Energy & Fuels 1998, 12, 1031-1039
1031
Packing and Viscosity of Concentrated Polydisperse Coal-Water Slurries Boris Veytsman, Joel Morrison, Alan Scaroni, and Paul Painter* Energy Institute, The Pennsylvania State University, University Park, Pennsylvania 16802 Received May 12, 1998. Revised Manuscript Received July 7, 1998
The viscosity of polydisperse slurries close to the packing limit is discussed. It is shown that the divergence of the viscosity at the close packing limit causes the dependence of the slurry viscosity on loading to be universal. Ways of increasing the maximal loading of polydisperse slurries are described. A new theory of packing of powders based on a generalization of the Furnas “telescopic tube method” is proposed. Unlike the original Furnas model, this theory allows the calculation of the maximal packing for powders with an arbitrary size distribution of particles. The application of the theory to the problem of reducing the viscosity of coal-water slurries is discussed.
1. Introduction The transport of slurries is an important problem arising in numerous industrial applications, ranging from construction engineering to combustion processes.1 It is often necessary to operate with concentrated slurries, that is, with slurries having high solids content (loading). The transport of such slurries is impeded by the fact that their viscosity grows drastically with increased loading. Clearly, it would be advantageous to lower the viscosity of a slurry at a given loading or maximize the loading for a given viscosity. In principle, this can be achieved by specifying the particle size distribution of the system. Common grinding processes produce polydisperse powders that form polydisperse slurries. Our aim is to control this process and produce an “optimal” blend of particles of different sizes. Consider the following situation that is typically found in the coal-water slurry field. We have two different mills that produce particles of different sizes (“fine” and “coarse” powders). Both outputs are polydisperse, but the particle size distributions are different. We wish to determine how to blend these outputs to produce a slurry of the lowest possible viscosity at a given loading. Of course, the answer to this question depends on the shape of the particles, the details of their interactions, etc., and is a prohibitively complex problem that probably cannot be solved exactly. Nevertheless, it should be possible to construct a relatively simple model that would give the correct trend of the dependence of viscosity on the slurry composition. Such a theory would allow an operator of a plant to choose good blending ratios for slurry formation without thinking too much about the complexity of the underlying hydrodynamic processes. Thus, motivated by this application to the flow of coal-water slurries, we will discuss the case of highly loaded systems where the content of the solid-phase approaches the close packing limit. The hydrodynamics of slurries is a well-studied subject with about a 70 year history.2 However, most
of the research in this field is devoted to dilute systems,3 where the concentration of solid particles is small enough to allow the use of a virial expansion of the interparticle interactions. Most work on concentrated systems is confined to the problem of monodisperse solutions of spheres. Polydisperse systems were discussed in the pioneering work of Farris,4 however, and this approach was based on an effective medium assumption. Unfortunately, this assumption is valid only if the difference in size of the constituent particles is very large. The work by Probstein et al.5 treats polydisperse systems by mapping them into “effective bimodal” systems with adequately chosen particle diameters; this approach is also valid only for large differences in sizes. Other recent experimental and theoretical works6-9 are devoted to bimodal polydisperse systems. Unfortunately, thus far there is not a comprehensive theory of the rheology of polydisperse concentrated systems. An approach to such a theory is put forward in this paper. We will argue that the main factor determining the viscosity of highly concentrated slurries is the random close packing ratio, that is, the maximal possible loading of a given slurry. A way to achieve minimal viscosity is to, therefore, increase the close packing ratio. The problem of increasing the maximal packing ratio of powders is well-known due to its importance in the construction industry and other applications (see ref (1) Shook, C. A.; Rocko, M. C. Slurry Flow. Principles and Practice; Butterworth-Heinemann: Boston, London, Oxford, Singapore, Sydney, Toronto, 1991. (2) Lamb, H. Hydrodynamics, 6th ed.; Dover Publications: New York, 1945. (3) Happel, J.; Brenner, H. Low Reynolds Number Hydrodynamics, with Special Applications to Particulate Media; Prentice-Hall: Englewood Cliffs, NJ, 1965. (4) Farris, R. J. Trans. Soc. Rheol. 1968, 12, 281-301. (5) Probstein, R. F.; Sengun, M. Z.; Tseng, T.-C. J. Rheol. 1994, 38, 811-829. (6) Sengun, M. Z.; Probstein, R. F. Rheol. Acta 1989, 28, 382-393. (7) Sengun, M. Z.; Probstein, R. F. Rheol. Acta 1989, 28, 394-401. (8) Shapiro, A. P.; Probstein, R. F. Phys. Rev. Lett. 1992, 68, 14221425. (9) D’Haene, P.; Mewis, J. Rheol. Acta 1994, 33, 165-174.
