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Investigation of nanofluid bubble characteristics under non-equilibrium conditions
Chemical Engineering and Processing 86 (2014) 116–124
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Chemical Engineering and Processing: Process Intensification journal homepage: www.elsevier.com/locate/cep
Investigation of nanofluid bubble characteristics under nonequilibrium conditions Saeid Vafaei a, *, Theodorian Borca-Tasciuc b, *, Dongsheng Wen c, * a
Department of Mechanical, Materials and Manufacturing, University of Nottingham, Nottingham, UK Department of Mechanical, Aerospace and Nuclear Engineering, Rensselaer Polytechnic Institute, Troy, NY, USA c School of Chemical and Process Engineering, University of Leeds, Leeds, UK b
A R T I C L E I N F O
A B S T R A C T
Article history: Received 10 June 2014 Received in revised form 14 October 2014 Accepted 15 October 2014 Available online 22 October 2014
We report experimental and theoretical investigations of the bubble characteristics during the oscillatory growth period for several nanofluids. The nanoparticles were found to affect liquid–gas and solid surface tensions, which modulated the bubble contact angle, radius of triple line, bubble volume and the dynamics of bubble growth. To increase the accuracy of the Young–Laplace equation predictions during the bubble growth in the oscillatory period, a new method multi-section bubble (MSB) approach was developed. In this method, the bubble was divided into n sections (i.e., n = 1:N) and the Young–Laplace equation was solved for each section individually. As N increases, within each section the effects of inertia force and viscosity become reduced comparing to that of the liquid-gas surface tension. Unlike the conventional Young–Laplace approach (i.e., N = 1), the new approach is able to predict the bubble characteristics reliably in the following cases: (a) the oscillatory period when bubble is fluctuating; (b) the departure period when bubble is stretched upward, right before departure; and (c) the high shear stress condition when gas velocity is relatively high. ã 2014 Elsevier B.V. All rights reserved.
Keywords: Bubble Triple line Contact angle Solid surface tensions Nanoparticle Nanofluids Young–Laplace equation Surface wettability
1. Introduction The Young–Laplace equation has been used to predict the droplet shape and contact angle on horizontal [1] and inclined substrates. The Young–Laplace equation is also able to predict the bubble shape when both phases are in equilibrium at the interface and the effect of shear stress is negligible. However, the applicability of the Young–Laplace equation becomes invalid when: (a) the bubble is in the departure period [2], (b) the shear stress is relatively high and (c) the bubble is fluctuating [3]. One application where these conditions may occur is during boiling of fluids on hot surfaces, which has received a strong research interest with the invention of nanofluids. To address these challenges, one possible avenue would be to develop and solve a set of equations containing the additional effects of high shear and inertial forces. Here we propose an alternative solution based on using Young–Laplace equations over different sections of the bubble, in order to increase the accuracy of the prediction of bubble shape and enable the fast determination of important bubble
* Corresponding authors. E-mail addresses: [email protected] (S. Vafaei), [email protected] (T. Borca-Tasciuc), [email protected] (D. Wen). http://dx.doi.org/10.1016/j.cep.2014.10.010 0255-2701/ ã 2014 Elsevier B.V. All rights reserved.
characteristics such as volume and contact angle. The technique is applied to study the bubble characteristics inside nanofluids during the oscillatory period. It has been reported that nanoparticles are able to modify the thermal conductivity [4], viscosity [5], liquid–gas [6], and solid surface tensions, s sg s sl, [7], and possibly the gas–liquid– solid interactions at the triple line. The variation of liquid–gas and solid surface tensions would affect the force balance and consequently the dynamics of the triple line, which has a significant role on the bubble growth [3,8], and boiling heat transfer coefficient and critical heat flux [9,10]. The schematics of liquid–gas and solid surface tensions at the triple line are exhibited in Fig. 1, for bubbles and droplets. The dynamics of triple line on a substrate [8,11] or a hot plate [12–15], also depend on the nature of the gas and liquid phases, the characteristics of the solid surface [7,16,17], and the receding and advancing contact angles [18]. Nanofluids have been also found to induce different spreading and thinning (layering) behavior at the triple region compared to the pure liquids. The nanoparticles were observed to spread stepwise inside the triple region on a smooth hydrophilic glass surface. The number of layers of nanoparticles decreases in a stepwise pattern as nanoparticles gets closer to the edge of triple line. Apparently the layering phenomenon of the nanoparticles at the triple region depends on many factors such as concentration
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Nomenclature g Ro R1,R2 rd V Qn u
acceleration of gravity [m/s2] radius of curvature at apex [m] radius of curvature [m] radius of contact line [m] bubble volume [m3] nominal gas flow rate [ml/min] gas velocity [m/s]
Greek Symbols d height of apex [m] uo bubble contact angle [Deg.] ue droplet contact angle [Deg.] us asymptotic contact angle [Deg.] rl liquid density [kg/m3] rg gas density [kg/m3] s lg liquid–gas surface tension [N/m] s sg solid–gas surface tension [N/m] s sl solid–liquid surface tension [N/m] s lg n liquid–gas surface tension of nanofluids [N/m]
and characteristics of nanoparticles, nanoparticle charge, solid–liquid–gas materials and film size [19,20]. It has been shown theoretically that particles can spread the triple line to a distance of 20–50 times of the particle diameter through a structural disjoining pressure by self-ordering of particles in a confined wedge. However, the structural disjoining force only becomes significant at relatively high particle concentrations, i.e., over 20 vol.% [21]. In this paper we report the effects of different nanoparticles on the dynamics of triple line and bubble growth. Using a new multi section bubble (MSB) method presented here, the Young–Laplace equation was solved to predict the bubble shape and was compared with experimental data for bubble growth inside water, gold, silver and alumina nanofluids throughout the bubble formation period including the oscillatory period. The paper is organized as the following: Section 2 presents the new method for bubble shape prediction; Section 3 reports the experimental setup; Section 4 discusses the results and comparison between the model prediction and the experiment, and the final conclusions are summarized in Section 5 lines. 2. Prediction of the profile of bubble shape Mathematically, the Young–Laplace equation can be interpreted as a mechanical equilibrium condition between two
Fig. 2. Illustration of schematic of the bubble shape where bubble is divided into several.
fluids separated by an interface. It gives the pressure difference across the interface as a function of the product of the curvature multiplied by the gas–liquid surface tension 1 1 þ Þs R1 R2 lg
Dp ¼ ð
(1)
where R1 and R2 are the radii of the interface curvature, i.e., R1 is the radius of curvature, describing the latitude as it rotates and R2 is the radius of curvature in a vertical section of the bubble describing the longitude as it rotates. The centers of R1 and R2 are on the same line, vertical to the interface, but different location. Dp is the pressure difference between gas, pg, and liquid, pl (see Fig. 2). For bubbles, the gas pressure as a function of z is given by pg ðzÞ ¼
2s lg þ po þ rg gz Ro
(2)
The first term on the right hand side is the pressure difference at the bubble apex, po is liquid pressure at the apex, and last term is the hydrostatic gas pressure. Similarly the liquid pressure can be written as pl ðzÞ ¼ po þ rl gz
(3)
The radii of curvature are R1 ¼
ds r andR2 ¼ du sinu
(4)
where u , r, and s are, respectively, the bubble contact angle, the radius of the bubble and the length of bubble perimeter at the location of z (see Fig. 2). Ro is radius of curvature at apex. Substituting Eqs. (2)–(4) into Eq. (1), the Young–Laplace equation for bubbles becomes du 2 gz sinu ¼ ðr rg Þ ds Ro s lg l r
(5)
Similarly, the Young–Laplace equation for droplets can be written as du 2 gz sinu ¼ þ ðr rg Þ ds Ro s lg l r
Fig. 1. Schematic of forces at the bubble/droplet triple.
