Measurement of thermal conductivity with the needle probe

Transcription

1 High Temperatures ^ High Pressures, 2003/2004, volume 35/36, pages 127 ^ 138 DOI: /htjr099 Measurement of thermal conductivity with the needle probe Neil Lockmuller, John Redgrove National Physical Laboratory, Queens Road, Teddington, Middlesex, TW11 0LW, UK; neil.lockmuller@npl.co.uk L'udov t Kubic a r Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia; fyzikubo@savba.sk Presented at the 16th European Conference on Thermophysical Properties, Imperial College, London, England, 1 ^ 4 September 2002 Abstract. The thermal conductivity of a range of materials has been measured by a transient needle probe technique. These include graphite, Perspex, AFS testing sand, and silica microspheres. Particular attention is paid to measurement on powders, for which packing density is important. This work forms part of an NPL project to validate transient contact-probe methods (needle probe and transient plane source) for measurement of the thermal conductivity of soft solids such as foods, powders, slurries, and biological materials. 1 Introduction Industry needs reliable, flexible, and preferably rapid methods for measuring the thermophysical properties of materials. There are many techniques available, depending on specimen geometry, measuring regime, surrounding atmosphere, and temperature range. In this paper we present progress on development of validated measurements of the thermal conductivity of materials in the form of soft solids such as powders, slurries, sludges, pastes, or highly viscous fluids. We employ the transient needle probe (Blackwell 1953, 1954), which is similar to the transient hot-wire method (Davis 1984). Part of the validation is to be achieved by measuring the thermal conductivity of reference materials and comparison with results obtained by other methods. 2 Theory The principle of the needle probe is similar to that of the hot-wire in that the specimen is heated at a constant heat input with a long, straight, and ideally uniform heat source, and the temperature rise at a given point is measured as a function of time. The probe consists of a long, thin steel sheath containing a wire heating element, thermometer, and insulating material. For short heating times (defined by t 4 v 2 =a, where v is the needle probe sheath radius and a is the specimen thermal diffusivity) the temperature rise Dy vs natural logarithm of time obeys the relation below (Blackwell 1953; Bruijn et al 1983; Asher et al 1985; Xie and Cheng 2001): D E ln t=s Dy ˆ A ln t=s B, (1) t=s where t is the time from power switch-on and A, B, C, D are coefficients dependent on probe geometry, heater power per unit length, probe/specimen thermophysical properties, and the thermal contact resistance between the sample and the probe. This model assumes a probe of infinite length that is embedded in an infinite medium, and therefore that the isotherms formed within the specimen represent perfect cylinders of infinite length with continually increasing radii. For sufficiently large times, equation (1) approximates to a simple straight-line equation: Dy ˆ A ln t=s B. (2)

2 128 N Lockmuller, J Redgrove, L Kubic a r It is the above linear portion of the Dy vs ln (t=s) curve that is selected and used in the calculation of thermal conductivity by means of equation (3) below (Liang 1995): l ˆ Q 4pS, (3) where Q is the power dissipated per unit length and S is the slope [ A in equation (2)] of the linear portion of the Dy vs (t=s) curve. Figure 1 shows the general shape of the temperature rise of a needle probe during a specimen measurement. Temperature rise Region 1 Region 2 Region 3 ln t=s Figure 1. General shape of probe temperature rise. The curve can be divided into three regions: Region 1. This region corresponds to small heating times during which the shape of the curve is influenced by the thermophysical properties of the probe. It can be described, except for very small heating times, by the Jaeger model (Carslaw and Jaeger 1996). Region 2. In this region most of the heat from the probe is passing into the specimen and the curve is approximately linear, as described by equation (2). Region 3. In the ideal situation, with an infinitely long probe and infinitely large specimen, region 2 would continue indefinitely, but in reality the finite size of the specimen and finite length of the probe cause deviations from a straight line as time increases. Factors affecting the shape of this part of the curve include: (i) heat reaching the outer surface of the specimen, and (ii) deviation of isotherms within the specimen from cylindrical geometry due to the finite length of the probe and conductive heat loss through the probe leads. Region 2 is characterised mainly by the specimen thermal properties, whereas regions 1 and 3 are more strongly influenced by the probe or boundary of the specimen. Therefore, for evaluation of the specimen thermal conductivity, data are analysed for times that lie within region 2 (the evaluation time window). Factor (i) above is characterised by the penetration depth h pen : h pen ˆ at 1=2, (4) where h pen is the distance into the specimen that the applied heat diffuses after time t (Gustafsson 1991). It helps to have region 2 as wide as possible, thus giving plenty of data that represent the sample bulk thermal behaviour. To achieve this requires: (a) a probe length-to-diameter ratio as large as possible to approximate an `infinitely long and infinitesimally thin wire' (in practice, a probe length-to-diameter ratio >50 is often used and in the current work the ratio was 100); (b) as thin a needle probe as possible, thus allowing smaller specimens (for a large probe diameter a large heating time and thus a large specimen is required); (c) minimal thermal contact resistance between the specimen and the probe.

