1 Introduction

The need for new heat transfer agent for many applications, namely in the consumer electronics industry, requires electrical conducting materials, liquid at room temperature, with high thermal conductivity [1, 2]. In addition, the prohibition to use liquid mercury in many applications, namely in some thermometers and electronic switches, due to the toxic parameters and their danger to human health [3], led to the search of convenient metals to replace it. From the different possibilities, Galinstan, a eutectic alloy of gallium, indium, and tin with a melting point (Tm = 283.4 K) [2] has been proposed for many applications, namely for replacing mercury, a toxic element used for many years [3, 4].

Within the methods that can be used to measure the thermal conductivity ionic melts, including electrically conducting liquids, at high temperatures, conveniently described in a recent publication [5], the transient methods have been found to achieve, if applied according to the best knowledge of their mathematical descriptions, to be considered as primary methods. As approved by the CCQM—Comité Consultatif pour la Quantité de Matière, BIPM, in 1995—a primary method of measurement is a method having the highest metrological qualities, the operation of which can be completely described and understood, for which a complete uncertainty statement can be written down in terms of SI units, the results of which are, therefore, accepted without reference to a standard of the quantity being measured [6]. From these, the transient hot wire, is capable of attaining and expanded uncertainty smaller than 1%, if a careful application of method is done, and an optimal design of the instrument is performed [7, 8].

As one of the most promising and robust methods for high-temperature applications, the transient hot strip, is potentially a primary method, if the 3D heat transfer equation is solved for the geometries involved (a metal thin film deposited in a ceramic substrate), which is not the case yet. Efforts to develop instrumentation capable are an important step to achieve a primary method status. In our laboratory, and following the developments already presented to use the transient hot strip method to determine the thermal conductivity of gaseous mixtures up to 800 °C [9], we are currently developing an instrument to measure molten metals and high-temperature molten salts, namely molten carbonates for concentrated solar power stations. This paper shows the “proof of concept” for these new applications, by constructing a sensor to measure the thermal conductivity of molten Galinstan, a conductive and corrosive alloy, from room temperature up to 103 °C, the temperature interval of the major part of the application of this alloy, prone to surface oxidation at higher temperatures.

2 Materials and Methods

2.1 Material

Galinstan was produced by Haines & Maassen Metallhandelsgesellschaft mbH, Bonn, Germany, with product name Gallium/Indium/Tin Eutectic (NL-011) and received from ILK, Dresden, in PE bottles (liquid), with a stated density of 6440 kg·m−3. The composition of the eutectic alloy was given by the manufacturer, as Ga68.4In21.5Sn10, although there is the presence of some components like Cu and Pb (around 0.0035 wt%) and Al and Zn (< 0.00005 wt%). The total composition is presented in Table 1.

Table 1 Composition of Ga68.4In21.5Sn10 eutectic alloy, in mass %
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The samples were handled with care, following the recommendations present in MSDS, as they are corrosive and irritant.

2.2 Method

The theory of the transient-hot strip was developed by Gustafsson and co-workers [9, 10] for an infinitely long and infinitely thin strip. Assuming that the “end effects” are non-existing, the main solution for the temperature rise in the hot strip ΔTs can be approximated by Eq. 1:

$$\tau =\frac{2\sqrt{{k}_{f}t}}{w}{ k}_{f}=\frac{\lambda }{\rho {C}_{P}},$$
(1)
$$\Delta {T}_{s}=T\left(t\right)-{T}_{0}=\frac{{q}_{s}}{2\lambda \sqrt{\pi }}f\left(\tau \right),$$
$$f\left(\tau \right)=\tau {\text{erf}}\left(\frac{1}{\tau }\right)-\frac{{\tau }^{2}}{\sqrt{4\pi }}\left[1-\mathit{exp}\left(-\frac{1}{{\tau }^{2}}\right)\right]+\frac{1}{\sqrt{4\pi }}{E}_{1}\left(\frac{1}{{\tau }^{2}}\right),$$

where kf is the thermal diffusivity of the media, qs is the heat dissipated per unit length of the strip, w is the width of the strip, τ is a dimensionless time, and λ, ρ, and CP are, respectively, the thermal conductivity, density, and heat capacity of the fluid media. The mathematical functions erf(y) and E1(z) are, respectively, the error function and the exponential integral, obtainable from rational approximations and easily programmed [11]. From Eq. 1, the ratio ΔT/f(τ) is inversely proportional to the thermal conductivity and directly proportional to the heat dissipation per unit length, and therefore, it will be possible to use this solution as the base of the application of this model to calculate the thermal conductivity of the fluid. In the paper by Queirós et al. [8] we have shown that, for arbitrary values of 0 < τ < 465, the function f(τ) can be translated as Eq. 2, a fact that simplifies data processing:

$$f\left(\tau \right)=\text{0.56417847}\mathit{ln}\left(\tau \right)+\text{0.6835194}.$$
(2)

