Settling Velocity
Content
Pergamon
Chemical Engineering Science, Vol. 53, No. 2, pp. 315 323, 1998
PII: S0009-2509(97)00285-6
~; 1997 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0009-2509/98 $19.00 + 0.00
A model of settling velocity
Terence N. Smith School of Chemical Engineering, Curtin University of Technology, Perth 6001, Australia
(Received 30 January 1997; accepted 16 May 1997) Abstract--Attention is drawn to the lack of a successful theoretical model for calculation of the effect of solids concentration on the settling velocity of particles from suspension. The principal challenge to formulation of a model is solution of the equations of fluid motion through an ensemble of solid particles which are randomly distributed to space. A new theoretical model which addresses this difficulty is presented. It is based on permeation of the fluid through a structure composed of cells each of which consists of a single spherical particle and an associated body of fluid. The quantity of fluid in each cell is a random variable. Using this model, the settling velocity of a suspension at substantial volume fraction can be calculated without resort to empirically determined factors. Good agreement with experimental settling velocities is obtained. © 1997 Elsevier Science Ltd
Keywords: Settling velocity; particles; suspension; random array.
INTRODUCTION The settling velocity of solid particles from suspension in fluids is a quantity of fundamental importance to the design of engineering processes and plant. Regrettably, this importance is not matched by the availability of analytically based formulae for calculation of the velocity except in the simplest of cases. Classical analysis provides the result for settling of a single spherical particle in slow flow. It does not, however, extend to the emergence of inertial effects in flow of the fluid at higher Reynolds Number. Estimates of settling velocity must be obtained from correlations of experimental data. Neither does analysis extend successfully to the effect of substantial volume fraction of the solids on the settling velocity of a particle. To estimate the settling velocities of particles from concentrated suspensions, it is necessary, even in slow flow, to resort to empirical correlations. This situation does not constitute a serious deprivation for simple applications such as the design of equipment for the settling of uniformly sized particles. The correlations available are quite adequate for this purpose. For more complex cases, however, such as the settling of multidisperse suspensions, the lack of a fundamental theory is a serious limitation. While several correlations and procedures have been proposed, they do not enjoy the security of a mechanistic basis and, consequently, may not be applied with confidence beyond the range of the particular experimental data from which they have been derived. This paper presents a model from which a settling velocity for a monodisperse suspension at substantial volume fraction can be calculated without resort to
empirically determined factors. It is restricted, in this development, to slow flow but is, perhaps, illustrative of an approach with more general applicability. SINGLE PARTICLEIN SLOW FLOW A single particle of solid material settling under gravity through a fluid reaches a steady value of velocity determined by the balance between its weight and the resistance to its motion through the fluid. In slow flow, where the magnitude of the inertial effects is negligible in comparison with that of the viscous effects, the equation of motion of a simple, Newtonian fluid relative to the particle is VP = #V2u. (1)
Solution of this equation for flow of fluid relative to an isolated spherical particle of diameter d yields a pattern of normal and shear stresses on the surface of the sphere which may be integrated to give the traction as the familiar Stokes force
F = 3nd#w
(2)
in which w is the velocity of the sphere relative to that of the undisturbed fluid. The terminal settling velocity of the sphere is given by the formula
Wo d2(pe -- PF)g
18#
(3)
Beyond the limiting value, about 0.5, of the Reynolds Number of flow to which this analytical result applies, the terminal velocity may be reckoned 315
316 from the formula [~ 1 d ( p e - P F ) g ] 1/2 CD PF A
T. N. Smith the velocity of displaced liquid as (4)
Wo :
WR
W
--
WC
W
1 -
c
1 -
c
(7)
in which the drag coefficient, Co, is an empirically defined function of the Reynolds Number of the flow of fluid relative to the particle. For a particle which is not spherical, CD is a function also of the shape of the particle and of the orientation of the particle to the direction of the flow of fluid.
EXPECTATIONS OF EFFECT OF CONCENTRATION
This value is identical with the relative velocity of fluid flowing at superficial velocity w through a bed of solids at concentration c.
Flow through porous beds
An alternative form of correlation of velocities in fixed and fluidized beds of solid particles is based on the formula of Kozeny (1927) for slow flow through fine pores. Expressed in terms of the particle size of the solids and of the porosity of the bed formed by them, the velocity of the fluid is related to the motive pressure gradient by v
The results in eqs (3) and (4) apply to the movement of a single particle through a body of fluid of infinite extent, undisturbed by any other motion. Where particles of solid are situated in close proximity, as in the settling of a suspension or in the fluidization of a bed of particles, the pattern of flow of fluid about the individual particle is affected by those of its neighbours. In consequence, the settling velocity or, more generally, the relative velocity between the fluid and the particle differs from the result represented by eq. (3) or eq. (4). It is changed by a factor which depends on the volume fraction of the solid particles in the suspension. It can be supposed that the factor might be obtained as a function of the volume fraction and the Reynolds Number of flow of the single particle such that
W
-
=!1-. a_2e)2]vP 3 7 K L36(1 -
(8)
in which K is an experimentally determined constant.
THEORETICAL MODELS
Particle-particle interactions
There have been many attempts to develop theoretical models which account for the effect of volume fraction on settling velocity. The approaches followed by various workers in formulation of the model and in solution of the equations of motion are discussed in some detail by Happel and Brenner (1965) and by Batchelor (1972). A particular complication in formulation of a model is the necessity to incorporate a realistic spatial distribution of the particles. Settling is a dynamic process. Freely settling particles move relatively to each other. Visual observations and statistical analysis by Smith (1968) of the locations of spheres settling in slow flow at a volume fraction of 0.025 show random occupation of elements of space by the spheres. Most of the models which deal with this complication are restricted in application to sparse concentrations of particles. The flow of fluid relative to the particles is visualized as a composite of the interacting fields of flow about the widely separated, individual spheres. Proceeding from a specified distribution of the particles to space, techniques of linear superposition of the flows about each particle are applied to express the interactions and so to obtain approximate solutions. The limitation to very small concentrations arises because the fundamental solution to the flow about a particle in an infinite medium is unbounded. In accounting for the interactions of many particles, the effects of those at great distances must eventually be discounted. For a random dispersion of particles to space, Batchelor (1972) finds the result w = Wo(1 - 6.55c) correct to the order of the concentration, c. (9)
=fn[NRE, C].
