Effect of thermocapillary instability on liquid film breakdown

International Journal of Heat and Mass Transfer 145 (2019) 118692

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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt

Effect of thermocapillary instability on liquid film breakdown E.A. Chinnov a,b, E.N. Shatskiy a,b,⇑, V.V. Semionov a a b

Kutateladze Institute of Thermophysics SB RAS, 630090 Novosibirsk, Russia Novosibirsk State University, 630090 Novosibirsk, Russia

a r t i c l e

i n f o

Article history: Received 24 July 2018 Received in revised form 3 September 2019 Accepted 3 September 2019

Keywords: Thermocapillary instability Regular structures Liquid film breakdown

a b s t r a c t The film of water flowing down along a vertical surface with a heater was studied experimentally at Re = 10–50. The initial temperature of the water film varied from 15 to 70 °C and heat fluxes on the heater varied from 0 to 6.5 W/cm2. Simultaneous measurements of the film thickness and surface temperature carried out. The effect of development of thermocapillary instability type A on wave amplitudes, deformation of the liquid film surface, and formation of the first stable dry spot on the heater was investigated. It is shown that when the longitudinal temperature gradients reach values larger then 7–10 K/mm formation of thermocapillary structures begins. At the leading edge of the heater, X/mm < 10–15, the thermocapillary structures in the form of a series of rivulets with a thin film between them are formed on the surface of residual liquid film after wave front propagation. The distance between the rivulets is k/mm = 10. It is shown that the formation of the first stainable dry spots occurs in areas where deformation liquid film reaches its maximum value, and the value of the wave’s amplitudes decreases. The interaction of waves with thermocapillary structure type A leads to an increase in the critical heat flux corresponding to the liquid film breakdown on 75% in comparison with the data known in literature. A new mechanism of action on the film flow was first identified and studied in detail. Ó 2019 Elsevier Ltd. All rights reserved.

1. Introduction The film flows are widely used in various industrial technologies and apparatuses, and understanding the processes occurring in flowing films is an important task for implementation of such devices. Currently, active investigation of these processes and search for the methods of heat and mass transfer enhancement are in progress [1–3]. An important factor affecting the heat transfer intensity and film stability to breakdown is interaction of waves on the interface with thermocapillary structures. Hydrodynamic two-dimensional waves in the isothermal liquid films are unstable to three-dimensional perturbations. The wavelength of instability to transverse three-dimensional perturbations decreases with increasing Reynolds number [4]. It is determined that the transition from regular two-dimensional structures to three-dimensional flow is accompanied by essential redistribution of liquid in the longitudinal direction [5,6]. Characteristic forms of three-dimensional structures developing during the transition are described. It is found out that the predominant structures on the film surface at Re > 50 are the short-lived rivulets, presented by

⇑ Corresponding author. E-mail address: [email protected] (E.N. Shatskiy). https://doi.org/10.1016/j.ijheatmasstransfer.2019.118692 0017-9310/Ó 2019 Elsevier Ltd. All rights reserved.

the chains of not less than 5–8 waves with close values of the transverse coordinate [7]. When the liquid films flow along the heated surfaces, thermocapillary instabilities of various types associated with liquid transport along the interface due to the surface tension gradient are formed together with hydrodynamic instability leading to development of the wave flow [8]. When the structures were formed under regime A, the high temperature gradients of up to 10–15 K/mm were observed in the upper part of the heater. When the threshold heat flux density was achieved, considerable deformations were formed on the film surface in the upper part of the heater, and the flow was divided into vertical rivulets with certain wavelength K. At low Reynolds numbers, high thermocapillary stresses directed against the flow led to the film thickening in the form of a horizontal roller [9]. To date, structures A have been studied in detail experimentally and theoretically at Re
5 [10–13], and at higher Reynolds numbers they were observed at Re = 5–12 [14], Re = 50 [15], and Re = 150 [16] (Re = C/qm, where C is specific mass flow rate of liquid, q is liquid density, and m is kinematic viscosity of liquid). In regime B, the rivulet flow formed gradually with increasing heat flux and distance from the upper edge of the heater. Nonuniformities in the liquid film thickness across the flow led to formation of temperature non-uniformities on the liquid film surface.