(6)
In case of nanofluids, the liquid–gas surface tension, s lg, has to change to the liquid–gas surface tension of nanofluids, s lg n. The Young–Laplace equation can be solved [1,22–30], with following system of ordinary differential equations for axisymmetric
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interfaces, to obtain the bubble shape dr ¼ cosu ds
(7)
dz ¼ sinu ds
(8)
dV ¼ pr2 sinu ds
(9)
where V is bubble volume. This system of ordinary differential equations avoids the singularity problem at the bubble apex, since sinu 1 ¼ r s¼0 Ro
(10)
The Young–Laplace equation has been solved to predict the shape of axisymmetric liquid pendants and sessile drops on an ideal solid surface [1,25], as well as bubble shapes under low Capillary and Webber numbers when the effects of inertia force and viscose shear are negligible comparing to that of liquid–gas surface tension. In these studies, the Young–Laplace has been applied to the whole bubble, i.e., the whole bubble shape has been predicted by one set of equations, i.e., Eqs. (5) and (7)–(9). For clarity, here we call this method the uni-section bubble (USB) approach. The accuracy of the prediction has been examined by experimental data [25,26], and numerical simulations based on the volume of fluid method and level-set method [27,28], where good agreement has been observed. These studies confirm the validity of the Young–Laplace equation in predicting the shape of droplets [1,6,7,16] and bubbles [2,8,11,23,24,31] under an equilibrium condition between gas and liquid. Solving the Young–Laplace equation, using the USB method has been unsuccessful in predicting the whole bubble shape in other conditions [3,8,11] such as: (a) the shear stress between gas and liquid is relatively high, (b) the last few milliseconds before bubble departure where the effect of viscosity is dominated, and (c) when bubble starts fluctuating. The weak equilibrium between gas and liquid at the interface is most-likely responsible for the lack of accuracy of USB method.In this study, a new method called the multi-section bubble (MSB) approach is employed, to help predict the bubble shape even under non-equilibrium situations through an approach similar to the differentiation and subsequent integration method. Here, rather than using just one section as the conventional Young–Laplace equation, the bubble shape is divided into n parts (n = 1:N) with the starting point for each bubble section (such as the height and radius) measured experimentally. Even if the equilibrium for the whole bubble cannot be met, a quasi-equilibrium status would be assumed for each section of the bubble and the Young–Laplace equations, i.e., Eqs. (5) and (7)–(9), could be applied. In this work, the systems of Eqs. (5) and (7)–(9) are solved by the Runge–Kutta methods for each individual part by using two experimental data, the radius, r, and the height, z, as the inputs at the end of the each part (see Fig. 2). Starting from the first part, n = 1, from the apex (point A), with the following initial boundary conditions rð0Þ ¼ zð0Þ ¼ u ð0Þ ¼ Vð0Þ ¼ 0
from the end of the first part. So the initial boundary conditions of the second part, n = 2, such as radius, r, height, z, contact angle, u, and volume, V, are identical with those of at the end of the first part, n = 1. However, the radius and height of the second section is provided from experimental data at the location of point B (see Fig. 2). Under such initial conditions, the same sets of ordinary differential Eqs. (5) and (7)–(9) can be then solved to provide the bubble shape at n = 2. Such a process continues subsequently until it reaches the final division (i.e., n = N). Once the bubble shape is predicted, the volume, and contact angle, can be easily extracted from the modeled shape. As the number of parts, N, increases, or size of each part, Ds, decreases (see Fig. 2), the effects of inertia force and the viscosity decrease comparing to that of the liquid–gas surface tension, which satisfy the applicability of the Young– Laplace equation. The effects of inertial force and shear stress, respectively, depend on the volume and area of each section which is changing with Ds. However, the effect of liquid–gas surface tension depends on the perimeter of top or bottom sides of each section and does not change significantly with Ds. The MSB approach is able to predict the bubble shape right before departure, solving the Young–Laplace equation where bubble is stretched upward and effect of viscosity is dominant. The USB approach is not valid for the departure period. In this work, the systems of Eqs. (5) and (7)–(9) were solved for each individual part by using two experimental inputs, which enables us to predict the bubble shape, bubble volume and bubble contact angle. It shall be noted that non-symmetrical bubble were not considered, and most of current experiments showed axisymmetric bubble profiles under given conditions. The air absorption into the base liquid was considered to be negligible since the bubble formation period (45 ms and 166 ms) was short and contact area between the air and liquid was small. The bubble shape prediction by the MSB method were compared with experimental data, and insights into bubble characteristics were extracted for bubble growth inside several nanofluids. parts. 3. Experimental setup Fig. 3 illustrates the schematic of experimental setup for capturing the formation of bubbles inside water and nanofluids. Two orifices with diameters, dn, of 0.2 mm and 0.4 mm were
(11)
the first section of the bubble shape can be predicted. Because of the continuity of the bubble shape, the second part, n = 2, starts
Fig. 3. Schematic of the experimental.
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Fig. 4. An example of bubble growth inside silver nanofluid for a nominal gas flow rate of 0.5 ml/min, from 0.4 mm substrate nozzle.
fabricated in the center of a stainless steel substrate plate, which was submerged into a transparent square-sized glass container with a 20 mm by 20 mm base and 72 mm height. The substrate nozzle was large enough to allow the triple line to expand freely and was polished to reduce roughness (with average value of the peaks and valleys, Ra = 0.021 mm, and largest difference from peak-to-valley, Rz = 0.03 mm). The glass container was filled with deionized water or nanofluids to a height of 20 mm and was open to the atmosphere. The gas flow was supplied by compressed air in a cylinder, connected to a gas flow controller (model F-200CV002 of Bronkhorst) through a pressure reduction valve. Nominal gas flow rates, Qn, in the range of 0.1–0.7 ml/min were used with an accuracy of 0.5%. Details of bubble formation, especially the dynamics of triple line, were captured by a high speed camera (Photosonics Phantom V4.3, 1200 frames/s) equipped with an optical microscope head (10X Navitar Macro zoom 7000). The resolution of the camera is 5 mm per pixel. Fig. 4 shows an example of bubble growth inside silver nanofluid for a nominal gas flow rate of 0.5 ml/min, from 0.4 mm substrate nozzle. The profile of bubble is obtained from the captured images by fast camera, equipped with a microscope during the bubble growth inside water and nanofluids from 0.2 mm and 0.4 mm substrate nozzles. Since the bubble formation period is longer for the 0.4 mm substrate nozzle, more information is available for this nozzle than for the 0.2 mm nozzle, for a given camera speed. Gold, silver and alumina nanoparticles, with a narrow size distribution averaged at 5 nm for gold and silver and 25 nm for alumina, were dispersed into deionized water without any surfactant (see Fig. 5). The gold and silver nanoparticle concentrations were controlled at 0.01% by weight, and the concentration of the alumina nanofluid was 0.0037% by weight. A drop of nanofluid was left on top of a stainless steel surface to dry slowly and the dried material was analyzed by an Energy Dispersive X-ray spectroscopy (EDX), which confirmed the purity of the nanoparticles. Images of bubbles or droplets captured at steady state were analyzed with a Drop Shape Analysis System (KRUSS, DSA 100) to obtain the gas–liquid and solid surfaces tensions of water, and nanofluids. However, the bubble growth images in the oscillatory period or when away from steady state were analyzed using the method described in Section 2. The accuracy of the surface tension measurements is 0.01 mN/m. The gas–liquid surface tensions of pure water and gold nanofluids were 0.07238 0.0041 and 0.06753 0.0066 mN/m, respectively, as determined by the instrument using six different bubble images [8,11]. The liquid–gas surface tension of alumina nanofluids at a low concentration of 0.001 vol.% was similar to that of water [10]. The liquid–gas surface tension of silver nanofluids was 0.05373 0.005 mN/m, from six different droplet images. The
details of the experimental set up can be found in references [3,8,11,22] setup. To determine the solid surface tensions, the DSA system was employed to measure the radius of the triple line and the height of six small droplets of each sample liquid, which were used as the two inputs to solve the systems of Eqs. (6)–(9). As the liquid droplets on the substrate were in stationary and equilibrium conditions, the uni-section bubble (USB) approach was applied to obtain the characteristics of droplets. Having the characteristics of droplet such as radius of droplet, rd, contact angle, ue, and droplet volume, V, the difference between the solid surface tensions was found from, nanoparticles.
s lg cosus ¼ s sg s sl " rd sinue ¼
(12)
3V
pð2 þ cosus Þð1 cosus Þ2
#1=3 sin2 us
(13)
where us is asymptotic contact angle. The experimental uncertainties for prediction of the droplet volume and contact angle are estimated to be less than 0.9%. The solid surface tensions, s sg s sl,
Fig. 5. Transmission electron microscopy (TEM) pictures of silver.
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of water, gold and silver nanofluids on stainless steel substrate, respectively, are 0.00415 0.0008 N/m, 0.0124 0.001 N/m and 0.0027 0.0004 N/m. The accuracy and details of the solid surface tensions measurement is given clearly in reference [1,7]. The gas–liquid and solid surface tension values are then imported in the MSB simulations and along with the experimentally measured section height and radius are used to accurately predict the bubble shape.
804.919 ms
3.0E-03
805.752 ms 806.586 ms 808.252 ms 810.752 ms 814.085 ms 2.0E-03
We report experimental investigations of the bubble characteristics in the oscillatory growth period for several nanofluids. A new method was developed to predict the bubble shape, using the Young–Laplace equation, when liquid and gas phases are not in equilibrium or shear stress is relatively high. In this method, the bubble is divided into n sections (i.e., n = 1:N), and n is gradually increased until the entire bubble shape can be predicted. Fig. 6 compares the experimental data with predicted shapes of two bubbles: one near the beginning and the other near the departure stage inside silver nanofluid. The Young–Laplace equation using N = 1, called here the USB approach, could not predict the bubble shapes because the effects of shear stress and viscosity acting on the entire bubble were strong compared to that of the liquid–gas surface tension. However, these effects are less important when smaller sections are modeled with the Young–Laplace equation and key characteristics (i.e., radius and height) are experimentally provided for the endpoints of these sections. As shown in Fig. 6, the small bubble needed to be divided into at least 3 parts (N = 3) and the larger bubble needed to be divided into minimum 4 parts (N = 4) for the method to converge and provide a shape that closely matches the experiment. The larger bubble required division into more parts, likely due to increased upward stretching before the departure. An example of applying the MSB method for the entire set of bubble images is shown in Fig. 7 for bubbles growing in silver nanofluid from 0.2 mm substrate nozzle for 0.1 ml/min nominal air flow rate. The figure shows the evolution of bubble growth and demonstrates the short bubble development period. For a given 3.E-03
Exp
z (m)
4. Results and discussion
1.0E-03
0.0E+00 0.0E+00
5.0E-04 1.0E-03 r (m)
1.5E-03
Fig. 7. Variation of bubble shape with time, predicted by the Young–Laplace equation silver nanofluids from a 0.2 mm stainless steel substrate nozzle for nominal air flow rate of 0.1 ml/min.
gas flow rate, the lack of bubble fluctuation from 0.2 mm substrate nozzle might be attributed to short bubble development period and high gas velocity. As gas velocity increases, the inertial force becomes less important compared to the dynamic pressure. As a result, the bubble fluctuation would not take place. shapes. To highlight the advancement of the MSB approach, a comparison of the bubble contact angles and bubble volumes extracted using MSB and USB methods is shown in Fig. 8 for
90
1th part
3th part 2.E-03 z (m)
4th part
70 θo .
