New Liquid Fraction Measurement Methodology for Phase Change Material Analysis Based on X-Ray Computed Tomography

New liquid fraction measurement methodology for phase change material analysis based on X-ray computed tomography Dario Guarda a , b , * , Jorge Martinez-Garcia b , Benjamin Fenk b , David Schiffmann b , Damian Gwerder b , Anastasia Stamatiou b , J¨ org Worlitschek b , Simone Mancin a , Philipp Schuetz b a Department of Management and Engineering, University of Padova, Vicenza, 36100, Italy b Competence Centre for Thermal Energy Storage, Lucerne University of Applied Sciences and Arts, 6048, Horw, Switzerland ARTICLE INFO Keywords: Phase change materials Latent thermal energy storage X-ray computed tomography Optimisation Liquid-solid phase change ABSTRACT To tackle climate change, it is of fundamental importance to develop more efficient systems to convert and manage energy. Latent Thermal Energy Storage (LTES) systems, which are based on Phase Change Materials (PCMs), present many interesting features that can lead us to achieve energy sustainability. In fact, these systems can be employed in many applications (e.g., HVAC, buildings thermal management, thermal energy recovery). To use PCMs, it is crucial to understand and monitor their phase-change behaviour. However, the evaluation of two-phase systems is still difficult because of the intrinsic complexity and transient behaviour. In the present study, a new methodology to measure liquid fraction is proposed. X-ray Computed Tomography (XCT) is used to follow the solidification process of eicosane. XCT and the acquisition settings were optimised to ensure maximum contrast between the phases. Then, the solidification process of eicosane was dynamically studied and the volumetric liquid fraction was extracted. Finally, via the Ansys Fluent software, a CFD model, based on the enthalpy-porosity method, was implemented to simulate the solidification of eicosane. The numerical results were compared against the experimental data, showing excellent agreement. This novel experimental methodology opens new possibilities to characterise and understand the solid-liquid phase transition process of PCMs, by directly observing and quantifying the solid/liquid phase evolution in time, the volumetric shrinkage, the presence of voids and the different phases that might occur during the phase-change process. 1. Introduction Global warming is an imminent problem [1]. The United Nations in 2015 defined 17 Sustainable Development Goals [2] for a better future. In particular, there are two goals (affordable and clean energy; climate action) that are linked with energy production, consumption and management. To reach these goals, many scientists are working to better exploit energy by introducing Latent Thermal Energy Storage (LTES) systems, based on the so-called Phase Change Materials (PCMs). From 1990s to 2010s, the number of articles published on PCMs has increased by a factor of around 18 times, i.e., from around 713 to 13319 (source: Scopus 2022). At the beginning of the 2020s (2020, 2021 and 2022), already 8887 articles have been published on this topic. In fact, the possibility of storing thermal energy in a nearly isothermal way [3] and with a favourable storage density [4] (LTES can store from 5 to 14 times the energy of a sensible system [3]) is very attractive. The advantages of PCMs open many potential application fields, such as Heating, Ventilation and Air Conditioning (HVAC) systems [5], electrical energy generation [6], buildings thermal management [7], thermal energy recovery [8,9], refrigerated transport in the cold chain [10] or thermal management of electronic devices and electric vehicle batteries [11–14]. The use of LTES decouples energy availability and demand [15]. Since renewable sources are intermittent, PCMs represent a fundamental technology to achieve future energy sustainability. PCMs comprehend materials that can be used to store thermal energy through the solid-liquid phase-change process. They can be classified into three categories [3]: organics, inorganics, and eutectics, the latter being mixtures of the first two. For analysing these materials, it is crucial to understand their phase-change behaviour to control LTES systems [16]. However, the two-phase processes are complex, and, in the case of the solid/liquid phase transition, it is difficult to track the melting front * Corresponding author. Department of Management and Engineering, University of Padova, Vicenza, 36100, Italy. E-mail address: [email protected] (D. Guarda). [17] as the solid phase is often opaque. For example, organic and inorganic PCMs have different phasechange mechanisms, especially paraffins versus salt hydrates. Paraffins present a more conventional behaviour, while salt hydrates [18,19] have a phase change that is actually a hydration and dehydration process. Depending on the kind of salt under analysis, they can melt congruently or not, resulting in the phase-separation phenomenon called segregation [18,19]. Some salt hydrates also present supercooling. Thus, if it is already hard to describe phase change of these PCMs, a mixture of organics and inorganics could lead to even more complex behaviours. Liquid fraction is one of