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Differential settling of a binary mixture
Powder Technology 92 (1997) 171–178
Differential settling of a binary mixture Terence N. Smith U School of Chemical Engineering, Curtin University of Technology, GPO Box U1987, Perth, WA 6845, Australia Received 6 November 1996; revised 28 February 1997; accepted 7 March 1997
Abstract A model for the differential settling of particles of various species is presented. It satisfies conditions specified for matching of pressure gradients and velocity profiles in the flow of fluid about the particles in the settling system. Its form is based on an analytical solution of the equations of motion for slow flow of fluid within a spherical particle–envelope system. The expression for calculation of the settling velocity contains two functional terms. The first of these represents a mean settling velocity for all of the species in the system while the second provides an adjustment for the differential velocity of the particular species. Application of the model to the estimation of the settling velocities of particles of two sizes from suspension is illustrated. Coefficients of the functional terms are obtained from experimental results. The first term shows close correspondence with the formula of Richardson and Zaki for the settling of particles of uniform size. Because of its essentially mechanistic derivation, this model is expected to extend usefully to the description of differential settling in complex systems containing particles of several sizes and densities. Keywords: Settling; Suspensions; Velocity profiles; Differential settling; Binary system modelling
1. Introduction
Differential settling of solid particles through a liquid is the fundamental process utilized in the operation of hydraulic classifiers for the separation of solid species according to size or density. Several models which offer procedures for the estimation of settling velocities are available. Most of these models are, however, empirical rather than mechanistic in structure and so may not be applied with confidence beyond the range of the data over which they have been tested. For systems of particles of equal density, Lockett and AlHabbooby [1] propose a model in which all particles settle against a constant interstitial fluid velocity. The velocity of a particle of given size relative to the fluid is calculated from the single-particle settling velocity of that particle and the total volume fraction of solids in the suspension using the correlation of Richardson and Zaki [2] for the settling velocity of uniformly sized particles. Mirza and Richardson [3] offer an adjustment which improves the performance of this model. Treatment of systems containing particles of different densities is sometimes based on the concept that each particle settles through a fluid which is modified in density or viscosU
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ity by the presence of other settling particles. Masliyah [4] proposes a variation of the Lockett and Al-Habbooby [1] model in which settling velocity is directly dependent on the difference between the densities of the solid species and that of the suspension. Patwardhan and Tien [5] also take this approach. In the model of Selim [6], a particle is considered to be settling through a fluid consisting of the liquid and all particles of smaller size. From a fundamental point of view, modification of the properties of the liquid in the equation of motion of the fluid relative to the solid particles is not permissible. Each particle in a settling system moves at a velocity determined by the balance between the motive force of its weight and the resistive force exerted on it by the liquid in contact with its surface. This resistive force is derived from stresses induced in the fluid continuum by gravitational and hydrodynamic effects. Only those solid particles which are very small and do not have substantial settling velocities may be regarded as part of the fluid continuum. To give some promise of general utility, a model of differential settling should have a justifiable mechanistic basis. It is the purpose of this paper to present a model of differential settling in a form which expresses the salient features of the physical interaction between settling particles. It is analytically based but utilizes numerical coefficients evaluated from experimental data on binary systems.
0032-5910/97/$17.00 q 1997 Elsevier Science S.A. All rights reserved PII S 0 0 3 2 - 5 9 1 0 ( 9 7 ) 0 3 2 3 6 - 1
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2. Motion of particles and fluid It is well established by observation of the settling of uniformly sized particles that variations in velocity do occur as neighbouring particles approach and separate [7] and that these perturbations to velocity operate on scales of length related to the spacing of particles [8]. Where several kinds of particles are settling from suspension, variations in the spacing and, consequently, in the velocities of particles are also caused by the approach, passage and recession of other kinds of particles moving at different average velocities. In models of settling, the motion of a particle is regarded as quasi-steady. Its progress at the mean value of its settling velocity is viewed as the result of an interaction with the surrounding fluid which can be expressed in terms of mean values of determining functions of all variables in the system. Fundamental differences between models arise from the various ways in which this interaction is defined. In the construction of models of concentrated systems, a simplification which is sometimes made is that the interaction between the solid particle and the fluid is localized within an envelope of fluid surrounding the particle. This envelope has a volume with a mean value specific to that type of particle in the settling system. Fluid beyond this volume interacts with neighbouring particles. This approach to modelling is referred to as the cell model. In this model, it is necessary that each particle–envelope system be independent in the sense that no force be exerted on that system by the neighbouring systems. Contact between the fluid envelopes must be stress free. Fluid may pass through the cell but there must be no discontinuity of flow of the fluid as it passes from envelope to envelope. These and other conditions governing the motion of particles and fluid in a system containing several species of settling particles can be defined in terms of the variables pertinent to the particle–envelope model. Particular specifications can be made and corresponding statements of relationships between the variables can be identified for: (i) pressure gradient in the fluid; (ii) spatial volume balance; (iii) continuity and balance of flow of fluid and particles; (iv) matching of profiles of fluid velocity about adjacent particles. 2.1. Pressure gradient The mean vertical pressure gradient in the fluid must balance the weight of the suspension. The part of that gradient imposed by the weight of the solids is the dynamic pressure gradient which drives the motion of the fluid relative to the solid particles. This is dp s8(riyr)cig dz
(1)
in which r is the mass density of the fluid, ri is the mass density of the solid of type i and ci is its volume fraction in the suspension. This same pressure gradient must be exerted in the vicinity of each type of particle so that no sustained lateral flow of fluid is induced. 2.2. Volume balance Where several types of particles are settling through fluid, each type of particle has a certain average volume of associated fluid. Volume balance over the system requires that the sum of the volume fractions of the solid species and of the associated volumes of fluid be unity. This requirement may be expressed by the equation ci 8 s1 ki
(2)
in which ki is the volume fraction of species i within its particular fluid envelope. 2.3. Continuity statement The flows of fluid and solid particles through any zone must satisfy the governing continuity statement. In batch settling of a suspension, there is no net flow across the horizontal plane. Accordingly, the continuity statement requires that 8
ci [kiviq(1yki)(aiqvi)]s0 ki
(3)
in which ai is the average velocity of fluid within the envelope relative to the particle and vi is the velocity of the particle. 2.4. Velocity profiles The profile of fluid velocity between the surface of a particle and that of its neighbour must be continuous. The maximum in this velocity profile marks the extent of influence of each particle and the boundary between the envelopes. Determination of the detail of the pattern of flow of fluid induced about a particle by the mean pressure gradient permits identification of two characteristic values. The velocity a, as used in Eq. (3), is the average velocity of fluid within the envelope relative to the particle. The velocity b of the fluid relative to the particle at a specific point on the surface of the fluid envelope is the effective maximum in the profile. The condition for matching of fluid velocities between contiguous particle–envelope systems is given by biqvisb1qv1
(4)
An effective model of differential settling must satisfy the conditions specified by Eqs. (1)–(4). The pattern of flow of the fluid relative to the particle it surrounds must be found by solution of the equations of motion under the pertinent bound-
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ary conditions. Values of a and b for insertion in the equations above follow from the particular solution obtained.
