CHARLES L. TUCKER and hrdustrial Engineering, University IL 61801, U.S.A.
III of Illinois at Urbana-Champaign,
(Received 29 August 1980; accepted 31 March
between the statistical properties of a mixture, the variance in concentration among a number of samples taken from the mixture, and the size and geometry of the samples are considered, using a continuum model. Sample variance is shown to depend on an integral of the product of the mixture correlation function and a function depending only on the sample shape. Analytical and numerical methods for evaluating the sample shape function are presented, as well as a simple, general approximation valid for small distances. Some familiar results relating sample variance to sample size are shown to be special cases of the more general result. Theoretical aspects of mixture correlograms are also considered. An argument is made for assigning correolgram shapes on a physical basis, and some examples are given. All mixtures in which molecular diffusion is negligible are shown to have a correlogram whose initial slope is determined by the interfacial area per unit volume. The complete correlogram for a mixture with randomly arranged clumps is derived, and a model correlogram for layered mixtures with no long-range order is proposed. The theory is compared to experimental results to show that it correctly describes real mixtures. Abstract-Relationships
The extent to which two or more substances are mixed controls important features of many products. Mixing often determines such features as texture, extent of chemical reaction, appearance, uniformity among successive units and physical properties in a host of manufactured goods. Thus, the measurement and control of mixing and the predictive design of mixing devices are subjects of considerable importance. Much of mixing theory, especially for liquids mixed in laminar flow, uses deterministic descriptions of mixing such as interfacial area or striation thicknessIl-41. But real mixtures are often difficult to compare to theory because they have a more complex structure, which is best dealt with statistically. One convenient statistical measure of mixing is the variance in concentration of one mixture component among a number of small samples taken from the mixture. This sample variance technique is easily applied, since almost any means can be used to measure the concentration in the samples. Light transmittance[5,6], chemical analysis  and electrical conductivity[S, 91 have been used, and many other methods are possible. This paper is concerned with relating the convenient sample variance measurement of mixing to statistical features of the mixture and to some of the common measures used in mixing theory. The results can be used to relate experimental measurements to fundamental characteristics of the mixture, which can in turn be related to details of the mixing process. This makes it possible to quantitatively compare theoretical predictions of mixing to experiments. Literature on the statistical description of mixtures can be divided into two subsets according to the basic approach taken. One set of workers has used a continuum model of the mixture where the concentration of tach component can be defined at a point and the concen-
tration profile is a continuous
function of space. The well paper by Danckwerts[lO] is the chief example of this group. More recently, Scott and Bridgewater[ll] derived the relationship between sample variance and sample geometry using this approach. They pointed out that the continuum approach is the most general, being particularly suited to liquid mixtures but fully capable of dealing with mixtures of solid particles as well. However, when dealing with solid particles the continuum approach requires that particles be sliced apart during sampling if the sample boundary passes through the particle. It is seldom possible to sample particulate mixtures this way, so a second group of workers has dealt with solids mixtures by taking concentration to be a discrete function of space. In these models the composition of the mixture is defined only at certain points, using either individual particles or a small unit sample as the discrete unit. Examples of this approach include the work of Bourne[l2-141, Kristensen[lS-191 and Cooke and Bridgewater. These workers have also derived expressions for the variance among samples. Kristenscn has compared some of the principal results of the two approaches and shown that they are consistent. He points out that while the continuum model is more general, either type of model might be preferred according to the type of mixture and actual sampling technique used. One common result from both approaches is that a sample variance measurement will depend on both the statistical properties of the mixture and the sample size and geometry. This paper explores both the effect of sample geometry and mixture statistics on sample variance. Primary emphasis is on liquid-liquid mixing, so the more general continuum approach is used. A new derivation is given for the variance among samples drawn from a mixture. The derivation makes it possible to generalize the effect of sample size and shape, and to easily identify known
C. L. TUCKER
the cases when only sample size is important. The results lead to a rational derivation of the correlograms of clumpy and layered random mixtures. A general relationship between the correlogram shape and the ratio of interfacial area to mixture volume is also given. Comparisons with experiments from the literature are used to show that the results describe actual mixtures. EFFFCT
It is sufficient to consider a two-component mixture consisting of components A and E. The analysis may be generalized to multi-component mixtures by letting one component be A and all others (including free space, in a mixture of solid particles) be 8; for a mixture of n components the analysis must be performed (n-l) times. The volume fractions of A and B at any point are denoted by a and b and the averages over the entire mixture are d and 6 The following analysis assumes that the statistical properties of the mixture are uniform, e.g. that there is no long-range variation in d Methods of dealing with long-range segregation are available in the literature [9,18,19]. are used in statistical description of Two quantities mixtures. One is the variance in concentration among points, Us’:
a.2 = (0.