3 Measurement of thermal conductivity with the needle probe Experimental method 3.1 Apparatus The probe used in the experiments was supplied by Dr Izaa«k van Haneghem of Wageningen University in the Netherlands. It consists of a 1 mm outer diameter stainless steel sheath of length 100 mm containing separate Constantan wire heating element, a differential thermocouple (DTC) and silicone rubber insulation compound. One junction of the DTC is adjacent to the heater wire, and approximately halfway along it, whilst the other is positioned at the tip of the sheath (figure 2). The heater wire falls short of the tip by approximately 40 mm. barrel stainless steel thermocouple heating wire silicone rubber compound tube or sheath hot junction thermocouple cold reference junction Figure 2. The needle probe. 30 mm 40 mm 100 mm The DTC, with its hot and cold reference junctions positioned as shown approximately in figure 2, is used to determine the temperature rise inside the probe at a point between the heater wire and inner wall of the sheath. This is on condition that (i) prior to heating, the probe and specimen are at the same uniform steady temperature, and (ii) during a measurement the DTC cold junction temperature remains constant. This arrangement is more accurate than one which uses a single thermocouple junction or other single sensor, for two reasons. First, the output is proportional to temperature rise only, ie it does not appear on top of a larger signal at high and low temperatures. Second, having the reference and hot junction in the needle minimises the sensitivity to temperature changes of the medium. The needle probe apparatus consists of the following: ö1 mm diameter 100 mm length needle probe with Type T differential thermocouple (DTC), öscanner with isothermal connector block and internal amplifiers for each channel, öpower supply (1 A at 20 V, stability 0.01%), öjunction box housing various electrical connections and 0.1 O standard resistor (SR), öcomputer with analog-to-digital converter (ADC) card connected to scanner, ötemperature-controlled fluid circulator and fluid bath ( 40 8Cto 150 8C), öspecimen cell (inner diameter 85 mm, inner depth 250 mm). The apparatus and needle probe are shown in figures 3 and 4, respectively. The specimen container is filled with specimen material and a lid with a fluid-tight seal is bolted to the top. The probe is inserted into the specimen through a hole in the lid and sealed with an O-ring. The specimen container is placed in the fluid bath, the temperature of which is controlled by having the fluid pumped through the temperaturecontrolled fluid circulator. The scanner, standard resistor, and ADC card are used to measure the power supplied to the needle probe heater wire and voltage of the differential thermocouple.

4 130 N Lockmuller, J Redgrove, L Kubic a r specimen container fluid bath junction box scanner voltage source Figure 3. Needle probe apparatus. probe temperaturecontrolled fluid circulator computer with ADC IEEE Figure 4. Needle probe with 1 mm diameter stainless steel sheath. 3.2 Setting measurement parameters For all suitable specimen types, the experimental parameters are set as follows (see Kubic är 1990, for a general discussion of the methodology for pulse methods). Most of the parameter values recommended below are based on the authors' experience of contact-probe measurements on various materials. (1) There should be a minimum stabilisation time between measurement runs, dependent on sample size and thermal properties. Figure 5 is a general guide for the probe approach based on experimentation on various materials and helps in determining the required stabilisation time as a function of specimen thermal conductivity. After a measurement, we monitor the DTC output and its rate of change until they are acceptably stable (typically 0.2 K min 1 and 2.5 mk min 1, respectively). This led in practice to stabilisation times of about 30 min for graphite and 60 min for Perspex and silica microspheres. (2) To avoid both damage to the probe and a temperature-dependent change in sample thermal properties, the power output and measuring time are chosen so as to limit the temperature rise to no more than 4 K. (In this paper, we report measurements on dry materials only. For moist materials a rise of 1 ^ 2 K would be more appropriate as a 4 K rise would generally be too large, causing both mass and heat transfer.) (3) An initial test is performed for approximately 1000 s with a scanning interval of 1 s, setting a heater power as indicated by figure 6. (As with figure 5, figure 6 is based on experimentation with various materials.) An analysis of the slope vs middle time (figure 8) gives an optimal measuring time (see section 3.4 for a fuller explanation).

