Velocity small particles, linear

This reaction is carried out in tall fluidized beds of high L/dt ratio. Pressures up to 200 kPa are used at temperatures around 300°C. The copper catalyst is deposited onto the surface of the silicon metal particles. The product is a vapor-phase material and the particulate silicon is gradually consumed. As the particle diameter decreases the minimum fluidization velocity decreases also. While the linear velocity decreases, the mass velocity of the fluid increases with conversion. Therefore, the leftover small particles with the copper catalyst and some debris leave the reactor at the top exit. 

The smaller the particle size, the faster the rate of generating theoretical plate (HETP) per unit of time. Figure 14.8 shows a plot of HETP versus linear carrier velocity u for small particles. It indicates that the smaller the particle size, the lower the HETP. It is also important to note that small particles provide nearly the same HETP over a wider range of flow rate.14... 

The approach of representing the fluid and particle motion by their component frequencies is only valid if drag is a linear function of relative velocity and acceleration, i.e., if the particle Reynolds number is low. This is the reason for the restriction on small particles noted earlier. The terminal velocity of the particle relative to the fluid is superimposed on the turbulent fluctuations and is unaffected by turbulence if Re is low (see Chapter 11). 

Fluidized Beds Fluidized beds are reactors in which small particles (with average size below 0.1 mm) are fluidized by the reactant gases or liquids. When the linear velocity is above the minimum required for fluidization, a dense fluidized bed is obtained. As the superficial velocity increases, the bed expands and becomes increasingly dilute. At a high enough linear velocity, the smallest particles entrain from the bed and have to be separated from the exhaust gases and recycled. 

The elimination or estimation of the axial dispersion contribution presents a more difficult problem. Established correlations for the axial dispersion coefficient are notoriously unreliable for small particles at low Reynolds number(17,18) and it has recently been shown that dispersion in a column packed with porous particles may be much greater than for inert non-porous particles under similar hydrodynamic conditions(19>20). one method which has proved useful is to make measurements over a range of velocities and plot (cj2/2y ) (L/v) vs l/v2. It follows from eqn. 6 that in the low Reynolds number region where Dj. is essentially constant, such a plot should be linear with slope Dj, and intercept equal to the mass transfer resistance term. Representative data for several systems are shown plotted in this way in figure 2(21). CF4 and iC io molecules are too large to penetrate the 4A zeolite and the intercepts correspond only to the external film and macropore diffusion resistance which varies little with temperature. 

Ostergaard91 found that for very small sand particles, i.e., 40 through 60 and 60 through 80 mesh, the gas (nitrogen) holdup was independent of particle size and liquid (water) velocity, but increased linearly with nominal gas velocity. The gas holdup in the gas -liquid (solids-free) system was higher than that in a gas-liquid fluidized bed. 

Case II refers to situations where the particle-wall interactions are purely repulsive. The particles are separated from the wall by a thin layer of solvent, even in the absence of any motion. Slip is thus possible for very slow flows, indicating that the sticking yield stress is vanishingly small. The residual film thickness for weak flows corresponds to a balance between the osmotic forces and the short-range repulsive forces, independently of any elastohydrodynamic contribution. This is clearly reflected in Fig. 16c, d, where we observe that the particle facet is nearly flat and symmetric. Since tire pressure in the leading and rear regions of the facet are equal and opposite, the lift force is very small. The film thickness, which is set by the balance of the short-range forces, is constant so that the stress/velocity relationship is linear. 

The fact that the van Deemter plots of small particles are generally flat beyond the optimum linear velocity means that higher linear velocity would not compromise column efficiency significantly. Therefore, in practice, rapid gradients and high flow rates are usually applied in small particle columns to shorten the runtime and increase sample throughput. The column temperature also affects separation speed and column efficiency, and is used to control the selectivity of the separation as well. An increase in temperature leads to a decrease in retention in reversed phase UHPLC. It is generally considered that the A term... 

The low exponent of dp may seem a little surprising. However, if one considers a boundary layer mechanism, particle suspension will depend on the linear velocity which the particle sees. As this velocity fails rapidly across the boundary layer to zero at the base (approximating to a turbulent boundary layer profile), large particles experience relatively large forces and small particles, small ones. Thus the effect of size nearly cancels out and this is found too in pipe flow. 

Most applications in ultrasound deal with rather small particle velocities v. This can be described by the acoustic Mach number M = v/cl. A 20-kHz high-intensity step horn with peak amplitude of 100 pm creates a Mach number of about 0.01. For low Mach numbers, the acoustic approximation of incompressible liquids is applicable. In certain cases, for example focused sound fields, much higher Mach numbers are present and lead to distortion of the sinusoidal wave form. Density and sound pressure no longer satisfy a simple linear equation with a fixed speed of sound c. A pressure-dependent sound speed is observed (material nonlinearity). Convective terms in the equation of motion cannot be neglected at higher Mach numbers and lead to a more complicated situation (convective nonlinearity) where the real speed of the wave front consists of two parts, the speed... 

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