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The dual-mode heat flow meter technique: A versatile method for characterizing thermal conductivity
International Journal of Heat and Mass Transfer 53 (2010) 5581–5586
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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
The dual-mode heat flow meter technique: A versatile method for characterizing thermal conductivity Nayandeep K. Mahanta, Alexis R. Abramson * Department of Mechanical and Aerospace Engineering, Case Western Reserve University, 10900 Euclid Avenue, Cleveland, OH 44106-7222, USA
a r t i c l e
i n f o
Article history: Received 27 October 2009 Accepted 24 May 2010 Available online 9 August 2010 Keywords: Thermal conductivity Steady-state Heat flow meter Anisotropic materials Conductors Insulators
a b s t r a c t The dual-mode heat flow meter technique was developed for steady-state thermal characterization by optimizing the curve fit between the experimental temperature profile and the prediction from a simple analytical solution taking into account conductive as well as radiative heat transfer. The validation results were seen to be in good agreement with published literature values and demonstrated the versatility of the method. Moreover, the technique is ideally suited for characterization of anisotropic materials without necessitating any additional information about the nature of anisotropy and lends itself as an attractive alternative for characterization of novel materials with engineered transport properties. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction Efficient and accurate thermal characterization is critical to the continued development of materials with engineered thermal transport properties for applications such as thermal management, thermoelectrics, thermal insulation and more. Thermal characterization techniques should be simple and reliable despite being constrained by the unpredictable nature of the materials being tested. For instance, the measurements should not be affected by the fragility of the samples, their unexpectedly extreme thermal properties (either very low or very high), their constrained geometries, and most importantly, their limited availability. Conventional steady-state thermal conductivity measurement techniques like the heat flow meter [1] are simple and easy to use but have been limited to certain ranges of the spectrum of thermal conductivity, often require specific geometries for testing, and even minor heat losses can significantly influence accuracy. In contrast, transient techniques, such as laser-flash or transient hot-strip, are known to be reliable over the entire range but conductivity measurements typically involve relatively complicated data analysis and are contingent upon the accurate determination of specific heat capacity and density [2,3]. Additionally, most conventional techniques, namely laser-flash or three-omega, may only provide an effective thermal conductivity when used for characterizing materials with anisotropic thermal properties unless modifications are applied [2,4]. Thus, a simple and versatile characterization technique that * Corresponding author. Tel.: +1 216 368 4191; fax: +1 216 368 6445. E-mail address: [email protected] (A.R. Abramson). 0017-9310/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijheatmasstransfer.2010.05.063
is capable of measuring the thermal conductivity in a particular direction without requiring any knowledge or assumption about the properties in the other directions is desirable. In this work, we report on a steady-state thermal characterization technique which can be used for measurement of insulators as well as conductors and is particularly suitable for thin, anisotropic materials. In essence, the method was developed from the standard ASTM C 518 based heat flow meter technique [1], which compares the steady-state temperature gradient in the sample to that of a reference material to determine the thermal conductivity of the unknown sample. The conventional heat flow meter technique assumes one-dimensional conduction to be the only mode of heat transfer and works very well for determining through-thickness thermal conductivity of thin thermal insulators as long as the requirement that one-dimensional heat flow with negligible losses is strictly maintained. The validity of the assumption for onedimensional conduction to be the only mode of heat transfer relies on using test specimens that are short along the direction of heat flow and have a large cross-section. This works very well for low thermal conductivity materials [5,6]. Owing to their short length, the surface area available for convection and/or radiation heat losses from the test specimen becomes negligible compared to the large cross-section available for conduction heat transfer through the specimen. Alternatively for highly conductive materials, establishing a measurable temperature drop is critical and therefore longer specimens may be required for the measurement. However, due to a large surface area, the effects of convection and radiation may no longer be negligible, thereby casting doubt on the reliability and accuracy of the technique. Previous works on use of
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Nomenclature A ACu As AsCu Bi Bir hr hrCu k kCu L Lc LCu P Q Qc Qr Rc Rr T Tavg T avgCu Tce The T1 t
area of cross-section of sample area of cross-section of the copper reference surface area of sample surface area of the copper reference Biot number Biot number for radiation coefficient of radiation heat transfer from sample coefficient of radiation heat transfer from copper reference thermal conductivity of sample thermal conductivity of copper reference length of sample characteristic length length of copper reference perimeter of sample total heat flow heat flow due to conduction heat flow due to radiation resistance to conduction heat transfer resistance to radiation heat transfer temperature average temperature average temperature of copper reference cold end temperature hot end temperature ambient temperature thickness
the standard technique did not consider the effects of the convection and/or radiation losses [5–7], perhaps because the heat flow meter had primarily been used for testing materials with thermal conductivities less than 20 W/m-K [5,6]. More importantly, these materials had provided the researchers with a test specimen such that the thickness and width were larger compared to the length in the direction along which the thermal conductivity was desired to be estimated. However, a situation where such a specimen may not be obtainable is likely, especially with the widespread research on novel composite materials, including nanocomposites. In other words, it is likely that a material could be thin and anisotropic while being required to be tested for thermal conductivity in the direction perpendicular to its thickness. To overcome these obstacles, the dual-mode heat flow meter, which considers both conduction and radiation modes of heat transfer along a specimen of relatively long length was conceived to broaden the applicability of the heat flow meter technique while maintaining its inherent simplicity. This technique accurately and reliably measures thermal conductivity along the direction of one-dimensional heat flow. Convection heat loss can be minimized by conducting the experiments under conditions of very high vacuum (pressures less than 105 Torr).
w x
width Cartesian coordinate along the direction of conduction heat transfer
Greek symbols DT temperature difference in sample DTCu temperature difference in copper reference Dx distance between two points on the sample DxCu distance between two points on the copper reference e emissivity of sample eCu emissivity of copper r Stefan–Boltzmann constant x dimensional ratio of heat transfer due to radiation and conduction Subscripts avg average Cu copper ce cold end he hot end r radiation s surface 1 ambient Subscript to subscripts Cu copper
sary for this method can be achieved with various experimental arrangements, all differing slightly from one another. Copper (Alloy 110 ASTM B-152) having a thermal conductivity of 388 W/m-K was used as the reference material for the purpose of this work, but other materials may be equally suitable. All the test specimens used in this work were about 1 mm in thickness, which ensured that the temperature remains uniform at each cross-section at a particular location along the length of the test specimen. Two small strip heaters (15 mm wide) were placed above and below the copper strip at one end. The entire assembly was suspended between clamps mounted on two end supports and then placed inside a bell-jar vacuum chamber (15 inches in diameter and 18 inches in height) such that the test specimen is 4 inches above the base plate. It is worth noting that a vacuum environment serves to keep heat losses from the specimen to a minimum as it not only avoids convection but also prevents heat conduction which would adversely affect the experiment if the test specimen was merely surrounded by another insulating material. The conduction into the insulation would be particularly significant if the material to be tested is also a thermal insulator. The stainless steel base plate (21 inches in diameter and 0.5 inches in thickness) of the vacuum chamber was in thermal contact with the end
2. Experimental details The typical experimental configuration of the dual-mode heat flow meter includes a thin specimen (5 mm 50 mm, width length) in line with and contacting a reference material of similar width as shown schematically in Fig. 1. The sample and the reference material were held together by two pieces of thin kapton tape (65 lm thick) that were of similar width as the specimen. Please note that only a diagrammatic representation is used for describing the arrangement of the sample and the reference material; as the underlying assumptions (discussed later) neces-
Fig. 1. Schematic of the experimental setup for the dual-mode heat flow meter technique. The thermocouple used for measuring the ambient temperature is not shown in the schematic.
