Experimental investigation of the evaporation rate of

International Communications in Heat and Mass Transfer 77 (2016) 190–194

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Experimental investigation of the evaporation rate of supercooled water droplets at constant temperature and varying relative humidity☆ S. Ruberto ⁎, J. Reutzsch, B. Weigand Institute of Aerospace Thermodynamics, University of Stuttgart, Pfaffenwaldring 31, 70569 Stuttgart, Germany

a r t i c l e

i n f o

Available online 16 August 2016 Keywords: Experiment Optical levitation Evaporation Supercooled Water droplet

a b s t r a c t Supercooled water droplets are found in clouds at high altitude. They are exposed to very low temperatures and high relative humidity. The phase change of supercooled water droplets is an interesting heat and mass transfer problem. It is of paramount interest to understand droplet dynamics in clouds and hence, rain, snow and hail generating mechanisms. Therefore, in this work freely suspended supercooled water droplets are investigated experimentally. We present the evaporation rate at a constant temperature of 268.15 K and six different relative humidities (28 % − 89 %). It is found, that the evaporation rate is linear dependent on the relative humidity. © 2016 Elsevier Ltd. All rights reserved.

1. Introduction Water droplets in clouds at high altitude are exposed to very low temperatures and high relative humidity as noted by Pruppacher and Klett [1]. Despite subzero temperatures, the droplets can still be liquid and are then called supercooled. The phase change of supercooled water droplets (SWDs) and the interaction between liquid and already frozen droplets is of paramount interest to understand rain, snow and hail generating mechanisms. For this purpose, freely suspended SWDs are investigated experimentally. This has been done by using a levitation technique, where the droplet is trapped in a test chamber. In literature different levitation techniques can be found. SWDs at different subzero temperatures were investigated by Roth et al. [2] with optical levitation and recently by Tong et al. [3] in an electrodynamic trap. The latter present results of evaporating droplets in mostly dry ambient. In this work we present optically levitated single water droplets. The optical levitation is a stable trap first investigated by Ashkin [4]. During the levitation the scattered light, also known as Mie scattering, in the forward hemisphere of SWDs is observed. A benefit of the optical levitation is that no further laser needs to illuminate the droplet. With this non-intrusive measurement technique, described by Roth et al. [2] and Wilms [5] the droplet diameter and the evaporation rate can be derived. The investigated size of SWDs in our study is around 50 μm and is in the magnitude of droplet diameters appearing in clouds mentioned by Pruppacher and Klett [1]. In the experiments carried out in this work, the influence of the relative humidity on the evaporation of SWDs is systematically investigated. To the best of the authors'

☆ Communicated by W.J. Minkowycz. ⁎ Corresponding author. E-mail address: [email protected] (S. Ruberto).

http://dx.doi.org/10.1016/j.icheatmasstransfer.2016.08.005 0735-1933/© 2016 Elsevier Ltd. All rights reserved.

knowledge, the only experimental observation of evaporating SWDs with known relative humidity is given by Tong et al. [3]. However, their experiments were done in dry ambient and at different subzero temperatures and so, the experimental results cannot be compared directly. Therefore, our paper is focused on the effect of humidity on the evaporation of SWD. 2. Material and methods This chapter begins with a description of the experimental setup for trapping SWDs, as well as the measurement setup. It is followed by a section describing the evaluation methods and ends with a brief comment on how to calculate the evaporation rate. 2.1. Experimental setup A schematic of the experimental setup is shown in Fig. 1. The observation chamber, with its inner dimensions 9 × 14 mm2, is cooled down with a cryostat (Huber, Unistat 815w). A controlled flow (Bronkhorst, El-Flow-MFC) of almost dry nitrogen (Air Liquid, 99.999 %), provided by a pressurised gas cylinder, passes from the top to the bottom of the chamber. It is important to have an aerosol-free flow, because otherwise freezing could be triggered. The humidity of the nitrogen flow can be adjusted with an arrangement of precise mass flow meters and an evaporator (Bronkhorst, CEM). The flow rate in the presented work is about V˙ ¼ 40 mln =min, leading to a velocity of u = 0.0055 m/s in the chamber. This results in very low Reynolds numbers in the chamber of ReD = 0.02. This flow is only needed in order to guarantee that the chamber conditions are always similar and evaporated mass is transferred out of the chamber, preventing saturated chamber conditions. A single water droplet is generated with an in-house build dropleton-demand generator [6]. The droplets in this study have an initial

