Modeling the effect of liquid viscosity and surface tension on bubble formation

MODELING THE EFFECT OF LIQUID VISCOSITY AND SURFACE TENSION ON BUBBLE FORMATION ZHANG YALI NATIONAL UNIVERSITY OF SINGAPORE 2004 MODELING THE EFFECT OF LIQUID VISCOSITY AND SURFACE TENSION ON BUBBLE FORMATION ZHANG YALI (B ENG, HUT) A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF ENGINEERING DEPARTMENT OF CHEMICAL AND BIOMOLECULAR ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2004 ACKNOWLEDGEMENT I would like to express my deep appreciation to my supervisor, Associate Professor Reginald B H Tan, for his invaluable advice, patient and continuous encouragement throughout the project Particular thanks to Dr Deng Rensheng for his assistance in the programming, Mr Xiao Zongyuan, Miss Xie Shuyi for their supportive discussion on this work I extremely appreciate my family for their deep love and support for me during the whole study process Finally I would like to give my thanks to National University of Singapore for supporting me to complete my work i TABLE OF CONTENTS Acknowledgement i Table of contents ii Summary vi Nomenclature viii List of figures xiii List of tables Chapter xv Introduction 1.1 Significance and objective for study of single bubble formation 1.2 Factors affecting the bubble formation at a submerged orifice 1.3 Organization of thesis Chapter Literature Review 2.1 Introduction 2.2 Overview of the literature models and forces introduced 2.3 Spherical model 2.3.1 The model of Davidson and Schüler 2.3.2 The models of Hayes et al and Sullivan et al 2.3.3 The model of Swope 10 ii 2.3.4 The model of Ramakrishnan et al 11 2.3.5 The model of Tsuge and Hibino 13 2.3.6 The model of Miyahara et al 15 2.3.7 The model of Gaddis and Vogelpohl 16 2.3.8 The model of Deshpande et al 18 2.4 Pseudo-spherical models 19 2.4.1 The model of Pinczewski 19 2.4.2 The model of Terasaka and Tsuge 21 2.4.3 The model of Yoo et al 24 2.5 Non-spherical models 24 2.5.1 The model of Marmur and Rubin 25 2.5.2 The model of Hooper 26 2.5.3 The model of Tan and Harris 27 2.5.4 The model of Liow and Gray 29 2.6 Summary Chapter 31 Theoretical Model Development 32 3.1 Introduction 32 3.2 Bubbling system and assumptions 32 3.3 Equations of motion 34 3.3.1 Force analysis based on the interfacial elements 34 3.3.2 Calculation of the virtual mass 38 3.4 Thermodynamics of the bubbling system 40 iii 3.5 Summary Chapter 42 Numerical Solution 43 4.1 Introduction 43 4.2 Initial conditions 43 4.3 Boundary conditions 45 4.4 Finite time-difference procedure 45 4.4.1 Finite difference versions of equations of motion 45 4.4.2 Finite difference versions of thermodynamic equations 47 4.5 Calculation of interfacial coordinates 50 4.6 Simulation of bubble growth process 51 Chapter Results and Discussion 54 5.1 Bubble growth curve and bubble shape during formation 54 5.2 Effect of viscosity on the bubble volume 59 5.3 Effect of surface tension 62 5.4 Comparison of experimental and simulated values of bubble volume 64 5.5 Analysis on modified Reynolds number 66 5.5.1 Expression for modified Reynolds number 66 5.5.2 Values comparison of modified Reynolds number 67 5.5.3 Conclusion 71 iv Chapter Conclusions and Recommendations 72 6.1 Conclusions 72 6.2 Recommendations for future work 72 References 74 v SUMMARY Many physical and chemical engineering processes involve heat or mass transfer across an interface at which two immiscible fluids contact In such operations a large interfacial area per unit volume is necessary to bring about efficient mass and heat transfer between the two phases The method of gas dispersion through submerged nozzles, orifices or slots is the simplest and the most common, which permits simple design and leads to reasonably large interfacial areas Due to the extremely complicated phenomena involved in this process, a somewhat simplified starting point has been to consider bubble formation from a single submerged orifice beneath the liquid, which has been the subject of study by many investigators An improved non-spherical model for bubble formation and detachment at a submerged orifice has been developed The model is based on the interfacial element approach of Tan and Harris (1986), and is modified to include the influence of viscosity in a Newtonian liquid via a viscous drag force on each interfacial element The gas-liquid interface is divided into a finite number of differential elements, and equations of motion are applied to each element to calculate the instantaneous coordinates constituting the bubble shape during its motion One powerful advantage