Volume 1 Number 12 Dennis R. Dinger 1 October 2003
An Update
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The topic is one that I think is very useful. Jim Funk published it in the Ceramic Bulletin about twenty years ago, so many of you may not realize it is out there. The figures in this article will test your abilities to visualize and your abilities to do mental gymnastics. I tried to make them as simple as possible, but the transformations can still be confusing.
Using Irreversible Dilatometry to Design a Firing Curve
The reference for the original publication of this subject is:
Funk, James E., "Designing the Optimum Firing Curve for Porcelains," Ceramic Bulletin, 61[6] 632-635 (1982).
If you are interested in this subject, you can dig this paper out of the archives, dust it off, and put it to good use.
The Reasoning Behind This Approach
To keep costs to a minimum, firing schedules should be as short as possible. Engineers are frequently asked to shorten their firing curves. Where should one begin? The irreversible thermal expansion (ITE) curves are a good starting point for such changes.
Jim Funk came up with a relatively easy way to use TE curves for this purpose. As far as I know, he 'invented' and then published this procedure.
As ceramics are heated they expand and contract at a variety of rates. The presence of reactions, decompositions, phase changes, etc., affects the rates at which the wares expand and contract during firing. The cross-sectional thicknesses of the wares also affect the kiln temperature ranges over which the reactions take place. There are no such reactions over wide ranges in temperature in pure alumina wares, so alumina thermal expansion traces are relatively linear over those temperature ranges. As temperatures rise, wares expand. When sintering and vitrification take place, wares contract. Thermal dilatometers measure the rates and amounts of expansion/contraction as wares are fired. So the question is: How does one use thermal expansion data to design firing curves?
In the temperature ranges in which the rates of expansion or contraction are very fast, wares may crack (or shatter) if they are heated or cooled too quickly. When the outside of the ware heats and expands, but the inside is still at much lower temperatures, stresses build up and the ware may shatter. In the temperature ranges in which there is little thermal expansion or contraction, however, the fundamental sizes of the outside (at higher temperature) and the inside (at lower temperature) of the ware will remain similar in size. In such temperature ranges, the wares can be heated quickly without harm. Why? When little expansion takes place over a temperature range, stresses due to size differences in structure will be small. When the structures at two points within the ware are approximately the same size, even though the temperatures at the two points differ by 100oC, internal stresses will be minimal. Obviously, kilns require finite lengths of time to increase temperatures by 100oC, but in such cases, the temperature range can be covered as quickly as allowed by the kiln. The outside of the ware will heat quickly, the inside will lag behind, but when there is no structural size differential over the temperature range, the temperature can be raised quickly. (This obviously is a best case example.)
On the other hand, when expansion and contraction rates are high, kiln temperatures should be raised slowly to minimize internal stresses.
This method for designing firing curves uses the principles described in the preceding paragraphs: When expansion or contraction rates are high, heating and cooling rates must be slow; and when expansion or contraction rates are low, heating and cooling rates can be fast. Several mirrored transformations of the irreversible thermal expansion (ITE) curve (an ITE analysis begins with a dry, unfired sample of the ware) will produce the fundamental shape of the firing curve. Consider the ITE trace shown as Figure 1.
Figure 1. A Sample Irreversible Dilatometer Trace
This is an actual ITE trace to which an extra V-shaped peak and valley (in the 600-700oC range) have been added. The extra peak and valley will help show the procedures used with this method. If you print this figure and flip it along the 45o diagonal from lower left to upper right, temperature will then be the Y-axis, and thermal expansion will be the X-axis. You will need to hold the sheet of paper up to the light and look through the paper to see this. This can be done with any printed dilatometer trace.
After this first transformation, horizontal lines on the ITE trace (no thermal expansion) become vertical lines (fast temperature changes). And near-vertical lines on the ITE trace (fast thermal expansion) become horizontal lines (slow temperature changes) on the transformed curve. After this one transformation, the temperature axis (the new Y axis) is in the proper orientation for a firing curve. The new X axis, which is Thermal Expansion, will be changed to time (for periodic kilns) or length (for tunnel kilns) to produce the completed firing curve.
When playing with an ITE curve in a spreadsheet program, instead of plotting thermal expansion vs temperature (which is the normal form for ITE data), plotting temperature vs thermal expansion will accomplish this first transformation.