V is the volume of an individual sample, and W(r) is a weighting function or sample shape function which contains the influence of sample shape. To find W(r) one must first find a function W*(w, r) which depends on Y, the location of a point within the sample volume, as well as r. For a spherical shell of radius r and thickness dr centered at x, W*(x, r) is the fraction of the volume of that she11 lying within the sample volume. The sample shape function is the average of W* over the sample volume:
As an example, the sample shape function for a sphere of radius r, is
Overbars are used here to imply averaging entire mixture. Note that o+b=l
in Appendix I. There it is shown that for samples of identical size and shape the variance among the concentration of one component in the sample, u=‘. is given by
Danckwerts[lO] points out that u.,’ is related to the extent to which molecular diffusion has erased concentration differences in the mixture. If there has been no diffusion, then
This can be combined with eqn (5) and with tHe correlogram for any mixture to find the sample variance when that mixture is tested using spherical samples. By definition W(r) is identically zero for r greater than the longest chord contained in the sample, so the integral in eqn (5) will always be finite despite the infinite upper limit. Equation (5) is equivalent to Scott and Bridgewater’s eqn (9), though the derivation is somewhat different. For volume samples, the sample shape function W(r) is related to Scott and Bridgewater’s density function N(r) by N(r)/V=4r?W(r).
This will always be true in mixtures of solid particles and in solid/liquid mixtures. It is also a good approximation for mixtures of highly viscous liquids such as polymer melts, and for liquid mixtures which are formed rapidly and examined soon-thereafter. Appendix 4 shows how soon a mixture must be examined for diffusion to be negligible, depending on the diffusion coefficient and the degree of mixing. In these cases ca2 is completely determined by the mixing ratio, a result which is very useful when applying the theory which follows. ‘The second statistical quantity is the correlation coefficient, p(r). If a, and a, are the concentrations at two points separated by a distance r. then
Several useful results can be easily derived from this new form of the sample shape function. First, the shape function at small values of r for any sample depends only on its surface to volume ratio. To see this, consider values of r smaller than the smallest dimensions of the sample. Divide the sample into two regions, a boundary region of points a distance of r or less from the sample surface, and an interior region comprising the rest of the sample. If the vector x is replaced by a scalar X, defined as the distance between a point and the sample surface, then in the boundary region W*(x, r) = (x + r)/2r
p(r) = (a, - G)(ar - E)/u*‘.
The correlation coefficient is always equal to unity for r equals zero, and it falls to zero when the relation between concentrations is random. The derivation of the relationship between sample variance, sample geometry and mixture statistics is given
while in the interior region W* equals unity. To average W* over the sample volume. note that du =A,dx
where A. is the surface area of the sample. Performing
the averaging in accordance
with eqn (6) gives the result
W(r) = 1 - A,r/4 V
contribution to the integral for sample variance if the transverse dimension of the sample is small compared to c. For the proper size samples, eqn (15) reduces to Danckwert’s formula
for small values of r. This allows at least part of the sample shape function to be estimated very easily. AF’PLICATION
with S, being the linear scale of segregation given by
An interesting application of eqn (II) is to mixtures with no long-range regular structure. For this type of mixture, the correlogram equals zero for values of r greater than some value, say l. If t is on the order of the smallest dimensions of the sample or less, then eqn (I 1) can be used to find the sample variance without calculating the complete sampIe shape function.- If 5 is small compared to the distance (4V/A,), then W(r) will be very nearly unity whenever p(r) is non-zero, and the actual shape of the sample ceases to be important (though its volume is still important). In this case, eqn (5) reduces to a result derived by Danckwerts[lO],
Sr. = &(r)dr.
The two equations apply to mixtures with no long-range correlation. In a similar manner, area samples such as squares can be viewed as volume samples having one dimension much smaller than the other two. If A is the area of the sample and P the perimeter, the approximate sample shape function is W(r) = (V/2Ar)[ 1 - (PtfA)].
u,2/cr0== 2S”/ v
where Sv is the volume scale of segregation.
sv = 2a
Danckwerts’ scale of segregation is only defined for mixtures with correlograms which are non-negative and go to zero for I greater than <, but the more general eqn (5) can be used for any correlogram regardless of shape. Note that the approximate sample shape function can be used to make accurate calculations of the error in eqn (12). Highly elongated line samples and flat area samples have previously been treated as separate cases[ll]. But in the continuum treatment, they may be thought of as limiiing cases of volume samples. For example, a slender light beam used in a Kght transmittance measurement may be thought of as a highly elongated cylinder with a finite cross section, while a mixture spread in a tray and divided into squares still gives a sample with finite thickness and finite volume. For line samples, the sample shape function is given by
This approximation is good for r greater than the thickness but less than the other linear dimensions of the sample. Substituting this into eqn (5) will show that, in the right range of sample size, variance is inversely proportional to sample area. One can now build a coherent picture of the way in which sample variance changes with sample size for mixtures where the volume scale of segregation is defined. When the sample is very small it will tend to consist entirely of eithel one component or the other, and uCczwill be very nearly equal to Us’. Very large samples (compared to the scale of segregation) will follow eqn (12), and the variance will be inversely proportional to sample volume. Deviation from this equation will arise as the sample size is decreased, and can first be detected using the approximate form of the sample shape function (eqn tl) in eqn (5). The curve of transition between small and large samples must be calculated using eqn (5) and the complete sample shape function. If the sample is highly elongated in one direction, then a portion of the curve will resemble the curve for a pure line sample (eqn 16), though for sufficiently large samples the variance will still be inversely proportional to sample volume.
