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Calculator for viscosity mixing based on the Lederer - Roegiers equation

Posted By: Gokhan

Calculator for viscosity mixing based on the Lederer - Roegiers equation - 06/23/19 10:34 AM

I just realized that the Widman viscosity-mixing calculator and similar calculators on the Internet are totally useless. They are very inaccurate.

After some brief reading, it looks like the Lederer - Roegiers equation works well. The only caveat is that there is an adjustable parameter alpha that accounts for the intermolecular cohesion energy, which ideally needs to be determined empirically.

Here is the article I read:

Viscosity-blending equations
Boris Zhmud, Ph.D., Assoc.Prof., MRSC
Sveacon Consulting, Stockholm, Sweden
Lube-Tech, No: 93, Page 1
http://www.lube-media.com/wp-content/uploads/2017/11/Lube-Tech093-ViscosityBlendingEquations.pdf

I created a spreadsheet that you can plug in your values to calculate the blend viscosity:

[Linked Image]

Here is the link for the calculator. Enjoy!

Gokhan's viscosity-mixing calculator based on the Lederer - Roegiers equation
Posted By: Gokhan

Re: Calculator for viscosity mixing based on the Lederer - Roegiers equation - 06/23/19 01:17 PM

Some clarification on alpha:

alpha = 1.0 for mixing similar oils.

Typical alpha values for mixing some different oil types are listed in the image as well as the spreadsheet.

Clarification on dynamic vs. kinematic viscosity:

You need to use the dynamic viscosity. However, if the density values are not available, you can use the kinematic viscosity with reduced accuracy.

dynamic viscosity (DV) = density * kinematic viscosity (KV)

Density varies as a function of the temperature. You can use this table to estimate it for different temperatures (the exact rate of change with temperature varies with the oil):

Code
T (°C)   density(T)/density(15.6 °C)

15.6	 1.000 
20	 0.997 
30	 0.990 
40	 0.983 
50	 0.976 
60	 0.969 
70	 0.962 
80	 0.955 
90	 0.948 
100	 0.941 
110	 0.934 
120	 0.926 
130	 0.919 
140	 0.912 
150	 0.905 
160	 0.898 
170	 0.891 
180	 0.883 
190	 0.876 
200	 0.869

[Linked Image]
Posted By: Tjbouwhu

Re: Calculator for viscosity mixing based on the Lederer - Roegiers equation - 06/23/19 01:55 PM

Thank you, I have a fuel diluting GDI....this would be helpful for blending in some 0W-30!
Posted By: JAG

Re: Calculator for viscosity mixing based on the Lederer - Roegiers equation - 06/23/19 03:46 PM

Cool, thanks. That might account for the strange behavior of a mix of a relatively viscous polymer ester and paraffinic base oils in which the mix viscosity is much less than the simple calculators predict. Do you now need to update your spreadsheet that shows the HTFS viscosity or is it close enough?
Posted By: Gokhan

Re: Calculator for viscosity mixing based on the Lederer - Roegiers equation - 06/23/19 11:04 PM

Originally Posted by JAG
Cool, thanks. That might account for the strange behavior of a mix of a relatively viscous polymer ester and paraffinic base oils in which the mix viscosity is much less than the simple calculators predict. Do you now need to update your spreadsheet that shows the HTFS viscosity or is it close enough?

You're welcome! The calculation for the HTFS viscosity for a single oil doesn't use the viscosity-mixing calculator. I was asked to calculate the HTHS viscosity of two oils mixed/blended together and I wrote this calculator so that I can use it in that calculation:

https://www.bobistheoilguy.com/foru...erature-full-shear-viscosity#Post5141467
Posted By: Gokhan

Re: Calculator for viscosity mixing based on the Lederer - Roegiers equation - 06/24/19 12:11 AM

In addition to the DV calculator, which can be used at any temperature, I have now included calculators for KV40 and KV100 as well.

