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Mixing Measures
7
Mixing Measures Ica Manas-Zloczower and Miron Kaufman
This chapter gives an overview of the most commonly used mixing measures for both dispersive and distributive mixing. Main emphasis is placed on the use of entropic measures for distributive mixing characterization. The application of entropic measures for systems involving multiple species is also discussed.
7.1
Introduction
In polymer processing operations the quality of the final product is highly dependent on mixing. A significant number of theoretical and experimental efforts were undertaken to predict the dependence of mixing quality on material parameters, design, and processing conditions [1]. Yet, a big challenge remains finding criteria which would allow for in situ or predictive mixing optimization. Manas-Zloczower and Tadmor [2] looked at the distribution of number of passes over the flights in single screw extruders as a way to assess dispersive mixing performance. Residence time distributions have also been studied in conjunction with their use in evaluating the distributive mixing in processing equipment. Gao et al. modeled and analyzed the mean residence time and residence time distribution in a twin screw extruder [3]. They and Gasner et al. found a correlation between percent drag flow and residence time [3, 4]. That is, a larger percentage of drag flow meant the fraction of residence time in the mixing elements decreased, resulting in poorer mixing quality. Vainio et al. used the shape of the residence time distribution to evaluate dispersive mixing efficiency [5]. Mixing performance is strongly affected by specific characteristics of the flow field. Several researchers have adopted criteria based on the evaluation of the flow strength and/or shear stress distributions [6–15]. In general, the dispersive mixing efficiency of a flow field can be characterized in terms that account for the elongational flow contribution and the magnitude of stresses generated. One simple way to quantify the elongational flow components is to compare the relative magnitudes of the rate of deformation, |D|, and the vorticity, |ω|, tensors. The parameter , defined as: =
|D| |D| + |ω |
(7.1)
can be used as a basic measure of the mixing efficiency when assessing machine design and/ or operating conditions. is equal to one for pure elongation, 0.5 for simple shear, and zero
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for pure rotation. Therefore, the closer the value of to one, the better the dispersive mixing efficiency is. A more rigorous method to quantify the elongational flow components is by employing a flow strength parameter, Sf, that is frame invariant: Sf =
2 (tr D2 )2 o2
tr D
(7.2)
o2
where D is the Jaumann time derivative of D, i.e., the time derivative of D with respect to a frame of reference rotating with the same angular velocity as the fluid element. The parameter Sf ranges from zero for pure rotational flow to infinite for pure elongational flow. A simple shear flow corresponds to a Sf value of one. For consistency the flow strength parameter can be normalized according to: new =
Sf 1 + Sf
(7.3)
Like , new is equal to one for pure elongation, 0.5 for simple shear, and zero for pure rotation. Flow field calculations allow computation of volumetric distributions of shear stress and the parameters and new. Calculations for new require the second derivatives of the velocities, and are therefore more sensitive to mesh design. A higher uniform mesh density has to be used in order to reduce numerical errors. Although useful, especially in terms of defining trends, this type of global characterization does not reflect the actual dynamics of the dispersive mixing process. Passive tracers have often been used to help evaluate mixing dynamics in a variety of processing equipment. These passive tracers are assumed not to affect the flow field and not to interact with each other. Wong and Manas-Zloczower [16] studied distributive mixing in internal mixers by tracking the motion of passive tracers. Distributive mixing was quantified in terms of the probability density function of a pairwise correlation function. Avalosse used a similar technique to study mixing in a stirred tank [17]. Mackley and Saraiva used kinematic mixing rates and concentration distributions of passive tracers to look at mixing in oscillatory flow within baffled tubes [18]. Yoshingaga et al. numerically simulated distributive mixing in a twin screw extruder using residence time distributions and distribution of length stretch between tracers [19]. Li and Manas-Zloczower [20, 21] and Cheng and Manas-Zloczower [22] have also used tracers to study the dynamics of the mixing process and proposed length and area stretch, pairwise correlation function, and volume fraction of islands as criteria to quantify distributive mixing. Shearer and Tzonganakis used reactive polymer tracers as a microscopic probe of the interfacial surface area between two polymer melts and found a non-linear relationship between screw speed and mixing performance [23]. Wang and ManasZloczower [24] used the tracers’ flow history to calculate temporal shear rate and flow strength parameter distributions. Temporal distributions are histograms displaying the flow history of a number of particles in the system, up to a given time. These temporal distributions can characterize the overall dispersive mixing efficiency of the mixer. The histories can also be used with a kinetic model of agglomerate dispersion or droplet breakup to calculate minor
7.2 Entropic Measures
253
component size distribution and examine the dispersive mixing capability of a mixing device for a specific system.
7.2
Entropic Measures
7.2.1
Shannon Entropy
The entropy is the rigorous measure of homogeneity or lack of information. The information entropy for a particular experimental condition with a set of M possible outcomes is: M
Sinformation = − ∑ p j ln p j j =1
(7.4)
where pj is the relative frequency of outcome #j. Equation (7.4) provides the standard measure [25, 26] of lack of information as it is the unique function that possesses the following desired features: •• the entropy depends only on the probability distribution; •• the lowest entropy (S = 0) corresponds to one of the probabilities being unity and the rest being zero (i.e., total information); •• the largest value for the entropy (S = lnM) is achieved when all probabilities are equal to each other (i.e., the absence of any information); and •• S is additive over partitions of the outcomes. To calculate the Shannon entropy S the region of study is divided into equal size bins and tracer densities in each bin are used as estimators of probabilities. Selection of the number of bins in the system is an important part of mixing characterization as it determines the scale of segregation. An analysis of the variation of entropy S with the number of bins enables one to determine the quality of mixing dependence on the scale of segregation.