Figure 2 shows the original ITE trace in dark blue and the transformed trace (which is a reflection about the 45o line) in magenta. The 45o reflection axis is also shown in the figure. Ignore the vertical line and the arrow for the time being. Note that the origins in the next four figures are the points where the two curves touch. The origins correspond to room temperature and zero thermal expansion or zero time in each case. The new axis labels are shown in the later figures.
Figure 2. The original ITE trace mirrored about the diagonal.
To produce a firing curve, the new X axis (for the magenta line) must be declared to be Time instead of Thermal Expansion. And a proper firing curve must continually increase to the right with time. But the magenta line in Figure 2 does not constantly increase to the right. Therefore, this first transformation of the ITE trace (the magenta line in Figure 2) is not an acceptable firing curve. The vertical line in Figure 2 is the reflection axis for the second transformation. The first point at which the newly transformed curve turns back towards the Y axis is shown by the arrow. All temperatures above the point where the newly transformed trace touches this line (all temperatures above the arrow) must be mirrored across this vertical line. Figure 3 shows the result of the second transformation applied to the ITE curve. Note that the high temperature part of the curve in Figure 2, which points to the upper left corner of the diagram, is transformed so it points towards the upper right corner of Figure 3.
Figure 3. Result of the 2nd transformation of the ITE curve.
In Figure 3, the vertical reflection axis has been moved to the right to the new point (shown by the new arrow) at which this transformation of the original ITE curve once again turns back towards the Temperature axis. All points at temperatures above this point must once again be mirrored across this vertical axis. The result of this third transformation is shown in Figure 4. Once again, the high temperature part of the curve points towards the upper left corner of the diagram in Figure 4. This means the curve must be transformed yet again.
Figure 4. Result of the 3rd transformation.
The new vertical transformation line, and the new temperature above which all points must again be transformed (shown by the new arrow), are shown in Figure 4.
In an automatic calculation procedure, one must check the results after each transformation. Every time the newly transformed trace turns back towards the temperature axis, another transformation must be performed. In each case, the mirror transformation must be performed for all data at temperatures above the lowest temperature at which the new trace turns back towards the temperature axis. Eventually after a sufficient number of transformations, all points on the transformed trace will have a positive slope. At that point, the mirror transformations are complete.
Figure 5 shows the result of the 4th (and final) transformation required in this example.
Figure 5. The Finished Transformation Showing The Heating Curve
The transformed trace in Figure 5 is in the form of a proper heating curve. Temperature is the Y-axis and Time is the X-axis. All points on the transformed (magenta) trace increase in temperature with time. The magenta line no longer appears to resemble the original TE trace, but as shown in the figures above, the two are precisely related. For the ITE trace in this example, only 4 transformations were needed. Depending on the ITE curve, many more transformations may be required. But they are easily calculated automatically in common spreadsheet programs. MS Excel®, which was used to produce the figures in this article, can be programmed to automatically perform all required transformations.
Figure 5 finally shows a heating profile calculated from ITE data. No time scale is shown because that is something that needs to be determined experimentally (or by experience). Temperature in Figure 5 goes from 30oC to a firing temperature of 1100oC. A soak can be added at the firing temperature and similar transformations can be applied to the cooling curve of the thermal expansion run to obtain the cooling profile for the overall firing schedule. The cooling curve of the firing schedule can also be produced from the Reversible Thermal Expansion data (which is produced by running a sample of fired ware in the dilatometer).
The heating curve, shown by the magenta line in Figure 5 should be considered to be a starting point. If the firing curve is being calculated for a production ware, the calculated heating curve from this procedure should be compared to the heating curve in the production kiln.
Figure 6 shows a complete firing curve formed by using this heating curve, a soak, and a cooling curve. Figure 6 shows a 10 hour heating schedule, a one hour soak, and a 9 hour cooling schedule. No significance should be placed on these times in this example. They were chosen to show how the heating schedule calculated by this procedure fits into the total firing schedule.
