Anisotropic heat transfer prediction of

Chemical Physics Letters 555 (2013) 239–246 Contents lists available at SciVerse ScienceDirect Chemical Physics Letters journal homepage: www.elsevier.com/locate/cplett Anisotropic heat transfer prediction of multiscale wires using pulse laser thermal relaxation technique Jin W Tan, Yue Cheng, Denis S.G Yap, Feng Gong, Son T Nguyen, Hai M Duong ⇑ Department of Mechanical Engineering, National University of Singapore, Singapore a r t i c l e i n f o Article history: Received 27 June 2012 In final form October 2012 Available online 24 October 2012 a b s t r a c t A theoretical model is developed that predicts the thermal characterization of multiscale wires using the pulse laser thermal relaxation technique This into account anisotropic heat transfer and radiation heat lost to the surroundings The simulation results have better agreement with the experimental results than previous models Using the validated model, the heat transfer characteristics of multiscale wires with various morphologies (radius of 15–25 lm, length of 500–1000 lm, radial thermal conductivities of 0.05–5.00 W/mK, axial thermal conductivities of 10–2000 W/mK) and experimental conditions (laser power outputs of 20–50 kW and laser pulse width of 5–9 ns) are studied Ó 2012 Elsevier B.V All rights reserved Introduction Technology has evolved significantly in recent years and more technological products are focusing on being smaller, lighter and better In the electronic industry, heat dissipation is gaining increased importance due to the increased levels of dissipated power [1] Hence, it is important to source for materials which are able to conduct heat well This justifies the recent significant research efforts on materials with excellent thermal conductivities such as carbon nanotubes (CNTs) to replace current materials CNTs have been extensively studied since they were first discovered by Iijima in 1991 [2] CNTs have great strength, light weight, high stability and excellent electrical and heat conductivities [1,3–5] Due to their size, CNTs are ideal materials for nano-scale devices In this Letter, specific focus is made on the excellent thermal conductivity of CNTs The thermal conductivity of CNTs and CNT bundles has been reported to range from to 3000 W/mK [1,3–15] The wide range of thermal conductivity is due to the morphology of the CNTs (multi-walled or single-walled, length, diameter of bundle) There are various methods for measuring the thermal conductivity of the CNT One of the earliest experimental methods of measuring their thermal conductivity is the 3x technique [6–8,10] A CNT sample is connected between two metal bases forming a bridge structure A constant-amplitude AC current is then passed through the CNT which will create a temperature fluctuation at 2x, where x is the frequency of the AC current Subsequently, a third-harmonic voltage signal would be induced by the temperature fluctuation and this voltage signal would be recorded For this ⇑ Corresponding author E-mail address: mpedhm@nus.edu.sg (H.M Duong) 0009-2614/$ - see front matter Ó 2012 Elsevier B.V All rights reserved http://dx.doi.org/10.1016/j.cplett.2012.10.021 method, the thermal conductivity of the CNT can be determined by selecting an optimal x whereby the voltage signal can be clearly distinguished However, there are several limitations of the 3x technique The 3x method requires the CNT to have a linear current to voltage relationship within the applied AC voltage range It is also difficult to carry out the 3x method for wire samples with low thermal conductivity as the heat transfer between the two bases will become less accurate to measure [12,15] To overcome the limitations of the 3x method, the transient electrothermal technique (TET) has been developed [9,12,15] For the TET technique, a CNT sample is suspended between two copper electrodes which are excellent heat sinks A step DC current is then applied through the CNT which results in a temperature rise due to electrical heating This temperature change would result in a resistance change and hence, a change in the measured voltage The change in voltage is directly proportional to the temperature change In this way, the thermal conductivity of the CNT can be determined from the voltage evolution curve which has the same profile as the temperature evolution curve However, the TET technique is not accurate as the slow rising time would make it difficult for experimentalists to measure short wires with relatively high thermal conductivity like CNTs [13] This is due to the short