Computational study on anisotropic therm

International Journal of Thermal Sciences 77 (2014) 165e171 Contents lists available at ScienceDirect International Journal of Thermal Sciences journal homepage: www.elsevier.com/locate/ijts Computational study on anisotropic thermal characterization of 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, Engineering Drive 1, Singapore 117576, Singapore a r t i c l e i n f o a b s t r a c t Article history: Received 21 June 2013 Received in revised form 22 October 2013 Accepted 25 October 2013 Available online Numerical models predicting anisotropic heat transfer of multi-scale wires using the transient electrothermal (TET) technique were successfully developed Compared to previous models, the developed models are more realistic and accurate by taking into account the anisotropic thermal conduction in both axial 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 to the wire, its temperature increases and eventually reaches the steady state The temperature evolution is probed by measuring the variation of voltage/resistance over the wire, which is then used to determine the axial and radial thermal diffusivities of the wire For the first time, the developed models are solved using implicit finite difference method, giving more accurate predictions than the previous models using Green’s function The obtained results are in excellent agreement with the experimental data Using the validated 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 the thermal characterization of the wires were also quantified Our results are beneficial to experimentalists on optimization of measurement conditions of the experiments characterizing the thermal properties of multi-scale wires such as carbon-based microfibers Ó 2013 Elsevier Masson SAS All rights reserved Keywords: Carbon nanotube Nanowire Anisotropic heat transfer Radiation heat loss Transient electrothermal technique Finite difference method Introduction In order to promote potential engineering applications of micro/ nanoscale materials, tremendous efforts have been put into research to understand better the fundamental properties of micro/ nanoscale materials However, the research of thermal transport in the micro/nanoscale structures has been crucial Several techniques [1e6] have been developed to study thermophysical properties of wires/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 36 method was first developed to replace conventional techniques [7e 19] and measure thermal properties of carbon nanotubes (CNTs) and thin films at micro/nanoscale By minimizing heat loss on measuring probes, experimental results can be more accurate Compared to the 36 method, the TET technique produces a higher signal to noise ratio and the experiment time is significantly reduced Moreover, the TET technique can be used to measure * Corresponding author E-mail address: mpedhm@nus.edu.sg (H.M Duong) 1290-0729/$ e see front matter Ó 2013 Elsevier Masson SAS All rights reserved http://dx.doi.org/10.1016/j.ijthermalsci.2013.10.018 thermophysical properties of conductive, semiconductive, and nonconductive wires The PLTR technique has been developed subsequently but still have some limits compared to the TET technique In the PLTR technique, pulse laser with duration of several nanoseconds is used to heat the to-be-measured wire However, due to the laser reflection at the wire surface and laser transmission through the wire, the thermal energy absorbed by the sample is difficult to be determined in the PLTR technique When measuring the thermal conductivities of semiconductive and nonconductive wires, thin films are coated on the wire to make the samples conductive and hence, the laser reflection on the wire surfaces will increase due to the coated metal thin films on the wires [3] In addition, the thermal heating of the supply current in PLTR technique is ignored which will result in some inaccuracies of the measured results [3] Moreover, laser equipment used in the PLTR technique is normally expensive and the experiment systems are always relatively complex Due to these limitations and inaccuracies of the 36 method and the PLTR technique, the TET technique is more preferred in this work In the TET technique [4], sample wire is suspended between two electrodes as shown in Fig Heat loss to the ambient through convection in vacuum environment is ignored When experiment 166 F Gong et al / International Journal of Thermal Sciences 77 (2014) 165e171 Nomenclature cp kr l L0 qloss Q_ r r0 R Re TN specific heat capacity (J/kg K) radial thermal conductivity (W/K m) dimensionless length length (m) heat loss flux (W/m2) heat generation rate (W/m3) radius (m) maximal radius of wire (m) dimensionless radius electric resistance (U) ambient temperature (K) starts, a