IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 21, NO. 1, JANUARY 2006 425

Investigation of the Thermal TransferCoefficient by the Energy Balance of Fault

Arcs in Electrical InstallationsXiang Zhang, Gerhard Pietsch, Member, IEEE, and Ernst Gockenbach, Fellow, IEEE

AbstractIn order to determine the pressure rise due to faultarcs in electrical installations, the portion of energy heating thesurrounding gas of the fault arc has to be known. The ratio ofthe portion of energy to the electrical energy, the thermal transfercoefficient, well known in literature as -factor, is adopted here.This paper presents a theoretical approach to calculate the thermaltransfer coefficient and to determine the pressure rise in an elec-trical installation. It is based on the solution of the fundamentalhydro- and thermodynamic conservation equations taking into ac-count melting and evaporation of metals as well as chemical re-actions with the surrounding gas of the fault arc. The results forclosed arc chambers show that factors such as the kinds of insu-lating gas and of electrode material, the size of the test vessel, andthe gas density considerably influence the thermal transfer coeffi-cient and thus the pressure rise. Furthermore it is demonstrated,with an example of a short-circuit in a compact medium-voltagestation with heavy metal evaporation, that the mathematical ap-proach is a reliable tool to assess the development of pressure.

Index TermsChemical reaction, electrical installation, energybalance, fault arc, hydro- and thermodynamics, melting and evap-oration, pressure, relative purity, theoretical approach, thermaltransfer coefficient.

I. INTRODUCTION

I F a fault arc in an electrical installation occurs, it may en-danger the maintenance personnel and seriously damage theelectrical equipmentandeven thebuildingof the installation.Oneof the main effects of fault arcs is the pressure stress on the me-chanical parts of the installation and on the walls of the building.

In order to determine the pressure rise , the portion of en-ergy heating the surrounding gas of the fault arc in an electricalinstallation has to be known. To simulate the energy transfer fromthe fault arc to its surroundinggas in theelectrical installation, it isassumed that the thermal transfer coefficient is the ratio of theinternal energy of the surrounding gas to the electrical energy

of the fault arc and can be expressed by [1][5]

(1)

where and arethevolumeofthegasspaceunderconsiderationand the adiabatic coefficient of the surrounding gas, respectively.

Manuscript received September 21, 2004; revised March 14, 2005. Paper no.TPWRD-00445-2004.

X. Zhang and E. Gockenbach are with the Institute of Electrical Power Sys-tems, Division of High Voltage Engineering, University of Hanover, Hanover30167, Germany (e-mail: zhang@si.uni-hannover.de).

G. Pietsch is with the Department of Electrical Engineering and GasDischarge Technology, Aachen University of Technology, Aachen 52056,Germany.

Digital Object Identifier 10.1109/TPWRD.2005.852274

In the past, several calculation methods have been developedto simulate the pressure rise in the surroundings of fault arcs[2], [3]. In those approaches, the thermal transfer coefficienthas been experimentally determined in a closed vessel applying(1). The application of the measured -values is limited to thespecial boundary conditions of those experiments and to the as-sumptions of the gas model used in (1).

In the case of fault arcs, the thermal transfer coefficient andthe pressure rise depend on several parameters such as thekinds of insulating gas and of electrode material, the size ofthe test vessel, and the gas density. For a general estimation ofthe pressure rise, the influences of all these parameters on thepressure rise can be investigated by experiments, which are,however, sometimes difficult to execute. In some cases, stationsare included in existing buildings without the possibility ofdetermining the pressure stress by tests on site.

In the following Section II the energy balance of a faultarc in an electrical installation is generally described by thecorrelative transfer coefficients, and the influences of eachenergy transfer on the gas pressure are specified. In Section IIIthe term relative purity of the gas status is introduced and thethermal transfer coefficient at relatively pure conditions ismeasured to be constant for the insulating gases. Thus the thermaltransfer coefficient for any conditions can be characterized bythe thermal transfer coefficient at relatively pure conditionstogether with other transfer coefficients related to melting andevaporation as well as chemical reactions. In Section IV themathematical descriptions of the changes of gas density andtransfer energy which are related to melting and evaporation aswell as chemical reactions are given. In Section V the equationof state and the conservation equations based on the hydro-and thermodynamics are applied for calculation of the pressuredevelopment. The change of density resulting from Section IVconcerning evaporation and chemical reactions is applied forthe continuity equation. With the special expression of meltingand evaporation as well as chemical reactions in Section IV thegeneral physical description of the energy balance in Section IIis changed into a mathematical energy equation. In Section VIthe thermal transfer coefficient and the development of pressureare calculated for a closed test vessel and a medium-voltagestation, respectively. Section VII reaches a conclusion.

