Sustained drag reduction and thermo-hydraulic performance enhancement in textured hydrophobic microchannels

International Journal of Heat and Mass Transfer 119 (2018) 551–563

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Sustained drag reduction and thermo-hydraulic performance enhancement in textured hydrophobic microchannels D. Dilip, S. Vijay Kumar, M.S. Bobji ⇑, Raghuraman N. Govardhan Department of Mechanical Engineering, Indian Institute of Science, Bangalore 560 012, India

a r t i c l e

i n f o

Article history: Received 18 June 2017 Received in revised form 16 October 2017 Accepted 17 November 2017

Keywords: Textured hydrophobic surface Trapped air bubbles Pressure drop Drag reduction Heat transfer Thermo-hydraulic performance index

a b s t r a c t Drag reduction obtained on flow over textured hydrophobic surfaces has been ascribed to the presence of entrapped air within the surface micro-texture. To sustain the drag reduction, it is important that the entrapped air be maintained on the surface. However, the entrapped air bubbles tend to shrink with time and finally disappear, causing the drag reduction also to reduce and eventually vanish. Recent research shows that by controlling the absolute pressure of water, it is possible to sustain the entrapped air bubbles on the surface and hence the drag reduction for extended periods of time. In this paper, we explore the possibility of sustaining the entrapped air by varying the absolute temperature of water in the vicinity of the textured surface. For this, the textured surface is externally heated and the evolution in the size of trapped air bubbles with time is observed. Simultaneous pressure and temperature measurements are made along with the visualization, to study the concomitant effects on drag and heat transfer. We find that, varying the absolute temperature influences the trapped air bubble dynamics appreciably, which in turn affects the measured pressure drop across the channel. By varying the external heat input, it was found that the trapped air bubbles can be maintained on the surface for prolonged periods of time, at an optimum size suitable for drag reduction, such that sustained and maximized drag reduction can be achieved. The presence of trapped air bubbles is found to inhibit the heat transfer across the surface. However, when the pressure drop reduction achieved due to the presence of air bubbles is significant enough, the combined thermo-hydraulic performance is found to be enhanced. The results, not only provides important inputs towards achieving sustained drag reduction from textured hydrophobic surfaces, but also ascertains the feasibility of using such surfaces in micro-scale heat transfer applications. Ó 2017 Elsevier Ltd. All rights reserved.

1. Introduction Superhydrophobic surfaces, prepared through a combination of surface texturing and hydrophobic surface chemistry [1–7], have been found to be useful in lowering the liquid flow resistance (drag) in microfluidic passages [8–16]. When a textured hydrophobic surface is immersed in water, an underwater Cassie state is established on the surface, where the penetration of water into the surface cavities is prevented by the entrapped air within [17– 19]. Water comes into contact only with a fraction of the solid surface and the rest of the area is covered by the trapped air pockets. When water flows past such a surface, the trapped air bubbles act as near shear free regions reducing flow resistance [20]. In addition to the fractional coverage of air, the size of the air pockets also play a prominent role in determining the quantum of drag reduction achievable [21,22]. For achieving drag reduction, various kinds of ⇑ Corresponding author. E-mail address: [email protected] (M.S. Bobji). https://doi.org/10.1016/j.ijheatmasstransfer.2017.11.093 0017-9310/Ó 2017 Elsevier Ltd. All rights reserved.

micro-textures like pillars/posts [10], ridges [11], holes [15,16] and random surface features [14,23] have been investigated and some of these studies have reported significant reductions in drag of the order of 30–40% [10,14]. Recent research indicates that the drag reduction achieved by using textured hydrophobic surfaces is short lived due to the gradual dissolution of entrapped air into the flowing water [14,29]. The rate of dissolution is dependent on the concentration gradient across the air-water interface (i.e. the difference in concentration of air inside bubble and the concentration of dissolved air in water) and also the convective effects caused by the flow [15,16,23]. The sustainability of trapped air on superhydrophobic surfaces and the factors leading to the breakdown of underwater Cassie state have recently been investigated both numerically and experimentally [23–28]. In order to increase the longevity of drag reduction, it is important that the air bubbles be maintained on the surface. For this, air has to be supplied to the trapped air pockets continuously. Different methods like direct supply of air [30,31] and electrolysis [32,33] have been tried out, to keep the surface cavities replete

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Nomenclature A As Cp C1 Cs D Di Dh DH f h hm Hair I k kf _ m _a m n Nu Nuo P Pe q q” Q Re

area of the surface surface area of bubble specific heat concentration of air in water saturation concentration of air in water diameter of bubble initial diameter of bubble diameter of hole channel hydraulic diameter degree of saturation heat transfer coefficient convective mass transfer coefficient Henrys constant for air current, amp diffusivity of air in water thermal conductivity of fluid mass flow rate of water convective mass transfer rate number of moles Nusselt number reference Nusselt number pressure Peclet number heat transfer heat flux volume flow rate of water Reynolds number

with air. However these methods use a self-regulating mechanism which involves intricate arrangements with active feedback control. Very recently liquid infused surfaces with trapped air (LISTA) have been proposed to reduce the drag force and to counter the air depletion problem [34]. In our previous studies [15,16], we have experimentally demonstrated a simple method to sustain the air bubbles on the cavities, by locally changing the absolute pressure within the channel. By altering the pressure, the solubility of water flowing through the channel was altered. Thus by making the flowing water supersaturated with air, the concentration gradient causing transport of air across the water-air interface was reversed, forcing the dissolved air in water to migrate into the air pockets causing them to grow in size. It was demonstrated that by carefully controlling the solubility of water through pressure, the trapped air bubbles can be sustained for extended periods of time. To locally supersaturate the water flowing over the surface, it is required that a pressure lower than the atmospheric pressure be maintained within the channel. In many applications, maintaining sub-atmospheric pressures within the channel is often difficult or inconvenient. Moreover in pressure driven flows, the flowing water will be generally undersaturated. Since the solubility of water is a function of temperature as well, an alternate method to control the solubility of water is by locally heating the water as it flows over the textured surface. This would increase the temperature of water flowing over the surface, thus locally supersaturating the water. Hence by local heating, it would be possible to realize the same effect, equivalent to that of reduction in pressure. In this paper, the possibility of sustaining the drag reduction by controlling the local absolute temperature of water is investigated. For this the textured surface is externally heated by means of a resistance heater. By varying the power input to the heater, it is possible to precisely control the local temperature of water near the surface. The consequent effect of this local heating of water on the dynamics of air bubbles and the pressure drop across the channel is systematically studied.

s Sh T Ti To Ts Tf THPI u U V x y

parameter defined by Eq. (17) Sherwood number temperature fluid inlet temperature fluid outlet temperature textured surface temperature mean fluid temperature thermo hydraulic performance index average velocity of flow velocity of flow relative to bubble voltage height of the channel width of the channel

Greek Symbols q density of water l dynamic viscosity h standard temperature DP pressure drop DPo reference pressure drop Subscripts O oxygen N nitrogen

