1 Introduction

The physics of supercritical fluids is extremely complex and till today not fully understood. This is due to a number of concurrent factors, which are briefly discussed hereafter. First of all, a new theoretical framework is currently under development that links closely the dynamics and thermodynamics of supercritical fluids. This approach dates back to the pioneering work of Gorelli et al. [6, 7], who investigated the propagation of density fluctuations in supercritical fluids as a function of pressure and temperature. The authors found a direct correlation between the damping of acoustic waves and the occurrence of extrema in thermodynamic response functions (e.g. specific heat capacity \(c_{p}\), thermal diffusivity \(D_{T}\) and kinematic shear \(\nu _s\) and volume \(\upmu _{b}\) viscosities). It was found that the maxima of \(c_{p}\), defining the so called Widom line, divide the supercritical region in a liquid-like and in a gas-like sub-region. In the gas-like area, the adiabatic propagation of sound waves was observed. Instead, in the liquid-like region, positive sound dispersion was detected, as result of a complex interplay of acoustic and thermal waves [6]. Macroscopically, this occurs when the measured acoustic damping rate \(\Gamma \) deviates from the classical counterpart \(\Gamma _c\) due to non negligible volume viscosities, as typically found in liquids [17]. Second, reliable experimental data for the supercritical region are still missing, thus significantly hampering the validation of theoretical and thermodynamic models. This is particularly true with respect to the determination of fluid temperature and composition in two-phase mixing regimes. Third, the physical mechanisms controlling the transition from two-phase to single-phase mixing are still not understood. In our previous investigations [9, 10], we analysed different theoretical models and assessed their capability to predict the onset of single-phase mixing. The analysis clearly showed that, in presence of large temperature and/or concentration gradients, the role of evaporation cannot be neglected.

These preliminary considerations lie out the research path of the present work, which is summarised in the following sections. First, we have focused on the study of a single droplet at near-critical conditions. Contrary to sprays, droplets provide a simplified configuration, for which analytical solutions can be obtained to describe the simultaneous exchange of mass and energy at high-pressure conditions. Second, we have built a high-pressure, temperature-controlled test rig, equipped with an electrical droplet generator, to perform droplet experiments under controlled conditions. Third, several laser diagnostic methods have been developed and improved to enable the measurement of mean droplet temperatures and composition in the wake of an evaporating droplet. Moreover, laser induced thermal acoustics (LITA) has been extended to enable the measurement of the acoustic damping rates in the supercritical region of pure fluids. Following the approach proposed by Mysik [17], these measurements enable us, for the first time, to assess the importance of volume viscosity in the liquid-like supercritical region and may lead to improved closure models for modelling momentum transport and dissipation effects in supercritical fluids. Finally, the temperatures derived from non-resonant, spontaneous Raman scattering data and laser induced fluorescence and phosphorescence thermometry (LIFP) data provided by Preusche et al. [23] are employed to validate the theoretical prediction of an improved evaporation model. It is important to emphasise, that only the temporal evolution of the droplet temperature can be validated, since the presented measurements do not allow droplet size investigations. Additionally, the concentration field measured using non-resonant Raman scattering is compared to direct numerical simulation of a free falling evaporating droplet at high pressure and temperature conditions.

2 Experimental Setup

Experimental investigations are performed using two heatable high pressure, high temperature chambers. Two chambers are used since the investigations are performed at two different laboratories. One chamber is optimized for spectroscopic investigations, while the other is used for phenomenological investigations as well as measurements, using laser induced thermal acoustics. To distinguish between both chambers the first chamber located at the Technical University of Darmstadt (TUDA) is referred to as TUDA-chamber, whilst the second one located at the University of Stuttgart (US) is named US-chamber. Both chambers are designed for investigations of free-falling droplets in a near–critical environment. The droplet generator on top of the high pressure, high temperature chamber can be replaced with a closed lid for operation as homogeneous flow reactor for calibration purposes. The experimental setups are operated as continuous–flow reactors. The inlet mass flow is controlled using a Coriolis–based mass flow controller for carbon dioxide and carbon dioxide mixtures or a heat–capacity based mass flow controller for other fluids. Pressurized fluids are supplied through an annular orifice on top of the chambers. The pressure is controlled using a pneumatic valve at the system exhaust. The pressure inside the chamber is measured at the chamber exhaust by a temperature–compensated pressure transducer with an uncertainty rated at \(\pm 0.05\) MPa (US-chamber) or \(\pm 0.03\) MPa (TUDA-chamber). Temperature measurements inside the chamber are located at three different heights using resistance thermometers (US-chamber) or type–K thermocouples (TUDA-chamber) penetrating the metal core. Due to the temperature dependent uncertainty of the resistance thermometers the measurement uncertainties are calculated for each condition separately. The uncertainty of the type–K thermocouples is rated at \(\pm 1\) K. Both temperature and pressure are logged continuously. To ensure no contamination from previous investigations, the chamber is carefully evacuated before each set of experiments.