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A transverse temperature gradient (up to 1 K/mm), leading to an increase in deformation of the liquid film surface, appeared [17– 19]. Natural evolution of the developed three-dimensional waves into the thermocapillary-wave rivulets at heating a vertically flowing water film under the conditions of low temperature gradients was studied in [20] for Re = 10. The fact of deformation of the horseshoe-like hydrodynamic waves during their propagation through the heated region is registered. It is shown that temperature non-uniformities appear at the front of three-dimensional wave, when it moves along the heater. These non-uniformities led to liquid film deformation and appearance of rivulets due to the action of thermocapillary forces. Investigation of the Marangoni effect in a heated liquid film with developed wave motion is described in [21]. The temperature fields of the film were visualized by infrared thermography, and the film thickness was measured by confocal microscopy. The technique of simultaneous measurement of interface temperature and velocity was resented in [22]. In [23] the hydrodynamic characteristics of the falling liquid films were studied using the measurement techniques based on the planar laser-induced fluorescence and particle images. Heat transfer in the flow of a dielectric liquid film on the local heaters was studied experimentally in [24]. When the threshold heat flux was achieved, regular structures were formed on the film surface. With a decrease in the plate inclination, the wavelength of regular structures increased. At that, a change in the plate inclination did not significantly affect heat transfer at heat fluxes below the threshold ones. The film breakdown occurred either after or at the time of regular structure formation. It was shown that the heat flux, corresponding to formation of the first stable dry spot, reduced with decreasing inclination of the plate and increases with decreasing length of the heater along the flow. Heat transfer and breakdown in a falling liquid film on the extended heaters were studied experimentally in [25–27]. Water with initial temperature of 20 °C was used as a working liquid. A fiber-optic sensor was used to control the film thickness. Formation of the rivulet flow was detected. Dry spots formed in the region of a thin film between the rivulets. It was found out that the distance between a nozzle and heater determines hydrodynamics of liquid at relatively low heat fluxes, but it does not have a noticeable effect on the heat flux, corresponding to the film breakdown [26]. As the heat flux increases, the film thickness between the rivulets decreases gradually, and when a certain critical thickness is reached, the film breaks spontaneously. The critical film thickness is almost independent on the Reynolds number and plate inclination; and it is approximately 60 lm. The process of breakdown involves two stages: (1) sharp film thinning until a thin residual layer is formed; (2) breakdown and drying of the residual film. The heat flux necessary to break the film is almost independent on plate inclination, but it increases with the Reynolds number [27]. The influence of liquid properties and Reynolds number on the thermocapillary breakdown of the film flowing along a vertical plate with the heater of 150  150 mm is studied in [28]. It is determined that liquid viscosity has a significant effect on the threshold value of specific heat flux corresponding to the film breakdown. To take into account the influence of liquid properties, the traditional fracture criterion was modified. This allowed to generalize successfully all the data. This work is aimed at experimental investigation of the passage of hydrodynamic waves through a heater, their deformation when interacting with thermocapillary structures that form on the heater, as well as the effect of this interaction on stability of the heated water film flow to breakdown.