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10).10 The pioneering work in this field was done by Furnas11 as early as 1931. He proposed a semiempirical theory of the packing of spheres of different radii which works best for binary mixtures of spherical particles with a large difference in size. However, it has a number of difficulties when applied to the case of continuous particle size distributions, as we will discuss below. Moreover, it predicts only the “optimal” composition of a blend but gives no information about blends having any other composition. Another approach to the problem of the packing of powders is based on the recent theory of Oakeshott and Edwards.12 The advantage of this latter approach is that it starts with small deviations from the reference system of monodisperse spheres. This is different from the Furnas starting point of a binary mixture with infinitely different particle sizes. Accordingly, the Edwards approach may provide a more self-consistent transition to the case of continuously polydisperse powders. However, as of today, the Edwards calculations concern only powders with a polydispersity small enough to allow the use of a perturbation technique. The generalization of this method to the more realistic case of powders with a broad polydispersity remains an unsolved but very interesting and important problem. In this paper, we propose a new theory for the packing of particles based on a refinement of the original ideas of Furnas. This theory starts from a blend of particles with close sizes, thus ensuring a logical transition to continuous distributions. In the framework of this theory, we propose a scheme for the calculation of packing ratios of polydisperse powders. Then we use the results of these calculations to determine the viscosity of highly concentrated polydisperse slurries. And finally, we describe applications of the theory to the flow of coal-water slurries in a combustion power plant. We will commence with a brief review of the viscosity of slurries and the model developed by Furnas. 2. Viscosity of Highly Concentrated Slurries The viscosity of complex systems such as concentrated slurries is highly sensitive to how this property is defined and measured and also on the rate at which stress is applied. Here we will discuss only the simplest variantsthe effective shear viscosity at zero-frequency shear rate. Nevertheless, we believe that analogous results can be obtained at fixed shear stress or shear rate. For dilute suspensions the classical result of Einstein3 shows that the effective viscosity depends only on the loading
5 η ) η0 1 + φ , φ , 1 2
(
)
(1)
where η is the effective viscosity, η0 is the viscosity of the solvent, and φ is the loading (volume fraction of the solid phase). As loading is increased, the universality of eq 1 is lost and the effective viscosity depends on the details of interparticle interactions, particle shapes, etc. (10) Cumberland, D. J.; Crawford, R. J. The Packing of Particles; Elsevier: Amsterdam, New York, 1987. (11) Furnas, C. C. Ind. Eng. Chem. 1931, 23, 1052-1058. (12) Oakeshott, R. B. S.; Edwards, S. F. Phys. A 1994, 302, 482498.
The surprising fact is that in the limit of large φ, universality is recovered. This was first recognized by Dougherty and Krieger (see ref 13 and references therein), and we will show the validity of this conclusion here in a slightly more general framework than originally proposed. The viscosity of a suspension is usually expressed in the form of a power law, which takes account of the fact that beyond a certain value of the loading, φmax, the viscosity of the slurry is effectively infinite and flow ceases. Dougherty and Krieger obtained an equation of the following form
(
η ) η0 1 -
)
φ φmax
-[η(φ)]φmax
(2)
where [η(φ)] is the intrinsic viscosity and is supposed to be a “well-behaved” function of φ. If φ is close enough to φmax, the main contribution to eq 2 is from the singularity at φ f φmax. Accordingly, we will substitute the unknown function [η(φ)] by the intrinsic viscosity [η] ) [η(φmax)]. To determine the error of such an approximation, we can take the log of eq 2 and differentiate to obtain
(
)
[η(φ)] φ d[η(φ)] 1 dη - φmax ln 1 ) φ η dφ φmax dφ 1φmax
(3)
The first term in eq 3 describes the influence of the divergent factor 1 - φ/φmax in eq 2, while the second term describes the influence of the variation in the function [η(φ)]. If we substitute [η(φ)] by the constant value [η] ) [η(φmax)], we make an error of the order of the ratio of the second and first terms in eq 3. This error is negligible as long as
|
| [ (
)(
)]
φ φ 1 d[η(φ)] , - φmax 1 ln 1 φmax φmax [η(φ)] dφ
-1
(4)
Experimental data13 indicate that when φ changes from 0 to φmax, [η(φ)] changes from 2.5 to 2.7. Therefore, |d[η(φ)]/dφ|/[η(φ)] ≈ 0.1. On the other hand, the minimal value of the right-hand side of eq 4 is e/φmax ≈ 4 and is achieved at φ ) φmaxe/(e - 1) ) 0.63φmax. We see that the condition described by eq 4 is always satisfied. Therefore, we can write down
(
η ) η0 1 -
φ
)
φmax
-[η]φmax
(5)
where the intrinsic viscosity [η] is a constant coefficient approximately equal to 2.7. This simple argument shows why the DoughertyKrieger correlation, eq 5, is so successful for concentrated slurries13,14 and why other equations suggested for the concentration dependence of viscosity η(φ) give essentially the same result.1 Since eq 5 has the simplest form, this is what we will use for the calculation of viscosity. If the slurry is polydisperse, [η] and φmax depend on the distribution of sizes and shapes of the solid particles. (13) Krieger, I. M. Adv. Colloid Interface Sci. 1972, 3, 111-136. (14) Allain, C.; Cloitre, M.; Lacoste, B.; Marsone, I. J. Chem. Phys. 1994, 100, 4537-4542.
