PS05 - MECH 411 Introduction to Fluid Mechanics American...

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MECH 411 Introduction to Fluid Mechanics Dec 25, 2004 American University of Beirut, Fall 2004-2005 Handout # PS5 Problem Set 5 Solution Textbook: 5.30, 5.38, 5.45, 5.67, 3.150, 3.162, 3.175, 3.176 1
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Problem 5.30 Variables: [ τ ] = ML - 1 T - 2 , [ ρ ] = ML - 3 , [ μ ] = ML - 1 T - 1 , [Ω] = T - 1 , [ R ] = L , r ] = L . Wall shear stress τ w = f ( ρ,μ, Ω ,R, Δ r ). Using ( ρ, Ω ,R ) as repeating variables. Number of dimensionless groups = 6 - 3 = 3. Π 1 = τ w 1 2 ρ Ω 2 R 2 = C f Π 2 = Δ r R Π 3 = ρ Ω R 2 μ = Re Therefore, in dimensionless form, τ w 1 2 ρ Ω 2 R 2 = f p Δ r R , ρ Ω R 2 μ P 2
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Problem 5.38 Variables: [ d ] = L , [ D ] = L , [ U ] = LT - 1 , [ ρ ] = ML - 3 , [ μ ] = ML - 1 T - 1 , [ Y ] = MT - 2 . The number of dimensionless groups is 6 - 3 = 3. Using D,ρ and U as the repeating variables, the following dimensionless groups are obtained: Π 1 = ρUD μ = Re Π 2 = d D Π 3 = ρU 2 D Y = We Therefore, the relation in dimensionless form is d D = f (Re , We) 3
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Problem 5.45 Model diferential equation For chemical reaction dynamics in a plug reactor is u ∂C ∂x = D 2 C ∂x 2 - kC - ∂C ∂t (a) The above equation is physically homogeneous so that all terms have the units oF s - 1 . Then k has the unit oF s - 1 and D has the unit oF m 2 /s . (b) De±ning u * = u/V , x * = x/L , and t * = t/ ( L/V ) then the above equation is written as V L u * ∂C ∂x * = D L 2 2 C ∂x * 2 - kC - V L ∂C ∂t * u * ∂C ∂x * = D V L 2 C ∂x * 2 - kL V C - ∂C ∂t * The pi groups that appear are D V L which is ratio oF mass difusion efect to inertia efect and kL/V which is the ratio oF reaction time scale to inertia time scale.
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PS05 - MECH 411 Introduction to Fluid Mechanics American...

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