cancel
Showing results for 
Search instead for 
Did you mean: 
cancel
Showing results for 
Search instead for 
Did you mean: 

Community Tip - Have a PTC product question you need answered fast? Chances are someone has asked it before. Learn about the community search. X

Curve Fitting - F1

wzelik
12-Amethyst

Curve Fitting - F1

Dear Collab Users

I need urgently fitting equation for nonlinear curve.
Could you advice me?
I tried to find similar exapmples into the history of forum unfortunately without success.

W. Zelik

ZMR
POLAND
64 REPLIES 64

Conventional methods are tried, then discarded in favor of a spline method.
Konwencjonalne wyposa�enie modele s� pr�by, a nast�pnie wyrzuci� na rzecz prosta metoda cspline, kt�ra dzia�a doskonale.

What is the physical process behind this data? The data look vaguely like an exponential rise followed by an exponential decay. But there seems to be a lot of noise in the data, and a major irregularity around 2 to 4.

Here are a couple of somewhat arbitrary fits. But fitting functions that do not have a basis in physical reality tends to be an exercise in futility.

__________________
� � � � Tom Gutman
RichardJ
19-Tanzanite
(To:TomGutman)

On 12/29/2009 4:24:38 PM, Tom_Gutman wrote:
> and a major
>irregularity around 2 to 4.

Look at columns 4, 5, and 6. It looks like the irregularity is real, and the data is not as simple as two exponential curves. It almost looks discontinuous, with three or four segments.

Richard
wzelik
12-Amethyst
(To:TomGutman)

Thank's for yours help, I didn't expect so quick response.

This data comes from thermochemistry system which is used by me. It is result of calculation between MgO, CaO, SiO2 and Al2O3 in the 1600 degC temperature.
SL - means amount of liquid phase. Columns (0) is CaO/SiO2 ratio. Columns 1 to 8 is slag (liquid)depending on the Al2O3 contents in the system cols(1)=0%Al2O3,...cols(8)=35%Al2O3.

It is first step what I looking.
I try to find equation which describe slag amount depending on the CaO/SiO2 ratio and const Al2O3.
Next (if my knowlage will be enough) I will try to find equation SLAG=f(CaO/SiO2, Al2O3)(surface).

Once again big thank you!.

W. Zelik
RichardJ
19-Tanzanite
(To:wzelik)

On 12/29/2009 5:23:15 PM, wzelik wrote:
>Thank's for yours help, I
>didn't expect so quick
>response.
>
>This data comes from
>thermochemistry system which
>is used by me. It is result of
>calculation between MgO, CaO,
>SiO2 and Al2O3 in the 1600
>degC temperature.

Calculation? Do you mean the data is purely theoretical? Or is it derived from experimental data?

>SL - means amount of liquid
>phase. Columns (0) is CaO/SiO2
>ratio. Columns 1 to 8 is slag
>(liquid)depending on the Al2O3
>contents in the system
>cols(1)=0%Al2O3,...cols(8)=35%
>Al2O3.

OK. The data looks to be discontinuous. Is this real? For example, for 0% Al2O3 there appears to be discontinuities at 0.579 and 1.1143 (and also arguably at 1.5, where it goes to all zeros). At 25% Al2O3 (column 6) there appear to be sudden changes around 0.25 and 3.286. If these are real (for example phase boundaries in the solid phase) it may be better to fit the data in segments, with the ends of the curves in each segment constrained to take the same value at the transition point(s).

>It is first step what I
>looking.
>I try to find equation which
>describe slag amount depending
>on the CaO/SiO2 ratio and
>const Al2O3.
>Next (if my knowlage will be
>enough) I will try to find
>equation SLAG=f(CaO/SiO2,
>Al2O3)(surface).

What is the end purpose for the equation? What do you intend to use it for (this might make a difference to the best approach to the problem)?

Richard


After a considerable amount of study, it was found that the regress fitting was decidedly unstable and it was deleted. That left only the spline and a rational ortho-polynomial fit to use. To keep the worksheet functional when the y-date had a lot of zeros (0s), a small amount of selected smoothing was provided making a trifling change in the data. This was necessary in order to use a spline method as well as the rational polynomial.