7.2.2
Renyi Entropies
If the last axiom, the additivity one, is relaxed to consider only statistically independent partitions, Renyi [27] found that the information entropy is replaced by a one-variable function: M ln ∑ p j j =1 S( ) = 1−
(7.5)
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In the limit → 1, Eq. (7.5) reduces to Eq. (7.4). If > 0, the function S() defined in Eq. (7.5) is maximized when all p’s are equal to each other. It follows that 0 ≤ S() ≤ ln(M) provided β ≥ 0. Thus the relative entropy, a name coined by Shannon, S()/ln(M) constitutes an index of homogeneity: it is 100% for total homogeneity and is small for high order of segregation. By varying β the Renyi entropies can be tailored to specific needs. When ≤ 1 one focuses on low concentration regions, while when 1 areas of high concentration are probed. 7.2.2.1 Applications Wang, Manas-Zloczower, and Kaufman [28] have applied Renyi entropies to a twin flight single screw extruder by following trajectories of tracers advected by the flow. The Renyi entropy was determined as a function of the axial distance (for cross-sections perpendicular on the extruder axis) and for different values of the Renyi parameter, . A comparison of the entropy for = 2 to traditional correlations sum shows good agreement (Fig. 7.1). The entropic measure offers a couple of advantages over the correlations sum method: •• probing the concentrations of tracers in fixed bins renders the entropic measure less dependent on geometric details and •• the calculation time is shorter when estimating entropies.
Figure 7.1: Distributive mixing characterization in a twin flight single screw extruder. The evolution of the correlation dimensions from correlation sums, and Renyi entropy of = 2, for 655 (initially clustered in 5 groups evenly distributed in the cross section), 5770 (initially clustered into 10 groups evenly distributed in the cross section), and 10940 (initially clustered into 20 groups evenly distributed in the cross section particles using 405, 2880, and 6480 bins, respectively for the Renyi entropy. The particles are continuously injected at their initial positions. (Reprinted with permission from [28])
7.2 Entropic Measures
255
Figure 7.2: Distributive mixing characterization in a twin flight single screw extruder operated at various throttle ratios. The evolution of the relative Renyi entropies for throttle ratios of –0.5, 0, and +0.5 for (a) = 0, (b) = 1, (c) = 2, (d) = 10. (Reprinted with permission from [29])
Wang, Manas-Zloczower, and Kaufman [29] have used Renyi entropies to study the dependence of distributive mixing on processing conditions in a screw extruder. In Fig. 7.2, the entropy versus distance along the extruder is plotted for three processing conditions determined by the throttle ratio Qp/Qd. For negative values of the throttle ratio one gets a maximum in the entropy for a particular optimum extruder length. Wang, Manas-Zloczower, and Kaufman [30] have used Renyi entropies to study the dependence of distributive mixing on initial positions of the tracers. As apparent from Fig. 7.3, tracers located close to the screw exhibit better distributive mixing due to the fact that they move slower and the flow has greater circulation.
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Figure 7.3: Influence of initial particle positions on distributive mixing in a twin flight single screw extruder. Tracers are initially placed in the various regions of the cross-section (as shown in the insert above) and their distribution is recorded at the exit of the 3rd pitch; lines are color-coded based on the initial zone. The best mixing is achieved by tracers starting close to the screw. (Reprinted with permission from [30])
7.2.3
Multi-Component Shannon Entropy
The characterization of the quality of mixing for systems in which tracers are specified not only by their locations but also by some other physical property such as size, color, conductance, transparency, etc. is the focus of this section. Again, the space of interest is divided into M bins labeled j = 1, 2, … M and there are C species of tracers/particles present in the system and they are labeled c = 1, 2, … C. The overall quality of mixing is described by the Shannon entropy of Eq. (7.4), adapted to reflect the presence of more than one species in the system:
7.2 Entropic Measures
C
S = −∑
M
∑ pc, j lnpc, j
c =1 j =1
257
(7.6)
In Eq. (7.6), pc, j is the joint probability for a particle to be of species “c” and in bin “j” and is estimated by the fraction of particles of species c located in bin j out of all particles. Equation (7.6) can be further expanded, using the following conditional probabilities needed to express conditional entropies: pc/j is the probability of finding a particle of species c conditional on being in bin j, pj is the probability for bin j, pj/c is probability of finding a particle in bin j conditional on being of species c, and pc is the probability for species c. In view of the additivity axiom in Section 7.2.1, the Shannon entropy can be expressed as: S = S (locations) + Slocations (species)
(7.7)
where: Slocations (species) =
M
∑ [ p j S j (species)] j =1
C
S j (species) = − ∑ [ pc / j ln pc / j ] c =1
M
S (locations) = − ∑ [ p j ln p j ] j =1
(7.8)
(7.9) (7.10)
Sj (species) is the entropy of mixing the species at the location of bin j and S (locations) is the entropy associated with the overall spatial distribution of particles irrespective of species. Slocations (species) is a spatial average of the entropy of mixing of species conditional on location and because Slocations (species) ≤ ln(C), one can normalize the entropy by ln(C) to get the relative entropy, which takes values between 0 and 1. In view of the additivity axiom, one can also express the Shannon entropy as: S = S (species) + Sspecies (locations)
(7.11)
where: C
S (species) = − ∑ [ pc ln pc ] c =1
M
Sc (locations) = − ∑ [ p j / c ln p j / c ] j =1
Sspecies (locations) =
C
∑ [ pc Sc (locations)] c =1
(7.12) (7.13)
(7.14)
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7 Mixing Measures