Nevertheless, there seems to be very little error in using the DV calculator for KV mixing. So, if you don't bother, use the DV calculator (first calculator) for KV mixing as well. If you're worried about the accuracy, use the KV40 and KV100 calculators (second and third calculators) with the respective KV values.

Gokhan's DV and KV viscosity-mixing calculators based on the Lederer - Roegiers equation

[Linked Image]
Posted By: Gokhan

Re: Calculator for viscosity mixing based on the Lederer - Roegiers equation - 07/02/19 01:58 AM

Here are the equations for viscosity mixing/blending.

___________________________________________________

Arrhenius equation:

n = n1^x1 * n2^x2

___________________________________________________

Lederer - Roegiers equation:

n = n1^x1_effective * n2^x2_effective

with

x1_effective = x1 / (x1 + x2 * alpha)
x2_effective = x2 * alpha / (x1 + x2 * alpha)

___________________________________________________

These are the descriptions of the variables:

^ represents exponentiation: a^b = a raised to the power of b.

n = dynamic viscosity of the mix/blend
n1 = dynamic viscosity of oil 1
n2 = dynamic viscosity of oil 2

Note that the dynamic viscosities should be measured at the same temperature for both oils and the mix/blend.

x1 = fraction of oil 1 (between 0 and 1)
x2 = fraction of oil 2 (between 0 and 1)

Note that x1 + x2 = 1.

alpha = coefficient to account for the relative intermolecular cohesion energy of oil 2 with respect to oil 1

alpha = 1 if oil 1 and oil 2 have similar molecular structures; otherwise, alpha < 1 or alpha > 1 depending on how oil 2 differs from oil 1.
If alpha = 1 (oil 1 and oil 2 have similar molecular structures), Lederer - Roegiers equation reduces to the Arrhenius equation.

x1_effective = effective fraction of oil 1 correcting for the difference in the intermolecular cohesion energy of the two oils
x2_effective = effective fraction of oil 2 correcting for the difference in the intermolecular cohesion energy of the two oils

Note that x1_effective + x2_effective = 1.

Example:

If you have a 50/50 blend of an oil with a viscosity 16.0 cP with an oil with a viscosity of 8.0 cP, then the resulting viscosity of the mix/blend is:

16.0^0.5 * 8.0^0.5 = square root(16.0 * 8.0) = 11.3 cP

In other words for a 50/50 mix/blend of similar oils, the viscosity of the mix/blend is the geometric average of the two viscosities (11.3 cP), not the arithmetic average of the two viscosities (12.0 cP).

Finally note that while these equations are for the dynamic viscosities in cP units, they can also be used with good accuracy for the kinematic viscosities in cSt units. (dynamic viscosity = kinematic viscosity * density, with the density measured at the same temperature as the viscosity.)
Posted By: Garak

Re: Calculator for viscosity mixing based on the Lederer - Roegiers equation - 07/10/19 06:26 AM

Originally Posted by Gokhan
I just realized that the Widman viscosity-mixing calculator and similar calculators on the Internet are totally useless. They are very inaccurate.

Are you going to go back and correct all the posts you made that were premised on them being nigh on the level of physical constants? On a constructive note, an equation renderer might be of value here.
Posted By: edhackett

Re: Calculator for viscosity mixing based on the Lederer - Roegiers equation - 07/10/19 06:43 PM

Originally Posted by Garak
Originally Posted by Gokhan
I just realized that the Widman viscosity-mixing calculator and similar calculators on the Internet are totally useless. They are very inaccurate.

Are you going to go back and correct all the posts you made that were premised on them being nigh on the level of physical constants? On a constructive note, an equation renderer might be of value here.


As an analytical chemist(retired) Gokhan's equation raised a lot of red flags as to potential inaccuracies. He's taking a formula that is specifically for a binary mixture and applying it to a mixture of two multi component mixtures, which is definitely not a binary mixture. The equations obtain their accuracy by empirically deriving alpha for each binary mixture at a single mixing ratio. Gokhan is guessing alpha.