It is not difficult to obtain an orderly worksheet if the second SL column is not present. However, for completeness� sake, it is included with choices marked.
wzelik
12-Amethyst
(To:RichardJ)

As I wrote I try to find eqation which desribe amount of slag as a function of CaO/SiO2 ratio and Al2O3 content in the system. This is a first purpose, second will be a surface equation (CaO/SiO2, and Al2O3)
It was made in the FACTSAGE ver 6.1 which contain very rich database of pure substances and non ideal solution as well.
I took a reaction with parameter:
<30-A>g CaO + g SiO2 + g Al2O3 + 70 g MgO.
A:1,1...30, B=0 to 35 % separately (in column).
In the first column we have 0 % Al2O3 when high CaO/SiO2 ratio is observed we don't have liguid phase becouse free lime and high calcium silicate don't react with MgO in this temperature.

W. Zelik
wzelik
12-Amethyst
(To:wzelik)

I'm sorry for problem copy-paste.
Reaction is:
(30-A)gram CaO + (A)gram SiO2 + (B) gram Al2O3 + 70 gram MgO.
A:=1,1...30, B:=0,5,10,15,...35 (in column)
To complete previous information: For column (0) we have observed peak, becouse as I wrote for higher CaO/SiO2 liquid phase doesn't appear for low value of CaO/SiO2 ratio Forsterite (Mg2SiO4, mp=1890degC) is formed.


W. Zelik

I have changed my philosophy to accommodate those views and run automatically on any data from the SL table. The entire program has been simplified. To avoid curve fitting difficulties, a conditional section has been added to process zero-values when interpolating. Program is slow.
RichardJ
19-Tanzanite
(To:wzelik)

On 12/30/2009 1:45:38 AM, wzelik wrote:
>As I wrote I try to find eqation which
>desribe amount of slag as a function of
>CaO/SiO2 ratio and Al2O3 content in the
>system. This is a first purpose, second
>will be a surface equation (CaO/SiO2,
>and Al2O3)

I understand that you want to find the equation. What do you want to do with the equation once you have it though? If it's just an empirical equation you obviously can't use it to get some sort of physical insight into what is happening. Do you want it so you can calculate intermediate values? Or so that you can plug the equations into some other piece of software for further calculations? What you want to do with the equations could make a big difference as to the best approach.

Richard
wzelik
12-Amethyst
(To:RichardJ)

Dear Richard
I'm refractory engineer. My factory have produced
basic refractory material, mainly for steel
industry. I'm interesting in corrosion model and
direct dissolution in the higher temperature is
very important factor. Corrosion of refractory
materials is very complex problem depending on
many factors and I try to describe one of them.
For example in the steel ladle we have slag from
CaO-Al2O3-SiO2 system. Typically in the slag zone
we recommend MgO-C materials. Because inside the
bricks we have carbon, infiltration of slag is
limited. Chemical reaction between material and
slag is rather surfacing (of course when we have
small oxygen partial pressure) and direct
dissolution has great importance.
Forgive me question, but have you ever analyzed
ternary phase diagram?

Besides i'm sorry for my english.
W. Zelik

I have completely revised the previous worksheets on this thread to use Programming and Boolean operators for automatic operation. Running time is slow. Posted in MC11.
RichardJ
19-Tanzanite
(To:wzelik)

On 12/30/2009 1:31:00 PM, wzelik wrote:
>Dear Richard
>I'm refractory engineer. My
>factory have produced
>basic refractory material,
>mainly for steel
>industry. I'm interesting in
>corrosion model and
>direct dissolution in the
>higher temperature is
>very important factor.
>Corrosion of refractory
>materials is very complex
>problem depending on
>many factors and I try to
>describe one of them.
>For example in the steel ladle
>we have slag from
>CaO-Al2O3-SiO2 system.
>Typically in the slag zone
>we recommend MgO-C materials.
>Because inside the
>bricks we have carbon,
>infiltration of slag is
>limited. Chemical reaction
>between material and
>slag is rather surfacing (of
>course when we have
>small oxygen partial pressure)
>and direct
>dissolution has great
>importance.

