Ceramic Processing E-zine Volume 1 Number 2 1 December 2002

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My new book, Rheology for Ceramists, is finished. If everything continues to go well, a downloadable version of it should be available for purchase on my web site within the next few days (early December 2002.) A free downloadable preview is available on the website now. The preview contains a PDF version of the Table of Contents, Preface, and Chapter 3 in a self-extracting zip file. Information about the book is also available on the website. A paperback edition, which is my ultimate goal, will take several more months to produce. Anyway ... for those of you who are interested, check out the free preview chapter, and keep watching for the PDF format downloadable e-version of the book which should be available soon.

The Microsoft Excel® spreadsheet program utilizes Solver® routines to perform optimizations. This optimization routine is an implementation of a 'Simplex' routine. Solver® is a very handy tool to install and know how to use. It is an add-in contained in the Excel® distribution package, but it is not automatically installed in most cases. Because it is not automatically available for use, many ceramists are not familiar with it. Since ceramists may need to perform optimizations, this article will describe its use and give a simple handy example.

If Solver® has already been installed in Excel® on your computer, it is available by selecting the Tools menu and clicking on Solver. If it is not listed in the Excel® Tools menu, it needs to be installed. To install the Solver® routine into Excel®, select the Tools menu, select Add-ins, and then check the box in front of the Solver Add-in. The Excel® distribution CD will need to be inserted in the drive to complete the installation. Excel® should then take care of the details and install this routine so it will be available for use whenever Excel® runs.

Optimizations are frequently performed using a mathematical algorithm known as a Simplex algorithm. Simplex routines allow one value (in this case, the contents of one cell) to be maximized, minimized, or set equal to a particular value, by changing a set of variables consistent with a set of well-defined constraints. These routines are commonly used for financial planning, for example, to maximize sales or to minimize expenses. Such routines are also very handy in ceramic processing, for example, when several particle size distributions are to be blended to match another distribution. The best fit, in such cases, is defined by minimizing an error value. Optimization routines work well when trying to determine the percentages of several raw materials required to best produce the properties of a known target body. These routines also work well to perform least-squares-type calculations. When the desired body composition has several known (measurable) properties, optimization routines can calculate the percentage of each candidate raw material to best produce the body. Optimization routines are also very useful in a Predictive Process Control (PPC) environment.

If a solution to the optimization is not possible using the input data, the optimization routine simply finishes with a message that says 'no solution is possible.' The non-solution case is not specific just to the Solver® environment -- it is characteristic of Simplex routines in general. Strictly speaking, the 'no solution possible' case is not a problem. It is a valid and useful result from the optimization calculation. Engineers might consider it to be a problem because it provides no suggestions concerning the direction to move, or changes to be made, to find a valid solution. When the 'no solution possible' message pops up in a body composition calculation, for example, more candidate raw materials and/or wider constraints are usually required.

Think back to the days when you were studying algebra and the solutions to simultaneous equations. (Yes, I know ... Yuk!) The Simplex method (on which optimization routines are based) is one that solves X simultaneous equations in Y unknowns -- and you always need at least one more equation than the number of unknowns to be solved. This holds true in the Solver® environment as well. It's very easy to define a problem with 42 constraints and only 3 candidate variables. Such conditions favor the 'no solution possible' result. The more candidate variables available, the more likely the routine will find a valid solution. It is just too easy to define too many constraints, to define them too narrowly, or even to define them so they conflict, that the problems are impossible to solve. Keep this in mind when using optimization routines. If you're trying to match a body composition by mixing constituent ingredients, the more ingredients available for the optimization routine to use, the more likely a valid solution can be achieved.

The easiest way to explain optimization routines is to provide an example. Although it would be easy to attach an Excel® file containing the example, you won't learn how to use such programs without actually trying it yourself. The example, therefore, will be confined to the pages of this E-zine. I encourage you to try this problem in your own computer.

Someone recently asked if my add-in programs could be used to match particle size distributions. The answer is "No," because the spreadsheet program and Solver® can easily perform this task. The sample problem to follow will be to match (as well as possible) a target particle size distribution using three ingredient distributions. The following table shows the setup required in the main spreadsheet.

Table 1. Sample Calculation to Match a Particle Size Distribution Using Solver®

	A	B	C	D	E	F	G	H	I
1		Fraction:	0.2	0.3	0.5	Sum Fr:	1.0	SumSqs:	266.2
2		Target	X	Y	Z		Squares
3	Size	CPFT	CPFT	CPFT	CPFT	Blend	of Devs:
4	100	100	100	100	100	100	0.0
5	84	94	92	96	98	96	4.8
6	71	88	84	92	96	92	19.4
7	59	82	76	88	94	89	43.6
8	50	76	68	84	92	85	77.4
9	42	70	60	80	90	81	121

Print a copy of this section of the E-zine and follow along, step by step, in your Excel® spreadsheet program.