The accuracy of the Widman calculator and Gokhan's equation can be easily and economically tested. All we need to do is compare the results of the two methodologies to measured values. To that end, I contacted Blackstone and they will do viscosity cSt @100C for $10 each. I'm willing to buy 5-7 oils and pay for 12 analysis. That should give us a reasonable idea how things fall out of the equations.

My plan would be to do mixes of oils of at least two SAE grades apart and of differing base stock mixtures. The mixes would be be biased toward differing brands, wide viscosity differences, and ratios other than 1:1, but would include mixes of the same brand line and mixes of similar oils from different brands.

Gokan would supply data from the methodology proposed in this thread, Garak could supply the Widman calculations, and I will provide the measured values. Is everyone game?

Ed
Posted By: MolaKule

Re: Calculator for viscosity mixing based on the Lederer - Roegiers equation - 07/10/19 07:59 PM

Sounds like a plan, but I think we need to stipulate that the equations and constants discussed in

posts #5141416 and #5149034 be frozen until the Blackstone results have materialized.
Posted By: Gokhan

Re: Calculator for viscosity mixing based on the Lederer - Roegiers equation - 07/10/19 11:45 PM

Clarification:

alpha is an adjustable empirical constant that you can only determine by mixing/blending two oils and then actually measuring the viscosity of the mix/blend. You then adjust alpha until the theory agrees with the experiment. Once you know alpha for the two oils you are mixing/blending, you can use it to calculate the viscosity of the mix/blend for different fractions of oil 1 and oil 2.

If you don't know alpha empirically, you can assume that alpha = 1 (as the Widman mixing/blending calculator does but it also does some strange things I don't understand) and go from there. This assumes that oil 1 and oil 2 have similar base oils with viscosities in the same order of magnitude. For two oils with greatly differing base-oil types or viscosities, you can use the table to guess a better alpha but it will only be a rough guess.
Posted By: Garak

Re: Calculator for viscosity mixing based on the Lederer - Roegiers equation - 07/11/19 08:08 AM

Originally Posted by edhackett
Gokan would supply data from the methodology proposed in this thread, Garak could supply the Widman calculations, and I will provide the measured values. Is everyone game?

I'm on board!
Posted By: edhackett

Re: Calculator for viscosity mixing based on the Lederer - Roegiers equation - 07/26/19 04:05 PM

I have oils and a plan. The order for sample bottles apparently fell though the cracks at Blackstone. I called this morning and bottles are now on the way.

I'll be doing the mixing using Class A graduated cylinders. What? You don't have Class A measuring equipment in your bar kit?! grin

Ed
Posted By: edhackett

Re: Calculator for viscosity mixing based on the Lederer - Roegiers equation - 08/19/19 02:55 AM

Blackstone has had the samples for a week. No data yet. I'm leaving for a camping trip on Wednesday. If I don't get the data by Tuesday night, I won't get it posted until the following Tuesday.

I'm looking forward to getting the results of this experiment. It took close to three hours to do the mixing as getting a precision that I was comfortable with took long drain times from the cylinder. If you want an oil that will cling like crazy, Redline 50WT Race Oil is the cat's pajamas!

Ed
Posted By: Garak

Re: Calculator for viscosity mixing based on the Lederer - Roegiers equation - 08/22/19 02:49 AM

Set it to drip, go on a camping trip, then come back. wink
Posted By: edhackett

Re: Calculator for viscosity mixing based on the Lederer - Roegiers equation - 08/28/19 01:49 AM

Here are the data for the viscosity calculations. I tried to pick combinations that replicate common mixing practices seen on BITOG. The first is the classic 50:50 Mobil 1 0w-40 and 0W20. The rest simulated adding a quart of heavier oil to thicken things a bit, or a quart of thinner oil to thin things down a bit. There is also the practice of adding a quart of Redline as an “additive”. Other mixes are intended to test the equations with differing base stocks and viscosity ranges.