Fig. 1
figure 1

Horizontal and vertical sections of both chambers. Upper left: Vertical section through US-chamber. Upper centre (section B—B): Horizontal section through US-chamber at centre of first window. Upper right (section C—C): Horizontal section through US-chamber at fluid inlet and annular orifice. Lower left: Vertical section through TUDA-chamber. Lower right (section D—D): Horizontal section through TUDA-chamber at point of temperature measurement. A: annular orifice; EI: electrical insulation; F: fluid inlet; G: graphite gaskets; H: heating cartridges; I: thermal insulation; O: FFKM O–rings; S: spark plug; T: resistance thermometer; V: vermiculite plate; W: quartz windows; WR: Willis O–rings

The US-chamber depicted in the upper part of Fig. 1 is built with heat–resistant stainless steel (EN-1.4913). It is designed for pressures up to 8 MPa and temperatures up to 773 K. For optical accessibility eight ultra-violett (UV)–transparent quartz windows are placed at two different heights with an angle of \(90^\circ \) to each other. The temperature is controlled using eight heating cartridges in the chamber body and a heating plate with four cartridges below the chamber. The temperature in the cartridges and the chamber body is hereby measured using type–K thermocouples. The chamber encloses a cylindrical core with a height of 240 mm and a diameter of 40 mm. A mineral–based silicate is used for thermal insulation. Additionally, the bottom of the heating plate is insulated using a vermiculite plate. The TUDA-chamber depicted in the lower part of Fig. 1 is designed for pressures up to 6 MPa and temperatures up to 553 K. The chamber is built with temperature–resistant stainless steel (EN-1.4571) and is optically accessible through four UV–transparent quartz windows placed at an angle of \(90^\circ \) to each other. The temperature is controlled using six heating cartridges in the chamber body. The temperature in the cartridges is hereby measured using type–J thermocouples. The elevated pressure chamber encloses a cylindrical core with a height of 85 mm and a diameter of 48 mm. A detailed description of the droplet generator and its periphery as well as an investigation on the reproducibility of the droplet detachment, can be found in the works of Weckenmann et al. [32], Oldenhof et al. [18] and Lamanna et al. [9]. Ouedraogo et al. [20] presented a numerical study of the detachment principle as well as the underlying physics.

3 Experimental Methods

3.1 2D Raman Spectroscopy for Mixture Fraction Determination

Spontaneous Raman scattering can be used to identify different substance molecules due to their spectral response to common incident light [5]. In order to measure the binary mixture fractions in the wake of the evaporating droplets, the Raman Stokes shift response of the involved substances is used. The liquid injected substances feature more complex Raman scattering responses compared to the ambient gas phase nitrogen (N\(_2\)), which exhibits one vibrational stretching response at a wavenumber shift of around 2330 cm\(^{-1}\). However, only the carbon hydrogen (C-H) stretching region of their response is evaluated at a wavenumber shift in the region of around 2700–3000 cm\(^{-1}\) for acetone, n-heptane, n-hexane and di-ethyl-ether. For the fluoroketone FK-5-1-12, the filters are changed to include the wavenumbers of 400–1400 cm\(^{-1}\), which includes the C-H\(_2\) bending response. Generally, the spectral peaks are not temperature or pressure invariant and shift in their wavelength slightly with changes in those conditions (single digit wavenumbers) [30], but selecting broader filter and integration bands can mitigate this.

The detection system, starting from the pressure vessel, uses an achromatic lens, followed by a optic relay with two objective lenses. The relay is used to focus to infinity and parallelize the incoming light as much as possible to minimize angle dependent filtering. This leads into the filter section. First, the Rayleigh scattering at 532 nm is blocked, subsequently the image is split spectrally by the image splitter. Two additional filters on the two camera paths each then restrict the remaining signals further spectrally, before they are imaged through objective lenses by two charge coupled devices (CCD) cameras. Due to the used binning, best possible resolution is approximately \(33\,\upmu \textrm{m} \, \times 130 \, \upmu \textrm{m}\) (vertical \(\times \) horizontal). A more detailed description of the optical setup is presented in [1]. The N\(_2\) and C-H filter regions are the integrated signals for two recording channels, \(S_\textrm{sp}\) and \(S_\textrm{amb}\), that are used to measure the mixture fraction.

A pulsed neodymium-doped yttrium aluminium garnet (Nd:YAG) laser is used as illumination source. Two 1 \(\upmu \)s apart, 7.5 ns long laser pulses are shaped into a 130 \(\upmu \)m thick (e\(^{-2}\)), 12 mm high vertical laser sheet. This sheet is subsequently trimmed by an aperture edge to a height of approximately 7 mm and forms the image plane above the falling droplets as shown in Fig. 2. The intensity profile of the laser is changed to approximately \(\left( M^2\right) \) of 6–7, which produces a larger diffraction limited focus and prevents laser breakthroughs in the dense gas while simultaneously allowing a wider beam at the pressure vessel windows. The pulse energy is around 300 mJ each.

The Raman cross-section of the substances represents an interaction probability of the photons scattering off the substance molecules. They vary between substances and their individual Raman response bands, as well as with changing temperature [14]. This necessitates calibration of the relevant mixture fractions at variable conditions, as the scattering intensities are proportional to the cross-sections. However, the temperature range in the experiments is relatively low (393–553 K), and calibration results showed no discernible temperature and pressure dependence for mixing with nitrogen. The method is only shortly introduced. A more in depth explanation can be read in Bork et al. [1].