2th part
50
30
1.E-03 10 0.0E+00
4.0E-09
8.0E-09
1.2E-08
3
Volume (m )
0.E+00 0.E+00
1.E-03 r (m)
2.E-03
Fig. 6. Comparison of experimental data with the predictions from the Young– Laplace equation during bubble growth inside silver nanofluids from a 0.2 mm stainless steel substrate nozzle for nominal air flow rate of 0.1 ml/min. (The small bubble was near the beginning of bubble formation, t = 805.752 ms, and the large bubble is just before bubble departure, t = 814.085 ms.) We found that the smaller bubble needs to be split into 3 parts while the large bubble into 4 parts in order for the model to correctly reproduce the bubble.
Fig. 8. Variation of bubble contact angle with volume inside water from a 0.4 mm stainless steel substrate nozzle at 0.7 ml/min nominal air flow rate. The full line shows the prediction of bubble contact angle by conventional method where the Young–Laplace equation is accurate enough for N = 1. The dashed line demonstrates the prediction of bubble contact angle in the oscillatory period where MSB approach was applied after splitting the bubbles into two sections. Large deviations are observed for volumes > than 6 109 m3, except for the cases when the bubble expansion rate is near zero. When bubble expanding upward and bubble volume expansion rate is zero, the bubble is in a steady state condition, and USB and MSB approaches are valid. However, when bubble expanding downward and bubble volume expansion rate is zero, the USB approach is not valid, because the bubble is stretched laterally and effect of viscosity is important.
S. Vafaei et al. / Chemical Engineering and Processing 86 (2014) 116–124
120
1.E-06
121
0.1 ml/min
8.E-07
100
0.5 ml/min
80
0.7 ml/min
6.E-07
80
4.E-07
60
60
2.E-07 0.E+00 0.0E+00
θo.
100
θo
Bubble Volume Expansion Rate (m 3/sec) .
120
40
40
5.0E-09
1.0E-08
20 0.0E+00
20 1.5E-08
3
Volume (m ) Fig. 9. Variation of bubble volume expansion rate and bubble contact angle with volume inside silver nanofluids from a 0.4 mm stainless steel substrate nozzle, for a 0.1 ml/min nominal air flow rate. (The diamond and cross symbols depict respectively the variation of contact angle and bubble volume expansion rate.)
bubbles growing inside water from 0.4 mm stainless steel substrate nozzle under a nominal air flow rate of 0.7 ml/min. The solid curve is the USB prediction while the dashed line illustrates the MSB prediction, which provide more accurate prediction of variation of bubble contact angle with time in this case. For the MSB results, the bubble was divided into two parts, N = 2, the upper part starting from point A and ending at point C and second part starting from point C and ending at the triple line (point E) (see Fig. 2). The bubble volume, V, and contact angle, uo, were obtained by solving the system of ordinary differential Eqs. (5) and (7)–(9) for the upper and lower parts separately. While for the smaller bubble volumes, both methods yield similar results, the USB method is not able to correctly yield the oscillations of the contact angle with volume, for volumes >6 109 m3, except when the bubble volume expansion rate is near zero. The major uncertainties associated with bubble volume and contact angle calculations are related to the two experimental inputs, the radius, r, and height, z, of bubble, or for each sections of the bubble. The uncertainties of these two parameters are half pixel size. The experimental uncertainties of calculation of the bubble volume and contact angle were estimated to be less than 0.15%. 0.1 ml/min. Figs. 9 and 10 demonstrate the variation of bubble volume expansion rate, contact angle and radius of contact line with 3.5E-04
5.0E-09 1.0E-08 3 Volume (m )
1.5E-08
Fig. 11. Variation of bubble contact angle and nominal air flow rate with volume inside silver nanofluids from 0.4 mm stainless steel substrate nozzle. The bubbles with lower nominal air flow rate are more prone to fluctuations, as indicated by the increased number of fluctuations before departure and the smaller threshold volume that indicates when the contact angle starts increasing after the initial decrease. The departure volume and contact angle does not seem to be affected by the flow.
volume inside silver nanofluid from a 0.4 mm stainless steel substrate nozzle, under a nominal air flow rate of 0.1 ml/min, as extracted using the MSB method. At the very beginning when bubble volume was in range of (0 V 2.676 E – 9 m3), the effect of buoyancy force was negligible. So, as bubble volume increased, the bubble expanded laterally; the radius of triple line increased, and the bubble contact angle decreased. As the bubble volume expanded further (2.676E-9 m3 V 6.48 E – 9 m3), the buoyancy force became more important, which lifted the bubble upward, resulting in an increase in bubble contact angle and a reduction in the radius of triple line. As the bubble volume increased further (6.48 E – 9 m3 V 8.24 E – 9 m3), the bubble started fluctuating and expanding laterally again, consequently the bubble contact angle decreased and sometimes the radius of contact line increased. The bubble volume expansion rate became zero twice when the bubble volume was in range of 6.48 E – 9 m3 V 8.24 E – 9 m3. The first time it happened when the bubble height stop expanding vertically and the second time occurred when the bubble height was start developing vertically again. The bubble continued to expand similarly till the departure point (8.24 E – 9 m3 V 1.298 E – 9 m3) important. Figs. 11 and 12 demonstrate that the fluctuations of the bubble contact angle and radius of triple line with volume inside the silver nanofluid for the entire air flow rates, i.e., 0.1–0.7 ml/min, from a
120 3.5E-04
0.1 ml/min
100 3.0E-04
60
2.5E-04
40 2.0E-04 0.0E+00
5.0E-09
1.0E-08
20 1.5E-08
3
Volume (m ) Fig. 10. Variation of bubble contact angle and radius of contact line with volume inside silver nanofluids from 0.4 mm stainless steel substrate nozzle, for 0.1 ml/min nominal air flow rate. (The diamond and cross symbols depict respectively the variation of contact angle and radius of triple.)
rd (m).
θo
rd (m) .
0.5 ml/min
80
3.0E-04
0.7 ml/min
2.5E-04
2.0E-04 0.0E+00
1.0E-08 5.0E-09 3 Volume (m )
1.5E-08
Fig. 12. Variation of radius of contact line and nominal air flow rate with volume inside silver nanofluids from 0.4 mm stainless steel substrate.
