Cut point diameter

The cumulative analyses of feed, overflow, and product are plotted in Fig. 30.3. The cut-point diameter is the mesh size of the sereen, which from Table 30.1 is 1.651 mm. Also from Table 30.1, for this screen,... 

For these reasons we obtain a smooth, s-shaped, grade-efficiency curve in practice. The cut size, or x-50 cut-point (often referred to as the d-50 cut-point diameter ) is taken as the size that is separated with a fractional efficiency of 0.5 X50. 

Thus, although this model was derived by considering the gas flow pattern in the cyclone, it relates, in its final form, the separation cut-point diameter, X50, to the pressure drop. Hence, the pressure drop needs to be predicted to use the model. A good pressure drop model for this purpose is that of Shepherd and Lapple, Eq. (4.3.18). 

Fig. 6.1.1). This boundary layer flow can vary from about 4% to 16% of Q but a good, average value for calculation purposes is 10%. As a consequence, approximately 90% of the incoming flow Q directly participates in the flow along the walls and in the formation of the inner vortex. This is the reason for the factor 0.9 in Eq. (6.1.5) and in Eqs. (6.2.3) and (6.4.2) below. As we will see below, the inner vortex flow has a major influence on the cut-point diameter, X50.

A very fundamental characteristic of any lightly-loaded cyclone is its cut-point diameter or cut size, X50, produced by the spin of the inner vortex. This is the particle diameter that has a 50% probability of capture. As discussed elsewhere in this book, the cut size is analogous to the screen openings of an ordinary sieve or screen although, with a cyclone, the separation is not as sharp as that of a sieve. 

Knowing Ar we are now in a position to compute vqcs from Eq. (6.2.1), which is needed in the computation of the cut-point diameter of the inner vortex ... 

In this chapter we shall derive and present relationships or formulae that will allow us to predict a cyclone s cut-point diameter, grade-efficiency curve, overall or gross efficiency, and pressure drop on the basis of measurements taken on a geometrically similar cyclone. These formulae should also allow us to evaluate the performance of an operating cyclone and, if necessary, assist us in troubleshooting its design, mechanical condition, or mode of operation. 

Figure 8.2.1 includes data from cyclones of different geometries. Although we can clearly see the trend in the figure, there is considerable scatter. For Rein > 2 X 10, Stkinso) varies by a factor of 4, and so does the cut-point diameter. Trying to predict the performance of all cyclones, irrespective of the geometry, from a Stk o-Re plot is therefore not a worthwhile exercise. Even so, the plot does give a ball-park estimate of the Stkinso of cyclones. 

We note from the experimental data reported above that the model s cut-point diameter, X50, is 0.98 yum. We do not have data at the same Re for model and prototype, so we will make use of the approximations mentioned in the main text. Rein is large enough in both model and prototype to assume that the grade-efficiency is about the same in the two cyclones for the same value of Stk, and that their Eu values are the same. 

The 1.72 m diameter cyclone described in Appendices 4.A and 5.A is to be operated at an absolute pressure of 0.1 atmosphere. We wish to compute the X50 cut-point diameter. We shall assume, for the sake of simplicity, that there is no change in the total friction factor as a result of the decrease in operating pressure. 

The all-important cut size or cut-point diameter of the irmer vortex may now be computed directly from Eq. (6.2.3) or, if necessary, Eq. (6.2.6) where the particle density, pp, now refers to the density of the liquid phase. 

If Co < CoL, then there is no mass loading effect and the comparatively simple method for computing the cyclone s separation performance, as described in Sects. 13.5.1 and 13.5.2, applies. Conditions that may lead to this scenario include a low liquid loading, Co, a very fine feed drop size distribution, and a large inner vortex cut-point diameter, X50 ... 

Dry Process. Ten kg each of the ground field pea and fababean were passed through an Alpine Pin Mill model 250 CW (Alpine American Corp., Natick, MA) (Figure 1). Two passes through the mill reduced the particle size to less than 325-mesh. The pin-milled flours were then fractionated into light and dense particles by a single pass through the Alpine Air Classifier Type 132 MP at a cut point of 15 microns (800-mesh) diameter between the two fractions (9 ), followed by a reclassification of the dense fraction (20). The two protein fractions were combined. 

There are several different approaches that are commonly used to determine particle size distributions in air. One of them, impaction, has been discussed earlier. Multistage impactors with different cut points are used extensively to obtain both mass and chemical composition data as a function of size for particles with diameters > 0.2 /xm. Others, including methods based on optical properties, electrical or aerodynamic mobility, and diffusion speeds, are described briefly in the following section. The condensation particle counter (CPC) is used as a detector in combination with some of these size-sorting methods. 

