Measurement of Thermal Conductivity and Diffusivity of Electrically

5th European Thermal-Sciences Conference 18-22 May 2008, Eindhoven, the Netherlands

Measurement of Thermal Conductivity and Diffusivity of Electrically Conducting and Highly Corrosive Liquids from Small Samples with a New Transient Hot-Wire Instrument A. J.-J. Kadjo1, 3, J.-P. Maye1, J. Saillard2, G. Thévenot2, J.-P. Garnier1, and S. Martemianov1 1

Laboratoire d'Etudes Thermiques, UMR CNRS 6608, ESIP-ENSMA, 40 Avenue du Recteur Pineau, 86022 Poitiers, France 2 Commissariat à l'Energie Atomique, CEA/Le Ripault, France 3 To whom correspondence should be addressed. e-mail: [email protected]

Abstract The transient hot-wire technique is widely used for measurements of the thermal conductivity and thermal diffusivity of most fluids. However, for some particular liquids like concentrated nitric acid solutions or similar nitric mixtures , the thermal properties of which are important to know for industrial or security applications , this technique had not still been used (except for a complex particular apparatus using a hot-wire embedded in a glass capillary tube for conductivity measurement), essentially on account of a technological incompatibility between measurement probe materials and highly electrically conducting and corrosive liquids ; moreover , the possible highly energetic character (explosive) of these liquids requires minimum volume liquid samples and safety measurement devices and processes. It is the purpose of this paper to report on a new patented instrument, based on a tantalum short hot-wire probe technology, which responds to the precedent requirements and allows safety automatic and quasi simultaneous conductivity and diffusivity measurements of concentrated nitric acid solutions and similar nitric mixtures from liquid samples less than 2 cm3, with uncertainties less than 5% for conductivity and 12% for diffusivity.

1. Introduction The transient hot-wire (THW) method has been widely used for measuring the thermal conductivity of materials; for liquids, in particular, this technique has now become the leading method for most applications. It generally involves the measurement of the temperature response of a thin metallic wire immersed in the liquid and subjected to a step heat power; the thermal conductivity is determined from the theoretical and ideal model of an infinitely long line heat source in an infinite liquid medium and eventual corrections taking into account of the non-idealities of the experimental apparatus. In most favorable conditions, thermal conductivities of liquids can be determined with claimed accuracies better than 0.5% [Beiraos and al., 2006; Healy and al., 1976]. Simultaneous measurements of the thermal conductivity and diffusivity by the THW method have also been proposed by several authors [Nieto de Castro and al., 1988; Watanabe, 1997; Zhang and Fujji, 2000; etc.] with lower accuracies and more complex experimental procedures. For electrically conducting liquids, the THW method is not usable with bare metal wires because leakage of electric current and polarization of the liquid at the surface of the wire can occur, leading to uncontrolled errors in the wire temperature and heating power measurements. For such liquids, the wire must then be electrically insulated by coating with a thin insulation layer. A number of coating techniques have been employed for this purpose (polyester coating [Nagasaka and Nagashima, 1981], silica coating [Yamasue and al., 2002], aluminum oxide coated wire [Fukuyama and al., 2006; Zhang and Fujii, 2000], tantalum oxide coated wire [Alloush and al., 1982;


