Cyclones equation

For gas-liquid separators designed as in Fig. 3.2-6, the separation can be estimated using a common cyclone equation ... 

Cyclone Efficiency. Most cyclone manufacturers provide grade-efficiency curves to predict overall collection efficiency of a dust stream in a particular cyclone. Many investigators have attempted to develop a generalized grade-efficiency curve for cyclones, eg, see (159). One problem is that a cyclone s efficiency is affected by its geometric design. Equation 15 was proposed to calculate the smallest particle size collectable in a cyclone with 100% efficiency (157). 

For smaller particles, the theory indicates that efficiency decreases according to the dotted line of Figure 7. Experimental data (134) (sofld line of Eig. 7) for a cyclone of Eig. 9 dimensions show that equation 15 tends to overstate collection efficiency for moderately coarse particles and understate efficiency for the finer fraction. The concept of particle cut-size, defined as the size of particle collected with 50% mass efficiency, determined by equation 16 has been proposed (134). 

This equation is for Eigure 9 cyclone dimension ratios. The term the effective number of spirals the gas makes in the cyclone, was found to be approximately 5 for Lapple s system (134). The soHd line grade-efficiency curve of Eigure 7 is also used with Lapple s cyclone, which is a somewhat taller, less compact cyclone than many commercial designs. 

Cyclone Pressure Drop. Typical cyclone pressure drops range from 250 to 2000 Pa. Most data are reported for clean air flowing through the cyclone and these data are conservative for design purposes. Many investigators have unsuccessfully attempted to relate pressure drops to inlet and oudet dimension ratios. Manufacturers caUbration curves or experimental measurements on cyclones of similar dimension should be used where possible. If a rehable experimental measurement is available, however, the pressure drop at other conditions can be estimated by first evaluating the constant i in equation 17. 

Some empirical equations to predict cyclone pressure drop have been proposed (165,166). One (166) rehably predicts pressure drop under clean air flow for a cyclone having the API model dimensions. Somewhat surprisingly, pressure drop decreases with increasing dust loading. One reasonable explanation for this phenomenon is that dust particles approaching the cyclone wall break up the boundary layer film (much like spoiler knobs on an airplane wing) and reduce drag forces. 

As with dust cyclones, no reliable pressure-drop equations exist (see Sec. 17), although many have been published. A part of the problem is that there is no standard cyclone geometry. Calvert (R-12) experimentally obtained AP = 0.000513 J Q /hiWi) 2.8hiWi/dl), where AP is in cm of water Pg is the gas density, g/cm is the gas volumetric flow rate, cmVs hj and Wj are cyclone inlet height and width respectively, cm and is the gas outlet diameter, cm. This equation is in the same form as that proposed by Shepherd and Lapple [Ind. Eng. Chem, 31, 1246 (1940)] but gives only 37 percent as much pressure drop. 

The equation for DptK, the theoretical size particle removed by the cyclone, i... 

When consistent units are used, the particle size will either be in meters or feet. The equation contains effects of cyclone size, velocity, viscosity, and density of solids. In practice, a design curve as given in Fig. 17-39 uses Dptk the size at which 50 percent of sohds of a given size are collected by the cyclone. The material entering the cyclone is divided into fractional sizes, and the collecdion efficiency for each size is determined. The total efficiency of coUection is the sum of the col-lecdion efficiencies of the cuts. 

Djo,. (base) is the mieron size that a standard eyelone can achieve operating under the base conditions. This is given in Figure 54, as computed from the equation below. For example, a 25.4 cm (10 in.) Diameter cyclone has a bas Djq point of... 

The steady, laminar, incompressible fluid flow in cyclone collectors is governed by the Navier-Stokes equations ... 

However, more accurate predictions for the spin velocity may be obtained with allowance made for the effect of viscosity in the governing equation for the spin velocity. According to the experimental data of Keisall," which indicates that the spin velocity in a cyclone is a function of R only, the A component of Eq. (13.1) in the cylindrical polar coordinate sy stem may reduce to... 

For a given particle of size d, from the point M where the equilibrium line meets the line of zero vertical velocity (see Fig. 13.4), the critical path of the particle may be defined. All particles of this size between points D and G are entrained in the downward stream and are collected. The remaining particles of this size join in the upward-moving stream of fluid and penetrate the cyclone. The point D may be obtained by tracking back the particle trajectory from the point M using the equation of the particle trajectory, which is given by... 

