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Torricelli s Law and Draining Pipes

Question. How is the rate at which water exits a draining container affected by various factors?The only force at work is gravity.Water exits faster with more water in the container.Exit velocity, height, and volume are all functions of time.What are these functions?. The System. Parametersr: radius of the exit holeR: inner radius of the pipeh0: height of the tube (from bottom of exit hole)f: distance from bottom of exit hole to groundFunctionsh(t): height of the water columnV(t):32887

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Torricelli s Law and Draining Pipes

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    1. Torricelli’s Law and Draining Pipes MATH 2413 Professor McCuan Steven Lansel, Brandon Luders December 10, 2002

    2. Question How is the rate at which water exits a draining container affected by various factors? The only force at work is gravity. Water exits faster with more water in the container. Exit velocity, height, and volume are all functions of time. What are these functions?

    3. The System

    4. The System (continued) Initial conditions: h0, V0, v0 when t=0 Cross-sectional area: Exit hole: a = pr2 Pipe: A = pR2 Four pipes: Two values for r Two values for R Each pipe has a different set of initial conditions

    5. The Pipes

    6. Finding v(t) For each pipe: Fill with water to the top with the hole plugged. Elevate the pipe while keeping it vertical. Let the water drain. Start keeping time. When the trajectory hits a predetermined horizontal distance, stop timing. For initial velocity, measure the farthest point the trajectory reaches. For draining time, see how long it takes to drain the pipe completely.

    7. Data Tables

    8. Projectile Motion

    9. Velocity Plots

    10. Linear Regressions

    12. Analysis All of our data was strongly linear (linear correlation factors were all between -0.99 and -1). v(t) is most likely a linear function. Let v(t) = a – bt.

    13. Deriving h(t) Overview: Find two descriptions for dV/dt. Set them equal to each other. Find a formula for dh/dt. Plug in v(t). Solve for h(t).

    14. What is dh/dt? Outflow from the pipe: Chain rule: Set them equal!

    15. Deriving h(t), Part 2

    16. Height Slope Fields

    17. Height Graphs

    19. Deriving Torricelli’s Law Overview: Reinsert v into the equation, eliminating t. Solve for a. Express v in terms of h.

    22. Torricelli’s Law Ideal law: Experimental factors cause decrease in effectiveness Rotational motion Viscosity More appropriate law: Would a better value for alpha work? We can use this to theoretically describe the motion of the pipes!

    23. Physical Proof of Torricelli’s Law Bernoulli’s equation for ideal fluid: Let point a be at the top of the container, and point b at the hole

    24. What is alpha?

    25. Theoretical h(t) Overview: Plug in Torricelli for v(t), not a – bt. Integrate with respect to dt. Solve for h(t).

    27. Theoretical h(t) Equations hBB = 0.26598t^2 - 7.66689t + 55.25, t < 14.4126 hBS = 0.01662t^2 - 1.97398t + 58.6, t < 59.3726 hSB = 4.25565t^2 - 31.2586t + 57.4, t < 3.67259 hSS = 0.26598t^2 - 7.69805t + 55.7, t < 14.4712

    28. Comparing h(t) Graphs

    29. Special Case: Draining Times How long does it take to drain each pipe? Pipe Ideal Actual Percent Error BB 14.4 s 13.3 s 7.6% BS 59.4 s 50.4 s 15.2% SB 3.7 s 3.8 s 2.7% SS 14.5 s 13.1 s 9.7% This is based on alpha being 0.84. It is too low. How did we derive ideal draining times?

    30. Deriving Draining Time Solve for when h(t) = 0.

    32. Modeling Other Functions We can use this to also model the velocity and height:

    33. Extensions More complicated systems

    34. Equilibrium Points The height of the water column is affected by two factors: Water leaving through hole (variable rate) Water entering through top (constant rate) Equilibrium when those two are equal

    35. Finding Equilibrium (Experimental) Keep the pipe (pipe BS) unplugged and fill it with water coming from a constant source of water (for example, a showerhead). After about four minutes, plug the hole. Measure the time it takes for the equilibrium water column to drain. Use this to find the height of the water column.

    36. Experimental Results

    37. Differential Equation This is not solvable by typical ODE methods. Slope fields and Euler’s method can be used to numerically interpret this ODE.

    38. Equilibrium Slope Field

    39. Finding Equilibrium (Theory)

    40. What is b (or not 2b)? b is the rate at which water enters the pipe and can be determined experimentally. A container (bucket) with known volume was filled by the water source. The time it took to fill the source was recorded.

    41. Finding Equilibrium (Theory, Part 2)

    42. Two-Hole System (no inflow)

    43. Two-Hole Slope Field

    44. Experimental Calculations

    45. Two-Hole System (inflow)

    46. Two-Hole Inflow Slope Field

    47. Qualitative Analysis A qualitative view of the system showed that the equilibrium point was between the second and third holes of the four-hole system (between 14 and 28 inches). The numerical solution was 19.8 inches.

    48. Applying Torricelli’s Law

    49. Applications with Inflow Suppose water flows into Tank 1 at rate b.

    50. Calculus of Variations Given a constant volume, what is the shape of the container that drains it in the shortest amount of time? Two scenarios: Actual shape? Degenerative case?

    51. Conclusions For a draining cylindrical container, Height and volume decrease quadratically. Exit velocity decreases linearly. The container drains in finite time. Torricelli’s Law is obeyed for a non-ideal value of alpha near 1. Equilibrium can be achieved if there is a constant inflow. Multiple holes increases the rate of decrease and decreases emptying time.

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