A MODIFIED METHOD TO GENERATE STABLE DROPLET CLUSTERS LEVITATING ABOVE THE LOCALLY HEATED WATER SURFACE

Transcription

1 A MODIFIED METHOD TO GENERATE STABE DROPET CUSTERS EVITATING ABOVE THE OCAY HEATED WATER SURFACE Alexander A. Fedorets a and eonid A. Dombrovsky b a Tyumen State University, Semakov 0, Tyumen, 65003, Russia b Joint Institute for High Temperatures, Krasnokazarmennaya 7A, Mosco, 6, Russia Abstract A modified laboratory installation to generate levitating clusters of droplets above the heated ater surface is presented. The laser-heated sitall plate ith an opaque graphite coating is used as a substrate for the ater layer hich produces a levitating droplet cluster. The cluster is stabilized ith the use of infrared heating as as recently demonstrated by the authors. A combined experimental and computational method is developed to obtain the -D quasi-steady temperature field in sitall plate and ater and also to calculate the radial profile of heat flux at the ater surface. Both the temperature and heat flux at the ater surface are important boundary conditions for the upcoming computational modeling of cluster formation. Introduction The levitating droplets hich form the clusters of regularly positioned spherical droplets above the heated ater surface ere observed for the first time about telve years ago []. To typical photographs of droplet clusters are presented in Fig.. Note that the cluster can move along the ater surface as a hole (ithout any changes in its structure). Figure. The photographs of the levitating droplet cluster (vie from above) taken ith the time interval about s beteen shots. Corresponding author. Tel ldombr@yandex.ru 4-

2 The behavior of the levitating single droplets and their clusters in the upcoming flo of vapor of various liquids and entrained air has been studied experimentally and the laboratory observations. The stages of this study ith the relevant archival publications are not described in the present paper. One can find additional references to early papers in []. At the same time, more recent papers based on specific infrared properties of ater should be mentioned. First of all, the absorption band of ater at the avelength about 3 μm [3, 4] as used in paper [5] to retrieve the surface temperature of the upper surface of various ater droplets from the measurements of their brightness temperature. The next step related ith infrared heating of the cluster droplets enabled us to transfer from the passive observation to managing the process [6, 7]. Moreover, it as experimentally demonstrated that ide-range infrared irradiation can be used to stabilize the droplet cluster by preventing the ordinary groth of single droplets because of predominant steam condensation on their surface. The objective of the present study is to-fold: () to suggest a modified experimental method to generate stable droplet clusters above the locally heated ater surface and () to develop a combined experimental and computational procedure to obtan the radial distribution of heat flux to the ater surface. Experimental procedure to generate levitating droplets The schematic presentation of experimental installation presented in Fig. a includes the folloing designations: is the droplet cluster, is the horizontal ater layer, 3 is the cylindrical cavity in the massive duralumin plate ith the central orifice, 4 is the sitall plate/substrate ith an absorbing coating containing fine graphite particles at the irradiated side, 5 is the annular groove filled ith epoxy glue for fixing the substrate, 6 is the laser beam used to heat the substrate, 7 is the special plate to reflect a part of laser radiation, and 8 are the sources of infrared radiation used to stabilize the cluster. a b Figure. (a) The schematic of the experimental installation and (b) the radiative flux distribution across the laser beam. 4-

3 The laser heating suggested in [6] has several advantages as compared ith the electric heating used in previous studies [,, 5]. Much loer inertia of heating and very smooth surface of sitall substrate are the main of these advantages. Note that smooth surface enabled one to avoid unfavorable centers of nucleation and the resulting collapse of the droplet cluster due to pop up gas bubbles []. A semiconductor laser ith the avelength of 808±0 nm as used in the experiments. The laser poer can be regulated from zero to W = 600 mw. A reflected part of the laser beam as used to measure the current laser poer during the experiment. The reflecting plate as also used before the experiment to analyze a distribution of radiative flux across the beam. Figure b indicates that cross section of the beam looks like a square ith the side length of δ = 0.6 mm, and the spatial distribution of radiative flux is almost uniform. To simplify the calculations, an equivalent circular beam of the same poer as considered. The beam radius as estimated as follos: r = δ π mm () = Infrared heating of cluster as made by four radiation sources EK-850 produced by the firm Helioorks and characterized by almost blackbody radiation at temperature T =3K. To arrange relatively uniform irradiation of the droplet cluster, these radiation sources ere placed symmetrically ith respect to the vertical axis of the cluster and the radiative flux as directed at the angle 45 to the axis [7]. The stereomicroscope Zeiss Discovery.V8 equipped by a video camera as used in continuous observations of the ater cluster behavior. The thickness of ater layer, d, as controlled ith error ± µm using the laser triangular sensor RF603-5/ made by the company Riftec. The value of d = 50 µm as carefully supported to be constant during the experiment. Note that in the experiments ith infrared stabilization of the cluster (Fig. 4) the value of d = 300 µm as used. The distilled ater containing natural micro admixtures of surface-active substances as used in the experiments. The effect of these admixtures is favorable because they prevent thermal capillary flos on ater surface [8]. T, o C W 08.6 mw 4.4 mw Figure 3. Typical temperature profiles at the open ater surface. Radial profile of temperature at the ater surface as measured using the thermal imager Flir A655sc ith the avelength range from 7.5 to 4 μm, the matrix containing

