Modelling of particle mixing and segregation in the transverse plane of a rotary kiln

Pergamon Chemical En#ineerin9 Science, Vol. 51, No. 17, pp. 4167 4181, 1996 Copyright ~l: 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved P I I : S0009-2509(96)00250-3 0009 2509/96 $15.00 + 0.00 M O D E L L I N G OF PARTICLE MIXING A N D SEGREGATION IN THE TRANSVERSE PLANE OF A ROTARY KILN A. A. BOATENG* and P. V. BARR Department of Metals and Materials Engineering, The University of British Columbia, Vancouver, B.C. Canada, V6T 1Z4 (First received 17 February 1994; accepted 29 January 1996) Abstract--Thorough mixing of particles in the transverse plane of a rotary kiln or drier is essential to the uniform heating or cooling of the charge and ultimately, to the generation of a homogeneous product. However, differencesin particle size and density result in a de-mixing process whereby smaller or denser particles segregate to form an inner core or kidney of segregated material which may never reach the bed surface to be exposed to freeboard temperatures. A model has been developed to predict the preferential movement of particles in the shearing active layer. This model determines the extent of fine particle segregation and is based on the principle of percolation in the active layer, whereby fines sift through the matrix of the bed to form the segregated core. Incorporating results of a granular flow model developed for this purpose, it has been possible to establish the dimensions of the segregated core as well as fines (jetsam) concentration in the rest of the bed cross-section. This result is necessary in assessing the effect of segregation on bed temperature nonuniformities in rotary kilns. Copyright ~ 1996 Elsevier Science Ltd Keywords: Rotary kiln, jetsam-flotsam concentration, mixing, segregation, granular temperature, diffusion, percolation. INTRODUCTION The rheological properties of the bed material can be expected to change during the passage of charge through a rotary kiln and changes in properties, such as particle size, shape and surface character, may result in distinct changes in bed behavior. Depending on the rhcological properties and also rotation rate and the degree of fill of the cylinder, the various types of bed motion that may result in the transverse plane of a rotary cylinder have been characterized (Henein et al., 1983) as centrifuging, cataracting, cascading, rolling, slumping and slipping, all of which arc shown in Fig. 1. Centrifuging, which occurs at very high speeds of revolution, is an extreme condition in which all the bed material rotates with the kiln wall. Cascading, which also occurs at relatively high rates of rotation, is a condition in which the height of the leading edge (called the shear wedge) rises above the bed surface and particles cascade or 'shower' down on the free surface as depicted in the figure. Neither of these conditions is generally considered to be desirable in commercial rotary kiln operation and will not be considered. Starting, however, at the other extreme, i.e. at very low rates of rotation, and moving to progressively higher rates, the bed will typically move from slipping, in which the bulk of the bed material, en-bloc, slips against the wall, to slumping, whereby a piece of the bulk material at the shear wedge *Corresponding author. Present address: Solite Corporation, P.O. Box 27211, Richmond, VA 23261, U.S.A. becomes unstable, yields and empties down the incline, to rolling which involves a steady discharge onto the bed surface. In the slumping mode the dynamic angle of repose varies in a cyclical manner while in the rolling mode the angle of repose remains constant. The most desirable bed motion is usually one that follows the rolling mode shown in Fig. 2, since this promotes good mixing of particles along with rapid surface renewal at the exposed bed. For this mode the bed material is characterized by two distinct regions; the relatively thinner active layer which is formed as the granular material flows down the sloping upper bed surface and the much thicker 'plug flow' region where the material is carried upward by the rotating wall of the kiln. Thus in this mode the energy imparted by the kiln's rotation is continuously fed into the plug flow region as potential energy which is subsequently released and dissipated in the active layer. The active layer itself is characterized by vigorous mixing of particles and hence a high rate of surface renewal which promotes heat transfer from the freeboard and, ultimately, promotes the generation of a homogeneous product during processing applications. This good mixing assumes particles are evenly sized which effectively means that, statistically, exposure to freeboard will virtually be the same. Unfortunately, when significant variation in particle size occurs, superimposed on this mixing will be the tendency of small particles in the active layer to sift downwards through the matrix of larger particles. Thus, the bed motion tends to concentrate finer material within the core (see Fig. 3), and material within the core, because it has very little chance of reaching the 4167 A. A. BOATENGand P. V. BARR 4168 Slipping Cascading Slumping Cataracting Rolling 0 Centrifuging Fig. 1. Bed behavior modes. Fig. 2. Rolling bed motion depicting two regions. Top section: active layer; bottom section: plug flow region. Fig. 3. Schematic of radial segregation of different size particles in a rotary kiln. exposed bed surface for direct heat transfer from the freeboard, tends to have a lower temperature than the surrounding material. Thus segregation can counteract advective transport of energy and hence promote temperature gradients within the bed. However, the net effect is not necessarily negative; for an industrial process such as limestone calcination whereby smaller particles react faster than larger ones (at the same temperature) segregation of fines to the cooler core may be essential in obtaining uniform calcination of all particles. This suggests that particle size distribution in the feed material might be optimized, which points out the need for developing our predictive capabilities for the material mixing and segregation. The objective of this work was to construct a mathematical model that would describe the phenomenon of segregation and to predict the extent and dimensions of the segregated core. It is worth pointing out that, although segregation in drum mixers is a well known