Dispersion of solids in nonhomogeneous viscous flows

PII:

Chemical Engineering Science, Vol. 53, No. 10, pp. 1803—1817, 1998 ( 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain S0009–2509(98)00010–4 0009—2509/98 $19.00#0.00

Dispersion of solids in nonhomogeneous viscous flows S. Hansen, D. V. Khakhars and J. M. Ottino* Laboratory for Fluid Mechanics, Chaos and Mixing, Department of Chemical Engineering, Northwestern University, Evanston, IL 60208, U.S.A. (Received 13 January 1997; accepted 9 January 1998) Abstract—Dispersion of powdered solids in viscous liquids is the result of the interaction between a complex flow and incompletely understood phenomena — rupture and erosion of solids — occurring at agglomerate length scales. Breakup of solid clusters in nonhomogeneous flows is studied by dynamic modeling. Complex flow behavior is simulated by means of a chaotic flow; breakup is characterized by the ‘fragmentation number’, Fa, which is the ratio of deforming viscous forces to resisting cohesive forces. A condition for rupture, Fa'Fa , 4%1 developed using a two fragment model cast in terms of a microstructural vector model, is presented. Clusters rupture and erode causing the population to evolve in space and time; conditions based on the magnitude of Fa determine whether or not rupture occurs, and the probability of erosion. Results are analyzed by means of fragmentation theory. It is shown that the polydispersity is not constant, that the cluster size distribution resulting from dispersion is not self-similar, and that erosion in a nonhomogeneous flow leads to a wider size distribution than predicted by mean-field approaches. It is shown as well that regardless of the mixing, that the mass fraction of ultimate size clusters can be predicted by a polynomial relation derived via fragmentation theory and that the overall rates of erosion in both poorly mixed or well mixed flows can be described by a power law. ( 1998 Elsevier Science Ltd. All rights reserved. Keywords: Mixing; dispersion; chaos.

INTRODUCTION

Dispersion of powdered solids in viscous liquids is important in many practical applications. Polymer processing may be singled out as a significant example; in this case the nature and the quality of dispersion of solids greatly affect the properties of the resulting composite (Manas-Zloczower, 1994). The objective of mixing of solids — or dispersion, as it is generally called — is to break agglomerates to aggregate size and then distribute them throughout the flow, the process giving rise to broad, time evolving cluster size distributions. Powders are said to consist of agglomerates — a collection of aggregates — which in turn are composed of ultimate particles or crystals. The terminology is defined by structure; however, agglomerates break due to flow, and the denser aggregates do not. In the case of carbon black, for example, aggregates are of the order of 0.1 km and agglomerates are of the order of 100 km and larger (Manas-Zloczower, 1994). Aggregates and agglomerates, are often referred to as clusters, a cluster being

* Corresponding author. s Permanent address: Department of Chemical Engineering, Indian Institute of Technology, Bombay, Powai, Bombay 400076, India.

composed of particles. The size of a cluster is given by the number of particles composing the cluster. We adopt this terminology here. We mention as well that the entire process of dispersion of a powder into a liquid involves several stages — wetting, impregnation, rupture, etc. — which may be occurring with some degree of overlap; several scenarios have been proposed (see, for example, Parfitt’s 1992). We shall not enter into those aspects here. Our objective here is to highlight the interplay of breakup processes and the underlying flow. The current level of understanding of how agglomerates form and break in complex flows is not up to par with droplet breakup and coalescence (see, for example, Stone 1994; Tjahjadi and Ottino, 1991). The reasons for this discrepancy are many; potential forces, which are quite susceptible to contamination effects, multi-body interactions between aggregates, and effects due to irregular shapes are all imperfectly understood. To our knowledge, there are no studies of fragmentation and erosion in non-homogenous flows and certainly none in chaotic flows. It is the objective of this paper to provide a starting point for such analysis. There is however a large body of theoretical work dealing with general aspects of fragmentation; this provides a useful point of departure. The term

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fragmentation theory is commonly used in the physics literature to refer to a broader class of processes involving breakup of solids, such as rocks. Much of this literature can be adapted to the dispersion of agglomerates of interest here. Fragmentation theory recognizes two main modes of breakup: rupture and erosion; rupture referring to the breakage of a cluster into several fragments of comparable size, erosion to the gradual shearing off of small fragments from larger clusters [Another mode, of less direct interest here is shattering (McGrady and Ziff, 1987); this refers to the case when the probability of breaking small clusters is greater than that of breaking larger clusters.] Excellent introductions to this area are provided by Redner (1990), and Cheng and Redner (1990). There are several inter-related objectives of this paper: to demonstrate that fragmentation theory can be used to analyze dispersion of solids — and in doing so we present new results and extend the theory; to present a relatively a simple model of dispersion; and, lastly, to highlight the effects of mixing on dispersion. We focus primarily on dispersive mixing in chaotic flows [concepts relevant to extensive mixing can be found in Ottino (1990)]. Two modes of breakup are simulated: erosion, resulting in a bimodal size distribution with coarse and fine particles, and rupture, resulting in fragments of comparable size. The results of the simulations are interpreted in terms of fragmentation theory. The paper is organized as follows. We first review dispersion and concepts important to modeling; this is followed by an introduction to elements of fragmentation theory which are later used to analyze the data. Then we describe the flow system and computational studies. The results are divided into two parts: rupture and erosion. The last section summarizes the conclusions and recommendations. DISPERSION

Breakup by hydrodynamic forces may be characterized by the Fragmentation number, Fa, which is the ratio of the viscous shear stress to the strength of the agglomerate (see for example, Rwei et al., 1990, 1991), kc5 Fa" ¹