The entropy of species, Sspecies (locations), describes the distribution for particles of that particular species only. Note that since Sspecies (locations) ≤ ln(M), one can normalize the entropy by ln(M) to get the relative entropy, which takes values between 0 and 1. 7.2.3.1 Application: Simultaneous Dispersive and Distributive Mixing Index As an example of using the entropy Sspecies (locations) for system characterization, Alemaskin, Manas-Zloczower, and Kaufman [31] employed the entropy of species to develop an index reflecting the dispersion and distribution of cohesive agglomerates in a single screw extruder. In the example presented, dispersion occurs by an erosion mechanism and involves the formation of different size species in the system. The index developed characterizes simultaneously the spatial mixing of each species present in the system, but also a material property, as reflected in a preference for a certain size species. This index is based on replacing the pc’s in Eq. (7.14) with Gaussian distribution weights centered on the preferred size: I =
C 1 f c Sc (locations) ∑ ln(M ) c =1
(7.15)
The index I is a weighted average of the entropies associated with different species, with weights fc that add up to unity, which reflect the relative importance of different sizes for the physical specifications of the final product. For Gaussian weights:
Figure 7.4: Comparison between the mixing efficiency of different helix angles extruders. The case studied involved agglomerates of silica of initial radius 0.5 mm dispersing by an erosion process in a Newtonian fluid of viscosity 1160 Pa s. The preferred size is represented by agglomerates of radius 0.039 mm. (Reprinted with permission from [31])
7.2 Entropic Measures
(rc − ropt )2 f c = A exp − 2 2 C (rc − ropt )2 A = ∑ exp − 2 2 c =1
−1
259
(7.16)
The favored size is ropt and the width of the distribution function is . Alemaskin, ManasZloczower, and Kaufman [31] have explored using the index I for agglomerates dispersing and distributing in a single screw extruder by Newtonian and non-Newtonian fluids. Different extruder designs (different helix angles) were considered. Figure 7.4 shows the effect of extruder design on mixing efficiency when an “intermediate” particle size is the desired outcome. The evolution of the mixing index I along the extruder length also points to an optimum extruder length when a preferential particle size is the goal.
7.2.4
Modified Multi-Component Shannon Entropy
Perfect mixing of several components is achieved when at each spatial location the fraction of each component is the same as in the whole system. Motivated by this characterization of perfect mixing, Camesasca, Kaufman, and Manas-Zloczower [32] modified the multicomponent Shannon entropy as described below. The probabilities pj, c are estimated by using frequencies: n j, c p j, c =
M
Pc C
ni , c
∑∑P i =1 c =1
(7.17)
c
where nj, c is the number of particles of species c in bin #j and Pc is the relative concentration of species c in the whole system. The entropy achieves its maximum when all pj, c are equal to each other. In this case, Eq. (7.17) implies: C
n j ,1 P1
=
n j, 2 P2
= ... =
n j, C PC
=
∑ n j, c c =1 C
∑ Pc
for j = 1, 2, , M
(7.18)
c =1
The significance of Eq. (7.18) is that the modified multi-component Shannon entropy is maximized when at each location j the fraction of each component is the same as in the whole system, which is the main characteristic of perfect mixing. The pj, c represents the joint probability to find a complex of size Pc in species c in bin #j.
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7.2.4.1 Applications to Extruders Camesasca, Manas-Zloczower, and Kaufman [33] have compared the quality of mixing of two miscible polymeric melts distinguished by color in three different design extruders: single screw, single screw with ridges on the channel bottom, and single screw with pins on the channel bottom, operated under the same conditions. The species entropy conditional on locations versus axial distance for the three extruders is shown in Fig. 7.5. The best mixer was found to be the single screw extruder with pins. The role of the scale of segregation is also illustrated: at smaller scale, e.g., higher number of bins used in the analysis, the mixing quality decreases. Mixing Entropy vs. distance from Inlet - 120 bins
1
simple ridge pin
0.9 0.8
0.8 0.7 Mixing Entropy
0.6 0.5 0.4
0.6 0.5 0.4
0.3
0.3
0.2
0.2
0.1
0.1
(a)
0
simple ridge pin
0.9
0.7 Mixing Entropy
Mixing Entropy vs. distance from Inlet - 480 bins
1
0
20
40 60 80 Axial Distance From Inlet (10 -2 m)
100
(b)
0
0
20
40 60 80 Axial Distance From Inlet (10 -2 m)
100
Mixing Entropy vs. distance from Inlet - 969 bins
1
simple ridge pin
0.9 0.8
Mixing Entropy
0.7 0.6 0.5 0.4 0.3 0.2 0.1
(c)
0
0
20
40 60 80 Axial Distance From Inlet (10 -2 m)
100
Figure 7.5: The species entropy conditional on locations, normalized by ln(2), versus the axial distance from the inlet at different scales of observation. (a) 120 bins; (b) 480 bins; (c) 969 bins. (Reprinted with permission from [33])
7.2 Entropic Measures
261
Alemaskin, Manas-Zloczower, and Kaufman [34] have used the modified entropy to evaluate the color homogeneity of a mixture of resin pellets of two different colors. Seven cuts from the extrudate in a “screw crash” experiment, performed at Dow Chemicals Company, were analyzed using the RGB representation of colored pictures. For each pixel of a grayscale picture there is an intensity n, varying from 0 for black to 255 for white. In the Alemaskin et al. scheme [34], at each pixel there are n white and (255 – n) black particles. The modified entropy Slocation (species), with black and white being the possible species, can be calculated after deciding the ideal concentrations as described in full detail in [34]. The extrudate sample and the variation of the index of color homogeneity for the seven cross-sections examined is shown in Fig. 7.6. The same authors [35] have adapted their methodology to measure color uniformity in numerical simulations of flow in an extruder when advecting light pellets (tracers) of two colors.
