1)
where Pl is the thermal power deriving from and through the element and remaining in the elementary part dx; Pr the thermal power due to Joule shelf-heating of the elementary part dx; Pc the thermal power, due to heat capacity, remaining in the elementary part dx; and Pa the thermal power dissipated to the surrounding area by the external surface of the elementary part dx. The radiation is neglected, as in nominal current operation the temperature differences are relatively low [11]. For Pl, Pr, Pc and Pa the following equations [12,13] are valid: Pc = FCm
dDT dx dt
Pa = sSdxDT Pl = uF
d DT dx dx 2
(2) (3)
2
dx Pr = I 2z(1+ hDT) F
(4) (5)
Eqs. (1)–(5) lead to: FCm
dDT d2DT I 2z + sSDT =uF + (1+ hDT) dt dx F
d2DT 1 dDT + m 2DT − + c= 0 2 dx A dt
(6) (7)
Eq. (7) can be employed in order to calculate D? as a function of the distance x, and the operating time t, of the system (see Fig. 1). The limit conditions are formulated as follows: (a) at the instance t= 0, there is no current flow, so:
C.S. Psomopoulos, C.G. Karagiannopoulos / Electric Power Systems Research 61 (2002) 161–167
DT(x, t) t = 0 = 0
(8)
(b) for x= 9 l, the temperature change is zero, independent of time, as was mentioned, and, therefore:
)
dDT(x, t) =0 dt x 9l
(9)
(c) due to the symmetry of the model it can be easily assumed that in the element’s middle, x =0, temperature reaches the maximum value. Hence the limit condition at x = 0 is:
)
dDT(x, t) =0 dx x=0
(10)
(d) for t= the temperature across the element in nominal (or lower) current operation is stable, and, therefore: DT(x, t) t = =const
(11)
To solve differential Eq. (7) the Laplace transform is used, that is: (12) (13)
or
m2−
163
p cA D= A p(p− m 2A)
(20)
From relations Eqs. (15), (17) and (20) it follows that: 2
[(x, p)= b1e − x m
− p/A
2
+ b2e x m
− p/A
+
cA p(p−m 2A) (21)
where i1 and i2 are determined by the use of the limit conditions. Thus, relation Eq. (21) becomes: [(x, p) =
cA cA cos h[(x/ A) p −m 2A] − p(p− m 2A) p(p− m 2A) cos h[(l/ A) p −m 2A] (22)
To calculate DT(x, t) from [(x, p), the inverse Laplace transform is used, and finally, at each position x of the fuse element, DT can be given by two different equations (the solution has two forms): First form of DT: DT(x, t) =
16cl 2 (− 1)n + 1 (2n − 1)^x % cos ^ n = 1 2n−1 2l
n
2
e[m A − (2n − 1) ^ A/4l ]t − 1 [4l 2m 2 − (2n − 1)2^ 2] 2
2
2
(23)
Second form of DT: Based on the limit condition (a), it ensues that:
DT(x, t)=
c m2At (e − 1) m2
Based on relations 12 and 13, differential Eq. (7) becomes: d2[(x, p) p c + m 2 − [(x, p) + = 0 2 dx A p
cos
(15)
The solution of this differential equation has two terms: (a) the solution of the homogeneous [1(x, p):
d [1(x, p) p = − m 2 − [1(x, p) 2 dx A 2
(16)
2 − p/A
2 − p/A
+b2 ex m
(17)
(2n − 1)^x 2l
2
e[m A − (2n − 1) ^ A/4l ]t − 1 4m 2l 2 − (2n − 1)2^ 2
+
that is: [1(x, p)=b1 e − x m
4c^ % (− 1)n + 2(2n − 1) m 2 n=1
+
2
e[m
2
2
2A − (2n − 1)2^2A/4l2]t
(2n − 1)2^ 2
n
2At
− em
(24)
Comparing the two forms it is obvious that the first form is the simpler, as the resulting mathematical series is simpler. Hence, the first form will be used in the following application.