Local heating of water, as it flows over the textured surface, requires that heat be transferred across the textured hydrophobic surface into water. While most of the studies on textured hydrophobic surfaces in microchannels have focused on its drag reducing characteristics, there have been only limited studies on the thermal transport across such surfaces. In many applications at the micro scale, such as electronic cooling, in addition to the pressure drop reduction, achieving higher rates of heat transfer is also important. In the past decade, there have been a number of theoretical and experimental studies that have reported heat transfer enhancement in microchannels by using rough/wavy surfaces as the channel/tube walls [35–39]. These studies, however, have also reported large increase in the pressure drop across the channel, caused by the modified geometry of the surface. Since textured hydrophobic surfaces are capable of delivering substantial pressure drop reduction across the channel even in the presence of roughness, it appears to be a promising candidate for applications at the micro-scale where both heat transfer enhancement and pressure drop reduction are important. However the entrapped air on the surface is likely to have a detrimental effect on the thermal transport performance of the surface, as air is an insulator. Recent studies on thermal transport have shown that in general hydrophobic surfaces deliver lower thermal transport performance (i.e. lower Nusselt number (Nu)) when compared to hydrophilic ones [40,41]. Surfaces with high contact angles exhibits a decrease in the pressure drop but also display an associated reduction in the heat transfer performance [41]. The thermal transport performance on textured hydrophobic surfaces decreases as the size of the shear free regions is increased [42–44]. An effective medium approach [45] which treats textured surface with trapped air as a single substrate revealed that even though convective heat transport increases due to the modified fluid velocity profile on the surface, the decrease in the thermal conductivity of the substrate due to the presence of air inhibits the thermal transport appreciably.

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Since the presence of air on the surface helps to reduce drag, but impedes the thermal transport, there have been some studies which have focused on the combined outcome of these two contradictory effects. Through numerical simulations Maynes et al. [42] and Cheng et al. [46] have shown that the combined thermohydraulic performance (ratio of the Nusselt number to the frictional resistance) increases over the theoretical value. It was shown by Enrights et al. [47] that the thermo-hydraulic performance for longitudinal ridges is higher compared to pillars and transverse ridges. However the improvement in the combined thermo-hydraulic performance shown by simulations has not been confirmed by experiments [48]. In the present study, we experimentally evaluate the usefulness of the textured hydrophobic surfaces from a combined thermo-hydraulic perspective. In this work, in addition to investigating the possibility of sustaining air bubbles through heat transfer, we also experimentally investigate the effect of the trapped air bubbles on the combined thermo-hydraulic performance of the channel. For this we define a Thermo-Hydraulic Performance Index (THPI) which takes into account the effect of air presence both on the pressure drop as well as on the thermal transport such that THPI > 1, indicates an improvement in the overall performance. We show that by using a textured hydrophobic surface inside a microchannel, an improvement in the overall performance can be achieved. The results show that textured hydrophobic surfaces hold immense potential to be used in micro-scale heat transfer applications. 2. Experimental methods The experimental surface was prepared by photo etching a smooth brass metal sheet of dimensions 100 mm  30 mm  0.8 mm [15,16]. An ultra violet light sensitive photo resist, is laminated onto one side of the brass sheet and the negative of the desired pattern printed on a photo mask is placed closely in contact over it. Exposure to UV light makes the exposed areas to become hardened. The unexposed resist is washed away to reveal the raw metal while the exposed resist remains on the surface. A heated solution of cupric chloride is then used to etch away the unprotected metal. The required depth of texture is achieved by controlling the duration of etching. Fig. 1(a) shows the brass surface after etching. The texture consists of a regular array of blind holes and was chosen for experimentation based on the observations made by Bobji et al. [29]. Fig. 1(b) shows the zoomed in view of the blind holes. The surface was characterized using a 3D non-contact optical profilometer and the salient dimensions of the texture are found to be hole diameter 300 mm, depth 160 mm and pitch 370 mm and the variations in these dimensions is found to be less than about 10 mm across the surface. The surface is made hydrophobic through coating of a self

Fig. 1. (a) Textured surface patterned with a regular array of blind holes. (b) Zoomed in view of the holes. (c) Image of a water drop on the SAM coated surface showing the apparent contact angle (125°).

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assembled monolayer (SAM). For this, the textured surface is kept submerged in a 50 mM solution of 1-dodecanethiol in ethanol for about two hours [15,16,49,50]. After taking out, the surface is dried in an oven at 60 °C. To measure the contact angle, a 10 ll drop was placed on the surface and its optical image was obtained using an optical microscope. Fig. 1(c) shows the image of the water droplet resting on the surface. The contact angle was found to be 125°. Experiments are carried out in a rectangular microchannel whose walls are constituted of a textured hydrophobic surface on one side and a hydrophilic flat transparent glass surface on the other as shown in Fig. 2. The width of the channel is 1 mm, which is maintained by means of a stainless steel spacer. Water flow through the channel is purely by gravity and this is accomplished by keeping the inlet water tank at a height greater than that of the channel. The absolute pressure within the channel can be set by varying the height of the water tank with respect to the channel. The water used in the experiments is exposed to atmospheric air in a large tank for about 12 h prior to the commencement of the experiment. Before each experiment, the channel was dried and flushed with dry nitrogen for about 30 min. The channel was then carefully filled with water, so that air is trapped inside the holes uniformly over the entire surface. The air bubbles on the textured surface were visualized using a total internal reflection based technique which has been previously used in visualizing trapped air bubbles [14–16,29,51]. For this, a beam of light at an angle greater than the critical angle required for total internal reflection is incident on the surface so that it undergoes total internal reflection at the surface making the air bubbles trapped on the surface to appear as bright spots. The schematic of the visualization setup is also shown in Fig. 2. The images of the trapped air bubbles are recorded at regular intervals of time using an optical microscope (Olympus LG-PS2) with a camera attached to it. From the images of the bubbles, bubble diameter is measured to an accuracy of ±1% using an image processing algorithm. For bubble diameters of about 300 lm, which is the case in our experiments, the accuracy of measurement will be ±3 lm. Along with the air bubble visualization, pressure drop measurements are simultaneously made across the channel length by means of a wet/wet differential pressure transmitter (Dwyer instruments Inc. USA, Series 655A 316) of range 0–300 H2O and with an accuracy: ±0.25% of FS, to understand the effect of bubble dynamics on drag. Flow rate is measured accurately by filling a beaker with water for a known length of time and measuring the weight of water with a precision digital balance (Essae, Model DS-852) of range 0.01–3 kg with an accuracy of ±0.3%. Concentration of air in water is obtained by using a dissolved oxygen meter (Hanna Instruments Model HI 9146) having a range of 0–45 ppm and with an accuracy of ±1.5% FS. The textured surface is externally heated by using a Nichrome resistance heater to control the local temperature of water as it flows over the surface (Fig. 2). In order to minimize variations of temperature across the surface and to help maintain the temperature, a thermal mass made of Aluminium was placed between the textured hydrophobic surface and the resistance heater. Intimate contact was ensured between the surfaces of the heater, thermal mass and the textured surface and a thermal conducting paste was used to eliminate any air gaps present. The heater and the surface were insulated from the surroundings by using glass wool insulation in order to reduce the heat loss to the surroundings as much as possible. The channel was sealed off to prevent water leakage to the surroundings by means of a silica gel sealant. The temperature of water at the inlet and outlet of the microchannel as well as of the textured hydrophobic surface is measured by means of T type (copper-constantan) thermocouples (range 200 °C to 350 °C) with an accuracy of ±0.5 °C. A multi channel temperature indicator (Heatcon Instruments Inc.) with a resolution