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0.4 mm nozzle. After an initial drop in the contact angle with volume, seen at all flow rates, the bubble contact angle starts increasing at threshold volumes of, 5.94 E – 9 m3, 7.07 E – 9 m3 and 7.81 E – 9 m3 for 0.1, 0.5 and 0.7 ml/min flow rates, respectively. These results indicate that the bubbles with lower nominal air flow rate have less resistance against fluctuations, and as a result they fluctuate more before departure. While the fluctuations are better observed in the contact angle, they also affect the radius of contact line, as seen in Fig. 12. The contact line increased first with the flow rate, then decreased for 0.5 ml/min and 0.7 ml/min cases, and one more cycle was observed at the lower flow rate of 0.1 ml/min. It should be noted that the minimum contact line radius is limited by the nozzle radius. As the flow rate increased, the initial radius of the contact line increased, most likely an artifact due to the limitations in the speed of the camera. On the other hand, the departure volume and departure contact angle were not affected much by the flow rate, indicating that for a given nanofluid the major role in these effects was played by other parameters such as surface wettability, liquid–gas surface tension and buoyancy force line). The fluctuation of bubbles during growth has been attributed to the large initial bubble volume expansion rate [3] which increased the downward inertial rapidly compressing the air inside the bubbles. Followed by expansion, the cycle was repeated till bubble departure. As the volumetric flow rate increases, the inertial force becomes less dominant vs. pressure, so the number of fluctuations decreases. That's why the effect of bubble fluctuation on radius of triple line can be seen only for low gas flow rates, e.g., 0.1 ml/min rate. Fig. 13 shows the variation of bubble contact angle with time inside water, gold, silver, and alumina nanofluids from 0.4 mm stainless steel substrate nozzle, for 0.5 ml/min nominal air flow rate. Fig. 13 clearly demonstrates the fluctuation of bubbles during bubble growth. The fluctuation of bubble has been attributed to the large initial bubble volume expansion rate [3], which increased the downward inertial force rapidly. The large increase of inertial force in a short period of time pushed the air bubble downward which might compress the air inside bubble and causing bubble fluctuation. The bubble fluctuation could affect the dynamics of the radius of triple line especially under high inertial force condition [3]. For the same conditions, the bubble fluctuation was not observed from needle nozzle, since the maximum inertial force was effective in a longer time with a lower value. The mechanisms of bubble fluctuation have been discussed in details in Ref. [3]. More fluctuation can be detected inside alumina nanofluid, since
the bubble growth has longer bubble formation period. Fig. 13 illustrates that the amplitude of fluctuations of bubble contact angle is damped over time, because of viscous damping. nozzle. Figs. 14 and 15 show variation of radius of triple line and bubble height with time inside water, gold, silver, and alumina nanofluids from 0.4 mm stainless steel substrate nozzle, for 0.5 ml/min nominal air flow rate. These figures illustrate that bubble height and contact angle would be usually affected by bubble fluctuations while the radius of triple line would only be affected at low gas flow rate (see Fig. 12). It is clear that nanoparticles change the waiting time, bubble formation time, bubble frequency and bubble contact angle. Here the duration between the departure point and the initiation of next bubble is defined as the waiting time and the duration between the initiation and the departure point is called the bubble formation time. The total bubble formation time is the summation of waiting and bubble formation times. rate. The expansion of radius of triple line is related to the force balance between forces that push the triple line toward the gas and liquid phases. The triple line spreads towards the gas phase more, as the solid surface tensions, s sg s sl, increases. Among water, gold, and silver nanofluids, gold nanofluids has the maximum solid surface tensions and the minimum radius of triple line while the silver nanofluid has the other extreme. For a given droplet volume, the bubble contact angle of water was observed to be smaller than
Fig. 13. Variation of bubble contact angle with time inside water, gold, silver, and alumina nanofluids from 0.4 mm stainless steel substrate nozzle, for 0.5 ml/min nominal air flow.
Fig. 15. Variation of bubble height with time inside water, gold, silver, and alumina nanofluids from 0.4 mm stainless steel substrate nozzle, for 0.5 ml/min nominal air flow.
Fig. 14. Variation of radius of triple line with time inside water, gold, silver and alumina nanofluids from 0.4 mm stainless steel substrate nozzle, for 0.5 ml/min nominal air flow rate.
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that of the alumina nanofluid [10]. As a result, the solid surface tensions of alumina nanofluids is lower than that of water as the liquid–gas surface tension of water has a similar value to that of alumina nanofluid (00.073 N/m–0.0745 N/m) at a low concentration of 0.001 vol.% [10]. Clearly, as the solid surface tensions decreases, the triple line would like to expand more toward the liquid phase. The resistance forces against the expansion of triple line inside silver nanofluid were weaker. Consequently, the silver nanofluids had maximum radius of triple line. The material of nanoparticles has a significant role on the behavior of triple line inside nanofluids. In fact, the behavior of nanofluid triple line such as variation of contact angle and radius of triple line depends on base liquid, gas, solid materials, concentration and characteristics of nanoparticles. These parameters affect the liquid–gas and solid surface tensions and consequently change the force balance at the triple line, resulting different contact angles and radius of triple line. At the detachment point, the main effective forces are the upward buoyancy force, (rl rg)gV, and the downward surface tension force, 2s lgrdp sin uo. As the vertical component of surface tension force increases, the buoyancy force and consequently the departure bubble volume increases. The vertical component of surface tension force depends on the liquid–gas surface tension, contact angle and radius of the triple line. It has reported that as the gas–liquid surface tension reduces, the downward surface tension force decreases, which requires less buoyancy force for departure and resulting reduction of bubble departure volume [32–34]. However a few other studies reported that there is no effect of surface tension on bubble departure volume [35,36], whereas others showed that the bubble departure volume increased with the decrease of surface tension as bubbles were allowed to grow further into liquid [37]. There are difficulties to assess the role of gas–liquid surface tension alone as the bubble departure volume is modulated by many other experimental variables such as orifice diameter, wettability, radius of triple line, gas–liquid–solid physical properties and gas flow rate. Clearly both experiments and simulation in this work shows that nanoparticles suspended in the liquid could have profound effect on bubble dynamics, which would have important implications to boiling heat transfer with nanofluids. The effects of nanoparticles during boiling are two sides: surface effect and bulk effect. It would modify the characteristics of the heating surface due to nanoparticle deposition [38–40] and affect the dynamics of bubble growth and bubble departure, because of variation of the liquid–gas and solid surface tensions with concentration of nanoparticles [6,7]. The evaporation of nanofluids inside the microlayer beneath the bubble would increase the concentration of nanoparticles in the microlayer region, so the possibility of collisions, agglomeration and deposition of nanoparticles in the region increases. Deposited nanoparticles would form a tiny porous layer of agglomerated nanoparticles on heated surface. The characteristics of heated substrate would be modified continuously [41,42] as the thickness of deposited nanoparticles increases over time during the boiling process. The deposited nanoparticles may also modify the solid surface tensions, force balance at the triple line, dynamics of triple line, surface roughness and nucleation site density [38,39] rate. The presence of nanoparticle inside the nanofluid would change the liquid–gas and solid surface tensions [6,7], as results, the force balance in the bubble triple line would change (see Fig. 1a) which, as shown in this work, has significant impacts on the dynamics of triple line during the bubble formation and bubble departure periods. Consequently the bubble formation method would provide opportunities to explore the bulk effect of nanoparticles on the nanofluid boiling heat transfer, irrespective of the nanoparticle surface deposition effect.
123
5. Conclusions A new method based on the division of bubble was developed to raise the accuracy of the prediction of the characteristics and shape of bubble when: (a) the bubble is stretched upward during the departure period, (b) the shear stress is relatively high and (c) the gas–liquid equilibrium at the interface is relatively weak when bubble fluctuates. The new approach raises the accuracy of bubble shape prediction by solving the Young–Laplace equation in each individual section separately. This study clearly demonstrated that the conventional method (N = 1) can not be employed when bubbles are under non-equilibrium conditions. The comparison of the new approach in bubble prediction with the experiments show that the applicability of the Young–Laplace equation can be extended to the entire bubble formation period, including the oscillatory and departure periods, and can predict accurately bubbles for all nozzle size and different nanofluids. Experiments show that different silver, gold, and alumina nanoparticles have different effects on liquid–gas and solid surface tensions, which results in different triple line dynamics. It was observed that as the solid surface tensions, s sg s sl, increases, the radius of the triple line expands towards the gas phase. Among all fluids tested, gold nanofluid has the maximum solid surface tensions and the minimum radius of triple line, whereas silver nanofluid has the other extreme. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36]
S.R.V. Vafaei, M.Z. Podowski, Adv. Colloid Interface Sci. 113 (2005) 133–146. S. Vafaei, T. Borca-Tasciuc, D. Wen, Colloids Surf. A 369 (2010) 11–19. S. Vafaei, P. Angeli, D. Wen, Colloids Surf. A 387 (2011) 240–247. J. Buongiorno, J. Appl. Phys. 106 (2009) 0943121–0943214. L. Godson, B. Raja, D. Mohan Lal, S. Wongwises, Exp. Heat Transfer 23 (2010) 317–332. S. Vafaei, A. Purkayastha, A. Jain, G. Ramanath, T. Borca-Tasciuc, Nanotechnology 20 (2009) 185702–185708. S. Vafaei, D. Wen, T. Borca-Tasciuc, Langmuir 27 (2011) 2211–2218. S. Vafaei, D. Wen, Langmuir 26 (2010) 6902–6907. E. Forrest, E. Williamson, J. Buongiorno, L.W. Hu, M. Rubner, R. Cohen, Int. J. Heat Mass Transfer 53 (2010) 58–67. S.J. Kim, I.C. Bang, J. Buongiorno, L.W. Hu, Int. J. Heat Mass Transfer 50 (2007) 4105–4116. S. Vafaei, D. Wen, Adv. Colloid Interface Sci. 159 (2010) 72–93. K. Sefiane, Appl. Phys. Lett. 89 (2006) 441061–441063. J.R. Moffat, K. Sefiane, M.E.R. Shanahan, J. Phys. Chem. B 113 (2009) 8860–8866. R.V. Craster, O.K. Matar, K. Sefiane, Langmuir 25 (2009) 3601–3609. K. Sefiane, R. Bennacer, Adv. Colloid Interface Sci. 147–148 (2009) 263–271. S. Vafaei, T. Borca-Tasciuc, M.Z. Podowski, A. Purkayastha, G. Ramanath, P.M. Ajayan, Nanotechnology 17 (2006) 2523–2527. L. Gao, T.J. McCarthy, Langmuir 24 (2009) 14105–14115. K. Sefiane, J. Skilling, J. MacGillivray, Adv. Colloid Interface Sci. 138 (2008) 101–120. D.T. Wasan, A.D. Nikolov, Nature 423 (2003) 156–159. A. Nikolov, K. Kondiparty, D.T. Wasan, Langmuir 26 (2010) 7665–7670. A. Chengara, D. Nikolov, D.T. Wasan, A. Trokhymchuk, J. Colloid Interface Sci. 280 (2004) 192–201. S. Vafaei, D. Wen, J. Colloid Interface Sci. 362 (2011) 285–291. J.F. Padday, Phil. Trans. Roy. Soc. London Ser. A: Math. Phys. Sci. 269 (1971) 265–293. A. Marmur, E. Rubin, Chem. Eng. Sci. 28 (1973) 1455–1464. O.I. del Rio, A.W. Neumann, J. Colloid Interface Sci. 196 (1997) 136. D. Gerlach, G. Biswas, F. Durst, V. Kolobaric, Int. J. Heat Mass Transfer 48 (2005) 425–438. D. Gerlach, G. Tomar, G. Biswas, F. Durst, Int. J. Heat Mass Transfer 49 (2006) 740–754. I. Chakraborty, B. Ray, G. Biswas, F. Durst, A. Sharma, P.S. Ghoshdastidar, Phys. Fluids 21 (2009) 0621031–06210317. M.S. Longuet-Higgins, B.R. Kerman, K. Lunde, J. Fluid Mech. 230 (1991) 365–390. Y. Chen, M. Groll, Int. J. Heat Mass Transfer 49 (2006) 1115–1128. S. Vafaei, D. Wen, J. Colloid Interface Sci. 343 (2010) 291–297. R.L. Datta, D.H. Napier, D.M. Newitt, Trans. Inst. Chem. Eng. 28 (1950) 14–26. R.L. Benzing, J.E. Myers, Ind. Eng. Chem. 47 (1955) 2087–2090. P.D. Coppock, G.T. Meiklejohn, Trans. Inst. Chem. Eng. 29 (1951) 75–86. C.J. Quigley, A.I. Johnson, B.L. Harris, Chem. Eng. Prog. Symp. Ser. 16 (1955) 31–44. J.F. Davidson, B.O.G. Schuler, Trans. Inst. Chem. Eng. 38 (1960) 335–342.