(14) Here, A, is the cross-sectional area of the sample and L the sample length. Substituting this into eqn (5) gives the sample variance for line samples: u<2tLra== (2/L) [
(I - r/Up(r)
This equation and eqn (14) are the limiting forms as the sample cross section becomes very small. For samples with finite volume, eqn (11) still gives the sample shape function for small r, but this will make only a very small
All of these features can be seen in Fig. 1, which shows variance vs sample size for several different sample shapes. A linear correlogram is used here, taking
Ozzr5I r > 1.
The sample shapes used in this figure start with a cube with sides of length L. One dimension of the cube is kept constant and the other two decreased to produce elongated samples with square cross sections. The limit of a perfect line sample is calculated using eqn (15). Notice that even when the sample is a hundred times longer than it is wide. there is only about a decade of sample size
Fig. I. Effect of sample shape on sample variance. Samples have length L and square cross section. The ratios indicated on the figure are the ratio of sample length to transverse dimension. A linear
where the usual approximation for a line sample (eqn 16) applies. The sample shape functions used to produce Fig. 1 were evaluated numerically instead of using the definition in eqn (6). A computer program divides the sample into a large number of equal unit volumes with the centers of adjacent volumes separated by a distance Ar. All possible pairs of two volumes are considered and the distance between them computed vectorially. The distance is rounded to the nearest integer multiple of Ar, then the number of pairs of volumes for each distance is counted and divided by the tota number of pairs considered. This produces a normalized weighting function An(r) which is related to the sample shape function by
= 47rr2 W(r)Ari V. can then be computed kmsX ,& (kAr)
all of the data. I-rli;
Fig. 2. Comparison of correlogram frey to bilinear form used in
to M~xhxe SC&,
of eqn (5).
To verify that this theory does indeed represent real behavior, it was compared to some data taken on actual mixtures. Hall and Godfrey extruded white and colored clay through various mixing plates, allowed the mixture to solidfy and examined cross sections of the mixture. For each of seven mixtures they reported a c&relogram as well as sample variance for several different sample sizes. Because their correlograms and sample variances were based on the entire mixture, their data contains no statistical errors of the type that would be present in a typical sampling experiment. The correlograms were somewhat different for each of the mixtures but the general shapes were very similar. It was found that a bilinear correlogram could be used to
; - ariaI; 1
by Hall and their data.
has the form
rll <(P - [email protected]
(p - a/3lI(P - a) 5 rll 5 p rll > P
- ff) (22)
where (Y and @ are adjustable parameters and I can be varied to alter the scale of segregation. Values of (I = 0.3 and p =5 were used and I was varied to match this shape to the individual correlograms. Figure 2 compares one of Hall and Godfrey’s correlograms with the corresponding bilinear approximation. Hall and Godfrey used square samples, for which the shape function was calculated numerically. Figure 3 shows the comparison between the theoretical curve and the data. Note that the data for all of the mixtures fall on the same normalized plot with very good agreement. The few areas where the fit is mediocre are due to errors in approximating the correlograms. It is clear that as long as one has free reign to choose any shape of correlogram, nearly any set of data can be fitted. The goodness of fit in Fig. 3 can be achieved with a relatively crude fit to the correlograms because the sample variance depends on a weighted integral of the correlogram and is not sensitive to small details. One could just as well choose another form of the correlogram. In fact, this same data has been fit very nicely using an exponential correlogram[201. If this type of measurement and interpretation is to be useful, one must have a good reason for choosing the shape of the correlogram.
Fig. 3. Data from Hall and Godfrey compared to theory.
FORMS OF CORRELOCRAMS
The potential difficulty is that these variances would be very close to a.’ and accurate measurements might be difficult. Alternately, one can use eqn (24) to get an accurate measure of the interfacial area per unit volume by measuring only the initial part of the correlogram. While interfacial surface area determines the initial portion of the correlogram, it will now be shown that the shape of the correlogram for larger values of r is determined by the mixture pattern.
