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Air Pocket

Air Pocket. 8th problem. The Problem. A vertical air jet from a straw produces a cavity on a water surface. What parameters determine the volume and the depth of the cavity?. The Apparatus. We used a square-shaped aquarium and a compressor to reproduce the phenomenon.

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Air Pocket

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  1. Air Pocket 8th problem

  2. The Problem A vertical air jet from a straw produces a cavity on a water surface. What parameters determine the volume and the depth of the cavity?

  3. The Apparatus • We used a square-shaped aquarium and a compressor to reproduce the phenomenon. • We measured the depth and the width at the surface • We changed the velocity of the air jet and the height of the straw • Two pipes with different cross-sections were used

  4. Speed of Air Jet Bernoulli’s equation is: Neglecting the difference between the heights of the two points:

  5. The Cavity • We approximated the shape of the cavity with a paraboloid • A paraboloid is a parabola rotated about the y axis • We measured cavity width, depth, as the function of pipe height and air speed

  6. Cavity Parameters as Functions of Pipe Height

  7. Cavity Parameters as Functions of Air Speed

  8. Cavities with different pipe cross-sections

  9. Volume of the Cavity • A paraboloid is a parabola rotated about the y axis. • Its volume equals its inverse function’s volume rotated about the x axis • We need to calculate the volume of a solid of revolution Inverse function of the parabola

  10. Volume of the Cavity

  11. Volume of the cavity • Parameter ‘A’ contains the rate of the width and height of the cavity, in the following way:

  12. Volume of the cavity • The volume of a solid of revolution:The volume of the rotated inverse parabola: • This is the volume of the original paraboloid, too

  13. Measured Volumes

  14. Work Needed for the Formation of the Cavity • We calculated this work in two ways • First method: • Using that ,where F is the buoyancy and • Like dipping the paraboloid from the surface to h0 depth continually:

  15. Work Needed for the Formation of the Cavity • We get • Substituting back :

  16. Another Method for Determining the Work • When the cavity is created, the mass centre of the water which filled the cavity will move to the surface of the water, which means that its potential energy will rise:The needed work equals this change of energy:

  17. Another Method for Determining the Work • The problem is then, to determine Δh • Assuming that ρ is constant, Δh is the difference of the h0 (height of the water surface) and the hm mass centre of the paraboloid: • hm can be calculated from the geometry of the paraboloid

  18. Another Method for Determining the Work • We calculated the centre of mass of the paraboloid by this formula: • We got:

  19. Another Method for Determining the Work • Substituting back to the equation with the work:

  20. Calcuating Efficiency • Efficiency is the rate of input work and and useful work • Input work is the mechanical energy change of air slowing down at the surface • Useful work is the work done while creating the cavity

  21. Calculating Efficiency • Efficiency is the rate of the two: • Air fill the cavity continously; its mass is

  22. Efficiencies • Substituting:

  23. Volumes and efficiencies of different cross-sections

  24. Conclusions

  25. The measurements • Cavity width, depth • As the function of: • Speed of the air jet • Height of the pipe • We measured two types of pipem, with different cross-sections

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