Column A contains particle class sizes. Column B contains the cumulative percent finer than (CPFT) values of the target distribution. Columns C, D, and E contain the corresponding CPFT values of the candidate distributions. Each of these distributions must be measured at each of the same sizes listed in Column A for the target distribution. So the sample problem is this: We have three distributions, X, Y, and Z, and we want to calculate the blend of these three that best matches the Target distribution.

The fractions of each ingredient (X, Y, and Z) to be used are shown in the yellow cells C1, D1, and E1. These values could be located elsewhere on the spreadsheet, but this is a handy location for them.

The green cells labeled 'Blend' in Column F, contain the CPFT values of the mixture of the three candidate distributions. These cells must each contain equations to calculate the resulting distribution values. The CPFT values of the 'Blend' are calculated by summing the fractions of each distribution to be used (C1, D1, and E1) times the CPFT values in each size class (C4-C9, D4-D9, and E4-E9). The equation to be entered in F4, for example, should be:

This cell can then be highlighted and dragged (using the cell's fill handle in the lower right corner of the highlighted cell) down the column to fill in the other CPFT cells. This equation will be copied into the other cells in this column. The dollar signs force all of the equations to point to the fractional values in the first row, as the C, D, and E CPFT cell locations change from row 4 through row 9 during the drag operation. After this equation has been dragged into rows 5 through 9, the equation in cell F9 should be:

To make this into a least squares calculation, the squares of the deviations for each row will be defined in the blue cells in Column G. The deviation is the difference between each calculated CPFT value in Column F and the Target CPFT value in Column B. To calculate the square of the deviation for the size represented by row 4, enter the following equation in G4:

Drag this entry down the column to fill in Cells G5 through G9. The sum of the squares of the deviations (the sum of these entries in Column G) will be placed in the orange cell I1. Enter the following equation into cell I1:

One more cell needs to be set up, and that is the yellow cell G1 which contains the sum of the fractional values in cells C1, D1, and E1. For any Blend size distribution to be valid, the sum of the fractional values of the constituents should always equal 1.0. Enter the following into cell G1:

Now the spreadsheet is set up sufficiently so the Solver® routine can be used. If you are using a different spreadsheet program than Excel®, the optimization routines available with other spreadsheet programs should work similarly to Solver®. They may look different, but they should contain the same functionality.

Click on the Tools menu, and then the Solver item, and the 'Solver Parameters' window should pop up. The first entry in this window defines the cell to be maximized, minimized, or set equal to a value. In the 'Set Target Cell:' box, enter $I$1 (or just click in the 'Set Target Cell:' box to move the cursor there, and then click cell I1 on the main spreadsheet.) On the next line in the 'Solver Parameters' window, click on the 'Min' check box to show that you want this cell minimized. Next is a box labeled 'By Changing Cells:' Click inside this box to bring the cursor here, and then select the boxes C1, D1, and E1 on the main spreadsheet. Point to C1, click and hold the cursor button down while dragging it to highlight all three boxes, and then release the cursor button. The entry in this box should then appear as '$C$1:$E$1'.

Finally, constraints need to be added one at a time into the 'Subject to the Constraints:' window. We want each of the fractional values for the candidate distributions to be equal to or greater than zero, and we want the sum of the three fractional values to equal 1.0. To do this, add four constraints. For each constraint, start by clicking the Add button and another window will pop up to allow you to define the constraint. When asked for the 'Cell Reference:' for the first constraint, click in that box to move the cursor to it, and then click on cell C1 in the spreadsheet. Then select '>=', enter 0 in the 'Constraint:' box, and click OK. Repeat this procedure for cells D1 and E1. To set the sum to 1.0 for the fourth constraint, click Add, click in the 'Cell Reference:' box to move the cursor to it, point to and click on cell G1 on the main spreadsheet, select '=', enter a 1 for the value of the constraint, and click OK.

         Set Target Cell:            $I$1
         Equal to:                      Min
         By Changing Cells:       $C$1:$E$1
         Subject to the Constraints:
                $C$1 >= 0
                $D$1 >= 0
                $E$1 >= 0
                $G$1 = 1

This says you want to minimize the value in the cell containing the sum of the squares of the deviations by changing the fractional values of each distribution, subject to the constraints that each fraction is zero or positive, and the sum of the fractions always equals 1 (100%).