characteristic time of heat transfer which is comparable to the rising time of the electric current Thus, the pulse laser thermal relaxation (PLTR) technique is developed to complement the TET technique [13] The experiment setup of the PLTR method is similar to that of the TET method The main difference will be the usage of a laser, operating in a pulsed mode, to heat the sample instead of the electrical heating in the TET technique Nonetheless, a small DC current would still be fed through the CNT so that the temperature change would lead to a change in the resistance The thermal conductivity of the CNT is 240 J.W Tan et al / Chemical Physics Letters 555 (2013) 239–246 determined from the temperature relaxation curve, which occurs right after the pulse heating While the PLTR technique appears to be an excellent method to determine the thermal conductivity of the CNT, the costs of purchasing the experimental equipments, especially the laser equipment, are relatively high Hence, this letter aims to develop more accurate and realistic numerical models on the thermal characterization of the CNT with various morphology (radius of 15–25 lm, length of 500–1000 lm, radial thermal conductivities of 0.05– 5.00 W/mK, axial thermal conductivities of 10–2000 W/mK) and measurement conditions (laser power outputs of 20–50 kW and laser pulse width of 5–9 ns) using the PLTR method Through simulation results in this letter, experimentalists can first optimize the experiment conditions using the PLTR technique before purchasing the required equipments Compared to a similar previous model [13], the developed model is more accurate by taking into account the anisotropic heat transfer (axial and radial directions) and heat loss from the CNT to the surroundings The previous models predicting the thermal conductivity of CNT often took into account only an axial heat transfer of the CNT [1,3–14] Radial heat transfer and radiation heat loss to the surroundings is often neglected or not mentioned Huang et al [14] studied the effect of radiation heat loss on the thermal properties of the CNT They reported that the radiation heat loss must be taken into consideration when measuring CNTs, with high aspect ratio, at high temperatures (>800 K) Hence, this signifies the importance of anisotropic heat transfer and radiation heat loss to the surroundings In this letter, a mathematical model is developed to predict the heat transfer within the CNT using the PLTR technique By incorporating the anisotropic heat transfer and eliminating the assumption of negligible heat loss, the developed model in this letter is a significant improvement over previous models [13] The numerical model is validated by experimental results In order to widen the scope of the developed model, further simulations are carried out under different operating conditions, including different CNT diameters and lengths, different radial and axial thermal conductivities, different laser powers, and different laser pulse widths The developed model ignores the thermal boundary resistance between the to-be-measured wire and the electrodes The to-bemeasured wire is assumed to be a solid cylindrical tube And the radiation used in the developed model is assumed not to depend on the phonon path T Tmax R r¼ R0 L l¼ L0 at s ¼ L2 L0 hẳ 2ị 3ị 4ị 5ị where h is the non-dimensional temperature, r is the non-dimensional radius, l is the non-dimensional length and s is the nondimensional time aL is the thermal diffusivity of the CNT in the axial direction and it is defined as aL ¼ qKCLp Substitution of Eqs (2)–(5) into Eq (1) yields: !  2 @h aR L0 @h @ h @2h L20 Q1 L2 T ỵ ỵ 2ỵ ẳ max @ s aL R0 r @r @r T max aL qC p aL @l  r h j qC p R0 rẳ1 6ị At the start of the experiment, no heating takes place and the CNT is at room temperature As the temperature term in Eq (1) is the temperature distribution within the CNT due to pulse laser heating, the initial condition of the study is: T ¼ 0K at t ¼ s for In this letter, anisotropic heat transfer and radiation heat loss from the CNT to the ambient environment is taken into account (by treating the CNT as a black body, last term in Eq (1)) The rate of thermal energy accumulation within the CNT due to pulse laser heating (first term in Eq (1)) is the summation of the heat conduction in the radial (second term in Eq (1)) and axial direction (third term in Eq (1)) of the CNT and the laser power used to heat the sample (fourth term in Eq (1)) In cylindrical coordinates, the governing equation is: 2rTjR0 @T kR @ @T @2T ẳ R ị ỵ K L ỵ q_ @t R @R @R R0 @L ð1Þ where q and