While a few investigators have actually measured the correlograms of mixtures(6,16,18,23], this measurement is normally quite tedious. In practice, one often prefers to measure only sample variance, perhaps for a few different sample sizes. Though it is theoretically possible to calculate the correlogram from a curve of sample variance vs sample size, Scott and Bridgewater have pointed out that the correlogram is related to a derivative of the variance-sample size curve. Given the experimental errors in sample variance which come from using a finite number of samples, this differentiation cannot produce accurate results. It is far more convenient to assume a certain shape for the correlogram and use sample variance measurements to determine the scale of mixing. Previous theoretical studies have made wide use of the linear correlogram (eqn 19) and the exponential correlogram
mixtures might reasonably be assumed to consist of clumps or regions with some correlation between points within the same clump but no correlation between points in different clumps. The clumps Sight comprise individual particles or even a region with some fairly complex substructure. Let the correlation for two points in the same clump be known so that
p(r) = emTI’.
p(r) = p*(r)
These correlograms are convenient in that they have finite volume scales of segregation, but there appears to be no other reason for choosing them except that they are somewhat similar to the few correlograms that have actually been measured. A much more attractive procedure would be to choose a correlogram shape for physical reasons. For example, laminar mixing of viscous liquids produces a striated mixture, which pattern in turn determines the shape of the correlogram. In this section, several results are derived which make this approach possible. All of the derivations which follow apply to mixtures in which no significant molecular diffusion has taken place. As mentioned before, such an assumption is good for highly viscous liquids, mixtures where at ieast one component consists of solid particles, and liquid-liquid mixtures which are mixed rapidly and sampled soon thereafter. All such mixtures share one common feature: the initial portion of the correlogram is linear and the slope is determined by the interfacial area per unit volume. This result is derived in Appendix 2 where it is shown that p(r) = 1- r/2&A
for small values of r. A is defined for any mixture using the total interfacial area, A,,,, and the volume of the mixture, V, : A = 2 VJA,.
In regular layered mixtures, A is the thickness of Ihe repeating unit (the striation thickness). Equation (24) is particularly useful since it places a concise physical interpretation on the slope of the initial portion of any correlogram. It also suggests that the interfacial area per unit volume might be measured without knowing the rest of the correlogram by choosing small enough samples. Then, only the portion of the correlogram approximated by eqn (24) would be included in the integral of eqn (5).
for two points in the same clump but p(r) = 0
for two points in different clumps. If all of the clumps have the same size and shape, then the correlation coefficient of the entire mixture can he computed by averaging over a single representative clump. The first point is chosen from anywhere in the clump, designated by the vector x, and the second point from anywhere in the mixture. If the volume of the region is V,, then this gives p(r) =(1/V,)
dx, r) dv.
But the only contributions to p(x. r) are from second points lying inside the sample volume so that p(r) = (l/V,)
“, p*(r) WY%, r) dv
where W* is defined exactly as before. Now eqn (6) can be applied to give p(r) = p*(r) W(r)
where W(r) is the sample shape function for the shape of the clumps. That is, the correlation coefficient for a mixture with random clumps equals the product of the correlation coefficient within the clumps and the sample shape function for the clumps. When each clump is uniform in concentration, p*(r) equals unity and the correlogram equals the shape function for the clumps. This situation might occur, e.g. when polymer pellets of two colors are dry blended to a random mixture and then compacted and melted. Each particle then constitutes a clump and the clumps will be roughly spherical. For perfectly spherical clumps the correlogram is given by the sample shape function for a
Fig. 4. Comparison of linear and exponential correlograms to correlogram derived for mixture with random spherical clumps. Ail mixtures have the same interfacial area per unit volume. sphere (eqn 7). Figure 4 compares this form to the linear and exponential forms used before in the literature, with all three correlograms having the same interfacial area per unit volume. Obviously, the correlogram of a clumpy mixture must fall to zero for r equal to the longest chord which can be drawn through a clump. The linear correlogram goes to zero too quickly to represent any real clumpy mixture, since no shape is more compact than a sphere. The exponential correlogram has a tail but it is too long to represent finite sized clumps. The spherical shape serves as a good starting point for the correlogram of clumpy mixtures. Correlograms for other clump shapes can be calculated numerically and one can average over different shapes and/or sizes of clumps as well. This procedure is far more attractive than choosing a simple correlogram on a purely ad hoc basis, but the analysis does show why the linear and exponential correlogram have proven so useful in the past.