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Repeating the preceding considerations, we conclude that eq 5 is still valid but the parameters [η] and φmax are now some functions of these factors. We will neglect the variations in [η], concentrating on the contribution of the 1 - φ/φmax term. The recipe for the calculation of slurry viscosity is therefore the following: (1) Calculate the maximal random packing loading φmax. (2) Use the generalized Dougherty-Krieger correlation eq 5 to calculate η at a given loading φ. Now it is evident that the only way to decrease the viscosity at a given loading is to increase φmax. This is the problem of optimal packing of particles and is discussed in the following sections. 3. An Aside on Definitions In the following paragraphs we will use the expressions “optimal packing” (or loading) and “maximal packing” (or loading) frequently. It is important to make our meaning clear. For a monodisperse system of particles, the maximal packing corresponds to the close packing limit, φmax ) φ0 ≈ 0.637. If we have a binary mixture of large and small particles, we can have different loadings depending on the arrangement of large and small spheres. One such arrangement will give the greatest loading at a given composition of the mixture. We will call such loading the maximal packing φmax. If the fraction of small particles is low enough, the maximal loading is achieved by placing the small particles in the voids between the large ones. It is obvious that in the general case
φmax g φ0
(6)
Suppose that we are now allowed to change the composition of the binary mixture. The maximal loading φmax will change with the composition. There is a special optimal composition at which φmax achieves the greatest value φoptimal. We will call this value the optimal loading. We will argue below that the optimal binary mixture has “just enough” small particles to fit in the voids between the large ones. In the general case of polydisperse mixtures, our nomenclature will be analogous. We will call the maximal loading φmax the greatest loading achievable at a given composition of the mixture. In a typical situation, we are allowed to change the composition of the mixture subject to some constraints. We will call the mixture that satisfies these constraints and has the greatest φmax the optimal mixture and the corresponding loading the optimal loading. 4. Furnas Theory Revisited The Furnas theory of the packing of polydisperse particles has been a major tool in particle packing science for about 63 years. Whether in its original form11 or with some modifications (see ref 1), it has been used to find the optimal size distribution for particles in many industrial applications. However, there are a number of problems with Furnas’ theory. We will see that these deficiencies are caused mostly by the fact that Furnas started from the limiting case of a binary mixture with an (effectively) infinite size ratio. Consequently, the transition to a smooth size distribution
is neither simple nor convincing. We will present in this paper another approach to the packing of particles based on ideas close to those of Furnas. For clarity we will first present a brief summary of Furnas’ theory. Furnas starts with a binary mixture of two kinds of spheres of diameters DL and DS such that
DS ,1 DL
(7)
If we had only monodisperse large spheres randomly packed in a container, their maximal loading would be φ0 ≈ 0.637.10 We can increase this loading by placing smaller spheres in the voids between the larger ones, however. If the condition described by eq 7 is satisfied, then the volume fraction of the space accessible to the small spheres is 1 - φ0. If we fill all this space with small spheres, their volume fraction with respect to the original volume would be φ0(1 - φ0). The resulting binary mixture will have a composition ratio (volume ratio of large spheres to small spheres) of 1:(1 - φ0) and a solid loading of 1 - (1 - φ0)2. Any deviation from this composition ratio will lead to a looser packing. If, for example, the content of smaller spheres is less than 1 - φ0, there would be unfilled space between the large spheres. If this content is greater than 1 - φ0, then some small spheres will not fit between the large ones: they will be located in an “outside” region. The volume fraction of the solid phase in the “outside” region will be φ0. Therefore, the overall content of the solid phase would be less than 1 - φ0. We see that the composition 1:(1 - φ0) is the optimal composition for binary mixtures of spheres satisfying the condition of eq 7. The loading 1 - (1 - φ0)2 is the maximal loading for such mixtures. The extension to n-component systems with diameters satisfying the condition
DL ) D1 . D2 . ... . Dn ) DS
(8)
can be performed using Furnas’ “telescopic tube method”, which means that to obtain the optimal distribution we fill the available volume step by step. First, we place the largest spheres up to the maximal monodisperse loading φ0. Then we fill the space between the largest spheres by the particles of diameter D2. Then we repeat this procedure with spheres of diameter D3 and continue until the smallest particles are placed. By extending the arguments presented above for a binary system, it is easy to see that the resulting blend would have a composition defined by the ratios (largest to smallest)
1:(1 - φ0):(1 - φ0)2: ... :(1 - φ0)n-1
(9)
and the loading will be 1 - (1 - φ0)n. It is clear that this composition is optimal and the calculated loading is maximal. So far we have discussed only spherical particles, but these considerations should also apply to particles of any shape, as long as they are geometrically alike and the condition in eq 8 is satisfied. For nonspherical particles, we must interpret Di as some characteristic size. The only parameter affected by the nonsphericity is the close packing ratio φ0.
