Plotting Viscosity Data and making the ASTM D341 Graph (Logarithms)

[andyhessler]

Hello all,
There's been a few discussions on the Walther equation for plotting the viscosity/temperature relationship of lubricants. I'm trying to basically make my own plot...but I am having trouble understanding the vertical scale to use to get the linear relationship. I think I've puzzled out how to make a log scale. But, the log log needed for this equation isn't working out.
The ASTM d341 chart uses this scale. And while I could download the graph...that's not going to tell me how the vertical scale was created. I sure appreciate the help.
(the "why" is a much longer story)
Thanks,
Andy

 

[chicopee]

show us the equation for which you want a log-log graph paper.

 

[romke]

actually if you look more carefully you will notice that the horizontal scale is a log scale (in K, but usually values are shown in degrees C) and the vertical scale is a loglog scale. only then you can get a straight line based on the Walther equation.

with standard spreadsheet software you will not be able to make a chart like that, because you will not be able to get the vertical scaling needed.

 

[andyhessler]

Walther formula[edit]
The Walther formula[1] is typically written in the form

log_{10}[\log_{10}(\nu+\lambda)] = A - B\,\log_{10}(T)

where λ is a shift constant, and A, B are empirical parameters.

http://en.wikipedia.org/wiki/Temperature_dependence_of_liquid_viscosity

 

[25362]

Walther's formula:

(ν+a) = bd^1/Tc​

where:
a, b, c, d are constants
ν is the kinematic viscosity (m²/s)
T is the absolute temperature (K)

It has been found that if ν is in cS, then a [≅] 0.6. By taking d=10 and log twice one gets:

log₁₀log₁₀(ν_cS+0.6) = log₁₀(log₁₀b+1/T^c)​

Although an incorrect derivation of the above equation, the ASTM viscosity-temperature chart ewhere the ordinate is log₁₀log₁₀(ν_cS +0.6) and the abscissa is log₁₀T, is quite successful when used on oils (mineral and synthetic) under "normal" conditions.

 

[electricpete]

I'm using the post above to refresh my memory.
You could build a table:
T, v, X, Y, X1, Y1.
T is the independent variable Ttemperature (in the correct units)
v = b*10^*(1/T) - 0.6
Y = log(log(v + 0.6)
X is log(logb + 1/T) (plug in whatever b is… I'm working from what is posted above).
Y1 = (log(v + 0.6)
X1 is (logb + 1/T) (plug in whatever b is… I'm working from what is posted above).

Then if you a simple excel linear plot Y vs X or log log excel plot based on Y1 vs X1 I think you would have a straight line on your graph (slope 1 on the linear plot). The only problem is the axes are not v and T. You could take out all auto-generated gridlines and tickmarks and build your own vertical gridlines using extra series with constant values of T and your own horizontal gridlines with constant values of v.

Labeling those gridlines would be the challenge. Two approaches:
1 - You could add labels manually (for example in powerpoint) and get a nice pretty chart.
2 - If you wanted excel to do it you might keep it a small number of gridlines to keep it manageable and label the gridlines using the normal series label feature. Then to read the scale you wouldn't look at each gridline and then at the legend to figure out the values associated with the gridlines. Easier for you to build but not as pretty when you're done.


=====================================
(2B)+(2B)' ?

 

[electricpete]

(plug in whatever b is… I'm working from what is posted above).

hmmm, maybe not. I'm remembering now given two sets of (T,v) you can solve the constants and plot a curve in the T,v plane (which is not a straight line) and read off any value you want. With computers that seems like the easy way. You don't need the ASTM graph paper anymore unless you don't have access to computer.

=====================================
(2B)+(2B)' ?

 

[electricpete]

Here is a spreadsheet I built that does all the viscosity / temperature calcs I need based on Walther equation.

=====================================
(2B)+(2B)' ?

 

[andyhessler]

Thanks for sharing your spreadsheet. I am beginning to think the brute force approach is going to be needed...where I plot all of this out in AutoCAD. Wish me luck!