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ELECTROTEHNICĂ, ELECTRONICĂ, AUTOMATICĂ, 60 (2012), nr. 314

1. Introduction

From time immemorial, the maximumcontinuous operating performance andcurrent loading capacity of cables and wiresis important criterion in the design of cablesystems (materials, geometry, construction)and design of power electrical installation(e.g. deposit method).

The passing electrical current by meansof Joule losses heats up the conductor andtherefore its temperature and temperature ofits electrical insulation system exceeds theambient temperature. This temperaturedepends quadratically on the passingcurrent. Long-term exceeding of projectedoperating temperatures of cable conductorscauses the thermal degradation [1],especially in cooperation with electric fieldcauses significantly faster aging of power

cables [2] or HV cables [3, 4] insulation,increased corrosion of cable cores andterminals [5] in extreme cases anddeterioration of mechanical properties of

1 Michal VÁRY, Dipl. Ing., PhD., Department of Materials andTechnologies, Institute of Power and Applied ElectricalEngineering, Slovak University of Technology, Faculty ofElectrical Engineering And Information Technologies,Bratislava 812 19, Slovak Republic, [email protected]

2 Vladimír GOGA, Dipl. Ing., PhD., Department of Mechanics,Institute of Power and Applied Electrical Engineering,Slovak University of Technology, Faculty of ElectricalEngineering And Information Technologies, Bratislava 812

19, Slovak Republic, [email protected] Juraj PAULECH, Dipl. Ing., Department of Mechanics,Institute of Power and Applied Electrical Engineering,Slovak University of Technology, Faculty of ElectricalEngineering And Information Technologies, Bratislava 81219, Slovak Republic, [email protected]

insulation and interfacial agingperformance [6], especially in compositematerial systems [7]. Also synergiccooperation of various degradation factorshave to be considered [8, 9].

In recent years, there has been rapiddevelopment of computational methods,models and simulations, which allows, as willbe shown, to determine the steady state

temperature of the current passing the bareconductor. In this paper, two numericalmethods will be presented – the FiniteElement Method (FEM), represented bysimulation in ANSYS Workbench and theFinite Volume Method (FVM) represented byanalysis in ANSYS CFX.

2. Materials and Methods

The proposed elementary model consistsof horizontally arranged bare electric copper

conductor (thermal conductivity401 Wm-1K-1) with diameter of 1,48 mm. Theconductor carried AC current according tomeasurement order from 5 to 30 A RMS.Cooling of the conductor was only due tofree convection and radiation effects(ambient temperature was 22 °C) [10].

Experimental measurements wereperformed to determine reference surfacetemperatures of the conductor under steadythermal-electric state. These values werecompared to the temperatures obtained from

analytical calculation and numericalsimulations.In next chapters, four approaches to

determine cooling of electric conductor via

Experimental, Analytical and Computational Approaches toBare Electric Wire Loading Characteristics

Michal VÁRY1, Vladimír GOGA2, Juraj PAULECH3

Abstract

This article describes cooling analyses of horizontally arranged bare electric conductor using analytical andnumerical methods. Analytical solution results from Fourier differential equation of thermal conductivityextended by radiation and convection effect. Two different numerical approaches will be considered. Results ofthese analyses will be compared to the results obtained from experimental measurements. The proposedelementary model consists of horizontally arranged bare electric copper conductor with diameter of 1,48 mm.The conductor was loaded with AC current from 5 to 30 A RMS.

Keywords: electro-thermal analysis, free convection, radiation, cooling of a bare horizontal conductor, analyticalsolution, ANSYS Workbench, ANSYS CFX, experimental measurement

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ELECTROTEHNICĂ, ELECTRONICĂ, AUTOMATICĂ, 60 (2012), nr. 3 15

free convection will be shown – analytical,two numerical solutions and experimentalmeasurement.

3. Results

The experimental results proposed in thispaper were achieved by loadingcharacteristic measurements for horizontalCu bare conductor, diameter 1,48 mm (seeFigure 1 for connection details).