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Suppose now that we have a mixture of particles whose size difference is not as large, so eq 8 must be substituted by the weaker condition
DL ) D1 > D2 > ... > Dn ) DS
(10)
Also suppose that the diameters of the largest and smallest spheres, DL and DS, are fixed and we want to achieve the maximal loading of the blend by arbitrarily choosing the blend composition. Furnas proposed that the best choice is given by an equal spacing of D1, D2, ..., Dn in a log scale so that
(11)
Figure 1. Correlations for the function y(x). (s) Furnas correlation eq 12, (- - -) new correlation eq 23, (...) experimental data.11
where K is equal to DL/DS (note that Furnas’ equations appear slightly different than those given here because he considered a system of n + 1 different particle sizes while here we consider n). The composition of this Furnas blend is proposed to be the same as an “ideal” blend satisfying eq 8, namely eq 9. The only free parameter left is the number of “divisions” n or, equivalently, the ratio of the subsequent diameters x. A naive suggestion would be to take n as large as possible: in the ideal given by case eq 11, loading increases continuously with n. However, for large n, Furnas’ distribution becomes more and more narrow, going to a δ-like peak at n f ∞. For this limiting distribution the loading is φ0 instead of 1 as predicted by the ideal blend theory. The cause of this discrepancy is the fact that as n increases the subsequent sizes Di become closer, thus violating the basic assumption described by eq 8. We will call this a screening effect. To circumvent this difficulty, Furnas introduces an empirical factor y that characterizes the ratio of the actual gain in volume upon mixing to the ideal gain (i.e., when the assumption in eq 8 is valid). He assumes that for mixtures with a composition described by eq 11, this ratio depends only on the ratios of the sizes x but not on the number n of these different sizes. An experimental estimation11,15 for the function y(x) gives the following empirical expression:
Figure 2. Maximal loadings for binary blends. (s) Furnas recipe, eq 19, (- - -) optimal recipe, eq 24.
Dn D2 D3 ) ) ... ) ) x ) K1/(n-1) D1 D2 Dn-1
y(x) ) 1.0 - 2.62x + 1.62x2
(12)
What apparently has not been noticed over the years, or at least has not been remarked upon, is that this relationship is unphysical, becoming negative at values of y(x) slightly greater than 0.6, as shown in Figure 1 (in all fairness to Furnas, modern desktop computers that allow an easy fitting to data were not available in 1931 and he apparently fitted only the available data points, then extrapolated by hand the curve joining the last data point to the value of y(x) ) 0 at x ) 1). Also shown in this figure is a line describing a relationship we will derive later. The increase in loading with n is now affected by two factors: an increase in the “ideal” loading and a decrease of the ratio y. Calculating the maximal loading as function of n, Furnas suggested a rather complicated equation for determination of optimal number of sizes (see ref 11). This equation can be numerically solved (15) Anderegg, F. O. Ind. Eng. Chem. 1931, 23, 1058-1064.
