Liquid droplets on hot surfaces

Transcription

1 University of Ljubljana Faculty of mathematics and physics Seminar Liquid droplets on hot surfaces Author: Gregor Kostevc Mentor: doc. dr. Dušan Babič Abstract In this seminar behaviour of liquid droplets on a very hot surface is presented and a simple model is introduced to describe droplet evaporation. Special cases where droplet dynamics is involved are mentioned, important processes and their qualitative description is presented. This seminar supplements a previous seminar - Leidenfrost effect, presented in December 2011.

2 Contents 1 Introduction 2 2 Droplet-surface interaction 4 3 Dynamic Leidenfrost effect 6 4 Droplets on a liquid surface 9 5 Conclusion 12 1 Introduction Water droplet deposited on a hot metal surface eventually evaporates. Time needed for the droplet to evaporate decreases with increasing temperature of the metal surface until at some point it starts to become longer, although the plate is hotter and heat transfer is expected to be faster. That phenomenon was first reported in 1732 by Hermann Boerhaave, but it was Johann Gottlob Leidenfrost (1756) who first investigated it in more detail with water droplets on a very hot iron spoon, and so his name is associated with the phenomenon. Figure 1 shows lifetime of droplets in relation to metal surface temperature. The longest lifetime corresponds to Leidenfrost temperature. When water droplet hits the hot metal surface which temperature is slightly above the boiling temperature of water, it wets the surface and due to full contact of liquid and metal plate, heat transfer between water and plate is high so water eventually boils and evaporates. This happens faster at higher temperatures since heat transfer depends on temperature difference. In the process of boiling vapour bubbles form at the bottom of a droplet and travel upward. At higher temperatures they become abundant and coalesce, columns of vapour are formed. This process is called nucleate boiling which is schematically represented on figure 1 with other boiling phases. With further increase of temperature vapour does not form only on places at the bottom of droplet, forming columns, but much of the droplet s bottom transforms into vapour. This layer of vapour between water and metal acts as an insulator and has a great impact on the rate of heat transfer. Raising the temperature yet decreases heat transfer until at some point the whole droplet is supported by a layer of vapour. A temperature when heat transfer rate reaches its minimum is defined as Leidenfrost temperature. From this point, raising the temperature only affects temperature gradient and thus increases the heat transfer rate. 2

3 Figure 1: Boiling of a water droplet on a metal surface has four stages. Figure represents rate of heat transfer between droplet and hot plate in dependence of a temperature difference between temperature of a hotplate and boiling temperature of water. In single phase regime no vapour bubbles are formed and evaporation happens only on the dropletair boundary. In nucleate boiling regime vapour bubbles form that travel upward and at higher temperatures form columns of vapour. In transition boiling regime droplet s bottom is partially covered with vapour film. Leidenfrost point is reached when the whole droplet levitates on a vapour film [1, 2]. 3

4 Leidenfrost effect is also at the heart of some popular tricks - walking over hot coals, dipping wet fingers into molten lead and its counterpart dipping fingers into liquid nitrogen. 2 Droplet-surface interaction Let us now look in more detail what happens to a water droplet on a hot metal surface with temperature high above the water boiling temperature. When a drop is deposited on a surface, the liquid directly exposed to the surface evaporates almost instantaneously and a thin vapour film is created over which the droplet floats. The whole process is complicated and depends on many parameters. Most important of those are droplet and surface temperature, surface roughness, droplet impact velocity and size. For the beginning let us examine a sessile droplet, that is spherically shaped droplet with typical size of 1 mm resting on an above-leidenfrost temperature surface. An experiment with sessile drop on a 400 C shows that vapour layer supporting a droplet has a thickness of approximately 30 µm [3]. Assuming we know exact values of the temperature of metal surface and initial volume of the droplet, next parameter that can be easily measured is droplet s lifetime. We will present a simple theoretical model, with known initial parameters, which correctly estimates droplet s lifetime. In the model the following assumptions will be used: temperature T p of metal surface is constant, droplet s temperature at water boiling temperature T b (neglecting droplet pressure change due to surface tension) is constant and uniform, vapour layer is at average temperature of T b and T p, droplet s surface in contact with vapour layer is πr 2 0, where r 0 is radius of the droplet, vaporization and radiation from the remaining droplet s surface (3πr 2 0) are neglected. Two equations describe behaviour of the droplet, derived from energy balance and force balance. Energy balance, comprising of conductive, radiative and latent heat transfer gives the equation λ v δ (T p T b ) + σɛ(tp 4 Tb 4 T p T b ) = Φ m q L + Φ m c pv, (1) 2 where δ is the thickness of vapour cushion between the droplet and hot plate (as shown on figure 2), λ v thermal conductivity of vapour, σ Stefan-Boltzmann constant, ɛ emissivity, q L latent heat of evaporation, c pv specific heat of vapour and Φ m mass flow of evaporating liquid from the bottom surface of the droplet. Left side of the equation can be thought of as net heat flux from hot plate to droplet and right hand side as heat flux used for vaporization and increasing vapours temperature. For a droplet to be supported by vapour layer, the pressure, causing vapour flow beneath the drop, integrated over the contact surface must counteract the weight of the drop [4] (ρ l ρ v )V g = 2π 4 r0 0 prdr, (2)

