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International Journal of Thermal Sciences 77 (2014) 165e171

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International Journal of Thermal Sciences

journal homepage: www.elsevier .com/locate/ i j ts

Computational study on anisotropic thermal characterizationof multi-scale wires using transient electrothermal technique

Feng Gong, Yue Cheng, Jin Wen Tan, Soon Ghee Denis Yap, Son Truong Nguyen,Hai M. Duong*

Department of Mechanical Engineering, National University of Singapore, 9 Engineering Drive 1, Singapore 117576, Singapore

a r t i c l e i n f o

Article history:Received 21 June 2013Received in revised form22 October 2013Accepted 25 October 2013Available online

Keywords:Carbon nanotubeNanowireAnisotropic heat transferRadiation heat lossTransient electrothermal techniqueFinite difference method

* Corresponding author.E-mail address: mpedhm@nus.edu.sg (H.M. Duong

1290-0729/$ e see front matter � 2013 Elsevier Mashttp://dx.doi.org/10.1016/j.ijthermalsci.2013.10.018

a b s t r a c t

Numerical models predicting anisotropic heat transfer of multi-scale wires using the transient electro-thermal (TET) technique were successfully developed. Compared to previous models, the developedmodels are more realistic and accurate by taking into account the anisotropic thermal conduction in bothaxial and radial directions and the radiation heat loss from the wire surface to the measurement ambient.In the TET technique, the to-be-measured wire is placed between two electrodes. By feeding a step DC tothe wire, its temperature increases and eventually reaches the steady state. The temperature evolution isprobed by measuring the variation of voltage/resistance over the wire, which is then used to determinethe axial and radial thermal diffusivities of the wire. For the first time, the developed models are solvedusing implicit finite difference method, giving more accurate predictions than the previous models usingGreen’s function. The obtained results are in excellent agreement with the experimental data. Using thevalidated models, the effects of various wire morphologies (radius of 10e200 mm, length of 5e20 mm),and experimental conditions (DC supply of 5e50 mA and ambient temperature of 0e25 �C) on thethermal characterization of the wires were also quantified. Our results are beneficial to experimentalistson optimization of measurement conditions of the experiments characterizing the thermal properties ofmulti-scale wires such as carbon-based microfibers.

� 2013 Elsevier Masson SAS. All rights reserved.

1. Introduction

In order to promote potential engineering applications of micro/nanoscale materials, tremendous efforts have been put intoresearch to understand better the fundamental properties of micro/nanoscale materials. However, the research of thermal transport inthe micro/nanoscale structures has been crucial. Several techniques[1e6] have been developed to study thermophysical properties ofwires/tubes at micro/nanoscale, such as the 36 method [1,2,5,6],the pulsed laser-assisted thermal relaxation (PLTR) technique [3]and the transient electrothermal (TET) technique [4]. The 36methodwas first developed to replace conventional techniques [7e19] and measure thermal properties of carbon nanotubes (CNTs)and thin films at micro/nanoscale. By minimizing heat loss onmeasuring probes, experimental results can be more accurate.

Compared to the 36 method, the TET technique produces ahigher signal to noise ratio and the experiment time is significantlyreduced. Moreover, the TET technique can be used to measure

).

son SAS. All rights reserved.

thermophysical properties of conductive, semiconductive, andnonconductive wires. The PLTR technique has been developedsubsequently but still have some limits compared to the TETtechnique. In the PLTR technique, pulse laser with duration ofseveral nanoseconds is used to heat the to-be-measured wire.However, due to the laser reflection at the wire surface and lasertransmission through the wire, the thermal energy absorbed by thesample is difficult to be determined in the PLTR technique. Whenmeasuring the thermal conductivities of semiconductive andnonconductive wires, thin films are coated on the wire to make thesamples conductive and hence, the laser reflection on the wiresurfaces will increase due to the coated metal thin films on thewires [3]. In addition, the thermal heating of the supply current inPLTR technique is ignored which will result in some inaccuracies ofthe measured results [3]. Moreover, laser equipment used in thePLTR technique is normally expensive and the experiment systemsare always relatively complex. Due to these limitations and inac-curacies of the 36 method and the PLTR technique, the TET tech-nique is more preferred in this work.