The thermal conductivity of the ideal hot strip, immersed in a fluid media, can then be obtained from Eq. 3:

$${\Delta T}_{s}=\frac{{q}_{s}}{2\lambda \sqrt{\pi }}\left[0.56417847\mathit{ln}\left(\tau \right)+0.6835194\right]=a\mathit{ln}\left(\tau \right)+b,$$
(3)
$$\lambda =\frac{{q}_{s}}{2\sqrt{\pi }}\frac{0.56417847}{a},$$

with a = 0.56417847 and b = 0.6835194.

The real model that will describe the sensor, needs several modifications, as the strip is not infinitely long or infinitely thin, neither it is floating in space, and has finite properties; therefore, it can store heat while being warmed. In addition, the platinum metal film has to be electrically insulated from electrically conducting media.

However, if

  1. i)

    the time length is adequately chosen to be greater than the time, then strip needs to be warmed (heat wave reaches the media), as explained before [8] and,

  2. ii)

    two strips differing only in the length are used to compensate for the end effects, these factors can be minimized. Therefore, in our equipment, a long and a short strip are used, and the “strip difference” properties are measured.

  3. iii)

    a ceramic sheet supports the strips, and an insulating coating has to be built. Therefore, and following Gustafsson development of the theory [12], the measured thermal conductivity is an effective property, λeff given by Eq. 4:

    $$\lambda_{eff} = (\lambda_{fluid} \times \lambda_{ceramic} )^{1/2} ,$$
    (4)

where λceramic is the thermal conductivity of the ceramic support.

Having obtained the slope a of the temperature rise of the strip difference of our sensor as a function of τ (or t), we can obtain the estimated thermal conductivity. The reference temperature of the measurements was calculated assuming that it was equal to that of a measurement with a transient hot wire [4], as explained below.

3 Measuring Instrument

3.1 The Sensor

Inspired in the procedure developed many years ago to minimize the “end effects,” when in the real system we have a finite length strip, ceramic/thin metal-film-based sensors with two strips were constructed as described before [13, 14] and detailed for the current sensor [7], with physical dimensions given by hot strip length, lL = 15 mm, short strip lS = 5 mm, strip width w = 110 μm, and thickness t = 0.34 μm. Details of the construction process were described elsewhere and will not be repeated here [8, 13, 14].

The main obstacle was the coating development. First, we have used a sensor coated with a thin film of ceramic, deposited by PVD, as previously reported. However, we arrived to the conclusion that, after some years due to aging factors and manufacturers modifications, this coating was porous to water and Galinstan, breaking the electrical insulation and impeding measurements. This solution was very elegant but did not work. Several other products, like alumina pastes, alumina sprays, and alumina paints did not work also. As a final solution, we tried to use a halogen-free, flexible, flame retardant heat-shrinkable tubing, KTG-CB-HFT25.4, made by CYG (Changyuan Electronics, Dongguan Co., Ltd., China), supporting 125 °C, which could fit the sensor as a sleeve, and with good radial and longitudinal shrinking properties. The material details are not disclosed by the company, just irradiated cross-linked modified polyolefin. On shrinking the tube, we tried to avoid bubbles staying inside the sensor.

Figure 1 shows the sensor used, after the measurements with Galinstan.

Fig. 1
figure 1

The thermal conductivity sensor after heat-shrinkable tubing application. Gray shadows were caused by Galinstan melt wetting. Ruler in cm and inches

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The instrument was completed by vertical tubular furnace, with temperature control (EUROTHERM 91e) up to 0.2 K, a ceramic crucible (Alumina), a platinum resistance thermometer (ASL, WIKA, GTH 7000), calibrated, with a standard uncertainty of 0.01 °C, and a supporting structure.

3.2 The Data Acquisition System

The data acquisition system was fully described in [8]. The sensor is part of a Wheatstone bridge circuit, which compensates for the end effects, a differential amplifier and Picolog 1612 unit (a 16 Channel Data logger of PICO TECHNOLOGY), equipped with a USB interface.