II. ENERGY BALANCE OF FAULT ARCS

If a fault arc in an electrical installation occurs, the electricalenergy of the arc plasma is transferred to its surroundings via

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426 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 21, NO. 1, JANUARY 2006

Fig. 1. Simplified energy balance of a fault arc in an electrical installation.

different mechanisms of interaction (Fig. 1). The energy inputinto the fault arc by Joule heat is balanced by severalenergy exchanges like heat conduction and radiation

by the interactions of the arc column with the electrodesand the walls of the electrical installation. They are absorbedby the insulating gas outside the electrical installation and donot contribute to the pressure rise in the electrical installation.Furthermore, evaporated metal from the arc root points togetherwith chemical reactions play an important role in the energytransfer from the fault arc to the surrounding gas. The metalevaporation enhances the quantities of gas and further increasesthe pressure rise. The necessary energy for melting andevaporation is to be considered in the energy balance [1], [6].Due to chemical reactions between the evaporated materialsand the surrounding gas, the oxidation of the evaporated metalswith air leads to a decrease of particles in the surrounding gasaround the fault arc and the heat of the chemical reactionsmakes a considerable contribution to the energy balance of thefault arc [1], [7]. If openings of the arc chamber (e.g., pressurerelief flaps in switchboards) are present, gas convectionhas to be considered additionally. The convective transfer ofheat and the mass of the gas cause a change in the internalheat of the surrounding gas and therefore a pressure rise, too[2], [3], [8].

Based on the above discussions, an energy balance of the faultarc can be formulated, dividing each energy portion by the elec-trical energy

(2)

where , and are the coefficientsrepresenting the energy of heat conduction, radiation, convec-tion, melting and evaporation as well as chemical reaction re-spectively (with the symbols + for an endothermic reaction and for an exothermic reaction).

The internal energy is composed of those energy por-tions contributing to the surrounding gas of the fault arc

), convection by pressure relief flaps as wellas metallic vapors and their chemical reactions .

According to the experimental results from [1], [2], [3], [9]for both closed and open vessels, the radiation coefficientremains almost unaffected and the conduction coefficientis relatively small, thus its value is negligible. For these reasons,

the coefficients contributing to the surrounding gas of the faultarc, i.e., , can be assumed to be constant

(3)and thus (2) can be simplified by

(4)

III. CONCEPT OF RELATIVE PURITYIn order to understand the following, it is of advantage to

introduce the term relative purity of the gas status. If a gassurrounding the fault arc in a closed vessel is not contaminatedby impurities from the electrodes or the interaction of the faultarc with the walls of the vessel, it is named pure. If, at a givenelectrical energy of the fault arc, the gas density in a closedvessel is high enough so that the particle concentration of theimpurities generated by the fault arc is negligible, it is namedrelatively pure.

From (4) we can find that is just the thermal transfer co-efficient at relatively pure conditions because other coef-ficients have no contribution to (4). The thermal transfer coeffi-cient at relatively pure conditions can be determined from(1) by the measurement of the pressure in a closed vessel at highgas densities, which means a fixed substance constituent and acomparatively low average temperature of the surrounding gas.The thermal transfer coefficient at a relatively pure con-dition of an insulating gas, whose value depends on the kind ofinsulating gas and is independent on the gas model to a large ex-tent, is found to be constant at high gas densities. The -valueof as insulating gas is found to be [3]

(5)With a relatively pure insulating gas, the thermal transfer

coefficient depends on the internal energy of the gas, whichis proportional to the specific heat at constant volume ofthe insulating gas if the change of the temperature of the sur-rounding gas at high gas densities is low and the temperatureof the surrounding gas is accepted as the ambient temperature.Thus the -values of another insulation gases can easily bedetermined by [7]

(6)Due to the heat transfer at the arc roots, metal electrodes

or walls of the test vessel may melt and evaporate to a cer-tain extent. Metallic vapors can even dominate the gas com-positions. In this case, the condition of relative purity willnot be fulfilled. The vaporized particles will directly influencethe pressure stress. Furthermore, the vapors may react with thesurrounding gas in endothermic or exothermic reactions, influ-encing the energy balance of the fault arc and the density ofthe surrounding gas (if e.g., chemical reactions lead to powdersextracting particles from the surrounding gas). In this case, ma-terial evaporation and chemical reactions have to be consideredin the thermodynamic system because they will influence thepressure development.

