midterm review - Midterm 1 Review Problem 1 Over a century...

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Midterm 1 Review Problem 1) Over a century ago, waterclocks were often used to tell time on cloudy days when sundials were useless. A waterclock consists of a box which holds a certain amount of water that would drain in one hour. Shown below is a waterclock with the relevant dimensional parameters along with the initial water height, H, the time to empty the waterclock, t, water density, ρ , water viscosity, μ , and gravitational acceleration, g. a) Using the Buckingham Pi theorem, determine what you can about the time it takes to empty the waterclock, T, as a function of relevant dimensionless groups. Use density, gravitational acceleration, and initial water height as the core variables. b) If L, d, and h are all much smaller than H, then the dimensionless groups involving L, d, and h are nearly zero and the functional dependence on these dimensionless groups is nearly constant. If a new fluid is used in place of water with a viscosity twice that of water and density three times that of water, how should the initial fluid height level change to maintain similarity? How long would this water clock hold fluid for?
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Solution t H l h μ d ρ g M 0 0 0 0 1 0 1 0 L 0 1 1 1 ‐1 1 ‐3 1 T 1 0 0 0 ‐1 0 0 ‐2 8 parameters – 3 dimensions = 5 dimensionless groups. π 1 = t ρ a g b H c = T M L 3 a L T 2 b L 1 c a = 0, b = 1/2, c = 1/2 π 1 = tg 1/ 2 H 1/ 2 π 2 = l ρ d g e H f = L M L 3 d L T 2 e L 1 f d = 0, e = 0, c = 1 π 2 = lH 1 π 3 = d
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  • Spring '10
  • Muller
  • Trigraph, open channel, Buckingham Pi Theorem, dimensionless groups, initial water height, dimensionless groups.

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