(5)
Wo
EMPIRICAL CORRELATIONS
Fluidization and settling
Several empirical formulae have been derived to describe the relationship between settling velocity and volume fraction. Richardson and Zaki (1954) offer the relatively simple form
w = wo(1 -
c) u
(6)
in which N is a function of the Reynolds Number of flow, dwop/#. This correlation is commonly used for estimation of velocities in settling and fluidization of solids at substantial spatial concentrations. It is appropriate, for developments which follow, to establish the equivalence between the settling velocity of a suspension of solid particles and the superficial velocity of the fluid through a bed of fluidized particles. This is best shown by consideration of the value of the relative velocity between fluid and solid in each case. In batch settling, that is, settling of a suspension in a vessel with a closed bottom, there is, across any fixed horizontal plane, an upward flow of fluid equal to the downward flow of solids. The velocity, wn, of the solids relative to the fluid is given by the difference between the downward velocity of the particles and
A model of settling velocity
Labyrinths and cell models
317
Evaluation of the interactions between the flows about individual particles becomes very difficult at substantial concentrations. First order interactions are no longer sufficient to express the effects on the flows of the profiles of neighbouring particles. To obtain solutions of the equation of motion over the profile of each particle, it is necessary to impose a degree of regularity on the structure of the dispersion. This allows boundary conditions to be prescribed at the interfaces between the units or cells of the structure so that closed solutions can be obtained for the flow about each particle or group of particles. Several such models of flow through labyrinthine or cellular structures have been developed. Happel (1958) utilizes a structure in which the solid, spherical particle is at the centre of a spherical envelope of fluid. The geometry of the structure is such that, while the spacing is even, there is no requirement for alignment of adjacent particles. A condition of zero tangential shear stress is prescribed on the surface of the cell. This specification precludes interaction between adjacent cells by such stresses and confines the effects between the particle and the fluid to the cell. The slow flow equation, VP = #V2u (10)
With the assumption that the spherical fluid envelopes may distort in order to fill space completely without substantial effect on the settling velocity, eq. (11) gives a result for the settling velocity of a suspension in which the particles are evenly spaced but have no specific spatial order. DEFICIENCIESOF MODELS
Random allocation to space
is then solved using spherical harmonics to obtain the settling velocity as v
-
3 - (9/2)7 + (9/2)75 - 31,6
=
w0
3 + 275
(11)
in which 7 is the ratio of the diameter of the solid particle to the diameter of the cell. That is,
7 = cl/3.
(12)
While the models for very small concentrations have been applied to dispersions in which the particles are randomly located, those for substantial concentrations have not been successful in dealing with geometrical irregularity. The elegant model of Happel (1958) shows the product of this deficiency. It yields settling velocities which are much slower than observed values. Figure 1 compares the theoretical settling velocity calculated from eq. (11) with eq. (6), the correlation of experimental settling velocities of Richardson and Zaki (1954). Only at the largest concentrations does the model reflect actual behaviour. The theoretical and experimental velocities in Fig. 1 reflect the difference between the resistances to flow of fluid presented by particles in a regular array and by particles situated randomly in space. For smaller concentrations, an explanation may be offered in terms of lessened resistance when particles approach to form clusters. For greater concentrations, an explanation may be given in terms of a widening of the passages through which fluid flows when particles are displaced from regularity of spacing. It is well known that the resistance of a pair of particles in close proximity is less than twice that of a single particle. Stimson and Jeffrey (1926) obtain a solution for the motion of two touching spheres along the line of centres which gives a reduction of
1 0,9 0.8 --'O--R & Z 0.7
I
.>,
0.6
>
•--13-- HAPPEL ]
0.5
0.4
(n
0.3 0.2 0.1 0 0 0,1 0.2 0.3 Concentration 0.4 0.5
Fig. 1. Settling of suspensions with random and regular spatial distributions.
318
T. N. Smith
A NEW MODEL
resistance by a factor of 0.645. Wakiya (1957) obtains 0.694 for motion across the line of centres. Resistances of pairs of spheres separated by various distances along axes orientated at angles to the direction of motion are analysed and discussed by Happel and Brenner (1965). These authors also discuss the procedures for obtaining solutions for the resistances of groupings of more than two spheres and refer to some approximate results. The resistance of each sphere decreases with increasing number in the group. The rate of flow of liquid through a pipe or pore of radius r is given by the Hagen-Poiseuille formula
7[?"4
Q = -~-p VP.
(13)
If the total sectional area of a porous medium with n uniformly sized pores, n~r2/4, is redistributed by displacement of the solid particles forming the wails of the pores so that the areas of some pores are decreased and the areas of others are increased, the effective value of the term r 4 in eq. (13) is increased. Consequently, the rate of flow through the medium is increased by this disproportionate contribution of the larger pores.
The model proposed in this paper is applicable to the settling of spherical particles of uniform size dispersed randomly in space. The flow of fluid relative to the particles is a permeation under slow flow through the interstitial volume. Each solid particle is located at the centre of a spherical cell in which it presents the resistance to flow of fluid through the cell. The flow of fluid through the cell is driven by the pressure gradient imposed on the suspension by the weight of the solid particles. The flow through each cell depends on the permeability of the cell itself and also on that of the surrounding medium. The superficial velocity of fluid through the ensemble of cells is identified with the settling velocity of the suspension of particles.
Spatial distribution of particles
The allocation of particles to space is based on a model of the probability of intrusion upon the province of a neighbouring particle as successively more distant elements of fluid in the space surrounding an object particle are searched. Figure 2 depicts the space surrounding a particle as a series of shells composed of small, equal elements of volume. While Fig. 2 shows a placement of elements of fluid space around the object particle, there is no requirement for a particular geometry. The probability is that of association of an element with the particle, not of occupation of a specific location. The probability of no encounter of a second particle in successive inspections of t elements of fluid volume is the probability that the space between neighbouring particles consists of a volume at least as great as that. It is given by the single-event Poisson distribution or exponential distribution as
Increase in velocity at low concentrations
Models such as that of Batchelor (1972) for settling of particles at low concentrations do account, to some degree, for the effects of random dispersion to space. There is evidence, however, that the scale of the allowance expressed by eq. (9) is insufficient. Experiments reported by Kaye and Boardman (1962), by Johne (1966) and by Koglin (1971, 1973) show that the settling velocity of particles at very low volume fractions may actually exceed the single-particle value. The settling velocity of spherical particles in slow flow is found to rise from Wo for the single particle to reach a maximum value of about 1.5Wo at a volume fraction of about 0.01. Koglin (1971) concludes that this behaviour arises from dynamic effects which cause transitory formations of particles such as those observed by Jayaweera et al. (1964) and by Crowley (1971) rather than from the exercise of interparticular forces which might cause permanent aggregations. The evidence shows that, as the volume fraction of a suspension increases from zero, the settling velocity of particles first increases with volume fraction owing to a 'clustering' effect but then decreases as the effect of resistance to flow of fluid between the particles becomes dominant. The argument supporting this observation of settling velocities which exceed the value for a single particle is that, at very small concentrations, the effect of the proximity of a neighbouring particle is to decrease the resistance of the pair of particles to a greater degree than the increase in resistance apparent in a regular array at that concentration. It is the random displacement or deviation from the regular position and spacing which gives rise to this result.
p(t) = e x p ( - 2 t )
(14)
in which 2 is the mean probability of encounter at a single inspection. Evidently, p=l for t = 0 , p=0 for t ~ o c .
The interpretation in this model of an encounter with a neighbouring particle after the addition of t elements
1111111111111
,,llmll,,
Q
IIIIIIIIIII IIIIIII Ill
Fig. 2. Volume surrounding a particle.