(1)

where c5 is J2D : D, or the magnitude of the rate-ofdeformation tensor, D"[+v#(+v)T]/2, v being the velocity of the flow. The term ¹ denotes the characteristic cohesive strength of the agglomerate and plays a role analogous to the surface tension stress (p/R) in the definition of the Capillary number of liquid drops (Rallison, 1981; Stone, 1994). Unlike surface tension, however, the agglomerate strength is not a material property but depends on the internal structure, density (degree of compaction), moisture, etc. The cohesive strength of an agglomerate owes its origins to interparticle bonds due to electrostatic charges, van der Waals forces, or moisture. Several models have been developed. The simplest models are

based on two particles; an example of this class is that of Bolen and Colwell (1958) (cf. Tadmor and Gogos, 1979). Other models, e.g. Rumpf ’s (1962), consider the agglomerate as a collection of spherical particles of radius, a, occupying a volume fraction, u, bonded to each other via cohesive forces. According to this model, the tensile strength of an agglomerate is given by 9 u ¹" n F 32 na2 b

(2)

where F represents the average binding force of a single bond and n the average number of bonds per b particle. This model holds provided several conditions are met: (i) there is a large number of bonds in the section to be broken, (ii) the bonds and particles are randomly distributed through out the agglomerate, and (iii) the mean cohesive force for a bond is homogeneous throughout. It is also important to note that the model, by construction, does not consider fractal structures. Kendall’s model (1988), on the other hand, assumes that breakage occurs at a strength limiting flaw, and concludes that the strength of the agglomerate is given by u4!5@6 !1@6 c , (3) Jl a f where ! and ! are the fracture surface energy and the c equilibrium surface energy, and l is the flaw size. f While cohesive forces hold agglomerates together, hydrodynamic forces breakup agglomerates into aggregates. The analysis of the hydrodynamic stress on an agglomerate is complex owing to the structure of the agglomerate. However, simplifying assumptions make the problem solvable. For instance, Bagster and Tomi (1974) assume that an agglomerate is a nonporous homogeneous sphere, and determine that the tensile stress acting normal to a surface and the shear stress acting on the surface within the sphere are both independent of the size of the sphere. The model by Alder and Mills (1979), on the other hand, considers breakup of porous agglomerates. They describe the flow outside the agglomerate with the Stokes equation, and the flow inside the agglomerate with the Brinkman equation, allowing for permeability. Models which neglect the internal length scale of porous agglomerates lead to critical fragmentation numbers that are independent of the size of the agglomerate. There are exceptions. Sonntag and Russel (1987), address this issue, and model the breakup of a porous fractal agglomerate in shear flow. The agglomerate is taken to be a porous sphere with its porosity decreasing with radial distance from the center, and the flow is modeled by the Stokes and Brinkman equations (Alder and Mills, 1979). The stresses in the agglomerate are described by linear elasticity, and the stability of the agglomerate is described by the Mises yield criteria. Breakup occurs when the local energy of distortion in the agglomerate (defined in Sonntag and Russel (1987), is greater than ¹"11.03

Dispersion of solids in nonhomogeneous viscous flows

some critical energy. Sonntag and Russel, (1987) find that in the limit of nondraining agglomerates the relationship between the radius of the agglomerate and the shear rate is RJcR ~1@nD

(4)

where n"2E(3!D)/D(D!1), and E is a constant which depends on the elastic nature of the solid. Experiments by Sonntag and Russel (1986) indicate that D+2.48 and E+4.45 for clusters of polystyrene microspheres, which gives nD+3. Horwatt et al. (1992) also consider the size of fragments produced by the breakup of fractal clusters. Their study, which computationally examines the breakup of diffusionlimited aggregated (DLA) clusters, also shows a power-law correlation between agglomerate radius and the applied stress with the exponent increasing with fractal dimension. The above models — with the exception of Horwatt et al. (1992) — do not consider the effects of heterogeneities. Feke (1991) specifically addresses this issue. Kendall (1988) shows that the strength of a porous solid depends strongly on the presence of flaws and cracks in the solid [see eq. (3)]. Following Kendall’s lead, Feke assumes that breakup occurs at one strength limiting flaw which is planar and divides the spherical cluster into two pieces. Since the stresses on the flaw (or rupture plane) are dependent on the orientation of the flaw relative to the flow, flows which reorient the agglomerate, like simple shear, are more efficient at dispersing solids, i.e. more likely to breakup agglomerates. The role of flow type on the rupture of agglomerates was considered by Manas-Zloczower and Feke (1988) based on a simple model for the process, in which the fragments formed on rupture are assumed to be equal sized and spherical. The fragments, initially held together by van der Waals forces, separate from each other if the hydrodynamic forces are large enough. The model is thus qualitatively similar to that of Feke (1991) — the strength limiting flaw is the bond between the two spherical fragments. We revisit this model below from a slightly different viewpoint. The velocity of one fragment relative to the other, taking hydrodynamic and potential interactions between the two fragments into account, is

G

C DH

rr rr v"D ) r#u]r! A(r) #B(r) I! r2 r2 C(r) # F 6nkR c

)D)r (5)

where r is a vector joining the centers of the two spheres, D is the rate of strain tensor, u is the vorticity of the driving flow, A(r) and B(r) are known functions (Batchelor and Green, 1972), and I is the identity matrix. In the last term, F is the physicochemical c force between the particles, k is the fluid viscosity, R is the radius of the spheres, and C(r) is a function which accounts for the particle proximity effect on drag (Spielman, 1970). The first three terms in the above

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equation give the relative velocity between the spheres in a linear flow field under the influence of hydrodynamic interactions, and the last term gives the relative velocity due to the physicochemical forces between the spheres. The underlying physics is revealed more clearly by recasting eq. (5) in the following dimensionless form: 1 dr F "(1!A (r))D : mm!C(r) c r dt Fa