phenomenon and has been characterized in many ways, quantitative prediction of the composition of the core has been lacking. Fan and Wang (1975) reviewed over 30 forms of mixing indices that have appeared in the literature to characterize particulate mixing and segregation in drum mixers. Most of these characterization parameters are probabilistic or statistical in nature and, although often helpful, tend to conceal the details of the phenomenon and yield little Modelling of particle mixing and segregation information on, for example, the effect of material properties on flow and hence on segregation (Bridgwater et al., 1985). Scaling factors are also difficult to evaluate without a good grasp of the physical phenomena which drive segregation. The segregation model developed in the present work considers a binary mixture of small and large particles in the continuously shearing active layer of the kiln bed. Continuum equations are employed to describe the mixing and segregation rates in the transverse plane of the bed which result from both particle percolation and diffusional mixing. The diffusion coefficients and the convective terms for material concentration in the continuum equations used in describing mixing and segregation were obtained from a granular flow model developed specifically for this model and described by Boateng (1993). The percolation velocities were calculated using existing models which relate percolation to void formation in shear planes (Savage and Lun, 1988). Model predictions for segregation in the radial direction are compared to experimental results and the adequacy of the model is discussed. MODEL DESCRIPTION The main causes of segregation are differences in particle size, density, shape, roughness and resilience (Williams and Khan, 1973). Although any of these may produce segregation under certain circumstances, most rotary kiln segregation arises from differences in particle size (Pollard and Henein, 1989) and the current work is focused on this phenomenon. The mechanisms by which size segregation occurs are well founded (Williams and Khan, 1973; Bridgwater et al., 1985). These include: (i) Trajectory segregation: This is due to the fact that, for certain modes of kiln operation, particles being discharged from the plug flow region into the active layer may be projected horizontally from the apex onto the exposed bed surface. This situation may apply in the slumping, rolling and cataracting modes whereby different sized particles are emptied onto the surface during surface renewal. It has been suggested that the distance that these particles travel is proportional to the square of the particle diameter (Bridgwater, 1976) which means that finer particles will tend to be concentrated at mid-chord section. (ii) Percolation: When a bed of particles is disturbed so that rearrangement takes place (rapid shearing), the probability that a particle will find a void into which to fall depends on the size of the particles (Savage and Lun, 1988). Thus smaller particles will tend to filter downwards through a bed of flowing granular material while large particles will simultaneously tend to be displaced upwards. Trajectory segregation has been identified (Bridgwater et al., 1985) as the main cause of axial segregation or 'banding' whereby particles of different sizes are selectively collected into bands occurring over the 4169 kiln length. This axial segregation is not considered in the present work and therefore not critically reviewed; rather, attention is focused on segregation in the transverse plane, specifically, percolation. A review of the literature pertaining to segregation of the bed material in rotary kilns leads to a few conclusions which include the following: (i) Radial segregation proceeds very rapidly from a condition of uniform mixing of particles within the bed and is fully implemented within 2 to 10 kiln revolutions I-see e.g. Rogers and Clements (1971) and Pollard and Henein (1989)]. The mechanism of segregation can therefore be considered as a steady-state problem. (ii) The segregation process is continuous and there is a constant discharge of fines from the plug flow region into the active layer. This discharge of fines occurs in the upper part of the bed toward the apex and is followed by percolation normal to the bed surface as material is sheared in the active layer. (iii) The 'kidney' (or 'tongue') formed by the segregated core does not consist entirely of fine material; it also contains some small amounts of coarse particles I-i.e.there are concentration gradients within the core itself; see Henein et al. (1983)]. (iv) The bulk velocity distribution in the active layer does not change with addition of fines and the bed behavior (e.g. rolling, slumping, etc.) remains unchanged with fines (Henein, 1980). (v) The percolation velocity of fine particles depends on the size of the voids formed in an underlying layer of particles; these voids are formed in a random manner (Savage and Lun, 1988). (vi) For particles below some critical size, spontaneous percolation may also occur in the plug flow region thereby resulting in a possible collection of fines near the bed-wall interface (Bridgwater and Ingram, 1971). (vii) Downward movement of segregating particles in the active layer is compensated by an equal volumetric upward movement of bulk particles in the active layer I-squeeze expulsion mechanism, Savage and Lun (1988)]. Based on the foregoing information, a credible mathematical model would be one which accounts for the mass conservation of the sinking and/or floating particles in a control volume and whose analysis can be restricted to the active layer since, owing to (vi) above, the probability of fines moving through the plug flow region is very low compared with the dilated shearing flow in the active layer. In modelling this phenomenon the plug flow region can therefore be assumed impermeable thereby serving only as the circulation path by which particles are fed back to the active layer. This assumption precludes spontaneous percolation from the model. The situation to model is shown schematically in Fig. 4. The coordinate system used here is one which allocates a Cartesian system to the active layer such that 0 < x < 2L where 2L is the chord length, and the 4170 A.A. BOATENGand P. V. BARR Segregation Diffusion-like mixing Drift Drift = A Diffusion-like ' Segregation mixing Circulation through plug flow / kCj(y.w,)[1-Cj(,.