Elsevier

Chemical Engineering Science

Volume 173, 14 December 2017, Pages 37-48
Chemical Engineering Science

A novel numerical approach for investigation of the gas bubble characteristics in stagnant liquid using Young-Laplace equation

https://doi.org/10.1016/j.ces.2017.07.018Get rights and content

Highlights

  • The Young-Laplace equation with MSB method is developed to predict the gas bubble shape.

  • The bubble height is obtained from experiments in deionized water and SiO2 nanofluids.

  • Bubble size is compared with conventional Young-Laplace method and experiments.

  • Bubble characteristics such as volume, center of gravity, instantaneous contact angle, and aspect ratio are obtained.

  • Sensitivity of bubble characteristics to the Bond number has been considered.

Abstract

In the present study, the Young-Laplace equation was applied to simulate the adiabatic gas bubble growth from a submerged needle in stagnant liquid column. In order to solve the Young-Laplace equation the axisymmetric bubble height was used as input from experimental data. To increase the accuracy of Young-Laplace equations’ prediction during the bubble growth, the bubble was divided into four sections with the same height, and Young-Laplace equation was solved for each section individually. By dividing the bubble into four sections, the effects of viscosity and inertia forces within each section were reduced as compared to that of buoyancy and liquid-gas surface tension. Unlike the conventional Young-Laplace approach (one Young-Laplace equation for the entire bubble), the new approach was able to predict bubble characteristics reliably during the growth cycle. The bubble growth was investigated in a column of liquid with a triple contact line that fixed to the needle perimeter. To validate the numerical results, the bubble profiles that predicted by numerical simulation were compared with the experimental results. Experiments were performed by injection of air at constant gas flow rate of 600 ml/h in the quiescent deionized water and SiO2 nanofluid. The nanoparticle concentrations were 0.05, 0.1 and 0.2 wt%, and air flow injected from G14 and G17 standard needles. Eventually, evaluation of bubble characteristics, such as the bubble volume, the center of gravity, the instantaneous contact angle, and the bubble aspect ratio were investigated, and the effects of variation of liquid properties on the bubble characteristics were discussed. The results show that the present method can predict the bubble shape during 97.5% of growth time with mean absolute error of 6%. Furthermore, the results revealed that the bubble size decreased with increment of Bond number. Also, bubble instantaneous contact angle and bubble aspect ratio were almost irrelative to Bond number during the growth cycle.

Introduction

Combustion, chemical reactions, boiling, and petroleum industry are examples of industrial applications of multiphase flow phenomenon. Boiling is one of the most important multiphase subjects which is used in many industries. Bubble dynamic has an important role in boiling that includes formation, growth, and detachment of bubble. The adiabatic process of gas injection into the liquid, from a needle or an orifice, is one of the easiest ways to simulate the bubble formation process in boiling. Due to the importance of the bubble formation, growth, and detachment process, many experimental (Jamialahmadi et al., 2001, Di Bari and Robinson, 2013, Dietrich et al., 2013, Xie et al., 2012, Zhu et al., 2014, Snabre and Magnifotcham, 1998) and numerical studies have been carried out to investigate bubble's characteristics.

Lesage and Marois (2013) investigated the bubble detachment characteristics numerically and experimentally. They considered some correlations for prediction of bubble detachment characteristics, such as height, vertical position of the center of gravity, the apex principle radius of curvature, and the bubble width.

In the case of numerical studies on bubble dynamics, different methods have been applied. Lattice Boltzmann (Qiu et al., 2015), Potential Flow Theory (Zhang and Tan, 2000), Boundary Integral (Oguz and Prosperetti, 1993, Wong et al., 1998, Higuera, 2005, Xiao and Tan, 2005, Higuera and Medina, 2006), Finite Volume (Di Bari et al., 2013), Finite Element (Simmons et al., 2015), Volume of Fluid (VOF) (Zahedi et al., 2014, Hanafizadeh et al., 2015, Biń et al., 2004), and Combined Level Set and Volume of Fluid (CLSVOF) (Buwa et al., 2007, Gerlach et al., 2007, Albadawi et al., 2013a) methods are the examples of applied numerical methods in bubble phenomenon simulations. In a comprehensive work, Albadawi et al. (2013b) numerically evaluated the growth and detachment of an isolated air bubble, injected through a single orifice, by using four different interface capturing methods including, algebraic VOF, geometric VOF, LS, and CLSVOF.