Cp are the density and the specific heat capacity of the CNT, respectively kR and kL are the thermal conductivities of the CNT in the radial and axial direction, respectively q is the rate of thermal energy generation due to pulse laser heating as we are only concerned with the temperature evolution due to laser heating Hence, T in Eq (1) represents the temperature of the CNT due to pulse laser heating only It is important to note that the maximum R R0 and L L0 ð7aÞ where R0 is the radius of the CNT and L0 is the length of the CNT Alternatively, in dimensionless terms, the initial condition can be expressed as: h ¼ 0K at s ¼ for r and l ð7bÞ For this letter, we assumed that the copper electrodes used to hold the CNT in place are excellent heat sinks We shall further assume that the outer surface of the CNT is completely heated up within the laser pulse width Although only a portion of the CNT surface might be heated up by the laser, this assumption is reasonable as heat transfer on the surface of the CNT will happen instantly from the heated side to the rest of the CNT due to its small size Thus, the boundary conditions for this system are: T ¼ 0K at L ¼ 0m for Simulation methodology qC p temperature attained by the CNT should not exceed its burning temperature in air, which is typically 400–600 °C [16–22].For simplicity, Eq (1) is normalized using the following non-dimensional terms: R R0 and t > 0s ð8aÞ T ¼ 0K at L ¼ L0 for R R0 and t > 0s _ laser pulse qt at t ¼ tlaser pulse and R ¼ R0 for T ẳ T max ẳ qC p 8bị L L0 ð8cÞ where tlaser pulse is the laser pulse width Similarly, in dimensionless terms, the boundary conditions are: h ¼ at l ¼ for r and s>0 h ¼ at l ¼ for r and s > aLtlaser pulse and r ¼ for h ¼ at s ¼ L20 ð8dÞ ð8eÞ l ð8fÞ The dimensionless model equation is solved using the finite difference technique A uniform two-dimensional structured grid is used in this letter where Dr is set to be equal to Dl (see Fig 1(a)) The grid covers only half of the CNT as the temperature distribution within the CNT is identical for both halves To begin the simulation, user inputs of the CNT morphology (length, radius, density and specific heat capacity), thermal conductivities of the CNT (radial and axial) and laser parameters (power output and 241 J.W Tan et al / Chemical Physics Letters 555 (2013) 239–246 Figure (a) PLTR experimental setup and computational grid The dark bold arrows represent the pulse laser heating and the wavy arrows represent the heat loss from the CNT to the surroundings (b) Model validation using experimental data by Guo et al [13] laser pulse width) are required Subsequently, the CNT is heated up to the maximum temperature within the laser pulse width and allowed to cool down The simulation will stop when the CNT has cooled down to a temperature close to K In order to investigate the anisotropic heat transfer of the wire sample, the root-mean-square error (RMSE) was used to quantify the agreement between the experiment data and the developed model with various thermal conductivity ratios, kkRL ¼ À 1000 The root-mean-square error is defined as: Simulation results and discussions 3.1 Validation of developed model The original experimental data of Guo et al [13] was used to validate our developed model as their experimental set-up is similar to the model development There is an excellent agreement between the experimental data and the simulation results (see Fig 1(b)) The predictions of the proposed model are also better than those of Guo et al [13], especially during the initial cooling period (0–1 ms) This happens as the previous authors’ models [13,23] only took into account an axial-dimensional heat transfer along the CNT axial and neglected radiation heat loss The developed model eliminated the assumptions of the previous models by taking into account anisotropic heat transfer (axial and radial directions) within the CNT, with the radiation heat lost to the surroundings Table Summary of the root-mean-square error (RMSE) calculated by the proposed model and Guo et al [13]’s model.a kL (W/mK) Ratiob 10 100 1000 9.0 9.3 9.4 9.56 9.8 10 11 0.024018653 0.021964111 0.021820129 0.022149067 0.023735017 0.025853119 0.040906308 0.024019130 0.021964017 0.021819726 0.022149067 0.023733877 0.025852933 0.040904505 0.024019180 0.021963997 0.021819682 0.022128921 0.023733747 0.025852774 0.040904285 0.024019185 0.021963995 0.021819678 0.022149067c 0.023733734 0.025852758 0.040904263 a a L0 = 566.4 lm [13]; R0 = 11.65 lm [13]; q = 1900 kg/m3 [21]; CP = 470 J/kgK [21]; laser power = 14.5 kW; laser pulse width = ns [13]; Stefan–Boltzmann constant, r = 5.67  10À8 J/sm2 K4 b Ratioofconductiv ities ¼ kkRL c RMSE value for Guo et al [13]’s model 242 J.W Tan et al / Chemical Physics Letters 555 (2013) 239–246 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi PN i¼1 hi;experiment hi;predict ị RMSE ẳ N overcoming experimental limits and determining the axial and radial thermal conductivities of larger and longer wires ð9Þ 3.2 Effects of wire morphologies and thermal properties on the heat transfer mechanism The smaller RMSE gives a better model prediction Table shows a summary of the values of the RMSE calculated by Guo et al [13] model and the proposed model The proposed model can give 3% more accurate axial thermal conductivity kL = 9.4 W/ mK (bold in Table 1) as well as the radial thermal conductivity, kR = 0.01 W/mK (with a thermal conductivity ratio of 1000) for the first time Although there is insignificant difference in the RMSE values in Table for nanoscale wires when their thermal conductivity ratios are increased from 10 to 1000, these values and the RMSE method may be useful for experimentalists Using the validated heat transfer model, further computer simulations were conducted to investigate the heat transfer of the wire with various morphologies (radius, length and different axial and radial thermal conductivities) under different operating conditions (laser power and the laser pulse width) The further simulation results are very useful for experimentalists to optimize preliminary measurement conditions and equipment purchase Table summarizes the wire parameters and the operating conditions obtained from experimental data [14] and the developed model as a base case It is important to note that for stability and higher accuracy of simulation results, a radius grid, Dr was set to be equal to an longitudinal grid, Dl for all the cases In the finite different technique, the non-dimensional time step, Ds, must satisfy the condition: Table Simulation parameters (base case) Simulation parameters Simulation values Nanowire morphology Radius, R0 [lm]a Length heated by the laser, L0 [lm]a Density, q [kg/m3] [21] Specific heat capacity, CP [J/kgK] [21] 17.0 500.0 1900 470 Thermal conductivities of nanowire Radial thermal conductivity, kR [W/mK] Axial thermal conductivity, kL [W/mK] [3,11] 200 Laser parameters Power output [kW]b Pulse width, tlaser pulse [ns] 40 Other parameters Temperature coefficient of resistance, e [KÀ1] [24–26] Stefan–Boltzmann constant, r [J/sm2 K4] Non-dimensional time step, Ds Number of radial grids, Drc Number of longitudinal grids, Dlc 2.0  10À3 5.67  10À8 2.0  10À7 50 1471 (  2  2 ) 1 1 Ds ¼ ; Number of Dr Number of Dl ð10Þ Table summarizes the different operating conditions to study the heat transfer characteristics of the wires with different morphologies For most of the simulations, the thermal conductivity ratio was kept constant at 100 because the kkRL ratios of 10, 100, 1000 not affect the result significantly as shown in Table The range of parameters in Table was chosen to ensure the wire is not burnt by the laser energy Figures 2–4 present the graphs of the absolute temperatures and voltage ratios against time The only noticeable difference for the normalized temperature curve occurs when the length (Simulation runs 4–6) and the axial thermal conductivity, kL, (Simulation runs 10–13) of the wires is varied However, this difference is not critical and thus, we decided to exclude the graph of the normalized temperature in this Letter By plotting the voltage ratios and absolute temperatures against time in the same plot, experimentalists who measure the voltage change on the wire can determine a Typical dimensions of CNT [9,14] Chosen such that the maximum temperature attained by the CNT will not exceed its burning temperature c Refer to Figure 1(a) for definitions b Table Effects of various wire morphologies and measurement conditions on the PLTR technique.a–c Simulation run 10 11 12 13 14 15 16 17 18 a b c d e Nanowire morphology Thermal conductivities of nanowire Laser parameters R0 (lm) L0 (lm) kR (W/mK) kR (W/mK) Power output (kW) Pulse width (ns) 15 17 25 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 500 500 500 500 850 1000 500 500 500 500 500 500 500 500 500 500 500 500 2.00 2.00 2.00 2.00 2.00 2.00 0.05 2.00 5.