The thermal transfer coefficient corresponding to the in-ternal energy of the gas impurities is characterized by the mu-

ZHANG et al.: INVESTIGATION OF THE THERMAL TRANSFER COEFFICIENT 427

tual dependency of the coefficients , anddescribed by (4), whereas is obtained by (6) and the energyof convection is derived from the well-known hydrody-namics (see Section V).

In order to determine the internal energy and the pressurerise of the surrounding gas as well as their correspondingthermal transfer coefficient , the processes of melting andevaporation as well as chemical reactions mustbe investigated particularly.

IV. EVAPORATION AND CHEMICAL REACTIONSDuring a fault arc, additional particles resulting from metal

vapors (e.g., copper, aluminum or iron) lead to a change in thedensity of the surrounding gas in an electrical installation [1]

(7)where is time, the effective value of the short-circuit current,

the order number, the components of the gas mixture,the density for , corresponding to various com-ponents, and the specific mass loss per charge of the -thcomponent, which is proportional to the -coefficient [1],[6].

The energy for melting and evaporation is to be con-sidered in the energy balance, i.e., [6]

Me solid state Me gaseous state (8.0)where Me is a symbol for different metals, e.g., Al, Cu, and Feetc.

If metals (Al, Cu or Fe) are evaporated in air, the followingchemical reactions have to be considered

(8.1)(8.2)(8.3)(8.4)(8.5)

Due to chemical reactions, a change of the gas density resultsfrom the contribution of the reaction rates [7]

(9)where is the order number, the chemical reactions,the stoichiometric coefficient of the -th component in the -thchemical reaction, the molecular weight for ,corresponding to various components, and the chemical reac-tion rate for , corresponding to various chemicalreactions.

The energy for melting and evaporation and the heatof the chemical reaction can be represented by the den-

sity and the specific enthalpy of the generated or consumedgases for [7]

(10)

It is reasonable to suppose that the rate of chemical reactionsis faster than that of metal evaporation.

V. GOVERNING EQUATIONSWith the considerations of melting and evaporation as well

as chemical reactions, the relevant mathematical models are de-veloped, for which the equation of state and the conservationequations (i.e., continuity-, momentum- and energy equations)based on the hydro- and thermodynamics [7] are applied.

Continuity equation

(11)where and are the diffusion coefficient and the turbulentPrandtl number of the -th components, and are the ve-locity and the turbulent viscosity, is the mass fraction of the-th component by the gas mixtures

(12)

Momentum equation

(13)with the bulk viscosity and the dynamic viscosity .

With the special expression of melting and evaporation aswell as chemical reactions, the general physical description ofthe energy balance (4) is changed into a mathematical energyequation

(14)

with the thermal conductivity and the temperature .In (11), the density gradient of the -th component

from gas compositions, which participate in material meltingand evaporation as well as chemical reactions described by (7)and (9), appears in (11) as the source term of the density change.The enthalpy gradient of the -th components from

gas compositions is considered as the source term of (14),which describes the change of the generated or consumed en-ergy or characterized by (10).

According to thermodynamics [7], the change of the internalenergy per volume unit can be specified by the gradient ofthe enthalpy and of the pressure in (14). Together with theother terms for the change of the internal energy at rel-atively pure conditions in a volume unit, convectionand conduction (its value is relatively small, thus neg-ligible), (14) is in accordance with the energy balance (4) usingthe transfer coefficients.

With the use of (5) to (14) and the equation of state(15)

the pressure rise, the gas density, the temperature and the flowvelocity as well as other hydro- and thermodynamic items in the

428 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 21, NO. 1, JANUARY 2006

TABLE ITEST CONDITIONS

Fig. 2. Test container. 1, 2: connectors, 3: pressure transmitters, 4: fault arc,5: electrodes.

electrical installation can be calculated with the sole assumptionof (5).