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[41] S.M. Kwark, R. Kumar, G. Moreno, J. Yoo, S.M. You-Int, J. Int, Heat Mass Transfer 53 (2010) 972–981. [42] H.T. Phan, N. Caney, P. Marty, S.P. Colasson, J. Gavillet, NanoSc. Microsc. Therm. Eng. 14 (2010) 229–244.
Contents lists available at ScienceDirect
Chemical Engineering and Processing: Process Intensification journal homepage: www.elsevier.com/locate/cep
Investigation of nanofluid bubble characteristics under nonequilibrium conditions Saeid Vafaei a, *, Theodorian Borca-Tasciuc b, *, Dongsheng Wen c, * a
Department of Mechanical, Materials and Manufacturing, University of Nottingham, Nottingham, UK Department of Mechanical, Aerospace and Nuclear Engineering, Rensselaer Polytechnic Institute, Troy, NY, USA c School of Chemical and Process Engineering, University of Leeds, Leeds, UK b
A R T I C L E I N F O
A B S T R A C T
Article history: Received 10 June 2014 Received in revised form 14 October 2014 Accepted 15 October 2014 Available online 22 October 2014
We report experimental and theoretical investigations of the bubble characteristics during the oscillatory growth period for several nanofluids. The nanoparticles were found to affect liquid–gas and solid surface tensions, which modulated the bubble contact angle, radius of triple line, bubble volume and the dynamics of bubble growth. To increase the accuracy of the Young–Laplace equation predictions during the bubble growth in the oscillatory period, a new method multi-section bubble (MSB) approach was developed. In this method, the bubble was divided into n sections (i.e., n = 1:N) and the Young–Laplace equation was solved for each section individually. As N increases, within each section the effects of inertia force and viscosity become reduced comparing to that of the liquid-gas surface tension. Unlike the conventional Young–Laplace approach (i.e., N = 1), the new approach is able to predict the bubble characteristics reliably in the following cases: (a) the oscillatory period when bubble is fluctuating; (b) the departure period when bubble is stretched upward, right before departure; and (c) the high shear stress condition when gas velocity is relatively high. ã 2014 Elsevier B.V. All rights reserved.
Keywords: Bubble Triple line Contact angle Solid surface tensions Nanoparticle Nanofluids Young–Laplace equation Surface wettability
1. Introduction The Young–Laplace equation has been used to predict the droplet shape and contact angle on horizontal [1] and inclined substrates. The Young–Laplace equation is also able to predict the bubble shape when both phases are in equilibrium at the interface and the effect of shear stress is negligible. However, the applicability of the Young–Laplace equation becomes invalid when: (a) the bubble is in the departure period [2], (b) the shear stress is relatively high and (c) the bubble is fluctuating [3]. One application where these conditions may occur is during boiling of fluids on hot surfaces, which has received a strong research interest with the invention of nanofluids. To address these challenges, one possible avenue would be to develop and solve a set of equations containing the additional effects of high shear and inertial forces. Here we propose an alternative solution based on using Young–Laplace equations over different sections of the bubble, in order to increase the accuracy of the prediction of bubble shape and enable the fast determination of important bubble
* Corresponding authors. E-mail addresses: [email protected] (S. Vafaei), [email protected] (T. Borca-Tasciuc), [email protected] (D. Wen). http://dx.doi.org/10.1016/j.cep.2014.10.010 0255-2701/ ã 2014 Elsevier B.V. All rights reserved.
characteristics such as volume and contact angle. The technique is applied to study the bubble characteristics inside nanofluids during the oscillatory period. It has been reported that nanoparticles are able to modify the thermal conductivity [4], viscosity [5], liquid–gas [6], and solid surface tensions, s sg s sl, [7], and possibly the gas–liquid– solid interactions at the triple line. The variation of liquid–gas and solid surface tensions would affect the force balance and consequently the dynamics of the triple line, which has a significant role on the bubble growth [3,8], and boiling heat transfer coefficient and critical heat flux [9,10]. The schematics of liquid–gas and solid surface tensions at the triple line are exhibited in Fig. 1, for bubbles and droplets. The dynamics of triple line on a substrate [8,11] or a hot plate [12–15], also depend on the nature of the gas and liquid phases, the characteristics of the solid surface [7,16,17], and the receding and advancing contact angles [18]. Nanofluids have been also found to induce different spreading and thinning (layering) behavior at the triple region compared to the pure liquids. The nanoparticles were observed to spread stepwise inside the triple region on a smooth hydrophilic glass surface. The number of layers of nanoparticles decreases in a stepwise pattern as nanoparticles gets closer to the edge of triple line. Apparently the layering phenomenon of the nanoparticles at the triple region depends on many factors such as concentration
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Nomenclature g Ro R1,R2 rd V Qn u
acceleration of gravity [m/s2] radius of curvature at apex [m] radius of curvature [m] radius of contact line [m] bubble volume [m3] nominal gas flow rate [ml/min] gas velocity [m/s]
Greek Symbols d height of apex [m] uo bubble contact angle [Deg.] ue droplet contact angle [Deg.] us asymptotic contact angle [Deg.] rl liquid density [kg/m3] rg gas density [kg/m3] s lg liquid–gas surface tension [N/m] s sg solid–gas surface tension [N/m] s sl solid–liquid surface tension [N/m] s lg n liquid–gas surface tension of nanofluids [N/m]
and characteristics of nanoparticles, nanoparticle charge, solid–liquid–gas materials and film size [19,20]. It has been shown theoretically that particles can spread the triple line to a distance of 20–50 times of the particle diameter through a structural disjoining pressure by self-ordering of particles in a confined wedge. However, the structural disjoining force only becomes significant at relatively high particle concentrations, i.e., over 20 vol.% [21]. In this paper we report the effects of different nanoparticles on the dynamics of triple line and bubble growth. Using a new multi section bubble (MSB) method presented here, the Young–Laplace equation was solved to predict the bubble shape and was compared with experimental data for bubble growth inside water, gold, silver and alumina nanofluids throughout the bubble formation period including the oscillatory period. The paper is organized as the following: Section 2 presents the new method for bubble shape prediction; Section 3 reports the experimental setup; Section 4 discusses the results and comparison between the model prediction and the experiment, and the final conclusions are summarized in Section 5 lines. 2. Prediction of the profile of bubble shape Mathematically, the Young–Laplace equation can be interpreted as a mechanical equilibrium condition between two
Fig. 2. Illustration of schematic of the bubble shape where bubble is divided into several.
fluids separated by an interface. It gives the pressure difference across the interface as a function of the product of the curvature multiplied by the gas–liquid surface tension 1 1 þ Þs R1 R2 lg
Dp ¼ ð
(1)
where R1 and R2 are the radii of the interface curvature, i.e., R1 is the radius of curvature, describing the latitude as it rotates and R2 is the radius of curvature in a vertical section of the bubble describing the longitude as it rotates. The centers of R1 and R2 are on the same line, vertical to the interface, but different location. Dp is the pressure difference between gas, pg, and liquid, pl (see Fig. 2). For bubbles, the gas pressure as a function of z is given by pg ðzÞ ¼
2s lg þ po þ rg gz Ro
(2)
The first term on the right hand side is the pressure difference at the bubble apex, po is liquid pressure at the apex, and last term is the hydrostatic gas pressure. Similarly the liquid pressure can be written as pl ðzÞ ¼ po þ rl gz
(3)
The radii of curvature are R1 ¼
ds r andR2 ¼ du sinu
(4)
where u , r, and s are, respectively, the bubble contact angle, the radius of the bubble and the length of bubble perimeter at the location of z (see Fig. 2). Ro is radius of curvature at apex. Substituting Eqs. (2)–(4) into Eq. (1), the Young–Laplace equation for bubbles becomes du 2 gz sinu ¼ ðr rg Þ ds Ro s lg l r
(5)
Similarly, the Young–Laplace equation for droplets can be written as du 2 gz sinu ¼ þ ðr rg Þ ds Ro s lg l r
Fig. 1. Schematic of forces at the bubble/droplet triple.