of this model is that there is no need for an empirical detachment criterion because the vi instant of detachment occurs naturally as a consequence of bubble growth and shape evolution vii NOMENCLATURE Symbol Description Unit a0 cross-sectional area of orifice C orifice flow coefficient dimensionless C' effective orifice coefficient dimensionless c0 velocity of sound in the gas CD drag coefficient dimensionless D orifice diameter m Db diameter of bubble in Equation (2.1) m De equivalent diameter of bubble ( De = Dm maximum horizontal diameter of bubble in Equation (2.15) m F upward force in Equation (2.13) N Fb buoyancy force N FD viscous drag force N Fep excess pressure force N Fi inertial force N Fm force due to the momentum of gas N m2 m/s 6Vb π ) m viii Results and Discussion 5.5 Analysis on modified Reynolds number 5.5.1 Expression for modified Reynolds number The values of the modified Reynolds number Re ' for all the experimental data from the literature in the present work were investigated Re ' can be calculated through the ratio of the gas inertial force through the orifice and viscous force resistance, both of which are expressed as follows: v Fi = a0 ρ c FD = µ l a0 v0 D (5.1) (5.2) where a is the cross-sectional area of the orifice, D is the orifice diameter, v0 is the velocity of gas through the orifice Dividing equation (5.1) by (5.2), The expression for Re ' can be abstained as, Re ' = ρ c Qg πr0 µ l (5.3) where: Q g is the constant gas flow rate into the chamber r0 is the radius of the orifice 66 Results and Discussion 5.5.2 Values comparison of modified Reynolds number The results of Re ' for corresponding experimental conditions are calculated by Equation (5.3) and shown in Table 5.1 to 5.4, respectively, except for the value of Re ' for the experiment in Figure 5.4 is show below due to only one gas flow rate being employed Simulation of bubble formation for the following conditions (Terasaka and Tsuge, 1990): System: N2-92wt%glycerol Vc = 42.5 cm3 and 97.5 cm3 r0 = 0.0735 cm µ = 0.154 Pa.s Q g = 1.1 cm3/s The value of Re ' corresponding to the given gas flow rate is calculated as Re ' = 0.0068 Table 5.1 Values of modified Reynolds number I Simulation of bubble formation for the following conditions (Terasaka and Tsuge, 1990): System: N2-90wt%glycerol Vc = 34.1 cm3, 75 cm3 and 286 cm3 67 Results and Discussion r0 = 0.0765 cm µ = 0.118 Pa.s Results for the values of Re ' corresponding to the gas flow rate are present below: Q g (cm3/s) 0.15 0.20 0.25 0.33 0.50 0.65 0.80 1.00 1.10 1.81 2.60 3.40 5.50 6.50 10.00 Re ' 0.0012 0.0015 0.0019 0.0025 0.0039 0.0050 0.0062 0.0078 0.0085 0.0140 0.0200 0.0260 0.0430 0.0500 0.0780 Table 5.2 Values of modified Reynolds number II Values of Re ' for the experimental conditions (Ramakrishnan et al., 1969): System: air-glycerol solution Vc = 50 cm3 r0 = 0.352 cm 68 Results and Discussion are shown below µ =0.045Pa.s Q g (cm /s) 3 10 15 20 30 40 60 µ =0.302Pa.s Re ' 0.0088 0.0130 0.0180 0.0220 0.0310 0.0440 0.0660 0.0880 0.1320 0.1770 0.2650 Qg (cm3/s) 10 15 20 30 40 50 60 Re ' 0.0095 0.0130 0.0160 0.0250 0.0320 0.0470 0.0630 0.0950 0.1260 0.1580 0.1900 Table 5.3 Values for modified Reynolds number III Values of Re ' are presented in the following table r0 = 0.352 cm Q g (cm /s) 3 10 15 20 30 40 60 r0 =0.298 cm ' Re 0.0088 0.0130 0.0180 0.0220 0.0310 0.0440 0.0660 0.0880 0.1320 0.1770 0.2650 Q g (cm /s) 1.7 2.5 2.5 3.5 4.5 6.5 8.5 15 30 40 60 r0 =0.184 cm ' Re 0.0089 0.0130 0.0130 0.0180 0.0240 0.0340 0.0440 0.0780 0.1570 0.2100 0.3140 Q g (cm3/s) 1.5 2.5 5.5 7.5 10 15 30 45 60 Re ' 0.0127 0.0211 0.0338 0.0465 0.0633 0.0845 0.1270 0.2530 0.3800 0.5070 69 Results and Discussion Related to the corresponding conditions (Ramakrishnan et al., 1969) as shown below: System: air-glycerol solution Vc = 50 cm3 µ = 0.045 Pa.s Table 5.4 Values for modified Reynolds number IV Values of Re ' in Table 5.4 Qg (cm3/s) 2.00 3.00 4.40 5.20 6.00 7.00 8.00 9.00 10.77 12.30 13.23 18.46 28.75 38.30 47.50 Re ' 0.47 0.71 1.03 1.22 1.41 1.65 1.88 2.12 2.53 2.89 3.11 4.34 6.76 9.01 11.17 For conditions corresponding to Figure 5.8 (Ramakrishnan et al., 1969) System: air-water ( σ = 71.7 mN/m) air-glycerol solution ( σ = 41.4 mN/m) Vc = 50 cm3 70 Results and Discussion r0 = 0.298 cm µ = 0.001 Pa.s, 5.5.3 Conclusion From the values of Re ' tabulated in section 5.5.2 it can be seen that the experimental points from Terasaka and Tsuge(1990) have a very low values of Re ' ([...]