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Incident ray

Reflected ray Water flow through channel

Entrapped air

Textured Surface

Incident ray Stationary water for TIR

Reflected ray Hydrophilic Glass

Microchannel θ = 53o

Water in

To Water out

Ti

Thermal mass

Textured surface To pressure transducer

Ts1

Insulation Thermocouple

Resistance Ts2 heater

To pressure transducer

Fig. 2. Schematic of the experimental setup used for heating. The textured surface is heated by using an electrical resistance heater. To facilitate uniform heating of the surface a thermal mass is provided between the heater and the textured surface. Glass wool insulation is provided to prevent heat losses to the surroundings. The entrapped air pockets on the textured surface are visualized using total-internal reflection of light. Pressure drop is measured by using a pressure transducer. Temperature of water at inlet (Ti) and outlet (To) and that of the surface at two different locations along the surface (Ts1 and Ts2) are measured using T type thermocouples.

of 0.1 °C was used to acquire the temperature data. The heat supplied is provided by a strip heater powered by a variable power supply unit of range 0–30 V and 0–5 A. The error in measuring the current is ±1% and voltage is ±1%. The heat supplied is determined by taking the product of voltage and current given to the heater. The error in the heat flux measured is obtained from the error in measurements of voltage and current and is found to have a variation of less than ±2%.

3. Bubble growth during heating In the case of experiments discussed in this paper, it may be noted that the absolute pressure within the channel is always maintained greater than the atmospheric pressure. At this condition, the water flowing through the channel becomes undersaturated with air due to the increased solubility of air in water at pressures higher than the atmospheric pressure. The trapped air bubbles in this case, will shrink with time and eventually disappear; the time period of this being dependant on the absolute pressure within the channel and also on the flow rate as explained in our previous studies [15,16]. To prevent the shrinkage of air bubbles at a pressure higher than atmospheric pressure, the textured surface is externally heated by a resistance heater while maintaining the absolute pressure at the same value. As a result of this heat supply, the local temperature of water near the surface increases. By varying the heat supplied, the local temperature of water can be effectively controlled to vary the degree of supersaturation of the water locally. The effect of heat transfer on the behavior of trapped air bubbles is illustrated in Fig. 3 which shows the time sequence of images of the bubbles when the absolute pressure within the channel is maintained at 12 kPa above the atmospheric pressure and the flow rate

through the channel is 3 ml/s. For each image a corresponding schematic is also shown to illustrate the shape of the air-water interface at different instants of time. Fig. 3 (a) shows the initial condition of the air bubbles. Since the pressure within the channel is above the atmospheric pressure, the water flowing through the channel is undersaturated. Since the flowing water is undersaturated air trapped inside on the textured surface will migrate across the airwater interface into the flowing water. As a result the entrapped air bubbles shrink with time. In order to make the water supersaturated at this pressure, the heater was turned on at about 7 min after immersion. The power input to the heater was maintained constant at 50 W. This external heating raises the temperature of the textured surface, which in turn heats the water locally near the surface, thus altering its air solubility. For the initial 3 min after the heater is turned on, the air bubbles continue to shrink, because the temperature rise is not sufficient enough to make the water supersaturated at the high pressure (12 kPa above atmospheric) maintained within the channel. However when the heating is continued, the temperature of water rises further thus making the water near the surface to become locally supersaturated with air. This causes the dissolved air in the flowing water to migrate into the air pockets thus enabling growth of the trapped air bubbles on the surface. Fig. 3(b) shows the condition of the bubbles just before the commencement of bubble growth. As the heating is continued, the air bubble size gradually increases (Fig. 3(c)–(d)). In Fig. 3(d), the air bubbles have grown bigger and appear to be protruding out of the holes. If the heating is continued further, the bubbles grow even bigger and start merging with neighboring bubbles to form bigger bubbles which eventually get detached from the surface. The growth, merging and detachment of the bubbles is found to be similar to our previous study [16] for the case when water was supersaturated by changing the pressure. From the images, such as those in Fig. 3, the diameter of the air bubbles in the streamwise direction (D) was measured. Fig. 4

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Fig. 3. Time sequence of images visualizing trapped air bubbles on a hydrophobic surface with blind holes. The pressure in the channel (Pchannel) is maintained at 12 kPa above the atmospheric pressure. The flow rate is 3 ml/s. The times after immersion are (a) t = 0 min, (b) t = 10 min, (c) t = 35 min, (d) t = 61 min. Heat input is provided at 7 min after immersion which cause the air bubbles to grow with time. The schematics shown indicate the air-water interface corresponding to each of the time sequence of images. Till heat is supplied air bubbles decay with time.

1.1 + 12 kPa

(d)

1.3

Q: 3 mL /s

+ 12 kPa

1.2 1.1

D/D i

shows the variation of the measured diameter (D) of the bubbles normalized by the initial bubble diameter (Di). It can be seen that initially, when the heater is not in operation, there is decrease in the size of the bubbles as seen in the images in Fig. 3(a) and (b). The heater is switched on at time th (Fig. 4). However the bubble diameter D/Di continues to decrease for a very short period of time before the growth of the bubble ensues at time tg, after which D/Di gradually increases with time (Fig. 3(c) and (d)). Variation in the bubble size (D) with time for different power inputs of 15 W, 21 W, 30 W and 50 W given to the heater is shown in Fig. 5. Variation in bubble size for the 0 W case (no heat supplied) is also shown. When no heat is supplied (0 W case) the entrapped air bubbles shrink in size with time because the flowing water is undersaturated at the higher pressure of 12 kPa (gauge) maintained within the channel. In the case of 15 W heat supply, the growth of air bubbles does not occur and with time the air bubbles gradually shrunk in size. This is because, even when the power

1

0.9 0W 15 W 21 W 30 W 50 W

0.8 0.7 0

20

40

60

80

100

120

140

t-tg (minutes) Fig. 5. Bubble growth with time at different power inputs of 0 W, 15 W, 21 W, 30 W and 50 W to the heater. The bubble diameter (D) at any instant has been normalized with the initial diameter of the bubble (Di). At 0 W, the air bubble size rapidly decreases with time. At 15 W, the air bubble size decreases with time. At higher power inputs of 21 W, 30 W and 50 W there is an increase in size of the bubbles as heat is supplied. At higher power inputs to the heater, the rate of growth of air bubbles is higher.