6y9gq9g 5y6g 6 as (1yg 3)(6q4g 5)
(11) (12)
3. Settling of two sizes
6y4.5gy7.5g 3q9g 5y3g 6 bs 6q4g 5
Application of the particle–envelope model to the settling of spherical particles of different sizes and densities at small Reynolds numbers is presented in earlier papers [9,10]. The specifications in Eqs. (1)–(4) are invoked and an analytical solution of the flow field is used to obtain values of a and b. For the purposes of the development presented in this paper, the solution for settling of a suspension of particles of two sizes and equal density is required. Simplifications of the equations above can be made in this case. Since the pressure gradient in the fluid must be uniform through both types of particle–envelope systems, the volume fraction of the cell occupied by the particle in each case must be the same. It must be the total volume fraction of solids in the suspension. Thus k1sk2sc1qc2sk
(5)
The continuity statement may now be written as c1 [kv1q(1yk)(a1qv1)] k q
c2 [kv2q(1yk)(a2qv2)]s0 k
(6)
or c1[v1q(1yk)a1]qc2[v2q(1yk)a2]s0
(7)
Solution of this equation together with that expressing the matching of velocities b1qv1sb2qv2
(8)
gives the values of the settling velocities v1 and v2. The analytical solution of the flow field utilizes the cell model of Happel [11] for slow flow of a spherical particle within a spherical fluid envelope on the surface of which the shear stress is zero. In the particular form of solution required for application to this settling problem, the force on the particle is specified as its net weight in the fluid and velocities of flow of fluid through the envelope relative to the particle to produce this force are obtained. These may be expressed in the form a1syau01, a2syau02
(9)
b1sybu01, b2sybu02
(10)
where u0 is the single-particle settling velocity and a and b are functions of the ratio of particle diameter to cell diameter. Because the volume fraction of the solid particle within each type of cell is the same in this case, the values of a and b are also the same for each. The actual functions of k, given in terms of gsk1/3, are
Insertion of the expressions for a and b from Eqs. (9) and (10) into the statements of continuity and equal velocity leads to v1ya
1yk c2 (c1u01qc2u02)yb (u01yu02)s0 k k
(13)
v2ya
1yk c1 (c1u01qc2u02)yb (u02yu01)s0 k k
(14)
Comparison of velocities obtained from this model with experimental settling velocities reflects the trends of velocity with the governing variables but does not show good correspondence with actual values [9]. The explanation for the deviation is essentially the same as that for the behaviour of the Happel [11] model of singlespecies settling. Solution of the cell model for a system in which the particles are regularly spaced yields a settling velocity which is less than that of a system in which there are variations in spacing.
4. Derivation of velocity functions While the model as proposed above does not perform satisfactorily with insertion of the analytical functions for relative velocities, the essential form of it satisfies the conditions specified in Eqs. (1)–(4) and so merits further attention. An obvious development of the model is to evaluate the functions a and b from experimental data. This process is facilitated by isolation of the two functions by manipulation of Eqs. (13) and (14) to give (c1v1qc2v2)ya(1yk)(c1u01qc2u02)s0
(15)
(v1yv2)yb(u01yu02)s0
(16)
Figs. 1 and 2 show values of a and b calculated from the experimental data of Story [12] and Thomson [13] cited in the earlier work [9]. These settling tests cover combinations of the sizes 252 and 185 mm, 252 and 129 mm, 252 and 60 mm and 185 and 60 mm of glass beads in various ratios of volume fraction settling in a mixture of glycerol and water. The data from the tests are tabulated in Appendix A. Expressions to describe a and b as functions of k may be obtained by fits to the data shown in these figures. Derived in the form of simple polynomials, the results are
as1y3.85kq3.90k 2
(17)
2
(18)
bs1y3.87kq3.72k
The first of these correlations, for the average velocity of the fluid relative to the particles, may be compared with that
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Fig. 1. Derivation of a from experimental data: n, 252:185; h, 252:129; e, 252:60; s, 185:60.
A likely explanation is that for these two sizes, the closest of the combinations tested, the dynamic effects between the particles approach the scale of the interactions between particles in a monodisperse system. The masses of the particles and the sizes of the associated fluid envelopes are comparable. In these circumstances, there is a tendency for particles of the two sizes to travel together and so to diminish the process of differential settling. Argument that some mechanism of this type, rather than experimental error, is responsible for the deviation of the data for this combination of sizes from the function b is supported by the comparatively good coverage of data for the same combination by the function a in Fig. 1. The correlation for a is, like that for b, derived with omission of the data points for the tests with mixtures of 252 and 185 mm particles. Delineation of the relationship between the ratio of particle sizes and the function b requires further experimental work with very sharply defined size fractions.
5. Discussion
Fig. 2. Derivation of b from experimental data: n, 252:185; h, 252:129; e, 252:60; s, 185:60.
obtained by Richardson and Zaki [2] for the settling of monodisperse suspensions. The form of that correlation is indicated in Eq. (19) in which the settling velocity of the particles relative to the fluid is presented as a function of the single-particle settling velocity and of the concentration of solids: v1su01(1yk)Nsu01a(1yk)
(19)
The function a, calculated from this formula using a value of 4.65 for the exponent N, is plotted as a broken line on Fig. 1 for comparison with a as derived from this analysis. While there is a difference in the shapes of the two functions, there is a correspondence of values in the lower part of the range of the experimental data. In fitting the experimental settling velocities to find the function b, the data points for the combination of 252 and 185 mm particles are ignored. These results are clearly in deviation from those for the other combinations shown in Fig. 2. They show smaller values of b. This indicates that the differences between the settling velocities of the two sizes are less than might be expected from the model.