When these values are okay, click the Solve button. The 'Solver Results' window then shows whether it has achieved a valid solution, and if so, it asks if you want those values placed on the spreadsheet, or if you want the spreadsheet to restore the original values. You can see the solution values on the spreadsheet behind the 'Solver Results' window. If you don't like what you see, you can check the 'Restore Original Values' box instead. Then, when Solver® exits, it will restore the values in the spreadsheet that were present before the Solver® calculation started. The default setting is to 'Keep Solver Solution'. If you don't click the 'Restore Original Values' check box, the new solution values will remain on the main spreadsheet when you click OK to close the 'Solver Results' window.

For this example, the solution should be approximately 0.5 X, 0.5 Y, and 0.0 Z. These three values in C1, D1, and E1 will produce the target distribution. Instead of a simple 0.0, Solver® frequently produces weird numbers that are extremely small and essentially zero. For example, my result showed 4.83E-07 for the fraction of Z in the solution. This is a zero. If this happens, you can manually change the values in C1, D1, and E1, to 0.5, 0.5, and 0, respectively. This is not necessary, but it does look a lot cleaner with a 0 showing in cell E1.

The form of the spreadsheet in this example can be extended to include many more rows containing particle sizes and many more columns containing candidate distributions. When adding more distributions, just remember that each new fractional value must be constrained to >= 0 for Solver® to produce a useful result.

Also, notice in the 'Solver Parameters' window that there is an Options button. There are several parameters that can be defined to control how the Simplex routine works. When it is a large calculation, it can take quite a while to calculate the solution. The time you are willing to wait can be defined as one of these options.

Hopefully, I've given you enough to get you started using Solver® to help with your calculations. I leave it to each of you to read through the Help feature (as necessary) which explains the intricacies of Solver® and its options.

The packing potential of powders in suspension is an important concept to understand. In this article, we will consider some of the highlights of the subject of particle packing and the effects particle packing potential has on the rheological properties of suspensions and forming bodies.

When the expression 'packing potential' is used, it refers to the maximum packing factor to which a system of particles can pack when all particles are ideally positioned. In actual practice, systems of particles will never achieve such high values. The packing potential should be considered an upper limit -- one that will never be reached in practice, but one that is theoretically possible.

The minimum porosity that will be contained in this pack is simply the difference between 100% and the maximum packing factor. For example, when the maximum packing factor is 66%, the minimum porosity expected is 34%.

A monodispersion occurs when all particles are exactly the same size. A carton full of BBs, for example, is a monodispersion. A monodispersion of spherical particles has a typical packing potential of about 60vol%. If each sphere was placed carefully, by hand, into its ideal location, the packing potential could be as high as 74.04% (a perfect close-packed arrangement). A perfect simple cubic arrangement would produce a packing potential of 52.36%. But when a volume of spheres is simply dumped into a container, the packing potential of a random dense pack is expected to be about 60%.

Monodispersions of particles pack poorly. Packing factors that are 60% or lower are indicative of poor packing. Fortunately, most ceramic bodies are not monodispersions.

Most ceramic bodies contain a range of particle sizes and the packing potentials are usually much greater than 60%. Generally speaking, the broader a particle size distribution, the denser it should be able to pack. A broad particle size distribution contains particles covering a wide range of sizes. A narrow distribution contains particles that are mostly all the same size. The rule of thumb 'the broader the particle size distribution, the better the packing' isn't always true because there can be broad distributions that don't pack well. For example, when most of the particles reside in the coarsest size class and they set up a structure similar to that of a monodispersion, then the nature and width of the remainder of the distribution (no matter how fine the particles contained) no longer matters. Generally speaking, however, this rule of thumb is a reasonable guide to improving packing potentials.

It is advantageous for body particle size distributions to pack well. Experience has shown that distributions that can pack well usually also have excellent rheological properties. Although the purpose of this article is to discuss the influence of packing potential on rheological properties, other considerations remain that need to be reviewed first.

If a particle size distribution has a packing potential of 90vol%, which is easily possible in a broad distribution, one should not expect to achieve 90% packing in the ware after forming. Why? Packing potential tells how well a distribution could pack if each particle achieved its ideal location in a dense pack. This can only happen in a production environment if the body slip is totally deflocculated so all particles traveled as individuals at all times. Even then, it is unlikely that all particles will find their ideal locations. If a slip with a 90vol% packing potential is dewatered in a filter press, it is possible that the cake might have an actual packing factor as high as 85%. Probably, the actual packing factor will be even lower.

