Let me go back to my analogy and enhance a bit of it.
Same scenario; marbles into water in the bathtub. But we're going to add a component of bleach to represent degradable additives. Marbles represent contamination; they go in at reasonably consistent rate (1 marble/min) and the bleach represents additives that will degrade over time. (I chose bleach for two reasons for this example: 1) it degrades in strength with time, so it will alter it's presence and 2) it's fully soluble in water, unlike the marbles).
OK - after 24 minutes, there would be 24 marbles in the tub. If you are keeping the water level consistent (at 12 quarts), then you'd have 2 marbles / quart at that 24 minute mark. (aka 2m/qrt for shorthand).
Because you're using bleach water in this example, the bleach has a lifespan that degrades with time; it looses it's ability to "clean" (disinfect) as the chlorine oxidizes. The marbles will increase in count, because they do not degrade. Bleach (a product that typically uses a form of chlorine) will loose it's function with time as it degrades, so it must be replenished to have the same effect of killing germs.
There are three situations that are possible here regarding the tub volume:
1) volume is increased
2) volume stays consistent
3) volume decreases
In #1, you would be adding bleach water to the tub, and because there was no water loss, the volume would go up. That increase in total volume does have an effect on the contamination ratio. Perhaps you add 1 quart every 12 minutes, so after 24 minutes you'd have 24 marbles and 14 quarts. Your contamination ratio is 24/14 = 1.7m/qrt. (it also increases the chlorine count; we've not defined a rate though for this example).
In #2, you would be not adding or subtracting any water; simple formula as it's 24/12 = 2m/qrt. (bleach will degrade over time; we've not defined it's rate, but the effect is lesser as time goes on).
In #3, you are losing water and not replacing it. Perhaps you've lost 2 qrts, and so now your ratio is now 24/10 = 2.4m/qrt. (bleach still degrades, and has the double effect of volume loss and degradation over time).
The bleach-water added to the system will affect the ratio of the main sump's bleach to water, because the bleach will change it's presence and effect with time. That is not true of the contamination; it's there for the full life of the sump until it's either filtered out or drained out. It's completely appropriate for AHarmon to adjust his formula for certain additives that are present not only in the sump, but also present in the amount of lube being added. But it is not needed or even correct to try to adjust the aspects that are not directly affected by the addition of the new lube. Because we hold our engine sumps at a reasonably consistent level, and because the input of more lube does not bring in more contamination, the correct mathematical methodology is to adjust the numerator with the measured presence of the contamination element (Fe, Al, Pb, etc) and hold the denominator steady as the constant. As long as we hold the volume steady in the sump, the effort to calculate a concentration of contaminants and metals is pretty simple; it's variable over constant (changing numerator / steady denominator).
In the tub water and marble example, it's really hard to know your bleach effectiveness of the chlorine because you have to know the "at the moment" concentration, plus a rate of time-effect degradation, plus the amount added in make-up water. So, when we turn to oils, the additives in lubes (those that can be detected by elemental analysis, and those that are present but cannot been seen in a UOA) all have different reactions in the crankcase. Some will degrade in effect due to contamination, some by consumption, some by evaporation, etc. Some will be affected in a greater sense, others less so. I would have to divert the conversation to a chemist or lube engineer (like Mola) to let us know how each additive changes with time/distance exposure; that's way past my bailiwick. But the point to know is that they see their concentration ratio always altered not only by the simple operation of the engine, but also the addition of lube. It's much harder to know how to calculate a correction factor for things like Ca and Mg, because although they are present and seen in a UOA, we have a difficult time knowing how much of the Ca and Mg are free to work, versus busy already engaged with soot and other insolubles, and then we throw in the topic of "fresh" oil? You can see the difficulty here in how the math must be worked. You not only have to know what you have a minute "x", but also how much is no longer free to do work even though it's present, versus how much left the system out the tailpipe, and then how much more you're putting in. Gets VERY difficult to calculate. A UOA can tell us how much Ca and Mg are in the sample, but they cannot tell us how much of those amounts are actually free to work, or busy already engaged with contamination such as soot and insolubles. When we measure TBN/TAN, that would generally tell us how much has be taken from the available mass, but as we've all seen before, the degradation of TBN, and/or the inversion of TBN/TAN in ratio, does not in most cases have any correlation to wear rates. So using that TBN/TAN count really does not mean you'd see a shift in wear; it's only a predictor that the sump may change in the future and so you should pay closer attention to subsequent UOAs and maybe even UOA more often. But in and of itself, we've go no proof that the presence of something in a UOA has any ability to help us define an effect, because no correlation exists. Without correlation, there can be no causation. Additionally, things like Ca and Mg are multi-purpose; they are detergents and yet also help with lubricity and other desirable characteristics. Really gets complicated in how to define an effect that you, at best, can only hope for secondary or tertiary links in correlation.
But it all sums up this way ....
- If you hold your sump at a reasonably consistent level, then you don't adjust the ratio with the denominator, if the measurable you're tracking does not present more of the variable in the make-up volume.
Adding lube does not induce more contaminants (Fe, Al, Cr, Pb, Cu, Si etc). Do not adjust your formula. Just keep the sump reasonably level and take your UOA samples when the sump is near the desired level.
- If the make-up volume does bring in more of the measured variable, then AHarmon's general math is as reasonable as any other.
Adding lube does bring in more additives (Ca, Mg, Ti, Boron, etc). Adjust your formula. How you adjust it depends upon known (and presumed, if unknown) degradation rates and mathematical model chosen.
Make sense?