(6)
In case of nanofluids, the liquid–gas surface tension, s lg, has to change to the liquid–gas surface tension of nanofluids, s lg n. The Young–Laplace equation can be solved [1,22–30], with following system of ordinary differential equations for axisymmetric
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interfaces, to obtain the bubble shape dr ¼ cosu ds
(7)
dz ¼ sinu ds
(8)
dV ¼ pr2 sinu ds
(9)
where V is bubble volume. This system of ordinary differential equations avoids the singularity problem at the bubble apex, since sinu 1 ¼ r s¼0 Ro
(10)
The Young–Laplace equation has been solved to predict the shape of axisymmetric liquid pendants and sessile drops on an ideal solid surface [1,25], as well as bubble shapes under low Capillary and Webber numbers when the effects of inertia force and viscose shear are negligible comparing to that of liquid–gas surface tension. In these studies, the Young–Laplace has been applied to the whole bubble, i.e., the whole bubble shape has been predicted by one set of equations, i.e., Eqs. (5) and (7)–(9). For clarity, here we call this method the uni-section bubble (USB) approach. The accuracy of the prediction has been examined by experimental data [25,26], and numerical simulations based on the volume of fluid method and level-set method [27,28], where good agreement has been observed. These studies confirm the validity of the Young–Laplace equation in predicting the shape of droplets [1,6,7,16] and bubbles [2,8,11,23,24,31] under an equilibrium condition between gas and liquid. Solving the Young–Laplace equation, using the USB method has been unsuccessful in predicting the whole bubble shape in other conditions [3,8,11] such as: (a) the shear stress between gas and liquid is relatively high, (b) the last few milliseconds before bubble departure where the effect of viscosity is dominated, and (c) when bubble starts fluctuating. The weak equilibrium between gas and liquid at the interface is most-likely responsible for the lack of accuracy of USB method.In this study, a new method called the multi-section bubble (MSB) approach is employed, to help predict the bubble shape even under non-equilibrium situations through an approach similar to the differentiation and subsequent integration method. Here, rather than using just one section as the conventional Young–Laplace equation, the bubble shape is divided into n parts (n = 1:N) with the starting point for each bubble section (such as the height and radius) measured experimentally. Even if the equilibrium for the whole bubble cannot be met, a quasi-equilibrium status would be assumed for each section of the bubble and the Young–Laplace equations, i.e., Eqs. (5) and (7)–(9), could be applied. In this work, the systems of Eqs. (5) and (7)–(9) are solved by the Runge–Kutta methods for each individual part by using two experimental data, the radius, r, and the height, z, as the inputs at the end of the each part (see Fig. 2). Starting from the first part, n = 1, from the apex (point A), with the following initial boundary conditions rð0Þ ¼ zð0Þ ¼ u ð0Þ ¼ Vð0Þ ¼ 0
from the end of the first part. So the initial boundary conditions of the second part, n = 2, such as radius, r, height, z, contact angle, u, and volume, V, are identical with those of at the end of the first part, n = 1. However, the radius and height of the second section is provided from experimental data at the location of point B (see Fig. 2). Under such initial conditions, the same sets of ordinary differential Eqs. (5) and (7)–(9) can be then solved to provide the bubble shape at n = 2. Such a process continues subsequently until it reaches the final division (i.e., n = N). Once the bubble shape is predicted, the volume, and contact angle, can be easily extracted from the modeled shape. As the number of parts, N, increases, or size of each part, Ds, decreases (see Fig. 2), the effects of inertia force and the viscosity decrease comparing to that of the liquid–gas surface tension, which satisfy the applicability of the Young– Laplace equation. The effects of inertial force and shear stress, respectively, depend on the volume and area of each section which is changing with Ds. However, the effect of liquid–gas surface tension depends on the perimeter of top or bottom sides of each section and does not change significantly with Ds. The MSB approach is able to predict the bubble shape right before departure, solving the Young–Laplace equation where bubble is stretched upward and effect of viscosity is dominant. The USB approach is not valid for the departure period. In this work, the systems of Eqs. (5) and (7)–(9) were solved for each individual part by using two experimental inputs, which enables us to predict the bubble shape, bubble volume and bubble contact angle. It shall be noted that non-symmetrical bubble were not considered, and most of current experiments showed axisymmetric bubble profiles under given conditions. The air absorption into the base liquid was considered to be negligible since the bubble formation period (45 ms and 166 ms) was short and contact area between the air and liquid was small. The bubble shape prediction by the MSB method were compared with experimental data, and insights into bubble characteristics were extracted for bubble growth inside several nanofluids. parts. 3. Experimental setup Fig. 3 illustrates the schematic of experimental setup for capturing the formation of bubbles inside water and nanofluids. Two orifices with diameters, dn, of 0.2 mm and 0.4 mm were
(11)
the first section of the bubble shape can be predicted. Because of the continuity of the bubble shape, the second part, n = 2, starts
Fig. 3. Schematic of the experimental.
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Fig. 4. An example of bubble growth inside silver nanofluid for a nominal gas flow rate of 0.5 ml/min, from 0.4 mm substrate nozzle.
fabricated in the center of a stainless steel substrate plate, which was submerged into a transparent square-sized glass container with a 20 mm by 20 mm base and 72 mm height. The substrate nozzle was large enough to allow the triple line to expand freely and was polished to reduce roughness (with average value of the peaks and valleys, Ra = 0.021 mm, and largest difference from peak-to-valley, Rz = 0.03 mm). The glass container was filled with deionized water or nanofluids to a height of 20 mm and was open to the atmosphere. The gas flow was supplied by compressed air in a cylinder, connected to a gas flow controller (model F-200CV002 of Bronkhorst) through a pressure reduction valve. Nominal gas flow rates, Qn, in the range of 0.1–0.7 ml/min were used with an accuracy of 0.5%. Details of bubble formation, especially the dynamics of triple line, were captured by a high speed camera (Photosonics Phantom V4.3, 1200 frames/s) equipped with an optical microscope head (10X Navitar Macro zoom 7000). The resolution of the camera is 5 mm per pixel. Fig. 4 shows an example of bubble growth inside silver nanofluid for a nominal gas flow rate of 0.5 ml/min, from 0.4 mm substrate nozzle. The profile of bubble is obtained from the captured images by fast camera, equipped with a microscope during the bubble growth inside water and nanofluids from 0.2 mm and 0.4 mm substrate nozzles. Since the bubble formation period is longer for the 0.4 mm substrate nozzle, more information is available for this nozzle than for the 0.2 mm nozzle, for a given camera speed. Gold, silver and alumina nanoparticles, with a narrow size distribution averaged at 5 nm for gold and silver and 25 nm for alumina, were dispersed into deionized water without any surfactant (see Fig. 5). The gold and silver nanoparticle concentrations were controlled at 0.01% by weight, and the concentration of the alumina nanofluid was 0.0037% by weight. A drop of nanofluid was left on top of a stainless steel surface to dry slowly and the dried material was analyzed by an Energy Dispersive X-ray spectroscopy (EDX), which confirmed the purity of the nanoparticles. Images of bubbles or droplets captured at steady state were analyzed with a Drop Shape Analysis System (KRUSS, DSA 100) to obtain the gas–liquid and solid surfaces tensions of water, and nanofluids. However, the bubble growth images in the oscillatory period or when away from steady state were analyzed using the method described in Section 2. The accuracy of the surface tension measurements is 0.01 mN/m. The gas–liquid surface tensions of pure water and gold nanofluids were 0.07238 0.0041 and 0.06753 0.0066 mN/m, respectively, as determined by the instrument using six different bubble images [8,11]. The liquid–gas surface tension of alumina nanofluids at a low concentration of 0.001 vol.% was similar to that of water [10]. The liquid–gas surface tension of silver nanofluids was 0.05373 0.005 mN/m, from six different droplet images. The
details of the experimental set up can be found in references [3,8,11,22] setup. To determine the solid surface tensions, the DSA system was employed to measure the radius of the triple line and the height of six small droplets of each sample liquid, which were used as the two inputs to solve the systems of Eqs. (6)–(9). As the liquid droplets on the substrate were in stationary and equilibrium conditions, the uni-section bubble (USB) approach was applied to obtain the characteristics of droplets. Having the characteristics of droplet such as radius of droplet, rd, contact angle, ue, and droplet volume, V, the difference between the solid surface tensions was found from, nanoparticles.
s lg cosus ¼ s sg s sl " rd sinue ¼
(12)
3V
pð2 þ cosus Þð1 cosus Þ2
#1=3 sin2 us
(13)
where us is asymptotic contact angle. The experimental uncertainties for prediction of the droplet volume and contact angle are estimated to be less than 0.9%. The solid surface tensions, s sg s sl,
Fig. 5. Transmission electron microscopy (TEM) pictures of silver.