common cases and the directional correlogram simply averaged over all directions to get an overall correlogram. This can certainly be done when the samples are spherical and give no preference to any direction. Other sample shapes can also be used with an average correlogram when the relative orientation between the samples and the layers is random. This could come about either because of some long-range swirling or bending of layers in the mixture or because the samples are oriented at random. The complete correlogram of a regular layered mixture in the direction normal to the interfaces is periodic with period A and is shown in Fig. S(a). Appendix 3 shows how this can be used to derive the correlogram in any direction, and how the results are integrated to get a three-dimensional average correlogram for a regular layered mixture. The average correlogram is shown in Fig. 5(b). Three-dimensional averaging damps out the oscillations and gives a correlogram whose initial portion agrees with eqn (24), but the correlogram still is negative in places and in fact damps out so slowly that the volume scale of segregation is undefined for this mixture. The oscillations in this correlogram cause violent oscillations in the curve of variance vs sample size. For example, with a spherical sample the variance drops by more than three orders of magnitude with only a 60% increase in sample size, and then rises by almost the same amount with another 30% increase in sample size. This behavior is not unexpected, since for a perfectly ordered mixture there will be some sample sizes for which every sample
Another important class of mixtures for which there are no previous statistical models is layered mixtures. Layered or lamellar mixtures have long been used in modeling laminar mixing of liquids [ I-31, and recently have been used in turbulent mixing models as well [4,24]. As will be seen, the problem in modeling layered mixtures lies in finding reasonable ways of incorporating short- and longrange randomness. The most common lamellar model is the regular layered mixture, comprising alternate flat layers of the two mixture components. The layers of any component all have the same thickness, which may be different from the thickness of the other component layers. Clearly, such a mixture is anisotropic and has long-range order. Nadav and Tadmor derived the initial part of the correlogram of a regular layered mixture in the direction normal to the interfaces. They defined a scale of segregation by integrating the correlogram only up to the point of the first zero crossing. Their analysis was well suited to their particular experimental study, but does not apply to bulk sampling. First, eqn (5) requires the entire product of the correlogram and shape function to be integrated regardless of whether it is positive or negative. Also, the correlogram for the entire mixture must be used, not just the correlogram for a single direction. The anisotropy of the mixture complicates the problem, but this complication can be ignored for many
Fig. 5. Correlograms for regular layered mixtures. d is the volume fraction of the minor component. (a) One-dimensional correlograms in direction normal to interfaces. (h) Three-dimensional average cnrrelogram.
Sample variance measurement
has very nearly the same composition, even when the samples are quite small. While the regular layered mixture is convenient for mixing theoreticians, it seems unlikely that the longrange regularity of this structure would exist in practice. Rather, variations in layer thickness and orientation of the layers would bring about some randomness, keeping the actual variance from ever going to zero. In many situations it is even unreasonable to expect the local structure to be too regular. For example, if the random mixture of pellets mentioned in the last section were melted by conduction and given a large shear strain, each pellet would be deformed into a thin layer with thickness 1. Where adjacent layers and different compositions the boundary would be an interface, but because of the original random structure of the mixture there would be no correlation between the composition of adjacent layers. For this type of structure, the correlogram in the direction normal to the interfaces is the linear correlogram (eqn 19). This can be averaged over all directions using the method shown in Appendix 3 to get the three-dimensional average correlogram: (30) This mixture has short-range randomness in the sense that the correlogram is always positive, but still retains long-range structure in that the correlogram never goes to zero. Long-range randomness can be imposed by assuming that the mixture is clumpy, with the correlation coefficient within the clumps given by eqn (30). If spherical clumps are used and it is assumed that the radius of the clump is (al), then the correlogram will be 1 -(l/2 + 3/4u)(r/l) + (3/8a)(r/I)Z +(1116n3)(r/I)3-(1/32u3)(rl[)4; Osr
Curves of sample variance vs sample size are shown in Fig. 6 for a random layered mixture and spherical samples. Note that all of the curves for finite Q reach asymptotes as predicted by eqn (12) but that the volume scale of segregation is highly dependent on (Y. For LY equals infinity the correlogram is given by eqn (30) and for large samples the variance curve shows the slope that might ordinarily be expected from a line sample. This happens with layers of infinite extent because the concentration varies in one direction across the sample but not at all in the two orthogonal directions. Thus, for infinite layers the volume sample looks like a line sample with a distorted cross section, and the variance is inversely proportional to the linear dimensions of the sample instead of to the volume. As (Y is increased, the sample variance curves come closer to this limit over more of their length, but all curves for finite 4 eventually fall away to the volume limit when the sample becomes as large as the clumps in the mixture. The random layered mixture shows the type of sampling behavior that would be expected of a real mixture, and should be a useful model for dealing with layered mixtures. APPLlCATlON
As an example of the application of the random layered mixture model, we now consider experiments by Rotz and Suh[S] on laminar flow liquid-liquid mixing between concentric cylinders. They studied the geometry shown in Fig. 7 with both smooth and grooved inner cylinders and showed that the addition of V-grooves improved mixing performance. This improvement is due
This type of mixture mixture.
will be called a random
Size 10 Mixture Scde,
variance for random pled with sphersal
layered mixture samples.
mixer geometry Sub.
studied by Ralz and
C. L. TUCKER
to secondary motions, similar to Taylor vortices, induced by the grooves. The kinematics of these flows are undeniably complex, but the experimental results can be rationalized with the random layered mixture model within a constant, using a very simple mixing analysis. The mixer is assumed to operate by giving the fluid a large shear strain whose characteristic value is y. Assuming that axial Bow through the mixer is slow and has a small effect on the deformation field, y will be proportional to the product of the angular velocity of the inner cylinder, w, and the fluid residence time ?,: y = Kwi,.
which depends on the inner cylinHere K is a constant der geometry and rotational Reynolds number. For convenience the inlet manifold is assumed to provide an initial striation thickness A,, and a spatially uniform orientation of interfaces. Under these assumptions and providing that y is large, the outlet mixture has a striation thickness A which is[Z] A = 2Aoly.