1.05

(a)

D/D i

1

(c)

tg 0.95

(b)

0.9

Heater on Q: 3 mL /s

0.85

0

th

15

30

45

60

Time (minutes) Fig. 4. Variation in bubble size (D) with time at a heat input of 50 W. The channel pressures (Pchannel) is maintained at 12 kPa above the atmospheric pressure. The bubbles show gradual shrinkage at initial times. At time th, the heater is switched on. On heating, the size of the air bubbles increase with time. The time at which air bubbles start growing is marked as tg. Data is normalized by the initial bubble diameter (Di). , Heater off; , Heater on.

input to the heater is 15 W, the local temperature rise of water is not sufficient enough to make the water supersaturated at the pressure of 12 kPa above atmospheric pressure maintained inside the channel. However it may be noted that, the addition of heat, even if it is in small amounts, will have a salutary effect on the degree of saturation of water. Hence in the case of 15 W power input, even though the bubbles shrink in size with time, the rate of shrinkage is lesser than the case when no power is supplied. For the higher power inputs of 21 W, 30 W and 50 W, the qualitative trends for the bubble size variation with time were nearly the same. In all the three cases, the air bubbles gradually grow in size with time. The change in size of air bubbles for 21 W, 30 W and 50 W cases is shown from time (tg), when growth of air bubbles begin to occur. A distinct difference is observed on the rate of growth of the bubbles for the three cases, with the bubble size (D) increasing more rapidly at higher heat inputs as shown in Fig. 5. This is because, as the power input is increased, the local supersaturation

D. Dilip et al. / International Journal of Heat and Mass Transfer 119 (2018) 551–563

of water also increases, leading to higher rates of growth at higher power inputs. In all experiments discussed so far, the pressure within the channel was maintained at 12 kPa above the atmospheric pressure. At this pressure, the flowing water becomes highly undersaturated which results in rapid shrinkage of the air bubbles as soon as the flow is commenced. Even when heat is supplied, it takes a finite amount of time to raise the temperature of water to a stage when it will be supersaturated with air. During this time lag the air bubbles would continue to shrink. As a result even for the experiments in which the heater was on, right from the start, the bubbles had shrunk to small size, before the temperature increase of water due to heating reversed the trend as seen in the 21 W, 30 W, and 50 W cases. In order to correlate the growth rate of bubbles with the temperature rise of water, the temperature difference of water between the inlet and outlet of the channel ðT o  T i Þ is plotted against time for various heat inputs, as shown in Fig. 6. In all the cases, the temperature difference ðT o  T i Þ initially increases and then tends to stabilize to a steady value at later times. This is expected because, after the initial transients, the heat transfer reaches a steady state, and hence ðT o  T i Þ remains at a constant value at later times. For the 15 W case, the measured maximum difference ðT o  T i Þ is about 0.3 °C, whereas in the 21 W, 30 W and 50 W case the maximum temperature difference is about 0.5 °C, 0.8 °C and 1.2 °C respectively. From the temperature traces and bubble observations, it is clear that, even minor changes in temperature of water are sufficient enough to cause the growth of air bubbles. From Figs. 5 and 6, it can be seen that the rate of growth of the bubble is strongly dependent on the temperature. Larger the difference in temperature ðT o  T i Þ, faster is the rate of growth of the bubble. The variation in the bubble size that happens over time doesn’t seem to have a significant impact on the temperature difference ðT o  T i Þ. From Figs. 5 and 6, it can be seen that by controlling the water temperature through heat transfer, the rate of growth of the air bubbles on the surface can be controlled. The increase in water temperature results from the heat transferred from the heater to the water across the textured surface. When power input is given to the heater, it heats up the textured surface. The heat transfer from the textured surface to water is caused by the temperature difference between the textured surface temperature (Ts), and the mean fluid temperature T f ¼ ðT i þ T o Þ=2. The variation in the temperature difference ðT s  T f Þ plotted against time is shown in Fig. 7. With time,

+ 12 kPa

1.4

Q: 3mL /s 50 W

1.2

To-Ti (oC )

1 30 W

0.8 0.6

21 W

0.4

15 W

0.2 0

0

50

100

150

t-th (minutes) Fig. 6. Variation in the temperature difference of water ðT o  T i Þ across the channel as a function of time at a various input power of 15 W, 21 W, 30 W and 50 W to the heater. ðT o  T i Þ gradually increases with time till a steady state value is reached. Larger the heat input, larger is the temperature difference of water across the channel.

2.4 + 12 kPa

Q: 3mL /s

50 W

2 1.6

Ts-Tf (oC)

556

30 W

1.2 21 W

0.8 15 W

0.4 0

0

50

100

150

t-th (minutes) Fig. 7. Variation in the temperature difference of water ðT s  T f Þ as a function of time at a input power of 15 W, 21 W, 30 W, 50 W to the heater. With time, ðT s  T f Þ gradually increases and reaches a steady value. Larger the heat input, larger is the temperature difference.

ðT s  T f Þ gradually increases and tends to reach a steady value at larger times. The rate of increase in ðT s  T f Þ is higher at higher power inputs. The steady state value to which ðT s  T f Þ tends to, is higher at higher heat inputs. For a power input of 15 W, the maximum temperature ðT s  T f Þ is about 0.55 °C, whereas at 21 W, 30 W, and 50 W, it is 0.9 °C, 1.35 °C and 2 °C, respectively. It may be noted that temperature increase of the surface required to make the air bubbles to grow is very minimal and in our case it is in the range of 0.2–2.5 °C which is well below the range of temperature increase required for boiling. 4. Effect of heating on pressure drop Localized heating of water flowing over the textured hydrophobic surface causes the trapped air bubbles on the cavities to grow as seen from Figs. 4 and 5; the rate of growth of the bubbles being dependent on the rate at which heat is supplied (Fig. 5). It would be interesting to see how the pressure drop (DP) across the channel, varies as a result of the change in size of air bubbles, consequent to heating. For this, the pressure drop for all the different cases of power input is measured and compared with that of the reference case i.e. in Wenzel state. First the pressure drop across the channel was measured when the holes were completely filled with water for various flow rates across the channel. This was used as the reference pressure drop in our experiments, and the measured values as a function of flow rate are shown in Fig. 8 from a set of about 3 runs. As expected at low flow rates within the laminar range, the pressure drop shows a linear variation with flow rate. All pressure drop measurements shown with trapped air later in this chapter (DP) are normalized by the reference pressure drop (DPo) shown in Fig. 8, for the corresponding flow rate (Q). The time variation of pressure drop (DP) for a case where the input power to the heater is 50 W is shown in Fig. 9. The data corresponds to the bubble visualization images shown in Fig. 3, with the time instants corresponding to the images (a–d) in Fig. 3 also marked on the pressure drop plot. Initially, when no heat is supplied the bubbles gradually shrink (as in Fig. 3(a) and (b)) resulting in a decrease in pressure drop. The heater was turned on after about 7 min. A further reduction in pressure drop was observed for a few minutes after the heater was switched on. At about 10 min after immersion, the minimum pressure drop was achieved (Fig. 3(b)). Beyond 10 min, because of the rise in temperature caused by the heat transfer, the air bubbles begin to grow in size (Fig. 3(c) and (d)), causing a progressive increase in the pressure drop. With time, as the air bubbles grow, the pressure drop further

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300

1.1 + 12 kPa

1 200

0.95

P/ Po

Pressure drop P (Pa)

Q: 3 mL /s

1.05

250

150

0.9 0.85

100

15 W 21 W 30 W 50 W

0.8 50 0

0.75 0

2

4

6

8

10

0.7

0

20

Flow rate Q (mL/s)

40

60

80

100

120

140

160

180

t-th (minutes)

Fig. 8. Pressure drop measurements for the microchannel with no trapped air on the textured surface. The pressure drop (DP) shows a linear variation with time (Q) as expected in the laminar regime.

a

b

1.04 50 W

+ 12 kPa

1.02

(d)

1

Fig. 10. Measured normalized pressure drop (DP/DPo) with time corresponding to variations in the input heat supplied. The important difference caused by input power variation is the rate of increase in pressure drop, with more rapid growth of the bubbles occurring at higher heat inputs. The schematic shown indicate the airwater interface at (DP/DPo)  1 for (a) 21 W, 30 W and 50 W cases where the air bubbles have grown and is protruding out of the holes, and (b) 15 W case where the air bubbles have shrunk and completely disappeared; the cavity is fully filled with water.