The resistances of the long strip (RL) and short strip (RS) were calibrated as a function of temperature in a temperature calibration facility in CIENCIAS ULISBOA Laboratory between room temperature and 373 K. The results were found to have a linear dependence on temperature, exemplified for the resistance of the strip length difference, RW = RL − RS, given by Eq. 5:

$${R}_{\mathrm{W}}\left(T\right)={R}_{\mathrm{W}}\left({T}_{0}\right)\left[1+{\alpha }_{\mathrm{RW}}\left(T-{T}_{0}\right)\right],$$
(5)

where αRW, the mean coefficient of variation of the resistance with the temperature, was found to be 0.001270 ± 0.000001. In this equation, T0 = 293.15 K. The values for the individual strips were found to be, respectively, αRL = 0.001263 ± 0.00009, αRS = 0.001251 ± 0.00003, the ratio of the resistances being independent of temperature. These agreements make us believe that both strips have very similar electrical properties, and the success of the “end-effect” compensation can be assured. Figure 2 shows the resistances of the strips and the “ideal” floating strip (the strip length difference), as a function of temperature.

Fig. 2
figure 2

Strip resistances (RL, RS, and RW) variation with temperature

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The transient resistance variation and temperature rise in the Transient Hot Strip (THS) sensor can be calculated from the bridge configuration, as described in [8], and given by Eq. 6:

$${R}_{\mathrm{W}}\left(t\right)={R}_{\mathrm{W}}\left(0\right)\left[1+{\alpha }_{\mathrm{RW}}\Delta {T}_{\mathrm{W}}\right],$$
$$\Delta {T}_{\mathrm{W}}=\frac{\Delta {R}_{W}\left(t\right)}{{\alpha }_{\mathrm{RW}}{R}_{\mathrm{W}}\left(0\right)}.$$
(6)

Keeping in mind that for t = 0, the temperature of the strip is assumed to be in equilibrium at the fluid temperature, Tb. In addition, using two strips, a short and long one, with lengths lS and lL, the heat dissipated in the strip, per unit length, qw can be evaluated from the applied voltage, for a strip of length lW = lL − lS, by Eq. 7:

$${q}_{w}=\frac{{V}^{2}{R}_{w}}{{\left({R}_{L}+{R}_{S}+{R}_{3}\right)}^{2}}\frac{1}{\left({l}_{L}-{l}_{S}\right)},$$
(7)

where V is the applied voltage and R3 = 270 Ω.

As explained in the previous application, the ideal model for the hot strip given by Eq. 3, has to be demonstrated to occur for the real application, and in the complex geometry of our system, only when the heat wave dissipated in the strips reaches the coating/fluid interface. Figure 3 shows one run (T28) with Galinstan, at Tb = 348.01 K, the linear part of the full run in the inset, and from which the thermal conductivity was calculated. Deviations of the experimental points in the liner region to the fit do not amount to more than ± 1.5%, as shown in Fig. 4.

Fig. 3
figure 3

The experimental transient temperature rises in the “strip difference,” ΔTW, as a function of ln τ, for a run with Galinstan at Tb = 348.01 K. The inset shows the detail of the regression line, in the linear operational region

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Fig. 4
figure 4

Percentages deviations of the experimental temperature rise from the linear regression (the ideal model), as a function of ln τ

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4 Results and Discussion

To obtain the thermal conductivity, we need to estimate the effect of the coating and ceramic support in the transient heat transfer in the sensor. The coating is made from a heat-shrinkable tubing, as explained above, which has an estimated thermal conductivity of 0.19 W·m−1·K−1 to 0.25 W·m−1·K−1 (manufacturers data, before shrinking).

No data exist for the shrank tube. The ceramic support, made from Rubalit 708S CeramTec, Marktredwitz, Germany), for which the value of the thermal conductivity, between 296 K and 1000 K, was measured by NETZSCH, Selb, Germany, and the values can be found in Lourenço PhD thesis (1998) [15] and used in references [8, 13, 14]. The value varies between 25.5 W·m−1·K−1 at 296 K and 22.3 W·m−1·K−1 at 373 K, about 100 times greater than the estimated value for the heat-shrinkable tube. Therefore, the thermal resistance of the ceramic support (about the same thickness of the tube coating) is 100 times smaller than the that of this coating, which will control the rate of heat transfer, the thermal resistance (and therefore, the thermal conductivity) of the system ceramic + thin platinum film + coating can be assumed to be that of the heat-shrinkable tubing (considering the resistance of the platinum film also negligible) and consequently

$${R}_{\mathrm{sensor}}={R}_{\mathrm{coating}}+ {R}_{\mathrm{film}}+{R}_{\mathrm{ceramic}} \sim {R}_{\mathrm{coating}},$$
(8)

and Eq. 4 can be used, replacing λceramic by λcoating.