In the TET technique [4], samplewire is suspended between twoelectrodes as shown in Fig. 1. Heat loss to the ambient throughconvection in vacuum environment is ignored. When experiment

Nomenclature

cp specific heat capacity (J/kg K)kr radial thermal conductivity (W/Km)l dimensionless lengthL0 length (m)qloss heat loss flux (W/m2)_Q heat generation rate (W/m3)r radius (m)r0 maximal radius of wire (m)R dimensionless radiusRe electric resistance (U)TN ambient temperature (K)

T maximal temperature (K)Tr0 wire surface temperature (K)x axial parameter (m)

Greek symbolsax axial thermal diffusivity (m2/s)ar radial thermal diffusivity (m2/s)r density (kg/m3)s StefaneBoltzmann constant (W/m3 K4)q dimensionless temperatureqr0 dimensionless temperature at surfaces dimensionless time

F. Gong et al. / International Journal of Thermal Sciences 77 (2014) 165e171166

starts, a step DC is supplied to the wire to introduce electricalheating. Upon heating, the temperature of the sample wire willincrease and eventually reach the steady state. The time required toreach the steady state strongly depends on the morphologies andthermophysical properties of the wires. With same thermophysicalproperties and experimental conditions, a longer wire requires alonger time to reach the steady state. Similarly, with same wirelength and experimental conditions, a wire with a larger thermaldiffusivity/conductivity will take shorter time to reach its steadythermal state. The temperature evolution during the heating pro-cess can be probed by measuring the voltage/resistance variationover the wire [4].

Guo et al. [4] applied the TET technique to measure the thermaldiffusivities of single-wall carbon nanotube (SWNT) bundles andpolyester fibers. A 25.4 mm thick platinum wire was used as thereference sample to verify this technique. The thermal diffusivitiesof SWNT bundles obtained from the temperature change profilewere determined by applying linear fitting at the initial stage ofelectrical heating when 0 < t<Dt, where Dt is very small and thetemperature gradient along the wire is also small. The temperaturechangewith timewas described as DT¼ q0/rcpDt, where q0, r and cpare the heat generation rate per unit volume, the density andspecific heat capacity of the wire, respectively. However, theexisting models of Guo et al. [4] were solved by Green’s function,which only considered one-dimensional (axial direction) heattransfer and ignored the radial heat conduction and heat loss

Fig. 1. Schematic of the transient electrothermal (TET) technique principle and thedeveloped anisotropic heat transfer model with heat loss. The to-be-measured samplewire is suspended between two electrodes, a step DC is fed to the sample to provideelectrical heating. The temperature evolution is probed by measuring the variation ofvoltage over the wire.

through radiation to the ambient. Due to the model assumptionsand the limits of the Green’s function, these models used in Guoet al.’s work might not predict accurate thermal conductivities ofSWNT bundles and polyester fibers [5].

In this paper, modified computational models were developed.The developed models eliminate several model assumptions ofprevious works by taking into account the anisotropic heat transferand the radiation heat loss from the wire surface to the experi-mental ambient. Using an implicit finite difference method to solvethe developedmodels is more powerful than the integral of Green’sfunction [5]. As the developed models consider heat transfer inboth axial and radial directions, as well as heat loss from surface ofthe wire through radiation, the heat transfer phenomena and thetemperature distribution in the wire are more accurately predicted.Implementing global fitting method to analyze the temperatureevolution, more accurate thermophysical properties of the to-be-measured wires can be derived. To validate our developedmodels, experimental data of SWCNT bundles [4] were used onlydue to the limitation of experimental data. The simulation resultsgave a better agreement with the experimental data than Guoet al.’s models [4]. For helping experimentalists to optimize theirmeasurement conditions, effects of various wire morphologies(radius of 10e200 mm, length of 5e20 mm) and experimentalconditions (DC supply of 5e50 mA and ambient temperature of 0e25 �C) on the heat transfer characteristics of multi-scale wires werealso quantified by using our validated models.