Figure 6. Firing Curve Using Heating Curve From
Figure 5,
1 hr Soak at 1100oC, Plus Cooling Curve
The nature of the firing schedule in Figure 6, with temperature versus time, applies to periodic kilns. When the kiln to be used is a tunnel kiln, then the Time scale should be changed to position (in feet or meters) along the length of the kiln. The firing zone in a tunnel kilns will be fixed at a certain position within the kiln. This will help to fix the heating rate. The temperature in a tunnel kiln starts at 200-300oC at the entrance, and the temperature must reach the firing temperature at the firing zone of the kiln. The soak period will also be related to the length of the firing zone, and the cooling curve will necessarily fill the remaining length of the kiln. The calculated heating, soak, and cooling curves can each be scaled to properly fit the required lengths in each tunnel kiln.
Periodic kilns have no fixed time requirements (corresponding to tunnel kiln length requirements) to reach the firing temperature, to soak, or to cool again to room temperature. They are totally flexible.
General Approach To Modifying A Firing Curve
Once a firing curve exists (such as the one shown in Figure 6), one must decide the total duration of the firing (for a periodic kiln) or one must obtain the specific lengths (for a tunnel kiln.) If for instance, a 10 hour cycle is desired in a periodic kiln, the time axis from the curve in Figure 6 should be re-scaled to fit a 10 hour cycle. If the curve is to fit a 100m tunnel kiln, in which the firing zone is positioned 60-70m from the entrance, the heating curve from Figure 6 should be scaled from the entrance temperature to the firing temperature in the first 60m, the soak should be from 60-70m, and the cooling curve should be scaled to fit from 70m to the end of the kiln at 100m. The time axis should be replaced by position (distance along the tunnel kiln) and the new X-axis should be scaled to these specific kiln values.
As mentioned above, this point in each case should be considered a starting point for all modifications. What modifications should be made? Generally speaking, all modifications should be to lower the heating and cooling rates in certain ranges of the calculated firing curve. The assumption is made that the calculated firing curve will initially be fast (too fast), so all modifications will take place in the direction of process speed reductions. Heating and cooling rates covering the temperature ranges of all body reactions should be reduced in most cases. For example, a decision may be made to set the rate over the quartz inversion range to 100oC/hour. Similar decisions should be made for all other reaction temperature ranges. One might also decide to lower the highest heating rate on the new firing curve to equal the highest heating rate possible in the particular kiln to be used.
Discussions of a variety of considerations that apply to firing curve modifications will be included in the next sections.
Adjustments to Firing Curves
Critical Heating Rate
At each temperature on a firing curve, there is a critical heating rate. Proceeding too fast through any particular temperature can cause losses. If there are no losses (or minimal losses) produced by any successful firing curve, that does not mean that the firing curve has been optimized. It still may be possible to shorten the firing schedule. As firing schedules are shortened, one point on each curve will be primarily responsible for new losses. When losses can be associated with specific temperature ranges in the heating or cooling curves, heating and cooling rates in those zones must be reduced accordingly. As the critical heating rate in each zone is identified and fixed, the rates in other regions of the firing curve can be increased until the next problem occurs. Then, the process can be repeated to fix the new heating or cooling rate for the new problem, etc.
If a body contains quartz, or the ware contains other minerals which transform through several phases during heating and cooling, heating rates frequently need to be slower passing through those transformation temperatures. 573oC, for example, is the quartz inversion temperature. The quartz inversion temperature is not as critical on heating (because the quartz is still free powder) as it is on cooling (when quartz may be surrounded by glassy phase). In porcelains, the heating rates during dehydroxylation and carbon burnout ranges should also be considered. Heating too quickly through these ranges can cause problems because reactions may not be complete at the center of the ware when the outside of the ware moves into the next temperature ranges.
In vitrified wares, the glass transition temperature on cooling is also critically important. Jim Funk taught that many problems occur at the quartz inversion temperature on the cooling curve, NOT necessarily because cooling is too quick through the 600-550oC quartz inversion range but because the cooling curve is too steep through the glass transition (Tg) temperature at higher temperatures. When cooling through Tg is too fast, structures can be frozen into place before they have sufficient chance to relax. This produces a buildup of internal stresses. If internal stresses resulting from fast cooling through Tg are sufficient, the quartz inversion may be sufficient to cause the body to fail. Such problems may occur at the 573oC inversion temperature, but they may be the direct result of high cooling rates at higher temperatures.