VI. SIMULATION RESULTSIn order to investigate the influences of some relevant param-

eters (e.g., electrode material, insulating gas, gas density andthe volume of the test vessel) on the pressure rise, a fault arcwith an electrical power of 100 kW during 80 ms has been pro-vided for the experiment, which occurred inside a closed vesselof different volumes ( m and m ) underthe conditions given in Table I.

The four-flanged container illustrated in Fig. 2 was used asa single-pole gas-insulated test vessel. During a flashover, thefault arc was ignited between two electrodes 5 introduced intothe test vessel through covers. By removing the lateral cover ofthe pressure transmitter 3, the volume of the container can bevaried. The test container is connected to a compressor via twoconnectors 1 and 2. By activating the compressor, it is possibleto fill the test container and to set different initial pressures andthe corresponding filling densities in such an enclosure.

First, the pressure rise due to the energy of the fault arc hasbeen experimentally measured. The measured thermal transfercoefficient can be derived from (1). In the same way the cal-culated thermal transfer coefficient can be obtained by (1) aswell, after the development of pressure is calculated from (5)to (15). By the comparison of the measured and the calculatedthermal transfer coefficients, the influences of the relevant pa-rameters on the pressure rise can be investigated and the ap-proach for the pressure calculation can be verified, too.

Fig. 3. Calculated (without test) and measured (with test) k -values forAl-electrodes in displayed versus the gas density.

From (1), it is known that the thermal transfer coefficientdepends on the pressure rise and also on the adiabatic

coefficient . In order to compare the -values without beinginfluenced by different -values (which depend on the temper-ature and on the kind of surrounding gas), instead of comparingthe pressure stresses from measurement and calculation, the

-values of different insulating gases at ambient temperature(300 K) have been used for the assessment of the thermaltransfer coefficient .

Metal evaporation as well as the chemical reactions betweenthe insulating gas air and the electrode materials aluminum andcopper are considered in (7) to (10) respectively. Whenis used as insulating gas, only metal evaporation is taken intoaccount.

The calculated and measured results of the thermal transfercoefficient for the closed container are presented in Figs. 3to 6 as a function of the gas density (initial pressure). A rathergood agreement between both calculated and measured resultscan be observed.

In Figs. 3 and 4 the results depict the insulating gas air and theelectrode materials aluminum and copper, respectively. For theinitial pressures of air above 0.1 MPa (corresponding to a densityof 1.2 kg/m ), the -values are in the range of 0.60 to 0.80. Forinitialpressuresbelow0.1Mpa(1.2kg/m ), the -valuesdeclineto about 0.30 to 0.50 with the falling pressure. In Figs. 5 and 6 thethermal transfer coefficients are displayed for the insulatinggas and the electrode materials aluminum and copper. Withan initial gas pressure above 0.1 MPa (corresponding to a densityof 6.1 kg/m ), the calculated -values are almost constant atabout 0.50 to 0.70 in both test volumes of the container. Below theinitial pressure of 0.1 MPa (6.1 kg/m ), the -values rise up toabout 1.0 and 1.2 depending on the test volume of the container inthe case of aluminum electrodes. In the case of copper electrodesthe -values remain almost unchanged.

Based on the specific mass loss of metals per charge unitof the -th component in (7) and the test conditions given inTable I, the mass fractions of copper vapors in at the initialpressure of 0.3 MPa (18.3 kg/m ) are 0.068% and 0.035% in thetest volumes of 0.07 m and 0.14 m , respectively. Calculatingthe overpressure in the test container, the average temperaturesof 333 to 354 K of the surrounding gas have been found. That

ZHANG et al.: INVESTIGATION OF THE THERMAL TRANSFER COEFFICIENT 429

Fig. 4. Calculated (without test) and measured (with test) k -values forCu-electrodes in air displayed versus the gas density.

Fig. 5. Calculated (without test) and measured (with test) k -values forAl-electrodes in SF displayed versus the gas density.

Fig. 6. Calculated (without test) and measured (with test) k -values forCu-electrodes in SF displayed versus the gas density.

is why the surrounding gas of the fault arc is indeed relativelypure in this case, i.e., the influences of the metal vapors andchemical reactions are negligible.