The experimental data from the binary systems are depicted by the model in terms of the functions a and b of the solids volume fraction. Within the model, these functions have distinct physical significances associated with satisfaction of conditions both of continuity of flow, expressed by Eq. (7), and of matching of velocity profiles, expressed by Eq. (8). The actual values of the two functions, as described by Eqs. (17) and (18), are close. Justification for retention of distinct identities for a and b may be drawn from the observation concerning the tests with mixtures of 252 and 185 mm particles. Deviation of the results of these tests from those with mixtures having a greater ratio of sizes is obvious in the values of b but is not substantial in the values of a. It is clear that a is an expression of the mean velocity of settling of the suspension and that b is an expression of the effect of differential settling of the two species of particles. These connections are evident in the formulae for the settling velocities, Eqs. (13) and (14). Since the functions a and b in the particle–envelope model represent, respectively, the average velocity of the fluid relative to the particle and the velocity at the boundary of the envelope, it is to be expected that the value of b should exceed that of a. This is not actually the case. For the values derived from the settling data, a is somewhat greater than b. An explanation of this occurrence may be given in terms of dynamic effects during settling. As already observed, particles settling through fluid do not maintain a fixed spacing but continually approach and recede from one another. The dynamic clusters so formed have greater settling velocities than those of uniformly spaced particles [7]. Within a system containing two species of particles, such aggregations increase the settling velocities of both species of particles. This leads to values of the average velocity relative to the
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fluid, expressed by the factor a, which are greater than those for a regular array of particles. Within any aggregation, however, the relative velocity between neighbouring particles may remain small so that the factor b, which is used to generate velocities required for the matching of profiles of velocity between adjacent particles, may remain comparable with the value that it would have in a regular array. So it is arguable that a might exceed b. It is useful to compare the performance of this model with that of Lockett and Al-Habbooby [1]. The settling velocity of particles relative to the fluid is given by that model as v1su01(1yc1yc2)N
(20)
in which N is the exponent in the Richardson and Zaki [2] correlation for single-species settling. In terms of the conditions identified above for matching of the flows about the two species, this result corresponds to the specifications a1sb1, a2sb2
Fig. 4. 252 and 129 mm particles in the ratio 1:2: h, experimental; ———, model; =, Lockett and Al-Habbooby [1].
(21)
or, in terms of the formulation in Eqs. (9) and (10),
asb
(22)
The matching condition is that the average interstitial velocity is the same for both species of particles. In view of the arguments presented here concerning the distinct identities of a and b, this equality should be regarded as an approximation or a coincidence rather than as a justifiable specification. While the model of Lockett and Al-Habbooby performs reasonably well, the model proposed here gives results which are in better accordance with the settling data utilized in this work. Figs. 3–6 show comparisons of the performances of the two models with experimental results for particles of sizes 252 and 129 mm in proportions 4:1, 252 and 129 mm in proportions 1:2, 252 and 60 mm in proportions 1:1 and 185 and 60 mm in proportions 1:3, respectively. Settling velocities given by the Lockett and Al-Habbooby model are too great for the larger particles and too small for the smaller particles.
Fig. 5. 252 and 60 mm particles in the ratio 1:1: h, experimental; ———, model; =, Lockett and Al-Habbooby [1].
Fig. 6. 185 and 60 mm particles in the ratio 1:3: h, experimental; ———, model; =, Lockett and Al-Habbooby [1].
Fig. 3. 252 and 129 mm particles in the ratio 4:1: h, experimental; ———, model; =, Lockett and Al-Habbooby [1].
It is, of course, to be expected that a model of differential settling containing two parameters should be more effective than one containing a single parameter. The advantage of this model is not in the number of parameters but in the mechanisms that they represent. The functions a and b in Eqs. (13) and (14) are identified with terms derived from a solution of the equation of motion of the fluid
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relative to the particle and so have direct physical significance. They define velocities which are included in the statements of compliance with the conditions governing differential settling specified in Eqs. (1)–(4). Because this model is based on a solution of the equation of motion, it promises some flexibility in extension to systems in which several species of particles are settling. The conditions expressed in Eqs. (1)–(4) remain valid. In particular, for systems containing particles of different densities, Eq. (1) requires that the gradient of pressure in the fluid be uniform. This specification is essential so as to avoid discontinuities in the pressure field or conditions which would generate transverse flows of fluid. The volume fraction ki of each species of solid particle within its particle–envelope cell is then given by Eq. (2). Values of the functions a and b of ki given by Eqs. (17) and (18) may then be inserted in Eqs. (3) and (4) to define the behaviour of the system. While these functions have been derived from an analysis of data from the settling of particles of different sizes but of the same density, the application of them is expected to be more general. This expectation is, of course, subject to delineation of the effects, particularly on b, of ratios of particle sizes and densities. The utility of this particular form of model is restricted to systems in which the particles are settling at small values of Reynolds number. It is based on a solution of the equations of motion for slow flow. Eqs. (13) and (14) are linear in the single-particle velocities. So simple a form cannot describe the behaviour of systems in which inertial effects in the flow are substantial. In the correlation of Richardson and Zaki [2] for settling of a monodisperse suspension, allowance for the emergence of inertial effects with increasing Reynolds number is in the form of a dependence of the exponent N in Eq. (19) on the single-particle Reynolds number. This form of adjustment is used by Lockett and Al-Habbooby in their model of two-species settling. No correspondingly simple variation of the functions in Eqs. (17) and (18) is permissible.
6. Conclusions
A model for estimation of the settling velocities of particles of two sizes from suspension is presented. It satisfies conditions specified for matching of pressure gradients and velocity profiles in the flow of fluid about the particles and is based on an analytical solution of the equations of motion for slow flow. The expression for calculation of the settling velocity contains two functional terms. The first of these represents a mean settling velocity for all of the species in the system while the second provides an adjustment for the differential velocity of the particular species. Numerical coefficients for the terms in the governing formulae, Eqs. (13) and (14), are
obtained from experimental results and are expressed by Eqs. (17) and (18) as functions of the concentrations of the species in the suspension. The function in Eq. (18) is the manifestation of the requirement for matching of fluid velocity profiles between particles of different species. It is well identified from experimental data but shows a loss of influence as the difference between the sizes of the particles becomes small. Definition of this effect requires further experimental work with very sharply defined size fractions. Because of its essentially mechanistic derivation, this model is expected to extend usefully to the description of differential settling in more complex systems containing particles of several sizes and densities.
7. List of symbols
a b c g k N p u0 v z
average velocity of fluid within the envelope relative to the particle maximum in the profile of velocity of flow relative to the particle volume fraction of a solid species in the suspension gravitational acceleration volume fraction of a solid species within its fluid envelope exponent in Eq. (19) dynamic pressure in fluid terminal settling velocity of a single particle settling velocity of a solid species vertical coordinate
Greek letters
a b g r
coefficient in Eq. (9) coefficient in Eq. (10) function in Eq. (11) density of fluid or solid
Subscripts 1 2 i
first solid species second solid species ith solid species
Appendix A The data of Story [12] and Thomson [13] for glass spheres settling in aqueous glycerol are given in Table A1.