5 Stabilisation time=min Measurement of thermal conductivity with the needle probe Figure 5. Suggested minimum specimen pre-measurement stablisation times as a function of thermal conductivity. 10 Power output=w m Figure 6. Recommended heater power vs specimen thermal conductivity. Measured thermal conductivity Useable power range Heater power per unit length Figure 7. Illustrative plot of specimen thermal conductivity as a function of heater power. (4) A set of measurements is made with various heater powers to find the range of powers within which the measured thermal conductivity is constant and for which the temperature rise does not exceed 4 K (figure 7). (5) A convenient number of data points (heater current and DTC voltage) to be acquired during a measurement is 250 to 300. The scanning interval is equal to the total measuring time divided by the chosen number of points to be acquired.

6 132 N Lockmuller, J Redgrove, L Kubic a r 3.3 Measurement method Electrical power is supplied to the needle probe heater element, which is connected in series with a standard resistor for measurement of current. The power dissipated per unit length is calculated from the current and the resistance per unit length of the heater element. The latter is determined from a specimen of wire from the same batch from which the element was made. The outputs from the standard resistor and differential thermocouple are connected via a junction box to a programmable scanner, which relays the signals to the ADC in the computer. Each channel on the scanner has its own two-stage amplifier, the gain of which is set for the required sensitivity. Specimens are placed in a suitable container and then immersed in the fluid bath. The fluid is circulated continuously between the bath and temperature-controlled fluid circulator (figure 3). Non-solid specimens, such as powders, are put into the copper specimen cell, which is then placed in the bath in a special supporting frame. To prevent fluid from leaking into the cell, a lid is attached and secured with a seal and nuts and bolts. The needle probe is inserted into the specimen through a feed-through in the lid. Solid specimens are placed inside polythene sacks before being immersed in the fluid bath. The measurement automation software (in LabVIEW) performs the following: öremote programming of the needle probe power supply unit (PSU), scanner, and ADC, ödata acquisition, processing, and logging, öautomated measurement runs according to a schedule set by the operator, ötests for specimen thermal equilibrium according to criteria set by the operator. Only the fluid bath temperature is set manually. Prior to measurement, the user inputs to the computer program the required stability criteria, the experimental details, eg specimen identification, and a schedule of measurements, including powers, times, sampling rates, and number of repeat runs. All results are saved to a text file or an MS Excel spreadsheet as they are received. Thermal stability is determined by measuring Dy and rate of change of Dy with no heater power applied. Temperature rise is calculated from the DTC output. To quantify the precision of the measurements (typically a few per cent), they are repeated. Typically a minimum of five measurements are made at a given temperature. 3.4 Analysis of experimental results The data analysis method involves plotting the slope of the Dy vs ln (t=s) data against either ln (t=s) or absolute time and using that to determine which portion of the data is most suitable for fitting to a straight line. Our implementation of this is, at present, still partly subjective but has scope for a well-defined analysis procedure with objectively defined data selection criteria. The procedure for estimation of the time window is as follows (Kubic är and Boha c 2000): the temperature rise data are plotted as a function of ln (t=s), where the time values range from t ˆ 0 to t ˆ t max. A fixed time interval D is chosen and we define a `middle time' t m which corresponds to the time values between t ˆ D=2 and t ˆ t max D=2. For each t m value, a straight line is fitted to the temperature-rise data in the time interval (t m D=2, t m D=2), and the resulting slope value is plotted versus t m, as in figure 8. A time interval D corresponding to 25 scans was used, on the assumption that the measuring time is represented by 250 scans. The three regions of the plot in figure 8, ie the central window and the regions either side of it, correspond to the three regions in figure 1. The range of points in figure 8 that lie in the flattest part of the data are selected, and the mean slope value S obtained, from which l can be determined by using the relation l ˆ Q=4pS [equation (3)].

7 Measurement of thermal conductivity with the needle probe 133 Slope time window Middle time, t m Figure 8. Determination of evaluation time window from plot of slope vs middle time. The limits shown in figure 8 were chosen by eye and selected on the basis of maximum flatness and minimal degree of data scattering. 4 Specimen details Four materials have been measured: graphite (supplied by Morganite Special Carbons), Perspex, silica microspheres, and AFS testing sand. The solid specimens (Perspex and graphite) were in the form of a pair of rectangular bricks measuring 230 mm 100 mm 100 mm. One of the largest faces of each half-specimen was machined flat and the two halves placed together, with each face having a 1 mm diameter circular groove cut into it to accommodate the needle probe (figure 9). This is a similar arrangement to that adopted for the hot-wire measurement technique. The specimens were held together with tape. groove half-specimen needle probe half-specimen groove Figure 9. Assembly of solid specimens and needle probe. In preparing the powder specimens, a different arrangement is needed and it is important to pay careful attention to specimen packing density. The thermal conductivity of powders is very sensitive to particle packing density, which is easily changed when the specimen is disturbed. Densities of powders are therefore specified according to how they are packed into a container, the two conditions in the current work being poured bulk density and tap density (Svarkovsky 1987). To achieve a poured bulk density the powder is poured gently into a container whilst avoiding mechanical vibrations or shock.