2. Materials and methods The setup was a closed circulation circuit, including a reservoir with a pump, working section, filter, rotameters, pipelines and shut-off valves. The working section consisted of a carrier plate, where a film former, heat stabilizer and heater were mounted. The working liquid (water with a dye, q/(kg/m3) = 997, t * 106/ (m2/s) = 0.912, r * 103/(N/m2 K) = 0.191) was fed by the pump to the film former, which included a storage chamber, dispenser and nozzle with a calibrated flat slit. Liquid flowed down the plate and along the connecting channels under the action of gravity, and returned to the reservoir with the electric pump. The initial temperature of the water film varied from 15 to 70 °C. Heat fluxes on the heater varied from 0 to 6.5 W/cm2. A copper flat heat exchanger 150 mm wide and 100 mm long was used as a heating element. Inside heat exchanger, heated liquid was pumped through the rectangular channels. The detailed description of setup can be found in [15]. The critical heat flux corresponding to the film breakdown was determined by formation of the first stable dry spot along in center of the heater. Due to the finite width of the heater, dried areas appeared along the lateral boundaries, which appeared at large temperature gradients at the textolite-heater boundary. With increasing heat flux, the size of the dried areas increased, and the greatest width of the dried areas was reached to the bottom of the heater. At the same time, the size of the drained zones for the largest heat fluxes did not exceed 50% of the width of the heater. According to the fluorescent method, the narrowing of the film flow did not lead to a change in the flow in the central part of the heater (where all measurements were taken). Changes in the flow occurred along drained areas, where 2 rivulet were formed with a greater thickness and flow rate. The temperature on the surface of vertically falling liquid film and thickness distribution in the heating region were measured synchronously, and the mechanism of liquid film breakdown under the conditions of complex interaction of wave and thermocapillary instabilities at Re = 10–50 was systematically studied in experiments. The thickness field was measured by the fluorescence method. To excite the fluorophore, a laser with diode pumping RLM-532-2000, continuously illuminating the area of 120  120 mm, was applied. Light, reradiated by fluorophore, was registered by the digital camera PCO 1200 h, with digital capacity of 10 bits, frequency of up to 500 Hz in the full frame regime (1280  1024 pix.). A red filter was installed in front of the camera to cut off the reflected laser light. At measurements on the area of 100  100 mm, the system provided spatial resolution of 0.1 mm. Rhodamine 6G was used in the experiments as the fluorescent dye. In the range of concentrations of 10–100 lmol/L, rhodamine 6G has virtually no effect on the surface tension of water [29]. The dependence of fluorescent properties of its working solutions of different concentrations on the temperature was determined experimentally in the temperature range from 20 °C to 90 °C [18]. To obtain a sufficient dynamic range of fluorescent liquid brightness, the solution with concentration of 20 mg/l (44.4 lmol/L) was chosen for the experiments on the heater, since the temperature change in brightness of solution fluorescence with such concentration at the temperature of 90 °C did not exceed 1% in comparison with brightness of solution at 20 °C [18]. The temperature field was measured by an infrared scanner Titanium 570 M. The heat flux from the heater surface was calculated by the difference in temperatures of pumped liquid at the heat exchanger inlet and outlet for a given mass flow rate. The methods and experimental technique are described in detail in [15,30].

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3. Results and discussion Formation of thermocapillary structures in the upper part of the heater at Re = 33 and q = 5.4 W/cm2 is shown in Fig. 1. At the initial time (t/s = 0, Fig. 1a), in the upper part of the heater (0 < X/ mm < 10) one can observe the unperturbed residual layer after propagation of the wave front, whose crest is located at X/ mm = 15–20. Temperature and thickness distributions can be considered uniform here (Fig. 2, t/s = 0). At the next moment (t/ s = 0.02), temperature non-uniformity appears on the surface of the residual layer (Fig. 2 a, t/s = 0.02). The temperature difference between the maximum (Z/mm = 79) and minimum (Z/mm = 85) is 11 °C. At that, there is no noticeable surface deformation (Fig. 2b, t/s = 0.02). With further consideration, the surface of the residual layer is deformed due to the action of thermocapillary forces (Fig. 1c), and thermocapillary structures with distance between rivulets of k/mm = 11 are formed in the upper part of the heater (Fig. 1c) which corresponds to the distance between the structures in regime A. The formation of thermocapillary structures begins when the longitudinal temperature gradient reaches values larger then 7–10 K/mm, which occurs when q/(W/cm2) = 4 is exceeded. Fig. 3 shows the dependence of wave amplitude in the rivulet and in-between rivulets on the path length of the film along the

heater for Re = 33 (A is wave amplitude, X is coordinate along the heater, and X = 0 is heater beginning). The Z coordinates for the rivulet (where the thickness of the film in the longitudinal direction is maximum) and the in-between rivulets region (where the thickness of the film in the longitudinal direction is minimum) were determined on the field thickness averaged over 600 frames. After that along the selected lines, the coordinates of the local maxima of the thickness and their value were determined for each of the 600 frames. It can be seen that the wave amplitude in the upper part of the heater is higher than in the lower part and it increases with increasing heat flux density. When the high longitudinal temperature gradients are achieved, what occurs at the heat flux density in the range from 3 to 3.4 W/cm2, formation of thermocapillary structures begins. At the leading edge of the heater, X/mm < 10–15, the thermocapillary structures in the form of a series of rivulets with a thin film between them are formed on the surface of residual liquid film after wave front propagation. The distance between the rivulets is k/mm = 10, which corresponds to the distance between the structures investigated earlier in regime A at other Reynolds numbers. When a new front inflows onto such structures, periodic nonuniformity of the wave front across the thickness appears. Such perturbation of the front leads to formation of three-dimensional waves with a transverse distance between the ridges of 10 mm.

a

b

c

d

Fig. 1. Distribution of film temperature and thickness during formation of thermocapillary structures at moments (a) t/s = 0, (b) t/s = 0.02, (c) t/s = 0.04, (d) t/s = 0.08, Re = 33, q/(W/cm2) = 5.4.