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Table A1 Coarse 252 mm Fine 129 mm
Coarse 252 mm Fine 185 mm
u0s4.19 mm/s u0s1.09 mm/s
u0s4.19 mm/s u0s2.27 mm/s
cCqcF
cC
cF
vC (mm/s)
vF (mm/s)
cCqcF
cC
cF
vC (mm/s)
vF (mm/s)
0.062 0.125 0.187 0.250 0.312 0.374
0.050 0.100 0.150 0.200 0.250 0.300
0.012 0.025 0.037 0.050 0.062 0.075
3.23 2.12 1.68 0.82 0.55 0.29
0.63 0.46 0.30 0.16 0.12 0.07
0.062 0.125 0.187 0.250 0.312 0.374
0.050 0.100 0.150 0.200 0.250 0.300
0.012 0.025 0.037 0.050 0.062 0.075
2.46 1.91 0.98 0.62 0.33
1.49 1.09 0.62 0.42 0.23
0.066 0.131 0.197 0.262 0.328 0.393
0.044 0.087 0.131 0.175 0.218 0.262
0.022 0.044 0.066 0.087 0.109 0.131
2.32 1.38 0.83 0.53 0.29
0.51 0.24 0.17 0.12 0.07
0.066 0.131 0.197 0.262 0.328 0.393
0.044 0.087 0.131 0.175 0.218 0.262
0.022 0.044 0.066 0.087 0.109 0.131
2.36 1.68 0.82 0.46 0.25
1.64 1.09 0.55 0.34 0.19
0.062 0.125 0.187 0.250 0.312 0.374
0.031 0.062 0.094 0.125 0.156 0.187
0.031 0.062 0.094 0.125 0.156 0.187
3.98 2.15 1.86 1.01 0.66
1.13 0.53 0.39 0.21 0.14
0.062 0.125 0.187 0.250 0.312
0.031 0.062 0.094 0.125 0.156
0.031 0.062 0.094 0.125 0.156
2.97 2.14 1.36 1.04 0.56
2.22 1.55 1.02 0.77 0.43
0.066 0.131 0.197 0.262 0.328 0.393
0.022 0.044 0.066 0.087 0.109 0.131
0.044 0.087 0.131 0.175 0.218 0.262
2.48 1.38 1.06 0.44 0.25
0.61 0.30 0.23 0.10 0.08
0.066 0.131 0.197 0.262 0.328 0.393
0.022 0.044 0.066 0.087 0.109 0.131
0.044 0.087 0.131 0.175 0.218 0.262
2.22 1.35 0.77 0.51 0.29
1.51 0.94 0.55 0.38 0.23
0.062 0.125 0.187 0.250 0.312 0.374
0.012 0.025 0.037 0.050 0.062 0.075
0.050 0.100 0.150 0.200 0.250 0.300
0.062 0.125 0.187 0.250 0.312 0.374
0.012 0.025 0.037 0.050 0.062 0.075
0.050 0.100 0.150 0.200 0.250 0.300
2.06 1.39 0.93 0.56 0.34
1.22 0.96 0.60 0.38 0.26
Coarse 252 mm Fine 60 mm
1.35
0.33
0.63 0.31
0.15 0.09
Coarse 185 mm Fine 60 mm
u0s4.19 mm/s u0s0.24 mm/s
u0s2.27 mm/s u0s0.24 mm/s
cCqcF
cC
cF
vC (mm/s)
vF (mm/s)
cCqcF
cC
cF
vC (mm/s)
vF (mm/s)
0.062 0.125 0.187 0.250 0.312 0.374
0.047 0.094 0.140 0.187 0.234 0.281
0.016 0.031 0.047 0.062 0.078 0.094
2.07 1.39 0.85 0.54 0.28
y0.18 y0.22 y0.22 y0.15 y0.09
0.062 0.125 0.187 0.250 0.312 0.374
0.047 0.094 0.140 0.187 0.234 0.281
0.016 0.031 0.047 0.062 0.078 0.094
1.22 0.81 0.61 0.29 0.15
y0.04 y0.05 y0.07 y0.04 y0.02
0.062 0.125 0.187 0.250 0.312 0.374 0.468
0.031 0.062 0.094 0.125 0.156 0.187 0.234
0.031 0.062 0.094 0.125 0.156 0.187 0.234
3.26 2.32 1.67 0.91 0.51 0.25
0.26 0.03 y0.16 y0.11 y0.08 0.09
0.062 0.125 0.187 0.250 0.312 0.374
0.031 0.062 0.094 0.125 0.156 0.187
0.031 0.062 0.094 0.125 0.156 0.187
1.11 0.79 0.52 0.29 0.15
0.04 y0.01 y0.02 y0.02 y0.01
0.062 0.125 0.187 0.250 0.312 0.374 0.437
0.016 0.031 0.047 0.062 0.078 0.094 0.109
0.047 0.094 0.140 0.187 0.234 0.281 0.328
y0.02 y0.04 y0.02
0.062 0.125 0.187 0.250 0.312 0.374
0.016 0.031 0.047 0.062 0.078 0.094
0.047 0.094 0.140 0.187 0.234 0.281
1.15 0.85 0.55 0.28 0.19
0.08 0.04 0.02 0.00 0.01
0.88 0.73 0.24
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References [1] M.J. Lockett and H.M. Al-Habbooby, Trans. Inst. Chem. Eng., 51 (1973) 281. [2] J.F. Richardson and W.N. Zaki, Trans. Inst. Chem. Eng., 32 (1954) 35. [3] S. Mirza and J.F. Richardson, Chem. Eng. Sci., 34 (1979) 447. [4] J.H. Masliyah, Chem. Eng. Sci., 34 (1979) 1166. [5] V.S. Patwardhan and C. Tien, Chem. Eng. Sci., 40 (1985) 1051. [6] M.S. Selim, A.C. Kothari and R.M. Turian, AIChE J., 29 (1983) 1029.
[7] K.O.L.F. Jayaweera, B.J. Mason and G.W. Slack, J. Fluid Mech., 20 (1964) 121. [8] R. Johne, Chem. Ing. Tech., 38 (1966) 428. [9] T.N. Smith, Trans. Inst. Chem. Eng., 43 (1965) T69. [10] T.N. Smith, Trans. Inst. Chem. Eng., 45 (1967) T311. [11] J. Happel, AIChE J., 4 (1958) 197. [12] M.J. Story, B.E. (Hons.) Thesis, University of Adelaide, Australia, 1963. [13] R.W. Thomson, B.E. (Hons.) Thesis, University of Adelaide, Australia, 1963.
