I've posted this in the past, but this is the basic cycle:
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The TBN/TAN relationship is more about the oil's ability to neutralize acids. Depending on the design and propensity toward contaminants (blow-by for example) your posit is quite correct, that it may meet its saturation point with contaminants before the TBN is depleted at which point those contaminants will fall out of suspension and plate on parts, creating varnish. You add moisture to that mix and you'll get sludge.