5 Figure 2: A droplet deposited on a hot plate with temperature T p levitates on a thin film of vapour with thickness δ. Temperature of a droplet is assumed to have a uniform temperature equal to liquid s boiling temperature. Arrows show radial vapour flow that is driven by the pressure of the droplet s gravitational force. assuming a contact surface between droplet and metal plate is written as ds d = 2πrdr, where r 0 is radius of the contact surface, ρ l density of water and ρ v density of vapour. Pressure p can be obtained by assuming that vapour flow beneath the drop is laminar and viscous and can be approximated by radial flow between parallel surfaces separated by a distance δ, since sessile drop, despite its surface tension, flattens to some extent at the bottom (as shown in figure 4). Thus pressure at the bottom has the form [4]: p(r) = 3 Φ mη ρ v δ 3 (r2 0 r 2 ), (3) where Φ m vapour is mass flux, η viscosity of vapour, r 0 radius of the contact surface and r coordinate measured from the center of droplet. Substituting above result into equation 4 and integrating gives [4]: (ρ l ρ v )V g = 3πΦ mηr 2 0 2ρ v δ 3. (4) To have the full set of equations we now write continuity equation as ρ l dv dt = πr2 0Φ m. (5) In order to get analytical solution, we make two additional approximations - vapour pressure does not change with droplet volume and radiation is neglected [4]. Substituting equation 1 into equations 4 and 5 then combining the two we get vaporization rate of the droplet dv dt = π [ ] 8λ v (T p T b ) 3 1/4 ( ) 7/12 ρ v g 3V 9ηρ 3 l (q. L + c pv (T p T b )/2) 4π As mentioned before we neglected contribution of radiation which is, for T p 250 C and λ v 0.04W/mK, approximately ten times smaller than that of conduction. 5

6 Figure 3: Lifetime of droplets is measured with respect to the temperature difference between hot plate temperature and boiling temperature of the water. Experimental results are shown as dots. Analytical solution of lifetime (equation 6) is plotted along with the numerical solution, which includes radiation and vaporization on the top surface of the droplet. [4]. Integrating above equation for V (t = 0) = V 0 gives lifetime of a droplet τ = ( ) ( 12 4π 5π 3 ) 7/12 [ 9ηρ 3 l (q L + c pv (T p T b )/2) 8λ v (T p T b ) 3 ρ v g ] 1/4 V 5/12 0 (6) Equation 6 gives a good qualitative approximation (plotted on figure 3) for the lifetime of sessile droplets despite the approximations. Taking into account contributions of radiation and vaporization including the top surface, leads to even better description of the vaporization rate which fits to the experimental data as shown on figure 3. 3 Dynamic Leidenfrost effect Sessile droplets deposited on a flat hot surface rest on that surface until complete evaporation. But droplets can also be released onto a surface from heights greater than thickness of vapour layer, thus gaining significant momentum before the impact with hot surface. Right after the impact a drop undergoes a deformation that depends on the relation between droplet s kinetic and surface energy. Processes as bouncing, droplet spreading and atomization take place and are shown on figure 4. We will examine the parameters those processes depend on and try to extend the qualitative description presented in previous section to the impacting droplets in dynamic Leidenfrost effect. As with the sessile droplets, the temperature of the metal surface must be high enough for the impacting droplet not to come into contact with the surface. The parameter characterizing impacting droplet is the Weber number which is the measure of relative 6