S. Ruberto et al. / International Communications in Heat and Mass Transfer 77 (2016) 190–194

Nomenclature BY D ,D0 cp D12,R f n Nu

Y

−Y

1;s Spalding's mass transfer number, BY ¼ 1;∞ Y 1s −1 [–] Diameter [m] Specific heat capacity [J/(kg K)] Diffusion coefficient at reference temperature [m²/s] Focal length [m] Real part of refractive index [–] Nusselt number, Nu ¼ αkD [–]

M1 , M2 Molar mass of droplet liquid and ambient gas [mol/kg] p Ambient pressure [Pa] p1,s ,p1,∞ Pressure at droplet surface and far away from droplet [Pa] Saturation vapour pressure [Pa] pv Re

Reynolds number, Re ¼ ρ uη D [–]

Sh t T Tw ,T∞ TR ,Ts

Sherwood number, Sh ¼ βD12D [–] Time [s] Temperature [K] Wet-bulb temperature and ambient temperature [K] Reference temperature and temperature at droplet surface [K] Velocity [m/s] Flow rate at standard conditions T = 268.15 K, p = 1013.25 hPa [ln/min] Molar fraction at droplet surface and far away from droplet [–] Mass fraction at droplet surface and far away from droplet [–]

u V_ X1,s , X1,∞ Y1,s , Y1,∞

Greek symbols β Evaporation rate [m²/s] Δ Accuracy, range [–] Δθ Angular fringe distance [°] θ Mounting angle of camera [°] λ Laser wavelength [m] Density gas vapour mixture and density of liquid dropρR ,ρl let [kg/m³] pH O φ Ambient relative humidity, φ ¼ p2 100 % [%]

191

of about 10 mm. Here, a SWD is trapped and its position is monitored by a Position Sensitive Device (PSD). A humidity sensor (Sensirion SHT-71, Δφ ±3 %, ΔT ± 0.5 K) in the observation chamber controls the temperature and relative humidity φ at the entry. In addition, the relative humidity is measured at the levitation position. Therefore, a second equal humidity sensor is driven into the chamber before and at the end of a measurement series. In doing so, it can be assured that the ambient conditions are stable during the experiment. While the droplet is levitated, it scatters the laser light. As shown in Fig. 1 the Mie scattering is recorded by a CCD line camera (E2V, AVIIVA SM2, 2048 pixels) in the forward hemisphere under an angle of observation of θ = 60° in a range of ± 10°. Therefore, the light is collected by a cylindrical lens (f = 0.08 m), which is adjusted right in the focal plane of the camera in order to provide a Fourier transformation. This means, rays scattered under a certain angle will nearly be independent of the position of the droplet in the chamber and hence always illuminate the same camera pixel. Then, with a previous made calibration, the obtained recordings of intensity over pixel are transformed to intensity over scattering angle. According to Fig. 1, the camera is connected to a frame grabber card (Matrox, Helios CL) and triggered externally (LabSmith, LC880) with 5000 Hz. Once a droplet reaches the observation position, an oscilloscope, which samples the PSD, starts the recording. Thereby, a defined, arbitrary delay of 150 ms is introduced, so that the drop is trapped stable and cooled down to ambient temperature. A current recording contains 60,000 lines, which corresponds to a measurement duration of about 12 s. Besides the Mie scattering, also the droplet position is saved. This is necessary for the evaluation, to ensure that the droplet is exposed to a constant temperature. 2.2. Evaluation methods The following section describes the evaluation of the scattered light in the forward hemisphere. This method for droplet sizing is of advantage, as it is non-intrusive. In the forward hemisphere the scattered light has a regular pattern, consisting of bright and dark fringes as shown in Fig. 2. According to Glantschnig and Chen [8], this fringe spacing is mainly dependent on the droplet diameter, while being less dependent on the refractive index. Hence, based on geometrical optics they derived a relation for the diameter

v

D¼ diameter D0 = 50 μm ± 6 μm. To avoid freezing of the droplets due to nucleation by impurities, water for laboratory analysis (Merck LiChrosolv®, spec. conductance ≤1 μS/cm) is used. With a Nd:YAG laser (Laser Quantum, OPUS 532 nm, 5 W), which is directed through the chamber, single droplets are optically levitated. Therefore, a lens with a focal length of f = 0.1 m, which can be driven in height (PI, Linear Positioning Stages), focusses the laser. Thus, the droplet can be positioned within the chamber by moving the focusing lens. It is important, that the light is not absorbed by the investigated substance. So the corresponding absorption coefficient of water for the chosen laser wavelength is low [7]. In fact, it is almost the minimum in the absorption spectrum of water. The droplet should not heat up notably when considering additionally the small size. In order to get a droplet levitated it has to fall along the centre of the laser beam. Therefore, two precise motion stages (PI, linear positioning stages) adjust the position of the droplet generator. Typically, the laser power in the presented experiments is about 2.4 W±0.2 W. To avoid a heating of the observation chamber, it is necessary to cool down the laser trap. For the investigations, the ambient conditions in the chamber are controlled as follows. A thermocouple (Omega, Type K, unsheathed, diameter 0.125 mm, measurement accuracy ΔT ± 0.1 K) is driven into the chamber to measure the temperature along the height. A U-shaped temperature profile develops, which has a constant temperature region with a length