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Fig. 2. Distribution of film temperature (a) and thickness (b) along line X/mm = 10 during formation of thermocapillary structures, Re = 33, q/(W/cm2) = 5.4.

Fig. 3. Wave amplitude in the rivulet (a) and in-between rivulets (b) vs. film path length along the heater for different values of heat flux (q/(W/cm2)) on the heater, Re = 33.

However, at X/mm > 23, where the liquid film is heated, its temperature is close to the heater temperature and temperature gradients on its surface decrease sharply, this instability wavelength ceases to be decisive, the crests of three-dimensional waves merge in the lower part of the heater and the thermocapillary structures with an average distance between rivulets of 15 mm form in regime B. The similar pattern was observed for other Reynolds numbers. Thus, for Re = 50, when heat flux density q/(W/cm2) = 4 is reached, the temperature gradients increase to 10 K/mm, which is typical for the occurrence of structures of type A. Periodically, these structures were clearly visible. They were formed in the residual layer ahead of the oncoming wave front near the leading edge of the heater. The average distance between the rivulets in the transverse direction corresponded to the average length of the instability wave in regime A at lower values of Re. It was found that the thickness of the residual film layer in the region between the rivulets could be reduced significantly. Three-dimensional hydrodynamic waves decayed in the upper part of the heater. Thermocapillary instability in the upper part of the heater had a determining effect on the process.

Dependences of film surface deformation on longitudinal coordinate along the heater for different values of the heat flux are shown in Fig. 4. Criterion Def=(hriv-hval)/h0, defined as the ratio of difference between the average film thicknesses on the rivulet crest (hriv) and in-between rivulets (hval) to the initial mean film thickness at X/mm = 0 (h0) is used to describe quantitatively the transverse deformations in the liquid film. As can be seen from the diagrams, the character of dependence varies at high heat flux densities. This is caused by the fact that when the threshold heat flux density is reached at the upper edge of the heater (0 < X/ mm < 30), the thermocapillary structures in the form of rivulets with a thin film between them form in the residual layer of liquid between the wave fronts. Interaction of waves with these structures leads to an increase in deformation at the beginning of the heater, increase in the wave amplitudes. In the lower part of the heater deformations decrease. At that, a rivulet flow with an inter-rivulet distance of about 15 mm is formed in the lower part of the heater, as well as at smaller Re. Formation of structures A and waves interaction with a stable dry spot at different times are shown in Fig. 5 for Re = 33,

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8

qcr, W/cm2

7 6 5

1

4

2

3 2 1 0 0.00

10.00

20.00

30.00

40.00

50.00

60.00

80.00 T0, °С

70.00

Fig.6. Dependence of the heat flux corresponding to formation of the first stable dry spot on the initial film temperature. 1 – Re = 33, 2 – Re = 50.

Fig. 4. Dependence of film surface deformation on longitudinal coordinate along the heater for different heat flux densities.

q/(W/cm2) = 5.83 and initial film temperature T0/°C = 23. The dry spot boundary is indicated by a black line. The moment when the wave front has not yet reached the dry spot boundary is shown in Fig. 5a. The top edge of the dry spot is 30 mm from the heater beginning. According to Fig. 1, at this distance, the front has been already deformed, therefore, by the time when the wave begins interacting with the spot (Figs. 1c and 5b), the bulk of liquid is concentrated in the forming rivulets. Liquid hold in the bridge flows onto the spot and deforms it, but there is no complete washing (Fig. 5c and d). The effect of initial liquid temperature on heat flux density corresponding to the film breakdown (formation of the first stable spot) qcr for the Reynolds numbers of 33 and 50 (Fig. 6) was investigated. The initial temperature of the water film varied from 15 to

70 °C. It was found that the value of the critical heat flux density, corresponding to breakdown, decreases substantially with increasing initial temperature of the liquid film. Experimental data on the heat flux, corresponding to film breakdown, were generalized in the form of dependence between criterion Kmcr on Re (Fig. 7). Criterion Kmcr includes specific heat power Wcr/B, released on the heater for film heating.

