Reliability of Data

In a previous entry I showed that the basic concepts of quality control, which depends upon the laws of probability (statistics), are surprisingly simple. All that we are trying to do is measure lengths of lines. The equations used to calculate the mean and standard deviation are those that describe only two lines so that no matter how many samples are tested, the calculations of those parameters result in just those two lines which are independent of each other. While “n” data points occupy “n” dimensions, the mean and standard deviation occupy only two. We can use the standard deviation as the ruler to measure the lengths of interest.

What makes things difficult is the fuzziness of those lines. In quality control the first thing we want to determine is the length of the distance from the measured length (sample mean) to some desired length. To do that we use a ruler in which the standard deviation is set to be one. For convenience, and because the standard deviation is defined as the second moment around the mean, the targeted mean is subtracted from the data points so that the resulting length of the data vector is reduced to the difference between the sample mean and the target. That length is then divided by the standard deviation. The resulting length is then measured not in inches or millimeters but rather in units of the standard deviation ruler. As an example, assume that 100 was the target value, the measured mean was 85 and the standard deviation was 10. We are not interested in what the actual measured mean is, but rather how close it is to the target, based upon the standard deviation ruler:

1. (100-85)/10 = a distance of 1.5 SD units. In some cases the measurement is not from the desired target, but to upper and lower limits.

However, the mean value is fuzzy and the standard deviation may or may not be fuzzy. The data generated in calculating the mean make up a random variable (X= (x1, x2, —, xn)) in vector space. How fuzzy it is depends upon the length of the SD, and the type of distribution. While there are many distributions, if the SD is not fuzzy, what is called the normal distribution is often used. Because of the uncertainty in the mean, the distribution function tells us the chances of the mean actually being somewhere else.  In example 1 with only the mean being fuzzy, and using the normal distribution, we can say that there is a 6.68% chance that the true mean of the data is the desired mean.

Unfortunately, the SD often is fuzzy too and is thus also a random variable. The square of the SD is called the variance, and has its own distribution function called the chi squared distribution. While the normal distribution is independent of the number of data points defining the random variable, the form of the chi squared distribution depends upon the degrees of freedom. The chi square distribution with one degree of freedom is the square of the normal distribution. That distribution may be used to determine whether two measured standard deviations are really the same.

How the fuzziness or uncertainty is handled will be covered later. Although the mathematics gets more complex, especially when multivariate sets of data must be considered, the goal is still to simply measure lengths with a specific ruler.



Superiority of the AR Grading System


AR Grading. The Asphalt Residue (AR) grading system used in the Western part of the United States for decades grew out of the fact that the asphalts in this area differed greatly. While various grades were in use, the workhorse grade was AR 4000 which meant that the asphalt in the pavement, irrespective of crude source, would have the same consistency. AR 4000 meant that the viscosity at 60° C of the asphalt after the RTFO test would be 3000 (2500 in Washington) to 5000 poises. A viscosity of 4000 poise was selected as it was found that at 4000 poises tenderness in oversanded mixes was easier to handle.  60° C is used as in most cases that is about the highest temperature the pavement reaches although in the deserts it can reach considerably higher temperatures. On the other hand, the viscosity at 60° C from the RTFO of equivalent asphalts graded by the AC grading system (2000 ± 400 poises based on original viscosity at 60° F) or by penetration grading system (85/100 based on penetration at 25° C) can vary greatly. For the 85/100 penetration grade, the range of the 60° C viscosity after the RTFO of those asphalts evaluated during the development of the AR grading system varied from about 1600 to over 7000 poises. For an AC 2000 grade asphalt, the probable viscosity after the RTFO aging would range over about 4000-8000 poises, depending on the crude source. The equivalent PG grade is PG 64-XX.

PG Grading. There is an astounding number of PG grades, 7, and up to 6 subgrades within each grade, based upon low temperature properties. If there was consistency within the grades it might make sense, but we have regressed even back beyond the AC grading system. These grades were set up primarily to control tenderness and rutting even while leaving the gradation specification so open that gradations that would allow grievous rutting are included. The equivalent PG grade is based upon the Dynamic Shear test of G*/sinδ of 1.00 kPa at 64° C with no maximum. For a sinδ of 1.00 (close to that of unmodified asphalt) the viscosity is G*· sinδ or 1000 poises. The G*/sinδ value from the RTFO test would be 2.20 kPa min or 2200 poises with sinδ = 1.00 and again there is no maximum. Sinδ for modified asphalts is less than one thus that drops the specification minimum viscosity below that of non-modified asphalt.` In other words, for the asphalt as placed in the pavement, the AR 4000 specification is 3000-5000 poises at 60° C. For the PG 64-XX , the-in place viscosity at 64° C can vary from somewhat less than 2200 poises to as high as one wishes.



Philosophical Inconsistency of the PG Grading System. I am only addressing the grading system, not the value of the low temperature specification. I am not suggesting that there is anything wrong with the use of the DSR, as it is a handy tool. I am suggesting that the grading should have been based upon the consistency of the RTFO residue whether viscosity tubes are used or the DSR. The value of the DSR data is that we can get information about the effect of polymer modification from the phase angle, sigma (δ).

We have shown above that the range of the allowed viscosity from the RTFO test of any particular PG grade is greater than that of any previous grading system even though there is are 7 specific grades in order to control rutting. The implication is that controlling rutting requires fine tuning. Yet, at the same time there is a movement to use warm mixes, one of the benefits of which is that the asphalt will have a considerably lower viscosity than the intention of the grade.

Controlling Rutting. The prime control of tenderness and rutting should be with aggregate gradation.  As long as the gradation specification allows badly oversanded mixes, rutting will be a problem.

Robert L. Dunning, chemistdunning@gmail.com, www.petroleumsciences.com