8 134 N Lockmuller, J Redgrove, L Kubic a r We shall refer hereafter to powders in this state as loose-fill specimens, and the powder density as loose-fill density. For the tap density condition, the powder grains are normally compacted by dropping a powder-filled container repeatedly onto a hard surface from a fixed height until the density attains a constant value. This procedure is described in ASTM standard B527 ^ 93. Measurements on powders were performed under both loose-fill and tap density conditions. For measurements performed under loose-fill conditions the specimens (silica or sand) were poured carefully into a cylindrical copper cell (inner diameter 85 mm, depth 250 mm) and levelled off prior to insertion of the needle probe (figure 10). In the case of measurements performed under tap density conditions the specimen cell was tapped repeatedly with a hammer with varying degrees of force prior to levelling the powder off. Compaction was achieved in this wayöwhich is an approximation to the recommended methodöbecause the apparatus required by the procedure outlined in ASTM standard B527 ^ 93 was not available. The measurements under loose-fill conditions were performed three times, with the specimen decanted and poured back into the specimen cell for each measurement, to check for reproducibility of the specimen bulk thermal conductivity value. O-ring seal bolt O-ring seal needle probe Figure 10. Specimen cell. 5 Results Figures 11, 12, and 13 and table 1 show the results of the tests to determine the optimum power per unit length for the specimens of graphite, Perspex, silica powder, and AFS sand. In each case the maximum temperature rise was kept within the 4 K limit, with the exception of graphite, for which temperature rises of up to 12 K were used. This was necessary because of the small slope of the linear portion of the graph with respect to power and overall temperature rise. The graphite thermal conductivity results (figure 11) are somewhat dependent on power output. The Perspex results (table 1), apart from one result, are almost independent of heater power for powers of 0.85 W m 1 upwards. (It is for this reason that the Perspex results are tabulated rather than shown graphically.) The silica microspheres results (figure 12) do not vary significantly with heater power across the range of heater powers used. For AFS sand (figure 13) the measured thermal conductivity is practically independent of heater power for powers between 0.6 and 1.2 W m 1.

9 Measurement of thermal conductivity with the needle probe Power output=w m 1 Figure 11. Measured thermal conductivity vs heat output for graphite. Table 1. Measured thermal conductivity, l, vs heat output, Q, for Perspex. Q=W m 1 l=w m 1 K 1 Q=W m 1 l=w m 1 K Power output=w m 1 Figure 12. Measured thermal conductivity vs heat output for loose-fill silica powder. Table 2. Thermal conductivity results, l, compared with NPL reference values, l NPL,forgraphite (NPL 1998) and Perspex (Salmon 1999). l l Material Q=W m 1 Dy=K l=w m 1 K 1 SD=% l NPL =Wm 1 K 1 NPL % l NPL Graphite Graphite Perspex Perspex Perspex

10 136 N Lockmuller, J Redgrove, L Kubic a r 0.27 Thermnal conductivity=w m 1 K Power output=w m 1 Figure 13. Measured thermal conductivity vs heat output for loose-fill AFS sand. After the above testing, final definitive measurements were made on specimens of graphite, Perspex, AFS sand, and silica microspheres. Measurements were performed on the graphite and Perspex reference materials at a number of different powers per unit length (table 2). The graphite reference values, obtained with the hot-wire, are known to within 8% and the Perspex values, obtained with the NPL guarded hot-plate, to within 3% (Salmon 1999). Note that results most different from the reference values were obtained with powers that lie outside the optimum range shown in figure 6. However, overall, the probe results on the reference materials are in good agreement with the reference values Tap density=kg m 3 Figure 14. Thermal conductivity of the AFS testing sand as a function of tap density Tap density=kg m 3 Figure 15. Thermal conductivity of the silica microspheres as a function of tap density.