3.1.1 Evaluation

For calibration, the mixture fractions are set with mass flows and evaporated into the pressure vessel via an evaporator and temperature controlled tubing to prevent condensation. The target vapor mixture fraction inside the vessel can then be determined with [1, 2]

$$\begin{aligned} x_\textrm{sp,set}&= \frac{N_\textrm{sp}}{N_\textrm{sp}+N_\textrm{amb}}&= \frac{S_{\textrm{sp}}}{S_{\textrm{sp}}(1-k_{2})+k_{1} S_{\textrm{amb}}}, \end{aligned}$$

where \(x_\textrm{sp,set}\) is the mass flow determined molar mixture fraction of the injected fluid substance, \(N_\textrm{sp}\) is the molecule number density of the injected fluid substance and \(N_\textrm{amb}\) is the molecule number density of the ambient gas phase substance. The scattering intensity of nitrogen is proportional to its Raman cross-section, which differs from the other substances. In order to correlate the measured signals (\(S_\textrm{sp}\), \(S_\textrm{amb}\)) to the number of molecules, a scaling factor is introduced between the measured injected substances and nitrogen, \(k_1\). An additional factor, \(k_2\), deals with the cross-talk between the two channels and is therefore subtracted proportionally with \(S_\textrm{sp}\) from the ambient signal. This is evaluated per image pixel to account for local deviations of the imaging system.

The two parameters \(k_1\) and \(k_2\) are fit to minimize deviations against \(x_\textrm{set}\) using non-linear least squares regression. The pressures 2, 4 and 6 MPa and variable temperatures ranging from 393 to 513 K (depending on substance) are calibrated, with mixture fractions increased until set point stability cannot be maintained or onset of condensation. Comparing the factors \(k_1\) between the substances with C-H stretching response shows that the scattering intensities are approximately proportional to the number of C-H bonds using this detection method [2].

Finally, during the measurement of evaporating droplets, the wakes are imaged identically to the calibration. The results are evaluated for \(x_{\textrm{sp}}\) and a modified mixture fraction result that is described in the following.

3.1.2 Relative Density and Adiabatic Mixing Temperature Estimation

The other way to interpret the wake behind the falling droplet involves the gas phase further away from the droplet wake, but still inside the imageable field of view. There, the mixture fraction of the injected liquid is very low or below the detection threshold. Due to the calibration of the parameter \(k_1\), there is a correlation between the signal strength of nitrogen and the injected fluid. Additionally, since the injected fluid presence is low, the cross-talk influence is low as well. These specific conditions are used to create an approximate maximum signal value of this fluid in the following way

$$\begin{aligned} S_\textrm{sp, max}&= k_1S_\mathrm {amb,x_\textrm{sp}=0}. \end{aligned}$$

\(S_\textrm{sp, max}\) is the approximate maximum count value in the image of the injected fluid, if it had the number density of the pure nitrogen that is actually imaged at that position. In other words, the pure nitrogen signal \(S_\mathrm {amb,x_\textrm{sp}=0}\) at the edges of the camera field of view is used to derive a pure injected fluid signal at this reference number density. Referring to Fig. 2, this information would be derived from beyond the left image border (from where the laser sheet enters), as the wake results are cropped to the droplet wake. With this information, the alternative mixture fraction value \(x_{\textrm{sp},l}\) is derived

$$\begin{aligned} x_{\textrm{sp,l}}&= \frac{S_\textrm{sp}}{S_\textrm{sp, max}}, \end{aligned}$$

where the mixture fraction is calculated as the fraction of the injected fluid species signal \(S_\textrm{sp}\) and its’ approximated maximum \(S_\textrm{sp, max}\). This concentration result is erroneous, because the number density reference is that of pure nitrogen at the pressure vessel conditions. The fraction of the locally correct concentration and this result is used as a measure of number density increase due to mixing and temperature change

$$\begin{aligned} \frac{x_{\textrm{sp}}}{x_{\textrm{sp,l}}}&= \frac{N}{N_\infty }. \end{aligned}$$

Using the Virial equation of state (EOS) together with Tsonopoulos’ model for mixing and the second coefficient, as well as Orbey and Vera’s model for the third coefficient [16, 19, 22, 31], the nitrogen and injected fluid mixture properties are derived using an adiabatic mixing approach from the respective pure substance data sourced from Refprop [12]. As reference points, the to be mixed pure nitrogen is always at the relevant conditions of the pressure vessel. The pressure of the injected substance is known, but not its temperature, since the injector temperature does not represent the true fluid temperature at the time of the wake recording.

Using non-linear least squares regression, the mixed fluid’s relative number density is repeatedly re-evaluated as a function of mixture fraction using different starting enthalpies for the injected substance. This process finds a theoretical, pure liquid starting enthalpy, that minimizes the difference between modeled and the experimentally derived relative number density. The mixture is then adiabatically mixed using this pure liquid enthalpy and the enthalpy from the ambient gas at pressure vessel conditions. Finally, temperatures for each given mixture fraction can be approximated using the EOS. So there is a correlation between the mixture fraction \(x_\textrm{SP}\) and the temperature of the vapour in the droplet wake. This can be seen in Fig. 2 in Section 4.1. The actually measured vapour concentration never reaches the saturation condition, so temperatures for mole fractions past the measurement derived densities are extrapolated using these EOS results. An example can be seen in Fig. 5 in Section 5.

