Acceleration radial

Since the radial acceleration functions simply as an amplified gravitational acceleration, the particles settle toward the bottom -that is, toward the circumference of the rotor-if the particle density is greater than that of the supporting medium. A distance r from the axis of rotation, the radial acceleration is given by co r, where co is the angular velocity in radians per second. The midpoint of an ultracentrifuge cell is typically about 6.5 cm from the axis of rotation, so at 10,000, 20,000, and 40,000 rpm, respectively, the accelerations are 7.13 X 10, 2.85 X 10 , and 1.14 X 10 m sec" or 7.27 X 10, 2.91 X 10, and 1.16 X 10 times the acceleration of gravity (g s). 

The force of a molecule subject to radial acceleration is given by Newton s second law ... 

A buoyant force is given by the product of the volume V of the particle, the density p of the solution, and the radial acceleration ... 

Radial acceleration Rate of velocity change with respect to time in a radial direction. 

Consider an open bucket of water resting on a turntable that is rotating at an angular velocity on (see Fig. 4-5). The (inward) radial acceleration due to the rotation is o r, which results in a corresponding radial pressure gradient at all points in the fluid, in addition to the vertical pressure gradient due to gravity. Thus the pressure differential between any two points within the fluid separated by dr and dz is... 

For very small particles or low density solids, the terminal velocity may be too low to enable separation by gravity settling in a reasonably sized tank. However, the separation can possibly be carried out in a centrifuge, which operates on the same principle as the gravity settler but employs the (radial) acceleration in a rotating system (o r) in place of the vertical gravitational... 

Because of the extreme importance of the ultracentrifuge, it seems appropriate to describe it in some detail. Although the particulars differ from instrument to instrument, the essential features are present in all ultracentrifuges. The actual sedimentation takes place in a cell mounted within an aluminum or titanium rotor. The cell is sector shaped its side walls converge toward the center along radial lines. Since the radial acceleration is proportional to the distance from the axis of rotation, we see that this quantity varies from top to bottom in the cell. Although this variation is considered explicitly in a section below, it is sufficient for the present to consider the average acceleration at the midpoint of the cell, which is typically located 6.5 10 2 m from the center of the rotor. For speeds of 10,000, 20,000, and 40,000 rpm, accelerations of... 

Besides pumping, centripetal acceleration is created. A maximum fluid rotational velocity of up to 12 m/s, and a corresponding radial acceleration in excess of 106 g have been produced within a diamond-shaped microchamber (55 x 55 im). This notch chamber was constmcted along the side wall of an otherwise straight channel (30 pm wide, 30 pm deep) which was fabricated on a PDMS chip. This microstructure caused flow detachment at the opening of the notch, leading to recirculating flow of microvortex inside the notch [384]. 

In converging or diverging sections of annular dies, the fluid elements are subjected to axial and radial accelerations. Neglecting the radial special accelerations (for small tapers), the -component equation of motion reduces to... 

In Figure 6, section 1 is immediately downstream of the expansion where radial acceleration of fluid is small so pressure is uniform across the cross-section (and area is A2 not Af. Section 2 is sufficiently far downstream for flow to be parallel and pressure constant across the cross-section. 

These equations as well as Eqs (2-49) and (2-50) are difficult to solve, but following Lapple and Shepherd (1940) we may obtain a solution for a special case by assuming that the particle moves at the same speed as the fluid (vt = v = vot) and that the radial acceleration dvr/dt = 0. Thus vf = vr and Eq (2-49) reduces to... 

The cyclone or centrifugal separator is a device utilizing radial acceleration for separating partides suspended in a gas stream. It con-... 

Lapple and Shepherd (1940) have given the general equations for motion of particles in a cyclone. These equations cannot be solved except by the method of approximations. If the tangential and radial accelerations are neglected, the radial velocity of the particle is given by the equation... 

When a particle is moving in a circular path around a point a distance r away with an angular velocity of co, it experiences a radial acceleration of... 

One way to remove these particles from air is to subject them to high centrifugal forces. If an aerosol particle is caused to move in a circular path, it will have a radial acceleration given by Eq. 7.1. This radial acceleration can be likened to the acceleration due to gravity in a gravitational field. By rotating the aerosol, accelerations many times that of gravity can be achieved. 

Example 8.1 A centrifuge has a radius of 50 cm and is operated at 500 r/min. Determine the ratio of radial acceleration to gravitational acceleration in this case. 

As can be seen from Example 8.1, quite large radial accelerations are possible, indicating that very small particles can be removed in this manner. 

At the terminal velocity, the outward radial acceleration, d r/dP = 0 so that this equation becomes ... 

The situation is much more complicated for calculation of the hydrodynamic regularities upon suspension in stirred tanks, because here the radial acceleration may not be ignored. (This is evident from the fact that the specific stirrer power required for suspension decreases with increasing tank diameter ) Zehner developed a theoretical concept, which took into consideration the radial acceleration in the stirred tank and in this way realized relationships, which described the measured in tanks of different sizes well. 

Problem 12-6. The Linear Stability of a Spherically Symmetric Fluid Interface to Radial Accelerations. The classical Rayleigh Taylor analysis that is described in Section B examines the stability of a plane interface between two fluids of different density to accelerations normal to the interface and shows that the interface is unstable or stable, depending on whether the acceleration is directed from the heavier fluid to the lighter fluid, or vice versa. In this problem, we consider the related problem of a spherically symmetric interface that is subjected to radial accelerations. This is a generalization of the problem of an expanding or contracting gas bubble that was considered in Chap. 4. 

The uniform rotation results in an (inward) radial acceleration equal to mh. This gives rise to a... 

Fig. 7.2.2 Segment of Fig. 7.2.1 the mass Mn oscillates radial to the pivot S with the frequency coD yaw rate Qz d2-dn=working stroke ar, Coriolis acceleration at, radial acceleration vt, tangential speed... 

The formula shows that the radial acceleration is shifted by 90° out of phase to the desired signal at(t) and that it increases quadratically with coD. 

The suppression of the radial acceleration is realized mechanically by using acceleration sensors that are not sensitive to lateral accelerations, and by keeping an orientation of 90 ° between the drive and detection direction as precisely as possible. 

Fig. 2.7 Particle depletion in centrifugal fields due to radial acceleration and radial trajectories...

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