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I need to do viscosity measurements at different temperatures in order to fit a model of the form (Flucher equation): $$\log_{10}{\eta}=A+\frac{B}{T-T_0}$$

Where $T_0$ is constant and known, $T$ is the temperature at which I'm making the measurement, $\eta$ is me the measured quantity, and $A$ and $B$ are the constants I'm looking for.

How to know the minimum amount of measurements I need to do to fit the model?

corcholatacolormarengo
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  • Well, you should take at least two measurements to hope to solve for the two unknowns, but the more measurements the better. What kind of answer are you hoping for? Why not just take a lot of measurements? – symplectomorphic Feb 25 '19 at 23:39
  • @symplectomorphic "at least two measurements to hope to solve for the two unknowns" - Maybe that's the answer I'm looking for. I mean...You say that more measurements is better? Why would that be the case? Why not just those two? – corcholatacolormarengo Feb 25 '19 at 23:41
  • Because your measurements will contain error. It's extremely, extremely unlikely that your two measurements will lie perfectly on a curve of the exact form you have specified. Instead the measurements will lie "roughly close" to that curve, and you need several measurements to estimate the parameters $A$ and $B$ by calculating the "curve of best fit." – symplectomorphic Feb 25 '19 at 23:43
  • @symplectomorphic I see. So what I got was: If I was able to measure perfectly I would need two, but since I cant, more points will be needed. And the minimum number of points is given by the unknowns in the equation, regardless of its form. – corcholatacolormarengo Feb 25 '19 at 23:50
  • If you were able to measure perfectly and your model of the phenomenon were exactly right, you’d only need two measurements. But of course neither of these things is true. (Even if your measurements were errorless, you wouldn’t be able to solve for the unknowns unless the points really did fall exactly on the curve.) – symplectomorphic Feb 26 '19 at 00:39
  • It might be helpful to consider the case where the model is a straight line with a slope and intercept parameter. Imagine this model is correct but you have some measurement error. Then after a bunch of measurements you scatter plot will be long “cloud” centered on a best fit line that is close to the line of the correct model. If you had instead just taken two points, there would of course be a unique line passing through them, but just picking em two random points in the cloud, the slope and intercept of the line between them might be nowhere near the true values. – spaceisdarkgreen Feb 26 '19 at 01:05
  • (Google image search “linear regression” for a picture of what I described in words.) – spaceisdarkgreen Feb 26 '19 at 01:08
  • If, at a time, you start getting a few values, let me know. – Claude Leibovici Feb 26 '19 at 15:50

1 Answers1

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First small remark : Fulcher and not Flucher (to be more precise, this is Vogel-Fulcher-Tammann correlation).

As said in comments, since you have two parameters $A,B$, two points would be sufficient.

However, you precised that this is for viscosity and the problem is that the measurements of this specific property have never been very accurate. On the other hand, viscosity is very important in many areas such as design or rating of heat exchangers, pipelines , trays in distillation columns,and so on.

So, my answer would be : as many measurements as you can afford.

On the other hand, you must remember that, typically, $T_0$ is not known; it is one parameter to be adjusted.

My last point would be that this model extrapolates very poorly. So, try to really cover the range of temperature you will be concerned by.

Claude Leibovici
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