That's all very interesting, but I think you misunderstand what I am asking about. It is possible to provide a curve, described by an equation of some sort, that models your data. There are many ways to do that though, and the best choice depends on what you want to use the resulting equation for. In Study's worksheet there are a couple of pertinent examples. One is to interpolate using a cubic spline. That will produce a curve that is guaranteed to go through every point (it is an interpolation, not a least squares fit). However, it has as many coefficients as there are points in the curve, so although you have an equation that can be used (for example) to find values at intermediate CaO/SiO2 ratios, you do not have some simple looking equation that you can easily write on a piece of paper. Splines actually don't like discontinuities in the data either, and interpolated values close to them may be way off. What Study has posted is a good start, but it would probably need to be modified. Another approach is to fit a rational polynomial to the data (rational polynomials can handle discontinuities very well). That is a least squares fit, and will not pass exactly through every point. You might get a decent fit with relatively few coefficients though, so you might get a fairly simple expression. Neither of the above equations will tell you anything about the physics or chemistry of the system, but they may be useful to you anyway, depending on what you intend to use the equation for. But that's the question: what are you intending to use it for? Do you just want to interpolate the existing data (spline or linear interpolation are probably the best bets)? Or find an equation that you could somehow use in a process control system (splines would probably have too many coefficients, but the rational polynomials might work)? Or to try to find the phase boundaries from the equations (in which case this is probably not be the best approach at all)? Or something else?

>Forgive me question, but have
>you ever analyzed
>ternary phase diagram?

I've used them. What exactly do you mean by "analyzed" though? Analyzed in what way?

>Besides i'm sorry for my
>english.

It seems good enough to me 🙂

Richard

Thanks, Richard. Due to the irregular shape of any row of data, it is next to impossible to approximate it with a power series. In fact, several smoothing fruitions won�t run on one. A spline and a rational fraction are all that I can find to fit that data in one row. Other programming difficulties come from Column # 1 with a bunch of zeros, which has to be conditionally evaluated. Interpolation is a success, however, but the submiteer can not export any empirical formula except the rational fraction, whose coefficients are long and clumsy.

I give up and I am leaving this thread. I have read scores of ternary subjects. The SL data does not say what the physical terms are for the various rows and columns. Just a wild mess of a matrix to wrestle with.

I give up and I am leaving this thread. I have read scores of ternary subjects. The SL data does not say what the physical terms are for the various rows and columns. Just a wild mess of a matrix to wrestle with.

Corrected Programming Section slightly.

Thank you Study once again for your help.
Yours examples are very interesting.

W.Zelik
wzelik
12-Amethyst
(To:RichardJ)

Thank you Richard for your exhaustive answer. What
I looking for is equation as simple as posible
which will able to describe with good accuracy of
calculation data. I'm asking you about ternary
diagram because for sure you realize that it is
rather complicate and time consuming process,
besides we are limit to the three components for
practical using.
I think that if I will able to describe for
example saturation solubility of MgO as a function
of CaO/SiO2 and Al2O3 will be great. We can use it
this equation for estimate to tendency to direct
dissolution our material.
Thank’s Study, You and Tom I have examples for
thinking.
By the way, Do you thing that searching of
equation of surface as a function of Cao/SiO2,
Al2O3 does make sens?

W. Zelik

wzelik
12-Amethyst
(To:wzelik)

I enclose data from the same calculation. This time
it is saturation solubility of MgO from posted
reaction. You can see that as previous data are
irregular. But I'm interesting especially what do
you think about surface equation fitting as a
function (CaO/SiO2, Al2O3).