N.K. Mahanta, A.R. Abramson / International Journal of Heat and Mass Transfer 53 (2010) 5581–5586
support clamping the sample and served as the heat sink for the experiment because of its huge thermal capacity in comparison to the test specimen. Temperature data was obtained from measurements from eight thermocouples (Omega Engineering, Inc. Type K, Size 40, 80 lm diameter) attached using kapton tape: four thermocouples on the sample, two thermocouples (typically 3 mm apart) on the copper strip for measuring the amount of heat flowing into the sample, one for measuring the heat sink temperature and one attached to the bell-jar for measuring the ambient temperature (note that the vacuum chamber used can accommodate a maximum of eight thermocouple feedthroughs). A data acquisition module was used to acquire 1000 samples/s and then average them to yield one temperature reading per second, which ensured that the data was free of any electrical noise.
T avg ¼
such that,
T 2 þ T 21 ðT þ T 1 Þ 4T 3avg
T 4 T 41 ¼ 4T 2avg ðT T 1 Þ
dx
2
erP T 4 T 41 ¼ 0
ð1Þ
where k stands for thermal conductivity, A is the area of cross-section, P is the perimeter of the exposed surface, T1 is the ambient temperature, e is the emissivity and r (= 5.67 108 W/m2-K4) is the Stefan–Boltzmann constant. It is to be noted that the temperature of the bell-jar was taken as the value for T1. In addition, Eq. (1) is based on the assumption of hemispherical radiation from the test specimen. For T T1 (i.e. the heater power is maintained low enough such that the temperatures are within 10 °C of ambient), the above equations can be made linear in T by making the following approximation using The and Tce, which represent the approximate hot and cold end temperatures of the sample, respectively. The detailed process for obtaining an approximate linear differential equation is described as follows:
T 4 T 41 ¼ T 2 þ T 21 ðT þ T 1 ÞðT T 1 Þ
ð2dÞ
We then substitute the average of The and Tce for T in Eq. (2b) to obtain
T avg ¼
2
2
ð2cÞ
thus yielding
d T The heat transfer through the sample in the experimental arrangement described above can be analyzed by modeling it as a finite slab of constant cross-section with radiation from its surfaces. As shown in Fig. 2, the steady-state heat conduction through the region of interest, 0 6 x 6 L, is governed by the following equation:
d T
ð2bÞ
T he þT ce 2
þ T1 2
ð2eÞ
Substituting the result from (2d) into (1) yields:
3. System modeling and data analysis
kA
T þ T1 2
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ð2aÞ
For
T T1 we can define an average temperature as follows:
Fig. 2. Basis for forming a thermal resistance network to determine the relative magnitudes of conduction and radiation. A portion of the amount of heat flowing through the copper strip is lost to the surroundings by radiation.
2
dx
hr P ðT T 1 Þ ¼ 0 kA
ð3Þ
where hr ¼ 4erT 3avg is the coefficient of radiation heat transfer. Eq. (3) is the well known ‘‘pin-fin” heat transfer equation with a coefficient for radiation in place of the usual convection heat transfer. The temperature distribution in the unknown sample is then obtained by solving Eq. (3) using a set of appropriate boundary conditions. In this case, one end of the sample (x = L) is maintained at the temperature of the heat sink, Tce. The other end (x = 0), in contact with the copper reference, receives a constant amount of heat due to conduction through the copper strip. Note that the use of both temperature and heat flux boundary conditions at respective ends (versus two temperature boundary conditions) guarantees a higher accuracy result when thermal conductivity is finally determined. Thus, even though it is possible to use a constant temperature boundary condition at x = 0, it was seen that using a constant heat flux condition ensures that the analytical prediction for temperature profile in the unknown sample is, among all parameters, most sensitive to its thermal conductivity, which is desirable for a thermal characterization technique, especially when testing thermal insulators. On the contrary, we observed that using constant temperature boundary conditions at both ends caused the analytical temperature profile thus obtained to be most sensitive to changes in the end temperatures. In other words, the analytical solution becomes insensitive to variations in thermal conductivity, which is not favorable for the curve-fitting routine (as will be described later) where the sample thermal conductivity is to be varied to obtain the best-fit between the analytical and experimental temperature profiles. In addition, the calculations for determining thermal conductivity remain unaffected by the contact resistance present between the copper reference and the unknown sample, thus yielding an accurate result. This is because, the heat flux calculated based on the temperature readings from the two thermocouples on copper already takes into account the contact resistance (if any) between the sample and the copper reference. With knowledge of the thermal conductivity of the copper reference, kCu, as well as the temperature gradient measured by the two closely placed thermocouples on the copper reference, Fourier Law can be used to determine the total amount of heat flowing, Q, at this location. It is important to note, however, that radiation heat losses from the portion of copper extending from the location of the two thermocouples to the reference/sample interface (shown as distance LCu in Fig. 2) could lead to the actual amount of heat conduction into the sample, Qc, to differ significantly from Q calculated above. In fact, the ratio Qc/Q depends strongly on the ratio Rr/Rc with Rr and Rc, representing the thermal resistance due to radiation over the length, LCu, and the thermal resistance due to conduction through the sample, respectively (see thermal resistance network shown in Fig. 2). Note that it was not experimentally feasible to avoid
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placing the two thermocouples some measurable distance away from the reference/sample interface, and therefore radiation losses were, at times, significant; however, placing the two thermocouples on the copper in very close proximity to each other minimizes radiation losses at this location, facilitating the use of Fourier Law here only. For thin and/or thermally insulating samples, Rc can be significantly higher than Rr, which leads to significant radiation losses in the copper reference material. Since Qc is the required boundary condition in order to solve Eq. (3), it must be determined by subtracting an amount, Qr, from Q, where Qr is the radiation heat loss over the length, LCu. In summary, the following equations are employed to determine Qc:
Q ¼ kCu ACu
DT Cu DxCu
4. Comparison to the conventional heat flow meter
ð4aÞ
Rr ¼ 1=hrCu AsCu Q r ¼ T avgCu T 1 Rr
ð4bÞ
Qc ¼ Q Qr
ð4dÞ
ð4cÞ
where ACu is the cross-sectional area of copper with DTCu and DxCu, respectively, representing the temperature difference and distance between the two thermocouples on copper. It is worth mentioning at this point that the possible heat loss due to radiation from the portion of copper between the two thermocouples was found to be minimal owing to (a) the close proximity of the thermocouples resulting in a small surface area and (b) high thermal conductivity and low emissivity of copper along with only moderately high experimental temperatures (325 K at the most), which result in the resistance to radiation being significantly higher than the resistance to conduction. The coefficient of heat transfer due to radiation from copper is represented by hrCu ¼ 4eCu rT 3avgCu while AsCu and T avgCu , respectively, being the surface area and average temperature of copper over the length of LCu. The expression obtained for the temperature distribution in the sample using Qc calculated above as the second boundary condition is then of the following form:
c ðT ce T 1 Þ coshðxxÞ þ kAQ x sinhfxðL xÞg þ T1 coshðxLÞ rffiffiffiffiffiffiffiffi hr P where x ¼ kA
TðxÞ ¼
with varying sample thermal conductivities until the coefficient-ofdetermination reaches its maximum value, which is achieved when the assumed thermal conductivity is close to the actual thermal conductivity of the sample. It is worth mentioning at this point that the thin (1 mm) and long (>50 mm) test specimens allowed us to safely assume uniform temperatures across any cross-section along the length of the specimen and ensured the validity of the assumption of one-dimensional heat conduction through the test specimen. In addition, this meant that the technique can be safely applied to characterize anisotropic materials as it directly provides the thermal conductivity in the direction of heat flow.