7 Figure 4: Top: Droplet released on a hot surface from some height spreads and rebounds after the impact. The figure shows a droplet with 1 mm radius and Weber number 11, impacting on a surface with temperature 200 C [3]. Bottom: For high We droplet s kinetic energy is high relative to surface tension and droplet breaks into smaller droplets - the process is called atomization. importance of droplet s momentum and surface tension [3] We = ρv2 R σ where ρ is water density, σ surface tension, R droplet radius and v impact velocity. Weber number is calculated with the properties of water at the boiling temperature since we assume that upon impact a droplet reaches the boiling temperature very rapidly. Weber number of droplets used when observing dynamic Leidenfrost effect are usually between 10 and 100. For We < 30 the surface tension dominates and the droplet recoils and rebounds after the impact. When We is between 30 and 80 a droplet recoils and rebounds, although in the later stage the droplet may disintegrate into smaller droplets - the process is called atomization. For We > 80 droplet has so much kinetic energy that surface tension cannot prevent droplet from splashing already in the spreading stage thus forming tiny droplets [5]. Temperature required for a stable vapour film is obviously not the same as in the case of sessile droplets. If the liquid locally touches the surface, it starts boiling and small vapour bubbles formed inside the drop rise and burst on the drop surface, creating small droplets. The process is known as secondary atomization. Thus a dynamic Leidenfrost temperature can be defined as the lowest temperature for which the vapour cushion causes drop bouncing without secondary atomization [6]. Its dependence on We is shown on figure 5. A quantity of interest is maximum radius R max to which the droplet spreads during the impact. In a simple model kinetic energy converts to surface energy while spreading and experiments show that R max depends on Weber number and dependence takes the form R max R 0 We 0.3, where R 0 is the initial radius of the droplet. Maximum radius 7

8 Figure 5: Left: Dynamic Leidenfrost temperature T LD is defined as a temperature above which impacting droplets do not come into contact with the metal surface. Consequently T LD depends on dynamic properties of droplets which are characterized by a Weber number thus it is convenient to show T LD dependence on We [7]. Right: During the spreading of the droplet, kinetic energy is transformed into surface energy. Maximum diameter of the droplet is shown, relative to We in logarithmic scale [8]. of the droplet after impingement is important for heat transfer during the recoil stage. Relation between maximum radius and Weber number is shown on figure 5. A droplet recoils after achieving the maximum radius due to surface tension forces and bounces of the plate repeating the oscillatory movement which is shown on figure 6a. Figure 6b shows an example of oscillations of a droplet with We 20. Impacting drops were recorded with high-speed camera at the rate of 1000 frames per second. Data shown are height of the centre of mass from the impact surface, horizontal (diameter) and vertical (height) dimension of a droplet. It seems droplets have two oscillation modes: one due to rebound on the impact surface, and one corresponding to free oscillations. Each time a droplet impacts on a surface it is forced to spread and the minimum in height of the droplet approximately every 40 ms corresponds to spreading. Impact also induces oscillations on droplet s surface. These oscillations are known as capillary waves. Measured characteristic time of free oscillations is about 20 ms, which corresponds to theoretical oscillation period of a droplet s capillary waves in fundamental mode [8] τ = 2π ρr 3 0 γ. (7) Here ρ is density of water, γ its surface tension and R 0 droplets mean radius. Free oscillations of Leidenfrost drops depend only on the balance between kinetic and surface energy. 8