A model of settling velocity of volume is that these elements are shared equally with the neighbour. Each of the particles is surrounded by t/2 elements of volume. This process creates two cells of the same concentration simultaneously. It does not, however, lead to bias since each trial is independent and the final result is the creation of two complete, identical populations. The maximum value of the volume fraction is that at which the solid spheres make contact with each other. The number of elements of volume occupied by the particle and its associated fluid at this maximum concentration, CM, is m. The volume fraction of solid in the cell consisting of the sphere, its associated fluid at maximum concentration and the t/2 additional elements of volume is given by c
CMm
319
which is then used to obtain the fraction of space occupied as
f(c)/c f (s) = Z [ f (c)/c~] "
(22)
The suspension is composed of particles of solid contained in cells distributed randomly to space with the frequency distribution described by eq. (20).
m + t/2"
(15)
The number of elements of volume, t, is related to volume fraction by t = 2 cM m - 2m.
C
(16)
The individual cell The cell containing the individual solid particle is of the kind developed by Happel (1958). The solid, spherical particle is at the centre of a spherical cell and the fluid passes through the cell in the x-direction as illustrated in Fig. 3. A condition of zero tangential shear stress is prescribed on the surface of the cell. This specification precludes interaction between adjacent cells by such stresses and ensures hydrodynamic balance between the particle and the fluid within its containing cell. The term Wo in eq. (11) is the single-particle settling velocity. For more general application of this solution of this equation of motion, Wo may be substituted from eq. (3) expressed in terms of the pressure gradient VP = (pv - pe)gc
(23)
The value of t corresponding to the mean value, g, of the concentration of the solids in suspension provides the value of 2, the frequency parameter in eq. (14), from the identity
1
2
f=2
C,I
""m
2m.
(17)
induced in the fluid to balance the weight of the settling particles. This gives the single-particle velocity as d 2 VP Wo = - - - - . (24) 18# c Equation (11) may then be written as v= 3 - (9/2)7 + (9/2)75 - 376 d 2 -VP. 3 + 275 18#c (25)
The value of 2t to provide the probability associated with the concentration c is then 2t and that probability is p(c)=exp[
(CM/C)- 1 (CM/g)- 1
(18)
(CM/g)(CM/C)--~]"
(19)
Permeability of a cell The objective of this development is to obtain a value of a permeability to flow of fluid of the medium formed by the randomly located solid particles.
.... ..........
.
The contact concentration, cu, in this formula must be given a value appropriate to the circumstances. The maximum value attainable is 0.7405, corresponding to rhombohedral close packing of spheres. A simple cubic array gives 0.5236. The probability p(c) is a cumulative frequency of particles with cell concentrations less than or equal to the value c. The frequency distribution is obtained by differentiation of this function. Equation (20) gives the frequency of the individual particles or of the cells containing them which fall within the discrete interval of concentration 6c centred on the value c as
....'"
""..,,,
y'"
"',,,
0i
".,,,
,' j... ......... ...
i
i
J ©
",
"', '.. ...
,,,"
i
.............
'..,.
Shear stress at surface of cell is zero
f (c) = p(c + 6c/2) - p(c - 6c/2).
(20)
The space in the suspension occupied by cells within this particular band of concentration is found from
.........................Flow of fluidthrough cell Fig. 3. Contiguous cells.
s(c) = f (c)/c
(21)
320
T. N. Smith
Settlin9 velocity It is appropriate now to review the model in terms of its validity as a reflection of the physical processes to which it is to be applied. Justification is sought for its use in the calculation of velocities in flow of fluids through porous media, in fluidization of beds of particles and in settling of particles from suspension. Flow of a fluid through the cells of the medium is represented by an average value of the superficial velocity. The velocity through any individual cell depends on its permeability. Larger cells, in which the fraction of volume occupied by the solid particle is small, have larger permeabilities and allow greater velocities. In smaller cells, where the solid occupies a larger fraction of the volume, the permeabilities are smaller and the velocities are smaller. The solid particles remain stationary and the fluid passes through each cell at a velocity which varies with the size of the cell. The model presented in this paper'is evidently applicable to the case of flow through a porous medium. The solid particles in a homogeneously fluidized bed are not stationary but are in constant motion. The movements are, however, restricted to a relatively short scale of length by collisions with other particles in the closely populated space of the bed. There is no sustained passage of individual particles through the bed. There is a dynamic equilibrium of short motions of the particles which presents to the stream of fluid a field of particles randomly allocated to space. This corresponds with the nature of the suggested model. In settling of particles through a fluid, there is also a dynamic equilibrium in the spatial arrangement. As observed by Kaye and Boardman (1962), there is a transient formation of pairs and larger groups of particles which settle with greater velocities than individual particles. This effect leads to settling, at small volume concentrations of solids, at velocities which exceed the single-particle value. With increasing concentration, the effect is less significant because of the greater frequency of collisions with neighbouring particles. Indeed, at concentrations approaching those of fluidized beds of solids, there can be no sustained differentiation of settling velocities between particles. The interaction between fluid and particles is of the kind embodied in the suggested model. To estimate the settling velocity using this model, the distribution of cell sizes for the particular concentration of the suspension and the corresponding values of f(s) and k must be calculated from eqs (18)-(20), (22) and (27). Values of k' and k are then found using eqs (28) and (29). Then the velocity is calculated from eq. (26) using the vaue of k and the pressure gradient from eq. (23) using the mean concentration to give
Restricting the model to slow flow, the permeability, k, is defined by the relationship
v = kVP/l~
(26)
in which v is the superficial velocity of the fluid of viscosity/~ induced by VP, the motive pressure gradient. The permeability of the independent cell is then obtained from eqs (25) and (26) as k= 3 - (9/2)7 + (9/2)7 s - 376 d 2 3 + 2y s 18~-~" (27)
The shape of the cell for which the permeability is obtained is spherical. Spheres do not, of course, pack to fill space entirely. It is assumed that, with only a minor effect on the permeability, the spherical surface of any cell can be subjected to a degree of distortion sufficient to allow space to be filled by a group of contiguous cells. Because of the specification of zero shear stress at the fluid boundary of the cell, this requirement is considered to be reasonable. To proceed with development of the model, a permeability for each cell in the medium is required. These must then be combined so as to define the composite permeability of the whole medium. The flow through an individual cell within a heterogeneous medium composed of several types of cells is not simply the product of the permeability of the cell and the general pressure gradient in the medium. It depends also on the permeability of the surrounding medium since fluid must pass to and from the cell through that medium. By analogy with the result given by Carslaw and Jaeger (1959) for the thermal or magnetic flux through a spherical cell of permeability k embedded in a medium of permeability k, the effective permeability k' of a cell is obtained as k' 3k k. 2k + k (28)
It is evident from eq. (28) that the permeability of the surrounding medium restricts the flow which can be attained through the individual cell. The maximum value of the effective permeability is 3k, three times the permeability of the medium.
Flow through the medium The permeability of a medium composed of an ensemble of randomly placed cells with various values of permeability is obtained by summing the products of the permeability and the fraction of space occupied by each of the various species of cells so that = Zf(s)k'.