Kawamata and al., 1988; Perkins and al., 2000; Ramires and al., 1993, Ramires and al., 1994]. In these works, the liquid sample volumes have been chosen large enough to verify the ideal infinite medium condition around the sensor, without objective of sample miniaturization ; these liquid sample volumes, rarely precised by the THW users, are generally larger than 40 cm3 (50 cm3 in [Ramires and al., 1994], 80 cm3 in [Alloush and al., 1982], etc.). A variant of the classical THW method, using sinusoidal alternating current, has also been developed [Griesinger and al., 1999; Griesinger and al., 1997], with the claimed advantage of smaller (13 cm3) liquid volume samples; the experimental set up has however only been tested for water and glycerin and the method is said to be unsuitable for low thermal conductors. A particular category of electrically conductive and highly corrosive liquids, called special in the present work, is represented by some nitric acid compounds. These technically important liquids can be very energetic and explosive during experimentations; so, they have usually to be tested and operated in specific security rooms if the working samples are not small enough. Up to the present moment, no attempt had been done to expand the THW method to these special liquids for their thermal properties measurements. The aim of the present work was the development of a TSHW method adapted to direct thermal conductivity and diffusivity measurements in small (< 2 cm3) special liquid samples. In comparison with the other THW methods previously applied to thermal conductivity measurements in liquids or even with the “ flash “ method used in [Roy, 1997] for thermal diffusivity measurements, this objective corresponds to an important liquid sample reduction; moreover, for applications to special liquids as defined previously, the tantalum hot-wire technology has been chosen because tantalum metal has well known electrical and thermal properties [Cardonne and al., 1995; Milosevic and al., 1999], has the higher chemical resistance to corrosion with most concentrated strong acids (nitric, sulfuric, hydrochloric, hydrobromic,…) [Cardarelli and al., 1996] and can be manufactured in thin wires (25 µm minimum diameter) which can easily be reduced in diameter by a specific electrochemical treatment and easily insulated from the electrically conducting liquids by a tantalum pentoxide anodic film of controlled nanometric thickness [Alloush and al., 1982; Ramires and al., 1993; Vermilyea, 1953].

2. Principle of the THW Method 2.1. Conductivity Measurement The THW method is a transient dynamic technique based on the measurement of the temperature rise of a linear heat source (hot wire) embedded in the test material. For an infinitely long metallic wire (radius: r0) heated at time t > 0 with a constant heat flux q per unit length and immersed in an infinite homogeneous liquid medium (thermal conductivity and diffusivity: λ and a, respectively) with uniform initial temperature T0, the instantaneous temperature T(t) of the wire is given by Healy and al. (1976)

⎛ q ⎞ ⎛ 4 Fo ⎞ T ( t ) − T0 = ΔT ( t ) = ⎜ ⎟ ln ⎜ ⎟ ⎝ 4πλ ⎠ ⎝ C ⎠

(1)

where γ is the Euler's constant (γ = 0.5772..) with C = eγ = 1.781 and FO is the Fourier number defined by Fo =

at r02

(2)


Equation (1) is the analytical solution of an ideal thermal conductive model, only valid for FO >> 1 and over a limited time interval (without convective transfers in the liquid medium). In practice, the measured response e (t) of the wire to an overheating ΔT (t), resulting from the Joule effect in consequence of a constant current i, is given by e ( t ) = R ( t ) i = R0 (1 + β 0 ΔT ( t ) ) i

(3)

where R(t) is the instantaneous electrical wire resistance (corresponding to a wire temperature T(t)) and β0 the wire temperature coefficient (defined for R(T0) = R0) deduced from an experimental calibration. Taking account of (1) and (3), the thermal conductivity λ can be determined from ⎛ qR β i ⎞ ⎛ de ( t ) ⎞ λ = ⎜ 0 0 ⎟ ⎜⎜ ⎟ ⎝ 4π ⎠ ⎝ d ( ln t ) ⎟⎠

−1

(4)

where de ( t ) / d ( ln ( t ) ) is a numerical constant deduced from experimental data for t values witch satisfy the condition FO >> 1 and without thermal convection.

