Question on recursive dilution (formula needed)

Because I have a certain amount of mechanical ability and because I’m cheap I am doing 50,000 mile maintenance on my truck. I’ve come to the point where I’m going to “change” the transmission fluid–which is not actually DUE for another 100,000 miles according to Ford. (Sure. 150,000 mile service intervals. That tranny will last forever, right?)

Looking at what’s required to actually flush all the fluid, I’ve decided that (since I’m doing it really early after all) I can just drop the pan, change that, and then drive for a few hundred miles and repeat. (Changing the filter, too.)

On changing the fluid in the pan, I will get about 1/2 the total fluid according to my Specifications chart. That means that I’ll have 50% new fluid until I fire up the engine and the new fluid starts circulating with the old 50%. (I’ll still have 50% new at this point, but I can’t separate them any longer.)

After the fluid has thoroughly mixed, I’ll drain the fluid in the pan and refill with new. What percentage of fluid have I replaced at this point? Can it even be calculated? I’m sure it can, but I don’t know how. It seems that 75% has been replaced, but I feel that isn’t right. I also feel that there’s a formula and that someone here can tell me what it is.

After thoroughly mixing, what percent remains if I drain and fill yet again? If I can get a formula, I can decide how many times to do this (and at what cost) to make me feel things are good until the future 150,000 mile dealer flush and refill. As part of the initial drain I’m adding a drain plug to the pan to make future draining easier.

TLDR; If I replace 50% of a solution with a different fluid, then replace 50% of THAT solution, how much of the original solution remains? What if 50% is replaced again? Again?

Assuming a thoroughly random distribution of new fluid in the old, you’ll get a mix ratio of 0.5n, where n is the number of transmission fluid changes in succession, and also presuming no meaningful degree of oil aging.

ie: At mix 1, you’ll get 0.5 old oil, 0.5 new oil (read as 50/50).
At mix 2, it’s 0.25 old oil, 0.75 new oil (3/1 new)
At mix 3, it’s 0.125, etc.
At 10 mixes, you should statistically have less than 1% old lubricant remaining.

In practice, if the old oil is denser and the truck is allowed to sit, the older oil is more likely to drip down into the pan by some amount, so your ratios ought to be slightly better than 50% each time.

As usual when I ask a question like this, the formula is very simple. Thank you.

I take it that the 0.5 is the percent changed so if I only changed 25% of the fluid I would substitute 0.25 for the 0.5?

Since you’re interested in the amount of old oil remaining, the ratio should be oil_remaining:total_oil.

So if you’re only changing 25%, you’d swap in .75 in your equation. (It’s 1 - changed-fraction).

D’oh! Thank you again.

Not sure if there’s a typo here or if you’re just being very conservative, but at 10 mixes the fraction of old lubricant would be 1/1024 or about 0.1%. It would take only 7 mixes to get the old lubricant to be less than 1% of the total.


One more thing: if you don’t feel like bashing out tables and want to get straight to how many rinse cycles you’ll need:

log (desired cleanliness) divided by log (mix ratio) will give you how many “wash cycles” you’ll need.

So let’s suppose you want to get to that 1% old oiliness (0.01) and you can do 25% at a time:

log (0.01)/log (0.75) is a hair over 16 changes. That’s a bit much, but if you were able to change 75% of the oil at a time, you could do it in just 3.3 runs.

I realize this is mainly about the formula(s) for dilution, but at what point does repeated rounds of transmission fluid and filter become more expensive than just going to a shop to have the transmission and torque converter power-flushed, which uses approximately twice the full fill of fluid and just one filter?

Well, in gallon size the fluid is just under $5/quart. The filter doesn’t need to be changed each time I cycle it and, actually, the shop manual says it can be “re-used unless contaminated.”* I’m not even going to try to justify that, but remember that I’m doing some extra changes to forestall the idea of going the full 150,000 miles before doing anything.

A transmission shop quoted a friend of mine $250 to do a flush. I think I’m still ahead by doing my preventive something instead of just following the manufacturer’s suggestion.

*= Regarding the filter, the transmission shop told my friend that the tranny filter, unlike an oil filter, isn’t meant to get the same granularity of contaminants out. And, when I changed it, the inside of old filter looks like the new one. I don’t see yuck like when I change an oil filter. That, coupled with the service manual’s suggestion to re-use it plus the 150,000 mile interval makes me think I’m ok.

I was recently puzzling out the same issue – drain+fill vs. $250 flush. I came across this calculator, which – even in view-only mode – gives you a pretty good idea (for those trying to avoid the math):

Incidentally, that spreadsheet can be downloaded with formulas in place, and modified to suit your situation, if desired.

File > Download > Microsoft Excel (.xlsx)

That’s cool. I can definitely see a point of diminishing returns for my investment.