(29)
Since the value of k is required for calculation of each of the individual effective permeabilities, eqs (28) and (29) must be solved simultaneously to obtain this result. The superficial velocity of flow of fluid through the medium is found by insertion of the permeability from eq. (29) into eq. (26).
w --
k(Pl" -- P~)gc kt
(30)
A model of settling velocity PERFORMANCE OF MODEL
321
Theoretical settlin 9 velocity
Settling velocities calculated from the model for a range of concentrations up to 0.5, which may be regarded as the practical limit in settling, are presented in Table 1. These are obtained using a value of cM in eq. (19) of 0.7405, the figure for rhombohedral close packing, to generate the spatial distribution. This figure is chosen because it allows the probability that neighbouring particles might approach to ultimate proximity. Substitution of smaller values, down to 0.5, has only a very slight effect on the shape of the lowconcentration end of the spatial distribution. It is this part of the distribution which presents the greatest permeability to flow of fluid. Accordingly, the computed value of settling velocity has very little sensitivity to the value of CM. Also shown in Table 1 are velocities calculated from the theoretical model of Happel (1958) as presented in
eq. (11). This model may be regarded as equivalent to the new model but with a regular spacing of particles rather than a random dispersion to space. The difference between the two models is obvious. The resistance to flow of fluid through a matrix of solid particles is maximized by spacing the particles evenly. The velocities for the model of random placement reflect the reduction of the individual resistances of particles by the mechanisms previously discussed.
Comparison with experiments
Figure 4 shows a comparison of the calculated settling velocities with experimental values as represented by the correlation of Richardson and Zaki (1954) given in eq. (6). Correspondence of the results of the model with experimental settling velocities is good for concentrations of 0.05 and greater. This is the range of settling and fluidization experiments from which the correlation is derived.
Table 1. Theoretical settling velocities Concentration 0.0 0.01 0.02 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 Settling velocity model 1.701 1.222 1.086 0.841 0.596 0.432 0.313 0.225 0.160 0.111 0.075 0.050 0.032
1.8 1.6 1.4
>,
Velocity at small volume fraction
For concentrations less than 0.05, the particular experiments of Kaye and Boardman (1962), Johne (1966) and Koglin (1971, 1973) must be accepted as a more appropriate depiction of the relationship between settling velocity and concentration. The results of Koglin (1973) are indicated in Fig. 5 together with the correlation of Richardson and Zaki from eq. (6) and those of the model. As the volume fraction of solids falls below 0.03, the model shows an elevation of settling velocity above the single-particle value. The emergence of this effect at concentrations at this level is in correspondence with the experimental observations. However, as the concentration reduces to zero, the model maintains a settling velocity of about 1.7 times greater than that of the single particle. This deviation is associated with
Settling velocity equation (11) 1.000 0.677 0.594 0.453 0.322 0.237 0.177 0.133 0.099 0.073 0.053 0.038 0.026
1.2 1
"-'O--R & Z ] MO DEL
> 0t
0.8 0.6 0.4 0.2 0 0.1 0.2 0.3 0.4 0.5 Concentration
¢1
Fig. 4. Theoretical and experimental settling velocities.
322
1.8
T.N. Smith
I
--"O--" R & Z
MODEL ---O-- E X P T
1.6
1,4
>,~ 1,2
.S
0.8
0.6
0.4 0.2
0
0.01
0.02
0.03
0.04
0.05
Concentration
Fig. 5. Settling velocity at very small concentration.
a departure of the model from reality at very small concentrations. Where particles are very widely spaced, differential velocities between neighbouring particles can be maintained over long distances and times. The model does not permit this. Fluid passes through a matrix of cells each of which contains a particle. The cells do not move relatively to one another. Accordingly, the model is appropriate only when the concentration of solids is sufficiently great to present impediment by collision to the sustained relative movement of particles. At a concentration of 0.1, the distance between centres of particles is 2.15 particle diameters. At a concentration of 0.01, the distance between centres of particles is 4.64 particle diameters and at a concentration of 0.001, the distance is 10.00 particle diameters. It may be inferred from these figures that the model would be effective at a concentration of 0.1, approximate at 0.01 but ineffective at 0.001. The model does, however, clearly reflect the experimentally observed tendency to settling velocities greater than the single-particle value at small concentrations.
CONCLUSION
The model also reflects the observed tendency to settling velocities greater than the single-particle settling velocity at concentrations of about 0.01. The model does, however, lose validity at very small concentrations where the mean distance between particles is several times the particle diameter so that sustained differential movement of neighbouring particles becomes possible.
NOTATION
A theoretical model which permits calculation of the velocity of uniformly sized solid particles settling in slow flow through fluids is presented. It is based on permeation of the fluid through the interstitial space between spherical particles randomly distributed to space. The model is constructed with governing equations which require no substitution of empirically determined factors. Velocities obtained from the model correspond well with correlations of experimental settling velocities and fluidization velocities in the practically important range of concentration from 0.1 to the contact concentration of the particles.
volume fraction of solids in the suspension mean volume fraction cM maximum value of volume fraction C~ drag coefficient of the particle in the fluid d diameter of the spherical particle f(c) fraction of cells of concentration c f(s) fraction of spatial volume occupied by cells with concentration c F drag force on a particle 9 gravitational acceleration k permeability to flow permeability of the composite medium k' effective permeability of a cell K Kozeny constant m number of elements of fluid volume n number of pores N exponent in Richardson and Zaki formula NRE Reynolds of flow of a single particle p probability p(c) probability that concentration does not exceed c p(t) probability of no encounter in t trials VP gradient of pressure Q rate of flow through a pore r radius of pore s(c) volume of space occupied by cells of concentration c t number of elements of fluid volume f mean value of t
c
A model of settling velocity u v w w0
WR
323
vector velocity of fluid superficial velocity of flow of fluid settling velocity of particles single-particle settling velocity velocity of settling particles relative to fluid ratio of particle diameter to cell diameter porosity parameter in probability function viscosity of fluid density of fluid density of solid particle
REFERENCES
Greek letters
7 e 2 #
PF
pp
Batchelor, G. K. (1972) J. Fluid Mech. 52, 245. Carslaw, H. S. and Jaeger, J. C. (1959) Conduction of Heat in Solids, 2nd Edn, p. 426. Oxford University Press, London, U.K. Crowley, J. M. (1971) J. Fluid Mech. 45, 151.
Happel, J. (1958) A.I.Ch.E.J. 4, 197. Happel, J. and Brenner, H. (1965) Low Reynolds Number Hydrodynamics, Prentice-Hall, Englewood Cliffs, NJ, U.S.A. Jayaweera, K. O. L. F., Mason, B. J. and Slack, G. W. (1964) J. Fluid Mech. 20, 121. Johne, R. (1966) Chem. lng. Technik 38, 428. Kaye, B. H. and Boardman, R. P. (1962) Symposium on Interaction between Fluids and Particles, p. 17. Instn Chem. Engrs, London, U.K. Koglin, B. (1971) Chem. Ing. Technik 43, 761. Koglin, B. (1973) In: Proc. 1st Int. Conf. Particle Technol., p. 266, IIT Research Institute, Chicago. Kozeny, J. (1927) Ber. Akad. Wiss. Wien Abt. Ila 136, 271. Richardson, J. F. and Zaki, W. N. (1954) Trans. Instn Chem. Engrs 32, 35. Smith, T. N. (1968) J. Fluid Mech. 32, 203. Stimson, M. and Jeffrey, G. B. (1926) Proc. Roy. Soc. A 111, 110. Wakiya, S. (1957) Res. Report No. 6, Niigata Univ. Coll. Eng.