The Tg temperature is visible on the Reversible Thermal Expansion trace. It is frequently in the 750-850oC range. Some kilns are not designed to go slowly through this temperature range because the thinking at the time they were constructed was that the only critical range on cooling was the quartz inversion range. In such kilns, one may simply need to go more slowly through both zones (which may require one to go slowly throughout the whole tunnel kiln).
Thickness of the Ware
A major concern for a firing curve is the thickness of the ware. The thinner the cross-section of the ware, the faster the firing can be. Ceramic tiles are relatively thin and they can be fired in 30-40 minutes cold to cold. Sanitaryware is thicker. Bricks and refractories are thicker still. As thicknesses increase, firing curves must necessarily lengthen.
The reason for the longer duration firing curves in thick ware is the fact that the heat has to be conducted from the ware surfaces to the center of the ware. The first problem in firing is to transfer the heat from the burners to the surfaces of the ware. If temperatures are not uniform throughout a kiln, this can be a firing or setting problem. But when heat is uniformly distributed throughout the kiln, it still has to be transferred to the ware, and then it must flow within the ware to reach all internal points. Heat transfer to the center of the ware is especially slow when the ware is thick. The surface temperatures of ceramic wares should not to be too different from the center temperatures of the ware. How different is too different? Each product will have a different delta T (from surface to center) that it can tolerate. In some wares, 50oC may be OK. In other wares, 10oC is too much.
Type of Kiln
Another consideration is the type of kiln used for the firing. Periodic kilns lined with heavy refractories will not heat quickly. Heavy refractory linings soak up heat as they rise in temperature along with the wares. The extra thermal load of heavy refractories will necessarily make heating and cooling proceed slowly. Heavy refractories, however, work well in tunnel kilns which are run under steady-state conditions. The only ceramics to be heated in tunnel kilns are the wares to be fired, kiln furniture, and the refractories in the kiln cars. In roller kilns, only the wares need to be heated. Everything else in such kilns is already at the required temperatures. For this reason, roller kilns can achieve especially fast firing curves.
For periodic kilns to proceed quickly, good burners (high velocity burners work well) used with fiber linings can produce fast heating rates. This is the case because the fiber linings have very low heat contents. Very little heat is required to bring a fiber lining up to temperature. Therefore, for fiber-lined periodic kilns, the main heat load is the ware and the kiln furniture. Heavy refractories in a periodic kiln cause firing schedules to be slow. In such cases, the kiln will heat no faster than is required to bring the refractories and the ware up to temperature. Cooling in such kilns will be slow as well for similar reasons.
Where to Make Adjustments
My suggestion is to pick what you consider to be the most critical temperature on the heating curve. Match the heating rate of the calculated firing curve to the heating rate of the production firing curve. Do the same on the cooling side. Then study the differences between the calculated and production schedules.
Just because a kiln has a 24 hour firing schedule does not mean all heating and cooling rates are sufficiently slow. I have seen 12 hour schedules that cooled more slowly through Tg and quartz inversion temperatures than the cooling rates in their 24 hour counterparts. Sometimes, total firing times can be reduced considerably, while still producing slow cooling rates through the critical temperature ranges.
Summary
This is a fairly simple method to quickly arrive at a reasonable firing curve from a very common instrument -- a thermal dilatometer. This method indicates the temperature ranges in which one can heat and cool quickly and the temperature ranges in which one must proceed slowly. It should be widely applicable to all ceramic products and processes.
Miscellany
I didn't want to just rewrite what Jim said in his original article, so I didn't reread his paper before writing this article. He and I always explained things differently anyway, so this E-zine article plus the original from the Ceramic Bulletin should provide you with two different explanations of the same subject. I trust that if any of you are interested in this subject, you will get a copy of Jim's original paper from your local ceramic library. If any of you are interested in an MS Excel® program that performs this calculation, send me an e-mail to let me know. I have a working MS Excel® program to perform this calculation, but it is not in a distributable form. If there is enough interest, I will fix it up and make it available.
I continue to look for suggested topics for future columns. I have had some great suggested topics, but they happened to be in areas in which I have no experience. Keep the suggestions coming. I look forward to hearing from you.
Until next time ...
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