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 200 200 200 200 200 200 200 200 200 10 200 2000 200 200 200 200 200 200 40 40 40 40 40 40 40 40 40 40 40 40 20 40 50 40 40 40 7 7 7 7 7 7 7 7 Tmax (°C) Dsd (Â10À7) No Dre No Dle 614.16 417.70 46.38 417.70 133.29 72.35 417.70 417.70 417.70 417.70 417.70 417.70 72.35 417.70 590.37 220.36 417.70 615.04 2.0 2.0 2.0 2.0 1.0 0.5 2.0 2.0 2.0 0.1 2.0 20.0 2.0 2.0 2.0 2.0 2.0 4.0 40 50 50 50 30 30 50 50 50 50 50 10 50 50 50 50 50 30 1333 1471 1000 1471 1500 1765 1471 1471 1471 1471 1471 294 1471 1471 1471 1471 1471 882 The density and the specific heat capacity are kept constant (i.e similar to the base case [q = 1900 kg/m3, CP = 470 J/kgK]) Bold entries indicate changed values with respect to the base case Highlighted entries indicate the base case Non-dimensional time step Refer to Figure 1(a) for definitions J.W Tan et al / Chemical Physics Letters 555 (2013) 239–246 243 Figure (a) Predicted thermal energy distribution within the CNT with different wire radius, R0 Insert: temperature rise of the CNT up to 20 ns (b) Predicted thermal energy distribution within the CNT as the length, L0, is varied Insert: temperature rise the wire temperature and vice versa The Figures 2–4 also present the upper limitations of the measurement conditions without burning the wire samples The voltage ratio is related to the absolute temperature, Tabs of the wire by the following expression: Voltage ratio; DV ¼ eT abs V ð11Þ where e is the temperature coefficient of resistance of the wire and it is assumed to be a constant value of 0.002 KÀ1 [24–26] in this letter Figure 2(a) presents how the heat transfer of the wire sample is affected by varying the wire radius (15–25 lm) As the laser pulse width (7 ns) is much smaller than the total computational time, an inset is included in Figure 2(a) to depict the temperature rise inside 244 J.W Tan et al / Chemical Physics Letters 555 (2013) 239–246 Figure (a) Effects of varying the radial thermal conductivity, kR, on the predicted temperature evolution for the CNT of the base case dimensions Insert: exploded view of the temperature rise (b) Effects of varying the axial thermal conductivity, kL, on the predicted temperature evolution for the CNT of the base case dimensions Insert: temperature rise of the CNT the wire within the laser pulse width As seen in Figure 2(a) and Table 3, the maximum temperature of the wire decreases with increased wire radius The boundary condition in Eq (8c) explains this phenomenon as the maximum temperature is directly proportional to the rate of thermal energy generation due to pulse _ which is inversely proportional to the wire vollaser heating ðqÞ, ume/surface Figure 2(b) presents the effects of the wire length, L0 as similar to those of the wire radius An increase of the wire J.W Tan et al / Chemical Physics Letters 555 (2013) 239–246 245 Figure (a) Predicted temperature evolution of CNT, with base case dimensions, as the laser power is varied Insert: temperature rise of the CNT up to 20 ns (b) Predicted heat transfer characteristic of CNT, with base case dimensions, as the laser pulse width is varied Insert: temperature rise of the CNT length would also decrease the maximum temperature of the wire after pulse laser heating Besides the difference on the maximum temperature, the longer wire requires a longer time to cool down to 1% of its maximum temperature When the wire length is increased, it would naturally take a longer time for the thermal energy to be dissipated within the wire Also, doubling the wire length would increase the total time by fourfold The effects of the radial thermal conductivity of the wire, kR, on its heat transfer characteristics are shown in Figure 3(a) and simulations 7–9 Increasing the radial thermal conductivity of the wire sample does not affect the heat transfer within the wire significantly This observation is consistent with the validation process mentioned in Section 3.1 where the thermal conductivity ratios of 10–1000 not affect the accuracy of the simulation results significantly However, the radial thermal conductivity of the wire is still important Its heat transfer characteristics as thermal energy, regardless of its magnitude, are still dissipated in the radial direction and this is accountable for the overall heat transfer of the wire Since experimental methods determining the radial thermal conductivity of the wire have been very limited, the developed model in this letter can give the radial thermal conductivity of the wire by using the measured voltage change over the experimental time The axial thermal conductivities of the wire, kL, are varied in simulations 10–12 The axial thermal conductivity is the most common thermal property of a wire determined by experimentalists [1,3–14] This is because the wire length is usually two to three 246 J.W