Measurement of transient heat transfer in vicinity of

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International Communications in Heat and Mass Transfer 89 (2017) 57–63

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International Communications in Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ichmt

Measurement of transient heat transfer in vicinity of gas–liquid interface using high-speed phase-shifting interferometer

MARK

Yuki Kandaa,⁎, Eita Shojib, Lin Chenc, Junnosuke Okajimad, Atsuki Komiyad, Shigenao Maruyamad,e a

Graduate School of Engineering, Tohoku University, 6-6, Aramaki Aza Aoba, Aoba-ku, Sendai, Miyagi 980-8579, Japan Department of Chemical Engineering, Tohoku University, 6-6, Aramaki Aza Aoba, Aoba-ku, Sendai, Miyagi 980-8579, Japan c Department of Aerospace Engineering, Tohoku University, 6-6, Aramaki Aza Aoba, Aoba-ku, Sendai, Miyagi 980-8579, Japan d Institute of Fluid Science, Tohoku University, 2-1-1, Katahira, Aoba-ku, Sendai 980-8577, Japan e National Institute of Technology, Hachinohe College, Tamonoki, Hachinohe 039-1192, Japan b

A R T I C L E I N F O

A B S T R A C T

Keywords: Visualization Gas–liquid interface Interferometry Heat transfer High-speed measurement

The heat transfer process and initial stage of coupled convection at a gas–liquid interface are observed with high temporal and spatial resolutions in view of understanding phase transition dynamics such as evaporation or condensation for energy technologies. A high-speed phase-shifting interferometer is used to precisely measure the transient heat conduction and convection processes near the gas–liquid interface of a small water droplet. In the present study, the transient heat conduction around a water droplet interface during the adiabatic expansion process before the appearance of convection is visualized and examined. In the visualization experiment, transient density variations due to heat conduction in the vicinity of the gas–liquid interface are observed with temporal and spatial resolutions of 1 ms and 8.83 μm/pixel, respectively. It is determined that convection appears at approximately t = 0.25 s in a fast depressurization process, while transitions in both temperature and pressure are observed. In addition, the transient density variations and distributions of the gas phase before convection are compared with numerical simulations as an optical path length difference, and there is good agreement between the simulations and experimental results. The measurement methods developed in this study can be applied in the measurement of interfacial heat and mass transfers with high temporal and spatial resolutions.

1. Introduction The quantitative measurement of heat and mass transfer at a gas–liquid interface in processes such as the heating and evaporation of water droplets [1] or fuels [2] is necessary for engineering and industrial applications. Knowledge of gas temperature and concentration at the interface and the distribution or gradient near the interface is important to understand the energy balance and the boundary conditions at the interface during phase transitions. Detailed information at or near the interface of a thermal event such as droplet evaporation can aid in improving the performance of thermal management systems. Such information can utilized to study vapor–water mist in a cooling system [3] and spray cooling for electric devices [4] to improve their performance and also in the field of firefighting [5]. The droplet evaporation process is a complex phenomenon arising due to heat and mass transfer between different phases. The droplet evaporation rate has been evaluated by means of several theoretical



Corresponding author. E-mail address: [email protected] (Y. Kanda).

http://dx.doi.org/10.1016/j.icheatmasstransfer.2017.09.020

Available online 21 October 2017 0735-1933/ © 2017 Elsevier Ltd. All rights reserved.

modeling approaches including numerical studies [6]. In most cases, these proposed models have assumed quasi-steady-state conditions. However, this assumption is only feasible for certain liquids when the evaporation is slow and the liquid is not volatile. Experimental studies have also assumed that heat and mass transfer are not transient during evaporation, and therefore, very few measurements have been conducted to evaluate the evaporation process. In this context, Lopes et al. [7] identified transient heat transfer at the gas–liquid interface, where it governs the evolution of the local temperature distribution on the droplet and its evaporation rate. From the above discussion, it becomes obvious that an understanding of transient heat transfer at the interface is crucial. In addition, experimental systems to measure the interfacial phenomena are urgently required. The study of the impact (or impingement) and evaporation of a droplet on a solid surface is very challenging. In this context, Bhardwaj et al. [8] suggested that droplet deposition measurement requires a disparate range of time scales from 1 ms to several minutes due to the

