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Formula Sheet CHE 312 Fluid Mechanics (Fall 2010) Final Exam, December 20, 9-11 am Print this document on a single sheet of paper and bring it to the exam; there will be no spare sheets at the exam. You are allowed to add information on the side of the sheet you printed on; the reverse side should be blank Newtonian fluids: τ xy = µ Hydrostatics ∂p = −ρ g ∂z Fz = V ρ f g dvx dy τ xy shear stress, µ viscosity, dvx velocity gradient dy p hydrostatic pressure, ρ density, g gravitational acceleration, z points up Fz buoyancy force, V displaced fluid volume, ρ f fluid density dm total mass m is a conserved quantity = φm ,in − φm ,out dt Steady state mechanical energy balance (Bernoulli equation) 1 p Wɺnf ∆ v 2 + gz + = − e fr ∆ is “out minus in”; Wɺnf work rate not related to flow; e fr frictional loss. 2 ρ φ m Frictional loss 4A L In a straight pipe or channel with length L and (hydraulic) diameter Dh ≡ : e fr = 2 fU 2 . W Dh Total mass balance: U= φV A : average velocity; A: cross sectional area; W: wetted perimeter. f: Fanning friction factor, f is a function of Re = Special cases: laminar flow in a round tube: f = Loss coefficients K: e fr = K ρUDh ε , and the relative wall roughness . µ Dh 16 ; turbulent flow with smooth walls: 4 f = 0.316 Re −1/ 4 . Re U2 . 2 Momentum balance d ( mv ) = φm,in v in − φm,out v out + ∑ F dt v is the velocity vector; ∑ F is the sum of forces acting on the system. Drag force FD on a particle moving relative to fluid 1 FD = CD ρ f A⊥ ∆v ∆v CD drag coefficient; A⊥ area ‘seen’ by the flow; ∆v = v f − v p . 2 ρ d ∆v CD depends on the shape of the particle and Re = f p , d p is the particle size. Spherical particles (diameter d p , and therefore A⊥ = µ π 4 d p2 ) For Re<1 Stokes law applies: FD = 3π d p µ ∆v , which is the same as CD = ( 24 . Re ) 24 1 + 0.15 Re0.687 if Re < 1000; CD = 0.44 if Re ≥ 1000 . Re Ergun equation (frictional loss in a packed bed with porosity ε made of spheres all having diameter d p ) A more general correlation: CD = vsup µ (1 − ε ) φ 1 2 e fr = 1.75 L vsup + 150 L vsup = V ; L is the bed length; A its cross sectional area. 3 2 3 dp ε ρ d pε A Darcy equation & permeability for flow in a porous medium d ( p + ρ gz ) µ =− v k is permeability (units m2); v is superficial velocity. dx k (1 − ε ) 2