However, the found injected substance temperature is not physical with respect to the experiment, as the injection into the vessel is not an adiabatic process, furthermore this consideration neglects all dynamic processes [1, 10]. Rather, the intersection of the vapour-liquid equilibrium (VLE) of the mixtures and the adiabatic mixing line is used to estimate the liquid phase temperature. Since the third Virial coefficient derivation of Orbey and Vera [19] isn’t valid for polar fluids, this method wasn’t strictly suited for acetone mixing. Furthermore, the employed enthalpy mixing rule neglected excess enthalpy. To measure the liquid phase of the droplet independently from this extrapolation, laser-induced fluorescence and phosphorescence thermometry was attempted with acetone. Additionally, a different EOS is then used to account for polarity and excess mixing enthalpy.

3.2 Laser Induced Fluorescence and Phosphorescence Thermometry

The fluorescence of a substance is the emission of a photon after the absorption of another photon by the involved molecule. In contrast to scattering processes, the molecule is now resonant to the incoming photon and enters a higher energy state. In order to return to the ground state, the molecule may emit the photon from this higher state. The two radiative processes that can happen are the fluorescence and the phosphorescence. The fluorescence is a fast process, where the radiation is given off in the order of nanoseconds. The phosphorescence emits from an intermediate lower energy state. Time is needed to transition to this intermediate state and to emit the photon, which now has less energy and is therefore, in general, red-shifted compared to the fluorescence emission. The phosphorescence generally transpires in the range of 10\(^{-6}\)\(10^0\) seconds. Acetone can be excited with UV light and then emits both fluorescence and phosphorescence as described. The spectral overlap between the two radiations is significant. The temporal overlap is small, and through time gating these two emissions can be separated sufficiently [2, 3, 24, 25].

In order to estimate the droplet temperature of acetone, the temperature dependent quenching of the phosphorescence is exploited. This is normalized with the combined fluorescence and phosphorescence response to reference the local UV energy area density inside the droplet. However, the phosphorescence signal is relatively weak, so the fluorescence signal is dominant in this normalization. The energy area density dependent emission of the fluorescence is linear with respect to the employed measurement energy (\(\ge 1.5\) mJ cm\(^{-2}\)). The phosphorescence response, however, exhibits saturation effects starting below 1 mJ cm\(^{-2}\) [2, 3]. This necessitates an energy dependent calibration.

3.2.1 Evaluation

For calibration, the pressure vessel is filled with acetone and a vertical rectangular UV profile of approximately 8 mm\(\times 2\) mm is imaged into the acetone using an optic 4f relay and 320 nm dye laser emission source. The absoprtion cross-section of acetone is relatively low at this wavelength [34], which is advantageous due to the very low absorption in the vapour phase. The UV light is therefore negligibly absorbed in the nitrogen acetone vapour mixture towards the liquid droplet.

Similar to the Raman detection setup, a two CCD camera system is used. One camera records the combined non-intensified fluorescence and phosphorescence response and the other camera the intensified phosphorescence response, which is gated temporally from the fluorescence using the intensifier gate. A 650 nm short pass spectral filter is responsible for blocking the trigger laser (emitting at 675 nm), which is used to start the recording chain on droplet crossing. The image splitter is now a 30% transmission 70% reflection mirror. The stronger reflection signal is guided towards an objective lens with image intensifier for the phosphorescence signal. This reduces the necessary amplification. The fluorescence phosphorescence combination is imaged using the 30% transmission signal, now imaged by an identical objective lens and camera, but without image intensification. The time gate delay for the phosphorescence separation is set using the steep edge fall-off of the initial fluorescence signal. It is set to 20 ns after the 50% count level compared on the camera, compared to the maximum. This maximum is reached when the entire fluorescence peak is imaged from it’s beginning. A more detailed description of the setup is available in [23].

The acetone is now imaged in two different ways at variable temperatures and pressures. The temperature is limited by either evaporation, both in the calibration setup and when failing to generate droplets in the experiment, or the critical point around 508 K at 6 MPa. The acetone then becomes supercritical during calibration, which can not be achieved during droplet injection. Additionally, beyond 500 K the phosphorescence signal becomes very weak and imaging it becomes difficult even with an image intensifier. The measured pressures are chosen to reproduce those of the acetone and nitrogen droplet wake mixture fraction Raman experiment at 2, 4 and 6 MPa. First, the timing is set so that the phosphorescence is recorded separately, gated by the intensifier. This yields

$$\begin{aligned} V&= \frac{\sum S_{\textrm{LIP,I,ROI}}}{\sum S_{\mathrm {LIP+LIF,ROI}}}. \end{aligned}$$