2.2. Diffusivity Measurement Considering the previous (Sect. 2.1) linear heat source (hot wire) immersed in an infinite liquid medium, the instantaneous liquid temperature T(r, t) at a distance r from the heat source and with the hypothesis of a thermal conducting regime is given by Carslaw and Jaeger (1959) ΔT ( r , t ) = T ( r , t ) − T0 = −

⎛ −r 2 ⎞ Ei ⎜ ⎟ 4πλ ⎝ 4at ⎠ q

(5)

where Ei is the exponential integral. Equation (5) reduces to (1) for r = r0 when Fo >> 1. When a constant heat pulse of duration t0 is applied on the wire, the temperature rise becomes [Bristow and al., 2001; Kluitenberg and al., 1993; de Vries, 1952] ΔT ( r , t ) =

⎛ −r 2 ⎞ ⎤ q ⎡ ⎛ −r 2 ⎞ ⎢ Ei ⎜ ⎟ − Ei ⎜ ⎟⎥ 4πλ ⎣ ⎝ 4a(t − t0 ) ⎠ ⎝ 4at ⎠ ⎦

(6)

If the distance r is fixed, the ΔT(r, t) function takes a maximum value ΔTm (corresponding to a time tm) for

∂ΔT =0 ∂t

(7)

The thermal diffusivity a can then be deduced from measurement of t0, tm and r according to the expression (8) which has already been used for determination of a in soils [Bilskie and al., 1998; Bristow and al., 2001] a=

r2 4

⎡1/ ( tm − t0 ) − 1/ tm ⎤ ⎢ ⎥ ⎢⎣ ln ( tm / ( tm − t0 ) ) ⎥⎦

(8)


3. Experimental Apparatus The whole experimental apparatus is composed of an electrical measurement system (Sect. 3.3) associated to a specific stainless steel vessel (Sect. 3.2) equipped with an independent tantalum hotwire probe (Sect. 3.1). The device constituted by vessel and probe is a patented device by Thevenot and al. (2006).

3.1. Tantalum Two-Wires Probe The measuring probe (Figure 1) is composed of two parallel tantalum wires (rough lengths 15 mm and 2 mm), each wire spot-welded on two 1 mm diameter tantalum prongs. The long wire (diameter 25 μm) is used for the λ determination (Sect. 2.1) and the a determination (as a linear source supplied by a constant current of intensity I during a time t0) while the short wire (diameter 5 μm) is only used for the a determination (Sect. 2.2) as a resistance thermometric sensor (supplied by a constant current of intensity i ≈ 1 mA). The two tantalum wires (purity: 99.9%) are spot welded at their ends on 1mm diameter tantalum prongs partially embedded (Figure 1) in a 4 mm diameter ceramic rod to constitute the two-wires probe. The tantalum elements (wires and prongs), which have to be immersed in special liquids, are electrically insulated in situ (after spot-welding of the hot-wires on the prongs) with a coating layer of tantalum pentoxide (Ta2O5), according to a procedure described by Alloush and al. (1982). This coating layer, with a thickness of about 70 nm, is formed by anodization and has been proven to be strongly adherent and electrically efficient throughout a series of temperature cycles, with temperature values up to 550 K, for thermal conductivity measurements in toluene [Ramires and al., 1994].

Figure 1 : Two-wires probe

3.2. Measuring Cell Figure 2 shows the measuring cell (perspective view (Figure 2.a) and vertical section (Figure 2.b)). The cell is composed of a 2 cm3 liquid content stainless steel vessel with the tantalum wires probe mounted on the top with a Teflon gasket. The cell is designed to work in the 0.1-1 Mpa pressure range and the 0 °C to 80 °C temperature range with a vessel immersed in a liquid thermostatic bath (LAUDA Ecoline RE312) at controlled temperature with a precision better than 0.1 °C. The cell is also designed to allow easy filling and emptying of the liquid in the vessel which includes, for this purpose, two specific liquid inputs (outputs).


a) Perspective view

b) Vertical section according to the plan I-I of figure 2a.