Figure 1. Schematic illustration of the experimentalsetup

Current in the measurement circuit (5, 10,15, 20, 25 and 30 Amps AC) was regulatedby autotransformer (AT) and currenttransformer (PT) and measured by analogammeter (A) (precision 0.5 %). Conductorsurface temperature was measured byK-type thermocouple (chromel-alumel) [11]and real time logged into computer by Fluke289 multimeter (MT) and support logging

software Fluke View Forms (PC) [12]. Thetime logging interval was set to 1 second.The temperature measurement precisionwas 1 %. Ambient temperature duringmeasurement was 22 °C (see Figure 2 fortime evolution).

Figure 2. Time evolution of conductor surfacetemperature (loading currents: 5, 10, 15, 20,25, 30 A)

The results of measurements are showenin Table 1.

Table 1. Conductor surface temperature –measurement

I[A] Tmeasure[°C]5 24,60

10 30,0015 38,3020 50,4025 65,6230 84,24

The analytical solution results fromFourier differential equation of thermalconductivity that was extended by radiationand convection effect [13]. Convectioncoefficient was calculated according to

criterion equations and it was set up astemperature dependent variable. The resultsfrom analytical solution are showen inTable 2.

Table 2. Conductor surface temperature – theanalytical solution

I [A] Tanalytical [°C]5 24,1410 30,1515 39,5420 52,2925 68,80

30 89,84

To obtain numerical solution ANSYSWorkbench program was used, wheresteady-state electro-thermal (abbr. E-T)simulation was performed. The model and allboundary conditions were created accordingto the analytical solution. This approach wasrecently used for LV and MV cableevaluation [14] and simulation of dielectriclosses in power cables [15]. Results from

steady state E-T analysis are in Table 3.Table 3. Conductor surface temperature – the

numerical solution

I [A] TE-T [°C]5 24,1010 30,1215 39,5120 52,3225 68,8530 89,80

The last approach of conductor coolingpresented in this paper is fluid CFD analysis

using ANSYS CFX code. This software isleading programme for fluid analysis andthermal distribution. For CFX analysis thereis no need to build analytical equations forthermal field – convection, radiation,conduction in conductor. The solvingprocess is very hardware and time-intensive,because it is necessary to model a relativelylarge area of air around the conductor,where air flow is simulated and thecomputational calculation is iterative

process. The final surface temperaturesresults for ANSYS CFX analysis are inTable 4.

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Table 4. Conductor surface temperature – CFXsimulation

I[A] TCFX [°C]5 25,25

10 32,4515 42,65

20 55,6525 71,8530 90,35

Figure 3 shows the calculated andmeasured surface temperatures in graphicform.

Figure 3. Dependence of conductor surfacetemperature for measurement and individualsolutions

In Table 5, there are differences oftemperature for the individual solutions incomparison with measurement results.

Table 5. Differences of conductor surfacetemperatures for the individual approaches

I[A]

Tmeasure

[°C]∆∆∆∆Tanalytical [%] ∆∆∆∆TE-T [%] ∆∆∆∆TCFX

[%]5 24,60 -1,89 -2,03 2,6410 30,00 0,49 0,40 8,1715 38,30 3,23 3,15 11,3620 50,40 3,75 3,81 10,4225 65,62 4,85 4,92 9,4930 84,24 6,65 6,60 7,25

4. Discussion

Analytical solutionBoundary conditions were Joule heat

generated in conductor by electric losses,radiation and convection conditions. Finally,surface temperature of the conductor wasobtained.