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of water, gold and silver nanofluids on stainless steel substrate, respectively, are 0.00415 0.0008 N/m, 0.0124 0.001 N/m and 0.0027 0.0004 N/m. The accuracy and details of the solid surface tensions measurement is given clearly in reference [1,7]. The gas–liquid and solid surface tension values are then imported in the MSB simulations and along with the experimentally measured section height and radius are used to accurately predict the bubble shape.
804.919 ms
3.0E-03
805.752 ms 806.586 ms 808.252 ms 810.752 ms 814.085 ms 2.0E-03
We report experimental investigations of the bubble characteristics in the oscillatory growth period for several nanofluids. A new method was developed to predict the bubble shape, using the Young–Laplace equation, when liquid and gas phases are not in equilibrium or shear stress is relatively high. In this method, the bubble is divided into n sections (i.e., n = 1:N), and n is gradually increased until the entire bubble shape can be predicted. Fig. 6 compares the experimental data with predicted shapes of two bubbles: one near the beginning and the other near the departure stage inside silver nanofluid. The Young–Laplace equation using N = 1, called here the USB approach, could not predict the bubble shapes because the effects of shear stress and viscosity acting on the entire bubble were strong compared to that of the liquid–gas surface tension. However, these effects are less important when smaller sections are modeled with the Young–Laplace equation and key characteristics (i.e., radius and height) are experimentally provided for the endpoints of these sections. As shown in Fig. 6, the small bubble needed to be divided into at least 3 parts (N = 3) and the larger bubble needed to be divided into minimum 4 parts (N = 4) for the method to converge and provide a shape that closely matches the experiment. The larger bubble required division into more parts, likely due to increased upward stretching before the departure. An example of applying the MSB method for the entire set of bubble images is shown in Fig. 7 for bubbles growing in silver nanofluid from 0.2 mm substrate nozzle for 0.1 ml/min nominal air flow rate. The figure shows the evolution of bubble growth and demonstrates the short bubble development period. For a given 3.E-03
Exp
z (m)
4. Results and discussion
1.0E-03
0.0E+00 0.0E+00
5.0E-04 1.0E-03 r (m)
1.5E-03
Fig. 7. Variation of bubble shape with time, predicted by the Young–Laplace equation silver nanofluids from a 0.2 mm stainless steel substrate nozzle for nominal air flow rate of 0.1 ml/min.
gas flow rate, the lack of bubble fluctuation from 0.2 mm substrate nozzle might be attributed to short bubble development period and high gas velocity. As gas velocity increases, the inertial force becomes less important compared to the dynamic pressure. As a result, the bubble fluctuation would not take place. shapes. To highlight the advancement of the MSB approach, a comparison of the bubble contact angles and bubble volumes extracted using MSB and USB methods is shown in Fig. 8 for
90
1th part
3th part 2.E-03 z (m)
4th part
70 θo .
2th part
50
30
1.E-03 10 0.0E+00
4.0E-09
8.0E-09
1.2E-08
3
Volume (m )
0.E+00 0.E+00
1.E-03 r (m)
2.E-03
Fig. 6. Comparison of experimental data with the predictions from the Young– Laplace equation during bubble growth inside silver nanofluids from a 0.2 mm stainless steel substrate nozzle for nominal air flow rate of 0.1 ml/min. (The small bubble was near the beginning of bubble formation, t = 805.752 ms, and the large bubble is just before bubble departure, t = 814.085 ms.) We found that the smaller bubble needs to be split into 3 parts while the large bubble into 4 parts in order for the model to correctly reproduce the bubble.
Fig. 8. Variation of bubble contact angle with volume inside water from a 0.4 mm stainless steel substrate nozzle at 0.7 ml/min nominal air flow rate. The full line shows the prediction of bubble contact angle by conventional method where the Young–Laplace equation is accurate enough for N = 1. The dashed line demonstrates the prediction of bubble contact angle in the oscillatory period where MSB approach was applied after splitting the bubbles into two sections. Large deviations are observed for volumes > than 6 109 m3, except for the cases when the bubble expansion rate is near zero. When bubble expanding upward and bubble volume expansion rate is zero, the bubble is in a steady state condition, and USB and MSB approaches are valid. However, when bubble expanding downward and bubble volume expansion rate is zero, the USB approach is not valid, because the bubble is stretched laterally and effect of viscosity is important.
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120
1.E-06
121
0.1 ml/min
8.E-07
100
0.5 ml/min
80
0.7 ml/min
6.E-07
80
4.E-07
60
60
2.E-07 0.E+00 0.0E+00
θo.
100
θo
Bubble Volume Expansion Rate (m 3/sec) .
120
40
40
5.0E-09
1.0E-08
20 0.0E+00
20 1.5E-08
3
Volume (m ) Fig. 9. Variation of bubble volume expansion rate and bubble contact angle with volume inside silver nanofluids from a 0.4 mm stainless steel substrate nozzle, for a 0.1 ml/min nominal air flow rate. (The diamond and cross symbols depict respectively the variation of contact angle and bubble volume expansion rate.)
bubbles growing inside water from 0.4 mm stainless steel substrate nozzle under a nominal air flow rate of 0.7 ml/min. The solid curve is the USB prediction while the dashed line illustrates the MSB prediction, which provide more accurate prediction of variation of bubble contact angle with time in this case. For the MSB results, the bubble was divided into two parts, N = 2, the upper part starting from point A and ending at point C and second part starting from point C and ending at the triple line (point E) (see Fig. 2). The bubble volume, V, and contact angle, uo, were obtained by solving the system of ordinary differential Eqs. (5) and (7)–(9) for the upper and lower parts separately. While for the smaller bubble volumes, both methods yield similar results, the USB method is not able to correctly yield the oscillations of the contact angle with volume, for volumes >6 109 m3, except when the bubble volume expansion rate is near zero. The major uncertainties associated with bubble volume and contact angle calculations are related to the two experimental inputs, the radius, r, and height, z, of bubble, or for each sections of the bubble. The uncertainties of these two parameters are half pixel size. The experimental uncertainties of calculation of the bubble volume and contact angle were estimated to be less than 0.15%. 0.1 ml/min. Figs. 9 and 10 demonstrate the variation of bubble volume expansion rate, contact angle and radius of contact line with 3.5E-04
5.0E-09 1.0E-08 3 Volume (m )
1.5E-08
Fig. 11. Variation of bubble contact angle and nominal air flow rate with volume inside silver nanofluids from 0.4 mm stainless steel substrate nozzle. The bubbles with lower nominal air flow rate are more prone to fluctuations, as indicated by the increased number of fluctuations before departure and the smaller threshold volume that indicates when the contact angle starts increasing after the initial decrease. The departure volume and contact angle does not seem to be affected by the flow.
volume inside silver nanofluid from a 0.4 mm stainless steel substrate nozzle, under a nominal air flow rate of 0.1 ml/min, as extracted using the MSB method. At the very beginning when bubble volume was in range of (0 V 2.676 E – 9 m3), the effect of buoyancy force was negligible. So, as bubble volume increased, the bubble expanded laterally; the radius of triple line increased, and the bubble contact angle decreased. As the bubble volume expanded further (2.676E-9 m3 V 6.48 E – 9 m3), the buoyancy force became more important, which lifted the bubble upward, resulting in an increase in bubble contact angle and a reduction in the radius of triple line. As the bubble volume increased further (6.48 E – 9 m3 V 8.24 E – 9 m3), the bubble started fluctuating and expanding laterally again, consequently the bubble contact angle decreased and sometimes the radius of contact line increased. The bubble volume expansion rate became zero twice when the bubble volume was in range of 6.48 E – 9 m3 V 8.24 E – 9 m3. The first time it happened when the bubble height stop expanding vertically and the second time occurred when the bubble height was start developing vertically again. The bubble continued to expand similarly till the departure point (8.24 E – 9 m3 V 1.298 E – 9 m3) important. Figs. 11 and 12 demonstrate that the fluctuations of the bubble contact angle and radius of triple line with volume inside the silver nanofluid for the entire air flow rates, i.e., 0.1–0.7 ml/min, from a
120 3.5E-04
0.1 ml/min
100 3.0E-04
60
2.5E-04
40 2.0E-04 0.0E+00
5.0E-09
1.0E-08
20 1.5E-08
3
Volume (m ) Fig. 10. Variation of bubble contact angle and radius of contact line with volume inside silver nanofluids from 0.4 mm stainless steel substrate nozzle, for 0.1 ml/min nominal air flow rate. (The diamond and cross symbols depict respectively the variation of contact angle and radius of triple.)
rd (m).