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λ θ ρ ω ωp ω0 ψ

Nomenclature a cp E I i, j k Δl n p r T x, y, z

integration interval specific heat at constant pressure, J/(kg·K) integrated data along optical path beam intensity indexing pixel positions thermal conductivity, W/(m·K) optical path length difference, m reference index of gas pressure, Pa radius, m temperature, K Cartesian coordinates

wavelength, nm phase-shifting angle, rad density, kg/m3 beam frequency, Hz plasma frequency, Hz eigen frequency of molecule, Hz phase difference

Subscripts 0 i p ref t

under standard conditions initial pressure constant reference test

Greek symbols ε

refractive index difference along radial direction

image-processing technique that affords several benefits. With this technique, digital data such as 8-bit images, which are commonly employed in many applications, can be generated using certain phaseshifted interferograms. The technique allows accurate high-resolution measurements because the information corresponding to each pixel of the imaging detector can be utilized. In addition, digital images exhibit color depth, and the acquired phase and spatial information are magnified by a factor of over 100 with respect to those of a conventional interferogram. In this context, Torres et al. [20] proposed phase-shifting with the use of a quarter-wave plate and rotating polarizer with a stepper motor. Takahashi et al. [21] applied this measurement technique to study supersonic heat transfer flows in microchannels. However, such a measurement system can only be applied to study static or “slow” processes. Subsequently, Shoji et al. [22–24] developed a highspeed phase-shifting interferometer with an Arbaa prism, and their system was proved to have a simpler and more sophisticated optical structure than other previous designs. In this paper, we propose a new approach to measure the non-stationary interfacial heat transport phenomena around a water droplet subjected to a depressurization process with the use of a high-speed phase-shifting interferometer. The optical path length difference caused by variation in the density field due to the temperature variations is detected with the interferometer and subsequently quantitatively evaluated. The experimental results are analyzed with respect to numerical optical information calculated using the heat conduction equation considering pressure change. The results and implications of the accurate measurement of heat transfer at the interface are also discussed in detail in this study.

time difference of a jetted drop oscillation and its evaporation, and thus, they proposed high-speed temperature measurements to investigate the impact and evaporation of droplets on a heated solid substrate using a laser-based thermoreflectance method with temporal and spatial resolutions of 100 μs and 20 μm, respectively. For the measurement of transient heat transfer near the interface, such highspeed evaluations allow the obtainment of detailed information at the moment of heat transportation. Meanwhile, thermocouples have formed the prevalent device to measure the temperature at gas–liquid interfaces. In this context, Gatapova et al. [9] measured the temperature profiles across a gas–liquid interface using a micro-thermocouple and obtained the detailed temperature field near the interface. However, sometimes the thermocouple disturbs the fluid and heat transfer due to physical contact with the interface; further, the thermocouple offers lower spatial resolutions because it only affords point measurements. Hence, a non-intrusive temperature measurement technique with large temporal and spatial resolutions is required. For non-contact measurement, optical measurement systems offer a better solution over conventional methods. In this regard, Jung et al. [10] visualized the dynamics of a droplet impacting a surface and acquired the instantaneous temperature distribution of the collision surface simultaneously using an IR camera. However, the IR camera has limited application because the emissivity detected by the camera depends on the surface materials and their temperature changes. In this context, Yi et al. [11] measured instantaneous temperature fields by temporal measurement of the phosphorescence from a Mg4FGeO6:Mn doped hot plate cooled by an oblique impinging air jet. The use of thermographic phosphors is also restrictive due to their limited operating temperature range and the characteristic of continuous decay of phosphorescence. Meanwhile, interferometry has become a useful non-contact measurement approach to study heat and mass transfer phenomena via examining variations in optical parameters such as the refractive index [12,13]. The Mach–Zehnder interferometer is one of the most commonly used devices for the real-time visualization of changes in flow, temperature, and concentration fields [14,15]. In this context, Gryzagoridis [16] investigated natural convection from an isothermal horizontal downward-facing surface using the interferometric technique. Yadav et al. [17] applied interferometry for combustion experiments conducted with methanol droplets. However, the operating principle of the Mach–Zehnder interferometer restricts the spatial resolution of measurements for less fringe numbers, and additional information is often required to determine the physical qualities accurately. The phase-shifting technique for interferometry was developed [18,19] to solve the abovementioned problem. Phase-shifting is an