In order to validate this hypothesis, we have made measurements of the thermal conductivity of water, within the range of the present measurements, to estimate the thermal conductivity of the shrank tube. The value of the thermal conductivity of water was obtained from the standard reference data of IUPAC, published by Ramires et al. [16]. In Fig. 5, we show the validation test for water at 297 K, from which we obtained a value of λcoating = 0.175 W·m−1·K−1. Other tests with water at higher temperatures (up to 70 °C) showed no temperature dependence, so an average value of 0.18 ± 0.01 W·m−1·K−1 was assumed (a standard uncertainty of 0.022, in relative terms). The calculation of the uncertainty of the measurements was made using ISO_GUM guidelines, expressing the contributions of all variables in the measurement, due to the sensor geometry (length, width), resistance calibration, voltage, and time measurement in the data acquisition system. These contributions to the standard uncertainty amounted to 0.015. Using a propagation factor k = 2, we obtain Ur(λ) = 2 × (0.022^2 + 0.015^2) = 0.053. Consequently, the estimated expanded relative uncertainty at a 95% confidence level (k = 2), Ur(λ) = 0.06, was assumed for the experimental thermal conductivity data of Galinstan.

Fig. 5
figure 5

A typical run for the validation of the experimental method with water, at 297 K, where the transient hot strip model is verified

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The results obtained for the thermal conductivity of Galinstan (Ga68.4In21.5Sn10) are presented in Table 2.

Table 2 Thermal conductivity (uncertainty (95% confidence level), Ur(λ) = 0.06), λexp, of Galinstan (Ga68.4In21.5Sn10) as a function of temperature
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The experimental data show a linear dependence on temperature in this temperature interval, given by Eq. 9:

$$\frac{\lambda }{\mathrm{W}\cdot {\mathrm{m}}^{-1}\cdot {\mathrm{K}}^{-1}}={a}_{0}+ {a}_{1}\left(\frac{T}{\mathrm{K}}\right).$$
(9)

In this equation, a0 = (21.61485 ± 0.090160) W·m−1·K−1, a1 = (6.0182 ± 0.13561) × 10−2 W·m−1·K−2 and a root-mean-square deviation of 0.12 W·m−1·K−1. Deviations of the experimental points from the regression line are never bigger than 0.46%, as shown in Fig. 6, consistent with the estimated uncertainty of the reported data.

Fig. 6
figure 6

Deviations, in %, of the experimental points from Eq. 9. Dashed lines represent the root-mean-square deviation of the fit (0.46%)

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The thermal conductivity of Galinstan has been measured by several workers. However, the composition is not always the eutectic composition, but “quasi” eutectic, and this might explain some of the differences found. In Table 3, we show the available data in the literature, identifying the alloy composition, the method of measurement, temperature interval, and estimated uncertainty, for available data.

Table 3 Available data for the thermal conductivity of Galinstan
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In Fig. 7, we show all the available data, plotted for the temperature range 290 K to 410 K. It is clear that the data of Prohkorenko et al. [17,18,19] is far apart from the other sets of data, including the theoretical calculations, using the Bridgman model (for the molecular liquid thermal conductivity) and the Wiedemann–Franz law and the metal Lorentz number), for the electronic component [21]. This fact was already noticed by Plevachuck et al. [1], stating that these discrepancy (also observed at higher temperatures), citing “Experimental difficulties related to convection, uncertain wetting conditions, an insufficient level of purity, or oxidation effects can significantly reduce the accuracy of thermal conductivity measurements. Indeed, the aforementioned problems could be responsible for such a large variance between different data sets.”

Fig. 7
figure 7

The thermal conductivity of Galinstan. Present data and literature data, including theoretical calculations (Bridgmann model and Wiedmann–Franz law) [21]. The error bars in our data display our estimated uncertainty (6%)

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The other three sets of data agree within their mutual uncertainty, our data being slightly smaller than Plevachuck et al. [1] and Laube et al. [19], but having almost identical temperature coefficients. The small difference in the alloy composition might explain these differences, as our alloy is poorer in tin and richer in gallium.

Finally, we would like to say that we cannot assure that no air was kept inside the shrank sleeve. Air bubbles can represent an additional heat transfer resistance/loss that might decrease the temperature rise in the strips. This can be systematic, and therefore, does not contribute to the slope and can be neglected, or depending on temperature rise/time, and affect the thermal conductivity reported, possibly explaining the smaller value measured for the thermal conductivity of Galinstan. Attempts to diminish this effect would be done in a near future.

5 Conclusions

New data on the thermal conductivity of molten Galinstan have been presented, by using the transient hot strip (THS) method. The sensors now developed, can be improved in the future, namely by using a different coating material, in order to extend the temperature range of the measurements. The measuring bridge is also under revision to improve signal to noise ratio. The theory of the method is approximate, and there are still improvements, at an analytical method, by developing a 3D model heat transfer model for the system. Application to measure the thermal conductivity of gallium and molten carbonate salt mixtures is under study.