Kmcr ¼ qcr ðrT =ðcq2 g 2=3 m5=3 ÞÞL=lm ¼ qcr LrT =ðcq2 g mlm Þ 2

¼ W cr rT =ðBcq2 g mlm Þ; 2

where B is heater width, c is heat capacity of liquid, g is acceleration of gravity, L is heater length, lm is scale of viscous-gravitational interaction = (m2/g)1/3, Pr is Prandtl number = m/a¼ lcp =k, qcr is heat flux density, corresponding to liquid breakdown, k is heat conductivity of liquid, l is dynamic viscosity of liquid, m is kinematic viscosity of liquid, rT is temperature derivative of surface tension coefficient = @ r/oT, q is density of liquid.

T °C

T °C

a

c

T °C

ð1Þ

b

T °C

d

Fig.5. Temperature distribution on the film surface at formation of structures A and interaction of oncoming front with a dry spot. Dark line shows the dry spot boundary, arrow indicates flow direction. (a) t = 0, (b) t/s = 0.02, (c) t/s = 0.04, (d) t/s = 0.08, Re = 33, T0/°C = 23, q/(W/cm2) = 5.83.

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Kmcr*

30000

and, it exceeds significantly (on 75%) data on breakdown without thermocapillary structures of type A. It can be concluded that a new mechanism of action on the film flow was first identified and studied in detail. The result is important since the discovered method for increasing the liquid film stability can be used at industrial installations.

25000 20000

1 2

15000

Declaration of Competing Interest

3 4

10000

The authors declared that there is no conflict of interest.

5 5000

Acknowledgement Re

0 0

50

100

150

200

Fig.7. Generalization of experimental data for the heat flux corresponding to film breakdown. 1 – water during formation of structures B [33], 2 – liquid FC-72 during formation of structures B [33], 3 – MD-3F during formation of structures B [33], 4 – water during formation of structures B at different angles of plate inclination [33], 5 – water during formation of structure A.

The dimensionless parameter Km was used in [31] to construct a map of the flow regimes of MD3-F and a 25% solution of ethyl alcohol in water on a vertical surface with a 6.5  13 mm heater. The parameter Km is an analogue of the Marangoni number and can also be considered as the ratio of the scale of thermocapillary tangential stress on the film surface ssur = (or/oT)/(oT/ox) to the scale of the tangential stress on the wall with a purely gravitational flow sW = qgh0 = q(gm)2/3(3Re)1/3. In this case, the characteristic temperature gradient on the surface of the film is estimated by the heat balance @T/ox = q/(cplRe). In [32], the parameter Km was successfully used to summarize the data on wave phenomena in a falling film with local heating. To take into account the influence of liquid properties and size of the heater Km was modified by multiplying by ratio L/lm. Data on the liquid film breakdown at formation of structures A and B are presented in Fig. 7. When only structures B are formed on the liquid film surface, the breakdown always occurred in the lower part of the heater between the formed rivulets. With an increase in the heat flux density, a sharp increase in deformation of the liquid film occurred in the lower part of the heater (Fig. 4), and this led to its breakdown. Data for different types of liquid are shown in Fig. 7. Data for breakdown in regime B are generalized by dependence [33]

KmcrB ¼ 165Re

ð2Þ

When the structures were formed on the liquid film surface in regime A, the heat flux density corresponding to breakdown increased substantially. The nature of liquid film breakdown changed, and this is explained by a change in film deformation. A significant influence of initial temperature of the liquid film relates to a change in viscosity, included into criterion Km, depending on temperature.

KmcrA ¼ 290Re

ð3Þ

4. Conclusion Thus, it is shown that formation of metastable thermocapillary structures of type A in the upper part of the heater affects the value of the critical heat flux corresponding to the liquid film breakdown. Interaction of three-dimensional waves with thermocapillary structure leads to an increase in wave amplitudes, their transformation into rivulets. At that, deformation of the liquid film in the middle part of the heater increases. It is shown that the value of the critical heat flux increases with the film Reynolds number

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