θo
rd (m) .
0.5 ml/min
80
3.0E-04
0.7 ml/min
2.5E-04
2.0E-04 0.0E+00
1.0E-08 5.0E-09 3 Volume (m )
1.5E-08
Fig. 12. Variation of radius of contact line and nominal air flow rate with volume inside silver nanofluids from 0.4 mm stainless steel substrate.
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0.4 mm nozzle. After an initial drop in the contact angle with volume, seen at all flow rates, the bubble contact angle starts increasing at threshold volumes of, 5.94 E – 9 m3, 7.07 E – 9 m3 and 7.81 E – 9 m3 for 0.1, 0.5 and 0.7 ml/min flow rates, respectively. These results indicate that the bubbles with lower nominal air flow rate have less resistance against fluctuations, and as a result they fluctuate more before departure. While the fluctuations are better observed in the contact angle, they also affect the radius of contact line, as seen in Fig. 12. The contact line increased first with the flow rate, then decreased for 0.5 ml/min and 0.7 ml/min cases, and one more cycle was observed at the lower flow rate of 0.1 ml/min. It should be noted that the minimum contact line radius is limited by the nozzle radius. As the flow rate increased, the initial radius of the contact line increased, most likely an artifact due to the limitations in the speed of the camera. On the other hand, the departure volume and departure contact angle were not affected much by the flow rate, indicating that for a given nanofluid the major role in these effects was played by other parameters such as surface wettability, liquid–gas surface tension and buoyancy force line). The fluctuation of bubbles during growth has been attributed to the large initial bubble volume expansion rate [3] which increased the downward inertial rapidly compressing the air inside the bubbles. Followed by expansion, the cycle was repeated till bubble departure. As the volumetric flow rate increases, the inertial force becomes less dominant vs. pressure, so the number of fluctuations decreases. That's why the effect of bubble fluctuation on radius of triple line can be seen only for low gas flow rates, e.g., 0.1 ml/min rate. Fig. 13 shows the variation of bubble contact angle with time inside water, gold, silver, and alumina nanofluids from 0.4 mm stainless steel substrate nozzle, for 0.5 ml/min nominal air flow rate. Fig. 13 clearly demonstrates the fluctuation of bubbles during bubble growth. The fluctuation of bubble has been attributed to the large initial bubble volume expansion rate [3], which increased the downward inertial force rapidly. The large increase of inertial force in a short period of time pushed the air bubble downward which might compress the air inside bubble and causing bubble fluctuation. The bubble fluctuation could affect the dynamics of the radius of triple line especially under high inertial force condition [3]. For the same conditions, the bubble fluctuation was not observed from needle nozzle, since the maximum inertial force was effective in a longer time with a lower value. The mechanisms of bubble fluctuation have been discussed in details in Ref. [3]. More fluctuation can be detected inside alumina nanofluid, since
the bubble growth has longer bubble formation period. Fig. 13 illustrates that the amplitude of fluctuations of bubble contact angle is damped over time, because of viscous damping. nozzle. Figs. 14 and 15 show variation of radius of triple line and bubble height with time inside water, gold, silver, and alumina nanofluids from 0.4 mm stainless steel substrate nozzle, for 0.5 ml/min nominal air flow rate. These figures illustrate that bubble height and contact angle would be usually affected by bubble fluctuations while the radius of triple line would only be affected at low gas flow rate (see Fig. 12). It is clear that nanoparticles change the waiting time, bubble formation time, bubble frequency and bubble contact angle. Here the duration between the departure point and the initiation of next bubble is defined as the waiting time and the duration between the initiation and the departure point is called the bubble formation time. The total bubble formation time is the summation of waiting and bubble formation times. rate. The expansion of radius of triple line is related to the force balance between forces that push the triple line toward the gas and liquid phases. The triple line spreads towards the gas phase more, as the solid surface tensions, s sg s sl, increases. Among water, gold, and silver nanofluids, gold nanofluids has the maximum solid surface tensions and the minimum radius of triple line while the silver nanofluid has the other extreme. For a given droplet volume, the bubble contact angle of water was observed to be smaller than
Fig. 13. Variation of bubble contact angle with time inside water, gold, silver, and alumina nanofluids from 0.4 mm stainless steel substrate nozzle, for 0.5 ml/min nominal air flow.
Fig. 15. Variation of bubble height with time inside water, gold, silver, and alumina nanofluids from 0.4 mm stainless steel substrate nozzle, for 0.5 ml/min nominal air flow.
Fig. 14. Variation of radius of triple line with time inside water, gold, silver and alumina nanofluids from 0.4 mm stainless steel substrate nozzle, for 0.5 ml/min nominal air flow rate.
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that of the alumina nanofluid [10]. As a result, the solid surface tensions of alumina nanofluids is lower than that of water as the liquid–gas surface tension of water has a similar value to that of alumina nanofluid (00.073 N/m–0.0745 N/m) at a low concentration of 0.001 vol.% [10]. Clearly, as the solid surface tensions decreases, the triple line would like to expand more toward the liquid phase. The resistance forces against the expansion of triple line inside silver nanofluid were weaker. Consequently, the silver nanofluids had maximum radius of triple line. The material of nanoparticles has a significant role on the behavior of triple line inside nanofluids. In fact, the behavior of nanofluid triple line such as variation of contact angle and radius of triple line depends on base liquid, gas, solid materials, concentration and characteristics of nanoparticles. These parameters affect the liquid–gas and solid surface tensions and consequently change the force balance at the triple line, resulting different contact angles and radius of triple line. At the detachment point, the main effective forces are the upward buoyancy force, (rl rg)gV, and the downward surface tension force, 2s lgrdp sin uo. As the vertical component of surface tension force increases, the buoyancy force and consequently the departure bubble volume increases. The vertical component of surface tension force depends on the liquid–gas surface tension, contact angle and radius of the triple line. It has reported that as the gas–liquid surface tension reduces, the downward surface tension force decreases, which requires less buoyancy force for departure and resulting reduction of bubble departure volume [32–34]. However a few other studies reported that there is no effect of surface tension on bubble departure volume [35,36], whereas others showed that the bubble departure volume increased with the decrease of surface tension as bubbles were allowed to grow further into liquid [37]. There are difficulties to assess the role of gas–liquid surface tension alone as the bubble departure volume is modulated by many other experimental variables such as orifice diameter, wettability, radius of triple line, gas–liquid–solid physical properties and gas flow rate. Clearly both experiments and simulation in this work shows that nanoparticles suspended in the liquid could have profound effect on bubble dynamics, which would have important implications to boiling heat transfer with nanofluids. The effects of nanoparticles during boiling are two sides: surface effect and bulk effect. It would modify the characteristics of the heating surface due to nanoparticle deposition [38–40] and affect the dynamics of bubble growth and bubble departure, because of variation of the liquid–gas and solid surface tensions with concentration of nanoparticles [6,7]. The evaporation of nanofluids inside the microlayer beneath the bubble would increase the concentration of nanoparticles in the microlayer region, so the possibility of collisions, agglomeration and deposition of nanoparticles in the region increases. Deposited nanoparticles would form a tiny porous layer of agglomerated nanoparticles on heated surface. The characteristics of heated substrate would be modified continuously [41,42] as the thickness of deposited nanoparticles increases over time during the boiling process. The deposited nanoparticles may also modify the solid surface tensions, force balance at the triple line, dynamics of triple line, surface roughness and nucleation site density [38,39] rate. The presence of nanoparticle inside the nanofluid would change the liquid–gas and solid surface tensions [6,7], as results, the force balance in the bubble triple line would change (see Fig. 1a) which, as shown in this work, has significant impacts on the dynamics of triple line during the bubble formation and bubble departure periods. Consequently the bubble formation method would provide opportunities to explore the bulk effect of nanoparticles on the nanofluid boiling heat transfer, irrespective of the nanoparticle surface deposition effect.
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5. Conclusions A new method based on the division of bubble was developed to raise the accuracy of the prediction of the characteristics and shape of bubble when: (a) the bubble is stretched upward during the departure period, (b) the shear stress is relatively high and (c) the gas–liquid equilibrium at the interface is relatively weak when bubble fluctuates. The new approach raises the accuracy of bubble shape prediction by solving the Young–Laplace equation in each individual section separately. This study clearly demonstrated that the conventional method (N = 1) can not be employed when bubbles are under non-equilibrium conditions. The comparison of the new approach in bubble prediction with the experiments show that the applicability of the Young–Laplace equation can be extended to the entire bubble formation period, including the oscillatory and departure periods, and can predict accurately bubbles for all nozzle size and different nanofluids. Experiments show that different silver, gold, and alumina nanoparticles have different effects on liquid–gas and solid surface tensions, which results in different triple line dynamics. It was observed that as the solid surface tensions, s sg s sl, increases, the radius of the triple line expands towards the gas phase. Among all fluids tested, gold nanofluid has the maximum solid surface tensions and the minimum radius of triple line, whereas silver nanofluid has the other extreme. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36]
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