2. Computational model development

There have been several physical models developed in previousstudies to derive the thermal diffusivity of micro/nanoscale wiresusing the temperature evolution history generated from experi-ments [3e5]. In order to make reported models more realistic, andthus more accurate, our developed models considered the heattransfer both in axial direction and in radial direction as well as theheat loss to the ambient through radiation.

A schematic configuration of the physical model considered inthis study is shown in Fig. 1. The wire is mounted between twoelectrode blocks and a step DC is supplied to introduce electricalheating of the wire. During the heating process, heat is consideredto be transported in both radial direction and axial direction, anddue to the vacuum ambient, heat is only lost to the ambient throughradiation from the surface (denoted by arrows in Fig. 1). In thefigure, ax and ar denote the axial and radial thermal diffusivities,respectively.

Considering the geometry of micro/nanoscale wires, the cylin-drical coordinate system was adopted. The transient heat equationfor the anisotropic model in cylindrical coordinate, can bewritten as:

F. Gong et al. / International Journal of Thermal Sciences 77 (2014) 165e171 167

vT ¼ av2T þ a

�1 v

�rvT

��þ

_Q �2εs T4r0 � T4N

(1)

vt x

vx2 r r vr vr rcp

� �rcpr0

where s¼ 5.6703�10�8 W/m2 K4 is the StefaneBoltzmann con-stant. ε is the emissivity of the wire sample. Owing to the carbon-based wires utilized as the sample in the current study, ε isassumed to be unity for simplicity in the following equation [20]. r,cp and _Q are the density, the specific heat capacity at constantpressure and the energy generated rate per unit volume of thewire,respectively. Tr0 and TN are the temperature at the surface of thewire and the temperature of the ambient, repectively.

In the experiment, the heat source _Q is the result of electricalheating due to supplied current I. Using Ohm’s Law, the rate of heatsource per unit volume can be expressed as:

_Q ¼ I2Repr20L0

(2)

where Re, L0, and r0 are the resistance, the length and the radius ofthe wire, respectively.

Eq. (1) is normalized using the following dimensionless terms:

q ¼ T � TNTm � TN

(3)

s ¼ axtL20

(4)

R ¼ rr0

(5)

l ¼ xL0

(6)

where q, s, R and l are the non-dimensional temperature, time,radius and length, respectively. TN, T and ax are the ambient tem-perature, the maximal temperature and the thermal diffusivity ofthe wire in axial direction, respectively. Substitution of Eqs. (3)e(6)into Eq. (1) yields:

vq

vs¼ v2q

vl2þ ar

al

L20r20

1Rvq

vRþ v2q

vR2

" #þ L20al Tm � TNð Þ

_QrCp

� 2L20saxrcpr0

Tm � TNð Þ3q4r0 þ 4TN Tm � TNð Þ2q3r0h

þ6T2N Tm � TNð Þq2r0 þ 4T3Nqr0 � (7)

where qr0 is the dimensionless temperature at the surface of thesample.

The physical boundary conditions are yet to be defined in orderto obtain a unique solution to the governing equations. At the startof the experiment, t¼ 0, the samplewire and the two electrodes areat the ambient temperature TN, which is kept constant at 25 �C(298 K) in this paper. The initial condition of this study is:

Tðr;0Þ ¼ 298 K and Tðx;0Þ ¼ 298 K for 0� r � r0 and 0� x� L0(8)

Similarly, in the dimensionless terms, the initial condition canbe expressed as:

qðR;0Þ ¼ 0 and qðl;0Þ ¼ 0 for 0 � R � 1 and 0 � l � 1 (9)

In the experiment, the wire is mounted between two electrodeblocks. Since the thermal diffusivity of the electrode is very high,

the temperature of the electrode remains at the ambient value TNduring the course of experiment. Therefore, the temperatures at thetwo ends of the wire, x¼ 0 and x¼ L0, are fixed at the ambienttemperature TN. This condition is represented as the Dirichletboundary condition of prescribed temperature at x¼ 0 and x¼ L0.The Dirichlet boundary conditions can be expressed mathemati-cally as:

Tð0; tÞ ¼ TN and TðL0; tÞ ¼ TN for 0 � r � r0 and t > 0(10)

Alternatively, using the dimensionless terms, the boundaryconditions in Eq. (10) can be expressed as:

qð0; sÞ ¼ 0 and qð1; sÞ ¼ 0 for 0 � R � 1 and s > 0 (11)

In the proposed physical model, heat is lost to the ambientthrough radiation from the wire surface at radius r¼ r0. Consid-ering a small section of length vx, the radiation heat loss qloss isgiven by:

qloss ¼ s T4 � T4N� �

$ 2pr0vxð Þ (12)

The heat conduction _qr in the radial direction for the samesection is given by

_qr ¼ �krvTvr

jr¼r0

$ 2pr0vxð Þ (13)

where kr is the thermal conductivity in the radial direction.Assuming a thermal equilibrium at the outer surface, the radiationheat loss equals to the radial heat conduction. Substituting Eq. (13)into Eq. (12), we obtain the following expression for the tempera-ture gradient as a function of heat loss at the surface as following:

vTvr

jr¼r0

¼ �s

krT4 � T4N

� �(14)

Eq. (14) defines a Neumann boundary condition of prescribedtemperature gradient at the surface. Substituting the dimensionlessterms into Eq. (14) yields Neumann boundary condition indimensionless:

vq

vR

����R¼1

¼ � sr0arrcp

�ðTm � TNÞ3q4r0 þ 4TNðTm � TNÞ2q3r0

þ 6T2NðTm � TNÞq2r0 þ 4T3Nqr0

�(15)

The dimensionless equations of the developed models aresolved using the implicit finite difference method. Various param-eters of wire morphologies (radius, length of 5e20 mm, axial andradial thermal diffusivities) and experimental conditions (DC sup-ply and ambient temperatures) are quantified. Subsequently, thewire is heated gradually until the thermal steady state. The tem-perature of the wire sample is calculated and recorded every0.00025 s. The simulation will stop when the temperature of wiresample no longer change with time, which indicates the thermalsteady state. All calculations described above were programmedand solved using the Fortran 90 language.

3. Simulation results and discussions

3.1. Model validation

A range of thermal diffusivity was tested and the trial valuegiving the best fit (minimum error) of the experiment data wastaken as the sample’s thermal diffusivity. The tested axial thermal

Table 1Base case simulation parameters.

Simulation parameters Simulation values

Nanowire morphologyRadius, r0[mm]a 32.5Length heated by the laser, L0 [mm]a 1.31Density, r [kg/m3] [18] 1900Specific heat capacity, cp [J/kg K] [18] 470

Thermal diffusivity of nanowireRadial thermal diffusivity ar, [m2/s] 2.72 � 10�6

Axial thermal diffusivity, ax [m2/s]a 2.72 � 10�5

Other parametersDC supply [mA] 8.039Ambient temperature [�C] 25StefaneBoltzmann constant, s [W/m2 K4] 5.67� 10�8

a Used in the TET experiment conducted by Guo [4].

F. Gong et al. / International Journal of Thermal Sciences 77 (2014) 165e171168

diffusivity values ranged from 1.5 �10�5 to 3.9� 10�5 m2/s with aninterval of 0.1 �10�5 m2/s, while the axial to radial diffusivity ratio,ax/ar, was kept at 10. The other parameters used for these base casesimulations canbe found inTable 1. Thefitting errorwas determinedusing the R2 coefficient determination and the root mean squaremethod [21,22] as described in Eqs. (16) and (17), respectively.

