Problem 16 Rising bubble

Transcrição

1 Problem 16 Rising bubble reporter: Ibraim Rebouças

2 Problem 16 Rising bubble A vertical tube is filled with a viscous fluid. On the bottom of the tube, there is a large air bubble. Study the bubble rising from the bottom to the surface. Reporter: Ibraim Rebouças 2

3 Contents Theoretical introduction Kinds of bubble Viscosity Drag forces Lateral Flow Rise velocity Smaller spherical cap Influence of the temperature Experiments Tests Comparison between theory and experiments Conclusion Reporter: Ibraim Rebouças 3

4 Kinds of bubbles Spherical bubble Very viscous fluids. Governed by stokes low Spherical cap bubble Stable to moderated viscous fluids. Taylor s bubble Height bigger than the diameter of the pipe. Constant velocity for all volumes. Reporter: Ibraim Rebouças 4

5 The phenomenon Reporter: Ibraim Rebouças 5

6 Viscosity It s the measurement of the resistance to a shear stress. The viscosity is measured by a coefficient. is called absolute viscosity coefficient. Kinematic viscosity is de density of the fluid Reporter: Ibraim Rebouças 6

7 Forces acting on the bubble Buoyancy Drag forces Weight Reporter: Ibraim Rebouças 7

8 Forces acting on the bubble! 1 2 %& ' ( "bv Both are wrong, because they do not consider the boundary effects. Reporter: Ibraim Rebouças 8

9 Border of the bubble Pressure We must have conservation of mass for the liquid ahead and behind the bubble. * ) )* + constant20 Buoyancy " )+ ) 4 2 1" * 4 "4 54 Reporter: Ibraim Rebouças 9

10 Large spherical cap or Taylor bubble velocity For Taylor bubble or large spherical cap bubbles, the velocity can be described by the following equation: Taylor bubble Large spherical cap The velocity is constant and does not depend on the volume of the bubble. Terminal velocity (m/s) 67 " : : ; For low viscosity fluids(e.g. Water), the Diameter of the equation can be written as: bubble Reporter: Ibraim Rebouças 10

11 Spherical cap bubbles Diameter smaller than the tube Lateral oscillation Wave length = > 2?@ > A Δ6 Capillary > A Δ6 Surface tension coefficient Difference of density (air-liquid) Drag coefficient D Drag force? D 7 2 D Reporter: Ibraim Rebouças 11

12 Rise velocity of spherical cap At the equilibrium: B? 2 D G6E? 2 D 6E Finally: E Reporter: Ibraim Rebouças 12

13 Influence of temperature The temperature acts changing the viscosity of the fluid and the density of the air inside the bubble. Viscosity parameter Reference temperature in Kelvin HI*+ J KLMN JLMN "1 K Temperature in kelvin Accuracy of O6% for temperature range: 02J2100Q Reference viscosity Reporter: Ibraim Rebouças 13

14 Experimental description Experiment 1 Experiment 2 Experiment 3 Experiment 4 Vary the volume of the bubble Vary the temperature Vary the diameter of the tube Influence of surfactant Reporter: Ibraim Rebouças 14

15 Materials Burette 50 ml Pipette 10 ml Glass tube 250 ml Syringe 10 ml Rubber tube Camera Computer software's Reporter: Ibraim Rebouças 15

16 Experiment 1: Variation of the volume Volume Tube diameter: 15 mm Syringe: 20 ml Fluid: Glycerin Temperature: 20Q Temperature Diameter Source of errors: Measuring ruler and conversion scale: O0.5TT Volume of the syringe: O0.5 TU Reporter: Ibraim Rebouças 16

17 Experiment 1: Variation of the volume 9ml 8ml Volume Temperature Position (cm) Position (cm) Diameter Time (s) Time (s) ET m/s ET m/s Reporter: Ibraim Rebouças 17

18 Experiment 1: Variation of the volume 7ml 6ml Volume Temperature Position (cm) Position (cm) Diameter Time(s) Time(s) ET m/s ET m/s Reporter: Ibraim Rebouças 18

