Stokes_Derivation

Stokes_Derivation - 492 PHYSICAL TREATMENT METHODS 0 E O N...

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Unformatted text preview: 492 PHYSICAL TREATMENT METHODS 0 E O N - .0 .5 Outlet zone Spherical particle FIGURE 12.15 Definition sketch for a spheried particle settling in (a) m idediud rectangular horizontd-flow settling tank and (b) a fluid whim to the form of buoyancy, gravity, and drag. balance can be written for a discrete particle that is settling: d , mffif=rg—I;—rb 022» where m p = mass of settling particle, kg vs = particle settling velocity, m/s FG = gravitational force, N F, = buoyant force, N F D = drag force, N The net gravitational force is given by Fa—Fa= (10,,- Pw)ng (12.23) where pp = density of particle, kg/m3 pw = density of water, kg/m3 g = acceleration due to gravity, m/s2 VP = volume of particle (1rd 3 /6), m3 d p = diameter of particle, m The drag force is a function of the cross-sectional area of the particle, the settling velocity of the particle, the liquid density, and the coeflicient of drag: 2 5=g¢ag (um 12.7 SEDIMENTATION 493 where CD = coeflicient of drag A p = cross-sectional area of particle (adj/4), m2 pw = density of water, kg/m3 vs = particle settling velocity, m/s For spherical particles, the coefficient of drag can be estimated using the following relationship: CD=24.+ 3 Nkfiv: + 0.34 (1225) where N R = Reynolds number, dimensionless v: d p pw p. p. = liquid viscosity, kg/ m - s and other terms are as defined previously. In the ideal system, the terminal settling velocity is attained quickly, and the acceleration term can be assumed to be negligible. Thus, Eq. (12.22) can be rewritten as F0 — FB = FD (12.26) Substituting Eq. (12.23) for F6 — F3 and Eq. (12.24) for FD and solving for the settling velocity 03 yields u, = — —— (12.27) where d p is the diameter of the particle, in meters, and the other terms are as defined previously. When NR < 0.3, the first term of Eq. (12.25) predominates, and the discrete particle settling rate becomes (Stokes’ law) g( p, - Md} Us _ 18” (12.28) where pp = particle density, kg/m3 )1 = liquid viscosity, kg/ m - s Particle densities in water and wastewater treatment vary consider- ably, but fall into distinct bands. Organic materials (bacteria, food particles, and fecal particles) usually have densities in the 1030 to 1100 kg/m3 range. Chemical flocs produced in precipitation reactions have WATER QUALITY MANAGEMENT WATER] [Mil W CHARACTERISTICS - MODELING - MODIFICATION George Tchobanoglous Edward D. Schroeder UNIVERSITY OF CALIFORNIA AT DAVIS Addison I a] W isley 1 Longman Reading, Massachusetts I Menlo Park, California Don Mills, Ontario I Wokingham, England I Amsterdam I Sydney Singapore I Tokyo I Mexico City I Bogota Santiago I San Juan ...
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This note was uploaded on 05/20/2010 for the course EVEG 3110 taught by Professor Malone during the Spring '10 term at LSU.

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Stokes_Derivation - 492 PHYSICAL TREATMENT METHODS 0 E O N...

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