But most ceramic slips are flocculated, or at least partially flocculated. Under flocculated conditions, gel structures form by pulling particles into the structure. Open structures are produced as colloids and larger particles are immobilized into the structure. Shear-thinning and thixotropic rheological properties are produced as shear breaks down such structures. When the shear is removed, gelation rebuilds the structures. The open structures produced by gelation phenomena have little if any relationship to predicted, potential packing factors of distributions. Even when gel structures are densified by dewatering in filter pressing and slip casting operations, the actual packing factors in the cakes and casts will still be much lower than the packing potential values.

Actual packing factors are usually much lower than the values of the maximum packing potentials. This being the case, the question that everyone is probably asking is: What good, then, is the calculated packing potential? The answer to this question lies in the realm of suspension rheological properties. The first step in the treatment of this subject is to review the functions of the carrier fluid and the carrier fluid's relationship to packing.

There are two main functions of carrier fluid that apply to this discussion. Surely there are many other functions of the carrier fluid that could be listed, but two apply directly to this topic: (1) The carrier fluid fills pores; and (2) Non-pore carrier fluid separates particles.

One reason the packing potential of a distribution is important is that it defines the upper limit of solids content that allows the 'suspension' to have fluid properties. A 'suspension' that doesn't have fluid properties is of no value to anyone. Some might define it as an oxymoron. True. When the packing potential of a powder is not considered, one might try to make a 70vol% solids suspension using a particle size distribution that has a maximum packing potential of only 65vol%. This won't work. If the powders to be used in the suspension are characterized by a relatively narrow particle size distribution which has a maximum packing potential of 65vol%, a 70vol% solids 'suspension' won't contain enough carrier fluid to fill the pores. 30vol% fluid can't fill 35vol% pores, nor will it provide any fluidity.

Sufficient carrier fluid must be present in any suspension to fill all pores. The packing potential defines the minimum amount of fluid (by difference from 100%) required to fill all of the pores in a dense pack of the body powders. This is the first function of the carrier fluid.

To be useful, sufficient additional carrier fluid must also be present to perform the second function, which is to separate the particles and impart fluidity. Filling only the pores doesn't separate particles nor will it impart fluidity. Sufficient carrier fluid must be present to do both: fill the pores, and separate particles.

As particle size distributions improve from a packing point of view, less carrier fluid is tied up in the pore filling function, and more carrier fluid is available to separate particles, impart fluidity, and reduce suspension viscosities. The greater the average distance between particles in suspension, the lower will be the measured viscosities.

Packing potential, obviously, is defined by the overall particle size distribution of the body powders. Particle packing is a large subject that will be treated more fully at another time. Daily fluctuations in the particle size distribution of the body powders directly affect packing potentials; packing potential directly affects the functionality of carrier fluids because it defines the amount of fluid required to fill pores and the amount of fluid remaining to separate particles; and the amount of fluid separating particles defines the average distance between particles (the InterParticle Spacing, IPS) which directly affects the viscous and rheological properties of the suspension.

Consider the solids content routinely used in your production body. Consider also the control viscosity used for this production body. What is the rheology target of this body at this solids content when tuned to this control viscosity? (1) Is the body extremely shear-thinning because it is highly flocculated? (2) Is it only somewhat shear-thinning because it is partially flocculated? (3) Is it minimally shear-thinning with dilatant properties that appear at relatively low shear rates because it is mostly deflocculated? (4) Is it severely dilatant because it is severely deflocculated? The most probable answers should be one of the first two, possibly the third, but certainly not the fourth.

All four of these conditions are possible in any production body as a result of particle size distribution fluctuations (and the corresponding packing potential fluctuations) from batch to batch. When body solids contents are fixed, which is typical, and viscosities are tuned to a target value at a single shear rate (a single rpm on the viscometer), all of these conditions are possible.

Let's assume that the target suspension properties were selected for a body in which packing potential was okay (not really great nor poor) and the target viscosity was achieved by adding small amounts of flocculant to produce a partially flocculated suspension (#2 above). The rheology was shear-thinning and dilatancy was not a problem. Let's assume this suspension is the one that works best in the process, and this solids content, this target viscosity, and this rheology are the properties the batch house personnel are trying to match from batch to batch and day to day.

Now let's consider what happens to packing, rheological properties, and process properties as the particle size distribution changes a little from batch to batch and day to day. The fluctuations that will be discussed here are more severe than one should experience in a well-controlled production environment, but the natures and directions of the fluctuations show the possible results.