the key parameters for the control of LTESs, because it is proportional to the stored latent energy (if sensible heat is neglected). It represents how much liquid phase is present with respect to the whole amount of PCM employed in the storage. However, it is difficult to accurately quantify this value from experimental data. Temperature probes and video imaging have been proposed to monitor liquid fraction during the phase change. The temperature probe works as a local indicator [17]; in fact, when the melting front passes through the probe, the temperature increases, resulting in information on the solid-liquid interface. This information could be meaningful if the temperature probes are uniformly distributed in the PCM. Thus, many temperature probes are needed. Seddegh et al. [20] used 16 thermocouples, and Kamkari and Shokouhmand [21] used 32, 30 or 26 depending on the geometry under study. However, the use of a large number of probes could be challenging, or they could even interfere with the phase-change process, acting, for example, as nucleation sites during the solidification process or sources for external heat. In any case, other authors employed a similar technique. Among them, Righetti et al. [22, 23] extensively analysed two PCMs embedded in periodic porous structures realised with additive manufacturing via T-type thermocouples. Landini et al. [11] analysed a Thermal Management System TMS for a Li-ion pouch cell with a PT100 Resistance Temperature Detector (RTD). The use of video recording gives a few more details, but requires the presence of transparent windows, which modify the thermal boundary conditions, as highlighted by Zhang et al. [24,25]. Moreover, the use of video imaging is not able to provide quantitative information about the entire volume but only about the observed surface of transparent PCMs. In fact, the solid phase is usually opaque; hence, the view can be obstructed. Mancin et al. [21,26–28] and Kamkari and Groulx [29,30] used both thermocouples and video cameras to obtain not only temperature data, but also liquid fraction information. This brief overview highlights the need for a non-destructive and quantitative technique enabling a direct evaluation of the volumetric liquid fraction. The results would become of paramount interest to develop accurate models of solid-liquid phase change process as well as control algorithms to manage LTES systems. At the same time, the results could also be used to validate numerical models, which usually use the average or local temperature, against a measured liquid fraction curve. To date, the enthalpy-porosity method [31] has only been validated against temperature [32,33] or simplified liquid fraction obtained from video recordings [27,28,32,34]. In the present study, the new methodology proposed by MartinezGarcia et al. [35,36] to measure volumetric liquid fraction from time-lapse X-ray Computed Tomography (XCT) imaging measurements is applied to eicosane. The method is based on XCT, a non-destructive technique which allows obtaining a digital volumetric representation of the sample reconstructed from multiple projection images collected from different directions. XCT has been deployed in many fields, from medical testing [37,38] to engineering [39–41]. Here, it is used to track the solidification process of eicosane. The work is divided into three parts. At first, X-rays and measurement settings were optimised to achieve the maximum contrast between the liquid and solid phases. Secondly, the solidification process of eicosane was dynamically analysed in the XCT measurement with the optimised parameters. Finally, the results obtained from a numerical model were compared to experimental data showing an excellent match. 1.1. Novelty To the best of the authors’ knowledge, the XCT approach for analysing solid-liquid phase change is fairly new to the field of non-metallic phase change materials [35,42,43]. Kohler et al. [42] applied XCT to study the formation and segregation of magnesium chloride hexahydrate phases enclosed in a tube-in-tube heat exchanger. Martinez-Garcia et al. [35,43] and Guarda et al. [44] started developing an XCT approach to precisely quantify the evolution of the liquid volume fraction during solid-liquid phase change in PCMs. The authors analysed the ice melting process [35,44] and the solidification of calcium chloride hexahydrate PCMs [35]. Regarding the work on ice melting, the authors also compared the results from a numerical model against the experimental liquid fraction from the XCT, showing an excellent agreement. The current work also aims at validating the enthalpy-porosity method against the measured liquid fraction profile obtained by time-lapse XCT scan during the solidification process and this, to the best of the authors’ knowledge, has not been done before. Finally, the possibility to inspect the internal microstructure of the PCMs, while solid-liquid phase change Nomenclature A mush Mushy zone constant [kg m -3 s -1 ] c p Specific heat [J kg -1 K -1 ] g Gravitational acceleration [m s -2 ] h ref Specific reference enthalpy at T ref [J kg -1 ] H Specific total enthalpy (sensible and latent) [J kg -1 ] I/I 0 Transmission [-] k Thermal conductivity [W m -1 K -1 ] L Latent