As you can see, the measured viscosity varied significantly from the published “typical” viscosity in some cases. The data came from U.S.A. region PDS sheets published directly on the oil manufacture’s web site. In the case of Castrol, the sheet for the GTX 20W-50 was dated 2012. The data for the Valvoline SAE 30 ND was from 2006, which was current when the oil used was purchased. The other oil’s data are current.

These are the calculations I’d like to see run at a minimum:

Widman measured cSt.
Widman PDS cSt.

Gokhan’s calculator measured dynamic viscosity, estimated alpha from tables in calculator(state number used).
Gohkan’s calculator measured cSt., alpha 1.0 for all.
Gokhan’s calculator measured cSt., estimated alpha from tables in calculator(state number used).
Gokhan’s calculator PDS cSt., alpha 1.0 for all.
Gokhan’s calculator PDS cSt., estimated alpha from tables in calculator(state number used).

Those combinations will allow us to see how much error is introduced by the difference in a measured viscosity vs, typical, the calculation methods, and the effect of adding an estimated alpha into the equation. Once all the data has been posted I’ll post the measured values for the mixes. At that point, someone who likes to play with Excel can wrap it all up in a neat, informative table.

The oils:
Oil (Measured [email protected]) [PDS [email protected], PDS density g/[email protected]]

Mobil 1 0W-20 EP(8.18) [8.6, 0.839]
Mobil 1 0W-40 FS (11.73) [12.9, 0.8456]
Castrol GTX 5W-30 ULTRACLEAN (9.56) [11.0, 0.862]
Castrol GTX 20W-50 (17.21) [18.09, 0.884]
Redline 50WT Race Oil (17.29) [19.2, 0.897]
Valvoline SAE 30 ND (10.86) [11.0, 0.889]
Rotella T5 10W-30 (11.34) [12, 0.869]

The mixes:
Oil 1(fraction) - Oil 2(fraction)

1. Mobil 1 0W-20(0.5) - Mobil 1 0W-40(0.5)
2. Mobil 1 0W-20(0.2) - Mobil 1 0W40 (0.8)
3. Mobil 1 0W-20(0.8) - Mobil 1 0W40(0.2)
4. Castrol GTX 5W-30((0.2) - Castrol GTX 20W-50(0.8)
5. Castrol GTX 5W-30((0.8) - Castrol GTX 20W-50(0.2)
6. Mobil 1 0W-20(0.5) - Redline 50WT Race(0.5)
7. Mobil 1 0W-20(0.2) - Redline 50WT Race(0.8)
8. Mobil 1 0W-20(0.8) - Redline 50WT Race(0.2)
9. Mobil 1 0W-40(0.2) - Redline 50WT Race(0.8)
10. Mobil 1 0W-40(0.8) - Redline 50WT Race(0.2)
11. Castrol GTX 5W-30(0.5) - Rotella T5 10W30(0.5)
12. Mobil 1 0W-20(0.2) - Castrol GTX 20W-50(0.8)
13. Redline 50WT Race(0.5) - Valvoline SAE30 ND(0.5)

Ed


Posted By: edhackett

Re: Calculator for viscosity mixing based on the Lederer - Roegiers equation - 08/29/19 08:54 PM

I received a PM from Gokhan and he indicated that he would like someone else to do the calculations using his calculator at this time. I'll excuse myself from doing the calculations as I've seen the results and don't want to chance introducing any unintentional bias into choosing a value for alpha.

Garak, MolaKule, Shannow, or anyone else who would like to plug in the numbers, please feel free to do so.

Ed
Posted By: Gokhan

Re: Calculator for viscosity mixing based on the Lederer - Roegiers equation - 09/01/19 03:48 AM

Great work, Ed!

I was able to get to it briefly.

