Dimensionless numbers, mathematical modelling & VI

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In physics, as it's often impossible to model something full scale, mathematical models are produced, which can predict
a) how a system behaves; and
b) how a model will behave when smaller than the machine to be designed.

A key tool in this modelling is "dimensionless numbers", which are ratios of parameters that end up not having a length, velocity, time, mass component.

Super simple example is a "scale" model. Take a 1:24 model, and it's dimensions are Length/Length (L/L cancells, and thus has no "dimension")...Multiplied by a length, it then gives the dimension to the length of the scale model.

Pretty simple, but when modelling complex systems, the dimensionless numbers start to take into effect other things, e.g. velocity has a length and time component, mass can come into it (density x length x length x length)...etc.

A commonly used dimensionless number in hydraulic systems is Reynolds Number, equal to
density X Velocity X Length/dynamic viscosity

It works for ducts, pumps, pipes, skin frictions, determining laminar and turbulent flow regimes...

As an example, if you wanted to do a scale model of say a pipe, running water at a constant temperature, if you halved the pipe size, doubled the velocity, you would get the same modeled behaviour when you scaled it up...you could also predict behaviour from empirical models on something that you haven't built.

Bearings use another number, Sommerfeld's Number...
So= (r/c)^2 X uN/P
r = shaft radius
c = diametrical clearance (r/c becomes dimensionless)
u = absolute viscosity (Units PaS, or (Force Seconds/Length^2)
N = rotational speed (Units Radians per second, radians being dimensionless, the units are 1/s)
P = projected bearing pressure (Force/length^2)

So the uN/P becomes also dimensionless...the number is dimensionless, and is used to model bearings.

What it tells you is that to change anything on a correctly designed bearing, you have to look at what has to change also to keep So the same.

Double the bearing load, and you can do any of the following to keep So the same :
* double bearing area (brings P back to the same);
* double the rotational speed;
* increase the journal size (note, this has to iteratively play back into the bearing width to balance P and r);
* reduce bearing radial clearance by 29.3%;
* double the viscosity; or
* similar to the journal size, manipulate all of the above to keep within safe parameters.

Study the number, and get a feel for what the variables do in bearing behaviour.

Which gets me back to the title, the dimensionless number that is Viscosity Index...it's a dimensionless number useful for comparisons for sure...for comparing the viscosity indices of oil.

It is NOT, however, one of those dimensionless numbers that can be used to predict anything whatsoever about lubricant behaviour, and any such analysis would be back to the basics of actual viscosity at the operating temperature being examined...

As such, IMO, VI is not one of the most important determinants in a lubricant.
 
But since we know the viscosity at 40 and 100c, can't VI be used to estimate and compare viscosities above 100c? So given a similar 100c viscosity, we could say that the oil with highest VI is better for higher loads at higher temperatures than 100c?
 
That's not using "VI" as the defining oil factor as some posit on the site.

Yours is an If/Then argument...

If KV100 (or HTHS) is good for that condition, then
High VI is less wrong at other temperatures.

VI is meaningless without a thumbtack to nail it to an operating point.
 
This is the kind of informed discussion I like to see on BITOG. Once you have calculated a Sommerfeld number, what do you compare it to? Are there accepted values that must be achieved to have good bearing life?
 
Very few fully understand the affect an oil's viscosity index has on operational viscosity.
HTHSV is the most important viscosity measure at normal operating temp's and higher. If two oils have the same HTHSV @150C spec's, the oil with the higher VI will be progressively lighter at temp's lower than 150C and progressively heavier at higher temp's.
The following link explains why VI should not be ignored when selecting a lubricant:

http://www.machinerylubrication.com/Read/28956/lubricant-viscosity-index
 
Originally Posted By: A_Harman
This is the kind of informed discussion I like to see on BITOG. Once you have calculated a Sommerfeld number, what do you compare it to? Are there accepted values that must be achieved to have good bearing life?


If you go into full blown analysis, you can place your bearing parameters anywhere that you like.

Take the last half of the Sommerfeld number, and it's it's own number, and can within a given design (drop off the r/c component), can show frictional characteristics (and wear relationships)...eg.

lub_3.gif


You can see that a reduction in viscosity will reduce friction (by reducing operating film thickness), even to the point of boundary, when additives are doing the work.

http://www.roymech.co.uk/Useful_Tables/Tribology/Liquid_Lubrication.htm
 
Shannow, nice graph.

The trick is to know what operational viscosity represents that hydrodynamic/boundary. And if you found the combination of oil pressure/oil temp's that it approximates you'd never want to spend any time there since you have no safety margin.
Still the info would be great to know so one can choose just how much viscosity safety margin you're prepared to live with.
 
And in the 0w16 world of the future, what will be the impact of another 12% reduction in the HTHS (2.6 to 2.3) be on hydrodynamic film thicknesses and the amount of time spent in boundary lubrication? From the Sommerfeld equation, all other factors being equal, that would mean a 12% reduction in the So number. To get the 12% back, the factors could be varied: 12% increase in operating speed, or 12% decrease in bearing unit load. But neither of these seems appropriate for better fuel economy or the trend toward highly loaded, downsized engines.