for any given DL and DS. Unfortunately, in most cases, the solution is not an integer value so additional adjustments are necessary. If we have m > n sizes, another ad hoc adjustment must be made. After this sketch of the Furnas theory, we can see its advantages and flaws. It works best for polydisperse blends, where only a finite number of components are present and the sizes of the components are very different. If the sizes are close, the theory does not work in a self-consistent manner. Rather it uses a number of ad hoc adjustments and corrections. It is not clear why the number of consecutive diameters is affected by the “screening” effect but not the spacing of sizes nor the composition of the blend. Most important for our purposes, this theory gives the “optimal” composition but does not predict the loading if the composition is other than “optimal”. Here we will propose a new theory. Like the Furnas model, it is based on the “telescopic tubes” conceptual approach. However, unlike Furnas, we will start from a blend of particles of close diameters and will also consider the packing of an arbitrary blend. 5. Modified Theory of Packing of Polydisperse Particles Binary Mixtures. In the previous section, we saw that the main disadvantage of Furnas’ theory is caused by the fact that it is based on conclusions drawn from an unrealistic “ideal case”, namely, from a binary mixture of spheres of infinitely different radii. We will start an alternative approach from the “opposite end”, a binary mixture of particles of close radii. Let us discuss a volume V filled by particles with sizes DL and
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Energy & Fuels, Vol. 12, No. 5, 1998 1035
DS such that DL > DS. Suppose we first completely filled the space with the large particles. Their overall (solids) volume is V1 ) φ0V, where φ0 is the maximal loading for the monodisperse mixture. Let us now add the smaller particles. If the overall volume of the smaller particles is smaller than some value Vmax, they will fit in the space between the larger particles. Note what we are saying here. There will be a distribution of void sizes between the large particles. A proportion of these will be large enough to accommodate particles of size DS. The total volume of these particular voids is the quantity Vmax, and this will be proportional to the volume of voids (1 - φ0)V and will also depend in some unknown fashion on the ratio x ) DS/DL (for a simple binary mixture x ) K, see eq 11). We can write this relationship as
Vmax ) φ0(1 - φ0)VY(x)
(13)
where Y(x) is an unknown function. As long as the size of the container is large enough to neglect surface effects, Y(x) will not depend on V. In this case, Y(x) is a universal function (for a given particle shape) of the size ratio x. If x ) 0, Y(x) ) 1; this is the “ideal” case of infinitely different sizes, where Vmax ) φ0(1 - φ0)V. At the other extreme, if x ) 1, Y(x) ) 0; we have close packing of particles of diameter DL and cannot fit any more particles of the same radius into the volume V. It is obvious that the function Y(x) introduced here is closely related to Furnas’ function y(x). Actually, we will prove below that it coincides with Furnas’ function y(x). We will now calculate the maximal loading φmax of a binary mixture of particles having a fraction fL of large particles and a fraction fS ) 1 - fL of small particles. Returning to the container discussed above, we see that it contains a volume V1 ) φ0V of large particles and a volume V2 ) V1 fS/fL ) φ0VfS/fL of small particles. Therefore, the total volume of the solid phase is
Vsolid )
1 φV fL 0
(14)
If the total volume of all the smaller particles V2 is less than Vmax, then all smaller particles fit between the larger ones. Therefore, the total volume of the blend (including solid phase and voids) is V and the maximal loading is
φmax )
Vsolid 1 ) φ0, V2 e Vmax V fL
Vtotal e V +
(16)
As a crude estimation of Vtotal, we can then use the right-hand side of eq 16. Actually, this estimation is not so crude as it seems at first glance and can adequately describe some modes of the “random” packing of particles. So we have
φmax )
Vsolid ) Vtotal
φ0 fL Vmax 1φ0 V
V2 g Vmax
(17)
Using eqs 13, 15, and 17, we have for the loading:
{
φmax ) φ0 1 - fS
fS e φ0
1 - (1 - fS)(1 - φ0)Y(x)
fS g
(1 - φ0)Y(x) 1 + (1 - φ0)Y(x) (1 - φ0)Y(x) 1 + (1 - φ0)Y(x) (18)
We will now show that the function Y(x) coincides with the Furnas function y(x). Suppose that following the Furnas treatment we took closely packed large particles in a container of volume V and closely packed small particles in a volume (1 - φ0)V and mixed them. The maximal loading of such mixture, according to eq 18 is