P/ Po

0.98 (a) 0.96 0.94

(c)

0.92 0.9

(b)

Q: 3 mL/s

Heater on

0.88 0

th

10

20

30

40

50

60

Time (minutes) Fig. 9. Measured pressure drop (DP) with time at a power input of 50 W supplied to the heater. The data shown here corresponds to the bubble visualizations shown in Fig. 3, with the time instants corresponding to images (a–d) also being indicated here. The pressure drop decreases initially due to shrinkage of the bubbles. Heater is turned on at time th. On heating, the pressure drop increases as the bubble grows, and becomes greater than the reference value. , Heater off; , Heater on; - - -, Reference.

increases reaching values that are greater than the reference pressure drop, DPo. If the heating is continued, the pressure drop will increase further as the growing bubbles protrude more and more into the flow, thus partially blocking the flow of water through the channel. The growth of the air bubbles will lead to eventual merging and detachment from the surface. Bubble merging and detachment caused by heating, would also influence the measured pressure drop which is expected to follow the same qualitative variation shown in our previous study [16] where the bubble was grown by controlling the absolute pressure of water. Variation of the pressure drop (DP) with time for different power inputs of 15 W, 21 W, 30 W and 50 W provided to the heater is shown in Fig. 10. As seen from the plot, for all the cases, the pressure drop gradually rises and becomes equal to that of the reference pressure drop. At a heat input of 50 W, the air bubbles grow rapidly with time and hence the pressure drop increases at a higher rate to reach the reference pressure drop in about 50 min. When the heat input was decreased to 30 W, the time taken for the pressure drop to reach the reference pressure drop was twice as that of the case of 50 W. At a heat input of 21 W, the growth of air bubbles occurs at a slower rate compared to 30 W. At the initial condition when the heating was started (t = th), the bubbles were small. On heating, the pressure drop across the channel reduced with time

till it reached a minimum pressure drop. However when the heating was continued, the pressure drop gradually increased. From the visualized images of the air bubbles, it was observed that at 21 W, the bubble size remained almost the same for a long period of time delivering substantial drag reduction throughout this period (0 < t – th < 140 min) (Fig. 10). However, at a heat input of 15 W, it was found that the heat input was not sufficient enough to prevent the air bubbles from shrinking and eventually disappearing. As a result of the gradual air bubble shrinkage, the pressure drop gradually increased to become equal to that of the reference pressure drop. Hence even through the pressure drop in all the 4 cases eventually increased to reach the reference pressure drop, the reason for the increase is different for the 15 W case. The schematics shown in Fig. 10 clearly indicates this difference with (a) showing the air bubble protruding out of the holes at (DP/DPo)  1 which is the case for 21 W, 30 W and 50 W, whereas (b) shows the cavity devoid of air and completely filled with water at (DP/DPo)  1 which is the case for 15 W. Nevertheless, it may be noted that even if the heat input is not sufficient enough to cause growth of air bubbles, as in the 15 W case, the time taken for the pressure drop to increase and reach the reference pressure drop would still be larger for such cases than the case without any heating at all. This is because any heat input would increase the temperature of water thus changing its air solubility thereby enabling the air bubbles to stay on the surface for longer periods of time. The rate of increase in pressure drop is found to increase at higher power inputs to the heater because the water undergoes larger changes in temperature at higher inputs as shown in Fig. 6 causing the bubbles to grow more rapidly. From Fig. 10, it is seen that, it is possible to sustain the air bubbles and hence the drag reduction for long periods of time by varying the input power to the heater or in effect the local temperature of water. For example, in Fig. 10, at 21 W the bubbles remained on the surface for a longer period of time. Therefore it is clear that by

D. Dilip et al. / International Journal of Heat and Mass Transfer 119 (2018) 551–563

providing an optimum heat input, the bubbles can be sustained for long. The experimental results for the case of 21 W input power, in which the maximum pressure drop was obtained (Fig. 10) for a longer period of time is subjected to further analysis to understand the effect of trapped air bubbles on pressure drop and heat transfer. Fig. 11 shows the variation in the pressure drop with the normalized bubble diameter (D/Dh), Dh being the diameter of the hole. When the bubble diameter is 93% of the hole diameter, the pressure drop is found to be about 10% lower than the reference pressure drop. The bubble size is smaller than the hole size (D/Dh < 1), because at the higher pressure maintained inside the channel, the water is undersaturated and as the heater is not in operation, the bubbles have shrunk in size and the air-water interface has slightly moved into the hole. As heat supply commences, the bubble size gradually increases, and the pressure drop is found to decrease and reach a minimum when D/Dh  1, i.e. when the bubbles size had increased and become almost flush with the surface. As bubbles further grows, the pressure drop increases to reach the reference pressure drop when the bubble size is about 114% of the hole size. At this condition the air bubbles are slightly protruding out of the holes (D/Dh > 1). It may be noted that when heating is continued further, the protrusion of air bubbles into the flow become more and more pronounced, and the protruding bubbles will cause a partial blockage of flow through the channel. At larger bubble sizes, greater is the blockage of the flow leading to larger pressure drops. When bubble size is larger than the size of the hole, due to blockage, there is an immobilized layer of liquid formed near the wall leading to an increased pressure drop. A pressure difference would be created between the front and rear of the bubbles, and the resulting force acting on the bubble also contributes to the larger pressure drop across the channel. This means that, the presence of the air bubbles on the surface, does not essentially reduce drag, but it may even increase it if the bubble sizes are large. Whether drag reduction can be achieved or not is thus found to be strongly dependant on the curvature of the air-water interface, with the maximum drag reduction being obtained when the normalized bubble size is close to 1. This observation is consistent with recent analytical and theoretical studies [52–55] and also the experimental findings from our previous studies [15,16] when bubble size was varied by controlling pressure. The maximum pressure drop reduction obtained at the optimum bubble size is observed to be about 15%. It may be noted that the maximum drag reduction of 15%, even though significant, is much lower compared to shear free fraction of 55% imparted to the textured surface, because the maximum slip velocity is attained only at the centre of the shear free regions and will be much lesser on either sides with the velocity gradually reducing from the centre to zero at the adjacent noslip zones. It may also be noted that in the present experiments, the heat input is kept at a constant value in each case and was not altered to compensate for any atmospheric temperature variations. In a practical system, however, through a feed back control, it is possible to adjust the input based on the requirement, so that minute control of the bubble size can be realized. Hence, air bubbles can be made to remain on the surfaces at an optimum size, thus delivering maximum and sustained drag reduction for longer periods of time.

