In the context of complex industrial processes such as casting, welding, or additive manufacturing, there is an increasing need for knowledge of thermophysical properties of liquid metals at high temperature. The specific heat is a key factor in heat transfer and there is a lack of data in literature, particularly beyond the melting point. Several methods have been developed to measure properties of liquid metals. For example, Razouk et al [1] have adapted an induction furnace to realize a drop calorimetry facility. The sample and its container are heated in the furnace, then they are dropped in a calorimeter. The variation of enthalpy is measured and allows computation of the specific heat. Although this method was applied to solid specimen, liquid copper has been used during the calibration process, and the results showed good agreement with the data from literature. However, the accuracy of the results is difficult to assess due to the uncertain temperature measurement. Moreover, the contact with a crucible raises the question of chemical diffusion. Another approach consists in using contactless methods, which offer the advantage of reaching high temperatures while avoiding sample contamination. In the early 1990s, Pottlacher et al [2] have developed a pulsed ohmic heating device that can measure electrical conductivity, density, and enthalpy. The sample is a thin wire traversed by a strong current for a very short time, creating a pulse. The specific heat is calculated from the variation of enthalpy as a function of temperature. This device allows measurements at high pressure (up to 3 800 bar [3]) under water, and at high temperatures (up to 10 000 K). More recently, Pichler et al [4] measured the specific heat and latent heat of fusion using the same device and also by employing a conventional differential scanning calorimeter to extend the measurements to the solid state.

Another way to perform non-contact measurements is to use levitation devices. Fecht and Johnson [5] conducted a theoretical study in which thermal capacity was measured using an energy balance on an electromagnetically levitated drop. In their study, a modulated calorimetry method is used: the sample receives sinusoidal heating power. It appears that the oscillation frequency of the power can reduce measurement error if it is correctly chosen. Schetelat [6] also studied the modulated calorimetry method by developing a finite volume model (using the commercial software Fluent) and comparing it with analytical results. This study highlights the importance of having a homogeneous temperature in the sample studied, and therefore a Biot number lower than 0.1. For higher Biot numbers, Schetelat [7] proposed another method based on the model developed by Fecht [5] and Wunderlich [8]. This method considers two regions of the sample, and the temperature is measured at both regions. The heating is a pseudo-white noise modulation that allows to determine the transfers functions of the system. The method was validated by a numerical model but not on experimental data. Fukuyama et al [9] and then Watanabe et al [10] proposed a similar method using an electromagnetic levitation (EML) facility but assisted by a strong magnetic field (4 T and up to 10 T for thermal conductivity measurement) to stop the movement of the liquid and achieve more stable and accurate measurements. To measure the specific heat, laser modulation is added to the heat induced by the levitator, and the temperature of the sample is measured by a pyrometer. It should be noted that the experiment lasts 60 minutes, which may result in the evaporation or oxidation of the sample.

Rhim et al [11] used an electrostatic levitation (ESL) facility to determine the ratio of specific heat and hemispherical total emissivity during the radiative cooling of a silicon sample. The value of specific heat at melting temperature is taken from literature and used to determine the value of emissivity. Then, the emissivity is assumed to be constant during the cooling for computing the specific heat. They show a non-linear variation of specific heat according to temperature. Ishikawa et al [12], [13] also used ESL to measure the hemispherical total emissivity and the heat capacity of molten nickel, rhodium, and platinum. The emissivity is measured with spectrometers and then the specific heat is deduced from the cooling curve. These measurements are realized only at the melting temperature.

Another approach was presented by Sun et al [14], [15], who used an aerodynamic levitation device (ADL). The experiments are carried out with two different levitation gases and only the cooling curves are used. The convective heat losses are first determined for both gases using a platinum sphere, which emissivity and specific heat are given by Ishikawa et al [13]. Then, the emissivity and the specific heat capacity are determined for the sample from the cooling phase with both gases. However, this technique assumes that emissivity is independent of temperature and equal to the emissivity at the melting temperature. For this reason, measurements were only conducted within a temperature range of 50 K around the melting temperature of the sample.

The above-mentioned studies therefore allow the measurement of specific heat, but impose constraints on the shape of the sample [2], its material (EML), the duration of the experiment [9], [10], the temperature measurement [12], [13], [14], [15] and the hypothesis of constant emissivity [11], [12]. Independently, the number of papers discussing liquid materials is rather low.