ð5Þ ð6Þ
Buried in Eqs. (4b) and (6), in the heat transfer coefficient terms, hrCu and hr, respectively, is the potentially unknown parameter of emissivity for copper and the sample. There are various ways to determine the emissivity of a material with the most simple approach being the heating of a material to a known temperature (i.e. using a separate temperature probe to confirm), and then employing a precise infrared thermometer with a tunable emissivity input that can be adjusted until the known temperature value is read. Our analysis demonstrated that error in the emissivity measurements of up to 15% does not significantly influence the results, and therefore this procedure was suitable for the method. Such a behavior can be predicted intuitively from Eqs. (5) and (6) as the emissivity term is seen to be first under a square-root and then inside hyperbolic sine and cosine functions, thus resulting in the relative insensitivity of the temperature profile to the emissivity of the material. If an exact emissivity value is unknown, the curve-fitting procedure may be run over a range of emissivity values varying between ±0.15 on either side of an estimated mean value, and an average thermal conductivity calculated. Once all parameters are known, a ‘‘guess” of sample thermal conductivity is then used in Eq. (5) to calculate the temperature distribution in the sample. This analytical temperature profile is then compared with the one obtained experimentally, and the coefficient-of-determination (commonly referred to as the R-squared or R2 value) is computed. This process is repeated
A careful examination of Eq. (3) indicates that the relative influence of radiation and conduction is determined by the magnitude of the coefficient of the dependent variable. In essence, this coefficient represents the ratio of heat transfer by radiation to conduction, and can be expressed non-dimensionally as follows: 3
Bir ¼
hr Lc 8reT avg ¼ k k
wþt 2 L wt
ð7Þ
where, the characteristic dimension, Lc, is a function of the length (L), width (w) and thickness (t) of the sample as shown for this case. This dimensionless ratio, Bir, may be referred to as the Biot number for radiation and is similar to the conventional Biot number, Bi, which gives the ratio of the magnitude of conduction to convection heat transfer of a solid surrounded by a fluid. This parameter may be employed to assess the magnitude of radiation versus conduction heat transfer in the experiment to determine if the dual-mode heat flow meter described herein must be employed rather than a more conventional heat flow meter setup to ensure accuracy. Comparing the temperature profile specified in Eq. (5) to a simple Fourier law relation (in which conduction is the only mode of heat transfer) we found that if Bir does not exceed 0.3, then the conventional heat flow meter (in vacuum) may be employed; however, if this value is surpassed, then the dual-mode heat flow meter (which additionally considers radiation losses) is a viable alternative. As seen from Fig. 3, the temperature distribution in hypothetical samples with Bir > 0.3 can become highly nonlinear. The temperature profiles shown in Fig. 3 were obtained for L = 20 mm, w = 5 mm, t = 10 lm, Qc = 25 mW and Tce = T1 = 25 °C. The thermal conductivity, k, was varied to have values of 100 W/m-K, 500 W/m-K, 1000 W/m-K and 1500 W/m-K to obtain Bir values of 4.47, 0.89, 0.45 and 0.29, respectively. The difference between the two techniques was found to be appreciable only for thermal insulators and/or very thin samples but was not significantly affected by variations in emissivity.
Fig. 3. Temperature distribution in hypothetical samples with different values for the Biot number for radiation. Temperature profiles become highly nonlinear as the Biot number increases beyond 0.3.
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5. Uncertainty analysis Determination of thermal conductivity based on a curve-fitting technique discussed above is influenced by uncertainties associated with each experimental data point as well as prediction uncertainty [8], which is the uncertainty associated with the theoretical solution. For this work, the prediction uncertainty specifically results from modeling uncertainties [8] due to the assumptions in the mathematical representation of the process and the incorporation of other random uncertainties arising from experimental data, such as the position of the thermocouples, length of the sample, etc., in the solution. Thus, in reality, the analytical solution used for comparison with the experimental data should not appear as a fine line, but rather as an area or a ‘‘band” around the experimental data points [8]. These potential inaccuracies lead to a maximum of 12% error in the final results, which was determined by computing the standard deviation in the results obtained from a set of at least five different measurements on a particular sample and expressing it as a percentage of the mean. 6. Results and discussion Table 1 shows the thermal conductivities of some standard materials as measured with the dual-mode heat flow meter technique. All the measurements were conducted on test specimens with standard dimensions specified earlier (5 mm 50 mm, width length). The thickness, however, varied from one material to another. Also, an emissivity value of 0.25 was used for the copper reference, which was measured using an infrared thermometer as described earlier. The aluminum, brass and graphite-based materials (HT-1210, HT-710, SS-600 and SS-1500) enabled validation experiments to be conducted on a wide range of relatively high thermal conductivity materials. The more insulating materials such as CirlexÒ and TeflonÒ were tested to demonstrate the extension of the method to low thermal conductivity materials. It is interesting to note the discrepancy between the actual thermal conductivity and the value that would be obtained without considering the radiation effects, especially for CirlexÒ and TeflonÒ. Fig. 4 shows a comparison of the experimental and analytical temperature profiles obtained for CirlexÒ, a polymer material. Note that the analytical solution has been plotted for two very different values of emissivity to demonstrate that a highly accurate determination of emissivity is not necessary since the analytical solution is not significantly influenced by small errors in this parameter. The thermal conductivities obtained for the two curves corresponding to emissivity values of 0.55 and 0.85 were 0.14 W/m-K and 0.2 W/m-K, respectively, with the ratio of radiation heat loss to conduction heat transfer being 4.5. The uncertainties in the temperature measurements and the location of thermocouples are, respectively, represented by vertical and horizontal error bars. Additional uncertainties involved in the analytical predictions
Fig. 4. Experimental and analytical temperature profiles for CirlexÒ. The vertical and horizontal error bars indicate uncertainties in measurements of temperature and position, respectively. The modeling uncertainties in the analytical solution are not quantifiable and hence not shown. These can be visualized to widen each of the analytical temperature profiles corresponding to a particular emissivity into a band.