5 cm
Figure 7.6: Top: Extrudate samples and the 7 cross-sections analyzed; Bottom: Evolution of color homogeneity index based on the grayscale entropic analysis. (Reprinted with permission from [34])
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7.2.4.2 Applications to Micromixers Camesasca, Manas-Zloczower, and Kaufman [36] have expanded their work on simulating and measuring mixing of fluids to microfluidics. In particular, by using the Slocations (species) they demonstrate quantitatively that the staggered herringbone mixer developed by Whitesides’ group [37] is a better mixer than the straight diagonal ridges micromixer. Camesasca, Kaufman, and Manas-Zloczower have recently found [38] that replacing the periodic patterning of the walls with self-similar (fractal) patterning yields higher levels of mixing. This is shown in Fig. 7.7, where three microchannels designed using a fractal protocol of patterning are compared with the classical staggered herringbone mixer. The entropic measures developed by Kaufman, Manas-Zloczower, and their students have been adopted by several other groups researching micromixing [39, 40, 41]. ELQVPPUHVROXWLRQ
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Figure 7.7: Mixing in microchannels as described by the mixing entropy at different scales of obser vation (from 50 to 2.5 μm resolution). Mixing efficiency at all scales of observation show the mixing efficiency ranging W1.25 > W1.50 > mSHM > W1.75. The symbol W# stands for the classical Weierstrass function with a fractal dimension “#” used in the patterning. (Reprinted with permission from [38])
7.2 Entropic Measures
263
7.2.5 Renyi Generalized Entropies and Fractal Properties The fractal properties of microstructures generated as a result of mixing can be also described in terms of multifractal dimensions. Consider the system completely covered with M equal volume boxes, M = –D, where = a/L is the ratio of the linear size of the box, a, to the linear size of the system, L, and D is the dimension of the embedding space. Parameter measures the scale of segregation in the analysis. As the number of boxes M increases, or the scale of segregation decreases, S() ≈ –d() ln(), where d() are labeled as the Renyi generalized dimensions. Thus, at small scales of segregation, 1, the relative entropy is approximately the ratio of the generalized dimension to the embedding dimension: S( )/ln(M ) ≈ d( )/ D
(7.19)
The interpretation of d() as a dimension originates in the case → 0. According to Eq. (7.19), S(0) = ln(number of occupied boxes). But the fractal (Hausdorff) dimension is: dHausdorff = ln (number of occupied boxes)/–ln() = S(0)/–ln() = DS(0)/lnM. One defines generalized dimensions in the same manner for all values of > 0. For = 1, one gets the information dimension as S(1) is the Shannon entropy. For = 2, one gets the correlation dimension as S(2), which is related to the logarithm of the correlation sum. 7.2.5.1 Applications Kaufman, Camesasca, and Manas-Zloczower [42] determined the multifractal dimensions for the published [37] images of mixing of a fluorescent and a non-fluorescent fluid in a microchannel. Camesasca, Kaufman, and Manas-Zloczower [38] have recently demonstrated that characterizing the interface of two fluids with multifractal dimensions provides a quantification of mixing that complements the entropy. In Fig. 7.8, one can see the increase in the topological, information, and correlation dimensions as the mixing progresses. Camesasca [43] has also calculated entropies and fractal dimensions for the baker’s transformation [44] showing the dimension of the interface between the two species is evolving from 1 (line) to 2 (surface) as the degree of mixing increases.
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7 Mixing Measures
W-1.25
W-1.50
2.0
2.0
Correlation Information Topological
1.6
1.4
1.2
1.0
Correlation Information Topological
1.8
Genralized Dimensions
Genralized Dimensions
1.8
1.6
1.4
1.2
0
1000
2000
3000
Distance From Inlet [Pm]
4000
1.0
5000
0
1000
2000
W-1.75
5000
3000
4000
5000
2.0
Correlation Information Topological
Correlation Information Topological
1.8
Genralized Dimensions
1.8
Genralized Dimensions
4000
mSHM
2.0
1.6
1.4
1.2
1.0
3000
Distance From Inlet [Pm]
1.6
1.4
1.2
0
1000
2000
3000
Distance From Inlet [Pm]
4000
5000
1.0
0
1000
2000
Distance From Inlet [Pm]
Figure 7.8: Generalized dimensions D0, D1, D2 calculated for the systems W1.25, W1.50, W1.7,5 and mSHM. The common trend is an increase from D = 1 at the inlet (before starting the mixing action) where the interface is a simple line to D > 1 as mixing progresses; a substantial increase in all of the mixers is present. (Reprinted with permission from [38])
7.3
Summary
We have demonstrated through several examples (polymer extrusion, microfluidics) that entropy can be tailored to efficiently evaluate mixing in a variety of applications. The procedures developed are versatile in the sense that they can be applied to both numerical work and to experimental results via image processing. The entropic mixing measures have the potential to be useful for optimization of equipment design and processing conditions. Studies of the plausible correlation between the multifractal dimensions of the microstructures resulting from the mixing processes and the physical properties of the material, such as electrical conductivity, may facilitate the design and realization of materials with tailored properties.
References
265
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[33] Camesasca, M., Manas-Zloczower, I., Kaufman, M. Plastics, Rubber and Composites: Macro molecular Engineering, 33, 372 (2005). [34] Alemaskin, K., Manas-Zloczower, I., Kaufman, M., Polym. Eng. Sci., 45, 1031 (2005). [35] Alemaskin, K., Manas-Zloczower, I., Kaufman, M., Polym. Eng. Sci., 45, 1011 (2005). [36] Camesasca, M., Manas-Zloczower, I., Kaufman, M., J. Micromech. Microeng., 15, 2038 (2005). [37] Strook, A. D., Dertinger, S. K. W., Ajdari, A., Mezic, I., Stone, H. A., Whitesides, G. M., Science, 295, 647 (2002). [38] Camesasca, M., Kaufman, M., Manas-Zloczower, I., J. Micromech. Microeng., 16, 2298 (2006). [39] Kang, T. G., Kwon, T. H., J. Micromech. Microeng., 14, 891 (2004). [40] Chang, C.-C., Yang, R-J., J. Micromech. Microeng., 16, 1453(2006). [41] Hoffmann, M., Schluter, M., Rabiger, N., Chem. Eng. Scie., 61, 29618 (2006). [42] Kaufman, M., Camesasca, M., Manas-Zloczower, I., Nanotech 2006, NSTI, Nanotechnology Con ference and Trade Show Technical Proceedings, Vol. 2, 578 Boston (2006). [43] Camesasca, M., Ph. D. Thesis, Case Western Reserve University, 2006. [44] Wiggins, S., Ottino, J. M., Phil. Trans. R. Soc. Lond. A, 362, 937 (2004).