(6)

dm "[(1!B(r))D!)] ) m!(1!B(r))(D : mm)m. dt (7) Here r is the center-to-center distance, and m is a unit vector along the line joining the centers of the two fragments, so that r"rm. Distances are made dimensionless with respect to the radius of a fragment (R), shear rates and time with respect to the characteristic shear rate c5"J2D : D, and F "DF D"HR/(24z2) c c with respect to H/R, where H is the Hamaker constant, and z"(r!2R). The above equations are identical in form to those for the breakup for a slender drop (Khakhar and Ottino, 1986) and form part of a more general class of deformation and breakup processes referred to as the linear vector model (Olbricht et al., 1982). Several conclusions result from the above equations: They reveal that the dimensionless parameter of the system is the fragmentation number (Fa) with the characteristic strength given by H ¹" . 6nR3

(8)

The rate of separation of the fragments depends on the functions A(r), C(r), F , and the fragmentac tion number, while the rate of rotation depends only on the function B(r). Further, it is apparent that the separation between the fragments increases only when the hydrodynamic force exceeds the binding physicochemical force. The pair of fragments rotates as a material element in a ‘flow’ with an effective velocity gradient tensor M"D#)/(1!B).

(9)

Since B is positive the ‘flow’ appears to have a higher vorticity relative to the strain rate than the imposed flow. Based on the above discussion, a criterion for the separation of the fragments is easily obtained. If at least one eigenvalue of the tensor M is positive, the pair orients along the corresponding principal axis, and the critical Fragmentation number for separation is given by J2 1 (10) Fa " 4%1 ej 12z2 0 where ej /J2 is the asymptotic value of D : mm (i.e. the largest positive eigenvalue of M), and z is the dimen0 sionless equilibrium separation. For Fa'Fa the 4%1 fragments separate indefinitely. In arriving at the

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above equation the physicochemical force was estimated for a pair of equal sized spheres as F "1/12z2 c 0 (Rumpf, 1962), and the result (1!A)/C"1 for small separations was used. Special cases of this result were presented by Manas-Zloczower and Feke (1988). For purely extensional flows ()"0), we have ej"1/J2 for planar flows and ej"J2/3 for axisymmetric flows. In the case of simple shear flow, eigenvalues are imaginary, hence no asymptotic orientation exists and the separation distance oscillates with time as the fragments rotate in the flow. We note that the dependence of the critical shear rate on the agglomerate size for this model is similar to that found by Sonntag and Russel (1987) [eq. (4) with nD"1/3], though the basic assumptions of the model are very different (set Fa "const. at incipient breakup [eq. (10)] and 4%1 equate eqs (1) and equation (8); this gives the !1/3 with the shear rate). It was only recently that dispersive mixing was shown to be characterized by two models of breakup, rupture and erosion (Rwei et al., 1990, 1991). Rupture refers to the breakage of a cluster into several fragments of comparable size, erosion to the gradual shearing off of small fragments from the surface of larger clusters. The main qualitative difference between the two mechanisms is energy input: low for erosion, high for rupture. Evidently, erosion dominates dispersive mixing when the hydrodynamic stresses are low, such as in the case of dispersion of titanium dioxide (Lee et al., 1993). However, as noted by Rwei et al. (1990), erosion may be important to dispersion even when rupture is present. The kinetics of erosion, for compact spherical structures in shear flows with shear rate c5 and short times, can be described by R

!R(t) */*5 JcR t R */*5

Zloczower et al., 1985). Let us now turn to fragmentation theory and how it can be used to interpret results from computer simulations. FRAGMENTATION THEORY

The irreversible, continuous breakup of clusters in a well-mixed medium is described by the linear fragmentation equation (Redner 1990), Lc(x, t) "!a(x)c(x, t) Lt

P

#

a(y) f (xDy)c(y, t) dy

(12)

x

where c(x, t) is the concentration of clusters of mass x at time t, a(x) is the overall rate of fragmentation of clusters of mass x, and f (xDy), referred to as the relative rate of breakup, is the expected number of fragments of size x produced from the breakage of a cluster of size y. The breakup kernels, a(x) and f (xDy), determine the kinetics of the fragmentation process. Relatively little can be said about general characteristics; these are highly process-specific. However, rather general conclusions may be drawn about the overall fragmentation process even in the absence of an exact specification of the kernels. Typically, homogeneous kernels are considered, in which the overall rate of breakup is characterized by a(x)Jxj, where j is known as the homogeneity index. Note that, in particular, this form allows for breakup to be proportional to surface area. The relative rate of breakup is assumed to be homogeneous,

AB

1 x f (xDy)" b . y y

(11)

where R is the radius of the parent agglomerate, and R is the initial radius. This relation implies that the */*5 rate of erosion is proportional to the surface area of the agglomerate. Studies (Rwei et al., 1990, 1991) show that the critical fragmentation number for erosion is more than an order of magnitude less than the fragmentation number for rupture. Some simulations, involving various degrees of complexity, have attempted to model the dispersion process. For instance, Manas-Zloczower et al. (1982, 1984) model dispersive mixing as a well-mixed tank with a recycling stream which passes through a highshear zone. When an agglomerate passes through the high-shear zone, it breaks into two equal sized pieces if the hydrodynamic forces on the agglomerate exceed the cohesive forces holding the agglomerate together, i.e. Fa'1. In this simulation, the critical shear stress is independent of the size of the cluster. As the simulation time increases, the size distribution broadens, and the mode size decreases. A similar study considers dispersion in roll-mills, batch dispersive mixers with closely spaced cylinders as mixing elements (Manas-

=

(13)

Thus, a homogeneous relative rate of breakup implies that the size of the resulting fragment relative to the size of the original cluster, x/y, is the important parameter in characterizing a single breakup event rather than the absolute size of the cluster itself. The function b(x/y) defines the average number of particles produced on a single breakup event, i,