11 Measurement of thermal conductivity with the needle probe 137 The measured thermal conductivities of AFS testing sand and silica microspheres, respectively, are shown in figures 14 and 15 as a function of tap density. The thermal conductivity of silica microspheres increased by approximately 1.1% for each percent increase in packing density, whilst for AFS sand the increase was approximately 2.4%. Tables 3 and 4 show the results for three repeat measurements of the thermal conductivity of silica microspheres and AFS sand respectively. The specimen was decanted from the specimen cell and re-filled each time. Each result is the average of ten successive measurements on an undisturbed specimen. Table 3. Reproducibility of the measured thermal conductivity value for loose-fill silica microspheres (Dy ˆ 2:7 K). Measurement l=w m 1 K 1 SD=% Loose-fill density/kg m Table 4. Reproducibility of the measured thermal conductivity value for loose-fill AFS testing sand (Dy ˆ 2:1 K). Measurement l=w m 1 K 1 SD=% Loose-fill density/kg m SD=% Figure 16. Standard deviation (SD) as a function of thermal conductivity. Repeated results for powders under loose-fill conditions, where the specimens were decanted and re-filled each time, show that the thermal conductivities are reproducible to within 0:1% for silica microspheres and 0:5% for AFS sand. Figure 16 shows the standard deviation (SD) values vs thermal conductivity for all four specimens, both solids and powders. It can be seen that the standard deviation is significantly larger for the high-conductivity specimen (graphite) than for the lower conductivity specimens (Perspex and the two powders). 6 Conclusion A needle probe apparatus and a methodology are presented for validated measurements of the thermal conductivity of soft solids. The apparatus was tested on solid Perspex

12 138 N Lockmuller, J Redgrove, L Kubic a r and graphite and silica microspheres and AFS sand. Deviations from the reference values for Perspex and graphite fell within 2% for those measurements conducted with power outputs lying within the suggested range (figure 6). The standard deviation for Perspex is within 1% while for graphite a standard deviation of 5% was established. The reason for the comparatively high 5% standard deviation in the graphite results may be due to thermal contact resistance between the probe and specimen; this could be reduced by use of a thermal-contact paste. Measurements of thermal conductivity of silica powder and AFS sand are presented as a function of packing density with additional results under loose-fill conditions. On the basis of the high reproducibility of the thermal conductivity on being decanted and re-filled, the powders tested merit consideration as candidate reference materials. The more easily controllable density for silica microspheres and lower percentage increase in thermal conductivity with increasing density suggest that this is the better potential reference material. Satisfactory results with the needle probe can be obtained for materials with thermal conductivities lying between at least 0.2 and 6.0 W m 1 K 1 : Acknowledgments. The authors are grateful to Jenny Morrison (Oxford University vacation student) for her help with the needle probe measurements. This work formed part of the Thermal Metrology Programme, 2001 ^ 4, funded by the National Measurement System Policy Unit of the UK Department of Trade and Industry. References Asher G B, Sloan E D, Graboski M S, 1985 Int. J. Thermophys ^ 294 ASTM Standard B527 ^ 93 (reapproved 1997), ``Standard test method for determination of tap density of metallic powders and compounds'', pp 419 ^ 420 Blackwell J H, 1953 Can. J. Phys ^ 417 Blackwell J H, 1954 J. Appl. Phys ^ 144 Bruijn P J, Van Haneghem I A, Schenk J, 1983 High Temp. ^ High Press ^ 366 Carslaw H S, Jaeger J C, 1996 Conduction of Heat in Solids (Oxford: Clarendon Press) Davis W R, 1984, in Compendium of Thermophysical Property Measurement Methods Volume 1, Eds K D Magl c, A Cezairlyan, V E Peletsky (New York: Plenum Press) pp 231 ^ 254 Gustafsson S E, 1991 Rev. Sci. Instrum ^ 804 Kubic a r L', 1990 ``Pulse method of measuring basic thermophysical parameters'', in Comprehensive Analytical Chemsitry Volume XII Thermal Analysis Part E, Ed. G Svehla (Amsterdam: Elsevier) Kubic a r L', Bohäc V, 2000 Meas. Sci. Technol ^ 258 Liang X-G, 1995 Meas. Sci. Technol ^ 471 NPL, 1998 ``NPL hot-wire measurements. Thermal conductivity of graphite specimen measured at NPL by transient hot-wire method in resistive wire mode (April 1998)'', results not previously published Salmon D R, 1999, personal communication, National Physical Laboratory, Teddington TW11 0LW, UK Svarkovsky L, 1987 Powder Testing Guide: Methods of Measuring the Physical Properties of Bulk Powders (Amsterdam: Elsevier) Xie H, Cheng S, 2001 Meas. Sci. Technol ^ 62 ß 2003 a Pion publication