Other assumptions about the inlet which are possible but simply alter eqn (33) by a constant factor and do not affect the argument. The mixer performance is now given by A = 2A,jKwf,.
In the sense that there are distributions of strain and residence time in the mixer, eqn (34) may be regarded as a scaling argument based on appropriate average values. Rotz and Suh used a hypodermic needle and syringe to extract samples, a procedure giving roughly a spherical sample. Note that mixing and diffusion within a single sample does not affect the results. The only quantity used in sample variance calculations is the total amount of one component in each sample. The hypodermic needle and syringe cause some homogenization of the sample, but as long as the sample concentration is measured correctly this has no effect on sample variance. The shape function for a spherical sample was used along with the random layered mixture correlogram to calculate a curve of sample variance v.9 predicted mixing performance. In these experiments, sample volume was a constant 1cm3 and the variation in mixing comes from different values of fluid residence time. Figure 8 compares the data to theory using a equal to ten. It is not clear what value should be assigned IO A,, nor what the K values should be for the different mixers. Since the goal was to see if the random layered model could be used to represent real mixtures, the data set for each mixer was simply translated horizontally to line up with the theoretical curve, a procedure equivalent to adjusting the value AJK. There is substantial scatter in this data because of the small number of samples used in each measurement_ so confidence limits are indicated on the figure. Within these limits agreement is quite good, showing that these data are in accordance with simple
Fig. 8. Data of Rotz and Suh fitted by simple mixing theory and random layered mixture model with a = 10. Points falling within the dashed lines have 95% confidence limits which fall across the theoretical curve.
and sample variance mixture model.
The variance in concentration of a number of samples taken from a mixture has been shown to be determined by an integral of the product of the mixture correlation function and a function depending on the sample shape. This sample shape function can be evaluated analyticalty for simple shapes and numerically for complicated shapes. A general approximation valid for small distances gives the sample shape function from only the surface area to volume ratio of the sample. These results are in accordance with previous results and identify clearly the region of sample size where only the volume of the sample and the volume scale of segregation of the mixture determine sample variance. The general treatment for volume samples includes line and area samples when these are assumed to have finite volume. Familiar results for line and area samples apply over a certain range of sample size but fail for sufficiently large samples. Then the volume of the samples must be considered. An argument has been made for assigning correlogram shapes to mixtures on a physical basis. All mixtures in which molecular diffusion is negligible have a correlogram whose initial portion is linear, with a slope depending on the ratio of interfacial area to volume. Mixtures with random clumps have correlograms which are determined by the sample shape function for the clumps. For spherical clumps, the actual correlogram is intermediate between the linear and cxponcntial forms used in previous studies. A correlogram has also been proposed for studying layered mixtures and has been shown to accurately represent actual mixtures. Arknowledgemenf-The aurhor gratefully acknowledges Prof. C. A. Rotz of the University of Texas for making available the original data on his grooved
cylinder mixing experiments.
Sample variance measurement NOTATION
concentration of component A at a point in the mixture average concentration of component A component of mixture; area of flat sample cross sectional area of elongated “line” sample total interfacial surface area in mixture A, AS surface area of sample b concentration of component B at a point in the mixture 8- average concentration of component B B component of mixture c concentration defect at a point in the mixture (difference between actual and average concentration of A) C concentration of A in sample used to measure degree of mixing D molecular diffusivity i,j,k,m integer index variables K proportionality constant relating strain, rotational speed, and residence time in concentric cylinder mixer I characteristic length scale of mixing, has different interpretation for different correlogram shapes length of elongated “line” sample length of side of square sample number of samples in entire mixture in mixture number of components function used in numerical evaluation of sample shape function sample shape function of Scott and BridgeN(r) water perimeter