On the other hand, in the case of aluminum electrodes themass fraction of aluminum vapors in air at the initial pressureof 0.01 MPa (0.12 kg/m ) is 19.6% in the smaller test volume.The calculated temperature of the surrounding gas under these

conditions has been found to be 3813 K. In such a case, the gasstatus is far from the relative purity.

Comparing the shapes of the -curves versus the gas den-sity for air and , some differences are obvious. Whereas the

-factors in air decrease with falling density, they increase orare nearly constant in . These appearances result from therising importance of the consumption of O and Al particles inchemical reactions, if air is used as the insulating gas. In the caseof as insulating gas, such a consumption is not possible. Onthe contrary, a strong evaporation of aluminum electrodes hap-pens, which increases the pressure considerably. The evapora-tion of copper is less important.

Comparing the -values of air and at high gas densitiesthe -values of air tend to increase. This behavior results fromthe difference in the specific heat of the insulating gases at con-stant volume, which is higher for air.

The results for air as they are presented in Figs. 3 and 4 areless different. The -values of copper electrodes at high gasdensities appear to be slightly lower than those of aluminumelectrodes. The reason lies in the higher specific heat at constantvolume of Al particles in air due to the evaporation of aluminumcompared to that of Cu particles.

The -values differ only slightly in the range of the low den-sity depending on the test volume of the container. At an initialpressure of 0.01 MPa (1.2 kg/m ) in the larger test volume, theweight of O fraction is 3.26 g and the O molecules are so nu-merous that chemical reactions as they are initiated here will notconsume all of them. In this case the evaporated Al or Cu parti-cles of 3.42 g or 1.71 g will consume the O molecules of 3.05 gor 0.58 g respectively to meet the requirements of the chemicalreactions described by (8.1)to (8.3). In chemical reactions theO particles are consumed, thus the thermal transfer coefficient

is smaller.However, if chemical reactions are absent because of the lack

of O molecules (e.g., the weight of O fraction in the smallervolume is 1.63 g only), the gas products may result from aboosting evaporation of aluminum electrode material, whichmay raise the -value as shown in Fig. 3.

At the smaller gas densities, the surrounding gas is heated toa higher temperature by the same electrical energy of the faultarc. In this case the mass fraction of the metal evaporation inthe surrounding gas is enlarged to raise the pressure on the onehand. Chemical reactions on the other hand, which yield thesolid products of chemical reactions, lead to a consumption ofgas and by that to a decrease of the pressure. In general, theadditional density of generated or consumed particles plays adecisive role.

The effect is important in electrical installations with pres-sure relief openings as well. By the gas stream through pressurerelief flaps, the density of the surrounding gas in the electricalinstallation decreases. In order to take the decrease of densityinto account for the pressure calculation the melting and evap-oration as well as chemical reactions must be investigated bythe calculation models, that reflect on the variation of the ac-tual gas density in electrical installations. It is expected that theexact calculation of pressure depends on the amount of energyreleased in electrical installations at least at low gas densities.

430 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 21, NO. 1, JANUARY 2006

Fig. 7. Compact medium-voltage station under consideration.

TABLE IIGEOMETRICAL CONDITIONS

As an example for the calculation of pressure, a fault arc in acompact medium-voltage station with severe metal evaporationis considered in Fig. 7. The relevant geometrical conditions aregiven in Table II.

A short-circuit current of 16 kA was initiated in one of threecable compartments

- -, and

-, which were linked

with each other by the openings-

. Although the switchchamber was provided with a pressure relief opening, itdid not act as desired during the experiment. The fault arcburned between the copper electrodes, which developed to athree-phase fault arc with a duration of 1.0 s. The measuredelectrical power during fault arc is displayed in Fig. 8.

During the fault arc, an iron sheet of the housing of about 1 kgand copper electrodes of about 100 g in the cable compartmentswere evaporated, as may be seen in Fig. 9. The set of chemicalreactions between air, iron and copper have been considered in(8.1), (8.2), (8.4), and (8.5).

In Figs. 10 and 11 the calculated and measured curves of thepressure in the middle of the cable compartment

-and in

the middle of the transformer compartment are depicted. Arather good agreement between both calculated and measuredresults can be reached.