and in acquiring experimental data have been discussed in the previous section on Uncertainty Analysis. The results are seen to closely match (within 15% of) previously reported values [10–16] and confirm the applicability of the dual-mode heat flow meter technique for simple, reliable and accurate characterization of materials over a wide range of thermal conductivities. Moreover, this technique offers an additional advantage over the transient methods (also capable of testing materials with wide ranging thermal properties) because density and specific heat capacity are not required for the analysis. In addition, the technique was validated with materials having highly anisotropic thermal properties (HT-1210, HT-710, SS-600 and SS-1500) and can be seen to require no knowledge or assumption about the nature of anisotropy; the thermal conductivity determined corresponds directly to the direction of heat flow. This is in contrast to most transient techniques, which give an effective thermal conductivity when characterizing anisotropic materials. As stated earlier, the capability of accurately characterizing anisotropic materials is only due to the particular experimental configuration, specifically the size and shape of the test specimens, which ensure the validity of the underlying assumptions used for data analysis. Finally, in comparison to other variations on the heat flow meter, the experimental configuration does not require the samples to be of very specific dimensions and/or meet specific limitations placed due to the thermal resistance of the test specimen, thus establishing the method to be more versatile than various other techniques. This is evident from the wide variation in the Bir values from CirlexÒ to SS-1500. In
Table 1 Thermal conductivity of standard materials obtained using the dual-mode heat flow meter setup. The validation results indicate the versatility of the characterization technique. Material (thickness in mm)
CirlexÒ (0.67 mm) FEP TeflonÒ (1.2 mm) Aluminum (1100–H19 Foil) (0.3 mm) Brass (230 Brass OSO15) (0.55 mm) HT-1210 (0.05 mm) HT-710 (0.75 mm) SS-600 (0.15 mm) SS-1500 (0.03 mm)
Thermal conductivity (W/m-K) Measured (mean ± SD)
Literature
0.16 ± 0.01 0.17 ± 0.02 214 ± 15 170 ± 11 106 ± 5 216 ± 8 560 ± 34 1522 ± 74
0.17 [9] 0.20 [10] 218 [11] 159 [12] 108 [13,14] 217 [13,14] 585 [13–16] 1536 [13–16]
Percentage error (%)
Emissivity range
Biot number (Bir) range
Thermal conductivity calculated without considering radiation (W/m-K)
+5.88 +15.0 +1.84 -6.92 +1.85 +0.46 +4.27 +0.91
0.55–0.85 0.45–0.75 0.05–0.15 0.15–0.45 0.6–0.9 0.6–0.9 0.6–0.9 0.6–0.9
1188–1194 1005–1008 0.94–0.97 1.26–1.27 2.12–2.26 1.07–1.15 0.37–0.38 0.16–0.18
1.72 1.02 224 171 134 246 598 1578
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summary, the dual-mode heat flow meter is capable characterizing thermal insulators a well as conductors by using test specimens of the exact same dimensions. 7. Conclusion The dual-mode heat flow meter reported in this work is a simple, reliable and versatile method for characterizing thermal conductivity of solids. The method preserves the inherent simplicities of a steady-state technique while providing a wide range of applicability; usually associated with only the more intricate, transient thermal characterization techniques. Thus, this technique can be seen to combine the pros of the conventional heat flow meter with that of the transient techniques while managing to avoid the cons of either. Moreover, the ability for accurate characterization of anisotropic materials without requiring any knowledge about the nature of anisotropy makes it an attractive choice for research on novel engineered materials. Acknowledgements The authors acknowledge Helen K. Mayer, John Chang, Julian Norley and John Southard of Graftech International Holdings, Inc. for providing the graphite-based materials used in this work. This paper was prepared with financial support from the state of Ohio. The content reflects the views of Case Western Reserve University and does not purport to reflect the views of the state of Ohio. References [1] ASTM C 518-04, Standard test method for steady-state thermal transmission properties by means of the heat flow meter apparatus, ASTM International, 2004.
[2] W.J. Parker, R.J. Jenkins, C.P. Butler, G.L. Abbott, Flash method for determining thermal diffusivity, heat capacity and thermal conductivity, J. Appl. Phys. 32 (9) (1961) 1679–1684. [3] S.E. Gustafsson, E. Karawacki, M.N. Khan, Transient hot-strip method for simultaneously measuring thermal conductivity and thermal diffusivity of solids and liquids, J. Phys. D Appl. Phys. 12 (9) (1979) 1411–1421. [4] D.G. Cahill, Thermal conductivity measurement from 30 to 750 K: the 3x method, Rev. Sci. Instrum. 61 (2) (1990) 802–808. [5] A. Tleoubaev, A. Brzezinski, Errors of heat flow meter method caused by thermal contact resistance, Therm. Conduct. 29: Therm. Expansion 17 (2008) 465–473. [6] S.R. Hostler, A.R. Abramson, M.D. Gawryla, S.A. Bandi, D.A. Schiraldi, Thermal conductivity of clay-based aerogel, Int. J. Heat Mass Transfer 52 (3–4) (2009) 665–669. [7] A. Tleoubaev, A. Brzezinski, Fast measurements of absolute thermal conductivity excluding thermal contact resistance errors, Therm. Conduct. 28: Therm. Expansion 16 (2006) 339–347. [8] H.W. Coleman, W.G. Steele, Experimentation and Uncertainty Analysis for Engineers, Wiley–Interscience, New York, 1999. [9] Fralock – A Division of Lockwood Industries, Inc., CirlexÒ technical data sheet, 2008. [10] McMaster-CarrÒ Document No. 8545KAC, More about high-performance plastics, 2008. [11] The ASM International Handbook Committee, Properties and selection: nonferrous alloys and special-purpose materials, Metals Handbook, vol. 2, 10th ed., ASM International, 1990. [12] J.R. Davis, Copper and copper alloys, Davis and Associates ASM Specialty Handbook, ASM International, 2001. [13] H.K. Mayer, J.C. Chang, Private communication, GrafTech International Limited, Parma, OH, USA, 2009. [14] J.C. Chang, H.K. Mayer, M.D. Smalc, A.R. Abramson, N.K. Mahanta, Thermal conductivity measurements of graphite and nanocomposite materials, Extended Abstract and Poster, ID No. 249, Carbon’09: The Annual World Conference on Carbon, Biarritz, France, 2009. [15] Technical bulletin 321: Data sheet for eGraf spreadershield™ products. GrafTech International Limited, Parma, OH, USA, 2009. Available from:. [16] Y. Xiong, M. Smalc, J. Norely, J. Chang, H. Mayer, P. Skandakumaran, B. Reis, Thermal tests and analysis of thin graphite heat spreader for hot spot reduction in handheld devices, in: 11th IEEE Intersociety Conference on Thermal and Thermomechanical Phenomena in Electronic Systems, Orlando, FL, USA, 1–3, 2008, pp. 583–590.