of flat “area” samples P that concentrations at two points P(i, i) probability have values i and j probability that two points have P(i, j/k, conditional i and j given that the first concentrations point lies in region k distance between two points, or radial coordinatc in spherical coordinate system radius of spherica samples linear scale of segregation defined in eqn (17) volume scale of segregation defined in eqn (13) fluid residence time in concentric cylinder mixer sample volume clump volume in clumpy random mixture volume of entire mixture sample shape function defined in eqn (6) fraction of spherical shell of radius r with center at x which lies inside sample volume scalar position variable position vector referred to coordinate system attached to sample volume position of reference point for ith sample parameter in bilinear correlogram; parameter in random layered mixture model parameter in bilinear correlogram a
of mixing Y 6
strain imparted to fluid slab thickness in diffusion model c distance at which correlogram falls to zero in random mixture in spherical coordinate tr angle coordinate system striation thickness defined in eqn (25) initial striation thickness at the beginning of mixing coefficient defined in eqn (4) p(r) correlation correlation coefficient for clumpy mixture P*(r) when both points lie within the same clump variance in concentration among all points in mixture variance in concentration among samples taken from the mixture coordinate in spherical coordinate a5 angular system angular velocity of rotating element in conw centric cylinder mixer
REFEPENCES [I] Spencer R. S. and Wiley R. M., 1. Co/l. Sci. 1951 6 133. 121 Mohr W. D.. Saxton R. L. and Jepron C. H., Ind. &gng Ckem. 1957 49 1855.  Erwin L., Polym. Engng Sci. 1978 18 1044.  Ottino J. M., Ranz W. E. and MacoskoC. W., Chem. Engng Sci. 1979 34 877. 151 Ratz C. A. and Suh N. I’.. Polym. Engng SC;. 1976 16 664.  Nadav N. and Tadmor Z., Chem. Engng Sci. 1973 28 2115. 171 Malguarnera S. C. and Suh N. P.. PoIym. Engng Sci. 1977 17 111. 181 Tucker C. L. and Sub N. P., Polym. Engng Sci. 1980 20 875. [9l Tucker C. L. and Sub N. P.. Polym. Engng Sci. 1980 20 887. [lo] Danckwerts P. V.. Appt. Scienf. Res. 1952 A3 279. ill] Scott A. M. and Bridgewater J.. Chem. Engng Sci. 1974 29 1798. 1121 Bourne J. R.. Chem. Engng (London) I%5 CE198. 1131 Bourne J. R.. Chem. Engng Sci. 1967 22 693. 1141 Bourne J. R.. Chem. Engng Sci. 1968 23 339. 1151 Kristensen H. G.. Powd. Tech. 1973 7 249. 1161 Kristensen H. G., Powd. Tech. 1973 7 149.  Kristensen H. G.. Arch. Phorm. Chem. Sci. Ed. 1973 1 102. [IS] Kristensen H. G., Arch. Phorm. Cbem. Sci. Ed. 1973 1 121. 1191 Kristensen H. G., Arch. Pharm Chem. Sci. Ed. 1973 1 145. [XI] Cooke M. H. and Bridgewater J., Chem. Engng Sci. 1977 32 13.53. [211 Kristensen H. G., Chem. Engng Sci. 1978 33 555. 1221 Bertholf W. M.. ASTM Spec. Tech. Pub. NO. 114 1951. I231 Hall K. R. and Godfrey J. C.. Instn.Chrm. Engngs. London,
1965 10 71. 1241 Lee L. J.. Ottino J. M.. Ram W. E. and Macoskb C. W., SPE Tech. Papers 1979 25 439. 1251 Rotz C. A. and Sub N. P.. I. Fluids Engng 1979 101 186.  Rohsenow W. M. and Hartnett J. P., Handbook of Heal Transfer. McGraw-Hill 1973.
Derivation of basic result Consider a mixture which is sampled with identically shaped samples which have volume V. For each sample one measures C. the volume fraction of component A. C equals the average of the concentration of A over the sample.
while the average value of C will depend on the average
uu’p(r) dv’ =
(A21 For convenience,
we define the quantity
that the deviation sample is
of C from
its average for any particular
This double integral is taken over all pairs of incremental volumes within the sample. If the squared deviation of C is averaged over all samples in the mixture, one gets the sample variance, oc2: f&z = (C - @.
samples in the mixture. If we take yi to a reference point in the ith sample, and coordinate system with its origin at yi to the sample, then eqns (AS) and (A6)
The order of the integration and summation can be reversed, bringing the summation inside the integral. Then, inspection of cqn (4) shows that (l/M) I is simply x and x’
,= Equation correlation
Here the order of integration has been changed and terms depending only on r brought outside the inner integral. The inner integral depends only on the sample geometry and becomes the sample shape function, W(r). W(r)=(IIV)jv
c(x)&‘) II” V’
Let M be the number of be the spatial location of if x is referred to a local and oriented with respect can be combined to give
The infinite upper limit is used on r because W* will be zero when r is so great that the entire shell lies outside the sample. Substituting back into eqn (AIO) and rearranging gives
of C is now
C-~=(liV+c.,d~. The square
aazp(r) W*(x, r&r+
y;) = u**p(r) of the distance
(A7) can now be expressed as the coefficient over a single representative
(A% average of the sample,
so thar the sample
to eqn (5).