=) ] dy b.,,) Fig. 4. The mechanism of percolation and the calculation domain for the segregation problem. origin, 0, is at the apex of the bed. As mentioned earlier, only a simple binary system of two particle sizes, each of the same density, is considered. Since the calculation domain is restricted to the active layer, the fine particles are assumed to be larger than the critical size which causes spontaneous percolation. This occurs when the diameter ratio of the small to large particles exceeds a critical value, which, for closely packed bed, may be given as (Savage and Lun, 1988) dps a = ~ ~< 0.1547 (1) where dp~ and dpt are, respectively, the sizes of the small and large particles for the binary system. By choosing the size ratio to be greater than the critical value, it is implicitly assumed that percolation will occur only when the voids formed are larger than the smaller size particle, dps , of the binary mixture. For a continuously shearing active layer of the rotary kiln, it will be further assumed that void formation is a random occurrence and follows a probability law. For generality the model is developed on the basis of the terminology usually employed for fluidized-bed segregation whereby sinking particles, either due to size or density differences are 'jetsam' and floating particles are 'flotsam' [see e.g. Gibilaro and Rowe (1974)]. The model is derived on a volume balance basis and the concentration terms refer to volumetric fraction of jetsam in a given volume of solids. The relationship between the volume fraction jetsam concentration and the number of particles in the control volume can, therefore, be expressed as ~/tr3 Cj = 9 - 1 + qa 3 (2) where .9 is the solids concentration, q is the particle number ratio, ns/nr, with nj and nF being the respective jetsam and flotsam number particles. ~ wCj (x+d~y) wCj (x.y) T _rd_ C / ' dy ky~.~) , .................... KC,,,.., [1-%..., ] ! ii =..- . . . . . . . . . . . . . . . . . 2 Fig. 5. Control volume for the material conservation in the active layer. Following on with the continuum assumption, the control volume required for the material balance in the active layer is that shown in Fig. 5. The equilibrium concentration of jetsam within the control volume depends upon the interaction of three phenomena; i.e. (i) convection (drift) caused by the bulk velocity of materials, (ii) diffusion-like mixing, and (iii) segregation associated with movement through voids. The various mechanisms by which jetsam is spread over the cross-section is schematically depicted in Fig. 5. Of the three mechanisms shown, segregation is the only one that distinguishes jetsam from flotsam and it depends on the percolation of jetsam into the underlying layer of particles and subsequent displacement of flotsam from beneath as a compensation. Because the upward flow of material that compensates percolation of jetsam may itself contain jetsam, the rate of jetsam concentration due to the segregation mechanism is represented by a nonlinear concentration gradient. GOVERNING EQUATIONS The governing equations for mixing and segregation are derived by considering an equilibrium balance of material for the control volume shown in Fig. 5. Firstly, particles drift into the control volume by convection as a result of the bulk velocity in the active layer. The rate of jetsam dispersion in and out Modelling of particle mixing and segregation of the control volume may be represented, respectively, as AuCstxm and AuCatx+d~.y),where A is the area normal to the bulk flow, and u is the bulk velocity. Secondly, the rate of diffusional mixing is proportional to the concentration gradient and the effect of this component in the x-wise direction of the active layer may be neglected relative to the large advection term. The rate of diffusion-like mixing at each x-position in the active layer is therefore given as - ~(OCj/Sy), where F is the proportionality constant equal to the product of the diffusion coefficient and the participating area in the control volume, i.e. DrA(m2/s)(m2). Thirdly the rate of segregation for jetsam particles is given by a non-linear quantity, Cj(1 - Ca), which is the product of the area and the percolation velocity, i.e. Av v (m3/s). By employing the Taylor series expansion, the rates of jetsam outflow from the control volume may be expressed as follows: Bulk flow: Au(y) Cj I.... = Au(y) t32 ICJb, + ~---~(Cs)dx+~x2 + ...] (3) Diffusion: c~yl,_ar----------~ -- j dy + ... (4) Segregation: ~ c j ...... [1 - cjl,.~] = ~[1 - cji, x] {CjI,.~+~---f(Cs)dY+ ""} kC j,,~ E1 - Cdlr_ar.~] = kC JIr,~ [1 }] ,,, By expanding the terms given in eqs (3)-(5) and substituting the rate of jetsam inflow of particles to the control volume, the net change of jetsam concentration becomes Dy02C'/dx dy dz + vv(1 0y2 = u(y) ~ -- 2C j) ~ dx dy dz dxdydz (6) and the differential equation describing movement of jetsam in the active layer may be written as D,-ST-y ~2C ~ J+ v,(1 - 2Cz)~-u(Y)~xS = O. (7) In arriving at eq. (7), the boundary layer condition whereby ux >> ur has been imposed; the y-component of the species convection term has been ignored and thus the vertical movement of jetsam occurs only by percolation or diffusion. This assumption was justified by velocity measurements and observations made 4171 in a 1 m rotary drum containing granular solids (Boateng, 1993). Flow model applied to the active layer In order to solve for eq. (7) the velocity profile, u(y), the percolation velocity, vp, and the diffusion coefficient, Dr, must be available. As mentioned earlier, the velocity parallel to the bed surface, Ux(y), and the diffusion coefficient were obtained from a granular flow model developed specifically for the determination of segregation and heat transfer in rotary drums following physical simulations as described in Boateng (1993). Experiments performed in a 1 m diameter rotary drum showed that the two regions of material flow in the transverse plane of the drum (i.e. the active or shear layer and the plug flow region) can be modelled separately. In the plug flow region particles follow a rigid body motion and the velocity there vary linearly with radius and rotation rate, i.e. u(y) = ogr.While the plug flow region is entirely rigid, the stress tensor in the active layer may have frictional as well as collisional contributions depending on the rotation rate. For the drum speeds studied in the experimental campaign (2-5 rpm) the velocity profile shapes