Curve fitting pump curves 10

[Patrick02]

Dear all

Can anyone tell me which curve fit is the best fit to use when working with head versus flow rate data and head versus efficiency data on centrifugal slurry pumps with a low specific speed (100-700 or so)?

I am a Mechanical Engineer fresh out of university and am currently working for a slurry pump manufacturing company.

One of the projects I have been assigned is to help fix up pump selection software that the company currently employees in producing sales quotes. In its current form, the software tends to misquote on certain efficiency and %QBEP values when working with certain pump curves.

I believe that part of the problem lies with the way the software plots a curve through the pump input data it is given. Currently the software will accept 7 data points per speed setting (since these are slurry pumps, the impellers are not trimmed but rather belts are used to adjust the pump speeds). The data points inputted per speed setting consist of the following:

7 flow rate values
7 corresponding head values
7 corresponding efficiency values

Once these data points have been entered into the program it plots a 4th order polynomial through the data and generates curves of head versus flow rate and efficiency versus flow rate.

I have contested that plotting a 4th order polynomial through the data does not make the generated curves more accurate. I cannot remember from theory where I saw 4th order terms in any pumping related equations. I argue that a 2nd order curve is far more likely to represent the statistically most likely value of either head or efficiency that a customer may obtain for a given flow rate should they test the pump, even though the actual data points inputted into the program deviate from the curve more than they would for a 4th order curve.

Thus I recommended a 2nd order curve fit for both head versus flow rate data and efficiency versus flow rate data and increasing the number of data points per curve from 7 to 10 so as to get more data values throughout the range of flow rates for a particular curve.

I have been challenged in this regard by the software developer, who is a senior mechanical engineer, and who tells me a 4th order curve will describe the data perfectly. He holds the position that a 4th order curve will fit irregularities in the actual performance of the pump that my university theory cannot account for and has dismissed my arguments on the grounds of inexperience in the field.

I have searched far and wide in literature for some guidance, including curve fitting techniques, correlation coefficients, residual plots etc but have not found anything worthwhile yet. I would appreciate the advice of anyone who has had experience with this type of problem.

Kind Regards,

Patrick

 

[UmeshMathur]

Hi, electricpete:

First, many thanks for your intense effort. I think your effort has demonstrated clearly that concocting a function of the type you used (power and exponential functions with 9 constants) would prove to be a painful undertaking for all but the most intrepid of practitioners. Recall that this was necessary for a relatively innocuous data set.

As you suspected, I did read off the values manually from the manufacturer's graph, hence the slight jitter. This is most often what users are reduced to doing as vendors invariably fail to provide either numerical values of the coordinates or an analytical equation for their curves.

The spline interpolation routines available in any decent software library can easily be required to provide any desired value for the terminal slope. If you know what the stonewall flow is (recall that this is not really a measurable quantity), you should add it to the data set and then impose whatever terminal slope you desire. Numerically, a very high slope would do the job, as a computer cannot "as yet" handle infinity!

For those curious about splines, the end conditions merely constrain the array of spline coefficient values so that the final curve behaves the way you would like it to, aesthetically speaking, if you were using a draftsman’s flexible strip to do the job.

 

[electricpete]

I would still say each method has it’s benefits.

Advantages of least squares fit:

- It gives a smoother curve in general

- If you do a real good job of least-square fitting a smooth curve to the noisy data, it will tend to be more accurate (not compared against the datapoints, but compared to the unknown “true” curve which presumably is a smooth curve through the "middle" of the data).

- It’s available in excel – a widely available program. Spline is not (although I think if you select the “smoothed curve” option on a chart, excel draws a cubic spline through your data? But the equation is not accessible?)

Advantage of cubic spline fit:

- If a terminal slope is known, that is an advantage to the spline method (can’t be put into the least squared method).

- When I zoomed out I see the form of the my least-squares fit function extends fairly smoothly outwards to 0 DP units. Unfortunately it only extends left to 0.5 in flow rate units due to my choice of fit functions. I could have avoided this by eliminating the “zeroshifter” parameter in the first fit (yhat1). Lesson learned by trial and error.

- Spline takes less time. Once you decide cubic spline there are no more decisions to be made and the computer gives you the result. The least-squared fit involves a little trial and error (although if you repeatedly fit pump curves I think it wouldn’t take long to know which terms to use). It took me just a little over an hour to do this least squares fit (and I have never fit a pump curve before). Then some additional time to write it up and post it on the board.