1� R2 ¼P

iðai � fiÞ2Piðai � aÞ2

(16)

εrms ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPN

i ½ai � fi�2N

s(17)

Error analysis was carried out for each trial and presented inFig. 2. It can be seen that the minimum error point was between2.70 � 10�5 and 3.14 � 10�5 m2/s. To find the axial diffusivity valuecorresponding to the minimum error, more simulations are con-ducted within this range with smaller discretization of0.02 � 10�5 m2/s. Suggested by both the R2 coefficient determina-tion and the root mean squaremethod, the best fit simulation curveoccurs when the error value is minimum at the thermal diffusivityof 2.92� 10�5 m2/s. Therefore, the thermal diffusivity of the samplewire is determined to be 2.92 � 10�5 m2/s. Comparing with2.72 � 10�5 m2/s which is the thermal diffusivity generated usingthe previous model [4], current anisotropic simulation model isable to increase the accuracy by 7%. This happens as the previousmodels only considered the one-dimension heat transfer alongaxial direction and ignored the heat loss to the surrounding me-dium through radiation [4]. This proposed model also gives 4%more accuracy comparing with our previous work using the PLTRtechnique [23], where similar method was used and the accuracywas increased by 3% comparedwith the previousmodel [4]. Furtherdiscussion in Section 3.2 also proved that the radial heat conduc-tion should not be ignored in the previous models.

The same global fitting method was implemented to determinethe best ax to ar ratio. Fig. 3 shows the error analysis using both theR2 coefficient determination and the root mean square methods. Itcan be observed that from both analysis methods, the minimumerror occurs when ax/ar is 40. Hence, the predicted values for axialand radial diffusivities are 2.92 � 10�5 and 7.3 � 10�7 m2/s,respectively. The simulation results performed with these diffu-sivity values are highly consistent with the experimental data,which is shown in Fig. 4. The axial to radial diffusivity ratio of 40 isused for further simulations.

Fig. 2. Variation of error with diffusivity. Error analysis is carried out using both the R2

coefficient determinant and the root mean square method. A common diffusivity valueis suggested at the minimum error point which is then taken as the prediction result.

3.2. Effects of anisotropic thermal property on heat transfermechanism

Fig. 5 presents the temperature profiles of the sample wire as afunction of time at different ax/ar ratios varying from 1 to 10,000.The results are obtained from the simulations numbered from 1 to5 as shown in Table 2. When ax/ar is 10,000 or larger, the simulatedmodel represents a very long wire where radial diffusivity is small.So less heat is conducted in the radial direction and heat loss to theambient through the radiation is negligible. On the other hand,when the ax/ar ratio is much smaller than 10,000 heat transferoccurs in both axial and radial directions. It can be observed in Fig. 5that when ax/ar is 10,000, the steady state temperature of the wireis higher than those observed for the wires with smaller ax/ar ra-tios. This means the anisotropic heat transfer and heat lost toambient should not be ignored like previous models’ assumptionswhen the ax/ar is less than 10,000. This observation agrees wellwith our previous discussion where very large ax/ar ratio is used tosimulate one directional heat transfer without heat loss throughradiation.

3.3. Effects of DC supply and wire morphology on heat conductionmeasurement

Other than analyzing the thermophysical properties of a wirebased on the experimental data collected, thismodel can be applied

Fig. 3. Error analysis is carried out to determine the best ax to ar ratio using the RootMean Square method (black) and the R2 coefficient determinant (red). Both erroranalysis methods indicate that the minimum error occurs when ax/ar is 40. (Forinterpretation of the references to color in this figure legend, the reader is referred tothe web version of this article.)

Fig. 4. Comparison of simulation results with experiment data [4]. Simulation inputsare shown in Table 1. Since the simulation result agrees well with the experiment data,the simulation model is validated.

F. Gong et al. / International Journal of Thermal Sciences 77 (2014) 165e171 169

to predict how heat conducts within a wire under various condi-tions. Based on the simulation results, the effect of each experimentcondition on the heat transfer mechanism can be determined. Forinstance, a longer wire takes longer time to reach its steady thermalstate and a higher DC supply will heat up the wire to a highertemperature over a longer period of time.