observed were mixed mimicking, in some instances, what might be used to describe pseudoplastic (concave from bed surface) or Newtonian (linear) or dilatant (convex) type flows. One component value of the velocity fluctuation, T, a measure of particle collisions called granular temperature which was measured, ranged from a low of 2.5 to a high of 70 cm2/s 2 depending upon the location in the active layer or drum speed. At the free surface ~/gdp was in the range 0.11-0.24. These values lead to the conclusion that, for the drum speeds studied, the flow in the active layer was not very rapid; rather, it fell within a general body of granular flows where shearing and unshearing flows may coexist within the same flow field [see e.g. Campbell and Brennen (1983) and Johnson and Jackson (1987)]. In modelling such flows the constitutive equations developed by Lun et al. (1984) may be adopted but must be modified to take cognisance of the participating role of the frictional and collisional stress generations [see e.g. Johnson and Jackson (1987)]. Although the medium is granular the flow model employed is based on the assumption of a continuum similar to viscous flows except that the equilibrium states are not states of hydrostatic pressure but are, rather, governed by a yield criterion. The flow model takes advantage of the thinness of the active layer and compares material flow in this layer to other thin flows, for example, boundary layer flows with a generalized equation, Pd-'xX (U 2 -- u~u)dy = pgsin ~ + g2(v)ppdpT 1/2 du dy" (8) This model provides the possibility of predicting the active layer depth and the velocity distribution by 4172 A.A. BOATENGand P. V. BARR choosing an appropriate velocity profile and applying the constitutive equations of Johnson and Jackson (1987) for the stress tensor. Although the condition modelled does not constitute a very rapid flow situation the advantage of using these granular flow theories over others such as plastic flow models [see e.g. Mandl and Luque (1970)] is that the former case allows for the calculation of the additional fluid property, granular temperature. This field variable is defined as the kinetic energy per unit mass contained in a random motion of particles [see e.g. Zhang and Campbell (1992)] and hence may be used to determine kinetic diffusion however small the value may be. The solution of eq. (8) makes use of the experimental results for the free surface as the first boundary condition due to the mixed profiles observed for these slow flows, i.e. us = CocoR. At the interface between the active layer and the plug flow region, u = ux, o, r = r~, and the Coulomb yield criterion, Pxy = Pyr tan q~ (q~ being the friction angle), were employed, respectively, as the second and third boundary conditions. Since tan ~b is constant, the latter implies that the ratio between the shear and normal stresses at the interface must be constant. The solution is closed by satisfying the criterion p,, fo u,t(x, y) dy = ppf i (9) upy (r) dr it being equality between mass flux in and out of the active layer. The diffusion flux in the active layer occurs as a result of particle interactions in a continuously shearing active layer. Therefore, once the granular temperature is available [typically by an iteration procedure on eq. (8)] the kinetic diffusion coefficient, Dr, may be computed as (Savage, 1993) Or = dPx~ where ep is the coefficient of restitution of particles and go(oa) is a radial distribution function from collision theory (Lun et al., 1984; Johnson and Jackson, 1987). The procedure to determine u(y) and ~ using eq. (8) is detailed in Boateng (1993) and will not be repeated here. Typical velocity profiles (calculated and measured) as a function of depth and rotation rate for a 1 m rotary drum are shown in Fig. 6 [from Boateng (1993)]. Also shown are the diffusion coefficients computed as a function of bed surface position, x, and drum rotation rate. Any discrepancy between the model and experiment may be attributed, in part, to the degree of applicability of the constitutive relations used, an assumed isotropy in T for the entire active layer depth, or the accuracy associated with ep values. This approach simplifies the model without much penalty since any inaccuracy of the flow model would not obscure the particle rearrangement process. The velocity field does not affect the steady-state particle segregation profiles but, rather, the rate at which segregation is accomplished. At this point the percolation velocity is the only remaining unknown component required for the solution of the segregation problem. In order to determine this velocity, the model developed by Savage and Lun (1988) for segregation in inclined chute flow was adopted. The justification in doing so is the fact that void formation in any underlying layer of particles in the shear region (active layer) is still a probable event irrespective of the continuum assumption. The model considers the probability for formation of a void in an underlying layer of particles with a size large enough to capture the smaller particles within the overlying layer. The net percolation velocity for the smaller particles in the neighborhood has been determined by Savage and Lun (1988) to be =d (10) (du~ 1 (1l) 8(% + 1),99o(,9) 5 E tj CA .,,1 I 1.4 i 0 I 1.2 -5 -10 ~ -15 predicted 0 I measured -20 0.8 E ,,.o 0.6 0.4 -25 0.2 -30 0.0 0 50 100 Velocity, em/s 50 -0.2 -20 L / I 1.0 "7 03 -35 -50 I // t i \ .- .... I - -D~IO ---Dx100 \ I;/" -"'-.. ",X, / t / .f,-'-'-~-.,,--.....__ - ~ " ~ ' ~ 2:o " 4.0 5.O I 4 r i l I 0 20 40 60 80 100 Distance F r o m Apex, c m Fig. 6. Predicted and measured profiles as a function of depth and rotational rate for 1 m rotary drum (Boateng, 1993): (a) velocity, (b) diffusion coefficient. 20 4173 Modelling of particle mixing and segregation with the percolation velocities for smaller (jetsam), vp~, and larger (flotsam) particles, vp~, being given by the following expressions: I)ps = dpl ~ y G(~, o) E -- E m --]- 1 + - ~ - ~ ) l ( 1 x exp {(1 + r/)a/(1 +q)-E.} vv,=dp, ~yy G(q,a) E - E,, + I + (il +~-a)l x exp {(l +,7)/(1 +.)- ~.