step DC is supplied to the wire to introduce electrical heating Upon heating, the temperature of the sample wire will increase and eventually reach the steady state The time required to reach the steady state strongly depends on the morphologies and thermophysical properties of the wires With same thermophysical properties and experimental conditions, a longer wire requires a longer time to reach the steady state Similarly, with same wire length and experimental conditions, a wire with a larger thermal diffusivity/conductivity will take shorter time to reach its steady thermal state The temperature evolution during the heating process can be probed by measuring the voltage/resistance variation over the wire [4] Guo et al [4] applied the TET technique to measure the thermal diffusivities of single-wall carbon nanotube (SWNT) bundles and polyester fibers A 25.4 mm thick platinum wire was used as the reference sample to verify this technique The thermal diffusivities of SWNT bundles obtained from the temperature change profile were determined by applying linear fitting at the initial stage of electrical heating when < t (10) Alternatively, using the dimensionless terms, the boundary conditions in Eq (10) can be expressed as: q0; sị ẳ and q1; sị ẳ for R and s > In the proposed physical model, heat is lost to the ambient through radiation from the wire surface at radius r ¼ r0 Considering a small section of length vx, the radiation heat loss qloss is given by:   $ð2pr0 vxÞ qloss ¼ s T À TN x L0 (6) where q, s, R and l are the non-dimensional temperature, time, radius and length, respectively TN, T and ax are the ambient temperature, the maximal temperature and the thermal diffusivity of the wire in axial direction, respectively Substitution of Eqs (3)e(6) into Eq (1) yields: " # L20 vq v2 q ar L20 vq v2 q Q_ ỵ ẳ 2ỵ þ al r02 R vR vR2 al ðTm À TN Þ rCp vs vl vT vr j r¼r0 $ð2pr0 vxÞ 2L20 s h ðT À TN Þ3 qr0 þ 4TN ðTm À TN Þ2 qr0 ax rcp r0 m (7) where qr0 is the dimensionless temperature at the surface of the sample The physical boundary conditions are yet to be defined in order to obtain a unique solution to the governing equations At the start of the experiment, t ¼ 0, the sample wire and the two electrodes are at the ambient temperature TN, which is kept constant at 25  C (298 K) in this paper The initial condition of this study is: r0 and x L0 (13) where kr is the thermal conductivity in the radial direction Assuming a thermal equilibrium at the outer surface, the radiation heat loss equals to the radial heat conduction Substituting Eq (13) into Eq (12), we obtain the following expression for the temperature gradient as a function of heat loss at the surface as following: vT vr j r¼r0  s 4 ¼ À T À TN kr (14) Eq (14) defines a Neumann boundary condition of prescribed temperature gradient at the surface Substituting the dimensionless terms into Eq (14) yields Neumann boundary condition in dimensionless: sr0  vq Tm TN ị3 qr0 ỵ 4TN Tm TN ị2 qr0 ẳ ar rcp vRRẳ1  2 qr0 ỵ 6TN Tm TN ịqr0 ỵ 4TN (12) q_ r ẳ kr l ẳ qr0 Tm TN ịqr0 ỵ 4TN ỵ6TN (11) (4) (5) r T0; tị ẳ TN and TL0 ; tị ẳ TN The heat conduction q_ r in the radial direction for the same section is given by r r0 Tr; 0ị ẳ 298 K and Tx; 0ị ẳ 298 K for the temperature of the electrode remains at the ambient value TN during the course of experiment Therefore, the temperatures at the two ends of the wire, x ¼ and x ¼ L0, are fixed at the ambient temperature TN This condition is represented as the Dirichlet boundary condition of prescribed temperature at x ¼ and x ¼ L0 The Dirichlet boundary conditions can be expressed mathematically as: (3) R ¼ À 167 (15) The dimensionless equations of the developed models are solved using the implicit finite difference method Various parameters of wire morphologies (radius, length of 5e20 mm, axial and radial thermal diffusivities) and experimental conditions (DC supply and ambient temperatures) are quantified Subsequently, the wire is heated gradually until the thermal steady state The temperature of the wire sample is calculated and recorded every 0.00025 s The simulation will stop when the temperature of wire sample no longer change with time, which indicates the thermal steady state All calculations described above were programmed and solved using the Fortran 90 language (8) Similarly, in the dimensionless terms, the initial condition can be expressed as: qR; 0ị ẳ and ql; 0ị ¼ for R and l (9) In the experiment, the wire is mounted between two electrode blocks Since the thermal diffusivity of the electrode is very high, Simulation results and discussions 3.1 Model validation A range of thermal diffusivity was tested and the trial value giving the best fit (minimum error) of the experiment data was taken as the sample’s thermal diffusivity