Figure 2 : Patented measuring cell [Thevenot and al., 2006]

3.3. Measurement System The whole experimental system is composed of the measuring probe, a source meter (Keithley 2400), a nanovoltmeter (Keithley 2182), a Wheatstone bridge (Dantec 55M01), a scope (Tektronix TDS 3014) and a microcomputer which links and controls the different elements. The source meter and nanometer are used both for λ and a determinations while the a determination requires also the use of the Wheatstone bridge, a specific device for temperature measurements with a cold wire; the scope is possibly used for visualization of the thermal response (given by the short wire) to the heat–pulse (given by the long wire). The source meter 2400 (from Keithley) can supply and measure very precisely the voltage or current by means of a four-wire technique. The precision of the sourcing current in the working range (0 to 100 mA) is 0.034% of the reading + offset. On account of the limited measurement capabilities of the meter, it was necessary to add a nanovoltmeter 2182 (from Keithley) to ensure better precision. On the (0 to 100 mV) range, the resolution of this meter is 10 nV and the accuracy over one year is better than 30 ppm of reading. The speed of acquisition of the source meter and nanovoltmeter can vary; depending on the choice of the integration time NPLC (number of power line cycles). In order to provide the best signal-to-noise ratio, the NPLC value is often chosen corresponding to a 20 ms sampling period and typically an acquisition frequency of 10 Hz. The different elements are connected to a microcomputer via a GPIB interface and a LabView® program (National Instruments), specifically written for this application, controls the instruments and reads the data.

4. Experimental Results 4.1. Introduction Preliminary validation experiments have been carried out for measurement of thermal conductivity and diffusivity of water and toluene, the liquid most commonly used as references [Bilskie and al., 1998; Bristow and al., 2001; Healy and al., 1976; Ramires and al., 1993; Perkins and al., 2000; Wakeham and Zalaf, 1987] and also of Galden® liquid, a dielectric perfluorinated liquid with lower thermal conductivity. An original application of the method to thermal conductivity and diffusivity measurements in pure nitric acid is presented and discussed in Sect. 4.2 and Sect. 4.3.


4.2. Thermal Conductivity Measurement Nitric acid is very commonly used in industrial chemistry, in particular for fabrication of many propellants. The lack of physical properties knowledge, like thermal conductivity and diffusivity, results of its very oxidative and corrosive character. However, some conductivity values of nitric acid solutions, with concentrations up to 50%, were published by Van Der Held and Van Drunen (1949) from experiments with a lacquered wire embedded in a glass capillary tube. The conductivity values of concentrated nitric acid at 10% and 50% presented by these authors are respectively 0.552 W m-1 K-1 and 0.412 W m-1 K-1 (with a liquid temperature T = 17 °C); moreover, they show a linear dependence of thermal conductivity with the nitric acid concentration. Figure 4 shows the transient responses of the long wire (Sect. 3.1) in a pure nitric acid at 0 °C for five current values (80, 70, 60, 50, and 40 mA). On this figure, E represents the potential difference between the potentials at unspecified and initial times. The median corresponding value of the thermal conductivity is 0.320 W m-1 K-1 at 0 °C. This value is coherent with the previous measured values of Van Der Held and Van Drunen (1949) but higher than the theoretical extrapolated value (λ = 0.237 W m-1 K-1) corresponding to pure nitric acid at 17 °C; however, this last value is also different of the extrapolated value (λ = 0.285 W m-1 K-1) corresponding to pure nitric acid given by a calculation software (with nitric acid concentration validity range: 2 mass % to 90 mass %) based on the German VDI Waermeatlas (10th edition) and thus there is a need, to explain these differences, of more systematic λ measurements versus concentration and temperature. 1800 1600 1400 E (µ V)

1200 1000 800

i (mA) 80 70 60 50 40 λ (W m-1 K-1) 0.321 0.320 0.319 0.317 0.321

600 400 200 0 -2,5 -2 -1,5 -1 -0,5

0 0,5 ln (t)

1

1,5

2

2,5

Figure 3 : Typical transient responses of a hot-wire in pure nitric acid at 0 °C for i = 80mA, 70 mA, 60 mA, 50 mA, and 40 mA (, , U, S, ) and corresponding measured λ values