Mixing Measures Ica Manas-Zloczower and Miron Kaufman
This chapter gives an overview of the most commonly used mixing measures for both dispersive and distributive mixing. Main emphasis is placed on the use of entropic measures for distributive mixing characterization. The application of entropic measures for systems involving multiple species is also discussed.
7.1
Introduction
In polymer processing operations the quality of the final product is highly dependent on mixing. A significant number of theoretical and experimental efforts were undertaken to predict the dependence of mixing quality on material parameters, design, and processing conditions [1]. Yet, a big challenge remains finding criteria which would allow for in situ or predictive mixing optimization. Manas-Zloczower and Tadmor [2] looked at the distribution of number of passes over the flights in single screw extruders as a way to assess dispersive mixing performance. Residence time distributions have also been studied in conjunction with their use in evaluating the distributive mixing in processing equipment. Gao et al. modeled and analyzed the mean residence time and residence time distribution in a twin screw extruder [3]. They and Gasner et al. found a correlation between percent drag flow and residence time [3, 4]. That is, a larger percentage of drag flow meant the fraction of residence time in the mixing elements decreased, resulting in poorer mixing quality. Vainio et al. used the shape of the residence time distribution to evaluate dispersive mixing efficiency [5]. Mixing performance is strongly affected by specific characteristics of the flow field. Several researchers have adopted criteria based on the evaluation of the flow strength and/or shear stress distributions [6–15]. In general, the dispersive mixing efficiency of a flow field can be characterized in terms that account for the elongational flow contribution and the magnitude of stresses generated. One simple way to quantify the elongational flow components is to compare the relative magnitudes of the rate of deformation, |D|, and the vorticity, |ω|, tensors. The parameter , defined as: =
|D| |D| + |ω |
(7.1)
can be used as a basic measure of the mixing efficiency when assessing machine design and/ or operating conditions. is equal to one for pure elongation, 0.5 for simple shear, and zero
252
7 Mixing Measures
for pure rotation. Therefore, the closer the value of to one, the better the dispersive mixing efficiency is. A more rigorous method to quantify the elongational flow components is by employing a flow strength parameter, Sf, that is frame invariant: Sf =
2 (tr D2 )2 o2
tr D
(7.2)
o2
where D is the Jaumann time derivative of D, i.e., the time derivative of D with respect to a frame of reference rotating with the same angular velocity as the fluid element. The parameter Sf ranges from zero for pure rotational flow to infinite for pure elongational flow. A simple shear flow corresponds to a Sf value of one. For consistency the flow strength parameter can be normalized according to: new =
Sf 1 + Sf
(7.3)
Like , new is equal to one for pure elongation, 0.5 for simple shear, and zero for pure rotation. Flow field calculations allow computation of volumetric distributions of shear stress and the parameters and new. Calculations for new require the second derivatives of the velocities, and are therefore more sensitive to mesh design. A higher uniform mesh density has to be used in order to reduce numerical errors. Although useful, especially in terms of defining trends, this type of global characterization does not reflect the actual dynamics of the dispersive mixing process. Passive tracers have often been used to help evaluate mixing dynamics in a variety of processing equipment. These passive tracers are assumed not to affect the flow field and not to interact with each other. Wong and Manas-Zloczower [16] studied distributive mixing in internal mixers by tracking the motion of passive tracers. Distributive mixing was quantified in terms of the probability density function of a pairwise correlation function. Avalosse used a similar technique to study mixing in a stirred tank [17]. Mackley and Saraiva used kinematic mixing rates and concentration distributions of passive tracers to look at mixing in oscillatory flow within baffled tubes [18]. Yoshingaga et al. numerically simulated distributive mixing in a twin screw extruder using residence time distributions and distribution of length stretch between tracers [19]. Li and Manas-Zloczower [20, 21] and Cheng and Manas-Zloczower [22] have also used tracers to study the dynamics of the mixing process and proposed length and area stretch, pairwise correlation function, and volume fraction of islands as criteria to quantify distributive mixing. Shearer and Tzonganakis used reactive polymer tracers as a microscopic probe of the interfacial surface area between two polymer melts and found a non-linear relationship between screw speed and mixing performance [23]. Wang and ManasZloczower [24] used the tracers’ flow history to calculate temporal shear rate and flow strength parameter distributions. Temporal distributions are histograms displaying the flow history of a number of particles in the system, up to a given time. These temporal distributions can characterize the overall dispersive mixing efficiency of the mixer. The histories can also be used with a kinetic model of agglomerate dispersion or droplet breakup to calculate minor
7.2 Entropic Measures
253
component size distribution and examine the dispersive mixing capability of a mixing device for a specific system.