I have a question – what is innocous about this set of data compared to typical pump curve data? (bearing in mind the least-squares method is well equipped to deal with noisy data)

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[electricpete]

Also - 9 constants yes, but only 3 terms: a power-law and two exponentials

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[UmeshMathur]

Hi, electricpete:

First, I believe that the curve fitting method anyone chooses is always subjective (driven by personal experience and belief). My reason to enter this discussion was simply to mention that cubic splines are a much overlooked tool that can do the job very efficiently and without much controversy as to the quality of the fit.

I used the word innocuous to describe my data set simply because (a) despite not containing any weird kinks or discontinuities, it is incapable of being fit properly by any polynomial up to at least order 5, and (b) in all cases, the polynomial extrapolations are truly horrible. This would be a nightmare for most people who don’t have much experience or knowledge beyond polynomial regression.

Your own very creditable effort demonstrated that concocting a well-behaved function took quite a bit of mathematical ingenuity, if not cunning. I’m sorry to say that most engineers do not have the time or patience to do this, in my experience, even if they have the requisite skills.

 

[electricpete]

Yes, that's a good point. I think if the data is smooth then spline will do the job very well and least-square curve fitting could waste a lot of time. If done poorly, curve fitting gives worse results. To do well may take a lot of time especially considering extrapolation requirements.

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[spciesla]

You may be interested in the following curve-fitting program. It has a table/spreadsheet interface, and data can easily be copied from excel.

http://curveexpert.webhop.biz/

 

[Patrick02]

Hi again,

I have been monitoring the conversation between electricpete and UmeshMathur. It is interesting to note the outcome of trying to fit integer polynomial order fits to data versus spline fitting. I have also been evaluating the data I have been given and comparing the results of the various fits. I am still assessing which fit is "best" for my situation.

Essentially I have found through searching through our current database that I have 7 data points in general to describe a relationship between flow rate and head produced and flow rate and efficiency achieved. I initially hoped that I could increase the amount of data available to me and possibly get multiple sets of data for a single condition, thus increasing my confidence in the curve I eventually fit to the data. Since I am dealing with an entire database of over 200 pumps, retesting each pump is just not viable (My engineering manager would simply not allow for it). So for the immediate and practical future I have 7 data points to relate 2 variables, in general. These data points were extracted from the present pump selection software database. Due to a lack of documentation I also have no idea whether they represent test data or data that has been inputted from a previous published curve.

One of my fundamental problems is assessing the reliability of the data (but as the situation would have it – it is all that I have to work with).

I must ensure that the curves that are eventually fitted to the data represent the most likely relationship between the input and output variables in a typical situation for the pump. I have assumed points of inflection and turning points are not representative of the physical situation in terms of relating flow to both efficiency and pump head. Forgive the crude drawings but essentially the relationships are depicted below. Thus I do not expect any sudden turning points and so on within the data (as could happen when a 4th order polynomial tries to accommodate an outliner such as a datum with a high noise probability). Sometime the 0 flow condition on Curve 1 does not correspond to the highest point on the curve (as depicted in this case by the 0 on Curve 1) and Curve 1 can “hump” at a certain flow rate, but the relationship should not produce any sudden points of inflection. The same holds true for Curve 2.

Ix
H I x
I0 x
I x
I x
I x
I x
I_________________________ (Curve 1)
Flow

I
Eff I x
I x x
I x x
I
I x
I
Ix_____________________________ (Curve 2)
Flow

I think one of my core problems is that I have little faith in the input data values. That does not mean to say I think they are non meaningful but rather I have no reliable assessment of just how meaningful they may or may not be. If I had reliable experimental data (including some error analysis) I would happily use the curve fitting techniques as well as the programs suggested to me by the respondents in this forum to obtain the “best” curve fit for my situation.