4 640х480 pixels, and the temperature resolution of 50 mk. The lens Close-up.9x as sufficient to obtain infrared images ith 50х50 μm size of a pixel. The frequency of the recording as equal to 50 frames per second. To typical temperature profiles are presented in Fig. 3b. The use of external infrared heating to obtain stable cluster of droplets The effect of infrared irradiation observed in [7] is illustrated in Fig. 4a here time variation of the droplet surface area, S = 4πa at laser poer of W = 63 mw and variable poer of infrared radiation W is presented. The duration of every video record as equal to 60 s including () the initial period of t < 0 s ithout infrared irradiation, () the active period of 0 < t < 40 s ith infrared heating the droplet cluster, and (3) the last period of 40 < t < 60s ithout infrared heating. 3.0 S _ Figure 4. Time variation of relative area of droplet surface S at the initial droplet radius of a = 7. µm: W 0, 9. 3mW, 3 9. mw; d 300 μm [6]. = t, s = Every part of the time dependence ( t) S in Fig. 4 is almost linear and the rates of the droplet groth at the first and third periods are the same. It means that the droplet groth can be ell described by the knon d-squared la [9] or its modification called the elliptic la [0]. This appears to be true also in the case of infrared irradiation but the irradiation leads to a significant decrease in the groth rate of ater droplets. The generalized experimental results can be found in recent paper [6]. It as shon that there is a simple linear dependence of ( ) W S 0 and one can obtain the value of W hich gives zero value of S 0. It as shon that the ratio W W is eakly sensitive to the laser poer and can be estimated as follos [6]: 0 0 W = W W 6% () Approximate relation () is convenient to obtain the required infrared radiation poer to stabilize the levitating clusters at different conditions of the laser heating of ater layer. 4-4

5 Temperature field calculations It is difficult to formulate correctly thermal boundary conditions on the open surface of the evaporating ater layer. Therefore, the temperature measurements for this surface are used to complete the problem statement. The suggested approach can be treated as a combined method hich enables one to obtain the temperature field in sitall plate and ater layer and also the radial distribution of heat flux at the open ater surface. The steady-state temperature field in the computational region including the central parts of both the sitall substrate and the ater layer is described by the folloing axisymmetric boundary-value conduction problem: T k z z z = 0, + T kr r r r ( T T ) = 0 T h e + qrad, r < r k = z 0, r r < R T r = R, = 0 r < r < R 0 < z < D = d + d (3a) exper z = D, T = T ( r) (3b) T r = 0, = 0 r here R = 6 mm is the radius of the computational region, ds = mm is the thickness of sitall plate, the subscrips s and refer to sitall and ater, respectively. The thermal conductivity is desribed by the folloing step-ise function: k ( T ) k = k s ( T ), ( T ), 0 < z < ds d < z < D The thermal conductivity of sitall as assumed to be independent of temperature and equal to k =. s 4 Wm- K - []. The dependence k ( T ) as obtained by a linear interpolation beteen the knon values of k = Wm- K - at T = 73 К, Wm - K - at 33К and Wm - K - at 373К. Obviously, the laser irradiation takes place in the narro region of r < r only. It as shon that the use of constant value of the radiative flux, qrad = W ( πr ), is a good approach to the experimental conditions. The value of h = 5 Wm- K - obtained as an estimate for the stable stratification of air under the heated coating of the sitall plate as used in the calculations. Note that possible error in the h value has a negligible effect on the computational results. The heat flux to be determined is calculated as follos: s z= D s (3c) (3d) T q ( r, D) = k (4) z The ordinary time-dependent technique ith the uniform initial temperature T = 0 95 K and the so-called alternating-direction implicit finite-difference method [ 4] ere employed to solve numerically the problem (3a-d). The same home code has been

6 recently used in calculations of temperature field in the case of volumetric absorption of near-infrared radiation in human tissues [5]. The uniform rectangular grid ith 70 axial and 00 radial intervals as used in numerical calculations. The numerical results are presented in Figs One can see in Fig. 6 that calculated temperature of ater at the surface of sitall plate is very close to the saturation temperature Ts = 373 K. It means that the ater layer thickness should be reduced by using use the laser poer greater than about 400 mw. The temperature fields (Fig. 5) and the radial profiles of temperature and heat flux at the ater surface (Figs. 3 and 7) at different values of laser poer are practically similar to each other. These observations make reasonable to consider possible simplification of estimates at intermediate values of W. 6 a 6 b 5 4 sitall plate ater layer T, o C sitall plate ater layer T, o C x, mm x, mm Figure 5. Calculated temperature fields at (a) W = mw and (b) W = 4. 4 mw. 4-6