Chemical Engineering Science, Vol. 53, No. 2, pp. 315 323, 1998
PII: S0009-2509(97)00285-6
~; 1997 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0009-2509/98 $19.00 + 0.00
A model of settling velocity
Terence N. Smith School of Chemical Engineering, Curtin University of Technology, Perth 6001, Australia
(Received 30 January 1997; accepted 16 May 1997) Abstract--Attention is drawn to the lack of a successful theoretical model for calculation of the effect of solids concentration on the settling velocity of particles from suspension. The principal challenge to formulation of a model is solution of the equations of fluid motion through an ensemble of solid particles which are randomly distributed to space. A new theoretical model which addresses this difficulty is presented. It is based on permeation of the fluid through a structure composed of cells each of which consists of a single spherical particle and an associated body of fluid. The quantity of fluid in each cell is a random variable. Using this model, the settling velocity of a suspension at substantial volume fraction can be calculated without resort to empirically determined factors. Good agreement with experimental settling velocities is obtained. © 1997 Elsevier Science Ltd
Keywords: Settling velocity; particles; suspension; random array.
INTRODUCTION The settling velocity of solid particles from suspension in fluids is a quantity of fundamental importance to the design of engineering processes and plant. Regrettably, this importance is not matched by the availability of analytically based formulae for calculation of the velocity except in the simplest of cases. Classical analysis provides the result for settling of a single spherical particle in slow flow. It does not, however, extend to the emergence of inertial effects in flow of the fluid at higher Reynolds Number. Estimates of settling velocity must be obtained from correlations of experimental data. Neither does analysis extend successfully to the effect of substantial volume fraction of the solids on the settling velocity of a particle. To estimate the settling velocities of particles from concentrated suspensions, it is necessary, even in slow flow, to resort to empirical correlations. This situation does not constitute a serious deprivation for simple applications such as the design of equipment for the settling of uniformly sized particles. The correlations available are quite adequate for this purpose. For more complex cases, however, such as the settling of multidisperse suspensions, the lack of a fundamental theory is a serious limitation. While several correlations and procedures have been proposed, they do not enjoy the security of a mechanistic basis and, consequently, may not be applied with confidence beyond the range of the particular experimental data from which they have been derived. This paper presents a model from which a settling velocity for a monodisperse suspension at substantial volume fraction can be calculated without resort to
empirically determined factors. It is restricted, in this development, to slow flow but is, perhaps, illustrative of an approach with more general applicability. SINGLE PARTICLEIN SLOW FLOW A single particle of solid material settling under gravity through a fluid reaches a steady value of velocity determined by the balance between its weight and the resistance to its motion through the fluid. In slow flow, where the magnitude of the inertial effects is negligible in comparison with that of the viscous effects, the equation of motion of a simple, Newtonian fluid relative to the particle is VP = #V2u. (1)
Solution of this equation for flow of fluid relative to an isolated spherical particle of diameter d yields a pattern of normal and shear stresses on the surface of the sphere which may be integrated to give the traction as the familiar Stokes force
F = 3nd#w
(2)
in which w is the velocity of the sphere relative to that of the undisturbed fluid. The terminal settling velocity of the sphere is given by the formula
Wo d2(pe -- PF)g
18#
(3)
Beyond the limiting value, about 0.5, of the Reynolds Number of flow to which this analytical result applies, the terminal velocity may be reckoned 315
316 from the formula [~ 1 d ( p e - P F ) g ] 1/2 CD PF A
T. N. Smith the velocity of displaced liquid as (4)
Wo :
WR
W
--
WC
W
1 -
c
1 -
c
(7)
in which the drag coefficient, Co, is an empirically defined function of the Reynolds Number of the flow of fluid relative to the particle. For a particle which is not spherical, CD is a function also of the shape of the particle and of the orientation of the particle to the direction of the flow of fluid.
EXPECTATIONS OF EFFECT OF CONCENTRATION
This value is identical with the relative velocity of fluid flowing at superficial velocity w through a bed of solids at concentration c.
Flow through porous beds
An alternative form of correlation of velocities in fixed and fluidized beds of solid particles is based on the formula of Kozeny (1927) for slow flow through fine pores. Expressed in terms of the particle size of the solids and of the porosity of the bed formed by them, the velocity of the fluid is related to the motive pressure gradient by v
The results in eqs (3) and (4) apply to the movement of a single particle through a body of fluid of infinite extent, undisturbed by any other motion. Where particles of solid are situated in close proximity, as in the settling of a suspension or in the fluidization of a bed of particles, the pattern of flow of fluid about the individual particle is affected by those of its neighbours. In consequence, the settling velocity or, more generally, the relative velocity between the fluid and the particle differs from the result represented by eq. (3) or eq. (4). It is changed by a factor which depends on the volume fraction of the solid particles in the suspension. It can be supposed that the factor might be obtained as a function of the volume fraction and the Reynolds Number of flow of the single particle such that
W
-
=!1-. a_2e)2]vP 3 7 K L36(1 -
(8)
in which K is an experimentally determined constant.
THEORETICAL MODELS
Particle-particle interactions
There have been many attempts to develop theoretical models which account for the effect of volume fraction on settling velocity. The approaches followed by various workers in formulation of the model and in solution of the equations of motion are discussed in some detail by Happel and Brenner (1965) and by Batchelor (1972). A particular complication in formulation of a model is the necessity to incorporate a realistic spatial distribution of the particles. Settling is a dynamic process. Freely settling particles move relatively to each other. Visual observations and statistical analysis by Smith (1968) of the locations of spheres settling in slow flow at a volume fraction of 0.025 show random occupation of elements of space by the spheres. Most of the models which deal with this complication are restricted in application to sparse concentrations of particles. The flow of fluid relative to the particles is visualized as a composite of the interacting fields of flow about the widely separated, individual spheres. Proceeding from a specified distribution of the particles to space, techniques of linear superposition of the flows about each particle are applied to express the interactions and so to obtain approximate solutions. The limitation to very small concentrations arises because the fundamental solution to the flow about a particle in an infinite medium is unbounded. In accounting for the interactions of many particles, the effects of those at great distances must eventually be discounted. For a random dispersion of particles to space, Batchelor (1972) finds the result w = Wo(1 - 6.55c) correct to the order of the concentration, c. (9)
=fn[NRE, C].
