Signing in again...
I'd like to say that I value to input of Greg.L and it's clear you're no dope. The discussion here is valuable and I certainly was in the wrong with my original assumptions regarding pressures. Partial pressure is a dificult concept for me to grasp but the systems I deal with almost exclusively deal with flow conditions. Where flow doesn't exist, the principles change.
I completely accept and have accepted (after being corrected) that oxygen will get in. The matter remains though of headspace [volume] affecting a brew. Greg.L's comments are that oxygen is the enemy with regards to infections, and I'll take his word for it (though I can't find much about it online. Sanitation, yes, which is probably why so many members here are more concerned about that because by and large that's the message you get when sitting behind a keyboard.)
I'll disregard the area of a fermenter which is making contact with the wort. The rate of transfer will be constant regardless for a small brew.
The larger the surface area, the greater the rate of transfer of oxygen (as the permiability coefficient is [quantity x thickness] / [area x time x Δp] )
Doing some maths on paper the relationship between fermenter area : headspace is -
Surface area : headspace
d + 4·l : d·l
Where l is the height of the headspace and d is the diameter of the drum/fermenter.
- For a case where the headspace is very, very low, the surface area is much greater than the voume
- For a case where the headspace is infinitely high, surface area will be 4 x the volume.
So the greater the headspace, the lower the surface area in respect to the volume. In layman's terms, it will take
longer for oxygen to acheive the sample proportion by volume with a larger headspace due to ingress via permiability. This favours larger headspace.
What DrSmurto is talking about is managing the oxygen proportion within a headspace. The smaller the headspace, the easier this would be to manage. In addition if the vessel (cask, conical SS fermenter etc.) has any sort of taper at the top then by topping it up you are reducing the surface area of the liquid to the gas about it, which is favourable. This isn't typical of your standard home brew bucket or Coopers fermenter.
I completely understand that minimising headspace is ideal to manage the proportion of oxygen within that headspace.
As homebrewers with simple fermenters and no real means for managing the oxygen content, I'm still confident larger headspace won't increase the proportion of oxygen inside the fermenter's headspace. Quantity yes, proportion no.
All that aside the most important message I've got from all this discussion is to be careful about leaving fermented beer in an HDPE container for extended periods. This is a habit I have for lagers and is something I'll look at changing. Especially considering Greg.L's comments about the increased risk of infection with oxidation.
Attached is the maths for above for those interested. Not signing out.
View attachment 3791_001.pdf