P

i"

1

b(r) dr

(14)

0

and obeys

P

1

rb(r) dr"1

(15)

0

due to conservation of mass. Frequently, breakup processes exhibit self-similarity. In such processes the distribution of clusters becomes time invariant when appropriately scaled. For homogeneous kernels, if the mass is scaled with respect to an average cluster size, the size distribution

Dispersion of solids in nonhomogeneous viscous flows

at long times has the form

relative rate of breakup — is given by

c(x, t)"s~2/(l), l"x/s(t)

(16)

where s(t) is the average cluster size and / is referred to as the scaling distribution. Such scaling may be used to reduce the number of variables from two (x and t) to one (x/s). The exponent of s in eq. (16) results from conservation of mass. This self-similar form is approached asymptotically and is independent of the initial conditions. Size distributions can be characterized in terms of moments. The moments of the ‘bare’ and ‘scaled’ size distributions are defined as

P

M (t), a

=

xa c(x, t) dx

(17)

0

and

P

=

la/(l) dl (18) 0 respectively. Moments have physical significance; for example, M is the total mass per unit volume. The 1 number average cluster size is, s (t)"M /M , and n 1 0 the weight average cluster size is, s (t)"M /M . w 2 1 Finally, the polydispersity, which characterizes the width of the distribution is m a,

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M M P" 2 0 . M 2 1

(19)

If the scaling ansatz holds, then M (t)"s(t)a~1 m , a a and, hence, the polydispersity is independent of time, and s (t)Js (t). n w The use of the self-similar form of the size distribution allows the determination of the decay of s(t). When j'0, the average cluster size decays algebraically as s(t)&t~1@j.

AB

c(x, t)"

g j c(g, 0)#g(t)h(x), x

(21)

where g"[x1~j#(1!j)Kt]1@(1~j), K is a rate constant, h(x) is the distribution of sizes of primary particles (or aggregates), and g(t) is determined by conservation of mass. Equation (21) cannot be described by the scaling (16), and does not have an asymptotic form, i.e. it is always dependent upon the initial conditions. THE JOURNAL BEARING FLOW

Let us now consider briefly the flow system used in our studies, the so-called journal bearing flow ( j.b.f.). A more complete description can be found in Swanson and Ottino (1990). A particularly important aspect of this flow is that it can be realized experimentally and manipulated to produce both regular and chaotic flows. Fluid is contained between two eccentric cylinders and is driven by the rotation of one or both of the cylinders. Flow parameters and streamlines relevant to this study are shown in Fig. 1. Analytic stream functions, which allow for tractable computations can be found in Wannier (1950). At low Reynolds and Strouhal numbers, the position of a fluid element is determined by the angular displacement of the boundaries — the actual velocity of the cylinders being unimportant since inertia is negligible. The time to achieve steady flow at the start of each period of the flow is thus negligibly small. The most important aspects of the mixing process can be captured by examination of two protocols for the motion

(20)

When j'0, smaller clusters break more easily than larger clusters resulting in loss of mass to infinitesimal size clusters termed ‘shattering’. The scaling (16) is not valid when j(0 (Filippov, 1961). Furthermore, scaling allows for determination of the tails of the size distribution — limits of small mass, x/s;1, and large mass, x/s<1 (see Redner, 1990). Only limiting forms of the size distribution can be obtained if the breakup kernels are left unspecified. The early work of Kolmogorov (1941), considering that the size of fragments produced on breakage are random, showed that the size distribution is log-normal. A few solutions corresponding to specific homogeneous breakup kernels have also been studied (Ziff and McGrady 1985, 1986; McGrady and Ziff, 1987; Ziff, 1991; Williams, 1990). Many of these solutions have a physical basis but all results pertain to rupture. One recent result focuses on erosion. Hansen and Ottino (1996a) determine that the size distribution for erosion — which is described by a nonhomogeneous

Fig. 1. (a) Journal bearing geometry with relevant parameters. (b) Streamlines in the journal bearing when the inner cylinder is rotating. (c) Streamlines when the outer cylinder is rotating.

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of the boundaries, both counter-rotating, i.e. the inner cylinder is rotated clockwise and the outer is rotated counterclockwise. ‘Crossing of streamlines’, a necessary condition for chaos in two-dimensional flows, is achieved by alternating periodic motion of the cylinders (Ottino, 1990). In our particular case the inner is rotated for a quarter-period, the outer is rotated for a half-period, then, again, the inner is rotated for a quarter-period, and so on. In the protocols we consider, R # "R # , where R denotes the radius, o o i i # is the total angular displacement per period and subscripts i and o designate inner and outer cylinder, respectively. One mixing protocol produces a globally chaotic system and is considered well mixed, while the other produces a regular region separated from a chaotic region and is thus referred to as poorly mixed. In the well mixed system, the angular displacement of the outer cylinder is 720° per period, and, in the poorly

Fig. 2. Poincare´ sections corresponding to (a) the ‘poorly mixed’ case (angular displacement of the outer cylinder # "180° per period of the flow), and (b) the ‘well-mixed’ 0 case (angular displacement of the outer cylinder # "720° 0 per period of the flow). The geometry of the flow is specified in Fig. 1(a).

mixed system the angular displacement of the outer cylinder is 180° per period. The Poincare´ sections produced by these protocols are symmetric about the center line connecting the axes of the cylinders (see Fig. 2). MODELING