I tested both spreadsheet calculators in my original post -- the first one using only the kinematic viscosities (KV) and the second one also using the densities. The error in using the simpler calculator was between +0.00% and +0.06% -- always very slightly overestimating the KV of the mix -- and it's obviously entirely negligible given the accuracy of the KV values. Therefore, from now on, I will only use the Lederer - Roegiers formula, which is

n = n1^x1_effective * n2^x2_effective

with

x1_effective = x1 / (x1 + x2 * alpha)
x2_effective = x2 * alpha / (x1 + x2 * alpha)

so that I plug in the KV values into the n's instead of bothering to convert them into the dynamic viscosity (DV).

Using alpha = 1, here are the results for the KV100 of the mix in cSt. I calculated them both using the Blackstone values (first column) and PDS values (second column).

Code
Mix #    Lab      PDS

 1	 9.80    10.53
 2	10.91    11.90
 3	 8.79     9.33
 4	15.30    16.38
 5	10.75    12.15
 6	11.89    12.85
 7	14.89    16.35
 8	 9.50    10.10
 9	16.00    17.73
10	12.68    13.97
11	10.41    11.49
12	14.83    15.59
13	13.70    14.53


As far as the alpha is concerned, it can only be determined empirically. When the experimental results are presented, we can calculate the alpha to see if it makes sense.

A concern that remains is the accuracy of the Blackstone KV measurements. Typical commercial viscometers seem to have an accuracy around ±3%. Therefore, we shouldn't expect an agreement with the theory beyond about ±4%, which is about ±0.4 cSt, when two different oils with measured viscosities are mixed. In fact Wear Check told me that the error in a single KV measurement is "less than ±0.5 cSt," which is consistent with this observation: accuracy around ±0.3 cSt for a single oil and ±0.3 cSt × √2 ~ ±0.4 cSt for two different oils mixed together.
Posted By: edhackett

Re: Calculator for viscosity mixing based on the Lederer - Roegiers equation - 09/02/19 04:18 PM

Originally Posted by Gokhan

Using alpha = 1, here are the results for the KV100 of the mix in cSt. I calculated them both using the Blackstone values (first column) and PDS values (second column).

Code
Mix #    Lab      PDS

 1	 9.80    10.53
 2	10.91    11.90
 3	 8.79    9.33
 4	15.30    16.38
 5	10.75    12.15
 6	11.89    12.85
 7	14.89    16.35
 8	 9.50    10.10
 9	16.00    17.73
10	12.68    13.97
11	10.41    11.49
12	14.83    15.59
13	13.70    14.53


Here's the Widman data including the lab values for the mixes:

Code
Mix #    Lab   PDS      Lab Actual

 1	9.77    10.49   10.14
 2	10.89   11.86   11.11
 3	8.77    9.30    8.95
 4	15.21   16.31   15.08
 5	10.69   12.10   10.61
 6	11.73   12.64   11.38
 7	14.75   16.17   14.66
 8	9.42    10.00   9.37
 9	15.96   17.68   14.86
10	12.65   13.93   12.20
11	10.40   11.49   10.12
12	14.69   15.45   14.44
13	13.63   14.42   13.12


OK, you statisticians, have fun!

Ed
Posted By: edhackett

Re: Calculator for viscosity mixing based on the Lederer - Roegiers equation - 09/02/19 09:06 PM

Darn it, friggin dyslexia! Even after checking the table twice, I managed a Typo. The measured viscosity of mix #7 should read 14.75 not 17.75. Could one of the moderators please correct the table?

Ed
Posted By: Gokhan

Re: Calculator for viscosity mixing based on the Lederer - Roegiers equation - 09/12/19 06:46 AM

I have finally got to calculate the empirical alpha values using the Blackstone KV100 for Oil 1, Oil 2, and Mix.

My conclusions in summary:

(1) Lederer - Roegiers equation when used with the proper alpha value works!

(2) Arrhenius equation and the Widman mixing calculator do not work!

