Reynolds number cyclone

Reentrainment is generally reduced by lower inlet gas velocities. Calvert (R-12) reviewed the hterature on predicting the onset of entrainment and found that of Chien and Ibele (ASME Pap. 62-WA170) to be the most reliable. Calvert applies their correlation to a liquid Reynolds number on the wall of the cyclone, Nrcl = 4QilhjVi, where is the volumetric liquid flow rate, cmVs hj is the cyclone inlet height, cm and Vi is the Idnematic liquid viscosity, cmVs. He finds that the onset of entrainmeut occurs at a cyclone inlet gas velocity V i, m/s, in accordance with the relationship in = 6.516 — 0.2865 lu A Re,L ... 

Thus if the feed flow rate to such a cyclone is altered the new diaracteristic velocity can be calculated iom Equation (8.38), Reynolds number Rom (8.39), Euler number ftom (8.42) and pressure drop from (8.40). The new cut size can be calculated from Equation (8.37) after using (8.41) to give the new Stokes-50 number. However, Equations (8.41) and (8.42) are vaUd only for cyclones of Reitema s optimum geometry. 

The Reynolds number defines flow features of the system and, in the case of hydrocyclones, the characteristic dimension may be taken as the cyclone body diameter D ... 

Contraction coefficient for flow from the freeboard to the cyclone inlet Contraction coefficient for flow from the cyclone barrel to the cyclone outlet Solids loading, kg of sohds/m of gas Vortex length below gas outlet tube, m Cyclone inlet width, m Effective number of soUd spirals in a cyclone Reynolds number at the cyclone inlet based on... 

Recovery of underflow from a cyclone Reynolds number... 

The second term on the right-hand side in (2.2.1) can be simplified. In gas cyclones we are concerned with small particles (small x) moving in a fluid of low density (small p), so that the particle Reynolds number ... 

Very central to cyclone technology is the dynamically equivalent particle diameter. This is the diameter of an equi-dense sphere that has the same terminal velocity as the actual particle. Calculating this can be difficult in the range of intermediate Reynolds numbers, or when the Cunningham correction is significant. In the region where Stokes drag law applies, we call it the Stokesian diameter. 

Muschelknautz (1972, 1980) gave expressions for / based on experiment. He found that, in a cyclone without dust, / decreases with the Reynolds number, as it does in normal pipe flow, but that the variation is somewhat different. He also found that / increases with increasing dust load due to the relatively slow movement of the separated dust along the cyclone wall. This dust forms a spiral-shaped dust helix or strand and acts as an additional component augmenting the dust-free wall friction. Such a dust helix can be seen in the laboratory cyclone shown in Fig. 4.2.4. He expressed / in two additive parts one, which we will call fair for a dust free cyclone, and the other, fdusti accounting for the effect of the dust. 

We compare the predictions of the Barth and the Muschelknautz models for pressure loss across the vortex tube with those of Eq. (4.3.17) in Table 4.3.1. In the models of Barth and Muschelknautz, Eux is related to the velocity ratio vqcs in a semi-empirical way, based on a curve-fit to experimental data. Muschelknautz and Krambrock (1970) also observed that Eux was somewhat dependent upon the vortex tube Reynolds number, Rcx but that it had little effect for Rcx > 5 x 10. Since the vast majority of practical cyclones operate well above this value, we do not attempt to fine tune the model to account for Rcx effects herein. We also want to point out that Muschelknautz and Krambrock s data automatically include the pressure loss associated with gas entering the vortex tube since the upstream static pressure they used to derive their vortex tube loss coefficient (Eux) was measured at a point ahead of the vortex tube. 

In order to compute certain key cyclone characteristics, such as the internal spin velocity, vocs, or the particle cut size in the inner vortex core, X50, it is necessary to first compute the gas-phase and total gas-plus-solids wall friction factors, fair and /, respectively. Gas-phase wall friction factors for both cylindrical and conical cyclones as a function of body Reynolds number and relative wall roughness are presented in Fig. 6.1.3. Muschelknautz and Trefz define the cyclone body Reynolds number (compare with Eq. 4.2.8) as ... 

Figure 6.1.3 is useful in showing thow the (solids free) gas friction factor in conical- and cylindrical-bodied cyclones varies with cyclone Reynolds number and relative wall roughness, that is fair = f kg/R,Rep). Even so, if we wish to incorporate it into a cyclone computer model, we need to express this functional relationship in equation form. Although the dependency between the variables shown in Fig. 6.1.3 is very nonlinear, and difficult to fit , the authors have developed a set of equations that fit the entire range of fair, kg/R and Rep values shown in Fig. 6.1.3 for both conical- and cylindrical-bodied cyclones. These empirical equations have a maximum error of about 20 to 22% relative to the data points shown in Fig. 6.1.3. This error decreases, of course, with increasing solids loading. The gas phase friction factors computed with the empirical curve fits shown below have proven sufficiently accurate for most design applications.

An additional problem in achieving Reynolds-number similarity is that, when comparing the performance of one industrial cyclone with that of another, obtaining data at the same Re for the two is often not possible. 

But is it really necessary to scale-up cyclones on the basis of Re similarity A redeeming feature is that, in many cases, Reynolds number similarity is not very critical. This has long been known, but the issue has only been studied quantitatively recently, by Overcamp and Scarlett (1993), among others. They deflned Re and Stk in terms of the inlet velocity. We shall use the symbols Rein and Stkm, respectively. Figure 8.2.1 shows a plot of the square root of Stkin50 against Re for a wide range of cyclones, taken from their paper. 

This is the formal requirement for dynamic similarity, and is consistent with the results of the classical dimensional analysis in the main text. As we mentioned there, experience teaches us that over a wide range of operating conditions Reynolds number similarity is not all that critical for Stokes number similarity between cyclones, and this indicates that, in this range, it is not all that critical for dynamic similarity. 

Determine the velocity at which we would have to operate a 6 (152 mm) diameter model cyclone to obtain Reynolds number similarity to a 48 (1220 mm) industrial cyclone. 

In most applications of interest absolute pressure is of sufficient magnitude to cause the mean free path of the gas molecules to be much smaller than the particles feeding the cyclone. This mean free path is the average distance a gas molecule travels between collisions with another molecule. Under such conditions the gas behaves as a continuum and, if the particle Reynolds number is sufficiently small (less than 1 in any case), the familiar Stokes law may, as discussed in Chap. 2, express the drag force acting on a particle moving through the gas... 

The Reynolds number of the film, which is swirling around the wall of a cyclone, was calculated as ... 

In a simple cyclone diffuser performance test conducted by the writers, however, a much larger decrease in cyclone pressure loss was observed. The apparatus tested was a 6-inch (153 mm) cylindrical bodied cyclone shown in Fig. 15.1.15. The cyclone Reynolds number for these tests were in the 800 to 1000 range (per Eq. 4.2.8) which, for cyclones, makes the Eu value very insensitive to the Reynolds number. (See gas phase friction factor charts in Ch. 6, for example.)... 

The mean axial velocity in the cyclone body v was used to evaluate the Reynolds and Euler numbers. This is signified with the subscript b. His plot, featuring the line representing Eq. (8.2.7) and his supporting data, is shown in Fig. 8.2.2. 

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