This paper proposes a new method for determining the specific heat of liquid metal by aerodynamic levitation. This device has the advantages of being applicable to any material (adapted to ADL), having a heating system independent of levitation, and not being limited in temperature. However, two important specificities are required: the knowledge of the absorbed part of the heating by the sample and its temperature homogeneity during the cooling phase of the experiment. In this article, Section II presents the experimental device and the methodology to obtain the specific heat. A new analysis of two experimental phases allows to determine both the heat losses and then the specific heat. Section III is dedicated to the method to determine the absorb part of the heating. A new specific three-color pyrometer is presented to measure the temperature and the absorptivity, thereby allowing the temperature dependence of absorptivity to be accounted for in the calculation of specific heat. The laser power is carefully measured to ensure a good knowledge of the heat absorbed by the sample. Section IV concerns the validity of the temperature homogeneity of the sample though numerical simulation. Finally, Section V presents experimental results on molten iron and nickel in order to compare the method with literature. Results on molten zirconium are also presented as only few data on this material are available in the literature.

Samples used for measurements are cut from a bulk material (99.95% pure iron from Goodfellow, 99.99% pure nickel from Neyco, 99.20% pure zirconium from Goodfellow). The mass of the samples is set at (30 ± 2) mg to ensure a liquid sphere with a radius of around 1 mm once melted. The samples are weighed before and after experiment with a precision scale (Mettler Toledo XS205DU).

The ADL device developed at the laboratory IRDL is presented in Fig. 1. The sample is placed in an inert chamber, above a copper levitation nozzle. The chamber has four viewports enabling measurements and visualization during experiment. An ytterbium fiber laser (IPG PHOTONICS YLR 300/3000 QCW) with a 1 070 nm wavelength heats the sample from the top, and a three-color infrared pyrometer is used to measure the temperature and the absorptivity at 1 070 nm (see Section III). The chamber is evacuated with a vacuum pump up to less than 1 mbar and then filled with inert gas (pure argon AirLiquide Alphagaz2). This operation is repeated twice to limit oxidation of the sample. Two numerical flowmeters control the gas flow through the levitation nozzle and through a “cross-jet” nozzle. The cross-jet is used to prevent metal vapors from polluting the viewport and interfere with the laser and the pyrometer. The levitation flow is set at 1 ·min^− 1. A power meter (PRIMES cube M) is used before each experiment to measure the laser power through the window. A high-speed camera is used to monitor the sample behavior but not directly used in this study. More details about the experimental device are available in previous papers [16], [17], [18].

The sample is melted to obtain a spherical shape, then it is heated to reach several temperature plateaus, and finally cooled when the laser is switched off. Figure 2 shows the temperature of the sample during a test, measured by the three-color infrared pyrometer used. During the cooling, the sample reaches a supercooled state before solidification. The melting temperature is identified by the increase in temperature during recalescence.

As shown in Fig. 3, the main idea is to use a temperature plateau and the cooling phase at the same temperature. Figure 3 shows the temperature of the sample during a test and the methodology applied. During a plateau, the temperature remains constant, and we assume that the absorbed power is equal to the power lost, Eq. (1). This power lost includes both convective and radiative heat losses.

where is the absorptivity at laser wavelength (1 070 nm), the laser power reaching the sample (W), the total heat losses (W), the mean plateau temperature (K). In Eq. (1), the total heat losses are computed for each plateaus Pi (Fig. 3). Knowing the absorbed power (see Section III) and the temperature, the total heat losses can be determined for this specific level of temperature.

Then, the cooling phase at the same temperature is considered. As this is the same temperature, the heat losses are assumed the same as on the plateau. The heat transfer equation with the lumped body assumption is applied on the sample:

where is the mass of the sample (kg), the specific heat at constant pressure (J·kg^− 1·K^−‍‍1), the time (s). To apply the lumped body assumption, it is important to have a uniform temperature throughout the sample, which is the subject of Section IV. The variation of temperature can be measured during the cooling and the heat capacity is directly computed according to Eq. (2). The mass of the sample is measured before and after the experiment to evaluate the amount of vaporized matter. This quantity remains below 1% for all tests. Therefore, the surface S is assumed to be constant along the whole experiment. To compute the specific heat, the mass after experiment is used as the vaporization occurs before the cooling phase.