φFurnas )
φ0(2 - φ0) 2 - φ0 - (1 - φ0)Y(x)
(19)
The volume of the solids in this mixture is φ0(2 - φ0)V, and the total volume of the mixture is
VFurnas )
φ0(2 - φ0) V φFurnas
(20)
Accordingly, the volume gain upon mixing is
∆V ) V + (1 - φ0)V - VFurnas ) (2 - φ0)(φFurnas - φ0) V (21) φFurnas The ideal volume gain is ∆Videal ) (1 - φ0)V. The Furnas function y(x) is, therefore given, by
(15)
If the volume of the small particles is greater than Vmax, they will not all fit between the larger particles. Therefore, the total volume of the blend Vtotal is greater than V. It is difficult to obtain an a priori prediction for Vtotal. However, we can estimate a value. Obviously, the volume Vtotal of the closely packed mixture is smaller than the volume of the following two-layered hypothetical structure: first layersall larger particles and those small particles that can fit between the larger ones; second layersthe remainder of the smaller particles packed with a loading φ0. It is easy to calculate the total volume of this twolayered structure, obtaining the following inequality:
fS 1 V - Vmax fL φ0
y(x) )
(2 - φ0)(φFurnas - φ0) ∆V ) ∆Videal φFurnas(1 - φ0)
(22)
Substituting φFurnas from eq 19, we obtain, after some algebra, y(x) ) Y(x), and we can use the notation y(x) for our empirical function Y(x), describing the maximal fraction of small particles that fit between the large ones. We will need an analytical expression for the function y(x). It is tempting to use Furnas’ correlation eq 12. Unfortunately, as we mentioned above, the correlation described by eq 12 has an unphysical region of negative y at x g 0.62 (Figure 1). Using the same data as Furnas, we obtained the following correlation for the function y(x) (Figure 1):
1036 Energy & Fuels, Vol. 12, No. 5, 1998
y(x) )
Veytsman et al.
1-x 1 + 2.68x + 3.98x2
(23)
This has the correct behavior at x ) 0 and x ) 1 and has the advantage of being monotonic in the interval 0 e x e 1. The correlation in eq 23, as well as the original correlation (eq 12), strictly speaking, only applies to spherical particles, but it is customary to neglect the nonuniversality of y(x) for particles close to spheres. We will, therefore, assume that the correlation in eq 23, is universal. Now we can consider the optimal mixing ratio for binary mixtures. Furnas suggests the composition 1:(1 - φ0), with the loading given by eq 19. However, eq 18 suggests that the optimal composition is 1:(1 - φ0)y(x), and the loading for this composition is
φoptimal ) φ0[1 + (1 - φ0)y(x)]
(24)
In the limit x f 0, y(x) f 1, these formulas coincide, giving φoptimal ) φFurnas ) φ0(2 - φ0). In the other limit, x f 1, y(x) f 0, they both give φoptimal ) φFurnas ) φ0. However, between these limits they differ and eq 24 always gives a higher maximal loading than Furnas’ recipe (see Figure 2). The difference is quite small (and this explains the success of Furnas’ recipe over the last 60 years), but we will see that it becomes important for polydisperse mixtures. We believe that this new theory is potentially more useful than the classical Furnas theory. Its main advantage is not just the higher loading predicted by eq 24 than by the Furnas recipe eq 19, the gain is quite small. The most important feature is that it can predict maximal loading for any composition. Polydisperse Blends. We now turn our attention to the more common situation of a polydisperse blend. Suppose we have n sets of particles of consecutive sizes DL ) D1, D2, ..., Dn ) DS that satisfy the condition described by eq 10. Let the volume of solids in each of these groups be v1, v2, ..., vn. The volume fractions of the particles of the corresponding sizes are f1, f2, ..., fn, where
fi )
vi
(25)
n vk ∑k)1
We want to calculate the maximal volume fraction of solids after having mixed our particles. We will solve this problem iteratively. Suppose we have mixed particles in the groups 1, 2, ..., i and want to add particles of the group i + 1. Also suppose the mixture has, after the ith, step, the maximal loading φi. The volume of solids in the resulting blend, the total volume of the blend, and the volume of voids are i
) Vsolid i
∑ k)1
vk, Vtotal ) i
1
i
∑ φ k)1 i
vk, Vvoids ) i
( ) 1
φi
-1