5. Heat transfer across the textured hydrophobic surface In the previous sections, it has been shown that textured hydrophobic surfaces can be useful in delivering substantial drag reduction in microfluidic applications. We also showed that, by controlling the local temperature, the size of the air bubbles can be controlled and hence sustained and maximum drag reduction

1.05 + 12 kPa

1

P/ Po

558

0.95

0.9

0.85 21 W

0.8 0.93

0.96

Q: 3 mL /S

0.99

1.02

1.05

1.08

1.11

1.14

D/D h Fig. 11. Pressure drop versus normalized bubble diameter in flow through a textured hydrophobic microchannel with schematics showing the corresponding water-air interface. The absolute pressure within the channel is higher than the atmospheric pressure (Pchannel > Patm). On heating the temperature of water increases and the bubble size gradually increases with time. The maximum pressure drop is obtained when the bubble is flush with the surface. The power input to the channel is 21 W.

can be obtained for longer periods of time. However, in many applications at the micro scale, such as electronic cooling, in addition to the pressure drop reduction, achieving higher rates of heat transfer is also equally important. So far, there have been only very limited research on the potential of using hydrophobic surfaces for applications involving heat transfer. The main disadvantage of using hydrophobic surfaces in such applications is that the trapped air being an insulator tends to reduce the heat transfer across the surface. However, if substantial amount of drag reduction can be achieved by using textured hydrophobic surfaces, it may be possible to achieve a higher heat transfer rate for the same pressure drop. So it is very important to understand the thermal transport characteristics of hydrophobic surfaces relative to the amount of drag reduction that can be achieved. 5.1. Heat transfer calculations The suitability of textured hydrophobic surfaces for heat transfer applications can be ascertained only by determining the effect of trapped air bubbles on the heat transfer rate. To this end, temperature measurements were carried out at salient points in the microchannel as shown in Fig. 2 to evaluate the heat transfer coefficient when the holes on the textured surface were filled with bubbles and compared to the case when the holes were devoid of air bubbles. For various heat inputs, the variation of the heat transfer coefficient with time is determined. The step by step method and equations used for the determination of the heat transfer coefficient are given below. The heat flux q00 (W/m2) to the water is given by

q00 ¼

q A

ð1Þ

where q is the heat transferred to water in Watts given by q ¼ ðVI  heatlossesÞ, V being the voltage and I, the current supplied, and A is the test surface area. As the heat supplied to water is used to raise the water temperature from the channel inlet to outlet, the heat transferred to water can be expressed as:

_ p ðT o  T i Þ; q ¼ mC

ð2Þ

_ is the mass flow rate of water in kg/s, C p is the specific where m heat of water in kJ/kg K, and Ti and To are the inlet and outlet

D. Dilip et al. / International Journal of Heat and Mass Transfer 119 (2018) 551–563

temperatures of water respectively. The heat transfer coefficient h (W/m2 K) is given by

h¼

q ; ADT

ð3Þ

where DT ¼ ðT s  T f Þ; Ts is the average surface temperature given by T s ¼ ðT s1 þ T s2 Þ=2 and Tf is the mean fluid temperature given by T f ¼ ðT i þ T o Þ=2: From Eqs. (2) and (3), the heat transfer coefficient h can be evaluated as:

h¼

_ p ðT o  T i Þ mC : A DT

ð4Þ

The hydraulic diameter of the channel,

Dh ¼

2xy ; xþy

ð5Þ

where x and y are the height and width of the channel respectively, and the corresponding Reynolds number is

Re ¼

quDH ; l

ð6Þ

where q is the density of water in kg/m3, u is the average velocity of water flow through the channel. The Nusselt number is given by:

Nu ¼

hDH ; kf

ð7Þ

where kf is the thermal conductivity of water (W/m K). The uncertainty in Nu was estimated to be less than ±2.8%, with the largest contribution to the uncertainty coming from the temperature measurements. 5.2. Effect of trapped air bubbles on thermal transport The time variation in the non dimensional heat transfer coefficient (Nu) at various power inputs to the channel plotted against time is shown in Fig. 12. With time, the Nusselt number gradually increases and then stabilizes at nearly constant value at large times, when steady state is attained. For all the different cases of power input, the same qualitative trend was observed. Nusselt number for different power inputs showed minor variations in its steady state value, with higher power inputs resulting in marginally higher Nusselt number. The steady state Nusselt number varied from 3.6 for the 15 W case to 3.9, when the power input to the heater was 50 W.

5 + 12 kPa

Q: 3 mL /s

4.5

To assess the combined hydraulic and thermal performance of the channel, results of the 21 W heat input case was chosen, since the maximum pressure drop reduction was obtained for the 21 W case. The variation of the temperature difference of water between the inlet and outlet of the channel ðT o  T i Þ with time for the 21 W is shown in Fig. 6 whereas the temperature difference between the surface and the mean water temperature Tf, is shown in Fig. 7. In the time variation of the non dimensional heat transfer coefficient, Nu for the case of 21 W power input, from Fig. 12, it is seen that there is an initial increase in Nusselt number at small times after the heater is on, but after about 40 min, the Nusselt number (Nu) attains a constant value. It is found that throughout the experiment, the Nusselt number is consistently lower than that of the reference value of the Nusselt number (Nuo) shown by the dashed line (Fig. 12) which corresponds to the value of Nusselt number for the 21 W case at large times, when the holes are completely filled with water and are devoid of any air bubbles. The reduction in Nusselt number observed for the case with air bubbles compared to the case without air bubbles, is expected, because air has a low thermal conductivity and the presence of trapped air bubbles on the surface acts as an insulating mechanism which hinders the heat transfer across the surface. The variations in the bubble size, caused by the heating, does not seem to influence the value of the Nusselt number appreciably, which is evident from the near constant value of Nusselt number after 40 min seen in Fig. 12, even though the bubble size is found to gradually increase even beyond 40 min as seen in Fig. 5. However it may be noted that, in the present case, the change in size of the bubble was limited to a small range (Fig. 5), and when the air bubble size becomes very large, the Nusselt number is likely to decrease. From Figs.10 and 12, it is clear that, even though using a textured hydrophobic surface inside a microchannel will result in a reduction in the pressure drop, the thermal transport performance is compromised due to the presence of air bubbles on the surface. In order to assess the suitability of textured hydrophobic surfaces for micro scale cooling applications, such as electronic cooling, it is thus required to define an overall performance index which evaluates the heat transfer performance relative to the pumping cost, with the maximum value of the index being optimal from the cooling point of view. The combined effect of reduction in Nusselt number and pressure drop in flow over textured hydrophobic surfaces may be assessed by comparing the ratio of Nusselt number and pressure drop between the cases when the holes on the surface are filled with air bubbles (Cassie state) and when the holes are completely filled with water (reference Wenzel state). In the case of the surface holes filled with bubbles, this ratio is denoted by (Nu/DP) whereas for the case when the holes are filled with water (reference state), the ratio is denoted by (Nuo/DPo). The combined effect of Nusselt number and pressure drop reduction is then given o by the index, ðNu Þ=ðNu Þ. We define this index as the ThermoDP DP o Hydraulic Performance Index (THPI) which can also be written in the form:

4

Nu

3.5

THPI ¼ 3 15 W 21 W 30 W 50 W

2.5 2 1.5

559

0

20

40

60

80

100

120

140

160

180

t-th (minutes) Fig. 12. Variation of Nusselt number (Nu) with time with air bubbles trapped on the surface. The power input to the heater are 15 W, 21 W, 30 W and 50 W. With time, Nu increases and attains a steady state value for all cases. Nu for the reference surface (holes filled with water) for the 21 W case is shown by dashed lines.