A specific three-color infrared pyrometer has been developed to measure the temperature and the absorptivity at the laser wavelength required in Eq. (1) and Eq. (2). It allows to measure the radiance at three wavelengths thanks to two dichroic mirrors and Gaussian pass-band filters (± 10 nm): = 950 nm, = 1 070 nm, and = 1 250 nm. Wavelengths and are used to compute the temperature and wavelength is used to compute the emissivity-absorptivity at the exact laser wavelength (according to Kirchhoff’s law).

The signal (A) measured by the pyrometer at the wavelength is expressed as:

where is a specific function of the pyrometer, is the normal spectral emissivity of a sample at the wavelength , and is the radiance of a blackbody at the wavelength :

where and are constants and the temperature. By defining the ratio of signals at wavelengths and and from the Eq. (4) the following equation is obtained:

For a blackbody, the emissivity ratio is equal to one and Eq. (5) can be written as:

Constants and are determined with Eq. (6) through calibration with a graphite blackbody heated by laser and controlled by a commercial 2-color pyrometer. Once and are known and based on the knowledge of the melting temperatures of the tested materials (pure iron and zirconium), the emissivity ratio is defined. This ratio is assumed to remain constant over the whole temperature range. Finally, once , and are known, the temperature of the sample can be computed for the rest of the test using Eq. (7).

The same methodology is applied to signal 2 at the wavelength 1 070 nm, using Eq. (8):

The constants and are determined by calibration, and the temperature is computed as explained with Eq. (7). Eq. (8) allows to compute the normal spectral emissivity with respect to temperature, and then the absorptivity at 1 070 nm according to Kirchhoff’s law. Unlike some previous works in literature, the emissivity variation will be measured and integrated to the specific heat determination.

The previous methodology developed in Section II requires a homogeneous temperature within the sample to be validated. To check this requirement, a numerical model of the experiment is used. This 2D axisymmetric model is developed with the finite elements software Comsol Multiphysics and models both the gas flow around the spherical sample and the liquid movements inside the sample as well as all the heat transfers. More details on the model are available in previous work [18], [19]. Figure 4 shows the difference between the maximum temperature and the minimum temperature inside the sample during a 3 seconds laser heating and a cooling. Figure 5 shows the numerical field of temperature in argon gas and in the sample, and the velocity vectors in the argon, at 3 seconds. During the heating, the average temperature reaches 2 000 K and the difference between maximum and minimum temperatures remains around 35 K. When the laser is switched off, this difference quickly decreases and remains around 2.2 K. These differences are small enough to consider the sample as a lumped body, especially during the cooling.

Moreover, several phenomena are neglected in this model: the thermal conductivity of liquid iron depends on temperature [20]. Here it is set at 40 W·m^− 1·K^− 1, which may be lower than real conductivity. The Marangoni effect due to surface tension gradient and the rotation of the sample on its axis are not modeled. All these phenomena would tend to increase mixing in the sample and therefore reduce the temperature difference. The temperature difference is then overestimated in this model and yet within an acceptable range to proceed (< 2% at 2 000 K).

Signals from the pyrometer are processed to compute, first, the temperature of the sample and, second, the absorptivity as a function of temperature. Figure 6 presents an example of absorptivity at 1 070 nm for liquid zirconium. These values are then used to compute heat losses, based on the temperature plateau and Eq. 1. Finally, the specific heat is determined based on the cooling section and Eq. 2.