2. Experimental system 2.1. Visualization cell The visualization experiment of transient heat transfer around a water droplet during depressurization was conducted with the use of the visualization cell shown in Fig. 1. As indicated in Fig. 1, the experimental cell was made of aluminum. The cell was equipped with two 37-mm-diameter quartz glasses (Sigmakoki Co., Ltd., Japan) operating as windows at the test zone when using interferometer. The volume of the observational area was 4.5 mL. To eliminate heat transfer though the substance supporting the water droplet, a thin polyvinylidene chloride film (0.011 mm, Asahi Kasei Home Products Co., Japan, k < 0.13 W / (m·K)) was used as the substrate for the water droplet. A drop of purified water with a volume of 50 μL was placed on the film using a micropipette (Pipetman P100, 58

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Fig. 1. Schematic of experimental visualization cell.

GILSON, Inc.). Fig. 2 shows an overview of the experimental apparatus including the visualization cell. The experimental system consisted of a gas reservoir tank, gas cylinder filled with nitrogen with gas purity of 99.99995% (Taiyo Nippon Sanso Co., Japan), pressure regulator (JETH200 CR, Nissan Tanaka Co., Japan), vacuum pump (N816.3KT.45.18, KNF Japan Co., Ltd., Japan), and the cell. The tank and cell were connected via flow control valves, which included two needle valves and a fine-flow adjustment valve. The other components were connected via stainless tubes and valves.

Fig. 3. Schematic of high-speed phase-shifting interferometer (the arrows indicate the polarization direction of light).

converted into circularly polarized ones by means of a quarter-wave plate, and thus, that time no interferograms appeared. The beam was collimated by another plano-convex lens because the Arbaa prism needs a collimated incident beam to eliminate optical errors arising from the shape of the output beam and the film characteristic of the prism. The incident beam was split into four parallel beams by the Arbaa prism, which was equipped with three polarizers and one compensator at the output. Therefore, after the incident circularly polarized beam passed through the Arbaa prism, three phaseshifted interferograms and a shadowgraph image could be obtained. The output beams were subsequently detected by an imaging lens and a high-speed camera detector. A phase-shifting technique was applied to the interferometer used in the study. In this approach, three phase-shifted interferograms were converted to 8-bit gray–scale images (called phase-shifted data), and the image intensity ranged from 0 to 255 (0 and 255 mean black and white color, respectively). In the study, a three-step algorithm was used for the conversion of the interferograms into the required parameters. The phase-shifted data were derived from three interferograms, which can be expressed by Eq. (1):