As mentioned earlier, we consider two modes of dispersion: the first model highlights rupture and the second erosion. The first set of simulations describe rupture; agglomerates, convected by the flow, instantaneously break into two or more pieces when the local shear-rate is above some critical value determined by the mass of the agglomerate (i.e. when Fa'1). Geometrical aspects are neglected; clusters are envisioned as fractal structures occupying a spherical region in the flow. In the second set of simulations, we model erosion. In both sets of simulations, we assume that the particles are passive, i.e. they move as fluid elements. The Pe´clet number is taken to be large (i.e. Brownian motion is not considered, a good assumption in the context of polymer processing) and hydrodynamic and other interactions between clusters are neglected as is aggregation (a more questionable assumption if loadings are high). Our results therefore apply to dilute systems. To track the particles, velocity profiles, obtained by differentiation of analytic streamlines, are numerically integrated by a fourth-order Runge— Kutta method. Since we assume that the clusters are advected passively, and we neglect aggregation (or flocculation), clusters in our computer experiments may at times overlap. Rupture In simulations modeling rupture, a large agglomerate of radius 0.1R consisting of 30,000 aggregates is o placed in the flow — various initial positions of the agglomerate are considered. The radius of an aggregate is a"1.6]10~3 R . This agglomerate is advected o in the flow and proceeds to break into several smaller agglomerates. We assume that each agglomerate consists of aggregates of unit mass — which do not break — such that each agglomerate has a mass equal to an integer number of aggregates. Assuming that rupture can be described by eq. (4), the largest agglomerate stable at a point in the flow is xJcR ~1@nD. Hence, we define the Fragmentation number, Fa"c5xnD/a, where a is an adjustable parameter. Agglomerates rupture instantaneously into two or more pieces when Fa'1; deformation of the agglomerates by the flow is neglected. Such conditions would be valid for brittle agglomerates. The shear rate (c5) is the magnitude of the rate-of-deformation tensor, which is determined analytically by differentiation of the velocity field. Here, the rate-of-deformation is that of the bulk flow at the center of mass of the agglomerate; a similar assumption is used in drop breakup studies. This fragmentation number becomes independent of the mass of the agglomerate as the fractal dimension of

Dispersion of solids in nonhomogeneous viscous flows

the cluster approaches three [see eq. (4)], and, thus, this model reduces to that of Manas-Zloczower et al. (1982) when the agglomerates are not fractal (or D"3). In the simulations of rupture presented here the fractal dimension is set at D"2.5, E"4.5 so that nD"3, and a is chosen so that 2% of the rates of deformation in the flow will disperse agglomerates consisting of two aggregates. We have investigated other values of a, but the results are not very revealing: when a is much higher than the value used, breakup into single particles is almost immediate; on the other hand, when a is lower, the smallest possible agglomerates in the flow increases from a single aggregate to clusters containing more than one aggregate, with the results otherwise remaining similar to those presented here. After a cluster reaches the breakup limit, it is necessary to determine the size and position of the resulting fragments. In order to determine the size of the fragments, we define a relative rate of breakup. One of the simplest relative rates of breakup is binary breakup into two equal-sized pieces (Manas-Zloczower et al., 1982). We examine the role of the relative rate of breakup by contrasting two simulations: one allows only for binary breakup; the other allows for breakup into several fragments. We allow agglomerates to breakup into clusters made up of any integer number of aggregates. In the case of binary breakup, an agglomerate of mass y breaks to form, an agglomerate of mass z, where z is a random integer between one and y!1, and an agglomerate of mass y!z. Determining the size of fragments produced on a single breakage into i fragments is more complex if i is greater than two. In these type of simulations, a homogeneous relative rate of breakup is used; the size of the product relative to the parent agglomerate defines the process, provided the average agglomerate size is much greater than one aggregate. If the mass of the breaking agglomerate is larger than i aggregates, the size of the fragments is determined as follows: A random number, o , is chosen for each fragment. j The masses of the first through (i!1)th fragments are determined by x j"o j y

C

D

i ~1 + o k k/1

(22)

where y is the initial mass of the rupturing agglomerate, and all fractions are rounded off. The mass of the ith fragment is determined via conservation of mass. If the mass of the breaking agglomerate is less than i aggregates, then the agglomerate merely breaks into its constituent aggregates. As a simplification, the number of clusters formed on breakup are assumed to be independent of the local shear rate. Other more complex relative rates of breakup are possible (Hill and Ng, 1996), however eq. (22) is amenable to our simulations. Furthermore, eq. (22) provides a good starting point for the study of different relative rates of breakup.

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The positions of new agglomerates are determined, after the sizes are calculated. The radius of each cluster is defined as R"ax1@D, where a is the radius of an aggregate. In the binary breakup simulations, the two fragments of an agglomerate are placed in the flow such that the weighted average of the two fragments is at the center of mass of the unruptured agglomerate at the point of breakup. Initially after breakup, the clusters are in contact, i.e. the distance between the two centers of mass is equal to the sum of the radii of the agglomerates. The angle between the streamline at the point of rupture and the line connecting the centers of mass of the two new agglomerates is 45°, with the faster moving agglomerate placed downstream from the slower moving agglomerate. This direction essentially corresponds to the principal axis of stretching, approximating the local flow to be a simple shear flow, and is in accordance with the theory for fragment separation discussed above. Hence, the two new agglomerates initially move away from each other after breakup, as would be expected. Parts of agglomerates may at times overlap, because a fragment from one rupturing agglomerate may be placed partially over another agglomerate already in the flow — one can imagine that all the clusters do not remain in the same plane after breakup. In simulations which allow for breakup into more than two fragments, the locations of the fragments formed on rupture are difficult to determine since they depend on the position of each fragment on the original cluster, the local flow, etc., and would require a detailed micromechanical analysis. Here we determine the positions of the fragments using the following simplified procedure. The position of the largest agglomerate resulting from the rupture is found by setting the weighted average of the centers of mass of the product to the point of rupture. The other agglomerates are placed randomly within a circle with a radius equal to the radius of the parent agglomerate and a center at the point of breakup. Again, the fragments are allowed to overlap. The process of rupture necessarily induces some cross-streamline motion since the fragments, in general, are not placed after rupture on the streamline corresponding to mother agglomerate. The mixing produced by such random displacements (of the order of the size of the agglomerate) is expected to be small considering the small size of the agglomerates relative to the flow domain. Erosion Erosion is modeled as binary breakup into an aggregate of unit mass and an agglomerate of y!1, where y is the mass of the agglomerate before erosion. In this case we are interested in the erosion of several agglomerates with various sizes. Initially, 10,000 agglomerates are distributed in the flow such that no area has a statistically higher density than any other area. This is accomplished by placing the clusters on a regular rectangular grid a distance * apart, where * is the square root of the average area of an