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
The dual-mode heat flow meter technique: A versatile method for characterizing thermal conductivity Nayandeep K. Mahanta, Alexis R. Abramson * Department of Mechanical and Aerospace Engineering, Case Western Reserve University, 10900 Euclid Avenue, Cleveland, OH 44106-7222, USA
a r t i c l e
i n f o
Article history: Received 27 October 2009 Accepted 24 May 2010 Available online 9 August 2010 Keywords: Thermal conductivity Steady-state Heat flow meter Anisotropic materials Conductors Insulators
a b s t r a c t The dual-mode heat flow meter technique was developed for steady-state thermal characterization by optimizing the curve fit between the experimental temperature profile and the prediction from a simple analytical solution taking into account conductive as well as radiative heat transfer. The validation results were seen to be in good agreement with published literature values and demonstrated the versatility of the method. Moreover, the technique is ideally suited for characterization of anisotropic materials without necessitating any additional information about the nature of anisotropy and lends itself as an attractive alternative for characterization of novel materials with engineered transport properties. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction Efficient and accurate thermal characterization is critical to the continued development of materials with engineered thermal transport properties for applications such as thermal management, thermoelectrics, thermal insulation and more. Thermal characterization techniques should be simple and reliable despite being constrained by the unpredictable nature of the materials being tested. For instance, the measurements should not be affected by the fragility of the samples, their unexpectedly extreme thermal properties (either very low or very high), their constrained geometries, and most importantly, their limited availability. Conventional steady-state thermal conductivity measurement techniques like the heat flow meter [1] are simple and easy to use but have been limited to certain ranges of the spectrum of thermal conductivity, often require specific geometries for testing, and even minor heat losses can significantly influence accuracy. In contrast, transient techniques, such as laser-flash or transient hot-strip, are known to be reliable over the entire range but conductivity measurements typically involve relatively complicated data analysis and are contingent upon the accurate determination of specific heat capacity and density [2,3]. Additionally, most conventional techniques, namely laser-flash or three-omega, may only provide an effective thermal conductivity when used for characterizing materials with anisotropic thermal properties unless modifications are applied [2,4]. Thus, a simple and versatile characterization technique that * Corresponding author. Tel.: +1 216 368 4191; fax: +1 216 368 6445. E-mail address: [email protected] (A.R. Abramson). 0017-9310/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijheatmasstransfer.2010.05.063
is capable of measuring the thermal conductivity in a particular direction without requiring any knowledge or assumption about the properties in the other directions is desirable. In this work, we report on a steady-state thermal characterization technique which can be used for measurement of insulators as well as conductors and is particularly suitable for thin, anisotropic materials. In essence, the method was developed from the standard ASTM C 518 based heat flow meter technique [1], which compares the steady-state temperature gradient in the sample to that of a reference material to determine the thermal conductivity of the unknown sample. The conventional heat flow meter technique assumes one-dimensional conduction to be the only mode of heat transfer and works very well for determining through-thickness thermal conductivity of thin thermal insulators as long as the requirement that one-dimensional heat flow with negligible losses is strictly maintained. The validity of the assumption for onedimensional conduction to be the only mode of heat transfer relies on using test specimens that are short along the direction of heat flow and have a large cross-section. This works very well for low thermal conductivity materials [5,6]. Owing to their short length, the surface area available for convection and/or radiation heat losses from the test specimen becomes negligible compared to the large cross-section available for conduction heat transfer through the specimen. Alternatively for highly conductive materials, establishing a measurable temperature drop is critical and therefore longer specimens may be required for the measurement. However, due to a large surface area, the effects of convection and radiation may no longer be negligible, thereby casting doubt on the reliability and accuracy of the technique. Previous works on use of
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Nomenclature A ACu As AsCu Bi Bir hr hrCu k kCu L Lc LCu P Q Qc Qr Rc Rr T Tavg T avgCu Tce The T1 t
area of cross-section of sample area of cross-section of the copper reference surface area of sample surface area of the copper reference Biot number Biot number for radiation coefficient of radiation heat transfer from sample coefficient of radiation heat transfer from copper reference thermal conductivity of sample thermal conductivity of copper reference length of sample characteristic length length of copper reference perimeter of sample total heat flow heat flow due to conduction heat flow due to radiation resistance to conduction heat transfer resistance to radiation heat transfer temperature average temperature average temperature of copper reference cold end temperature hot end temperature ambient temperature thickness
the standard technique did not consider the effects of the convection and/or radiation losses [5–7], perhaps because the heat flow meter had primarily been used for testing materials with thermal conductivities less than 20 W/m-K [5,6]. More importantly, these materials had provided the researchers with a test specimen such that the thickness and width were larger compared to the length in the direction along which the thermal conductivity was desired to be estimated. However, a situation where such a specimen may not be obtainable is likely, especially with the widespread research on novel composite materials, including nanocomposites. In other words, it is likely that a material could be thin and anisotropic while being required to be tested for thermal conductivity in the direction perpendicular to its thickness. To overcome these obstacles, the dual-mode heat flow meter, which considers both conduction and radiation modes of heat transfer along a specimen of relatively long length was conceived to broaden the applicability of the heat flow meter technique while maintaining its inherent simplicity. This technique accurately and reliably measures thermal conductivity along the direction of one-dimensional heat flow. Convection heat loss can be minimized by conducting the experiments under conditions of very high vacuum (pressures less than 105 Torr).
w x
width Cartesian coordinate along the direction of conduction heat transfer
Greek symbols DT temperature difference in sample DTCu temperature difference in copper reference Dx distance between two points on the sample DxCu distance between two points on the copper reference e emissivity of sample eCu emissivity of copper r Stefan–Boltzmann constant x dimensional ratio of heat transfer due to radiation and conduction Subscripts avg average Cu copper ce cold end he hot end r radiation s surface 1 ambient Subscript to subscripts Cu copper
sary for this method can be achieved with various experimental arrangements, all differing slightly from one another. Copper (Alloy 110 ASTM B-152) having a thermal conductivity of 388 W/m-K was used as the reference material for the purpose of this work, but other materials may be equally suitable. All the test specimens used in this work were about 1 mm in thickness, which ensured that the temperature remains uniform at each cross-section at a particular location along the length of the test specimen. Two small strip heaters (15 mm wide) were placed above and below the copper strip at one end. The entire assembly was suspended between clamps mounted on two end supports and then placed inside a bell-jar vacuum chamber (15 inches in diameter and 18 inches in height) such that the test specimen is 4 inches above the base plate. It is worth noting that a vacuum environment serves to keep heat losses from the specimen to a minimum as it not only avoids convection but also prevents heat conduction which would adversely affect the experiment if the test specimen was merely surrounded by another insulating material. The conduction into the insulation would be particularly significant if the material to be tested is also a thermal insulator. The stainless steel base plate (21 inches in diameter and 0.5 inches in thickness) of the vacuum chamber was in thermal contact with the end
2. Experimental details The typical experimental configuration of the dual-mode heat flow meter includes a thin specimen (5 mm 50 mm, width length) in line with and contacting a reference material of similar width as shown schematically in Fig. 1. The sample and the reference material were held together by two pieces of thin kapton tape (65 lm thick) that were of similar width as the specimen. Please note that only a diagrammatic representation is used for describing the arrangement of the sample and the reference material; as the underlying assumptions (discussed later) neces-
Fig. 1. Schematic of the experimental setup for the dual-mode heat flow meter technique. The thermocouple used for measuring the ambient temperature is not shown in the schematic.