(3) alpha values ranged from 0.42 to 1.46 in Ed's samples, greatly deviating from the Arrhenius equation and Widman mixing calculator, which assume alpha = 1.

(4) You need to have at least one measured sample to calculate the alpha empirically, from which you can calculate the viscosity for different mix ratios for the same two oils.

If you want to calculate the alpha for a given sample directly without trial and error, here is the formula:

beta = [ln(n / n2)] / [ln (n1 / n2)]

alpha = [x1 * (1 - beta)] / [ (1 - x1) * beta]

Note the parentheses. n1, n2, and n are the viscosities of Oil 1, Oil 2, and Mix, respectively. x1 is the fraction of Oil 1. Once you know the alpha for a given sample, you can calculate the viscosity for different fractions of the same two oils using:

n = n1^x1_effective * n2^x2_effective

x1_effective = x1 / (x1 + x2 * alpha)
x2_effective = 1 - x1_eff

Here are the average alpha values for Ed's samples, with the info in the parenthesis showing the approximate oil composition:

Code
Oil 1                             Oil 2                                    alpha

M1 EP 0W-20 (~ PAO)               M1 FS 0W-40 (~ GTL)                      1.40
Castrol GTX UC 5W-30 (~ Gr II)    Castrol GTX 20W-50 (~ Gr II)             0.86
M1 EP 0W-20 (~ PAO)               Redline 50WT Race (~ PAO)                0.85
M1 FS 0W-40 (~ GTL)               Redline 50WT (~ PAO)                     0.42
Castrol GTX UC 5W-30 (~ Gr II)    Rotella T5 10W-30 (~ Group II + GTL ?)   0.50
M1 EP 0W-20 (~PAO)                Castrol GTX 20W-50 (~ Group II)          0.81
Redline 50WT Race (~ PAO)         Valvoline SAE 30 ND (~ Group II)         1.46

Note that similar base oils result in alpha ~ 1 as expected but when you mix a PAO base oil even with a GTL base oil, alpha differs from 1 significantly.

Use these alpha values along with the spreadsheet provided in the original post for a lot more accurate estimates of the viscosity of the mix. Note that the order of the oils matters when you use a given alpha value -- do not exchange Oil 1 and Oil 2.

The actual empirical alpha values using the Blackstone KV100 for Oil 1, Oil 2, and Mix:

Code
Sample #  alpha_empirical   Name	        Average alpha_empirical

1	1.47	M1 EP 0W-20 - M1 FS 0W-40	1.40
2	1.41	M1 EP 0W-20 - M1 FS 0W-40	
3	1.33	M1 EP 0W-20 - M1 FS 0W-40	
4	0.86	GTX 5W-30 UC - GTX 20W-50	0.86
5	0.86	GTX 5W-30 UC - GTX 20W-50	
6	0.79	M1 EP 0W-20 - Redline 50WT	0.85
7	0.88	M1 EP 0W-20 - Redline 50WT	
8	0.89	M1 EP 0W-20 - Redline 50WT	
9	0.39	M1 FS 0W-40 - Redline 50WT	0.42
10	0.45	M1 FS 0W-40 - Redline 50WT	
11	0.50	GTX 5W-30 UC - Rotel. T5 10W-30	0.50
12	0.81	M1 EP 0W-20 - GTX 20W-50	0.81
13	1.46	Redline 50WT - Valvoline 30 ND	1.46

Last but not least, thank you very much, edhackett, for this great work! It's much appreciated!
Posted By: Lowflyer

Re: Calculator for viscosity mixing based on the Lederer - Roegiers equation - 09/13/19 08:07 AM

Originally Posted by Gokhan
I just realized that the Widman viscosity-mixing calculator and similar calculators on the Internet are totally useless. They are very inaccurate.
I believe all of this, no problem. But may I ask how you realized, they are very inaccurate? The Counter-test, was?
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