2.2. High-speed phase-shifting interferometer The high-speed phase-shifting interferometer consists of a polarizing Mach–Zehnder interferometer, an Arbaa prism, and a high-speed camera (FASTCAM SA3, Photron Ltd., Japan). The schematic of the optical configuration of the interferometer is shown in Fig. 3. As an optical source, a linearly polarized He–Ne laser (05-LHP-991, IDEX Co., Melles Griot, USA) with an output power of 10 mW was used. In the setup, a neutral density (ND) filter was positioned immediately after the beam output from the laser, and the beam intensity was controlled by synchronizing the shutter speed of the high-speed camera. The polarization angle of the beam was adjusted to 45° by means of a half-wave plate. A spatial filter consisting of an objective lens and a pinhole was mounted to expand the laser beam and reduce its spatial noise. The beam was collimated by means of a plano-convex lens and split into the test and reference beams with the use of a polarizing beam splitter. With this arrangement, the difference in the polarizing angle between the test and reference beams was π/2, and their intensities were set to be identical via adjusting the half-wave plate. Both beams were reflected at flat mirrors, and the test beam was made to travel through the test section. Subsequently, another polarizing beam splitter superposed both beams. The resulting beam was directed through a plano-convex lens to change its magnification. In addition, the polarized beams were

I (i, j, 2π /3) − I (i, j, 4π /3) ⎞⎟ ψ ⎛⎜i, j⎞⎟ = arctan⎜⎛ 3 ( , , 2 I i j π /3) − 2I (i, j, 0) + I (i, j, 4π /3) ⎠ ⎝ ⎝ ⎠

(1)

where ψ(i, j) represents the phase difference between the test and Fig. 2. Schematic of experimental system (a) Experimental flow loop design and (b) photograph of the cell.

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images were captured, they were converted to phase-shifted data by an image-processing unit. The experiment was conducted three times under the same conditions.

reference beams. This phase difference can be converted into density, temperature, and other physical quantities. Indices i and j represent indexing pixel positions, and I (i, j, θ) denotes the intensity of the beam. In this study, the intensity information of the three phase-shifted interferograms was obtained from the images detected by the high-speed camera. The symbol θ denotes a chosen phase-shifting angle. For the three-step algorithm used in this study, values of 0, 2π∕ 3, and 4π ∕ 3 were set as the phase-shifting angles. Finally, the phase difference was calculated using Eq. (1) at each pixel of the obtained image. Details regarding the algorithm [19] and the applications of the algorithm can be found in our previous reports on interferometer designs [20–26].

2.4. Measurement uncertainties Measurement uncertainty analysis of the experimental data was carried out to determine the accuracy of the results. The temperature error of the thermocouple was determined as ± 1 K and the measurement error of the data logger was ± 1.2 K, and therefore, the temperature measurement error in the cell was ± 2 K. The uncertainly of the pressure sensor was ± 0.05 MPa within a range of 0.00–10.0 MPa and the error of the data logger was ± 0.026 MPa. Thus, the uncertainly of pressure was estimated as ± 0.06 MPa. The uncertainty of the interferometry measurement arose due to the nonlinearity of the CMOS sensor and optical devices as well as the errors in the data processing procedures. In the experimental set-up, the main factors of bias error are measurement and intensity errors. The resolution of the measured position on the camera was approximately 8.83 μm/pixel, corresponding to the actual length of one pixel in the bitmap image. The CMOS sensor of the high-speed camera detects the distribution of the optical path length difference with great precision (a measurement error of < 1 pixel). Therefore, the maximum bias error due to the measurement error of the location of 1 pixel was 7.49 × 10− 8 m, which corresponds to an error of 5.64% with respect to the actual measurement value. As regards the measurement error of the intensity, the resolution of the detected intensity datum is 1.24 × 10− 9 m, which corresponds to an error of ± 0.066% for the maximum measurement value of the optical path length difference in this study. More details on the theoretical analysis of uncertainties in interferometers can be found in our previous reports [20–26].

