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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 ρ , and (4) fluid viscosity, μ . 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 ρ µ = ( ) ( ) ( ) ( ) 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 µ ρ ρ ρ µ µ ρ µ ρ ρ µ µ ⎛ ⎞ ⇒ = = ⎜ ⎟ ⎝ ⎠ ⎛ ⎞ ⇔ = ⎜ ⎟ ⎝ ⎠ 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 µ eff (cp) 12 20 26 42 410 740 4650 23500 HEC polymer µ 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
- Shear Stress, Bingham, shear st ress