Journal: PTEC (Powder Technology)
Article: 3289
Differential settling of a binary mixture Terence N. Smith U School of Chemical Engineering, Curtin University of Technology, GPO Box U1987, Perth, WA 6845, Australia Received 6 November 1996; revised 28 February 1997; accepted 7 March 1997
Abstract A model for the differential settling of particles of various species is presented. It satisfies conditions specified for matching of pressure gradients and velocity profiles in the flow of fluid about the particles in the settling system. Its form is based on an analytical solution of the equations of motion for slow flow of fluid within a spherical particle–envelope system. The expression for calculation of the settling velocity contains two functional terms. The first of these represents a mean settling velocity for all of the species in the system while the second provides an adjustment for the differential velocity of the particular species. Application of the model to the estimation of the settling velocities of particles of two sizes from suspension is illustrated. Coefficients of the functional terms are obtained from experimental results. The first term shows close correspondence with the formula of Richardson and Zaki for the settling of particles of uniform size. Because of its essentially mechanistic derivation, this model is expected to extend usefully to the description of differential settling in complex systems containing particles of several sizes and densities. Keywords: Settling; Suspensions; Velocity profiles; Differential settling; Binary system modelling
1. Introduction
Differential settling of solid particles through a liquid is the fundamental process utilized in the operation of hydraulic classifiers for the separation of solid species according to size or density. Several models which offer procedures for the estimation of settling velocities are available. Most of these models are, however, empirical rather than mechanistic in structure and so may not be applied with confidence beyond the range of the data over which they have been tested. For systems of particles of equal density, Lockett and AlHabbooby [1] propose a model in which all particles settle against a constant interstitial fluid velocity. The velocity of a particle of given size relative to the fluid is calculated from the single-particle settling velocity of that particle and the total volume fraction of solids in the suspension using the correlation of Richardson and Zaki [2] for the settling velocity of uniformly sized particles. Mirza and Richardson [3] offer an adjustment which improves the performance of this model. Treatment of systems containing particles of different densities is sometimes based on the concept that each particle settles through a fluid which is modified in density or viscosU
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ity by the presence of other settling particles. Masliyah [4] proposes a variation of the Lockett and Al-Habbooby [1] model in which settling velocity is directly dependent on the difference between the densities of the solid species and that of the suspension. Patwardhan and Tien [5] also take this approach. In the model of Selim [6], a particle is considered to be settling through a fluid consisting of the liquid and all particles of smaller size. From a fundamental point of view, modification of the properties of the liquid in the equation of motion of the fluid relative to the solid particles is not permissible. Each particle in a settling system moves at a velocity determined by the balance between the motive force of its weight and the resistive force exerted on it by the liquid in contact with its surface. This resistive force is derived from stresses induced in the fluid continuum by gravitational and hydrodynamic effects. Only those solid particles which are very small and do not have substantial settling velocities may be regarded as part of the fluid continuum. To give some promise of general utility, a model of differential settling should have a justifiable mechanistic basis. It is the purpose of this paper to present a model of differential settling in a form which expresses the salient features of the physical interaction between settling particles. It is analytically based but utilizes numerical coefficients evaluated from experimental data on binary systems.
0032-5910/97/$17.00 q 1997 Elsevier Science S.A. All rights reserved PII S 0 0 3 2 - 5 9 1 0 ( 9 7 ) 0 3 2 3 6 - 1
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2. Motion of particles and fluid It is well established by observation of the settling of uniformly sized particles that variations in velocity do occur as neighbouring particles approach and separate [7] and that these perturbations to velocity operate on scales of length related to the spacing of particles [8]. Where several kinds of particles are settling from suspension, variations in the spacing and, consequently, in the velocities of particles are also caused by the approach, passage and recession of other kinds of particles moving at different average velocities. In models of settling, the motion of a particle is regarded as quasi-steady. Its progress at the mean value of its settling velocity is viewed as the result of an interaction with the surrounding fluid which can be expressed in terms of mean values of determining functions of all variables in the system. Fundamental differences between models arise from the various ways in which this interaction is defined. In the construction of models of concentrated systems, a simplification which is sometimes made is that the interaction between the solid particle and the fluid is localized within an envelope of fluid surrounding the particle. This envelope has a volume with a mean value specific to that type of particle in the settling system. Fluid beyond this volume interacts with neighbouring particles. This approach to modelling is referred to as the cell model. In this model, it is necessary that each particle–envelope system be independent in the sense that no force be exerted on that system by the neighbouring systems. Contact between the fluid envelopes must be stress free. Fluid may pass through the cell but there must be no discontinuity of flow of the fluid as it passes from envelope to envelope. These and other conditions governing the motion of particles and fluid in a system containing several species of settling particles can be defined in terms of the variables pertinent to the particle–envelope model. Particular specifications can be made and corresponding statements of relationships between the variables can be identified for: (i) pressure gradient in the fluid; (ii) spatial volume balance; (iii) continuity and balance of flow of fluid and particles; (iv) matching of profiles of fluid velocity about adjacent particles. 2.1. Pressure gradient The mean vertical pressure gradient in the fluid must balance the weight of the suspension. The part of that gradient imposed by the weight of the solids is the dynamic pressure gradient which drives the motion of the fluid relative to the solid particles. This is dp s8(riyr)cig dz
(1)
in which r is the mass density of the fluid, ri is the mass density of the solid of type i and ci is its volume fraction in the suspension. This same pressure gradient must be exerted in the vicinity of each type of particle so that no sustained lateral flow of fluid is induced. 2.2. Volume balance Where several types of particles are settling through fluid, each type of particle has a certain average volume of associated fluid. Volume balance over the system requires that the sum of the volume fractions of the solid species and of the associated volumes of fluid be unity. This requirement may be expressed by the equation ci 8 s1 ki
(2)
in which ki is the volume fraction of species i within its particular fluid envelope. 2.3. Continuity statement The flows of fluid and solid particles through any zone must satisfy the governing continuity statement. In batch settling of a suspension, there is no net flow across the horizontal plane. Accordingly, the continuity statement requires that 8
ci [kiviq(1yki)(aiqvi)]s0 ki
(3)
in which ai is the average velocity of fluid within the envelope relative to the particle and vi is the velocity of the particle. 2.4. Velocity profiles The profile of fluid velocity between the surface of a particle and that of its neighbour must be continuous. The maximum in this velocity profile marks the extent of influence of each particle and the boundary between the envelopes. Determination of the detail of the pattern of flow of fluid induced about a particle by the mean pressure gradient permits identification of two characteristic values. The velocity a, as used in Eq. (3), is the average velocity of fluid within the envelope relative to the particle. The velocity b of the fluid relative to the particle at a specific point on the surface of the fluid envelope is the effective maximum in the profile. The condition for matching of fluid velocities between contiguous particle–envelope systems is given by biqvisb1qv1
(4)
An effective model of differential settling must satisfy the conditions specified by Eqs. (1)–(4). The pattern of flow of the fluid relative to the particle it surrounds must be found by solution of the equations of motion under the pertinent bound-
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ary conditions. Values of a and b for insertion in the equations above follow from the particular solution obtained.