} " E -- E,, (13) The function G(r/, a), in eqs (12) and (13), relates the packing of particles around a void to particle size ratio tr, and particle number ratio r/, and is given by the expression [see Savage and Lun (1988)] G(t/, a) = 4k~T(M/N)(1 + rla) ~(1 + q){(1 + ~/)(1 + qa)/(1 + qa)2 + E2/kav(M/N)} (14) where E is the mean void diameter ratio and E . is the minimum possible void diameter ratio when spontaneous percolation occurs. All the terminology in the forgoing equations is consistent with that of Savage and Lun (1988). It must be noted that the parameters M/N, E,,, and kay are constants which depend on particle packing. Numerical values for cases such as cubic array and closest packing are available [see e.g. Savage and Lun (1988)]. In applying such a model to the rotary kiln, it must be pointed out that, as a result of jetsam segregation, the values of M/N, Era, and k~v would be susceptible to changes because of rearrangement of the particle ensemble. Nevertheless, it is possible to alter these constants dynamically with respect to both time and space (e.g. for each kiln revolution or material turn-around in the cross-section) by recalculating solids fraction in the granular flow model (Boateng, 1993). BOUNDARY CONDITIONS The calculation domain for jetsam segregation and the percolation process is shown in Fig. 4. Owing to kiln rotation, an initially well mixed binary mixture will follow a specific path in the plug flow (passive) region until it crosses the yield line (the demarcation between the active layer and the plug flow region) into the active layer. For the active layer, material enters from the plug flow region with a given jetsam concentration and then travels down the incline plane in a streaming flow. During the journey, jetsam particles sink when the voids in the underlying layer are large enough for the particles to percolate. If this does not occur then these particles will pass the yield line again and recirculate. The plug flow region is nonshearing and serves only as an 'escalator'. Particles within this region neither mix nor percolate unless they are small enough to undergo spontaneous percolation; a condition which is precluded from the model simply by size selection. The percolation process in the active layer is repeated for each material turn-around, and as the jetsam content in the core increases, fines will no longer be visible at the exposed bed surface. Henein (1980) has observed that the only situation where fine particles are observed at the exposed bed surface is when the drum is loaded with 40-50% fines. The boundary conditions for eq. (7) will, therefore, depend on the operational conditions. For a dilute mixture of jetsam particles, for example, the boundary conditions will be as follows: x=O Cs=Go y=0 G=0 Y=6x Cs(l - CJ) = 0 (15) where Cjo is the influx of jetsam particles from the apex (bed-wall boundary) at the beginning of rotation. The second condition indicates that, at the free surface, there are no jetsam particles since all the fines in such a dilute mixture will percolate to the core region, whereas the third condition is the result of the nonlinear concentration term which will render pure jetsam (Cj = 1) at the yield line [see e.g. Gibilaro and Rowe (1974)]. This supports the assumption that the yield line is impermeable to flotsam-jetsam percolation. The percolation process described above allows particles at the interface to be replaced by those 'escalated' by the plug flow and, as a result, the most appropriate boundary condition for the interface will rather be ~Cs(x, 6x)/Oy = 0 (16) and this was the boundary condition used in the model. SOLUTION OF THE GOVERNINGEQUATIONS The basic expression describing segregation, eq. (7), with the appropriate boundary conditions, eqs (15) and (16), can be solved when the bulk velocity, the percolation velocity and the diffusion coefficients are all determined a priori. The solution of the differential equation can be achieved by considering the problem in terms of several particular cases [-see e.g. Gibilaro and Rowe (1974) and Savage and Lun (1988)]. Strongly segregating system (Case I). For a strongly segregated binary mixture of different size particles, the diffusion of jetsam particles in the vertical plane can be ignored. This situation is pertinent to a very dilute mixture where q--* 0 and, as can be deduced from eq. (2), although the gradient does not go to zero, it follows that Cj--* 0. The differential equation for segregation in such a case becomes v, --~y OCj - ~Cj Ux(y) - ~ x = o (17) A. A. BOATENGand P. V. BARR 4174 with boundary conditions given in eqs (15) and (16). It should be pointed out that eq. (17) is the same as that employed to describe segregation on inclined planes (chute flows) and can be solved analytically by the method of characteristics [-see e.g. Bridgwater (1976) and Savage and Lun (1988)]. Radial mixing (Case II). When the system contains mono-sized particles (i.e. particles are identified only by color differences) of uniform density, the percolation term in the differential equation can be ignored and the problem reduces to that of diffusional mixing with drift. In this case, Cj = C (color) and the resulting differential equation may be written as clude their ultimate exploitation for various reasons, e.g. a recirculation term is required to furnish jetsam particles from the plug flow region into the active layer as shown in Fig. 4. Therefore, in the present work, the derivative terms in the governing equations were replaced by finite difference approximations and the resulting algebraic equations were solved numerically. The discretized equations employed in the solution are as follows [see e.g. Anderson et al. (1984)]: For eq. (17) (23) (~2C If')C D,,--a),2 - u x ( y ) 7 ~x~- = o (18) For eq. (18) 2Dy, which is a linear diffusion problem of Graetz [see e.g. Arpaci (1966)]. An analytical solution to eq. (18) exists for several boundary conditions. By employing the boundary condition discussed above, i.e. q C(x, 0) = 0 2 ~ - ~ ,~0: (-2,1) exp { - ),,2x/2s} cos 2,y - n = 0, 1, 2, ... (21) Mixing and segregation (Case III). This is the complete solution to the mixing and segregation problem [eq. (7)] that describes the movement of jetsam particles by the combined mechanism of mixing and segregation in the transverse plane of a rotary kiln. The differential equation, given earlier as eq. (7), is 0C2 Oy ~ i,j--I - (24) ~Ci+l,j. A_ Equation (22) may be solved numerically by linearizing the nonlinear term and discretizing the resulting equation as (2n + 1)~z 26~ 2Cj) C + { [ A y s 2D~, l l}Ci,)+l i + Ayj +l] Ayj+ (20) where s = u/2Dy and v~,(l - 1 (19) the solution for the diffusional mixing may be given as (Arpaci, 1966) + 2Dy, 1 bli,j] - - x + S-;x ~) c ,. j [Ayj_ 1 + Ayj+ 1] Ayj+ , + {[Ayj ~2D~+Ayj+ 1]}~ aC(x, ~x)/~y = o D ~2 Cj 1 [Ayj_ ; +-Ayj+ 1] Ayj 1 C(O, y) = Cjo C(x, y) ui- t,J c. .