nas’ “telescopic tube” blend. Let us take particles of size D1 and fill the volume V with them. The effective minimal diameter of the resulting system is D1. Then we will add exactly as many particles of diameter D2 as will fit in the voids between the larger particles. The effective minimal diameter of the resulting blend is D2. Then we add exactly as many particles of diameter D3 that can be fit between the previously placed particles. The effective minimal diameter of the blend after this becomes D3. We can continue this process with particles of gradually decreasing sizes, and the effective diameter of the blend will decrease accordingly. Now we must make an assumption about the packing of these particles. We will call this assumption the basic hypothesis. We will assume that the amount of particles that can be added to any blend without increasing its total volume depends only on three parameters: the volume of voids, the added particle size, and the effective minimal diameter of the blend. This assumption means that all information about a polydisperse blend can be (for our purposes) mapped into two numbers: the volume fraction of voids and the minimal effective diameter. Actually, this assumption (as well as many other things in the packing of particles) was originally made implicitly but not explicitly by Furnas. It allowed him to use data obtained on binary blends for polydisperse mixtures. An explicit formulation of this underlying assumption allows one to exploit its deep consequences. The maximal volume of particles of the (i + 1)th group that can be fitted into the blend after the ith step can now be written as
∑ vi)y k)1
Di+1 Dieff
(27)
and the total volume of the particles in the system after the (i + 1)th step is
Vi+1total )
{
Vitotal Vitotal +
vi+1 e Vi+1max 1 (v - Vi+1max) φ0 i+1
vi+1 g Vi+1max
(28)
We can now calculate Di+1eff provided that we know Dieff. If vi+1 g Vi+1max, the result will be Di+1eff ) Dieff. Let us consider the case vi+1 e Vi+1max. To calculate Di+1eff, we note that the volume of particles of size Di+1 that can be fitted in the system without changing its volume after the (i + 1)th step is ∆Vi+1 ) Vi+1max - vi+1. On the other hand, according to the basic hypothesis, this volume is i+1 ∆Vi+1 ) φ0(Vitotal - ∑k)1 vi)y(Di+1/Di+1eff). Therefore, we have
i
∑ vk k)1
Di+1eff
(26)
We will call Dieff the effective minimal diameter of the mixture, if it is impossible to add any particles of size Dieff and larger without changing the volume of the mixture. To understand the meaning of Dieff, we will consider a special case of polydisperse mixtures: Fur-
( )
i
Vi+1max ) φ0(Vitotal -
{
Di+1
) x(yi+1) Di+1
vi+1 e Vi+1max
(29)
vi+1 g Vi+1max
where
yi+1 )
Vi+1max - vi+1 φ0(Vitotal -
i+1 vi) ∑k)1
(30)
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Energy & Fuels, Vol. 12, No. 5, 1998 1037
Figure 3. Packing of bidisperse slurries. (s, - - -) Theoretical predictions for “dry packing”, size ratios 4:1 and 2:1, correspondingly, (+, ×) data from ref 8.
Figure 4. Packing of bidisperse slurries. (+, ×) Theoretical predictions for “viscosity packing limit”, size ratios 4:1 and 2:1, correspondingly, (- - -, s) data from ref 8.
and x(y) is the inverse of the function y(x). Using the correlation in eq 23 for the function y(x) we can obtain the following analytical formula for the function x(y):
x(y) )
1-y 1 + 2.68y + 3.68y2
(31)
Equations 27-30 allow one to calculate the maximal loading of the given blend if the initial conditions V1total, D1eff are known. These initial conditions are
V1total )
v1 φ0
D1eff ) D1
(32)
Now the problem of computing the maximal loading of a given blend is solved. Indeed, for any given composition v1, v2, ..., vn of a blend with particle sizes D1, D2, ..., Dn, we can use the following step-by-step algorithm starting from the initial values of eq 32. (1) Provided that Dieff and Vitotal are known, calculate Vi+1max and yi+1 using eqs 27 and 30. (2) With these values of Vi+1max and yi+1, calculate Vi+1max and Di+1eff from eqs 28 and 29. (3) Return to step 1. Having calculated Vntotal, we can obtain the resulting maximal loading as
φmax )
n vi ∑k)1
Vntotal
(33)
Actually, our calculations depend only on the ratios of the partial volumes vi, so we can substitute in formulas 27-30 and 32 the corresponding volume fractions fi instead of the volumes vi. 6. Comparison with Experiment In this section, we discuss a comparison of this theory with experimental data. Since Andregg,15 there have been many papers dealing with experimental data on polydisperse slurries. One of the most fundamental studies, on bidisperse slurries, was published by Shapiro and Probstein.8 They measured packing limits and the viscosity of suspensions of glass spheres in glycerine. To interpret their data one should note that they found two distinct packing limits: the so-called “dry packing” limit (about 0.624 of solids) and the “viscosity limit” equal to 0.524. It could be speculated that the
Figure 5. Viscosity for mono- and bidisperse suspensions of glass beads in glycerine. (s) Theoretical prediction for monodisperse suspension, (- - -) theoretical prediction for the 50/50 mixture with size ratio 4:1, (+, ×) data from ref 8.