The maximum pressures of about 120 and 105 kPa in themiddle of the cable compartment

-and in the middle of the

transformer compartment are reached about 0.02 and 0.04 s,respectively, after the ignition of the fault arc. The pressure os-cillation during the first pressure peak in the transformer com-partment is believed to have resulted from the impact of theexterior environment on the opening of O . After the first pres-sure peak, the pressure relief opening O causes a strong decayof pressure to about 103 kPa in both compartments.

After about 0.2 s, a smaller increase of the electrical poweroccurs as may be seen in Fig. 8. This results from the burning-

Fig. 8. Measured electrical power of a short-circuit current of 16 kA.

Fig. 9. Burned cable compartments of the compact medium-voltage station.(a) Cable compartment R

-. (b) Cable compartments R

-and R

-.

through of the housing of the cable compartments and is con-nected with the heavy evaporation of iron. About 90% of the gasmixture in the cable compartments now consists of iron vapors.The heavy evaporation follows an increase in the arc voltage andin the pressure, which lasts about 0.35 s. After that, the ambientpressure is reached up to the extinction of the fault arc.

ZHANG et al.: INVESTIGATION OF THE THERMAL TRANSFER COEFFICIENT 431

Fig. 10. Calculated (full line) and measured (dotted line) pressuredevelopments in the cable compartment R

-.

Fig. 11. Calculated (full line) and measured (dotted line) pressuredevelopment in the transformer compartment R .

During the last 0.2 s a further pressure oscillation in measure-ment and calculation is detected, which belongs to an enforcedoscillation of energy resulting from the arc movement.

VII. CONCLUSIONIn order to be able to predict the pressure rise in electrical in-

stallations due to fault arcs, it is necessary to know the portionof electrical energy of the causative fault arc corresponding tothe thermal transfer coefficient , which causes the pressurerise. With the determination of the thermal transfer coefficient

, it is possible to calculate the pressure rise in electrical in-stallations (canceled), even for severe conditions under whichmeasurements cannot be performed.

In the evolution of the energy balance for the calculation ofthe pressure in an electrical installation, the thermal transfer co-efficient at relatively pure conditions is introduced, whichhas been experimentally proven to be constant for the insulatinggas . On the assumption of , the thermal transfer co-efficient for any conditions is then evolved. In the study ofthe thermal transfer coefficient melting and evaporation ofmaterials as well as chemical reactions which contribute to thedevelopment of pressure, are taken into consideration.

It has been proven that a satisfactory agreement between bothcalculated and measured results of the thermal transfer coeffi-cient can be reached. The calculated and measured results showthat the development of pressure and its corresponding thermal

transfer coefficient are dependent on the kinds of insulating gasand electrode material, the size of the test vessel, and the gasdensity. In general, the proposed method can be used especiallyin smaller arcing chambers, where melting and vaporization ofelectrode materials as well as chemical reactions are important.

Furthermore it is demonstrated that the developed approach isable to calculate the development of pressure accurately for anexample of a short-circuit in a compact medium-voltage stationwith heavy metal evaporation. Thus, it is possible to calculatethe development of pressure in dependence on the gas density inelectrical installations with pressure relief openings in a reliableway.

ACKNOWLEDGMENT

This work was finished at Aachen University of Technology,Aachen, Germany.

REFERENCES[1] A. Dasbach and G. J. Pietsch, Calculation of pressure wave in substa-

tion buildings due to arcing faults, IEEE Trans. Power Del., vol. 5, no.4, pp. 17601765, Oct. 1990.

[2] M. Schumacher and G. J. Pietsch, Analysis of pressure phenomena ingas insulated switchgear installations, in Proc. Symp. Electrical Dis-charges in Gases, 1991, pp. 16.

[3] G. Friberg and G. J. Pietsch, Calculation of pressure rise due to arcingfault, IEEE Trans. Power Del., vol. 14, no. 2, pp. 365370, Apr. 1999.

[4] F. Lutz and G. J. Pietsch, The calculation of overpressure in metalen-closed switchgear due to internal arcing, IEEE Trans. Power App. Syst.,vol. PAS-101, no. 11, pp. 42304236, Nov. 1982.

[5] H. Kuwahara, K. Yoshinaga, S. Sakuma, T. Yamauchi, and T. Myamoto,Fundamental investigation on internal arcs in SF gas-filled en-closure, IEEE Trans. Power App. Syst., vol. PAS-101, no. 10, pp.39773987, Oct. 1982.