2.3. Experimental conditions and procedure The experimental system was installed in a temperature-controlled room maintained at 294 K. First, a 50-μL purified water droplet was placed on the polyvinylidene chloride film. After setting the flanges holding the two quartz glasses, the cell was set on the support of the interferometer. The camera was focused on the water droplet interface, and vacuum conditions were achieved in 5 min with the use of the vacuum pump in the cell and reservoir tank. Next, the inside of the cell was purged by nitrogen gas. The pressure inside was increased to a preset value of ~3 MPa at first, and subsequently, the cell was depressurized to half that pressure. This operation was repeated three times, after which the inside of the cell was assumed to be filled with nitrogen. After the cell reached the thermal steady-state condition, it was depressurized. Depressurization experiments were performed over the absolute pressure range from 3.11 MPa to 2.12 MPa. The depressurization was controlled by a ball valve connected to the cell by use of the differential pressure between the cell and tank. The differential pressure was controlled by adjusting the pressure of the pressurized reservoir tank with valves. After adjusting the pressure of the reservoir tank to about 2.09 MPa, the ball valve was opened and the pressure of the cell was decreased. The pressure values in the cell and gas reservoir tank were measured by means of a piezoelectric pressure sensor (601CA, Kistler Japan Co., Ltd., Japan) and a charge amplifier (5070A, Kistler Japan Co., Ltd., Japan). The temperature of the gas phase in the cell was measured with a T-type thermocouple (T/T-T-40-2, Ishikawa Trading Co., Ltd., Japan), while the temperature in the gas reservoir tank was measured by means of a PT100 thermal resistance thermometer (R003, CHINO Co., Japan). The pressure and temperature data were collected using a data logger with a sampling rate of 100 ms (NR600, KEYENCE Co., Japan). The transient heat transfer during the depressurization process was visualized using the high-speed phaseshifting interferometer. The frame rate of the camera was set to 1000 fps, and the shutter speed was 1/50000 s. After the interferogram

3. Results and discussion 3.1. Measurement of transient heat transfer by high-speed phase-shifting interferometer 3.1.1. Basic interferograms obtained during depressurization experiment Snapshots of the phase-shifted data obtained by the high-speed phase-shifting interferometer after depressurization are shown in Fig. 4. The visualization area was approximately 1.32 mm in width and 5.84 mm in height, while the spatial resolution was approximately 8.83 μm/pixel. The frame rate was 1000 fps, corresponding to a measurement time resolution of 1 ms. The snapshots in Fig. 4 show temporal changes in the interference fringes due to the transient density distribution occurring in the vicinity of the gas–liquid interface, caused by heat conduction near the Fig. 4. Sequential phase-shifted data around water droplet during depressurization from 3.11 MPa to 2.12 MPa (frame rate of 1000 fps).

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optical datasets at each point were averaged during all three experimental trials, and the maximum sample standard deviation was estimated as 4 ms. 3.2. Quantitative validation of optical path length difference by density variation 3.2.1. Optical path length difference The optical path length difference can be obtained as the integration of the refractive index difference between the test and reference beams. The refractive index variation in the test section is produced by a density change in the cell. The optical path length difference can be obtained from the intensity distribution of the image. The phase shift image has an intensity distribution in the range from 0 to 255 (8 bits) for each interference fringe, and there is a discontinuity point of intensity between the respective interference fringes. The intensity data were fitted at the discontinuous point by adding the intensity (by adding a value of 255) to obtain the continuous intensity distribution. In addition, they were converted into the optical path length difference in the range of 0–λ/2. The optical path length difference is an integral that can be expressed as follows:

Δl = Fig. 5. Two–dimensional axisymmetric numerical model.

+∞

∫−∞

[nt (x , y, z ) − nref ]dz

(2)

where Δl denotes the optical path length difference between the reference and test beams and n the reference index of the gas. Subscripts t and ref denote “test” and “reference”, respectively.

interface. After heat conduction, convection was observed near the interface at approximately t = 0.25 s. Thus, the thermal boundary layer due to pure diffusion by thermal conduction was disturbed after t = 0.25 s in this experimental system. In addition, an interference fringe appeared at the top of visualization area due to the temperature difference between the wall of the cell and depressurized gas. From these measurement results, transient heat conduction around the water droplet interface during adiabatic expansion before convection could be visualized with a high time-resolution of 1 ms.