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support clamping the sample and served as the heat sink for the experiment because of its huge thermal capacity in comparison to the test specimen. Temperature data was obtained from measurements from eight thermocouples (Omega Engineering, Inc. Type K, Size 40, 80 lm diameter) attached using kapton tape: four thermocouples on the sample, two thermocouples (typically 3 mm apart) on the copper strip for measuring the amount of heat flowing into the sample, one for measuring the heat sink temperature and one attached to the bell-jar for measuring the ambient temperature (note that the vacuum chamber used can accommodate a maximum of eight thermocouple feedthroughs). A data acquisition module was used to acquire 1000 samples/s and then average them to yield one temperature reading per second, which ensured that the data was free of any electrical noise.
T avg ¼
such that,
T 2 þ T 21 ðT þ T 1 Þ 4T 3avg
T 4 T 41 ¼ 4T 2avg ðT T 1 Þ
dx
2
erP T 4 T 41 ¼ 0
ð1Þ
where k stands for thermal conductivity, A is the area of cross-section, P is the perimeter of the exposed surface, T1 is the ambient temperature, e is the emissivity and r (= 5.67 108 W/m2-K4) is the Stefan–Boltzmann constant. It is to be noted that the temperature of the bell-jar was taken as the value for T1. In addition, Eq. (1) is based on the assumption of hemispherical radiation from the test specimen. For T T1 (i.e. the heater power is maintained low enough such that the temperatures are within 10 °C of ambient), the above equations can be made linear in T by making the following approximation using The and Tce, which represent the approximate hot and cold end temperatures of the sample, respectively. The detailed process for obtaining an approximate linear differential equation is described as follows:
T 4 T 41 ¼ T 2 þ T 21 ðT þ T 1 ÞðT T 1 Þ
ð2dÞ
We then substitute the average of The and Tce for T in Eq. (2b) to obtain
T avg ¼
2
2
ð2cÞ
thus yielding
d T The heat transfer through the sample in the experimental arrangement described above can be analyzed by modeling it as a finite slab of constant cross-section with radiation from its surfaces. As shown in Fig. 2, the steady-state heat conduction through the region of interest, 0 6 x 6 L, is governed by the following equation:
d T
ð2bÞ
T he þT ce 2
þ T1 2
ð2eÞ
Substituting the result from (2d) into (1) yields:
3. System modeling and data analysis
kA
T þ T1 2
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ð2aÞ
For
T T1 we can define an average temperature as follows:
Fig. 2. Basis for forming a thermal resistance network to determine the relative magnitudes of conduction and radiation. A portion of the amount of heat flowing through the copper strip is lost to the surroundings by radiation.
2
dx
hr P ðT T 1 Þ ¼ 0 kA
ð3Þ
where hr ¼ 4erT 3avg is the coefficient of radiation heat transfer. Eq. (3) is the well known ‘‘pin-fin” heat transfer equation with a coefficient for radiation in place of the usual convection heat transfer. The temperature distribution in the unknown sample is then obtained by solving Eq. (3) using a set of appropriate boundary conditions. In this case, one end of the sample (x = L) is maintained at the temperature of the heat sink, Tce. The other end (x = 0), in contact with the copper reference, receives a constant amount of heat due to conduction through the copper strip. Note that the use of both temperature and heat flux boundary conditions at respective ends (versus two temperature boundary conditions) guarantees a higher accuracy result when thermal conductivity is finally determined. Thus, even though it is possible to use a constant temperature boundary condition at x = 0, it was seen that using a constant heat flux condition ensures that the analytical prediction for temperature profile in the unknown sample is, among all parameters, most sensitive to its thermal conductivity, which is desirable for a thermal characterization technique, especially when testing thermal insulators. On the contrary, we observed that using constant temperature boundary conditions at both ends caused the analytical temperature profile thus obtained to be most sensitive to changes in the end temperatures. In other words, the analytical solution becomes insensitive to variations in thermal conductivity, which is not favorable for the curve-fitting routine (as will be described later) where the sample thermal conductivity is to be varied to obtain the best-fit between the analytical and experimental temperature profiles. In addition, the calculations for determining thermal conductivity remain unaffected by the contact resistance present between the copper reference and the unknown sample, thus yielding an accurate result. This is because, the heat flux calculated based on the temperature readings from the two thermocouples on copper already takes into account the contact resistance (if any) between the sample and the copper reference. With knowledge of the thermal conductivity of the copper reference, kCu, as well as the temperature gradient measured by the two closely placed thermocouples on the copper reference, Fourier Law can be used to determine the total amount of heat flowing, Q, at this location. It is important to note, however, that radiation heat losses from the portion of copper extending from the location of the two thermocouples to the reference/sample interface (shown as distance LCu in Fig. 2) could lead to the actual amount of heat conduction into the sample, Qc, to differ significantly from Q calculated above. In fact, the ratio Qc/Q depends strongly on the ratio Rr/Rc with Rr and Rc, representing the thermal resistance due to radiation over the length, LCu, and the thermal resistance due to conduction through the sample, respectively (see thermal resistance network shown in Fig. 2). Note that it was not experimentally feasible to avoid
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placing the two thermocouples some measurable distance away from the reference/sample interface, and therefore radiation losses were, at times, significant; however, placing the two thermocouples on the copper in very close proximity to each other minimizes radiation losses at this location, facilitating the use of Fourier Law here only. For thin and/or thermally insulating samples, Rc can be significantly higher than Rr, which leads to significant radiation losses in the copper reference material. Since Qc is the required boundary condition in order to solve Eq. (3), it must be determined by subtracting an amount, Qr, from Q, where Qr is the radiation heat loss over the length, LCu. In summary, the following equations are employed to determine Qc:
Q ¼ kCu ACu
DT Cu DxCu
4. Comparison to the conventional heat flow meter
ð4aÞ
Rr ¼ 1=hrCu AsCu Q r ¼ T avgCu T 1 Rr
ð4bÞ
Qc ¼ Q Qr
ð4dÞ
ð4cÞ
where ACu is the cross-sectional area of copper with DTCu and DxCu, respectively, representing the temperature difference and distance between the two thermocouples on copper. It is worth mentioning at this point that the possible heat loss due to radiation from the portion of copper between the two thermocouples was found to be minimal owing to (a) the close proximity of the thermocouples resulting in a small surface area and (b) high thermal conductivity and low emissivity of copper along with only moderately high experimental temperatures (325 K at the most), which result in the resistance to radiation being significantly higher than the resistance to conduction. The coefficient of heat transfer due to radiation from copper is represented by hrCu ¼ 4eCu rT 3avgCu while AsCu and T avgCu , respectively, being the surface area and average temperature of copper over the length of LCu. The expression obtained for the temperature distribution in the sample using Qc calculated above as the second boundary condition is then of the following form:
c ðT ce T 1 Þ coshðxxÞ þ kAQ x sinhfxðL xÞg þ T1 coshðxLÞ rffiffiffiffiffiffiffiffi hr P where x ¼ kA
TðxÞ ¼
with varying sample thermal conductivities until the coefficient-ofdetermination reaches its maximum value, which is achieved when the assumed thermal conductivity is close to the actual thermal conductivity of the sample. It is worth mentioning at this point that the thin (1 mm) and long (>50 mm) test specimens allowed us to safely assume uniform temperatures across any cross-section along the length of the specimen and ensured the validity of the assumption of one-dimensional heat conduction through the test specimen. In addition, this meant that the technique can be safely applied to characterize anisotropic materials as it directly provides the thermal conductivity in the direction of heat flow.
