CYCLONES Consider a force balance on a particle that moves in rotating flow as illustrated in the drawing below. The outward, centrifugal force on the particle, Fc, will be balanced by the inward, drag force on the particle, FD, as it moves through the fluid. The relative velocity between the particle and the fluid that determines drag depends on both the particle’s radial velocity outward, dr/dt, and on the inward velocity of the gas, Vr, as it moves to the cyclone axis. VT, gas tangential velocityParticle’s radial position, r Vr, gas inward velocity dr/dt, particle’s outward radial velocity particle’s actual path Next, we will make some assumptions about the behavior of the particle and the fluid: 1. The particle moves with a tangential velocity that is the same as the gas tangential velocity at radial position “r”; that is, the particle does not “slip” tangentially, 2. The “positive” direction is radially outward. Now we can construct a balance for the forces acting on the particle in the radial direction using Newton’s second law, which says that the sum of these forces will equal mass times acceleration. ∑= amF (1) FC – FD = m a (2) 22p3rC2Tp3dtrd6dVdtdrCd3rV6d⎟⎟⎠⎞⎜⎜⎝⎛ρπ=⎟⎠⎞⎜⎝⎛+µπ−⎟⎟⎠⎞⎜⎜⎝⎛⎟⎟⎠⎞⎜⎜⎝⎛ρπ (3)which, using the definition that µρ=τ18CdCp2 reduces to 0rVVdtdr1dtrd2Tr22=⎟⎟⎠⎞⎜⎜⎝⎛−τ+τ+ . (4) This is a second order, non-linear differential equation. Note that gas tangential velocity, VT, varies with radial position, r. If nwallwall,TnTrVttanconsrV == , (5) where the subscript “wall” corresponds to conditions at the wall of the cyclone as has been shown from experiments, then Eq (4) becomes 0rrVVdtdr1dtrd1n2n2wall2wall,Tr22=⎟⎟⎠⎞⎜⎜⎝⎛−τ+τ++ . (6) Collection Efficiency: Barth Solution Equation (6) can be solved in different ways, depending on the assumptions made. Barth’s solution, the “static particle approach”, considers a particle for which the outward centrifugal force just balances the inward drag force. This particle is “static”, because it moves neither outward nor inward. In this case, 0dtdrdtrd22== (7) so that Eq (6) reduces to Cp2TrCVrV18dρµ= (8) where the values of Vr, r, and VT are taken at the radial position of maximum tangential velocity. Iozia and colleagues used this approach along with measurements that determined the location of maximum tangential velocity to find dc, the particle size collected with 50% efficiency. 2max,tcpcvZQ9dρπµ= (9)where dc is particle size collected with 50% efficiency, µ is gas viscosity, Q is gas flow ρp is particle density, Zc is height of the control surface, vt,max is the gas tangential velocity on the control surface The height of the control surface depends on the radius of the control surface, rc: 53.125.02cDDeDba2D52.0r⎟⎠⎞⎜⎝⎛⎟⎠⎞⎜⎝⎛⎟⎠⎞⎜⎝⎛=−, (10) an empirical equation for which r2 = 0.889 After rc is known, the proper equation for Zc can be used Br2ifSHZcc<−= (11a) ()()()Br2if1Br21BDhHSHZccc>⎟⎠⎞⎜⎝⎛−−−−−= (11b) The value for maximum tangential velocity on the control surface can be determined from: 33.074.061.02inletmax,tDHDDeDbav1.6v−−⎟⎠⎞⎜⎝⎛⎟⎠⎞⎜⎝⎛⎟⎠⎞⎜⎝⎛=, (12) an empirical equation for which r2 = 0.983, where vinlet is the gas velocity through the inlet to the cyclone, baQvinlet= . (13) Once the particle size that is collected with 50% efficiency, dc, has been calculated, the efficiency for that cyclone on particles of any size can then be determined from: β⎟⎠⎞⎜⎝⎛+=ηdd11c (14) where the slope parameter, β, is given by: 222cDbaln05.1Dbaln21.5dln87.062.0ln⎥⎦⎤⎢⎣⎡⎟⎠⎞⎜⎝⎛+⎟⎠⎞⎜⎝⎛+−=β (15)for which r2 = 0.833 In Eq. (15) above, dc must be given in units of centimeters. Pressure Drop in a Cyclone Recall that: ⎟⎟⎠⎞⎜⎜⎝⎛ρ⎟⎠⎞⎜⎝⎛ρ∆=∆g1v21HPL2g (16) where ∆P is pressure drop in cm of water, or whatever fluid is used in the manometer ∆H is the number of velocity pressures of pressure drop for the cyclone (see below) ρg is gas density v is gas inlet velocity ρL is the density of the fluid in the manometer, and g is the acceleration of gravity. If no value for ∆H is available from experiments, then theory can be used to calculate it. The Dirgo approach is probably best. ()()()3/12eDBDhDHDSDba20H⎥⎥⎦⎤⎢⎢⎣⎡=∆ (17) References: Iozia, Donna Lee and David Leith, “Effect of Cyclone Dimensions on Gas Flow Pattern and Collection Efficiency”, Aerosol Science and Technology, 10: 491 (1989). Iozia, Donna Lee and David Leith, “The Logistic Function and Cyclone Fractional Efficiency”, Aerosol Science and Technology, 12: 598-606 (1990). Ramachandran, G., David Leith, John Dirgo, and Henry Feldman, “Cyclone Optimization Based on a New Empirical Model for Pressure Drop”, Aerosol Science and Technology, 15: 135-148