3.2.2. Numerical simulation by finite element method To assess the reliability of the experimental results for transient heat conduction as the difference in the optical path length, a numerical simulation of the two-dimensional (2D) axisymmetric heat conduction in the gas and liquid phases was performed using the finite element method by means of the COMSOL Multiphysics Version 5.2a package. The calculation was time-dependent and the time step was set to 0.05 s; the model geometry is shown in Fig. 5. In this calculation, the film was not considered because the heat transfer through the film was negligibly small. In addition, convection and evaporation of water were not considered in the calculation. The governing equation of the heat transfer can be expressed as follows:

3.1.2. Initial-state discussions In this study, the time of the start of depressurization was assumed as the time immediately before a slight intensity gradient observed at the interface. No interference fringes were observed under the initial conditions, that is, the thermal steady-state condition around the water droplet before depressurization. However, in this initial state, the obtained phase-shifted images exhibit a slight intensity distribution. This distribution corresponds to the initial noise of the image due to the optical components, and it is not an artifact of the transport phenomenon. Therefore, these initial distributions should be subtracted from a subsequent image with an actual density distribution after a certain time. However, intensity inversion was generated by the noise (that is, the black and white components of the image were switched). Therefore, in order to uniformly subtract the initial distribution, it was needed to select images with equivalent ambient brightness, as shown in the sequential images in Fig. 4. In the study, the ambient color was set as white (the intensity value was 229 as averaged over three experimental trials), and images before decompression were selected as images with the initial distribution. This image was defined as the image just before depressurization (t = 0.000 s, as noted in Fig. 4) and it was not always the same image as that of the start of depressurization (as mentioned in the previous paragraph) and the maximum time difference between the two images was < 5 ms. Further, this initial distribution process may cause a temporal error when selecting images with uniform elapsed times in the three experimental trials. The images were selected every 50 ms. However, when attempting to subtract the selected intensity gradient of the initial image, it was realized that the time difference of the selected image was not just 50 ms. The time instances of these

ρcp

dT T ∂p ⎛ ∂ρ ⎞ = ∇⋅(k∇T ) − dt ρ ∂t ⎝ ∂T ⎠ p

(3)

Here, ρ denotes the density, cp the specific heat at constant pressure, T the temperature, p the pressure, and k the thermal conductivity. In this calculation, the pressure variation was also considered. The details of the pressure variation are described later. The initial temperatures of the water droplet and gas phase were set as Ti = 294 K, and the boundary temperature of the wall was assumed to be isothermal and set to the same temperature as the initial temperature of Ti = 294 K. In this calculation, nitrogen was assumed an ideal gas, and the specific heat ratio was constant at 1.4. Thermal properties such as the specific heat at constant pressure and thermal conductivity of nitrogen and water were assumed as a function of temperature only. As regards the calculation mesh, the maximum element size was set to 0.01 mm and the minimum element size was set to 0.0002 mm in the interior of the droplet and the vicinity of around 3.5 mm of the gas–liquid interface. For the other parts, the maximum element size was set to 0.05 mm and the minimum element size was 0.0002 mm. Finally, with the numerical simulation, the 2D axisymmetric temperature distribution at the gas phase was estimated. The 1D temperature distribution along the axis of the water droplet was extracted from the 2D temperature distribution. This temperature was converted into density information by means of the gas state equation. The 61

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interval of the extracted density values was in accordance with the maximum element size (0.01 mm). Here, it is to be noted that the element size affects the accuracy of numerical density values. In the calculation, the maximum element size in the interior of the droplet and the vicinity of the interface was set to 0.01 mm and the corresponding density was compared with those of 0.025 mm and 0.05 mm. The maximum error was − 0.0173% compared with the density value corresponding to the element size of 0.01 mm. 3.2.3. Lorentz oscillation model The optical information needs to be converted from the density information. The density value can be converted to refractive index by means of the Lorentz oscillation model (developed from the original theory of Lorenz (1869) and Lorentz (1878), which relates the refractive index of a material to its polarizability) [27,28]. The relation between the density and refractive index n can be expressed as follows: 2

1/2

ρ ⎛ ωp ⎞ ⎤ ⎡ n = ⎢1 + ρ0 ⎜⎝ ω02 − ω2 ⎟⎠ ⎥ ⎣ ⎦

Fig. 6. Distribution of optical path length difference in gas phase under various pressures.