(b) and a partial solution [2(x, p). 3. Results and discussion
If it is assumed that: [2(x, p)=D
(18)
substituting this in Eq. (15) yields:
m2−
p c D + =0 A p
(19)
By the use of appropriate software, which was created for the analytical solution of Eq. (23), calculations of the temperature rise along the length of the fuse element were made, and characteristics DT = f(x) and DT = f(t) were drawn, for different parameters of the
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Fig. 2. (a) Temperature distribution lengthwise, and (b) temperature rise to the mid-point, as a function of time, of a thin copper wire simulating a fuse element with current value as parameter. Experimental characteristic DT= f(t) for a current value 40 A is also illustrated with the dashed and dotted line. Geometric characteristics of the copper wire: cross-section 0.55 mm2, and length 0.06 m.
model (current values and element’s geometric characteristics). Fig. 2a and b, Fig. 3a and b show results for copper elements with length 6 cm and cross-section 0.55 and 0.7 mm2, respectively, for different current values. The lengths and the cross-sections of the elements were selected to correspond to low and medium voltage fuses, which are employed in the distribution network. In order to evaluate the relation given in Eq. (23), measurements were carried out on thin wires with the above geometric characteristics, simulating fuse elements of practical applications. Fig. 4 shows the sim-
plified schematic diagram of the experimental set-up. A stabilized current supply provided undistorted 50 Hz sinusoidal current waveforms, even for short-circuit operating conditions. The sensors placed along the thin wires, which simulated the fuse element, were Pt-1000 resistances as well as Fe– CuNi thermocouples. Temperature measurements could be taken simultaneously using a digital recorder, thereby allowing temperature monitoring during the experiment. Potential measurements were performed at the ends of the element, implementing a digital voltage tracing system. As a first
C.S. Psomopoulos, C.G. Karagiannopoulos / Electric Power Systems Research 61 (2002) 161–167
step, indicating measurements were performed and experimental characteristics DT =f(t) (for constant current value) were drawn. Those characteristics are presented in Fig. 2b and Eq. (3)b. Fig. 5a shows theoretical results, DT = f(x), for copper elements with length 20 cm and cross-section 0.7 mm2, with the current value as the variable parameter. Fig. 5b shows measured and calculated steady-state temperatures at the mid-point of the fuse wire as functions of the current, DT =f(I), for copper elements with the above geometric characteristics. In the same figure, the experimental results of M.S. Agarwal et al. [11] are presented.
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As can be seen in Fig. 2b Fig. 3b Fig. 5b, the theoretical results are in good agreement with the experimental measurements that were carried out, as well as with the experimental results of other authors. Small differences can be observed, which probably exist because of the simplifications, which were made during the model’s development. These simplifications will be discussed further. The thermal dissipation capability is a linear function of the heat transfer coefficient (s) and of the surface that dissipates the heat [12,13]. The insulating materials that support the conductors (insulators, etc.) do not have the capability to conduct, and hence dissipate,
Fig. 3. (a) Temperature distribution lengthwise; and (b) temperature rise to the mid-point, as a function of time, of a thin copper wire simulating a fuse element with current value as parameter. Experimental characteristic DT= f(t) for a current value 50 A is also illustrated with the dashed and dotted line. Geometric characteristics of the copper wire: cross-section 0.7 mm2 and length 0.06 m.
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C.S. Psomopoulos, C.G. Karagiannopoulos / Electric Power Systems Research 61 (2002) 161–167
Fig. 4. Simplified schematic diagram of the experimental arrangement. 1: current stabilizer; 2: thin copper wire; 3: digital temperature recorder; 4: digital voltage tracing system.
heat through their surface. Therefore, it can be reasonably assumed that the heat produced on the fuse element dissipates to the environment only through the fuse end tags, fuse caps and conductors. In addition, the clamping of the conductors at the terminals of the fuse-carriers must meet the specification requirements concerning the suitability of the cross-section, and the size of the bindings and the coupling bolts. Moreover, when the clamping of the two parts (terminal/conductor) takes place after appropriate preparatory work [14], contact heating losses do not appear in practice. As regards the stationary contacts between fuses and fuse-carriers, the contact resistance’s value is of the order of 10 − 4 –10 – 5 V [14]. These heat sources [12,13] in the suggested model have been neglected. This seems to be appropriate and realistic because the produced
Fig. 5. (a) Temperature distribution lengthwise with parameter the current value; and temperature rise to the mid-point, as a function of the current value of a thin copper wire simulating a fuse element. Experimental characteristics DT=f(I) are also illustrated. Geometric characteristics of the copper wire: cross-section 0.7 mm2, and length 0.20 m.