heat of fusion [J kg -1 ] N pixel start Number of PCM pixels at the start of the experiment [-] N pixel end Number of PCM pixels at the end of the experiment [-] p Pressure [Pa] S Source term of the energy equation [W m -3 ] t Time [s] T Temperature [K] T ref Reference temperature [K] T L Liquidus temperature, T liquidus [K] T S Solidus temperature T solidus [K] v Velocity [m s -1 ] x Sample thickness along the beam direction [m] z Contrast [-] Greek Letters β Thermal expansion coefficient [K -1 ] γ Liquid fraction [-] μ Attenuation coefficient [m -1 ] μ L Mean grey value of liquid phase [-] μ S Mean grey value of solid phase [-] ρ Density [kg m -3 ] ρ 0 Reference density for the Boussinesq approximation [kg m - 3 ] σ L Grey value standard deviation of the liquid phase [-] σ S Grey value standard deviation of the solid phase [-] τ  Stress tensor [Pa] occurs, opens unprecedented opportunities in this field of research. It will help to better understand the phase-change process as well as to clarify the phenomena underpinning the main issues of particular PCMs, such as the segregation of salt hydrates or the sedimentation of nanoparticles in nano-PCMs. 2. Materials and methods The PCM analysed in this study is eicosane (CAS number: 112-95-8), an organic paraffin with a melting point around 36.7 ◦C [45]. The XCT measurement system is a Diondo D2 system from Diondo, Hattingen, Germany. The measurements were conducted with the X-ray source XWT-225 TCHE+. A 4343 CT detector from Varex, Salt Lake City, U.S.A, with a pixel size of 150 μ m, was used. In all the experiments, the amplification capacity was set to 8 pF, without pixel binning, and 3 frame binning. The reconstruction was performed with the Cera software package from Siemens Healthineers, Erlangen, Germany. In Fig. 1, a picture of the XCT is displayed. On the left side of the picture, the X-ray source is presented with the different available filters. In the middle, the sample stands on the rotating plate and on the right side of the photo, one can see the detector with the black foil. The X-rays are generated by the source and focused towards the detector. They pass through the filter, which is used to ensure that only higher energy photons are transmitted. Then, they encounter the sample which absorbs part of the X-rays depending on its size and on the attenuation coefficient (which depends on the atomic number and density of the material) [46]. Eq. (1) reports the Beer-Lambert law of attenuation for a monochromatic incident beam with a constant attenuation coefficient [46]. In the equation, I 0 is the incident X-ray intensity, and I is the intensity of the transmitted beam (i.e., after the interaction with the sample), μ is the attenuation coefficient and x is the sample thickness along the beam direction. I(x) I0 =e− μ x(1) For the sake of brevity, no additional details are reported here, interested readers may refer to Ref. [46] for further information. While the sample rotates, the detector receives the signals. To quantify I in Eq. (1), the detector measures the optical photons created by the X-ray photons based on a scintillation process. Since I 0 and x are known quantities, the output radiographies obtained are basically a map of the attenuation coefficients. For the sake of simplicity, in the discussion presented in the current work, the attenuation coefficient can be considered only a function of the density. A denser object absorbs and deflects more X-rays, resulting in higher attenuation. Every single radiography acquired is a projection. It is possible to acquire many projections while the sample rotates. With all these projections, it is eventually possible to reconstruct a 3D model of the analysed sample: this process is called reconstruction and is generally based on the Felkamp-Davis-Kress algorithm (FDK), [47]. However, this algorithm presents a few limitations which could lead to the generation of different types of artefacts. A static XCT scan is the process of acquiring different projections, reconstructing them together, generating a 3D image stack. A dynamic XCT experiment consists of many static scans run in subsequent moments. This generates multiple 3D image stacks. Ideally, each stack is representative of a certain instant in a time-dependent experiment, as the solid-liquid phase change occurs. 2.1. Voltage and filter optimisation The XCT experiments started by optimising the X-ray acceleration voltage and filter to identify the best settings to reach the maximum contrast between liquid and solid phases. For this reason, two eicosane samples were prepared: one in solid phase and one in liquid phase (melted in an oven at 50 ◦C). A sample consists of 26 g of PCM (eicosane, solid/liquid density of 856/772 kg m −3 , see Table 2) inserted into a glass vial, Fig. 2a. Glass was chosen as the container material because its density is significantly different from that of either liquid or solid eicosane (the density of borosilicate glass is 2230 kg m −3 [48]). The two samples were inserted into an extruded polystyrene cylinder (XPS, from Swisspor, Boswil, Switzerland) that acted as insulation material (thermal conductivity of 0.034 W m −1 K −1 [49]). In Fig. 2b and c the sample inserted into the XPS is