To investigate the effect of DC supply on the heat conduction ofthewire, simulations numbered from 6 to 11 in Table 2 were carriedout by varying the DC supply from 5 to 50 mA. The temperaturechanges of the wire as a function of heating time with different DCsupplies are shown in Fig. 6. It is observed that the steady statetemperature as well as the time required to reach the state bothincrease when increasing the DC supply. The wire’s temperaturechanges are very small at low DC values of 5 and 10 mA, so DCvalues lower than 10 mA are not recommended for thermal mea-surement. When the DC supply is bigger, the steady state temper-ature is significant. For example, steady state temperatures of 370,450, 570 and 730 K are reached after 0.015, 0.020, 0.023 and 0.031 s

Fig. 5. Absolute temperature evolution profile with varying ax/ar values 1, 10, 100,1000 and 10,000. It can be seen that when ax/ar is 10,000, the steady state temperatureof the wire is higher than the rest. This can be explained by considering it as only onedirectional heat transfer without heat loss to the ambient. The significance of aniso-tropic heat transfer is observed.

with DC values of 20, 30, 40 and 50 mA, respectively. During theexperiment, it is important to keep the sample below its burningtemperature. For carbon nanowires, the average burning temper-ature ranges from 673 to 873 K (400e600 �C) [24e30]. Therefore,the current supply must be well controlled. At a DC of 50 mA, thesteady state temperature is 730 K, higher than the burning point ofcarbon nanowires and thus, lower DC supply should be used for theheating. As presented, this model is helpful in determining a suit-able DC supply for a specific wire.

To investigate the effects of wire morphologies on heat trans-port phenomena, simulations numbered from 12 to 14 were con-ducted with the parameters shown in Table 2. In these simulations,sample lengths of 5 mm, 10 mm, and 20 mm were used. Thetemperature evolution produced by the simulation is shown inFig. 7. From the temperature profile, it can be seen that the longerthe wire, the longer it takes to reach its thermal steady state, andthe higher the steady state temperature it stabilizes. In addition,another set of simulations numbered from 18 to 21 in Table 2 wasconducted to study the effect of the radius of the sample on theheat transfer mechanism. The obtained temperature evolutionhistory is shown in Fig. 8. It is observed that with all otherexperiment conditions fixed, a wire with smaller diameter has ahigher steady state temperature. These observations are reason-able because similar to the characteristics of a very thin wire, along wire can be assumed to have only one directional heattransfer. In other words, it simulates a condition where heattransfer in radial direction is negligible, so is the heat loss throughradiation. As a result, a longer or thinner wire settles at a highersteady state temperature given all other conditions unchanged.From Fig. 8, the temperature profiles of the wires with radii of100 mm and 200 mm are unsuitable for experiments as the tem-perature rises are very small. However, this insufficient tempera-ture rise could be solved by increasing the DC supply. Caution mustbe taken so that the increased DC supply will not cause over-heating of the sample.

3.4. Effects of ambient temperatures on the wire heat transfer

The radiation heat loss of the wire is taken into account in thecurrent model which is related to the ambient temperatures. Theeffects of different ambient temperatures are obvious according tothe StefaneBoltzmann law. The higher the ambient temperature,the less heat is lost through radiation. In other words, a lowerambient temperature will result in more heat loss through radia-tion. Given that radiation is ignored in conventional models, thecomputational error is even greater. Since our developed modelsimulates anisotropic heat flow and takes into account of the heatloss to the experiment surroundings, the shape of the temperatureprofile should not vary for different ambient temperatures. Toprove this, simulations numbered from 15 to 17 in Table 2 werecarried out. The absolute temperature evolution history is shown inFig. 9 while the normalized temperature evolution history is shownin Fig. 10. From the absolute temperature evolution history in Fig. 9,it can be seen that each temperature profile starts from its givenambient temperature and rises till the thermal steady state reaches.Fig. 10 presents that despite their different starting temperatures,the normalized temperature profiles coincide with each other. Thisobservation agrees with our previous discussion, and hence,demonstrates the high accuracy of our proposed anisotropicsimulation model. From the simulation results, it is evident that toobtain a suitable temperature profile for any given wires, theexperiment setup must be well designed, and all contributingfactors need to be taken into consideration. With the help of thesimulation model, appropriate experiment conditions can be ob-tained before conducting the experiments.

