Not mixing the liquid under the FB with the wort in the grain bed will reduce your lauter efficiency a small amount. How much depends on what fraction of the total liquid volume is under the FB (more under the FB leads to lower lauter efficiency), and just how much diffusion of sugar occurs from the bulk of the mash into the dilute wort under the FB. The efficiency loss can be reduced by doing several vorlauf rounds near the end of the mash.
Edit: On thinking about it a little more, the optimal way to deal with a false bottom w/o pump might be to slowly draw off a volume equal to the volume under the FB at the end of the mash, and then pour the collected liquid back over the grain. You would do this at the end of the mash, and only once. If anyone is really interested I can explain the rationale for this approach.
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I would love a more detailed explanation! Sometimes I do that kind of modified vorlauf but it has never made a terribly huge difference on my system. My false bottom has fully 1 gallon under it and my typical mash volume is 3.5 gallons to end up with 2.5 gallons of beer.
Full disclosure: I am not the type of brewer that worries about hitting all my numbers and eking every iota of efficiency out of my system. I get the appeal of it but it's almost more fun for me to see how lazy I can be and still make damn good beer.
Lauter efficiency is the fraction of sugar created in the mash that is collected in the BK. The formula is:
Lauter Efficiency = Amount of Sugar into BK / Amount of Sugar Created in Mash
Since all the sugar created in the mash ends up in either the BK, or left in the wort absorbed by the spent grain, this can be rewritten as:
Lauter Efficiency = 1 - Amount of Sugar Left in Grain / Amount of Sugar Created in Mash
So, the more sugar in the wort absorbed by the spent grain, the lower the lauter efficiency.
With a false bottom, and no recirculation, the concentration of sugar in wort in the grain bed will be higher than the concentration of sugar in the wort under the FB. Now if you pull the bag/basket, all the low concentration wort stays in the BK, and the wort absorbed by the grain is the highest concentration wort, meaning more sugar is held in the spent grain. Thus, reduced lauter efficiency.
On the other hand, if you drain the volume under the FB into a bucket, you collect the low concentration wort, and it is replaced by higher concentration wort under the FB. The wort in the grain bed flows towards the bottom of the grain bed, and this is high concentration wort. Now if you slowly pour the drained lower concentration wort back on top of the grain bed, the bottom of the grain bed will have higher concentration wort, and the top of the bed will be lower concentration wort. Pulling the bag/basket will drain most of the higher concentration wort, and lower concentration wort will be left behind in the grain. Thus you get an increase in lauter efficiency.
The above effect is not perfect, as there will be some mixing of the higher and lower concentration worts when draining. And the difference in lauter efficiency is not huge, probably just a few percent. But, if you are struggling with low efficiency, and it bothers you, then this is something you can do to help.
Brew on