Here, the images are summed across the pixels of the region of interest (ROI) which have count values above the background offset plus detection noise. The signal with the subscript LIP,I is the laser induced phosphorescence response alone imaged by the intensified camera, while LIP+LIF is both fluorescence and phosphorescence as recorded by the non-intensified camera. In order to mitigate intensifier amplification and liquid density based changes to the emission intensity, an amplification ratio between the two cameras is recorded with the time gating on the intensifier changed to include the entire fluorescence emission as well

$$\begin{aligned} V_\textrm{amp}&= \frac{\sum S_{\mathrm {LIP+LIF,I,ROI},i}}{\sum S_{\mathrm {LIP+LIF,ROI},i}}&V_{\textrm{LIFP},i}&=\frac{V_i}{\overline{V}_\textrm{amp}}. \end{aligned}$$

The amplification measurement is averaged over all frames i to generate the set-point amplification factor \(\overline{V}_\textrm{amp}\). The calibration measurement value is \(V_{\textrm{LIFP},i}\) and the UV energy area density \(E_\textrm{UV}\). \(V_{\textrm{LIFP},i}\) is fit with a first order polynomial against the energy area density for every pressure-temperature combination

$$\begin{aligned} V_\textrm{LIFP,E}&= AE_\textrm{UV}+B&T_\textrm{LIFP}&=\frac{CV_{\textrm{LIFP,E}}(E_\textrm{UV})+D}{GV_\textrm{LIFP,E}(E_\textrm{UV})^2+HV_\textrm{LIFP,E}(E_\textrm{UV})+I}. \end{aligned}$$

The fit parameters A and B are determined for every calibration set point. The non-linear fit parameters CDGHI are fit for each queried \(E_\textrm{UV}\). The droplet measurement value of \(\overline{V}_{\textrm{LIFP},i}\) is then put into the equation for the estimated temperature using LIFP \(T_\textrm{LIFP}\), with the fit parameters for an equivalent energy density \(E_\textrm{UV}\) to determine the temperature. This calibration method was found to be pressure independent within discernible precision and accuracy [23].

Note that the temperature is only evaluated for the average of 500 droplets. This is done to mitigate the influence of changing droplet topology and signal lensing of the liquid phase. Additionally, signals that cross certain count thresholds are discarded or lead to the entire image pair of the channels to be discarded. Otherwise, the lensing, leading to higher local excitation energy densities, introduces strong systematic errors into the temperature evaluation. The uncertainty was determined to be of the order of \(\pm 3\) K [23].

3.3 Laser Induced Thermal Acoustics

Laser induced thermal acoustics (LITA) also more generally known as laser induced (transient) grating spectroscopy (LIGS) utilizes the non-linear interaction of matter with an optical interference pattern to measure independently and simultaneously speed of sound, acoustic damping rates as well as thermal diffusivities. The optical interference pattern is generated by two short pulsed excitation laser beams from a pulsed Nd:YAG laser with an excitation wavelength \(\lambda _{exc} = 1064\) nm, a laser pulse length of \(\tau _{pulse} = 10\) ns and a 30 GHz line–width. The excitation beams are crossed with the same direction of linear polarization to produce a spatially periodic modulated polarization/light intensity distribution. A third input wave is provided using a continuous wave DPSS laser with a wavelength of \(\lambda _{int} = 532\) nm and a 5 MHz line–width to interrogate the resulting changes in the optical properties of the investigated fluids and is scattered by the spatially periodic perturbations within the measurement volume. All beams are focused into the probe volume using an AR–coated lens with a focal length of \(f = 1000\) mm at 532 nm. The scattered signal beam is spatially and spectrally filtered using a coupler and single–mode/multi-mode fibres and detected by an avalanche detector. The voltage signal is logged with 20 GS/s by a 1 GHz bandwidth digital oscilloscope.

A detailed description of the optical arrangement used for the presented investigations as well as the spatial resolution of approximately 312 \(\mathrm {\upmu m}\) in diameter and less than 2 mm in length in the x-direction can be found in the work of Steinhausen et al. [28]. For a theoretical description of the generation of laser induced gratings the reader is referred to Cummings et al. [4], Schlamp et al. [26] as well as Stampanoni-Panariello et al. [27].

3.3.1 LITA Post-rocessing

The speed of sound can be extracted from a LITA signal using direct Fourier transformation (DFT). Note that the speed of sound data is hereby obtained from the frequency domain of the LITA signal using only geometrical parameters of the optical setup. Hence, no modelling assumptions or equations of states are necessary. As described by Hemmerling et al. [8] the speed of sound \(c_{s}\) can be estimated as follows

$$\begin{aligned} c_{s} = \frac{\nu \Lambda }{n}. \end{aligned}$$

The dominating frequency of the LITA signal is \(\nu \) and the constant n indicates the resonant behaviour of the fluid. For resonant fluid behaviour \(n=1\), whereas in case of non-resonant fluid behaviour \(n=2\). The grid spacing of the optical interference pattern is denoted by \(\Lambda \) and is a calibration parameter for the optical setup. \(\Lambda \) can be expressed as referred in Stampanoni-Panariello et al. [27] as

$$\begin{aligned} \Lambda = \frac{\lambda _{exc}}{2\sin \left( \Theta /2\right) }, \end{aligned}$$

where \(\lambda _{exc}\) denotes the wavelength of the excitation beams and \(\Theta \) is the crossing angle of the excitation beams.