(4) maximum element size was 0.01 mm. This result indicates that the Abel transform depends on the element size (that is, the interval of the extracted density value). As the maximum element size decreases, the path difference converges to a certain value. Therefore, the optical path length differences at the interface as obtained by the Abel transform for maximum element sizes of 0.025 mm, 0.05 mm, and, 0.01 mm were plotted. A polynomial approximation curve of the second degree was estimated with respect to this value. The intercept of the curve corresponds to the value of the optical path length difference when the maximum element size is infinitely small, which is the convergence value of the calculation. The maximum difference between this convergence value and the calculated value when the maximum element size was 0.01 mm was −30.1 nm at 0.05 s, and the error of the calculated value with respect to the convergence value was − 4.13%.

Here, ρ0 represents the density of nitrogen gas under standard conditions (0.1013 MPa, 273 K), ω the beam frequency obtained from the wavelength of the He–Ne laser (wavelength: 632.8 nm), ωp the plasma frequency, and ω0 the eigen frequency of the molecule. For nitrogen gas, ρ0, ωp, ω0, and ω are 1.251 kg/m3, 8.255 × 1013 Hz, 3.417 × 1015 Hz, and 4.741 × 1014 Hz, respectively [29,30]. By using Eq. (4) for the 1D density distribution, the refractive index was calculated. Subtracting this value from those of the reference and test beams, the refractive index difference corresponding to the term within the integral in Eq. (2) was calculated. 3.2.4. Abel transform In order to compare the numerical values with the experimental results, the numerical value has to be transformed to the integrated value, which is the optical path length difference, because the measurement data are integrated along the optical axis. Here, the Abel transform, based on the Radon transform was applied to the numerical refractive index difference. The Abel transform is expressed as follows [31]:

E (y ) = 2

∫y

a

ε (r ) r r 2 − y2

3.2.5. Comparisons of experimental and numerical results The experimental distributions of the optical path length differences in the gas phases along the observation line are shown in Fig. 6, and these are compared with the numerical results. The experimental data are thinned out for every three data points in the graph. In the figure, the abscissa values reflect the distance from the interface of the droplet, while the ordinate denotes the optical path length difference. The path difference was the relative difference from the ambient area, which is a gas phase region remote from the interface. The optical path length difference was averaged by the five lines surrounding the observation line for all three experimental runs (15 data points were averaged). The results in Fig. 6 show the density distributions in the gas phase (in the form of the optical path difference). The uncertainty of the measurement data was evaluated by means of the sample standard deviation, and the maximum error was 7.03 × 10− 8 m. The experimental transient variations in the optical path length difference indicate that a transient density distribution existed just before the depressurization. These density changes were caused by heat conduction near the gas–liquid interface by depressurization. The pressure variation in the heat conduction process during depressurization affected the density value in the case of both experiment and numerical calculations. Therefore, the numerical simulation results were corrected by using the assumption of the pressure drop process, as noted in the legend of Fig. 6; path length difference values corresponding to pressures from 0.05 s to 0.20 s were assumed to match the corresponding experimental results, and the most matching pressure change was applied (for e.g., the gas-phase pressure reached 2.21 MPa at 0.20 s), and this pressure was compared with the experimental points for each curve. The presumed pressure change was different from that of the measurement value obtained by means of pressure sensor. In this study, the

dr (5)