7.2
Entropic Measures
7.2.1
Shannon Entropy
The entropy is the rigorous measure of homogeneity or lack of information. The information entropy for a particular experimental condition with a set of M possible outcomes is: M
Sinformation = − ∑ p j ln p j j =1
(7.4)
where pj is the relative frequency of outcome #j. Equation (7.4) provides the standard measure [25, 26] of lack of information as it is the unique function that possesses the following desired features: •• the entropy depends only on the probability distribution; •• the lowest entropy (S = 0) corresponds to one of the probabilities being unity and the rest being zero (i.e., total information); •• the largest value for the entropy (S = lnM) is achieved when all probabilities are equal to each other (i.e., the absence of any information); and •• S is additive over partitions of the outcomes. To calculate the Shannon entropy S the region of study is divided into equal size bins and tracer densities in each bin are used as estimators of probabilities. Selection of the number of bins in the system is an important part of mixing characterization as it determines the scale of segregation. An analysis of the variation of entropy S with the number of bins enables one to determine the quality of mixing dependence on the scale of segregation.
7.2.2
Renyi Entropies
If the last axiom, the additivity one, is relaxed to consider only statistically independent partitions, Renyi [27] found that the information entropy is replaced by a one-variable function: M ln ∑ p j j =1 S( ) = 1−
(7.5)
254
7 Mixing Measures
In the limit → 1, Eq. (7.5) reduces to Eq. (7.4). If > 0, the function S() defined in Eq. (7.5) is maximized when all p’s are equal to each other. It follows that 0 ≤ S() ≤ ln(M) provided β ≥ 0. Thus the relative entropy, a name coined by Shannon, S()/ln(M) constitutes an index of homogeneity: it is 100% for total homogeneity and is small for high order of segregation. By varying β the Renyi entropies can be tailored to specific needs. When ≤ 1 one focuses on low concentration regions, while when 1 areas of high concentration are probed. 7.2.2.1 Applications Wang, Manas-Zloczower, and Kaufman [28] have applied Renyi entropies to a twin flight single screw extruder by following trajectories of tracers advected by the flow. The Renyi entropy was determined as a function of the axial distance (for cross-sections perpendicular on the extruder axis) and for different values of the Renyi parameter, . A comparison of the entropy for = 2 to traditional correlations sum shows good agreement (Fig. 7.1). The entropic measure offers a couple of advantages over the correlations sum method: •• probing the concentrations of tracers in fixed bins renders the entropic measure less dependent on geometric details and •• the calculation time is shorter when estimating entropies.
Figure 7.1: Distributive mixing characterization in a twin flight single screw extruder. The evolution of the correlation dimensions from correlation sums, and Renyi entropy of = 2, for 655 (initially clustered in 5 groups evenly distributed in the cross section), 5770 (initially clustered into 10 groups evenly distributed in the cross section), and 10940 (initially clustered into 20 groups evenly distributed in the cross section particles using 405, 2880, and 6480 bins, respectively for the Renyi entropy. The particles are continuously injected at their initial positions. (Reprinted with permission from [28])
7.2 Entropic Measures
255
Figure 7.2: Distributive mixing characterization in a twin flight single screw extruder operated at various throttle ratios. The evolution of the relative Renyi entropies for throttle ratios of –0.5, 0, and +0.5 for (a) = 0, (b) = 1, (c) = 2, (d) = 10. (Reprinted with permission from [29])
Wang, Manas-Zloczower, and Kaufman [29] have used Renyi entropies to study the dependence of distributive mixing on processing conditions in a screw extruder. In Fig. 7.2, the entropy versus distance along the extruder is plotted for three processing conditions determined by the throttle ratio Qp/Qd. For negative values of the throttle ratio one gets a maximum in the entropy for a particular optimum extruder length. Wang, Manas-Zloczower, and Kaufman [30] have used Renyi entropies to study the dependence of distributive mixing on initial positions of the tracers. As apparent from Fig. 7.3, tracers located close to the screw exhibit better distributive mixing due to the fact that they move slower and the flow has greater circulation.
256
7 Mixing Measures
Figure 7.3: Influence of initial particle positions on distributive mixing in a twin flight single screw extruder. Tracers are initially placed in the various regions of the cross-section (as shown in the insert above) and their distribution is recorded at the exit of the 3rd pitch; lines are color-coded based on the initial zone. The best mixing is achieved by tracers starting close to the screw. (Reprinted with permission from [30])
7.2.3
Multi-Component Shannon Entropy
The characterization of the quality of mixing for systems in which tracers are specified not only by their locations but also by some other physical property such as size, color, conductance, transparency, etc. is the focus of this section. Again, the space of interest is divided into M bins labeled j = 1, 2, … M and there are C species of tracers/particles present in the system and they are labeled c = 1, 2, … C. The overall quality of mixing is described by the Shannon entropy of Eq. (7.4), adapted to reflect the presence of more than one species in the system:
7.2 Entropic Measures
C
S = −∑
M
∑ pc, j lnpc, j
c =1 j =1
257
(7.6)
In Eq. (7.6), pc, j is the joint probability for a particle to be of species “c” and in bin “j” and is estimated by the fraction of particles of species c located in bin j out of all particles. Equation (7.6) can be further expanded, using the following conditional probabilities needed to express conditional entropies: pc/j is the probability of finding a particle of species c conditional on being in bin j, pj is the probability for bin j, pj/c is probability of finding a particle in bin j conditional on being of species c, and pc is the probability for species c. In view of the additivity axiom in Section 7.2.1, the Shannon entropy can be expressed as: S = S (locations) + Slocations (species)
(7.7)
where: Slocations (species) =
M
∑ [ p j S j (species)] j =1
C
S j (species) = − ∑ [ pc / j ln pc / j ] c =1
M
S (locations) = − ∑ [ p j ln p j ] j =1
(7.8)
(7.9) (7.10)
