Velocity gravitational

Drag Coefficient, Cd g(p - /o ) pu L characteristic length dimension of object p density of object u relative velocity gravitational force inertial force Free settling... 

Minimal efficiency at an intermediate air velocity occurs because different forces act to collect airborne particles at different velocities. At low velocities, gravitational, diffusional, and electrostatic forces act on the particle. Their effect is inversely proportional to air velocity. At high velocities, inertia forces come into play, and they are directly proportional to air velocity. The nature of inertial effects is such that below a certain air velocity, collection due to inertial forces is zero. 

Figure 9.5a shows a portion of a cylindrical capillary of radius R and length 1. We measure the general distance from the center axis of the liquid in the capillary in terms of the variable r and consider specifically the cylindrical shell of thickness dr designated by the broken line in Fig. 9.5a. In general, gravitational, pressure, and viscous forces act on such a volume element, with the viscous forces depending on the velocity gradient in the liquid. Our first task, then, is to examine how the velocity of flow in a cylindrical shell such as this varies with the radius of the shell.

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). 

Falling ball viscometers are based on Stokes law, which relates the viscosity of a Newtonian fluid to the velocity of the falling sphere. If a sphere is allowed to fall freely through a fluid, it accelerates until the viscous force is exactly the same as the gravitational force. The Stokes equation relating viscosity to the fall of a soHd body through a Hquid may be written as equation 34, where ris the radius of the sphere and d are the density of the sphere and the hquid, respectively g is the gravitational force and p is the velocity of the sphere. 

Fluid statics, discussed in Sec. 10 of the Handbook in reference to pressure measurement, is the branch of fluid mechanics in which the fluid velocity is either zero or is uniform and constant relative to an inertial reference frame. With velocity gradients equal to zero, the momentum equation reduces to a simple expression for the pressure field, Vp = pg. Letting z be directed vertically upward, so that g, = —g where g is the gravitational acceleration (9.806 mVs), the pressure field is given by... 

A particle falling under the action of gravity will accelerate until the drag force balances gravitational force, after which it falls at a constant terminal or free-settling velocity given by... 

From the standpoint of collector design and performance, the most important size-related property of a dust particfe is its dynamic behavior. Particles larger than 100 [Lm are readily collectible by simple inertial or gravitational methods. For particles under 100 Im, the range of principal difficulty in dust collection, the resistance to motion in a gas is viscous (see Sec. 6, Thud and Particle Mechanics ), and for such particles, the most useful size specification is commonly the Stokes settling diameter, which is the diameter of the spherical particle of the same density that has the same terminal velocity in viscous flow as the particle in question. It is yet more convenient in many circumstances to use the aerodynamic diameter, which is the diameter of the particle of unit density (1 g/cm ) that has the same terminal settling velocity. Use of the aerodynamic diameter permits direct comparisons of the dynamic behavior of particles that are actually of different sizes, shapes, and densities [Raabe, J. Air Pollut. Control As.soc., 26, 856 (1976)]. 

Gravitational Sedimentation Methods In gravitational sedimentation methods, particle size is determined from settling velocity... 

In most solids, the sound speed is an increasing function of pressure, and it is that property that causes a compression wave to steepen into a shock. The situation is similar to a shallow water wave, whose velocity increases with depth. As the wave approaches shore, a small wavelet on the trailing, deeper part of the wave moves faster, and eventually overtakes similar disturbances on the front part of the wave. Eventually, the water wave becomes gravitationally unstable and overturns. 

For a shock wave in a solid, the analogous picture is shown schematically in Fig. 2.6(a). Consider a compression wave on which there are two small compressional disturbances, one ahead of the other. The first wavelet moves with respect to its surroundings at the local sound speed of Aj, which depends on the pressure at that point. Since the medium through which it is propagating is moving with respect to stationary coordinates at a particle velocity Uj, the actual speed of the disturbance in the laboratory reference frame is Aj - -Ui- Similarly, the second disturbance advances at fl2 + 2- Thus the second wavelet overtakes the first, since both sound speed and particle velocity increase with pressure. Just as a shallow water wave steepens, so does the shock. Unlike the surf, a shock wave is not subject to gravitational instabilities, so there is no way for it to overturn. 

The distance above the catalyst bed in which the flue gas velocity has stabilized is refened to as the transport disengaging height (TDH). At this distance, there is no further gravitation of catalyst. The center-line of the first-stage cyclone inlets should be at TDH or higher otherwise, excessive catalyst entrainment will cause extreme catalyst losses. 

To determine the size of a particle having a velocity, w, in the gravitational field, both sides of equation 63 are multiplied by the complex Re/ArK ... 

Separation Rates in Tubular-and Solid-Bowl Centrifuges To evaluate the radial velocity of a particle moving toward a centrifuge wall, the expression for particle settling in a gravitational field is applied ... 

Depending on the particle diameter and properties of the liquid, the radial motion of particles will be laminar, turbulent or transitional. The motion of large particles at Re > 500 is turbulent. Therefore, their settling velocity in a gravitational field may be expressed as ... 

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