For these studies, we entered separately the compositions of "fine", "coarse", and "total" particles. "Fine" particles are defined as those of diam <2.5 pm, the cut-point of most dichotomous samplers. "Coarse" particles are those with diameters between 2.5 pm and the maximum collected (usually 15-20 pm). Many of the data were taken with different size cuts, so it was necessary to group the data in different ways and interpolate around the 2.5-pm region. "Total" particles are either those collected without size segregation, or else the sum of the collected size fractions weighted by the relative mass loadings of each. 

Instead of using the virtual impactor approach, North American air monitoring programs in the 1980s and later have adopted simpler reference methods that use the weighing of filters in the laboratory. The filters are obtained from samplers equipped with an inlet device that provides for a sharp cut-point in particle entry for samples of particles < 10 xm diameter or <2.5 [im diameter, which are operated over a fixed time period of 24 hours. The inlet fractionation is facilitated either by a carefully designed cyclone or by an impactor. The combination of the two samplers can give estimates of mass concentration for fine-particle and coarse-particle concentrations. 

Due to the viscosity of the fluid, a parabolic velocity front exists which is flattened in the case of large diameter tubes. The cut is not sharp therefore since the upward force on the particle depends upon its axial position in the tube. Roller [11] showed that the effect of the uneven cut is the removal of some coarse above the theoretical cut-point, while leaving behind some of the fines. Thus, while the separate fractions are not accurately sized, the final mass fraction is often reasonably close to the correct value. This was confirmed by Stairmand [12] who pointed out that the method was not applicable to bimodal distributions. 

Hydrocyclones, also called hydroclones, employ self-generated mild centrifugal forces to separate the particles into groups of predominantly small and predominantly large ones. Because of bypassing, the split of sizes is not sharp. The characteristic diameter of the product is taken as rfso, the diameter than which 50 wt % of the material is greater or less. The key elements of a hydrocyclone are identified on Figure 12.3(h). A typical commercial unit has an inlet area about 1% of the cross-sectional area between the vessel wall and the vortex finder, a vortex finder with diameter 35 40% that of the vessel, and an apex diameter not less than 25% that of the vortex finder. For such a unit, the equation for the cut point is... 

A cut point can be defined as the diameter of that particle which just reaches one-half the distance between ri and rj. If is the cut diameter, a particle of this size moves a distance y — (r2 — r jl during the settling time allowed. If a particle of diameter is to be removed, it must reach the bowl wall in the available time. Thus rg = / a and = (r, -I- Equation (30.77) then becomes... 

A cut point or critical diameter can be defined as the diameter of a particle that reaches -j the distance between rj and rj. This particle moves a distance of half the liquid layej or (rj — r,)/2 during the time this particle is in the centrifuge. The integration is then between / = (/ , + r )/ at t = 0 and r = rj at t =. Then we obtain... 

Assuming that half of all those particles present in the feed with a particular diameter are removed during their transit through the bowl, those particles with diameters greater than x will be mostly removed from the liquid, whereas those particles with diameters smaller than x will be likely to remain in the liquid. In this context as here defined is known as the "cut-point" or "critical" diameter. 

A viscous suspension contains particles of a solid of 1461 kg/m density, and is to be clarified by centrifugation. The liquid has a density of 801 kg/m and a viscosity of 100 cp. A clarifying centrifuge with bowl 44.5 diameter and 197 mm height is used. During operation an annular air core of 14.23 mm diameter is formed. Calculate the cut point for a capacity of 169.92 L/h, when the centrifuge runs at 23,000 rpm speed. 

Estimate the cut point when treating 1 L/s of suspension in a hydrocyclone of 2 in diameter. The concentration of the suspension is 15% by volume, its density is 1250 kg/m , and behaves as non-Newtonian with determined values of the flow behavior index of 1.28 and the fluid consistency index of 2.03 x 10 . The density of the suspended solids is 2800 kg/m . 

Stage Nominal cut-point (tun) As calibrated cut-point (pm) Nozzle diameter (cm) Number of nozzles 5/W > P/Po Nozzle Reynolds number... 

Ambler (1952) introduced the cut point concept in the sedimentation centrifuge separation. It is defined as the diameter of the particle that just reaches one-half the distance between rj and 2. If a solid particle is to be removed from the fluid, it must travel the distance ( 2 i)/2 to the bowl wall in the available residence... 

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