The tested axial thermal 168 F Gong et al / International Journal of Thermal Sciences 77 (2014) 165e171 diffusivity values ranged from 1.5  10À5 to 3.9  10À5 m2/s with an interval of 0.1  10À5 m2/s, while the axial to radial diffusivity ratio, ax/ar, was kept at 10 The other parameters used for these base case simulations can be found in Table The fitting error was determined using the R2 coefficient determination and the root mean square method [21,22] as described in Eqs (16) and (17), respectively P ða fi ị2 R2 ẳ Pi i i aị rms s PN i ẵai À fi Š ¼ N (16) (17) Error analysis was carried out for each trial and presented in Fig It can be seen that the minimum error point was between 2.70  10À5 and 3.14  10À5 m2/s To find the axial diffusivity value corresponding to the minimum error, more simulations are conducted within this range with smaller discretization of 0.02  10À5 m2/s Suggested by both the R2 coefficient determination and the root mean square method, the best fit simulation curve occurs when the error value is minimum at the thermal diffusivity of 2.92  10À5 m2/s Therefore, the thermal diffusivity of the sample wire is determined to be 2.92  10À5 m2/s Comparing with 2.72  10À5 m2/s which is the thermal diffusivity generated using the previous model [4], current anisotropic simulation model is able to increase the accuracy by 7% This happens as the previous models only considered the one-dimension heat transfer along axial direction and ignored the heat loss to the surrounding medium through radiation [4] This proposed model also gives 4% more accuracy comparing with our previous work using the PLTR technique [23], where similar method was used and the accuracy was increased by 3% compared with the previous model [4] Further discussion in Section 3.2 also proved that the radial heat conduction should not be ignored in the previous models The same global fitting method was implemented to determine the best ax to ar ratio Fig shows the error analysis using both the R2 coefficient determination and the root mean square methods It can be observed that from both analysis methods, the minimum error occurs when ax/ar is 40 Hence, the predicted values for axial and radial diffusivities are 2.92  10À5 and 7.3  10À7 m2/s, respectively The simulation results performed with these diffusivity values are highly consistent with the experimental data, which is shown in Fig The axial to radial diffusivity ratio of 40 is used for further simulations Fig 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 value is suggested at the minimum error point which is then taken as the prediction result Table Base case simulation parameters Simulation parameters Simulation values Nanowire morphology Radius, r0[mm]a Length heated by the laser, L0 [mm]a Density, r [kg/m3] [18] Specific heat capacity, cp [J/kg K] [18] 32.5 1.31 1900 470 Thermal diffusivity of nanowire Radial thermal diffusivity ar, [m2/s] Axial thermal diffusivity, ax [m2/s]a 2.72  10À6 2.72  10À5 Other parameters DC supply [mA] Ambient temperature [ C] StefaneBoltzmann constant, s [W/m2 K4] 8.039 25 5.67  10À8 a Used in the TET experiment conducted by Guo [4] 3.2 Effects of anisotropic thermal property on heat transfer mechanism Fig presents the temperature profiles of the sample wire as a function of time at different ax/ar ratios varying from to 10,000 The results are obtained from the simulations numbered from to as shown in Table When ax/ar is 10,000 or larger, the simulated model represents a very long wire where radial diffusivity is small So less heat is conducted in the radial direction and heat loss to the ambient through the radiation is negligible On the other hand, when the ax/ar ratio is much smaller than 10,000 heat transfer occurs in both axial and radial directions It can be observed in Fig that when ax/ar is 10,000, the steady state temperature of the wire is higher than those observed for the wires with smaller ax/ar ratios This means the anisotropic heat transfer and heat lost to ambient should not be ignored like previous models’ assumptions when the ax/ar is less than 10,000 This observation agrees well with our previous discussion where very large ax/ar ratio is used to simulate one directional heat transfer without heat loss through radiation 3.3 Effects of DC supply and wire morphology on heat conduction measurement Other than analyzing the thermophysical properties of a wire based on the experimental data collected, this model can be applied Fig Error analysis is carried out to determine the best ax to ar ratio using the Root Mean Square method (black) and the R2 coefficient determinant (red) Both error analysis methods indicate that the minimum error occurs when ax/ar is 40 (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) F Gong et al / International Journal of Thermal Sciences 77 (2014) 165e171 Fig Comparison of simulation results with experiment data [4] Simulation inputs are shown in Table Since the simulation result agrees well with the experiment data, the simulation model is validated to predict how heat conducts within a wire under various conditions Based on the simulation results, the effect of each experiment condition on the heat transfer mechanism can be determined For instance, a longer wire takes longer time to reach its steady thermal state and a higher DC supply will heat up the wire to a higher temperature over a longer period of time To investigate the effect of DC supply on the heat conduction of the wire, simulations numbered from to 11 in Table were carried out by varying the DC supply from to 50 mA The temperature changes of the wire as a function of heating time with different DC supplies are shown in Fig It is observed that the steady state temperature as well as the time required to reach the state both increase when increasing the DC supply The wire’s temperature changes are very small at low DC values of and 10 mA, so DC values lower than 10 mA are not recommended for thermal measurement When the DC supply is bigger, the steady state temperature 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 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 temperature of the wire is higher than the rest This can be explained by considering it as only one directional heat transfer without heat loss to the ambient The significance of anisotropic heat transfer is observed 169 with DC values of 20, 30, 40 and 50 mA, respectively During the experiment, it is important to keep the sample below its burning temperature For carbon nanowires, the average burning temperature ranges from 673 to 873 K (400e600  C) [24e30] Therefore, the current supply must be well controlled At a DC of 50 mA, the steady state temperature is 730 K, higher than the burning point of carbon nanowires and thus, lower DC supply should be used for the heating As presented, this model is helpful in determining a suitable DC supply for a specific wire To investigate the effects of wire morphologies on heat transport phenomena, simulations numbered from 12 to 14 were conducted with the parameters shown in Table In these simulations, sample lengths of mm, 10 mm, and 20 mm were used The temperature evolution produced by the simulation is shown in Fig From the temperature profile, it can be seen that the longer the wire, the longer it takes to reach its thermal steady state, and the higher the steady state temperature it stabilizes In addition, another set of simulations numbered from 18 to 21 in Table was conducted to study the effect of the radius of the sample on the heat transfer mechanism The obtained temperature evolution history is shown in Fig It is observed that with all other experiment conditions fixed, a wire with smaller diameter has a higher steady state temperature These observations are reasonable because similar to the characteristics of a very thin wire, a long wire can be assumed to have only one directional heat transfer In other words, it simulates a condition where heat transfer in radial direction is negligible, so is the heat loss through radiation As a result, a longer or thinner wire settles at a higher steady state temperature given all other conditions unchanged From Fig 8, the temperature profiles of the wires with radii of 100 mm and 200 mm are unsuitable for experiments as the temperature rises are very small However, this insufficient temperature rise could be solved by increasing the DC supply Caution must be taken so that the increased DC supply will not cause overheating 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 the current model which is related to the ambient temperatures The effects of different ambient temperatures are obvious according to the StefaneBoltzmann law The higher the ambient temperature, the less heat is lost through radiation In other words, a lower ambient temperature will result in more heat loss through radiation Given that radiation is ignored in conventional models, the computational error is even greater Since our developed model simulates anisotropic heat flow and takes into account of the heat loss to the experiment surroundings, the shape of the temperature profile should not vary for different ambient temperatures To prove this, simulations numbered from 15 to 17 in Table were carried out The absolute temperature evolution history is shown in Fig while the normalized temperature evolution history is shown in Fig 10 From the absolute temperature evolution history in Fig 9, it can be seen that each temperature profile starts from its given ambient 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 