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agglomerate, and then giving random displacements, uniform between 0 and *, in the positive x and positive y directions. The initial size distribution of the agglomerates is uniform between 400 and 500 aggregates per agglomerate, i.e. there are approximately 100 agglomerates of every mass from 400 to 500 aggregates, but there are none smaller than 400 aggregates and none larger than 500 aggregates. Like the simulations of rupture, erosion is characterized by a Fragmentation number. There is evidence that the rate of erosion is proportional to the shear rate and surface area of the agglomerate (Powell and Mason, 1982; Rwei et al., 1991) — eq. (11) is a consequence of this relationship between the rate of erosion and surface area. Therefore, in these simulations, the probability that an agglomerate erodes is proportional to Fa"c5x2@3/a, instead of zero when Fa is less than unity and one when Fa is greater than unity. After an aggregate is eroded off the surface of an agglomerate, only the mass of the agglomerate changes. The positions of the aggregates are not determined, since aggregates cannot break. Since the number of agglomerates does not increase with time, we can simulate erosion of significantly more agglomerates than in the case of rupture. A problem common to both sets of studies is the discretization of time by the simulation and a critical issue is the frequency of checks for breakup. An appropriate frequency of checks must be used, since a low frequency of checks may result in missing when breakup occurs, and a high frequency will result in simulations which are too long. A similar problem occurs when modeling aggregation (Hansen and Ottino, 1996b). The model of erosion is not as sensitive to this frequency of checks, as we can increase or lower the probability of erosion on each check, by decreasing or increasing a, to coincide with lower or higher frequencies of checks, respectively. The frequency of checks is chosen so that the rate of breakup does not increase by more than 0.5% when the frequency of checks is doubled. RESULTS

Rupture The relative sizes of fragments and the number of fragments produced on each rupture event both affect the resulting size distribution. Figure 3 shows the size distributions resulting from binary breakup into equal size fragments and binary breakup into random size fragments. As expected, the size distribution resulting from breakup into equal size pieces is much narrower. Figure 4 shows the cumulative weight fractions resulting from a simulation which allows for binary breakup, and three simulations which allow for breakup into several clusters. Here, we see that largest agglomerate present in the flow is independent of the relative rate of breakup. The number of small agglomerates present is dependent on the number of fragments produced on a breakup event. As the expected

Fig. 3. The cumulative mass fraction of agglomerates less than mass x, after one period in the ‘well-mixed’ flow. The broken line shows the distribution of agglomerates which break into two equal size fragments. The solid line shows the distribution of agglomerates which break into two random size fragments.

Fig. 4. The cumulative mass fraction of agglomerates less than mass x, after one period in the ‘well-mixed’ flow, for simulations with different relative rates of breakup. The number of child particles produced on a breakup event, i, is, from right to left: two, five, ten, and 20.

number of fragments increases, the number of small agglomerates increases. As stated earlier, if the scaling (16) holds we expect the average cluster size to decay algebraically as s(t)&t~1@j. However, the average cluster size in both the well-mixed system and poorly mixed system does not decay algebraically; it decays in discrete intervals (see Fig. 5). This is a result of the periodic driving of the j.b.f. Each time the cylinder driving the flow changes s (t) drops sharply. This is because the velocity n field, and consequently the rate of deformation field,

Dispersion of solids in nonhomogeneous viscous flows

Fig. 5. The evolution of the number average agglomerate size is the simulations of rupture. The solid line is the poorly mixed flow, and the broken line is the well mixed flow. Note that time is the total displacement of the boundaries, with one time unit equivalent to a full revolution of the outer boundary, i.e. one is equivalent to one period of the ‘poorly mixed’ flow, and four is equivalent to one period of the ‘well-mixed’ flow, unless otherwise stated.

changes instantaneously at the start of each new flow period, as explained earlier. Agglomerates which are stable in a low shear zone when one cylinder is rotating may thus suddenly experience high shear rates and rupture when the flow changes. This results in an overall rate of breakup which is time varying, and has a periodicity which coincides with that of the flow. Also of interest is the polydispersity of the cluster size distribution. As shown in Fig. 6, the polydispersity changes with the average cluster size. Initially, the polydispersity is one; the number average and weight average cluster size are equal, since only one cluster is present. As breakup begins the size distribution widens and the polydispersity increases. Then, the polydispersity approaches one as the average cluster size approaches one. The polydispersity is not constant, because breakup occurs rapidly, and the size distribution does not evolve to a self-similar form. Rupture in a well-mixed system is considerably different from that in a poorly mixed system. In a well-mixed system, the initial placement of the cluster affects only the initial dispersion of the agglomerate. For instance, a cluster could initially be in a region of the flow with a low rate of deformation, and may not rupture. However, the cluster will eventually be advected into a region of the flow with a high rate of deformation. If the system is well-mixed, the end result of rupture will be independent of the initial placement of the agglomerate; however, the evolution will be dependent on the initial conditions. On the other hand, in a poorly mixed flow, a cluster in a region with low rates of deformation, may never leave that region and may never break. Hence, the final

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Fig. 6. The polydispersity of the weight average agglomerate size. The solid line is the poorly mixed flow, and the broken line is the well mixed flow. Time is defined as in Fig. 5.

