Rafal had asked, in a different thread, for more information on replenishing a fixer; this is my attempt to explain it.
What this is, and who should read it (or not): I wrote this as a primer on how a replenished fixer system works. If you want to understand this sort of thing, you may find it interesting. If you don't care, then it's not worth your time; there is nothing directly useful here. At best, you might read replenisher instructions elsewhere, and be excited to say, "Oh! Now I understand why they say to do that." I wish I could have made it interesting, but I hope I made it understandable.
I wanted to just explain how a two-stage fix system could be replenished, but after trying various approaches, I couldn't come up with a straightforward way to do it. So I decided to try it in two parts - the first is a primer on how replenishment works in a single tank. The second part will go into a 2-stage fixer system which gets more complicated because the tanks interact with each other.
Part 1, single tank fixing:
Replenishment of a single fixer tank is fairly simple to understand. When the paper goes into the fixer tank it also brings along some liquid from the previous tank. So the fixer gets slightly diluted from this "carry-in". When the paper is removed from the fixer, it takes a bit of the fixer ("carry-out") into the following tank (when I say "tank" it can also mean tray).
So two things happen, aside from the actual fixing operation: first, the fixer becomes slightly diluted, and second, some of the active fixer (and silver) is lost to the following tank. If the carry-in and carry-out are roughly balanced, the tank volume stays the same.
If we don't replenish, here's what CAN happen: the silver will gradually build up to some high level (which it cannot exceed because of the continuous dilution due to solution carry-in), and the fixing agent will gradually diminish in concentration (because of the dilution). The missing fixer (including silver) has gone into the following tank; if that tank is wash water to be discarded, all this silver will be lost.
In reality, the extreme situation above never happens because we decide when the fixer has been used enough; we dump it into the silver-recovery tank (or whatever else we choose to do), then we start again with fresh fixer.
If we DO use a replenisher, here's how it should work. Ideally the replenisher formula is overconcentrated slightly, just enough to overcome the dilution effect of the carry-in solution. Thus the fixer will always remain at normal strength. However, the volume will increase, so we should remove roughly the same volume as that of the added replenisher. (It should go into the silver-recovery tank.)
The ideal replenisher, when combined with the carry-in, would dilute the tank's silver concencentration to a specified aim point. Fortunately fixer is not very critical as to it's exact chemical makeup, so there is a lot of leeway to use whatever replenishment rate one needs.
A summary may be in order here. The main difference between a batch vs replenished system is that the batch system is always changin; it starts out "clean" and gradually becomes both diluted and exhausted. At which point it is dumped and the cyle restarts. A replenished system reaches an equilibrium, then stays there, potentially forever. In both cases, silver-bearing fixer is lost (via carry-out) to the following solution; the loss in volume is equal in both cases; the loss of silver is directly proportional to the silver concentration in the fixer tank.
*NOTE 1: there are actually a couple other things going on, which can be significant or not, depending on how you do things: evaporation will tend to concentrate the solution, and some of the preservative will be used up (typically sulfite oxidizes to sulfate).
*Note 2: to put the amount of dilution into perspective, if we use a 500 ml tank (1/2 liter; about 1/2 quart), and the carryover from a sheet of paper is 15 ml, each sheet of paper dilutes the fixer to about 97% of it's prior concentration. If you ran, for example, 5 sheets, the concentration would fall to (0.97 x 0.97 x 0.97 x 0.97 x 0.97) = 0.86, etc. (I am making up the carryover numbers; it could vary a lot.)
I think it's worth mentioning that this single tank system has two competing "goals." For low fixer costs, we want to use the heck out of our fixer, so that the silver reaches a high concentration. But for best print life, we want to keep the silver concentration below some specified limit (the reasons for this are discussed in other threads).
If your effluent is regulated, other issues come into play - the more you let your silver spread out, the more costly it's gonna be to get it back later.
Next will be part 2, covering a two-stage fixer with replenishment.