~ Ci,j + Ax ,- x,jS Ci,; = [v,/Ay's + ui 1.;/Ax] OCj u(y)-z - Ci, j = { ' [2A1 3- A2 3- A3] + A1Ci.j_IA3Ci.j_ 1 3- Z2Ci_l, j - d C 2 d y } (25) where A1= (22) Di/Ayj A2 = ui, j/Ax A3 = O. } {AtCi'j+l = Ui.j/Ayj. The nonlinear term OC2/t?y, is discretized as ECi,j Jr- C i,j 1J2 - [Ci, j+ 1 3- C i,j] 2 "t']Ci,j -'~ Ci, j- ll(Ci,j - Ci,j-1) -- 7[Ci,j+ l 3- Ci,jl(Ci, j+ l - Ci,j). -[ 4A yi 4Ays (26) Although eq. (22) represents a nonlinear partial differential equation, a solution can be found by functional transformation [see e.g. Ames (1965)]. Numerical solution The analytical methods suggested provide one avenue of approach to the solution of the governing equations. However, factors such as geometry pre- Although the medium is granular, the continuum assumption demands that, just as in the case of fluid flow, eq. (25) employs the appropriate upwinding technique and, as a result, eq. (26) represents upstream donor cell differencing whereby 7 = 1 gives a full upstream effect. For 7 = 0, the equation becomes numerically unstable as established in Anderson et al. (1984). Modelling of particle mixing and segregation It might be noted that in the preceding development eq. (23) is an explicit algebraic formulation because of the parabolic nature of the differential equation. Thus, once the mixture concentration at the apex is given, the jetsam concentration along the chord length can be computed by marching down the incline. Equation (24) is the algebraic form of a onedimensional diffusion-convection equation (Graetz problem) and may be solved numerically using the tri-diagonal method algorithm (TDMA) [see e.g. Anderson et al. (1984)]. Equation (25) is an implicit algebraic equation for the calculation of two-dimensional jetsam distribution in the cross-section; it may be solved by an iterative procedure, for example, the Gauss-Siedel method, whereby the nonlinear term, which is expressed by eq. (26), is computed using previous values of Ci.i. In all the above cases, a solution technique is employed whereby a set of calculations is carried out by marching from the apex to the base. The results of this set of calculations represent the concentration of jetsam particles for a single pass or one material turn-around in the cross-section of the kiln. Because there is no diffusion in the plug flow region, particles are escalated from the base to apex after the pass. The second set of calculations for the next pass is initiated with the convected concentration as a boundary condition at the new location on the interface. The calculation is repeated until the overall jetsam concentration in the cross-section equals the jetsam loading. Because the bed material circulates for about three or four times per each kiln revolution, this approach allows for the estimation of the number of revolutions required to accomplish a complete mixing or a complete segregation. The solution method may, therefore, represent a pseudo-transient solution for jetsam concentration in a two-dimensional plane. MODEL VALIDATION As was mentioned earlier, the objective of the model was to determine the extent and dimensions of the segregated core and, as a result, estimate the jetsam concentration gradient. In order to substantiate the validity of the calculations a validation of the model was carried out against the experimental data of Henein (1980). In that work, a 40cm ID drum loaded with a prescribed jetsam concentration was rotated for a desired number of times and then stopped. The bed was then sectioned using discs which were inserted normal to the drum axis. In each section fines concentrations were measured beginning from the apex to the base by sieving and weighing, or by simply counting, thereby mapping out a one-dimensional representation of jetsam concentrations as a function of chord length. In order to convert the two-dimensional model result developed in the present work into the one-dimensional representation in Henein's experiment, the jetsam concentration for all radial nodes at each x-location was averaged using the formula i,~ax /j~x Cj., = ~ Cj A i j Ai,i. 1= 1 " ] 1=1 (27) 4175 Validation of the model begins with Fig. 7 which shows the predicted and measured radial segregation patterns determined for the case of a strongly segregated system (Case I). It can be said that the predictions show good agreement with measured results. It is worth pointing out that the measurements in Fig. 7 were taken from different axial locations of the same experimental run, hence, although initial jetsam loading in the entire drum is the same, jetsam loading in each axial section (between discs) is different because of axial movement. Each of these represent a crosssection of a specified jetsam loading reported in the figures. Because of the low levels of these initial concentrations the calculations were approximated by the case of a strongly segregated system (i.e. Case I). It might also be noted that the ratio of the fine particle diameter to the coarse particle diameter used in the experiment was about 0.125 which is below the threshold for which spontaneous percolation could occur. The experimental results suggest that fines might have sifted through the matrix of the plug flow region down to the drum's wall as evidenced by the small but nonzero jetsam concentration reported at the apex. Since the model precludes spontaneous percolation, the boundary conditions for the numerical solution require that jetsam concentration at the apex be zero. Therefore, the difference between measured and predicted results are partly attributed to spontaneous percolation. APPLICATIONOF SEGREGATIONMODEL One of the first applications of the segregation model was to calculate particle concentration profiles as functions of radius at mid-chord plane of a 0.41 m drum (comparable to a pilot kiln at the University of British Columbia). Simulations were carried out for Cases I - I I I and the plausibility of each situation examined. The results of these calculations are depicted in Fig. 8 for the respective jetsam loadings of 20, 30 and 50%, using polyethylene pellets as the bed material. As seen from these figures the difference between a strongly segregated system (Case I) and combined mixing and segregation (Case III) is clearly apparent. The result shows that if diffusion is present then it will tend to spread jetsam concentration by moving fines towards the top; and when percolation ceases (Case II) the bed will be well mixed. Because Case II is for complete mixing, no further discussion on this case is carried out. Figure 8(c) shows that, perhaps, the boundary condition imposed at the top (i.e. Cs --=0) is not applicable to higher jetsam loading since the profile is forced to zero at the free surface. As was mentioned earlier, Henein (1980) had observed that fines begin to appear on top of the bed at jetsam loadings of 40%. None the less, the profiles show that, for higher jetsam loading, the strongly segregated case (Case I) is clearly no longer applicable, rather, Case III must give a more plausible result and the constraint Cs = 0, at the exposed bed surface, must be removed. 