cluster size distribution is dependent on the initial position of the agglomerate. The evolution of the cluster size distribution is dependent on the initial conditions even in the wellmixed system, because the rate of breakup is dependent on the rate of deformation which the cluster experiences. The rate of deformation which a specific agglomerate experiences changes as it is advected throughout the nonhomogeneous flow, and, thus, the overall rate of breakup is dependent on time. The nonconstant polydispersity and agglomerate size distribution which is not self-similar are results of the time varying rate of breakup. To further address this issue, we simulate the breakup of several agglomerates, O(102), which was determined by Hansen and Ottino (1996b), in the case of aggregation, to be the minimum number of agglomerates necessary for selfsimilarity. The agglomerates are initially uniformly distributed throughout the flow, as in the earlier described simulations of erosion. Thus, the rate of breakup may be nearly independent of time, since it is averaged over many agglomerates covering the wide variety of rates of deformation in the flow. The results match those of rupture starting with one agglomerate. The average cluster size decays in discrete jumps, and the polydispersity is not constant. The rate of breakup changes time periodically with the flow — this may be similar to an industrial mixer such as an extruder which has several different mixing zones, with different rates of deformation. Erosion As mentioned earlier, the rate of breakup is probably time periodic with the j.b.f. Figure 7 shows the rate of erosion for the well-mixed and poorly mixed systems, when the inner cylinder is rotating, and when the outer cylinder is rotating. Clearly, the rates of

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Fig. 7. The rate of erosion. The rate is defined as the number of breakage events of clusters of mass x, divided by the integral of c(x, t)dt. Triangles show the rate in the poorly mixed flow, while squares show the rate in the well mixed flow. Open symbols designate that the outer cylinder is rotating, and closed that the inner is rotating.

Fig. 8. The mass fraction of eroded aggregates. The lines show the best fits provided by eq. (24) — broken for the well-mixed flow, solid for the poorly mixed flow. The solid squares are for the well mixed flow, and the open triangles are for the poorly mixed flow. Time is defined as in Fig. 5.

erosion corresponding to inner and outer rotations are different. Furthermore, the homogeneity index resulting from these simulations is not 2/3, like the exponent n from the Fragmentation number. The rate of erosion of the larger clusters is slower than expected, because of the spatial variation in the rate of deformation. The large agglomerates in high-shear regions erode fast, but the only large agglomerates which remain in the system for a significant amount of time are in low-shear regions. However, the rates of erosion of agglomerates smaller than 450 aggregates are well fit by the power-law form of the homogeneous overall rate of breakup, with

G

j"

0.42$0.001 well mixed, inner rotating 0.62$0.001 well mixed, outer rotating 0.40$0.001 poorly mixed, inner rotating 0.58$0.001 poorly mixed, outer rotating.

Furthermore, the overall rate of erosion is dependent on the flow, which, in this case, changes periodically, and, therefore, the overall rate is also periodic. Often, the mass fraction of aggregates eroded off the agglomerates, that is the fraction solids which has reached the ultimate size, is of interest. According to eq. (21), this mass fraction is

P AB P x H

¼ "1! 0 f

x

x H

g j c(g, 0) dx x

.

(23)

xc(x, 0) dx

x L

Consider a specific example. If one assumes that j"2/3 and the concentration of clusters is uniform within the initial size range, then ¼ is given by f

Fig. 9. The cumulative mass fraction of agglomerates less than mass x, from the simulations of erosion in the wellmixed flow. The dotted line is the initial condition, the solid lines are the results from the simulations, and the dashed lines are the best fit provided by eq. (21), with j"2/3, and K determined by the fit in Fig. 8. From right to left, the times coinciding with the dashed and solid lines are: 1 period, 2 periods and 3 periods.

6(x5@3!x5@3) (x4@3!x4@3) L (Kt)! H L (Kt)2 ¼" H f 5(x2 !x2) 2(x2 !x2) H L H L 2(x !x ) H L (Kt)3 # 27(x2 !x2) H L

(24)

where K is unknown [from eq. (21)] and t(3x1@3 /K. L When t'3x1@3 /K, corrections must be made to eq. L (24) to account for the fraction of agglomerates which

Dispersion of solids in nonhomogeneous viscous flows

Fig. 10. The spatial variation of agglomerate sizes in simulations of erosion in the journal bearing flow. Initially, there are 10,000 agglomerates consisting of 400—500 aggregates. Colors: red — less than 200 aggregates, yellow — 200 to 300 aggregates, green — 300—400 aggregates, blue-more than 400 aggregates. (a) Poorly mixed flow after four periods, (b) well-mixed flow after one period.

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have reached the size of an aggregate and are no longer eroding; eq. (23), however, remains valid for all times. The mass fraction of agglomerates composed of just one aggregate resulting from our simulations can be fit with this polynomial (see Fig. 8). Having determined the rate of erosion (j"2/3, and K found by the fit in Fig. 8), we attempt to predict the evolution of the cluster size distribution in the globally mixed flow, by fitting our numerical data with eq. (21). Figure 9 shows the cumulative mass fraction predicted by eq. (21), and the cumulative mass fraction resulting from the simulation of erosion in the wellmixed system. Equation (21) predicts the mass of fines produced and the bimodal shape of the size distribution. However, the tails of the coarse product in our simulations differs from those predicted by eq. (21). The discrepancy is described as follows: In spite of the chaotic mixing, as evidence by Poincare´ sections, some agglomerates seem to remain in the high-shear regions longer than others. These agglomerates erode faster than what may be expected in terms of a rate of erosion averaged over the entire flow. In fact, Fig. 7 shows that the rate of erosion of some of the larger clusters is much lower than predicted by a power-law form of the overall rate of breakup. Likewise, some agglomerates do not spend a significant amount of time in the high-shear regions, and do not erode as fast as expected. Thus, the size range of coarse product in the j.b.f. is wider than predicted by eq. (21), which assumes perfect mixing. The difference between the evolution of the size distribution in the globally mixed flow and that expected by eq. (21) suggests that there may be spatial variations in the system. Figure 10 shows the distribution of agglomerates in the globally mixed and poorly mixed flows. Even in the globally mixed flow, the agglomerates are not uniformly distributed — larger clusters are in the regions with lower rates of deformations and vice versa. Also, the clusters are not uniformly distributed in the poorly mixed system. Hence, the initially uniform system evolves to a system with spatial fluctuations due to the variation in rate of deformation throughout the flow.