As descripted in our previous work [28] the evaluation of LITA signals proposed by Schlamp et al. [26] enables us after careful calibration to extract the speed of sound \(c_{s}\), the acoustic damping rate \(\Gamma \) as well as the thermal diffusivity \(D_{T}\) from the shape of the LITA signal. Using the assumptions presented by Steinhausen et al.  [28] the time-dependent diffraction efficiency \(\Psi (t)\) of a detected LITA signal shows the following dependencies

$$\begin{aligned} \Psi (t) = f \left( c_{s}, t, t_{0}, D_{T}, U_{eP}, U_{\Theta }, \bar{\eta }, \Lambda , \Gamma , \sigma , \omega \right) , \end{aligned}$$

where \(t_{0}\) indicates the time of the laser pulse, t the time, \(\bar{\eta }\) the beam misalignment in horizontal y-direction. \(U_{\Theta }\) and \(U_{eP}\) denote the approximate modulation depth of thermalisation and electrostriction gratings, respectively, and the Gaussian half–width of the excitation and interrogation beam in the focal point is expressed as \(\omega \) and \(\sigma \), respectively.

The calibration of the geometrical parameters of the optical arrangement, namely the grid spacing of the optical interference pattern \(\Lambda \) and the Gaussian half–width of the excitation beam \(\omega \), is done in well-known quiescent conditions. \(\Lambda \) is hereby calculated using measurements in a pure nitrogen and pure argon atmosphere. A DFT together with a von Hann window and a bandpass filter is used together with Eq. (8). For the calibration of the Gaussian half–width of the excitation beam \(\omega \) only investigations in a pure argon atmosphere with pressures up to 4 MPa are utilised. Argon is chosen as the fluid for calibration because of two reasons. First, argon shows non-resonant fluid behaviour leading to a simplified model of the LITA signal independent from the thermal diffusivity (see [26, 28]). Second, for gaseous argon the volume viscosity is negligible compared to the shear viscosity (see [13, 15]). Therefore, the acoustic damping rate can be estimated using the expression for the classic acoustic damping rate shown in Li et al. [13]

$$\begin{aligned} \Gamma _{c} = \frac{1}{2 \varrho } \left[ \frac{4}{3} \ \upmu _{s} + \left( \gamma -1\right) \ \frac{\kappa }{c_{p}} \right] . \end{aligned}$$

The fluid density is indicated by \(\varrho \), \(\gamma \) is the specific heat ratio, \(\upmu _{s}\) the shear viscosity, \(\kappa \) the thermal conductivity and \(c_{p}\) the specific isobaric heat capacity. Curve fitting of the LITA signal to the theoretical expression presented by Steinhausen et al. [28] is achieved using a robust non-linear least–absolutes fit with the Levenberg–Marquardt algorithm. For this purpose, the non-linear fit in Matlab R2018a (MathWorks) with the robust option Least Absolute Residuals (LAR) is utilised.

4 Results

4.1 Raman Mixture Fractions

The results of the Raman campaign are quantitative mixture fractions in droplet wakes of the measured substances, as well as the temperature estimations derived from measurement number density considerations. Figure 2 shows examples for n-hexane droplets in nitrogen atmosphere.

Fig. 2
figure 2

n-hexane droplets in nitrogen atmosphere. Left: single droplet mixture fraction result for nitrogen atmosphere temperature of \(T_\textrm{ch} = 533\) K, injector temperature of \(T_\textrm{inj} = 473\) K and pressure of \(p_\textrm{ch} = 6\) MPa. Horizontal x and vertical z distances are normalized to approximated droplet diameter d. Right: number density data estimated adiabatic mixing temperature curves dependent on mixture fraction. Varying ambient nitrogen temperature as shown, injector temperature fixed at 473 K and a pressure of 6 MPa

The adiabatic mixing temperature \(T_\mathrm {ad.mix}\) estimated from the experiment number density is put into context with the mixture VLE. The VLE is interpolated from isotherms calculated with the Peng-Robinson EOS, Huron-Vidal mixing rule, Mathias-Copeman alpha function and the non-random two liquid Gibbs free excess model (personal communication J. Vrabec, Thermodynamics and Thermal Separation Processes, University Berlin). The actual measured mixture fractions are always far below saturation in all conditions. In order to have temperature information beyond the actually measured mixture fractions, the EOS states are fitted to the experimental data and can be used as an extrapolation towards the dew line [1, 10]. Due to the involved assumptions and approximations, this temperature function becomes more erroneous with rising mixture fraction.

4.2 Laser Induced Fluorescence and Phosphorescence Thermometry

The number density data from the Raman experiments was fitted to an adiabatic mixing model using the PC-SAFT EOS. This differs from the approach for other fluids than nitrogen and acetone, where the previously described Virial EOS is used for mixing. However, the potential is there to improve the other temperature approximations as well using more advanced EOS. The advantages for using this EOS were the inclusion of excess properties for the adiabatic mixing, in addition to the EOS being fitted to experimental and molecular simulation data for the polar acetone and nitrogen mixture. The results show agreement within 9 K an can be read in Preusche et al. [23].

4.3 Laser Induced Thermal Acoustics

The validation of the calibration process of the grid spacing is depicted in Fig. 3. The relative distribution of the ratio between the speed of sound data extracted from the LITA signal using Eq. (8) and theoretical values taken from the National Institute of Standards and Technology (NIST) database by Lemmon et al. [11] is shown. Since the ratio between the measured and the theoretical value is chosen, a ratio of 1 indicates a perfect validation. Note that we omitted the distinction between the fluids argon, nitrogen as well as the fluid mixtures for clarity.