ðNu=Nuo Þ ðDP=DPo Þ

ð8Þ

where the denominator represents the reduction in the measured pressure drop due to the presence of entrapped air bubbles with respect to the reference pressure drop and the numerator indicates the reduction in heat transfer coefficient due the presence of air bubbles with respect to the reference case. THPI is similar to the performance indices that have been used earlier to analyze the combined thermal and hydraulic performance in wavy microchannels for electronic cooling [56,57]. THPI is also similar in character to the performance index used by Maynes et al. [42] and the ‘goodness factor’ used by Cheng et al. [46] in their numerical studies on

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flow and heat transfer over superhydrophobic surfaces. The uncertainty in THPI has been estimated from the uncertainties of Nu and DP, and was found to be within ±3%. When THPI > 1, it indicates the relative dominance of the pressure drop reduction in comparison with the reduction in heat transfer, and hence an enhancement in the overall performance. Fig. 13 shows the variation of THPI with time. It is found that THPI values are initially lesser than 1, indicating a decrease in the overall performance due to the presence of air bubbles. With time, THPI increases and at about 40 min becomes greater than 1, indicating a gain in the overall performance. However at later times, i.e. after about 130 min, the value of THPI is found to become less than 1, indicating a reduction in the overall performance. THPI is found to be greater than 1 for a time period of about 90 min. This shows that during this period, even though the presence of air bubbles causes the heat transfer coefficient to decrease, this effect is offset by the pressure drop reduction obtained due to the presence of bubbles. This is consistent with the numerical results of Maynes et al. [42] and Cheng et al. [46]. It may be noted that THPI becomes greater than 1 solely due to the large pressure drop obtained, when the size of the air bubbles are nearly flush with the surface, even when the Nusselt number consistently remained lower than the reference case and at a near constant value after 40 min as seen in Fig. 12. Hence it is clear that by controlling the local water temperature near the surface, the size of the trapped air bubbles can be maintained nearly flush with the surface and superior combined thermal-hydraulic performance can thus be obtained from a textured hydrophobic surface. In fact, these results provide the first experimental validation that the presence of air bubbles on the textured surface enhances the combined thermal-hydraulic performance of textured hydrophobic surfaces in microchannel flows. 6. Discussion The size evolution of entrapped air bubbles indicates that when the temperature of water is locally varied by heating the textured surface by sufficient amounts, air bubble size increases with time; the rate of increase being dependant on the quantum of heat input. In our experiments, the pressure within the channel (Pchannel) was maintained above the atmospheric pressure (Patm) so that water while flowing through the channel is undersaturated, when the heater is not in operation. The degree of saturation of water f is defined as (f = C1/Cs), where C1 is the concentration of air in water

1.2 21 W

+ 12.5 kPa

1.1

THPI

1

at any instant of time, and Cs is the saturation concentration. The saturation concentration of air in water (Cs) at a temperature T is governed by the Henry’s law given by:

Cs ¼

ð9Þ

where P is the partial pressure of gas at saturation and Hair is the Henrys constant which is a strong function of temperature. Henry’s constant for air can be obtained from the Henry’s constants for Oxygen and Nitrogen as

Hair ¼ where

nO 1 nair HO nO nair

1 þ nnN

and

air

nN nair

1 HN

ð10Þ

are the molar fractions of Oxygen and Nitrogen

respectively. The temperature dependence of the Henrys constant is given by the Van’t Hoff equation [58,59] as,


 1 1  Hair ðTÞ ¼ Hair ðhÞexp C T h

ð11Þ

where h is the standard temperature (298 K) and C is a constant in Kelvin. Any increase in temperature, will cause the Henry’s constant to increase which will cause the saturation concentration of air in water to decrease from its initial value, thus making the water supersaturated (f > 1) at the same pressure. This would cause some of the dissolved air in water to migrate into the air pockets resulting in growth of the air bubbles. The size evolution of a single stationary spherical air bubble suspended in an infinite medium can be expressed as a function of the degree of saturation f of water and is given by the EpsteinPlesset equation [60] as

" #
2 D 8kC s ðf  1Þ t ¼1þ Di qD2i

ð12Þ

where D is the diameter of the bubble at any instant t, Di is the initial diameter of the bubble, k is the diffusivity of air in water, q is the density of air, f is the degree of saturation (f = C1/Cs) and Cs is the saturation concentration of air in water. However, in the presence of flow, as in the case of the present experiments, the change in the bubble diameter with time, predicted by Eq. (12) is found to be much lower, when compared with the experimentally observed variation. This is because, Eq. (12) is based on the solution of the diffusion equation obtained by Epstein and Plesset [60] for dissolution of stationary spherical bubbles in a stationary infinite medium. In the present experiments, Peclet number (Pe = UD/k, U being the velocity of flow relative to the bubble) which gives the ratio of the contribution to mass transport by convection to that by diffusion is found to be large (Pe  1) and hence the removal of air from the cavities is primarily due to mass convection caused by the flow rather than by diffusion. The convective mass transfer rate from or to the bubble is then given by:

_ a ¼ hm As ðC s  C 1 Þ; m

0.9

ð13Þ

where ðC s  C 1 Þ, is the difference in concentration of dissolved air in water across the interface, As is the bubble surface area and hm the convective mass transfer coefficient. The non-dimensional mass transfer coefficient Sherwood number is given as:

0.8 0.7 Q: 3 mL /s

0.6

P ; Hair

0

50

100

150

200

t-th (minutes) Fig. 13. Variation of the Thermo-hydraulic performance index (THPI) with time, which represents the variation of heat transfer coefficient relative to the pressure drop with time. THPI is higher than 1 between 40 min and 130 min which indicates an enhancement in the combined thermal-hydraulic performance in this duration.

Sh ¼

hm D ; k

ð14Þ

where D is the bubble diameter and k is the diffusivity. For a spherical bubble exposed to flow, when Pe >> 1, the Sherwood number scales with the Peclet number [61] as

Sh ¼ aPe1=2

ð15Þ

D. Dilip et al. / International Journal of Heat and Mass Transfer 119 (2018) 551–563

where a is the proportionality constant. From Eqs. (13)–(15), it can be shown that the size evolution of the bubbles can be expressed in the functional form:


3=2 D ¼ 1 þ st Di

ð16Þ

where the parameter s is dependent on the degree of saturation f, and is given by:

s¼a

3C s

q

1=2

ðf  1ÞðkUÞ

3=2

ð17Þ

Di

where a is a proportionality constant from Eq. (15) and f is dependent on the temperature through Eqs. (9) and (11). For bubbles, exposed to flow, similar functional form has been previously obtained by Al-Hayes and Winterton [62] where the exponent of diameter D is 3/2. The functional form given by Eqs. (16) and (17), clearly shows the dependence of the size evolution of the bubbles on the degree of saturation f and the relative flow velocity U between the bubble and the flow. Since the relative flow velocity is a function of the flow rate Q, it follows that for a fixed flow rate through the channel, the growth or shrinkage of the bubble and the rate at which it occurs is dependent only on the degree of saturation f. When f > 1, it results in growth of the bubbles and when f < 1, shrinkage of the bubbles occur. It may be noted that since the air bubbles are in contact with the heated surface, it is likely that the air bubbles may undergo thermal expansion. However, since air is an ideal gas, by using ideal gas equation it can be shown that for the range of temperature increase of the surface discussed in this study, the change in diameter of the bubbles due to thermal expansion will be negligibly small. In Fig. 14, experimental data for change in the bubble diameter (D) during growth/shrinkage of air bubbles is plotted in the form of 1.4