... orifice The lift-off occurs continuously as a natural of consequence of the growth and rise of the bubble The detachment is assumed to happen as the vertical distance ( y ) between the center of the bubble and the orifice is equivalent to the final bubble radius ( r ' ) They concluded the viscosity has a major effect on bubble size For constant flow condition, the surface tension has no effect other than... solving the equations of motion based on the bubble surface 3 Introduction Detailed numerical solutions for bubble formation process will be given in chapter 4 In addition, this chapter describes the finite time difference forms for equations of motion as well as the thermodynamic equations Results and discussion will be presented in chapter 5 The effect of liquid viscosity and surface tension on bubble formation. .. through the orifice q (b) The velocity of liquid phase u (c) The submergence of the orifice below the liquid H (d) The pressure drop through the orifice ∆P 1.3 Objective and organization of thesis The present thesis aims to model the effect of liquid viscosity and surface tension on bubble formation through a single submerged orifice Chapter 2 presents a comprehensive review of theoretical and experimental... several liquids and correlated the formation of bubbles with the physical variables of the system by application of Newton’s second law of motion to the bubble at the instant just before release from the orifice, which is the base of models by Davidson and Schüler (1960a, b) Two types of bubble formation are described by both theoretical treatment and experimental research At low gas flow rates the volume... (1969) The authors experimentally investigated the influence of three factors including liquid viscosity, surface tension and liquid density on bubble formation They found that the values of bubble volume in two liquids with different surface tension were seen to be different at low gas flow rate but almost identical at higher flow rate, indicating that the contribution of surface tension to the bubble. .. applied to the motion of bubble near the instant of detachment from the orifice to study the air bubble formation with a wide range of liquids The surface tension of the liquids varied 9 Literature Review from 0.0178 to 0.0724 N/m, and the liquid viscosities ranged from 0.436 × 10 −3 to 0.713 Pa.s 2.3.3 The model of Swope Swope (1971) developed a mathematical model for slow formation of gas bubble at... the downward forces and the bubble accelerates The detachment occurs when the distance between the bubble center and the orifice plate is equal to the bubble radius at the end of the first stage 11 Literature Review such that the sequent expanding bubble does not coalesce with the previous one The final volume of the bubble is evaluated by the sum of the two stages Fig 2.2 Two-stage bubble formation. .. equation modified to include the hydrostatic and surface tension pressure was applied simultaneously to calculate the flow into the bubble 7 Literature Review r' y Fig 2.1 One-stage bubble formation model in viscous liquid by Davidson and Schüler (1960a) The initial conditions are taken as the bubble radius ( r ' ) equal to the orifice radius ( r0 ) and with the center of the bubble in the plane of the. .. respectively 2.3.4 The model of Ramakrishnan et al Ramakrishnan et al (1969) presented a theoretical model aimed at predicting the bubble size with the influence of both liquid viscosity and surface tension, and explained the discrepancies in the literature data for constant flow conditions Two stages of bubble formation namely the expansion stage and the detachment stage are employed in their model The proposed... time rate of change of the momentum of the gas entering the bubble, while at higher rates of gas flow the order of magnitude of the two forces was reversed Furthermore, the transition of the two regions occurred when the force due to the momentum is equal to the surface tension force, which can also be shown by the experiment Sullivan et al (1964) employed the similar mechanism of Newton’s second law ... mechanism of Newton’s second law applied to the motion of bubble near the instant of detachment from the orifice to study the air bubble formation with a wide range of liquids The surface tension of the. . .MODELING THE EFFECT OF LIQUID VISCOSITY AND SURFACE TENSION ON BUBBLE FORMATION ZHANG YALI (B ENG, HUT) A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF ENGINEERING DEPARTMENT OF CHEMICAL AND. .. integrating the vertical component of the liquid pressure over the surface of the bubble When F is negative (early stage of bubble growth) the bubble remains on the plate floor and the bubble surface