shown. The block in Fig. 2c was used for the voltage and filter optimisation, while the one in Fig. 2b was used for the final dynamic XCT experiment. The temperature in the XCT measurement chamber was kept at 20 ◦C, thus, XPS helped slowing down the solidification process of the liquid sample. It was estimated that the liquid fraction variation in a single XCT scan lasting 5 min is below 5–10%, enabling to realistically follow the liquid fraction evolution. In fact, in the present case, the increase in liquid fraction of 1% during 1 scan was estimated by considering the solidification time of eicosane (see section 2.4 for the initial and boundary conditions) given by a CFD simulation (500 min). With this value, it was possible to estimate the solidification rate as 0.002 min −1 , meaning that, for a XCT scan of 5 min, the liquid fraction change is 0.01, that is, 1%. The solid sample (the one at the bottom) was inserted upside down as shown in Fig. 2d to reduce Feldkamp artefacts [47]. The XPS-samples block was located inside the XCT device and scans of 500 projections with the rotating sample were taken with 8 different acceleration voltage and filter combinations, as listed in Table 1. To identify the proper combination of acceleration voltage and filter to maximise the contrast between solid and liquid, the study started from the parameters proposed by Martinez-Garcia et al. [35], who performed a similar analysis with ice and calcium chloride hexahydrate. In their study, the authors set the acceleration voltage at 160 kV, but no information was given regarding the filter. For this reason, a sensitivity analysis of the behaviour of the acceleration voltage and filter combination was performed to understand if there were any effects on the solid-liquid contrast and to identify the most suitable parameters. In general, the lower the acceleration voltage, the higher the contrast, but the longer the scan time and the smaller the sample that can be inspected. Moreover, the use of filters allows cutting lowerenergy X-rays, thus proper voltage-filter combination must be selected to avoid penalising the final image quality. Starting from the value of acceleration voltage proposed by Fig. 1. The different components of the XCT are highlighted. The X-rays are generated from the “source” and focused on the “detector”. The signal received by the detector depends on both the X-ray beam and the kind of object analysed, which rotates as radiographies are acquired. A physical filter can be used to consider only higher energy photons/X-rays; in the figure, the selected filter is the 1.0 mm of aluminium. Martinez-Garcia et al. [35], two additional values were investigated: 120 kV and 200 kV. Furthermore, different filters were considered to investigate their effects on contrast and scan time. In fact, the thicker and denser the filter, the longer the scan time. There are no reasons to test thick and dense filters with low voltage because the scan time would increase too much, rendering the dynamic measurement of the phase change impossible. Four cases were investigated, starting from the lightest to the heaviest, namely, unfiltered, 1.0 mm aluminium, 1.0 mm of aluminium +0.5 mm of copper and 0.5 mm of copper +0.5 mm of tin. The Source-Object Distance (SOD) was set to 379.5 mm and the Source-Detector Distance (SDD) was 600 mm. All reconstructed images have a dimension of 480 x 480 x 2500 pixels with cubic voxels of 95 μ m edge length. To compare the different stacks and choose the most appropriate voltage and filter combination, the contrast between solid and liquid phases as defined in Eq. (2) was considered. μ S and μ L represent the mean grey level of the pixels of the solid and liquid phases, respectively, while σ S and σ L their corresponding standard deviations. An optimum measurement parameter set maximises this contrast. The higher the difference between the mean grey values of solid and liquid, the higher the zvalue. The uncertainty in the estimation of the mean grey value was considered via the standard deviation, which appears in the denominator of the definition of z. The lower the uncertainty, the higher the contrast, z. With this definition, it is possible to avoid that images with large scattering, i.e. large σ S and σ L values, may display a large contrast, hence reducing the accuracy of the final measurement. z= μ S− μ L mean{ σ S; σ L}(2) Once all images were collected, for each stack, 5 zones of randomly selected solid and liquid regions were analysed, and from these, the mean grey levels and standard deviations ( μ S , μ L , σ S and σ L ) were calculated. This led to a unique z for each case in the proposed analysis. In the end, the combination number 2 (120 kV acceleration voltage and 1.0 mm Al filter) was chosen as optimal. In the Results section, further details are provided. 