A good analogy would be to imagine that I was tasked to determine the relationship between Force and Deflection in a spring. For arguments sake I will assume that in reality the spring is linear under the range at which I am testing it, that is to say F=kX, where k is a constant – but at the start of investigation assume I do not know this fact and have been told that any relationship may apply. All I have to work with is a few data points. I have no idea of their origin or the test conditions (if they were derived experimentally as opposed to being read off a previously published graph). The relationship may be follows:



I
I x
I
X I x
I x
I x
I x
I_________________________
F

I may know from university that the relationship should be linear over a certain range but I am told that in reality it is always slightly non linear. Since I find it hard to accept most things at face value without some proof I cannot accept this statement. If however I try to fit a curve (whether a polynomial, spline or other) to the data I find that, for arguments sake, a 3rd order polynomial fits the data very well (with corresponding R^2 vales and residual plots that look good). I find a linear fit does rather badly yet in my gut I simply feel it may more representative of the relationship. In the end it may turn out that I was only partly correct and that a 2nd order curve could the most representative (albeit with an extremely small coefficient for the x^2 term). But from the given data I do not know. I could simply say that I will work with what I have been given and whatever curve appears to fit best, I will implement but I am apprehensive of causing trouble down the line, especially since my generated curves are to be used as masters that will be used as references in an extrapolation process that finds data between two curves.

Back to the pump curves, since I have little confidence in the data but testing the entire range of pumps is not feasible I have decided to try test one or two pumps thoroughly (up and down the curve etc) and obtain experimental data. I will apply what seems to be the most representative fit for the data, taking into account all the above comments and suggestions and then look at the resulting curve function. I hope to use that as a template for which to decide how the rest of the data should actually be fit and apply that fit type to the rest of the data set, regardless of what the initial data values say.

Of course in doing this I am assuming that the pumps behave hydraulically in the same manner (most of them are scaled versions of each other so it may be reasonable to assume this). I run the possible risk of forcing a fit to a data set that may in reality differ from the curve type I obtained from testing but for the time being I am willing to take that risk. If there are any major deviations for any of my pumps I will look at that pump individually and reassess the fit, possibly even testing that pump (if allowed).

Does that sound reasonable?

Regards,

Patrick02

P.S: All The curve fitting programs I have downloaded are obviously similar in function but the one suggested by spciesla is quite user friendly.

 

[Patrick02]

Just a quick afterthought; I have data for multiple curves for the same pump as each curve as given in Curve 1 represents data at a particular speed setting and typically I have between 4 and 7 speed settings per pump. Thus I could convert all my equations to a non dimensional form and that would give me a lot more data on one curve. I could then try determine the best fit through that non-dimensional curve and use it as a template to generate my dimensional curves. This follows on from UmeshMathur’s comments on using the fan laws. The affinity laws are well known for certain parameters such as flow rate and pump head or pressure but I have never attempted to generate the non dimensional form of the relationship between flow and efficiency with respect to pump speed. It should be an interesting exercise to see if it is worthwhile or not. It’s worth a shot and I will know at the end whether it is useful to me or not. This does not alleviate my lack of faith in the data but it is another alternative to consider

 

[UmeshMathur]

Patrick02:

A few suggestions:
(1) The general shapes of both your curves are quite reasonable.

(2) Since you have data for several RPMs, I think you really have no choice but to use the non-dimensional form of the curves. By the way, for applying the similarity/affinity laws, the proper non-dimensional quantities when impeller diameter might also be a variable are: (a) Qs/N/D^3, and (b)g*Hp/N^2/D^2, where Qs=actual suction volumetric flow[m^3/s], N=speed[revolutions/s], D=diameter[m], g=gravitational constant[Newtons/kgf], and Hp=polytropic head[m*kgf/kg], in consistent metric units. (I thought I would mention the diameter in case some of your test data was for geometrically similar but different size pumps. Also, the quantities mentioned in previous posts are NOT dimensionless).

(2) Rest assured that the non-dimensional form for polytropic efficiency v/s Qs/N/D^3 is theoretically sound for pumps as well as compressors.

(3) Remember that these laws were developed for Newtonian fluids. Their use for mildly non-Newtonian fluids is OK, provided the viscosity is not very large and you use the proper effective viscosity. Another dimensionless quantity to check with varying fluid properties is (c) rho*N*D^2/mu, where rho=fluid density, and mu=dynamic viscosity.

(4) In case you see a systematic deviation in the non-dimensional curves for a particular RPM relative to the others, you might suspect an RPM measurement error (quite common) and consider applying an adjustment to the quoted RPM to make the values line up better with the rest of the curves.