6y9gq9g 5y6g 6 as (1yg 3)(6q4g 5)
(11) (12)
3. Settling of two sizes
6y4.5gy7.5g 3q9g 5y3g 6 bs 6q4g 5
Application of the particle–envelope model to the settling of spherical particles of different sizes and densities at small Reynolds numbers is presented in earlier papers [9,10]. The specifications in Eqs. (1)–(4) are invoked and an analytical solution of the flow field is used to obtain values of a and b. For the purposes of the development presented in this paper, the solution for settling of a suspension of particles of two sizes and equal density is required. Simplifications of the equations above can be made in this case. Since the pressure gradient in the fluid must be uniform through both types of particle–envelope systems, the volume fraction of the cell occupied by the particle in each case must be the same. It must be the total volume fraction of solids in the suspension. Thus k1sk2sc1qc2sk
(5)
The continuity statement may now be written as c1 [kv1q(1yk)(a1qv1)] k q
c2 [kv2q(1yk)(a2qv2)]s0 k
(6)
or c1[v1q(1yk)a1]qc2[v2q(1yk)a2]s0
(7)
Solution of this equation together with that expressing the matching of velocities b1qv1sb2qv2
(8)
gives the values of the settling velocities v1 and v2. The analytical solution of the flow field utilizes the cell model of Happel [11] for slow flow of a spherical particle within a spherical fluid envelope on the surface of which the shear stress is zero. In the particular form of solution required for application to this settling problem, the force on the particle is specified as its net weight in the fluid and velocities of flow of fluid through the envelope relative to the particle to produce this force are obtained. These may be expressed in the form a1syau01, a2syau02
(9)
b1sybu01, b2sybu02
(10)
where u0 is the single-particle settling velocity and a and b are functions of the ratio of particle diameter to cell diameter. Because the volume fraction of the solid particle within each type of cell is the same in this case, the values of a and b are also the same for each. The actual functions of k, given in terms of gsk1/3, are
Insertion of the expressions for a and b from Eqs. (9) and (10) into the statements of continuity and equal velocity leads to v1ya
1yk c2 (c1u01qc2u02)yb (u01yu02)s0 k k
(13)
v2ya
1yk c1 (c1u01qc2u02)yb (u02yu01)s0 k k
(14)
Comparison of velocities obtained from this model with experimental settling velocities reflects the trends of velocity with the governing variables but does not show good correspondence with actual values [9]. The explanation for the deviation is essentially the same as that for the behaviour of the Happel [11] model of singlespecies settling. Solution of the cell model for a system in which the particles are regularly spaced yields a settling velocity which is less than that of a system in which there are variations in spacing.
4. Derivation of velocity functions While the model as proposed above does not perform satisfactorily with insertion of the analytical functions for relative velocities, the essential form of it satisfies the conditions specified in Eqs. (1)–(4) and so merits further attention. An obvious development of the model is to evaluate the functions a and b from experimental data. This process is facilitated by isolation of the two functions by manipulation of Eqs. (13) and (14) to give (c1v1qc2v2)ya(1yk)(c1u01qc2u02)s0
(15)
(v1yv2)yb(u01yu02)s0
(16)
Figs. 1 and 2 show values of a and b calculated from the experimental data of Story [12] and Thomson [13] cited in the earlier work [9]. These settling tests cover combinations of the sizes 252 and 185 mm, 252 and 129 mm, 252 and 60 mm and 185 and 60 mm of glass beads in various ratios of volume fraction settling in a mixture of glycerol and water. The data from the tests are tabulated in Appendix A. Expressions to describe a and b as functions of k may be obtained by fits to the data shown in these figures. Derived in the form of simple polynomials, the results are
as1y3.85kq3.90k 2
(17)
2
(18)
bs1y3.87kq3.72k
The first of these correlations, for the average velocity of the fluid relative to the particles, may be compared with that
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Fig. 1. Derivation of a from experimental data: n, 252:185; h, 252:129; e, 252:60; s, 185:60.
A likely explanation is that for these two sizes, the closest of the combinations tested, the dynamic effects between the particles approach the scale of the interactions between particles in a monodisperse system. The masses of the particles and the sizes of the associated fluid envelopes are comparable. In these circumstances, there is a tendency for particles of the two sizes to travel together and so to diminish the process of differential settling. Argument that some mechanism of this type, rather than experimental error, is responsible for the deviation of the data for this combination of sizes from the function b is supported by the comparatively good coverage of data for the same combination by the function a in Fig. 1. The correlation for a is, like that for b, derived with omission of the data points for the tests with mixtures of 252 and 185 mm particles. Delineation of the relationship between the ratio of particle sizes and the function b requires further experimental work with very sharply defined size fractions.
5. Discussion
Fig. 2. Derivation of b from experimental data: n, 252:185; h, 252:129; e, 252:60; s, 185:60.
obtained by Richardson and Zaki [2] for the settling of monodisperse suspensions. The form of that correlation is indicated in Eq. (19) in which the settling velocity of the particles relative to the fluid is presented as a function of the single-particle settling velocity and of the concentration of solids: v1su01(1yk)Nsu01a(1yk)
(19)
The function a, calculated from this formula using a value of 4.65 for the exponent N, is plotted as a broken line on Fig. 1 for comparison with a as derived from this analysis. While there is a difference in the shapes of the two functions, there is a correspondence of values in the lower part of the range of the experimental data. In fitting the experimental settling velocities to find the function b, the data points for the combination of 252 and 185 mm particles are ignored. These results are clearly in deviation from those for the other combinations shown in Fig. 2. They show smaller values of b. This indicates that the differences between the settling velocities of the two sizes are less than might be expected from the model.