When everything is perfect and the packing potential of the body powder is really high (> 95vol%), little fluid will be required to fill the pores; lots of fluid will be available to separate particles; viscosities will be low because particles are so far apart; and the suspension will need to be more highly flocculated than normal to achieve the target viscosity. This should still produce a shear-thinning rheology and the forming properties should be quite good. Depending on the nature of the flocculants and the concentrations required, syneresis could become a problem. (Syneresis in the production body is not good. Syneresis is the extreme densification of the gel structure due to high flocculant concentrations. This subject will be covered at another time.)

On a day when particle size fluctuations cause the packing potential to decrease from normal, more carrier fluid will be required to fill pores; less will be available to separate particles; particles will be closer together; and the untuned viscosity will be higher. Under these conditions, it is possible that deflocculant (instead of the routinely used flocculant) will be necessary to achieve the target viscosity. When the target viscosity is achieved in a deflocculated suspension, the rheology will be less shear-thinning and dilatancy may begin to appear at the higher shear rates within the range of process shear conditions. (Dilatancy in the production body is also not good. Dilatancy is produced when particle/particle collisions dominate during suspension shear. This subject will be covered at another time also.)

On a day when packing potential is particularly bad, lots of carrier fluid will be required to fill pores; little will be available to separate particles; particles will be crowded; and untuned viscosities will be high. When high concentrations of deflocculant are required to approach target viscosities (if the viscosities can be achieved at all), suspensions will typically exhibit dilatant rheology (and sometimes extremely dilatant rheology). Processing will be difficult; rips can form where shear is applied; and differential shrinkage cracks may appear during drying. Extreme levels of deflocculation are undesirable, as are the poor processing characteristics that accompany them.

These examples show that when a fixed solids content plus a single target viscosity define a production slip or body, rheological properties will change as the particle size distribution and its maximum packing potential fluctuate from day to day. Fluctuations like these can happen in any production suspension.

All of the possible rheological consequences in these examples can occur as particle size distributions and packing potentials fluctuate from batch to batch. These examples are more extreme than should be present in most plants, but they show the types of rheological variations possible when packing potentials change. These examples may be extreme, but the extreme cases are possible (hopefully unlikely, but nevertheless possible).

The packing potentials of particle size distributions directly affect the rheological properties of production batches. As the particle size distributions fluctuate from batch to batch, packing potentials change; more or less carrier fluid is required to fill interparticle pores; less or more carrier fluid is available to separate particles; and at fixed suspension solids contents, the full range of additives from flocculants to deflocculants may be required to achieve target viscosities. The resulting rheological properties can vary from extremely shear-thinning (when highly flocculated) to extremely dilatant (when highly deflocculated) depending upon the natures and concentrations of the chemical additives required.

Packing potential is obviously important to those companies intent on maximizing the packing density of powders during their forming processes. But the packing potential of body powders is important to all other production processes as well because it directly affects the functionality of the interparticle fluid, which determines the nature and concentrations of chemical additives required to tune the suspension, which controls the measured rheological properties.

Are any of you interested in short courses? The PPC course that Jim Funk and I offered in the past could be offered again, or I could offer introductory courses on industrial combustion, or rheology, or other more specific topics. Think about it and send me your comments. Feel free to suggest topics you would like to see offered, the length you think would be appropriate for each course (1, 2, 3, or 4 days), and possible locations where the courses could be taught. Personally, I like Charlotte, NC, because it's easily accessible to many by car, and it's near a major airport. Send any suggestions and comments to QuestionsandComments@DingerCeramics.com .

Although I didn't use any of my add-in functions in the sample problem in this first article, the add-ins contain functions to produce the 4th root of 2 size series classification used in the table, as well as functions to produce square root of two and factor of two particle size series. Other functions that calculate numbers of particles, surfaces areas, and the packing potential discussed in the second article are also included in the add-in set.

I am aware that the first time the various rheological terms are used in these articles, the terminology may be unfamiliar to some. Wherever I was able to identify such a term, I included a brief definition. I will expand on the subject of rheology in future issues where some of these concepts will be discussed in more detail.

Paperback hard copies and electronic (PDF) copies of 'Particle Calculations for Ceramists,' the downloadable Excel® add-in functions, and other downloadable articles and explanations are available at my web site. If you're interested in any of those items, click HERE to visit that web page. Further details on the books, the add-in functions, and the subject of PPC (which I mentioned in the articles above), can be found at the web site as well. Links are available on the book page of the web site to get to any of this information.

If you can't find something on the web site that you think should be there, send me an e-mail at QuestionsandComments@DingerCeramics.com and I'll take care of it.

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