(5) All in all, I feel that a spline is probably your best bet in the circumstances you have described with a limited amount of data. Before doing any curve fits, however, do make sure that the curves for different RPMs were not generated from the curve for one RPM simply by using the affinity laws. The same check should be applied for different pump diameters.

(6) Lastly, since you are dealing with slurries, there could easily be some uncertainty as to how representative the measured values really might be, especially in case slurry density changes as a result of a differing solids content.

Do advise us when you have finished your assignment, as the results will be interesting to all who have participated in this thread.

Have fun, and cheers.

 

[EdStainless]

U-M, great comments about fitting. I will have to remember this when my next such task comes along.

Patrick, I thnk that early on in this discussion you hit on one of the root issues. It isn't the type of fit that is used, it is the uncertanty in the data. If you re-test the same pump over and over you will get different values (not to mention testing different pumps of the same model). That is what you need to explore. It is likely that some of your most basic measurements (shaft speed or pressure) are not nearly accuraate enough to support the conclusions that you are trying to make.

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[Patrick02]

Hi to all those who helped me during my query on this thread.

My apologies with the delay on my reply. I thought I would give you a concluding synopsis of what I had to do to fit these curves.

My immediate involvement in the project is now finished and it has been a learning curve for me.

Soon after the conclusion of this discussion I was informed by the marketing manager that the software selection program used by us only handles 4th order polynomial fits when generating data and nothing else. There was neither time nor money available to change this during the scope of the project. Why such a constraint was placed in the program in the first place is beyond me. Thus I was limited to using a 4th order fit to fit our data. The focus of the program as communicated to me and hence my assigned brief was to “fix” our database for the immediate future while a more comprehensive test program on the entire range of pumps was to be implemented. The input data source for the fixed database was our current published curve book. Essentially I had to ensure that what was in our selection program matched up to what was given in our curve book. My only guess was by inference this meant that what was in our curve book fairly represented the operational characteristics of the pump through customer witness tests (even though test data was not available to me). I devised a program to take digital images of our published curves, scale them and grab data points off them. Those points would be taken as what was seen in the curve book. My job was to fit 4th order polynomials through those points and generate an equation for the polynomial, which the software selection program could use. The whole process appeared a little bizarre to me but nevertheless it was my assigned brief. I could not successfully fit a 4th order curve through all my points, as discussed earlier, so I devised a program which would “add error” to certain points far to the left and right of the best efficiency point along a certain speed curve in order to increase the accuracy of the fit in the region close to the best efficiency point. After this was done for all speed curves on a particular pump curve I generated a so called “selectable region” where sales personal could safely make selections on a pump with errors between what they saw in a curve book and what the program told them of less than 1%. In regions outside of that the error increased dramatically. For example on some speed curves I was forced to add an error of up to 25% to the 0 flow condition (even though this is nonsensical) to make it fit well near the region of best efficiency. Thus if a salesman selected a pump at a flow condition near zero the program would tell him that he had a pump that was 35% efficient when in reality it was only 10% or less.

In this manner I managed to produce a short term solution to our problem. I did not, however, like the fundamental approach to this problem and method of “playing” with our data by adding and subtracting error in order to get a software program to work. Thus I documented where I added error and how I constructed the new curves so that sales personal were fully aware of the situation when making selections. I have since advocated a new testing program for our pumps with more documentation and control to avoid this type of situation. I may have fixed a problem but I did not consider it engineering. The one benefit to all of this is that we now have a lot more information on other aspects of our curves, which is documented electronically. This will help me when we eventually supplement this “fixed” database with up to date and controlled test data to increase our confidence in the data.

Thank you for all your help. This has been a learning experience for me.


Kind regards,

Patrick

 

[BigInch]

Sorry for the late entry into what is apparently a closing discussion, but I would like to know one thing.

Why is it important to know the efficiency of a pump that is flowing at near zero? Is anybody really going to operate a pump for extended periods of time at a near zero flowrate. Surely that the operator expects to pay a heavy penalty in efficiency when flows go lower than 50% BEP. Should not the object be to develop highly accurate efficiency predictions at reasonable flowrates, or failing that, design a better pump station configuration?