The experimental data from the binary systems are depicted by the model in terms of the functions a and b of the solids volume fraction. Within the model, these functions have distinct physical significances associated with satisfaction of conditions both of continuity of flow, expressed by Eq. (7), and of matching of velocity profiles, expressed by Eq. (8). The actual values of the two functions, as described by Eqs. (17) and (18), are close. Justification for retention of distinct identities for a and b may be drawn from the observation concerning the tests with mixtures of 252 and 185 mm particles. Deviation of the results of these tests from those with mixtures having a greater ratio of sizes is obvious in the values of b but is not substantial in the values of a. It is clear that a is an expression of the mean velocity of settling of the suspension and that b is an expression of the effect of differential settling of the two species of particles. These connections are evident in the formulae for the settling velocities, Eqs. (13) and (14). Since the functions a and b in the particle–envelope model represent, respectively, the average velocity of the fluid relative to the particle and the velocity at the boundary of the envelope, it is to be expected that the value of b should exceed that of a. This is not actually the case. For the values derived from the settling data, a is somewhat greater than b. An explanation of this occurrence may be given in terms of dynamic effects during settling. As already observed, particles settling through fluid do not maintain a fixed spacing but continually approach and recede from one another. The dynamic clusters so formed have greater settling velocities than those of uniformly spaced particles [7]. Within a system containing two species of particles, such aggregations increase the settling velocities of both species of particles. This leads to values of the average velocity relative to the
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fluid, expressed by the factor a, which are greater than those for a regular array of particles. Within any aggregation, however, the relative velocity between neighbouring particles may remain small so that the factor b, which is used to generate velocities required for the matching of profiles of velocity between adjacent particles, may remain comparable with the value that it would have in a regular array. So it is arguable that a might exceed b. It is useful to compare the performance of this model with that of Lockett and Al-Habbooby [1]. The settling velocity of particles relative to the fluid is given by that model as v1su01(1yc1yc2)N
(20)
in which N is the exponent in the Richardson and Zaki [2] correlation for single-species settling. In terms of the conditions identified above for matching of the flows about the two species, this result corresponds to the specifications a1sb1, a2sb2
Fig. 4. 252 and 129 mm particles in the ratio 1:2: h, experimental; ———, model; =, Lockett and Al-Habbooby [1].
(21)
or, in terms of the formulation in Eqs. (9) and (10),
asb
(22)
The matching condition is that the average interstitial velocity is the same for both species of particles. In view of the arguments presented here concerning the distinct identities of a and b, this equality should be regarded as an approximation or a coincidence rather than as a justifiable specification. While the model of Lockett and Al-Habbooby performs reasonably well, the model proposed here gives results which are in better accordance with the settling data utilized in this work. Figs. 3–6 show comparisons of the performances of the two models with experimental results for particles of sizes 252 and 129 mm in proportions 4:1, 252 and 129 mm in proportions 1:2, 252 and 60 mm in proportions 1:1 and 185 and 60 mm in proportions 1:3, respectively. Settling velocities given by the Lockett and Al-Habbooby model are too great for the larger particles and too small for the smaller particles.
Fig. 5. 252 and 60 mm particles in the ratio 1:1: h, experimental; ———, model; =, Lockett and Al-Habbooby [1].
Fig. 6. 185 and 60 mm particles in the ratio 1:3: h, experimental; ———, model; =, Lockett and Al-Habbooby [1].
Fig. 3. 252 and 129 mm particles in the ratio 4:1: h, experimental; ———, model; =, Lockett and Al-Habbooby [1].
It is, of course, to be expected that a model of differential settling containing two parameters should be more effective than one containing a single parameter. The advantage of this model is not in the number of parameters but in the mechanisms that they represent. The functions a and b in Eqs. (13) and (14) are identified with terms derived from a solution of the equation of motion of the fluid
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relative to the particle and so have direct physical significance. They define velocities which are included in the statements of compliance with the conditions governing differential settling specified in Eqs. (1)–(4). Because this model is based on a solution of the equation of motion, it promises some flexibility in extension to systems in which several species of particles are settling. The conditions expressed in Eqs. (1)–(4) remain valid. In particular, for systems containing particles of different densities, Eq. (1) requires that the gradient of pressure in the fluid be uniform. This specification is essential so as to avoid discontinuities in the pressure field or conditions which would generate transverse flows of fluid. The volume fraction ki of each species of solid particle within its particle–envelope cell is then given by Eq. (2). Values of the functions a and b of ki given by Eqs. (17) and (18) may then be inserted in Eqs. (3) and (4) to define the behaviour of the system. While these functions have been derived from an analysis of data from the settling of particles of different sizes but of the same density, the application of them is expected to be more general. This expectation is, of course, subject to delineation of the effects, particularly on b, of ratios of particle sizes and densities. The utility of this particular form of model is restricted to systems in which the particles are settling at small values of Reynolds number. It is based on a solution of the equations of motion for slow flow. Eqs. (13) and (14) are linear in the single-particle velocities. So simple a form cannot describe the behaviour of systems in which inertial effects in the flow are substantial. In the correlation of Richardson and Zaki [2] for settling of a monodisperse suspension, allowance for the emergence of inertial effects with increasing Reynolds number is in the form of a dependence of the exponent N in Eq. (19) on the single-particle Reynolds number. This form of adjustment is used by Lockett and Al-Habbooby in their model of two-species settling. No correspondingly simple variation of the functions in Eqs. (17) and (18) is permissible.
6. Conclusions
A model for estimation of the settling velocities of particles of two sizes from suspension is presented. It satisfies conditions specified for matching of pressure gradients and velocity profiles in the flow of fluid about the particles and is based on an analytical solution of the equations of motion for slow flow. The expression for calculation of the settling velocity contains two functional terms. The first of these represents a mean settling velocity for all of the species in the system while the second provides an adjustment for the differential velocity of the particular species. Numerical coefficients for the terms in the governing formulae, Eqs. (13) and (14), are
obtained from experimental results and are expressed by Eqs. (17) and (18) as functions of the concentrations of the species in the suspension. The function in Eq. (18) is the manifestation of the requirement for matching of fluid velocity profiles between particles of different species. It is well identified from experimental data but shows a loss of influence as the difference between the sizes of the particles becomes small. Definition of this effect requires further experimental work with very sharply defined size fractions. Because of its essentially mechanistic derivation, this model is expected to extend usefully to the description of differential settling in more complex systems containing particles of several sizes and densities.
7. List of symbols
a b c g k N p u0 v z
average velocity of fluid within the envelope relative to the particle maximum in the profile of velocity of flow relative to the particle volume fraction of a solid species in the suspension gravitational acceleration volume fraction of a solid species within its fluid envelope exponent in Eq. (19) dynamic pressure in fluid terminal settling velocity of a single particle settling velocity of a solid species vertical coordinate
Greek letters
a b g r
coefficient in Eq. (9) coefficient in Eq. (10) function in Eq. (11) density of fluid or solid
Subscripts 1 2 i
first solid species second solid species ith solid species
Appendix A The data of Story [12] and Thomson [13] for glass spheres settling in aqueous glycerol are given in Table A1.