9 Figure 6: Left: Figure shows gap between the bottom of the droplet and surface of the plate as it changes with time. Bouncing Leidenfrost droplets spread and rebound on the metal surface many times, each time losing energy which is seen from the figure above as the maximum height reached after each period decreases. Each time a droplet hits the surface (gap is 0), it does not bounce off immediately - some contact time can be observed [8]. Right: More thorough measurement of height of centre of mass (green), diameter (blue) and height of a droplet (red) reveals another oscillations independent from bouncing. These oscillations are known as capillary waves. 4 Droplets on a liquid surface Further dynamics of Leidenfrost droplets can be observed when a rigid metal plate is replaced by a viscous liquid. Such set up is achieved with liquid nitrogen droplets on a surface of a liquid at room temperature. Thermodynamics of such droplets is similar as for water droplets on a metal plate but showing richer variety of dynamic phenomena. When a small droplet of liquid nitrogen is deposited on an initially calm surface of viscous liquid at room temperature thin vapour layer forms separating a droplet and the liquid surface. The droplet and vapour layer are supported by surface tension of the underlying liquid. Because of gravity surface under the droplet deforms creating a nest that surrounds the droplet. The shapes of the droplet and the nest are determined by the surface tensions of the liquids. During the evaporation, flows in both liquids are created. In the droplet vortex flows form as shown in figure 7. They are induced by evaporating vapour flows around the droplet. Flows in the droplet are also present in droplets deposited on a metal plate, mentioned in previous sections, but their influence on dynamic behaviour is negligible until deposited on a liquid surface. Additionally, flows in the supporting liquid are observed where liquid just below the droplet sinks down while on the surface of the liquid strong converging flows are formed as indicated with white arrows on figure 7 (left). These circulating flows in the bulk of the liquid can be associated with thermal convection. Cold liquid just bellow the droplet sinks and is replaced by a warm liquid which flows on the surface. 9

10 Figure 7: Left: A liquid nitrogen droplet deposited on a liquid surface at room temperature is supported by its vapour layer which is supported by surface tension of the underlying liquid. Due to the gravity surface below the droplet deforms, creating a nest. Vapour flow around the droplet induces toroidal flows in the droplet. Convective flows in the supporting liquid are due to sinking of cold liquid just below the droplet. White arrows show convective flows on liquid s surface. Right: Time evolution of droplet radius is shown, where R 0 is initial radius of the droplet. To indicate the importance of convective flows on evaporation time the following supporting liquids were used: glycerol (squares), glycerol (76%)-ethanol (24%) mixture (triangles), ethanol (diamonds) and the following solid substrates: glass (stars) and copper (spheres) [9]. Convecting flows in the supporting liquid have an important role in the evaporation process since the flow keeps the liquid temperature below the droplet constant by removing cold and providing warm liquid. The experiment shows that droplet lifetimes are shorter when placed on more viscous liquids where flow rates are derogated by viscosity. The effect of convective flows in viscous liquids on the evaporation time of droplets can be seen on figure 7 (right). To show the effect of viscosity on the evaporation time, independently of thermal conductivity, glycerol and ethanol were used as supporting liquids with small difference in thermal conductivities (0.28 W/mK and 0.18 W/mK) while viscosity of glycerol is three orders higher than that of ethanol so the differences in evaporation rates, shown in figure 7 (right), are caused by the convective flow rates that depend on viscosity of liquid. Described internal dynamics also has an influence on droplet s movement on the liquid surface. We will take a look at two limiting cases - droplet on a low (ethanol) and high (glycerol) viscosity liquid. On low viscosity liquids the droplet glides on the surface. The droplet movement is the result of the asymmetrical convective flows around the droplet that alter heat transfer rates and create a difference in evaporation rate on the opposite sides of the droplet. The abrupt stops and resumption of droplet motion in random direction are due to internal dynamics of the droplet, driven by thermal fluctuations and oscillations induced when depositing a droplet on a liquid surface. 10