Sj (species) is the entropy of mixing the species at the location of bin j and S (locations) is the entropy associated with the overall spatial distribution of particles irrespective of species. Slocations (species) is a spatial average of the entropy of mixing of species conditional on location and because Slocations (species) ≤ ln(C), one can normalize the entropy by ln(C) to get the relative entropy, which takes values between 0 and 1. In view of the additivity axiom, one can also express the Shannon entropy as: S = S (species) + Sspecies (locations)
(7.11)
where: C
S (species) = − ∑ [ pc ln pc ] c =1
M
Sc (locations) = − ∑ [ p j / c ln p j / c ] j =1
Sspecies (locations) =
C
∑ [ pc Sc (locations)] c =1
(7.12) (7.13)
(7.14)
258
7 Mixing Measures
The entropy of species, Sspecies (locations), describes the distribution for particles of that particular species only. Note that since Sspecies (locations) ≤ ln(M), one can normalize the entropy by ln(M) to get the relative entropy, which takes values between 0 and 1. 7.2.3.1 Application: Simultaneous Dispersive and Distributive Mixing Index As an example of using the entropy Sspecies (locations) for system characterization, Alemaskin, Manas-Zloczower, and Kaufman [31] employed the entropy of species to develop an index reflecting the dispersion and distribution of cohesive agglomerates in a single screw extruder. In the example presented, dispersion occurs by an erosion mechanism and involves the formation of different size species in the system. The index developed characterizes simultaneously the spatial mixing of each species present in the system, but also a material property, as reflected in a preference for a certain size species. This index is based on replacing the pc’s in Eq. (7.14) with Gaussian distribution weights centered on the preferred size: I =
C 1 f c Sc (locations) ∑ ln(M ) c =1
(7.15)
The index I is a weighted average of the entropies associated with different species, with weights fc that add up to unity, which reflect the relative importance of different sizes for the physical specifications of the final product. For Gaussian weights:
Figure 7.4: Comparison between the mixing efficiency of different helix angles extruders. The case studied involved agglomerates of silica of initial radius 0.5 mm dispersing by an erosion process in a Newtonian fluid of viscosity 1160 Pa s. The preferred size is represented by agglomerates of radius 0.039 mm. (Reprinted with permission from [31])
7.2 Entropic Measures
(rc − ropt )2 f c = A exp − 2 2 C (rc − ropt )2 A = ∑ exp − 2 2 c =1
−1
259
(7.16)
The favored size is ropt and the width of the distribution function is . Alemaskin, ManasZloczower, and Kaufman [31] have explored using the index I for agglomerates dispersing and distributing in a single screw extruder by Newtonian and non-Newtonian fluids. Different extruder designs (different helix angles) were considered. Figure 7.4 shows the effect of extruder design on mixing efficiency when an “intermediate” particle size is the desired outcome. The evolution of the mixing index I along the extruder length also points to an optimum extruder length when a preferential particle size is the goal.
7.2.4
Modified Multi-Component Shannon Entropy
Perfect mixing of several components is achieved when at each spatial location the fraction of each component is the same as in the whole system. Motivated by this characterization of perfect mixing, Camesasca, Kaufman, and Manas-Zloczower [32] modified the multicomponent Shannon entropy as described below. The probabilities pj, c are estimated by using frequencies: n j, c p j, c =
M
Pc C
ni , c
∑∑P i =1 c =1
(7.17)
c
where nj, c is the number of particles of species c in bin #j and Pc is the relative concentration of species c in the whole system. The entropy achieves its maximum when all pj, c are equal to each other. In this case, Eq. (7.17) implies: C
n j ,1 P1
=
n j, 2 P2
= ... =
n j, C PC
=
∑ n j, c c =1 C
∑ Pc
for j = 1, 2, , M
(7.18)
c =1
The significance of Eq. (7.18) is that the modified multi-component Shannon entropy is maximized when at each location j the fraction of each component is the same as in the whole system, which is the main characteristic of perfect mixing. The pj, c represents the joint probability to find a complex of size Pc in species c in bin #j.
260
7 Mixing Measures
7.2.4.1 Applications to Extruders Camesasca, Manas-Zloczower, and Kaufman [33] have compared the quality of mixing of two miscible polymeric melts distinguished by color in three different design extruders: single screw, single screw with ridges on the channel bottom, and single screw with pins on the channel bottom, operated under the same conditions. The species entropy conditional on locations versus axial distance for the three extruders is shown in Fig. 7.5. The best mixer was found to be the single screw extruder with pins. The role of the scale of segregation is also illustrated: at smaller scale, e.g., higher number of bins used in the analysis, the mixing quality decreases. Mixing Entropy vs. distance from Inlet - 120 bins
1
simple ridge pin
0.9 0.8
0.8 0.7 Mixing Entropy
0.6 0.5 0.4
0.6 0.5 0.4
0.3
0.3
0.2
0.2
0.1
0.1
(a)
0
simple ridge pin
0.9
0.7 Mixing Entropy
Mixing Entropy vs. distance from Inlet - 480 bins
1
0
20
40 60 80 Axial Distance From Inlet (10 -2 m)
100
(b)
0
0
20
40 60 80 Axial Distance From Inlet (10 -2 m)
100
Mixing Entropy vs. distance from Inlet - 969 bins
1
simple ridge pin
0.9 0.8
Mixing Entropy
0.7 0.6 0.5 0.4 0.3 0.2 0.1
(c)
0
0
20
40 60 80 Axial Distance From Inlet (10 -2 m)
100
Figure 7.5: The species entropy conditional on locations, normalized by ln(2), versus the axial distance from the inlet at different scales of observation. (a) 120 bins; (b) 480 bins; (c) 969 bins. (Reprinted with permission from [33])
7.2 Entropic Measures
261
Alemaskin, Manas-Zloczower, and Kaufman [34] have used the modified entropy to evaluate the color homogeneity of a mixture of resin pellets of two different colors. Seven cuts from the extrudate in a “screw crash” experiment, performed at Dow Chemicals Company, were analyzed using the RGB representation of colored pictures. For each pixel of a grayscale picture there is an intensity n, varying from 0 for black to 255 for white. In the Alemaskin et al. scheme [34], at each pixel there are n white and (255 – n) black particles. The modified entropy Slocation (species), with black and white being the possible species, can be calculated after deciding the ideal concentrations as described in full detail in [34]. The extrudate sample and the variation of the index of color homogeneity for the seven cross-sections examined is shown in Fig. 7.6. The same authors [35] have adapted their methodology to measure color uniformity in numerical simulations of flow in an extruder when advecting light pellets (tracers) of two colors.
