Ok, if we assume a "to fermenter" volume of 3.0 gal, and 0.25 gal of trub left in the BK, then the required post-boil volume is 3.25 gal. Also, assuming a 1.0 gal/hr boil-off rate, a 90 minute boil would boil off 1.5 gal, so we need a pre-boil volume of 4.75 gal. Might as well use all 5.0 lb of your grain, and save the DME to boost the OG if your conversion efficiency is lower than expected. At 0.1 gal/lb grain absorption rate (typical for a moderately well drained bag, or a less well drained bag with a light squeeze after initial drain and sparge) would mean we lose 0.5 gal to grain absorption, so we need to start with 5.25 gal of total brewing water. Since you plan on doing a dunk sparge, 3.0 gal for mashing, and 2.25 gal for sparging should work nicely (and give you a reasonable mash thickness of 2.4 qt/lb.
Now if you get 100% conversion efficiency (all available starch converted to sugar during the mash) your end of mash wort SG should be ~1.054. And you will collect 2.5 gal of this wort. Sparging with 2.25 gal of water will result in a sparged wort SG of ~1.014. You should collect 2.25 gal of sparged wort, as no additional grain absorption occurs during the sparge (the grain is already "saturated.") Combining the initial and sparged wort will give you 4.75 gal of wort at an SG of ~1.035. Boiling this 4.75 gal down to 3.25 gal will result in an OG of ~1.051 which seems reasonable for a pilsner.
Now, I'm assuming that you do not have your own grain mill, and are using grain milled by your supplier. Supplier crushes tend to be coarser than ideal for BIAB, so that brewers who use traditional 3-vessel systems don't have stuck mash/sparge issues. With these coarser crushes, it is quite common not to achieve 100% conversion efficiency with the typical 60 minute mash, so you might want to consider doing a 90 minute mash to increase conversion. If you have a refractometer, you can monitor the SG of the wort during the mash, and stop mashing when you reach the max possible SG (1.054 in this case.) Or, you can just start checking the wort SG every 15 minutes after 60 minutes of mashing, and stop the mash when the SG of the wort stops increasing. (Your SG might be lower than predicted if your enzymes crap out, or your malt's potential is less than the typical 37 pts/lb assumed in the calculations above.)
I used
my mash and lauter prediction spreadsheet to do the above calculations.
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