7 T, o C W 08.6 mw 4.4 mw sitall plate ater layer x, mm Figure 6. Temperature profile along the axis of the computational region. q, kw/m 00 W 08.6 mw 4.4 mw Figure 7. Radial profiles of heat flux at the open ater surface. Possible use of approximate similarity of temperature fields in analytical estimates Strictly speaking, the conduction problem in not linear because of temperature dependence of thermal conductivity of ater and not quite similar temperature profiles of ater surface at various values of laser poer. Typical profiles and heat flux at the ater surface are compared in Fig. 8 using the folloing dimensionless functions: T ( ) ( r) T ( R) θ r = ( ) T ( 0) T ( R) q ( r) ( 0) ξ r = (5) q 4-7

8 .0 θ 0.9 a.0 ξ 0.8 b Figure 8. Profiles of dimensionless (a) temperature and (b) heat flux at the ater surface: W = mw, W = 4. 4 mw. One can see that the difference beteen the profiles of the above introduced dimensional profiles is rather small even at very different values of laser poer. It means that one can use an analytical estimate of ater temperature and heat flux for intermediate values of W using a linear interpolation beteen the profiles for minimum and maximum values of laser poer: min W W θ = θ ( ) min + θ max θ min (6a) P min max P P P W W min W W ( x ) max min max x = x min + x (6b) P P P P W W One needs also a relation beteen the maximum values of temperature and heat flux at max max the ater surface, T and q, and the laser poer to transfer from the dimensionless values of θ ( r) to the real temperature profile. An analysis of the experimental data enabled us to suggest the folloing analytical approximation for the dependence of max T W : ( ) T 954 min max = T W 0. W W W W = W 00mW (7) A similar relation can be easily obtained from the calculations for the dependence of max q ( W ). The errors of the resulting approximations are insignificant, and one can obtain reliable estimates of the laser poer effects on the key parameters of the process under investigation. = Conclusions A modified laboratory installation to generate levitating clusters of droplets above the heated ater surface as presented. The laser-heated sitall plate ith an opaque graphite coating as used as a substrate for the ater layer hich produces a levitating droplet cluster. The cluster as stabilized both thermally and mechanically ith the use of infrared heating as as recently demonstrated by the authors. A combined experimental and computational method as developed. The radial temperature profile at the open ater surface as measured and then used as a 4-8

9 boundary condition in the heat conduction calculations. The numerical data for the steady-state -D temperature field in sitall plate and ater layer as ell as the radial profile of heat flux at the ater surface ere obtained. An appoximate analytical relation as suggested to calculate the local values of temperature and heat flux at the ater surface. The data obtained for both the temperature and heat flux can be treated as important boundary conditions for the computational modeling to be developed for the cluster formation process. Acknoledgments The authors are grateful to the Russian Foundation for Basic Research (Grants No a and a) for the financial support of the present study. References. Fedorets AA: 'Droplet cluster'. JETP ett. 004, 79 (8), Fedorets AA, Marchuk IV, and Kabov OA: 'On the Role of Capillary Waves in the'. JETP ett. 04, 99 (5), Hale GM and Querry MP: 'Optical constants of ater in the 00nm to 00µm avelength region'. Appl. Optics 973, (3), Zolotarev VM and Dyomin AV.: 'Optical constants of ater in ide avelength range 0.Å m'. Opt. Spectr. 977, 43 (), Fedorets AA, Dombrovsky A, and Smirnov AM: 'The use of infrared selfemission measurements to retrieve surface temperature of levitating ater droplets'. Infrared Phys. Tech. 05, 69, Fedorets AA, Dombrovsky A, and Medvedev DN: 'An effect of infrared irradiation on suppression of condensation groth of ater droplets in levitating clusters'. JETP ett. 05, 0 (7), Dombrovsky A, Fedorets AA, and Medvedev DN: 'The use of infrared irradiation to stabilize levitating clusters of ater droplets, Infrared Phys. Tech. 06, 75, Scriven E and Sternling CV: 'The Marangoni effects'. Nature 960, 87 (4733), Sirignano WA: Fluid dynamics and transport of droplets and sprays, Cambridge (UK), Cambridge Univ. Press, Dombrovsky A and Sazhin SS: 'A simplified non-isothermal model for droplet heating and evaporation'. Int. Comm. Heat Mass Transfer 003, 30 (6), 'Technical task on polycor and sitall substrates'. Russian Federal Nuclear Center, Sarov (Russia), 05, vol., part.. Yanenko N: The method of fractional steps, Heidelberg, Springer-Verlag, Mitchel AR and Griffiths DF: The finite-difference method in partial differential equations, Chichester (UK), John Wiley & Sons, Smith GD: Numerical solution of partial differential equations: finite difference methods. Third edition, Oxford (UK), Clarendon Press, Dombrovsky A, Timchenko V, Pathak C, Piazena H, Müller W, and Jackson M: 'Radiative heating of superficial human tissues ith the use of ater-filtered infrared-a radiation: A computational modeling', Int. J. Heat Mass Transfer, 05, 85,