ð5Þ ð6Þ
Buried in Eqs. (4b) and (6), in the heat transfer coefficient terms, hrCu and hr, respectively, is the potentially unknown parameter of emissivity for copper and the sample. There are various ways to determine the emissivity of a material with the most simple approach being the heating of a material to a known temperature (i.e. using a separate temperature probe to confirm), and then employing a precise infrared thermometer with a tunable emissivity input that can be adjusted until the known temperature value is read. Our analysis demonstrated that error in the emissivity measurements of up to 15% does not significantly influence the results, and therefore this procedure was suitable for the method. Such a behavior can be predicted intuitively from Eqs. (5) and (6) as the emissivity term is seen to be first under a square-root and then inside hyperbolic sine and cosine functions, thus resulting in the relative insensitivity of the temperature profile to the emissivity of the material. If an exact emissivity value is unknown, the curve-fitting procedure may be run over a range of emissivity values varying between ±0.15 on either side of an estimated mean value, and an average thermal conductivity calculated. Once all parameters are known, a ‘‘guess” of sample thermal conductivity is then used in Eq. (5) to calculate the temperature distribution in the sample. This analytical temperature profile is then compared with the one obtained experimentally, and the coefficient-of-determination (commonly referred to as the R-squared or R2 value) is computed. This process is repeated
A careful examination of Eq. (3) indicates that the relative influence of radiation and conduction is determined by the magnitude of the coefficient of the dependent variable. In essence, this coefficient represents the ratio of heat transfer by radiation to conduction, and can be expressed non-dimensionally as follows: 3
Bir ¼
hr Lc 8reT avg ¼ k k
wþt 2 L wt
ð7Þ
where, the characteristic dimension, Lc, is a function of the length (L), width (w) and thickness (t) of the sample as shown for this case. This dimensionless ratio, Bir, may be referred to as the Biot number for radiation and is similar to the conventional Biot number, Bi, which gives the ratio of the magnitude of conduction to convection heat transfer of a solid surrounded by a fluid. This parameter may be employed to assess the magnitude of radiation versus conduction heat transfer in the experiment to determine if the dual-mode heat flow meter described herein must be employed rather than a more conventional heat flow meter setup to ensure accuracy. Comparing the temperature profile specified in Eq. (5) to a simple Fourier law relation (in which conduction is the only mode of heat transfer) we found that if Bir does not exceed 0.3, then the conventional heat flow meter (in vacuum) may be employed; however, if this value is surpassed, then the dual-mode heat flow meter (which additionally considers radiation losses) is a viable alternative. As seen from Fig. 3, the temperature distribution in hypothetical samples with Bir > 0.3 can become highly nonlinear. The temperature profiles shown in Fig. 3 were obtained for L = 20 mm, w = 5 mm, t = 10 lm, Qc = 25 mW and Tce = T1 = 25 °C. The thermal conductivity, k, was varied to have values of 100 W/m-K, 500 W/m-K, 1000 W/m-K and 1500 W/m-K to obtain Bir values of 4.47, 0.89, 0.45 and 0.29, respectively. The difference between the two techniques was found to be appreciable only for thermal insulators and/or very thin samples but was not significantly affected by variations in emissivity.
Fig. 3. Temperature distribution in hypothetical samples with different values for the Biot number for radiation. Temperature profiles become highly nonlinear as the Biot number increases beyond 0.3.
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5. Uncertainty analysis Determination of thermal conductivity based on a curve-fitting technique discussed above is influenced by uncertainties associated with each experimental data point as well as prediction uncertainty [8], which is the uncertainty associated with the theoretical solution. For this work, the prediction uncertainty specifically results from modeling uncertainties [8] due to the assumptions in the mathematical representation of the process and the incorporation of other random uncertainties arising from experimental data, such as the position of the thermocouples, length of the sample, etc., in the solution. Thus, in reality, the analytical solution used for comparison with the experimental data should not appear as a fine line, but rather as an area or a ‘‘band” around the experimental data points [8]. These potential inaccuracies lead to a maximum of 12% error in the final results, which was determined by computing the standard deviation in the results obtained from a set of at least five different measurements on a particular sample and expressing it as a percentage of the mean. 6. Results and discussion Table 1 shows the thermal conductivities of some standard materials as measured with the dual-mode heat flow meter technique. All the measurements were conducted on test specimens with standard dimensions specified earlier (5 mm 50 mm, width length). The thickness, however, varied from one material to another. Also, an emissivity value of 0.25 was used for the copper reference, which was measured using an infrared thermometer as described earlier. The aluminum, brass and graphite-based materials (HT-1210, HT-710, SS-600 and SS-1500) enabled validation experiments to be conducted on a wide range of relatively high thermal conductivity materials. The more insulating materials such as CirlexÒ and TeflonÒ were tested to demonstrate the extension of the method to low thermal conductivity materials. It is interesting to note the discrepancy between the actual thermal conductivity and the value that would be obtained without considering the radiation effects, especially for CirlexÒ and TeflonÒ. Fig. 4 shows a comparison of the experimental and analytical temperature profiles obtained for CirlexÒ, a polymer material. Note that the analytical solution has been plotted for two very different values of emissivity to demonstrate that a highly accurate determination of emissivity is not necessary since the analytical solution is not significantly influenced by small errors in this parameter. The thermal conductivities obtained for the two curves corresponding to emissivity values of 0.55 and 0.85 were 0.14 W/m-K and 0.2 W/m-K, respectively, with the ratio of radiation heat loss to conduction heat transfer being 4.5. The uncertainties in the temperature measurements and the location of thermocouples are, respectively, represented by vertical and horizontal error bars. Additional uncertainties involved in the analytical predictions
Fig. 4. Experimental and analytical temperature profiles for CirlexÒ. The vertical and horizontal error bars indicate uncertainties in measurements of temperature and position, respectively. The modeling uncertainties in the analytical solution are not quantifiable and hence not shown. These can be visualized to widen each of the analytical temperature profiles corresponding to a particular emissivity into a band.