Used physical quantities are:c specific heat of conductor [Jkg-1K-1]cp air air specific heat (constant pressure)

[Jkg-1K-1]d diameter of conductor [m]g gravity [ms-2]

Gr Grashoff number [-]I electric current [A]L characteristic dimension of

conductor [m]

Nu0 Nusselt number (starting value) [-]Nu Nusselt number [-]Pr Prandtl number [-]r radius of conductor [m]R conductor resistivity [Ω]Ra Rayleigh number [-]

S cross-section area of conductor [m2]t surface temperature of conductor

[°C]T surface temperature of conductor [K]tamb ambient temperature [°C]Tamb ambient temperature [K]∆T temperature difference between

conductor surface and ambient [K]V volume of conductor [m3]α convective heat transfer coefficient

[Wm-2K-1]βair air expansion coefficient [K-1]ε emissivity [-]λ thermal conductivity of conductor

material [Wm-1K-1]λair air thermal conductivity

[Wm-1K-1]µair air dynamic viscosity [Nsm-2]νair air kinematic viscosity [m2s-1]ρ density of conductor material [kgm-3]ρe electric resistivity [Ωm]ρair air density [kgm-3]σ Stephan-Boltzman constant

[5.6704×10-8 Wm-2K-4]τ time [s]

ϕ heat flux from the surface ofconductor [Wm-2]ϕK heat flux for convection [Wm-2]ϕR heat flux for radiation [Wm-2]ΦV heat generated in conductor [Wm-3]

General equation for heat transfer in solidmaterials is called Fourier-Kirchhoff law

(1)

For steady state temperature does notdepend on time

(2)

Applying (2) into the (1) we obtainPoisson equation

(3)

Our conductor that carries the electriccurrent is horizontally arranged cylinder, so itis necessary to transform (3) into thecylindrical coordinate system

(4)

Solution of this second-order differentialequation has form

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(5)

where temperature is primary unknown.Cross-section area of conductor is solid

circle, therefore r ranges from 0 to final

radius, therefore the logarithmic term ln(r)has to be eliminated. We determine c1 = 0.Then (5) is altering into (6).

(6)

Integration constant c2 is evaluatedaccording to known heat flow from surface ofconductor. This heat flow is defined by sumof convective and radiation heat flows fromconductor surface. For convection it isdefined:

(7)For radiation it is defined:

(8)

Heat transfer from conductor surface isdescribed by Fourier law

(9)

Boundary condition for surface of conductoris

(10)After some math operations we get

iterative rule for calculation surfacetemperature (indices [i], [i+1] representiterative steps)

(11)

Heat generated in conductor is calculatedaccording to Joule heat looses, so it isnecessary to calculate resistance ofconductor

(12)

Resistivity is temperature dependedvariable. For cooper conductor (ρe 20°C =1,69×10-8 Ω.m) we obtained temperaturedependency in following form [16]:

(13)

Then heat generated in conductor iscalculated as follows:

(14)

Calculation of convective heat transfercoefficient is more complicated. Thiscalculation is based on empirical equations

based on Nusselt number and it istemperature dependent variable:

(15)

Characteristic dimension for horizontal

cylinder is [17]:(16)

Nusselt number is the ratio of convectiveto conductive heat transfer across theboundary and it is calculated using nextequations [17]:

(17)

The Rayleigh number is defined as theproduct of the Grashof number, whichdescribes the relationship between buoyancyand viscosity within a fluid, and the Prandtlnumber, which describes the relationshipbetween momentum diffusivity and thermaldiffusivity. Hence the Rayleigh number itselfmay also be viewed as the ratio of buoyancyand viscosity forces times the ratio ofmomentum and thermal diffusivities. For freeconvection around horizontally arrangedcylindrical conductor these equations can beformulated:

(18)

Air kinematic viscosity is based on

dynamic viscosity and density of air and allthese material properties including airthermal conductivity are temperaturedependent variables

(19)

Air properties for t = 20 °C are presentedin Table 6.

Table 6. Air properties for t = 20 °C

cp air [Jkg-1K-1] Pr [-] βair [K-1]1.007×103 0.7083 3.43×10-3 νair [m2s-1] λair [Wm-1K-1] ρair [kgm-3]1.527×10-5 2.589×10-2 1.186µair [Nsm-2]1.811×10-5

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Temperature dependencies for these airproperties were calculated for specifiedtemperature range 22-127 °C (see Figure 4).

24

26

28

30

32

34

36

38

40

20 40 60 80 100 120 140

α

[ W m - 2 K - 1 ]

t [°C]

calculated

approximated

Figure 4. Calculated and approximated values of α

According to (15) convective heat transfercoefficient was calculated as temperaturedependent variable. Approximation functionfor αapprox was sufficiently achieved, indefined temperature range, by logarithmicregression (20).

(20)

Next unknown parameter in (11), thatdescribes conductor surface temperature, iscoefficient of emissivity ε. For our case, the

value of emissivity was chosen for cooperpolished surface ε = 0,07.Now it is possible to calculate iterative

rule (11) for conductor surface temperaturein chosen range of electric currents, seeTable 2 for results.

Numerical solution – ANSYS Workbench

The model and all boundary conditionswere created according to the analyticalsolution. Material properties of conductorand convective heat transfer coefficient were

set as temperature dependent variables [18].Mesh of finite electro-thermal 3D elementswas created in the software. Surfacetemperature of the conductor was obtaineddirectly from the software, see Figure 5.

Figure 5. Mesh of the conductor (left), temperature ofthe conductor (right)

Number of elements for this simulationwas 3 160. Results from this numericalsolution are in Table 3.

Numerical solution – ANSYS CFX code

Geometry for CFX analysis:– air area around the conductor: block

with dimensions 2000×1000×1 mm(this area represents room with theconductor, thickness 1 mm is due tosymmetry boundary conditionmentioned below),

– conductor geometry: cylindrical surfacewith diameter 1,48 mm and length 1mm,

– position of conductor: a = 1100 mm,see Figure 6.

Figure 6. Conductor position in air area

Mesh

We need to create mesh of volumeelements in the air area. The mesh aroundthe conductor surface and walls must bevery fine (called inflation) because of near-wall boundary flow has to be modeled, seeFigure 7. Final number of elements was208 664.

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Figure 7. Mesh of air area around the conductor

Boundry conditions

Boundaries of the air area were set toadiabatic walls conditions. Air propertieswere chosen according to ANSYS CFXmaterial library. The model was set up tosolve free convection.

The body of conductor itself was notmodeled. Joule heat generation wasconverted to heat flux from surface of theconductor into the air area. The reason whythe thickness of the model is only 1 mm isthat there are symmetry conditions on thefront and back surface of the model. That

means the model itself represents infiniteregion in this direction (infinite length of theconductor).

The boundary conditions for this modelwere:

– heat flux from the surface of conductorϕ based on electric losses (seeTable 7);

Table 7. Heat flux from conductor surface

I [A] ϕϕϕϕ [Wm-2]5 52,910 211,815 476,520 847,125 1323,630 1906,0

– emissivity coefficient on the surface ofconductor ε = 0.07;

– emissivity coefficient on the wallsεw = 0.1.

Solution:

The simulation was calculated as steady-state analysis using cluster computer

16 × 4.4 GHz cores, 64 GB of RAM. Iterativesolution took approximately 4,5 hours forevery load step. Final surface temperature of

the conductor was obtained as averagetemperature on the conductor surface (seeFigure 8).

Figure 8. Conductor surface temperature – ANSYSCFX

But this average temperature is notconstant value during fictive iteration time

(real behaviour of air flow around conductoris oscillating transient stream, called vonKármán stream) [19, 20], therefore theaverage temperature was obtainedaccording to chosen number of iterations atthe end of simulation (see Figure 9).

Figure 9. Temperature changes during iterativeprocess

5. ConclusionsResults from individual solutions were

compared to measured temperature data.Deviations between calculations and

measurement were in acceptable range.Analytical solution was relatively accurate

and simple but unusable for models withcomplex geometry.

More appropriate way to calculate surfacetemperature of the conductor is using theANSYS Workbench environment. There ispossibility to create complex geometry (if theconvective heat transfer coefficient ispossible to define adequately). The mostrobust solution is CFD analysis becausethere is no need to build analytical equationsand calculate convective heat transfercoefficient [21, 22]. Solving process isrelatively more hardware- and time-intensive

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than ANSYS Workbench solution, though.Results presented in this paper show that

there are acceptable differences betweenmeasurement data and results of analyticaland numerical solutions. That is the reason

for using numerical analyses for this purposefirst. The advantages of these analyses arein time- and cost-effectiveness and in thepossibility of optimization of given issue.Currently, this approach, based ondiagnostic measurement input data, can beused for thermal field simulations in cablesand electric machines, wchich results can beused for faulty states and spots evaluationand life time predictions [23].

It is important to mention that thiscomputational approaches may also have

limited application and relevance of results invery complex cases (cable with multipleloaded cores, vertical cable/wire) or in caseswhen the values of constants and theirtemperature dependancies for calculation ofheat transfer are not given or known(underground cables, cables with so calledairbag or similar). In this cases the loadingcapacitance could be better determined byexperimental approach.

6. Acknowledgements

This work was supported by GrantAgency KEGA, grant No. 015STU-4/2012and grant VEGA 1/0534/12.

7. References

[1] BHUIYAN M, MUSILEK P,HECKENBERGEROVA J, KOVAL D.”Evaluating thermal aging characteristics ofelectric power transmission lines”. InProceedings of the IEEE 23rd CanadianConference on Electrical and ComputerEngineering (CCECE 2010); 2010, May 2-5;

Calgary, Canada; p. 1–4.[2] SIMONI L, “A general phenomenological life

model for insulating materials undercombined stress”. In IEEE Transactions onDielectrics and Electrical Insulation. 1999,vol. 6 (no. 2): p. 250 – 258.

[3] MAZZANTI G, MONTANARI GC, SIMONI L,“Insulation Characterization in multistressConditions by Accelerated Life Tests: AnApplication to XLPE and EPR for HV Cables”.In IEEE Insulation Magazine. 1997, vol. 13(no. 6): p. 24 –34.

[4] MONTANARI GC, MAZZANTI G, SIMONI L,“Progress In Electrothermal Life Modeling ofElectrical Insulation during the LastDecades”. In IEEE Transactions on

Dielectrics and Electrical Insulation. 2002,vol. 9 (no. 5): p. 730 – 745.

[5] KOPČA M, POLJOVKA P, VÁRY M, LELÁKJ. ”Electrochemical Corrosion of ContactAreas of Electric Terminals”. In Proceedingsof 5th International Conference Study and

Control of Corrosion in the Perspective ofSustainable Development of UrbanDistribution Grids ; 2006, June 18-20; TárguMures, Romania; p. 119-123.

[6] TANAKA T, “Aging of Polymeric andComposite Insulating Materials”. In IEEETransaction on Dielectrics and ElectricalInsulation. 2002, vol. 9 (no. 5): p. 704-716.

[7] KOPČA M, VÁRY M, DOSOUDIL R,ĎURMAN V, LELÁK J, ”Long-Term Stabilityof Plastic-Ferrite Composites UsingAccelerated Thermal Ageing Method”. InElectrotehnica, Electronica, Automatica. 2011, vol. 59 (no. 3): p. 7-10.

[8] LELÁK J, ĎURMAN V, KOPČA M, PACKA J,VÁRY M, KOZA E. ”Effect of VariousDegradation Factors on the Properties ofPVC Insulated Power Cables”. InProceedings of 5th International ConferenceStudy and Control of Corrosion in thePerspective of Sustainable Development ofUrban Distribution Grids ; 2006, June 18-20;Tárgu Mures, Romania; p. 99-103.

[9] VERBICH O, SULOVÁ J, PACKA J,ĎURMAN V, LELÁK J, VÁRY M, ”Effect of aLong-Term Flooding on the DielectricProperties of Flame Retarding Cables”. InElectrotehnica, Electronica, Automatica. 2008, vol. 56 (no. 1-2): p. 14-16.