One of my rules of engineering is to take advantage of higher order effects whenever possible. So the (r/c)^2 factor becomes important. The 12% reduction could be offset by increasing bearing diameter by 6%, or by reducing bearing clearance by 6%. An increase in bearing diameter is directionally wrong for friction, so the least painful way to return Sommerfeld to happiness would be a reduction in bearing clearance.

Am I making sense here?
 
Originally Posted By: Shannow
As such, IMO, VI is not one of the most important determinants in a lubricant.

But the value of a high VI still cannot be ignored. A hypothetical, sufficiently high VI reverts u to a constant across the possible ranges of temperatures. There's obviously value to that from an engineering standpoint.

From a physics or mathematics standpoint, it's even better. Bazinga!
 
Shannow,

I think that you are working from the perspective that VI would be used alone in an oil discussion. I don't recall the discussion of VI without some other measure of viscosity (SAE grade or Viscosity at 40 or 100C). So, it seems that everyone here has pre-agreed with you since VI is not used without a second reference. For most applications, even Caterham would turn down an oil with a VI of 270 in an 5W20 application if the oil was an SAE 60.
 
Originally Posted By: dave1251
Agreed but at what point does the higher VI equate to performance that I can measure with my crude measurements?

Where you're located, in the absence of an oil pressure gauge, you might be rather hard pressed to notice a difference in service between an oil with an adequate VI versus a hypothetical one where the viscosity remains constant across all observed temperatures. Obviously, we're not losing engines left and right because of thermal stresses on lubrication at both ends of the temperature scale.

The lubricant with that hypothetically high VI would be fantastic for cold starts in Arctic conditions, extreme short tripping, racing, and extremely high heat applications. I gather that with switches to even lighter lubricants, it's going to be more important to avoid excessive thinning at elevated temperatures.
 
Originally Posted By: A_Harman
And in the 0w16 world of the future, what will be the impact of another 12% reduction in the HTHS (2.6 to 2.3) be on hydrodynamic film thicknesses and the amount of time spent in boundary lubrication? From the Sommerfeld equation, all other factors being equal, that would mean a 12% reduction in the So number. To get the 12% back, the factors could be varied: 12% increase in operating speed, or 12% decrease in bearing unit load. But neither of these seems appropriate for better fuel economy or the trend toward highly loaded, downsized engines.

One of my rules of engineering is to take advantage of higher order effects whenever possible. So the (r/c)^2 factor becomes important. The 12% reduction could be offset by increasing bearing diameter by 6%, or by reducing bearing clearance by 6%. An increase in bearing diameter is directionally wrong for friction, so the least painful way to return Sommerfeld to happiness would be a reduction in bearing clearance.

Am I making sense here?


Making perfect sense, and some of the Honda stuff that Buster has posted indicates that is what they are doing.

Stiffer cranks also implies larger diameter, shorter length journals, which as an aside have a greater side leakage to internal recirculation ratio.
 
Originally Posted By: CATERHAM
Very few fully understand the affect an oil's viscosity index has on operational viscosity.
HTHSV is the most important viscosity measure at normal operating temp's and higher. If two oils have the same HTHSV @150C spec's, the oil with the higher VI will be progressively lighter at temp's lower than 150C and progressively heavier at higher temp's.
The following link explains why VI should not be ignored when selecting a lubricant:

http://www.machinerylubrication.com/Read/28956/lubricant-viscosity-index


Thickening effectiveness of different types of polymeric viscosity index improvers varies with temperature such that as temperature rises above certain temperatures they can no longer significantly "open up", thereby causing them to not add as much viscosity per degree, as they do at lower temperatures. Obviously, VI is based only on two temps.: 40C and 100C. I think what you said is at least approximately true of Newtonian oils but not for most non-Newtonian oils. I think the farther above 100C the local oil temperature is, the less true your statement is for non-Newtonian oils and it is closer to true between 40C and 100C, since that is the range whose endpoints are the temps. that VI is based on. At the high end of the temperature range, the VI of the base oil mix becomes more dominant in determining the bulk VI than the combination of the base oil mix and the viscosity index improvers, relative to the contribution of the base oil mix to the bulk VI in the 40C to 100C temperature range. Most motor oils sold these days are non-Newtonian, mostly due to the presence of polymeric viscosity index improvers, making the understanding of their behavior particularly important.
 
Yes an oils viscosity index is based on the two 40C and 100C temp's. Extrapolating lower than 40C can be done to a certain extent; at least down to 0C if not -10C to -15C.

At higher temp's we fortunately have the HTHSV @150C spec' for accurate viscosity comparisons. We also know that viscosity change with temperature has lost the parabolic effect that occurs at low temp's and from 100C to 150C and higher, viscosity change with temperature is virtually linear.

We also know that the higher the VI of an oil with the same HTHSV of a lower VI oil, will be lighter at normal operating temp's (90C-100C) and of course progressively more so at lower temp's.
This effect is most noticeable with large VI differences as is the case between ultra high 0W-20 oils and typically low VI 5W-20 oils.
 
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