Tan et al / Chemical Physics Letters 555 (2013) 239–246 orders of magnitudes larger than its radius Hence, most experimentalists only consider a one-dimensional heat transfer which is insufficient in our opinion The results in Figure 3(b) show that by reducing the axial thermal conductivity by 20 times, the experimental time taken for the wire to cool down would be approximately 20 times longer As expected, the higher the axial thermal conductivity, the faster thermal energy is dissipated within the wire In Figure 3b, the maximum temperature attained by the wire due to pulse laser heating does not depend on its thermal conductivities in both the radial and axial directions The reason for this independence is that the maximum temperature calculated using Eq (8c) is only a function of the rate of thermal energy generation _ the laser pulse width ðt laser pulse Þ, the density q and specific heat ðqÞ, capacity Cp of the wire 3.3 Effect of operating conditions on the heat transfer mechanism The last simulations in this letter explore the effects of the laser parameters on the heat transfer characteristics of the wire Simulations 13–15 study the heat transfer characteristics of the wire _ when the laser power ðqovolume of CNTÞ is varied and other parameters (the wire morphologies and the pulse width of the laser) are kept constant Instinctively, a higher laser power would heat the wire to a higher temperature The predictions in Figure 4(a) confirm doubling the laser power would double the maximum temperature of the wire However, the cooling time of the wire is not affected by changing the laser power as the heat transfer characteristic is mainly governed by the thermal conductivities of the wire Simulations 16–8 concern the effects of the laser pulse width, tlaser pulse, on the heat transfer of the wire sample and the results are reported in Figure 4(b) Similar to the influence of the laser power, an increase of the laser pulse width would increase the maximum temperature of the wire but not affect the total time taken for the wire to cool down As mentioned before, the laser power and the laser pulse width are directly proportional to the maximum temperature of the wire as shown in the boundary condition in Eq (8c) Hence, with this proposed model, experimentalists will be able to optimize the experimental conditions and the relevant laser equipment to conduct the PLTR experiments without burning the wire fit better with experimental results The proposed anisotropic heat transfer model may be used to predict the heat transfer in both axial and radial directions of the wire over a wide range of wire morphologies, thermal conductivities and experimental conditions In particular, this model is able to provide economic benefits to experimentalists who wish to purchase expensive laser equipment used for conducting the PLTR experiment Acknowledgments The authors deeply acknowledge the Start up grant R-265-000361-133 and SERC 2011 Public Sector Research Funding (PSF) Grant R-265-000-424-305 for the funding support We would like to thank Prof Dimitrios Papavassiliou (School of Chemical, Biological and Materials Engineering, University of Oklahoma, USA) and Eric Markus (Blekinge Institute of Technology, Sweden) for useful advice and English correction of the manuscript References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] Conclusions A finite difference model was developed to study the heat transfer of the multiscale wires including the CNT using the PLTR method To the best of our knowledge, this letter could be the first (or one of the first) numerical studies on the anisotropic heat transfer of the wire, with the radiation heat loss to the surroundings being taken into account This model has been proven to be an improvement over the existing model by Guo et al 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C Chen, X Xiong, S Chao, A Busnaina, S Sridhar, M.R Dokmeci, in: Proc Seventh IEEE Int Conf Nanotechnol., 2007, p 1062  ... axial heat transfer of the CNT [1,3–14] Radial heat transfer and radiation heat loss to the surroundings is often neglected or not mentioned Huang et al [14] studied the effect of radiation heat. .. effects of the laser parameters on the heat transfer characteristics of the wire Simulations 13–15 study the heat transfer characteristics of the wire _ when the laser power ðqovolume of CNTÞ... axial-dimensional heat transfer along the CNT axial and neglected radiation heat loss The developed model eliminated the assumptions of the previous models by taking into account anisotropic heat transfer