I've often wondered when rinsing out a container, what the best strategy is to get it clean: Many rinses with a small amount of water, or a few with a large amount?
Every time I empty the container, there's a bit of liquid that stays due to surface tension. Then I fill the container again, lowering the concentration of the original contents, before dumping as much liquid as I can out. If I fill it to a volume
V, and empty out all but a volume
v, then after the first rinse, the concentration of the original liquid is
where
ρ0 is the full concentration of the liquid. Each rinse applies the same factor, so after
n rinses,
If we suppose that there's always the same amount left in the bottle, say 1%, we can try varying the other parameters to see how they relate.
Each line represents an amount we fill the container, from 2% to 100%, and the x-axis represents the number of times we've filled and dumped. More interesting (for an eco-friendly person anyway) might be the total amount of water required to dilute the liquid to a specific concentration. The total volume is simply
nV, so we can rewrite the equation above as
where ρ* is the target concentration. Plotting that looks like
What this says is that if you're looking to save water, you just have to rinse a lot of times with a little bit!
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