2.2. Dynamic XCT procedure for eicosane solidification measurement Once the optimum XCT settings were identified, the solidification process of eicosane was analysed using time-lapse XCT measurements. This means that the scans were acquired sequentially at the same time interval, without changing any other parameter. Thus, a liquid sample at an initial temperature of 50 ◦C was mounted inside the XCT device and solidified while keeping the room temperature at 20 ◦C. Differently from what was shown before for the optimisation process, only the top sample was located inside the XCT. The X-ray tube was set to a 120 kV acceleration voltage and a tube current of 270 μ A, resulting in a power of 32.4 W. The filter used was a 1.0 mm layer of aluminium. The detector was set at an amplification capacity of 8 pF, without pixel binning, an acquisition time of 120 ms was used and 3 frames were averaged. Projections were acquired via vertical scan field extension with 2 slices, meaning that two scans of 330 projections each were acquired, one for the top part of the sample and one for the bottom part; the scans were later combined during reconstruction. In the Cera software, used for reconstruction, a beam hardening correction of 1 was applied. All reconstructed images have a dimension of 320 x 320 x 656 pixels with cubic voxels of 103 μ m edge size. It is worth noting that, for comparison purposes, all the stacks were reconstructed with a fixed grey level scale. The source-object distance (SOD) was set to 155 mm, while the source-detector distance (SDD) was 462.6 mm. One scan lasted 4 min and 34 s. On average, a scan was recorded every 18 min. 2.3. Post-processing of the dynamic XCT experiment At the end of the solidification process, when all the scans were acquired, the images were post-processed in order to compute the volumetric liquid fraction, which is defined as the ratio of the pixel classified as “liquid” to the total number of pixels in the PCM (i.e., classified either Fig. 2. (a) 30 mm diameter glass vial used as PCM container. (b, c) XPS cylinder with vials inserted in the middle. (b) System used for the dynamic XCT experiment. (c) System used for the voltage and filter optimisation step. (d) Scheme of the XPS cylinder with the two glass vials analysed and XCT image with 200 kV acceleration voltage and 1.0 mm aluminium filtration. Table 1 Acceleration voltage and filter combinations analysed with additional information regarding the XCT acquisitions. # Voltage [kV] Filter Current [ μ A] Power [W] Integration time [ms] Scan time [mm: ss] 1 120 unfiltered 270 32.4 180 04:50 2 120 1.0 mm Al 334 40.08 180 04:50 3 160 unfiltered 250 40 150 04:04 4 160 1.0 mm Al 250 40 170 04:35 5 160 1.0 mm Al +0.5 mm Cu 250 40 350 09:10 6 200 1.0 mm Al 250 50 120 03:18 7 200 1.0 mm Al +0.5 mm Cu 250 50 220 05:51 8 200 0.5 mm Cu +0.5 mm Sn 250 50 510 13:15 as solid or liquid). To measure the volumetric liquid fraction profile over time, the image-based approach developed by Martinez-Garcia et al. [35] was applied. The approach was implemented through an in-house developed script written in Python 3.8.8, specifically designed to work efficiently with the measured image stacks. After importing all the image stacks, one for each datapoint, different greyscale threshold values were computed using the Otsu thresholding algorithm [50]. First, the background (black) and container thresholds were estimated, and once all the non-PCM pixels were removed from the images, the eicosane solid-liquid threshold was calculated. In Fig. 3, a histogram is presented to better illustrate the thresholding procedure. Fig. 3a shows the histogram of the XCT image stack that was recorded at 5 h from the beginning of the experiment. In this histogram, three peaks appear. Starting from lower grey values to higher ones (i.e., from lower density materials to higher ones), the first peak corresponds to the background and XPS pixels, the second and third ones (highlighted in Fig. 3b) denote the PCM and the last one represents the glass container. So, the first step of the algorithm is to compute the two green threshold values which separate the contributions of the XPS material (i. e., background) and the container from the PCM, going from left to right, respectively. The second step, which can be observed better in Fig. 3b, where only the solid-liquid eicosane pixels are reported in the histogram, consists of computing the threshold value that separates the liquid from the solid phase, i.e., the liquid-solid threshold. This threshold is represented by the red vertical line in the histogram. The green thresholds in Fig. 3 were computed taking into account all the stacks. In fact, for each stack, the background and container thresholds were calculated as previously described, via the Otsu algorithm [50]. Then, the maximum value of all the background thresholds (as well as the minimum of the container thresholds) was used in the further calculation of liquid volume fractions for all the stacks. On the other hand, the solid-liquid threshold was calculated from a single stack, in which, looking at the histogram, there were about half solid and half liquid PCM phases, as shown in Fig. 3. Once all the thresholds were computed, the total number of PCM pixels and the liquid ones were determined and used to calculate the volumetric liquid fraction profile over time as: Liquid Fraction =Number of pixels in liquid state Total number of PCM pixels at the beginning (3) 2.4. Numerical modelling The same setup was then numerically analysed using Ansys Fluent 18.2. To simulate the solid-liquid phase change, the enthalpy-porosity method proposed by Voller