(5)
Wo
EMPIRICAL CORRELATIONS
Fluidization and settling
Several empirical formulae have been derived to describe the relationship between settling velocity and volume fraction. Richardson and Zaki (1954) offer the relatively simple form
w = wo(1 -
c) u
(6)
in which N is a function of the Reynolds Number of flow, dwop/#. This correlation is commonly used for estimation of velocities in settling and fluidization of solids at substantial spatial concentrations. It is appropriate, for developments which follow, to establish the equivalence between the settling velocity of a suspension of solid particles and the superficial velocity of the fluid through a bed of fluidized particles. This is best shown by consideration of the value of the relative velocity between fluid and solid in each case. In batch settling, that is, settling of a suspension in a vessel with a closed bottom, there is, across any fixed horizontal plane, an upward flow of fluid equal to the downward flow of solids. The velocity, wn, of the solids relative to the fluid is given by the difference between the downward velocity of the particles and
A model of settling velocity
Labyrinths and cell models
317
Evaluation of the interactions between the flows about individual particles becomes very difficult at substantial concentrations. First order interactions are no longer sufficient to express the effects on the flows of the profiles of neighbouring particles. To obtain solutions of the equation of motion over the profile of each particle, it is necessary to impose a degree of regularity on the structure of the dispersion. This allows boundary conditions to be prescribed at the interfaces between the units or cells of the structure so that closed solutions can be obtained for the flow about each particle or group of particles. Several such models of flow through labyrinthine or cellular structures have been developed. Happel (1958) utilizes a structure in which the solid, spherical particle is at the centre of a spherical envelope of fluid. The geometry of the structure is such that, while the spacing is even, there is no requirement for alignment of adjacent particles. A condition of zero tangential shear stress is prescribed on the surface of the cell. This specification precludes interaction between adjacent cells by such stresses and confines the effects between the particle and the fluid to the cell. The slow flow equation, VP = #V2u (10)
With the assumption that the spherical fluid envelopes may distort in order to fill space completely without substantial effect on the settling velocity, eq. (11) gives a result for the settling velocity of a suspension in which the particles are evenly spaced but have no specific spatial order. DEFICIENCIESOF MODELS
Random allocation to space
is then solved using spherical harmonics to obtain the settling velocity as v
-
3 - (9/2)7 + (9/2)75 - 31,6
=
w0
3 + 275
(11)
in which 7 is the ratio of the diameter of the solid particle to the diameter of the cell. That is,
7 = cl/3.
(12)
While the models for very small concentrations have been applied to dispersions in which the particles are randomly located, those for substantial concentrations have not been successful in dealing with geometrical irregularity. The elegant model of Happel (1958) shows the product of this deficiency. It yields settling velocities which are much slower than observed values. Figure 1 compares the theoretical settling velocity calculated from eq. (11) with eq. (6), the correlation of experimental settling velocities of Richardson and Zaki (1954). Only at the largest concentrations does the model reflect actual behaviour. The theoretical and experimental velocities in Fig. 1 reflect the difference between the resistances to flow of fluid presented by particles in a regular array and by particles situated randomly in space. For smaller concentrations, an explanation may be offered in terms of lessened resistance when particles approach to form clusters. For greater concentrations, an explanation may be given in terms of a widening of the passages through which fluid flows when particles are displaced from regularity of spacing. It is well known that the resistance of a pair of particles in close proximity is less than twice that of a single particle. Stimson and Jeffrey (1926) obtain a solution for the motion of two touching spheres along the line of centres which gives a reduction of
1 0,9 0.8 --'O--R & Z 0.7
I
.>,
0.6
>
•--13-- HAPPEL ]
0.5
0.4
(n
0.3 0.2 0.1 0 0 0,1 0.2 0.3 Concentration 0.4 0.5
Fig. 1. Settling of suspensions with random and regular spatial distributions.
318
T. N. Smith
A NEW MODEL
resistance by a factor of 0.645. Wakiya (1957) obtains 0.694 for motion across the line of centres. Resistances of pairs of spheres separated by various distances along axes orientated at angles to the direction of motion are analysed and discussed by Happel and Brenner (1965). These authors also discuss the procedures for obtaining solutions for the resistances of groupings of more than two spheres and refer to some approximate results. The resistance of each sphere decreases with increasing number in the group. The rate of flow of liquid through a pipe or pore of radius r is given by the Hagen-Poiseuille formula
7[?"4
Q = -~-p VP.
(13)
If the total sectional area of a porous medium with n uniformly sized pores, n~r2/4, is redistributed by displacement of the solid particles forming the wails of the pores so that the areas of some pores are decreased and the areas of others are increased, the effective value of the term r 4 in eq. (13) is increased. Consequently, the rate of flow through the medium is increased by this disproportionate contribution of the larger pores.
The model proposed in this paper is applicable to the settling of spherical particles of uniform size dispersed randomly in space. The flow of fluid relative to the particles is a permeation under slow flow through the interstitial volume. Each solid particle is located at the centre of a spherical cell in which it presents the resistance to flow of fluid through the cell. The flow of fluid through the cell is driven by the pressure gradient imposed on the suspension by the weight of the solid particles. The flow through each cell depends on the permeability of the cell itself and also on that of the surrounding medium. The superficial velocity of fluid through the ensemble of cells is identified with the settling velocity of the suspension of particles.
Spatial distribution of particles
The allocation of particles to space is based on a model of the probability of intrusion upon the province of a neighbouring particle as successively more distant elements of fluid in the space surrounding an object particle are searched. Figure 2 depicts the space surrounding a particle as a series of shells composed of small, equal elements of volume. While Fig. 2 shows a placement of elements of fluid space around the object particle, there is no requirement for a particular geometry. The probability is that of association of an element with the particle, not of occupation of a specific location. The probability of no encounter of a second particle in successive inspections of t elements of fluid volume is the probability that the space between neighbouring particles consists of a volume at least as great as that. It is given by the single-event Poisson distribution or exponential distribution as
Increase in velocity at low concentrations
Models such as that of Batchelor (1972) for settling of particles at low concentrations do account, to some degree, for the effects of random dispersion to space. There is evidence, however, that the scale of the allowance expressed by eq. (9) is insufficient. Experiments reported by Kaye and Boardman (1962), by Johne (1966) and by Koglin (1971, 1973) show that the settling velocity of particles at very low volume fractions may actually exceed the single-particle value. The settling velocity of spherical particles in slow flow is found to rise from Wo for the single particle to reach a maximum value of about 1.5Wo at a volume fraction of about 0.01. Koglin (1971) concludes that this behaviour arises from dynamic effects which cause transitory formations of particles such as those observed by Jayaweera et al. (1964) and by Crowley (1971) rather than from the exercise of interparticular forces which might cause permanent aggregations. The evidence shows that, as the volume fraction of a suspension increases from zero, the settling velocity of particles first increases with volume fraction owing to a 'clustering' effect but then decreases as the effect of resistance to flow of fluid between the particles becomes dominant. The argument supporting this observation of settling velocities which exceed the value for a single particle is that, at very small concentrations, the effect of the proximity of a neighbouring particle is to decrease the resistance of the pair of particles to a greater degree than the increase in resistance apparent in a regular array at that concentration. It is the random displacement or deviation from the regular position and spacing which gives rise to this result.
p(t) = e x p ( - 2 t )
(14)
in which 2 is the mean probability of encounter at a single inspection. Evidently, p=l for t = 0 , p=0 for t ~ o c .
The interpretation in this model of an encounter with a neighbouring particle after the addition of t elements
1111111111111
,,llmll,,
Q
IIIIIIIIIII IIIIIII Ill
Fig. 2. Volume surrounding a particle.