APPENDIX 2 Partial correlogram for mixtures with no diffusion In deriving the correlogxam for mixtures with no diffusion, it is easiest to use Nadav and Tadmor’s expression for the correlation coefficient in terms of probabilities[61. Using the notation P(i.n to indicate the probability that (I, equals i and a2 equals j, this is p(r) = P(1, I)(&/& + P(O,O)(dlr)
0) - P(0,
Let the total volume of the mixture be V, and the total interfacial surface area be A,. Consider a specific value of r which in general must be small compared to the curvature of the interfaces and less than one-half of the shortest distance between any two interfaces. Divide the mixture into four regions. Region I lies wholly in component A and comprises points a distance r or less from the interface, while Region II consists of the remainder of A. Region III lies in component B and is the set of component points I or less from the interface, and Region IV is the rest of component B. The probabilities uf the various ways a dipole with length r might fall on this mixture can now be enumerated. The probability that the first point of the dipole will fall in Region I equals the volume of Region I divided by the volume of the mixture. P(I)
= AIrI V,.
Given that this occurs, there are two conditional probabilities. The first is that the second point also falls in material A. The outer integral in eqn (AlO) specifies an incremental dv, for which the inner integral is evaluated. This inner can be expressed in spherical coordinates,
“, o.*p(r) du’ =
d ~ ,o,*p(r)?sin~%drd8d# 111
l/1) = 3/4
point falls in material B.
is that the second P(l,O/I)
on r, 0 and 4 chosen to limit the integral to points volume. Note that r is simply the distance between the volume increment do’ and the increment do specified by the outer integral. Let I be the location of this latter incremental volume. To avoid the difficulty of specifying limits W*(r, I) as follows. on 6’ and d in eqn (Al I), define a function Consider a spherical shell of radius r and thickness dr, centered at II. W*(x. r) is the fraction of the volume of that shell lying inside the sample. Since the integrand in eqn (All) is a function of I only, one can easily evaluate the integrals with respect to 8 and b when the integrand is multiplied by W*. That is, with the limits
within the sample
P(I, and the second
These probabilities are found by drawing a sphere with radius r, centered a distance x from the interfaces. The probabilities correspond to the areas of the shell lying in materials A and B and are averaged over the range of x between 0 and r. Following this same notation, the other probabilities are as follows: For Region
Sample For Region
the average is given by PUII) P(0,
= A&V, = 3/4
l/III) = l/4.
= (&V-m - A,/)/
P(0, O/Iv) = 1. The various
can now be collected.
= P(1, l/I)P(Z)
If a coordinate system is chosen with the z-axis perpendicular to the interfaces in the mixture, then p will nat depend on 0 and, for any given direction 4. the concentrdtion profile will be identical to the profile in the r-direction but with period (Allcos 41). This means that the correlation coefficient in any direction can he written by simply replacing A with (A&OS 41) in eqn (Cl) (the one-dimensional correlogram). Doing this and integrating in accordance with eqn (C3) gives the average correlation coefficient for a regular layered mixture, *
P(I,l)=ri-r/ZA using A = 2 V,,,/A,.
The other probabilities
x sin (2n&A)
which is graphed
P(0. I) = r/2&
APPENDIX 4 Effect of diffusion on ra2 For a mixture which is formed quickly and then examined after some time has elapsed, the effect of diffusion on c.,~ can be estimated using a slab diffusion model. Consider a mixture consisting of alternating layers of the two components, the thickness of each layer being S. Let x be a coordinate direction normal to the interface, with origin such that at time equal zero
P(O,O) = 6P(l,O)=
S_ubstituting these into eqn (Bl) and using the relationship B = 1, the correlation coefficient is found to be p(r) = Iwhich corresponds
p(r) = (A/r)“T, [Cl-cos 27rrid)/Zn’7r’~~l
in Fig. S(b).
APPENDIX 3 Correlogram for regularlayered mixture The regular layered mixture consists of alternate slabs of mixtures A and B with thicknesses d and &, respectively. The first step in finding the three-dimensional average correlogram for this mixture is to find the one-dimensional directional correlogram in the direction normal to the interfaces. Again this is most easily done using probabilities (eqn Bl). Considering only distances up to h and enumerating the probabilities as in Appendix 2, one gets
Now allow diffusion to occur with constant solution to the diffusion equation is _ (a - Ii) = x (2/nm)exp _=1
x cos (Znnrl!4).
1; 0; -1:
can be computed
m=3,7,11,... m=2,4,6 ,... m=1.5,9....
Under the conditions mentioned in the text, sample variance may be computed using a three-dimensional average correlogram. If a spherical coordinate system is used, then the correlation coefficient is a function of B and 4 as well as r, and
p(r) = x [(I _-CDS 2?md)/n2&~] n=,
x cos (n?T*lG)F(rn)
The probabilities, and thus the correlogram, are periodic with period A, so fhe correlograms for various values of d have the shapes displayed in Fig. S(a). Taking advantage of this periodicity, the complete one-dimensional correlogram can be expressed as a Fourier cosine series, giving
%2) exp (- 2m2n2W6*).
Evaluating this expression numerically shows that a,* drops by 1.5% when (DTIGL) equals 0.1, and by less than 5% when (Dr/@) equals 0.3. In a typical polymer system one might have D on the order of 10~*cm2~sec. Even with d as small as 10pm one still has 30 set to take the sample before the results are significantly disturbed by diffusion.