Fig. 3
figure 3

Thermodynamic data for validation are taken from Lemmon et al. [11]

Relative distribution of the ratio between the measured speed of sound data and theoretical values for \(p_\textrm{ch} = 2\) to 8 MPa and temperatures up to \(T_\textrm{ch} = 700\) K (nitrogen \(-99.999\)% purity) and \(p_\textrm{ch} = 0.5\) to 8 MPa and temperatures up to \(T_\textrm{ch} = 600\) K (argon \(-99.998\)% purity and carbon dioxide \(-99.995\)% purity) as well as the investigated fluid mixtures argon-helium with mole fractions \(x_\textrm{Ar} = 0.9000\) and \(x_\textrm{Ar} = 0.2000\), argon-nitrogen (\(x_\textrm{Ar} = 0.8000\)), argon-carbon dioxide (\(x_\textrm{Ar} = 0.8000\)) and nitrogen-carbon dioxide (\(x_\textrm{N2} = 0.7999\)) for \(_\textrm{ch} = 0.5\) to 8 MPa and \(T_\textrm{ch} = 301\) K. The calibration resulted in a grid spacing of \(\Lambda = \Lambda _\textrm{cal} = 29.47 \pm 0.05\) \(\mathrm {\upmu m}\). The speed of sound is calculated using a DFT with Eq. (8).

The observed offset in the distribution for carbon dioxide is a result of residual moisture in the experimental rig as discussed in Steinhausen et al. [28]. For all other fluids and fluid mixtures the distribution in Fig. 3 shows a good agreement between the measured speed of sound data and the theoretical values. Note that the measurement uncertainty using a confidence interval of 95% of the acquired speed of sound for all investigated fluids is below 2% while the width of the distribution is approximately 3%. Therefore we estimate that the speed of sound can be extracted with a relative uncertainty rated at 3% (95% confidence interval).

Fig. 4
figure 4

Acoustic damping rate ratio \(\Gamma _{\textrm{LITA}}/\Gamma _{c,\textrm{NIST}}\) over chamber pressure \(p_\textrm{ch}\) for pure carbon dioxide (99.995% purity) at temperatures up to 600 K temperature. Classical acoustic damping rates are estimated using NIST database by Lemmon et al. [11]. Experimental and theoretical data are taken from Li et al. [13]. Curve fitting input parameters: \(\Lambda = \Lambda _{\textrm{cal}} = 29.47\) \(\mathrm {\upmu m}\); \(\sigma = \sigma _{\textrm{th}} = 177\) \(\mathrm {\upmu m}\); \(\omega _{\textrm{SM}} = \omega _{\textrm{SM,cal}} = 254\) \(\mathrm {\upmu m}\); \(\omega _{\textrm{MM}} = \omega _{\textrm{MM,cal}} = 225\) \(\mathrm {\upmu m}\)

Figure 4 depicts acoustic damping rate ratios \(\Gamma _{\textrm{LITA}}/\Gamma _{c,\textrm{NIST}}\) for pure carbon dioxide (99.995% purity) at various temperatures for pressures between 0.5 and 8 MPa. For curve fitting the calibrated value for the grid spacing \(\Lambda = \Lambda _{\textrm{cal}} = 29.47\) \(\mathrm {\upmu m}\) as well as the calibrated values for the Gaussian beam width of the excitation beam \(\omega _{\textrm{SM}} = \omega _{\textrm{SM,cal}} = 254\) \(\mathrm {\upmu m}\); \(\omega _{\textrm{MM}} = \omega _{\textrm{MM,cal}} = 225\) \(\mathrm {\upmu m}\) are used. For the latter a distinction between a single-mode fibre with diameter of \(4 \ \mathrm {\upmu m}\) (SM) and a multi-mode fibre with diameter of \(25 \ \mathrm {\upmu m}\) (MM) is implemented. The Gaussian beam width of the interrogation beam is set to the theoretical value based on the specification of the used laser source \(\sigma = \sigma _{\textrm{th}} = 177\) \(\mathrm {\upmu m}\). Values are compared to the experimental and theoretical investigations by Li et al. [13]. For pressures up to 1 MPa our experimental investigation shows good consensus with data by Li et al. [13]. At higher pressure the assumption of a linear pressure dependence seems to be not applicable and a monotonic increase in the acoustic damping rate ratios \(\Gamma _{\textrm{LITA}}/\Gamma _{c,\textrm{NIST}}\) is observed.

5 Discussion

The importance of volume viscosities for complex fluids, such as carbon dioxide, from sub- to supercritical fluid states is shown by the comparison of the measured acoustic damping rate \(\Gamma _{\textrm{LITA}}\) and the classic acoustic damping rate \(\Gamma _{c,\textrm{NIST}}\), which neglects the contribution of the volume viscosity, shown in Fig. 4. For carbon dioxide this is also true at low pressure conditions. As depicted, the acoustic damping rate ratio shows an non-linear, monotonic increase. At near to supercritical pressures (6 to 8 MPa) the measured acoustic damping rate \(\Gamma _{\textrm{LITA}}\) exceeds the classic acoustic damping rate \(\Gamma _{c,\textrm{NIST}}\) by two orders of magnitude, which highlights the importance of the consideration of the volume viscosity in supercritical fluid physics.