−1=2 −1 2λ : cosðθ=2Þ þ n sinðθ=2Þ 1 þ n2 −2 n cosðθ=2Þ Δθ

ð1Þ

Here the variables are the mounting angle of the camera θ, the laser wavelength λ, the angular fringe distance Δθ and finally the refractive index n. Duft and Leisner [9] measured the refractive index for a wide range of supercooled temperatures. According to their data the refractive index at T = 268.15 K is n = 1.333. Note, that only the real part of the refractive index is considered, as absorbance is neglected. Next, the implemented evaluation algorithm is briefly explained. To remind, the saved data is a matrix with the size 2048 (pixel) and 60,000 (amount of lines). Each single line contains the intensity distribution over the observed range of scattering angle. At first, every line is filtered and the angular distance Δθ between two neighbouring maxima is determined. As there are several maxima, an average angular distance Δθ is calculated. Then, Eq. (1) is applied to calculate the droplet diameter. It should be pointed out, that the time, at which the line is grabbed, is well known, because the camera is triggered externally. The accuracy of the diameter for the current evaluation is ± 2 % in the range 20 μmb D b 60 μm. For D b 20 μm the accuracy starts to decrease. This is due to less bright maxima and therefore increasing noise. Furthermore, the amount of maxima in the observed angular range decreases, leading to more uncertainties in the angular distance Δθ. Fig. 3 shows the diameter evolution over time, having apparently a parabolic trend. It can be noted that the diameter oscillates. This oscillating of the diameter is not physical, but is related to the appearance of morphology

192


Fig. 1. Schematic of experimental setup for trapping SWDs and recording the Mie scattering.

dependent resonances (MDR) [10], which get more pronounced for smaller diameters and decrease the accuracy additionally. For validation, the evaluation code has been adopted to calculations performed with the Mie scattering code presented by Bohren and Huffman [11]. Since the evaporation rate shall be derived, at this point it is assumed that the D²-law applies to the evaporation of a supercooled water droplet [2]. Then the evaporation rate β = − ∂D2/∂t describes the surface regression rate. Integrating the left hand side with the initial diameter D(t= 0 s) = D0 and dividing by the squared initial diameter D0 lead to

D D0

2 ¼ 1−β

t D0 2

:

ð2Þ

To remind, the measurement is started with a delay to ensure a stable position of the droplet. Hence, the initial diameter D0 is extrapolated from a linear regression through D2 over time t. Fig. 4 shows a linear regression according Eq. (2) to derive the evaporation rate.

instance Mills [12]. The assumptions of the model are mainly fulfilled in the current experimental setup. It can be assumed that a quiescent atmosphere is given, as the Reynolds number of the droplet is ReD ≪ 1 (Stokes flow). Also the droplet is of pure fluid, isolated and the process is isobaric. A Nusselt number Nu ≈ 2, as well as a Sherwood number Sh ≈ 2, is obtained with the correlation of Ranz and Marshall [13]. For the calculations reliable substance properties are crucial, as they have a major impact on the results. In literature several formulations and fittings to experimental data can be found. However, most of them are found to derive similar values. Thus, it is necessary to define a reference temperature TR ¼ Ts þ

1 ðT ∞ −T s Þ: 3

ð3Þ

Next the evaporation rate is calculated according to the D²-law. This model predicts that the square of the droplet diameter decreases linearly with time. The classical D²-law is described in many textbooks for

This equation follows the “1/3 scheme” proposed by Hubbard et al. [14]. Thereby, the subscript s denotes a state indefinitely close to the droplet surface, while ∞ is a condition far away from the droplet. During the evaporation the droplet cools down to the wet-bulb temperature. This temperature is below the ambient temperature and depends on the relative humidity and the temperature of the ambient. The wet-bulb temperature can be calculated as described by Mills [12] or Pruppacher and Klett [1]. In both cases the wet-bulb temperature is

Fig. 2. Recorded scattered light of a trapped SWD. The flickering of the maxima is caused by MDRs.