(a)

Q: 3 mL /s

q = 50 W

1.3

q = 30 W

(D/D i)3/2

1.2

q = 21 W

1.1 1 0.9

q= 0W

0.8

q = 15 W + 12 kPa

0.7

0

20

40

60

80

100

120

140

Time (minutes) 0.008

(b)

Q: 3mL /s

0.006 0.004

s

0.002 0 -0.002 -0.004 + 12 kPa

-0.006 0

10

20

30

40

50

60

q (Watts) Fig. 14. Rate of change of bubble size with time at different heat inputs to the channel. Variation in the initial slopes of the data (s) in (a), plotted against various power inputs is shown in (b).

561

(D/Di)3/2 with time. The data corresponds to heat inputs of 0 W, 15 W, 21 W, 30 W and 50 W. The bubble size data shows a nearly linear variation with time as seen from Fig. 14(a). The slope of the data at small times gives the parameter s in Eq. (16). The variation in s corresponding to Fig. 14(a) is shown in Fig. 14(b). The rate of change in size of the bubble in Fig. 14(a) increases linearly as power input q is increased. This means that by changing temperature, the rate of change in the bubble can be controlled. Thus by controlling the local temperature of water, through heat supply, the dissolution of entrapped air bubbles into water can be prevented even at pressures greater than the atmospheric pressure within the channel. The underwater Cassie state of wetting can thus be maintained on the surface for extended periods of time. It may be noted that even though the water used in the experiments was allowed to saturate at atmospheric pressure, the degree of saturation of water at the entrance to the channel may become higher than 1, because the heat supplied to the channel can affect the inlet temperature of water to the channel, thereby changing the degree of saturation (f) of water at inlet. In addition, heat losses from the channel through the transparent glass (which serves as one of the walls of the channel), along with the other heat losses from the channel tends to increase the temperature of water surrounding the channel, provided to facilitate the air-bubble visualization through TIR. Any increase in the surrounding water temperature is also bound to influence the temperature of water at the inlet to the channel and thus alter its degree of saturation from the expected value. This being the case, the local degree of saturation of water in the vicinity of the surface, is determined not only by the temperature increase of water from the inlet to the outlet, but also due to the increase in the inlet water temperature relative to the temperature of water at the inlet tank where the degree of saturation is 1. It may be noted that any increase in the temperature of water at the inlet to channel from the wet bulb temperature of ambient air can significantly influence the degree of saturation at inlet and thereby affect the evolution in the bubble size. The value of the average Nusselt number predicted by the classical theory for fully developed laminar flow in a one side insulated rectangular duct of high aspect ratio, for uniform heat flux in flow direction and uniform wall temperature at particular flow cross section is 5.385 [63]. In the present experiments, the estimated Nusselt number from the experimental data, when the holes on the textured surface were free from air bubbles (i.e. completely filled with water) is 4.107, which is reasonably close to the classical value. The Nusselt number estimated from the experimental data, when the surface holes are filled with air bubbles is found to be 3.65. It may be noted that, in the microchannel used in the present experiments, only one side is heated and the other side consists of a wall made of transparent flat glass which is not fully adiabatic. During experiments, the glass wall gets heated up, causing heat losses to occur through the glass wall, which eventually results in a decrease in the value of the Nusselt number from the classical value of 5.385 for a duct with one side adiabatic wall. Heat losses occurring from other parts of the channel may also contribute to the reduced value of the measured Nusselt number. Fig. 15 shows the variation of the Thermal hydraulic performance index (THPI) with the normalized diameter of the bubble (D/Dh), with schematics showing the water–air interface at a few time instants. THPI is greater than 1, when the normalized bubble diameter (D/Dh) falls between 0.97 and 1.05, indicating an overall performance enhancement. The increase in the overall performance is found to be more than 5% (Fig. 15) when the bubbles are maintained at the optimum size with the bubbles being almost flush with the surface. Hence, if the heat input is controlled in such a way that the bubble size is maintained nearly flush with the surface, then the textured hydrophobic surface can deliver an enhanced overall performance compared to the reference surface.

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1.2 21 W

+ 12.5 kPa

1.1

THPI

1 0.9 0.8 0.7 0.6 Q: 3 mL /S

0.5

0.96

1

1.04

1.08

1.12

1.16

D/D h Fig. 15. Variation of the Thermo-hydraulic performance index (THPI) with the normalized diameter of the bubble. Improvement in the overall performance results when the bubbles are almost flush with the surface.

Hence, for the same pressure drop across the channel the textured hydrophobic surface can deliver an enhancement in the thermal transport performance. 7. Conclusions Drag reduction obtained from textured hydrophobic surfaces is short-lived due to the gradual dissolution of trapped air bubbles on the surface micro-texture to flowing water. In this paper, we have presented a simple method to sustain the drag reduction in textured hydrophobic microchannels by controlling the concentration of dissolved air in water locally near the surface. For this temperature of water flowing over the surface was varied by heating the textured surface, and the consequent effect on the trapped air bubble dynamics, pressure drop across the channel and thermal transport across the surface was studied. We find that controlling the absolute temperature of water locally near the surface influences the trapped air bubble dynamics appreciably, which in turn has a significant effect on the pressure drop across the channel. It was shown that by varying the temperature, the size of the trapped air bubbles on the textured surface can be controlled, to deliver maximum drag reduction for extended periods of time. Maximum drag reduction was achieved when the trapped air bubbles were maintained nearly flush with the surface. Presence of trapped air bubbles on the surface cause a significant decrease in the Nusselt number. However, when the bubbles are maintained nearly flush with the surface, the thermo-hydraulic performance index is found to be enhanced. This is because the large pressure drop reduction obtained at optimum bubble size more than compensates for the reduction in the heat transfer. The results obtained have important implications for the design and use of such surfaces in micro-scale heat transfer applications. Conflict of interest The author declares that there is no conflict of interest. References [1] S. Shibuichi, T. Onda, N. Satoh, K. Tsujii, Super water-repellent surfaces resulting from fractal structure, J. Phys. Chem. 100 (1996) 19512–19517. [2] J. Bico, C. Marzolin, D. Quéré, Pearl drops, Europhys. Lett. 47 (2) (1999) 220– 226. [3] S. Herminghaus, Roughness-induced non-wetting, Europhys. Lett. 52 (2) (2000) 165–170. [4] Z. Yoshimitsu, A. Nakajima, T. Watanabe, K. Hashimoto, Effects of surface structure on the hydrophobicity and sliding behavior of water droplets, Langmuir 18 (2002) 5818–5822. [5] M. Callies, D. Quéré, On water repellency, Soft Matter 1 (2005) 55–61.

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