et al. [31] was implemented. The method considers the fluid domain as a porous medium, where the porosity is equal to the liquid fraction. The liquid fraction γ is defined in Eq. (4). When the system is fully solid, γ is 0, while it is 1 when it is fully liquid. In the melting range (i.e., between T solidus and T liquidus ) γ varies between 0 and 1 depending on temperature T. γ=⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ 0if T <Tsolidus T−Tsolidus Tliquidus −Tsolidus if Tsolidus <T<Tliquidus 1if T >Tliquidus (4) The transient governing equations represent the balances of mass (5), momentum (6), and energy (7). The gravitational acceleration was set to 9.81 m s −2 . ∂ρ ∂ t+ ∇ • ( ρ v →) = 0(5) ∂ ∂ t( ρ v →) + ∇ • ( ρ v →v →) = − ∇ p+ ∇ • ( τ ) + ρ (T)g →+(1−γ)2 (γ3+0.001)Amush v → (6) ∂ ∂ t( ρ H) + ∇ • ( ρ v →H) = ∇ • (k∇T) + S(7) The last addendum in the momentum equation (6) is a source term proportional to the velocity through the mushy zone constant A mush , which in the present simulations was set to its default value of 10 5 kg m −3 s −1 . S is the source term in the energy equation (Eq. (7)). The Boussinesq approximation was considered for buoyancy driven flows. The density, ρ , is constant in all the equations except for the gravitational term in the momentum equation, where its value varies depending on the temperature, as expressed in Eq. (8). ρ (T) = ρ 0(1−βΔT)(8) The enthalpy term in the energy equation is the sum of a sensible term and a latent one, as can be seen in Eq. (9). H=(href +∫T Tref cpdT)+ (γL)(9) The material properties used in the CFD simulations are listed in Table 2. Some of the properties were measured in the HSLU laboratory. ρ 0 of the Boussinesq approximation (8) is computed as the mean solidliquid density, i.e., 814 kg m −3 . The domain is axisymmetric and the geometry is presented in Fig. 4. Fig. 3. Histogram of a single image stack measured close to the middle of the solidification process, 5 h from the beginning of the test. (a) Full histogram with thresholds computed via the Otsu algorithm and (b) close-up histogram highlighting the PCM region. In the model, not only the PCM is simulated, but also the solid zones, where only the conduction model is applied. These zones are the XPS (density of 45 kg m −3 , specific heat of 1404 J kg −1 K −1 and thermal conductivity of 0.034 W m −1 K −1 [49]), the glass vials made of borosilicate glass (density of 2230 kg m −3 , specific heat of 800 J kg −1 K −1 and thermal conductivity of 1.2 W m −1 K −1 [48]), the polypropylene PP of the cap (density of 900 kg m −3 , specific heat of 1700 J kg −1 K −1 and thermal conductivity of 0.2 W m −1 K −1 [51]), and the air above the eicosane inside the container, which is supposed to be stagnant and thus considered as a solid layer (density of 1.139 kg m −3 , specific heat of 1006 J kg −1 K −1 and thermal conductivity of 0.027 W m −1 K −1 [52]). The bottom of the domain is assumed to be adiabatic, while a natural convection boundary condition (heat transfer coefficient of 10 W m −2 K −1 and free stream temperature of 20 ◦C) was assigned on the side and on the top exterior walls of the XPS. The test sample (glass vial and PCM) was initialised at 50 ◦C, the temperature at which eicosane was melted inside the oven. The XPS temperature was initialised at 20 ◦C. For pressure-velocity coupling, the SIMPLE algorithm was considered [53], while for spatial discretization, PRESTO! (PRessure STaggering Option) was used for pressure. The least-squares cell-based and the second-order upwind algorithm were used for gradient, and equations of momentum and energy balances [53]. The relaxation factor for the liquid fraction was set to its default value of 0.9. Time step and mesh sensitivity analysis were conducted in order to optimise the numerical model. Time steps of 0.05 s, 0.1 s, 0.25 s, 0.5 s, 0.8 s, and 1 s were analysed with a mesh of around 13k elements (12959). After choosing the optimal time step size of 0.1 s, meshes of 4k (3878), 6k (5845), 8k (7861), 10k (9878), 13k (12959) and 18k (17831) elements were studied. In the end, the mesh with 10k element was chosen as optimal. The reader is directed to the Results section for further details. 3. Results 3.1. Voltage and filter optimisation With reference to the experiments listed in Table 1 and the parameter z defined in Eq. (2), Fig. 5 presents the results of the voltage and filter optimisation. In Fig. 5a, the contrast values of all combinations are reported. In conclusion, the highest contrast z is obtained by combination 2 (i.e. acceleration voltage 120 kV with a filter of 1.0 mm of aluminium). Therefore, this parameter combination is chosen for further XCT analysis. It is worth noting that an increase in filtration leads to lower contrast values even if higher voltages, 160 kV and 200 kV, are used. Differently, with the lowest acceleration voltage, the use of the filter does not change the contrast z significantly. In addition, z shows a decreasing trend with acceleration voltage when the same filter (1.0 mm Al) is applied, see Fig. 5b. We can conclude that, for this kind of analysis, it is preferable, when possible, to use low acceleration voltages, because lower acceleration voltages result in higher grey level contrast between liquid/solid phases. This simplifies the post-processing and, in particular, the segmentation of XCT images and the quantification of the phases. As described above, 120 kV of acceleration voltage and 1.0 mm of aluminium as filter were chosen for the subsequent analyses. 