and in acquiring experimental data have been discussed in the previous section on Uncertainty Analysis. The results are seen to closely match (within 15% of) previously reported values [10–16] and confirm the applicability of the dual-mode heat flow meter technique for simple, reliable and accurate characterization of materials over a wide range of thermal conductivities. Moreover, this technique offers an additional advantage over the transient methods (also capable of testing materials with wide ranging thermal properties) because density and specific heat capacity are not required for the analysis. In addition, the technique was validated with materials having highly anisotropic thermal properties (HT-1210, HT-710, SS-600 and SS-1500) and can be seen to require no knowledge or assumption about the nature of anisotropy; the thermal conductivity determined corresponds directly to the direction of heat flow. This is in contrast to most transient techniques, which give an effective thermal conductivity when characterizing anisotropic materials. As stated earlier, the capability of accurately characterizing anisotropic materials is only due to the particular experimental configuration, specifically the size and shape of the test specimens, which ensure the validity of the underlying assumptions used for data analysis. Finally, in comparison to other variations on the heat flow meter, the experimental configuration does not require the samples to be of very specific dimensions and/or meet specific limitations placed due to the thermal resistance of the test specimen, thus establishing the method to be more versatile than various other techniques. This is evident from the wide variation in the Bir values from CirlexÒ to SS-1500. In
Table 1 Thermal conductivity of standard materials obtained using the dual-mode heat flow meter setup. The validation results indicate the versatility of the characterization technique. Material (thickness in mm)
CirlexÒ (0.67 mm) FEP TeflonÒ (1.2 mm) Aluminum (1100–H19 Foil) (0.3 mm) Brass (230 Brass OSO15) (0.55 mm) HT-1210 (0.05 mm) HT-710 (0.75 mm) SS-600 (0.15 mm) SS-1500 (0.03 mm)
Thermal conductivity (W/m-K) Measured (mean ± SD)
Literature
0.16 ± 0.01 0.17 ± 0.02 214 ± 15 170 ± 11 106 ± 5 216 ± 8 560 ± 34 1522 ± 74
0.17 [9] 0.20 [10] 218 [11] 159 [12] 108 [13,14] 217 [13,14] 585 [13–16] 1536 [13–16]
Percentage error (%)
Emissivity range
Biot number (Bir) range
Thermal conductivity calculated without considering radiation (W/m-K)
+5.88 +15.0 +1.84 -6.92 +1.85 +0.46 +4.27 +0.91
0.55–0.85 0.45–0.75 0.05–0.15 0.15–0.45 0.6–0.9 0.6–0.9 0.6–0.9 0.6–0.9
1188–1194 1005–1008 0.94–0.97 1.26–1.27 2.12–2.26 1.07–1.15 0.37–0.38 0.16–0.18
1.72 1.02 224 171 134 246 598 1578
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summary, the dual-mode heat flow meter is capable characterizing thermal insulators a well as conductors by using test specimens of the exact same dimensions. 7. Conclusion The dual-mode heat flow meter reported in this work is a simple, reliable and versatile method for characterizing thermal conductivity of solids. The method preserves the inherent simplicities of a steady-state technique while providing a wide range of applicability; usually associated with only the more intricate, transient thermal characterization techniques. Thus, this technique can be seen to combine the pros of the conventional heat flow meter with that of the transient techniques while managing to avoid the cons of either. Moreover, the ability for accurate characterization of anisotropic materials without requiring any knowledge about the nature of anisotropy makes it an attractive choice for research on novel engineered materials. Acknowledgements The authors acknowledge Helen K. Mayer, John Chang, Julian Norley and John Southard of Graftech International Holdings, Inc. for providing the graphite-based materials used in this work. This paper was prepared with financial support from the state of Ohio. The content reflects the views of Case Western Reserve University and does not purport to reflect the views of the state of Ohio. References [1] ASTM C 518-04, Standard test method for steady-state thermal transmission properties by means of the heat flow meter apparatus, ASTM International, 2004.
[2] W.J. Parker, R.J. Jenkins, C.P. Butler, G.L. Abbott, Flash method for determining thermal diffusivity, heat capacity and thermal conductivity, J. Appl. Phys. 32 (9) (1961) 1679–1684. [3] S.E. Gustafsson, E. Karawacki, M.N. Khan, Transient hot-strip method for simultaneously measuring thermal conductivity and thermal diffusivity of solids and liquids, J. Phys. D Appl. Phys. 12 (9) (1979) 1411–1421. [4] D.G. Cahill, Thermal conductivity measurement from 30 to 750 K: the 3x method, Rev. Sci. Instrum. 61 (2) (1990) 802–808. [5] A. Tleoubaev, A. Brzezinski, Errors of heat flow meter method caused by thermal contact resistance, Therm. Conduct. 29: Therm. Expansion 17 (2008) 465–473. [6] S.R. Hostler, A.R. Abramson, M.D. Gawryla, S.A. Bandi, D.A. Schiraldi, Thermal conductivity of clay-based aerogel, Int. J. Heat Mass Transfer 52 (3–4) (2009) 665–669. [7] A. Tleoubaev, A. Brzezinski, Fast measurements of absolute thermal conductivity excluding thermal contact resistance errors, Therm. Conduct. 28: Therm. Expansion 16 (2006) 339–347. [8] H.W. Coleman, W.G. Steele, Experimentation and Uncertainty Analysis for Engineers, Wiley–Interscience, New York, 1999. [9] Fralock – A Division of Lockwood Industries, Inc., CirlexÒ technical data sheet, 2008. [10] McMaster-CarrÒ Document No. 8545KAC, More about high-performance plastics, 2008. [11] The ASM International Handbook Committee, Properties and selection: nonferrous alloys and special-purpose materials, Metals Handbook, vol. 2, 10th ed., ASM International, 1990. [12] J.R. Davis, Copper and copper alloys, Davis and Associates ASM Specialty Handbook, ASM International, 2001. [13] H.K. Mayer, J.C. Chang, Private communication, GrafTech International Limited, Parma, OH, USA, 2009. [14] J.C. Chang, H.K. Mayer, M.D. Smalc, A.R. Abramson, N.K. Mahanta, Thermal conductivity measurements of graphite and nanocomposite materials, Extended Abstract and Poster, ID No. 249, Carbon’09: The Annual World Conference on Carbon, Biarritz, France, 2009. [15] Technical bulletin 321: Data sheet for eGraf spreadershield™ products. GrafTech International Limited, Parma, OH, USA, 2009. Available from: