Equilibrium settling velocities

Table 3.2 Equilibrium Settling Velocities for Spheres of Standard Density.35...

Tube Reynolds number, Re 2RpF p M Particle Reynolds number", Re 2a < p p P Particle Reynolds number in an unbounded stationary fluid, p Settling velocity-mean fluid velocity ratio. Eccentric equilibrium position of sphere center. 

The equilibrium orbit theory is based on the concept of the equilibrium radius, originally proposed by Dries sen" and Criner. According to this concept, particles of a given size attain an equilibrium radial orbit position in the cyclone where their terminal settling velocity is equal to the radial velocity of the liquid. Particles are therefore elutriated by the inward radial flow according to the balance of the centrifugal and drag forces, and Stokes law is usually assumed. 

The residence-time theory assumes non-equilibrium conditions and considers whether a particle will reach the cyclone wall in the residence time available. Rietema first proposed this theory and assumed homogeneous distribution of all particles across the inlet. The cut size will then be the size of the particle which, if entering precisely in the centre of the inlet pipe will just reach the wall in residence time T. In mathematical terms, this means that the particle radial settling velocity integrated with time should therefore be equal to half... 

While true sedimentary equilibrium is rare in nature, the assumption of equilibrium over short durations, e.g., minutes, enables one to use measured profiles C z ) to estimate the settling velocity of sediment. In this case we set dCjdt in Eq. (27.32) to zero, which results in... 

The measurement (indeed, the existence of) of the just suspension speed is subject to the assumption that there exists a speed where there is equilibrium between the particles lifted from the vessel base and those settling back onto the base. This assumption is reasonable with fast settling particles but becomes more and more dubious as the settling velocity of the particles decreases. 

On the analogy of hydraulic jump in open channel flow, the granular jump can occur in a horizontal pneumatic conveying, provided that for a given U the particle flow pattern belongs to the suspension flow accompanied with the settled layer and that the particles are introduced into the pipe with velocity hi er than the equilibrium particle velocity for U. Since the deceleration of particle is not observed in the upstream side of the jump, it is infared that the particle is discontinuously decelerated at the jump. 

Suspended particle transport is the result of dynamic equilibrium between particle detachment and deposition (sedimentation). A suspended particle will settle when the fluid shear stress drops below a critical level. Neglecting reaction, mass conservation of an insecticide associated with particles in settling velocity class i can be described by ... 

Erosion of noncohesive sediments depends on the equilibrium suspended load transport capacity and bed load flux relative to local conditions. The mode of transport for a noncohesive particle class k depends on the magnitude of the local shear velocity relative to the particle settling velocity (Ws ) and Shields critical shear velocity for initiation of motion ( cr, fc7 m/s) ... 

If the shear velocity exceeds the critical shear velocity and the particle settling velocity, eroded sediment will be at least partially transported as suspended load. For suspended load, the direction and magnitude of the water-bed sediment exchange flux (Jo, g/m /s) ultimately depends on the difference between the near-bed actual concentration (5 e, g/m ) and the near-bed equilibrium concentration (5eq, g/m ), as described by Equation 10.13. 

Determine the electrostatic velocity and the ratio of this velocity to the settling velocity for a 0.6-pm particle (e = 10) in a field strength of 1200 kV/m [12 kV/cm]. Assume that the particle is charged to the average Boltzmann equilibrium charge. 

Now that we have settled on a model, one needs to choose the appropriate algorithm. Three methods have been used to study polymers in the continuum Monte Carlo, molecular dynamics, and Brownian dynamics. Because the distance between beads is not fixed in the bead-spring model, one can use a very simple set of moves in a Monte Carlo simulation, namely choose a monomer at random and attempt to displace it a random amount in a random direction. The move is then accepted or rejected based on a Boltzmann weight. Although this method works very well for static and dynamic properties in equilibrium, it is not appropriate for studying polymers in a shear flow. This is because the method is purely stochastic and the velocity of a mer is undefined. In a molecular dynamics simulation one can follow the dynamics of each mer since one simply solves Newton s equations of motion for mer i,... 

The activity of the catalyst is roughly proportional to the surface area of nickel used. In an adiabatic prereformer, a high activity is desired to maximize the space velocity. In a tubular reformer the activity may be of less significance because the reactor volume is settled by mechanical criteria. Most industrial tubular reformers operate at space velocities of 2000-4000 hr However, equilibrium conversion can be... 

Stokes s Law. The force required to move a sphere of radius r at uniform velocity v through a medium of viscosity q is equal to 67cr )r. This law is applied in the determination of particle size by sedimentation, elutriation or centrifugation. For a particle of density d, settling in a medium of density g being the acceleration due to gravity, when equilibrium is reached the following equation applies ... 

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