0.2 i i i ~,~ i r 0.~ i Model Predictions Experimental (Henein, 1980) I 0.2 I ~===a o I 0 0.i O @ + co ~ 0 0 m°ee @ I I I I 0.1 c o.o.o 0.8 1.0 C Q ~~' O C.) O I Model Predictions Experimental (Henein, 1980) =- 0.2 O O I [ ~ Model Predictions Experimental (Henein, 1980) O 00 ~=~ + -0.1 ~ - 0 . 2 0.0 ~ ~ 0.2 0.4 , ~ , 0.6 ~ , 0.8 I --0.1 ,[' 1.0 ~ , ! I 1 , I ~ I t t - 0 . 2 0.0 0.2 0.4 0.6 0.8 1.0 Distance F r o m Apex, x / 2 L .2 - 0 . 2 0.0 .2 0.2 0.4 0.6 .2 Distance F r o m Apex, x/2L Distance F r o m Apex, x / 2 L Fig. 7. Predicted and measured profiles for jetsam concentration for a 40 cm drum: limestone with dpv/dpj = 8: (a) 3.11 rpm, 16% fill (dpe = 4.2 mm) Cjo = 5%; (b) 3.11 rpm, 16% fill (dee = 4.2 mm), Cao = 9%; (c) 3.19 rpm, 14% fill (dpv = 4.2 mm), C~o = 5%. I 0.2 ' I F 0.0 6 ' I I'-. -0.2 ' I I " c.;.," •< 0.0 I",, I >., ~''.. I 6 -0.2 \"J r,,3 = O3 I 1 I - -- Case II "-- Caselll -'t t 1 1~ - 0 . 6 fi - 0 . 6 -0.8 -0.8 0 t_ m 0.0 I , , 0.2 I 0.4 , I Concentration, [-] 0.8 .... .< Case III > -0.8 c "" - 1 . 2 , 0.6 I I II 0 -1.0 0.0 Jetsam -- iI ]'"'"........ o -1.0 - i o -0.6 O L I= ' , -0.2 I , t I h I ~ I 0.0 0.2 0.4 0.6 0.8 Jetsam C o n c e n t r a t i o n , [ - ] -1.0 t N -~ .2 I~1 1.0 - 0 . 2 0.0 0.2 0.4 0.6 © > Case Case 6 -0.2 CO O9 , I "~ - 0 . 4 -0.4 -0.4 -1.2 -0.2 I t b - cas°" I---- c.,o ~ "--.., 0.2 0.2 ' > .> 0.8 Jetsam Concentration, 1.0 .2 [-] Fig. 8. Predicted jetsam concentration in the active layer at the mid-chord position for the three cases described in the text: 0.41 m drum, 2 rpm at 12% fill; bed material is polyethylene pellets, dpe/d w = 2: (a) Cjo = 0.2, (b) Cjo = 0.3, (c) Cjo = 0.5. Modelling of particle mixing and segregation Figure 9 shows the radial profiles at mid-plane for polyethylene pellets at various jetsam loadings for the entire bed depth with thickness, H. Notice the symmetry between the concentration gradient in the active layer and that in the plug flow region for Case I [Fig. 9(a)]. This is the result of the 'escalator' role played by the plug flow region as was depicted in Fig. 4 and which symmetrically rearranges jetsam particles. This symmetry is distorted when Case III was applied [Fig. 9(b)] due to the effect of the diffusion term in the governing equations which tends to spread jetsam in the radial direction of the active layer (active layer mixing). The effect of kiln speed on segregation was examined using Cases I and III. For both cases there was very little effect of rotation rate on the concentration profiles. This is not surprising because the model seeks a steady-state solution and the result must converge to the jetsam loading. However, prior to convergence, and for each material circulation in the cross-section, coinciding with each calculation, the jetsam concentration gradients were different and depended on rotation rate. In order to show the segregated core in two-dimensional representation, contours of concentration gradients were plotted for the result shown in Fig. 7 (Fig. 10). As can be seen, the kidney is clearly depicted in these contour plots and their dimensions can be estimated for purposes such as bed thermal conductivity modelling. Also, to examine the segregation patterns which are likely to occur in industrial kilns, the model was tested for a hypothetical 2.5 m ID kiln. Figure 11 predicts the radial concentration as function of surface position as particles move from the apex to the base. It provides a comparison between predictions at the pilot scale (for which measurements were made) and the industrial scale. As seen from the results, the distribution of jetsam in the larger kiln tends to skew to the apex, depicting a more pronounced segregated kidney. The reason for the differ- I 0"2ti I ' -o.o : i ' i ' I ' l 4177 ence in jetsam distribution in the two geometries is attributed to the fact that, for the same degree of fill, the chord length in the larger kiln is over five times longer than that of the pilot kiln and, as a result, most of the percolation process occurs between the apex and the mid-chord. Discussion of mixing through density compensation It is evident from the forgoing results that the bed would be 'well mixed' either when jetsam particles are not present (Cj = 0) or when the net percolation velocity of jetsam is zero (vp = 0). Since the former case is unrealistic, it will not be considered further. However, further exploration of the latter might be helpful in addressing similar industrial problems. It has been shown elsewhere that (Alonso et al., 1991) if the mixture contains some denser particles, then these will percolate together with the fine particles as jetsam. Since larger particles (of same density) are flotsam, percolation may be prevented by introducing denser materials with large particle size into a mixture. Such weight compensation methods of minimizing free surface segregation have been discussed by Alonso et al. (1991). The model developed in the present work is used to explore the appropriate size and density ratio which will combine to eliminate the percolation velocity in order to achieve a well mixed bed. Equation (11) suggests that this can be accomplished by either setting the shear rate to zero (no shearing in the active layer; du/dy = 0) or making vp~ = vp~.The only option is the latter since the former case cannot be accomplished in a continuously shearing active layer. Therefore, for flotsam particles to behave as jetsam, the following condition for the flux of particles between layers must be satisfied: (28) Vpspj = vpvpF. Substitution ofeqs (12) and (13) into eq. (28) gives the density ratios for which flotsam particles will sink as 0,2 la I i , - i b I Ib . . . . . . . . . . . . . . . . . . . . . . -0.0 -0.2 6 ) -0.4 t ' : -0.4 ,1 r,f] 09 -0.6 0 -0.8 --I.0 C J0 = 0. i C J0 = 0.2 .... cJo -0.2 I I 0.0 CASE -0.8 ¢//" q) --1 .2 0 [ CASE I 0.2 ' I 0.4 ' I 0.6 ' = 0 c 0.3 I "d ' 0.8 Jetsam Concentration, [-] .0 F :_-_ o:= I -1.0 III -- o., I t -1.2 -0.2 L I 0.0 , I 0.2 , , 0.4 I 0.6 , I 0.8 Jetsam Concentration, [-] Fig. 9. Predicted jetsam concentration in both active layer and plug flow region shown at the mid-chord position for various jetsam loadings: (a) Case I and (b) Case II of model. .0 4178 A. A. BOATENGand P. V. BARR I . . . . I . . . . I . . . . I . . . . I . . . . I . . . . I . . . . I . . . . I -0.050 -0. I O0 -0.150 -0.200 -0.250 l , , , , l , , i , l , , , , l , , -0.150 -0.100 -0.050 0.000 I . . . . I . . . . I . . . . I 0.050 0.100 I I . . . . . . . . I 0.150 . . . . I'. -0.050 -0. I O0 -0.150 -0.200 -0.250 b I -0.150 ~ I [ I -0.I00 i I i [ ~ , -0.050 , , I , 0.000 , ~ , I , 0.050 ~ , , I , 0.100 , , ~ I 0.150 Fig. 10. Contour plots of jetsam concentration for limestone using operational conditions of Henein (1980); jetsam loading at 9%: (a) Case I and (b) Case IIl of model. a function of the size ratio in a mixture of a dilute system. This relationship is presented in Fig. 12 for two packing conditions, i.e. for closest packing and for a simple cubic array of particles. As can be seen, for a size ratio of flotsam to jetsam ranging between 1.5 and 2.0 the density ratio of the flotsam to jetsam particles for which the flotsam will behave as jetsam will range between 7 and 8 for the closest packing; and it increases exponentially at higher size ratios. The result indicates that eliminating size segregation by density compensation may not be practically feasible in most industrial operations except, perhaps, for 4179 Modelling of particle mixing and segregation . . . . I . . . . I . . . . I . . . . I . . . . I . . . . I 1 -0.50 -0.75 -1.00 -1.25 -1.50 a i i | , I -0.75 ' I . . . . , , I , , ~ ; I ~ ~ ~ ~ I ~ -0.50 -0.25 0.00 I I T . . . . . . . . , ~ ~ I . . . . 0.25 . . . . I I . . . . 0.50 . . . . I I , , 0.75 . . . . I ' ' -0.50 -0.75 -1.00 -1.25 -1.50 b -1.75 , I . . . . -0.75 I . . . . -0.50 [ I ' -0.25 ' ' I . . . . 0.00 I . . . . 0.25 I . . . . 0.50 I , . 0.75 Fig. 11. Contour plots of jetsam concentration in a 2.5 m drum at 12% fill and jetsam loading of 10%: (a) 2 rpm; (b) 5 rpm. a few applications, one being incineration of solid waste where size and density differences are likely to be widely spread. Thus the only option left in dealing with radial segregation in kilns would appear to be either removing jetsam from the feed system (which may not be practical) or use the mechanisms of segregation to the advantage of the process. For example, in heat treatment applications, loading of low heat capacity materials as fines may be a good strategy for achieving a homogeneous product. This can best be accomplished through an improved predictive capability of the segregation problem. CONCLUSIONS An analytical model which relates particle segregation rates to primary operating parameters such as kiln diameter, bed depth and rotation rate has been 4180 A. A. BOATENGand P. V. BARR 12 I J/ I g2(L9) r H i! t0 tl I :.,3 kay L ,ml M/N r~ 8 # - - - - Closest packing --Simple cubic 6 , 1.0 t 1.5 I I 2.0 , I 2.5 ,3.0 Fig. 12. Density compensation of particle size segregation [following Alonso (1991)]. Depicts density ratio required for flotsam to behave as jetsam in a mixture with specified size ratio. developed. The model predictions are in good agreement with experimental results of Henein (1980); and it can be applied to predict the size and extent of the segregated core as well as to show the effect of segregation on material mixing. The model has been used to establish that elimination of size segregation by weight compensation, as described by Alonso et al. (1991), is possible but not practicable for industrial kilns. The model may also be used to make an estimation of the effective thermal conductivity of the bed which, in a granular medium, is a function of the particle size distribution. Acknowledgements--The authors would like to thank Dr. J. R. Ferron, Department of Chemical Engineering, University of Rochester, for valuable discussions. A. A. Boateng was Assistant Professor of Engineering at Swarthmore College, PA, during preparation of the manuscript. Technical support provided to him by the college is acknowledged. Financial support was provided by Alcan Canada and the Natural Science and Engineering Research Council of Canada. E Em Fr 9 #o u(y) I Size R a t i o ; d p v / d p j A CF Cj Co dp Dr ep R NOTATION interfacial area, m 2 flotsam concentration, dimensionless jetsam concentration, dimensionless proportionality constant particle diameter, m diffusion coefficient, me/s coefficient of restitution of particles, dimensionless mean void diameter ratio void diameter ratio that results in spontaneous percolation rotational Froude n u m b e r ( = ~o2 Rg-1) dimensionless acceleration due to gravity, m/s 2 pair distribution function in collisional theory L,p a function in constitutive equation for granular flow bed depth, m segregation flux, me/s 2 ratio of mean voids projected area and mean projected total area distance from apex to mid-chord of bed cross-section ratio of n u m b e r of voids to number of particles in a layer diffusion flux, 1/s cylinder radius, m velocity parallel to bed surface, m/s tangential velocity of rotary drum, m/s percolation velocity, m/s Greek letters 6, fix active layer depth at distance, x, from apex, m A active layer depth at mid-chord, m q particle n u m b e r ratio ~9 solids volume concentration (solids fraction), dimensionless dynamic angle of repose, rad p bulk density, kg/m 3 pp particle density, kg/m 3 a particle size ratio, dimensionless 2P granular temperature (or grain temperature), m 2 s 2 q~ static angle of repose of material ~ angular velocity, l/s Subscripts al b J F pf active layer bed jetsam flotsam plug flow REFERENCES Alonso, M., Satoh, M. and Miyanami, K., 1991, Optimum combination of size ratio, density ratio and concentration to minimize free surface segregation. 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K., 1988, Particle size segregation in inclined chute flow of dry cohesionless granular solids. J. Fluid Mech. 189, 311 335. Zhang, Y. and Campbell, C. S., 1992, The interface between fluid-like and solid-like behaviour in two-dimensional granular flows. J. Fluid Mech. 237, 541 568.