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Table A1 Coarse 252 mm Fine 129 mm
Coarse 252 mm Fine 185 mm
u0s4.19 mm/s u0s1.09 mm/s
u0s4.19 mm/s u0s2.27 mm/s
cCqcF
cC
cF
vC (mm/s)
vF (mm/s)
cCqcF
cC
cF
vC (mm/s)
vF (mm/s)
0.062 0.125 0.187 0.250 0.312 0.374
0.050 0.100 0.150 0.200 0.250 0.300
0.012 0.025 0.037 0.050 0.062 0.075
3.23 2.12 1.68 0.82 0.55 0.29
0.63 0.46 0.30 0.16 0.12 0.07
0.062 0.125 0.187 0.250 0.312 0.374
0.050 0.100 0.150 0.200 0.250 0.300
0.012 0.025 0.037 0.050 0.062 0.075
2.46 1.91 0.98 0.62 0.33
1.49 1.09 0.62 0.42 0.23
0.066 0.131 0.197 0.262 0.328 0.393
0.044 0.087 0.131 0.175 0.218 0.262
0.022 0.044 0.066 0.087 0.109 0.131
2.32 1.38 0.83 0.53 0.29
0.51 0.24 0.17 0.12 0.07
0.066 0.131 0.197 0.262 0.328 0.393
0.044 0.087 0.131 0.175 0.218 0.262
0.022 0.044 0.066 0.087 0.109 0.131
2.36 1.68 0.82 0.46 0.25
1.64 1.09 0.55 0.34 0.19
0.062 0.125 0.187 0.250 0.312 0.374
0.031 0.062 0.094 0.125 0.156 0.187
0.031 0.062 0.094 0.125 0.156 0.187
3.98 2.15 1.86 1.01 0.66
1.13 0.53 0.39 0.21 0.14
0.062 0.125 0.187 0.250 0.312
0.031 0.062 0.094 0.125 0.156
0.031 0.062 0.094 0.125 0.156
2.97 2.14 1.36 1.04 0.56
2.22 1.55 1.02 0.77 0.43
0.066 0.131 0.197 0.262 0.328 0.393
0.022 0.044 0.066 0.087 0.109 0.131
0.044 0.087 0.131 0.175 0.218 0.262
2.48 1.38 1.06 0.44 0.25
0.61 0.30 0.23 0.10 0.08
0.066 0.131 0.197 0.262 0.328 0.393
0.022 0.044 0.066 0.087 0.109 0.131
0.044 0.087 0.131 0.175 0.218 0.262
2.22 1.35 0.77 0.51 0.29
1.51 0.94 0.55 0.38 0.23
0.062 0.125 0.187 0.250 0.312 0.374
0.012 0.025 0.037 0.050 0.062 0.075
0.050 0.100 0.150 0.200 0.250 0.300
0.062 0.125 0.187 0.250 0.312 0.374
0.012 0.025 0.037 0.050 0.062 0.075
0.050 0.100 0.150 0.200 0.250 0.300
2.06 1.39 0.93 0.56 0.34
1.22 0.96 0.60 0.38 0.26
Coarse 252 mm Fine 60 mm
1.35
0.33
0.63 0.31
0.15 0.09
Coarse 185 mm Fine 60 mm
u0s4.19 mm/s u0s0.24 mm/s
u0s2.27 mm/s u0s0.24 mm/s
cCqcF
cC
cF
vC (mm/s)
vF (mm/s)
cCqcF
cC
cF
vC (mm/s)
vF (mm/s)
0.062 0.125 0.187 0.250 0.312 0.374
0.047 0.094 0.140 0.187 0.234 0.281
0.016 0.031 0.047 0.062 0.078 0.094
2.07 1.39 0.85 0.54 0.28
y0.18 y0.22 y0.22 y0.15 y0.09
0.062 0.125 0.187 0.250 0.312 0.374
0.047 0.094 0.140 0.187 0.234 0.281
0.016 0.031 0.047 0.062 0.078 0.094
1.22 0.81 0.61 0.29 0.15
y0.04 y0.05 y0.07 y0.04 y0.02
0.062 0.125 0.187 0.250 0.312 0.374 0.468
0.031 0.062 0.094 0.125 0.156 0.187 0.234
0.031 0.062 0.094 0.125 0.156 0.187 0.234
3.26 2.32 1.67 0.91 0.51 0.25
0.26 0.03 y0.16 y0.11 y0.08 0.09
0.062 0.125 0.187 0.250 0.312 0.374
0.031 0.062 0.094 0.125 0.156 0.187
0.031 0.062 0.094 0.125 0.156 0.187
1.11 0.79 0.52 0.29 0.15
0.04 y0.01 y0.02 y0.02 y0.01
0.062 0.125 0.187 0.250 0.312 0.374 0.437
0.016 0.031 0.047 0.062 0.078 0.094 0.109
0.047 0.094 0.140 0.187 0.234 0.281 0.328
y0.02 y0.04 y0.02
0.062 0.125 0.187 0.250 0.312 0.374
0.016 0.031 0.047 0.062 0.078 0.094
0.047 0.094 0.140 0.187 0.234 0.281
1.15 0.85 0.55 0.28 0.19
0.08 0.04 0.02 0.00 0.01
0.88 0.73 0.24
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References [1] M.J. Lockett and H.M. Al-Habbooby, Trans. Inst. Chem. Eng., 51 (1973) 281. [2] J.F. Richardson and W.N. Zaki, Trans. Inst. Chem. Eng., 32 (1954) 35. [3] S. Mirza and J.F. Richardson, Chem. Eng. Sci., 34 (1979) 447. [4] J.H. Masliyah, Chem. Eng. Sci., 34 (1979) 1166. [5] V.S. Patwardhan and C. Tien, Chem. Eng. Sci., 40 (1985) 1051. [6] M.S. Selim, A.C. Kothari and R.M. Turian, AIChE J., 29 (1983) 1029.
[7] K.O.L.F. Jayaweera, B.J. Mason and G.W. Slack, J. Fluid Mech., 20 (1964) 121. [8] R. Johne, Chem. Ing. Tech., 38 (1966) 428. [9] T.N. Smith, Trans. Inst. Chem. Eng., 43 (1965) T69. [10] T.N. Smith, Trans. Inst. Chem. Eng., 45 (1967) T311. [11] J. Happel, AIChE J., 4 (1958) 197. [12] M.J. Story, B.E. (Hons.) Thesis, University of Adelaide, Australia, 1963. [13] R.W. Thomson, B.E. (Hons.) Thesis, University of Adelaide, Australia, 1963.
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