Figures 7, 8, and 9 show the specific heat for, respectively, liquid iron, liquid nickel, and liquid zirconium. To compare the results obtained with literature, the values for iron and nickel are converted into molar heat capacity (J·mol^− 1·K^− 1) using a molar mass of 55.845 g·mol^− 1 for iron, and 58.693 g·mol^− 1 for nickel. Figure 10 shows the values of specific heat in J·kg^− 1·K^− 1 obtained in this work for the three metals. The uncertainty computations are realized with the propagation law, based on Eq. (2) with more details in annex. The specific heat of liquid iron presented Fig. 7 is measured between 2 000 K (49.17 J·K^− 1·mol^− 1) and 2 337 K (47.69 J·K^− 1·mol^− 1). Another ADL facility is presented by Sun et al [14] and allows to measure the specific heat only close to melting temperature, but the method presented here allows measurements beyond this point. At 2 000 K, the result is close to the value obtained by Watanabe et al [21] by EML and modulated calorimetry. Beyond 2 000 K, the results are compared to the values obtained by Savvatimskiy et al [22] and Pottlacher et al [23] by the derivation of enthalpy against temperature. This method provides a unique value for specific heat for the entire temperature range, and the results of this work are included within their uncertainty range. The scattering of our results can be explained by the presence of oxides that can modify the value of emissivity/absorptivity measured, and by the beginning of vaporization at high temperature. The measurement of 38.48 J·K^− 1·mol^− 1 at 2 109 K, lower than the other points, presents an absorptivity of 0.288 which is 6% lower than the average absorptivity measured for iron at this temperature (between 2 091 K and 2 126 K). The large uncertainty of the point 47.69 J·K^−‍1·mol^−‍1 at 2 337 K is due to the uncertainty on the slope during the cooling. This point is the hottest reached during a test (highest plateau in Fig. 2), the cooling just started and the lumped body behavior is not entirely guaranteed. Indeed, during the initial stages of cooling, the sample cannot be considered as a lumped body. It takes around 30 ms for the temperature to become uniform throughout the sample. To improve the measurement, the sample must be heated to a higher temperature, allowing sufficient cooling to achieve lumped body behavior. However, this can cause the sample to start vaporizing.

The specific heat of liquid nickel, presented Fig. 8, is measured between 1 838 K (53.57 J·K^−‍1·mol⁻¹) and 2 162 K (44.06 J·K⁻¹·mol⁻¹). The results are higher than the values obtained by other levitation facilities [12], [14], [21], [26] and close to the values obtained by Pottlacher et al [23]. The specific heat of liquid zirconium presented Fig. 9 is measured between 2 283 K (396.6 J·kg⁻¹·K⁻¹) and 2 754 K (561.6 J·kg⁻¹·K⁻¹). The results show a very good agreement with the literature. A slightly increase of specific heat with the temperature is measured by Korobenko et al [27] and by the method presented here. Since the method used by Brunner et al [28] derives enthalpy on a large temperature range, only one value of specific heat is obtained and this increase is not measured. As for iron, the large uncertainties for the hottest measurements are due to the uncertainty on the slope during the cooling. Especially, the measurement at 2 735 K is realized on the last plateau of a test, and the cooling is not long enough for this temperature to have a precise measure. However, a more precise measurement is possible with this method as the vaporization temperature of zirconium is far hotter. It is then possible to redo another dedicated experiment to heat the sample at a higher temperature, in order to have a longer cooling and then a lumped body behavior at the desired temperature. This is done for the hottest point (2 754 K) and effectively reduces uncertainty. Nevertheless, measurements at these very high temperatures remain difficult because pyrometer’s photodiodes tend to saturate. The measurements for the three metals are presented Fig. 10. The data are more scattered for iron and nickel than for zirconium. Again, this can be explained by the much higher vaporization temperature in the case of zirconium.

A novel method for specific heat measurement has been developed with an ADL facility. The sample is heated by a laser to reach and maintain temperature plateaus and then the laser is switched off to let the sample cool. A three-color pyrometer has been developed to measure the temperature and the absorptivity at laser’s wavelength during the test. Heat losses are evaluated on a temperature plateau and the heat transfer equation with the lumped body assumption is applied on the cooling section to compute the specific heat. This method has been tested on iron, nickel, and zirconium and shows good agreement with the data available in the literature. This method has the advantages of reaching high temperatures above the melting point, being applicable for non-magnetic materials thanks to ADL and without requiring prior information on the emissivity-absorptivity and heat losses.

Nevertheless, the results are highly dependent on the absorptivity measurement and the evaluation of the slope . The lumped body assumption is required on the cooling, and its validity has been verified with a numerical model. Moreover, the beginning of sample vaporization can limit the temperature at which measurements are conducted. Preventing this phenomenon could improve the measurements. Even with these uncertainties, this methodology will be applied on various other unknown alloys and materials in the near future.