Another important numerical method for prediction of bubble characteristics is the Young-Laplace equations set (Vafaei et al., 2015, Vafaei et al., 2010, Yakhshi-Tafti et al., 2011, Gerlach et al., 2005, Mori and Baines, 2001). Lesage et al. (2013) used the Capillary (Young-Laplace) equations to predict the vapor bubble shape during the growth cycle. The results showed that the bubble shape evolution had dependence on the physical mechanisms quantified in the Bond number. In another research, Gerlach et al. (2005) investigated the static formation of air bubbles from a submerged orifice based on the force balance principles. They corroborated the Young-Laplace predictions with experimental results.

Vafaei and Wen (2010b) used the Young-Laplace equation to investigate the bubble formation from a submerged micro-nozzle. They used the air and water as gas and liquid phases, respectively. In their investigation, the experimentally obtained bubble height and triple line radius were used as inputs of Young-Laplace equation. They concluded that Young-Laplace equation was able to predict the bubble characteristics quite well until the last milliseconds before bubble detachment. In another studies (Chesters, 1977, Chesters, 1978) the Young-Laplace equation is solved with a perturbation approach in cylindrical coordinates, and Chen and Groll (2006) solved the Young-Laplace equation in arc-length coordinates by using the forth order Runge–Kutta method.

In both experimental and numerical investigations, effects of many parameters, such as electric and gravitational field (Di Marco et al., 2003), electro hydro dynamics (EHD) (Gao et al., 2013), injection gas types (Shahjahan et al., 2013), and micro-size injection needles (Dietrich et al., 2013) have been studied on bubble dynamics. One of the most important and effective parameters in bubble dynamics, which have attracted the attention of many scientists during the last decade, is nanoparticles' effect on this phenomenon (Su et al., 2009).

Wang and Wu (2015) simulated the growth and departure of a single bubble behavior in Al2O3/H2O nanofluid and pure water boiling process by an improved Moving Particle Semi-implicit method. In another research, Vafaei and Wen (2015) modified the Young-Laplace equation for prediction of bubble shape and investigation of nanoparticles' effect on bubble growth and departure. Also, Vafaei et al. (2014) studied the bubble characteristics during the oscillatory growth for deionized water and several nanofluids including gold, silver, and alumina, via the Young-Laplace equation. They divided the bubble into several sections and solved Young-Laplace equation for each section individually. To solve the Young-Laplace equation in each section, it was necessary to obtain the height and radius of starting point of each section from experiments. They concluded that multi section bubble (MSB) method was applicable for prediction of the shape and main features of bubble in non-equilibrium condition.

In another research, Vafaei and Wen (2010a) applied the Young-Laplace equation to predict bubble formation based on running angle and curvature system. They investigated the effects of gold nanoparticles on behavior of bubble triple line in comparison with bubble characteristics in deionized water. They used radius of contact line and height of bubble as inputs of Young-Laplace equations for prediction of bubble shape.

It is concluded that many experimental data should be available as the necessary inputs to solve the Young-Laplace equation in case of applying MSB method for prediction of bubble behavior. Hence, in the present work, a new approach is developed for prediction of bubble shape produced from submerged needle in deionized water and SiO2 nanofluid columns by applying the Young-Laplace equation. In order to increase the accuracy of prediction of bubble characteristics under non-equilibrium condition, the bubble is divided into four sections with same height, and Young-Laplace equation is solved for each section separately. Unlike the previous conventional MSB methods, the bubble height is the only input of Young-Laplace equation solver from experiments. In this study, the needle diameter is considered as the triple line radius of bubble, and the effect of liquid properties' variations on bubble dynamics that resulted from addition of SiO2 nanoparticles is studied numerically.