A model of settling velocity of volume is that these elements are shared equally with the neighbour. Each of the particles is surrounded by t/2 elements of volume. This process creates two cells of the same concentration simultaneously. It does not, however, lead to bias since each trial is independent and the final result is the creation of two complete, identical populations. The maximum value of the volume fraction is that at which the solid spheres make contact with each other. The number of elements of volume occupied by the particle and its associated fluid at this maximum concentration, CM, is m. The volume fraction of solid in the cell consisting of the sphere, its associated fluid at maximum concentration and the t/2 additional elements of volume is given by c
CMm
319
which is then used to obtain the fraction of space occupied as
f(c)/c f (s) = Z [ f (c)/c~] "
(22)
The suspension is composed of particles of solid contained in cells distributed randomly to space with the frequency distribution described by eq. (20).
m + t/2"
(15)
The number of elements of volume, t, is related to volume fraction by t = 2 cM m - 2m.
C
(16)
The individual cell The cell containing the individual solid particle is of the kind developed by Happel (1958). The solid, spherical particle is at the centre of a spherical cell and the fluid passes through the cell in the x-direction as illustrated in Fig. 3. A condition of zero tangential shear stress is prescribed on the surface of the cell. This specification precludes interaction between adjacent cells by such stresses and ensures hydrodynamic balance between the particle and the fluid within its containing cell. The term Wo in eq. (11) is the single-particle settling velocity. For more general application of this solution of this equation of motion, Wo may be substituted from eq. (3) expressed in terms of the pressure gradient VP = (pv - pe)gc
(23)
The value of t corresponding to the mean value, g, of the concentration of the solids in suspension provides the value of 2, the frequency parameter in eq. (14), from the identity
1
2
f=2
C,I
""m
2m.
(17)
induced in the fluid to balance the weight of the settling particles. This gives the single-particle velocity as d 2 VP Wo = - - - - . (24) 18# c Equation (11) may then be written as v= 3 - (9/2)7 + (9/2)75 - 376 d 2 -VP. 3 + 275 18#c (25)
The value of 2t to provide the probability associated with the concentration c is then 2t and that probability is p(c)=exp[
(CM/C)- 1 (CM/g)- 1
(18)
(CM/g)(CM/C)--~]"
(19)
Permeability of a cell The objective of this development is to obtain a value of a permeability to flow of fluid of the medium formed by the randomly located solid particles.
.... ..........
.
The contact concentration, cu, in this formula must be given a value appropriate to the circumstances. The maximum value attainable is 0.7405, corresponding to rhombohedral close packing of spheres. A simple cubic array gives 0.5236. The probability p(c) is a cumulative frequency of particles with cell concentrations less than or equal to the value c. The frequency distribution is obtained by differentiation of this function. Equation (20) gives the frequency of the individual particles or of the cells containing them which fall within the discrete interval of concentration 6c centred on the value c as
....'"
""..,,,
y'"
"',,,
0i
".,,,
,' j... ......... ...
i
i
J ©
",
"', '.. ...
,,,"
i
.............
'..,.
Shear stress at surface of cell is zero
f (c) = p(c + 6c/2) - p(c - 6c/2).
(20)
The space in the suspension occupied by cells within this particular band of concentration is found from
.........................Flow of fluidthrough cell Fig. 3. Contiguous cells.
s(c) = f (c)/c
(21)
320
T. N. Smith
Settlin9 velocity It is appropriate now to review the model in terms of its validity as a reflection of the physical processes to which it is to be applied. Justification is sought for its use in the calculation of velocities in flow of fluids through porous media, in fluidization of beds of particles and in settling of particles from suspension. Flow of a fluid through the cells of the medium is represented by an average value of the superficial velocity. The velocity through any individual cell depends on its permeability. Larger cells, in which the fraction of volume occupied by the solid particle is small, have larger permeabilities and allow greater velocities. In smaller cells, where the solid occupies a larger fraction of the volume, the permeabilities are smaller and the velocities are smaller. The solid particles remain stationary and the fluid passes through each cell at a velocity which varies with the size of the cell. The model presented in this paper'is evidently applicable to the case of flow through a porous medium. The solid particles in a homogeneously fluidized bed are not stationary but are in constant motion. The movements are, however, restricted to a relatively short scale of length by collisions with other particles in the closely populated space of the bed. There is no sustained passage of individual particles through the bed. There is a dynamic equilibrium of short motions of the particles which presents to the stream of fluid a field of particles randomly allocated to space. This corresponds with the nature of the suggested model. In settling of particles through a fluid, there is also a dynamic equilibrium in the spatial arrangement. As observed by Kaye and Boardman (1962), there is a transient formation of pairs and larger groups of particles which settle with greater velocities than individual particles. This effect leads to settling, at small volume concentrations of solids, at velocities which exceed the single-particle value. With increasing concentration, the effect is less significant because of the greater frequency of collisions with neighbouring particles. Indeed, at concentrations approaching those of fluidized beds of solids, there can be no sustained differentiation of settling velocities between particles. The interaction between fluid and particles is of the kind embodied in the suggested model. To estimate the settling velocity using this model, the distribution of cell sizes for the particular concentration of the suspension and the corresponding values of f(s) and k must be calculated from eqs (18)-(20), (22) and (27). Values of k' and k are then found using eqs (28) and (29). Then the velocity is calculated from eq. (26) using the vaue of k and the pressure gradient from eq. (23) using the mean concentration to give
Restricting the model to slow flow, the permeability, k, is defined by the relationship
v = kVP/l~
(26)
in which v is the superficial velocity of the fluid of viscosity/~ induced by VP, the motive pressure gradient. The permeability of the independent cell is then obtained from eqs (25) and (26) as k= 3 - (9/2)7 + (9/2)7 s - 376 d 2 3 + 2y s 18~-~" (27)
The shape of the cell for which the permeability is obtained is spherical. Spheres do not, of course, pack to fill space entirely. It is assumed that, with only a minor effect on the permeability, the spherical surface of any cell can be subjected to a degree of distortion sufficient to allow space to be filled by a group of contiguous cells. Because of the specification of zero shear stress at the fluid boundary of the cell, this requirement is considered to be reasonable. To proceed with development of the model, a permeability for each cell in the medium is required. These must then be combined so as to define the composite permeability of the whole medium. The flow through an individual cell within a heterogeneous medium composed of several types of cells is not simply the product of the permeability of the cell and the general pressure gradient in the medium. It depends also on the permeability of the surrounding medium since fluid must pass to and from the cell through that medium. By analogy with the result given by Carslaw and Jaeger (1959) for the thermal or magnetic flux through a spherical cell of permeability k embedded in a medium of permeability k, the effective permeability k' of a cell is obtained as k' 3k k. 2k + k (28)
It is evident from eq. (28) that the permeability of the surrounding medium restricts the flow which can be attained through the individual cell. The maximum value of the effective permeability is 3k, three times the permeability of the medium.
Flow through the medium The permeability of a medium composed of an ensemble of randomly placed cells with various values of permeability is obtained by summing the products of the permeability and the fraction of space occupied by each of the various species of cells so that = Zf(s)k'.