W. Zelik

If an empirical formula was wanted for export to another workup, only the rational fraction can be used. Assuming a single column SL amount of data was obtained from industrial experience, the expectation of any future results may be obtained from a statistical analysis. In general the most probable one-time result would be the mean of the data already accumulated. The lengthy document here explores this in detail.
Thus your work may be summarized by just giving the mean of each SL column without the interpolations. Is this what you wanted for an �equation?� The histogram shows what has happened step-wise without interpolation. Graphically.

This applies to the first posting you msde of SL data. On the new data you just posted, it gices you a start.

RichardJ
19-Tanzanite
(To:wzelik)

On 12/31/2009 8:56:58 AM, wzelik wrote:
>I enclose data from the same
>calculation. This time
>it is saturation solubility of
>MgO from posted
>reaction. You can see that as
>previous data are
>irregular. But I'm interesting
>especially what do
>you think about surface
>equation fitting as a
>function (CaO/SiO2, Al2O3).

How about this approach? The data is fitted using a number of low order polynomials, one for each phase region (I assume the sudden changes in slope are phase boundaries). The polynomials are constrained to take on the same values at the phase boundaries, and the position of the boundary is allowed to vary in the fit. I think this gives a simple set of equations that describe the curves, and is also a way to objectively find the positions of the phase boundaries.

This approach could be extended to a surface fit, but I'm not sure I have time to do that in the near future. I have to get back to "real" work, which may not be as interesting, but clients pay me for it!

Richard

Well, that is fast work. But a ternary de Finetti diagram can't be constructed on a computer, only that data points, one by one. Thst amount of work is staggering, using proability theory to calculate the lines. For example, one point can be found as the mean of each type of 3 components and marked on the de Finetti triangle. Then the outward paths are calculated from probability for each component.

I think that is a very substantial job and might take over 1000 calculations.

Manufacturers of Portland Cement using slag just pour it into the grinder.

A triangular plot cannot be found in the MathCad'store. This entire thread needs a custom worksheet that takes many calculations by any formula. Try ice/water/vapor for example. Some equations of state are based on (# of Variables) plus (# of Phases) equals (# of Components) plus 2.

Privately, I think getting a slag analysis is a bit of a nightmare, not for this Forum.

I have torn up the last posting on SL and reworked the data with no knowledge of which column is what. Used a power series fraction instad of Hermites. I assumed 2 columns were known and the job was to find any residue remaining after they were processed. For example, an analysis of coal removes the moistute and volatile matter with heat, leaving ash as the third comnponent.

Taking another approach, selected quantities of two components were added to a �mix,� and a third was marked as an �unknown� component. Then the mix was analyzed, and the total was compared to a second �mix� with the third component added. A ternary de Finetti diagram was prepared to show that the data cannot be represented on such a graph. All steps were vectorized. Throughout it is assumed the data are from a time series sequence.

What does the first SL column denote?

Thank you very much Study. You are amazing.
First column of SL data means CaO/SiO2 ratio.
Next a few columns means slag amount with increasing
contents of Al2O3 (second - 0 % Al2O3, third - 5 %
Al2O3, ... up to 35 % Al2O3. Of course as a function
CaO/SiO2 ratio also.
As I posted, I have tried to find algebraic
equation: SLAG_AMOUNT=f(CaO/SiO2,AL2O3=const)for
each column.
Thanks You, Richard and Tom I got answer.
Now I wondering, Is it possible to find surface
equation?: SLAG_AMOUNT=f(CaO/SiO2, Al2O3)

I'm not sure if I right understand your concept with
Finetti diagram!

W. Zelik

Well, now that I understand the meaning of each column and your proposition, here is an easy document that solves your problem. The importance of completely defining the physical meanings of the data was previously stressed by RLJackson. I have read your firm�s Home Page and realize you synthesize refractory materials. Reminds me of Mycalex in Schenectady, NY USA that made solid insulation from powdered mica and a heat setting binder.

I guess I understand Zelik�s proposition very well now and I have produced a new document that does it. About 2% errors in places.

Thank you once again.
I think that it is what I want.

W.Zelik

PS.
If you have any sugestion about surface fitting I will be apreciated.
Announcements

Top Tags