Section snippets

Force balance analysis

In this section, a vertical element of bubble is considered to apply the force balance analysis. According to the Fig. 1, the principle forces acting on the considered element are pressure, buoyancy and surface tension forces. The shear stress is negligible compared with the other forces in quasi-static condition. By using the force balance on the vertical element in z-direction, Eq. (1) is derived (Vafaei et al., 2015, Vafaei and Wen, 2010, Vafaei and Wen, 2015, Vafaei et al., 2014, Vafaei and

Numerical method

The Young-Laplace equation introduced in previous section is applied to simulate the bubble behavior and predict the bubble profile in growth time. Most of the previous studies on characteristics of gas bubble injected into liquid column have concentrated on low gas flow rates, small injector diameters, and low Bond numbers. The Young-Laplace equation is valid in the quasi-static condition of the bubble growth, due to elimination of shear stress force. In non-equilibrium conditions such as

Experimental setup

To validate the Young-Laplace equation prediction of the bubble profile in formation, growth and detachment, a series of experiments are designed. Fig. 4 illustrates the schematics of experimental setup for capturing the formation of air bubble in deionized water and water based SiO2 nanofluids. Standard needles of G14 and G17 are used as injectors of air bubble into the container. The needles are placed in the center of square-sized Pyrex-glass container with dimensions of 100mm(L)×100mm(W)×300

Numerical method validation

Air bubble profiles resulted from Young-Laplace equation are validated with two test cases corresponding to injection of air into water through standard needles of G14 and G17. To predict the bubble profiles, the experimentally measured height of bubbles are used as the only input of Young-Laplace equation solution. In previous investigations, the Young-Laplace equation have been applied for prediction of bubble shape in quasi-static condition. Whereas, the present solution method is able to

Results and discussion

The numerical solution of Young-Laplace equation is used to predict the shape of air bubble in growth stage in water and 0.05–0.2 wt% SiO2 nanofluid with a fixed bubble foot radius at terrestrial condition. The experimental height of bubble is used as input of numerical solution of Young-Laplace equation. The bubble height is divided into four sections with the same height, and Young-Laplace equation is solved for each section individually. The curvature radius of bubble apex is selected

Conclusion

A new approach was developed for prediction of bubble shape by applying the Young-Laplace equation. In order to increase the accuracy of prediction of bubble characteristics under non-equilibrium conditions arising from high gas flow rate, which stretches the bubble stretched upward, the bubble was divided into four sections with the same height. The Young-Laplace equation was solved for each section individually. Unlike the previous common MSB methods, the bubble height was the only input of

References (42)

Cited by (17)

  • Stabilization and performance of a novel viscoelastic N<inf>2</inf> foam for enhanced oil recovery

    2021, Journal of Molecular Liquids
    Citation Excerpt :

    Over time, the change of bubble size in SDS-EDAB foam was slower than SDS-HPAM foam, and the stability of the foam was stronger. According to the Young-Laplace Equation [43], due to the different capillary pressure in the adjacent bubbles, the gas in the small bubble diffused to the large bubble during the aging process of the foam. Moreover, with the continuous aging of the foam, the liquid film possessed the gravity drainage and capillary force at the Plateau boundary of the bubble, resulting in the thinning of the liquid film until the bubble burst.

  • Effects of nano-bubbles and constant/variable-frequency ultrasound-assisted freezing on freezing behaviour of viscous food model systems

    2021, Journal of Food Engineering
    Citation Excerpt :

    Based on this, the NBs in G20% and G30% were considered to be more stable than the others, which suggested that the NBs in real viscous foods like smoothie and fruit puree might better maintain their sizes. However, according to Young-Laplace equation (Gharedaghi et al., 2017) and Henry's law (Ritzoulis and Rhoades, 2013), NBs tended to shrink into smaller sizes, which can be below the detection limit and finally dissolve in the solution, making only the larger bubbles be included in the measurement range (Lafond et al., 2018; Zhu et al., 2018b). Therefore, the conclusion drawn from the bubble size analysis may not be reliable, and further cross-validation experiments need to be conducted.

View all citing articles on Scopus
View full text