19 Experiment 1: Variation of the volume 5ml 4ml Volume Temperature Position (cm) Position (cm) Diameter Time(s) Time(s) ET m/s ET m/s Reporter: Ibraim Rebouças 19

20 Experiment 1: Variation of the volume 3 ml 2 ml Volume Temperature Position (cm) Position (cm) Diameter Time(s) Time(s) ET m/s ET m/s Reporter: Ibraim Rebouças 20

21 Experiment 1: Variation of the volume Position x Time 25 Volume Temperature Diameter Position (cm) Precision: ±0.5 mm 9 ml 8 ml 7 ml 6 ml 5 ml 4 ml Time (s) 3 ml 2 ml Reporter: Ibraim Rebouças 21

22 Experiment 1: Variation of the volume Volume Temperature Diameter Volume (ml) Experimental velocity ( X Y) Theoretical velocity ( X Y) Average 0, Velocity (m/s) Volume (ml) Out of the regime Average error for all points: 4.62% Average error for [4;9]: 1.37% Theory Experiment Experiment agrees with the theory Reporter: Ibraim Rebouças 22

23 Experiment 2: Variation of the temperature 35 Position in function of time for different temperatures Volume Temperature Diameter Position(cm) C 45 C 35 C 25 C Time(s) Reporter: Ibraim Rebouças 23

24 Experiment 2: Variation of the temperature Volume Temperature Diameter Temperature T ( C) Calculated Viscosity (m²/s) \]^_ Predicted Velocity (m/s) Experimental velocity Reporter: Ibraim Rebouças 24

25 Experiment 2: Variation of temperature Volume Temperature Diameter Velocity (m/s) Velocity in function of temperature Temperature ( C) Average error: 10% Theoretical Experimental Reporter: Ibraim Rebouças 25

26 Experiment 3: Variation of the diameter Fluid: Soy bean oil Kinematic viscosity at20q :7.5 10^` 10 Volume 9 8 Experimental velocities Temperature Diameter Position (cm) Precision: ±0.5 mm 5mm 6.5mm 25mm 0,193 m/s 0,0703 m/s 0,0240 m/s Time (s) Reporter: Ibraim Rebouças 26

27 Experiment 3: Variation of the diameter Volume Diameter (mm) Calculated Velocity (m/s) Experimental Velocity (m/s) Temperature Diameter Velocity (m/s) Average error: 11% Theoretical Experimental Diameter (mm) Reporter: Ibraim Rebouças 27

28 Conclusion Diameter of the tube Viscosity Temperature Diameter Reporter: Ibraim Rebouças 28

29 References Reporter: Ibraim Rebouças 29

30 Thank you! Reporter: Ibraim Rebouças 30

31 Lateral Flow For bubble measures comparable to the tube. Reporter: Ibraim Rebouças 31

32 Pressure and volume variation P 0 P = ρ. g. h + P For the following parameters: Fluid: Water Length of beaker: 1m h = 0 P = P 0 P = ρ. g. h = 1.10³ Pa 0 ρ H O 0 2 = 10 5 P = 1,10 kg / m³ g = 9.81 m / s² 3 Pa P P. V P. V = V2 = 1. 1 T1 T2 V 1 Reporter: Ibraim Rebouças 32

33 The weight of the bubble Considering a bubble with a volume of 2x10-5 m³ (20mL): Buoyancy Weight B = lvg B = 10 ρ 4 W = ρ gvg = ρ g = m V = 2 10 W = mg Relation between them W B = = Conclusion W = 0.6% of B, sow 0 Reporter: Ibraim Rebouças 33

34 Reporter: Ibraim Rebouças 34

35 Fluid(U) C Fluid(U) C Ammonia 1.05 Freon Ethanol 5.72 Benzene 4.34 Gasoline 3.68 Mercury 1.07 Glycerin 28.0 Methanol 4.63 Reporter: Ibraim Rebouças 35