[6] C. J. Smithells and E. A. Brandes, Metals Reference Book. London,U.K., 1976.

[7] H. D. Baehr, Thermodynamik. Berlin, Germany: Springer Verlag,1990.

[8] S. V. Patankar and D. B. Spalding, Numerical Heat Transfer and FluidFlow. Bristol, PA: Hemisphere, 1980.

[9] M. Born, Optik. Berlin, Germany: Springer Verlag, 1999.

Xiang Zhang received the B.Sc. and M.Sc. degrees in electrical engineeringfrom Xian Jiaotong University, Xian, China, in 1989 and 1992, respectively,and the Ph.D. degree in electrical engineering from the Aachen University ofTechnology, Aachen, Germany, in 2002.

Currently, she is a Research Fellow on asset management of networks ofthe Schering-Institute of High Voltage Technology, University of Hanover,Hanover, Germany. From 1992 to 1997, she was a Research Engineer withXian High Voltage Apparatus Research Institute, Xian, China. Her main areasof interest include high-voltage apparatus, gas discharge, arc modeling, andasset management of network.

Gerhard Pietsch (M83) received the Diploma and Ph.D. degrees in physicsfrom the University of Kiel, Kiel, Germany, in 1967 and 1971, respectively.

From 1972 to 1974, he was with AEG High Voltage Laboratory, Kassel, Ger-many. Since 1975, he has been Professor of Electrical Engineering and Gas Dis-charge Technology at the Aachen University of Technology, Aachen, Germany.

Dr. Pietsch is a member of the German Association of Electrical Engineers(VDE) and the German Physical Society (DPG).

Ernst Gockenbach (M83SM88F01) received the Ph.D. degree in elec-trical engineering from the Technical University of Darmstadt, Darmstadt, Ger-many, in 1979.

Currently, he is Professor and Director of the Schering-Institute of HighVoltage Technology, University of Hanover, Hanover, Germany. From 1979 to1982, he was with Siemens AG, Berlin, Germany. From 1982 to 1990, he waswith E. Haefely AG, Basel, Switzerland.

Dr. Gockenbach is a member of VDE and CIGRE, chairman of CIGRE StudyCommittee D1 Materials and Emerging Technologies for Electrotechnology,and a member of national and international Working Groups (IEC, IEEE) forStandardization of High Voltage Test and Measuring Procedures.

tocInvestigation of the Thermal Transfer Coefficient by the Energy Xiang Zhang, Gerhard Pietsch, Member, IEEE, and Ernst GockenbachI. I NTRODUCTIONII. E NERGY B ALANCE OF F AULT A RCS

Fig.1. Simplified energy balance of a fault arc in an electricaIII. C ONCEPT OF R ELATIVE P URITYIV. E VAPORATION AND C HEMICAL R EACTIONSV. G OVERNING E QUATIONS

TABLE I T EST C ONDITIONSFig.2. Test container. 1, 2: connectors, 3: pressure transmitteVI. S IMULATION R ESULTS

Fig.3. Calculated (without test ) and measured (with test ) $k_Fig.4. Calculated (without test ) and measured (with test ) $k_Fig.5. Calculated (without test ) and measured (with test ) $k_Fig.6. Calculated (without test ) and measured (with test ) $k_Fig.7. Compact medium-voltage station under consideration.TABLE II G EOMETRICAL C ONDITIONSFig.8. Measured electrical power of a short-circuit current of Fig.9. Burned cable compartments of the compact medium-voltage Fig.10. Calculated (full line) and measured (dotted line) pressFig.11. Calculated (full line) and measured (dotted line) pressVII. C ONCLUSIONA. Dasbach and G. J. Pietsch, Calculation of pressure wave in suM. Schumacher and G. J. Pietsch, Analysis of pressure phenomena G. Friberg and G. J. Pietsch, Calculation of pressure rise due tF. Lutz and G. J. Pietsch, The calculation of overpressure in meH. Kuwahara, K. Yoshinaga, S. Sakuma, T. Yamauchi, and T. MyamotC. J. Smithells and E. A. Brandes, Metals Reference Book. LondonH. D. Baehr, Thermodynamik . Berlin, Germany: Springer Verlag, 1S. V. Patankar and D. B. Spalding, Numerical Heat Transfer and FM. Born, Optik . Berlin, Germany: Springer Verlag, 1999.