Going the Big Inch!

http://virtualpipeline.spaces.msn.com

 

[Patrick02]

That is perfectly correct BigInch. There is no reason for any of our customers to wish to operate a pump far off its design point. Anything less than 60% QBEP or greater than 120% is considered a no go area. This served as the basis for my reasoning that I could decrease the accuracy in the regions < 60% or >120% QBEP in order to improve the accuracy of the fit though the region within 60% to 120% QBEP since no one would wish to operate outside the region anyway. The fundimental problem I had with this had more to do with the methodology behind the data in that I should not have been grabbing it from a curve book but rather be using reliable and controlled test data. We have reasonable efficiency predictions at reasonable flowrates but I was hoping that we will in time gather more test data on all our curves.

Regards,

Patrick

 

[BigInch]

Patric,
Thanks for that confirmation. I couldn't imagine what all the fuss was about. I thought my petroleum concentric experience had blinded me to some strange slurry application that required extended running off BEP. IMO you made a good decision to concentrate your study towards the reasonable flow ranges. I don't consider any efficiency data outside the margins you mention as significant for operating purposes, and start-up/shut-downs are such limited times, it makes no difference what's happening there. If operating in those regions, there are problems that need looking after far worse than determining pump efficiency to the 4th order.

I always limit my pumps to running between 50% and your same 120% and will change a multiple pump configuration to accomodate operation within those ranges at all times. If flow conditions change, I will try changing the process operation first, but if that's impossible I will try ripping out the existing pumps and get some that will operate within the proper margins. The money saved in one year is considerable in relation to pump costs when pumps are operated at proper efficiencies and arn't being destroyed by poor operating practices.I

No matter what the published curves say or what performance is quoted, we will do shop flow tests to develop the as-built curves and then test in the field as proof of ultimate performance acceptability, so usually I get pretty good data, eventually.

For simulating pump performance, I will generally use 10 points, mainly because it makes an easy-read graph, but I will go to 16 points if extras are warranted for flowrates important to the objective of the simulation.

What I would like to be able to get to 4 signficant figures are the dry and wet moments of inertias. Where's my wish list?



Going the Big Inch!

http://virtualpipeline.spaces.msn.com

 

[Patrick02]

BigInch,

What you are doing sounds quite awesome and even though I work for a pump manufacturer, were I to work for a process company I would also perform field tests of published curves simply because normally it is not that difficult to do and can save a lot of time and money should a problem be picked up. Gaining information on dry and wet moments of inertia seems interesting and would help were you able to conduct tests to get those figures. My particular field of expertise (even though I am fairly new to the game) is vibration analysis. What I want to study is the effect of sump level on the vibration response of our long vertical spindle pumps. Currently no one knows just how different slurry types and densities affect the critical speeds of these pumps. All quotes on pump critical speeds for the vertical spindle range are based on "dry" calculations. Due to the damping effect slurries have on the rotordynamics of the system they are bound to lower the critical speeds.

Just going back to what you mentioned on good operating practices and cost savings, you may be interested to check out an article called "How to Justify Machinery Improvements Using Reliability Engineering Principles." It is also about how best practices can lower the life cycle costs of the pump and the authors use reliability engineering simulations to test the theory. It is quite an interesting read. Go to :

http://www.barringer1.com/Papers.htm

and scroll down to article 13. and download it. Hope you enjoy the article

Regards,

Patrick

 

[BigInch]

Pat, I know the Barringer website and did "pick up" that doc. Its a good site. I have pretty well cleaned it out. I like simulations that incorporate Monte Carlo techniques and have gleened a lot of good info from them. (#15)

I'm doing some consulting off/on for a mideast state oil co where the pumps are generally high capacity, high pressure and quite high cost, so they get them with all the bells and whistles. I've found that even the biggest pump mfgrs sometimes get it wrong and have seen a few cases where the as-built curves are a percent or two or three under the guoted curves ... over the entire Q range, and that was not for slurries. I can only imagine what you're trying to deal with. To expect them to be 4th order accurate way off BEP, just seemed to be too much to ask for in the first place, and in the second, not of much practical use at the end of the day. Critical speeds of shafts, now that's useful. Sounds like a high gray matter project. I don't know much about slurries, but from what I have studied, they seem to suffer some radical behavior changes with what seems like tiny variations in densities and flows too. At one point they're liquids, then shift to almost solids.... ugh! There must be some really strange effective mass dampening going on around the shafts when the density, surface speed, and flowrates change.

Would you mind dropping me an e-mail (profile secton on my webspace page)?

Going the Big Inch!

http://virtualpipeline.spaces.msn.com