Fig. 3. Evaluated droplet diameter D in μm over the recorded time t in s for T=268.15 K and φ=47 %. The zoom shows the scattering of the diameter caused by MDRs.

2.3. Calculation of the evaporation rate


Fig. 4. Dimensionless squared diameter (D/D0)2 over t/D20 at T=268.15 K and φ=47 % and linear regression to derive the evaporation rate β.

iterated whereby, the results differ less than 0.5 K. According to Wilms [5] the evaporation rate

193

Fig. 5. Evaluated evaporation rates for six relative humidities at T=268.15 K. The solid line is a weighted linear fit through the experimental data. Calculated values are plotted as dashed-line. The square represents the result of a numerical simulation with FS3D.

and the relative humidity φ. Finally, the evaporation rate β can be calculated from Eq. (4).

humidity in the chamber. Nevertheless, the Reynolds number of the droplet is still ReD b b 1 and the value apparently fits the trend of the others. The depicted values show a linear decrease with increasing relative humidity. As expected it leads to zero evaporation at φ = 100 % relative humidity. The error bars indicated in the graph show that the measurements are well reproducible. They have been calculated as the standard deviation of the mean evaporation rate and multiplied with a Student factor for 95 % confidence. The relative uncertainties range between ± 0.5 % and ± 3.0 % of the mean value. In Fig. 5a weighted linear fit (solid line), as well as the calculated values according to Section 2.4 (dashed line), are plotted. Although both lines do not match exactly, the calculations indicate that the evaluated evaporation rates are plausible. It should be pointed out, that the calculations are sensitive to the ambient temperature and especially the assumed wetbulb temperature on the droplet surface. In addition, a direct numerical simulation (DNS) at a relative humidity of φ =0 % was carried out with our in-house code FS3D, which uses the volume-of-fluid (VOF) method [18]. The simulation was set up according to the experiment and is depicted by the square in Fig. 5. The result is in good agreement with the experimental data (solid line). In future the experimental data in this research work will be used to further validate numerical simulations of FS3D. In Fig. 5 the experiments show a slightly higher evaporation rate β compared to the D2-model. However, there is no evidence that the observation chamber heats up during the experiment, which would alter the ambient temperature. In case it would, the evaporation rate within a measurement series should increase. This could not be stated, as the measurements show a normal scatter. Another explanation could be that the droplet heats up due to absorption. To remind, the absorption coefficient of water for the present wavelength and the droplet size are minimal. It was estimated, that a droplet at constant initial diameter could heat up by not more than 0.5 K due to absorption. Actually, this value represents an upper limit, because the absorption lowers, while the absorbing length, namely the diameter, decreases due to evaporation. Also the time between two measurements is sufficiently long for the ambient to restore, in order that a previous evaporating droplet does not alter the relative humidity.

3. Results and discussion

4. Conclusions

In the following the measurement results performed at a temperature T = 268.15 K are presented. Fig. 5 depicts the evaluated evaporation rate β for six experiments at the given relative humidity and temperature. The measurement series have been carried out on different days. Thereby, the circles represent the mean value of 10 measurements, at a flow rate of V˙ ¼ 40 mln =min. However, the evaporation rate for the relative humidity φ = 28 % was measured at a flow rate of V˙ ¼ 80 mln =min . This was done in order to achieve a lower relative

In this paper the experimentally derived evaporation rates of SWDs at constant temperature and six different relative humidities are presented. The evaporation rate depends linearly on the relative humidity. A comparison with calculations where the D²-law is assumed to be valid suggests that the data is plausible. Also a numerical simulation with the in-house DNS code FS3D was in good agreement with the D2-model and the experimental data. Further experiments are planned at lower temperatures, in order to get a parametrisation for

β¼

8ρR D12;R ln ð1 þ BY Þ ρl

ð4Þ

can be derived by applying Fick's first law of diffusion. Here ρR is the density of the mixture of the surrounding gas and vapour. The density of the mixture is calculated considering nitrogen as ideal gas at the reference temperature TR. An equation to calculate the diffusion coefficient D12,R of water vapour into air can be found in Pruppacher and Klett [1]. Supercooled water density ρl is given by Wölk and Strey [15]. The last variable is Spalding's mass transfer number BY ¼