Table 2Effects of wire morphologies and experimental conditions on heat transfer measured by the TET technique.

Simulationrun

Radius of wirer0 (mm)

Length of wireL0 (mm)

Axial thermaldiffusivity ax (m2 s�1)

Anisotropic heat transferratio ax/ar

DC supplyI (mA)

Ambient temperatureT0 (�C)

1 32.5 20 2.72 � 10�5 1 8.039 252 32.5 20 2.72 � 10�5 10 8.039 253 32.5 20 2.72 � 10�5 100 8.039 254 32.5 20 2.72 � 10�5 1000 8.039 255 32.5 20 2.72 � 10�5 10000 8.039 256 32.5 1.31 2.92 � 10�5 40 5 257 32.5 1.31 2.92 � 10�5 40 10 258 32.5 1.31 2.92 � 10�5 40 20 259 32.5 1.31 2.92 � 10�5 40 30 2510 32.5 1.31 2.92 � 10�5 40 40 2511 32.5 1.31 2.92 � 10�5 40 50 2512 32.5 5 2.92 � 10�5 40 8.039 2513 32.5 10 2.92 � 10�5 40 8.039 2514 32.5 20 2.92 � 10�5 40 8.039 2515 32.5 1.31 2.92 � 10�5 40 8.039 016 32.5 1.31 2.92 � 10�5 40 8.039 1517 32.5 1.31 2.92 � 10�5 40 8.039 2518 10 1.31 2.92 � 10�5 40 8.039 2519 32.5 1.31 2.92 � 10�5 40 8.039 2520 100 1.31 2.92 � 10�5 40 8.039 2521 200 1.31 2.92 � 10�5 40 8.039 25Base Case 32.5 1.31 2.72 � 10�5 1 8.039 25

Bold entries indicate changed values with respect to the base case.

F. Gong et al. / International Journal of Thermal Sciences 77 (2014) 165e171170

4. Conclusions

In this work, the computational models based on the transientelectrothermal (TET) technique are developed successfully for theanisotropic thermal characterization of multi-scale wires. The TETtechnique overcomes the limits and disadvantages of previousthermal characterization techniques such as the 36 method andthe PLTR technique. Implicit finite difference method is applied tosolve the transient heat equations in dimensionless, and to provideunconditional stability. The merits of the developed models overcurrent ones are as follows. (i) The developed models are morerealistic and can characterize more accurately the anisotropicthermal characteristics of the wires, (ii) the developed modelsprove that radial heat conduction and heat lost should not beignored in the previous models, and (iii) the simulation results areuseful for the experimentalists optimizing their thermal measure-ment conditions. By implementing the boundary conditions cor-responding to the prevailing experimental setup, the present study

Fig. 6. Absolute temperature evolution history for various DC supply. This simulationprofile helps to determine the suitable current supply range for the experiment, henceto prevent overheating of the wire. This step is extremely important for experimentplanning.

Fig. 7. Normalized temperature evolution history of wires with length 5 mm, 10 mm,and 20 mm. It is observed that the longer the wire, the longer it takes to reach itsthermal steady state, and the higher the thermal steady state temperature.

Fig. 8. Absolute temperature evolution history of wires with radius 10 mm, 32.5 mm,100 mm, and 200 mm.

Fig. 9. Absolute temperature evolution history at ambient temperature T0 equals to 0,15, and 25 �C.

Fig. 10. Normalized temperature evolution history for ambient temperature T0 equalsto 0, 15, and 25 �C. Though their absolute temperature evolution profiles are different,their normalized profiles coincide with each other. The simulation result agrees wellwith the theory.

F. Gong et al. / International Journal of Thermal Sciences 77 (2014) 165e171 171

can readily used in other problems such as the 36 method and thePLTR technique.

Appendix A. Supplementary data

Supplementary data related to this article can be found at http://dx.doi.org/10.1016/j.ijthermalsci.2013.10.018.

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