It should be noted that the appearance of free convection can easily be detected during the experimental run, when the observed voltage as a function of ln(t) departs from linearity. An analysis of this figure also shows that increasing the i values leads to earlier onset of convection according to the fact that the temperature rises more rapidly with higher currents; moreover, using the transient responses (Figure 4) and a temperature calibration of the probe, it should be deduced that, the maximum heating of the wire is about 3.9 °C for i = 80 mA and less than 1 °C for i ≤ 50 mA (inducing a negligible radiative heat transfer). With negligible errors in consequences of the radiative transfers and the convective transfers using the wire, the principal uncertainties on the λ measurement come from the end effects of the finite wire [Knibbe, 1986] and the uncertainty relative to the determination of q in (4). The final


uncertainty relative to λ is estimated to be less than 5%; this value is greater than uncertainties given by some authors (Sect. 1) for other liquids and larger samples but can be lowered from relative measurements using reference liquids (water, toluene, etc.), the thermal properties of which are claimed to be measured with a precision better than 0.5% [Ramires and al., 1993].

4.3. Thermal Diffusivity Measurement Figure 4 presents the voltage response E of the cold 5 µm wire (thermal sensor supplied by a constant intensity i = 1 mA) to a constant heat-pulse (I = 200 mA) of duration t0 = 10 ms applied to the parallel 25 µm hot-wire 110 µm apart (Figure 1); the corresponding measured diffusivity is determined from (8) according to a method described in [Bristow and al., 2001] and initially proposed for measuring soil thermal properties with specific multi-needle probes. The voltage response E (representative of the temperature response of the thermal sensor) shown on Figure 4 is given by the scope after amplification of the signal delivered by the Wheatstone bridge. From this response are determined t0 and tm used in (8): t0 (duration of the heat pulse transferred from the hot-wire) is deduced using the two electromagnetic peaks recorded by the cold wire at the beginning and the end of the heat pulse (Figure 4) while tm (corresponding to dE/dt = 0) is deduced from a polynomial fitting of the response E (t). The uncertainties on a determination from (8) arise principally from uncertainties on the measured wires distance r and also from the t0 and tm determinations. For a convenient r value (corresponding to a maximum value leading to a sufficient accuracy of the tm determination), the uncertainty on the a determination is less than 12%, a value slightly superior to those (7% to 9%) [Nieto de Castro and al., 1988] generally obtained with most THW methods and larger liquid samples. However, as for λ determination, this uncertainty on a measurement can be lowered from relative measurements using reference liquids with better known thermal diffusivities. -4 -4,1 -4,2 -4,3 E(V)

-4,4 -4,5

electromagnetics peaks

-4,6 -4,7 -4,8 t0 =10ms

-4,9

tm =39ms

-5 0

0,01 0,02

0,03 0,04

0,05

0,06 0,07

0,08 0,09

0,1

t (s)

Figure 4 : Typical transient response of the cold wire in pure nitric acid at 20 °C (corresponding thermal diffusivity value : a = 8.7 x 10-8 m2.s-1).

5. Conclusion The TSHW methods and the associated patented measuring cell presented in this paper have proven to be well suitable to determine the thermal conductivity λ and the thermal diffusivity a of electrically conducting and very aggressive liquids like pure nitric acid from small (< 2 cm3) liquid samples.


Measurements of λ and a, carried out independently and successively in such liquids with the same apparatus and according to an experimental procedure controlled by LabView, have shown that -

-

the thermal conductivity λ should be measured with an uncertainty less than 5% using a 25 µm diameter tantalum hot-wire, with a wire heating and a heat transfer to the liquid respectively less than t 1 °C and 10 mJ and a measuring time around 1 s. the thermal diffusivity a should be measured with an uncertainty less than 12% using a two parallel tantalum wires probe with a 25 µm diameter hot-wire heating and a heat transfer to the liquid respectively less than 9 °C and 6 mJ and a measuring time less than 0.1 s.

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