5 cm
Figure 7.6: Top: Extrudate samples and the 7 cross-sections analyzed; Bottom: Evolution of color homogeneity index based on the grayscale entropic analysis. (Reprinted with permission from [34])
262
7 Mixing Measures
7.2.4.2 Applications to Micromixers Camesasca, Manas-Zloczower, and Kaufman [36] have expanded their work on simulating and measuring mixing of fluids to microfluidics. In particular, by using the Slocations (species) they demonstrate quantitatively that the staggered herringbone mixer developed by Whitesides’ group [37] is a better mixer than the straight diagonal ridges micromixer. Camesasca, Kaufman, and Manas-Zloczower have recently found [38] that replacing the periodic patterning of the walls with self-similar (fractal) patterning yields higher levels of mixing. This is shown in Fig. 7.7, where three microchannels designed using a fractal protocol of patterning are compared with the classical staggered herringbone mixer. The entropic measures developed by Kaufman, Manas-Zloczower, and their students have been adopted by several other groups researching micromixing [39, 40, 41]. ELQVPPUHVROXWLRQ
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Figure 7.7: Mixing in microchannels as described by the mixing entropy at different scales of obser vation (from 50 to 2.5 μm resolution). Mixing efficiency at all scales of observation show the mixing efficiency ranging W1.25 > W1.50 > mSHM > W1.75. The symbol W# stands for the classical Weierstrass function with a fractal dimension “#” used in the patterning. (Reprinted with permission from [38])
7.2 Entropic Measures
263
7.2.5 Renyi Generalized Entropies and Fractal Properties The fractal properties of microstructures generated as a result of mixing can be also described in terms of multifractal dimensions. Consider the system completely covered with M equal volume boxes, M = –D, where = a/L is the ratio of the linear size of the box, a, to the linear size of the system, L, and D is the dimension of the embedding space. Parameter measures the scale of segregation in the analysis. As the number of boxes M increases, or the scale of segregation decreases, S() ≈ –d() ln(), where d() are labeled as the Renyi generalized dimensions. Thus, at small scales of segregation, 1, the relative entropy is approximately the ratio of the generalized dimension to the embedding dimension: S( )/ln(M ) ≈ d( )/ D
(7.19)
The interpretation of d() as a dimension originates in the case → 0. According to Eq. (7.19), S(0) = ln(number of occupied boxes). But the fractal (Hausdorff) dimension is: dHausdorff = ln (number of occupied boxes)/–ln() = S(0)/–ln() = DS(0)/lnM. One defines generalized dimensions in the same manner for all values of > 0. For = 1, one gets the information dimension as S(1) is the Shannon entropy. For = 2, one gets the correlation dimension as S(2), which is related to the logarithm of the correlation sum. 7.2.5.1 Applications Kaufman, Camesasca, and Manas-Zloczower [42] determined the multifractal dimensions for the published [37] images of mixing of a fluorescent and a non-fluorescent fluid in a microchannel. Camesasca, Kaufman, and Manas-Zloczower [38] have recently demonstrated that characterizing the interface of two fluids with multifractal dimensions provides a quantification of mixing that complements the entropy. In Fig. 7.8, one can see the increase in the topological, information, and correlation dimensions as the mixing progresses. Camesasca [43] has also calculated entropies and fractal dimensions for the baker’s transformation [44] showing the dimension of the interface between the two species is evolving from 1 (line) to 2 (surface) as the degree of mixing increases.
264
7 Mixing Measures
W-1.25
W-1.50
2.0
2.0
Correlation Information Topological
1.6
1.4
1.2
1.0
Correlation Information Topological
1.8
Genralized Dimensions
Genralized Dimensions
1.8
1.6
1.4
1.2
0
1000
2000
3000
Distance From Inlet [Pm]
4000
1.0
5000
0
1000
2000
W-1.75
5000
3000
4000
5000
2.0
Correlation Information Topological
Correlation Information Topological
1.8
Genralized Dimensions
1.8
Genralized Dimensions
4000
mSHM
2.0
1.6
1.4
1.2
1.0
3000
Distance From Inlet [Pm]
1.6
1.4
1.2
0
1000
2000
3000
Distance From Inlet [Pm]
4000
5000
1.0
0
1000
2000
Distance From Inlet [Pm]
Figure 7.8: Generalized dimensions D0, D1, D2 calculated for the systems W1.25, W1.50, W1.7,5 and mSHM. The common trend is an increase from D = 1 at the inlet (before starting the mixing action) where the interface is a simple line to D > 1 as mixing progresses; a substantial increase in all of the mixers is present. (Reprinted with permission from [38])
7.3
Summary
We have demonstrated through several examples (polymer extrusion, microfluidics) that entropy can be tailored to efficiently evaluate mixing in a variety of applications. The procedures developed are versatile in the sense that they can be applied to both numerical work and to experimental results via image processing. The entropic mixing measures have the potential to be useful for optimization of equipment design and processing conditions. Studies of the plausible correlation between the multifractal dimensions of the microstructures resulting from the mixing processes and the physical properties of the material, such as electrical conductivity, may facilitate the design and realization of materials with tailored properties.
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