C.S. Psomopoulos, C.G. Karagiannopoulos / Electric Power Systems Research 61 (2002) 161–167
heat in these contacts, in nominal current operation, is negligible compared with the heat produced in the thin fuse element (resistance of the order of 10 − 2 V). Furthermore, the main reason these sources are neglected is that the external surface of the conductors, the terminals of the fuse-carriers and the fuse end tags are much greater (by a factor of at least 1000) than the fuse element’s surface, resulting in a high heat dissipation rate. Besides that, the thermal capacity of the above, because of their great volume compared with the fuse element, stabilizes the temperature in continuous nominal operation. Thus, the assumption that in the fuse element’s edges there are heat sinks with good temperature stability seems to be reasonable. Fig. 2b Fig. 3b shows that in a short time period (a few seconds) the temperature on the element is stabilized, and it can also be observed that the highest temperature is reached in the center of the fuse element, as was expected. Besides, as can be seen by comparing Fig. 3a Fig. 5a, the length of the element seriously affects the maximum temperature of the element (for the same current values). Of course, the parameter that determines the fuse element’s length must be the nominal voltage at which the fuse operates. Arcing between the end tags of the fuse for a long time is the most undesirable event when a short-circuit is being interrupted. According to the above, the proposed model can estimate with satisfactory accuracy, the temperature distribution along the fuse element for different current values during nominal operation. The parameters needed for this calculation are: the fuse element’s geometric characteristics (length and cross-section), the element’s material and the surrounding agent. To our opinion, this model could be useful in the design of fuses, as many experimental tests concerning the estimation of the nominal current may be omitted.
4. Conclusions The analytic expression for the temperature of the fuse elements, under nominal current operation, seems to be the main advantage of this theoretical model compared with previous models. This mathematical expression give results that are in good agreement with the experimental measurements carried out, as well as with the work of others. The developed mathematical model should be useful in the design of fuses.
Appendix A. Nomenclature I
current through the fuse element (RMS value)
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sustained short-circuit current cross-section of the fuse element 2yR perimeter of the fuse element density of the fuse element specific heat of the fuse element coefficient of the heat transfer from the fuse element to the surrounding area u thermal conductivity of the fuse element z resistivity of the fuse element temperature coefficient of ? h t time since the beginning of the fuse element operation dx elementary part of the fuse element T temperature of the fuse element Th ambient temperature DT T−Ta A u/Cm thermal diffusivity of the fuse element m 2 −(sSF−I 2hz)/F 2u by definition c I 2z/F 2u by definition Id F S m C s
References [1] J.G. Leach, P.G. Newbery, A. Wright, Analysis of high-rupturing-capacity fuselinks prearcing phenomena by a finite difference method, Proc. IEE 120 (9) (1973) 987 – 993. [2] E. Jacks, High rupturing capacity fuses, Design and Applications for safety in electrical systems, E. & F. N. SPON Ltd, London, 1975. [3] A. Wright, P.G. Newbery, Electric fuses, Peter Peregrinus, London, 1982. [4] R. Wilkins, S. Wade, J.S. Floyd, A suite of interactive programs for fuse design and development, Proc. of Int. Conf. on Electric Fuses and their Applications, Trondheim, Norway, 1984, pp. 227 – 235. [5] R. Reinhold, Transient performance of electric power systems, MIT Press, 1970. [6] T. Chicata, Y. Ueda, Y. Murai, T. Miyamoto, Spectroscopic observations of arcs in current limiting fuse through sand, Proc. of Int. Conf. on Electric Fuses and their Applications, Liverpool, 1976, pp. 114 – 121. [7] V.N. Narancic, G. Fecteau, Arc energy and critical tests for HV current-limiting fuses, Ibid., 1975, pp. 236 – 251. [8] D.R. Barrow, A.F. Howe, A. Wrigth, Methods of determining fuse arc parameters, in: Int. Conf. on Electrical Contacts, Arcs Apparatus and their Applications, Xi’an Jiaotong University, Xi’an, China, 1989. [9] D.R. Barrow, A.F. Howe, N. Cook, The chemistry of electric fuse arcing, IEE Proc.-A 138 (1) (1991) 83 – 88. [10] L.A.V. Cheim, A.F. Howe, Spectroscopic observation of high breaking capacity fuse arcs, IEE Proc.-Sci. Meas. Technol. 141 (2) (1994) 123 – 128. [11] M.S. Agarwal, A.D. Stokest, P. Kovitas, Pre-arcing behaviour of open fuse wire, J. Phys. D: Appl. Phys. 20 (1987) 1237 – 1242. [12] Alan J. Chapman, Heat Transfer, 4th ed., Maxwell Macmillan International, New York, 1989. [13] J. Taine, J.-P. Petit, Heat Transfer, 1st ed., Prentice Hall International, UK, 1993. [14] R. Holm, Electric contacts theory and applications, 4th ed., Springer, Berlin, 1979.