Due to the very low volumetric concentration of the dispersed particles involved in the fluid flow for most cyclones, the presence of the particles does not have a significant effect on the fluid flow itself. In these circumstances, the fluid and the particle flows may be considered separately in the numerical simulation. A common approach is to first solve the fluid flow equations without considering the presence of particles, and then simulate the particle flow based on the solution of the fluid flow to compute the drag and other interactive forces that act on the particles. 

For steady, incompressible fluid flow in a cyclone separator, the governing Navier-Stokes equations of motion are given, in a Cartesian coordinate system, by ... 

The pressure drop in a typical cyclone is usually between 0.5 and 8 inches of water. It can be larger, but rarely exceeds 10 inches water for single units. The API study [7] summarizes the various factors. Lapple [13,16] gives calculation equations, but in general the most reliable pressure drop information is obtained from the manufacturer. 

In this equation, r is the cyclone radius and n is dependent on the coefficient of friction. Theoretically, in the absence of wall friction, n should equal 1.0. Actual measurements, however, indicate that n ranges from 0.5 to 0.7 over a large portion of the cyclone radius. The spiral velocity in a cyclone may reach a value several times the average inlet-gas velocity. 

The pressure drop in a cyclone will be due to the entry and exit losses, and friction and kinetic energy losses in the cyclone. The empirical equation given by Stairmand (1949) can be used to estimate the pressure drop ... 

The complex three-dimensional flow pattern within the cyclone is dominated by the radial (Fr) and tangential (V0) velocity components. The vertical component is also significant but plays only an indirect role in the separation. The tangential velocity in the vortex varies with the distance from the axis in a complex manner, which can be described by the equation... 

These equations can serve as a guide for estimating performance but cannot be expected to provide precise predicted behavior. However, they can be used effectively to scale experimental results for similar designs of different sizes operating under various conditions. For example, two cyclones of a given design should have the same efficiency when the value of Nst is the same for both. That is, if a given cyclone has a known efficiency for particles of diameter d, a similar cyclone will have the same efficiency for particles of diameter d2, where... 

These equations can be used to either predict the performance of a given cyclone or size the cyclone for given conditions. For example, if the definitions of AEu and NRc from Eqs. (12-57) and (12-58) are substituted into Eq. (12-67) and the result rearranged for D, the result is... 

The traditional (Rosin, et al., 1932) mechanistic approach equates the time necessary for a particle to settle at a Stokes law velocity across the width of a cyclone s inlet duct, to the available residence time of the carrier gas stream in its number of spiral traverses within the barrel. With reference to Fig. 1, this permits solving for the smallest particle size able to cross the entire width and reach the wall in the available time. 

If the fluid stream is a gas, the last term in the above equation is essentially unity. Unless the cyclone itself is rotating or, for example, located on another planet, g can be taken as 32.2. If the bulk solids angle of internal friction is unknown, then taking an average value of 62 degrees, the equation reduces to ... 

This equation is a result of the residence time theory of particle collection. In this theory, the time that it takes for a particle to reach the wall is balanced by the time that a particle spends in the cyclone. The particle size that makes it to the wall by the time that it exits the cyclone is the particle size collected at 50 percent collection efficiency, Dpth. 

The effects of temperature and pressure manifest themselves in how they affect the gas density and gas viscosity From the equation above, it can be seen that cyclone efficiency is theoretically related to gas density and gas viscosity as... 

It is found that ut0 is approximately equal to the velocity with which the gas stream enters the cyclone separator. If these values for ur and ut are now substituted into equation 1.50, the terminal falling velocity of the smallest particle which the separator will retain is given by ... 

The solid could also be withdrawn and returned to the reactor using a cyclone, and in this case (still assuming complete mixing) the complete equations for the solid phase will have to be solved because T is finite. 

The Reynolds stress model requires the solution of transport equations for each of the Reynolds stress components as well as for dissipation transport without the necessity to calculate an isotropic turbulent viscosity field. The Reynolds stress turbulence model yield an accurate prediction on swirl flow pattern, axial velocity, tangential velocity and pressure drop on cyclone simulation [7,6,13,10],... 

The finite volume methods have been used to discretised the partial differential equations of the model using the Simple method for pressure-velocity coupling and the second order upwind scheme to interpolate the variables on the surface of the control volume. The segregated solution algorithm was selected. The Reynolds stress turbulence model was used in this model due to the anisotropic nature of the turbulence in cyclones. Standard fluent wall functions were applied and high order discretisation schemes were also used. 

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