This observation agrees with our previous discussion, and hence, demonstrates the high accuracy of our proposed anisotropic simulation model From the simulation results, it is evident that to obtain a suitable temperature profile for any given wires, the experiment setup must be well designed, and all contributing factors need to be taken into consideration With the help of the simulation model, appropriate experiment conditions can be obtained before conducting the experiments 170 F Gong et al / International Journal of Thermal Sciences 77 (2014) 165e171 Table Effects of wire morphologies and experimental conditions on heat transfer measured by the TET technique Simulation run Radius of wire r0 (mm) Length of wire L0 (mm) Axial thermal diffusivity ax (m2 sÀ1) 10 11 12 13 14 15 16 17 18 19 20 21 Base Case 32.5 32.5 32.5 32.5 32.5 32.5 32.5 32.5 32.5 32.5 32.5 32.5 32.5 32.5 32.5 32.5 32.5 10 32.5 100 200 32.5 20 20 20 20 20 1.31 1.31 1.31 1.31 1.31 1.31 10 20 1.31 1.31 1.31 1.31 1.31 1.31 1.31 1.31 2.72 2.72 2.72 2.72 2.72 2.92 2.92 2.92 2.92 2.92 2.92 2.92 2.92 2.92 2.92 2.92 2.92 2.92 2.92 2.92 2.92 2.72                       10À5 10À5 10À5 10À5 10À5 10À5 10À5 10À5 10À5 10À5 10À5 10À5 10À5 10À5 10À5 10À5 10À5 10À5 10À5 10À5 10À5 10À5 Anisotropic heat transfer ratio ax/ar DC supply I (mA) Ambient temperature T0 ( C) 10 100 1000 10000 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40 8.039 8.039 8.039 8.039 8.039 10 20 30 40 50 8.039 8.039 8.039 8.039 8.039 8.039 8.039 8.039 8.039 8.039 8.039 25 25 25 25 25 25 25 25 25 25 25 25 25 25 15 25 25 25 25 25 25 Bold entries indicate changed values with respect to the base case Conclusions In this work, the computational models based on the transient electrothermal (TET) technique are developed successfully for the anisotropic thermal characterization of multi-scale wires The TET technique overcomes the limits and disadvantages of previous thermal characterization techniques such as the 36 method and the PLTR technique Implicit finite difference method is applied to solve the transient heat equations in dimensionless, and to provide unconditional stability The merits of the developed models over current ones are as follows (i) The developed models are more realistic and can characterize more accurately the anisotropic thermal characteristics of the wires, (ii) the developed models prove that radial heat conduction and heat lost should not be ignored in the previous models, and (iii) the simulation results are useful for the experimentalists optimizing their thermal measurement conditions By implementing the boundary conditions corresponding to the prevailing experimental setup, the present study Fig Normalized temperature evolution history of wires with length mm, 10 mm, and 20 mm It is observed that the longer the wire, the longer it takes to reach its thermal steady state, and the higher the thermal steady state temperature Fig Absolute temperature evolution history for various DC supply This simulation profile helps to determine the suitable current supply range for the experiment, hence to prevent overheating of the wire This step is extremely important for experiment planning Fig Absolute temperature evolution history of wires with radius 10 mm, 32.5 mm, 100 mm, and 200 mm F Gong et al / International Journal of Thermal Sciences 77 (2014) 165e171 Fig 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 equals to 0, 15, and 25  C Though their absolute temperature evolution profiles are different, their normalized profiles coincide with each other The simulation result agrees well with the theory can readily used in other problems such as the 36 method and the PLTR 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 References [1] T.Y Choi, D Poulikakos, J Tharian, U Sennhauser, Measurement of thermal conductivity of individual multiwalled carbon nanotubes by the 3-omega method, Appl Phys Lett 87 (2005) [2] T.Y Choi, D Poulikakos, J Tharian, U Sennhauser, Measurement of the thermal conductivity of individual carbon nanotubes by the four-point threeomega method, Nano Lett (2006) 1589e1593 [3] J.Q Guo, X.W Wang, D.B Geohegan, G Eres, C Vincent, Development of pulsed laser-assisted thermal relaxation technique for thermal characterization of microscale wires, J Appl Phys 103 (2008) [4] J.Q Guo, X.W Wang, T Wang, Thermal characterization of microscale conductive and nonconductive wires using transient electrothermal technique, J Appl Phys 101 (2007) 171 [5] J.B Hou, X.W Wang, P Vellelacheruvu, J.Q Guo, C 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multi-scale... Guo et al [4] were solved by Green’s function, which only considered one-dimensional (axial direction) heat transfer and ignored the radial heat conduction and heat loss T Tr0 x maximal temperature