[10] PYTLAK P, MUSILEK P, LOZOWSKI E,TOTH J, ”Modelling precipitation cooling ofoverhead conductors”. In Electric PowerSystems Research. 2011, vol. 81 (no. 12): p.2147-2154.

[11]The Omega Temperature MeasurementHandbook

® and Encyclopedia. Vol.

MMXIV™, 7th Edition. Omega EngineeringInc.; 2010.

[12] KOPČA M, VÁRY M, Bezpe č nos ť elektrických zariadení [Safety of ElectricalAppliances]. FEI STU in Bratislava, ISBN978-80-227-3462-2; 2011; p. 35 – 49.

[13] KALOUSEK M, HUČKO B, Prenos tepla[Heat transfer] . FEI STU in Bratislava, ISBN80-227-0881-X; 1996.

[14] ANDREOU GT, LABRIDIS DP, ”ExperimentalEvaluation of a Low-Voltage PowerDistribution Cable Model Based on a Finite-Element Approach”. In IEEE Transactions onPower Delivery. 2007, vol. 22 (no. 3): p. 1455 – 1460.

[15] IDIR N, WEENS Y, FRANCHAUD JJ, ”SkinEffect and Dielectric Loss Models of PowerCables”. In IEEE Transactions on Dielectricsand Electrical Insulation. 2009, vol. 16 (no.1): p. 147 – 154.

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[16] ANSYS, Theory manual for Workbench. 2011.

[17] BENNETT CO, Momentum, Heat and MassTransfer . McGraw-Hill, ISBN 0070046670;1962.

[18] DULUC MC, XIN S, LUSSEYRAN F, LE

QUÉRÉ P, ”Numerical and experimentalinvestigation of laminar free convectionaround a thin wire: Long time scalings andassessment of numerical approach”. InInternational Journal of Heat and Fluid Flow. 2008, vol. 29 (no 4): p 1125-1138.

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8. Biography

Michal VÁRY was born in Nitra(Slovakia), on December 9th,

1979.He graduated the SlovakUniversity of Technology inBratislava (Slovakia), Faculty ofElectrical Engineering and

Information Technologies, in 2004.He received the PhD degree in electricalengineering, materials and technologies, fromthe Slovak University of Technology in

Bratislava (Slovakia), in 2009. He is theassistant and teacher at Institute of Power andApplied Electrical Engineering, Department ofMaterials and Technologies, Slovak Universityof Technology in Bratislava (Slovakia).His research interests concern: cable insulation

systems aging, polarization processes indielectric materials, monitoring of photovoltaicsystems performance, aging of photovoltaicsystems encapsulation materials.

Vladimír GOGA was born inŠtúrovo (Slovakia), on March12th, 1981.He graduated the SlovakUniversity of Technology inBratislava (Slovakia), Faculty ofMechanical Engineering, in2004.

He received the PhD degree in appliedmechanics, from the Slovak University ofTechnology in Bratislava (Slovakia), in 2009.He is the assistant and teacher at Institute ofPower and Applied Electrical Engineering,Department of Applied Mechanics andMechatronics, Slovak University of Technologyin Bratislava (Slovakia).His research interests concern: dynamics ofmechanical systems, structural mechanics,material testing, computational mechanics andmechatronics.

Juraj Paulech was born inTrnava (Slovakia), onSeptember 25th, 1984.He graduated the SlovakUniversity of Technology inBratislava (Slovakia), Faculty of

Electrical Engineering and InformationTechnologies, in 2009.He is PhD. student at Institute of Power andApplied Electrical Engineering, Department ofApplied Mechanics and Mechatronics, SlovakUniversity of Technology in Bratislava

(Slovakia).His research interests concern: thermal-fluid flow dynamics, FEM elements forFGM materials, computer simulations forrenewable energy sources.

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