3.2. Numerical model: time step and mesh sensitivity analyses The results of the time step sensitivity analysis are reported in Fig. 6. Fig. 6a shows the liquid fraction: it appears that, for γ greater than 0.5, γ does not change significantly when increasing the time step size from 0.05 s to 1 s, but then, the different curves slightly deviate from each other. In Fig. 6b, the computational time of the simulations performed with different time steps is reported. The trend is hyperbolic, it decreases as the time step increases. When it comes to decide the proper time step for extremely long simulations, a trade-off between the computational effort and time incurs. In this case, to avoid to continuously increase the numerical error, a time step of 0.1 s was chosen, which ensures a proper limit of the time step noise while reducing the total simulation time. The results of the mesh sensitivity analysis are reported in Fig. 7. In Fig. 7a, the volumetric liquid fraction is reported; γ does not significantly change when the mesh size is increased from 4k to 18k. In Fig. 7b, the computational time of the simulations with different mesh sizes is reported. There is no clear trend. This lack of a trend is due to the fact that the main parameter governing these kinds of simulations is the time step. The optimal mesh chosen was the one with 10k elements because it had the lowest execution time between all the simulations performed. 3.3. 4D CT results and comparison with simulations Fig. 8 shows the results of the XCT experiments collected during the solidification of the eicosane within the vial sample. Sixteen 3D renderings of the eicosane cylinder are proposed to visualise the entire phase-change process. Those renderings were generated with GeoDict2022 software, developed by Math2Market GmbH, Kaiserslautern, Germany. In the current visualisations, the container and the background were removed in order to increase the visibility of the phase Table 2 Eicosane properties considered in the CFD model. Eicosane Source Density (solid/liquid) 856/772 kg m -3 [45]/Digital densimeter Mettler Toledo DM40 Specific heat (solid/liquid) 2250/2483 J kg −1 K −1 Mettler Toledo DSC 823e with FRS 5+sensor and Huber TC100 Intracooler Thermal conductivity (solid) 0.402 W m −1 K −1 Linseis THB-100 Dynamic viscosity 0.003747 Pa s Rheometer Anton Paar MCR 302 Thermal expansion coefficient 0.0032151 K −1 Estimated as the inverse of the melting point (assumed 38 ◦C) Latent heat of fusion 223.8 kJ kg −1 Mettler Toledo DSC 823e with FRS 5+sensor and Huber TC100 Intracooler Melting range (T solidus /T liquidus ) 37/39 ◦C Mettler Toledo DSC 823e with FRS 5+sensor and Huber TC100 Intracooler Fig. 4. Zones in the numerical domain analysed in the simulations. Dimensions are in millimetres. change material. The solid phase opacity was set at 90% to show the evolution of the liquid in the inner part of the sample. In the figure, the lighter region represents the liquid phase, which continuously decreases as the solidification process evolves from the walls towards the core. Moreover, two videos were also included to directly show the dynamics of the phase change evolution over time. Supplementary video related to this article can be found at https://doi.org/10.1016/j.ijthermalsci.2023.108585. The evolution of the phase change process can also be visualised using the diagram reported in Fig. 9, where the liquid fraction extracted as previously described, is plotted over time. The experimental profile also includes the experimental uncertainties (the error bars in the graph) that were calculated by changing the PCM threshold given by the Otsu algorithm between ±0.5%. Therefore, the experimental error was considered to be half of the difference between volumetric liquid fraction profiles computed with a PCM threshold of ±0.5% times the one given by the Otsu algorithm. The same diagram also reports the numerical results obtained by implementing the time step size (0.1 s) and mesh (10k elements) previously selected. Fig. 5. Impact study of acceleration voltage and filter optimisation results on the contrast z. (a) z value for all experiments and (b) z behaviour at different voltages with the same filter. Fig. 6. Time step sensitivity analysis. (a) Evolution of the liquid fraction over time at different time steps and (b) computational times of the corresponding simulations. Fig. 7. Mesh sensitivity analysis. (a) Evolution of the liquid fraction over time for simulations with different meshes, and (b) computational time as a function of the mesh size. The liquid fraction evolution estimated from the simulation agrees fairly well with the liquid fraction from the dynamic XCT, particularly in the central zone, where both the solid and the liquid phases are roughly equally present. This period of the experiment corresponds to the time interval in which the segmentation algorithm employed performs better. In any case, the present results allow for