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Unformatted text preview: Homework Set #1: SOLUTIONS Problem #1: Consider a spherical particle falling freely in a large fluid body. It is found experimentally that the terminal settling velocity u of the particle is dependent on: (1) particle diameter d ; (2) the buoyant weight of the particle W (the difference between the gravity and buoyant forces acting on the particle); (3) fluid density , and (4) fluid viscosity, . It is assumed that the velocity u can be mathematically expressed as products of powers of the above variables (i.e. similar to the example of dimensional analysis given in the previous lectures). Obtain the relationship for u using dimensional analysis. Suggestion : solving the set of linear equations in terms of the exponent of W. Answer: If a b c d u Cd W = ( ) ( ) ( ) ( ) b c d a-1-2-3 1-1 L t L MLt ML M L t C = Equating dimensions: M L 1 3 t -1 2 b c d a b c d b d = + + = + = Solving in terms of b: a = -1, c = (b-1), d = (1-2b) ( )( ) 2 2 2 (1/ ) / / b b b b b W u C d W C d d W u C = = = Problem #2: The Kelco Oil Field Group, Inc., has published the following data for two fluids, their XanvisTM drilling fluid and HEC polymer: dv x /dy (s-1 ) 1022 511 340 170 10.2 5.1 1.0 0.1 XanvisTM product eff (cp) 12 20 26 42 410 740 4650 23500 HEC polymer eff (cp) 65 100 130 200 950 1250 3500 5900 For both fluids, attempt to fit these data to both the Bingham and power-law models. For each fluid, determine which model fits the data best and attempt to estimate the model parameters for...
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- Spring '08