Y 1;∞ −Y 1;s : Y 1;s −1

ð5Þ

Y1 denotes the mass fraction of the water, with Y1,s being at the droplet surface and Y1,∞ being far away from the droplet. It is assumed that at the droplet surface the partial pressure of the water vapour p1,s equals the saturation vapour pressure pv. Therefore, it can be calculated as a function of the surface temperature Ts. Murphy and Koop [16] provide an equation for the saturation vapour pressure pv. According to Pruppacher and Klett [1] despite the small droplet size, the effect of curvature becomes important only starting from radii ≤0.1 μm, so in our case there is no need to correct the saturation pressure with the Kelvin p1;s p

equation. Next, by assuming an ideal gas, the molar fraction X 1;s ¼

follows from Dalton's law of partial pressure, with the ambient pressure p. Thus, the mass fraction is finally given by Ys ¼

X 1;s M 1 ; X 1;s M1 þ 1−X 1;s M2

ð6Þ

with M1 the molar mass of water and M2 the molar mass of nitrogen. The properties of nitrogen are calculated with NIST-REFPROP [17]. The mass fraction Y∞ is calculated likewise Eq. (6) with X 1;∞ ¼

p1;∞ ðT ∞ Þ p

φ

194


the dependency of the evaporation rate on temperature and relative humidity. Finally, the experimental data will be used to validate numerical simulations, which will be done with the DNS code FS3D [18]. Acknowledgments This work is done within the Collaborative Research Center Transregio 75 (SFB-TRR 75) “Droplet Dynamics Under Extreme Ambient Conditions”. The authors acknowledge the financial support of the German Research Foundation (DFG) TRR75/2. Also, the authors would like to thank N. Roth for implementing the Bohren and Huffman code and calculating the Mie scattering. References [1] H.R. Pruppacher, J.D. Klett, Microphysics of Clouds and Precipitation, second ed. Kluwer Academic Publishers, Dordrecht, 1997. [2] N. Roth, K. Anders, A. Frohn, Determination of size, evaporation rate and freezing of water droplets using light scattering and radiation pressure, Part. Part. Syst. Charact. 11 (3) (1994) 207–211. [3] H.J. Tong, B. Ouyang, N. Nikolovski, D.M. Lienhard, F.D. Pope, M. Kalberer, A new electrodynamic balance (EDB) design for low-temperature studies: application to immersion freezing of pollen extract bioaerosols, Atmos. Meas. Tech. 8 (3) (2015) 1183–1195. [4] A. Ashkin, History of optical trapping and manipulation of small-neutral particle, Atoms, and Molecules, IEEE J. Sel. Top. Quantum Electron. 6 (6) (2000) 841–856.

[5] J. Wilms, Evaporation of Multicomponent Droplets(PhD thesis) Universität Stuttgart, 2005. [6] A. Frohn, N. Roth, Dynamics of Droplets, Springer, Berlin, 2000 80–83. [7] G.M. Hale, M.R. Querry, Optical constants of water in the 200-nm to 200-Mm wavelength region, Appl. Opt. 12 (3) (1973) 555–563. [8] W.J. Glantschnig, S.H. Chen, Light scattering from water droplets in the geometrical optics approximation, Appl. Opt. 20 (14) (1981) 2499–2509. [9] D. Duft, T. Leisner, The index of refraction of supercooled solutions determined by the analysis of optical rainbow scattering from levitated droplets, Int. J. Mass Spectrom. 233 (1-3) (2004) 61–65. [10] B.R. Johnson, Theory of morphology-dependent resonances: shape resonances and width formulas, J. Opt. Soc. Am. A 10 (2) (1993) 343–352. [11] C.F. Bohren, D.R. Huffman, Absorption and Scattering of Light by Small Particles, Wiley, New York, 1998. [12] A.F. Mills, Mass Transfer, second ed. Prentice Hall, Upper Saddle River, New Jersey, 2001. [13] W.E. Ranz, W.R. Marshall, Evaporation from drops part I, Chem. Eng. Prog. 48 (3) (1952) 141–146. [14] G.L. Hubbard, V.E. Denny, A.F. Mills, Droplet evaporation: effects of transient and variable properties, Int. J. Heat Mass Transf. 18 (9) (1975) 1003–1008. [15] J. Wölk, R. Strey, Homogeneous nucleation of H2O and D2O in comparison: the isotope effect, J. Phys. Chem. B 105 (47) (2001) 11683–11701. [16] D.M. Murphy, T. Koop, Review of the vapour pressures of ice and supercooled water for atmospheric applications, Q. J. R. Meteorol. Soc. 131 (2005) 1539–1565. [17] E.W. Lemmon, M.L. Huber, M.O. McLinden, NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties-REFPROP, Version 9.1, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, 2013. [18] J. Schlottke, B. Weigand, Direct numerical simulation of evaporating droplets, J. Comput. Phys. 227 (10) (2008) 5215–5237.