difference is connected to the “coating” of glass balls by glycerol during their viscosity measurements. A comparison of our prediction for the maximal “dry packing” with their experimental findings is shown in Figure 3. It can be seen that the agreement is reasonably good, particularly as no adjustable parameters were used to calculate the curves shown in Figure 3. To obtain predictions for viscosity, we used the formalism described above but substituted φmax by its experimental value φmax ) 0.524. The results are shown in Figure 4. Once again, the agreement seems to be quite reasonable. Using the data for the packing limit, we calculated the viscosity of bidisperse slurries (Figure 5). The theory seems to work remarkably well. Let us now consider polydisperse slurries. Probstein et al.5 measured the viscosity and maximal packing fraction for glass beads with uniformly distributed sizes. The maximal packing fraction for a uniform distribution depends only on the ratio Dmax/Dmin, the maximal and minimal sizes in the distribution. In Figure 6 we plot the experimental prediction as well as the data from ref 5. The agreement between the theory and experiments seems to be very good. To interpret the viscosity data we should once again note that the maximal packing for dry slurries and the “viscosity limit” differed in the experiments5 by a factor of 1.19. Introducing this factor into our equation we calculated the viscosity of the suspensions with a linear size distribution (Figure 7). The predictions are again in good agreement with experiment.
1038 Energy & Fuels, Vol. 12, No. 5, 1998
Veytsman et al.
Figure 6. Maximal packing for uniform size distributions. (s) Theoretical result, (+) data from ref 5.
Figure 7. Viscosity for polydisperse suspensions of glass beads in glycerine. (s) Theoretical prediction for linear size distribution, Dmax/Dmin ) 5.7, (+) data from ref 8.
7. Practical Guide to Optimization of Slurries In this final section we will present a simple hypothetical example of how our equations work. We are presently undertaking experimental studies that will allow a more detailed comparison of theory and experiment. We will present these in a future publication. Consider the following problem. There are two mills producing “fine” and coarse “particles”. The product slurry is based on a mix of the particles produced by both mills. Given the size distribution of the particles from each mill, one must find the mixing ratio that gives the slurry with the lowest viscosity. The methods described in this paper allow one to solve this problem. The information about the slurries can be represented as a pair of histograms (f1fine, f2fine, ..., fnfine) and (f1coarse, f2coarse, ..., fncoarse) where fifine,coarse is the volume fraction of particles with sizes between Di - ∆D/2 and Di + ∆D/ 2, ∆D being a step in a histogram. This histogram is the natural result of the size distribution measuring device commonly used in industry. At any given composition ξ:(1 - ξ) of the resulting mix, we can calculate the histogram of the mixture as
fi ) ξfifine + (1 - ξ)ficoarse
(34)
Using the algorithm developed in the previous section, we can then determine the maximal loading φmax(ξ) at any given composition. Then we can calculate the viscosity η(ξ) of the mixed slurry using eq 5. The minimum of the function η(ξ) (that coincides with the maximum of the function φmax(ξ) gives the optimal slurry composition for this system.
Figure 8. Viscosity of a mix of coal powders obtained from two mills. (a) Histogram of “fine” powder, (b) histogram of “coarse” powder, (c) relative viscosity of the mixture at the loading φ ) 60.1%.
A typical histogram of a micronized coal after grinding is shown on Figure 8a. Since a typical loading used in the flow of coal-water slurries is above 60%, the viscosity of the slurry is about 30 times greater than the viscosity of pure water. To lower the overall viscosity at the same loading (61.7%), we considered adding a controlled amount of coarse coal. The histogram of the coarse component is shown in Figure 8b. A plot of the viscosity as a function of the composition of the mixture is shown in Figure 8c. It can be seen that the calculated viscosity of the mixture is much less that the viscosity of the components. We have developed a computer program Coal Viscosity Calculator that allows an optimization of the slurry composition in the working conditions of a power plant.
Polydisperse Coal-Water Slurries
We believe that this software can be used also for other industrial problems where powder packing optimization is needed. 8. Conclusion In this paper we propose a theory for the packing viscosity of concentrated slurries. Unlike other theories, it does not rest on the assumption of a big difference in particles size between components and easily copes with overlapping distributions.
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Due to the semiempirical character of the assumptions, we do not claim to predict the details of the behavior of slurries. Rather we predict trends in the viscosity dependence on composition. However, we believe that this method will be useful for industrial applications when quick results are required. Acknowledgment. The authors gratefully acknowledge the support of the U.S. Department of Energy under cooperative agreement No. DE-FC22-92PC92162. EF980120G