11 Figure 8: Left and center: Droplet on liquid with high viscosity turns into a pulsating starlike structure. Due to a simultaneous excitation of multiple surface deformation modes there is a switching in number of petals from 4 to 7 as seen on the plot showing wavelength of standing waves in relation to time. Switching is present only at the beginning - when droplets radius decreases with time N remains at the smaller value. Right: Droplet on a liquid with low viscosity glides on the surface. [9]. During the evaporation when droplet radius becomes smaller droplet velocity increases. As shown on figure 8 (right) droplet is leaving a trail of vapour behind making its motion reminiscent of a rapidly moving comet. On a highly viscous surface gliding behaviour is not observed. Instead a droplet turns into a pulsating star-like structure shown on figure 8 (left). The number of petals is roughly determined by the droplets radius and when the radius gets smaller during the evaporation the number of petals is reduced. A time evolution of number of petals is shown on figure 8 (center). Switching and oscillations are observed, reflecting a competition between unstable surface deformation modes when N varies from 4 to 7. Wavelength of standing-wave oscillations on droplet surface is determined by droplet radius R as [9] λ = 2πR N, (8) where N is the number of petals. Introducing multiple Leidenfrost droplets on a liquid surface leads to an interesting formation of a long living bound states shown on figure 9. Bound state is the sum of a long-range attraction between droplets due to a meniscus formed on a supporting liquid and strong short-ranged repulsion between droplets created by nitrogen vapour flows around droplets which prevent droplet coalescence. Eventually droplets coalesce due to thermal fluctuations and present dynamics. 11

12 Figure 9: Long-lived bound states are formed as the sum of long range attraction between droplets because of meniscus formed on a supporting liquid and short range repulsion due to evaporating vapour flows around droplets which prevent droplet coalescence. 5 Conclusion In the seminar I presented Leidenfrost effect - droplets on a hot surface floating on a thin vapour film. I have presented a simple model and showed that with introducing some dynamics and replacing rigid surface with a liquid interesting behaviour is observed. There is still a rich variety of experiments with Leidenfrost effect I haven t described in this seminar, including self-propelled droplets on a hot surface with asymmetric geometry [10] and inverted Leidenfrost case with hot droplets on a cold supporting liquid [11, 12], which can be achieved with water droplets initially at room temperature on a liquid nitrogen surface. In this case it is the supporting liquid that evaporates, thus thermodynamics as well as dynamics of such set up are different as in previous experiments described in this seminar. But Leidenfrost effect is not limited to laboratory observations. It plays an important role in industrial applications where its presence is usually undesired. An example is spray cooling where droplets instead of wetting the surface are floating on a vapour which acts as an insulator and reduced the heat transfer rate. Further, it has a negative effect in steel quenching where rapid cooling is of great importance in preventing unwanted phase transformations of the material structure to obtain desired properties of materials. In cooling systems in nuclear reactors, where temperatures reach nearly 1000 C, Leidenfrost effect can lead to overheated fuel plates and undesired consequences. It is avoided by increasing the pressure of the fluid and its flow rate. 12

13 References [1] linke/papers/walker leidenfrost essay.pdf (2011). [2] Bernardin, J. D. J. Heat Tran. 121, (1999). [3] Chatzikyriakou, D., Walker, S., Hewitt, G., Narayanan, C., Lakehal, D. Appl. Therm. Phys. (2008). [4] Gottfried, B. S., Lee, C. J., Bell, K. J. Int. J. Heat Mass Tran. 9, 1167 (1966). [5] Xie, H., Zhou, Z. Int. J. Heat Mass Tran. 50, (2007). [6] Bertola, V. Int. J. Heat Mass Tran. 52, (2009). [7] Bertola, V., Sefiane, K. Phys. Fluids 17, (2011). [8] J.-B., O., Bertola, V. European Conference on Liquid Atomiz. and Spray Systems 50, (2007). [9] Snezhko, A., Jacob, E. B., Aranson, I. S. New J. Phys. 10, (2008). [10] Linke, H., Aleman, B., Melling, L., Taormina, M., Francis, M., Dow-Hygelund, C., Narayanan, V. Phys. Rev. Lett. 96, (2006). [11] Kim, H. J. Kor. Phys. Soc. 49, (2006). [12] Kim, H., Lee, Y. H., Cho, H. J. Kor. Phys. Soc. 58, (2011). 13