The employed laser diagnostics methods, namely spontaneous Raman scattering and laser induced fluorescence and phosphorescence thermometry enable a quantitative comparison of droplet evaporation processes with theoretical evaporation models and direct numerical simulations. The Raman scattering investigations lead to a concentration-density field in the droplet wake. By applying the PC-SAFT EOS together with an adiabatic mixing assumption this concentration-density field can be used to estimate a concentration-temperature profile in the wake of the droplet. To compare the experimentally gained concentration-temperature profile of a n-hexane droplet in a nitrogen atmosphere with direct numerical simulations, we extracted the temperature and concentration of each numerical cell inside the region of interest from the numerical data; for more detail the reader is referred to Steinhausen et al. [29]. Figure 5 depicts the numerical data in red in a T,x-diagram. The concentration-temperature field estimated from the Raman scattering results are displayed in black. Furthermore, an extrapolated curve fit of the experimental concentration-temperature data is shown as a black line and the VLE is presented as a blue line. The latter is computed using the PC-SAFT EOS.

Fig. 5
figure 5

Image is taken from Steinhausen et al. [29]

Comparison of Raman scattering, direct numerical simulation and theoretical concentration-temperature profiles; Black line: extrapolated curve fit of the experimental concentration-temperature data using an adiabatic mixing assumption together with the PC-SAFT EOS; Blue line: VLE; Ambient nitrogen temperature 473 K, injector temperature 473 K and pressure of 6 MPa. Droplet diameter at \(t = 0.5\) s is 1.25 mm.

The presented comparison shows reasonably good agreement. Hence, our proposed method to extract temperature data from the Raman scattering results is supported by the direct numerical simulation. The extrapolated Raman temperature, proposed by Lamanna et al. [10], is the intersection of the PC-SAFT based fit of the experimental data with the VLE.

Fig. 6
figure 6

Comparison of the analytical model by Young [33] and modified Pinheiro et al. [21] with LIFP measurement, Raman extrapolated temperature and the theoretical global thermodynamic equilibrium. Ambient nitrogen temperature 513 K, injector temperature 473 K and pressure of 6 MPa. Droplet diameter at \(t = 1\) s is 1.35 mm

The performed spectroscopic investigations of free falling droplets are compared with the temperature evolution predicted by the evaporation model from Young [33]. In Fig. 6, the evolution of the droplet temperature of a preheated, free falling acetone droplet with an initial fluid temperature of 473 K in a nitrogen atmosphere with an temperature of 513 K at 6 MPa is depicted together with the estimated temperatures using LIFP and spontaneous Raman scattering. The temporal evolution is hereby estimated using an improved version of the evaporation model by Young [33]. To account for convective effects a convection correction from Pinheiro and Vedovoto [21] is applied. Additionally real gas effects are considered by taking the fugacity as well as solubility effects into account. Note that the evaporation model by Young [33] is a Langmuir type model following Onsager’s theory and heat conduction within the liquid phase is not considered. Hence, the surface temperature of the droplet equals the liquid phase temperature. For a detailed description and discussion regarding the selection criteria of evaporation model as well as the importance of non-equilibrium models the reader is referred to Lamanna et al. [10]. The results show a good agreement between the Raman derived temperature estimation (extrapolated Raman temperature) and the LIFP temperature results measured with a droplet detachment frequency of 1 Hz and the analytical investigations. Note, that the evolution of the droplet temperatures, measured (circle and cross) or analytical (dash-dotted), are significantly above the global equilibrium temperature (dashed line) for the given conditions [23]. In case of an acetone droplet in nitrogen atmosphere, the droplets would reach a wet bulb temperatures above the global equilibrium temperature. The difference to the equilibrium is explained by the strong interdependencies of energy and mass fluxes at high pressures droplet evaporation as has been shown by Lamanna et al. [10]. Therefore, the evolution over time is strongly dependent on substance parameters and dynamic conditions such as the forced advection due to the fall velocity.

6 Conclusions

In the presented work, an experimental setup as well as three laser diagnostic techniques for the investigation of relaxation phenomena in supercritical fluids and near-critical droplet evaporation was introduced. By applying laser-induced thermal acoustics in pure fluids from sub- to supercritical fluid condition the importance of volume viscosities was assessed. The experimental results clearly indicate that volume viscosities should be considered when studying supercritical fluid phenomena. Using spontaneous Raman scattering the concentration-density field of a free falling evaporating droplet was measured and converted in a temperature-concentration field and an estimated droplet temperature. Comparisons between the extracted temperature-concentration field and a direct numerical simulation yields reasonably good agreement. The estimated Raman temperature was validated using direct LIFP thermography. Finally, both droplet temperatures measurements were utilised to validate the theoretical prediction of an improved evaporation model regarding the temporal evolution of the droplet temperature. The validation shows an excellent agreement between the direct droplet temperature measurement (LIFP), the estimated Raman temperature as well as the theoretical predictions by the improved evaporation model.