one of the first direct experimental validations of one of the most implemented numerical models for solid/liquid phase change during solidification. The mean absolute error between the experimental and numerical liquid fraction is around 5.1%. Moreover, to show the promising capabilities of the XCT tool, a detailed analysis of the volume change during the phase change is proposed. In fact, since the solidification process is associated with a volumetric shrinkage, Fig. 10 shows how the XCT analysis enables to follow and estimate the volume variation by plotting the evolution of the number of PCM pixels during the phase change. At the beginning of the experiment, the number of pixels in the PCM liquid phase is (calculated with the computed threshold as shown in Fig. 3) 34321239, whereas at the end of the process, this value reduces to 29352847, resulting in a total volumetric shrinkage of 14.5% (see Eq. (10)). From the plot, it can be observed that the volumetric shrinkage varies almost linearly with time. volumetric shrinkage % =100 Npixel start −Npixel end Npixel start (10) Fig. 8. 3D renderings of the liquid (inner lighter region) and solid (outer nearly transparent region) eicosane phases. The container contribution was removed and it is not visible. Fig. 9. Experimental-numerical comparison for liquid fraction evolution. Fig. 11 provides a direct 3D visualisation of the volumetric shrinkage, which generates a large central cavity and many internal voids, which could not be accounted for without the XCT study. 4. Conclusions This study focuses on the development of a new experimental method to measure the volumetric liquid fraction evolution during the solid/liquid phase change processes of PCMs. This novel technique is based on dynamic X-ray computed tomography (XCT) image acquisition, allowing for a proper reconstruction of the entire process. Here, the solidification of eicosane is examined at room temperature inside the XCT, while scans are acquired. The collected results allowed to identify the most suitable XCT settings to achieve the optimum contrast between the solid and liquid PCM phases, which were used during the full solidification experiment. The transient volumetric liquid fraction is one of the most difficult parameters to be measured with non-intrusive sensors, which can affect the process by acting as additional nucleation sites/surfaces. This work can be considered as one of the first attempts in which the volumetric liquid fraction during the solidification of a PCM is measured without the introduction of any sensors inside the material. The experimentally determined liquid fraction curve was then used to successfully validate the numerical results obtained by employing the well-known enthalpy-porosity method. The present results demonstrate the promising features of the XCT measurement system, which permits to analyse solid-liquid phase change process, including the volume shrinkage phenomenon, allowing for a detailed study of the void generation and distribution. Future works will focus on the study of different PCMs, including salt hydrates, to use the XCT method to elucidate the underlying mechanisms for the segregation issue. Finally, this work can be considered as a reference for researchers who want to track the solid-liquid phase change in PCMs with X-ray computed tomography. The proposed experimental approach allows to select an optimised XCT setup enabling the highest possible contrast between liquid and solid PCM, and thus accurately monitoring the phases during the solid-liquid transition. In particular, for a given case, the procedure that is proposed and followed here consists of: (i) run a sensitivity analysis of the effects of the scanning parameters (acceleration voltage – filter) on the contrast; (ii) select the best voltage-filter combination to maximise the contrast while keeping the scan time at a reasonable value; (iii) run the dynamic experimental measurements; (iv) perform the reconstruction and evaluate the volumetric liquid fraction. Declaration of competing interest The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: Philipp Schuetz reports financial support was provided by Swiss National Science Foundation. Data availability Data will be made available on request. Acknowledgements The authors want to acknowledge the support of the Swiss National Science Foundation made available by grant 200021_201088 (“Investigation of Salt Hydrates Segregation with XCT”) and grant 206021_189608 (“FaCTES: Fast in-situ Computed Tomography for Energy Research and Space Biology”). The authors also wish to acknowledge the support and the computational resources made available by the High-Performance Computing Lab at the Department of Management and Engineering (DTG), cofunded by the University of Padova in the framework of the program “Scientific research instrumentation 2015”. 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Evolution of the number of PCM pixels along the solidification process of eicosane. Fig. 11. Vertical cross sections of the sample volume at the beginning (left) and at the end (right) of the XCT experiment are highlighted from the 3D rendering. It is possible to observe the volumetric shrinkage due to solidification.