(29)
Since the value of k is required for calculation of each of the individual effective permeabilities, eqs (28) and (29) must be solved simultaneously to obtain this result. The superficial velocity of flow of fluid through the medium is found by insertion of the permeability from eq. (29) into eq. (26).
w --
k(Pl" -- P~)gc kt
(30)
A model of settling velocity PERFORMANCE OF MODEL
321
Theoretical settlin 9 velocity
Settling velocities calculated from the model for a range of concentrations up to 0.5, which may be regarded as the practical limit in settling, are presented in Table 1. These are obtained using a value of cM in eq. (19) of 0.7405, the figure for rhombohedral close packing, to generate the spatial distribution. This figure is chosen because it allows the probability that neighbouring particles might approach to ultimate proximity. Substitution of smaller values, down to 0.5, has only a very slight effect on the shape of the lowconcentration end of the spatial distribution. It is this part of the distribution which presents the greatest permeability to flow of fluid. Accordingly, the computed value of settling velocity has very little sensitivity to the value of CM. Also shown in Table 1 are velocities calculated from the theoretical model of Happel (1958) as presented in
eq. (11). This model may be regarded as equivalent to the new model but with a regular spacing of particles rather than a random dispersion to space. The difference between the two models is obvious. The resistance to flow of fluid through a matrix of solid particles is maximized by spacing the particles evenly. The velocities for the model of random placement reflect the reduction of the individual resistances of particles by the mechanisms previously discussed.
Comparison with experiments
Figure 4 shows a comparison of the calculated settling velocities with experimental values as represented by the correlation of Richardson and Zaki (1954) given in eq. (6). Correspondence of the results of the model with experimental settling velocities is good for concentrations of 0.05 and greater. This is the range of settling and fluidization experiments from which the correlation is derived.
Table 1. Theoretical settling velocities Concentration 0.0 0.01 0.02 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 Settling velocity model 1.701 1.222 1.086 0.841 0.596 0.432 0.313 0.225 0.160 0.111 0.075 0.050 0.032
1.8 1.6 1.4
>,
Velocity at small volume fraction
For concentrations less than 0.05, the particular experiments of Kaye and Boardman (1962), Johne (1966) and Koglin (1971, 1973) must be accepted as a more appropriate depiction of the relationship between settling velocity and concentration. The results of Koglin (1973) are indicated in Fig. 5 together with the correlation of Richardson and Zaki from eq. (6) and those of the model. As the volume fraction of solids falls below 0.03, the model shows an elevation of settling velocity above the single-particle value. The emergence of this effect at concentrations at this level is in correspondence with the experimental observations. However, as the concentration reduces to zero, the model maintains a settling velocity of about 1.7 times greater than that of the single particle. This deviation is associated with
Settling velocity equation (11) 1.000 0.677 0.594 0.453 0.322 0.237 0.177 0.133 0.099 0.073 0.053 0.038 0.026
1.2 1
"-'O--R & Z ] MO DEL
> 0t
0.8 0.6 0.4 0.2 0 0.1 0.2 0.3 0.4 0.5 Concentration
¢1
Fig. 4. Theoretical and experimental settling velocities.
322
1.8
T.N. Smith
I
--"O--" R & Z
MODEL ---O-- E X P T
1.6
1,4
>,~ 1,2
.S
0.8
0.6
0.4 0.2
0
0.01
0.02
0.03
0.04
0.05
Concentration
Fig. 5. Settling velocity at very small concentration.
a departure of the model from reality at very small concentrations. Where particles are very widely spaced, differential velocities between neighbouring particles can be maintained over long distances and times. The model does not permit this. Fluid passes through a matrix of cells each of which contains a particle. The cells do not move relatively to one another. Accordingly, the model is appropriate only when the concentration of solids is sufficiently great to present impediment by collision to the sustained relative movement of particles. At a concentration of 0.1, the distance between centres of particles is 2.15 particle diameters. At a concentration of 0.01, the distance between centres of particles is 4.64 particle diameters and at a concentration of 0.001, the distance is 10.00 particle diameters. It may be inferred from these figures that the model would be effective at a concentration of 0.1, approximate at 0.01 but ineffective at 0.001. The model does, however, clearly reflect the experimentally observed tendency to settling velocities greater than the single-particle value at small concentrations.
CONCLUSION
The model also reflects the observed tendency to settling velocities greater than the single-particle settling velocity at concentrations of about 0.01. The model does, however, lose validity at very small concentrations where the mean distance between particles is several times the particle diameter so that sustained differential movement of neighbouring particles becomes possible.
NOTATION
A theoretical model which permits calculation of the velocity of uniformly sized solid particles settling in slow flow through fluids is presented. It is based on permeation of the fluid through the interstitial space between spherical particles randomly distributed to space. The model is constructed with governing equations which require no substitution of empirically determined factors. Velocities obtained from the model correspond well with correlations of experimental settling velocities and fluidization velocities in the practically important range of concentration from 0.1 to the contact concentration of the particles.
volume fraction of solids in the suspension mean volume fraction cM maximum value of volume fraction C~ drag coefficient of the particle in the fluid d diameter of the spherical particle f(c) fraction of cells of concentration c f(s) fraction of spatial volume occupied by cells with concentration c F drag force on a particle 9 gravitational acceleration k permeability to flow permeability of the composite medium k' effective permeability of a cell K Kozeny constant m number of elements of fluid volume n number of pores N exponent in Richardson and Zaki formula NRE Reynolds of flow of a single particle p probability p(c) probability that concentration does not exceed c p(t) probability of no encounter in t trials VP gradient of pressure Q rate of flow through a pore r radius of pore s(c) volume of space occupied by cells of concentration c t number of elements of fluid volume f mean value of t
c
A model of settling velocity u v w w0
WR
323
vector velocity of fluid superficial velocity of flow of fluid settling velocity of particles single-particle settling velocity velocity of settling particles relative to fluid ratio of particle diameter to cell diameter porosity parameter in probability function viscosity of fluid density of fluid density of solid particle
REFERENCES
Greek letters
7 e 2 #
PF
pp
Batchelor, G. K. (1972) J. Fluid Mech. 52, 245. Carslaw, H. S. and Jaeger, J. C. (1959) Conduction of Heat in Solids, 2nd Edn, p. 426. Oxford University Press, London, U.K. Crowley, J. M. (1971) J. Fluid Mech. 45, 151.
Happel, J. (1958) A.I.Ch.E.J. 4, 197. Happel, J. and Brenner, H. (1965) Low Reynolds Number Hydrodynamics, Prentice-Hall, Englewood Cliffs, NJ, U.S.A. Jayaweera, K. O. L. F., Mason, B. J. and Slack, G. W. (1964) J. Fluid Mech. 20, 121. Johne, R. (1966) Chem. lng. Technik 38, 428. Kaye, B. H. and Boardman, R. P. (1962) Symposium on Interaction between Fluids and Particles, p. 17. Instn Chem. Engrs, London, U.K. Koglin, B. (1971) Chem. Ing. Technik 43, 761. Koglin, B. (1973) In: Proc. 1st Int. Conf. Particle Technol., p. 266, IIT Research Institute, Chicago. Kozeny, J. (1927) Ber. Akad. Wiss. Wien Abt. Ila 136, 271. Richardson, J. F. and Zaki, W. N. (1954) Trans. Instn Chem. Engrs 32, 35. Smith, T. N. (1968) J. Fluid Mech. 32, 203. Stimson, M. and Jeffrey, G. B. (1926) Proc. Roy. Soc. A 111, 110. Wakiya, S. (1957) Res. Report No. 6, Niigata Univ. Coll. Eng.