Here, ε(r) denotes the refractive index difference along the radial direction of the droplet, which was calculated by numerical simulation. Further, a denotes the integration interval, and it is the apparent radius of the sphere corresponding to the symmetric function. Parameter E(y) denotes the integrated data along the optical path shown in Fig. 1. For the Abel transform, it was needed to calculate the apparent radius of the sphere. In this experiment, the shape of the water droplet was assumed as a partial sphere. From the interferometry images, the height of the droplet was estimated as 2.04 mm, and the apparent radius of the sphere was estimated as 4.50 mm, with a corresponding volume of 50.00 μL. Finally, Eq. (5) was applied to the numerical refractive index difference, and the optical path length difference was calculated. Here, it is noted that the Abel transform can be applied to axisymmetric distributions only. After about 0.25 s from the start of depressurization, the interference fringe was not axisymmetric due to the beginning of convection, and therefore, only the transient heat conduction was analyzed, and these calculations did not include convection. The element size of the numerical calculation can also affect the Abel transform. With regard to the optical path length difference obtained by means of the transform, as in the case of density, the path difference for the maximum element sizes of 0.025 mm and 0.05 mm was compared. In the vicinity of 0.5 mm from the interface, the maximum error was −13.8% as compared with the value when the 62

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Acknowledgement

depressurization was manually controlled by the ball valve, and the measurement delay of the pressure sensor was 11 μs. Further, the highspeed camera and pressure sensor were not synchronized, and the true decompression point was unknown. These factors may result in the delay of pressure transmission for the actual depressurization process inside the cell. The capabilities of measurement and synchronization for high-speed pressure variation are significant for further studies in this direction. A close agreement was observed between the experimental and numerical distributions of the optical path length difference for the assumed pressure variation. These results indicated that before convection (~ t = 0.25 s), the heat transfer by the conduction was purely through diffusion phenomena and that the heat conduction equation considering pressure change could be applied in these very short time intervals. From this result, using the proposed measurement system and numerical methods, the pressure variation in the gas phase during depressurization can be estimated. In addition, the transient heat transport phenomenon and temperature gradient near the interface can be reproduced and identified as optical values with the numerical calculation. In this study, the temperature of the gas phase was measured with a thermocouple. This measured temperature was used to monitor the temperature of the gas phase during the experiment and to determine the initial temperature of the experiment. In addition, unsteady temperature changes during decompression could also be measured with this thermometer. However, the measurement delay of this thermocouple is about 0.2 s (time constant is about 0.03 s), and it is difficult to measure the temperature change at the time scales measured in this experiment. Thus, it is believed that the measurement method proposed in this paper is effective for high-speed measurements of heat transport phenomena at or near the interface with respect to probe measurements.

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4. Conclusion In this study, the transient density distribution caused by heat conduction near a gas–liquid interface during depressurization was quantitatively measured by means of a high-speed phase-shifting interferometer. The time resolution of the proposed measurement system was 1 ms. The proposed approach allows for the measurement of highspeed transient heat conduction before convection in the vicinity of a gas–liquid interface. In the study, the experimental results agreed well with numerical simulations considering pressure variation. The following conclusions were drawn. (1) As per visualization of the transient density variation in the form of interferometric fringes, it was observed that the thermal boundary layer due to pure diffusion by thermal conduction was disturbed at around t = 0.25 s, and subsequently, convection near the interface was observed under experimental conditions. (2) The transient heat transfer was purely due to diffusion by thermal conduction at about 0.20 s after depressurization, and it was concluded that the heat conduction equation considering pressure change could be applicable to this experimental system. This conclusion was supported by the agreement between numerical calculation and experimental results. The proposed measurement system suffers from issues such as measurement and synchronization for high-speed pressure variation; these problems will form the focus of future studies. With these issues addressed, the proposed measurement and analytical method can enable the visualization of high-speed transport phenomena and quantitative measurements of the interfacial temperature and concentration variation with phase transitions for engineering and industrial applications such as energy conversion, bubble manipulation, and cooling systems. 63