CONCLUSIONS

Dispersion of powdered solids in viscous liquids is a complex process involving interactions between complex flow and phenomena — rupture and erosion of solids — occurring at agglomerate’s length scales. No model can faithfully describe all possible effects. The model presented here simulates complex flow behavior and mixing by means of a two-dimensional chaotic flow; breakup is described in terms of a parameter, Fa, that plays the same role as the capillary number in drop breakup and dispersion. The fragmentation number, Fa, is defined as the ratio of deforming viscous forces to resisting cohesive forces; the condition Fa(O(1) determines conditions where no fragmentation is possible. Clusters are advected by the flow and rupture and erode causing the population to

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evolve in space and time. A more complete model would allow more fragments to be formed as more energy is put into breaking an agglomerate, or as the rate of deformation experienced by an agglomerate increases. Also, this model does not allow for the clusters to deform, as the rate of breakup is dependent on the rate of deformation, not the deformation itself. The model serves as a first approximation and allows one to consider the effect of chaotic advection on dispersion. Furthermore, it may serve as a foundation for future studies of this complex problem. The effect of the relative rate of breakup used in the model is not intuitively obvious, because the local rate-of-deformation determines the stable size of fragments in the flow. An example may clarify this: consider two simulations, in (i) the agglomerates can only break into two pieces, and in (ii) the agglomerates can break into several fragments at once. In both simulations, agglomerates proceed to break until all the fragments reach a stable size. For instance, a large cluster passing through a high-shear zone breaks when Fa'1, and each resulting fragment will continue to break until its mass is below a stable size determined by the local rate of deformation. In simulation (i), an agglomerate may break several times before all the fragments reach a stable size, whereas all the child fragments of an agglomerate from simulation (ii) might be at stable size after only one breakage; however, an agglomerate from simulation (i) might break into the same size child fragments as an agglomerate from simulation (ii), because the rate of deformation, and not the number of fragments produced on each breakage, dictates the stable agglomerate size. These caveats notwithstanding, the simulations indicate that the agglomerates become smaller as the number of fragments per breakage increases. It is found, not too surprisingly, that the average cluster size in the simulations of rupture decays in discrete intervals with the periodic changes in the flow, that the polydispersity is not constant and that the cluster size distribution resulting is not self-similar. Erosion in a nonhomogeneous flow leads to a wider size distribution than predicted by a meanfield approach, such as eq. (21). An unexpected, and more far reaching result is that, regardless of the mixing, the mass fraction of ultimate size clusters can be predicted, or fit, by a mean-field polynomial [eq. (23)] based on fragmentation theory. Moreover, the overall rates of erosion in both poorly mixed or well mixed flows can be described by the homogeneous kernel, which is a power law. A nontrivial question is the likelihood of assembling a description involving both breakup and aggregation. In principle one may combine a model of aggregation, such as that of Hansen and Ottino (1996b), and this model of breakup. There are difficulties though. One problem is assigning the location of fragments after breakage. Another is instantaneous aggregation upon after breakup with other nearby particles. A judicious combination of a realistic

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representation of the most important physics and efficient computations will be needed to reach this goal. Acknowledgment This work was supported by the Office of Basic Energy Sciences of the Department of Energy.

m o j p /(n) u u )

ratio of fragment size to original particle size ("x/y) random number, eq. (22) interfacial tension scaling distribution volume fraction of solid in a single agglomerate vorticity vector ("+]v), eq. (5) vorticity tensor ["(+v!+vT)/2], eq. 7

NOTATION

a a(x) b(m) c(x, t) e j Fa Fa 4%1 f (xDy)

g(t) h(x) H i I K l f m a M M a n n b P r r R s(t) s n s w ¹ x z z 0

particle radius, eq. (2) overall rate of breakup, eq. (12) homogeneous breakup kernel, eq. (13) concentration of agglomerates of mass x at time t asymptotic value of D : mm eq. (10) fragmentation number critical fragmentation number for separation relative rate of breakup; number of agglomerates of mass x expected from breakup of an agglomerate of mass y function determined by conservation of mass, eq. (21) size distribution of primary particles Hamaker constant number of fragments produced by breakup of a single agglomerate identity tensor rate constant for erosion length scale of flaw size ath moment of the ‘scaled’ size distribution, eq. (18) effective velocity gradient tensor, eq. (9) ath moment of the ‘bare’ size distribution, eq. (17) material parameter for breakup, eq. (4) average number of bonds per particle polydispersity, ("s /s ) w n center-to-center vector; r, magnitude of r dummy variable, eqs (14) and (15) agglomerate radius average agglomerate size ("M /M ) a`1 a number average size ("M /M ) 1 0 weight average size ("M /M ) 2 1 characteristic cohesive strength of an agglomerate, eq. (1) agglomerate mass separation between two agglomerates ("r!2R) dimensionless equilibrium separation of two agglomerates

Greek letters c5 rate of strain, J2D : D ! equilibrium surface energy, eq. (3) ! fracture surface energy, eq. (3) c j homogeneity index; power-law index for a(x) k liquid phase viscosity l scaled size, ["x/s(t)]

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